Vector Space Of Functions Research Articles

Quasi-Weyl asymptotics of the spectrum of the vector Dirichlet problem

In a space of vector functions, we consider the spectral problem Open image in new window , where Open image in new window , and the aαjk and pjk are constants, x ∈ Ω, and Ω is a bounded open set. The boundary conditions correspond to the Dirichlet problem. Let N±(μ) be the positive and negative spectral counting functions. We establish the asymptotics N±(μ) ∼ (mesmΩ)φ±(μ) as μ → +0. The functions φ±(μ) are independent of Ω. In the nonelliptic case, these asymptotics are in general different from the classical (Weyl) asymptotics.

Forced flexural-and-torsional vibrations of a cantilever beam of constant cross section

The paper addresses the forced flexural-and-torsional vibrations of a cantilever beam of constant cross section. The relevant boundary-value problem is solved. The system of two partial differential equations of the fourth order that describes these vibrations is analyzed in a vector-function space and is subjected to an equivalent transformation to obtain one vector equation of the fourth order with two matrices as coefficients. One is an idempotent matrix; the other is a diagonal matrix. This makes it much easier to construct a Cauchy vector function as an analytic function of these matrices

QES SYSTEMS, INVARIANT SPACES AND POLYNOMIALS RECURSIONS

Let us denote [Formula: see text], the finite-dimensional vector spaces of functions of the form ψ(x) = pn(x)+f(x)pm(x), where pn(x) and pm(x) are arbitrary polynomials of degree at most n and m in the variable x while f(x) represents a fixed function of x. Conditions on m, n and f(x) are found such that families of linear differential operators exist which preserve [Formula: see text]. A special emphasis is accorded to the cases where the set of differential operators represents the enveloping algebra of some abstract algebra. These operators can be transformed into linear matrix valued differential operators. In the second part, such types of operators are considered and a connection is established between their solutions and series of polynomials-valued vectors obeying three terms recurrence relations. When the operator is quasi-exactly solvable, it possesses a finite-dimensional invariant vector space. We study how this property leads to the truncation of the polynomials series.

On the spectrum of a vector Schrödinger operator

We consider the asymptotics of the spectrum of a Sturm-Liouville operator acting on a space of vector functions and show that this asymptotics is affected by “rotation” of eigenvectors of the potential. A similar result is obtained for a vector Schrodinger operator.

Integral equations, large and small forcing functions: Periodicity

The defining property of an integral equation with resolvent R ( t , s ) is the relation between a ( t ) and ∫ 0 t R ( t , s ) a ( s ) d s for functions a ( t ) in a given vector space. We study the behaviour of a solution of an integral equation: x ( t ) = a 1 ( t ) + a 2 ( t ) − ∫ 0 t C ( t , s ) x ( s ) d s when a 1 ( t ) is periodic, C ( t + T , s + T ) = C ( t , s ) , while a 2 ( t ) is typified by ( t + 1 ) β with 0 < β < 1 . There is a resolvent, R ( t , s ) , so that x ( t ) = a 1 ( t ) + a 2 ( t ) − ∫ 0 t R ( t , s ) [ a 1 ( s ) + a 2 ( s ) ] d s . We show that the integral ∫ 0 t R ( t , s ) a 2 ( s ) d s so closely approximates a 2 ( t ) that the only trace of that large function, a 2 ( t ) , in the solution is an L p -function, p < ∞ . In short, that large function a 2 ( t ) has essentially no long-term effect on the solution which turns out to be the sum of a periodic function, a function tending to zero, and an L p -function. The noteworthy property here is that with great precision the integral ∫ 0 t R ( t , s ) a ( s ) d s can duplicate vector spaces of functions both large and small, both monotone and oscillatory; however, it cannot duplicate a given nontrivial periodic function a ( t ) other than k [ 1 + ∫ − ∞ t C ( t , s ) d s ] where k is constant. The integral ∫ 0 t R ( t , s ) sin ( s + 1 ) β d s is an L p approximation to sin ( t + 1 ) β for 0 < β < 1 , but contraction mappings show us that precisely at β = 1 that approximation fails and sin ( t + 1 ) − ∫ 0 t R ( t , s ) sin ( s + 1 ) d s approaches a nontrivial periodic function.

On the spectrum of a vector Schrödinger operator

We consider the asymptotics of the spectrum of a Sturm-Liouville operator acting on a space of vector functions and show that this asymptotics is affected by “rotation” of eigenvectors of the potential. A similar result is obtained for a vector Schrodinger operator.

Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations

This study focuses on the numerical resolution of backward stochastic differential equations with data dependent on a jump-diffusion process. We propose and analyse a numerical scheme based on iterative regression functions which are approximated by projections on vector spaces of functions, with coefficients evaluated using Monte Carlo simulations. Regarding the error, we derive explicit bounds with respect to the time step, the number of paths simulated and the number of functions: this allows us to optimally adjust the parameters to achieve a given accuracy. We also present numerical tests related to option pricing with differential interest rates and locally risk-minimizing strategies (Follmer- Schweizer decomposition).

Vector Image Segmentation by Piecewise Continuous Approximation

In this article we present an approach to the segmentation problem by a piecewise approximation of the given image with continuous functions. Unlike the common approach of Mumford and Shah in our formulation of the problem the number of segments is a parameter, which can be estimated. The problem can be stated as: Compute the optimal segmentation with a fixed number of segments, then reduce the number of segments until the segmentation result fulfills a given suitability. This merging algorithm results in a multi-objective optimization, which is not only resolved by a linear combination of the contradicting error functions. To constrain the problem we use a finite dimensional vector space of functions in our approximation and we restrict the shape of the segments. Our approach results in a multi-objective optimization: On the one hand the number of segments is to be minimized, on the other hand the approximation error should also be kept minimal. The approach is sound theoretically and practically: We show that for L 2-images a Pareto-optimal solution exists and can be computed for the discretization of the image efficiently.

The Generalized Moment Problem with Complexity Constraint

In this paper, we present a synthesis of our differentiable approach to the generalized moment problem, an approach which begins with a reformulation in terms of differential forms and which ultimately ends up with a canonically derived, strictly convex optimization problem. Engineering applications typically demand a solution that is the ratio of functions in certain finite dimensional vector space of functions, usually the same vector space that is prescribed in the generalized moment problem. Solutions of this type are hinted at in the classical text by Krein and Nudelman and stated in the vast generalization of interpolation problems by Sarason. In this paper, formulated as generalized moment problems with complexity constraint, we give a complete parameterization of such solutions, in harmony with the above mentioned results and the engineering applications. While our previously announced results required some differentiability hypotheses, this paper uses a weak form involving integrability and measurability hypotheses that are more in the spirit of the classical treatment of the generalized moment problem. Because of this generality, we can extend the existence and well-posedness of solutions to this problem to nonnegative, rather than positive, initial data in the complexity constraint. This has nontrivial implications in the engineering applications of this theory. We also extend this more general result to the case where the numerator can be an arbitrary positive absolutely integrable function that determines a unique denominator in this finite-dimensional vector space. Finally, we conclude with four examples illustrating our results.

Algebra schemes and their representations

The equivalence (Cartier duality) between the category of topologically flat formal k-groups and the category of flat bialgebras has been treated as a duality of continuous vector spaces (of functions) [G, Expose VIIB by P. Gabriel, 2.2.1]. This is owing to the fact that the reflexivity of vector spaces of infinite dimension does not hold if one does not provide them with a certain topology and does not consider the continuous dual. In this paper we obtain this duality without providing the vector spaces of functions with a topology. Let R be a commutative ring with unit. It is natural to consider R-modules as R-module functors in the following way: if M is an R-module, let M be the Rmodule functor defined by M(S) := M ⊗R S for every R-algebra S which belongs to the category CR of R-algebras. Now, if M is a functor of R-modules, its dual M∗ can be defined in a natural way as the functor of R-modules defined M∗(S) := HomS(M|S ,S). In this work we will prove that the functor defined by an R-module is reflexive: M ∼ →M∗∗, even in the case of R being a ring. We call the functors M∗ R-module schemes and if they are R-algebra functors too, we will say they are R-algebra schemes. In section 2 we study and characterize the vector space schemes (2.3, 2.17) and we characterize when the module scheme closure of an R-module functor M is equal to M∗∗ (2.8, 2.9). P. Gabriel [G, Expose VIIB , 1.3.5] proved that the category of topologically flat formal R-varieties is equivalent to to the category of flat cocommutative Rcoalgebras, where R is a pseudocompact ring. We prove (4.2) that the category of R-algebra schemes is equivalent to the category of R-coalgebras, where R is a ring. From this perspective, on the theory of algebraic groups and their representations R-module schemes appear in a necessary way, as also doR-algebra schemes as linear envelopes of groups. Let G = Spec A be an R-group and let G· be the functor of points of G, i.e., G·(S) = HomR−sch(SpecS,G) for all S ∈ CR, and let RG· be the “linear envelope of G·” (see section 3). We prove that the R-algebra scheme closure ofRG· is theR-algebra scheme A∗ (3.3, 5.4) and the category of G-modules is equal to the category of A∗-modules (5.5). So, the theory of linear representations of a group G = Spec A is a particular case of the theory of A∗-modules (5.7, 5.8, 6.4, etc). Moreover, there is a bijective correspondence between the R-rational points of A∗ and the multiplicative characters of G (5.6). When R is an algebraically closed field and G is smooth we prove that the completion of RG· by its ideal functors of finite codimension is also A∗ (3.5, 5.9).

Compact inverses of first-order normal differential operators

In this work, firstly we describe all normal extensions of a minimal operator generated by linear differential-operator expression of first order in the Hilbert space of vector functions in finite interval. Later on, we investigate discreteness of spectrum and asymptotical behavior of s-numbers of the inverses of these normal extensions.

(C, 1)-Summability of Fourier Series in Root Functions of Ordinary Linear Differential Operators in a Space of Vector Functions

(C, 1)-Summability of Fourier Series in Root Functions of Ordinary Linear Differential Operators in a Space of Vector Functions

On Causal Invertibility with Respect to a Cone of Integral-Difference Operators in Vector Function Spaces

Let \(\mathbb{S}\) be a cone in ℝn. A bounded linear operator T: Lp (ℝn) → Lp (ℝn) is said to be causal with respect to \(\mathbb{S}\) if the implication x(s) = 0 (s e W − \(\mathbb{S}\)) ⇒ (Tx) (s) = 0 (s e W − \(\mathbb{S}\)) is valid for any x e Lp (ℝn) and any open subset W\(\subseteq\) ℝn. The set of all causal operators is a Banach algebra. We describe the spectrum of the operator $$(Tx)(t) = \sum\limits_{n = 1}^\infty {a_n x(t - t_n )} + \int {\mathbb{S}g(s)x(t - s)ds,} \quad t \in \mathbb{R}^n ,$$ in this algebra. Here x ranges in a Banach space \(\mathbb{E}\), the an are bounded linear operators in \(\mathbb{E}\), and the function g ranges in the set of bounded operators in \(\mathbb{E}\).

Compact Multiplication Operators on Weighted Spaces of Vector-Valued Continuous Functions

In this note we characterize the compact multiplication operators Mπ on the weighted locally convex spaces CV0(X,E) of vector-valued continuous functions induced by the operator-valued mappings π : X → B(E). 0. Introduction. Let L(X,E) be a vector space of functions from a nonempty set X to a vector space E over the field C or R. Let T (E) be a set of linear transformations from E to itself. If φ : X → X and π : X → T (E) are mappings such that the weighted composite function π.f ◦ φ belongs to L(X,E), whenever f ∈ L(X,E), then the mapping taking f to π.f ◦φ is a linear transformation on L(X,E) and we denote it by Wπ,φ. In case L(X,E) is a topological vector space and the mapping Wπ,φ is continuous, we call Wπ,φ the weighted composition operator on L(X,E) induced by the symbol (π, φ). In case φ is the identity map, we call it the multiplication operator induced by π and we denote it by Mπ. For details on these operators we refer to [11]. The compact weighted composition operators on spaces of continuous functions have been studied extensively by many authors like Kamowitz [5], Feldman [3], Singh and Summers [13], Jamison and Rajagopalan [4], Takagi [14], Chan [2] and Singh and Manhas [12]. As we know, the class of weighted composition operators include the class of multiplication operators and the class of composition operators. One natural question arises: is it possible to get the behavior of the compact multiplication operators from the study of the compact weighted composition operators? In general, it is not possible since the conditions obtained earlier for a weighted composition operator to be compact is not satisfied by the identity map φ. So it motivates us to 1991 AMS Mathematics Subject Classification. Primary 47B38, 47B07, Secondary 47A56, 46E10, 46E40.

Exceptional Congruences for Powers of the Partition Function

In Journal of London Math. Soc.31 (1956), 350–359, Morris Newman studied vector spaces of functions arising from lifts to Γ0(p) of certain eta-products on the group Γ0(pQ), Q = pn. In this paper, the author considers vector spaces of modular functions obtained as lifts of more general eta-products from Γ0(pQ) to Γ0(p), (Q, p) = 1. Specifically considered are functions arising as lifts of functions of the form

Double Dirichlet series over function fields

We construct a finite-dimensional vector space of functions of two complex variables attached to a smooth algebraic curve C over a finite field , q odd, and a level. These functions collect the analytic information about the cohomology of the curve and its quadratic twists that is encoded in the corresponding L-functions; they are double Dirichlet series in two independent complex variables s and w. We prove that these series satisfy a finite, non-abelian group of functional equations in the two complex variables (s, w) and are rational functions in q-s and q-w with a specified denominator. The group is D6, the dihedral group of order 12.

Generalized Multiplicative Inequalities for Ideal Spaces

We study the problem of completely describing the domains that enjoy the generalized multiplicative inequalities of the embedding theorem type. We transfer the assertions for the Sobolev spaces L p 1(Ω) to the function classes that result from the replacement of L p (Ω) with an ideal space of vector-functions. We prove equivalence of the functional and geometric inequalities between the norms of indicators and the capacities of closed subsets of Ω. The most comprehensible results relate to the case of the rearrangement invariant ideal spaces.

Conditions for the Positivity and Coercive Solvability of the Matrix Schrödinger Operator in Banach Spaces of Vector Functions

necessarily imply that ∆u, qu ∈ Hp. The separability of the operator A with a Hermitian-valued potential q(x) = q∗(x) was studied in [1], and the separability ofA for the case in which l = 1 under the condition | Im q(x)| ≤M Re q(x) was investigated, e.g., in [2]. In the present paper, we study the Hp-separability of A under the assumption that the eigenvalues of the matrix q(x) lie in the sector

Some planar algebras related to graphs

Let X denote a finite nonempty set, and let W denote a matrix whose rows and columns are indexed by X and whose entries belong to some field K. We study three planar algebras related to W. Briefly, a planar algebra is a graded vector space ν = ∪ b ∈ Z+ ∪ {+,-} ν n which is closed under planar operators. The first planar algebra which we study, F W = ∪F W n , is defined by the group theoretic properties of W. For n E Z + , F W n is the vector space of functions from X to K which are constant on the?ut(W)-orbits of X n , and F W + , F W - are identified with K. The second planar algebra, P W = ∪P W n , is the planar algebra generated W. We define it combinatorially: P W n is spanned by functions from X to K defined via statistical mechanical sums on certain planar open graphs. The third planar algebra, O W = ∪O W n , differs from P W only in that the open graphs defining the functions need not be planar. It turns out that P W ⊆ O W ⊂ F W . We show that P W = O W if and only if P W 4 contains a single special function known as the transposition. We show that O W = F W whenever |X|! is not divisible by the characteristic of K.

Two-dimensional Fuchsian systems and the Chebyshev property

Let ( x( t), y( t)) ⊤ be a solution of a Fuchsian system of order two with three singular points. The vector space of functions of the form P( t) x( t)+ Q( t) y( t), where P, Q are real polynomials, has a natural filtration of vector spaces, according to the asymptotic behavior of the functions at infinity. We describe a two-parameter class of Fuchsian systems, for which the corresponding vector spaces obey the Chebyshev property (the maximal number of isolated zeros of each function is less than the dimension of the vector space). Up to now, only a few particular systems were known to possess such a non-oscillation property. It is remarkable that most of these systems are of the type studied in the present paper. We apply our results in estimating the number of limit cycles that appear after small polynomial perturbations of several quadratic or cubic Hamiltonian systems in the plane.