We consider transversally harmonic foliated maps between two Riemannian manifolds equipped with Riemannian foliations. We give various characterisations of such maps and we study the relation between the properties ‘harmonic’ and ‘transversally harmonic’ for a given map. We also consider these problems for particular classes of manifolds: manifolds with transversally almost Hermitian foliations and Riemannian flows.

Let T be a bounded operator on a complex Banach space X. Let V be an open subset of the complex plane. We give a condition sufficient for the mapping f(z)↦ (T−z)f(z) to have closed range in the Fréchet space H(V, X) of analytic X-valued functions on V. Moreover, we show that there is a largest open set U for which the map f(z)↦ (T−z)f(z) has closed range in H(V, X) for all V⊆U. Finally, we establish analogous results in the setting of the weak–* topology on H(V, X*).

We use a result of Y. Furuta to show that for almost all positive integers m, the cyclotomic field has an infinite Hilbert p-class field tower with high rank Galois groups at each step, simultaneously for all primes p of size up to about (log logm)1 + o(1). We also use a recent result of B. Schmidt to show that for infinitely many m there is an infinite Hilbert p-class field tower over for some p≥m0.3385 + o(1). These results have immediate applications to the divisibility properties of the class number of .

The paper deals with a criterion for the sum of a special series to be a transcendental number. The result does not make use of divisibility properties or any kind of equation and depends only on the random oscillation of convergence.

In this paper we shall show that Weyl's theorem holds for class A(k) operators T where k≥>1, via its hyponormal transform . Next we shall prove some applications of Weyl's theorem on class A(k) operators.

A regular semigroup (cf. [4, p. 38]) is a C0-semigroup T(⋅) that has an extension as a holomorphic semigroup W(⋅) in the right halfplane , such that ||W(⋅)|| is bounded in the ‘unit rectangle’ Q:=(0, 1]× [−1, 1]. The important basic facts about a regular semigroup T(⋅) are: (i) it possesses a boundary groupU(⋅), defined as the limit lims → 0+W(s+i⋅) in the strong operator topology; (ii) U(⋅) is a C0-group, whose generator is iA, where A denotes the generator of T(⋅); and (iii) W(s+it)=T(s)U(t) for all s+it ∈ (cf. Theorems 17.9.1 and 17.9.2 in [3]). The following converse theorem is proved here. Let A be the generator of a C0-semigroup T(⋅). If iA generates a C0-group, U(⋅), then T(⋅) is a regular semigroup, and its holomorphic extension is given by (iii). This result is related to (but not included in) known results of Engel (cf. Theorem II.4.6 in [2]), Liu [7] and the author [6] for holomorphic extensions into arbitrary sectors, of C0-semigroups that are bounded in every proper subsector. The method of proof is also different from the method used in these references. Criteria for generators of regular semigroups follow as easy corollaries.

We study quantum Schubert varieties from the point of view of regularity conditions. More precisely, we show that these rings are domains that are maximal orders and are AS-Cohen-Macaulay and we determine which of them are AS-Gorenstein. One key fact that enables us to prove these results is that quantum Schubert varieties are quantum graded algebras with a straightening law that have a unique minimal element in the defining poset. We prove a general result showing when such quantum graded algebras are maximal orders. Finally, we exploit these results to show that quantum determinantal rings are maximal orders.

The unit sum number u(R) of a ring R is the least k such that every element is the sum of k units; if there is no such k then u(R) is ω or ∞ depending whether the units generate R additively or not. If RM is a left R-module, then the unit sum number of M is defined to be the unit sum number of the endomorphism ring of M. Here we show that if R is a ring such that R/J(R) is semisimple and is not a factor of R/J(R) and if P is a projective R-module such that JP ≪ P, (JP small in P), then u(P)= 2. As a result we can see that if P is a projective module over a perfect ring then u(P)=2.

We relate the coefficients of the probabilistic zeta function of a finite monolithic group to those of an almost simple group.

A maximal symmetry group is a group of isomorphisms of a three-dimensional hyperbolic manifold of maximal order in relation to the volume of the manifold. In this paper we determine all maximal symmetry groups of the types PSL(2, q) and PGL(2, q). Depending on the prime p there are one or two such groups with q=pk and k always equals 1, 2 or 4.

We study the Friedrichs extensions of unbounded cyclic subnormals. The main result of the present paper is the identification of the Friedrichs extensions of certain cyclic subnormals with their closures. This generalizes as well as complements the main result obtained in [5]. Such characterizations lead to abstract Galerkin approximations, generalized wave equations, and bounded -functional calculi.

We make explicit Poincaré duality for the equivariant K-theory of equivariant complex projective spaces. The case of the trivial group provides a new approach to the K-theory orientation [3].

Some oscillation criteria are obtained for the damped PDE with p-Laplacian The results established here are extensions of some classical oscillation theorems due to Fite-Wintner and Kamenev for second order ordinary differential equations, and improve and complement recent results of Mařík and Usami.

In this paper, we present some geometric characterizations of the Moufang quadrangles of mixed type, i.e., the Moufang quadrangles all the points and lines of which are regular. Roughly, we classify generalized quadrangles with enough (to be made precise) regular points and lines with the property that the dual nets associated to the regular points satisfy the Axiom of Veblen-Young, or a very weak version of the Axiom of Desargues. As an application we obtain a geometric characterization and axiomatization of the generalized inversive planes arising from the Suzuki-Tits ovoids related to a polarity in a mixed quadrangle. In the perfect case this gives rise to a characterization with one axiom less than in a previous result by the second author.

Let be an arbitrary strictly increasing infinite sequence of positive integers. For an integer n≥1, let . Let r>0 be a real number and q≥ 1 a given integer. Let be the eigenvalues of the reciprocal power LCM matrix having the reciprocal power of the least common multiple of xi and xj as its i, j-entry. We show that the sequence converges and . We show that the sequence converges if and . We show also that if r> 1, then the sequence converges and , where t and l are given positive integers such that t≤l−1.

In this note, we find all the solutions of the Diophantine equation x2 + 5a 13b = yn in positive integers x, y, a, b, n≥ 3 with x and y coprime.