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101.
Vladimir Gurvich 《Discrete Applied Mathematics》2010,158(14):1496-302
Consider an electrical circuit, each edge e of which is an isotropic conductor with a monomial conductivity function . In this formula, ye is the potential difference and current in e, while μe is the resistance of e; furthermore, r and s are two strictly positive real parameters common for all edges. In particular, the case r=s=1 corresponds to the standard Ohm’s law.In 1987, Gvishiani and Gurvich [A.D. Gvishiani, V.A. Gurvich, Metric and ultrametric spaces of resistances, in: Communications of the Moscow Mathematical Society, Russian Math. Surveys 42 (6 (258)) (1987) 235-236] proved that, for every two nodes a,b of the circuit, the effective resistance μa,b is well-defined and for every three nodes a,b,c the inequality holds. It obviously implies the standard triangle inequality μa,b≤μa,c+μc,b whenever s≥r. For the case s=r=1, these results were rediscovered in the 1990s. Now, after 23 years, I venture to reproduce the proof of the original result for the following reasons:
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- It is more general than just the case r=s=1 and one can get several interesting metric and ultrametric spaces playing with parameters r and s. In particular, (i) the effective Ohm resistance, (ii) the length of a shortest path, (iii) the inverse width of a bottleneck path, and (iv) the inverse capacity (maximum flow per unit time) between any pair of terminals a and b provide four examples of the resistance distances μa,b that can be obtained from the above model by the following limit transitions: (i) r(t)=s(t)≡1, (ii) r(t)=s(t)→∞, (iii) r(t)≡1,s(t)→∞, and (iv) r(t)→0,s(t)≡1, as t→∞. In all four cases the limits μa,b=limt→∞μa,b(t) exist for all pairs a,b and the metric inequality μa,b≤μa,c+μc,b holds for all triplets a,b,c, since s(t)≥r(t) for any sufficiently large t. Moreover, the stronger ultrametric inequality μa,b≤max(μa,c,μc,b) holds for all triplets a,b,c in examples (iii) and (iv), since in these two cases s(t)/r(t)→∞, as t→∞.
- •
- Communications of the Moscow Math. Soc. in Russ. Math. Surveys were (and still are) strictly limited to two pages; the present paper is much more detailed.Although a translation in English of the Russ. Math. Surveys is available, it is not free in the web and not that easy to find.
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- The last but not least: priority.
102.
We propose a couple of general ways of constructing authentication schemes from actions of a semigroup on a set, without exploiting any specific algebraic properties of the set acted upon. Then we give several concrete realizations of this general idea, and in particular, we describe several authentication schemes with long-term private keys where forgery (a.k.a. impersonation) is NP-hard. Computationally hard problems that can be employed in these realizations include the Graph Colorability problem, the Diophantine problem, and many others. 相似文献
103.
We describe multivariable hypergeometric series in orthogonal polynomials. These series are solutions of special systems of Gelfand-type equations. The difference andq-analogs are also given. 相似文献
104.
We present some novel thermodynamic ideas based on the Maupertuis principle. By considering Hamiltonians written in terms of appropriate action-angle variables we show that thermal states can be characterized by the action variables and by their evolution in time when the system is nonintegrable. We propose dynamical definitions for the equilibrium temperature and entropy as well as an expression for the nonequilibrium entropy valid for isolated systems with many degrees of freedom. This entropy is shown to increase in the relaxation to equilibrium of macroscopic systems with short-range interactions, which constitutes a dynamical justification of the Second Law of Thermodynamics. Several examples are worked out to show that this formalism yields the right microcanonical (equilibrium) quantities. The relevance of this approach to nonequilibrium situations is illustrated with an application to a network of coupled oscillators (Kuramoto model). We provide an expression for the entropy production in this system finding that its positive value is directly related to dissipation at the steady state in attaining order through synchronization. 相似文献
105.
The Wigner function and the symplectic tomogram of an entangled quantum state, which is a superposition of the photon’s coherent
states (even and odd coherent states), is studied. Photon statistics and violation of Bell’s inequality for the photon state
are discussed. 相似文献
106.
Inhomogeneity and anisotropy are intrinsic characteristics of daytime and nighttime atmospheric turbulence. For example, turbulent eddies are often stretched in the direction of the mean wind, and the turbulence statistics depends on the height above the ground. Recent studies have shown that the log-amplitude and phase fluctuations of plane and spherical sound waves are significantly affected by turbulence inhomogeneity and anisotropy. The present paper is devoted to studies of the mean sound field and the coherence functions of plane and spherical sound waves propagating through inhomogeneous anisotropic turbulence with temperature and velocity fluctuations. These statistical moments of a sound field are important in many practical applications, e.g., for source detection, ranging, and recognition. Formulas are derived for the mean sound field and coherence function of initially arbitrary waveform. Using the latter formula, we also obtained formulas for the coherence functions of plane and spherical sound waves. All these formulas coincide with those known in the literature for two limiting cases: homogeneous isotropic turbulence with temperature and wind velocity fluctuations, and inhomogeneous anisotropic turbulence with temperature fluctuations only. Using the formulas obtained, we have numerically shown that turbulence inhomogeneity significantly affects the coherence functions of plane and spherical sound waves. 相似文献
107.
Daniel S. Covita Dimitrios F. Anagnostopoulos Hermann Fuhrmann Hubert Gorke Detlev Gotta Alexander Gruber Albert Hirtl Tomoichi Ishiwatari Paul Indelicato Thomas S. Jensen Eric-Olivier Le Bigot Valeri E. Markushin Michael Nekipelov Vladimir N. Pomerantsev Vladimir P. Popov Joaquim M. F. dos Santos Philipp Schmid Leopold M. Simons Marian Theisen Martino Trassinelli Joao F. C. A. Veloso Johann Zmeskal 《The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics》2018,72(4):72
108.
109.