Power Sums Related to Semigroups S(d<Subscript>1</Subscript>, d<Subscript>2</Subscript>, d<Subscript>3</Subscript>)

The explicit formulas for the sums of positive powers of the integers unrepresentable by the triple of integers , are derived.     

A Note on Polynomially Growing C<Subscript>0</Subscript>-Semigroups

We characterize polynomial growth of a C₀-semigroup in terms of the first power of the resolvent of its generator. We do this for a class of semigroups which includes C₀-semigroups on Hilbert spaces and analytic semigroups on Banach spaces. Furthermore, we characterize polynomial growth for discrete semigroups.     

New results on <Emphasis Type="Italic">C</Emphasis><Subscript>11</Subscript> and <Emphasis Type="Italic">C</Emphasis><Subscript>12</Subscript> lattices with applications to Grothendieck categories and torsion theories

In this paper, which is a continuation of our previous paper [T. Albu, M. Iosif, A. Tercan, The conditions (C _i ) in modular lattices, and applications, J. Algebra Appl. 15 (2016), http: dx.doi.org/10.1142/S0219498816500018], we investigate the latticial counterparts of some results about modules satisfying the conditions (C ₁₁) or (C ₁₂). Applications are given to Grothendieck categories and module categories equipped with hereditary torsion theories.     

The effect of C<Subscript>60</Subscript> fullerene on the toughening of epoxy compositions

The effect of C₆₀ fullerene on the mechanical properties of epoxy resins has been investigated. It is found that this filler affects the tensile modulus and tensile strength of epoxy compositions only slightly, but their impact strength at a 0.01–0.12 wt.% content of C₆₀ increases by about 100–200%. A molecular mechanism of the toughening effect of C₆₀ on epoxy resins is suggested.     

Perturbations of nuclear <Emphasis Type=Italic>C</Emphasis>*-algebras

Kadison and Kastler introduced a natural metric on the collection of all C*-subalgebras of the bounded operators on a separable Hilbert space. They conjectured that sufficiently close algebras are unitarily conjugate. We establish this conjecture when one algebra is separable and nuclear. We also consider one-sided versions of these notions, and we obtain embeddings from certain near inclusions involving separable nuclear C*-algebras. At the end of the paper we demonstrate how our methods lead to improved characterisations of some of the types of algebras that are of current interest in the classification programme.     

Stability of <Emphasis Type="Italic">C</Emphasis>-regularized Semigroups

Let T = (T(t))t≥0 be a bounded C-regularized semigroup generated by A on a Banach space X and R(C) be dense in X. We show that if there is a dense subspace Y of X such that for every x ∈ Y, σu(A, Cx), the set of all points λ ∈ iR to which (λ - A)^-1 Cx can not be extended holomorphically, is at most countable and σr(A) N iR = Ф, then T is stable. A stability result for the case of R(C) being non-dense is also given. Our results generalize the work on the stability of strongly continuous senfigroups.     

Perturbations of Generalized Mehler Semigroups and Applications to Stochastic Heat Equations with Levy Noise and Singular Drift

In this paper we solve the Kolmogorov equation and, as a consequence, the martingale problem corresponding to a stochastic differential equation of type dX _t =AX _t dt+b(X _t )dt+dY _t , on a Hilbert space E, where (Y _t ) _t0 is a Levy process on E,A generates a C ₀-semigroup on E and b:EE. Our main point is to allow unbounded A and also singular (in particular, non-continuous) b. Our approach is based on perturbation theory of C ₀-semigroups, which we apply to generalized Mehler semigroups considered on L ²(), where is their respective invariant measure. We apply our results, in particular, to stochastic heat equations with Levy noise and singular drift.     

Consistency Analysis of Operator Splitting Methods for C<Subscript>0</Subscript>-Semigroups Expression

The consistency of different splitting schemes has mostly been studied for bounded operators by means of the traditional Taylor series expansion. In this paper second order of consistency is proved for the symmetrically weighted sequential (SWS) splitting and the Marchuk-Strang (MS) splitting for unbounded generators of strongly continuous semigroups.     

Orientation ordering and the transition to the orientational glass state in fullerite C<Subscript>60</Subscript>

We propose a model for describing the low-temperature transition to the orientational glass state in solid molecular C₆₀ in the framework of a theory similar to spin glass theory. We find a replica symmetric solution and also break the replica symmetry. The obtained results agree with experimental data on the partial retention of the orientational long-range order in the glass phase and on the presence of a broad maximum on the curve corresponding to the temperature dependence of the orientational part of the heat capacity. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 2, pp. 356–368, May, 2008.     

The Cayley Transform of the Generator of a Bounded C<Subscript>0</Subscript>-Semigroup

Let A be the generator of a uniformly bounded C₀-semigroup on the Banach space X. We present sufficient conditions on the resolvent , under which the Cayley transform is a power-bounded operator, i.e., .     

Classification of amply regular graphs with <Emphasis Type="Italic">b</Emphasis><Subscript>1</Subscript> = 6

An undirected graph with v vertices in which the degrees of all vertices are equal to k, each edge is contained in exactly λ triangles, and the intersection of the neighborhoods of any two vertices at distance 2 contains exactly µ vertices is called amply regular with parameters (v, k, λ, µ). We complete the classification of amply regular graphs with b ₁ = 6, where b ₁ = k ? λ ? 1.     

An Isomorphic Version of Nakano’s Characterisation of <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>(Σ)

In 1941 Nakano gave a characterisation of real spaces C₀(Σ) as M-spaces satisfying an additional restriction on their norm. We give an isomorphic version of Nakano’s result which involves showing that for any Banach lattice (a suitably modified version of) Nakano’s condition is equivalent to the norm on the Dedekind completion being Fatou.     

Distance-Regular Shilla Graphs with <Emphasis Type="Italic">b</Emphasis><Subscript>2</Subscript> = <Emphasis Type="Italic">c</Emphasis><Subscript>2</Subscript>

A Shilla graph is defined as a distance-regular graph of diameter 3 with second eigen-value θ₁ equal to a₃. For a Shilla graph, let us put a = a₃ and b = k/a. It is proved in this paper that a Shilla graph with b₂ = c₂ and noninteger eigenvalues has the following intersection array:

$$\left\{ {\frac{{{b^2}\left( {b - 1} \right)}}{2},\frac{{\left( {b - 1} \right)\left( {{b^2} - b + 2} \right)}}{2},\frac{{b\left( {b - 1} \right)}}{4};1,\frac{{b\left( {b - 1} \right)}}{4},\frac{{b{{\left( {b - 1} \right)}^2}}}{2}} \right\}$$

If Γ is a Q-polynomial Shilla graph with b₂ = c₂ and b = 2r, then the graph Γ has intersection array

$$\left\{ {2tr\left( {2r + 1} \right),\left( {2r + 1} \right)\left( {2rt + t + 1} \right),r\left( {r + t} \right);1,r\left( {r + t} \right),t\left( {4{r^2} - 1} \right)} \right\}$$

and, for any vertex u in Γ, the subgraph Γ₃(u) is an antipodal distance-regular graph with intersection array

$$\left\{ {t\left( {2r + 1} \right),\left( {2r - 1} \right)\left( {t + 1} \right),1;1,t + 1,t\left( {2r + 1} \right)} \right\}$$

The Shilla graphs with b₂ = c₂ and b = 4 are also classified in the paper.

     

A solvable group isospectral to S<Subscript>4</Subscript>(3)

We construct a solvable group G of order 5 648 590 729 620 such that the set of element orders of G coincides with that of the simple group S⁴(3). This completes the determination of finite simple groups isospectral to solvable groups.     

Semigroups,Potential Spaces and Applications to (S)PDE

The paper studies perturbed semilinear parabolic partial (pseudo-) differential equations on σ-finite measure spaces under low smoothness assumptions. We obtain results on existence, uniqueness and regularity. The hypotheses are formulated in terms of the semigroup, regularity is measured by means of abstract potential spaces. Being a priori analytic, our results allow to investigate related stochastic partial differential equations in the almost sure pathwise sense. For example we can study (fractional) semilinear heat equations driven by fractional Brownian noises on metric measure spaces.     

Obstructions to the uniform stability of a <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>-semigroup

Let T _t : X → X be a C ₀-semigroup with generator A. We prove that if the abscissa of uniform boundedness of the resolvent s ₀(A) is greater than zero then for each nondecreasing function h(s): ?₊ → R ₊ there are x′ ∈ X′ and x ∈ X satisfying ∫ ₀ ^∞ h(|〈x′, T _x x〉|)dt = ∞. If i? ∩ Sp(A) ≠ Ø then such x may be taken in D(A ^∞).     

On strongly regular graphs with <Emphasis Type="Italic">b</Emphasis><Subscript>1</Subscript> &lt; 24

Let γ be a connected edge-regular graph with parameters (v, k, λ), and let b ₁ = k?λ?1. It is well known that, if b ₁ = 1, then Γ is either a polygon or a complete multipartite graph with parts of order 2. Graphs with b ₁ ≤ 4 were classified earlier. The investigation of graphs even in the case b ₁ = 5 involves great difficulties. However, for strongly regular graphs, the situation is much simpler. In this paper, we classify strongly regular graphs with b ₁ < 24.     

Free <Emphasis Type="Bold">C</Emphasis><Subscript>+</Subscript>-actions on <Emphasis Type="Bold">C</Emphasis><Superscript>3</Superscript> are translations

Let X be a smooth contractible three-dimensional affine algebraic variety with a free algebraic C₊-action on it such that S=X//C₊ is smooth. We prove that X is isomorphic to S×C and the action is induced by a translation on the second factor. As a consequence we show that any free algebraic C₊-action on C³ is a translation in a suitable coordinate system.