scholarly journals Detailed Proof of Classical Gagliardo–Nirenberg Interpolation Inequality with Historical Remarks

2021 ◽  
Vol 40 (2) ◽  
pp. 217-236
Author(s):  
Alberto Fiorenza ◽  
Maria Rosaria Formica ◽  
Tomáš Roskovec ◽  
Filip Soudský
Keyword(s):  
Interpolation Inequality ◽  
Detailed Proof ◽  
Historical Remarks

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

Atoms ◽  
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.

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

Geosciences ◽  
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.

scholarly journals 6. Researches in Thermometry

1880 ◽  
Vol 10 ◽  
pp. 562-564
Author(s):  
Edmund J. Mills
Keyword(s):  
Simple Method ◽  
Detailed Proof ◽  
Calibration Unit

AbstractHaving had occasion in the course of my work to investigate some of the properties of the mercurial thermometer, I have obtained a series of results which are comprised in a memoir now submitted to the Society. A brief summary of these is given in the following abstract.1. After describing a simple method of calibrating a thermometer, I give a detailed proof (following Pierre) that the calibration unit gradually undergoes a slight diminution in value. In the course of five years this may amount, in a new thermometer, to as much as ·21 per cent. The 0°–100° interval, therefore, requires periodic verification.

Strong extension axioms and Shelah's zero-one law for choiceless polynomial time

2003 ◽  
Vol 68 (1) ◽  
pp. 65-131 ◽  
Author(s):  
Andreas Blass ◽  
Yuri Gurevich
Keyword(s):  
Fixed Point ◽  
Polynomial Time ◽  
Order Logic ◽  
Complexity Class ◽  
First Order Logic ◽  
Infinitary Logic ◽  
Extensive Discussion ◽  
First Order ◽  
Detailed Proof ◽  
Strong Extension

AbstractThis paper developed from Shelah's proof of a zero-one law for the complexity class “choiceless polynomial time,” defined by Shelah and the authors. We present a detailed proof of Shelah's result for graphs, and describe the extent of its generalizability to other sorts of structures. The extension axioms, which form the basis for earlier zero-one laws (for first-order logic, fixed-point logic, and finite-variable infinitary logic) are inadequate in the case of choiceless polynomial time; they must be replaced by what we call the strong extension axioms. We present an extensive discussion of these axioms and their role both in the zero-one law and in general.

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