scholarly journals Estimates by gap potentials of free homotopy decompositions of critical Sobolev maps

2019 ◽  
Vol 9 (1) ◽  
pp. 1214-1250
Author(s):  
Jean Van Schaftingen
Keyword(s):  
Riemannian Manifold ◽  
Finite Number ◽  
Homotopy Group ◽  
Compact Riemannian Manifold ◽  
A Priori ◽  
Homotopy Classes ◽  
Continuous Map ◽  
Homotopy Decomposition ◽  
Sobolev Maps ◽  
Finite Set

Abstract A free homotopy decomposition of any continuous map from a compact Riemannian manifold 𝓜 to a compact Riemannian manifold 𝓝 into a finite number maps belonging to a finite set is constructed, in such a way that the number of maps in this free homotopy decomposition and the number of elements of the set to which they belong can be estimated a priori by the critical Sobolev energy of the map in Ws,p(𝓜, 𝓝), with sp = m = dim 𝓜. In particular, when the fundamental group π1(𝓝) acts trivially on the homotopy group πm(𝓝), the number of homotopy classes to which a map can belong can be estimated by its Sobolev energy. The estimates are particular cases of estimates under a boundedness assumption on gap potentials of the form $$\begin{array}{} \displaystyle \iint\limits_{\substack{(x, y) \in \mathcal{M} \times \mathcal{M} \\ d_\mathcal{N} (f (x), f (y)) \ge \varepsilon}} \frac{1}{d_\mathcal{M} (y, x)^{2 m}} \, \mathrm{d} y \, \mathrm{d}x. \end{array}$$ When m ≥ 2, the estimates scale optimally as ε → 0. When m = 1, the total variation of the maps appearing in the decomposition can be controlled by the gap potential. Linear estimates on the Hurewicz homomorphism and the induced cohomology homomorphism are also obtained.

scholarly journals Hearing the shape of a compact Riemannian manifold with a finite number of piecewise impedance boundary conditions

1997 ◽  
Vol 20 (2) ◽  
pp. 397-402 ◽  
Author(s):  
E. M. E. Zayed
Keyword(s):  
Riemannian Manifold ◽  
Boundary Conditions ◽  
Finite Number ◽  
Spectral Function ◽  
Compact Riemannian Manifold ◽  
Smooth Boundary ◽  
Beltrami Operator ◽  
Smooth Functions ◽  
Impedance Boundary ◽  
Impedance Boundary Conditions

The spectral functionΘ(t)=∑i=1∞exp(−tλj), where{λj}j=1∞are the eigenvalues of the negative Laplace-Beltrami operator−Δ, is studied for a compact Riemannian manifoldΩof dimension “k” with a smooth boundary∂Ω, where a finite number of piecewise impedance boundary conditions(∂∂ni+γi)u=0on the parts∂Ωi(i=1,…,m)of the boundary∂Ωcan be considered, such that∂Ω=∪i=1m∂Ωi, andγi(i=1,…,m)are assumed to be smooth functions which are not strictly positive.

The inverse problem of recovering the coefficients of a differential equations on a graph

2020 ◽  
Vol 28 (5) ◽  
pp. 727-738
Author(s):  
Victor Sadovnichii ◽  
Yaudat Talgatovich Sultanaev ◽  
Azamat Akhtyamov
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Finite Number ◽  
Characteristic Equation ◽  
Arbitrary Number ◽  
New Class ◽  
Finite Set

AbstractWe consider a new class of inverse problems on the recovery of the coefficients of differential equations from a finite set of eigenvalues of a boundary value problem with unseparated boundary conditions. A finite number of eigenvalues is possible only for problems in which the roots of the characteristic equation are multiple. The article describes solutions to such a problem for equations of the second, third, and fourth orders on a graph with three, four, and five edges. The inverse problem with an arbitrary number of edges is solved similarly.

On the non-regularity of tangent sphere bundles

1978 ◽  
Vol 82 (1-2) ◽  
pp. 13-17 ◽  
Author(s):  
David E. Blair
Keyword(s):  
Riemannian Manifold ◽  
Large Class ◽  
Constant Curvature ◽  
Positive Constant ◽  
Contact Structure ◽  
Compact Riemannian Manifold ◽  
Contact Structures ◽  
Sphere Bundle ◽  
Contact Manifolds ◽  
Tangent Sphere

SynopsisClassically the tangent sphere bundles have formed a large class of contact manifolds; their contact structures are not in general regular, however. Specifically we prove that the natural contact structure on the tangent sphere bundle of a compact Riemannian manifold of non-positive constant curvature is not regular.

EIGENVALUES OF GEOMETRIC OPERATORS RELATED TO THE WITTEN LAPLACIAN UNDER THE RICCI FLOW

2017 ◽  
Vol 59 (3) ◽  
pp. 743-751
Author(s):  
SHOUWEN FANG ◽  
FEI YANG ◽  
PENG ZHU
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Ricci Flow ◽  
Compact Riemannian Manifold ◽  
Laplacian Operator ◽  
Witten Laplacian ◽  
Normalized Ricci Flow ◽  
Geometric Operator

AbstractLet (M, g(t)) be a compact Riemannian manifold and the metric g(t) evolve by the Ricci flow. In the paper, we prove that the eigenvalues of geometric operator −Δφ + $\frac{R}{2}$ are non-decreasing under the Ricci flow for manifold M with some curvature conditions, where Δφ is the Witten Laplacian operator, φ ∈ C2(M), and R is the scalar curvature with respect to the metric g(t). We also derive the evolution of eigenvalues under the normalized Ricci flow. As a consequence, we show that compact steady Ricci breather with these curvature conditions must be trivial.

scholarly journals FEATURES OF SECTIONING THE TURNS OF THE TRANSFORMING ELEMENT OF THE TRANSFORMER-KEY ACTUATING STRUCTURE IN THE BOOST CHANNEL OF THE DC POWER SYSTEM

2020 ◽  
Vol 2020 (6) ◽  
pp. 25-31
Author(s):  
K.O. Lypkivskyi ◽  
◽  
A.G. Mozharovskyi ◽  
Keyword(s):  
High Efficiency ◽  
A Priori ◽  
Current Source ◽  
Voltage Regulation ◽  
Specific Class ◽  
Voltage Supply ◽  
Emergency Situations ◽  
Time Control ◽  
Finite Set ◽  
In Series

One of the effective ways to ensure the normalized operation of the electricity consumer with an unstable primary power source is the organization of the corresponding voltage supply channel. In a system with a direct current source, the voltage supply is implemented by introducing a rectifier semiconductor bridge in series with the load, into the diagonal of which AC energy is supplied, the voltage level of which is purposefully changed by a corresponding converter with a transformer- and-switches executive structure (TSES). To achieve high efficiency of the use of key elements of TSVS, it is proposed to assign the functions of rectification and voltage regulation to a specific class of TSES – a multilevel rectifier consisting of a transformer and a finite set of parallel connected pairs of serially connected thyristors, the common points of which are connected to the corresponding taps of the sectioned secondary turns of the transformer. By discrete-time control of thyristors, it is necessary to regulate voltage levels, it is attached. The linearity of the scale of these levels is ensured by the proposed transformer sectioning law. This power supply system is characterized by small energy losses in semiconductor elements (only two thyristors work at a time), and the a priori impossibility of emergency situations during transitions from one level to another. References 14, figures 3, tables 3.

The Palais–Smale conditions for the Yang–Mills functional

1988 ◽  
Vol 108 (3-4) ◽  
pp. 189-200
Author(s):  
D. R. Wilkins
Keyword(s):  
Riemannian Manifold ◽  
Principal Bundle ◽  
Compact Riemannian Manifold ◽  
Hilbert Manifold ◽  
Yang Mills ◽  
Palais Smale Condition ◽  
Dimension 2

SynopsisWe consider the Yang–Mills functional denned on connections on a principal bundle over a compact Riemannian manifold of dimension 2 or 3. It is shown that if we consider the Yang–Mills functional as being defined on an appropriate Hilbert manifold of orbits of connections under the action of the group of principal bundle automorphisms, then the functional satisfies the Palais–Smale condition.

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