ChemInform Abstract: BOUNDARY PHASE STABILITY AND CRITICAL PHENOMENA IN HIGHER, ORDER SOLID SOLUTION SYSTEMS PART 2, CONDITIONS FOR CRITICAL POINTS AND APPROXIMATE SOLUTIONS

1974 ◽  
Vol 5 (2) ◽  
Author(s):  
E. RUDY ◽  
G. J. THROOP
Keyword(s):  
Solid Solution ◽  
Critical Phenomena ◽  
Phase Stability ◽  
Critical Points ◽  
Approximate Solutions ◽  
Higher Order ◽  
Boundary Phase ◽  
Abstract Boundary

Phase diagrams and higher-order critical points

1975 ◽  
Vol 12 (1) ◽  
pp. 345-355 ◽  
Author(s):  
Robert B. Griffiths
Keyword(s):  
Phase Diagrams ◽  
Critical Points ◽  
Higher Order

scholarly journals Existence and Iterative Algorithms of Positive Solutions for a Higher Order Nonlinear Neutral Delay Differential Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-28 ◽  
Author(s):  
Zeqing Liu ◽  
Ling Guan ◽  
Sunhong Lee ◽  
Shin Min Kang
Keyword(s):  
Differential Equation ◽  
Positive Solutions ◽  
Delay Differential Equation ◽  
Iterative Algorithms ◽  
Approximate Solutions ◽  
Higher Order ◽  
Banach Fixed Point Theorem ◽  
Neutral Delay ◽  
Neutral Delay Differential Equation ◽  
Delay Differential

This paper is concerned with the higher order nonlinear neutral delay differential equation[a(t)(x(t)+b(t)x(t-τ))(m)](n-m)+[h(t,x(h1(t)),…,x(hl(t)))](i)+f(t,x(f1(t)),…,x(fl(t)))=g(t),for allt≥t0. Using the Banach fixed point theorem, we establish the existence results of uncountably many positive solutions for the equation, construct Mann iterative sequences for approximating these positive solutions, and discuss error estimates between the approximate solutions and the positive solutions. Nine examples are included to dwell upon the importance and advantages of our results.

CRITICAL DIMENSIONS OF SYSTEMS WITH JOINT MULTICRITICAL AND LIFSHITZ-POINT-LIKE BEHAVIOR

2011 ◽  
Vol 25 (22) ◽  
pp. 1839-1845 ◽  
Author(s):  
ARTEM V. BABICH ◽  
LESYA N. KITCENKO ◽  
VYACHESLAV F. KLEPIKOV
Keyword(s):  
Field Theory ◽  
Critical Phenomena ◽  
Critical Points ◽  
Order Parameter ◽  
Mean Field Theory ◽  
Mean Field ◽  
Critical Dimensions ◽  
Lifshitz Point ◽  
The Mean ◽  
Derivatives Of

In this article, we consider a model that allows one to describe critical phenomena in systems with higher powers and derivatives of order parameter. The systems considered have critical points with joint multicritical and Lifshitz-point-like properties. We assess the lower and upper critical dimensions of these systems. These calculation enable us to find the fluctuation region where the mean field theory description does not work.

scholarly journals Implications of the principle of effective stress

2020 ◽  
Author(s):  
Gerd Gudehus
Keyword(s):  
Critical Phenomena ◽  
Pore Water ◽  
Effective Stress ◽  
Critical Points ◽  
Pore Water Pressure ◽  
Numerical Models ◽  
Water Pressure ◽  
Shear Bands ◽  
Triaxial Tests ◽  
Water Saturated

AbstractWhile Terzaghi justified his principle of effective stress for water-saturated soil empirically, it can be derived by means of the neutrality of the mineral with respect to changes of the pore water pressure $$p_w$$ p w . This principle works also with dilating shear bands arising beyond critical points of saturated grain fabrics, and with patterns of shear bands as relics of critical phenomena. The shear strength of over-consolidated clay is explained without effective cohesion, which results also from swelling up to decay, while rapid shearing of water-saturated clay can lead to a cavitation of pore water. The $$p_w$$ p w -neutrality is also confirmed by triaxial tests with sandstone samples, while Biot’s relation with a reduction factor for $$p_w$$ p w is contestable. An effective stress tensor is heuristically legitimate also for soil and rock with relics of critical phenomena, particularly for critical points with a Mohr–Coulomb condition. Therein, the $$p_w$$ p w -neutrality of the solid mineral determines the interaction of solid fabric and pore water, but numerical models are questionable due to fractal features.

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