Congruence Classes Research Articles

On the Jacobson radical of a semiring

Based on the minimal and simple representations of semirings, we introduce two Jacobson-type Hoehnke radicals, namely, m-radical and s-radical, of a semiring [Formula: see text]. Every minimal (simple) [Formula: see text]-semimodule is a quotient of [Formula: see text] by a regular right congruence (maximal) [Formula: see text] on [Formula: see text] such that [Formula: see text] is a maximal [Formula: see text]-saturated right ideal in [Formula: see text]. Thus the m(s)-radical becomes an intersection of a class of regular congruences. The m(s)-radical of the direct product of two semirings is the product of their individual [Formula: see text]-radicals.

On Permutation Rho-Transitive in Permutation Rho-Algebras

Some of the novel algebraic topics explored in this paper, like Permutation -algebra, permutation edge, permutation -transitive, permutation -ideal, permutation -ideal, and permutation -subalgebra are discussed and looked into. Additionally, permutation -algebra, permutation -morphism, equivalence relation with its congruence classes and quotient permutation -algebras were defined with specific results relating to our unique notions that have been developed and examined.

Benford's law and random integer decomposition with congruence stopping condition

Benford's law is a statement about the frequency that each digit arises as the leading digit of numbers in a dataset. It is satisfied by various common integer sequences, such as the Fibonacci numbers, the factorials, and the powers of most integers. In this paper, we prove that integer sequences resulting from a random integral decomposition process (which we model as discrete “stick breaking”) subject to a certain congruence stopping condition approach Benford distribution asymptotically. We also show that our requirement on the number of congruence classes defining the congruence stopping condition is necessary for Benford behavior to occur and is a critical point; deviation from that would result in drastically different behavior.

Formations and i-Fitting classes of inverse semigroups, congruences and languages

Formations and i-Fitting classes of inverse semigroups, congruences and languages

Congruence classes for modular forms over small sets

Serre showed that for any integer [Formula: see text] for almost all [Formula: see text] where [Formula: see text] is the [Formula: see text] Fourier coefficient of any modular form with rational coefficients. In this paper, we consider a certain class of cuspforms and study [Formula: see text] over the set of integers with [Formula: see text] many prime factors. Moreover, we show that any residue class [Formula: see text] can be written as the sum of at most 13 Fourier coefficients, which are polynomially bounded as a function of [Formula: see text]

Triangles with one fixed side–length, a Furstenberg-type problem, and incidences in finite vector spaces

Abstract The first goal of this paper is to prove a sharp condition to guarantee of having a positive proportion of all congruence classes of triangles in given sets in 𝔽 q 2 {\mathbb{F}_{q}^{2}} . More precisely, for A , B , C ⊂ 𝔽 q 2 {A,B,C\subset\mathbb{F}_{q}^{2}} , if | A | ⁢ | B | ⁢ | C | 1 2 ≫ q 4 {|A||B||C|^{\frac{1}{2}}\gg q^{4}} , then for any λ ∈ 𝔽 q ∖ { 0 } {\lambda\in\mathbb{F}_{q}\setminus\{0\}} , the number of congruence classes of triangles with vertices in A × B × C {A\times B\times C} and one side-length λ is at least ≫ q 2 {\gg q^{2}} . In higher dimensions, we obtain similar results for k-simplex but under a slightly stronger condition. Compared to the well-known L 2 {L^{2}} method in the literature, our approach offers better results in both conditions and conclusions. When A = B = C {A=B=C} , the second goal of this paper is to give a new and unified proof of the best current results on the distribution of simplex due to Bennett, Hart, Iosevich, Pakianathan and Rudnev (2017) and McDonald (2020). The third goal of this paper is to study a Furstenberg-type problem associated to a set of rigid motions. The main ingredients in our proofs are incidence bounds between points and rigid motions. While the incidence bounds for large sets are due to the author and Semin Yoo (2023), the bound for small sets will be proved by using a point–line incidence bound in 𝔽 q 3 {\mathbb{F}_{q}^{3}} due to Kollár (2015).

Parts in k-indivisible partitions always display biases between residue classes

Let k,t be coprime integers, and let 1≤r≤t. We let Dk×(r,t;n) denote the total number of parts among all k-indivisible partitions (i.e., those partitions where no part is divisible by k) of n which are congruent to r modulo t. In previous work of the authors [3], an asymptotic estimate for Dk×(r,t;n) was shown to exhibit unpredictable biases between congruence classes. In the present paper, we confirm our earlier conjecture in [3] that there are no “ties” (i.e., equalities) in this asymptotic for different congruence classes. To obtain this result, we reframe this question in terms of L-functions, and we then employ a nonvanishing result due to Baker, Birch, and Wirsing [1] to conclude that there is always a bias towards one congruence class or another modulo t among all parts in k-indivisible partitions of n as n becomes large.

On 7-Dimensional Nilpotent Leibniz Algebras With 1-Dimensional Leib Ideal

Leibniz algebras are nonanticommutative versions of Lie algebras. Lie algebras have many applications in many scientific areas as well as mathematical areas. Scientists from different disciplines have used specific examples of Lie algebras according to their needs. However, we mathematicians are more interested in generality than in obtaining a few examples. The classification problem for Leibniz algebras has an intrinsically wild nature as in Lie algebras. In this article, the approach of congruence classes of bilinear forms is extended to classify certain subclasses of seven-dimensional nilpotent Leibniz algebras over complex numbers. Certain cases of seven-dimensional complex nilpotent Leibniz algebras of those with one-dimensional Leib ideal and derived algebra of codimension two are classified.

Ramanujan type congruences for quotients of Klein forms

In this work, Ramanujan type congruences modulo powers of primes p≥5 are derived for a general class of products that are modular forms of level p. These products are constructed in terms of Klein forms and subsume generating functions for t-core partitions known to satisfy Ramanujan type congruences for p=5,7,11. The vectors of exponents corresponding to products that are modular forms for Γ1(p) are subsets of bounded polytopes with explicit parameterizations. This allows for the derivation of a complete list of products that are modular forms for Γ1(p) of weights 1≤k≤5 for primes 5≤p≤19 and whose Fourier coefficients satisfy Ramanujan type congruences for all powers of the primes. For each product satisfying a congruence, cyclic permutations of the exponents determine additional products satisfying congruences. Common forms among the exponent sets lead to products satisfying Ramanujan type congruences for a broad class of primes, including p>19. Canonical bases for modular forms of level 5≤p≤19 are constructed by summing weight one Hecke Eisenstein series of levels 5≤p≤19 and expressing the result as a quotient of Klein forms. Generating sets for the graded algebras of modular forms for Γ1(p) and Γ(p) are formulated in terms of permutations of the exponent sets. A sieving process is described by decomposing the space of modular forms of weight 1 for Γ1(p) as a direct sum of subspaces of modular forms for Γ(p) of the form qr/pZ[[q]]. Since the relevant bases generate the graded algebra of modular forms for these groups, the weight one decompositions determine series dissections for modular forms of higher weight that lead to additional classes of congruences.

The 3-Isogeny Selmer Groups of the Elliptic Curves y2=x3+n2

Abstract Consider the family of elliptic curves $E_{n}:y^{2}=x^{3}+n^{2}$, where $n$ varies over positive cubefree integers. There is a rational $3$-isogeny $\phi $ from $E_{n}$ to $\hat {E}_{n}:y^{2}=x^{3}-27n^{2}$ and a dual isogeny $\hat {\phi }:\hat {E}_{n}\rightarrow E_{n}$. We show that for almost all $n$, the rank of $\operatorname {Sel}_{\phi }(E_{n})$ is $0$, and the rank of $\operatorname {Sel}_{\hat {\phi }}(\hat {E}_{n})$ is determined by the number of prime factors of $n$ that are congruent to $2\bmod 3$ and the congruence class of $n\bmod 9$.

Exploring congruent diversification histories with flexibility and parsimony

Abstract Using phylogenies of present‐day species to estimate diversification rate trajectories—speciation and extinction rates over time—is a challenging task due to non‐identifiability issues. Given a phylogeny, there exists an infinite set of trajectories that result in the same likelihood; this set has been coined a congruence class. Previous work has developed approaches for sampling trajectories within a given congruence class, with the aim to assess the extent to which congruent scenarios can vary from one another. Based on this sampling approach, it has been suggested that rapid changes in speciation or extinction rates are conserved across the class. Reaching such conclusions requires to sample the broadest possible set of distinct trajectories. We introduce a new method for exploring congruence classes that we implement in the R package CRABS. Whereas existing methods constrain either the speciation rate or the extinction rate trajectory, ours provides more flexibility by sampling congruent speciation and extinction rate trajectories simultaneously. This allows covering a more representative set of distinct diversification rate trajectories. We also implement a filtering step that allows selecting the most parsimonious trajectories within a class. We demonstrate the utility of our new sampling strategy using a simulated scenario. Next, we apply our approach to the study of mammalian diversification history. We show that rapid changes in speciation and extinction rates need not be conserved across a congruence class, but that selecting the most parsimonious trajectories shrinks the class to concordant scenarios. Our approach opens new avenues both to truly explore the myriad of potential diversification histories consistent with a given phylogeny, embracing the uncertainty inherent to phylogenetic diversification models, and to select among these different histories. This should help refining our inference of diversification trajectories from extant data.

A Structural Theorem for Sets with Few Triangles

AbstractWe show that if a finite point set $$P\subseteq {\mathbb {R}}^2$$ P ⊆ R 2 has the fewest congruence classes of triangles possible, up to a constant M, then at least one of the following holds. There is a $$\sigma >0$$ σ > 0 and a line l which contains $$\Omega (|P|^\sigma )$$ Ω ( | P | σ ) points of P. Further, a positive proportion of P is covered by lines parallel to l each containing $$\Omega (|P|^\sigma )$$ Ω ( | P | σ ) points of P. There is a circle $$\gamma $$ γ which contains a positive proportion of P. This provides evidence for two conjectures of Erdős. We use the result of Petridis–Roche–Newton–Rudnev–Warren on the structure of the affine group combined with classical results from additive combinatorics.

Regular semigroups weakly generated by one element

In this paper we study the regular semigroups weakly generated by a single element x, that is, with no proper regular subsemigroup containing x. We show there exists a regular semigroup F_{1} weakly generated by x such that all other regular semigroups weakly generated by x are homomorphic images of F_{1}. We define F_{1} using a presentation where both sets of generators and relations are infinite. Nevertheless, the word problem for this presentation is decidable. We describe a canonical form for the congruence classes given by this presentation, and explain how to obtain it. We end the paper studying the structure of F_{1}. In particular, we show that the ‘free regular semigroup {textrm{FI}}_2 weakly generated by two idempotents’ is isomorphic to a regular subsemigroup of F_{1} weakly generated by {xx',x'x}.

On Rational Algorithms for Recognition of Belonging to Congruence Classes

On Rational Algorithms for Recognition of Belonging to Congruence Classes

Unexpected biases between congruence classes for parts in k-indivisible partitions

For integers k,t≥2, and 1≤r≤t let Dk×(r,t;n) be the number of parts among all k-indivisible partitions of n (i.e., partitions where all parts are not divisible by k) of n that are congruent to r modulo t. Using Wright's circle method, we derive an asymptotic for Dk×(r,t;n) as n→∞ when k,t are coprime. The main term of this asymptotic does not depend on r, and so, in a weak asymptotic sense, the parts are equidistributed among congruence classes. However, inspection of the lower order terms indicates a bias towards different congruence classes modulo t. This induces an ordering on the congruence classes modulo t, which we call the k-indivisible ordering. We prove that for k≥6(t2−1)π2 the k-indivisible ordering matches the natural ordering. We also explore the properties of these orderings when k<6(t2−1)π2.

Estimating contact network properties by integrating multiple data sources associated with infectious diseases.

To effectively mitigate the spread of communicable diseases, it is necessary to understand the interactions that enable disease transmission among individuals in a population; we refer to the set of these interactions as a contact network. The structure of the contact network can have profound effects on both the spread of infectious diseases and the effectiveness of control programs. Therefore, understanding the contact network permits more efficient use of resources. Measuring the structure of the network, however, is a challenging problem. We present a Bayesian approach to integrate multiple data sources associated with the transmission of infectious diseases to more precisely and accurately estimate important properties of the contact network. An important aspect of the approach is the use of the congruence class models for networks. We conduct simulation studies modeling pathogens resembling SARS-CoV-2 and HIV to assess the method; subsequently, we apply our approach to HIV data from the University of California San Diego Primary Infection Resource Consortium. Based on simulation studies, we demonstrate that the integration of epidemiological and viral genetic data with risk behavior survey data can lead to large decreases in mean squared error (MSE) in contact network estimates compared to estimates based strictly on risk behavior information. This decrease in MSE is present even in settings where the risk behavior surveys contain measurement error. Through these simulations, we also highlight certain settings where the approach does not improve MSE.

Regular semigroups weakly generated by idempotents

A regular semigroup is weakly generated by a set [Formula: see text] if it has no proper regular subsemigroup containing [Formula: see text]. In this paper, we study the regular semigroups weakly generated by idempotents. We show there exists a regular semigroup [Formula: see text] weakly generated by [Formula: see text] idempotents such that all other regular semigroups weakly generated by [Formula: see text] idempotents are homomorphic images of [Formula: see text]. The semigroup [Formula: see text] is defined by a presentation [Formula: see text] and its structure is studied. Although each of the sets [Formula: see text], [Formula: see text], and [Formula: see text] is infinite for [Formula: see text], we show that the word problem is decidable as each congruence class has a “canonical form”. If [Formula: see text] denotes [Formula: see text] for [Formula: see text], we prove also that [Formula: see text] contains copies of all [Formula: see text] as subsemigroups. As a consequence, we conclude that (i) all regular semigroups weakly generated by a finite set of idempotents, which include all finitely idempotent generated regular semigroups, strongly divide [Formula: see text]; and (ii) all finite semigroups divide [Formula: see text].

Berkovich–Uncu Type Partition Inequalities Concerning Impermissible Sets and Perfect Power Frequencies

Recently, Rattan and the first author (Ann. Comb. 25 (2021) 697–728) proved a conjectured inequality of Berkovich and Uncu (Ann. Comb. 23 (2019) 263–284) concerning partitions with an impermissible part. In this article, we generalize this inequality upon considering t impermissible parts. We compare these with partitions whose certain parts appear with a frequency which is a perfect $$t^{th}$$ power. In addition, the partitions that we study here have smallest part greater than or equal to s for some given natural number s. Our inequalities hold after a certain bound, which for given t is a polynomial in s, a major improvement over the previously known bound in the case $$t=1$$ . To prove these inequalities, our methods involve constructing injective maps between the relevant sets of partitions. The construction of these maps crucially involves concepts from analysis and calculus, such as explicit maps used to prove countability of $$\mathbb {N}^t$$ , and Jensen’s inequality for convex functions, and then merge them with techniques from number theory such as Frobenius numbers, congruence classes, binary numbers and quadratic residues. We also show a connection of our results to colored partitions. Finally, we pose an open problem which seems to be related to power residues and the almost universality of diagonal ternary quadratic forms.

Block colourings of star systems

An e-star system of order n is a decomposition of the complete graph Kn into copies of the complete bipartite graph K1,e (or e-star). Such systems are known to exist if and only if n≥2e and e divides (n2). We consider block colourings of such systems, where each e-star is assigned a colour, and two e-stars which share a vertex receive different colours. We present a computer analysis of block colourings of small 3-star systems. Furthermore, we prove that: (i) for n≡0,1 mod 2e there exists either an n or (n−1)-block colourable e-star system of order n; and (ii) when e=3, the same result holds in the remaining congruence classes mod 6.

Rapidly changing speciation and extinction rates can be inferred in spite of nonidentifiability

The birth-death model is commonly used to infer speciation and extinction rates by fitting the model to phylogenetic trees with exclusively extant taxa. Recently, it was demonstrated that speciation and extinction rates are not identifiable if the rates are allowed to vary freely over time. The group of birth-death models that have the same likelihood is called a congruence class, and there is no statistical evidence to favor one model over the other. This issue has led researchers to question if and what patterns can reliably be inferred from phylogenies of only extant taxa and whether time-variable birth-death models should be fitted at all. We explore the congruence class in the context of several empirical phylogenies as well as hypothetical scenarios. For these empirical phylogenies, we assume that we inferred the true congruence class. Thus, our conclusions apply to any empirical phylogeny for which we robustly inferred the true congruence class. When we summarize shared patterns in the congruence class, we show that strong directional trends in speciation and extinction rates are shared among most models. Therefore, we conclude that the inference of strong directional trends is robust. Conversely, estimates of constant rates or gentle slopes are not robust and must be treated with caution. Interestingly, the space of valid speciation rates is narrower and more limited in contrast to extinction rates, which are less constrained. These results provide further evidence and insights that speciation rates can be estimated more reliably than extinction rates.