One-dimensional Operators Research Articles

Sturm–Liouville problem for a one-dimensional thermoelastic operator in Cartesian, cylindrical, and spherical coordinate systems

The problem of constructing eigenfunctions of a one-dimensional thermoelastic operator in Cartesian, cylindrical, and spherical coordinate systems is considered. The corresponding Sturm–Liouville problem is formulated using Fourier’s separation of variables applied to a coupled system of thermoelasticity equations, assuming that the heat transfer rate is finite. It is shown that the eigenfunctions of the one-dimensional thermoelastic operator are expressed in terms of well-known trigonometric, cylinder, and spherical functions. However, coupled thermoelasticity problems are solved analytically only under certain boundary conditions, whose form is determined by the properties of the eigenfunctions.

A Novel Approach to the Fractional Laplacian via Generalized Spherical Means

Although at least ten equivalent definitions of the fractional Laplacian exist in unbounded domains, we introduce an additional equivalent definition based on the generalized spherical mean-value operator—a Fourier multiplier operator involving the normalized Bessel function. Specifically, we demonstrate that this new definition allows us to reduce any n-dimensional fractional Laplacian to a one-dimensional operator, which simplifies computation and enhances efficiency. We propose two methods for computing the generalized spherical means of a given function: one by solving standard wave equations and the other by solving Darboux’s equations.

The fine spectra of a difference of weighted shift operators acting in some classical sequence spaces

In this paper we investigate the fine spectra, the ergodic properties and the linear dynamics of the operatorC(x)=(−nxn+n(n−1)xn−2)n∈N0,x=(xn)n∈N0∈CN0, and of its formal adjoint C′ acting in the classical sequence spaces s, s′, ω and φ. The operators C and C′ acting in s are conjugate to the one-dimensional Ornstein-Uhlenbeck differential operator Ou=u″−xu′ and to the differential operator O′u=u″+xu′+u acting in S(R), respectively.

Student understanding of eigenvalue equations in quantum mechanics: Symbolic blending and sensemaking analysis

As part of an effort to examine students’ mathematical sensemaking (MSM) in a spins-first quantum mechanics course during the transition from discrete (spin) to continuous (position) systems, students were asked to construct an eigenvalue equation for a one-dimensional position operator. A subset of responses took the general form of an eigenvalue equation written in Dirac notation. Symbolic blending, a combination of symbolic forms and conceptual blending, as well as a categorical framework for MSM, were used in the analysis. The data suggest two different symbolic forms for an eigenvalue equation that share a symbol template but have distinct conceptual schemata: A transformation that reproduces the original and to operate is to act. These symbolic forms, when blended with two sets of contextual knowledge, form the basis of three different interpretations of eigenvalue equations modeled here as conceptual blends. The analysis in this study serves as a novel example of, and preliminary evidence for, student engagement in sensemaking activities in the transition from discrete to continuous systems in a spins-first quantum mechanics course. Published by the American Physical Society 2024

Quintuple-based Representation Learning for Bipartite Heterogeneous Networks

Recent years have seen rapid progress in network representation learning, which removes the need for burdensome feature engineering and facilitates downstream network-based tasks. In reality, networks often exhibit heterogeneity, which means there may exist multiple types of nodes and interactions. Heterogeneous networks raise new challenges to representation learning, as the awareness of node and edge types is required. In this article, we study a basic building block of general heterogeneous networks, the heterogeneous networks with two types of nodes. Many problems can be solved by decomposing general heterogeneous networks into multiple bipartite ones. Recently, to overcome the demerits of non-metric measures used in the embedding space, metric learning-based approaches have been leveraged to tackle heterogeneous network representation learning. These approaches first generate triplets of samples, in which an anchor node, a positive counterpart, and a negative one co-exist, and then try to pull closer positive samples and push away negative ones. However, when dealing with heterogeneous networks, even the simplest two-typed ones, triplets cannot simultaneously involve both positive and negative samples from different parts of networks. To address this incompatibility of triplet-based metric learning, in this article, we propose a novel quintuple-based method for learning node representations in bipartite heterogeneous networks. Specifically, we generate quintuples that contain positive and negative samples from two different parts of networks. And we formulate two learning objectives that accommodate quintuple-based learning samples, a proximity-based loss that models the relations in quintuples by sigmoid probabilities and an angular loss that more robustly maintains similarity structures. In addition, we also parameterize feature learning by using one-dimensional convolution operators around nodes’ neighborhoods. Compared with eight methods, extensive experiments on two downstream tasks manifest the effectiveness of our approach.

On the number of closed gaps of discrete periodic one-dimensional operators

From the general inverse theory of periodic Jacobi matrices, it is known that a periodic Jacobi matrix of minimal period p≥2 may have at most p−2 closed spectral gaps. We discuss the maximal number of closed gaps for one-dimensional periodic discrete Schrödinger operators of period p. We prove nontrivial upper and lower bounds on this quantity for large p and compute it exactly for p≤6. Among our results, we show that a discrete Schrödinger operator of period four or five may have at most a single closed gap, and we characterize exactly which potentials may exhibit a closed gap. For period six, we show that at most two gaps may close. In all cases in which the maximal number of closed gaps is computed, it is seen to be strictly smaller than p−2, the bound guaranteed by the inverse theory. We also discuss similar results for purely off-diagonal Jacobi matrices.

Sturm–Liouville Problem for a One-Dimensional Thermoelastic Operator in Cartesian, Cylindrical, and Spherical Coordinate Systems

Sturm–Liouville Problem for a One-Dimensional Thermoelastic Operator in Cartesian, Cylindrical, and Spherical Coordinate Systems

Sturm-Liouville problems and global bounds by small control sets and applications to quantum graphs

We develop a Logvinenko–Sereda theory for one-dimensional vector-valued self-adjoint operators. We thus deliver upper bounds on L2-norms of eigenfunctions – and linear combinations thereof – in terms of their L2- and W1,2-norms on small control sets that are merely measurable and suitably distributed along each interval. An essential step consists in proving a Bernstein-type estimate for Laplacians with rather general vertex conditions. Our results carry over to a large class of Schrödinger operators with magnetic potentials; corresponding results are unknown in higher dimension. We illustrate our findings by discussing the implications in the theory of quantum graphs.

On the Inverse Spectral Problem for the One-dimensional Stark Operator on the Semiaxis

On the Inverse Spectral Problem for the One-dimensional Stark Operator on the Semiaxis

The spectrum of the $C^*$-algebra of singular integral operators with semi-almost periodic coefficients

A $C^*$-algebra generated by one-dimensional singular integral operators with semi-almost periodic coefficients is studied. The primitive spectrum of this algebra is described, that is, all of its primitive ideals are listed and the Jacobson topology is described. Bibliography: 27 titles.

RECONSTRUCTION THE DOMAIN OF DEFINITION OF SOMEDIFFERENTIAL OPERATOR ON A DIRECTED GRAPH

In this work it is proposed to study differential operators on a graph as an operator composed of differential operators on one-dimensional arcs and matrix operators on interior vertices of the graph. The work explores some questions concerning the theoretical side of ordinary differential equations with integro-differential conditions on stratified sets like graph. The attention will be paid to reconstruction of the domain of differential operator on directed graph. The reconstruction of the domain of differential operator means a simple specifying the boundary conditions from a known differential equations and its known eigenvalues. The paper studies the case of the second order differential equations with irregular boundary conditions on the vertices of directed graph. To achieve our goal we use the fact that finite set of eigenvalues serves as additional information for reconstruction of the domain of the differential operator on stratified set. The constructive algorithms for reconstructing the domain of definition of differential operator on directed graph are developed. All boundary functions from the spectral data are uniquely restored.

OPERATOR GROUP GENERATED BY A ONE-DIMENSIONAL DIRAC SYSTEM

In this paper, we construct a strongly continuous operator group generated by a one-dimensional Dirac operator acting in the space \(\mathbb{H} = {{\left( {{{L}_{2}}[0,\pi ]} \right)}^{2}}\). The potential is assumed to be summable. It is proved that this group is well-defined in the space \(\mathbb{H}\) and in the Sobolev spaces \(\mathbb{H}_{U}^{\theta }\), \(\theta 0\), with fractional index of smoothness \(\theta \) and under boundary conditions \(U\). Similar results are proved in the spaces \({{\left( {{{L}_{\mu }}[0,\pi ]} \right)}^{2}}\), \(\mu \in (1,\infty )\). In addition we obtain estimates for the growth of the group as \(t \to \infty \).

Numerical simulations of underwater explosions using a compressible multi-fluid model

We present a novel solver for simulating compressible multi-fluid multiphase flow in underwater explosions (UNDEXs). The developed solver uses a modified version of Saurel's six-equation model, which includes an additional total mixture energy equation to resolve discrepancies in the thermodynamic states predicted under shock conditions. Additionally, we integrate a more precise stiffened gas equation of state (SG-EOS) that is determined using a novel method to enhance the accuracy of predicting experimental data based on a shock Hugoniot curve. We also propose a solution procedure using the modified Saurel's six-equation model on a three-dimensional (3D) structured Cartesian grid system. This involves discretizing the equation system using a Godunov scheme with a two-fluid Harten-Lax-van Leer-Contact approximate Riemann solver and a MUSCL-Hancock primitive scheme with total-variation-diminishing limiters, achieving a second-order extension. Both the dimensional splitting and fractional-step methods are utilized to model one-dimensional (1D) operators, splitting them into sequential operators. The modified model is validated for 1D and 3D problems, including the water–air shock tube, cavitation, shock–bubble interaction, and UNDEX problems in a free field, near a free surface, and near a rigid dam. Our simulations accurately predict the shockwave propagation, shock and free-surface interactions, cavitation evolution, and water jetting impact characteristics, exhibiting satisfactory agreement with those of previous studies. The proposed solver provides insight into the effects of UNDEXs on rigid structures, with potential applications in engineering and defense. The proposed method for determining the SG-EOS parameters can be applied to other areas of research involving high-pressure multi-phase flows.

Fractional Integrals Associated with the One-Dimensional Dunkl Operator in Generalized Lizorkin Space

This paper explores the realm of fractional integral calculus in connection with the one-dimensional Dunkl operator on the space of tempered functions and Lizorkin type space. The primary objective is to construct fractional integral operators within this framework. By establishing the analogous counterparts of well-known operators, including the Riesz fractional integral, Feller fractional integral, and Riemann–Liouville fractional integral operators, we demonstrate their applicability in this setting. Moreover, we show that familiar properties of fractional integrals can be derived from the obtained results, further reinforcing their significance. This investigation sheds light on the utilization of Dunkl operators in fractional calculus and provides valuable insights into the connections between different types of fractional integrals. The findings presented in this paper contribute to the broader field of fractional calculus and advance our understanding of the study of Dunkl operators in this context.

Eigenvalues of one-dimensional Hamiltonian operators with an eigenparameter in the boundary condition

In this paper, one-dimensional Hamiltonian operators with spectral parameter-dependent boundary conditions are investigated. First, the eigenvalues of the problem under consideration are transformed into the eigenvalues of an operator in an appropriate Hilbert space. Then, some properties of the eigenvalues are given. Moreover, the continuity and differentiability of the eigenvalues of the problem are obtained, and the differential expressions of the eigenvalues concerning each parameter are also given. Finally, Green’s function is also involved.

Continuity for the one-dimensional centered Hardy-Littlewood maximal operator at the derivative level

We prove the continuity of the map f↦(Mf)′ from W1,1(R) to L1(R), where M is the centered Hardy-Littlewood maximal operator. This solves a question posed by Carneiro, Madrid and Pierce.

Regular versus singular solutions in quasilinear indefinite problems with sublinear potentials

The aim of this paper is analyzing existence, multiplicity, and regularity issues for the positive solutions of the quasilinear Neumann problem{−(u′/1+(u′)2)′=λa(x)f(u),0<x<1,u′(0)=u′(1)=0. Here, (u′/1+(u′)2)′ is the one-dimensional curvature operator, λ∈R is a parameter, which is generally taken positive, the weight a(x) changes sign, and, in most occasions, the function f(u) has a sublinear potential F(u) at ∞. Our discussion displays the manifold patterns occurring for these solutions, depending on the behavior of the potential F(u) at u=0, and, possibly, at infinity, and of the weight function a(x) at its nodal points.

Upper bounds on quantum dynamics in arbitrary dimension

Motivated by the research on upper bounds on the rate of quantum transport for one-dimensional operators, particularly, the recent works of Jitomirskaya–Liu and Jitomirskaya–Powell and the earlier ones of Damanik–Tcheremchantsev, we propose a method to prove similar bounds in arbitrary dimension. The method applies both to Schrödinger and to long-range operators.In the case of ergodic operators, one can use large deviation estimates for the Green function in finite volumes to verify the assumptions of our general theorem. Such estimates have been proved for numerous classes of quasiperiodic operators in one and higher dimension, starting from the works of Bourgain, Goldstein, and Schlag.One of the applications is a power-logarithmic bound on the quantum transport defined by a multidimensional discrete Schrödinger (or even long-range) operator associated with an irrational shift, valid for all Diophantine frequencies and uniformly for all phases. To the best of our knowledge, these are the first results on the quantum dynamics for quasiperiodic operators in dimension greater than one that do not require exclusion of a positive measure of phases. Moreover, and in contrast to localisation, the estimates are uniform in the phase.The arguments are also applicable to ergodic operators corresponding to other kinds of base dynamics, such as the skew-shift.

An Improved One-Dimensional Bending Angle Forward Operator for the Assimilation of Radio Occultation Profiles in the Lower Troposphere

Abstract Under very large vertical gradients of atmospheric refractivity, which are typical at the height of the planetary boundary layer, the assimilation of radio occultation (RO) observations into numerical weather prediction (NWP) models presents several serious challenges. In such conditions, the assimilation of RO bending angle profiles is an ill-posed problem, the uncertainty associated with the RO observations is higher, and the one-dimensional forward operator used to assimilate these observations has several theoretical deficiencies. As a result, a larger percentage of these RO observations are rejected at the NWP centers by existing quality control procedures, potentially limiting the benefits of this data type to improve weather forecasting in the lower troposphere. To address these problems, a new methodology that enables the assimilation of RO data to be extended to the lower moist troposphere has been developed. Challenges associated with larger atmospheric gradients of refractivity are partially overcome by a reformulation that has minimal effect at higher altitudes. As a first step toward this effort, this study presents both the theoretical development of this new methodology and a forecast impact assessment of it using the NCEP NWP system. Though using a conservative approach, benefits in the lower tropical troposphere are already noticeable. The encouraging results of this work open the potential for further exploitation and optimization of RO assimilation.

Two-stage melting of an intercomponent Potts long-range order in two dimensions

The interplay of topology and competing interactions can induce enriched phases and phase transitions at finite temperatures. We consider a weakly coupled two-dimensional hexatic-nematic XY model with a relative ${Z}_{3}$ Potts degrees of freedom, and apply the matrix product state method to solve this model rigorously. Since the partition function is expressed as a product of two-legged one-dimensional transfer matrix operator, an entanglement entropy of the eigenstate corresponding to the maximal eigenvalue of this transfer operator can be used as a stringent criterion to determine various phase transitions precisely. At low temperatures, the intercomponent ${Z}_{3}$ Potts long-range order (LRO) exists, indicating that the hexatic and nematic fields are locked together and their respective vortices exhibit quasi-LRO. In the hexatic regime, below the BKT transition of the hexatic vortices, the intercomponent ${Z}_{3}$ Potts LRO appears, accompanying the binding of nematic vortices. In the nematic regime, however, the intercomponent ${Z}_{3}$ Potts LRO undergoes a two-stage melting process. An intermediate Potts liquid phase emerges between the Potts ordered and disordered phases, characterized by an algebraic correlation with formation of charge-neutral pairs of both hexatic and nematic vortices. These two-stage phase transitions are associated with the proliferation of the domain walls and vortices of the relative ${Z}_{3}$ Potts variable, respectively. Our results thus provide a prototype example of two-stage melting of a two-dimensional LRO driven by multiple topological defects.