Cauchy-Schwarz Research Articles

Time-dependent and Caputo derivative order-dependent quasi-uniform synchronization on fuzzy neural networks with proportional and distributed delays

This paper focuses on studying the quasi-uniform (Q-U) synchronization on fractional-order fuzzy neural networks (FOFNNs) with the proportional and distributed delays. The impacts of the Caputo derivative order, coefficients of network system and control gain constants on the synchronization performance are taken into account. Two novel time-dependent and Caputo derivative order-dependent algebraic criteria on the Q-U synchronization of FOFNNs are established by using Cauchy–Schwartz inequality, Minkowski inequality, Cp inequality and Hölder inequality, which reveal the internal influences of the time and fractional derivative order on Q-U synchronization with the derivative order interval (0,2). The simulation examples further confirm the effectiveness and practicability of the algebraic criteria in terms of the MATLAB toolbox.

A Convex Form That Is Not a Sum of Squares

Every convex homogeneous polynomial (or form) is nonnegative. Blekherman has shown that there exist convex forms that are not sums of squares via a nonconstructive argument. We provide an explicit example of a convex form of degree 4 in 272 variables that is not a sum of squares. The form is related to the Cauchy-Schwarz inequality over the octonions. The proof uses symmetry reduction together with the fact (due to Blekherman) that forms of even degree that are near-constant on the unit sphere are convex. Using this same connection, we obtain improved bounds on the approximation quality achieved by the basic sum-of-squares relaxation for optimizing quaternary quartic forms on the sphere. Funding: This work was supported by the Australian Research Council [Grant DE210101056].

The Existence of Weak Solutions for the Vorticity Equation Related to the Stratosphere in a Rotating Spherical Coordinate System

In this paper, based on the Euler equation and mass conservation equation in spherical coordinates, the ratio of the stratospheric average width to the planetary radius and the ratio of the vertical velocity to the horizontal velocity are selected as parameters under appropriate boundary conditions. We establish the approximate system using these two small parameters. In addition, we consider the time dependence of the system and establish the governing equations describing the atmospheric flow. By introducing a flow function to code the system, a nonlinear vorticity equation describing the planetary flow in the stratosphere is obtained. The governing equations describing the atmospheric flow are transformed into a second-order homogeneous linear ordinary differential equation and a Legendre’s differential equation by applying the method of separating variables based on the concepts of spherical harmonic functions and weak solutions. The Gronwall inequality and the Cauchy–Schwartz inequality are applied to priori estimates for the vorticity equation describing the stratospheric planetary flow under the appropriate initial and boundary conditions. The existence and non-uniqueness of weak solutions to the vorticity equation are obtained by using the functional analysis technique.

Statistical inference of dependent competing risks from Marshall–Olkin bivariate Burr-XII distribution under complex censoring

Dealing with competing risks is an important problem in reliability analysis and attracts much attention from scholars. It is more practical to consider competing risks with dependent failure causes in reality. In this article, statistical inference of the Marshall–Olkin bivariate Burr-XII distribution under adaptive type-II progressive hybrid censoring is discussed to show the procedure of dependent competing risks analysis in the complex data structure. The maximum likelihood estimation and lognormal approximation confidence intervals of parameters are computed. The existence and uniqueness of solutions are proved with Cauchy-Schwarz inequality. The Bayesian method with Gamma-Dirichlet prior and Metropolis-Hastings algorithm are further considered to find satisfied estimation of parameters. In addition, dynamic cumulative residual entropy is derived to quantify the information uncertainty of data. We finally compare the performance of various methods by conducting a simulation study and real data analysis.

Robust Design for RIS-Assisted Anti-Jamming Communications With Imperfect Angular Information: A Game-Theoretic Perspective

This paper utilizes a reconfigurable intelligent surface (RIS) to enhance the anti-jamming performance of wireless communications, due to its powerful capability of constructing smart and reconfigurable radio environment. In order to establish the practical interactions between the base station (BS) and the jammer, a Bayesian Stackelberg game is formulated, where the BS is the leader and the jammer acts as the follower. Specifically, with the help of a RIS-assisted transmitter, the BS attempts to reliably convey information to users with maximum utilities, whereas the smart jammer tries to interfere the signal reception of users with desired energy efficiency (EE) threshold. Since the BS and the jammer are not cooperative parties, the practical assumption that neither side can obtain the other's strategies is adopted in the proposed game, and the angular information based imperfect channel state information (CSI) is also considered. After tackling the practical assumption by using Cauchy-Schwarz inequality and the imperfect angular information by using the discretization method, the closed-form solution of both sides can be obtained via the duality optimization theory, which constitutes the unique Stackelberg equilibrium (SE). Numerical results demonstrate the superiority and validity of our proposed robust schemes over the existing approaches.

The Schwartz Inequality and the Schwartz Formula for A-Analytic Functions

In this paper, we study A-analytic functions. We provide main fundamental theorems of the theory of A-analytic functions and prove analogs of the Schwartz inequality, the Schwartz formula, and the Poisson formula for A-analytic functions.

A note on the strong Schwarz inequality in Hilbert $A$-modules

A note on the strong Schwarz inequality in Hilbert $A$-modules

On a mixed problem for Hilfer type differential equation of higher order

The study considers the solvability of a mixed problem for a Hilfer type partial differential equation of the even order with initial value conditions and small positive parameters in mixed derivatives in threedimensional domain. It studies the solution to this fractional differential equation of higher order in the class of regular functions. The case, when the order of fractional operator is 1 < α < 2, is examined. During this study the authors use the Fourier series method and obtain a countable system of ordinary differential equations. The initial value problem is integrated as an ordinary differential equation and the integrated constants find by the aid of given initial value conditions. Using the Cauchy–Schwarz inequality and the Bessel inequality, it is proved the absolute and uniform convergence of the obtained Fourier series. The stability of the solution to the mixed problem on the given functions is studied.

Quantum Thermodynamic Uncertainties in Nonequilibrium Systems from Robertson-Schrödinger Relations.

Thermodynamic uncertainty principles make up one of the few rare anchors in the largely uncharted waters of nonequilibrium systems, the fluctuation theorems being the more familiar. In this work we aim to trace the uncertainties of thermodynamic quantities in nonequilibrium systems to their quantum origins, namely, to the quantum uncertainty principles. Our results enable us to make this categorical statement: For Gaussian systems, thermodynamic functions are functionals of the Robertson-Schrödinger uncertainty function, which is always non-negative for quantum systems, but not necessarily so for classical systems. Here, quantum refers to noncommutativity of the canonical operator pairs. From the nonequilibrium free energy, we succeeded in deriving several inequalities between certain thermodynamic quantities. They assume the same forms as those in conventional thermodynamics, but these are nonequilibrium in nature and they hold for all times and at strong coupling. In addition we show that a fluctuation-dissipation inequality exists at all times in the nonequilibrium dynamics of the system. For nonequilibrium systems which relax to an equilibrium state at late times, this fluctuation-dissipation inequality leads to the Robertson-Schrödinger uncertainty principle with the help of the Cauchy-Schwarz inequality. This work provides the microscopic quantum basis to certain important thermodynamic properties of macroscopic nonequilibrium systems.

A note on the Cauchy–Schwarz inequality for expectations

In this short communication we remark that the well-known Cauchy–Schwarz inequality for expectations of random variables is a consequence of Jensen’s inequality, which does not seem to have appeared previously in the literature.

Some new Gaussian product inequalities

The Gaussian product inequality is a long-standing conjecture. In this paper, we investigate the three-dimensional inequality E[X12X22m2X32m3]≥E[X12]E[X22m2]E[X32m3] for any centered Gaussian random vector (X1,X2,X3) and m2,m3∈N. First, we show that this inequality is implied by a combinatorial inequality. The combinatorial inequality can be verified directly for small values of m2 and arbitrary m3. Hence the corresponding cases of the three-dimensional inequality are proved. Second, we show that the three-dimensional inequality is equivalent to an improved Cauchy-Schwarz inequality. This observation leads us to derive some novel moment inequalities for bivariate Gaussian random variables.

Quantum-correlated photons generated by nonlocal electron transport

Since the realization of high-quality microwave cavities coupled to quantum dots, one can envisage the possibility to investigate the coherent interaction of light and matter in semiconductor quantum devices. Here we study a parallel double quantum dot device operating as single-electron splitter interferometer, with each dot coupled to a local photon cavity. We explore how quantum correlation and entanglement between the two separated cavities are generated by the coherent transport of a single electron passing simultaneously through the two different dots. We calculate the covariance of the cavity occupations by use of a diagrammatic perturbative expansion based on Keldysh Green's functions to the fourth order in the dot-cavity interaction strength, taking into account vertex diagrams. In this way, we demonstrate the creation of entanglement by showing that the classical Cauchy-Schwarz inequality is violated if the energy levels of the two dots are almost degenerate. For large level detunings or a single dot coupled to two cavities, we show that the inequality is not violated.

Photon-pair generation from a chip-scale Cs atomic vapor cell.

The realization of a narrowband photonic quantum source based on an atomic device is considered essential in the practical development of photonic quantum information science and technology. In this study, we present the first step toward the development of a photon-pair source based on a microfabricated Cs atomic vapor cell. Time-correlated photon pairs from the millimeter-scale Cs vapor cell are emitted via the spontaneous four-wave mixing process of the cascade-type 6S1/2-6P3/2-8S1/2 transition of 133Cs. The maximum normalized cross-correlation value between the signal and idler photons is measured as 622(8) under a weak pump power of 10 µ;W. Our photon source violates the Cauchy-Schwartz inequality by a factor of >105. We believe that our approach has very important applications in the context of realizing practical scalable quantum networks based on atom-photon interactions.

Measuring linear correlation between random vectors

We introduce a new scalar coefficient to measure linear correlation between random vectors which preserves all the relevant properties of Pearson’s correlation in arbitrarily large dimensions. The new measure and its bounds are derived from a mass transportation approach in which the expected inner product of two random vectors is taken as a measure of their covariance and then standardized by the maximal attainable value given their marginal covariance matrices. The new correlation is maximized when the average squared Euclidean distance between the random vectors is minimal and attains value one when, additionally, it is possible to establish an affine relationship between the vectors. In several simulative studies we show the limiting distribution of the empirical estimator of the newly defined index and of the corresponding rank correlation.A comparative study based on financial data shows that our proposed correlation, though derived from a novel approach, behaves similarly to some of the multivariate dependence notions recently introduced in the literature.Throughout the paper, we also give some auxiliary results of independent interest in matrix analysis and mass transportation theory, including an improvement to the Cauchy–Schwarz inequality for positive definite covariance matrices.

Further Study on C-Eigenvalue Inclusion Intervals for Piezoelectric Tensors

The C-eigenpair of piezoelectric tensors finds applications in the area of the piezoelectric effect and converse piezoelectric effect. In this paper, we provide some characterizations of C-eigenvectors by exploring the structure of piezoelectric tensors, and establish sharp C-eigenvalue inclusion intervals via Cauchy–Schwartz inequality. Further, we propose the lower and upper bounds of the largest C-eigenvalue and evaluate the efficiency of the best rank-one approximation of piezoelectric tensors. Numerical examples are proposed to verify the efficiency of the obtained results.

An Empirical Examination of VIX Market Fluctuations

Abstract This study investigates the impacts of the Standard & Poor’s (S&P) 500 Index, the NASDAQ Volatility Index (VXN), the spot prices of gold and silver, and the U.S. Dollar Index (DXY) on the CBOE Volatility Index (VIX). Specifically, the proposed arbitrage pricing theory (APT) model, programmed by Python, performs multi-factor regression analysis that analyzes the degree of impact measured by the beta coefficient of systematic risks on each factor. Our samples include daily transaction data from 2007 to 2018, divided into three periods to assess the VIX’s impact for each sub-period. The three periods are entitled as the “Global Financial Crisis Period” (2007–2009), the “European Debt Crisis Period” (2010–2012), and the “Follow-up Period” (2013–2018). Empirical results show that the major factors are the S&P 500 index and the VXN, with the DXY being a minor factor. Moreover, both gold and silver spot prices have a significant impact on the VIX. Keywords: VIX, Global financial crisis, European debt crisis, Systematic risk, Arbitrage pricing theory, Multi-factor regression model, Schwarz's inequality, Triangle inequality.

Room-temperature biphoton source with a spectral brightness near the ultimate limit

Biphotons, generated from a hot atomic vapor via the process of spontaneous four-wave mixing (SFWM), have the following merits: Stable and tunable frequencies as well as linewidth. Such merits are very useful in the application of long-distance quantum communication. However, the hot-atom SFWM biphoton sources previously had far lower values of generation rate per linewidth, i.e., spectral brightness, as compared with the sources of biphotons generated by the spontaneous parametric down-conversion process. Here we report a hot-atom SFWM source of biphotons with a linewidth of 960 kHz and a generation rate of $3.7\ifmmode\times\else\texttimes\fi{}{10}^{5}$ pairs/s. The high generation rate, together with the narrow linewidth, results in a spectral brightness of $3.8\ifmmode\times\else\texttimes\fi{}{10}^{5}$ pairs/s/MHz, which is 17 times the previous best result with atomic vapors and also better than all known results with all kinds of media. The all-co-propagating scheme together with a large optical depth (OD) of the atomic vapor is the key improvement, enabling the achieved spectral brightness to be about one quarter of the ultimate limit. Furthermore, this biphoton source had a signal-to-background ratio (SBR) of 2.7, which violated the Cauchy-Schwartz inequality for classical light by about 3.6-fold. Although an increasing spectral brightness usually leads to a decreasing SBR, our systematic study indicates that both of the present spectral brightness and SBR can be enhanced by further increasing the OD. This work demonstrates a significant advancement and provides useful knowledge in quantum technology using photons.

Ulam–Hyers Stability for a Class of Caputo-Type Fractional Stochastic System with Delays

In this study, we explore the Caputo-type fractional stochastic differential equations (FSDEs) with delays. Firstly, the existence and uniqueness of the solutions for the considered system are obtained under the non-Lipschitz condition by using the Carath e ´ odory approximation method with the aid of Fatous Lemma, It o ^ isometry, Cauchy–Schwarz inequality, and Gronwall’s inequality. Secondly, we study the Ulam–Hyers (U–H) stability of the addressed equations with the aid of the Cauchy–Schwarz inequality and Gronwall’s inequality. Finally, an example with numerical simulation is presented to support the validity of our theory.

Nonclassical correlated deterministic single-photon pairs for a trapped atom in bimodal cavities

Single photons and single-photon pairs, inherently nonclassical in their nature, are fundamental elements of quantum sciences and technologies. Here, we propose to realize the nonclassical correlated deterministic photon pairs at the single-photon level for a single atom trapped in bimodal cavities. It is shown that the photon emissions for bimodal cavities exhibit a high single photon purity and relatively large intracavity photon numbers, which are ascribed to the combination of optical Stark shift enhanced energy-spectrum anharmonicity and quantum interference suppressed two-photon excitation. Our scheme generates photon-photon pairs with strong photon blockade for cavity modes via constructing a highly tunable cavity-enhanced deterministic parametric down-conversion process. Furthermore, we show that the cross-correlation function between the two cavity modes vastly violates the Cauchy-Schwarz inequality of the classical boundary, which unambiguously demonstrates the nonclassicality of the single-photon pairs. Our result reveals a prominent strategy to generate the high-quality two-mode single-photon sources with strong nonclassical correlation, which could provide versatile applications in distribution of quantum networks, long distance quantum communication, and fundamental tests of quantum physics.

An experimental setup to generate narrowband bi-photons via four-wave mixing in cold atoms

We present our recently-built experimental setup designed to generate near-infrared and narrow-band correlated photon pairs by inducingfour-wave mixing in a cold gas of 87Rb atoms confined in a magneto-optical trap. The experimental setup and its automation and control approach are described in detail. A characterization of the optical density of the atomic ensemble as well as the basic statistical measurements of the generated light are reported. The non-classical nature of the photons pairs is confirmed by observing a violation of Cauchy-Schwarz inequality by a factor of 5.6 × 10 5 in a Hanbury Brown – Twiss interferometer. A 1/e coherence time for the heralded, idler photons of 4.4 ± 0.1 ns is estimated from our observations. We are able to achieve a value of 104 s−1pair-detection-rate, which results in a spectral brightness of 280 (MHz s)−1. The combination of high brightness and narrow-band spectrum makes this photon-pair source a viable tool in fundamental studies of quantum states and opens the door to use them in quantum technologies.