Exact solutions to the angular Teukolsky equation with s ≠ 0

We first convert the angular Teukolsky equation under the special condition of τ ≠ 0, s ≠ 0, m=0 into a confluent Heun differential equation (CHDE) by taking different function transformation and variable substitution. And then according to the characteristics of both CHDE and its analytical solution expressed by a confluent Heun function (CHF), we find two linearly dependent solutions corresponding to the same eigenstate, from which we obtain a precise energy spectrum equation by constructing a Wronskian determinant. After that, we are able to localize the positions of the eigenvalues on the real axis or on the complex plane when τ is a real number, a pure imaginary number, and a complex number, respectively and we notice that the relation between the quantum number l and the spin weight quantum number s satisfies the relation l=∣s∣+ n, n=0, 1, 2···. The exact eigenvalues and the corresponding normalized eigenfunctions given by the CHF are obtained with the aid of Maple. The features of the angular probability distribution (APD) and the linearly dependent characteristics of two eigenfunctions corresponding to the same eigenstate are discussed. We find that for a real number τ, the eigenvalue is a real number and the eigenfunction is a real function, and the eigenfunction system is an orthogonal complete system, and the APD is asymmetric in the northern and southern hemispheres. For a pure imaginary number τ, the eigenvalue is still a real number and the eigenfunction is a complex function, but the APD is symmetric in the northern and southern hemispheres. When τ is a complex number, the eigenvalue is a complex number, the eigenfunction is still a complex function, and the APD in the northern and southern hemispheres is also asymmetric. Finally, an approximate expression of complex eigenvalues is obtained when n is greater than ∣s∣.     

Two-loop quark self-energy in a new formalism (I) Overlapping divergences

A new integration technique for multi-loop Feynman integrals, called the matrix method, is developed and then applied to the divergent part of the overlapping two-loop quark self-energy function iΣ in the light-cone gauge n · A^a(x) = 0, n² = 0. It is shown that the coefficient of the double-pole term is strictly local, even off mass-shell, while the coefficient of the single-pole term contains local as well as nonlocal parts. On mass-shell, the single-pole part is local, of course. It is worth noting that the original overlapping self-energy integral reduces eventually to 10 covariant and 38 noncovariant-gauge integrals. We were able to verify explicitly that the divergent parts of the 10 double covariant-gauge integrals agreed precisely with those currently used to calculate radiative corrections in the Standard Model.

Our new technique is amazingly powerful, being applicable to massive and massless integrals alike, and capable of handling both covariant-gauge integrals and the more difficult noncovariant-gauge integrals. Perhaps the most important feature of the matrix method is the ability to execute the 4ω-dimensional momentum integrations in a single operation, exactly and in analytic form. The method works equally well for other axial-type gauges, notably the temporal gauge (n² > 0) and the pure axial gauge (n² < 0).     

Differential operators on a Riemann surface with projective structure

Let X be a Riemann surface equipped with a projective structure and a theta characteristic on X, or in other words, is a holomorphic line bundle equipped with a holomorphic isomorphism with the holomorphic cotangent bundle Ω_X. The complement of the zero section in the total space of the line bundle has a natural holomorphic symplectic structure, and using , this symplectic structure has a canonical quantization. Using this quantization, holomorphic differential operators on X are constructed. The main result is the construction of a canonical isomorphism

, n≥0, provided i[−2(k−1),0].     

Superconformal invariants and extended supersymmetry

The superconformal invariants in analytic superspace are found. Superconformal invariance is shown to imply that the Green's functions of analytic operators are invariant holomorphic sections of a line bundle on a product of certain harmonic superspaces. It is argued that the correlation functions for a class of sufficiently low dimension gauge invariant operators in N = 2 and N = 4 supersymmetric Yang-Mills theory can be evaluated up to constants.     

Differential operators and immersions of a Riemann surface into a Grassmannian

We consider equivariant holomorphic immersions of a universal cover of a compact Riemann surface X into a Grassmannian satisfying a nondegeneracy condition. The equivariance condition says that there is a homomorphism ρ of the Galois group to that takes the natural action of the Galois group on to the action of the Galois group on defined using ρ. We prove that the space of such embeddings are in bijective correspondence with the space of all holomorphic differential operators of order two on a rank n vector bundle over X with the property that the symbol of the operator is an isomorphism.     

Connes’ spectral triple and U(1) gauge theory on finite sets

We first apply Connes’ noncommutative geometry to a finite point set. The explicit form of the action functional of U(1) gauge field on this n-point set is obtained. We then construct the U(1) gauge theory on a disconnected manifold consisting of n copies of a given manifold. In this case, the explicit action functional of U(1) gauge field is also obtained.     

Quantum partial least squares regression algorithm for multiple correlation problem

Partial least squares (PLS) regression is an important linear regression method that efficiently addresses the multiple correlation problem by combining principal component analysis and multiple regression. In this paper, we present a quantum partial least squares (QPLS) regression algorithm. To solve the high time complexity of the PLS regression, we design a quantum eigenvector search method to speed up principal components and regression parameters construction. Meanwhile, we give a density matrix product method to avoid multiple access to quantum random access memory (QRAM) during building residual matrices. The time and space complexities of the QPLS regression are logarithmic in the independent variable dimension n, the dependent variable dimension w, and the number of variables m. This algorithm achieves exponential speed-ups over the PLS regression on n, m, and w. In addition, the QPLS regression inspires us to explore more potential quantum machine learning applications in future works.     

Dynamics of spinor Bose–Einstein condensate subject to dissipation

We investigate the internal dynamics of the spinor Bose–Einstein condensates subject to dissipation by solving the Lindblad master equation. It is shown that for the condensates without dissipation its dynamics always evolve along a specific orbital in the phase space of(n_0, θ) and display three kinds of dynamical properties including Josephson-like oscillation, self-trapping-like oscillation, and ‘running phase'. In contrast, the condensates subject to dissipation will not evolve along the specific dynamical orbital. If component-1 and component-(-1) dissipate at different rates, the magnetization m will not conserve and the system transits between different dynamical regions. The dynamical properties can be exhibited in the phase space of(n_0, θ, m).     

On generalized Bäcklund transformations for equations describing pseudo-spherical surfaces

It is shown that each one-parameter subgroup of SL(2,R) gives rise to a local correspondence theorem between suitably generic solutions of arbitrary scalar equations describing pseudo-spherical surfaces. Thus, if appropriate genericity conditions are satisfied, there exist local transformations between any two solutions of scalar equations arising as integrability conditions of sl(2,R)-valued linear problems.

A complete characterization of evolution equations u_t=K(x,t,u,u_x,…,u_x^k) which are of strictly pseudo-spherical type is also provided.     

Intersecting M-branes

We present the magnetic duals of Güven's electric-type solutions of D = 11 supergravity preserving 1/4 or 1/8 of the D = 11 supersymmetry. We interpret the electric solutions as n orthogonal intersecting membranes and the magnetic solutions as n orthogonal intersecting 5-branes, with n = 2, 3; these cases obey the general rule that p-branes can self-intersect on (p − 2)-branes. On reduction to D = 4 these solutions become electric or magnetic dilaton black holes with dilaton coupling constant a = 1 (for n = 2) or (for n = 3). We also discuss the reduction to D = 10.     

Non-linear multi-plane wave solutions of self-dual Yang-Mills theory

New solutions of self-dual Yang-Mills (SDYM) equations are constructed in Minkowski space-time for the gauge groupSL(2, ). After proposing a Lorentz covariant formulation of Yang's equations, a set of Ansätze for exact non-linear multiplane wave solutions are proposed. The gauge fields are rational functions ofe ^x·ki (K _i ² =0, 1iN) for these Ansätze. At least, three families of multisoliton type solutions are derived explicitly. Their asymptotic behaviour shows that non-linear waves scatter non-trivially in Minkowski SDYM.On leave from LPTHE Université Paris VI, 4, Place Jussieu, Tour 16, ler étage, F-75230 Paris Cedex 05, France     

Nonperturbative dynamics of noncommutative gauge theory

We present a nonperturbative lattice formulation of noncommutative Yang–Mills theories in arbitrary even dimension. We show that lattice regularization of a noncommutative field theory requires finite lattice volume which automatically provides both an ultraviolet and an infrared cutoff. We demonstrate explicitly Morita equivalence of commutative U(p) gauge theory with p·n_f flavours of fundamental matter fields on a lattice of size L with twisted boundary conditions and noncommutative U(1) gauge theory with n_f species of matter on a lattice of size p·L with single-valued fields. We discuss the relation with twisted large N reduced models and construct observables in noncommutative gauge theory with matter.     

Inverse full state hybrid projective synchronization for chaotic maps with different dimensions

A new synchronization scheme for chaotic(hyperchaotic) maps with different dimensions is presented.Specifically,given a drive system map with dimension n and a response system with dimension m,the proposed approach enables each drive system state to be synchronized with a linear response combination of the response system states.The method,based on the Lyapunov stability theory and the pole placement technique,presents some useful features:(i) it enables synchronization to be achieved for both cases of n m and n m;(ii) it is rigorous,being based on theorems;(iii) it can be readily applied to any chaotic(hyperchaotic) maps defined to date.Finally,the capability of the approach is illustrated by synchronization examples between the two-dimensional H′enon map(as the drive system) and the three-dimensional hyperchaotic Wang map(as the response system),and the three-dimensional H′enon-like map(as the drive system) and the two-dimensional Lorenz discrete-time system(as the response system).     

T型石墨烯及其衍生物的结构与电子特性

采用基于密度泛函理论的第一原理方法研究了T型石墨烯及其衍生物-n(n=1—5)的结构稳定性和电子结构性质.T型石墨烯是一种拥有四角形环的二维碳材料同素异构体,通过改变连接四角形环的碳链上的碳原子个数n,可以得到一系列的sp-sp~2杂化结构,称其为T型石墨烯衍生物-n.计算结果表明:这些材料的结构稳定性、化学键类型和电子结构性质都依存于n的奇偶性.其中T型石墨烯(n=0)的结构最稳定,并形成一个由8个碳原子构成的大环.声子谱计算的结果表明,n为偶数时的体系具有动力学稳定性,而n为奇数时的体系则是不稳定的.n为偶数时体系四角形环之间的碳链上的化学键呈单、三键交叉排列,体系显示为金属性特征,且随着n的增大,体系的金属性加强.n为奇数时体系四角形环之间的碳链上的化学键则为双键连续排列,体系呈金属性且具有磁性(n=1除外).研究表明该系列材料作为一种新的二维碳材料同素异构体,具有独特的结构和丰富的电子结构特性,很可能在纳米器件中得到广泛应用.     

Quantum algorithm for neighborhood preserving embedding

Neighborhood preserving embedding (NPE) is an important linear dimensionality reduction technique that aims at preserving the local manifold structure. NPE contains three steps, i.e., finding the nearest neighbors of each data point, constructing the weight matrix, and obtaining the transformation matrix. Liang et al. proposed a variational quantum algorithm (VQA) for NPE [Phys. Rev. A 101 032323 (2020)]. The algorithm consists of three quantum sub-algorithms, corresponding to the three steps of NPE, and was expected to have an exponential speedup on the dimensionality n. However, the algorithm has two disadvantages: (i) It is not known how to efficiently obtain the input of the third sub-algorithm from the output of the second one. (ii) Its complexity cannot be rigorously analyzed because the third sub-algorithm in it is a VQA. In this paper, we propose a complete quantum algorithm for NPE, in which we redesign the three sub-algorithms and give a rigorous complexity analysis. It is shown that our algorithm can achieve a polynomial speedup on the number of data points m and an exponential speedup on the dimensionality n under certain conditions over the classical NPE algorithm, and achieve a significant speedup compared to Liang et al.'s algorithm even without considering the complexity of the VQA.     

Action of the loop group on the self dual Yang-Mills equation

Recently Kac-Moody symmetry has played an important role in mathematical physics. Dolan and Chau, Ge and Wu discovered an infinitesimal action of the Kac-Moody Lie algebra on the space of solutions of SDYM. We have discovered an action of the loop group on the space of generalized solutions of SDYM, which exponentiates the Kac-Moody action. The group acts by adding a special type of source onto the solution. The action is a geometric construction using the twistor picture.     

The propagator of the Calogero-Moser system in an external quadratic potential

We obtain the Hamilton operator of the Calogero-Moser quantum system in an external quadratic potential by conjugating the radial part for the action of SO(n) by conjugacy of the Hamilton operator of the quantum harmonic oscillator on the Euclidean vector space of real symmetric matrices. Then, with Mehler's formula, we derive the propagator of the problem. We also investigate some schemes to change the interaction constant. For two-particle systems, we obtain explicit formulae, whereas for many-particle systems, we reduce the computation of the propagator to finding a definite integral. We give also the short time approximation, the energy levels and the trace of the propagation operator.     

Domain-wall supergravities from sphere reduction

Kaluza-Klein sphere reductions of supergravities that admit Ads × Sphere vacuum solutions are believed to be consistent. The examples include the S⁴ and S⁷ reductions of eleven-dimensional supergravity, and the S⁵ reduction of ten-dimensional type IIB supergravity . In this paper we provide evidence that sphere reductions of supergravities that admit instead Domain-wallxSphere vacuum solutions are also consistent, where the background can be viewed as the near-horizon structure of a dilatonic p-brane of the theory. The resulting lower-dimensional theory is a gauged supergravity that admits a domain wall, rather than AdS, as a vacuum solution. We illustrate this consistency by taking the singular limits of certain modulus parameters, for which the originalSⁿ compactifying spheres (n = 4, 5 or 7) becomes S^p × R^q, with p = n − q < n. The consistency of the S⁴, S⁷ reductions then implies the consistency of the ^Sp reductions of the lower-dimensional supergravities. In particular, we obtain explicit non-linear ansätze for the S³ reduction of type IIA and heterotic supergravities, restricting to the U(1)² subgroup of the SO(4) gauge group of S³. We also study the black-hole solutions in the lower-dimensional gauged supergravities with domain-wall backgrounds. We find new domain-wall black holes which are not the singular-modulus limits of the AdS black holes of the original theories, and we obtain their Killing spinors.     

Remarks on stochastic non-linear birth and death models

The first part of this paper is an attempt to formulate and motivate additional work on the important problem of obtaining global bounds applicable to the controlled truncation of the paper relates specifically to the linear birth, quadratic death model. Asymptotic results are given for the first finite difference ΔT_m where T_m is the exactly known mean time to extinction starting from state m (m= 0,1,…). These results are in terms of the environmental carrying capacity n^* taken to be large. For m near zero ΔT_me^n*/(n^*)²; whereas, for m near n^*ΔT_m ≈ (π/2)^1/2/(n^*)^3/2. This indicates the vastly different time scales in those two regions of state space - with considerably slower action near extinction than near n^*.