Introduction to Singularity Theory with Historical Remarks

Few-Cycle Infrared Pulse Evolving in FEL Oscillators and Its Application to High-Harmonic Generation for Attosecond Ultraviolet and X-ray Pulses

Atoms ◽

10.3390/atoms9010015 ◽

2021 ◽

Vol 9 (1) ◽

pp. 15

Author(s):

Ryoichi Hajima

Keyword(s):

Harmonic Generation ◽

Short Pulse ◽

High Harmonic ◽

High Harmonic Generation ◽

Pulse Generation ◽

Solid State Lasers ◽

Linear Accelerators ◽

X Ray ◽

Ultrafast Science ◽

Historical Remarks

Generation of few-cycle optical pulses in free-electron laser (FEL) oscillators has been experimentally demonstrated in FEL facilities based on normal-conducting and superconducting linear accelerators. Analytical and numerical studies have revealed that the few-cycle FEL lasing can be explained in the frame of superradiance, cooperative emission from self-bunched systems. In the present paper, we review historical remarks of superradiance FEL experiments in short-pulse FEL oscillators with emphasis on the few-cycle pulse generation and discuss the application of the few-cycle FEL pulses to the scheme of FEL-HHG, utilization of infrared FEL pulses to drive high-harmonic generation (HHG) from gas and solid targets. The FEL-HHG enables one to explore ultrafast science with attosecond ultraviolet and X-ray pulses with a MHz repetition rate, which is difficult with HHG driven by solid-state lasers. A research program has been launched to develop technologies for the FEL-HHG and to conduct a proof-of-concept experiment of FEL-HHG.

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Stochastic P-bifurcation in a bistable Van der Pol oscillator with fractional time-delay feedback under Gaussian white noise excitation

Advances in Difference Equations ◽

10.1186/s13662-019-2356-1 ◽

2019 ◽

Vol 2019 (1) ◽

Author(s):

Yajie Li ◽

Zhiqiang Wu ◽

Guoqi Zhang ◽

Feng Wang ◽

Yuancen Wang

Keyword(s):

Time Delay ◽

White Noise ◽

Gaussian White Noise ◽

Singularity Theory ◽

Van Der Pol Oscillator ◽

Order System ◽

Damping Force ◽

Van Der Pol ◽

Restoring Force ◽

White Noise Excitation

Abstract The stochastic P-bifurcation behavior of a bistable Van der Pol system with fractional time-delay feedback under Gaussian white noise excitation is studied. Firstly, based on the minimal mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of damping force and restoring force, and the original system is further simplified to an equivalent integer order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and the critical parametric conditions for stochastic P-bifurcation of system amplitude are determined according to the singularity theory. Finally, the types of stationary PDF curves of system amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical solutions and Monte Carlo simulation results verifies the theoretical analysis in this paper.

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Theory of Effective Stress in Soil and Rock and Implications for Fracturing Processes: A Review

Geosciences ◽

10.3390/geosciences11030119 ◽

2021 ◽

Vol 11 (3) ◽

pp. 119

Author(s):

Vincenzo Guerriero ◽

Stefano Mazzoli

Keyword(s):

Porous Media ◽

Effective Stress ◽

Biological Tissues ◽

Theory Of Elasticity ◽

Past Century ◽

Metal Powders ◽

Science Community ◽

Experimental Approaches ◽

Historical Remarks ◽

Open Issues

The effective stress principle (ESP) plays a basic role in geology and engineering problems as it is involved in fundamental issues concerning strain and failure of rock and soil, as well as of other porous materials such as concrete, metal powders, biological tissues, etc. Although since its introduction in the 1920s the main ESP aspects have been unravelled and theoretically derived, these do not appear to have been always entirely perceived by many in the science community dealing with ESP-related topics but having little familiarity with the complex theories of porous media and poroelasticity. The purpose of this review is to provide a guidance for the reader who needs an updated overview of the different theoretical and experimental approaches to the ESP and related topics over the past century, with particular reference to geological fracturing processes. We begin by illustrating, after some introductive historical remarks, the basic theory underlying the ESP, based on theory of elasticity methods. Then the different ESP-related theories and experimental results, as well as main interpretations of rock jointing and fracturing phenomena, are discussed. Two main classical works are then revisited, and a rigorous ESP proof is derived. Such a proof is aimed at geologists, engineers and geophysicists to become more familiar with theories of porous media and poroelasticity, being based on the classical theory of elasticity. The final part of this review illustrates some still open issues about faulting and hydraulic fracturing in rocks.

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Historical remarks on Suslin's problem

Set Theory, Arithmetic, and Foundations of Mathematics ◽

10.1017/cbo9780511910616.002 ◽

2011 ◽

pp. 1-12 ◽

Cited By ~ 1

Author(s):

Akihiro Kanamori

Keyword(s):

Historical Remarks

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Bifurcations of a Nonlinear Two-Degree-of-Freedom System Under Narrow-Band Stochastic Excitation

Journal of Vibration and Acoustics ◽

10.1115/1.2930228 ◽

1992 ◽

Vol 114 (1) ◽

pp. 24-31

Author(s):

R. Lin ◽

K. Huseyin ◽

C. W. S. To

Keyword(s):

Relaxation Time ◽

Correlation Time ◽

Averaging Method ◽

Narrow Band ◽

Singularity Theory ◽

Random Parameter ◽

Degree Of Freedom ◽

Stochastic Excitation ◽

Static Approach ◽

Two Degree Of Freedom

In this paper, bifurcations of a nonlinear two-degree-of-freedom system subjected to a narrow-band stochastic excitation are investigated. Under the assumption that the correlation time greatly exceeds the relaxation time, a quasi-static approach combined with averaging method is adopted to obtain the bifurcation equations, and the singularity theory is applied to analyze the bifurcations. It is demonstrated that bifurcation patterns jump from one to another due to the influence of a random parameter. The probabilities of the jumping bifurcation patterns are given.

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The Ermakov equation: A commentary

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm0802146l ◽

2008 ◽

Vol 2 (2) ◽

pp. 146-157 ◽

Cited By ~ 63

Author(s):

P.G.L. Leach ◽

S.K. Andriopoulos

Keyword(s):

Nonlinear Systems ◽

Differential Equations ◽

Ordinary Differential Equations ◽

English Translation ◽

Singularity Theory ◽

Discrete Math ◽

Previous Article ◽

Short History ◽

History Of ◽

Modern Context

We present a short history of the Ermakov equation with an emphasis on its discovery by thewest and the subsequent boost to research into invariants for nonlinear systems although recognizing some of the significant developments in the east. We present the modern context of the Ermakov equation in the algebraic and singularity theory of ordinary differential equations and applications to more divers fields. The reader is referred to the previous article (Appl. Anal. Discrete math., 2 (2008), 123-145) for an english translation of Ermakov's original paper.

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