Nonsymmetric Case Research Articles

Products and powers of principal symmetric ideals

Principal symmetric ideals were recently introduced by Harada et al. in [The minimal free resolution of a general principal symmetric ideal, preprint (2023), arXiv:2308.03141], where their homological properties are elucidated. They are ideals generated by the orbit of a single polynomial under permutations of variables in a polynomial ring. In this paper, we determine when a product of two principal symmetric ideals is principal symmetric and when the powers of a principal symmetric ideal are again principal symmetric ideals. We characterize the ideals that have the latter property as being generated by polynomials invariant up to a scalar multiple under permutation of variables. Recognizing principal symmetric ideals is an open question for the purpose of which we produce certain obstructions. We also demonstrate that the Hilbert functions of symmetric monomial ideals are not all given by symmetric monomial ideals, in contrast to the non-symmetric case.

A study on the hilbert space structure and positive operators

Let X be a real Banach space. We prove that the existence of an injective, positive, symmetric and not strictly operator from X into its dual implies that either X admits an equivalent Hilbertian norm or it contains a nontrivially complemented subspace which is isomorphic to a Hilbert space. We also treat the non-symmetric case.

Double-Deck Structure in a Fluid Flow Induced by a Uniformly Rotating Disk with Small Irregularities: the Nonsymmetric Case

Double-Deck Structure in a Fluid Flow Induced by a Uniformly Rotating Disk with Small Irregularities: the Nonsymmetric Case

Regular ellipsoids and a Blaschke-Santaló-type inequality for projections of non-symmetric convex bodies

It is shown that every not-necessarily symmetric convex body K in Rn has an affine image K˜ of K such that the covering numbers of K˜ by growing dilates of the unit Euclidean ball, as well as those of the unit Euclidean ball by growing dilates of K˜, decrease in a regular way. This extends to the non-symmetric case a famous theorem by Pisier, albeit with worse estimates on the rate of decrease of the covering numbers. The affine image K˜ can be chosen to have either barycentre or Santaló point at the origin.In the proof we use Pisier's theorem as a black box, as well as a suggested approach by Klartag and V. Milman. A key new ingredient is Blaschke-Santaló-type inequalities for projections of a body K˜ with Santaló point at the origin, which could be of independent interest. Unlike the application to covering, these (as well as the analogous inequalities for centred convex bodies that were already considered by Klartag and Milman [28]) can be shown to be optimal up to absolute constants.We also present an application to results around the mean norm of isotropic (not-necessarily symmetric) convex bodies.

Large densities in a competitive two-species chemotaxis system in the non-symmetric case

Large densities in a competitive two-species chemotaxis system in the non-symmetric case

On the diversity-multiplexing trade-off of hybrid diamond relay channels

The diversity-multiplexing trade-off (DMT) of a diamond relay channel in a quasi-static slow Rayleigh fading environment is studied, where one of the relays operates in half-duplex mode while the other one is full-duplex; hence, the term hybrid is used to refer to this channel. DMT is a known powerful tool that captures the inherent tension between the reliability and the transmission rate in a wireless communication network at high signal-to-noise ratios (SNR). In this study, we have assumed that the average SNRs of different links vary at different rates, each represented by an exponent at high SNR. These average SNR exponents are categorized into two different classes: In Class I, both of the relays are placed either in close proximity to the source or in close proximity to the destination, whereas in Class II one of the relays is closer to the source while the other one is closer to the destination. Subsequently, for Class I and for a special symmetric case of Class II, an explicit expression of the DMT of the static quantize-map-and-forward (SQMF) strategy and the optimal static schedule in the half-duplex relay, are derived. The non-symmetric case of Class II can numerically be studied using the optimization problem provided in the paper for any given parameters. As a corollary, our analysis shows that for Class I, the SQMF strategy in the proposed hybrid diamond relay channel reaches the ideal full-duplex DMT upper bound if the degrees of freedom provided by the path through the full-duplex relay is greater than or equal to that provided by the path through the half-duplex relay, or if the degrees of freedom provided by the former is less than that provided by the latter, but still large enough to support the desired multiplexing gain.

Quantum pseudointegrable Hamiltonian impact system.

Quantization of a toy model of a pseudointegrable Hamiltonian impact system is introduced, including Einstein-Brillouin-Keller quantization conditions, a verification of Weyl's law, the study of their wave functions, and a study of their energy levels properties. It is demonstrated that the energy level statistics are similar to those of pseudointegrable billiards. Yet, here, the density of wave functions which concentrate on projections of classical level sets to the configuration space does not disappear at large energies, suggesting that there is no equidistribution in the configuration space in the large energy limit; this is shown analytically for some limit symmetric cases and is demonstrated numerically for some nonsymmetric cases.

Confining integro-differential equations originating from evolutionary biology: Ground states and long time dynamics

<p style='text-indent:20px;'>We consider nonlinear mutation selection models, known as <i>replicator-mutator</i> equations in evolutionary biology. They involve a nonlocal mutation kernel and a confining fitness potential. We prove that the long time behaviour of the Cauchy problem is determined by the principal eigenelement of the underlying linear operator. The novelties compared to the literature on these models are about the case of symmetric mutations: we propose a new milder sufficient condition for the existence of a principal eigenfunction, and we provide what is to our knowledge the first quantification of the spectral gap. We also recover existing results in the non-symmetric case, through a new approach.</p>

Directional amplitude backscatter modulation with suppressed Doppler based on rotating resonant loop

The directional amplitude backscatter modulation with suppressed Doppler is demonstrated based on the scattering from a symmetrically rotating resonant loop. The concept is studied theoretically and experimentally with perfectly compatible results. The symmetrical rotation of the scatterer and the effect of radial resonance, as the two crucial points to realize the idea, are highlighted through the comparison between the symmetric and non-symmetric cases, and the results obtained for scatterers with and without radial resonance. The presented backscattering modulation technique provides an amplitude modulating waveform which is uniquely linked to the directional reradiation pattern of the rotating loop scatterer in a definite resonant mode. With the pure directional amplitude modulation (DAM) induced on the backscattered wave, the envelope waveform can be accurately retrieved form the received signal using the In-phase and Quadrature (IQ) representation. The contribution of the background in a real environment can be detected and removed to obtain the exact modulating waveform. This property of the proposed backscattering modulation method can be applied for sensing, localization, and identification purposes with high sensitivity, read range, and robustness.

Robust level-3 BLAS Inverse Iteration from the Hessenberg Matrix

Inverse iteration is known to be an effective method for computing eigenvectors corresponding to simple and well-separated eigenvalues. In the non-symmetric case, the solution of shifted Hessenberg systems is a central step. Existing inverse iteration solvers approach the solution of the shifted Hessenberg systems with either RQ or LU factorizations and, once factored, solve the corresponding systems. This approach has limited level-3 BLAS potential since distinct shifts have distinct factorizations. This paper rearranges the RQ approach such that data shared between distinct shifts can be exploited. Thereby the backward substitution with the triangular R factor can be expressed mostly with matrix–matrix multiplications (level-3 BLAS). The resulting algorithm computes eigenvectors in a tiled, overflow-free, and task-parallel fashion. The numerical experiments show that the new algorithm outperforms existing inverse iteration solvers for the computation of both real and complex eigenvectors.

Absorption caused by potassium molecules in astrophysical plasmas: Data needed for modeling

In this study, the processes of photo-dissociation in the cases of symmetric and strongly non-symmetric systems which involve hydrogen and potassium are investigated. The aim of this study was to calculate data for those small molecules and molecular state characterizations as well as to obtain the corresponding partial and average cross-sections. Moreover, the corresponding spectral absorption rate coefficients for symmetric and strongly non-symmetric cases i.e. for K2+ and KH+ cases are given in tables as a function of temperatures and wavelengths allowing potential modeling. The current research and presented results can be of interest for astrophysical plasma research and laboratory plasma investigation.

Infinitely many solutions for higher order impulsive equations without symmetry

We investigate the existence of multiple solutions of boundary value problems for higher order impulsive equations. By using variational and perturbative methods, we show the existence of infinitely many solutions in the nonsymmetric case.

Convergence of a Class of Delayed Neural Networks with Real Memristor Devices

Neural networks with memristors are promising candidates to overcome the limitations of traditional von Neumann machines via the implementation of novel analog and parallel computation schemes based on the in-memory computing principle. Of special importance are neural networks with generic or extended memristor models that are suited to accurately describe real memristor devices. The manuscript considers a general class of delayed neural networks where the memristors obey the relevant and widely used generic memristor model, the voltage threshold adaptive memristor (VTEAM) model. Due to physical limitations, the memristor state variables evolve in a closed compact subset of the space; therefore, the network can be mathematically described by a special class of differential inclusions named differential variational inequalities (DVIs). By using the theory of DVI, and the Lyapunov approach, the paper proves some fundamental results on convergence of solutions toward equilibrium points, a dynamic property that is extremely useful in neural network applications to content addressable memories and signal-processing in real time. The conditions for convergence, which hold in the general nonsymmetric case and for any constant delay, are given in the form of a linear matrix inequality (LMI) and can be readily checked numerically. To the authors knowledge, the obtained results are the only ones available in the literature on the convergence of neural networks with real generic memristors.

Equilibrium points in restricted problems on S2 and H2

In this paper, we consider the restricted four-body problem on S2 and the restricted three-body problem on H2. In the first case, the primary particles are considered to be rotating around the vertical axis. In the latter case, the primaries move at a hyperbolic relative equilibrium. If the primary particles locate at height z0 = 0 and they form an isosceles triangle on S2, then the number of symmetric equilibrium points of the restricted four-body problem depends on the angular velocity, the configuration, and the masses of the primaries. For this case, the number of equilibrium points is 0, 2, 4, 6, or 8. If the primaries are at height z0 ∈ (0, 1), then the primaries form an equilateral triangle and the number of symmetric equilibrium points is 8, 11, 14, 17, or 20. The number depends on the position of the primaries, and it is higher than the previous case due to the higher number of symmetry axes of the triangle formed by the primaries. On the other hand, for the restricted three-body problem on H2, the number of equilibrium points is 3, considering that the primaries move at a hyperbolic relative equilibrium. We analyzed the general (non-symmetric) case, showing the locations of the equilibrium points for the given positions of the primaries. We also studied the stability of these equilibrium points.

Planar central configurations of six bodies

Central configurations of the n-body problem play an important role in the study of celestial mechanics. In this paper, we study six-body planar central configurations having certain symmetries. Special attention is given to the existence results of such configurations. With analytic proofs, we show the existence of a new central configuration, which is convex but not strictly convex, a non-symmetric concave case of central configuration, and some cases of stacked central configurations, which are central configurations with subsets of the bodies also forming central configurations.

Exponential Convergence Towards Consensus for Non-Symmetric Linear First-Order Systems in Finite and Infinite Dimensions

We consider finite and infinite-dimensional first-order consensus systems with timeconstant interaction coefficients. For symmetric coefficients, convergence to consensus is classically established by proving, for instance, that the usual variance is an exponentially decreasing Lyapunov function. We investigate here the convergence to consensus in the non-symmetric case: we identify a positive weight which allows to define a weighted mean corresponding to the consensus, and obtain exponential convergence towards consensus. Moreover, we compute the sharp exponential decay rate.

A novel difference and derivative of linearly correlated fuzzy number-valued functions

In the present paper, the notion of the linearly correlated difference for linearly correlated fuzzy numbers is introduced. Especially, the linearly correlated difference and the generalized Hukuhara difference are coincident for interval numbers or even symmetric fuzzy numbers. Accordingly, an appropriate metric is induced by using the norm and the linearly correlated difference in the set of linearly correlated fuzzy numbers. Based on the symmetry of the basic fuzzy number, the linearly correlated derivative is proposed by the linearly correlated difference of linearly correlated fuzzy number-valued functions. In both non-symmetric and symmetric cases, the equivalent characterizations of the linearly correlated differentiability of a linearly correlated fuzzy number-valued function are established, respectively. Moreover, it is shown that the linearly correlated derivative is consistent with the generalized Hukuhara derivative for interval-valued functions.

An analytical study of collapse of locally corroded tubes under external pressure

This paper presents a perturbative analysis of bifurcation pressure of 2D corroded rings characterized by locally thinned regions under external pressure by asymptotically analyzing the Timoshenko differential equation and parametric analysis. First, for the case of one corrosion region, we formulate regular perturbative solution separately for anti-symmetric bifurcation and symmetric bifurcation cases. Concise formulation is presented by introducing a corrosion severity parameter and perturbative solution is presented. Second, we present a detailed analysis for the case where corrosion region has small angular extent by converting it into a singular perturbative problem and conduct various asymptotic analyses to illustrate the interesting interlacing behaviors. The derived explicit formula explains some numerical observation by Fatt in literature. Third, we conduct parametric analysis for the case of two interacting corrosion regions by classifying it into two cases: symmetric corrosion case and non-symmetric corrosion case. Finally, we present asymptotical analysis to investigate the extremely corroded case and show by Prufer transformation that anti-symmetric bifurcation pressure decreases when the distance of two corrosion regions is increasing. This paper serves to enhance the understanding of the collapse pressure of subsea pipes with local thickness reduction.

The location of a minimum variance squared distance functional

In this paper, we introduce a novel multivariate functional that represents a position where the intrinsic uncertainty of a system of mutually dependent risks is maximally reduced. The proposed multivariate functional defines the location of the minimum variance of squared distance (LVS) for some n-variate vector of risks X. We compute the analytical representation of LVS(X), which consists of the location of the minimum expected squared distance, LES(X), covariance matrix A, and a matrix B of the multivariate central moments of the third order of X. From this representation it follows that LVS(X) coincides with LES(X) when X has a multivariate symmetric distribution, but differs from it in the non-symmetric case. As LES(X) is often considered a neutral multivariate risk measure, we show that LVS(X) also possesses the important properties of multivariate risk measures: translation invariance, positive homogeneity, and partial monotonicity. We also study the mean-variance approach based on the balanced sum of an expectation and a variance of the square of the aforementioned Euclidean distance and control for the closeness of LES(X) and LVS(X). The proposed theory and the results are distribution free, meaning that we do not assume any particular distribution for the random vector X. The results are demonstrated with real data of Danish fire losses.

Bubbling, Bistable Limit Cycles and Quasi-Periodic Oscillations in Queues with Delayed Information

We consider a model describing the length of two queues that incorporates customer choice behavior based on delayed queue length information. The symmetric case, where the values of the time-delay parameter in each queue are the same, was recently studied. It was shown that under some conditions, the stable equilibrium solution becomes unstable as the common time delay passes a threshold value. This one-time stability switch occurs only at a symmetry-breaking Hopf bifurcation where a family of stable asynchronous limit-cycle solutions arise. In this paper, we examine the non-symmetric case, wherein the values of the time-delay parameter in each queue are different. We show that, in contrast to the symmetric case, the non-symmetric case allows bubbling, multiple stability switches and coexistence of distinct families of stable limit cycles. An investigation of the dynamical behavior of the non-symmetric system in a neighborhood of a double-Hopf bifurcation using numerical continuation explains the occurrence of the bistable limit cycles. Quasi-periodic oscillations were also observed due to the presence of torus bifurcations near the double-Hopf bifurcation. These identifications of the underlying mechanisms that cause unwanted oscillations in the system give a better understanding of the effects of providing delayed information and consequently help in better management of queues.