Sharp Conditions Research Articles

Existence and Uniqueness Theorems in the Inverse Problem of Recovering Surface Fluxes from Pointwise Measurements

Inverse problems of recovering surface fluxes on the boundary of a domain from pointwise observations are considered. Sharp conditions on the data ensuring existence and uniqueness of solutions in Sobolev classes are exposed. They are smoothness conditions on the data, geometric conditions on the location of measurement points, and the boundary of a domain. The proof relies on asymptotics of fundamental solutions to the corresponding elliptic problems and the Laplace transform. The inverse problem is reduced to a linear algebraic system with a nondegerate matrix.

On the extinction-extinguishing dichotomy for a stochastic Lotka–Volterra type population dynamical system

Applying the Foster–Lyapunov type criteria and a martingale method, we study a two-dimensional process (X,Y) arising as the unique nonnegative solution to a pair of stochastic differential equations driven by independent Brownian motions and compensated spectrally positive Lévy random measures. Both processes X and Y can be identified as continuous-state nonlinear branching processes where the evolution of Y is negatively affected by X. Assuming that process X extinguishes, i.e. it converges to 0 but never reaches 0 in finite time, and process Y converges to 0, we identify rather sharp conditions under which the process Y exhibits, respectively, one of the following behaviors: extinction with probability one, extinguishing with probability one or both extinction and extinguishing occurring with strictly positive probabilities.

Operator θ$\theta$‐Hölder functions with respect to ∥·∥p$\Vert \cdot \Vert _p$, 0<p⩽∞$0< p\leqslant \infty$

Let θ ∈ ( 0 , 1 ) $\theta \in (0,1)$ and ( M , τ ) $({\mathcal {M}},\tau )$ be a semifinite von Neumann algebra. We consider the function spaces introduced by Sobolev (J. London Math. Soc. (2) 95 (2017), 157–176; Geom. Funct. Anal. 27 (2017), 676–725) (denoted by S d , θ $S_{d,\theta }$ ), showing that there exists a constant d > 0 $d>0$ depending on p $p$ , 0 < p ⩽ ∞ $0<p\leqslant \infty$ , only such that every function f : R → C ∈ S d , θ $f:{\mathbb {R}}\rightarrow {\mathbb {C}}\in S_{d,\theta }$ is operator θ $\theta$ -Hölder with respect to ∥ · ∥ p $\Vert \cdot \Vert _p$ , that is, there exists a constant C p , f $C_{p,f}$ depending on p $p$ and f $f$ only such that the estimate f ( A ) − f ( B ) p ⩽ C p , f A − B θ p \begin{equation*} \hspace*{7pc}{\left\Vert f(A) -f(B)\right\Vert} _p \leqslant C_{p,f}{\left\Vert {\left| A-B \right|}^\theta \right\Vert} _p\hspace*{-7pc} \end{equation*} holds for arbitrary self-adjoint τ $\tau$ -measurable operators A $A$ and B $ B$ . In particular, we obtain a sharp condition such that a function f $f$ is operator θ $\theta$ -Hölder with respect to all quasi-norms ∥ · ∥ p $\Vert \cdot \Vert _p$ , 0 < p ⩽ ∞ $0<p\leqslant \infty$ , which complements the results on the case for 1 θ < p < ∞ $ \frac{1}{\theta }< p<\infty$ by Aleksandrov and Peller (J. Funct. Anal. 258 (2010), 3675–3724), and the case when p = ∞ $p=\infty$ treated by Aleksandrov and Peller (Adv. Math. 224 (2010), 910–966), and by Nikol ′ $^\prime$ skaya and Farforovskaya (Algebra i Analiz 22 (2010), 198–213 (Russian)). As an application, we show that this class of functions is operator θ $\theta$ -Hölder with respect to a wide class of symmetrically quasi-normed operator spaces affiliated with M ${\mathcal {M}}$ , which unifies the results on specific functions due to Birman, Koplienko and Solomjak (Izv. Vysš. Učebn. Zaved. Matematika 3 (1975), no. 154, 3–10; Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 178 (1989)), Bhatia (Comm. Math. Phys. 111 (1987), 33–39), Ando (Math. Z. 197 (1988), 403–409) and Ricard (Arch. Math. (Basel) 104 (2015), 37–45; Adv. Math. 333 (2018), 194–211) with significant extension. In addition, when θ > 1 $\theta >1$ , we obtain a reverse of the Birman–Koplienko–Solomjak inequality, which extends a couple of existing results on fractional powers t ↦ t θ $t\mapsto t^\theta$ by Ando et al.

Experimental comparison between various fitting approaches based on RMSE minimization for photovoltaic module parametric identification

One solution for online condition monitoring of photovoltaic (PV) modules is to identify single-diode model parameter values from measured current–voltage (I–V) curves. By this way, use of expensive thermal cameras and radiometric sensors utilized in traditional monitoring methods can be avoided. Unfortunately, most of the parameter identification methods require measurements of the operating conditions, i.e., irradiance and temperature. This article proposes a novel procedure for identification of the single-diode model parameter values along with the operating irradiance and temperature values from measured I–V curves without needing any other measurement. The only inputs of the proposed procedure are the I–V curve measurements at the actual operating conditions together with the parameter values of the module model in standard test conditions. The proposed procedure is experimentally validated using I–V curves of three PV module types measured from two different locations. Both the whole I–V curves or only a part of them, in a limited voltage range, are considered. Moreover, I–V curve measurements with an emulated increase of the series resistance are used to demonstrate the correctness of the identified series resistance values. It is shown that the procedure identifies the operating irradiance and temperature with high accuracy even during sharp irradiance transitions and low irradiance conditions and identifies series and shunt resistances very reliably under nearly constant high irradiance conditions. Moreover, for the first time, a comprehensive comparison of various fitting approaches based on root-mean-square error (RMSE) minimization, including two novel approaches, is presented. The results show that the different fitting approaches based on RMSE minimization affect the accuracy of the parameters identification in a different way, this meaning that the used fitting approach is a factor that should be considered when implementing model parameter identification by curve fitting.

Gradient estimates for Orlicz double phase problems with variable exponents

Optimal regularity estimates are established for the gradient of solutions to non-uniformly elliptic equations of Orlicz double phase with variable exponents type in divergence form under sharp conditions on such highly nonlinear operators for the Calderón–Zygmund theory.

A Survey on Sharp Oscillation Conditions for Delay Difference Equations

A Survey on Sharp Oscillation Conditions for Delay Difference Equations

Sharp growth conditions for boundedness of maximal function in generalized Orlicz spaces

We study sharp growth conditions for the boundedness of the Hardy-Littlewood maximal function in the generalized Orlicz spaces. We assume that the generalized Orlicz function \(\phi(x,t)\) satisfies the standard continuity properties (A0), (A1) and (A2). We show that if the Hardy-Littlewood maximal function is bounded from the generalized Orlicz space to itself then \(\phi(x,t)/t^p\) is almost increasing for large \(t\) for some \(p&gt;1\). Moreover we show that the Hardy-Littlewood maximal function is bounded from the generalized Orlicz space \(L^\phi(\mathbb{R}^n)\) to itself if and only if \(\phi\) is weakly equivalent to a generalized Orlicz function \(\psi\) satisfying (A0), (A1) and (A2) for which \(\psi(x,t)/t^p\) is almost increasing for all \(t&gt;0\) and some \(p&gt;1\).&#x0D;

From cuts to [formula omitted]-trees

We provide sharp conditions under which a collection of separators A of a connected topological space Z leads to a canonical R-tree T. Any group acting on Z by homeomorphisms will act by homeomorphisms on T.

An optimal Liouville theorem for the porous medium equation

Under a sharp asymptotic growth condition at infinity, we prove a Liouville type theorem for the inhomogeneous porous medium equation, provided it stays universally close to the heat equation. Additionally, for the homogeneous equation, we show that for the conclusion to hold, it is enough to assume the sharp asymptotic growth at infinity only in the space variable. The results are optimal, meaning that the growth condition at infinity cannot be weakened.

An optimal transport problem with backward martingale constraints motivated by insider trading

We study a single-period optimal transport problem on R2 with a covariance-type cost function c(x,y)=(x1−y1)(x2−y2) and a backward martingale constraint. We show that a transport plan γ is optimal if and only if there is a maximal monotone set G that supports the x-marginal of γ and such that c(x,y)=minz∈Gc(z,y) for every (x,y)∈suppγ. We obtain sharp regularity conditions for the uniqueness of an optimal plan and for its representation in terms of a map. Our study is motivated by a variant of the classical Kyle model of insider trading from (Rev. Econ. Stud. 61 (1994) 131–152).

On Distributed Nonconvex Optimization: Projected Subgradient Method for Weakly Convex Problems in Networks

The stochastic subgradient method is a widely-used algorithm for solving large-scale optimization problems arising in machine learning. Often these problems are neither smooth nor convex. Recently, Davis et al. [1-2] characterized the convergence of the stochastic subgradient method for the weakly convex case, which encompasses many important applications (e.g., robust phase retrieval, blind deconvolution, biconvex compressive sensing, and dictionary learning). In practice, distributed implementations of the projected stochastic subgradient method (stoDPSM) are used to speed-up risk minimization. In this paper, we propose a distributed implementation of the stochastic subgradient method with a theoretical guarantee. Specifically, we show the global convergence of stoDPSM using the Moreau envelope stationarity measure. Furthermore, under a so-called sharpness condition, we show that deterministic DPSM (with a proper initialization) converges linearly to the sharp minima, using geometrically diminishing step-size. We provide numerical experiments to support our theoretical analysis.

On the complexity of local-equitable coloring of graphs

An equitable k-partition (k≥2) of a vertex set S is a partition of S into k disjoint subsets (may be empty sets) such that the sizes of any two subsets of S differ by at most one. A local-equitable k-coloring of G is an assignment of k colors to the vertices of G such that, for every maximal clique of G, the coloring of this clique forms an equitable k-partition of itself. The local-equitable coloring of G is a stronger version of clique-coloring of graphs. Chordal graphs are 2-clique-colorable but not necessarily local-equitably 2-colorable. In this paper, we prove that it is NP-complete to decide the local-equitable 2-colorability in chordal graphs and even in split graphs. In addition, we prove that claw-free split graphs are local-equitably k-colorable when k≤4, but not necessarily local-equitably k-colorable when k≥5. A sufficient and sharp condition of local-equitably k-colorability is also given in claw-free split graphs. Secondly, we show that, given a split graph G, deciding the local-equitable k-colorability of G is solvable in polynomial time when k=ω(G)−1, where ω(G) is the clique number of G. At last, we prove that the decision problem of local-equitable 2-coloring of planar graphs is solvable in polynomial time.

A sharp critical threshold for a traffic flow model with look-ahead dynamics

We study a nonlocal traffic flow model with an Arrhenius type look-ahead interaction. We show a sharp critical threshold condition on the initial data which distinguishes the global smooth solutions and finite time wave break-down.

Numerical simulation of electrohydrodynamic convection induced by fast oscillating field emission

The paper presents a mathematical model of electrohydrodynamic convection in a mixture of benzene and chlorobenzene subjected to highly inhomogeneous alternating electric field. The field inhomogeneity is determined by the shape of the electrodes, of which one is a flat electrode, and the other is a tip oriented perpendicular to the plane of the first electrode. Recent experiments carried out with a field oscillating at a frequency of 50 Hz have shown that the excitation of an averaged convective flow at a given maximum voltage between the electrodes (8 kV) is observed only at a mole fraction of chlorobenzene of at least 0.4. The flow arises in a threshold manner and is a narrow intense jet directed from the sharp electrode towards the plane closed by an axisymmetric return flow. We propose a model that explains this phenomenon by analyzing the interaction between two fields of charged particles. The sharp electrode injects electrons every half period due to cold field emission. We assume that a chlorobenzene molecule has an uncompensated dipole moment, i.e., in a strong electric field, a chlorobenzene molecule gains an electron and forms a negative ion. However, an electron can exist in a polar dielectric solution for a long time due to the effect of its salvation by the medium. Therefore, the ionization process occurs both at the injector electrode and in the bulk of the medium. The mathematical model includes the Navier–Stokes equation, the transport equations for electrons and ionized chlorobenzene molecules, and the equation of alternating electric field. The sharp electrode boundary conditions are formulated to take into account the cold emission of electrons, which is described by a piecewise linear approximation of the Fowler–Nordheim law. The study of the performance of the proposed model was carried out by the method of direct numerical simulations of non-stationary physical fields. It is shown that if the presence of electrons in the volume of the medium is ignored, the averaged electrohydrodynamic convection does not occur, even if the sharp electrode continues to inject chlorobenzene molecules. The proposed model explains most of the experimental facts. As in the experiment, the excitation of convection critically depends on the mole fraction of chlorobenzene in the binary mixture. The calculated and experimentally observed structures of the convective flow are in good qualitative and quantitative agreement. The results of numerical experiments provide a good basis for the development of an averaged theory of electrohydrodynamic convection.

Instability, index theorem, and exponential trichotomy for Linear Hamiltonian PDEs

Consider a general linear Hamiltonian system∂tu=JLu\partial _{t}u=JLuin a Hilbert spaceXX. We assume that L:X→X∗\ L:X\rightarrow X^{\ast }induces a bounded and symmetric bi-linear form⟨L⋅,⋅⟩\left \langle L\cdot ,\cdot \right \rangleonXX, which has only finitely many negative dimensionsn−(L)n^{-}(L). There is no restriction on the anti-self-dual operatorJ:X∗⊃D(J)→XJ:X^{\ast }\supset D(J)\rightarrow X. We first obtain a structural decomposition ofXXinto the direct sum of several closed subspaces so thatLLis blockwise diagonalized andJLJLis of upper triangular form, where the blocks are easier to handle. Based on this structure, we first prove the linear exponential trichotomy ofetJLe^{tJL}. In particular,etJLe^{tJL}has at most algebraic growth in the finite co-dimensional center subspace. Next we prove an instability index theorem to relaten−(L)n^{-}\left ( L\right )and the dimensions of generalized eigenspaces of eigenvalues of JL\ JL, some of which may be embedded in the continuous spectrum. This generalizes and refines previous results, where mostlyJJwas assumed to have a bounded inverse. More explicit information for the indexes with pure imaginary eigenvalues are obtained as well. Moreover, when Hamiltonian perturbations are considered, we give a sharp condition for the structural instability regarding the generation of unstable spectrum from the imaginary axis. Finally, we discuss Hamiltonian PDEs including dispersive long wave models (BBM, KDV and good Boussinesq equations), 2D Euler equation for ideal fluids, and 2D nonlinear Schrödinger equations with nonzero conditions at infinity, where our general theory applies to yield stability or instability of some coherent states.

Global asymptotic stability of a scalar delay Nicholson's blowflies equation in periodic environment

This paper is considered with a scalar delay Nicholson's blowflies equation in periodic environments. By taking advantage of some novel differential inequality techniques and the fluctuation lemma, we set up the sharp condition to characterize the global asymptotic stability of positive periodic solutions on the addressed equation. The obtained results improve and supplement some existing ones in recent literature, and then give a more perfect answer to an open problem proposed by Berezansky et al. in [Appl. Math. Model. 34(2010) 1405–1417]. In particular, two numerical examples are provided to verify the reliability and feasibility of the theoretical findings.

Sharp conditions for a class of nonlinear Schrödinger equations.

This paper studies a class of nonlinear Schrödinger equations in two space dimensions. By constructing a variational problem and the so-called invariant manifolds of the evolution flow, we get a sharp condition for global existence and blow-up of solutions.

Global population dynamics of a single species structured with distinctive time-varying maturation and self-limitation delays

&lt;p style='text-indent:20px;'&gt;We consider the classical Nicholson's blowflies model incorporating two distinctive time-varying delays. One of the delays corresponds to the length of the individual's life cycle, and another corresponds to the specific physiological stage when self-limitation feedback takes place. Unlike the classical formulation of Nicholson's blowflies equation where self-regulation appears due to the competition of the productive adults for resources, the self-limitation of our considered model can occur at any developmental stage of an individual during the entire life cycle. We aim to find sharp conditions for the global asymptotic stability of a positive equilibrium. This is a significant challenge even when both delays are held at constant values. Here, we develop an approach to obtain a sharp and explicit criterion in an important situation where the two delays are asymptotically apart. Our approach can be also applied to the non-autonomous Mackey-Glass equation to provide a partial solution to an open problem about the global dynamics.&lt;/p&gt;

Cross-constrained variational method and nonlinear Schrödinger equation with partial confinement

&lt;p style='text-indent:20px;'&gt;In this paper, we study the nonlinear Schrödinger equation with a partial confinement. By applying the cross-constrained variational arguments and invariant manifolds of the evolution flow, the sharp condition for global existence and blowup of the solution is derived.&lt;/p&gt;

On a system of nonlinear Schrödinger equations with quadratic interaction and L2-critical growth

In this paper, we study the dynamical behavior of solutions of nonlinear Schrödinger equations with quadratic interaction and [Formula: see text]-critical growth. We give sharp conditions under which the existence of global and blow-up solutions are deduced. We also show the existence, stability, and blow-up behavior of normalized solutions of this system.