Asymptotic Form Research Articles

A general form for precise asymptotics for complete convergence under sublinear expectation

<abstract><p>Let $ \{X_n, n\geq 1\} $ be a sequence of independent and identically distributed random variables in a sublinear expectation $ (\Omega, \mathcal H, {\mathbb {\widehat{E}}}) $ with a capacity $ {\mathbb V} $ under $ {\mathbb {\widehat{E}}} $. In this paper, under some suitable conditions, I show that a general form of precise asymptotics for complete convergence holds under sublinear expectation. It can describe the relations among the boundary function, weighted function, convergence rate and limit value in studies of complete convergence. The results extend some precise asymptotics for complete convergence theorems from the traditional probability space to the sublinear expectation space. The results also generalize the known results obtained by Xu and Cheng <sup>[<xref ref-type="bibr" rid="b34">34</xref>]</sup>.</p></abstract>

Short time asymptotics for American maximum options with a dividend-paying asset

<abstract><p>We investigate the asymptotic behaviors of American maximum options with dividend-paying assets near maturity. Using the exercise conditions of American options, we obtain the asymptotic forms of the two boundaries with respect to time-to-maturity. Furthermore, we derive the matched asymptotic expansion for the rescaled value function of American maximum option. The all results are provided with detailed computations and derivations. Numerical examples show that the asymptotic value function and exercise boundaries can provide an efficient alternative for the true ones, respectively.</p></abstract>

Trees with power-like height dependent weight

We consider planar rooted random trees whose distribution is even for fixed height h and size N and whose height dependence is given by a power function hα. Defining the total weight for such trees of fixed size to be ZN, a detailed analysis of the analyticity properties of the corresponding generating function is provided. Based on this, we determine the asymptotic form of ZN and show that the local limit at large size is identical to the Uniform Infinite Planar Tree, independent of the exponent α of the height distribution function.

A new quantum mechanical pseudo material: Application in cryptography

Recently relativistic Dirac material such as graphene and the topological material as topological insulators have played prominent role in providing some of the materials which incorporate more mathematical and relativistic physical nature in general. Here we suggest a material which is quantum mechanical in nature but is a pseudo material. An interacting quantum many-particle system in one dimension representing the prime numbers on the x-axis through the asymptotic form of the prime counting function is formulated. The effective quantum and classical mechanics of a single particle of the system are studied by deriving the potential function that a single particle experiences in this system. We express our classical Hamiltonian in terms of the angle coordinate to illustrate certain interesting mathematical properties related to the prime numbers. Since it is a mathematical material having prime numbers as the atoms interacting with each other like ordinary mater, this material can have real and important application in cryptography. The elliptic curves that appear in the study of this system are related to primality test and are heavily used in communication cryptography. This analysis opens a new way of looking at materials which shows that simulating a material through a potential is a productive one.

The possibilities to estimate hydroloosening efficiency by asymptotic form of barodynamic curve

The possibilities to estimate hydroloosening efficiency by asymptotic form of barodynamic curve

Generation of waves due to bottom disturbances in a viscous fluid

The generation of two-dimensional surface waves due to various types of bottom disturbances such as underwater explosions, earthquakes, or volcanic eruptions is investigated here. Assuming linear theory the present problem is formulated as an initial value problem for the wave potential function ϕ and Stokes stream function ψ. Viscosity is considered. The physical model is illustrated by a sketch. Fourier and Laplace transform techniques are applied in the mathematical analysis to obtain the form of the free surface in terms of a multiple infinite integral. This integral is evaluated asymptotically by the method of steepest descent. The asymptotic form of the free surface is depicted graphically in some figures for different values of the viscosity and different types of ground disturbances. Appropriate conclusions are made.

Specific neutral and charged black holes inf(R)gravitational theory

With the successes of $f(R)$ theory as a neutral modification of Einstein's general relativity (GR), we continue our study in this field and attempt to find general %natural { neutral} and charged black hole (BH) solutions. In the previous papers \cite{Nashed:2020mnp,Nashed:2020tbp}, we applied the field equation of the $f(R)$ gravity to a spherically symmetric space-time $ds^2=-U(r)dt^2+\frac{dr^2}{V(r)}+r^2 \left( d\theta^2+\sin^2\theta d\phi^2 \right)$ with unequal metric potentials $U(r)$ and $V(r)$ and with/without electric charge. {Then we have obtained equations which include all the possible static solutions with spherical symmetry.} To ensure the closed form of system of the resulting differential equations in order to obtain specific solutions, we assumed the derivative of the $f(R)$ with respect to the scalar curvature $R$ to have a form %$F_1(r)=\frac{df(R(r))}{dR(r)} \propto %{\color{red} 1 +} %\frac{c}{r^n}$ but in case $n>2$, the resulting black hole solutions with/without charge do not %generate asymptotically GR BH solutions in the limit $c\rightarrow 0$ which means that the only case that can generate GR BHs is $n=2$. %In this paper, we assume another form, i.e., $F_1(r) {=\frac{df(R(r))}{dR(r)} } = 1-\frac{F_0-\left(n-3\right)}{r^n}$ with a constant $F_0$ and show that we can generate asymptotically GR BH solutions for $n>2$ but we show that the $n=2$ case is not allowed. This form of $F_1(r)$ could be the most acceptable physical form that we can generate from it physical metric potentials that can have a well-known asymptotic form and we obtain the metric of the Einstein general relativity in the limit of $F_0\to n-3$. We show that the form of the electric charge depends on $n$ and that $n\neq 2$.}

Black hole perturbations in modified gravity

We study the linear perturbations about nonrotating black holes in the context of degenerate higher-order scalar-tensor (DHOST) theories, using a systematic approach that extracts the asymptotic behaviour of perturbations (at spatial infinity and near the horizon) directly from the first-order radial differential system governing these perturbations. For axial (odd-parity) modes, this provides an alternative to the traditional approach based on a second-order Schr\"odinger-like equation with an effective potential, which we also discuss for completeness. For polar (even-parity) modes, which contain an additional degree of freedom in DHOST theories, and are thus more complex, we use a direct treatment of the four-dimensional first-order differential system (without resorting to a second order reformulation). We illustrate our study with two specific types of black hole solutions: 'stealth' Schwarzschild black holes, with a non trivial scalar hair, as well as a class of non-stealth black holes whose metric is distinct from Schwarzschild. The knowledge of the asymptotic behaviours of the perturbations enables us to compute numerically quasi-normal modes, as we show explicitly for the non-stealth solutions. Finally, the asymptotic form of the modes also signals some pathologies in the stealth and non-stealth solutions considered here.

Lifetime optimization of dense wireless sensor networks using continuous ring-sector model

Wireless sensor networks (WSNs) are becoming increasingly utilized in applications that require remote collection of data on environmental conditions. In particular dense WSNs are emerging as an important sensing platforms for the Internet of Things (IoT). WSNs are able to generate huge volumes of raw data, which require network structuring and efficient collaboration between nodes to ensure efficient transmission. In order to reduce the amount of data carried in the network, data aggregation is used in WSNs to define a policy of data fusion and compression. In this paper, we investigate a model for data aggregation in a dense WSN with a single sink. The model divides a circular coverage region centered at the sink into patches which are intersections of sectors of concentric rings, and data in each patch is aggregated at a single node before transmission. Nodes only communicate with other nodes in the same sector. Based on these assumptions, we formulate a linear programming problem to maximize system lifetime by minimizing the maximum proportionate energy consumption over all nodes. Under a wide variety of conditions, the optimal solution employs two transmissions mechanisms: direct transmission, in which nodes send information directly to the sink; and stepwise transmission, in which nodes transmit information to adjacent nodes. An exact formula is given for the proportionate energy consumption rate of the network. Asymptotic forms of this exact solution are also derived, and are verified to agree with the linear programming solution. We investigate three strategies for improving system lifetime: nonuniform energy and information density; iterated compression; and modifications of rings. We conclude that iterated compression brings the greatest increase in system lifetime, but at the cost of possible information loss.

Polaron Models with Regular Interactions at Strong Coupling

We study a class of polaron-type Hamiltonians with sufficiently regular form factor in the interaction term. We investigate the strong-coupling limit of the model, and prove suitable bounds on the ground state energy as a function of the total momentum of the system. These bounds agree with the semiclassical approximation to leading order. The latter corresponds here to the situation when the particle undergoes harmonic motion in a potential well whose frequency is determined by the corresponding Pekar functional. We show that for all such models the effective mass diverges in the strong coupling limit, in all spatial dimensions. Moreover, for the case when the phonon dispersion relation grows at least linearly with momentum, the bounds result in an asymptotic formula for the effective mass quotient, a quantity generalizing the usual notion of the effective mass. This asymptotic form agrees with the semiclassical Landau–Pekar formula and can be regarded as the first rigorous confirmation, in a slightly weaker sense than usually considered, of the validity of the semiclassical formula for the effective mass.

Error Exponents for Asynchronous Multiple Access Channels, Controlled Asynchronism May Outperform Synchronism

Exponential error bounds achievable by universal coding and decoding are derived for frame-asynchronous discrete memoryless multiple access channels with two senders, via the method of subtypes, a refinement of the method of types. An empirical entropy decoder is employed. A key tool is an improved packing lemma, that overcomes the technical difficulty caused by codeword repetitions via an induction based new argument. The asymptotic form of the bounds admits numerical evaluation. This demonstrates that error exponents achievable by synchronous transmission can be superseded via controlled asynchronism, i.e. a deliberate shift of the codewords.

Analysis of ULA Spacing Threshold and Capacity Reduction in Distributed Satellite MIMO Scenarios

In this letter, we consider a distributed satellite MIMO scenario that includes two GEO satellites and a uniform linear array (ULA). The effect of geometrical arrangement of ULA on channel capacity is investigated, and it is proved that the optimal capacity can always be obtained as long as its antenna spacing is larger than a threshold. The closed-form expressions of this threshold and its asymptotic form are derived. Considering the threshold may be too large to satisfy when a low frequency is used or two satellites are close, the exact capacity reduction when the threshold cannot be satisfied is also analyzed, which can help to reach a compromise between capacity and antenna spacing.

Full-Dimensional Potential Energy Surface for Ro-vibrationally Inelastic Scattering between H2 Molecules.

We report a new full-dimensional potential energy surface (PES) for the inelastic scattering between ro-vibrationally excited H2 molecules. The new PES is based on 39,462 multi-reference configuration interaction points in dynamically relevant regions. The analytic form of the PES consists of a short-range term fit with the permutational invariant polynomial-neural network method and a long-range term with a physically correct asymptotic functional form accounting for both electrostatic and dispersion terms, which are connected smoothly with a switching function. The PES compares favorably with existing accurate PESs near the H2 equilibrium geometries but covers a much larger configuration space for H2 with up to 10 vibrational quanta. Full-dimensional quantum scattering calculations on the new PES reproduce the recent Stark-induced adiabatic Raman passage results for the HD(v = 1) + H2 scattering near 1 K, validating its accuracy. These calculations also revealed significant differences with existing PESs in describing scattering of vibrationally excited molecules, underscoring the ability of the new PES in handling such dynamics.

Circulant L-ensembles in the thermodynamic limit

L-ensembles are a class of determinantal point processes which can be viewed as a statistical mechanical systems in the grand canonical ensemble. Circulant L-ensembles are the subclass which are locally translationally invariant and furthermore subject to periodic boundary conditions. Existing theory can very simply be specialised to this setting, allowing for the derivation of formulas for the system pressure, and the correlation kernel, in the thermodynamic limit. For a one-dimensional domain, this is possible when the circulant matrix is both real symmetric, or complex Hermitian. The special case of the former having a Gaussian functional form for the entries is shown to correspond to free fermions at finite temperature, and be generalisable to higher dimensions. A special case of the latter is shown to be the statistical mechanical model introduced by Gaudin to interpolate between Poisson and unitary symmetry statistics in random matrix theory. It is shown in all cases that the compressibility sum rule for the two-point correlation is obeyed, and the small and large distance asymptotics of the latter are considered. Also, a conjecture relating the asymptotic form of the hole probability to the pressure is verified.

Semi-infinite moving crack between two orthotropic strips

The problem of a semi-infinite moving crack at the interface of two bonded dissimilar strips of finite width has been solved. The crack is subjected to two constant normal displacements applied on the boundaries. The problem has been converted to one that is suitable for applying the Wiener–Hopf technique. Fourier transform technique has been used to reduce the mixed boundary value problem to the standard form of the Wiener–Hopf equation which has been solved in asymptotic form to obtain the analytical expressions of stress intensity factor and crack opening displacement. Finally the numerical values of stress intensity factor and crack opening displacement have been plotted graphically against various parameters to show the effects of these parameters on stress intensity factor and crack opening displacement.

Instantons on multi-Taub-NUT spaces I: Asymptotic form and index theorem

We study finite action anti-self-dual Yang-Mills connections on the multi-Taub-NUT space. We establish the curvature and the harmonic spinors decay rates and compute the index of the associated Dirac operator. This is the first in a series of papers proving the completeness of the bow construction of instantons on multi-Taub-NUT spaces and exploring it in detail.

Real-space modeling for complex structures based on small-angle X-ray scattering.

A three-dimensional real-space model has been created for hierarchical materials by matching observed and simulated small-angle X-ray scattering patterns. The simulation is performed by arranging the positions of small primary particles and constructing an aggregate structure in a finite-sized cell. In order to avoid the effect of the finite size of the cell, the cell size is extended to infinity by introducing an asymptotic form of the long-range correlations among the primary particles. As a result, simulations for small-angle X-ray scattering patterns can be performed correctly in the low-wavenumber regime (<0.1 nm-1), allowing the model to handle hundred-nanometre-scale structures composed of primary particles of a few nanometres in size. An aerogel structure was determined using this model, resulting in an excellent match with the experimental scattering pattern. The resultant three-dimensional model can generate cross-sectional images similar to those obtained by transmission electron microscopy, and the calculated pore-size distribution is in accord with that derived from the gas adsorption method.

Mimetic Euler-Heisenberg theory, charged solutions, and multihorizon black holes

We construct several new classes of black hole (BH) solutions in the context of the mimetic Euler-Heisenberg theory. We separately derive three differently charged BH solutions and their relevant mimetic forms. We show that the asymptotic form of all BH solutions behaves like a flat spacetime. These BHs, either with/without cosmological constant, have the non constant Ricci scalar, due to the contribution of the Euler-Heisenberg parameter, which means that they are not solution to standard or mimetic $f(R)$ gravitational theory without the Euler-Heisenberg non-linear electrodynamics and at the same time they are not equivalent to the solutions of the Einstein gravity with a massless scalar field. Moreover, we display that the effect of the Euler-Heisenberg theory makes the singularity of BH solutions stronger compared with that of BH solutions in general relativity. Furthermore, we show that the null and strong energy conditions of those BH solutions are violated, which is a general trend of mimetic gravitational theory. The thermodynamics of the BH solutions are satisfactory although there appears a negative Hawking temperature under some conditions. Additionally, these BHs obey the first law of thermodynamics. We also study the stability, using the geodesic deviation, and derive the stability condition analytically and graphically. Finally, for the first time and under some conditions, we derived multi-horizon BH solutions in the context of the mimetic Euler-Heisenberg theory and study their related physics.

Distribution Amplitudes of K^{*} and ϕ at the Physical Pion Mass from Lattice QCD.

We present the first lattice QCD calculation of the distribution amplitudes of longitudinally and transversely polarized vector mesons K^{*} and ϕ using large momentum effective theory. We use the clover fermion action on three ensembles with 2+1+1 flavors of highly improved staggered quarks action, generated by the MIMD Lattice Computation Collaboration, at physical pion mass and {0.06,0.09,0.12} fm lattice spacings and choose three different hadron momenta P_{z}={1.29,1.72,2.15} GeV. The resulting lattice matrix elements are nonperturbatively renormalized in a recently proposed hybrid scheme. An extrapolation to the continuum and infinite momentum limit is carried out. We find that, while the longitudinal distribution amplitudes tend to be close to the asymptotic form, the transverse ones deviate rather significantly from the asymptotic form. Our final results provide crucial abinitio theory inputs for analyzing pertinent exclusive processes.

Anisotropic compact stars in higher-order curvature theory

In this paper we shall consider spherically symmetric spacetime solutions describing the interior of stellar compact objects, in the context of higher-order curvature theory of the {{mathrm {f(R)}}} type. We shall derive the non-vacuum field equations of the higher-order curvature theory, without assuming any specific form of the {{mathrm {f(R)}}} theory, specifying the analysis for a spherically symmetric spacetime with two unknown functions. We obtain a system of highly non-linear differential equations, which consists of four differential equations with six unknown functions. To solve such a system, we assume a specific form of metric potentials, using the Krori–Barua ansatz. We successfully solve the system of differential equations, and we derive all the components of the energy–momentum tensor. Moreover, we derive the non-trivial general form of {{mathrm {f(R)}}} that may generate such solutions and calculate the dynamic Ricci scalar of the anisotropic star. Accordingly, we calculate the asymptotic form of the function {mathrm {f(R)}}, which is a polynomial function. We match the derived interior solution with the exterior one, which was derived in [1], with the latter also resulting to a non-trivial form of the Ricci scalar. Notably but rather expected, the exterior solution differs from the Schwarzschild one in the context of general relativity. The matching procedure will eventually relate two constants with the mass and radius of the compact stellar object. We list the necessary conditions that any compact anisotropic star must satisfy and explain in detail that our model bypasses all of these conditions for a special compact star textit{Her X--1}, which has an estimated mass and radius (mass = 0.85 pm 0.15M_{circledcirc } and radius = 8.1 pm 0.41~text {km}). Moreover, we study the stability of this model by using the Tolman–Oppenheimer–Volkoff equation and adiabatic index, and we show that the considered model is different and more stable compared to the corresponding models in the context of general relativity.