$\mathcal {}$-Matrix-valued Wave Packet Frames in L2(ℝd,ℂs×r)$L^{2}(\mathbb {R}^{d}, \mathbb {C}^{s\times r})$

We study frame properties of a matrix-valued wave packet system in the matrix-valued function space \(L^{2}(\mathbb {R}^{d}, \mathbb {C}^{s\times r})\), where the lower frame condition is controlled by a bounded linear operator \(\mathcal {K}\) on \(L^{2}(\mathbb {R}^{d}, \mathbb {C}^{s\times r})\) (lower \(\mathcal {K}\)-frame condition, in short). There are many differences between ordinary frames and \(\mathcal {K}\)-frames. The lower \(\mathcal {K}\)-frame condition for matrix-valued wave packet Bessel sequences in \(L^{2}(\mathbb {R}^{d},\mathbb {C}^{s\times r})\) in terms of operators; a trace functional associated with a bounded linear operator on \(L^{2}(\mathbb {R}^{d}, \mathbb {C}^{s\times r})\); and a series associated with a matrix-valued Bessel sequence is presented. It is shown that matrix-valued wave packet frames are stable under small perturbation with respect to wave packet window functions.     

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).     

Noether symmetry approach in $$f(\mathcal {G},T)$$ gravity

We explore the recently introduced modified Gauss–Bonnet gravity (Sharif and Ikram in Eur Phys J C 76:640, 2016), \(f(\mathcal {G},T)\) pragmatic with \(\mathcal {G}\), the Gauss–Bonnet term, and T, the trace of the energy-momentum tensor. Noether symmetry approach has been used to develop some cosmologically viable \(f(\mathcal {G},T)\) gravity models. The Noether equations of modified gravity are reported for flat FRW universe. Two specific models have been studied to determine the conserved quantities and exact solutions. In particular, the well known deSitter solution is reconstructed for some specific choice of \(f(\mathcal {G},T)\) gravity model.     

Uniform in N global well-posedness of the time-dependent Hartree–Fock–Bogoliubov equations in $$\mathbb {R}^{1+1}$$

We prove the global well-posedness of the time-dependent Hartree–Fock–Bogoliubov (TDHFB) equations in \(\mathbb {R}^{1+1}\) with two-body interaction potential of the form \(N^{-1}v_N(x) = N^{\beta -1} v(N^\beta x)\) where \(v\ge 0\) is a sufficiently regular radial function, i.e., \(v \in L^1(\mathbb {R})\cap C^\infty (\mathbb {R})\). In particular, using methods of dispersive PDEs similar to the ones used in Grillakis and Machedon (Commun Partial Differ Equ 42:24–67, 2017), we are able to show for any scaling parameter \(\beta >0\) the TDHFB equations are globally well-posed in some Strichartz-type spaces independent of N, cf. (Bach et al. in The time-dependent Hartree–Fock–Bogoliubov equations for Bosons, 2016. arXiv:1602.05171).     

Braidings of Tensor Spaces

Let V be a braided vector space, i.e., a vector space together with a solution \({\hat{R}\in {{End}}(V\otimes V)}\) of the Yang–Baxter equation. Denote \({T(V):=\bigoplus_k V^{\otimes k}}\) . We associate to \({\hat{R}}\) a one-parameter family of solutions \({T(\hat{R})\in {\rm End}(T(V)\otimes T(V))}\) of the Yang–Baxter equation on the tensor space T (V). Main ingredients of the solution are braid analogues of the binomial coefficients and of the Pochhammer symbols. The association \({\hat{R}\rightsquigarrow T(\hat{R})}\) is functorial with respect to V.     

Searching for cosmological preferred axis using cosmographic approach

Recent released Planck data and other astronomical observations show that the universe may be anisotropic on large scales. This hints a cosmological privileged axis in our anisotropic expanding universe. This paper proceeds a modified redshift in anisotropic cosmological model as \( 1+\tilde{z}(t,\hat{\mathbf{p }})=\frac{a(t_{0)}}{a(t)}(1-A(\hat{\mathbf{n }}.\hat{\mathbf{p }}))\) (where A is the magnitude of anisotropy, \(\hat{\mathbf{n }}\) is the direction of privileged axis, and \(\hat{\mathbf{p }}\) is the direction of each SNe Ia sample to galactic coordinates) along with anisotropic parameter \(\delta =\frac{A(\hat{\mathbf{n }}.\hat{\mathbf{p }})}{1+A(\hat{\mathbf{n }}.\hat{\mathbf{p }})}\). The luminosity distance is expanded with model-independent cosmographic parameters as a function of modified redshift \(\tilde{z}\). As the transformation matrix \(M(n\times n)\) is obtained to convert the Taylor series coefficients of isotropic luminosity distance to corresponding anisotropic parameters. These results culminate the magnitude of anisotropy about \(\mid A\mid \simeq 10^{-3}\) and the direction of preferred axis as \((l,b)=\left( 297^{-34}_{+34},3^{-28}_{+28}\right) \), which are consistent with other studies in \(1-\sigma \) confidence level.     

Shock Fluctuations for the Hammersley Process

We consider the Hammersley interacting particle system starting from a shock initial profile with densities \(\rho ,\lambda \in {\mathbb R}\) (\( \rho < \lambda \)). The microscopic shock is taken as the position of a second-class particle initially at the origin, and the main results are: (i) a central limit theorem for the shock; (ii) the variance of the shock equals \(2[\lambda \rho (\lambda - \rho )]^{-1}t + O(t^{2/3})\). By using the same method of proof, we also prove similar results for first-class particles.     

Signals of leptophilic dark matter at the ILC

Adopting a model independent approach, we constrain the various effective interactions of leptophilic DM particles with the visible world from the WMAP and Planck data. The thermally averaged indirect DM annihilation cross section and the DM–electron direct-detection cross section for such a DM candidate are observed to be consistent with the respective experimental data. We study the production of cosmologically allowed leptophilic DM in association with \(Z\, (Z\rightarrow f\bar{f})\), \(f\equiv q,\,e^-,\, \mu ^-\) at the ILC. We perform the \(\chi ^2\) analysis and compute the 99% C.L. acceptance contours in the \(m_\chi \) and \(\varLambda \) plane from the two-dimensional differential distributions of various kinematic observables obtained after employing parton showering and hadronisation to the simulated data. We observe that the dominant hadronic channel provides the best kinematic reach of 2.62 TeV (\(m_\chi \) = 25 GeV), which further improves to \(\sim \)3 TeV for polarised beams at \(\sqrt{s} = 1\) TeV and an integrated luminosity of 1 ab\(^{-1}\).     

On the Local Equivalence Between the Canonical and the Microcanonical Ensembles for Quantum Spin Systems

We study a quantum spin system on the d-dimensional hypercubic lattice \(\Lambda \) with \(N=L^d\) sites with periodic boundary conditions. We take an arbitrary translation invariant short-ranged Hamiltonian. For this system, we consider both the canonical ensemble with inverse temperature \(\beta _0\) and the microcanonical ensemble with the corresponding energy \(U_N(\beta _0)\). For an arbitrary self-adjoint operator \(\hat{A}\) whose support is contained in a hypercubic block B inside \(\Lambda \), we prove that the expectation values of \(\hat{A}\) with respect to these two ensembles are close to each other for large N provided that \(\beta _0\) is sufficiently small and the number of sites in B is \(o(N^{1/2})\). This establishes the equivalence of ensembles on the level of local states in a large but finite system. The result is essentially that of Brandao and Cramer (here restricted to the case of the canonical and the microcanonical ensembles), but we prove improved estimates in an elementary manner. We also review and prove standard results on the thermodynamic limits of thermodynamic functions and the equivalence of ensembles in terms of thermodynamic functions. The present paper assumes only elementary knowledge on quantum statistical mechanics and quantum spin systems.     

Geodesic Deviation Equation in ΛCDM f(T,T) Gravity

The geodesic deviation equation has been investigated in the framework of \(f(T,\mathcal {T})\) gravity, where T denotes the torsion and \(\mathcal {T}\) is the trace of the energy-momentum tensor, respectively. The FRW metric is assumed and the geodesic deviation equation has been established following the General Relativity approach in the first hand and secondly, by a direct method using the modified Friedmann equations. Via fundamental observers and null vector fields with FRW background, we have generalized the Raychaudhuri equation and the Mattig relation in \(f(T,\mathcal {T})\) gravity. Furthermore, we have numerically solved the geodesic deviation equation for null vector fields by considering a particular form of \(f(T,\mathcal {T})\) which induces interesting results susceptible to be tested with observational data.     

Gibbs Measures Over Locally Tree-Like Graphs and Percolative Entropy Over Infinite Regular Trees

Consider a statistical physical model on the d-regular infinite tree \(T_{d}\) described by a set of interactions \(\Phi \). Let \(\{G_{n}\}\) be a sequence of finite graphs with vertex sets \(V_n\) that locally converge to \(T_{d}\). From \(\Phi \) one can construct a sequence of corresponding models on the graphs \(G_n\). Let \(\{\mu _n\}\) be the resulting Gibbs measures. Here we assume that \(\{\mu _{n}\}\) converges to some limiting Gibbs measure \(\mu \) on \(T_{d}\) in the local weak\(^*\) sense, and study the consequences of this convergence for the specific entropies \(|V_n|^{-1}H(\mu _n)\). We show that the limit supremum of \(|V_n|^{-1}H(\mu _n)\) is bounded above by the percolative entropy \(H_{\textit{perc}}(\mu )\), a function of \(\mu \) itself, and that \(|V_n|^{-1}H(\mu _n)\) actually converges to \(H_{\textit{perc}}(\mu )\) in case \(\Phi \) exhibits strong spatial mixing on \(T_d\). When it is known to exist, the limit of \(|V_n|^{-1}H(\mu _n)\) is most commonly shown to be given by the Bethe ansatz. Percolative entropy gives a different formula, and we do not know how to connect it to the Bethe ansatz directly. We discuss a few examples of well-known models for which the latter result holds in the high temperature regime.     

New General Approach for Normally Ordering Coordinate-Momentum Operator Functions

By virtue of integration technique within ordered product of operators and Dirac’s representation theory we find a new general formula for normally ordering coordinate-momentum operator functions, that is \(f(g\hat {{Q}}+h\hat {P})= :\exp [\textstyle {g^{2}+h^{2} \over 4}\textstyle {{\partial ^{2}} \over {\partial (g\hat {{Q}}+h\hat {P})^{2}}}]f(g\hat {{Q}}+h\hat {P})\):, where \(\hat {Q}\) and \(\hat {P}\) are the coordinate operator and momentum operator respectively, the symbol :: denotes normal ordering. Using this formula we can derive a series of new relations about Hermite polynomial and Laguerre polynomial, as well as some new differential relations.     

Spinor Field at the Phase Transition Point of Reissner-Nordström de Sitter Space

The radial parts of Dirac equation between the outer black hole horizon and the cosmological horizon are solved in Reissner-Nordström de Sitter (RNdS) space when it is at the phase transition point. We use an accurate polynomial approximation to approximate the modified tortoise coordinate \(\hat{r}_{*}\) in order to get the inverse function \(r=r(\hat{r}_{*})\) and the potential \(V(\hat{r}_{*})\). Then we use a quantum mechanical method to solve the wave equation numerically. We consider two cases, one is when the two horizons are lying close to each other, the other is when the two horizons are widely separated.     

Supersymmetric deformations of 3D SCFTs from tri-Sasakian truncation

We holographically study supersymmetric deformations of \(N=3\) and \(N=1\) superconformal field theories in three dimensions using four-dimensional \(N=4\) gauged supergravity coupled to three-vector multiplets with non-semisimple \(SO(3)\ltimes (\mathbf {T}^3,\hat{\mathbf {T}}^3)\) gauge group. This gauged supergravity can be obtained from a truncation of 11-dimensional supergravity on a tri-Sasakian manifold and admits both \(N=1,3\) supersymmetric and stable non-supersymmetric \(AdS_4\) critical points. We analyze the BPS equations for SO(3) singlet scalars in detail and study possible supersymmetric solutions. A number of RG flows to non-conformal field theories and half-supersymmetric domain walls are found, and many of them can be given analytically. Apart from these “flat” domain walls, we also consider \(AdS_3\)-sliced domain wall solutions describing two-dimensional conformal defects with \(N=(1,0)\) supersymmetry within the dual \(N=1\) field theory while this type of solutions does not exist in the \(N=3\) case.     

Leptoquark mechanism of neutrino masses within the grand unification framework

We demonstrate the viability of the one-loop neutrino mass mechanism within the framework of grand unification when the loop particles comprise scalar leptoquarks (LQs) and quarks of the matching electric charge. This mechanism can be implemented in both supersymmetric and non-supersymmetric models and requires the presence of at least one LQ pair. The appropriate pairs for the neutrino mass generation via the up-type and down-type quark loops are \(S_3\)–\(R_2\) and \(S_{1,\,3}\)–\(\tilde{R}_2\), respectively. We consider two distinct regimes for the LQ masses in our analysis. The first regime calls for very heavy LQs in the loop. It can be naturally realized with the \(S_{1,\,3}\)–\(\tilde{R}_2\) scenarios when the LQ masses are roughly between \(10^{12}\) and \(5 \times 10^{13}\) GeV. These lower and upper bounds originate from experimental limits on partial proton decay lifetimes and perturbativity constraints, respectively. Second regime corresponds to the collider accessible LQs in the neutrino mass loop. That option is viable for the \(S_3\)–\(\tilde{R}_2\) scenario in the models of unification that we discuss. If one furthermore assumes the presence of the type II see-saw mechanism there is an additional contribution from the \(S_3\)–\(R_2\) scenario that needs to be taken into account beside the type II see-saw contribution itself. We provide a complete list of renormalizable operators that yield necessary mixing of all aforementioned LQ pairs using the language of SU(5). We furthermore discuss several possible embeddings of this mechanism in SU(5) and SO(10) gauge groups.     

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.     

Star product on complex sphere $$\mathbb {S}^{2n}$$

We construct a \(U_q\bigl (\mathfrak {s}\mathfrak {o}(2n+1)\bigr )\)-equivariant local star product on the complex sphere \(\mathbb {S}^{2n}\) as a non-Levi conjugacy class \(SO(2n+1)/SO(2n)\).     

Monte Carlo Glauber model with meson cloud: predictions for 5.44 TeV Xe + Xe collisions

We study, within the Monte-Carlo Glauber model, centrality dependence of the midrapidity charged multiplicity density \(dN_{ch}/d\eta \) and of the anisotropy coefficients \(\varepsilon _{2,3}\) in Pb + Pb collisions at \(\sqrt{s}=5.02\) TeV and in Xe + Xe collisions at \(\sqrt{s}=5.44\) TeV. Calculations are performed for versions with and without nucleon meson cloud. The fraction of the binary collisions, \(\alpha \), has been fitted to the data on \(dN_{ch}/d\eta \) in Pb + Pb collisions. We obtain \(\alpha \approx 0.09(0.13)\) with (without) meson cloud. The effect of meson cloud on the \(dN_{ch}/d\eta \) is relatively small. For Xe + Xe collisions for 0–5% centrality bin we obtain \(dN_{ch}/d\eta \approx 1149\) and 1134 with and without meson cloud, respectively. We obtain \(\varepsilon _2(\mathrm {Xe})/\varepsilon _2(\mathrm {Pb})\sim 1.45\) for most central collisions, and \(\varepsilon _2(\mathrm {Xe})/\varepsilon _2(\mathrm {Pb})\) close to unity at Open image in new window . We find a noticeable increase of the eccentricity in Xe + Xe collisions at small centralities due to the prolate shape of the Xe nucleus. The triangularity in Xe + Xe collisions is bigger than in Pb + Pb collisions at Open image in new window . We obtain \(\varepsilon _3(\text{ Xe })/\varepsilon _3(\text{ Pb })\sim 1.3\) at Open image in new window .     

On the Non-existence of Two-Valued Lattice Homomorphisms of Quantum Logic

We show that there exists no two-valued lattice homomorphism from the lattice \(\mathscr{L}(\mathscr{H})\) of all closed linear subspaces of a Hilbert space \(\mathscr{H}\) with \(\dim (\mathscr{H}) \geq 2\) by using the notion of prime filters.