We axiomatically define (pre-)Hilbert categories. The axioms resemble those for monoidal Abelian categories with the addition of an involutive functor. We then prove embedding theorems: any locally small pre-Hilbert category whose monoidal unit is a simple generator embeds (weakly) monoidally into the category of pre-Hilbert spaces and adjointable maps, preserving adjoint morphisms and all fini...

in this paper we proved that every g-riesz basis for hilbert space $h$ with respect to $k$ by adding a condition is a riesz basis for hilbert $b(k)$-module $b(h,k)$. this is an extension of [a. askarizadeh,m. a. dehghan, {em g-frames as special frames}, turk. j. math., 35, (2011) 1-11]. also, we derived similar results for g-orthonormal and orthogonal bases. some relationships between dual fram...

The toric Hilbert scheme is a parameter space for all ideals with the same multi-graded Hilbert function as a given toric ideal. Unlike the classical Hilbert scheme, it is unknown whether toric Hilbert schemes are connected. We construct a graph on all the monomial ideals on the scheme, called the flip graph, and prove that the toric Hilbert scheme is connected if and only if the flip graph is ...

Hilbert geometry is a metric geometry that extends the hyperbolic Cayley-Klein geometry. In this video, we explain the shape of balls and their properties in a convex polygonal Hilbert geometry. First, we study the combinatorial properties of Hilbert balls, showing that the shapes of Hilbert polygonal balls depend both on the center location and on the complexity of the Hilbert domain but not o...

Hybrid registration schemes are a powerful alternative to fully automatic registration algorithms. Current methods for hybrid registration either include the landmark information as a hard constraint, which is too rigid and leads to difficult optimization problems, or as a soft-constraint, which introduces a difficult to tune parameter for the landmark accuracy. In this paper we model the defor...

In this article, the convergence of the often used iterative softthresholding algorithm for the solution of linear operator equations in infinite dimensional Hilbert spaces is analyzed in detail. As main result we show that the algorithm converges with linear rate as soon as the underlying operator satisfies the so-called finite basis injectivity property. This result quantifies the experience ...

A new iterative algorithm for the solution of minimization problems in infinitedimensional Hilbert spaces which involve sparsity constraints in form of `p-penalties is proposed. In contrast to the well-known algorithm considered by Daubechies, Defrise and De Mol, it uses hard instead of soft shrinkage. It is shown that the hard shrinkage algorithm is a special case of the generalized conditiona...

It is well-known that the support vector machine paradigm is equivalent to solving a regularization problem in a reproducing kernel Hilbert space. The squared norm penalty in the standard support vector machine controls the smoothness of the classification function. We propose, under the framework of smoothing spline ANOVA models, a new type of regularization to conduct simultaneous classificat...

Coordinate formalism on Hilbert manifolds developed in [1] is reviewed. The results of [1] are applied to the simpliest case of a Hilbert manifold: the abstract Hilbert space. In particular, functional transformations preserving properties of various linear operators on Hilbert spaces are found. Any generalized solution of an arbitrary linear differential equation with constant coefficients is ...

Motivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifold M to be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that every Sol 3–manifold is diffeomorphic to a cusp cross-section of a (generalize...