Search Results (37)

Semiclassical (SC) approximations for various representations of a quantum state are constructed on a single (Lagrangian) surface in the phase space but such surface is not available for chaotic systems. An analogous evolution surface underlies SC representations of the evolution operator, albeit in a doubled phase space. Here, it is shown that corresponding to the Fourier transform on a unitary operator, represented as a Green function or spectral Wigner function, a Legendre transform generates a resolvent surface as the classical basis for SC representations of the resolvent operator in the double-phase space, independently of the integrable or chaotic nature of the system. This surface coincides with derivatives of action functions (or generating functions) depending on the choice of appropriate coordinates, and its growth departs from the energy shell following trajectories in the double-phase space. In an initial study of the resolvent surface based on its caustics, its complex nature is revealed to be analogous to a multidimensional sponge. Resummation of the trace of the resolvent in terms of linear combinations of periodic orbits, known as pseudo orbits or composite orbits, provides a cutoff to the SC sum at the Heisenberg time. Here, it is shown that the corresponding actions for higher times can be approximately included within true secondary periodic orbits, in which heteroclinic orbits join multiple windings of relatively short periodic orbits into larger circuits. Full article

This study focuses on the numerical solution and dynamics analysis of fractional governing equations related to double-layer nanoplates based on the shifted Legendre polynomials algorithm. Firstly, the fractional governing equations of the complicated mechanical behavior for bilayer nanoplates are constructed by combining the Fractional Kelvin–Voigt (FKV) model with the Caputo fractional derivative and the theory of nonlocal elasticity. Then, the shifted Legendre polynomial is used to approximate the displacement function, and the governing equations are transformed into algebraic equations to facilitate the numerical solution in the time domain. Moreover, the systematic convergence analysis is carried out to verify the convergence of the ternary displacement function and its fractional derivatives in the equation, ensuring the rigor of the mathematical model. Finally, a dimensionless numerical example is given to verify the feasibility of the proposed algorithm, and the effects of material parameters on plate displacement are analyzed for double-layer plates with different materials. Full article

The variance function (VF) is central to natural exponential family (NEF) theory. Prompted by an online query about whether, beyond the classical normal NEF, other real-line NEFs with bounded VFs exist, we establish three complementary sets of sufficient conditions that yield many such families. One set imposes a polynomial-growth bound on the NEF’s generating measure, ensuring rapid tail decay and a uniformly bounded VF. A second set relies on the Legendre duality, requiring a uniform positive lower bound on the second derivative of the generating function, which likewise ensures a bounded VF. The third set starts from the standard normal distribution and constructs an explicit sequence of NEFs whose Laplace transforms and VFs remain bounded. Collectively, these results reveal a remarkably broad class of NEFs whose Laplace transforms are not expressible in elementary form (apart from those stemming from the standard normal case), yet can be handled easily using modern symbolic and numerical software. Worked examples show that NEFs with bounded VFs are far more varied than previously recognized, offering practical alternatives to the normal and other classical models for real-data analysis across many fields. Full article

The analysis of resilience indicators was based on daily milk yields recorded from 3347 lactations of 3080 Holstein cows located on 10 farms between 2022 and 2024. Six farms used an automatic milking system. A random regression function with a fourth-degree Legendre polynomial was used to predict the lactation curve. The indicators were the natural log-transformed variance (LnVar), lag-1 autocorrelation (r-auto), and skewness (skew) of daily milk yield (DMY) deviations from the predicted lactation curve, as well as the log-transformed variance of DMY (Var). The single-step genomic prediction method (ssGBLUP) was used for genomic evaluation. A total of 9845 genotyped animals and 36,839 SNPs were included. Heritability estimates were low (0.02–0.13). The strongest genetic correlation (0.87) was found between LnVar and Var. The genetic correlation between r-auto and skew was also strong but negative (−0.73). Resilience indicators showed a negative correlation with milk yield per lactation and a positive correlation with fat and protein contents. The negative correlation between fertility and two resilience indicators may be due to the evaluation period (50th–150th day of lactation) being when cows are most often bred after calving, and a decrease in production may accompany a significant oestrus. The associations between resilience indicators and health traits (clinical mastitis, claw health) were weak but mostly favourable. Full article

The principal aim of this paper is to introduce the extended Voigt-type function ${\mathbb{V}}_{\mu ,\nu }(x,y)$ and its counterpart extension ${\mathbb{W}}_{\mu ,\nu }(x,y)$, involving the Neumann function $Y_{\nu }$ in the kernel of the representing integral. The newly defined integral reduces to the classical Voigt functions $K(x,y)$ and $L(x,y)$, and to their generalizations by Srivastava and Miller, by the unification of Klusch. Following an approach by Srivastava and Pogány, we also present the multiparameter and multivariable versions ${\mathbb{V}}_{\mu ,\nu }^{\left(r\right)}(x,y),{\mathbb{W}}_{\mu ,\nu }^{\left(r\right)}(x,y)$ and the r positive integer of the initial extensions ${\mathbb{V}}_{\mu ,\nu }(x,y),{\mathbb{W}}_{\mu ,\nu }(x,y)$. Several computable series expansions are obtained for the discussed Voigt-type functions in terms of Humbert confluent hypergeometric functions ${\Psi }_2^{\left(r\right)}$. Furthermore, by transforming the input extended Voigt-type functions by the Grünwald–Letnikov fractional derivative, we establish representation formulae in terms of the associated Legendre functions of the second kind $Q_{\eta }^{-\nu }$ in the two-parameter and two-variable cases. Finally, functional bounding inequalities are given for ${\mathbb{V}}_{\mu ,\nu }(x,y)$ and ${\mathbb{W}}_{\mu ,\nu }(x,y)$. Particularly interesting results are presented for the Neumann function $Y_{\nu }$ and for the Struve ${\mathbf{H}}_{\nu }$ function in the form of several functional bounds. The article ends with a thorough discussion and closing remarks. Full article

Borehole pulsed eddy-current (PEC) systems based on uniform linear multicoil arrays (ULMAs) perform efficient nondestructive evaluations (NDEs) of metal casings. However, the limited physical space of the borehole restricts the degrees of freedom (DoFs) of ULMAs to be less than the number of constraints, which leads to the difficulty of compensating for the differences in signals acquired by different receivers with different transmitting-to-receiving distances (TRDs), and thus limits the effectiveness of the ULMA system. To solve this problem, this paper proposes sparse linear constraint minimum variance (S-LCMV) for NDEs of downhole casings employing ULMAs. By transforming and characterizing the original PEC signal, it was observed that the signal power dramatically decreased with increasing Legendre polynomial stage, confirming that the signal was sparsely distributed over the Gauss–Legendre stages. Using this property, the S-LCMV cost function with reduced constraints was constructed to provide enough DoFs to accurately calculate the weight coefficients, thus improving the detection performance. The effectiveness of the proposed method was verified through field experiments on an 8-element ULMA installed in a borehole PEC system for NDEs of oil-well casings. The results demonstrate that the proposed method could improve the weighting effect by reducing the number of constraints by 70% while ensuring the approximation accuracy, which effectively improved the signal-to-noise ratio of the measured signals and reduced the computational cost by about 87.9%. Full article

We compare a relativistic and a nonrelativistic version of Ostrogradsky’s method for higher-time derivative theories extended to scalar field theories and consider as an alternative a multi-field variant. We apply the schemes to space–time rotated modified Korteweg–de Vries systems and, exploiting their integrability, to Hamiltonian systems built from space–time rotated inverse Legendre transformed higher-order charges of these systems. We derive the equal-time Poisson bracket structures of these theories, establish the integrability of the latter theories by means of the Painlevé test and construct exact analytical period benign solutions in terms of Jacobi elliptic functions to the classical equations of motion. The classical energies of these partially complex solutions are real when they respect a certain modified CPT-symmetry and complex when this symmetry is broken. The higher-order Cauchy and initial-boundary value problem are addressed analytically and numerically. Finally, we provide the explicit quantization of the simplest mKdV system, exhibiting the usual conundrum of having the choice between having to deal with either a theory that includes non-normalizable states or spectra that are unbounded from below. In our non-Hermitian system, the choice is dictated by the correct sign in the decay width. Full article

A household consumption and optimal portfolio problem pertinent to life insurance (LI) in a continuous time setting is examined. The family receives a random income before the parents’ retirement date. The price of the risky asset is driven by the exponential Ornstein–Uhlenbeck (O-U) process, which can better reflect the state of the financial market. If the parents pass away prior to their retirement time, the children do not have any work income and LI can be purchased to hedge the loss of wealth due to the parents’ accidental death. Meanwhile, utility functions (UFs) of the parents and children are individually taken into account in relation to the uncertain lifetime. The purpose of the family is to appropriately maximize the weighted average of the corresponding utilities of the parents and children. The optimal strategies of the problem are achieved using a dynamic programming approach to solve the associated Hamilton–Jacobi–Bellman (HJB) equation by employing the convex dual theory and Legendre transform (LT). Finally, we aim to examine how variations in the weight of the parents’ UF and the coefficient of risk aversion affect the optimal policies. Full article

The notion of k-harmonic curves is associated with the kth-order variational problem defined by the k-energy functional. The present paper gives a geometric formulation of this higher-order variational problem on a Riemannian manifold M and describes a generalized Legendre transformation defined from the kth-order tangent bundle $T^{k}M$ to the cotangent bundle $T^{*}{T}^{k-1}M$. The intrinsic version of the Euler–Lagrange equation and the corresponding Hamiltonian equation obtained via the Legendre transformation are achieved. Geodesic and cubic polynomial interpolation is covered by this study, being explored here as harmonic and biharmonic curves. The relationship of the variational problem with the optimal control problem is also presented for the case of biharmonic curves. Full article

Legendre coefficients of an integrable function $f\left(x\right)$ are proved to coincide with the Fourier coefficients with a nonnegative index of a suitable Abel-type transform of the function itself. The numerical computation of N Legendre coefficients can thus be carried out efficiently in $\mathcal{O}\left(NlogN\right)$ operations by means of a single fast Fourier transform of the Abel-type transform of $f\left(x\right)$. Symmetries associated with the Abel-type transform are exploited to further reduce the computational complexity. The dual problem of calculating the sum of Legendre expansions at a prescribed set of points is also considered. We prove that a Legendre series can be written as the Abel transform of a suitable Fourier series. This fact allows us to state an efficient algorithm for the evaluation of Legendre expansions. Finally, some numerical tests are illustrated to exemplify and confirm the theoretical results. Full article

This study focuses on the development of a time-variable regional geo-potential model for Antarctica using the spherical cap harmonic analysis (SCHA) basis functions. The model is derived from line-of-sight gravity difference (LGD) measurements obtained from the GRACE-Follow-On (GFO) mission. The solution of a Laplace equation for the boundary values over a spherical cap is used to expand the geo-potential coefficients in terms of Legendre functions with a real degree and integer order suitable for regional modelling, which is used to constrain the geo-potential coefficients using LGD measurements. To validate the performance of the SCHA, it is first utilized with LGD data derived from a L2 JPL (Level 2 product of the Jet Propulsion Laboratory). The obtained LGD data are used to compute the local geo-potential model up to K_max = 20, corresponding to the SH degree and order up to 60. The comparison of the radial gravity on the Earth’s surface map across Antarctica with the corresponding radial gravity components of the L2 JPL is carried out using local geo-potential coefficients. The results of this comparison provide evidence that these basis functions for K_max = 20 are valid across the entirety of Antarctica. Subsequently, the analysis proceeds using LGD data obtained from the Level 1B product of GFO by transforming these LGD data into the SCHA coordinate system and applying them to constrain the SCHA harmonic coefficients up to K_max = 20. In this case, several independent LGD profiles along the trajectories of the satellites are devised to verify the accuracy of the local model. These LGD profiles are not employed in the inverse problem of determining harmonic coefficients. The results indicate that using regional harmonic basis functions, specifically spherical cap harmonic analysis (SCHA) functions, leads to a close estimation of LGD compared to the L2 JPL. The regional harmonic basis function exhibits a root mean square error (RMSE) of 3.71 × 10⁻⁴ mGal. This represents a substantial improvement over the RMSE of the L2 JPL, which is 6.36 × 10⁻⁴ mGal. Thus, it can be concluded that the use of local geo-potential coefficients obtained from SCHA is a reliable method for extracting nearly the full gravitational signal within a spherical cap region, after validation of this method. The SCHA model provides significant realistic information as it addresses the mass gain and loss across various regions in Antarctica. Full article

We discuss the generalized quantum Lyapunov exponents $L_q$, defined from the growth rate of the powers of the square commutator. They may be related to an appropriately defined thermodynamic limit of the spectrum of the commutator, which plays the role of a large deviation function, obtained from the exponents $L_q$ via a Legendre transform. We show that such exponents obey a generalized bound to chaos due to the fluctuation–dissipation theorem, as already discussed in the literature. The bounds for larger q are actually stronger, placing a limit on the large deviations of chaotic properties. Our findings at infinite temperature are exemplified by a numerical study of the kicked top, a paradigmatic model of quantum chaos. Full article

The density of states of a quantum system can be calculated from its definition, but, in some cases, this approach is quite cumbersome. Alternatively, the density of states can be deduced from the microcanonical entropy or from the canonical partition function. After discussing the relationship among these procedures, we suggest a simple numerical method, which is equivalent in the thermodynamic limit to perform a Legendre transformation, to obtain the density of states from the Helmholtz free energy. We apply this method to determine the many-body density of states of the unitary Fermi gas, a very dilute system of identical fermions interacting with a divergent scattering length. The unitary Fermi gas is highly symmetric due to the absence of any internal scale except for the average distance between two particles and, for this reason, its equation of state is called universal. In the last part of the paper, by using the same thermodynamical techniques, we review some properties of the density of states of a Schwarzschild black hole, which shares the problem of finding the density of states directly from its definition with the unitary Fermi gas. Full article

The technique of functional Legendre transforms is used to develop an effective method for calculating the characteristics of critical phenomena in quantum field theory models in the Euclidean space of dimension d. Based on the diagrammatic representation of the second Legendre transform in the theory with a cubic interaction potential, the construction of self-consistent equations is carried out, the solution of which makes it possible to find the dimensions not only of the main fields, but also of the quadratic on the composite operators within the $1/n$-expansion. Application of the proposed methods in the model F has given the opportunity to calculate in the main approximation by $1/n$ the anomalous dimensions of both scalar and tensor composite operators quadratic on the fields $\varphi$. For them, as functions of the spatial dimension d, we obtained explicit analytical expressions in the form of relations of two polynomials with integer coefficients. Full article

In this paper, based on the Euler equation and mass conservation equation in spherical coordinates, the ratio of the stratospheric average width to the planetary radius and the ratio of the vertical velocity to the horizontal velocity are selected as parameters under appropriate boundary conditions. We establish the approximate system using these two small parameters. In addition, we consider the time dependence of the system and establish the governing equations describing the atmospheric flow. By introducing a flow function to code the system, a nonlinear vorticity equation describing the planetary flow in the stratosphere is obtained. The governing equations describing the atmospheric flow are transformed into a second-order homogeneous linear ordinary differential equation and a Legendre’s differential equation by applying the method of separating variables based on the concepts of spherical harmonic functions and weak solutions. The Gronwall inequality and the Cauchy–Schwartz inequality are applied to priori estimates for the vorticity equation describing the stratospheric planetary flow under the appropriate initial and boundary conditions. The existence and non-uniqueness of weak solutions to the vorticity equation are obtained by using the functional analysis technique. Full article