Lipschitz extensions of definable p‐adic functions

In this paper, we prove a definable version of Kirszbraun's theorem in a non‐Archimedean setting for definable families of functions in one variable. More precisely, we prove that every definable function , where and , that is λ‐Lipschitz in the first variable, extends to a definable function that is λ‐Lipschitz in the first variable.     

All λ‐Designs with Small λ are Type‐1

A λ‐design is a family of subsets of such that for all and not all are of the same size. Ryser's and Woodall's λ‐design conjecture states that each λ‐design can be obtained from a symmetric block design by a certain complementation procedure. Our main result is that the conjecture is true when λ < 63. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 408–431, 2012     

Cell decomposition and definable functions for weak p‐adic structures

We develop a notion of cell decomposition suitable for studying weak p‐adic structures (reducts of p‐adic fields where addition and multiplication are not (everywhere) definable). As an example, we consider a structure with restricted addition.     

Group theory and p‐adic valued models of swarm behaviour

The swarm behaviour can be controlled by different localizations of attractants (food pieces) and repellents (dangerous places), which, respectively, attract and repel the swarm propagation. If we assume that at each time step, the swarm can find out not more than p ?1 attractants ( ), then the swarm behaviour can be coded by p ‐adic integers, ie, by the numbers of the ring Z _p . Each swarm propagation has the following 2 stages: (1) the discover of localizations of neighbour attractants and repellents and (2) the logistical optimization of the road system connecting all the reachable attractants and avoiding all the neighbour repellents. In the meanwhile, at the discovering stage, the swarm builds some direct roads and, at the logistical stage, the transporting network of the swarm gets loops (circles) and it permanently changes. So, at the first stage, the behaviour can be expressed by some linear p ‐adic valued strings. At the second stage, it is expressed by non‐linear modifications of p ‐adic valued strings. The second stage cannot be described by conventional algebraic tools; therefore, I have introduced the so‐called non‐linear group theory for describing both stages in the swarm propagation.     

Q5‐factorization of λKn

Q_k is the simple graph whose vertices are the k‐tuples with entries in {0, 1} and edges are the pairs of k‐tuples that differ in exactly one position. In this paper, we proved that there exists a Q₅‐factorization of λK_n if and only if (a) n ≡ 0(mod 32) if λ ≡ 0(mod 5) and (b) n ≡ 96(mod 160) if λ ? 0(mod 5).     

Flag‐Transitive Point‐Primitive Symmetric (ν,κ,λ) Designs With λ at Most 100

Let be a symmetric (ν,κ,λ) design with λ ≤ 100. If G is a flag‐transitive and point‐primitive automorphism group of , then G must be an affine or almost simple group.     

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.     

Approximation properties of λ‐Bernstein‐Kantorovich operators with shifted knots

In the present article, Kantorovich variant of λ‐Bernstein operators with shifted knots are introduced. The advantage of using shifted knot is that one can do approximation on [0,1] as well as on its subinterval. In addition, it adds flexibility to operators for approximation. Some basic results for approximation as well as rate of convergence of the introduced operators are established. The r^th order generalization of the operator is also discussed. Further for comparisons, some graphics and error estimation tables are presented using MATLAB.     

Exceptions and characterization results for type‐1 λ‐designs

Let $X$ be a finite set with $v$ elements, called points and $\beta$ be a family of subsets of $X$, called blocks. A pair $\left(X,\beta \right)$ is called $\lambda$‐design whenever $\mid \beta \mid =\mid X\mid$ and

• 1. for all $B_i,B_j\in \beta ,i\ne j,\mid B_i\cap B_j\mid =\lambda$;
• 2. for all $B_j\in \beta ,\mid B_j\mid =k_j>\lambda$, and not all $k_j$ are equal.

The only known examples of $\lambda$‐designs are so‐called type‐1 designs, which are obtained from symmetric designs by a certain complementation procedure. Ryser and Woodall had independently conjectured that all $\lambda$‐designs are type‐1. Let $r,r^{*}?(r>r^{*})$ be replication numbers of a $\lambda$‐design $D=\left(X,\beta \right)$ and $g=\text{gcd}\left(r?1,r^{*}?1\right),m=\text{gcd}\left(\left(r?r^{*}\right)∕g,\lambda \right)$, and $m\prime =m$, if $m$ is odd and $m\prime =m∕2$, otherwise. For distinct points $x$ and $y$ of $D$, let $\lambda \left(x,y\right)$ denote the number of blocks of $X$ containing $x$ and $y$. We strengthen a lemma of S.S. Shrikhande and N.M. Singhi and use it to prove that if $\frac{r\left(r?1\right)}{\left(v?1\right)}?\frac{k\left(r?r^{*}\right)}{m\prime \left(v?1\right)}$ are not integers for $k=1,2,\text{…},m\prime ?1$, then $D$ is type‐1. As an application of these results, we show that for fixed positive integer $\theta$ there are finitely many nontype‐1 $\lambda$‐designs with $r=r^{*}+\theta$. If $r?r^{*}=27$ or $r?r^{*}=4p$ and $r^{*}\ne {\left(p?1\right)}^2$, or $v=7p+1$ such that $p?1,13\left(\mathrm{mod}21\right)$ and $p?4,9,19,24\left(\mathrm{mod}35\right)$, where $p$ is a positive prime, then $D$ is type‐1. We further obtain several inequalities involving $\lambda (x,y)$, where equality holds if and only if D is type‐1.     

Some results on λ‐designs with two block sizes

A λ‐design is a family ?? = {B₁, B₂, …, B_v} of subsets of X = {1, 2, …, v} such that |B_i∩B_j| = λ for all i≠jand not all B_i are of the same size. The only known example of λ‐designs (called type‐1 designs) are those obtained from symmetric designs by a certain complementation procedure. Ryser [J Algebra 10 (1968), 246–261] and Woodall [Proc London Math Soc 20 (1970), 669–687] independently conjectured that all λ‐designs are type‐1. Let g = gcd(r ? 1, r^* ? 1), where rand r^* are the two replication numbers. Ionin and Shrikhande [J Combin Comput 22 (1996), 135–142; J Combin Theory Ser A 74 (1996), 100–114] showed that λ‐designs with g = 1, 2, 3, 4 are type‐1 and that the Ryser–Woodall conjecture is true for λ‐designs on p + 1, 2p + 1, 3p + 1, 4p + 1 points, where pis a prime. Hein and Ionin [Codes and Designs—Proceedings of Conference honoring Prof. D. K. Ray‐Chaudhuri on the occasion of his 65th birthday, Ohio State University Mathematical Research Institute Publications, 10, Walter de Gruyter, Berlin, 2002, pp. 145–156] proved corresponding results for g = 5 and Fiala [Codes and Designs—Proceedings of Conference honoring Prof. D. K. Ray‐Chaudhuri on the occasion of his 65th birthday, Ohio State University Mathematical Research Institute Publications, 10, Walter de Gruyter, Berlin, 2002, pp. 109–124; Ars Combin 68 (2003), 17–32; Ars Combin, to appear] for g = 6, 7, and 8. In this article, we consider λ designs with exactly two block sizes. We show that in this case, the conjecture is true for g = 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, and for g = 10, 14, 18, 22 with v≠4λ ? 1. We also give two results on such λ‐designs on v = 9p + 1 and 12p + 1 points, where pis a prime. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:95‐110, 2011     

Enclosings of λ‐Fold 5‐Cycle Systems: Adding One Vertex

A k‐cycle system of a multigraph G is an ordered pair (V, C) where V is the vertex set of G and C is a set of k‐cycles, the edges of which partition the edges of G. A k‐cycle system of is known as a λ‐fold k‐cycle system of order V. A k‐cycle system of (V, C) is said to be enclosed in a k‐cycle system of if and . We settle the difficult enclosing problem for λ‐fold 5‐cycle systems with u = 1.     

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)     

Linear extension operators for continuous functions on definable sets in the p‐adic context

Let E be a subset of . A linear extension operator is a linear map that sends a function on E to its extension on some superset of E . In this paper, we show that if E is a semi‐algebraic or subanalytic subset of , then there is a linear extension operator such that is semi‐algebraic (subanalytic) whenever f is semi‐algebraic (subanalytic).     

The λ-complete multigraphs

Let E₁……E_m be the edges of a hypergraph. Assume each element in the union of the edges occurs in at least two of E₁……E_m. Assume further that every pair of distinct edges, E₁, and E₁, intersect in at most one element and that for each such pair there are exactly λ other edges E_k such that E₁ and E₁ both intersect E_k. We characterize the hypergraphs for which each |E₁| ≤ 2. These are the λ-complete multigraphs.     

On certain character sums over p ‐adic rings and their L ‐functions

In this paper we present a new method for evaluating exponential sums associated to a restricted power series in one variable modulo p^l , a power of a prime. We show that for sufficiently large l, these sums can be expressed in terms of Gauss sums. Moreover, we study the associated L ‐functions; we show that they are rational, then we determine their degrees and the weights as Weil numbers of their reciprocal roots and poles. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sufficient conditions for λ′‐optimality in graphs with girth g

For a connected graph the restricted edge‐connectivity λ′(G) is defined as the minimum cardinality of an edge‐cut over all edge‐cuts S such that there are no isolated vertices in G–S. A graph G is said to be λ′‐optimal if λ′(G) = ξ(G), where ξ(G) is the minimum edge‐degree in G defined as ξ(G) = min{d(u) + d(v) ? 2:uv ∈ E(G)}, d(u) denoting the degree of a vertex u. A. Hellwig and L. Volkmann [Sufficient conditions for λ′‐optimality in graphs of diameter 2, Discrete Math 283 (2004), 113–120] gave a sufficient condition for λ′‐optimality in graphs of diameter 2. In this paper, we generalize this condition in graphs of diameter g ? 1, g being the girth of the graph, and show that a graph G with diameter at most g ? 2 is λ′‐optimal. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 73–86, 2006     

Peters type polynomials and numbers and their generating functions: Approach with p‐adic integral method

The aim of this paper is to introduce and investigate some of the primary generalizations and unifications of the Peters polynomials and numbers by means of convenient generating functions and p‐adic integrals method. Various fundamental properties of these polynomials and numbers involving some explicit series and integral representations in terms of the generalized Stirling numbers, generalized harmonic sums, and some well‐known special numbers and polynomials are presented. By using p‐adic integrals, we construct generating functions for Peters type polynomials and numbers (Apostol‐type Peters numbers and polynomials). By using these functions with their partial derivative eqautions and functional equations, we derive many properties, relations, explicit formulas, and identities including the Apostol‐Bernoulli polynomials, the Apostol‐Euler polynomials, the Boole polynomials, the Bernoulli polynomials, and numbers of the second kind, generalized harmonic sums. A brief revealing and historical information for the Peters type polynomials are given. Some of the formulas given in this article are given critiques and comments between previously well‐known formulas. Finally, two open problems for interpolation functions for Apostol‐type Peters numbers and polynomials are revealed.