A limit theorem for matrix-solutions of Hamiltonian systems

The following limit theorem on Hamiltonian systems (resp. corresponding Riccati matrix equations) is shown: Given(N, N)-matrices,A, B, C andn ∈ {1,…, N} with the following properties:A and kemelB(x) are constant, rank(I, A, …, A ^n?1) B(x)≠N,B(x)∈C _n(R), andB(x)(A ^T)^j-1 C(x)∈C _n-j(R) forj=1, …, n. Then \(\mathop {\lim }\limits_{x \to x_0 } \eta _1^T \left( x \right)V\left( x \right)U^{ - 1} \left( x \right)\eta _2 \left( x \right) = d_1^T \left( {x_0 } \right)U\left( {x_0 } \right)d_2 \) forx ₀∈R, whenever the matricesU(x), V(x) are a conjoined basis of the differential systemU′=AU + BV, V′=CU?A ^TV, and whenever η_i(x)∈R ^N satisfy η_i(x ₀)=U(x ₀)d _i ∈ imageU(x ₀) η′_i-Aη_ni(x) ∈ imageB(x),B(x)(η′_i(x)-Aη_i(x)) ∈C _n-1 R fori=1,2.     

Some results on starlike trees and sunlike graphs

A tree is called starlike if it has exactly one vertex of degree greater than two. In [4] it was proved that two starlike treesG andH are cospectral if and only if they are isomorphic. We prove here that there exist no two non-isomorphic Laplacian cospectral starlike trees. Further, letG be a simple graph of ordern with vertex setV(G)={1,2, …,n} and letH={H ₁,H ₂, ...H _n } be a family of rooted graphs. According to [2], the rooted productG(H) is the graph obtained by identifying the root ofH _i with thei-th vertex ofG. In particular, ifH is the family of the paths $P_{k_1 } , P_{k_2 } , ..., P_{k_n } $ with the rooted vertices of degree one, in this paper the corresponding graphG(H) is called the sunlike graph and is denoted byG(k ₁,k ₂, …,k _n ). For any (x ₁,x ₂, …,x _n ) ∈I _* ⁿ , whereI _*={0,1}, letG(x ₁,x ₂, …,x _n ) be the subgraph ofG which is obtained by deleting the verticesi ₁, i₂, …,i _j ∈ V(G) (0≤j≤n), provided that $x_{i_1 } = x_{i_2 } = ... = x_{i_j } = 0$ . LetG(x ₁,x ₂,…, x _n] be the characteristic polynomial ofG(x ₁,x ₂,…, x _n ), understanding thatG[0, 0, …, 0] ≡ 1. We prove that $$G[k_1 , k_2 ,..., k_n ] = \Sigma _{x \in ^{I_ * ^n } } \left[ {\Pi _{i = 1}^n P_{k_i + x_i - 2} (\lambda )} \right]( - 1)^{n - (\mathop \Sigma \limits_{i = 1}^n x_i )} G[x_1 , x_2 , ..., x_n ]$$ where x=(x ₁,x ₂,…,x _n );G[k ₁,k ₂,…,k _n ] andP _n (γ) denote the characteristic polynomial ofG(k ₁,k ₂,…,k _n ) andP _n , respectively. Besides, ifG is a graph with λ₁(G)≥1 we show that λ₁(G)≤λ₁(G(k ₁,k ₂, ...,k _n )) < for all positive integersk ₁,k ₂,…,k _n , where λ₁ denotes the largest eigenvalue.     

Group pursuit in Pontryagin’s nonstationary example

On the interval [t ₀, ∞), we consider the following group pursuit problem with one evader: 1 $$ z_i^{(l)} + a_1 (t)z_i^{(l - 1)} + a_2 (t)z_i^{(l - 2)} + \cdots + a_l (t)z_i = u_i - v, u_i ,v \in V, z_i^{(q)} (t_0 ) = z_i^q , $$ where z _i , u _i , v ∈ R ^v , (v ≥ 2), V is a strictly convex compact set in R ^v , the functions a ₁(t), a ₂(t), …, a _l (t) are continuous, i = 1, 2, …, n and q = 0, 1, …, l ? 1. Let ? _q (t, s) be the solution of the Cauchy problem $$ \begin{gathered} \omega ^{(l)} + a_1 (t)\omega ^{(l - 1)} + a_2 (t)\omega ^{(l - 2)} + \cdots + a_l (t)\omega = 0, \omega ^{(q)} (s) = 1, \hfill \\ \omega ^{(r)} (s) = 0, r = 0, \ldots q - 1,q + 1, \ldots ,l - 1, \hfill \\ \end{gathered} $$ and let $$ \xi _\iota (t) = \varphi _0 (t,t_0 )Z_i^0 + \varphi _1 (t,t_0 )Z_i^1 + \cdots + \varphi _{l - 1} (t,t_0 )Z_i^{l - 1} . $$ We prove that if there exist continuous functions α _i (t) and ξ _i ¹ (t) such that the ξ _i ¹ (t) are Bohr almost periodic on [t ₀, ∞), α _i (t) > 0 for all t ≥ t ₀, lim _t→∞(ξ _i ¹ (t) ? α _i (t)ξ _i (t)) = 0, lim _t→∞(min _i α _i (t) ∝ _t0 ^t |? _l?1(t, s)| ds) = ∞, and there exist points h _i ⁰ ∈ H _i ¹ = {ξ _i ¹ (t), t ∈ [0, ∞)} such that 0 ∈ Int co{h _i ⁰ }, then the pursuit problem with evader discrimination is solvable.     

Signed k-independence in graphs

Let k ≥ 2 be an integer. A function f: V(G) → {?1, 1} defined on the vertex set V(G) of a graph G is a signed k-independence function if the sum of its function values over any closed neighborhood is at most k ? 1. That is, Σ _x∈N[v] f(x) ≤ k ? 1 for every v ∈ V(G), where N[v] consists of v and every vertex adjacent to v. The weight of a signed k-independence function f is w(f) = Σ _v∈V(G) f(v). The maximum weight w(f), taken over all signed k-independence functions f on G, is the signed k-independence number α _s ^k (G) of G. In this work, we mainly present upper bounds on α _s ^k (G), as for example α _s ^k (G) ≤ n ? 2?(Δ(G) + 2 ? k)/2?, and we prove the Nordhaus-Gaddum type inequality $\alpha _S^k \left( G \right) + \alpha _S^k \left( {\bar G} \right) \leqslant n + 2k - 3$ , where n is the order, Δ(G) the maximum degree and $\bar G$ the complement of the graph G. Some of our results imply well-known bounds on the signed 2-independence number.     

Automorphisms of free groups and the mapping class groups of closed compact orientable surfaces

Let N be the stabilizer of the word w = s ₁ t ₁ s ₁ ^?1 t ₁ ^?1 … s _g t _g s _g ^?1 t _g ^?1 in the group of automorphisms Aut(F _2g ) of the free group with generators ?ub;s _i, t _i?ub; _i=1,…,g . The fundamental group π₁(Σ_g) of a two-dimensional compact orientable closed surface of genus g in generators ?ub;s _i, t _i?ub; is determined by the relation w = 1. In the present paper, we find elements S _i, T _i ∈ N determining the conjugation by the generators s _i, t _i in Aut(π₁(Σ_g)). Along with an element β ∈ N, realizing the conjugation by w, they generate the kernel of the natural epimorphism of the group N on the mapping class group M _g,0 = Aut(π₁(Σ_g))/Inn(π₁(Σ_g)). We find the system of defining relations for this kernel in the generators S ₁, …, S _g, T ₁, …, T _g, α. In addition, we have found a subgroup in N isomorphic to the braid group B _g on g strings, which, under the abelianizing of the free group F _2g , is mapped onto the subgroup of the Weyl group for Sp(2g, ?) consisting of matrices that contain only 0 and 1.     

On the equivalence of the Euler-Pommier operators in spaces of analytic functions

In the space A (θ) of all one-valued functions f(z) analytic in an arbitrary region G ? ? (0 ∈ G) with the topology of compact convergence, we establish necessary and sufficient conditions for the equivalence of the operators L ₁=α _n z ⁿ Δ ⁿ + ... + α₁ zΔ+α₀ E and L ₂= z ⁿ a _n (z)Δ ⁿ + ... + za ₁(z)Δ+a ₀(z)E, where δ: (Δ?)(z)=(f(z)-?(0))/z is the Pommier operator in A(G), n ∈ ?, α _n ∈ ?, a _k (z) ∈ A(G), 0≤k≤n, and the following condition is satisfied: Σ _j=s ^n?1 α _j+1 ∈ 0, s=0,1,...,n?1. We also prove that the operators z ^s+1Δ+β(z)E, β(z) ∈ A _R , s ∈ ?, and z ^s+1 are equivalent in the spaces A _R, 0?R?-∞, if and only if β(z) = 0.     

Solvability of a Multi-Point Boundary Value Problem at Resonance

Let f: [0, 1] × R² → R be a function satisfying Caratheodory’s conditions and e(t) ∈ L¹[0, 1]. Let η_i ∈ (0, 1), i = 1, …, k, with 0 s< η₁ < … < η_k < 1, be given. This paper is concerned with the problem of existence of a solution for the multi-point boundary value problem . Conditions for the existence of a solution for the above boundary value problem are given using Leray Schauder Continuation theorem.     

Arithmetic-progression-weighted subsequence sums

Let G be an abelian group, let s be a sequence of terms s ₁, s ₂, …, s _n ∈ G not all contained in a coset of a proper subgroup of G, and let W be a sequence of n consecutive integers. Let $$W \odot S = \left\{ {w_1 s_1 + \cdots + w_n s_n :w_i a term of W,w_i \ne w_j for i \ne j} \right\},$$ which is a particular kind of weighted restricted sumset. We show that |W ⊙ S| ≥ min{|G| ? 1, n}, that W ⊙ S = G if n ≥ |G| + 1, and also characterize all sequences S of length |G| with W ⊙ S ≠ G. This result then allows us to characterize when a linear equation $$a_1 x_1 + \cdots + a_r x_r \equiv \alpha mod n,$$ where α, a ₁, …, a _r ∈ ? are given, has a solution (x ₁, …, x _r ) ∈ ? ^r modulo n with all x _i distinct modulo n. As a second simple corollary, we also show that there are maximal length minimal zero-sum sequences over a rank 2 finite abelian group $G \cong C_{n_1 } \oplus C_{n_2 }$ (where n ₁ |n ₂ and n ₂ ≥ 3) having k distinct terms, for any k ε [3, min{n ₁ + 1, exp(G)}]. Indeed, apart from a few simple restrictions, any pattern of multiplicities is realizable for such a maximal length minimal zero-sum sequence.     

Minimal decompositions of graphs into mutually isomorphic subgraphs

SupposeG _n={G ₁, ...,G _k } is a collection of graphs, all havingn vertices ande edges. By aU-decomposition ofG _n we mean a set of partitions of the edge setsE(G _t ) of theG _i , sayE(G _t )== \(\sum\limits_{j = 1}^r {E_{ij} } \) E _ij , such that for eachj, all theE _ij , 1≦i≦k, are isomorphic as graphs. Define the functionU(G _n) to be the least possible value ofr aU-decomposition ofG _n can have. Finally, letU _k (n) denote the largest possible valueU(G) can assume whereG ranges over all sets ofk graphs havingn vertices and the same (unspecified) number of edges. In an earlier paper, the authors showed that $$U_2 (n) = \frac{2}{3}n + o(n).$$ In this paper, the value ofU _k (n) is investigated fork>2. It turns out rather unexpectedly that the leading term ofU _k (n) does not depend onk. In particular we show $$U_k (n) = \frac{3}{4}n + o_k (n),k \geqq 3.$$      

Finiteness of graded generalized local cohomology modules

We consider two finitely generated graded modules over a homogeneous Noetherian ring $R = \oplus _{n \in \mathbb{N}_0 } R_n$ with a local base ring (R ₀, m₀) and irrelevant ideal R ₊ of R. We study the generalized local cohomology modules H _b ⁱ (M,N) with respect to the ideal b = b₀ + R ₊, where b₀ is an ideal of R ₀. We prove that if dimR ₀/b₀ ≤ 1, then the following cases hold: for all i ≥ 0, the R-module H _b ⁱ (M,N)/a₀ H _b ⁱ (M,N) is Artinian, where $\sqrt {\mathfrak{a}_0 + \mathfrak{b}_0 } = \mathfrak{m}_0$ ; for all i ≥ 0, the set $Ass_{R_0 } \left( {H_\mathfrak{b}^i \left( {M,N} \right)_n } \right)$ is asymptotically stable as n→?∞. Moreover, if H _b ⁱ (M,N) _n is a finitely generated R ₀-module for all n ≤ n ₀ and all j < i, where n ₀ ∈ ? and i ∈ ?₀, then for all n ≤ n ₀, the set $Ass_{R_0 } \left( {H_\mathfrak{b}^i \left( {M,N} \right)_n } \right)$ is finite.     

2-Reducible Two Paths and an Edge Constructing a Path in (2k + 1)-Edge-Connected Graphs

Let k ≥ 5 be an odd integer and G = (V(G), E(G)) be a k-edge-connected graph. For ${X\subseteq V(G),e(X)}$ denotes the number of edges between X and V(G) ? X. We here prove that if ${\{s_i,t_i\}\subseteq X_i\subseteq V(G)(i=1,2),f}$ is an edge between s ₁ and ${s_2,X_1\cap X_2=\emptyset,e(X_1)\le 2k-3,e(X_2)\le 2k-2}$ , and e(Y) ≥ k + 1 for each ${Y\subseteq V(G)}$ with ${Y\cap\{s_1,t_1,s_2,t_2\}=\{s_1,t_2\}}$ , then there exist paths P ₁ and P ₂ such that P _i joins s _i and ${t_i,V(P_i)\subseteq X_i}$ (i = 1, 2) and ${G-f-E(P_1\cup P_2)}$ is (k ? 2)-edge-connected, and in fact we give a generalization of this result.     

Presentations of alternating semigroups

One presentation of the alternating groupA _n hasn?2 generatorss ₁,…,s_n?2 and relationss ₁ ³ =s _i ² =(s_1?1s_i)³=(s_js_k)²=1, wherei>1 and |j?k|>1. Against this backdrop, a presentation of the alternating semigroupA _n ^c )A _n is introduced: It hasn?1 generatorss ₁,…,S _n?2,e, theA _n-relations (above), and relationse ²=e, (es ₁)⁴, (es _j)²=(es _j)⁴,es _i=s _i s ₁ ^-1 es ₁, wherej>1 andi≥1.     

The height of minimal Hilbert bases

For an integral polyhedral cone C = pos{a¹,…,a^m, a ⁱ ∈ ?^d, a subset $\cal B$ (C) ? C ∩ ?^d is called a minimal Hilbert basis of C iff (i) each element of C∩?^d can be written as a non-negative integral combination of elements of $\cal B$ (C) and (ii) $\cal B$ (C) has minimal cardinality with respect to all subsets of C ∩ ?^d for which (i) holds. We give a tight bound for the so-called height of an element of the basis which improves on former results.     

Commuting and centralizing generalized derivations on lie ideals in prime rings

Let R be a noncommutative prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, and L a noncentral Lie ideal of R. If F and G are generalized derivations of R and k ≥1 a fixed integer such that [F(x), x] _k x ? x[G(x), x] _k = 0 for any x ∈ L, then one of the following holds:

1. either there exists an a ∈ U and an α ∈ C such that F(x) = xa and G(x) = (a + α)x for all x ∈ R
2. or R satisfies the standard identity s ₄(x ₁, …, x ₄) and one of the following conclusions occurs

1. there exist a, b, c, q ∈ U, such that a ?b + c ?q ∈ C and F(x) = ax + xb, G(x) = cx + xq for all x ∈ R
2. there exist a, b, c ∈ U and a derivation d of U such that F(x) = ax+d(x) andG(x) = bx+xc?d(x) for all x ∈ R, with a + b ? c ∈ C.

     

Resolvability in circulant graphs

A set W of the vertices of a connected graph G is called a resolving set for G if for every two distinct vertices u, v ∈ V (G) there is a vertex w ∈ W such that d(u, w) ≠ d(v, w). A resolving set of minimum cardinality is called a metric basis for G and the number of vertices in a metric basis is called the metric dimension of G, denoted by dim(G). For a vertex u of G and a subset S of V (G), the distance between u and S is the number min s∈S d(u, s). A k-partition Π = {S 1 , S 2 , . . . , S k } of V (G) is called a resolving partition if for every two distinct vertices u, v ∈ V (G) there is a set S i in Π such that d(u, Si )≠ d(v, Si ). The minimum k for which there is a resolving k-partition of V (G) is called the partition dimension of G, denoted by pd(G). The circulant graph is a graph with vertex set Zn , an additive group of integers modulo n, and two vertices labeled i and j adjacent if and only if i-j (mod n) ∈ C , where CZn has the property that C =-C and 0 ■ C. The circulant graph is denoted by Xn, Δ where Δ = |C|. In this paper, we study the metric dimension of a family of circulant graphs Xn, 3 with connection set C = {1, n/2 , n-1} and prove that dim(Xn, 3 ) is independent of choice of n by showing that dim(Xn, 3 ) ={3 for all n ≡ 0 (mod 4), 4 for all n ≡ 2 (mod 4). We also study the partition dimension of a family of circulant graphs Xn,4 with connection set C = {±1, ±2} and prove that pd(Xn, 4 ) is independent of choice of n and show that pd(X5,4 ) = 5 and pd(Xn,4 ) ={3 for all odd n ≥ 9, 4 for all even n ≥ 6 and n = 7.     

On oscillation of solutions of a nonautonomous quasilinear second-order equation

Sufficient conditions are obtained for the initial values of nontrivial oscillating (for t=ω) solutions of the nonautonomous quasilinear equation $$y'' \pm \lambda (t)y = F(t,y,y'),$$ wheret ∈ Δ=[a, ω[,-∞ <a < ω ≤+ ∞, λ(t) > 0, λ(t) ∈ C _Δ ⁽¹⁾ , |F((t,x,y))|≤L(t)(|x|+|y|)^1+α, L(t) ≥-0, α ∈ [0,+∞[, F: Δ × R² →R,F ∈C _Δ×R ²,R is the set of real numbers, and R² is the two-dimensional real Euclidean space.     

Some Bistar Bipartite Ramsey Numbers

For bipartite graphs G ₁, G ₂, . . . ,G _k , the bipartite Ramsey number b(G ₁, G ₂, . . . , G _k ) is the least positive integer b so that any colouring of the edges of K _b,b with k colours will result in a copy of G _i in the ith colour for some i. A tree of diameter three is called a bistar, and will be denoted by B(s, t), where s ≥ 2 and t ≥ 2 are the degrees of the two support vertices. In this paper we will obtain some exact values for b(B(s, t), B(s, t)) and b(B(s, s), B(s, s)). Furtermore, we will show that if k colours are used, with k ≥ 2 and s ≥ 2, then \({b_{k}(B(s, s)) \leq \lceil k(s - 1) + \sqrt{(s - 1)^{2}(k^{2} - k) - k(2s - 4)} \rceil}\) . Finally, we show that for s ≥ 3 and k ≥ 2, the Ramsey number \({r_{k}(B(s, s)) \leq \lceil 2k(s - 1)+ \frac{1}{2} + \frac{1}{2} \sqrt{(4k(s - 1) + 1)^{2} - 8k(2s^{2} - s - 2)} \rceil}\) .     

Leavitt Path Algebras of Edge-Colored Graphs

Given any edge-colored graph G and any commutative unital ring R, we construct a generalized Leavitt path algebra L _R (G).We show that L _R (G) is a certain free product of L _R (G _i ), where G _i s are 1-colored subgraphs of G. We also show that L _R (G) may be written as a free product of simpler algebras. In the end, we define a natural ${\mathbb{Z}}$ -grading for L _R (G) and give four necessary conditions for simplicity of L _R (G).     

A Functional Equation Of J.A. Baker Involving Integral Means

The functional equation $$f(x)={1\over 2}\int^{x+1}_{x-1}f(t)\ dt\ \ \ {\rm for}\ \ \ x\ \in\ {\rm R}$$ has the linear functions ?(x) = a + bx (a, b ∈ ?) as trivial solutions. It is shown that there are two kinds of nontrivial solutions, (i) ?(x) = eλ_i ^x (i = 1, 2, …), where the λ_i∈ ? are the fixed points of the map z ? sinh z, and (ii) C^∞-solutions ? for which the values in the interval [?1,1] can be prescribed arbitrarily, but with the provision that ?^(j)(? 1) = ?^(j)(0) = ?^(j)(1) = 0 for all j = 0, 1, 2 …     

The maximal Fejér operator on the spacesH p (R×···×R)

It is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from thed-dimensional Hardy spaceH _p (R×···×R) toL _p (R ^d ) (1/2<p<∞) and is of weak type (H ₁ ^?i ,L ₁) (i=1,…,d), where the Hardy spaceH ₁ ^?i is defined by a hybrid maximal function. As a consequence, we obtain that the Fejér means of a functionf ∈H ₁ ^?i ?L(logL) ^d?1 converge a.e. to the function in question. Moreover, we prove that the Fejér means are uniformly bounded onH _p (R×···×R) whenever 1/2<p<∞. Thus, in casef ∈H _p (R×···×R) the Fejér means converge tof inH _p (R×···×R) norm. The same results are proved for the conjugate Fejér means, too.