Lower Bounds for the Degree of a Branched Covering of a Manifold

The problem of finding new lower bounds for the degree of a branched covering of a manifold in terms of the cohomology rings of this manifold is considered. This problem is close to M. Gromov’s problem on the domination of manifolds, but it is more delicate. Any branched (finite-sheeted) covering of manifolds is a domination, but not vice versa (even up to homotopy). The theory and applications of the classical notion of the group transfer and of the notion of transfer for branched coverings are developed on the basis of the theory of n-homomorphisms of graded algebras.The main result is a lemma imposing conditions on a relationship between the multiplicative cohomology structures of the total space and the base of n-sheeted branched coverings of manifolds and, more generally, of Smith–Dold n-fold branched coverings. As a corollary, it is shown that the least degree n of a branched covering of the N-torus T ^N over the product of k 2-spheres and one (N ? 2k)-sphere for N ≥ 4k + 2 satisfies the inequality n ≥ N ? 2k, while the Berstein–Edmonds well-known 1978 estimate gives only n ≥ N/(k + 1).     

Span of a DL-algebra

Unless otherswise specified, all objects are defined over a field k of characteristic 0. Let K be a field. The unessentialness of an extension of the algebra Der K by means of a splittable semisimple Lie algebra is established. Let D _K be the category of differential Lie algebras (DL-algebras) (g;K). In this paper for an extension L/K the functor η:D _K → D _L , defining the tensor product L ? _K of vector spaces and the homomorphism of Lie algebras, is constructed. If the extension L/K is algebraic, then η is unique. The results will be required for strengthening the progress on Gelfand–Kirillov problem and weakened conjecture [1, 2].     

The asymptotic distribution of a single eigenvalue gap of a Wigner matrix

We show that the distribution of (a suitable rescaling of) a single eigenvalue gap $\lambda _{i+1}(M_n)-\lambda _i(M_n)$ of a random Wigner matrix ensemble in the bulk is asymptotically given by the Gaudin–Mehta distribution, if the Wigner ensemble obeys a finite moment condition and matches moments with the GUE ensemble to fourth order. This is new even in the GUE case, as prior results establishing the Gaudin–Mehta law required either an averaging in the eigenvalue index parameter $i$ , or fixing the energy level $u$ instead of the eigenvalue index. The extension from the GUE case to the Wigner case is a routine application of the Four Moment Theorem. The main difficulty is to establish the approximate independence of the eigenvalue counting function $N_{(-\infty ,x)}(\tilde{M}_n)$ (where $\tilde{M}_n$ is a suitably rescaled version of $M_n$ ) with the event that there is no spectrum in an interval $[x,x+s]$ , in the case of a GUE matrix. This will be done through some general considerations regarding determinantal processes given by a projection kernel.     

Modelling the motion of a three-link manipulator based on a plane with a time-dependent inclination

This work deals with the modelling of a three-link manipulator mounted on a plane with a time-dependent inclination. Two cases are considered.

• (i)The plane is part of a rigid body.
• (ii)The plane is in a moored ship.

     

Conformality of a quasiconformal mapping at a point

Letf be a quasiconformal mapping with the complex dilation μ. A new condition on μ is introduced for the conformality off at a pointz. The result extends the classical Teichmüller-Wittich-Belinskiî regularity theorem.     

Simultaneous decomplexification of a pair of complex matrices by a similarity transformation

The following question is treated: Under what conditions can complex n-by-n matrices A and B be made real by the same similarity transformation? It is shown that if the algebra generated by A and B contains a matrix with a simple real spectrum, then the problem of the simultaneous decomplexification of a matrix pair can be reduced to the decomplexification of a single matrix by a diagonal similarity transformation. From this result, sufficient conditions are derived for the possibility of simultaneous decomplexification. An example illustrating these conditions is given.     

Bound states of a two-boson system on a two-dimensional lattice

We consider a Hamiltonian of a two-boson system on a two-dimensional lattice Z². The Schrödinger operator H(k₁, k₂) of the system for k₁ = k₂ = π, where k = (k₁, k₂) is the total quasimomentum, has an infinite number of eigenvalues. In the case of a special potential, one eigenvalue is simple, another one is double, and the other eigenvalues have multiplicity three. We prove that the double eigenvalue of H(π,π) splits into two nondegenerate eigenvalues of H(π, π ? 2β) for small β > 0 and the eigenvalues of multiplicity three similarly split into three different nondegenerate eigenvalues. We obtain asymptotic formulas with the accuracy of β² and also an explicit form of the eigenfunctions of H(π, π ?2β) for these eigenvalues.     

Approximation of a system of singularly perturbed reaction-diffusion parabolic equations in a rectangle

The Dirichlet problem for a system of singularly perturbed reaction-diffusion parabolic equations in a rectangle is considered. The higher order derivatives of the equations are multiplied by a perturbation parameter ?², where ? takes arbitrary values in the interval (0, 1]. When ? vanishes, the system of parabolic equations degenerates into a system of ordinary differential equations with respect to t. When ? tends to zero, a parabolic boundary layer with a characteristic width ? appears in a neighborhood of the boundary. Using the condensing grid technique and the classical finite difference approximations of the boundary value problem, a special difference scheme is constructed that converges ?-uniformly at a rate of O(N ^?2ln² N + N ₀ ^?1 , where \(N = \mathop {\min }\limits_s N_s \), N ^s + 1 and N ₀ + 1 are the numbers of mesh points on the axes x _s and t, respectively.     

Hochschild cohomology ring modulo nilpotence of a one point extension of a quiver algebra defined by two cycles and a quantum-like relation

We consider a one point extension algebra B of a quiver algebra A _q over a field k defined by two cycles and a quantum-like relation depending on a nonzero element q in k. We determine the Hochschild cohomology ring of B modulo nilpotence and show that if q is a root of unity, then B is a counterexample to Snashall-Solberg’s conjecture.     

Approximate solution of a parabolic equation with the use of a rational approximation to the operator exponential

For the abstract parabolic equation \(\dot x = Bx + bv\left( t \right)\) with an unbounded self-adjoint operator B, where b is a vector and v(t) is a scalar function, we suggest a solution method based on the evaluation of some rational function of the operator B. We obtain a priori estimates of the approximation error for the output function y(t) = <x(t), l>, where l is a given vector. The results of a numerical experiment for the inhomogeneous heat equation are presented.     

Deviation of elements of a Banach space from a system of subspaces

We prove that if X is a real Banach space, Y ₁ ? Y ₂ ? ... is a sequence of strictly embedded closed linear subspaces of X, and d ₁ ≥ d ₂ ≥ ... is a nonincreasing sequence converging to zero, then there exists an element x ∈ X such that the distance ρ(x, Y _n ) from x to Y _n satisfies the inequalities d _n ≤ ρ(x, Y _n ) ≤ 8d _n for n = 1, 2, ....     

On a generalization of semisimple modules

Let R be a ring with identity. A module \(M_R\) is called an r-semisimple module if for any right ideal I of R, MI is a direct summand of \(M_R\) which is a generalization of semisimple and second modules. We investigate when an r-semisimple ring is semisimple and prove that a ring R with the number of nonzero proper ideals \(\le \)4 and \(J(R)=0\) is r-semisimple. Moreover, we prove that R is an r-semisimple ring if and only if it is a direct sum of simple rings and we investigate the structure of module whenever R is an r-semisimple ring.     

An approximation scheme for a problem of search for a vector subset

We consider the following clustering problem: Given a vector set, find a subset of cardinality k and minimum square deviation from its mean. The distance between the vectors is defined by the Euclideanmetric. We present an approximation scheme (PTAS) that allows us to solve this problem with an arbitrary relative error ? in time O(n ^2/?+1(9/?)^3/? d), where n is the number of vectors of the input set and d denotes the dimension of the space.     

Equations of motion of a charged particle in a five-dimensional model of the general theory of relativity with a nonholonomic four-dimensional velocity space

The equations of motion of a charged particle in a five-dimensional model of the general theory of relativity with a nonholonomic four-dimensional velocity space are considered. A nonholonomic distribution defined by the differential form ω = A ₀ dx ⁰ + A ₁ dx ¹ + A ₂ dx ² + A ₃ dx ³ + dx ⁴ on a five-dimensional smooth Lorentzian manifold is studied. By means of the Pontryagin maximum principle, it is proved that the equations of horizontal geodesics for this distribution are the same as the equations of motion of a charged particle in the general theory of relativity. Thus, a model of the Kaluza-Klein theory is built by means of the sub-Lorentzian geometry. Finally, the geodesic sphere, which appears in a constant magnetic field, is studied, as well as its singular points.     

Compact Perturbations of a Strongly Positive Operator

Let X be a real Hilbert space and \(X^{*}\) its dual space. Let A be a strongly positive operator from X to \(X^{*}\). Sufficient conditions are provided to assert that a compact perturbation of A reaches a fixed value at least once and at most finitely many times. When the compact perturbation is linear, the value is reached just once. The same conclusions are obtained when operators map the space X into itself. As an application of our results two examples are given related to some types of integral equations. The proof of results is constructive and is based upon a continuation method.     

H-convex standard fundamental domain of a subgroup of a modular group

The two major ways of obtaining fundamental domains for discrete subgroups of SL(2,?) are the Dirichlet Polygon construction (see Lehner in Discontinuous Groups and Automorphic Functions, American Mathematical Society, Providence, 1964) and Ford’s construction (see Ford in Automorphic Functions, McGraw–Hill, New York, 1929). Each of these two methods yield a hyperbolically convex fundamental domain for any discrete subgroup of SL(2,?).However, the Dirichlet polygon construction and Ford’s construction are not well adapted for the actual construction of a hyperbolically convex fundamental domain due to their nature of construction and their reliance on knowing almost all elements of the group under discussion.

A third-and most important and practical-method of obtaining a fundamental domain is through the use of a right coset decomposition as described below. Let Γ₂ be a subgroup of Γ₁ and

$\Gamma_{1}=\Gamma_{2}\cdot \{L_{1},L_{2},\ldots,L_{m}\}.$

If \(\mathbb{F}\) is a fundamental domain of the bigger group Γ₁, then the set

$\mathcal{R}_{\Gamma}=\Biggl(\overline{\bigcup_{k=1}^{m}L_{k}(\mathbb{F})}\,\Biggr)^{o}$

(1)

is a fundamental domain of Γ₂. One can ask at this juncture, is it possible to choose the right cosets suitably so that the set ?_Γ is hyperbolically convex? We will answer this question affirmatively for

$\Gamma_{1}=\Gamma(1)\quad \mbox{and}\quad \mathbb{F}=\biggl\{\tau \in \mathbb{H}:|\tau|>1\ \&;\ |\mathrm{Re}(\tau)|<\frac{1}{2}\biggr\}.$

     

On the representation of substitutions as products of a transposition and a full cycle

A method of solving equations of the form \( {g^{{y_1}}} \cdot h \cdot {g^{{y_2}}} \cdot h \cdot \ldots \cdot {g^{{y_1}}} \cdot h \cdot {g^{{y_{l + 1}}}} = \sigma \) in the symmetric group S _n is proposed, where h is a transposition, g is a full cycle, and σ × S _n . The method is based on building all sets of generalized inversions of the bottom line of the substitution σ by means of a system of Boolean equations associated with σ. An example of solving an equation in a group S₆ is given.     

The logarithm of the modulus of a holomorphic function as a minorant for a subharmonic function. II. The complex plane

Let u ? ?∞be a subharmonic function in the complex plane. We establish necessary and/or sufficient conditions for the existence of a nonzero entire function f for which the modulus of the product of each of its kth derivative k = 0, 1,..., by any polynomial p is not greater than the function Ce ^u in the entire complex plane, where C is a constant depending on k and p. The results obtained significantly strengthen and develop a number of results of Lars Hörmander (1997).