In this article we develop a test to determine whether a sextic polynomial that is irreducible over $\mathbb{Q}$ has Galois group isomorphic to the alternating group ${{A}_{4}}$. This test does not involve the computation of resolvents, and we use this test to construct several infinite families of such polynomials.

Let $V$ be an algebraic $\text{K}3$ surface defined over a number field $K$. Suppose $V$ has Picard number two and an infinite group of automorphisms $\mathcal{A}\,=\,\text{Aut(}V/K\text{)}$. In this paper, we introduce the notion of a vector height $\mathbf{h}:\,V\,\to \,\text{Pic(}V\text{)}\,\otimes \,\mathbb{R}$ and show the existence of a canonical vector height $\mathbf{\hat{h}}$ with the following properties:

$$\widehat{\mathbf{h}}\,\left( \sigma P \right)\,=\,{{\sigma }_{*}}\widehat{\mathbf{h}}\left( P \right)$$

$${{h}_{D}}(P)\,=\,\mathbf{\hat{h}}(P)\,\cdot \,D\,+\,O(1),$$

where $\sigma \,\in \,\mathcal{A},\,{{\sigma }_{*}}$ is the pushforward of $\sigma $ (the pullback of ${{\sigma }^{-1}}$), and ${{h}_{D}}$ is a Weil height associated to the divisor $D$. The bounded function implied by the $O(1)$ does not depend on $P$. This allows us to attack some arithmetic problems. For example, we show that the number of rational points with bounded logarithmic height in an $\mathcal{A}$-orbit satisfies

$${{N}_{\mathcal{A}(P)}}(t,\,D)\,=\,\#\{Q\,\in \,\mathcal{A}(P)\,:\,{{h}_{D}}(Q)\,<\,t\}\,=\,\frac{\mu (P)}{s\,\log \,\omega }\,\log t\,+\,O\left( \log \left( \mathbf{\hat{h}}(P)\,\cdot \,D\,+\,2 \right) \right).$$

Here, $\mu (P)$ is a nonnegative integer, $s$ is a positive integer, and $\omega $ is a real quadratic fundamental unit.

Let FGn,k denote the Grassmann manifold of all k-dimensional (left) F-vector subspace of Fn for F = ℝ, the reals, C, the complex numbers, or H the quaternions. The problem of determining which of the Grassmannians bound was addressed by the author in [4]. Partial results were obtained in [4] for the case F = ℝ, including a sufficient condition, due to A. Dold, on n and k for ℝ Gn,k to bound. Here, we show that Dold's condition is also necessary, and obtain a new proof of sufficiency using the methods of this paper, which cover the complex and quaternionic cases as well.

We consider rings as in the title and find the precise obstacle for them not to be Quasi-Frobenius, thus shedding new light on an old open question in Ring Theory. We also find several partial affirmative answers for that question.

In this short note we introduce a notion called quantum injectivity of locally compact quantum groups, and prove that it is equivalent to amenability of the dual. In particular, this provides a new characterization of amenability of locally compact groups.

Under the assumptions of case of theorem 1 we derive from (3.32) the matrix equation

so that there corresponds the matrix B to the bilinear form

4.1

on the linear space

4.2

and fP,μ, is symmetric if ɛ = (-1)μ+1, anti-symmetric if ɛ = (-1)μ.

The last statement remains true in the case a) if P is symmetric irreducible because in that case fP,μ is 0.

The basic theory of the Hausdorff means for double sequences was developed some thirty - three years ago by C.R. Adams [1], and independently by F. Hallenbach [3], Yet today, many of the properties of these means remain largely uninvestigated. The calculations here, although clearly more complex, for the most part break down into obvious modifications of the calculations in the one dimensional case.

This note proves two propositions on infinite doubly stochastic matrices, both of which already appear in the literature: one with an unnecessarily sophisticated proof (Kendall [2]) and the other with the incorrect assertion that the proof is trivial (Isbell [l]). Both are purely algebraic; so we are, if you like, in the linear space of all real doubly infinite matrices A = (aij).

Proposition 1. Every extreme point of the convex set of ail doubly stochastic matrices is a permutation matrix.

Kendall's proof of this depends on an ingenious choice of a topology and the Krein-Milman theorem for general locally convex spaces [2]. The following proof depends on practically nothing: for example, not on the axiom of choice.

Let g(k) denote the least integer such that every graph , with n vertices and n+g(k) edges, contains at least k edge-disjoint cycles; let h(k) be similarly defined for planar graphs. Loops and multiple edges (i.e., cycles of length one and two) are permitted in both cases.

If R is a ring, a right (left) ideal of R is said to be large if it has non-zero intersection with each non-zero right (left) ideal of R [8]. If S is a set, let |S| be the cardinal number of S. We say a right (left) ideal I of a ring R is very large if |R/I| < < No. If a is an element of a ring R such that (a)r = {r ∊ R|ar = 0} is very large then we say a is very singular. The set of all very singular elements of a ring R is a two sided ideal of R. If R is a prime ring, then 0 is the only very singular element of R and a very large right (left) ideal of R is indeed large provided that R is not finite.

We give a simple proof of a theorem by Andrunakievič and Rjabuhin which states that a reduced ring is a subdirect product of entire rings. Our proof makes no use of m-systems and is in some sense similar to the proof of the corresponding theorem in the commutative case due to Krull.

A reduced ring is a ring without non-zero nilpotent elements. It is wellknown that if a reduced ring is commutative, then it is a subdirect product of integral domains [2]. This result has been generalized to arbitrary reduced rings [1]. The proof in the general case is somewhat complicated. We present a simple proposition which leads to a simple proof of the general case.

Necessary and sufficient conditions are proved for the existence of maximal and minimal positive solutions of the semilinear differential equation Δu = -ƒ(x, u) in exterior domains of Euclidean n-space. The hypotheses are that ƒ(x, u) is nonnegative and Hölder continuous in both variables, and bounded above and below by ugi(| x |, u), i = 1, 2, respectively, where each gi(r, u) is monotone in u for each r > 0.

For any group G and integer n, let Pn:G→G be the function defined by Pn(g) = gn for all g ϵ G. If G is abelian then Pn is a homomorphism for all n. Conversely, it is well known (and easy to show) that if P2 or P-1 is a homomorphism then G is abelian. As the groups Gn described below show, for every n other than 2 and -1 there exist non-abelian groups for which Pn is a homomorphism.

The Bishop-Phelps theorem guarantees the existence of support points and support functionals for a nonempty closed convex subset of a Banach space; equivalently, it guarantees the existence of subdifferentials and points of subdifferentiability of a proper lower semicontinuous convex function on a Banach space. In this note we show that most of these results cannot be extended to pairs of convex sets or functions, even in Hilbert space. For instance, two proper lower semicontinuous convex functions need not have a common point of subdifferentiability nor need they have a subdifferential in common. Negative answers are also obtained to certain questions concerning density of support points for the closed sum of two convex subsets of Hilbert space.

In this paper we discuss the existence of mild and classical solutions for a class of abstract non-autonomous neutral functional differential equations. An application to partial neutral differential equations is considered.

Recently, F. H. Clarke and Y. Ledyaev established a multidirectional mean value theorem applicable to lower semi-continuous functions on Hilbert spaces, a result which turns out to be useful in many applications. We develop a variant of the result applicable to locally Lipschitz functions on certain Banach spaces, namely those that admit a C1-Lipschitz continuous bump function.

Generalized boundary value problems are considered for hyperbolic equations of the form utt — uss + λp(s, t)u = 0. By constructing symmetric Green's functions appropriate to such problems the existence of eigenvalues is established.

If f(x) is a polynomial with integral coefficients then the integer r is said to be a residue of f(x) modulo an integer m if the congruence

is soluble for x; otherwise r is termed a non-residue. When m is a prime p, Mord ell [4] has shown that the least nonnegative residue ℓ of f(x) (mod p) satisfies

In [2] Brungs shows that every ring T between a principal (right and left) ideal domain R and its quotient field is a quotient ring of R. In this note we obtain similar results without assuming the ascending chain conditions. For a (right and left) Bezout domain R we show that T is a quotient ring of R which is again a Bezout domain; furthermore Tis a valuation domain if and only if T is a local ring.

We extend Maxwell’s representation of harmonic polynomials to $h$-harmonics associated to a reflection invariant weight function ${{h}_{k}}$. Let ${{\mathcal{D}}_{i}},\,1\,\le \,i\,\le \,d$, be Dunkl’s operators associated with a reflection group. For any homogeneous polynomial $P$ of degree $n$,we prove the polynomial ${{\left| x \right|}^{2\gamma +d-2+2n}}P\left( \mathcal{D} \right)\left\{ 1/{{\left| x \right|}^{2\gamma +d-2}} \right\}$ is a $h$-harmonic polynomial of degree $n$, where $\gamma \,=\,\sum \,ki$ and $\mathcal{D}\,=\,\left( {{\mathcal{D}}_{1}},\ldots ,{{\mathcal{D}}_{d}} \right)$. The construction yields a basis for $h$-harmonics. We also discuss self-adjoint operators acting on the space of $h$-harmonics.