Scalar curvature of systems with fractal distribution functions

Starting with the relative entropy for two close statistical states we define the metric and calculate the scalar curvature R for systems with classical, boson and fermion fractal distribution functions with moment order parameter q  . In particular, we find that for q≠1$q\ne 1$ the scalar curvature is closer to zero implying that the fractal bosonic and fermionic systems are more stable than the standard ones.     

Constant scalar curvature and warped product globally null manifolds

This paper deals with the curvature properties of a class of globally null manifolds (M,g) which admit a global null vector field and a complete Riemannian hypersurface. Using the warped product technique we study the fundamental problem of finding a warped function such that the degenerate metric g admits a constant scalar curvature on M. Our work has an interplay with the static vacuum solutions of the Einstein equations of general relativity.     

LEFT-INVARIANT PSEUDO-EINSTEIN METRICS ON LIE GROUPS

In this article, we focus on left-invariant pseudo-Einstein metrics on Lie groups. To begin with, we give some examples of pseudo-Einstein metrics on Lie groups. Also we calculate the Levi-civita connection, and then Ricci tensor associated with left-invariant pseudo-Riemannian metrics on the unimodular Lie groups of dimension three. Furthermore, we show that the left-invariant pseudo-Einstein metric on SL(2) is unique up to a constant. At last, we study the left-invariant pseudo-Riemannian metrics on compact Lie groups and classify the pseudo-Einstein metrics on the low-dimensional compact Lie groups.     

Geometric curvature and phase of the Rabi model

We study the geometric curvature and phase of the Rabi model. Under the rotating-wave approximation (RWA), we apply the gauge independent Berry curvature over a surface integral to calculate the Berry phase of the eigenstates for both single and two-qubit systems, which is found to be identical with the system of spin-1/2 particle in a magnetic field. We extend the idea to define a vacuum-induced geometric curvature when the system starts from an initial state with pure vacuum bosonic field. The induced geometric phase is related to the average photon number in a period which is possible to measure in the qubit–cavity system. We also calculate the geometric phase beyond the RWA and find an anomalous sudden change, which implies the breakdown of the adiabatic theorem and the Berry phases in an adiabatic cyclic evolution are ill-defined near the anti-crossing point in the spectrum.     

Exact Scalar Field Inflationary Solution in Rainbow Universe

Using the Friedmann equation in rainbow Universe, we obtain an exact scalar field Inflationary Solution, which is a modification of the exact scalar field with negative potential −V ₀+m ² φ ²/2. Because the rainbow metric is Finsler metric, the result in this paper implies that the research of Finsler geometry in Cosmology should lead to several new physics theories.     

Scalar Flat Metrics of Eguchi-Hanson Type

We use a new method to construct a class of asymptotically locally flat, scalar flat metrics. These metrics were constructed via algebraic geometry method by LeBrun before and provide counterexamples to the generalized positive action conjecture of Hawking and Pope.     

Scalar Flat Metrics of Eguchi- Hanson Type

We use a new method to construct a class of asymptotically locally flat, scalar flat metrics. These metrics were constructed via algebraic geometry method by LeBrun before and provide counterexamples to the generalized positive action conjecture of Hawking and Pope.     

ECG Classification Based on Wasserstein Scalar Curvature

Electrocardiograms (ECG) analysis is one of the most important ways to diagnose heart disease. This paper proposes an efficient ECG classification method based on Wasserstein scalar curvature to comprehend the connection between heart disease and the mathematical characteristics of ECG. The newly proposed method converts an ECG into a point cloud on the family of Gaussian distribution, where the pathological characteristics of ECG will be extracted by the Wasserstein geometric structure of the statistical manifold. Technically, this paper defines the histogram dispersion of Wasserstein scalar curvature, which can accurately describe the divergence between different heart diseases. By combining medical experience with mathematical ideas from geometry and data science, this paper provides a feasible algorithm for the new method, and the theoretical analysis of the algorithm is carried out. Digital experiments on the classical database with large samples show the new algorithm’s accuracy and efficiency when dealing with the classification of heart disease.     

Scalar curvature and holomorphy potentials

A holomorphy potential is a complex valued function whose complex gradient, with respect to some Kähler metric, is a holomorphic vector field. Given k$k$ holomorphic vector fields on a compact complex manifold, form, for a given Kähler metric, a product of the following type: a function of the scalar curvature multiplied by functions of the holomorphy potentials of each of the vector fields. It is shown that the stipulation that such a product be itself a holomorphy potential for yet another vector field singles out critical metrics for a particular functional. This may be regarded as a generalization of the extremal metric variation of Calabi, where k=0$k=0$ and the functional is the square of the L²$L^2$-norm of the scalar curvature. The existence question for such metrics is examined in a number of special cases. Examples are constructed in the case of certain multifactored product manifolds. For the SKR metrics investigated by Derdzinski and Maschler and residing in the complex projective space, it is shown that only one type of nontrivial criticality holds in dimension three and above.     

Null lifts and projective dynamics

We describe natural Hamiltonian systems using projective geometry. The null lift procedure endows the tangent bundle with a projective structure where the null Hamiltonian is identified with a projective conic and induces a Weyl geometry. Projective transformations generate a set of known and new dualities between Hamiltonian systems, as for example the phenomenon of coupling-constant metamorphosis. We conclude outlining how this construction can be extended to the quantum case for Eisenhart–Duval lifts.     

Quantization and contact structure on manifolds with projective structure

We consider complex manifolds with a class of holomorphic coordinate functions satisfying the condition that each transition function is given by the standard action on of some element in . We show that such a manifold has a natural contact structure. Given any contact manifold, one can associate with it a symplectic manifold. It is shown that the symplectic manifolds arising from complex manifolds with special coordinate functions of the above type admit a canonical quantization.     

Scalar length scales and spatial averaging effects in turbulent piloted methane/air jet flames

The effects of spatial averaging in measurements of scalar variance and scalar dissipation in three piloted methane/air jet flames (Sandia flames C, D, and E) are investigated. Line imaging of Raman scattering, Rayleigh scattering, and laser-induced CO fluorescence is applied to obtain simultaneous single-shot measurements of temperature, the mass fractions of all major species, and mixture fraction, ξ, along 7-mm segments. Spatial filters are applied to ensembles of instantaneous profiles to quantify effects of spatial averaging on the Favre mean and variance of mixture fraction and scalar dissipation at several locations in the three flames. The radial contribution to scalar dissipation, χ_r = 2D_ξ (∂ξ/∂r)², is calculated from the filtered instantaneous profiles. The variance of mixture fraction tends to decrease linearly with increasing filter width, while the mean and variance of scalar dissipation are observed to follow an exponential dependence. In each case, the observed functional dependence is used to extrapolate to zero filter width, yielding estimates of the “fully resolved” profiles of measured quantities. Length scales for resolution of scalar variance and scalar dissipation are also extracted from the spatial filtering analysis and compared with length scales obtained from spatial autocorrelations. These results provide new insights on the small scale structure of turbulent jet flames and on the spatial resolution requirements for measurements of scalar variance and scalar dissipation.     

标量衍射理论的非傍轴近似及其有效性

当光束束腰（或衍射孔孔径）可与波长相比拟或光束具有较大的发散角时,傍轴近似不再成立.在标量瑞利.索末菲衍射积分的基础上,进一步研究了衍射场的非傍轴近似解,并详细分析了解的有效性.以平面波圆孔衍射为例,对衍射场的精确解、非傍轴近似解以及菲涅耳近似解进行了详细的数值计算和比较研究.结果表明,非傍轴近似对微小孔衍射非常精确、有效.     

Scalar analysis of arbitrarily shaped optical waveguides by full guided mode expansion method and optimized sine method

This paper shows that the full guided mode expansion method and the optimized sine method, with a rather rigorous procedure of determing the dimensions of reference waveguides, provide simple, fast and accurate way for the scalar analysis of arbitrarily shaped optical waveguides. Use of a smaller matrix results in good agreement with previous works.     

Plane-Symmetric Solitons of Spinor and Scalar Fields

We consider a system of nonlinear spinor and scalar fields with minimal coupling in general relativity. The nonlinearity in the spinor field Lagrangian is given by an arbitrary function of the invariants generated from the bilinear spinor forms S= and P=i⁵; the scalar Lagrangian is chosen as an arbitrary function of the scalar invariant = _,^,, that becomes linear at 0. The spinor and the scalar fields in question interact with each other by means of a gravitational field which is given by a plane-symmetric metric. Exact plane-symmetric solutions to the gravitational, spinor and scalar field equations have been obtained. Role of gravitational field in the formation of the field configurations with limited total energy, spin and charge has been investigated. Influence of the change of the sign of energy density of the spinor and scalar fields on the properties of the configurations obtained has been examined. It has been established that under the change of the sign of the scalar field energy density the system in question can be realized physically iff the scalar charge does not exceed some critical value. In case of spinor field no such restriction on its parameter occurs. In general it has been shown that the choice of spinor field nonlinearity can lead to the elimination of scalar field contribution to the metric functions, but leaving its contribution to the total energy unaltered.     

Spatial curvature and large scale Lorentz violation

The tension between the Hubble constant values obtained from local measurements and cosmic microwave background (CMB) measurements has motivated us to consider the cosmological model beyond ΛCDM. We investigate the cosmology in the large scale Lorentz violation model with a non-vanishing spatial curvature. The degeneracy among spatial curvature, cosmological constant, and cosmological contortion distribution makes the model viable in describing the known observational data. We obtain some constraints on the spatial curvature by comparing the relationship between measured distance modulus and red-shift with the predicted one, the evolution of matter density over time, and the evolution of effective cosmological constant. The implications of the large scale Lorentz violation model with the non-vanishing spatial curvature under these constrains are discussed.     

Generalized and projective synchronization in modulated time-delayed systems

In this Letter, we propose a method for generalized and projective synchronization in modulated time-delayed systems using nonlinear active control. Sufficient condition for generalized synchronization is calculated analytically by Krasovskii-Lyapunov stability theory. The validity of the proposed algorithm has been confirmed by simulation results. The proposed method helps to find the explicit form of the functional relation between the synchronized variables, for which very few formulations are known at the present moment. Both usual and lag-anticipatory cases can be treated on the same footing.     

Outer curvature and conformal geometry of an imbedding

The differential geometry of an imbedded (e.g. string or membrane world sheet) surface in a higher-dimensional background is shown to be conveniently describable (except in the null limit case) in terms of what are designated as its first, second, and third fundamental tensors, which will have the respective symmetry properties η_μν = η_(μν) as a trivial algebraic identity, K_μν^ρ = K_(μν)^ρ as the “generalised Weingarten identity”, which is the (Frobenius type) integrability condition for the imbedding, and Ξ_λμν^ρ = Ξ_(λμν)^ρ as a “generalised Codazzi equation”, which depends on the background geometry being flat or of constant curvature, needing replacement by a more complicated expression for a generic value of the background curvature B_κλ^μ_ν. The “generalised Gauss equation” expressing the dependence on this background curvature of the internal curvature tensor R_κλ^μ_ν of the imbedded surface is converted into terms of the first and second fundamental tensors, and it is thereby demonstrated that the vanishing of the (conformally invariant) “conformation tensor”, i.e. the trace free part C_μν^ρ of the second fundamental tensor K_μv^ρ, is a sufficient condition for conformal flatness of the imbedded surface (and thus in particular for the vanishing of its (Weyl type) conformal curvature tensor C_κλ^μ_ν) provided the background is itself conformally flat. In a trio of which the first two members are the generalised Gauss and Codazzi equations, the “third” member is shown to give an expression in terms of C_μν^ρ for the (trace free, conformally invariant) “outer curvature” tensor Ω_κλ^μ_ν whose vanishing is the condition for feasibility of the natural generalisation of the Walker frame transportation ansatz. The vanishing of C_μν^ρ is shown to be sufficient in a conformally flat background for the vanishing also of Ω_κλ^μ_ν.     

The Scalar Fields with Negative Kinetic Energy, Dark Matter and Dark Energy

The inhomogeneous cosmological model with generalized nonstatic Majumdar-Papapetrou metric is considered. The scalar field with negative kinetic energy and some usual matter sources of the gravitational field such as two-component nonlinear sigma model and perfect fluid are presented. Some exact solutions in these models are obtained and analyzed. In particular it is shown that the latent mass effect and effect of accelerating expansion (quintessence) of the Universe exist in these models. The 5-dimensional generalization of the model is presented, too.