Congruence Properties Research Articles

Some Congruence Properties of a Restricted Bipartition Function cN(n)

Let cN(n) denote the number of bipartitions (λ,μ) of a positive integer n subject to the restriction that each part of μ is divisible by N. In this paper, we prove some congruence properties of the function cN(n) for N=7, 11, and 5l, for any integer l≥1, by employing Ramanujan’s theta-function identities.

Presupposition cancellation: explaining the ‘soft–hard’ trigger distinction

Some presuppositions are easier to cancel than others in embedded contexts. This contrast has been used as evidence for distinguishing two fundamentally different kinds of presuppositions, ‘soft’ and ‘hard’. ‘Soft’ presuppositions are usually assumed to arise in a pragmatic way, while ‘hard’ presuppositions are thought to be genuine semantic presuppositions. This paper argues against such a distinction and proposes to derive the difference in cancellation from inherent differences in how presupposition triggers (and the sentences that contain them) interact with the context: their focus sensitivity, anaphoricity, and question–answer congruence properties. The paper also aims to derive the presuppositions of additive particles such as too, also, again, and of it-clefts.

Multiple Spectra of Bernoulli Convolutions

AbstractLet μλ be the Bernoulli convolution associated with λ ∈ (0, 1). The well-known result of Jorgensen and Pedersen shows that if λ = 1/(2k) for some k ∈ ℕ, then μ1/(2k) is a spectral measure with spectrum Γ(1/(2k)). The recent research on the spectrality of μλ shows that μλ is a spectral measure only if λ = 1/(2k) for some k ∈ ℕ. Moreover, for certain odd integer p, the multiple set pΓ(1/(2k)) is also a spectrum for μ1/(2k). This is surprising because some spectra for the measure μ1/(2k) are thinning. In this paper we mainly characterize the number p that has the above property. By applying the properties of congruences and the order of elements in the finite group, we obtain several conditions on p such that pΓ(1/(2k)) is a spectrum for μ1/(2k).

Spacetimes of Weyl and Ricci type N in higher dimensions

We study the geometrical properties of null congruences generated by an aligned null direction of the Weyl tensor (WAND) in spacetimes of Weyl and Ricci type N (possibly with a non-vanishing cosmological constant) in an arbitrary dimension. We prove that a type N Ricci tensor and a type III or N Weyl tensor have to be aligned. In such spacetimes, the multiple WAND has to be geodetic. For spacetimes with type N aligned Weyl and Ricci tensors, the canonical form of the optical matrix in the twisting and non-twisting cases is derived and the dependence of the Weyl and the Ricci tensors and Ricci rotation coefficients on the affine parameter of the geodetic null congruence generated by the WAND is obtained.

A Sorted Semantic Framework for Applied Process Calculi

Applied process calculi include advanced programming constructs such as type systems, communication with pattern matching, encryption primitives, concurrent constraints, nondeterminism, process creation, and dynamic connection topologies. Several such formalisms, e.g. the applied pi calculus, are extensions of the the pi-calculus; a growing number is geared towards particular applications or computational paradigms. Our goal is a unified framework to represent different process calculi and notions of computation. To this end, we extend our previous work on psi-calculi with novel abstract patterns and pattern matching, and add sorts to the data term language, giving sufficient criteria for subject reduction to hold. Our framework can directly represent several existing process calculi; the resulting transition systems are isomorphic to the originals up to strong bisimulation. We also demonstrate different notions of computation on data terms, including cryptographic primitives and a lambda-calculus with erratic choice. Finally, we prove standard congruence and structural properties of bisimulation; the proof has been machine-checked using Nominal Isabelle in the case of a single name sort.

Combinatorial Properties of Rogers-Ramanujan Type Identities Arising from Hall-Littlewood Polynomials

Here we consider the q-series coming from the Hall-Littlewood polynomials, $$\begin{array}{l}{R_{v} (a, b; q) = {\sum_{\mathop {\lambda}\limits_{\lambda_1 \leq a}}} q^{c | \lambda |} P_{2\lambda}(1, q, q^{2}, \ldots ; q^{2b+d}).}\end{array}$$ These series were defined by Griffin, Ono, and Warnaar in their work on the framework of the Rogers-Ramanujan identities. We devise a recursive method for computing the coefficients of these series when they arise within the Rogers-Ramanujan framework. Furthermore, we study the congruence properties of certain quotients and products of these series, generalizing the famous Ramanujan congruence $$p(5n+4) \equiv 0\quad ({\rm mod}\, 5).$$

Congruence properties for a certain kind of partition functions

In light of the modular equations of fifth and seventh order, we derive some congruence properties for a certain kind of partition functions a(n) which satisfy ∑n=0∞a(n)qn≡(q;q)∞k(modm), where k is a positive integer with 1≤k≤24 and m=2,3. In view of these properties, we obtain many infinite families of congruences for cϕk(n), the number of generalized Frobenius partitions of n with k colors, and cϕk‾(n), the number of generalized Frobenius partitions of n with k colors whose order is k under cyclic permutation of the k colors. Meanwhile, we also apply the main theorems to some other kinds of partition functions.

A moving approach for the Vector Hysteron Model

A moving approach for the VHM (Vector Hysteron Model) is here described, to reconstruct both scalar and rotational magnetization of electrical steels with weak anisotropy, such as the non oriented grain Silicon steel. The hysterons distribution is postulated to be function of the magnetization state of the material, in order to overcome the practical limitation of the congruency property of the standard VHM approach. By using this formulation and a suitable accommodation procedure, the results obtained indicate that the model is accurate, in particular in reproducing the experimental behavior approaching to the saturation region, allowing a real improvement respect to the previous approach.

Congruence property in conformal field theory

The congruence subgroup property is established for the modular representations associated to any modular tensor category. This result is used to prove that the kernel of the representation of the modular group on the conformal blocks of any rational, C_2-cofinite vertex operator algebra is a congruence subgroup. In particular, the q-character of each irreducible module is a modular function on the same congruence subgroup. The Galois symmetry of the modular representations is obtained and the order of the anomaly for those modular categories satisfying some integrality conditions is determined.

Influence of geometry variations on the gravitational focusing of timelike geodesic congruences

We derive a set of equations describing the linear response of the convergence properties of a geodesic congruence to arbitrary geometry variations. It is a combination of equations describing the deviations from the standard Raychaudhuri-type equations due to the geodesic shifts and an equation describing the geodesic shifts due to the geometry variations. In this framework, the geometry variations, which can be chosen arbitrarily, serve as probes to investigate the gravitational contraction processes from various angles. We apply the obtained framework to the case of conformal geometry variations, characterized by an arbitrary function $f(x)$, and see that the formulas get simplified to a great extent. We investigate the response of the convergence properties of geodesics in the latest phase of gravitational contractions by restricting the class of conformal geometry variations to the one satisfying the strong energy condition. We then find out that in the final stage, $f$ and $ {\bf D}\cdot {\bf D} f$ control the overall contraction behavior and that the contraction rate gets larger when $f$ is negative and $|f|$ is so large as to overwhelm $|{\bf D} \cdot {\bf D} f|$. (Here ${\bf D} \cdot {\bf D}$ is the Laplacian operator on the spatial hypersurfaces orthogonal to the geodesic congruence in concern.) To get more concrete insights, we also apply the framework to the time-reversed Friedmann-Robertson-Walker model as the simplest case of the singularity formations.

Psi-Calculi in Isabelle

This paper presents a mechanisation of psi-calculi, a parametric framework for modelling various dialects of process calculi including (but not limited to) the pi-calculus, the applied pi-calculus, and the spi calculus. psi-calculi are significantly more expressive, yet their semantics is as simple in structure as the semantics of the original pi-calculus. Proofs of meta-theoretic properties for psi-calculi are more involved, however, not least because psi-calculi (unlike simpler calculi) utilise binders that bind multiple names at once. The mechanisation is carried out in the Nominal Isabelle framework, an interactive proof assistant designed to facilitate formal reasoning about calculi with binders. Our main contributions are twofold. First, we have developed techniques that allow efficient reasoning about calculi that bind multiple names in Nominal Isabelle. Second, we have adopted these techniques to mechanise substantial results from the meta-theory of psi-calculi, including congruence properties of bisimilarity and the laws of structural congruence. To our knowledge, this is the most extensive formalisation of process calculi mechanised in a proof assistant to date.

Dwork congruences and reflexive polytopes

We show that the coefficients of the power series expansion of the principal period of a Laurent polynomial satisfy strong congruence properties. These congruences play key role in the explicit p-adic analytic continuation of the unit-root. The methods we use are completely elementary.

Congruences and d -filters of principal p -algebras

Abstract The concept of d -filters is introduced in p -algebras. Some properties of d -filters are studied. It is proved that the class F d ( L ) of all d -filters of a p -algebra L is a bounded complete lattice. A characterization of d -filters of a principal p -algebra is given. Also many properties of congruences induced by the d -filters are derived. A relationship between the d -filters of a principal p -algebra L and the congruences in [ Φ , ∇] is established.

Congruence properties of induced representations and their applications

Congruence properties of the representations $U_\alpha :=U^{\mathrm {PSL} (2,\mathbb {Z})}_{\chi _\alpha }$ are studied for the projective modular group $\mathrm {PSL} (2,\mathbb {Z})$ induced by a family $\chi _\alpha$ of characters for the Hecke congruence subgroup $\Gamma _0(4)$, basically introduced by A. Selberg. The interest in the representations $U_\alpha$ stems from their presence in the transfer operator approach to Selberg’s zeta function for this Fuchsian group and the character $\chi _\alpha$. Hence, the location of the nontrivial zeros of this function and therefore also the spectral properties of the corresponding automorphic Laplace–Beltrami operator $\Delta _{\Gamma ,\chi _\alpha }$ are closely related to their congruence properties. Even if, as expected, these properties of the $U_\alpha$ are easily shown to be equivalent to those well-known for the characters $\chi _\alpha$, surprisingly, both the congruence and the noncongruence groups determined by their kernels are quite different: those determined by $\chi _\alpha$ are character groups of type I of the group $\Gamma _0(4)$, whereas those determined by $U_\alpha$ are character groups of the same kind for $\Gamma (4)$. Furthermore, unlike infinitely many of the groups $\ker \chi _\alpha$, whose noncongruence properties follow simply from Zograf’s geometric method together with Selberg’s lower bound for the lowest nonvanishing eigenvalue of the automorphic Laplacian, such arguments do not apply to the groups $\ker U_\alpha$, for the reason that they can have arbitrary genus $g\geq 0$, unlike the groups $\ker \chi _\alpha$, which all have genus $g=0$.

Some congruences for Siegel theta series

We discuss an arithmetic approach to some congruence properties of Siegel theta series of even positive definite unimodular quadratic forms.

Material-symmetries congruency in transversely isotropic and orthotropic hyperelastic materials

In this work we present the material-symmetries congruency property for anisotropic hyperelastic materials. A transversely isotropic or orthotropic material model holding this property guarantees the recovery of the isotropic behavior when, for example, the volume of reinforcement in a composite material vanishes. A material presenting material-symmetries congruency must show it not only from an analytical, theoretical point of view, but also from a numerical, practical point of view. The former may be obtained from construction of the stored energy function, whereas the latter must be obtained guaranteeing that the parameter-fitting procedure yields material parameters which combined with the specific model results in an isotropic behavior prediction for isotropic materials. We show that some anisotropic models do not present either analytical or numerical material-symmetries congruency.

Congruence properties of multiplicative functions on sumsets and monochromatic solutions of linear equations

Letting $\Omega(n)$ denote the number of prime factors of $n$ counted with multiplicity, Rivat, Sarkozy and Stewart (1999) proved a result regarding maximal cardinalities of sets ${\cal A},{\cal B}\subset\{1,\ldots,N\}$ so that for every $a\in{\cal A}$ and $b\in{\cal B}$, $\Omega(a+b)$ is even. This paper extends their work in several directions. The role of $\lambda(n)=(-1)^{\Omega(n)}$ is generalized to all non-constant completely multiplicative functions $f:\mathbb{N}\rightarrow \{-1,1\}$. Rather than just $\Omega$ being even on ${\cal A}+{\cal B}$, we extend the result to all possible parities of $\Omega$ on ${\cal A}$, ${\cal B}$, and ${\cal A}+{\cal B}$. Furthermore, we prove that many such pairs $({\cal A},{\cal B})$ exist. Results from Ramsey theory and extremal graph theory are used.

REDEFINED FUZZY CONGRUENCES ON SEMIGROUPS

We redefine a fuzzy congruence, discuss some properties of the fuzzy congruences, find the fuzzy congruence generated by a fuzzy relation on a semigroup, and give some lattice theoretic properties of the fuzzy congruence on semigroups.

An Inverted Preisach Model With Analytical Weight Function and Its Numerical Discrete Formulation

This paper proposes a numerical discrete formulation for an inverted hysteresis model, which is based on the classic Preisach model, but applies an inverted switch operator. The inverted switch operator is modified from the switch operator of the classic Preisach model. By applying this inverted switch operator, the inverted model retains the wiping-out and congruency properties. The distribution of the weight function can be approximated with an analytical function. By means of a quasi-Newton method algorithm, the small number of parameters in this function can be directly determined from major loops. Iterative procedure is circumvented in the implementation of this model in finite-element analysis. The simulated hysteresis loops, such as minor loops and reversal curves, show good agreement to the measurement data. To realize efficient numerical calculation, the integral expression of the inverted model is rewritten in a discrete form. The double integration of the weight function is numerically approximated in the precalculation stage. In this way, it costs low memory usage and short calculation time during the calculation procedure.

Selection of an instrument to evaluate the organizational environment of nurses working in intensive care: an integrative review

Objective: To determine an appropriate survey instrument to evaluate the impact of organizational structures on the work environment of intensive care nurses. Background: Internationally the demand for intensive care is increasing. Solely increasing bed capacity is not sustainable. Large capacity multi-specialty Intensive Care Units are emerging as the preferred organizational model with benefits resulting from optimizing operational synergies and economies of scale. The impact of this organizational transition on intensive care nurses is not well understood. An appropriate survey instrument for intensive care nurses is required. Design: Integrative literature review. Data Sources: CINAHL, PubMed, EMBASE and OVID Nursing databases searched for studies published between 2005 and 2013. Review methods: An integrative review and quality assessment of the studies was undertaken to select nurse outcome measures associated with organizational structures across a range of acute and critical care settings. Congruence between nurse outcome measures and nurse survey instruments tested in the literature was assessed to select instruments for further psychometric evaluation. Results: Thirty-one cross sectional quantitative studies, from fourteen countries, were reviewed. Twenty one nurse outcome measures associated with organizational factors were identified and a total of twenty five survey instruments used in the studies reviewed. Assessment of congruence and psychometric properties determined that a combination of two instruments is required to comprehensively assess the organizational environment of nurses working in intensive care units. Conclusion: The environment of nurses working in intensive care is effectively evaluated with an instrument that combines subscales from the Practice Environment Scale-Nurse Work Index and Maslach’s Burnout Inventory.