Finite Number Of Functions Research Articles

Reusing Topological Nexttime Logic

In this paper, a particular extension of the constitutive bi-modal logic for single-agent subset spaces will be provided. That system, which originally was designed for revealing the intrinsic relationship between knowledge and topology, has been developed in several directions in recent years, not least towards a comprehensive knowledge-theoretic formalism. This line is followed here to the extent that subset spaces are supplied with a finite number of functions which shall represent certain knowledge-enabling actions. Due to the corresponding functional modalities, another basic system for subset spaces, topological nexttime logic, comes into play. The resulting merge of logics can, for example, be applied to comparing the different effects of those actions in respect of knowledge. Subsequently, the completeness and the decidabilty of the basic combined system and of a certain extension thereof will be proved.

Affine, quasi-affine and co-affine frames on local fields of positive characteristic

The concept of quasi-affine frame in Euclidean spaces was introduced to obtain translation invariance of the discrete wavelet transform. We extend this concept to a local field K of positive characteristic. We show that the affine system generated by a finite number of functions is an affine frame if and only if the corresponding quasi-affine system is a quasi-affine frame. In such a case the exact frame bounds are equal. This result is obtained by using the properties of an operator associated with two such affine systems. We characterize the translation invariance of such an operator. A related concept is that of co-affine system. We show that there do not exist any co-affine frame in L2(K).

Filtrations in Dyson–Schwinger equations: Next-to[formula omitted]-leading log expansions systematically

Dyson–Schwinger equations determine the Green functions Gr(α,L) in quantum field theory. Their solutions are triangular series in a coupling constant α and an external scale parameter L for a chosen amplitude r, with the order in L bounded by the order in the coupling.Perturbation theory calculates the first few orders in α. On the other hand, Dyson–Schwinger equations determine next-to{j}-leading log expansions, Gr(α,L)=1+∑j=0∞∑MpjMαjM(u). ∑M sums for any finite j a finite number of functions M in u. Here, u is the one-loop approximation to Gr, for example, for the (inverse) propagator in massless Yukawa theory, u=αL/2.The leading logs come then from the trivial representation, e.g. M(u)=1−1−2u in massless Yukawa theory. The respective period at j=0 is p0M(u)=1. All non-leading logs are organized by corresponding suppressions in powers αj.We describe an algebraic method to derive all next-to{j}-leading log terms from the knowledge of the first (j+1) terms in perturbation theory and their filtrations. This implies the calculation of the functions M(u) and periods pjM.In the first part of our paper, we investigate the structure of Dyson–Schwinger equations and develop a method to filter their solutions. Applying renormalized Feynman rules maps each filtered term to a certain power of α and L in the log-expansion.Based on this, the second part derives the next-to{j}-leading log expansions. Our method is general. Here, we exemplify it using the examples of the propagator in Yukawa theory and the photon self-energy in quantum electrodynamics. In particular, we give explicit formulas for the leading log, next-to-leading log and next-to-next-to-leading log orders in terms of at most three-loop Feynman integrals. The reader may apply our method to any (set of) Dyson–Schwinger equation(s) appearing in renormalizable quantum field theories.

Semi-orthogonal wavelet frames on local fields

AbstractWe investigate semi-orthogonal wavelet frames on local fields of positive characteristic and provide a characterization of frame wavelets by means of some basic equations in the frequency domain. The theory of frame multiresolution analysis recently proposed by Shah [J. Operators (2015), Article ID 216060] on local fields is used to establish equivalent conditions for a finite number of functions

Convex Envelopes of Some Quadratic Functions over the n-Dimensional Unit Simplex

In this paper we discuss convex envelopes over the unit simplex for some quadratic functions. The discussion exploits the combinatorics related to quadratic functions over the unit simplex. After recalling the definition of the underlying convex graph of a quadratic function, i.e., the graph whose nodes are the vertices of the unit simplex, and whose edges correspond to the edges of the unit simplex along which the function is strictly convex, it is shown that the analytical formula of the convex envelope for quadratic functions whose underlying graph is triangle-free, is the maximum of a finite number of functions, each of which is related to a spanning forest of the graph. The special case of an underlying star graph is discussed, and computational experiments, based on a decomposition of a general quadratic function into a sum of quadratic functions with an underlying star graph, are presented.

Group Update Method for Sparse Minimax Problems

A group update algorithm is presented for solving minimax problems with a finite number of functions, whose Hessians are sparse. The method uses the gradient evaluations as efficiently as possible by updating successively the elements in partitioning groups of the columns of every Hessian in the process of iterations. The chosen direction is determined directly by the nonzero elements of the Hessians in terms of partitioning groups. The local $$q$$q-superlinear convergence of the method is proved, without requiring the imposition of a strict complementarity condition, and the $$r$$r-convergence rate is estimated. Furthermore, two efficient methods handling nonconvex case are given. The global convergence of one method is proved, and the local $$q$$q-superlinear convergence and $$r$$r-convergence rate of another method are also proved or estimated by a novel technique. The robustness and efficiency of the algorithms are verified by numerical tests.

The NC-proximal average for multiple functions

The NC-proximal average is a parametrized function used to continuously transform one proper, lsc, prox-bounded function into another. Until now, it has been defined for two functions. The purpose of this article is to redefine it so that any finite number of functions may be used. The layout generally follows that of [11], extending those results to the more general case and in some instances giving alternate proofs by using techniques developed after the publication of that paper. We conclude with an example examining the discontinuity of the minimizers of the NC-proximal average.

Choice of free parameters in expansions of discrete-time Volterra models using Kautz functions

This work tackles the problem of modeling nonlinear systems using Volterra models based on Kautz functions. The drawback of requiring a large number of parameters in the representation of these models can be circumvented by describing every kernel using an orthonormal basis of functions, such as the Kautz basis. The resulting model, so-called Wiener/Volterra model, can be truncated into a few terms if the Kautz functions are properly designed. The underlying problem is how to select the free-design complex poles that fully parameterize these functions. A solution to this problem has been provided in the literature for linear systems, which can be represented by first-order Volterra models. A generalization of such strategy focusing on Volterra models of any order is presented in this paper. This problem is solved by minimizing an upper bound for the error resulting from the truncation of the kernel expansion into a finite number of functions. The aim is to minimize the number of two-parameter Kautz functions associated with a given series truncation error, thus reducing the complexity of the resulting finite-dimensional representation. The main result is the derivation of an analytical solution for a sub-optimal expansion of the Volterra kernels using a set of Kautz functions.

A note on the optimal expansion of Volterra models using Laguerre functions

This work tackles the problem of expanding Volterra models using Laguerre functions. A strict global optimal solution is derived when each multidimensional kernel of the model is decomposed into a set of independent orthonormal bases, each of which parameterized by an individual Laguerre pole intended for representing the dominant dynamic of the kernel along a particular dimension. It is proved that the solution derived minimizes the upper bound of the squared norm of the error resulting from the practical truncation of the Laguerre series expansion into a finite number of functions. This is an extension of the results in Campello, Favier and Amaral [(2004). Optimal expansions of discrete-time Volterra models using Laguerre functions. Automatica, 40, 815–822.], where an optimal solution was obtained for the usual yet particular case in which a single Laguerre pole is used for expanding a given kernel along all its dimensions. It is also proved that the particular and extended solutions are equivalent to each other when the Volterra kernels are symmetric.

Simultaneous greedy approximation in Banach spaces

We study nonlinear m-term approximation with regard to a redundant dictionary D in a Banach space. It is known that in the case of Hilbert space H the pure greedy algorithm (or, more generally, the weak greedy algorithm) provides for each f ∈ H and any dictionary D an expansion into a series f = ∑ j = 1 ∞ c j ( f ) ϕ j ( f ) , ϕ j ( f ) ∈ D , j = 1 , 2 , … with the Parseval property: ∥ f ∥ 2 = ∑ j | c j ( f ) | 2 . The orthogonal greedy algorithm (or, more generally, the weak orthogonal greedy algorithm) has been introduced in order to enhance the rate of convergence of greedy algorithms. Recently, we have studied analogues of the PGA and WGA for a given finite number of functions f 1 , … , f N with a requirement that the dictionary elements ϕ j of these expansions are the same for all f i , i = 1 , … , N . We have studied convergence and rate of convergence of such expansions which we call simultaneous expansions. The goal of this paper is twofold. First, we work in a Hilbert space and enhance the convergence of the simultaneous greedy algorithms by introducing an analogue of the orthogonalization process, and we give estimates on the rate of convergence. Then, we study simultaneous greedy approximation in a more general setting, namely, in uniformly smooth Banach spaces.

Multiresolution of L p Spaces

Multiresolution analysis plays a major role in wavelet theory. In this paper, multiresolution of L p spaces is studied. Let S be a shift-invariant subspace of L p ( R s ) (1≤ p≤∞) generated by a finite number of functions with compact support, and let S k be the 2 k -dilate of S for each integer k∈ Z . It is shown that the intersection of S k ( k∈ Z ) is always trivial. It is more difficult to deal with the problem whether the union of S k ( k∈ Z ) is dense in L p ( R s ). The case p=1 or 2 can be solved by Wiener′s density theorem. Under the assumption that S is refinable, it is proved in this paper that the union of S k ( k∈ Z ) is dense in L p ( R s ), provided s=1, 2, or 2≤ p<∞. The same is true for p=∞, if L ∞( R s ) is replaced by C 0( R s ). Counterexamples are given to demonstrate that for s≥3 and 1<p<2, the aforementioned results on density are no longer true in general.

On Approximation by Translates of Globally Supported Functions

We consider L p -approximation (1 ≤ p ≤ ∞) from the dilates of a space generated by a finite number of functions that have mild polynomial decay at infinity. In particular, the local-controlled density order of such a family of approximating spaces is characterized in terms of the Strang-Fix condition.

A block spin construction of ondelettes. Part I: Lemari� functions

Using block spin assignments, we construct anL2-orthonormal basis consisting of dyadic scalings and translates of just a finite number of functions. These functions have exponential localization, and for even values of a construction parameterM one can make them classCM−1 with vanishing moments up to orderM inclusive. Such a basis has an important application to phase cell cluster expansions in quantum field theory.

On the quasidifferential of a continual maximum function

A formula for the quasidifferential of a maximum function over a continual set of functions is presented, which in case of a finite number of functions may differ from Dem'Yanov's formula [2].

Field operators and their spectral properties in finite-dimensional quantum field theory

In Ref. 1 we have considered the finite-dimensional quantum mechanics. There the quantum mechanical space of states wasV=C r. It is known that the second quantization of this space is the space of square-summable functions of finite number of variables(L 2(Rr,dx)) (Segal isomorphism). Creation and annihilation operators were introduced in Ref. 1, and the former coincided with the usual position and momentum operators in the conventional quantum mechanics. In this paper we shall investigate the spectral properties of field operators. We shall show that the isomorphism between the exponential ofV andL 2(Rr,dx) can be understood as the decomposition by generalized eigenvectors of field operators (“Fourier transform”).

A special basis for 𝐶([0,1

This paper constructs a basis for C ( [ 0 , 1 ] ) C([0,1]) which converges weakly to zero whose elements are nevertheless norm bounded away from zero.

Some results concerning proofs of statements about programs

A programming language is viewed as a language for expressing “instructions” for a computation to be performed by a particular machine. A class of abstract machines (which includes universal machines) is defined. These machines are viewed as devices which execute “instructions” expressed in programming languages. Using this model and an appropriate definition of a programming language, it is shown that there is at least one system of logic which has the following properties for all machines in this class. o (1) For three concepts of the equivalence of computations and of programs, this system can be used to show that two computations or programs are or are not equivalent. (2) Given a program and a finite number of functions, this system can be used to show that the program does or does not specify the computation of these functions. That is, it is shown that certain relations of equivalence among programs and the relation of a program to the functions whose computation it specifies probably obey the law of excluded middle in this system of logic.

On a theorem of Goffman concerning Schauder series

1. A classical question posed by Lusin asks whether it is possible to find for each measurable function defined on [0, 2ir] a corresponding trigonometric series, with coefficients converging to zero, that converges almost everywhere to the function. This question was answered affirmatively by Men'sov [4] for real-valued functions, but the general question remains unanswered. Thus, it is of interest to inquire whether there be any Schauder bases for LP, p> l, with respect to which every measurable function has a pointwise representation. Although Talalyan and Arutyanyan [8] have shown that a prime candidate, the Haar system, does not have this property, Gundy [3 ] has proved that systems of the specified type do exist. The Schauder functions are total in each of the LP spaces, although they in no case constitute a basis. In [1], Goffman solved Lusin's problem for this system by way of a sequence of careful estimates culminating in a construction of the required series. An interesting aspect of this work is that not all of the Schauder functions are required for the construction. Indeed, it is clear from a superficial examination of the arguments employed in [1] that any finite number of functions could be deleted from the system and the work carried through with no resulting essential modification of the demonstrations. In analogy with work of Talalyan [6], and Goffman and Waterman [2], it is appropriate to ask whether infinitely many Schauder functions could be discarded in such a way that the above-mentioned result of Goffman would remain in force for the abbreviated system. In the present note it is shown that this is the case, and a characterization is given of those subsystems for which the Goffman theorem holds. The result suggests that there may be lurking in the background some very general form of the Muintz theorem.

Transformation of the Linear Equations of Magnetoelastic Interactions for an Infinitely Long Prism

Brown's linear equations of magnetostriction in a homogeneous cubic crystal and for an infinitely long prism along the applied external magnetic field are transformed into a new system of equations. In the transformed system, the two components of the magnetostatic scalar potential, the elasticity potential, and the three components of the elastic displacement vector, satisfy separate differential equations. The general solution is a linear combination of a finite number of functions, each one being a solution of the Helmholtz equation in two independent variables.

Basic sequences in the space of measurable functions

1. In a topological vector space X, a basic sequence [xn] is one whose finite linear combinations are dense in X. In a recent work, [l], A. A. Talalyan has observed that the space of measurable functions has a distinctly different character, with respect to the behavior of basic sequences, from, for example, the Lp spaces, p^l. A striking result of Talalyan is the fact that if { „} is basic, i.e., for every measurable , there are finite linear combinations of the , then if any finite number of functions is deleted from { „}, the remaining sequence is basic. This readily implies the existence of universal expansions, and the existence of a subsequence \<pnk which is basic even though the complement of the sequence {«*} is infinite. The proof given by Talalyan necessitates the use of considerable machinery from the theory of orthonormal systems in L2, and is quite involved. Our purpose is to show that the result follows almost immediately from the fact that the space M of measurable functions, with the topology of convergence in measure, has a trivial dual.