Arbitrary Linear Combination Research Articles

Безмассовое поле Штюкельберга, точные решения в декартовых координатах и калибровочные степени свободы

In the paper we examine the massless Stueckelberg field. Among the eleven field function components, the antisymmetric tensor represents the gauge variables, whereas the scalar and vector correspond to physically observable quantities. It is shown that in Cartesian coordinates the Stueckelberg equations permit the existence of five independent solutions which describe the different states of the field. We have derived an expression for the energy-momentum tensor of the massless Stueckelberg field. We find its explicit form for arbitrary linear combination of five established solutions. We have found four combinations of five solutions which do not contribute to energy-momentum tensor, therefore they correspond to purely gauge states. There exists only one solution which corresponds to nonvanishing energy-momentum tensor, it relates to physically observable states of the massless Stueckelberg field.

Wronskian solutions and linear superposition of rational solutions to B-type Kadomtsev–Petviashvili equation

The Wronskian solutions to the B-type Kadomtsev–Petviashvili (BKP) equation are discussed based on the Plücker relation. Rational solutions, positon solutions, negaton solutions, and complexiton solutions to the BKP equation are directly constructed. The Wronskian formulation is employed to generate rational solutions in the form of determinants. A polynomial identity is demonstrated that an arbitrary linear combination of two Wronskian polynomial solutions of different orders is again a solution to the bilinear BKP equation. The validity of the linear superposition principle can be inferred for two Wronskian rational solutions to certain equations under specific conditions.

Asymptotics for the twisted eta-product and applications to sign changes in partitions

We prove asymptotic formulas for the complex coefficients of (zeta q;q)_infty ^{-1}, where zeta is a root of unity, and apply our results to determine secondary terms in the asymptotics for p(a, b, n), the number of integer partitions of n with number of parts congruent a modulo b. Our results imply that, as n rightarrow infty , the difference p(a_1,b,n)-p(a_2,b,n) for a_1 ne a_2 oscillates like a cosine when renormalized by elementary functions. Moreover, we give asymptotic formulas for arbitrary linear combinations of {p(a,b,n)}_{1 le a le b}.

Nonlinear quantum error correction

We introduce a theory of quantum error correction (QEC) for a subclass of states within a larger Hilbert space. In the standard theory of QEC, the set of all encoded states is formed by an arbitrary linear combination of the codewords. However, this can be more general than required for a given quantum protocol which may only traverse a subclass of states within the Hilbert space. Here we propose the concept of nonlinear QEC (NLQEC), where the encoded states are not necessarily a linear combination of codewords. We introduce a sufficiency criterion for NLQEC with respect to the subclass of states. The new criterion gives a more relaxed condition for the formation of a QEC code, such that under the assumption that the states are within the subclass of states, the errors are correctable. This allows us, for instance, to effectively circumvent the no-go theorems regarding optical QEC for Gaussian states and channels, for which we present explicit examples.

Occupation time of a renewal process coupled to a discrete Markov chain

A semi-Markov process is one that changes states in accordance with a Markov chain but takes a random amount of time between changes. We consider the generalisation to semi-Markov processes of the classical Lamperti law for the occupation time of a two-state Markov process. We provide an explicit expression in Laplace space for the distribution of an arbitrary linear combination of the occupation times in the various states of the process. We discuss several consequences of this result. In particular, we infer the limiting distribution of this quantity rescaled by time in the long-time scaling regime, as well as the finite-time corrections to its moments.

Heavy-Traffic Analysis of Queueing Systems with No Complete Resource Pooling

We study the heavy-traffic limit of the generalized switch operating under MaxWeight, without assuming that the complete resource pooling condition is satisfied and allowing for correlated arrivals. The main contribution of this paper is the steady-state mean of linear combinations of queue lengths in heavy traffic. We showcase the generality of our result by presenting various stochastic networks as corollaries, each of which is a contribution by itself. In particular, we study the input-queued switch with correlated arrivals, and we show that, if the state space collapses to a full-dimensional subspace, the correlation among the arrival processes does not matter in heavy traffic. We exemplify this last case with a parallel-server system, an [Formula: see text]-system, and an ad hoc wireless network. Whereas these results are obtained using the drift method, we additionally present a negative result showing a limitation of the drift method. We show that it is not possible to obtain the individual queue lengths using the drift method with polynomial test functions. We do this by presenting an alternate view of the drift method in terms of a system of linear equations, and we use this system of equations to obtain bounds on arbitrary linear combinations of the queue lengths.

The extension of a functional equation of the Lambert W function

The principal branch of the Lambert W function satisfies a well-known functional equation with respect to the sum of two of its values, but this equation is valid only on a restricted set. In this paper, we extend this equation to all the branches of W and to the arbitrary complex linear combination of two values. The results give an application for the unwinding number, too.

Approximation by linear combinations of translates of a single function

We study approximation by arbitrary linear combinations of $n$ translates of a single function of periodic functions. We construct some linear methods of this approximation for univariate functions in the class induced by the convolution with a single function, and prove upper bounds of the $L^p$-approximation convergence rate by these methods, when $n \to \infty$, for $1 \leq p \leq \infty$. We also generalize these results to classes of multivariate functions defined the convolution with the tensor product of a single function. In the case $p=2$, for this class, we also prove a lower bound of the quantity characterizing best approximation of by arbitrary linear combinations of $n$ translates of arbitrary function.

Master functions and equations for perturbations of vacuum spherically symmetric spacetimes

Perturbation theory of vacuum spherically-symmetric spacetimes is a crucial tool to understand the dynamics of black hole perturbations. Spherical symmetry allows for an expansion of the perturbations in scalar, vector, and tensor harmonics. The resulting perturbative equations are decoupled for modes with different parity and different harmonic numbers. Moreover, for each harmonic and parity, the equations for the perturbations can be decoupled in terms of (gauge-invariant) master functions that satisfy 1+1 wave equations. By working in a completely general perturbative gauge, in this paper we study what is the most general master function that is linear in the metric perturbations and their first-order derivatives and satisfies a wave equation with a potential. The outcome of the study is that for each parity we have two branches of solutions with similar features. One of the branches includes the known results: In the odd-parity case, the most general master function is an arbitrary linear combination of the Regge-Wheeler and the Cunningham-Price-Moncrief master functions whereas in the even-parity case it is an arbitrary linear combination of the Zerilli master function and another master function that is new to our knowledge. The other branch is very different since it includes an infinite collection of potentials which in turn lead to an independent collection master of functions which depend on the potential. The allowed potentials satisfy a non-linear ordinary differential equation. Finally, all the allowed master functions are gauge invariant and can be written in a fully covariant form.

Applications in random matrix theory of a PIII′ τ-function sequence from Okamoto’s Hamiltonian formulation

We consider the singular linear statistic of the Laguerre unitary ensemble (LUE) consisting of the sum of the reciprocal of the eigenvalues. It is observed that the exponential generating function for this statistic can be written as a Toeplitz determinant with entries given in terms of particular [Formula: see text] Bessel functions. Earlier studies have identified the same determinant, but with the [Formula: see text] Bessel functions replaced by [Formula: see text] Bessel functions, as relating to the hard edge scaling limit of a generalized gap probability for the LUE, in the case of non-negative integer Laguerre parameter. We show that the Toeplitz determinant formed from an arbitrary linear combination of these two Bessel functions occurs as a [Formula: see text]-function sequence in Okamoto’s Hamiltonian formulation of Painlevé III[Formula: see text], and consequently the logarithmic derivative of both Toeplitz determinants satisfies the same [Formula: see text]-form Painlevé III[Formula: see text] differential equation, giving an explanation of a fact which can be observed from earlier results. In addition, some insights into the relationship between this characterization of the generating function, and its characterization in the [Formula: see text] limit, both with the Laguerre parameter [Formula: see text] fixed, and with [Formula: see text] (this latter circumstance being relevant to an application to the distribution of the Wigner time delay statistic), are given.

Probability Error Bounds for Approximation of Functions in Reproducing Kernel Hilbert Spaces

We find probability error bounds for approximations of functions f in a separable reproducing kernel Hilbert space H with reproducing kernel K on a base space X , firstly in terms of finite linear combinations of functions of type K x i and then in terms of the projection π x n on span K x i i = 1 n , for random sequences of points x = x i i in X . Given a probability measure P , letting P K be the measure defined by d P K x = K x , x d P x , x ∈ X , our approach is based on the nonexpansive operator L 2 X ; P K ∋ λ ↦ L P , K λ ≔ ∫ X λ x K x d P x ∈ H , where the integral exists in the Bochner sense. Using this operator, we then define a new reproducing kernel Hilbert space, denoted by H P , that is the operator range of L P , K . Our main result establishes bounds, in terms of the operator L P , K , on the probability that the Hilbert space distance between an arbitrary function f in H and linear combinations of functions of type K x i , for x i i sampled independently from P , falls below a given threshold. For sequences of points x i i = 1 ∞ constituting a so-called uniqueness set, the orthogonal projections π x n to span K x i i = 1 n converge in the strong operator topology to the identity operator. We prove that, under the assumption that H P is dense in H , any sequence of points sampled independently from P yields a uniqueness set with probability 1. This result improves on previous error bounds in weaker norms, such as uniform or L p norms, which yield only convergence in probability and not almost certain convergence. Two examples that show the applicability of this result to a uniform distribution on a compact interval and to the Hardy space H 2 D are presented as well.

Linear superposition of Wronskian rational solutions to the KdV equation

A linear superposition is studied for Wronskian rational solutions to the KdV equation, which include rogue wave solutions. It is proved that it is equivalent to a polynomial identity that an arbitrary linear combination of two Wronskian polynomial solutions with a difference two between the Wronskian orders is again a solution to the bilinear KdV equation. It is also conjectured that there is no other rational solutions among general linear superpositions of Wronskian rational solutions.

On the generalization of moyal equation for an arbitrary linear quantization

For an arbitrary linear combination of quantizations, the kernel of the inverse operator is constructed. An equation for the evolution of the Wigner function for an arbitrary linear quantization is derived and it is shown that only for Weyl quantization this equation does not contain a source of quasi-probability. Stationary solutions for the Wigner function of a harmonic oscillator are constructed, depending on the characteristic function of the quantization rule. In the general case of Hermitian linear quantization these solutions are real but not positive. We found the representation of Weyl quantization in the form of the limit of a sequence of linear Hermitian quantizations, such that for each element of this sequence the stationary solution of the Moyal equation is positive.

A polynomial conjecture connected with rogue waves in the KdV equation

A polynomial conjecture, associated with rational solutions including rogue wave solutions of the KdV equation, is presented. The conjecture can be used to show that for the bilinear KdV equation, an arbitrary linear combination of two Wronskian polynomial solutions with a difference two between the Wronskian orders will again be a solution.

Fundamental Limits of Caching for Demand Privacy Against Colluding Users

This article investigates the problem of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">demand privacy against colluding users</i> for shared-link coded caching systems, where no subset of users can learn any information about the demands of the remaining users. The notion of privacy used here is stronger than similar notions adopted in past work and is motivated by the practical need to insure privacy regardless of the file distribution. Two scenarios are considered: <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Single File Retrieval</i> (SFR) and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Linear Function Retrieval</i> (LFR), where in the latter case each user demands an arbitrary linear combination of the files. This paper’s main contributions are a novel achievable scheme for LFR, referred as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">privacy key scheme</i> , and a new information theoretic converse bound for SFR. Clearly, being SFR a special case of LFR, an achievable scheme for LFR works for SFR as well, and a converse for SFR is a valid converse for LFR as well. By comparing the performance of the achievable scheme with the converse bound derived in this article (for the small cache size regime) and existing converse bounds without privacy constraints (in the remaining memory regime), the communication load of the privacy key scheme turns out to be optimal to within a constant multiplicative gap in all parameter regimes. Numerical results show that the new privacy key scheme outperforms in some regime known schemes based on the idea of virtual users, which also satisfy the stronger notion of user privacy against colluding users adopted here. Moreover, the privacy key scheme enjoys much lower subpacketization than known schemes based on virtual users.

Key Superposition Simultaneously Achieves Security and Privacy in Cache-Aided Linear Function Retrieval

This work investigates the problem of cache-aided content Secure and demand Private Linear Function Retrieval (SP-LFR), where three constraints are imposed on the system: (a) each user is interested in retrieving an arbitrary linear combination of the files in the server’s library; (b) the content of the library must be kept secure from a wiretapper who obtains the signal sent by the server; and (c) no subset of colluding users together can obtain information about the demands of the remaining users. A procedure is proposed to derive an SP-LFR scheme from a given Placement Delivery Array (PDA), which is known to give coded caching schemes with low subpacketization for systems with neither security nor privacy constraints. This procedure uses the superposition of security keys and privacy keys, in both the cache placement and transmitted signal, to guarantee content security and demand privacy, respectively. In particular, among all PDA-based SP-LFR schemes, the memory-load pairs achieved by the PDA describing the Maddah-Ali and Niesen’s scheme are Pareto optimal and have the lowest subpacketization. Moreover, the achieved load-memory tradeoff is optimal to within a constant multiplicative gap, except for the small memory regime (i.e., when the cache size is between 1 and 2) and the number of files is smaller than the number of users. Remarkably, the memory-load tradeoff does not worsen compared to the best known schemes that guarantee either only content security in all regimes or only demand privacy in the regime mentioned above.

Necessary and sufficient conditions for arbitrary linear combinations of solutions of a class of nonlinear partial differential equations to satisfy the differential equation

Necessary and sufficient conditions for arbitrary linear combinations of solutions of a class of nonlinear partial differential equations to satisfy the differential equation

Uncommonly accurate energies for the general quartic oscillator

AbstractRecent advances in the asymptotic analysis of energy levels of potentials produce relative errors in eigenvalue sums of order 10−34, but few non‐trivial potentials have been solved numerically to such accuracy. We solve the general quartic potential (arbitrary linear combination of x2 and x4) beyond this level of accuracy using a basis of several hundred oscillator states. We list the lowest 20 eigenvalues for 9 such potentials. We confirm the known asymptotic expansion for the levels of the pure quartic oscillator, and extract the next two terms in the asymptotic expansion. We give analytic formulas for expansion in up to three even basis states. We confirm the virial theorem for the various energy components to similar accuracy. The sextic oscillator levels are also given. These benchmark results should be useful for extreme tests of approximations in several areas of chemical physics and beyond.

Local times and sample path properties of the Rosenblatt process

Let Z=(Zt)t≥0 be the Rosenblatt process with Hurst index H∈(1∕2,1). We prove joint continuity for the local time of Z, and establish Hölder conditions for the local time. These results are then used to study the irregularity of the sample paths of Z. Based on analogy with similar known results in the case of fractional Brownian motion, we believe our results are sharp. A main ingredient of our proof is a rather delicate spectral analysis of arbitrary linear combinations of integral operators, which arise from the representation of the Rosenblatt process as an element in the second chaos.

Sum-Set Inequalities From Aligned Image Sets: Instruments for Robust GDoF Bounds

We present sum-set inequalities specialized to the generalized degrees of freedom (GDoF) framework. These are information theoretic lower bounds on the entropy of bounded density linear combinations of discrete, power-limited dependent random variables in terms of the joint entropies of arbitrary linear combinations of new random variables that are obtained by power level partitioning of the original random variables. These bounds generalize the aligned image sets approach, and are useful instruments to obtain GDoF characterizations for wireless networks, especially with multiple antenna nodes, subject to arbitrary channel strength and channel uncertainty levels. To demonstrate the utility of these bounds, we consider various examples of interference and broadcast channels for which we obtain tight GDoF characterizations with the aid of sum-set inequalities.