Reconstructible and Half-Reconstructible Tournaments: Application to Their Groups of Hemimorphisms

Let T and T¹ be tournaments with n elements, E a basis for T, E′ a basis for T′, and k ≥ 3 an integer. The dual of T is the tournament T” of basis E defined by T(x, y) = T(y, x) for all x, y ε E. A hemimorphism from T onto T′ is an isomorphism from T onto T” or onto T. A k-hemimorphism from T onto T′ is a bijection f from E to E′ such that for any subset X of E of order k the restrictions T/X and T¹/f(X) are hemimorphic. The set of hemimorphisms of T onto itself has group structure, this group is called the group of hemimorphisms of T. In this work, we study the restrictions to n – 2 elements of a tournament with n elements. In particular, we prove: Let k ≥ 3 be an integer, T a tournament with n elements, where n ≥ k + 5. Then the following statements are equivalent: (i) All restrictions of T to subsets with n – 2 elements are k-hemimorphic. (ii) All restrictions of T to subsets with n – 2 elements are 3-hemimorphic. (iii) All restrictions of T to subsets with n – 2 elements are hemimorphic. (iv) All restrictions of T to subsets with n – 2 elements are isomorphic, (v) Either T is a strict total order, or the group of hemimorphisms of T is 2-homogeneous.     

On the location of fixed points on pairs of spaces

Let f: (X, A)→(X, A) be an admissible selfmap of a pair of metrizable ANR's. A Nielsen number of the complement Ñ(f; X, A) and a Nielsen number of the boundary ñ(f; X, A) are defined. Ñ(f; X, A) is a lower bound for the number of fixed points on C1(X - A) for all maps in the homotopy class of f. It is usually possible to homotope f to a map which is fixed point free on Bd A, but maps in the homotopy class of f which have a minimal fixed point set on X must have at least ñ(f; X, A) fixed points on Bd A. It is shown that for many pairs of compact polyhedra these lower bounds are the best possible ones, as there exists a map homotopic to f with a minimal fixed point set on X which has exactly Ñ(f; X - A) fixed points on C1(X−A) and ñ(f; X, A) fixed points on Bd A. These results, which make the location of fixed points on pairs of spaces more precise, sharpen previous ones which show that the relative Nielsen number N(f; X, A) is the minimum number of fixed points on all of X for selfmaps of (X, A), as well as results which use Lefschetz fixed point theory to find sufficient conditions for the existence of one fixed point on C1(X−A).     

Graphs and ranks of monoids

For a finite monoid S, let ν(S) (ν_d(S)) denote the least number n such that there exists a graph (directed graph) Γ of order n with End(Γ)?S. Also let rank(S) be the smallest number of elements required to generate S. In this paper, we use Cayley digraphs of monoids, to connect lower bounds of ν(S) (ν_d(S)) to the lower bounds of rank(S). On the other hand, we connect upper bounds of rank(S) to upper bounds of ν(S) (ν_d(S)).     

Existence of Lipschitzian solutions to the classical problem of the calculus of variations in the autonomous case

Under general growth assumptions, that include some cases of linear growth, we prove existence of Lipschitzian solutions to the problem of minimizing ∫_a^bL(x(s),x′(s)) ds with the boundary conditions x(a)=A, x(b)=B.     

To the problem of a strong differentiability of integrals along different directions

It is proved that for any given sequence (σ _n ,n ∈ ℕ)=Γ₀ ⊂ Γ, where Γ is the set of all directions in ℝ² (i.e., pairs of orthogonal straight lines) there exists a locally integrable functionf on ℝ² such that: (1) for almost all directionsσ ∈ Γ\Γ₀ the integral ∫f is differentiable with respect to the familyB _2σ of open rectangles with sides parallel to the straight lines fromσ: (2) for every directionσ _n ∈ Γ₀ the upper derivative of ∫f with respect toB _{2σ n} equals +∞; (3) for every directionσ ∈ Γ the upper derivative of ∫ |f| with respect toB _2σ equals +∞.     

Characterizations of degree one bivariate measures of concordance

We call a measure of concordance κ of an ordered pair (X,Y) of two continuous random variables a bivariate measure of concordance. This κ may be considered to be a function κ(C) of the copula C associated with (X,Y). κ is considered to be of degree n if, given any two copulas A and B, the value of their convex sum, κ(tA+(1−t)B), is a polynomial in t of degree n. Examples of bivariate measures of concordance are Spearman’s rho, Blomqvist’s beta, Gini’s measure of association, and Kendall’s tau. The first three of these are of degree one, but Kendall’s tau is of degree two. We exhibit three characterizations of bivariate measures of concordance of degree one.     

The Vector Space Kinna‐Wagner Principle is Equivalent to the Axiom of Choice

We show that the axiom of choice AC is equivalent to the Vector Space Kinna‐Wagner Principle, i.e., the assertion: “For every family 𝒱= {V_i : i ∈ k} of non trivial vector spaces there is a family ℱ = {F_i : i ∈ k} such that for each i ∈ k, F_i is a non empty independent subset of V_i”. We also show that the statement “every vector space over ℚ has a basis” implies that every infinite well ordered set of pairs has an infinite subset with a choice set, a fact which is known not to be a consequence of the axiom of multiple choice MC.     

Characterization of Generating Ideals in Some Rings of Entire Functions

Let E be a ring of entire functions on with the operation of pointwise multiplication, and let f ₁,...,f _m be a set of nonzero elements in E. The ideal E(f ₁,...,f _m ) in E with generators f ₁,...,f _m is said to be generating if E(f ₁,...,f _m ) = E. The generating ideals in rings of entire functions on determined by the growth of their maximum moduli are characterized in terms of the distribution of the zero sets of their generators. Under the additional condition of rapid variation of the weight sequences determining the ring, criteria for generating ideals are established; they are stated in terms of d(z) max _{1 j m} d _j (z), where d _j (z) is the distance from a point to the zero set of f _j for 1 j m. It is shown that, in rings of entire functions of finite or minimal type with respect to a given order, a similar characterization (i.e., in terms of d(z)) cannot be given.     

Investigating derivatives by means of combinatorial analysis of the components of the function

Given a composite function of the form h(x) = f(g(x)), difficulties are often encountered in calculating the value of the nth derivative at some point x = x₀ when one attempts to determine whether its nth derivative becomes zero at this point, or attempts to find the sign of the nth derivative by differentiating it n times and substituting x₀.

This present paper offers an alternative method that allows the investigation of the nth derivative of function h(x) based on the investigation of functions f?(x) and g(x) only.

Several examples are given, which implement the conclusions on the properties of the relation.     

Moments of the maximum of normed partial sums of ρ^--mixing random variables

Let {Xn,n ≥ 1} be a sequence of identically distributed ρ^--mixing random variables and set Sn =∑i^n=1 Xi,n ≥ 1,the suffcient and necessary conditions for the existence of moments of supn≥1 ｜Sn/n^1/r｜^p（0 〈 r 〈 2,p 〉 0） are given,which are the same as that in the independent case.     

Approximation of infinitely differentiable multivariate functions is intractable

We prove that L_∞-approximation of C^∞-functions defined on [0,1]^d is intractable and suffers from the curse of dimensionality. This is done by showing that the minimal number of linear functionals needed to obtain an algorithm with worst case error at most ε(0,1) is exponential in d. This holds despite the fact that the rate of convergence is infinite.     

On the rank of 3×3×3-tensors

Let U, V and W be finite dimensional vector spaces over the same field. The rank of a tensor τ in U???V???W is the minimum dimension of a subspace of U???V???W containing τ and spanned by fundamental tensors, i.e. tensors of the form u???v???w for some u in U, v in V and w in W. We prove that if U, V and W have dimension three, then the rank of a tensor in U???V???W is at most six, and such a bound cannot be improved, in general. Moreover, we discuss how the techniques employed in the proof might be extended to prove upper bounds for the rank of a tensor in U???V???W when the dimensions of U, V and W are higher.     

TheD 2-triangulation for simplicial homotopy algorithms for computing solutions of nonlinear equations

A triangulation of arbitrary refinement of grid sizes of (0, 1] × ⁿ is proposed for simplicial homotopy algorithms for computing solutions of nonlinear equations. On each level the new triangulation, called theD ₂-triangulation, subdivides ⁿ into simplices according to theD ₁-triangulation. We prove that theD ₂-triangulation is superior to theK ₂-triangulation andJ ₂-triangulation in the number of simplices. Numerical tests show that the simplicial homotopy algorithm based on theD ₂-triangulation indeed is much more efficient.     

Unique maximal rings of functions

For several classes of groups G, we characterize when the near-ring M₀(G) of 0-preserving selfmaps on G contains a unique maximal ring. Definitive results are obtained for finite Abelian, finite nilpotent, and finite permutation groups. As an application, we determine those finite groups G such that all rings in M₀(G) are commutative.     

Automorphisms of Models of True Arithmetic: Subgroups which Extend to a Maximal Subgroup Uniquely

We show that if M is a countable recursively saturated model of True Arithmetic, then G = Aut(M) has nonmaximal open subgroups with unique extension to a maximal subgroup of Aut(M). Mathematics Subject Classification: 03C62, 03C50.     

Unified Treatment of Algebraic and Geometric Difference by a New Difference Scheme and its Continuity Properties

We introduce a new operation for the difference of two sets A and C of R ⁿ depending on a parameter . This new operation may yield as special cases the classical difference and the Minkowski difference, if the sets A and C are closed, convex sets, if int(C) is nonempty, and if A or C bounded. Continuity properties with respect to both the operands and the parameter of this operation are studied. Lipschitz properties of the Minkowski difference between two sets of a normed vector space are proved in the bounded case as well as in the unbounded case without condition on the dimension of the space.     

Additive problems for integers with a given number of prime divisors

The asymptotics of sums of the form Στ(|bn−a|) (summation overn<N, ω(n)=k) is studied, whereω(n) is the number of distinct prime divisors ofn, andτ(n) is the number of all divisors. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 749–762, May, 1998. In conclusion, the author wishes to express his gratitude to Professor N. M. Timofeev for valuable advice. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00502.     

A note on exponent of convergence of zeros of entire functions

If f(z) is an entire function with ρ ₁ > 0 as its exponent of convergence of zeros and if 0 ≤ α < ρ ₁, then we prove the existence of entire functions each having α as its exponent of convergence of zeros.      

Finitely Presented Functors and Degenerations#

Given two d-dimensional Λ-modules M and N, then M degenerates to N if and only if there exists an exact sequence of the form 0→ U→ U ⊕ M→ N→ 0 for some U ? mod Λ (Zwara, 1998 Zwara , G. ( 1998 ). A degeneration-like order for modules . Arch. Math. 71 : 437 – 444 . [CSA] [CROSSREF]  [Google Scholar]). Having this as a starting point, in this article we give a characterization of degenerations by the existence of a certain finitely presented functor. This gives new information about U in the sequence above. We show how this new information can be used to prove that M   _deg N even when M ⊕ X ≤  _deg N ⊕ X for some X for modules over Λ _q  = k〈 x,y〉/〈 x ²,y ²,xy + qyx〉, q ≠ 0 in k, where k is an algebraically closed field.