Dimensional Case Research Articles

Approximation of basins of attraction for bistable dynamical system for olive disease control

In this paper the bistability of a mathematical model for the control of an olive tree disease is studied. The basin of stability values were computed for three different pairs of bistable equilibrium points by using the software bSTAB. Moreover an extension of the software functionalities is made, first by approximating the shape of the attractors in three dimensions and second by extending the sensitivity study with respect to some important parameters of the numerical scheme, e.g. hyperparameters, to the two dimensional case. Analogously, the bifurcation diagram of the basin of stability values with respect to one parameter of the model was extended to the two dimensional case where two parameters of the model can vary simultaneously. Finally an approximation of the surfaces of the sensitivity analysis and bifurcation diagrams were made.

Canonical equivariant cohomology classes generating zeta values of totally real fields

It is known that the special values at nonpositive integers of a Dirichlet L L -function may be expressed using the generalized Bernoulli numbers, which are defined by a generating function. The purpose of this article is to consider the generalization of this classical result to the case of Hecke L L -functions of totally real fields. Hecke L L -functions may be expressed canonically as a finite sum of zeta functions of Lerch type. By combining the non-canonical multivariable generating functions constructed by Shintani [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), pp. 393–417], we newly construct a canonical class, which we call the Shintani generating class, in the equivariant cohomology of an algebraic torus associated to the totally real field. Our main result states that the specializations at torsion points of the derivatives of the Shintani generating class give values at nonpositive integers of the zeta functions of Lerch type. This result gives the insight that the correct framework in the higher dimensional case is to consider higher equivariant cohomology classes instead of functions.

Absolute points of correlations of PG(5,qn)

The sets of the absolute points of (possibly degenerate) polarities of a projective space are well known. The sets of the absolute points of (possibly degenerate) correlations, different from polarities, of PG(2,qn) have been completely determined by B.C. Kestenband in 11 papers from 1990 to 2014, for non-degenerate correlations and by J. D'haeseleer and N. Durante in [3] for degenerate correlations. The sets of the absolute points of degenerate correlations, different from degenerate polarities, of PG(3,qn) and PG(4,qn) have been classified respectively in [8] and [10]. In this paper, we consider the five dimensional case and determine the sets of the absolute points of degenerate correlations, different from degenerate polarities, of PG(5,qn). Moreover, we show that some of these sets are related to some ovoids of PG(5,qn).

Growth of log-analytic functions

We show that unary log-analytic functions are polynomially bounded. In the higher dimensional case, globally a log-analytic function can have exponential growth. We show that a log-analytic function is polynomially bounded on a definable set which contains the germ of every ray at infinity.

Efficient random subspace decision forests with a simple probability dimensionality setting scheme

Random subspace decision forests are commonly used machine learning methods in a wide range of application domains. How to set the random subspace dimensionality ds in decision forests is a considerable issue that impacts classification quality and efficiency, especially for high dimensional cases. To obtain effective and efficient decision forests that are generally suitable for various classification cases, this paper proposes a novel framework, named Efficient Random Subspace decision forest (ERS). A Half-Range Discrete Uniform distribution-based Varied Dimensionality setting (HRDUVD) method is provided for determining the random subspace dimensionality, and the ERS is formed based on the HRDUVD method. In more detail, a simple discrete uniform distribution in a specific range is employed to set with a given probability the number of randomly selected features for each tree in random subspace decision forests. The HRDUVD method removes the hesitation which appropriate ds value one should preset for different datasets, while also achieving adequate classification performance along with a relatively short running time. Therefore, setting ds using the discrete uniform distribution is a highly useful strategy for the proposed ERS.

Slowly rotating and accelerating α′ -corrected black holes in four and higher dimensions

We consider the low-energy effective action of string theory at order $\alpha '$, including $R^2$-corrections to the Einstein-Hilbert gravitational action and non-trivial dilaton coupling. By means of a convenient field redefinition, we manage to express the theory in a frame that enables us to solve its field equations analytically and perturbatively in $\alpha ' $ for a static spherically symmetric ansatz in an arbitrary number of dimensions. The set of solutions we obtain is compatible with asymptotically flat geometries exhibiting a regular event horizon at which the dilaton is well-behaved. For the 4-dimensional case, we also derive the stationary black hole configuration at first order in $\alpha '$ and in the slowly rotating approximation. This yields string theory modifications to the Kerr geometry, including terms of the form $a$, $a^2$, $\alpha '$ and $a\alpha '$. In addition, we obtain the first $\alpha'$ correction to the C-metrics, which accommodates accelerating black holes. We work in the string frame and discuss the connection to the Einstein frame, for which rotating black holes have already been obtained in the literature.

Mass-, Energy-, and Momentum-Preserving Spectral Scheme for Klein-Gordon-Schrödinger System on Infinite Domains

For the nonlinear conservative system, how to design an efficient scheme to preserve as manyinvariants as possible is a challenging task. The aim of this paper is to construct the finite difference/spectral method for the Klein–Gordon–Schrödinger (KGS) system on infinite domains ( , and 3) to conserve three kinds of the most important invariants, namely, the mass, the energy, and the momentum. Regarding the mass- and momentum-conservation laws as globally physical constraints, we elaborately combine the exponential scalar auxiliary variable (ESAV) approach and Lagrange multiplier approach to construct the ESAV/Lagrange multiplier reformulation of the KGS system to preserve its original energy-conservation law. When solving the ESAV/Lagrange multiplier reformulation, we employ the Hermite–Galerkin spectral method for the spatial approximation and apply the Crank–Nicolson scheme for the temporal discretization. At each time level, we only need to solve linearly algebraic systems with constant coefficients and a set of quadratic algebraic equations which can be efficiently solved by Newton iteration. We establish the conservation properties of the proposed method at the fullydiscrete level, which indicates that our scheme can preserve conserved quantities including mass, energy, and momentums of the KGS system. Numerical experiments are carried out to demonstrate the accuracy and conservation properties of the proposed scheme. As the applications of our scheme, we simulate the nonlinear interactions of vector solitons for KGS system in 2 dimensional/3 dimensional cases.

Relation of stability and bifurcation properties between continuous and ultradiscrete dynamical systems via discretization with positivity: One dimensional cases

The stability and bifurcation properties of one-dimensional discrete dynamical systems with positivity, which are derived from continuous ones by tropical discretization, are studied. The discretized time interval is introduced as a bifurcation parameter in the discrete dynamical systems, and the emergence condition of an additional bifurcation, flip bifurcation, is identified. The correspondence between the discrete dynamical systems with positivity and the ultradiscrete ones derived from them is discussed. It is found that the derived ultradiscrete max-plus dynamical systems can retain the bifurcations of the original continuous ones via tropical discretization and ultradiscretization.

Second Order Two-Species Systems with Nonlocal Interactions: Existence and Large Damping Limits

We study the mathematical theory of second order systems with two species, arising in the dynamics of interacting particles subject to linear damping, to nonlocal forces and to external ones, and resulting into a nonlocal version of the compressible Euler system with linear damping. Our results are limited to the 1 space dimensional case but allow for initial data taken in a Wasserstein space of probability measures. We first consider the case of smooth nonlocal interaction potentials, not subject to any symmetry condition, and prove existence and uniqueness. The concept of solutions relies on a stickiness condition in case of collisions, in the spirit of previous works in the literature. The result uses concepts from classical Hilbert space theory of gradient flows (cf. Brezis, Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, 1973) and a trick used in Brenier et al. (J. Math. Pures Appl. 99(5):577–617, 2013). We then consider a large-time and large-damping scaled version of our system and prove convergence to solutions to the corresponding first order system. Finally, we consider the case of Newtonian potentials - subject to symmetry of the cross-interaction potentials - and external convex potentials. After showing existence in the sticky particles framework in the spirit of Brenier et al. (J. Math. Pures Appl. 99(5):577–617, 2013), we prove convergence for large times towards Dirac delta solutions for the two densities. All the results share a common technical framework in that solutions are considered in a Lagrangian framework, which allows to estimate the behavior of solutions via L^{2} estimates of the pseudo-inverse variables corresponding to the two densities. In particular, due to this technique, the large-damping result holds under a rather weak condition on the initial data, which does not require well-prepared initial velocities. We complement the results with numerical simulations.

Square Root Convexity of Fisher Information along Heat Flow in Dimension Two

Recently, Ledoux, Nair, and Wang proved that the Fisher information along the heat flow is log-convex in dimension one, that is d2dt2log(I(Xt))≥0 for n=1, where Xt is a random variable with density function satisfying the heat equation. In this paper, we consider the high dimensional case and prove that the Fisher information is square root convex in dimension two, that is d2dt2IX≥0 for n=2. The proof is based on the semidefinite programming approach.

Error estimate of a consistent splitting GSAV scheme for the Navier-Stokes equations

We carry out a rigorous error analysis of the first-order semi-discrete (in time) consistent splitting scheme coupled with a generalized scalar auxiliary variable (GSAV) approach for the Navier-Stokes equations with no-slip boundary conditions. The scheme is linear, unconditionally stable, and only requires solving a sequence of Poisson type equations at each time step. By using the build-in unconditional stability of the GSAV approach, we derive optimal global (resp. local) in time error estimates in the two (resp. three) dimensional case for the velocity and pressure approximations.

An Intrinsically Hydrodynamic Approach to Multidimensional QHD Systems

In this paper we consider the multi-dimensional Quantum Hydrodynamics (QHD) system, by adopting an intrinsically hydrodynamic approach. The present paper continues the analysis initiated in Antonelli et al. (Commun Math Phys 383:2113–2161, 2021) where the one dimensional case was investigated. Here we extend the analysis to the multi-dimensional problem, in particular by considering two physically relevant classes of solutions. First of all we consider two-dimensional initial data endowed with point vortices; by assuming the continuity of the mass density and a quantization rule for the vorticity, we are able to study the Cauchy problem and provide global finite energy weak solutions. The same result can be obtained also by considering spherically symmetric initial data in the multi-dimensional setting. For rough solutions with finite energy, we are able to provide suitable dispersive estimates, which also apply to a more general class of Euler–Korteweg equations. Moreover we are also able to show the sequential stability of weak solutions with positive density. Analogously to the one dimensional case, this is achieved through the a priori bounds given by a new functional, first introduced in Antonelli et al. (2021).

Practical computational advantage from the quantum switch on a generalized family of promise problems

The quantum switch is a quantum computational primitive that provides computational advantage by applying operations in a superposition of orders. In particular, it can reduce the number of gate queries required for solving promise problems where the goal is to discriminate between a set of properties of a given set of unitary gates. In this work, we use Complex Hadamard matrices to introduce more general promise problems, which reduce to the known Fourier and Hadamard promise problems as limiting cases. Our generalization loosens the restrictions on the size of the matrices, number of gates and dimension of the quantum systems, providing more parameters to explore. In addition, it leads to the conclusion that a continuous variable system is necessary to implement the most general promise problem. In the finite dimensional case, the family of matrices is restricted to the so-called Butson-Hadamard type, and the complexity of the matrix enters as a constraint. We introduce the “query per gate'' parameter and use it to prove that the quantum switch provides computational advantage for both the continuous and discrete cases. Our results should inspire implementations of promise problems using the quantum switch where parameters and therefore experimental setups can be chosen much more freely.

Optimal feedback controls of stochastic linear quadratic control problems in infinite dimensions with random coefficients

It is a longstanding unsolved problem to characterize the optimal feedback controls for general linear quadratic optimal control problem of stochastic evolution equation with random coefficients. A solution to this problem is given in [22] under some assumptions which can be verified for interesting concrete models, such as controlled stochastic wave equations, controlled stochastic Schrödinger equations, etc. More precisely, the authors establish the equivalence between the existence of an optimal feedback operator and the solvability of the corresponding operator-valued, backward stochastic Riccati equations. However, their result cannot cover some important stochastic partial differential equations, such as stochastic heat equations, stochastic stokes equations, etc. A key contribution of the current work is to relax the C0-group assumption of unbounded linear operator A in [22] and using the contraction semigroup assumption instead. Therefore, our result can be well applicable in the linear quadratic problem of stochastic parabolic equations. To this end, we introduce a suitable notion to the aforementioned Riccati equation, and some delicate techniques which are even new in the finite dimensional case.

Long time behaviour of the discrete volume preserving mean curvature flow in the flat torus

We show that the discrete approximate volume preserving mean curvature flow in the flat torus $$\mathbb {T}^N$$ starting near a strictly stable critical set E of the perimeter converges in the long time to a translate of E exponentially fast. As an intermediate result we establish a new quantitative estimate of Alexandrov type for periodic strictly stable constant mean curvature hypersurfaces. Finally, in the two dimensional case a complete characterization of the long time behaviour of the discrete flow with arbitrary initial sets of finite perimeter is provided.

Nonlinear wave interactions in geochemical modeling

This paper is concerned with the study of the main wave interactions in a system of conservation laws in geochemical modeling. We study the modeling of the chemical complexes on the rock surface. The presence of stable surface complexes affects the relative permeability. We add terms representing surface complexes to the accumulation function in the model presented in [14]. This addition allows to take into account the interaction of ions with the rock surface in the modeling of the oil recovery by the injection of carbonated water. Compatibility hypotheses with the modeling are made on the coefficients of the system to obtain meaningful solutions. We developed a Riemann solver taking into account the complexity of the interactions and bifurcations of nonlinear waves. Such bifurcations occur at the inflection and resonance surfaces. We present the solution of a generalized eigenvalue problem in a (n+1)-dimensional case, which allows the construction of rarefaction curves. A method to find the discontinuous solutions is also presented. We find the solution path for some examples.

Gradient blowup behavior for a viscous Hamilton-Jacobi equation with degenerate gradient nonlinearity

The paper is concerned with gradient blowup behavior for a semilinear parabolic equationut−Δu=δm(x)|∇u|p+ΛinΩ×(0,T) with the zero Dirichlet condition. Two results with p>m+2 and m⩾0 are established. One of which is that any gradient blowup solution follows a global ODE type behavior, with domination of normal derivatives over the tangential derivatives. The other is about time-increasing solutions. Zhang and Hu (2010) [50] obtained the precise gradient blowup rate in one-dimensional case, but the higher dimensional case was left as an open problem. Here we solve this problem by establishing the gradient blowup rate, for any small γ>0,C(T−t)−m+1p−m−2⩽‖∇u‖∞⩽Cγ(T−t)−m+1p−m−2−γ for suitable ranges of p and m, which extends the result of (Attouchi and Souplet (2020) [3]) to the case m≠0. As an important by-product which is of independent interests itself, the gradient estimate near boundary for the corresponding elliptic equation is derived under weaker assumptions on the inhomogeneous term.

On a preserver problem concerning Kubo-Ando means of positive operators

We study bijective transformations ϕ of the set of positive operators on a Hilbert space such that given a Kubo-Ando mean σ with certain properties, ‖AσB‖=‖ϕ(A)σϕ(B)‖ holds for every pair A,B of positive operators. It is shown that whenever the Hilbert space is finite dimensional, then ϕ is necessarily conjugation by a fixed unitary or antiunitary operator. Coupled with earlier results, this completely resolves the above preserver problem in the finite dimensional case. Most ideas of the proof carry over to the infinite dimensional setting, thus some partial results are obtained in that case as well.

A Brief Essay on "Newtonian Gravity", Escape Velocity and "Dark Matter" for Galaxies and Star Clusters.

What is attempted in this article is to be introduced the "Newtonian gravity" as a non-relativistic theory which can for instance be applied in mass formations such as the stars, the globular and open star clusters and the galaxies. In this paper the attention is focused on clusters and galaxies that are more easy to handle since their "dynamics" depends solely on matter [no gas, fermion and radiation pressure]. Initially the field eqs. for a vector gravity field ( ) are introduced rather than tensor field . Then the gradient of a scalar field is found which is related to the vector field ( ) in the 3-D (dimensional) case. Since the case of a galaxy is actually 2-D with its thickness being ( )th of its Diameter, the 2-D eqs. are solved toward the solution of the mass distribution with radial symmetry. The next step which is done is to show compatibility (but only an approximate one) of the field eqs. for NN "objects" (NN»1) consisting a Galaxy. Subsequently the Effective Potential is introduced by using an "object" not as one of the NN-Newton ones, but as an object moving in the effective potential. With this “equipment” (one-object-potential) one can easily define the escape velocity of an object found in position r far from the center of a galaxy. A discussion of “dark matter” is done. Finally the radial and angular eqs. of motion are attempted to be solved and an integral of time with respect to radial distance is given which unfortunately cannot be inverted. However when the motion is bounded then a condition of periodicity can be found.

Higher dimensional Shimura varieties in the Prym loci of ramified double covers

AbstractIn this paper, we construct Shimura subvarieties of dimension bigger than one of the moduli space of δ‐polarized abelian varieties of dimension p, which are generically contained in the Prym loci of (ramified) double covers. The idea is to adapt the techniques already used to construct Shimura curves in the Prym loci to the higher dimensional case, namely, to use families of Galois covers of . The case of abelian covers is treated in detail, since in this case, it is possible to make explicit computations that allow to verify a sufficient condition for such a family to yield a Shimura subvariety of .