The duality theorem for locally compact Abelian <Emphasis Type="Italic">n</Emphasis>-groups

We obtain a generalization of the Pontryagin-Van Campen theorem to the case of locally compact topological n-groups. We also consider the convolutions of measures and the Fourier transform on locally compact topological n-groups.     

Grade zero part of forced graded algebras

This paper concerns a certain subcategory of the category of representations for a semisimple algebraic group G in characteristic p, which arises from the semisimple modules for the corresponding quantum group at a p-th root of unity. The subcategory, thus, records the cohomological difference between quantum groups and algebraic groups. We define translation functors in this category and use them to obtain information on the irreducible characters for G when the Lusztig character formula does not hold.     

The Quaternion Domain Fourier Transform and its Properties

So far quaternion Fourier transforms have been mainly defined over \({\mathbb{R}^2}\) as signal domain space. But it seems natural to define a quaternion Fourier transform for quaternion valued signals over quaternion domains. This quaternion domain Fourier transform (QDFT) transforms quaternion valued signals (for example electromagnetic scalar-vector potentials, color data, space-time data, etc.) defined over a quaternion domain (space-time or other 4D domains) from a quaternion position space to a quaternion frequency space. The QDFT uses the full potential provided by hypercomplex algebra in higher dimensions and may moreover be useful for solving quaternion partial differential equations or functional equations, and in crystallographic texture analysis. We define the QDFT and analyze its main properties, including quaternion dilation, modulation and shift properties, Plancherel and Parseval identities, covariance under orthogonal transformations, transformations of coordinate polynomials and differential operator polynomials, transformations of derivative and Dirac derivative operators, as well as signal width related to band width uncertainty relationships.     

On the Probabilistic Nature of Quantum Mechanics and the Notion of Closed Systems

The notion of “closed systems” in Quantum Mechanics is discussed. For this purpose, we study two models of a quantum mechanical system P spatially far separated from the “rest of the universe” Q. Under reasonable assumptions on the interaction between P and Q, we show that the system P behaves as a closed system if the initial state of P ∨ Q belongs to a large class of states, including ones exhibiting entanglement between P and Q. We use our results to illustrate the non-deterministic nature of quantum mechanics. Studying a specific example, we show that assigning an initial state and a unitary time evolution to a quantum system is generally not sufficient to predict the results of a measurement with certainty.     

Generalized <Emphasis Type="Italic">p</Emphasis>-Adic Fourier Transform and Estimates of Integral Modulus of Continuity in Terms of This Transform

We consider a new class of functions on the p-adic linear space ? _p ⁿ for which a Fourier transform can be defined.We prove equalities of Parseval type, an inversion formula and a sufficient condition for a function to be represented as this Fourier transform. Also we give a sharp estimate of the L²(? _p ⁿ ) modulus of continuity in terms of Fourier transform generalizing the result of S. S. Platonov in the case n = 1. Finally we prove a generalization of this result and its converse for L^q(? _p ⁿ ) with appropriate q.     

Properly discontinuous group actions on affine homogeneous spaces

Let G be a real algebraic group, H ≤ G an algebraic subgroup containing a maximal reductive subgroup of G, and Γ a subgroup of G acting on G/H by left translations. We conjecture that Γ is virtually solvable provided its action on G/H is properly discontinuous and ΓG/H is compact, and we confirm this conjecture when G does not contain simple algebraic subgroups of rank ≥2. If the action of Γ on G/H (which is isomorphic to an affine linear space Aⁿ) is linear, our conjecture coincides with the Auslander conjecture. We prove the Auslander conjecture for n ≤ 5.     

Algebraic <Emphasis Type="Italic">K</Emphasis>-theory of Gorenstein projective modules

We introduce the Gorenstein algebraic K-theory space and the Gorenstein algebraic K-group of a ring, and show the relation with the classical algebraic K-theory space, and also show the ‘resolution theorem’ in this context due to Quillen. We characterize the Gorenstein algebraic K-groups by two different algebraic K-groups and by the idempotent completeness of the Gorenstein singularity category of the ring. We compute the Gorenstein algebraic K-groups along a recollement of the bounded Gorenstein derived categories of CM-nite Gorenstein algebras.     

Involutions and trivolutions on second dual of algebras related to locally compact groups and topological semigroups

We investigate involutions and trivolutions in the second dual of algebras related to a locally compact topological semigroup and the Fourier algebra of a locally compact group. We prove, among the other things, that for a large class of topological semigroups namely, compactly cancellative foundation \(*\)-semigroup S when it is infinite non-discrete cancellative, \(M_a(S)^{**}\) does not admit an involution, and \(M_a(S)^{**}\) has a trivolution with range \(M_a(S)\) if and only if S is discrete. We also show that when G is an amenable group, the second dual of the Fourier algebra of G admits an involution extending one of the natural involutions of A(G) if and only if G is finite. However, \(A(G)^{**}\) always admits trivolution.     

Universal geometrical equivalence of the algebraic structures of common signature

This article is a part of our effort to explain the foundations of algebraic geometry over arbitrary algebraic structures [1–8]. We introduce the concept of universal geometrical equivalence of two algebraic structures A and B of a common language L which strengthens the available concept of geometrical equivalence and expresses the maximal affinity between A and B from the viewpoint of their algebraic geometries. We establish a connection between universal geometrical equivalence and universal equivalence in the sense of equality of universal theories.     

ON ALGEBRAIC VOLUME DENSITY PROPERTY

A smooth affine algebraic variety X equipped with an algebraic volume form ω has the algebraic volume density property (AVDP) if the Lie algebra generated by complete algebraic vector fields of ω-divergence zero coincides with the space of all algebraic vector fields of ω-divergence zero. We develop an effective criterion of verifying whether a given X has AVDP. As an application of this method we establish AVDP for any homogeneous space X = G/R that admits a G-invariant algebraic volume form where G is a linear algebraic group and R is a closed reductive subgroup of G.     

Flat coordinates for Saito Frobenius manifolds and string theory

We investigate the connection between the models of topological conformal theory and noncritical string theory with Saito Frobenius manifolds. For this, we propose a new direct way to calculate the flat coordinates using the integral representation for solutions of the Gauss–Manin system connected with a given Saito Frobenius manifold. We present explicit calculations in the case of a singularity of type A _n . We also discuss a possible generalization of our proposed approach to SU(N) _k /(SU(N) _k+1 × U(1)) Kazama–Suzuki theories. We prove a theorem that the potential connected with these models is an isolated singularity, which is a condition for the Frobenius manifold structure to emerge on its deformation manifold. This fact allows using the Dijkgraaf–Verlinde–Verlinde approach to solve similar Kazama–Suzuki models.     

On branching index of algebraic extension spectrum points

The paper studies the branching A-index and the topological branching index of points of the spectrum of Arens-Hoffman algebraic extension of a semisimple commutative Banach algebras A.     

Quadratic differentials (A(z − a)(z − b)/(z − c)2) dz2 and algebraic Cauchy transform

We discuss the representability almost everywhere (a.e.) in C of an irreducible algebraic function as the Cauchy transform of a signed measure supported on a finite number of compact semi-analytic curves and a finite number of isolated points. This brings us to the study of trajectories of the particular family of quadratic differentials A(z ? a)(z ? b)×(z ? c)^?2 dz². More precisely, we give a necessary and sufficient condition on the complex numbers a and b for these quadratic differentials to have finite critical trajectories. We also discuss all possible configurations of critical graphs.     

QUANDLE VARIETIES,GENERALIZED SYMMETRIC SPACES,AND <Emphasis Type="Italic">φ</Emphasis>-SPACES

We define a quandle variety as an irreducible algebraic variety Q endowed with an algebraically defined quandle operation ?. It can also be seen as an analogue of a generalized affine symmetric space or a regular s-manifold in algebraic geometry.Assume that Q is normal as an algebraic variety and that the action of its inner automorphism group Inn(Q) has a dense orbit. Then we show that there is an algebraic group G acting on Q with the same orbits as Inn(Q) such that each G-orbit is isomorphic to the quandle (G/H, ?φ) associated to the group G, an automorphism φ of G and a subgroup H of Gφ.     

Qualitative uncertainty principles for the hypergeometric Fourier transform

We prove an L ^p version of the Donoho–Stark’s uncertainty principle for the hypergeometric Fourier transform on \({\mathbb{R}^d}\). Next, using the ultracontractive properties of the semigroups generated by the Heckman–Opdam Laplacian operator, we obtain an L ^p Heisenberg–Pauli–Weyl uncertainty principle for the hypergeometric Fourier transform on \({\mathbb{R}^d}\).     

On subelliptic manifolds

A smooth complex quasi-affine algebraic variety Y is flexible if its special group SAut(Y) of automorphisms (generated by the elements of one-dimensional unipotent subgroups of Aut(Y)) acts transitively on Y, and an algebraic variety is stably flexible if its product with some affine space is flexible. An irreducible algebraic variety X is locally stably flexible if it is a union of a finite number of Zariski open sets each of which is stably flexible. The main result of this paper states that the blowup of a locally stably flexible variety along a smooth algebraic subvariety (not necessarily equidimensional or connected) is subelliptic, and, therefore, it is an Oka manifold.     

Parity sheaves on the affine Grassmannian and the Mirković–Vilonen conjecture

We prove the Mirkovi?–Vilonen conjecture: the integral local intersection cohomology groups of spherical Schubert varieties on the affine Grassmannian have no p-torsion, as long as p is outside a certain small and explicitly given set of prime numbers. (Juteau has exhibited counterexamples when p is a bad prime.) The main idea is to convert this topological question into an algebraic question about perverse-coherent sheaves on the dual nilpotent cone using the Juteau–Mautner–Williamson theory of parity sheaves.     

Few distinct distances implies no heavy lines or circles

We study the structure of planar point sets that determine a small number of distinct distances. Specifically, we show that if a set \(\mathcal{P}\) of n points determines o(n) distinct distances, then no line contains Ω(n ^7/8) points of \(\mathcal{P}\) and no circle contains Ω(n ^5/6) points of \(\mathcal{P}\).We rely on the partial variant of the Elekes-Sharir framework that was introduced by Sharir, Sheffer, and Solymosi in [19] for bipartite distinct distance problems. To prove our bound for the case of lines we combine this framework with a theorem from additive combinatorics, and for our bound for the case of circles we combine it with some basic algebraic geometry and a recent incidence bound for plane algebraic curves by Wang, Yang, and Zhang [20].A significant difference between our approach and that of [19] (and of other related results) is that instead of dealing with distances between two point sets that are restricted to one-dimensional curves, we consider distances between one set that is restricted to a curve and one set with no restrictions on it.     

Hamiltonian operators in differential algebras

We develop a previously proposed algebraic technique for a Hamiltonian approach to evolution systems of partial differential equations including constrained systems and propose a defining system of equations (suitable for computer calculations) characterizing the Hamiltonian operators of a given form. We demonstrate the technique with a simple example.     

Scalar Field Analysis over Point Cloud Data

Given a real-valued function f defined over some metric space \(\mathbb{X}\), is it possible to recover some structural information about f from the sole information of its values at a finite set \(L\subseteq\mathbb{X}\) of sample points, whose locations are only known through their pairwise distances in \(\mathbb{X}\)? We provide a positive answer to this question. More precisely, taking advantage of recent advances on the front of stability for persistence diagrams, we introduce a novel algebraic construction, based on a pair of nested families of simplicial complexes built on top of the point cloud L, from which the persistence diagram of f can be faithfully approximated. We derive from this construction a series of algorithms for the analysis of scalar fields from point cloud data. These algorithms are simple and easy to implement, they have reasonable complexities, and they come with theoretical guarantees. To illustrate the genericity and practicality of the approach, we also present some experimental results obtained in various applications, ranging from clustering to sensor networks.