Formality Conjectures and Deformation

Let M be a Poisson manifold. Kontsevich proved that star products exist on M and he gave a classification. To relate his classification with other classifications, one could try to extend the Connes–Flato–Sternheimer invariant to a general Poisson manifold. We show how generalization of this invariant is related to the formality conjecture for chains. Finally, we show how to prove those conjectures step by step. Our approach, different from Tamarkin's, will give explicit formulas but doesn't yet solve the general conjecture.     

Feynman Graphs,Rooted Trees,and Ringel-Hall Algebras

We construct symmetric monoidal categories of rooted forests and Feynman graphs. These categories closely resemble finitary abelian categories, and in particular, the notion of Ringel-Hall algebra applies. The Ringel-Hall Hopf algebras of , are dual to the corresponding Connes-Kreimer Hopf algebras on rooted trees and Feynman diagrams. We thus obtain an interpretation of the Connes-Kreimer Lie algebras on rooted trees and Feynman graphs as Ringel-Hall Lie algebras.     

Inflationary Cosmology with Quantum Gravitational Effects and Swampland Conjectures *

Recently proposed two swampland criteria that arising from string theory landscape leads to the important challenge of the realization of single-field inflationary models. Especially one of swampland criteria which implies a large tensor-to-scalar ratio is strongly in tension with recent observational results. In this paper, we explore the possibility the swampland conjectures could be compatible with single-field inflationary scenarios if the effects due to the quantum theory of gravity are considered. We show that the quantum gravitational effects due to the nonlinear dispersion relation provides significant modifications on the amplitude of both the scalar and tensor perturbation spectra. Such modifications could be either raise or reduce the perturbation spectra depending on the values of the parameters in the nonlinear terms of the dispersion relations. Therefore, these effects can reduce the tensor-to-scalar ratio to a smaller value, which helps to relax the tension between the swampland conjecture and observational data.     

Macdonald Polynomials in Superspace: Conjectural Definition and Positivity Conjectures

We introduce a conjectural construction for an extension to superspace of the Macdonald polynomials. The construction, which depends on certain orthogonality and triangularity relations, is tested for high degrees. We conjecture a simple form for the norm of the Macdonald polynomials in superspace and a rather non-trivial expression for their evaluation. We study the limiting cases q =?0 and q =???, which lead to two families of Hall?CLittlewood polynomials in superspace. We also find that the Macdonald polynomials in superspace evaluated at q =?t =?0 or q =?t = ?? seem to generalize naturally the Schur functions. In particular, their expansion coefficients in the corresponding Hall?CLittlewood bases appear to be polynomials in t with nonnegative integer coefficients. More strikingly, we formulate a generalization of the Macdonald positivity conjecture to superspace: the expansion coefficients of the Macdonald superpolynomials expanded into a modified version of the Schur superpolynomial basis (the q = t =?0 family) are polynomials in q and t with nonnegative integer coefficients.     

Probability on Graphs: Random Processes on Graphs and Lattices,by G. Grimmett

The 12 member states of the European high energy physics laboratory CERN are considering the construction of a huge new accelerator. This article outlines our understanding of the fundamental forces of nature and the subnuclear structure of matter, and describes the accelerator that will enable some of their mysteries to be explained.     

光波辐射下正多边形异向介质柱聚焦特性分析

基于时域有限差分方法,建立了光波辐射下正三角形、正四边形和正六边形无耗异向介质柱的二维电磁模型.分析了其近场特性,并对模型的有效性进行了验证.仿真结果表明:即便对于εr和μr都很小且无耗的正多边形异向介质柱,当平面光波沿正多边形顶角角平分线方向传播时,在异向介质柱内不会发生电磁聚焦.     

The Cut Metric,Random Graphs,and Branching Processes

In this paper we study the component structure of random graphs with independence between the edges. Under mild assumptions, we determine whether there is a giant component, and find its asymptotic size when it exists. We assume that the sequence of matrices of edge probabilities converges to an appropriate limit object (a kernel), but only in a very weak sense, namely in the cut metric. Our results thus generalize previous results on the phase transition in the already very general inhomogeneous random graph model introduced by the present authors in Random Struct. Algorithms 31:3–122 (2007), as well as related results of Bollobás, Borgs, Chayes and Riordan (Ann. Probab. 38:150–183, 2010), all of which involve considerably stronger assumptions. We also prove corresponding results for random hypergraphs; these generalize our results on the phase transition in inhomogeneous random graphs with clustering (Random Struct. Algorithms, 2010, to appear).     

Continuity and Anomalous Fluctuations in Random Walks in Dynamic Random Environments: Numerics,Phase Diagrams and Conjectures

We perform simulations for one dimensional continuous-time random walks in two dynamic random environments with fast (independent spin-flips) and slow (simple symmetric exclusion) decay of space-time correlations, respectively. We focus on the asymptotic speeds and the scaling limits of such random walks. We observe different behaviors depending on the dynamics of the underlying random environment and the ratio between the jump rate of the random walk and the one of the environment. We compare our data with well known results for static random environment. We observe that the non-diffusive regime known so far only for the static case can occur in the dynamical setup too. Such anomalous fluctuations give rise to a new phase diagram. Further we discuss possible consequences for more general static and dynamic random environments.     

Trivalent zirconium and hafnium ions in yttrium oxide ceramics

An analysis of the electron spin resonance (ESR) spectrum of transparent ceramics composed of yttrium oxide with zirconium and hafnium additives has revealed the presence of signals (with similar parameters) from Zr³⁺ and Hf³⁺ ions, which have a similar electron configurations of the ground states: [Kr]4d ¹ and [Xe]5d ¹, respectively. It is shown that the pulsed cathodoluminescence spectra of these ions consist of two bands peaking at λ ≈ 818 and 900 nm.     

Graphs and reflection groups

It is shown that graphs that generalize the ADE Dynkin diagrams and have appeared in various contexts of two-dimensional field theory may be regarded in a natural way as encoding the geometry of a root system. After recalling what are the conditions satisfied by these graphs, we define a bilinear form on a root system in terms of the adjacency matrices of these graphs and undertake the study of the group generated by the reflections in the hyperplanes orthogonal to these roots. Some non-integrally laced graphs are shown to be associated with subgroups of these reflection groups. The empirical relevance of these graphs in the classification of conformal field theories or in the construction of integrable lattice models is recalled, and the connections with recent developments in the context of supersymmetric theories and topological field theories are discussed. A. ClaudeDedicated to: A. Claude     

Random Knots in 3-Dimensional 3-Colour Percolation: Numerical Results and Conjectures

Three-dimensional three-colour percolation on a lattice made of tetrahedra is a direct generalization of two-dimensional two-colour percolation on the triangular lattice. The interfaces between one-colour clusters are made of bicolour surfaces and tricolour non-intersecting and non-self-intersecting curves. Because of the three-dimensional space, these curves describe knots and links. The present paper presents a construction of such random knots using particular boundary conditions and a numerical study of some invariants of the knots. The results are sources of precise conjectures about the limit law of the Alexander polynomial of the random knots.

     

Givental Graphs and Inversion Symmetry

Inversion symmetry is a very non-trivial discrete symmetry of Frobenius manifolds. It was obtained by Dubrovin from one of the elementary Schlesinger transformations of a special ODE associated to a Frobenius manifold. In this paper, we review the Givental group action on Frobenius manifolds in terms of Feynman graphs and obtain an interpretation of the inversion symmetry in terms of the action of the Givental group. We also consider the implication of this interpretation of the inversion symmetry for the Schlesinger transformations and for the Hamiltonians of the associated principle hierarchy.     

On Conjectures of A. Eremenko and A. Gabrielov for Quasi-Exactly Solvable Quartic

We study polynomials p(x) satisfying a differential equation of the form p′′?h′p′ + Hp = 0, where h = x ³/3 + ax and H is a polynomial. We prove a conjecture of A. Eremenko and A. Gabrielov.     

Spectroscopic and Fluorescence Studies on the Trivalent Ce,Eu,Nd and La Metal Ions Rhodamine C Florescent Dye Complexes

The four isolates solid complexes: [La(RHC)(NO₃)₂]·3H₂O, [Nd(RHC)(NO₃)₂]·4H₂O, [Eu(RHC)(NO₃)₂]·2H₂O, and [Ce(RHC)(NO₃)₂]·5H₂O that obtained by the reaction of the nitrate salts of the Ce(Ⅲ), Eu(Ⅲ), Nd(Ⅲ) and La(Ⅲ) ions and rhodamie C (RHC) ligand were interpretative using elemental analysis (C, H and N), molar conductivity, infrared, electronic, fluorescence and 1H-NMR spectra to achieve the speculated suitable formula. The low molar conductance values of the synthesized RHC complexes concluded the non-electrolytic behavior. The infrared spectra recorded the absence of stretching vibration ν(OH) of the -COOH and presence of two new vibration bands at 1 597～1 601 and 1 383～1 399 cm-1 which were assigned to ν_as(COO-) and ν_s(COO-). The difference between them revealed that the carboxylate group acts as a bidentate ligand. 1HNMR spectra of Europium and lanthanum(Ⅲ) complexes were supported the FTIR results based on the absent of proton of the carboxylic group. Therefore, the microanalytical and spectroscopic results deduced that RHC acts as a monobasic bidentate ligand, and coordinated to the central metal(Ⅲ) ions via the two oxygen atoms of deprotonated carboxylic group. Fluorescence studies were performed on the metal complexes of Ce3+, Tb3+, Th4+, Gd3+ and La3+, that referred a quenching in the fluorescene intensity of rhodamine C in the aqueous state after complexation. The antimicrobial assessment against some kind of bacteria and fungi were also checked and recorded enhancement in case of their complexes.