By D. Arrowsmith, C.M. Place

This article discusses the qualitative houses of dynamical platforms together with either differential equations and maps, The process taken is based seriously on examples (supported by way of huge workouts, tricks to recommendations and diagrams to strengthen the fabric together with a remedy of chaotic behaviour. The unheard of renowned curiosity proven in recent times within the chaotic behaviour of discrete dynamic structures together with such issues as chaos and fractals has had its impression at the undergraduate and graduate curriculum. The e-book is geared toward classes in dynamics, dynamical structures and differential equations and dynamical platforms for complex undergraduates and graduate scholars. functions in physics, engineering and biology are thought of and advent to fractal imaging and mobile automata are given.

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2 ) has the meaning of superlluid density p,. The superfluid density p, is equal to the full density p = P i. at zero temperature. Taking this into account, we will replace p;. by p,, if T #- 0. 2), we took into account only the field functions l/f 0(x r), lf/0(x, r), which had no zeros in the x, r space (more precisely 1/10(x, r)l 2 = p(x, r) was close to p0(k0)). Now let us consider the case when l/f 0(x, r) may vanish at some discrete set of points in the x plane (for a fixed If we go around such a point, the phase variable cp(x, r) acquires an increment 2rrn, where n is an integer.

T plane. The considerations determining the main diagram series for the self energy parts of Green's functions A, B are also applicable if we want to find the main diagram series for the pressure p. 20) where p0 is the pressure of the ideal Bose gas, C0 is the constant (C0 = ). 5), and D is the sum of all connected (vacuum) diagrams. ned factors (1) and (2): D ':I:;; + + 8 O C> + + + c:x:=> + + ... 2 1) + �+ b ... c 6 Low-density Bose gas 35 These diagrams contain either two vertices of the second-order or two independent loops, or one independent loop with either one second-order vertex or two third-order vertices.

In r0a where a is the vortex core radius. This formula is valid for r0 » a. We now express the contribution of those x, y for which I x - y l r0 as a fu nctional integral over the new vector potential function 2mc Ii fJr. np. = (dli(x), dl i(y)) -+ I x - yI : • _ YI < ,0 = = - -, > (8. <>(o;A;)CT;dA; ; J exp ( - J d3x(curl A)2 )nx<>(o;A;)CT;dA; c (mc2 ) 1/2 - np"-- i ,k Ix k q= 2n p ' - (8. 16) (8. 1 7) is a coupling constant. We can prove (8. 16) by making a shift transform ation which cancels the linear term J(jA) d 3 x in the exponent of the integrand in the numerator on the right-hand side of(8.