On the Divisibility a! + b! | (a + b)!

We introduce a class of new integrable lattice models labeled by a pair of positive integers N and r. The integrable model is obtained from the Gauge/YBE correspondence, which states the equivalence of the 4d \(\mathcal {N} =1\) \(S^{1}\times S^{3}/ \mathbb {Z} _{r}\) index of a large class of SU(N) quiver gauge theories with the partition function of 2d classical integrable spin models. The integrability of the model (star-star relation) is equivalent with the invariance of the index under the Seiberg duality. Our solution to the Yang-Baxter equation is one of the most general known in the literature, and reproduces a number of known integrable models. Our analysis identifies the Yang-Baxter equation with a particular duality (called the Yang-Baxter duality) between two 4d \(\mathcal {N} =1\) supersymmetric quiver gauge theories. This suggests that the integrability goes beyond 4d lens indices and can be extended to the full physical equivalence among the IR fixed points.

Let F q \mathbb {F}_q be a finite field of q q elements. E. Howe has shown that there is a natural correspondence between the isogeny classes of two-dimensional ordinary abelian varieties over F q \mathbb {F}_q which do not contain a principally polarized variety and pairs of positive integers ( a , b ) (a,b) satisfying q = a 2 + b q = a^2 + b , where gcd ( q , b ) = 1 \gcd (q,b)=1 and all prime divisors ℓ \ell of b b are in the arithmetic progression ℓ ≡ 1 ( mod 3 ) \ell \equiv 1 \pmod 3 . This arithmetic criterion allows us to give good upper bounds, and for many finite fields good lower bounds, for the frequency of occurrence of isogeny classes of varieties having this property.

Let $G$ be a nontrivial connected graph with vertex set $V(G)$. A set $S \subseteq V(G)$ is a geodetic hop dominating set of $G$ if the following two conditions hold for each $x\in V(G)\setminus S$: $(1)$ $x$ lies in some $u$-$v$ geodesic in $G$ with $u,v\in S$, and $(2)$ $x$ is of distance $2$ from a vertex in $S$. The minimum cardinality $\gamma_{hg}(G)$ of a geodetic hop dominating set of $G$ is the geodetic hop domination number of $G$. A geodetic hop dominating set $S$ is a minimal geodetic hop dominating set if $S$ does not contain a proper subset that is itself geodetic hop dominating set. The maximum cardinality of a minimal geodetic hop dominating set in $G$ is the \emph{upper geodetic hop domination number} of $G$, and is denoted by $\gamma^{+}_{hg}(G)$. This paper initiates the study of the minimal geodetic hop dominating set and the corresponding upper geodetic hop domination number of nontrivial connected graphs. Interestingly, every pair of positive integers $a$ and $b$ with $2\le a\le b$ is realizable as the geodetic domination number and the upper geodetic hop domination number, respectively, of some graph. Furthermore, this paper investigates the concept in the join, corona and lexicographic product of graphs.

Given a pair of positive integers and such that , for integer the quantity , called the partition function is considered; this by definition is equal to the cardinality of the set The properties of and its asymptotic behaviour as are studied. A geometric approach to this problem is put forward. It is shown that for sufficiently large , where and are positive constants depending on and , and and are characteristics of the exponential growth of the partition function. For some pair the exponents and are calculated as the logarithms of certain algebraic numbers; for other pairs the problem is reduced to finding the joint spectral radius of a suitable collection of finite-dimensional linear operators. Estimates of the growth exponents and the constants and are obtained.

On [formula omitted] broadcast domination numbers of grids

Globally regular (i.e., asymptotically flat and regular interior) spherically symmetric and localized (“particlelike”) solutions of the coupled Einstein Yang–Mills (EYM) equations with gauge group SU(2) have been known for more than 20 years, yet their properties are still not well understood. Spherically symmetric Yang–Mills fields are classified by a choice of isotropy generator and SO(5) is distinguished as the simplest gauge group having a model with a non-Abelian residual (little) group, SU(2)×U(1), which admits globally regular particlelike solutions. We exhibit an algebraic gauge condition which normalizes the residual gauge freedom to a finite number of discrete symmetries. This generalizes the well-known reduction to the real magnetic potential w(r,t) in the original SU(2) YM model. Reformulating using gauge-invariant polynomials dramatically simplifies the system and makes numerical search techniques feasible. We find three families of embedded SU(2) EYM equations within the SO(5) system, one of which was first detected only within the gauge-invariant polynomial reduced system. Numerical solutions representing mixtures of the three SU(2) subsystems are found, classified by a pair of positive integers.

Providing an algebraic proof that φ is indeed a bijection is an instructive exercise. By exploiting the multiplicative structure of the codomain, we can construct a map ψ : Z+ × Z+ → Z+ which is immediately recognized as a bijection. (No need to resort to algebraic calculation or a pictorial argument with diagonals.) For each pair of positive integers m and n, let ψ(m, n) = 2m−1(2n − 1). Bijectivity of ψ is equivalent to the fact that every positive integer has a unique representation as the product of an odd positive integer and a non-negative integer power of 2. As one referee noted, this fact is also key to Glaisher’s bijection between partitions of a positive integer into odd parts and partitions with distinct parts [1, Ex. 2.2.6; 4, p. 12].

Let G = (V, E) be a connected graph with at least three vertices. A set S Í E is called an edge-to-edge geodetic dominating set of G if S is both an edge-to-edge geodetic set of G and an edge dominating set of G. The edge-to- edge geodetic domination number ¡gee(G) of G is the minimum cardinality of its edge-to-edge geodetic dominating sets and any edge-to-edge geodetic dominating set of minimum cardinality is said to be a gee- set of G. Some general properties satisfied by this concept are studied. Connected graphs of size m?2 with edge-to-geodetic domination number 2 or m or m-1 are charaterized. We proved that if G is a connected graph of size m ? 3 and G­ is also connected,then 4 ?¡gee(G) + ¡gee(G­) ? 2m -2. Moreover we characterized graphs for which the lower and the upper bounds are sharp. It is shown that, for every pair of positive integers a and b with 2 ?a ? b, there exists a connected graph G with gee(G) = a and ¡gee(G) = b. Also it is shown that, for every pair of positive integers a and b with 2 < a ? b, there exists a connected graph G with ¡e(G) = a and¡ gee(G) = b, where ¡e(G) is the edge domination number of G and gee(G) is the edge-to-edge geodetic number of G.

On the interval recurrence property of ( N, d)-Ornstein-Uhlenbeck processes

In this paper we study epimorphisms, dominions and related properties for some classes of structurally (n,m)-regular semigroups for any pair (n,m) of positive integers. In Section 2, after a brief introduction of these semigroups, we prove that the class of structurallly (n,m)-generalized inverse semigroups is closed under morphic images. We then prove the main result of this section that the class of structurally (n,m)-generalized inverse semigroups is saturated and, thus, in the category of all semigroups, epimorphisms in this class are precisely surjective morphisms. Finally, in the last section, we prove that the variety of structurally (o, n)-left regular bands is saturated in the variety of structurally (o, k)-left regular bands for all positive integers k and n with 1 ≤ k ≤ n.

In this short note we show that for any pair of positive integers (d, n) with n > 2 and d > 1 or n = 2 and d > 4, there always exist projective varieties X ⊆ ℙ N of dimension n and degree d and an integer s 0 such that Hilb s (X) is reducible for all s ≥ s 0. X will be a projective cone in ℙ N over an arbitrary projective variety Y ⊆ ℙ N−1. In particular, we show that, opposite to the case of smooth surfaces, there exist projective surfaces with a single isolated singularity which have reducible Hilbert scheme of points.

In this short paper, it is proved that every pair of large positive odd integers satisfying some necessary conditions can be represented in the form of a pair of one prime, two prime squares and k powers of 2. In particular, we have k = 332.

For a given ‐partition of the vertices of a (di)graph , we study properties of the spanning bipartite subdigraph of induced by those arcs/edges that have one end in each . We determine, for all pairs of nonnegative integers , the complexity of deciding whether has a 2‐partition such that each vertex in (for ) has at least (out‐)neighbours in . We prove that it is ‐complete to decide whether a digraph has a 2‐partition such that each vertex in has an out‐neighbour in and each vertex in has an in‐neighbour in . The problem becomes polynomially solvable if we require to be strongly connected. We give a characterisation of the structure of ‐complete instances in terms of their strong component digraph. When we want higher in‐degree or out‐degree to/from the other set, the problem becomes ‐complete even for strong digraphs. A further result is that it is ‐complete to decide whether a given digraph has a ‐partition such that is strongly connected. This holds even if we require the input to be a highly connected eulerian digraph.

We investigate algebraic sets over certain finitely generated torsion-free metabelian groups. The class of groups under consideration is the class of so-called ρ-groups. It consists of all wreath products of finitely generated free abelian groups and their subgroups. In particular, it includes all free metabelian groups of finite rank. Our main result is a characterization of certain irreducible algebraic sets over ρ-groups. More precisely, we consider irreducible algebraic sets which are determined by a system of equations in n indeterminates. For their coordinate groups, we introduce a discrete invariant called the relative characteristic. This is an ordered pair of non-negative integers. We determine the structure of the coordinate group of the n-dimensional affine space, and show that its relative characteristic is (n, n). Then we characterize the irreducible algebraic sets of relative characteristic (n, n) and (0, k ) where 0 ≤ k ≤ n . We also obtain some examples of somewhat unusual algebraic sets over ρ-groups, thus demonstrating that algebraic sets over these groups are much more varied and complicated than, say, algebraic sets over free groups.

We show that, for every ordered pair of nonnegative integers (k 1, k 2), there exists a unique (up to equivalence) one-weight Z4-linear code of type 4 k 12 k 2. We derive an upper bound and a lower bound on the highest minimum distance between some extended one-weight Z4-linear codes and the Reed-Muller codes of order 1 and same lengths.