Universal volume of groups and anomaly of Vogel’s symmetry

We show that integral representation of universal volume function of compact simple Lie groups gives rise to six analytic functions on \({ CP }^2\), which transform as two triplets under group of permutations of Vogel’s projective parameters. This substitutes expected invariance under permutations of universal parameters by more complicated covariance. We provide an analytic continuation of these functions and calculate their change (anomaly) under permutations of parameters (Vogel’s symmetry). This last relation is universal generalization, for an arbitrary simple Lie group and, moreover, to an arbitrary point in Vogel’s plane, of the Kinkelin’s reflection relation on Barnes’ \(G(1+N)\) function. Kinkelin’s relation gives asymmetry of the \(G(1+N)\) function (which is essentially reciprocal of the volume function for \({ SU }(N)\) groups) under \(N\leftrightarrow -N\) transformation (which is equivalent of the permutation of Vogel’s parameters for \({ SU }(N)\) groups), and coincides with above-mentioned anomaly of permutations at the \({ SU }(N)\) line on Vogel’s plane. Our results also give an anomaly of Vogel’s symmetry of the universal partition function of Chern–Simons theory on three-dimensional sphere. This effect is analogous to modular covariance, instead of invariance, of partition functions of appropriate gauge theories under modular transformation of couplings.     

Confinement of atoms with Robin’s condition: Spontaneous symmetry breaking

The properties of the ground state of the hydrogen atom in a spherical vacuum cavity with general boundary “not going out” conditions (i.e., when the probability current through the boundary vanishes) are studied. It is shown that in contrast to the confinement of an atom by a potential barrier, in this case depending on the parameters of the cavity, the atom could be in stable equilibrium at the center of the cavity or shift towards its periphery: spontaneous breaking of spherical symmetry occurs. The phase diagram of the shift and the dependence of the shift value and the binding energy of the ground state of the atom on the cavity parameters are presented. At the same time, the deformation properties of the electron wave function (WF) for an asymmetric distortion are so nontrivial that a non-zero shift occurs even when an electron is repulsed from the cavity boundary.     

A Physical Approach to Tsirelson’s Problem

Tsirelson’s problem deals with how to model separate measurements in quantum mechanics. In addition to its theoretical importance, the resolution of Tsirelson’s problem could have great consequences for device independent quantum key distribution and certified randomness. Unfortunately, understanding present literature on the subject requires a heavy mathematical background. In this paper, we introduce quansality, a new theoretical concept that allows to reinterpret Tsirelson’s problem from a foundational point of view. Using quansality as a guide, we recover all known results on Tsirelson’s problem in a clear and intuitive way.     

Whether the Sun’s supergranulation possesses a scaling?

The Sun’s supergranulation is considered as a collective effect provided by a statistical ensemble of narrow upflow jets. The responsibility for their occurrence rests with the Rayleigh-Taylor blow-up instability. It is this mechanism that is a trigger for the collapse of thermal irregularities and leads to the formation of such jets. The scaling laws of turbulence generated by these jets are discussed.     

The corrections to scaling within Mazenko’s theory in the limit of low and high dimensions

We consider corrections to scaling within an approximate theory developed by Mazenko for nonconserved order parameter in the limit of low (d → 1) and high (d → ∞) dimensions. The corrections to scaling considered here follows from the departures of the initial condition from the scaling morphology. Including corrections to scaling, the equal time correlation function has the form: C(r, t) = f ₀(r/L)+L ^−ω f ₁(r/L)+…, where L is a characteristic length scale (i.e. domain size). The correction-to-scaling exponent ω and the correction-to-scaling functions f ₁(x) are calculated for both low and high dimensions. In both dimensions the value of ω is found to be ω = 4 similar to 1D Glauber model and OJK theory (the theory developed by Ohta, Jasnow and Kawasaki).     

Mei symmetry and Mei conserved quantity of Nielsen equations for a non-holonomic system of Chetaev’s type with variable mass

Mei symmetry and Mei conserved quantity of Nielsen equations for a non-holonomic, non-conservative system of Chetaev's type with variable mass are studied. The differential equations of motion of the Nielsen equation for the system, the definition and criterion of Mei symmetry, and the condition and the form of Mei conserved quantity deduced directly by Mei symmetry for the system are obtained. An example is given to illustrate the application of the results.     

Is space-time symmetry a suitable generalization of parity-time symmetry?

We discuss space-time symmetric Hamiltonian operators of the form H=H₀+igH^′$H=H_0+ig{H}^{\prime }$, where H₀$H_0$ is Hermitian and g$g$ real. H₀$H_0$ is invariant under the unitary operations of a point group G$G$ while H^′$H^{\prime }$ is invariant under transformation by elements of a subgroup G^′$G^{\prime }$ of G$G$. If G$G$ exhibits irreducible representations of dimension greater than unity, then it is possible that H$H$ has complex eigenvalues for sufficiently small nonzero values of g$g$. In the particular case that H$H$ is parity-time symmetric then it appears to exhibit real eigenvalues for all 0<g<g_c$0<g<g_c$, where g_c$g_c$ is the exceptional point closest to the origin. Point-group symmetry and perturbation theory enable one to predict whether H$H$ may exhibit real or complex eigenvalues for g>0$g>0$. We illustrate the main theoretical results and conclusions of this paper by means of two- and three-dimensional Hamiltonians exhibiting a variety of different point-group symmetries.     

Berry’s phase for an electron in a periodic potential without inversion symmetry

Berry’s phase is calculated for an electron in a simple one-dimensional solid. The model used is a generalized Kronig-Penney potential, parameterized so that it may or may not have inversion symmetry. It is shown that the Berry’s phase as a function of an asymmetry parameter evolves from a linear to a non-linear form as inversion symmetry is broken. The functional form of the Berry’s phase is seen to be band-dependent in a simple way, suggesting that it can be used to identify the band in question.     

Relativistic generalization of Bell’s inequalities in Wigner’s form

A relativistic generalization of Bell’s inequalities in Wigner’s form was obtained for the decays of a pseudoscalar and a scalar particle to two particles having a nonzero spin (fermions and photons). Both inequalities involving a full anticorrelation of final-particle spins and having a nonrelativistic analog and inequalities involving a full correlation of spins are considered. It is shown that Bohr’s complementarity principle may be tested experimentally in the relativistic region.     

Dynamics of Maxwell’s pendulum

The stability of motion of Maxwell’s pendulum is investigated in a uniform gravity field. By means of several canonic transforms of the equations of pendulum motion and the method of the surfaces of Poincaré sections, the problem is reduced to investigation of the immobile-point stability retaining the area of mapping of the plane into itself. In the space of dimensionless parameters, the stability and instability regions are singled out.     

Newton’s Metaphysics of Space as God’s Emanative Effect

In several of his writings, Isaac Newton proposed that physical space is God’s “emanative effect” or “sensorium,” revealing something interesting about the metaphysics underlying his mathematical physics. Newton’s conjectures depart from Plato and Aristotle’s metaphysics of space and from classical and Cambridge Neoplatonism. Present-day philosophical concepts of supervenience clarify Newton’s ideas about space and offer a portrait of Newton not only as a mathematical physicist but an independent-minded rationalist philosopher.     

Bogolyubov’s theory of superfluidity

The Bogolyubov model of liquid helium is considered. We derive sufficient conditions which ensure an appearance of the Bose condensate in the model. For some temperatures and some positive values of the chemical potential there is the gapless Bogolyubov spectrum of elementary excitations, leading to the proper microscopic interpretation of the superfluidity.     

Fluctuation scaling and covariance matrix of constituents’ flows on a bipartite graph

We investigate an association between a power-law relationship of constituents’ flows (mean versus standard deviation) and their covariance matrix on a directed bipartite network. We propose a Poisson mixture model and a method to infer states of the constituents’ flows on such a bipartite network from empirical observation without a priori knowledge on the network structure. By using a proposed parameter estimation method with high frequency financial data we found that the scaling exponent and simultaneous cross-correlation matrix have a positive correspondence relationship. Consequently we conclude that the scaling exponent tends to be 1/2 in the case of desynchronous (specific dynamics is dominant), and to be 1 in the case of synchronous (common dynamics is dominant).     

Spatial symmetry of head-related transfer function

Methods for analyzing the spatial symmetry of head-related transfer function (HRTF) are proposed. The influences of anatomical structures on the symmetry of HRTF are investigated using HRTFs measured on KEMAR mannequin and human subjects. Results show that for KEMAR mannequin, pinnae destroy the front-back symmetry of HRTF above 5 to 6 kHz, while for human subjects the frequency reduces to 2.5 kHz because of the locations of ears. Furthermore, at low and median frequencies, HRTF is approximately left-right symmetrical. While as frequencies increase, the asymmetry caused by the fine anatomical leftright differences appears. The starting frequency and the extent of the left-right asymmetry in HRTF depend on individuals. The analyses demonstrate the spatial symmetrical characteristics of HRTF and the frequency ranges in which the current binaural models are valid.     

Extensions of Lieb’s Concavity Theorem

The operator function (A,B)→ Trf(A,B)(K ^*)K, defined in pairs of bounded self-adjoint operators in the domain of a function f of two real variables, is convex for every Hilbert Schmidt operator K, if and only if f is operator convex. We obtain, as a special case, a new proof of Lieb’s concavity theorem for the function (A,B)→ TrA ^p K ^* B ^q K, where p and q are non-negative numbers with sum p+q ≤ 1. In addition, we prove concavity of the operator function

A Generalization of Wigner’s Law

We present a generalization of Wigner’s semicircle law: we consider a sequence of probability distributions , with mean value zero and take an N × N real symmetric matrix with entries independently chosen from p _N and analyze the distribution of eigenvalues. If we normalize this distribution by its dispersion we show that as N → ∞ for certain p _N the distribution weakly converges to a universal distribution. The result is a formula for the moments of the universal distribution in terms of the rate of growth of the k ^th moment of p _N (as a function of N), and describe what this means in terms of the support of the distribution. As a corollary, when p _N does not depend on N we obtain Wigner’s law: if all moments of a distribution are finite, the distribution of eigenvalues is a semicircle.     

A model of Jupiter’s magnetodisk

An axially symmetric equilibrium model of Jupiter’s magnetodisk is developed in the MHD approximation that takes the plasma corotation and the centrifugal force into account. The model is constructed for two cases: (1) the magnetodisk plasma is assumed to have a uniform temperature; (2) the plasma pressure is assumed to be an adiabatic function of density. Analytical expressions for the magnetic field, current density, and magnetodisk temperature and thickness distributions are obtained as functions of the system parameters, viz., the radial distribution of plasma pressure in the equatorial plane, the transverse magnetic field in the center of the layer, and the angular velocity of the plasma rotation.