Congruence Classes Research Articles

On equiangular lines in $17$ dimensions and the characteristic polynomial of a Seidel matrix

For $e$ a positive integer, we find restrictions modulo $2^e$ on the coefficients of the characteristic polynomial $\chi_S(x)$ of a Seidel matrix $S$. We show that, for a Seidel matrix of order $n$ even (resp. odd), there are at most $2^{\binom{e-2}{2}}$ (resp. $2^{\binom{e-2}{2}+1}$) possibilities for the congruence class of $\chi_S(x)$ modulo $2^e\mathbb Z[x]$. As an application of these results, we obtain an improvement to the upper bound for the number of equiangular lines in $\mathbb R^{17}$, that is, we reduce the known upper bound from $50$ to $49$.

Arity Shape of Polyadic Algebraic Structures

Concrete two-set (module-like and algebra-like) algebraic structures are investigated from the viewpoint that the initial arities of all operations are arbitrary. Relations between operations arising from the structure definitions, however, lead to restrictions which determine their possible arity shapes and lead us to the partial arity freedom principle. In this manner, polyadic vector spaces and algebras, dual vector spaces, direct sums, tensor products and inner pairing spaces are reconsidered. As one application, elements of polyadic operator theory are outlined: multistars and polyadic analogs of adjoints, operator norms, isometries and projections are introduced, as well as polyadic C*-algebras, Toeplitz algebras and Cuntz algebras represented by polyadic operators. Another application is connected with number theory, and it is shown that congruence classes are polyadic rings of a special kind. Polyadic numbers are introduced (see Definition 6.16), and Diophantine equations over these polyadic rings are then considered. Polyadic analogs of the Lander-Parkin-Selfridge conjecture and Fermat's last theorem are formulated. For nonderived polyadic ring operations (on polyadic numbers) neither of these statements holds, and counterexamples are given. Finally, a procedure for obtaining new solutions to the equal sums of like powers equation over polyadic rings by applying Frolov's theorem to the Tarry-Escott problem is presented.

Quotientopes

For any lattice congruence of the weak order on $\mathfrak{S}_n$, N. Reading proved that glueing together the cones of the braid fan that belong to the same congruence class defines a complete fan. We prove that this fan is the normal fan of a polytope.

A penteract partition by means of the optimal subdivision of cells

Freudental's algorithm obtained way back in early forties have been traditionally used for simplicial triangulating of the hypercube. The main advantage of this algorithm is that it only generates one congruence class. Unfortunately, Freudental's algorithm is not optimal with respect to the measure of degeneracy. The multigrid methods require the degeneracy measure to be as small as possible. The minimal subdivision in the 3-dimensional case and the uniform tesseract corner subdivision in the 4-dimensional case are optimal in regards the measure of degeneracy and multigrid applications. The question about the optimal refinement strategy in more dimensional cases is still an open problem. This paper deals with a penteract subdivision with degeneracy measure much better than one obtained by the Freudental algorithm.

Refinement strategies related to cubic tetrahedral meshes

The paper deals with refinement techniques suitable for application of finite element multigrid methods with cubic trial functions. The 27-refinement strategy by Edelsbrunner and Grayson has not been studied from computational point of view up to now. This refinement strategy is said to be dark red refinement strategy (DRRS) by analogy with the red refinement strategy in the quadratic case. A detail analysis of DRRS is an aim of this paper. The dependence of the DRRS on the numbering of vertex nodes requires an algorithm development. Freudenthal partition of the cube has been widely used by researchers for obtaining hierarchical tetrahedral triangulations. In the cubic case, the refinement of the first Sommerville tetrahedron is of considerable practical importance. Since the DRRS generates one class of similarity only with a particular numbering of nodes, we have developed a detailed algorithm for proper implementation of this numbering. The situation is much more complicated when an arbitrary tetrahedron is refined. A large volume of computations are necessary in order to be established a correct numbering of the vertex nodes. If the best numbering of nodes is not chosen for all elements in all levels, the measure of degeneracy tends to infinity when the number of levels grows up unlimited. To avoid these difficulties a new canonical refinement strategy (CRS) is obtained. The CRS is superior compared to the 27-partition technique with respect to the degeneracy measure. It generates only regular tetrahedra and canonical simplices in all levels and for all mixed domains. These two tetrahedra are the most convenient from a computational point of view. Moreover, for all mixed domains the CRS essentially reduces the number of congruence classes. The new refinement strategy does not need an algorithm for correct numbering of the vertex nodes. The DRRS generates a very complicated refinement tree and a higher measure of degeneracy in combination with the face centered partition of the cube. On the contrary the Z refinement strategy obtained by the authors is superior than the DRRS in regard to the number of congruence classes and the measure of degeneracy. Some results on boundary value problems in changing domains are discussed. The advantages of the CRS are demonstrated by solving an anisotropic diffusion problem in a shrinking domain. The behavior of discretization and truncation errors in approximate finite element solutions is illustrated by varying sequences of finite element triangulations.

A family of congruences for Rogers–Ramanujan subpartitions

In 2015 Choi, Kim, and Lovejoy studied a weighted partition function, A1(m), which counted subpartitions with a structure related to the Rogers–Ramanujan identities. They conjectured the existence of an infinite class of congruences for A1(m), modulo powers of 5. We give an explicit form of this conjecture, and prove it for all powers of 5.

Finite Point Configurations in the Plane, Rigidity and Erdős Problems

For a finite point set E ⊂ ℝd and a connected graph G on k + 1 vertices, we define a G-framework to be a collection of k + 1 points in E such that the distance between a pair of points is specified if the corresponding vertices of G are connected by an edge. We consider two frameworks the same if the specified edge-distances are the same. We find tight bounds on such distinct-distance drawings for rigid graphs in the plane, deploying the celebrated result of Guth and Katz. We introduce a congruence relation on a wider set of graphs, which behaves nicely in both the real-discrete and continuous settings. We provide a sharp bound on the number of such congruence classes. We then make a conjecture that the tight bound on rigid graphs should apply to all graphs. This appears to be a hard problem even in the case of the nonrigid 2-chain. However, we provide evidence to support the conjecture by demonstrating that if the Erd˝os pinned-distance conjecture holds in dimension d, then the result for all graphs in dimension d follows.

Minimal Conformally Flat Hypersurfaces

We study conformally flat hypersurfaces $$f:M^{3} \rightarrow {\mathbb {Q}}^{4}(c)$$ with three distinct principal curvatures and constant mean curvature H in a space form with constant sectional curvature c. First we extend a theorem due to Defever when $$c=0$$ and show that there is no such hypersurface if $$H\ne 0$$ . Our main results are for the minimal case $$H=0$$ . If $$c\ne 0$$ , we prove that if $$f:M^{3} \rightarrow {\mathbb {Q}}^{4}(c)$$ is a minimal conformally flat hypersurface with three distinct principal curvatures then $$f(M^3)$$ is an open subset of a generalized cone over a Clifford torus in an umbilical hypersurface $${\mathbb {Q}}^{3}({\tilde{c}})\subset {\mathbb {Q}}^4(c)$$ , $$\tilde{c}>0$$ , with $${\tilde{c}}\ge c$$ if $$c>0$$ . For $$c=0$$ , we show that, besides the cone over the Clifford torus in $${{\mathbb {S}}}^3\subset {{\mathbb {R}}}^4$$ , there exists precisely a one-parameter family of (congruence classes of) minimal isometric immersions $$f:M^3 \rightarrow {{\mathbb {R}}}^4$$ with three distinct principal curvatures of simply connected conformally flat Riemannian manifolds.

English

English

Hopf algebras on decorated noncrossing arc diagrams

Noncrossing arc diagrams are combinatorial models for the equivalence classes of the lattice congruences of the weak order on permutations. In this paper, we provide a general method to endow these objects with Hopf algebra structures. Specific instances of this method produce relevant Hopf algebras that appeared earlier in the literature.

GOLDBACH REPRESENTATIONS IN ARITHMETIC PROGRESSIONS AND ZEROS OF DIRICHLET L ‐FUNCTIONS

Assuming a conjecture on distinct zeros of Dirichlet L-functions we get asymptotic results on the average number of representations of an integer as the sum of two primes in arithmetic progression. On the other hand the existence of good error terms gives information on the the location of zeros of L-functions and possible Siegel zeros. Similar results are obtained for an integer in a congruence class expressed as the sum of two primes.

Algebraic curves uniformized by congruence subgroups of triangle groups

We construct certain subgroups of hyperbolic triangle groups which we call \congruence subgroups. These groups include the classical congruence subgroups of SL2(Z), Hecke triangle groups, and 19 families of arithmetic triangle groups associated to Shimura curves. We determine the eld of moduli of the curves associated to these groups and thereby realize the groups PSL2(Fq) and PGL2(Fq) regularly as Galois groups.

On the fourth power moment of the error term for the divisor problem with congruence conditions

Let [Formula: see text] denote the number of factorizations [Formula: see text], where each of the factors [Formula: see text] belongs to a prescribed congruence class [Formula: see text]. Let [Formula: see text] be the error term of the asymptotic formula of [Formula: see text]. In this paper, we establish an asymptotic formula of the fourth power moment of [Formula: see text] and prove that [Formula: see text] with [Formula: see text], which improves the previous value [Formula: see text] of Liu [On higher-power moments of the error term for the divisor problem with congruence conditions, Monatsh. Math. 163(2) (2011) 175–195].

Privacy enabled disjoint and dynamic address auto-configuration protocol for 6Lowpan

In unsecured 6LoWPANs, the nodes can be easily identified by their IPv6 as well as MAC addresses. An adversary can snoop (and later, spoof) these addresses, thereby posing a major threat against the node’s identity and communication integrity. Such threats necessitate enabling privacy by obscuring the node’s addresses. This study proposes a protocol for dynamic, auto-configuring and conflict-free IPv6 addressing scheme that attempts to ensure privacy of nodes. In the proposed protocol, each node obtains a three-level hierarchical IPv6 address space which is dynamically generated on basis of congruence classes. Use of congruence classes, along with hierarchical addressing, facilitates generation of inter-leaved (and hence, disjoint) and non-fragmented address space for each node, resulting in conflict free address auto-generation. Nodes auto-configure their address sets independently with congruence seeds shared by routers, potentially reducing router complexity. To ensure the MAC address privacy, MAC address also changes when IPv6 address changes and it is derived from the interface identification (IID) part of the IPv6 address. The proposed protocol runs on Contiki operating system, simulated in Cooja. Simulated results highlight lower latency and optimal communication costs when compared with existing protocols.

On the Moduli Space of Isometric Surfaces with the Same Mean Curvature in 4-Dimensional Space Forms

We study the moduli space of congruence classes of isometric surfaces with the same mean curvature in 4-dimensional space forms. Having the same mean curvature means that there exists a parallel vector bundle isometry between the normal bundles that preserves the mean curvature vector fields. We prove that if both Gauss lifts of a compact surface to the twistor bundle are not vertically harmonic, then there exist at most three non-trivial congruence classes. We show that surfaces with a vertically harmonic Gauss lift possess a holomorphic quadratic differential, yielding thus a Hopf-type theorem. We prove that such surfaces allow locally a one-parameter family of isometric deformations with the same mean curvature. This family is trivial only if the surface is superconformal. For such compact surfaces with non-parallel mean curvature, we prove that the moduli space is the disjoint union of two sets, each one being either finite, or a circle. In particular, for surfaces in $$\mathbb {R}^4$$ we prove that the moduli space is a finite set, under a condition on the Euler numbers of the tangent and normal bundles.

On congruence properties of the restricted partition functions $$p_{\omega }(n,k)$$ p ω ( n , k ) and $$p_{\nu }(n,k)$$ p ν ( n , k )

In a recent work by Andrews, Dixit, and Yee the partition functions $$p_{\omega }(n)$$ and $$p_{\nu }(n)$$ were introduced in connection with the third order mock theta functions $$\omega (q)$$ and $$\nu (q)$$ , respectively. The function $$p_{\omega }(n)$$ counts the number of partitions of n in which each odd part is less than twice the smallest part, and $$p_{\nu }(n)$$ counts the number of partitions of n under the same conditions as $$p_{\omega }(n)$$ and having all parts distinct. In this paper, we consider restrictions of these functions, namely $$p_{\omega }(n,k)$$ and $$p_{\nu }(n,k)$$ , where k is the number of parts. We present congruence properties for these restricted partition functions and we obtain classes of Ramanujan-type congruences for them.

On the Semigroup Whose Elements Are Subgraphs of a Complete Graph

Let K n be a complete graph on n vertices. Denote by S K n the set of all subgraphs of K n . For each G , H ∈ S K n , the ring sum of G and H is a graph whose vertex set is V ( G ) ∪ V ( H ) and whose edges are that of either G or H, but not of both. Then S K n is a semigroup under the ring sum. In this paper, we study Green’s relations on S K n and characterize ideals, minimal ideals, maximal ideals, and principal ideals of S K n . Moreover, maximal subsemigroups and a class of maximal congruences are investigated. Furthermore, we prescribe the natural order on S K n and consider minimal elements, maximal elements and covering elements of S K n under this order.

A study on (strong) order-congruences in ordered semihypergroups

In this paper, we introduce the concepts of order-congruences and strong order-congruences on an ordered semihypergroup $S,$ and obtain the relationship between strong order-congruences and pseudoorders on $S.$ Furthermore, we characterize the (strong) order-congruences by the $\rho$-chains, where $\rho$ is a (strong) congruence on $S.$ Moreover, we give a method of constructing order-congruences, and prove that every hyperideal $I$ of an ordered semihypergroup $S$ is congruence class of one order-congruence on $S$ if and only if $I$ is convex. Finally, we define and study the strong order-congruence generated by a strong congruence. As an application of the results of this paper, we solve an open problem on ordered semihypergroups given by Davvaz et al.

Families of even non-congruent numbers with prime factors in each odd congruence class modulo eight

New families of even non-congruent numbers with arbitrarily many distinct prime factors are presented. Methods for generating even non-congruent numbers from existing non-congruent numbers are also given. These results yield new even non-congruent numbers whose factorizations contain primes in all four odd congruence classes modulo eight.

Congruence formulas for Legendre modular polynomials

Let p≥5 be a prime number. We generalize the results of E. de Shalit [4] about supersingular j-invariants in characteristic p. We consider supersingular elliptic curves with a basis of 2-torsion over F‾p, or equivalently supersingular Legendre λ-invariants. Let Fp(X,Y)∈Z[X,Y] be the p-th modular polynomial for λ-invariants. A simple generalization of Kronecker's classical congruence shows that R(X):=Fp(X,Xp)p is in Z[X]. We give a formula for R(λ) if λ is supersingular. This formula is related to the Manin–Drinfeld pairing used in the p-adic uniformization of the modular curve X(Γ0(p)∩Γ(2)). This pairing was computed explicitly modulo principal units in a previous work of both authors. Furthermore, if λ is supersingular and is in Fp, then we also express R(λ) in terms of a CM lift (which is shown to exist) of the Legendre elliptic curve associated to λ.