On strongly stable normal operators in Krein spaces

For bounded normal operators in Krein spaces we give a necessary and sufficient condition for strong stability. The same result for unitary operators was obtained by M.G.Krein [1] (see also [2]). For selfadjoint operators we refer to the papers of P.Jonas, H.Langer [3] and H.Langer [4].     

Parametric study of elastic mechanical properties of graphene nanoribbons by a new structural mechanics approach

The elastic mechanical behavior of different sized zigzag and armchair graphene nanoribbons is numerically investigated and predicted using a new structural mechanic approach. According to the proposed method three dimensional, two nodded spring elements of three degrees of freedom per node, which remain straight when deformed, are combined in order to simulate realistically the interatomic interactions appearing within the graphene nanostructure. The computed variations of mechanical elastic properties are fitted by appropriate size dependent non-linear functions of two independent variables i.e. length and width, in order to express the analytical rules governing the elastic behavior of graphene nanoribbons within specific dimension limits. The numerical results, which are compared with corresponding data given in the open literature, demonstrate thoroughly the important influence of size and chirality of a narrow graphene monolayer on its elastic behavior.     

Some inequalities about mixed volumes

We prove inequalities about the quermassintegralsV _k (K) of a convex bodyK in ℝ ⁿ (here,V _k (K) is the mixed volumeV((K, k), (B _n ,n − k)) whereB _n is the Euclidean unit ball). (i) The inequality

Ultraheavy Yukawa-bound states of fourth-generation at Large Hadron Collider

A study of bound states of the fourth-generation quarks in the range of 500?C700 GeV is presented, where the binding energies are expected to be mainly of Yukawa origin, with QCD subdominant. Near degeneracy of their masses exhibits a new ??isospin??. The production of a colour-octet, isosinglet vector meson via $q\bar q \to \omega_8$ is the most interesting. Its leading decay modes are $\pi_8^\pm W^\mp$ , $\pi_8^0Z^0$ , and constituent quark decay, with $q\bar q$ and $t\bar t'$ and $b\bar b'$ subdominant. The colour octet, isovector pseudoscalar ?? ₈ meson decays via constituent quark decay, or to Wg. This work calls for more detailed study of fourth-generation phenomena at LHC.     

A proportional Dvoretzky-Rogers factorization result

If is an -dimensional normed space and , there exists , such that the formal identity can be written as , with . This is proved as a consequence of a Sauer-Shelah type theorem for ellipsoids.

     

On the Volume Ratio of Two Convex Bodies

Let K and L be two convex bodies in Rⁿ. The volume ratio vr(K,L) of K and L is defined by vr(K, L = inf(|K|/|T(L)|)^1/n, wherethe infimum is over all affine transformations T of Rⁿ for whichT(L) K. It is shown in this paper that vr(K, L) , where c > 0 is an absolute constant. This isoptimal up to the logarithmic term. 2000 Mathematics SubjectClassification 52A40, 46B07 (primary); 52A21, 52A20 (secondary).     

Lower Bound for the Maximal Number of Facets of a 0/1 Polytope

Let $f_{n-1}(P)$ denote the number of facets of a polytope $P$ in ${\Bbb R}^n$. We show that there exist 0/1 polytopes $P$ with $$f_{n-1}(P)\geq\left (\frac{cn}{\log^2 n}\right )^{n/2},$$ where $c>0$ is an absolute constant. This improves earlier work of Barany and Por on a question of Fukuda and Ziegler.     

John's Theorem for an Arbitrary Pair of Convex Bodies

We provide a generalization of John's representation of the identity for the maximal volume position of L inside K, where K and L are arbitrary smooth convex bodies in ⁿ . From this representation we obtain Banach–Mazur distance and volume ratio estimates.     

On the average volume of sections of convex bodies

The average section functional as(K) of a star body in Rn is the average volume of its central hyperplane sections: \(as\left( k \right) = \int_{{S^{n - 1}}} {\left| {K \cap {\xi ^ \bot }} \right|} d\sigma \left( \xi \right)\). We study the question whether there exists an absolute constantC > 0 such that for every n, for every centered convex body K in R ⁿ and for every 1 ≤ k ≤ n ? 2,

$$as\left( K \right) \leqslant {C^k}{\left| K \right|^{\frac{k}{n}}}\mathop {\max }\limits_{|E \in G{r_{n - k}}} {\kern 1pt} as\left( {K \cap E} \right)$$

. We observe that the case k = 1 is equivalent to the hyperplane conjecture. We show that this inequality holds true in full generality if one replaces C by CL _K orCd_ovr(K, BP _k ⁿ ), where L _K is the isotropic constant of K and dovr(K, BP _k ⁿ ) is the outer volume ratio distance of K to the class BP _k ⁿ of generalized k-intersection bodies. We also compare as(K) to the average of as(K ∩ E) over all k-codimensional sections of K. We examine separately the dependence of the constants on the dimension when K is in some classical position. Moreover, we study the natural lower dimensional analogue of the average section functional.

     

A note on subgaussian estimates for linear functionals on convex bodies

We give an alternative proof of a recent result of Klartag on the existence of almost subgaussian linear functionals on convex bodies. If is a convex body in with volume one and center of mass at the origin, there exists such that

for all , where is an absolute constant. The proof is based on the study of the -centroid bodies of . Analogous results hold true for general log-concave measures.