Article
Mathematics
Jean-Pierre Antoine, Camillo Trapani
Summary: In this article, we examine the spectral behavior of a self-adjoint operator A in a Hilbert space H when it is expressed in terms of generalized eigenvectors. By utilizing the formalism of Gel'fand distribution bases, we investigate the conditions for the generalized eigenspaces to be one-dimensional, indicating a simple spectrum for A.
Article
Physics, Multidisciplinary
Davide Lonigro
Summary: In this study, we investigate the phenomena of bound states and resonances in a system composed of two-level systems interacting with a one-dimensional boson field. We evaluate the self-energy of the model and provide an analytic expression that is applicable to a wide range of dispersion relations and coupling functions. Specifically, we analyze the case of identical two-level systems, distinguishing between dominant and suppressed contributions to the self-energy, and examine the phenomenology of bound states in the presence of a single dominant contribution.
EUROPEAN PHYSICAL JOURNAL PLUS
(2022)
Article
Optics
Philip Caesar Flores, Eric A. Galapon
Summary: This paper studies the relativistic version of the Aharonov-Bohm time-of-arrival operator for spin-0 particles, and provides insights beyond the original results by taking its rigged Hilbert space extension. Time-of-arrival distributions are constructed using eigenfunctions that exhibit unitary arrival, and the expectation value is calculated, showing that particles can arrive earlier or later than expected classically. The constructed time-of-arrival distribution and expectation value are also consistent with special relativity.
Article
Physics, Multidisciplinary
Dean Alvin L. Pablico, Eric A. Galapon
Summary: In this paper, we provide a complete description of a time of arrival (TOA) operator, which is conjugate with the system Hamiltonian, by explicitly solving all the terms in the expansion. We interpret the terms beyond the leading term as quantum corrections to the Weyl quantization of the classical arrival time, expressed as integrals of the interaction potential. We investigate the properties of these quantum corrections in detail and find that they always vanish for linear systems but are nonvanishing for nonlinear systems. Finally, we consider the example of an anharmonic oscillator potential.
EUROPEAN PHYSICAL JOURNAL PLUS
(2023)
Article
Mathematics
Enrico Celeghini, Manuel Gadella, Mariano A. del Olmo
Summary: This paper reviews the generalization of Euclidean and pseudo-Euclidean groups in quantum mechanics. The study finds that these groups give rise to a more general family of groups, with Euclidean and pseudo-Euclidean groups as subgroups. The paper also constructs generalized Hermite functions on multidimensional spaces and investigates their transformation laws under Fourier transform.
Article
Mathematics, Applied
Thomas Jahn, Christian Richter
Summary: The existence of best coapproximations in certain dimensions of vector spaces with gauge measurements implies that the gauge is a norm or even a Hilbert space norm. Furthermore, the coproximinality of closed subspaces of a fixed dimension also implies coproximinality of subspaces of lower dimensions.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Mathematics, Applied
Ozlem Baksi
Summary: In this work, we derive the asymptotic formulas for the sum of negative eigenvalues smaller than -epsilon (epsilon > 0) of a self-adjoint operator L, defined by a specific differential expression with a boundary condition.
Article
Engineering, Electrical & Electronic
Xingchao Jian, Wee Peng Tay
Summary: This research considers statistical graph signal processing in a generalized framework where each vertex of a graph is associated with an element from a Hilbert space. The concept of joint wide-sense stationarity is introduced in this framework to characterize a graph random process as a combination of uncorrelated oscillation modes across both the vertex and Hilbert space domains. Numerical experiments demonstrate that the generalized approach achieves better estimation performance compared to traditional methods.
IEEE TRANSACTIONS ON SIGNAL PROCESSING
(2022)
Article
Operations Research & Management Science
Simeon Reich, Truong Minh Tuyen
Summary: This paper studies the generalized Fermat-Torricelli problem and the split feasibility problem with multiple output sets in Hilbert spaces. It introduces the generalized Fermat-Torricelli problem and proposes and analyzes a subgradient algorithm for solving this problem. Then it studies the convergence of variants of the proposed algorithm for solving the split feasibility problem with multiple output sets. Our algorithms for solving this problem are completely different from previous ones because we do not use the least squares sum method.
JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
(2023)
Article
Mathematics, Applied
Songxiao Li, Jizhen Zhou
Summary: In this paper, we characterize the boundedness and compactness of the Hankel operator H-μ from Bloch type spaces to BMOA and the Bloch space. Moreover, we obtain the essential norm of H-μ from B-α to B and BMOA.
Article
Mathematics, Applied
Athanasios Kouroupis
Summary: We investigate composition operators on the Hardy space 7-12 of Dirichlet series with square summable coefficients. The main result of this study is the necessary condition, expressed in terms of a Nevanlinna-type counting function, for a certain class of composition operators to be compact on 7-12. To achieve this, we extend our notions to a Hardy space 7-12 of generalized Dirichlet series, induced naturally by a sequence of Beurling's primes.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2023)
Article
Mathematics
Ezgi Erdogan, Enrique A. Sanchez Perez
Summary: We present a new stochastic approach to approximate (nonlinear) Lipschitz operators in normed spaces using their eigenvectors. We propose different integral representations for these approximations depending on the properties of the operators, whether they are locally constant, (almost) linear, or convex. We introduce the notion of eigenmeasure and focus on extending a function with known eigenvectors to the entire space. We provide information on natural error bounds, giving tools to measure the extent to which the map can be considered diagonal with minimal errors. In particular, we show an approximate spectral theorem for Lipschitz operators that have certain convexity properties.
Article
Mathematics, Applied
Di Yang, Chunhui Zhou
Summary: This paper introduces the generalized Brezin-Gross-Witten (BGW) tau function for the Gelfand-Dickey hierarchy, which depends on (r-1) constant parameters d(1), ..., d(r-1). It is shown that this tau function satisfies a family of linear equations called the W-constraints of the second kind. The operators giving rise to these linear equations also depend on (r-1) constant parameters. It is proved that there is a one-to-one correspondence between the two sets of parameters.
Article
Mathematics
Imran Ali, Haider Abbas Rizvi, Ramakrishnan Geetha, Yuanheng Wang
Summary: In this article, a nonlinear system of generalized ordered XOR-inclusion problems in Hilbert space is introduced and studied. Initially, the resolvent operator related to the (a,?)-XOR-weak-ANODD multivalued mapping is defined. The results of existence are demonstrated using the fixed point technique. To make the suggested system more realistic, the S-iterative algorithm is created and it is shown that the sequence generated through this technique strongly converges with the proposed system's solution. One example is provided in support of the existence result as well.
Article
Mathematics, Applied
Tamara Bottazzi, Cristian Conde
Summary: This article examines the inequalities fulfilled by orthogonal projection operator P defined on an inner product space H, and generalizations of these inequalities for certain families of bounded linear operators defined on H. Several new inequalities involving the norm and numerical radius of an operator are also established.
Article
Mathematics
Matija Bucic, Richard Montgomery
Summary: This article improves upon previous research by showing that any n-vertex graph can be decomposed into O(n log* n) cycles and edges.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Laurentiu G. Maxim, Jose Israel Rodriguez, Botong Wang, Lei Wu
Summary: The paper investigates the relationship between linear optimization degree and geometric structure. By analyzing the geometric structure of the conormal variety of an affine variety, the Chern-Mather classes of the given variety can be completely determined. Additionally, the paper shows that these bidegrees coincide with the linear optimization degrees of generic affine sections.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
William Chan, Stephen Jackson, Nam Trang
Summary: Under the determinacy hypothesis, this paper completely characterizes the existence of nontrivial maximal almost disjoint families for specific cardinals kappa, considering the ideals of bounded subsets and subsets of cardinality less than kappa.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Zhenguo Liang, Zhiyan Zhao, Qi Zhou
Summary: This paper investigates the reducibility of the one-dimensional quantum harmonic oscillator perturbed by a time quasi-periodic quadratic form. It provides a description and upper bound for the growth of the Sobolev norms of the solution, and demonstrates the optimality of the upper bound.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Zhao Yu Ma, Yair Shenfeld
Summary: This study provides a new approach to understanding the extremal cases of Stanley's inequalities by establishing a connection between the combinatorics of partially ordered sets and the geometry of convex polytopes.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Laurent Laurent, Rosa M. Miro-Roig
Summary: This paper discusses the problem of constructing matrices of linear forms of constant rank by focusing on vector bundles on projective spaces. It introduces important examples of classical Steiner bundles and Drezet bundles, and uses the classification of globally generated vector bundles to describe completely the indecomposable matrices of constant rank up to six.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Nicoletta Cantarini, Fabrizio Caselli, Victor Kac
Summary: In this paper, we construct a duality functor in the category of continuous representations to study the Lie superalgebra E(4, 4). By constructing a specific type of Lie conformal superalgebra, we obtain that E(4, 4) is its annihilation algebra. Furthermore, we also obtain an explicit realization of E(4, 4) on a supermanifold in the process of studying.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Rotem Assouline, Bo'az Klartag
Summary: This article studies the horocyclic Minkowski sum of two subsets in the hyperbolic plane and its properties. It proves an inequality relating the area of the subsets when they are Borel-measurable, and provides a connection to other inequalities.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Alessio Porretta
Summary: This article discusses Fokker-Planck equations driven by Levy processes in the entire Euclidean space, under the influence of confining drifts, similar to the classical Ornstein-Ulhenbeck model. A new PDE method is introduced to obtain exponential or sub-exponential decay rates of zero average solutions as time goes to infinity, under certain diffusivity conditions on the Levy process, including the fractional Laplace operator as a model example. The approach relies on long-time oscillation estimates of the adjoint problem and applies to both local and nonlocal diffusions, as well as strongly or weakly confining drifts.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Weichao Qian, Yong Li, Xue Yang
Summary: In this paper, we investigate the persistence of resonant invariant tori in Hamiltonian systems with high-order degenerate perturbation, and prove a quasiperiodic Poincare theorem under high degeneracy, answering a long-standing conjecture on the persistence of resonant invariant tori in general situations.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Julius Ross, David Witt Nystroem
Summary: This article extends Prekopa's Theorem and the Brunn-Minkowski Theorem from convexity to F-subharmonicity, and applies it to the interpolation problem of convex functions and convex sets, introducing a new notion of harmonic interpolation.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Airi Takeuchi, Lei Zhao
Summary: In this article, we explore the connection between several integrable mechanical billiards in the plane through conformal transformations. We discuss the equivalence of free billiards and central force problems, as well as the correspondence between integrable Hooke-Kepler billiards. We also investigate the integrability of Kepler billiards and Stark billiards, and the relationship between billiard systems and Euler's two-center problems.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Damiano Rossi
Summary: In this study, we prove new results in generalised Harish-Chandra theory by providing a description of the Brauer-Lusztig blocks using the p-adic cohomology of Deligne-Lusztig varieties. We then propose new conjectures for finite reductive groups by considering geometric analogues of the p-local structures. Our conjectures coincide with the counting conjectures for large primes, thanks to a connection established between p-structures and their geometric counterparts. Finally, we simplify our conjectures by reducing them to the verification of Clifford theoretic properties.
ADVANCES IN MATHEMATICS
(2024)
