Quantum surfaces, special functions, and the tunneling effect

The notion of quantum embedding is considered for two classes of examples: quantum coadjoint orbits in Lie coalgebras and quantum symplectic leaves in spaces with non-Lie permutation relations. A method for constructing irreducible representations of associative algebras and the corresponding trace formulas over leaves with complex polarization are obtained. The noncommutative product on the leaves incorporates a closed 2-form and a measure which (in general) are different from the classical symplectic form and the Liouville measure. The quantum objects are related to some generalized special functions. The difference between classical and quantum geometrical structures could even occur to be exponentially small with respect to the deformation parameter. That is interpreted as a tunneling effect in the quantum geometry.Comment: 51 pages, Latex-file, added remark

On [L]-homotopy groups

Some properties of [L]-homotopy group for finite complex L are investigated. It is proved that for complex L whose extension type lying between Sn and Sn+1 n-th [L]-homotopy group of Sn is isomorphic to Z

Coherent transforms and irreducible representations corresponding to complex structures on a cylinder and on a torus

We study a class of algebras with non-Lie commutation relations whose symplectic leaves are surfaces of revolution: a cylinder or a torus. Over each of such surfaces we introduce a family of complex structures and Hilbert spaces of antiholomorphic sections in which the irreducible Hermitian representations of the original algebra are realized. The reproducing kernels of these spaces are expressed in terms of the Riemann theta-function and its modifications. They generate quantum K\"ahler structures on the surface and the corresponding quantum reproducing measures. We construct coherent transforms intertwining abstract representations of an algebra with irreducible representations, and these transforms are also expressed via the theta-function.Comment: 21 pages, Amstex-fil

Topological groups that realize homogeneity of topological spaces

We present results on simplifying an acting group preserving properties of actions: transitivity, being a coset space and preserving a fixed equiuniformity in case of a $G$-Tychonoff space

Local multiplicity of continuous maps between manifolds

Let $M$ and $N$ be smooth (real or complex) manifolds, and let $M$ be equipped with some Riemannian metric. A continuous map $f\colon M\longrightarrow N$ admits a local $k$-multiplicity if, for every real number $\omega >0$, there exist $k$ pairwise distinct points $x_1,\ldots,x_k$ in $M$ such that $f(x_1)=\cdots=f(x_k)$ and \diam\{x_1,\ldots,x_k\}<\omega. In this paper we systematically study the existence of local $k$-mutiplicities and derive criteria for the existence of local $k$-multiplicity in terms of Stiefel--Whitney classes and Chern classes of the vector bundle $f^*\tau N\oplus(-\tau M)$. For example, as a corollary of one criterion we deduce that for $k\geq 2$ a power of $2$, $M$ a compact smooth manifold with the integer $s:=\max\{\ell : \bar{w}_{\ell}(M)\neq 0\}$, and $N$ a parallelizable smooth manifold, if $s\geq \dim N-\dim M+1$ and $\bar{w}_{s}(M)^{k-1}\neq 0$, any continuous map $M\longrightarrow N$ admits a local $k$-multiplicity. Furthermore, as a special case of this corollary we recover, when $k=2$, the classical criterion for the non-existence of an immersion $M\looparrowright N$ between manifolds $M$ and $N$

Partitions of nonzero elements of a finite field into pairs

In this paper we prove that the nonzero elements of a finite field with odd characteristic can be partitioned into pairs with prescribed difference (maybe, with some alternatives) in each pair. The algebraic and topological approaches to such problems are considered. We also give some generalizations of these results to packing translates in a finite or infinite field, and give a short proof of a particular case of the Eliahou--Kervaire--Plaigne theorem about sum-sets

A center transversal theorem for an improved Rado depth

A celebrated result of Dol'nikov, and of \v{Z}ivaljevi\'c and Vre\'cica, asserts that for every collection of $m$ measures $\mu_1,\dots,\mu_m$ on the Euclidean space $\mathbb R^{n + m - 1}$ there exists a projection onto an $n$-dimensional vector subspace $\Gamma$ with a point in it at depth at least $\tfrac{1}{n + 1}$ with respect to each associated $n$-dimensional marginal measure $\Gamma_*\mu_1,\dots,\Gamma_*\mu_m$. In this paper we consider a natural extension of this result and ask for a minimal dimension of a Euclidean space in which one can require that for any collection of $m$ measures there exists a vector subspace $\Gamma$ with a point in it at depth slightly greater than $\tfrac{1}{n + 1}$ with respect to each $n$-dimensional marginal measure. In particular, we prove that if the required depth is $\tfrac{1}{n + 1} + \tfrac{1}{3(n + 1)^3}$ then the increase in the dimension of the ambient space is a linear function in both $m$ and $n$.Comment: v.2: Corrections in Sections 3 and 4 implemented, not affecting the course of the proof; v.3: Replaced with a joint paper by 3 authors with a stronger result; v.4: Final version, accepted to Discrete Comp. Geo

On commutative and non-commutative C*-algebras with the approximate n-th root property

We say that a C*-algebra X has the approximate n-th root property (n\geq 2) if for every a\in X with ||a||\leq 1 and every \epsilon>0 there exits b\in X such that ||b||\leq 1 and ||a-b^n||<\epsilon. Some properties of commutative and non-commutative C*-algebras having the approximate n-th root property are investigated. In particular, it is shown that there exists a non-commutative (resp., commutative) separable unital C*-algebra X such that any other (commutative) separable unital C*-algebra is a quotient of X. Also we illustrate a commutative C*-algebra, each element of which has a square root such that its maximal ideal space has infinitely generated first Cech cohomology.Comment: 17 page

More bisections by hyperplane arrangements

In 2017 Barba, Pilz \& Schnider considered particular and modified cases of the following hyperplane measure partition problem: For the given collection of $j$ measures on $\mathbb R^d$ find a $k$-element affine hyperplane arrangement that bisects each of them into equal halves simultaneously. They solved the problem affirmatively in the case when $d=k=2$ and $j=4$. Furthermore, they conjectured that every collection of $j$ measures on $\mathbb R^d$ can be bisected with a $k$-element affine hyperplane arrangement provided that $d\geq \lceil j/k \rceil$. The conjecture was confirmed in the case when $d\geq j/k=2^a$ by Hubard and Karasev in 2018. In this paper we give a different proof of the Hubard and Karasev result using the framework of Blagojevi\'c, Frick, Haase \& Ziegler (2016), based on the equivariant relative obstruction theory, that was developed for handling the Gr\"unbaum--Hadwiger--Ramos hyperplane measure partition problem. Furthermore, this approach allowed us to prove even more, that for every collection of $2^a(2h+1)+\ell$ measures on $\mathbb R^{2^a+\ell}$ there exists a $(2h+1)$-element affine hyperplane arrangement that bisects them simultaneously

Tunnel catch from potential wells and energy detection

We consider the one-dimensional Schr\"{o}dinger operator in the semiclassical regime assuming that its double-well potential is the sum of a finite "physically given" well and a square shape probing well whose width or depth can be varied (tuned). We study the dynamics of initial state localized in the physical well. It is shown that if the probing well is not too close to the physical one and if its parameters are specially tuned, then the {\it tunnel catch effect} appears, i.e. the initial state starts tunneling oscillations between the physical and probing wells. The asymptotic formula for the probability of finding the state in the probing well is obtained. We show that the observation of the tunnel catch effect can be used to determine the energy level of the initial state, and we obtain the corresponding asymptotic formula for the initial state energy. We also calculate the leading term of the tunneling splitting of energy levels in this double well potential.Comment: 11 pages, 4 figure