Real Constant Coefficients Research Articles

Capacity Bounds for the $K$ -User Gaussian Interference Channel

The capacity region of the $K$ -user Gaussian interference channel (GIC) is a long-standing open problem and even capacity outer bounds are little known in general. A significant progress on degrees-of-freedom (DoF) analysis, a first-order capacity approximation, for the $K$ -user GIC has provided new important insights into the problem of interest in the high signal-to-noise ratio (SNR) limit. However, such capacity approximation has been observed to have some limitations in predicting the capacity at finite SNR. In this paper, we develop a new upper-bounding technique that utilizes a new type of genie signal and applies time sharing to genie signals at $K$ receivers. Based on this technique, we derive new upper bounds on the sum capacity of the three-user GIC with constant, complex channel coefficients and then generalize to the $K$ -user case to better understand sum-rate behavior at finite SNR. We also provide closed-form expressions of our upper bounds on the capacity of the $K$ -user symmetric GIC easily computable for any $K$ . From the perspectives of our results, some sum-rate behavior at finite SNR is in line with the insights given by the known DoF results, while some others are not. In particular, the well-known $K/2$ DoF achievable for almost all constant real channel coefficients turns out to be not embodied as a substantial performance gain over a certain range of the cross-channel coefficient in the $K$ -user symmetric real case especially for large $K$ . We further investigate the impact of phase offset between the direct-channel coefficient and the cross-channel coefficients on the sum-rate upper bound for the three-user complex GIC. As a consequence, we aim to provide new findings that could not be predicted by the prior works on DoF of GICs.

Algorithms for verification of characteristic sets of discrete autonomous nD systems with n ≥ 2

An important question for multidimensional systems is to characterize the minimum information required to uniquely specify a ‘trajectory’. For a discrete nD system described by partial difference equations with real constant coefficients, the notion of a characteristic set formalizes this idea. The issue of characteristic sets for n ≥ 3 has largely remained open since Valcher’s seminal work for n = 2 (Valcher (2000)). In this paper, we provide a necessary and sufficient condition for a given subset (either a cone or a sub-lattice) in Zn to be a characteristic set for a discrete scalar autonomous nD system. This necessary and sufficient condition enables us to formulate an algebraic test for verifying whether a given cone or a sub-lattice is a characteristic set for a given discrete scalar autonomous nD system. We then provide algorithms - that are implementable using standard computational algebra packages - for doing this check. This is achieved by first converting the above-mentioned necessary and sufficient condition to another equivalent algebraic condition that is more suited for applying Grobner bases theory. We further pursue the question of ‘minimality’ of characteristic sets. We show how the idea of minimal characteristic sub-lattices is related with the notion of autonomy degree and Krull dimension. Thus, we provide a complete solution to the problem of determining if a given cone or a sub-lattice is a characteristic set for a scalar autonomous discrete nD system.

Stability radii for real linear Hamiltonian systems with perturbed dissipation

We study linear dissipative Hamiltonian (DH) systems with real constant coefficients that arise in energy based modeling of dynamical systems. We analyze when such a system is on the boundary of the region of asymptotic stability, i.e., when it has purely imaginary eigenvalues, or how much the dissipation term has to be perturbed to be on this boundary. For unstructured systems the explicit construction of the real distance to instability (real stability radius) has been a challenging problem. We analyze this real distance under different structured perturbations to the dissipation term that preserve the DH structure and we derive explicit formulas for this distance in terms of low rank perturbations. We also show (via numerical examples) that under real structured perturbations to the dissipation the asymptotical stability of a DH system is much more robust than for unstructured perturbations.

Boundary value problem with normal derivatives for a higher-order elliptic equation on the plane

For an elliptic operator of order 2l with constant (and only leading) real coefficients, we consider a boundary value problem in which the normal derivatives of order (k j −1), j = 1,..., l, where 1 ≤ k 1 < ··· < k l, are specified. It becomes the Dirichlet problem for kj = j and the Neumann problem for k j = j + 1. We obtain a sufficient condition for the Fredholm property of which problem and derive an index formula.

On stability sets for numerical discretizations of neutral delay differential equations

Abstract The paper discusses the -method discretization of the neutral delay differential equation y'(t) = ay (t) + by (t - τ) + cy' (t - τ), t &gt; 0, where a, b, c are real constant coefficients and is a positive real lag. Using recent developments on stability of appropriate delay difference equations we give a complete description of stability sets for this discretization. Some of their properties and related comparisons with the stability set for the underlying neutral differential equation are discussed as well.

Every discrete 2D autonomous system admits a finite union of parallel lines as a characteristic set

In this paper, we show that every discrete 2D autonomous system, that is described by a set of linear partial difference equations with constant real coefficients, admits a finite union of parallel lines as a characteristic set. In order to prove our claim, we first look at a special class of scalar discrete 2D systems and provide such characteristic sets for systems in this class. This special class has the property that systems in this class have their quotient rings to be finitely generated modules over a one-variable Laurent polynomial subring of the original two-variable Laurent polynomial ring in the shift operators. We show that such systems always admit a finite collection of horizontal lines for a characteristic set. We then extend this result to non-scalar discrete 2D autonomous systems. We achieve this in two steps: first, we show that every scalar discrete 2D system can be converted into a system in the above-mentioned class by a coordinate transformation on the independent variables set, $$\mathbb {Z}^2$$Z2. Using this we then show that characteristic sets for the original system can be found by applying the inverse coordinate transformation on characteristic sets of the transformed system. Since the transformed system, by virtue of being in the special class, admits a finite union of horizontal lines as a characteristic set, the original system is guaranteed to admit a characteristic set that is a coordinate transformation applied to a finite union of horizontal lines. The coordinate transformation maps this union of horizontal lines to a union of parallel, but possibly tilted, lines. In the next step, we generalize the scalar case to the general vector case: that is, systems with more than one dependent variables. The main motivation for studying characteristic sets that are unions of finitely many parallel lines is that, arguably, such sets can be called "thin" in $$\mathbb {Z}^2$$Z2 in comparison to the prevalent notions of convex cones and half-spaces as characteristic sets (see "Appendix 1").

Strongly elliptic linear operators without coercive quadratic forms. I. Constant coefficient operators and forms

A family of linear homogeneous 4th order elliptic differential operators L with real constant coefficients, and bounded nonsmooth convex domains \Omega are constructed in \mathbb{R}^6 so that the L have no constant coefficient coercive integro-differential quadratic forms over the Sobolev spaces W^{2,2}(\Omega) .

New features of stability of linear systems of differential equations with constant coefficients

We consider stability of linear systems of differential equations with constant real coefficients whose matrices are off-diagonally non-negative. The results are applied to arbitrary linear systems of differential equations with constant complex coefficients.

Gaussian bounds for higher-order elliptic differential operators with Kato type potentials

Let P(D) be a nonnegative homogeneous elliptic operator of order 2m with real constant coefficients on Rn and V be a suitable real measurable function. In this paper, we are mainly devoted to establish the Gaussian upper bound for Schrödinger type semigroup e−tH generated by H=P(D)+V with Kato type perturbing potential V, which naturally generalizes the classical result for Schrödinger semigroup e−t(Δ+V) as V∈K2(Rn), the famous Kato potential class. Our proof significantly depends on the analyticity of the free semigroup e−tP(D) on L1(Rn). As a consequence of the Gaussian upper bound, the Lp-spectral independence of H is concluded.

A formula substituting the undetermined coefficients and the annihilator methods

This article presents an easy formula for obtaining a particular solution of a linear ordinary differential equation with constant real coefficients, , where f is a function given by a linear combination of polynomials, trigonometrical and exponential real function products. The desired solution is obtained through solving straightforward lower triangular linear systems with constant coefficients. Three examples are given.

A Family of Fundamental Solutions of Elliptic Partial Differential Operators with Real Constant Coefficients

We present a construction of a family of fundamental solutions for elliptic partial differential operators with real constant coefficients. The elements of such a family are expressed by means of jointly real analytic functions of the coefficients of the operators and of the spatial variable. The aim is to write detailed expressions for such functions. Such expressions are then exploited to prove regularity properties in the frame of Schauder spaces and jump properties of the corresponding single layer potentials.

Design with L-splines

We recently obtained a criterion to decide whether a given space of parametrically continuous piecewise Chebyshevian splines (i.e., splines with pieces taken from different Extended Chebyshev spaces) could be used for geometric design. One important field of application is the class of L-splines, that is, splines with pieces taken from the null space of some fixed real linear differential operator, generally investigated under the strong requirement that the null space should be an Extended Chebyshev space on the support of each possible B-spline. In the present work, we want to show the practical interest of the criterion in question for designing with L-splines. With this in view, we apply it to a specific class of linear differential operators with real constant coefficients and odd/even characteristic polynomials. We will thus establish necessary and sufficient conditions for the associated splines to be suitable for design. Because our criterion was achieved via a blossoming approach, shape preservation will be inherent in the obtained conditions. One specific advantage of the class of operators we consider is that hyperbolic and trigonometric functions can be mixed within the null space on which the splines are based. We show that this produces interesting shape effects.

Characterizations of Hardy spaces associated to higher order elliptic operators

In this paper, the authors first show that the classical Hardy space H1(Rn) can be characterized by the non-tangential maximal functions and the area integrals associated with the semigroups e−tP and e−tP, respectively, where P is an elliptic operator with real constant coefficients of homogeneous order 2m (m⩾1). Moreover, the authors also prove that H1(Rn) can be characterized by the Riesz transforms ∇mP−1/2 if and only if m is an odd integer. In the main part of this paper, the authors develop a theory of Hardy space associated with L, where L is a higher order divergence form elliptic operator with complex bounded measurable coefficients. The authors set up a molecular Hardy space HL1(Rn) and give its characterizations by area integrals related to the semigroups e−tL and e−tL, respectively. Finally, authors give the (HL1,L1) boundedness of Riesz transforms, square functions and maximal functions associated with L.

Systems of second-order linear ODE’s with constant coefficients and their symmetries II. The case of non-diagonal coefficient matrices

We complete the analysis of the symmetry algebra L for systems of n second-order linear ODEs with constant real coefficients, by studying the case of coefficient matrices having a non-diagonal Jordan canonical form. We also classify the Levi factor (maximal semisimple subalgebra) of L , showing that it is completely determined by the Jordan form. A universal formula for the dimension of the symmetry algebra of such systems is given. As application, the case n = 5 is analyzed.

Approximation by local ℒ-splines that are exact on subspaces of the kernel of a differential operator

We construct local ℒ-splines with uniform knots that preserve subsets from the kernel of a linear differential operator ℒ of order r with constant real coefficients and pairwise distinct roots of the characteristic polynomial. Pointwise estimates are found for the error of approximation by the constructed ℒ-splines on classes of functions defined by differential operators of orders smaller than r.

Systems of second-order linear ODE’s with constant coefficients and their symmetries

Starting from the study of the symmetries of systems of 4 second-order linear ODEs with constant real coefficients, we determine the dimension and generators of the symmetry algebra for systems of n equations described by a diagonal Jordan canonical form. We further prove that some dimensions between the lower and upper bounds cannot be attained in the diagonal case, and classify the Levi factors of the symmetry algebras.

Monotony of solutions of some difference and differential equations

In this contribution motivated by some analysis of the first author concerning bounds of topological entropy it is shown that a well known sufficient condition for a difference and differential equation with constant real coefficients to possess strictly monotone solution appears to be also necessary. Transparent proofs of adequate generalizations to Banach space analogs are presented.

Local ℒ-splines preserving the differential operator kernel

Odd-order local ℒ-splines with uniform nodes are constructed. The splines preserve basis functions from the kernel of the linear differential operator ℒ with constant real coefficients and pairwise different roots of the characteristic polynomial. A pointwise error of approximation by the constructed splines on appropriate classes of differentiable functions is given.

Transmission Eigenvalues for Operators with Constant Coefficients

In this paper we study the interior transmission problem and transmission eigenvalues for multiplicative perturbations of linear partial differential operator of order $\geq2$ with constant real coefficients. Under suitable growth conditions on the symbol of the operator and the perturbation, we show the discreteness of the set of transmission eigenvalues and derive sufficient conditions on the existence of transmission eigenvalues. We apply these techniques to the case of the biharmonic operator and the Dirac system. In the hypoelliptic case we present a connection to scattering theory.

On a boundary value problem for a higher-order elliptic equation

For an elliptic 2lth-order equation with constant (and only leading) real coefficients, we consider the boundary value problem in which the (kj − 1)st normal derivatives, j = 1,..., l, are specified, where 1 ≤ k1 < ... < kl. If kj = j, then it becomes the Dirichlet problem; and if kj = j + 1, then it becomes the Neumann problem. We obtain a sufficient condition for this problem to be Fredholm and present a formula for the index of the problem.