Local Monotonicity Research Articles

On the Positive Definiteness of Limiting Coderivative for Set-Valued Mappings

This paper concerns the interconnection between the positive definiteness of limiting coderivative and the local strong maximal monotonicity of set-valued mappings suspected in Mordukhovich and Nghia (SIAM J. Optim. 26, 1032–1059, 2016, Conjecture 3.6). We disprove the conjecture by a counterexample and provide some special classes at which it is true. However, the positive definiteness of limiting coderivative characterizes a new property called nearly strong monotonicity. Consequently, we show that the strong metric regularity of set-valued mappings could be obtained under the positive definiteness of limiting coderivative.

Local monotonicity coefficients in Orlicz sequence spaces equipped with the p-Amemiya norm

In this paper, the monotonicity is investigated with respect to Orlicz sequence space l_{varPhi , p} equipped with the p-Amemiya norm, and the necessary and sufficient condition is obtained to guarantee the uniform monotonicity, locally uniform monotonicity, and strict monotonicity for l_{varPhi , p}. This completes the results of the paper (Cui et al. in J. Math. Anal. Appl. 432:1095–1105, 2015) which were obtained for the non-atomic measure space. Local upper and lower coefficients of monotonicity at any point of the unit sphere are calculated, l_{varPhi , p} is calculated.

Well-posedness of backward stochastic partial differential equations with Lyapunov condition

Abstract In this paper we show the existence and uniqueness of strong solutions for a large class of backward SPDEs, where the coefficients satisfy a specific type Lyapunov condition instead of the classical coercivity condition. Moreover, based on the generalized variational framework, we also use the local monotonicity condition to replace the standard monotonicity condition, which is applicable to various quasilinear and semilinear BSPDE models.

Well posedness, large deviations and ergodicity of the stochastic 2D Oldroyd model of order one

In this work, we establish the unique global solvability of the stochastic two dimensional viscoelastic fluid flow equations, arising from the Oldroyd model for the non-Newtonian fluid flows perturbed by multiplicative Gaussian noise. A local monotonicity property of the linear and nonlinear operators and a stochastic generalization of the Minty–Browder technique are exploited in the proofs. The Laplace principle for the strong solution of the stochastic system is established in a suitable Polish space using a weak convergence approach. The Wentzell–Freidlin large deviation principle is proved using the well known results of Varadhan and Bryc. The large deviations for shot time are also considered. We also establish the existence of a unique ergodic and strongly mixing invariant measure for the stochastic system with additive Gaussian noise, using the exponential stability of strong solutions.

Uniformly locally o-minimal structures and locally o-minimal structures admitting local definable cell decomposition

We define and investigate a uniformly locally o-minimal structure of the second kind in this paper. All uniformly locally o-minimal structures of the second kind have local monotonicity, which is a local version of monotonicity theorem of o-minimal structures. We also demonstrate a local definable cell decomposition theorem for definably complete uniformly locally o-minimal structures of the second kind. We define dimension of a definable set and investigate its basic properties when the given structure is a locally o-minimal structure which admits local definable cell decomposition.

On the three dimensional Kelvin-Voigt fluids: global solvability, exponential stability and exact controllability of Galerkin approximations

In this work, we consider the three-dimensional viscoelastic fluid flow equations, arising from the motion of Kelvin-Voigt fluids in bounded and unbounded domains. We investigate the global solvability results, asymptotic behavior and also address some control problems of such viscoelastic fluid flow equations with fading and memory of length \begin{document}$ \tau $\end{document} . A local monotonicity property of the linear and nonlinear operators and a localized version of the Minty-Browder technique are used to obtain global solvability results. Since we are not using compactness arguments in the proofs, the global solvability results are also valid in unbounded domains like Poincare domains. We also remark that using an \begin{document}$ m $\end{document} -accretive quantization of the linear and nonlinear operators, one can establish the existence and uniqueness of strong solutions for the Navier-Stokes-Voigt equations and avoid the tedious Galerkin approximation scheme. We examine the asymptotic behavior of the stationary solutions and also establish the exponential stability results. Finally, under suitable assumptions on the Galerkin basis, we consider the controlled Galerkin approximated 3D Kelvin-Voigt fluid flow equations. Using the Hilbert uniqueness method combined with Schauder's fixed point theorem, the exact controllability of the finite dimensional Galerkin approximated system is established.

Deterministic and stochastic equations of motion arising in Oldroyd fluids of order one: existence, uniqueness, exponential stability and invariant measures

In this work, we consider the two-dimensional viscoelastic fluid flow equations, arising from the Oldroyd model for the non-Newtonian fluid flows. We investigate the well-posedness of such models in two-dimensional bounded and unbounded (Poincaré domains) domains, both in deterministic and stochastic settings. The existence and uniqueness of weak solution in the deterministic case is proved via a local monotonicity property of the linear and nonlinear operators and a localized version of the Minty-Browder technique. Some results on the exponential stability of stationary solutions are also established. The global solvability results for the stochastic counterpart are obtained by a stochastic generalization of the Minty-Browder technique. The exponential stability results in the mean square as well as in the pathwise (almost sure) sense are also discussed. Using the exponential stability results, we finally prove the existence of a unique invariant measure, which is ergodic and strongly mixing.

Coercivity condition for higher moment a priori estimates for nonlinear SPDEs and existence of a solution under local monotonicity

Higher moment a priori estimates for solutions to nonlinear SPDEs governed by locally-monotone operators are obtained under appropriate coercivity condition. These are then used to extend known existence and uniqueness results for nonlinear SPDEs under local monotonicity conditions to allow derivatives in the operator acting on the solution under the stochastic integral.

Characterizations of Centrality in {varvec{C}}^*-algebras via Local Monotonicity and Local Additivity of Functions

In this paper, we give two characterizations of central elements in a C^*-algebra mathcal {A} in terms of local properties of maps on mathcal {A} given by the function calculus. We prove that for a strictly convex increasing function f defined on an open interval which is unbounded from above, an element ain mathcal {A} is central if and only if f is locally monotone at a. That result significantly improves similar theorems by Ogasawara, Pedersen, Wu, Molnár and Virosztek. An analogous statement on local additivity is also presented.

Anomaly Detection Based on Mining Six Local Data Features and BP Neural Network

Key performance indicators (KPIs) are time series with the format of (timestamp, value). The accuracy of KPIs anomaly detection is far beyond our initial expectations sometimes. The reasons include the unbalanced distribution between the normal data and the anomalies as well as the existence of many different types of the KPIs data curves. In this paper, we propose a new anomaly detection model based on mining six local data features as the input of back-propagation (BP) neural network. By means of vectorization description on a normalized dataset innovatively, the local geometric characteristics of one time series curve could be well described in a precise mathematical way. Differing from some traditional statistics data characteristics describing the entire variation situation of one sequence, the six mined local data features give a subtle insight of local dynamics by describing the local monotonicity, the local convexity/concavity, the local inflection property and peaks distribution of one KPI time series. In order to demonstrate the validity of the proposed model, we applied our method on 14 classical KPIs time series datasets. Numerical results show that the new given scheme achieves an average F1-score over 90%. Comparison results show that the proposed model detects the anomaly more precisely.

Variational Convexity and the Local Monotonicity of Subgradient Mappings

It is well known that the subgradient mapping associated with a lower semicontinuous function is maximal monotone if and only if the function is convex, but what characterization can be given for the case in which a subgradient mapping is only maximal monotone locally instead of globally? That question is answered here in terms of a condition more subtle than local convexity. Applications are made to the tilt stability of a local minimum and to the local execution of the proximal point algorithm in optimization.

Geometry of Orlicz spaces equipped with norms generated by some lattice norms in mathbb {R}^{2}

In Orlicz spaces generated by convex Orlicz functions a family of norms generated by some lattice norms in mathbb {R}^{2} are defined and studied. This family of norms includes the family of the p-Amemiya norms (1le ple infty ) studied in Cui et al. (Nonlinear Anal. 69:1796–1816, 2008; Nonlinear Anal. 71:6343–6364, 2009; J. Math. Anal. Appl. 432:1095–1105, 2015; Nonlinear Anal. 75:3973–3993, 2012) and He et al. (Fixed Point Theory Appl. 2013:1–18, 2013). Criteria for strict monotonicity, lower and upper local uniform monotonicities and uniform monotonicities of Orlicz spaces and their subspaces of order continuous elements, equipped with these norms, are given in terms of the generating Orlicz functions, and the lattice norms in mathbb {R}^{2}. The problems of strict convexity and of the existence of order almost isometric as well as order isometric copies in these spaces are also discussed.

Stochastic Navier–Stokes equations perturbed by Lévy noise with hereditary viscosity

In this paper, we study the stochastic Navier–Stokes equations (SNSE) perturbed by Lévy noise in three dimensions with a hereditary viscous term which depends on the past history. We establish the local solvability of the Cauchy problem for such systems. The local monotonicity property of the nonlinear term of the cutoff problem and a stochastic generalization of the Minty–Browder technique are exploited in the proofs. Finally, we show that the global solvability results hold under smallness condition on the initial data and suitable assumptions on the noise coefficients.

Geometric properties of F-normed Orlicz spaces

The paper deals with F-normed functions and sequence spaces. First, some general results on such spaces are presented. But most of the results in this paper concern various monotonicity properties and various Kadec–Klee properties of F-normed Orlicz functions and sequence spaces and their subspaces of elements with order continuous norm, when they are generated by monotone Orlicz functions on {mathbb {R}}_{+} and equipped with the classical Mazur–Orlicz F-norm. Strict monotonicity, lower (and upper) local uniform monotonicity and uniform monotonicity in the classical sense as well as their orthogonal counterparts are considered. It follows from the criteria that are presented for these properties that all the above classical monotonicity properties except for uniform monotonicity differ from their orthogonal counterparts [in contrast to Köthe spaces (see Hudzik et al. in Rocky Mt J Math 30(3):933–950, 2000)]. The Kadec–Klee properties that are considered in this paper correspond to various kinds of convergence: convergence locally in measure and convergence globally in measure for function spaces, uniform convergence and coordinatewise convergence in the case of sequence spaces.

Local approach to monotonicity properties in $$\psi $$ ψ -direct sums of Köthe spaces

The points of upper and lower monotonicity of absolute norms in \(\mathbb {R} ^{2}\) are characterized. These results are used to calculate the characteristic of monotonicity of the space \(\left\langle \mathbb {R} ^{2},\Vert \cdot \Vert _{\psi }\right\rangle .\) Moreover, criteria for points of upper and lower monotonicity as well as points of upper and lower local monotonicity in direct sums of Kothe spaces are given. Some global monotonicity properties of direct sums of Kothe spaces are established.

Local energy inequalities for mean curvature flow into evolving ambient spaces

We establish a local monotonicity formula for mean curvature flow into a curved space whose metric is also permitted to evolve simultaneously with the flow, extending the work of Ecker (Ann Math (2) 154(2):503–525, 2001), Huisken (J Differ Geom 31(1):285–299, 1990), Lott (Commun Math Phys 313(2):517–533, 2012), Magni, Mantegazza and Tsatis (J Evol Equ 13(3):561–576, 2013) and Ecker et al. (J Reine Angew Math 616:89–130, 2008). This formula gives rise to a monotonicity inequality in the case where the target manifold’s geometry is suitably controlled, as well as in the case of a gradient shrinking Ricci soliton. Along the way, we establish suitable local energy inequalities to deduce the finiteness of the local monotone quantity.

Optimal insurance design under background risk with dependence

In this paper, we revisit the problem of optimal insurance under a general criterion that preserves stop-loss order when the insured faces two mutually dependent risks: background risk and insurable risk. According to the local monotonicity of conditional survival function, we derive the optimal contract forms in different types of interval. Because the conditional survival function reflects the dependence between background risk and insurable risk, the dependence structure between the two risks plays a critical role in the insured’s optimal insurance design. Furthermore, we obtain the optimal insurance forms explicitly under some special dependence structures. It is shown that deductible insurance is optimal and the Mossin’s Theorem is still valid when background risk is stochastically increasing in insurable risk, which generalizes the corresponding results in Lu et al. (2012). Moreover, we show that an individual will purchase no insurance when the sum of the two risks is stochastically decreasing in insurable risk.

Efficient Multi-Strategy Intra Prediction for Quality Scalable High Efficiency Video Coding.

As an extension of High Efficiency Video Coding (HEVC), the Scalable High Efficiency Video Coding (SHVC) introduces multiple layers with inter-layer predictions, which greatly increases the complexity on top of the already complicated HEVC encoder. In Intra prediction for Quality SHVC, Coding Tree Unit (CTU) allows recursive splitting into four depth levels, which considers 35 Intra prediction modes and interlayer reference (ILR) mode to determine the best possible mode at each depth level. This achieves the highest coding efficiency but incurs a substantially high computational complexity. In this paper, we propose a novel Intra prediction scheme to effectively speed up the enhancement layer Intra-coding in Quality SHVC. The new features of the proposed framework include: First, spatial correlation and its correlation degree are combined to predict most probable depth level candidates. Second, for a given depth candidate, based on the probabilities of the ILR mode, we check the ILR mode by examining the residual distribution based on skewness and kurtosis to determine whether the residuals follow a Gaussian distribution. In that case, the Intra prediction comparisons, which require a high complexity, are skipped. Third, during Intra prediction selection from 35 Intra prediction modes, spatial and inter-layer correlations are combined with the local monotonicity of the Hadamard costs associated with the modes in a small neighborhood, to examine only a portion of Intra prediction modes. Finally, a hypothesis testing on the currently selected depth level is performed to examine whether the residuals present significant differences within their block to early terminate depth selection. The proposed multi-step multistrategy scheme aims to minimize the number of depth selections while greatly reducing the mode decision complexity for a depth candidate in a hierarchical fashion. Our experimental results demonstrate that the proposed scheme can achieve a speedup gain of more than 75% in average on the test video sequences, while maintaining almost the same coding efficiency. .

Heat ball formulæ for k-forms on evolving manifolds

Abstract We establish a local monotonicity identity for vector bundle-valued differential k-forms on superlevel sets of appropriate heat kernel-like functions. As a consequence, we obtain new local monotonicity formulæ for the harmonic map and Yang–Mills heat flows on evolving manifolds. We also show how these methods yield local monotonicity formulæ for the Yang–Mills–Higgs flow.

ЛОКАЛЬНАЯ МОНОТОННАЯ ИНТЕРПОЛЯЦИЯ И ОДНОПАРАМЕТРИЧЕСКИЕ ГРУППЫ

ЛОКАЛЬНАЯ МОНОТОННАЯ ИНТЕРПОЛЯЦИЯ И ОДНОПАРАМЕТРИЧЕСКИЕ ГРУППЫ