Illustrating the Central Limit Theorem
Corcoran, Mimi
2016-01-01
Statistics is enjoying some well-deserved limelight across mathematics curricula of late. Some statistical concepts, however, are not especially intuitive, and students struggle to comprehend and apply them. As an AP Statistics teacher, the author appreciates the central limit theorem as a foundational concept that plays a crucial role in…
A Randomized Central Limit Theorem
Eliazar, Iddo; Klafter, Joseph
2010-05-01
The Central Limit Theorem (CLT), one of the most elemental pillars of Probability Theory and Statistical Physics, asserts that: the universal probability law of large aggregates of independent and identically distributed random summands with zero mean and finite variance, scaled by the square root of the aggregate-size (√{n}), is Gaussian. The scaling scheme of the CLT is deterministic and uniform - scaling all aggregate-summands by the common and deterministic factor √{n}. This Letter considers scaling schemes which are stochastic and non-uniform, and presents a "Randomized Central Limit Theorem" (RCLT): we establish a class of random scaling schemes which yields universal probability laws of large aggregates of independent and identically distributed random summands. The RCLT universal probability laws, in turn, are the one-sided and the symmetric Lévy laws.
A Randomized Central Limit Theorem
The Central Limit Theorem (CLT), one of the most elemental pillars of Probability Theory and Statistical Physics, asserts that: the universal probability law of large aggregates of independent and identically distributed random summands with zero mean and finite variance, scaled by the square root of the aggregate-size (√(n)), is Gaussian. The scaling scheme of the CLT is deterministic and uniform - scaling all aggregate-summands by the common and deterministic factor √(n). This Letter considers scaling schemes which are stochastic and non-uniform, and presents a 'Randomized Central Limit Theorem' (RCLT): we establish a class of random scaling schemes which yields universal probability laws of large aggregates of independent and identically distributed random summands. The RCLT universal probability laws, in turn, are the one-sided and the symmetric Levy laws.
Visualizing the Central Limit Theorem through Simulation
Ruggieri, Eric
2016-01-01
The Central Limit Theorem is one of the most important concepts taught in an introductory statistics course, however, it may be the least understood by students. Sure, students can plug numbers into a formula and solve problems, but conceptually, do they really understand what the Central Limit Theorem is saying? This paper describes a simulation…
Central Limit Theorem for Coloured Hard Dimers
Maria Simonetta Bernabei
2010-01-01
Full Text Available We study the central limit theorem for a class of coloured graphs. This means that we investigate the limit behavior of certain random variables whose values are combinatorial parameters associated to these graphs. The techniques used at arriving this result comprise combinatorics, generating functions, and conditional expectations.
Central Limit Theorem for Nonlinear Hawkes Processes
Zhu, Lingjiong
2012-01-01
Hawkes process is a self-exciting point process with clustering effect whose jump rate depends on its entire past history. It has wide applications in neuroscience, finance and many other fields. Linear Hawkes process has an immigration-birth representation and can be computed more or less explicitly. It has been extensively studied in the past and the limit theorems are well understood. On the contrary, nonlinear Hawkes process lacks the immigration-birth representation and is much harder to analyze. In this paper, we obtain a functional central limit theorem for nonlinear Hawkes process.
A Central Limit Theorem for Punctuated Equilibrium
Bartoszek, Krzysztof
2016-01-01
Current evolutionary biology models usually assume that a phenotype undergoes gradual change. This is in stark contrast to biological intuition, which indicates that change can also be punctuated - the phenotype can jump. Such a jump can especially occur at speciation, i.e. dramatic change occurs that drives the species apart. Here we derive a central limit theorem for punctuated equilibrium. We show that, if adaptation is fast, for weak convergence to hold, dramatic change has to be a rare e...
Central limit theorem and deformed exponentials
The central limit theorem (CLT) can be ranked among the most important ones in probability theory and statistics and plays an essential role in several basic and applied disciplines, notably in statistical thermodynamics. We show that there exists a natural extension of the CLT from exponentials to so-called deformed exponentials (also denoted as q-Gaussians). Our proposal applies exactly in the usual conditions in which the classical CLT is used. (fast track communication)
Sticky central limit theorems on open books
Hotz, Thomas; Le, Huiling; Marron, J Stephen; Mattingly, Jonathan C; Miller, Ezra; Nolen, James; Owen, Megan; Patrangenaru, Vic; Skwerer, Sean
2012-01-01
Given a probability distribution on an open book (a metric space obtained by gluing a disjoint union of copies of a half-space along their boundary hyperplanes), we define a precise concept of when the Fr\\'echet mean (barycenter) is "sticky". This non-classical phenomenon is quantified by a law of large numbers (LLN) stating that the empirical mean eventually almost surely lies on the (codimension 1 and hence measure 0) "spine" that is the glued hyperplane, and a central limit theorem (CLT) stating that the limiting distribution is Gaussian and supported on the spine. We also state versions of the LLN and CLT for the cases where the mean is nonsticky (that is, not lying on the spine) and partly sticky (that is, on the spine but not sticky).
A central limit theorem for a new statistic on permutations
Chatterjee, Sourav; Diaconis, Persi
2016-01-01
This paper does three things: It proves a central limit theorem for a novel permutation statistic, the number of descents plus the number of descents in the inverse. It provides a clear illustration of a new approach to proving central limit theorems more generally. It gives us an opportunity to acknowledge the work of our teacher and friend B. V. Rao.
Central limit theorem for reducible and irreducible open quantum walks
Sadowski, Przemysław; Pawela, Łukasz
2016-04-01
In this work we aim at proving central limit theorems for open quantum walks on {{Z}}^d . We study the case when there are various classes of vertices in the network. In particular, we investigate two ways of distributing the vertex classes in the network. First, we assign the classes in a regular pattern. Secondly, we assign each vertex a random class with a transition invariant distribution. For each way of distributing vertex classes, we obtain an appropriate central limit theorem, illustrated by numerical examples. These theorems may have application in the study of complex systems in quantum biology and dissipative quantum computation.
Central limit theorem for reducible and irreducible open quantum walks
Sadowski, Przemysław; Pawela, Łukasz
2016-07-01
In this work we aim at proving central limit theorems for open quantum walks on {mathbb {Z}}^d. We study the case when there are various classes of vertices in the network. In particular, we investigate two ways of distributing the vertex classes in the network. First, we assign the classes in a regular pattern. Secondly, we assign each vertex a random class with a transition invariant distribution. For each way of distributing vertex classes, we obtain an appropriate central limit theorem, illustrated by numerical examples. These theorems may have application in the study of complex systems in quantum biology and dissipative quantum computation.
Central limit theorem for Fourier transform of stationary processes
Peligrad, Magda
2009-01-01
We consider asymptotic behavior of Fourier transforms of stationary ergodic sequences with finite second moments. We establish the central limit theorem (CLT) for almost all frequencies and also the annealed CLT. The theorems hold for all regular sequences. Our results shed new light on the foundation of spectral analysis and on the asymptotic distribution of periodogram, and it provides a nice blend of harmonic analysis, theory of stationary processes and theory of martingales.
Improving Conceptions in Analytical Chemistry: The Central Limit Theorem
Rodriguez-Lopez, Margarita; Carrasquillo, Arnaldo, Jr.
2006-01-01
This article describes the central limit theorem (CLT) and its relation to analytical chemistry. The pedagogic rational, which argues for teaching the CLT in the analytical chemistry classroom, is discussed. Some analytical chemistry concepts that could be improved through an understanding of the CLT are also described. (Contains 2 figures.)
Central Limit Theorems for Multicolor Urns with Dominated Colors
Berti, Patrizia; Pratelli, Luca; Rigo, Pietro
2009-01-01
An urn contains balls of d colors. At each time, a ball is drawn and then replaced together with a random number of balls of the same color. Assuming that some colors are dominated by others, we prove central limit theorems. Some statistical applications are discussed.
The Central Limit Theorem for the Smoluchovski Coagulation Model
Kolokoltsov, Vassili
2007-01-01
The general model of coagulation is considered. For basic classes of unbounded coagulation kernels the central limit theorem (CLT) is obtained for the fluctuations around the dynamic law of large numbers (LLN). A rather precise rate of convergence is given both for LLN and CLT.
Almost sure central limit theorems for random functions
LU Chuanrong; QIU Jin; XU Jianjun
2006-01-01
Let {Xn,-∞＜ n ＜∞} be a sequence of independent identically distributed be a random function such that Tn = ASn+ Rn,where supnE|Rn|＜∞ and Rn = o(√n)a.s.,or Rn = O(n1/2-2γ) a.s.,0 ＜γ＜ 1/8.In this paper,we prove the almost sure central limit theorem (ASCLT) and the function-typed almost sure central limit theorem (FASCLT) for the random function Tn.As a consequence,it can be shown that ASCLT and FASCLT also hold for U-statistics,Von-Mises statistics,linear processes,moving average processes,error variance estimates in linear models,power sums,product-limit estimators of a continuous distribution,product-limit estimators of a quantile function,etc.
Central limit theorem of linear regression model under right censorship
HE; Shuyuan(何书元); HUANG; Xiang(Heung; Wong)(黄香)
2003-01-01
In this paper, the estimation of joint distribution F(y,z) of (Y, Z) and the estimation in thelinear regression model Y = b′Z + ε for complete data are extended to that of the right censored data. Theregression parameter estimates of b and the variance of ε are weighted least square estimates with randomweights. The central limit theorems of the estimators are obtained under very weak conditions and the derivedasymptotic variance has a very simple form.
Functional central limit theorems for single-stage samplings designs
Boistard, Hélène; Lopuhaä, Hendrik P.; Ruiz-Gazen, Anne
2015-01-01
For a joint model-based and design-based inference, we establish functional central limit theorems for the Horvitz-Thompson empirical process and the H\\'ajek empirical process centered by their finite population mean as well as by their super-population mean in a survey sampling framework. The results apply to single-stage unequal probability sampling designs and essentially only require conditions on higher order correlations. We apply our main results to a Hadamard differentiable statistica...
FAST TRACK COMMUNICATION: Central limit theorem and deformed exponentials
Vignat, C.; Plastino, A.
2007-11-01
The central limit theorem (CLT) can be ranked among the most important ones in probability theory and statistics and plays an essential role in several basic and applied disciplines, notably in statistical thermodynamics. We show that there exists a natural extension of the CLT from exponentials to so-called deformed exponentials (also denoted as q-Gaussians). Our proposal applies exactly in the usual conditions in which the classical CLT is used.
Two-parameter Non-commutative Central Limit Theorem
Blitvić, Natasha
2012-01-01
The non-commutative Central Limit Theorem (CLT) introduced by Speicher in 1992 states that given almost any sequence of non-commutative random variables that commute or anti-commute pair-wise, the *-moments of the normalized partial sum S_N=(b_1+...+ b_N)/\\sqrt{N} are given by a Wick-type formula refined to count the number of crossings in the underlying pair-partitions. When coupled with explicit matrix models, the theorem yields random matrix models for creation and annihilation operators on the q-Fock space of Bozejko and Speicher. In this paper, we derive a non-commutative CLT when the pair-wise commutation coefficients are real numbers (as opposed to signs). The statistics of the limiting random variable are a second-parameter refinement of those above, jointly indexing the number of crossings and nestings in the underlying pair-partitions. Coupled with analogous matrix constructions, the theorem yields random matrix models for creation and annihilation operators on the recently introduced (q,t)-Fock spa...
Continuous-variable entanglement distillation and noncommutative central limit theorems
Campbell, Earl T.; Genoni, Marco G.; Eisert, Jens
2013-04-01
Entanglement distillation transforms weakly entangled noisy states into highly entangled states, a primitive to be used in quantum repeater schemes and other protocols designed for quantum communication and key distribution. In this work, we present a comprehensive framework for continuous-variable entanglement distillation schemes that convert noisy non-Gaussian states into Gaussian ones in many iterations of the protocol. Instances of these protocols include (a) the recursive-Gaussifier protocol, (b) the temporally reordered recursive-Gaussifier protocol, and (c) the pumping-Gaussifier protocol. The flexibility of these protocols gives rise to several beneficial trade-offs related to success probabilities or memory requirements, which can be adjusted to reflect experimental demands. Despite these protocols involving measurements, we relate the convergence in this protocol to new instances of noncommutative central limit theorems, in a formalism that we lay out in great detail. Implications of the findings for quantum repeater schemes are discussed.
Central limit theorem behavior in the skew tent map
In this paper we study and establish central limit theorem behavior in the skew (generalized) tent map transformation T: Y →Y originally considered by Billings and Bollt [Billings L, Bollt EM. Probability density functions of some skew tent maps. Chaos, Solitons and Fractals 2001; 12: 365-376] and Ito et al. [Ito S, Tanaka S, Nakada H. On unimodal linear transformations and chaos. II. Tokyo J Math 1979; 2: 241-59]. When the measure ν is invariant under T, the transfer operator PT:L1(ν)→L1(ν) governing the evolution of densities f under the action of the skew tent map, as well as the unique stationary density, are given explicitly for specific transformation parameters. Then, using this development, we solve the Poisson equation f=PTf+φ for two specific integrable observables φ and explicitly calculate the variance σ(φ)2=∫Yφ2(y)ν(dy)
Central limit theorem for products of toral automorphisms
Conze, Jean-Pierre; Roger, Mikaël
2010-01-01
Let $(\\tau_n)$ be a sequence of toral automorphisms $\\tau_n : x \\rightarrow A_n x \\hbox{mod}\\ZZ^d$ with $A_n \\in {\\cal A}$, where ${\\cal A}$ is a finite set of matrices in $SL(d, \\mathbb{Z})$. Under some conditions the method of "multiplicative systems" of Koml\\`os can be used to prove a Central Limit Theorem for the sums $\\sum_{k=1}^n f(\\tau_k \\circ \\tau_{k-1} \\cdots \\circ \\tau_1 x)$ if $f$ is a H\\"older function on $\\mathbb{T}^d$. These conditions hold for $2\\times 2$ matrices with positive coefficients. In dimension $d$ they can be applied when $A_n= A_n(\\omega)$, with independent choices of $A_n(\\omega)$ in a finite set of matrices $\\in SL(d, \\mathbb{Z})$, in order to prove a "quenched" CLT.
An almost Sure Central Limit Theorem for the Weight Function Sequences of NA Random Variables
Qunying Wu
2011-08-01
Consider the weight function sequences of NA random variables. This paper proves that the almost sure central limit theorem holds for the weight function sequences of NA random variables. Our results generalize and improve those on the almost sure central limit theorem previously obtained from the i.i.d. case to NA sequences.
A multivariate central limit theorem for continuous local martingales
Zanten, van, M.
1998-01-01
A theorem on the weak convergence of a properly normalized multivariate continuous local martingale is proved. The time-change theorem used for this purpose allows for short and transparent arguments.
A GUE Central Limit Theorem and Universality of Directed First and Last Passage Site Percolation
Baik, Jinho; Suidan, Toufic M.
2004-01-01
We prove a GUE central limit theorem for random variables with finite fourth moment. We apply this theorem to prove that the directed first and last passage percolation problems in thin rectangles exhibit universal fluctuations given by the Tracy-Widom law.
Vitesse dans le theoreme limite central pour certains processus stationnaires fortement decorreles
Borgne, Stephane Le; Pene, Francoise
2003-01-01
We prove a central limit theorem with speed $n^{-1/2}$ for stationary processes satisfying a strong decorrelation hypothesis. The proof is a modification of the proof of a theorem of Rio. It is elementary but quite long and technical.
Central limit theorems for multivariate semi-Markov sequences and processes, with applications
Ball, Frank
1999-01-01
In this paper, central limit theorems for multivariate semi-Markov sequences and processes are obtained, both as the number of jumps of the associated Markov chain tends to infinity and, if appropriate, as the time for which the process has been running tends to infinity. The theorems are widely applicable since many functions defined on Markov or semi-Markov processes can be analysed by exploiting appropriate embedded multivariate semi-Markov sequences. An application to a ...
Central limit theorem for the Banach-valued weakly dependent random variables
The central limit theorem (CLT) for the Banach-valued weakly dependent random variables is proved. In proving CLT convergence of finite-measured (i.e. cylindrical) distributions is established. A weak compactness of the family of measures generated by a certain sequence is confirmed. The continuity of the limiting field is checked
Central limit theorems for smoothed extreme value estimates of Poisson point processes boundaries
Girard, Stéphane; Menneteau, Ludovic
2011-01-01
In this paper, we give sufficient conditions to establish central limit theorems for boundary estimates of Poisson point processes. The considered estimates are obtained by smoothing some bias corrected extreme values of the point process. We show how the smoothing leads Gaussian asymptotic distributions and therefore pointwise confidence intervals. Some new unidimensional and multidimensional examples are provided.
Central limit theorems for smoothed extreme value estimates of point processes boundaries
Girard, Stéphane; Menneteau, Ludovic
2005-01-01
In this paper, we give sufficient conditions to establish central limit theorems for boundary estimates of Poisson point processes. The considered estimates are obtained by smoothing some bias corrected extreme values of the point process. We show how the smoothing leads Gaussian asymptotic distributions and therefore pointwise confidence intervals. Some new unidimensional and multidimensional examples are provided.
The Power of Doing: A Learning Exercise That Brings the Central Limit Theorem to Life
Price, Barbara A.; Zhang, Xiaolong
2007-01-01
This article demonstrates an active learning technique for teaching the Central Limit Theorem (CLT) in an introductory undergraduate business statistics class. Groups of students carry out one of two experiments in the lab, tossing a die in sets of 5 rolls or tossing a die in sets of 10 rolls. They are asked to calculate the sample average of each…
Local Central Limit Theorem for diffusions in a degenerate and unbounded Random Medium
Chiarini, Alberto; Deuschel, Jean-Dominique
2015-01-01
We study a symmetric diffusion $X$ on $\\mathbb{R}^d$ in divergence form in a stationary and ergodic environment, with measurable unbounded and degenerate coefficients. We prove a quenched local central limit theorem for $X$, under some moment conditions on the environment; the key tool is a local parabolic Harnack inequality obtained with Moser iteration technique.
A Functional Central Limit Theorem for a Class of Urn Models
Gopal K Basak; Amites Dasgupta
2005-11-01
We construct an independent increments Gaussian process associated to a class of multicolor urn models. The construction uses random variables from the urn model which are different from the random variables for which central limit theorems are available in the two color case.
Central Limit Theorems for a Class of Irreducible Multicolor Urn Models
Gopal K Basak; Amites Dasgupta
2007-11-01
We take a unified approach to central limit theorems for a class of irreducible multicolor urn models with constant replacement matrix. Depending on the eigenvalue, we consider appropriate linear combinations of the number of balls of different colors. Then under appropriate norming the multivariate distribution of the weak limits of these linear combinations is obtained and independence and dependence issues are investigated. Our approach consists of looking at the problem from the viewpoint of recursive equations.
Central limit theorems for directional and linear random variables with applications
García-Portugués, Eduardo; Crujeiras, Rosa M.; González-Manteiga, Wenceslao
2014-01-01
A central limit theorem for the integrated squared error of the directional-linear kernel density estimator is established. The result enables the construction and analysis of two testing procedures based on squared loss: a nonparametric independence test for directional and linear random variables and a goodness-of-fit test for parametric families of directional-linear densities. Limit distributions for both test statistics, and a consistent bootstrap strategy for the goodness-of-fit test, a...
Central limit theorems for the excursion sets volumes of weakly dependent random fields
Bulinski, Alexander; Timmermann, Florian
2010-01-01
The multivariate central limit theorems (CLT) for the volumes of excursion sets of stationary quasi-associated random fields on $\\mathbb{R}^d$ are proved. Special attention is paid to Gaussian and shot noise fields. Formulae for the covariance matrix of the limiting distribution are provided. Statistical versions of the CLT are considered as well. They employ three different estimators of the asymptotic covariance matrix. Some numerical results are also discussed.
A Central Limit Theorem for Convolution Equations and Weakly Self-Avoiding Walks
Bolthausen, Erwin; Ritzmann, Christine
2001-01-01
The main result of this paper is a general central limit theorem for distributions defined by certain renewal type equations. We apply this to weakly self-avoiding random walks. We give good error estimates and Gaussian tail estimates which have not been obtained by other methods. We use the lace expansion and at the same time develop a new perspective on this method: We work with a fixed point argument directly in the x-space without using Laplace or Fourier transformation.
On the Effect of Random Norming on the Rate of Convergence in the Central Limit Theorem
Hall, Peter
1988-01-01
It is shown that "studentizing," i.e., normalizing by the sample standard deviation rather than the population standard deviation, can improve the rate of convergence in the central limit theorem. This provides concise confirmation of one feature of the folklore that a studentized sum is in some sense more robust than a normed sum. The case of infinite population standard deviation is also examined.
Central limit theorem for first-passage percolation time across thin cylinders
Chatterjee, Sourav
2009-01-01
We prove that first-passage percolation times across thin cylinders of the form $[0,n]\\times [-h_n,h_n]^{d-1}$ obey Gaussian central limit theorems as long as $h_n$ grows slower than $n^{1/(d+1)}$. It is an open question as to what is the fastest that $h_n$ can grow so that a Gaussian CLT still holds. We conjecture that $n^{1/(d+1)}$ is the right answer for $d\\ge 2$.
Zipf's law is not a consequence of the central limit theorem
Troll, G.; Beim Graben, P.
1998-02-01
It has been observed that the rank statistics of string frequencies of many symbolic systems (e.g., word frequencies of natural languages) follows Zipf's law in good approximation. We show that, contrary to claims in the literature, Zipf's law cannot be realized by the central limit theorem(s). The observation that a log-normal distribution of string frequencies yields an approximately Zipf-like rank statistics is actually misleading. Indeed, Zipf's law for the rank statistics is strictly equivalent to a power law distribution of frequencies. There are two natural ways to perform the infinite size limit for the vocabulary. The first one is the method of choice in the literature; it makes the upper word length bound tend to infinity and leads in the case of a multistate Bernoulli process via a central limit theorem to a log-normal frequency distribution. An alternative and for text samples actually better realizable way is to make the lower frequency bound tend to zero. This limit procedure leads to a power law distribution and hence to Zipf's law-at least for Bernoulli processes and to a very good approximation for natural languages where it passes the χ2 test. For the Bernoulli case we will give a heuristic proof.
Bodnar, Taras; Mazur, Stepan; Parolya, Nestor
2016-01-01
In this paper we consider the asymptotic distributions of functionals of the sample covariance matrix and the sample mean vector obtained under the assumption that the matrix of observations has a matrix variate general skew normal distribution. The central limit theorem is derived for the product of the sample covariance matrix and the sample mean vector. Moreover, we consider the product of an inverse covariance matrix and the mean vector for which the central limit theorem is established a...
Taming systematic uncertainties at the LHC with the central limit theorem
Fichet, Sylvain
2016-01-01
We study the simplifications occurring in any likelihood function in the presence of a large number of small systematic uncertainties. We find that the marginalisation of these uncertainties can be done analytically by means of second-order error propagation, error combination, the Lyapunov central limit theorem, and under mild approximations which are typically satisfied for LHC likelihoods. The outcomes of this analysis are i) a very light treatment of systematic uncertainties ii) a convenient way of reporting the main effects of systematic uncertainties such as the detector effects occuring in LHC measurements.
A central limit theorem for the determinant of a Wigner matrix
Tao, Terence; Vu, Van
2011-01-01
We establish a central limit theorem for the log-determinant $\\log|\\det(M_n)|$ of a Wigner matrix $M_n$, under the assumption of four matching moments with either the GUE or GOE ensemble. More specifically, we show that this log-determinant is asymptotically distributed like $N(\\log \\sqrt{n!} - 1/2 \\log n, 1/2 \\log n)_\\R$ when one matches moments with GUE, and $N(\\log \\sqrt{n!} - 1/4 \\log n, 1/4 \\log n)_\\R$ when one matches moments with GOE.
Juan Carlos Aquino
2013-06-01
Full Text Available The application of different unit root statistics is by now a standard practice in empirical work. Even when it is a practical issue, these statistics have complex nonstandard distributions depending on functionals of certain stochastic processes, and their derivations represent a barrier even for many theoretical econometricians. These derivations are based on rigorous and fundamental statistical tools which are not (very well known by standard econometricians. This paper aims to fill this gap by explaining in a simple way one of these fundamental tools: namely, the Functional Central Limit Theorem. To this end, this paper analyzes the foundations and applicability of two versions of the Functional Central Limit Theorem within the framework of a unit root with a structural break. Initial attention is focused on the probabilistic structure of the time series to be considered. Thereafter, attention is focused on the asymptotic theory for nonstationary time series proposed by Phillips (1987a, which is applied by Perron (1989 to study the effects of an (assumed exogenous structural break on the power of the augmented Dickey-Fuller test and by Zivot and Andrews (1992 to criticize the exogeneity assumption and propose a method for estimating an endogenous breakpoint. A systematic method for dealing with efficiency issues is introduced by Perron and Rodriguez (2003, which extends the Generalized Least Squares detrending approach due to Elliot et al. (1996. An empirical application is provided.
Corrections to the Central Limit Theorem for Heavy-Tailed Probability Densities
Lam, Henry; Burch, Damian; Bazant, Martin Z
2011-01-01
Classical Edgeworth expansions provide asymptotic correction terms to the Central Limit Theorem (CLT) up to an order that depends on the number of moments available. In this paper, we provide subsequent correction terms beyond those given by a standard Edgeworth expansion in the general case of regularly varying distributions with diverging moments (beyond the second). The subsequent terms can be expressed in a simple closed form in terms of certain special functions (Dawson's integral and parabolic cylinder functions), and there are qualitative differences depending on whether the number of moments available is even, odd or not an integer, and whether the distributions are symmetric or not. If the increments have an even number of moments, then additional logarithmic corrections must also be incorporated in the expansion parameter. An interesting feature of our correction terms for the CLT is that they become dominant outside the central region and blend naturally with known large-deviation asymptotics when ...
Sanov and central limit theorems for output statistics of quantum Markov chains
In this paper, we consider the statistics of repeated measurements on the output of a quantum Markov chain. We establish a large deviations result analogous to Sanov’s theorem for the multi-site empirical measure associated to finite sequences of consecutive outcomes of a classical stochastic process. Our result relies on the construction of an extended quantum transition operator (which keeps track of previous outcomes) in terms of which we compute moment generating functions, and whose spectral radius is related to the large deviations rate function. As a corollary to this, we obtain a central limit theorem for the empirical measure. Such higher level statistics may be used to uncover critical behaviour such as dynamical phase transitions, which are not captured by lower level statistics such as the sample mean. As a step in this direction, we give an example of a finite system whose level-1 (empirical mean) rate function is independent of a model parameter while the level-2 (empirical measure) rate is not
A necessary moment condition for the fractional functional central limit theorem
Johansen, Søren; Nielsen, Morten Ørregaard
2012-01-01
innovation sequence. When d is close to -1/2 this moment condition is very strong. Our main result is to show that when -1/2
Central limit theorem for a class of globally correlated random variables
Budini, Adrián A.
2016-06-01
The standard central limit theorem with a Gaussian attractor for the sum of independent random variables may lose its validity in the presence of strong correlations between the added random contributions. Here, we study this problem for similar interchangeable globally correlated random variables. Under these conditions, a hierarchical set of equations is derived for the conditional transition probabilities. This result allows us to define different classes of memory mechanisms that depend on a symmetric way on all involved variables. Depending on the correlation mechanisms and statistics of the single variables, the corresponding sums are characterized by distinct probability densities. For a class of urn models it is also possible to characterize their domain of attraction, which, as in the standard case, is parametrized by the probability density of each random variable. Symmetric and asymmetric q -Gaussian attractors (q <1 ) arise in a particular two-state case of these urn models.
A central-limit theorem for a single-false match rate
Dietz, Zachariah; Schuckers, Michael E.
2010-04-01
In this paper, we present a central limit theorem (CLT) for the estimation of a false match rate for a single matching system. The false match rate is often a significant factor in an evaluation of such a matching system. To achieve the main result here we utilize the covariance/correlation structure for matching proposed by Schuckers. Along with the main result we present an illustration of the methodology here on biometric authentication data from Ross and Jain. This illustration is from resampling match decisions on three different biometric modalities: hand geometry, fingerprint and facial recognition and shows that as the number of matching pairs grows the sampling distribution for an FMR approaches a Gaussian distribution. These results suggest that statistical inference for a FMR based upon a Gaussian distribution is appropriate.
WANG YUEBAO; YANG YANG; ZHOU HAIYANG
2003-01-01
A random functional central limit theorem is obtained for processes of partial sums andproduct sums of linear processes generated by non-stationary martingale differences. It devel-ops and improves some corresponding results on processes of partial sums of linear processesgenerated by strictly stationary martingale differences, which can be found in [5].
PENG ShiGe
2009-01-01
This is a survey on normal distributions and the related central limit theorem under sublinear expectation. We also present Brownian motion under sublinear expectations and the related stochastic calculus of Ito's type. The results provide new and robust tools for the problem of probability model uncertainty arising in financial risk, statistics and other industrial problems.
2009-01-01
This is a survey on normal distributions and the related central limit theorem under sublinear expectation.We also present Brownian motion under sublinear expectations and the related stochastic calculus of It?’s type.The results provide new and robust tools for the problem of probability model uncertainty arising in financial risk,statistics and other industrial problems.
A necessary moment condition for the fractional functional central limit theorem
Johansen, Søren; Nielsen, Morten Ørregaard
the innovation sequence. When d is close to -1/2 this moment condition is very strong. Our main result is to show that under some relatively weak conditions on u_{t}, the existence of q≥max(2,(d+1/2)^{-1}) is in fact necessary for the FCLT for fractionally integrated processes and that q>max(2,(d+1......We discuss the moment condition for the fractional functional central limit theorem (FCLT) for partial sums of x_{t}=Delta^{-d}u_{t}, where d in (-1/2,1/2) is the fractional integration parameter and u_{t} is weakly dependent. The classical condition is existence of q>max(2,(d+1/2)^{-1}) moments of....../2)^{-1}) moments are necessary and sufficient for more general fractional processes. Davidson and de Jong (2000) presented a fractional FCLT where only q>2 finite moments are assumed, which is remarkable because it is the only FCLT where the moment condition has been weakened relative to the earlier condition. As...
A Necessary Moment Condition for the Fractional Functional Central Limit Theorem
Johansen, Søren; Nielsen, Morten Ørregaard
innovation sequence. When d is close to -1/2 this moment condition is very strong. Our main result is to show that under some relatively weak conditions on u(t), the existence of q=max(2,(d+1/2)-¹) is in fact necessary for the FCLT for fractionally integrated processes and that q>max(2,(d+1/2)-¹) moments are......We discuss the moment condition for the fractional functional central limit theorem (FCLT) for partial sums of x(t)=¿^(-d)u(t), where d ¿ (-1/2,1/2) is the fractional integration parameter and u(t) is weakly dependent. The classical condition is existence of q>max(2,(d+1/2)-¹) moments of the...... necessary and sufficient for more general fractional processes. Davidson and de Jong (2000) presented a fractional FCLT where only q>2 finite moments are assumed, which is remarkable because it is the only FCLT where the moment condition has been weakened relative to the earlier condition. As a corollary to...
Martingale approximation and optimality of some conditions for the central limit theorem
Volný, Dalibor
2009-01-01
Let $(X_i)$ be a stationary and ergodic Markov chain with kernel $Q$, $f$ an $L^2$ function on its state space. If $Q$ is a normal operator and $f = (I-Q)^{1/2}g$ (which is equivalent to the convergence of $\\sum_{n=1}^\\infty \\frac{\\sum_{k=0}^{n-1}Q^kf}{n^{3/2}}$ in $L^2$), we have the central limit theorem (cf\\. \\cite{D-L 1}, \\cite{G-L 2}). Without assuming normality of $Q$, the CLT is implied by the convergence of $\\sum_{n=1}^\\infty \\frac{\\|\\sum_{k=0}^{n-1}Q^kf\\|_2}{n^{3/2}}$, in particular by $\\|\\sum_{k=0}^{n-1}Q^kf\\|_2 = o(\\sqrt n/\\log^q n)$, $q>1$ by \\cite{M-Wu} and \\cite{Wu-Wo} respectively. We shall show that if $Q$ is not normal and $f\\in (I-Q)^{1/2} L^2$, or if the conditions of Maxwell and Woodroofe or of Wu and Woodroofe are weakened to $\\sum_{n=1}^\\infty c_n\\frac{\\|\\sum_{k=0}^{n-1}Q^kf\\|_2}{n^{3/2}}<\\infty$ for some sequence $c_n\\searrow 0$, or by $\\|\\sum_{k=0}^{n-1}Q^kf\\|_2 = O(\\sqrt n/\\log n)$, the CLT need not hold.
Assessment of the statistics of the Strehl ratio: predictions of central limit theorem analysis
Tyler, Glenn A.
2006-11-01
For a beam propagating through turbulence, the statistics of the Strehl ratio are determined by recognizing that the real and imaginary parts of the on-axis far-field pattern can be represented as the sum of many contributions from the aperture. With this in mind, the central limit theorem (CLT) can be used to develop the statistics of the real and imaginary parts of the optical field, which through the appropriate mathematical manipulations as described here can then be used to develop the probability distribution of the far-field irradiance. The results obtained in this way (which we call the CLT theory or analysis) provide an analytic expression that agrees with the results of detailed wave-optics simulations. This provides an approach by which the statistics of the Strehl ratio can be rapidly determined. A key feature of this work is that the analytic results depend on the values of a few relevant turbulence parameters that include r0,fG, and σ2l. Therefore, a measurement of these parameters at various sites of interest allows us to rapidly assess the detailed nature of the statistical fluctuations of the far-field irradiance that will be experienced at these locations.
Central Limit Theorem for Partial Linear Eigenvalue Statistics of Wigner Matrices
Bao, Zhigang; Pan, Guangming; Zhou, Wang
2013-01-01
In this paper, we study the complex Wigner matrices Mn=1/sqrt{n}Wn whose eigenvalues are typically in the interval [-2,2]. Let λ 1≤ λ 2⋯≤ λ n be the ordered eigenvalues of M n . Under the assumption of four matching moments with the Gaussian Unitary Ensemble (GUE), for test function f 4-times continuously differentiable on an open interval including [-2,2], we establish central limit theorems for two types of partial linear statistics of the eigenvalues. The first type is defined with a threshold u in the bulk of the Wigner semicircle law as mathcal{A}n[f; u]=sum_{l=1}nf(λl)mathbf{1}_{\\{λl≤ u\\}}. And the second one is mathcal{B}n[f; k]=sum_{l=1}kf(λl) with positive integer k= k n such that k/ n→ y∈(0,1) as n tends to infinity. Moreover, we derive a weak convergence result for a partial sum process constructed from mathcal{B}n[f; lfloor ntrfloor]. The main difficulty is to deal with the linear eigenvalue statistics for the test functions with several non-differentiable points. And our main strategy is to combine the Helffer-Sjöstrand formula and a comparison procedure on the resolvents to extend the results from GUE case to general Wigner matrices case. Moreover, the results on mathcal{A}n[f;u] for the real Wigner matrices will also be briefly discussed.
M. Gharib
1996-09-01
Full Text Available In this paper a uniform estimate is obtained for the remainder term in the central limit theorem (CLT for a sequence of random vectors forming a homogeneous Markov chain with arbitrary set of states. The result makes it possible to estimate the rate of convergence in the CLT without assuming the finiteness of the absolute third moment of the transition probabilities. Some consequences are also proved.
Occupancy of phase space, extensivity of Sq, and q-generalized central limit theorem
Tsallis, Constantino
2006-06-01
Increasing the number N of elements of a system typically makes the entropy to increase. The question arises on what particular entropic form we have in mind and how it increases with N. Thermodynamically speaking it makes sense to choose an entropy which increases linearly with N for large N, i.e., which is extensive. If the N elements are probabilistically independent (no interactions) or quasi-independent (e.g., short-range interacting), it is known that the entropy which is extensive is that of Boltzmann-Gibbs-Shannon, SBG≡-k∑i=1Wpilnpi. If they are, however, globally correlated (e.g., through long-range interactions), the answer depends on the particular nature of the correlations. There is a large class of correlations (in one way or another related to scale-invariance) for which an appropriate entropy is that on which nonextensive statistical mechanics is based, i.e., Sq≡k(1-∑i=1Wpiq)/q-1 ( S1=SBG), where q is determined by the specific correlations. We briefly review and illustrate these ideas through simple examples of occupation of phase space. A very similar scenario emerges with regard to the central limit theorem (CLT). If the variables that are being summed are independent (or quasi-independent, in the sense that they gradually become independent if N→∞), two basic possibilities exist: if the variance of the random variables that are being composed is finite, the N→∞ attractor in the space of distributions is a Gaussian, whereas if it diverges, it is a Lévy distribution. If the variables that are being summed are however globally correlated, there is no reason to expect the usual CLTs to hold. The N→∞ attractor is expected to depend on the nature of the correlations. That class of correlations (or part of it) that makes Sq to be extensive for q≠1 is expected to have a qe-Gaussian as its N→∞ attractor, where qe depends on q [ qe(q) such that qe(1)=1], and where qe-Gaussians are proportional to [1-(1-qe)β x2] ( β>0; qe<3
Moen, David H.; Powell, John E.
2008-01-01
Using Microsoft® Excel, several interactive, computerized learning modules are developed to illustrate the Central Limit Theorem's appropriateness for comparing the difference between the means of any two populations. These modules are used in the classroom to enhance the comprehension of this theorem as well as the concepts that provide the…
Limit theorems for 2D invasion percolation
Damron, Michael
2010-01-01
We prove limit theorems and variance estimates for quantities related to ponds and outlets for 2D invasion percolation. We first exhibit several properties of a sequence (O(n)) of outlet variables, the n-th of which gives the number of outlets in the box centered at the origin of side length 2^n. The most important of these properties describe the sequence's renewal structure and exponentially fast mixing behavior. We use these to prove a central limit theorem and strong law of large numbers for (O(n)). We then show consequences of these limit theorems for the pond radii and outlet weights.
In this Letter we analyse the behaviour of the probability density function of the sum of N deterministic variables generated from the triangle map of Casati-Prosen. For the case in which the map is both ergodic and mixing the resulting probability density function quickly concurs with the Normal distribution. However, when the map is weakly chaotic, and fuzzily not mixing, the resulting probability density functions are described by power-laws. Moreover, contrarily to what it would be expected, as the number of added variables N increases the distance to Gaussian distribution increases. This behaviour goes against standard central limit theorem. By extrapolation of our finite size results we preview that in the limit of N going to infinity the distribution has the same asymptotic decay as a Lorentzian (or a q=2-Gaussian).
Stable convergence and stable limit theorems
Häusler, Erich
2015-01-01
The authors present a concise but complete exposition of the mathematical theory of stable convergence and give various applications in different areas of probability theory and mathematical statistics to illustrate the usefulness of this concept. Stable convergence holds in many limit theorems of probability theory and statistics – such as the classical central limit theorem – which are usually formulated in terms of convergence in distribution. Originated by Alfred Rényi, the notion of stable convergence is stronger than the classical weak convergence of probability measures. A variety of methods is described which can be used to establish this stronger stable convergence in many limit theorems which were originally formulated only in terms of weak convergence. Naturally, these stronger limit theorems have new and stronger consequences which should not be missed by neglecting the notion of stable convergence. The presentation will be accessible to researchers and advanced students at the master's level...
Limit Theorems for Dispersing Billiards with Cusps
Bálint, P.; Chernov, N.; Dolgopyat, D.
2011-12-01
Dispersing billiards with cusps are deterministic dynamical systems with a mild degree of chaos, exhibiting "intermittent" behavior that alternates between regular and chaotic patterns. Their statistical properties are therefore weak and delicate. They are characterized by a slow (power-law) decay of correlations, and as a result the classical central limit theorem fails. We prove that a non-classical central limit theorem holds, with a scaling factor of {sqrt{nlog n}} replacing the standard {sqrt{n}} . We also derive the respective Weak Invariance Principle, and we identify the class of observables for which the classical CLT still holds.
Central limit theorem for integrated square error of kernel estimators of spherical density
ZHAO; Lincheng
2001-01-01
［1］ Cruzeiro, A. B., Malliavin, P., Renormalized differential geometry on path spaces: Structural equation, curvature, J. Funct. Anal., 1996, 139: 119-181.［2］ Stroock, D. W., Some thoughts about Riemannian structures on path spaces, preprint, 1996.［3］ Driver, B., A Cameron-Martin type quasi-invariance theorem for Brownian motion on a compact manifold, J. Funct. Anal., 1992, 109: 272-376.［4］ Enchev, O., Stroock, D. W., Towards a Riemannian geometry on the path space over a Riemannian manifold, J. Funct. Anal., 1995, 134: 392-416.［5］ Hsu, E., Quasi-invariance of the Wiener measure on the path space over a compact Riemannian manifold, J. Funct. Anal., 1995, 134: 417-450.［6］ Lyons, T. J., Qian, Z. M., A class of vector fields on path space, J.Funct. Anal., 1997, 145: 205-223.［7］ Li, X. D., Existence and uniqueness of geodesics on path spaces, J. Funct. Anal., to be published.［8］ Driver, B., Towards calculus and geometry on path spaces, in Proc. Symp. Pure and Appl. Math. 57 (ed. Cranston, M., Pinsky, M.), Cornell: AMS, 1993, 1995.
See, Lai-Chu; Huang, Yu-Hsun; Chang, Yi-Hu; Chiu, Yeo-Ju; Chen, Yi-Fen; Napper, Vicki S.
2010-01-01
This study examines the timing using computer-enriched instruction (CEI), before or after a traditional lecture to determine cross-over effect, period effect, and learning effect arising from sequencing of instruction. A 2 x 2 cross-over design was used with CEI to teach central limit theorem (CLT). Two sequences of graduate students in nursing…
Central Limit Theorems for Cavity and Local Fields of the Sherrington-Kirkpatrick Model
Chen, Wei-Kuo
2010-01-01
One of the remarkable applications of the cavity method is to prove the Thouless-Anderson-Palmer (TAP) system of equations in the high temperature analysis of the Sherrington-Kirkpatrick (SK) model. This naturally leads us to the important study of the limit laws for cavity and local fields. The first quantitative results for both fields based on Stein's method were studied by Chatterjee. Although Stein's method provides us an efficient search for the limiting distributions, the nature of this method in some way restricts the exploration for optimal and general results. In this paper, our study based on Gaussian interpolation obtains the CLT for cavity fields. With the help of this result, we conclude the CLT for local fields. In both cases, better quantitative results are given.
Kelson, Daniel D
2014-01-01
Star-formation rates (SFR) of disk galaxies strongly correlate with stellar mass, with a small dispersion in SSFR at fixed mass, sigma~0.3 dex. With such small scatter this star-formation main sequence (SFMS) has been interpreted as deterministic and fundamental. Here we demonstrate that it is a simple consequence of the central limit theorem. Our derivation begins by approximating in situ stellar mass growth as a stochastic process, much like a random walk (where the expectation of SFR at any time is equal to the SFR at the previous time). We then derive expectation values for median SSFR of star-forming disks and their scatter over time. We generalize the results for stochastic changes in SFR that are not independent of each other but are correlated over time. For unbiased samples of (disk) galaxies, we derive an expectation that should be independent of mass, decline as 1/T, and have a relative scatter that is independent of mass and time. The derived SFMS and its evolution matches published data to z=10 ...
Limit theorems for sequences of random trees
Balding, David; Ferrari, Pablo A.; Fraiman, Ricardo; Sued, Mariela
2004-01-01
We consider a random tree and introduce a metric in the space of trees to define the ``mean tree'' as the tree minimizing the average distance to the random tree. When the resulting metric space is compact we have laws of large numbers and central limit theorems for sequence of independent identically distributed random trees. As application we propose tests to check if two samples of random trees have the same law.
Pluchino, Alessandro; Rapisarda, Andrea; Tsallis, Constantino
2008-05-01
We give a closer look at the Central Limit Theorem (CLT) behavior in quasi-stationary states of the Hamiltonian Mean Field model, a paradigmatic one for long-range-interacting classical many-body systems. We present new calculations which show that, following their time evolution, we can observe and classify three kinds of long-standing quasi-stationary states (QSS) with different correlations. The frequency of occurrence of each class depends on the size of the system. The different microscopic nature of the QSS leads to different dynamical correlations and therefore to different results for the observed CLT behavior.
Limit Theorems in Free Probability Theory I
Chistyakov, G. P.; Götze, F.
2006-01-01
Based on a new analytical approach to the definition of additive free convolution on probability measures on the real line we prove free analogs of limit theorems for sums for non-identically distributed random variables in classical Probability Theory.
Functional limit theorems for generalized variations of the fractional Brownian sheet
Pakkanen, Mikko; Réveillac, Anthony
2016-01-01
We prove functional central and non-central limit theorems for generalized variations of the anisotropic d-parameter fractional Brownian sheet (fBs) for any natural number d. Whether the central or the non-central limit theorem applies depends on the Hermite rank of the variation functional and o...
Limit theorems for fragmentation processes with immigration
Knobloch, Robert
2012-01-01
In this paper we extend two limit theorems which were recently obtained for fragmentation processes to such processes with immigration. More precisely, in the setting with immigration we consider a limit theorem for the process counted with a random characteristic as well as the asymptotic behaviour of an empirical measure associated with the stopping line corresponding to the first blocks, in their respective line of descent, that are smaller than a given size. In addition, we determine the asymptotic decay rate of the size of the largest block in a homogeneous fragmentation process with immigration. The techniques used to proves these results are based on submartingale arguments.
Limit theorems for bifurcating integer-valued autoregressive processes
Blandin, Vassili
2012-01-01
We study the asymptotic behavior of the weighted least squares estimators of the unknown parameters of bifurcating integer-valued autoregressive processes. Under suitable assumptions on the immigration, we establish the almost sure convergence of our estimators, together with the quadratic strong law and central limit theorems. All our investigation relies on asymptotic results for vector-valued martingales.
Limit theorems for Markov random fields
Markov Random Fields (MRF's) have been extensively applied in Statistical Mechanics as well as in Bayesian Image Analysis. MRF's are a special class of dependent random variables located at the vertices of a graph whose joint distribution includes a parameter called the temperature. When the number of vertices of the graph tends to infinity, the normalized distribution of statistics based on these random variables converge in distribution. It can happen that for certain values of the temperature, that the rate of growth of these normalizing constants change drastically. This feature is generally used to explain the phenomenon of phase transition as understood by physicist. In this dissertation the author will show that this drastic change in normalizing constants occurs even in the relatively smooth case when all the random variables are Gaussian. Hence any image analytic MRF ought to be checked for such discontinuous behavior before any analysis is performed. Mixed limit theorems in Bayesian Image Analysis seek to replace intensive simulations of MRF's with limit theorems that approximate the distribution of the MRF's as the number of sites increases. The problem of deriving mixed limit theorems for MRF's on a one dimensional lattice graph with an acceptor function that has a second moment has been studied by Chow. A mixed limit theorem for the integer lattice graph is derived when the acceptor function does not have a second moment as for instance when the acceptor function is a symmetric stable density of index 0 < α < 2
Limit Theorems for the Sample Entropy of Hidden Markov Chains
Han, Guangyue
2011-01-01
The Shannon-McMillan-Breiman theorem asserts that the sample entropy of a stationary and ergodic stochastic process converges to the entropy rate of the same process almost surely. In this paper, we focus our attention on the convergence behavior of the sample entropy of a hidden Markov chain. Under certain positivity assumption, we prove that a central limit theorem (CLT) with some Berry-Esseen bound for the sample entropy of a hidden Markov chain, and we use this CLT to establish a law of iterated logarithm (LIL) for the sample entropy.
Limit theorems for self-similar tilings
Bufetov, Alexander I
2012-01-01
We study deviation of ergodic averages for dynamical systems given by self-similar tilings on the plane and in higher dimensions. The main object of our paper is a special family of finitely-additive measures for our systems. An asymptotic formula is given for ergodic integrals in terms of these finitely-additive measures, and, as a corollary, limit theorems are obtained for dynamical systems given by self-similar tilings.
Limit theorems for extremes with random sample size
Silvestrov, Dmitrii S.; Teugels, Jozef L.
1998-01-01
This paper is devoted to the investigation of limit theorems for extremes with random sample size under general dependence-independence conditions for samples and random sample size indexes. Limit theorems of weak convergence type are obtained as well as functional limit theorems for extremal processes with random sample size indexes.
Goedel incompleteness theorems and the limits of their applicability. I
This is a survey of results related to the Goedel incompleteness theorems and the limits of their applicability. The first part of the paper discusses Goedel's own formulations along with modern strengthenings of the first incompleteness theorem. Various forms and proofs of this theorem are compared. Incompleteness results related to algorithmic problems and mathematically natural examples of unprovable statements are discussed. Bibliography: 68 titles.
大数定律及中心极限定理在保险中的应用%Applications of Law of Large Numbers and Central Limit Theorem in Insurance
王丙参; 魏艳华; 林朱
2011-01-01
文中研究了大数定律及中心极限定理的含义及关系，阐述了它们在制定保费及自留额、拟定保险单位数及减少保险个人平均危险值等方面的应用．%It discussed the meanings and relationships of the law of large numbers and Central Limit Theorem, stud- ied the applications in the formulating premium and retention, the insurance units, reducing the average individual risk values, etc.
Reflexivity and the diagonal argument in proofs of limitative theorems
Młynarski, Kajetan
2011-01-01
This paper discusses limitations of reflexive and diagonal arguments as methods of proof of limitative theorems (e.g. G\\"odel's theorem on Entscheidungsproblem, Turing's halting problem or Chaitin-G\\"odel's theorem). The fact, that a formal system contains a sentence, which introduces reflexitivity, does not imply, that the same system does not contain a sentence or a proof procedure which solves this problem. Second basic method of proof - diagonal argument (i.e. showing non-eqiunumerosity o...
Goedel incompleteness theorems and the limits of their applicability. I
Beklemishev, Lev D [Steklov Mathematical Institute, Russian Academy of Sciences, Moscow (Russian Federation)
2011-01-25
This is a survey of results related to the Goedel incompleteness theorems and the limits of their applicability. The first part of the paper discusses Goedel's own formulations along with modern strengthenings of the first incompleteness theorem. Various forms and proofs of this theorem are compared. Incompleteness results related to algorithmic problems and mathematically natural examples of unprovable statements are discussed. Bibliography: 68 titles.
Some scaled limit theorems for an immigration super-Brownian motion
2008-01-01
In this paper,the small time limit behaviors for an immigration super-Brownian motion are studied,where the immigration is determined by Lebesgue measure.We first prove a functional central limit theorem,and then study the large and moderate deviations associated with this central tendency.
Some Limit Theorems for Negatively Associated Random Variables
Yu Miao; Wenfei Xu; Shanshan Chen; Andre Adler
2014-08-01
Let $\\{X_n,n≥ 1\\}$ be a sequence of negatively associated random variables. The aim of this paper is to establish some limit theorems of negatively associated sequence, which include the $L^p$-convergence theorem and Marcinkiewicz–Zygmund strong law of large numbers. Furthermore, we consider the strong law of sums of order statistics, which are sampled from negatively associated random variables.
The Energy Transformation Limit Theorem for Gas Flow Systems
Volov, V T
2011-01-01
The limit energy theorem which determines the possibility of transformation the energy flow in power systems in the absence of technical work is investigated and proved for such systems as gas lasers and plasmatrons, chemical gas reactors, vortex tubes, gas-acoustic and other systems, as well as a system of close stars. In the case of the same name ideal gas in the system the maximum ratio of energy conversion effectiveness is linked to the Carnot theorem, which in its turn is connected with the Nernst theorem. However, numerical analyses show that the class of flow energy systems is non-carnot one. The ratio of energy conversion effectiveness depends on the properties of the working medium; a conventional cycle in open-circuit is essentially irreversible. The proved theorem gives a more strongly worded II law of thermodynamics for the selected class of flow energy systems. Implications for astrophysical thermodynamic systems and the theory of a strong shock wave are discussed.
Noncentral limit theorem and the bootstrap for quantiles of dependent data
Sharipov, Olimjon S.; Wendler, Martin
2012-01-01
We will show under minimal conditions on differentiability and dependence that the central limit theorem for quantiles holds and that the block bootstrap is weakly consistent. Under slightly stronger conditions, the bootstrap is strongly consistent. Without the differentiability condition, quantiles might have a non-normal asymptotic distribution and the bootstrap might fail.
Limit theorems and inequalities via martingale methods
Chazottes Jean-René
2014-01-01
Full Text Available In these notes, we first give a brief overwiew of martingales methods, from Paul Lévy (1935 untill now, to explain why these methods have become a central tool in probability, statistics and ergodic theory. Next, we present some recent results for/or based on martingales: exponential bounds for super-martingales, concentration inequalities for Lipschitz functionals of dynamical systems, oracle inequalities for the Cox model in a high dimensional setting, and invariance principles for stationary sequences.
Riedler, Martin G; Wainrib, Gilles
2011-01-01
In the present paper we present limit theorems for a sequence of Piecewise Deterministic Markov Processes taking values in Hilbert spaces. This class of processes provides a rigorous framework for stochastic spatial models in which discrete random events are globally coupled with continuous space-dependent variables solving partial differential equations, e.g., stochastic hybrid models of excitable membranes. We derive a law of large numbers which establishes a connection to deterministic macroscopic models and a martingale central limit theorem which connects the stochastic fluctuations to diffusion processes. As a prerequisite we carry out a thorough discussion of Hilbert space valued martingales associated to the PDMPs. Furthermore, these limit theorems provide the basis for a general Langevin approximation to PDMPs, i.e., certain stochastic partial differential equations that are expected to be similar in their dynamics to PDMPs. We apply these results to compartmental-type models of spatially extended ne...
A Trotter-Kato theorem for quantum Markov limits
Bouten, Luc [BTA VOF, Beringe (Netherlands); Gohm, Rolf; Gough, John [Aberystwyth University, Dept. for Mathematics and Physics, Wales (United Kingdom); Nurdin, Hendra [UNSW Australia, School of Electrical Engineering and Telecommunications, Sydney, NSW (Australia)
2015-04-30
Using the Trotter-Kato theorem we prove the convergence of the unitary dynamics generated by an increasingly singular Hamiltonian in the case of a single field coupling. The limit dynamics is a quantum stochastic evolution of Hudson-Parthasarathy type, and we establish in the process a graph limit convergence of the pre-limit Hamiltonian operators to the Chebotarev-Gregoratti-von Waldenfels Hamiltonian generating the quantum Ito evolution. (orig.)
Strong Limit Theorems for Arbitrary Fuzzy Stochastic Sequences
FEI Wei-yin
2008-01-01
Based on fuzzy random variables, the concept of fuzzy stochastic sequences is defined. Strong limit theorems for fuzzy stochastic sequences are established. Some known results in non-fuzzy stochastic sequences are extended. In order to prove results of this paper, the notion of fuzzy martingale difference sequences is also introduced.
The Energy Transformation Limit Theorem for Gas Flow Systems
Volov, V. T.
2011-01-01
The limit energy theorem which determines the possibility of transformation the energy flow in power systems in the absence of technical work is investigated and proved for such systems as gas lasers and plasmatrons, chemical gas reactors, vortex tubes, gas-acoustic and other systems, as well as a system of close stars. In the case of the same name ideal gas in the system the maximum ratio of energy conversion effectiveness is linked to the Carnot theorem, which in its turn is connected with ...
Transportation Distance and the Central Limit Theorem
Ekisheva, S.; Houdré, C.
2006-01-01
For probability measures on a complete separable metric space, we present sufficient conditions for the existence of a solution to the Kantorovich transportation problem. We also obtain sufficient conditions (which sometimes also become necessary) for the convergence, in transportation, of probability measures when the cost function is continuous, non-decreasing and depends on the distance. As an application, the CLT in the transportation distance is proved for independent and some dependent ...
Uniform Central Limit Theorems for Multidimensional Diffusions
Rohde, Angelika
2010-01-01
It has recently been shown that there are grave differences in the regularity behavior of the empirical process based on scalar diffusions as compared to the classical empirical process, due to the existence of diffusion local time. Besides establishing strong parallels to classical theory such as Ossiander's bracketing CLT and the general Gin\\'e-Zinn CLT for uniformly bounded families of functions, we find increased regularity also for multivariate ergodic diffusions, assuming that the invariant measure is finite with Lebesgue density $\\pi$. The effect is diminishing for growing dimension but always present. The fine differences to the classical iid setting are worked out using exponential inequalities for martingales and additive functionals of continuous Markov processes as well as the characterization of the sample path behavior of Gaussian processes by means of the generic chaining bound. To uncover the phenomenon, we study a smoothed version of the empirical diffusion process. It turns out that uniform ...
Fluctuation limit theorems for age-dependent critical binary branching systems
Murillo-Salas Antonio
2011-03-01
Full Text Available We consider an age-dependent branching particle system in ℝd, where the particles are subject to α-stable migration (0 < α ≤ 2, critical binary branching, and general (non-arithmetic lifetimes distribution. The population starts off from a Poisson random field in ℝd with Lebesgue intensity. We prove functional central limit theorems and strong laws of large numbers under two rescalings: high particle density, and a space-time rescaling that preserves the migration distribution. Properties of the limit processes such as Markov property, almost sure continuity of paths and generalized Langevin equation, are also investigated.
Limit theorems for stationary increments Lévy driven moving averages
Basse-O'Connor, Andreas; Lachièze-Rey, Raphaël; Podolskij, Mark
kernel function g at 0. First order asymptotic theory essentially comprise three cases: stable convergence towards a certain infinitely divisible distribution, an ergodic type limit theorem and convergence in probability towards an integrated random process. We also prove the second order limit theorem...
Böinghoff, Christian; Kersting, Götz
2012-01-01
Intermediately subcritical branching processes in random environment are at the borderline between two subcritical regimes and exhibit a particularly rich behavior. In this paper, we prove a functional limit theorem for these processes. It is discussed together with two other recently proved limit theorems for the intermediately subcritical case and illustrated by several computer simulations.
A Strong Limit Theorem on Generalized Random Selection for m-valued Random Sequences
WANGZhong-zhi; XUFu-xia
2003-01-01
In this paper, a strong limit theorem on gambling strategy for binary Bernoulli sequence, i.e.irregularity theorem, is extended to random selection for dependent m-valued random variables, via using a new method-differentiability on net. Furthermore, by allowing the selection function to take value in finite interval [-M, M], the conception of random selection is generalized.
Strong limit theorems in noncommutative L2-spaces
Jajte, Ryszard
1991-01-01
The noncommutative versions of fundamental classical results on the almost sure convergence in L2-spaces are discussed: individual ergodic theorems, strong laws of large numbers, theorems on convergence of orthogonal series, of martingales of powers of contractions etc. The proofs introduce new techniques in von Neumann algebras. The reader is assumed to master the fundamentals of functional analysis and probability. The book is written mainly for mathematicians and physicists familiar with probability theory and interested in applications of operator algebras to quantum statistical mechanics.
Central limit behavior of deterministic dynamical systems
Tirnakli, Ugur; Beck, Christian; Tsallis, Constantino
2007-04-01
We investigate the probability density of rescaled sums of iterates of deterministic dynamical systems, a problem relevant for many complex physical systems consisting of dependent random variables. A central limit theorem (CLT) is valid only if the dynamical system under consideration is sufficiently mixing. For the fully developed logistic map and a cubic map we analytically calculate the leading-order corrections to the CLT if only a finite number of iterates is added and rescaled, and find excellent agreement with numerical experiments. At the critical point of period doubling accumulation, a CLT is not valid anymore due to strong temporal correlations between the iterates. Nevertheless, we provide numerical evidence that in this case the probability density converges to a q -Gaussian, thus leading to a power-law generalization of the CLT. The above behavior is universal and independent of the order of the maximum of the map considered, i.e., relevant for large classes of critical dynamical systems.
Mixing rates and limit theorems for random intermittent maps
Bahsoun, Wael; Bose, Christopher
2016-04-01
We study random transformations built from intermittent maps on the unit interval that share a common neutral fixed point. We focus mainly on random selections of Pomeu-Manneville-type maps {{T}α} using the full parameter range 0CLT and stable laws, depending on the parameters chosen in the range 0<α <1 ) for the associated skew product; these are detailed in theorem 3.2. Since our estimates in theorem 1.1 also hold for 1≤slant α <∞ we study a second class of random transformations derived from piecewise affine Gaspard-Wang maps, prove existence of an infinite (σ-finite) invariant measure and study the corresponding correlation asymptotics. To the best of our knowledge, this latter kind of result is completely new in the setting of random transformations.
Limit theorems for von Mises statistics of a measure preserving transformation
Denker, Manfred
2011-01-01
For a measure preserving transformation $T$ of a probability space $(X,\\mathcal F,\\mu)$ we investigate almost sure and distributional convergence of random variables of the form $$x \\to \\frac{1}{C_n} \\sum_{i_1
Limit theorems for Markov processes indexed by continuous time Galton-Watson trees
Bansaye, Vincent; Marsalle, Laurence; Tran, Viet Chi
2009-01-01
We study the evolution of a particle system whose genealogy is given by a supercritical continuous time Galton-Watson tree. The particles move independently according to a Markov process and when a branching event occurs, the offspring locations depend on the position of the mother and the number of offspring. We prove a law of large numbers for the empirical measure of individuals alive at time $t$. This relies on a probabilistic interpretation of its intensity by mean of an auxiliary process. This latter has the same generator as the Markov process along the branches plus additional branching events, associated with jumps of accelerated rate and biased distribution. This comes from the fact that choosing an individual uniformly at time $t$ favors lineages with more branching events and larger offspring number. The central limit theorem is considered on a special case. Several examples are developed, including applications to splitting diffusions, cellular aging, branching L\\'evy processes and ancestral line...
Multi-channel sampling theorems for band-limited signals with fractional Fourier transform
2008-01-01
Multi-channel sampling for band-limited signals is fundamental in the theory of multi-channel parallel A/D environment and multiplexing wireless communication environment. As the fractional Fourier transform has been found wide applications in signal processing fields, it is necessary to consider the multi-channel sampling theorem based on the fractional Fourier transform. In this paper, the multi-channel sampling theorem for the fractional band-limited signal is firstly proposed, which is the generalization of the well-known sampling theorem for the fractional Fourier transform. Since the periodic nonuniformly sampled signal in the fractional Fourier domain has valuable applications, the reconstruction expression for the periodic nonuniformly sampled signal has been then obtained by using the derived multi-channel sampling theorem and the specific space-shifting and phase-shifting properties of the fractional Fourier transform. Moreover, by designing different fractional Fourier filters, we can obtain reconstruction methods for other sampling strategies.
A Fluctuation Limit Theorem of Branching Processes with Immigration and Statistical Applications
Ma, Chunhua
2009-01-01
We prove a general fluctuation limit theorem for Galton-Watson branching processes with immigration. The limit is a time-inhomogeneous OU type process driven by a spectrally positive Levy process. As applications of this result, we obtain some asymptotic estimates for the conditional least-squares estimator of the offspring mean.
A scaling limit theorem for the parabolic Anderson model with exponential potential
Lacoin, Hubert
2010-01-01
The parabolic Anderson problem is the Cauchy problem for the heat equation with random potential and localized initial condition. In this paper we consider potentials which are constant in time and independent exponentially distributed in space. We study the growth rate of the total mass of the solution in terms of weak and almost sure limit theorems, and the spatial spread of the mass in terms of a scaling limit theorem. The latter result shows that in this case, just like in the case of heavy tailed potentials, the mass gets trapped in a single relevant island with high probability.
Conditional limit theorems for regulated fractional Brownian motion
Awad, Hernan; 10.1214/09-AAP605
2009-01-01
We consider a stationary fluid queue with fractional Brownian motion input. Conditional on the workload at time zero being greater than a large value $b$, we provide the limiting distribution for the amount of time that the workload process spends above level $b$ over the busy cycle straddling the origin, as $b\\to\\infty$. Our results can be interpreted as showing that long delays occur in large clumps of size of order $b^{2-1/H}$. The conditional limit result involves a finer scaling of the queueing process than fluid analysis, thereby departing from previous related literature.
Some inequalities and limit theorems under sublinear expectations
Hu, Ze-Chun; Yang, Yan-Zhi
2012-01-01
In this note, we study inequality and limit theory under sublinear expectations. We mainly prove Doob's inequality for submartingale and Kolmogrov's inequality. By Kolmogrov's inequality, we obtain a special version of Kolmogrov's law of large numbers. Finally, we present a strong law of large numbers for independent and identically distributed random variables under one-order type moment condition.
Yafaev, D. R.
2010-01-01
We obtain two-sided bounds on kinetic and potential energies of a bound state of a quantum particle in the semiclassical limit, as the Planck constant $\\hbar\\ri 0$. Proofs of these results rely on the generalized virial theorem obtained in the paper as well as on a decay of eigenfunctions in the classically forbidden region.
HEAVY TRAFFIC LIMIT THEOREMS IN FLUID BUFFER MODELS
YIN Gang; ZHANG Hanqin
2004-01-01
A fluid buffer model with Markov modulated input-output rates is considered.When traffic intensity is near its critical value, the system is known as in heavy traffic.It is shown that a suitably scaled sequence of the equilibrium buffer contents has a weakor distributional limit under heavy traffic conditionsThis weak limit is a functional of adiffusion process determined by the Markov chain modulating the input and output rates.The first passage time of the reflected process is examinedIt is shown that the mean firstpassage time can be obtained via a solution of a Dirichlet problemThen the transitiondensity of the reflected process is derived by solving the Kolmogorov forward equation witha Neumann boundary conditionFurthermore, when the fast changing part of the generatorof the Markov chain is a constant matrix, the representation of the probability distributionof the reflected process is derivedUpper and lower bounds of the probability distributionare also obtained by means of asymptotic expansions of standard normal distribution.
Limit Theorems for Competitive Density Dependent Population Processes
Parsons, Todd L
2010-01-01
Near the beginning of the century, Wright and Fisher devised an elegant, mathematically tractable model of gene reproduction and replacement that laid the foundation for contemporary population genetics. The Wright-Fisher model and its extensions have given biologists powerful tools of statistical inference that enabled the quantification of genetic drift and selection. Given the utility of these tools, we often forget that their model - for mathematical, and not biological reasons - makes assumptions that are violated in most real-world populations. In this paper, I consider an alternative framework that merges P. A. P. Moran's continuous-time Markov chain model of allele frequency with the density dependent models of ecological competition proposed by Gause, Lotka and Volterra, that, unlike Moran's model allow for a stochastically varying -- but bounded -- population size. I require that allele numbers vary according to a density-dependent population process, for which the limiting law of large numbers is a...
Almost Sure Central Limit Theorem for a Nonstationary Gaussian Sequence
Qing-pei Zang
2010-01-01
Let be a standardized non-stationary Gaussian sequence, and let denote , . Under some additional condition, let the constants satisfy as for some and , for some , then, we have almost surely for any , where is the indicator function of the event and stands for the standard normal distribution function.
Central limit theorems for multiple stochastic integrals and Malliavin calculus
Nualart, David; Ortiz, Salvador
2007-01-01
We give a new characterization for the convergence in distribution to a standard normal law of a sequence of multiple stochastic integrals of a fixed order with variance one, in terms of the Malliavin derivatives of the sequence. We extend our result to the multidimensional case and prove a weak convergence result for a sequence of square integrable random variables.
The Central Limit Theorem for Exchangeable Random Variables Without Moments
Klass, Michael; Teicher, Henry
1987-01-01
If $\\{X_n, n \\geq 1\\}$ is an exchangeable sequence with $(1/b_n(\\sum^n_1X_i - a_n)) \\rightarrow N(0, 1)$ for some constants $a_n$ and $0 < b_n \\rightarrow \\infty$ then $b_n/n^\\alpha$ is slowly varying with $\\alpha = 1$ or $\\frac{1}{2}$ and necessary conditions (depending on $\\alpha$) which are also sufficient, are obtained. Three such examples are given, one with infinite mean, one with no positive moments, and the third with almost all conditional distributions belonging to no domain of attr...
Mathematical statistics and limit theorems Festschrift in honour of Paul Deheuvels
Mason, David; Pfeifer, Dietmar; Steinebach, Josef
2015-01-01
This Festschrift in honour of Paul Deheuvels’ 65th birthday compiles recent research results in the area between mathematical statistics and probability theory with a special emphasis on limit theorems. The book brings together contributions from invited international experts to provide an up-to-date survey of the field. Written in textbook style, this collection of original material addresses researchers, PhD and advanced Master students with a solid grasp of mathematical statistics and probability theory.
A weak limit theorem for numerical approximation of Brownian semi-stationary processes
Podolskij, Mark; Thamrongrat, Nopporn
In this paper we present a weak limit theorem for a numerical approximation of Brownian semi-stationary processes studied in [14]. In the original work of [14] the authors propose to use Fourier transformation to embed a given one dimensional (Levy) Brownian semi-stationary process into a two-par....../drift process needs to be numerically simulated. In particular, weak approximation errors for smooth test functions can be obtained from our asymptotic theory.......In this paper we present a weak limit theorem for a numerical approximation of Brownian semi-stationary processes studied in [14]. In the original work of [14] the authors propose to use Fourier transformation to embed a given one dimensional (Levy) Brownian semi-stationary process into a two......-parameter stochastic field. For the latter they use a simple iteration procedure and study the strong approximation error of the resulting numerical scheme given that the volatility process is fully observed. In this work we present the corresponding weak limit theorem for the setting, where the volatility...
Anomalous scaling due to correlations: limit theorems and self-similar processes
We derive theorems which outline explicit mechanisms by which anomalous scaling for the probability density function of the sum of many correlated random variables asymptotically prevails. The results characterize general anomalous scaling forms, explain their universal character, and specify universality domains in the spaces of joint probability density functions of the summand variables. These density functions are assumed to be invariant under arbitrary permutations of their arguments. Examples from the theory of critical phenomena are discussed. The novel notion of stability implied by the limit theorems also allows us to define sequences of random variables whose sum satisfies anomalous scaling for any finite number of summands. If regarded as developing in time, the stochastic processes described by these variables are non-Markovian generalizations of Gaussian processes with uncorrelated increments, and provide, e.g., explicit realizations of a recently proposed model of index evolution in finance
Day, Troy
2012-04-01
The process of evolutionary diversification unfolds in a vast genotypic space of potential outcomes. During the past century, there have been remarkable advances in the development of theory for this diversification, and the theory's success rests, in part, on the scope of its applicability. A great deal of this theory focuses on a relatively small subset of the space of potential genotypes, chosen largely based on historical or contemporary patterns, and then predicts the evolutionary dynamics within this pre-defined set. To what extent can such an approach be pushed to a broader perspective that accounts for the potential open-endedness of evolutionary diversification? There have been a number of significant theoretical developments along these lines but the question of how far such theory can be pushed has not been addressed. Here a theorem is proven demonstrating that, because of the digital nature of inheritance, there are inherent limits on the kinds of questions that can be answered using such an approach. In particular, even in extremely simple evolutionary systems, a complete theory accounting for the potential open-endedness of evolution is unattainable unless evolution is progressive. The theorem is closely related to Gödel's incompleteness theorem, and to the halting problem from computability theory. PMID:21849390
Functional limit theorems for L\\'evy processes satisfying Cram\\'er's condition
Barczy, Matyas
2011-01-01
We consider a L\\'evy process that starts from $x<0$ and conditioned on having a positive maximum. When Cram\\'er's condition holds, we provide two weak limit theorems as $x\\to -\\infty$ for the law of the (two-sided) path shifted at the first instant when it enters $(0,\\infty)$, respectively shifted at the instant when its overall maximum is reached. The comparison of these two asymptotic results yields some interesting identities related to time-reversal, insurance risk, and self-similar Markov processes.
A limit theorem for moments in space of the increments of Brownian local time
Campese, Simon
2015-01-01
We proof a limit theorem for moments in space of the increments of Brownian local time. As special cases for the second and third moments, previous results by Chen et al. (Ann. Prob. 38, 2010, no. 1) and Rosen (Stoch. Dyn. 11, 2011, no. 1), which were later reproven by Hu and Nualart (Electron. Commun. Probab. 14, 2009; Electron. Commun. Probab. 15, 2010) and Rosen (S\\'eminaire de Probabilit\\'es XLIII, Springer, 2011) are included. Furthermore, a conjecture of Rosen for the fourth moment is s...
On Almost Sure Max-limit Theorems of Complete and Incomplete Samples from Stationary Sequences
Bin TONG; Zuo Xiang PENG
2011-01-01
Let Mn denote the partial maximum of a strictly stationary sequence (Xn). Suppose that some of the random variables of (Xn) can be observed and let (M~)n stand for the maximum of observed random variables from the set {Xi,..., Xn}. In this paper, the almost sure limit theorems related to random vector ((M~),Mn) are considered in terms of i.i.d. case. The related results are also extended to weakly dependent stationary Gaussian sequence as its covariance function satisfies some regular conditions.
Kulik, Rafal
2011-01-01
Long Memory Stochastic volatility (LMSV) models capture two standardized features of financial data: the log-returns are uncorrelated, but their squares, or absolute values are (highly) dependent and they may have heavy tails. EGARCH and related models were introduced to model leverage, i.e. negative dependence between previous returns and future volatility. Limit theorems for partial sums, sample variance and sample covariances are basic tools to investigate the presence of long memory and heavy tails and their consequences. In this paper we extend the existing literature on the asymptotic behaviour of the partial sums and the sample covariances of long memory stochastic volatility models in the case of infinite variance. We also consider models with leverage, for which our results are entirely new in the infinite variance case. Depending on the nterplay between the tail behaviour and the intensity of dependence, wo types of convergence rates and limiting distributions can arise. In articular, we show that t...
Heck, Richard G
2011-01-01
Frege's Theorem collects eleven essays by Richard G Heck, Jr, one of the world's leading authorities on Frege's philosophy. The Theorem is the central contribution of Gottlob Frege's formal work on arithmetic. It tells us that the axioms of arithmetic can be derived, purely logically, from a single principle: the number of these things is the same as the number of those things just in case these can be matched up one-to-one with those. But that principle seems so utterlyfundamental to thought about number that it might almost count as a definition of number. If so, Frege's Theorem shows that a
Limits to the validity of the quantum-mechanical Nyquist theorem
It is suggested that the conventional form of the quantum-mechanical Nyquist theorem, derived from the fluctuation-dissipation theorem, may only be approximately correct when the effects of energy states with finite lifetimes are taken into account. These effects arise in the formulation of dissipation processes for a general thermodynamic system and similar breakdowns are possible for other applications of the fluctuation-dissipation theorem. The breakdown of the quantum-mechanical Nyquist theorem is analogous to a similar result obtained by Theimer and Dirk in 1982 for the conventional, classical Nyquist theorem. Our new expression for the theorem was developed by using procedures similar to those employed in the seminal derivation by Callen and Welton in 1951; in contrast to this work, however, it is utilized by us the Wigner-Weisskopf theory of natural line width to provide a more realistic description of energy dissipation processes. Our new expression for the quantum-mechanical Nyquist theorem contains terms not found in the conventional form involving the observation time and energy state lifetimes. A possible application of our new result for describing laser light scattering by vibrational and polariton modes in dielectric crystals is also discussed
We investigate the consequences of Birkhoff's theorem in general relativity (GR) and in modified Newtonian dynamics (MOND). We study, in particular, the system of a finite-mass test particle inside a spherical shell. In both GR and MOND, we find nonvanishing acceleration for that test particle. The direction of the acceleration is such that it pushes the test particle toward the center of the shell. In GR, the acceleration is found analytically in the case of a small test mass with a small displacement from the center of the shell. In MOND, the acceleration is found analytically in the limit of large test mass and small displacement, and a comparison to numerical values is made. Numerical simulations are done for more general cases with parameters that mimic the system of a galaxy in a cluster. In GR, the acceleration is highly suppressed and physically insignificant. In MOND, on the contrary, the acceleration of the point particle can be a significant fraction of the field just outside of the spherical shell.
Central limit behavior in the Kuramoto model at the “edge of chaos”
Miritello, Giovanna; Pluchino, Alessandro; Rapisarda, Andrea
2009-12-01
We study the relationship between chaotic behavior and the Central Limit Theorem (CLT) in the Kuramoto model. We calculate sums of angles at equidistant times along deterministic trajectories of single oscillators and we show that, when chaos is sufficiently strong, the Pdfs of the sums tend to a Gaussian, consistently with the standard CLT. On the other hand, when the system is at the “edge of chaos” (i.e. in a regime with vanishing Lyapunov exponents), robust q-Gaussian-like limit distributions naturally emerge, consistently with recently proved generalizations of the CLT.
Central limit approximations for Markov population processes with countably many types
Barbour, A D
2012-01-01
When modelling metapopulation dynamics, the influence of a single patch on the metapopulation depends on the number of individuals in the patch. Since there is usually no obvious natural upper limit on the number of individuals in a patch, this leads to systems in which there are countably infinitely many possible types of entity. Analogous considerations apply in the transmission of parasitic diseases. In this paper, we prove central limit theorems for quite general systems of this kind, together with bounds on the rate of convergence in an appropriately chosen weighted $\\ell_1$ norm.
LIMIT THEOREMS FOR KERNEL DENSITY ESTIMATORS IN SPACES OF ARBITRARY NATURE
Orlov A. I.
2015-04-01
Full Text Available Some estimators of the probability density function in spaces of arbitrary nature are used for various tasks in statistics of non-numerical data. Systematic exposition of the theory of such estimators had a start in our work [2]. This article is a direct continuation of the article [2]. We will regularly use references to conditions and theorems of the article [2], in which we introduced several types of nonparametric estimators of the probability density. We studied more linear estimators. In this article we consider particular cases - kernel density estimates in spaces of arbitrary nature. When estimating the density of the one-dimensional random variable, kernel estimators become the Parzen-Rosenblatt estimators. Asymptotic behavior of kernel density estimators in the general case of an arbitrary nature spaces are devoted to Theorem 1 - 8. Under different conditions we prove the consistency and asymptotic normality of kernel density estimators. We have studied uniform convergence. We have introduced the concept of "preferred rate differences" and studied nuclear density estimators based on it. We have also introduced and studied natural affinity measures which are used in the analysis of the asymptotic behavior of kernel density estimators. We have found the asymptotic behavior of dispersions of kernel density estimators and considered the examples including kernel density estimators in finite-dimensional spaces and in the space of square-integrable functions
Sunny Chauhan
2013-11-01
Full Text Available In this paper, we utilize the notion of common limit range property in Non-Archimedean Menger PM-spaces and prove some fixed point theorems for two pairs of weakly compatible mappings. Some illustrative examples are furnished to support our results. As an application to our main result, we present a common fixed point theorem for four finite families of self mappings. Our results improve and extend several known results existing in the literature.
Universal central extensions of direct limits of Lie superalgebras
Neher, Erhard
2011-01-01
We show that the universal central extension of a direct limit of perfect Lie superalgebras L_i is (isomorphic to) the direct limit of the universal central extensions of L_i. As an application we describe the universal central extensions of some infinite rank Lie superalgebras.
Herbrand's Fundamental Theorem - an encyclopedia article
Wirth, Claus-Peter
2015-01-01
Herbrand's Fundamental Theorem provides a constructive characterization of derivability in first-order predicate logic by means of sentential logic. Sometimes it is simply called "Herbrand's Theorem", but the longer name is preferable as there are other important "Herbrand theorems" and Herbrand himself called it "Th\\'eor\\`eme fondamental". It was ranked by Bernays [1957] as follows: "In its proof-theoretic form, Herbrand's Theorem can be seen as the central theorem of predicate logic. It exp...
Functional limit theorems for generalized variations of the fractional Brownian sheet
Pakkanen, Mikko; Réveillac, Anthony
the smallest component of the Hurst parameter vector of the fBs. The limiting process in the former result is another fBs, independent of the original fBs, whereas the limit given by the latter result is an Hermite sheet, which is driven by the same white noise as the original fBs. As an application...
Ivanov, Alexander V.; Leonenko, Nikolai; Ruiz-Medina, María D.; Savich, Irina N.
2013-01-01
The limit Gaussian distribution of multivariate weighted functionals of nonlinear transformations of Gaussian stationary processes, having multiple singular spectra, is derived, under very general conditions on the weight function. This paper is motivated by its potential applications in nonlinear regression, and asymptotic inference on nonlinear functionals of Gaussian stationary processes with singular spectra.
Limit theorems for splitting trees with structured immigration and applications to biogeography
Richard, Mathieu
2010-01-01
We consider a branching process with Poissonian immigration where individuals have inheritable types. At rate $\\theta$, new individuals singly enter the total population and start a new population which evolves like a supercritical, homogeneous, binary Crump-Mode-Jagers process: individuals have i.i.d. lifetimes durations (non necessarily exponential) during which they give birth independently at constant rate b. First, using spine decomposition, we relax previously known assumptions required for a.s. convergence of total population size. Then, we consider three models of structured populations: either all immigrants have a different type, or types are drawn in a discrete spectrum or in a continuous spectrum. In each model, the vector (P_1,P_2,...) of relative abundances of surviving families converges a.s. In the first model, the limit is the GEM distribution with parameter $\\theta/b$.
Nonergodicity and central-limit behavior for long-range Hamiltonians
Pluchino, A.; Rapisarda, A.; Tsallis, C.
2007-10-01
We present a molecular dynamics test of the Central-Limit Theorem (CLT) in a paradigmatic long-range-interacting many-body classical Hamiltonian system, the HMF model. We calculate sums of velocities at equidistant times along deterministic trajectories for different sizes and energy densities. We show that, when the system is in a chaotic regime (specifically, at thermal equilibrium), ergodicity is essentially verified, and the Pdfs of the sums appear to be Gaussians, consistently with the standard CLT. When the system is, instead, only weakly chaotic (specifically, along longstanding metastable Quasi-Stationary States), nonergodicity (i.e., discrepant ensemble and time averages) is observed, and robust q-Gaussian attractors emerge, consistently with recently proved generalizations of the CLT.
A feasible central limit theory for realised volatility under leverage
Barndorff-Nielsen, Ole Eiler; Shephard, Neil
In this note we show that the feasible central limit theory for realised volatility and realised covariation recently developed by Barndor-Nielsen and Shephard applies under arbitrary diusion based leverage eects. Results from a simulation experiment suggest that the feasible version of the limit...
Variances and covariances in the Central Limit Theorem for the output of a transducer
Heuberger, Clemens; Kropf, Sara; Wagner, Stephan
2015-01-01
We study the joint distribution of the input sum and the output sum of a deterministic transducer. Here, the input of this finite-state machine is a uniformly distributed random sequence. We give a simple combinatorial characterization of transducers for which the output sum has bounded variance, and we also provide algebraic and combinatorial characterizations of transducers for which the covariance of input and output sum is bounded, so that the two are asymptotically independent. Our results are illustrated by several examples, such as transducers that count specific blocks in the binary expansion, the transducer that computes the Gray code, or the transducer that computes the Hamming weight of the width-w non-adjacent form digit expansion. The latter two turn out to be examples of asymptotic independence. PMID:27087727
A central limit theorem for random walk in random environment on marked Galton-Watson trees
Faraud, Gabriel
2008-01-01
In this article we focus on a general model of random walk on random marked trees. We prove a recurrence criterion, analogue to the recurrence criterion proved by R. Lyons and Robin Pemantle (1992) in a slightly different model. In the critical case, we obtain a criterion for the positive/null recurrence. Several regimes appear, as proved (in a similar model), by Y. Hu and Z. Shi (2007). We focus on the "diffusive" regime and improve their result in this case, by obtaining a functional Centra...
Central limit theorem for biased random walk on multi-type Galton-Watson trees
Dembo, Amir; Sun, Nike
2010-01-01
Let T be a rooted supercritical multi-type Galton-Watson (MGW) tree with types coming from a finite alphabet, conditioned to non-extinction. The lambda-biased random walk (X_t, t>=0) on T is the nearest-neighbor random walk which, when at a vertex v with d(v) offspring, moves closer to the root with probability lambda/[lambda+d(v)], and to each of the offspring with probability 1/[lambda+d(v)]. This walk is recurrent for lambda>=rho and transient for 0
Berry-Esseen's central limit theorem for non-causal linear processes in Hilbert space
Machkouri, Mohamed EL
2010-01-01
Let $H$ be a real separable Hilbert space and $(a_k)_{k\\in\\mathbb{Z}}$ a sequence of bounded linear operators from $H$ to $H$. We consider the linear process $X$ defined for any $k$ in $\\mathbb{Z}$ by $X_k=\\sum_{j\\in\\mathbb{Z}}a_j(\\varepsilon_{k-j})$ where $(\\varepsilon_k)_{k\\in\\mathbb{Z}}$ is a sequence of i.i.d. centered $H$-valued random variables. We investigate the rate of convergence in the CLT for $X$ and in particular we obtain the usual Berry-Esseen's bound provided that $\\sum_{j\\in\\mathbb{Z}}\\vert j\\vert\\|a_j\\|_{\\mathcal{L}(H)}<+\\infty$ and $\\varepsilon_0$ belongs to $L_H^{\\infty}$.
A Central Limit Theorem for the Volumes of High Excursions of Stationary Associated Random Fields
Vadim Demichev
2015-05-01
Full Text Available We prove that under certain conditions the excursion sets volumes of stationary positively associated random fields converge after rescaling to the normal distribution as the excursion level and the size of the observation window grow. In addition, we provide a number of examples.
Del Barrio, Eustasio; Lescornel, Hélène; Loubes, Jean-Michel
2016-01-01
Wasserstein barycenters and variance-like criterion using Wasserstein distance are used in many problems to analyze the homo-geneity of collections of distributions and structural relationships between the observations. We propose the estimation of the quantiles of the empirical process of the Wasserstein's variation using a bootstrap procedure. Then we use these results for statistical inference on a distribution registration model for general deformation functions. The tests are based on th...
Carlen, E
2011-01-01
We prove for the rescaled convolution map $f\\to f\\circledast f$ propagation of polynomial, exponential and gaussian localization. The gaussian localization is then used to prove an optimal bound on the rate of entropy production by this map. As an application we prove the convergence of the CLT to be at the optimal rate $1/\\sqrt{n}$ in the entropy (and $L^1$) sense, for distributions with finite 4th moment.
Central limit theorem for biased random walk on multi-type Galton-Watson trees
Dembo, Amir
2010-01-01
Let T be a rooted multi-type Galton-Watson (MGW) tree of finitely many types with at least one offspring at each vertex, and an offspring distribution with exponential tails. The lambda-biased random walk (X_t, t>=0) on T is the nearest-neighbor random walk which, when at a vertex v with d(v) offspring, moves closer to the root with probability lambda/(lambda+d(v)), and to each of the offspring with probability 1/(lambda+d(v)). This walk is recurrent for lambda >= rho and transient for 0 <= lambda < rho, with rho the Perron-Frobenius eigenvalue for the (assumed) irreducible matrix of expected offspring numbers. We prove the following quenched CLT for the critical value lambda = rho: for almost every T, the process |X_{floor(nt)}|/sqrt{n} converges in law as n tends to infinity to a deterministic positive multiple of a reflected Brownian motion. Following the approach of Peres and Zeitouni (2008) for Galton-Watson trees, our proof is based on a new explicit description of a reversing measure for the walk...
Occupancy of phase space, extensivity of Sq, and q-generalized central limit theorem
Tsallis, Constantino
2005-01-01
Increasing the number $N$ of elements of a system typically makes the entropy to increase. The question arises on {\\it what particular entropic form} we have in mind and {\\it how it increases} with $N$. Thermodynamically speaking it makes sense to choose an entropy which increases {\\it linearly} with $N$ for large $N$, i.e., which is {\\it extensive}. If the $N$ elements are probabilistically {\\it independent} (no interactions) or quasi-independent (e.g., {\\it short}-range interacting), it is ...
Noether's theorem relates symmetries and conservation laws of Hamiltonians systems. Arnol'd's theorem uses those integrals of motion for the construction of sufficient stability conditions of hydrodynamical problems, which are Hamiltonian with a singular Poisson bracket. Finally, Andrews' theorem imposes restriction on the existence of Arnol'd stable solutions of symmetric systems. It is shown that denial of Andrews'theorem implies the divergence of the velocity component normal to the symmetric coordinate. This proof by reductio ad absurdum may be used to determine the strength of the symmetry breaking elements, necessary to overcome the limitations imposed by this theorem (Author)
Bell's Theorem from Moore's Theorem
Fields, Chris
2012-01-01
It is shown that the restrictions of what can be inferred from classically-recorded observational outcomes that are imposed by the no-cloning theorem, the Kochen-Specker theorem and Bell's theorem also follow from restrictions on inferences from observations formulated within classical automata theory. Similarities between the assumptions underlying classical automata theory and those underlying universally-unitary quantum theory are discussed.
Schleimer, Saul
2009-01-01
This note is an exposition of Waldhausen's proof of Waldhausen's Theorem: the three-sphere has a single Heegaard splitting, up to isotopy, in every genus. As a necessary step we also give a sketch of the Reidemeister-Singer Theorem.
Range-limited Centrality Measures in Complex Networks
Ercsey-Ravasz, Maria; Chawla, Nitesh V; Toroczkai, Zoltan
2011-01-01
Here we present a range-limited approach to centrality measures in both non-weighted and weighted directed complex networks. We introduce an efficient method that generates for every node and every edge its betweenness centrality based on shortest paths of lengths not longer than $\\ell = 1,...,L$ in case of non-weighted networks, and for weighted networks the corresponding quantities based on minimum weight paths with path weights not larger than $w_{\\ell}=\\ell \\Delta$, $\\ell=1,2...,L=R/\\Delta$. These measures provide a systematic description on the positioning importance of a node (edge) with respect to its network neighborhoods 1-step out, 2-steps out, etc. up to including the whole network. We show that range-limited centralities obey universal scaling laws for large non-weighted networks. As the computation of traditional centrality measures is costly, this scaling behavior can be exploited to efficiently estimate centralities of nodes and edges for all ranges, including the traditional ones. The scaling ...
A trick discovered by Vaidman, Aharanov, and Albert permitting retrodiction of the outcomes of more measurements than one would naively have thought possible is extended to a case in which the retrodicted observables are forbidden all to have values by a Bell-Kochen-Specker theorem. A rather peculiar analysis shows that an even better trick that retrodicts the outcomes of more informative measurements of these same observables is impossible
The large central charge limit of conformal blocks
Fateev, Vladimir
2011-01-01
We study conformal blocks of conformal field theories with a W3 symmetry algebra in the limit where the central charge is large. In this limit, we compute the four-point block as a special case of an sl3-invariant function. In the case when two of the four fields are semi-degenerate, we check that our results agree with the block's combinatorial expansion as a sum over Young diagrams. We also show that such a block obeys a sixth-order differential equation, and that it has an unexpected singularity at z=-1, in addition to the expected singularities at z=0,1,infinity.
Almost Subadditive Extensions of Kingman's Ergodic Theorem
Schurger, Klaus
1991-01-01
Based on two notions of almost subadditivity which were introduced by Derriennic and Schurger, two a.s. limit theorems are proved which both generalize Kingman's subadditive ergodic theorem. These results, being valid under weak moment conditions, are obtained by short proofs. One of these proofs is completely elementary and does not even make use of Birkhoff's ergodic theorem which, instead, is obtained as a by-product. Finally, an improvement of Liggett's a.s. limit theorem is given.
Noether's theorem attains its maximum simplicity and depth when formulated in curved space-time, gravitation being included. Extension to curved space-times is here made simple by the use of a formulation, for the flat case, due to Jackiw. The exposition purports to be pedagogical. (Author)
Sustaining Rural Afghanistan under Limited Central Government Influence
John William Groninger
2013-06-01
Full Text Available Land and water access insecurity, land grabbing, and unstable common property status of critical local resources continue to drive conflicts, rural landlessness and environmental problems throughout many areas of Afghanistan where formal government is weak or entirely absent. In contrast to traditional development strategies that favor infrastructure enhancement and backed by enforced national policies, we offer Afghan-specific strategies based on resource conservation and increased capacity of local resource management institutions that can function when and where central government cannot be relied upon to assume or maintain a supportive role. Resource conservation and building local capacity are key components of existing and proposed future efforts to increase stability. However, support for these efforts, whether government or community-based, has been limited in portions of rural Afghanistan , apparently due to low stakeholder confidence in retaining access to improved land, water and other critical resources when international forces withdraw. Powerful individuals and groups, operating outside local community structures, are increasingly impacting land use practices. We suggest a thorough assessment of the present and likely future social environment, including awareness of likely conflicts resulting from agricultural or natural resource improvements, before any tangible actions are taken.
Limiting Central Bank Credit to the Government; Theory and Practice
Carlo Cottarelli
1993-01-01
This paper examines central bank independence with reference to the constraints on central bank credit to the government, focusing on how such credit should be regulated. It discusses why credit should be contsrained, and in which forms, and how to implement those constraints.
The relativistic virial theorem and scale invariance
Gaite, Jose
2013-01-01
The virial theorem is related to the dilatation properties of bound states. This is realized, in particular, by the Landau-Lifshitz formulation of the relativistic virial theorem, in terms of the trace of the energy-momentum tensor. We construct a Hamiltonian formulation of dilatations in which the relativistic virial theorem naturally arises as the condition of stability against dilatations. A bound state becomes scale invariant in the ultrarelativistic limit, in which its energy vanishes. However, for very relativistic bound states, scale invariance is broken by quantum effects and the virial theorem must include the energy-momentum tensor trace anomaly. This quantum field theory virial theorem is directly related to the Callan-Symanzik equations. The virial theorem is applied to QED and then to QCD, focusing on the bag model of hadrons. In massless QCD, according to the virial theorem, 3/4 of a hadron mass corresponds to quarks and gluons and 1/4 to the trace anomaly.
Hydrogeological mapping of north-central Madagascar using limited data
Davies, Jeffrey
2009-01-01
North-central Madagascar is well endowed with surface-water. Due to soil erosion and pollution, rivers can no longer provide year round clean water supplies. Shallow aquifers are being developed to provide sustainable rural and urban water supplies. A survey of water sources located 2760 boreholes, wells, ponds and springs in the area, but understanding of groundwater occurrence remains poor. The area comprises four hydrogeological zones: the dry Central High Plateau with erosi...
The relativistic virial theorem and scale invariance
Gaite, Jose
2013-01-01
The virial theorem is related to the dilatation properties of bound states. This is realized, in particular, by the Landau-Lifshitz formulation of the relativistic virial theorem, in terms of the trace of the energy-momentum tensor. We construct a Hamiltonian formulation of dilatations in which the relativistic virial theorem naturally arises as the condition of stability against dilatations. A bound state becomes scale invariant in the ultrarelativistic limit, in which its energy vanishes. H...
Del Barrio, Eustasio; Lescornel, Hélène; Loubes, Jean-Michel
2016-01-01
Wasserstein barycenters and variance-like criterion using Wasser-stein distance are used in many problems to analyze the homogeneity of collections of distributions and structural relationships between the observations. We propose the estimation of the quantiles of the empirical process of the Wasserstein's variation using a bootstrap procedure. Then we use these results for statistical inference on a distribution registration model for general deformation functions. The tests are based on th...
Central limit theorem for germination-growth models in R^{ d } with non-Poisson locations
Chiu, S. N.; Quine, M. P.
2001-01-01
Seeds are randomly scattered in Rd according to an m-dependent point process. Each seed has its own potential germination time. From each seed that succeeds in germinating, a spherical inhibited region grows to prohibit germination of any seed with later potential germination time. We show that under certain conditions on the distribution of the potential germination time, the number of germinated seeds in a large region has an asymptotic normal distribution.
Soft theorems from anomalous symmetries
Huang, Yu-tin
2015-01-01
We discuss constraints imposed by soft limits for effective field theories arising from symmetry breaking. In particular, we consider those associated with anomalous conformal symmetry as well as duality symmetries in supergravity. We verify these soft theorems for the dilaton effective action relevant for the a-theorem, as well as the one-loop effective action for N=4 supergravity. Using the universality of leading transcendental coefficients in the alpha' expansion of string theory amplitudes, we study the matrix elements of operator R^4 with half maximal supersymmetry. We construct the non-linear completion of R^4 that satisfies both single and double soft theorems up to seven points. This supports the existence of duality invariant completion of R^4.
Virial theorem for radiating accretion discs
Mach, Patryk
2011-01-01
A continuum version of the virial theorem is derived for a radiating self-gravitating accretion disc around a compact object. The central object is point-like, but we can avoid the regularization of its gravitational potential. This is achieved by applying a modified Pohozaev-Rellich identity to the gravitational potential of the disk only. The theorem holds for general stationary configurations, including discontinuous flows (shock waves, contact discontinuities). It is used to test numerica...
We examine a soft scalar theorem which has proved useful in the evaluation of certain Feynman graphs. The use of this theorem is described in connection with the determination of the Λnphi coupling in a unified model of weak and electromagnetic interactions. (author)
Smith, Michael D.
2016-01-01
The Parity Theorem states that any permutation can be written as a product of transpositions, but no permutation can be written as a product of both an even number and an odd number of transpositions. Most proofs of the Parity Theorem take several pages of mathematical formalism to complete. This article presents an alternative but equivalent…
Saa, Diego
2005-01-01
Goedel's results have had a great impact in diverse fields such as philosophy, computer sciences and fundamentals of mathematics. The fact that the rule of mathematical induction is contradictory with the rest of clauses used by Goedel to prove his undecidability and incompleteness theorems is proved in this paper. This means that those theorems are invalid.
Double Soft Theorem for Perturbative Gravity
Saha, Arnab Priya
2016-01-01
Following up on the recent work of Cachazo, He and Yuan \\cite{arXiv:1503.04816 [hep-th]}, we derive the double soft graviton theorem in perturbative gravity. We show that the double soft theorem derived using CHY formula precisely matches with the perturbative computation involving Feynman diagrams. In particular, we find how certain delicate limits of Feynman diagrams play an important role in obtaining this equivalence.
Integral fluctuation theorem for the housekeeping heat
The housekeeping heat Qhk is the dissipated heat necessary to maintain the violation of detailed balance in nonequilibrium steady states. By analysing the evolution of its probability distribution, we prove an integral fluctuation theorem (exp[-βQhk]) = 1 valid for arbitrary-driven transitions between steady states. We discuss Gaussian limiting cases and the difference between the new theorem and both the Hatano-Sasa and the Jarzynski relation. (letter to the editor)
Integral fluctuation theorem for the housekeeping heat
Speck, T.; Seifert, U.
2005-01-01
The housekeeping heat $Q\\hk$ is the dissipated heat necessary to maintain the violation of detailed balance in nonequilibrium steady states. By analyzing the evolution of its probability distribution, we prove an integral fluctuation theorem $\\mean{\\exp[-\\beta Q\\hk]}=1$ valid for arbitrary driven transitions between steady states. We discuss Gaussian limiting cases and the difference between the new theorem and both the Hatano-Sasa and the Jarzynski relation.
Kreutzer, Stephan
2009-01-01
Algorithmic meta-theorems are general algorithmic results applying to a whole range of problems, rather than just to a single problem alone. They often have a "logical" and a "structural" component, that is they are results of the form: every computational problem that can be formalised in a given logic L can be solved efficiently on every class C of structures satisfying certain conditions. This paper gives a survey of algorithmic meta-theorems obtained in recent years and the methods used to prove them. As many meta-theorems use results from graph minor theory, we give a brief introduction to the theory developed by Robertson and Seymour for their proof of the graph minor theorem and state the main algorithmic consequences of this theory as far as they are needed in the theory of algorithmic meta-theorems.
Tight closure and vanishing theorems
Tight closure has become a thriving branch of commutative algebra since it was first introduced by Mel Hochster and Craig Huneke in 1986. Over the past few years, it has become increasingly clear that tight closure has deep connections with complex algebraic geometry as well, especially with those areas of algebraic geometry where vanishing theorems play a starring role. The purpose of these lectures is to introduce tight closure and to explain some of these connections with algebraic geometry. Tight closure is basically a technique for harnessing the power of the Frobenius map. The use of the Frobenius map to prove theorems about complex algebraic varieties is a familiar technique in algebraic geometry, so it should perhaps come as no surprise that tight closure is applicable to algebraic geometry. On the other hand, it seems that so far we are only seeing the tip of a large and very beautiful iceberg in terms of tight closure's interpretation and applications to algebraic geometry. Interestingly, although tight closure is a 'characteristic p' tool, many of the problems where tight closure has proved useful have also yielded to analytic (L2) techniques. Despite some striking parallels, there had been no specific result directly linking tight closure and L∼ techniques. Recently, however, the equivalence of an ideal central to the theory of tight closure was shown to be equivalent to a certain 'multiplier ideal' first defined using L2 methods. Presumably, deeper connections will continue to emerge. There are two main types of problems for which tight closure has been helpful: in identifying nice structure and in establishing uniform behavior. The original algebraic applications of tight closure include, for example, a quick proof of the Hochster-Roberts theorem on the Cohen-Macaulayness of rings of invariants, and also a refined version of the Brianqon-Skoda theorem on the uniform behaviour of integral closures of powers of ideals. More recent, geometric
Quantum Rate Distortion, Reverse Shannon Theorems, and Source-Channel Separation
Datta, Nilanjana; Hsieh, Min-Hsiu; Wilde, Mark M.
2013-01-01
We derive quantum counterparts of two key theorems of classical information theory, namely, the rate distortion theorem and the source-channel separation theorem. The rate-distortion theorem gives the ultimate limits on lossy data compression, and the source-channel separation theorem implies that a two-stage protocol consisting of compression and channel coding is optimal for transmitting a memoryless source over a memoryless channel. In spite of their importance in the classical domain, the...
The hypervirial and Hellmann-Feynman theorems are used in the methods of 1/N expansion to construct Rayleigh-Schroedinger perturbation expansion for bound-state energy eigenvalues of spherical symmetric potentials. A new iteration procedure of calculating correction terms of arbitrarily high orders is obtained for any kind of 1/N expansion. The recurrence formulas for three variants of the 1/N expansion are considered in this work, namely, the 1/n expansion, the shifted and unshifted 1/N expansions which are applied to the Gaussian and Patil potentials. As a result, their credibility could be reliably judged when account is taken of high order terms of the eigenenergies. It is also found that there is a distinct advantage in using the shifted 1/N expansion over the two other versions. However, the shifted 1/N expansion diverges for s states and in certain cases is not applicable as far as complicated potentials are concerned. In an effort to solve these problems we have incorporated the principle of minimal sensitivity in the shifted 1/N expansion as a first step toward extending the scope of applicability of that technique, and then we have tested the obtained approach to some unfavorable cases of the Patil and Hellmann potentials. The agreement between our numerical calculations and reference data is quite satisfactory. (author)
Intersection homology Kunneth theorems
Friedman, Greg
2008-01-01
Cohen, Goresky and Ji showed that there is a Kunneth theorem relating the intersection homology groups $I^{\\bar p}H_*(X\\times Y)$ to $I^{\\bar p}H_*(X)$ and $I^{\\bar p}H_*(Y)$, provided that the perversity $\\bar p$ satisfies rather strict conditions. We consider biperversities and prove that there is a K\\"unneth theorem relating $I^{\\bar p,\\bar q}H_*(X\\times Y)$ to $I^{\\bar p}H_*(X)$ and $I^{\\bar q}H_*(Y)$ for all choices of $\\bar p$ and $\\bar q$. Furthermore, we prove that the Kunneth theorem...
A novel sampling theorem on the sphere
McEwen, J D
2011-01-01
We develop a novel sampling theorem on the sphere and corresponding fast algorithms by associating the sphere with the torus through a periodic extension. The fundamental property of any sampling theorem is the number of samples required to represent a band-limited signal. To represent exactly a signal on the sphere band-limited at L, all sampling theorems on the sphere require O(L^2) samples. However, our sampling theorem requires less than half the number of samples of other equiangular sampling theorems on the sphere and an asymptotically identical, but smaller, number of samples than the Gauss-Legendre sampling theorem. The complexity of our algorithms scale as O(L^3), however, the continual use of fast Fourier transforms reduces the constant prefactor associated with the asymptotic scaling considerably, resulting in algorithms that are fast. Furthermore, we do not require any precomputation and our algorithms apply to both scalar and spin functions on the sphere without any change in computational comple...
To string together six theorems of physics by Pythagoras theorem
Cui, H. Y.
2002-01-01
In this paper, we point out that there are at lest six theorems in physics sharing common virtue of Pythagoras theorem, so that it is possible to string these theorems together with the Pythagoras theorem for physics teaching, the six theorems are Newton's three laws of motion, universal gravitational force, Coulomb's law, and the formula of relativistic dynamics. Knowing the internal relationships between them, which have never been clearly revealed by other author, will benefit the logic of...
Virial Theorem and Hypervirial Theorem in a spherical geometry
Li, Yan; Zhang, Fu-Lin; Chen, Jing-Ling
2010-01-01
The Virial Theorem in the one- and two-dimensional spherical geometry are presented, in both classical and quantum mechanics. Choosing a special class of Hypervirial operators, the quantum Hypervirial relations in the spherical spaces are obtained. With the aid of the Hellmann-Feynman Theorem, these relations can be used to formulate a \\emph{perturbation theorem without wave functions}, corresponding to the Hypervirial-Hellmann-Feynman Theorem perturbation theorem of Euclidean geometry. The o...
Zapletal, Jindrich
2005-01-01
I prove preservation theorems for countable support iteration of proper forcing concerning certain classes of capacities and submeasures. New examples of forcing notions and connections with measure theory are included.
Energy turnover in European hares is centrally limited during early, but not during peak lactation
Valencak, Teresa G.; Ruf, Thomas
2009-01-01
We investigated metabolizable energy intake (MEI) and milk energy output in European hares throughout gestation and lactation in females raising three young, i.e., close to maximum litter size in this precocial species. We hypothesized that herbivorous hares may face a central limitation of energy turnover during lactation, imposed by maximum capacity of the gastrointestinal tract. Females were provided with low-energy or high-energy diets, either continually, or during lactation only. Unexpe...
Limitations and potentials of dual-purpose cow herds in Central Coastal Veracruz, Mexico
Absalón-Medina, Victor Antonio; Blake, Robert W.; Fox, Danny Gene; Juárez-Lagunes, Francisco I.; Charles F. Nicholson; Canudas-Lara, Eduardo G.; Rueda-Maldonado, Bertha L.
2011-01-01
Feed chemical and kinetic composition and animal performance information was used to evaluate productivity limitations and potentials of dual-purpose member herds of the Genesis farmer organization of central coastal Veracruz, Mexico. The Cornell Net Carbohydrate and Protein System model (Version 6.0) was systematically applied to specific groups of cows in structured simulations to establish probable input–output relationships for typical management, and to estimate probable outcomes from al...
Voting, Lobbying, and the Decentralization Theorem
Lockwood, Benjamin
2007-01-01
This paper revisits the fiscal "decentralization theorem", by relaxing the role of the assumption that governments are benevolent, while retaining the assumption of policy uniformity. If instead, decisions are made by direct majority voting, (i) centralization can welfare-dominate decentralization even if there are no externalities and regions are heterogenous; (ii) decentralization can welfare-dominate centralization even if there are positive externalities and regions are hom...
Navier Stokes Theorem in Hydrology
Narayanan, M.
2005-12-01
In a paper presented at the 2004 AGU International Conference, the author outlined and stressed the importance of studying and teaching certain important mathematical techniques while developing a course in Hydrology and Fluid Mechanics. The Navier-Stokes equations are the foundation of fluid mechanics, and Stokes' theorem is used in nearly every branch of mechanics as well as electromagnetics. Stokes' Theorem also plays a vital role in many secondary theorems such as those pertaining to vorticity and circulation. Mathematically expressed, Stokes' theorem can be expressed by considering a surface S having a bounding curve C. Here, V is any sufficiently smooth vector field defined on the surface and its bounding curve C. In an article entitled "Corrections to Fluid Dynamics" R. F. Streater, (Open Systems and Information Dynamics, 10, 3-30, 2003.) proposes a kinetic model of a fluid in which five macroscopic fields, the mass, energy, and three components of momentum, are conserved. The dynamics is constructed using the methods of statistical dynamics, and results in a non-linear discrete-time Markov chain for random fields on a lattice. In the continuum limit he obtains a non-linear coupled parabolic system of field equations, showing a correction to the Navier-Stokes equations. In 2001, David Hoff published an article in Journees Equations aux derivees partielles. (Art. No. 7, 9 p.). His paper is entitled : Dynamics of Singularity Surfaces for Compressible Navier-Stokes Flows in Two Space Dimensions. In his paper, David Hoff proves the global existence of solutions of the Navier-Stokes equations of compressible, barotropic flow in two space dimensions with piecewise smooth initial data. These solutions remain piecewise smooth for all time, retaining simple jump discontinuities in the density and in the divergence of the velocity across a smooth curve, which is convected with the flow. The strengths of these discontinuities are shown to decay exponentially in time
The Goldstone boson equivalence theorem with fermions
Durand, Loyal; Riesselmann, Kurt
1995-01-01
The calculation of the leading electroweak corrections to physical transition matrix elements in powers of $M_H^2/v^2$ can be greatly simplified in the limit $M_H^2\\gg M_W^2,\\, M_Z^2$ through the use of the Goldstone boson equivalence theorem. This theorem allows the vector bosons $W^\\pm$ and $Z$ to be replaced by the associated scalar Goldstone bosons $w^\\pm$, $z$ which appear in the symmetry breaking sector of the Standard Model in the limit of vanishing gauge couplings. In the present pape...
Raychaudhuri equation and singularity theorems in Finsler spacetimes
Minguzzi, E.
2015-09-01
The Raychaudhuri equation and its consequences for chronality are studied in the context of Finsler spacetimes. It is proved that the notable singularity theorems of Lorentzian geometry extend to the Finslerian domain. Indeed, so do the theorems by Hawking, Penrose, Hawking and Penrose, Geroch, Gannon, Tipler and Kriele, and also the Topological Censorship theorem and so on. It is argued that the notable results in causality theory connected to achronal sets, future sets, domains of dependence, limit curve theorems, length functional, Lorentzian distance and geodesic connectedness, extend to the Finslerian domain. Results concerning the spacetime asymptotic structure, horizons differentiability and conformal transformations are also included.
Stability of Gas Clouds in Galactic Nuclei: An Extended Virial Theorem
Chen, Xian; Amaro-Seoane, Pau; Cuadra, Jorge
2016-03-01
Cold gas entering the central 1-102 pc of a galaxy fragments and condenses into clouds. The stability of the clouds determines whether they will be turned into stars or can be delivered to the central supermassive black hole (SMBH) to turn on an active galactic nucleus (AGN). The conventional criteria to assess the stability of these clouds, such as the Jeans criterion and Roche (or tidal) limit, are insufficient here, because they assume the dominance of self-gravity in binding a cloud, and neglect external agents, such as pressure and tidal forces, which are common in galactic nuclei. We formulate a new scheme for judging this stability. We first revisit the conventional Virial theorem, taking into account an external pressure, to identify the correct range of masses that lead to stable clouds. We then extend the theorem to further include an external tidal field, which is equally crucial for the stability in the region of our interest—in dense star clusters, around SMBHs. We apply our extended Virial theorem to find new solutions to controversial problems, namely, the stability of the gas clumps in AGN tori, the circum-nuclear disk in the Galactic Center, and the central molecular zone of the Milky Way. The masses we derive for these structures are orders of magnitude smaller than the commonly used Virial masses (equivalent to the Jeans mass). Moreover, we prove that these clumps are stable, contrary to what one would naively deduce from the Roche (tidal) limit.
D'Agostini, G
2005-01-01
It is curious to learn that Enrico Fermi knew how to base probabilistic inference on Bayes theorem, and that some influential notes on statistics for physicists stem from what the author calls elsewhere, but never in these notes, {\\it the Bayes Theorem of Fermi}. The fact is curious because the large majority of living physicists, educated in the second half of last century -- a kind of middle age in the statistical reasoning -- never heard of Bayes theorem during their studies, though they have been constantly using an intuitive reasoning quite Bayesian in spirit. This paper is based on recollections and notes by Jay Orear and on Gauss' ``Theoria motus corporum coelestium'', being the {\\it Princeps mathematicorum} remembered by Orear as source of Fermi's Bayesian reasoning.
Converse Barrier Certificate Theorem
Wisniewski, Rafael; Sloth, Christoffer
2013-01-01
This paper presents a converse barrier certificate theorem for a generic dynamical system.We show that a barrier certificate exists for any safe dynamical system defined on a compact manifold. Other authors have developed a related result, by assuming that the dynamical system has no singular...... points in the considered subset of the state space. In this paper, we redefine the standard notion of safety to comply with generic dynamical systems with multiple singularities. Afterwards, we prove the converse barrier certificate theorem and illustrate the differences between ours and previous work by...
Poutiainen, H. (Hayley)
2015-01-01
Group theory is a mathematical domain where groups and their properties are studied. The evolution of group theory as an area of study is said to be the result of the parallel development of a variety of different studies in mathematics. Sylow’s Theorems were a set of theorems proved around the same time the concept of group theory was being established, in the 1870s. Sylow used permutation groups in his proofs which were then later generalized and shown to hold true for all finite groups....
Kartavtsev, Alexander
2014-01-01
According to the Goldstone theorem a scalar theory with a spontaneously broken global symmetry contains strictly massless states. In this letter we identify a loophole in the current-algebra proof of the theorem. Therefore, the question whether in models with Mexican hat potential the tangential excitations are strictly massless or are just almost massless as compared to the radial ones remains open. We also argue that mass of the tangential excitations approaches zero even if the symmetry is not spontaneously broken but a combination of the field components invariant under the symmetry transformations acquires a large vacuum expectation value.
Bressler, Paul; Gorokhovsky, Alexander; Nest, Ryszard;
2015-01-01
The main result of the present paper is an analogue of Kontsevich formality theorem in the context of the deformation theory of gerbes. We construct an L∞L∞ deformation of the Schouten algebra of multi-vectors which controls the deformation theory of a gerbe.......The main result of the present paper is an analogue of Kontsevich formality theorem in the context of the deformation theory of gerbes. We construct an L∞L∞ deformation of the Schouten algebra of multi-vectors which controls the deformation theory of a gerbe....
Maximum principle and convergence of central schemes based on slope limiters
Mehmetoglu, Orhan
2012-01-01
A maximum principle and convergence of second order central schemes is proven for scalar conservation laws in dimension one. It is well known that to establish a maximum principle a nonlinear piecewise linear reconstruction is needed and a typical choice is the minmod limiter. Unfortunately, this implies that the scheme uses a first order reconstruction at local extrema. The novelty here is that we allow local nonlinear reconstructions which do not reduce to first order at local extrema and still prove maximum principle and convergence. © 2011 American Mathematical Society.
Normal Limits, Nonnormal Limits, and the Bootstrap for Quantiles of Dependent Data
Sharipov, O. Sh.; Wendler, M.
2012-01-01
We will show under very weak conditions on differentiability and dependence that the central limit theorem for quantiles holds and that the block bootstrap is weakly consistent. Under slightly stronger conditions, the bootstrap is strongly consistent. Without the differentiability condition, quantiles might have a non-normal asymptotic distribution and the bootstrap might fail.
Virial Theorem and Scale Transformations.
Kleban, Peter
1979-01-01
Discussed is the virial theorem, which is useful in classical, quantum, and statistical mechanics. Two types of derivations of this theorem are presented and the relationship between the two is explored. (BT)
Dalen, D. van
2008-01-01
The following pages make form a new chapter for the book Logic and Structure. This chapter deals with the incompleteness theorem, and contains enough basic material for the treatment of the required notions of computability, representability and the like. This chapter will appear in the next edition of Logic and Structure. Comments are welcome.
Dalen, D. van
2008-01-01
The following pages make form a new chapter for the book Logic and Structure. This chapter deals with the incompleteness theorem, and contains enough basic material for the treatment of the required notions of computability, representability and the like. This chapter will appear in the next edition
An Improved Subadditive Ergodic Theorem
Liggett, Thomas M.
1985-01-01
A new version of Kingman's subadditive ergodic theorem is presented, in which the subadditivity and stationarity assumptions are relaxed without weakening the conclusions. This result applies to a number of situations that were not covered by Kingman's original theorem. The proof involves a rather simple reduction to the additive case, where Birkhoff's ergodic theorem can be applied.
Abramovitz, Buma; Berezina, Miryam; Berman, Abraham; Shvartsman, Ludmila
2009-01-01
In this article we describe the process of studying the assumptions and the conclusion of a theorem. We tried to provide the students with exercises and problems where we discuss the following questions: What are the assumptions of a theorem and what are the conclusions? What is the geometrical meaning of a theorem? What happens when one or more…
Sampling theorems and compressive sensing on the sphere
McEwen, J D; Thiran, J -Ph; Vandergheynst, P; Van De Ville, D; Wiaux, Y
2011-01-01
We discuss a novel sampling theorem on the sphere developed by McEwen & Wiaux recently through an association between the sphere and the torus. To represent a band-limited signal exactly, this new sampling theorem requires less than half the number of samples of other equiangular sampling theorems on the sphere, such as the canonical Driscoll & Healy sampling theorem. A reduction in the number of samples required to represent a band-limited signal on the sphere has important implications for compressive sensing, both in terms of the dimensionality and sparsity of signals. We illustrate the impact of this property with an inpainting problem on the sphere, where we show superior reconstruction performance when adopting the new sampling theorem.
Virial theorem and hypervirial theorem in a spherical geometry
Li Yan; Chen Jingling [Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin 300071 (China); Zhang Fulin, E-mail: flzhang@tju.edu.cn, E-mail: chenjl@nankai.edu.cn [Physics Department, School of Science, Tianjin University, Tianjin 300072 (China)
2011-09-09
The virial theorem in the one- and two-dimensional spherical geometry are presented in both classical and quantum mechanics. Choosing a special class of hypervirial operators, the quantum hypervirial relations in the spherical spaces are obtained. With the aid of the Hellmann-Feynman theorem, these relations can be used to formulate a perturbation theorem without wavefunctions, corresponding to the hypervirial-Hellmann-Feynman theorem perturbation theorem of Euclidean geometry. The one-dimensional harmonic oscillator and two-dimensional Coulomb system in the spherical spaces are given as two sample examples to illustrate the perturbation method. (paper)
Carnot's theorem as Noether's theorem for thermoacoustic engines
Onset in thermoacoustic engines, the transition to spontaneous self-generation of oscillations, is studied here as both a dynamical critical transition and a limiting heat engine behavior. The critical transition is interesting because it occurs for both dissipative and conservative systems, with common scaling properties. When conservative, the stable oscillations above the critical point also implement a reversible engine cycle satisfying Carnot's theorem, a universal conservation law for entropy flux. While criticality in equilibrium systems is naturally associated with symmetries and universal conservation laws, these are usually exploited with global minimization principles, which dynamical critical systems may not have if dissipation is essential to their criticality. Acoustic heat engines furnish an example connecting equilibrium methods with dynamical and possibly even dissipative critical transitions: A reversible engine is shown to map, by a change of variables, to an equivalent system in apparent thermal equilibrium; a Noether symmetry in the equilibrium field theory implies Carnot's theorem for the engine. Under the same association, onset is shown to be a process of spontaneous symmetry breaking and the scaling of the quality factor predicted for both the reversible and irreversible engines is shown to arise from the Ginzburg-Landau description of the broken phase. copyright 1998 The American Physical Society
Fixed points, intersection theorems, variational inequalities, and equilibrium theorems
Sehie Park
2000-07-01
Full Text Available From a fixed point theorem for compact acyclic maps defined on admissible convex sets in the sense of Klee, we first deduce collectively fixed point theorems, intersection theorems for sets with convex sections, and quasi-equilibrium theorems. These quasi-equilibrium theorems are applied to give simple and unified proofs of the known variational inequalities of the Hartman-Stampacchia-Browder type. Moreover, from our new fixed point theorem, we deduce new variational inequalities which can be used to obtain fixed point results for convex-valued maps. Finally, various general economic equilibrium theorems are deduced in the forms of the Nash type, the Tarafdar type, and the Yannelis-Prabhakar type. Our results are stated for not-necessarily locally convex topological vector spaces and for abstract economies with arbitrary number of commodities and agents. Our new results extend a lot of known works with much simpler proofs.
Weak Characterizations of Stochastic Integrability and Dudley's Theorem in Infinite Dimensions
Ondreját, Martin; Veraar, M.
2014-01-01
Roč. 27, č. 4 (2014), s. 1350-1374. ISSN 0894-9840 R&D Projects: GA ČR GAP201/10/0752 Institutional support: RVO:67985556 Keywords : stochastic integration in Banach spaces * almost sure limit theorems * Dudley representation theorem * universal representation theorem * weak characterization of stochastic integrability * Doob representation theorem Subject RIV: BA - General Mathematics Impact factor: 0.857, year: 2014 http://library.utia.cas.cz/separaty/2013/SI/ondrejat-0392394.pdf
Thermal Tachyons and the "g"-Theorem
Chaudhuri, Shyamoli
2002-01-01
We give a pedagogical introduction to Affleck and Ludwig's g-theorem, distinguishing its applications in field theory vs string theory. We clarify the recent proposal that the vacuum degeneracy $g$ of a noncompact worldsheet sigma model with a continuous spectrum of scaling dimensions is lowered under renormalization group flow while preserving the central charge. As an illustration we argue that the IR stable endpoint of the relevant flow of the worldsheet RG induced by a thermal tachyon in ...
Jeffrey J. Opperman
2010-03-01
Full Text Available Hydropower dam construction is expanding rapidly in Central America because of the increasing demand for electricity. Although hydropower can provide a low-carbon source of energy, dams can also degrade socially valued riverine and riparian ecosystems and the services they provide. Such degradation can be partially mitigated by the release of environmental flows below dams. However, environmental flows have been applied infrequently to dams in Central America, partly because of the lack of information on the ecological, social, and economic aspects of rivers. This paper presents a case study of how resource and information limitations were addressed in the development of environmental flow recommendations for the Patuca River in Honduras below a proposed hydroelectric dam. To develop flow recommendations, we applied a multistep process that included hydrological analysis and modeling, the collection of traditional ecological knowledge (TEK during field trips, expert consultation, and environmental flow workshops for scientists, water managers, and community members. The final environmental flow recommendation specifies flow ranges for different components of river hydrology, including low flows for each month, high-flow pulses, and floods, in dry, normal, and wet years. The TEK collected from local and indigenous riverine communities was particularly important for forming hypotheses about flow-dependent ecological and social factors that may be vulnerable to disruption from dam-modified river flows. We show that our recommended environmental flows would have a minimal impact on the dam's potential to generate electricity. In light of rapid hydropower development in Central America, we suggest that environmental flows are important at the local scale, but that an integrated landscape perspective is ultimately needed to pursue hydropower development in a manner that is as ecologically sustainable as possible.
Taylor, Marika
2016-01-01
The F theorem states that, for a unitary three dimensional quantum field theory, the F quantity defined in terms of the partition function on a three sphere is positive, stationary at fixed point and decreases monotonically along a renormalization group flow. We construct holographic renormalization group flows corresponding to relevant deformations of three-dimensional conformal field theories on spheres, working to quadratic order in the source. For these renormalization group flows, the F quantity at the IR fixed point is always less than F at the UV fixed point, but F increases along the RG flow for deformations by operators of dimension $3/2 < \\Delta < 5/2$. Therefore the strongest version of the F theorem is in general violated.
N V Rao
2003-02-01
The general theme of this note is illustrated by the following theorem: Theorem 1. Suppose is a compact set in the complex plane and 0 belongs to the boundary . Let $\\mathcal{A}(K)$ denote the space of all functions on such that is holomorphic in a neighborhood of and (0) = 0. Also for any given positive integer , let $\\mathcal{A}(m, K)$ denote the space of all such that is holomorphic in a neighborhood of and $f(0) = f'(0) = \\cdots = f^{(m)}(0) = 0$. Then $\\mathcal{A}(m, K)$ is dense in $\\mathcal{A}(K)$ under the supremum norm on provided that there exists a sector $W = \\{re^{i}; 0 ≤ r ≤ , ≤ ≤ \\}$ such that $W \\cap K = \\{0\\}$. (This is the well-known Poincare's external cone condition).} We present various generalizations of this result in the context of higher dimensions replacing holomorphic with harmonic.
2012-01-01
At present,there are many factors limiting large area centralized,rapid development,and moderately large-scale land operation in China.These factors include(i) the existing land utilization system is still at adaptation stage,and it lacks universal agreement of people on large-scale land operation;(ii) farmers’ dependence on land is great;(iii) it is difficult to transfer surplus labor;(iv) there is no positive connection between promotion of moderately large-scale land operation and realization of increase of farmers’ income;(v) it remains to be proved whether large-scale operation can become a stable rural occupation and whether big farming households can grow to professional farmers;(vi) large-scale land operation in rural areas may lead to waste of resources;(vii) the promotion of large-scale land operation may cause other social contradictions.
A New Simple Approach for Entropy and Carnot Theorem
Entropy and Carnot theorem occupy central place in the typical Thermodynamics courses at the university level. In this work, we suggest a new simple approach for introducing the concept of entropy. Using simple procedure in TV plane, we proved that for reversible processes ∫dQ/T=0 and it is sufficient to define entropy. And also, using reversible processes in TS plane, we give an alternative simple proof for Carnot theorem
Stephen A. Ross
2011-01-01
We can only estimate the distribution of stock returns but we observe the distribution of risk neutral state prices. Risk neutral state prices are the product of risk aversion - the pricing kernel - and the natural probability distribution. The Recovery Theorem enables us to separate these and to determine the market's forecast of returns and the market's risk aversion from state prices alone. Among other things, this allows us to determine the pricing kernel, the market risk premium, the pro...
Vela Velupillai, K.
2011-01-01
Takashi Negishi's remarkable youthful contribution to welfare economics, general equilibrium theory and, with the benefit of hindsight, also to one strand of computable general equilibrium theory, all within the span of six pages in one article, has become one of the modern classics of general equilibrium theory and mathematical economics. Negishi's celebrated theorem and what has been called Negishi's Method have formed one foundation for computable general equilibrium theory. In this paper ...
Coevolution. Extending Prigogine Theorem
Leon, Antonio
2006-01-01
The formal consideration of the concept of interaction in thermodynamic analysis makes it possible to deduce, in the broadest terms, new results related to the coevolution of interacting systems, irrespective of their distance from thermodynamic equilibrium. In this paper I prove the existence of privileged coevolution trajectories characterized by the minimum joint production of internal entropy, a conclusion that extends Prigogine theorem to systems evolving far from thermodynamic equilibri...
No ghost theorem and cohomology theorem for strings in arbitrary static backgrounds
This paper considers a string moving in an arbitrary time-independent background given by an arbitrary conformal field theory of appropriate central charge (e.g., c = 25 for bosonic string) and one flat time-like dimension. The authors show that the physical subspace of the Hilbert space is positive semi-definite (no ghost theorem) and that the cohomology of the BRST operator is trivial except for the ghost number one (for open bosonic string) sector (cohomology theorem). Both the proofs are reductio ad absurdum proofs based on the corresponding theorems for the strings moving in flat background. In cases where there is an extra flat space-like dimension (besides the flat time-like one), the transverse subspace with positive-definite norm can be constructed
Bourgain's discretization theorem
Giladi, Ohad; Schechtman, Gideon
2011-01-01
Bourgain's discretization theorem asserts that there exists a universal constant $C\\in (0,\\infty)$ with the following property. Let $X,Y$ be Banach spaces with $\\dim X=n$. Fix $D\\in (1,\\infty)$ and set $\\d= e^{-n^{Cn}}$. Assume that $\\mathcal N$ is a $\\d$-net in the unit ball of $X$ and that $\\mathcal N$ admits a bi-Lipschitz embedding into $Y$ with distortion at most $D$. Then the entire space $X$ admits a bi-Lipschitz embedding into $Y$ with distortion at most $CD$. This mostly expository article is devoted to a detailed presentation of a proof of Bourgain's theorem. We also obtain an improvement of Bourgain's theorem in the important case when $Y=L_p$ for some $p\\in [1,\\infty)$: in this case it suffices to take $\\delta= C^{-1}n^{-5/2}$ for the same conclusion to hold true. The case $p=1$ of this improved discretization result has the following consequence. For arbitrarily large $n\\in \\N$ there exists a family $\\mathscr Y$ of $n$-point subsets of ${1,...,n}^2\\subseteq \\R^2$ such that if we write $|\\mathscr ...
The Non-Signalling theorem in generalizations of Bell's theorem
Walleczek, Jan; Groessing, Gerhard
2014-01-01
Does "epistemic non-signalling" ensure the peaceful coexistence of special relativity and quantum nonlocality? The possibility of an affirmative answer is of great importance to deterministic approaches to quantum mechanics given recent developments towards generalizations of Bell's theorem. By generalizations of Bell's theorem we here mean efforts that seek to demonstrate the impossibility of any deterministic theories to obey the predictions of Bell's theorem, including even nonlocal hidden...
Tau leaping of stiff stochastic chemical systems via local central limit approximation
Stiffness manifests in stochastic dynamic systems in a more complex manner than in deterministic systems; it is not only important for a time-stepping method to remain stable but it is also important for the method to capture the asymptotic variances accurately. In the context of stochastic chemical systems, time stepping methods are known as tau leaping. Well known existing tau leaping methods have shortcomings in this regard. The implicit tau method is far more stable than the trapezoidal tau method but underestimates the asymptotic variance. On the other hand, the trapezoidal tau method which estimates the asymptotic variance exactly for linear systems suffers from the fact that the transients of the method do not decay fast enough in the context of very stiff systems. We propose a tau leaping method that possesses the same stability properties as the implicit method while it also captures the asymptotic variance with reasonable accuracy at least for the test system S1↔S2. The proposed method uses a central limit approximation (CLA) locally over the tau leaping interval and is referred to as the LCLA-τ. The CLA predicts the mean and covariance as solutions of certain differential equations (ODEs) and for efficiency we solve these using a single time step of a suitable low order method. We perform a mean/covariance stability analysis of various possible low order schemes to determine the best scheme. Numerical experiments presented show that LCLA-τ performs favorably for stiff systems and that the LCLA-τ is also able to capture bimodal distributions unlike the CLA itself. The proposed LCLA-τ method uses a split implicit step to compute the mean update. We also prove that any tau leaping method employing a split implicit step converges in the fluid limit to the implicit Euler method as applied to the fluid limit differential equation
Tautenhahn, Susanne; Lichstein, Jeremy W; Jung, Martin; Kattge, Jens; Bohlman, Stephanie A; Heilmeier, Hermann; Prokushkin, Anatoly; Kahl, Anja; Wirth, Christian
2016-06-01
Fire is a primary driver of boreal forest dynamics. Intensifying fire regimes due to climate change may cause a shift in boreal forest composition toward reduced dominance of conifers and greater abundance of deciduous hardwoods, with potential biogeochemical and biophysical feedbacks to regional and global climate. This shift has already been observed in some North American boreal forests and has been attributed to changes in site conditions. However, it is unknown if the mechanisms controlling fire-induced changes in deciduous hardwood cover are similar among different boreal forests, which differ in the ecological traits of the dominant tree species. To better understand the consequences of intensifying fire regimes in boreal forests, we studied postfire regeneration in five burns in the Central Siberian dark taiga, a vast but poorly studied boreal region. We combined field measurements, dendrochronological analysis, and seed-source maps derived from high-resolution satellite images to quantify the importance of site conditions (e.g., organic layer depth) vs. seed availability in shaping postfire regeneration. We show that dispersal limitation of evergreen conifers was the main factor determining postfire regeneration composition and density. Site conditions had significant but weaker effects. We used information on postfire regeneration to develop a classification scheme for successional pathways, representing the dominance of deciduous hardwoods vs. evergreen conifers at different successional stages. We estimated the spatial distribution of different successional pathways under alternative fire regime scenarios. Under intensified fire regimes, dispersal limitation of evergreen conifers is predicted to become more severe, primarily due to reduced abundance of surviving seed sources within burned areas. Increased dispersal limitation of evergreen conifers, in turn, is predicted to increase the prevalence of successional pathways dominated by deciduous hardwoods
NONE
1998-12-01
From 1996, the Norwegian and Swedish power markets were joined and a common power exchange was established. The two countries deal differently with bottlenecks (transmission obstruction) in their central networks. This report compares methods for dealing with such bottlenecks and looks at the alternatives. It emphasises the efficiency of pricing and incentives and the possibility of exercising market power under the different methods. Norway uses a method of price regions, or bottleneck tax. Prices are determined for the various price regions so as to keep the power flow below specified bounds. A surplus region is assigned a lower price than a deficit region and the bottleneck tax is the difference in price between two such price regions. The Swedish system is based on a counter purchase concept. In his offer to the spotmarket, the supplier has bound himself to provide a certain amount to the current system price regardless of network limitations. Up-regulation means that he produces more than this amount. Down-regulation means that he is paid for supplying less than he had offered to the current system price. In up- or down-regulation, compensation is given as the difference between the system price and the price on the counter purchase market. The main conclusions are: (1) Counter purchase is unsuitable as the main strategy for Norway. (2) Counter purchase may be suitable with short-lived and unpredicted bottlenecks; price regions may be suitable for long-lasting and predicted bottlenecks. Time is a central factor. (3) Present-day models for bottleneck management in Norway and Sweden do not give the optimum short-term load distribution on the network. In general, the current Norwegian system works fairly well, although it might be worthwhile to consider a system that approaches node pricing. 3 refs., 34 figs., 3 tabs.
Food limitation of sea lion pups and the decline of forage off central and southern California.
McClatchie, Sam; Field, John; Thompson, Andrew R; Gerrodette, Tim; Lowry, Mark; Fiedler, Paul C; Watson, William; Nieto, Karen M; Vetter, Russell D
2016-03-01
California sea lions increased from approximately 50 000 to 340 000 animals in the last 40 years, and their pups are starving and stranding on beaches in southern California, raising questions about the adequacy of their food supply. We investigated whether the declining sea lion pup weight at San Miguel rookery was associated with changes in abundance and quality of sardine, anchovy, rockfish and market squid forage. In the last decade off central California, where breeding female sea lions from San Miguel rookery feed, sardine and anchovy greatly decreased in biomass, whereas market squid and rockfish abundance increased. Pup weights fell as forage food quality declined associated with changes in the relative abundances of forage species. A model explained 67% of the variance in pup weights using forage from central and southern California and 81% of the variance in pup weights using forage from the female sea lion foraging range. A shift from high to poor quality forage for breeding females results in food limitation of the pups, ultimately flooding animal rescue centres with starving sea lion pups. Our study is unusual in using a long-term, fishery-independent dataset to directly address an important consequence of forage decline on the productivity of a large marine predator. Whether forage declines are environmentally driven, are due to a combination of environmental drivers and fishing removals, or are due to density-dependent interactions between forage and sea lions is uncertain. However, declining forage abundance and quality was coherent over a large area (32.5-38° N) for a decade, suggesting that trends in forage are environmentally driven. PMID:27069651
Stability of Gas Clouds in Galactic Nuclei: An Extended Virial Theorem
Chen, Xian; Cuadra, Jorge
2015-01-01
Cold gas entering the central $1$ to $10^2$ pc of a galaxy fragments and condenses into clouds. The stability of the clouds determines whether they will be turned into stars or can be delivered to the central supermassive black hole (SMBH) to turn on an active galactic nucleus (AGN). The conventional criteria to assess the stability of these clouds, such as the Jeans criterion and Roche (or tidal) limit, are insufficient here, because they assume the dominance of self-gravity in binding a cloud, and neglect external agents, such as pressure and tidal forces, which are common in galactic nuclei. We formulate a new scheme for judging this stability. We first revisit the conventional Virial theorem, taking into account an external pressure, to identify the correct range of masses that lead to stable clouds. We then extend the theorem to include an external tidal field, crucial for the stability in the region of interest -- in dense star clusters, around SMBHs. We apply our extended Virial theorem to find the cor...
A Limit Theorem for Double Arrays
Rosalsky, Andrew; Teicher, Henry
1981-01-01
The main result establishes that row sums $S_n$ of a double array of rowwise independent, infinitesimal (or merely uniformly asymptotically constant) random variables satisfying $\\lim \\sup |S_n - M_n| \\leq M_0 < \\infty$ a.c. (for some choice of constants $M_n$), obey a weak law of large numbers, i.e., $S_n - \\operatorname{med} S_n$ converges in probability to 0. No moment assumptions are imposed on the individual summands and zero-one laws are unavailable. As special cases, a new result for w...
A new VLA/e-MERLIN limit on central images in the gravitational lens system CLASS B1030+074
Quinn, Jonathan; Jackson, Neal; Tagore, Amitpal; Biggs, Andrew; Birkinshaw, Mark; Chapman, Scott; De Zotti, Gianfranco; McKean, John; Pérez-Fournon, Ismael; Scott, Douglas; Serjeant, Stephen
2016-07-01
We present the new Very Large Array 22 GHz and extended Multi-Element Remote-Linked Interferometer Network 5 GHz observations of CLASS B1030+074, a two-image strong gravitational lens system whose background source is a compact flat-spectrum radio quasar. In such systems we expect a third image of the background source to form close to the centre of the lensing galaxy. The existence and brightness of such images is important for investigation of the central mass distributions of lensing galaxies, but only one secure detection has been made so far in a galaxy-scale lens system. The noise levels achieved in our new B1030+074 images reach 3 μJy beam-1 and represent an improvement in central image constraints of nearly an order of magnitude over previous work, with correspondingly better resulting limits on the shape of the central mass profile of the lensing galaxy. Simple models with an isothermal outer power-law slope now require either the influence of a central supermassive black hole (SMBH), or an inner power-law slope very close to isothermal, in order to suppress the central image below our detection limit. Using the central mass profiles inferred from light distributions in Virgo galaxies, moved to z = 0.5, and matching to the observed Einstein radius, we now find that 45 per cent of such mass profiles should give observable central images, 10 per cent should give central images with a flux density still below our limit, and the remaining systems have extreme demagnification produced by the central SMBH. Further observations of similar objects will therefore allow proper statistical constraints to be placed on the central properties of elliptical galaxies at high redshift.
Arrow's Theorem in Judgement Aggregation
Franz Dietrich; Christian List
2005-01-01
In response to recent work on the aggregation of individual judgements on logically connected propositions into collective judgements, it is often asked whether judgement aggregation is a special case of Arrowian preference aggregation. We argue the opposite. After proving a general impossibility theorem, we construct an embedding of preference aggregation into judgement aggregation and prove Arrow's theorem as a corollary of our result. Although we provide a new proof of Arrow's theorem, our...
Perspectives on the CAP Theorem
Gilbert, Seth; Lynch, Nancy Ann
2012-01-01
Almost twelve years ago, in 2000, Eric Brewer introduced the idea that there is a fundamental trade-off between consistency, availability, and partition tolerance. This trade-off, which has become known as the CAP Theorem, has been widely discussed ever since. In this paper, we review the CAP Theorem and situate it within the broader context of distributed computing theory. We then discuss the practical implications of the CAP Theorem, and explore some general techniques for coping with the i...
A theorem in relativistic electronics
Yongjian, Yu
1990-04-01
This paper presents a theorem that connects the dispersion relation of the Electron Cyclotron Maser' and the oscillation equation of the Gyromonotron. This theorem gives us a simple way of obtaining the osscillating characteristics of the Gyromonotron provided that dispersion relation of the ECRM is given. Though the theorem is proved only with the case of ECRM and Gyromonotron, it holds for other kinds of Electron Masers, FEL4etc. and corresponding osscillators.
We in the nuclear power industry consider ourselves to be at the forefront of civilised progress. Yet, all too often, even we ourselves don't believe our public relations statements about nuclear power. Why is this? Let us approach the question by considering Godel's Theorem. Godel's Theorem is extremely complicated mathematically, but for our purposes can be simplified to the maxim that one cannot validate a system from within that system. Scientists, especially those in the fields of astronomy and nuclear physics, have long realised the implications of Godel's Theorem. The people to whom we must communicate look to us, who officially know everything about our industry, to comfort and reassure them. And we forget that we can only comfort them by addressing their emotional needs, not by demonstrating our chilling objectivity. Let us try something completely new in communication. Instead of looking for incremental rules which will help us marginally differentiate the way we communicate about minor or major incidents, let us leapfrog across 'objectivity' to meaning and relevance. If we truly believe that nuclear energy is a good thing, this leap should not be difficult. Finally, if we as communicators are not prepared to be meaningful and relevant - not prepared to leapfrog beyond weasel terms like 'minor incident' - what does that say about the kinds of people we believe the nuclear community to be? Are nuclear people a group apart, divisible from the rest of the human race by their evil? In fact the nuclear community is a living, laughing, normal part of a whole society; and is moreover a good contributor to the technological progress that society demands. When we ourselves recognise this, we will start to communicate nuclear issues in the same language as the rest of society. We will start to speak plainly and convincingly, and our conviction will leapfrog our audience into being able to believe us
Cobham's theorem for substitutions
Durand, Fabien
2010-01-01
The seminal theorem of Cobham has given rise during the last 40 years to a lot of works around non-standard numeration systems and has been extended to many contexts. In this paper, as a result of fifteen years of improvements, we obtain a complete and general version for the so-called substitutive sequences. Let $\\alpha$ and $\\beta$ be two multiplicatively independent Perron numbers. Then, a sequence $x\\in A^\\mathbb{N}$, where $A$ is a finite alphabet, is both $\\alpha$-substitutive and $\\beta$-substitutive if and only if $x$ is ultimately periodic.
Fluctuation theorem in spintronics
Microscopic reversibility is a key in deriving the Onsager relation. It even leads a new exact relationship that would be valid far from equilibrium, called fluctuation theorem (FT). The FT provides a precise statement for the second law of thermodynamics; and remarkably, reproduces the linear response theory. We consider the FT in the spin-dependent transport and derive universal relations among nonlinear spin and charge transport coefficients. We apply the relations to a quantum dot embedded in a two-terminal Aharonov-Bohm interferometer and check that the relations are satisfied.
Reiher, Christian
2012-01-01
Tur\\'{a}n's theorem is a cornerstone of extremal graph theory. It asserts that for any integer $r \\geq 2$ every graph on $n$ vertices with more than ${\\tfrac{r-2}{2(r-1)}\\cdot n^2}$ edges contains a clique of size $r$, i.e., $r$ mutually adjacent vertices. The corresponding extremal graphs are balanced $(r-1)$-partite graphs. The question as to how many such $r$-cliques appear at least in any $n$-vertex graph with $\\gamma n^2$ edges has been intensively studied in the literature. In particula...
Limits to physiological plasticity of the coral Pocillopora verrucosa from the central Red Sea
Ziegler, M.
2014-07-26
Many coral species display changing distribution patterns across coral reef depths. While changes in the underwater light field and the ability to associate with different photosynthetic symbionts of the genus Symbiodinium explain some of the variation, the limits to physiological plasticity are unknown for most corals. In the central Red Sea, colonies of the branching coral Pocillopora verrucosa are most abundant in shallow high light environments and become less abundant in water depths below 10 m. To further understand what determines this narrow distribution, we conducted a cross-depths transplant experiment looking at physiological plasticity and acclimation in regard to depth. Colonies from 5, 10, and 20 m were collected, transplanted to all depths, and re-investigated after 30 and 210 d. All coral colonies transplanted downward from shallow to deep water displayed an increase in photosynthetic light-harvesting pigments, which resulted in higher photosynthetic efficiency. Shallow-water specimens transplanted to deeper water showed a significant decrease in total protein content after 30 and 210 d under low light conditions compared to specimens transplanted to shallow and medium depths. Stable isotope data suggest that heterotrophic input of carbon was not increased under low light, and consequently, decreasing protein levels were symptomatic of decreasing photosynthetic rates that could not be compensated for through higher light-harvesting efficiency. Our results provide insights into the physiological plasticity of P. verrucosa in changing light regimes and explain the observed depth distribution pattern. Despite its high abundance in shallow reef waters, P. verrucosa possesses limited heterotrophic acclimation potential, i.e., the ability to support its mainly photoautotrophic diet through heterotrophic feeding. We conclude that P. verrucosa might be a species vulnerable to sudden changes in underwater light fields resulting from processes such as
Limits to physiological plasticity of the coral Pocillopora verrucosa from the central Red Sea
Ziegler, Maren; Roder, Cornelia M.; Büchel, Claudia; Voolstra, Christian R.
2014-12-01
Many coral species display changing distribution patterns across coral reef depths. While changes in the underwater light field and the ability to associate with different photosynthetic symbionts of the genus Symbiodinium explain some of the variation, the limits to physiological plasticity are unknown for most corals. In the central Red Sea, colonies of the branching coral Pocillopora verrucosa are most abundant in shallow high light environments and become less abundant in water depths below 10 m. To further understand what determines this narrow distribution, we conducted a cross-depths transplant experiment looking at physiological plasticity and acclimation in regard to depth. Colonies from 5, 10, and 20 m were collected, transplanted to all depths, and re-investigated after 30 and 210 d. All coral colonies transplanted downward from shallow to deep water displayed an increase in photosynthetic light-harvesting pigments, which resulted in higher photosynthetic efficiency. Shallow-water specimens transplanted to deeper water showed a significant decrease in total protein content after 30 and 210 d under low light conditions compared to specimens transplanted to shallow and medium depths. Stable isotope data suggest that heterotrophic input of carbon was not increased under low light, and consequently, decreasing protein levels were symptomatic of decreasing photosynthetic rates that could not be compensated for through higher light-harvesting efficiency. Our results provide insights into the physiological plasticity of P. verrucosa in changing light regimes and explain the observed depth distribution pattern. Despite its high abundance in shallow reef waters, P. verrucosa possesses limited heterotrophic acclimation potential, i.e., the ability to support its mainly photoautotrophic diet through heterotrophic feeding. We conclude that P. verrucosa might be a species vulnerable to sudden changes in underwater light fields resulting from processes such as
In this communication we establish stochastic limit laws leading from Zipf's law to Pareto's and Heaps' laws. We consider finite ensembles governed by Zipf's law and study their asymptotic statistics as the ensemble size tends to infinity. A Lorenz-curve analysis establishes three types of limit laws for the ensembles' statistical structure: 'communist', 'monarchic', and Paretian. Further considering a dynamic setting in which the ensembles grow stochastically in time, a functional central limit theorem analysis establishes a Gaussian approximation for the ensembles' stochastic growth. The Gaussian approximation provides a generalized and corrected formulation of Heaps' law. (fast track communication)
Theorems on Positive Data: On the Uniqueness of NMF
Lauerberg, Hans; Christensen, Mads Græsbøll; Pumbley, Mark; Hansen, Lars Kai; Jensen, Søren Holdt
2008-01-01
We investigate the conditions for which nonnegative matrix factorization (NMF) is unique and introduce several theorems which can determine whether the decomposition is in fact unique or not. The theorems are illustrated by several examples showing the use of the theorems and their limitations. W...... have shown that corruption of a unique NMF matrix by additive noise leads to a noisy estimation of the noise-free unique solution. Finally, we use a stochastic view of NMF to analyze which characterization of the underlying model will result in an NMF with small estimation errors....
Theorems on Positive Data: On the Uniqueness of NMF
Mads Græsbøll Christensen
2008-05-01
Full Text Available We investigate the conditions for which nonnegative matrix factorization (NMF is unique and introduce several theorems which can determine whether the decomposition is in fact unique or not. The theorems are illustrated by several examples showing the use of the theorems and their limitations. We have shown that corruption of a unique NMF matrix by additive noise leads to a noisy estimation of the noise-free unique solution. Finally, we use a stochastic view of NMF to analyze which characterization of the underlying model will result in an NMF with small estimation errors.
Experimental studies of the transient fluctuation theorem using liquid crystals
Soma Datta; Arun Roy
2009-05-01
In a thermodynamical process, the dissipation or production of entropy can only be positive or zero, according to the second law of thermodynamics. However, the laws of thermodynamics are applicable to large systems in the thermodynamic limit. Recently a fluctuation theorem, known as the transient fluctuation theorem (TFT), which generalizes the second law of thermodynamics to small systems has been proposed. This theorem has been tested in small systems such as a colloidal particle in an optical trap. We report for the first time an analogous experimental study of TFT in a spatially extended system using liquid crystals.
Weinberg Soft Theorems from Weinberg Adiabatic Modes
Mirbabayi, Mehrdad
2016-01-01
Soft theorems for the scattering of low energy photons and gravitons and cosmological consistency conditions on the squeezed-limit correlation functions are both understood to be consequences of invariance under large gauge transformations. We apply the same method used in cosmology -- based on the identification of an infinite set of "adiabatic modes" and the corresponding conserved currents -- to derive flat space soft theorems for electrodynamics and gravity. We discuss how the recent derivations based on the asymptotic symmetry groups (BMS) can be continued to a finite size sphere surrounding the scattering event, when the soft photon or graviton has a finite momentum. We give a finite distance derivation of the antipodal matching condition previously imposed between future and past null infinities, and explain why all but one radiative degrees of freedom decouple in the soft limit. In contrast to earlier works on BMS, we work with adiabatic modes which correspond to large gauge transformations that are $...
OTTER, Resolution Style Theorem Prover
1 - Description of program or function: OTTER (Other Techniques for Theorem-proving and Effective Research) is a resolution-style theorem-proving program for first-order logic with equality. OTTER includes the inference rules binary resolution, hyper-resolution, UR-resolution, and binary para-modulation. These inference rules take as small set of clauses and infer a clause. If the inferred clause is new and useful, it is stored and may become available for subsequent inferences. Other capabilities are conversion from first-order formulas to clauses, forward and back subsumption, factoring, weighting, answer literals, term ordering, forward and back demodulation, and evaluable functions and predicates. 2 - Method of solution: For its inference process OTTER uses the given-clause algorithm, which can be viewed as a simple implementation of the set of support strategy. OTTER maintains three lists of clauses: axioms, sos (set of support), and demodulators. OTTER is not automatic. Even after the user has encoded a problem into first-order logic or into clauses, the user must choose inference rules, set options to control the processing of inferred clauses, and decide which input formulae or clauses are to be in the initial set of support and which, if any, equalities are to be demodulators. If OTTER fails to find a proof, the user may try again different initial conditions. 3 - Restrictions on the complexity of the problem - Maxima of: 5000 characters in an input string, 64 distinct variables in a clause, 51 characters in any symbol. The maxima can be changed by finding the appropriate definition in the header.h file, increasing the limit, and recompiling OTTER. There are a few constraints on the order of commands
A volume-limited sample of X-ray galaxy groups and clusters: III. Central abundance drops
Panagoulia, Electra K; Fabian, Andy C
2014-01-01
We present the results of a search and study of central abundance drops in a volume-limited sample (z<=0.071) of 101 X-ray galaxy groups and clusters. These are best observed in nearby, and so best resolved, groups and clusters, making our sample ideal for their detection. Out of the 65 groups and clusters in our sample for which we have abundance profiles, 8 of them have certain central abundance drops, with possible central abundance drops in another 6. All sources with central abundance drops have X-ray cavities, and all bar one exception have a central cooling time <=1 Gyr. These central abundance drops can be generated if the iron injected by stellar mass loss processes in the core of these sources is in grains, which then become incorporated in the central dusty filaments. These, in turn, are dragged outwards by the bubbling feedback process in these sources. We find that data quality significantly affects the detection of central abundance drops, inasmuch as a higher number of counts in the centr...
Abelian theorems for Whittaker transforms
Richard D. Carmichael
1987-01-01
Full Text Available Initial and final value Abelian theorems for the Whittaker transform of functions and of distributions are obtained. The Abelian theorems are obtained as the complex variable of the transform approaches 0 or ∞ in absolute value inside a wedge region in the right half plane.
Geometry of the Adiabatic Theorem
Lobo, Augusto Cesar; Ribeiro, Rafael Antunes; Ribeiro, Clyffe de Assis; Dieguez, Pedro Ruas
2012-01-01
We present a simple and pedagogical derivation of the quantum adiabatic theorem for two-level systems (a single qubit) based on geometrical structures of quantum mechanics developed by Anandan and Aharonov, among others. We have chosen to use only the minimum geometric structure needed for the understanding of the adiabatic theorem for this case.…
A continuous mapping theorem for the smallest argmax functional
Seijo, Emilio; Sen, Bodhisattva
2011-01-01
This paper introduces a version of the argmax continuous mapping theorem that applies to M-estimation problems in which the objective functions converge to a limiting process with multiple maximizers. The concept of the smallest maximizer of a function in the d-dimensional Skorohod space is introduced and its main properties are studied. The resulting continuous mapping theorem is applied to three problems arising in change-point regression analysis. Some of the results proved in connection t...
Energy Budget and the Virial Theorem in Interstellar Clouds
Vazquez-Semadeni, Enrique
1997-01-01
The Virial Thoerem is a mathematical expression obtained from the equation of motion for a fluid, which describes the energy budget of particular regions within the flow. This course reviews the basic theory leading to the Virial Theorem, discusses its applicability and limitations, and then summarizes observational results concerning the physical and statistical properties of interstellar clouds which are normally understood in terms of the Virial Theorem, in particular the so-called ``Larso...
A limited role for suppression in the central field of individuals with strabismic amblyopia.
Brendan T Barrett
Full Text Available BACKGROUND: Although their eyes are pointing in different directions, people with long-standing strabismic amblyopia typically do not experience double-vision or indeed any visual symptoms arising from their condition. It is generally believed that the phenomenon of suppression plays a major role in dealing with the consequences of amblyopia and strabismus, by preventing images from the weaker/deviating eye from reaching conscious awareness. Suppression is thus a highly sophisticated coping mechanism. Although suppression has been studied for over 100 years the literature is equivocal in relation to the extent of the retina that is suppressed, though the method used to investigate suppression is crucial to the outcome. There is growing evidence that some measurement methods lead to artefactual claims that suppression exists when it does not. METHODOLOGY/RESULTS: Here we present the results of an experiment conducted with a new method to examine the prevalence, depth and extent of suppression in ten individuals with strabismic amblyopia. Seven subjects (70% showed no evidence whatsoever for suppression and in the three individuals who did (30%, the depth and extent of suppression was small. CONCLUSIONS: Suppression may play a much smaller role in dealing with the negative consequences of strabismic amblyopia than previously thought. Whereas recent claims of this nature have been made only in those with micro-strabismus our results show extremely limited evidence for suppression across the central visual field in strabismic amblyopes more generally. Instead of suppressing the image from the weaker/deviating eye, we suggest the visual system of individuals with strabismic amblyopia may act to maximise the possibilities for binocular co-operation. This is consistent with recent evidence from strabismic and amblyopic individuals that their binocular mechanisms are intact, and that, just as in visual normals, performance with two eyes is better than
Andreev's Theorem on hyperbolic polyhedra
Roeder, R K W; Dunbar, W D; Roeder, Roland K. W.; Hubbard, John H.; Dunbar, William D.
2004-01-01
In 1970, E. M. Andreev published a classification of all three-dimensional compact hyperbolic polyhedra having non-obtuse dihedral angles. Given a combinatorial description of a polyhedron, $C$, Andreev's Theorem provides five classes of linear inequalities, depending on $C$, for the dihedral angles, which are necessary and sufficient conditions for the existence of a hyperbolic polyhedron realizing $C$ with the assigned dihedral angles. Andreev's Theorem also shows that the resulting polyhedron is unique, up to hyperbolic isometry. Andreev's Theorem is both an interesting statement about the geometry of hyperbolic 3-dimensional space, as well as a fundamental tool used in the proof for Thurston's Hyperbolization Theorem for 3-dimensional Haken manifolds. It is also remarkable to what level the proof of Andreev's Theorem resembles (in a simpler way) the proof of Thurston. We correct a fundamental error in Andreev's proof of existence and also provide a readable new proof of the other parts of the proof of And...
Some Theorems on Generalized Basic Hypergeometric Series
A. D. Wadhwa
1972-07-01
Full Text Available In an earlier paper the author has established two theorems on generalized hypergeometric functions. In each theorem a numerator differs from a denominator by a positive integer. These theorems were further used to prove some theorems on the sums of Kampe de Feriet functions. Here, we have established the theorems which are the basic analogues of the theorems proved in the earlier paper.
Combinatorial Reciprocity Theorems
Beck, Matthias
2012-01-01
A common theme of enumerative combinatorics is formed by counting functions that are polynomials evaluated at positive integers. In this expository paper, we focus on four families of such counting functions connected to hyperplane arrangements, lattice points in polyhedra, proper colorings of graphs, and $P$-partitions. We will see that in each instance we get interesting information out of a counting function when we evaluate it at a \\emph{negative} integer (and so, a priori the counting function does not make sense at this number). Our goals are to convey some of the charm these "alternative" evaluations of counting functions exhibit, and to weave a unifying thread through various combinatorial reciprocity theorems by looking at them through the lens of geometry, which will include some scenic detours through other combinatorial concepts.
2010-07-02
... Economic Zone Off Alaska; Central Gulf of Alaska License Limitation Program; Amendment 86 AGENCY: National... the Gulf of Alaska (FMP) to NMFS for review. If approved, Amendment 86 would add a Pacific cod... of Amendment 86 to the Fishery Management Plan for Groundfish of the Gulf of Alaska, and...
POTENTIAL AND LIMITS OF RENEWABLE ENERGY IN THE CENTRAL AND SOUTH-EAST EUROPE REGION
CIRLEA Filip; Iancu, Iulian
2012-01-01
Renewable energy sources (solar power, wind power, hydroenergy, biomass, biofuels) with energy efficiency contribute to increasing security of electricity supply, competitiveness and sustainable development. The countries of the Central and South-East Europe region must to develop a focus on alternative energy sources and on energy efficiency and energy saving. Developing the renewable energy sector in a sustainable manner in the Central and South-East Europe region would enhance security of ...
The Weinberg-Witten theorem on massless particles: an essay
In this essay we deal with the Weinberg-Witten theorem which imposes limitations on massless particles. First we motivate a classification of massless particles given by the Poincare group as the symmetry group of Minkowski spacetime. We then use the fundamental structure of the background in the form of Poincare covariance to derive restrictions on charged massless particles known as the Weinberg-Witten theorem. We address possible misunderstandings in the proof of this theorem motivated by several papers on this topic. In the last section the consequences of the theorem are discussed. We treat it in the context of known particles and as a constraint for emergent theories. (Abstract Copyright [2008], Wiley Periodicals, Inc.)
Complex integration and Cauchy's theorem
Watson, GN
2012-01-01
This brief monograph by one of the great mathematicians of the early twentieth century offers a single-volume compilation of propositions employed in proofs of Cauchy's theorem. Developing an arithmetical basis that avoids geometrical intuitions, Watson also provides a brief account of the various applications of the theorem to the evaluation of definite integrals.Author G. N. Watson begins by reviewing various propositions of Poincaré's Analysis Situs, upon which proof of the theorem's most general form depends. Subsequent chapters examine the calculus of residues, calculus optimization, the
-Dimensional Fractional Lagrange's Inversion Theorem
F. A. Abd El-Salam
2013-01-01
Full Text Available Using Riemann-Liouville fractional differential operator, a fractional extension of the Lagrange inversion theorem and related formulas are developed. The required basic definitions, lemmas, and theorems in the fractional calculus are presented. A fractional form of Lagrange's expansion for one implicitly defined independent variable is obtained. Then, a fractional version of Lagrange's expansion in more than one unknown function is generalized. For extending the treatment in higher dimensions, some relevant vectors and tensors definitions and notations are presented. A fractional Taylor expansion of a function of -dimensional polyadics is derived. A fractional -dimensional Lagrange inversion theorem is proved.