ON FUNDAMENTAL FOURIER COEFFICIENTS OF SIEGEL CUSP FORMS OF DEGREE 2

Let F be a Siegel cusp form of degree $2$ , even weight $k \ge 2$ , and odd square-free level N. We undertake a detailed study of the analytic properties of Fourier coefficients $a(F,S)$ of F at fundamental matrices S (i.e., with $-4\det (S)$ equal to a fundamental discriminant). We prove that as S varies along the equivalence classes of fundamental matrices with $\det (S) \asymp X$ , the sequence $a(F,S)$ has at least $X^{1-\varepsilon }$ sign changes and takes at least $X^{1-\varepsilon }$ ‘large values’. Furthermore, assuming the generalized Riemann hypothesis as well as the refined Gan–Gross–Prasad conjecture, we prove the bound $\lvert a(F,S)\rvert \ll _{F, \varepsilon } \frac {\det (S)^{\frac {k}2 - \frac {1}{2}}}{ \left (\log \lvert \det (S)\rvert \right )^{\frac 18 - \varepsilon }}$ for fundamental matrices S.

A numerical scheme for the solution of neutral integro-differential equations including variable delay

In this study, an effective numerical technique has been introduced for finding the solutions of the first-order integro-differential equations including neutral terms with variable delays. The problem has been defined by using the neutral integro-differential equations with initial value. Then, an alternative numerical method has been introduced for solving these type of problems. The method is expressed by fundamental matrices, Laguerre polynomials with their matrix forms. Besides, the solution has been obtained by using the collocation points with regard to the reduced system of algebraic equations and Laguerre series.

On the Benefits of Populations for the Exploitation Speed of Standard Steady-State Genetic Algorithms

It is generally accepted that populations are useful for the global exploration of multi-modal optimisation problems. Indeed, several theoretical results are available showing such advantages over single-trajectory search heuristics. In this paper we provide evidence that evolving populations via crossover and mutation may also benefit the optimisation time for hillclimbing unimodal functions. In particular, we prove bounds on the expected runtime of the standard ($$\mu +1$$ μ + 1 ) GA for OneMax that are lower than its unary black box complexity and decrease in the leading constant with the population size up to $$\mu =o\left( \sqrt{\log n}\right) $$ μ = o log n . Our analysis suggests that the optimal mutation strategy is to flip two bits most of the time. To achieve the results we provide two interesting contributions to the theory of randomised search heuristics: (1) A novel application of drift analysis which compares absorption times of different Markov chains without defining an explicit potential function. (2) The inversion of fundamental matrices to calculate the absorption times of the Markov chains. The latter strategy was previously proposed in the literature but to the best of our knowledge this is the first time is has been used to show non-trivial bounds on expected runtimes.

Lamb waves in multilayered media: 3D and 6D formalisms

A mathematical model for analyzing Lamb waves propagating in stratified media with arbitrary elastic anisotropy is worked out. The model incorporates a combined Fundamental Matrix (FM) and Modified Transfer Matrix (MTM) methods. Multilayered unbounded plates with different types of boundary conditions imposed on the outer surfaces are considered. Closed form fundamental matrices and secular equations for dispersion relations are derived.

An Efficient and Flexible Solution for Camera Autocalibration from N≥3 Views

This paper presents an efficient and flexible solution for camera autocalibration from N≥3 views, given image correspondences and zero (or known) skew only. The knowledge is not required on camera motion, 3D information, scene, or internal constraints. Our method is essentially based only on the fundamental matrices and its main virtues are threefold. Firstly, it is shown that, in the center-oriented metric coordinates, the focal length and aspect ratio can be estimated independent of considerable principle point shift (PPs). Thus, our method includes recursive steps: estimating focal length and aspect ratio and then calculating the PPs. Secondly, the optimal geometric constraints are selected for calibration by using error propagation analyses. Thirdly, the Levenberg–Marquardt algorithm is adopted for the fast final refinement of four internal parameters. Our method is fast and efficient to derive a unique calibration. Besides, this method can be applied to calibrate the focal length from two views, without requiring the prior knowledge of PPs. Good performance of our method is evaluated and confirmed in both the simulation experiments and the practical tests.