A Generalized Representation Formula for Systems of Tensor Wave Equations

In this paper, we generalize the Kirchhoff-Sobolev parametrix of Klainerman and Rodnianski (Hyperbolic Equ. 4(3):401–433, 2007) to systems of tensor wave equations with additional first-order terms. We also present a different derivation, which better highlights that such representation formulas are supported entirely on past null cones. This generalization of (Hyperbolic Equ. 4(3):401–433, 2007) is a key component for extending Klainerman and Rodnianski’s breakdown criterion result for Einstein-vacuum spacetimes in (J. Amer. Math. Soc. 23(2):345–382, 2009) to Einstein-Maxwell and Einstein-Yang-Mills spacetimes.     

A Proof of Factorization Formula for Critical Percolation

We give mathematical proofs to a number of statements which appeared in the series of papers by Simmons et al. (Phys Rev E 76(4):041106, 2007; J Stat Mech Theory Exp 2009(2):P02067, 33, 2009) where they computed the probabilities of several percolation events.     

Non-uniqueness Results for Critical Metrics of Regularized Determinants in Four Dimensions

The regularized determinant of the Paneitz operator arises in quantum gravity [see Connes in (Noncommutative geometry, 1994), IV.4.γ]. An explicit formula for the relative determinant of two conformally related metrics was computed by Branson in (Commun Math Phys 178:301–309, 1996). A similar formula holds for Cheeger’s half-torsion, which plays a role in self-dual field theory [see Juhl in (Families of conformally covariant differential operators, q-curvature and holography. Progress in Mathematics, vol 275, 2009)], and is defined in terms of regularized determinants of the Hodge laplacian on p-forms (p < n/2). In this article we show that the corresponding actions are unbounded (above and below) on any conformal four-manifold. We also show that the conformal class of the round sphere admits a second solution which is not given by the pull-back of the round metric by a conformal map, thus violating uniqueness up to gauge equivalence. These results differ from the properties of the determinant of the conformal Laplacian established in (Commun Math Phys 149:241–262, 1992), (Ann Math 142:171–212, 1995), (Commun Math Phys 189:655–665, 1997).     

On Consistency of the Quantum-Like Representation Algorithm

In this paper we continue to study so-called “inverse Born’s rule problem”: to construct a representation of probabilistic data of any origin by a complex probability amplitude which matches Born’s rule. The corresponding algorithm—quantum-like representation algorithm (QLRA)—was recently proposed by A. Khrennikov (Found. Phys. 35(10):1655–1693, 2005; Physica E 29:226–236, 2005; Dokl. Akad. Nauk 404(1):33–36, 2005; J. Math. Phys. 46(6):062111–062124, 2005; Europhys. Lett. 69(5):678–684, 2005). Formally QLRA depends on the order of conditioning. For two observables (of any origin, e.g., physical or biological) a and b, b|a- and a|b conditional probabilities produce two representations, say in Hilbert spaces H ^b|a and H ^a|b . In this paper we prove that under “natural assumptions” (which hold, e.g., for quantum observables represented by operators with nondegenerate spectra) these two representations are unitary equivalent. This result proves the consistency of QLRA.     

A Macroscopic Model for a System of Swarming Agents Using Curvature Control

In this paper, we study the macroscopic limit of a new model of collective displacement. The model, called PTWA, is a combination of the Vicsek alignment model (Vicsek et al. in Phys. Rev. Lett. 75(6):1226–1229, 1995) and the Persistent Turning Walker (PTW) model of motion by curvature control (Degond and Motsch in J. Stat. Phys. 131(6):989–1021, 2008; Gautrais et al. in J. Math. Biol. 58(3):429–445, 2009). The PTW model was designed to fit measured trajectories of individual fish (Gautrais et al. in J. Math. Biol. 58(3):429–445, 2009). The PTWA model (Persistent Turning Walker with Alignment) describes the displacements of agents which modify their curvature in order to align with their neighbors. The derivation of its macroscopic limit uses the non-classical notion of generalized collisional invariant introduced in (Degond and Motsch in Math. Models Methods Appl. Sci. 18(1):1193–1215, 2008). The macroscopic limit of the PTWA model involves two physical quantities, the density and the mean velocity of individuals. It is a system of hyperbolic type but is non-conservative due to a geometric constraint on the velocity. This system has the same form as the macroscopic limit of the Vicsek model (Degond and Motsch in Math. Models Methods Appl. Sci. 18(1):1193–1215, 2008) (the ‘Vicsek hydrodynamics’) but for the expression of the model coefficients. The numerical computations show that the numerical values of the coefficients are very close. The ‘Vicsek Hydrodynamic model’ appears in this way as a more generic macroscopic model of swarming behavior as originally anticipated.     

Black Hole Entropy: From Shannon to Bekenstein

In this note we have applied directly the Shannon formula for information theory entropy to derive the Black Hole (Bekenstein-Hawking) entropy. Our analysis is semi-classical in nature since we use the (recently proposed Banerjee in Int. J. Mod. Phys. D 19:2365–2369, 2010 and Banerjee and Majhi in Phys. Rev. D 81:124006, 2010; Phys. Rev. D 79:064024, 2009; Phys. Lett. B 675:243, 2009) quantum mechanical near horizon mode functions to compute the tunneling probability that goes in to the Shannon formula, following the general idea of Brillouin (Science and Information Theory, Dover, New York, 2004). Our framework conforms to the information theoretic origin of Black Hole entropy, as originally proposed by Bekenstein.     

Towards Quantum Computational Logics

Quantum computational logics have recently stirred increasing attention (Cattaneo et al. in Math. Slovaca 54:87–108, 2004; Ledda et al. in Stud. Log. 82(2):245–270, 2006; Giuntini et al. in Stud. Log. 87(1):99–128, 2007). In this paper we outline their motivations and report on the state of the art of the approach to the logic of quantum computation that has been recently taken up and developed by our research group.     

Large scale synthesis of silicon nanowires

Artificial nanostructures (Samuelson et al., Physica E 21:560–567, 2004; Xia et al., Adv Mater 15:353–389, 2003) show promise for the organization of functional materials (Huck and Samuelson, Nanotechnology 14:NIL_5–NIL_8, 2003) to create nanoelectronic (Mizuta and Oda, Science 279:208–211, 2008) or nano-optical devices (Mazur et al.; Tanemura et al., Synthesis, Optical Properties and Functional Applications of ZnO Nano-materials: A Review, 1–3:58–63, 2008). However, in most manufacturing recipes described so far, nanostructures are synthesized in solution and/or uncontrolled deposition results in random arrangements; this makes it difficult to measure the properties of attached nanodevices or to integrate them with conventionally fabricated microcircuitry. Here, we describe a fully CMOS compatible process technology for mass manufacture of polysilicon nanowires by the CVD (chemical vapor deposition) method. The large scale production of nanowires could successfully be synthesized on silicon (100) substrates. However, the method presented here can successfully be employed with all technologically useful substrates with good adhesion for silicon such as SiO₂, diamond-like carbon or III–V semiconductors. This opens up the possibility for the fabrication of strain-sensitive and defect-sensitive optoelectronic devices on the optimum III–V substrate (Fonstad et al.). Finally, scanning electron microscopy (SEM) was used to characterize the as-synthesized nanowires and energy-filtered transmission electron microscopy (EFTEM) and scanning transmission electron microscopy (STEM) analysis were used to determine the nanowire composition.     

Probability of inflation in loop quantum cosmology

Inflationary models of the early universe provide a natural mechanism for the formation of large scale structure. This success brings to forefront the question of naturalness: Does a sufficiently long slow roll inflation occur generically or does it require a careful fine tuning of initial parameters? In recent years there has been considerable controversy on this issue (Hollands and Wald in Gen Relativ Gravit, 34:2043, 2002; Kofman et al. in J High Energy Phys 10:057, 2002); (Gibbons and Turok in Phys Rev D 77:063516, 2008). In particular, for a quadratic potential, Kofman et al. (J High Energy Phys 10:057, 2002) have argued that the probability of inflation with at least 65 e-foldings is close to one, while Gibbons and Turok (Phys Rev D 77:063516, 2008) have argued that this probability is suppressed by a factor of ~10⁻⁸⁵. We first clarify that such dramatically different predictions can arise because the required measure on the space of solutions is intrinsically ambiguous in general relativity. We then show that this ambiguity can be naturally resolved in loop quantum cosmology (LQC) because the big bang is replaced by a big bounce and the bounce surface can be used to introduce the structure necessary to specify a satisfactory measure. The second goal of the paper is to present a detailed analysis of the inflationary dynamics of LQC using analytical and numerical methods. By combining this information with the measure on the space of solutions, we address a sharper question than those investigated in Kofman et al. (J High Energy Phys 10:057, 2002), Gibbons and Turok (Phys Rev D 77:063516, 2008), Ashtekar and Sloan (Phys Lett B 694:108, 2010): What is the probability of a sufficiently long slow roll inflation which is compatible with the seven year WMAP data? We show that the probability is very close to 1. The material is so organized that cosmologists who may be more interested in the inflationary dynamics in LQC than in the subtleties associated with measures can skip that material without loss of continuity.     

Singular Harmonic Maps and Applications to General Relativity

The study of axially symmetric stationary multi-black-hole configurations and the force between co-axially rotating black holes involves, as a first step, an analysis on the “boundary regularity” of the so-called reduced singular harmonic maps. We carry out this analysis by considering those harmonic maps as solutions to some homogeneous divergence systems of partial differential equations with singular coefficients. Our results extend previous works by Weinstein (Comm Pure Appl Math 43:903–948, 1990; Comm Pure Appl Math 45:1183–1203, 1992) and by Li and Tian (Manu Math 73(1):83–89, 1991; Commun Math Phys 149:1–30, 1992; Differential geometry: PDE on manifolds, vol 54, pp. 317–326, 1993). This paper is based on the Ph.D. thesis of the author (Singular harmonic maps into hyperbolic spaces and applications to general relativity, PhD thesis, The State University of New Jersey, Rutgers, 2009).     

On Steady Distributions of Kinetic Models of Conservative Economies

We analyze the large-time behavior of various kinetic models for the redistribution of wealth in simple market economies introduced in the pertinent literature in recent years. As specific examples, we study models with fixed saving propensity introduced by Chakraborti and Chakrabarti (Eur. Phys. J. B 17:167–170, 2000), as well as models involving both exchange between agents and speculative trading as considered by Cordier et al. (J. Stat. Phys. 120:253–277, 2005) We derive a sufficient criterion under which a unique non-trivial stationary state exists, and provide criteria under which these steady states do or do not possess a Pareto tail. In particular, we prove the absence of Pareto tails in pointwise conservative models, like the one in (Eur. Phys. J. B 17:167–170, 2000), while models with speculative trades introduced in (J. Stat. Phys. 120:253–277, 2005) develop fat tails if the market is “risky enough”. The results are derived by a Fourier-based technique first developed for the Maxwell-Boltzmann equation (Gabetta et al. in J. Stat. Phys. 81:901–934, 1995; Bisi et al. in J. Stat. Phys. 118(1–2):301–331, 2005; Pareschi and Toscani in J. Stat. Phys. 124(2–4):747–779, 2006) and from a recursive relation which allows to calculate arbitrary moments of the stationary state.     

Quantum-Like Model for Decision Making Process in?Two Players Game A Non-Kolmogorovian Model

In experiments of games, players frequently make choices which are regarded as irrational in game theory. In papers of Khrennikov (Information Dynamics in Cognitive, Psychological and Anomalous Phenomena. Fundamental Theories of Physics, Kluwer Academic, Norwell, 2004; Fuzzy Sets Syst. 155:4–17, 2005; Biosystems 84:225–241, 2006; Found. Phys. 35(10):1655–1693, 2005; in QP-PQ Quantum Probability and White Noise Analysis, vol. XXIV, pp. 105–117, 2009), it was pointed out that statistics collected in such the experiments have “quantum-like” properties, which can not be explained in classical probability theory. In this paper, we design a simple quantum-like model describing a decision-making process in a two-players game and try to explain a mechanism of the irrational behavior of players. Finally we discuss a mathematical frame of non-Kolmogorovian system in terms of liftings (Accardi and Ohya, in Appl. Math. Optim. 39:33–59, 1999).     

Thomae Formula for General Cyclic Covers of $${{\mathbb {CP}}^1}$$

Let X be a general cyclic cover of \mathbbCP¹{\mathbb{CP}^{1}} ramified at m points, λ₁... λ _m . we define a class of non-positive divisors on X of degree g −1 supported in the pre images of the branch points on X, such that the Riemann theta function does not vanish on their image in J(X). We generalize the results of Bershadsky and Radul (Commun Math Phys 116:689–700, 1988), Nakayashiki (Publ Res Inst Math Sci 33(6):987–1015, 1997) and Enolskii and Grava (Lett Math Phys 76(2–3):187–214, 2006) and prove that up to a certain determinant of the non-standard periods of X, the value of the Riemann theta function at these divisors raised to a high enough power is a polynomial in the branch point of the curve X. Our approach is based on a refinement of Accola’s results for 3 cyclic sheeted cover (Accola, in Trans Am Math Soc 283:423–449, 1984) and a generalization of Nakayashiki’s approach explained in Nakayashiki (Publ Res Inst Math Sci 33(6):987–1015, 1997) for general cyclic covers.     

Evaluating Quasilocal Energy and Solving Optimal Embedding Equation at Null Infinity

We study the limit of quasilocal energy defined in Wang and Yau (Phys Rev Lett 102(2):021101, 2009; Commun Math Phys 288(3):919–942, 2009) for a family of spacelike 2-surfaces approaching null infinity of an asymptotically flat spacetime. It is shown that Lorentzian symmetry is recovered and an energy-momentum 4-vector is obtained. In particular, the result is consistent with the Bondi–Sachs energy-momentum at a retarded time. The quasilocal mass in Wang and Yau (Phys Rev Lett 102(2):021101, 2009; Commun Math Phys 288(3):919–942, 2009) is defined by minimizing quasilocal energy among admissible isometric embeddings and observers. The solvability of the Euler-Lagrange equation for this variational problem is also discussed in both the asymptotically flat and asymptotically null cases. Assuming analyticity, the equation can be solved and the solution is locally minimizing in all orders. In particular, this produces an optimal reference hypersurface in the Minkowski space for the spatial or null exterior region of an asymptotically flat spacetime.     

A Dark Energy Model in a Scale Covariant Theory of Gravitation

An axially symmetric Bianchi type-I space time with variable equation of state (EoS) parameter and constant deceleration parameter has been investigated in scale covariant theory of gravitation formulated by Canuto et al. (Phys. Rev. Lett. 39:429, 1977). With the help of special law of variation for Hubble’s parameter proposed by Bermann (Nuovo Cimento 74B:182, 1983) a dark energy cosmological model is obtained in this theory. Some physical and kinematical properties of the model are also discussed.     

The Maxwell Electromagnetic Equations and the Lorentz Type Force Derivation—The Feynman Approach Legacy

R. Feynman’s “heretical” approach (Dyson in Am. J. Phys. 58:209–211, 1990; Dyson in Phys. Today 42(2):32–38, 1989) to deriving the Lorentz force based Maxwell electromagnetic equations is revisited, the its complete legacy is argued both by means of the geometric considerations and its deep relation with the vacuum field theory approach devised (Prykarpatsky et al. in Int. J. Theor. Phys. 49:798–820, 2010; Prykarpatsky et al. in Preprint ICTP, 2008, ). Being completely classical, we reanalyze the Feynman’s derivation from the classical Lagrangian and Hamiltonian points of view and construct its nontrivial relativistic generalization compatible with the vacuum field theory approach.     

An Example of a Nearly Integrable Hamiltonian System with a Trajectory Dense in a Set of Maximal Hausdorff Dimension

The famous ergodic hypothesis suggests that for a typical Hamiltonian on a typical energy surface nearly all trajectories are dense. KAM theory disproves it. Ehrenfest (The Conceptual Foundations of the Statistical Approach in Mechanics. Ithaca, NY: Cornell University Press, 1959) and Birkhoff (Collected Math Papers. Vol 2, New York: Dover, pp 462–465, 1968) stated the quasi-ergodic hypothesis claiming that a typical Hamiltonian on a typical energy surface has a dense orbit. This question is wide open. Herman (Proceedings of the International Congress of Mathematicians, Vol II (Berlin, 1998). Doc Math 1998, Extra Vol II, Berlin: Int Math Union, pp 797–808, 1998) proposed to look for an example of a Hamiltonian near with a dense orbit on the unit energy surface. In this paper we construct a Hamiltonian which has an orbit dense in a set of maximal Hausdorff dimension equal to 5 on the unit energy surface.     

Enzyme-induced growth of silver nanoparticles studied on single particle level

Based on their interesting properties, metal nanoparticles show the potential as an analytical tool in electronic (Burmeister et al. 2004), optical (Yguerabide and Yguerabide 1998), and catalytic applications (Liu 2006). Their characteristics depend on the composition, shape, and size of the single particles. These various properties are utilized in many different approaches such as optics, magnetics (Lang et al. 2007), and laser technology (Csaki et al. 2007). We investigated an alternative method for the synthesis of nanoparticles. In this case, an enzyme, horseradish peroxidase, induces a silver deposition and replaces a metal nanoparticle as the reaction seed. Depending on the reaction time, we could obtain particles in a range of few nanometers up to more than 250 nm. For a better understanding of the enzymatic silver deposition process, the silver particles produced by this process were analyzed by SEM, TEM, and atomic force microscopy (AFM) on a single particle level after different enhancement times. The AFM images were utilized for the characterization of particle height and volume to study the enzyme kinetics, i.e., the particle growth process. Thereby, two different phases are described: a first growth phase probably induced by the enzyme-related growth, and a second, more unspecific growth based on the metal deposition onto the silver deposits. These findings may help to use the enzyme-induced silver deposition in a quantitative manner for bioanalytical applications.