Congruence Properties Research Articles

CONGRUENCES MODULO SQUARES OF PRIMES FOR FU'S k DOTS BRACELET PARTITIONS

In 2007, Andrews and Paule introduced the family of functions Δk(n) which enumerate the number of broken k-diamond partitions for a fixed positive integer k. In that paper, Andrews and Paule proved that, for all n ≥ 0, Δ1(2n+1) ≡ 0 (mod 3) using a standard generating function argument. Soon after, Shishuo Fu provided a combinatorial proof of this same congruence. Fu also utilized this combinatorial approach to naturally define a generalization of broken k-diamond partitions which he called k dots bracelet partitions. He denoted the number of k dots bracelet partitions of n by 𝔅k(n) and proved various congruence properties for these functions modulo primes and modulo powers of 2. In this note, we extend the set of congruences proven by Fu by proving the following congruences: For all n ≥ 0, [Formula: see text] We also conjecture an infinite family of congruences modulo powers of 7 which are satisfied by the function 𝔅7.

On the arithmetic and geometry of binary Hamiltonian forms

Given an indefinite binary quaternionic Hermitian form f with coefficients in a maximal order of a definite quaternion algebra over Q, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational integers with absolute value at most s by f , as s tends to +∞. We compute the volumes of hyperbolic 5-manifolds constructed by quaternions using Eisenstein series. In the Appendix, V. Emery computes these volumes using Prasad’s general formula. We use hyperbolic geometry in dimension 5 to describe the reduction theory of both definite and indefinite binary quaternionic Hermitian forms.

Congruence Properties of Binary Partition Functions

Let $${\mathcal{A}}$$ be a finite subset of $${\mathbb{N}}$$ containing 0, and let f (n) denote the number of ways to write n in the form $${\sum \varepsilon _{j}2^{j}}$$ , where $${\varepsilon _{j} \epsilon \mathcal{A}}$$ . We show that there exists a computable $${T = T (\mathcal{A})}$$ so that the sequence (f (n) mod 2) is periodic with period T. Variations and generalizations of this problem are also discussed.

Multipliers of Hypersemilattices

The notion of multipliers is introduced in a hypersemilattice and some properties of multipliers are studied. In addition, a set of equivalent conditions are established for two multipliers of a hypersemilattice to be equal in the sense of mappings. Further, the properties of invariance subsets and invariance congruences are studied with respect to multipliers.

Partitions and the grand canonical ensemble

Two disconnected remarks about partitions. First, a pedagogical remark connecting pure mathematics with statistical physics. The grand canonical ensemble of statistical mechanics is applied to the counting of partitions. This picture borrowed from physics gives a simple approximation to the exact calculation of the partition function by Hardy and Ramanujan. Second, an exact formula is guessed for the function NS(m,n) defined in a recent paper by Andrews, Garvan, and Liang. The formula was subsequently proved by Garvan. We hope that it may lead to a better understanding of the beautiful new congruence properties of partitions discovered by Andrews.

Weak Markovian Bisimulation Congruences and Exact CTMC-Level Aggregations for Concurrent Processes

We have recently defined a weak Markovian bisimulation equivalence in an integrated-time setting, which reduces sequences of exponentially timed internal actions to individual exponentially timed internal actions having the same average duration and execution probability as the corresponding sequences. This weak Markovian bisimulation equivalence is a congruence for sequential processes with abstraction and turns out to induce an exact CTMC-level aggregation at steady state for all the considered processes. However, it is not a congruence with respect to parallel composition. In this paper, we show how to generalize the equivalence in a way that a reasonable tradeoff among abstraction, compositionality, and exactness is achieved for concurrent processes. We will see that, by enhancing the abstraction capability in the presence of concurrent computations, it is possible to retrieve the congruence property with respect to parallel composition, with the resulting CTMC-level aggregation being exact at steady state only for a certain subset of the considered processes.

Analysis of the designs of iraca hats made in Colón - Genova, Nariño

This document presents an analysis of the interlaces used in the configuration of nine straw hat styles produced by the Colon - Genova community, Narino, Colombia. These hats are known under the names of Comun, Pintao, Gallineto- Granizo, Fino, Cuadros, Calado or Huecos, Costeno or Vueltiao, Bandera and Bird's-Eye. The interpretation of the geometric thought employed in the design of these hats had like reference, between other authors, the methodologies proposed by Gerdes (1999) and Aroca (2009), that have been applied in the African continent and in South America. Besides, the existence of a logic of the construction became established, based in count strategies and organization that become evident in the crossed of the iracas or chullas. The evolution techniques were evidenced, which imply modifications in the basic fabric bearing to changes in the logic of the construction, resulting in the generation of new designs that search to fulfill to the needs of the present-day society. Craftsmen discuss the symmetry in their designs applying complex processes of estimate and count, which are tried to show. The symmetric figures correspond to triangles, quadrilateral and hexagons organized in stripes that fulfill properties of congruence, similarity and parallelism, being this a language of the researchers interpretation validated upon showing the results to academic communities.

Congruence on Regular Semigroups

This paper deals with the properties of congruence on regular semigroups. The motivation to prove the theorems in this paper is due to the results of J. M. Howie (2).

Sporadic sequences, modular forms and new series for 1/π

Two new sequences, which are analogues of six sporadic examples of D. Zagier, are presented. The connection with modular forms is established and some new series for 1/π are deduced. The experimental procedure that led to the discovery of these results is recounted. Proofs of the main identities will be given, and some congruence properties that appear to be satisfied by the sequences will be stated as conjectures.

On the number of partitions of n into k different parts

We study the number of partitions of n into k different parts by constructing a generating function. As an application, we will prove mysterious identities involving convolution of divisor functions and a sum over partitions. By using a congruence property of the overpartition function, we investigate values of a certain convolution sum of two divisor functions modulo 8.

Lattice properties of congruences for stochastic relations

We have a look at the set of congruences for a stochastic relation; conditions under which the infimum or the supremum of two congruences is a congruence again are investigated. Congruences are based on smooth equivalence relations, and consequences of the observation that the supremum of two smooth relations may fail to be smooth are discussed: analytic spaces are not closed under pushouts, and the set of countably generated σ-algebras is not closed under finite intersections.

An Efficient Key Management Scheme for Wireless Sensor Networks

Wireless sensor networks (WSNs) can be used in a wide range of environments. Due to the inherent characteristics of wireless communications, WSNs are more vulnerable to be attacked than conventional networks. Authentication and data confidentiality are critical in these settings. It is necessary to design a useful key management scheme for WSNs. In this paper, we propose a novel key management scheme called MAKM (modular arithmetic based key management). The proposed MAKM scheme is based on the congruence property of modular arithmetic. Each member sensor node only needs to store a key seed. This key seed is used to compute a unique shared key with its cluster head and a group key shared with other nodes in the same cluster. Thus, MAKM minimizes the key storage space. Furthermore, sensor nodes in the network can update their key seeds very quickly. Performance evaluation and simulation results show that the proposed MAKM scheme outperforms other key-pool-based schemes in key storage space and resilience against nodes capture. MAKM scheme can also reduce time delay and energy consumption of key establishment in large-scale WSNs.

Arithmetic properties of overpartition pairs

Abstract. Bringmann and Lovejoy introduced a rank for overpartition pairs and investigated its role in congruence properties of pp(n), the number of overpartition pairs of n. In particular, they applied the theory of Klein forms to show that there exist many Ramanujan-type congruences for pp(n). In this paper, we derive two Ramanujantype identities and some explicit congruences for pp(n). Moreover, we find three ranks as combinatorial interpretations of the fact that pp(n) is divisible by three for any n. We also construct infinite families of congruences for pp(n) modulo 3 and 5, and two congruence relations modulo 9.

On the representation of integers by indefinite binary Hermitian forms

Given an indefinite binary quaternionic Hermitian form $f$ with coefficients in a maximal order of a definite quaternion algebra over $\mathbb Q$, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational integers with absolute value at most $s$ by $f$, as $s$ tends to $+\infty$. We compute the volumes of hyperbolic 5-manifolds constructed by quaternions using Eisenstein series. In the Appendix, V. Emery computes these volumes using Prasad's general formula. We use hyperbolic geometry in dimension 5 to describe the reduction theory of both definite and indefinite binary quaternionic Hermitian forms.

Minimum Group Congruences on Eventually Regular Semigroups

An eventually regular semigroup is a semigroup in which some power of any element is regular. The minimum group congruence on an eventually regular semigroup is investigated by means of weak inverse. Furthermore, some properties of the minimum group congruence on an eventually regular semigroup are characterized.

A note on the properties of a generalized Prandtl–Ishlinskii model

A generalized Prandtl–Ishlinskii (PI) model has been proposed in Al Janaideh et al (2009Smart Mater. Struct. 18 045001) to describe the asymmetric hysteresis in smart actuators.Because the wiping out and congruency properties are essential for the validity of hysteresismodels and have not been mentioned in Al Janaideh et al (2009 Smart Mater. Struct. 18045001), in this note these properties for this generalized model are investigated andconfirmed in detail.

Congruences for Hermitian modular forms of degree 2

We give two congruence properties of Hermitian modular forms of degree 2 over Q(−1) and Q(−3). The one is a congruence criterion for Hermitian modular forms which is generalization of Sturmʼs theorem. Another is the well-definedness of the p-adic weight for Hermitian modular forms.

A novel rate-independent hysteresis model of a piezostack actuator using the congruencyproperty

This paper presents a novel hysteresis prediction model for a piezostack actuator. Themodel proposed in this work is a type of rate-independent hysteresis and is formulatedusing the inherent congruency property which exists in most piezoelectric materials.Specifically, the model is established by exploiting the fact that the high-order hystereticcurve segment is congruent with its first-order one that is limited by the same consecutivemaximum and minimum values of input. Thus, in order to successfully implement thismodel two discretized first-order datasets of the ascending and descending curves need tobe experimentally identified in advance. Using both the identified datasets and thecongruency property, a systematic approach for predicting the hysteresis of thepiezostack actuator is then obtained in two cases of input voltage: monotonicascending and monotonic descending. The developed model is experimentallyrealized in order to demonstrate the effectiveness on the hysteresis prediction. In theexperiment, three waveforms of input excitation schemes—a triangular waveformof decreasing amplitude, a triangular waveform of increasing amplitude and amulti-extremes triangular waveform—are applied to the proposed model. The hysteresischaracteristics of the piezostack actuator predicted from the proposed model arecompared with those obtained from the classical Preisach model. It is shownthat the proposed model gives better accuracy, less computation time for thehysteresis prediction and more feasibility to realize than the classical Preisach model.

Supercongruences for Apéry-like numbers

It is known that the numbers which occur in Apéryʼs proof of the irrationality of ζ ( 2 ) have many interesting congruence properties while the associated generating function satisfies a second order differential equation. We prove supercongruences for a generalization of numbers which arise in Beukersʼ and Zagierʼs study of integral solutions of Apéry-like differential equations.

Design of high-resolution radar waveforms for multi-radar and dense target environments

This study addresses the waveform design problem of high-resolution radar, especially for dense target environments when the detection of small targets in the neighbourhood of large target is more important. Such a practical application demands an ambiguity function with a sharp central spike and large clear area around it, which is free from the sidelobes. This study investigates the sidelobe-pushing property of the Linear Congruence (LC) Codes that can produce the desired sidelobe-free area around the mainlobe. Minimum Sequence Length Criteria for achieving the desired sidelobe-free area are also described. In order to achieve the sharp central mainlobe in the delay–Doppler plane for resolving closely spaced targets, modifications are suggested to the LC codes and such modified sequences are called ‘Modified Linear Congruence Codes’. The designed codes are highly suitable in dense target scenarios as well as in multi-radar environments.