Locally uniformly convex norms in Banach spaces and their duals
Haydon, Richard
2006-01-01
It is shown that a Banach space with locally uniformly convex dual admits an equivalent norm which is itself locally uniformly convex. It follows that on any such space all continuous real-valued functions may be uniformly approximated by C^1 functions.
Zone diagrams in compact subsets of uniformly convex normed spaces
Kopecká, Eva; Reich, Simeon
2010-01-01
A zone diagram is a relatively new concept which has emerged in computational geometry and is related to Voronoi diagrams. Formally, it is a fixed point of a certain mapping, and neither its uniqueness nor its existence are obvious in advance. It has been studied by several authors, starting with T. Asano, J. Matousek and T. Tokuyama, who considered the Euclidean plane with singleton sites, and proved the existence and uniqueness of zone diagrams there. In the present paper we prove the existence of zone diagrams with respect to finitely many pairwise disjoint compact sites contained in a compact and convex subset of a uniformly convex normed space. The proof is based on the Schauder fixed point theorem, the Curtis-Schori theorem regarding the Hilbert cube, and on recent results concerning the characterization of Voronoi cells as a collection of line segments and their geometric stability with respect to small changes of the corresponding sites. Along the way we obtain the continuity of the Dom mapping as wel...
Uniformly convex and strictly convex Orlicz spaces
Masta, Al Azhary
2016-02-01
In this paper we define the new norm of Orlicz spaces on ℝn through a multiplication operator on an old Orlicz spaces. We obtain some necessary and sufficient conditions that the new norm to be a uniformly convex and strictly convex spaces.
赋pAmemiya范数Orlicz空间的局部凸点%Local convex point of Orlicz space with pAmemiya norm
Institute of Scientific and Technical Information of China (English)
王晓燕
2013-01-01
运用几何方法给出赋p-Amemiya范数 Orlicz 空间具有局部一致凸点和弱局部一致凸点的必要条件。%Applied geometric methods to give the necessary conditions for the existence of local uniform convex point and the weakly local uniform convex point of Orlicz space with p-Amemiya norm.
Uniformly Convex Metric Spaces
Kell Martin
2014-01-01
In this paper the theory of uniformly convex metric spaces is developed. These spaces exhibit a generalized convexity of the metric from a fixed point. Using a (nearly) uniform convexity property a simple proof of reflexivity is presented and a weak topology of such spaces is analyzed. This topology called co-convex topology agrees with the usualy weak topology in Banach spaces. An example of a $CAT(0)$-spaces with weak topology which is not Hausdorff is given. This answers questions raised b...
Institute of Scientific and Technical Information of China (English)
夏萍; 吐尔德别克
2011-01-01
讨论了复拟Banach空间的复凸性,给出了复凸性的另一种新的等价刻画,即分别应用取值于复拟Banach空间中的Hardy鞅和解析鞅的弱Orlicz空间范数不等式刻划了解析q一致凸性和q一致PL凸性.%In this paper , we discuss the convexity of complex quasi-Banach space , and give another new equivalent description of the complex conrexity , namely , using weak Orlicz space norm inequalities of Hardy martingales and analytic martingales with values in complex quasi-Banach space, we describe q-uniform convexity and q-uniform PL-convexity of the complex space.
Browder's Fixed Point Theorem and Some Interesting Results in Intuitionistic Fuzzy Normed Spaces
Directory of Open Access Journals (Sweden)
Cancan M
2010-01-01
Full Text Available We define and study Browder's fixed point theorem and relation between an intuitionistic fuzzy convex normed space and a strong intuitionistic fuzzy uniformly convex normed space. Also, we give an example to show that uniformly convex normed space does not imply strongly intuitionistic fuzzy uniformly convex.
Parametric R-norm directed-divergence convex function
Garg, Dhanesh; Kumar, Satish
2016-06-01
In this paper, we define parametric R-norm directed-divergence convex function and discuss their special cases and prove some properties similar to Kullback-Leibler information measure. From R-norm divergence measure new information measures have also been derived and their relations with different measures of entropy have been obtained and give its application in industrial engineering.
A class of free locally convex spaces
Sipacheva, O. V.
2003-04-01
Stratifiable spaces are a natural generalization of metrizable spaces for which Dugundji's theorem holds. It is proved that the free locally convex space of a stratifiable space is stratifiable. This means, in particular, that the space of finitely supported probability measures on a stratifiable space is a retract of a locally convex space, and that each stratifiable convex subset of a locally convex space is a retract of a locally convex space.
Uniformly convex-transitive function spaces
Rambla-Barreno, Fernando; Talponen, Jarno
2009-01-01
We introduce a property of Banach spaces called uniform convex-transitivity, which falls between almost transitivity and convex-transitivity. We will provide examples of uniformly convex-transitive spaces. This property behaves nicely in connection with some Banach-valued function spaces. As a consequence, we obtain new examples of convex-transitive Banach spaces.
Complex Convexity of Orlicz Modular Sequence Spaces
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Lili Chen
2016-01-01
Full Text Available The concepts of complex extreme points, complex strongly extreme points, complex strict convexity, and complex midpoint locally uniform convexity in general modular spaces are introduced. Then we prove that, for any Orlicz modular sequence space lΦ,ρ, lΦ,ρ is complex midpoint locally uniformly convex. As a corollary, lΦ,ρ is also complex strictly convex.
Compactly convex sets in linear topological spaces
Banakh, T; Ravsky, O
2012-01-01
A convex subset X of a linear topological space is called compactly convex if there is a continuous compact-valued map $\\Phi:X\\to exp(X)$ such that $[x,y]\\subset\\Phi(x)\\cup \\Phi(y)$ for all $x,y\\in X$. We prove that each convex subset of the plane is compactly convex. On the other hand, the space $R^3$ contains a convex set that is not compactly convex. Each compactly convex subset $X$ of a linear topological space $L$ has locally compact closure $\\bar X$ which is metrizable if and only if each compact subset of $X$ is metrizable.
Barrelled locally convex spaces
PÃ©rez Carreras, P
1987-01-01
This book is a systematic treatment of barrelled spaces, and of structures in which barrelledness conditions are significant. It is a fairly self-contained study of the structural theory of those spaces, concentrating on the basic phenomena in the theory, and presenting a variety of functional-analytic techniques.Beginning with some basic and important results in different branches of Analysis, the volume deals with Baire spaces, presents a variety of techniques, and gives the necessary definitions, exploring conditions on discs to ensure that they are absorbed by the barrels of the sp
On Uniform Convexity of Banach Spaces
Institute of Scientific and Technical Information of China (English)
Qing Jin CHENG; Bo WANG; Cui Ling WANG
2011-01-01
This paper gives some relations and properties of several kinds of generalized convexity in Banach spaces. As a result, it proves that every kind of uniform convexity implies the Banach-Sakes property, and several notions of uniform convexity in literature are actually equivalent.
Fixed Points of Non-expansive Operators on Weakly Cauchy Normed Spaces
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Sahar M. Ali
2007-01-01
Full Text Available We proved the existence of fixed points of non-expansive operators defined on weakly Cauchy spaces in which parallelogram law holds, the given normed space is not necessarily be uniformly convex Banach space or Hilbert space, we reduced the completeness and the uniform convexity assumptions which imposed on the given normed space.
Institute of Scientific and Technical Information of China (English)
CHENG LIXIN; TENG YANMEI
2005-01-01
This paper presents a type of variational principles for real valued w* lower semicon tinuous functions on certain subsets in duals of locally convex spaces, and resolve a problem concerning differentiability of convex functions on general Banach spaces. They are done through discussing differentiability of convex functions on nonlinear topological spaces and convexification of nonconvex functions on topological linear spaces.
Convex analysis and optimization in Hadamard spaces
Bacak, Miroslav
2014-01-01
This book gives a first systematic account on the subject of convex analysis and optimization in Hadamard spaces. It is primarily aimed at both graduate students and researchers in analysis and optimization.
Deformation in locally convex topological linear spaces
Institute of Scientific and Technical Information of China (English)
DING; Yanheng
2004-01-01
We are concerned with a deformation theory in locally convex topological linear spaces. A special "nice" partition of unity is given. This enables us to construct certain vector fields which are locally Lipschitz continuous with respect to the locally convex topology. The existence, uniqueness and continuous dependence of flows associated to the vector fields are established. Deformations related to strongly indefinite functionals are then obtained. Finally, as applications, we prove some abstract critical point theorems.
A further characteristic of abstract convexity structures on topological spaces
Xiang, Shu-Wen; Xia, Shunyou
2007-11-01
In this paper, we give a characteristic of abstract convexity structures on topological spaces with selection property. We show that if a convexity structure defined on a topological space has the weak selection property then satisfies H0-condition. Moreover, in a compact convex subset of a topological space with convexity structure, the weak selection property implies the fixed point property.
Uniformly convex subsets of the Hilbert space with modulus of convexity of the second order
Balashov, Maxim V.; Repovš, Dušan,
2011-01-01
We prove that in the Hilbert space every uniformly convex set with modulus of convexity of the second order at zero is an intersection of closed balls of fixed radius. We also obtain an estimate of this radius.
On fixed points and uniformly convex spaces
Gelander, Tsachik
2008-01-01
The purpose of this note is to present two elementary, but useful, facts concerning actions on uniformly convex spaces. We demonstrate how each of them can be used in an alternative proof of the triviality of the first $L_p$-cohomology of higher rank simple Lie groups, proved in [BFGM].
ON THE PRODUCT OF GATEAUX DIFFERENTIABILITY LOCALLY CONVEX SPACES
Institute of Scientific and Technical Information of China (English)
Shen Xisheng; Cheng Lixin
2005-01-01
A locally convex space is said to be a Gateaux differentiability space (GDS)provided every continuous convex function defined on a nonempty convex open subset D of the space is densely Gateaux differentiable in D.This paper shows that the product of a GDS and a family of separable Frechet spaces is a GDS,and that the product of a GDS and an arbitrary locally convex space endowed with the weak topology is a GDS.
ON Lp-MATRICIALLY NORMED SPACES
Institute of Scientific and Technical Information of China (English)
Turdebek N. Bekjan
2005-01-01
It is proved that there is only one Lp-matricially normed space of dimension 1 and that quotient spaces of Lp-matricially normed spaces are also Lp-matricially normed spaces. Some properties of Lp-matricially normed spaces are given.
Convexity conditions and normal structure of Banach spaces
Saejung, Satit
2008-08-01
We prove that F-convexity, the property dual to P-convexity of Kottman, implies uniform normal structure. Moreover, in the presence of the WORTH property, normal structure follows from a weaker convexity condition than F-convexity. The latter result improves the known fact that every uniformly nonsquare space with the WORTH property has normal structure.
A Generalization of Uniformly Extremely Convex Banach Spaces
Suyalatu Wulede; Wurichaihu Bai; Wurina Bao
2016-01-01
We discuss a new class of Banach spaces which are the generalization of uniformly extremely convex spaces introduced by Wulede and Ha. We prove that the new class of Banach spaces lies strictly between either the classes of k-uniformly rotund spaces and k-strongly convex spaces or classes of fully k-convex spaces and k-strongly convex spaces and has no inclusive relation with the class of locally k-uniformly convex spaces. We obtain in addition some characterizations and properties of this ne...
Measuring Voting Power in Convex Policy Spaces
Directory of Open Access Journals (Sweden)
Sascha Kurz
2014-03-01
Full Text Available Classical power index analysis considers the individual’s ability to influence the aggregated group decision by changing its own vote, where all decisions and votes are assumed to be binary. In many practical applications we have more options than either “yes” or “no”. Here we generalize three important power indices to continuous convex policy spaces. This allows the analysis of a collection of economic problems like, e.g., tax rates or spending that otherwise would not be covered in binary models.
Institute of Scientific and Technical Information of China (English)
王纯; 潘思明
2011-01-01
In this paper, in order to investigate the convergences of Mann-type iterative scheme for a countable family of strict pseudocontraction mappings in a uniformly convex Banach space with the Frechet differentiable norm, we used the results obtained by Marino-Xu, Zhou, Osilike-Udomene, Zhang-Guo and the corresponding results to ex tend some conclusions obtained by Chidume-Shahzad to the real uniformly convex Banach space with the Frechet dif ferentiable norm under the countable strictly pseudocontraction mappings. And the proof of the weak convergences of Mann-type iterative scheme for a countable family of strict pseudocontraction mappings in this Banach space with the Frechet differentiable norm was given.%为了研究在具有Fréchet可微范数的实一致凸Banach空间中的可数的严格伪压缩映射族Mann型迭代方案的收敛性,利用Marino-Xu,Zhou,Osilike-Udomene,Zhang-Guo的结论以及其它相关的结论,在已有结论的基础上,将Chidume-Shahzad的某些结论推广到具有Fréhet可微范数的实一致凸Banach空间的无限严格伪压缩映射族的情形下,并给出了在具有Fréchet可微范数的实一致凸Banach空间中的可数严格伪压缩映射族的Mann型迭代方案弱收敛性的证明.
Bayoumi, A
2003-01-01
All the existing books in Infinite Dimensional Complex Analysis focus on the problems of locally convex spaces. However, the theory without convexity condition is covered for the first time in this book. This shows that we are really working with a new, important and interesting field. Theory of functions and nonlinear analysis problems are widespread in the mathematical modeling of real world systems in a very broad range of applications. During the past three decades many new results from the author have helped to solve multiextreme problems arising from important situations, non-convex and
Continuity of Extremal Elements in Uniformly Convex Spaces
Ferguson, Timothy
2013-01-01
In this paper, we study the problem of finding the extremal element for a linear functional over a uniformly convex Banach space. We show that a unique extremal element exists and depends continuously on the linear functional, and vice versa. Using this, we simplify and clarify Ryabykh's proof that for any linear functional on a uniformly convex Bergman space with kernel in a certain Hardy space, the extremal functional belongs to the corresponding Hardy space.
On a-order-convexity of Fuzzy Syntopogenous Spaces
Institute of Scientific and Technical Information of China (English)
WANG Hong
2007-01-01
In this paper,we combine L-fuzzy syntopogenous structure on X with algebraic structure on X.First,the *-increasing and *-decreasing spaces have been studied.Second,we define a-order-convexity on syntopogenous structures (X,S,≤).some important properties of a-order-convexity have been obtained.
Properties of distance functions on convex surfaces and Alexandrov spaces
Rataj, Jan
2009-01-01
If $X$ is a convex surface in a Euclidean space, then the squared (intrinsic) distance function $\\dist^2(x,y)$ is d.c. (DC, delta-convex) on $X\\times X$ in the only natural extrinsic sense. For the proof we use semiconcavity (in an intrinsic sense) of $\\dist^2(x,y)$ on $X \\times X$ if $X$ is an Alexandrov space with nonnegative curvature. Applications concerning $r$-boundaries (distance spheres) and the ambiguous locus (exoskeleton) of a closed subset of a convex surface are given.
Convexity and the "Pythagorean" metric of space(-time)
Kalogeropoulos, Nikos
2016-01-01
We address the question about the reasons why the "Wick-rotated", positive-definite, space-time metric obeys the Pythagorean theorem. An answer is proposed based on the convexity and smoothness properties of the functional spaces providing the kinematic framework for the statistical or quantum treatments of gravity. We rely on particular moduli of convexity and smoothness which are extremized by Hilbert spaces. In the spirit of the variational principles employed in classical and quantum Physics, such Hilbert spaces dominate in a functional integral approach. The "Pythagorean" metric of space(-time) is then induced by such Hilbert spaces.
Topological Properties of Real Normed Space
Directory of Open Access Journals (Sweden)
Nakasho Kazuhisa
2014-09-01
Full Text Available In this article, we formalize topological properties of real normed spaces. In the first part, open and closed, density, separability and sequence and its convergence are discussed. Then we argue properties of real normed subspace. Then we discuss linear functions between real normed speces. Several kinds of subspaces induced by linear functions such as kernel, image and inverse image are considered here. The fact that Lipschitz continuity operators preserve convergence of sequences is also refered here. Then we argue the condition when real normed subspaces become Banach’s spaces. We also formalize quotient vector space. In the last session, we argue the properties of the closure of real normed space. These formalizations are based on [19](p.3-41, [2] and [34](p.3-67.
Institute of Scientific and Technical Information of China (English)
2008-01-01
In general normed spaces,we consider a multiobjective piecewise linear optimization problem with the ordering cone being convex and having a nonempty interior.We establish that the weak Pareto optimal solution set of such a problem is the union of finitely many polyhedra and that this set is also arcwise connected under the cone convexity assumption of the objective function.Moreover,we provide necessary and suffcient conditions about the existence of weak(sharp) Pareto solutions.
Moduli spaces of convex projective structures on surfaces
DEFF Research Database (Denmark)
Fock, V. V.; Goncharov, A. B.
2007-01-01
We introduce explicit parametrisations of the moduli space of convex projective structures on surfaces, and show that the latter moduli space is identified with the higher Teichmüller space for defined in [V.V. Fock, A.B. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, ma...
The extension of the Krein-Smulian theorem for Orlicz sequence spaces and convex sets
Granero, Antonio S.
2007-02-01
If X is a Banach space and C[subset of]X** a convex subset, for x**[set membership, variant]X** and A[subset of]X** let be the distance from x** to C and . In this paper we prove that if [phi] is an Orlicz function, I an infinite set and X=l[phi](I) the corresponding Orlicz space, equipped with either the Luxemburg or the Orlicz norm, then for every w*-compact subset K[subset of]X** we have if and only if [phi] satisfies the [Delta]2-condition at 0. We also prove that for every Banach space X, every nonempty convex subset C[subset of]X and every w*-compact subset K[subset of]X** then and, if K[intersection]C is w*-dense in K, then .
Airborne gravimetry data sparse reconstruction via L1-norm convex quadratic programming
Yang, Ya-Peng; Wu, Mei-Ping; Tang, Gang
2015-06-01
In practice, airborne gravimetry is a sub-Nyquist sampling method because of the restrictions imposed by national boundaries, financial cost, and database size. In this study, we analyze the sparsity of airborne gravimetry data by using the discrete Fourier transform and propose a reconstruction method based on the theory of compressed sensing for large-scale gravity anomaly data. Consequently, the reconstruction of the gravity anomaly data is transformed to a L1-norm convex quadratic programming problem. We combine the preconditioned conjugate gradient algorithm (PCG) and the improved interior-point method (IPM) to solve the convex quadratic programming problem. Furthermore, a flight test was carried out with the homegrown strapdown airborne gravimeter SGA-WZ. Subsequently, we reconstructed the gravity anomaly data of the flight test, and then, we compared the proposed method with the linear interpolation method, which is commonly used in airborne gravimetry. The test results show that the PCG-IPM algorithm can be used to reconstruct large-scale gravity anomaly data with higher accuracy and more effectiveness than the linear interpolation method.
Bounded cohomology with coefficients in uniformly convex Banach spaces
Bestvina, Mladen; Bromberg, Ken; Fujiwara, Koji
2013-01-01
We show that for acylindrically hyperbolic groups $\\Gamma$ (with no nontrivial finite normal subgroups) and arbitrary unitary representation $\\rho$ of $\\Gamma$ in a (nonzero) uniformly convex Banach space the vector space $H^2_b(\\Gamma;\\rho)$ is infinite dimensional. The result was known for the regular representations on $\\ell^p(\\Gamma)$ with $1
-Boundedness and -Compactness in Finite Dimensional Probabilistic Normed Spaces
Indian Academy of Sciences (India)
Reza Saadati; Massoud Amini
2005-11-01
In this paper, we prove that in a finite dimensional probabilistic normed space, every two probabilistic norms are equivalent and we study the notion of -compactness and -boundedness in probabilistic normed spaces.
Convexity and the Euclidean Metric of Space-Time
Directory of Open Access Journals (Sweden)
Nikolaos Kalogeropoulos
2017-02-01
Full Text Available We address the reasons why the “Wick-rotated”, positive-definite, space-time metric obeys the Pythagorean theorem. An answer is proposed based on the convexity and smoothness properties of the functional spaces purporting to provide the kinematic framework of approaches to quantum gravity. We employ moduli of convexity and smoothness which are eventually extremized by Hilbert spaces. We point out the potential physical significance that functional analytical dualities play in this framework. Following the spirit of the variational principles employed in classical and quantum Physics, such Hilbert spaces dominate in a generalized functional integral approach. The metric of space-time is induced by the inner product of such Hilbert spaces.
A Unified Fixed Point Theory in Generalized Convex Spaces
Institute of Scientific and Technical Information of China (English)
Sehie PARK
2007-01-01
Let β be the class of 'better' admissible multimaps due to the author.We introduce newconcepts of admissibility (in the sense of Klee) and of Klee approximability for subsets of G-convexuniform spaces and show that any compact closed multimap in β from a G-convex space into itselfwith the Klee approximable range has a fixed point.This new theorem contains a large number ofknown results on topological vector spaces or on various subclasses.of the class of admissible G-convexspaces.Such subclasses are those of C-spaces,sets of the Zima-Hadzic type,locally G-convex spaces,and LG-spaces.Mutual relations among those subclasses and some related results are added.
VARIOUS NOTIONS OF ORTHOGONALITY IN NORMED SPACES
Institute of Scientific and Technical Information of China (English)
N.B. OKELO; J.O. AGURE; P.O. OLECHE
2013-01-01
In this paper, we present various notions and aspects of orthogonality in normed spaces. Characterizations and generalizations of orthogonality are also consid-ered. Results on orthogonality of the range and the kernel of elementary operators and the operators implementing them also are given.
Directory of Open Access Journals (Sweden)
Lu-Chuan Ceng
2013-01-01
Full Text Available We introduce composite implicit and explicit iterative algorithms for solving a general system of variational inequalities and a common fixed point problem of an infinite family of nonexpansive mappings in a real smooth and uniformly convex Banach space. These composite iterative algorithms are based on Korpelevich's extragradient method and viscosity approximation method. We first consider and analyze a composite implicit iterative algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space and then another composite explicit iterative algorithm in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literatures.
Directory of Open Access Journals (Sweden)
Rafa Espínola
2010-01-01
Full Text Available We study the existence of fixed points and convergence of iterates for asymptotic pointwise contractions in uniformly convex metric spaces. We also study the existence of fixed points for set-valued nonexpansive mappings in the same class of spaces. Our results do not assume convexity of the metric which makes a big difference when studying the existence of fixed points for set-valued mappings.
Asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces
Kohlenbach, Ulrich; Leuştean, Laurentiu
2007-01-01
This paper provides a fixed point theorem for asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces as well as new effective results on the Krasnoselski-Mann iterations of such mappings. The latter were found using methods from logic and the paper continues a case study in the general program of extracting effective data from prima-facie ineffective proofs in the fixed point theory of such mappings.
Zone diagrams in compact subsets of uniformly convex normed spaces
Kopecká, E. (Eva); Reem, D.; Reich, S.
2012-01-01
A zone diagram is a relatively new concept which has emerged in computational geometry and is related to Voronoi diagrams. Formally, it is a fixed point of a certain mapping, and neither its uniqueness nor its existence are obvious in advance. It has been studied by several authors, starting with T. Asano, J. Matousek and T. Tokuyama, who considered the Euclidean plane with singleton sites, and proved the existence and uniqueness of zone diagrams there. In the present paper we prove the exist...
Zone diagrams in compact subsets of uniformly convex normed spaces
2010-01-01
A zone diagram is a relatively new concept which has emerged in computational geometry and is related to Voronoi diagrams. Formally, it is a fixed point of a certain mapping, and neither its uniqueness nor its existence are obvious in advance. It has been studied by several authors, starting with T. Asano, J. Matousek and T. Tokuyama, who considered the Euclidean plane with singleton sites, and proved the existence and uniqueness of zone diagrams there. In the present paper we prove the exist...
赋β-范空间中的最佳逼近问题%The Problems of Best Approximation in β-Normed Spaces(0＜β＜1)
Institute of Scientific and Technical Information of China (English)
王见勇
2008-01-01
This paper deals with the problems of best approximation in β-normed spaces.With the tool of conjugate cone introduced in [1] and via the Hahn-Banach extension theorem of β-subseminorm in [2],the characteristics that an element in a closed subspace is the best approximation are given in Section 2.It is obtained in Section 3 that all convex sets or subspaces of a β-normed space are semi-Chebyshev if and only if the space is itself strictly convex.The fact that every finite dimensional subspace of a strictly convex β-normed space must be Chebyshev is proved at last.
Skala, Vaclav
2016-06-01
There are many space subdivision and space partitioning techniques used in many algorithms to speed up computations. They mostly rely on orthogonal space subdivision, resp. using hierarchical data structures, e.g. BSP trees, quadtrees, octrees, kd-trees, bounding volume hierarchies etc. However in some applications a non-orthogonal space subdivision can offer new ways for actual speed up. In the case of convex polygon in E2 a simple Point-in-Polygon test is of the O(N) complexity and the optimal algorithm is of O(log N) computational complexity. In the E3 case, the complexity is O(N) even for the convex polyhedron as no ordering is defined. New Point-in-Convex Polygon and Point-in-Convex Polyhedron algorithms are presented based on space subdivision in the preprocessing stage resulting to O(1) run-time complexity. The presented approach is simple to implement. Due to the principle of duality, dual problems, e.g. line-convex polygon, line clipping, can be solved in a similarly.
Differential calculus in normed linear spaces
Mukherjea, Kalyan
2007-01-01
This book presents Advanced Calculus from a geometric point of view: instead of dealing with partial derivatives of functions of several variables, the derivative of the function is treated as a linear transformation between normed linear spaces. Not only does this lead to a simplified and transparent exposition of "difficult" results like the Inverse and Implicit Function Theorems but also permits, without any extra effort, a discussion of the Differential Calculus of functions defined on infinite dimensional Hilbert or Banach spaces.The prerequisites demanded of the reader are modest: a sound understanding of convergence of sequences and series of real numbers, the continuity and differentiability properties of functions of a real variable and a little Linear Algebra should provide adequate background for understanding the book. The first two chapters cover much of the more advanced background material on Linear Algebra (like dual spaces, multilinear functions and tensor products.) Chapter 3 gives an ab ini...
A common fixed point for operators in probabilistic normed spaces
Energy Technology Data Exchange (ETDEWEB)
Ghaemi, M.B. [Faculty of Mathematics, Iran University of Science and Technology, Narmak, Tehran (Iran, Islamic Republic of)], E-mail: mghaemi@iust.ac.ir; Lafuerza-Guillen, Bernardo [Department of Applied Mathematics, University of Almeria, Almeria (Spain)], E-mail: blafuerz@ual.es; Razani, A. [Department of Mathematics, Faculty of Science, I. Kh. International University, P.O. Box 34194-288, Qazvin (Iran, Islamic Republic of)], E-mail: razani@ikiu.ac.ir
2009-05-15
Probabilistic Metric spaces was introduced by Karl Menger. Alsina, Schweizer and Sklar gave a general definition of probabilistic normed space based on the definition of Menger [Alsina C, Schweizer B, Sklar A. On the definition of a probabilistic normed spaces. Aequationes Math 1993;46:91-8]. Here, we consider the equicontinuity of a class of linear operators in probabilistic normed spaces and finally, a common fixed point theorem is proved. Application to quantum Mechanic is considered.0.
Convex analysis and monotone operator theory in Hilbert spaces
Bauschke, Heinz H
2017-01-01
This reference text, now in its second edition, offers a modern unifying presentation of three basic areas of nonlinear analysis: convex analysis, monotone operator theory, and the fixed point theory of nonexpansive operators. Taking a unique comprehensive approach, the theory is developed from the ground up, with the rich connections and interactions between the areas as the central focus, and it is illustrated by a large number of examples. The Hilbert space setting of the material offers a wide range of applications while avoiding the technical difficulties of general Banach spaces. The authors have also drawn upon recent advances and modern tools to simplify the proofs of key results making the book more accessible to a broader range of scholars and users. Combining a strong emphasis on applications with exceptionally lucid writing and an abundance of exercises, this text is of great value to a large audience including pure and applied mathematicians as well as researchers in engineering, data science, ma...
Space Junk Norms: US Advantages in Creating a Debris Reducing Outer Space Norm
2011-05-01
represented by the debate over space weaponization. Stephen Krasner defines norms as “standards of behavior defined in terms of rights and...designer of the 20 Stephen Clark, “Air Force Spaceplane is an Odd Bird with a Twisted Path...the Sea. New., rev. ed. Manchester: University Press, 1988. Clark, Stephen . “Air Force Spaceplane is an Odd Bird with a Twisted Path,” Space.com, 2
Mathematical methods linear algebra normed spaces distributions integration
Korevaar, Jacob
1968-01-01
Mathematical Methods, Volume I: Linear Algebra, Normed Spaces, Distributions, Integration focuses on advanced mathematical tools used in applications and the basic concepts of algebra, normed spaces, integration, and distributions.The publication first offers information on algebraic theory of vector spaces and introduction to functional analysis. Discussions focus on linear transformations and functionals, rectangular matrices, systems of linear equations, eigenvalue problems, use of eigenvectors and generalized eigenvectors in the representation of linear operators, metric and normed vector
Various Expressions for Modulus of Random Convexity
Institute of Scientific and Technical Information of China (English)
Xiao Lin ZENG
2013-01-01
We first prove various kinds of expressions for modulus of random convexity by using an Lo(F,R)-valued function's intermediate value theorem and the well known Hahn-Banach theorem for almost surely bounded random linear functionals,then establish some basic properties including continuity for modulus of random convexity.In particular,we express the modulus of random convexity of a special random normed module Lo(F,X) derived from a normed space X by the classical modulus of convexity of X.
关于Orlicz空间中p一致凸性的刻画%Characteristics of p-uniform convexity in Orlicz spaces
Institute of Scientific and Technical Information of China (English)
许立滨; 鄂明川; 于继杰
2013-01-01
众所周知,p一致凸性是Banach空间重要的几何性质,分别对赋p-Amemiya范数、Luxemburg范数及Orlicz范数的Orlicz空间的p一致凸性做了细致的研究,得到了相关的一些结果和结论.%It is well known that p-uniform convexity is an important geometric propery in Banach spaces.The p-uniform convexity of Orlicz spaces which were respectively equipped with p-Amemiya norm,Luxemburg norm and Orlicz norm were discussed in detail,and some relative results and conclusions were obtained.
The best simultaneous approximation in linear 2-normed spaces
Acikgoz, Mehmet
2012-01-01
In this paper, we shall investigate and analyse a new study on the best simultaneous approximation in the context of linear 2-normed spaces inspired by Elumalai and his coworkers in Elumalai. The basis of this investigation is to extend and refinement the definition of the classical aproximation, best approximation and some related concepts to linear 2-normed spaces.
Remarks on quasi-isometric non-embeddability into uniformly convex Banach spaces
Nowak, Piotr W.
2005-01-01
We construct a locally finite graph and a bounded geometry metric space which do not admit a quasi-isometric embedding into any uniformly convex Banach space. Connections with the geometry of $c_0$ and superreflexivity are discussed.
A proximal point method for nonsmooth convex optimization problems in Banach spaces
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Y. I. Alber
1997-01-01
Full Text Available In this paper we show the weak convergence and stability of the proximal point method when applied to the constrained convex optimization problem in uniformly convex and uniformly smooth Banach spaces. In addition, we establish a nonasymptotic estimate of convergence rate of the sequence of functional values for the unconstrained case. This estimate depends on a geometric characteristic of the dual Banach space, namely its modulus of convexity. We apply a new technique which includes Banach space geometry, estimates of duality mappings, nonstandard Lyapunov functionals and generalized projection operators in Banach spaces.
A Generalized Mazur-Ulam Theorem for Fuzzy Normed Spaces
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J. J. Font
2014-01-01
Full Text Available We introduce fuzzy norm-preserving maps, which generalize the concept of fuzzy isometry. Based on the ideas from Vogt, 1973, and Väisälä, 2003, we provide the following generalized version of the Mazur-Ulam theorem in the fuzzy context: let X, Y be fuzzy normed spaces and let f:X→Y be a fuzzy norm-preserving surjection satisfying f(0=0. Then f is additive.
A RANDOM FIXED POINT ITERATION FOR THREE RANDOM OPERATORS ON UNIFORMLY CONVEX BANACH SPACES
Institute of Scientific and Technical Information of China (English)
Binayak S. Choudhury
2003-01-01
In the present paper we introduce a random iteration scheme for three random operators defined on a closed and convex subset of a uniformly convex Banach space and prove its convergence to a common fixed point of three random operators. The result is also an extension of a known theorem in the corresponding non-random case.
NEW COLLECTIVELY FIXED POINT THEOREMS AND APPLICATIONS IN G-CONVEX SPACES
Institute of Scientific and Technical Information of China (English)
DING Xie-ping(丁协平); Park Jong-yeoul
2002-01-01
By applying the technique of continuous partition of unity and Tychonoff's fixed point theorem, some new collectively fixed point theorems for a family of set-valued mappings defined on the product space of noncompact G-convex spaces are proved. As applications, some nonempty intersetion theorems of Ky Fan type for a family of subsets of the product space of G-convex spaces are proved; An existence theorem of solutions for a system of nonlinear inequalities is given in G-convex spaces and some equilibrium existence theorems of abstract economies are also obtained in G-convex spaces. Our theorems improve, unify and generalized many important known results in recent literature.
GENERALIZED VECTOR QUASI-EQUILIBRIUM PROBLEMS IN LOCALLY G-CONVEX SPACES
Institute of Scientific and Technical Information of China (English)
DING Xie-ping
2005-01-01
Some classes of generalized vector quasi-equilibrium problems (in short,GVQEP) are introduced and studied in locally G-convex spaces which includes most of generalized vector equilibrium problems, generalized vector variational inequality problems,quasi-equilibrium problems and quasi-variational inequality problems as special cases. First,an equilibrium existence theorem for one person games is proved in locally G-convex spaces.As applications, some new existence theorems of solutions for the GVQEP are established in noncompact locally G-convex spaces. These results and argument methods are new and completely different from that in recent literature.
BERGMAN TYPE OPERATOR ON MIXED NORM SPACES WITH APPLICATIONS
Institute of Scientific and Technical Information of China (English)
RENGUANGBIN; SHIJIHUAI
1997-01-01
The authors investigate the conditions for the botmdedness of Bergman type operators Pa,t in mixed norm space Lp,q(μ) on the unit ball of Cn (n ≥ 1), and obtain a sufficient conditionand a necessary condition for general normal function μ, and a sufficient and necessary condition for μ(r)=(1-r2)α logβ(2(1-r)-1)(α>0,β≥0), This generalizes the result of Forelli-Rudin[3] on Bergman operator in Bergman space. As applications,a more natural method is given to compute the duality of the mixed norm space,solve the Gleason's problem for mixed norm space and obtain the characterization of mixed norm space in terms of partial derivatives.Moreover,it is proved that f∈L(0) ∞,q(μ) for holomorphic function f,1≤q≤∞.
Bredies, Kristian
2009-01-01
We consider the task of computing an approximate minimizer of the sum of a smooth and a non-smooth convex functional, respectively, in Banach space. Motivated by the classical forward-backward splitting method for the subgradients in Hilbert space, we propose a generalization which involves the iterative solution of simpler subproblems. Descent and convergence properties of this new algorithm are studied. Furthermore, the results are applied to the minimization of Tikhonov-functionals associated with linear inverse problems and semi-norm penalization in Banach spaces. With the help of Bregman-Taylor-distance estimates, rates of convergence for the forward-backward splitting procedure are obtained. Examples which demonstrate the applicability are given, in particular, a generalization of the iterative soft-thresholding method by Daubechies, Defrise and De Mol to Banach spaces as well as total-variation-based image restoration in higher dimensions are presented.
Institute of Scientific and Technical Information of China (English)
武希琳; 国起
2011-01-01
Uniform convexity in every direction in locally convex spaces is introduced and several equivalent definitions are given.Every bounded closed convex set in a uniformly convex in every direction space is proved to have a normal structure.%引进了局部凸空间中方向一致凸的概念,给出了相关的几个等价定义,证明了方向一致凸的局部凸空间的任一有界闭凸集具有正规结构。
Ishikawa iteration process with errors for nonexpansive mappings in uniformly convex Banach spaces
Deng Lei; Li Shenghong
2000-01-01
We shall consider the behaviour of Ishikawa iteration with errors in a uniformly convex Banach space. Then we generalize the two theorems of Tan and Xu without the restrictions that C is bounded and limsupnsn
Cyclic pairs and common best proximity points in uniformly convex Banach spaces
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Gabeleh Moosa
2017-06-01
Full Text Available In this article, we survey the existence, uniqueness and convergence of a common best proximity point for a cyclic pair of mappings, which is equivalent to study of a solution for a nonlinear programming problem in the setting of uniformly convex Banach spaces. Finally, we provide an extension of Edelstein’s fixed point theorem in strictly convex Banach spaces. Examples are given to illustrate our main conclusions.
Invariant and semi-invariant probabilistic normed spaces
Energy Technology Data Exchange (ETDEWEB)
Ghaemi, M.B. [School of Mathematics Iran, University of Science and Technology, Narmak, Tehran (Iran, Islamic Republic of)], E-mail: mghaemi@iust.ac.ir; Lafuerza-Guillen, B. [Departamento de Estadistica y Matematica Aplicada, Universidad de Almeria, Almeria E-04120 (Spain)], E-mail: blafuerz@ual.es; Saiedinezhad, S. [School of Mathematics Iran, University of Science and Technology, Narmak, Tehran (Iran, Islamic Republic of)], E-mail: ssaiedinezhad@yahoo.com
2009-10-15
Probabilistic metric spaces were introduced by Karl Menger. Alsina, Schweizer and Sklar gave a general definition of probabilistic normed space based on the definition of Menger . We introduce the concept of semi-invariance among the PN spaces. In this paper we will find a sufficient condition for some PN spaces to be semi-invariant. We will show that PN spaces are normal spaces. Urysohn's lemma, and Tietze extension theorem for them are proved.
Ayupov, Sh A
2011-01-01
In the present article we prove a fixed point theorem for reflections of compact convex sets and give a new characterization of state space of JB-algebras among compact convex sets. Namely they are exactly those compact convex sets which are strongly spectral and symmetric.
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Messaoud Bounkhel
2015-01-01
Full Text Available The present paper is devoted to the study of the generalized projection πK:X∗→K, where X is a uniformly convex and uniformly smooth Banach space and K is a nonempty closed (not necessarily convex set in X. Our main result is the density of the points x∗∈X∗ having unique generalized projection over nonempty close sets in X. Some minimisation principles are also established. An application to variational problems with nonconvex sets is presented.
Messaoud Bounkhel
2015-01-01
The present paper is devoted to the study of the generalized projection πK:X∗→K, where X is a uniformly convex and uniformly smooth Banach space and K is a nonempty closed (not necessarily convex) set in X. Our main result is the density of the points x∗∈X∗ having unique generalized projection over nonempty close sets in X. Some minimisation principles are also established. An application to variational problems with nonconvex sets is presented.
Modified Block Iterative Algorithm for Solving Convex Feasibility Problems in Banach Spaces
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Kim JongKyu
2010-01-01
Full Text Available The purpose of this paper is to use the modified block iterative method to propose an algorithm for solving the convex feasibility problems for an infinite family of quasi- -asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in uniformly smooth and strictly convex Banach spaces with Kadec-Klee property. The results presented in the paper improve and extend some recent results.
Modified Block Iterative Algorithm for Solving Convex Feasibility Problems in Banach Spaces
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Shih-sen Chang
2010-01-01
Full Text Available The purpose of this paper is to use the modified block iterative method to propose an algorithm for solving the convex feasibility problems for an infinite family of quasi-ϕ-asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in uniformly smooth and strictly convex Banach spaces with Kadec-Klee property. The results presented in the paper improve and extend some recent results.
Widder-Arendt theorem and integrated semigroups in locally convex space
Institute of Scientific and Technical Information of China (English)
肖体俊; 梁进
1996-01-01
A well-known result established by Arendt in 1987 with regard to the Laplace transforms in Banach spaces is developed. A Widder-Arendt theorem in the setting of sequentially complete locally convex spaces is set up (Theorem 1.1). Moreover, integrated semigroups in such spaces are introduced and generation theorems and some basic properties for semigroups of this type are obtained. As examples, elliptic differential operators on certain classes of function spaces with locally convex topology are shown to be the generators of integrated semigroups under some conditions.
On k-nearly uniform convex property in generalized CesÃƒÂ ro sequence spaces
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Suthep Suantai
2003-10-01
Full Text Available We define a generalized CesÃƒÂ ro sequence space ces(p, where p=(pk is a bounded sequence of positive real numbers, and consider it equipped with the Luxemburg norm. The main purpose of this paper is to show that ces(p is k-nearly uniform convex (k-NUC for kÃ¢Â‰Â¥2 when limnÃ¢Â†Â’Ã¢ÂˆÂžinfpn>1. Moreover, we also obtain that the CesÃƒÂ ro sequence space cesp(whereÃ¢Â€Â‰1
The Fermat-Torricelli problem in normed planes and spaces
Martini, Horst; Weiss, Gunter
2002-01-01
We investigate the Fermat-Torricelli problem in d-dimensional real normed spaces or Minkowski spaces, mainly for d=2. Our approach is to study the Fermat-Torricelli locus in a geometric way. We present many new results, as well as give an exposition of known results that are scattered in various sources, with proofs for some of them. Together, these results can be considered to be a minitheory of the Fermat-Torricelli problem in Minkowski spaces and especially in Minkowski planes. This demonstrates that substantial results about locational problems valid for all norms can be found using a geometric approach.
NORMALLY DISTRIBUTED PROBABILITY MEASURE ON THE METRIC SPACE OF NORMS
Institute of Scientific and Technical Information of China (English)
Á.G. HORVÁTH
2013-01-01
In this paper we propose a method to construct probability measures on the space of convex bodies. For this purpose, first, we introduce the notion of thinness of a body. Then we show the existence of a measure with the property that its pushforward by the thinness function is a probability measure of truncated normal distribution. Finally, we improve this method to find a measure satisfying some important properties in geometric measure theory.
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Cui Yunan
2011-01-01
Full Text Available Abstract Uniformly convex W-hyperbolic spaces with monotone modulus of uniform convexity are a natural generalization of both uniformly convexnormed spaces and CAT(0 spaces. In this article, we discuss the existence of fixed points and demiclosed principle for mappings of asymptotically non-expansive type in uniformly convex W-hyperbolic spaces with monotone modulus of uniform convexity. We also obtain a Δ-convergence theorem of Krasnoselski-Mann iteration for continuous mappings of asymptotically nonexpansive type in CAT(0 spaces. MSC: 47H09; 47H10; 54E40
Midpoint locally uniformly convexity on locally convex spaces%关于局部凸空间的中点局部一致凸性
Institute of Scientific and Technical Information of China (English)
陈利国; 罗成
2011-01-01
The notions of（weakly） midpoint locally uniformly convexity on locally convex spaces are introduced.It is proved that the dual property between（weakly） midpoint locally uniformly convexity and（weakly） midpoint locally uniformly smoothness,and disscuss the relationship between them and other convexity.Corresponding notions and results in Banach space is generalized.%给出局部凸空间的（弱）中点局部一致凸性,证明了它与（弱）中点局部一致光滑性具有对偶性质,讨论它们与其它凸性之间的关系,推广了Banach空间相应概念和结果.
Difference equations in normed spaces stability and oscillations
Gil, Michael
2007-01-01
Difference equations appear as natural descriptions of observed evolution phenomena because most measurements of time evolving variables are discrete. They also appear in the applications of discretization methods for differential, integral and integro-differential equations. The application of the theory of difference equations is rapidly increasing to various fields, such as numerical analysis, control theory, finite mathematics, and computer sciences. This book is devoted to linear and nonlinear difference equations in a normed space. The main methodology presented in this book is based on a combined use of recent norm estimates for operator-valued functions with the following methods and results: The freezing methodThe Liapunov type equationThe method of majorantsThe multiplicative representation of solutionsDeals systematically with difference equations in normed spaces Considers new classes of equations that could not be studied in the frameworks of ordinary and partial difference equationsDevelops ...
Institute of Scientific and Technical Information of China (English)
李佳霖; 崔云安
2013-01-01
含有k-一致凸性单调模的k-一致凸弱双曲空间是包含一致凸赋范空间和CAT(0)空间的一致凸弱双曲空间的自然归纳.针对这个空间,我们研究讨论k一致凸弱双曲空间以及其上渐近非扩张映射的不动点存在性问题.这些结果也同样推广了一致凸弱双曲空间和CAT(0)空间上相应的近期的一些理论结果.%K-uniformly convex W-hyperbolic spaces with monotone modulus of k-uniform convexity are natural generalization of uniformly convex W-hyperbolic spaces which contain both of uniformly convex normed spaces and CAT(0) spaces.We discuss some properties of K-uniformly convex W-hyperbolic spaces and the existence of fixed points for asymptotically nonexpansive mappings.These results extend the corresponding recent results in uniformly convex W-hyperbolic spaces and CAT(0) spaces.
K-Fuzzy赋范空间与WF-Fuzzy赋范空间%K-Fuzzy Normed Space and WF-Fuzzy Normed Space
Institute of Scientific and Technical Information of China (English)
方锦暄; 郭翀琦
2000-01-01
The relations between the K-fuzzy norme d sp ace and the WF-fuzzy normed space are further considered.The Hausdorff K -fuzzy normed space and the WF-fuzzy normed space are essentially consis tent.%研究了K-fuzzy赋范空间与WF-fuzzy赋范空间之间的关系,证明了Hausdorff的K-fuzzy赋范空间与WF-fuzzy赋范空间本质上是一致的.
Convex functions, monotone operators and differentiability
Phelps, Robert R
1993-01-01
The improved and expanded second edition contains expositions of some major results which have been obtained in the years since the 1st edition. Theaffirmative answer by Preiss of the decades old question of whether a Banachspace with an equivalent Gateaux differentiable norm is a weak Asplund space. The startlingly simple proof by Simons of Rockafellar's fundamental maximal monotonicity theorem for subdifferentials of convex functions. The exciting new version of the useful Borwein-Preiss smooth variational principle due to Godefroy, Deville and Zizler. The material is accessible to students who have had a course in Functional Analysis; indeed, the first edition has been used in numerous graduate seminars. Starting with convex functions on the line, it leads to interconnected topics in convexity, differentiability and subdifferentiability of convex functions in Banach spaces, generic continuity of monotone operators, geometry of Banach spaces and the Radon-Nikodym property, convex analysis, variational princ...
A Fixed Point Theorem for Set-Valued Mapping in Abstract Convex Space with Application
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FAN Xiao Dong; XIANG Shu Wen
2009-01-01
A new fixed point theorem and the selection property for upper semi-continuous setvalued mappings in abstract convexity space are established. As their applications the existence of Nash equilibrium for n-person non-cooperative generalized games is proved.
Continuous solutions for fractional integral inclusion in locally convex topological space
Institute of Scientific and Technical Information of China (English)
Rabha W. Ibrahim
2009-01-01
The existence of continuous solutions for fractional integral inclusion via its singlevalued problem and fixed point theorem for set-valued function in locally convex topological spaces is discussed. The proof of the single-valued problem will be based on the Leray- Schauder fixed point theorem. Moreover, the controllability of this solution is studied.
Non-expansive Mappings and Iterative Methods in Uniformly Convex Banach Spaces
Institute of Scientific and Technical Information of China (English)
Hai Yun ZHOU
2004-01-01
In this article, we will investigate the properties of iterative sequence for non-expansive mappings and present several strong and weak convergence results of successive approximations to fixed points of non-expansive mappings in uniformly convex Banach spaces. The results presented in this article generalize and improve various ones concerned with constructive techniques for the fixed points of non-expansive mappings.
Weighted composition operators between growth spaces on circular and strictly convex domain
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Shayesteh Rezaei
2015-06-01
Full Text Available Let $Omega_X$ be a bounded, circular and strictly convex domain of a Banach space $X$ and $mathcal{H}(Omega_X$ denote the space of all holomorphic functions defined on $Omega_X$. The growth space $mathcal{A}^omega(Omega_X$ is the space of all $finmathcal{H}(Omega_X$ for which $$|f(x|leqslant C omega(r_{Omega_X}(x,quad xin Omega_X,$$ for some constant $C>0$, whenever $r_{Omega_X}$ is the Minkowski functional on $Omega_X$ and $omega :[0,1rightarrow(0,infty$ is a nondecreasing, continuous and unbounded function. Boundedness and compactness of weighted composition operators between growth spaces on circular and strictly convex domains were investigated.
Local Uniform Convexity and Kadec-Klee Type Properties in K-interpolation spaces II
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Peter G. Dodds
2004-01-01
Full Text Available We study local uniform convexity and Kadec-Klee type properties in K-interpolation spaces of Lorentz couples. We show that a wide class of Banach couples of (commutative and non-commutative Lorentz spaces possess the (so-alled (DGL-property originally introduced by Davis, Ghoussoub and Lindenstrauss in the context of renorming order continuous Banach latties. This property is used as a key tool to show that local uniform convexity and certain Kadec-Klee type properties in non-commutative symmetric spaces of measurable operators may be inferred from corresponding properties of the parameter space of the K-interpolation method. Further applications are given to renorming properties of separable symmetric Banach function spaces and their non-commutative counterparts.
Institute of Scientific and Technical Information of China (English)
贺飞; 刘德; 罗成
2007-01-01
在局部凸空间框架下,我们利用Drop定理,Phelps引理和Ekeland交分原理的赋范线性空间的形式对其分别进行了推广.并且阐述了这些定理之间以及和它们赋范线性空间的形式之间是等价的.%In locally convex spaces, we extend drop theorem, Phelps' lemma and Ekeland's principle by using their own normed linear spaces versions. Moreover we show that these theorems are equivalent to each other and to their normed linear spaces counterparts.
On growth and covering theorems of quasi-convex mappings in the unit ball of a complex Banach space
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张文俊; 刘太顺
2002-01-01
A class of biholomorphic mappings named "quasi-convex mapping" is introduced in the unitball of a complex Banach space. It is proved that this class of mappings is a proper subset of the class ofstarlike mappings and contains the class of convex mappings properly, and it has the same growth and coveringtheorems as the convex mappings. Furthermore, when the Banach space is confined to Cn, the "quasi-convexmapping" is exactly the "quasi-convex mapping of type A" introduced by K. A. Roper and T. J. Suffridge.
On intrinsic geometry of surfaces in normed spaces
Burago, Dmitri; Ivanov, Sergei
2010-01-01
We prove three facts about intrinsic geometry of surfaces in a normed (Minkowski) space. When put together, these facts demonstrate a rather intriguing picture. We show that (1) geodesics on saddle surfaces (in a space of any dimension) behave as they are expected to: they have no conjugate points and thus minimize length in their homotopy class; (2) in contrast, every two-dimensional Finsler manifold can be locally embedded as a saddle surface in a 4-dimensional space; and (3) geodesics on c...
Statistical Λ-Convergence in Probabilistic Normed Spaces
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M. Aldhaifallah
2017-01-01
Full Text Available The main objective of the study was to understand the notion of Λ-convergence and to study the notion of probabilistic normed (PN spaces. The study has also aimed to define the statistical Λ-convergence and statistical Λ-Cauchy in PN-spaces. The concepts of these approaches have been defined by some examples, which have demonstrated the concepts of statistical Λ-convergence and statistical Λ-Cauchy in PN-spaces. Previous studies have also been used to understand similar terminologies and notations for the extraction of outcomes.
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唐献秀; 林尤武; 吴建功
2011-01-01
给出局部凸空间平均一致凸性的一些等价刻画与某些凸性的关系.%We obtained some necessary and sufficient conditions for average uniform convexity in locally converx spaces.At the same time,we discussed the relationship between this convexity and some other convexity.
On Ideal Convergence of Double Sequences in Probabilistic Normed Spaces
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Vijay KUMAR; Bernardo LAFUERZA-GUILL(E)N
2012-01-01
The notion of ideal convergence is a generalization of statistical convergence which has been intensively investigated in last few years.For an admissible ideal ∮ (∈) N × N,the aim of the present paper is to introduce the concepts of ∮-convergence and ∮*-convergence for double sequences on probabilistic normed spaces (PN spaces for short).We give some relations related to these notions and find condition on the ideal ∮ for which both the notions coincide.We also define ∮-Cauchy and ∮*-Cauchy double sequences on PN spaces and show that ∮-convergent double sequences are ∮-Cauchy on these spaces.We establish example which shows that our method of convergence for double sequences on PN spaces is more general.
The free abelian topological group and the free locally convex space on the unit interval
Leiderman, A G; Pestov, V G
1992-01-01
We give a complete description of the topological spaces $X$ such that the free abelian topological group $A(X)$ embeds into the free abelian topological group $A(I)$ of the closed unit interval. In particular, the free abelian topological group $A(X)$ of any finite-dimensional compact metrizable space $X$ embeds into $A(I)$. The situation turns out to be somewhat different for free locally convex spaces. Some results for the spaces of continuous functions with the pointwise topology are also obtained. Proofs are based on the classical Kolmogorov's Superposition Theorem.
The free abelian topological group and the free locally convex space on the unit interval
Leiderman, A. G.; Morris, S. A.; Pestov, V. G.
1992-01-01
We give a complete description of the topological spaces $X$ such that the free abelian topological group $A(X)$ embeds into the free abelian topological group $A(I)$ of the closed unit interval. In particular, the free abelian topological group $A(X)$ of any finite-dimensional compact metrizable space $X$ embeds into $A(I)$. The situation turns out to be somewhat different for free locally convex spaces. Some results for the spaces of continuous functions with the pointwise topology are also...
Tao, L; Nicholson, C
2004-07-07
Brain extracellular space (ECS) constitutes a porous medium in which diffusion is subject to hindrance, described by tortuosity, lambda = (D/D*)1/2, where D is the free diffusion coefficient and D* is the effective diffusion coefficient in brain. Experiments show that lambda is typically 1.6 in normal brain tissue although variations occur in specialized brain regions. In contrast, different theoretical models of cellular assemblies give ambiguous results: they either predict lambda-values similar to experimental data or indicate values of about 1.2. Here we constructed three different ECS geometries involving tens of thousands of cells and performed Monte Carlo simulation of 3-D diffusion. We conclude that the geometrical hindrance in the ECS surrounding uniformly spaced convex cells is independent of the cell shape and only depends on the volume fraction alpha (the ratio of the ECS volume to the whole tissue volume). This dependence can be described by the relation lambda = ((3-alpha)/2)1/2, indicating that the geometrical hindrance in such ECS cannot account for lambda > 1.225. Reasons for the discrepancy between the theoretical and experimental tortuosity values are discussed.
On nonlinear stability in various random normed spaces
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Saadati Reza
2011-01-01
Full Text Available Abstract In this article, we prove the nonlinear stability of the quartic functional equation 1 6 f ( x + 4 y + f ( 4 x - y = 3 0 6 9 f x + y 3 + f ( x + 2 y (1 + 1 3 6 f ( x - y - 1 3 9 4 f ( x + y + 4 2 5 f ( y - 1 5 3 0 f ( x (2 (3 in the setting of random normed spaces Furthermore, the interdisciplinary relation among the theory of random spaces, the theory of non-Archimedean space, the theory of fixed point theory, the theory of intuitionistic spaces and the theory of functional equations are also presented in the article.
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Laowang Worawut
2011-01-01
Full Text Available Abstract Abkar and Eslamian (Nonlinear Anal. TMA, 74, 1835-1840, 2011 prove that if K is a nonempty bounded closed convex subset of a complete CAT(0 space X, t : K → K is a single-valued quasi-nonexpansive mapping and T : K → KC(K is a multivalued mapping satisfying conditions (E and (Cλ for some λ ∈ (0, 1 such that t and T commute weakly, then there exists a point z ∈ K such that z = t(z ∈ T(z. In this paper, we extend this result to the general setting of uniformly convex metric spaces. Nevertheless, condition (E of T can be weakened to the strongly demiclosedness of I - T.
Alabiso, Carlo
2015-01-01
This book is an introduction to the theory of Hilbert space, a fundamental tool for non-relativistic quantum mechanics. Linear, topological, metric, and normed spaces are all addressed in detail, in a rigorous but reader-friendly fashion. The rationale for an introduction to the theory of Hilbert space, rather than a detailed study of Hilbert space theory itself, resides in the very high mathematical difficulty of even the simplest physical case. Within an ordinary graduate course in physics there is insufficient time to cover the theory of Hilbert spaces and operators, as well as distribution theory, with sufficient mathematical rigor. Compromises must be found between full rigor and practical use of the instruments. The book is based on the author's lessons on functional analysis for graduate students in physics. It will equip the reader to approach Hilbert space and, subsequently, rigged Hilbert space, with a more practical attitude. With respect to the original lectures, the mathematical flavor in all sub...
Botelho, Fabio
2014-01-01
This book introduces the basic concepts of real and functional analysis. It presents the fundamentals of the calculus of variations, convex analysis, duality, and optimization that are necessary to develop applications to physics and engineering problems. The book includes introductory and advanced concepts in measure and integration, as well as an introduction to Sobolev spaces. The problems presented are nonlinear, with non-convex variational formulation. Notably, the primal global minima may not be attained in some situations, in which cases the solution of the dual problem corresponds to an appropriate weak cluster point of minimizing sequences for the primal one. Indeed, the dual approach more readily facilitates numerical computations for some of the selected models. While intended primarily for applied mathematicians, the text will also be of interest to engineers, physicists, and other researchers in related fields.
On Hereditarily Indecomposable Banach Spaces
Institute of Scientific and Technical Information of China (English)
Li Xin CHENG; Huai Jie ZHONG
2006-01-01
This paper shows that every non-separable hereditarily indecomposable Banach space admits an equivalent strictly convex norm, but its bi-dual can never have such a one; consequently, every non-separable hereditarily indecomposable Banach space has no equivalent locally uniformly convex norm.
Institute of Scientific and Technical Information of China (English)
田有先; 张石生
2002-01-01
Some convergence theorems of Ishikawa type iterative sequence with errors for nonlinear general quasi-contractive mapping in convex metric spaces are proved. The results not only extend and improve the corresponding results of L. B . Ciric , Q. H. Liu , H. E.Rhoades and H. K . Xu , et al., but also give an affirmative answer to the open question of Rhoades-Naimpally-Singh in convex metric spaces.
A G-KKM type theorem and its applications to minimax inequalities on G-convex spaces
Directory of Open Access Journals (Sweden)
Mohammad S. R. Chowdhury
1998-01-01
Full Text Available A G-KKM type theorem is obtained on G-convex spaces. As application, a generalization of Ky Fan's minimax inequality to non-compact sets on G-convex spaces is first obtained. As special cases of this minimax inequality, some new minimax inequalites are obtained. Four fixed point theorems and four equivalent formulations of the second minimax inequality are also obtained.
A Note on The Convexity of Chebyshev Sets
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Sangeeta
2009-07-01
Full Text Available Perhaps one of the major unsolved problem in Approximation Theoryis: Whether or not every Chebyshev subset of a Hilbert space must be convex. Many partial answers to this problem are available in the literature. R.R. Phelps[Proc. Amer. Math. Soc. 8 (1957, 790-797] showed that a Chebyshev set in an inner product space (or in a strictly convex normed linear space is convex if the associated metric projection is non-expansive. We extend this result to metricspaces.
Kashima, Yohei
2011-01-01
Subdifferentials of a singular convex functional representing the surface free energy of a crystal under the roughening temperature are characterized. The energy functional is defined on Sobolev spaces of order -1, so the subdifferential mathematically formulates the energy's gradient which formally involves 4th order spacial derivatives of the surface's height. The subdifferentials are analyzed in the negative Sobolev spaces of arbitrary spacial dimension on which both a periodic boundary condition and a Dirichlet boundary condition are separately imposed. Based on the characterization theorem of subdifferentials, the smallest element contained in the subdifferential of the energy for a spherically symmetric surface is calculated under the Dirichlet boundary condition.
Stability of Various Functional Equations in Non-Archimedean Intuitionistic Fuzzy Normed Spaces
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Syed Abdul Mohiuddine
2012-01-01
Full Text Available We define and study the concept of non-Archimedean intuitionistic fuzzy normed space by using the idea of t-norm and t-conorm. Furthermore, by using the non-Archimedean intuitionistic fuzzy normed space, we investigate the stability of various functional equations. That is, we determine some stability results concerning the Cauchy, Jensen and its Pexiderized functional equations in the framework of non-Archimedean IFN spaces.
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Aunyarat Bunyawat
2012-01-01
Full Text Available We introduce an iterative method for finding a common fixed point of a countable family of multivalued quasi-nonexpansive mapping {Ti} in a uniformly convex Banach space. We prove that under certain control conditions, the iterative sequence generated by our method is an approximating fixed point sequence of each Ti. Some strong convergence theorems of the proposed method are also obtained for the following cases: all Ti are continuous and one of Ti is hemicompact, and the domain K is compact.
Institute of Scientific and Technical Information of China (English)
Wen ZHANG
2012-01-01
By a ball-covering B of a Banach space X,we mean thatB is a collection of open (or closed)balls off the origin whose union contains the unit sphere of X; and X is said to have the ball-covering property provided it admits a ball-covering of countably many balls.This paper shows that universal finite representability and B-convexity of X can be characterized by properties of ball-coverings of its finite dimensioral subspaces.
Institute of Scientific and Technical Information of China (English)
ISAC G.; LI Jin-lu
2005-01-01
The notion of"exceptional family of elements (EFE)" plays a very important role in solving complementarity problems. It has been applied in finite dimensional spaces and Hilbert spaces by many authors. In this paper, by using the generalized projection defined by Alber, we extend this notion from Hilbert spaces to uniformly smooth and uniformly convex Banach spaces,and apply this extension to the study of nonlinear complementarity problems in Banach spaces.
The Arc Distortion in QH Inner Ψ-uniform (or Convex) Domains in Real Banach Spaces
Institute of Scientific and Technical Information of China (English)
Man Zi HUANG; Xian Tao WANG
2011-01-01
Let D and D' be domains in real Banach spaces of dimension at least 2.The main aim of this paper is to study certain arc distortion properties in the quasihyperbolic metric defined in real Banach spaces.In particular,when D' is a QH inner Ψ-uniform domain with Ψ being a slow (or a convex domain),we investigate the following:For positive constants c,h,C,M,suppose a homeomorphism f:D → D' takes each of the 10-neargeodesics in D to (c,h)-solid in D'.Then f is C-coarsely MLipschitz in the quasihyperbolic metric.These are generalizations of the corresponding result obtained recently by V(a)is(a)l(a).
A maximin characterization of the escape rate of nonexpansive mappings in metrically convex spaces
Gaubert, Stephane
2010-01-01
We establish a maximin characterization of the linear escape rate of the orbits of a nonexpansive mapping on a complete (hemi-)metric space, under a mild form of Busemann's nonpositive curvature condition (we require a distinguished family of geodesics with a common origin to satisfy a convexity inequality). This characterization, which involves horofunctions, generalizes the Collatz-Wielandt characterization of the spectral radius of a nonnegative matrix. It yields as corollaries a theorem of Kohlberg and Neyman (1981), concerning nonexpansive maps in Banach spaces, a variant of a Denjoy-Wolff type theorem of Karlsson (2001), together with a refinement of a theorem of Gunawardena and Walsh (2003), concerning order-preserving positively homogeneous self-maps of symmetric cones.
Fixed Point of Generalized Eventual Cyclic Gross in Fuzzy Norm Spaces for Contractive Mappings
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S. A. M. Mohsenialhosseini
2015-01-01
Full Text Available We define generalized eventual cyclic gross contractive mapping in fuzzy norm spaces, which is a generalization of the eventual cyclic gross contractions. Also we prove the existence of a fixed point for this type of contractive mapping on fuzzy norm spaces.
Stability of Pexiderized Quadratic Functional Equation in Random 2-Normed Spaces
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Mohammed A. Alghamdi
2015-01-01
Full Text Available The aim of this paper is to investigate the stability of Hyers-Ulam-Rassias type theorems by considering the pexiderized quadratic functional equation in the setting of random 2-normed spaces (RTNS, while the concept of random 2-normed space has been recently studied by Goleţ (2005.
Rathee, Savita; Dhingra, Kusum; Kumar, Anil
2016-01-01
Here, we extend the notion of (E.A.) property in a convex metric space defined by Kumar and Rathee (Fixed Point Theory Appl 1-14, 2014) by introducing a new class of self-maps which satisfies the common property (E.A.) in the context of convex metric space and ensure the existence of common fixed point for this newly introduced class of self-maps. Also, we guarantee the existence of common best proximity points for this class of maps satisfying generalized non-expansive type condition. We furnish an example in support of the proved results.
Institute of Scientific and Technical Information of China (English)
陈利国; 罗成
2011-01-01
首先引入局部凸空间的k-一致极凸性和k-一致极光滑性这一对对偶概念,它们既是Banach空间κ-一致极凸性和k-一致极光滑性推广,又是局部凸空间一致极凸性和一致极光滑性的自然推广.其次讨论它们与其它k-凸性(k-光滑性)之间的关系.最后,在p-自反的条件下给出它们之间的等价对偶定理.%The dual notions of fc-uniformly extreme convexity and fc-uniformly extreme smoothness are introduced In locally convex spaces, which are generalization of both k-uniformly extreme convexity (fc-uniformly extreme smoothness) in Banach spaces and uniformly extreme convexity (uniformly extreme smoothness) In locally convex spaces. The relationship between them and the other convexity (smoothness) are discussed. In addition, on the conditions of P-reflexivity, we obtain further the equivalent dual theorem of between fc-uniformly extreme convexity and k- uniformly extreme smoothness.
Lp stability for entropy solutions of scalar conservation laws with strict convex flux
Adimurthi; Ghoshal, Shyam Sundar; Veerappa Gowda, G. D.
Here we consider the scalar convex conservation laws in one space dimension with strictly convex flux which is in C1. Existence, uniqueness and L1 contractivity were proved by Kružkov [14]. Using the relative entropy method, Leger showed that for a uniformly convex flux and for the shock wave solutions, the L2 norm of a perturbed solution relative to the shock wave is bounded by the L2 norm of the initial perturbation. Here we generalize the result to Lp norm for all 1⩽p<∞. Also we show that for the non-shock wave solution, Lp norm of the perturbed solution relative to the modified N-wave is bounded by the Lp norm of the initial perturbation for all 1⩽p<∞.
2-一致凸Banach空间的特征不等式%A Character Inequality of the 2-Uniformly Convex Banach Space
Institute of Scientific and Technical Information of China (English)
李婷婷; 乌日娜; 苏雅拉图
2012-01-01
In this paper,the character inequality in the 2-uniformly convex Banach space is studied. By the method of Banach space theory, this problem is discussed in the 2- uniformly convex Banach space, a character inequality of the 2-uniformly convex Banach space is obtained by combining the definition of 2- uniformly convex Banach space. Moreover, a corresponding result in the locally 2-uniformly convex Banach space is established.%研究了2-一致凸Banach空间的特征不等式问题.在2-一致凸Banach空间中,利用Banach空间理论的方法,结合2-一致凸Banach空间的定义,给出了2-一致凸Banach空间X的一个特征不等式,并将此结果推广到局部2-一致凸Banach空间的情形.
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Zhi-Ang Zhou
2013-01-01
Full Text Available A new notion of the ic-cone convexlike set-valued map characterized by the algebraic interior and the vector closure is introduced in real ordered linear spaces. The relationship between the ic-cone convexlike set-valued map and the nearly cone subconvexlike set-valued map is established. The results in this paper generalize some known results in the literature from locally convex spaces to linear spaces.
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M. Eshaghi Gordji
2011-01-01
Full Text Available We prove the generalized Hyers-Ulam-Rassias stability of a general system of Euler-Lagrange-type quadratic functional equations in non-Archimedean 2-normed spaces and Menger probabilistic non-Archimedean-normed spaces.
Institute of Scientific and Technical Information of China (English)
胡正平; 王玲丽
2012-01-01
In order to construct a high-dimensional data approximate model in the purpose of the best coverage of the distribution of high-dimensional samples, the classification algorithm of multiple observation samples based on LI norm convex hull data description is proposed. The convex hull for each class in the train set and multiple observation samples in the test set is constructed as the first step. So the classification of multiple observation samples is transformed to the similarity of convex hulls. If the test convex hull and every train hull are not overlapping, LI norm distance measure is used to solve the similarity of convex hulls. Otherwise, LI norm distance measure is used to solve the similarity of reduced convex hulls. Then the nearest neighbor classifier is used to solve the classification of multiple observation samples. Experiments on three types of databases show that the proposed method is valid and efficient.%为建立高维空间样本分布的最佳覆盖为目标来实现覆盖分类,该文提出基于L1范数凸包数据描述的多观测样本分类算法.首先对训练集的每个类别以及测试集的多观测样本分别构造凸包模型,这样多观测样本的分类就转化为凸包模型的相似性度量问题.若测试集的凸包模型与训练集无重叠,采用L1范数距离测度进行凸包模型之间的相似性度量;若有重叠,采用L1范数距离测度进行收缩凸包(reduced convex hulls)之间的相似性度量.然后采用最近邻准则作为多观测样本的分类决策.在3个数据库上进行的实验结果,表明该文提出方法对于多观测样本分类具有可行性和有效性.
Vector-Valued almost Convergence and Classical Properties in Normed Spaces
Indian Academy of Sciences (India)
A Aizpuru; R Armario; F J Garcia-Pacheco; F J Perez-Fernandez
2014-02-01
In this paper we study the almost convergence and the almost summability in normed spaces. Among other things, spaces of sequences defined by the almost convergence and the almost summability are proved to be complete if the basis normed space is so. Finally, some classical properties such as completeness, reflexivity, Schur property, Grothendieck property, and the property of containing a copy of 0 are characterized in terms of the almost convergence.
Institute of Scientific and Technical Information of China (English)
Kai Ting WEN
2011-01-01
In this paper,a new Ky Fan matching theorem is established in noncompact L-convex spaces.As applications,a fixed point theorem and equilibrium existence theorems for systems of general quasiequilibrium problems and systems of quasiequilibrium problems in noncompact L-convex spaces are obtained.
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Gurucharan Singh Saluja
2010-01-01
Full Text Available In this paper, we give some necessary and sufficient conditions for an implicit iteration process with errors for a finite family of asymptotically quasi-nonexpansive mappings converging to a common fixed of the mappings in convex metric spaces. Our results extend and improve some recent results of Sun, Wittmann, Xu and Ori, and Zhou and Chang.
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Plern Saipara
2017-03-01
Full Text Available In this paper, we suggest the modified random S-iterative process and prove the common random fixed point theorems of a finite family of random uniformly quasi-Lipschitzian operators in a generalized convex metric space. Our results improves and extends various results in the literature.
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Meng Wen
2012-01-01
Full Text Available We introduce a new iterative scheme with Meir-Keeler contractions for an asymptotically nonexpansive mapping in -uniformly smooth and strictly convex Banach spaces. We also proved the strong convergence theorems of implicit and explicit schemes. The results obtained in this paper extend and improve many recent ones announced by many others.
A Convex Formulation for Magnetic Particle Imaging X-Space Reconstruction.
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Justin J Konkle
Full Text Available Magnetic Particle Imaging (mpi is an emerging imaging modality with exceptional promise for clinical applications in rapid angiography, cell therapy tracking, cancer imaging, and inflammation imaging. Recent publications have demonstrated quantitative mpi across rat sized fields of view with x-space reconstruction methods. Critical to any medical imaging technology is the reliability and accuracy of image reconstruction. Because the average value of the mpi signal is lost during direct-feedthrough signal filtering, mpi reconstruction algorithms must recover this zero-frequency value. Prior x-space mpi recovery techniques were limited to 1d approaches which could introduce artifacts when reconstructing a 3d image. In this paper, we formulate x-space reconstruction as a 3d convex optimization problem and apply robust a priori knowledge of image smoothness and non-negativity to reduce non-physical banding and haze artifacts. We conclude with a discussion of the powerful extensibility of the presented formulation for future applications.
Institute of Scientific and Technical Information of China (English)
林尤武; 魏文展; 唐献秀; 吴建功
2011-01-01
引进局部凸空间平均局部一致凸性的概念,给出其对偶的定义,即局部凸空间平均局部一致光滑性,并在p-自反的条件下得到它们之间的对偶定理,即(X,P)是平均局部一致凸(平均局部一致光滑)的当且仅当(X＇,P′)是平均局部一致光滑(平均局部一致凸)的.%The definition of average locally uniform convexity in locally convex spaces is introduced, and we give the definition of average locally uniform smoothness in locally convex spaces as the dual of average locally uniform convexity. In addition,on the conditions of P - reflexivity, we obtained the dual relations between average locally uniform convexity and average locally u-niform smoothness. Also, (X,P) is average locally uniform convexity (average locally uniform smoothness) if and only if (X ,P ) is average locally uniform smoothness (average local uniform convexity).
Bornological Locally Convex Cones
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Davood Ayaseh
2014-10-01
Full Text Available In this paper we define bornological and b-bornological cones and investigate their properties. We give some characterization for these cones. In the special case of locally convex topological vector space both these concepts reduce to the known concept of bornological spaces. We introduce and investigate the convex quasiuniform structures U_{tau}, U_{sigma}(P,P* and \\U_{beta}(P,P* on locally convex cone (P,U.
Nedjar, Sebastien; Cicchetti, Rosine; Lakhal, Lotfi; 10.3166/isi.11.6.11-31
2010-01-01
In various approaches, data cubes are pre-computed in order to answer efficiently OLAP queries. The notion of data cube has been declined in various ways: iceberg cubes, range cubes or differential cubes. In this paper, we introduce the concept of convex cube which captures all the tuples of a datacube satisfying a constraint combination. It can be represented in a very compact way in order to optimize both computation time and required storage space. The convex cube is not an additional structure appended to the list of cube variants but we propose it as a unifying structure that we use to characterize, in a simple, sound and homogeneous way, the other quoted types of cubes. Finally, we introduce the concept of emerging cube which captures the significant trend inversions. characterizations.
Fast inference of ill-posed problems within a convex space
Fernandez-de-Cossio-Diaz, J.; Mulet, R.
2016-07-01
In multiple scientific and technological applications we face the problem of having low dimensional data to be justified by a linear model defined in a high dimensional parameter space. The difference in dimensionality makes the problem ill-defined: the model is consistent with the data for many values of its parameters. The objective is to find the probability distribution of parameter values consistent with the data, a problem that can be cast as the exploration of a high dimensional convex polytope. In this work we introduce a novel algorithm to solve this problem efficiently. It provides results that are statistically indistinguishable from currently used numerical techniques while its running time scales linearly with the system size. We show that the algorithm performs robustly in many abstract and practical applications. As working examples we simulate the effects of restricting reaction fluxes on the space of feasible phenotypes of a genome scale Escherichia coli metabolic network and infer the traffic flow between origin and destination nodes in a real communication network.
A New Section Theorem in L-convex Spaces and Its Applications%建立在L-凸空间上的截口定理及其应用
Institute of Scientific and Technical Information of China (English)
金彩云; 程曹宗
2007-01-01
In this paper, the author gives a new section theorem in L-convex spaces. And as its applications, the author proves a coincident theorem and a two-functional minimax theorem established in L-convex spaces.
DERIVATIVES OF HARMONIC MIXED NORM AND BLOCH-TYPE SPACES IN THE UNIT BALL OF Rn
Institute of Scientific and Technical Information of China (English)
Tang Xiaomin; Hu Zhangjian; Lu Xiaofen
2011-01-01
Let H(B) be the set of all harmonic functions f on the unit ball B of Rn.For 0 ＜ p,q ≤ ∞ and a normal weight φ, the mixed norm space Hp,q,φ(B) consists of all functions f in H(B) for which the mixed norm ||·||p,q,φ ＜ ∞. In this article, we obtain some characterizations in terms of radial, tangential, and partial derivative norms in Hp,q,φ(B).The parallel results for the Bloch-type space are also obtained. As an application, the analogous problems for polyharmonic functions are discussed.
A New Representation and Algorithm for Constructing Convex Hulls in Higher Dimensional Spaces
Institute of Scientific and Technical Information of China (English)
吕伟; 梁友栋
1992-01-01
This paper presents a new and simple scheme to describe the convex hull in Rd,which only uses three kinds of the faces of the convex hull.i.e.,the d-1-faces,d-2-faces and 0-faces.Thus,we develop and efficient new algorithm for constructing the convex hull of a finite set of points incrementally.This algorithm employs much less storage and time than that of the previously-existing approaches.The analysis of the runniing time as well as the storage for the new algorithm is also theoretically made.The algorithm is optimal in the worst case for even d.
Institute of Scientific and Technical Information of China (English)
Li WEI; Rui Lin TAN; Hai Yun ZHOU
2011-01-01
In this paper, we introduce a new iterative scheme for finding the common element of the set of solutions of an equilibrium problem, the set of solutions of variational inequalities for an e-inversely strongly monotone operator and the set of fixed points of relatively nonexpansive mappings in a real uniformly smooth and 2-uniformly convex Banach space. Some weak convergence theorems are obtained, to extend the previous work.
van de Vel, MLJ
1993-01-01
Presented in this monograph is the current state-of-the-art in the theory of convex structures. The notion of convexity covered here is considerably broader than the classic one; specifically, it is not restricted to the context of vector spaces. Classical concepts of order-convex sets (Birkhoff) and of geodesically convex sets (Menger) are directly inspired by intuition; they go back to the first half of this century. An axiomatic approach started to develop in the early Fifties. The author became attracted to it in the mid-Seventies, resulting in the present volume, in which graphs appear si
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E. U. Ofoedu
2009-01-01
a new iterative sequence for a countably infinite family of m-accretive mappings and prove strong convergence of the sequence to a common zero of these operators in uniformly convex real Banach space. Consequently, we obtain strong convergence theorems for a countably infinite family of pseudocontractive mappings. Our theorems extend and improve some important results which are announced recently by various authors.
Generalized Contraction and Invariant Approximation Results on Nonconvex Subsets of Normed Spaces
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Mujahid Abbas
2014-01-01
Full Text Available Wardowski (2012 introduced a new type of contractive mapping and proved a fixed point result in complete metric spaces as a generalization of Banach contraction principle. In this paper, we introduce a notion of generalized F-contraction mappings which is used to prove a fixed point result for generalized nonexpansive mappings on star-shaped subsets of normed linear spaces. Some theorems on invariant approximations in normed linear spaces are also deduced. Our results extend, unify, and generalize comparable results in the literature.
A Convex Optimization Approach to pMRI Reconstruction
Zhang, Cishen
2013-01-01
In parallel magnetic resonance imaging (pMRI) reconstruction without using estimation of coil sensitivity functions, one group of algorithms reconstruct sensitivity encoded images of the coils first followed by the magnitude only image reconstruction, e.g. GRAPPA, and another group of algorithms jointly compute the image and sensitivity functions by regularized optimization which is a non-convex problem with local only solutions. For the magnitude only image reconstruction, this paper derives a reconstruction formulation, which is linear in the magnitude image, and an associated convex hull in the solution space of the formulated equation containing the magnitude of the image. As a result, the magnitude only image reconstruction for pMRI is formulated into a two-step convex optimization problem, which has a globally optimal solution. An algorithm based on split-bregman and nuclear norm regularized optimizations is proposed to implement the two-step convex optimization and its applications to phantom and in-vi...
Parthood and Convexity as the Basic Notions of a Theory of Space
DEFF Research Database (Denmark)
Robering, Klaus
, a ``pregeometry'' is described in which only the notion of convexity but no further axiom is added to that background framework. Pregeometry is extended to the full system in three steps. First the notion of a line segment is explained as the convex hull of the mereological sum of two points. In a second step two...... axioms are added which describe what it means for a thus determined line segment to be ``straight''. In the final step we deal with the order of points on a line segment and define the notion of a line. The presentation of the geometric system is concluded with a brief consideration of the geometrical...
弱*局部一致凸空间的一些性质%Some Properties of Weak* Locally Uniformly Convex Banach Spaces
Institute of Scientific and Technical Information of China (English)
高继
2001-01-01
Some equivalent conditions and the properties of Weak* locally uniformly convex Banach spaces are given,and the inheritances of Weak* local uniform convexity in a product and ultraproduct spaces are discussed.%讨论了弱*局部一致凸空间的一些等价定义和性质，以及乘积空间的弱*凸部一致凸的传递性.
DEFF Research Database (Denmark)
M. Gaspar, Raquel; Murgoci, Agatha
2010-01-01
of particular importance to practitioners: yield convexity adjustments, forward versus futures convexity adjustments, timing and quanto convexity adjustments. We claim that the appropriate way to look into any of these adjustments is as a side effect of a measure change, as proposed by Pelsser (2003...
DEFF Research Database (Denmark)
Lauritzen, Niels
-Motzkin elimination, the theory is developed by introducing polyhedra, the double description method and the simplex algorithm, closed convex subsets, convex functions of one and several variables ending with a chapter on convex optimization with the Karush-Kuhn-Tucker conditions, duality and an interior point...
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Majid Abrishami-Moghaddam
2014-06-01
Full Text Available In this paper we establish a generalization of the triangle inequality in quasi 2-normed linear spaces. Moreover, we obtained a new characterization of 2-inner product spaces by comparing the 2-angular distance and 2-skew-angular distance with each other.
Midpoint sets contained in the unit sphere of a normed space
Swanepoel, Konrad J
2010-01-01
The midpoint set M(S) of a set S of points is the set of all midpoints of pairs of points in S. We study the largest cardinality of a midpoint set M(S) in a finite-dimensional normed space, such that M(S) is contained in the unit sphere, and S is outside the closed unit ball. We show in three dimensions that this maximum (if it exists) is determined by the facial structure of the unit ball. In higher dimensions no such relationship exists. We also determine the maximum for euclidean and sup norm spaces.
Carleson Type Measures for Harmonic Mixed Norm Spaces with Application to Toeplitz Operators
Institute of Scientific and Technical Information of China (English)
Zhangjian HU; Xiaofen LV
2013-01-01
Let Ω be a bounded domain in Rn with a smooth boundary,and let hp,q be the space of all harmonic functions with a finite mixed norm.The authors first obtain an equivalent norm on hp,q,with which the definition of Carleson type measures for hp,q is obtained.And also,the authors obtain the boundedness of the Bergman projection on hp,q which turns out the dual space of hp,q.As an application,the authors characterize the boundedness (and compactness) of Toeplitz operators Tμ on hp,q for those positive finite Borel measures μ.
Institute of Scientific and Technical Information of China (English)
陈利国; 罗成
2009-01-01
Let X be a real linear space,P be a family of separated seminorms on X,(X,T_P) denote the locally convex space generated by P,and the dual (X,P) denote the space X that has the topolopy T_P generated by the seminorm family P.The notions of uniformly extreme convex and uniformly extreme smooth for dual (X,P) are introduced,and their dual relationship is proved.The relationship with the other convexity (smoothness) are discussed.In addition,on the conditions of P-reflexivity,the dual theorem between uniformly extreme convexity and uniformly extreme smoothness is obtained,and thus the notions and results are generalized in Banach spaces.%设X是一个实线性空间,P是X上的一可分离的半范数族,(X,T_P)表示由P生成的局部凸空间,(X,P)为一个偶对.引入偶对(X,P)为一致极凸和一致极光滑的概念,并证明它们具有对偶关系,讨论了与其它几种凸性(光滑性)之间的关系,另外,在P-自反的条件下给出它们之间的对偶定理,从而推广了Banach空间相应概念和结果.
Institute of Scientific and Technical Information of China (English)
Mohammad Janfada; Abolfazl Nezhadali Baghan
2012-01-01
In this paper using the concept of Felbin-type fuzzy 2-norm ‖.,.‖ on a vector space,two Ⅰ-topologies τ‖.,.‖ and τ*‖.,.‖ is constructed.After making our elementary observations on this fuzzy Ⅰ-topologies,the continuity of vector space operations is discussed and it is proved that the vector space with Ⅰ-topology τ‖.,.‖ is not Ⅰ-topological vector space but with τ*‖.,.‖ is Ⅰ-topological vector space.Next we study the relationship between this two Ⅰ-topologies and it is proved that τ‖.,.‖(∈)τ‖.,.‖.
ON TOPOLOGICAL LINEAR CONTRACTIONS ON NORMED SPACES AND APPLICATION
Institute of Scientific and Technical Information of China (English)
SHIHMAU-HSIAN; TAMPING-KWAN; TANKOK-KEONG
1999-01-01
Shlnultaneous contractificttions, simultaneous proper contractification8 and scxnlgroup(countable family or finite family) of commuting operators and of non-commuting operatorsare first given. Characterizations are given for single bounded llner operator being a topo-logical proper contraction. By using complexification of a real Banach space and by applying afixed point theorem of Edelstein, it is shown that every compact topological strict contractionon a Banach space is a topological proper contraction. Finally, results on simultaneous propercontractification are applied to study the stability of a common fixed point of maps which areFr6chet differentiable at that point.
Relations between Lipschitz functions and convex functions
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
[1]Zajicek, J., On the differentation of convex functions in finite and infinite dimensional spaces, Czech J. Math.,1979, 29: 340-348.[2]Hu, T. C., Klee, V. L., Larman, D. G., Optimization of globally convex functions, SIAM J. Control Optim., 1989,27: 1026-1047.[3]Cepedello Boiso, M., Approximation of Lipschitz functions by △-convex functions in Banach spaces, Israel J.Math., 1998, 106: 269-284.[4]Asplund, E., Frechet differentiability of convex functions, Acta Math., 1968, 121: 31-47.[5]Johnson, J. A., Lipschitz spaces, Pacific J. Math, 1974, 51: 177-186.[6]Stromberg, T., The operation of infimal convolution, Dissert. Math., (Rozprawy Mat.), 1996, 325: 58.[7]Kadison, R. V., Ringrose, J. R., Fundamentals of the theory of operator algebras, volume Ⅰ: Elementary Theory,Graduate Studies in Math., vol. 15, Amer. Math. Soc., 1997.[8]Phelps, R. R., Convex functions,monotone operators and differentiability, Lect. Notes in Math., vol. 1364,Springer-Verlag, 1977.[9]Lindenstrauss, J., On operators which attain their norm, Israel J. Math., 1963, 1: 139-148.[10]Press, D., Gateaux differentiable functions are somewhere Frechet differentiable, Rend. Circ. Mat. Palermo,1984, 33: 122-133.[11]Press, D., Differentiability of Lipschitz functions on Banach spaces, J. Funct. Anal., 1990, 91:312-345.[12]Lindenstrauss, J., Press, D., On Frechet differentiability of Lipschitz maps between Banach spaces, Annals of Math., 2003, 157: 257-288.[13]Press, D., Gateaux differentiable Lipschitz functions need not be Frechet differentiable on a residual set, Supplemento Rend. Circ. Mat. Palermo, Serie Ⅱ, 1982, 2: 217-222.
On the Weak* Drop Property for Polar of Closed Bounded Convex Sets%关于有界闭凸集的极上的弱滴性
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张子厚
2004-01-01
We define and study the weak* drop property for the polar of a closed bounded convex set in a Banach space which is both a generalization of the weak* drop property for dual norm in a Banach space and a characterization of the sub-differential mappingx→αp(x) from S(X) into 2S(X*) that is norm upper semi-countinuous and norm compact-valued.
Banach Spaces which are Dual to k Nearly Uniformly Convex Spaces%k接近一致凸空间的对偶空间
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苏雅拉图; 乌敦其其格; 包来友
2011-01-01
In this paper, we introduce two new classes of Banach spaces: k nearly uniformly smooth spaces and ω nearly uniformly smooth spaces, which are the dual notions to k nearly uniformly convex spaces and ω nearly uniformly convex spaces respectively, introduced by Denka Kutzarova. We give some characters and properties of these two kinds of Banach spaces. Apart from that we really distinguish the k uniformly smooth spaces, k nearly uniformly smooth spaces, ω nearly uniformly smooth spaces, fully k smooth spaces and nearly uniformly smooth spaces.%该文引入了两类新的Banach空间,即k接近一致光滑空间和ω接近一致光滑空间,它们分别是Denka Kutzarova所引入的k接近一致凸空间和ω接近一致凸空间的对偶空间.作为主要结果,得到了这两类Banach空间的特征刻画及一些性质,弄清了k一致光滑空间、k接近一致光滑空间、ω接近一致光滑空间,完全k光滑空间和接近一致光滑空间的蕴涵关系.
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Margherita Fochi
2013-01-01
Full Text Available Let be a real normed space with dimension greater than 2 and let be a real functional defined on . Applying some ideas from the studies made on the conditional Cauchy functional equation on the restricted domain of the vectors of equal norm and the isosceles orthogonal vectors, the conditional quadratic equation and the D’Alembert one, namely, and , have been studied in this paper, in order to describe their solutions. Particular normed spaces are introduced for this aim.
Global approximation of convex functions
Azagra, D
2011-01-01
We show that for every (not necessarily bounded) open convex subset $U$ of $\\R^n$, every (not necessarily Lipschitz or strongly) convex function $f:U\\to\\R$ can be approximated by real analytic convex functions, uniformly on all of $U$. In doing so we provide a technique which transfers results on uniform approximation on bounded sets to results on uniform approximation on unbounded sets, in such a way that not only convexity and $C^k$ smoothness, but also local Lipschitz constants, minimizers, order, and strict or strong convexity, are preserved. This transfer method is quite general and it can also be used to obtain new results on approximation of convex functions defined on Riemannian manifolds or Banach spaces. We also provide a characterization of the class of convex functions which can be uniformly approximated on $\\R^n$ by strongly convex functions.
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SuLinning
1994-01-01
In this paper, the conditions for the non lcally convex topological vector space to have the H,-B. extension property is discussed, and the following three results are proved; (1)A closed subspace E0 of a linear topological space E to have the H.-B. property if and only if for every closed hyperplane of E0 is weakly closed, (2) A locally bounded linear topological space (E,τo)to have the H.-B extension property if and only if for every closed subspace E0 of E, the weak topology σ(E0,E*0)属于τ1|E0, where τ1 is the finest locally convex topology on E which is coarser then τ0. (3)Let E be separated and let E be the completion of E. If every closed subspace E0 of E is the complete hull of E0∩E,then E has H.-B. extension property if and only if E has H.-B. extension property.
Fixed and periodic points in the probabilistic normed and metric spaces
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Ghaemi, M.B. [Department of Mathematics, Faculty of Science, Imam Khomeini International University, P.O. Box 34194-288, Qazvin (Iran, Islamic Republic of) and Faculty of Mathematics, Iran University of Science and Technology, Tehran (Iran, Islamic Republic of)]. E-mail: M-Ghaemi@sbu.ac.ir; Razani, Abdolrahman [Department of Mathematics, Faculty of Science, Imam Khomeini International University, P.O. Box 34194-288, Qazvin (Iran, Islamic Republic of) and Department of Mathematics, Faculty of Science, Imam Khomeini International University, P.O. Box 34194-288, Qazvin (Iran, Islamic Republic of)]. E-mail: razani@ikiu.ac.ir
2006-06-15
In this paper, a fixed point theorem is proved, i.e., if A is a C-contraction in the Menger space (S,F) and E-bar S be such that A(E)-bar is compact, then A has a fixed point. In addition, under the same condition, the existence of a periodic point of A is proved. Finally, a fixed point theorem in probabilistic normed spaces is proved.
{\\lambda}-statistical convergent function sequences in intuitionistic fuzzy normed spaces
Karakaya, Vatan; Ertürk, Müzeyyen; Gürsoy, Faik
2011-01-01
Fuzzy logic was introduced by Zadeh in 1965. Since then, the importance of fuzzy logic has come increasingly to the present.There are many applications of fuzzy logic in the field of science and engineering, e.g. population dynamics (Barros), chaos control (Feng,Fradkov), computer programming (Giles), nonlinear dynamical systems (Hong), etc. The concept of intuitionistic fuzzy set, as a generalization of fuzzy logic, was introduced by Atanassov in 1986. Quite recently Park has introduced the concept of intuitionistic fuzzy metric space, and Saadati and Park studied the notion of intuitionistic fuzzy normed space. Intuitionistic fuzzy analogues of many concept in classical analysis was studied by many authors (Mursaleen, Rsaadati, Jebril, Dinda, etc.). The concept of statistical convergence was introduced by Fast. Mursaleen defined {\\lambda}-statistical convergence in Muhammed. Also the concept of statistical convergence was studied in intuitionistic fuzzy normed space in Karakus..Quite recently, Karakaya et a...
DEFF Research Database (Denmark)
Jacob, Riko
We determine the computational complexity of the dynamic convex hull problem in the planar case. We present a data structure that maintains a finite set of n points in the plane under insertion and deletion of points in amortized O(log n) time per operation. The space usage of the data structure...... is O(n). The data structure supports extreme point queries in a given direction, tangent queries through a given point, and queries for the neighboring points on the convex hull in O(log n) time. The extreme point queries can be used to decide whether or not a given line intersects the convex hull......, and the tangent queries to determine whether a given point is inside the convex hull. The space usage of the data structure is O(n). We give a lower bound on the amortized asymptotic time complexity that matches the performance of this data structure....
On the stability of an AQCQ-functional equation in random normed spaces
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Jang Sun Young
2011-01-01
Full Text Available Abstract In this paper, we prove the Hyers-Ulam stability of the following additive-quadratic-cubic-quartic functional equation f ( x + 2 y + f ( x - 2 y = 4 f ( x + y + 4 f ( x - y - 6 f ( x (1 + f ( 2 y + f ( - 2 y - 4 f ( y - 4 f ( - y (2 (3 in random normed spaces. 2010 Mathematics Subject Classification: 46S40; 39B52; 54E70
Powers of Convex-Cyclic Operators
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Fernando León-Saavedra
2014-01-01
Full Text Available A bounded operator T on a Banach space X is convex cyclic if there exists a vector x such that the convex hull generated by the orbit Tnxn≥0 is dense in X. In this note we study some questions concerned with convex-cyclic operators. We provide an example of a convex-cyclic operator T such that the power Tn fails to be convex cyclic. Using this result we solve three questions posed by Rezaei (2013.
DEFF Research Database (Denmark)
Brodal, Gerth Stølfting; Jacob, Rico
2002-01-01
In this paper we determine the computational complexity of the dynamic convex hull problem in the planar case. We present a data structure that maintains a finite set of n points in the plane under insertion and deletion of points in amortized O(log n) time per operation. The space usage...... of the data structure is O(n). The data structure supports extreme point queries in a given direction, tangent queries through a given point, and queries for the neighboring points on the convex hull in O(log n) time. The extreme point queries can be used to decide whether or not a given line intersects...... the convex hull, and the tangent queries to determine whether a given point is inside the convex hull. We give a lower bound on the amortized asymptotic time complexity that matches the performance of this data structure....
Niemi, Antti H.
2013-12-01
We investigate the application of the discontinuous Petrov-Galerkin (DPG) finite element framework to stationary convection-diffusion problems. In particular, we demonstrate how the quasi-optimal test space norm improves the robustness of the DPG method with respect to vanishing diffusion. We numerically compare coarse-mesh accuracy of the approximation when using the quasi-optimal norm, the standard norm, and the weighted norm. Our results show that the quasi-optimal norm leads to more accurate results on three benchmark problems in two spatial dimensions. We address the problems associated to the resolution of the optimal test functions with respect to the quasi-optimal norm by studying their convergence numerically. In order to facilitate understanding of the method, we also include a detailed explanation of the methodology from the algorithmic point of view. © 2013 Elsevier Ltd. All rights reserved.
Common Fixed Point Theorems in Uniformly Convex Metric Space%一致凸度量空间的公共不动点定理
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曾秀华; 邓磊
2015-01-01
利用一致凸度量空间中的凸性模和自映象对的次相容性，讨论了一类4个自映象的公共不动点的存在性和唯一性问题，得到了一个公共不动点定理。该结果改进和推广了近期的相关结果。%Using the sub‐compatibility of convex modulus and self‐mapping pair in uniformly convex metric spaces ,we discuss the existence and uniqueness of some common fixed points with four self‐mappings in this paper .A new common fixed point theorem is obtained ,which largely improves and extends some re‐lated results that have been published recently in uniformly convex metric spaces .
On the Uniformly Extremely Convex Spaces and Uniformly Extremely Smooth Spaces%关于一致极凸空间与一致极光滑空间
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苏雅拉图
2016-01-01
给出一致极凸空间与一致极光滑空间的若干特征刻画，指出一致极光滑空间是既严格介于一致光滑空间和强光滑空间之间，又严格介于完全k-光滑空间和强光滑空间之间的一类 Banach空间，而一致极光滑空间和接近一致极光滑空间之间不存在任何蕴含关系。%In this article,some characteristic descriptions of uniformly extremely convex spaces and uniformly extremely smooth spaces are obtained.It is pointed out that the class of uniformly extremely smooth spaces lies strictly between either the class of uniformly smooth spaces and strongly smooth spaces or the class of fully k-smooth spaces and strongly smooth spaces.The relations between uniformly extremely smooth space with nearly uniformly smooth space are given also.
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Sasaki, Makoto [Iwate Medical University, Department of Radiology, Morioka (Japan); Honda, Satoshi [St. Luke' s International Hospital, Department of Radiology, Tokyo (Japan); Yuasa, Tatsuhiko; Iwamura, Akihide [Kohnodai Hospital, National Center of Neurology and Psychiatry, Department of Neurology, Ichikawa (Japan); Shibata, Eri [Iwate Medical University, Department of Neuropsychiatry, Morioka (Japan); Ohba, Hideki [Iwate Medical University, Department of Neurology, Morioka (Japan)
2008-02-15
The aim of this study was to determine the performance of axial and coronal magnetic resonance imaging (MRI) in detecting the narrowing of the cerebrospinal fluid (CSF) space at the high convexity and high midline areas, which is speculated to be one of the clinical characteristics of idiopathic normal pressure hydrocephalus (iNPH). We retrospectively examined axial and coronal T1-weighted images of 14 iNPH patients and 12 age-matched controls. The narrowness of the CSF space at the high convexity/midline was blindly evaluated by five raters using a continuous confidence rating scale for receiver operating characteristic (ROC) analysis. Axial and coronal imaging accurately determined the presence of the narrow cisterns/sulci at the high convexity/midline and was capable of predicting probable/definite iNPH with a high degree of accuracy. there were also no significant differences in the detection of this finding between the axial and coronal images. Both axial and coronal T1-weighted MRI can detect the narrow CSF space at the high convexity/midline accurately and may therefore facilitate clinicians in choosing a management strategy for iNPH patients. (orig.)
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Bapurao Chandrabahan Dhage
2014-11-01
Full Text Available In this paper, the author introduces a notion of partially condensing mappings in a partially ordered normed linear space and proves some hybrid fixed point theorems under certain mixed conditions of algebra, analysis and topology. The applications of abstract results presented here are given to some nonlinear functional integral equations for proving the existence as well as global attractivity of the comparable solutions under certain monotonicity conditions. The abstract theory presented here is very much useful to develop the algorithms for the solutions of some nonlinear problems of analysis and allied areas of mathematics. A realization of of our hypotheses is also indicated by a numerical example.
GENERALIZED Ⅰ-NONEXPANSIVE MAPS AND INVARIANT APPROXIMATION RESULTS IN p-NORMED SPACES
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N. Hussain
2006-01-01
We extend the concept of R-subcommuting maps due to Shahzad[17,18] to the case of non-starshaped domain and obtain a common fixed point result for this class of maps on non-starshaped domain in the setup of p-normed spaces. As applications, we establish noncommutative versions of various best approximation results for generalized I-nonexpansive maps on non-starshaped domain. Our results unify and extend that of Al-Thagafi, Dotson, Habiniak, Jungck and Sessa, Latif, Sahab, Khan and Sessa and Shahzad.
Fréchet Envelopes of Nonlocally Convex Variable Exponent Hörmander Spaces
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Joaquín Motos
2016-01-01
Full Text Available We show that the dual Bp·locΩ′ of the variable exponent Hörmander space Bp(·loc(Ω is isomorphic to the Hörmander space B∞c(Ω (when the exponent p(· satisfies the conditions 0 space hp-. Finally, some questions are proposed.
DEFF Research Database (Denmark)
Johnsen, Jon
The article deals with a simplified proof of the Sobolev embedding theorem for Triebel-Lizorkin spaces (that contain the $L_p$-Sobolev spaces $H^s_p$ as special cases). The method extends to a proof of the corresponding fact for general Triebel–Lizorkin spaces based on mixed $L_p$-norms. In this ......_p$-norms. In this context a Nikol’skij– Plancherel-Polya inequality for sequences of functions satisfying a geometric rectangle condition is proved. The results extend also to anisotropic spaces of the quasi-homogeneous type....
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Tong-Huei Chang
2009-01-01
Full Text Available We use a concept of abstract convexity to define the almost S-KKM𝒞 property, al-S-KKM𝒞(X,Y family, and almost Φ-spaces. We get some new approximate fixed point theorems and fixed point theorems in almost Φ-spaces. Our results extend some results of other authors.
Locative and Directional Prepositions in Conceptual Spaces: The Role of Polar Convexity
Zwarts, J.; Gärdenfors, Peter
2016-01-01
We approach the semantics of prepositions from the perspective of conceptual spaces. Focusing on purely spatial locative and directional prepositions, we analyze both types of prepositions in terms of polar coordinates instead of Cartesian coordinates. This makes it possible to demonstrate that the
Narici, Lawrence
2011-01-01
BackgroundTopology Valuation Theory Algebra Linear Functionals Hyperplanes Measure Theory Normed SpacesCommutative Topological GroupsElementary ConsiderationsSeparation and Compactness Bases at 0 for Group Topologies Subgroups and Products Quotients S-Topologies Metrizability CompletenessCompleteness Function Groups Total BoundednessCompactness and Total Boundedness Uniform Continuity Extension of Uniformly Continuous Maps CompletionTopological Vector SpacesAbsorbent and Balanced Sets Convexity-AlgebraicBasic PropertiesConvexity-Topological Generating Vector Topologies A Non-Locally Convex Spa
Parthood and Convexity as the Basic Notions of a Theory of Space
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Robering, Klaus
. The geometry developed within this framework roughly corresponds to the "line spaces'' known from the literature. The basic ideas of the system are presented in the article's "Introduction" within a historical context. After a brief presentation of the logical and mereological framework adopted...... principles known by the names of Peano and Pasch. Two additional topics are treated in short sections at the end of the article: (1) the introduction of coordinates and (2) the idea of a ``geometrical algebra''....
Niemi, Antti
2013-05-01
We revisit the finite element analysis of convection-dominated flow problems within the recently developed Discontinuous Petrov-Galerkin (DPG) variational framework. We demonstrate how test function spaces that guarantee numerical stability can be computed automatically with respect to the optimal test space norm. This makes the DPG method not only stable but also robust, that is, uniformly stable with respect to the Péclet number in the current application. We employ discontinuous piecewise Bernstein polynomials as trial functions and construct a subgrid discretization that accounts for the singular perturbation character of the problem to resolve the corresponding optimal test functions. We also show that a smooth B-spline basis has certain computational advantages in the subgrid discretization. The overall effectiveness of the algorithm is demonstrated on two problems for the linear advection-diffusion equation. © 2011 Elsevier B.V.
The New Fixed Point Theorems in Abstract Convex Uniform Spaces%抽象凸一致空间中的新型不动点定理
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刘学文
2009-01-01
The new concepts of better admissible class BG and of Klee approximability for subsets of abstract convex uniform spaces are introduced. Some new fixed point theorems for better admissible set-valued mappings are proved in abstract convex uniform spaces. These results generalize some known results in literature.%在抽象凸一致空间中定义了一组新的较佳容许集值映射组BG,并在抽象凸一致空间的子集上引入了Klee邻接性质.利用此定义和性质,在抽象凸一致空间中证明了一组涉及较佳容许集值映射组的新的不动点定理.
Institute of Scientific and Technical Information of China (English)
Yongjie Piao∗
2015-01-01
A classΦof 5-dimensional functions was introduced and an existence and uniqueness of common fixed points for a family of non-self mappings satisfying aφi-quasi-contractive condition and a certain boundary condition was given on complete metrically convex metric spaces, and from which, more general unique common fixed point theorems were obtained.Our main results generalize and improve many same type common fixed point theorems in references.
Institute of Scientific and Technical Information of China (English)
朴勇杰
2005-01-01
In this paper, we use the well known KKM type theorem for generalized convex spaces due to Park (Elements of the KKM theory for generalized convex spaces, Korean J. Comp. Appl. Math., 7(2000), 1-28) to obtain an almost fixed point theorem for upper [resp., lower] semicontinuous multimaps in locally G-convex spaces, and then give a fixed point theorem for upper semicontinuous multimap with closed Г-convex values.
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段丽芬; 左明霞; 崔云安
2014-01-01
Using convex theories of Banach spaces,the authors studied locally uniform rotundity and weakly locally uniform rotundity of the Orlicz function spaces equipped with the generalized Orlicz norm and obtained sufficient and necessary conditions so as to make them be locally uniform rotund and weakly locally uniform rotund.%利用Banach空间凸性理论研究赋广义 Orlicz 范数 Orlicz 函数空间的局部一致凸和弱局部一致凸问题，得到了 Orlicz函数空间关于广义 Orlicz范数局部一致凸和弱局部一致凸的条件。
Stabilities of Cubic Mappings in Various Normed Spaces: Direct and Fixed Point Methods
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H. Azadi Kenary
2012-01-01
Full Text Available In 1940 and 1964, Ulam proposed the general problem: “When is it true that by changing a little the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true?”. In 1941, Hyers solved this stability problem for linear mappings. According to Gruber (1978 this kind of stability problems are of the particular interest in probability theory and in the case of functional equations of different types. In 1981, Skof was the first author to solve the Ulam problem for quadratic mappings. In 1982–2011, J. M. Rassias solved the above Ulam problem for linear and nonlinear mappings and established analogous stability problems even on restricted domains. The purpose of this paper is the generalized Hyers-Ulam stability for the following cubic functional equation: (++(−=(++(−+2(3−(,≥2 in various normed spaces.
Institute of Scientific and Technical Information of China (English)
于亚璇; 刘德; 罗成
2007-01-01
设X是实线性空间,P是X上的一族分离半范数,且TP是X上由P生成的局部凸分离拓扑.证明了半范数族P和它的每一个S-最简形式具有相同的凸性和光滑性.在P-自反的条件下,得到偶对(X,P)是一致光滑的(一致凸的)当且仅当它的强对偶(X',P')是一致凸的(一致光滑的).对其它的凸性和光滑性也有类似结果.%Let X be a real linear space,P a family of separated seminorms on X and TP the locally convex separated topology on X generated by P.The convexity and smoothness in the locally convex spaces are studied.It is proved that the seminorm family P and every its S-simplest form have the same convexity and smoothness.Under the condition of P-reflexivity,a dual pair (X,P) is uniformly smooth (uniformly convex) if and only if its strong dual pair (X',P') is uniformly convex (uniformly smooth).And similar results for the other convexity and smoothness are obtained.
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Luis Ángel Gutiérrez Méndez
2013-01-01
Full Text Available We prove that the cardinality of the space ℋ([a,b] is equal to the cardinality of real numbers. Based on this fact we show that there exists a norm on ℋ([a,b] under which it is a Banach space. Therefore if we equip ℋ([a,b] with the Alexiewicz topology then ℋ([a,b] is not K-Suslin, neither infra-(u nor a webbed space.
Klee, Victor; Ziegler, Günter
2003-01-01
"The appearance of Grünbaum's book Convex Polytopes in 1967 was a moment of grace to geometers and combinatorialists. The special spirit of the book is very much alive even in those chapters where the book's immense influence made them quickly obsolete. Some other chapters promise beautiful unexplored land for future research. The appearance of the new edition is going to be another moment of grace. Kaibel, Klee and Ziegler were able to update the convex polytope saga in a clear, accurate, lively, and inspired way." (Gil Kalai, The Hebrew University of Jerusalem) "The original book of Grünbaum has provided the central reference for work in this active area of mathematics for the past 35 years...I first consulted this book as a graduate student in 1967; yet, even today, I am surprised again and again by what I find there. It is an amazingly complete reference for work on this subject up to that time and continues to be a major influence on research to this day." (Louis J. Billera, Cornell University) "The or...
Fixed point theorems in locally convex spacesÃ¢Â€Â”the Schauder mapping method
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S. Cobzaş
2006-03-01
Full Text Available In the appendix to the book by F. F. Bonsal, Lectures on Some Fixed Point Theorems of Functional Analysis (Tata Institute, Bombay, 1962 a proof by Singbal of the Schauder-Tychonoff fixed point theorem, based on a locally convex variant of Schauder mapping method, is included. The aim of this note is to show that this method can be adapted to yield a proof of Kakutani fixed point theorem in the locally convex case. For the sake of completeness we include also the proof of Schauder-Tychonoff theorem based on this method. As applications, one proves a theorem of von Neumann and a minimax result in game theory.
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文开庭
2008-01-01
In this paper,new existence theorems for maximal elements are established in L-convex spaces.As application,existence theorems of equilibrium for qualitative games and abstract economies are obtained in L-convex spaces.%在L-凸空间中建立了新的极大元定理.作为应用,获得了L-凸空间中抽象经济和定性对策的平衡存在定理.
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陈利国; 罗成; 王君
2013-01-01
The dual notions of midpoint locally k-uniformly convexity and midpoint locally A:-uniformly smooth on Locally Convex Spaces are introduced , which are generalizations of both midpoint locally k-u-niformly convexity (midpoint locally k-uniformly smoothness) in Banach spaces and midpoint locally uniformly convexity (midpoint locally uniformly smoothness) in locally convex spaces. The relationship between them and the other convexity (smoothness) are discussed.%引入局部凸空间的中点局部k-一致凸性和中点局部k-一致光滑性这一对对偶概念,它们既是Banach空间中点局部k-一致凸性和中点局部k-一致光滑性推广,又是局部凸空间中点局部一致凸性和中点局部一致光滑性的自然推广.讨论它们与其它k-凸性(k-光滑性)之间的关系.
A direct proof of Sobolev embeddings for quasi-homogeneous Lizorkin–Triebel spaces with mixed norms
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Jon Johnsen
2007-01-01
Full Text Available The article deals with a simplified proof of the Sobolev embedding theorem for Lizorkin–Triebel spaces (that contain the Lp-Sobolev spaces Hps as special cases. The method extends to a proof of the corresponding fact for general Lizorkin–Triebel spaces based on mixed Lp-norms. In this context a Nikol' skij–Plancherel–Polya inequality for sequences of functions satisfying a geometric rectangle condition is proved. The results extend also to anisotropic spaces of the quasi-homogeneous type.
Salt-body Inversion with Minimum Gradient Support and Sobolev Space Norm Regularizations
Kazei, V.V.
2017-05-26
Full-waveform inversion (FWI) is a technique which solves the ill-posed seismic inversion problem of fitting our model data to the measured ones from the field. FWI is capable of providing high-resolution estimates of the model, and of handling wave propagation of arbitrary complexity (visco-elastic, anisotropic); yet, it often fails to retrieve high-contrast geological structures, such as salt. One of the reasons for the FWI failure is that the updates at earlier iterations are too smooth to capture the sharp edges of the salt boundary. We compare several regularization approaches, which promote sharpness of the edges. Minimum gradient support (MGS) regularization focuses the inversion on blocky models, even more than the total variation (TV) does. However, both approaches try to invert undesirable high wavenumbers in the model too early for a model of complex structure. Therefore, we apply the Sobolev space norm as a regularizing term in order to maintain a balance between sharp and smooth updates in FWI. We demonstrate the application of these regularizations on a Marmousi model, enriched by a chunk of salt. The model turns out to be too complex in some parts to retrieve its full velocity distribution, yet the salt shape and contrast are retrieved.
Covering Numbers for Convex Functions
Guntuboyina, Adityanand
2012-01-01
In this paper we study the covering numbers of the space of convex and uniformly bounded functions in multi-dimension. We find optimal upper and lower bounds for the $\\epsilon$-covering number of $\\C([a, b]^d, B)$, in the $L_p$-metric, $1 \\le p 0$, and $\\C([a,b]^d, B)$ denotes the set of all convex functions on $[a, b]^d$ that are uniformly bounded by $B$. We summarize previously known results on covering numbers for convex functions and also provide alternate proofs of some known results. Our results have direct implications in the study of rates of convergence of empirical minimization procedures as well as optimal convergence rates in the numerous convexity constrained function estimation problems.
Recovery of Sparse Probability Measures via Convex Programming
Pilanci, Mert; El Ghaoui, Laurent; Chandrasekaran, Venkat
2012-01-01
We consider the problem of cardinality penalized optimization of a convex function over the probability simplex with additional convex constraints. The classical ℓ_1 regularizer fails to promote sparsity on the probability simplex since ℓ_1 norm on the probability simplex is trivially constant. We propose a direct relaxation of the minimum cardinality problem and show that it can be efficiently solved using convex programming. As a first application we consider recovering a spa...
Luis Ángel Gutiérrez Méndez; Juan Alberto Escamilla Reyna; Maria Guadalupe Raggi Cárdenas; Juan Francisco Estrada García
2013-01-01
We prove that the cardinality of the space $ℋ\\mathrm{}\\left(\\left[a,b\\right]\\right)$ is equal to the cardinality of real numbers. Based on this fact we show that there exists a norm on $ℋ\\mathrm{}\\left(\\left[a,b\\right]\\right)$ under which it is a Banach space. Therefore if we equip $ℋ\\mathrm{}\\left(\\left[a,b\\right]\\right)$ with the Alexiewicz topology then $ℋ\\mathrm{}\\left(\\left[a,b\\right]\\right)$ is not K-Suslin, neither infra-( $u$ ) nor a webbed space.
On Convex Quadratic Approximation
den Hertog, D.; de Klerk, E.; Roos, J.
2000-01-01
In this paper we prove the counterintuitive result that the quadratic least squares approximation of a multivariate convex function in a finite set of points is not necessarily convex, even though it is convex for a univariate convex function. This result has many consequences both for the field of
On Convex Quadratic Approximation
den Hertog, D.; de Klerk, E.; Roos, J.
2000-01-01
In this paper we prove the counterintuitive result that the quadratic least squares approximation of a multivariate convex function in a finite set of points is not necessarily convex, even though it is convex for a univariate convex function. This result has many consequences both for the field of
一种特殊赋范空间中的Borsuk问题%Borsuk's Problem in a Special Normed Space
Institute of Scientific and Technical Information of China (English)
徐常青; 苑立平; 丁仁
2004-01-01
Borsuk's problem is a famous problem in combinatorial geometry. It deals with the problem of partitioning a set into parts of smaller diameter. The problem was posed by the well-known Polish mathematician K. Borsuk in 1933. Many results have been obtained since then. In this paper, we discuss the Borsuk's problem in the normed space R2 with regular hexagon as its unit sphere ∑ and obtain some new results.
Institute of Scientific and Technical Information of China (English)
苏永福
2001-01-01
文[4]把文[3]的主要结果从Hilbert空间推广到一致凸Banach空间,证明了一致凸Banach空间中文上从有界闭凸集到自身的渐近非扩张映象的迭代序列收敛定理.本文将有界闭凸集的条件减弱为闭凸集,从而推广了文[4]的相应结果.%In paper [4], the relative result of Jiirgen schu is extended to a uniformly convex Banach space, and the convergence of iterative sequence in an uniformly conves Banach space for asymptotically non - expanstive mapping is proved.In paper [4], T is asymptotically non - expanstive mapping with sequence {Kn} in a bounded closed convex subset C of uniformly convex Banach space.In this paper, we let only C is closed convex subset of uniformlly convex Banach space. But convergence theorms of iterative sequences for asymptotically non-expanstive mapping was also proved.
Niemi, Antti H.
2011-05-14
We revisit the finite element analysis of convection dominated flow problems within the recently developed Discontinuous Petrov-Galerkin (DPG) variational framework. We demonstrate how test function spaces that guarantee numerical stability can be computed automatically with respect to the so called optimal test space norm by using an element subgrid discretization. This should make the DPG method not only stable but also robust, that is, uniformly stable with respect to the Ṕeclet number in the current application. The e_ectiveness of the algorithm is demonstrated on two problems for the linear advection-di_usion equation.
Institute of Scientific and Technical Information of China (English)
林强
2001-01-01
在Hausdorff局部凸线性拓扑空间中讨论集值系统x∈F(x,y),y∈G(x,y)解的存在性，推广了Rzepechi B在一致凸Banach空间中相应的结论。%This paper deals with existence of solutions for set-valued system x∈F(x,y)，y∈G(x,y) in local convex Hausdorff topological vector space. The results obtained are generalization of Rzepechi theorem in uniform convex Banach space.
Greedy vs. L1 convex optimization in sparse coding
DEFF Research Database (Denmark)
Ren, Huamin; Pan, Hong; Olsen, Søren Ingvor;
2015-01-01
, such as face and action recognition, a comparative study of codes in abnormal event detection is less studied and hence no conclusion is gained on the effect of codes in detecting abnormalities. We constrict our comparison in two types of the above L0-norm solutions: greedy algorithms and convex L1-norm...
Uniform convexity and the splitting problem for selections
Balashov, Maxim V; 10.1016/j.jmaa.2009.06.045
2009-01-01
We continue to investigate cases when the Repov\\v{s}-Semenov splitting problem for selections has an affirmative solution for continuous set-valued mappings. We consider the situation in infinite-dimensional uniformly convex Banach spaces. We use the notion of Polyak of uniform convexity and modulus of uniform convexity for arbitrary convex sets (not necessary balls). We study general geometric properties of uniformly convex sets. We also obtain an affirmative solution of the splitting problem for selections of certain set-valued mappings with uniformly convex images.
Minimum d-convex partition of a multidimensional polyhedron with holes
Directory of Open Access Journals (Sweden)
Ion Băţ
2008-11-01
Full Text Available In a normed space Rn over the field of real numbers R, which is an α-space [36, 39], one derives the formula expressing the minimum number of d-convex pieces into which a geometric n-dimensional polyhedron with holes can be partitioned. The problem of partitioning a geometric n-dimensional polyhedron has many theoretical and practical applications in various fields such as computational geometry, image processing, pattern recognition, computer graphics, VLSI engineering, and others [5, 10, 11, 19, 21, 28, 29, 31, 43]. Mathematics Subject Classification: 68U05, 52A30, 57Q05
Institute of Scientific and Technical Information of China (English)
Tie Xin GUO; Xiao Lin ZENG
2012-01-01
Let (Ω,F,P) be a probability space and L0(F,R) the algebra of equivalence classes of realvalued random variables on (Ω,F,P).When L0(F,R) is endowed with the topology of convergence in probability,we prove an intermediate value theorem for a continuous local function from L0(F,R) to L0(F,R).As applications of this theorem,we first give several useful expressions for modulus of random convexity,then we prove that a complete random normed module (S,‖· ‖) is random uniformly convex itf LF(S) is uniformly convex for each fixed positive number p such that 1 ＜ p ＜ +oo.
On the convexity of N-Chebyshev sets
Borodin, Petr A.
2011-10-01
We define N-Chebyshev sets in a Banach space X for every positive integer N (when N=1, these are ordinary Chebyshev sets) and study conditions that guarantee their convexity. In particular, we prove that all N-Chebyshev sets are convex when N is even and X is uniformly convex or N\\ge 3 is odd and X is smooth uniformly convex.
Relations between Lipschitz functions and convex functions
Institute of Scientific and Technical Information of China (English)
RUAN Yingbin
2005-01-01
We discuss the relationship between Lipschitz functions and convex functions.By these relations, we give a sufficient condition for the set of points where Lipschitz functions on a Hilbert space is Frechet differentiable to be residual.
若干近凸空间和近光滑空间的注记%Remarks on Some Nearly Convex and Nearly Smooth Banach Spaces
Institute of Scientific and Technical Information of China (English)
苏雅拉图; 莎仁格日乐; 乌敦其其格
2012-01-01
This paper points out that two definitions of near-strong convexity （resp. near-very convexity, locally near uniformly smoothness） are equavlent. Some reslults about near-strong con-vexity （resp. near-very convexity, locally near uniformly smoothness） are unified.%指出了关于近-强凸（近-非常凸、局部近-一致光滑）的两种不同形式的定义实质上是等价的,从而统一了有关文献中的一些结果.
On Some Properties of Hyperconvex Spaces
Directory of Open Access Journals (Sweden)
Dev Phulara
2010-01-01
Full Text Available We are going to answer some open questions in the theory of hyperconvex metric spaces. We prove that in complete ℝ-trees hyperconvex hulls are uniquely determined. Next we show that hyperconvexity of subsets of normed spaces implies their convexity if and only if the space under consideration is strictly convex. Moreover, we prove a Krein-Milman type theorem for ℝ-trees. Finally, we discuss a general construction of certain complete metric spaces. We analyse its particular cases to investigate hyperconvexity via measures of noncompactness.
On Some Properties of Hyperconvex Spaces
Directory of Open Access Journals (Sweden)
Bugajewski Dariusz
2010-01-01
Full Text Available We are going to answer some open questions in the theory of hyperconvex metric spaces. We prove that in complete -trees hyperconvex hulls are uniquely determined. Next we show that hyperconvexity of subsets of normed spaces implies their convexity if and only if the space under consideration is strictly convex. Moreover, we prove a Krein-Milman type theorem for -trees. Finally, we discuss a general construction of certain complete metric spaces. We analyse its particular cases to investigate hyperconvexity via measures of noncompactness.
Institute of Scientific and Technical Information of China (English)
陈亮; 吐尔德别克
2011-01-01
用Hardy鞅的原子分解刻画了复拟Banach空间的解析(q一致凸性,并用原子分解证明了两个Hardy鞅空间的嵌入关系.%The analytic q-uniform convexity of complex Quasi-Banach space is described by atomic decomposition of Hardy martingales,the embedding relationship between two Hardy martingale spaces is discussed by the method of atomic decomposition.
Institute of Scientific and Technical Information of China (English)
朴勇杰
2009-01-01
The definitions of S-KKM property and F-invariable property for multi-valued mapping are established, and by which, a new almost fixed point theorem and several fixed point theorems on Haudorff locally G-convex uniform space are obtained, and a quasi-variational in-equality theorem for acyclic map on Hausdorff φ-space is proved. Our results improve and generalize the corresponding results in recent literatures.
Institute of Scientific and Technical Information of China (English)
丁协平
2004-01-01
By applying continuous selection theorem and collectively fixed point theorem for a family of set-valued mappings on a product space of locally G-convex uniform spaces, some new coincidence theorems for two families of setvalued mappings defined on the product G-convex spaces are proved. Tnese theorems improve, unify and generalize many important coincidence theorems in the recent literature.%利用连续选择定理和乘积局部G-凸一致空间上的聚合不动点定理,对定义在乘积G-凸空间上的两个集值映象簇证明某些新的重合点定理.这些定理改进,统一和推广了最近文献中的很多重合点定理.
Minimizing convex functions by continuous descent methods
Directory of Open Access Journals (Sweden)
Sergiu Aizicovici
2010-01-01
Full Text Available We study continuous descent methods for minimizing convex functions, defined on general Banach spaces, which are associated with an appropriate complete metric space of vector fields. We show that there exists an everywhere dense open set in this space of vector fields such that each of its elements generates strongly convergent trajectories.
Linearization functors on real convex sets
Velasco, Mauricio
2012-01-01
We prove that linearizing certain families of polynomial optimization problems leads to new functorial operations in real convex sets. We show that under some conditions these operations can be computed or approximated in ways amenable to efficient computation. These operations are convex analogues of Hom functors, tensor products, symmetric powers, exterior powers and general Schur functors on vector spaces and lead to novel constructions even for polyhedra.
A Note on Convex Renorming and Fragmentability
Indian Academy of Sciences (India)
A K Mirmostafaee
2005-05-01
Using the game approach to fragmentability, we give new and simpler proofs of the following known results: (a) If the Banach space admits an equivalent Kadec norm, then its weak topology is fragmented by a metric which is stronger than the norm topology. (b) If the Banach space admits an equivalent rotund norm, then its weak topology is fragmented by a metric. (c) If the Banach space is weakly locally uniformly rotund, then its weak topology is fragmented by a metric which is stronger than the norm topology.
Indian Academy of Sciences (India)
Oscar Valero
2006-05-01
Given a normed cone (, ) and a subcone , we construct and study the quotient normed cone $(X/Y,\\tilde{p})$ generated by . In particular we characterize the bicompleteness of $(X/Y,\\tilde{p})$ in terms of the bicompleteness of (, ), and prove that the dual quotient cone $((X/Y)^∗,\\|\\cdot\\|_{\\tilde{p},u})$ can be identified as a distinguished subcone of the dual cone $(X^∗,\\|\\cdot\\|_{\\tilde{p},u})$. Furthermore, some parts of the theory are presented in the general setting of the space $CL(X,Y)$ of all continuous linear mappings from a normed cone (, ) to a normed cone (, ), extending several well-known results related to open continuous linear mappings between normed linear spaces.
On convexity in complex networks
Marc, Tilen
2016-01-01
Metric graph properties lie in the heart of the analysis of complex networks, while in this paper we study their convexity. We analyze the expansion of convex subsets of nodes in empirical networks and also convexity of small subgraphs known as graphlets. We demonstrate that convexity is an inherent property of complex networks not present in a random graph. According to our perception of convexity, a convex network is such in which every connected subset of nodes induces a convex subgraph. Especially convex are technological networks and social collaboration graphs, whereas food webs are the only networks studied that are truly non-convex. Many other networks can be divided into a non-convex core surrounded by a convex periphery. We interpret convexity in terms of redundancy of shortest paths in a network and discuss possible applications.
On Approximation of Function Classes in Lorentz Spaces with Anisotropic Norm
Institute of Scientific and Technical Information of China (English)
G.Akishev
2013-01-01
In this paper, Lorentz space of functions of several variables and Besov’s class are considered. We establish an exact approximation order of Besov’s class by partial sums of Fourier ’s series for multiple trigonometric system.
Mixed Norm Estimate for Radon Transform on Weighted $L^p$ Spaces
Indian Academy of Sciences (India)
Ashisha Kumar; Swagato K Ray
2010-09-01
We will discuss about the mapping property of Radon transform on $L^p$ spaces with power weight. It will be shown that the Pitt’s inequality together with the weighted version of Hardy–Littlewood–Sobolev lemma imply weighted inequality for the Radon transform.
Computing the stretch factor and maximum detour of paths, trees, and cycles in the normed space
DEFF Research Database (Denmark)
Wulff-Nilsen, Christian; Grüne, Ansgar; Klein, Rolf;
2012-01-01
The stretch factor and maximum detour of a graph G embedded in a metric space measure how well G approximates the minimum complete graph containing G and the metric space, respectively. In this paper we show that computing the stretch factor of a rectilinear path in L 1 plane has a lower bound of Ω......(n log n) in the algebraic computation tree model and describe a worst-case O(σn log 2 n) time algorithm for computing the stretch factor or maximum detour of a path embedded in the plane with a weighted fixed orientation metric defined by σ ... compute the stretch factor or maximum detour of trees and cycles in O(σn log d+1 n) time. We also obtain an optimal O(n) time algorithm for computing the maximum detour of a monotone rectilinear path in L 1 plane. © 2012 World Scientific...
Convex bodies in Euclidean and Weil-Petersson geometries
Yamada, Sumio
2011-01-01
On a convex body in a Euclidean space, we introduce a new variational formulation for its Funk metric, a Finsler metric compatible with the tautological Finsler structure of the convex body. We generalize the metric on Teichmuller spaces with the Weil-Petersson distance function. A set of similarities the resulting metric structure shares with Thurston's asymmetric metric is noted.
Convergence in a generalized fuzzy normed space%广义模糊赋范空间中的收敛性
Institute of Scientific and Technical Information of China (English)
杨萍; 曹怀信; 司海燕
2010-01-01
Aim To prove some properties about convergence in a generalized fuzzy normed space. Methods By introducing the definitions of generalized fuzzy normed space, fuzzy convergence, fuzzy boundedness, Cauchy sequence and completeness, several convergence theorems of sequences in a generalized fuzzy normed space are proved. Moreover the relation between this kind of completeness and completeness in a normed space is considered. Results The following.results are obtained: limit of a fuzzy convergent sequence is unique; each subsequence of a fuzzy convergent sequence converges to the limit of the sequence; each fuzzy convergent sequence is a Cauchy sequence; each Cauchy se-quence is fuzzy bounded and each Cauchy sequence which has a fuzzy convergent subsequence is fuzzy convergent and there exist incomplete generalized fuzzy normed spaces. Conclusion It has been shown that some concepts and results in a normed space can be similarly established in a generalized fuzzy normed space.%目的 证明广义模糊赋范空间中关于收敛的一些性质.方法 定义了广义模糊赋范空间,模糊收敛性,模糊有界性,柯西列和完备性.借助这些定义,证明了广义模糊赋范空间中序列的若干收敛定理.而且考虑了这种完备性和赋范空间中的完备性的关系.结果 证明了以下结果:模糊收敛序列的极限是唯一的;模糊收敛序列的任一子列模糊收敛到此序列的极限;模糊收敛的序列是柯西列;柯西列是模糊有界的;任一有模糊收敛子列的柯西列是模糊收敛的;存在不完备的广义模糊赋范空间.结论 说明赋范空间中的一些概念和结果可类似的在广义模糊赋范空间中建立.
Viscosity Approximation Methods for Two Accretive Operators in Banach Spaces
Directory of Open Access Journals (Sweden)
Jun-Min Chen
2013-01-01
Full Text Available We introduced a viscosity iterative scheme for approximating the common zero of two accretive operators in a strictly convex Banach space which has a uniformly Gâteaux differentiable norm. Some strong convergence theorems are proved, which improve and extend the results of Ceng et al. (2009 and some others.
On the Rate of Structural Change in Scale Spaces
DEFF Research Database (Denmark)
Gustafsson, David Karl John; Pedersen, Kim Steenstrup; Lauze, Francois Bernard;
2009-01-01
We analyze the rate in which image details are suppressed as a function of the regularization parameter, using first order Tikhonov regularization, Linear Gaussian Scale Space and Total Variation image decomposition. The squared L2-norm of the regularized solution and the residual are studied...... as a function of the regularization parameter. For first order Tikhonov regularization it is shown that the norm of the regularized solution is a convex function, while the norm of the residual is not a concave function. The same result holds for Gaussian Scale Space when the parameter is the variance...
Peña-Casanova, Jordi; Quintana-Aparicio, María; Quiñones-Ubeda, Sonia; Aguilar, Miquel; Molinuevo, José Luis; Serradell, Mónica; Robles, Alfredo; Barquero, María Sagrario; Villanueva, Clara; Antúnez, Carmen; Martínez-Parra, Carlos; Frank-García, Anna; Aguilar, María Dolores; Fernández, Manuel; Alfonso, Verónica; Sol, Josep M; Blesa, Rafael
2009-06-01
This study forms part of the Spanish Multicenter Normative Studies (NEURONORMA project). Normative data for people aged over 49 years are presented for selected tasks of the visual object and space perception battery (VOSP) and for the judgment of line orientation (JLO) test. Age-adjusted norms were derived from a sample of 341 participants who are cognitively normal and community-dwelling. Age- and education-adjusted norms are also provided. Years of education were modeled on age-scaled scores to derive regression equations that were applied for further demographic adjustments. The normative information provided here should prove useful for characterizing and interpreting individual test performances as well as comparing the scores from these tests with any other test using NEURONORMA norms.
Institute of Scientific and Technical Information of China (English)
赵吕慧子; 孙经先
2011-01-01
The definition of a class of new operators, convex-power 1-set-contraction operators in Banach spaces is giv en , and the existence of fixed points of this class of operators is studied. By using methods of approximation by opera tors, the fixed point theorems of Rothe and Altman type convex-power 1-set-contraction operators is obtained, which generalize fixed point theorems of 1-set-contraction operators.%在Banach空间中给出了一类新算子——凸幂1集压缩算子的定义,研究了这类新算子不动点的存在性问题,利用算子逼近的方法,获得了Rothe及Altman型凸幂1集压缩算子的不动点定理,推广了1集压缩算子的不动点定理.
Directory of Open Access Journals (Sweden)
Ofoedu EU
2008-01-01
Full Text Available Abstract Let be a real reflexive and strictly convex Banach space which has a uniformly Gâteaux differentiable norm and be a closed convex nonempty subset of . Strong convergence theorems for approximation of a common zero of a countably infinite family of -accretive mappings from to are proved. Consequently, we obtained strong convergence theorems for a countably infinite family of pseudocontractive mappings.
Directory of Open Access Journals (Sweden)
E. U. Ofoedu
2008-03-01
Full Text Available Let E be a real reflexive and strictly convex Banach space which has a uniformly GÃƒÂ¢teaux differentiable norm and C be a closed convex nonempty subset of E. Strong convergence theorems for approximation of a common zero of a countably infinite family of m-accretive mappings from C to E are proved. Consequently, we obtained strong convergence theorems for a countably infinite family of pseudocontractive mappings.
Statistical properties of convex clustering
Tan, Kean Ming; Witten, Daniela
2015-01-01
In this manuscript, we study the statistical properties of convex clustering. We establish that convex clustering is closely related to single linkage hierarchical clustering and $k$-means clustering. In addition, we derive the range of the tuning parameter for convex clustering that yields a non-trivial solution. We also provide an unbiased estimator of the degrees of freedom, and provide a finite sample bound for the prediction error for convex clustering. We compare convex clustering to so...
Exact and Approximate Sizes of Convex Datacubes
Nedjar, Sébastien
In various approaches, data cubes are pre-computed in order to efficiently answer Olap queries. The notion of data cube has been explored in various ways: iceberg cubes, range cubes, differential cubes or emerging cubes. Previously, we have introduced the concept of convex cube which generalizes all the quoted variants of cubes. More precisely, the convex cube captures all the tuples satisfying a monotone and/or antimonotone constraint combination. This paper is dedicated to a study of the convex cube size. Actually, knowing the size of such a cube even before computing it has various advantages. First of all, free space can be saved for its storage and the data warehouse administration can be improved. However the main interest of this size knowledge is to choose at best the constraints to apply in order to get a workable result. For an aided calibrating of constraints, we propose a sound characterization, based on inclusion-exclusion principle, of the exact size of convex cube as long as an upper bound which can be very quickly yielded. Moreover we adapt the nearly optimal algorithm HyperLogLog in order to provide a very good approximation of the exact size of convex cubes. Our analytical results are confirmed by experiments: the approximated size of convex cubes is really close to their exact size and can be computed quasi immediately.
Supremum norm differentiability
Directory of Open Access Journals (Sweden)
I. E. Leonard
1983-01-01
Full Text Available The points of Gateaux and Fréchet differentiability of the norm in C(T,E are obtained, where T is a locally compact Hausdorff space and E is a real Banach space. Applications of these results are given to the space ℓ∞(E of all bounded sequences in E and to the space B(ℓ1,E of all bounded linear operators from ℓ1 into E
A Mean Point Based Convex Hull Computation Algorithm
Directory of Open Access Journals (Sweden)
Digvijay Singh
2016-11-01
Full Text Available The optimal solution of a Linear Programming problem (LPP is a basic feasible solution and all basic feasible solutions are extreme or boundary points of a convex region formed by the constraint functions of the LPP. In fact, the feasible solution space is not always a convex set so the verification of extreme points for optimality is quite difficult. In order to cover the non-convex feasible points within a convex set, a convex hull is imagined so that the extreme or boundary points may be checked for evaluation of the optimum solution in the decision-making process. In this article a computer assisted convex hull computation algorithm using the Mean Point and Python code verified results of the designed algorithm are discussed.
Hörmander, Lars
1994-01-01
The first two chapters of this book are devoted to convexity in the classical sense, for functions of one and several real variables respectively. This gives a background for the study in the following chapters of related notions which occur in the theory of linear partial differential equations and complex analysis such as (pluri-)subharmonic functions, pseudoconvex sets, and sets which are convex for supports or singular supports with respect to a differential operator. In addition, the convexity conditions which are relevant for local or global existence of holomorphic differential equations are discussed, leading up to Trépreau’s theorem on sufficiency of condition (capital Greek letter Psi) for microlocal solvability in the analytic category. At the beginning of the book, no prerequisites are assumed beyond calculus and linear algebra. Later on, basic facts from distribution theory and functional analysis are needed. In a few places, a more extensive background in differential geometry or pseudodiffer...
Finding approximately rank-one submatrices with the nuclear norm and l1 norm
Doan, Xuan Vinh
2010-01-01
We propose a convex optimization formulation with the nuclear norm and $\\ell_1$-norm to find a large approximately rank-one submatrix of a given nonnegative matrix. We develop optimality conditions for the formulation and characterize the properties of the optimal solutions. We establish conditions under which the optimal solution of the convex formulation has a specific sparse structure. Finally, we show that, under certain hypotheses, with high probability, the approach can recover the rank-one submatrix even when it is corrupted with random noise and inserted as a submatrix into a much larger random noise matrix.
Convex Geometry and Stoichiometry
Jer-Chin,
2011-01-01
We demonstrate the benefits of a convex geometric perspective for questions on chemical stoichiometry. We show that the balancing of chemical equations, the use of "mixtures" to explain multiple stoichiometry, and the half-reaction for balancing redox actions all yield nice convex geometric interpretations. We also relate some natural questions on reaction mechanisms with the enumeration of lattice points in polytopes. Lastly, it is known that a given reaction mechanism imposes linear constraints on observed stoichiometries. We consider the inverse question of deducing reaction mechanism consistent with a given set of linear stoichiometric restrictions.
Bertilsdotter Rosqvist, Hanna; Arnberg, Klara
2015-01-01
Within sexual geographies, sexual struggles over urban public spaces are frequently explored. Less common is research on sexual struggles within sexually shared spaces and gay spaces. The aim of the article is to examine discursive struggles of meanings of gay male identity enacted in discussions of commodification/capitalism, disclosure, and space in Swedish gay press during 1969-1986. We trace the ambivalent feelings or the emergence of a new gay male norm situated between commercialism and non-commercialism within the Swedish gay press back to the 1970s. In the article we show how a monosexualization process was taking place in both the Swedish gay press as well as within sexual spaces. We explore rhetorical struggles between two competing discursive meanings of (ideal homonormative) male homosexuality, gay culture, and space: one wider (inclusive) and one narrower (exclusive).
Institute of Scientific and Technical Information of China (English)
夏顺友
2013-01-01
利用抽象凸空间满足的H0条件和紧集的有限覆盖及与之相应的单位分解构造标准单纯形上的连续映射,从而由Brouwer不动点定理证明了抽象凸锥度量空间上具有邻域抽象凸值的锥度量上半连续集值映射的一个锥度量逼近连续选择定理,并由此得到具有邻域抽象凸值的锥度量上半连续集值映射的一个不动点定理,然后将此不动点定理应用于博弈论,通过构造锥度量上半连续最优反应集值映射得到抽象凸锥度量策略空间上的n人非合作广义博弈Nash平衡的一个存在性结果.%Constructing a continuity map on standard simplex by using H0 condition of abstract convex space and the partition of unity subordinate to the finite covering of a compact set,a cone metric approximate continuity selection for cone metric upper semi-continuous set-valued maps in abstract convex cone metric spaces is proved by employing Brouwer fixed point theorem.Then a fixed point theorem for cone metric upper semi-continuous maps is derived.As an application,constructing cone metric upper semi-continuous best reflect map,the existence of Nash equilibrium of n-person non-cooperative generalized game with abstract convex cone metric strategy space is proved.
Tropicalized Lambda Lengths, Measured Laminations and Convexity
DEFF Research Database (Denmark)
C. Penner, R.
cell decomposition of a surface is discovered in the limit. Finally, the tropical analogue of the convex hull construction in Minkowski space is formulated as an explicit algorithm that serially simplifies a triangulation with respect to a fixed lamination and has its own independent applications....
Tropicalized Lambda Lengths, Measured Laminations and Convexity
DEFF Research Database (Denmark)
C. Penner, R.
This work uncovers the tropical analogue for measured laminations of the convex hull construction of decorated Teichmueller theory, namely, it is a study in coordinates of geometric degeneration to a point of Thurston's boundary for Teichmueller space. This may offer a paradigm for the extension...
Regularization methods for a class of variational inequalities in banach spaces
Buong, Nguyen; Phuong, Nguyen Thi Hong
2012-11-01
In this paper, we introduce two regularization methods, based on the Browder-Tikhonov and iterative regularizations, for finding a solution of variational inequalities over the set of common fixed points of an infinite family of nonexpansive mappings on real reflexive and strictly convex Banach spaces with a uniformly Gateaux differentiate norm.
On the fixed points of nonexpansive mappings in direct sums of Banach spaces
Wiśnicki, Andrzej
2011-01-01
We show that if a Banach space X has the weak fixed point property for nonexpansive mappings and Y has the generalized Gossez-Lami Dozo property or is uniformly convex in every direction, then the direct sum of X and Y with a strictly monotone norm has the weak fixed point property. The result is new even if Y is finite-dimensional.
A Variable Kernel Convex Spaces Expected Revenue Balance Control Model%一种变量核凸空间内期望收益均衡控制模型
Institute of Scientific and Technical Information of China (English)
杨玉杰; 李慧敏
2015-01-01
Through the construction of the mathematical model for the Bias game and economic growth index prediction, achieve the desired control earnings balance on the regional economy in the variable kernel convex spaces, the traditional expected revenue balance control model with nonlinear step nonlinear model into the optimization method, the optimization speed is slow, and the ultimate convergence error is large. By introducing the expected profit function and the expected equilibrium method, the fairness of the rational exchange protocol are described, the construction of a variable kernel con-vex spaces expected revenue balance control model, to improve the control accuracy of the mathematical model, variable kernel convex space is given within the expected revenue equalization control conditions, finally carries on the proof of con-vergence and global stability of expected equilibrium point the linear sequence, realizes the variable kernel convex spaces expected revenue balance control, and the control model is globally stable.%通过对贝叶斯博弈经济增长指数的数学模型构建和预测,在变量核凸空间内实现对区域经济的期望收益均衡控制,传统的期望收益均衡控制模型采用非线性模型下的非线性步进寻优方法,寻优速度慢,且最终收敛误差较大.通过引入期望收益函数和期望均衡的方法,给出理性交换协议的公平性描述,构建一种变量核凸空间内期望收益均衡控制模型,提高数学模型的控制准确性,给出变量核凸空间内期望收益均衡控制条件,最后进行期望均衡点线性数列的收敛性和全局稳定性证明,实现了变量核凸空间内期望收益均衡控制,且控制模型是全局稳定的.
The Knaster-Kuratowski-Mazurkiewicz theorem and abstract convexities
Cain, George L., Jr.; González, Luis
2008-02-01
The Knaster-Kuratowski-Mazurkiewicz covering theorem (KKM), is the basic ingredient in the proofs of many so-called "intersection" theorems and related fixed point theorems (including the famous Brouwer fixed point theorem). The KKM theorem was extended from Rn to Hausdorff linear spaces by Ky Fan. There has subsequently been a plethora of attempts at extending the KKM type results to arbitrary topological spaces. Virtually all these involve the introduction of some sort of abstract convexity structure for a topological space, among others we could mention H-spaces and G-spaces. We have introduced a new abstract convexity structure that generalizes the concept of a metric space with a convex structure, introduced by E. Michael in [E. Michael, Convex structures and continuous selections, Canad. J. MathE 11 (1959) 556-575] and called a topological space endowed with this structure an M-space. In an article by Shie Park and Hoonjoo Kim [S. Park, H. Kim, Coincidence theorems for admissible multifunctions on generalized convex spaces, J. Math. Anal. Appl. 197 (1996) 173-187], the concepts of G-spaces and metric spaces with Michael's convex structure, were mentioned together but no kind of relationship was shown. In this article, we prove that G-spaces and M-spaces are close related. We also introduce here the concept of an L-space, which is inspired in the MC-spaces of J.V. Llinares [J.V. Llinares, Unified treatment of the problem of existence of maximal elements in binary relations: A characterization, J. Math. Econom. 29 (1998) 285-302], and establish relationships between the convexities of these spaces with the spaces previously mentioned.
SMOOTHING BY CONVEX QUADRATIC PROGRAMMING
Institute of Scientific and Technical Information of China (English)
Bing-sheng He; Yu-mei Wang
2005-01-01
In this paper, we study the relaxed smoothing problems with general closed convex constraints. It is pointed out that such problems can be converted to a convex quadratic minimization problem for which there are good programs in software libraries.
Reconstruction of convex bodies from surface tensors
DEFF Research Database (Denmark)
Kousholt, Astrid; Kiderlen, Markus
2016-01-01
We present two algorithms for reconstruction of the shape of convex bodies in the two-dimensional Euclidean space. The first reconstruction algorithm requires knowledge of the exact surface tensors of a convex body up to rank s for some natural number s. When only measurements subject to noise...... of surface tensors are available for reconstruction, we recommend to use certain values of the surface tensors, namely harmonic intrinsic volumes instead of the surface tensors evaluated at the standard basis. The second algorithm we present is based on harmonic intrinsic volumes and allows for noisy...
Reconstruction of convex bodies from surface tensors
DEFF Research Database (Denmark)
Kousholt, Astrid; Kiderlen, Markus
We present two algorithms for reconstruction of the shape of convex bodies in the two-dimensional Euclidean space. The first reconstruction algorithm requires knowledge of the exact surface tensors of a convex body up to rank s for some natural number s. The second algorithm uses harmonic intrinsic...... volumes which are certain values of the surface tensors and allows for noisy measurements. From a generalized version of Wirtinger's inequality, we derive stability results that are utilized to ensure consistency of both reconstruction procedures. Consistency of the reconstruction procedure based...
NORM management; Gerencia de NORM
Energy Technology Data Exchange (ETDEWEB)
Reis, Rocio dos
2016-07-01
In the chapter 9 concepts and examples for helping to solve the NORM question in the industries are presented. The challenge is to handle with radioactivity questions and many industries do not know that are producing radioactivity material besides having to learn and match the nuclear concepts and legislation with the conventional pollutants. The risks associated to NORM and a methodology to handle with the question are mentioned. The need for establishing responsibilities is also highlighted. Finally, a planning to manage NORM is suggested. The equation for determination os minerals activity and concentrates in secular equilibrium is annexed in A.
Institute of Scientific and Technical Information of China (English)
白小净; 魏文展
2015-01-01
This paper introduces the concept of smoothness and uniform smoothness and discusses the relationship that strictly convex and smoothness ,uniformly convex and uniform smoothness be‐tween Z-spaces and B-Z-spaces ,and gives the inference equivalence in reflexive B -Z -spaces .At the same time ,it introduces the concept of weak convexity and (E) property ,resulting in the relevant theorer of weak convexity and (E) property .%引入了Z－空间中光滑性与一致光滑性的概念，讨论了在Z －空间及B －Z －空间中严格凸性与光滑性、一致凸性与一致光滑性之间的关系，并且给出了它们在自反的B－Z －空间下等价的若干结果：还引入了B －Z－空间上的弱凸性与（E ）性质的概念，得出了弱凸性及（E ）性质的相关定理。
Egalitarianism in Convex Fuzzy Games
Brânzei, R.; Dimitrov, D.A.; Tijs, S.H.
2002-01-01
In this paper the egalitarian solution for convex cooperative fuzzy games is introduced.The classical Dutta-Ray algorithm for finding the constrained egalitarian solution for convex crisp games is adjusted to provide the egalitarian solution of a convex fuzzy game.This adjusted algorithm is also a f
Average Convexity in Communication Situations
Slikker, M.
1998-01-01
In this paper we study inheritance properties of average convexity in communication situations. We show that the underlying graph ensures that the graphrestricted game originating from an average convex game is average convex if and only if every subgraph associated with a component of the underlyin
Convex bodies of states and maps
Grabowski, Janusz; Ibort, Alberto; Kuś, Marek; Marmo, Giuseppe
2013-10-01
We give a general solution to the question of when the convex hulls of orbits of quantum states on a finite-dimensional Hilbert space under unitary actions of a compact group have a non-empty interior in the surrounding space of all density operators. The same approach can be applied to study convex combinations of quantum channels. The importance of both problems stems from the fact that, usually, only sets with non-vanishing volumes in the embedding spaces of all states or channels are of practical importance. For the group of local transformations on a bipartite system we characterize maximally entangled states by the properties of a convex hull of orbits through them. We also compare two partial characteristics of convex bodies in terms of the largest balls and maximum volume ellipsoids contained in them and show that, in general, they do not coincide. Separable states, mixed-unitary channels and k-entangled states are also considered as examples of our techniques.
Efficient Approximation of Convex Recolorings
Moran, Shlomo; Snir, Sagi
2005-01-01
A coloring of a tree is convex if the vertices that pertain to any color induce a connected subtree; a partial coloring (which assigns colors to some of the vertices) is convex if it can be completed to a convex (total) coloring. Convex coloring of trees arise in areas such as phylogenetics, linguistics, etc. eg, a perfect phylogenetic tree is one in which the states of each character induce a convex coloring of the tree. Research on perfect phylogeny is usually focused on finding a tree so t...
Solórzano, Pedro
2010-01-01
A holonomic space $(V,H,L)$ is a normed vector space, $V$, a subgroup, $H$, of $Aut(V, \\|\\cdot\\|)$ and a group-norm, $L$, with a convexity property. We prove that with the metric $d_L(u,v)=\\inf_{a\\in H}\\{\\sqrt{L^2(a)+\\|u-av\\|^2}\\}$, $V$ is a metric space which is locally isometric to a Euclidean ball. Given a Sasaki-type metric on a vector bundle $E$ over a Riemannian manifold, we prove that the triplet $(E_p,Hol_p,L_p)$ is a holonomic space, where $Hol_p$ is the holonomy group and $L_p$ is the length norm defined within. The topology on $Hol_p$ given by the $L_p$ is finer than the subspace topology while still preserving many desirable properties. Using these notions, we introduce the notion of holonomy radius for a Riemannian manifold and prove it is positive. These results are applicable to the Gromov-Hausdorff convergence of Riemannian manifolds.
Subaperture Stitching Interferometry for Large Convex Aspheric Surfaces Project
National Aeronautics and Space Administration — The size and accuracy specifications of telescope mirrors are ever more demanding. This is particularly true for secondary mirrors, as they are convex and thus...
2010-12-02
evaluating the function ΘP (A) for any fixed A,P is equivalent to solving the so-called Quadratic Assignment Problem ( QAP ), and thus we can employ various...tractable linear programming, spectral, and SDP relaxations of QAP [40, 11, 33]. In particular we discuss recent work [14] on exploiting group...symmetry in SDP relaxations of QAP , which is useful for approximately computing elementary convex graph invariants in many interesting cases. Finally in
赋范线性空间中的Hilbert型积分不等式%Hilbert-type Integral Inequalities in Normed Linear Spaces
Institute of Scientific and Technical Information of China (English)
匡继昌
2013-01-01
基于利用一个积分恒等式的新技巧,建立了赋范线性空间中新的Hilbert型积分不等式.这些新的结果包含了n维欧氏空间中n重积分的Hilbert型积分不等式作为其特殊情形.%In this paper we employ a new technique based on an integral identity to study some new Hilbert-type integral inequalities in normed linear spaces. These new results include the corresponding multiple Hilbert-type integral inequalities in Rn as special cases.
Directory of Open Access Journals (Sweden)
Tian Zhou Xu
2010-01-01
Full Text Available Using the fixed point methods, we prove the generalized Hyers-Ulam stability of the general mixed additive-quadratic-cubic-quartic functional equation f(x+ky+f(x−ky=k2f(x+y+k2f(x−y+2(1−k2f(x+((k4−k2/12[f(2y+f(−2y−4f(y−4f(−y] for a fixed integer k with k≠0,±1 in non-Archimedean normed spaces.
Mixed-norm estimates for the M/EEG inverse problem using accelerated gradient methods.
Gramfort, Alexandre; Kowalski, Matthieu; Hämäläinen, Matti
2012-04-07
Magneto- and electroencephalography (M/EEG) measure the electromagnetic fields produced by the neural electrical currents. Given a conductor model for the head, and the distribution of source currents in the brain, Maxwell's equations allow one to compute the ensuing M/EEG signals. Given the actual M/EEG measurements and the solution of this forward problem, one can localize, in space and in time, the brain regions that have produced the recorded data. However, due to the physics of the problem, the limited number of sensors compared to the number of possible source locations, and measurement noise, this inverse problem is ill-posed. Consequently, additional constraints are needed. Classical inverse solvers, often called minimum norm estimates (MNE), promote source estimates with a small ℓ₂ norm. Here, we consider a more general class of priors based on mixed norms. Such norms have the ability to structure the prior in order to incorporate some additional assumptions about the sources. We refer to such solvers as mixed-norm estimates (MxNE). In the context of M/EEG, MxNE can promote spatially focal sources with smooth temporal estimates with a two-level ℓ₁/ℓ₂ mixed-norm, while a three-level mixed-norm can be used to promote spatially non-overlapping sources between different experimental conditions. In order to efficiently solve the optimization problems of MxNE, we introduce fast first-order iterative schemes that for the ℓ₁/ℓ₂ norm give solutions in a few seconds making such a prior as convenient as the simple MNE. Furthermore, thanks to the convexity of the optimization problem, we can provide optimality conditions that guarantee global convergence. The utility of the methods is demonstrated both with simulations and experimental MEG data.
Energy Technology Data Exchange (ETDEWEB)
Gray, P. [ed.
1997-02-01
The author reviews the question of regulation for naturally occuring radioactive material (NORM), and the factors that have made this a more prominent concern today. Past practices have been very relaxed, and have often involved very poor records, the involvment of contractors, and the disposition of contaminated equipment back into commercial service. The rationale behind the establishment of regulations is to provide worker protection, to exempt low risk materials, to aid in scrap recycling, to provide direction for remediation and to examine disposal options. The author reviews existing regulations at federal and state levels, impending legislation, and touches on the issue of site remediation and potential liabilities affecting the release of sites contaminated by NORM.
Introducing the Adaptive Convex Enveloping
Yu, Sheng
2011-01-01
Convexity, though extremely important in mathematical programming, has not drawn enough attention in the field of dynamic programming. This paper gives conditions for verifying convexity of the cost-to-go functions, and introduces an accurate, fast and reliable algorithm for solving convex dynamic programs with multivariate continuous states and actions, called Adaptive Convex Enveloping. This is a short introduction of the core technique created and used in my dissertation, so it is less formal, and misses some parts, such as literature review and reference, compared to a full journal paper.
Convex polytopes and quantum states
Energy Technology Data Exchange (ETDEWEB)
Wilmott, Colin; Kampermann, Hermann; Bruss, Dagmar [Institut fuer Theoretische Physik III, Heinrich-Heine-Universitaet Duesseldorf (Germany)
2010-07-01
A convex polytope is defined as the convex hull of a finite non-empty set of vectors. We present a theorem of Rado (1952) which characterizes the convex hull of the collection of all permutations of a given real d-tuple in terms of the Hardy-Littlewood-Polya spectral order relation prec. We give a necessary and sufficient condition to construct a d-dimensional convex polytope which utilizes Rado's original (d-1)-dimensional characterization, and we describe how the resulting polytope may be placed in a quantum mechanical framework.
Decomposability of Abstract and Path-Induced Convexities in Hypergraphs
Directory of Open Access Journals (Sweden)
Malvestuto Francesco Mario
2015-08-01
Full Text Available An abstract convexity space on a connected hypergraph H with vertex set V (H is a family C of subsets of V (H (to be called the convex sets of H such that: (i C contains the empty set and V (H, (ii C is closed under intersection, and (iii every set in C is connected in H. A convex set X of H is a minimal vertex convex separator of H if there exist two vertices of H that are separated by X and are not separated by any convex set that is a proper subset of X. A nonempty subset X of V (H is a cluster of H if in H every two vertices in X are not separated by any convex set. The cluster hypergraph of H is the hypergraph with vertex set V (H whose edges are the maximal clusters of H. A convexity space on H is called decomposable if it satisfies the following three properties:
Remarks on the Operator Norm Localization Property
Institute of Scientific and Technical Information of China (English)
Xianjin WANG
2011-01-01
The author studies the metric spaces with operator norm localization property. It is proved that the operator norm localization property is coarsely invariant and is preserved under certain infinite union. In the case of finitely generated groups, the operator norm localization property is also preserved under the direct limits.
Minimax-optimal rates for sparse additive models over kernel classes via convex programming
Raskutti, Garvesh; Yu, Bin
2010-01-01
Sparse additive models are families of $d$-variate functions that have the additive decomposition \\mbox{$f^* = \\sum_{j \\in S} f^*_j$,} where $S$ is a unknown subset of cardinality $s \\ll d$. We consider the case where each component function $f^*_j$ lies in a reproducing kernel Hilbert space, and analyze a simple kernel-based convex program for estimating the unknown function $f^*$. Working within a high-dimensional framework that allows both the dimension $d$ and sparsity $s$ to scale, we derive convergence rates in the $L^2(\\mathbb{P})$ and $L^2(\\mathbb{P}_n)$ norms. These rates consist of two terms: a \\emph{subset selection term} of the order $\\frac{s \\log d}{n}$, corresponding to the difficulty of finding the unknown $s$-sized subset, and an \\emph{estimation error} term of the order $s \\, \
Plane geometry and convexity of polynomial stability regions
Henrion, Didier
2008-01-01
The set of controllers stabilizing a linear system is generally non-convex in the parameter space. In the case of two-parameter controller design (e.g. PI control or static output feedback with one input and two outputs), we observe however that quite often for benchmark problem instances, the set of stabilizing controllers seems to be convex. In this note we use elementary techniques from real algebraic geometry (resultants and Bezoutian matrices) to explain this phenomenon. As a byproduct, we derive a convex linear matrix inequality (LMI) formulation of two-parameter fixed-order controller design problem, when possible.
Institute of Scientific and Technical Information of China (English)
朴勇杰
2006-01-01
我们应用已有的一般化凸空间上的KKM型定理得到Ky Fan型重合点定理,然后作为应用给出截口定理和择一性问题.我们的主要结果对文[1-5]中的相应的结果进行了改进和一般化.%We use the well-known KKM type theorem on generalized convex spaces to obtain Ky Fan type coincidence point theorem and then give section theorem and alternative problem as applications. Our main results generalize and improve the corresponding results in [1-5].
广义凸空间上重合点定理和极大极小不等式%Coincidence Point Theorems and Minimax Inequalities on Generalized Convex Spaces
Institute of Scientific and Technical Information of China (English)
朴勇杰; 朴东哲
2011-01-01
利用已知的KKM型定理,在广义空间上得到若干个新的重合点定理和推广的Fan-Browder型不动点定理,并且讨论了Von Newmann-Sion型极大极小不等式.主要结果改进和推广了文献中的相应结论.%By using well-known KKM type theorems, some new Ky Fan type coincidence theorems and generalized Fan-Browder type fixed point theorems on generalized convex spaces were obtained, and then Von Neumann-Sion type minimax inequlaties were discussed. The main results improve and generalize the corresponding results in the recent literature.
Constructive techniques for zeros of monotone mappings in certain Banach spaces.
Diop, C; Sow, T M M; Djitte, N; Chidume, C E
2015-01-01
Let E be a 2-uniformly convex real Banach space with uniformly Gâteaux differentiable norm, and [Formula: see text] its dual space. Let [Formula: see text] be a bounded strongly monotone mapping such that [Formula: see text] For given [Formula: see text] let [Formula: see text] be generated by the algorithm: [Formula: see text]where J is the normalized duality mapping from E into [Formula: see text] and [Formula: see text] is a real sequence in (0, 1) satisfying suitable conditions. Then it is proved that [Formula: see text] converges strongly to the unique point [Formula: see text] Finally, our theorems are applied to the convex minimization problem.
Global optimization over linear constraint non-convex programming problem
Institute of Scientific and Technical Information of China (English)
ZHANG Gui-Jun; WU Ti-Huan; YE Rong; YANG Hai-qing
2005-01-01
A improving Steady State Genetic Algorithm for global optimization over linear constraint non-convex programmin g problem is presented. By convex analyzing, the primal optimal problem can be converted to an equivalent problem, in which only the information of convex extremes of feasible space is included, and is more easy for GAs to solve. For avoiding invalid genetic operators, a redesigned convex crossover operator is also performed in evolving. As a integrality, the quality of two problem is proven, and a method is also given to get all extremes in linear constraint space. Simulation result show that new algorithm not only converges faster, but also can maintain an diversity population, and can get the global optimum of test problem.
Gradient of the Value Function in Parametric Convex Optimization Problems
Baotić, Mato
2016-01-01
We investigate the computation of the gradient of the value function in parametric convex optimization problems. We derive general expression for the gradient of the value function in terms of the cost function, constraints and Lagrange multipliers. In particular, we show that for the strictly convex parametric quadratic program the value function is continuously differentiable at every point in the interior of feasible space for which the Linear Independent Constraint Qualification holds.
Convex Games versus Clan Games
Brânzei, R.; Dimitrov, D.A.; Tijs, S.H.
2006-01-01
In this paper we provide characterizations of convex games and total clan games by using properties of their corresponding marginal games.We show that a "dualize and restrict" procedure transforms total clan games with zero worth for the clan into monotonic convex games.Furthermore, each monotonic
Convex Optimization without Projection Steps
Jaggi, Martin
2011-01-01
We study the general problem of minimizing a convex function over a compact convex domain. We will investigate a simple iterative approximation algorithm that does not need projection steps in order to stay inside the optimization domain. Instead of a projection step, the linearized problem defined by a current subgradient is solved, which gives a step direction that will naturally stay in the domain. The approach generalizes the sparse greedy algorithm of Clarkson (and the low-rank SDP solver by Hazan) to arbitrary convex domains, and to using subgradients for the case of non-differentiable convex functions. Analogously, we give a convergence proof guaranteeing {\\epsilon}-small duality gap after O(1/{\\epsilon}) iterations. The framework allows us understand the sparsity of approximate solutions for any l1-regularized convex optimization problem, expressed as a function of the approximation quality. We obtain matching upper and lower bounds of {\\Theta}(1/{\\epsilon}) for the sparsity for l1-problems. The same ...
Institute of Scientific and Technical Information of China (English)
文开庭
2009-01-01
In this paper,a new GLKKM type theorem is established for noncompact complete L-convex metric spaces.As applications,the properties of the solution set of variational in-equalities,intersection point sets,Ky Fan sections and maximal element sets are shown,and a Fan-Browder fixed point theorem is obtained.
On convex relaxation of graph isomorphism.
Aflalo, Yonathan; Bronstein, Alexander; Kimmel, Ron
2015-03-10
We consider the problem of exact and inexact matching of weighted undirected graphs, in which a bijective correspondence is sought to minimize a quadratic weight disagreement. This computationally challenging problem is often relaxed as a convex quadratic program, in which the space of permutations is replaced by the space of doubly stochastic matrices. However, the applicability of such a relaxation is poorly understood. We define a broad class of friendly graphs characterized by an easily verifiable spectral property. We prove that for friendly graphs, the convex relaxation is guaranteed to find the exact isomorphism or certify its inexistence. This result is further extended to approximately isomorphic graphs, for which we develop an explicit bound on the amount of weight disagreement under which the relaxation is guaranteed to find the globally optimal approximate isomorphism. We also show that in many cases, the graph matching problem can be further harmlessly relaxed to a convex quadratic program with only n separable linear equality constraints, which is substantially more efficient than the standard relaxation involving n2 equality and n2 inequality constraints. Finally, we show that our results are still valid for unfriendly graphs if additional information in the form of seeds or attributes is allowed, with the latter satisfying an easy to verify spectral characteristic.
Brasco, Lorenzo
2012-01-01
We investigate some basic properties of the {\\it heart} $\\heartsuit(\\mathcal{K})$ of a convex set $\\mathcal{K}.$ It is a subset of $\\mathcal{K},$ whose definition is based on mirror reflections of euclidean space, and is a non-local object. The main motivation of our interest for $\\heartsuit(\\mathcal{K})$ is that this gives an estimate of the location of the hot spot in a convex heat conductor with boundary temperature grounded at zero. Here, we investigate on the relation between $\\heartsuit(\\mathcal{K})$ and the mirror symmetries of $\\mathcal{K};$ we show that $\\heartsuit(\\mathcal{K})$ contains many (geometrically and phisically) relevant points of $\\mathcal{K};$ we prove a simple geometrical lower estimate for the diameter of $\\heartsuit(\\mathcal{K});$ we also prove an upper estimate for the area of $\\heartsuit(\\mathcal{K}),$ when $\\mathcal{K}$ is a triangle.
Institute of Scientific and Technical Information of China (English)
唐玉超; 刘理蔚
2007-01-01
本文研究了在一致凸Banach空间中定义在闭凸集C上渐近非扩张映象T不动点的迭代问题,我们的讨论去掉了在刘和薛[2]中C是有界的假设.%In this paper,we approximate fixed point of asymptotically nonexpansive mapping T on a closed,convex subset C of a uniformly convex Banach space.Our argument removes the boundedness assumption on C,generalizing theorems of Liu and Xue.
Directory of Open Access Journals (Sweden)
Satit Saejung
2005-01-01
Full Text Available We prove that the moduli of U-convexity, introduced by Gao (1995, of the ultrapower X˜ of a Banach space X and of X itself coincide whenever X is super-reflexive. As a consequence, some known results have been proved and improved. More precisely, we prove that uX(1>0 implies that both X and the dual space X∗ of X have uniform normal structure and hence the “worth” property in Corollary 7 of Mazcuñán-Navarro (2003 can be discarded.
Directory of Open Access Journals (Sweden)
Wangkeeree Rabian
2010-01-01
Full Text Available For a countable family of strictly pseudo-contractions, a strong convergence of viscosity iteration is shown in order to find a common fixed point of in either a p-uniformly convex Banach space which admits a weakly continuous duality mapping or a p-uniformly convex Banach space with uniformly Gâteaux differentiable norm. As applications, at the end of the paper we apply our results to the problem of finding a zero of accretive operators. The main result extends various results existing in the current literature.
High-Dimensional Analysis of Convex Optimization-Based Massive MIMO Decoders
Ben Atitallah, Ismail
2017-04-01
A wide range of modern large-scale systems relies on recovering a signal from noisy linear measurements. In many applications, the useful signal has inherent properties, such as sparsity, low-rankness, or boundedness, and making use of these properties and structures allow a more efficient recovery. Hence, a significant amount of work has been dedicated to developing and analyzing algorithms that can take advantage of the signal structure. Especially, since the advent of Compressed Sensing (CS) there has been significant progress towards this direction. Generally speaking, the signal structure can be harnessed by solving an appropriate regularized or constrained M-estimator. In modern Multi-input Multi-output (MIMO) communication systems, all transmitted signals are drawn from finite constellations and are thus bounded. Besides, most recent modulation schemes such as Generalized Space Shift Keying (GSSK) or Generalized Spatial Modulation (GSM) yield signals that are inherently sparse. In the recovery procedure, boundedness and sparsity can be promoted by using the ℓ1 norm regularization and by imposing an ℓ∞ norm constraint respectively. In this thesis, we propose novel optimization algorithms to recover certain classes of structured signals with emphasis on MIMO communication systems. The exact analysis permits a clear characterization of how well these systems perform. Also, it allows an automatic tuning of the parameters. In each context, we define the appropriate performance metrics and we analyze them exactly in the High Dimentional Regime (HDR). The framework we use for the analysis is based on Gaussian process inequalities; in particular, on a new strong and tight version of a classical comparison inequality (due to Gordon, 1988) in the presence of additional convexity assumptions. The new framework that emerged from this inequality is coined as Convex Gaussian Min-max Theorem (CGMT).
Watkins, N. W.; Chau, Y.; Chapman, S. C.
2010-12-01
The idea of describing animal movement by mathematical models based on diffusion and Brownian motion has a long heritage. It has thus been natural to account for those aspects of motion that depart from the Brownian by the use of models incorporating long memory & subdiffusion (“the Joseph effect”) and/or heavy tails & superdiffusion (“the Noah effect”). My own interest in this problem was originally from a geoscience perspective, and was triggered by the need to model time series in space physics where both effects coincide. Subsequently I have been involved in animal foraging studies [e.g. Edwards et al, Nature, 2007]. I will describe some recent work [Watkins et al, PRE, 2009] which studies how fixed-timestep and variable-timestep formulations of anomalous diffusion are related in the presence of heavy tails and long range memory (stable processes versus the CTRW). Quantities for which different scaling relations are predicted between the two approaches are of particular interest, to aid testability. I will also present some of work in progress on the convex hull of anomalously diffusing walkers, inspired by its possible relevance to the idea of home range in biology, and by Randon-Furling et al’s recent analytical results in the Brownian case [PRL, 2009].
Convex functions, monotone operators and differentiability
Phelps, Robert R
1989-01-01
These notes start with an introduction to the differentiability of convex functions on Banach spaces, leading to the study of Asplund spaces and their intriguing relationship to monotone operators (and more general set-values maps) and Banach spaces with the Radon-Nikodym property. While much of this is classical, some of it is presented using streamlined proofs which were not available until recently. Considerable attention is paid to contemporary results on variational principles and perturbed optimization in Banach spaces, exhibiting their close connections with Asplund spaces. An introductory course in functional analysis is adequate background for reading these notes which can serve as the basis for a seminar of a one-term graduate course. There are numerous excercises, many of which form an integral part of the exposition.
Decision Problems For Convex Languages
Brzozowski, Janusz; Xu, Zhi
2008-01-01
In this paper we examine decision problems associated with various classes of convex languages, studied by Ang and Brzozowski (under the name "continuous languages''). We show that we can decide whether a given language L is prefix-, suffix-, factor-, or subword-convex in polynomial time if L is represented by a DFA, but that the problem is PSPACE-hard if L is represented by an NFA. In the case that a regular language is not convex, we prove tight upper bounds on the length of the shortest words demonstrating this fact, in terms of the number of states of an accepting DFA. Similar results are proved for some subclasses of convex languages: the prefix-, suffix-, factor-, and subword-closed languages, and the prefix-, suffix-, factor-, and subword-free languages.
On Fuzzy Simplex and Fuzzy Convex Hull
Institute of Scientific and Technical Information of China (English)
Dong QIU; Wei Quan ZHANG
2011-01-01
In this paper,we discuss fuzzy simplex and fuzzy convex hull,and give several representation theorems for fuzzy simplex and fuzzy convex hull.In addition,by giving a new characterization theorem of fuzzy convex hull,we improve some known results about fuzzy convex hull.
The Convex Coordinates of the Symmedian Point
Boyd, J. N.; Raychowdhury, P. N.
2006-01-01
In this note, we recall the convex (or barycentric) coordinates of the points of a closed triangular region. We relate the convex and trilinear coordinates of the interior points of the triangular region. We use the relationship between convex and trilinear coordinates to calculate the convex coordinates of the symmedian point of the triangular…
Convexity Adjustments for ATS Models
DEFF Research Database (Denmark)
Murgoci, Agatha; Gaspar, Raquel M.
Practitioners are used to value a broad class of exotic interest rate derivatives simply by preforming for what is known as convexity adjustments (or convexity corrections). We start by exploiting the relations between various interest rate models and their connections to measure changes. As a re......Practitioners are used to value a broad class of exotic interest rate derivatives simply by preforming for what is known as convexity adjustments (or convexity corrections). We start by exploiting the relations between various interest rate models and their connections to measure changes....... As a result we classify convexity adjustments into forward adjustments and swaps adjustments. We, then, focus on affine term structure (ATS) models and, in this context, conjecture convexity adjustments should be related of affine functionals. In the case of forward adjustments, we show how to obtain exact...... formulas. Concretely for LIBOR in arrears (LIA) contracts, we derive the system of Riccatti ODE-s one needs to compute to obtain the exact adjustment. Based upon the ideas of Schrager and Pelsser (2006) we are also able to derive general swap adjustments useful, in particular, when dealing with constant...
Small sets in convex geometry and formal independence over ZFC
Directory of Open Access Journals (Sweden)
Menachem Kojman
2005-01-01
Full Text Available To each closed subset S of a finite-dimensional Euclidean space corresponds a σ-ideal of sets (S which is σ-generated over S by the convex subsets of S. The set-theoretic properties of this ideal hold geometric information about the set. We discuss the relation of reducibility between convexity ideals and the connections between convexity ideals and other types of ideals, such as the ideals which are generated over squares of Polish space by graphs and inverses of graphs of continuous self-maps, or Ramsey ideals, which are generated over Polish spaces by the homogeneous sets with respect to some continuous pair coloring. We also attempt to present to nonspecialists the set-theoretic methods for dealing with formal independence as a means of geometric investigations.
Linear System Identification via Atomic Norm Regularization
Shah, Parikshit; Tang, Gongguo; Recht, Benjamin
2012-01-01
This paper proposes a new algorithm for linear system identification from noisy measurements. The proposed algorithm balances a data fidelity term with a norm induced by the set of single pole filters. We pose a convex optimization problem that approximately solves the atomic norm minimization problem and identifies the unknown system from noisy linear measurements. This problem can be solved efficiently with standard, freely available software. We provide rigorous statistical guarantees that explicitly bound the estimation error (in the H_2-norm) in terms of the stability radius, the Hankel singular values of the true system and the number of measurements. These results in turn yield complexity bounds and asymptotic consistency. We provide numerical experiments demonstrating the efficacy of our method for estimating linear systems from a variety of linear measurements.
Systematization of problems on ball estimates of a convex compactum
Dudov, S. I.
2015-09-01
We consider a class of finite-dimensional problems on the estimation of a convex compactum by a ball of an arbitrary norm in the form of extremal problems whose goal function is expressed via the function of the distance to the farthest point of the compactum and the function of the distance to the nearest point of the compactum or its complement. Special attention is devoted to the problem of estimating (approximating) a convex compactum by a ball of fixed radius in the Hausdorff metric. It is proved that this problem plays the role of the canonical problem: solutions of any problem in the class under consideration can be expressed via solutions of this problem for certain values of the radius. Based on studying and using the properties of solutions of this canonical problem, we obtain ranges of values of the radius in which the canonical problem expresses solutions of the problems on inscribed and circumscribed balls, the problem of uniform estimate by a ball in the Hausdorff metric, the problem of asphericity of a convex body, the problems of spherical shells of the least thickness and of the least volume for the boundary of a convex body. This makes it possible to arrange the problems in increasing order of the corresponding values of the radius. Bibliography: 34 titles.
Convergence on Composite Iterative Schemes for Nonexpansive Mappings in Banach Spaces
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Jung JongSoo
2008-01-01
Full Text Available Abstract Let be a reflexive Banach space with a uniformly Gâteaux differentiable norm. Suppose that every weakly compact convex subset of has the fixed point property for nonexpansive mappings. Let be a nonempty closed convex subset of , a contractive mapping (or a weakly contractive mapping, and nonexpansive mapping with the fixed point set . Let be generated by a new composite iterative scheme: , , . It is proved that converges strongly to a point in , which is a solution of certain variational inequality provided that the sequence satisfies and , for some and the sequence is asymptotically regular.
Convex Aspherical Surface Testing Using Catadioptric Partial Compensating System
Wang, Jingxian; Hao, Qun; Hu, Yao; Wang, Shaopu; Li, Tengfei; Tian, Yuhan; Li, Lin
2016-01-01
Aspheric optical components are the indispensable part of modern optics systems. With the constant development of aspheric optical fabrication technique, the systems with large aperture convex aspheric optical components are widely used in astronomy and space optics. Thus, the measurement of the figure error of the whole convex aspherical surface with high precision comes to be a challenge in the area of optical surface manufacture, and surface testing method is also very important. This paper presents a new partial compensating system by the combination of a refractive lens and a reflective mirror for testing convex aspherical surface. The refractive lens is used to compensate the aberration of the tested convex asphere partially. The reflective mirror is a spherical mirror which is coaxial to the refractive lens and reflecting the lights reflected by the tested convex asphere back to the convex asphere itself. With the long focal length and large aperture system we can realize a lighter and more compact system than the refractive partial compensating system because the spheric reflective mirror is more easily to realize and can bending the light conveniently.
Institute of Scientific and Technical Information of China (English)
陈雪; 马建文
2007-01-01
The Householder transformation-norm structure function in L2 vector space of linear algebra is introduced, and the edge enhancement for remote sensing images is realized. The experiment result is compared with traditional Laplacian and Sobel edge enhancements and it shows that the effect of the new method is better than that of the traditional algorithms.
Iterative Methods for Pseudocontractive Mappings in Banach Spaces
Directory of Open Access Journals (Sweden)
Jong Soo Jung
2013-01-01
Full Text Available Let E a reflexive Banach space having a uniformly Gâteaux differentiable norm. Let C be a nonempty closed convex subset of E, T:C→C a continuous pseudocontractive mapping with F(T≠∅, and A:C→C a continuous bounded strongly pseudocontractive mapping with a pseudocontractive constant k∈(0,1. Let {αn} and {βn} be sequences in (0,1 satisfying suitable conditions and for arbitrary initial value x0∈C, let the sequence {xn} be generated by xn=αnAxn+βnxn-1+(1-αn-βnTxn, n≥1. If either every weakly compact convex subset of E has the fixed point property for nonexpansive mappings or E is strictly convex, then {xn} converges strongly to a fixed point of T, which solves a certain variational inequality related to A.
Stable 1-Norm Error Minimization Based Linear Predictors for Speech Modeling
DEFF Research Database (Denmark)
Giacobello, Daniele; Christensen, Mads Græsbøll; Jensen, Tobias Lindstrøm;
2014-01-01
In linear prediction of speech, the 1-norm error minimization criterion has been shown to provide a valid alternative to the 2-norm minimization criterion. However, unlike 2-norm minimization, 1-norm minimization does not guarantee the stability of the corresponding all-pole filter and can generate...... of the shift operator associated with the particular prediction problem considered. The second method uses the alternative Cauchy bound to impose a convex constraint on the predictor in the 1-norm error minimization. These methods are compared with two existing methods: the Burg method, based on the 1-norm...... minimization of the forward and backward prediction error, and the iteratively reweighted 2-norm minimization known to converge to the 1-norm minimization with an appropriate selection of weights. The evaluation gives proof of the effectiveness of the new methods, performing as well as unconstrained 1-norm...
Institute of Scientific and Technical Information of China (English)
刘斌斌; 徐刚
2012-01-01
Without any hypothesis of continuity,this paper examines existence theorems of Nash equilibrium of a class of games with non-empty,convex and compact players' strategy spaces by using partition of unity and Browder fixed point theorem. At least one Nash equilibrium can be guaranteed in such games under some mild conditions on players' reaction correspondences, which generalizes the previous results in this field.%利用单位划分原理和Browder不动点定理,在对博弈局中人收益函数没有做任何连续性假设条件下,讨论了一类非空、紧凸空间上博弈纳什均衡存在性问题.当局中人的最优反应函数满足一些基本条件时,得到了若干纳什均衡存在性定理,是对现有讨论纳什均衡存在性问题的一个重要拓展和完善.
The genealogy of convex solids
Domokos, Gabor; Szabó, Timea
2012-01-01
The shape of homogeneous, smooth convex bodies as described by the Euclidean distance from the center of gravity represents a rather restricted class M_C of Morse-Smale functions on S^2. Here we show that even M_C exhibits the complexity known for general Morse-Smale functions on S^2 by exhausting all combinatorial possibilities: every 2-colored quadrangulation of the sphere is isomorphic to a suitably represented Morse-Smale complex associated with a function in M_C (and vice versa). We prove our claim by an inductive algorithm, starting from the path graph P_2 and generating convex bodies corresponding to quadrangulations with increasing number of vertices by performing each combinatorially possible vertex splitting by a convexity- preserving local manipulation of the surface. Since convex bodies carrying Morse-Smale complexes isomorphic to P_2 exist, this algorithm not only proves our claim but also defines a hierarchical order among convex solids and general- izes the known classification scheme in [35], ...
The Fixed Point Property in c0 with an Equivalent Norm
Directory of Open Access Journals (Sweden)
Berta Gamboa de Buen
2011-01-01
Full Text Available We study the fixed point property (FPP in the Banach space c0 with the equivalent norm ‖⋅‖D. The space c0 with this norm has the weak fixed point property. We prove that every infinite-dimensional subspace of (c0,‖⋅‖D contains a complemented asymptotically isometric copy of c0, and thus does not have the FPP, but there exist nonempty closed convex and bounded subsets of (c0,‖⋅‖D which are not ω-compact and do not contain asymptotically isometric c0—summing basis sequences. Then we define a family of sequences which are asymptotically isometric to different bases equivalent to the summing basis in the space (c0,‖⋅‖D, and we give some of its properties. We also prove that the dual space of (c0,‖⋅‖D over the reals is the Bynum space l1∞ and that every infinite-dimensional subspace of l1∞ does not have the fixed point property.
NP-completeness of weakly convex and convex dominating set decision problems
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Joanna Raczek
2004-01-01
Full Text Available The convex domination number and the weakly convex domination number are new domination parameters. In this paper we show that the decision problems of convex and weakly convex dominating sets are \\(NP\\-complete for bipartite and split graphs. Using a modified version of Warshall algorithm we can verify in polynomial time whether a given subset of vertices of a graph is convex or weakly convex.
The generalized vector convex-like mappings on real linear spaces%实线性空间中的广义向量类凸性
Institute of Scientific and Technical Information of China (English)
余国林; 刘三阳
2008-01-01
By using the concept of vector closure,a property of the generalized vector convexlike maps on real linear spaces is presented and it is showed that the Slater constraint qualification for vector optimization holds if the constraint mapping is the generalized vector convexlike. Furthermore, characterization of proper effi-ciency is established in terms of saddle-point criterion.%作者利用向量闭包给出了实线性空间上广义向量类凸映射的一个性质,并依此说明在向量优化问题中,如果约束映射是广义向量类凸的则Slater约束品性成立.另外,作者利用鞍点准则刻画了实线性空问中的真有效性.
Local Routing in Convex Subdivisions
DEFF Research Database (Denmark)
Bose, Prosenjit; Durocher, Stephane; Mondal, Debajyoti;
2015-01-01
In various wireless networking settings, node locations determine a network’s topology, allowing the network to be modelled by a geometric graph drawn in the plane. Without any additional information, local geometric routing algorithms can guarantee delivery to the target node only in restricted...... classes of geometric graphs, such as triangulations. In order to guarantee delivery on more general classes of geometric graphs (e.g., convex subdivisions or planar subdivisions), previous local geometric routing algorithms required Θ(logn) state bits to be stored and passed with the message. We present...... the first local geometric routing algorithm using only one state bit to guarantee delivery on convex subdivisions and the first local geometric memoryless routing algorithm that guarantees delivery on edge-augmented monotone subdivisions (including all convex subdivisions) when the algorithm has knowledge...
On Convex Hull of Orthogonal Scalar Spectral Functions of a Carleman Operator
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S. M. Bahri
2008-11-01
Full Text Available In this paper we describe the closed convex hull of orthogonal resolvents of an abstract symmetric operator of defect indices (1; 1, then we study the convex hull of orthogonal spectral functions of a Carleman operator in the Hilbert space L^2(X;mu.
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Simon Larson
2016-04-01
Full Text Available Abstract We prove geometric $$L^p$$ L p versions of Hardy’s inequality for the sub-elliptic Laplacian on convex domains $$\\Omega $$ Ω in the Heisenberg group $$\\mathbb {H}^n$$ H n , where convex is meant in the Euclidean sense. When $$p=2$$ p = 2 and $$\\Omega $$ Ω is the half-space given by $$\\langle \\xi , \
Extreme properties of quermassintegrals of convex bodies
Institute of Scientific and Technical Information of China (English)
LENG; Gangsong
2001-01-01
［1］Ball,K.,Shadows of convex bodies,Trans.Amer.Math.Soc.,1991,327:891-901.［2］Lutwak,E.,Mixed projection inequalities,Trans.Amer.Math.Soc.,1985,287:92-106.［3］Bourgain,J.,Lindenstrauss,J.,Projection bodies,Israel Seminar (G.A.F.A) 1986-1987,Lecture Notes in Math.Vol.1317,Berlin-New York:Springer-Verlag,1988,250-269.［4］Chakerian,G.D.,Lutwak,E.,Bodies with similar projections,Trans.Amer.Math.Soc.,1997,349:1811-1820.［5］Schneider,R.,Weil,W.,Zonoids and related topics,Convexity and its Applications (eds.Gruber,P.M.,Wills,J.M.),Basel:Birkhuser,1983,296-316.［6］Schneider,R.,Convex Bodies:the Brunn-Minkowski Theory,Cambridge:Cambridge University Press,1993.［7］Schneider,R.,On the determination of convex bodies by projection and girth functions,Result Math.,1998,33:155-160.［8］Thompson,A.C.,Minkowski Geometry,Cambridge:Cambridge University Press,1996.［9］Petty,C.M.,Projection bodies,in Proceedings,Coll Convexity,Copenhagen,1965,Kbenhavns Univ.Mat.Inst.,1967,234-241.［10］Schneider,R.,Zu einem problem von Shephard über die projectionen konvexer kirper,Math.Z.,1967,101:71-81.［11］Ball,K.,Volume ratios and a reverse isoprimetric inequalitity,J.London Math.Soc.,1991,44(2):351-359.［12］Gardner,R.J.,Intersection bodies and the Busemann-Petty problem,Trans.Amer.Math.Soc.,1994,342:435-445.［13］Gardner,R.J.,A positive answer to the Busemann-petty problem in three dimensions,Annals of Math.,1994,140:435-447.［14］Grinberg,E.L.,Isoperimetric inequalities and identities fork-dimensional cross-sections of convex bodies,Math.Ann.,1991,291:75-86.［15］Goodey,P.,Schneider,R.,Weil,W.,On the determination of convex bodies by projection functions,Bull.London Math.Soc.,1997,29:82-88.［16］Lutwak,E.,Intersection bodies and dual mixed volumes,Adv.Math.,1988,71:232-261.［17］Zhang,G.,Centered bodies and dual mixed volumes,Trans.Amer.Soc.,1994,345:777-801.［18］Zhang,G.,Dual Kinematic formulas,Trans.Amer.Soc.,1999,351:985-995.［19
Institute of Scientific and Technical Information of China (English)
文开庭
2012-01-01
A new GLKKM theorem is established in noncompact L-convex spaces. As applications, existence theorems of the solution for generalized equilibrium problems with lower and upper bounds are obtained.%在非紧L-凸空间中建立了一个新的GLKKM定理。作为应用,获得了带上下界的广义平衡问题的解的存在定理.
Some properties of Ba spaces%Ba空间的一些性质
Institute of Scientific and Technical Information of China (English)
魏文展; 董鸽
2005-01-01
讨论由一列线性赋范空间生成的Ba空间的几个性质.首先证明Ba空间既是囿空间又是桶形空间,并指出若X是线性赋范空间,则X*是Ba空间,接着又给出Ba空间等度连续、可分、局部一致凸等的几个充要条件.%Several properties of Ba spaces, which are produced by a sequence of linearly normed spaces, are discussed. It is shown that Ba spaces are both Bornological spaces and barrel spaces, and if X is a normed space, then X* is a Ba space. Thereby, some necessary and sufficient conditions under which Ba spaces are equicontinuous, separable and locally uniformly convex are given, respectively.
Weighted norm inequalities and indices
Directory of Open Access Journals (Sweden)
Joaquim Martín
2006-01-01
Full Text Available We extend and simplify several classical results on weighted norm inequalities for classical operators acting on rearrangement invariant spaces using the theory of indices. As an application we obtain necessary and sufficient conditions for generalized Hardy type operators to be bounded on ?p(w, ?p,8(w, Gp(w and Gp,8(w.
Greedy vs. L1 Convex Optimization in Sparse Coding
DEFF Research Database (Denmark)
Ren, Huamin; Pan, Hong; Olsen, Søren Ingvor
Sparse representation has been applied successfully in many image analysis applications, including abnormal event detection, in which a baseline is to learn a dictionary from the training data and detect anomalies from its sparse codes. During this procedure, sparse codes which can be achieved...... and action recognition, a comparative study of codes in abnormal event detection is less studied and hence no conclusion is gained on the effect of codes in detecting abnormalities. We constrict our comparison in two types of the above L0-norm solutions: greedy algorithms and convex L1-norm solutions....... Considering the property of abnormal event detection, i.e., only normal videos are used as training data due to practical reasons, effective codes in classification application may not perform well in abnormality detection. Therefore, we compare the sparse codes and comprehensively evaluate their performance...
Directory of Open Access Journals (Sweden)
Haiqing Wang
2012-01-01
Full Text Available Let be a uniformly convex Banach space and ={(∶0≤0((≠∅. Consider the iterative method that generates the sequence {} by the algorithm +1=(++(1−−(1/∫0(,≥0, where {}, {}, and {} are three sequences satisfying certain conditions, ∶→ is a contraction mapping. Strong convergence of the algorithm {} is proved assuming either has a weakly continuous duality map or has a uniformly Gâteaux differentiable norm.
Institute of Scientific and Technical Information of China (English)
邓磊
2006-01-01
New classes of Gθ -correspondences and Gθ -majorized mappings without open lower sections are introduced in G-convex spaces. Some existence theorems of maximal elements for Gθ -correspondences and Gθ -majorized mappings are obtained under nonparacompact setting of G-convex spaces. As applications, some new equilibrium existence theorems for qualitative games and generalized games with infinite set of players and Gθ -majorized preference correspondences are established under nonparacompact setting of G-convex spaces.%给出了不具有开原象的Gθ-对应和Gθ-优化映象的概念;在非仿紧的G-凸空间中证明了关于Gθ-对应和Gθ-优化映象的极大元存在定理.作为应用,在非仿紧的G-凸空间中建立了具有无限个选手和Gθ-优化选择对应的定性博弈和广义博弈的平衡存在定理.
ON A GENERALIZED MODULUS OF CONVEXITY AND UNIFORM NORMAL STRUCTURE
Institute of Scientific and Technical Information of China (English)
Yang Changsen; Wang Fenghui
2007-01-01
In this article, the authors study a generalized modulus of convexity, δ(α)(∈).Certain related geometrical properties of this modulus are analyzed. Their main result is that Banach space X has uniform normal structure if there exists ∈, 0 ≤∈≤1, such that δ(α)(1 + ∈) ＞ (1 - α)∈.
On a convex combination of solutions to elliptic variational inequalities
Directory of Open Access Journals (Sweden)
Sergiu Aizicovici
2007-02-01
Full Text Available We consider continuous descent methods for the minimization of convex functions defined on a general Banach space. In our previous work we showed that most of them (in the sense of Baire category converged. In the present paper we show that convergent continuous descent methods are stable under small perturbations.
Norms of Random Submatrices and Sparse Approximation
2008-07-28
usual Hilbert space operator norm; the `1 to `2 operator norm ‖·‖1→2 computes the maximum `2 norm of a column; and ‖·‖max returns the maximum absolute...estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining...the spectral norm of a random column submatrix. Its proof is analogous with that of Theorem 3.2 but relies on a sharp noncommutative Khintchine
A Characterization of Generalized Monotone Normed Cones
Institute of Scientific and Technical Information of China (English)
S.ROMAGUERA; E.A.S(A)NCHEZ-P(E)REZ; O.VALERO
2007-01-01
Let C be a cone and consider a quasi-norm p defined on it. We study the structure of the couple (C, p) as a topological space in the case where the function p is also monotone. We characterize when the topology of a quasi-normed cone can be defined by means of a monotone norm. We also define and study the dual cone of a monotone normed cone and the monotone quotient of a general cone.We provide a decomposition theorem which allows us to write a cone as a direct sum of a monotone subcone that is isomorphic to the monotone quotient and other particular subcone.
Quantum information and convex optimization
Energy Technology Data Exchange (ETDEWEB)
Reimpell, Michael
2008-07-01
This thesis is concerned with convex optimization problems in quantum information theory. It features an iterative algorithm for optimal quantum error correcting codes, a postprocessing method for incomplete tomography data, a method to estimate the amount of entanglement in witness experiments, and it gives necessary and sufficient criteria for the existence of retrodiction strategies for a generalized mean king problem. (orig.)
Bounding the errors for convex dynamics on one or more polytopes
Tresser, Charles
2007-09-01
We discuss the greedy algorithm for approximating a sequence of inputs in a family of polytopes lying in affine spaces by an output sequence made of vertices of the respective polytopes. More precisely, we consider here the case when the greed of the algorithm is dictated by the Euclidean norms of the successive cumulative errors. This algorithm can be interpreted as a time-dependent dynamical system in the vector space, where the errors live, or as a time-dependent dynamical system in an affine space containing copies of all the original polytopes. This affine space contains the inputs, as well as the inputs modified by adding the respective former errors; it is the evolution of these modified inputs that the dynamical system in affine space describes. Scheduling problems with many polytopes arise naturally, for instance, when the inputs are from a single polytope P, but one imposes the constraint that whenever the input belongs to a codimension n face, the output has to be in the same codimension n face (as when scheduling drivers among participants of a carpool). It has been previously shown that the error is bounded in the case of a single polytope by proving the existence of an arbitrary large convex invariant region for the dynamics in affine space: A region that is simultaneously invariant for several polytopes, each considered separately, was also constructed. It was then shown that there cannot be an invariant region in affine space in the general case of a family of polytopes. Here we prove the existence of an arbitrary large convex invariant set for the dynamics in the vector space in the case when the sizes of the polytopes in the family are bounded and the set of all the outgoing normals to all the faces of all the polytopes is finite. It was also previously known that starting from zero as the initial error set, the error set could not be saturated in finitely many steps in some cases with several polytopes: Contradicting a former conjecture, we show
Widths of some classes of convex functions and bodies
Konovalov, V. N.; Maiorov, Vitalii E.
2010-02-01
We consider classes of uniformly bounded convex functions defined on convex compact bodies in \\mathbb{R}^d and satisfying a Lipschitz condition and establish the exact orders of their Kolmogorov, entropy, and pseudo-dimension widths in the L_1-metric. We also introduce the notions of pseudo-dimension and pseudo-dimension widths for classes of sets and determine the exact orders of the entropy and pseudo-dimension widths of some classes of convex bodies in \\mathbb{R}^drelative to the pseudo-metric defined as the d-dimensional Lebesgue volume of the symmetric difference of two sets. We also find the exact orders of the entropy and pseudo-dimension widths of the corresponding classes of characteristic functions in L_p-spaces, 1\\le p\\le\\infty.
Institute of Scientific and Technical Information of China (English)
Chuan Qiang CHEN; Bo Wen HU
2013-01-01
We study microscopic spacetime convexity properties of fully nonlinear parabolic partial differential equations.Under certain general structure condition,we establish a constant rank theorem for the spacetime convex solutions of fully nonlinear parabolic equations.At last,we consider the parabolic convexity of solutions to parabolic equations and the convexity of the spacetime second fundamental form of geometric flows.
Revisiting separation properties of convex fuzzy sets
Separation of convex sets by hyperplanes has been extensively studied on crisp sets. In a seminal paper separability and convexity are investigated, however there is a flaw on the definition of degree of separation. We revisited separation on convex fuzzy sets that have level-wise (crisp) disjointne...
A Note on Permutationally Convex Games
van Velzen, S.; Hamers, H.J.M.; Norde, H.W.
2005-01-01
In this paper we generalise marginal vectors and permutational convexity.We show that if a game is generalised permutationally convex, then the corresponding generalised marginal vector is a core element.Furthermore we refine the concept of permutational convexity and show that this refinement yield
Computing farthest neighbors on a convex polytope
Cheong, O.; Shin, C.S.; Vigneron, A.
2002-01-01
Let N be a set of n points in convex position in R3. The farthest-point Voronoi diagram of N partitions R³ into n convex cells. We consider the intersection G(N) of the diagram with the boundary of the convex hull of N. We give an algorithm that computes an implicit representation of G(N) in expecte
Computing farthest neighbors on a convex polytope
Cheong, O.; Shin, C.S.; Vigneron, A.
2002-01-01
Let N be a set of n points in convex position in R3. The farthest-point Voronoi diagram of N partitions R³ into n convex cells. We consider the intersection G(N) of the diagram with the boundary of the convex hull of N. We give an algorithm that computes an implicit representation of G(N) in
Generalized geometrically convex functions and inequalities.
Noor, Muhammad Aslam; Noor, Khalida Inayat; Safdar, Farhat
2017-01-01
In this paper, we introduce and study a new class of generalized functions, called generalized geometrically convex functions. We establish several basic inequalities related to generalized geometrically convex functions. We also derive several new inequalities of the Hermite-Hadamard type for generalized geometrically convex functions. Several special cases are discussed, which can be deduced from our main results.
Firey linear combinations of convex bodies
Institute of Scientific and Technical Information of China (English)
XIONG Ge; XIAO Qi-ming; CHEUNG Wing-Sum
2009-01-01
For convex bodies, the Firey linear combinations were introduced and studied in several papers. In this paper the mean width of the Firey linear combinations of convex bodies is studied, and the lower bound of the mean width of the Firey linear combinations of convex body and its polar body is given.
Relations for certain symmetric norms and anti-norms before and after partial trace
Rastegin, Alexey E
2012-01-01
Changes of some unitarily invariant norms and anti-norms under the operation of partial trace are examined. The norms considered form a two-parametric family, including both the Ky Fan and Schatten norms as particular cases. The obtained results concern operators acting on the tensor product of two finite-dimensional Hilbert spaces. For any such operator, we obtain lower bounds on norms of its partial trace in terms of the corresponding dimensionality and norms of this operator. Similar inequalities, but in the opposite direction, are obtained for certain anti-norms of positive matrices. Applications of the results to generalized quantum entropies are discussed. We derive inequalities between the unified entropies of a composite quantum system and one of its subsystems, where the traced-out dimensionality is involved as well.
Constrained spacecraft reorientation using mixed integer convex programming
Tam, Margaret; Glenn Lightsey, E.
2016-10-01
A constrained attitude guidance (CAG) system is developed using convex optimization to autonomously achieve spacecraft pointing objectives while meeting the constraints imposed by on-board hardware. These constraints include bounds on the control input and slew rate, as well as pointing constraints imposed by the sensors. The pointing constraints consist of inclusion and exclusion cones that dictate permissible orientations of the spacecraft in order to keep objects in or out of the field of view of the sensors. The optimization scheme drives a body vector towards a target inertial vector along a trajectory that consists solely of permissible orientations in order to achieve the desired attitude for a given mission mode. The non-convex rotational kinematics are handled by discretization, which also ensures that the quaternion stays unity norm. In order to guarantee an admissible path, the pointing constraints are relaxed. Depending on how strict the pointing constraints are, the degree of relaxation is tuneable. The use of binary variables permits the inclusion of logical expressions in the pointing constraints in the case that a set of sensors has redundancies. The resulting mixed integer convex programming (MICP) formulation generates a steering law that can be easily integrated into an attitude determination and control (ADC) system. A sample simulation of the system is performed for the Bevo-2 satellite, including disturbance torques and actuator dynamics which are not modeled by the controller. Simulation results demonstrate the robustness of the system to disturbances while meeting the mission requirements with desirable performance characteristics.
Institute of Scientific and Technical Information of China (English)
王娴; 何震
2004-01-01
Xu和Norr已经证明了建立在一致凸Banach空间的一个非空有界闭凸子集上的渐进非扩张映射的三步迭代的收敛定理问题.引入(L-α)一致李普希兹的概念,然后在一些已有结果的基础上,证明一致凸Banach空间的紧子集上的(L-α)一致李普希兹渐进非扩张映射的三步迭代序列的收敛问题.这个结论是对Xu和Noor的相应结果的推广.%Xu and Noor had proved the theorem on convergence of three-step iterations for asymptotically nonexpansive mapping on nonempty closed, bounded, and convex subset of uniformly convex Banach space. Based on some results given by K Tan and H K Xu[1] proved, the convergence of three-step iterations of (L-α) uniformly Lipschitz asymptotically nonexpansive mapping on a compact subset of a uniform convex Banach space had proved. The results presented extended the corresponding of Xu and Noor[5].
Institute of Scientific and Technical Information of China (English)
胡正平; 路亮; 冯春生
2011-01-01
The goal of one-class classification model is to design the covering model for the target class. The model should ideally cover the objects of target class, and reject all other non-targets. The support vector data description (SVDD) aims to find the minimum ball enclosed all the target objects. However, this model couldn't perform better for the data with irregular and complex distribution. The convex hull data description (CHDD) based tight covering model is presented in this paper. The model is a non-parametric classifier which covers the irregular data adaptively in the feature spaces. By the introduction of kernel functions, the stronger ability of nonlinear classification could be obtained. When the training set of target class contains outliers, the model can be made more robust by rejecting a fraction of the training objects. The experimental results show that the presented method performs better by comparing its results on the UCI, MNIST and MIT-CBCL face data sets with other one-class cl assification methods.%一类分类问题的研究目标是设计目标类样本的覆盖函数,理想情况下使得目标类样本被接受,所有非目标类的样本被拒绝.经典SVDD覆盖模型寻找包含训练数据的最小半径超球对其进行覆盖,该模型对非规则复杂分布的数据描述存在较多的冗余区域.本文提出一种基于训练集样本凸壳数据描述(Convex Hull Data Description,CHDD)的紧致覆盖模型.该模型无须参数设置,可实现对样本非规则复杂分布的自适应覆盖,并可通过利用核函数方法获得更强的非线性分类能力.当训练集包含噪声样本时,通过拒绝一定比例的目标类样本可获得更为鲁棒的凸壳边界描述.在UCI数据库、MNIST手写体数据库和MIT-CBCL人脸识别数据库上的实验结果表明了本文方法的有效性,相比现有一类分类算法,CHDD取得更好的分类效果.
HOLOMORPHIC MANIFOLDS ON LOCALLY CONVEX SPACES
Institute of Scientific and Technical Information of China (English)
Tsoy-Wo Ma
2005-01-01
Based on locally compact perturbations of the identity map similar to the Fredholm structures on real Banach manifolds, complex manifolds with inverse mapping theorem as part of the defintion are proposed. Standard topics including holomorphic maps, morphisms, derivatives, tangent bundles, product manifolds and submanifolds are presented. Although this framework is elementary, it lays the necessary foundation for all subsequent developments.
A noncommutative convexity in C*-bimodules
Directory of Open Access Journals (Sweden)
Mohsen Kian
2017-02-01
Full Text Available Let A and B be C*-algebras. We consider a noncommutative convexity in Hilbert A-B-bimodules, called A-B-convexity, as a generalization of C*-convexity in C*-algebras. We show that if X is a Hilbert A-B-bimodule, then Mn(X is a Hilbert Mn(A-Mn(B-bimodule and apply it to show that the closed unit ball of every Hilbert A-B-bimodule is A-B-convex. Some properties of this kind of convexity and various examples have been given.
Expanding Norms for Narration.
Johnson, Cynthia J.
1995-01-01
This article addresses issues in determining normal development of narrative skills in school-age children and adolescents. It considers the purpose and use of norms in this area and research on currently available norms. It focuses on difficulties in compiling norms, including children's wide ability range and effects of situational variables,…
Basic Topological and Geometric Properties of Cesàro–Orlicz Spaces
Indian Academy of Sciences (India)
Yunan Cui; Henryk Hudzik; Narin Petrot; Suthep Suantai; Alicja Szymaszkiewicz
2005-11-01
Necessary and sufficient conditions under which the Cesàro–Orlicz sequence space $\\mathrm{ces}_$ is nontrivial are presented. It is proved that for the Luxemburg norm, Cesàro–Orlicz spaces $\\mathrm{ces}_$ have the Fatou property. Consequently, the spaces are complete. It is also proved that the subspace of order continuous elements in $\\mathrm{ces}_$ can be defined in two ways. Finally, criteria for strict monotonicity, uniform monotonicity and rotundity (= strict convexity) of the spaces $\\mathrm{ces}_$ are given.
Evaluating convex roof entanglement measures.
Tóth, Géza; Moroder, Tobias; Gühne, Otfried
2015-04-24
We show a powerful method to compute entanglement measures based on convex roof constructions. In particular, our method is applicable to measures that, for pure states, can be written as low order polynomials of operator expectation values. We show how to compute the linear entropy of entanglement, the linear entanglement of assistance, and a bound on the dimension of the entanglement for bipartite systems. We discuss how to obtain the convex roof of the three-tangle for three-qubit states. We also show how to calculate the linear entropy of entanglement and the quantum Fisher information based on partial information or device independent information. We demonstrate the usefulness of our method by concrete examples.
Extreme Points of the Convex Set of Joint Probability Distributions with Fixed Marginals
Indian Academy of Sciences (India)
K R Parthasarathy
2007-11-01
By using a quantum probabilistic approach we obtain a description of the extreme points of the convex set of all joint probability distributions on the product of two standard Borel spaces with fixed marginal distributions.
Institute of Scientific and Technical Information of China (English)
Yekini SHEHU
2014-01-01
Let K be a nonempty, closed and convex subset of a real reflexive Banach space E which has a uniformly Gˆateaux differentiable norm. Assume that every nonempty closed con-vex and bounded subset of K has the fixed point property for nonexpansive mappings. Strong convergence theorems for approximation of a fixed point of Lipschitz pseudo-contractive map-pings which is also a unique solution to variational inequality problem involving φ-strongly pseudo-contractive mappings are proved. The results presented in this article can be applied to the study of fixed points of nonexpansive mappings, variational inequality problems, con-vex optimization problems, and split feasibility problems. Our result extends many recent important results.
Convex Hulls of Algebraic Sets
Gouveia, João
2010-01-01
This article describes a method to compute successive convex approximations of the convex hull of a set of points in R^n that are the solutions to a system of polynomial equations over the reals. The method relies on sums of squares of polynomials and the dual theory of moment matrices. The main feature of the technique is that all computations are done modulo the ideal generated by the polynomials defining the set to the convexified. This work was motivated by questions raised by Lov\\'asz concerning extensions of the theta body of a graph to arbitrary real algebraic varieties, and hence the relaxations described here are called theta bodies. The convexification process can be seen as an incarnation of Lasserre's hierarchy of convex relaxations of a semialgebraic set in R^n. When the defining ideal is real radical the results become especially nice. We provide several examples of the method and discuss convergence issues. Finite convergence, especially after the first step of the method, can be described expl...
Institute of Scientific and Technical Information of China (English)
PENG Jia-yin
2011-01-01
The notions of norm and distance in BCI-algebras are introduced,and some basic properties in normed BCI-algebras are given.It is obtained that the isomorphic(homomorphic)image and inverse image of a normed BCI-algebra are still normed BCI-algebras.The relations of normaled properties between BCI-algebra and Cartesian product of BCIalgebras are investigated.The limit notion of sequence of points in normed BCI-algebras is introduced,and its related properties are investigated.
Directory of Open Access Journals (Sweden)
Yves Biollay
1979-01-01
Full Text Available We show in this paper that the sequence {max|uk|}, where the uk are the eigenfunctions of the problem Δu+λu=0 in D⊂Rn and u=0 on ∂D, is not bounded generally if one imposes the norm ∫Du2p(xdx=1, p=(1,2,3,…. The same holds with the norm ∫D|gradu|2pdx=1 when n>4p−1. On the other hand, if D⊂R2, resp. R3 the norm ∫D|gradu|2dx=1 implies max|uk|→k→∞0, resp. max|uk|=0(1.
non-Lipschitzian mappings without convexity
Directory of Open Access Journals (Sweden)
G. Li
1999-01-01
real Hilbert space H, and ℑ={Tt:t∈G} a representation of G as asymptotically nonexpansive type mappings of C into itself. Let L(x={z∈H:infs∈Gsupt∈G‖Tts x−z‖=inft∈G‖Tt x−z‖} for each x∈C and L(ℑ=∩x∈C L(x. In this paper, we prove that ∩s∈Gconv¯{Tts x:t∈G}∩L(ℑ is nonempty for each x∈C if and only if there exists a unique nonexpansive retraction P of C into L(ℑ such that PTs=P for all s∈G and P(x∈conv¯{Ts x:s∈G} for every x∈C. Moreover, we prove the ergodic convergence theorem for a semitopological semigroup of non-Lipschitzian mappings without convexity.
The problem of convexity of Chebyshev sets
Balaganskii, V. S.; Vlasov, L. P.
1996-12-01
Contents Introduction §1. Definitions and notation §2. Reference theorems §3. Some results Chapter I. Characterization of Banach spaces by means of the relations between approximation properties of sets §1. Existence, uniqueness §2. Prom approximate compactness to 'sun'-property §3. From 'sun'-property to approximate compactness §4. Differentiability in the direction of the gradient is sufficient for Fréchet and Gâteaux differentiability §5. Sets with convex complement Chapter II. The structure of Chebyshev and related sets §1. The isolated point method §2. Restrictions of the type \\vert\\overline{W}\\vert Klee (discrete Chebyshev set) §4. A survey of some other results Conclusion Bibliography
Use of Convexity in Ostomy Care
Salvadalena, Ginger; Pridham, Sue; Droste, Werner; McNichol, Laurie; Gray, Mikel
2017-01-01
Ostomy skin barriers that incorporate a convexity feature have been available in the marketplace for decades, but limited resources are available to guide clinicians in selection and use of convex products. Given the widespread use of convexity, and the need to provide practical guidelines for appropriate use of pouching systems with convex features, an international consensus panel was convened to provide consensus-based guidance for this aspect of ostomy practice. Panelists were provided with a summary of relevant literature in advance of the meeting; these articles were used to generate and reach consensus on 26 statements during a 1-day meeting. Consensus was achieved when 80% of panelists agreed on a statement using an anonymous electronic response system. The 26 statements provide guidance for convex product characteristics, patient assessment, convexity use, and outcomes. PMID:28002174
CONVEX CLASS OF STARLIKE FUNCTIONS
Gupta, V. P.
1984-01-01
Let ＄S＄ denote the class of functions of the form ＄f(z)=z-￥sum_{n=2}^{￥infty}|a_{n}|z^{n}＄ that are analytic and univalent in the unit disk ＄U＄. Let ＄S(￥alpha, ￥beta)＄ and ＄K(￥alpha, ￥beta)＄ denote the subclasses of ＄S＄ consisting respectively, of starlike and close-to-convex functions of order ＄￥alpha(0￥leqq￥alpha
Finite dimensional convexity and optimization
Florenzano, Monique
2001-01-01
The primary aim of this book is to present notions of convex analysis which constitute the basic underlying structure of argumentation in economic theory and which are common to optimization problems encountered in many applications. The intended readers are graduate students, and specialists of mathematical programming whose research fields are applied mathematics and economics. The text consists of a systematic development in eight chapters, with guided exercises containing sometimes significant and useful additional results. The book is appropriate as a class text, or for self-study.
Directory of Open Access Journals (Sweden)
Litvinov V. A.
2013-01-01
Full Text Available The article deals with the problem of norm and normative approach to the diachronic language study. It identifies specificity of the normative approach to linguistic means within various linguistic traditions and determines the main features of the linguistic norm. The author points out that specific norms appear at each stage of language development as the result of correlation of the existing language means.
Institute of Scientific and Technical Information of China (English)
周海云; 高改良; 陈东青
2003-01-01
In the present paper, by virtue of new analysis technique, we have established a new strong convergence theorem for the modified Mann iteration scheme for a class of asymptotically nonexpansive mappings in uniformly convex Banach spaces. Our results improve the recent ones announced by Schu, Rhoades and others.%通过使用新的分析技巧,建立了(关于)渐近非扩展映象的修正的迭代格式的强收敛定理.所得结果改进了Schu,Rhoades以及其他作者相关的结果.
Directory of Open Access Journals (Sweden)
Kim JongKyu
2009-01-01
Full Text Available We study the strong convergence of two kinds of viscosity iteration processes for approximating common fixed points of the pseudocontractive semigroup in uniformly convex Banach spaces with uniformly Gâteaux differential norms. As special cases, we get the strong convergence of the implicit viscosity iteration process for approximating common fixed points of the nonexpansive semigroup in Banach spaces satisfying some conditions. The results presented in this paper extend and generalize some results concerned with the nonexpansive semigroup in (Chen and He, 2007 and the pseudocontractive mapping in (Zegeye et al., 2007 to the pseudocontractive semigroup in Banach spaces under different conditions.
Fast Bundle-Level Type Methods for Unconstrained and Ball-Constrained Convex Optimization
2014-12-01
of half- spaces , hence it is convex and closed. Therefore, the subproblem (3.4) always has a unique solution as long as Qk is non-empty. To finish the...pixels in the image. The ‖u‖TV is convex and non-smooth. Table 5.1 Uniformly distributed QP instances A : n = 4000,m = 3000, L = 2.0e6, e0 = 2.89e4 Alg...generation. Mathematical pro- gramming, 118(1):177–206, 2009. [14] G. Lan. Bundle-level type methods uniformly optimal for smooth and non-smooth convex
A simple view on convex analysis and its applications
J. Brinkhuis (Jan); V. Tikhomirov
2005-01-01
textabstractOur aim is to give a simple view on the basics and applications of convex analysis. The essential feature of this account is the systematic use of the possibility to associate to each convex object---such as a convex set, a convex function or a convex extremal problem--- a cone, without
Entropy coherent and entropy convex measures of risk
Laeven, R.J.A.; Stadje, M.
2013-01-01
We introduce two subclasses of convex measures of risk, referred to as entropy coherent and entropy convex measures of risk. Entropy coherent and entropy convex measures of risk are special cases of φ-coherent and φ-convex measures of risk. Contrary to the classical use of coherent and convex measur
Institute of Scientific and Technical Information of China (English)
MENG Zhiqing; HU Yuda
2000-01-01
In this paper, we introduce the concepts of the cone-weak subdifferential and the cone-weak direction derivative of convex set-valued mapping in a locally convex topological vector space. We study the relationship between them and obtain some important results.
STRONG CONVERGENCE OF APPROXIMATED SEQUENCES FOR NONEXPANSIVE MAPPINGS IN BANACH SPACES
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
This paper studies the convergence of the sequence defined by x0∈C,xn+1=αnu+(1-αn)Txn,n=0,1,2,…, where 0 ≤αn ≤ 1, limn→∞αn = 0, ∑∞n=0 αn = ∞, and T is a nonexpansive mapping from a nonempty closed convex subset C of a Banach space X into itself. The iterative sequence {xn} converges strongly to a fixed point of T in the case when X is a uniformly convex Banach space with a uniformly Gateaux differentiable norm or a uniformly smooth Banach space only. The results presented in this paper extend and improve some recent results.
Institute of Scientific and Technical Information of China (English)
梁嘉宁; 黎永锦
2011-01-01
Let X be a q-normed space, and let T be a Zamfirescu operator. The sequence of Ishikawa iterations with T converges to the fixed point of T is proved, and certain Mann and Ishikawa iterative schemes for Zamfirescu operator T are equivalent.%设X为q-赋范空间,T为Zamfireseu算子.证明了Zamfirescu算子T的Ishikawa迭代收敛于T的不动点,并且某些Ishikawa迭代序列与Mann迭代序列的收敛性是等价的.
DEFF Research Database (Denmark)
Flockhart, Trine
The volume investigates how state socialization of the Euro-Atlantic constitutive norm set has taken place from a number of European and transatlantic international organizations into the 'New Europe'. The volume utilizes a new framework for norms transfer called 'Complex Socialization'....
'Global' norms and 'local' agency
DEFF Research Database (Denmark)
Björkdahl, Annika; Gusic, Ivan
2015-01-01
This article explores how the 'liberal democratic peace package' is received in post-conflict spaces. As such, it is part of a critical peace research agenda that raises critical questions concerning the quality of peace in many post-conflict societies. A close reading of the peace-building process...... in post-conflict Kosovo provides the backdrop for the theoretical discussion that identifies friction in norm diffusion processes and the different agencies that are generated through encounters between global norms and local practices. We unpack the interplay between the 'global' and the 'local......' in peacebuilding and, through the lens of friction, we reveal the diverse and unequal encounters that produce new power relations. By foregrounding agency, we theorise different agentive subjects in the post-conflict setting, and map local agency from various segments of society that may localise, co-opt or reject...
Efficient Line Searching for Convex Functions
den Boef, E.; den Hertog, D.
2004-01-01
In this paper we propose two new line search methods for convex functions. These new methods exploit the convexity property of the function, contrary to existing methods.The worst method is an improved version of the golden section method.For the second method it is proven that after two evaluations
Introduction to Convex and Quasiconvex Analysis
J.B.G. Frenk (Hans); G. Kassay
2004-01-01
textabstractIn the first chapter of this book the basic results within convex and quasiconvex analysis are presented. In Section 2 we consider in detail the algebraic and topological properties of convex sets within Rn together with their primal and dual representations. In Section 3 we apply the re
Stochastic Dominance: Convexity and Some Efficiency Tests
A.M. Lizyayev (Andrey)
2009-01-01
textabstractThis paper points out the importance of Stochastic Dominance (SD) efficient sets being convex. We review classic convexity and efficient set characterization results on SD efficiency of a given portfolio relative to a diversified set of assets and generalize them in the following
Convex trace functions of several variables
DEFF Research Database (Denmark)
Hansen, Frank
2002-01-01
We prove that the function (x1,...,xk)¿Tr(f(x1,...,xk)), defined on k-tuples of symmetric matrices of order (n1,...,nk) in the domain of f, is convex for any convex function f of k variables. The matrix f(x1,...,xk) is defined by the functional calculus for functions of several variables, and it ...
1990-01-01
to Convex Bodies, Geometriae Dedicata 2" (1973) 225-248. 10. H. Guggenheimer, "The Analytic Geometry of the Unsymmetric Minkowski Plane," Lecture...Mathematics, Vol. 58, No. 2, 1975. 19. E. Lutwak, "On Cross-Sectional Measures of Polar Reciprocal Convex Bodies," Geometriae Dedicata 5, (1976) 79-80
Differential analysis of matrix convex functions II
DEFF Research Database (Denmark)
Hansen, Frank; Tomiyama, Jun
2009-01-01
We continue the analysis in [F. Hansen, and J. Tomiyama, Differential analysis of matrix convex functions. Linear Algebra Appl., 420:102--116, 2007] of matrix convex functions of a fixed order defined in a real interval by differential methods as opposed to the characterization in terms of divide...
Swanson, David
2011-01-01
We give elementary proofs of formulas for the area and perimeter of a planar convex body surrounded by a band of uniform thickness. The primary tool is a integral formula for the perimeter of a convex body which describes the perimeter in terms of the projections of the body onto lines in the plane.
Strictly convex functions on complete Finsler manifolds
Indian Academy of Sciences (India)
YOE ITOKAWA; KATSUHIRO SHIOHAMA; BANKTESHWAR TIWARI
2016-10-01
The purpose of the present paper is to investigate the influence of strictly convex functions on the metric structures of complete Finsler manifolds. More precisely we discuss the properties of the group of isometries and the exponential maps on a complete Finsler manifold admitting strictly convex functions.
Toric geometry of convex quadrilaterals
Legendre, Eveline
2009-01-01
We provide an explicit resolution of the Abreu equation on convex labeled quadrilaterals. This confirms a conjecture of Donaldson in this particular case and implies a complete classification of the explicit toric K\\"ahler-Einstein and toric Sasaki-Einstein metrics constructed in [6,23,14]. As a byproduct, we obtain a wealth of extremal toric (complex) orbi-surfaces, including K\\"ahler-Einstein ones, and show that for a toric orbi-surface with 4 fixed points of the torus action, the vanishing of the Futaki invariant is a necessary and sufficient condition for the existence of K\\"ahler metric with constant scalar curvature. Our results also provide explicit examples of relative K-unstable toric orbi-surfaces that do not admit extremal metrics.
Convex analysis and global optimization
Tuy, Hoang
2016-01-01
This book presents state-of-the-art results and methodologies in modern global optimization, and has been a staple reference for researchers, engineers, advanced students (also in applied mathematics), and practitioners in various fields of engineering. The second edition has been brought up to date and continues to develop a coherent and rigorous theory of deterministic global optimization, highlighting the essential role of convex analysis. The text has been revised and expanded to meet the needs of research, education, and applications for many years to come. Updates for this new edition include: · Discussion of modern approaches to minimax, fixed point, and equilibrium theorems, and to nonconvex optimization; · Increased focus on dealing more efficiently with ill-posed problems of global optimization, particularly those with hard constraints;
Reconstruction of convex bodies from surface tensors
DEFF Research Database (Denmark)
Kousholt, Astrid; Kiderlen, Markus
The set of all surface tensors of a convex body K (Minkowski tensors derived from the surface area measure of K) determine K up to translation, and hereby, the surface tensors of K contain all information on the shape of K. Here, shape means the equivalence class of all convex bodies...... that are translates of each other. An algorithm for reconstructing an unknown convex body in R 2 from its surface tensors up to a certain rank is presented. Using the reconstruction algorithm, the shape of an unknown convex body can be approximated when only a finite number s of surface tensors are available....... The output of the reconstruction algorithm is a polytope P, where the surface tensors of P and K are identical up to rank s. We establish a stability result based on a generalization of Wirtinger’s inequality that shows that for large s, two convex bodies are close in shape when they have identical surface...
A Note on Upper Convex Density
Institute of Scientific and Technical Information of China (English)
YIN JIAN-DONG; ZHOU ZUO-LING
2010-01-01
For a self-similar set E satisfying the open set condition,upper convex density is an important concept for the computation of its Hausdorff measure,and it is well known that the set of relative interior points with upper convex density 1has a full Hausdorff measure.But whether the upper convex densities of E at all the relative interior points are equal to 1? In other words,whether there exists a relative interior point of E such that the upper convex density of E at this point is less than 1?In this paper,the authors construct a self-similar set satisfying the open set condition,which has a relative interior point with upper convex density less than 1.Thereby,the above problem is sufficiently answered.
Generalized convexity, generalized monotonicity recent results
Martinez-Legaz, Juan-Enrique; Volle, Michel
1998-01-01
A function is convex if its epigraph is convex. This geometrical structure has very strong implications in terms of continuity and differentiability. Separation theorems lead to optimality conditions and duality for convex problems. A function is quasiconvex if its lower level sets are convex. Here again, the geo metrical structure of the level sets implies some continuity and differentiability properties for quasiconvex functions. Optimality conditions and duality can be derived for optimization problems involving such functions as well. Over a period of about fifty years, quasiconvex and other generalized convex functions have been considered in a variety of fields including economies, man agement science, engineering, probability and applied sciences in accordance with the need of particular applications. During the last twenty-five years, an increase of research activities in this field has been witnessed. More recently generalized monotonicity of maps has been studied. It relates to generalized conve...
Norm and anti-norm inequalities for positive semi-definite matrices
Bourin, Jean-Christophe
2010-01-01
Some subadditivity results involving symmetric (unitarily invariant) norms are obtained. For instance, if $g(t)=\\sum_{k=0}^m a_kt^k$ is a polynomial of degree $m$ with non-negative coefficients, then, for all positive operators $A,\\,B$ and all symmetric norms, $\\|g(A+B)\\|^{1/m} \\le \\|g(A)\\|^{1/m} + \\|g(B)\\|^{1/m}$. To give parallel superadditivity results, we investigate anti-norms, a class of functionals containing the Schatten $q$-norms for $q\\in(0,1]$ and $q<0$. The results are extensions of the Minkowski determinantal inequality. A few estimates for block-matrices are derived. For instance, let $f:[0,\\infty) \\to [0,\\infty)$ be concave and $p\\in(1,\\infty)$. If $f^p(t)$ is superadditive, then $Tr f(A) \\ge (\\sum_{i=1}^m f^p(a_{ii}))^{1/p}$ for all positive $m\\times m$ matrix $A=[a_{ij}]$. Furthermore, for the normalized trace $\\tau$, we consider functions $\\phi(t)$ and $f(t)$ for which the functional $A\\mapsto\\phi\\circ\\tau\\circ f(A)$ is convex or concave, and obtain a simple analytic criterion.
Zachos, Anastasios
2010-01-01
We obtain the plasticity equations for convex quadrilaterals on a complete convex surface with bounded specific curvature and derive a plasticity principle which states that: Given four shortest arcs which meet at the weighted Fermat-Torricelli point P_F and their endpoints form a convex quadrilateral, an increase of the weight that corresponds to a shortest arc causes a decrease to the two weights that correspond to the two neighboring shortest arcs and an increase to the weight that corresponds to the opposite shortest arc. We show a connection between the plasticity of convex quadrilaterals on a complete convex surface with bounded specific curvature with the plasticity of generalized convex quadrilaterals on a manifold which is composed by triangles located on a complete convex surface of bounded specific curvature and triangles located on a two dimensional sphere whose constant Gaussian curvature equals to the infimum or supremum of the specific curvature. Furthermore, we give some cases of geometrizatio...
Directory of Open Access Journals (Sweden)
Xue-song Li
2009-01-01
Full Text Available We study the strong convergence of two kinds of viscosity iteration processes for approximating common fixed points of the pseudocontractive semigroup in uniformly convex Banach spaces with uniformly Gâteaux differential norms. As special cases, we get the strong convergence of the implicit viscosity iteration process for approximating common fixed points of the nonexpansive semigroup in Banach spaces satisfying some conditions. The results presented in this paper extend and generalize some results concerned with the nonexpansive semigroup in (Chen and He, 2007 and the pseudocontractive mapping in (Zegeye et al., 2007 to the pseudocontractive semigroup in Banach spaces under different conditions.
Institute of Scientific and Technical Information of China (English)
丁协平
2002-01-01
In this paper, by using a collectively fixed point theorem due to author, some equilibrium existence theorems for qualitative games and generalized games with an infinite number of agents, with noncompact strategy spaces and with U-majorized correspondences are proved in locally G-convex uniform spaces. These theorems improve some known results in the literatures.%应用作者得到的一个聚合不动点定理,在局部G-凸一致空间内对具有无限多个经济人,具有非紧策略空间和具有U-优化对应的定性对策和广义对策,证明了某些平衡存在性定理.这些定理改进和推广了文献中某些已知结果.
U.S. Department of Health & Human Services — RxNorm provides normalized names for clinical drugs and links its names to many of the drug vocabularies commonly used in pharmacy management and drug interaction...
Singh, M.P.; Arrott, M.; Balke, T.; Chopra, A.; Christiaanse, R.M.J.; Cranefield, S.; Dignum, F.; Eynard, D.; Farcas, E.; Fornare, N.; Gandon, F.; Governatori, G...; Dam, H.K.; Hulstijn, J.; Krueger, I.; Lam, H.P.; Meisinger, M.; Noriega, P.; Tony, B.; Savarimuthu, R.; Tadanki, K.; Verhagen, H.; Villata, S.
2013-01-01
This chapter presents a variety of applications of norms. These applications include governance in sociotechnical systems, data licensing and data collection, understanding software development teams, requirements engineering, assurance, natural resource allocation, wireless grids, autonomous vehicl
Bundles of Norms About Teen Sex and Pregnancy.
Mollborn, Stefanie; Sennott, Christie
2015-09-01
Teen pregnancy is a cultural battleground in struggles over morality, education, and family. At its heart are norms about teen sex, contraception, pregnancy, and abortion. Analyzing 57 interviews with college students, we found that "bundles" of related norms shaped the messages teens hear. Teens did not think their communities encouraged teen sex or pregnancy, but normative messages differed greatly, with either moral or practical rationalizations. Teens readily identified multiple norms intended to regulate teen sex, contraception, abortion, childbearing, and the sanctioning of teen parents. Beyond influencing teens' behavior, norms shaped teenagers' public portrayals and post hoc justifications of their behavior. Although norm bundles are complex to measure, participants could summarize them succinctly. These bundles and their conflicting behavioral prescriptions create space for human agency in negotiating normative pressures. The norm bundles concept has implications for teen pregnancy prevention policies and can help revitalize social norms for understanding health behaviors.
Convex integration theory solutions to the h-principle in geometry and topology
Spring, David
1998-01-01
This book provides a comprehensive study of convex integration theory in immersion-theoretic topology. Convex integration theory, developed originally by M. Gromov, provides general topological methods for solving the h-principle for a wide variety of problems in differential geometry and topology, with applications also to PDE theory and to optimal control theory. Though topological in nature, the theory is based on a precise analytical approximation result for higher order derivatives of functions, proved by M. Gromov. This book is the first to present an exacting record and exposition of all of the basic concepts and technical results of convex integration theory in higher order jet spaces, including the theory of iterated convex hull extensions and the theory of relative h-principles. A second feature of the book is its detailed presentation of applications of the general theory to topics in symplectic topology, divergence free vector fields on 3-manifolds, isometric immersions, totally real embeddings, u...
Araújo, Luís
2000-01-01
In an economy where there is no double coincidence of wants and without recordkeeping of past transactions, money is usually seen as the only mechanism that can support exchange. In this paper, we show that, as long as the population is finite and agents are sufficiently patient, a social norm establishing gift-exchange can substitute for money. Notwithstanding, for a given discount factor, the growth of the population size eventually leads to the breakdown of the social norm, ...
Quasi-convex Functions in Carnot Groups
Institute of Scientific and Technical Information of China (English)
Mingbao SUN; Xiaoping YANG
2007-01-01
In this paper, the authors introduce the concept of h-quasiconvex functions on Carnot groups G. It is shown that the notions of h-quasiconvex functions and h-convex sets are equivalent and the L∞ estimates of first derivatives of h-quasiconvex functions are given. For a Carnot group G of step two, it is proved that h-quasiconvex functions are locally bounded from above. Furthermore, the authors obtain that h-convex functions are locally Lipschitz continuous and that an h-convex function is twice differentiable almost everywhere.
On the vertex index of convex bodies
Bezdek, Karoly
2011-01-01
We introduce the vertex index, vein(K), of a given centrally symmetric convex body K, which, in a sense, measures how well K can be inscribed into a convex polytope with small number of vertices. This index is closely connected to the illumination parameter of a body, introduced earlier by the first named author, and, thus, related to the famous conjecture in Convex Geometry about covering of a d-dimensional body by 2^d smaller positively homothetic copies. We provide asymptotically sharp estimates (up to a logarithmic term) of this index in the general case. Also, we provide sharp estimates in dimensions 2 and 3.
Weak Convergence Theorems for a Countable Family of Strict Pseudocontractions in Banach Spaces
Directory of Open Access Journals (Sweden)
Cholamjiak Prasit
2010-01-01
Full Text Available We investigate the convergence of Mann-type iterative scheme for a countable family of strict pseudocontractions in a uniformly convex Banach space with the Fréchet differentiable norm. Our results improve and extend the results obtained by Marino-Xu, Zhou, Osilike-Udomene, Zhang-Guo and the corresponding results. We also point out that the condition given by Chidume-Shahzad (2010 is not satisfied in a real Hilbert space. We show that their results still are true under a new condition.
Designing Camera Networks by Convex Quadratic Programming
Ghanem, Bernard
2015-05-04
In this paper, we study the problem of automatic camera placement for computer graphics and computer vision applications. We extend the problem formulations of previous work by proposing a novel way to incorporate visibility constraints and camera-to-camera relationships. For example, the placement solution can be encouraged to have cameras that image the same important locations from different viewing directions, which can enable reconstruction and surveillance tasks to perform better. We show that the general camera placement problem can be formulated mathematically as a convex binary quadratic program (BQP) under linear constraints. Moreover, we propose an optimization strategy with a favorable trade-off between speed and solution quality. Our solution is almost as fast as a greedy treatment of the problem, but the quality is significantly higher, so much so that it is comparable to exact solutions that take orders of magnitude more computation time. Because it is computationally attractive, our method also allows users to explore the space of solutions for variations in input parameters. To evaluate its effectiveness, we show a range of 3D results on real-world floorplans (garage, hotel, mall, and airport).
Revising incompletely specified convex probabilistic belief bases
CSIR Research Space (South Africa)
Rens, G
2016-04-01
Full Text Available International Workshop on Non-Monotonic Reasoning (NMR), 22-24 April 2016, Cape Town, South Africa Revising Incompletely Specified Convex Probabilistic Belief Bases Gavin Rens CAIR_, University of KwaZulu-Natal, School of Mathematics, Statistics...
Some integral inequalities for logarithmically convex functions
Directory of Open Access Journals (Sweden)
Mevlüt Tunç
2014-07-01
Full Text Available The main aim of the present note is to establish new Hadamard like integral inequalities involving log-convex function. We also prove some Hadamard-type inequalities, and applications to the special means are given.
Closed Graph and Open Mapping Theorems for Normed Cones
Indian Academy of Sciences (India)
Oscar Valero
2008-05-01
A quasi-normed cone is a pair (, ) such that is a (not necessarily cancellative) cone and is a quasi-norm on . The aim of this paper is to prove a closed graph and an open mapping type theorem for quasi-normed cones. This is done with the help of appropriate notions of completeness, continuity and openness that arise in a natural way from the setting of bitopological spaces.
The convexity radius of a Riemannian manifold
Dibble, James
2014-01-01
The ratio of convexity radius over injectivity radius may be made arbitrarily small within the class of compact Riemannian manifolds of any fixed dimension at least two. This is proved using Gulliver's method of constructing manifolds with focal points but no conjugate points. The approach is suggested by a characterization of the convexity radius that resembles a classical result of Klingenberg about the injectivity radius.
Convex-Faced Combinatorially Regular Polyhedra of Small Genus
Directory of Open Access Journals (Sweden)
Jörg M. Wills
2011-12-01
Full Text Available Combinatorially regular polyhedra are polyhedral realizations (embeddings in Euclidean 3-space E3 of regular maps on (orientable closed compact surfaces. They are close analogues of the Platonic solids. A surface of genus g ≥ 2 admits only finitely many regular maps, and generally only a small number of them can be realized as polyhedra with convex faces. When the genus g is small, meaning that g is in the historically motivated range 2 ≤ g ≤ 6, only eight regular maps of genus g are known to have polyhedral realizations, two discovered quite recently. These include spectacular convex-faced polyhedra realizing famous maps of Klein, Fricke, Dyck, and Coxeter. We provide supporting evidence that this list is complete; in other words, we strongly conjecture that in addition to those eight there are no other regular maps of genus g, with 2 ≤ g ≤ 6, admitting realizations as convex-faced polyhedra in E3. For all admissible maps in this range, save Gordan’s map of genus 4, and its dual, we rule out realizability by a polyhedron in E3.
Gauss images of hyperbolic cusps with convex polyhedral boundary
Fillastre, François
2009-01-01
We prove that a 3--dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by its Gauss image. Furthermore, any spherical metric on the torus with cone singularities of negative curvature and all closed contractible geodesics of length greater than $2\\pi$ is the metric of the Gauss image of some convex polyhedral cusp. This result is an analog of the Rivin-Hodgson theorem characterizing compact convex hyperbolic polyhedra in terms of their Gauss images. The proof uses a variational method. Namely, a cusp with a given Gauss image is identified with a critical point of a functional on the space of cusps with cone-type singularities along a family of half-lines. The functional is shown to be concave and to attain maximum at an interior point of its domain. As a byproduct, we prove rigidity statements with respect to the Gauss image for cusps with or without cone-type singularities. In a special case, our theorem is equivalent to existence of a circle pattern on the torus, with prescrib...
Entropy Coherent and Entropy Convex Measures of Risk
Laeven, R.J.A.; Stadje, M.A.
2011-01-01
We introduce two subclasses of convex measures of risk, referred to as entropy coherent and entropy convex measures of risk. We prove that convex, entropy convex and entropy coherent measures of risk emerge as certainty equivalents under variational, homothetic and multiple priors preferences, respe
Mollborn, Stefanie; Domingue, Benjamin W; Boardman, Jason D
2014-06-01
Researchers seeking to understand teen sexual behaviors often turn to age norms, but they are difficult to measure quantitatively. Previous work has usually inferred norms from behavioral patterns or measured group-level norms at the individual level, ignoring multiple reference groups. Capitalizing on the multilevel design of the Add Health survey, we measure teen pregnancy norms perceived by teenagers, as well as average norms at the school and peer network levels. School norms predict boys' perceived norms, while peer network norms predict girls' perceived norms. Peer network and individually perceived norms against teen pregnancy independently and negatively predict teens' likelihood of sexual intercourse. Perceived norms against pregnancy predict increased likelihood of contraception among sexually experienced girls, but sexually experienced boys' contraceptive behavior is more complicated: When both the boy and his peers or school have stronger norms against teen pregnancy he is more likely to contracept, and in the absence of school or peer norms against pregnancy, boys who are embarrassed are less likely to contracept. We conclude that: (1) patterns of behavior cannot adequately operationalize teen pregnancy norms, (2) norms are not simply linked to behaviors through individual perceptions, and (3) norms at different levels can operate independently of each other, interactively, or in opposition. This evidence creates space for conceptualizations of agency, conflict, and change that can lead to progress in understanding age norms and sexual behaviors.
Mollborn, Stefanie; Domingue, Benjamin W.; Boardman, Jason D.
2014-01-01
Researchers seeking to understand teen sexual behaviors often turn to age norms, but they are difficult to measure quantitatively. Previous work has usually inferred norms from behavioral patterns or measured group-level norms at the individual level, ignoring multiple reference groups. Capitalizing on the multilevel design of the Add Health survey, we measure teen pregnancy norms perceived by teenagers, as well as average norms at the school and peer network levels. School norms predict boys’ perceived norms, while peer network norms predict girls’ perceived norms. Peer network and individually perceived norms against teen pregnancy independently and negatively predict teens’ likelihood of sexual intercourse. Perceived norms against pregnancy predict increased likelihood of contraception among sexually experienced girls, but sexually experienced boys’ contraceptive behavior is more complicated: When both the boy and his peers or school have stronger norms against teen pregnancy he is more likely to contracept, and in the absence of school or peer norms against pregnancy, boys who are embarrassed are less likely to contracept. We conclude that: (1) patterns of behavior cannot adequately operationalize teen pregnancy norms, (2) norms are not simply linked to behaviors through individual perceptions, and (3) norms at different levels can operate independently of each other, interactively, or in opposition. This evidence creates space for conceptualizations of agency, conflict, and change that can lead to progress in understanding age norms and sexual behaviors. PMID:25104920
Jiménez, A
2012-01-01
An $n-1$--dimensional tropical simplex $\\TT_A$ is the set of points tropically spanned by $n$ points in $n-1$--dimensional space, when they are not contained in any tropical hyperplane. The coordinates of the points are written as the columns of an $n\\times n$ real matrix $A$. In theorem \\ref{thm:convexity}, we show that convexity of $\\TT_A$ is equivalent to normality and tropical idempotency of $A$. A description of $\\TT_A$ by $n(n-1)$ linear inequalities is immediate from $A$. Set $n=4$. We study \\textbf{tropical tetrahedra} which are \\textbf{convex} and \\textbf{maximal} (i.e., having the maximal number of extremal points, which is 20, and maximal number of facets, which is 12). By tropicality, the facets in $\\TT_A$ are $m$--gons, with $m=3,4,5,6$. In corollary \\ref{cor:no(0,0,12,0)ni(0,1,10,1)}, we show that a polyhedron $\\TT_A$ combinatorially equivalent to the regular dodecahedron does not occur, i.e., the polygon--vector $(f_3,f_4,f_5,f_6)$ of $\\TT_A$ (with $12=f_3+f_4+f_5+f_6$) cannot be $(0,0,12,0)$. ...
Modified Ishikawa Iterative Process with Errors in Normed Linear Spaces%赋范线性空间中修改的具误差的Ishikawa迭代程序
Institute of Scientific and Technical Information of China (English)
姚永红; 陈汝栋; 周海云
2007-01-01
Ishikawa iterative sequences with errors different from the iterative sequences introduced by Liu and Xu are given. Moreover, the problem of approximating the fixed points of (ψ)-hemicontractive mapping in normed linear spaces by the modified Ishikawa iterative sequences with errors is investigated. The results presented in this paper improve and extend the results of the others.%本文给出了有别于刘立山以及徐洪坤的意义下的具误差的Ishikawa迭代程序.进一步,还研究了赋范线性空间中(ψ)-半压缩映象不动点的具误差的Ishikawa迭代逼近问题.所得的结果改进和推广了许多相应的结果.
DEFF Research Database (Denmark)
Beran, Eric Bengt
1996-01-01
This paper describes the basic nature of the InducedNorm Control Toolbox (INCT). The toolbox is a set of Matlab-filesusing LMITOOL and the Semidefinite Programming package(SP). Thetoolbox is public domain. The INCT provides a series of analysisand synthesis tools for continuous time-invariant lin......This paper describes the basic nature of the InducedNorm Control Toolbox (INCT). The toolbox is a set of Matlab-filesusing LMITOOL and the Semidefinite Programming package(SP). Thetoolbox is public domain. The INCT provides a series of analysisand synthesis tools for continuous time...
Italian Word Association Norms.
1966-07-01
Ricerche Istituto Nazionale I Psicologia , Roma Italy. A different, but not unrelated, approach is to use word association norms to study other types of... Psicologia , 1961 , 55, 1,13-155. Chiari, S. 11 comportamento associative nell’eta ovolutiva, Rivista di Psicologia , 1 96 1b , 55, 175-189. Cofer, C.N...and Russell, VI.A. Systematic changes in word association norms: 1910-1952. Journal of Abnormal and Social Psychology, 19C0, 60, 293-303. lilb Kurez, I
Iterative Schemes for Convex Minimization Problems with Constraints
Directory of Open Access Journals (Sweden)
Lu-Chuan Ceng
2014-01-01
Full Text Available We first introduce and analyze one implicit iterative algorithm for finding a solution of the minimization problem for a convex and continuously Fréchet differentiable functional, with constraints of several problems: the generalized mixed equilibrium problem, the system of generalized equilibrium problems, and finitely many variational inclusions in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another implicit iterative algorithm for finding a fixed point of infinitely many nonexpansive mappings with the same constraints, and derive its strong convergence under mild assumptions.
Variation, structure and norms
DEFF Research Database (Denmark)
Harder, Peter
2014-01-01
that an evolutionary account can reintegrate the opposed fragments into a whole picture that puts each of them in their ‘ecological position’ with respect to each other. Empirical usage facts should be seen in the context of operational norms in relation to which actual linguistic choices represent adaptations...
McBreen, J.; Tosto, Di G.; Dignum, F.; Hofstede, G.J.
2011-01-01
The goal of this paper is to propose a method of modelling the evolution of social norms in different cultural settings. We analyse the role of culture in shaping agents' normative reasoning and hence their behaviour. The general notion of 'value' is discussed from the perspective of the BDI framewo
McBreen, J.; Tosto, Di G.; Dignum, F.; Hofstede, G.J.
2011-01-01
The goal of this paper is to propose a method of modelling the evolution of social norms in different cultural settings. We analyse the role of culture in shaping agents' normative reasoning and hence their behaviour. The general notion of 'value' is discussed from the perspective of the BDI
Combining norms to prove termination
DEFF Research Database (Denmark)
Genaim, S.; Codish, M.; Gallagher, John Patrick;
2002-01-01
of deriving automatically a candidate norm with which to prove termination. Instead of deriving a single, complex norm function, it is sufficient to determine a collection of simpler norms, some combination of which, leads to a proof of termination. We propose that a collection of simple norms, one for each...... of the recursive data-types in the program, is often a suitable choice. We first demonstrate the power of combining norm functions and then the adequacy of combining norms based on regular types....
Spectral calibration for convex grating imaging spectrometer
Zhou, Jiankang; Chen, Xinhua; Ji, Yiqun; Chen, Yuheng; Shen, Weimin
2013-12-01
Spectral calibration of imaging spectrometer plays an important role for acquiring target accurate spectrum. There are two spectral calibration types in essence, the wavelength scanning and characteristic line sampling. Only the calibrated pixel is used for the wavelength scanning methods and he spectral response function (SRF) is constructed by the calibrated pixel itself. The different wavelength can be generated by the monochromator. The SRF is constructed by adjacent pixels of the calibrated one for the characteristic line sampling methods. And the pixels are illuminated by the narrow spectrum line and the center wavelength of the spectral line is exactly known. The calibration result comes from scanning method is precise, but it takes much time and data to deal with. The wavelength scanning method cannot be used in field or space environment. The characteristic line sampling method is simple, but the calibration precision is not easy to confirm. The standard spectroscopic lamp is used to calibrate our manufactured convex grating imaging spectrometer which has Offner concentric structure and can supply high resolution and uniform spectral signal. Gaussian fitting algorithm is used to determine the center position and the Full-Width-Half-Maximum（FWHM）of the characteristic spectrum line. The central wavelengths and FWHMs of spectral pixels are calibrated by cubic polynomial fitting. By setting a fitting error thresh hold and abandoning the maximum deviation point, an optimization calculation is achieved. The integrated calibration experiment equipment for spectral calibration is developed to enhance calibration efficiency. The spectral calibration result comes from spectral lamp method are verified by monochromator wavelength scanning calibration technique. The result shows that spectral calibration uncertainty of FWHM and center wavelength are both less than 0.08nm, or 5.2% of spectral FWHM.
Directory of Open Access Journals (Sweden)
Edward L. Rubin
2017-06-01
Full Text Available Objective to revisit the debate about rational choice theory from the legal cultural and historical perspectives. Methods dialectic approach to the cognition of social phenomena allowing to analyze them in their historical development and functioning in the context of the integrity of subjective and objective factors this determines the choice of the research methods systemicstructural formallegal and comparative. Results The first part of this chapter will explain the way in which people in societies different from our own were subject to other motivations in situations where selfinterest would tend to dominate in our society. The reasoning is based on three examples one drawn from the history of Ancient Rome one from the High Middle Ages of the European society and one from a contemporary nonWestern culture. The second part of the chapter analyzes the reason why material selfinterest maximizing became a dominant motivation in the modern Western society. The works on historical sociology attribute this development to Calvinism but this hypothesis suffers from some serious defects. In the article we prove that the modern sensibility resulted from much longeracting trends specifically secularization urbanization and commercialization. The final section of the chapter explores the relationship between the Westrsquos prevailing norm of selfinterest maximization and the particular norms that have been discussed in microeconomic theory. It argues that some of these norms are internal to the prevailing one and are thus explicable in terms of material selfinterest but that others reflect additional norms in the general society that exist alongside and sometimes in competition with the prevailing norm of selfinterest maximization. The historicallybased view that selfinterest maximizing is a prevailing norm rather than a human universal allows these other norms to be acknowledged in a plausible and realistic manner rather than being explained away by a
非常极凸空间的推广及其对偶概念%ON SOME GENERALIZATION OF VERY EXTREME CONVEXITY AND DUAL CONCEPT
Institute of Scientific and Technical Information of China (English)
李广利; 苏雅拉图; 刘瑞兰
2011-01-01
本文研究了k非常极凸和k非常极光滑空间的问题.利用Banach空间理论的方法,证明了k非常极凸空间和k非常极光滑空间是一对对偶概念,并且k非常极凸空间(k非常极光滑空间)是严格介于k一致极凸空间和k非常凸空间(k一致极光滑空间和k非常光滑空间)之间的一类新的Banach空间,得到了k非常极凸空间和k非常极光滑空间的若干等价刻画以及k非常极凸(k非常极光滑性)与其它凸性(光滑性)之间的蕴涵关系,推广了非常极凸空间和非常极光滑空间,完善了k非常极凸空间及其对偶空间的研究.%In this article, we study the problems of about fc-very extreme convex spaces and fc-very extreme smooth spaces. Using the method of Banach spaces theory, we show that the fc-very extreme convex space is just the duality of fc-very extreme smooth space and lies strictly between fc-uniformly extreme convex spaces and fc-very convex spaces. We obtain some characteristic descriptions of fc-very extreme convex spaces or fc-very extreme smooth space, and study the relation between fc-very extreme convexity (resp. Fc-very extreme smoothness) with various convexity (resp. Various smoothness). The results generalize the very extreme convexity and very extreme smoothness, and perfect the research on fc-very extreme convex space and its dual space.
Fractional norm regularization: learning with very few relevant features.
Kaban, Ata
2013-06-01
Learning in the presence of a large number of irrelevant features is an important problem in high-dimensional tasks. Previous studies have shown that L1-norm regularization can be effective in such cases while L2-norm regularization is not. Furthermore, work in compressed sensing suggests that regularization by nonconvex (e.g., fractional) semi-norms may outperform L1-regularization. However, for classification it is largely unclear when this may or may not be the case. In addition, the nonconvex problem is harder to solve than the convex L1 problem. In this paper, we provide a more in-depth analysis to elucidate the potential advantages and pitfalls of nonconvex regularization in the context of logistic regression where the regularization term employs the family of Lq semi-norms. First, using results from the phenomenon of concentration of norms and distances in high dimensions, we gain intuition about the working of sparse estimation when the dimensionality is very high. Second, using the probably approximately correct (PAC)-Bayes methodology, we give a data-dependent bound on the generalization error of Lq-regularized logistic regression, which is applicable to any algorithm that implements this model, and may be used to predict its generalization behavior from the training set alone. Third, we demonstrate the usefulness of our approach by experiments and applications, where the PAC-Bayes bound is used to guide the choice of semi-norm in the regularization term. The results support the conclusion that the optimal choice of regularization depends on the relative fraction of relevant versus irrelevant features, and a fractional norm with a small exponent is most suitable when the fraction of relevant features is very small.
The Iterative Reweighted Mixed-Norm Estimate for Spatio-Temporal MEG/EEG Source Reconstruction.
Strohmeier, Daniel; Bekhti, Yousra; Haueisen, Jens; Gramfort, Alexandre
2016-10-01
Source imaging based on magnetoencephalography (MEG) and electroencephalography (EEG) allows for the non-invasive analysis of brain activity with high temporal and good spatial resolution. As the bioelectromagnetic inverse problem is ill-posed, constraints are required. For the analysis of evoked brain activity, spatial sparsity of the neuronal activation is a common assumption. It is often taken into account using convex constraints based on the l1-norm. The resulting source estimates are however biased in amplitude and often suboptimal in terms of source selection due to high correlations in the forward model. In this work, we demonstrate that an inverse solver based on a block-separable penalty with a Frobenius norm per block and a l0.5-quasinorm over blocks addresses both of these issues. For solving the resulting non-convex optimization problem, we propose the iterative reweighted Mixed Norm Estimate (irMxNE), an optimization scheme based on iterative reweighted convex surrogate optimization problems, which are solved efficiently using a block coordinate descent scheme and an active set strategy. We compare the proposed sparse imaging method to the dSPM and the RAP-MUSIC approach based on two MEG data sets. We provide empirical evidence based on simulations and analysis of MEG data that the proposed method improves on the standard Mixed Norm Estimate (MxNE) in terms of amplitude bias, support recovery, and stability.
Directory of Open Access Journals (Sweden)
Yanhui Li
2014-01-01
Full Text Available This paper investigates the Hankel norm filter design problem for stochastic time-delay systems, which are represented by Takagi-Sugeno (T-S fuzzy model. Motivated by the parallel distributed compensation (PDC technique, a novel filtering error system is established. The objective is to design a suitable filter that guarantees the corresponding filtering error system to be mean-square asymptotically stable and to have a specified Hankel norm performance level γ. Based on the Lyapunov stability theory and the Itô differential rule, the Hankel norm criterion is first established by adopting the integral inequality method, which can make some useful efforts in reducing conservativeness. The Hankel norm filtering problem is casted into a convex optimization problem with a convex linearization approach, which expresses all the conditions for the existence of admissible Hankel norm filter as standard linear matrix inequalities (LMIs. The effectiveness of the proposed method is demonstrated via a numerical example.
Optimal convex shapes for concave functionals
Bucur, Dorin; Lamboley, Jimmy
2011-01-01
Motivated by a long-standing conjecture of Polya and Szeg\\"o about the Newtonian capacity of convex bodies, we discuss the role of concavity inequalities in shape optimization, and we provide several counterexamples to the Blaschke-concavity of variational functionals, including capacity. We then introduce a new algebraic structure on convex bodies, which allows to obtain global concavity and indecomposability results, and we discuss their application to isoperimetriclike inequalities. As a byproduct of this approach we also obtain a quantitative version of the Kneser-S\\"uss inequality. Finally, for a large class of functionals involving Dirichlet energies and the surface measure, we perform a local analysis of strictly convex portions of the boundary via second order shape derivatives. This allows in particular to exclude the presence of smooth regions with positive Gauss curvature in an optimal shape for Polya-Szeg\\"o problem.
On the convexity of Relativistic Ideal Magnetohydrodynamics
Ibáñez, José-María; Aloy, Miguel-Ángel; Martí, José-María; Miralles, Juan-Antonio
2015-01-01
We analyze the influence of the magnetic field in the convexity properties of the relativistic magnetohydrodynamics system of equations. To this purpose we use the approach of Lax, based on the analysis of the linearly degenerate/genuinely non-linear nature of the characteristic fields. Degenerate and non-degenerate states are discussed separately and the non-relativistic, unmagnetized limits are properly recovered. The characteristic fields corresponding to the material and Alfv\\'en waves are linearly degenerate and, then, not affected by the convexity issue. The analysis of the characteristic fields associated with the magnetosonic waves reveals, however, a dependence of the convexity condition on the magnetic field. The result is expressed in the form of a generalized fundamental derivative written as the sum of two terms. The first one is the generalized fundamental derivative in the case of purely hydrodynamical (relativistic) flow. The second one contains the effects of the magnetic field. The analysis ...
A generalization of the convex Kakeya problem
Ahn, Heekap
2012-01-01
We consider the following geometric alignment problem: Given a set of line segments in the plane, find a convex region of smallest area that contains a translate of each input segment. This can be seen as a generalization of Kakeya\\'s problem of finding a convex region of smallest area such that a needle can be turned through 360 degrees within this region. Our main result is an optimal Θ(n log n)-time algorithm for our geometric alignment problem, when the input is a set of n line segments. We also show that, if the goal is to minimize the perimeter of the region instead of its area, then the optimum placement is when the midpoints of the segments coincide. Finally, we show that for any compact convex figure G, the smallest enclosing disk of G is a smallest-perimeter region containing a translate of any rotated copy of G. © 2012 Springer-Verlag Berlin Heidelberg.
Cost Allocation and Convex Data Envelopment
DEFF Research Database (Denmark)
Hougaard, Jens Leth; Tind, Jørgen
This paper considers allocation rules. First, we demonstrate that costs allocated by the Aumann-Shapley and the Friedman-Moulin cost allocation rules are easy to determine in practice using convex envelopment of registered cost data and parametric programming. Second, from the linear programming...... such as Data Envelopment Analysis (DEA). The convexity constraint of the BCC model introduces a non-zero slack in the objective function of the multiplier problem and we show that the cost allocation rules discussed in this paper can be used as candidates to allocate this slack value on to the input (or output...... problems involved it becomes clear that the allocation rules, technically speaking, allocate the non-zero value of the dual variable for a convexity constraint on to the output vector. Hence, the allocation rules can also be used to allocate inefficiencies in non-parametric efficiency measurement models...
Non-convex multi-objective optimization
Pardalos, Panos M; Žilinskas, Julius
2017-01-01
Recent results on non-convex multi-objective optimization problems and methods are presented in this book, with particular attention to expensive black-box objective functions. Multi-objective optimization methods facilitate designers, engineers, and researchers to make decisions on appropriate trade-offs between various conflicting goals. A variety of deterministic and stochastic multi-objective optimization methods are developed in this book. Beginning with basic concepts and a review of non-convex single-objective optimization problems; this book moves on to cover multi-objective branch and bound algorithms, worst-case optimal algorithms (for Lipschitz functions and bi-objective problems), statistical models based algorithms, and probabilistic branch and bound approach. Detailed descriptions of new algorithms for non-convex multi-objective optimization, their theoretical substantiation, and examples for practical applications to the cell formation problem in manufacturing engineering, the process design in...
Institute of Scientific and Technical Information of China (English)
ZHANG Yan-hu; YAN Wen-jun; LU Jian-ning; ZHAO Guang-zhou
2005-01-01
Multi-objective robust state-feedback controller synthesis problems for linear discrete-time uncertain systems are addressed. Based on parameter-dependent Lyapunov functions, the Gl2 and GH2 norm expressed in terms of LMI (Linear Matrix Inequality) characterizations are further generalized to cope with the robust analysis for convex polytopic uncertain system.Robust state-feedback controller synthesis conditions are also derived for this class of uncertain systems. Using the above results,multi-objective state-feedback controller synthesis procedures which involve the LMI optimization technique are developed and less conservative than the existing one. An illustrative example verified the validity of the approach.
Non-convex onion peeling using a shape hull algorithm
Fadili, Jalal M.; Melkemi, Mahmoud; Elmoataz, Abderrahim
2004-01-01
International audience; The convex onion-peeling of a set of points is the organization of these points into a sequence of interpolating convex polygons. This method is adequate to detect the shape of the “center” of a set of points when this shape is convex. However it reveals inadequate to detect non-convex shapes. Alternatively, we propose an extension of the convex onion-peeling method. It consists in representing a set of points with a sequence of non-convex polylines which are computed ...
Cost Allocation and Convex Data Envelopment
DEFF Research Database (Denmark)
Hougaard, Jens Leth; Tind, Jørgen
problems involved it becomes clear that the allocation rules, technically speaking, allocate the non-zero value of the dual variable for a convexity constraint on to the output vector. Hence, the allocation rules can also be used to allocate inefficiencies in non-parametric efficiency measurement models...... such as Data Envelopment Analysis (DEA). The convexity constraint of the BCC model introduces a non-zero slack in the objective function of the multiplier problem and we show that the cost allocation rules discussed in this paper can be used as candidates to allocate this slack value on to the input (or output...
Convex functions and the rolling circle criterion
1995-01-01
Given 0≤R1≤R2≤∞, CVG(R1,R2) denotes the class of normalized convex functions f in the unit disc U, for which ∂f(U) satisfies a Blaschke Rolling Circles Criterion with radii R1 and R2. Necessary and sufficient conditions for R1=R2, growth and distortion theorems for CVG(R1,R2) and rotation theorem for the class of convex functions of bounded type, are found.
A Complete Characterization of the Gap between Convexity and SOS-Convexity
Ahmadi, Amir Ali
2011-01-01
Our first contribution in this paper is to prove that three natural sum of squares (sos) based sufficient conditions for convexity of polynomials via the definition of convexity, its first order characterization, and its second order characterization are equivalent. These three equivalent algebraic conditions, henceforth referred to as sos-convexity, can be checked by semidefinite programming whereas deciding convexity is NP-hard. If we denote the set of convex and sos-convex polynomials in $n$ variables of degree $d$ with $\\tilde{C}_{n,d}$ and $\\tilde{\\Sigma C}_{n,d}$ respectively, then our main contribution is to prove that $\\tilde{C}_{n,d}=\\tilde{\\Sigma C}_{n,d}$ if and only if $n=1$ or $d=2$ or $(n,d)=(2,4)$. We also present a complete characterization for forms (homogeneous polynomials) except for the case $(n,d)=(3,4)$ which is joint work with G. Blekherman and is to be published elsewhere. Our result states that the set $C_{n,d}$ of convex forms in $n$ variables of degree $d$ equals the set $\\Sigma C_{...
Fixed point variational solutions for uniformly continuous pseudocontractions in banach spaces
Directory of Open Access Journals (Sweden)
2006-01-01
Full Text Available Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm, let K be a nonempty closed convex subset of E , and let T:K→K be a uniformly continuous pseudocontraction. If f:K→K is any contraction map on K and if every nonempty closed convex and bounded subset of K has the fixed point property for nonexpansive self-mappings, then it is shown, under appropriate conditions on the sequences of real numbers { α n } , { μ n } , that the iteration process z 1 ∈K , z n+1 = μ n ( α n T z n +( 1− α n z n +( 1− μ n f( z n , n∈ℕ , strongly converges to the fixed point of T , which is the unique solution of some variational inequality, provided that K is bounded.
Directory of Open Access Journals (Sweden)
Yali Li
2008-01-01
Full Text Available Let K be a nonempty closed convex subset of a reflexive and strictly convex Banach space E with a uniformly Gâteaux differentiable norm, ℱ={T(h:h≥0} a generalized asymptotically nonexpansive self-mapping semigroup of K, and f:K→K a fixed contractive mapping with contractive coefficient β∈(0,1. We prove that the following implicit and modified implicit viscosity iterative schemes {xn} defined by xn=αnf(xn+(1−αnT(tnxn and xn=αnyn+(1−αnT(tnxn, yn=βnf(xn−1+(1−βnxn−1 strongly converge to p∈F as n→∞ and p is the unique solution to the following variational inequality: 〈f(p−p,j(y−p〉≤0 for all y∈F.
Best approximation in quotient spaces with application to the finishing of optical surfaces
Bennett, Therese Lynn
The finishing of axisymmetric optical surfaces relies upon the use of Computer Numerically Controlled (CNC) machines. Porsching and Hall developed a mathematical model for a general class of such machines, which can be used to generate material removal rates. These removal rates give the amount of material removed at each radial point on the workpiece per unit time. In this thesis, we use these material removal rates to generate two different material removal strategies for Operator Controlled (OC) finishing. Given an initial error profile which is an element of the quotient space C[a, b]IK where K is the space of constant functions, we seek a best approximation to this error profile by elements of a closed convex set. Some other nonstandard features of this best approximation problem are nonnegativity constraints on the variables, the nondifferentiability of the approximating functions, and the fact that the approximating functions do not form a Haar space. We explore best approximation on a quotient space and show that many of the standard existence and uniqueness theorems can be extended. For the first removal strategy, we solve the best approximation problem using the L2-norm, which leads to a quadratic programming problem (QPP). This QPP is solved after first using a modified Gram-Schmidt process to obtain a set of orthogonal removal rates. Our second strategy employs best minimax approximation. Since we do not have a Haar space, the standard characterizations theorems do not apply, although similar theorems are established. Using two definitions of the infinity norm on the quotient space C[a, b]IK, the problem can be reformulated as both a semi-infinite programming problem and as a convex programming problem. Computer programs which implement the least squares algorithm and two minimax algorithms which solve the related convex programming problems are described. Finally, we generate and discuss numerical examples which illustrate the theory and simulate the
Convex Hulls of Orbits and Orientations of a Moving Protein Domain
Longinetti, Marco; Sottile, Frank
2007-01-01
We study the facial structure and Carath\\'eodory number of the convex hull of an orbit of the group of rotations in R^3 acting on the space of pairs of anisotropic symmetric 3\\times 3 tensors. This is motivated by the problem of determining the structure of some proteins in aqueous solution.
Short, Lindsey A; Hatry, Alexandra J; Mondloch, Catherine J
2011-02-01
The current research investigated the organization of children's face space by examining whether 5- and 8-year-olds show race-contingent aftereffects. Participants read a storybook in which Caucasian and Chinese children's faces were distorted in opposite directions. Before and after adaptation, participants judged the normality/attractiveness of expanded, compressed, and undistorted Caucasian and Chinese faces. The method was validated with adults and then refined to test 8- and 5-year-olds. The 5-year-olds were also tested in a simple aftereffects paradigm. The current research provides the first evidence for simple attractiveness aftereffects in 5-year-olds and for race-contingent aftereffects in both 5- and 8-year-olds. Evidence that adults and 5-year-olds may possess only a weak prototype for Chinese children's faces suggests that Caucasian adults' prototype for Chinese adult faces does not generalize to child faces and that children's face space undergoes a period of increasing differentiation between 5 and 8 years of age.
Some Characterizations of Convex Interval Games
Brânzei, R.; Tijs, S.H.; Alparslan-Gok, S.Z.
2008-01-01
This paper focuses on new characterizations of convex interval games using the notions of exactness and superadditivity. We also relate big boss interval games with concave interval games and obtain characterizations of big boss interval games in terms of exactness and subadditivity.
A generalization of the convex Kakeya problem
Ahn, Heekap
2013-09-19
Given a set of line segments in the plane, not necessarily finite, what is a convex region of smallest area that contains a translate of each input segment? This question can be seen as a generalization of Kakeya\\'s problem of finding a convex region of smallest area such that a needle can be rotated through 360 degrees within this region. We show that there is always an optimal region that is a triangle, and we give an optimal Θ(nlogn)-time algorithm to compute such a triangle for a given set of n segments. We also show that, if the goal is to minimize the perimeter of the region instead of its area, then placing the segments with their midpoint at the origin and taking their convex hull results in an optimal solution. Finally, we show that for any compact convex figure G, the smallest enclosing disk of G is a smallest-perimeter region containing a translate of every rotated copy of G. © 2013 Springer Science+Business Media New York.
Robust Utility Maximization Under Convex Portfolio Constraints
Energy Technology Data Exchange (ETDEWEB)
Matoussi, Anis, E-mail: anis.matoussi@univ-lemans.fr [Université du Maine, Risk and Insurance institut of Le Mans Laboratoire Manceau de Mathématiques (France); Mezghani, Hanen, E-mail: hanen.mezghani@lamsin.rnu.tn; Mnif, Mohamed, E-mail: mohamed.mnif@enit.rnu.tn [University of Tunis El Manar, Laboratoire de Modélisation Mathématique et Numérique dans les Sciences de l’Ingénieur, ENIT (Tunisia)
2015-04-15
We study a robust maximization problem from terminal wealth and consumption under a convex constraints on the portfolio. We state the existence and the uniqueness of the consumption–investment strategy by studying the associated quadratic backward stochastic differential equation. We characterize the optimal control by using the duality method and deriving a dynamic maximum principle.
Dynamic Matchings in Convex Bipartite Graphs
DEFF Research Database (Denmark)
Brodal, Gerth Stølting; Georgiadis, Loukas; Hansen, Kristoffer Arnsfelt
2007-01-01
We consider the problem of maintaining a maximum matching in a convex bipartite graph G = (V,E) under a set of update operations which includes insertions and deletions of vertices and edges. It is not hard to show that it is impossible to maintain an explicit representation of a maximum matching...
Differential analysis of matrix convex functions
DEFF Research Database (Denmark)
Hansen, Frank; Tomiyama, Jun
2007-01-01
We analyze matrix convex functions of a fixed order defined in a real interval by differential methods as opposed to the characterization in terms of divided differences given by Kraus [F. Kraus, Über konvekse Matrixfunktionen, Math. Z. 41 (1936) 18-42]. We obtain for each order conditions for ma...
Estimates for oscillatory integrals with convex phase
Energy Technology Data Exchange (ETDEWEB)
Chakhkiev, M A [Moscow State Social University, Moscow (Russian Federation)
2006-02-28
We consider methods for estimating one-dimensional oscillatory integrals with convex phase and amplitudes of bounded variation or Lipschitz class amplitudes. In particular, we improve the estimate for the Piercey integral with near-caustic parameter values, and also consider estimation methods for n-dimensional oscillatory integrals.
Some Characterizations of Convex Interval Games
Brânzei, R.; Tijs, S.H.; Alparslan-Gok, S.Z.
2008-01-01
This paper focuses on new characterizations of convex interval games using the notions of exactness and superadditivity. We also relate big boss interval games with concave interval games and obtain characterizations of big boss interval games in terms of exactness and subadditivity.
Directional Convexity and Finite Optimality Conditions.
1984-03-01
system, Necessary Conditions for optimality. Work Unit Number 5 (Optimization and Large Scale Systems) *Istituto di Matematica Applicata, Universita...that R(T) is convex would then imply x(u,T) e int R(T). Cletituto di Matematica Applicata, Universita di Padova, 35100 ITALY. Sponsored by the United
Convexity properties of Hamiltonian group actions
Guillemin, Victor
2005-01-01
This is a monograph on convexity properties of moment mappings in symplectic geometry. The fundamental result in this subject is the Kirwan convexity theorem, which describes the image of a moment map in terms of linear inequalities. This theorem bears a close relationship to perplexing old puzzles from linear algebra, such as the Horn problem on sums of Hermitian matrices, on which considerable progress has been made in recent years following a breakthrough by Klyachko. The book presents a simple local model for the moment polytope, valid in the "generic&rdquo case, and an elementary Morse-theoretic argument deriving the Klyachko inequalities and some of their generalizations. It reviews various infinite-dimensional manifestations of moment convexity, such as the Kostant type theorems for orbits of a loop group (due to Atiyah and Pressley) or a symplectomorphism group (due to Bloch, Flaschka and Ratiu). Finally, it gives an account of a new convexity theorem for moment map images of orbits of a Borel sub...
Subset Selection by Local Convex Approximation
DEFF Research Database (Denmark)
Øjelund, Henrik; Sadegh, Payman; Madsen, Henrik
1999-01-01
least squares criterion. We propose an optimization technique for the posed probelm based on a modified version of the Newton-Raphson iterations, combined with a backward elimination type algorithm. THe Newton-Raphson modification concerns iterative approximations to the non-convex cost function...
Conference on Convex Analysis and Global Optimization
Pardalos, Panos
2001-01-01
There has been much recent progress in global optimization algo rithms for nonconvex continuous and discrete problems from both a theoretical and a practical perspective. Convex analysis plays a fun damental role in the analysis and development of global optimization algorithms. This is due essentially to the fact that virtually all noncon vex optimization problems can be described using differences of convex functions and differences of convex sets. A conference on Convex Analysis and Global Optimization was held during June 5 -9, 2000 at Pythagorion, Samos, Greece. The conference was honoring the memory of C. Caratheodory (1873-1950) and was en dorsed by the Mathematical Programming Society (MPS) and by the Society for Industrial and Applied Mathematics (SIAM) Activity Group in Optimization. The conference was sponsored by the European Union (through the EPEAEK program), the Department of Mathematics of the Aegean University and the Center for Applied Optimization of the University of Florida, by th...
Characterization of the minimal penalty of a convex risk measure with applications to Levy processes
Hernández-Hernández, Daniel
2012-01-01
The minimality of the penalization function associated with a convex risk measure is analyzed in this paper. First, in a general static framework, we provide necessary and sufficient conditions for a penalty function defined in a convex and closed subset of the absolutely continuous measures with respect to some reference measure $\\mathbb{P}$ to be minimal. When the probability space supports a L\\'{e}vy process, we establish results that guarantee the minimality property of a penalty function described in terms of the coefficients associated with the density processes. The set of densities processes is described and the convergence of its quadratic variation is analyzed.
Bot, Radu Ioan
2012-01-01
The aim of this paper is to develop an efficient algorithm for solving a class of unconstrained nondifferentiable convex optimization problems in finite dimensional spaces. To this end we formulate first its Fenchel dual problem and regularize it in two steps into a differentiable strongly convex one with Lipschitz continuous gradient. The doubly regularized dual problem is then solved via a fast gradient method with the aim of accelerating the resulting convergence scheme. The theoretical results are finally applied to an l1 regularization problem arising in image processing.
Fast approximate convex decomposition using relative concavity
Ghosh, Mukulika
2013-02-01
Approximate convex decomposition (ACD) is a technique that partitions an input object into approximately convex components. Decomposition into approximately convex pieces is both more efficient to compute than exact convex decomposition and can also generate a more manageable number of components. It can be used as a basis of divide-and-conquer algorithms for applications such as collision detection, skeleton extraction and mesh generation. In this paper, we propose a new method called Fast Approximate Convex Decomposition (FACD) that improves the quality of the decomposition and reduces the cost of computing it for both 2D and 3D models. In particular, we propose a new strategy for evaluating potential cuts that aims to reduce the relative concavity, rather than absolute concavity. As shown in our results, this leads to more natural and smaller decompositions that include components for small but important features such as toes or fingers while not decomposing larger components, such as the torso, that may have concavities due to surface texture. Second, instead of decomposing a component into two pieces at each step, as in the original ACD, we propose a new strategy that uses a dynamic programming approach to select a set of n c non-crossing (independent) cuts that can be simultaneously applied to decompose the component into n c+1 components. This reduces the depth of recursion and, together with a more efficient method for computing the concavity measure, leads to significant gains in efficiency. We provide comparative results for 2D and 3D models illustrating the improvements obtained by FACD over ACD and we compare with the segmentation methods in the Princeton Shape Benchmark by Chen et al. (2009) [31]. © 2012 Elsevier Ltd. All rights reserved.
The Identification of Convex Function on Riemannian Manifold
Directory of Open Access Journals (Sweden)
Li Zou
2014-01-01
Full Text Available The necessary and sufficient condition of convex function is significant in nonlinear convex programming. This paper presents the identification of convex function on Riemannian manifold by use of Penot generalized directional derivative and the Clarke generalized gradient. This paper also presents a method for judging whether a point is the global minimum point in the inequality constraints. Our objective here is to extend the content and proof the necessary and sufficient condition of convex function to Riemannian manifolds.
Optimal placement of convex polygons to maximize point containment
Energy Technology Data Exchange (ETDEWEB)
Dickerson, M. [Middlebury College, VT (United States); Scharstein, D. [Cornell Univ., Ithaca, NY (United States)
1996-12-31
Given a convex polygon P with m vertices and a set S of n points in the plane, we consider the problem of finding a placement of P that contains the maximum number of points in S. We allow both translation and rotation. Our algorithm is self-contained and utilizes the geometric properties of the containing regions in the parameter space of transformations. The algorithm requires O(nk{sup 2} m{sup 2} log(mk)) time and O(n + m) space, where k is the maximum number of points contained. This provides a linear improvement over the best previously known algorithm when k is large ({Theta}(n)) and a cubic improvement when k is small. We also show that the algorithm can be extended to solve bichromatic and general weighted variants of the problem.
Allometric relationships between traveltime channel networks, convex hulls, and convexity measures
Tay, Lea Tien; Sagar, B. S. Daya; Chuah, Hean Teik
2006-06-01
The channel network (S) is a nonconvex set, while its basin [C(S)] is convex. We remove open-end points of the channel connectivity network iteratively to generate a traveltime sequence of networks (Sn). The convex hulls of these traveltime networks provide an interesting topological quantity, which has not been noted thus far. We compute lengths of shrinking traveltime networks L(Sn) and areas of corresponding convex hulls C(Sn), the ratios of which provide convexity measures CM(Sn) of traveltime networks. A statistically significant scaling relationship is found for a model network in the form L(Sn) ˜ A[C(Sn)]0.57. From the plots of the lengths of these traveltime networks and the areas of their corresponding convex hulls as functions of convexity measures, new power law relations are derived. Such relations for a model network are CM(Sn) ˜ ? and CM(Sn) ˜ ?. In addition to the model study, these relations for networks derived from seven subbasins of Cameron Highlands region of Peninsular Malaysia are provided. Further studies are needed on a large number of channel networks of distinct sizes and topologies to understand the relationships of these new exponents with other scaling exponents that define the scaling structure of river networks.
Reverse convex problems: an approach based on optimality conditions
Directory of Open Access Journals (Sweden)
Ider Tseveendorj
2006-01-01
Full Text Available We present some results concerning reverse convex problems. Global optimality conditions for the problems with a nonsmooth reverse convex constraint are established and convergence of an algorithm in the case of linear program with an additional quadratic reverse convex constraint is studied.
On Quasi E-Convex Bilevel Programming Problem
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E. A. Youness
2005-01-01
Full Text Available Bilevel programming problems involve two optimization problems where the data of the first one is implicity determined by the solution of the second. This study introduces the notions of E-convexity and quasi E-convexity in bilevel programming problems to generalize quasi convex bilevel programming problems.
Reverse convex problems: an approach based on optimality conditions
Ider Tseveendorj
2006-01-01
We present some results concerning reverse convex problems. Global optimality conditions for the problems with a nonsmooth reverse convex constraint are established and convergence of an algorithm in the case of linear program with an additional quadratic reverse convex constraint is studied.
Fixed point theorems for generalized Lipschitzian semigroups in Banach spaces
Directory of Open Access Journals (Sweden)
Balwant Singh Thakur
1999-01-01
Full Text Available Fixed point theorems for generalized Lipschitzian semigroups are proved in p-uniformly convex Banach spaces and in uniformly convex Banach spaces. As applications, its corollaries are given in a Hilbert space, in Lp spaces, in Hardy space Hp, and in Sobolev spaces Hk,p, for 1
A two-layer recurrent neural network for nonsmooth convex optimization problems.
Qin, Sitian; Xue, Xiaoping
2015-06-01
In this paper, a two-layer recurrent neural network is proposed to solve the nonsmooth convex optimization problem subject to convex inequality and linear equality constraints. Compared with existing neural network models, the proposed neural network has a low model complexity and avoids penalty parameters. It is proved that from any initial point, the state of the proposed neural network reaches the equality feasible region in finite time and stays there thereafter. Moreover, the state is unique if the initial point lies in the equality feasible region. The equilibrium point set of the proposed neural network is proved to be equivalent to the Karush-Kuhn-Tucker optimality set of the original optimization problem. It is further proved that the equilibrium point of the proposed neural network is stable in the sense of Lyapunov. Moreover, from any initial point, the state is proved to be convergent to an equilibrium point of the proposed neural network. Finally, as applications, the proposed neural network is used to solve nonlinear convex programming with linear constraints and L1 -norm minimization problems.
Soft-tissue cephalometric norms for the Lambada population in Telangana Region of Andhra Pradesh
Directory of Open Access Journals (Sweden)
Mayuri Thomas
2012-01-01
Conclusion: Although established Caucasian norms are applicable to the Lambada ethnic tribe, few parameters like basic upper lip thickness (ULT and upper lip strain (ULS, skeletal convexity, inferior sulcus to H line (LS-H were significantly different. The male group exhibited straighter profile, thicker lips, prominent nose, deep mentolabial sulcus, and a prominent chin than females. The differences could be considered in diagnosis and treatment planning for orthodontic practice and for orthognathic surgery.
Higher-order principal component pursuit via tensor approximation and convex optimization
Institute of Scientific and Technical Information of China (English)
Sijia Cai; Ping Wang; Linhao Li; Chuhan Zhang
2014-01-01
Recovering the low-rank structure of data matrix from sparse errors arises in the principal component pursuit (PCP). This paper exploits the higher-order generalization of matrix recovery, named higher-order principal component pursuit (HOPCP), since it is critical in multi-way data analysis. Unlike the convexification (nuclear norm) for matrix rank function, the tensorial nuclear norm is stil an open problem. While existing preliminary works on the tensor completion field provide a viable way to indicate the low complexity estimate of tensor, therefore, the paper focuses on the low multi-linear rank tensor and adopt its convex relaxation to formulate the convex optimization model of HOPCP. The paper further propose two algorithms for HOPCP based on alternative minimization scheme: the augmented Lagrangian alternating di-rection method (ALADM) and its truncated higher-order singular value decomposition (ALADM-THOSVD) version. The former can obtain a high accuracy solution while the latter is more efficient to handle the computational y intractable problems. Experimental re-sults on both synthetic data and real magnetic resonance imaging data show the applicability of our algorithms in high-dimensional tensor data processing.
A New Interpolation Approach for Linearly Constrained Convex Optimization
Espinoza, Francisco
2012-08-01
In this thesis we propose a new class of Linearly Constrained Convex Optimization methods based on the use of a generalization of Shepard\\'s interpolation formula. We prove the properties of the surface such as the interpolation property at the boundary of the feasible region and the convergence of the gradient to the null space of the constraints at the boundary. We explore several descent techniques such as steepest descent, two quasi-Newton methods and the Newton\\'s method. Moreover, we implement in the Matlab language several versions of the method, particularly for the case of Quadratic Programming with bounded variables. Finally, we carry out performance tests against Matab Optimization Toolbox methods for convex optimization and implementations of the standard log-barrier and active-set methods. We conclude that the steepest descent technique seems to be the best choice so far for our method and that it is competitive with other standard methods both in performance and empirical growth order.
Rationally convex sets on the unit sphere in ℂ2
Wermer, John
2008-04-01
Let X be a rationally convex compact subset of the unit sphere S in ℂ2, of three-dimensional measure zero. Denote by R( X) the uniform closure on X of the space of functions P/ Q, where P and Q are polynomials and Q≠0 on X. When does R( X)= C( X)? Our work makes use of the kernel function for the bar{δ}b operator on S, introduced by Henkin in [5] and builds on results obtained in Anderson Izzo Wermer [3]. We define a real-valued function ɛ X on the open unit ball int B, with ɛ X ( z, w) tending to 0 as ( z, w) tends to X. We give a growth condition on ɛ X ( z, w) as ( z, w) approaches X, and show that this condition is sufficient for R( X)= C( X) (Theorem 1.1). In Section 4, we consider a class of sets X which are limits of a family of Levi-flat hypersurfaces in int B. For each compact set Y in ℂ2, we denote the rationally convex hull of Y by widehat{Y}. A general reference is Rudin [8] or Aleksandrov [1].
Directory of Open Access Journals (Sweden)
Yan Tang
2013-01-01
Full Text Available Suppose that C is a nonempty closed convex subset of a real reflexive Banach space E which has a uniformly Gateaux differentiable norm. A viscosity iterative process is constructed in this paper. A strong convergence theorem is proved for a common element of the set of fixed points of a finite family of pseudocontractive mappings and the set of solutions of a finite family of monotone mappings. And the common element is the unique solution of certain variational inequality. The results presented in this paper extend most of the results that have been proposed for this class of nonlinear mappings.
Norms for environmentally responsible behaviour
DEFF Research Database (Denmark)
Thøgersen, John
The currently used concept of personal or moral norms is ambiguous with regard to its motivational content. Therefore, a revision of the norm taxonomy is suggested, implying a distinction between three types of personal norms, called introjected, identified, and integrated norms. A preliminary...... assessment of the taxonomy is carried out based of a survey of a random sample of Danish residents 18 years or older. A range of norm constructs were measured with regard to four environmentally relevant behaviours: buying organic milk, buying energy saving light bulbs, source-separating compostable kitchen...... is also supported, with the reservation that the different behavioural references are more than just different methods of measuring the same latent construct(s). People evidently hold different norms for different environmentally responsible behaviours....
The Mackey convergence condition for spaces with webs
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Thomas E. Gilsdorf
1988-01-01
Full Text Available If each sequence converging to 0 in a locally convex space is also Mackey convergent to 0, that space is said to satisfy the Mackey convergence condition. The problem of characterizing those locally convex spaces with this property is still open. In this paper, spaces with compatible webs are used to construct both a necessary and a sufficient condition for a locally convex space to satisfy the Mackey convergence condition.
Coalescence between two convex liquid surfaces
Yang, Fan; Jian, Zhen; Li, Erqiang; Thoroddsen, S. T.
2015-11-01
We study the coalescence of two convex surfaces of the same liquid. One of the convex free surfaces is formed at a circular opening of a closed tank by imposing a negative pressure difference. The other surface is a droplet of larger curvature, which is pendant from a concentric nozzle. The coalescence starts from near-zero velocity, so the configuration can be characterized by two dimensionless numbers: the Ohnesorge number Oh = μ /√{ ργL } and the radius ratio between the two surfaces α =rd /rs . We use high-speed video, PIV and numerical simulations, using the Gerris program, to study the dynamics of the coalescence. Our focus is on the interface shapes, the growth-rate of the neck connecting the two surfaces and the formation of a vortex ring. The growth-rate is compared to earlier models for the coalescence of drops or bubbles.
Convex Modeling of Interactions with Strong Heredity
Haris, Asad; Witten, Daniela; Simon, Noah
2015-01-01
We consider the task of fitting a regression model involving interactions among a potentially large set of covariates, in which we wish to enforce strong heredity. We propose FAMILY, a very general framework for this task. Our proposal is a generalization of several existing methods, such as VANISH [Radchenko and James, 2010], hierNet [Bien et al., 2013], the all-pairs lasso, and the lasso using only main effects. It can be formulated as the solution to a convex optimization problem, which we solve using an efficient alternating directions method of multipliers (ADMM) algorithm. This algorithm has guaranteed convergence to the global optimum, can be easily specialized to any convex penalty function of interest, and allows for a straightforward extension to the setting of generalized linear models. We derive an unbiased estimator of the degrees of freedom of FAMILY, and explore its performance in a simulation study and on an HIV sequence data set. PMID:28316461
Convex Arrhenius plots and their interpretation
Truhlar, Donald G.; Kohen, Amnon
2001-01-01
This paper draws attention to selected experiments on enzyme-catalyzed reactions that show convex Arrhenius plots, which are very rare, and points out that Tolman's interpretation of the activation energy places a fundamental model-independent constraint on any detailed explanation of these reactions. The analysis presented here shows that in such systems, the rate coefficient as a function of energy is not just increasing more slowly than expected, it is actually decreasing. This interpretation of the data provides a constraint on proposed microscopic models, i.e., it requires that any successful model of a reaction with a convex Arrhenius plot should be consistent with the microcanonical rate coefficient being a decreasing function of energy. The implications and limitations of this analysis to interpreting enzyme mechanisms are discussed. This model-independent conclusion has broad applicability to all fields of kinetics, and we also draw attention to an analogy with diffusion in metastable fluids and glasses. PMID:11158559
On the convexity of Relativistic Hydrodynamics
Ibáñez, José María; Martí, José María; Miralles, Juan Antonio; 10.1088/0264-9381/30/5/057002
2013-01-01
The relativistic hydrodynamic system of equations for a perfect fluid obeying a causal equation of state is hyperbolic (Anile 1989 {\\it Relativistic Fluids and Magneto-Fluids} (Cambridge: Cambridge University Press)). In this report, we derive the conditions for this system to be convex in terms of the fundamental derivative of the equation of state (Menikoff and Plohr 1989 {\\it Rev. Mod. Phys.} {\\bf 61} 75). The classical limit is recovered.
Coefficient inequalities for starlikeness and convexity
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Ali Rosihan M.
2013-06-01
Full Text Available For an analytic function $f(z=z+\\sum_{n=2}^\\infty a_n z^n$ satisfying the inequality $\\sum_{n=2}^\\infty n(n-1|a_n|\\leq \\beta$, sharp bound on $\\beta$ is determined so that $f$ is either starlike or convex of order $\\alpha$. Several other coefficient inequalities related to certain subclasses are also investigated.
Tensor norms and operator ideals
Defant, A; Floret, K
1992-01-01
The three chapters of this book are entitled Basic Concepts, Tensor Norms, and Special Topics. The first may serve as part of an introductory course in Functional Analysis since it shows the powerful use of the projective and injective tensor norms, as well as the basics of the theory of operator ideals. The second chapter is the main part of the book: it presents the theory of tensor norms as designed by Grothendieck in the Resumé and deals with the relation between tensor norms and operator ideals. The last chapter deals with special questions. Each section is accompanied by a series of exer
When is multidimensional screening a convex program?
Figalli, Alessio; McCann, Robert J
2009-01-01
A principal wishes to transact business with a multidimensional distribution of agents whose preferences are known only in the aggregate. Assuming a twist (= generalized Spence-Mirrlees single-crossing) hypothesis and that agents can choose only pure strategies, we identify a structural condition on the preference b(x,y) of agent type x for product type y -- and on the principal's costs c(y) -- which is necessary and sufficient for reducing the profit maximization problem faced by the principal to a convex program. This is a key step toward making the principal's problem theoretically and computationally tractable; in particular, it allows us to derive uniqueness and stability of the principal's optimum strategy -- and similarly of the strategy maximizing the expected welfare of the agents when the principal's profitability is constrained. We call this condition non-negative cross-curvature: it is also (i) necessary and sufficient to guarantee convexity of the set of b-convex functions, (ii) invariant under r...
Logical, algebraic, analytic and probabilistic aspects of triangular norms
Klement, Erich Peter
2005-01-01
This volume gives a state of the art of triangular norms which can be used for the generalization of several mathematical concepts, such as conjunction, metric, measure, etc. 16 chapters written by leading experts provide a state of the art overview of theory and applications of triangular norms and related operators in fuzzy logic, measure theory, probability theory, and probabilistic metric spaces.Key Features:- Complete state of the art of the importance of triangular norms in various mathematical fields- 16 self-contained chapters with extensive bibliographies cover both the theoretical ba
Convexity at finite temperature and non-extensive thermodynamics
Alexandre, J.
2016-09-01
Assuming that tunnel effect between two degenerate bare minima occurs, in a scalar field theory at finite volume, this article studies the consequences for the effective potential, to all loop orders. Convexity is achieved only if the two bare minima are taken into account in the path integral, and a new derivation of the effective potential is given, in the large volume limit. The effective potential then has a universal form, it is suppressed by the space time volume, and does not feature spontaneous symmetry breaking as long as the volume is finite. The finite temperature analysis leads to surprising thermal properties, following from the non-extensive expression for the free energy. Although the physical relevance of these results is not clear, the potential application to ultra-light scalar particles is discussed.
Entropies from Coarse-graining: Convex Polytopes vs. Ellipsoids
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Nikos Kalogeropoulos
2015-09-01
Full Text Available We examine the Boltzmann/Gibbs/Shannon SBGS and the non-additive Havrda-Charvát/Daróczy/Cressie-Read/Tsallis Sq and the Kaniadakis κ-entropy Sκ from the viewpoint of coarse-graining, symplectic capacities and convexity. We argue that the functional form of such entropies can be ascribed to a discordance in phase-space coarse-graining between two generally different approaches: the Euclidean/Riemannian metric one that reflects independence and picks cubes as the fundamental cells in coarse-graining and the symplectic/canonical one that picks spheres/ellipsoids for this role. Our discussion is motivated by and confined to the behaviour of Hamiltonian systems of many degrees of freedom. We see that Dvoretzky’s theorem provides asymptotic estimates for the minimal dimension beyond which these two approaches are close to each other. We state and speculate about the role that dualities may play in this viewpoint.
Stochastic Integration in Abstract Spaces
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J. K. Brooks
2010-01-01
-valued process (∫ called the stochastic integral. The Lebesgue space of these integrable processes is studied and convergence theorems are given. Extensions to general locally convex spaces are presented.
Distributed Controllers for Norm Enforcement
Testerink, B.J.G.; Dastani, M.M.; Bulling, Nils
2016-01-01
This paper focuses on computational mechanisms that control the behavior of autonomous systems at runtime without necessarily restricting their autonomy. We build on existing approaches from runtime verification, control automata, and norm-based systems, and define norm-based controllers that
Directory of Open Access Journals (Sweden)
Madjid Mirzavaziri
2007-01-01
norms ‖⋅‖1 and ‖⋅‖2 on ℂn such that N(A=max{‖Ax‖2:‖x‖1=1, x∈ℂn} for all A∈ℳn. This may be regarded as an extension of a known result on characterization of minimal algebra norms.
Distributed Controllers for Norm Enforcement
Testerink, B.J.G.; Dastani, M.M.; Bulling, Nils
2016-01-01
This paper focuses on computational mechanisms that control the behavior of autonomous systems at runtime without necessarily restricting their autonomy. We build on existing approaches from runtime verification, control automata, and norm-based systems, and define norm-based controllers that enforc
Mutational analysis a joint framework for Cauchy problems in and beyond vector spaces
Lorenz, Thomas
2010-01-01
Ordinary differential equations play a central role in science and have been extended to evolution equations in Banach spaces. For many applications, however, it is difficult to specify a suitable normed vector space. Shapes without a priori restrictions, for example, do not have an obvious linear structure. This book generalizes ordinary differential equations beyond the borders of vector spaces with a focus on the well-posed Cauchy problem in finite time intervals. Here are some of the examples: - Feedback evolutions of compact subsets of the Euclidean space - Birth-and-growth processes of random sets (not necessarily convex) - Semilinear evolution equations - Nonlocal parabolic differential equations - Nonlinear transport equations for Radon measures - A structured population model - Stochastic differential equations with nonlocal sample dependence and how they can be coupled in systems immediately - due to the joint framework of Mutational Analysis. Finally, the book offers new tools for modelling.
Norms of certain Jordan elementary operators
Zhang, Xiaoli; Ji, Guoxing
2008-10-01
Let be a complex Hilbert space and let denote the algebra of all bounded linear operators on . For , the Jordan elementary operator UA,B is defined by UA,B(X)=AXB+BXA, . In this short note, we discuss the norm of UA,B. We show that if and ||UA,B||=||A||||B||, then either AB* or B*A is 0. We give some examples of Jordan elementary operators UA,B such that ||UA,B||=||A||||B|| but AB*[not equal to]0 and B*A[not equal to]0, which answer negatively a question posed by M. Boumazgour in [M. Boumazgour, Norm inequalities for sums of two basic elementary operators, J. Math. Anal. Appl. 342 (2008) 386-393].
Inductive limits and geometry of Banach spaces
Taskinen, Jari
1999-01-01
One of the main problems in the theory of inductive limits of Banach spaces is the projective description problem, finding a reasonable representation for the continuous seminorms. The problem is nontrivial even in the simplest cases. Recall that given, for example, an increasing sequence of Banach spaces (Yk)[infty infinity]k=1 with continuous embeddings Yk[hookrightarrow A: rt arrow-hooked]Yk+1 the inductive limit is the space Y=[cup B: union or logical sum]kYk endowed with the finest locally convex topology [tau] such that every embedding Yk[hookrightarrow A: rt arrow-hooked](Y, [tau]) becomes continuous. It is possible to give abstract definitions for families of continuous seminorms generating the topology [tau], but the connection with the norms of the step spaces Yk is not necessarily very close. For example, if the spaces Yk are Banach spaces of continuous functions endowed with weighted sup-norms, it is not clear if the continuous seminorms of the inductive limit are of the same type.We mention that inductive limits of spaces of continuous and holomorphic functions occur in many areas of analysis like linear partial differential operators, convolution equations [BD1], [E], complex and Fourier analysis and distribution theory. The projective description problem in these spaces has been thoroughly studied in [BMS1, BB1, BB2, BB3, BT, BM1, BM2], to mention some examples. We refer to the survey articles [BM1,BMS2, BB3]. The present work is also connected with the factorization problems which are treated in the book [Ju].
Convex functions and optimization methods on Riemannian manifolds
Udrişte, Constantin
1994-01-01
This unique monograph discusses the interaction between Riemannian geometry, convex programming, numerical analysis, dynamical systems and mathematical modelling. The book is the first account of the development of this subject as it emerged at the beginning of the 'seventies. A unified theory of convexity of functions, dynamical systems and optimization methods on Riemannian manifolds is also presented. Topics covered include geodesics and completeness of Riemannian manifolds, variations of the p-energy of a curve and Jacobi fields, convex programs on Riemannian manifolds, geometrical constructions of convex functions, flows and energies, applications of convexity, descent algorithms on Riemannian manifolds, TC and TP programs for calculations and plots, all allowing the user to explore and experiment interactively with real life problems in the language of Riemannian geometry. An appendix is devoted to convexity and completeness in Finsler manifolds. For students and researchers in such diverse fields as pu...
Asymptotic Performance of Sparse Signal Detection Using Convex Programming Method
Institute of Scientific and Technical Information of China (English)
LEI Chuan; ZHANG Jun
2012-01-01
The detection of sparse signals against background noise is considered.Detecting signals of such kind is difficult since only a small portion of the signal carries information.Prior knowledge is usually assumed to ease detection.In this paper,we consider the general unknown and arbitrary sparse signal detection problem when no prior knowledge is available.Under a Neyman-Pearson hypothesis-testing framework,a new detection scheme is proposed by combining a generalized likelihood ratio test (GLRT)-like test statistic and convex programming methods which directly exploit sparsity in an underdetermined system of linear equations.We characterize large sample behavior of the proposed method by analyzing its asymptotic performance.Specifically,we give the condition for the Chernoff-consistent detection which shows that the proposed method is very sensitive to the (e)2 norm energy of the sparse signals.Both the false alarm rate and the miss rate tend to zero at vanishing signal-to-noise ratio (SNR),as long as the signal energy grows at least logarithmically with the problem dimension.Next we give a large deviation analysis to characterize the error exponent for the Neyman-Pearson detection.We derive the oracle error exponent assuming signal knowledge.Then we explicitly derive the error exponent of the proposed scheme and compare it with the oracle exponent.We complement our study with numerical experiments,showing that the proposed method performs in the vicinity of the likelihood ratio test (LRT) method in the finite sample scenario and the error probability degrades exponentially with the number of observations.
Convex and Radially Concave Contoured Distributions
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Wolf-Dieter Richter
2015-01-01
Full Text Available Integral representations of the locally defined star-generalized surface content measures on star spheres are derived for boundary spheres of balls being convex or radially concave with respect to a fan in Rn. As a result, the general geometric measure representation of star-shaped probability distributions and the general stochastic representation of the corresponding random vectors allow additional specific interpretations in the two mentioned cases. Applications to estimating and testing hypotheses on scaling parameters are presented, and two-dimensional sample clouds are simulated.
Width Distributions for Convex Regular Polyhedra
Finch, Steven R
2011-01-01
The mean width is a measure on three-dimensional convex bodies that enjoys equal status with volume and surface area [Rota]. As the phrase suggests, it is the mean of a probability density f. We verify formulas for mean widths of the regular tetrahedron and the cube. Higher-order moments of f_tetra and f_cube have not been examined until now. Assume that each polyhedron has edges of unit length. We deduce that the mean square width of the regular tetrahedron is 1/3+(3+sqrt(3))/(3*pi) and the mean square width of the cube is 1+4/pi.
Trace-Inequalities and Matrix-Convex Functions
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Ando Tsuyoshi
2010-01-01
Full Text Available Abstract A real-valued continuous function on an interval gives rise to a map via functional calculus from the convex set of Hermitian matrices all of whose eigenvalues belong to the interval. Since the subpace of Hermitian matrices is provided with the order structure induced by the cone of positive semidefinite matrices, one can consider convexity of this map. We will characterize its convexity by the following trace-inequalities: for . A related topic will be also discussed.
Lower Bound for Convex Hull Area and Universal Cover Problems
Khandhawit, Tirasan; Sriswasdi, Sira
2011-01-01
In this paper, we provide a lower bound for an area of the convex hull of points and a rectangle in a plane. We then apply this estimate to establish a lower bound for a universal cover problem. We showed that a convex universal cover for a unit length curve has area at least 0.232239. In addition, we show that a convex universal cover for a unit closed curve has area at least 0.0879873.
Restricted strong convexity and weighted matrix completion: Optimal bounds with noise
Negahban, Sahand
2010-01-01
We consider the matrix completion problem under a form of row/column weighted entrywise sampling, including the case of uniform entrywise sampling as a special case. We analyze the associated random observation operator, and prove that with high probability, it satisfies a form of restricted strong convexity with respect to weighted Frobenius norm. Using this property, we obtain as corollaries a number of error bounds on matrix completion in the weighted Frobenius norm under noisy sampling and for both exact and near low-rank matrices. Our results are based on measures of the "spikiness" and "low-rankness" of matrices that are less restrictive than the incoherence conditions imposed in previous work. Our technique involves an $M$-estimator that includes controls on both the rank and spikiness of the solution, and we establish non-asymptotic error bounds in weighted Frobenius norm for recovering matrices lying with $\\ell_q$-"balls" of bounded spikiness. Using information-theoretic methods, we show that no algo...
Approximation of Convex Bodies by Convex Bodies%凸体间的逼近
Institute of Scientific and Technical Information of China (English)
国起; Sten Kaijser
2003-01-01
For the affine distance d(C,D)between two convex bodies C,D(∈)Rn,which reduces to the Banach-Mazur distance for symmetric convex bodies, the bounds of d(C,D)have been studied for many years. Some well known estimates for the upper-bounds are as follows: F. John proved d(C,D)≤n1/2 if one is an ellipsoid and another is symmetric,d(C,D)≤n if both are symmetric, and fromF. John's result and d(C1,C2)≤d(C1,C3)d(C2,C3) one has d(C,D)≤n2 for general convex bodies;M.Lassak proved d(C,D)≤(2n-1) if one of them is symmetric.In this paper we get an estimate which includes all the results above as special cases and refines some of them in terms of measures of asymmetry for convex bodies.
Stable anisotropic plasma confinement in magnetic configurations with convex-concave field lines
Tsventoukh, M. M.
2014-02-01
It is shown that a combination of the convex and the concave part of a field line provides a strong stabilizing action against convective (flute-interchange) plasma instability (Tsventoukh 2011 Nucl. Fusion 51 112002). This results in internal peaking of the stable plasma pressure profile that is calculated from the collisionless kinetic stability criterion for any magnetic confinement system with combination of mirrors and cusps. Connection of the convex and concave field line parts results in a reduction of the space charge that drives the unstable E × B motion, as there is an opposite direction of the particle drift in a non-uniform field at convex and concave field lines. The pressure peaking arises at the minimum of the second adiabatic invariant J that takes place at the ‘middle’ of a tandem mirror-cusp transverse cross-section. The position of the minimum in J varies with the particle pitch angle that results in a shift of the peaking position depending on plasma anisotropy. This allows one to improve a stable peaked pressure profile at a convex-concave field by changing the plasma anisotropy over the trap cross-section. Examples of such anisotropic distribution functions are found that give an additional substantial enhancement in the maximal central pressure. Furthermore, the shape of new calculated stable profiles has a wide central plasma layer instead of a narrow peak.
Error bound results for convex inequality systems via conjugate duality
Bot, Radu Ioan
2010-01-01
The aim of this paper is to implement some new techniques, based on conjugate duality in convex optimization, for proving the existence of global error bounds for convex inequality systems. We deal first of all with systems described via one convex inequality and extend the achieved results, by making use of a celebrated scalarization function, to convex inequality systems expressed by means of a general vector function. We also propose a second approach for guaranteeing the existence of global error bounds of the latter, which meanwhile sharpens the classical result of Robinson.
Institute of Scientific and Technical Information of China (English)
Shi Sheng ZHANG; Chi Kin CHAN; H.W. JOSEPH LEE
2012-01-01
The purpose of this paper is by using the modified block iterative method to propose an algorithm for finding a common element in the intersection of the set of common fixed points of an infinite family of quasi-φ-asymptotically nonexpansive and the set of solutions to an equilibrium problem and the set of solutions to a variational inequality.Under suitable conditions some strong convergence theorems are established in 2-uniformly convex and uniformly smooth Banach spaces.As applications we utilize the results presented in the paper to solving the convex feasibility problem (CFP) and zero point problem of maximal monotone mappings in Banach spaces.The results presented in the paper improve and extend the corresponding results announced by many authors.
Long Wave Dynamics along a Convex Bottom
Didenkulova, Ira; Soomere, Tarmo
2008-01-01
Long linear wave transformation in the basin of varying depth is studied for a case of a convex bottom profile in the framework of one-dimensional shallow water equation. The existence of travelling wave solutions in this geometry and the uniqueness of this wave class is established through construction of a 1:1 transformation of the general 1D wave equation to the analogous wave equation with constant coefficients. The general solution of the Cauchy problem consists of two travelling waves propagating in opposite directions. It is found that generally a zone of a weak current is formed between these two waves. Waves are reflected from the coastline so that their profile is inverted with respect to the calm water surface. Long wave runup on a beach with this profile is studied for sine pulse, KdV soliton and N-wave. Shown is that in certain cases the runup height along the convex profile is considerably larger than for beaches with a linear slope. The analysis of wave reflection from the bottom containing a s...
Molecular Graphics of Convex Body Fluids.
Gabriel, Adrian T; Meyer, Timm; Germano, Guido
2008-03-01
Coarse-grained modeling of molecular fluids is often based on nonspherical convex rigid bodies like ellipsoids or spherocylinders representing rodlike or platelike molecules or groups of atoms, with site-site interaction potentials depending both on the distance among the particles and the relative orientation. In this category of potentials, the Gay-Berne family has been studied most extensively. However, conventional molecular graphics programs are not designed to visualize such objects. Usually the basic units are atoms displayed as spheres or as vertices in a graph. Atomic aggregates can be highlighted through an increasing amount of stylized representations, e.g., Richardson ribbon diagrams for the secondary structure of proteins, Connolly molecular surfaces, density maps, etc., but ellipsoids and spherocylinders are generally missing, especially as elementary simulation units. We fill this gap providing and discussing a customized OpenGL-based program for the interactive, rendered representation of large ensembles of convex bodies, useful especially in liquid crystal research. We pay particular attention to the performance issues for typical system sizes in this field. The code is distributed as open source.
Naturally Occurring Radioactive Materials (NORM)
Energy Technology Data Exchange (ETDEWEB)
Gray, P. [ed.
1997-02-01
This paper discusses the broad problems presented by Naturally Occuring Radioactive Materials (NORM). Technologically Enhanced naturally occuring radioactive material includes any radionuclides whose physical, chemical, radiological properties or radionuclide concentration have been altered from their natural state. With regard to NORM in particular, radioactive contamination is radioactive material in an undesired location. This is a concern in a range of industries: petroleum; uranium mining; phosphorus and phosphates; fertilizers; fossil fuels; forestry products; water treatment; metal mining and processing; geothermal energy. The author discusses in more detail the problem in the petroleum industry, including the isotopes of concern, the hazards they present, the contamination which they cause, ways to dispose of contaminated materials, and regulatory issues. He points out there are three key programs to reduce legal exposure and problems due to these contaminants: waste minimization; NORM assesment (surveys); NORM compliance (training).
Consistency of trace norm minimization
Bach, Francis
2007-01-01
Regularization by the sum of singular values, also referred to as the trace norm, is a popular technique for estimating low rank rectangular matrices. In this paper, we extend some of the consistency results of the Lasso to provide necessary and sufficient conditions for rank consistency of trace norm minimization with the square loss. We also provide an adaptive version that is rank consistent even when the necessary condition for the non adaptive version is not fulfilled.
Civilsamfundets ABC: N for Norm
DEFF Research Database (Denmark)
Lund, Anker Brink; Meyer, Gitte
2016-01-01
Bogstaveligt talt: Hvad er civilsamfundet? Anker Brink Lund og Gitte Meyer fra CBS Center for Civil Society Studies gennemgår civilsamfundet bogstav for bogstav. Vi er nået til N for Norm.......Bogstaveligt talt: Hvad er civilsamfundet? Anker Brink Lund og Gitte Meyer fra CBS Center for Civil Society Studies gennemgår civilsamfundet bogstav for bogstav. Vi er nået til N for Norm....
Optimization in function spaces with stability considerations in Orlicz spaces
Kosmol, Peter
2011-01-01
This is an essentially self-contained book on the theory of convex functions and convex optimization in Banach spaces, with a special interest in Orlicz spaces. Approximate algorithms based on the stability principles and the solution of the corresponding nonlinear equationsaredeveloped in this text.A synopsis of the geometry of Banach spaces, aspects of stability and the duality of different levels of differentiability and convexity is developed. And it isprovided a novel approach to the fundamental theorems of Variational Calculus based on the principle of pointwise minimization of the Lagra
Making Norms to Tackle Global Challenges
DEFF Research Database (Denmark)
Nilsson, Adriana
2017-01-01
This paper argues that Intergovernmental Organisations (IGOs) can play a significant role in the processes of system transformation required by Grand Challenges. The reason is their potential to influence socio-technical regimes connected to policy areas in which they have authority. Supported...... public and private actors, IGOs implement and protect novel practices that reinforce the new norms, from legally binding agreements to the creation of new spaces for international collaboration. These processes are examined here in the field of global health, where outside pressure directed...
Directory of Open Access Journals (Sweden)
Steven G. Krantz
2010-01-01
Full Text Available We treat the classical concept of domain of holomorphy in ℂn when the holomorphic functions considered are restricted to lie in some Banach space. Positive and negative results are presented. A new view of the case n=1 is considered.
Directory of Open Access Journals (Sweden)
Siwaporn Saewan
2010-01-01
Full Text Available We introduce a modified block hybrid projection algorithm for solving the convex feasibility problems for an infinite family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings and the set of solutions of the generalized equilibrium problems. We obtain a strong convergence theorem for the sequences generated by this process in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in this paper improve and extend some recent results.
The uniqueness-of-norm problem for Calkin algebras
Skillicorn, Richard
2015-01-01
We examine the question of whether the Calkin algebra of a Banach space must have a unique complete algebra norm. We present a survey of known results, and make the observation that a recent Banach space construction of Argyros and Motakis (preprint, 2015) provides the first negative answer. The parallel question for the weak Calkin algebra also has a negative answer; we demonstrate this using a Banach space of Read (J. London Math. Soc. 1989).
Convexity-preserving Bernstein–Bézier quartic scheme
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Maria Hussain
2014-07-01
Full Text Available A C1 convex surface data interpolation scheme is presented to preserve the shape of scattered data arranged over a triangular grid. Bernstein–Bézier quartic function is used for interpolation. Lower bound of the boundary and inner Bézier ordinates is determined to guarantee convexity of surface. The developed scheme is flexible and involves more relaxed constraints.
Finding sets of points without empty convex 6-gons
Overmars, M.H.
2001-01-01
Erdös asked whether every large enough set of points in general position in the plane contains six points that form a convex 6-gon without any points from the set in its interior. In this note we show how a set of 29 points was found that contains no empty convex 6-gon. To this end a fast
Convergence of Algorithms for Reconstructing Convex Bodies and Directional Measures
DEFF Research Database (Denmark)
Gardner, Richard; Kiderlen, Markus; Milanfar, Peyman
2006-01-01
We investigate algorithms for reconstructing a convex body K in Rn from noisy measurements of its support function or its brightness function in k directions u1, . . . , uk. The key idea of these algorithms is to construct a convex polytope Pk whose support function (or brightness function) best ...
In-vivo Convex Array Vector Flow Imaging
DEFF Research Database (Denmark)
Jensen, Jørgen Arendt; Brandt, Andreas Hjelm; Nielsen, Michael Bachmann
2014-01-01
In-vivo VFI scans obtained from the abdomen of a human volunteer using a convex array transducers and trans- verse oscillation vector flow imaging (VFI) are presented. A 3 MHz BK Medical 8820e (Herlev, Denmark) 192-element convex array probe is used with the SARUS experimental ultrasound scanner....
Transverse-Mode Control of VCSELs With Convex Mirror
Institute of Scientific and Technical Information of China (English)
无
2003-01-01
We propose the transverse-mode control of vertical-cavity surface-emitting lasers (VCSELs) with a convex mirror. A possibility of improvements on single-mode output power and higher-order mode suppression is presented by optimizing a convex mirror.
Infinitesimal nonrigidity of convex surfaces with planar boundary
Institute of Scientific and Technical Information of China (English)
LI Chunhe; HONG Jiaxing
2005-01-01
In the present paper infinitesimal nonrigidity of a class of convex surfaces with planar boundary is given. This result shows that if the image of the Gauss map of an evolution convex surface with planar boundary covers some hemisphere, this surface may be of infinitesimal nonrigidity for the isometric deformation of planar boundary.
Homotopy Method for Non-convex Programming in Unbonded Set
Institute of Scientific and Technical Information of China (English)
徐庆; 于波
2005-01-01
In the past few years, much and much attention has been paid to the method for solving non-convex programming. Many convergence results are obtained for bounded sets. In this paper, we get global convergence results for non-convex programming in unbounded sets under suitable conditions.
(Average-) convexity of common pool and oligopoly TU-games
Driessen, T.S.H.; Meinhardt, H.
2000-01-01
The paper studies both the convexity and average-convexity properties for a particular class of cooperative TU-games called common pool games. The common pool situation involves a cost function as well as a (weakly decreasing) average joint production function. Firstly, it is shown that, if the rele
Simple sufficient conditions for starlikeness and convexity for meromorphic functions
Directory of Open Access Journals (Sweden)
Goswami Pranay
2016-01-01
Full Text Available In this paper we investigate some extensions of sufficient conditions for meromorphic multivalent functions in the open unit disk to be meromorphic multivalent starlike and convex of order α. Our results unify and extend some starlikeness and convexity conditions for meromorphic multivalent functions obtained by Xu et al. [2], and some interesting special cases are given.
PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming
Candes, Emmanuel J; Voroninski, Vladislav
2011-01-01
Suppose we wish to recover a signal x in C^n from m intensity measurements of the form ||^2, i = 1, 2,..., m; that is, from data in which phase information is missing. We prove that if the vectors z_i are sampled independently and uniformly at random on the unit sphere, then the signal x can be recovered exactly (up to a global phase factor) by solving a convenient semidefinite program---a trace-norm minimization problem; this holds with large probability provided that m is on the order of n log n, and without any assumption about the signal whatsoever. This novel result demonstrates that in some instances, the combinatorial phase retrieval problem can be solved by convex programming techniques. Finally, we also prove that our methodology is robust vis a vis additive noise.
Convex Set Theory for Reliability Assessment of Steel Beam with Bounded Uncertainty
Institute of Scientific and Technical Information of China (English)
Li-Zhe Jia; Yi-Ming Duan
2014-01-01
Probabilistic reliability model established by insufficient data is inaccessible. The convex model was applied to model the uncertainties of variables. A new non-probabilistic reliability model was proposed based on the robustness of system to uncertainty. The non-probabilistic reliability model, the infinite norm model, and the probabilistic model were used to assess the reliability of a steel beam, respectively. The results show that the resistance is allowed to couple with the action effect in the non-probabilistic reliability model. Additionally, the non-probabilistic reliability model becomes the same accurate as probabilistic model with the increase of the bounded uncertain information. The model is decided by the available data and information.
Waardes as norme en as meta-norme/beginsels
Directory of Open Access Journals (Sweden)
P. J. Van Niekerk
1987-03-01
Full Text Available The fruitfulness and the necessity of the distinction between values and norms and as principles are investigated by way of the theory of political development, the legal philosophical issue surrounding natural law and positivism and the views of Habermas. In political developmental theory questions centring on value gained legitimate relevancy under the influ ence of the post-behaviorist approach. The quest for cultural universalia or values as principles became important in this sphere because it seems to be the only way to escape from the syndrome of modernity. Through the rejection of the oppositions and one-sidedness of legal positivism and natural law and with the aid of the distinction between values, norms and principles the productive contribution of this spurious dilemma is high lighted and a clearer delineation is given of the concepts legal develop ment and structural violence. In conclusion Habermas's distinction between norms and meta-norms is investigated critically and immanent contradictions in his views are pointed out. The central place which this issue has in his thought can be seen as a confirmation of the importance of this distinction. It is relevant for all the normative disciplines which - in contrast to the natural sciences - focus on the role of linguistic, social, ethical, legal and artistic norms valid for human societies.
The inverse moment problem for convex polytopes
Gravin, Nick; Pasechnik, Dmitrii; Robins, Sinai
2011-01-01
The goal of this paper is to present a general and novel approach for the reconstruction of any convex d-dimensional polytope P, from knowledge of its moments. In particular, we show that the vertices of an N-vertex polytope in R^d can be reconstructed from the knowledge of O(DN) axial moments (w.r.t. to an unknown polynomial measure od degree D) in d+1 distinct generic directions. Our approach is based on the collection of moment formulas due to Brion, Lawrence, Khovanskii-Pukhikov, and Barvinok that arise in the discrete geometry of polytopes, and what variously known as Prony's method, or Vandermonde factorization of finite rank Hankel matrices.
DIFFERENTIAL SUBORDINATIONS AND α-CONVEX FUNCTIONS
Institute of Scientific and Technical Information of China (English)
Jacek DZIOK; Ravinder Krishna RAINA; Janusz SOK(O)L
2013-01-01
This article presents some new results on the class SLMα of functions that are analytic in the open unit discu ={z:[z[＜ 1} satisfying the conditions that f(0)=0,f'(0)=1,and α (1+ zf"(z)/f'(z)) + (1-α)zf'(z)/f(x) ∈(p)(u)for all z ∈ u,where α is a real number and (p)(z) =1 + τ2z2/ 1-τz-τ2z2 (z ∈ u).The number τ =(1-√5)/2 is such that τ2 =1 + T.The class SLMα introduced by J.Dziok,R.K.Raina,and J.Sokól [3,Appl.Math.Comput.218 (2011),996-1002] is closely related to the classes of starlike and convex functions.The article deals with several ideas and techniques used in geometric function theory and differential subordinations theory.
Quantification of small, convex particles by TEM
Energy Technology Data Exchange (ETDEWEB)
Andersen, Sigmund J. [SINTEF Materials and Chemistry, Department of Synthesis and Properties, Material Physics, NO-7465 Trondheim (Norway)], E-mail: sigmund.j.andersen@sintef.no; Holme, Borge [SINTEF Materials and Chemistry, P.O. Box 124, Blindern, NO-0314 Oslo (Norway); Marioara, Calin D. [SINTEF Materials and Chemistry, Department of Synthesis and Properties, Material Physics, NO-7465 Trondheim (Norway)
2008-07-15
It is shown how size distributions of arbitrarily oriented, convex, non-overlapping particles extracted from conventional transmission electron microscopy (TEM) images may be determined by a variation of the Schwartz-Saltykov method. In TEM, particles cut at the surfaces have diminished projections, which alter the observed size distribution. We represent this distribution as a vector and multiply it with the inverse of a matrix comprising thickness-dependent Scheil or Schwartz-Saltykov terms. The result is a corrected size distribution of the projections of uncut particles. It is shown how the real (3D) distribution may be estimated when particle shape is considered. Computer code to generate the matrix is given. A log-normal distribution of spheres and a real distribution of pill-box-shaped dispersoids in an Al-Mg-Si alloy are given as examples. The errors are discussed in detail.
Weighted composition operators and locally convex algebras
Institute of Scientific and Technical Information of China (English)
Edoardo Vesentini
2005-01-01
The Gleason-Kahane-Zelazko theorem characterizes the continuous homomorphism of an associative, locally multiplicatively convex, sequentially complete algebra A into the field C among all linear forms on A. This characterization will be applied along two different directions. In the case in which A is a commutative Banach algebra, the theorem yields the representation of some classes of continuous linear maps A: A → A as weighted composition operators, or as composition operators when A is a continuous algebra endomorphism. The theorem will then be applied to explore the behaviour of continuous linear forms on quasi-regular elements, when A is either the algebra of all Hilbert-Schmidt operators or a Hilbert algebra.
Convex Decomposition Based Cluster Labeling Method for Support Vector Clustering
Institute of Scientific and Technical Information of China (English)
Yuan Ping; Ying-Jie Tian; Ya-Jian Zhou; Yi-Xian Yang
2012-01-01
Support vector clustering (SVC) is an important boundary-based clustering algorithm in multiple applications for its capability of handling arbitrary cluster shapes. However,SVC's popularity is degraded by its highly intensive time complexity and poor label performance.To overcome such problems,we present a novel efficient and robust convex decomposition based cluster labeling (CDCL) method based on the topological property of dataset.The CDCL decomposes the implicit cluster into convex hulls and each one is comprised by a subset of support vectors (SVs).According to a robust algorithm applied in the nearest neighboring convex hulls,the adjacency matrix of convex hulls is built up for finding the connected components; and the remaining data points would be assigned the label of the nearest convex hull appropriately.The approach's validation is guaranteed by geometric proofs.Time complexity analysis and comparative experiments suggest that CDCL improves both the efficiency and clustering quality significantly.
ANALYSIS TO NEYMAN-PEARSON CLASSIFICATION WITH CONVEX LOSS FUNCTION
Institute of Scientific and Technical Information of China (English)
Min Han; Dirong Chen; Zhaoxu Sun
2008-01-01
Neyman-Pearson classification has been studied in several articles before.But they all proceeded in the classes of indicator functions with indicator function as the loss function,which make the calculation to be difficult.This paper investigates NeymanPearson classification with convex loss function in the arbitrary class of real measurable functions.A general condition is given under which Neyman-Pearson classification with convex loss function has the same classifier as that with indicator loss function.We give analysis to NP-ERM with convex loss function and prove it's performance guarantees.An example of complexity penalty pair about convex loss function risk in terms of Rademacher averages is studied,which produces a tight PAC bound of the NP-ERM with convex loss function.
Introducing convex layers to the Traveling Salesman Problem
Liew, Sing
2012-01-01
In this paper, we will propose convex layers to the Traveling Salesman Problem (TSP). Firstly, we will focus on human performance on the TSP. Experimental data shows that untrained humans appear to have the ability to perform well in the TSP. On the other hand, experimental data also supports the hypothesis of convex hull i.e. human relies on convex hull to search for the optimal tour for the TSP. Secondly, from the paper published by Bonabeau, Dorigo and Theraulaz, social insect behavior would be able to help in some of the optimizing problems, especially the TSP. Thus, we propose convex layers to the TSP based on the argument that, by the analogy to the social insect behavior, untrained humans' cognition should be able to help in the TSP. Lastly, we will use Tour Improvement algorithms on convex layers to search for an optimal tour for a 13-cities problem to demonstrate the idea.