Cyclic codes and the Frobenius automorphism

Using the theory of cyclic codes the following problem of K. Burde' on characterizing finite fields GF (qⁿ) is solved:¶ Consider GF (qⁿ) as a vector space over GF (q). For which GF (qⁿ) exists for any k = 0, . . . ,n exactly one subspace C of dimension k and which is invariant under the Frobenius automorphism?     

Every Invariant Measure Semigroup Contains an Ideal which is Embeddable in a Group

We show that every invariant measure semigroup S with associated invariant measure mu contains an ideal S₀ which is embeddable as an open subsemigroup in a locally compact abelian group G in such a way that the restriction to S₀ of mu coincides with the restriction to S₀ of a Haar measure on G. This is a positive answer to a question posed by J.H Williamson. As a consequence the generalization of Pontryagin's duality theorem for S is obtained.     

L^p -uniqueness for Dirichlet operators with singular potentials

We study the problem of strong uniqueness in L^p for the Dirichlet operator perturbed by a singular complex-valued potential. First we construct the generator -H_p of a C₀-semigroup in L^p, with H_p extending the restriction of the perturbed Dirichlet operator to the set of smooth functions. The corresponding sesquilinear form in L² is not assumed to be sectorial. Then we reveal sufficient conditions on the logarithmic derivative # of the measure rdx \rho dx and the potential q which ensure that -H_p is the only extension of D+b·?-q \upharpoonright_C0^￥ \Delta +\beta \cdot \nabla -q \upharpoonright_{C_0^{\infty}} which generates a C₀-semigroup on L^p. The method of a priori estimates of solutions to corresponding differential equations is employed.     

Adapting Some Ideas from Stochastic Control Theory to Studying the Heat Equation in Closed Smooth Domains

Sufficient conditions on bounded domains D ² R ^d={(t,x)} of class C⁴ are given under which solutions of the heat equation u_t=j u+f in D have continuous second-order derivatives with respect to (t,x) in D^- . The equation is supplemented with C⁴ boundary data and it is assumed that f] C² .     

A characterization of infinite 3-abelian groups

In this note we prove that every infinite group G is 3-abelian (i.e. (ab)³ = a³b³ for all a, b in G) if and only if in every two infinite subsets X and Y of G there exist x ? Xx\in X and y ? Yy\in Y such that (xy)³ = x³y³.     

The Problem of Freeness for Euler Monoids and Möbius Groups

The Euler monoid E_n = {(a,b,t) epsilon Z³ : a² + b² = tⁿ, n S 1, is free if and only if n is odd (Theorem 1). We extend the results of Lyndon and Ullman, and Beardon concerning the set of those rational numbers mu epsilon (-2,2) for which the matrix Möbius group G_mu generated by A= and B = is not free (Theorems 2, 3, 4).     

Regular orbits in powers of permutation representations

Let (Q,G) be a faithful permutation representation of a finite group G. Suppose that the G-set Q has t distinct non-zero marks. In a permutation representation analogue of a theorem of Brauer on linear representations, it is shown that the direct power (Q,G)^t of (Q,G) contains a regular orbit. As a corollary, the probability that a random element of Q^r lies in a regular orbit of (Q,G)^r is shown to tend to 1 exponentially fast as r tends to \infin\infin. Further, knowledge of the rate of convergence is equivalent to knowledge of the second largest value of the character of the linear permutation representation.     

Set of matrices with common Lyapunov solution

We present necessary and sufficient conditions for a set of matrices to coincide with a set of the form {A: HA + A^*H > 0} or {T : T^*HT < H} for some (hidden) positive definite matrix H.     

Second Neighborhood via First Neighborhood in Digraphs

Let D be a simple digraph without loops or digons. For any v ? V(D) v\in V(D) , the first out-neighborhood N⁺(v) is the set of all vertices with out-distance 1 from v and the second neighborhood N⁺⁺(v) of v is the set of all vertices with out-distance 2 from v. We show that every simple digraph without loops or digons contains a vertex v such that |N⁺⁺(v)| 3 g|N⁺(v)| |N^{++}(v)|\geq\gamma|N^+(v)| , where % = 0.657298... is the unique real root of the equation 2x³ + x² -1 = 0.     

Homology of the double loop space of the homogeneous space SU(n)/SO(n)

We study the mod 2 homology of the double loop space of SU(n)/SO(n) using the Serre spectral sequence along with the Eilenberg-Moore spectral sequence. Then we also get the homology of the double loop space of the set of all Lagrangian subspaces of the symplectic vector space R²ⁿ.     

On f-injective modules

In this paper, the notions of f-injective and f^*-injective modules are introduced. Elementary properties of these modules are given. For instance, a ring R is coherent iff any ultraproduct of f-injective modules is absolutely pure. We prove that the class S^* \Sigma^* of f^*-injective modules is closed under ultraproducts. On the other hand, S^* \Sigma^* is not axiomatisable. For coherent rings R, S^* \Sigma^* is axiomatisable iff every c₀ \chi_0 -injective module is f^*-injective. Further, it is shown that the class S \Sigma of f-injective modules is axiomatisable iff R is coherent and every c₀ \chi_0 -injective module is f-injective. Finally, an f-injective module H, such that every module embeds in an ultraprower of H, is given.     

Kodaira dimensions and hyperbolicity of nonpositively curved compact Kähler manifolds

In this article, we prove that a compact Kähler manifold M ⁿ with real analytic metric and with nonpositive sectional curvature must have its Kodaira dimension, its Ricci rank and the codimension of its Euclidean de Rham factor all equal to each other. In particular, M ⁿ is of general type if and only if it is without flat de Rham factor. By using a result of Lu and Yau, we also prove that for a compact Kähler surface M ² with nonpositive sectional curvature, if M ² is of general type, then it is Kobayashi hyperbolic.     

Symplectic invariants of elliptic fixed points

To the germ of an area--preserving diffeomorphism at an elliptic fixed point, we associate the germ of Mather's minimal action. This yields a strictly convex function which is symplectically invariant and comprises the classical Birkhoff invariants as the Taylor coefficients of its convex conjugate. In addition, however, the minimal action contains information about the local dynamics near the fixed point; for instance, it detects the C⁰--integrability of the diffeomorphism. Applied to the Reeb flow, this leads to new period spectrum invariants for three--dimensional contact manifolds; a particular case is the geodesic flow on a two--dimensional Riemannian manifold, where the period spectrum is the classical length spectrum.     

A pinching theorem for minimal hypersurfaces in a sphere

We study compact minimal hypersurfaces Mⁿ in Sⁿ⁺¹S^{n+1} with two distinct principal curvatures and prove that if the squared norm S of the second fundamental form of Mⁿ satisfies S \geqq nS \geqq n, then S o nS \equiv n and Mⁿ is a minimal Clifford torus.     

A note on compact elements in the lattice of radicals

An example of radical compact ring A whose Dorroh extension A¹ is not radical compact is constructed.     

Existence of invariant measures for transcendental subexpanding functions

We consider the problem of the existence of absolutely continuous invariant measures for transcendental meromorphic functions. We prove sufficient conditions for a subexpanding meromorphic function f to have a C-finite absolutely continuous invariant measure 7 and we find a class of functions satisfying these assumptions.     

On subtractive varieties,V: congruence modularity and the commutators

In a congruence modular subtractive variety there are both the commutator of ideals and the commutator of congruences. We prove that, if I^' is the smallest congruence having an ideal I as a congruence class, then [I,J] = 0 /[I^', J^']. The general identity [0/ !,0 / #] = 0/[!,#] for !, # congruences, does not always hold; we give several conditions equivalent to this identity and sufficient conditions for it to hold. In the meantime, we get some other characterizations of the commutator of ideals. We also deal with the equational definability of principal commutators in a subtractive variety and with the extension property of the commutator from ideals of a subalgebra to the commutator of ideals of the whole algebra.     

The centroid of a weak*-closed inner ideal in a JBW*-triple

It is shown that there exists a ^*-homomorphism from the continuous centroid L^b (A){\cal L}^b (A) of a JBW^*-triple A onto the continuous centroid L^b (J){\cal L}^b (J) of an arbitrary weak^*-closed inner ideal J in A.     

On Ricci curvature of isotropic and Lagrangian submanifolds in complex space forms

Let M ⁿ be a Riemannian n-manifold. Denote by S(p) and [`(Ric)](p)\overline {Ric}(p) the Ricci tensor and the maximum Ricci curvature on M <sup>n</sup> at a point p ? M<sup>n</sup>p\in M^n, respectively. First we show that every isotropic submanifold of a complex space form [(M)\tilde]<sup>m</sup>(4 c)\widetilde M^m(4\,c) satisfies S ￡ ((n-1)c+ [(n<sup>2</sup>)/4] H<sup>2</sup>)gS\leq ((n-1)c+ {n^2 \over 4} H^2)g, where H<sup>2</sup> and g are the squared mean curvature function and the metric tensor on M <sup>n</sup>, respectively. The equality case of the above inequality holds identically if and only if either M <sup>n</sup> is totally geodesic submanifold or n = 2 and M <sup>n</sup> is a totally umbilical submanifold. Then we prove that if a Lagrangian submanifold of a complex space form [(M)\tilde]<sup>m</sup>(4 c)\widetilde M^m(4\,c) satisfies [`(Ric)] = (n-1)c+ [(n²)/4] H²\overline {Ric}= (n-1)c+ {n^2 \over 4} H^2 identically, then it is a minimal submanifold. Finally, we describe the geometry of Lagrangian submanifolds which satisfy the equality under the condition that the dimension of the kernel of second fundamental form is constant.