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In this paper, we establish the following reverse Stein-Weiss inequality, namely the reversed weighted Hardy-Littlewood-Sobolev inequality, in Rn: ... for any nonnegative functions ... and ... such that .... We derive the existence of extremal functions for the above inequality. Moreover, some asymptotic behaviors are obtained for the corresponding Euler-Lagrange system. For an analogous weighted system, we prove necessary conditions of existence for any positive solutions by using the Pohozaev identity. Finally, we also obtain the corresponding Stein-Weiss and reverse Stein-Weiss inequalities on the ndimensional sphere Sn by using the stereographic projections. Our proof of the reverse Stein-Weiss inequalities relies on techniques in harmonic analysis and differs from those used in the proof of the reverse (non-weighted) Hardy-Littlewood-Sobolev inequalities. [ABSTRACT FROM AUTHOR]

A compact hyperbolic surface of genus g is said to be extremal if it admits an extremal disc, a disc of the largest radius determined only by g. We discuss how many extremal discs are embedded in non-orientable extremal surfaces of genus 6. This is the final genus in our interest because it is already known for g = 3, 4, 5, or g > 6. We show that non-orientable extremal surfaces of genus 6 admit at most two extremal discs. The locus of extremal discs is also obtained for each surface. Consequently non-orientable extremal surfaces of arbitrary genus g ≧. 3 admit at most two extremal discs. Furthermore we determine the groups of automorphisms of non-orientable extremal surfaces of genus 6 with two extremal discs. [ABSTRACT FROM AUTHOR]

We identify the exposed points of the unit ball of the dual space of the ball algebra. As a corollary, we show that the predual of H∝(Bd) has no extreme points in its unit ball. [ABSTRACT FROM AUTHOR]

Measure differential equations can model non-classical problems like the quantum effects. In this paper we will solve extremal problems for eigenvalues of measure differential equations by exploiting the approximation of general measures by smooth measures and the continuity results of eigenvalues in weak* topology of measures. [ABSTRACT FROM AUTHOR]

Barycentric interpolation is arguably the method of choice for numerical polynomial interpolation. The polynomial interpolant is expressed in terms of function values using the so-called barycentric weights, which depend on the interpolation points. Few explicit formulae for these barycentric weights are known. In [H. Wang and S. Xiang, Math. Comp., 81 (2012), 861—877], the authors have shown that the barycentric weights of the roots of Legendre polynomials can be expressed explicitly in terms of the weights of the corresponding Gaussian quadrature rule. This idea was subsequently implemented in the Chebfun package [L. N. Trefethen and others, The Chebfun Development Team, 2011] and in the process generalized by the Chebfun authors to the roots of Jacobi, Laguerre and Hermite polynomials. In this paper, we explore the generality of the link between barycentric weights and Gaussian quadrature and show that such relationships are related to the existence of lowering operators for orthogonal polynomials. We supply an exhaustive list of cases, in which all known formulae are recovered and also some new formulae are derived, including the barycentric weights for Gauss-Radau and Gauss-Lobatto points. Based on a fast O(n) algorithm for the computation of Gaussian quadrature, due to Hale and Townsend, this leads to an O(n) computational scheme for barycentric weights.Barycentric interpolation is arguably the method of choice for numerical polynomial interpolation. The polynomial interpolant is expressed in terms of function values using the so-called barycentric weights, which depend on the interpolation points. Few explicit formulae for these barycentric weights are known. In [H. Wang and S. Xiang, Math. Comp., 81 (2012), 861—877], the authors have shown that the barycentric weights of the roots of Legendre polynomials can be expressed explicitly in terms of the weights of the corresponding Gaussian quadrature rule. This idea was subsequently implemented in the Chebfun package [L. N. Trefethen and others, The Chebfun Development Team, 2011] and in the process generalized by the Chebfun authors to the roots of Jacobi, Laguerre and Hermite polynomials. In this paper, we explore the generality of the link between barycentric weights and Gaussian quadrature and show that such relationships are related to the existence of lowering operators for orthogonal polynomials. We supply an exhaustive list of cases, in which all known formulae are recovered and also some new formulae are derived, including the barycentric weights for Gauss-Radau and Gauss-Lobatto points. Based on a fast O(n) algorithm for the computation of Gaussian quadrature, due to Hale and Townsend, this leads to an O(n) computational scheme for barycentric weights. [ABSTRACT FROM AUTHOR]

Given an irreducible discrete time Markov chain on a finite state space, we consider the largest expected hitting time T(ɑ) of a set of stationary measure at least ɑ for ɑ ϵ (0, 1). We obtain tight inequalities among the values of T(ɑ) for different choices of ɑ. One consequence is that T(ɑ) = T(1/2)/ɑ for all ɑ < 1/2. As a corollary we have that if the chain is lazy in a certain sense as well as reversible, then T(1/2) is equivalent to the chain's mixing time, answering a question of Peres. We furthermore demonstrate that the inequalities we establish give an almost everywhere pointwise limiting characterisation of possible hitting time functions T(ɑ) over the domain ɑ ϵ (0, 1/2]. [ABSTRACT FROM AUTHOR]

Let X ⊂ PN be a nondegenerate irreducible closed subvariety of dimension n over the field of complex numbers and let SX ⊂ PN be its secant variety. X ⊂ PN is called 'secant defective' if dim(SX) is strictly less than the expected dimension 2n + 1. In a 1993 paper, F.L. Zak showed that for a secant defective manifold it is necessary that N ≤ (nn+2 - 1 and that the Veronese variety ν2(Pn) is the only boundary case. Recently R. Muñoz, J. C. Sierra, and L. E. Solá Conde classified secant defective varieties next to this extremal case. In this paper, we will consider secant defective manifolds X ⊂ PN of dimension n with N = (nn+2) - 1 - ∊ for ∊ ≥ 0. First, we will prove that X is an LQEL-manifold of type δ = 1 for ∊ ≤ n - 2 by showing that the tangential behavior of X is good enough to apply the Scorza lemma. Then we will completely describe the above manifolds by using the classification of conicconnected manifolds given by Ionescu and Russo. Our method generalizes previous results by Zak, and by Muñoz, Sierra, and Solá Conde. [ABSTRACT FROM AUTHOR]

We determine extremal entire functions for the problem of majorizing, minorizing, and approximating the Gaussian function e-p?x2 by entire functions of exponential type. This leads to the solution of analogous extremal problems for a wide class of even functions that includes most of the previously known examples, plus a variety of new interesting functions such as |x|α for -1 < α; log((x² + α²)/(x² + ß²)), for 0 = α < ß; log((x² + α²); and x2n log x² , for n ? N. Further applications to number theory include optimal approximations of theta functions by trigonometric polynomials and optimal bounds for certain Hilbert-type inequalities related to the discrete Hardy-Littlewood-Sobolev inequality in dimension one. [ABSTRACT FROM AUTHOR]

We characterize those valued fields for which the image of the valuation ring under every polynomial in several variables contains an element of maximal value or zero. [ABSTRACT FROM AUTHOR]

We consider the problem of finding the extremal function $ f $norm for the class of non-vanishing functions whose first $ n$We define an analytic function $ K$ and show that the functions $ K$ satisfy a certain differential equation. This equation yields a set of relationships between the area moments and the circle moments of $ \vert f\vert^2$is a polynomial of degree at most $ n$ [ABSTRACT FROM AUTHOR]

The article examines the maximum principle and convergence of second order central schemes for scalar hyperbolic conservation laws. It explores the development of a numerical algorithm through nonlinear slope reconstructions with minmod limiters. It further provides a mathematical proof for maximum principle and convergence without reducing the first order at local extrema.

We consider the $ L_p$-Gaussian variables ( $ -1\leq q\leq 1$ the $ L_p$ $ 1\leq p \leq 2$), whilst the $ L_p$ $ 2\leq p \leq \infty$-dependence. Moreover, the extremal cases $ q = \pm 1$ [ABSTRACT FROM AUTHOR]

This paper studies the CR-version of Obata theorem on a pseudo-Hermitian CR-manifold $ (M,\theta)$ with contact form $ \theta=\frac{1}{2i} (\partial \rho_A - \bar{\partial} \rho_A)$ $ \rho_A(z)=\vert z\vert^2 + \mathrm{Re} \sum^n_{j=1}A_jz^2_j-1$ $ A_j \in (-1,1).$ [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the next extremal case for a Castelnuovo-type bound $ \mathrm{reg} V \le \lceil (\deg V - 1)/ {\mbox{\rm codim} } V \rceil + 1$. A Buchsbaum variety with the maximal regularity is known to be a divisor on a variety of minimal degree if the degree of the variety is large enough. We show that a Buchsbaum variety satisfying $ \mathrm{reg} V = \lceil (\deg V - 1)/ {\mbox{\rm codim} } V \rceil$ $ \deg V \gg 0$ [ABSTRACT FROM AUTHOR]

We will study the dependence of eigenvalues of the one-dimensional $ p$ spaces, $ 1\le \gamma\le \infty$ norms. As applications, we will study some extremal problems of eigenvalues by developing some analytical methods. [ABSTRACT FROM AUTHOR]

The article examines the asymptotic properties of special types of extremal point configurations called greedy energy points. It states that the extremal point configurations are generated by a greedy algorithm which is an energy minimizing construction. It notes that the behavior of greedy points significantly differs from that of minimal energy points in some situations.

Motivated by the equation satisfied by the extremals of certain Hardy-Sobolev type inequalities, we show sharp $ L^q$ $ \mathbb{R}^n$ Hardy-Sobolev inequality. [ABSTRACT FROM AUTHOR]

Let $ T$-skeleton of $ T$ by attaching new edges and vertices and subdividing its faces. Such refinements provide a mechanism of convergence of the discrete triangulation to the classical surface. We will prove a bound on the distortion of the discrete extremal lengths of path families on $ T$. In particular, the result does not require bounded degree. [ABSTRACT FROM AUTHOR]

We study a disc formula for the relative extremal function for a subset of a complex manifold and apply it to give a description of pluripolar sets and polynomial hulls. [ABSTRACT FROM AUTHOR]

The article provides the classification of ternary extremal self-dual codes of length 28. It reveals that all ternary self-dual codes are important class of linear codes for both theoretical and practical reasons, and it is fundamental problem to classify self-dual codes of modest length and determine the largest minimum weight among self-dual codes of that length. It also presents the self-dual codes and unimodular lattices.

A compact hyperbolic surface of genus $g$ is said to be extremal if it admits an extremal disc, a disc of the largest radius determined by $g$. We know how many extremal discs are embedded in a non-orientable extremal surface of genus $g=3$ or $g>6$. We show in the present paper that there exist $144$ non-orientable extremal surfaces of genus $4$, and find the locations of all extremal discs in those surfaces. As a result, each surface contains at most two extremal discs. Our methods used here are similar to those in the case of $g=3$. [ABSTRACT FROM AUTHOR]

In this paper, we study the problem of finding the extremal element for a linear functional over a uniformly convex Banach space. We show that a unique extremal element exists and depends continuously on the linear functional, and vice versa. Using this, we simplify and clarify Ryabykh's proof that for any linear functional on a uniformly convex Bergman space with kernel in a certain Hardy space, the extremal function belongs to the corresponding Hardy space. [ABSTRACT FROM AUTHOR]

Let $a in mathbb {C} setminus [-1,1]$ be given. We consider the problem of finding $sup |p(a)|$ among all polynomials $p$ with complex coefficients of degree less than or equal to $n$ with $max _{-1leq x leq 1}|p(x)| leq 1$. We derive an asymptotic expression for the extremal polynomial and for the extremal value in terms of elementary functions. The solution is based on the description of Zolotarev polynomials with respect to square root polynomial weights. [ABSTRACT FROM AUTHOR]

In this paper we settle a conjecture of Ball that uniform quasiconvexity and uniform positivity of the second variation are sufficient for a $C^{1}$ extremal to be a strong local minimizer. Our result holds for a class of variational functionals with a power law behavior at infinity. The proof is based on the decomposition of an arbitrary variation of the dependent variable into its purely strong and weak parts. We show that these two parts act independently on the functional. The action of the weak part can be described in terms of the second variation, whose uniform positivity prevents the weak part from decreasing the functional. The strong part ``localizes'', i.e. its action can be represented as a superposition of ``Weierstrass needles'', which cannot decrease the functional either, due to the uniform quasiconvexity conditions. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to analyse a class of quadratic extremal problems defined on various Hilbert spaces of analytic functions, thereby generalizing an extremal problem on the Dirichlet space which was solved by S.D. Fisher. Each extremal problem considered here is shown to be connected with a system of orthogonal polynomials. The orthogonal polynomials then determine properties of the extremal function, and provide information about the existence of extremals. [ABSTRACT FROM AUTHOR]

Canonical divisors in Bergman spaces can be found as solutions of extremal problems. We derive a formula for certain extremal functions in the weighted Bergman spaces $A^p_{\alpha}$ for $\alpha > -1$ and $1 \leq p < \infty$. This leads to a study of the zeros of a specific family of hypergeometric functions. [ABSTRACT FROM AUTHOR]

We investigate divisorial contractions of extremal rays from smooth fourfolds. When the exceptional divisor is contracted to a curve, we prove that the divisor is a $\mathbb{P}^{2}$-bundle or quadric bundle over a smooth curve and the contraction is the blowing up along the curve. Furthermore we determine the local analytic structure of the contraction. [ABSTRACT FROM AUTHOR]

We show that Teichmüller's classical lower bound for the capacity of a ring domain, obtained by circular symmetrization, can be replaced by an explicit one which is almost always better. The proof is based on a duplication formula for the solution of an associated extremal problem. Some inequalities are obtained for conformal invariants. [ABSTRACT FROM AUTHOR]

Let X ⋐ Pn, n ≥ 3, be an integral and non-degenerate curve. For any P ∈ Pn the X-rank rX(P) of P is the minimal cardinality of a set S ⋐ Y such that P is in the linear span of S. Landsberg and Teitler proved that rX(P) ≤ n for any X and any P. Here we classify the pairs (X,Q), Q ∈ Xreg, such that all points of the tangent line TQX (except Q) have X-rank n: X ≅ P¹ and TQX has order of contact deg(X) + 2 - n with X at Q. [ABSTRACT FROM AUTHOR]