A New Development of the Classical Single Ladder Problem via Converting the Ladder to a Staircase

Our purpose is to shed some new light on problems arising from a study of the classical Single Ladder Problem (SLP). The basic idea is to convert the SLP to a corresponding Single Staircase Problem. The main result (Theorem 1) shows that this idea works fine and new results can be obtained by just calculating rational solutions of an algebraic equation. Some examples of such concrete calculations are given and examples of new results are also given. In particular, we derive a number of integer SLPs with congruent ladders, where a set of rectangular boxes with integer sides constitutes a staircase along a common ladder. Finally, the case with a regular staircase along a given ladder is investigated and illustrated with concrete examples.

A note on the paper "Optimality conditions for nonsmooth interval-valued and multiobjective semi-infinite programming"

A nonsmooth semi-infinite interval-valued vector programming problem is solved in the paper by Jennane et all. (RAIRO-Oper. Res. doi: 10.1051/ro/2020066, 2020). The necessary optimality condition obtained by the authors, as well as its proof, is false. Some counterexamples are given to refute some results on which the main result (Theorem 4.5) is based. For the convinience of the reader, we correct the faulty in those results, propose a correct formulation of Theorem 4.5 and give also a short proof.

On a Periodic Boundary Value Problem for a Fractional–Order Semilinear Functional Differential Inclusions in a Banach Space

We consider the periodic boundary value problem (PBVP) for a semilinear fractional-order delayed functional differential inclusion in a Banach space. We introduce and study a multivalued integral operator whose fixed points coincide with mild solutions of our problem. On that base, we prove the main existence result (Theorem ). We present an example dealing with existence of a trajectory for a time-fractional diffusion type feedback control system with a delay satisfying periodic boundary value condition.

Space of chord diagrams on spherical curves

In this paper, we give a definition of [Formula: see text]-valued functions from the ambient isotopy classes of spherical/plane curves derived from chord diagrams, denoted by [Formula: see text]. Then, we introduce certain elements of the free [Formula: see text]-module generated by the chord diagrams with at most [Formula: see text] chords, called relators of Type (I) ((SI[Formula: see text]I), (WI[Formula: see text]I), (SI[Formula: see text]I[Formula: see text]I), or (WI[Formula: see text]I[Formula: see text]I), respectively), and introduce another function [Formula: see text] derived from [Formula: see text]. The main result (Theorem 1) shows that if [Formula: see text] vanishes for the relators of Type (I) ((SI[Formula: see text]I), (WI[Formula: see text]I), (SI[Formula: see text]I[Formula: see text]I), or (WI[Formula: see text]I[Formula: see text]I), respectively), then [Formula: see text] is invariant under the Reidemeister move of type RI (strong RI[Formula: see text]I, weak RI[Formula: see text]I, strong RI[Formula: see text]I[Formula: see text]I, or weak RI[Formula: see text]I[Formula: see text]I, respectively) that is defined in [N. Ito and Y. Takimura, [Formula: see text] and weak [Formula: see text] homotopies on knot projections, J. Knot Theory Ramifications 22 (2013) 1350085 14 pp].

Derivations of polynomial semirings

The aim of this paper is the investigation of derivations in semiring of polynomials over idempotent semiring. For semiring [Formula: see text], where [Formula: see text] is a commutative idempotent semiring we construct derivations corresponding to the polynomials from the principal ideal [Formula: see text] and prove that the set of these derivations is a non-commutative idempotent semiring closed under the Jordan product of derivations — Theorem 3.3. We introduce generalized inner derivations defined as derivations acting only over the coefficients of the polynomial and consider [Formula: see text]-derivations in classical sense of Jacobson. In the main result, Theorem 5.3, we show that any derivation in [Formula: see text] can be represented as a sum of a generalized inner derivation and an [Formula: see text]-derivation.

Analysis of a sliding frictional contact problem with unilateral constraint

We consider a mathematical model that describes the equilibrium of an elastic body in frictional contact with a moving foundation. The contact is modeled with a multivalued normal compliance condition with unilateral constraint, associated with a sliding version of Coulomb’s law of dry friction. We present a description of the model, list the assumptions on the data and derive its primal variational formulation, in terms of displacement. Then we prove an existence and uniqueness result, Theorem 3.1. We proceed with a penalization method in the study of the contact problem for which we present a convergence result, Theorem 4.1. Finally, under additional hypotheses, we consider a variational formulation of the problem in terms of the stress, the so-called dual variational formulation, and prove an equivalence result, Theorem 5.3. The proofs of the theorems are based on arguments of monotonicity, compactness, convexity and lower semicontinuity.

‘POSITIVELY HOMOGENEOUS LATTICE HOMOMORPHISMS BETWEEN RIESZ SPACES NEED NOT BE LINEAR’

This note furnishes an example showing that the main result (Theorem 4) in Toumi [‘When lattice homomorphisms of Archimedean vector lattices are Riesz homomorphisms’, J. Aust. Math. Soc. 87 (2009), 263–273] is false.

Finite groups with given systems of weakly S-propermutable subgroups

A subgroupOur main goal here is to prove the following result (Theorem 1.2):This theorem generalizes many known results and also gives the positive answer to Question 18.91 (b) in Kourovka Notebook [

The estimation of solution of Fredholm integral equation in three variables

The aim of the present paper is to establish an existence result (Theorem 2.1) as well as an estimation result (Theorem 2.2) for a functional equation in three variables.

Nonlinear Problems with p(·)-Growth Conditions and Applications to Antiplane Contact Models

We consider a general boundary value problem involving operators of the form div(a(·, ∇u(·)) in which a is a Carathéodory function satisfying a p(·)-growth condition. We are interested on the weak solvability of the problem and, to this end, we start by introducing the Lebesgue and Sobolev spaces with variable exponent, together with their main properties. Then, we state and prove our main existence and uniqueness result, Theorem 3.1. The proof is based on a Weierstrass-type argument. We also consider two antiplane contact problems for nonhomogenous elastic materials of Hencky-type. The contact is frictional and it is modelled with a regularized version of Tresca’s friction law and a power-law friction, respectively. We prove that the problems cast in the abstract setting, then we use Theorem 3.1 to deduce their unique weak solvability.