On Osgood theorem in Banach spaces

Let X be a real Banach space, ω : [0, +∞) → ? be an increasing continuous function such that ω(0) = 0 and ω(t + s) ≤ ω(t) + ω(s) for all t, s ∈ [0, +∞). According to the infinite dimensional analog of the Osgood theorem if ∫¹₀ (ω(t))^?1 dt = ∞, then for any (t₀, x₀) ∈ ?×X and any continuous map f : ?×X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all t ∈ ?, x, y ∈ X, the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has a unique solution in a neighborhood of t₀. We prove that if X has a complemented subspace with an unconditional Schauder basis and ∫¹₀ (ω(t))^?1 dt < ∞ then there exists a continuous map f : ? × X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all (t, x, y) ∈ ? × X × X and the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has no solutions in any interval of the real line.     

Cleavages of graphs: the spectral radius

A cleavage of a finite graph G is a morphism f : H → G of graphs such that if P is the m × n characteristic matrix defined as P _ik = 1 if i ∈ f ^?1(k), otherwise = 0, then A(H)P ≤ PA(G), where A(G) and A(H) are the adjacency matrices of G and H, respectively. This concept generalizes induced subgraphs, quotients of graphs, Galois covers, path-tree graphs and others. We show that for spectral radii we have the inequality ρ(H) ≤ ρ(G). Equality holds only in case f : H → G is an equivariant quotient and H has isoperimetric constant i(H) = 0.     

Constrained Ramsey numbers of graphs

Given two graphs G and H, let f(G,H) denote the minimum integer n such that in every coloring of the edges of K_n, there is either a copy of G with all edges having the same color or a copy of H with all edges having different colors. We show that f(G,H) is finite iff G is a star or H is acyclic. If S and T are trees with s and t edges, respectively, we show that 1+s(t?2)/2≤f(S,T)≤(s?1)(t²+3t). Using constructions from design theory, we establish the exact values, lying near (s?1)(t?1), for f(S,T) when S and T are certain paths or star‐like trees. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 1–16, 2003     

Odd‐angulated graphs and cancelling factors in box products

Under what conditions is it true that if there is a graph homomorphism G □ H → G □ T, then there is a graph homomorphism H→ T? Let G be a connected graph of odd girth 2k + 1. We say that G is (2k + 1)‐angulated if every two vertices of G are joined by a path each of whose edges lies on some (2k + 1)‐cycle. We call G strongly (2k + 1)‐angulated if every two vertices are connected by a sequence of (2k + 1)‐cycles with consecutive cycles sharing at least one edge. We prove that if G is strongly (2k + 1)‐angulated, H is any graph, S, T are graphs with odd girth at least 2k + 1, and ?: G□ H→S□T is a graph homomorphism, then either ? maps G□{h} to S□{t_h} for all h∈V(H) where t_h∈V(T) depends on h; or ? maps G□{h} to {s_h}□ T for all h∈V(H) where s_h∈V(S) depends on h. This theorem allows us to prove several sufficient conditions for a cancelation law of a graph homomorphism between two box products with a common factor. We conclude the article with some open questions. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:221‐238, 2008     

The minimum degree of Ramsey‐minimal graphs

We write H → G if every 2‐coloring of the edges of graph H contains a monochromatic copy of graph G. A graph H is G‐minimal if H → G, but for every proper subgraph H′ of H, H′ ? G. We define s(G) to be the minimum s such that there exists a G‐minimal graph with a vertex of degree s. We prove that s(K_k) = (k ? 1)² and s(K_a,b) = 2 min(a,b) ? 1. We also pose several related open problems. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 167–177, 2007     

A stability result for abstract evolution problems

Consider an abstract evolution problem in a Hilbert space H (1) where A(t) is a linear, closed, densely defined operator in H with domain independent of t ≥ 0 and G(t,u) is a nonlinear operator such that ‖G(t,u)‖a(t) ‖u‖^p, p = const > 1, ‖f(t)‖ ≤ b(t). We allow the spectrum of A(t) to be in the right half‐plane Re(λ) < λ₀(t), λ₀(t) > 0, but assume that lim_{t → ∞}λ₀(t) = 0. Under suitable assumptions on a(t) and b(t), the boundedness of ‖u(t)‖ as t → ∞ is proved. If f(t) = 0, the Lyapunov stability of the zero solution to problem (1) with u₀ = 0 is established. For f ≠ 0, sufficient conditions for the Lyapunov stability are given. The novel point in our study of the stability of the solutions to abstract evolution equations is the possibility for the linear operator A(t) to have spectrum in the half‐plane Re(λ) < λ₀(t) with λ₀(t) > 0 and lim_{t → ∞}λ₀(t) = 0 at a suitable rate. The new technique, proposed in the paper, is based on an application of a novel nonlinear differential inequality. Copyright © 2012 John Wiley & Sons, Ltd.     

Asymptotic Behaviour of C0–Semigroups with Bounded Local Resolvents

Let {T(t)}_t≥0 be a C₀–semigroup on a Banach space X with generator A, and let H^∞_T be the space of all x ∈ X such that the local resolvent λ ↦ R(λ, A)x has a bounded holomorphic extension to the right half–plane. For the class of integrable functions ϕ on [0, ∞) whose Fourier transforms are integrable, we construct a functional calculus ϕ ↦ T_ϕ, as operators on H^∞_T. Weshow that each orbit T(·)T_ϕx is bounded and uniformly continuous, and T(t)T_ϕx → 0 weakly as t → ∞, and we give a new proof that ∥T(t)R(μ, A)x∥ = O(t). We also show that ∥T(t)T_ϕx∥ → 0 when T is sun –reflexive, and that ∥T(t)R(μ, A)x∥ = O(ln t) when T is a positive semigroup on a normal ordered space X and x is a positive vector in H^∞_T.     

Roman and Total Domination

Abstract

A set S of vertices is a total dominating set of a graph G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set is the total domination number ^γt(G). A Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u with f (u)=0 is adjacent to at least one vertex v of G for which f (v)=2. The minimum of f (V (G))=∑_{u ∈ V (G)} f (u) over all such functions is called the Roman domination number γR (G). We show that γt(G) ≤ γR (G) with equality if and only if ^γt(G)=2γ(G), where γ(G) is the domination number of G. Moreover, we characterize the extremal graphs for some graph families.     

On the Logistic Equation in the Complex Plane

The famous logistic differential equation is studied in the complex plane. The method used is based on a functional analytic technique which provides a unique solution of the ordinary differential equation (ODE) under consideration in H ₂(𝔻) or H ₁(𝔻) and gives rise to an equivalent difference equation for which a unique solution is established in ?₂ or ?₁. For the derivation of the solution of the logistic differential equation this discrete equivalent equation is used. The obtained solution is analytic in {z ∈ ?: |z| <T}, T > 0. Numerical experiments were also performed using the classical 4th order Runge–Kutta method. The obtained results were compared for real solutions as well as for solutions of the form y(t) = u(t) + iv(t), t ∈ ?. For t ∈ ? the solution derived by the present method, seems to have singularities, that is, points where it ceases to be analytic, in certain sectors of the complex plane. These sectors, depending on the values of the involved parameters, can move at different directions, join forming common sectors, or pass through each other and continue moving independently. Moreover, the real and imaginary part of the solution seem to exhibit oscillatory behavior near these sectors.     

On the measurability of functions with quasi-continuous and upper semi-continuous vertical sections

Let f: ?² → ? be a function with upper semicontinuous and quasi-continuous vertical sections f _x (t) = f(x, t), t, x ∈ ?. It is proved that if the horizontal sections f ^y (t) = f(t, y), y, t ∈ ?, are of Baire class α (resp. Lebesgue measurable) [resp. with the Baire property] then f is of Baire class α + 2 (resp. Lebesgue measurable and sup-measurable) [resp. has Baire property].     

Constrained Ramsey numbers for rainbow matching

The Ramsey number R_k(G) of a graph G is the minimum number N, such that any edge coloring of K_N with k colors contains a monochromatic copy of G. The constrained Ramsey number f(G, T) of the graphs G and T is the minimum number N, such that any edge coloring of K_N with any number of colors contains a monochromatic copy of G or a rainbow copy of T. We show that these two quantities are closely related when T is a matching. Namely, for almost all graphs G, f(G, tK₂) = R_{t ? 1}(G) for t≥2. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:91‐95, 2011     

On the Propagation of Regularity and Decay of Solutions to the k-Generalized Korteweg-de Vries Equation

We study special regularity and decay properties of solutions to the IVP associated to the k-generalized KdV equations. In particular, for datum u ₀ ∈ H ^3/4+ (?) whose restriction belongs to H ^l ((b, ∞)) for some l ∈ ?⁺ and b ∈ ? we prove that the restriction of the corresponding solution u(·, t) belongs to H ^l ((β, ∞)) for any β ∈ ? and any t ∈ (0, T). Thus, this type of regularity propagates with infinite speed to its left as time evolves.     

Fourier type 2 operators with respect to locally compact abelian groups

A linear and bounded operator T between Banach spaces X and Y has Fourier type 2 with respect to a locally compact abelian group G if there exists a constant c > 0 such that∥T ∥₂ ≤ c∥f∥₂ holds for all X‐valued functions f ∈ L^X₂(G) where is the Fourier transform of f. We show that any Fourier type 2 operator with respect to the classical groups has Fourier type 2 with respect to any locally compact abelian group. This generalizes previous special results for the Cantor group and for closed subgroups of ?ⁿ. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A remark on Schrödinger operators

We study the almost everythere convergence to the initial dataf(x)=u(x, 0) of the solutionu(x, t) of the two-dimensional linear Schrödinger equation Δu=i? _t u. The main result is thatu(x, t) →f(x) almost everywhere fort → 0 iff ∈H ^p (R²), wherep may be chosen <1/2. To get this result (improving on Vega’s work, see [6]), we devise a strategy to capture certain cancellations, which we believe has other applications in related problems.     

Inverse Problems for Time-Dependent Singular Heat Conductivities: Multi-Dimensional Case

We consider an inverse boundary value problem for the heat equation ? _t u = div (γ? _x u) in (0, T) × Ω, u = f on (0, T) × ?Ω, u| _t=0 = u ₀, in a bounded domain Ω ? ? ⁿ , n ≥ 2, where the heat conductivity γ(t, x) is piecewise constant and the surface of discontinuity depends on time: γ(t, x) = k ² (x ∈ D(t)), γ(t, x) = 1 (x ∈ Ω?D(t)). Fix a direction e* ∈ 𝕊 ^n?1 arbitrarily. Assuming that ?D(t) is strictly convex for 0 ≤ t ≤ T, we show that k and sup {e*·x; x ∈ D(t)} (0 ≤ t ≤ T), in particular D(t) itself, are determined from the Dirichlet-to-Neumann map : f → ?_ν u(t, x)|_{(0, T)×?Ω}. The knowledge of the initial data u ₀ is not used in the proof. If we know min_0≤t≤T (sup _x∈D(t) x·e*), we have the same conclusion from the local Dirichlet-to-Neumann map. Numerical examples of stationary and moving circles inside the unit disk are shown. The results have applications to nondestructive testing. Consider a physical body consisting of homogeneous material with constant heat conductivity except for a moving inclusion with different conductivity. Then the location and shape of the inclusion can be monitored from temperature and heat flux measurements performed at the boundary of the body. Such a situation appears for example in blast furnaces used in ironmaking.     

Edge‐face chromatic number and edge chromatic number of simple plane graphs

Given a simple plane graph G, an edge‐face k‐coloring of G is a function ? : E(G) ∪ F(G) → {1,…,k} such that, for any two adjacent or incident elements a, b ∈ E(G) ∪ F(G), ?(a) ≠ ?(b). Let χ_e(G), χ_ef(G), and Δ(G) denote the edge chromatic number, the edge‐face chromatic number, and the maximum degree of G, respectively. In this paper, we prove that χ_ef(G) = χ_e(G) = Δ(G) for any 2‐connected simple plane graph G with Δ (G) ≥ 24. © 2005 Wiley Periodicals, Inc. J Graph Theory     

A remark on divisibility of definable groups

We show that if G is a definably compact, definably connected definable group defined in an arbitrary o‐minimal structure, then G is divisible. Furthermore, if G is defined in an o‐minimal expansion of a field, k ∈ ? and p_k : G → G is the definable map given by p_k (x ) = x^k for all x ∈ G , then we have |(p_k )^–1(x )| ≥ k^r for all x ∈ G , where r > 0 is the maximal dimension of abelian definable subgroups of G . (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bounded factorization property for Fréchet spaces

An operator T ∈ L(E, F) factors over G if T = RS for some S ∈ L(E, G) and R ∈ L(G, F); the set of such operators is denoted by L^G(E, F). A triple (E, G, F) satisfies bounded factorization property (shortly, (E, G, F) ∈ ???) if L^G(E, F) ? LB(E, F), where LB(E, F) is the set of all bounded linear operators from E to F. The relationship (E, G, F) ∈ ??? is characterized in the spirit of Vogt's characterisation of the relationship L(E, F) = LB(E, F) [23]. For triples of K?othe spaces the property ??? is characterized in terms of their K?othe matrices. As an application we prove that in certain cases the relations L(E, G₁) = LB(E, G₁) and L(G₂, F) = LB(G₂, F) imply (E, G, F) ∈ ??? where G is a tensor product of G₁ and G₂.     

A wavelike functional equation of Pexider type

Summary. Let (G, +) and (H, +) be abelian groups such that the equation 2u = v 2u = v is solvable in both G and H. It is shown that if f₁, f₂, f₃, f₄, : G ×G ? H f_1, f_2, f_3, f_4, : G \times G \longrightarrow H satisfy the functional equation f₁(x + t, y + s) + f₂(x - t, y - s) = f₃(x + s, y - t) + f₄(x - s, y + t) for all x, y, s, t ? G x, y, s, t \in G , then f₁, f₂, f₃, and f₄ are given by f₁ = w + h, f₂ = w - h, f₃ = w + k, f₄ = w - k where w : G ×G ? H w : G \times G \longrightarrow H is an arbitrary solution of f (x + t, y + s) + f (x - t, y - s) = f (x + s, y - t) + f (x - s, y + t) for all x, y, s, t ? G x, y, s, t \in G , and h, k : G ×G ? H h, k : G \times G \longrightarrow H are arbitrary solutions of D_y,t³g(x,y) = 0 \Delta_{y,t}^{3}g(x,y) = 0 and D_x,t³g(x,y) = 0 \Delta_{x,t}^{3}g(x,y) = 0 for all x, y, s, t ? G x, y, s, t \in G .     

Construction of a unique self-adjoint generator for a symmetric local semigroup

A definition is given of a symmetric local semigroup of (unbounded) operators P(t) (0 ? t ? T for some T > 0) on a Hilbert space $\text{H}$, such that P(t) is eventually densely defined as t → 0. It is shown that there exists a unique (unbounded below) self-adjoint operator H on $\text{H}$ such that P(t) is a restriction of e^?tH. As an application it is proven that H₀ + V is essentially self-adjoint, where e^?tH₀ is an L^p-contractive semigroup and V is multiplication by a real measurable function such that V ∈ L^{2 + ε} and e^?δV ∈ L¹ for some ε, δ > 0.