scholarly journals MORSE INEQUALITIES ON CERTAIN INFINITE 2-COMPLEXES

2007 ◽  
Vol 49 (2) ◽  
pp. 155-165 ◽  
Author(s):  
R. AYALA ◽  
L. M. FERNÁNDEZ ◽  
J. A. VILCHES
Keyword(s):  
Finite Number ◽  
Morse Function ◽  
Dimensional Case ◽  
Morse Inequalities ◽  
Discrete Morse Function

AbstractUsing the notion of discrete Morse function introduced by R. Forman for finite cw-complexes, we generalize it to the infinite 2-dimensional case in order to get the corresponding version of the well-known discrete Morse inequalities on a non-compact triangulated 2-manifold without boundary and with finite homology. We also extend them for the more general case of a non-compact triangulated 2-pseudo-manifold with a finite number of critical simplices and finite homology.

scholarly journals Discrete Morse Inequalities on Infinite Graphs

10.37236/127 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Rafael Ayala ◽  
Luis M. Fernández ◽  
José A. Vilches
Keyword(s):  
Morse Function ◽  
Special Kind ◽  
Infinite Graphs ◽  
Morse Inequalities ◽  
Morse Functions ◽  
Critical Elements ◽  
Finite Case ◽  
Discrete Morse Function

The goal of this paper is to extend to infinite graphs the known Morse inequalities for discrete Morse functions proved by R. Forman in the finite case. In order to get this result we shall use a special kind of infinite subgraphs on which a discrete Morse function is monotonous, namely, decreasing rays. In addition, we shall use this result to characterize infinite graphs by the number of critical elements of discrete Morse functions defined on them.

scholarly journals WITTEN DEFORMATION FOR NONCOMPACT MANIFOLDS WITH BOUNDED GEOMETRY

2021 ◽  
pp. 1-38
Author(s):  
Xianzhe Dai ◽  
Junrong Yan
Keyword(s):  
Essential Role ◽  
Morse Function ◽  
Noncompact Manifold ◽  
Morse Inequalities ◽  
Bounded Geometry ◽  
Noncompact Manifolds ◽  
Witten Laplacian ◽  
Relative Cohomology ◽  
Witten Deformation ◽  
Small Eigenvalues

Abstract Motivated by the Landau–Ginzburg model, we study the Witten deformation on a noncompact manifold with bounded geometry, together with some tameness condition on the growth of the Morse function f near infinity. We prove that the cohomology of the Witten deformation $d_{Tf}$ acting on the complex of smooth $L^2$ forms is isomorphic to the cohomology of the Thom–Smale complex of f as well as the relative cohomology of a certain pair $(M, U)$ for sufficiently large T. We establish an Agmon estimate for eigenforms of the Witten Laplacian which plays an essential role in identifying these cohomologies via Witten’s instanton complex, defined in terms of eigenspaces of the Witten Laplacian for small eigenvalues. As an application, we obtain the strong Morse inequalities in this setting.

scholarly journals Some examples of minimally degenerate Morse functions

1987 ◽  
Vol 30 (2) ◽  
pp. 289-293 ◽  
Author(s):  
Frances Kirwan
Keyword(s):  
Riemannian Manifold ◽  
Critical Points ◽  
Morse Function ◽  
Compact Riemannian Manifold ◽  
Connected Components ◽  
Morse Inequalities ◽  
Poincaré Polynomial ◽  
Morse Functions ◽  
Image Position

Let X be a compact Riemannian manifold. If f:X→ℝ is a nondegenerate Morse function in the sense of Bott [2] then one has Morse inequalities which can be expressed in the formwhere Pt(X) is the Poincaré polynomial Σtidim Hi(X;ℚ of X ann {Cβ|β ∈B} are the connected components of the set of critical points for f For any polynomial Q(t)∈ℤ[t] we write Q(t)≧0 if all the coefficients of Q are nonnegative.

scholarly journals Expected sizes of Poisson–Delaunay mosaics and their discrete Morse functions

2017 ◽  
Vol 49 (3) ◽  
pp. 745-767 ◽  
Author(s):  
Herbert Edelsbrunner ◽  
Anton Nikitenko ◽  
Matthias Reitzner
Keyword(s):  
Point Process ◽  
Poisson Point Process ◽  
Probabilistic Models ◽  
Morse Function ◽  
Discrete Point ◽  
Expected Number ◽  
Low Dimensions ◽  
Morse Functions ◽  
Point Set ◽  
Discrete Morse Function

AbstractMapping every simplex in the Delaunay mosaic of a discrete point set to the radius of the smallest empty circumsphere gives a generalized discrete Morse function. Choosing the points from a Poisson point process in ℝn, we study the expected number of simplices in the Delaunay mosaic as well as the expected number of critical simplices and nonsingular intervals in the corresponding generalized discrete gradient. Observing connections with other probabilistic models, we obtain precise expressions for the expected numbers in low dimensions. In particular, we obtain the expected numbers of simplices in the Poisson–Delaunay mosaic in dimensionsn≤ 4.

scholarly journals Min-Max Theory for Cell Complexes

2020 ◽  
Vol 27 (03) ◽  
pp. 447-454
Author(s):  
Lacey Johnson ◽  
Kevin Knudson
Keyword(s):  
Morse Function ◽  
Critical Values ◽  
Discrete Version ◽  
Mountain Pass Lemma ◽  
Smooth Functions ◽  
Alternate Proof ◽  
Cell Complexes ◽  
Lusternik Schnirelmann ◽  
Regular Cell Complex ◽  
Discrete Morse Function

In the study of smooth functions on manifolds, min-max theory provides a mechanism for identifying critical values of a function. We introduce a discretized version of this theory associated to a discrete Morse function on a (regular) cell complex. As applications we prove a discrete version of the mountain pass lemma and give an alternate proof of a discrete Lusternik–Schnirelmann theorem.

scholarly journals Ascending and descending regions of a discrete Morse function

2009 ◽  
Vol 42 (6-7) ◽  
pp. 639-651 ◽  
Author(s):  
Gregor Jerše ◽  
Neža Mramor Kosta
Keyword(s):  
Morse Function ◽  
Discrete Morse Function

scholarly journals A Discrete Morse Perspective on Knot Projections and a Generalised Clock Theorem

10.37236/9979 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Daniele Celoria ◽  
Naya Yerolemou
Keyword(s):  
Morse Function ◽  
Finite Sequence ◽  
Graph Laplacian ◽  
Perfect Matchings ◽  
Closed Formula ◽  
Admissibility Condition ◽  
Spanning Forests ◽  
Discrete Morse Function ◽  
Complete Characterisation

We obtain a simple and complete characterisation of which matchings on the Tait graph of a knot diagram induce a discrete Morse function (dMf) on the two sphere, extending a construction due to Cohen. We show these dMfs are in bijection with certain rooted spanning forests in the Tait graph. We use this to count the number of such dMfs with a closed formula involving the graph Laplacian. We then simultaneously generalise Kauffman's Clock Theorem and Kenyon-Propp-Wilson's correspondence in two different directions; we first prove that the image of the correspondence induces a bijection on perfect dMfs, then we show that all perfect matchings, subject to an admissibility condition, are related by a finite sequence of click and clock moves. Finally, we study and compare the matching and discrete Morse complexes associated to the Tait graph, in terms of partial Kauffman states, and provide some computations.

scholarly journals ON THE MAXIMUM NUMBER OF PERIOD ANNULI FOR SECOND ORDER CONSERVATIVE EQUATIONS

2021 ◽  
Vol 26 (4) ◽  
pp. 612-630
Author(s):  
Armands Gritsans ◽  
Inara Yermachenko
Keyword(s):  
Differential Equation ◽  
Finite Number ◽  
Potential Function ◽  
Critical Points ◽  
Upper Bound ◽  
Morse Function ◽  
Second Order

We consider a second order scalar conservative differential equation whose potential function is a Morse function with a finite number of critical points and is unbounded at infinity. We give an upper bound for the number of nonglobal nontrivial period annuli of the equation and prove that the upper bound obtained is sharp. We use tree theory in our considerations.

scholarly journals Tunnel effect and symmetries for Kramers–Fokker–Planck type operators

2011 ◽  
Vol 10 (3) ◽  
pp. 567-634 ◽  
Author(s):  
Frédéric Hérau ◽  
Michael Hitrik ◽  
Johannes Sjöstrand
Keyword(s):  
Finite Number ◽  
Morse Function ◽  
Semiclassical Limit ◽  
Tunnel Effect ◽  
Local Minima ◽  
Semiclassical Asymptotics ◽  
Additional Assumptions ◽  
Exponentially Small ◽  
Fokker Planck

AbstractWe study operators of Kramers–Fokker–Planck type in the semiclassical limit, assuming that the exponent of the associated Maxwellian is a Morse function with a finite number n0 of local minima. Under suitable additional assumptions, we show that the first n0 eigenvalues are real and exponentially small, and establish the complete semiclassical asymptotics for these eigenvalues.

Criteria and Methods for Reliable Three-Dimensional Image Reconstruction

1974 ◽  
Vol 32 ◽  
pp. 330-331
Author(s):  
R. A. Crowther
Keyword(s):  
Image Reconstruction ◽  
Finite Number ◽  
Density Distribution ◽  
Electron Micrographs ◽  
Mathematical Problem ◽  
Three Dimensional ◽  
Ideal Case ◽  
Dimensional Image ◽  
General Object ◽  
The Ideal

The reconstruction of a three-dimensional image of a specimen from a set of electron micrographs reduces, under certain assumptions about the imaging process in the microscope, to the mathematical problem of reconstructing a density distribution from a set of its plane projections.In the absence of noise we can formulate a purely geometrical criterion, which, for a general object, fixes the resolution attainable from a given finite number of views in terms of the size of the object. For simplicity we take the ideal case of projections collected by a series of m equally spaced tilts about a single axis.

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