Search

An abstract system of congruences describes a way of partitioning a space into finitely many pieces satisfying certain congruence relations. Examples of abstract systems of congruences include paradoxical decompositions and $n$-divisibility of actions. We consider the general question of when there are realizations of abstract systems of congruences satisfying various measurability constraints. We completely characterize which abstract systems of congruences can be realized by nonmeager Baire measurable pieces of the sphere under the action of rotations on the $2$-sphere. This answers a question of Wagon. We also construct Borel realizations of abstract systems of congruences for the action of $\mathsf{PSL}_2(\mathbb{Z})$ on $\mathsf{P}^1(\mathbb{R})$. The combinatorial underpinnings of our proof are certain types of decomposition of Borel graphs into paths. We also use these decompositions to obtain some results about measurable unfriendly colorings., minor corrections

In this paper, using the well-known Karlsson-Minton formula, we mainly establish two divisibility results concerning truncated hypergeometric series. Let $n>2$ and $q>0$ be integers with $2\mid n$ or $2\nmid q$. We show that $$\sum_{k=0}^{p-1}\frac{(q-\frac{p}{n})_k^n}{(1)_k^n}\equiv0\pmod{p^3} $$ and $$p^n\sum_{k=0}^{p-1}\frac{(1)_k^n}{(\frac{p}{n}-q+2)_k^n}\equiv0\pmod{p^3}$$ for any prime $p>\max\{n,(q-1)n+1\}$, where $(x)_k$ denotes the Pochhammer symbol defined by $$ (x)_k=\begin{cases}1,\quad &k=0,\\ x(x+1)\cdots(x+k-1),\quad &k>0.\end{cases}$$ Let $n\geq4$ be an even integer. Then for any prime $p$ with $p\equiv-1\pmod{n}$, the first congruence above implies that $$\sum_{k=0}^{p-1} \frac{(\frac{1}{n})_k^n}{(1)_k^n}\equiv0\pmod{p^3}. $$ This confirms a recent conjecture of Guo., 11 pages. Some typos have been corrected

We prove that families of polar codes with multiple kernels over certain symmetric channels can be viewed as polar decreasing monomial-Cartesian codes, offering a unified treatment for such codes, over any finite field. We define decreasing monomial-Cartesian codes as the evaluation of a set of monomials closed under divisibility over a Cartesian product. Polar decreasing monomial-Cartesian codes are decreasing monomial-Cartesian codes whose sets of monomials are closed respect a partial order inspired by the recent work of Bardet, Dragoi, Otmani, and Tillich ["Algebraic properties of polar codes from a new polynomial formalism," 2016 IEEE International Symposium on Information Theory (ISIT)]. Extending the main theorem of Mori and Tanaka ["Source and Channel Polarization Over Finite Fields and Reed-Solomon Matrices," in IEEE Transactions on Information Theory, vol. 60, no. 5, pp. 2720--2736, May 2014], we prove that any sequence of invertible matrices over an arbitrary field satisfying certain conditions polarizes any symmetric over the field channel. In addition, we prove that the dual of a decreasing monomial-Cartesian code is monomially equivalent to a decreasing monomial-Cartesian code. Defining the minimal generating set for a set of monomials, we use it to describe the length, dimension and minimum distance of a decreasing monomial-Cartesian code.

In [3], we derived three results in additive combinatorics for function fields. The proofs of these results depended on a recent bound for the large sieve with sparse sets of moduli for function fields by the first and third-named authors in [1]. Unfortunately, they discovered an error in this paper and demonstrated in [2] that this result cannot hold in full generality. In the present paper, we formulate a plausible conjecture under which the said three results in [3] remain true and the method of proof goes through using the same arguments. However, these results are now only conditional and still await a full proof.

We provide new families of divisibility and strong divisibility sequences based on some factorization properties of Chebyshev polynomials.

The predicate of divisibility on two consecutive numbers DW(x,y) ÷ x | y ∧ 1+x | y was introduced by L. van den Dries and A. Wilkie when they studied some properties of subsets of natural numbers, existentially definable with unit, addition and divisibility. Undecidability of the existential theory of natural numbers with multiplication and DW and definability of addition and multiplication in terms of DWand divisibility is proved in the paper. Then the definability of multiplication in the terms of addition and DW is proved. Some definability questions for order and DW are also considered in the paper.

When Dedekind introduced the notion of a module, he also defined their divisibility and related arithmetical notions (e.g. the LCM of modules). The introduction of notations for these notions allowed Dedekind to state new theorems, now recognized as the modular laws in lattice theory. Observing the dualism displayed by the theorems, Dedekind pursued his investigations on the matter. This led him, 20 years later, to introduce Dualgruppen, equivalent to lattices Dedekind [Uber Zerlegungen von Zahlen durch ihre grosten gemeinsamen Teiler. In Dedekind (1930–1932), volume II, 1897, 103–147; Uber die von drei Moduln erzeugte Dualgruppe. In Dedekind (1930–1932), volume II, 1900, 236–271]. After a brief exposition of the basic elements of Dualgruppe theory, and with the help of his Nachlass, I show how Dedekind gradually built his theory through layers of computations, often repeated in slight variations and attempted generalizations. I study the tools he devised to help and accompany him in his computations. I h...

Let p be a prime number and let S be a countable set. Let us consider the collection ${Div}_{p}^{S}$ of all subsets of S whose cardinalities are multiples of p and the complements of such sets. Then the collection ${Div}_{p}^{S}$ constitutes a (set-representable) quantum logic (i.e., ${Div}_{p}^{S}$ is an orthomodular poset). We show in this note that each state on ${Div}_{p}^{S}$ can be extended over the Boolean algebra expS of all subsets of S. We also prove that all pure states on ${Div}_{p}^{S}$ are two-valued. (If we lend to a main result a possible interpretation in terms of quantum entities, the logics ${Div}_{p}^{S}$ have higher degree of noncompatibility but somewhat classical states.)

This paper attempts to rethink difference and divisibility as conditions of (im)possibility for love and survival in the wake of Derrida's newly discovered—and just recently published—Geschlecht III. I argue that Derrida's deconstruction of what he calls ‘the grand logic of philosophy’ allows us to think love and survival without positing unicity as a sine qua non. This hypothesis is tested in and through a deconstructive reading of Heidegger's second essay on Trakl in On the Way to Language, where Heidegger's phonocentrism and surreptitious nationalism converge in an effort to ‘save the earth’ from a ‘degenerate’ Geschlecht that cannot survive the internal diremption between Geschlechter. I show that one way of problematizing Heidegger's claim is to point to the blank spaces in the ‘E i n’ of Trakl's ‘E i n Geschlecht’, an internal fissuring in the very word Heidegger mobilizes in order to secure the future of mankind.

It is well known that every MV-algebra can be converted into a residuated lattice satisfying divisibility and the double negation law. In a previous paper the first author and J. Kühr introduced the concept of an NMV-algebra which is a non-associative modification of an MV-algebra. The natural question arises if an NMV-algebra can be converted into a residuated structure, too. Contrary to MV-algebras, NMV-algebras are not based on lattices but only on directed posets and the binary operation need not be associative and hence we cannot expect to obtain a residuated lattice but only an essentially weaker structure called a conditionally residuated poset. Considering several additional natural conditions we show that every NMV-algebra can be converted in such a structure. Also conversely, every such structure can be organized into an NMV-algebra. Further, we study an a bit more stronger version of an algebra where the binary operation is even monotone. We show that such an algebra can be organized into a residuated poset and, conversely, every residuated poset can be converted in this structure.

For nonnegative integers $j$ and $n$ let $\Theta(j,n)$ be the number of entries in the $n$-th row of Pascal's triangle that are not divisible by $2^{j+1}$. In this paper we prove that the family $j\mapsto\Theta(j,n)$ usually follows a normal distribution. The method used for proving this theorem involves the computation of first and second moments of $\Theta(j,n)$, and uses asymptotic analysis of multivariate generating functions by complex analytic methods, building on earlier work by Drmota (1994) and Drmota, Kauers and Spiegelhofer (2016)., Comment: 15 pages

A graph $G$ has a $C_k$-decomposition if its edge set can be partitioned into cycles of length $k$. We show that if $\delta(G)\geq 2|G|/3-1$, then $G$ has a $C_4$-decomposition, and if $\delta(G)\geq |G|/2$, then $G$ has a $C_{2k}$-decomposition, where $k\in \mathbb{N}$ and $k\geq 4$ (we assume $G$ is large and satisfies necessary divisibility conditions). These minimum degree bounds are best possible and provide exact versions of asymptotic results obtained by Barber, Kuhn, Lo and Osthus. In the process, we obtain asymptotic versions of these results when $G$ is bipartite or satisfies certain expansion properties.

The Department of Basic Education in South Africa has identified factorisation as a problem area in Mathematics teaching. Learners in earlier grades are exposed to mathematical tasks, involving con...

If p is a prime and G is a finite solvable group, we prove that there is a McKay correspondence between the irreducible characters of G of degree not divisible by p and those of a p-Sylow normalizer of G which respects divisibility. This phenomenon does not happen outside solvable groups.

Let [Formula: see text] with [Formula: see text] and [Formula: see text] be an odd prime. We find a supercongruence for [Formula: see text] and related sums of powers of binomial coefficients. These results complement prior results for [Formula: see text] with [Formula: see text] obtained recently by the author.

In this paper we prove non-existence of nontrivial partial difference sets in Abelian groups of order $$8p^3$$ , where $$p\ge 3$$ is a prime number. These groups seemed to have the potential of admitting at least two infinite families of PDSs, and even the smallest case, $$p=3$$ , had been open for twenty years until settled recently by the authors and E. Neubert. Here, using the integrality and divisibility conditions for PDSs, we first describe all hypothetical parameter sets of nontrivial partial difference sets in these groups. Then we prove the non-existence of a PDS for each of these hypothetical parameter sets by combining a recent local multiplier result with some geometry and elementary number theory.

Summary Even and odd numbers appear early in history of mathematics [9], as they serve to describe the property of objects easily noticeable by human eye [7]. Although the use of parity allowed to discover irrational numbers [6], there is a common opinion that this property is “not rich enough to become the main content focus of any particular research” [9]. On the other hand, due to the use of decimal system, divisibility by 2 is often regarded as the property of the last digit of a number (similarly to divisibility by 5, but not to divisibility by any other primes), which probably restricts its use for any advanced purposes. The article aims to extend the definition of parity towards its notion in binary representation of integers, thus making an alternative to the articles grouped in [5], [4], and [3] branches, formalized in Mizar [1], [2].

The paper investigates the divisibility of the permanent function of (1,−1)-matrices by different powers of 2. It is shown that the Krauter–Seifter bound is the best possible one for generic (1,−1)-matrices.

In this paper, we propose a new characterization of non-Markovian quantum evolution based on the covariance matrix. The fundamental properties of covariance matrices are elucidated. The measure captures quite directly the characteristics of non-Markovianity from the perspective of uncertainty. We consider several typical examples and compare the covariance matrix characterization of quantum non-Markovianity with Fisher-information matrix, divisibility and the Breuer-Laine-Piilo characterization of quantum non-Markovianity.

Let $p_1, p_2, \cdots$ be the sequence of all primes in ascending order. Using explicit estimates from the prime number theory, we show that if $ k \geq5 $, then (p_{k+1}-1)! \mid(\tfrac12 (p_{k +1} - 1))! p_ k!, which improves a previous result of the second author.

The Kendrick mass defect (KMD) analysis of multiply charged polymeric distributions has recently revealed a surprising isotopic split in their KMD plots-namely a 1/z difference between KMDs of isotopes of an oligomer at charge state z. Relying on the KMD analysis of actual and simulated distributions of poly(ethylene oxide) (PEO), the isotopic split is mathematically accounted for and found to go with an isotopic misalignment in certain cases. It is demonstrated that the divisibility (resp. indivisibility) of the nominal mass of the repeating unit (R) by z is the condition for homolog ions to line up horizontally (resp. misaligned obliquely) in a KMD plot. Computing KMDs using a fractional base unit R/z eventually corrects the misalignments for the associated charge state while using the least common multiple of all the charge states as the divisor realigns all the points at once. The isotopic split itself can be removed by using either a new charge-dependent KMD plot compatible with any fractional base unit or the remainders of KM (RKM) recently developed for low-resolution data all found to be linked in a unified theory. These original applications of the fractional base units and the RKM plots are of importance theoretically to satisfy the basics of a mass defect analysis and practically for a correct data handling of single stage and tandem mass spectra of multiply charged homo- and copolymers. Graphical Abstract ᅟ.

We develop a new approach to Mobius inversion by considering the spectral problem for certain symmetric, positive-definite matrices closely connected to the divisibility relation. This spectral pro...

In recent years, we have witnessed numerous new results on divisibility properties concerning binomial sums. Many conjectures and results were proposed by Z.-W. Sun. Some of these conjectures were proved by himself, and also by some scholars such as V.J.W. Guo, G.S. Mao, B. He, and Y.P. Mu. Motivated by their work, we study the WZ method and its applications to prove divisibility properties of binomial sums. In this paper, we propose a method which can be used to explore more such divisibility properties. Additionally, we give some generalized divisibility properties which can imply those divisibility properties established by Z.-W. Sun and B. He.

We characterize tenable Polya urn schemes via a decomposition of their replacement matrices into small submatrices with certain conditions on the determinants. The characterization also involves an interplay between the submatrices and the initial conditions, as well as certain divisibility conditions.

We estimate the number of monotone discrete functions related to the divisibility of numbers.

For a given odd integer $n>1$, we provide some families of imaginary quadratic number fields of the form $\mathbb{Q}(\sqrt{x^2-t^n})$ whose ideal class group has a subgroup isomorphic to $\mathbb{Z}/n\mathbb{Z}$., 10 pages, accepted for publication in Journal of Number Theory (2017)

El propósito de este trabajo consiste en destacar el papel central que tiene el problema de la divisibilidad del alma en los dos enfoques bajo los que se presenta la investigación psicológica aristotélica: el hilemórfico y el cardiocéntrico. Mientras que el primero sostiene que el alma es la forma o esencia del cuerpo entero, el segundo aboga por la localización del alma en el corazón, pues asegura que allí se manifiestan los principios de las partes o facultades anímicas. A simple vista, podría suponerse que existe una incompatibilidad entre ambos enfoques, ya que no tendría mucho sentido afirmar que el alma se halla en todo el cuerpo y, al mismo tiempo, decir que se encuentra exclusivamente en uno de sus órganos específicos. Entonces, la tesis de este trabajo es que el pneuma ofrece la clave para la conciliación de estos dos enfoques, dado que es el instrumento empleado por las partes del alma para desplegar sus funciones por todo el cuerpo. El corazón, como sede material de dichas partes, irradia el pneuma en forma de sangre por todo el organismo; y éste, hasta que no se extingue esa operación vital, preserva su esencia, causa formal o razón de ser de su existir. De este modo, el hilemorfismo resulta sustentado por el cardiocentrismo.

We construct an infinite family of pairs of imaginary quadratic fields Q ( D ) and Q ( D + 1 ) with D ∈ Z whose class numbers are both divisible by 3.

Recently Peter Keevash solved asymptotically the existence question for Steiner systems by showing that $S(t,k,n)$ exists whenever the necessary divisibility conditions on the parameters are satisfied and $n$ is sufficiently large in terms of $k$ and $t$. The purpose of this paper is to make a conjecture which if true would be a significant extension of Keevash's theorem, and to give some theoretical and computational evidence for the conjecture. We phrase the conjecture in terms of the notions (which we define here) of synchronization and separation for association schemes. These definitions are based on those for permutation groups which grow out of the theory of synchronization in finite automata. In this theory, two classes of permutation groups (called \emph{synchronizing} and \emph{separating}) lying between primitive and $2$-homogeneous are defined. A big open question is how the permutation group induced by $S_n$ on $k$-subsets of $\{1,\ldots,n\}$ fits in this hierarchy; our conjecture would give a solution to this problem for $n$ large in terms of $k$. We prove the conjecture in the case $k=4$: our result asserts that $S_n$ acting on $4$-sets is separating for $n\ge10$ (it fails to be synchronizing for $n=9$)., Comment: Error in previous version corrected

The project portfolio selection problem considering divisibility is a new research problem rising in recent years. However, two deficiencies are discovered in current divisible project portfolio selection research, one is that researchers always ignore the already started exiting projects when selecting a project portfolio, and the other is that the project parameters are all considered as exact values which are not consistent with practice situation. Under this circumstance, the paper first discusses the dynamic project portfolio selection problem with project divisibility. Meanwhile, due to the lack of correlative historical data, some project parameters are given by experts’ estimates and are treated as uncertain variables. Therefore, a mean-variance mixed integer nonlinear optimal selection model is first developed in this paper to deal with the uncertain dynamic project portfolio selection problem with divisibility. For the convenience of computations, an equivalent mixed integer linear programming representation is proposed. Numerical examples with two scenarios are presented to shed light on the characteristics of the proposed model.

Let (Fn)n be the sequence of Fibonacci numbers. The order of appearance (in the Fibonacci sequence) of a positive integer n is defined as z(n)=min{k≥1:n∣Fk}. Very recently, Trojovská and Venkatachalam proved that, for any k≥1, the number z(n) is divisible by 2k, for almost all integers n≥1 (in the sense of natural density). Moreover, they posed a conjecture that implies that the same is true upon replacing 2k by any integer m≥1. In this paper, in particular, we prove this conjecture.

F. Harary, R.W. Robinson and N.C. Wormald studied a complete tripartite graph K(m, n, s), and putted forward the conjecture, for t|(mn + ms + ns), t > 1 and t is even, then G has a t isomorphic fac...

In this paper, we derive some identities on Pell, Pell-Lucas, and balancing numbers and the relationships between them. We also deduce some formulas on the sums, divisibility properties, perfect squares, Pythagorean triples involving these numbers. Moreover, we obtain the set of positive integer solutions of some specific Pell equations in terms of the integer sequences mentioned in the text.

The paper discusses a discrete model of hepatic plate histogenesis based on the dissymmetry distribution times of cell cycles. The model is based on the assumption of the finality of cell time as an internal parameter of a single cycle of biological spacetime. Another assumption is the divisibility of maternal cell time in half between daughter cells, which makes it possible to consider the stem cell macrocycle in histogenesis as a system of nested cell cycles with progressively decreasing time. In addition, it was expected that at each step of cell division in homogeneous histogenesis, a asymmetric distribution of reference of cell time management by mitosis, which is related to the intron region of the genome. The obtained numerical model agrees with the characteristics of the architectonics of the liver, the limited number of cell divisions, and the frequency distribution of mitosis and aging as the extinction process of cells with short cycles.

We mainly introduce two new kinds of numbers given by $$R_n=\sum_{k=0}^n\binom nk\binom{n+k}k\frac1{2k-1}\quad\ (n=0,1,2,...)$$ and $$S_n=\sum_{k=0}^n\binom nk^2\binom{2k}k(2k+1)\quad\ (n=0,1,2,...).$$ We find that such numbers have many interesting arithmetic properties. For example, if $p\equiv1\pmod 4$ is a prime with $p=x^2+y^2$ (where $x\equiv1\pmod 4$ and $y\equiv0\pmod 2$), then $$R_{(p-1)/2}\equiv p-(-1)^{(p-1)/4}2x\pmod{p^2}.$$ Also, $$\frac1{n^2}\sum_{k=0}^{n-1}S_k\in\mathbb Z\ \ {and}\ \ \frac1n\sum_{k=0}^{n-1}S_k(x)\in\mathbb Z[x]\quad\text{for all}\ n=1,2,3,...,$$ where $S_k(x)=\sum_{j=0}^k\binom kj^2\binom{2j}j(2j+1)x^j$. For any positive integers $a$ and $n$, we show that, somewhat surprisingly, $$\frac1{n^2}\sum_{k=0}^{n-1}(2k+1)\binom{n-1}k^a\binom{-n-1}k^a\in\mathbb Z\ \ {and} \ \ \frac 1n\sum_{k=0}^{n-1}\frac{\binom{n-1}k^a\binom{-n-1}k^a}{4k^2-1}\in\mathbb Z.$$ We also solve a conjecture of V.J.W. Guo and J. Zeng, and pose several conjectures for further research., 32 pages

We examine an elementary problem on prime divisibility of binomial coefficients. Our problem is motivated by several related questions on alternating groups., Comment: 16 pages, 1 figure; ancillary attachments include computer code and output. v2 has improved asymptotic density estimates and other minor fixes. v3 has minor fixes for publication

An accepted conclusion for modeling a physiological neuron is its divisibility into two mathematical processes. These are the digital process and the analog process. In a 1943 paper McCulloch and Pitts presented a neural model performing switching logic (an IT digital process). Their model represents the basis of the model used in the well-known Perceptron, devised by Rosenblatt in 1957. The Perceptron is a neural network of models that included varying weights, which corresponds to a neuron’s analog process. This talk will define the digital and the analog processes. These processes will be associated with a physiological neuron’s anatomy. The Perceptron’s neuron model continues to be used in current neural networks despite a serious limitation of only realizing linearly separable switching functions. Linearly separable switching functions will also be defined. The currently used “linear” neuron model will be shown to have severely limited switching logic realizations. Linear separability will also be defined. A general non-linear neuron model will be defined, which is capable of performing all possible switching logic realizations. Examples of both a linear neuron model’s and nonlinear neuron models’ switching logic realizations will be included. The proposed nonlinear neuron model, which more accurately takes into account actual neurophysiologic mechanisms, can form the basis of modeling network processes at all levels of the nervous system, including the brain.

Hyperspectral data has the characteristics of large number of bands, narrow band width and large amount of data, which brings great difficulties to the further interpretation of the image. The partition optimal band selection(POBS) is proposed to select a certain number of bands according to the actual needs. The weight calculation formula of the band includes three parameters: standard deviation, information entropy and correlation coefficient, so it can be guaranteed that the selected band subset contains a large amount of information and low correlation. In addition, the whole band space is evenly divided into several band subspaces, and then the band with the largest weight is selected in each band subspace. The bands can be more dispersed through the uniform partition of the band space. It can ensure that the bands in the band subset are more representative. The classification test is carried out on two hyperspectral datasets by KNN method, and the classification results show that POBS selects a large amount of band information, and the divisibility of the ground object is good.

Let $$p \ge 5$$ be a prime number. We find all the possible subgroups G of $$\mathrm{GL}_2 ({\mathbb Z}/ p {\mathbb Z})$$ such that there exist a number field k and an elliptic curve $${\mathcal {E}}$$ defined over k such that the $$\mathrm{Gal}(k ({\mathcal {E}}[p])/k)$$ -module $${\mathcal {E}}[p]$$ is isomorphic to the G-module $$({\mathbb Z}/ p {\mathbb Z})^2$$ and there exists $$n \in {\mathbb N}$$ such that the local–global divisibility by $$p^n$$ does not hold over $${\mathcal {E}}(k)$$ .

We consider algebras over a field K with generators x 1 , x 2 , … , x n subject to ( n 2 ) quadratic relations of the form x i x j = x k x l with ( i , j ) ≠ ( k , l ) and, moreover, every monomial x i x j appears at most once in one of the defining relations. If these relations are non-degenerate then it is shown that the algebra is left and right Noetherian, satisfies a polynomial identity and has Gelfand–Kirillov dimension at most n. In case the defining relations are square-free this was already established by Gateva-Ivanova, Jespers and Okninski. To prove these results we investigate the structure of the underlying monoid S, defined by the same presentation. It is called a quadratic monoid. We show that there is a strong link with the divisibility monoids and monoids of I-type (also referred to as YB-monoids). Monoids of I-type are examples of non-degenerate quadratic monoids. They are a monoid interpretation of non-degenerate involutive set-theoretic solutions of the Yang–Baxter equation and have been studied quite intensively in recent years, by Gateva-Ivanova and Van den Bergh, Etingof, Schedler and Soloviev, Jespers and Okninski. Divisibility monoids have been introduced by Kuske. These are cancellative monoids that include the class of monoids of I-type. We show that they have a presentation with at most ( n 2 ) relations and if they have precisely ( n 2 ) defining relations then they are monoids of I-type.

Educational software has significantly changed how mathematics is taught and learned. One challenge for educators is choosing the most appropriate software among numerous options. Therefore, we mapped free mathematics education software according to number and operation content. The study was carried out with public elementary school teachers (grades 6 - 9). The teachers watched a presentation on the features of each software type and filled out a checklist about the software and its content. The results showed that 63% of the 32 software titles were appropriate for developing numeric and algebraic reasoning. According to the teachers, these titles were appropriate for developing and consolidating concepts related to the number system, operations and properties of natural and whole numbers, numeric expressions, divisibility, prime numbers, decomposition into prime factors, GCD, LCM, operations with rational numbers in fraction and decimal form, comparison and operations on equivalent fractions, first degree equations, and first and second degree polynomial functions. https://doi.org/10.26803/ijlter.16.11.3

Our main purpose is to extend several results of interest that have been proved for modules over integral domains to modules over arbitrary commutative rings $R$ with identity. The classical ring of quotients $Q$ of $R$ will play the role of the field of quotients when zero-divisors are present. After discussing torsion-freeness and divisibility (Sections 2–3), we study Matlis-cotorsion modules and their roles in two category equivalences (Sections 4–5). These equivalences are established via the same functors as in the domain case, but instead of injective direct sums $\oplus Q$ one has to take the full subcategory of $Q$-modules into consideration. Finally, we prove results on Matlis rings, i.e. on rings for which $Q$ has projective dimension $1$ (Theorem 6.4).

For every finite Coxeter group Γ, each positive braid in the corresponding braid group admits a unique decomposition as a finite sequence of elements of Γ, the so-called Garside-normal form. The study of the associated adjacency matrix Adj ( Γ ) allows to count the number of Garside-normal form of a given length. In this paper we prove that the characteristic polynomial of Adj ( B n ) divides the one of Adj ( B n + 1 ) . The key point is the use of a Hopf algebra based on signed permutations. A similar result was already known for the type A. We observe that this does not hold for type D. The other Coxeter types (I, E, F and H) are also studied.

An R-module M is said to satisfy ACCd (resp. DCCd) on submodules if for every ascending (resp. descending) chain {Mi} of submodules of M, Mi=φi(Mi+1) (resp. Mi+1=φi(Mi)) for some φi∈EndR(M) for i≫0. A nonzero module with ACCd or DCCd on submodules contains an essential submodule which is a direct sum of uniform submodules almost all noetherian. We show that if M is a finitely generated self-projective and self-injective R-module with ACCd or DCCd on submodules, then M is a finite direct sum of uniform submodules. It is shown that a regular right self-injective ring with ACCd or DCCd on right ideals must be semisimple artinian. We also prove that if M is a nonzero nonsingular module over a right noetherian ring and E(M)(ℕ) satisfies ACCd or DCCd on submodules, then M is semisimple. Next we consider some conditions for modules with ACCd (resp. DCCd) on submodules to satisfy ACC (resp. DCC) on some families of prime submodules. Finally, we show that a commutative ring with DCCd on ideals has dimensio...

We generalize Cooper's method of quantifier elimination for classical Presburger arithmetic to give a new proof that all parametric Presburger families (as defined by Kevin Woods) are definable by formulas with polynomially bounded quantifiers in an expanded language with predicates for divisibility by f(t) for every polynomial f over the integers. In fact, this quantifier bounding method works more generally in expansions of Presburger arithmetic with multiplication by scalars from any ring of functions into Z., 13 pages