By a quantum metric space we mean a C∗-algebra (or more generally an order-unit space) equipped with a generalization of the usual Lipschitz seminorm on functions which one associates to an ordinary metric. We develop for compact quantum metric spaces a version of Gromov–Hausdorff distance. We show that the basic theorems of the classical theory have natural quantum analogues. Our main example ...

We study algebraic neural networks (AlgNNs) with commutative algebras which unify diverse architectures such as Euclidean convolutional networks, graph and group under the umbrella of signal processing. An AlgNN is a stacked layered information processing structure where each layer conformed by an algebra, vector space homomorphism between algebra endomorphisms space. Signals are modeled elemen...

A spectral function on a formally real Jordan algebra is a real-valued function which depends only on the eigenvalues of its argument. One convenient way to create them is to start from a function f : R 7→ R which is symmetric in the components of its argument, and to define the function F (u) := f(λ(u)) where λ(u) is the vector of eigenvalues of u. In this paper, we show that this construction...

in this paper, we consider a class of time-dependent neutral stochastic evolution equations with the infinite delay and a fractional brownian motion in a hilbert space. we establish the existence and uniqueness of mild solutions for these equations under non-lipschitz conditions with lipschitz conditions being considered as a special case. an example is provided to illustrate the theory

‎in this work we prove malliavin differentiability for the solution to an sde with locally lipschitz and semi-monotone drift‎. ‎to prove this formula‎, ‎we construct a sequence of sdes with globally lipschitz drifts and show that the $p$-moments of their malliavin derivatives are uniformly bounded‎.

Our aim in this paper is to prove an analog of Younis's Theorem on the image under the Jacobi transform of a class functions satisfying a generalized Dini-Lipschitz condition in the space $mathrm{L}_{(alpha,beta)}^{p}(mathbb{R}^{+})$, $(1< pleq 2)$. It is a version of Titchmarsh's theorem on the description of the image under the Fourier transform of a class of functions satisfying the Dini-Lip...

Starting with a spectral triple on unital C⁎-algebra A an action of discrete group G, if the is uniformly bounded (in Lipschitz sense) reduced crossed product A⋊rG constructed in [23]. The main instrument Kasparov external product. We note that this construction still works for twisted products when appropriate sense. Under suitable assumptions we discuss some basic properties resulting triples...

Abstract In this paper, we study the groups of isometries and set bi‐Lipschitz automorphisms spectral triples from a metric viewpoint, in propinquity framework Latrémolière. particular, prove that these sets are compact automorphism group triple ‐algebra with respect to Monge–Kantorovich metric, which induces topology pointwise convergence. We then necessary sufficient condition for convergence...