Class of Recursive Wideband Digital Differentiators and Integrators

New designs of recursive digital differentiators are obtained by optimizing a general fourth-order recursive digital filter over different Nyquist bands. In addition, another design of recursive digital differentiator is also obtained by optimizing the specified pole-zero locations of existing recursive digital differentiator of second-order system. Further, new designs of recursive digital integrators are obtained by inverting the transfer functions of designed recursive digital differentiators with suitable modifications. Thereafter, the zero-reflection approach is discussed and then applied to improve the phase responses of designed recursive digital differentiators and integrators. The beauty of finally obtained recursive digital differentiators and integrators is that they have nearly linear phase responses over wideband and also provide the choice of suitable recursive digital differentiator and integrator according to the importance of accuracy, bandwidth and the system simplicity

The m−Order Linear Recursive Quaternions

This study considers the m−order linear recursive sequences yielding some well-known sequences (such as the Fibonacci, Lucas, Pell, Jacobsthal, Padovan, and Perrin sequences). Also, the Binet-like formulas and generating functions of the m−order linear recursive sequences have been derived. Then, we define the m−order linear recursive quaternions, and give the Binet-like formulas and generating functions for them

Recursive Approximation of the High Dimensional max Function

An alternative smoothing method for the high dimensional max functionhas been studied. The proposed method is a recursive extension of thetwo dimensional smoothing functions. In order to analyze the proposedmethod, a theoretical framework related to smoothing methods has beendiscussed. Moreover, we support our discussion by considering someapplication areas. This is followed by a comparison with analternative well-known smoothing method.n dimensional max function;recursive approximation;smoothing methods;vertical linear complementarity (VLCP)

Non-polynomial Worst-Case Analysis of Recursive Programs

We study the problem of developing efficient approaches for proving worst-case bounds of non-deterministic recursive programs. Ranking functions are sound and complete for proving termination and worst-case bounds of nonrecursive programs. First, we apply ranking functions to recursion, resulting in measure functions. We show that measure functions provide a sound and complete approach to prove worst-case bounds of non-deterministic recursive programs. Our second contribution is the synthesis of measure functions in nonpolynomial forms. We show that non-polynomial measure functions with logarithm and exponentiation can be synthesized through abstraction of logarithmic or exponentiation terms, Farkas' Lemma, and Handelman's Theorem using linear programming. While previous methods obtain worst-case polynomial bounds, our approach can synthesize bounds of the form $\mathcal{O}(n\log n)$ as well as $\mathcal{O}(n^r)$ where $r$ is not an integer. We present experimental results to demonstrate that our approach can obtain efficiently worst-case bounds of classical recursive algorithms such as (i) Merge-Sort, the divide-and-conquer algorithm for the Closest-Pair problem, where we obtain $\mathcal{O}(n \log n)$ worst-case bound, and (ii) Karatsuba's algorithm for polynomial multiplication and Strassen's algorithm for matrix multiplication, where we obtain $\mathcal{O}(n^r)$ bound such that $r$ is not an integer and close to the best-known bounds for the respective algorithms.Comment: 54 Pages, Full Version to CAV 201

Recurrence relations for the number of solutions of a class of Diophantine equations

Recursive formulas are derived for the number of solutions of linear and quadratic Diophantine equations with positive coefficients. This result is further extended to general non-linear additive Diophantine equations. It is shown that all three types of the recursion admit an explicit solution in the form of complete Bell polynomial, depending on the coefficients of the power series expansion of the logarithm of the generating functions for the sequences of individual terms in the Diophantine equations.Comment: 11 pages, Latex. Dedicated to the 70-th anniversary of Apolodor Radut

Free-cut elimination in linear logic and an application to a feasible arithmetic

International audienceWe prove a general form of 'free-cut elimination' for first-order theories in linear logic, yielding normal forms of proofs where cuts are anchored to nonlogical steps. To demonstrate the usefulness of this result, we consider a version of arithmetic in linear logic, based on a previous axiomatisation by Bellantoni and Hofmann. We prove a witnessing theorem for a fragment of this arithmetic via the 'witness function method', showing that the provably convergent functions are precisely the polynomial-time functions. The programs extracted are implemented in the framework of 'safe' recursive functions, due to Bellantoni and Cook, where the ! modality of linear logic corresponds to normal inputs of a safe recursive program