Finite Linear Combination Research Articles

Some Aspects of Identification of Dynamic Objects under Incorrect Observation Conditions

We solve the problems of numerical-analytical representation of the solution of an equation describing the dynamic object and its measurable output, as well as optimal computation of the values of continuous linear functionals (numerical characteristics) of measurable functions based on incorrect data containing not only fluctuation error but also singular interference. The method provides the maximum possible decomposition of computational procedures, does not require to carry out traditional linearization operations and the choice of initial approximations, and also is not related to the computation of spectral coefficients in finite linear combinations (with given basic functions) that describe integral curves, measurable functions, and singular interference.

A new approach to the Thomas–Fermi boundary-value problem

Given the Thomas–Fermi equation $$\sqrt{x}\varphi ''=\varphi ^{3 \over 2}$$ , this paper changes first the dependent variable by defining $$y(x) \equiv \sqrt{x \varphi (x)}$$ . The boundary conditions require that y(x) must vanish at the origin as $$\sqrt{x}$$ , whereas it has a fall-off behaviour at infinity proportional to the power $${1 \over 2}(1-\chi )$$ of the independent variable x, $$\chi $$ being a positive number. Such boundary conditions lead to a 1-parameter family of approximate solutions in the form $$\sqrt{x}$$ times a ratio of finite linear combinations of integer and half-odd powers of x. If $$\chi $$ is set equal to 3, in order to agree exactly with the asymptotic solution of Sommerfeld, explicit forms of the approximate solution are obtained for all values of x. They agree exactly with the Majorana solution at small x, and remain very close to the numerical solution for all values of x. Remarkably, without making any use of series, our approximate solutions achieve a smooth transition from small-x to large-x behaviour. Eventually, the generalized Thomas–Fermi equation that includes relativistic, non-extensive and thermal effects is studied, finding approximate solutions at small and large x for small or finite values of the physical parameters in this equation.

Numerical analysis of sparse initial data identification for parabolic problems

In this paper we consider a problem of initial data identification from the final time observation for homogeneous parabolic problems. It is well-known that such problems are exponentially ill-posed due to the strong smoothing property of parabolic equations. We are interested in a situation when the initial data we intend to recover is known to be sparse, i.e. its support has Lebesgue measure zero. We formulate the problem as an optimal control problem and incorporate the information on the sparsity of the unknown initial data into the structure of the objective functional. In particular, we are looking for the control variable in the space of regular Borel measures and use the corresponding norm as a regularization term in the objective functional. This leads to a convex but non-smooth optimization problem. For the discretization we use continuous piecewise linear finite elements in space and discontinuous Galerkin finite elements of arbitrary degree in time. For the general case we establish error estimates for the state variable. Under a certain structural assumption, we show that the control variable consists of a finite linear combination of Dirac measures. For this case we obtain error estimates for the locations of Dirac measures as well as for the corresponding coefficients. The key to the numerical analysis are the sharp smoothing type pointwise finite element error estimates for homogeneous parabolic problems, which are of independent interest. Moreover, we discuss an efficient algorithmic approach to the problem and show several numerical experiments illustrating our theoretical results.

Harmonic Interpolating Wavelets in a Ring

Complementing the authors' earlier joint papers on the application of orthogonal wavelets to represent solutions of Dirichlet problems with the Laplace operator and its powers in a disk and a ring and of interpolating wavelets for the same problem in a disk, we develop a technique of applying periodic interpolating wavelets in a ring for the Dirichlet boundary value problem. The emphasis is not on the exact representation of the solution in the form of (double) series in a wavelet system but on the approximation of solutions with any given accuracy by finite linear combinations of dyadic rational translations of special harmonic polynomials; these combinations are constructed with the use of interpolating wavelets. The obtained approximation formulas are simply calculated, especially if the squared Fourier transform of the Meyer scaling function with the properties described in the paper is explicitly defined in terms of the corresponding elementary functions.

Decomposition of graded local cohomology tables

Let R be a polynomial ring over a field. We describe the extremal rays and the facets of the cone of local cohomology tables of finitely generated graded R-modules of dimension at most two. Moreover, we show that any point inside the cone can be written as a finite linear combination, with positive rational coefficients, of points belonging to the extremal rays of the cone. We also provide algorithms to obtain decompositions in terms of extremal points and facets.

The Second-Moment Phenomenon for Monochromatic Subgraphs

What is the chance that among a group of $n$ friends, there are $s$ friends all of whom have the same birthday? This is the celebrated birthday problem which can be formulated as the existence of a monochromatic $s$-clique $K_s$ ($s$-matching birthdays) in the complete graph $K_n$, where every vertex of $K_n$ is uniformly colored with 365 colors (corresponding to birthdays). More generally, for a general connected graph $H$, let $T(H, G_n)$ be the number of monochromatic copies of $H$ in a uniformly random coloring of the vertices of the graph $G_n$ with $c_n$ colors. In this paper we show that $T(H, G_n)$ converges to ${Pois}(\lambda)$ whenever $\mathbb{E} T(H, G_n) \rightarrow \lambda$ and ${Var} T(H, G_n) \rightarrow \lambda$, that is, the asymptotic Poisson distribution of $T(H, G_n)$ is determined just by the convergence of its mean and variance. Moreover, this condition is necessary if and only if $H$ is a star-graph. In fact, the second-moment phenomenon is a consequence of a more general theorem about the convergence of $T(H,G_n)$ to a finite linear combination of independent Poisson random variables. As an application, we derive the limiting distribution of $T(H, G_n)$, when $G_n\sim G(n, p)$ is the Erdös--Rényi random graph. Multiple phase transitions emerge as $p$ varies from 0 to 1, depending on whether the graph $H$ is balanced or unbalanced.

Some arithmetic properties of the values of entire functions of finite order and their first derivatives

We describe a class of entire functions of finite order which, together with their first derivative, take sufficiently many algebraic values (with certain restrictions on the growth of the degree and height of these values). We show that, under certain conditions, any such function is a rational function of special form of an exponential. For entire functions of finite order which are not representable in the form of a finite linear combination of exponentials, we obtain an estimate for the number of points (in any fixed disc) at which the values of the function itself and its first derivative are algebraic numbers of bounded degree and height. Bibliography: 8 titles.

Compact linear combination of composition operators on Bergman spaces

Motivated by the question of Shapiro and Sundberg raised in 1990, study on linear combinations of composition operators has been a topic of growing interest. In this paper, we completely characterize the compactness of any finite linear combination of composition operators with general symbols on the weighted Bergman spaces in two classical terms: one is a function theoretic characterization of Julia-Carathéodory type and the other is a measure theoretic characterization of Carleson type. Our approach is completely different from what has been known so far.

0-Cycles on Grassmannians as Representations of Projective Groups

Let F be an infinite division ring, V be a left F-vector space, \(r\ge 1\) be an integer. We study the structure of the representation of the linear group \(\mathrm {GL}_F(V)\) in the vector space of formal finite linear combinations of r-dimensional vector subspaces of V with coefficients in a field. This gives a series of natural examples of irreducible infinite-dimensional representations of projective groups. These representations are non-smooth if F is locally compact and non-discrete.

On the controllability of a Boussinesq system for two-way propagation of dispersive waves

In this paper, we are concerned with a Boussinesq system of Benjamin–Bona–Mahony type equation, posed on a bounded interval, modeling the two-way propagation of surface waves in a uniform horizontal channel filled with an irrotational, incompressible and inviscid liquid under the influence of gravitation. The main focus is on the boundary controllability property, which corresponds to the question of whether the solutions can be driven to a given state at a given final time by means of controls acting at one endpoint of the interval. We first show that the system is not spectrally controllable. This means that no finite linear combination of eigenfunctions associated with the state equations, other than zero, can be steered to zero. Although the system is not spectrally controllable, it can be shown that it is approximately controllable, i.e., any state can be steered arbitrarily close to another state.

Existence of common fixed points for linear combinations of contractive maps in enhanced probabilistic metric spaces

In this paper, we introduce the concept of enhanced probabilistic metric space (briefly EPM-space) as a type of probabilistic metric space. Also, we investigate the existence of fixed points for a (finite or infinite) linear combination of different types of contractive mappings in EPM-spaces. Furthermore, we investigate about the convergence of sequences (generated by a finite or infinite family of contractive mappings) to a common fixed point. The useful application of this research is the study of the stability of switched dynamic systems, where we study the conditions under which the iterative sequences generated by a (finite or infinite) linear combination of mappings (contractive or not), converge to the fixed point. Also, some examples are given to support the obtained results. In the end, a number of figures give us an overview of the examples.

Spectral theorems associated with the directional short-time Fourier transform

In this paper, we are interested in the directional short-time Fourier transform by means of which the notion of a generalized two-wavelet multiplier is investigated. The boundedness and compactness of the generalized two-wavelet multipliers are studied on $$L^{p}({\mathbb {R}}^{d})$$, $$1 \le p \le \infty $$. After wards, we introduce the generalized Landau–Pollak–Slepian operator and we give its trace formula. We show that the generalized two-wavelet multiplier is unitary equivalent to a scalar multiple of the generalized Landau–Pollak–Slepian operator. As applications, we prove an uncertainty principle of Donoho–Stark type involving $$\varepsilon $$-concentration of the generalized two-wavelet multipliers. Moreover we study functions whose time–frequency content are concentrated in a region with finite measure in phase space using the phase space restriction operators as a main tool. We obtain approximation inequalities for such functions using a finite linear combination of eigenfunctions of these operators.

Two-Wavelet Multipliers on the Dual of the Laguerre Hypergroup and Applications

In this paper, we are interested in the Laguerre hypergroup $$\mathbb {K} = [0,\infty )\times {\mathbb {R}}$$ which is the fundamental manifold of the radial function space for the Heisenberg group. So, we consider the generalized shift operator generated by the dual of the Laguerre hypergroup $$\widehat{\mathbb {K}}$$ which can be topologically identified with the so-called Heisenberg fan, the subset of $${\mathbb {R}}^{2}$$ $$\begin{aligned} \cup _{j\in {\mathbb {N}}}\left\{ (\lambda ,\mu )\in {\mathbb {R}}^{2}:\mu =|\lambda |(2j+\alpha +1), \; \lambda \ne 0\right\} \cup \left\{ (0,\mu )\in {\mathbb {R}}^{2}:\mu \ge 0\right\} , \end{aligned}$$ by means of which the notion of a generalized two-wavelet multiplier is investigated. The boundedness and compactness of the generalized two-wavelet multipliers are studied on $$L^{p}_{\alpha }(\mathbb {K})$$ , $$1 \le p \le \infty $$ . Afterwards, we introduce the generalized Landau–Pollak–Slepian operator and we give its trace formula. We show that the generalized two-wavelet multiplier is unitary equivalent to a scalar multiple of the generalized Landau–Pollak–Slepian operator. As applications, we prove an uncertainty principle of Donoho–Stark type involving $$\varepsilon $$ -concentration of the generalized two-wavelet multiplier operators. Moreover, we study functions whose time–frequency content is concentrated in a region with finite measure in phase space using the phase space restriction operators as a main tool. We obtain approximation inequalities for such functions using a finite linear combination of eigenfunctions of these operators.

Monochromatic subgraphs in randomly colored graphons

Let T(H,Gn) be the number of monochromatic copies of a fixed connected graph H in a uniformly random coloring of the vertices of the graph Gn. In this paper we give a complete characterization of the limiting distribution of T(H,Gn), when {Gn}n≥1 is a converging sequence of dense graphs. When the number of colors grows to infinity, depending on whether the expected value remains bounded, T(H,Gn) either converges to a finite linear combination of independent Poisson variables or a normal distribution. On the other hand, when the number of colors is fixed, T(H,Gn) converges to a (possibly infinite) linear combination of independent centered chi-squared random variables. This generalizes the classical birthday problem, which involves understanding the asymptotics of T(Ks,Kn), the number of monochromatic s-cliques in a complete graph Kn (s-matching birthdays among a group of n friends), to general monochromatic subgraphs in a network.

Robust Regressor-Free Control of Rigid Robots Using Function Approximations

This paper develops a novel regressor-free robust controller for rigid robots whose dynamics can be described using the Euler–Lagrange equations of motion. The function approximation technique (FAT) is used to represent the robot’s inertia matrix, the Coriolis matrix, and the gravity vector as finite linear combinations of orthonormal basis functions. The proposed controller establishes a robust FAT control framework that uses a fixed control structure. The control objectives are to track reference trajectories in worst case scenarios where the robot dynamics are too costly to develop or otherwise unavailable. Detailed stability analysis via Lyapunov functions, the passivity property, and continuous switching laws shows uniform ultimate boundedness of the closed-loop dynamics. The simulation results of a three-degree-of-freedom (DOF) robot when the robot parameters are perturbed from their nominal values show good robustness of the proposed controller when compared with some well-established control methods. We also demonstrate success in the real-time experimental implementation of the proposed controller, which validates practicality for real-world robotic applications.

Poincaré and Eisenstein series for Jacobi forms of lattice index

Poincaré and Eisenstein series are building blocks for every type of automorphic forms. We define Poincaré series for Jacobi forms of lattice index and show that they reproduce Fourier coefficients of cusp forms under the Petersson scalar product. We compute the Fourier expansions of Poincaré and Eisenstein series and give an explicit formula for the Fourier coefficients of the trivial Eisenstein series in terms of values of Dirichlet L-functions at negative integers. For even weight and fixed index, finite linear combinations of Fourier coefficients of non-trivial Eisenstein series are equal to finite linear combinations of Fourier coefficients of the trivial one. This result is used to obtain formulas for the Fourier coefficients of Eisenstein series associated with isotropic elements of small order.

Type alternative for Frostman measures

For a finite positive Borel measure μ on R its exponential type, Tμ, is defined as the infimum of a>0 such that finite linear combinations of complex exponentials with frequencies between 0 and a are dense in L2(μ). The definition can be easily extended from finite to broader classes of measures. In this paper we prove a new formula for Tμ and use it to study growth and additivity properties of measures with finite positive type. As one of the applications, we show that Frostman measures on R may only have type zero or infinity.

Special relativistic Fourier transformation and convolutions

In this paper, we use the steerable special relativistic (space‐time) Fourier transform (SFT) and relate the classical convolution of the algebra for space‐time Cl(3,1)‐valued signals over the space‐time vector space , with the (equally steerable) Mustard convolution. A Mustard convolution can be expressed in the spectral domain as the point wise product of the SFTs of the factor functions. In full generality do we express the classical convolution of space‐time signals in terms of finite linear combinations of Mustard convolutions and vice versa the Mustard convolution of space‐time signals in terms of finite linear combinations of classical convolutions.

Weighted boundedness of multilinear maximal function using Dirac deltas

In this article we extend a method of Miguel de Guzman involving boundedness properties of maximal functions using Dirac deltas to multilinear setting. This method involves estimating maximal functions over finite linear combination of Dirac deltas. As an application, we obtain end-point weighted boundedness of the multilinear Hardy–Littlewood fractional maximal function with respect to multilinear weights.

A consistent estimator for spectral density matrix of a discrete time periodically correlated process

In this article, we introduce a weighted periodogram in the class of smoothed periodograms as a consistent estimator for the spectral density matrix of a periodically correlated process. We derive its limiting distribution that appears to be a certain finite linear combination of Wishart distribution. We also provide numerical derivations for our smoothed periodogram and exhibit its asymptotic consistency using simulated data.