A Novel and Efficient Method for Computing the Resistance Distance

The resistance distance is an intrinsic metric on graphs that have been extensively studied by many physicists and mathematicians. The resistance distance between two vertices of a simple connected graph $G$ is equal to the resistance between two equivalent points on an electrical network, constructed to correspond to $G$ , with each edge being replaced by a unit resistor. Hypercube $Q_{n}$ is one of the most efficient and versatile topological structures of the interconnection networks, which received much attention over the past few years. The folded $n$ -cube graph is obtained from hypercube $Q_{n}$ by merging vertices of the hypercube $Q_{n}$ that are antipodal, i.e., lie at a distance $n$. Folded $n$ -cube graphs have been studied in parallel computing as a potential network topology. The folded $n$ -cube has the same number of vertices but half the diameter as compared to hypercubes which play an important role in analyzing the efficiency of interconnection networks. We intend is to minimize the diameter. In this study, we will compute the resistance distance between any two vertices of the folded $n$ -cube by using the symmetry method and classic Kirchhoff's equations. This method is beneficial for distance-transitive graphs. As an application, we will also give an example and compute the resistance distance in the Biggs-Smith graph, which shows the competency of the proposed method.This work was supported by the Qatar National Library.Scopu

Discrete Mathematics and Symmetry

Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group

Lieb–Robinson Bounds for Multi–Commutators and Applications to Response Theory

We generalize to multi-commutators the usual Lieb–Robinson bounds for commutators. In the spirit of constructive QFT, this is done so as to allow the use of combinatorics of minimally connected graphs (tree expansions) in order to estimate time-dependent multi-commutators for interacting fermions. Lieb–Robinson bounds for multi-commutators are effective mathematical tools to handle analytic aspects of the dynamics of quantum particles with interactions which are non-vanishing in the whole space and possibly time-dependent. To illustrate this, we prove that the bounds for multi-commutators of order three yield existence of fundamental solutions for the corresponding non-autonomous initial value problems for observables of interacting fermions on lattices. We further show how bounds for multi-commutators of an order higher than two can be used to study linear and non-linear responses of interacting fermions to external perturbations. All results also apply to quantum spin systems, with obvious modifications. However, we only explain the fermionic case in detail, in view of applications to microscopic quantum theory of electrical conduction discussed here and because this case is technically more involved.FAPESP under Grant 2013/13215-5 Basque Government through the grant IT641-13 SEV-2013-0323 MTM2014-5385

Development of Computer Modelling Techniques for Microwave Thermography

Microwave thermography obtains information about the temperature of internal body tissues by a spectral measurement of the intensity of the natural thermally generated radiation emitted by the body tissues. At the lower microwave frequencies radiation can penetrate through tissue for distances useful for a range of medical applications. Radiation from inside the body may be detected and measured non-invasively at the skin surface by a microwave thermography system consisting of a suitable antenna to detect the radiation and a radiometer receiver to measure its intensity. In the microwave region the radiative power emitted per unit bandwidth is proportional to the temperature of the emitting tissue and the total radiative power received from the body tissues, P, is a weighted volume average of temperature P = kB ∫w(r) T(r) dV where k is Boltzmann's constant, B is the bandwidth, T(r) is the temperature at the position r and w(r) is the weighting function. The weighting function depends on the structure and dielectric properties of the tissues being viewed, the measurement frequency and the characteristics of the antenna. The Glasgow developed microwave thermography system operates at a central frequency of 3.2 GHz, chosen to give the optimum compromise between the depth from which radiation may be received, which decreases with increasing frequency, and the lateral spatial resolution which increases with increasing frequency. A Dicke configuration radiometer receiver and a cylindrical low-impedance waveguide antenna, which operates in contact with the skin surface, are used. The output from the radiometer is calibrated to degrees Celsius to give a "microwave temperature" of the tissues being viewed. The tissue temperature distribution, T(r), reflects the vascular and metabolic state of the tissue. Diseases which affect these physiological functions will result in changes in the tissue temperature and hence in the measured microwave temperature. It is not possible, however, to solve the indirect problem of retrieval of the temperature distribution in the tissue from a single frequency measurement of microwave temperature. It is therefore necessary to model the temperature distribution in the tissue and, from this, solve the direct problem of calculation of the microwave temperature. Measured microwave temperatures may then be compared with those modelled to indicate the physiological state of the tissue. Pennes (1948) The temperature distribution in the tissue may be determined by solution of the steady-state heat transfer equation KV2T +Wbcb(Ta -T) + Q = 0 where K is the thermal conductivity of the tissue, Wb is the perfusion rate of blood through the tissue, cb is the specific heat capacity of the blood, Ta is the arterial blood temperature and Q is the rate of metabolic heat generation in the tissue. The boundary condition of heat loss at the skin surface is governed by the equation K dT/dn= h(T-Te ) where Te is the ambient temperature and h is the heat transfer coefficient due to the combined effects of heat loss by radiation, convection and evaporation. The microwave temperature may be calculated from the modelled temperature distribution and use of plane wave theory to determine the weighting function, with an increased power attenuation constant to account for the response of the antenna. The modelling of the tissue is simplified by the fact that both the tissue thermal conductivity and the microwave dielectric properties of the tissue depend primarily on the water content of the tissue. This thermal and electromagnetic modelling has been carried out to determine the expected microwave temperature profiles across the female breast. Microwave and infra-red temperature measurements were made on a group of young, normal women and a group of older, post-menopausal women with breast disease. In general the younger women will have higher water content breast tissue than that of the older women due to the higher proportion of glandular and connective tissue and the smaller proportion of low water content fat tissue

Ahlfors circle maps and total reality: from Riemann to Rohlin

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein's ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler's planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem. Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree controls than available in Ahlfors' era. Accordingly, our partisan belief is that much remains to be clarified regarding the foundation and optimal control of Ahlfors circle maps. The game of sharp estimation may look narrow-minded "Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to contemplate how conformal and algebraic geometry are fighting together for the soul of Riemann surfaces. A second part explores the connection with Hilbert's 16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by including now Rohlin's theory (v.2