Universality of the Stochastic Airy Operator

We introduce a new method for studying universality of random matrices. Let T-n be the Jacobi matrix associated to the Dyson beta ensemble with uniformly convex polynomial potential. We show that after scaling, Tn converges to the stochastic Airy operator. In particular, the top edge of the Dyson beta ensemble and the corresponding eigenvectors are universal. As a byproduct, these ideas lead to conjectured operator limits for the entire family of soft edge distributions. (C) 2015 Wiley Periodicals, Inc.

Isolated singularities of polyharmonic operator in even dimension

We consider the equation Delta(2)u = g(x, u) >= 0 in the sense of distribution in Omega' = Omega\textbackslash {0} where u and -Delta u >= 0. Then it is known that u solves Delta(2)u = g(x, u) + alpha delta(0) - beta Delta delta(0), for some nonnegative constants alpha and beta. In this paper, we study the existence of singular solutions to Delta(2)u = a(x) f (u) + alpha delta(0) - beta Delta delta(0) in a domain Omega subset of R-4, a is a nonnegative measurable function in some Lebesgue space. If Delta(2)u = a(x) f (u) in Omega', then we find the growth of the nonlinearity f that determines alpha and beta to be 0. In case when alpha = beta = 0, we will establish regularity results when f (t) <= Ce-gamma t, for some C, gamma > 0. This paper extends the work of Soranzo (1997) where the author finds the barrier function in higher dimensions (N >= 5) with a specific weight function a(x) = |x|(sigma). Later, we discuss its analogous generalization for the polyharmonic operator.

Iron(III) salicylates of dipicolylamine bases showing photo-induced anticancer activity and cytosolic localization

An iron(III) salicylate having a dipicolylamine base (andpa) with a photoactive anthracenyl moiety is prepared, characterized, and studied for its photo-induced anticancer activity and cellular localization in HeLa and MCF-7 cells. Its phenyl analogue is structurally characterized by X-ray crystallography. The complex has a ternary structure in which the dipicolylamine ligand and salicylic acid in dianionic form (sal) display respective tridentate and bidentate mode of coordination in Fe(sal)(phdpa)Cl] (1). Complex Fe(sal)(andpa)Cl] (2) having a pendant anthracenyl moiety shows significant photocytotoxicity in visible light (400-700 nm) giving IC50 values of 8.6 +/- 0.7 and 3.4 +/- 0.9 mu M in HeLa and MCF-7 cells, while being essentially nontoxic in the dark (IC50 > 100 mu M). The complex shows cytosolic localization in the cancer cells. Formation of hydroxyl radicals ((OH)-O-center dot) as the reactive oxygen species is evidenced from the pUC19 DNA photocleavage studies. (C) 2015 Elsevier Ltd. All rights reserved.

Trace Formulae in Operator Theory

In this article, we survey several kinds of trace formulas that one encounters in the theory of single and multi-variable operators. We give some sketches of the proofs, often based on the principle of finite-dimensional approximations to the objects at hand in the formulas.

Action of Hopf on the twisted modular Hecke operator

Let Gamma subset of SL2(Z) be a principal congruence subgroup. For each sigma is an element of SL2(Z), we introduce the collection A(sigma)(Gamma) of modular Hecke operators twisted by sigma. Then, A(sigma)(Gamma) is a right A(Gamma)-module, where A(Gamma) is the modular Hecke algebra introduced by Connes and Moscovici. Using the action of a Hopf algebra h(0) on A(sigma)(Gamma), we define reduced Rankin-Cohen brackets on A(sigma)(Gamma). Moreover A(sigma)(Gamma) carries an action of H 1, where H 1 is the Hopf algebra of foliations of codimension 1. Finally, we consider operators between the levels A(sigma)(Gamma), sigma is an element of SL2(Z). We show that the action of these operators can be expressed in terms of a Hopf algebra h(Z).

Remarkable photo-induced anticancer activity of vitamin-S6 Schiff base copper(II) complexes of pyridyl bases

Vitamin-B6 (VB6) Schiff base (H2L) copper(II) complexes of pyridyl bases, viz. Cu(bpy)(L)] (1), Cu(phen)(L)] (2) and Cu(dppz)(L)] (3), where bpy is 2,2'-bipyridine, phen is 1,10-phenanthroline and dppz is dipyrido3,2-a:2',3'c]phenazine are synthesized, characterized and their phto-induced anticancer activity studied. The non-electrolytic one electron paramagnetic complexes exhibit a d-d band near 700 nm in DMF. The dppz complex intercalatively binds to calf-thymus DNA with binding constant (K-b) values of similar to 10(6) M-1. This complex exhibits low chemical nuclease activity but excellent DNA photocleavage activity when irradiated with red light of 705 nm forming (OH)-O-center dot radical. It displays remarkable photocytotoxicity in human cervical cancer cells (HeLa) giving IC50 value of 0.9 mu M in visible light (400-700 nm) while being less toxic in darkness (IC50 : 23 mu M). The cellular uptake of the complexes seems to be via VB6 transporting membrane carrier mediated diffusion pathway. Photo-induced cell death follows apoptotic pathway involving photo-generated intracellular reactive oxygen species.