Hilbert’s invariant integral

2009 ◽  
pp. 91-100
Author(s):  
Mike Mesterton-Gibbons
Keyword(s):  
Invariant Integral

scholarly journals Some Identities of Symmetry for the Generalized Bernoulli Numbers and Polynomials

2009 ◽  
Vol 2009 ◽  
pp. 1-8 ◽  
Author(s):  
Taekyun Kim ◽  
Seog-Hoon Rim ◽  
Byungje Lee
Keyword(s):  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Bernoulli Numbers And Polynomials ◽  
Power Sums ◽  
Invariant Integral ◽  
Generalized Bernoulli Polynomials ◽  
Symmetric Properties

By the properties ofp-adic invariant integral onℤp, we establish various identities concerning the generalized Bernoulli numbers and polynomials. From the symmetric properties ofp-adic invariant integral onℤp, we give some interesting relationship between the power sums and the generalized Bernoulli polynomials.

scholarly journals The mechanics and physics of fracturing: application to thermal aspects of crack propagation and to fracking

2015 ◽  
Vol 373 (2038) ◽  
pp. 20140119 ◽  
Author(s):  
Genady P. Cherepanov
Keyword(s):  
Shale Gas ◽  
Fractured Rock ◽  
Local Region ◽  
Gas Reservoirs ◽  
Moving Crack ◽  
Horizontal Drilling ◽  
Rock Drill ◽  
Invariant Integral ◽  
Set Up ◽  
Energy Conservation Law

By way of introduction, the general invariant integral (GI) based on the energy conservation law is presented, with mention of cosmic, gravitational, mass, elastic, thermal and electromagnetic energy of matter application to demonstrate the approach, including Coulomb's Law generalized for moving electric charges, Newton's Law generalized for coupled gravitational/cosmic field, the new Archimedes’ Law accounting for gravitational and surface energy, and others. Then using this approach the temperature track behind a moving crack is found, and the coupling of elastic and thermal energies is set up in fracturing. For porous materials saturated with a fluid or gas, the notion of binary continuum is used to introduce the corresponding GIs. As applied to the horizontal drilling and fracturing of boreholes, the field of pressure and flow rate as well as the fluid output from both a horizontal borehole and a fracture are derived in the fluid extraction regime. The theory of fracking in shale gas reservoirs is suggested for three basic regimes of the drill mud permeation, with calculating the shape and volume of the local region of the multiply fractured rock in terms of the pressures of rock, drill mud and shale gas.

scholarly journals Ways of application of the provisions of mechanics of bodies with cracks to the calculation of asphalt concrete on strength and plasticity

2018 ◽  
Vol 239 ◽  
pp. 05018 ◽  
Author(s):  
Anatoly Aleksandrov ◽  
Natalya Aleksandrova ◽  
Vasiliy Chusov ◽  
Aleksandr Riabov
Keyword(s):  
Fracture Mechanics ◽  
Brittle Fracture ◽  
Laboratory Test ◽  
Test Data ◽  
Asphalt Concrete ◽  
Invariant Integral ◽  
Accumulation Of Damage ◽  
Strength And Plasticity

The report discusses the principles of two major theories of fracture mechanics of bodies with cracks, which include the theory of accumulation of damage Kachanov–Rabotnov and theory of brittle fracture Griffith–Irwin, including the invariant integral Cherepanov–Rice, describing the criterion of growth the crack. To assess the application of these theories to the calculation of asphalt concrete, laboratory test data are given and based on their analysis the appropriate conclusions.

A local generalized Hilbert invariant integral

2001 ◽  
Vol 47 (1) ◽  
pp. 399-410
Author(s):  
Urszula Ledzewicz ◽  
Andrzej Nowakowski ◽  
Heinz Schaettler
Keyword(s):  
Invariant Integral

scholarly journals C-projection and monopole condensation in QCD

2014 ◽  
Vol 29 ◽  
pp. 1460210
Author(s):  
Y. M. Cho
Keyword(s):  
Effective Action ◽  
Gauge Invariance ◽  
Integral Expression ◽  
Gauge Invariant ◽  
Invariant Integral ◽  
Infra Red ◽  
The Stability ◽  
The One

We show that the monopole condensation is responsible for the confinement. To demonstrate this we present a new gauge invariant integral expression of the one-loop QCD effective action which has no infra-red divergence, and show that the color reflection invariance ("the C-projection") assures the gauge invariance and the stability of the monopole condensation.

The contraction of gravitating spheres

1964 ◽  
Vol 281 (1384) ◽  
pp. 39-48 ◽  
Keyword(s):  
General Relativity ◽  
Constant Density ◽  
Field Equations ◽  
Newtonian Theory ◽  
Full Field ◽  
Invariant Integral ◽  
Density Relation ◽  
Particle Paths ◽  
Rigorous Method ◽  
Adiabatic Contraction

Earlier ideas associating an invariant integral of the energy invariant with the number of nucleons in a gravitating body are shown to be fallacious, and thus do not provide a means of following through the contraction of such a body. It is shown how the full field equations of general relativity give a feasible and rigorous method of examining contracting models. Schwarzschild-type co-ordinates are introduced and are used to examine the slow adiabatic contraction of a sphere of constant density. The particle paths are found and the pressure-density relation permitting such slow adiabatic contraction is examined. It is shown that the simple 4/3 power law of Newtonian theory has to be replaced by a steeper dependence of pressure on density for high gravitational potentials. Radiation co-ordinates are introduced to examine radiating contracting systems, and equations fully specifying such a system are obtained. A simple example is given in outline to illustrate the method.

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