Analytical Condition Research Articles

Closed-form solutions for several classes of singular integral equations with convolution and Cauchy operator

The main goal of this paper is to study the existence of solutions for several new classes of singular convolution integral equations containing the Cauchy operator in the normal type case. To investigate the solutions of such equations, we establish the Noethericity theory of solvability. By means of the properties of complex Fourier transforms, we transform these equations into Riemann boundary value problems with nodes. The analytic solutions and conditions of solvability are obtained via using Riemann–Hilbert approach. Moreover, we also discuss the asymptotic property of solutions near nodes. Thus, this paper generalizes the theories of complex analysis, functional analysis, integral equations and the classical Riemann boundary value problems.

Fractional Besov Trace/Extension-Type Inequalities via the Caffarelli–Silvestre Extension

Let $$u(\cdot ,\cdot )$$ be the Caffarelli–Silvestre extension of f. The first goal of this article is to establish the fractional trace-type inequalities involving the Caffarelli–Silvestre extension $$u(\cdot ,\cdot )$$ of f. In doing so, firstly, we establish the fractional Sobolev/logarithmic Sobolev/Hardy trace inequalities in terms of $$\nabla _{(x,t)}u(x,t).$$ Then, we prove the fractional anisotropic Sobolev/logarithmic Sobolev/Hardy trace inequalities in terms of $$ {\partial _{t} u(x,t)}$$ or $$(-\varDelta )^{-\gamma /2}u(x,t)$$ only. Moreover, based on an estimate of the Fourier transform of the Caffarelli–Silvestre extension kernel and the sharp affine weighted $$L^p$$ Sobolev inequality, we prove that the $${\dot{H}}^{-\beta /2}({\mathbb {R}}^n)$$ norm of f(x) can be controlled by the product of the weighted $$L^p$$ -affine energy and the weighted $$L^p$$ -norm of $${\partial _{t} u(x,t)}.$$ The second goal of this article is to characterize non-negative measures $$\mu $$ on $${\mathbb {R}}^{n+1}_+$$ such that the embeddings $$\begin{aligned} \Vert u(\cdot ,\cdot )\Vert _{L^{q_0,p_0}_{\mu }({\mathbb {R}}^{n+1})}\lesssim \Vert f\Vert _{{\dot{\varLambda }}^{p,q}_\beta ({\mathbb {R}}^n)} \end{aligned}$$ hold for some $$p_0$$ and $$q_0$$ depending on p and q which are classified in three different cases: (1) $$p=q\in (n/(n+\beta ),1];$$ (2) $$(p,q)\in (1,n/\beta )\times (1,\infty );$$ (3) $$(p,q)\in (1,n/\beta )\times \{\infty \}.$$ For case (1), the embeddings can be characterized in terms of an analytic condition of the variational capacity minimizing function, the iso-capacitary inequality of open balls, and other weak-type inequalities. For cases (2) and (3), the embeddings are characterized by the iso-capacitary inequality for fractional Besov capacity of open sets.

Magnetic resonance of plasmonic modulation and switchable routing in gated graphene waveguides and their robustness and broadband tunability

We analytically derived an explicit condition for high-contrast plasmonic modulation and switchable routing in coupled graphene waveguides consisting of two coupled gated graphene sheets sandwiching a gyrotropic medium. The analytical condition intuitively predicts that by manipulating the graphene chemical potential and the gyration of the in-between medium we can combine robustness of the plasmonic routing to the deviation of design and material parameters due to imperfect fabrication and harsh environment with the extreme high contrast. Also, it predicts broadband tunability of the high-contrast plasmonic routing. The intuitive analytical condition reveals that the high contrast is achievable at a resonance gyration due to a magnetic resonance of nonreciprocal directional coupling. The magnetic resonance of nonreciprocity breaks a currently held preconception that the stronger magnetic field induces the stronger magnetically induced nonreciprocity. Numerical simulations demonstrate the robustness of the magnetic resonance of nonreciprocity and its tunability over a one-octave spanning spectral range. In practice, the chemical potential and the gyration can be manipulated by controlling the gate voltage of the graphene and the external magnetic field, respectively. Our findings would provide a robust and broadband tunable tool for high-contrast plasmonic modulator and switchable router and for probing of ultrafast magnetization dynamics.

A green ultrasonic-assisted micellar extraction coupled with ultra-high performance liquid chromatography with photodiode array method for quantitative analysis of active ingredients in Yangxinshi Tablet

An efficient and green ultrasonic-assisted micellar extraction method coupled with ultra-high performance liquid chromatography with photodiode array detection (UHPLC-PDA) was developed for the multi-ingredients quantitative analysis of Yangxinshi Tablet (YXST). The active ingredients were extracted from YXST using trehalose lipid biosurfactant solution as an environmentally friendly extraction solution. The response surface methodology (RSM) based on Box-Behnken design (BBD) was utilized to seek for the optimum extraction conditions of target analytes. When the concentration of trehalose lipid solution was 7 mg/mL, the liquid to solid was 125:1 (mL:g) and the extraction time was 40 min, the total extraction yield of eleven compounds (including puerarin, daidzin, ferulic acid, calycosin-7-O-β-D-glucoside, tetrahydropalmatine, coptisine, epiberberine, jatrorrhizine, berberine, palmatine chloride and icariin) obtained the maximum value. The relative standard deviations (RSD) of intra-day and inter-day precision were all less than 5.0%. The recoveries of all analytes were in the range of 95.1%−104% with the RSDs were all below 3.0%. Consequently, the ultrasonic-assisted micellar extraction coupled with UHPLC-PDA method could be successfully and efficiently applied to the extraction and quantitative analysis of target components in YXST.

A force explosion condition for spherically symmetric core-collapse supernovae

ABSTRACT Understanding which stars explode leaving behind neutron stars and which stars collapse forming black holes remains a fundamental astrophysical problem. We derive an analytic explosion condition for spherically symmetric core-collapse supernovae. The derivation starts with the exact governing equations, considers the balance of integral forces, includes the important dimensionless parameters, and includes an explicit set of self-consistent approximations. The force explosion condition is $\tilde{L}_\nu \tau _g - 0.06 \tilde{\kappa } \gt 0.38$, and only depends upon two dimensionless parameters. The first compares the neutrino power deposited in the gain region with the accretion power, $\tilde{L}_\nu \tau _g = L_{\nu } \tau _g R_{\rm NS}/ (G \dot{M} M_{\rm NS})$. The second, $\tilde{\kappa } = \kappa \dot{M} / \sqrt{G M_{\rm NS} R_{\rm NS}}$, parametrizes the neutrino optical depth in the accreted matter near the neutron star surface. Over the years, many have proposed approximate explosion conditions: the critical neutrino-luminosity, ante-sonic, and time-scale conditions. We are able to derive these other conditions from the force explosion condition, which unifies them all. Using numerical, steady-state and fully hydrodynamic solutions, we test the explosion condition. The success of these tests is promising in two ways. One, the force explosion condition helps to illuminate the underlying physics of explosions. Two, this condition may be a useful explosion diagnostic for more realistic, three-dimensional radiation hydrodynamic core-collapse simulations.

Abstract 2029: Multiomic analysis of whole transcriptome and high plex protein assays on a single FFPE slide

Abstract The GeoMx࣪ Digital Spatial Profiler (DSP) enables high-plex, high-throughput spatial profiling and quantification from a single slide for either protein or RNA. To fully understand the interplay between RNA and protein in tissue, we have developed a novel multi analyte assay that allows for the profiling of both analytes from the area of interest (AOI) on a single slide. Here we describe the development and performance of the multi analyte assay on cell pellet array (CPA) and various tissues including tonsil and colorectal cancer (CRC) using a high plex GeoMx Protein Panel and the GeoMx Human Whole Transcriptome Atlas (WTA). We successfully identified conditions to detect RNA and protein simultaneously, with comparable sensitivity and specificity to the single analyte conditions. The data accurately matched cell lines in an unbiased correlation analysis between GeoMx WTA and the entire CCLE RNAseq database. Furthermore, the assay performance was maintained in tissue samples. Tonsil and colorectal cancer biopsies were profiled for RNA (whole transcriptome) and protein from the same AOI, allowing simultaneous measurements of transcriptional and translational regulation of key biological pathways. This multi-analyte profiling enables deeper characterization of precious biological samples that are available in limited quantities. Citation Format: Shilah Bonnett, Nicholas Confuorto, Kit Fuhrman, Giang Ong, Alyssa Rosenbloom, John Lyssand, Gary Geiss, Joseph Beechem. Multiomic analysis of whole transcriptome and high plex protein assays on a single FFPE slide [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2022; 2022 Apr 8-13. Philadelphia (PA): AACR; Cancer Res 2022;82(12_Suppl):Abstract nr 2029.

Unveiling atom-photon quasi-bound states in hybrid plasmonic-photonic cavity

Abstract Dissipation, often associated with plasmons, leads to decoherence and is generally considered fatal for quantum nonlinearities and entanglement. Counterintuitively, by introducing a dissipative plasmonic nanoantenna into a typical cavity quantum electrodynamics (QED) system, we unveil the wide existence of the atom-photon quasi-bound state (qBS), a kind of exotic eigenstate with anomalously small decay, in the hybrid plasmonic-photonic cavity. To derive the analytical condition of atom-photon qBS, we formulate a quantized two-mode model of the local density of states by connecting the interacting uncoupled cavity modes to the macroscopic QED. With resonant plasmon-photon coupling, we showcase the single-atom qBS that improves the efficiency of single-photon generation over one order of magnitude; and the two-atom qBS that significantly enhances spontaneous entanglement generation compared with a bare photonic cavity. Notably, such single-atom and multi-atom qBS can be simultaneously accessed in realistic plasmonic-photonic cavities, providing a versatile platform for advanced quantum technologies, such as quantum light sources, quantum computation, and quantum information.

An Analytic and Computational Condition for the Finite Degree-of-Freedom of Linkages, and Its Relation to Lie Group Methods

Abstract The finite degree of freedom (DOF) of a mechanism is determined by the number of independent loop constraints. In this paper, a method to determine the maximal number of loop closure constraints (which is independent of a specific configuration) of multiloop linkages is introduced and applied to calculate the finite DOF. It rests on an algebraic condition on the joint screws, which stems from the analytic condition that minors of certain rank and their higher derivatives vanish. A computational algorithm is presented to determine the maximal rank of the constraint Jacobian in an arbitrary (possibly singular) reference configuration. This algorithm involves screw products of constant screw vectors only. Unlike the Lie group methods for estimating the DOF of so-called exceptional linkages, this method does not rely on partitioning kinematic loops into partial kinematic chains, and it is applicable to multiloop linkages. The DOF computed with this method is at least as accurate as the DOF computed with the Lie group methods. It gives the correct DOF for any (possibly overconstrained) linkage where the constraint Jacobian has maximal rank in regular configurations. The so determined maximal rank has further significance for classifying linkages as being exceptional or paradoxical but also for detecting singularities and shaky linkages.

Singular Perturbation and Small-Signal Stability for Inverter Networks

This article examines small-signal stability of electrical networks composed dominantly of three-phase grid-following inverters. We show that the mere existence of a high-voltage power flow solution does not necessarily imply small-signal stability; this motivates us to develop a framework for stability analysis that systematically acknowledges inverter dynamics. We identify a suitable time-scale decomposition for the inverter dynamics, and using singular perturbation theory, obtain an analytic sufficient condition to verify small-signal stability. Compared to the alternative of performing an eigenvalue analysis of the full-order network dynamics, our analytic sufficient condition reduces computational complexity and yields insights on the role of network topology and constitution as well as inverter-filter and control parameters in small-signal stability. Numerical simulations for a radial network validate the approach and illustrate the efficiency of our analytic conditions for designing and monitoring grid-tied inverter networks.

The Bloch oscillations of spin-orbit coupled Bose-Einstein condensates in deep optical lattices

We study the ground state and the Bloch oscillations of the spin-orbit coupled Bose-Einstein condensates in one-dimensional deep optical lattices. The phase transition between the zero-momentum phase and the plane-wave phase in untilted deep optical lattices is studied analytically. The Bloch oscillations in tilted lattices started in different ground state phases are analyzed. It is found that the Bloch dynamics is harmonic (anharmonic) in the zero-momentum phase (plane-wave phase). Furthermore, the modification of the Bloch oscillations induced by spin-orbit coupling, tunnelling coefficient and the external force (the inclination of the tilt) is revealed. In addition, the analytical condition for emerging the long-lived Bloch oscillations is obtained and the corresponding diagram in parameter space is provided. Particularly, we find that the spin-orbit coupling can extend the duration of the Bloch oscillations in the plane-wave phase. We provide an effective method to manipulate the Bloch dynamics in the deep optical lattices.

Wedge Domains in Compactly Causal Symmetric Spaces

AbstractMotivated by constructions in Algebraic Quantum Field Theory we introduce wedge domains in compactly causal symmetric spaces $M=G/H$, which includes in particular anti-de Sitter space in all dimensions and its coverings. Our wedge domains generalize Rindler wedges in Minkowski space. The key geometric structure we use is the modular flow on $M$ defined by an Euler element in the Lie algebra of $G$. Our main geometric result asserts that three seemingly different characterizations of these domains coincide: the positivity domain of the modular vector field, the domain specified by a KMS-like analytic extension condition for the modular flow, and the domain specified by a polar decomposition in terms of certain cones. In the second half of the article we show that our wedge domains share important properties with wedge domains in Minkowski space. If $G$ is semisimple, there exist unitary representations $(U,{\mathcal {H}})$ of $G$ and isotone covariant nets of real subspaces $\textsf {H}({\mathcal {O}}) \subseteq {\mathcal {H}}$, defined for any open subset ${\mathcal {O}} \subseteq M$, which assign to connected components of the wedge domains a standard subspace whose modular group corresponds to the modular flow on $M$. This corresponds to the Bisognano–Wichmann property in Quantum Field Theory. We also show that the set of $G$-translates of the connected components of the wedge domain provides a geometric realization of the abstract wedge space introduced by the first author and V. Morinelli.

Steady critical accretion onto black holes: Self-gravity and sonic point characteristics

The spherically symmetric steady accretion of polytropic perfect fluids onto a black hole is the simplest flow model that can demonstrate effects of backreaction (selfgravity). It has been discovered 16 years ago that backreaction does not influence some ("intensive") characteristics of sonic points, under suitable conditions. Herein we consider a wider class of equations of state, with polytropic indices in the range (1,2], and establish detailed boundary conditions that allow one to prove this fact. We find also numerical examples showing limits of our analytic criteria - if suitable analytic conditions are not satisfied, then selfgravity influences all characteristics of sonic points. That fact constrains the applicability of the recent proposal of Baumgarte and Shapiro to estimate the lifetime of black holes within compact stellar objects.

Impulsive feedback control of birhythmicity: Theory and experiment

We study the dynamic control of birhythmicity under an impulsive feedback control scheme where the feedback is made ON for a certain rather small period of time and for the rest of the time, it is kept OFF. We show that, depending on the height and width of the feedback pulse, the system can be brought to any of the desired limit cycles of the original birhythmic oscillation. We derive a rigorous analytical condition of controlling birhythmicity using the harmonic decomposition and energy balance methods. The efficacy of the control scheme is investigated through numerical analysis in the parameter space. We demonstrate the robustness of the control scheme in a birhythmic electronic circuit where the presence of noise and parameter fluctuations are inevitable. Finally, we demonstrate the applicability of the control scheme in controlling birhythmicity in diverse engineering and biochemical systems and processes, such as an energy harvesting system, a glycolysis process, and a p53-mdm2 network.

Dynamical analysis arising from the Willamowski-Rössler model

We study a three-dimensional Willamowski-Rössler model that is described as a spatially confined reaction system. To better restrict the parameter space of the chaotic dynamics of the system, the dynamics of boundary equilibria and subsystems of this model are investigated. According to the categories of the parameters, we classify two cases to discuss the relevant dynamics. For the extinction of this model, we obtain not only the local stability of the solutions but also global asymptotic stability. Furthermore, some amenable sufficient conditions for the uniform persistence for the model are given. Moreover, our analytic transversality conditions on the existence of periodic solutions via Hopf bifurcation are constructed, which is helpful to provide some theoretical analysis for exploring the parameter regions for the chaotic dynamics of the Willamowski-Rössler model.

What Digital Immunoassays Can Learn from Ambient Analyte Theory: A Perspective.

Immunoassays are important tools for clinical diagnosis as well as environmental and food analysis because they enable highly sensitive and quantitative measurements of analyte concentrations. In the 1980s, Roger Ekins suggested to improve the sensitivity of immunoassays by employing microspot assays, which are carried out under ambient analyte conditions and do not change the bulk analyte concentration of a sample during a measurement. More recently, the measurement of single analyte molecules has additionally attracted wide research interest. Although the ability to detect a single analyte molecule is not synonymous with the highest analytical sensitivity, single-molecule detection makes new routes accessible to avoiding background noise. This perspective follows the development of solid-phase immunoassays from the design of label techniques to single-molecule (digital) assays against the backdrop of Ekins's fundamental work on immunoassay theory. The essential aspects of both ambient analyte and digital assay approaches are presented as a guideline to finding a balance between the speed, sensitivity, and precision of immunoassays.

Chaotic Threshold of a Nonlinear Zener Systems Based on the Melnikov Method

Regarding a nonlinear Zener model with a viscoelastic Maxwell element as the research object, the complicated dynamic behaviors such as homoclinic bifurcation and chaos under harmonic excitation are investigated. At first, the analytically necessary condition for chaos in the sense of Smale horseshoe is derived based on the Melnikov method. Then, the system parameters that meet the analytical condition and the main resonance condition are selected for the numerical simulation. From the bifurcation diagrams and the largest Lyapunov exponents, it is found that the homoclinic orbit breaks, and the system goes to chaos in a crisis way when the excitation amplitude passes the first threshold. The system enters another new chaotic state in the form of period-doubling bifurcation with the increase of the excitation amplitude. At last, the effects of nonlinear term, stiffness coefficient and damping coefficient of Maxwell element on the analytically necessary condition for chaos are analyzed, respectively, and the correctness of the analytical result is proved by numerical simulation. The research result shows that the critical excitation amplitude decreases with the increase of nonlinear term. In addition, the chaotic threshold increases first and then tends to remain unchanged with the raise of stiffness coefficient. The chaotic threshold increases first and then decreases with the enhancement of damping. These results provide a theoretical basis for the research of nonlinear viscoelastic system in the future.

Transfer-Entropy-Regularized Markov Decision Processes

We consider the framework of transfer-entropy-regularized Markov decision process (TERMDP) in which the weighted sum of the classical state-dependent cost and the transfer entropy from the state random process to the control input process is minimized. Although TERMDPs are generally formulated as nonconvex optimization problems, an analytical necessary optimality condition can be expressed as a finite set of nonlinear equations, based on which an iterative forward–backward computational procedure similar to the Arimoto–Blahut algorithm is developed. It is shown that every limit point of the sequence generated by the proposed algorithm is a stationary point of the TERMDP. Applications of TERMDPs are discussed in the context of networked control systems theory and nonequilibrium thermodynamics. The proposed algorithm is applied to an information-constrained maze navigation problem, whereby we study how the price of information qualitatively alters the optimal decision polices.

Current-Limiting Virtual Synchronous Control and Stability Analysis Considering DC-Link Dynamics Under Normal and Faulty Grid Conditions

An improved nonlinear virtual synchronous control for three-phase grid-connected inverters, that can maintain a reliable operation under both normal and faulty grid conditions, i.e. balanced grid voltage sags, is proposed. The proposed controller can ensure a desired RMS current limitation at all times, provide virtual inertia and damping via the DC-link voltage and AC system frequency coupling, and realize the desired real and reactive power regulation without requiring accurate knowledge of the system parameters. Opposed to the conventional methods that use saturated PI controllers with or without anti-windup techniques to limit the reference value of the inverter current, the proposed controller includes a nonlinear bounded integrator, which limits the actual value (instead of the reference) of the inverter RMS current and leads to a fast system recovery even after significant grid voltage sags. The closed-loop stability of entire system is rigorously proven using nonlinear singular perturbation theory. Moreover, analytic conditions for the controller parameter selection to guarantee the stability of entire inverter system with the DC-link dynamics are provided. To prove the effectiveness of proposed controller and its superior performance compared to the traditional approaches, extensive Matlab/Simulink-based simulations are performed, followed by Typhoon-HIL hardware-in-the loop implementation using a TI microcontroller.

Analytic Investigation for Synchronous Firing Patterns Propagation in Spiking Neural Networks

Based on the moment closure method and mean field theory, a Gaussian random field is constructed to quantitatively and analytically characterize the dynamics of a random point field. The approach provides a theoretical tool to investigate synchronized spike propagation in a feedforward or recurrent spiking neural network. We show that the balance between the excitation and inhibition postsynaptic potentials is required for the occurrence of synfire chains. In particular, with a balanced network, the critical packet size of invasion and annihilation is observed. We also derive a sufficient analytic condition for the synchronization propagation in an asynchronous environment, which further allows us to disclose the possibility of spatial synaptic structure to sustain a stable synfire chain. Our findings are in good agreement with simulations and help us understand the propagation of spatio-temporal patterns in a random point field.

Single axis pointing for underactuated spacecraft with a residual angular momentum

The problem of aiming a generic body-fixed axis along an inertially fixed direction is dealt with for an underactuated spacecraft in the presence of a non-zero residual angular momentum, when only two reaction wheels can exchange angular momentum with the spacecraft platform. An analytical condition for the feasibility of the desired pointing is derived first, together with a closed-form solution for the corresponding attitude with zero platform angular rate. A nonlinear controller is then developed in the framework of singular perturbation theory, enforcing a two-timescale response to the system. Convergence to the desired attitude, when the pointing direction falls within admissible limits, is then proved for rest-to-rest maneuvers and randomly generated initial tumbling conditions for a configuration representative of a small-size satellite.