arXiv subjects
Publications and source records attributed to Chai Wah Wu.
The greatest common divisor (GCD) of $\binom{2n}{2k}$ for $1\leq k<n$ is known to be some power of 2 times the product of all odd primes p such that $2n=p^i+p^j$. We complete the analysis of this GCD by showing that this power of 2 is either 1 or 0 and relates it to Mersenne primes. We also show how to efficiently compute $GCD \{\binom{mn}{mk}: 1\leq k<n\}$ when n and m satisfy certain conditions.
In modern usage the Bernoulli numbers and Bernoulli polynomials follow Euler's approach and are defined using generating functions. Originally, they were derived by Bernoulli while characterizing Faulhaber's formula for the sum of consecutive powers. These equations have many consequences and applications in various areas of mathematics. We consider yet another application by studying the functional equation $f(x)+x^k=f(x+1)$ and show that a solution of this equation can be derived from Faulhaber's formula. We then use these results to study the totatives of n, i.e. numbers less than n that are coprime to n. In particular, we look at sums of powers of totatives of n that are less than n/2. We show that the sum of powers of this half of the totatives can also be expressed in the same structural form as the sum of powers of all totatives and provide explicit formulas for this sum. As an application of these results, we obtain a formula for the total area of all rectangles with coprime width and length and semiperimeter n.
We devise a control-theoretic reinforcement learning approach to support direct learning of the optimal policy. We establish various theoretical properties of our approach, such as convergence and optimality of our analog of the Bellman operator and Q-learning, a new control-policy-variable gradient theorem, and a specific gradient ascent algorithm based on this theorem within the context of a specific control-theoretic framework. We empirically evaluate the performance of our control theoretic approach on several classical reinforcement learning tasks, demonstrating significant improvements in solution quality, sample complexity, and running time of our approach over state-of-the-art methods.
Today, the vast majority of the world's digital information is represented using the fundamental assumption, introduced by Claude Shannon in 1948, that ``...the semantic aspects of communication are irrelevant to the engineering problem (of the design of communication systems)...''. Consider, nonetheless, the observation that we often combine a message with other information in order to deduce new facts, thereby expanding the value of such a message. It is noteworthy that to-date, no rigorous theory of communication has been put forth which postulates the existence of deductive capabilities on the receiver's side. The purpose of this paper is to present such a theory. We formally model such deductive capabilities using logic reasoning, and present a rigorous theory which covers the following generic scenario: Alice and Bob each have knowledge of some logic sentence, and they wish to communicate as efficiently as possible with the shared goal that, following their communication, Bob should be able to deduce a particular logic sentence that Alice knows to be true, but that Bob currently cannot prove. Many variants of this general setup are considered in this article; in all cases we are able to provide sharp upper and lower bounds. Our contribution includes the identification of the most fundamental requirements that we place on a logic and associated logical language for all of our results to apply. Practical algorithms that are in some cases asymptotically optimal are provided, and we illustrate the potential practical value of the design of communication systems that incorporate the assumption of deductive capabilities at the receiver using experimental results that suggest significant possible gains compared to classical systems.
Finding the $n$-th positive square number is easy, as it is simply $n^2$. But how do we find the complementary sequence, i.e., the $n$-th positive non-square number? For this case there is an explicit formula. However, for general constraints on numbers, a formula is harder to find. In this paper, we study how to compute the $n$-th integer that does (or does not) satisfy a certain condition. In particular, we consider it as a fixed point problem, relate it to the iterative method of Lambek and Moser, study a bisection approach to this problem, and provide novel formulas for various complementary sequences including the non-$k$-gonal numbers, non-$k$-gonal-pyramidal numbers, non-$k$-simplex numbers, non-sum-of-$k$-th-powers, and non-$k$-th-powers. For example, we show that the $n$-th non $k$-gonal number is given by $n+\text{round}\left(\sqrt{\frac{2n-2+\left\lfloor\frac{k+1}{4}\right\rfloor}{k-2}}\right)$ and that the $n$-th non-second-hexagonal number is $n+\left\lceil\sqrt{\frac{n}{2}}\right\rceil-1$.
Iterative solvers for large-scale linear systems such as Krylov subspace methods can diverge when the linear system is ill-conditioned, thus significantly reducing the applicability of these iterative methods in practice for high-performance computing solutions of such large-scale linear systems. To address this fundamental problem, we propose general algorithmic frameworks to modify Krylov subspace iterative solution methods which ensure that the algorithms are stable and do not diverge. We then apply our general frameworks to current implementations of the corresponding iterative methods in SciPy and demonstrate the efficacy of our stable iterative approach with respect to numerical experiments across a wide range of synthetic and real-world ill-conditioned linear systems.
Iterative refinement (IR) is a popular scheme for solving a linear system of equations based on gradually improving the accuracy of an initial approximation. Originally developed to improve upon the accuracy of Gaussian elimination, interest in IR has been revived because of its suitability for execution on fast low-precision hardware such as analog devices and graphics processing units. IR generally converges when the error associated with the solution method is small, but is known to diverge when this error is large. We propose and analyze a novel enhancement to the IR algorithm by adding a line search optimization step that guarantees the algorithm will not diverge. Numerical experiments verify our theoretical results and illustrate the effectiveness of our proposed scheme.
In this paper we consider the problem of computing the stationary distribution of nearly completely decomposable Markov processes, a well-established area in the classical theory of Markov processes with broad applications in the design, modeling, analysis and optimization of computer systems and applications. We devise a general mathematical framework of numerical solution methods that exploits forms of mixed-precision computation to significantly reduce computation times and that exploits forms of iterative approximate computing approaches to mitigate the impact of inaccurate computations, further reduce computation times, and ensure convergence. Then we derive a mathematical analysis that establishes theoretical properties of our general algorithmic framework including results on approximation errors, convergence behaviors, and other algorithmic characteristics. Numerical experiments demonstrate that our general algorithmic framework provides significant improvements in computation times over the most-efficient existing numerical methods.
Pandigital and penholodigital numbers are numbers that contain every digit or nonzero digit respectively. We study properties of pandigital or penholodigital numbers that are also square, oblong or prime.
Information theory has provided foundations for the theories of several application areas critical for modern society, including communications, computer storage, and AI. A key aspect of Shannon's 1948 theory is a sharp lower bound on the number of bits needed to encode and communicate a string of symbols. When he introduced the theory, Shannon famously excluded any notion of semantics behind the symbols being communicated. This semantics-free notion went on to have massive impact on communication and computing technologies, even as multiple proposals for reintroducing semantics in a theory of information were being made, notably one where Carnap and Bar-Hillel used logic and reasoning to capture semantics. In this paper we present, for the first time, a Shannon-style analysis of a communication system equipped with a deductive reasoning capability, implemented using logical inference. We use some of the most important techniques developed in information theory to demonstrate significant and sometimes surprising gains in communication efficiency availed to us through such capability, demonstrated also through practical codes. We thus argue that proposals for a semantic information theory should include the power of deductive reasoning to magnify the value of transmitted bits as we strive to fully unlock the inherent potential of semantics.
In an increasingly interconnected world, a key scientific challenge is to examine mechanisms that lead to the widespread propagation of contagions, such as misinformation and pathogens, and identify risk factors that can trigger large-scale outbreaks. Underlying both the spread of disease and misinformation epidemics is the evolution of the contagion as it propagates, leading to the emergence of different strains, e.g., through genetic mutations in pathogens and alterations in the information content. Recent studies have revealed that models that do not account for heterogeneity in transmission risks associated with different strains of the circulating contagion can lead to inaccurate predictions. However, existing results on multi-strain spreading assume that the network has a vanishingly small clustering coefficient, whereas clustering is widely known to be a fundamental property of real-world social networks. In this work, we investigate spreading processes that entail evolutionary adaptations on random graphs with tunable clustering and arbitrary degree distributions. We derive a mathematical framework to quantify the epidemic characteristics of a contagion that evolves as it spreads, with the structure of the underlying network as given via arbitrary {\em joint} degree distributions of single-edges and triangles. To the best of our knowledge, our work is the first to jointly analyze the impact of clustering and evolution on the emergence of epidemic outbreaks. We supplement our theoretical finding with numerical simulations and case studies, shedding light on the impact of clustering on contagion spread.
We study the square of opposition and its various geometric generalizations from an algebraic viewpoint. In particular, we show how the various shapes of oppositions can be framed under an algebraic framework and we illustrate this approach with algebraic structures beyond the traditional logical structures.
Numerical computation is essential to many areas of artificial intelligence (AI), whose computing demands continue to grow dramatically, yet their continued scaling is jeopardized by the slowdown in Moore's law. Multi-function multi-way analog (MFMWA) technology, a computing architecture comprising arrays of memristors supporting in-memory computation of matrix operations, can offer tremendous improvements in computation and energy, but at the expense of inherent unpredictability and noise. We devise novel randomized algorithms tailored to MFMWA architectures that mitigate the detrimental impact of imperfect analog computations while realizing their potential benefits across various areas of AI, such as applications in computer vision. Through analysis, measurements from analog devices, and simulations of larger systems, we demonstrate orders of magnitude reduction in both computation and energy with accuracy similar to digital computers.
Between the poker hands of straight, flush, and full house, which hand is more common? In standard 5-card poker, the order from most common to least common is straight, flush, full house. The same order is true for 7-card poker such as Texas hold'em. However, is the same true for n-card poker for larger n? We study the probability of obtaining these various hands for n-card poker for various values of $n\geq 5$. In particular, we derive equations for the probability of flush, straight and full house and show that the probability of flush is less than a straight when $n\leq 11$, and is more than a straight when n>11. Similarly, we show that the probability of full house is less than a straight when $n\leq 19$, and is more than a straight when n>19. This means that for games such as Big Two where the ordering of 13-card hands depends on the ordering in 5-card poker, the ranking ordering does not follow the occurrence probability ordering, contrary to what intuition suggests.
The rearrangement inequality states that the sum of products of permutations of 2 sequences of real numbers are maximized when the terms are similarly ordered and minimized when the terms are ordered in opposite order. We show that similar inequalities exist for multi-valued logic with the multiplication and addition operation replaced with various T-norms and T-conorms respectively. For instance, we show that the rearrangement inequality holds when the T-norms and T-conorms are derived from Archimedean copulas.
Masks have remained an important mitigation strategy in the fight against COVID-19 due to their ability to prevent the transmission of respiratory droplets between individuals. In this work, we provide a comprehensive quantitative analysis of the impact of mask-wearing. To this end, we propose a novel agent-based model of viral spread on networks where agents may either wear no mask or wear one of several types of masks with different properties (e.g., cloth or surgical). We derive analytical expressions for three key epidemiological quantities: the probability of emergence, the epidemic threshold, and the expected epidemic size. In particular, we show how the aforementioned quantities depend on the structure of the contact network, viral transmission dynamics, and the distribution of the different types of masks within the population. Through extensive simulations, we then investigate the impact of different allocations of masks within the population and trade-offs between the outward efficiency and inward efficiency of the masks. Interestingly, we find that masks with high outward efficiency and low inward efficiency are most useful for controlling the spread in the early stages of an epidemic, while masks with high inward efficiency but low outward efficiency are most useful in reducing the size of an already large spread. Lastly, we study whether degree-based mask allocation is more effective in reducing the probability of epidemic as well as epidemic size compared to random allocation. The result echoes the previous findings that mitigation strategies should differ based on the stage of the spreading process, focusing on source control before the epidemic emerges and on self-protection after the emergence.
A key scientific challenge during the outbreak of novel infectious diseases is to predict how the course of the epidemic changes under different countermeasures that limit interaction in the population. Most epidemiological models do not consider the role of mutations and heterogeneity in the type of contact events. However, pathogens have the capacity to mutate in response to changing environments, especially caused by the increase in population immunity to existing strains and the emergence of new pathogen strains poses a continued threat to public health. Further, in light of differing transmission risks in different congregate settings (e.g., schools and offices), different mitigation strategies may need to be adopted to control the spread of infection. We analyze a multi-layer multi-strain model by simultaneously accounting for i) pathways for mutations in the pathogen leading to the emergence of new pathogen strains, and ii) differing transmission risks in different congregate settings, modeled as network-layers. Assuming complete cross-immunity among strains, namely, recovery from any infection prevents infection with any other (an assumption that will need to be relaxed to deal with COVID-19 or influenza), we derive the key epidemiological parameters for the proposed multi-layer multi-strain framework. We demonstrate that reductions to existing network-based models that discount heterogeneity in either the strain or the network layers can lead to incorrect predictions for the course of the outbreak. In addition, our results highlight that the impact of imposing/lifting mitigation measures concerning different contact network layers (e.g., school closures or work-from-home policies) should be evaluated in connection with their effect on the likelihood of the emergence of new pathogen strains.
We study properties of functions of binomial coefficients mod 2 and derive a set of recurrence relations for sums of products of binomial coefficients mod 2. We show that they result in sequences that are the run length transforms of well known basic sequences. In particular, we obtain formulas for the run length transform of the positive integers, Fibonacci numbers, extended Lucas numbers and Narayana's cows sequence.