Fast Fuzzy Arithmetic Operations
Hampton, Michael; Kosheleva, Olga
1997-01-01
In engineering applications of fuzzy logic, the main goal is not to simulate the way the experts really think, but to come up with a good engineering solution that would (ideally) be better than the expert's control, In such applications, it makes perfect sense to restrict ourselves to simplified approximate expressions for membership functions. If we need to perform arithmetic operations with the resulting fuzzy numbers, then we can use simple and fast algorithms that are known for operations with simple membership functions. In other applications, especially the ones that are related to humanities, simulating experts is one of the main goals. In such applications, we must use membership functions that capture every nuance of the expert's opinion; these functions are therefore complicated, and fuzzy arithmetic operations with the corresponding fuzzy numbers become a computational problem. In this paper, we design a new algorithm for performing such operations. This algorithm is applicable in the case when negative logarithms - log(u(x)) of membership functions u(x) are convex, and reduces computation time from O(n(exp 2))to O(n log(n)) (where n is the number of points x at which we know the membership functions u(x)).
Arithmetic Operators for Pairing-Based Cryptography
Beuchat, Jean-Luc; Brisebarre, Nicolas; Detrey, Jérémie; Okamoto, Eiji
2007-01-01
Since their introduction in constructive cryptographic applications, pairings over (hyper)elliptic curves are at the heart of an ever increasing number of protocols. Software implementations being rather slow, the study of hardware architectures became an active research area. In this paper, we first study an accelerator for the eta_T pairing over F_3[x]/(x^97 + x^12 + 2). Our architecture is based on a unified arithmetic operator which performs addition, multiplication, and cubing over F_3^9...
Genetic Programming with Smooth Operators for Arithmetic Expressions
Ursem, Rasmus Kjær; Krink, Thiemo
This paper introduces the smooth operators for arithmetic expressions as an approach to smoothening the search space in Genetic Programming (GP). Smooth operator GP interpolates between arithmetic operators such as * and /, thereby allowing a gradual adaptation to the problem. The suggested...
A novel operation associated with Gauss' arithmetic-geometric means
Tanimoto, Shinji
2007-01-01
The arithmetic mean is the mean for addition and the geometric mean is that for multiplication. Then what kind of binary operation is associated with the arithmetic-geometric mean (AGM) due to C. F. Gauss? If it is possible to construct an arithmetic operation such that AGM is the mean for this operation, it can be regarded as an intermediate operation between addition and multiplication in view of the meaning of AGM. In this paper such an operation is introduced and several of its algebraic ...
Adams operations on higher arithmetic K-theory
Feliu, Elisenda
2010-01-01
We construct Adams operations on the rational higher arithmetic K-groups of a proper arithmetic variety. The de¿nition applies to the higher arithmetic K-groups given by Takeda as well as to the groups suggested by Deligne and Soulé, by means of the homotopy groups of the homotopy ¿ber of the reg......We construct Adams operations on the rational higher arithmetic K-groups of a proper arithmetic variety. The de¿nition applies to the higher arithmetic K-groups given by Takeda as well as to the groups suggested by Deligne and Soulé, by means of the homotopy groups of the homotopy ¿ber...... of the regulator map. They are compatible with the Adams operations on algebraic K-theory. The de¿nition relies on the chain morphism representing Adams operations in higher algebraic K-theory given previously by the author. It is shown that this chain morphism commutes strictly with the representative...
ARITHMETICAL OPERATIONS OF EXPONENTIAL FUZZY NUMBERS USING THE FUNCTION PRINCIPLE
SHAN-HUO CHEN; CHIEN-CHUNG WANG; SHU MAN CHANG
2006-01-01
Fuzzy numbers with exponential membership function are quite common in real world cases. In this paper, we applied Graded Mean Integration Representation Method to compute the representation of exponential fuzzy numbers. Then the Function Principle is applied to generate the arithmetical operations of exponential fuzzy numbers. By the way, some properties of operation under the Function Principle are proved. Finally, an application of the lifetime of the lamps of projectors is proposed.
Fractal surfaces from simple arithmetic operations
García-Morales, Vladimir
2016-04-01
Fractal surfaces ('patchwork quilts') are shown to arise under most general circumstances involving simple bitwise operations between real numbers. A theory is presented for all deterministic bitwise operations on a finite alphabet. It is shown that these models give rise to a roughness exponent H that shapes the resulting spatial patterns, larger values of the exponent leading to coarser surfaces.
Memristor-based Circuits for Performing Basic Arithmetic Operations
Merrikh-Bayat, Farnood
2010-01-01
In almost all of the currently working circuits, especially in analog circuits implementing signal processing applications, basic arithmetic operations such as multiplication, addition, subtraction and division are performed on values which are represented by voltages or currents. However, in this paper, we propose a new and simple method for performing analog arithmetic operations which in this scheme, signals are represented and stored through a memristance of the newly found circuit element, i.e. memristor, instead of voltage or current. Some of these operators such as divider and multiplier are much simpler and faster than their equivalent voltage-based circuits and they require less chip area. In addition, a new circuit is designed for programming the memristance of the memristor with predetermined analog value. Presented simulation results demonstrate the effectiveness and the accuracy of the proposed circuits.
Independence of basic arithmetic operations: evidence from cognitive neuropsychology
María P. Salguero-Alcañiz
2013-10-01
Full Text Available The cases described in literature evidence that arithmetical operations can function independently, which allows to infer that the cognitive processes involved in the different operations might be different. Objective of that work is to determine the different processes involved in the resolution of arithmetical operations: addition, subtraction and multiplication. Method: Instrument: Assesment of Numeric Processing and Calculation Battery (Salguero & Alameda, 2007, 2011. Subjects. Patients of acquired cerebral injury. Results and conclusions: The patient MNL preserves the addition and the multiplication but he presents altered the subtraction. On the contrary, the patient PP shows alterations in addition and multiplication but he conserves the skills for the subtraction. ISR presents a selective deficit for multiplication with intact addition and substraction. Finally, ACH preserves the addition but presents deficit for substraction and multiplication. This double dissociation confirms the postulates of the anatomical functional model of Dehaene and Cohen (1995, 1997 that consider a double route for the resolution of arithmetical simple operations: linguistic route, for numerical information learned automatically (of memory and would be used for the operations of addition and multiplication, on the other hand the semantic elaboration would be for substraction.
Interval arithmetic operations for uncertainty analysis with correlated interval variables
Jiang, Chao; Fu, Chun-Ming; Ni, Bing-Yu; Han, Xu
2016-08-01
A new interval arithmetic method is proposed to solve interval functions with correlated intervals through which the overestimation problem existing in interval analysis could be significantly alleviated. The correlation between interval parameters is defined by the multidimensional parallelepiped model which is convenient to describe the correlative and independent interval variables in a unified framework. The original interval variables with correlation are transformed into the standard space without correlation, and then the relationship between the original variables and the standard interval variables is obtained. The expressions of four basic interval arithmetic operations, namely addition, subtraction, multiplication, and division, are given in the standard space. Finally, several numerical examples and a two-step bar are used to demonstrate the effectiveness of the proposed method.
Chapman, Olive
2007-01-01
Mathematical tasks, centered on arithmetic word problems, are discussed as the basis of an approach to facilitate preservice elementary teachers' development of mathematical knowledge for teaching arithmetic operations. The approach consists of three groups of tasks that allow students to reflect on their initial knowledge, explore arithmetic word…
Second grade students understanding of the arithmetic operations
Slapar, Ana
2012-01-01
People face Mathematics at a very early stage in their lives since it is a part of various everyday activities. M. Jurkovič (2009) came to a conclusion in her research that Mathematics in the first grades of elementary schools is one of the most popular subjects, despite of the fact that a lot of pupils have difficulties with it. In my diploma I tried to asses how the pupils who have difficulties with Mathematics understand arithmetic operations of adding and subtracting, which calculati...
Evaluating the Main Battle Tank Using Fuzzy Number Arithmetic Operations
Deng Yong
2006-04-01
Full Text Available Since the descriptions and judgements on weapon systems are usually linguistic and fuzzy,it is more realistic to evaluate weapon systems in the framework of fuzzy sets theory. In thispaper, a new method to evaluate the best main battle tank is proposed. It can be seen that theproposed method is more efficient due to the fact that, by canonical representation of arithmeticoperation on fuzzy numbers, simple arithmetic operations on crisp numbers are used, instead ofcomplicated fuzzy numbers operations. In addition, the final scores of each alternative can berepresented as crisp numbers. As a result, the order of alternatives can be determined withoutthe procedure of ranking fuzzy numbers. Finally, a numerical example to evaluate the best mainbattle tanks is used to illustrate the efficiency of the proposed method.
Understanding and Using Principles of Arithmetic: Operations Involving Negative Numbers
Prather, Richard W.; Alibali, Martha W.
2008-01-01
Previous work has investigated adults' knowledge of principles for arithmetic with positive numbers (Dixon, Deets, & Bangert, 2001). The current study extends this past work to address adults' knowledge of principles of arithmetic with a negative number, and also investigates links between knowledge of principles and problem representation.…
Numeral words and arithmetic operations in the Alor-Pantar languages
Corbett, GG; Klamer, M; Schapper, A; Holton, G.; Kratochvil, F.; Robinson, L.
2014-01-01
The indigenous numerals of the AP languages, as well as the indigenous structures for arithmetic operations are currently under pressure from Indonesian, and will inevitably be replaced with Indonesian forms and structures. This chapter presents a documentary record of the forms and patterns currently in use to express numerals and arithmetic operations in the Alor-Pantar languages. We describe the structure of cardinal, ordinal and distributive numerals, and how operations of ...
Arithmetic Operations and Factorization using Asynchronous P Systems
Takayuki Murakawa
2012-07-01
Full Text Available
In the present paper, we consider the asynchronous parallelism in membrane computing, and propose asynchronous P systems that perform two basic arithmetic operations and factorization. Since there is no restrictive assumption for application of rules, sequential and maximal parallel executions are allowed on the asynchronous P system.
We first propose a P system that computes addition of two binary numbers of m bits. The P system works in O(m sequential and parallel steps using O(m types of objects. We next propose a P system for multiplication of the two binary numbers of m bits, and show that the P system works in O(m log m parallel steps or O(m^{3} sequential steps using O(m^{2} types of objects. Finally, we propose a P system for factorization of a positive integer of $m$ bits using the above P system as a sub-system. The P system computes the factorization in O(m log m parallel steps or O(4^{m}
Data Encryption and Decryption Algorithm Using Hamming Code and Arithmetic Operations
Kurapati Sundar Teja; Shanmukha Mallikarjuna Bandaru
2015-01-01
This paper explains the implementation of data encryption and decryption algorithm using hamming code and arithmetic operations with the help of Verilog HDL. As the days are passing the old algorithms are not remained so strong cryptanalyst are familiar with them. Hamming code is one of forward error correcting code which has got many applications. In this paper hamming code algorithm was discussed and the implementation of it was done with arithmetic operations. For high security...
Operation-Specific Encoding in Single-Digit Arithmetic
Zhou, Xinlin
2011-01-01
Solving simple arithmetic problems involves three stages: encoding the problem, retrieving or calculating the answer, and reporting the answer. This study compared the event-related potentials elicited by single-digit addition and multiplication problems to examine the relationship between encoding and retrieval/calculation stages. Results showed…
Data Encryption and Decryption Algorithm Using Hamming Code and Arithmetic Operations
Kurapati Sundar Teja
2015-08-01
Full Text Available This paper explains the implementation of data encryption and decryption algorithm using hamming code and arithmetic operations with the help of Verilog HDL. As the days are passing the old algorithms are not remained so strong cryptanalyst are familiar with them. Hamming code is one of forward error correcting code which has got many applications. In this paper hamming code algorithm was discussed and the implementation of it was done with arithmetic operations. For high security some arithmetic operations are added with hamming code process. A 3-bit data will be encrypted as 14-bit and using decryption process again we will receives 3-bit original data. The implemented design was tested on Spartan3A FPGA kit.
Contribution to the design of a micro-programmed floating point arithmetical operator
This report is intended for a presentation of the implementation of an arithmetical operator. Fast microprocessor together with microprogramming techniques were used for development. For clarity, this report is shared in three parts following the different steps of design and development. The first part relates the preliminary study stage, setting down the outlines of the project: tentative data, choice of components and architecture of operator. The second part is devoted to the development step. It deals with implementation aid systems and computation algorithms for arithmetical functions. Results and conclusions are the subject of the third part. (author)
Purpura, David J.; Lonigan, Christopher J.
2013-01-01
Validating the structure of informal numeracy skills is critical to understanding the developmental trajectories of mathematics skills at early ages; however, little research has been devoted to construct evaluation of the Numbering, Relations, and Arithmetic Operations domains. This study was designed to address this knowledge gap by examining…
Numeral words and arithmetic operations in the Alor-Pantar languages
Schapper, Antoinette; Holton, Gary; Klamer, Marian; Kratochvíl, František; Robinson, Laura; Klamer, Marian
2014-01-01
The indigenous numerals of the AP languages, as well as the indigenous structures for arithmetic operations are currently under pressure from Indonesian, and will inevitably be replaced with Indonesian forms and structures. This chapter presents a documentary record of the forms and patterns current
Reversible arithmetic logic unit
zhou, Rigui; Shi, Yang; Zhang, Manqun
2011-01-01
Quantum computer requires quantum arithmetic. The sophisticated design of a reversible arithmetic logic unit (reversible ALU) for quantum arithmetic has been investigated in this letter. We provide explicit construction of reversible ALU effecting basic arithmetic operations. By provided the corresponding control unit, the proposed reversible ALU can combine the classical arithmetic and logic operation in a reversible integrated system. This letter provides actual evidence to prove the possib...
A Modern Advanced Hill Cipher Involving a Permuted Key and Modular Arithmetic Addition Operation
V.U.K. Sastry
2011-05-01
Full Text Available In this paper we have devoted our attention to the study of a block cipher by generalizing advanced Hill cipher by including a permuted key. In this analysis we find that the iteration process, the mix operation and the modular arithmetic operation involved in the cipher mixes the binary bits of the key and the plaintext in a thorough manner. The avalanche effect and the cryptanalysis markedly indicate that the cipher is a strong one.
Operator priming and generalization of practice in adults' simple arithmetic.
Chen, Yalin; Campbell, Jamie I D
2016-04-01
There is a renewed debate about whether educated adults solve simple addition problems (e.g., 2 + 3) by direct fact retrieval or by fast, automatic counting-based procedures. Recent research testing adults' simple addition and multiplication showed that a 150-ms preview of the operator (+ or ×) facilitated addition, but not multiplication, suggesting that a general addition procedure was primed by the + sign. In Experiment 1 (n = 36), we applied this operator-priming paradigm to rule-based problems (0 + N = N, 1 × N = N, 0 × N = 0) and 1 + N problems with N ranging from 0 to 9. For the rule-based problems, we found both operator-preview facilitation and generalization of practice (e.g., practicing 0 + 3 sped up unpracticed 0 + 8), the latter being a signature of procedure use; however, we also found operator-preview facilitation for 1 + N in the absence of generalization, which implies the 1 + N problems were solved by fact retrieval but nonetheless were facilitated by an operator preview. Thus, the operator preview effect does not discriminate procedure use from fact retrieval. Experiment 2 (n = 36) investigated whether a population with advanced mathematical training-engineering and computer science students-would show generalization of practice for nonrule-based simple addition problems (e.g., 1 + 4, 4 + 7). The 0 + N problems again presented generalization, whereas no nonzero problem type did; but all nonzero problems sped up when the identical problems were retested, as predicted by item-specific fact retrieval. The results pose a strong challenge to the generality of the proposal that skilled adults' simple addition is based on fast procedural algorithms, and instead support a fact-retrieval model of fast addition performance. (PsycINFO Database Record PMID:26461035
Introduction to Projective Arithmetics
Burgin, Mark
2010-01-01
Science and mathematics help people to better understand world, eliminating many inconsistencies, fallacies and misconceptions. One of such misconceptions is related to arithmetic of natural numbers, which is extremely important both for science and everyday life. People think their counting is governed by the rules of the conventional arithmetic and thus other kinds of arithmetics of natural numbers do not exist and cannot exist. However, this popular image of the situation with the natural numbers is wrong. In many situations, people have to utilize and do implicitly utilize rules of counting and operating different from rules and operations in the conventional arithmetic. This is a consequence of the existing diversity in nature and society. To correctly represent this diversity, people have to explicitly employ different arithmetics. To make a distinction, we call the conventional arithmetic by the name Diophantine arithmetic, while other arithmetics are called non-Diophantine. There are two big families ...
Bruno, Adriano; Yasaki, Dan
2008-01-01
The arithmetic of the natural numbers can be extended to arithmetic operations on planar binary trees. This gives rise to a non-commutative arithmetic theory. In this exposition, we describe this arithmetree, first defined by Loday, and investigate prime trees.
All-Optical Logic and Arithmetic Operation using Soliton Control for Tree Architecture Use
S. Mitatha
2012-01-01
Full Text Available This study has proposed a new design system for basic all-optical logic and arithmetic operation using dark-bright soliton conversion control base on tree architecture. Optical tree architecture and nonlinear materials can provide a significant role for contribution in optical interconnecting network and ultrafast photonic computing system. In operation, the input data for all-optical binary logic can be formed by dark and bright soliton pulses for logic ‘0’ and ‘1’, respectively. By using the dark-bright soliton conversion behaviors, the conversion between dark and bright soliton pulses can be obtained and formed the logic pulse by a π/2 phase shifted device (i.e., an optical coupler. In application, the proposed scheme can be recognized as a simple and flexible system for forming the logic switching system, which can be used for advanced logical system.
Implementation of Arithmetic Operations With Time-Free Spiking Neural P Systems.
Liu, Xiangrong; Li, Ziming; Liu, Juan; Liu, Logan; Zeng, Xiangxiang
2015-09-01
Spiking neural P systems (SN P systems) are a class of distributed parallel computing devices inspired from the way neurons communicate by means of spikes. In most applications of SN P systems, synchronization plays a key role which means the execution of a rule is completed in exactly one time unit (one step). However, such synchronization does not coincide with the biological fact: in biological nervous systems, the execution times of spiking rules cannot be known exactly. Therefore, a "realistic" system called time-free SN P systems were proposed, where the precise execution time of rules is removed. In this paper, we consider building arithmetical operation systems based on time-free SN P systems. Specifically, adder, subtracter, multiplier, and divider are constructed by using time-free SN P systems. The obtained systems always produce the same computation result independently from the execution time of the rules. PMID:26335555
Hatim Zaini
2004-12-01
Full Text Available paper introduces a novel method for complex number representation. The proposed Redundant Complex Binary Number System (RCBNS is developed by combining a Redundant Binary Number and a complex number in base (-1+j. Donald [1] and Walter Penny [2,3] represented complex numbers using base –j and (-1+j in the classified algorithmic models. A Redundant Complex Binary Number System consists of both real and imaginary-radix number systems that form a redundant integer digit set. This system is formed by using complex radix of (-1+j and a digit set of á= 3, where á assumes a value of -3, -2, -1, 0, 1, 2, 3. The arithmetic operations of complex numbers with this system treat the real and imaginary parts as one unit. The carry-free addition has the advantage of Redundancy in number representation in the arithmetic operations. Results of the arithmetic operations are in the RCBNS form. The two methods for conversion from the RCBNS form to the standard binary number form have been presented. In this paper the RCBNS reduces the number of steps required to perform complex number arithmetic operations, thus enhancing the speed.
Raveh, Ira; Koichu, Boris; Peled, Irit; Zaslavsky, Orit
2016-01-01
In this article we present an integrative framework of knowledge for teaching the standard algorithms of the four basic arithmetic operations. The framework is based on a mathematical analysis of the algorithms, a connectionist perspective on teaching mathematics and an analogy with previous frameworks of knowledge for teaching arithmetic…
Schmalz, Mark S.
1993-09-01
The processing of Boolean imagery compressed by runlength encoding (RLE) frequently exhibits greater computational efficiency than the processing of uncompressed imagery, due to the data reduction inherent in RLE. In a previous publication, we outlined general methods for developing operators that compute over RLE Boolean imagery. In this paper, we present sequential and parallel algorithms for a variety of operations over RLE imagery, including the customary arithmetic and logical Hadamard operations, as well as the global reduce functions of image sum and maximum. RLE neighborhood-based operations, as well as the more advanced RLE operations of linear transforms, connected component labelling, and pattern recognition are presented in the companion paper.
Garay, Mauricio
2012-01-01
Arithmetic class are closed subsets of the euclidean space which generalise arithmetical conditions encoutered in dynamical systems, such as diophantine conditions or Bruno type conditions. I prove density estimates for such sets using Dani-Kleinbock-Margulis techniques.
Sullivan, Margaret A.
2008-01-01
This thesis examines two problems in on-board computing for space vehicles and develops rules for applying Reduced Precision Redundancy (RPR) as a new method of fault tolerance in Field Programmable Gate Arrays against Single Event Effects due to radiation on orbit. RPR was discovered by Snodgrass in 2006 and was first demonstrated using the single-input CORDIC algorithm. This research applies RPR to elementary multiple-input arithmetic operations (addition, subtraction, multiplication, divi...
Reversible arithmetic logic unit for quantum arithmetic
Thomsen, Michael Kirkedal; Glück, Robert; Axelsen, Holger Bock
2010-01-01
This communication presents the complete design of a reversible arithmetic logic unit (ALU) that can be part of a programmable reversible computing device such as a quantum computer. The presented ALU is garbage free and uses reversible updates to combine the standard reversible arithmetic...... ALU for a programmable computing device is possible and that the V-shape design is a very versatile approach to the design of quantum networks....... and logical operations in one unit. Combined with a suitable control unit, the ALU permits the construction of an r-Turing complete computing device. The garbage-free ALU developed in this communication requires only 6n elementary reversible gates for five basic arithmetic-logical operations on two n...
Conceptual Knowledge of Fraction Arithmetic
Siegler, Robert S.; Lortie-Forgues, Hugues
2015-01-01
Understanding an arithmetic operation implies, at minimum, knowing the direction of effects that the operation produces. However, many children and adults, even those who execute arithmetic procedures correctly, may lack this knowledge on some operations and types of numbers. To test this hypothesis, we presented preservice teachers (Study 1),…
Guthormsen, Amy M; Fisher, Kristie J; Bassok, Miriam; Osterhout, Lee; DeWolf, Melissa; Holyoak, Keith J
2016-04-01
Research on language processing has shown that the disruption of conceptual integration gives rise to specific patterns of event-related brain potentials (ERPs)-N400 and P600 effects. Here, we report similar ERP effects when adults performed cross-domain conceptual integration of analogous semantic and mathematical relations. In a problem-solving task, when participants generated labeled answers to semantically aligned and misaligned arithmetic problems (e.g., 6 roses + 2 tulips = ? vs. 6 roses + 2 vases = ?), the second object label in misaligned problems yielded an N400 effect for addition (but not division) problems. In a verification task, when participants judged arithmetically correct but semantically misaligned problem sentences to be "unacceptable," the second object label in misaligned sentences elicited a P600 effect. Thus, depending on task constraints, misaligned problems can show either of two ERP signatures of conceptual disruption. These results show that well-educated adults can integrate mathematical and semantic relations on the rapid timescale of within-domain ERP effects by a process akin to analogical mapping. PMID:25864403
Pankovic, Vladan; Predojevic, Milan
2006-01-01
In this work we define an universal arithmetical algorithm, by means of the standard quantum mechanical formalism, called universal qm-arithmetical algorithm. By universal qm-arithmetical algorithm any decidable arithmetical formula (operation) can be decided (realized, calculated. Arithmetic defined by universal qm-arithmetical algorithm called qm-arithmetic one-to-one corresponds to decidable part of the usual arithmetic. We prove that in the qm-arithmetic the undecidable arithmetical formu...
Reversible arithmetic logic unit for quantum arithmetic
This communication presents the complete design of a reversible arithmetic logic unit (ALU) that can be part of a programmable reversible computing device such as a quantum computer. The presented ALU is garbage free and uses reversible updates to combine the standard reversible arithmetic and logical operations in one unit. Combined with a suitable control unit, the ALU permits the construction of an r-Turing complete computing device. The garbage-free ALU developed in this communication requires only 6n elementary reversible gates for five basic arithmetic-logical operations on two n-bit operands and does not use ancillae. This remarkable low resource consumption was achieved by generalizing the V-shape design first introduced for quantum ripple-carry adders and nesting multiple V-shapes in a novel integrated design. This communication shows that the realization of an efficient reversible ALU for a programmable computing device is possible and that the V-shape design is a very versatile approach to the design of quantum networks. (fast track communication)
One-parameter generalizations of the logarithmic and exponential functions have been obtained as well as algebraic operators to retrieve extensivity. Analytical expressions for the successive applications of the sum or product operators on several values of a variable are obtained here. Applications of the above formalism are considered
S.G.Srikantaswamy
2011-12-01
Full Text Available The process of exchanging information is called Communication. The basic Communication system involvestransmitter, receiver and the channel. The data transmitted by the sender reaches receiver through thechannel. The unauthorized parties (cracker,hacker, eavesdropper, or attacker should not be able to access the information at the channel. Therefore transmitting data securely from the sender to the receiver is a very important aspect. A cryptographic system is unconditionally secure if the cipher text produced by the system does not contain enough information to determine uniquely the corresponding plaintext, no matter how much cipher text is available. A cryptographic system is said to be computationally secure if the cost of breaking the cipher exceeds the value of the encrypted information and the time required to break the cipher exceeds the useful lifetime of the content. One time pad system can be called as unconditionally secure algorithm, if the keys (pad usedare truly random in nature. In this paper, we are demonstrating that one-time pad can be used as an efficient encryption scheme by involving arithmetic and logical operations. Here we proposed a new key generation technique, to generate a key of any length just by providing a seed value, to encrypt the message. The problem generating key value has been solved by the use of key generation algorithm.
Markov, Svetoslav; Hayes, Nathan
2010-01-01
An approximate number is an ordered pair consisting of a (real) number and an error bound, briefly error, which is a (real) non-negative number. To compute with approximate numbers the arithmetic operations on errors should be well-known. To model computations with errors one should suitably define and study arithmetic operations and order relations over the set of non-negative numbers. In this work we discuss the algebraic properties of non-negative numbers starting from fa...
Arithmetic the foundation of mathematics
2015-01-01
Arithmetic factors into our lives on a daily basis, so it's hard to imagine a world without the six basic operations: addition, subtraction, multiplication, division, raising to powers, and finding roots. Readers will get a solid overview of arithmetic, while offering useful examples of how they are used in routine activities, such as social media applications. It reinforces Common Core math standards, including understanding basic math concepts and how they apply to students' daily lives and challenges. A history of arithmetic helps provide a contextual framework for the course of its develop
Burgos Gil, José Ignacio; Feliu, E.
2009-01-01
We give a new construction of higher arithmetic Chow groups for quasi-projective arithmetic varieties over a field. Our definition agrees with the higher arithmetic Chow groups defined by Goncharov for projective arithmetic varieties over a field. These groups are the analogue, in the Arakelov context, of the higher algebraic Chow groups defined by Bloch. The degree zero group agrees with the arithmetic Chow groups of Burgos. Our new construction is shown to be a contravariant functor and is ...
Trace formulae for arithmetical systems
For quantum problems on the pseudo-sphere generated by arithmetic groups there exist special trace formulae, called trace formulae for Hecke operators, which permit the reconstruction of wave functions from the knowledge of periodic orbits. After a short discussion of this subject, the Hecke operators trace formulae are presented for the Dirichlet problem on the modular billiard, which is a prototype of arithmetical systems. The results of numerical computations for these semiclassical type relations are in good agreement with the directly computed eigenfunctions. (author) 23 refs.; 2 figs
New technological design of arithmetics
There are illogical and irrational rules in numbers writing and pronunciation in almost of languages. A part of the aim is to show the electronic applications possibility of logical and systematic rules which are proposed by Raoelina Andriambololona to write and pronounce numbers; we had studied and created the arithmetic operations representation corresponding in binary basis and in hexadecimal basis. The brand new found concept corresponds as well as the method which uses the matrix product calculation, in according with the writing and the pronunciation of numbers. It was shown how to concept the arithmetic operators in digital electronics; and we proposed and assumed to make headway and to do amelioration for technical conception of calculator and arithmetic unite those are at the basic function of all computers and almost domestic sophisticated machine. The left hand side- right hand side and increasing order writing of number is exploited to build a new computer programming for a scientific calculator.
ASIC For Complex Fixed-Point Arithmetic
Petilli, Stephen G.; Grimm, Michael J.; Olson, Erlend M.
1995-01-01
Application-specific integrated circuit (ASIC) performs 24-bit, fixed-point arithmetic operations on arrays of complex-valued input data. High-performance, wide-band arithmetic logic unit (ALU) designed for use in computing fast Fourier transforms (FFTs) and for performing ditigal filtering functions. Other applications include general computations involved in analysis of spectra and digital signal processing.
Zariski decompositions on arithmetic surfaces
Moriwaki, Atsushi
2009-01-01
In this paper, we establish the Zariski decompositions of arithmetic R-divisors of continuous type on arithmetic surfaces and investigate several properties. We also develop the general theory of arithmetic R-divisors on arithmetic varieties.
We incorporate the string theory into the number theoretic formulation based on arithmetic geometry. The string theory is generalized p-adically and interpreted on an arithmetic surface. A p-adic multi-loop scattering amplitude is constructed. (orig.)
Rastegar, Arash
2015-01-01
By Grothendieck's anabelian conjectures, Galois representations landing in outer automorphism group of the algebraic fundamental group which are associated to hyperbolic smooth curves defined over number-fields encode all the arithmetic information of these curves. The Goal of this paper is to develop an arithmetic Teichmuller theory, by which we mean, introducing arithmetic objects summarizing the arithmetic information coming from all curves of the same topological type defined over number-...
Sugahara, K.; Weng, L
2015-01-01
We first introduce global arithmetic cohomology groups for quasi-coherent sheaves on arithmetic varieties, adopting an adelic approach. Then, we establish fundamental properties, such as topological duality and inductive long exact sequences, for these cohomology groups on arithmetic surfaces. Finally, we expose basic structures for ind-pro topologies on adelic spaces of arithmetic surfaces. In particular, we show that these adelic spaces are topologically self-dual.
No Arithmetic Cyclic Quadrilaterals
Beauregard, Raymond A.
2006-01-01
A quadrilateral is arithmetic if its area is an integer and its sides are integers in an arithmetic progression, and it is cyclic if it can be inscribed in a circle. The author shows that no quadrilateral is both arithmetic and cyclic.
Oller-Marcén, Antonio M.
2012-01-01
An integer $n$ is said to be \\textit{arithmetic} if the arithmetic mean of its divisors is an integer. In this paper, using properties of the factorization of values of cyclotomic polynomials, we characterize arithmetic numbers. As an application, in Section 2, we give an interesting characterization of Mersenne numbers.
Some new theorems have been propounded for the numbers (M-1), as they relate to other numerals, through the basic arithmetical operations, at different bases M. For some reason, we give the proof of the theorems for the case M=10 using mathematical induction, and by Peano's fifth axiom make our generalizations. Comments are made in respect of the numbers (M-1), (in this case 9). Apart from our theorems facilitating mathematical operations, evidences have also been given, from different sources of the interesting properties of this class of numbers, represented in our own case by the numeral 9. The theorems neither violate the divisibility rule for 9 nor are they a consequence of it. From symmetry, a suggestion is made in respect of the possible origin of the numeration in base 10, and the case of a ten dimensional Universe reconsidered. (author). 18 refs, 1 fig., 4 tabs
A Mathematical Basis for an Interval Arithmetic Standard
Bohlender, Gerd; Kulisch, Ulrich
2010-01-01
Basic concepts for an interval arithmetic standard are discussed in the paper. Interval arithmetic deals with closed and connected sets of real numbers. Unlike floating-point arithmetic it is free of exceptions. A complete set of formulas to approximate real interval arithmetic on the computer is displayed in section 3 of the paper. The essential comparison relations and lattice operations are discussed in section 6. Evaluation of functions for interval arguments is studied ...
李美蓉; 彭勃
2013-01-01
By utilizing some operational laws of pure linguistic variables, this paper proposes an aggregation operator called Induced Pure Linguistic Order Weighted Arithmetic Averaging (IPLOWAA) operator and studies some desirable properties of the IPLOWAA operator. Also, it develops a method of group decision making based on IPLOWAA and PLOWAA operators under multiplicative linguistic preference relations, which is straightforward and takes the known linguistic information into account sufficiently. An illustrative application to risk investment is given to verify the developed approach and to demonstrate its practicality and effectiveness.%利用纯语言变量的运算规律,提出了一种信息集成算子:导出的纯语言有序加权算术平均(IPLOWAA)算子,同时对IPLOWAA算子的性质进行了研究.在群体语言偏好关系下,给出了基于IPLOWAA算子和PLOWAA算子的一种群决策方法,该方法计算简洁方便且能充分利用已有的决策信息.将该方法应用在风险投资中,说明该方法的实用性和有效性.
Adelic divisors on arithmetic varieties
Moriwaki, Atsushi
2013-01-01
In this article, we generalize several fundamental results for arithmetic divisors, such as the continuity of the volume function, the generalized Hodge index theorem, Fujita's approximation theorem for arithmetic divisors and Zariski decompositions for arithmetic divisors on arithmetic surfaces, to the case of the adelic arithmetic divisors.
Gil, J. I. Burgos; Feliu, Elisenda
2012-01-01
context, of the higher algebraic Chow groups defined by Bloch. For projective varieties the degree zero group agrees with the arithmetic Chow groups defined by Gillet and Soulé, and in general with the arithmetic Chow groups of Burgos. Our new construction is shown to be a contravariant functor and is...... endowed with a product structure, which is commutative and associative....
Curiosities of arithmetic gases
Statistical mechanical systems with an exponential density of states are considered. The arithmetic analog of parafermions of arbitrary order is constructed and a formula for boson-parafermion equivalence is obtained using properties of the Riemann zeta function. Interactions (nontrivial mixing) among arithmetic gases using the concept of twisted convolutions are also introduced. Examples of exactly solvable models are discussed in detail
A general purpose arithmetic logic unit
A fast arithmetic and logic unit (ALU) has been constructed as a single CAMAC unit. This device has been designed to provide both arithmetic and logical operations on two 16-bit data fields. The ALU will be put into practical use in the energy trigger of the L3 experiment at LEP, CERN. Due to its simplicity and flexibility the circuit may have applications in other high energy physics experiments. In this paper we describe the details of this circuit. (orig.)
How to be Brilliant at Mental Arithmetic
Webber, Beryl
2010-01-01
How to be Brilliant at Mental Arithmetic addresses the twin pillars of mental arithmetic - mental recall and mental agility. Mental recall depends on familiarity with number bonds and plenty of opportunity to practise. Mental agility depends more on confidence with the number system and the four operations. Using the worksheets in this book, students will learn about: tens and units; addition, subtraction, multiplication and division; addition shortcuts; product squares; quick recall; number se
Memory Updating and Mental Arithmetic
Han, Cheng-Ching; Yang, Tsung-Han; Lin, Chia-Yuan; Yen, Nai-Shing
2016-01-01
Is domain-general memory updating ability predictive of calculation skills or are such skills better predicted by the capacity for updating specifically numerical information? Here, we used multidigit mental multiplication (MMM) as a measure for calculating skill as this operation requires the accurate maintenance and updating of information in addition to skills needed for arithmetic more generally. In Experiment 1, we found that only individual differences with regard to a task updating num...
FAST TRACK COMMUNICATION: Reversible arithmetic logic unit for quantum arithmetic
Kirkedal Thomsen, Michael; Glück, Robert; Axelsen, Holger Bock
2010-09-01
This communication presents the complete design of a reversible arithmetic logic unit (ALU) that can be part of a programmable reversible computing device such as a quantum computer. The presented ALU is garbage free and uses reversible updates to combine the standard reversible arithmetic and logical operations in one unit. Combined with a suitable control unit, the ALU permits the construction of an r-Turing complete computing device. The garbage-free ALU developed in this communication requires only 6n elementary reversible gates for five basic arithmetic-logical operations on two n-bit operands and does not use ancillae. This remarkable low resource consumption was achieved by generalizing the V-shape design first introduced for quantum ripple-carry adders and nesting multiple V-shapes in a novel integrated design. This communication shows that the realization of an efficient reversible ALU for a programmable computing device is possible and that the V-shape design is a very versatile approach to the design of quantum networks.
Precision Arithmetic: A New Floating-Point Arithmetic
Wang, Chengpu
2006-01-01
A new deterministic floating-point arithmetic called precision arithmetic is developed to track precision for arithmetic calculations. It uses a novel rounding scheme to avoid excessive rounding error propagation of conventional floating-point arithmetic. Unlike interval arithmetic, its uncertainty tracking is based on statistics and the central limit theorem, with a much tighter bounding range. Its stable rounding error distribution is approximated by a truncated normal distribution. Generic...
Dominici, Diego
2011-01-01
This work introduces a distance between natural numbers not based on their position on the real line but on their arithmetic properties. We prove some metric properties of this distance and consider a possible extension.
Ganea, Mihai
2009-01-01
Relations between some theories of semigroups (also known as theories of strings or theories of concatenation) and arithmetic are surveyed. In particular Robinson's arithmetic Q is shown to be mutually interpretable with TC, a weak theory of concatenation introduced by Grzegorczyk. Furthermore, TC is shown to be interpretable in the theory F studied by Tarski and Szmielewa, thus confirming their claim that F is essentially undecidable.
Nonstandard arithmetic and reverse mathematics
Keisler, H. Jerome
2006-01-01
We show that each of the five basic theories of second order arithmetic that play a central role in reverse mathematics has a natural counterpart in the language of nonstandard arithmetic. In the earlier paper [3] we introduced saturation principles in nonstandard arithmetic which are equivalent in strength to strong choice axioms in second order arithmetic. This paper studies principles which are equivalent in strength to weaker theories in second order arithmetic.
Kazhdan's Theorem on Arithmetic Varieties
Milne, J S
2001-01-01
Define an arithmetic variety to be the quotient of a bounded symmetric domain by an arithmetic group. An arithmetic variety is algebraic, and the theorem in question states that when one applies an automorphism of the field of complex numbers to the coefficients of an arithmetic variety the resulting variety is again arithmetic. This article simplifies Kazhdan's proof. In particular, it avoids recourse to the classification theorems. It was originally completed on March 28, 1984, and distribu...
Design of optimized Interval Arithmetic Multiplier
Rajashekar B.Shettar
2011-07-01
Full Text Available Many DSP and Control applications that require the user to know how various numericalerrors(uncertainty affect the result. This uncertainty is eliminated by replacing non-interval values withintervals. Since most DSPs operate in real time environments, fast processors are required to implementinterval arithmetic. The goal is to develop a platform in which Interval Arithmetic operations areperformed at the same computational speed as present day signal processors. So we have proposed thedesign and implementation of Interval Arithmetic multiplier, which operates with IEEE 754 numbers. Theproposed unit consists of a floating point CSD multiplier, Interval operation selector. This architectureimplements an algorithm which is faster than conventional algorithm of Interval multiplier . The costoverhead of the proposed unit is 30% with respect to a conventional floating point multiplier. Theperformance of proposed architecture is better than that of a conventional CSD floating-point multiplier,as it can perform both interval multiplication and floating-point multiplication as well as Intervalcomparisons
K. Anup Kumar
2012-07-01
Full Text Available In this investigation, we have modified the Feistel cipher by taking the plaintext in the form of a pair of square matrices. Here we have introduced the operation multiplication with the key matrices and the modular arithmetic addition in encryption. The modular arithmetic inverse of the key matrix is introduced in decryption. The cryptanalysis carried out in this paper clearly indicate that this cipher cannot be broken by the brute force attack and the known plaintext attack.
The Arithmetic of Supersymmetric Vacua
Bourget, Antoine
2016-01-01
We provide explicit formulas for the number of vacua of four-dimensional pure N=1 super Yang-Mills theories on a circle, with any simple gauge algebra and any choice of center and spectrum of line operators. These form a key ingredient in the semi-classical calculation of the number of massive vacua of N=1* gauge theories with gauge algebra su(n) compactified on a circle. Using arithmetic, we express that number in an SL(2,Z) duality invariant manner. We confirm our tally of massive vacua of the N=1* theories by a count of inequivalent extrema of the exact superpotential.
Numerical characterization of nef arithmetic divisors on arithmetic surfaces
Moriwaki, Atsushi
2012-01-01
In this paper, we give a numerical characterization of nef arithmetic R-Cartier divisors of C^0-type on an arithmetic surface. Namely an arithmetic R-Cartier divisor D of C^0-type is nef if and only if D is pseudo-effective and deg(D^2) = vol(D).
An "Arithmetic" Thinker Tackles Algebra
Armstrong, Alayne C.
2006-01-01
Working from Carolyn Kieran's categorization of "arithmetic" and "algebraic" thinkers, the article describes one eighth-grade "arithmetic" thinker's progress as she attempts to solve one- and two-step equations.
FPGA Based Quadruple Precision Floating Point Arithmetic for Scientific Computations
Mamidi Nagaraju
2012-09-01
Full Text Available In this project we explore the capability and flexibility of FPGA solutions in a sense to accelerate scientific computing applications which require very high precision arithmetic, based on IEEE 754 standard 128-bit floating-point number representations. Field Programmable Gate Arrays (FPGA is increasingly being used to design high end computationally intense microprocessors capable of handling floating point mathematical operations. Quadruple Precision Floating-Point Arithmetic is important in computational fluid dynamics and physical modelling, which require accurate numerical computations. However, modern computers perform binary arithmetic, which has flaws in representing and rounding the numbers. As the demand for quadruple precision floating point arithmetic is predicted to grow, the IEEE 754 Standard for Floating-Point Arithmetic includes specifications for quadruple precision floating point arithmetic. We implement quadruple precision floating point arithmetic unit for all the common operations, i.e. addition, subtraction, multiplication and division. While previous work has considered circuits for low precision floating-point formats, we consider the implementation of 128-bit quadruple precision circuits. The project will provide arithmetic operation, simulation result, hardware design, Input via PS/2 Keyboard interface and results displayed on LCD using Xilinx virtex5 (XC5VLX110TFF1136 FPGA device.
Connecting Arithmetic to Algebra
Darley, Joy W.; Leapard, Barbara B.
2010-01-01
Algebraic thinking is a top priority in mathematics classrooms today. Because elementary school teachers lay the groundwork to develop students' capacity to think algebraically, it is crucial for teachers to have a conceptual understanding of the connections between arithmetic and algebra and be confident in communicating these connections. Many…
Dominici, Diego
2009-01-01
What is the distance between 11 (a prime number) and 12 (a highly composite number)? If your answer is 1, then ask yourself "is this reasonable?" In this work, we will introduce a distance between natural numbers based on their arithmetic properties, instead of their position on the real line.
Calculation Methodology for Flexible Arithmetic Processing
García Chamizo, Juan Manuel; Mora Pascual, Jerónimo Manuel; Mora Mora, Higinio; Signes Pont, María Teresa
2003-01-01
A new operation model of flexible calculation that allows us to adjust the operation delay depending on the available time is presented. The operation method design uses look-up tables and progressive construction of the result. The increase in the operators’ granularity opens up new possibilities in calculation methods and microprocessor design. This methodology, together with the advances in technology, enables the functions of an arithmetic unit to be implemented on the basis of techniques...
Time-Precision Flexible Arithmetic Unit
García Chamizo, Juan Manuel; Mora Pascual, Jerónimo Manuel; Mora Mora, Higinio; Signes Pont, María Teresa
2003-01-01
A new conception of flexible calculation that allows us to adjust an operation depending on the available time computation is presented. The proposed arithmetic unit is based on this principle. It contains a control operation module that determines the process time of each calculation. The operation method design uses precalculated data stored in look-up tables, which provide, above all, quality results and systematization in the implementation of low level primitives that set parameters for ...
Arithmetic of Complex Manifolds
Lange, Herbert
1989-01-01
It was the aim of the Erlangen meeting in May 1988 to bring together number theoretists and algebraic geometers to discuss problems of common interest, such as moduli problems, complex tori, integral points, rationality questions, automorphic forms. In recent years such problems, which are simultaneously of arithmetic and geometric interest, have become increasingly important. This proceedings volume contains 12 original research papers. Its main topics are theta functions, modular forms, abelian varieties and algebraic three-folds.
Several ways of local timing of the Josephson-junction RSFQ (Rapid Single Flux Quantum) logic elements are proposed, and their peculiarities are discussed. Several examples of serial and parallel pipelined arithmetic blocks using various types of timing are suggested and their possible performance is discussed. Serial devices enable one to perform n-bit functions relatively slowly but using integrated circuits of a moderate integration scale, while parallel pipelined devices are more hardware-wasteful but promise extremely high productivity
Towards an arithmetical logic the arithmetical foundations of logic
Gauthier, Yvon
2015-01-01
This book offers an original contribution to the foundations of logic and mathematics, and focuses on the internal logic of mathematical theories, from arithmetic or number theory to algebraic geometry. Arithmetical logic is the term used to refer to the internal logic of classical arithmetic, here called Fermat-Kronecker arithmetic, and combines Fermat’s method of infinite descent with Kronecker’s general arithmetic of homogeneous polynomials. The book also includes a treatment of theories in physics and mathematical physics to underscore the role of arithmetic from a constructivist viewpoint. The scope of the work intertwines historical, mathematical, logical and philosophical dimensions in a unified critical perspective; as such, it will appeal to a broad readership from mathematicians to logicians, to philosophers interested in foundational questions. Researchers and graduate students in the fields of philosophy and mathematics will benefit from the author’s critical approach to the foundations of l...
Arithmetic functions in torus and tree networks
Bhanot, Gyan; Blumrich, Matthias A.; Chen, Dong; Gara, Alan G.; Giampapa, Mark E.; Heidelberger, Philip; Steinmacher-Burow, Burkhard D.; Vranas, Pavlos M.
2007-12-25
Methods and systems for performing arithmetic functions. In accordance with a first aspect of the invention, methods and apparatus are provided, working in conjunction of software algorithms and hardware implementation of class network routing, to achieve a very significant reduction in the time required for global arithmetic operation on the torus. Therefore, it leads to greater scalability of applications running on large parallel machines. The invention involves three steps in improving the efficiency and accuracy of global operations: (1) Ensuring, when necessary, that all the nodes do the global operation on the data in the same order and so obtain a unique answer, independent of roundoff error; (2) Using the topology of the torus to minimize the number of hops and the bidirectional capabilities of the network to reduce the number of time steps in the data transfer operation to an absolute minimum; and (3) Using class function routing to reduce latency in the data transfer. With the method of this invention, every single element is injected into the network only once and it will be stored and forwarded without any further software overhead. In accordance with a second aspect of the invention, methods and systems are provided to efficiently implement global arithmetic operations on a network that supports the global combining operations. The latency of doing such global operations are greatly reduced by using these methods.
Sets with Prescribed Arithmetic Densities
Luca, F.; Pomerance, C.; Porubský, Štefan
2008-01-01
Roč. 3, č. 2 (2008), s. 67-80. ISSN 1336-913X R&D Projects: GA ČR GA201/07/0191 Institutional research plan: CEZ:AV0Z10300504 Keywords : generalized arithmetic density * generalized asymptotic density * generalized logarithmic density * arithmetic al semigroup * weighted arithmetic mean * ratio set * R-dense set * Axiom A * delta-regularly varying function Subject RIV: BA - General Mathematics
Yablo's Paradox And Arithmetical Incompleteness
Leach-Krouse, Graham
2011-01-01
In this short paper, I present a few theorems on sentences of arithmetic which are related to Yablo's Paradox as G\\"odel's first undecidable sentence was related to the Liar paradox. In particular, I consider two different arithemetizations of Yablo's sentences: one resembling G\\"odel's arithmetization of the Liar, with the negation outside of the provability predicate, one resembling Jeroslow's undecidable sentence, with negation inside. Both kinds of arithmetized Yablo sentence are undecida...
Graph colorings, flows and arithmetic Tutte polynomial
D'Adderio, Michele; Moci, Luca
2011-01-01
We introduce the notions of arithmetic colorings and arithmetic flows over a graph with labelled edges, which generalize the notions of colorings and flows over a graph. We show that the corresponding arithmetic chromatic polynomial and arithmetic flow polynomial are given by suitable specializations of the associated arithmetic Tutte polynomial, generalizing classical results of Tutte.
Introduction to cardinal arithmetic
Holz, M; Weitz, E
1999-01-01
This book is an introduction into modern cardinal arithmetic in the frame of the axioms of Zermelo-Fraenkel set theory together with the axiom of choice (ZFC). A first part describes the classical theory developed by Bernstein, Cantor, Hausdorff, König and Tarski between 1870 and 1930. Next, the development in the 1970s led by Galvin, Hajnal and Silver is characterized. The third part presents the fundamental investigations in pcf theory which have been worked out by Shelah to answer the questions left open in the 1970s.Reviews:'The authors aim their text at beginners in set theory. They start
Arithmetic Data Cube as a Data Intensive Benchmark
Frumkin, Michael A.; Shabano, Leonid
2003-01-01
Data movement across computational grids and across memory hierarchy of individual grid machines is known to be a limiting factor for application involving large data sets. In this paper we introduce the Data Cube Operator on an Arithmetic Data Set which we call Arithmetic Data Cube (ADC). We propose to use the ADC to benchmark grid capabilities to handle large distributed data sets. The ADC stresses all levels of grid memory by producing 2d views of an Arithmetic Data Set of d-tuples described by a small number of parameters. We control data intensity of the ADC by controlling the sizes of the views through choice of the tuple parameters.
FPGA Based Quadruple Precision Floating Point Arithmetic for Scientific Computations
Mamidi Nagaraju; Geedimatla Shekar
2012-01-01
In this project we explore the capability and flexibility of FPGA solutions in a sense to accelerate scientific computing applications which require very high precision arithmetic, based on IEEE 754 standard 128-bit floating-point number representations. Field Programmable Gate Arrays (FPGA) is increasingly being used to design high end computationally intense microprocessors capable of handling floating point mathematical operations. Quadruple Precision Floating-Point Arithmetic is important...
Finite Field Arithmetic and its Application in Cryptography
Ansari, Bijan
2012-01-01
The groundbreaking idea of public key cryptography and the rapid expansion of the internetin the 80s opened a new research area for finite field arithmetic. The large size of fields incryptography demands new algorithms for efficient arithmetic and new metrics for estimatingfinite field operation performance. The area, power, and timing constraints on hand-heldand embedded devices necessitate accurate models to achieve expected goals. Additionally,cryptosystems need to protect their secrets a...
Crystallization of space: Space-time fractals from fractal arithmetic
Aerts, Diederik; Czachor, Marek; Kuna, Maciej
2016-02-01
Fractals such as the Cantor set can be equipped with intrinsic arithmetic operations (addition, subtraction, multiplication, division) that map the fractal into itself. The arithmetics allows one to define calculus and algebra intrinsic to the fractal in question, and one can formulate classical and quantum physics within the fractal set. In particular, fractals in space-time can be generated by means of homogeneous spaces associated with appropriate Lie groups. The construction is illustrated by explicit examples.
Crystallization of space: Space-time fractals from fractal arithmetics
Aerts, Diederik; Kuna, Maciej
2016-01-01
Fractals such as the Cantor set can be equipped with intrinsic arithmetic operations (addition, subtraction, multiplication, division) that map the fractal into itself. The arithmetics allows one to define calculus and algebra intrinsic to the fractal in question, and one can formulate classical and quantum physics within the fractal set. In particular, fractals in space-time can be generated by means of homogeneous spaces associated with appropriate Lie groups. The construction is illustrated by explicit examples.
Design of Floating Point Arithmetic Logic Unit with Universal Gate
Shraddha N. Zanjat; Dr.S.D.Chede; Prof.B.J.Chilke
2014-01-01
A floating point arithmetic and logic unit design using pipelining is proposed. By using pipeline with ALU design, ALU provides a high performance. With pipelining plus parallel processing concept ALU execute multiple instructions simultaneously. Floating point ALU unit is formed by combination of arithmetic modules (addition, subtraction, multiplication, division), Universal gate module. Each module is divided into sub-module. Bits selection determines which operation takes place at a partic...
MCNPX graphics and arithmetic tally upgrades
MCNPX tallies and cross-sections are plotted using the MCPLOT package. We report on an assortment of upgrades to MCPLOT that are intended to improve the appearance of two-dimensional tally and cross-section plots. We have also expanded the content and versatility of the MCPLOT 'help' command. Finally, we describe the initial phase of capability implementation to post-process tally data using arithmetic operations. These improvements will enable users to better display and manipulate simulation results. (authors)
Coherent states, pseudodifferential analysis and arithmetic
Basic questions regarding families of coherent states include describing some constructions of such and the way they can be applied to operator theory or partial differential equations. In both questions, pseudodifferential analysis is important. Recent developments indicate that they can contribute to methods in arithmetic, especially modular form theory. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’. (paper)
Coinductive Formal Reasoning in Exact Real Arithmetic
Niqui, Milad
2008-01-01
In this article we present a method for formally proving the correctness of the lazy algorithms for computing homographic and quadratic transformations -- of which field operations are special cases-- on a representation of real numbers by coinductive streams. The algorithms work on coinductive stream of M\\"{o}bius maps and form the basis of the Edalat--Potts exact real arithmetic. We use the machinery of the Coq proof assistant for the coinductive types to present the formalisation. The form...
Efficient Unified Arithmetic for Hardware Cryptography
SAVAŞ, Erkay; Savas, Erkay; Koç, Çetin Kaya; Koc, Cetin Kaya
2008-01-01
The basic arithmetic operations (i.e. addition, multiplication, and inversion) in finite fields, GF(q), where q = pk and p is a prime integer, have several applications in cryptography, such as RSA algorithm, Diffie-Hellman key exchange algorithm [1], the US federal Digital Signature Standard [2], elliptic curve cryptography [3, 4], and also recently identity based cryptography [5, 6]. Most popular finite fields that are heavily used in cryptographic applications due to elliptic curve based s...
Valuations on arithmetic surfaces
XU Ning
2009-01-01
In this paper,we give the definition of the height of a valuation and the definition of the big field Cp,G,where p is a prime and G R is an additive subgroup containing 1.We conclude that Cp,G is a field and Cp,G is algebraically closed.Based on this the author obtains the complete classification of valuations on arithmetic surfaces.Furthermore,for any m ≤ n ∈ Z,let Vm,n be an R-vector space of dimension n - m + 1,whose coordinates are indexed from rn to n.We generalize the definition of Cp,G,where p is a prime and G C Vm,n is an additive subgroup containing 1.We also conclude that Cp,G is a field if m ≤ 0 ≤ n.
Valuations on arithmetic surfaces
无
2009-01-01
In this paper, we give the definition of the height of a valuation and the definition of the big field Cp,G, where p is a prime and GR is an additive subgroup containing 1. We conclude that Cp,G is a field and Cp,G is algebraically closed. Based on this the author obtains the complete classification of valuations on arithmetic surfaces. Furthermore, for any m ≤n∈ Z, let Vm,n be an R-vector space of dimension n-m + 1, whose coordinates are indexed from m to n. We generalize the definition of Cp,G, where p is a prime and GVm,n is an additive subgroup containing 1. We also conclude that Cp,G is a field if m ≤0 ≤n.
New properties of arithmetic groups
Platonov, Vladimir P [Scientific Research Institute for System Studies of RAS (Russian Federation)
2011-01-25
New substantial results including the solutions of a number of fundamental problems have been obtained in the last decade or so: the first and rather unexpected examples of arithmetic groups with finite extensions that are not arithmetic were constructed; a criterion for arithmeticity of such extensions was found; deep rigidity theorems were proved for arithmetic subgroups of algebraic groups with radical; a theorem on the finiteness of the number of conjugacy classes of finite subgroups in finite extensions of arithmetic groups was proved, leading to numerous applications, in particular, this theorem made it possible to solve the Borel-Serre problem (1964) on the finiteness of the first cohomology of finite groups with coefficients in an arithmetic group; the problem posed more than 30 years ago on the existence of finitely generated integral linear groups that have infinitely many conjugacy classes of finite subgroups was solved; the arithmeticity question for solvable groups was settled. Similar problems were also solved for lattices in Lie groups with finitely many connected components. This paper is a survey of these results. Bibliography: 27 titles.
New properties of arithmetic groups
New substantial results including the solutions of a number of fundamental problems have been obtained in the last decade or so: the first and rather unexpected examples of arithmetic groups with finite extensions that are not arithmetic were constructed; a criterion for arithmeticity of such extensions was found; deep rigidity theorems were proved for arithmetic subgroups of algebraic groups with radical; a theorem on the finiteness of the number of conjugacy classes of finite subgroups in finite extensions of arithmetic groups was proved, leading to numerous applications, in particular, this theorem made it possible to solve the Borel-Serre problem (1964) on the finiteness of the first cohomology of finite groups with coefficients in an arithmetic group; the problem posed more than 30 years ago on the existence of finitely generated integral linear groups that have infinitely many conjugacy classes of finite subgroups was solved; the arithmeticity question for solvable groups was settled. Similar problems were also solved for lattices in Lie groups with finitely many connected components. This paper is a survey of these results. Bibliography: 27 titles.
Periodic orbits in arithmetical chaos
Length spectra of periodic orbits are investigated for some chaotic dynamical systems whose quantum energy spectra show unexpected statistical properties and for which the notion of arithmetical chaos has been introduced recently. These systems are defined as the unconstrained motions of particles on two dimensional surfaces of constant negative curvature whose fundamental groups are given by number theoretical statements (arithmetic Fuchsian groups). It is shown that the mean multiplicity of lengths l of periodic orbits grows asymptotically like c x el/2/l, l → ∞. Moreover, the constant c (depending on the arithmetic group) is determined. (orig.)
Arakelov theory of noncommutative arithmetic surfaces
Borek, Thomas
2008-01-01
The purpose of this paper is to initiate Arakelov theory in a noncommutative setting. More precisely, we are concerned with noncommutative arithmetic surfaces. We introduce a version of arithmetic intersection theory on noncommutative arithmetic surfaces and we prove an arithmetic Riemann-Roch theorem in this setup.
On the concavity of the arithmetic volumes
Ikoma, Hideaki
2013-01-01
In this note, we study the differentiability of the arithmetic volumes along arithmetic R-divisors, and give some equality conditions for the Brunn-Minkowski inequality for arithmetic volumes over the cone of nef and big arithmetic R-divisors.
Some studies on arithmetical chaos in classical and quantum mechanics
Several aspects of classical and quantum mechanics applied to a class of strongly chaotic systems are studied. The latter consists of single particles moving without external forces on surfaces of constant negative Gaussian curvature whose corresponding fundamental groups are supplied with an arithmetic structure. It is shown that the arithmetical features of the considered systems lead to exceptional properties of the corresponding spectra of lengths of closed geodesics (periodic orbits). The most significant one is an exponential growth of degeneracies in these geodesic length spectra. Furthermore, the arithmetical systems are distinguished by a structure that appears as a generalization of geometric symmetries. These pseudosymmetries occur in the quantization of the classical arithmetic systems as Hecke operators, which form an infinite algebra of self-adjoint operators commuting with the Hamiltonian. The statistical properties of quantum energies in the arithmetical systems have previously been identified as exceptional. They do not fit into the general scheme of random matrix theory. It is shown with the help of a simplified model for the spectral form factor how the spectral statistics in arithmetical quantum chaos can be understood by the properties of the corresponding classical geodesic length spectra. A decisive role is played by the exponentially increasing multiplicities of lengths. The model developed for the level spacings distribution and for the number variance is compared to the corresponding quantities obtained from quantum energies for a specific arithmetical system. Finally, the convergence properties of a representation for the Selberg zeta function as a Dirichlet series are studied. It turns out that the exceptional classical and quantum mechanical properties shared by the arithmetical systems prohibit a convergence of this important function in the physically interesting domain. (orig.)
Remarks on the arithmetic restricted volumes and the arithmetic base loci
Ikoma, Hideaki
2014-01-01
In this paper, we collect some fundamental properties of the arithmetic restricted volumes (or the arithmetic multiplicities) of the adelically metrized line bundles. The arithmetic restricted volume has the concavity property and characterizes the arithmetic augmented base locus as the null locus. We also show a generalized Fujita approximation for the arithmetic restricted volume.
Yablo's Paradox And Arithmetical Incompleteness
Leach-Krouse, Graham
2011-01-01
In this short paper, I present a few theorems on sentences of arithmetic which are related to Yablo's Paradox as G\\"odel's first undecidable sentence was related to the Liar paradox. In particular, I consider two different arithemetizations of Yablo's sentences: one resembling G\\"odel's arithmetization of the Liar, with the negation outside of the provability predicate, one resembling Jeroslow's undecidable sentence, with negation inside. Both kinds of arithmetized Yablo sentence are undecidable, and connected to the consistency sentence for the ambient formal system in roughly the same manner as G\\"odel and Jeroslow's sentences. Finally, I consider a sentence which is related to the Henkin sentence "I am provable" in the same way that first two arithmetizations are related to G\\"odel and Jeroslaw's sentences. I show that this sentence is provable, using L\\"ob's theorem, as in the standard proof of the Henkin sentence.
Inequalities for semistable families of arithmetic varieties
Kawaguchi, Shu; Moriwaki, Atsushi
1997-01-01
In this paper, we will consider a generalization of Bogomolov's inequality and Cornalba-Harris-Bost's inequality to semistable families of arithmetic varieties under the idea that geometric semistability implies a certain kind of arithmetic positivity. The first one is an arithmetic analogue of the relative Bogomolov's inequality proved by the second author. We also establish the arithmetic Riemann-Roch formulae for stable curves over regular arithmetic varieties and generically finite morphi...
Primitive geodesic lengths and (almost) arithmetic progressions
Lafont, Jean-François; McReynolds, D. B.
2014-01-01
In this article, we investigate when the set of primitive geodesic lengths on a Riemannian manifold have arbitrarily long arithmetic progressions. We prove that in the space of negatively curved metrics, a metric having such arithmetic progressions is quite rare. We introduce almost arithmetic progressions, a coarsification of arithmetic progressions, and prove that every negatively curved, closed Riemannian manifold has arbitrarily long almost arithmetic progressions in its primitive length ...
Neuropsychology of childhood arithmetic disorders.
Batchelor, E S
1989-01-01
The arithmetic learning disability literature was reviewed and critiqued. Due to the paucity of research in this area, few conclusions may be inferred. In general, the available research has provided tentative hypotheses about the nature of arithmetic disabilities. A variety of psychosocial variables notwithstanding, childhood arithmetic disability may directly result from cerebral dysfunction, poor motivation, and emotional/behavioral disturbance. However, further research is necessary in order to clarify the effects of maturation on arithmetic skills acquisition. Indeed, one approach to identification of the disorder would consider individual differences in neuropsychological development and performance affecting arithmetic achievement. It was concluded that a more comprehensive approach to investigating and diagnosing childhood arithmetic disability is needed. Reformulations and methods of study were articulated. Six related lines of research were outlined. A diagnostic rating scale was suggested which would account for type and severity of disorder. Diagnostic criteria were recommended based on the degree and definition of disability. Needs for remediation research were briefly explored. PMID:2485827
Study and realisation of the arithmetic unit of an information processing machine
After having defined the arithmetic unit of an information processing machine, and its role, and described the main characteristics of two types of machine (fixed or varying word length), the author of this research thesis reports the study of a decimal adder, describes the operation of a synchronous arithmetic unit for a varying word length machine, reports the technological study of the arithmetic unit (electronic components and circuits, printed circuits), and finally presents multiplication and division subroutines
Using fuzzy arithmetic in containment event trees
The use of fuzzy arithmetic is proposed for the evaluation of containment event trees. Concepts such as improbable, very improbable, and so on, which are subjective by nature, are represented by fuzzy numbers. The quantitative evaluation of containment event trees is based on the extension principle, by which operations on real numbers are extended to operations on fuzzy numbers. Expert knowledge is considered as state of the base variable with a normal distribution, which is considered to represent the membership function. Finally, this paper presents results of an example calculation of a containment event tree for the CAREM-25 nuclear power plant, presently under detailed design stage at Argentina. (author)
A New Fast Modular Arithmetic Method in Public Key Cryptography
WANG Bangju; ZHANG Huanguo
2006-01-01
Modular arithmetic is a fundamental operation and plays an important role in public key cryptosystem. A new method and its theory evidence on the basis of modular arithmetic with large integer modulus-changeable modulus algorithm is proposed to improve the speed of the modular arithmetic in the presented paper. For changeable modulus algorithm, when modular computation of modulo n is difficult, it can be realized by computation of modulo n-1 and n-2 on the perquisite of easy modular computations of modulo n-1 and modulo n-2. The conclusion is that the new method is better than the direct method by computing the modular arithmetic operation with large modulus. Especially, when computations of modulo n-1 and modulo n-2 are easy and computation of modulo n is difficult, this new method will be faster and has more advantages than other algorithms on modular arithmetic. Lastly, it is suggested that the proposed method be applied in public key cryptography based on modular multiplication and modular exponentiation with large integer modulus effectively
Relativity of arithmetics as a fundamental symmetry of physics
Czachor, Marek
2014-01-01
Arithmetic operations can be defined in various ways, even if one assumes commutativity and associativity of addition and multiplication, and distributivity of multiplication with respect to addition. In consequence, whenever one encounters `plus' or `times' one has certain freedom of interpreting this operation. This leads to some freedom in definitions of derivatives, integrals and, thus, practically all equations occurring in natural sciences. A change of realization of arithmetics, without altering the remaining structures of a given equation, plays the same role as a symmetry transformation. An appropriate construction of arithmetics turns out to be particularly important for dynamical systems in fractal space-times. Simple examples from classical and quantum, relativistic and nonrelativistic physics are discussed.
McNeil, Nicole M.; Rittle-Johnson, Bethany; Hattikudur, Shanta; Petersen, Lori A.
2010-01-01
This study examined if solving arithmetic problems hinders undergraduates' accuracy on algebra problems. The hypothesis was that solving arithmetic problems would hinder accuracy because it activates an operational view of equations, even in educated adults who have years of experience with algebra. In three experiments, undergraduates (N = 184)…
Counting arithmetic lattices and surfaces
Belolipetsky, Mikhail; Gelander, Tsachik; Lubotzky, Alexander; Shalev, Aner
2010-01-01
We give estimates on the number $AL_H(x)$ of arithmetic lattices $\\Gamma$ of covolume at most $x$ in a simple Lie group $H$. In particular, we obtain a first concrete estimate on the number of arithmetic 3-manifolds of volume at most $x$. Our main result is for the classical case $H=PSL(2,R)$ where we compute the limit of $\\log AL_H(x) / x\\log x$ when $x\\to\\infty$. The proofs use several different techniques: geometric (bounding the number of generators of $\\Gamma$ as a function of its covolu...
Predicting Arithmetic Abilities: The Role of Preparatory Arithmetic Markers and Intelligence
Stock, Pieter; Desoete, Annemie; Roeyers, Herbert
2009-01-01
Arithmetic abilities acquired in kindergarten are found to be strong predictors for later deficient arithmetic abilities. This longitudinal study (N = 684) was designed to examine if it was possible to predict the level of children's arithmetic abilities in first and second grade from their performance on preparatory arithmetic abilities in…
An arithmetic Lefschetz-Riemann-Roch theorem
Tang, Shun
2015-01-01
In this article, we consider regular arithmetic schemes in the context of Arakelov geometry, endowed with an action of the diagonalisable group scheme associated to a finite cyclic group. For any equivariant and proper morphism of such arithmetic schemes, which is smooth over the generic fibre, we define a direct image map between corresponding higher equivariant arithmetic K-groups and we discuss its transitivity property. Then we use the localization sequence of higher arithmetic K-groups a...
Is the conventional interval-arithmetic correct?
Andrzej Piegat; Marek Landowski
2012-01-01
Interval arithmetic as part of interval mathematics and Granular Computing is unusually important for development of science and engineering in connection with necessity of taking into account uncertainty and approximativeness of data occurring in almost all calculations. Interval arithmetic also conditions development of Artificial Intelligence and especially of automatic thinking, Computing with Words, grey systems, fuzzy arithmetic and probabilistic arithmetic. However, the mostly used con...
Differential forms on arithmetic jet spaces
Borger, James; Buium, Alexandru
2009-01-01
We study derivations and differential forms on the arithmetic jet spaces of smooth schemes, relative to several primes. As applications we give a new interpretation of arithmetic Laplacians and we discuss the de Rham cohomology of some specific arithmetic jet spaces.
On Volumes of Arithmetic Line Bundles
Yuan, Xinyi
2008-01-01
We show an arithmetic generalization of the recent work of Lazarsfeld-Mustata which uses Okounkov bodies to study linear series of line bundles. As applications, we derive a log-concavity inequality on volumes of arithmetic line bundles and an arithmetic Fujita approximation theorem for big line bundles.
Heights for line bundles on arithmetic surfaces
Jahnel, Joerg
1995-01-01
For line bundles on arithmetic varieties we construct height functions using arithmetic intersection theory. In the case of an arithmetic surface, generically of genus g, for line bundles of degree g equivalence is shown to the height on the Jacobian defined by the Theta divisor.
Solutions to Arithmetic Convolution Equations
Glöckner, H.; Lucht, L.G.; Porubský, Štefan
2007-01-01
Roč. 135, č. 6 (2007), s. 1619-1629. ISSN 0002-9939 R&D Projects: GA ČR GA201/04/0381 Institutional research plan: CEZ:AV0Z10300504 Keywords : arithmetic functions * Dirichlet convolution * polynomial equations * analytic equations * topological algebras * holomorphic functional calculus Subject RIV: BA - General Mathematics Impact factor: 0.520, year: 2007
Arithmetic theory of brick tilings
A new, 'arithmetic', approach to the algebraic theory of brick tilings is developed. This approach enables one to construct a simple classification of brick tilings in Zd and to find new proofs of several classical results on brick packing and tilings in Zd. In addition, possible generalizations of results on integer brick packing to the Euclidean plane R2 are investigated
The Development of Arithmetical Abilities
Butterworth, Brian
2005-01-01
Background: Arithmetical skills are essential to the effective exercise of citizenship in a numerate society. How these skills are acquired, or fail to be acquired, is of great importance not only to individual children but to the organisation of formal education and its role in society. Method: The evidence on the normal and abnormal…
Perceiving fingers in single-digit arithmetic problems
Ilaria eBerteletti
2015-03-01
Full Text Available In this study, we investigate in children the neural underpinnings of finger representation and finger movement involved in single-digit arithmetic problems. Evidence suggests that finger representation and finger-based strategies play an important role in learning and understanding arithmetic. Because different operations rely on different networks, we compared activation for subtraction and multiplication problems in independently localized finger somatosensory and motor areas and tested whether activation was related to skill. Brain activations from children between 8 and 13 years of age revealed that only subtraction problems significantly activated finger motor areas, suggesting reliance on finger-based strategies. In addition, larger subtraction problems yielded greater somatosensory activation than smaller problems, suggesting a greater reliance on finger representation for larger numerical values. Interestingly, better performance in subtraction problems was associated with lower activation in the finger somatosensory area. Our results support the importance of fine-grained finger representation in arithmetical skill and are the first neurological evidence for a functional role of the somatosensory finger area in proficient arithmetical problem solving, in particular for those problems requiring quantity manipulation. From an educational perspective, these results encourage investigating whether different finger-based strategies facilitate arithmetical understanding and encourage educational practices aiming at integrating finger representation and finger-based strategies as a tool for instilling stronger numerical sense.
Optimization Approaches for Designing Quantum Reversible Arithmetic Logic Unit
Haghparast, Majid; Bolhassani, Ali
2016-03-01
Reversible logic is emerging as a promising alternative for applications in low-power design and quantum computation in recent years due to its ability to reduce power dissipation, which is an important research area in low power VLSI and ULSI designs. Many important contributions have been made in the literatures towards the reversible implementations of arithmetic and logical structures; however, there have not been many efforts directed towards efficient approaches for designing reversible Arithmetic Logic Unit (ALU). In this study, three efficient approaches are presented and their implementations in the design of reversible ALUs are demonstrated. Three new designs of reversible one-digit arithmetic logic unit for quantum arithmetic has been presented in this article. This paper provides explicit construction of reversible ALU effecting basic arithmetic operations with respect to the minimization of cost metrics. The architectures of the designs have been proposed in which each block is realized using elementary quantum logic gates. Then, reversible implementations of the proposed designs are analyzed and evaluated. The results demonstrate that the proposed designs are cost-effective compared with the existing counterparts. All the scales are in the NANO-metric area.
A Short Survey on Arithmetic Transforms and the Arithmetic Hartley Transform
Cintra, R. J.; de Oliveira, H. M.
2015-01-01
Arithmetic complexity has a main role in the performance of algorithms for spectrum evaluation. Arithmetic transform theory offers a method for computing trigonometrical transforms with minimal number of multiplications. In this paper, the proposed algorithms for the arithmetic Fourier transform are surveyed. A new arithmetic transform for computing the discrete Hartley transform is introduced: the Arithmetic Hartley transform. The interpolation process is shown to be the key element of the a...
Computer arithmetic and validity theory, implementation, and applications
Kulisch, Ulrich
2013-01-01
This is the revised and extended second edition of the successful basic book on computer arithmetic. It is consistent with the newest recent standard developments in the field. The book shows how the arithmetic capability of the computer can be enhanced. The work is motivated by the desire and the need to improve the accuracy of numerical computing and to control the quality of the computed results (validity). The accuracy requirements for the elementary floating-point operations are extended to the customary product spaces of computations including interval spaces. The mathematical properties
Uncertainty propagation in fault trees using a quantile arithmetic methodology
A methodology based on Quantile Arithmetic, the probabilistic analog to Interval Analysis (Dempster 1969), is proposed for the computation of uncertainty propagation in Fault Tree Analysis (Apostolakis 1977). The basic events' continuous probability density functions are represented by equivalent discrete distributions through dividing them into a number of quantiles N. Quantile Arithmetic is then used to perform the binary arithmetical operations corresponding to the logical gates in the Boolean expression for the Top Event of a given Fault Tree. The computational characteristics of the proposed methodology as compared with the exact analytical solutions are discussed for the cases of the summation of M normal variables. It is further compared with the Monte Carlo method through the use of the efficiency ratio defined as the product of the labor and error ratios. (orig./HP)
Nonsymbolic, Approximate Arithmetic in Children: Abstract Addition Prior to Instruction
Barth, Hilary; Beckmann, Lacey; Spelke, Elizabeth S.
2008-01-01
Do children draw upon abstract representations of number when they perform approximate arithmetic operations? In this study, kindergarten children viewed animations suggesting addition of a sequence of sounds to an array of dots, and they compared the sum to a second dot array that differed from the sum by 1 of 3 ratios. Children performed this…
Arithmetic Operations Beyond Floating Point Number Precision
Wang, Chih-Yueh; Yin, Chen-Yang; Chen, Hong-Yu; Chen, Yung-Ko
2010-01-01
In basic computational physics classes, students often raise the question of how to compute a number that exceeds the numerical limit of the machine. While technique of avoiding overflow/underflow has practical application in the electrical and electronics engineering industries, it is not commonly utilized in scientific computing, because scientific notation is adequate in most cases. We present an undergraduate project that deals with such calculations beyond a machine's numerical limit, kn...
Arithmetic Operations on Trapezoidal Fuzzy Numbers
J. Vahidi
2013-10-01
Full Text Available In this paper, several new algebraic mathematics for positive fuzzy numbers of type $(\\overline{a}, \\overline{\\overline{a}}, \\overline{\\overline{\\overline{a}}}, \\overline{\\overline{\\overline{\\overline{a}}}}$ are devised and do not need the computation of $\\alpha$-cut of the fuzzy number. Direct mathematical expressions to evaluate exponential, square root, logarithms, inverse exponential etc. of positive fuzzy numbers of type $(\\overline{a}, \\overline{\\overline{a}}, \\overline{\\overline{\\overline{a}}}, \\overline{\\overline{\\overline{\\overline{a}}}}$ are obtained using the basic analytical principles of algebraic mathematics and Taylor series expansion. At the end, Various numerical examples are also solved to demonstrate the use of contrived expressions.
Arithmetical chaos and violation of universality in energy level statistics
A class of strongly chaotic systems revealing a strange arithmetical structure is discussed whose quantal energy levels exhibit level attraction rather than repulsion. As an example, the nearest-neighbour level spacings for Artin's billiard have been computed in a large energy range. It is shown that the observed violation of universality has its root in the existence of an infinite number of hermitian operators (Hecke operators) which commute with the Hamiltonian and generate mongeneric correlations in the eigenfunctions. (orig.)
Coinductive Formal Reasoning in Exact Real Arithmetic
Niqui, Milad
2008-01-01
In this article we present a method for formally proving the correctness of the lazy algorithms for computing homographic and quadratic transformations -- of which field operations are special cases-- on a representation of real numbers by coinductive streams. The algorithms work on coinductive stream of M\\"obius maps and form the basis of the Edalat--Potts exact real arithmetic. We use the machinery of the Coq proof assistant for the coinductive types to present the formalisation. The formalised algorithms are only partially productive, i.e., they do not output provably infinite streams for all possible inputs. We show how to deal with this partiality in the presence of syntactic restrictions posed by the constructive type theory of Coq. Furthermore we show that the type theoretic techniques that we develop are compatible with the semantics of the algorithms as continuous maps on real numbers. The resulting Coq formalisation is available for public download.
Dark energy as a manifestation of nontrivial arithmetic
Czachor, Marek
2016-01-01
Arithmetic operations (addition, subtraction, multiplication, division), as well as the calculus they imply, are non-unique. The examples of four-dimensional spaces, $\\mathbb{R}_+^4$ and $(-L/2,L/2)^4$, are considered where different types of arithmetic and calculus coexist simultaneously. In all the examples there exists a non-Diophantine arithmetic that makes the space globally Minkowskian, and thus the laws of physics are formulated in terms of the corresponding calculus. However, when one switches to the `natural' Diophantine arithmetic and calculus, the Minkowskian character of the space is lost and what one effectively obtains is a Lorentzian manifold. I discuss in more detail the problem of electromagnetic fields produced by a pointlike charge. The solution has the standard form when expressed in terms of the non-Diophantine formalism. When the `natural' formalsm is used, the same solution looks as if the fields were created by a charge located in an expanding universe, with nontrivially accelerating e...
Arithmetical chaos and quantum cosmology
In this paper, we present the formalism to start a quantum analysis for the recent billiard representation introduced by Damour, Henneaux and Nicolai in the study of the cosmological singularity. In particular we use the theory of Maass automorphic forms and recent mathematical results about arithmetical dynamical systems. The predictions of the billiard model give precise automorphic properties for the wavefunction (Maass-Hecke eigenform), the asymptotic number of quantum states (Selberg asymptotics for PSL(2,Z)), the distribution for the level spacing statistics (the Poissonian one) and the absence of scarred states. The most interesting implication of this model is perhaps that the discrete spectrum is fully embedded in the continuous one.
Arithmetic geometry and number theory
Weng, Lin
2006-01-01
Mathematics is very much a part of our culture; and this invaluable collection serves the purpose of developing the branches involved, popularizing the existing theories and guiding our future explorations.More precisely, the goal is to bring the reader to the frontier of current developments in arithmetic geometry and number theory through the works of Deninger-Werner in vector bundles on curves over p-adic fields; of Jiang on local gamma factors in automorphic representations; of Weng on Deligne pairings and Takhtajan-Zograf metrics; of Yoshida on CM-periods; of Yu on transcendence of specia
A Bertini-type theorem for free arithmetic linear series
Ikoma, Hideaki
2015-01-01
In this paper, we prove a version of the arithmetic Bertini theorem asserting that there exists a strictly small and generically smooth section of a given arithmetically free graded arithmetic linear series.
Arithmetic of quantum entropy function
Quantum entropy function is a proposal for computing the entropy associated with the horizon of a black hole in the extremal limit, and is related via AdS/CFT correspondence to the dimension of the Hilbert space in a dual quantum mechanics. We show that in N = 4 supersymmetric string theories, quantum entropy function formalism naturally explains the origin of the subtle differences between the microscopic degeneracies of quarter BPS dyons carrying different torsion, i.e. different arithmetical properties. These arise from additional saddle points in the path integral - whose existence depends on the arithmetical properties of the black hole charges - constructed as freely acting orbifolds of the original AdS2 x S2 near horizon geometry. During this analysis we demonstrate that the quantum entropy function is insensitive to the details of the infrared cutoff used in the computation, and the details of the boundary terms added to the action. We also discuss the role of the asymptotic symmetries of AdS2 in carrying out the path integral in the definition of quantum entropy function. Finally we show that even though quantum entropy function is expected to compute the absolute degeneracy in a given charge and angular momentum sector, it can also be used to compute the index. This can then be compared with the microscopic computation of the index.
From Arithmetic Sequences to Linear Equations
Matsuura, Ryota; Harless, Patrick
2012-01-01
The first part of the article focuses on deriving the essential properties of arithmetic sequences by appealing to students' sense making and reasoning. The second part describes how to guide students to translate their knowledge of arithmetic sequences into an understanding of linear equations. Ryota Matsuura originally wrote these lessons for…
Weak Theories of Concatenation and Arithmetic
Horihata, Yoshihiro
2012-01-01
We define a new theory of concatenation WTC which is much weaker than Grzegorczyk's well-known theory TC. We prove that WTC is mutually interpretable with the weak theory of arithmetic R. The latter is, in a technical sense, much weaker than Robinson's arithmetic Q, but still essentially undecidable. Hence, as a corollary, WTC is also essentially undecidable.
Some results on uniform arithmetic circuit complexity
Frandsen, Gudmund Skovbjerg; Valence, Mark; Barrington, David A. Mix
1994-01-01
polynomial-size unbounded fan-in arithmetic circuits satisfying a natural uniformity constraint (DLOGTIME-uniformity). A 1-input and 1-output arithmetic function over the fieldsF2n may be identified with ann-input andn-output Boolean function when field elements are represented as bit strings. We prove that...
Szemeredi's theorem and problems on arithmetic progressions
Szemeredi's famous theorem on arithmetic progressions asserts that every subset of integers of positive asymptotic density contains arithmetic progressions of arbitrary length. His remarkable theorem has been developed into a major new area of combinatorial number theory. This is the topic of the present survey.
Numerical Magnitude Representations Influence Arithmetic Learning
Booth, Julie L.; Siegler, Robert S.
2008-01-01
This study examined whether the quality of first graders' (mean age = 7.2 years) numerical magnitude representations is correlated with, predictive of, and causally related to their arithmetic learning. The children's pretest numerical magnitude representations were found to be correlated with their pretest arithmetic knowledge and to be…
Error-correcting codes in computer arithmetic.
Massey, J. L.; Garcia, O. N.
1972-01-01
Summary of the most important results so far obtained in the theory of coding for the correction and detection of errors in computer arithmetic. Attempts to satisfy the stringent reliability demands upon the arithmetic unit are considered, and special attention is given to attempts to incorporate redundancy into the numbers themselves which are being processed so that erroneous results can be detected and corrected.
Prevalence of Combined Reading and Arithmetic Disabilities
Dirks, Evelien; Spyer, Ginny; van Lieshout, Ernest C. D. M.; de Sonneville, Leo
2008-01-01
This study assesses the prevalence of combined reading and arithmetic disabilities in 799 Dutch schoolchildren using standardized school achievement tests. Scores of arithmetic, word recognition, reading comprehension, and spelling of children in fourth and fifth grade were used. The main interest involved the co-occurrence of word recognition and…
Patrick Lemaire
2010-10-01
Full Text Available In this paper, we provide an overview of three important issues regarding working-memory/executive functions (WM/EF, strategies, and cognitive development in the domain of arithmetic. One goal of this overview is to bring some lights on the depth and breadth of the most valuable contributions that André Vandierendonck and his collaborators made on these issues. First, we consider strategic aspects of arithmetic performance and strategic development in arithmetic. Second, the role of WM/EF on arithmetic performance and arithmetic strategies is discussed. Finally, some data are reported on how age-related changes in WM/EF affect strategic development in arithmetic. For each of these issues, we highlight how the works carried out by André Vandierendonck and his colleagues, when integrated in the broader context of research on cognitive arithmetic, contributed to our further understanding of participants' performance and age-related changes in this performance.
Level statistics in arithmetical and pseudo-arithmetical chaos
We investigate a long-standing riddle in quantum chaos, posed by certain fully chaotic billiards with constant negative curvature whose periodic orbits are highly degenerate in length. Depending on the boundary conditions for the quantum wavefunctions, the energy spectra either have uncorrelated levels usually associated with classical integrability or conform to the 'universal' Wigner-Dyson type although the classical dynamics in both cases is the same. The resolution turns out surprisingly simple. The Maslov indices of orbits within multiplets of degenerate length either yield equal phases for the respective Feynman amplitudes (and thus Poissonian level statistics) or give rise to amplitudes with uncorrelated phases (leading to Wigner-Dyson level correlations). The recent semiclassical explanation of spectral universality in quantum chaos is thus extended to the latter case of 'pseudo-arithmetical' chaos. (fast track communication)
Nondigital implementation of the arithmetic of real numbers by means of quantum computer media
Litvinov, Grigori; Maslov, Viktor; Shpiz, Grigori
1999-01-01
In the framework of a model for quantum computer media, a nondigital implementation of the arithmetic of the real numbers is described. For this model, an elementary storage "cell" is an ensemble of qubits (quantum bits). It is found that to store an arbitrary real number it is sufficient to use four of these ensembles and the arithmetic operations can be implemented by fixed quantum circuits.
Simplification of integrity constraints with aggregates and arithmetic built-ins
Martinenghi, Davide
2004-01-01
Both aggregates and arithmetic built-ins are widely used in current database query languages: Aggregates are second-order constructs such as CNT and SUM of SQL; arithmetic built-ins include relational and other mathematical operators that apply to numbers, such as ... time, simplified versions of such integrity constraints that can be tested before the execution of any update. In this way, virtually no time is spent for optimization or rollbacks at run time. Both set and bag semantics are considered....
Plain Polynomial Arithmetic on GPU
As for serial code on CPUs, parallel code on GPUs for dense polynomial arithmetic relies on a combination of asymptotically fast and plain algorithms. Those are employed for data of large and small size, respectively. Parallelizing both types of algorithms is required in order to achieve peak performances. In this paper, we show that the plain dense polynomial multiplication can be efficiently parallelized on GPUs. Remarkably, it outperforms (highly optimized) FFT-based multiplication up to degree 212 while on CPU the same threshold is usually at 26. We also report on a GPU implementation of the Euclidean Algorithm which is both work-efficient and runs in linear time for input polynomials up to degree 218 thus showing the performance of the GCD algorithm based on systolic arrays.
Urszula eMihulowicz
2014-05-01
Full Text Available Different specific mechanisms have been suggested for solving single-digit arithmetic operations. However, the neural correlates underlying basic arithmetic (multiplication, addition, subtraction are still under debate. In the present study, we systematically assessed single-digit arithmetic in a group of acute stroke patients (n=45 with circumscribed left- or right-hemispheric brain lesions. Lesion sites significantly related to impaired performance were found only in the left-hemisphere damaged group. Deficits in multiplication and addition were related to subcortical/white matter brain regions differing from those for subtraction tasks, corroborating the notion of distinct processing pathways for different arithmetic tasks. Additionally, our results further point to the importance of investigating fiber pathways in numerical cognition.
Arithmetic area for m planar Brownian paths
Desbois, Jean
2012-01-01
We pursue the analysis made in [1] on the arithmetic area enclosed by m closed Brownian paths. We pay a particular attention to the random variable S{n1,n2, ...,n} (m) which is the arithmetic area of the set of points, also called winding sectors, enclosed n1 times by path 1, n2 times by path 2, ...,nm times by path m. Various results are obtained in the asymptotic limit m->infinity. A key observation is that, since the paths are independent, one can use in the m paths case the SLE information, valid in the 1-path case, on the 0-winding sectors arithmetic area.
Hardware Implementations of GF (2m Arithmetic Using Normal Basis
Turki F. Al-Somani
2006-01-01
Full Text Available This study presents a survey of algorithms used in field arithmetic over GF (2m using normal basis and their hardware implementations. These include the following arithmetic field operations: addition, squaring, multiplication and inversion. This study shows that the type II Sunar-Koc multiplier is the best multiplier with a hardware complexity of m2 AND gates + XOR gates and a time complexity of TA+ (1+ l log2 (m l Tx. The study also show that the Itoh-Tsujii inversion algorithm was the best inverter and it requires almost log2 (m-1 multiplications.
Critical Path Reduction of Distributed Arithmetic Based FIR Filter
Sunita Badave
2016-03-01
Full Text Available Operating speed, which is reciprocal of critical path computation time, is one of the prominent design matrices of finite impulse response (FIR filters. It is largely affected by both, system architecture as well as technique used to design arithmetic modules. A large computation time of multipliers in conventionally designed multipliers, limits the speed of system architecture. Distributed arithmetic is one of the techniques, used to provide multiplier-free multiplication in the implementation of FIR filter. However suffers from a sever limitation of exponential growth of look up table (LUT with order of filter. An improved distributed arithmetic technique is addressed here to design for system architecture of FIR filter. In proposed technique, a single large LUT of conventional DA is replaced by number of smaller indexed LUT pages to restrict exponential growth and to reduce system access time. It also eliminates the use of adders. Selection module selects the desired value from desired page, which leads to reduce computational time of critical path. Trade off between access times of LUT pages and selection module helps to achieve minimum critical path so as to maximize the operating speed. Implementations are targeted to Xilinx ISE, Virtex IV devices. FIR filter with 8 bit data width of input sample results are presented here. It is observed that, proposed design perform significantly faster as compared to the conventional DA and existing DA based designs.
Electro-Photo-Sensitive Memristor for Neuromorphic and Arithmetic Computing
Maier, P.; Hartmann, F.; Emmerling, M.; Schneider, C.; Kamp, M.; Höfling, S.; Worschech, L.
2016-05-01
We present optically and electrically tunable conductance modifications of a site-controlled quantum-dot memristor. The conductance of the device is tuned by electron localization on a quantum dot. The control of the conductance with voltage and low-power light pulses enables applications in neuromorphic and arithmetic computing. As in neural networks, applying pre- and postsynaptic voltage pulses to the memristor allows us to increase (potentiation) or decrease (depression) the conductance by tuning the time difference between the electrical pulses. Exploiting state-dependent thresholds for potentiation and depression, we are able to demonstrate a memory-dependent induction of learning. The discharging of the quantum dot can further be induced by low-power light pulses in the nanowatt range. In combination with the state-dependent threshold voltage for discharging, this enables applications as generic building blocks to perform arithmetic operations in bases ranging from binary to decimal with low-power optical excitation. Our findings allow the realization of optoelectronic memristor-based synapses in artificial neural networks with a memory-dependent induction of learning and enhanced functionality by performing arithmetic operations.
Quality of Arithmetic Education for Children with Cerebral Palsy
Jenks, Kathleen M.; de Moor, Jan; van Lieshout, Ernest C. D. M.; Withagen, Floortje
2010-01-01
The aim of this exploratory study was to investigate the quality of arithmetic education for children with cerebral palsy. The use of individual educational plans, amount of arithmetic instruction time, arithmetic instructional grouping, and type of arithmetic teaching method were explored in three groups: children with cerebral palsy (CP) in…
Patrick Lemaire
2010-01-01
In this paper, we provide an overview of three important issues regarding working-memory/executive functions (WM/EF), strategies, and cognitive development in the domain of arithmetic. One goal of this overview is to bring some lights on the depth and breadth of the most valuable contributions that André Vandierendonck and his collaborators made on these issues. First, we consider strategic aspects of arithmetic performance and strategic development in arithmetic. Second, the role of WM/EF on...
Obstacle problem for Arithmetic Asian options
Laura Monti; Andrea Pascucci
2009-01-01
We prove existence, regularity and a Feynman-Ka\\v{c} representation formula of the strong solution to the free boundary problem arising in the financial problem of the pricing of the American Asian option with arithmetic average.
L(2)-cohomology of arithmetic varieties.
Saper, L; Stern, M
1987-08-01
The L(2)-cohomology of arithmetic quotients of bounded symmetric domains is studied. We establish the conjecture of Zucker equating the L(2)-cohomology of these spaces to the intersection cohomology of their Baily-Borel compactifications. PMID:16593866
L2-cohomology of arithmetic varieties
Saper, Leslie; Stern, Mark
1987-01-01
The L2-cohomology of arithmetic quotients of bounded symmetric domains is studied. We establish the conjecture of Zucker equating the L2-cohomology of these spaces to the intersection cohomology of their Baily-Borel compactifications. PMID:16593866
Recursive formula for arithmetic Asian option prices
Kyungsub Lee
2013-01-01
We derive a recursive formula for arithmetic Asian option prices with finite observation times in semimartingale models. The method is based on the relationship between the risk-neutral expectation of the quadratic variation of the return process and European option prices. The computation of arithmetic Asian option prices is straightforward whenever European option prices are available. Applications with numerical results under the Black-Scholes framework and the exponential L\\'evy model are...
Complete Program Synthesis for Linear Arithmetic
Mayer, Mikael
2010-01-01
Synthesis of programs or their fragments is a way to write programs by providing only their meaning without worrying about the implementation details. It avoids the drawback of writing sequential code, which might be difficult to check, error-prone or tedious. Our contribution is to provide complete program synthesis algorithms with unbounded data types in decidable theories. We present synthesis algorithms for Linear Rational Arithmetic, Linear Integer Arithmetic and Parametrized Linear Inte...
Herbrand consistency of some arithmetical theories
Salehi, Saeed
2012-01-01
G\\"odel's second incompleteness theorem is proved for Herbrand consistency of some arithmetical theories with bounded induction, by using a technique of logarithmic shrinking the witnesses of bounded formulas, due to Z. Adamowicz [Herbrand consistency and bounded arithmetic, \\textit{Fundamenta Mathematicae} 171 (2002) 279--292]. In that paper, it was shown that one cannot always shrink the witness of a bounded formula logarithmically, but in the presence of Herbrand consistency, for theories ...
Quantile arithmetic methodology for uncertainty propagation in fault trees
A methodology based on quantile arithmetic, the probabilistic analog to interval analysis, is proposed for the computation of uncertainties propagation in fault tree analysis. The basic events' continuous probability density functions (pdf's) are represented by equivalent discrete distributions by dividing them into a number of quantiles N. Quantile arithmetic is then used to performthe binary arithmetical operations corresponding to the logical gates in the Boolean expression of the top event expression of a given fault tree. The computational advantage of the present methodology as compared with the widely used Monte Carlo method was demonstrated for the cases of summation of M normal variables through the efficiency ratio defined as the product of the labor and error ratios. The efficiency ratio values obtained by the suggested methodology for M = 2 were 2279 for N = 5, 445 for N = 25, and 66 for N = 45 when compared with the results for 19,200 Monte Carlo samples at the 40th percentile point. Another advantage of the approach is that the exact analytical value of the median is always obtained for the top event
Grabner, Roland H; Rütsche, Bruno; Ruff, Christian C; Hauser, Tobias U
2015-07-01
The successful acquisition of arithmetic skills is an essential step in the development of mathematical competencies and has been associated with neural activity in the left posterior parietal cortex (PPC). It is unclear, however, whether this brain region plays a causal role in arithmetic skill acquisition and whether arithmetic learning can be modulated by means of non-invasive brain stimulation of this key region. In the present study we addressed these questions by applying transcranial direct current stimulation (tDCS) over the left PPC during a short-term training that simulates the typical path of arithmetic skill acquisition (specifically the transition from effortful procedural to memory-based problem-solving strategies). Sixty participants received either anodal, cathodal or sham tDCS while practising complex multiplication and subtraction problems. The stability of the stimulation-induced learning effects was assessed in a follow-up test 24 h after the training. Learning progress was modulated by tDCS. Cathodal tDCS (compared with sham) decreased learning rates during training and resulted in poorer performance which lasted over 24 h after stimulation. Anodal tDCS showed an operation-specific improvement for subtraction learning. Our findings extend previous studies by demonstrating that the left PPC is causally involved in arithmetic learning (and not only in arithmetic performance) and that even a short-term tDCS application can modulate the success of arithmetic knowledge acquisition. Moreover, our finding of operation-specific anodal stimulation effects suggests that the enhancing effects of tDCS on learning can selectively affect just one of several cognitive processes mediated by the stimulated area. PMID:25970697
Dynamic mental number line in simple arithmetic.
Yu, Xiaodan; Liu, Jie; Li, Dawei; Liu, Hang; Cui, Jiaxin; Zhou, Xinlin
2016-05-01
Studies have found that spatial-numerical associations could extend to arithmetic. Addition leads to rightward shift in spatial attention while subtraction leads to leftward shift (e.g., Knops et al. 2009; McCrink et al. 2007; Pinhas & Fischer 2008), which is consistent with the hypothesis of static mental number line (MNL) for arithmetic. The current investigation tested the hypothesis of dynamic mental number line which was shaped by the relative magnitudes of two operands in simple arithmetic. Horizontal and vertical electrooculograms (HEOG and VEOG) during simple arithmetic were recorded. Results showed that the direction of eye movements was dependent on the relative magnitudes of two operands. Subtraction was associated with larger rightward eye movements than addition (Experiment 1), and smaller-operand-first addition (e.g., 2+9) was associated with larger rightward eye movement than larger-operand-first addition (e.g., 9+2) only when the difference of two operands was large (Experiment 2). The results suggest that the direction of the mental number line could be dynamic during simple arithmetic, and that the eyes move along the dynamic mental number line to search for solutions. PMID:26645825
Arithmetic Self-Similarity of Infinite Sequences
Hendriks, Dimitri; Endrullis, Joerg; Dow, Mark; Klop, Jan Willem
2012-01-01
We define the arithmetic self-similarity (AS) of a one-sided infinite sequence sigma to be the set of arithmetic progressions through sigma which are a vertical shift of sigma. We classify the AS of several well-known sequences, such as the Thue-Morse sequence, the period doubling sequence, and the regular paperfolding sequence. The latter two are examples of (completely) additive sequences as well as of Toeplitz words. We investigate the intersection of these families. We give a complete characterization of single-gap patterns that yield additive Toeplitz words, and classify their AS. Moreover, we show that every arithmetic progression through a Toeplitz word generated by a one-gap pattern is again a Toeplitz word. Finally, we establish that generalized Morse sequences are specific sum-of-digits sequences, and show that their first difference is a Toeplitz word.
Arithmetic geometry over global function fields
Longhi, Ignazio; Trihan, Fabien
2014-01-01
This volume collects the texts of five courses given in the Arithmetic Geometry Research Programme 2009–2010 at the CRM Barcelona. All of them deal with characteristic p global fields; the common theme around which they are centered is the arithmetic of L-functions (and other special functions), investigated in various aspects. Three courses examine some of the most important recent ideas in the positive characteristic theory discovered by Goss (a field in tumultuous development, which is seeing a number of spectacular advances): they cover respectively crystals over function fields (with a number of applications to L-functions of t-motives), gamma and zeta functions in characteristic p, and the binomial theorem. The other two are focused on topics closer to the classical theory of abelian varieties over number fields: they give respectively a thorough introduction to the arithmetic of Jacobians over function fields (including the current status of the BSD conjecture and its geometric analogues, and the con...
Learning, Realizability and Games in Classical Arithmetic
Aschieri, Federico
2010-01-01
In this dissertation we provide mathematical evidence that the concept of learning can be used to give a new and intuitive computational semantics of classical proofs in various fragments of Predicative Arithmetic. First, we extend Kreisel modified realizability to a classical fragment of first order Arithmetic, Heyting Arithmetic plus EM1 (Excluded middle axiom restricted to Sigma^0_1 formulas). We introduce a new realizability semantics we call "Interactive Learning-Based Realizability". Our realizers are self-correcting programs, which learn from their errors and evolve through time. Secondly, we extend the class of learning based realizers to a classical version PCFclass of PCF and, then, compare the resulting notion of realizability with Coquand game semantics and prove a full soundness and completeness result. In particular, we show there is a one-to-one correspondence between realizers and recursive winning strategies in the 1-Backtracking version of Tarski games. Third, we provide a complete and fully...
Arithmetic in large GF(2(exp n))
Cameron, Kelly
1993-01-01
The decoding of Reed Solomon (BCH) codes usually requires large numbers of calculations using GF(2(exp n)) arithmetic. Though efficient algorithms and corresponding circuits for performing basic Galois field arithmetic are known, many of these techniques either become very slow or else require an inordinate amount of circuitry to implement when the size of the Galois field becomes much larger than GF(2(exp 8)). Consequently, most currently available Reed-Solomon decoders are built using small fields, such as GF(2(exp 8)) or GF(2(exp 10)), even though significant coding efficiencies could often be obtained if larger symbol sizes, such as GF(2(exp 16)) or GF(2(exp 32)), were used. Algorithms for performing the basic arithmetic required to decode Reed-Solomon codes have been developed explicitly for use in these large fields. They are discussed in detail.
Arithmetic area for m planar Brownian paths
We pursue the analysis made in Desbois and Ouvry (2011 J. Stat. Mech. P05024) on the arithmetic area enclosed by m closed Brownian paths. We pay particular attention to the random variable Sn1,n2,...,nm(m), which is the arithmetic area of the set of points, also called winding sectors, enclosed n1 times by path 1, n2 times by path 2,..., and nm times by path m. Various results are obtained in the asymptotic limit m→∞. A key observation is that, since the paths are independent, one can use in the m-path case the SLE information, valid in the one-path case, on the zero-winding sectors arithmetic area
Arithmetic area for m planar Brownian paths
Desbois, Jean; Ouvry, Stéphane
2012-05-01
We pursue the analysis made in Desbois and Ouvry (2011 J. Stat. Mech. P05024) on the arithmetic area enclosed by m closed Brownian paths. We pay particular attention to the random variable Sn1, n2,..., nm(m), which is the arithmetic area of the set of points, also called winding sectors, enclosed n1 times by path 1, n2 times by path 2,..., and nm times by path m. Various results are obtained in the asymptotic limit m\\to \\infty . A key observation is that, since the paths are independent, one can use in the m-path case the SLE information, valid in the one-path case, on the zero-winding sectors arithmetic area.
p-adic path set fractals and arithmetic
Abram, William; Lagarias, Jeffrey C.
2012-01-01
This paper considers a class C(Z_p) of closed sets of the p-adic integers obtained by graph-directed constructions analogous to those of Mauldin and Williams over the real numbers. These sets are characterized as collections of those p-adic integers whose p-adic expansions are describeed by paths in the graph of a finite automaton issuing from a distinguished initial vertex. This paper shows that this class of sets is closed under the arithmetic operations of addition and multiplication by p-...
Design of Parity Preserving Logic Based Fault Tolerant Reversible Arithmetic Logic Unit
Rakshith Saligram
2013-07-01
Full Text Available Reversible Logic is gaining significant consideration as the potential logic design style for implementationin modern nanotechnology and quantum computing with minimal impact on physical entropy .FaultTolerant reversible logic is one class of reversible logic that maintain the parity of the input and theoutputs. Significant contributions have been made in the literature towards the design of fault tolerantreversible logic gate structures and arithmetic units, however, there are not many efforts directed towardsthe design of fault tolerant reversible ALUs. Arithmetic Logic Unit (ALU is the prime performing unit inany computing device and it has to be made fault tolerant. In this paper we aim to design one such faulttolerant reversible ALU that is constructed using parity preserving reversible logic gates. The designedALU can generate up to seven Arithmetic operations and four logical operations.
Design of Parity Preserving Logic Based Fault Tolerant Reversible Arithmetic Logic Unit
Rakshith Saligram1
2013-06-01
Full Text Available Reversible Logic is gaining significant consideration as the potential logic design style for implementation in modern nanotechnology and quantum computing with minimal impact on physical entropy .Fault Tolerant reversible logic is one class of reversible logic that maintain the parity of the input and the outputs. Significant contributions have been made in the literature towards the design of fault tolerant reversible logic gate structures and arithmetic units, however, there are not many efforts directed towards the design of fault tolerant reversible ALUs. Arithmetic Logic Unit (ALU is the prime performing unit in any computing device and it has to be made fault tolerant. In this paper we aim to design one such fault tolerant reversible ALU that is constructed using parity preserving reversible logic gates. The designed ALU can generate up to seven Arithmetic operations and four logical operations
2010-07-01
... 40 Protection of Environment 6 2010-07-01 2010-07-01 false How do I convert my 1-hour arithmetic... SOURCES Operator Training and Qualification Monitoring § 60.2943 How do I convert my 1-hour arithmetic... emissions at 7 percent oxygen. (b) Use Equation 2 in § 60.2975 to calculate the 12-hour rolling averages...
Arithmetic algorithms for error-coded operands.
Avizienis, A.
1973-01-01
Arithmetic algorithms for separate and nonseparate codes are considered. The nonseparate AN code is formed when an uncoded operand X is multiplied by the check modulus A to give the coded operand AX. The separate codes are the residue code, and the inverse-residue code, which has significant advantages in fault detection of repeated-use faults. A set of algorithms for low-cost AN-coded operands is discussed together with questions of their implementation in a byte-organized arithmetic processor. Algorithms for inverse-residue coded operands of the STAR computer are also examined.
Arithmetic area for m planar Brownian paths
Desbois, Jean; Ouvry, Stephane
2012-01-01
We pursue the analysis made in [1] on the arithmetic area enclosed by m closed Brownian paths. We pay a particular attention to the random variable S{n1,n2, ...,n} (m) which is the arithmetic area of the set of points, also called winding sectors, enclosed n1 times by path 1, n2 times by path 2, ...,nm times by path m. Various results are obtained in the asymptotic limit m->infinity. A key observation is that, since the paths are independent, one can use in the m paths case the SLE informatio...
Derivations and Generating Degrees in the Ring of Arithmetical Functions
Alexandru Zaharescu; Mohammad Zaki
2007-05-01
In this paper we study a family of derivations in the ring of arithmetical functions of several variables over an integral domain, and compute the generating degrees of the ring of arithmetical functions over the kernel of these derivations.
Training of Attention in Children With Low Arithmetical Achievement.
Maria Guarnera; Antonella D’Amico
2014-01-01
This study focuses on the role of attentional processes in arithmetical skills and examines if training of basic attentive skills may improve also working memory abilities reducing arithmetic difficulties. In order to study the efficacy of attentional treatment in arithmetic achievement and in enhancing working memory abilities a test-treatment-retest quasi experimental design was adopted. The research involved 14 children, attending fourth and fifth grades, with Arithmetical Learning Disabil...
Training of Attention in Children With Low Arithmetical Achievement
Maria Guarnera; Antonella D’Amico
2014-01-01
This study focuses on the role of attentional processes in arithmetical skills and examines if training of basic attentive skills may improve also working memory abilities reducing arithmetic difficulties. In order to study the efficacy of attentional treatment in arithmetic achievement and in enhancing working memory abilities a test-treatment-retest quasi experimental design was adopted. The research involved 14 children, attending fourth and fifth grades, with Arithmetical Learning Disabil...
Personal Experience and Arithmetic Meaning in Semantic Dementia
Julien, Camille L.; Neary, David; Snowden, Julie S.
2010-01-01
Arithmetic skills are generally claimed to be preserved in semantic dementia (SD), suggesting functional independence of arithmetic knowledge from other aspects of semantic memory. However, in a recent case series analysis we showed that arithmetic performance in SD is not entirely normal. The finding of a direct association between severity of…
Intersection pairing for arithmetic cycles with degenerate Green currents
Moriwaki, Atsushi
1998-01-01
In this note, we would like to propose a suitable extension of the arithmetic Chow group of codimension one, in which the Hodge index theorem holds. We also prove an arithmetic analogue of Bogomolov's instability theorem for rank 2 vector bundles on arbitrary regular projective arithmetic varieties.
Price, Gavin R; Mazzocco, Michèle M M; Ansari, Daniel
2013-01-01
Do individual differences in the brain mechanisms for arithmetic underlie variability in high school mathematical competence? Using functional magnetic resonance imaging, we correlated brain responses to single digit calculation with standard scores on the Preliminary Scholastic Aptitude Test (PSAT) math subtest in high school seniors. PSAT math scores, while controlling for PSAT Critical Reading scores, correlated positively with calculation activation in the left supramarginal gyrus and bilateral anterior cingulate cortex, brain regions known to be engaged during arithmetic fact retrieval. At the same time, greater activation in the right intraparietal sulcus during calculation, a region established to be involved in numerical quantity processing, was related to lower PSAT math scores. These data reveal that the relative engagement of brain mechanisms associated with procedural versus memory-based calculation of single-digit arithmetic problems is related to high school level mathematical competence, highlighting the fundamental role that mental arithmetic fluency plays in the acquisition of higher-level mathematical competence. PMID:23283330
Mathematical Fuzzy Logic and Axiomatic Arithmetic
Hájek, Petr
Linz : Johannes Kepler Universität, 2010 - (Cintula, P.; Klement, E.; Stout, L.). s. 63-63 [Linz Seminar on Fuzzy Set Theory /31./. 03.02.2010-07.02.2010, Linz] Institutional research plan: CEZ:AV0Z10300504 Keywords : mathematical fuzzy logic * axiomatic arithmetic Subject RIV: BA - General Mathematics
Goldbach Conjecture and First-Order Arithmetic
Revilla, Fernando
2007-01-01
Using the concepts of Hyperbolic Classification of Natural Numbers, Essential Regions and Goldbach Conjecture Function we prove that the existence of a proof of the Goldbach Conjecture in First-Order Arithmetic would imply the existence of another proof in a certain extension that would not be valid in all states of time associated to natural numbers created by means of adequate dynamic processes.
Circuit lower bounds in bounded arithmetics
Pich, Ján
2015-01-01
Roč. 166, č. 1 (2015), s. 29-45. ISSN 0168-0072 R&D Projects: GA AV ČR IAA100190902 Keywords : bounded arithmetic * circuit lower bounds Subject RIV: BA - General Mathematics Impact factor: 0.548, year: 2014 http://www.sciencedirect.com/science/article/pii/S0168007214000888
Intuitionistic fixed point theories over Heyting arithmetic
Arai, Toshiyasu
2010-01-01
In this paper we show that an intuitionistic theory for fixed points is conservative over the Heyting arithmetic with respect to a certain class of formulas. This extends partly the result of mine. The proof is inspired by the quick cut-elimination due to G. Mints.
Backgrounds of arithmetic and geometry an introduction
Miron, Radu
1995-01-01
The book is an introduction to the foundations of Mathematics. The use of the constructive method in Arithmetic and the axiomatic method in Geometry gives a unitary understanding of the backgrounds of geometry, of its development and of its organic link with the study of real numbers and algebraic structures.
Approximate counting by hashing in bounded arithmetic
Jeřábek, Emil
2009-01-01
Roč. 74, č. 3 (2009), s. 829-860. ISSN 0022-4812 R&D Projects: GA AV ČR IAA1019401 Institutional research plan: CEZ:AV0Z10190503 Keywords : bounded arithmetic * approximate counting * universal hashing Subject RIV: BA - General Mathematics Impact factor: 0.631, year: 2009
Improved closeness centrality using arithmetic mean approach
Ruslan, Nuraimi; Sharif, Shamshuritawati
2015-12-01
In this paper, we improved the mathematical formulation of closeness centrality measure for weighted network. The proposed measure is used arithmetic mean approach and the performance is successfully better than the existing closeness centrality. This measure can be used as a measure of influential nodes.
Modular arithmetic weight and cyclic shifting.
Hartman, W. F.
1972-01-01
This note shows that the modular arithmetic weight of an integer is invariant to the cyclic shifts of its radix-2 form. This result leads to a reduced search for the minimum weight codeword in a cyclic AN-code as well as to a better understanding of previous work.
Non-commutative arithmetic circuits with division
Hrubeš, Pavel; Wigderson, A.
2015-01-01
Roč. 11, Article 14 (2015), s. 357-393. ISSN 1557-2862 EU Projects: European Commission(XE) 339691 - FEALORA Institutional support: RVO:67985840 Keywords : arithmetic circuits * non-commutative rational function * skew field Subject RIV: BA - General Mathematics http://theoryofcomputing.org/articles/v011a014/
Fuzzy Logic and Arithmetical Hierarchy III
Hájek, Petr
2001-01-01
Roč. 68, č. 1 (2001), s. 129-142. ISSN 0039-3215 R&D Projects: GA AV ČR IAA1030004 Institutional research plan: AV0Z1030915 Keywords : fuzzy logic * basic fuzzy logic * Lukasiewicz logic * Godel logic * product logic * arithmetical hierarchy Subject RIV: BA - General Mathematics
Non-commutative arithmetic circuits with division
Hrubeš, Pavel; Wigderson, A.
2015-01-01
Roč. 11, Article 14 (2015), s. 357-393. ISSN 1557-2862 EU Projects: European Commission(XE) 339691 - FEALORA Institutional support: RVO:67985840 Keywords : arithmetic circuits * non-commutative rational function * skew field Subject RIV: BA - General Mathematics http://theoryofcomputing.org/ articles /v011a014/
Fuzzy Logic and Arithmetical Hierarchy IV
Hájek, Petr
Berlin : Logos Verlag, 2004 - ( Hendricks , V.; Neuhaus, F.; Pedersen, S.; Scheffler, U.; Wansing, H.), s. 107-115 ISBN 3-8325-0475-3 R&D Projects: GA AV ČR IAA1030004 Institutional research plan: CEZ:AV0Z1030915 Keywords : fuzzy logic * arithmetical hierarchy Subject RIV: BA - General Mathematics
Arithmetic and Cognitive Contributions to Algebra
Cirino, Paul T.; Tolar, Tammy D.; Fuchs, Lynn S.
2013-01-01
Algebra is a prerequisite for access to STEM careers and occupational success (NMAP, 2008a), yet algebra is difficult for students through high school (US DOE, 2008). Growth in children's conceptual and procedural arithmetical knowledge is reciprocal, although conceptual knowledge has more impact on procedural knowledge than the reverse…
Retrieval-Induced Forgetting of Arithmetic Facts
Campbell, Jamie I. D.; Thompson, Valerie A.
2012-01-01
Retrieval-induced forgetting (RIF) is a widely studied phenomenon of human memory, but RIF of arithmetic facts remains relatively unexplored. In 2 experiments, we investigated RIF of simple addition facts (2 + 3 = 5) from practice of their multiplication counterparts (2 x 3 = 6). In both experiments, robust RIF expressed in response times occurred…
Kühn, Ulf; Müller, Jan Steffen
2012-01-01
We give an explicitly computable lower bound for the arithmetic self-intersection number of the dualizing sheaf on a large class of arithmetic surfaces. If some technical conditions are satisfied, then this lower bound is positive. In particular, these technical conditions are always satisfied for minimal arithmetic surfaces with simple multiplicities and at least one reducible fiber, but we have also used our techniques to obtain lower bounds for some arithmetic surfaces with non-reduced fib...
Language-specific memory for everyday arithmetic facts in Chinese-English bilinguals.
Chen, Yalin; Yanke, Jill; Campbell, Jamie I D
2016-04-01
The role of language in memory for arithmetic facts remains controversial. Here, we examined transfer of memory training for evidence that bilinguals may acquire language-specific memory stores for everyday arithmetic facts. Chinese-English bilingual adults (n = 32) were trained on different subsets of simple addition and multiplication problems. Each operation was trained in one language or the other. The subsequent test phase included all problems with addition and multiplication alternating across trials in two blocks, one in each language. Averaging over training language, the response time (RT) gains for trained problems relative to untrained problems were greater in the trained language than in the untrained language. Subsequent analysis showed that English training produced larger RT gains for trained problems relative to untrained problems in English at test relative to the untrained Chinese language. In contrast, there was no evidence with Chinese training that problem-specific RT gains differed between Chinese and the untrained English language. We propose that training in Chinese promoted a translation strategy for English arithmetic (particularly multiplication) that produced strong cross-language generalization of practice, whereas training in English strengthened relatively weak, English-language arithmetic memories and produced little generalization to Chinese (i.e., English training did not induce an English translation strategy for Chinese language trials). The results support the existence of language-specific strengthening of memory for everyday arithmetic facts. PMID:26265429
Ray system in lasers, nonlinear arithmetic pyramid and nonlinear arithmetic triangles
Yurkin, Alexander
2013-01-01
The paper describes a system of rays declining at small angles in lasers. The correlation between a group of rays and binomial coefficients is shown. The correlation of distribution of rays in the system of numbers placed in a three-dimensional table, the nonlinear arithmetic pyramid is shown. Two types of nonlinear arithmetic triangles are considered. Various types of partitions of integers is described.
Learning, Realizability and Games in Classical Arithmetic
Aschieri, Federico
2010-12-01
In this dissertation we provide mathematical evidence that the concept of learning can be used to give a new and intuitive computational semantics of classical proofs in various fragments of Predicative Arithmetic. First, we extend Kreisel modified realizability to a classical fragment of first order Arithmetic, Heyting Arithmetic plus EM1 (Excluded middle axiom restricted to Sigma^0_1 formulas). We introduce a new realizability semantics we call "Interactive Learning-Based Realizability". Our realizers are self-correcting programs, which learn from their errors and evolve through time. Secondly, we extend the class of learning based realizers to a classical version PCFclass of PCF and, then, compare the resulting notion of realizability with Coquand game semantics and prove a full soundness and completeness result. In particular, we show there is a one-to-one correspondence between realizers and recursive winning strategies in the 1-Backtracking version of Tarski games. Third, we provide a complete and fully detailed constructive analysis of learning as it arises in learning based realizability for HA+EM1, Avigad's update procedures and epsilon substitution method for Peano Arithmetic PA. We present new constructive techniques to bound the length of learning processes and we apply them to reprove - by means of our theory - the classic result of Godel that provably total functions of PA can be represented in Godel's system T. Last, we give an axiomatization of the kind of learning that is needed to computationally interpret Predicative classical second order Arithmetic. Our work is an extension of Avigad's and generalizes the concept of update procedure to the transfinite case. Transfinite update procedures have to learn values of transfinite sequences of non computable functions in order to extract witnesses from classical proofs.
The structure and software of the arithmetical module for the multi-microprocessor intelligent graphics terminal designed for realization of the world coordinate two-dimensional transformation are described. The module performs the operations like coordinate system displacement, scaling and rotation as well as transformations for window/viewport separation
Rodic, Maja; Zhou, Xinlin; Tikhomirova, Tatiana; Wei, Wei; Malykh, Sergei; Ismatulina, Victoria; Sabirova, Elena; Davidova, Yulia; Tosto, Maria Grazia; Lemelin, Jean-Pascal; Kovas, Yulia
2015-01-01
The present study evaluated 626 5-7-year-old children in the UK, China, Russia, and Kyrgyzstan on a cognitive test battery measuring: (1) general skills; (2) non-symbolic number sense; (3) symbolic number understanding; (4) simple arithmetic--operating with numbers; and (5) familiarity with numbers. Although most inter-population differences were…
Recursive double-size fixed precision arithmetic
Chabot, Christophe; Fousse, Laurent; Giorgi, Pascal
2011-01-01
This work is a part of the SHIVA (Secured Hardware Immune Versatile Architecture) project whose purpose is to provide a programmable and reconfigurable hardware module with high level of security. We propose a recursive double-size fixed precision arithmetic called RecInt. Our work can be split in two parts. First we developped a C++ software library with performances comparable to GMP ones. Secondly our simple representation of the integers allows an implementation on FPGA. Our idea is to consider sizes that are a power of 2 and to apply doubling techniques to implement them efficiently: we design a recursive data structure where integers of size 2^k, for k>k0 can be stored as two integers of size 2^{k-1}. Obviously for k<=k0 we use machine arithmetic instead (k0 depending on the architecture).
Dictionary of algebra, arithmetic, and trigonometry
Krantz, Steven G
2000-01-01
Clear, rigorous definitions of mathematical terms are crucial to good scientific and technical writing-and to understanding the writings of others. Scientists, engineers, mathematicians, economists, technical writers, computer programmers, along with teachers, professors, and students, all have the need for comprehensible, working definitions of mathematical expressions. To meet that need, CRC Press proudly introduces its Dictionary of Algebra, Arithmetic, and Trigonometry- the second published volume in the CRC Comprehensive Dictionary of Mathematics. More than three years in development, top academics and professionals from prestigious institutions around the world bring you more than 2,800 detailed definitions, written in a clear, readable style, complete with alternative meanings, and related references.From Abelian cohomology to zero ring and from the very basic to the highly advanced, this unique lexicon includes terms associated with arithmetic, algebra, and trigonometry, with natural overlap into geom...
Arithmetic Operand Ordering for Equivalence Checking
WENG Yanling; GE Haitong; YAN Xiaolang; Ren Kun
2007-01-01
An information extraction-based technique is proposed for RTL-to-gate equivalence checking. Distances are calculated on directed acyclic graph (AIG). Multiplier and multiplicand are distinguished on multiplications with different coding methods, with which the operand ordering/grouping information could be extracted from a given implementation gate netlist, helping the RTL synthesis engine generate a gate netlist with great similarity. This technique has been implemented in an internal equivalence checking tool, ZD_VIS. Compared with the simple equivalence checking, the speed is accelerated by at least 40% in its application to a class of arithmetic designs, addition and multiplication trees. The method can be easily incorporated into existing RTL-to-gate equivalence checking frameworks, increasing the robustness of equivalence checking for arithmetic circuits.
Floating-Point Arithmetic on Round-to-Nearest Representations
Kornerup, Peter; Panhaleux, Adrien
2012-01-01
Recently we introduced a class of number representations denoted RN-representations, allowing an un-biased rounding-to-nearest to take place by a simple truncation. In this paper we briefly review the binary fixed-point representation in an encoding which is essentially an ordinary 2's complement representation with an appended round-bit. Not only is this rounding a constant time operation, so is also sign inversion, both of which are at best log-time operations on ordinary 2's complement representations. Addition, multiplication and division is defined in such a way that rounding information can be carried along in a meaningful way, at minimal cost. Based on the fixed-point encoding we here define a floating point representation, and describe to some detail a possible implementation of a floating point arithmetic unit employing this representation, including also the directed roundings.
HIGH SPEED POINT ARITHMETIC ARCHITECTURE FOR ECC ON FPGA
Rahila Bilal,
2010-09-01
Full Text Available Elliptic curve cryptography plays a crucial role in networking and communication security. ECC have evolved in the recent past as an important alternative to established systems like RSA. This paper describes the implementation of an elliptic curve coprocessor based on the FPGA , which can provide a significant speedup for these cryptosystems. The FPGA configuration file is synthesized from VHDL code applying different hardware synthesis products. The implementation of ECC lies in three levels: scalar multiplication, point addition/doubling and finite field modular arithmetic. In this paper, we present a novel fast architecture for the point addition/doubling level in the projective coordinate. The proposed Architecture is based on Binary Field. The Design performs multiplication using Polynomial Basis. Analysis shows that, with reasonable hardware overhead, our architecture can achieve a high speedup for the point addition operation and point Doubling operation.Furthermore, the architecture is parameterized for different data widths to evaluate the optimal resource utilization.
Recursive double-size fixed precision arithmetic
Chabot, Christophe; Dumas, Jean-Guillaume; Fousse, Laurent; Giorgi, Pascal
2011-01-01
This work is a part of the SHIVA (Secured Hardware Immune Versatile Architecture) project whose purpose is to provide a programmable and reconfigurable hardware module with high level of security. We propose a recursive double-size fixed precision arithmetic called RecInt. Our work can be split in two parts. First we developped a C++ software library with performances comparable to GMP ones. Secondly our simple representation of the integers allows an implementation on FPGA. Our idea is to co...
Arithmetic Properties of the Ramanujan Function
Florian Luca; Igor E Shparlinski
2006-02-01
We study some arithmetic properties of the Ramanujan function (), such as the largest prime divisor ( ()) and the number of distinct prime divisors (()) of () for various sequences of . In particular, we show that ( ()) ≥ $(\\log n)^{33/31+(1)}$ for infinitely many , and $$P((p)(p^2)(p^3))>(1+(1))\\frac{\\log\\log p\\log\\log\\log p}{\\log\\log\\log\\log p}$$ for every prime with $(p)≠ 0$.
Set Theory and Arithmetic in Fuzzy Logic
Běhounek, Libor; Haniková, Zuzana
Cham : Springer, 2015 - (Montagna, F.), s. 63-89 ISBN 978-3-319-06232-7. - (Outstanding Contributions to Logic. 6) R&D Projects: GA ČR GPP103/10/P234; GA ČR GAP202/10/1826 Institutional support: RVO:67985807 Keywords : fuzzy set theory * fuzzy logic * naive comprehension * non-classical arithmetic Subject RIV: BA - General Mathematics
Randomness, pseudorandomness and models of arithmetic
Pudlák, P.
2013-01-01
Pseudorandmness plays an important role in number theory, complexity theory and cryptography. Our aim is to use models of arithmetic to explain pseudorandomness by randomness. To this end we construct a set of models $\\cal M$, a common element $\\iota$ of these models and a probability distribution on $\\cal M$, such that for every pseudorandom sequence $s$, the probability that $s(\\iota)=1$ holds true in a random model from $\\cal M$ is equal to 1/2.
Real closures of models of weak arithmetic
Jeřábek, Emil; Kolodziejczyk, L.. A.
2013-01-01
Roč. 52, 1-2 (2013), s. 143-157. ISSN 0933-5846 R&D Projects: GA AV ČR IAA100190902; GA MŠk(CZ) 1M0545 Institutional support: RVO:67985840 Keywords : bounded arithmetic * real-closed field * recursive saturation Subject RIV: BA - General Mathematics http://link.springer.com/article/10.1007%2Fs00153-012-0311-x
A sorting network in bounded arithmetic
Jeřábek, Emil
2011-01-01
Roč. 162, č. 4 (2011), s. 341-355. ISSN 0168-0072 R&D Projects: GA AV ČR IAA1019401; GA MŠk(CZ) 1M0545 Institutional research plan: CEZ:AV0Z10190503 Keywords : bounded arithmetic * sorting network * proof complexity * monotone sequent calculus Subject RIV: BA - General Mathematics Impact factor: 0.450, year: 2011 http://www.sciencedirect.com/science/article/pii/S0168007210001272
A common Misconception about the Categorical Arithmetic
Raguní, Giuseppe
2016-01-01
Although the categorical arithmetic is not effectively axiomatizable, the belief that the incompleteness Theorems can be apply to it is fairly common. Furthermore, the so-called "essential" (or "inherent") semantic incompleteness of the second-order Logic that can be deduced by these same Theorems does not imply the standard semantic incompleteness that can be derived using the Loewenheim-Skolem or the compactness Theorem. This state of affairs has its origins in an incorrect and misinterpret...
Set Theory and Arithmetic in Fuzzy Logic
Běhounek, Libor; Haniková, Zuzana
Cham: Springer, 2015 - (Montagna, F.), s. 63-89. (Outstanding Contributions to Logic. 6). ISBN 978-3-319-06232-7 R&D Projects: GA ČR GPP103/10/P234; GA ČR GAP202/10/1826 Institutional support: RVO:67985807 Keywords : fuzzy set theory * fuzzy logic * naive comprehension * non-classical arithmetic Subject RIV: BA - General Mathematics
Residues : The gateway to higher arithmetic I
Siebeneicher, Christian
2012-01-01
Residues to a given modulus have been introduced to mathematics by Carl Friedrich Gauss with the definition of congruence in the `Disquisitiones Arithmeticae'. Their extraordinary properties provide the basis for a change of paradigm in arithmetic. By restricting residues to remainders left over by divison Peter Gustav Lejeune Dirichlet - Gauss's successor in G\\"ottingen - eliminated in his `Lectures on number theory' the fertile concept of residues and attributed with the number-theoretic ap...
Duverne, Sandrine; Lemaire, Patrick; Michel, Bernard François
2003-08-01
Three groups of healthy younger adults, healthy older adults, and probable AD patients, performed an addition/number comparison task. They compared 128 couples of additions and numbers (e.g., 4 + 9 15) and had to identify the largest item for each problem by pressing one of two buttons located under each item. Manipulations of problem characteristics (i.e., problem difficulty and splits between correct sums and proposed numbers) enabled us to examine strategy selection and specific arithmetic fact retrieval processes. Results showed that arithmetic facts retrieval processes, which were spared with aging, were impaired in AD patients. However, AD patients were able to switch between strategies across trials according to problem characteristics as well as healthy older adults, and less systematically than healthy younger adults. We discuss implications of these findings for further understanding AD-related differences in arithmetic in particular, and problem solving in general. PMID:12907175
Are individual differences in arithmetic fact retrieval related to inhibition?
Bellon, Elien
2016-01-01
Although it has been proposed that inhibition is related to individual differences in mathematical achievement, it is not clear how it is related to specific aspects of mathematical skills, such as arithmetic fact retrieval. The present study therefore investigated the association between inhibition and arithmetic fact retrieval and further examined the unique role of inhibition in individual differences in arithmetic fact retrieval, in addition to numerical magnitude processin...
Sets of integers that do not contain long arithmetic progressions
O'Bryant, Kevin
2008-01-01
In 1946, Behrend gave a construction of dense finite sets of integers that do not contain 3-term arithmetic progressions. In 1961, Rankin generalized Behrend's construction to sets avoiding k-term arithmetic progressions, and in 2008 Elkin refined Behrend's 3-term construction. In this work, we combine Elkin's refinement and Rankin's generalization. Arithmetic progressions are handled as a special case of polynomial progressions. In 1946, Behrend gave a construction of dense finite sets of in...
Implicit Learning of Arithmetic Regularities Is Facilitated by Proximal Contrast
Prather, Richard W.
2012-01-01
Natural number arithmetic is a simple, powerful and important symbolic system. Despite intense focus on learning in cognitive development and educational research many adults have weak knowledge of the system. In current study participants learn arithmetic principles via an implicit learning paradigm. Participants learn not by solving arithmetic equations, but through viewing and evaluating example equations, similar to the implicit learning of artificial grammars. We expand this to the symbo...
Arithmetic on a Distributed-Memory Quantum Multicomputer
Van Meter, R; Nemoto, K; Itoh, K M; Meter, Rodney Van; Nemoto, Kae; Itoh, Kohei M.
2006-01-01
We evaluate the performance of quantum arithmetic algorithms run on a distributed quantum computer (a quantum multicomputer). We vary the node capacity and I/O capabilities, and the network topology. The tradeoff of choosing between gates executed remotely, through ``teleported gates'' on entangled pairs of qubits (telegate), versus exchanging the relevant qubits via quantum teleportation, then executing the algorithm using local gates (teledata), is examined. We show that the teledata approach performs better, and that carry-ripple adders perform well when the teleportation block is decomposed so that the key quantum operations can be parallelized. A node size of only a few logical qubits performs adequately provided that the nodes have two transceiver qubits. A linear network topology performs acceptably for a broad range of system sizes and performance parameters. We therefore recommend pursuing small, high-I/O bandwidth nodes and a simple network. Such a machine will run Shor's algorithm for factoring lar...
Iglesias-Sarmiento, Valentín; Deaño, Manuel
2016-01-01
This study analyzed the cognitive functioning underlying arithmetical difficulties and explored the predictors of arithmetic achievement in the last three grades of Spanish Primary Education. For this purpose, a group of 165 students was selected and divided into three groups of arithmetic competence: Mathematical Learning Disability group (MLD, n = 27), Low Achieving group (LA, n = 39), and Typical Achieving group (TA, n = 99). Students were assessed in domain-general abilities (working memory and PASS cognitive processes), and numerical competence (counting and number processing) during the last two months of the academic year. Performance of children from the MLD group was significantly poorer than that of the LA group in writing dictated Arabic numbers (d = -0.88), reading written verbal numbers (d = -0.84), transcoding written verbal numbers to Arabic numbers (-0.75) and comprehension of place value (d = -0.69), as well as in simultaneous (d = -0.62) and successive (d = -0.59) coding. In addition, a specific developmental sequence was observed in both groups, the implications of which are discussed. Hierarchical regression analysis revealed simultaneous coding (β = .47, t(155) = 6.18, p processing (β = .23, t(155) = 3.07, p < .01) as specific predictors of arithmetical performance. PMID:27320030
A Geometric Characterization of Arithmetic Varieties
Kapil Hari Paranjape
2002-08-01
A result of Belyi can be stated as follows. Every curve defined over a number field can be expressed as a cover of the projective line with branch locus contained in a rigid divisor. We define the notion of geometrically rigid divisors in surfaces and then show that every surface defined over a number field can be expressed as a cover of the projective plane with branch locus contained in a geometrically rigid divisor in the plane. The main result is the characterization of arithmetically defined divisors in the plane as geometrically rigid divisors in the plane.
Aztec arithmetic: positional notation and area calculation.
Harvey, H R; Williams, B J
1980-10-31
Texcocan-Aztec peoples in the Valley of Mexico used both picture symbols and lines and dots for numerical notation. Decipherment and analysis of mid-16th-century native pictorial land documents from the Texcocan region indicate that the line-and-dot system incorporated a symbol for zero and used position to ascribe values. Positional line-and-dot notation was used to record areas of agricultural fields, and analysis of the documentary data suggests that areas were calculated arithmetically. These findings demonstrate that neither positional notation nor the zero were unique to the Maya area, and they imply an equally sophisticated mathematical development among the Aztecs. PMID:17841389
Arithmetic, mutually unbiased bases and complementary observables
Complementary observables in quantum mechanics may be viewed as Frobenius structures in a dagger monoidal category, such as the category of finite dimensional Hilbert spaces over the complex numbers. On the other hand, their properties crucially depend on the discrete Fourier transform and its associated quantum torus, requiring only the finite fields that underlie mutually unbiased bases. In axiomatic topos theory, the complex numbers are difficult to describe and should not be invoked unnecessarily. This paper surveys some fundamentals of quantum arithmetic using finite field complementary observables, with a view considering more general axiom systems.
Functional verification of floating point arithmetic unit
For continuous real-time reactivity monitoring of PFBR reactivity safety channel, a FPGA based reactivity meter has been developed by Electronics Division, BARC. Verification of designs involved in Safety Critical systems is very important and necessary. Functional verification of this design is presently carried out by EID, IGCAR. In Reactivity meter, Floating Point Arithmetic Unit (FPAU) is a major and very important sub module, which needs to be completely verified first. Two types of verifications are possible: Functional verification and Formal verification. This paper discusses and shares the experiences of functional verification of FPAU module for all special floating point numbers. (author)
Arithmetic fundamental groups and moduli of curves
This is a short note on the algebraic (or sometimes called arithmetic) fundamental groups of an algebraic variety, which connects classical fundamental groups with Galois groups of fields. A large part of this note describes the algebraic fundamental groups in a concrete manner. This note gives only a sketch of the fundamental groups of the algebraic stack of moduli of curves. Some application to a purely topological statement, i.e., an obstruction to the subjectivity of Johnson homomorphisms in the mapping class groups, which comes from Galois group of Q, is explained. (author)
ARITHMETIC PROGRESSIONS FOR COUNTING PRIME NUMBERS
V.J.DEVASIA
2014-11-01
Full Text Available In this paper two arithmetic progressions are proposed for listing and counting the prime numbers less than or equal to a given integer. From these progressions, how one can filter out prime numbers is the topic of discussion in this paper. An easy to implement formula is presented to compute the number of primes by eliminating the number of composite numbers in an iterative and recursive manner. Numerical examples are presented to demonstrate how the procedure works in an efficient and simple way.
Stock, Pieter; Desoete, Annemie; Roeyers, Herbert
2010-01-01
In a 3-year longitudinal study, 471 children were classified, based on their performances on arithmetic tests in first and second grade, as having persistent arithmetic disabilities (AD), persistent low achieving (LA), persistent typical achieving, inconsistent arithmetic disabilities (DF1), or inconsistent low achieving in arithmetic. Significant…
Outer Billiards, Arithmetic Graphs, and the Octagon
Schwartz, Richard Evan
2010-01-01
Outer Billiards is a geometrically inspired dynamical system based on a convex shape in the plane. When the shape is a polygon, the system has a combinatorial flavor. In the polygonal case, there is a natural acceleration of the map, a first return map to a certain strip in the plane. The arithmetic graph is a geometric encoding of the symbolic dynamics of this first return map. In the case of the regular octagon, the case we study, the arithmetic graphs associated to periodic orbits are polygonal paths in R^8. We are interested in the asymptotic shapes of these polygonal paths, as the period tends to infinity. We show that the rescaled limit of essentially any sequence of these graphs converges to a fractal curve that simultaneously projects one way onto a variant of the Koch snowflake and another way onto a variant of the Sierpinski carpet. In a sense, this gives a complete description of the asymptotic behavior of the symbolic dynamics of the first return map. What makes all our proofs work is an efficient...
Arithmetic intersection on a Hilbert modular surface and the Faltings height
Yang, Tonghai
2013-01-01
In this paper, we prove an explicit arithmetic intersection formula between arithmetic Hirzebruch-Zagier divisors and arithmetic CM cycles in a Hilbert modular surface over $\\mathbb Z$. As applications, we obtain the first `non-abelian' Chowla-Selberg formula, which is a special case of Colmez's conjecture; an explicit arithmetic intersection formula between arithmetic Humbert surfaces and CM cycles in the arithmetic Siegel modular variety of genus two; Lauter's conjecture about the denominat...
Guest Editors' Introduction: Special Section on Computer Arithmetic
Nannarelli, Alberto; Seidel, Peter-Michael; Tang, Ping Tak Peter
2014-01-01
The articles in this special issue focus on current trends and developments in the field of computer arithmetic. This is a field that encompasses the definition and standardization of arithmetic system for computers. The field also deals with issues of hardware and software implementations and...
Higher Arithmetic Sequence and Its Implicit Common Difference
Wu Qu
2015-12-01
Full Text Available The concept of k-order sequence of first order arithmetic sequence has been defined by mathematical induction based on finite difference theory. It has been proved this sequence is higher arithmetic sequence. Meanwhile the sum formula and the derivation of its implicit common difference have been given.
Higher Arithmetic Sequence and Its Implicit Common Difference
Wu Qu
2015-01-01
The concept of k-order sequence of first order arithmetic sequence has been defined by mathematical induction based on finite difference theory. It has been proved this sequence is higher arithmetic sequence. Meanwhile the sum formula and the derivation of its implicit common difference have been given.
The behaviour of eigenstates of arithmetic hyperbolic manifolds
In this paper we study some problems arising from the theory of Quantum Chaos, in the context of arithmetic hyperbolic manifolds. We show that there is no strong localization (''scarring'') onto totally geodesic submanifolds. Arithmetic examples are given, which show that the random wave model for eigenstates does not apply universally in 3 degrees of freedom. (orig.)
24 CFR Appendix E to Part 3500 - Arithmetic Steps
2010-04-01
... 24 Housing and Urban Development 5 2010-04-01 2010-04-01 false Arithmetic Steps E Appendix E to Part 3500 Housing and Urban Development Regulations Relating to Housing and Urban Development...—Arithmetic Steps I. Example Illustrating Aggregate Analysis: ASSUMPTIONS: Disbursements: $360 for...
Children's Acquisition of Arithmetic Principles: The Role of Experience
Prather, Richard; Alibali, Martha W.
2011-01-01
The current study investigated how young learners' experiences with arithmetic equations can lead to learning of an arithmetic principle. The focus was elementary school children's acquisition of the Relation to Operands principle for subtraction (i.e., for natural numbers, the difference must be less than the minuend). In Experiment 1, children…
The Arithmetic Tie Effect Is Mainly Encoding-based.
Blankenberger, Sven
2001-01-01
Examined two possible explanations for the arithmetic tie effect: faster encoding of tie problems versus faster access to arithmetic facts. Found that the tie effect vanished with heterogeneous addition problems, and for seven out of eight participants, the effect vanished with heterogeneous multiplication problems. Concludes that the tie effect…
A novel chaotic encryption scheme based on arithmetic coding
In this paper, under the combination of arithmetic coding and logistic map, a novel chaotic encryption scheme is presented. The plaintexts are encrypted and compressed by using an arithmetic coder whose mapping intervals are changed irregularly according to a keystream derived from chaotic map and plaintext. Performance and security of the scheme are also studied experimentally and theoretically in detail
PaCAL: A Python Package for Arithmetic Computations with Random Variables
Marcin Korze?
2014-05-01
Full Text Available In this paper we present PaCAL, a Python package for arithmetical computations on random variables. The package is capable of performing the four arithmetic operations: addition, subtraction, multiplication and division, as well as computing many standard functions of random variables. Summary statistics, random number generation, plots, and histograms of the resulting distributions can easily be obtained and distribution parameter ?tting is also available. The operations are performed numerically and their results interpolated allowing for arbitrary arithmetic operations on random variables following practically any probability distribution encountered in practice. The package is easy to use, as operations on random variables are performed just as they are on standard Python variables. Independence of random variables is, by default, assumed on each step but some computations on dependent random variables are also possible. We demonstrate on several examples that the results are very accurate, often close to machine precision. Practical applications include statistics, physical measurements or estimation of error distributions in scienti?c computations.
Computer Arithmetic Algorithms for Mega-Digit Floating Point Numbers' Precision
Musbah J. Aqel; Mohammed H. Saleh
2007-01-01
IEEE standard 754 floating point is the most common representation used for floating point numbers, and many computer arithmetic algorithms are developed for basic operations on this standard. In this study, new computer algorithms are proposed to increase the precision range and to solve some problems that are available while using these algorithms. However, these algorithms provide an optional range of required accuracy (Mega-Digit precision) to meet new computer's applications.
Is integer arithmetic fundamental to mental processing?: the mind's secret arithmetic.
Snyder, A.W.; Mitchell, D.J.
1999-01-01
Unlike the ability to acquire our native language, we struggle to learn multiplication and division. It may then come as a surprise that the mental machinery for performing lightning-fast integer arithmetic calculations could be within us all even though it cannot be readily accessed, nor do we have any idea of its primary function. We are led to this provocative hypothesis by analysing the extraordinary skills of autistic savants. In our view such individuals have privileged access to lower ...
Lucimar M.F. de Carvalho
2008-06-01
Full Text Available OBJECTIVE: To investigate different fuzzy arithmetical operations to support in the diagnostic of epileptic events and non epileptic events. METHOD: A neuro-fuzzy system was developed using the NEFCLASS (NEuro Fuzzy CLASSIfication architecture and an artificial neural network with backpropagation learning algorithm (ANNB. RESULTS: The study was composed by 244 patients with a bigger frequency of the feminine sex. The number of right decisions at the test phase, obtained by the NEFCLASS and ANNB was 83.60% and 90.16%, respectively. The best sensibility result was attained by NEFCLASS (84.90%; the best specificity result were attained by ANNB with 95.65%. CONCLUSION: The proposed neuro-fuzzy system combined the artificial neural network capabilities in the pattern classifications together with the fuzzy logic qualitative approach, leading to a bigger rate of system success.OBJETIVO: Investigar diferentes operações aritméticas difusas para auxíliar no diagnóstico de eventos epilépticos e eventos não-epilépticos. MÉTODO: Um sistema neuro-difuso foi desenvolvido utilizando a arquitetura NEFCLASS (NEuro Fuzzy CLASSIfication e uma rede neural artificial com o algoritmo de aprendizagem backpropagation (RNAB. RESULTADOS: A amostra estudada foi de 244 pacientes com maior freqüência no sexo feminino. O número de decisões corretas na fase de teste, obtidas através do NEFCLASS e RNAB foi de 83,60% e 90,16%, respectivamente. O melhor resultado de sensibilidade foi obtido com o NEFCLASS (84,90%; o melhor resultado de especificidade foi obtido com a RNAB (95,65%. CONCLUSÃO: O sistema neuro-difuso proposto combinou a capacidade das redes neurais artificiais na classificação de padrões juntamente com a abordagem qualitativa da logica difusa, levando a maior taxa de acertos do sistema.
The arithmetic basis of special relativity
Under relatively general particle and rocket frame motions, it is shown that, for special relativity, the basic concepts can be formulated and the basic properties deduced using only arithmetic. Particular attention is directed toward velocity, acceleration, proper time, momentum, energy, and 4-vectors in both space-time and Minkowski space, and to relativistic generalizations of Newton's second law. The resulting mathematical simplification is not only completely compatible with modern computer technology, but it yields dynamical equations that can be solved directly by such computers. Particular applications of the numerical equations, which are either Lorentz invariant or are directly related to Lorentz-invariant formulas, are made to the study of a relativistic harmonic oscillator and to the motion of an electric particle in a magnetic field. (author)
Pinel, Philippe; Dehaene, Stanislas
2010-01-01
Language and arithmetic are both lateralized to the left hemisphere in the majority of right-handed adults. Yet, does this similar lateralization reflect a single overall constraint of brain organization, such an overall "dominance" of the left hemisphere for all linguistic and symbolic operations? Is it related to the lateralization of specific…
On Arithmetic Densities of Sets of Generalized Integers
Porubský, Štefan
Book 4, Volume 1. Kyiv : Institute of Mathematics, NAS of Ukraine, 2008, s. 132-136. ISBN 978-966-02-4891-5. [International Conference on Analytic Number Theory and Spatial Tessellation /4./. Kyiv (UA), 22.09.2008-28.09.2008] R&D Projects: GA ČR GA201/07/0191 Institutional research plan: CEZ:AV0Z10300504 Keywords : asymptotic density * logarithmic density * weighted means * arithmetical semigroup * arithmetic function * generalized arithmetic density * topological density Subject RIV: BA - General Mathematics
Frege, Dedekind, and Peano on the foundations of arithmetic
Gillies, Donald
2013-01-01
First published in 1982, this reissue contains a critical exposition of the views of Frege, Dedekind and Peano on the foundations of arithmetic. The last quarter of the 19th century witnessed a remarkable growth of interest in the foundations of arithmetic. This work analyses both the reasons for this growth of interest within both mathematics and philosophy and the ways in which this study of the foundations of arithmetic led to new insights in philosophy and striking advances in logic. This historical-critical study provides an excellent introduction to the problems of the philosop