The proof on the conjecture of extremal graphs for the kth eigenvalues of trees

We consider the only remaining unsolved case n≡0 (mod k) for the largest kth eigenvalue λ_k.of trees with n vertices. In this paper, the conjecture for this problem in [Shao Jia-yu, On the largest kth eignevalues of trees, Linear Algebra Appl. 221 (1995) 131] is proved and (from this) the complete solution to this problem, the best upper bound and the extremal trees of λ_k, is given in general cases above.     

A family of orthonormal wavelet bases with dilation factor 4

In this paper, we study a method for the construction of orthonormal wavelet bases with dilation factor 4. More precisely, for any integer M>0, we construct an orthonormal scaling filter mM(ξ) that generates a mother scaling function ?M, associated with the dilation factor 4. The computation of the different coefficients of ²|mM(ξ)| is done by the use of a simple iterative method. Also, this work shows how this construction method provides us with a whole family of compactly supported orthonormal wavelet bases with arbitrary high regularity. A first estimate of α(M), the asymptotic regularity of ?M is given by α(M)∼0.25M. Examples are provided to illustrate the results of this work.     

Lattice tensor products. II

G. Grätzer and F. Wehrung has recently introduced the lattice tensor product, A?B, of the lattices A and B. In this note, for a finite lattice A and an arbitrary lattice B, we compute the ideal lattice of A?B, obtaining the isomorphism Id(A?B)≌A?Id B. This generalizes an earlier result of G. Grätzer and F. Wehrung proving this isomorphism for A = M_3 and B n-modular. We prove this isomorphism by utilizing the coordinatization of A?B introduced in Part I of this paper.     

Two-point Taylor expansions in the asymptotic approximation of double integrals. Application to the second and fourth Appell functions

The main difficulty in Laplace's method of asymptotic expansions of double integrals is originated by a change of variables. We consider a double integral representation of the second Appell function F₂(a,b,b^′,c,c^′;x,y) and illustrate, over this example, a variant of Laplace's method which avoids that change of variables and simplifies the computations. Essentially, the method only requires a Taylor expansion of the integrand at the critical point of the phase function. We obtain in this way an asymptotic expansion of F₂(a,b,b^′,c,c^′;x,y) for large b, b^′, c and c^′. We also consider a double integral representation of the fourth Appell function F₄(a,b,c,d;x,y). We show, in this example, that this variant of Laplace's method is uniform when two or more critical points coalesce or a critical point approaches the boundary of the integration domain. We obtain in this way an asymptotic approximation of F₄(a,b,c,d;x,y) for large values of a,b,c and d. In this second example, the method requires a Taylor expansion of the integrand at two points simultaneously. For this purpose, we also investigate in this paper Taylor expansions of two-variable analytic functions with respect to two points, giving Cauchy-type formulas for the coefficients of the expansion and details about the regions of convergence.     

On the semiring of cone preserving maps

If K is a proper cone in Rⁿ, then the cone of all linear operators that preserve K, denoted by π(K), forms a semiring under usual operator addition and multiplication. Recently J.G. Horne examined the ideals of this semiring. He proved that if K₁, K₂ are polyhedral cones such that π(K₁) and π(K₂) are isomorphic as semirings, then K₁ and K₂ are linearly isomorphic. The study of this semiring is continued in this paper. In Sec. 3 ideals of π(K) which are also faces are characterized. In Sec. 4 it is shown that π(K) has a unique minimal two-sided ideal, namely, the dual cone of π(K¹), where K¹ is the dual cone of K. Extending Horne's result, it is also proved that the cone K is characterized by this unique minimal two-sided ideal of π(K). The set of all faces of π(K) inherits a quotient semiring structure from π(K). Properties of this face-semiring are given in Sec. 5. In particular, it is proved that this face-semiring admits no nontrivial congruence relation iff the duality operator of π(K) is injective. In Sec. 6 the maximal one-sided and two-sided ideals of π(K) are identified. In Sec. 8 it is shown that π(K) never satisfies the ascending-chain condition on principal one-sided ideals. Some partial results on the question of topological closedness of principal one-sided ideals of π(K) are also given.     

Approximating earliest arrival flows with flow-dependent transit times

For the earliest arrival flow problem one is given a network G=(V,A) with capacities u(a) and transit times τ(a) on its arcs a∈A, together with a source and a sink vertex s,t∈V. The objective is to send flow from s to t that moves through the network over time, such that for each time θ∈[0,T) the maximum possible amount of flow up to this time reaches t. If, for each θ∈[0,T), this flow is a maximum flow for time horizon θ, then it is called earliest arrival flow. In practical applications a higher congestion of an arc in the network often implies a considerable increase in transit time. Therefore, in this paper we study the earliest arrival problem for the case that the transit time of each arc in the network at each time θ depends on the flow on this particular arc at that time θ.For constant transit times it has been shown by Gale that earliest arrival flows exist for any network. We give examples, showing that this is no longer true for flow-dependent transit times. For that reason we define a relaxed version of this problem where the objective is to find flows that are almost earliest arrival flows. In particular, we are interested in flows that, for each θ∈[0,T), need only α-times longer to send the maximum flow to the sink. We give both constant lower and upper bounds on α; furthermore, we present a constant factor approximation algorithm for this problem.     

An optimal filtering method for stable analytic continuation

In this paper, we consider the problem of numerical analytic continuation of an analytic function f(z)=f(x+iy) on a strip domain Ω⁺={z=x+iy∈C∣x∈R,0<y<y₀}, where the data is given approximately only on the real axis y=0. This problem is severely ill-posed: the solution does not depend continuously on the given data. A novel method (filtering) is used to solve this problem and an optimal error estimate with Hölder type is proved. Numerical examples show that this method works effectively.     

The uniqueness of 1-systems of W 5(q) satisfying the BLT-property, with q odd

In the symplectic polar space W ₅(q) every 1-system which satisfies the BLT-property (and then q is odd) defines a generalized quadrangle (GQ) of order (q ²,q ³). In this paper, we show that this 1-system is unique, so that the only GQ arising in this way is isomorphic to the classical GQ H(4,q ²), q odd.     

Un semi-groupe régularisé pour un modèle de Lebowitz–RubinowA regularized semigroup for a Lebowitz–Rubinow's model

In this Note, we complete [1] and we study the Lebowitz–Rubinow's model with the biological law of perfect memory. In this model, each cell is characterized by its cell cycle length l (0?l₁<l<l₂<∞) and its age a (0<a<l). If l₁>0, a complete study of this model can be found in [1]. Here we show that if l₁=0 then this model becomes ill-posed. We use the theory of generalized semigroups to remedy to this model. To cite this article: M. Boulanouar, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 865–868.     

Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition II

Lee, Kim and Suh (2012) gave a characterization for real hypersurfaces M of Type (A) in complex two plane Grassmannians G ₂(? ^m+2) with a commuting condition between the shape operator A and the structure tensors φ and φ ₁ for M in G ₂(? ^m+2). Motivated by this geometrical notion, in this paper we consider a new commuting condition in relation to the shape operator A and a new operator φφ ₁ induced by two structure tensors φ and φ ₁. That is, this commuting shape operator is given by φφ ₁ A = A φφ ₁. Using this condition, we prove that M is locally congruent to a tube of radius r over a totally geodesic G ₂(? ^m+1) in G ₂(? ^m+2).     

A note on Vizing's independence number conjecture of edge chromatic critical graphs

In 1968, Vizing conjectured that, if G is a Δ-critical graph with n vertices, then α(G)?n/2, where α(G) is the independence number of G. In this note, we verify this conjecture for n?2Δ.     

The endomorphism ring of a Δ-module over a right noetherian ring

LetR be a right noetherian ring. A moduleM _R is called a Δ-module providedR satisfies the descending chain condition for annihilators of subsets ofM. For a Δ-module, a series 0?M ₁?M ₂?...?M _n =M can be constructed in which the factorsM _i /M _i?1 are sums of, α _i -semicritical modules where α₁≦α₂≦...≦α _n . In this paper we utilize this series in studying Λ=End(M _R ). It is shown that ifN={f∈Λ|Kerf is essential inM}, thenN is nilpotent. Specific bounds on the index of nilpotency are given in terms of this series. Further ifM is injective and α-smooth, the annihilators of the factors of this series are used to provide necessary and sufficient conditions for EndM _R to be semisimple.     

A logarithm type mean value theorem of the Riemann zeta function

For any integer K?2 and positive integer h, we investigate the mean value of |ζ(σ+it)|^2k×logh|ζ(σ+it)| for all real number 0<k<K and all σ>1−1/K. In case K=2, h=1, this has been studied by Wang in [F.T. Wang, A mean value theorem of the Riemann zeta function, Quart. J. Math. Oxford Ser. 18 (1947) 1-3]. In this note, we give a new brief proof of Wang's theorem, and, with this method, generalize it to the general case naturally.     

An oblique projection iterative method to compute Drazin inverse and group inverse

In this paper, an oblique projection iterative method is presented to compute matrix equation AXA=A of a square matrix A with ind(A)=1. By this iterative method, when taken the initial matrix X₀=A, the group inverse A_g can be obtained in absence of the roundoff errors. If we use this iterative method to the matrix equation A^kXA^k=A^k, a group inverse (A^k)_g of matrix A^k is got, then we use the formulae A_d=A^k-1(A^k)_g, the Drazin inverse A_d can be obtained.     

An efficient linearization technique for mixed 0-1 polynomial problem

This paper addresses a new and efficient linearization technique to solve mixed 0-1 polynomial problems to achieve a global optimal solution. Given a mixed 0-1 polynomial term z=c_tx₁x₂…x_ny, where x₁,x₂,…,x_n are binary (0-1) variables and y is a continuous variable. Also, c_t can be either a positive or a negative parameter. We transform z into a set of auxiliary constraints which are linear and can be solved by exact methods such as branch and bound algorithms. For this purpose, we will introduce a method in which the number of additional constraints is decreased significantly rather than the previous methods proposed in the literature. As is known in any operations research problem decreasing the number of constraints leads to decreasing the mathematical computations, extensively. Thus, research on the reducing number of constraints in mathematical problems in complicated situations have high priority for decision makers. In this method, each n-auxiliary constraints proposed in the last method in the literature for the linearization problem will be replaced by only 3 novel constraints. In other words, previous methods were dependent on the number of 0-1 variables and therefore, one auxiliary constraint was considered per 0-1 variable, but this method is completely independent of the number of 0-1 variables and this illustrates the high performance of this method in computation considerations. The analysis of this method illustrates the efficiency of the proposed algorithm.     

On a functional equation concerning affine transformations

In this paper we prove that, if the functional equation u(Rx + p) = α(R, p) u(x) + β(R, p) has a nonconstant solution u, then α and β have to be of a very special form. Finally, we obtain, under suitable hypotheses, the general solution of this equation.     

Pseudo-linear superposition principle for the Monge-Ampère equation based on generated pseudo-operations

Through this paper, we consider generated pseudo-operations of the following form: x⊕y=g⁻¹(g(x)+g(y)), x⊙y=g⁻¹(g(x)g(y)), where g is a continuous generating function. Pseudo-linear superposition principle, i.e., the superposition principle with this type of pseudo-operations in the core, for the Monge-Ampère equation is investigated.     

On the Lipschitz constant of the RSK correspondence

We view the RSK correspondence as associating to each permutation π∈Sn a Young diagram λ=λ(π), i.e. a partition of n. Suppose now that π is left-multiplied by t transpositions, what is the largest number of cells in λ that can change as a result? It is natural refer to this question as the search for the Lipschitz constant of the RSK correspondence.We show upper bounds on this Lipschitz constant as a function of t. For t=1, we give a construction of permutations that achieve this bound exactly. For larger t we construct permutations which come close to matching the upper bound that we prove.     

Weights of multipartitions and representations of Ariki-Koike algebras

We consider representations of the Ariki-Koike algebra, a q-deformation of the group algebra of the complex reflection group C_r?S_n. The representations of this algebra are naturally indexed by multipartitions of n, and for each multipartition λ we define a non-negative integer called the weight of λ. We prove some basic properties of this weight function, and examine blocks of small weight.     

Connected matroids with a small circumference

Lemos and Oxley proved that if M is a connected matroid with |E(M)|⩾3r(M), then M has a circuit C such that M⧹C is connected. In this paper, we shall improve this result proving that for a simple and connected matroid M, if r(M)⩾7 and |E(M)|⩾3r(M)−3, then M has a circuit C such that M⧹C is connected. To prove this result, we shall construct all the connected matroids having circumference at most five, with the exception of those which are 3-connected and have rank five.