A generalized Fourier–Hermite method for the Vlasov–Poisson system

A generalized Fourier–Hermite semi-discretization for the Vlasov–Poisson equation is introduced. The formulation of the method includes as special cases the symmetrically-weighted and asymmetrically-weighted Fourier–Hermite methods from the literature. The numerical scheme is formulated as a weighted Galerkin method with two separate scaling parameters for the Hermite polynomial and the exponential part of the new basis functions. Exact formulas for the error in mass, momentum, and energy conservation of the method depending on the parameters are devised and $$L^2$$ L 2 stability is discussed. The numerical experiments show that an optimal choice of the additional parameter in the generalized method can yield improved accuracy compared to the existing methods, but also reveal the distinct stability properties of the symmetrically-weighted method.

A New Hierarchy for Automaton Semigroups

We define a new strict and computable hierarchy for the family of automaton semigroups, which reflects the various asymptotic behaviors of the state-activity growth. This hierarchy extends that given by Sidki for automaton groups, and also gives new insights into the latter. Its exponential part coincides with a notion of entropy for some associated automata. We prove that the Order Problem is decidable whenever the state-activity is bounded. The Order Problem remains open for the next level of this hierarchy, that is, when the state-activity is linear. Gillibert showed that it is undecidable in the whole family. We extend the aforementioned hierarchy via a semi-norm making it more coarse but somehow more robust and we prove that the Order Problem is still decidable for the first two levels of this alternative hierarchy.

Response Characteristics of Systems With Parametric Excitation Through Damping and Stiffness

Floquet theory is combined with harmonic balance to study parametrically excited systems with combination of both time varying damping and stiffness. An approximated solution having an exponential part with unknown exponents and a periodic term consisting of a truncated Fourier series is considered. When applied to a system with parametric damping and stiffness the analysis shows that combination of parametric damping and stiffness alters stability characteristics, particularly in the primary and superharmonic instabilities comparing to the system with only parametric damping or stiffness. We also look at the initial conditions response and its frequency content. The second excitation harmonic in the system with parametric damping is seen to disrupt the coexistence phenomenon which is observed in the parametric damping case.

A Formula for the Möbius Function of the Permutation Poset Based on a Topological Decomposition

International audience The poset P of all permutations ordered by pattern containment is a fundamental object of study in the field of permutation patterns. This poset has a very rich and complex topology and an understanding of its Möbius function has proved particularly elusive, although results have been slowly emerging in the last few years. Using a variety of topological techniques we present a two term formula for the Mo ̈bius function of intervals in P. The first term in this formula is, up to sign, the number of so called normal occurrences of one permutation in another. Our definition of normal occurrences is similar to those that have appeared in several variations in the literature on the Möbius function of this and other posets, but simpler than most of them. The second term in the formula is (still) complicated, but we conjecture that it equals zero for a significant proportion of intervals. We present some cases where the second term vanishes and others where it is nonzero. Computing the Möbius function recursively from its definition has exponential complexity, whereas the computation of the first term in our formula is polynomial and the exponential part is isolated to the second term, which seems to often vanish. This is thus the first polynomial time formula for the Möbius function of what appears to be a large proportion of all intervals of P.

Improvement of Antipodal Vivaldi Antenna Performance for Wireless Application

This paper discusses about miniaturized size of the antipodal Vivaldi antenna by using triangular and circular slots on the Vivaldi antenna. A normal antipodal Vivaldi antenna using FR4 epoxy substrate with a dielectric constant of 4.4 and thickness 1.6mm has been designed. Then, the triangular and the circular slots are added at the upper layer of the miniaturized size of the antenna, the triangular slot plays a major role in reduction of size of the antenna and also results in the better improvement of return loss, gain and bandwidth operated at a frequency in the range of 3GHz to 25GHz and the circular slot placed at the exponential part of the antenna helps to reduce excessive mutual coupling between different slots of the antenna. The feeding technique used for designing of this antenna is micro strip feedline. The proposed antenna results in different applications by incrementing the frequency. The results of simulation are been realized using HFSS 13.0, a high frequency simulator structure program. The structured antipodal Vivaldi radio wire and the triangular and round openings antipodal Vivaldi receiving wire are manufactured. The arrival misfortune reaction and the increase of the manufactured radio wires are estimated and contrasted and the re-enactment results

Response Characteristics of Systems With Two-Harmonic Parametric Excitation

Floquet theory is combined with harmonic balance to study parametrically excited systems with two harmonics of excitation, where the second harmonic has twice the frequency of the first one. An approximated solution composed of an exponential part with unknown exponents and a periodic term consisting of a truncated Fourier series is considered. When applied to a two-harmonic Mathieu equation the analysis shows that the second harmonic alters stability characteristics, particularly in the primary and superharmonic instabilities. We also look at the initial conditions response and its frequency content. The second excitation harmonic in the system with parametric damping is seen to disrupt the coexistence phenomenon which is observed in the single-harmonic case.

Approximate Floquet Analysis of Parametrically Excited Multi-Degree-of-Freedom Systems With Application to Wind Turbines

General responses of multi-degrees-of-freedom (MDOF) systems with parametric stiffness are studied. A Floquet-type solution, which is a product between an exponential part and a periodic part, is assumed, and applying harmonic balance, an eigenvalue problem is found. Solving the eigenvalue problem, frequency content of the solution and response to arbitrary initial conditions are determined. Using the eigenvalues and the eigenvectors, the system response is written in terms of “Floquet modes,” which are nonsynchronous, contrary to linear modes. Studying the eigenvalues (i.e., characteristic exponents), stability of the solution is investigated. The approach is applied to MDOF systems, including an example of a three-blade wind turbine, where the equations of motion have parametric stiffness terms due to gravity. The analytical solutions are also compared to numerical simulations for verification.

Floquet-Based Analysis of General Responses of the Mathieu Equation

Solutions to the linear unforced Mathieu equation, and their stabilities, are investigated. Floquet theory shows that the solution can be written as a product between an exponential part and a periodic part at the same frequency or half the frequency of excitation. In the current work, an approach combining Floquet theory with the harmonic balance method is investigated. A Floquet solution having an exponential part with an unknown exponential argument and a periodic part consisting of a series of harmonics is assumed. Then, performing harmonic balance, frequencies of the response are found and stability of the solution is examined over a parameter set. The truncated solution is consistent with an existing infinite series solution for the undamped case. The truncated solution is then applied to the damped Mathieu equation and parametric excitation with two harmonics.

Quasi-analytical method for frequency-to-time conversion in CSEM applications

Frequency-to-time transformations are of interest to controlled-source electromagnetic methods when time-domain data are inverted for a subsurface resistivity model by numerical frequency-domain modeling at a selected, small number of frequencies whereas the data misfit is determined in the time domain. We propose an efficient, Prony-type method using frequency-domain diffusive-field basis functions for which the time-domain equivalents are known. Diffusive fields are characterized by an exponential part whose argument is proportional to the square root of frequency and a part that is polynomial in integer powers of the square root of frequency. Data at a limited number of frequencies suffice for the transformation back to the time. In the exponential part, several diffusion-time values must be chosen. Once a suitable range of diffusion-time values are found, the method is quite robust in the number of values used. The highest power in the polynomial part can be determined from the source and receiver type. When the frequency-domain data are accurately approximated by the basis functions, the time-domain result is also accurate. This method is accurate over a wider time range than other methods and has the correct late-time asymptotic behavior. The method works well for data computed for layered and 3D subsurface models.