The Aluffi algebra of hypersurfaces with isolated singularities

The Alu? algebra is an algebraic definition of a characteristic cycle of a hypersurface in intersection theory. In this paper, we study the Alu? algebra of quasi-homogeneous and locally Eulerian hypersurfaces with only isolated singularities. We prove that the Jacobian ideal of an a?ne hypersurface with isolated singularities is of linear type if and only if it is locally Eulerian. We show that the gradient ideal of a projective hypersurface with only isolated singularities is of linear type if and only if the a?ne curve in each a?ne chart associated to singular points is locally Eulerian. We show that the gradient ideal of Nodal and Cuspidal projective plane curves are of linear type.     

Dynamical behavior and solution of nonlinear difference equation via Fibonacci sequence

In this paper, we study the behavior of the difference equation $x_{n+1}=ax_{n}+\dfrac{bx_{n}x_{n-1}}{cx_{n-1}+dx_{n-2}},~n=0,1,\ldots,$ where the initial conditions $x_{-2},\ x_{-1},\ x_{0}$ are arbitrary positive real numbers and $a,b,c,d$ are positive constants. Also, we give the solution of some special cases of this equation.     

Iterative methods for solving proximal split minimization problems

In this paper, we propose two iterative algorithms for finding the minimum-norm solution of a split minimization problem. We prove strong convergence of the sequences generated by the proposed algorithms. The iterative schemes are proposed in such a way that the selection of the step-sizes does not need any prior information about the operator norm. We further give some examples to numerically verify the efficiency and implementation of our new methods and compare the two algorithms presented. Our results act as supplements to several recent important results in this area.     

A hybrid proximal point algorithm for finding minimizers and fixed points in CAT(0) spaces

We introduce a hybrid proximal point algorithm and establish its strong convergence to a common solution of a proximal point of a lower semi-continuous mapping and a fixed point of a demicontractive mapping in the framework of a CAT(0) space. As applications of our new result, we solve variational inequality problems for these mappings on a Hilbert space. Illustrative example is given to validate theoretical result obtained herein.     

On shadowing: Ordinary and ergodic

The aim of this paper is to introduce the notion of ergodic shadowing for a continuous onto map which is equivalent to the map being topologically mixing and has the ordinary shadowing property. In particular, we deduce the chaotic behavior of a map with ergodic shadowing property. Moreover, we define some kind of specification property and investigate its relation to the ergodic shadowing property.     

On the completeness of normed spaces

Let $(G,+)$ be an abelian group and let $E$ be a normed space. A mapping $f:G\to E$ is called $\epsilon$-quadratic if for a given $\epsilon >0$ it satisfies $6f(x+y)+f(x?y)?2f\left(x\right)?2f\left(y\right)6\le \epsilon$ for all $x,y\in G$. In this work, we show that $E$ is complete if every $\epsilon$-quadratic mapping $f:X\to E$ can be estimated by a quadratic mapping, where $X$ is ${\mathbb{N}}_0$ or a finitely generated free abelian group.     

A differential operator and weak topology for Lipschitz maps

We show that the Scott topology induces a topology for real-valued Lipschitz maps on Banach spaces which we call the L-topology. It is the weakest topology with respect to which the L-derivative operator, as a second order functional which maps the space of Lipschitz functions into the function space of non-empty weak^∗ compact and convex valued maps equipped with the Scott topology, is continuous. For finite dimensional Euclidean spaces, where the L-derivative and the Clarke gradient coincide, we provide a simple characterization of the basic open subsets of the L-topology. We use this to verify that the L-topology is strictly coarser than the well-known Lipschitz norm topology. A complete metric on Lipschitz maps is constructed that is induced by the Hausdorff distance, providing a topology that is strictly finer than the L-topology but strictly coarser than the Lipschitz norm topology. We then develop a fundamental theorem of calculus of second order in finite dimensions showing that the continuous integral operator from the continuous Scott domain of non-empty convex and compact valued functions to the continuous Scott domain of ties is inverse to the continuous operator induced by the L-derivative. We finally show that in dimension one the L-derivative operator is a computable functional.     

Discrimination of acoustic and turbulent components from aeroacoustic wall pressure field

The investigation of the wall pressure excitations over transportation vehicle panel is of great interest to improve the knowledge of vehicle interior noise transmission and also for future noise reduction strategies. A particularly useful task concerns the characterization and the separation of both acoustic and turbulent components of the wall pressure excitation. A new application of the Proper Orthogonal Decomposition (POD) is tested from two different databases: (i) wall pressure fields synthesized from theoretical average models and (ii) wall pressure fields obtained from Lattice Boltzmann Method numerical simulation. In each case, POD application leads to an energetic partitioning of the wall pressure field that permits to well decouple both acoustic and turbulent fields, especially for mid and high frequencies under interest. To validate such separation and to demonstrate the effectiveness of the POD method, the wavenumber spectrum analysis as well as phase analysis is successively performed. Such a new splitting method provides an instantaneous acoustic–turbulent separation of an inhomogeneous wall pressure field, suggesting many useful future applications.     

On necessary and sufficient conditions for near-optimal singular stochastic controls

In this paper we discuss the necessary and sufficient conditions for near-optimal singular stochastic controls for the systems driven by a nonlinear stochastic differential equations (SDEs in short). The proof of our result is based on Ekeland’s variational principle and some delicate estimates of the state and adjoint processes. It is well known that optimal singular controls may fail to exist even in simple cases. This justifies the use of near-optimal singular controls, which exist under minimal conditions and are sufficient in most practical cases. Moreover, since there are many near-optimal singular controls, it is possible to choose suitable ones, that are convenient for implementation. This result is a generalization of Zhou’s stochastic maximum principle for near-optimality to singular control problem.     

A GN model for thermoelastic interaction in an unbounded fiber-reinforced anisotropic medium with a circular hole

In this work, we have constructed the equations for generalized thermoelasticity of an unbounded fiber-reinforced anisotropic medium with a circular hole. The formulation is applied in the context of Green and Naghdi (GN) theory. The thermoelastic interactions are caused by (I) a uniform step in stress applied to the boundary of the hole with zero temperature change and (II) a uniform step in temperature applied to the boundary of the hole which is stress-free. The solutions for displacement, temperature and stresses are obtained with the help of the finite element procedure. The effects of the reinforcement on temperature, stress and displacement are studied. Results obtained in this work can be used for designing various fiber-reinforced anisotropic elements under mechanical or thermal load to meet special engineering requirements.