1 Hilbert modular surfaces 3 1.1 The Hilbert modular group . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 The Baily-Borel compactification . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.1 Siegel domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Hilbert modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Mk(Γ) is finite dimensional...

In a previous paper we defined the concept of an affinized projective variety and its associated Hilbert series. We computed the Hilbert series for varieties associated to quadratic monomial ideals. In this paper we show how to apply these results to affinized flag varieties. We discuss various examples and conjecture a correspondence between the Hilbert series of an affinized flag variety and ...

Noncommutative algebras, defined by the generators and relations, are considered. The definition and main results connected with the Gröbner basis, Hilbert series and Anick’s resolution are formulated. Most attention is paid to universal enveloping algebras. Four main examples illustrate the main concepts and ideas. Algorithmic problems arising in the calculation of the Hilbert series are inves...

Analogues for Hilbert C∗-modules of classical results of Fourier series theory in Hilbert spaces are considered. Relations between different properties of orthogonal and orthonormal systems for Hilbert C∗-modules are studied with special attention paid on the differences with the well-known Hilbert space situation.

A multi-dimensional ensemble empirical mode decomposition (MEEMD) for multidimensional data (such as images or solid with variable density) is proposed here. The decomposition is based on the applications of ensemble empirical mode decomposition (EEMD) to slices of data in each and every dimension involved. The final reconstruction of the corresponding intrinsic mode function (IMF) is based on ...

The vanishing ideal I of a subspace arrangement V 1 ∪ V 2 ∪ · · · ∪ V m ⊆ V is an intersection I 1 ∩ I 2 ∩ · · · ∩ I m of linear ideals. We give a formula for the Hilbert polynomial of I if the subspaces meet transversally. We also give a formula for the Hilbert series of the product ideal J = I 1 I 2 · · · I m without any assumptions on the subspace arrangement. It turns out that the Hilbert s...

‎This paper presents the numerical solution for a class of fractional differential equations. The fractional derivatives are described in the Caputo cite{1} sense. We developed a reproducing kernel method (RKM) to solve fractional differential equations in reproducing kernel Hilbert space. This method cannot be used directly to solve these equations, so an equivalent transformation is made by u...

Semi-nonparametric (SNP) models are models where only a part of the model is parametrized, and the non-specified part is an unknown function which is represented by an infinite series expansion. Therefore, SNP models are in essence models with infinitely many parameters. The theoretical foundation of series expansions of functions is Hilbert space theory, in particular the properties of Hilbert...

motivating the perturbations of frames in hilbert and banach spaces, in this paper we introduce the invariance of fr'echet frames under perturbation. also we show that for any fr'echet spaces, there is a fr'echet frame and any element in these spaces  has a series expansion.