Combinatorial characterization of a certain class of words and a conjectured connection with general subclasses of phylogenetic tree-child networks

The combinatorial study of phylogenetic networks has attracted much attention in recent times. In particular, one class of them, the so-called tree-child networks, are becoming the most prominent ones. However, their combinatorial properties are largely unknown. In this paper we address the problem of exactly counting them. We conjecture a relationship with the cardinality of a certain class of words. By solving the counting problem for the words, and on the basis of the conjecture, several simple recurrence formulas for general cases arise. Moreover, a precise asymptotic analysis is provided. Our results coincide with all current formulas in the literature for particular subclasses of tree-child networks, as well as with numerical results obtained for small networks. We expect that the study of the relationship between the newly defined words and the networks will lead to further combinatoric characterizations of this class of phylogenetic networks.

Maximums of the Additive Differential Probability of Exclusive-Or

At FSE 2004, Lipmaa et al. studied the additive differential probability adp⊕(α,β → γ) of exclusive-or where differences α,β,γ ∈ Fn2 are expressed using addition modulo 2n. This probability is used in the analysis of symmetric-key primitives that combine XOR and modular addition, such as the increasingly popular Addition-Rotation-XOR (ARX) constructions. The focus of this paper is on maximal differentials, which are helpful when constructing differential trails. We provide the missing proof for Theorem 3 of the FSE 2004 paper, which states that maxα,βadp⊕(α,β → γ) = adp⊕(0,γ → γ) for all γ. Furthermore, we prove that there always exist either two or eight distinct pairs α,β such that adp⊕( α,β → γ) = adp⊕(0,γ → γ), and we obtain recurrence formulas for calculating adp⊕. To gain insight into the range of possible differential probabilities, we also study other properties such as the minimum value of adp⊕(0,γ → γ), and we find all γ that satisfy this minimum value.

One-Kind Hybrid Power Means of the Two-Term Exponential Sums and Quartic Gauss Sums

The main purpose of this article is using the analytic methods and the properties of the classical Gauss sums to study the calculating problem of the hybrid power mean of the two-term exponential sums and quartic Gauss sums and then prove two interesting linear recurrence formulas. As applications, some asymptotic formulas are obtained.

A Unifying Framework for Perturbative Exponential Factorizations

We propose a framework where Fer and Wilcox expansions for the solution of differential equations are derived from two particular choices for the initial transformation that seeds the product expansion. In this scheme, intermediate expansions can also be envisaged. Recurrence formulas are developed. A new lower bound for the convergence of the Wilcox expansion is provided, as well as some applications of the results. In particular, two examples are worked out up to a high order of approximation to illustrate the behavior of the Wilcox expansion.

Estimates in the Taylor series method for polynomial total systems of PDEs

Many of total systems of PDEs can be reduced to the polynomial form. As was shown by various authors, one of the best methods for the numerical solution of the initial value problem for ODE systems is the Taylor Series Method (TSM). In the article, the authors consider the Cauchy problem for the total polynomial PDE system, obtain the recurrence formulas for Taylor coefficients, and then formulate and prove a theorem on the accuracy of its solutions by TSM.

Explicit formulas of alternating multiple zeta star values $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $

<abstract><p>In a recent paper ^{[<xref ref-type="bibr" rid="b4">4</xref>]}, Xu studied some alternating multiple zeta values. In particular, he gave two recurrence formulas of alternating multiple zeta values $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $. In this paper, we will give the closed forms representations of $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $ in terms of single zeta values and polylogarithms.</p></abstract>

Recursion formulas for poly-Bernoulli numbers and their applications

Recurrence formulas for generalized poly-Bernoulli polynomials are given. The formula gives a positive answer to a question raised by Kaneko. Further, as applications, annihilation formulas for Arakawa-Kaneko type zeta-functions and a counting formula for lonesum matrices of a certain type are also discussed.

Image Reconstruction Based on Novel Sets of Generalized Orthogonal Moments

In this work, we have presented a general framework for reconstruction of intensity images based on new sets of Generalized Fractional order of Chebyshev orthogonal Moments (GFCMs), a novel set of Fractional order orthogonal Laguerre Moments (FLMs) and Generalized Fractional order orthogonal Laguerre Moments (GFLMs). The fractional and generalized recurrence relations of fractional order Chebyshev functions are defined. The fractional and generalized fractional order Laguerre recurrence formulas are given. The new presented generalized fractional order moments are tested with the existing orthogonal moments classical Chebyshev moments, Laguerre moments, and Fractional order Chebyshev Moments (FCMs). The numerical results show that the importance of our general framework which gives a very comprehensive study on intensity image representation based GFCMs, FLMs, and GFLMs. In addition, the fractional parameters give a flexibility of studying global features of images at different positions and scales of the given moments.

SOME RECURRENCE FORMULAS FOR A NEW CLASS OF SPECIAL POLYNOMIALS AND SPECIAL FUNCTIONS

In this paper we used a new class of special functions and special polynomials which are solutions different Sturm Liouvile differential equations of second order. These functions form a basis of a space of square integrable functions over set of a real numbers. We investigated some properties of these polynomials and established some recurrence formulas. Using a new class of special functions, we obtained some useful summation formulas and recurrence formulas.