Uniformities with the same Hausdorff hypertopology

Two uniformities U and V on a set X are said to be H-equivalent if their corresponding Hausdorff uniformities on the set of all non-empty subsets of X induce the same topology. The uniformity U is said to be H-singular if no distinct uniformity on X is H-equivalent to U. The self-explanatory concepts of H-coarse, H-minimal and H-maximal uniformities are defined similarly.It is well known that not all uniformities are H-singular. We show here that there is a property which obstructs H-singularity: Every H-minimal uniformity has a base of finite-dimensional uniform coverings. Besides, we provide an intrinsic characterization of H-minimal uniformities and show that they are H-coarse. This characterization of H-minimality becomes a criterion for H-singularity for all uniformities that are either complete, uniformly locally precompact or proximally fine (e.g., metrizable ones). Some relevant properties which insure H-singularity are introduced and investigated in some aspect.     

Inequivalent surface-knots with the same knot quandle

We have a knot quandle and a fundamental class as invariants for a surface-knot. These invariants can be defined for a classical knot in a similar way, and it is known that the pair of them is a complete invariant for classical knots. In surface-knot theory the situation is different: There exist arbitrarily many inequivalent surface-knots of genus g with the same knot quandle, and there exist two inequivalent surface-knots of genus g with the same knot quandle and with the same fundamental class.     

Generalized Bowen-Franks groups of integral matrices with the same zeta function

In this paper the relation between the zeta function of an integral matrix and its generalized Bowen-Franks groups is studied. Suppose that A and B are nonnegative integral matrices whose invertible part is diagonalizable over the field of complex numbers and A and B have the same zeta function. Then there is an integer m, which depends only on the zeta function, such that, for any prime q such that gcd(q,m)=1, for any g(x)∈Z[x] with g(0)=1, the q-Sylow subgroup of the generalized Bowen-Franks group BF_g(x)(A) and BF_g(x)(B) are the same. In particular, if m=1, then zeta function determines generalized Bowen-Franks groups.     

On division algebras having the same maximal subfields

We show that if a field K of characteristic ≠ 2 satisfies the following property (*) for any two central quaternion division algebras D ₁ and D ₂ over K, the fact that D ₁ and D ₂ have the same maximal subfields implies that D ₁ ? D ₂ over K, then the field of rational functions K(x) also satisfies (*). This, in particular, provides an alternative proof for the result of S. Garibaldi and D. Saltman that the fields of rational functions k(x ₁, . . . , x _r ), where k is a number field, satisfy (*). We also show that K = k(x ₁, . . . , x _r ), where k is either a totally complex number field with a single dyadic place (e.g. ${k = \mathbb{Q}(\sqrt{-1})}$ ) or a finite field of characteristic ≠ 2, satisfies the analog of (*) for all central division algebras having exponent two in the Brauer group Br(K).     

Enumerating pairs of permutations with the same up-down form

For each sequence q = {q_i} = ± 1, i = 1, …, n−1 let N_q = the number of permutations σ of 1, 2, …, n with up-down sequence sgn(σ_i+1 − σ_i) = q_i, i = 1,…, n−1. Clearly Σ_q (N_q/n!) =1 but what is the probability p_n = Σ_q (N_q/n!)² that two random permutations have the same up-down sequence? We show that p_n = (Kⁿ⁻¹1,1) where 1 = 1(x, y) ≡ 1 and Kⁿ⁻¹ is the iterated integral operator with Kφ(x, y) = ∫₀¹∫₀¹ K(x, y; x′, y′)φ(x′, y′) dx′ dy′ on L²[0, 1] × [0, 1] where K(x, y; x′, y′) is 1 if (x− x′)(y − y′) > 0 otherwise, and (f, g) = ∫₀¹∫₀¹fg. The eigenexpression of K yeilds p_n ∼ cαⁿ as n → ∞, where c ≈ 1.6, α ≈ 0.55.We also give a recursion formula for a polynomial whose coefficients are the frequencies of all the possible forms.     

On nonabelian simple groups having the same prime graph as an alternating group

All instances of coincidence between the prime graphs of nonabelian simple groups G and S are found, where G is an alternating group of degree n ≥ 5 and S is a nonabelian finite simple group. The precise bound of the maximal number of pairwise nonisomorphic nonabelian simple groups with the same prime graph is given in the case that one of these groups is an alternating group.     

Groups with the same non-commuting graph

The non-commuting graph Γ_G of a non-abelian group G is defined as follows. The vertex set of Γ_G is G−Z(G) where Z(G) denotes the center of G and two vertices x and y are adjacent if and only if xy≠yx. It has been conjectured that if G and H are two non-abelian finite groups such that Γ_G≅Γ_H, then |G|=|H| and moreover in the case that H is a simple group this implies G≅H. In this paper, our aim is to prove the first part of the conjecture for all the finite non-abelian simple groups H. Then for certain simple groups H, we show that the graph isomorphism Γ_G≅Γ_H implies G≅H.     

Finite groups whose noncentral class sizes have the same p-part for some prime p

We describe the structure of the finite groups in which the sizes of noncentral conjugacy classes have the same p-part, for some prime p. We prove that they are solvable, they have a normal p-complement and their Fitting length is at most three.     

Ratios of harmonic functions with the same zero set

We study the ratio of harmonic functions u,v which have the same zero set Z in the unit ball \({B\subset \mathbb{R}^n}\). The ratio \({f=u/v}\) can be extended to a real analytic nowhere vanishing function in B. We prove the Harnack inequality and the gradient estimate for such ratios in any dimension: for a given compact set \({K\subset B}\) we show that \({\sup_K|f|\le C_1\inf_K|f|}\) and \({\sup_K\left|\nabla f\right|\le C_2 \inf_K|f|}\), where C ₁ and C ₂ depend on K and Z only. In dimension two we specify the dependence of the constants on Z in these inequalities by showing that only the number of nodal domains of u, i.e. the number of connected components of \({B\setminus Z}\), plays a role.     

Entire functions that have the same fixed points with their first derivatives

Let f be a nonconstant entire function, and let k (?2) be an integer. We denote by the set consisting of all the fixed points of f. This paper proves that if f and f′ have the same fixed points, namely, E_f(z)=E_f′(z), and if f^(k)(z)=z whenever f(z)=z, then f≡f′.     

Cost minimization of m simultaneous projects that require the same scarce resource

A company is responsible for m projects which are carried out concurrently. All m projects require the use of the same scarce resource for some time periods. An Integer Programming model is here presented to solve the problem of optimally allocating the scarce resource to the m projects so that the sum of total costs of carrying out the m projects is minimized.     

Next-fit packs a list and its reverse into the same number of bins

Suppose the next-fit algorithm packs {x₁, x₂, …, x_n} into k identical bins. Under modest assumptions about what fits into a bin, we prove that next-fit also packs {x_n, …, x₂, x₁} into k bins. Thus, the next-fit decreasing algorithm uses the same number of bins as a next-fit increasing algorithm.     

Optimal control of longitudinal vibrations of composite rods with the same wave propagation time in each part

We consider longitudinal elastic vibrations of a composite rod and find closedform expressions that describe optimal boundary controls bringing the rod from the quiescent state into a state with given displacement function φ(t) and velocity function ψ(t) in time T. We assume that the wave propagation time through each part of the rod is the same and T is a multiple of that time.     

Partitions of natural numbers with the same representation functions

For a set A of nonnegative integers the representation functions R₂(A,n), R₃(A,n) are defined as the number of solutions of the equation n=a+a^′,a,a^′∈A with a<a^′, a?a^′, respectively. Let D(0)=0 and let D(a) denote the number of ones in the binary representation of a. Let A₀ be the set of all nonnegative integers a with even D(a) and A₁ be the set of all nonnegative integers a with odd D(a). In this paper we show that (a) if R₂(A,n)=R₂(N?A,n) for all n?2N−1, then R₂(A,n)=R₂(N?A,n)?1 for all n?12N²−10N−2 except for A=A₀ or A=A₁; (b) if R₃(A,n)=R₃(N?A,n) for all n?2N−1, then R₃(A,n)=R₃(N?A,n)?1 for all n?12N²+2N. Several problems are posed in this paper.     

Extremal digraphs avoiding distinct walks of length 3 with the same endpoints

In this paper, we determine the maximum size of digraphs on n vertices which avoid two distinct walks of length 3 with the same initial vertex and the same terminal vertex. The digraphs attaining this maximum size are also characterized. Combining this with previous results, we obtain a full solution to a problem proposed by X. Zhan in 2007.     

On the space curves with the same image under the gauss maps

From an irreducible complete immersed curveX in a projective space ? other than a line, one obtains a curveX ^′ in a Graasmann manifoldG of lines in ? that is the image ofX under the Gauss map, which is defined by the embedded tangents ofX. The main result of this article clarifies in case of positive characteristic what curvesX have the sameX′: It is shown thatX is uniquely determined byX′ ifX, or equivalentlyX′, has geometric genus at least two, and that for curvesX ₁ andX ₂ withX ₁ ≠X ₂ in ?, ifX′₁ =X′₂ inG and eitherX ₁ orX ₂ is reflexive, then bothX ₁ andX ₂ are rational or supersingular elliptic; moreover, examples of smoothX ₁ andX ₂ in that case are given.     

Probabilistic assignment of indivisible objects when agents have the same preferences except the ordinal ranking of one object

Bogomolnaia and Moulin (2001) show that there is no rule satisfying equal treatment of equals, stochastic dominance efficiency, and stochastic dominance strategyproofness for a probabilistic assignment problem of indivisible objects. Kasajima (2013) shows that the incompatibility result still holds when agents are restricted to have single-peaked preferences. In this paper, we further restrict the domain and investigate the existence of rules satisfying the three properties. As it turns out, the three properties are still incompatible even if all agents have the same preferences except the ordinal ranking of one object.     

Transformation of the triple series of Gelfand, Graev, and Retakh into a series of the same type and related problems

A transformation of the triple series T related to the GrassmanianG _2,4 into a series of the same structure type is obtained. This transformation generalizes the reduction formula of Gelfand, Graev, and Retakh taking the series T to the Gauss function under two additional conditions and two more general reduction formulas taking the series T to the Appell function F ₁ and to the Horn function G ₂ under one of the additional conditions. The approach used to analyze the series T is based on the representation of the initial series T in terms of series with convenient computational properties.     

Families of simply connected 4-manifolds with the same Seiberg-Witten invariants

This article presents several new constructions of infinite families of smooth 4-manifolds with the property that any two manifolds in the same family are homeomorphic. While the construction gives strong evidence that any two of these manifolds of are not diffeomorphic, they cannot be distinguished by Seiberg-Witten invariants. Whether these manifolds are, or are not, diffeomorphic seems to be a very difficult question to answer. For one of these constructions, each member of the family is symplectic with the further property that each contains nullhomologous tori with the property that infinitely many log transformations on these tori yield nonsymplectic 4-manifolds. This is detected by calculations of Seiberg-Witten invariants. The surgery in question can be performed on any 4-manifold which contains as a codimension 0 submanifold a punctured surface bundle over a punctured surface and a nontrivial loop in the base which has trivial monodromy. A starting point for another class of examples in this paper is a family of examples which show that the Parshin-Arakelov theorem for holomorphic Lefschetz fibrations is false in the symplectic category. Such families are constructed by means of knot surgery on ellipitic surfaces. It is shown that for a fixed homeomorphism type X (of a simply connected elliptic surface) and a fixed integer g?3, there are infinitely many genus g Lefschetz fibrations on nondiffeomorphic 4-manifolds, all homeomorphic to X.