Vector Space Of Functions Research Articles

Maximal sectorial differential operators for first order

In this study, the maximal sectorial linear relations are described. Then, the discreteness of the spectrum of the linear maximal sectorial operators and asymptotical behavior of the eigenvalues of such operators in terms of the eigenvalues of its real part are investigated. Moreover, one result for the differential operators for first order in the Hilbert space of vector functions in finite interval is obtained.

Distribution of the eigenvalues of singular differential operators in a space of vector-functions

Distribution of the eigenvalues of singular differential operators in a space of vector-functions

Orthogonality and Minimality in the Homology of Locally Finite Graphs

Given a finite set $E$, a subset $D\subseteq E$ (viewed as a function $E\to \mathbb F_2$) is orthogonal to a given subspace $\mathcal F$ of the $\mathbb F_2$-vector space of functions $E\to \mathbb F_2$ as soon as $D$ is orthogonal to every $\subseteq$-minimal element of $\mathcal F$. This fails in general when $E$ is infinite.However, we prove the above statement for the six subspaces $\mathcal F$ of the edge space of any $3$-connected locally finite graph that are relevant to its homology: the topological, algebraic, and finite cycle and cut spaces. This solves a problem of Diestel (2010, arXiv:0912.4213).

On non-self-adjoint Sturm-Liouville operators in the space of vector functions

In this article we obtain asymptotic formulas for the eigenvalues and eigenfunctions of the non-self-adjoint operator generated in L2m [0, 1] by the Sturm-Liouville equation with m × m matrix potential and the boundary conditions which, in the scalar case (m = 1), are strongly regular. Using these asymptotic formulas, we find a condition on the potential for which the root functions of this operator form a Riesz basis.

Spectrums of Solvable Pantograph Differential-Operators for First Order

By using the methods of operator theory, all solvable extensions of minimal operator generated by first order pantograph-type delay differential-operator expression in the Hilbert space of vector-functions on finite interval have been considered. As a result, the exact formula for the spectrums of these extensions is presented. Applications of obtained results to the concrete models are illustrated.

The tensor product of function semimodules

Given two domains of functions with values in a field, the canonical map from the algebraic tensor product of the vector spaces of functions on the two domains to the vector space of functions on the product of the two domains is well known to be injective, but not generally surjective. By constructing explicit examples, we show that the corresponding map for semimodules of semiring-valued functions is in general not even injective. This impacts the formulation of topological quantum field theories over semirings. We also confirm the failure of surjectivity for functions with values in complete, additively idempotent semirings by describing a large family of functions that do not lie in the image.

Wiener algebra of operators on the lattice (μ ℤ) n depending on the small parameter μ>0

This article is devoted to the problems of invertibility and Fredholmness of operators in the weighted Wiener algebra acting in the weighted spaces of vector-functions on the lattice (μℤ) n = {x∈ℝ n : x = μy,y∈ℤ n } for small μ>0. We consider the discrete Schrödinger operators on (μℤ) n as applications and study the relation between the spectra of Schrödinger operators on ℝ n and its discrete models on (μℤ) n for small μ>0.

Homological techniques for the analysis of the dimension of triangular spline spaces

The spline space Ckr(Δ) attached to a subdivided domain Δ of Rd is the vector space of functions of class Cr which are polynomials of degree ⩽k on each piece of this subdivision. Classical splines on planar rectangular grids play an important role in Computer Aided Geometric Design, and spline spaces over arbitrary subdivisions of planar domains are now considered for isogeometric analysis applications. We address the problem of determining the dimension of the space of bivariate splines Ckr(Δ) for a triangulated region Δ in the plane. Using the homological introduced by Billera (1988), we number the vertices and establish a formula for an upper bound on the dimension. There is no restriction on the ordering and we obtain more accurate approximations to the dimension than previous methods and furthermore, in certain cases even an exact value can be found. The construction makes also possible to get a short proof for the dimension formula when k⩾4r+1, and the same method we use in this proof yields the dimension straightaway for many other cases.

Compactness of linear integral operators in ideal spaces of vector functions

Estimates for the measure of noncompactness of linear integral operators of vector functions in ideal spaces are obtained. When the kernel function is compact, no additional uniformity or measurability hypotheses are needed; however, also noncompact nonmeasurable kernel functions are treated.

Invariant subspaces in some function spaces on the light cone in

For certain topological vector spaces of functions on the light cone in we obtain a complete description of all the closed linear subspaces which are invariant with respect to the natural quasiregular representation of the group . In particular, we give a description of irreducible and indecomposable invariant subspaces. Among the function spaces we consider we include, in particular, the spaces and of continuous and infinitely differentiable functions on and also function spaces formed by functions with exponential growth on .Bibliography: 32 titles.

Vertex operators and quasimodularity of Chern numbers on the Hilbert scheme

In Carlsson and Okounkov (preprint) [7], Okounkov and the author defined a family of vertex operators on the equivariant cohomology groups of the Hilbert scheme of points on a smooth quasi-projective surface as a characteristic class of certain canonical bundles on Hilb⁎S×Hilb⁎S. We then proved a bosonization formula in terms of Nakajimaʼs Heisenberg operators (Nakajima, 1997 [23]). In this paper we apply this operator in the special case when S=C2 with a particular action of a torus, and prove that the generating functions of equivariant Chern numbers on Hilbn over n, are quasimodular forms in the generating variable. This property determines the answer up to a finite-dimensional vector space of functions of the generating variable, q. These generating functions can be thought of as the analogous correlation functions to Nekrasovʼs partition function in rank 1.We present a different proof of the bosonization formula which is based on the proof of a more general formula in K-theory given in an upcoming paper by Nekrasov, Okounkov and the author (in preparation) [6], but specialized to the surface of interest. By altering a certain bundle that appears in Carlsson et al. (in preparation) [6], and specializing the surface, the proof actually reduces to a much simpler self-contained application of the infinite wedge representation. This picture is consistent with both the original introduction of this operator in Nekrasov and Okounkov (2006) [26] and with Haimanʼs character theory of the Bridgeland, King and Reid isomorphism (Haiman, 2003 [11]).

Singular Sturm-Liouville operators with distribution potential on spaces of vector functions

Singular Sturm-Liouville operators with distribution potential on spaces of vector functions

Stochastic Receding Horizon Control With Bounded Control Inputs: A Vector Space Approach

We design receding horizon control strategies for stochastic discrete-time linear systems with additive (possibly) unbounded disturbances while satisfying hard bounds on the control actions. We pose the problem of selecting an appropriate optimal controller on vector spaces of functions and show that the resulting optimization problem has a tractable convex solution. Under marginal stability of the zero-control and zero-noise system we synthesize receding horizon polices that ensure bounded variance of the states while enforcing hard bounds on the controls. We provide examples that illustrate the effectiveness of our control strategies, and how quantities needed in the formulation of the resulting optimization problems can be calculated off-line.

Spectral synthesis in some topological vector spaces of functions

Spectral synthesis in some topological vector spaces of functions

On the completeness of systems of eigenfunctions and associated functions of differential operators of orders 2 − ε and 1 − ε

In the space of vector-functions, we consider a boundary-value problem for differential operators of fractional orders (2 − e) and (1 − e) and prove the completeness of the system of eigenfunctions and associated functions of this problem in the space \(L_1 \left( {\left[ {0,1} \right],\,\mathbb{C}^p } \right)\).

Some Partial Differential Equations of Multivariable Vector Functions and the Integral Equations of Multivariable Vector Functions

The relations between the partial differential equations of multivariable vector functions and the integral equations of multivariable vector functions which are correspond to them are discussed. The partial differential linear equations of multivariable vector functions can be transformed into the integral linear equations of multivariable vector functions by using the continuous wavelet transforms of multivariable vector function spaces. The result that in the weak topology the partial differential equations of multivariable vector functions are equivalent to the integral equations of multivariable vector functions which are correspond to them is obtained.

Asymptotics of the spectrum of a singular nonsemibounded fourth-order vector differential operator in a vector function space

Asymptotics of the spectrum of a singular nonsemibounded fourth-order vector differential operator in a vector function space

Multigrid in a weighted space arising from axisymmetric electromagnetics

Consider the space of two-dimensional vector functions whose components and curl are square integrable with respect to the degenerate weight given by the radial variable. This space arises naturally when modeling electromagnetic problems under axial symmetry and performing a dimension reduction via cylindrical coordinates. We prove that if the original three-dimensional domain is convex, then the multigrid V-cycle applied to the inner product in this space converges, provided certain modern smoothers are used. For the convergence analysis, we first prove several intermediate results, e.g., the approximation properties of a commuting projector in weighted norms, and a superconvergence estimate for a dual mixed method in weighted spaces. The uniformity of the multigrid convergence rate with respect to mesh size is then established theoretically and illustrated through numerical experiments.

On the deficiency indices of a singular differential operator of fourth order in the space of vector functions

On the deficiency indices of a singular differential operator of fourth order in the space of vector functions

Discreteness of the spectrum of a polar differential operation in the space of vector functions

Sufficient conditions for the discreteness of the spectrum of a polar differential operation in the space of vector functions are obtained.