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Goong Chen

Publications and source records attributed to Goong Chen.

11 recordsLinked to original sources
arXiv2025

A new complete proof of the random Brouwer fixed point theorem and its implied consequences of unification

We first establish a general random Sperner lemma by presenting a completely new approach for the theory of $L^{0}$-simplicial subdivisions of $L^{0}$-simplexes. Based on this, we are able to achieve a new complete proof of the random Brouwer fixed theorem in random Euclidean spaces, which can provide a solid foundation for various contemporary applications of interest. Afterward, we unify the works currently available and closely related to the random Brouwer fixed theorem: we first prove that the stochastic Brouwer fixed point theorem occurring elsewhere in stochastic analysis is equivalent to a special case of our random Brouwer fixed theorem, and then prove a general random Borsuk theorem and its equivalence with the random Brouwer fixed theorem. Finally, we conclude this paper with commentaries on recent state of study of the famous Schauder conjecture.

math.FA↗
arXiv2021

The noncompact Schauder fixed point theorem in random normed modules and its applications

Motivated by the randomized version of the classical Bolzano--Weierstrass theorem, in this paper we first introduce the notion of a random sequentially compact set in a random normed module and develop the related theory systematically. From these developments, we prove the corresponding Schauder fixed point theorem: let $E$ be a random normed module and $G$ a random sequentially compact $L^0$--convex set of $E$, then every $\sigma$--stable continuous mapping from $G$ to $G$ has a fixed point, which unifies all the previous random generalizations of the Schauder fixed point theorem. As one of the applications of the theorem, we prove the existence of Nash equilibrium points in the context of conditional information. It should be pointed out that the main challenge in this paper lies in overcoming noncompactness since a random sequentially compact set is generally noncompact.

math.FA↗
arXiv2018

Light, the universe, and everything -- 12 Herculean tasks for quantum cowboys and black diamond skiers

The Winter Colloquium on the Physics of Quantum Electronics (PQE) has been a seminal force in quantum optics and related areas since 1971. It is rather mindboggling to recognize how the concepts presented at these conferences have transformed scientific understanding and human society. In January, 2017, the participants of PQE were asked to consider the equally important prospects for the future, and to formulate a set of questions representing some of the greatest aspirations in this broad field. The result is this multi-authored paper, in which many of the world's leading experts address the following fundamental questions: (1) What is the future of gravitational wave astronomy? (2) Are there new quantum phases of matter away from equilibrium that can be found and exploited - such as the time crystal? (3) Quantum theory in uncharted territory: What can we learn? (4) What are the ultimate limits for laser photon energies? (5) What are the ultimate limits to temporal, spatial, and optical resolution? (6) What novel roles will atoms play in technology? (7) What applications lie ahead for nitrogen-vacancy centers in diamond? (8) What is the future of quantum coherence, squeezing, and entanglement for enhanced superresolution and sensing? (9) How can we solve (some of) humanity's biggest problems through new quantum technologies? (10) What new understanding of materials and biological molecules will result from their dynamical characterization with free electron lasers? (11) What new technologies and fundamental discoveries might quantum optics achieve by the end of this century? (12) What novel topological structures can be created and employed in quantum optics?

quant-ph↗
arXiv2016

The virial theorem and ground state energy estimate of nonlinear Schr\"odinger equations in $\mathbb{R}^2$ with square root and saturable nonlinearities in nonlinear optics

The virial theorem is a nice property for the linear Schrodinger equation in atomic and molecular physics as it gives an elegant ratio between the kinetic and potential energies and is useful in assessing the quality of numerically computed eigenvalues. If the governing equation is a nonlinear Schrodinger equation with power-law nonlinearity, then a similar ratio can be obtained but there seems no way of getting any eigenvalue estimate. It is surprising as far as we are concerned that when the nonlinearity is either square-root or saturable nonlinearity (not a power-law), one can develop a virial theorem and eigenvalue estimate of nonlinear Schrodinger (NLS) equations in R2 with square-root and saturable nonlinearity, respectively. Furthermore, we show here that the eigenvalue estimate can be used to obtain the 2nd order term (which is of order $ln\Gamma$) of the lower bound of the ground state energy as the coefficient $\Gamma$ of the nonlinear term tends to infinity.

math.AP↗
arXiv2012

Ground state of nonlinear Schrodinger systems with saturable nonlinearity

We prove the existence of ground state in a multidimensional nonlinear Schrodinger model of paraxial beam propagation in isotropic local media with saturable nonlinearity. Such ground states exist in the form of bright counterpropagating solitons. From the proof, a general threshold condition on the beam coupling constant for the existence of such fundamental solitons follows.

math-ph↗
arXiv2005

The two electron molecular bond revisited: from Bohr orbits to two-center orbitals

In this review we first discuss extension of Bohr's 1913 molecular model and show that it corresponds to the large-D limit of a dimensional scaling (D-scaling) analysis, as developed by Herschbach and coworkers. In a separate but synergetic approach to the two-electron problem, we summarize recent advances in constructing analytical models for describing the two-electron bond. The emphasis here is not maximally attainable numerical accuracy, but beyond textbook accuracy as informed by physical insights. We demonstrate how the interplay of the cusp condition, the asymptotic condition, the electron-correlation, configuration interaction, and the exact one electron two-center orbitals, can produce energy results approaching chemical accuracy. Reviews of more traditional calculational approaches, such as Hartree-Fock, are also given. The inclusion of electron correlation via Hylleraas type functions is well known to be important, but difficult to implement for more than two electrons. The use of the D-scaled Bohr model offers the tantalizing possibility of obtaining electron correlation energy in a non-traditional way.

physics.chem-ph↗
arXiv2000

Quantum Multi-object Search Algorithm with the Availability of Partial Information

Consider the unstructured search of an unknown number l of items in a large unsorted database of size N. The multi-object quantum search algorithm consists of two parts. The first part of the algorithm is to generalize Grover's single-object search algorithm to the multi-object case and the second part is to solve a counting problem to determine l. In this paper, we study the multi-object quantum search algorithm (in continuous time), but in a more structured way by taking into account the availability of partial information. The modeling of available partial information is done simply by the combination of several prescribed, possibly overlapping, information sets with varying weights to signify the reliability of each set. The associated statistics is estimated and the algorithm efficiency and complexity are analyzed. Our analysis shows that the search algorithm described here may not be more efficient than the unstructured (generalized) multi-object Grover search if there is ``misplaced confidence''. However, if the information sets have a ``basic confidence'' property in the sense that each information set contains at least one search item, then a quadratic speedup holds on a much smaller data space, which further expedite the quantum search for the first item.

quant-ph↗
arXiv2000

The Universality of the Quantum Fourier Transform in Forming the Basis of Quantum Computing Algorithms

The quantum Fourier transform (QFT) is a powerful tool in quantum computing. The main ingredients of QFT are formed by the Walsh-Hadamard transform H and phase shifts P(.), both of which are 2x2 unitary matrices as operators on the two-dimensional 1-qubit space. In this paper, we show that H and P(.) suffice to generate the unitary group U(2) and, consequently, through controlled-U operations and their concatenations, the entire unitary group U(2^n) on n-qubits can be generated. Since any quantum computing algorithm in an n-qubit quantum computer is based on operations by matrices in U(2^n), in this sense we have the universality of the QFT.

quant-ph↗
arXiv2000

Generalization of Grover's Algorithm to Multiobject Search in Quantum Computing, Part I: Continuous Time and Discrete Time

L. K. Grover's search algorithm in quantum computing gives an optimal, quadratic speedup in the search for a single object in a large unsorted database. In this paper, we generalize Grover's algorithm in a Hilbert-space framework for both continuous and discrete time cases that isolates its geometrical essence to the case where more than one object satisfies the search criterion.

quant-ph↗
arXiv2000

Generalization of Grover's Algorithm to Multiobject Search in Quantum Computing, Part II: General Unitary Transformations

There are major advantages in a newer version of Grover's quantum algorithm utilizing a general unitary transformation in the search of a single object in a large unsorted database. In this paper, we generalize this algorithm to multiobject search. We show the techniques to achieve the reduction of the problem to one on an invariant subspace of dimension just equal to two.

quant-ph↗