We introduce certain n X n matrices with integral elements that constitute a free semigroup with identity and generate the n-dimensional unimodular group. In terms of these matrices we define a certain sequence of n-dimensional vectors, which we show is the natural generalization to n dimensions of the Fibonacci sequence. Connections between the generalized Fibonacci sequences and certain Jacobi polynomials are found. The various basic identities concerning the Fibonacci numbers are shown to have natural extensions to n dimensions, and in some cases the proofs are rendered quite brief by the use of known theorems on matrices.

A group P is said to be a CEF-group if, for every countable group G, there is a factor group of P which contains a subgroup isomorphic to G. It was shown by Higman, Neumann, and Neumann [5] that the free group of rank two is a CEF-group. More recently, Levin [6] proved that if P is the free product of two cyclic groups, not both of order two, then P is a CEF-group. Later, Hall [3] gave an alternative proof of Levin's result.

In this paper we give a new proof of Levin's result (Theorem 2). The proof given yields information about the factor group H of P in which a given countable group G is embedded; for example, if G is given by a recursive presentation (this concept is denned in [4]), then a recursive presentation is obtained for H, and certain decision problems (in particular, the word problem) are solvable for the recursive presentation obtained for H if and only if they are solvable for the given recursive presentation of G.

The ideal generation conjecture has recently been proved for general points in , (k a field) [1], [6]. The proof in [1] is by induction. One of the starting points for the induction (called A(5) in [1]) is proved in [6]. The theoretical proof of A(5) in [6] seems to be very difficult, apparently even more difficult than the induction. Because of this, and also because [6] is not publically available, I feel it is worth knowing that A(5) can be proved numerically with modest readily available computing facilities. In this note I discuss the computation involved, and give a few explicit examples. In the course of working out these examples I found 26 points in , that satisfy the ideal generation conjecture, but which cannot be extended to 27 or 28 points satisfying the ideal generation conjecture. This phenomenon can be interpreted combinatorially, leading to an infinite number of similar examples.

Let J be an integral domain (i.e., a commutative ring without divisors of zero) with unit element, F its quotient field and J[x] the integral domain of polynomials with coefficients from J . The domain J is called integrally closed if every root of a monic polynomial over J which is in F also is in J.

In this paper, we consider the quantum version of a Hamiltonian model describing friction. This model consists of a particle which interacts with a bosonic reservoir representing a homogeneous medium through which the particle moves. We show that if the particle is confined, then the Hamiltonian admits a ground state if and only if a suitable infrared condition is satisfied. The latter is violated in the case of linear friction, but satisfied when the friction force is proportional to a higher power of the particle speed.

Let $p$ be an odd prime. We study the variation of the $p$-rank of the Selmer groups of Abelian varieties with complex multiplication in certain towers of number fields.

In the present note we introduce a straightforward algebraic generalization of inner product spaces, which we appropriately name generalized inner product (GIP) spaces. In the same fashion in which different topologies :an be introduced in inner product spaces, adequate topologies can be introduced in GIP spaces in such a manner that topological vector spaces are obtained. We enumerate and derive some fundamental properties of different topologies in GIP spaces, having primarily in mind their possible later application to quantum physics.

The desirability of having in quantum physics more general structures than Hilbert spaces (in which quantum mechanics is usually formulated) is suggested by Dirac's formalism (2), which deals with “unnormalizable” vectors. Unfortunately, although this formalism is very elegant from the point of view of the facili ty of dealing with its symbolism, it completely lacks in mathematical rigour.

Let $\mathcal{A}$ and $\mathcal{B}$ be large subsets of $\{1,\,.\,.\,.\,,\,N\}$. We study the number of pairs $\left( a,b \right)\,\in \,\mathcal{A}\,\times \,\mathcal{B}$ such that the sum of binary digits of $a\,+\,b$ is fixed.

A group G is said to have the subnormal join property (s.j.p.) if the join of two (and hence of finitely many) subnormal subgroups of G is always subnormal in G. Following Robinson [6], we shall denote the class of groups having this property by . A particular subclass of is , consisting of those groups G in which the join of two subnormals is again subnormal in G and has defect bounded in terms of the defects of the constituent subgroups (for a more precise definition see Section 7 of [6]).

In [16], Wielandt showed that groups which satisfy the maximal condition for subnormal subgroups have the s.j.p. Many further results on groups with the s.j.p. were proved in [6] and [7]. In Sections 2 and 3 of this paper, it will be shown that several of these results can be exhibited as corollaries of a few rather more general theorems on the classes , .

Let denote the variety of all groups which are extensions of a nilpotent-of-class-c group by a nilpotent-of-class-d group, and let denote the variety of all metabelian groups. The main result of this paper is the following theorem.

THEOREM. Let be a subvariety of which does not contain . Then every -group is an extension of a group of finite exponent by a nilpotent group by a group of finite exponent. In particular, a finitely generated torsion-free -group is a nilpotent-by-finite group.

This generalizes the main theorem of Ŝmel′kin [4], where the same result is proved for subvarieties of , where is the variety of abelian groups. See also Lewin and Lewin [2] for a related discussion.

The ultraspherical function

1.1

for |1 — x| < 2 is a solution of the differential equation

1.2

This equation has two independent solutions; of the two, only Pn(λ)(x) is analytic at x = 1, aside for some special values of λ, which we shall not consider.

Let ƒ be a map from a topological space X into a topological space F. We say that ƒ is proper in case ƒ is closed continuous and ƒ-1(y) is compact for all y ∊ Y. Proper maps have been extensively studied, see for example (3, Chapter I, §10) or (6). (The definition of a proper map given above is different from, but equivalent to, that given by Bourbaki in (3). In (6) only surjective proper maps are considered and these maps are called ƒitting maps.) It is known that if ƒ is a proper map, then X is compact if and only if ƒ(X) is compact, and X is paracompact if and only if ƒ(X) is paracompact.

We shall discuss solutions of linear partial differential equations of the form

1

where Ψ is an ordinary differential operator of order s with respect to t. Our first theorem gives a solution of (1) for the Cauchy data;

2

j = 1,2, ߪ,s − 1,

whenever the function P is annihilated by a finite iteration of the operator Φ.

A lattice L is finitely presented (or presentable) if and only if it can be described with finitely many generators and finitely many relations. Equivalently, L is the lattice freely generated by a finite partial lattice A, in notation, L = F(A). (For more detail, see Section 1.5 of [6].)

It is an old “conjecture” of lattice theory that in a finitely presented (or presentable) lattice the elements behave “freely” once we get far enough from the generators.

In this paper we prove a structure theorem that could be said to verify this conjecture.

THEOREM 1. Let L be a finitely presentable lattice. Then there exists a congruence relation θ such that L/θ is finite and each congruence class is embeddable in a free lattice.

The Schur or projective Schur group of a field consists of the classes of central simple algebras which occur in the decomposition of a group algebra or a twisted group algebra. For number fields, the projective Schur group has been determined in [8], whereas the Schur group is extensively studied in [25]. Recently, some authors have generalized these concepts to commutative rings. One then studies the classes of Azumaya algebras which are epimorphic images of a group ring or a twisted group ring. Though several properties of the Schur or projective Schur group defined in this way have been obtained, they remain rather obscure objects.

Let $\mathcal{T}$ be the ${{C}^{*}}$-algebra generated by the Toeplitz operators $\left\{ {{T}_{\varphi }}:\varphi \in {{L}^{\infty }}\left( S,d\sigma \right) \right\}$ on the Hardy space ${{H}^{2}}\left( S \right)$ of the unit sphere in ${{C}^{n}}$. It is well known that $\mathcal{T}$ is contained in the essential commutant of $\left\{ {{T}_{\varphi }}:\varphi \in \text{VMO}\cap {{L}^{\infty }}\left( S,d\sigma \right) \right\}$. We show that the essential commutant of $\left\{ {{T}_{\varphi }}:\varphi \in \text{VMO}\cap {{L}^{\infty }}\left( S,d\sigma \right) \right\}$ is strictly larger than $\mathcal{T}$.

Is there some absolute $\unicode[STIX]{x1D700}>0$ such that for any claw-free graph $G$, the chromatic number of the square of $G$ satisfies $\unicode[STIX]{x1D712}(G^{2})\leqslant (2-\unicode[STIX]{x1D700})\unicode[STIX]{x1D714}(G)^{2}$, where $\unicode[STIX]{x1D714}(G)$ is the clique number of $G$? Erdős and Nešetřil asked this question for the specific case where $G$ is the line graph of a simple graph, and this was answered in the affirmative by Molloy and Reed. We show that the answer to the more general question is also yes, and, moreover, that it essentially reduces to the original question of Erdős and Nešetřil.

For a given de Branges space $\mathcal{H}(E)$ we investigate de Branges subspaces defined in terms of majorants on the real axis. If $\omega $ is a nonnegative function on $\mathbb{R}$, we consider the subspace

$${{\mathcal{R}}_{\omega }}(E)=\text{Clo}{{\text{s}}_{\mathcal{H}(E)}}\{F\in \mathcal{H}(E):\text{there exists }C>0:|{{E}^{-1}}F|\le C\omega \,on\,\mathbb{R}\}.$$

We show that ${{\mathcal{R}}_{\omega }}(E)$ is a de Branges subspace and describe all subspaces of this form. Moreover, we give a criterion for the existence of positive minimal majorants.

Given a faithful action of a finite group $G$ on an algebraic curve $X$ of genus $gx\,\ge \,2$, we give explicit criteria for the induced action of $G$ on the Riemann–Roch space ${{H}^{0}}\left( X,\,{{\mathcal{O}}_{X}}\left( D \right) \right)$ to be faithful, where $D$ is a $G$-invariant divisor on $X$ of degree at least ${{2}_{gX}}\,-\,2$. This leads to a concise answer to the question of when the action of $G$ on the space ${{H}^{0}}\left( X,\,\Omega _{X}^{\otimes m} \right)$ of global holomorphic polydifferentials of order $m$ is faithful. If $X$ is hyperelliptic, we provide an explicit basis of ${{H}^{0}}\left( X,\,\Omega _{X}^{\otimes m} \right)$. Finally, we give applications in deformation theory and in coding theory and discuss the analogous problem for the action of $G$ on the first homology ${{H}_{1}}\left( X,\,\mathbb{Z}/m\mathbb{Z} \right)$ if $X$ is a Riemann surface.

In 1999 V. Arnol’d introduced the local contact algebra: studying the problem of classification of singular curves in a contact space, he showed the existence of the ghost of the contact structure (invariants which are not related to the induced structure on the curve). Our main result implies that the only reason for existence of the local contact algebra and the ghost is the difference between the geometric and (defined in this paper) algebraic restriction of a 1-form to a singular submanifold. We prove that a germ of any subset $N$ of a contact manifold is well defined, up to contactomorphisms, by the algebraic restriction to $N$ of the contact structure. This is a generalization of the Darboux-Givental’ theorem for smooth submanifolds of a contact manifold. Studying the difference between the geometric and the algebraic restrictions gives a powerful tool for classification of stratified submanifolds of a contact manifold. This is illustrated by complete solution of three classification problems, including a simple explanation of V. Arnold's results and further classification results for singular curves in a contact space. We also prove several results on the external geometry of a singular submanifold $N$ in terms of the algebraic restriction of the contact structure to $N$. In particular, the algebraic restriction is zero if and only if $N$ is contained in a smooth Legendrian submanifold of $M$.