Search

The classical result of L. Székelyhidi states that (under some assumptions) every solution of a general linear equation must be a polynomial function. It is known that Székelyhidi's result may be generalized to equations where some occurrences of the unknown functions are multiplied by a linear combination of the variables. In this paper we study the equations where two such combinations appear. The simplest nontrivial example of such a case is given by the equation F (x + y) - F (x) - F (y) = y f (x) + x f (y) considered by Fechner and Gselmann (Publ Math Debrecen 80(1–2):143–154, 2012). In the present paper we prove several results concerning the systematic approach to the generalizations of this equation. [ABSTRACT FROM AUTHOR]

The isospectral deformations of the Frobenius–Stickelberger–Thiele (FST) polynomials introduced in Spiridonov et al. (Commun Math Phys 272:139–165, 2007) are studied. For a specific choice of the deformation of the spectral measure, one is led to an integrable lattice (FST lattice), which is indeed an isospectral flow connected with a generalized eigenvalue problem. In the second part of the paper the spectral problem used previously in the study of the modified Camassa–Holm (mCH) peakon lattice is interpreted in terms of the FST polynomials together with the associated FST polynomials, resulting in a map from the mCH peakon lattice to a negative flow of the finite FST lattice. Furthermore, it is pointed out that the degenerate case of the finite FST lattice unexpectedly maps to the interlacing peakon ODE system associated with the two-component mCH equation studied in Chang et al. (Adv Math 299:1–35, 2016). [ABSTRACT FROM AUTHOR]

In this paper, we consider a system of two viscoelastic equations with Dirichlet boundary conditions. For certain class of relaxation functions and initial data, we establish general and optimal decay results. This result extends earlier one of Liu (Nonlinear Anal. TMA 71:2257–2267, 2010), in which only the usual exponential and polynomial decay rates are considered. The conditions of the relaxation functions g 1 (t) and g 2 (t) in our work appeared first in Messaoudi and Khulaifi (Appl. Math. Lett. 66:16–22, 2017). [ABSTRACT FROM AUTHOR]

Resultants are important special functions used to describe nonlinear phenomena. The resultant $$R_{r_1 \ldots r_n }$$ determines a consistency condition for a system of n homogeneous polynomials of degrees r, ..., r in n variables in precisely the same way as the determinant does for a system of linear equations. Unfortunately, there is a lack of convenient formulas for resultants in the case of a large number of variables. In this paper we use Cauchy contour integrals to obtain a polynomial formula for resultants, which is expected to be useful in applications. [ABSTRACT FROM AUTHOR]

In this paper we consider the following viscoelastic equation in a bounded domain, and establish a general decay result which depends on the behavior of α and g. This work generalizes and improves earlier results in the literature. [ABSTRACT FROM AUTHOR]

A new view on the Kowalevski top and the Kowalevski integration procedure is presented. For more than a century, the Kowalevski 1889 case, has attracted full attention of a wide community as the highlight of the classical theory of integrable systems. Despite hundreds of papers on the subject, the Kowalevski integration is still understood as a magic recipe, an unbelievable sequence of skillful tricks, unexpected identities and smart changes of variables. The novelty of our present approach is based on our four observations. The first one is that the so-called fundamental Kowalevski equation is an instance of a pencil equation of the theory of conics which leads us to a new geometric interpretation of the Kowalevski variables w, x1, x2 as the pencil parameter and the Darboux coordinates, respectively. The second is observation of the key algebraic property of the pencil equation which is followed by introduction and study of a new class of discriminantly separable polynomials. All steps of the Kowalevski integration procedure are now derived as easy and transparent logical consequences of our theory of discriminantly separable polynomials. The third observation connects the Kowalevski integration and the pencil equation with the theory of multi-valued groups. The Kowalevski change of variables is now recognized as an example of a two-valued group operation and its action. The final observation is surprising equivalence of the associativity of the two-valued group operation and its action to the n = 3 case of the Great Poncelet Theorem for pencils of conics. [ABSTRACT FROM AUTHOR]

Formulas of n variables of Łukasiewicz sentential calculus can be represented, via McNaughton’s theorem, by piecewise linear functions, with integer coefficients, from hypercube [0,1]n to [0,1], called McNaughton functions. As a consequence of the McNaughton representation of a formula it is obtained a canonical form of a formula. Indeed, up to logical equivalence, any formula can be written as an infimum of finite suprema of formulas associated to McNaughton functions which are truncated functions to $[0,1]$ of the restriction to [0,1]n of single hyperplanes, for short, called simple McNaughton functions. In the present paper we will concern with the problem of presenting formulas of Lukasiewicz sentential calculus in normal form. Here we list the main results we obtained: a) we give an axiomatic description of some classes of formulas having the property to be canonically mapped one-to-one onto the class of simple Mc Naughton functions; b) we provide normal forms for Lukasiewicz sentential calculus, making use of formulas defined in a); c) we prove the polynomial complexity of formulas, in normal form, coming from a certain class described as in a); d) we extend the results described in a), b) and c) to Rational Lukasiewicz logic. [ABSTRACT FROM AUTHOR]

Earlier, D. Yu. Grigoriev and N. N. Vorob'yov obtained an upper bound $d^\{O(\{n+d\; \backslash choose\; n\})\}$ for the number of vectors of multiplicities for solutions of systems of the form $g\_1=\; \backslash cdots\; =\; g\_n=0$ (assuming that the system has a finite number of solutions). In the system above, $g\_1,\; \backslash dots\; ,\; g\_n\; \backslash in\; F[X\_1,\; \backslash dots\; ,\; X\_n]$ are polynomials of degrees $\backslash deg\; (g\_i)\; \backslash leq\; d$. In the present paper, we show that the order of growth of this bound is close to the exact one. In particular, in the case $d=n$, the construction provides a doubly-exponential (in n) number of vectors of multiplicities. Bibliography: 4 titles. [ABSTRACT FROM AUTHOR]