Narrow Results

In this paper, we provide a simple pedagogical proof of the existence of covariant renormalizations in Euclidean perturbative quantum field theory on closed Riemannian manifolds, following the Epstein–Glaser philosophy. We rely on a local method that allows us to extend a distribution defined on an open set $\mathrm{\Omega }\subseteq M$ to the whole manifold $M$.

The collaboration network is an example of a social network which has both non-trivial temporal and spatial dependence. Based on the observations of collaborations in Physical Review Letters, a model of collaboration network is proposed which correctly reproduces the time evolution of the link length distributions, clustering coefficients, degree distributions, and assortative property of real data to a large extent.

The distribution of the "mixing time" or the "time to stationarity" in a discrete time irreducible Markov chain, starting in state i, can be defined as the number of trials to reach a state sampled from the stationary distribution of the Markov chain. Expressions for the probability generating function, and hence the probability distribution of the mixing time, starting in state i, are derived and special cases explored. This extends the results of the author regarding the expected time to mixing [Hunter, JJ (2006). Mixing times with applications to perturbed Markov chains. Linear Algebra and Its Applications, 417, 108–123] and the variance of the times to mixing, [Hunter, JJ (2008). Variances of first passage times in a Markov chain with applications to mixing times. Linear Algebra and Its Applications, 429, 1135–1162]. Some new results for the distribution of the recurrence and the first passage times in a general irreducible three-state Markov chain are also presented.

We review recent theoretical progress in evaluating higher order QCD corrections to Higgs boson differential distributions at hadron–hadron colliders.

The present paper focuses on the tapered planar waveguide solution for an initial given wave form. The algorithm is constructed in distributions space such that the calculations can be actually computed by taking some hypothesis regarding the mode series which appear. The whole argument leads to the conclusion that the wave is compressed towards the waveguide in the direction of tilting, leading thereafter to a focusing phenomena. We present two alternative constructions in order to compute the convolution which gives the wave inside the waveguide. The hypothesis are providing results in good approximation with the real evolution of the field within the definition domain.

Due to the lack of regularity of the solutions to the hydrostatic approximation of Navier–Stokes equations, an energy identity cannot be deduced. By including certain nonlinear perturbations to the hydrostatic approximation equations, the solutions to the perturbed problem are smooth enough so that they satisfy the corresponding energy identity. The perturbations considered in this paper are of the monotone class. Three kinds of problems are then studied. To do that, we introduce a functional setting and show in every case that the set of smooth functions with compact support is dense in the space where we search for solutions. When the perturbations are small enough in a certain sense, the solutions of the perturbed problem are close to those of the original one. As a result, this gives a new proof of the existence of solutions to the hydrostatic approximation of Navier–Stokes equations. Finally, this regularization technique has been applied to the analysis of a one-equation hydrostatic turbulence model.

We discuss the use of the ordered weighted averaging (OWA) operator in multi-criteria decision problems as a means of aggregating the individual criteria satisfactions. We emphasize the need for ordering the arguments, the criteria satisfactions, when using the OWA operator. We consider the situation where the criteria satisfactions have some uncertainty, are finite probability distributions and note the requirement of having to order probability distributions. We introduce the idea of using pairwise stochastic dominance to provide the necessary ordering relationship over the probability distributions. We note that while this approach is appropriate, it is often not possible, since the presence of a stochastic dominance relationship between all pairs of probability distributions is not always the case, the relationship is not complete. To circumvent this problem we introduce an approach called the probabilistic exceedance method (PEM), which allows us to provide a surrogate for the OWA aggregation of probability distributions that doesn't require a linear ordering over the probability distributions. We look at this in both the cases in which the criteria have equal and unequal importances.

The applicability of the fatigue-life distribution of Birnbaum–Saunders for modeling the components of partially prestressed concrete girder bridges is argued by the agreement of the calculated distribution with a large data set comprised of many small censored samples, each put into standardized form, of the fatigue-lives of bridge components. Also, the Weibull and lognormal distributions are compared by the appropriateness of the assumptions in their derivation to the cumulative damage to bridge components.

The maximum likelihood estimates of the parameters of the fatigue lives of the prestressing steel, mild steel, and concrete are found from randomly censored samples, for several censored data-sets under various stress regimes. The parameters are postulated to be log-linear functions of the maximum stress and stress regimes. Weighted least-square regression is applied to determine the unknown coefficients. To compensate for differences in the precision of the estimates obtained under each stress regime the variances are calculated by the bootstrap statistical method. This procedure gives an estimated distribution of the fatigue life with its parameters functions of the stress regime which is applicable under virtually any realistic condition.

In recent years, a considerable number of physicists have started applying physics concepts and methods to understand economic phenomena. The term "Econophysics" is sometimes used to describe this work. Economic fluctuations can have many repercussions, and understanding fluctuations is a topic that many physicists have contributed to in recent years. Further, economic systems are examples of complex interacting systems for which a huge amount of data exist and it is possible that the experience gained by physicists in studying fluctuations in physical systems might yield new results in economics. Much recent work in econophysics is focused on understanding the peculiar statistical properties of price fluctuations in financial time series. In this talk, we discuss three recent results. The first result concerns the probability distribution of stock price fluctuations. This distribution decreases with increasing fluctuations with a power-law tail well outside the Lévy stable regime and describes fluctuations that differ by as much as 8 orders of magnitude. Further, this nonstable distribution preserves its functional form for fluctuations on time scales that differ by 3 orders of magnitude, from 1 min up to approximately 10 days. The second result concerns the accurate quantification of volatility correlations in financial time series. While price fluctuations themselves have rapidly decaying correlations, the volatility estimated by using either the absolute value or the square of the price fluctuations has correlations that decay as a power-law and persist for several months. The third result bears on the application of random matrix theory to understand the correlations among price fluctuations of any two different stocks. We compare the statistics of the cross-correlation matrix constructed from price fluctuations of the leading 1000 stocks and a matrix with independent random elements, i.e., a random matrix. Contrary to first expectations, we find little or no deviation from the universal predictions of random matrix theory for all but a few of the largest eigenvalues of the cross-correlation matrix.

In 1954, it was proved that if f is infinitely differentiable in the interval I and some derivatives (of order depending on x) vanish at each x, then f is a polynomial. Later, it was generalized for the multivariable case. A further extension for distributions is possible. If Ω ⊆ Rⁿ is a non-empty connected open set, and for every , there exists m(φ) ∈ N such that (D^αu)(φ) = 0 for all multi-indices α satisfying ‖α‖ = m(φ), then u is a polynomial (in distributional sense).

In this paper, the inversion formula of the continuous Bessel wavelet transform of distributions in ${\beta }_{\mu }^{\prime }$-space and its associated results with Bessel wavelet kernel containing non-vanishing moments of even orders weighted by $x^{\mu +\frac{1}{2}}$ are investigated by exploiting the theory of Hankel transform.

In this paper, the inversion formula and the continuity of the continuous Bessel wavelet transform of distributions in $D_{L^2}^{\prime }$- type space are investigated. For the uniqueness of the inversion formula, the Bessel wavelet kernel of nonvanishing moments is considered.

New higher dimensional distributions are introduced in the framework of Clifford analysis. They complete the picture already established in previous work, offering unity and structural clarity. Amongst them are the building blocks of the principal value distribution, involving spherical harmonics, considered by Horváth and Stein.

Around the central theme of "square root" of the Laplace operator it is shown that the classical Riesz potentials of the first and of the second kind allow for an explicit expression of so-called Hilbert-Dirac convolution operators involving natural and complex powers of the Dirac operator.

In this paper, we evaluate some products of distributions in Colombeau algebra of generalized functions. In the classical theory of Schwartz distributions, multiplication of distributions is not defined for two arbitrary singular distributions. The properties of the Colombeau algebra allow us to calculate products of singular distributions which are not defined in the classical theory. The notion of association in Colombeau algebra of generalized functions allows us the results obtained in this way to be considered as products in the classical theory of distributions. The definition of the Colombeau product of distributions can be considered as generalization of their classical product in Schwartz theory.

Tauberian-type results characterizing the quasiasymptotic behavior of polynomial multiplication of Lizorkin distributions in terms of their Stockwell transforms are obtained. Some Abelian-type results relating the quasiasymptotic behavior and quasiasymptotic boundedness of Lizorkin distributions to the asymptotic behavior of their Stockwell transforms are given. Several Abelian-type results for the generalized wavelet transform are also presented.

The text and associated Supplemental Materials contribute internally consistent and therefore entirely comparable regional, temporal, and sectoral risk profiles to a growing literature on regional economic vulnerability to climate change. A large collection of maps populated with graphs of Monte-Carlo simulation results support a communication device in this regard — a convenient visual that we hope will make comparative results tractable and credible and resource allocation decisions more transparent. Since responding to climate change is a risk-management problem, it is important to note that these results address both sides of the risk calculation. They characterize likelihood distributions along four alternative emissions futures (thereby reflecting the mitigation side context); and they characterize consequences along these transient trajectories (which can thereby inform planning for the iterative adaptation side). Looking across the abundance of sectors that are potentially vulnerable to some of the manifestations of climate change, the maps therefore hold the potential of providing comparative information about the magnitude, timing, and regional location of relative risks. This is exactly the information that planners who work to protect property and public welfare by allocating scarce resources across competing venues need to have at their disposal — information about relative vulnerabilities across time and space and contingent on future emissions and future mitigation. It is also the type of information that integrated assessment researchers need to calibrate and update their modeling efforts — scholars who are exemplified by Professor Nordhaus who created and exercised the Dynamic Integrated Climate-Economy and Regional Integrated Climate-Economy models.

The text and associated Supplemental Materials contribute internally consistent and therefore entirely comparable regional, temporal, and sectoral risk profiles to a growing literature on regional economic vulnerability to climate change. A large collection of maps populated with graphs of Monte-Carlo simulation results support a communication device in this regard — a convenient visual that we hope will make comparative results tractable and credible and resource allocation decisions more transparent. Since responding to climate change is a risk-management problem, it is important to note that these results address both sides of the risk calculation. They characterize likelihood distributions along four alternative emissions futures (thereby reflecting the mitigation side context); and they characterize consequences along these transient trajectories (which can thereby inform planning for the iterative adaptation side). Looking across the abundance of sectors that are potentially vulnerable to some of the manifestations of climate change, the maps therefore hold the potential of providing comparative information about the magnitude, timing, and regional location of relative risks. This is exactly the information that planners who work to protect property and public welfare by allocating scarce resources across competing venues need to have at their disposal — information about relative vulnerabilities across time and space and contingent on future emissions and future mitigation. It is also the type of information that integrated assessment researchers need to calibrate and update their modeling efforts — scholars who are exemplified by Professor Nordhaus who created and exercised the Dynamic Integrated Climate-Economy and Regional Integrated Climate-Economy models.

Changes of field variables may lead to multivalued fields which do not satisfy the Schwarz integrability conditions. Their quantum field theory needs special care as is illustrated here in applications to superfluid and superconducting phase transitions. Extending the notions that first qantization governs fluctuating orbits while second quantization deals with fluctuating field, the theory of multivalued fields may be considered as a theory of third quantization. The lecture is an introduction to my new book on this subject.

A flat Friedmann universe filled with a mixture of anti-Chaplygin gas and dust-like matter evolves into a future soft singularity, where despite infinite tidal forces the geodesics can be continued. In the singularity the pressure of the anti-Chaplygin gas diverges, while its energy density is zero. The dust energy density however does not vanish, neither does the Hubble parameter, which implies further expansion, if its evolution is to be continuous. If so, the energy density and the pressure of the anti-Chaplygin gas would become ill-defined, hence only a contraction would be allowed. Paradoxically, the universe in this cosmological model would have to expand and contract simultaneously. The paradox can be avoided by redefining the anti-Chaplygin gas in a distributional sense. Then the Hubble parameter could be mirrored to have a jump at the singularity, allowing for a subsequent contraction. With this modification the set of Friedmann, Raychaudhuri and continuity equations are all obeyed both at the singularity and in its vicinity.