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Victor Alexandrov

Publications and source records attributed to Victor Alexandrov.

At least 37 records · Page 2Linked to original sources
arXiv2015

An analogue of a van der Waerden's theorem and its application to two-distance preserving mappings

The van der Waerden's theorem reads that an equilateral pentagon in Euclidean 3-space $\Bbb E^3$ with all diagonals of the same length is necessarily planar and its vertex set coincides with the vertex set of some convex regular pentagon. We prove the following many-dimensional analogue of this theorem: for $n\geqslant 2$, every $n$-dimensional cross-polytope in $\Bbb E^{2n-2}$ with all diagonals of the same length and all edges of the same length necessarily lies in $\Bbb E^n$ and hence is a convex regular cross-polytope. We also apply our theorem to the study of two-distance preserving mappings of Euclidean spaces.

math.MG↗
arXiv2014

End states in a 1-D topological Kondo insulator

To gain further insight into the properties of interacting topological insulators, we study a 1 dimensional model of topological Kondo insulators which can be regarded as the strongly interacting limit of the Tamm-Shockley model. Treating the model in a large $N$ expansion, we find a number of competing ground-state solutions, including topological insulating and valence bond ground-states. One of the effects to emerge in our treatment is a reconstruction of the Kondo screening process near the boundary of the material ("Kondo band bending"). Near the boundary for localization into a valence bond state, we find that the conduction character of the edge state grows substantially, leading to states that extend deeply into bulk. We speculate that such states are the one-dimensional analog of the light f-electron surface states which appear to develop in the putative topological Kondo insulator, SmB6.

cond-mat.str-el↗
arXiv2013

The first acquaintance with the tensor

The main concepts of the theory of tensors are presented. The emphasis is on the basic notions of tensor algebra and practical skills in culculations involving the Kronecker delta and Levi-Civita symbol. Sixty routine exercises are included. The article is intended for junior students of physical, mathematical, and geophysical departments of universities.

math.HO↗
arXiv2013

An analytical approach to the Rational Simplex Problem

In 1973, J. Cheeger and J. Simons raised the following question that still remains open and is known as the Rational Simplex Problem: Given a geodesic simplex in the spherical 3-space so that all of its interior dihedral angles are rational multiples of $π$, is it true that its volume is a rational multiple of the volume of the 3-sphere? We propose an analytical approach to the Rational Simplex Problem by deriving a function $f(t)$, defined as an integral of an elementary function, such that if there is a rational $t$, close enough to zero, such that the value $f(t)$ is an irrational number then the answer to the Rational Simplex Problem is negative.

math.MG↗
arXiv2013

Cubic Topological Kondo Insulators

Current theories of Kondo insulators employ the interaction of conduction electrons with localized Kramers doublets originating from a tetragonal crystalline environment, yet all Kondo insulators are cubic. Here we develop a theory of cubic topological Kondo insulators involving the interaction of spin quartets with a conduction sea. The spin quartets greatly increase the potential for strong topological insulators, entirely eliminating the weak-topological phases from the diagram. We show that the relevant topological behavior in cubic Kondo insulators can only reside at the lower symmetry X or M points in the Brillouin zone, leading to a three Dirac cones with heavy quasiparticles.

cond-mat.str-el↗
arXiv2013

Continuous deformations of polyhedra that do not alter the dihedral angles

We prove that, both in the hyperbolic and spherical 3-spaces, there exist nonconvex compact boundary-free polyhedral surfaces without selfintersections which admit nontrivial continuous deformations preserving all dihedral angles and study properties of such polyhedral surfaces. In particular, we prove that the volume of the domain, bounded by such a polyhedral surface, is necessarily constant during such a deformation while, for some families of polyhedral surfaces, the surface area, the total mean curvature, and the Gauss curvature of some vertices are nonconstant during deformations that preserve the dihedral angles. Moreover, we prove that, in the both spaces, there exist tilings that possess nontrivial deformations preserving the dihedral angles of every tile in the course of deformation.

math.MG↗
arXiv2012

Around the A.D. Alexandrov's theorem on a characterization of a sphere

This is a survey paper on various results relates to the following theorem first proved by A.D. Alexandrov: \textit{Let $S$ be an analytic convex sphere-homeomorphic surface in $\mathbb R^3$ and let $k_1(\boldsymbol{x})\leqslant k_2(\boldsymbol{x})$ be its principal curvatures at the point $\boldsymbol{x}$. If the inequalities $k_1(\boldsymbol{x})\leqslant k\leqslant k_2(\boldsymbol{x})$ hold true with some constant $k$ for all $\boldsymbol{x}\in S$ then $S$ is a sphere.} The imphases is on a result of Y. Martinez-Maure who first proved that the above statement is not valid for convex $C^2$-surfaces. For convenience of the reader, in addendum we give a Russian translation of that paper by Y. Martinez-Maure originally published in French in \textit{C. R. Acad. Sci., Paris, Sér. I, Math.} {\bf 332} (2001), 41--44.

math.DG↗
arXiv2012

Spin and holographic metals

In this paper we discuss two-dimensional holographic metals from a condensed matter physics perspective. We examine the spin structure of the Green's function of the holographic metal, demonstrating that the excitations of the holographic metal are "chiral", lacking the inversion symmetry of a conventional Fermi surface, with only one spin orientation for each point on the Fermi surface, aligned parallel to the momentum. While the presence of a Kramer's degeneracy across the Fermi surface permits the formation of a singlet superconductor, it also implies that ferromagnetic spin fluctuations are absent from the holographic metal, leading to a complete absence of Pauli paramgnetism. In addition, we show how the Green's function of the holographic metal can be regarded as a reflection coefficient in anti-de-Sitter space, relating the ingoing and outgoing waves created by a particle moving on the external surface.

cond-mat.str-el↗
arXiv2010

On a differential test of homeomorphism, found by N.V. Efimov

In the year 1968 N.V. Efimov has proven the following remarkable theorem: \textit{Let $f:\mathbb R^2\to\mathbb R^2\in C^1$ be such that $\det f'(x)<0$ for all $x\in\mathbb R^2$ and let there exist a function $a=a(x)>0$ and constants $C_1\geqslant 0$, $C_2\geqslant 0$ such that the inequalities $|1/a(x)-1/a(y)|\leqslant C_1 |x-y|+C_2$ and $|\det f'(x)|\geqslant a(x)|{\rm curl\,}f(x)|+a^2(x)$ hold true for all $x, y\in\mathbb R^2$. Then $f(\mathbb R^2)$ is a convex domain and $f$ maps $\mathbb R^2$ onto $f(\mathbb R^2)$ homeomorhically.} Here ${\rm curl\,}f(x)$ stands for the curl of $f$ at $x\in\mathbb R^2$. This article is an overview of analogues of this theorem, its generalizations and applications in the theory of surfaces, theory of functions, as well as in the study of the Jacobian conjecture and global asymptotic stability of dynamical systems.

math.DG↗
arXiv2010

Algebra versus analysis in the theory of flexible polyhedra

Two basic theorems of the theory of flexible polyhedra were proven by completely different methods: R. Alexander used analysis, namely, the Stokes theorem, to prove that the total mean curvature remains constant during the flex, while I.Kh. Sabitov used algebra, namely, the theory of resultants, to prove that the oriented volume remains constant during the flex. We show that none of these methods can be used to prove the both theorems. As a by-product, we prove that the total mean curvature of any polyhedron in the Euclidean 3-space is not an algebraic function of its edge lengths.

math.MG↗
arXiv2009

New manifestations of the Darboux's rotation and translation fields of a surface

We show how the rotation and translation fields of a surface, introduced by G. Darboux, may be used to obtain short proofs of a well-known theorem (that reads that the total mean curvature of a surface is stationary under an infinitesimal bending) and a new theorem (that reads that every infinitesimal flex of any simply connected closed surface is orthogonal to the surface at least at two points).

math.DG↗
arXiv2009

Simplices with equiareal faces

We study simplices with equiareal faces in the Euclidean 3-space by means of elementary geometry. We present an unexpectedly simple proof of the fact that, if such a simplex is non-degenerate, than every two of its faces are congruent. We show also that this statement is wrong for degenerate simplices and find all degenerate simplices with equiareal faces.

math.MG↗
arXiv2009

Flexible suspensions with a hexagonal equator

We construct a flexible (non immersed) suspension with a hexagonal equator in Euclidean 3-space and study its properties related to the Strong Bellows Conjecture which reads as follows: if an immersed polyhedron $\Cal P$ in Euclidean 3-space is obtained from another immersed polyhedron $\Cal Q$ by a continuous flex then $\Cal P$ and $\Cal Q$ are scissors congruent.

math.MG↗
arXiv2009

The Dehn invariants of the Bricard octahedra

We prove that the Dehn invariants of any Bricard octahedron remain constant during the flex and that the Strong Bellows Conjecture holds true for the Steffen flexible polyhedron.

math.MG↗
arXiv2008

On the total mean curvature of non-rigid surfaces

Using Green's theorem we reduce the variation of the total mean curvature of a smooth surface in the Euclidean 3-space to a line integral of a special vector field and obtain the following well-known theorem as an immediate consequence: the total mean curvature of a closed smooth surface in the Euclidean 3-space is stationary under an infinitesimal flex.

math.DG↗
arXiv2007

Materializing Superghosts

We construct the off-shell BV realization of N=1, d=10 SYM with 7 auxillary fields. This becomes possible due to materialized ghost phenomenon. Namely, supersymmetry ghosts are coordinates on a manifold B of 10-dimensional spinors with pure spinors cut out. Auxillary fields are sections of a bundle over B, and supersymmetry transformations are nonlinear in ghosts. By integrating out axillary fields we obtain on-shell supersymmetric BV action with terms quadratic in antifields. Exactly this on-shell BV action was obtained in our previous paper after integration out of auxiliary fields in the framework of Pure Spinor Superfield Formalism.

hep-th↗
arXiv2007

On Pure Spinor Superfield Formalism

We show that a certain superfield formalism can be used to find an off-shell supersymmetric description for some supersymmetric field theories where conventional superfield formalism does not work. This "new" formalism contains even auxiliary variables in addition to conventional odd super-coordinates. The idea of this construction is similar to the pure spinor formalism developed by N.Berkovits. It is demonstrated that using this formalism it is possible to prove that the certain Chern-Simons-like (Witten's OSFT-like) theory can be considered as an off-shell version for some on-shell supersymmetric field theories. We use the simplest non-trivial model found in [2] to illustrate the power of this pure spinor superfield formalism. Then we redo all the calculations for the case of 10-dimensional Super-Yang-Mills theory. The construction of off-shell description for this theory is more subtle in comparison with the model of [2] and requires additional Z_2 projection. We discover experimentally (through a direct explicit calculation) a non-trivial Z_2 duality at the level of Feynman diagrams. The nature of this duality requires a better investigation.

hep-th↗