Narrow Results

Let $NT(r,k,n)$ count the total number of parts among partitions of $n$ with rank congruent to $r$ modulo $k$ and let $M_{\omega }(r,k,n)$ count the total appearances of ones among partitions of $n$ with crank congruent to $r$ modulo $k$. We provide a list of over 70 congruences modulo $5$, $7$, $11$ and $13$ involving $NT(r,k,n)$ and $M_{\omega }(r,k,n)$, which are known as congruences of Andrews–Beck type. Some recent conjectures of Chan, Mao and Osburn are also included in this list.

In this paper, we survey the main known constructions of Ferrers diagram rank-metric codes, and establish new results on a related conjecture by Etzion and Silberstein. We also give a sharp lower bound on the dimension of linear rank-metric anticodes with a given profile. Combining our results with the multilevel construction, we produce examples of subspace codes with the largest known cardinality for the given parameters. We also apply results from algebraic geometry to the study of the analogous problem over an algebraically closed field, proving that the bound by Etzion and Silberstein can be improved in this case, and providing a sharp bound for full-rank matrices.

We in this paper derive necessary and sufficient conditions for the system of the periodic discrete-time coupled Sylvester matrix equations $A_k{X}_k+Y_k{B}_k=M_k,C_k{X}_{k+1}+Y_k{D}_k=N_k(k=1,2)$ over the quaternion algebra to be consistent in terms of ranks and generalized inverses of the coefficient matrices. We also give an expression of the general solution to the system when it is solvable. The findings of this paper generalize some known results in the literature.

Let be the semigroup of all partial transformations on X, and be the subsemigroups of of all full transformations on X and of all injective partial transformations on X, respectively. Given a non-empty subset Y of X, let , and . In 2008, Sanwong and Sommanee described the largest regular subsemigroup and determined Green's relations of . In this paper, we present analogous results for both and . For a finite set X with |X| ≥ 3, the ranks of , and are well known to be 4, 3 and 3, respectively. In this paper, we also compute the ranks of , and for any proper non-empty subset Y of X.

Recently, Andrews introduced the partition function ${\overline{C}}_{k,i}(n)$ as the number of overpartitions of $n$ in which no part is divisible by $k$ and only parts $\equiv \pm i(modk)$ may be overlined. He proved that ${\overline{C}}_{3,1}(9n+3)$ and ${\overline{C}}_{3,1}(9n+6)$ are divisible by $3$. Let ${\overline{A}}_l(n)$ be the number of overpartitions of $n$ into parts not divisible by $l$. In this paper, we call the overpartitions enumerated by the function ${\overline{A}}_l(n)$$l$-regular overpartitions. For ${\overline{A}}_3(n)$ and ${\overline{A}}_4(n)$, we obtain some explicit results on the generating function dissections. We also derive some congruences for ${\overline{A}}_l(n)$ modulo $3$, $6$ and $24$ which imply the congruences for ${\overline{C}}_{3,1}(n)$ proved by Andrews. By introducing a rank of vector partitions, we give a combinatorial interpretation of the congruences of Andrews for ${\overline{C}}_{3,1}(9n+3)$ and ${\overline{C}}_{3,1}(9n+6)$.

For $n\in \mathbb{N}$, let $X_n=\{a_1,a_2,\dots ,a_n\}$ be an $n$-element set and let $F=(X_n;{<}_f)$ be a fence, also called a zigzag poset. As usual, we denote by $I_n$ the symmetric inverse semigroup on $X_n$. We say that a transformation $\alpha \in I_n$ is fence-preserving if $x{<}_{f}y$ implies that $x\alpha {<}_{f}y\alpha$, for all $x,y$ in the domain of $\alpha$. In this paper, we study the semigroup $PF{I}_n$ of all partial fence-preserving injections of $X_n$ and its subsemigroup $I{F}_n=\{\alpha \in PF{I}_n:{\alpha }^{-1}\in PF{I}_n\}$. Clearly, $I{F}_n$ is an inverse semigroup and contains all regular elements of $PF{I}_n.$ We characterize the Green’s relations for the semigroup $I{F}_n$. Further, we prove that the semigroup $I{F}_n$ is generated by its elements with rank $\ge n-2$. Moreover, for $n\in 2\mathbb{N}$, we find the least generating set and calculate the rank of $I{F}_n$.

Let ${{𝒪}{𝒫}{ℰ}}_n$ be the monoid of all orientation-preserving and extensive full transformations on $\{1,2,\dots ,n\}$ ordered in the standard way. In this paper, we determine the minimum generating set and the minimum idempotent generating set of ${{𝒪}{𝒫}{ℰ}}_n$, and so the rank and the idempotent rank of ${{𝒪}{𝒫}{ℰ}}_n$ are obtained. Moreover, we describe maximal subsemigroups and maximal idempotent generated subsemigroups of ${{𝒪}{𝒫}{ℰ}}_n$ and completely obtain their classifications.

The variant of a semigroup $S$ with respect to an element $a\in S$, denoted $S^a$, is the semigroup with underlying set $S$ and operation ⋆ defined by $x\star y=xay$ for $x,y\in S$. In this paper, we study variants ${{𝒯}}_X^a$ of the full transformation semigroup ${{𝒯}}_X$ on a finite set $X$. We explore the structure of ${{𝒯}}_X^a$ as well as its subsemigroups $\mathrm{\text{Reg}}({{𝒯}}_X^a)$ (consisting of all regular elements) and ${{ℰ}}_X^a$ (consisting of all products of idempotents), and the ideals of $\mathrm{\text{Reg}}({{𝒯}}_X^a)$. Among other results, we calculate the rank and idempotent rank (if applicable) of each semigroup, and (where possible) the number of (idempotent) generating sets of the minimal possible size.

Let spt(n) denote the total number of appearances of smallest parts in the partitions of n. Recently, Andrews showed how spt(n) is related to the second rank moment, and proved some surprising Ramanujan-type congruences mod 5, 7 and 13. We prove a generalization of these congruences using known relations between rank and crank moments. We obtain explicit Ramanujan-type congruences for spt(n) mod ℓ for ℓ = 11, 17, 19, 29, 31 and 37. Recently, Bringmann and Ono proved that Dyson's rank function has infinitely many Ramanujan-type congruences. Their proof is non-constructive and utilizes the theory of weak Maass forms. We construct two explicit nontrivial examples mod 11 using elementary congruences between rank moments and half-integer weight Hecke eigenforms.

Let h₁, h₂,… be a sequence of elements in a free group and let H be the subgroup they generate. Let H′ be the subgroup generated by w₁, w₂, …, where each w_i is a word in h_i and possibly other h_j, such that the associated directed graph has the finite paths property. We show that rank H′≥ rank H. As a corollary we get that , where is the subgroup generated by the roots of the elements in H. If H₀ is finitely generated and the sequence of subgroups H₀, H₁, H₂, … satisfies then the sequence stabilizes, i.e. for some m, H_i=H_i+1 for every i≥ m. When applied to systems of equations in free groups, we give conditions on a transformation of the system such that the maximal rank of a solution (the inner rank) does not increase. In particular, we show that if in "Lyndon equation" the exponents a_i satisfy gcd(a₁,…,a_n)≠1 then the inner rank is ⌊ n/2⌋. The proofs are mostly elementary.

We give a thorough structural analysis of the principal one-sided ideals of arbitrary semigroups, and then apply this to full transformation semigroups and symmetric inverse monoids. One-sided ideals of these semigroups naturally occur as semigroups of transformations with restricted range or kernel.

Severe thunderstorms are a manifestation of deep convection. Conditional instability is known to be the mechanism by which thunderstorms are formed. The energy that drives conditional instability is convective available potential energy (CAPE), which is computed with radio sonde data at each pressure level. The purpose of the present paper is to identify the pattern or shape of CAPE required for the genesis of severe thunderstorms over Kolkata (22°32′N, 88°20′E) confined within the northeastern part (20°N to 24°N latitude, 85°E to 93°E longitude) of India. The method of chaotic graph theory is adopted for this purpose. Chaotic graphs of pressure levels and CAPE are formed for thunderstorm and non-thunderstorm days. Ranks of the adjacency matrices constituted with the union of chaotic graphs of pressure levels and CAPE are computed for thunderstorm and non-thunderstorm days. The results reveal that the rank of the adjacency matrix is maximum for non-thunderstorm days and a column with all zeros occurs very quickly on severe thunderstorms days. This indicates that CAPE loses connectivity with pressure levels very early on severe thunderstorm days, showing that for the genesis of severe thunderstorms over Kolkata short, and therefore broad, CAPE is preferred.

We study the number of unimodal sequences of weight n and rank m using a partial theta identity discovered by Ramanujan. We obtain rank difference identities as well as a congruence for the second rank moment.

New families of even non-congruent numbers with arbitrarily many distinct prime factors are presented. Methods for generating even non-congruent numbers from existing non-congruent numbers are also given. These results yield new even non-congruent numbers whose factorizations contain primes in all four odd congruence classes modulo eight.

In this paper, we consider endomorphisms of a finite directed path from monoid generators perspective. Our main aim is to determine the rank of the monoid $\mathrm{\text{wEnd}}{\overrightarrow{P}}_n$ of all weak endomorphisms of a directed path with $n$ vertices, which is a submonoid of the widely studied monoid ${{𝒪}}_n$ of all order-preserving transformations of an $n$-chain. Also, we describe the regular elements of $\mathrm{\text{wEnd}}{\overrightarrow{P}}_n$ and calculate its size and number of idempotents.

The monoid of all partial injections on a finite set (the symmetric inverse semigroup) is of particular interest because of the well-known Wagner–Preston Theorem. Let $n$ be a positive natural number and ${\mathrm{\text{PFI}}}_n$ be the semigroup of all fence-preserving partial one-to-one maps of $\{1,\dots ,n\}$ into itself with respect to composition of maps and the fence $1\prec 2\succ 3\prec \cdots n$. There is considered the inverse semigroup ${\mathrm{\text{IOF}}}_n^{\mathrm{\text{par}}}$ of all $\alpha \in {\mathrm{\text{PFI}}}_n$ such that $\alpha$ is regular in ${\mathrm{\text{PFI}}}_n$, order-preserving with respect to the order $1<2<\cdots <n$ and parity-preserving. According to the main result of the paper, it is $3n-6$ the least of the cardinalities of the generating sets of ${\mathrm{\text{IOF}}}_n^{\mathrm{\text{par}}}$ for $4\le n$. There is determined a concrete representation of a generating set of minimal size.

This is an introduction to a probabilistic model for the arithmetic of elliptic curves, a model developed in a series of articles of the author with Bhargava, Kane, Lenstra, Park, Rains, Voight, and Wood. We discuss the theoretical evidence for the model, and we make predictions about elliptic curves based on corresponding theorems proved about the model. In particular, the model suggests that all but finitely many elliptic curves over ℚ have rank ≤ 21, which would imply that the rank is uniformly bounded.

We simulate sytems up to 10 001 × 10 001 × 10 001 in three dimensions at the percolation threshold p_c of 0.31160. We find that the fractal dimension is ≃2.53±0.02, the cluster size distribution exponent, τ, is 2.186±0.002 and an exponent of 0.85 describing how the mass of the clusters scale with rank. Corrections-to-scaling exponents of ≃-0.7 are observed for n_s and for the mass of the largest cluster. We also check the percolation threshold and report good agreement with recent values.

We determine a large class of links, including some satellite and hyperbolic links, for each of which the tunnel number is the minimum possible, the number of its components minus one, and observe that the rank versus genus conjecture is valid for this class of links.

An R-module _RM is called morphic if M/im α ≅ ker α for every endomorphism α of M, that is, if the dual of the Noether isomorphism theorem holds. Mostly all morphic Z-modules are determined leaving open some classes of nonsplitting mixed groups, which actually cannot be completely characterized. However, for these classes, comprehensive information including negative examples is given.