Congruent Modulo Research Articles

An Algebraic Approach to the Solutions of the Open Shop Scheduling Problem

The open shop scheduling problem (OSSP) is one of the standard scheduling problems. It consists of scheduling jobs associated with a finite set of tasks developed by different machines. In this case, each machine processes at most one operation at a time, and the job processing order on the machines does not matter. The goal is to determine the completion times of the operations processed on the machines to minimize the largest job completion time, called Cmax. This paper proves that each OSSP has associated a path algebra called Brauer configuration algebra whose representation theory (particularly its dimension and the dimension of its center) can be given using the corresponding Cmax value. It has also been proved that the dimension of the centers of Brauer configuration algebras associated with OSSPs with minimal Cmax are congruent modulo the number of machines.

Generalized integer splines on arbitrary graphs

Generalized integer splines on a graph G with integer edge weights are integer vertex labelings such that if two vertices share an edge in G, the vertex labels are congruent modulo the edge weight. We introduce collapsing operations that reduce any simple graph to a single vertex, carrying with it the edge weight information. This corresponds to a sequence of surjective maps between the associated spline modules, leading to an explicit construction of a Z-module basis in terms of the edge weights.

Generalized Dissections and Monsky’s Theorem

Monsky’s celebrated equidissection theorem follows from his more general proof of the existence of a polynomial relation f among the areas of the triangles in a dissection of the unit square. More recently, the authors studied a different polynomial p, also a relation among the areas of the triangles in such a dissection, that is invariant under certain deformations of the dissection. In this paper we study the relationship between these two polynomials. We first generalize the notion of dissection, allowing triangles whose orientation differs from that of the plane. We define a deformation space of these generalized dissections and we show that this space is an irreducible algebraic variety. We then extend the theorem of Monsky to the context of generalized dissections, showing that Monsky’s polynomial f can be chosen to be invariant under deformation. Although f is not uniquely defined, the interplay between p and f then allows us to identify a canonical pair of choices for the polynomial f. In many cases, all of the coefficients of the canonical f polynomials are positive. We also use the deformation-invariance of f to prove that the polynomial p is congruent modulo 2 to a power of the sum of its variables.

Stieltjes continued fractions related to the paperfolding sequence and Rudin-Shapiro sequence

We investigate two Stieltjes continued fractions given by the paperfolding sequence and the Rudin-Shapiro sequence. By explicitly describing certain subsequences of the convergents Pn(x)/Qn(x) modulo 4, we give the formal power series expansions (modulo 4) of these two continued fractions and prove that they are congruent modulo 4 to algebraic series in Z[[x]]. Therefore, the coefficient sequences of the formal power series expansions are 2-automatic. Write Qn(x)=∑i≥0an,ixi. Then (Qn(x))n≥0 defines a two-dimensional coefficient sequence (an,i)n,i≥0. We prove that the coefficient sequences (an,imod4)n≥0 introduced by both (Qn(x))n≥0 and (Pn(x))n≥0 are 2-automatic for all i≥0. Moreover, the pictures of these two dimensional coefficient sequences modulo 4 present a kind of self-similar phenomenon.

Non-paritious Hilbert modular forms

The arithmetic of Hilbert modular forms has been extensively studied under the assumption that the forms concerned are “paritious”—all the components of the weight are congruent modulo 2. In contrast, non-paritious Hilbert modular forms have been relatively little studied, both from a theoretical and a computational standpoint. In this article, we aim to redress the balance somewhat by studying the arithmetic of non-paritious Hilbert modular eigenforms. On the theoretical side, our starting point is a theorem of Patrikis, which associates projectiveell -adic Galois representations to these forms. We show that a general conjecture of Buzzard and Gee actually predicts that a strengthening of Patrikis’ result should hold, giving Galois representations into certain groups intermediate between {{,mathrm{GL},}}_2 and {text {PGL}}_2; and we verify that the predicted Galois representations do indeed exist. On the computational side, we give an algorithm to compute non-paritious Hilbert modular forms using definite quaternion algebras. To our knowledge, this is the first time such a general method has been presented. We end the article with an example.

GENERALIZATIONS OF RAMANUJAN’S RANK FUNCTIONS COLLECTED FROM RAMANUJAN’S LOST NOTEBOOK

In1916, Srinivasa Ramanujan defined the Mock Theta functions in his lost notebook and unpublished papers. We prove the Mock Theta Conjectures with the help of Dyson’s rank and S. Ramanujan’s Mock Theta functions. These functions were quoted in Ramanujan’s lost notebook and unpublished papers. In1916, Ramanujan stated the theta series in x like A(x), B(x), C(x), D(x). We discuss the Ramanujan’s functions with the help of Dyson’s rank symbols. These functions are useful to prove the Mock Theta Conjectures. Now first Mock Theta Conjecture is “The number of partitions of 5n with rank congruent to 1 modulo 5 equals the number of partitions of 5n with rank congruent to 0 modulo 5 plus the number of partitions of n with unique smallest part and all other parts the double of the smallest part”, and Second Mock Theta Conjecture is “The double of the number of partitions of with rank congruent to 2 modulo 5 equals the sum of the number of partitions of with rank congruent to 0 and congruent to1 modulo 5, and the sum of one and the number of partitions of n with unique smallest part and all other parts  one plus the double of the smallest part”. This paper shows how to prove the Theorem 1.3 with the help of Dyson’s rank symbols N(0,5,5n+1), N(2,5, 5n+1) and shows how to prove the Theorem 1.4 with the help of Ramanujan’s theta series and Dyson’s rank symbols N(1,5, 5n+2), N(2,5, 5n+2) respectively.

Compatibility of arithmetic and algebraic local constants (the case )

We show that arithmetic local constants attached by Mazur and Rubin to pairs of self-dual Galois representations which are congruent modulo a prime number$p>2$are compatible with the usual local constants at all primes not dividing$p$and in two special cases also at primes dividing$p$. We deduce new cases of the$p$-parity conjecture for Selmer groups of abelian varieties with real multiplication (Theorem 4.14) and elliptic curves (Theorem 5.10).

Generalizations of Arnold’s version of Euler’s theorem for matrices

A recent result, conjectured by Arnold and proved by Zarelua, states that for a prime number p, a positive integer k, and a square matrix A with integral entries one has $${\textrm tr}(A^{p^k}) \equiv {\textrm tr}(A^{p^{k-1}}) ({\textrm mod}{p^k})$$ . We give a short proof of a more general result, which states that if the characteristic polynomials of two integral matrices A, B are congruent modulo p k then the characteristic polynomials of A p and B p are congruent modulo p k+1, and then we show that Arnold’s conjecture follows from it easily. Using this result, we prove the following generalization of Euler’s theorem for any 2 × 2 integral matrix A: the characteristic polynomials of A Φ(n) and A Φ(n)-ϕ(n) are congruent modulo n. Here ϕ is the Euler function, $$\prod_{i=1}^{l} p_i^{\alpha_i}$$ is a prime factorization of n and $$\Phi(n)=(\phi(n)+\prod_{i=1}^{l} p_i^{\alpha_i-1}(p_i+1))/2$$ .

A visible factor of the special L-value

Let~$A$ be a quotient of $J_0(N)$ associated to a newform $f$ such that the special $L$-value of $A$ (at $s=1$) is non-zero. We give a formula for the ratio of the special $L$-value to the real period of $A$ that expresses this ratio as a rational number. We extract an integer factor from the numerator of this formula; this factor is non-trivial in general and is related to certain congruences of $f$ with eigenforms of positive analytic rank. We use the techniques of visibility to show that, under certain hypotheses (which includes the first part of the Birch and Swinnerton-Dyer conjecture on rank), if an odd prime $q$ divides this factor, then $q$ divides either the order of the Shafarevich-Tate group or the order of a component group of $A$. Suppose $p$ is an odd prime such that $p^2$ does not divide $N$, $p$ does not divide the order of the rational torsion subgroup of $A$, and $f$ is congruent modulo a prime ideal over $p$ to an eigenform whose associated abelian variety has positive Mordell-Weil rank. Then we show that $p$ divides the factor mentioned above; in particular, $p$ divides the numerator of the ratio of the special $L$-value to the real period of $A$. Both of these results are as implied by the second part of the Birch and Swinnerton-Dyer conjecture, and thus provide theoretical evidence towards the conjecture.

A note on congruences for theta divisors

The classes of two theta divisors on an Abelian variety in the naive Grothendieck ring of varieties need not be congruent modulo the class of the affine line. To cite this article: F. Heinloth, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

The signature of a fibre bundle is multiplicative mod 4

We express the signature modulo 4 of a closed, oriented, $4k$-dimensional $PL$ manifold as a linear combination of its Euler characteristic and the new absolute torsion invariant defined in Korzeniewski [11]. Let $F \to E \to B$ be a $PL$ fibre bundle, where $F$, $E$ and $B$ are closed, connected, and compatibly oriented $PL$ manifolds. We give a formula for the absolute torsion of the total space $E$ in terms of the absolute torsion of the base and fibre, and then combine these two results to prove that the signature of $E$ is congruent modulo 4 to the product of the signatures of $F$ and $B$.

DIVISIBILITY OF DEDEKIND FINITE SETS

A Dedekind-finite set is said to be divisible by a natural number n if it can be partitioned into pieces of size n. We study several aspects of this notion, as well as the stronger notion of being partitionable into n pieces of equal size. Among our results are that the divisors of a Dedekind-finite set can consistently be any set of natural numbers (containing 1 but not 0), that a Dedekind-finite power of 2 cannot be divisible by 3, and that a Dedekind-finite set can be congruent modulo 3, to all of 0, 1, and 2 simultaneously. (In these results, 2 and 3 serve as typical examples; the full results are more general.)

The unification problem for confluent right-ground term rewriting systems

The unification problem for term rewriting systems (TRSs) is the problem of deciding, for a given TRS R and two terms M and N , whether there exists a substitution θ such that Mθ and Nθ are congruent modulo R (i.e., Mθ ↔ R * Nθ ). In this paper, the unification problem for confluent right-ground TRSs is shown to be decidable. To show this, the notion of minimal terms is introduced and a new unification algorithm for obtaining a substitution whose range consists of minimal terms is proposed. Our result extends the decidability of unification for canonical (i.e., terminating and confluent) right-ground TRSs given by Hullot [Proceedings of the 5th Conference on Automated Deduction, LNCS, vol. 87, 1980, p. 318] in the sense that the termination condition can be omitted.

A Constructive Approach to Spiegelung Relations between 3-Ranks of Absolute Ideal Class Groups and Congruent Ones Modulo (3)2 in Quadratic Fields

A Constructive Approach to Spiegelung Relations between 3-Ranks of Absolute Ideal Class Groups and Congruent Ones Modulo (3)2 in Quadratic Fields

On interpolation families of wavelet sets

The question of which groups are isomorphic to groups of interpolation maps for interpolation families of wavelet sets was raised by Dai and Larson. In this article it is shown that any finite group is isomorphic to a group of interpolation maps for some interpolation family of wavelet sets. A dyadic orthonormal (or orthogonal) wavelet is a function 4 E L2 (R) (Lebesgue measure), with the property that the set {20'(2't 1): n, l E 2} is an orthonormal basis for L2 (R). For certain measurable sets E C R the normalized characteristic function 72==7rXE is the Fourier transform of such a wavelet. There are several characterizations of such sets. In [DL] they are called wavelet sets, and the corresponding wavelet is called an s-elementary wavelet. In [FW],[HWW1],[HWW2] they are the support sets of MFS (minimal frequency supported) wavelets. In [DL], a method of operator-theoretic interpolation between certain special pairs of wavelets, and between single wavelets and special families of them is developed. The interpolation pairs of wavelet sets are extensively discussed and the method yields a new construction of a class of wavelets due to Meyer. An interpolation pair is the simplest case of an interpolation family (see definitions below). Affiliated to an interpolation family is a group (under composition) of measure preserving transformations of R called the interpolation maps. Problem E in [DL] asked for a characterization of all groups which are isomorphic to a group of interpolation maps for some interpolation family of wavelet sets. They showed, by example, that the cyclic groups 22 and 23 of order 2 and 3 could be realized in this way, but they proved no cases of other groups. The purpose of this paper is to prove that every finite group can be realized in this way. Before introducing the characterization of wavelet sets, we will need some more terminology. In [DL], two measurable sets E, F are called translation congruent modulo 2wr if there is a measurable bijection 0: E -> F such that s(s) s is an integral multiple of 2wr for each s E E. Analogously, two measurable sets G, H are called dilation congruent modulo 2 if there is a measurable bijection T: G -> H such that for each s E G there is an integer n depending on s, such that T (s) = 2s. We note in passing that if Ei and Fi are measurable subsets of R such that Ei and Received by the editors July 7, 1998 and, in revised form, November 24, 1998. 2000 Mathematics Subject Classification. Primary 42C40.

Congruence Relations for the Wiener Index of Hexagonal Chains

The Wiener index (W) of certain classes of hexagonal chains is examined. Necessary and sufficient conditions for their W-values to be congruent modulo 16 are established. As a corollary, a necessary condition for the coincidence of Wiener indices of hexagonal chains is formulated.

Thep-intersection number of a complete bipartite graph and orthogonal double coverings of a clique

Thep-intersection graph of a collection of finite sets {S i } i=1 n is the graph with vertices 1, ...,n such thati, j are adjacent if and only if |S i ∩S j |≥p. Thep-intersection number of a graphG, herein denoted θ p (G), is the minimum size of a setU such thatG is thep-intersection graph of subsets ofU. IfG is the complete bipartite graphK n,n andp≥2, then θ p (K n, n )≥(n 2+(2p−1)n)/p. Whenp=2, equality holds if and only ifK n has anorthogonal double covering, which is a collection ofn subgraphs ofK n , each withn−1 edges and maximum degree 2, such that each pair of subgraphs shares exactly one edge. By construction,K n has a simple explicit orthogonal double covering whenn is congruent modulo 12 to one of {1, 2, 5, 7, 10, 11}.

Wiener Number of Polyphenyls and Phenylenes

Abstract It is shown that the values which the Wiener numbers of isomeric polyphenyls may assume are all congruent modulo 36. Similarly, the Wiener numbers of isomeric phenylens are always congruent modulo 18. These findings provide a rationalization for the known fact that the isomer-discriminating power of the Wiener number is rather weak. Extensions of the present results to more general classes of graphs are also pointed out.

Congruence theorems for riemannian submanifolds

The aim of this paper is, essentially, to give sufficient conditions in terms of mean curvature for two submanifolds of a given Riemannian manifold to be congruent modulo a given 1-parameter group of transformations. The results obtained generalise those of several authors including M. Okumura and the present author [11].

A complete description for a monoid of deterministic bottom-up tree transformation classes

We consider the set M={ DB, LDB, NDB, LNDB, H, LH, NH} of which the elements are the class of deterministic bottom-up tree transformations (DB), the linear, the nondeleting and the linear, nondeleting subclasses of DB (LDB, NDB and LNDB); the class of homomorphism tree transformations (H), the linear and the nondeleting subclasses of H (LH and NH). The main aim of this paper is to obtain a finite generating system of the equations of the form X 1∘⋯∘ X m = Y 1∘⋯ Y n , where X i , Y j ϵM for 1⩽ i⩽ m, 1⩽ j⩽ n and ∘ is the composition of tree transformation classes. Therefore, we give a finite Thue system over M, where M is now considered as a seven-letter alphabet, such that two words X 1•⋯• X m and Y 1•⋯• Y n of the free monoid M ∗ are congruent modulo the Thue congruence ↔ T ∗ generated by T over M ∗ if and only if X 1∘⋯∘ X m = Y 1∘⋯∘ Y n . Moreover, we give a set N of representatives for the congruence ↔ T ∗ and an inclusion diagram of the tree transformation classes of the form Z 1∘⋯∘ Z k , where Z 1•⋯• Z k ϵN. We also present an algorithm, that to each word X 1•⋯•X mϵM ∗ produces the (unique) representative Z 1•⋯• Z k ϵN such that X 1•⋯• X m ↔ T ∗ Z 1•⋯• Z k . We prove that, using this algorithm and the inclusion diagram of the tree transformation classes represented by the elements of N, we can decide for any given tree transformation classes X 1∘⋯∘ X m and Y 1∘⋯∘ Y n obtained by composition from elements of M if X 1∘⋯∘ X m ⊆ Y 1∘⋯∘ Y n and, hence, if X 1∘⋯∘ X m = Y 1∘⋯∘ Y n .