Nonzero Integer Research Articles

Anomalous Exciton Condensation in Graphene Bilayers

In ordinary semiconductor bilayers, exciton condensates appear at total Landau-level filling factor nu{T}=1. We predict that similar states will occur in Bernal stacked graphene bilayers at many nonzero integer filling factors. For nu{T}=-3, 1 we find that the superfluid density of the exciton condensate vanishes and that a finite-temperature fluctuation-induced first order isotropic-smectic phase transition occurs when the layer densities are not balanced. These anomalous properties of bilayer graphene exciton condensates are due to the degeneracy of Landau levels with n=0 and n=1 orbital character.

CYCLIC SYSTEMS OF SIMULTANEOUS CONGRUENCES

This paper considers the cyclic system of n ≥ 2 simultaneous congruences [Formula: see text] for fixed nonzero integers (r, s) with r > 0 and (r, s) = 1. It shows there are only finitely many solutions in positive integers qi ≥ 2, with gcd (q1q2 ⋯ qn, s) = 1 and obtains sharp bounds on the maximal size of solutions for almost all (r, s). The extremal solutions for r = s = 1 are related to Sylvester's sequence 2, 3, 7, 43, 1807,…. If the positivity condition on the integers qi is dropped, then for r = 1 these systems of congruences, taken ( mod |qi|), have infinitely many solutions, while for r ≥ 2 they have finitely many solutions. The problem is reduced to studying integer solutions of the family of Diophantine equations [Formula: see text] depending on three parameters (r, s, m).

ON THE NUMBER OF IRREDUCIBLE COMPONENTS OF THE REPRESENTATION VARIETY OF A FAMILY OF ONE-RELATOR GROUPS

Let us consider the group G = 〈x, y | xm = yn〉 with m and n nonzero integers. The set R(G) of representations of G over SL(2, ℂ) is a four-dimensional algebraic variety which is an invariant of G. In this paper the number of irreducible components of R(G) together with their dimensions are computed. We also study the set of metabelian representations of this family of groups. Finally, the behavior of the projection t : R(G) → X(G), where X(G) is the character variety of the group, and some combinatorial aspects of R(G) are investigated.

Windmills and extreme $2$-cells

In this article we prove new results about the existence of 2-cells in disc diagrams which are extreme in the sense that they are attached to the rest of the diagram along a small connected portion of their boundary cycle. In particular, we establish conditions on a 2-complex X which imply that all minimal area disc diagrams over X with reduced boundary cycles have extreme 2-cells in this sense. The existence of extreme 2-cells in disc diagrams over these complexes leads to new results on coherence using the perimeter-reduction techniques we developed in an earlier article. Recall that a group is called coherent if all of its finitely generated subgroups are finitely presented. We illustrate this approach by showing that several classes of one-relator groups, small cancellation groups and groups with staggered presentations are collections of coherent groups. In this article we prove some new results about the existence of extreme 2-cells in disc diagrams which lead to new results on coherence. In particular, we combine the diagram results shown here with the theorems from [3] to establish the coherence of various classes of one-relator groups, small cancellation groups, and groups with relatively staggered presentations. The article is organized as follows: § 1 contains background definitions, § 2 recalls how extreme 2-cells lead to perimeter reductions and to coherent fundamental groups, § 3 introduces the concept of a windmill, § 4 uses windmills to prove that extreme 2-cells exist, and finally § 5 uses extreme 2-cells to prove that various groups are coherent. For instance, we obtain the following special case of Corollary 5.12: Corollary 0.1. Let G = 〈a1, . . . , ar, t |W 〉 where W has the form t W1t W2 . . . t Wk, N is arbitrary, and for each i, i is a nonzero integer and Wi is a reduced word in the ai. Suppose that {W1,W2, . . . ,Wk} freely generate a subgroup of the free group 〈a1, . . . , ar | −〉. Then G is coherent. 1. Basic Definitions In this section we review some basic definitions about 2-complexes and diagrams. Definition 1.1 (Combinatorial maps and complexes). A map Y → X between CW complexes is combinatorial if its restriction to each open cell of Y is a homeomorphism onto an open cell of X. A CW complex X is combinatorial provided Date: September 7, 2009. 2000 Mathematics Subject Classification. 20F06,20F67,57M07.

Some diophantine problems concerning equal sums of integers and their cubes

This paper gives a complete four-parameter solution of the simultaneous diophantine equations $x+y+z=u+v+w, x^3+y^3+z^3=u^3+v^3+w^3,$ in terms of quadratic polynomials in which each parameter occurs only in the first degree. This solution is much simpler than the complete solutions of these equations published earlier. This simple solution is used to obtain solutions of several related diophantine problems. For instance, the paper gives a parametric solution of the arbitrarily long simultaneous diophantine chains of the type $x^k_1+y^k_1+z^k_1=x^k_2+y^k_2+z^k_2=\ldots=x^k_n+y^k_n+z^k_n=\ldots,~~k=1,3.$ Further, the complete ideal symmetric solution of the Tarry-Escott problem of degree $4$ is obtained, and it is also shown that any arbitrarily given integer can be expressed as the sum of four distinct nonzero integers such that the sum of the cubes of these four integers is equal to the cube of the given integer.

Characterization of the torsion of the Jacobian of y2=x5+Ax and some applications

We characterize torsion subgroup of the Jacobian of the curve CA : y 2 = x5 + Ax, where A 6= 0 is 8th power free integer. As an application of our result we show that for any quadruple a1, a2, a3, a4 of pairwise distinct non-zero integers there exists an infinite set of integers D with the property that the Jacobian of CaiD is of positive rank for i = 1, 2, 3, 4.

Some Residually Solvable One-Relator Groups

This communication records some observations made in the course of studying one-relator groups from the point of view of residual solvability. As a contribution to clas- sication eorts we single out some relator types that render the corresponding one-relator groups residually solvable. a collection of facts and examples gathered while attempting to char- acterize the residually solvable one-relator groups in terms (of the form) of the (single) dening relator. In what follows we prove suf- ciency results for certain cases when the relator is a commutator, and then raise some questions. The class of one-relator groups shows a varied pattern of behavior with respect to residual properties. We begin with reviewing some of the literature that motivated our interest in the topic. G. Baum- slag in (3) showed that positive one-relator groups, which is to say that the relator has only positive exponents, are residually solvable. In the same paper he provided a specic example to demonstrate that not all one-relator groups are residually solvable. A free-by- cyclic group is necessarily residually solvable. As well are the free- by-solvable Baumslag{Solitar groups Bm;n (the groups with presen- tation ha;b;a 1 b m a = b n i for pairs of non-zero integers m;n), by a result of Peter Kropholler (15) who showed that in these groups the second derived subgroup is free. The Baumslag{Solitar groups

Series of reciprocal products with factors from linear recurrence sequences

In this article we study the values of power series ∑ z^n/\prod^j_{i = 0} W_{(n + im)k} at certain points of their domain of meromorphy from the arithmetical point of view. (W_n) is a sequence of non-zero integers satisfying a recurrence W_{n + 1} = pW_n + qW_{n − 1} with non-zero integers p, q such that the discriminant Δ = p^2 + 4q is positive but not a square. The main interest is to characterize the situations, where these values lie in the real quadratic number field ℚ(√Δ) or even in ℚ , but we also include some transcendence problems.

TheB-L/electroweak hierarchy in heterotic string and M-theory

E8 x E8 heterotic string and M-theory, when compactified on a Calabi-Yau threefold admitting an SU(4) vector bundle with Wilson lines, can give rise to the exact MSSM spectrum with three right-handed neutrino chiral superields, one per family. Rank preserving Wilson lines require that the standard model group be augmented by a gauged U(1)_B-L. Since there are no fields in this theory for which 3(B-L) is an even, non-zero integer, the gauged B-L symmetry must be spontaneously broken at a low scale, not too far above the electroweak scale. It is shown that in these heterotic standard models, the B-L symmetry can be broken, with a phenomenologically viable B-L/electroweak hierarchy, by at least one right-handed sneutrino acquiring a vacuum expectation value. This is explicitly demonstrated, in a specific region of parameter space, using a renormalization group analysis and soft supersymmetry breaking operators. The vacuum state is shown to be a stable, local minimum of the potential and the resultant hierarchy is explicitly presented in terms of tan[beta].

Extensions of the D(∓k 2)-triples {k 2, k 2 ± 1, 4k 2 ± 1}

Let n be a nonzero integer. A set of m distinct positive integers is called a D(n)-m-tuple if the product of any two of them increased by n is a perfect square. Let k be a positive integer. In this paper, we show that if {k2, k2 + 1, 4k2 + 1, d} is a D(−k2)-quadruple, then d = 1, and that if {k2 − 1, k2, 4k2 − 1, d} is a D(k2)-quadruple, then d = 8k2(2k2 − 1).

A RECURSIVE FORMULA FOR THE JONES POLYNOMIAL OF 2-BRIDGE LINKS AND APPLICATIONS

In this paper, we give a recursive formula for the Jones poly- nomial of a 2-bridge knot or link with Conway normal form C(i2n1, 2n2, i2n3,...,(i1) r 2nr) in terms of n1,n2,...,nr. As applications, we also give a recursive formula for the Jones polynomial of a 3-periodic link L (3) with rational quotient L = C(2,n1,i2,n2,..., nr,(i1) r 2) for any nonzero integers n1,n2,...,nr and give a formula for the span of the Jones polynomial of L(3) in terms of n1,n2,...,nr with ni 6 ±1 for all i = 1,2,...,r.

The statistics of particle trajectories in the inhomogeneous Sinai problem for a two-dimensional lattice

In connection with the two-dimensional model known as the `periodic Lorentz gas', we study the asymptotic behaviour of statistical characteristics of a free path interval of a point particle before its first occurrence in an -neighbourhood (a circle of radius ) of a non-zero integer point as given that the particle starts from the -neighbourhood of the origin. We evaluate the limit distribution function of the free path length and of the input aimed parameter (the distance from the trajectory to the integer point we are interested in) for a given value of the output aimed parameter. This problem was studied earlier for a particle starting from the origin (the homogeneous case).

Combinatorial aspects of the character variety of a family of one-relator groups

Let us consider the group G = 〈 x , y | x m = y n 〉 with m and n nonzero integers. In this paper, we study the character variety X ( G ) in SL ( 2 , C ) of the group G, obtaining by elementary methods an explicit primary decomposition of the ideal corresponding to X ( G ) in the coordinates X = t x , Y = t y and Z = t x y . As an easy consequence, a formula for computing the number of irreducible components of X ( G ) as a function of m and n is given. Finally we provide a combinatorial description of X ( G ) and we prove that in most cases it is possible to recover ( m , n ) from the combinatorial structure of X ( G ) .

A question about maximal non-valuation subrings

In this paper we pursue and deep the study of ring extensions \({R \subset S}\) such that R is a maximal non-valuation subring of S [Ben Nasr and Jarboui in Houston J Math, 2009 (in press)]. It is proved in Ben Nasr and Jarboui [Houston J Math, 2009 (in press), Theorem 3.2] that if R is integrally closed with finite Krull dimension, then R is a maximal non-valuation subring of qf (R) iff R is not local and |[R, qf (R)]| = dim(R) + 3. This result encourages us to pose the following question: Let n be a nonzero positive integer greater than 2 and let R be a finite-dimensional domain such that |[R, qf (R)]| = dim(R) + n, does there exists an overring S of R such that R is a maximal non-valuation subring of S? This paper deals mostly with this question. We solve this question in case R is integrally closed.

Uniqueness of roots up to conjugacy for some affine and finite type Artin groups

Let $G$ be one of the Artin groups of finite type ${\mathbf B}_n={\mathbf C}_n$, and affine type $\tilde{\mathbf A}_{n-1}$ and $\tilde{\mathbf C}_{n-1}$. In this paper, we show that if $\alpha$ and $\beta$ are elements of $G$ such that $\alpha^k=\beta^k$ for some nonzero integer $k$, then $\alpha$ and $\beta$ are conjugate in $G$. For the Artin group of type $\mathbf A_n$, this was recently proved by J. Gonz\'alez-Meneses. In fact, we prove a stronger theorem, from which the above result follows easily by using descriptions of those Artin groups as subgroups of the braid group on $n+1$ strands. Let $P$ be a subset of $\{1,...,n\}$. An $n$-braid is said to be \emph{$P$-pure} if its induced permutation fixes each $i\in P$, and \emph{$P$-straight} if it is $P$-pure and it becomes trivial when we delete all the $i$-th strands for $i\not\in P$. Exploiting the Nielsen-Thurston classification of braids, we show that if $\alpha$ and $\beta$ are $P$-pure $n$-braids such that $\alpha^k=\beta^k$ for some nonzero integer $k$, then there exists a $P$-straight $n$-braid $\gamma$ with $\beta=\gamma\alpha\gamma^{-1}$. Moreover, if $1\in P$, the conjugating element $\gamma$ can be chosen to have the first strand algebraically unlinked with the other strands. Especially in case of $P=\{1,...,n\}$, our result implies the uniqueness of root of pure braids, which was known by V. G. Bardakov and by D. Kim and D. Rolfsen.

Perfect powers expressible as sums of two cubes

Let n ⩾ 3 . This paper is concerned with the equation a 3 + b 3 = c n , which we attack using a combination of the modular approach (via Frey curves and Galois representations) with obstructions to the solutions that are of Brauer–Manin type. We shall show that there are no solutions in coprime, non-zero integers a, b, c, for a set of prime exponents n having Dirichlet density 28 219 44 928 ≈ 0.628 , and for a set of exponents n having natural density 1.

Two variants of the support problem for products of abelian varieties and tori

Let G be the product of an abelian variety and a torus defined over a number field K. Let P and Q be K-rational points on G. Suppose that for all but finitely many primes p of K the order of ( Q mod p ) divides the order of ( P mod p ) . Then there exist a K-endomorphism ϕ of G and a non-zero integer c such that ϕ ( P ) = c Q . Furthermore, we are able to prove the above result with weaker assumptions: instead of comparing the order of the points we only compare the radical of the order (radical support problem) or the ℓ-adic valuation of the order for some fixed rational prime ℓ ( ℓ-adic support problem).

A generalization of Meshulam’s theorem on subsets of finite abelian groups with no 3-term arithmetic progression

Let r 1, …, r s be non-zero integers satisfying r 1 + ⋯ + r s = 0. Let G $${\simeq \mathbb{Z} / k_1 \mathbb{Z}\oplus \cdots \oplus \mathbb{Z} / k_n \mathbb{Z}}$$ be a finite abelian group with k i |k i-1(2 ≤ i ≤ n), and suppose that (r i , k 1) = 1(1 ≤ i ≤ s). Let $${D_{\mathbf r}(G)}$$ denote the maximal cardinality of a set $${A \subseteq G}$$ which contains no non-trivial solution of r 1 x 1 + ⋯ + r s x s = 0 with $${x_i\,\in\,A (1 \le i \le s)}$$ . We prove that $${D_{\mathbf r}(G) \ll |G|/n^{s-2}}$$ . We also apply this result to study problems in finite projective spaces.

On trees with Laplacian eigenvalue one

Let T be a tree, and L(T) be its Laplacian matrix. Let p(T), q(T) be the number of pendant and quasipendant (adjacent to a pendant) vertices of T, respectively. We show that given any tree on n>1 vertices and any integer vector y of size n, there is a larger tree T ′ with p(T ′) = q(T ′) and a nonzero integer vector y′ such that L(T ′)y′ =y′ and y′ restricted to T is y. Let mT (1) denote the multiplicity of the Laplacian eigenvalue 1 of the tree T. Let T be a tree obtained by adding an edge between a vertex of a tree T 1 and a vertex of a tree T 2. We discuss the relationship among and mT (1) and give a condition under which which generalizes a known result of Grone, R. Merris, R. Sunder, VS. 1990. The Laplacian spectrum of a graph. SIAM J. Matrix Anal. Appl. 11(2) 218–238. We also give a sufficient condition under which . We give a complete characterization of trees that have 1 as the third smallest Laplacian eigenvalue. As an application we characterize trees in which spectral integral variation occurs in one place by adding an edge where the changed eigenvalue is the third smallest eigenvalue.

ON THE ASPHERICITY OF CERTAIN RELATIVE PRESENTATIONS OVER TORSION-FREE GROUPS

It is noteworthy to find whether a relative presentation with torsion-free coefficients is aspherical or not, since a nonaspherical relative presentation that defines a torsion-free group would provide a potential counterexample to the Kaplansky zero-divisor conjecture, see [7]. In this paper we investigate asphericity of the relative group presentation [Formula: see text] provided that m1, m2, …, mk are nonzero integers, k ≤ 3, each coefficient has infinite order and the relator is not a proper power.