Narrow Results

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.

Data envelopment analysis (DEA) is an excellent approach for evaluating the performance of decision making units (DMUs) that use multiple inputs to produce multiple outputs. This research utilizes the DEA to analyze the operating efficiency of Caspian cattle feedlot farms in Iran. The inputs utilized by the farms are number of calve, number of labors, total metabolizable energy intake, total crude protein intake and total cost of hygiene-treatment of calve; and the output considered is total live weigh gain of calve. By using two DEA models, the efficiency score for each farm is calculated and the efficient farms are ranked.

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.

Recently, Andrews proved two conjectures on a partition statistic introduced by Beck. Chern established some results on weighted rank and crank moments and proved many Andrews–Beck type congruences. Motivated by Andrews and Chern’s work, Lin, Peng and Toh introduced a partition statistic of k-colored partitions ${\mathrm{\text{NB}}}_k(r,m,n)$ which counts the total number of parts of the first component in each k-colored partition $\pi$ of n with ${\mathrm{\text{crank}}}_k(\pi )$ congruent to r modulo m and proved many congruences for ${\mathrm{\text{NB}}}_k(r,m,n)$. Very recently, Du and Tang proved a number of Andrews–Beck type congruences for ${\mathrm{\text{NB}}}_k(r,m,n)$ and confirmed all conjectures posed by Lin, Peng and Toh. Motivated by their work, we establish the generating functions of ${\mathrm{\text{NB}}}_2(r,5,n)-{\mathrm{\text{NB}}}_2(5-r,5,n)$ and prove several families of congruences modulo arbitrary powers of 5 for ${\mathrm{\text{NB}}}_2(r,5,n)$. In particular, we generalize a congruence modulo 5 for ${\mathrm{\text{NB}}}_2(r,5,n)$ due to Lin, Peng and Toh.

We present an elementary construction of an explicit two-parametric family of elliptic curves over $\mathbb{Q}$ of the form $y^2=x^3+k$ such that the rank of each member of the family is at least seven. The main new ingredient is Ramanujan’s identity for sums of rational cubes.

The concept of the spectral rank of an element in a ring is defined, and it is shown to be a genuine generalization of the same concept first studied in the setting of a Banach algebra. Furthermore, we prove that many of the desirable properties of this rank are still valid in the more abstract setting and give several examples to support and motivate the given definition. In particular, we are able to show that a nonzero idempotent of a semiprimitive and additively torsion-free ring is minimal if and only if it has a spectral rank of one. We also discover a precise connection between the spectral rank of an element in a ring and a purely algebraic definition of rank considered only recently by N. Stopar in [Rank of elements of general rings in connection with unit-regularity, J. Pure Appl. Algebra 224 (2020) 106211]. Specifically, we are able to show that if an element $a$ has a finite algebraic rank in a semiprimitive and additively torsion-free ring, then $a$ has the exact same spectral rank. An extra condition under which the converse holds true is also provided, and connections to the socle are identified. Finally, for both of these extended notions of rank considered in the setting of a ring, we prove a generalized Frobenius Inequality.

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 ${\overline{C}}_{3,1}(9n+6)$.

Let $X=\{1,\dots ,n\}$ be a finite chain and let ${𝒫}{𝒪}{ℐ}(X)$ be the semigroup of all injective order-preserving partial transformations on $X$. For any nonempty subset $Y$ of $X$, let ${𝒫}{𝒪}{ℐ}(X,Y)$ be the subsemigroup of ${𝒫}{𝒪}{ℐ}(X)$ of all transformations with range contained in $Y$. In this paper, we characterize Green’s relations on ${𝒫}{𝒪}{ℐ}(X,Y)$, and show that the semigroup ${𝒫}{𝒪}{ℐ}(X,Y)$ is left abundant but not right abundant when $Y$ is a proper subset of $X$. Moreover, the cardinality and the rank of the semigroup ${𝒫}{𝒪}{ℐ}(X,Y)$ is determined.

This paper is a continuation of our previous one under the same title. In both articles, we study the hyperelliptic curves $C_a:y^2=x^5+ax$ defined over $\mathbb{Q}$, and their Jacobians $J_a$ (without loss of generality a is a nonzero 8th power free integer). Previously, we considered the case when the polynomial $x^4+a$ is irreducible in $\mathbb{Q}[x]$ and obtained (under certain conditions on the quartic field $\mathbb{Q}(\sqrt[4]{-a})$) upper bounds for the $rank{J}_a(\mathbb{Q})$; in particular, we found infinite subfamilies of $J_a$ with rank zero. Now we consider all cases when $x^4+a$ is reducible in $\mathbb{Q}[x]$ and prove analogous results. First we obtain (under mild conditions on some quadratic fields) upper bounds for the ranks in a rather general situation, then we restrict to a several infinite subfamilies of $J_a$ (when 2a has two primes divisors) and get the best possible bounds or even the exact value of rank (if it is zero). We deduce as conclusions the complete lists of rational points on $C_a$ in such cases.

In this paper, we consider the monoid ${{𝒟}{𝒫}{𝒲}}_n$ of all partial isometries of a wheel graph $W_n$ with $n+1$ vertices. Our main objective is to determine the rank of ${{𝒟}{𝒫}{𝒲}}_n$. In the process, we also compute the ranks of three notable subsemigroups of ${{𝒟}{𝒫}{𝒲}}_n$. We also describe Green’s relations of ${{𝒟}{𝒫}{𝒲}}_n$ and of its three considered subsemigroups.

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.

Let R be a commutative ring with unity, and let $\mathrm{\Gamma }(R)$ denote the comaximal graph of R. The comaximal graph $\mathrm{\Gamma }(R)$ has vertex set as R, and any two distinct vertices x, y of $\mathrm{\Gamma }(R)$ are adjacent if $Rx+Ry=R$. Let ${\mathrm{\Gamma }}_2(R)$ denote the induced subgraph of $\mathrm{\Gamma }(R)$ on the set of all nonzero non-unit elements of R, and any two distinct vertices x, y of ${\mathrm{\Gamma }}_2(R)$ are adjacent if $Rx+Ry=R$. In this paper, we study the graphical structure as well the adjacency spectrum of ${\mathrm{\Gamma }}_2({\mathbb{Z}}_n)$, where $n\ge 4$ is a non-prime positive integer, and ${\mathbb{Z}}_n$ is the ring of integers modulo n. We show that for a given non-prime positive integer n with D number of positive proper divisors, the eigenvalues of ${\mathrm{\Gamma }}_2({\mathbb{Z}}_n)$ are $0$ with multiplicity $n-\phi (n)-D-1$, and remaining eigenvalues are contained in the spectrum of a symmetric $D\times D$ matrix. We further calculate the rank and nullity of ${\mathrm{\Gamma }}_2({\mathbb{Z}}_n)$. We also determine all the eigenvalues of ${\mathrm{\Gamma }}_2({\mathbb{Z}}_n)$ whenever ${\mathrm{\Gamma }}_2({\mathbb{Z}}_n)$ is a bipartite graph. Finally, apart from determining certain structural properties of ${\mathrm{\Gamma }}_2({\mathbb{Z}}_n)$, we conclude the paper by determining the metric dimension of ${\mathrm{\Gamma }}_2({\mathbb{Z}}_n)$.

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.

Let ${{𝒪}{𝒫}{ℰ}}_n$ be the monoid of all orientation-preserving and extensive full transformations on $\{1,\dots ,n\}$. In this paper, we compute the rank and the idempotent rank of the ideals of the monoid ${{𝒪}{𝒫}{ℰ}}_n$. Moreover, we determine the maximal subsemigroups as well as the maximal subsemibands of the ideals of the monoid ${{𝒪}{𝒫}{ℰ}}_n$. Our work extends previous results found in the literature.

The rank of adjacency matrix plays an important role in construction of linear codes from a directed strongly regular graph using different techniques, namely, code orthogonality, adjacency matrix determinant and adjacency matrix spectrum. The problem of computing the dimensions of such codes is an intriguing one. Several conjectures to determine the rank of adjacency matrix of a DSRG $\mathrm{\Gamma }$ over a finite field, keep researchers working in this area. To address the same to an extent, we have considered the problem of finding the rank over a finite field of the adjacency matrix of a DSRG $\mathrm{\Gamma }(v,k,t,\lambda ,\mu )$ with $\mu =1$, including some mixed Moore graphs and corresponding codes arising from them, in this paper.

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.

In this paper, we investigate the Fischer group $F_{i_{22}}$. This group is generated by a conjugacy class of involutions, any non-commuting pair of which has product of order 3. Such involutions are called transpositions and their conjugacy class is denoted by D. Subgroups generated by elements of D are called D-groups as they have been called by Enright [G. M. Enright, The structure and subgroups of the Fischer groups $F_{22}$ and $F_{23}$, Ph.D. thesis, University of Cambridge (1976)], Fischer embedded or 3-transposition groups. Here, we obtain the following main results:

• •The rank of the groups $F=3\cdot F_{i_{22}}$, the Weyl group W of type $E_6$ over fields of characteristic 2, the group M of shape $3^6⋊U_4(2)\cdot 2$ and the group $H_1=3{\mathrm{\Omega }}_7(3)$ are computed. If G is a 3-transposition group (Fischer embedded) for a class $D$ of transpositions, then $\text{rank}(G)$ is defined as $\text{max}\{|X\left|\right|X\subseteq D$, all elements in X commute$\}$.
• •We prove that the subgroups ${\mathrm{\Omega }}_8^{+}(2)\cdot S_3,3^5\cdot S_6,3\cdot U_4(3)\cdot 2$ and $S{U}_6(2)$ are Fischer embedded and we give an explicit construction for them. It is remarkable to mention that Enright studied certain D-groups under a very strong condition that is $O_2(G)\le Z(G)\ge O_3(G)$ and $G^{\prime }=G^{″}$ where G is the group generated by a proper subset E of D, $G^{\prime }$ is the derived subgroup of G and E is a single conjugacy class in G. Our study will be carried out without such conditions.

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.

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 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.