On QB ∞-Rings

In this article, we introduce a new class of rings, the QB _∞-rings. We establish various properties of this concept. These show that, in several respects, QB _∞-rings behave like QB-rings. We prove that the notion of QB _∞-rings is a Morita invariant property of rings and every finite subdirect product of QB _∞-rings is a QB _∞-ring. We also exhibit examples to point out that the class of QB _∞-rings is much larger than the class of QB-rings.     

Special domains and nonmanipulability

We investigate domains on which a nonmanipulable, nondictatorial social choice function exists, having at least three distinct values. We do not make the assumptions of Kalai and Muller (1977). We classify all such 2-person functions on the domain which is the cyclic group Z_m. We show that for any domain containing Z_m, existence for 2 voters and existence for some n > 2 voters are equivalent. We show that for an n-person, onto, nonmanipulable social choice function F on Z_m, F(P₁, P₂,…, P_n) {x₁, x₂,…, x_n} always, x_i being the most preferred alternative under preference P_i. We show that no domain containing the dihedral group admits such a social choice function. We show that there exists a domain on which all k-tuples are free for arbitrarily large k, for which such a social choice function does exist.     

Solutions to the XXX type Bethe ansatz equations and flag varieties

We consider a version of the A _N Bethe equation of XXX type and introduce a reporduction procedure constructing new solutions of this equation from a given one. The set of all solutions obtained from a given one is called a population. We show that a population is isomorphic to the sl _N +1 flag variety and that the populations are in one-to-one correspondence with intersection points of suitable Schubert cycles in a Grassmanian variety. We also obtain similar results for the root systems B _N and C _N . Populations of B _N and C _N type are isomorphic to the flag varieties of C _N and B _N types respectively.     

Intersecting self-similar Cantor sets

We define a self-similar set as the (unique) invariant set of an iterated function system of certain contracting affine functions. A topology on them is obtained (essentially) by inducing theC ¹-topology of the function space. We prove that the measure function is upper semi-continuous and give examples of discontinuities. We also show that the dimension is not upper semicontinuous. We exhibit a class of examples of self-similar sets of positive measure containing an open set. IfC ₁ andC ₂ are two self-similar setsC ₁ andC ₂ such that the sum of their dimensionsd(C ₁)+d(C ₂) is greater than one, it is known that the measure of the intersection setC ₂–C ₁ has positive measure for almost all self-similar sets. We prove that there are open sets of self-similar sets such thatC ₂–C ₁ has arbitrarily small measure.     

On some algebra of continuous linear operators

We present a criterion for an operator on L _p to belong to the set I _p of all sums of integral operators on L _p and multiplication operators by functions in L _∞. We describe the closure of I _p in the operator norm. We prove that the set L _p,1 of all sums of multiplication operators and operators on L _p mapping the unit ball of L _p into compact subsets of L ₁ is a Banach algebra.     

Approximation Numbers of Identity Operators between Symmetric Sequence Spaces

We prove asymptotic formulas for the behavior of approximation quantities of identity operators between symmetric sequence spaces. These formulas extend recent results of Defant, Masty o, and Michels for identities l_pⁿF_n with an n-dimensional symmetric normed space F_n with p-concavity conditions on F_n and 1p2. We consider the general case of identities E_nF_n with weak assumptions on the asymptotic behavior of the fundamental sequences of the n-dimensional symmetric spaces E_n and F_n. We give applications to Lorentz and Orlicz sequence spaces, again considerably generalizing results of Pietsch, Defant, Masty o, and Michels.     

The linear quadratic minimum-time problem for a class of discrete systems

We consider a linear discrete-time systems controlled by inputs on L ²([0, t _N ], U), where (t _i )_{1?≤?i?≤?N} is a given sequence of times. The final time t _N (or N) is considered to be free. Given an initial state x ₀ and a final one x _d , we investigate the optimal control which steers the system from x ₀ to x _d with a minimal cost J(N, u) that includes the final time and energy terms. We treat this problem for both infinte and finite dimensional state space. We use a method similar to the Hilbert Uniqueness Method. A numerical simulation is given.     

On the real roots of Euler polynomials

We show how are located the positive roots of the Euler polynomiale _n of degreen. We give an upper bound and a lower bound for the greatest root. This permits to determine an integerv _(n) such that the number of positive roots ofE _n is eitherv _(n) orv _(n) +2. We also study the behaviour of ther-th positive root ofE _n asn tends to infinity.     

A Class of Sparse Invertible Matrices and Their Use for Nonlinear Prediction of Nearly Periodic Time Series with Fixed Period

We introduce a class of sparse matrices U _m (A _{p 1} ) of order m by m, where m is a composite natural number, p ₁ is a divisor of m, and A _{p 1} is a set of nonzero real numbers of length p ₁. The construction of U _m (A _{p 1} ) is achieved by iteration, involving repetitive dilation operations and block-matrix operations. We prove that the matrices U _m (A _{p 1} ) are invertible and we compute the inverse matrix (U _m (A _{p 1} ))^?1 explicitly. We prove that each row of the inverse matrix (U _m (A _{p 1} ))^?1 has only two nonzero entries with alternative signs, located at specific positions, related to the divisors of m. We use the structural properties of the matrix (U _m (A _{p 1} ))^?1 in order to build a nonlinear estimator for prediction of nearly periodic time series of length m with fixed period.     

The A4‐structure of a graph

We define the A₄‐structure of a graph G to be the 4‐uniform hypergraph on the vertex set of G whose edges are the vertex subsets inducing 2K₂, C₄, or P₄. We show that perfection of a graph is determined by its A₄‐structure. We relate the A₄‐structure to the canonical decomposition of a graph as defined by Tyshkevich [Discrete Math 220 (2000) 201–238]; for example, a graph is indecomposable if and only if its A₄‐structure is connected. We also characterize the graphs having the same A₄‐structure as a split graph.     

On Overspill Principles and Axiom Schemes for Bounded Formulas

We study the theories I?_n, L?_n and overspill principles for ?_n formulas. We show that IE_n ? L?_n ? I?_n, but we do not know if I?_n L?_n. We introduce a new scheme, the growth scheme Crγ, and we prove that L?_n ? Cr?_n? I?_n. Also, we analyse the utility of bounded collection axioms for the study of the above theories. Mathematics Subject Classification: 03F30, 03H15.     

Tearing a strip off the plane

In this article we examine some homomorphic properties of certain subgraphs of the unit-distance graph. We define G_r to be the subgraph of the unit-distance graph induced by the subset (−∞, ∞) × [0, r] of the plane. The bulk of the article is devoted to examining the graphs G_r, when r is the minimum width such that G_r contains an odd cycle of given length. We determine for each odd n the minimum width r_n such that contains an n-cycle C_n, and characterize the embeddings of C_n in $G_{r_{n}}$. We then show that is homomorphically equivalent to C_n when n ≡ 3 (mod 4), but is a core when n ≡ 1 (mod 4). We begin by showing that G_r is homomorphically compact for each r ≥ 0, as defined in [1]. We conclude with some other interesting results and open problems related to the graphs G_r. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 17–33, 1998     

On the chromatic number of random graphs

We consider random graphs G_n,p with fixed edge-probability p. We refine an argument of Bollobás to show that almost all such graphs have chromatic number equal to n/{2 log_b n ? 2 log_b log_b n + O(1)} where b = 1/(1 ? p).     

Numerical Semigroups on Compound Sequences

We generalize the geometric sequence {a^p, a^p?1b, a^p?2b²,…, b^p} to allow the p copies of a (resp. b) to all be different. We call the sequence {a₁a₂a₃… a_p, b₁a₂a₃…a_p, b₁b₂a₃…a_p,…, b₁b₂b₃…b_p} a compound sequence. We consider numerical semigroups whose minimal set of generators form a compound sequence, and compute various semigroup and arithmetical invariants, including the Frobenius number, Apéry sets, Betti elements, and catenary degree. We compute bounds on the delta set and the tame degree.     

The Vector QD Algorithm for Smooth Functions (f, f′)

We deal with the functionz(f(z), f′(z)) wheref(z)=∑_i0 a_izⁱ, (a_i ) with lim_i→∞ a_i+1×a_i−1/(a_i)²=q. We investigate the convergence of the vector QD algorithm. We give the asymptotic behaviour of the generalized Hankel determinants. A convergence result on the vector orthogonal polynomials is proved.     

Schur spaces and weighted spaces of type H ∞

Abstract

We extend some results related to composition operators on H _υ(G) to arbitrary linear operators on H _υ0(G) and H _υ(G). We also give examples of rank-one operators on H _υ(G) which cannot be approximated by composition operators.     

Unitary units in modular group algebras of 2-groups

We study the unitary subgroupV _?(F₂ G) in the group algebras F₂ Gof 2-groups of maximal class over the field F₂of two elements. We show that there does not exist a normal complement to Gin V _?(F₂ G) if and only if Gis the dihedral group of order 2ⁿ(n≥ 5) or the semidihedral group of order 2ⁿ(n≥5). We also describe V _?(F₂ G) when the subgroup Ghas order 8 and 16 and in this case there exist a normal complement of Gin V _?(F₂ G).     

A Characteristic Martingale Related to the Counting Process of Records

Let (X _n ) be a sequence of nonnegative, integrable, independent and identically distributed random variables, with common distribution function F. We consider the problem of finding all distribution functions F such that N _n −cM _n is a discrete time martingale, where N _n is the counting process of upper records, M _n =max {X ₁,…,X _n } is the process of partial maxima and c is a positive constant. We solve the problem by explicitly giving the solution with finite support and using this for constructing the solution for the general case by a limiting process. We show that the set of solutions can be parameterized by their support and the mass at the leftmost point of the support.     

Sets that are connected in two random graphs

–We consider two random graphs G₁, G₂, both on the same vertex set. We ask whether there is a non‐trivial set of vertices S, so that S induces a connected subgraph both in G₁ and in G₂. We determine the threshold for the appearance of such a subset, as well as the size of the largest such subset. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 498–512, 2014