Gorenstein simplices with a given [formula omitted]-polynomial

Our goal in this paper is to compute the 2-rank of the class group of the fields K = ℚ ( − 2 , q , d ) \[\mathbb{K}=\mathbb{Q}\left( \sqrt{-2},\sqrt{q},\sqrt{d} \right)\] for any odd positive squarefree integer d and any prime integer q ≡ 5 (mod 8). Furthermore, we give the list of these fields whose 2-class groups are trivial, cyclic, of type (2, 2), (2, 4) or (2, 2, 2).

The problem of the semiclassical motion of an electron in a lattice subject to a strong magnetic field leads to a very interesting problem concerned with a low-dimensional ‘periodic’ topology [1], namely, the study of flat sections of two-dimensional surfaces in T3. Zorich [2] in the almost rational case and Dynnikov [3] in the generic case proved Novikov’s conjecture that open orbits have a strong asymptotic direction. On the basis of the analysis carried out by them, Novikov [4] deduced the following remarkable picture: for a fixed dispersion law, to each direction of the magnetic field giving rise to generic open fibres there corresponds a unique indivisible integer 2-cycle in T3 that is locally constant. This means that a countable family of pairwise disjoint open domains on the two-dimensional sphere is given, numbered by triples of relatively prime integers. The union of these zones is an open dense subset of the sphere, but it is still unclear whether or not it has full measure. In any case, if the energy is fixed, then the set is of full measure along with the set of directions for which all trajectories are closed (see below). Novikov and Mal’tsev [5] applied these results to the theory of conduction in normal metals, describing new universal topological phenomena, which can be observed in experiments. It is therefore very important, also from the physical point of view, to study in more detail the nongeneric directions of the magnetic field, which are certain to appear. Below we present a selection of the strongest assertions to date concerning the measure and properties of the above zones. These assertions are required for physical applications. They are based on a technique developed by Dynnikov, and have not been stated in earlier publications. At the time when the present paper was being prepared for print I was informed by Dynnikov that these results had been published in his recent paper [6].

In the present paper we analyze the representations in quantum mechanics of classical canonical transformations that are nonbijective, i.e., not one-to-one onto. We take as the central example the canonical transformation that changes the Hamiltonian of a one-dimensional oscillator of frequency κ−1 into one of frequency k−1 where κ, k are relatively prime integers. For the particular case k=1, the mapping of the original phase space (x,p) onto the new one (x̄, p̄) is κ to 1 and the equivalent points in (x,p) are related by a cyclic group 𝒞κ of linear canonical transformations. When formulating this problem in Bargmann Hilbert space, the canonical transformation can be related to the conformal transformation w=zκ, which again is κ to 1 and where a group 𝒞κ also appears. This cyclic group proves fundamental for the determination of representations of the conformal transformation in Bargmann Hilbert space. To begin with, it suggests that while we can take in the original Bargmann Hilbert space a single component function, in the new Bargmann Hilbert space we must take a κ component one. In this way we can map in a one-to-one fashion the states and operators in the old and new Bargmann Hilbert spaces. When translating these results to ordinary Hilbert space, we get in an unambiguous way the quantization of the observables appearing in the equations that determine the representation of the classical canonical transformation relating oscillators of frequencies κ−1 and k−1. Furthermore, we also get the solutions of these equations, and the resulting representation is unitary. While our discussion is restricted to the problem mentioned above, in the concluding section we indicate our surmise for deriving systematically the unitary representation in quantum mechanics of arbitrary canonical transformations.

In 1995, A. Wiles [1,2] announced a proof of Fermat's Last Theorem, which is stated as follows: If \(\pi\) is an odd prime and \(x\), \(y\), \(z\) are relatively prime positive integers, then \(z^\pi\not=x^\pi+y^\pi.\) In this note, a simpler proof of this theorem is offered. It is proved that if \(\pi\) is an odd prime and \(x\), \(y\), \(z\) are positive integers satisfying \(z^\pi=x^\pi+y^\pi,\) then \(x\), \(y,\) and \(z\) are each divisible by \(\pi\). The proof offered by Wiles uses cyclic groups. It is the purpose of this note to offer a proof of this result, with techniques within the grasp of Fermat.

In 1995, A, Wiles announced, using cyclic groups ( a subject area which was not available at the time of Fermat), a proof of Fermat's Last Theorem, which is stated as follows: If is an odd prime and x; y; z; are relatively prime positive integers, then z 6= x + y: In this note, an elegant proof of this result is given. It is proved, using elementary algebra, that if is an odd prime and x; y; z are positive integers satisfying z = x+y; then z; x; y are each divisible by :

Integers without large prime factors

We consider the exponential Diophantine equation $a^{x}+b^{y}=c^{z}$ in positive integers $x$, $y$ and $z$, where $a$, $b$ and $c$ are fixed pair-wise relatively prime positive integers greater than one. In this paper, we obtain several upper bounds for solutions $x$, $y$ and $z$ for which two of $x$, $y$ and $z$ are even. As their applications, we solve exponential Diophantine equations in which $a$, $b$ and $c$ are expressed as terms of linearly recurrence sequences.

A positroid is a matroid realized by a matrix such that all maximal minors are non-negative. Positroid polytopes are matroid polytopes of positroids. In particular, they are lattice polytopes. The Ehrhart polynomial of a lattice polytope counts the number of integer points in the dilation of that polytope. The Ehrhart series is the generating function of the Ehrhart polynomial, which is a rational function with the numerator called the \(h^\ast\)-polynomial. We compute the \(h^\ast\)-polynomials of an arbitrary positroid polytope by a family of shelling orders of it. We also compute the \(h^\ast\)-polynomial of any positroid polytope with some facets removed and we relate it to the descents of permutations. Our result generalizes that of Early, Kim, and Li for hypersimplices.Mathematics Subject Classifications: 05B35Keywords: Positroid, Ehrhart theory

Counting pure kcycles in sequences of Cayley graphs

Channel coding linear programming decoding (CC- LPD) and compressed sensing linear programming decoding (CS- LPD) are two setups that are formally tightly related. Recently, a connection between CC-LPD and CS-LPD was exhibited that goes beyond this formal relationship. The main ingredient was a lemma that allowed one to map vectors in the nullspace of some zero-one measurement matrix into vectors of the fundamental cone defined by that matrix. The aim of the present paper is to extend this connection along several directions. In particular, the above-mentioned lemma is extended from real measurement matrices where every entry is equal to either zero or one to complex measurement matrices where the absolute value of every entry is a non-negative integer. Moreover, this lemma and its generalizations are used to translate performance guarantees from CC-LPD to CS-LPD. In addition, the present paper extends the formal relationship between CC-LPD and CS-LPD with the help of graph covers. First, this graph-cover viewpoint is used to obtain new connec- tions between, on the one hand, CC-LPD for binary parity- check matrices, and, on the other hand, CS-LPD for complex measurement matrices. Secondly, this graph-cover viewpoint is used to see CS-LPD not only as a well-known relaxation of some zero-norm minimization problem but (at least in the case of real measurement matrices with only zeros, ones, and minus ones) also as a relaxation of a problem we call the zero-infinity operator minimization problem.

In the present paper we investigate commutativity of rings with unity satisfying any one of the properties (Error rendering LaTeX formula) , (Error rendering LaTeX formula) , and for some in where are non-negative integers. Finally, under different appopriate constraints on commutators, commutativity of R has been established.

In a recent paper [l] K . CORRADI and A. HAJNAL proved that if a finite graph without multiple edges contains at least 3k vertices and the valency of every vertex is at least 2k, where k is a positive integer, then the graph contains k independent circuits, i . e . the graph contains as a subgraph a set of k circuits no two of which have a vertex in common . The present paper contains extensions of this theorem. In a recent paper [2] P . ERDŐS and L. POSA proved, among other things, that if a finite graph with or without loops and multiple edges contains n vertices and at least n + 4 edges, then the graph contains two circuits without an edge in common . The present paper contains analogous results for planar graphs . We adopt the following notation : O k denotes a graph consisting of k independent circuits, kO denotes a graph consisting of k or more circuits no two of which have an edge in common . If q is a graph then 'V (q) denotes the set of vertices of q, `V i(q ) denotes the set of vertices of q having valency i in q (i being a non-negative integer), T 5 i (q), Ty i (q) denote the set of vertices of 4 having valency i and ' i, respectively, and &(q) denotes the set of edges of c~ . The valency of the vertex x in the graph will be denoted by v (x, C~) . I ? (() I will be denoted by V (q), J & (q) I by E(q) etc. In this notation the theorem Of CORRADI and HAJNAL quoted above states that if q is a finite graph without multiple edges and if V(q) 3k and T -2k_ 1 (4) =0, then q Ok ; and the theorem of ERDOS and POSA quoted above states that if is a finite graph and E(C) _V(q) + 4, then qD 2 O .

Let A be a semifinite and properly infinite von Neumann algebra of operators of a complex Hilbert space H. Call C e A compact (relative to A) if it is the limit in the norm of elements with finite support. The investigation of analytic properties of compact elements of A, generalizing welt-known classical theorems on compact operators, was started in [2] and is continued in the present paper. Let N i (resp. Ri) be the null (resp. range) projection of(1 C ) ~ where i is a natural number. Contrary to the classical case, the sequences N1 =R2 > R3 ~ "" need not be stationary, even if A is of type I. The question of the finiteness (resp. cofiniteness) of the projection Noo = sup Ni (resp. R~ = infRi) was left open in [2]. In the present paper it is shown that No~ is in fact always finite (and R~ is cofinite). It was shown already in [2] that if N® is finite, then inf(N®, Ro0) = 0 and sup(No~,R~)= 1. This is a decomposition of H in the sense of lattice theory or perhaps continuous geometry. In the present paper this result is refined by introducing the concepts of essentially closed subspaces and essentially topological direct sums. It is shown that the algebraic sum No~(H)+R®(H) is essentially topological direct and essentially closed. The operator 1 C decomposes into the sum of two closed operators affiliated with A, one summand being essentially nilpotent on Noo(H), the other one essentially regular on Roo(H). Throughout this paper H denotes a complex Hilbert space and A a properly infinite semifinite von Neumann algebra of linear operators of H. § 1. Preliminaries

A Sturmian sequence related to the uniqueness conjecture for Markoff numbers

Let $P$ and $Q$ be nonzero integers. Generalized Fibonacci and Lucas sequences are defined as follows: $U_{0}(P,Q)=0,U_{1}(P,Q)=1,$ and $ U_{n+1}(P,Q)=PU_{n}(P,Q)+QU_{n-1}(P,Q)$ for $n\geq 1$ and $ V_{0}(P,Q)=2,V_{1}(P,Q)=P,$ and $V_{n+1}(P,Q)=PV_{n}(P,Q)+QV_{n-1}(P,Q)$ for $n\geq 1,$ respectively. In this paper, we assume that $P$ and $Q$ are relatively prime odd positive integers and $P^{2}+4Q>0.$ We determine all indices $n$ such that $U_{n}=(P^{2}+4Q)x^{2}.$ Moreover, we determine all indices $n$ such that $(P^{2}+4Q)U_{n}=x^{2}.$ As a result, we show that the equation $V_{n}^{2}(P,1)+V_{n+1}^{2}(P,1)=x^{2}$ has solution only for $n=2,$ $P=1,$ $x=5$ and $V_{n+1}^{2}(P,-1)=V_{n}^{2}(P,-1)+x^{2}$ has no solutions. Moreover, we solve some Diophantine equations.