Sufficient Conditions Research Articles

Generalization Guarantees of Gradient Descent for Shallow Neural Networks

Abstract Significant progress has been made recently in understanding the generalization of neural networks (NNs) trained by gradient descent (GD) using the algorithmic stability approach. However, most of the existing research has focused on one-hidden-layer NNs and has not addressed the impact of different network scaling. Here, network scaling corresponds to the normalization of the layers. In this article, we greatly extend the previous work (Lei et al., 2022; Richards & Kuzborskij, 2021) by conducting a comprehensive stability and generalization analysis of GD for two-layer and three-layer NNs. For two-layer NNs, our results are established under general network scaling, relaxing previous conditions. In the case of three-layer NNs, our technical contribution lies in demonstrating its nearly co-coercive property by utilizing a novel induction strategy that thoroughly explores the effects of overparameterization. As a direct application of our general findings, we derive the excess risk rate of O(1/n) for GD in both two-layer and three-layer NNs. This sheds light on sufficient or necessary conditions for underparameterized and overparameterized NNs trained by GD to attain the desired risk rate of O(1/n). Moreover, we demonstrate that as the scaling factor increases or the network complexity decreases, less overparameterization is required for GD to achieve the desired error rates. Additionally, under a low-noise condition, we obtain a fast risk rate of O(1/n) for GD in both two-layer and three-layer NNs.

Necessary and sufficient conditions for rigidity of Pólya–Szegö inequality under Schwarz symmetrisation

Necessary and sufficient conditions for rigidity of Pólya–Szegö inequality under Schwarz symmetrisation

A Hybrid Fuzzy Evaluation Method for Quantitative Risk Classification of Barrier Lakes Based on an AHP Method Extended by D Numbers

The risk classification of barrier lakes is the key to conducting emergency treatment in a scientific manner. However, risk classification faces difficulties such as a short time for risk evaluation, complex evaluation indicators, difficulty in obtaining information quickly, and quantifying index weights. Based on this, this paper constructs a quantitative risk classification model for barrier lakes based on D-AHP. On the basis of studies on nearly 100 cases of barrier lakes, an eight-factor evaluation index system and quantitative classification are proposed. The methods of rapid calculation of reservoir capacity curve of barrier lakes and intelligent identification of particles on the surface of barrier bodies were developed, which realized the rapid acquisition of eight-factor evaluation index information in an emergency environment. The D-AHP method dealt with inconsistent weight assignment to evaluation factors by experts, which helped achieve weight quantification of eight factors. The risk assessment on 15 barrier lakes such as Tangjiashan barrier lake shows that the conclusions drawn for the risk classification method proposed in this paper are basically consistent with those of the traditional table-lookup method. However, the table-lookup method ignores cumulative loss impacts on the risk level of barrier lakes and considers the extremely severe loss of barrier lakes as a sufficient condition for the evaluation level to be grade I, and thus a deviation in the evaluation. The risk classification method proposed in this paper is more reasonable and reliable.

An active queue management method for multiple bottleneck networks based on distributed average tracking

The active queue management (AQM) is a key technology for the information infrastructure and the Internet. The AQM for multiple bottleneck networks is especially challenging than that for single-bottleneck networks because the stability analysis of multiple bottleneck networks is more complex. In this paper, the active queue management problem for multiple bottleneck networks is considered. A novel random early detection algorithm is designed based on distributed average tracking technology, which is used to estimate the global queue length and improve the network's performance. Sufficient conditions for global asymptotic stability of the closed-loop active queue management system are presented, and some simulation results with NS-2 are given. Compared with the RED algorithm, simulation results show that the presented algorithm has smaller oscillations at routers, which highlights its effectiveness.

Existence of solutions for time fractional semilinear parabolic equations in Besov–Morrey spaces

AbstractWe consider the Cauchy problem for a time fractional semilinear heat equation "Equation missing"where $$0<\alpha <1,\, \gamma >1,\, N\in \mathbb {Z}_{\geqslant 1}$$ 0 < α < 1 , γ > 1 , N ∈ Z ⩾ 1 and $$\mu (x)$$ μ ( x ) belongs to inhomogeneous/homogeneous Besov–Morrey spaces. The fractional derivative $$^{C}\partial ^{\alpha }_{t}$$ C ∂ t α is interpreted in the Caputo sense. We present sufficient conditions for the existence of local/global-in-time solutions to problem (P). Our results cover all existing results in the literature and can be applied to a large class of initial data.

Static voltage stability analysis of integrated smart energy systems

The analysis of multi-energy flows forms the cornerstone for the study of state estimation, safety assessment, and optimization in integrated smart energy systems (ISES). The interactions between various energy flows and the inherent variability of renewable energy sources often lead to significant challenges to the static stability of ISES. This paper investigates the static voltage stability of ISES under multi-energy coupling conditions through multi-energy flow analysis in electric-gas-thermal energy systems. First, a steady-state model of the ISES is constructed by representing the interconnected energy subsystems as equivalent sources and loads in the power grid. Subsequently, through the coupling elements of ISES, the power flows of the natural gas system (NGS) and the district heating system (DHS) are converted into active power in the electrical power system (EPS), resulting in an equivalent power flow equation for the ISES. Then, using Brouwer’s fixed-point theorem, the analytical sufficient conditions for solving the equivalent power flow equation are derived. Finally, a simulation model based on MATLAB/Simulink is established. The steady-state criterion for ISES is obtained in this paper, and the correctness and effectiveness of the proposed conclusions are verified by the simulation results.

CONHECIMENTOS SOBRE RASTREIO E DIAGNÓSTICO PRECOCE DE CÂNCER DE COLO UTERINO ENTRE AS MULHERES QUE FREQUENTAM UM AMBULATÓRIO DE UM CENTRO UNIVERSITÁRIO NA CIDADE DE PORTO VELHO – RO - BRASIL

Cervical cancer (CCU) is one of the most common types among women, making early diagnosis essential for better prognoses. It is characterized by slow progression, but it is amenable to screening and rapid detection, which can lead to effective treatment and a promising prognosis. Human papiloma virus (HPV) infection, especially by subtypes 16 and 18, is a necessary but not sufficient condition for the development of cervical cancer. The oncotic cytology test (Pap smear) aims to detect precursor lesions of CCU. This study investigated the level of knowledge about cervical cancer screening and early diagnosis among women attending an outpatient clinic at a University Center in Porto Velho, RO. A qualitative and descriptive approach was used through questionnaires to assess understanding of the preventive exam (Pap smear), risk factors, and preventive practices. The results showed: (i) high adherence to the cervical cancer preventive exam among participants; (ii) non-use of condoms (70%); (iii) educational actions on the preventive exam (60%); (iv) structural and personal barriers limiting access to the preventive exam; (v) 56.7% of participants had information about HPV; (vi) 83.3% of participants reported knowledge of cervical cancer; (vii) lack of knowledge about the frequency of the preventive exam in 55% of participants; (viii) 88.3% believe that early diagnosis improves prognosis and treatment success. The results of this study may help to develop effective educational interventions that encourage the preventive exam, contributing to reducing incidence and mortality rates associated with CCU, and improving women’s health in the region.

On the Quasi-diagonalization and Uncoupling of Gyroscopic Circulatory Multi-degree-of-freedom Systems

Abstract A new central result that gives the necessary and sufficient conditions for two n by n skew-symmetric matrices and one symmetric matrix to be simultaneously quasi-diagonalized by a real orthogonal congruence is proved. Based on this result, the decomposition of linear multi-degree-of-freedom dynamical systems with gyroscopic, circulatory, and potential forces is investigated through a real linear coordinate transformation generated by an orthogonal matrix. Several sets of conditions, applicable to real-life structural and mechanical systems arising in aerospace, civil, and mechanical engineering, under which such a coordinate transformation exists are found, thereby allowing these systems to be decomposed into independent, uncoupled subsystems, each with a maximum of two degrees of freedom. The conditions are expressed in terms of the coefficient matrices of the system. A specific form for the circulatory (gyroscopic) matrix is posited, and when the gyroscopic (circulatory) matrix is simple—a situation that commonly appears in real-life applications—it is shown that just a single necessary and sufficient condition is required for the decomposition of the multi-degree-of-freedom system. Numerical examples are provided throughout to demonstrate the analytical results.

Dynamic event-triggered adaptive security sliding mode control for singular systems with unknown deception attacks

This paper studies the problem of dynamic event-triggered adaptive security sliding mode control (SMC) for singular systems with deception attacks. First, due to limited network resources, a dynamic event-triggered scheme (DETS) is adopted to reduce signal transmission. Second, the deception attacks which make the system unstable are modeled as a nonlinear function. Third, some sufficient conditions are given to assure that the overall system is admissible with [Formula: see text] performance and a method is proposed to design the controller gain and the event-triggered parameter. Fourth, a novel adaptive SMC law is provided to ensure the reaching condition of the sliding surface. Finally, a practical example is employed to show the validity of the designed method.

Response of dissolved organic matter and disinfection by-product precursors to algal blooms and thermal stratification in deep reservoirs.

Algal bloom contribute substantially to dissolved organic matter (DOM) and disinfection by-product (DBP) precursors in deep reservoirs, threatening drinking water safety. However, the variations in DOM and DBP precursors in deep-water reservoirs during algal bloom remain unclear. UV and fluorescence spectroscopy and chlorination experiments were used to analyze the variations in DOM and DBP precursors during algal bloom in the Sanhekou Reservoir. Before algal bloom, the DOM and DBP precursors decreased due to biodegradation. After algal bloom, the DOM and DBP precursors increased by 48.3% and 86.9% due to algae producing protein-like compounds. Notably, the algal bloom produced a range of nitrogenous compounds that significantly promote the formation of trichloronitromethane, a major contributor to the mammalian cytotoxicity associated with DBPs. In addition, the heterogeneous matrix led to the stratification of DOM and DBP precursors. The surface water (0-5 m) was more vulnerable to algae, with protein-like components being much higher than in other layers, while humic and fulvic-like components were much lower. However, high temperatures and sufficient oxygen conditions accelerated the biodegradation of DOM and DBP precursors, resulting in significantly lower levels of DOM and DBP precursors in the surface water compared to other layers (p<0.05). This study provides insights into the variations and the drivers in DOM and DBP precursors during algal bloom, essential for developing water intake strategies in similar water reservoirs.

New design of robust H∞ controllers for descriptor discrete time-varying delay equations with bounded disturbances

In this paper, a linear matrix inequality (LMI)-based design of robust [Formula: see text] controllers for linear descriptor discrete-time delay equations is proposed. The existing works of FTS and [Formula: see text] control are extended to the linear descriptor equations subjected to time-varying delays and bounded disturbances. First, sufficient LMI-based conditions are given to ensure the admissibility of such system, and then the conditions are applied to design [Formula: see text] controllers. A numerical example with simulations is given to illustrate the proposed results.

Determining the N-Representability of a Reduced Density Matrix via Unitary Evolution and Stochastic Sampling.

The N-representability problem consists in determining whether, for a given p-body matrix, there exists at least one N-body density matrix from which the p-body matrix can be obtained by contraction, that is, if the given matrix is a p-body reduced density matrix (p-RDM). The knowledge of all necessary and sufficient conditions for a p-body matrix to be N-representable allows the constrained minimization of a many-body Hamiltonian expectation value with respect to the p-body density matrix and, thus, the determination of its exact ground state. However, the number of constraints that complete the N-representability conditions grows exponentially with system size, and hence, the procedure quickly becomes intractable for practical applications. This work introduces a hybrid quantum-stochastic algorithm to effectively replace the N-representability conditions. The algorithm consists of applying to an initial N-body density matrix a sequence of unitary evolution operators constructed from a stochastic process that successively approaches the reduced state of the density matrix on a p-body subsystem, represented by a p-RDM, to a target p-body matrix, potentially a p-RDM. The generators of the evolution operators follow the well-known adaptive derivative-assembled pseudo-Trotter method (ADAPT), while the stochastic component is implemented by using a simulated annealing process. The resulting algorithm is independent of any underlying Hamiltonian, and it can be used to decide whether a given p-body matrix is N-representable, establishing a criterion to determine its quality and correcting it. We apply the proposed hybrid ADAPT algorithm to alleged reduced density matrices from a quantum chemistry electronic Hamiltonian, from the reduced Bardeen-Cooper-Schrieffer model with constant pairing, and from the Heisenberg XXZ spin model. In all cases, the proposed method behaves as expected for 1-RDMs and 2-RDMs, evolving the initial matrices toward different targets.

Stability and l∞ Performance Analysis for Nabla Discrete Fractional Linear Positive System with Time-Varying Delays

In this paper, the stability and l∞-gain problem are investigated for the Nabla discrete fractional linear positive systems with bounded time-varying delays. First, a sufficient condition and a necessary condition are presented to ensure the system’s positivity. Then, based on the system’s positivity property, an asymptotically stable condition is established. Furthermore, it is demonstrated that the l∞-gain of such systems is determined by the system matrices and is independent of the magnitude of delays. Finally, numerical examples are provided to demonstrate the validity of the obtained results.

Square-mean S-Asymptotically $$\omega $$-Periodic Solutions for Some Stochastic Delayed Integrodifferential Inclusions

AbstractIn this work, by the resolvent operator theory, stochastic analysis theory and the fixed-point technique in Fréchet spaces, we present sufficient conditions for the existence of square-mean S-asymptotically $$\omega $$ ω -periodic mild solutions of abstract stochastic integrodifferential inclusions with infinite delay in separable Hilbert spaces and provide an example to illustrate the abstract results.

Boundedness of Some Commutators in Total Fofana Spaces

In this paper, we find necessary and sufficient conditions for the boundedness of the commutator of the Hardy-Littlewood maximal operator in total Fofana spaces. We also give in these spaces the boundedness of some sublinear operators and their commutators.

The generalized equity and exact credibility premiums with examples

We investigate a more generalized alternative to the conventional credibility premium that results from squared-error loss, namely, equitable credibility premium (Promislow and Young, 1999), based on exponential family, and we derive an accurate credibility for Bühlmann model with an example. We provide sufficient conditions for accurate credibility and expand a generalised formula for the credibility weight in specific circumstances. In this article, we expand upon the research conducted by Promislow and Young (2000) to include situations where an actuary is provided with data regarding a policyholder who has a more extensive exponential family. In the first section, we established our notation and reformulation of exact credibility for a Bühmann model. We derive the class of general equitable credibility premiums in the second section.

Pre-Compactness of Sets and Compactness of Commutators for Riesz Potential in Global Morrey-Type Spaces

In this paper, we establish sufficient conditions for the pre-compactness of sets in the global Morrey-type spaces GMpθw(·). Our main result is the compactness of the commutators of the Riesz potential b,Iα in global Morrey-type spaces from GMp1θ1w1(·) to GMp2θ2w2(·). We also present new sufficient conditions for the commutator b,Iα to be bounded from GMp1θ1w1(·) to GMp2θ2w2(·). In the proof of the theorem regarding the compactness of the commutator for the Riesz potential, we primarily utilize the boundedness condition for the commutator for the Riesz potential b,Iα in global Morrey-type spaces GMpθw(·), and the sufficient conditions derived from the theorem on pre-compactness of sets in global Morrey-type spaces GMpθw(·).

On a class of stochastic fractional heat equations

For the fractional heat equation ∂ ∂ t u ( t , x ) = − ( − Δ ) α 2 u ( t , x ) + u ( t , x ) W ˙ ( t , x ) \frac {\partial }{\partial t} u(t,x) = -(-\Delta )^{\frac {\alpha }{2}}u(t,x)+ u(t,x)\dot W(t,x) where the covariance function of the Gaussian noise W ˙ \dot W is defined by the heat kernel, we establish Feynman-Kac formulae for both Stratonovich and Skorohod solutions, along with their respective moments. In particular, we prove that d &gt; 2 + α d&gt;2+\alpha is a sufficient and necessary condition for the equation to have a unique square-integrable mild Skorohod solution. One motivation lies in the occurrence of this equation in the study of a random walk in random environment which is generated by a field of independent random walks starting from a Poisson field.

Matrices with hyperbolical Krein space numerical range

AbstractThis paper is devoted to matrices with hyperbolical Krein space numerical range. This shape characterizes the 2-by-2 case and persists for certain classes of matrices, independently of their size. Necessary and sufficient conditions for low dimensional tridiagonal matrices to have this shape are obtained involving only the matrix entries.

Coexistence states for a prey-predator model with self-diffusion and predator-taxis

This paper is concerned with the stationary problem of a prey-predator model with self-diffusion and predator-taxis. Under homogeneous Dirichlet boundary conditions, the bifurcation approach is used to derive the sufficient conditions for the existence of coexistence states. Our analysis shows that the coupling of self-diffusion and predator-taxis cause some difficulties in establishing a priori estimates for the solutions.