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Logical relations, defined inductively on the structure of types, provide a powerful tool to characterize higher-order functions. They often enable us to prove correctness of a program transformer written with higher-order functions concisely. This paper demonstrates that the technique of logical relations can be used to characterize call-by-value functions as well as delimited continuations. Based on the traditional logical relations for call-by-name functions, logical relations for call-by-value functions are first defined, whose CPS variant is used to prove the correctness of an offline specializer for the call-by-value λ- calculus. They are then modified to cope with delimited continuations and are used to establish the correctness of an offline specializer for the call-by-value λ- calculus with delimited continuation constructs, shift and reset. This is the first correctness proof for such a specializer. Along the development, correctness of the continuation-based and shift/reset-based let-insertion and A-normalization is established. [ABSTRACT FROM AUTHOR]

We describe in this chapter a Bat Algorithm and Genetic Algorithm (GA) conducting a performance comparison of the two algorithms Benchmark testing them in mathematical functions, parameters adjustment is done manually for both algorithms in 6 math functions, including some references on work done with the bat and area algorithm optimization with mathematical functions. [ABSTRACT FROM AUTHOR]

The Gibbs phenomenon poses a major obstruction in reconstructing a function with finite jumps by a partial Fourier sum or truncated orthogonal series expansion. In this paper, we present a Padé-type approach to treat the problem. To better reconstruct a function, we utilize an amplified Padétype approximation that takes into account the jump discontinuities of the function. The effectiveness of the proposed method is measured in terms of the Gibbs constant and steepness of the approximant at the location of the discontinuity. We present numerical examples to demonstrate the reconstruction. [ABSTRACT FROM AUTHOR]

In this paper, we present a generic differentiation rule that is applicable to all mathematical operators and illustrate how the generic differentiation rule is vital in deducing the derivatives of complex mathematical operations such as functional iteration. We also show how the generic rule offers more insight into the concept of differentiation and provides systematic proofs to differentiation rules that, otherwise, would have been proven using ad hoc approaches. Next, the generic rule is shown to yield interesting results including a proof that the first-degree approximation of the composite of iterated functions, at the limit where the iteration variable approaches zero, is the sum of the iterators of those functions. Consequently, at the vicinity of the specified limit, composition of iterated functions is remarkably symmetric. [ABSTRACT FROM AUTHOR]

Value function is usually used to deal with the reinforcement learning problems. In large or even continuous states, function approximation must be used to represent value function. Much of the current work carried out, however, has to design the structure of function approximation in advanced which cannot be adjusted during learning. In this paper, we propose a novel function approximation called Fuzzy CMAC (FCMAC) with automatic state partition (ASP-FCMAC) to automate the structure design for FCMAC. Based on CMAC (also known as the coding), ASP-FCMAC employs fuzzy membership function to avoid the setting of parameter in CMAC, and makes use of Bellman error to partition the state automatically so as to generate the structure of FCMAC. Empirical results in both mountain car and RoboCup Keepaway domains demonstrate that ASP-FCMAC can automatically generate the structure of FCMAC and agent using it can learn efficiently. [ABSTRACT FROM AUTHOR]

A combinational module for contiguous pattern matching tests whether the digits of a given pattern occur as a contiguous substring within a given search string. We present a unifying high-level synthesis of linear and tree-structured combinational networks for contiguous pattern matching based on the theory of list homomorphisms. From a predicate logic specification, we develop standard implementations for tree-structured modules processing the input digits of the search string in parallel. The tree-structured design is systematically specialized to iterative networks processing the search string sequentially from left to right or vice versa. The formal design is parameterized with the type of digits to be matched. Throughout the paper, we explicate functional methods for the structured description of combinational circuits with higher-order list functions. [ABSTRACT FROM AUTHOR]

Network calculus has successfully been applied to derive performance bounds for communication networks, whereas a number of issues still remain unsolved or are not well understood yet. Founded on min-plus convolution and de-convolution, network calculus obeys a strong analogy to system theory. However, system theory has been extended beyond the time domain, applying the Fourier transform and allowing for an efficient analysis in the frequency domain. A corresponding dual domain for network calculus has not been elaborated, so far. In this paper we show that in analogy to system theory such a dual domain for network calculus is given by convex/concave conjugates referred to also as Legendre transform. We provide solutions for dual operations and show that min-plus convolution and de-convolution become simple addition and subtraction in Legendre space. [ABSTRACT FROM AUTHOR]

This paper proposes a new selection method for Genetic Algorithms. The motivation behind the proposed method is to investigate the effect of different selection methods on the rate of convergence. The new method Dynamic Selection Method (DSM) is based on proportionate selection. DSM functions by continuously changing the criteria for parent selection (dynamic) based on the number of generations in a run and the current generation. Results show that by using DSM to maintain diversity in a population gives slower convergence, but, their overall performance was an improvement. Relationship between slower convergences, in GA runs, leading to better solutions, has been identified. [ABSTRACT FROM AUTHOR]

A chapter of the book "The Princeton Companion to Mathematics," edited by Timothy Gowers, is presented. It discusses different mathematical concepts including Bayesian analysis, Braid Groups and the Calculus of Variations. Class Field Theory, Quantum Computation, Riemann Surfaces and Riemann Zeta Function, are also discussed.

In evolution problems, one scalar variable, denoted by t and having a meaning of time, takes a special role, which is also reflected by mathematical analysis. In particular, here we first present a few useful assertions about spaces of abstract functions on a “time” interval I := [0, T ], introduced already in Section 1.5, but now possessing additionally derivatives with respect to time. Always, T will denote a fixed finite time horizon. [ABSTRACT FROM AUTHOR]

This chapter derives the positive sampling functions and their biorthogonal counterparts using Daubechies wavelets. Many advantages derive from the use of the sampling biorthogonal time domain (SBTD) method to replace the finite difference time domain (FDTD) scheme. [ABSTRACT FROM PUBLISHER]

This chapter introduces the simple replacement equations that are chosen in the Mathematica package. [ABSTRACT FROM PUBLISHER]

This chapter gives an overview on spheroidal wave functions as presented in the following nine chapters. [ABSTRACT FROM PUBLISHER]