Related papers: Minimal cubature rules on an unbounded domain
The nodes of certain minimal cubature rule are real common zeros of a set of orthogonal polynomials of degree $n$. They often consist of a well distributed set of points and interpolation polynomials based on them have desired convergence…
In his classical paper [5], Koornwinder studied a family of orthogonal polynomials of two variables, derived from symmetric polynomials. This family possesses a rare property that orthogonal polynomials of degree $n$ have $n(n+1)/2$ real…
Cubature rules on the triangle have been extensively studied, as they are of great practical interest in numerical analysis. In most cases, the process by which new rules are obtained does not preclude the existence of similar rules with…
A recent result in [2] on the non-existence of Gauss-Lobatto cubature rules on the triangle is strengthened by establishing a lower bound for the number of nodes of such rules. A method of constructing Lobatto type cubature rules on the…
Minimal cubature rules of degree $4n-1$ for the weight functions $$ W_{\a,\b,\pm \frac12}(x,y) = |x+y|^{2\a+1} |x-y|^{2\b+1} ((1-x^2)(1-y^2))^{\pm \frac12} $$ on $[-1,1]^2$ are constructed explicitly and are shown to be closed related to…
In this note, we prove that the infimum of the mean curvature on any disconnected boundary component of an unbounded mean convex domain in $\mathbb{R}^n$ must be zero.
For a family of near banded Toeplitz matrices, generalized characteristic polynomials are shown to be orthogonal polynomials of two variables, which include the Chebyshev polynomials of the second kind on the deltoid as a special case.…
The search for multivariate quadrature rules of minimal size with a specified polynomial accuracy has been the topic of many years of research. Finding such a rule allows accurate integration of moments, which play a central role in many…
In this short note we construct unbounded families of minimal surfaces of general type with canonical map of degree 4 such that the limits of the slopes assume countably many different values among 6+2/3 and 8.
Define a subset of the complex plane to be a Rolle's domain if it contains (at least) one critical point of every complex polynomial P such that P(-1)=P(1). Define a Rolle's domain to be minimal if no proper subset is a Rolle's domain. In…
We consider families of mappings with moduli inequalities, having different definition domains. Under some additional assumptions we have proved that such families are uniformly equicontinuous. We have considered four main cases: when…
As a complement to \cite{X12}, minimal cubature rules of degree $4m+1$ for the weight functions $$ \mathcal{W}_{\alpha,\beta ,\pm \frac12}(x,y) = |x+y|^{2\alpha+1} |x-y|^{2\beta+1} ((1-x^2)(1-y^2))^{\pm \frac12} $$ on $[-1,1]^2$ are shown…
We investigate the problem of constructing unextendible product bases in the qubit case - that is, when each local dimension equals 2. The cardinality of the smallest unextendible product basis is known in all qubit cases except when the…
This note presents absolute bounds on the size of the coefficients of the characteristic and minimal polynomials depending on the size of the coefficients of the associated matrix. Moreover, we present algorithms to compute more precise…
This is a preliminary note on a family of minimal surfaces in the 3-sphere defined by a compatible fourth order equation. The minimal surfaces are geometrically characterized either by having a surface of revolution like induced metric, or…
A ring has bounded factorizations if every cancellative nonunit $a \in R$ can be written as a product of atoms and there is a bound $\lambda(a)$ on the lengths of such factorizations. The bounded factorization property is one of the most…
We provide new families of minimal codes in any characteristic. Also, an inductive construction of minimal codes is presented.
Several recently discovered properties of multiple families of special polynomials (some orthogonal and some not) that satisfy certain differential, difference or q-difference equations are reviewed. A general method of construction of…
We give the lower bound for the growth of the maximum value for a solution to the minimal surface equation with 0 boundary values over an unbounded simply connected domain.
We introduce and study minimal (with respect to inclusion) solutions of systems of tropical linear differential equations. We describe the set of all minimal solutions for a single equation. It is shown that any tropical linear differential…