Monotonicity

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Tobias H Colding - One of the best experts on this subject based on the ideXlab platform.

  • new Monotonicity formulas for ricci curvature and applications i
    Acta Mathematica, 2012
    Co-Authors: Tobias H Colding
    Abstract:

    We prove three new Monotonicity formulas for manifolds with a lower Ricci curvature bound and show that they are connected to rate of convergence to tangent cones. In fact, we show that the derivative of each of these three monotone quantities is bounded from below in terms of the Gromov–Hausdorff distance to the nearest cone. The Monotonicity formulas are related to the classical Bishop–Gromov volume comparison theorem and Perelman’s celebrated Monotonicity formula for the Ricci flow. We will explain the connection between all of these. Moreover, we show that these new Monotonicity formulas are linked to a new sharp gradient estimate for the Green function that we prove. This is parallel to the fact that Perelman’s Monotonicity is closely related to the sharp gradient estimate for the heat kernel of Li–Yau. In [CM4] one of the Monotonicity formulas is used to show uniqueness of tangent cones with smooth cross-sections of Einstein manifolds. Finally, there are obvious parallelisms between our Monotonicity and the positive mass theorem of Schoen–Yau and Witten.

  • new Monotonicity formulas for ricci curvature and applications i
    arXiv: Differential Geometry, 2011
    Co-Authors: Tobias H Colding
    Abstract:

    We prove three new Monotonicity formulas for manifolds with a lower Ricci curvature bound and show that they are connected to rate of convergence to tangent cones. In fact, we show that the derivative of each of these three monotone quantities is bounded from below in terms of the Gromov-Hausdorff distance to the nearest cone. The Monotonicity formulas are related to the classical Bishop-Gromov volume comparison theorem and Perelman's celebrated Monotonicity formula for the Ricci flow. We will explain the connection between all of these. Moreover, we show that these new Monotonicity formulas are linked to a new sharp gradient estimate for the Green's function that we prove. This is parallel to that Perelman's Monotonicity is closely related to the sharp gradient estimate for the heat kernel of Li-Yau. In [CM4] we will use the Monotonicity formulas we prove here to show uniqueness of certain tangent cones of Einstein manifolds and in [CM3] we will prove a number of related Monotonicity formulas. Finally, there are obvious parallels between our Monotonicity and the positive mass theorem of Schoen-Yau and Witten.

Maciej Ciesielski - One of the best experts on this subject based on the ideXlab platform.

Paweł Kolwicz - One of the best experts on this subject based on the ideXlab platform.

  • Rotundity and Monotonicity properties of selected Cesàro function spaces
    Positivity, 2017
    Co-Authors: Tomasz Kiwerski, Paweł Kolwicz
    Abstract:

    We study rotundity, strict Monotonicity, lower local uniform Monotonicity and upper local uniform Monotonicity in some classes of Cesaro function spaces. We present full criteria of these properties in the Cesa ro–Orlicz function spaces \(Ces_{\varphi }\) (under some mild assumptions on the Orlicz function \(\varphi \)). Next, we prove a characterization of strict Monotonicity, lower local uniform Monotonicity and upper local uniform Monotonicity in the Cesaro–Lorentz function spaces \(C\Lambda _{\phi }\). We also show that the space \(C\Lambda _{\phi }\) is never rotund. Finally, we will prove that Cesaro–Marcinkiewicz function space \(CM_{\phi }^{(*)}\) is neither strictly monotone nor order continuous for any quasi-concave function \(\phi \).

  • A note on strict K-Monotonicity of some symmetric function spaces
    Commentationes Mathematicae, 2013
    Co-Authors: Maciej Ciesielski, Paweł Kolwicz, Ryszard Płuciennik
    Abstract:

    We discuss some sufficient and necessary conditions for strict K-Monotonicity of some important concrete symmetric spaces. The criterion for strict Monotonicity of the Lorentz space \(\Gamma _{p,w}\) with \(0\)

  • a note on strict k Monotonicity of some symmetric function spaces
    Commentationes Mathematicae, 2013
    Co-Authors: Maciej Ciesielski, Paweł Kolwicz, Ryszard Pluciennik
    Abstract:

    We discuss some sufficient and necessary conditions for strict K-Monotonicity of some important concrete symmetric spaces. The criterion for strict Monotonicity of the Lorentz space \(\Gamma _{p,w}\) with \(0\)<\(pandlt;\(\infty\) is given.

  • Monotonicity and rotundity of Lorentz spaces
    Nonlinear Analysis: Theory Methods & Applications, 2012
    Co-Authors: Maciej Ciesielski, Paweł Kolwicz, Anna Kamińska, Ryszard Płuciennik
    Abstract:

    Abstract Criteria for rotundity, strict Monotonicity, and lower local uniform Monotonicity of the Lorentz spaces Γ p , w of maximal functions are given under arbitrary nonnegative weight function w . Necessary conditions are also established for uniform Monotonicity of the spaces Γ p , w for 1 ≤ p ∞ . Moreover, the spaces Γ 1 , w that are uniformly monotone are characterized.

Amir M. Ben-amram - One of the best experts on this subject based on the ideXlab platform.

  • SAT-based termination analysis using Monotonicity constraints over the integers
    Theory and Practice of Logic Programming, 2011
    Co-Authors: Michael Codish, Amir M. Ben-amram, Igor Gonopolskiy, Carsten Fuhs, Jürgen Giesl
    Abstract:

    We describe an algorithm for proving termination of programs abstracted to systems of Monotonicity constraints in the integer domain. Monotonicity constraints are a nontrivial extension of the well-known size-change termination method. While deciding termination for systems of Monotonicity constraints is PSPACE complete, we focus on a well-defined and significant subset, which we call MCNP (for “Monotonicity constraints in NP”), designed to be amenable to a SAT-based solution. Our technique is based on the search for a special type of ranking function defined in terms of bounded differences between multisets of integer values. We describe the application of our approach as the back end for the termination analysis of Java Bytecode. At the front end, systems of Monotonicity constraints are obtained by abstracting information, using two different termination analyzers: AProVE and COSTA . Preliminary results reveal that our approach provides a good trade-off between precision and cost of analysis.

  • CAV - Size-Change Termination, Monotonicity Constraints and Ranking Functions
    Computer Aided Verification, 2009
    Co-Authors: Amir M. Ben-amram
    Abstract:

    Size-change termination involves deducing program termination based on the impossibility of infinite descent. To this end we may use a program abstraction in which transitions are described by Monotonicity constraints over (abstract) variables. When only constraints of the form x > y *** and x *** y *** are allowed, we have size-change graphs, for which both theory and practice are now more evolved then for general Monotonicity constraints. This work shows that it is possible to transfer some theory from the domain of size-change graphs to the general case, complementing and extending previous work on Monotonicity constraints. Significantly, we provide a procedure to construct explicit global ranking functions from Monotonicity constraints in singly-exponential time, which is better than what has been published so far even for size-change graphs. We also consider the integer domain, where general Monotonicity constraints are essential because the domain is not well-founded.

Dale R Durran - One of the best experts on this subject based on the ideXlab platform.

  • selective Monotonicity preservation in scalar advection
    Journal of Computational Physics, 2008
    Co-Authors: Peter N Blossey, Dale R Durran
    Abstract:

    An efficient method for scalar advection is developed that selectively preserves Monotonicity. Monotonicity preservation is applied only where the scalar field is likely to contain discontinuities as indicated by significant grid-cell-to-grid-cell variations in a smoothness measure conceptually similar to that used in weighted essentially non-oscillatory (WENO) methods. In smooth regions, the numerical diffusion associated with Monotonicity-preserving methods is avoided. The resulting method, while not globally Monotonicity preserving, allows the full accuracy of the underlying advection scheme to be achieved in smooth regions. The violations of Monotonicity that do occur are generally very small, as seen in the tests presented here. Strict positivity preservation may be effectively and efficiently obtained through an additional flux correction step. The underlying advection scheme used to test this methodology is a variant of the piecewise parabolic method (PPM) that may be applied to multi-dimensional problems using density-corrected dimensional splitting and permits stable semi-Lagrangian integrations using CFL numbers larger than one. Two methods for Monotonicity preservation are used here: flux correction and modification of the underlying polynomial reconstruction.