Higher Spin Dirac Operators Between Spaces of Simplicial Monogenics in Two Vector Variables

The higher spin Dirac operator \(\mathcal{Q}_{k,l}\) acting on functions taking values in an irreducible representation space for \(\mathfrak{so}(m)\) with highest weight \((k+\frac{1}{2},l+\frac{1}{2},\frac{1}{2},\ldots,\frac{1}{2})\), with k, l?∈?\(\mathbb{N}\) and \(k\geqslant l\), is constructed. The structure of the kernel space containing homogeneous polynomial solutions is then also studied.     

Quantifying Contextuality of Empirical Models in Terms of Trace-Distance

In order to quantify contextuality of empirical models, the quantity of contextuality (QoC) of empirical models is introduced in terms of the trace-distance. Let Q C(e) denote the QoC of an empirical model e. The following conclusions are proved. (i) An empirical model e is non-contextual if and only if Q C(e)=0, and then it is contextual if and only if Q C(e)>0; (ii) the QoC function QC is convex, contractive and continuous. Finally, the QoC of some famous models is computed, including PM-isotropic boxes P M ^α , M-isotropic boxes M ^α , C H _n -isotropic boxes \(CH_{n}^{\alpha }\) as well as K box, where α∈[0,1]. Moreover, P M ^α is non-contextual if and only if \(\alpha \in [\frac {1}{6},\frac {5}{6}]\); M ^α is non-contextual if and only if \(\alpha \in [0,\frac {4}{5}]\); when n is even, \(CH_{n}^{\alpha }\) is non-contextual if and only if \(\alpha \in [\frac {1}{n},\frac {n-1}{n}]\), and when n is odd, \(CH_{n}^{\alpha }\) is non-contextual if and only if \(\alpha \in [0,\frac {n-1}{n}]\). The most important thing is that it is very easy to compare the QoC of any two isotropic boxes discussed in the above.     

Two Phases of the Non-Commutative Quantum Mechanics with the Generalized Uncertainty Relations

We consider the quantum mechanics on the noncommutative plane with the generalized uncertainty relations \({\Delta } x_{1} {\Delta } x_{2} \ge \frac {\theta }{2}, {\Delta } p_{1} {\Delta } p_{2} \ge \frac {\bar {\theta }}{2}, {\Delta } x_{i} {\Delta } p_{i} \ge \frac {\hbar }{2}, {\Delta } x_{1} {\Delta } p_{2} \ge \frac {\eta }{2}\). We show that the model has two essentially different phases which is determined by \(\kappa = 1 + \frac {1}{\hbar ^{2} } (\eta ^{2} - \theta \bar {\theta })\). We construct a operator \(\hat {\pi }_{i}\) commuting with \(\hat {x}_{j} \) and discuss the harmonic oscillator model in two dimensional non-commutative space for three case κ > 0, κ = 0, κ < 0. Finally, we discuss the thermodynamics of a particle whose hamiltonian is related to the harmonic oscillator model in two dimensional non-commutative space.     

Heuristic Relative Entropy Principles with Complex Measures: Large-Degree Asymptotics of a Family of Multi-variate Normal Random Polynomials

Let \(z\in \mathbb {C}\), let \(\sigma ^2>0\) be a variance, and for \(N\in \mathbb {N}\) define the integrals

$$\begin{aligned} E_N^{}(z;\sigma ) := \left\{ \begin{array}{ll} {\frac{1}{\sigma }} \!\!\!\displaystyle \int _{\mathbb {R}}\! (x^2+z^2) \frac{e^{-\frac{1}{2\sigma ^2} x^2}}{\sqrt{2\pi }}dx&{}\quad \text{ if }\, N=1,\\ {\frac{1}{\sigma }} \!\!\!\displaystyle \int _{\mathbb {R}^N}\! \prod \prod \limits _{1\le k<l\le N}\!\! e^{-\frac{1}{2N}(1-\sigma ^{-2}) (x_k-x_l)^2} \prod _{1\le n\le N}\!\!\!\!(x_n^2+z^2) \frac{e^{-\frac{1}{2\sigma ^2} x_n^2}}{\sqrt{2\pi }}dx_n &{}\quad \text{ if }\, N>1. \end{array}\right. \!\!\! \end{aligned}$$

These are expected values of the polynomials \(P_N^{}(z)=\prod _{1\le n\le N}(X_n^2+z^2)\) whose 2N zeros \(\{\pm i X_k\}^{}_{k=1,\ldots ,N}\) are generated by N identically distributed multi-variate mean-zero normal random variables \(\{X_k\}^{N}_{k=1}\) with co-variance \(\mathrm{{Cov}}_N^{}(X_k,X_l)=(1+\frac{\sigma ^2-1}{N})\delta _{k,l}+\frac{\sigma ^2-1}{N}(1-\delta _{k,l})\). The \(E_N^{}(z;\sigma )\) are polynomials in \(z^2\), explicitly computable for arbitrary N, yet a list of the first three \(E_N^{}(z;\sigma )\) shows that the expressions become unwieldy already for moderate N—unless \(\sigma = 1\), in which case \(E_N^{}(z;1) = (1+z^2)^N\) for all \(z\in \mathbb {C}\) and \(N\in \mathbb {N}\). (Incidentally, commonly available computer algebra evaluates the integrals \(E_N^{}(z;\sigma )\) only for N up to a dozen, due to memory constraints). Asymptotic evaluations are needed for the large-N regime. For general complex z these have traditionally been limited to analytic expansion techniques; several rigorous results are proved for complex z near 0. Yet if \(z\in \mathbb {R}\) one can also compute this “infinite-degree” limit with the help of the familiar relative entropy principle for probability measures; a rigorous proof of this fact is supplied. Computer algebra-generated evidence is presented in support of a conjecture that a generalization of the relative entropy principle to signed or complex measures governs the \(N\rightarrow \infty \) asymptotics of the regime \(iz\in \mathbb {R}\). Potential generalizations, in particular to point vortex ensembles and the prescribed Gauss curvature problem, and to random matrix ensembles, are emphasized.

     

Global Regularity for the Navier-Stokes-Maxwell System with Fractional Diffusion

In this paper, we study the global regularity for the Navier-Stokes-Maxwell system with fractional diffusion. Existence and uniqueness of global strong solution are proved for \(\alpha \geqslant \frac {3}{2}\). When 0 < α < 1, global existence is obtained provided that the initial data \(\|u_{0}\|_{H^{\frac {5}{2}-2\alpha }}+\|E_{0}\|_{H^{\frac {5}{2}-2\alpha }}+\|B_{0}\|_{H^{\frac {5}{2}-2\alpha }}\) is sufficiently small. Moreover, when \(1<\alpha <\frac {3}{2}\), global existence is obtained if for any ε >?0, the initial data \(\|u_{0}\|_{H^{\frac {3}{2}-\alpha +\varepsilon }}+\|E_{0}\|_{H^{\frac {3}{2}-\alpha +\varepsilon }}+\|B_{0}\|_{H^{\frac {3}{2}-\alpha +\varepsilon }}\) is small enough.     

Fully constrained Majorana neutrino mass matrices using $$\varvec{\varSigma (72\times 3)}$$

In 2002, two neutrino mixing ansatze having trimaximally mixed middle (\(\nu _2\)) columns, namely tri-chi-maximal mixing (\(\text {T}\chi \text {M}\)) and tri-phi-maximal mixing (\(\text {T}\phi \text {M}\)), were proposed. In 2012, it was shown that \(\text {T}\chi \text {M}\) with \(\chi =\pm \,\frac{\pi }{16}\) as well as \(\text {T}\phi \text {M}\) with \(\phi = \pm \,\frac{\pi }{16}\) leads to the solution, \(\sin ^2 \theta _{13} = \frac{2}{3} \sin ^2 \frac{\pi }{16}\), consistent with the latest measurements of the reactor mixing angle, \(\theta _{13}\). To obtain \(\text {T}\chi \text {M}_{(\chi =\pm \,\frac{\pi }{16})}\) and \(\text {T}\phi \text {M}_{(\phi =\pm \,\frac{\pi }{16})}\), the type I see-saw framework with fully constrained Majorana neutrino mass matrices was utilised. These mass matrices also resulted in the neutrino mass ratios, \(m_1:m_2:m_3=\frac{\left( 2+\sqrt{2}\right) }{1+\sqrt{2(2+\sqrt{2})}}:1:\frac{\left( 2+\sqrt{2}\right) }{-1+\sqrt{2(2+\sqrt{2})}}\). In this paper we construct a flavour model based on the discrete group \(\varSigma (72\times 3)\) and obtain the aforementioned results. A Majorana neutrino mass matrix (a symmetric \(3\times 3\) matrix with six complex degrees of freedom) is conveniently mapped into a flavon field transforming as the complex six-dimensional representation of \(\varSigma (72\times 3)\). Specific vacuum alignments of the flavons are used to arrive at the desired mass matrices.     

Recent results on kaon decays by the NA48/2 experiment at CERN

The NA48/2 experiment reports the first observation of the rare decay K^± → π^±π⁰e⁺e^?, based on about 2000 candidates from 2003 data. The preliminary branching ratio in the full kinematic region is \(\mathcal {B}(K^{\pm } \to \pi ^{\pm }\pi ^{0}e^{+}e^{-})=(4.06\pm 0.17)\cdot 10^{-6}\). A sample of 4.687 × 10⁶\(K^{\pm }\to \pi ^{\pm }{\pi ^{0}_{D}}\) events collected in 2003/4 is analyzed to search for the dark photon (\(A^{\prime }\)) via the decay chain K^± → π^±π⁰, \(\pi ^{0}\to \gamma A^{\prime }\), \(A^{\prime }\to e^{+}e^{-}\). No signal is observed, limits in the plane mixing parameter ε² versus its mass \(m_{A^{\prime }}\) are reported.     

Electrical conductivity in single crystals of GaSe<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript>Te<Subscript>1 – <Emphasis Type="Italic">x</Emphasis></Subscript> solid solutions in strong electrical fields

This paper presents some results of studying the Poole–Frenkel effect with allowance for shielding in layered GaSe and GaTe single crystals and their solid solutions in strong electrical fields of up to 10⁵ V/cm at temperatures of 103–250 K. According to the relationship \(\left(\frac{\sigma}{\sigma(0)}\right)^{1/2}\) log\(\frac{\sigma}{\sigma(0)} = E\sqrt{\frac{\varepsilon}{4\pi n(0)kT}}\), there exists a linear dependence between \(\left(\frac{\sigma}{\sigma(0)}\right)^{1/2}\) log\(\frac{\sigma}{\sigma(0)}\) and the electrical field E (σ is the electrical conductivity in strong electrical fields, and σ(0) is the electrical conductivity in the ohmic region). The slopes of these lines have been determined at different temperatures (103–250 K) by estimating the concentration of current carriers n(0) = 3 × 10¹³–5 × 10¹⁵ cm^–3 in the ohmic region of the electrical conductivity of solid solutions of layered GaSe _x Te_1–x single crystals (x = 1.00, 0.95, 0.90, 0.80, 0.70, 0.30, 0.20, 0.10, 0).     

Fluctuation of Density of States for 1d Schrödinger Operators

We consider the 1d Schrödinger operator with random decaying potential and compute the 2nd term asymptotics of the density of states, which shows substantial differences between the cases \(\alpha > \frac{1}{2}\), \(\alpha < \frac{1}{2}\) and \(\alpha = \frac{1}{2}\).     

Universal Components of Random Nodal Sets

We give, as L grows to infinity, an explicit lower bound of order \({L^{\frac{n}{m}}}\) for the expected Betti numbers of the vanishing locus of a random linear combination of eigenvectors of P with eigenvalues below L. Here, P denotes an elliptic self-adjoint pseudo-differential operator of order \({m > 0}\), bounded from below and acting on the sections of a Riemannian line bundle over a smooth closed n-dimensional manifold M equipped with some Lebesgue measure. In fact, for every closed hypersurface \({\Sigma}\) of \({\mathbb{R}^n}\), we prove that there exists a positive constant \({p_\Sigma}\) depending only on \({\Sigma}\), such that for every large enough L and every \({x \in M}\), a component diffeomorphic to \({\Sigma}\) appears with probability at least \({p_\Sigma}\) in the vanishing locus of a random section and in the ball of radius \({L^{-\frac{1}{m}}}\) centered at x. These results apply in particular to Laplace–Beltrami and Dirichlet-to-Neumann operators.     

Synthesis and evaluation of bile acid amides of $$\alpha $$-cyanostilbenes as anticancer agents

A series of amino-substituted \(\alpha \)-cyanostilbene derivatives and their bile acid (cholic and deoxycholic acid) amides were designed and synthesized. A comparative study on the anticancer and antibacterial activity evaluation on the synthesized analogs was carried against the human osteosarcoma (HOS) cancer cell line, and two gram ?ve (E. coli and S. typhi) and two gram \(+\)ve (B. subtilis and S. aureus) bacterial strains. All the cholic acid \(\alpha \)-cyanostilbene amides showed an \(\hbox {IC}_{50}\) in the range 2–13 \(\upmu \hbox {M}\) against human osteosarcoma cells (HOS) with the most active analog (6g) possessing an \(\hbox {IC}_{50}\) of \(2\,\upmu \hbox {M}\). One of the amino-substituted \(\alpha \)-cyanostilbene, 4e, was found to possess an \(\hbox {IC}_{50}\) of \(3\,\upmu \hbox {M}\). An increase in the number of cells at the sub-\(\hbox {G}_{1}\) phase of the cell was observed in the in vitro cell cycle analysis of two most active compounds in the series (4e, 6g) suggesting a clear indication toward induction of apoptotic cascade. With respect to antibacterial screening, amino-substituted \(\alpha \)-cyanostilbenes were found to be more active than their corresponding bile acid amides. The synthesized compounds were also subjected to in silico study to predict their physiochemical properties and drug-likeness score.     

Equilibration in the Kac Model Using the GTW Metric $$d_2$$

We use the Fourier based Gabetta–Toscani–Wennberg metric \(d_2\) to study the rate of convergence to equilibrium for the Kac model in 1 dimension. We take the initial velocity distribution of the particles to be a Borel probability measure \(\mu \) on \(\mathbb {R}^n\) that is symmetric in all its variables, has mean \(\vec {0}\) and finite second moment. Let \(\mu _t(dv)\) denote the Kac-evolved distribution at time t, and let \(R_\mu \) be the angular average of \(\mu \). We give an upper bound to \(d_2(\mu _t, R_\mu )\) of the form \(\min \left\{ B e^{-\frac{4 \lambda _1}{n+3}t}, d_2(\mu ,R_\mu )\right\} ,\) where \(\lambda _1 = \frac{n+2}{2(n-1)}\) is the gap of the Kac model in \(L^2\) and B depends only on the second moment of \(\mu \). We also construct a family of Schwartz probability densities \(\{f_0^{(n)}: \mathbb {R}^n\rightarrow \mathbb {R}\}\) with finite second moments that shows practically no decrease in \(d_2(f_0(t), R_{f_0})\) for time at least \(\frac{1}{2\lambda }\) with \(\lambda \) the rate of the Kac operator. We also present a propagation of chaos result for the partially thermostated Kac model in Tossounian and Vaidyanathan (J Math Phys 56(8):083301, 2015).     

Distribution of Singular Values of Random Band Matrices; Marchenko–Pastur Law and More

We consider the limiting spectral distribution of matrices of the form \(\frac{1}{2b_{n}+1} (R + X)(R + X)^{*}\), where X is an \(n\times n\) band matrix of bandwidth \(b_{n}\) and R is a non random band matrix of bandwidth \(b_{n}\). We show that the Stieltjes transform of ESD of such matrices converges to the Stieltjes transform of a non-random measure. And the limiting Stieltjes transform satisfies an integral equation. For \(R=0\), the integral equation yields the Stieltjes transform of the Marchenko–Pastur law.     

Nature of the $$\Omega (2012)$$ through its strong decays

We extend our previous analysis on the mass of the recently discovered \(\Omega (2012)\) state by investigation of its strong decays and calculation of its width employing the method of light cone QCD sum rule. Considering two possibilities for the quantum numbers of \(\Omega (2012)\) state, namely 1P orbital excitation with \(J^P=\frac{3}{2}^-\) and 2S radial excitation with \(J^P=\frac{3}{2}^+\), we obtain the strong coupling constants defining the \(\Omega (1P/2S)\rightarrow \Xi K\) decays. The results of the coupling constants are then used to calculate the decay width corresponding to each possibility. Comparison of the obtained results on the total widths in this work with the experimental value and taking into account the results of our previous mass prediction on the \(\Omega (2012)\) state, we conclude that this state is 1P orbital excitation of the ground state \(\Omega \) baryon, whose quantum numbers are \(J^P=\frac{3}{2}^-\).     

Domain and Range of the Modified Wave Operator for Schrödinger Equations with a Critical Nonlinearity

We study the final problem for the nonlinear Schrödinger equation

$i{\partial }_{t}u+\frac{1}{2}\Delta u=\lambda|u|^{\frac{2}{n}}u,\quad (t,x)\in {\mathbf{R}}\times \mathbf{R}^{n},$

where\(\lambda \in{\bf R},n=1,2,3\). If the final data\(u_{+}\in {\bf H}^{0,\alpha }=\left\{ \phi \in {\bf L}^{2}:\left( 1+\left\vert x\right\vert \right) ^{\alpha }\phi \in {\bf L}^{2}\right\} \) with\(\frac{ n}{2} < \alpha < \min \left( n,2,1+\frac{2}{n}\right) \) and the norm\(\Vert \widehat{u_{+}}\Vert _{{\bf L}^{\infty }}\) is sufficiently small, then we prove the existence of the wave operator in L ². We also construct the modified scattering operator from H ^0,α to H ^0,δ with\(\frac{n}{2} < \delta < \alpha\).

     

The Largest Fragment of a Homogeneous Fragmentation Process

We show that in homogeneous fragmentation processes the largest fragment at time t has size

$$\begin{aligned} e^{-t \Phi '(\overline{p})}t^{-\frac{3}{2} (\log \Phi )'(\overline{p})+o(1)}, \end{aligned}$$

where \(\Phi \) is the Lévy exponent of the fragmentation process, and \(\overline{p}\) is the unique solution of the equation \((\log \Phi )'(\bar{p})=\frac{1}{1+\bar{p}}\). We argue that this result is in line with predictions arising from the classification of homogeneous fragmentation processes as logarithmically correlated random fields.

     

Excited state mass spectra of doubly heavy $$\Xi $$ baryons

In this paper, the mass spectra are obtained for doubly heavy \(\Xi \) baryons, namely, \(\Xi _{cc}^{+}\), \(\Xi _{cc}^{++}\), \(\Xi _{bb}^{-}\), \(\Xi _{bb}^{0}\), \(\Xi _{bc}^{0}\) and \(\Xi _{bc}^{+}\). These baryons consist of two heavy quarks (cc, bb, and bc) with a light (d or u) quark. The ground, radial, and orbital states are calculated in the framework of the hypercentral constituent quark model with Coulomb plus linear potential. Our results are also compared with other predictions, thus, the average possible range of excited states masses of these \(\Xi \) baryons can be determined. The study of the Regge trajectories is performed in (n, \(M^{2}\)) and (J, \(M^{2}\)) planes and their slopes and intercepts are also determined. Lastly, the ground state magnetic moments of these doubly heavy baryons are also calculated.     

On <Emphasis Type="Italic">τ</Emphasis>-Compactness of Products of <Emphasis Type="Italic">τ</Emphasis>-Measurable Operators

Let \(\mathcal {M}\) be a von Neumann algebra of operators on a Hilbert space \(\mathcal {H}\), τ be a faithful normal semifinite trace on \(\mathcal {M}\). We obtain some new inequalities for rearrangements of τ-measurable operators products. We also establish some sufficient τ-compactness conditions for products of selfadjoint τ-measurable operators. Next we obtain a τ-compactness criterion for product of a nonnegative τ-measurable operator with an arbitrary τ-measurable operator. We construct an example that shows importance of nonnegativity for one of the factors. The similar results are obtained also for elementary operators from \(\mathcal {M}\). We apply our results to symmetric spaces on \((\mathcal {M}, \tau )\). The results are new even for the *-algebra \(\mathcal {B}(\mathcal {H})\) of all linear bounded operators on \(\mathcal {H}\) endowed with the canonical trace τ = tr.     

Large Communities in a Scale-Free Network

We prove the existence of a large complete subgraph w.h.p. in a preferential attachment random graph process with an edge-step. That is, we consider a dynamic stochastic process for constructing a graph in which at each step we independently decide, with probability \(p\in (0,1)\), whether the graph receives a new vertex or a new edge between existing vertices. The connections are then made according to a preferential attachment rule. We prove that the random graph \(G_{t}\) produced by this so-called generalized linear preferential (GLP) model at time t contains a complete subgraph whose vertex set cardinality is given by \(t^\alpha \), where \(\alpha = (1-\varepsilon )\frac{1-p}{2-p}\), for any small \(\varepsilon >0\) asymptotically almost surely.