相关论文: On Function Theory in Quantum Disc: Invariant Kern…

Our previous work (math.QA/9808015) introduces the basic notions and announces some results on function theory in the quantum disc. The present paper establishes a relationship between those results and the quantum groups theory.

This work presents proofs of the main results of (math.QA/9808015), except those on q-Berezin transform to appear in a subsequent work. The notation and the results of (math.QA/9808037) and (math.QA/9808047) are used.

This work contains a proof of theorem 7.3 from math.QA/9808015. This theorem demonstrates the Berezin method to be applicable for producing a well known one-parameter deformation of the quantum disc.

This work considers the algebras of functions in the quantum matrix ball. An explicit formula for a positive invariant integral is presented.

The study of representations invariant to common transformations of the data is important to learning. Most techniques have focused on local approximate invariance implemented within expensive optimization frameworks lacking explicit…

The use of kernel functions is a common technique to extract important features from data sets. A quantum computer can be used to estimate kernel entries as transition amplitudes of unitary circuits. Quantum kernels exist that, subject to…

We find kernel functions of the $q$-Heun equation and its variants. We apply them to obtain $q$-integral transformations of solutions to the $q$-Heun equation and its variants. We discuss special solutions of the $q$-Heun equation from the…

We study unbounded invariant and covariant derivations on the quantum disk. In particular we answer the question whether such derivations come from operators with compact parametrices and thus can be used to define spectral triples.

Quantum kernels are reproducing kernel functions built using quantum-mechanical principles and are studied with the aim of outperforming their classical counterparts. The enthusiasm for quantum kernel machines has been tempered by recent…

One of the most natural connections between quantum and classical machine learning has been established in the context of kernel methods. Kernel methods rely on kernels, which are inner products of feature vectors living in large feature…

We consider positive semidefinite kernels valued in the $*$-algebra of adjointable operators on a VE-space (Vector Euclidean space) and that are invariant under actions of $*$-semigroups. A rather general dilation theorem is stated and…

In quantum mechanics, one can express the evolution operator and other quantities in terms of functional integrals. The main goal of this paper is to prove corresponding results in the geometric approach to quantum theory. We apply these…

An operator theoretic approach to invariant integration theory on non-compact quantum spaces is introduced on the example of the quantum (n,1)-matrix ball O_q(Mat_{n,1}). In order to prove the existence of an invariant integral, operator…

Invariance to nuisance transformations is one of the desirable properties of effective representations. We consider transformations that form a \emph{group} and propose an approach based on kernel methods to derive local group invariant…

The present work develops certain analytical tools required to construct and compute invariant kernels on the space of complex covariance matrices. The main result is the $\mathrm{L}^1$--Godement theorem, which states that any invariant…

This work deals with function theory on quantum complex hyperbolic spaces. The principal notions are expounded. We obtain explicit formulas for invariant integrals on `finite' functions on a quantum hyperbolic space and on the associated…

We study unitarity of the induced representations from coisotropic quantum subgroups which were introduced in math.QA/9804138. We define a real structure on coisotropic subgroups which determines an involution on the homogeneous space. We…

This is the first of a sequence of papers proving the quantum invariance under ordinary flops over an arbitrary smooth base. In this first part, we determine the defect of the cup product under the canonical correspondence and show that it…

It has been hypothesized that quantum computers may lend themselves well to applications in machine learning. In the present work, we analyze function classes defined via quantum kernels. Quantum computers offer the possibility to…

A differential calculus is set up on a deformation of the oscillator algebra. It is uniquely determined by the requirement of invariance under a seven-dimensional quantum group. The quantum space and its associated differential calculus are…