A NUMERICAL APPROACH TO THE CONTACT OF NOMINALLY FLAT SURFACES

Machined surfaces can be described by heights and wavelengths of the surface asperities that show a statistical variation. Considering that a regular wavy surface with a sinusoidal profile is the crudest model for a rough surface, studying the contact of regular wavy surfaces is a good approximation for the contact of nominally flat surfaces. Such contact problems exhibit periodicity that can be simulated with the aid of computational techniques derived for contact mechanics in the frequency domain. The displacement calculation, which is a necessary step in the resolution of the contact problem, is mathematically a convolution product that can be calculated in the frequency domain with increased computational efficiency. The displacement induced by a unit surface load can be expressed in the frequency domain by the frequency response functions, which are counterparts of the space domain solutions to half-space fundamental problems such as the Boussinesq problem. The displacement induced by a periodic pressure distribution can be computed by executing the convolution product between the frequency response function and pressure on a single period. It should be noted that the convolution calculation in the spectral domain implies that the contributions of all neighbouring pressure periods are accounted for. The need to treat numerically only a single period results in remarkable computational efficiency, allowing for high density meshes that can capture the essential features of any textured real surface. The displacement calculation promotes the solution of the contact problem by an iterative approach. The advanced method is benchmarked against existing analytical solutions for the 3D contact of surfaces possessing two-dimensional waviness. This essentially deterministic model, supported by a direct numerical solution that can be obtained for samples of real rough surfaces, presents itself as a worthy alternative to the existing statistical models for rough contact interaction.

Semi-Hyers–Ulam–Rassias Stability of a Volterra Integro-Differential Equation of Order I with a Convolution Type Kernel via Laplace Transform

In this paper, we investigate the semi-Hyers–Ulam–Rassias stability of a Volterra integro-differential equation of order I with a convolution type kernel. To this purpose the Laplace transform is used. The results obtained show that the stability holds for problems formulated with various functions: exponential and polynomial functions. An important aspect that appears in the form of the studied equation is the symmetry of the convolution product.

COMMUTATIVE NEUTRIX CONVOLUTION PRODUCT OF GENERALIZED FRESNEL COSINE INTEGRALS AND APPLICATIONS

The generalized Fresnel cosine integral $C_k(x)$ and its associated functions $C_{k+}(x)$ and $C_{k-}(x)$ are defined as locally summable functions on the real line. The generalized Fresnel cosine integrals have huge applications in physics, specially in optics and electromaghetics. In many diffraction problems the generalized Fresnel integrals plays an important role. In this paper are calculated the commutative neutrix convolutions of the generalized Fresnel cosine integral and its associated functions with $x^r, r=0,1,2,\dots$.

Certain Integral Operators of Analytic Functions

In this paper, two new integral operators are defined using the operator DRλm,n, introduced and studied in previously published papers, defined by the convolution product of the generalized Sălăgean operator and Ruscheweyh operator. The newly defined operators are used for introducing several new classes of functions, and properties of the integral operators on these classes are investigated. Subordination results for the differential operator DRλm,n are also obtained.

On the homological and algebraical properties of some Feichtinger algebras

Let G be a locally compact group (not necessarily abelian) and B be a homogeneous Banach space on G, which is in a good situation with respect to a homogeneous function algebra on G. Feichtinger showed that there exists a minimal Banach space B min in the family of all homogenous Banach spaces C on G, containing all elements of B with compact support. In this paper, the amenability and super amenability of B min with respect to the convolution product or with respect to the pointwise product are showed to correspond to amenability, discreteness or finiteness of the group G and conversely. We also prove among other things that B min is a symmetric Segal subalgebra of L 1(G) on an IN-group G, under certain conditions, and we apply our results to study pseudo-amenability and some other homological properties of B min on IN-groups. Furthermore, we determine necessary and sufficient conditions on A under which A min $\mathcal{A}_{\min}$ with the pointwise product is an abstract Segal algebra or Segal algebra in A, whenever A is a homogeneous function algebra with an approximate identity. We apply these results to study amenability of some Feichtinger algebras with respect to the pointwise product.

Central elements in affine mod p Hecke algebras via perverse -sheaves

Let $G$ be a split connected reductive group over a finite field of characteristic $p > 2$ such that $G_\text {der}$ is absolutely almost simple. We give a geometric construction of perverse $\mathbb {F}_p$ -sheaves on the Iwahori affine flag variety of $G$ which are central with respect to the convolution product. We deduce an explicit formula for an isomorphism from the spherical mod $p$ Hecke algebra to the center of the Iwahori mod $p$ Hecke algebra. We also give a formula for the central integral Bernstein elements in the Iwahori mod $p$ Hecke algebra. To accomplish these goals we construct a nearby cycles functor for perverse $\mathbb {F}_p$ -sheaves and we use Frobenius splitting techniques to prove some properties of this functor. We also prove that certain equal characteristic analogues of local models of Shimura varieties are strongly $F$ -regular, and hence they are $F$ -rational and have pseudo-rational singularities.

The Laguerre transform of a convolution product of vector-valued functions.

The Laguerre transform is applied to the convolution product of functions of a real argument (over the time axis) with values in Hilbert spaces. The main results have been obtained by establishing a relationship between the Laguerre and Laplace transforms over the time variable with respect to the elements of Lebesgue weight spaces. This relationship is built using a special generating function. The obtained dependence makes it possible to extend the known properties of the Laplace transform to the case of the Laguerre transform. In particular, this approach concerns the transform of a convolution of functions. The Laguerre transform is determined by a system of Laguerre functions, which forms an orthonormal basis in the weighted Lebesgue space. The inverse Laguerre transform is constructed as a Laguerre series. It is proven that the direct and the inverse Laguerre transforms are mutually inverse operators that implement an isomorphism of square-integrable functions and infinite squares-summable sequences. The concept of a q-convolution in spaces of sequences is introduced as a discrete analogue of the convolution products of functions. Sufficient conditions for the existence of convolutions in the weighted Lebesgue spaces and in the corresponding spaces of sequences are investigated. For this purpose, analogues of Young’s inequality for such spaces are proven. The obtained results can be used to construct solutions of evolutionary problems and time-dependent boundary integral equations.

Approximation by interpolation spectral subspaces of operators with discrete spectrum

The Laguerre transform is applied to the convolution product of functions of a real argument (over the time axis) with values in Hilbert spaces. The main results have been obtained by establishing a relationship between the Laguerre and Laplace transforms over the time variable with respect to the elements of Lebesgue weight spaces. This relationship is built using a special generating function. The obtained dependence makes it possible to extend the known properties of the Laplace transform to the case of the Laguerre transform. In particular, this approach concerns the transform of a convolution of functions. The Laguerre transform is determined by a system of Laguerre functions, which forms an orthonormal basis in the weighted Lebesgue space. The inverse Laguerre transform is constructed as a Laguerre series. It is proven that the direct and the inverse Laguerre transforms are mutually inverse operators that implement an isomorphism of square-integrable functions and infinite squares-summable sequences. The concept of a q-convolution in spaces of sequences is introduced as a discrete analogue of the convolution products of functions. Sufficient conditions for the existence of convolutions in the weighted Lebesgue spaces and in the corresponding spaces of sequences are investigated. For this purpose, analogues of Young’s inequality for such spaces are proven. The obtained results can be used to construct solutions of evolutionary problems and time-dependent boundary integral equations.

A New Support Vector Machine Based on Convolution Product

The support vector machine (SVM) and deep learning (e.g., convolutional neural networks (CNNs)) are the two most famous algorithms in small and big data, respectively. Nonetheless, smaller datasets may be very important, costly, and not easy to obtain in a short time. This paper proposes a novel convolutional SVM (CSVM) that has the advantages of both CNN and SVM to improve the accuracy and effectiveness of mining smaller datasets. The proposed CSVM adapts the convolution product from CNN to learn new information hidden deeply in the datasets. In addition, it uses a modified simplified swarm optimization (SSO) to help train the CSVM to update classifiers, and then the traditional SVM is implemented as the fitness for the SSO to estimate the accuracy. To evaluate the performance of the proposed CSVM, experiments were conducted to test five well-known benchmark databases for the classification problem. Numerical experiments compared favorably with those obtained using SVM, 3-layer artificial NN (ANN), and 4-layer ANN. The results of these experiments verify that the proposed CSVM with the proposed SSO can effectively increase classification accuracy.

The heat semigroup and equation related to a Bessel-type operators and the canonical Fourier Bessel transform

In this paper we study a translation operator associated with the canonical Fourier Bessel transform $\mathcal{F}_{\nu}^{\mathbf{m}}.$ We then use it to derive a convolution product and study some of its important properties. As a direct application, we introduce the heat semigroup generated by the Bessel-type operators $$\Delta_{\nu}^{\mathbf{m}^{-1}}=\frac{d^{2}}{dx^{2}}+\left( \frac{2\nu +1}{x}+2i \frac{a}{b} x\right) \frac{d}{dx}-\left( \frac{a^{2}}{b^{2}}x^{2}-2i\left( \nu +1\right) \frac{a}{b}\right) $$ and use it to solve the initial value problem for the heat equation governed by $\Delta_{\nu}^{\mathbf{m}^{-1}}.$