Weighted rational approximation in the complex plane

Given a triple (G, W, γ) of an open bounded set G in the complex plane, a weight function W(z) which is analytic and different from zero in G, and a number γ with 0 ≤ γ ≤ 1, we consider the problem of locally uniform rational approximation of any function ƒ(z), which is analytic in G, by weighted rational functions W^m_i+n_i(z)R_mi, n_i(z)_{i = 0}^∞, where R_mi, n_i(z) = P_mi(z)/Q_ni(z) with deg P_mi ≤ m_i and deg Q_ni ≤ n_i for all i ≥ 0 and where m_i + n_i → ∞ as i → ∞ such that lim m_i/[m_i + n_i] = γ. Our main result is a necessary and sufficient condition for such an approximation to be valid. Applications of the result to various classical weights are also included.     

Homology of certain algebras defined by graphs

Let K(G) for a finite graph G with vertices v₁,...,v_n denote the K-algebra with generators X₁,...,X_n and defining relations X_iX_j=X_jX_i if and only if v_i is not connected to v_j by an edge in G. We describe centralizers of monomials, show that the centralizer of a monomial is again a graph algebra, prove a unique factorization theorem for factorizations of monomials into commuting factors, compute the homology of K(G), and show that K(G) is the homology ring of a certain loop space. We also construct a K(π, 1) explicitly where π is the group with generators X₁,...,X_n and defining relations X_iX_j=X_jX_i if and only if v_i is not connected to v_j by an edge in G.     

Netlike partial cubes III. The median cycle property

A graph G has the Median Cycle Property (MCP) if every triple (u₀,u₁,u₂) of vertices of G admits a unique median or a unique median cycle, that is a gated cycle C of G such that for all i,j,k∈{0,1,2}, if x_i is the gate of u_i in C, then: {x_i,x_j}⊆I_G(u_i,u_j) if i≠j, and d_G(x_i,x_j)<d_G(x_i,x_k)+d_G(x_k,x_j). We prove that a netlike partial cube has the MCP if and only if it contains no triple of convex cycles pairwise having an edge in common and intersecting in a single vertex. Moreover a finite netlike partial cube G has the MCP if and only if G can be obtained from a set of even cycles and hypercubes by successive gated amalgamations, and equivalently, if and only if G can be obtained from K₁ by a sequence of special expansions. We also show that the geodesic interval space of a netlike partial cube having the MCP is a Pash-Peano space (i.e. a closed join space).     

On the existence of super P-groups

A finite group (G, ·) is said to be sequenceable if its elements can be arranged in a sequence a₀ = e, a₁, a₂,…, a_n?1 in such a way that the partial products b₀ = a₀, b₁ = a₀a₁, b₂ = a₀a₁a₂,…, b_n?1 = a₀a₁a₂ ··· a_n?1 are all distinct (and consequently are the elements of G in a new order). It is said to be R-sequenceable if its elements can be ordered in such a way that the partial products b₀ = a₀, b₁ = a₀a₁, b₂ = a₀a₁a₂,…, b_n?2 = a₀a₁a₂ ··· a_n?2 are all different and so that b_n?1 = a₀a₁a₂ ··· a_n?1 = b₀ = e. (in the first case, the ordering a₀,a₁,…,a_n?1 of the elements is said to be a sequencing of G and, in the second case, an R-sequencing of G.) It is a super P-group if every element of one particular coset hG′ of the derived group can be expressed as the product of the n elements of G in such a way that the orderings of the elements in these products are sequencings of G with the exception that, in the case that h = e, the element e of G′ must be expressed as a product of the n elements of G which forms an R-sequencing of G. It is proved that every non-Abelian group of order pq such that p has 2 as a primitive root, where p and q are distinct odd primes with p < q, is a super P-group. Also provided is evidence for the conjectures that all Abelian groups and all dihedral groups of doubly even order (except those of orders 4 and 8) are super P-groups.     

On a nonlinear p-adic dynamical system

We investigate the behavior of trajectories of a (3, 2)-rational p-adic dynamical system in the complex p-adic field ? _p , when there exists a unique fixed point x ₀. We study this p-adic dynamical system by dynamics of real radiuses of balls (with the center at the fixed point x ₀). We show that there exists a radius r depending on parameters of the rational function such that: when x ₀ is an attracting point then the trajectory of an inner point from the ball U _r (x ₀) goes to x ₀ and each sphere with a radius > r (with the center at x ₀) is invariant; When x ₀ is a repeller point then the trajectory of an inner point from a ball U _r (x ₀) goes forward to the sphere S _r (x ₀). Once the trajectory reaches the sphere, in the next step it either goes back to the interior of U _r (x ₀) or stays in S _r (x ₀) for some time and then goes back to the interior of the ball. As soon as the trajectory goes outside of U _r(x ₀) it will stay (for all the rest of time) in the sphere (outside of U _r(x ₀)) that it reached first.     

Optimal tristance anticodes in certain graphs

For z₁,z₂,z₃∈Zⁿ, the tristance d₃(z₁,z₂,z₃) is a generalization of the L₁-distance on Zⁿ to a quantity that reflects the relative dispersion of three points rather than two. A tristance anticodeA_d of diameter d is a subset of Zⁿ with the property that d₃(z₁,z₂,z₃)?d for all z₁,z₂,z₃∈A_d. An anticode is optimal if it has the largest possible cardinality for its diameter d. We determine the cardinality and completely classify the optimal tristance anticodes in Z² for all diameters d?1. We then generalize this result to two related distance models: a different distance structure on Z² where d(z₁,z₂)=1 if z₁,z₂ are adjacent either horizontally, vertically, or diagonally, and the distance structure obtained when Z² is replaced by the hexagonal lattice A₂. We also investigate optimal tristance anticodes in Z³ and optimal quadristance anticodes in Z², and provide bounds on their cardinality. We conclude with a brief discussion of the applications of our results to multi-dimensional interleaving schemes and to connectivity loci in the game of Go.     

On the syntomic regulator for K 1 of a surface

We consider elements of K ₁(S), where S is a proper surface over a p-adic field with good reduction, which are given by a formal sum ??(Z _i , f _i ) with Z _i curves in S and f _i rational functions on the Z _i in such a way that the sum of the divisors of the f _i is 0 on S. Assuming compatibility of pushforwards in syntomic and motivic cohomologies, our result computes the syntomic regulator of such an element, interpreted as a functional on H _dR ² (S), when evaluated on the cup product ????[??] of a holomorphic form ?? by the first cohomology class of a form of the second kind ??. The result is ?? _i ??F _?? , log(f _i ); F _?? ??_gl,Z _i , where F _?? and F _?? are Coleman integrals of ?? and ??, respectively, and the symbol in brackets is the global triple index, as defined in our previous work.     

Categoricity results for L∞κ

Let V denote a variety of algebras in a countable language. An algebra is said to be L_∞κ-free if it is L_∞κ-equivalent to a (V-) free algebra. If every L_∞ω1-free algebra of cardinality ω₁ is free, then for all infinite cardinals κ every L_∞κ-free algebra of cardinality κ is free. Further, assuming suitable set-theoretic hypotheses, if there is a non-free L_∞ω1-free algebra of cardinality ω₁, then for a proper class of cardinals κ there are non-free L_∞κ-free algebras of cardinality κ. The varieties in which the class of free algebras are definable by a sentence in L_ω1ω are characterized as the weak Schreier varieties in which every L_∞ω-free algebra of cardinality ω₁ is free. A weak Schreier variety is one in which every L_∞ω-elementary substructure of a free algebra is free. In fact, assuming suitable set-theortic hypotheses, for weak Schreier varieties the class of free algebras is definable in L_∞∞ iff it is definable in L_ω1ω. Varieties in uncountable languages are also considered.     

Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition II

Lee, Kim and Suh (2012) gave a characterization for real hypersurfaces M of Type (A) in complex two plane Grassmannians G ₂(? ^m+2) with a commuting condition between the shape operator A and the structure tensors φ and φ ₁ for M in G ₂(? ^m+2). Motivated by this geometrical notion, in this paper we consider a new commuting condition in relation to the shape operator A and a new operator φφ ₁ induced by two structure tensors φ and φ ₁. That is, this commuting shape operator is given by φφ ₁ A = A φφ ₁. Using this condition, we prove that M is locally congruent to a tube of radius r over a totally geodesic G ₂(? ^m+1) in G ₂(? ^m+2).     

An addition theorem for finite Abelian groups

If g₁, g₂, …, g_2n?1 is a sequence of 2n ? 1 elements in an Abelian group G of order n, it is known that there are n distinct indices i₁, i₂, …, i_n such that 0 = g_i1 + g_i2 + ? + g_in. In this paper a suitably general condition on the sequence is given which insures that every element g in G has a representation g = g_i1 + g_i2 + ? + g_in as the sum of n terms of the sequence.     

Tying up baric algebras

Given two baric algebras (A ₁, ?? ₁) and (A ₂, ?? ₂) we describe a way to define a new baric algebra structure over the vector space A ₁ ?? A ₂, which we shall denote (A ₁ ? A ₂, ?? ₁ ? ?? ₂). We present some easy properties of this construction and we show that in the commutative and unital case it preserves indecomposability. Algebras of the form A ₁ ? A ₂ in the associative, coutable-dimensional, zero-characteristic case are classified.     

Properties of Regular Uniform k—Stratified Graphs

A regular graph G = (V, E) is a k-stratified graph if V is partitioned into V₁, V₂, …, V_k subsets called strata. The stratification splits the degree d_v ∀v ϵ V into k-integers d_v1, d_v2, …, d_vk each one corresponding to a stratum. If d_v1 = d_v2 = … = d_vk ∀v ϵ V then G is called regular uniform k-stratified, RU^k_s(n, d) where n is the cardinality of the vertex set in each stratum and d is the degree of every vertex in each stratum. For every k, the class RU^k_s(n, d) has a unique graph generator class RU^l_s(n, d) derived by decomposition of graphs in RU^k_s(n, d). We investigate the minimization of the cardinality of V, the colorability, vertex coloring and the diameter of the graphs in the class. We also deal with complexity questions concerning RU^k_s(n, d). Some well-known computer network models such as barrel shifters and hypercubes are shown to belong in RU^k_s(n, d).     

Continuity of invariant measures for families of maps

Let {τ_r} be the family of maps from [0,1] into [0,1] with properties similar to those of τ_r(x) = rx(1 ? x), 0 ≤ r ≤ 4. The limiting behaviour of orbits {τ_r^j(x)}_j^∞ = 1 is a complicated and discontinuous function of the parameter r. The stochastic approximation to the difference equation x_{n + 1} = τ_r(x_n), x_{n + 1} = τ_r(x_n) + W, where W is a fixed random variable independent of r and x_n, is considered. It is shown that this Markov process admits a unique absolutely continuous invariant measure μ_r and, furthermore, that the map r → μ_r is continuous. Such a result is important in applications, since slight changes in the shape of τ_r no longer cause discontinuous consequences in the limiting behaviour of the system.     

Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition

In this paper, first we introduce a new notion of commuting condition that φφ ₁ A = A φ ₁ φ between the shape operator A and the structure tensors φ and φ ₁ for real hypersurfaces in G ₂(? ^m+2). Suprisingly, real hypersurfaces of type (A), that is, a tube over a totally geodesic G ₂(? ^m+1) in complex two plane Grassmannians G ₂(? ^m+2) satisfy this commuting condition. Next we consider a complete classification of Hopf hypersurfaces in G ₂(? ^m+2) satisfying the commuting condition. Finally we get a characterization of Type (A) in terms of such commuting condition φφ ₁ A = A φ ₁ φ.     

Sequences of Diophantine approximations

We sharpen a technique of Gelfond to show that, in a sense, the only possible gap-free sequences of “good” Diophantine approximations to a fixed α ∈ C are trivial ones. For example, suppose that a > 1 and that (δ_n)_n=1^∞ and (σ_n)_n=1^∞ are two positive, strictly increasing unbounded sequences satisfying δ_n+1 ≤ aδ_n and σ_n+1 ≤ aσ_n. If there is a sequence of nonzero polynomials P_n ∈ Z[x] with deg P_n ≤ δ_n, deg P_n + log height P_n ≤ σ_n, and ∣P_n(α)∣ ≤ e^{?(2a+1)δ_nσ_n}, then each P_n(α) = 0.     

A note on tunnel number of composite knots

Let K be a knot in a sphere S³. We denote by t(K) the tunnel number of K. For two knots K₁ and K₂, we denote by K₁?K₂ the connected sum of K₁ and K₂. In this paper, we will prove that if one of K₁ and K₂ has high distance while the other has distance at least 3 then t(K₁?K₂)=t(K₁)+t(K₂)+1.     

An approximate approach of global optimization for polynomial programming problems

Many methods for solving polynomial programming problems can only find locally optimal solutions. This paper proposes a method for finding the approximately globally optimal solutions of polynomial programs. Representing a bounded continuous variable x_i as the addition of a discrete variable d_i and a small variable ϵ_i, a polynomial term x_ix_i can be expanded as the sum of d_ix_j, d_jϵ_i; and ϵ_iϵ_j. A procedure is then developed to fully linearize d_ix_j and d_je_i, and to approximately linearize ϵ_iϵ_j with an error below a pre-specified tolerance. This linearization procedure can also be extended to higher order polynomial programs. Several polynomial programming examples in the literature are tested to demonstrate that the proposed method can systematically solve these examples to find the global optimum within a pre-specified error.     

Boundaries for algebras of holomorphic functions

Let A_u(B_G) be the Banach algebra of all complex valued functions defined on the closed unit ball B_G of a complex Banach space G which are uniformly continuous on B_G and holomorphic in the interior of B_G, endowed with the sup norm. A characterization of the boundaries for A_u(B_G) is given in case G belongs to a class of Banach spaces that includes the pre-dual of a Lorentz sequence space studied by Gowers in Israel J. Math. 69 (1990) 129-151. The non-existence of the Shilov boundary for A_u(B_G) is also proved.     

Enumeration of sequences of given specification according to rises,falls and maxima

Letn = (a₁.a₂ … a_N) denote a sequence of integers a_i={1.2.…n}. A rise is a part a_i.a_i+1 with a_i <a_i+1: a fall is a pair with a_ia_i+1: a level is a pair with a_i = a_i+1. A maximum is a triple a_i-1.a_ia_i+1 with a_i-1?a_i.a_i?a_i+1. If e_i is the number of a_j?n witha_j = i, then [e₁…e_n] is called the specification of n. In addition, a conventional rise is counted to the left of a₁ and a conventional fall to the right of a_N: ifa₁?a₂, then a₁ is counted as a conventional maximum, similarly if a_N-1 ? a_N thena_N is a conventional maximum. Simon Newcomb's problem is to find the number of sequences n with given specification and r rises; the refined problem of determining the number of sequences of given specification with r rises and s falls has also been solved recently. The present paper is concerned with the problem of finding the number of sequences of given specification with r rises, s falls. λ levels and λ maxima. A generating function for this enumerant is obtained as the quotient of two continuants. In certain special cases this result simplifies considerably.     

Mann iterative process for pseudocontractive mappings

Let K be a nonempty, closed and convex subset of a real Banach space E. Let T:K→K be a strictly pseudocontractive map. For a fixed x ₀∈K, define a sequence {x _n } by x _n+1=(1?α _n )x _n +α _n Tx _n , where {α _n } is a real sequence defined in [0,1] satisfying the following conditions (i) $\sum_{n=0}^{\infty }\alpha _{n}=\infty $ , (ii) lim? _n→∞ α _n =0. Then lim?inf? _n→∞‖x _n ?Tx _n ‖=0. If, in addition, T is demicompact, then {x _n } converges strongly to some fixed point of T. Remark 8 is important.