Cauchy-Schwarz Research Articles

Sampled-data stabilization analysis of neural-network-based control systems: A discontinuous bilateral looped-functional approach

This work is engaged in investigating the problem of stabilization of neural-network-based control systems (NNBCSs) via sampled-data (SD) control. Firstly, a time-dependent discontinuous bilateral looped-functional (DBLF) approach is proposed by fully considering the sampling state information, which relaxes the restriction of monotonicity. Then, a new lemma regarding sampled-data-based Lyapunov stability result is established for nonlinear systems, which is strictly proved by the Cauchy-Schwarz inequality. Moreover, the proposed lemma improves the conventional Lyapunov theorem and weakens the asymptotic stability conditions of continuous-time systems into discrete conditional expression. In addition, a SD three-layer fully connected feed-forward neural network (TLFCFFNN) controller is considered to reach the maximum interval of sampling. Finally, two numerical examples are presented to illustrate the effectiveness of the theoretical results.

Positivity and Schwarz inequality for Banach *-algebras

Positivity and Schwarz inequality for Banach *-algebras

Hyperbolic geometry of shapes of convex bodies

We use the intrinsic area to define a distance on the space of homothety classes of convex bodies in the $n$-dimensional Euclidean space, which makes it isometric to a convex subset of the infinite dimensional hyperbolic space. The ambient Lorentzian structure is an extension of the intrinsic area form of convex bodies, and Alexandrov–Fenchel inequality is interpreted as the Lorentzian reversed Cauchy–Schwarz inequality. We deduce that the space of similarity classes of convex bodies has a proper geodesic distance with curvature bounded from below by $-1$ (in the sense of Alexandrov). In dimension 3, this space is homeomorphic to the space of distances with non-negative curvature on the 2-sphere, and this latter space contains the space of flat metrics on the 2-sphere considered by W. P. Thurston. Both Thurston’s and the area distances rely on the area form. So the latter may be considered as a generalization of the “real part” of Thurston’s construction.

Lyapunov‐based iterative learning feedback control design for parabolic MIMO PDEs with time‐varying delays

This paper presents an iterative learning control approach to achieve precise tracking control for a class of repeatable parabolic multiple-input–multiple-output (MIMO) partial differential equations (PDEs) with time-varying delays over a finite time interval. Feedback control is utilised in the iterative learning control system to improve the convergence speed of the repeatable system. A Lyapunov–Krasovskii functional is constructed to deal with the time-varying delay problem caused by the model uncertainty of MIMO PDEs. Collocated piecewise control and piecewise measurement is considered based on the distributions of actuators and sensors to generate multiple inputs and multiple outputs. By using the mean value theorem for integrals, variants of the Poincaré–Wirtinger inequality in the 1D space, Cauchy–Schwartz inequality and Gronwall–Bellman lemma, sufficient conditions that guarantee the convergence of the iterative learning feedback control system are presented. Numerical simulation experiments and comparisons verify the effectiveness and advantages of the proposed method.

Less Mundane Applications of the Most Mundane Functions

Summary This note shows that the mere notion of the gradient of a linear function can lead to conceptually simple solutions to some familiar problems, such as finding certain distances, proving the Cauchy-Schwartz inequality, and visualizing the GM-AM inequality.

Observation of Nonclassical Correlations in Biphotons Generated from an Ensemble of Pure Two-Level Atoms.

We report the experimental verification of nonclassical correlations for an unfiltered spontaneous four-wave mixing process in an ensemble of cold two-level atoms, confirming theoretical predictions by Du etal. in 2007 for the violation of a Cauchy-Schwarz inequality in the system, and obtaining R=(1.98±0.03)≰1. Quantum correlations are observed in a nanoseconds timescale in the interference between the central exciting frequency and sidebands dislocated by the detuning to the atomic resonance. They prevail over the noise background coming from Rayleigh scattering from the same optical transition. These correlations are fragile with respect to processes that disturb the phase of the atomic excitation, but are robust to variations in number of atoms and to increasing light intensities.

Practical Verification of Quantum Properties in Quantum-Approximate-Optimization Runs

In order to assess whether quantum resources can provide an advantage over classical computation, it is necessary to characterize and benchmark the non-classical properties of quantum algorithms in a practical manner. In this paper, we show that using measurements in no more than 3 out of the possible $3^N$ bases, one can not only reconstruct the single-qubit reduced density matrices and measure the ability to create coherent superpositions, but also possibly verify entanglement across all $N$ qubits participating in the algorithm. We introduce a family of generalized Bell-type observables for which we establish an upper bound to the expectation values in fully separable states by proving a generalization of the Cauchy-Schwarz inequality, which may serve of independent interest. We demonstrate that a subset of such observables can serve as entanglement witnesses for QAOA-MaxCut states, and further argue that they are especially well tailored for this purpose by defining and computing an entanglement potency metric on witnesses. A subset of these observables also certify, in a weaker sense, the entanglement in GHZ states, which share the $\mathbb{Z}_2$ symmetry of QAOA-MaxCut. The construction of such witnesses follows directly from the cost Hamiltonian to be optimized, and not through the standard technique of using the projector of the state being certified. It may thus provide insights to construct similar witnesses for other variational algorithms prevalent in the NISQ era. We demonstrate our ideas with proof-of-concept experiments on the Rigetti Aspen-9 chip for ansatze containing up to 24 qubits.

Cardiac electro-mechanical activity in a deforming human cardiac tissue: modeling, existence-uniqueness, finite element computation and application to multiple ischemic disease.

In this study, the cardiac electro-mechanical model in a deforming domain is taken with the addition of mechanical feedback and stretch-activated channel current coupled with the ten Tusscher human ventricular cell level model that results in a coupled PDE-ODE system. The existence and uniqueness of such a coupled system in a deforming domain is proved. At first, the existence of a solution is proved in the deformed domain. The local existence of the solution is proved using the regularization and the Faedo-Galerkin technique. Then, the global existence is proved using the energy estimates in appropriate Banach spaces, Gronwall lemma, and the compactness procedure. The existence of the solution in an undeformed domain is proved using the lower semi-continuity of the norms. Uniqueness is proved using Young's inequality, Gronwall lemma, and the Cauchy-Schwartz inequality. For the application purpose, this model is applied to understand the electro-mechanical activity in ischemic cardiac tissue. It also takes care of the development of active tension, conductive, convective, and ionic feedback. The Second Piola-Kirchoff stress tensor arising in Lagrangian mapping between reference and moving frames is taken as a combination of active, passive, and volumetric components. We investigated the effect of varying strength of hyperkalemia and hypoxia, in the ischemic subregions of human cardiac tissue with local multiple ischemic subregions, on the electro-mechanical activity of healthy and ischemic zones. This system is solved numerically using the [Formula: see text] finite element method in space and the implicit-explicit Euler method in time. Discontinuities arising with the modeled multiple ischemic regions are treated to the desired order of accuracy by a simple regularization technique using the interpolating polynomials. We examined the cardiac electro-mechanical activity for several cases in multiple hyperkalemic and hypoxic human cardiac tissue. We concluded that local multiple ischemic subregions severely affect the cardiac electro-mechanical activity more, in terms of action potential (v) and mechanical parameters, intracellular calcium ion concentration [Formula: see text], active tension ([Formula: see text]), stretch ([Formula: see text]) and stretch rate ([Formula: see text]), of a healthy cell in its vicinity, compared to a single Hyperkalemic or Hypoxic subregion. The four moderate hypoxically generated ischemic subregions affect the waveform of the stretch along the fiber and the stretch rate more than a single severe ischemic subregion.

K Positivity and Schwarz inequality for two linear maps

k Positivity and Schwarz inequality for two linear maps

Improved unitary uncertainty relations

We derive strong variance-based uncertainty relations for arbitrary two and more unitary operators by re-examining the mathematical foundation of the uncertainty relation. This is achieved by strengthening the celebrated Cauchy–Schwarz inequality using a method of brackets and convex functions. The unitary uncertainty relations outperform several strong unitary uncertainty relations, notably better than some recent best lower bounds such as [Phys. Rev. Lett. 120, 230402 (2018)] and [Phys. Rev. A. 100, 022116 (2019)].

Buzano, Kreĭn and Cauchy-Schwarz inequalities

The Cauchy-Schwarz, Buzano and Kre\u{\i}n inequalities are three inequalities about inner product. The main goal of this article is to present refinements of Buzano and Cauchy-Schwarz inequalities, and to present a new proof of a refined version of a Kre\u{\i}n-type inequality. Applications that include Buzano-type inequalities for certain operators, operator norm and numerical radius inequalities of Hilbert space operators will be presented.

Impulsive conformable fractional stochastic differential equations with Poisson jumps

<p style='text-indent:20px;'>In this article, periodic averaging method for impulsive conformable fractional stochastic differential equations with Poisson jumps are discussed. By using stochastic analysis, fractional calculus, Doob's martingale inequality and Cauchy-Schwarz inequality, we show that the solution of the conformable fractional impulsive stochastic differential equations with Poisson jumps converges to the corresponding averaged conformable fractional stochastic differential equations with Poisson jumps and without impulses.</p>

A Cubic Convolution Interpolation-Based Chroma Subsampling Method for Bayer and RGBW CFA Raw Images

The Bayer and RGBW color filter array (CFA) raw images, denoted by <inline-formula> <tex-math notation="LaTeX">$I^{Bayer}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$I^{RGB}$ </tex-math></inline-formula>, respectively, have been widely used in consumer markets. In the demosaicking-first compression scheme, chroma subsampling is necessary prior to compressing <inline-formula> <tex-math notation="LaTeX">$I^{RGB}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$I^{RGBW}$ </tex-math></inline-formula>. Several linear interpolation-based chroma subsampling methods have been developed for <inline-formula> <tex-math notation="LaTeX">$I^{Bayer}$ </tex-math></inline-formula>, but no nonlinear interpolation-based chroma subsampling methods have targeted the above two CFA image types simultaneously. In this paper, we first propose a nonlinear interpolation-based, namely the cubic convolution interpolation-based (CCI-based), <inline-formula> <tex-math notation="LaTeX">$2\times 2$ </tex-math></inline-formula> block-distortion function for each <inline-formula> <tex-math notation="LaTeX">$2\times 2$ </tex-math></inline-formula> CFA block <inline-formula> <tex-math notation="LaTeX">$B^{CbCr}$ </tex-math></inline-formula>. Next, using the Cauchy-Schwarz inequality, we prove that the proposed block-distortion function is a convex function in the real domain, which further serves as the base of the initial subsampled chroma solution. Then, a CCI-based iterative method is proposed to improve the initial subsampled chroma solution. The results of comprehensive experimental tests using Bayer and RGBW CFA images created from the three RGB full-color datasets, namely IMAX, SCI (screen content images), and CI (classical images), demonstrate that on the Versatile Video Coding (VVC) platform VTM-11.0, the proposed method achieves substantial quality enhancement and quality-bitrate tradeoff merits of the reconstructed Bayer and RGBW CFA images compared with existing chroma subsampling methods.

Performance Analysis of SWIPT System with Imperfect Channel State Information

This paper studies the system performance of the multiple-input-single-output (MISO) system using simultaneous wireless information and power transfer (SWIPT) technology under imperfect channel state information. A power split strategy is used in the receiver, which divides the received radio frequency (RF) signal into two parts, one part is directly used for signal decoding, and the other part is used as energy harvesting. This paper studies the optimization of system channel capacity under the constraints of harvested energy and the transmission rate. The resulting optimization problem is a two-layer non-convex optimization one, which is simplified by Cauchy-Schwarz inequality. Through the linear search of the energy division ratio in its value range, it becomes a one-dimensional optimization problem only related to the transmission power, which can be solved by CVX. The simulation results verify the effectiveness of the algorithm.

Resolvable Group Target Tracking via Multi-Bernoulli Filter and Its Application to Sensor Control Scenario

In this paper, we address the problem of resolvable group target tracking (RGTT) and the application of RGTT in the sensor control scenarios. The aim of our approach is to further improve the tracking performance of multi-Bernoulli (MB) filter by exploiting the group structure information. Firstly, a single-target state transition function (SSTF) is derived from the direct integral solution of the single-target state stochastic differential equation (SDE), which is used to revise the prediction process of the standard MB filter. Then, combing the Gaussian mixture (GM) implementation, the prediction process of the proposed SSTF-based MB (SSTF-MB) filter is detailed. Next, considering the benefits and importance of group structure in the sensor control scenarios, the proposed SSTF-MB filter is applied into a mobile sensor platform whose control actions (e.g., motion direction) are uncertain. Specifically, the proposed SSTF-MB filter is merged into a sensor control strategy where the Cauchy-Schwarz (CS) divergence is selected as the objective function to compute the discrepancy between two first-order moment approximations of MBs. Finally, the efficiency and performance of the proposed SSTF-MB filter and its application to sensor control scenario is well demonstrated in the simulation experiments.

Fin-TS and Fix-TS on fractional quaternion delayed neural networks with uncertainty via establishing a new Caputo derivative inequality approach

This paper investigates the finite time synchronization (Fin-TS) and fixed time synchronization (Fix-TS) issues on Caputo quaternion delayed neural networks (QDNNs) with uncertainty. A new Caputo fractional differential inequality is constructed, then Fix-TS settling time of the positive definite function is estimated, which is very convenient to derive Fix-TS condition to Caputo QDNNs. By designing the appropriate self feedback and adaptive controllers, the algebraic discriminant conditions to achieve Fin-TS and Fix-TS on Caputo QDNNs are proposed based on quaternion direct method, Lyapunov stability theory, extended Cauchy Schwartz inequality, Jensen inequality. Finally, the correctness and validity of the presented results under the different orders are verified by two numerical examples.

On incidence bounds with Möbius hyperbolae in positive characteristic

We prove new incidence bounds between a plane point set, which is a Cartesian product, and a set of translates H of the hyperbola xy=λ≠0, over a field of asymptotically large positive characteristic p. They improve recent bounds by Shkredov, which are based on using explicit incidence estimates in the early terminated procedure of repeated applications of the Cauchy-Schwarz inequality, underlying many qualitative results related to growth and expansion in groups. The improvement – both quantitative, plus we are able to deal with a general H, rather than a Cartesian product – is mostly due to a non-trivial “intermediate” bound on the number of k-rich Möbius hyperbolae in positive characteristic. In addition, we make an observation that a certain energy-type quantity in the context of H can be bounded via the L2-moment of the Minkowski distance in H and can therefore fetch the corresponding estimates apropos of the Erdős distinct distance problem.

Some Properties of Cone Inner Product Spaces over Banach Algebra

The aim of this paper is to introduce a concept of a cone inner product space over Banach algebras. This is done by replacing the co-domain of the classical inner product space by an ordered Banach algebra. Some properties such as Cauchy-Schwarz inequality, parallelogram identity and Pythagoras theorem are established in this setting. Similarly, the notion of cone normed algebra was introduced. Some illustrative examples are given to support our findings.

Turán‐type inequalities for the supertrigonometric functions

This paper is devoted to the study of Turán‐type inequalities for some well‐known special functions such as supersine and supercosine which are derived by using a new form of the Cauchy–Bunyakovsky–Schwarz inequality.

Joint optimization scheme for intelligent reflecting surface aided multi‐relay networks

The latest research on hybrid networks of decode-and-forward (DF) relay and intelligent reflecting surface (IRS) is limited to just a single relay. To harvest multiple relay gains, this paper proposes a hybrid communication networks that combines multiple DF relays with an IRS. The authors formulate and address the joint optimization problem to maximize the end-to-end transmission rate by jointly optimizing the reflect beam forming, the best relay selection and the power allocation, subject to total transmit power constraint. Moreover, to better exploit the degrees of freedom, it is proposed that the source also transmits signals in the second time slot. While the optimization problem is difficult to solve, the authors propose some efficient approaches to make the problem tractable. First, power allocation problem is simplified by applying Cauchy-Schwarz inequality and introducing equivalent channel gain, second, non-convex rank-one constraint is overcome by utilizing semidefinite relaxation (SDR) approach, then the slacked non-convex constraint is overcome by applying first-order Taylor expansion approximation. The simulation results demonstrate that our proposed joint optimization scheme for hybrid networks of multiple DF relays with an IRS has significant performance improvement over the other optimization schemes, such as the optimization scheme for hybrid networks of single DF relay and an IRS.