Mislin’s theorem for fusion systems through Mackey functors

We state and prove a fusion system version of Mislin’s theorem [9 Mislin, G. On group homomorphisms inducing mod-p cohomology isomorphisms. Comment. Math. Helv. 65(3):454–461.[Web of Science ®] , [Google Scholar]] on cohomology and control of fusion using Mackey functors. The issue of an algebraic proof is also discussed.     

Alperin’s fusion theorem for localities

In these notes we give a version of the Alperin–Goldschmidt fusion theorem for localities.     

On Quillen?s Theorem A for posets

A theorem of McCord of 1966 and Quillen?s Theorem A of 1973 provide sufficient conditions for a map between two posets to be a homotopy equivalence at the level of complexes. We give an alternative elementary proof of this result and we deduce also a stronger statement: under the hypotheses of the theorem, the map is not only a homotopy equivalence but a simple homotopy equivalence. This leads then to stronger formulations of the simplicial version of Quillen?s Theorem A, the Nerve Lemma and other known results. In particular we establish a conjecture of Kozlov on the simple homotopy type of the crosscut complex and we improve a well-known result of Cohen on contractible mappings.     

Jacobi’s bound for systems of algebraic differential equations

This review paper is devoted to the Jacobi bound for systems of partial differential polynomials. We prove the conjecture for the system of n partial differential equations in n differential variables which are independent over a prime differential ideal \mathfrakp\mathfrak{p}. On the one hand, this generalizes our result about the Jacobi bound for ordinary differential polynomials independent over a prime differential ideal \mathfrakp\mathfrak{p} and, on the other hand, the result by Tomasovic, who proved the Jacobi bound for linear partial differential polynomials.     

Pontryagin’s maximum principle for dynamic systems on time scales

In this work, an analogue of Pontryagin’s maximum principle for dynamic equations on time scales is given, combining the continuous and the discrete Pontryagin maximum principles and extending them to other cases ‘in between’. We generalize known results to the case when a certain set of admissible values of the control is not necessarily closed (but convex) and the attainable set is not necessarily convex. At the same time, we impose an additional condition on the graininess of the time scale. For linear systems, sufficient conditions in the form of the maximum principle are obtained.     

He’s homotopy perturbation method for systems of integro-differential equations

In this article, the homotopy perturbation method [He JH. Homotopy perturbation technique. Comput Meth Appl Mech Eng 1999;178:257–62; He JH. A coupling method of homotopy technique and perturbation technique for nonlinear problems. Int J Non-Linear Mech 2000;35(1):37–43; He JH. Comparison of homotopy perturbation method and homotopy analysis method. Appl Math Comput 2004;156:527–39; He JH. Homotopy perturbation method: a new nonlinear analytical technique. Appl Math Comput 2003;135:73–79; He JH. The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl Math Comput 2004;151:287–92; He JH. Application of homotopy perturbation method to nonlinear wave equations Chaos, Solitons & Fractals 2005;26:695–700] is applied to solve linear and nonlinear systems of integro-differential equations. Some nonlinear examples are presented to illustrate the ability of the method for such system. Examples for linear system are so easy that has been ignored.     

NP-complete problems for systems of Linear polynomial’s values divisibilities

The paper studies the algorithmic complexity of subproblems for satisfiability in positive integers of simultaneous divisibility of linear polynomials with nonnegative coefficients. In the general case, it is not known whether this problem is in the class NP, but that it is in NEXPTIME is known. The NP-completeness of two series of restricted versions of this problem such that a divisor of a linear polynomial is a number in the first case, and a linear polynomial is a divisor of a number in the second case is proved in the paper. The parameters providing the NP-completeness of these problems have been established. Their membership in the class P has been proven for smaller values of these parameters. For the general problem SIMULTANEOUS DIVISIBILITY OF LINEAR POLYNOMIALS, NP-hardness has been proven for its particular case, when the coefficients of the polynomials are only from the set {1, 2} and constant terms are only from the set {1, 5}.     

On Baxter’s difference systems

We study the asymptotics of solutions of a difference system introduced by Baxter by using the general method for the asymptotic representation of such solutions due to Benzaid and Lutz. Some results of Tauberian type are obtained in the case when the spectral parameter belongs to the unit circle.     

An approximation algorithm for quadratic dynamic systems based on N. Chomsky’s grammar for Taylor’s formula

Single-step methods for the approximate solution of the Cauchy problem for dynamic systems are discussed. It is shown that a numerical integration algorithm with a high degree of accuracy based on Taylor’s formula can be proposed in the case of quadratic systems. An explicit estimate is given for the remainder. The algorithm is based on N. Chomsky’s generative grammar for the language of terms of Taylor’s formula.     

Addendum to Pontryagin’s maximum principle for dynamic systems on time scales

This note is an addendum to [L. Bourdin and E. Trélat, SIAM J. Cont. Optim., 2013] and [M. Bohner, K. Kenzhebaev, O. Lavrova and O. Stanzhytskyi, J. Differ. Equ. Appl., 2017], pointing out the differences between these papers and raising open questions.     

Error bounds for infinite systems of convex inequalities without Slater’s condition

The feasible set of a convex semi–infinite program is described by a possibly infinite system of convex inequality constraints. We want to obtain an upper bound for the distance of a given point from this set in terms of a constant multiplied by the value of the maximally violated constraint function in this point. Apart from this Lipschitz case we also consider error bounds of H?lder type, where the value of the residual of the constraints is raised to a certain power.?We give sufficient conditions for the validity of such bounds. Our conditions do not require that the Slater condition is valid. For the definition of our conditions, we consider the projections on enlarged sets corresponding to relaxed constraints. We present a condition in terms of projection multipliers, a condition in terms of Slater points and a condition in terms of descent directions. For the Lipschitz case, we give five equivalent characterizations of the validity of a global error bound.?We extend previous results in two directions: First, we consider infinite systems of inequalities instead of finite systems. The second point is that we do not assume that the Slater condition holds which has been required in almost all earlier papers. Received: April 12, 1999 / Accepted: April 5, 2000?Published online July 20, 2000     

Residue integrals and Waring’s formulas for algebraic or even transcendental systems

The present article is focused on the study of a special class of systems of non-linear transcendental equations for which classical algebraic and symbolic methods are inapplicable. For the purpose of study of such systems we develop a method for computing residue integrals with integration over certain cycles. We describe conditions under which the mentioned residue integrals coincide with power sums of the inverses to the roots of a system of equations (i.e. multidimensional Waring’s formulas). Based on this, we develop an algorithm that computes such power sums without computing the roots. As an application of the suggested method, we consider a problem of finding sums of multi-variable number series.     

Simplicial complexes and Macaulay’s inverse systems

Let Δ be a simplicial complex on V = {x ₁, . . . , x _n }, with Stanley–Reisner ideal ${I_{\Delta}\subseteq R=k[x_1,\ldots, x_n]}Let Δ be a simplicial complex on V = {x ₁, . . . , x _n }, with Stanley–Reisner ideal I_D í R=k[x₁,?, x_n]{I_{\Delta}\subseteq R=k[x_1,\ldots, x_n]} . The goal of this paper is to investigate the class of artinian algebras A=A(D,a₁,?,a_n) = R/(I_D,x₁^a₁,?,x_n^a_n){A=A(\Delta,a_1,\ldots,a_n)= R/(I_{\Delta},x_1^{a_1},\ldots,x_n^{a_n})} , where each a _i ≥ 2. By utilizing the technique of Macaulay’s inverse systems, we can explicitly describe the socle of A in terms of Δ. As a consequence, we determine the simplicial complexes, that we will call levelable, for which there exists a tuple (a ₁, . . . , a _n ) such that A(Δ, a ₁, . . . , a _n ) is a level algebra.     

Green’s function for elliptic systems: existence and Delmotte–Deuschel bounds

This paper is divided into two parts: In the main deterministic part, we prove that for an open domain \(D \subset \mathbb {R}^d\) with \(d \ge 2\), for every (measurable) uniformly elliptic tensor field a and for almost every point \(y \in D\), there exists a unique Green’s function centred in y associated to the vectorial operator \(-\nabla \cdot a\nabla \) in D. This result implies the existence of the fundamental solution for elliptic systems when \(d>2\), i.e. the Green function for \(-\nabla \cdot a\nabla \) in \(\mathbb {R}^d\). In the second part, we introduce a shift-invariant ensemble \(\langle \cdot \rangle \) over the set of uniformly elliptic tensor fields, and infer for the fundamental solution G some pointwise bounds for \(\langle |G(\cdot ; x,y)|\rangle \), \(\langle |\nabla _x G(\cdot ; x,y)|\rangle \) and \(\langle |\nabla _x\nabla _y G(\cdot ; x,y)|\rangle \). These estimates scale optimally in space and provide a generalisation to systems of the bounds obtained by Delmotte and Deuschel for the scalar case.