Let K:= Q(α) be an algebraic number field which is given by specifying the minimal polynomial f(X) for α over Q. We describe a procedure for finding the subfields L of K by constructing pairs (w(X), g(X)) of polynomials over Q such that L= Q(w(α)) and g(X) is the minimal polynomial for w(α). The construction uses local information obtained from the Frobenius-Chebotarev theorem about the Galois group Gal(f), and computations over p-adic extensions of Q.

We exhibit a sequence (un) which is not uniformly distributed modulo one even though for each fixed integer k ≥ 2 the sequence (kun) is u.d. (mod 1). Within the set of all such sequences, we characterize those with a well-behaved asymptotic distribution function. We exhibit a sequence (un) which is u.d. (mod 1) even though no subsequence of the form (ukn + j) is u.d. (mod 1) for any k ≥ 2. We prove that, if the subsequences (ukn) are u.d. (mod 1) for all squarefree k which are products of primes in a fixed set P, then (un) is u.d. (mod I) if the sum of the reciprocals of the primes in P diverges. We show that this result is the best possible of its type.

Let G be a connected reductive linear algebraic group over the complex numbers. For any element A of the Lie algebra of G, there is an action of the Weyl group W on the cohomology Hi(BA) of the subvariety BA (see below for the definition) of the flag variety of G. We study this action and prove an inequality for the multiplicity of the Weyl group representations which occur ((4.8) below). This involves geometric data. This inequality is applied to determine the multiplicity of the reflection representation of W when A is a nilpotent element of “parabolic type”. In particular this multiplicity is related to the geometry of the corresponding hyperplane complement.

Matching theorems, fixed point theorems and minimax inequalities are obtained in H-spaces which generalize the corresponding results of Bae-Kim-Tan, Browder, Fan, Horvath, Kim, Ko-Tan, Shih-Tan, Takahashi, Tan and Tarafdar to non-compact and/or non-convex settings.

Let G be a (not necessarily finite) group and ρ a finite dimensional faithful irreducible representation of G over an arbitrary field; write ρ¯ for ρ viewed as a projective representation. Suppose that ρ is not induced (from any proper subgroup) and that ρ¯ is not a tensor product (of projective representations of dimension greater than 1). Let K be a noncentral subgroup which centralizes all its conjugates in G except perhaps itself, write H for the normalizer of K in G, and suppose that some irreducible constituent, σ say, of the restriction p↓K is absolutely irreducible. It is proved that then (ρ is absolutely irreducible and) ρ¯ is tensor induced from a projective representation of H, namely from a tensor factor π of ρ¯↓H such that π↓K = σ¯ and ker π is the centralizer of K in G.

This paper considers analogs of results on integral operators studied by Hörmander. Using the sharp function introduced by Fefferman and Stein, we prove weighted norm inequalities on kernel operators which map an Lp space into an Lq space, with q not equal to p. The techniques recover known results about fractional integral operators and apply to multiplier operators which satisfy a generalization of the Hörmander multiplier condition.

This paper considers (left) near-rings which satisfy the left self distributive (LSD) identity: abc = abac. This is exactly the class of near-rings for which each left multiplication mapping, τa: x → ax, is a near-ring endomorphism. Simple and subdirectly irreducible ones are classified and semidirect sum decompositions into reduced and nilpotent pieces are given. LSD near-rings with restrictive conditions on nilpotent elements or annihilating sets are considered. Type 1 prime (semiprime) ideals in an LSD near-ring are completely prime (semiprime). Further results on prime and maximal ideals are given. Numerous examples are given to illuminate the theory and to illustrate its limitations. Some analogous theory for right self distributive near-rings is given (those satisfying the identity: abc = acbc).

Let λ0 be a semisimple eigenvalue of an operator T0. Let Γ0 be a circle with centre λs0 containing no other spectral value of T0. Some lower bounds are obtained for the convergence radius of the power series for the spectral projection P(t) and for trace T(t)P(t) associated with linear perturbation family T(t) = T0 + tV0 and the circle Γ0. They are useful when T0 is a member of a sequence (Tn) which approximates an operator T in a collectively compact manner. These bounds result from a modification of Kato's method of majorizing series, based on an idea of Redont. I λ0 is simple, it is shown that the same lower bound are valid for the convergence radius of a power series yielding an eigenvector of T(t).

The aim of this paper is is to establish Hadamard's type three-circles theorems for fully nonlinear elliptic and parabolic inequalities.

The following question is discussed and evidence for and against it is advanced: is it true that if F is an arbitrary finite subgroup of an arbitrary non-linear simple locally finite group G, then CG(F) is infinite? The following points to an affirmative answer.

Theorem A. Let F be an arbitrary finite subgroup of a non-linear simple locally finite group G. Then there exist subgroups D ◃ C ≤ G such that F centralizes C/D, F∩C ≤ D, and C/D is a direct product of finite alternating groups of unbounded orders. In particular, F centralizes an infinite section of G.

Theorem A is deduced from a “local” version, namely

Theorem B. There exists an integer valued function f(n, r) with the following properties. Let H be a finite group of order at most n, and suppose that H ≤ S, where S is either an alternating group of degree at least f = f(n, r) or a finite simple classical group whose natural projective representation has degree at least f. Then there exist subgroups D ◃ C ≤ S such that (i) [H, C] ≤ D, (ii) H ∩ C ≤ D, (iii) C/D ≅ Alt(r), (iv) D = 1 if S is alternating, and D is a p-group of class at most 2 and exponent dividing p2 if S is a classical group over a field of characteristic p.

The natural “local version” of our main question is however definitely false.

Proposition C. Let p be a given prime. Then there exists a finite group H that can be embedded in infinitely many groups PSL(n, p) as a subgroup with trivial centralizer.

Let X be a complex Banach space, G a compact abelian group and Λ a subset of Ĝ, the dual group pf G. Then LΛ1(G, X) has the Radon-Nikodym property if and only if X has the Radon-Nikodym property and Λ is Riesz set. In particular, H1 (T, X) has the Radon-Nikodym property if and only if X has the Radon-Nikodym property. This solves a problem of Hensgen.

Let g1, g2, …, gn be a sequence of mutually independent, normally distributed, random variables with mathematical expectation zero and variance unity. In this work, we obtain the average number of real zeros of the random algebraic equations Σnk=1 Kσ gk(ω)tk = C, where C is a constant independent of t and not necessarily zero. This average is (1/π) log n, when n is large and σ is non-negative.

For a polynomial f(x) over a finite field Fq, denote the polynomial f(y)−f(x) by ϕf(x, y). The polynomial ϕf has frequently been used in questions on the values of f. The existence is proved here of a polynomial F over Fq of the form F = Lr, where L is an affine linearized polynomial over Fq, such that f = g(F) for some polynomial g and the part of ϕf which splits completely into linear factors over the algebraic closure of Fq is exactly φF. This illuminates an aspect of work of D. R. Hayes and Daqing Wan on the existence of permutation polynomials of even degree. Related results on value sets, including the exhibition of a class of permutation polynomials, are also mentioned.

Two different ordinary differential operators L1 and L2 (not of the same order) defined on two adjacent intervals I1 and I2, respectively, with certain mixed conditions at the interface are considered. These problems are encountered in the study of ‘acoustic waveguides in ocean’, ‘transverse vibrations in nonhomogeneous strings’, etc. A complete set of physical conditions on the system give rise to three types of (selfadjoint) boundary value problems associated with the pair (L1, L2). In a series of papers, a systematic study of these new classes of problems is being developed. In the present paper, we construct the fundamental systems and exhibit the forms of solutions of nonhomogeneous problems associated with the pair (L1, L2).

We prove the higher integrability of nonnegative decreasing functions, verifying a reverse inequality, and we calculate the optimal integrability exponent for these functions.