On initial value problems for quaternionic valued functions

In [2], [6], [7], methods are discussed for solving initial value problems

A class of homogenization problems in the calculus of variations

We study a class of integral functionals for which the integrand f^e(x, u, ?u) is an oscillatory function of both x and u. Our method is based on the concept of Γ-convergenee. Technical difficulties arise because f^e(x, u, ?u) is not convex or equi-continuous in u with respect to e. Two somewhat different approaches, based respectively on abstract convergence theorems and the study of affine functions, are exploited together to overcome these technical difficulties. As an application, we give another proof of a homogenization result of P. L. Lions, G. Papanicolaou, and S. R. S. Varadhan for Hamilton-Jacobi equations.     

Bäcklund autotransformation for the equationu xt=eu−e−2u

The system of differential relations that arises in connection with the Bullough-Dodd-Zhiber-Shabat equationu _xt=e^u–e^–2u is considered. The consistency of this system is established, and it is shown that the system realizes a Bäcklund autotransformation for the equationu _xt=e^u–e^–2u. The associated three-dimensional dynamical systems, which are compatible on a two-dimensional invariant submanifold, are investigated, and a construction of their general solution, which gives the explicit form of the three-parameter soliton for the equationu _xt=e^u–e^–2u, is proposed.Bashkir State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 1, pp. 146–159, April, 1993.     

Regularized inner products of distorted plane waves and near-forward inverse scattering

Regularized inner products, r(s, e₁, e₂,) of the distorted plane waves fór the plasma wave equation u_u ? Δu + qu = 0 are introduced for potentials with compact support. The regularized inner product may be represented in terms of the far-field pattern. The use of the WKB approximation shows that the behaviour of r as e₂ → ? e₁ is closely related to the Radon transform of q. This observation indicates the possibility of finding the Radon transform of q in terms of r(s, e₁, e₂) and then of recovering q by means of the inverse Radon transform.     

On regularity criterion to the 3D axisymmetric incompressible MHD equations

This paper is concerned with the regularity criterion for a class of axisymmetric solutions to 3D incompressible magnetohydrodynamic equations. More precisely, for the solutions that have the form of u = u_re_r+u_θe_θ+u_ze_z and b = b_θe_θ, we prove that if |ru(x,t)|≤C holds for ?1≤t < 0, then (u,b) is regular at time zero. This result can be thought as a generalization of recent results in for the 3D incompressible Navier‐Stokes equations. Copyright © 2016 John Wiley & Sons, Ltd.     

New edge neighborhood graphs

Let G be an undirected simple connected graph, and e = uv be an edge of G. Let N _G(e) be the subgraph of G induced by the set of all vertices of G which are not incident to e but are adjacent to u or v. Let N _e be the class of all graphs H such that, for some graph G,N _G (e) H for every edge e of G. Zelinka [3] studied edge neighborhood graphs and obtained some special graphs in N _e. Balasubramanian and Alsardary [1] obtained some other graphs in N _e. In this paper we given some new graphs in N _e.     

On the structure of linear graphs

Denote byG(n; m) a graph ofn vertices andm edges. We prove that everyG(n; [n ²/4]+1) contains a circuit ofl edges for every 3 ≦l<c ₂ n, also that everyG(n; [n ²/4]+1) contains ak _e(u _n, u_n) withu _n=[c ₁ logn] (for the definition ofk _e(u _n, u_n) see the introduction). Finally fort>t ₀ everyG(n; [tn ^3/2]) contains a circuit of 2l edges for 2≦l<c ₃ t ². This work was done while the author received support from the National Science Foundation, N.S.F. G.88.     

Maximum matchings in sparse random graphs: Karp–Sipser revisited

We study the average performance of a simple greedy algorithm for finding a matching in a sparse random graph G_{n, c/n}, where c>0 is constant. The algorithm was first proposed by Karp and Sipser [Proceedings of the Twenty-Second Annual IEEE Symposium on Foundations of Computing, 1981, pp. 364–375]. We give significantly improved estimates of the errors made by the algorithm. For the subcritical case where c<e we show that the algorithm finds a maximum matching with high probability. If c>e then with high probability the algorithm produces a matching which is within n^1/5+o(1) of maximum size. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12, 111–177, 1998     

On edge star sets in trees

Let A be a Hermitian matrix whose graph is G (i.e. there is an edge between the vertices i and j in G if and only if the (i,j) entry of A is non-zero). Let λ be an eigenvalue of A with multiplicity m_A(λ). An edge e=ij is said to be Parter (resp., neutral, downer) for λ,A if m_A(λ)−m_A−e(λ) is negative (resp., 0, positive ), where A−e is the matrix resulting from making the (i,j) and (j,i) entries of A zero. For a tree T with adjacency matrix A a subset S of the edge set of G is called an edge star set for an eigenvalue λ of A, if |S|=m_A(λ) and A−S has no eigenvalue λ. In this paper the existence of downer edges and edge star sets for non-zero eigenvalues of the adjacency matrix of a tree is proved. We prove that neutral edges always exist for eigenvalues of multiplicity more than 1. It is also proved that an edge e=uv is a downer edge for λ,A if and only if u and v are both downer vertices for λ,A; and e=uv is a neutral edge if u and v are neutral vertices. Among other results, it is shown that any edge star set for each eigenvalue of a tree is a matching.     

Characterization of generalized convex functions by their best approximation in sign-monotone norms

Let {u₀, u₁,… u_{n − 1}} and {u₀, u₁,…, u_n} be Tchebycheff-systems of continuous functions on [a, b] and let f ε C[a, b] be generalized convex with respect to {u₀, u₁,…, u_{n − 1}}. In a series of papers ([1], [2], [3]) D. Amir and Z. Ziegler discuss some properties of elements of best approximation to f from the linear spans of {u₀, u₁,…, u_{n − 1}} and {u₀, u₁,…, u_n} in the L_p-norms, 1 p ∞, and show (under different conditions for different values of p) that these properties, when valid for all subintervals of [a, b], can characterize generalized convex functions. Their methods of proof rely on characterizations of elements of best approximation in the L_p-norms, specific for each value of p. This work extends the above results to approximation in a wider class of norms, called “sign-monotone,” [6], which can be defined by the property: ¦ f(x)¦ ¦ g(x)¦,f(x)g(x) 0, a x b, imply f g . For sign-monotone norms in general, there is neither uniqueness of an element of best approximation, nor theorems characterizing it. Nevertheless, it is possible to derive many common properties of best approximants to generalized convex functions in these norms, by means of the necessary condition proved in [6]. For {u₀, u₁,…, u_n} an Extended-Complete Tchebycheff-system and f ε C⁽ⁿ⁾[a, b] it is shown that the validity of any of these properties on all subintervals of [a, b], implies that f is generalized convex. In the special case of f monotone with respect to a positive function u₀(x), a converse theorem is proved under less restrictive assumptions.     

Homogenization of monotone operators in divergence form with <Emphasis Type="Italic">x</Emphasis>-dependent multivalued graphs

In a previous paper [4], we proved the existence of solutions to −div a(x, grad u) = f , together with appropriate boundary conditions, whenever a(x, e) belongs, for every fixed x, to a certain class of maximal monotone graphs in e. Here, we derive the corresponding homogenization result, letting a(x, e) depend upon a parameter ε, and imposing adequate ε-uniform boundedness and coercivity properties. The resulting homogenized graphs belong to the same class of maximal monotone graphs. Our results do not assume any kind of periodicity.      

The blow-up rate for a system of heat equations with neumann boundary conditions

This paper deals with the blow-up properties of solutions to a system of heat equations u _t=Δu, v _t=Δv in B _R×(0, T) with the Neumann boundary conditions εu/εη=e ^v, εv/εη=e ^u on S _R×[0, T). The exact blow-up rates are established. It is also proved that the blow-up will occur only on the boundary. This work is supported by the National Natural Science Foundation of China     

Liouville theorem for harmonic maps with potential

Let M, N be complete manifolds, u:M→N be a harmonic map with potential H, namely, a critical point of the functional , where e(u) is the energy density of u. We will give a Liouville theorem for u with a class of potentials H's. Received: Received: 10 July 1997     

Existence of unbiased estimate of regression parameters in simple linear EV models

It is well known that for one-dimensional normal EV regression model X = x u,Y =α βx e, where x, u, e are mutually independent normal variables and Eu=Ee=0, the regression parameters a and βare not identifiable without some restriction imposed on the parameters. This paper discusses the problem of existence of unbiased estimate for a and βunder some restrictions commonly used in practice. It is proved that the unbiased estimate does not exist under many such restrictions. We also point out one important case in which the unbiased estimates of a and βexist, and the form of the MVUE of a and βare also given.     

Total Dual Integrality of Matching Forest Constraints

Let G=(V, E, A) be a mixed graph. That is, (V, E) is an undirected graph and (V, A) is a directed graph. A matching forest (introduced by R. Giles) is a subset F of EèAE\cup A such that F contains no circuit (in the underlying undirected graph) and such that for each v ? Vv\in V there is at most one e ? Fe\in F such that v is head of e. (For an undirected edge e, both ends of e are called head of e.) Giles gave a polynomial-time algorithm to find a maximum-weight matching forest, yielding as a by-product a characterization of the inequalities determining the convex hull of the incidence vectors of the matching forests. We prove that these inequalities form a totally dual integral system. It is equivalent to an ``all-integer' min-max relation for the maximum weight of a matching forest. Our proof is based on an exchange property for matching forests, and implies Giles' characterization.     

On the rainbow matching conjecture for 3-uniform hypergraphs

Aharoni and Howard and, independently, Huang et al. (2012) proposed the following rainbow version of the Erdős matching conjecture: For positive integers n, k and m with n ⩾ km, if each of the families \(F_{1},\ldots,F_{m}\subseteq\left(\begin{array}{c}[n]\\ k\end{array}\right)\) has size more than \(\max\{\left(\begin{array}{c}n\\ k\end{array}\right)-\left(\begin{array}{c}n-m+1\\ k\end{array}\right),\left(\begin{array}{c}km-1\\ k\end{array}\right)\}\), then there exist pairwise disjoint subsets e₁,…,e_m such that e_i ∈ F_i for all i ∈ [m]. We prove that there exists an absolute constant n₀ such that this rainbow version holds for k = 3 and n ⩾ n₀. We convert this rainbow matching problem to a matching problem on a special hypergraph H. We then combine several existing techniques on matchings in uniform hypergraphs: Find an absorbing matching M in H; use a randomization process of Alon et al. (2012) to find an almost regular subgraph of H − V(M); find an almost perfect matching in H − V(M). To complete the process, we also need to prove a new result on matchings in 3-uniform hypergraphs, which can be viewed as a stability version of a result of Łuczak and Mieczkowska (2014) and might be of independent interest.

     

Doyen–Wilson theorem for perfect hexagon triple systems

The graph consisting of the three 3-cycles (or triples) (a,b,c), (c,d,e), and (e,f,a), where a,b,c,d,e and f are distinct is called a hexagon triple. The 3-cycle (a,c,e) is called an inside 3-cycle; and the 3-cycles (a,b,c), (c,d,e), and (e,f,a) are called outside 3-cycles. A hexagon triple system of order v is a pair (X,C), where C is a collection of edge disjoint hexagon triples which partitions the edge set of 3K_v. Note that the outside 3-cycles form a 3-fold triple system. If the hexagon triple system has the additional property that the collection of inside 3-cycles (a,c,e) is a Steiner triple system it is said to be perfect. In 2004, Küçükçifçi and Lindner had shown that there is a perfect hexagon triple system of order v if and only if and v≥7. In this paper, we investigate the existence of a perfect hexagon triple system with a given subsystem. We show that there exists a perfect hexagon triple system of order v with a perfect sub-hexagon triple system of order u if and only if v≥2u+1, and u≥7, which is a perfect hexagon triple system analogue of the Doyen–Wilson theorem.     

Nondeterminism of Linear Operators and Lower Entropy Estimates

Let u be a (bounded, linear) operator from a Hilbert space ℋ into the Banach space C(T), the space of continuous functions on the compact metric space T. We introduce and investigate numbers τ _n (u), n≥1, measuring the degree of determinism of the operator u. The slower τ _n (u) decreases, the less determined are functions in the range of u by their values on a certain set of points. It is shown that n ^−1/2 τ _n (u)≤2e _n (u), where e _n (u) are the (dyadic) entropy numbers of u. Furthermore, we transform the notion of strong local nondeterminism from the language of stochastic processes into that of linear operators. This property, together with a lower entropy estimate for the compact space T, leads to a lower estimate for τ _n (u), hence also for e _n (u). These results are used to prove sharp lower entropy estimates for some integral operators, among them, Riemann–Liouville operators with values in C(T) for some fractal set T. Some multi-dimensional extensions are treated as well.      

Computation of Topological Indices of Some Graphs

Let G=(V,E) be a simple connected graph with vertex set V and edge set E. The Wiener index of G is defined by W(G)=∑_{x,y}⊆V d(x,y), where d(x,y) is the length of the shortest path from x to y. The Szeged index of G is defined by Sz(G)=∑ _e=uv∈E n _u (e|G)n _v (e|G), where n _u (e|G) (resp. n _v (e|G)) is the number of vertices of G closer to u (resp. v) than v (resp. u). The Padmakar–Ivan index of G is defined by PI(G)=∑ _e=uv∈E [n _eu (e|G)+n _ev (e|G)], where n _eu (e|G) (resp. n _ev (e|G)) is the number of edges of G closer to u (resp. v) than v (resp. u). In this paper we find the above indices for various graphs using the group of automorphisms of G. This is an efficient method of finding these indices especially when the automorphism group of G has a few orbits on V or E. We also find the Wiener indices of a few graphs which frequently arise in mathematical chemistry using inductive methods.     

Identification of the Time Derivative Coefficient in a Fast Diffusion Degenerate Equation

In this paper, we deal with the identification of the space variable time derivative coefficient u in a degenerate fast diffusion differential inclusion. The function u is vanishing on a subset strictly included in the space domain Ω. This problem is approached as a control problem (P) with the control u. An approximating control problem (P _ε ) is introduced and the existence of an optimal pair is proved. Under certain assumptions on the initial data, the control is found in W ^2,m (Ω), with m>N, in an implicit variational form. Next, it is shown that a sequence of optimal pairs (u_e^*,y_e^*)(u_{\varepsilon }^{\ast },y_{\varepsilon }^{\ast }) of (P _ε ) converges as ε goes to 0 to a pair (u ^*,y ^*) which realizes the minimum in (P), and y ^* is the solution to the original state system.