An explicit calculation of Weil-Petersson forms

In this pedagogical review, we summarize the mathematical basis and practical hints for the explicit analytical computation of spectral sums that involve the eigenvalues of the Laplace operator in simple domains such as [Formula: see text]-dimensional balls (with [Formula: see text]), an annulus, a spherical shell, right circular cylinders, rectangles and rectangular cuboids. Such sums appear as spectral expansions of heat kernels, survival probabilities, first-passage time densities, and reaction rates in many diffusion-oriented applications. As the eigenvalues are determined by zeros of an appropriate linear combination of a Bessel function and its derivative, there are powerful analytical tools for computing such spectral sums. We discuss three main strategies: representations of meromorphic functions as sums of partial fractions, Fourier–Bessel and Dini series, and direct evaluation of the Laplace-transformed heat kernels. The major emphasis is put on a pedagogic introduction, the practical aspects of these strategies, their advantages and limitations. The review gathers many summation formulas for spectral sums that are dispersed in the literature.

We study the areal Mahler measure of the two-variable, $k$-parameter family $x+y+k$ and prove explicit formulas that demonstrate its relation to the standard Mahler measure of these polynomials. The proofs involve interpreting the areal Mahler measure as a random walk in the complex plane and utilizing the areal analogue of the Zeta Mahler function to arrive at the result. Using similar techniques, we also present formulas for a three-variable family $(x+1)(y+1)+kz$ in terms of the standard Mahler measure, along with terms that involve certain hypergeometric functions. For both families, we show that its areal Mahler measure is, up to elementary functions, a linear combination of the normal Mahler measure and the volume of the Deninger cycle of the corresponding family.

Creep affects the long-term deformation of concrete structures. Nonlinear creep further overestimates the safety factor of structures and affects the safety service performance. The coupling of creep and a damage model considering the rate effect is conducive to accurate prediction of nonlinear creep, but the iterative process of strain makes the calculation method more complex. The purpose of this study is to propose a nonlinear creep explicit method that considers the damage evolution of concrete under compression. Two groups of axial compression members with compressive stresses of 0.2 fc and 0.4 fc were made. Considering the law of concrete damage evolution under uniaxial compression, coupled with elastic creep and damage incremental strain, the lower limit of the medium stress level that gives rise to nonlinear creep is analyzed. The concrete nonlinear creep amplification coefficient with a loading age of 28 days and loading duration of 360 days is studied with consideration for the uncertainty of relative humidity and the theoretical thickness of the component. On this basis, the explicit calculation formula of the nonlinear creep amplification coefficient related to the concrete axial compressive strength and stress level is given. The results indicate that the nonlinear creep amplification coefficient increases nonlinearly with an increase in the stress level, and, when the compressive stress level ratio is higher than 0.6, the nonlinear creep amplification coefficient increases significantly; when the stress level is determined, the creep amplification coefficient decreases gradually with an increase in the compressive strength of the concrete. It is suggested that a stress level range of 0.35~0.75 should be used for the study of a nonlinear creep amplification factor under the medium stress state.

Based on the Kripke structure, linear temporal logic and generalized possibility measure, this article studies the model checking problems of generalized possibilistic fuzzy linear temporal logic (GPoFTL). The generalized possibilistic Kripke structure is introduced to describe the system model. The syntax of GPoFTL, which includes fuzzy temporal operators such as ``soon'', ``presently'', ``gradually'', ``within'', ``last'', ``nearly always'', ``almost alwayss'', ``in the long distant future'', ``in the middle of'', ``nearly until'' and ``almost until'', and its language semantics and path semantics under generalized possibility measure are given. Next, we explain the semantics of fuzzy temporal operators through some examples and prove that GPoFTL is an extension of generalized possibilistic linear temporal logic in fuzzy time temporal logic. Furthermore, the GPoFTL model checking algorithm is given by explicit calculation formulas one by one for any GPoFTL formula involving fuzzy temporal operators using fuzzy matrix operations. Last, the algorithm of necessary threshold model checking of GPoFTL is studied using automata theory and its time complexity is discussed.

A large part of research in noncommutative geometry is devoted to various far-reaching generalizations of the Atiyah–Singer index theory of elliptic operators on compact manifolds. Following Kasparov [K1], the notion of an elliptic operator on a manifold is replaced by that of a Fredholm module (or K-cycle) over an algebra of operators on Hilbert space. A number of index theorems for Fredholm modules associated to geometric data have been obtained by Kasparov [K1], Connes (see [C3] and the literature cited therein) and others. In order to handle concrete index problems it is not only necessary to dispose of an index theorem but also to provide an index formula which allows the explicit calculation of indices. The classical index formula of Atiyah and Singer is obtained from the index theorem by applying the Chern character in topological K-homology. This motivates the search for character formulas of Fredholm modules (K-cycles) that define a Chern character on the K-homology groups of an algebra of operators. It was this search for a Chern character in K-homology which led A. Connes to the invention of cyclic cohomology [C2]. He obtained various explicit character formulas [C1,2,3], which depend on the degree of analytic regularity (summability) of the given Fredholm module. Whereas the classical index problems of Atiyah–Singer are all finitely summable in the sense of [C2] there are many examples of Fredholm modules over (noncommutative) algebras which are infinite dimensional (not finitely summable). Characters of Fredholm modules have been calculated in many finitely summable cases, but as far as the author knows the character of a θ-summable, infinitedimensional (unbounded) Fredholm module [C1] has not yet been determined in a single case.

Solitary waves are commonly used to model long waves, such as tsunamis. Limited research is available for such waves propagating in converging water bodies, such as fjords, bays, and estuaries, where reflection and large amplified waves can be observed. Therefore, an extended analytical solution is presented to calculate the non-dimensional asymptotic amplified wave amplitude Aw and its trajectory for the oblique interaction of two distinct weakly non-linear solitary waves based on line-soliton solutions of the Kadomtsev–Petviashvili equation. Furthermore, an analytical model is developed for wave propagation in a converging geometry featuring Mach reflection. An oblique angle ψ̂ of Mach stem and a general stem angle φ are defined, along with their explicit calculation formulas. The Simulating WAves till SHore model based on the non-hydrostatic, non-linear shallow water equations, is used to validate the solutions and the analytical model numerically. The simulation results for Aw and ψ̂ show asymptotic behavior and converge toward the prediction of the extended solutions. At an average oblique angle of 17° between two solitary waves, the discrepancy between the numerical and analytical results for Aw is up to −9.0% for the first-order and +8.3% for the second-order solutions, respectively. For the most asymmetrical converging geometry examined, the simulated result of φ=14.2° remains 92.0% of the theoretical value. These findings indicate that the proposed solutions can readily and accurately predict the angles and amplitudes of amplified waves resulting from the interaction of weakly non-linear solitary waves in extensive converging water bodies.

The Turing patterns of reaction-diffusion equations defined over a square region are more complex because of the D4-symmetry of the spatial region. This leads to the occurrence of multiple equivariant Turing bifurcations. In this paper, taking the FHN model as an example, we give a explicit calculation formula of normal form for the simple and double Turing bifurcation of the reaction-diffusion equation with Dirichlet boundary conditions and defined on a square space, and we also obtain a method for the calculation of the existence of spatially inhomogeneous steady-state solutions. This paper provides a theoretical basis for exploring and predicting the pattern formation of spatial multimode interaction.

In this paper, a sequential quadratically constrained quadratic programming method of feasible directions is proposed for the optimization problems with nonlinear inequality constraints. At each iteration of the proposed algorithm, a feasible direction of descent is obtained by solving only one subproblem which consist of a convex quadratic objective function and simple quadratic inequality constraints without the second derivatives of the functions of the discussed problems, and such a subproblem can be formulated as a second-order cone programming which can be solved by interior point methods. To overcome the Maratos effect, an efficient higher-order correction direction is obtained by only one explicit computation formula. The algorithm is proved to be globally convergent and superlinearly convergent under some mild conditions without the strict complementarity. Finally, some preliminary numerical results are reported.

Up to now, the attempts to give a complete probabilistic description of estimated reflection coefficients have been developed under simplifying assumptions, being these either long data records, allowing us to assume the estimates to be Gaussianly distributed, or white noise signals. On the contrary, the purpose of this paper is to present explicit formulae for the marginal probability density functions of the estimated reflection coefficients of an arbitrary second-order stationary Gaussian autoregressive process; such expressions are consistent with the asymptotic distributions of the estimates, and are empirically verified to be also valid for short records (some tenths of data points). The probabilistic model involves a parametric family of distributions obtained by using a classical result in Statistical Theory; its parameters are then redefined so as to achieve the desired asymptotic behaviour of the first and second order moments of the distributions.

We study random recursive constructions in which the contracting vectors have different distributions at different stages. With such constructions, the one parameter family of martingales are introduced and the probabilistic behaviours of the limit random objects (not identically distributed) are discussed. We prove that the random fractal associated with such construction has a constant Hausdorff dimension almost surely and give an explicit formula to determine it. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Let k be a field of characteristic zero, a k-Lie algebra, the symmetrization map. The PBW quantization is the one parameter family of associative products: where is the homogeneous component of degree of the map , . In this paper we give an explicit formula for B. As an application, we prove that for each , is a bidifferential operator of order .

We derive an explicit formula for the asymptotic slope of the Aubin–Yau functional along a Bergman geodesic on a surface of complex dimension 2, extending the work of Phong–Sturm (On asymptotics for the Mabuchi energy functional, World Science Publishing, River Edge, 2004) on Riemann surfaces. This is equivalent to an explicit calculation of the Donaldson–Futaki invariant of a test configuration. The slope is given as a rational linear combination of period integrals of rational functions that sum to a rational number. The result gives a way to check directly whether a two-dimensional projective variety is K-stable.

Explicit computation of some families of Hurwitz numbers

We study aspects of the Wasserstein distance in the context of self-similar measures. Computing this distance between two measures involves minimising certain moment integrals over the space of couplings, which are measures on the product space with the original measures as prescribed marginals. We focus our attention on self-similar measures associated to equicontractive iterated function systems consisting of two maps on the unit interval and satisfying the open set condition. We are particularly interested in understanding the restricted family of self-similar couplings and our main achievement is the explicit computation of the 1st and 2nd moment integrals for such couplings. We show that this family is enough to yield an explicit formula for the 1st Wasserstein distance and provide non-trivial upper and lower bounds for the 2nd Wasserstein distance for these self-similar measures.

Chern classes and characteristic cycles of determinantal varieties