Our aim is to prove: If a, b, c are positive integers, if ab > 1, if a, b, c are relatively prime in pairs, and all are free of squares, if — ab is a quadratic residue of c, bc of a, ca of b, and if F(x, y, z) = ax2 + by2 — cz2, we have non-trivial solutions of

with the inequalities

Let $K$ be a finite Galois extension of the field of rational numbers with unit rank greater than 3. We prove that the ring of integers of $K$ is a Euclidean domain if and only if it is a principal ideal domain. This was previously known under the assumption of the generalized Riemann hypothesis for Dedekind zeta functions. We now prove this unconditionally.

Let k and t be positive integers; let q1, q2, . . . , qt be distinct prime numbers; and let ζ1, ζ2, ... , ζt be kth roots of unity, not necessarily primitive. Recent investigations on consecutive kth power residues have led to the following question: Under what conditions do there exist primes p that have a kth power character X such that

The purpose of this paper is to establish the following theorem :

Theorem 1. Let be a finite group with abelian Sylow 2-subgroups in which the centralizer of every involution is solvable. Then either is solvable or else /O is isomorphic to a subgroup of PΓL(2, q) containing PSL(2, q), where either q = 3 or 5 (mod 8), q ≥ 5, or q = 2n, n ≥ 2.

In the following paper ring will always mean commutative ring which may or may not have an identity. We use the letter N exclusively for nilpotents of the ring A.

A ring such that, given any two ideals L and M with L ⊆ M there exists an ideal Q such that L = QM is called a multiplication ring. For references to early papers on multiplication rings by Krull and Mori the reader is referred to [2]. A ring in which every regular ideal is invertible is called a Dedekind ring.

We study the structure of generalized Verma modules over a semi-simple complex finite-dimensional Lie algebra, which are induced from simple modules over a parabolic subalgebra. We consider the case when the annihilator of the starting simple module is a minimal primitive ideal if we restrict this module to the Levi factor of the parabolic subalgebra. We show that these modules correspond to proper standard modules in some parabolic generalization of the Bernstein-Gelfand-Gelfand category $\mathcal{O}$ and prove that the blocks of this parabolic category are equivalent to certain blocks of the category of Harish-Chandra bimodules. From this we derive, in particular, an irreducibility criterion for generalized Verma modules. We also compute the composition multiplicities of those simple subquotients, which correspond to the induction from simple modules whose annihilators are minimal primitive ideals.

Numerous combinatorial enumeration problems may be reduced to equivalent problems of enumerating sequences with prescribed restrictions. For example, the expression, given by Tutte [38], for the number of planar maps may be derived (see Cori and Richard [12]) by essentially a sequence enumeration technique. The correspondence between a set of configurations which are to be enumerated and an appropriate set of sequences is often complicated. Indeed, the existence of such a correspondence has occasionally only been discovered fortuitously by observing the equality of two counting series (see, for example, Klarner [25]).

Heawood (3) showed that for h > 1 the countries of any map on a surface of connectivity h = 3 − x can be coloured using at most

colours. In a previous paper (1)1 proved the following

The purpose of this paper is to extend various invariant subspace theorems for the circle group to multiply connected domains. Such attempts are not new. Actually, Sarason (4) studied the invariant subspaces of annulus operators acting on L2 and showed certain parallelisms between the unit disk case and the annulus case. Voichick (8) observed analytic functions on a finite Riemann surface and generalized the Beurling theorem on the closed invariant subspaces of H2 as well as the Beurling–Rudin theorem on the closed ideals of the disk algebra. Here we shall consider LP(Γ) and C(Γ) defined on the boundary Γ of a finite orientable Riemann surface R.

A net of degree k and order n is a set of n2 points and nk designated sets of points, called lines, such that

1. (1) The lines fall into k disjoint parallel classes, i.e. each line occurs in exactly one parallel class.

2. (2) Lines in the same parallel class have no points in common; lines in different parallel classes have exactly one point in common.

3. (3) Each point lies on exactly one line of each parallel class.

Let R be an associative ring with identity, and let denote the category of unital left R-modules. It is known that if R is a commutative, Noetherian ring, then the maximal torsion radicals of correspond to the minimal prime ideals of R. In fact, Nӑstӑsescu and Popescu [15, Proposition 2.7] have given a more general result valid for arbitrary commutative rings. This paper investigates maximal torsion radicals over rings not necessarily commutative.

We explore the geometric notion of prolongations in the setting of computational algebra, extending results of Landsberg and Manivel which relate prolongations to equations for secant varieties. We also develop methods for computing prolongations that are combinatorial in nature. As an application, we use prolongations to derive a new family of secant equations for the binary symmetric model in phylogenetics.

Although the axiom of complete regularity ought to be a separation axiom, in none of its usual forms does it look like an intrinsic separation axiom. Our aim in this paper is to establish such characterizations of complete regularity which naturally fit in between regularity and normality and which already have proved to be fundamental and useful. This can simply be achieved by replacing the family of all open sets (as used in the formulation of the separation axioms) by some suitable (sub)base of open sets. For the sake of simplicity, we assume all our spaces to be T1 and we shall operate with (sub)bases of closed sets (instead of open sets). It is appropriate to define the notion of a screening.

Two subsets A and B of a set X are said to be screened by the pair (C, D) if C ∪ D = X, A ∩ D = ∅ and C ∩ B = 0. (Consequently, A ⊂ C and B ⊂ D.)

Our motivation for this note originates with consideration of a subset A of Euclidean w-space, Rn, which contains only part of its boundary. The part contained is t h a t part of the closure of A which cannot be “bettered“ within A with respect to the preference associated with a fixed closed convex cone Γ. Here b is preferred to a if and only if a — b ∊ Γ; if, for instance, Γ is the non-negative orthant of Rn, this preference is ordinary vector inequality.

Let denote the set of points x = (x1, . . . , xn) with integral co-ordinates in Euclidean n-space. For any fixed integer m ≥ 2, let C = C(m) be the set of such points in the cube 0 ≤ xi < m (i = 1, 2, . . . , n) and let be any subset of C. Suppose that f(x) is any single-valued, integral-valued function, defined for all x ∊ . We consider solutions x ∊ of the congruence

This paper is concerned with the ideal theory of a commutative ring R. We say R has Property (α) if each primary ideal in R is a power of its (prime) radical; R is said to have Property (δ) provided every ideal in R is an intersection of a finite number of prime power ideals. In (2, Theorem 8, p. 33) it is shown that if D is a Noetherian integral domain with identity and if there are no ideals properly between any maximal ideal and its square, then D is a Dedekind domain. It follows from this that if D has Property (α) and is Noetherian (in which case D has Property (δ)), then D is Dedekind.

Speyer and Sturmfels associated Gröbner toric degenerations $\text{G}{{\text{r}}_{2}}{{({{\mathbb{C}}^{n}})}^{\mathcal{T}}}$ of $\text{G}{{\text{r}}_{2}}{{({{\mathbb{C}}^{n}})}^{{}}}$ with each trivalent tree $\mathcal{T}$ having $n$ leaves. These degenerations induce toric degenerations $M_{r}^{\mathcal{T}}$ of ${{M}_{r}}$, the space of $n$ ordered, weighted (by $\mathbf{r}$) points on the projective line. Our goal in this paper is to give a geometric (Euclidean polygon) description of the toric fibers and describe the action of the compact part of the torus as “bendings of polygons”. We prove the conjecture of Foth and Hu that the toric fibers are homeomorphic to the spaces defined by Kamiyama and Yoshida.

The spectrum of the Laplacian or more generally of a Schrödinger operator on an open manifold may have possibly a complicated aspect. For example, a Cantor set in the real axis may appear as the spectrum even for an innocent looking potential on a standard Riemannian manifold (see J. Moser [10]). The fundamental result of the spectral theory of periodic Schrödinger operators, however, says that the picture of the spectrum of a Schrödinger operator on ℝn with a periodic potential is simple; indeed the spectrum consists of a series of closed intervals of the real axis without accumulation, separated in general by gaps outside the spectrum (see M. Reed and B. Simon [13] or M. M. Skriganov [15] for instance).

We observe that every self-dual ternary code determines a holomorphic $\mathcal N=1$ superconformal field theory. This provides ternary constructions of some well-known holomorphic $\mathcal N=1$ superconformal field theories (SCFTs), including Duncan’s “supermoonshine” model and the fermionic “beauty and the beast” model of Dixon, Ginsparg, and Harvey. Along the way, we clarify some issues related to orbifolds of fermionic holomorphic CFTs. We give a simple coding-theoretic description of the supersymmetric index and conjecture that for every self-dual ternary code this index is divisible by $24$ ; we are able to prove this conjecture except in the case when the code has length $12$ mod $24$ . Lastly, we discuss a conjecture of Stolz and Teichner relating $\mathcal N=1$ SCFTs with Topological Modular Forms. This conjecture implies constraints on the supersymmetric indexes of arbitrary holomorphic SCFTs, and suggests (but does not require) that there should be, for each k, a holomorphic $\mathcal N=1$ SCFT of central charge $12k$ and index $24/\gcd (k,24)$ . We give ternary code constructions of SCFTs realizing this suggestion for $k\leq 5$ .