The Hilbert Series of the Ring Associated to an Almost Alternating Matrix

We give an explicit formula for the Hilbert Series of an algebra defined by a linearly presented, standard graded, residual intersection of a grade three Gorenstein ideal.     

A Hilbert Transform for Matrix Functions on Fractal Domains

We consider H?lder continuous circulant (2 × 2) matrix functions G¹₂{{\bf G}^1_2} defined on the fractal boundary Γ of a Jordan domain Ω in \mathbbR²ⁿ{\mathbb{R}^{2n}}. The main goal is to establish a Hilbert transform for such functions, within the framework of Hermitian Clifford analysis. This is a higher dimensional function theory centered around the simultaneous null solutions of two first order vector valued differential operators, called Hermitian Dirac operators. In Brackx et al. (Bull Braz Math Soc 40(3): 395–416, 2009) a Hermitian Cauchy integral was constructed by means of a matrix approach using circulant (2 × 2) matrix functions, from which a Hilbert transform was derived in Brackx et al. (J Math Anal Appl 344: 1068–1078, 2008) for the case of domains with smooth boundary. However, crucial parts of the method are not extendable to the case where the boundary of the considered domain is fractal. At present we propose an alternative approach which will enable us to define a new Hermitian Hilbert transform in that case. As a consequence, we give necessary and sufficient conditions for the Hermitian monogenicity of a circulant matrix function G¹₂{{\bf G}^1_2} in the interior and exterior of Ω, in terms of its boundary value g¹₂=G¹₂|_G{{\bf g}^1_2={\bf G}^1_2|_\Gamma}, extending in this way also results of Abreu Blaya et al. (Bound. Value Probl. 2008: 2008) (article ID 425256), (article ID 385874), where Γ is required to be Ahlfors–David regular.     

Matrix approaches to approximate solutions of variational inequalities in Hilbert spaces

A matrix approach to approximating solutions of variational inequalities in Hilbert spaces is introduced. This approach uses two matrices: one for iteration process and the other for regularization. Ergodicity and convergence (both weak and strong) are studied. Our methods combine new or well-known iterative methods (such as the original Mann’s method) with regularized processes involved regular matrices in the sense of Toeplitz.     

Eigenfunctions of the hyperbolic composition operator

Letu inH ² be zero at one of the fixed points of a hyperbolic Möbius transform of the unit diskD. We will show, under some additional conditions onu, that the doubly cyclic subspaceS _u =V _n=– C ⁿ u contains nonconstant eigenfunctions of the composition operatorC . This implies that the cyclic subspace generated byu is not minimal. If there is an infinite dimensional minimal invariant subspace ofC (which is equivalent to the existance of an operator with only trivial invariant subspaces), then it is generated by a function with singularities at the fixed points of .     

Riemann–Hilbert Problems,Matrix Orthogonal Polynomials and Discrete Matrix Equations with Singularity Confinement

In this paper, matrix orthogonal polynomials in the real line are described in terms of a Riemann–Hilbert problem. This approach provides an easy derivation of discrete equations for the corresponding matrix recursion coefficients. The discrete equation is explicitly derived in the matrix Freud case, associated with matrix quartic potentials. It is shown that, when the initial condition and the measure are simultaneously triangularizable, this matrix discrete equation possesses the singularity confinement property, independently if the solution under consideration is given by the recursion coefficients to quartic Freud matrix orthogonal polynomials or not.     

Eigenfunctions and optimal orbits

For the algebraic structure ($\text{R}$, max, +) we study the continuous analogue of the eigenvector-eigenvalue problem and relate it to a minimal-cost orbit problem. An explicit solution is given for the concave-quadratic case.     

The Hilbert stack

Let π:X→S$\pi :X\to S$ be a morphism of algebraic stacks that is locally of finite presentation with affine stabilizers. We prove that there is an algebraic S-stack—the Hilbert stack—parameterizing proper algebraic stacks mapping quasi-finitely to X. This was previously unknown, even for a morphism of schemes.     

Estimates for Dirichlet Eigenfunctions

Estimates for the Dirichlet eigenfunctions near the boundaryof an open, bounded set in euclidean space are obtained. Itis assumed that the boundary satisfies a uniform capacitarydensity condition.     

误差估计中矩阵求逆条件数的最优性在Hilbert空间中的推广

本文利用Ｈｉｌｂｅｒｔ空间中可逆算子的极分解定理，将误差估计中矩阵求逆条件数的最优性在Ｈｉｌｂｅｒｔ空间中进行推广，证明了线性有界算子Ａ的求逆条件数Ｋ（Ａ）＝ＡＡ－１在求算子扰动逆（Ａ＋Ｅ）－１的相对误差界中的极小性质，指出了算子求逆条件数在误差估计中为仅与算子Ａ有关的最佳常数值．     

The Hilbert Series of the Clique Complex

For a graph G, we show a theorem that establishes a correspondence between the fine Hilbert series of the Stanley-Reisner ring of the clique complex for the complementary graph of G and the fine subgraph polynomial of G. We obtain from this theorem some corollaries regarding the independent set complex and the matching complex.     

On the Ranges of Eigenfunctions on Compact Manifolds

The aim of this note is to give a sharp upper bound on the ratio [formula] where is a nonconstant eigenfunction for the Laplace–Beltramioperator on a connected compact Riemannian manifold withoutboundary. This ratio is always positive, since max>0 andmin<0 for every nonconstant eigenfunction. We assume thatmax–min, in order to simplify the notation. For the caseof a two-dimensional manifold with nonnegative Ricci curvature,our theorem implies that the above ratio is less than the ratioof the maximum divided by the absolute value of the minimumof the Bessel function of order zero. The proof is based on a gradient estimate from a previous paperof the author (see [5]), which in turn was proved using themaximum principle technique. In contrast to the standard applicationsof gradient estimates, which are based on integration alonggeodesics, we arrive at a contradiction by integrating the gradientestimate over small spheres centred at a point where the absolutevalue of the eigenfunction attains its maximum. The main motivation for our work is that the ratio of the maximumand the minimum of an eigenfunction plays a role in estimatesof the corresponding eigenvalues (see [5] and [7]). More precisely,our theorem implies that there are minimizing sequences of compactmanifolds such that the first eigenvalues of the manifolds approachthe corresponding lower bound for the first eigenvalue obtainedin [5, Theorem 2] for every possible ratio of the maximum andthe minimum of the corresponding eigenfunction. 1991 MathematicsSubject Classification 58G25.     

Eigenfunctions of the Laplacian on rotationally symmetric manifolds

Eigenfunctions of the Laplacian on a negatively curved, rotationally symmetric manifold are constructed explicitly under the assumption that an integral of converges. This integral is the same one which gives the existence of nonconstant harmonic functions on

     

Rodriguez-type Representation for the Eigenfunctions of Durrmeyer-type Operators

In this paper we consider a generalization of different variants of Durrmeyer- type modifications of Baskakov and Meyer- König and Zeller operators. We prove a general result concerning the commutativity of these operators with certain differential operators. Prom this result a Rodriguez- type formula for the eigenfunctions follows as a corollary.     

The distortion of Hilbert space

The unit sphere of Hilbert space, ₂, is shown to contain a remarkable sequence of nearly orthogonal sets. Precisely, there exist a sequence of sets of norm one elements of ₂, (C _i ) _i=1 , and reals _i 0 so that a) each setC _i has nonempty intersection with every infinite dimensional closed subspace of ₂ and b) forij,xC, andyC _j , |x, y|<_{min(i, j)} E. Odell was partially supported by NSF and TARP. Th. Schlumprecht was partially supported by NSF and LEQSF.