Sufficient Conditions In Order Research Articles

On the one-dimensional Pompeiu problem

AbstractWe investigate the Pompeiu property for subsets of the real line, under no assumption of connectedness. In particular we focus our study on finite unions of bounded (disjoint) intervals, and we emphasize the different results corresponding to the cases where the function in question is supposed to have constant integral on all isometric images, or just on all the translation-images of the domain. While no set of the previous kind enjoys the Pompeiu property in the latter sense, we provide a necessary and sufficient condition in order a union of two intervals to have the Pompeiu property in the former sense, and we produce some examples to give an insight of the complexity of the problem for three-interval sets.

New second order sufficient optimality conditions for state constrained parabolic control problems

We study a control problem governed by a semilinear parabolic equation with pointwise control and state constraints imposed at every point of the space-time cylinder. We obtain second order sufficient optimality conditions for local optimality in the sense of L 2 ( Q ) . Our results are valid for spatial domains of dimension less than or equal to three.

On the Second Order Sufficient Optimality Conditions for a Problem of Mathematical Programming

On the Second Order Sufficient Optimality Conditions for a Problem of Mathematical Programming

Factorization in group algebras

Let [Formula: see text] be a field, and let [Formula: see text] be a torsion-free abelian group. In this paper, we study some factorization properties in the group algebra of [Formula: see text] over [Formula: see text], [Formula: see text]. Among other results, we obtain necessary and sufficient conditions in order that [Formula: see text] should be an idf-domain. As an application, we get various results concerning factorization in [Formula: see text].

Bilevel inverse problems in neuromorphic imaging

Event or neuromorphic cameras are novel biologically inspired sensors that record data based on the change in light intensity at each pixel asynchronously. They have a temporal resolution of microseconds. This is useful for scenes with fast moving objects that can cause motion blur in traditional cameras, which record the average light intensity over an exposure time for each pixel synchronously. This paper presents a bilevel inverse problem framework for neuromorphic imaging. Existence of solution to the inverse problem is established. Second order sufficient conditions are derived under special situations for this nonconvex problem. A second order Newton type solver is derived to solve the problem. The efficacy of the approach is shown on several examples.

Fractional Lotka–Volterra model with Holling type III functional response

In this paper, we study a generalization of a Lotka–Volterra system with Holling type III functional response, where the Caputo fractional derivative is considered. Applying a multistage homotopy perturbation method, we obtain an analytical solution for the system. Moreover, analyzing the eigenvalues of the Jacobian matrix around the equilibria, we find sufficient conditions in order to guarantee the local stability and we present several examples to illustrate the behavior of solutions.

When is (D + I, K +I) an S-Laskerian pair?

Let T be a strongly Laskerian domain containing a field K as a subring. Let I be a non-zero proper ideal of T. Let D be a subring of K. The aim of this article is to determine necessary and sufficient conditions in order that (D + I, K + I) to be an S-Laskerian pair.

Homogenization for polynomial optimization with unbounded sets

This paper considers polynomial optimization with unbounded sets. We give a homogenization formulation and propose a hierarchy of Moment-SOS relaxations to solve it. Under the assumptions that the feasible set is closed at infinity and the ideal of homogenized equality constraining polynomials is real radical, we show that this hierarchy of Moment-SOS relaxations has finite convergence, if some optimality conditions (i.e., the linear independence constraint qualification, strict complementarity and second order sufficient conditions) hold at every minimizer, including the one at infinity. Moreover, we prove extended versions of Putinar-Vasilescu type Positivstellensatz for polynomials that are nonnegative on unbounded sets. The classical Moment-SOS hierarchy with denominators is also studied. In particular, we give a positive answer to a conjecture of Mai, Lasserre and Magron in their recent work.

On a question of supports

We give a sufficient condition in order that n closed connected subsets in the n-dimensional real projective space admit a common multitangent hyperplane.

On the image set and reversibility of shift morphisms over discrete alphabets

In this paper we provide sufficient conditions in order to show that the set image of a continuous and shift-commuting map defined on a shift space over an arbitrary discrete alphabet is also a shift space; additionally, if such a map is injective, then its inverse is also continuous and shift-commuting.

Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control

In a separable Hilbert space X, we study the controlled evolution equation u′(t)+Au(t)+p(t)Bu(t)=0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u'(t)+Au(t)+p(t)Bu(t)=0, \\end{aligned}$$\\end{document}where Age -sigma I (sigma ge 0) is a self-adjoint linear operator, B is a bounded linear operator on X, and pin L^2_{loc}(0,+infty ) is a bilinear control. We give sufficient conditions in order for the above nonlinear control system to be locally controllable to the jth eigensolution for any jge 1. We also derive semi-global controllability results in large time and discuss applications to parabolic equations in low space dimension. Our method is constructive and all the constants involved in the main results can be explicitly computed.

A truncated Newton algorithm for nonconvex sparse recovery

Some second-order sufficient conditions are established for problems with ℓ1, SCAD and MCP penalties respectively and then an algorithm with active set strategy is proposed for solving ℓ1, SCAD and MCP penalties based on an approximation of the second order sufficient conditions. The active sets are estimated by an identification technique to identify accurately zero components in a neighborhood of a critical point of ℓ1, SCAD and MCP penalties. In the algorithm, a truncated Newton direction is used to update the free variables, while dk=−xk is used to update the active variables. To accelerate the convergence, a nonmonotone line search strategy is used to guarantee global convergence. Numerical results illustrate that the proposed algorithm is competitive with several well-known methods.

Factor relations between some summability methods

In the present paper, using the result of Bennett [1] on characterization of factorable matrices, we give necessary and sufficient conditions in order that Σλ_{n}x_{n} is summable |R,p_{n}|_{s} whenever Σμ_{n}x_{n} is summable |C,0|_{k}, and Σλ_{n}x_{n} is summable |C,0|_{s} whenever Σμ_{n}x_{n} is summable |R,p_{n}|_{r},. where 1

Coke advances catalytic route to bioplastics

We study a property of dynamic optimization (DO) problems (as those encountered in model predictive control and moving horizon estimation) that is known as exponential decay of sensitivity (EDS). This property indicates that the sensitivity of the solution at stage i against a data perturbation at stage j decays exponentially with |i-j|. Building upon our previous results, we show that EDS holds under uniform boundedness of the Lagrangian Hessian, a uniform second order sufficiency condition (uSOSC), and a uniform linear independence constraint qualification (uLICQ). Furthermore, we prove that uSOSC and uLICQ can be obtained under uniform controllability and observability. Hence, we have that uniform controllability and observability imply EDS. These results provide insights into how perturbations propagate along the horizon and enable the development of approximation and solution schemes. We illustrate the developments with numerical examples.

Carleson and Reverse Carleson Measures on Homogeneous Siegel Domains

In this paper we study Carleson and reverse Carleson measures for holomorphic function spaces on a homogeneous Siegel domain of Type II. We prove several necessary conditions and sufficient conditions in order for a measure \(\mu \) to be Carleson and reverse Carleson on mixed-normed weighted Bergman spaces.

A Priori Error Analysis for an Optimal Control Problem Governed by a Variational Inequality of the Second Kind

We consider an optimal control problem governed by an elliptic variational inequality of the second kind. The problem is discretized by linear finite elements for the state and a variational discrete approach for the control. Based on a quadratic growth condition we derive nearly optimal a priori error estimates. Moreover, we establish second order sufficient optimality conditions that ensure a quadratic growth condition. These conditions are rather restrictive, but allow us to construct a one-dimensional locally optimal solution with reduced regularity, which serves as an exact solution for numerical experiments.

New Sufficient Conditions to Ulam Stabilities for a Class of Higher Order Integro-Differential Equations

In this work, we present sufficient conditions in order to establish different types of Ulam stabilities for a class of higher order integro-differential equations. In particular, we consider a new kind of stability, the σ-semi-Hyers-Ulam stability, which is in some sense between the Hyers–Ulam and the Hyers–Ulam–Rassias stabilities. These new sufficient conditions result from the application of the Banach Fixed Point Theorem, and by applying a specific generalization of the Bielecki metric.

When is (D,K) anS-accr pair?

PurposeThe purpose of this article is to determine necessary and sufficient conditions in order that (D,K) to be anS-accr pair, whereDis an integral domain andKis a field which containsDas a subring andSis a multiplicatively closed subset ofD.Design/methodology/approachThe methods used are from the topic multiplicative ideal theory from commutative ring theory.FindingsLetSbe a strongly multiplicatively closed subset of an integral domainDsuch that the ring of fractions ofDwith respect toSis not a field. Then it is shown that (D,K) is anS-accr pair if and only ifKis algebraic overDand the integral closure of the ring of fractions ofDwith respect toSinKis a one-dimensional Prüfer domain. LetD,S,Kbe as above. If each intermediate domain betweenDandKsatisfiesS-strong accr*, then it is shown thatKis algebraic overDand the integral closure of the ring of fractions ofDwith respect toSis a Dedekind domain; the separable degree ofKoverFis finite andKhas finite exponent overF, whereFis the quotient field ofD.Originality/valueMotivated by the work of some researchers onS-accr, the concept ofS-strong accr* is introduced and we determine some necessary conditions in order that (D,K) to be anS-strong accr* pair. This study helps us to understand the behaviour of the rings betweenDandK.

Directional Necessary Optimality Conditions for Bilevel Programs

The bilevel program is an optimization problem in which the constraint involves solutions to a parametric optimization problem. It is well known that the value function reformulation provides an equivalent single-level optimization problem, but it results in a nonsmooth optimization problem that never satisfies the usual constraint qualification, such as the Mangasarian–Fromovitz constraint qualification (MFCQ). In this paper, we show that even the first order sufficient condition for metric subregularity (which is, in general, weaker than MFCQ) fails at each feasible point of the bilevel program. We introduce the concept of a directional calmness condition and show that, under the directional calmness condition, the directional necessary optimality condition holds. Although the directional optimality condition is, in general, sharper than the nondirectional one, the directional calmness condition is, in general, weaker than the classical calmness condition and, hence, is more likely to hold. We perform the directional sensitivity analysis of the value function and propose the directional quasi-normality as a sufficient condition for the directional calmness. An example is given to show that the directional quasi-normality condition may hold for the bilevel program.

State Error Estimates for the Numerical Approximation of Sparse Distributed Control Problems in the Absence of Tikhonov Regularization

In this paper, we analyze optimal control problems of semilinear elliptic equations, where the controls are distributed. Box constraints for the controls are imposed and the cost functional does not involve the control itself, except possibly for a non-differentiable sparsity-promoting term. Under appropriate second order sufficient optimality conditions, first we estimate the difference between the discrete and continuous optimal states. Next, under an additional assumption on the optimal adjoint state, we prove error estimates for the controls and improve the estimates for the states.