Symmetric Problem Research Articles

On the atypical solutions of the symmetric binary perceptron

We study the random binary symmetric perceptron problem, focusing on the behavior of rare high-margin solutions. While most solutions are isolated, we demonstrate that these rare solutions are part of clusters of extensive entropy, heuristically corresponding to non-trivial fixed points of an approximate message-passing algorithm. We enumerate these clusters via a local entropy, defined as a Franz–Parisi potential, which we rigorously evaluate using the first and second moment methods in the limit of a small constraint density α (corresponding to vanishing margin κ ) under a certain assumption on the concentration of the entropy. This examination unveils several intriguing phenomena: (i) we demonstrate that these clusters have an entropic barrier in the sense that the entropy as a function of the distance from the reference high-margin solution is non-monotone when κ⩽1.429−α/log⁡α , while it is monotone otherwise, and that they have an energetic barrier in the sense that there are no solutions at an intermediate distance from the reference solution when κ⩽1.239−α/log⁡α . The critical scaling of the margin κ in −α/log⁡α corresponds to the one obtained from the earlier work of Gamarnik et al (2022 (arXiv:2203.15667)) for the overlap-gap property, a phenomenon known to present a barrier to certain efficient algorithms. (ii) We establish using the replica method that the complexity (the logarithm of the number of clusters of such solutions) versus entropy (the logarithm of the number of solutions in the clusters) curves are partly non-concave and correspond to very large values of the Parisi parameter, with the equilibrium being reached when the Parisi parameter diverges.

A remark on a conjecture on the symmetric Gaussian problem

AbstractIn this paper, we study the functional given by the integral of the mean curvature of a convex set with Gaussian weight with Gaussian volume constraint. It was conjectured that the ball centred at the origin is the only minimizer of such a functional for certain values of the mass. We prove that this is the case in dimension 2 while in higher dimension the situation is different. In fact, for small values of mass, the ball centred at the origin is a local minimizer, while for larger values the ball is a maximizer among convex sets with a uniform bound on the curvature.

Novel dimensionless parameters in 2D wedge drop

ABSTRACT The aim of this research is to create new dimensionless relations in rigid wedge impact problems, which provides the possibility of calculating wedge kinematic parameters with acceptable accuracy. To accomplish this, the rigid wedge impact problem is simulated and validated using the Dynamic Fluid Body Interaction (DFBI) model in STAR-CCM+. In this study, new non-dimensional relations for the kinematical parameters of the wedge are derived utilising the Buckingham π theorem and data-driven analysis. It is observed that the use of appropriate dimensionless relations in any type of symmetric water impact problems involving a wedge shaped object will result in a unique diagram of each wedge kinematical parameter. This ultimately leads to the creation of two separate formulas to determine the maximum wedge acceleration value and its corresponding time. These formulas play a vital role in the design of high-speed crafts, and planing vessels.

Construction of 4 x 4 symmetric stochastic matrices with given spectra

Abstract The symmetric stochastic inverse eigenvalue problem (SSIEP) asks which lists of real numbers occur as the spectra of symmetric stochastic matrices. When the cardinality of a list is 4, Kaddoura and Mourad provided a sufficient condition for SSIEP by a mapping and convexity technique. They also conjectured that the sufficient condition is the necessary condition. This study presents the same sufficient condition for SSIEP, but we do it in terms of the list elements. In this way, we provide a different but more straightforward construction of symmetric stochastic matrices for SSIEP compared to those of Kaddoura and Mourad.

Variational approach to a symmetric boundary value problem generated by a system of equations and separated boundary conditions

This work provides some information on the eigenvalues and eigenfunctions of a problem which is constructed by a system of equations and symmetric boundary conditions that includes the ordinary second‐order Sturm–Liouville boundary value problem. In particular, we show that the problem has an infinite number of discrete eigenvalues with a greatest lower bound and the corresponding eigenfunctions are complete in mean and energy. We introduce the results using the variational approach that enables us to consider only continuous pair functions instead of absolutely continuous pair functions.

A characterization of sets realizable by compensation in the SNIEP

The symmetric nonnegative inverse eigenvalue problem (SNIEP) is the problem of characterizing all possible spectra of entry-wise nonnegative symmetric matrices of given dimension. A list of real numbers is said to be symmetrically realizable if it is the spectrum of some nonnegative symmetric matrix. One of the most general sufficient conditions for realizability is the so-called C-realizability, which amounts to some kind of compensation between the positive and negative entries of the list of real numbers whose realizability one is trying to decide. A combinatorial characterization of C-realizable lists with zero trace was given in [11]. In this paper we make use of a recursive method for constructing simmetrically realizable lists due to Ellard and Šmigoc [3] to extend this combinatorial characterization of C-realizability to general lists with nonnegative trace. One consequence of this characterization is that the set of nonnegative C-realizable lists is a union of polyhedral cones whose faces are described by equations involving only linear combinations with coefficients 1 and −1 of the entries in the list. Another remarkable consequence is the monotonicity of C-realizability, i.e., the operation of increasing any positive entry of a C-realizable list preserves C-realizability.

A New Method for the Analysis of Broadband Oscillation Mode of New Energy Stations

The accurate identification of the broadband oscillation mode is the premise of solving the resonance risk of new energy stations. Reviewing the traditional Prony algorithm, the problems of the high model order and poor noise immunity in broadband oscillation mode are identified. The accuracy and running time of the Variational mode decomposition (VMD) is a symmetric trade-off problem. An improved strategy based on VMD is proposed. Firstly, the optimal value of the number of modes and penalty factors obtained by a particle swarm optimization algorithm is input into VMD to decompose the signal into multiple modes. Then, combined with the energy threshold method, the denoising and signal reconstruction of each mode component after decomposition are carried out. Finally, the Prony algorithm is used to identify the oscillation mode of the original signal and the reconstructed signal, respectively. The Signal-to-noise ratio (SNR) and model order are compared and analyzed. Through the analysis of the example and simulation data, it is shown that the proposed method effectively solves the problem of poor engineering adaptability of the traditional Prony algorithm. It also can accurately obtain the time-domain characteristics of broadband oscillation, which has a promising future in the engineering application.

Preconditioned discontinuous Galerkin method and convection‐diffusion‐reaction problems with guaranteed bounds to resulting spectra

AbstractThis paper focuses on the design, analysis and implementation of a new preconditioning concept for linear second order partial differential equations, including the convection‐diffusion‐reaction problems discretized by Galerkin or discontinuous Galerkin methods. We expand on the approach introduced by Gergelits et al. and adapt it to the more general settings, assuming that both the original and preconditioning matrices are composed of sparse matrices of very low ranks, representing local contributions to the global matrices. When applied to a symmetric problem, the method provides bounds to all individual eigenvalues of the preconditioned matrix. We show that this preconditioning strategy works not only for Galerkin discretization, but also for the discontinuous Galerkin discretization, where local contributions are associated with individual edges of the triangulation. In the case of nonsymmetric problems, the method yields guaranteed bounds to real and imaginary parts of the resulting eigenvalues. We include some numerical experiments illustrating the method and its implementation, showcasing its effectiveness for the two variants of discretized (convection‐)diffusion‐reaction problems.

Unconditionally Stable Crank-Nicolson Algorithm with Enhanced Absorption for Rotationally Symmetric Multi-Scale Problems in Anisotropic Magnetized Plasma

Unconditionally Stable Crank-Nicolson Algorithm with Enhanced Absorption for Rotationally Symmetric Multi-Scale Problems in Anisotropic Magnetized Plasma

Not even 6 dB: Gaussian quantum illumination in thermal background

In analyses of target detection with Gaussian state transmitters in a thermal background, the thermal occupation is taken to depend on the target reflectivity in a way which simplifies the analysis of the symmetric quantum hypothesis testing problem. However, this assumption precludes comparison of target detection performance between an arbitrary transmitter and a vacuum state transmitter, i.e. ‘detection without illumination’, which is relevant in a bright thermal background because a target can be detected by its optical shadow or some other perturbation of the background. Using a target-agnostic thermal environment leads to the result that the oft-claimed 6 dB possible reduction in the quantum Chernoff exponent for a two-mode squeezed vacuum transmitter over a coherent state transmitter in high-occupation thermal background is an unachievable limiting value, only occurring in a limit in which the target detection problem is ill-posed. Further analyzing quantum illumination in a target-agnostic thermal environment shows that a weak single-mode squeezed transmitter performs worse than ‘no illumination’, which is explained by the noise-increasing property of reflected low-intensity squeezed light.

МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ СИМЕТРИЧНОЇ КРАЙОВОЇ ЗАДАЧІ ДЛЯ ПОСЛАБЛЕНОГО НАСКРІЗНИМ ОТВОРОМ ШАРУ З ПОКРИТИМИ ДІАФРАГМОЮ ТОРЦЯМИ

The paper presents a new mathematical model for the solution of a static symmetric boundary-value problem for a layer weakened by a through hole with ends covered with a diaphragm. A new method based on a system of three singular integral equations has been developed and tested numerically. As a result of a high-precision numerical study, it was found that with an increase in the thickness of the layer, an increase in the relative circumferential stress occurs. With a decrease in one of the radii of the elliptical hole, an increase in the relative circumferential stress is also observed. The paper presents the respective dependency graphs. Keywords: three-dimensional boundary-value problems, singular integral equations, numerical experiment, static stretching–compression, a through hole.

The Solution and Numerical Verification of the Two-Layer Spherical Symmetric Heat Conduction Problem

The Solution and Numerical Verification of the Two-Layer Spherical Symmetric Heat Conduction Problem

Hp-Robust multigrid solver on locally refined meshes for FEM discretizations of symmetric elliptic PDEs

In this work, we formulate and analyze a geometric multigrid method for the iterative solution of the discrete systems arising from the finite element discretization of symmetric second-order linear elliptic diffusion problems. We show that the iterative solver contracts the algebraic error robustly with respect to the polynomial degree p ≥ 1 and the (local) mesh size h. We further prove that the built-in algebraic error estimator which comes with the solver is hp-robustly equivalent to the algebraic error. The application of the solver within the framework of adaptive finite element methods with quasi-optimal computational cost is outlined. Numerical experiments confirm the theoretical findings.

Spherically Symmetric Problem of General Relativity for a Fluid Sphere

Spherically Symmetric Problem of General Relativity for a Fluid Sphere

Bisymmetric non-negative Jacobi matrix realizations

Within the symmetric inverse eigenvalue problem, the case of bisymmetric Jacobi matrices occupies a central place, since for any strictly monotone list of n real numbers there exists a unique bisymmetric Jacobi matrix realizing the list. Apart from their meaning in several issues such as physics, mechanics, statistics, to cite some of them, the families of this kind of matrices whose spectrum is known are used as models for testing the different algorithms to recover the entries of matrices from spectra data. However, the spectrum is known only for a few families of bisymmetric Jacobi matrices and the examples mainly refer to the case when the spectrum is given by a linear or quadratic function of the order and of the row index. In the first part of this paper, we join all known cases by proving a general result about bisymmetric Jacobi realizations of strictly monotone sequences that are quadratic at most. In the second part, we focus on the non-negative bisymmetric realizations, obtaining new necessary conditions for a given list to be realized by a non-negative bisymmetric Jacobi matrix. The main novelty in our techniques is considering the gaps between the eigenvalues instead of focusing on the eigenvalues themselves. In the last part of this paper, we explicitly obtain the bisymmetric realization of any list for order less or equal to 6.

PLANE PROBLEMS ABOUT THE ACTION OF OSCILLATING LOAD ON THE BOUNDARY OF AN ELASTIC ISOTROPIC LAYER IN THE PRESENCE OF SURFACE STRESSES

In this paper, symmetric and antisymmetric plane problems about the action of oscillating load on the boundary of an elastic isotropic nanothin layer are considered. The nanoscale layer thickness is considered by introducing surface stresses in accordance with the Gurtin-Murdoch theory. According to this theory, it is assumed that, in addition to external loads, surface stresses act on the layer boundaries, which are described by Hooke's “surface” law. As a result, the properties of the elastic material of the layer with nanoscale thickness become different from the properties of the material of a regular-sized body, which is typical for nanomechanics problems. A standard technique was used for the solution of formulated problems, including the application of limiting absorption principle, the Fourier transform over infinitely extended coordinate and the theory of residues for finding the inverse Fourier transform. It is shown how it is possible to obtain solutions in the form of series in natural waves, in which the wave numbers are defined as the roots of the corresponding dispersion equations. For a specific example, dispersion relations were studied and graphs of the first dispersion curves were plotted. The behavior of barrier frequencies, changes in wave numbers and zones of existence of backward waves at different nanoscale layer thicknesses are analyzed. The results of the analysis showed that for an ultrathin layer, surface effects have a significant impact on the dispersion relations, and the trends in the dispersion curves can differ significantly for different modes and layer thicknesses.

Quasilinear Simulation of the Development of Weibel Turbulence in Anisotropic Collisionless Plasma

A spectral quasilinear approach to the problem of TEM-Weibel instability in an anisotropic collisionless plasma is developed, which takes into account only the integral nonlinear interaction of modes through the joint variation of the spatially averaged particle velocity distribution induced by these modes. Within this approximation, a closed system of equations is obtained for the one- and two-dimensional evolution of spatial modes (harmonics) of the distribution function of particles and the electromagnetic field under conditions when the plasma anisotropy axis, the wave vector, and the magnetic field of the modes are orthogonal to each other. The numerical solution of this system of equations is compared with the available results of one-dimensional analytical quasilinear theory in the region of its applicability, as well as with the results of two-dimensional simulation by the particle-in-cell method, which also takes into account the direct four-wave interaction of modes. It is established that in the simplest cases of one-dimensional and axially symmetric two-dimensional problems for a bi-Maxwellian plasma, quasilinear phenomena play the leading role at a quite long stage of nonlinear development of turbulence. It is noted that at a later stage of decay of turbulence and in a more general formulation of the problem, in particular, in the presence of an external magnetic field, the direct nonlinear interaction of modes can manifest itself along with quasilinear phenomena. Based on the analysis carried out, the contribution of certain nonlinear effects to the evolution of the spatial spectrum of Weibel turbulence is revealed, and the properties of this turbulence are studied, including the self-similar character and qualitatively different stages of the dynamics of unstable modes.

Mechanical response of concrete pipe rehabilitated by liner under external load: Analytical solution

AbstractUsing pipe lines to rehabilitate concrete drainage pipes is a popular trenchless technique. In this paper, the concrete pipe and liner are equivalent to a symmetrical structural mechanics problem, and the mechanical solution of the composite structure is obtained using the force method. Subsequently, the analytical solutions are compared with the test and finite element results. The distribution law of axial force, bending moment, and shear force of the composite structure are obtained, and the circumferential strain law of the inner and outer surfaces of the host pipe is analyzed. Lastly, the influence of pipe diameter and liner thickness on the bearing capacity of the host pipe is examined. The results indicate that the axial and shear force distribution of the composite structure remains the same for different pipe diameters, as the distribution diagrams of shear and axial force along the circumferential direction coincide. While the elastic modulus and thickness of the liner have no effect on the internal force distribution of the composite structure, they do directly affect the stress state of the host pipe. Additionally, the bottom support causes abrupt changes in the bending moments and strains of the host pipe.

Linear Response Equations Revisited: A Simple and Efficient Iterative Algorithm.

We present an algorithm to solve the linear response equations for Hartree-Fock, Density Functional Theory, and the Multiconfigurational Self-Consistent Field method that is both simple and efficient. The algorithm makes use of the well-established symmetric and antisymmetric combinations of trial vectors but further orthogonalizes them with respect to the scalar product induced by the response matrix. This leads to a standard, symmetric block eigenvalue problem in the expansion subspace that can be solved by diagonalizing a symmetric, positive definite matrix half the size of the expansion space. Numerical tests showed that the algorithm is robust and stable.

Hyperelastic constitutive relations for soft elastomers with thermally-induced residual stress

Residual stress widely exists in soft materials. Besides growth, inhomogeneous thermal expansion is also a primary cause of residual stress. However, establishing a proper hyperelastic constitutive relation is a great challenge since the existing theories cannot capture the change of underlying mechanical responses triggered by temperature variations. In this paper, a general hyperelastic constitutive relation for soft elastomers with thermally-induced residual stress is developed. We first reveal the initial temperature dependence of conventional thermoelastic models. This property attributes the alteration of the underlying thermoelastic response to free thermal expansions. Then, a compatibility-broken curvature compensation (CBCC) framework is established based on finite thermoelasticity. It generates a free thermal expansion to eliminate the Riemannian curvatures of the virtual stress-free configuration derived from the isothermal stress release. Such a mechanism indicates the non-local effect of the residual stress, which fundamentally modifies the traditional view that invariant formulations cover all the possible functional dependence of residual stress. Also, the obtained governing equations are similar to Einstein field equations of the general theory of relativity. This similarity may deeply imply a standard mechanism concerning the curvature compensation leading to residual stress genesis. We finally conduct comparative analyses of the spherically symmetric and axisymmetric problems between the current constitutive relation and the existing models. The influences of adopting distinct residual stresses, the performance of the non-local effect, and the availability of the new constitutive relation are investigated in detail. This framework can shed some light on the constitutive modeling of soft materials.