An anisotropic, superconvergent nonconforming plate finite element

The classical finite element convergence analysis relies on the following regularity condition: there exists a constant c independent of the element K and the mesh such that h_K/ρ_Kc, where h_K and ρ_K are diameters of K and the biggest ball contained in K, respectively. In this paper, we construct a new, nonconforming rectangular plate element by the double set parameter method. We prove the convergence of this element without the above regularity condition. The key in our proof is to obtain the O(h²) consistency error. We also prove the superconvergence of this element for narrow rectangular meshes. Results of our numerical tests agree well with our analysis.     

On Rough ℐ2-Convergence of Double Sequences

In this study, we introduce the notion of rough ?₂-convergence and the set of rough ?₂-limit points of a double sequence and obtained two rough ?₂-convergence criteria associated with this set. Later, we proved that this set is closed and convex. Finally, we examined the relationships between the set of cluster points and the set of rough ?₂-limit points of a double sequence.     

Absolute Summability Factors and Absolute Tauberian Theorems for Double Series and Sequences

Let A and B be the linear methods of the summability of double series with fields of bounded summability MA _b ^' and B _b ^' , respectively. Let T be certain set of double series. The condition x T is called B _b-Tauberian for A if A _b ^' B _b ^' .Some theorems about summability factors enable one to find new B _b-Tauberian conditions for A from the already known B _b-Tauberian conditions for A.     

On Trees with Double Domination Number Equal to the 2-Outer-Independent Domination Number Plus One

A vertex of a graph is said to dominate itself and all of its neighbors.A double dominating set of a graph G is a set D of vertices of G,such that every vertex of G is dominated by at least two vertices of D.The double domination number of a graph G is the minimum cardinality of a double dominating set of G.For a graph G =(V,E),a subset D V(G) is a 2-dominating set if every vertex of V(G) \ D has at least two neighbors in D,while it is a 2-outer-independent dominating set of G if additionally the set V(G)\D is independent.The 2-outer-independent domination number of G is the minimum cardinality of a 2-outer-independent dominating set of G.This paper characterizes all trees with the double domination number equal to the 2-outer-independent domination number plus one.     

Coding into Ramsey sets

In [6] W. T. Gowers formulated and proved a Ramsey-type result which lies at the heart of his famous dichotomy for Banach spaces. He defines the notion of weakly Ramsey set of block sequences of an infinite dimensional Banach space and shows that every analytic set of block sequences is weakly Ramsey. We show here that Gowers’ result follows quite directly from the fact that all G_δ sets are weakly Ramsey, if the Banach space does not contain c₀, and from the fact that all F_σδ sets are weakly Ramsey, in the case of an arbitrary Banach space. We also show that every result obtained by the application of Gowers’ theorem to an analytic set can also be obtained by applying the Theorem to a F_σδ set (or to a G_δ set if the space does not contain c₀). This fact explains why the only known applications of this technique are based on very low-ranked Borel sets (open, closed, F_σ, or G_δ).     

A Note on the Double Affine Hecke Algebra of Type GL2

We express the double affine Hecke algebra ? associated to the general linear group GL₂(k) (here, k is a field with char(k) ≠ 2) as an amalgamated free product of quadratic extensions over the three-dimensional quantum torus 𝒪_q((k^×)³). With an eye towards proving ring-theoretic results pertaining to ?, a general treatment of amalgamated products of Ore and quadratic extensions is given. We prove an analogue of the Hilbert Basis Theorem for an amalgamated product Q of quadratic extensions and determine conditions for when the one-sided ideals of Q are principal or doubly-generated. Furthermore, we determine sufficient conditions which imply Q is a principal ideal ring. Finally, we construct an explicit isomorphism from ? to the amalgamated free product ring of quadratic extensions over 𝒪_q((k^×)³), a ring known to be noetherian. Therefore, it follows that ? is noetherian.     

Rewarding Maps: On Greedy Optimization of Set Functions

Given a set function, that is, a map ƒ: (E) → {−∞} from the set (E) of subsets of a finite set E into the reals including −∞, the standard greedy algorithm (GA) for optimizing ƒ starts with the empty set and then proceeds by enlarging this set greedily, element by element. A set function ƒ is said to be tractable if in this way a sequence x₀ , x₁, . . ., x_N E (N #E) of subsets with max(ƒ) {ƒ(x₀), ƒ(x₁), . . ., ƒ(x_N)} will always be found. In this note, we will reinterpret and transcend the traditions of classical GA-theory (cf., e.g., [KLS]) by establishing necessary and sufficient conditions for a set function ƒ not just to be tractable as it stands, but to give rise to a whole family of tractable set functions ƒ(η) : (E) → : x ƒ(x) + Σ_{e x}η(e), where η runs through all real valued weighting schemes η : E → , in which case ƒ will be called rewarding. In addition, we will characterize two important subclasses of rewarding maps, viz. truncatably rewarding (or well-layered) maps, that is, set functions ƒ such that [formula] is rewarding for every i = 1, . . ., N, and matroidal maps, that is, set functions ƒ such that for every η : E → and every ƒ_eta-greedy sequence x₀, x₁, . . ., x_N as above, one has max(ƒ_η) = ƒ_η(x_i) for the unique i {0, . . ., N} with ƒ_η(x₀) < ƒ_η(x₁) < ··· < ƒ_η(x_i) ≥ ƒ_η(x_{i + 1}).     

On Compact Graphs

Let G be a finite simple graph with adjacency matrix A, and let P（A） be the convex closure of the set of all permutation matrices commuting with A. G is said to be compact if every doubly stochastic matrix which commutes with A is in P（A）. In this paper, we characterize 3-regular compact graphs and prove that if G is a connected regular compact graph, G - v is also compact, and give a family of almost regular compact connected graphs.     

Borel ideals vs. Borel sets of countable relations and trees

For every μ < ω₁, let I_μ be the ideal of all sets S ω^μ whose order type is <ω^μ. If μ = 1, then I₁ is simply the ideal of all finite subsets of ω, which is known to be Σ⁰₂-complete. We show that for every μ < ω₁, I_μ is Σ⁰_2μ-complete. As corollaries to this theorem, we prove that the set WO_ωμ of well orderings Rω × ω of order type <ω^μ is Σ⁰_2μ-complete, the set LP_μ of linear orderings R ω × ω that have a μ-limit point is Σ⁰_2μ+1-complete. Similarly, we determine the exact complexity of the set LT_μ of trees T ^<ωω of Luzin height <μ, the set WR_μ of well-founded partial orderings of height <μ, the set LR_μ of partial orderings of Luzin height <μ, the set WF_μ of well-founded trees T ^<ωω of height <μ(the latter is an old theorem of Luzin). The proofs use the notions of Wadge reducibility and Wadge games. We also present a short proof to a theorem of Luzin and Garland about the relation between the height of ‘the shortest tree’ representing a Borel set and the complexity of the set.     

Indivisible plexes in latin squares

In this paper, we present two constructions of divisible difference sets based on skew Hadamard difference sets. A special class of Hadamard difference sets, which can be derived from a skew Hadamard difference set and a Paley type regular partial difference set respectively in two groups of orders v ₁ and v ₂ with |v ₁ − v ₂| = 2, is contained in these constructions. Some result on inequivalence of skew Hadamard difference sets is also given in the paper. As a consequence of Delsarte’s theorem, the dual set of skew Hadamard difference set is also a skew Hadamard difference set in an abelian group. We show that there are seven pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 3⁵ or 3⁷, and also at least four pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 3⁹. Furthermore, the skew Hadamard difference sets deduced by Ree-Tits slice symplectic spreads are the dual sets of each other when q ≤ 3¹¹.      

Double Ore Extensions Versus Iterated Ore Extensions

Motivated by the construction of new examples of Artin–Schelter regular algebras of global dimension four, Zhang and Zhang [6 Zhang , J. J. , Zhang , J. ( 2008 ). Double Ore extensions . J. Pure Appl. Algebra 212 ( 12 ): 2668 – 2690 .[Crossref], [Web of Science ®] , [Google Scholar]] introduced an algebra extension A _P [y ₁, y ₂; σ, δ, τ] of A, which they called a double Ore extension. This construction seems to be similar to that of a two-step iterated Ore extension over A. The aim of this article is to describe those double Ore extensions which can be presented as iterated Ore extensions of the form A[y ₁; σ₁, δ₁][y ₂; σ₂, δ₂]. We also give partial answers to some questions posed in Zhang and Zhang [6 Zhang , J. J. , Zhang , J. ( 2008 ). Double Ore extensions . J. Pure Appl. Algebra 212 ( 12 ): 2668 – 2690 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Scagnostics Distributions

Scagnostics is a Tukey neologism for the term scatterplot diagnostics. Scagnostics are characterizations of the 2D distributions of orthogonal pairwise projections of a set of points in multidimensional Euclidean space. These characterizations include such measures as density, skewness, shape, outliers, and texture. We introduce a set of scagnostics measures based on graph theory and we analyze their distributions and performance. Our analysis is based on a restrictive set of criteria that must be met in order to have scagnostics measures that can be used effectively in exploratory data analysis.     

The t-invariant of analytic set germs of dimension 2

Let be an analytic set germ of dimension 2. We study the invariant t(X₀) defined as the least integer t such that any open semianalytic set germ of X₀ can be written as a union of t basic open set germs. It is known that 2≤t(X₀)≤3. In this note we provide a geometric criterion to determine the exact value of t(X₀).     

Domination number in graphs with minimum degree two

A set D of vertices of a graph G = （V, E） is called a dominating set if every vertex of V not in D is adjacent to a vertex of D. In 1996, Reed proved that every graph of order n with minimum degree at least 3 has a dominating set of cardinality at most 3n/8. In this paper we generalize Reed＇s result. We show that every graph G of order n with minimum degree at least 2 has a dominating set of cardinality at most （3n ＋IV21）/8, where V2 denotes the set of vertices of degree 2 in G. As an application of the above result, we show that for k ≥ 1, the k-restricted domination number rk （G, γ） ≤ （3n＋5k）/8 for all graphs of order n with minimum degree at least 3.     

Maps on spaces of symmetric matrices preserving idempotence

Suppose F is an arbitrary field. Let |F| be the number of the elements of F. Let M_n(F) be the space of all n×n matrices over F, and let S_n(F) be the subset of M_n(F) consisting of all symmetric matrices. Let V{S_n(F),M_n(F)}, a map Φ:V→V is said to preserve idempotence if A-λB is idempotent if and only if Φ(A)-λΦ(B) is idempotent for any A,BV and λF. It is shown that: when the characteristic of F is not 2, |F|>3 and n4, Φ:S_n(F)→S_n(F) is a map preserving idempotence if and only if there exists an invertible matrix PM_n(F) such that Φ(A)=PAP^-1 for every AS_n(F) and P^tP=aI_n for some nonzero scalar a in F.     

Borel extractions of converging sequences in compact sets of Borel functions

It is well known by a classical result of Bourgain–Fremlin–Talagrand that if K is a pointwise compact set of Borel functions on a Polish space then given any cluster point f of a sequence (f_n)_nω in K one can extract a subsequence (f_nk)_kω converging to f. In the present work we prove that this extraction can be achieved in a “Borel way.” This will prove in particular that the notion of analytic subspace of a separable Rosenthal compacta is absolute and does not depend on the particular choice of a dense sequence.     

Maximizing the distance between center,centroid and characteristic set of a tree

Consider a tree P_n-g,g , n≥ 2, 1≤ g≤ n-1 on n vertices which is obtained from a path on [1,2,?…?,n-g] vertices by adding g pendant vertices to the pendant vertex n-g. We prove that over all trees on n?≥?5 vertices, the distance between center and characteristic set, centroid and characteristic set, and center and centroid is maximized by trees of the form P_n-g,g , 2?≤?g?≤?n-3. For n≥ 5, we also supply the precise location of the characteristic set of the tree P_n-g,g , 2?≤?g?≤?n-3.     

Decomposing Graphs into Long Paths

It is known that the edge set of a 2-edge-connected 3-regular graph can be decomposed into paths of length 3. W. Li asked whether the edge set of every 2-edge-connected graph can be decomposed into paths of length at least 3. The graphs C ₃, C ₄, C ₅, and K ₄−e have no such decompositions. We construct an infinite sequence {F _i } _i=0 ^∞ of nondecomposable graphs. On the other hand, we prove that every other 2-edge-connected graph has a desired decomposition. This revised version was published online in June 2006 with corrections to the Cover Date.     

Convergence of a Halpern-type iteration algorithm for a class of pseudo-contractive mappings

Let E be a real reflexive Banach space with uniformly Gâteaux differentiable norm. Let K be a nonempty bounded closed and convex subset of E. Let T:K→K be a strictly pseudo-contractive map and let L>0 denote its Lipschitz constant. Assume F(T){xK:Tx=x}≠0/ and let zF(T). Fix δ(0,1) and let δ^* be such that δ^*δL(0,1). Define , where δ_n(0,1) and limδ_n=0. Let {α_n} be a real sequence in (0,1) which satisfies the following conditions: . For arbitrary x₀,uK, define a sequence {x_n}K by x_n+1=α_nu+(1−α_n)S_nx_n. Then, {x_n} converges strongly to a fixed point of T.