Cauchy-Schwarz Research Articles

C*-module operators which satisfy the generalized Cauchy–Schwarz type inequality

Let L ( H ) denote the C ∗ -algebra of adjointable operators on a Hilbert C ∗ -module H . In this paper, we introduce the generalized Cauchy–Schwarz inequality for operators in L ( H ) . More precisely, an operator A ∈ L ( H ) is said to satisfy the generalized Cauchy–Schwarz inequality if there exists ν ∈ ( 0 , 1 ) such that ‖ ⟨ A x , y ⟩ ‖ ≤ ( ‖ A x ‖ ‖ y ‖ ) ν ( ‖ A y ‖ ‖ x ‖ ) 1 − ν ( x , y ∈ H ) . We investigate various properties of operators which satisfy the generalized Cauchy–Schwarz inequality. In particular, we prove that if A satisfies the generalized Cauchy–Schwarz inequality such that A has the polar decomposition, then A is paranormal. In addition, we show that if for A the equality holds in the generalized Cauchy–Schwarz inequality, then A is cohyponormal. Among other things, when A has the polar decomposition, we prove that A is semi-hyponormal if and only if ‖ ⟨ A x , y ⟩ ‖ ≤ ‖ | A | 1 / 2 x ‖ ‖ | A | 1 / 2 y ‖ for all x , y ∈ H .

Edge Computing Offloading Method Based on Deep Reinforcement Learning for Gas Pipeline Leak Detection

Traditional gas pipeline leak detection methods require task offload decisions in the cloud, which has low real time performance. The emergence of edge computing provides a solution by enabling offload decisions directly at the edge server, improving real-time performance; however, energy is the new bottleneck. Therefore, focusing on the gas transmission pipeline leakage detection scenario in real time, a novel detection algorithm that combines the benefits of both the heuristic algorithm and the advantage actor critic (AAC) algorithm is proposed in this paper. It aims at optimization with the goal of real-time guarantee of pipeline mapping analysis tasks and maximizing the survival time of portable gas leak detectors. Since the computing power of portable detection devices is limited, as they are powered by batteries, the main problem to be solved in this study is how to take into account the node energy overhead while guaranteeing the system performance requirements. By introducing the idea of edge computing and taking the mapping relationship between resource occupation and energy consumption as the starting point, the optimization model is established, with the goal to optimize the total system cost (TSC). This is composed of the node’s transmission energy consumption, local computing energy consumption, and residual electricity weight. In order to minimize TSC, the algorithm uses the AAC network to make task scheduling decisions and judge whether tasks need to be offloaded, and uses heuristic strategies and the Cauchy–Buniakowsky–Schwarz inequality to determine the allocation of communication resources. The experiments show that the proposed algorithm in this paper can meet the real-time requirements of the detector, and achieve lower energy consumption. The proposed algorithm saves approximately 56% of the system energy compared to the Deep Q Network (DQN) algorithm. Compared with the artificial gorilla troops Optimizer (GTO), the black widow optimization algorithm (BWOA), the exploration-enhanced grey wolf optimizer (EEGWO), the African vultures optimization algorithm (AVOA), and the driving training-based optimization (DTBO), it saves 21%, 38%, 30%, 31%, and 44% of energy consumption, respectively. Compared to the fully local computing and fully offloading algorithms, it saves 50% and 30%, respectively. Meanwhile, the task completion rate of this algorithm reaches 96.3%, which is the best real-time performance among these algorithms.

Bounds for the Electrical Resistance for Non-homogeneous Two-dimensional Conducting Body

A mathematical model is developed to determine the steady-state current flow through in non-homogeneous isotropic conductor whose shape is a two-dimensional hollow domain. The corresponding electric boundary value problem is formulated using Maxwell's theory of electricity. The determination of the two-dimensional motion of charges is based on the concept of the electrical conductance. The Cauchy-Schwarz inequality is used to get the lower and upper bounds for the electrical conductance. The derived upper and lower bound formulae are illustrated by numerical examples.

Temporally ultralong biphotons with a linewidth of 50 kHz

We report the generation of biphotons, with a temporal full width at the half maximum (FWHM) of 13.4 ± 0.3 µs and a spectral FWHM of 50 ± 1 kHz, via the process of spontaneous four-wave mixing with laser-cooled atoms. The temporal width is the longest, and the spectral linewidth is the narrowest to date. This is also the first biphoton result that obtains a linewidth below 100 kHz, reaching a new milestone. The very long biphoton wave packet has a signal-to-background ratio of 3.4, which violates the Cauchy–Schwarz inequality for classical light by 4.8 folds. Furthermore, we demonstrated a highly tunable-linewidth biphoton source and showed that while the biphoton source’s temporal and spectral width were controllably varied by about 24 folds, its generation rate only changed by less than 15%. A spectral brightness or generation rate per pump power per linewidth of 1.2× 106 pairs/(s mW MHz) was achieved at the temporal width of 13.4 µs. The above results were made possible by the low decoherence rate and high optical depth of the experimental system, as well as a novel scheme of classical fields’ and biphotons’ propagation directions in the experiment. This work has demonstrated a high-efficiency ultranarrow-linewidth biphoton source and has made substantial advancements in quantum technology utilizing heralded single photons.

ESG Scores - Is it the new way to build a European portfolio?

Abstract The study makes a comparison between the performance of equity portfolios characterized by high ESG score stocks and portfolios with low ESG score stocks. In particular, we analyze three ESG scores: MSCI ESG Rating, Sustainalytics scores and S&P DJI/Robeco ESG Scores, by examining the European stock market in two periods: medium/long term (five years) and short-term (one year). First of all, we associate each component of the index in relation to its MSCI ESG Rating, Sustainalytics score and S&P DJI/Robeco ESG Scores. We build two portfolios: • First quartile portfolio 1Q (according to MSCI; Sustainalytics; and S&P DJI/Robeco ESG Scores), including securities of companies with the highest ESG score, based on ESG best-in-class screening strategy. • Fourth quartile portfolio 4Q (according to MSCI; Sustainalytics; and S&P DJI/Robeco ESG Scores), including securities of companies with the lowest ESG scores. We aim to answer the following questions: a) do portfolios with higher ESG scores stocks lead to better performances than those including stocks with low ESG scores? b), Are there some sectors that drive the performance within the sector breakdown? Results show a divergence between the composition of the first quartile, whereas there is more homogeneity in the fourth quartile. JEL classification numbers: G10, G11, F65 Keywords: ESG factor; asset management; sustainability finance. Schwarz's inequality, Triangle inequality.

Bounds for the electrical resistance for non-homogeneous conducting body

Abstract A mathematical model is developed to determine the steady-state electric current flow through in non-homogeneous isotropic conductor whose shape has a three-dimensional hollow body. The equations of the Maxwell's theory of electric current flow in a non-homogeneous isotropic solid conductor body are used to formulate the corresponding electric boundary value problem. The determination of the steady motion of charges is based on the concept of the electrical conductance. The derivation of the upper and lower bound formulae for the electrical conductance is based on Cauchy-Schwarz inequality. Two numerical examples illustrate the applications of the derived upper and lower bound formulae.

On Deriving the Least Squares Estimates in Introductory Regression Courses

Introductory regression books typically begin their derivation of the least squares matrix estimation formula by considering the simple linear regression model. We suggest beginning with the zero-intercept model which has advantages. We provide two examples of this approach, one of which is a new, non-calculus derivation using the Cauchy-Schwarz inequality.

An improved multigrid solver for the p-hierarchical basis finite element method using a space decomposition smoother

An improved hierarchical basis finite element multigrid solver for the p -version is presented and analysed. The proposed solver combines the features of p -hierarchical bases, multigrid and space decomposition effectively. Unlike the existing p -version multigrid solvers that perform smoothing only on a subspace of the multigrid space, we adopt a global smoothing strategy at each multigrid level via space decomposition, in a multiplicative Schwarz manner. The finite element subspaces are constructed on locally uniform p -refinements, yielding efficient implementations on p -hierarchical bases. Moreover, the error operator has been analysed and the p -version strengthened Cauchy Buniakowskii Schwarz inequality constants have been established yielding the first convergence estimates. Furthermore, numerical examples are given confirming the effectiveness of the solver. The numerical experiments show that the proposed solver outperforms the benchmark p -version space decomposition solver in terms of the convergence rates as well as the computational complexity.

Quantum information with top quarks in QCD

Top quarks represent unique high-energy systems since their spin correlations can be measured, thus allowing to study fundamental aspects of quantum mechanics with qubits at high-energy colliders. We present here the general framework of the quantum state of a top-antitop (tt¯) quark pair produced through quantum chromodynamics (QCD) in a high-energy collider. We argue that, in general, the total quantum state that can be probed in a collider is given in terms of the production spin density matrix, which necessarily gives rise to a mixed state. We compute the quantum state of a tt¯ pair produced from the most elementary QCD processes, finding the presence of entanglement and CHSH violation in different regions of phase space. We show that any realistic hadronic production of a tt¯ pair is a statistical mixture of these elementary QCD processes. We focus on the experimentally relevant cases of proton-proton and proton-antiproton collisions, performed at the LHC and the Tevatron, analyzing the dependence of the quantum state with the energy of the collisions. We provide experimental observables for entanglement and CHSH-violation signatures. At the LHC, these signatures are given by the measurement of a single observable, which in the case of entanglement represents the violation of a Cauchy-Schwarz inequality. We extend the validity of the quantum tomography protocol for the tt¯ pair proposed in the literature to more general quantum states, and for any production mechanism. Finally, we argue that a CHSH violation measured in a collider is only a weak form of violation of Bell's theorem, necessarily containing a number of loopholes.

Local bounds for singular Brascamp–Lieb forms with cubical structure

We prove a range of \(L^p\) bounds for singular Brascamp–Lieb forms with cubical structure. We pass through sparse and local bounds, the latter proved by an iteration of Fourier expansion, telescoping, and the Cauchy–Schwarz inequality. We allow \(2^{m-1}<p\le \infty \) with m the dimension of the cube, extending an earlier result that required \(p=2^m\). The threshold \(2^{m-1}\) is sharp in our theorems.

Stratifications of the ray space of a tropical quadratic form by Cauchy-Schwartz functions

Classes of an equivalence relation on a module $V$ over a supertropical semiring, called rays, carry the underlying structure of 'supertropical trigonometry' and thereby a version of convex geometry which is compatible with quasilinearity. In this theory, the traditional Cauchy-Schwarz inequality is replaced by the CS-ratio, which gives rise to special characteristic functions, called CS-functions. These functions partite the ray space $\mathrm{Ray}(V)$ into convex sets and establish the main tool for analyzing varieties of quasilinear stars in $\mathrm{Ray}(V)$. They provide stratifications of $\mathrm{Ray}(V)$ and, therefore, a finer convex analysis that helps better understand geometric properties.

Non‐Classical Correlations between Single Photons and Magnons

AbstractA scheme is proposed to realize the non‐classical correlation between magnons and photons in a cavity optomagnonic system, which supports both photon modes and a magnon mode. Starting with the system being initially prepared in its ground state, two laser pulses successively drive corresponding optical mode. A two‐mode squeezing interaction between optical mode 1 and the magnon mode is created by the first pulse, which leads to a non‐classical correlation between photons and magnons. To verify this non‐classical correlation, the second pulse is utilized to transfer the magnon state to another optical mode, thus the correlated photon–photon pairs are generated out of the cavity. By discussing the violation of Cauchy–Schwarz inequality, based on numerical simulation, it is confirmed that non‐classical correlated photon–magnon pairs can be created in the weak coupling regime, which relaxes the requirements of experimental conditions. The result indicates that cavity optomagnonics can be a promising platform for studying magnon‐based quantum information processing.

Increasing Trends in Oceanic Surface Poleward Eddy Heat Flux Observed Over the Past Three Decades

AbstractMesoscale processes make the largest contribution to ocean variability and are significant agents in ocean heat transport. In this investigation, we provide evidence based on satellite observations for an increasing trend in surface transient eddy heat flux (EHF) over the period 1993–2020. The enhanced EHF is particularly prominent in western boundary currents and their extensions at mid‐latitudes, with increases of 20%–40% per decade. Additionally, we decompose the EHF trend into contributions from velocity variance, temperature variance and a coherence parameter using a Cauchy–Schwarz inequality. We find that each contributes to the spatial pattern of the trend in EHF, with the contribution from enhanced temperature variance dominating the global zonal mean EHF trend over the past few decades.

Optimization of adiabatic frequency conversion in an all-pass resonator

Adiabatic frequency conversion (AFC) is a promising alternative for shifting the frequency of optical signals. In this paper we present a comprehensive theoretical study of the energy efficiency of AFC in an all-pass resonator. Through the Cauchy-Schwarz inequality, we deduce the upper limit of the energy efficiency of the AFC process and analyze its dependence on the input pulse shape. For a fixed pulse shape, we discuss the dependence of the AFC efficiency on its relevant timescales. We show that maximum efficiency requires overcoupling of the resonator to a degree dependent on the input pulse duration. We compare and contrast the requirements for optimal AFC for pulsed and continuous-wave inputs.

Apodictic discourse and the Cauchy-Bunyakovsky- Schwarz inequality

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Outage Constrained Robust Beamforming Optimization for Multiuser IRS-Assisted Anti-Jamming Communications With Incomplete Information

Malicious jamming attacks have been regarded as a serious threat to Internet of Things (IoT) networks, which can significantly degrade the Quality of Service (QoS) of users. This article utilizes an intelligent reflecting surface (IRS) to enhance anti-jamming performance due to its capability in reconfiguring the wireless propagation environment via dynamically adjusting each IRS reflecting elements. To enhance the communication performance against jamming attacks, a robust beamforming optimization problem is formulated in a multiuser IRS-assisted anti-jamming communications scenario with or without imperfect jammer’s channel state information (CSI). In addition, we further consider the fact that the jammer’s transmit beamforming can not be known at BS. Specifically, with no knowledge of jammers transmit beamforming, the total transmit power minimization problems are formulated subject to the outage probability requirements of legitimate users with the jammer’s statistical CSI, and signal-to-interference-plus-noise ratio requirements of legitimate users without the jammer’s CSI, respectively. By applying the decomposition-based large deviation inequality, Bernstein-type inequality, Cauchy–Schwarz inequality, and penalty nonsmooth optimization method, we efficiently solve the initial intractable and nonconvex problems. Numerical simulations demonstrate that the proposed anti-jamming approaches achieve superior anti-jamming performance and lower power-consumption compared to the non-IRS scheme and reveal the impact of key parameters on the achievable system performance.

The Schwarz inequality and the Harnack’s Theorem For A (z) – Analytical Function

This article discusses the theory fundamental theorems of A (z) – analytic functions and establishes analogies for Schwarz inequality as well as Harnack’s theorem for A(z) – analytic functions.

Turań-type inequalities for two hypergeometric supertrigonometric functions associated with Kummer confluent hypergeometric series of first type

In the paper, on account of the Kummer confluent hypergeometric series of first type, we suggest the Turań-type inequalities for the hypergeometric supersine and supercosine functions by using the Cauchy–Bunyakovsky–Schwarz inequality.

Spatially entangled photon pairs from lithium niobate nonlocal metasurfaces.

Metasurfaces consisting of nanoscale structures are underpinning new physical principles for the creation and shaping of quantum states of light. Multiphoton states that are entangled in spatial or angular domains are an essential resource for many quantum applications; however, their production traditionally relies on bulky nonlinear crystals. We predict and demonstrate experimentally the generation of spatially entangled photon pairs through spontaneous parametric down-conversion from a metasurface incorporating a nonlinear thin film of lithium niobate covered by a silica meta-grating. We measure the correlations of photon pairs and identify their spatial antibunching through violation of the classical Cauchy-Schwarz inequality, witnessing the presence of multimode entanglement. Simultaneously, the photon-pair rate is strongly enhanced by 450 times as compared to unpatterned films because of high-quality-factor resonances. These results pave the way to miniaturization of various quantum devices by incorporating ultrathin metasurfaces functioning as room temperature sources of quantum-entangled photons.

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.