Temperature - time analogy for thermorheologically complex polymeric materials

Conclusions 1. The viscoelastic properties of mixtures of crystalline polymers with elastomers have been studied as functions of temperature. It is shown that, for mixtures of PP with PIB and TEP, the concept of thermorheologically simple behavior is applicable only within a narrow interval of temperature.2. Parameters of the temperature—time analogy for thermorheologically complex materials have been calculated. It is established that, at temperatures above a characteristic temperature T*, a strong dependence of the correction coefficient on time and a strong dependence of the relaxation function on the ratio of components exist.3. It is shown that use of the concept of thermorheologically complex behavior for prediction of the long-term deformability allows analysis of the contributions of specific components to the viscoelastic behavior of the mixture in different temperature intervals.Paper presented at the Third All-Union Conference on Polymer Mechanics, Riga, November 10–12, 1976.Okhtinsk Scientific-Industrial Association "Plastpolimer," Leningrad. Translated from Mekhanika Polimerov, No. 4, pp. 614–620, July–August, 1977.     

Some sufficient conditions for the existence of a 1-factor

The following theorem is proved: Let G be a graph of even order. Assume that there exists a connected spanning subgraph F of G such that for every set U of four vertices in G, if the subgraph of F induced by U is a star, then the subgraph of G induced by U is complete. Then G has a 1-factor. The above theorem is derived from another sufficient condition for the existence of a 1-factor, which is also proved in this paper (Lemma 1).     

Some conditions for the existence of f-factors

Let m, l, n be three odd integers such that m > l > n. It is proved that if a graph G has an m-factor and an n-factor, then it also has an l-factor. In addition, we obtain sufficient conditions for the existence of an f-factor, in terms of vertex-deleted subgraphs.     

Necessary conditions for the existence of divisible designs

BOSE and CONNOR [2] proved that a symmetric regular divisible design with w classes of sizes g and joining numbers ₁ and ₂ must satisfy for every prime p the arithmetic condition (d₁, (–1)^sw)_p(d₂,(–l)^tg^w)_p=1, where d₁=k²–v₂, d₂= k–₁ s=(w-1)(w-2)/2, t=(v-w)(v-w-1)/2 and (*,*) is the Hilbert symbol. We show that if in addition _{1 2} and the design is fully symmetric divisible then (d₁, (–1)^s w)_p=(d₂, (–1)^tg^w)=1. Our assumption is by a result of CONNOR [5] fulfilled, if d₁ and ₁–₂ are relatively prime. Thus, we can exclude parameters not accessible to the Bose-Connor-Theorem. Our result can be derived from a theorem of RAGHAVARAO [9], and we give the precise assumptions of this theorem. We also discuss arithmetic restrictions for divisible designs which satisfy diverse other rules for the intersection numbers and generalize a result of DEMBOWSKI [6; 2.1.11].Dedicated to Professor Benz on occasion of his sixtieth birthday     

Feasibility conditions for the existence of walk-regular graphs

A graph X is walk-regular if the vertex-deleted subgraphs of X all have the same characteristic polynomial. Examples of such graphs are vertex-transitive graphs and distance-regular graphs. We show that the usual feasibility conditions for the existence of a distance-regular graph with a given intersection array can be extended so that they apply to walk-regular graphs. Despite the greater generality, our proofs are more elementary than those usually given for distance-regular graphs. An application to the computation of vertex-transitive graphs is described.     

The necessary and sufficient conditions for the existence of periodic orbits in a Lotka-Volterra system

By extending Darboux method to three dimension, we present necessary and sufficient conditions for the existence of periodic orbits in three species Lotka-Volterra systems with the same intrinsic growth rates. Therefore, all the published sufficient or necessary conditions for the existence of periodic orbits of the system are included in our results. Furthermore, we prove the stability of periodic orbits. Hopf bifurcation is shown for the emergence of periodic orbits and new phenomenon is presented: at critical values, each equilibrium are surrounded by either equilibria or periodic orbits.     

Long time existence for a slightly perturbed vortex sheet

Consider a flat two-dimensional vortex sheet perturbed initially by a small analytic disturbance. By a formal perturbation analysis, Moore derived an approximate differential equation for the evolution of the vortex sheet. We present a simplified derivation of Moore's approximate equation and analyze errors in the approximation. The result is used to prove existence of smooth solutions for long time. If the initial perturbation is of size ? and is analytic in a strip |??m γ| < ρ, existence of a smooth solution of Birkhoff's equation is shown for time t < k2p, if ? is sufficiently small, with κ → 1 as ? → 0. For the particular case of sinusoidal data of wave length π and amplitude e, Moore's analysis and independent numerical results show singularity development at time t^c = |log ?| + O(log|log ?|. Our results prove existence for t < κ|log ?|, if ? is sufficiently small, with k κ → 1 as ? → 0. Thus our existence results are nearly optimal.     

Long time existence for the heat equation with a noise term

Summary We consider the equationu _t =u _xx +u W forx on a finite interval, with Dirichlet boundary conditions. W is spacetime white noise. The initial condition is continuous and nonnegative. We show existence and uniqueness for all time, provided 1 <3/2.Supported by an NSF grant     

Some conditions for the existence of the total differential

We give conditions for the almost everywhere existence of the total differential of the functions of the weighted classes ACL_p^μ in a domain. The proofs rest on the technique of the weighted modulus of a family of curves.     

Tensor conditions for the existence of a common solution to the Lyapunov equation

Necessary and sufficient conditions for the existence of a common solution to the Lyapunov equation for two stable complex matrices are derived. These conditions are applied to the cases when a common weak solution to the Lyapunov equation exists. Conditions for the existence of a common solution to the Lyapunov equation for two complex 2 × 2 and two complex 3 × 3 matrices are derived.     

Sufficient conditions for the existence of a graph with a given variety of balls

It is proved that for each positive integer d and each collection of integers $\bar \tau $ = (τ ₀, τ ₁ …, τ _d ) such that τ ₀ ? τ ₁ ? … ? τ _d = 1 and τ _d?1 ? d ² + 1, there exists a graph of diameter d whose variety vector of the balls is equal to $\bar \tau $ ; if d ? 3 then there is no graph of diameter d whose variety vector of balls (τ ₀, τ ₁, …, τ _d ) satisfies the condition τ ₀ = τ ₁ = … = τ _d?1 ? 2d ? 1.     

Topological conditions for the existence of absorbing Cantor sets

This paper deals with strange attractors of S-unimodal maps . It generalizes earlier results in the sense that very general topological conditions are given that either

i)

guarantee the existence of an absorbing Cantor set provided the critical point of is sufficiently degenerate, or

ii)

prohibit the existence of an absorbing Cantor set altogether.

As a by-product we obtain very weak topological conditions that imply the existence of an absolutely continuous invariant probability measure for .

     

Sufficient conditions for the existence of a common stabilizer for a family of linear nonstationary plants

We consider the problem on the existence of a common stabilizer for a finite family of continuous linear scalar nonstationary plants. To solve the problem, we suggest to use a topological approach based on finding a common stabilizer in the form of a convex combination of stabilizers for the individual plants and on an application of well-known assertions about the properties of sets on simplices.