Congruence Classes Research Articles

On Generalised Kummer Congruences and Higher Rank Iwasawa Theory at Arbitrary Weights

In recent work Burns, Kurihara and Sano introduced a natural notion of (generalised) Stark elements of arbitrary rank and weight and conjectured a precise congruence relation between Stark elements of a fixed rank and different weights. This conjecture was shown to simultaneously generalise both the classical congruences of Kummer and the explicit reciprocity law of Artin-Hasse and Iwasawa. In this article, we show that the congruence conjecture also implies that the Iwasawa theoretical `zeta element’ that is conjecturally associated to the multiplicative group has good interpolation properties at arbitrary even integers.

How to grow it? Strategies of mathematical development presented by the example of enumerating certain set partitions

We describe in the form of a dialogue a development of various reflections on the combinatorics of set partitions; among the topics we pursue are the number of ways of partitioning a finite set into a fixed number $$d$$ of subsets of odd or even size, into $$a$$ parts of odd and $$b$$ parts of even size, and into $$d$$ parts, each of which has a size in a certain congruence class modulo some natural number $$m$$ . To this end, pattern guessing, recursion and induction, combinatorial interpretation and generating functions are employed. The participants of the dialogue represent different perspectives on and approaches to mathematics.

The existence of square non-integer Heffter arrays

An integer Heffter array H ( m , n ; s , t ) is an m × n partially filled matrix with entries from the set { ± 1, ± 2, …, ± m s } such that i) each row contains s filled cells and each column contains t filled cells, ii) every row and column sums to 0 (in Z ), and iii) no two entries agree in absolute value. Heffter arrays are useful for embedding the complete graph K 2 m s + 1 on an orientable surface in such a way that each edge lies between a face bounded by an s -cycle and a face bounded by a t -cycle. In 2015, Archdeacon, Dinitz, Donovan and Yaz i c i constructed square (i.e. m = n ) integer Heffter arrays for many congruence classes. In this paper we construct square integer Heffter arrays for all the cases not found in that paper, completely solving the existence problem for square integer Heffter arrays.

Congruence classes of large configurations in vector spaces over finite fields

In [1], Bennett, Hart, Iosevich, Pakianathan, and Rudnev found an exponent $s<d$ such that any set $E\subset \mathbb{F}_q^d$ with $|E|\gtrsim q^s$ determines $\gtrsim q^{\binom{k+1}{2}}$ congruence classes of $(k+1)$-point configurations for $k\leq d$. Because congruence classes can be identified with tuples of distances between distinct points when $k\leq d$, and because there are $\binom{k+1}{2}$ such pairs, this means any such $E$ determines a positive proportion of all congruence classes. In the $k>d$ case, fixing all pairs of distnaces leads to an overdetermined system, so $q^{\binom{k+1}{2}}$ is no longer the correct number of congruence classes. We determine the correct number, and prove that $|E|\gtrsim q^s$ still determines a positive proportion of all congruence classes, for the same $s$ as in the $k\leq d$ case.

Congruences for {varvec{q}}-Binomial Coefficients

We discuss q-analogues of the classical congruence left( {begin{array}{c}ap bpend{array}}right) equiv left( {begin{array}{c}a bend{array}}right) pmod {p^3}, for primes p>3, as well as its generalisations. In particular, we prove related congruences for (q-analogues of) integral factorial ratios.

Gröbner–Shirshov bases for congruence classes of complex reflection groups

The aim of this paper is to obtain the solvability of the word problem over congruence classes of complex reflection groups [Formula: see text] and [Formula: see text]. To do that, we use Gröbner–Shirshov basis theory and get the normal form structure of elements of these group types.

English

English

Comparing Hecke coefficients of automorphic representations

We prove a number of unconditional statistical results of the Hecke coefficients for unitary cuspidal representations of GL ( 2 ) \text {GL}(2) over number fields. Using partial bounds on the size of the Hecke coefficients, instances of Langlands functoriality, and properties of Rankin–Selberg L L -functions, we obtain bounds on the set of places where linear combinations of Hecke coefficients are negative. Under a mild functoriality assumption we extend these methods to GL ( n ) \text {GL}(n) . As an application, we obtain a result related to a question of Serre about the occurrence of large Hecke eigenvalues of Maass forms. Furthermore, in the cases where the Ramanujan conjecture is satisfied, we obtain distributional results of the Hecke coefficients at places varying in certain congruence or Galois classes.

Classification of the congruence classes of $$A_n^5$$ (n ⩾ 6) with 2-torsion free homology

In this paper, we classify the congruence classes of ${\boldsymbol~F}^5_{n(2)}$-polyhedra, i.e., $(n-1)$-connected, at most $(n+5)$-dimensional polyhedra with 2-torsion free homology. The proof relies on the matrix problem technique which was developed in the classification of representations of algebras and applied to the homotopy theory by Baues and Drozd (1999).

Congruence relations on almost distributive lattices-II

For a given ideal [Formula: see text] of an almost distributive lattice [Formula: see text], we study the smallest and the largest congruence relation on [Formula: see text] having [Formula: see text] as a congruence class.

Integer complexity: the integer defect

Define $\|n\|$ to be the complexity of $n$, the smallest number of ones needed to write $n$ using an arbitrary combination of addition and multiplication. John Selfridge showed that $\|n\|\ge 3\log_3 n$ for all $n$, leading this author and Zelinsky to define the defect of $n$, $\delta(n)$, to be the difference $\|n\|-3\log_3 n$. Meanwhile, in the study of addition chains, it is common to consider $s(n)$, the number of small steps of $n$, defined as $\ell(n)-\lfloor\log_2 n\rfloor$, an integer quantity. So here we analogously define $D(n)$, the integer defect of $n$, an integer version of $\delta(n)$ analogous to $s(n)$. Note that $D(n)$ is not the same as $\lceil \delta(n) \rceil$. We show that $D(n)$ has additional meaning in terms of the defect well-ordering considered in [3], in that $D(n)$ indicates which powers of $\omega$ the quantity $\delta(n)$ lies between when one restricts to $n$ with $\|n\|$ lying in a specified congruence class modulo $3$. We also determine all numbers $n$ with $D(n)\le 1$, and use this to generalize a result of Rawsthorne [18].

Characterizations of Majority Categories

In universal algebra, it is well known that varieties admitting a majority term admit several Mal’tsev-type characterizations. The main aim of this paper is to establish categorical counterparts of some of these characterizations for regular categories. We prove a categorical version of Bergman’s Double-projection Theorem: a regular category is a majority category if and only if every subobject S of a finite product $$A_1 \times A_2 \times \cdots \times A_n$$ is uniquely determined by its twofold projections. We also establish a categorical counterpart of the Pairwise Chinese Remainder Theorem for algebras, and characterize regular majority categories by the classical congruence equation $$\alpha \cap (\beta \circ \gamma ) = (\alpha \cap \beta ) \circ (\alpha \cap \gamma )$$ due to A.F. Pixley.

Dual skew Heyting almost distributive lattices

Abstract In this paper, we introduce the concept of dual skew Heyting almost distributive lattices (dual skew HADLs) and characterise it in terms of dual HADL. We define an equivalence relation θ on a dual skew HADL L and prove that θ is a congruence relation on the equivalence class [x]θ so that each congruence class is a maximal rectangular subalgebra and the quotient [y]θ/θ is a maximal lattice image of [x]θ for any y ∈ [x]θ. Moreover, we show that if the set PI (L) of all the principal ideals of an ADL L with 0 is a dual skew Heyting algebra then L becomes a dual skew HADL. Further we present different conditions on which an ADL with 0 becomes a dual skew HADL.

CARTAN SUBALGEBRAS IN DIMENSION DROP ALGEBRAS

AbstractWe completely classify Cartan subalgebras of dimension drop algebras with coprime parameters. More generally, we classify Cartan subalgebras of arbitrary stabilised dimension drop algebras that are non-degenerate in the sense that the dimensions of their fibres in the endpoints are maximal. Conjugacy classes by an automorphism are parametrised by certain congruence classes of matrices over the natural numbers with prescribed row and column sums. In particular, each dimension drop algebra admits only finitely many non-degenerate Cartan subalgebras up to conjugacy. As a consequence of this parametrisation, we can provide examples of subhomogeneous$\text{C}^{\ast }$-algebras with exactly$n$Cartan subalgebras up to conjugacy. Moreover, we show that in many dimension drop algebras two Cartan subalgebras are conjugate if and only if their spectra are homeomorphic.

On the Class of Subsets of Residuated lattice which induces a Congruence Relation

In this manuscript, we study the class of special subsets connected with a subset in a residuated lattice and investigate some related properties. We describe the union of elements of this class. Using the intersection of all special subsets connected with a subset, we give a necessary and sufficient condition for a subset to be a filter. Finally, by defining some operations, we endow this class with a residuated lattice structure and prove that it is isomorphic to the set of all congruence classes with respect to a filter.

Associations between personality traits, substance use, and depression in Romanian medical students

In 2016, Jang and Kim stated a conjecture about the norms of indecomposable integers in real quadratic number fields Q(D) where D>1 is a squarefree integer. Their conjecture was later disproved by Kala for D≡2mod4. We investigate such indecomposable integers in greater detail. In particular, we find the minimal D in each congruence class D≡1,2,3mod4 that provides a counterexample to the Jang-Kim Conjecture; provide infinite families of such counterexamples; and state a refined version of the Jang-Kim Conjecture. Lastly, we prove a slightly weaker version of our refined conjecture that is of the correct order of magnitude, showing the Jang-Kim Conjecture is only wrong by at most O(D).

Balanced factor congruences of double MS-algebras

In this paper, we introduce the concepts of Stone elements, central elements and Birkhoof central elements of a double MS-algebra and study their related properties. We observe that the center C(L) of a double MS -algebra L is precisely the Birkhoof center BC(L) of L. A complete description of factor congruences on a double MS-algebra is given by means of the central elements. A characterization of balanced factor congruences of double MS-algebra is obtained. A one-to-one correspondence between the class of all balanced factor congruences of a double MS-algebra L and the central elements of L is obtained.

Frobenius powers of some monomial ideals

In this paper, we characterize the (generalized) Frobenius powers and critical exponents of two classes of monomial ideals of a polynomial ring in positive characteristic: powers of the homogeneous maximal ideal, and ideals generated by positive powers of the variables. In doing so, we effectively characterize the test ideals and F-jumping exponents of sufficiently general homogeneous polynomials, and of all diagonal polynomials. Our characterizations make these invariants computable, and show that they vary uniformly with the congruence class of the characteristic modulo a fixed integer. Moreover, we confirm that for a diagonal polynomial over a field of characteristic zero, the test ideals of its reduction modulo a prime agree with the reductions of its multiplier ideals for infinitely many primes.

On the stable Andreadakis problem

Let Fn be the free group on n generators. Consider the group IAn of automorphisms of Fn acting trivially on its abelianization. There are two canonical filtrations on IAn: the first one is its lower central series Γ⁎; the second one is the Andreadakis filtration A⁎, defined from the action on Fn. In this paper, we establish that the canonical morphism between the associated graded Lie rings L(Γ⁎) and L(A⁎) is stably surjective. We then investigate a p-restricted version of the Andreadakis problem. A calculation of the Lie algebra of the classical congruence group is also included.

Checkerboard incircular nets: Laguerre geometry and parametrisation

We present a procedure which allows one to integrate explicitly the class of checkerboard IC-nets which has recently been introduced as a generalisation of incircular (IC) nets. The latter class of privileged congruences of lines in the plane is known to admit a great variety of geometric properties which are also present in the case of checkerboard IC-nets. The parametrisation obtained in this manner is reminiscent of that associated with elliptic billiards. Connections with discrete confocal coordinate systems and the fundamental QRT maps of integrable systems theory are made. The formalism developed in this paper is based on the existence of underlying pencils of conics and quadrics which is exploited in a Laguerre geometric setting.