For any irrational θ and rational number p/q such that q|qθ−p| < 1, a projection e of trace q|qθ−p| is constructed in the the irrational rotation algebra Aθ that is invariant under the Fourier transform. (The latter is the order four automorphism U 7→ V 7→ U, where U, V are the canonical unitaries generating Aθ .) Further, the projection e is approximately central, the cut down algebra eAθe con...

We generalize the classical universal approximation theorem for neural networks to case of complex-valued networks. Precisely, we consider feedforward with a complex activation function σ:C→C in which each neuron performs operation CN→C,z↦σ(b+wTz) weights w∈CN and bias b∈C, σ applied componentwise. completely characterize those functions associated have property, meaning that they can uniformly...

We develop some aspects of the homological algebra persistence modules, in both one-parameter and multi-parameter settings, considered as either sheaves or graded modules. The two theories are different. consider module sheaf tensor product Hom bifunctors well their derived functors, Tor Ext, give explicit computations for interval a classification injective, projective, flat state Kunneth theo...

This paper presentS a special case of Explanation-Based Generalization (EBG), Goal Generalization. Goal generalization. which cries to find a proof for the most general solvable ve~ion of a specific goal while solving the specific goal, is an application of EBG methods in automatic theorem proving and is a general technique applicable to many goal-oriented theorem proving systems. We will descr...

The effect of a small amount of noise on the standard mapping is considered. Whenever the standard mapping possesses accelerator modes (where the action increases approximately linearly with time), the diffusion coefficient contains a term proportional to the reciprocal of the variance of the noise term. At large values of the stochasticity parameter, the accelerator modes exhibit a universal b...

We consider the statistics of the scattering coefficient S of a chaotic microwave cavity coupled to a single port. We remove the nonuniversal effects of the coupling from the experimental S data using the radiation impedance obtained directly from the experiments. We thus obtain the normalized scattering coefficient whose probability density function (PDF) is predicted to be universal in that i...

Let A be a unital separable amenable C∗-algebra and C be a unital C∗-algebra with certain infinite property. We show that two full monomorphisms h1, h2 : A → C are approximately unitarily equivalent if and only if [h1] = [h2] in KL(A,C). Let B be a non-unital but σ-unital C ∗-algebra for which M(B)/B has the certain infinite property. We prove that two full essential extensions are approximatel...

In this paper we establish a direct connection between stable approximate unitary equivalence for ∗-homomorphisms and the topology of the KK-groups which avoids entirely C*-algebra extension theory and does not require nuclearity assumptions. To this purpose we show that a topology on the Kasparov groups can be defined in terms of approximate unitary equivalence for Cuntz pairs and that this to...