scholarly journals Degeneration at E2 of certain spectral sequences

2016 ◽  
Vol 27 (14) ◽  
pp. 1650111 ◽  
Author(s):  
Dan Popovici
Keyword(s):  
Spectral Sequence ◽  
Differential Operators ◽  
Spectral Gap ◽  
Sufficient Conditions ◽  
Main Idea ◽  
Second Step ◽  
Compact Complex Manifold ◽  
Frölicher Spectral Sequence ◽  
Foliated Structure ◽  
Laplace Type

We propose a Hodge theory for the spaces [Formula: see text] featuring at the second step either in the Frölicher spectral sequence of an arbitrary compact complex manifold [Formula: see text] or in the spectral sequence associated with a pair [Formula: see text] of complementary regular holomorphic foliations on such a manifold. The main idea is to introduce a Laplace-type operator associated with a given Hermitian metric on [Formula: see text] whose kernel in every bidegree [Formula: see text] is isomorphic to [Formula: see text] in either of the two situations discussed. The surprising aspect is that this operator is not a differential operator since it involves a harmonic projection, although it depends on certain differential operators. We then use this Hodge isomorphism for [Formula: see text] to give sufficient conditions for the degeneration at [Formula: see text] of the spectral sequence considered in each of the two cases in terms of the existence of certain metrics on [Formula: see text]. For example, in the Frölicher case, we prove degeneration at [Formula: see text] if there exists an SKT metric [Formula: see text] (i.e. such that [Formula: see text]) whose torsion is small compared to the spectral gap of the elliptic operator [Formula: see text] defined by [Formula: see text]. In the foliated case, we obtain degeneration at [Formula: see text] under a hypothesis involving the Laplacians [Formula: see text] and [Formula: see text] associated with the splitting [Formula: see text] induced by the foliated structure.

scholarly journals On the degeneration of the Frölicher spectral sequence and small deformations

2020 ◽  
Vol 7 (1) ◽  
pp. 62-72
Author(s):  
Michele Maschio
Keyword(s):  
Spectral Sequence ◽  
Deformation Theory ◽  
Differential Operators ◽  
Complex Structure ◽  
Second Step ◽  
Complex Manifolds ◽  
Pseudo Differential Operators ◽  
Frölicher Spectral Sequence ◽  
Compact Complex Manifolds ◽  
Small Deformations

AbstractWe study the behavior of the degeneration at the second step of the Frölicher spectral sequence of a 𝒞∞ family of compact complex manifolds. Using techniques from deformation theory and adapting them to pseudo-differential operators we prove a result à la Kodaira-Spencer for the dimension of the second step of the Frölicher spectral sequence and we prove that, under a certain hypothesis, the degeneration at the second step is an open property under small deformations of the complex structure.

scholarly journals Higher-page Bott–Chern and Aeppli cohomologies and applications

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Dan Popovici ◽  
Jonas Stelzig ◽  
Luis Ugarte
Keyword(s):  
Positive Integer ◽  
Spectral Sequence ◽  
Complex Manifolds ◽  
Serre Duality ◽  
Frölicher Spectral Sequence ◽  
Compact Complex Manifolds

Abstract For every positive integer r, we introduce two new cohomologies, that we call E r {E_{r}} -Bott–Chern and E r {E_{r}} -Aeppli, on compact complex manifolds. When r = 1 {r\kern-1.0pt=\kern-1.0pt1} , they coincide with the usual Bott–Chern and Aeppli cohomologies, but they are coarser, respectively finer, than these when r ≥ 2 {r\geq 2} . They provide analogues in the Bott–Chern–Aeppli context of the E r {E_{r}} -cohomologies featuring in the Frölicher spectral sequence of the manifold. We apply these new cohomologies in several ways to characterise the notion of page- ( r - 1 ) {(r-1)} - ∂ ⁡ ∂ ¯ {\partial\bar{\partial}} -manifolds that we introduced very recently. We also prove analogues of the Serre duality for these higher-page Bott–Chern and Aeppli cohomologies and for the spaces featuring in the Frölicher spectral sequence. We obtain a further group of applications of our cohomologies to the study of Hermitian-symplectic and strongly Gauduchon metrics for which we show that they provide the natural cohomological framework.

scholarly journals The approximation of the solution of wave problems by spectral expansions connected with elliptic differential operators

2021 ◽  
Vol 86 (2) ◽  
pp. 101-106
Author(s):  
Abdulkasim Akhmedov ◽  
Mohd Zuki Salleh ◽  
Abdumalik Rakhimov
Keyword(s):  
Wave Propagation ◽  
Differential Operators ◽  
Sufficient Conditions ◽  
Elastic Membrane ◽  
Conductive Material ◽  
Spectral Expansions ◽  
Vibration Process ◽  
Thermally Conductive ◽  
Elliptic Differential Operators ◽  
Wave Problems

In this research, we investigate the spectral expansions connected with elliptic differential operators in the space of singular distributions, which describes the vibration process made of thin elastic membrane stretched tightly over a circular frame. The sufficient conditions for summability of the spectral expansions connected with wave problems on the disk are obtained by taking into account that the deflection of the membrane during the motion remains small compared to the size of the membrane and for wave propagation problems, the disk is made of some thermally conductive material.

scholarly journals Formal Analysis of SET and NSL Protocols Using the Interpretation Functions-Based Method

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Hanane Houmani ◽  
Mohamed Mejri
Keyword(s):  
Communication Channel ◽  
Formal Analysis ◽  
Sufficient Conditions ◽  
Main Idea ◽  
The Internet ◽  
Cryptographic Protocol ◽  
Cryptographic Primitives ◽  
Algebraic Properties ◽  
Security Properties ◽  
Set Protocol

Most applications in the Internet such as e-banking and e-commerce use the SET and the NSL protocols to protect the communication channel between the client and the server. Then, it is crucial to ensure that these protocols respect some security properties such as confidentiality, authentication, and integrity. In this paper, we analyze the SET and the NSL protocols with respect to the confidentiality (secrecy) property. To perform this analysis, we use the interpretation functions-based method. The main idea behind the interpretation functions-based technique is to give sufficient conditions that allow to guarantee that a cryptographic protocol respects the secrecy property. The flexibility of the proposed conditions allows the verification of daily-life protocols such as SET and NSL. Also, this method could be used under different assumptions such as a variety of intruder abilities including algebraic properties of cryptographic primitives. The NSL protocol, for instance, is analyzed with and without the homomorphism property. We show also, using the SET protocol, the usefulness of this approach to correct weaknesses and problems discovered during the analysis.

Resonance and non-resonance in terms of average values. II

2001 ◽  
Vol 131 (5) ◽  
pp. 1217-1235
Author(s):  
M. N. Nkashama ◽  
S. B. Robinson
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Nonlinear Term ◽  
Spectral Gap ◽  
Elliptic Boundary ◽  
Sufficient Conditions ◽  
Existence Results ◽  
Elliptic Boundary Value ◽  
Semilinear Elliptic ◽  
The Given

We prove existence results for semilinear elliptic boundary-value problems in both the resonance and non-resonance cases. What sets our results apart is that we impose sufficient conditions for solvability in terms of the (asymptotic) average values of the nonlinearities, thus allowing the nonlinear term to have significant oscillations outside the given spectral gap as long as it remains within the interval on the average in some sense. This work generalizes the results of a previous paper, which dealt exclusively with the ordinary differential equation (ODE) case and relied on ODE techniques.

Embedding theorems and the spectra of certain differential operators

1991 ◽  
Vol 434 (1892) ◽  
pp. 643-656 ◽  
Keyword(s):  
Differential Operators ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Independent Interest ◽  
Embedding Theorems ◽  
Necessary And Sufficient

This paper is concerned with problems of the form n Ʃ k =0 (─1) k ( ρ 2 k y ( k ))( k ) = λ r 2 y on R , ry ∈ L 2 ( R ) and gives conditions on the coefficients sufficient to ensure that the spectrum is discrete; necessary conditions are also given. In certain circumstances, necessary and sufficient conditions for discreteness are given, thus extending the celebrated Molcanov criterion. These results follow from embedding theorems which have independent interest.

Oscillation theorems for semilinear elliptic differential operators

1978 ◽  
Vol 82 (1-2) ◽  
pp. 135-151 ◽  
Author(s):  
Manabu Naito ◽  
Norio Yoshida
Keyword(s):  
Differential Operators ◽  
Sufficient Conditions ◽  
Unbounded Domains ◽  
Differential Inequalities ◽  
Elliptic Differential Operator ◽  
Semilinear Elliptic ◽  
All Solutions ◽  
Existence Of Positive Solutions ◽  
Elliptic Differential Operators ◽  
Oscillation Theorems

SynopsisThe semilinear elliptic differential operator L[u] = Δu + c(x, u) is studied and sufficient conditions are derived for all solutions of uL[u] ≦ 0 with suitable boundary conditions to be oscillatory in unbounded domains of Rn. Here, unbounded domains to be considered are cones, strips and cylinders in Rn. The results are based on the conditions for the non-existence of positive solutions of ordinary differential inequalities.

scholarly journals On the K-th Power of Stokes Operator

2020 ◽  
Vol 63 (1) ◽  
pp. 1-10
Author(s):  
Eleftherios Protopapas ◽  
Keyword(s):  
Stokes Equation ◽  
Differential Operators ◽  
Applied Mathematics ◽  
Sufficient Conditions ◽  
Spherical Geometry ◽  
Stokes Operator ◽  
Creeping Flow ◽  
Elliptic Type ◽  
Spherical Coordinate ◽  
Partial Differential Operators

Stokes operators, are well known partial differential operators of elliptic type, which are often used in Applied Mathematics. Stokes equation describes the irrotational, axisymmetric creeping flow and Stokes bi-stream equation denotes the rotational one, where Necessary and sufficient conditions for the separability and the R-separability of the equation have been proved recently. Moreover, the 0-eigenspace and the generalized 0-eigenspace of the operator have been derived in several coordinate systems. Specifically, the spherical coordinate system is employed in many problems taking into account that in many engineering applications, the solutions in spherical geometry seem to be adequate for solving a problem. In the present manuscript, it is shown that equation admits a solution of the form where are solutions of Stokes equation and r is the radial spherical variable. Additionally, we obtain the kernel of the k-th power of the Stokes operator, in the spherical geometry for every

scholarly journals An inverse spectral problem for Sturm-Liouville-type integro-differential operators with robin boundary conditions

2019 ◽  
Vol 50 (3) ◽  
pp. 207-221 ◽  
Author(s):  
Sergey Buterin
Keyword(s):  
Boundary Condition ◽  
Inverse Problem ◽  
Boundary Conditions ◽  
Differential Operators ◽  
A Priori ◽  
Sufficient Conditions ◽  
Robin Boundary Conditions ◽  
Sturm Liouville ◽  
Necessary And Sufficient ◽  
Robin Boundary

The perturbation of the Sturm--Liouville differential operator on a finite interval with Robin boundary conditions by a convolution operator is considered. The inverse problem of recovering the convolution term along with one boundary condition from the spectrum is studied, provided that the Sturm--Liouville potential as well as the other boundary condition are known a priori. The uniqueness of solution for this inverse problem is established along with necessary and sufficient conditions for its solvability. The proof is constructive and gives an algorithm for solving the inverse problem.

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