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971.
It is shown that the admissible solutions of the continuity and Bernoulli or Burgers' equations of a perfect one-dimensional liquid are conditioned by a relation established in 1949–1950 by Pauli, Morette, and Van Hove, apparently, overlooked so far, which, in our case, stipulates that the mass density is proportional to the second derivative of the velocity potential. Positivity of the density implies convexity of the potential, i.e., smooth solutions, no shock. Non-elementary and symmetric solutions of the above equations are given in analytical and numerical form. Analytically, these solutions are derived from the original Ansatz proposed in Ref. 1 and from the ensuing operations which show that they represent a particular case of the general implicit solutions of Burgers' equation. Numerically and with the help of an ad hoc computer program, these solutions are simulated for a variety of initial conditions called compatible if they satisfy the Morette–Van Hove formula and anti-compatible if the sign of the initial velocity field is reversed. In the latter case, singular behaviour is observed. Part of the theoretical development presented here is rephrased in the context of the Hopf–Lax formula whose domain of applicability for the solution of the Cauchy problem of the homogeneous Hamilton–Jacobi equation has recently been enlarged. 相似文献
972.
S. Esposito 《International Journal of Theoretical Physics》2002,41(12):2417-2426
We present a method for reducing the order of ordinary differential equations satisfying a given scaling relation (Majorana scale-invariant equations). We also develop a variant of this method, aimed to reduce the degree of nonlinearity of the lower order equation. Some applications of these methods are carried out and, in particular, we show that second-order Emden–Fowler equations can be transformed into first-order Abel equations. The work presented here is a generalization of a method used by Majorana in order to solve the Thomas–Fermi equation. 相似文献
973.
We study the effective conductivity
e
for a random wire problem on the d-dimensional cubic lattice
d
, d2 in the case when random conductivities on bonds are independent identically distributed random variables. We give exact expressions for the expansion of the effective conductivity in terms of the moments of the disorder parameter up to the 5th order. In the 2D case using the duality symmetry we also derive the 6th order expansion. We compare our results with the Bruggeman approximation and show that in the 2D case it coincides with the exact solution up to the terms of 4th order but deviates from it for the higher order terms. 相似文献
974.
Kh. D. Ikramov 《Mathematical Notes》2002,71(3-4):500-504
Solvability conditions are examined for the matrix equation
, which cannot be found in the well-known reference books on matrix theory. Methods for constructing solutions to this equation are indicated. 相似文献
975.
Oscillation and nonoscillation criteria for the higher order self-adjoint differential equation (-1)n(talphay(n))(n)+q(t)y = 0 (*) are established. In these criteria, equation (*) is viewed as a perturbation of the conditionally oscillatory equation (-1)n(talphay(n))(n) - µ,t2n-y = 0, where
n, is the critical constant in conditional oscillation. Some open problems in the theory of conditionally oscillatory, even order, self-adjoint equations are also discussed. 相似文献
976.
The authors consider the nonlinear difference equation xn+1=xn+xn-kf(xn-k),n=0,1,(0.1) where (0,1), k 0,1, and f C1[[0,), [0,)] with f(x) < 0.They give sufficient conditions for the unique positive equilibrium of (0.1) to be a global attractor of all positive solutions. The results here are somewhat easier to apply than those of other authors. An application to a model of blood cell production is given. 相似文献
977.
We list characters (one-dimensional representations) of the reflection equation algebra associated with the fundamental vector representation of the Drinfeld–Jimbo quantum group
q
(gl(n)). 相似文献
978.
The semi-infinite Toda lattice is the system of differential equations d
n
(t)/dt =
n
(t)(b
n+1(t) – b
n
(t)), db
n
(t)/dt = 2(
n
2(t) –
n–1
2(t)), n = 1, 2, ..., t > 0. The solution of this system (if it exists) is a pair of real sequences
n
(t), b
n
(t) which satisfy the conditions
n
(0) =
n
,, b
n
(0) = b
n
, where
n
> 0 and b
n
are given sequences of real numbers. It is well known that the system has a unique solution provided that both sequences
n
and b
n
are bounded. When at least one of the known sequences
n
and b
n
is unbounded, many difficulties arise and, to the best of our knowledge, there are few results concerning the solution of the system. In this letter we find a class of unbounded sequences
n
and b
n
such that the system has a unique solution. The results are illustrated with a typical example where the sequences
i
(t), b
i
(t), i = 1, 2, ... can be exactly determined. The connection of the Toda lattice with the semi-infinite differential-difference equation d2/dt
2 log h
n
= h
n+1 + h
n–1 – 2h
n
, n = 1, 2, ... is also discussed and the above results are translated to analogous results for the last equation. 相似文献
979.
In this Letter we study twisted traces of products of intertwining operators for quantum affine algebras. They are interesting special functions, depending on two weights ,, three scalar parameters q,,k, and spectral parameters z
1,...,z
N
, which may be regarded as q-analogs of conformal blocks of the Wess–Zumino–Witten model on an elliptic curve. It is expected that in the rank 1 case they essentially coincide with the elliptic hypergeometric functions defined by Felder and Varchenko. Our main result is that after a suitable renormalization the traces satisfy four systems of difference equations – the Macdonald–Ruijsenaars equation, the q-Knizhnik–Zamolodchikov–Bernard equation, and their dual versions. We also show that in the case when the twisting automorphism is trivial, the trace functions are symmetric under the permutation , k . Thus, our results generalize those of Etingof and Schiffmann, dealing with the case q=1, and Etingof, Varchenko, and Schiffmann, dealing with the finite-dimensional case. 相似文献
980.
Marco Matone 《Foundations of Physics Letters》2002,15(4):311-328
We suggest that quantum mechanics and gravity are intimately related. In particular, we investigate the quantum Hamilton–Jacobi equation in the case of two free particles and show that the quantum potential, which is attractive, may generate the gravitational potential. The investigation, related to the formulation of quantum mechanics based on the equivalence postulate, is based on the analysis of the reduced action. A consequence of this approach is that the quantum potential is always non-trivial even in the case of the free particle. It plays the role of intrinsic energy and may in fact be at the origin of fundamental interactions. We pursue this idea, by making a preliminary investigation of whether there exists a set of solutions for which the quantum potential can be expressed with a gravitational potential leading term which alone would remain in the limit 0. A number of questions are raised for further investigation. 相似文献