Nonzero Function Research Articles

An infinite family of chiral, non-slice, non-alternating hyperbolic knots with vanishing Upsilon invariants

For a knot in the 3-sphere, its Upsilon invariant is a continuous, piecewise linear function defined on the interval [0,2]. It is a concordance invariant, so vanishes for slice knots. Since it only changes the sign for the mirror image with reversed orientation, it also vanishes for amphicheiral knots. For alternating knots, more generally, quasi-alternating knots, the Upsilon invariant is determined only by the signature. In particular, it vanishes if such a knot has zero signature. On the other hand, it is known that the Upsilon invariant is a non-zero convex function for L-space knots. In this paper, we construct an infinite family of hyperbolic knots, each of which has zero Upsilon invariant, but is chiral, non-slice, non-alternating. To confirm that the Upsilon invariant vanishes, we calculate the full knot Floer complex.

Invariant subspaces of contractions with constant characteristic function

We describe all the invariant subspaces of a completely nonunitary contraction T c T_c with nonzero scalar constant characteristic function c c , where | c | > 1 |c|>1 . Moreover, we show that T c T_c and restriction to its invariant subspaces are essentially normal. Using this description of T c T_c -invariant subspaces, we then identify the hyperinvariant subspaces.

A Composition Formula for the Modified Analytic Function Space Fourier–Feynman Transform

Composition formula is one of the most important research topics in functional analysis theory. Various relationships can be obtained using the composition formula. Since the generalized Brownian motion process used in this paper has a non-zero mean function, there are many restrictions on obtaining a composition formula for the modified analytic function space Fourier–Feynman transform. This paper contains an idea of how the composition Formula (9) below is established for the modified analytic function space Fourier–Feynman transform on function space. Using this idea, we are able to solve a problem that had never been solved before.

Compressing the two-particle Green’s function using wavelets: Theory and application to the Hubbard atom

Precise algorithms capable of providing controlled solutions in the presence of strong interactions are transforming the landscape of quantum many-body physics. Particularly, exciting breakthroughs are enabling the computation of non-zero temperature correlation functions. However, computational challenges arise due to constraints in resources and memory limitations, especially in scenarios involving complex Green’s functions and lattice effects. Leveraging the principles of signal processing and data compression, this paper explores the wavelet decomposition as a versatile and efficient method for obtaining compact and resource-efficient representations of the many-body theory of interacting systems. The effectiveness of the wavelet decomposition is illustrated through its application to the representation of generalized susceptibilities and self-energies in a prototypical interacting fermionic system, namely the Hubbard model at half-filling in its atomic limit. These results are the first proof-of-principle application of the wavelet compression within the realm of many-body physics and demonstrate the potential of this wavelet-based compression scheme for understanding the physics of correlated electron systems.

Generic uniformly continuous mappings on unbounded hyperbolic spaces

We consider a complete, unbounded hyperbolic metric space X and a concave, nonzero and nondecreasing function ω:[0,+∞)→[0,+∞) with ω(0)=0 and study the space Cω(X) of uniformly continuous self-mappings on X whose modulus of continuity is bounded from above by ω. We endow Cω(X) with the topology of uniform convergence on bounded sets and prove that the modulus of continuity of a generic mapping in Cω(X), in the sense of Baire categories, is precisely ω. Some related results in spaces of bounded mappings and in the topology of pointwise convergence are also discussed. This note can be seen as a completion of various results due to F. Strobin, S. Reich, A.J. Zaslavski, C. Bargetz and D. Thimm.

The ring A[X,Y;λ] and the SFT property

Let A be a ring and λ : R ≥ 0 → R ≥ 0 be an increasing nonzero function. In this paper, we show that if A is a Noetherian ring with characteristic n ≠ 0 and lim k → + ∞ λ ( k + 1 ) λ ( k ) = + ∞ , then A [ X , Y ; λ ] is an SFT ring. This result allows us to construct a nonNoetherian SFT ring which has some Noetherian completion. Also, we use the composite ring extension to construct many examples of such rings. Many examples are provided.

Inducing Clusters Deep Kernel Gaussian Process for Longitudinal Data

We consider the problem of predictive modeling from irregularly and sparsely sampled longitudinal data with unknown, complex correlation structures and abrupt discontinuities. To address these challenges, we introduce a novel inducing clusters longitudinal deep kernel Gaussian Process (ICDKGP). ICDKGP approximates the data generating process by a zero-mean GP with a longitudinal deep kernel that models the unknown complex correlation structure in the data and a deterministic non-zero mean function to model the abrupt discontinuities. To improve the scalability and interpretability of ICDKGP, we introduce inducing clusters corresponding to centers of clusters in the training data. We formulate the training of ICDKGP as a constrained optimization problem and derive its evidence lower bound. We introduce a novel relaxation of the resulting problem which under rather mild assumptions yields a solution with error bounded relative to the original problem. We describe the results of extensive experiments demonstrating that ICDKGP substantially outperforms the state-of-the-art longitudinal methods on data with both smoothly and non-smoothly varying outcomes.

Some generic hypersurfaces in a Euclidean space

<abstract><p>In this paper, we find three nontrivial characterizations of Euclidean spheres. In the first result, we show that the existence of a nonzero nontrivial concircular vector field $ {\omega } $ on a compact and connected hypersurface $ N $ of the Euclidean space $ R^{m+1} $ with a mean curvature $ \alpha $ constant along the integral curves of $ {\omega } $ and a shape operator $ T $ satisfying $ T({\omega) = \alpha \omega } $ implies that $ \alpha $ is a constant and $ N $ is isometric to a sphere, and the converse also holds. In the second result, we show that the presence of a unit Killing vector field $ \mathbf{v} $ on a compact and connected hypersurface $ N $ of a Euclidean space $ R^{m+1} $ gives a nonzero function $ \sigma = g\left(T \mathbf{v}, \mathbf{v}\right) $ with shape operator $ T $, and the integral of the function $ m\alpha \sigma Ric\left(\mathbf{v}, \mathbf{v}\right) $ has a certain lower bound, and is isometric to an odd-dimensional sphere, and the converse holds too. Finally, we show that for a compact and connected hypersurface $ N $ with support $ \rho $ and basic vector field $ \mathbf{u} $, the integral of the Ricci curvature $ Ric\left(\mathbf{u}, \mathbf{u}\right) $ has a specific lower bound and is necessarily isometric to a sphere, and the converse also holds.</p></abstract>

Tunable polarization properties of charge, spin, and valley in Janus VSiGeZ4 (Z = N, P, As) monolayers.

Polarization, as an important characterization of the symmetry breaking systems, has attracted tremendous attention in two-dimensional (2D) materials. Due to their significant symmetry breaking, Janus 2D ferrovalley materials provide a desirable platform to investigate the charge, spin, and valley polarization, as well as their coupling effects. Herein, using first-principles calculations, the polarization properties of charge, spin, and valley in Janus VSiGeZ4 (Z = N, P, and As) monolayers are systematically studied. The mirror symmetry breaking leads to a non-zero dipole moment and surface work function difference, indicating the presence of out-of-plane charge polarization. Magnetic properties calculations demonstrate that VSiGeN4 is a 2D-XY magnet with a Berezinskii-Kosterlitz-Thouless temperature of 342 K, while VSiGeP4 and VSiGeAs4 have an out-of-plane magnetization with a Curie temperature below room temperature. The magnetization can be rotated by applying biaxial strain, allowing manipulation of the spin polarization via nonmagnetic means. The spontaneous valley polarization is predicted to be 46, 49, and 70 meV for VSiGeN4, VSiGeP4, and VSiGeAs4, respectively, whose physical origin can be elucidated by employing the model analysis. In particular, the biaxial strain can induce the valley polarization switching from the valence (conduction) band to conduction (valence) band, but it hardly changes the valley polarization strength. Meanwhile, the valley extremum is transformed from the K' (K) to K (K') points. The present work not only provides an underlying insight into the polarization properties of Janus VSiGeZ4 but also offers a class of promising materials for spintronic and valleytronic devices.

Quantitative Uncertainty Principles for the Canonical Fourier Bessel Transform

The aim of this paper is to prove new uncertainty principles for the Canonical Fourier Bessel transform. To do so we prove a quantitative uncertainty inequality about the essential supports of a nonzero function for this transformation.

EQUILIBRIUM OF A NORMALLY LOADED ELASTIC SHALLOW CYLINDRICAL SHELL WITH DISLOCATIONS AND DISCLINATIONS

We consider the problem of the equilibrium of a normally loaded elastic shallow rectangular circular cylindrical shell, which contains sources of internal stress in the form of fields of continuously distributed edge dislocations and wedge disclinations or other sources, for example, thermal ones. Based on the modified system of Karman equations for the equilibrium of an elastic plate with dislocations and disclinations, a system of nonlinear equilibrium equations for the shell was constructed. The resulting system of equations differs from the Karman system of equilibrium equations for an elastic shallow cylindrical shell under the action of a normal load by the presence on the right side of the continuity equation of a nonzero function, called the incompatibility function, which is expressed through the densities of edge dislocations and wedge disclinations. The edges of the shell are freely clamped or hinged. In the absence of internal stress sources, this system transforms into a system of equilibrium equations for a shallow shell, and in the case of an infinitely large radius of curvature (and the presence of internal stress sources) into a modified system of Karman equilibrium equations for a flexible elastic plate with dislocations and disclinations. To solve the system of nonlinear shell equilibrium equations, the Newton – Kantorovich method in combination with the difference method is proposed. For the case of small values of the normal load and small values of the incompatibility function, linearization is carried out with respect to the deflection and stress function. As a result, a linear boundary value problem was obtained in the form of a system of differential equations for the deflection and the stress function, which is solved by the difference method. The solution of the linearized system is used as an initial approximation in the implementation of the Newton – Kantorovich method. Examples of numerical calculations of deflection and stress function for given values of normal load and incompatibility function are given, and corresponding graphs are constructed.

An efficient Hermite-Galerkin spectral scheme for three-dimensional incompressible Hall-magnetohydrodynamic system on infinite domain

The Hall-magnetohydrodynamic (Hall-MHD) system plays a significant role in the fluid description of an astrophysical plasma that features fast magnetic reconnection. Compared to the classical MHD system, the appearance of Hall term brings much more challenges in the field of numerical treatment. The aim of this paper is to construct an efficient spectral scheme for incompressible Hall-MHD system on three-dimensional infinite domain R3. Combining with a novel (HN+2,HN)-Hermite-Galerkin spectral scheme for spatial approximation, a new non-zero constant function approach for the nonlinear terms, a second-order projection method for the momentum equations, and BDF2 scheme for the decoupled system, here we establish, for the first time, a fully decoupled, completely linearized, and unconditionally energy stable scheme with directly spatial approximation in R3 and second-order temporal accuracy for incompressible Hall-MHD system. Numerical results are presented to illustrate the features of the proposed scheme, including convergence/stability tests and simulations of O- and X-points appearing in fast magnetic reconnection within the Hall-MHD regime.

Phase‐space analysis of the anisotropies in Weyl integrable spacetime: Kantowski–Sachs cosmology

We consider anisotropic Kantowski–Sachs cosmology in Weyl integrable spacetime with an ideal gas. We perform a detailed analysis of the dynamics by investigating the stationary points and reconstruct the cosmological history. This analysis gives important information for the initial value problem in Weyl integrable spacetime. We study three cases, in which the scalar field mass depends only on the density of the ideal gas, with or without cosmological constant term, and when there exists an exponential scalar field potential. We conclude that the existence of the nonzero potential function is essential in order that a spatially flat FLRW universe which can describe inflation can be a future attractor.

Optimal Hardy‐weights for elliptic operators with mixed boundary conditions

AbstractWe construct families of optimal Hardy‐weights for a subcritical linear second‐order elliptic operator with degenerate mixed boundary conditions. By an optimal Hardy‐weight for a subcritical operator we mean a nonzero nonnegative weight function W such that is critical, and null‐critical with respect to W. Our results rely on a recently developed criticality theory for positive solutions of the corresponding mixed boundary value problem.

Hungry Lotka–Volterra lattice under nonzero boundaries, block‐Hankel determinant solution, and biorthogonal polynomials

AbstractIn this paper, we first extend the hungry Lotka–Volterra lattice to a case of nonzero boundary conditions and present its corresponding exact solution expressed in terms of a block‐Hankel determinant. Then, we establish a connection between this hungry Lotka–Volterra lattice under nonzero boundary conditions and a set of biorthogonal polynomials. It turns out that the hungry Lotka–Volterra lattice under nonzero boundary conditions possesses a Lax pair expressed in terms of the biorthogonal polynomials. Moreover, we consider two special cases of the hungry Lotka–Volterra lattice. For the case , it reduces to the Lotka–Volterra lattice under nonzero boundary condition, which has been discussed in literature. We also present the result for in detail, which extends a known result to a case of nonzero boundary functions. All these results are obtained by virtue of Hirota's bilinear method and determinant techniques.

The Set of p-Adic Continuous Functions not Satisfying the Luzin (N) Property

In this short note, we will study the existence of a vector space of continuous functions f:{mathbb {Z}}_prightarrow mathbb Q_p, where {mathbb {Z}}_p and {mathbb {Q}}_p are, respectively, the ring of p-adic integers and the field of p-adic numbers, such that each nonzero function does not satisfy the Luzin (N) property and the dimension of the vector space is the continuum.

Matrix-valued nonstationary frames associated with the Weyl–Heisenberg group and the extended affine group

We study nonstationary frames of matrix-valued Gabor systems and wavelet systems in the matrix-valued function space [Formula: see text]. First, we show that a diagonal matrix-valued window function constitutes a frame for [Formula: see text] whenever each diagonal entry constitutes a frame for the space [Formula: see text]. This is not true for arbitrary nonzero matrix-valued function. Using this, we prove the existence of nonstationary matrix-valued Gabor frames associated with the Weyl–Heisenberg group in terms of density of real numbers. We give a representation of the frame operator of matrix-valued nonstationary Gabor system. A necessary condition with explicit frame bounds for nonstationary matrix-valued Gabor frames associated with the Weyl–Heisenberg group is given. We discuss matrix-valued frame preserving maps in terms of adjointablity, with respect to the matrix-valued inner product, of bounded linear operators acting on [Formula: see text]. It is shown that the image of a matrix-valued Gabor frame under bounded, linear and invertible operator on [Formula: see text] may not be a frame for [Formula: see text]. In this direction, we give sufficient conditions on bounded linear operators which can preserve frame conditions. Finally, we give necessary and sufficient condition for the existence of nonstationary matrix-valued wavelet frames associated with the extended affine group.

Feng-Liu’s Approach to Fixed Point Results of Intuitionistic Fuzzy Set-Valued Maps

The applications of non-zero self distance function have recently been discovered in both symmetric and asymmetric spaces. With respect to invariant point results, the available literature reveals that the idea has only been examined for crisp mappings in either symmetric or asymmetric spaces. Hence, the aim of this paper is to introduce the notion of invariant points for non-crisp set-valued mappings in metric-like spaces. To this effect, the technique of κ-contraction and Feng-Liu’s approach are combined to establish new versions of intuitionistic fuzzy functional equations. One of the distinguishing ideas of this article is the study of fixed point theorems of intuitionistic fuzzy set-valued mappings without using the conventional Pompeiu–Hausdorff metric. Some of our obtained results are applied to examine their analogues in ordered metric-like spaces endowed with an order and binary relation as well as invariant point results of crisp set-valued mappings. By using a comparative example, it is observed that a few important corresponding notions in the existing literature are complemented, unified and generalized.

Characterizing non-totally geodesic spheres in a unit sphere

<abstract><p>A concircular vector field $ \mathbf{u} $ on the unit sphere $ \mathbf{S}^{n+1} $ induces a vector field $ \mathbf{w} $ on an orientable hypersurface $ M $ of the unit sphere $ \mathbf{S}^{n+1} $, simply called the induced vector field on the hypersurface $ M $. Moreover, there are two smooth functions, $ f $ and $ \sigma $, defined on the hypersurface $ M $, where $ f $ is the restriction of the potential function $ \overline{f} $ of the concircural vector field $ \mathbf{u} $ on the unit sphere $ \mathbf{S}^{n+1} $ to $ M $ and $ \sigma $ is defined as $ g\left(\mathbf{u}, N\right) $, where $ N $ is the unit normal to the hypersurface. In this paper, we show that if function $ f $ on the compact hypersurface satisfies the Fischer–Marsden equation and the integral of the squared length of the vector field $ \mathbf{w} $ has a certain lower bound, then a characterization of a small sphere in the unit sphere $ \mathbf{S}^{n+1} $ is produced. Additionally, we find another characterization of a small sphere using a lower bound on the integral of the Ricci curvature of the compact hypersurface $ M $ in the direction of the vector field $ \mathbf{w} $ with a non-zero function $ \sigma $.</p></abstract>

Stochastic differential game strategies in the presence of reinsurance and dividend payout

<p style='text-indent:20px;'>This paper presents and examines a problem in which two insurance companies apply non-proportional reinsurance to control risk. Additionally, each firm pays out dividends. The situation is modelled as a zero-sum stochastic differential game between the two companies. The goal of one company is to maintain business competitive advantage over the other by sustaining or increasing the difference between the respective liquid reserves of the two companies while the second company aims to minimise that difference. A verification theorem is formulated, proved and subsequently employed to derive the saddle point components. For the case of the payoff with a non-zero running cost function, we are able to solve explicitly the differential game. Numerical simulations are presented to illustrate the results as well as the economic interpretation.</p>