Strongly indexable graphs and applications

In 1990, Acharya and Hegde introduced the concept of strongly k-indexable graphs: A (p,q)-graph G=(V,E) is said to be strongly k-indexable if its vertices can be assigned distinct numbers 0,1,2,…,p−1 so that the values of the edges, obtained as the sums of the numbers assigned to their end vertices form an arithmetic progression k,k+1,k+2,…,k+(q−1). When k=1, a strongly k-indexable graph is simply called a strongly indexable graph. In this paper, we report some results on strongly k-indexable graphs and give an application of strongly k-indexable graphs to plane geometry, viz; construction of polygons of same internal angles and sides of distinct lengths.     

Accelerating gradient projection methods for -constrained signal recovery by steplength selection rules

We propose a new gradient projection algorithm that compares favorably with the fastest algorithms available to date for ℓ₁-constrained sparse recovery from noisy data, both in the compressed sensing and inverse problem frameworks. The method exploits a line-search along the feasible direction and an adaptive steplength selection based on recent strategies for the alternation of the well-known Barzilai–Borwein rules. The convergence of the proposed approach is discussed and a computational study on both well conditioned and ill-conditioned problems is carried out for performance evaluations in comparison with five other algorithms proposed in the literature.     

Maximum fractional factors in graphs

We prove that fractional k-factors can be transformed among themselves by using a new adjusting operation repeatedly. We introduce, analogous to Berge’s augmenting path method in matching theory, the technique of increasing walk and derive a characterization of maximum fractional k-factors in graphs. As applications of this characterization, several results about connected fractional 1-factors are obtained.     

Partitioning a graph of bounded tree-width to connected subgraphs of almost uniform size

Assume that each vertex of a graph G is assigned a nonnegative integer weight and that l and u are nonnegative integers. One wishes to partition G into connected components by deleting edges from G so that the total weight of each component is at least l and at most u. Such an “almost uniform” partition is called an (l,u)-partition. We deal with three problems to find an (l,u)-partition of a given graph; the minimum partition problem is to find an (l,u)-partition with the minimum number of components; the maximum partition problem is defined analogously; and the p-partition problem is to find an (l,u)-partition with a fixed number p of components. All these problems are NP-complete or NP-hard, respectively, even for series-parallel graphs. In this paper we show that both the minimum partition problem and the maximum partition problem can be solved in time O(u⁴n) and the p-partition problem can be solved in time O(p²u⁴n) for any series-parallel graph with n vertices. The algorithms can be extended for partial k-trees, that is, graphs with bounded tree-width.     

Strong convergence theorems by hybrid method for asymptotically -strict pseudo-contractive mapping in Hilbert space

In this paper, we introduce the hybrid method of modified Mann’s iteration for an asymptotically k-strict pseudo-contractive mapping. Then we prove that such a sequence converges strongly to P_F(T)x₀. This main theorem improves the result of Issara Inchan [I. Inchan, Strong convergence theorems of modified Mann iteration methods for asymptotically nonexpansive mappings in Hilbert spaces, Int. J. Math. Anal. 2 (23) (2008) 1135–1145] and concerns the result of Takahashi et al. [W. Takahashi, Y. Takeuchi, R. Kubota, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert space, J. Math. Anal. Appl. 341 (2008) 276–286], and many others.     

A semi-discretization method based on quartic splines for solving one-space-dimensional hyperbolic equations

In this paper, based on C³ quartic splines, a semi-discretization method containing two schemes is constructed to solve one-space-dimensional linear hyperbolic equations. It is shown that both schemes are unconditionally stable and their approximation orders are of O(k⁵+h⁴) and of O(k⁷+h⁴) with k and h being step sizes in time and space, respectively, which are much higher than those of other published schemes. A numerical example is presented and the results are compared with other published numerical results.     

CFD study of section characteristics of Formula Mazda race car wings

A great deal of research has been done on the aerodynamic characteristics of race cars competing in major racing series through out the world. Because of the competitive nature of motor sport, this research is usually not published until after it is obsolete. The teams operating at the minor league levels of the sport do not have the funding resources of the major series to perform aerodynamic research. In an effort to provide some information for teams competing in the minor league Formula Mazda racecar class, this study was conducted using the Star-CD CFD code to perform a turbulent simulation (using a k–ε model) of the airflow on the front and rear wings of a Formula Mazda car with different angles of attack and the effect of the ground on the front wing. Results are presented graphically, showing pressure and velocity distributions and lift (C_l) and drag coefficients (C_d) for the different cases. It was shown that the ground effect has a marked effect on the C_l and that the angle of attack has a significant effect on the lift and drag coefficients, and it was shown that an angle of 12 below the horizontal seems to indicate stalling conditions. It is suggested that this information, along with experimental validation, can be valuable for improving the optimum handling of these Formula Mazda race cars.     

The convergence of three spline methods for scattered data interpolation and fitting

The convergences of three L₁ spline methods for scattered data interpolation and fitting using bivariate spline spaces are studied in this paper. That is, L₁ interpolatory splines, splines of least absolute deviation, and L₁ smoothing splines are shown to converge to the given data function under some conditions and hence, the surfaces from these three methods will resemble the given data values.     

A filtration of -Catalan numbers

The operator of F. Bergeron, Garsia, Haiman and Tesler [F. Bergeron, A. Garsia, M. Haiman, G. Tesler, Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions, Methods Appl. Anal. 6 (1999) 363–420] acting on the k-Schur functions [L. Lapointe, A. Lascoux, J. Morse, Tableaux atoms and a new Macdonald positivity conjecture, Duke Math. J. 116 (2003) 103–146; L. Lapointe, J. Morse, Schur functions analogs for a filtration of the symmetric functions space, J. Combin. Theory Ser. A 101 (2003) 191–224; L. Lapointe, J. Morse, Tableaux on k+1-cores, reduced words for affine permutations and k-Schur expansion, J. Combin. Theory Ser. A 112 (2005) 44–81] indexed by a single column has a coefficient in the expansion which is an analogue of the (q,t)-Catalan number with a level k. When k divides n we conjecture a representation theoretical model in this case such that the graded dimensions of the module are the coefficients of the (q,t)-Catalan polynomials of level k. When the parameter t is set to 1, the Catalan numbers of level k are shown to count the number of Dyck paths that lie below a certain Dyck path with q counting the area of the path.     

Stability for parametric implicit vector equilibrium problems

In this paper, we consider a class of parametric implicit vector equilibrium problems in Hausdorff topological vector spaces where a mapping f and a set K are perturbed by parameters and λ, respectively. We establish sufficient conditions for the upper semicontinuity and lower semicontinuity of the solution set mapping S:Λ₁×Λ₂→2^X for such parametric implicit vector equilibrium problems.     

Uniformly Gâteaux smooth approximations on

Every Lipschitz mapping from c₀(Γ) into a Banach space Y can be uniformly approximated by Lipschitz mappings that are simultaneously uniformly Gâteaux smooth and C^∞-Fréchet smooth.     

Minimum number of below average triangles in a weighted complete graph

Let G be an edge weighted graph with n nodes, and let A(3,G) be the average weight of a triangle in G. We show that the number of triangles with weight at most equal to A(3,G) is at least (n−2) and that this bound is sharp for all n≥7. Extensions of this result to cliques of cardinality k>3 are also discussed.     

Minimum degree conditions for -linked graphs

For a fixed multigraph H with vertices w₁,…,w_m, a graph G is H-linked if for every choice of vertices v₁,…,v_m in G, there exists a subdivision of H in G such that v_i is the branch vertex representing w_i (for all i). This generalizes the notions of k-linked, k-connected, and k-ordered graphs.Given a connected multigraph H with k edges and minimum degree at least two and n7.5k, we determine the least integer d such that every n-vertex simple graph with minimum degree at least d is H-linked. This value D(H,n) appears to equal the least integer d^′ such that every n-vertex graph with minimum degree at least d^′ is b(H)-connected, where b(H) is the maximum number of edges in a bipartite subgraph of H.     

The behavior of best -approximations as . A counterexample of convergence

In the space of summable sequences we give an example of a one-dimensional affine subspace C such that the best L_p-approximations of 0 from C fail to converge as p↓1. We thus give an answer to this problem of convergence in infinite measure spaces.     

On reducibility of n-ary quasigroups

An n-ary operation Q:Σⁿ→Σ is called an n-ary quasigroup of order |Σ| if in the relation x₀=Q(x₁,…,x_n) knowledge of any n elements of x₀,…,x_n uniquely specifies the remaining one. Q is permutably reducible if Q(x₁,…,x_n)=P(R(x_σ(1),…,x_σ(k)),x_σ(k+1),…,x_σ(n)) where P and R are (n-k+1)-ary and k-ary quasigroups, σ is a permutation, and 1<k<n. An m-ary quasigroup S is called a retract of Q if it can be obtained from Q or one of its inverses by fixing n-m>0 arguments. We prove that if the maximum arity of a permutably irreducible retract of an n-ary quasigroup Q belongs to {3,…,n-3}, then Q is permutably reducible.     

Highly connected coloured subgraphs via the regularity lemma

For integers , n≥k and r≥s, let m(n,r,s,k) be the largest (in order) k-connected component with at most s colours one can find in any r-colouring of the edges of the complete graph K_n on n vertices. Bollobás asked for the determination of m(n,r,s,k).Here, bounds are obtained in the cases s=1,2 and k=o(n), which extend results of Liu, Morris and Prince. Our techniques use Szemerédi’s Regularity Lemma for many colours.We shall also study a similar question for bipartite graphs.     

Strong Haagerup inequalities with operator coefficients

We prove a Strong Haagerup inequality with operator coefficients. If for an integer d, denotes the subspace of the von Neumann algebra of a free group F_I spanned by the words of length d in the generators (but not their inverses), then we provide in this paper an explicit upper bound on the norm on , which improves and generalizes previous results by Kemp–Speicher (in the scalar case) and Buchholz and Parcet–Pisier (in the non-holomorphic setting). Namely the norm of an element of the form ∑_{i=(i1,…,id)}a_iλ(g_i1g_id) is less than , where M₀,…,M_d are d+1 different block-matrices naturally constructed from the family (a_i)_iI^d for each decomposition of I^dI^l×I^d−l with l=0,…,d. It is also proved that the same inequality holds for the norms in the associated non-commutative L_p spaces when p is an even integer, pd and when the generators of the free group are more generally replaced by *-free -diagonal operators. In particular it applies to the case of free circular operators. We also get inequalities for the non-holomorphic case, with a rate of growth of order d+1 as for the classical Haagerup inequality. The proof is of combinatorial nature and is based on the definition and study of a symmetrization process for partitions.     

The topological structure of fuzzy sets with endograph metric

For a non-degenerate convex subset Y of the n-dimensional Euclidean space Rⁿ, let be the family of all fuzzy sets ofRⁿ, which are upper-semicontinuous, fuzzy convex and normal with compact supports contained in Y. We show that the space with the topology of endograph metric is homeomorphic to the Hilbert cube Q=[-1,1]^ω iff Y is compact; and the space is homeomorphic to {(x_n)Q:sup|x_n|<1} iff Y is non-compact and locally compact.     

Existence of periodic solutions for a fourth-order -Laplacian equation with a deviating argument

In this paper, we study the existence of periodic solutions for a fourth-order p-Laplacian differential equation with a deviating argument as follows:

[φ_p(u^″(t))]^″+f(u^″(t))+g(u(t−τ(t)))=e(t).