2‐connected 7‐coverings of 3‐connected graphs on surfaces

An m‐covering of a graph G is a spanning subgraph of G with maximum degree at most m. In this paper, we shall show that every 3‐connected graph on a surface with Euler genus k ≥ 2 with sufficiently large representativity has a 2‐connected 7‐covering with at most 6k ? 12 vertices of degree 7. We also construct, for every surface F² with Euler genus k ≥ 2, a 3‐connected graph G on F² with arbitrarily large representativity each of whose 2‐connected 7‐coverings contains at least 6k ? 12 vertices of degree 7. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 26–36, 2003     

Destructibility of stationary subsets of Pκλ

For a regular cardinal κ with κ ^<κ = κ and κ ≤ λ , we construct generically (forcing by a < κ‐closed κ ⁺‐c. c. p. o.‐set ℙ₀) a subset S of {x ∈ P _κ λ : x ∩ κ is a singular ordinal} such that S is stationary in a strong sense (F ^IA_κ λ ‐stationary in our terminology) but the stationarity of S can be destroyed by a κ ⁺‐c. c. forcing ℙ* (in V ^ℙ) which does not add any new element of P _κ λ . Actually ℙ* can be chosen so that ℙ* is κ‐strategically closed. However we show that such ℙ* itself cannot be κ‐strategically closed or even <κ‐strategically closed if κ is inaccessible. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Non Deterministic Classical Logic: The λμ++ ‐calculus

In this paper, we present an extension of λμ‐calculus called λμ⁺⁺‐calculus which has the following properties: subject reduction, strong normalization, unicity of the representation of data and thus confluence only on data types. This calculus allows also to program the parallel‐or.     

Archimedean ϕ ‐tolerance graphs

Let ? be a symmetric binary function, positive valued on positive arguments. A graph G = (V,E) is a ?‐tolerance graph if each vertex υ ∈ V can be assigned a closed interval I_υ and a positive tolerance t_υ so that xy ∈ E ? | I_x ∩ I_y|≥ ? (t_x,t_y). An Archimedean function has the property of tending to infinity whenever one of its arguments tends to infinity. Generalizing a known result of [15] for trees, we prove that every graph in a large class (which includes all chordless suns and cacti and the complete bipartite graphs K_2,k) is a ?‐tolerance graph for all Archimedean functions ?. This property does not hold for most graphs. Next, we present the result that every graph G can be represented as a ?_G‐tolerance graph for some Archimedean polynomial ?_G. Finally, we prove that there is a ?universal”? Archimedean function ? ^* such that every graph G is a ?^*‐tolerance graph. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 179–194, 2002     

On the existence of (3,λ)‐semiframes of type 3u

A (k,λ)‐semiframe of type g^u is a (k,λ)‐group‐divisible design of type g^u (??, ??, ??), in which the collection of blocks ?? can be written as a disjoint union ??=??∪?? where ?? is partitioned into parallel classes of ?? and ?? is partitioned into holey parallel classes, each holey parallel class being a partition of ??\G_j for some G_j∈??. In this paper, we shall prove that the necessary conditions for (3,λ)‐semiframes of type 3^u are also sufficient with one exception. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 253–265, 2009     

Repulsive knot energies and pseudodifferential calculus for O’Hara’s knot energy family E(α), α ∈ [2, 3)

We develop a precise analysis of J. O’Hara’s knot functionals E^(α), α ∈ [2, 3), that serve as self‐repulsive potentials on (knotted) closed curves. First we derive continuity of E^(α) on injective and regular H² curves and then we establish Fréchet differentiability of E^(α) and state several first variation formulae. Motivated by ideas of Z.‐X. He in his work on the specific functional E⁽²⁾, the so‐called Möbius Energy, we prove C^∞‐smoothness of critical points of the appropriately rescaled functionals $\tilde{E}^{(\alpha )}= {\rm length}^{\alpha -2}E^{(\alpha )}$ by means of fractional Sobolev spaces on a periodic interval and bilinear Fourier multipliers.     

The wi‐club filter on 𝒫κ λ

We develop the theory of C_{κ, λ}ⁱ, a strongly normal filter over ??_κ λ for Mahlo κ. We prove a minimality result, showing that any strongly normal filter containing {x ∈ ??_κ λ: |x | = |x ∩ κ | and |x | is inaccessible} also contains C_{κ, λ}ⁱ. We also show that functions can be used to obtain a basis for C_{κ, λ}ⁱ (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Increasing η ‐representable degrees

In this paper we prove that any Δ₃⁰ degree has an increasing η ‐representation. Therefore, there is an increasing η ‐representable set without a strong η ‐representation (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A classification result on weighted {δ (p3 + 1), δ; 3, p3}‐minihypers

We classify all {δ (p³ + 1), δ; 3, p³}‐minihypers, , for a prime number p₀ ≥ 7, with excess e ≤ p³. Such a minihyper is a sum of lines and of possibly one projected subgeometry PG(5, p), or a sum of lines and a minihyper which is a projected subgeometry PG(5, p) minus one line. When p is a square, also (possibly projected) Baer subgeometries PG(3, p^3/2) can occur. © 2004 Wiley Periodicals, Inc.     

The spectrum of BSA(v, 3, λ; α) with α = 2,3

In this article, a kind of auxiliary design BSA^* for constructing BSAs is introduced and studied. Two powerful recursive constructions on BSAs from 3‐IGDDs and BSA^*s are exploited. Finally, the necessary and sufficient conditions for the existence of a BSA(v, 3, λ; α) with α = 2, 3 are established. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 61–76, 2007     

Finding paths between 3‐colorings

Given a 3‐colorable graph G together with two proper vertex 3‐colorings α and β of G, consider the following question: is it possible to transform α into β by recoloring vertices of G one at a time, making sure that all intermediate colorings are proper 3‐colorings? We prove that this question is answerable in polynomial time. We do so by characterizing the instances G, α, β where the transformation is possible; the proof of this characterization is via an algorithm that either finds a sequence of recolorings between α and β, or exhibits a structure which proves that no such sequence exists. In the case that a sequence of recolorings does exist, the algorithm uses O(|V(G)|²) recoloring steps and in many cases returns a shortest sequence of recolorings. We also exhibit a class of instances G, α, β that require Ω(|V(G)|²) recoloring steps. © 2010 Wiley Periodicals, Inc. J Graph Theory 67: 69‐82, 2011     

Ω‐estimates related to irreducible algebraic integers

We study large values of the remainder term E_K (x) in the asymptotic formula for the number of irreducible integers in an algebraic number field K. We show that E_K (x) = Ω_± (√(x)(log x)) for certain positive constant B_K, improving in that way the previously best known estimate E_K (x) = Ω_± (x^(1/2)‐ε) for every ε > 0, due to A. Perelli and the present author. Assuming that no entire L‐function from the Selberg class vanishes on the vertical line σ = 1, we show that E_K (x) = Ω_± (√(x)(log log x)^{D (K)‐1}(log x)^‐1), supporting a conjecture raised recently by the author. In particular, it follows that the last omega estimate is a consequence of the Selberg Orthonormality Conjecture (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A short proof of the preservation of the ωω‐bounding property

There are two versions of the Proper Iteration Lemma. The stronger (but less well‐known) version can be used to give simpler proofs of iteration theorems (e.g., [7, Lemma 24] versus [9, Theorem IX.4.7]). In this paper we give another demonstration of the fecundity of the stronger version by giving a short proof of Shelah's theorem on the preservation of the ω^ω‐bounding property. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Embedding vector‐valued Besov spaces into spaces of γ ‐radonifying operators

It is shown that a Banach space E has type p if and only for some (all) d ≥ 1 the Besov space B^{(1/p – 1/2)d} _p,p (?^d ; E) embeds into the space γ (L²(?^d ), E) of γ ‐radonifying operators L²(?^d ) → E. A similar result characterizing cotype q is obtained. These results may be viewed as E ‐valued extensions of the classical Sobolev embedding theorems. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On κ‐hereditary Sets and Consequences of the Axiom of Choice

We will prove that some so‐called union theorems (see [2]) are equivalent in ZF⁰ to statements about the transitive closure of relations. The special case of “bounded” union theorems dealing with κ‐hereditary sets yields equivalents to statements about the transitive closure of κ‐narrow relations. The instance κ = ω₁ (i. e., hereditarily countable sets) yields an equivalent to Howard‐Rubin's Form 172 (the transitive closure Tc(x) of every hereditarily countable set x is countable). In particular, the countable union theorem (Howard‐Rubin's Form 31) and, a fortiori, the axiom of countable choice imply Form 172.     

Categorical abstract algebraic logic: Gentzen π ‐institutions and the deduction‐detachment property

Given a π ‐institution I , a hierarchy of π ‐institutions I ^{(n )} is constructed, for n ≥ 1. We call I ^{(n )} the n‐th order counterpart of I . The second‐order counterpart of a deductive π ‐institution is a Gentzen π ‐institution, i.e. a π ‐institution associated with a structural Gentzen system in a canonical way. So, by analogy, the second order counterpart I ⁽²⁾ of I is also called the “Gentzenization” of I . In the main result of the paper, it is shown that I is strongly Gentzen , i.e. it is deductively equivalent to its Gentzenization via a special deductive equivalence, if and only if it has the deduction‐detachment property . (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Time decay rate for two 3D magnetohydrodynamics‐ α models

We consider the long time behavior of solutions to the magnetohydrodynamics‐ α model in three spatial dimensions. Time decay rate in L²‐norm of the solution is obtained. Similar results for a generalized Leray‐ α‐magnetohydrodynamics model are also established. As a by‐product, an optimal time decay rate for the Navier–Stokes‐ α model is achieved. Copyright © 2013 John Wiley & Sons, Ltd.     

On some sets of dictionaries whose ω ‐powers have a given

A dictionary is a set of finite words over some finite alphabet X. The ω ‐power of a dictionary V is the set of infinite words obtained by infinite concatenation of words in V. Lecomte studied in [10] the complexity of the set of dictionaries whose associated ω ‐powers have a given complexity. In particular, he considered the sets ??( Σ ⁰k) (respectively, ??( Π ⁰_k), ??( Δ ¹₁)) of dictionaries V ? 2* whose ω ‐powers are Σ ⁰_k‐sets (respectively, Π ⁰_k‐sets, Borel sets). In this paper we first establish a new relation between the sets ??( Σ ⁰₂) and ??( Δ ¹₁), showing that the set ??( Δ ¹₁) is “more complex” than the set ??( Σ ⁰₂). As an application we improve the lower bound on the complexity of ??( Δ ¹₁) given by Lecomte, showing that ??( Δ ¹₁) is in Σ ¹ ₂(2^2*)\ Π ⁰₂. Then we prove that, for every integer k ≥ 2 (respectively, k ≥ 3), the set of dictionaries ??( Π ⁰_k+1) (respectively, ??( Σ ⁰_{k +1})) is “more complex” than the set of dictionaries ??( Π ⁰_k) (respectively, ??( Σ ⁰_k)) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On ε‐biased generators in NC0

Cryan and Miltersen (Proceedings of the 26th Mathematical Foundations of Computer Science, 2001, pp. 272–284) recently considered the question of whether there can be a pseudorandom generator in NC⁰, that is, a pseudorandom generator that maps n‐bit strings to m‐bit strings such that every bit of the output depends on a constant number k of bits of the seed. They show that for k = 3, if m ≥ 4n + 1, there is a distinguisher; in fact, they show that in this case it is possible to break the generator with a linear test, that is, there is a subset of bits of the output whose XOR has a noticeable bias. They leave the question open for k ≥ 4. In fact, they ask whether every NC⁰ generator can be broken by a statistical test that simply XORs some bits of the input. Equivalently, is it the case that no NC⁰ generator can sample an ε‐biased space with negligible ε? We give a generator for k = 5 that maps n bits into cn bits, so that every bit of the output depends on 5 bits of the seed, and the XOR of every subset of the bits of the output has bias 2. For large values of k, we construct generators that map n bits to bits such that every XOR of outputs has bias . We also present a polynomial‐time distinguisher for k = 4,m ≥ 24n having constant distinguishing probability. For large values of k we show that a linear distinguisher with a constant distinguishing probability exists once m ≥ Ω(2^kn^?k/2?). Finally, we consider a variant of the problem where each of the output bits is a degree k polynomial in the inputs. We show there exists a degree k = 2 pseudorandom generator for which the XOR of every subset of the outputs has bias 2^?Ω(n) and which maps n bits to Ω(n²) bits. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006