In physics,our expectations for system behavior are often guided by intuitive arithmetic.For systems composed of identical units,we anticipate synergy of the contributions from these units,where 1+1=2.Conversely,for s...In physics,our expectations for system behavior are often guided by intuitive arithmetic.For systems composed of identical units,we anticipate synergy of the contributions from these units,where 1+1=2.Conversely,for systems built from opposing units,we expect cancellation of their contributions,where 1-1=0.This intuitive arithmetic has long underpinned our understanding of physical properties of materials,from electronic transport to optical responses.However,scientific breakthroughs often occur when nature reveals ways to circumvent these seemingly fundamental rules,opening new possibilities that challenge our deepest assumptions about material behavior.展开更多
In this article, we report the derivation of high accuracy finite difference method based on arithmetic average discretization for the solution of Un=F(x,u,u′)+∫K(x,s)ds , 0 x s < 1 subject to natural boundary co...In this article, we report the derivation of high accuracy finite difference method based on arithmetic average discretization for the solution of Un=F(x,u,u′)+∫K(x,s)ds , 0 x s < 1 subject to natural boundary conditions on a non-uniform mesh. The proposed variable mesh approximation is directly applicable to the integro-differential equation with singular coefficients. We need not require any special discretization to obtain the solution near the singular point. The convergence analysis of a difference scheme for the diffusion convection equation is briefly discussed. The presented variable mesh strategy is applicable when the internal grid points of the solution space are both even and odd in number as compared to the method discussed by authors in their previous work in which the internal grid points are strictly odd in number. The advantage of using this new variable mesh strategy is highlighted computationally.展开更多
A new arithmetic coding system combining source channel coding and maximum a posteriori decoding were proposed. It combines source coding and error correction tasks into one unified process by introducing an adaptive ...A new arithmetic coding system combining source channel coding and maximum a posteriori decoding were proposed. It combines source coding and error correction tasks into one unified process by introducing an adaptive forbidden symbol. The proposed system achieves fixed length code words by adaptively adjusting the probability of the forbidden symbol and adding tail digits of variable length. The corresponding improved MAP decoding metric was derived. The proposed system can improve the performance. Simulations were performed on AWGN channels with various noise levels by using both hard and soft decision with BPSK modulation.The results show its performance is slightly better than that of our adaptive arithmetic error correcting coding system using a forbidden symbol.展开更多
Efficient solvers for optimization problems are based on linear and semidefinite relaxations that use floating point arithmetic. However, due to the rounding errors, relaxation thus may overestimate, or worst, underes...Efficient solvers for optimization problems are based on linear and semidefinite relaxations that use floating point arithmetic. However, due to the rounding errors, relaxation thus may overestimate, or worst, underestimate the very global optima. The purpose of this article is to introduce an efficient and safe procedure to rigorously bound the global optima of semidefinite program. This work shows how, using interval arithmetic, rigorous error bounds for the optimal value can be computed by carefully post processing the output of a semidefinite programming solver. A lower bound is computed on a semidefinite relaxation of the constraint system and the objective function. Numerical results are presented using the SDPA (SemiDefinite Programming Algorithm), solver to compute the solution of semidefinite programs. This rigorous bound is injected in a branch and bound algorithm to solve the optimisation problem.展开更多
An approximately optimal adaptive arithmetic coding (AC) system using a forbidden symbol (FS) over noisy channels was proposed which allows one to jointly and adaptively design the source decoding and channel correcti...An approximately optimal adaptive arithmetic coding (AC) system using a forbidden symbol (FS) over noisy channels was proposed which allows one to jointly and adaptively design the source decoding and channel correcting in a single process, with superior performance compared with traditional separated techniques. The concept of adaptiveness is applied not only to the source model but also to the amount of coding redundancy. In addition, an improved branch metric computing algorithm and a faster sequential searching algorithm compared with the system proposed by Grangetto were proposed. The proposed system is tested in the case of image transmission over the AWGN channel, and compared with traditional separated system in terms of packet error rate and complexity. Both hard and soft decoding were taken into account.展开更多
The system of linear equations plays a vital role in real life problems such as optimization, economics, and engineering. The parameters of the system of linear equations are modeled by taking the experimental or obse...The system of linear equations plays a vital role in real life problems such as optimization, economics, and engineering. The parameters of the system of linear equations are modeled by taking the experimental or observation data. So the parameters of the system actually contain uncertainty rather than the crisp one. The uncertainties may be considered in term of interval or fuzzy numbers. In this paper, a detailed study of three solution techniques namely Classical Method, Extension Principle method and α-cuts and interval Arithmetic Method to solve the system of fuzzy linear equations has been done. Appropriate applications are given to illustrate each technique. Then we discuss the comparison of the different methods numerically and graphically.展开更多
New operators are presented to introduce “arithmetic calculus”, where 1) the operators are just obvious mathematical facts, and 2) arithmetic calculus refers to summing and subtracting operations without solving equ...New operators are presented to introduce “arithmetic calculus”, where 1) the operators are just obvious mathematical facts, and 2) arithmetic calculus refers to summing and subtracting operations without solving equations. The sole aim of this paper is to make a case for arithmetic calculus, which is lurking in conventional mathematics and science but has no identity of its own. The underlying thinking is: 1) to shift the focus from the whole sequence to any of its single elements;and 2) to factorise each element to building blocks and rules. One outcome of this emerging calculus is to understand the interconnectivity in a family of sequences, without which they are seen as discrete entities with no interconnectivity. Arithmetic calculus is a step closer towards deriving a “Tree of Numbers” reminiscent of the Tree of Life. Another windfall outcome is to show that the deconvolution problem is explicitly well-posed but at the same time implicitly ill-conditioned;and this challenges a misconception that this problem is ill-posed. If the thinking in this paper is not new, this paper forges it through a mathematical spin by presenting new terms, definitions, notations and operators. The return for these out of the blue new aspects is far reaching.展开更多
This paper presents a new method of lossless image compression. An image is characterized by homogeneous parts. The bit planes, which are of high weight are characterized by sequences of 0 and 1 are successive encoded...This paper presents a new method of lossless image compression. An image is characterized by homogeneous parts. The bit planes, which are of high weight are characterized by sequences of 0 and 1 are successive encoded with RLE, whereas the other bit planes are encoded by the arithmetic coding (AC) (static or adaptive model). By combining an AC (adaptive or static) with the RLE, a high degree of adaptation and compression efficiency is achieved. The proposed method is compared to both static and adaptive model. Experimental results, based on a set of 12 gray-level images, demonstrate that the proposed scheme gives mean compression ratio that are higher those compared to the conventional arithmetic encoders.展开更多
Here we explore some fundamental aspects of what may be termed as a new arithmetic function （defined notionally as theT function） together with its possible inverse （TI）, as it is not in frequent discussion among ...Here we explore some fundamental aspects of what may be termed as a new arithmetic function （defined notionally as theT function） together with its possible inverse （TI）, as it is not in frequent discussion among the number theorists.展开更多
The advances of digital arithmetic techniques permit computer designers to implement high speed application specific chips. The currently produced digital circuits have demonstrated high performance in terms of severa...The advances of digital arithmetic techniques permit computer designers to implement high speed application specific chips. The currently produced digital circuits have demonstrated high performance in terms of several criteria, such as, high clock rate, short input/output delay, small silicon area, and low power dissipation. In this paper, we implement several sinusoidal generation methods to optimize their performance and output using advanced digital arithmetic techniques. In this paper, the implementations of advanced digital oscillator structures with and without pipelining are proposed. The synthesis results of the implementation with pipelining have proven that it is superior to other sinusoidal generation methods in terms of the maximum frequency and signal resolution. Hence, this method is used in the design of the proposed digital oscillator chip.展开更多
In this research we are going to define two new concepts: a) “The Potential of Events” (EP) and b) “The Catholic Information” (CI). The term CI derives from the ancient Greek language and declares all the Catholic...In this research we are going to define two new concepts: a) “The Potential of Events” (EP) and b) “The Catholic Information” (CI). The term CI derives from the ancient Greek language and declares all the Catholic (general) Logical Propositions (<img src="Edit_5f13a4a5-abc6-4bc5-9e4c-4ff981627b2a.png" width="33" height="21" alt="" />) which will true for every element of a set A. We will study the Riemann Hypothesis in two stages: a) By using the EP we will prove that the distribution of events e (even) and o (odd) of Square Free Numbers (SFN) on the axis Ax(N) of naturals is Heads-Tails (H-T) type. b) By using the CI we will explain the way that the distribution of prime numbers can be correlated with the non-trivial zeros of the function ζ(s) of Riemann. The Introduction and the Chapter 2 are necessary for understanding the solution. In the Chapter 3 we will present a simple method of forecasting in many very useful applications (e.g. financial, technological, medical, social, etc) developing a generalization of this new, proven here, theory which we finally apply to the solution of RH. The following Introduction as well the Results with the Discussion at the end shed light about the possibility of the proof of all the above. The article consists of 9 chapters that are numbered by 1, 2, …, 9.展开更多
In order to decrease the calculation complexity of connectivity reliability of road networks, an improved recursive decomposition arithmetic is proposed. First, the basic theory of recursive decomposition arithmetic i...In order to decrease the calculation complexity of connectivity reliability of road networks, an improved recursive decomposition arithmetic is proposed. First, the basic theory of recursive decomposition arithmetic is reviewed. Then the characteristics of road networks, which are different from general networks, are analyzed. Under this condition, an improved recursive decomposition arithmetic is put forward which fits road networks better. Furthermore, detailed calculation steps are presented which are convenient for the computer, and the advantage of the approximate arithmetic is analyzed based on this improved arithmetic. This improved recursive decomposition arithmetic directly produces disjoint minipaths and avoids the non-polynomial increasing problems. And because the characteristics of road networks are considered, this arithmetic is greatly simplified. Finally, an example is given to prove its validity.展开更多
Counting has always been one of the most important operations for human be-ings. Naturally, it is inherent in economics and business. We count with the unique arithmetic, which humans have used for millennia. However,...Counting has always been one of the most important operations for human be-ings. Naturally, it is inherent in economics and business. We count with the unique arithmetic, which humans have used for millennia. However, over time, the most inquisitive thinkers have questioned the validity of standard arithmetic in certain settings. It started in ancient Greece with the famous philosopher Zeno of Elea, who elaborated a number of paradoxes questioning popular knowledge. Millennia later, the famous German researcher Herman Helmholtz (1821-1894) [1] expressed reservations about applicability of conventional arithmetic with respect to physical phenomena. In the 20th and 21st century, mathematicians such as Yesenin-Volpin (1960) [2], Van Bendegem (1994) [3], Rosinger (2008) [4] and others articulated similar concerns. In validation, in the 20th century expressions such as 1 + 1 = 3 or 1 + 1 = 1 occurred to reflect important characteristics of economic, business, and social processes. We call these expressions synergy arithmetic. It is common notion that synergy arithmetic has no meaning mathematically. However in this paper we mathematically ground and explicate synergy arithmetic.展开更多
“Arithmetic Calculus” (AC), introduced recently by the author, is explored further in this paper by giving a new lease of life to the age-old differences table by transforming it into a new kind of network. Any sequ...“Arithmetic Calculus” (AC), introduced recently by the author, is explored further in this paper by giving a new lease of life to the age-old differences table by transforming it into a new kind of network. Any sequence that can be laid out in this network can be classified into one of five types of sequences, which can be expressed by algebraic polynomial or exponential functions but AC can reveal information concealed by algebra. The paper defines these sequences and refers to each family member as the parent sequence. These sequences make up their own universe as: (i) the population of the terms in each parent sequence is infinite;(ii) each parent sequence forms a hierarchy, in which their number of levels can extend into infinity;and (ii) there are infinitely diverse parent sequences. A glimpse of AC is illustrated through examples but their analytical capability is applicable in their universe. The paper shows that small sets of building blocks are operated by the convolution theorem or its variations, which is embedded “all-pervasively” in each and every member of the universe. Sufficient details are presented to ensure the emergence of the mosaic image of AC through problems including: (i) differentiation (reducement), (ii) integration (conducement), (iii) diagonal operations (reminiscent of gradient methods), and (iv) structure of hierarchies. These operations reveal that the new network can parallel Cartesian coordinates and that for problems with no noise, the deconvolution problem is well-posed against common myth of it being ill-posed.展开更多
基金supported by the National Natural Science Foundation of China (Grant No.12374109)the National Key Research and Development Program of China (Grant No.2023YFA1406600)。
文摘In physics,our expectations for system behavior are often guided by intuitive arithmetic.For systems composed of identical units,we anticipate synergy of the contributions from these units,where 1+1=2.Conversely,for systems built from opposing units,we expect cancellation of their contributions,where 1-1=0.This intuitive arithmetic has long underpinned our understanding of physical properties of materials,from electronic transport to optical responses.However,scientific breakthroughs often occur when nature reveals ways to circumvent these seemingly fundamental rules,opening new possibilities that challenge our deepest assumptions about material behavior.
文摘In this article, we report the derivation of high accuracy finite difference method based on arithmetic average discretization for the solution of Un=F(x,u,u′)+∫K(x,s)ds , 0 x s < 1 subject to natural boundary conditions on a non-uniform mesh. The proposed variable mesh approximation is directly applicable to the integro-differential equation with singular coefficients. We need not require any special discretization to obtain the solution near the singular point. The convergence analysis of a difference scheme for the diffusion convection equation is briefly discussed. The presented variable mesh strategy is applicable when the internal grid points of the solution space are both even and odd in number as compared to the method discussed by authors in their previous work in which the internal grid points are strictly odd in number. The advantage of using this new variable mesh strategy is highlighted computationally.
基金The National Natural Science Foundation ofChina(No60332030)
文摘A new arithmetic coding system combining source channel coding and maximum a posteriori decoding were proposed. It combines source coding and error correction tasks into one unified process by introducing an adaptive forbidden symbol. The proposed system achieves fixed length code words by adaptively adjusting the probability of the forbidden symbol and adding tail digits of variable length. The corresponding improved MAP decoding metric was derived. The proposed system can improve the performance. Simulations were performed on AWGN channels with various noise levels by using both hard and soft decision with BPSK modulation.The results show its performance is slightly better than that of our adaptive arithmetic error correcting coding system using a forbidden symbol.
文摘Efficient solvers for optimization problems are based on linear and semidefinite relaxations that use floating point arithmetic. However, due to the rounding errors, relaxation thus may overestimate, or worst, underestimate the very global optima. The purpose of this article is to introduce an efficient and safe procedure to rigorously bound the global optima of semidefinite program. This work shows how, using interval arithmetic, rigorous error bounds for the optimal value can be computed by carefully post processing the output of a semidefinite programming solver. A lower bound is computed on a semidefinite relaxation of the constraint system and the objective function. Numerical results are presented using the SDPA (SemiDefinite Programming Algorithm), solver to compute the solution of semidefinite programs. This rigorous bound is injected in a branch and bound algorithm to solve the optimisation problem.
文摘An approximately optimal adaptive arithmetic coding (AC) system using a forbidden symbol (FS) over noisy channels was proposed which allows one to jointly and adaptively design the source decoding and channel correcting in a single process, with superior performance compared with traditional separated techniques. The concept of adaptiveness is applied not only to the source model but also to the amount of coding redundancy. In addition, an improved branch metric computing algorithm and a faster sequential searching algorithm compared with the system proposed by Grangetto were proposed. The proposed system is tested in the case of image transmission over the AWGN channel, and compared with traditional separated system in terms of packet error rate and complexity. Both hard and soft decoding were taken into account.
文摘The system of linear equations plays a vital role in real life problems such as optimization, economics, and engineering. The parameters of the system of linear equations are modeled by taking the experimental or observation data. So the parameters of the system actually contain uncertainty rather than the crisp one. The uncertainties may be considered in term of interval or fuzzy numbers. In this paper, a detailed study of three solution techniques namely Classical Method, Extension Principle method and α-cuts and interval Arithmetic Method to solve the system of fuzzy linear equations has been done. Appropriate applications are given to illustrate each technique. Then we discuss the comparison of the different methods numerically and graphically.
文摘New operators are presented to introduce “arithmetic calculus”, where 1) the operators are just obvious mathematical facts, and 2) arithmetic calculus refers to summing and subtracting operations without solving equations. The sole aim of this paper is to make a case for arithmetic calculus, which is lurking in conventional mathematics and science but has no identity of its own. The underlying thinking is: 1) to shift the focus from the whole sequence to any of its single elements;and 2) to factorise each element to building blocks and rules. One outcome of this emerging calculus is to understand the interconnectivity in a family of sequences, without which they are seen as discrete entities with no interconnectivity. Arithmetic calculus is a step closer towards deriving a “Tree of Numbers” reminiscent of the Tree of Life. Another windfall outcome is to show that the deconvolution problem is explicitly well-posed but at the same time implicitly ill-conditioned;and this challenges a misconception that this problem is ill-posed. If the thinking in this paper is not new, this paper forges it through a mathematical spin by presenting new terms, definitions, notations and operators. The return for these out of the blue new aspects is far reaching.
文摘This paper presents a new method of lossless image compression. An image is characterized by homogeneous parts. The bit planes, which are of high weight are characterized by sequences of 0 and 1 are successive encoded with RLE, whereas the other bit planes are encoded by the arithmetic coding (AC) (static or adaptive model). By combining an AC (adaptive or static) with the RLE, a high degree of adaptation and compression efficiency is achieved. The proposed method is compared to both static and adaptive model. Experimental results, based on a set of 12 gray-level images, demonstrate that the proposed scheme gives mean compression ratio that are higher those compared to the conventional arithmetic encoders.
文摘Here we explore some fundamental aspects of what may be termed as a new arithmetic function （defined notionally as theT function） together with its possible inverse （TI）, as it is not in frequent discussion among the number theorists.
文摘The advances of digital arithmetic techniques permit computer designers to implement high speed application specific chips. The currently produced digital circuits have demonstrated high performance in terms of several criteria, such as, high clock rate, short input/output delay, small silicon area, and low power dissipation. In this paper, we implement several sinusoidal generation methods to optimize their performance and output using advanced digital arithmetic techniques. In this paper, the implementations of advanced digital oscillator structures with and without pipelining are proposed. The synthesis results of the implementation with pipelining have proven that it is superior to other sinusoidal generation methods in terms of the maximum frequency and signal resolution. Hence, this method is used in the design of the proposed digital oscillator chip.
文摘In this research we are going to define two new concepts: a) “The Potential of Events” (EP) and b) “The Catholic Information” (CI). The term CI derives from the ancient Greek language and declares all the Catholic (general) Logical Propositions (<img src="Edit_5f13a4a5-abc6-4bc5-9e4c-4ff981627b2a.png" width="33" height="21" alt="" />) which will true for every element of a set A. We will study the Riemann Hypothesis in two stages: a) By using the EP we will prove that the distribution of events e (even) and o (odd) of Square Free Numbers (SFN) on the axis Ax(N) of naturals is Heads-Tails (H-T) type. b) By using the CI we will explain the way that the distribution of prime numbers can be correlated with the non-trivial zeros of the function ζ(s) of Riemann. The Introduction and the Chapter 2 are necessary for understanding the solution. In the Chapter 3 we will present a simple method of forecasting in many very useful applications (e.g. financial, technological, medical, social, etc) developing a generalization of this new, proven here, theory which we finally apply to the solution of RH. The following Introduction as well the Results with the Discussion at the end shed light about the possibility of the proof of all the above. The article consists of 9 chapters that are numbered by 1, 2, …, 9.
基金The National Key Technology R& D Program of Chinaduring the 11th Five-Year Plan Period (No.2006BAJ18B03).
文摘In order to decrease the calculation complexity of connectivity reliability of road networks, an improved recursive decomposition arithmetic is proposed. First, the basic theory of recursive decomposition arithmetic is reviewed. Then the characteristics of road networks, which are different from general networks, are analyzed. Under this condition, an improved recursive decomposition arithmetic is put forward which fits road networks better. Furthermore, detailed calculation steps are presented which are convenient for the computer, and the advantage of the approximate arithmetic is analyzed based on this improved arithmetic. This improved recursive decomposition arithmetic directly produces disjoint minipaths and avoids the non-polynomial increasing problems. And because the characteristics of road networks are considered, this arithmetic is greatly simplified. Finally, an example is given to prove its validity.
文摘Counting has always been one of the most important operations for human be-ings. Naturally, it is inherent in economics and business. We count with the unique arithmetic, which humans have used for millennia. However, over time, the most inquisitive thinkers have questioned the validity of standard arithmetic in certain settings. It started in ancient Greece with the famous philosopher Zeno of Elea, who elaborated a number of paradoxes questioning popular knowledge. Millennia later, the famous German researcher Herman Helmholtz (1821-1894) [1] expressed reservations about applicability of conventional arithmetic with respect to physical phenomena. In the 20th and 21st century, mathematicians such as Yesenin-Volpin (1960) [2], Van Bendegem (1994) [3], Rosinger (2008) [4] and others articulated similar concerns. In validation, in the 20th century expressions such as 1 + 1 = 3 or 1 + 1 = 1 occurred to reflect important characteristics of economic, business, and social processes. We call these expressions synergy arithmetic. It is common notion that synergy arithmetic has no meaning mathematically. However in this paper we mathematically ground and explicate synergy arithmetic.
文摘“Arithmetic Calculus” (AC), introduced recently by the author, is explored further in this paper by giving a new lease of life to the age-old differences table by transforming it into a new kind of network. Any sequence that can be laid out in this network can be classified into one of five types of sequences, which can be expressed by algebraic polynomial or exponential functions but AC can reveal information concealed by algebra. The paper defines these sequences and refers to each family member as the parent sequence. These sequences make up their own universe as: (i) the population of the terms in each parent sequence is infinite;(ii) each parent sequence forms a hierarchy, in which their number of levels can extend into infinity;and (ii) there are infinitely diverse parent sequences. A glimpse of AC is illustrated through examples but their analytical capability is applicable in their universe. The paper shows that small sets of building blocks are operated by the convolution theorem or its variations, which is embedded “all-pervasively” in each and every member of the universe. Sufficient details are presented to ensure the emergence of the mosaic image of AC through problems including: (i) differentiation (reducement), (ii) integration (conducement), (iii) diagonal operations (reminiscent of gradient methods), and (iv) structure of hierarchies. These operations reveal that the new network can parallel Cartesian coordinates and that for problems with no noise, the deconvolution problem is well-posed against common myth of it being ill-posed.
