Numerical radius and zero pattern of matrices

Let A be an n×n complex matrix and r be the maximum size of its principal submatrices with no off-diagonal zero entries. Suppose A has zero main diagonal and x is a unit n-vector. Then, letting ‖A‖ be the Frobenius norm of A, we show that

|〈Ax,x〉^|2?(1−1/2r−1/2n)‖A^‖2.     

Graphs with many r-cliques have large complete r-partite subgraphs

Let r2 and c>0. Every graph on n vertices with at least cn^rcliques on r vertices contains a complete r-partite subgraphwith r–1 parts of size c^rlog n and one part of size greaterthan n^1–cr–1. This result implies a quantitativeform of the Erdös–Stone theorem.     

A spectral condition for odd cycles in graphs

Let G be a graph of sufficiently large order n, and let the largest eigenvalue μ(G) of its adjacency matrix satisfies . Then G contains a cycle of length t for every t?n/320This condition is sharp: the complete bipartite graph T₂(n) with parts of size ⌊n/2⌋ and ⌈n/2⌉ contains no odd cycles and its largest eigenvalue is equal to .This condition is stable: if μ(G) is close to and G fails to contain a cycle of length t for some t?n/321, then G resembles T₂(n).     

Spectralizable Operators

We introduce the notion of spectralizable operators. A closed operator A in a Hilbert space is called spectralizable if there exists a non-constant polynomial p such that the operator p(A) is a scalar spectral operator in the sense of Dunford. We show that such operators belongs to the class of generalized spectral operators and give some examples where spectralizable operators occur naturally. Vladimir Strauss gratefully acknowledges support by DFG, Grant No. TR 903/3-1.     

Sesquitransitive and Localizing Operator Algebras

An algebra of operators on a Banach space X is said to be transitive if X has no nontrivial closed subspaces invariant under every member of the algebra. In this paper we investigate a number of conditions which guarantee that a transitive algebra of operators is “large” in various senses. Among these are the conditions of algebras being localizing or sesquitransitive. An algebra is localizing if there exists a closed ball B ∌ 0 such that for every sequence (x _n ) in B there exists a subsequence and a bounded sequence (A _k ) in the algebra such that converges to a non-zero vector. An algebra is sesquitransitive if for every non-zero z ∈ X there exists C > 0 such that for every x linearly independent of z, for every non-zero y ∈ X, and every there exists A in the algebra such that and ||Az|| ≤ C||z||. We give an algebraic version of this definition as well, and extend Jacobson’s density theorem to algebraically sesquitransitive rings. The second and the third authors were supported by NSERC.     

Highly efficient quenching of excimer fluorescence by perylene diimide in DNA

Pyrene excimer fluorescence is effectively quenched by non-nucleosidic perylene diimides upon DNA duplex formation.     

Cyclic oxonium derivatives of polyhedral boron hydrides and their synthetic applications

Cyclic oxonium derivatives of polyhedral boron hydrides are a relatively new class of boron compounds. They have great potential for the modification of various types of organic and bioorganic molecules and the synthesis of compounds that could be used in different fields from the treatment of nuclear wastes to the treatment of cancer. In the present Perspective we would like to present an overview of the results of the preparation and synthetic application of these compounds.     

Oxidation of guanine in G, GG, and GGG sequence contexts by aromatic pyrenyl radical cations and carbonate radical anions: relationship between kinetics and distribution of alkali-labile lesions

Oxidatively generated DNA damage induced by the aromatic radical cation of the pyrene derivative 7,8,9,10-tetrahydroxytetrahydrobenzo[a]pyrene (BPT), and by carbonate radicals anions, was monitored from the initial one-electron transfer, or hole injection step, to the formation of hot alkali-labile chemical end-products monitored by gel electrophoresis. The fractions of BPT molecules bound to double-stranded 20-35-mer oligonucleotides with noncontiguous guanines G and grouped as contiguous GG and GGG sequences were determined by a fluorescence quenching method. Utilizing intense nanosecond 355 nm Nd:YAG laser pulses, the DNA-bound BPT molecules were photoionized to BPT*+ radicals by a consecutive two-photon ionization mechanism. The BPT*+ radicals thus generated within the duplexes selectively oxidize guanine by intraduplex electron-transfer reactions, and the rate constants of these reactions follow the trend 5'-..GGG.. > 5'-..GG.. > 5'-..G... In the case of CO3*- radicals, the oxidation of guanine occurs by intermolecular collision pathways, and the bimolecular rate constants are independent of base sequence context. However, the distributions of the end-products generated by CO3*- radicals, as well as by BPT*+, are base sequence context-dependent and are greater than those in isolated guanines at the 5'-G in 5'-...GG... sequences, and the first two 5'- guanines in the 5'-..GGG sequences. These results help to clarify the conditions that lead to a similar or different base sequence dependence of the initial hole injection step and the final distribution of oxidized, alkali-labile guanine products. In the case of the intermolecular one-electron oxidant CO3*-, the rate constant of hole injection is similar for contiguous and isolated guanines, but the subsequent equilibration of holes by hopping favors trapping and product formation at contiguous guanines, and the sequence dependence of these two phenomena are not correlated. In contrast, in the case of the DNA-bound oxidant BPT*+, the hole injection rate constants, as well as hole equilibration, exhibit a similar dependence on base sequence context, and are thus correlated to one another.     

Optimization of enzymatic biochemical logic for noise reduction and scalability: how many biocomputing gates can be interconnected in a circuit?

We report an experimental evaluation of the "input-output surface" for a biochemical AND gate. The obtained data are modeled within the rate-equation approach, with the aim to map out the gate function and cast it in the language of logic variables appropriate for analysis of Boolean logic for scalability. In order to minimize "analog" noise, we consider a theoretical approach for determining an optimal set for the process parameters to minimize "analog" noise amplification for gate concatenation. We establish that under optimized conditions, presently studied biochemical gates can be concatenated for up to order 10 processing steps. Beyond that, new paradigms for avoiding noise buildup will have to be developed. We offer a general discussion of the ideas and possible future challenges for both experimental and theoretical research for advancing scalable biochemical computing.     

Structural, magnetic, high-frequency and high-field EPR investigation of double-stranded heterometallic [{Ni(en)2}2(micro-NCS)4Cd(NCS)2](n).nCH3CN polymer self-assembled from cadmium oxide, nickel thiocyanate and ethylenediamine

A novel heterometallic 1D coordination polymer [{Ni(en)2}2(micro-NCS)4Cd(NCS)2](n) x nCH3CN (en = ethylenediamine) has been prepared using the self-assembly process in a one-pot reaction of cadmium oxide, nickel and ammonium thiocyanates with an acetonitrile solution of ethylenediamine. The complex consists of an uncommon cis-Cd(SCN)4(NCS)2(4-) fragment and a rare combination of cis-Ni(en)2(2+) and trans-Ni(en)2(2+) building blocks linked by micro(1,3)-NCS bridges into a double-stranded zigzag chain structure. Each chain is comprised of [Ni2Cd2(micro-NCS-N,S)4)] macrocycles with chair-like and rectangular-like shapes arrayed alternately. The shortest intrachain CdCd separations are 9.535(1) and 10.868(2) A, while the nearest NiNi distances are 5.418(1) and 6.612(2) A. A network of weak N-HS hydrogen bonds, involving the terminal NCS ligands and NH2-groups of en, links the infinite chains and results in the formation of an extended supramolecular three-dimensional framework. Variable-temperature (1.8-300 K) magnetic susceptibilities show a slight change of the micro(B) value at low temperature, indicative of weak antiferromagnetic interactions (J = 1.55 cm-1) between magnetic canters. High-field, high-frequency (100-400 GHz) EPR spectra were simulated using S = 1 ground state for separate Ni2+ ions with the spin Hamiltonian parameters g = 2.165, D = 0.45 cm-1 and E = 0.03 cm(-1). According to DFT calculations, the D and E parameters are -0.35 cm(-1) and 0.049 cm-1 for the cis arrangement of Ni2+ and 0.58 cm(-1) and 0.012 cm(-1) for trans.