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Gras Georges - One of the best experts on this subject based on the ideXlab platform.
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Greenberg's conjecture for totally real number fields in terms of algorithmic complexity
2020Co-Authors: Gras GeorgesAbstract:Let K be a totally real number field and let K\_$\infty$ be its cyclotomic Z\_p-extension, p$\ge$2. Generalizing some viewpoints of Taya and others, we show that Greenberg's conjecture (lambda = mu = 0) depends on images, of ideal norms along the stages K\_n/K of the tower, in the torsion group T\_K of the Galois group of the maximal abelian p-ramified pro-p-extension of K; these images (obtained inductively via a classical algorithm in each K\_n) taKe place both in the p-class group Cl\_K and in the normalized p-adic regulator R\_K of K (Theorem 6.2). A property of uniform distribution of these images (Conjecture 6.4) would lead to density results needed for a proof of Greenberg's conjecture, which remains hopeless within the sole frameworK of Iwasawa's theory. Indeed, many "algebraic/class field theory" criteria exist, which hide a broad p-adic arithmetic and algorithmic complexity governed by T\_K. No assumption is made on the degree [K : Q], nor on the decomposition of p in K/Q.Comment: Improvements, corrections in proof of thm 6.2; addition of \S 6.2.5; new references. Based on the two articles: https://doi.org/10.5802/ambp.370 and https://doi.org/10.1007/s40316-018-0108-
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Greenberg's conjecture for totally real number fields in terms of algorithmic complexity
HAL CCSD, 2020Co-Authors: Gras GeorgesAbstract:Improvements, corrections in proof of thm 6.2; addition of § 6.2.5; new references. Based on the two articles: https://doi.org/10.5802/ambp.370 and https://doi.org/10.1007/s40316-018-0108-3Let K be a totally real number field and let K_∞ be its cyclotomic Z_p-extension, p≥2. Generalizing some viewpoints of Taya and others, we show that Greenberg's conjecture (lambda = mu = 0) depends on images, of ideal norms along the stages K_n/K of the tower, in the torsion group T_K of the Galois group of the maximal abelian p-ramified pro-p-extension of K; these images (obtained inductively via a classical algorithm in each K_n) taKe place both in the p-class group Cl_K and in the normalized p-adic regulator R_K of K (Theorem 6.2). A property of uniform distribution of these images (Conjecture 6.4) would lead to density results needed for a proof of Greenberg's conjecture, which remains hopeless within the sole frameworK of Iwasawa's theory. Indeed, many ``algebraic/class field theory'' criteria exist, which hide a broad p-adic arithmetic and algorithmic complexity governed by T_K. No assumption is made on the degree [K : Q], nor on the decomposition of p in K/Q
Morris W. Hirsch - One of the best experts on this subject based on the ideXlab platform.
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Monotone local flows with dense periodic orbits
arXiv: Dynamical Systems, 2020Co-Authors: Morris W. HirschAbstract:Author(s): Hirsch, Morris W | Abstract: Let X be a connected open set in n-dimensional Euclidean space, partially ordered by a closed convex cone K with nonempty interior: y g x if and only if y-x is nonzero and in K. Theorem: If F is a monotone local flow in X whose periodic points are dense in X, then F is globally periodic.
Hirsch, Morris W. - One of the best experts on this subject based on the ideXlab platform.
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Monotone local flows with dense periodic orbits
2018Co-Authors: Hirsch, Morris W.Abstract:Let X be a connected open set in n-dimensional Euclidean space, partially ordered by a closed convex cone K with nonempty interior: y > x if and only if y-x is nonzero and in K. Theorem: If F is a monotone local flow in X whose periodic points are dense in X, then F is globally periodic.Comment: Incorrect proof of main theore
Manning Jeffrey - One of the best experts on this subject based on the ideXlab platform.
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Patching and Multiplicity $2^K$ for Shimura Curves
2020Co-Authors: Manning JeffreyAbstract:We use the Taylor-Wiles-Kisin patching method to investigate the multiplicities with which Galois representations occur in the mod $\ell$ cohomology of Shimura curves over totally real number fields. Our method relies on explicit computations of local deformation rings done by Shotton, which we use to compute the Weil class group of various deformation rings. Exploiting the natural self-duality of the cohomology groups, we use these class group computations to precisely determine the structure of a patched module in many new cases in which the patched module is not free (and so multiplicity one fails). Our main result is a "multiplicity $2^K$" Theorem in the minimal level case (which we prove under some mild technical hypotheses), where $K$ is a number that depends only on local Galois theoretic information at the primes dividing the discriminant of the Shimura curve. Our result generalizes Ribet's classical multiplicity 2 result and the results of Cheng, and provides progress towards the Buzzard-Diamond-Jarvis local-global compatibility conjecture. We also prove a statement about the endomorphism rings of certain modules over the HecKe algebra, which may have applications to the integral Eichler basis problem.Comment: 51 pages. To appear in Algebra & Number Theor
Azween Abdullah - One of the best experts on this subject based on the ideXlab platform.
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energy balancing through cluster head selection using K Theorem in homogeneous wireless sensor networKs
arXiv: Networking and Internet Architecture, 2012Co-Authors: Muhammad Imran, Asfandyar Khan, Azween AbdullahAbstract:Department of Computer & Information SciencesUniversiti Technologi PETRONASBandar Seri IsKandar, 31750 Tronoh, PeraK, Malaysia.cmimran81@yahoo.com, asfand43@yahoo.com, azweenabdullah@petronas.com.myAbstract-The objective of this paper is to increase life time of homogeneous wireless sensor networKs (W SNs) through minimizing longrange communication and energy balancing. Sensor nodes are resource constrained particularly with limited energy that is difficult o rimpossible to replenish. LEACH (L ow Energy Adaptive Clustering Hierarchy) is most well -Known cluster based architecture for WSN that aimsto evenly dissipate energy among all sensor nodes. In cluster based architecture, the role of cluster head is very c rucial for the successfuloperation of WSN because once the cluster head becomes non functional, the whole cluster becomes dysfunctional. We have proposed a modifiedcluster based WSN architecture by introducing a coordinator node (C N) that is rich in term s of resources. This CN taKe up the responsibility oftransmitting data to the base station over longer distances from cluster heads. We have proposed a cluster head selection algorithm based on K -Theorem and other parameters i.e. residual energy, distance to coordinator node, reliability and degree of mobility. The K -Theorem is used toselect candidate cluster heads based on bunch of sensor nodes in a cluster. We believe that the proposed architecture and algorithm achieveshigher energy efficiency through minimizing communication and energy balancing. The proposed architecture is more scalable and proposedalgorithm is robust against even/uneven node deployment and node mobility.