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Homology of Powers of Ideals： Artin-Rees numbers of Syzygies and the Golod Property

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Algebra Colloquium 2016年第4期23卷 689-700页

作者： Jiirgen Herzog Volkmar Welker Siamak Yassemi Fachbereich Mathematik Universitat Duisburg-Essen Campus Essen 45117 Essen Germany Philipps-Universitat Marburg Fachbereich Mathematik und Informatik 35032 Marburg Germany School of Mathematics Statistics and Computer Science College of Science University of Tehran Tehran Iran

Let Ro be a Noetherian local ring and R a standard graded R0-algebra with maximal ideal ra and residue class field K = Rim. For a graded ideal I in R we show that for k 〉〉 0：（1） the Artin-Rees number of the syzygy modules of Ik as submodules of the free modules from a free resolution is constant, and thereby the Artin-Rees number can be presented as a proper replacement of regularity in the local situation; and （2） R is a polynomial ring over the regular Ro, the ring R/Ik is Golod, its Poincar4-betti series is rational and the betti numbers of the free resolution of K over R/I^k are polynomials in k of a specific degree. The first result is an extension of the work by Swanson on the regularity of I^k for k 〉〉 0 from the graded situation to the local situation. The polynomiality consequence of the second result is an analog of the work by Kodiyalam on the behaviour of betti numbers of the minimal free resolution of R/Ik over R.

关键词： Artin-Rees numbers syzygies Golod rings betti numbers deviations

Algebraic Characterization of SSC of Uni-Cyclic Multigraphs

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Algebra Colloquium 2023年第2期30卷 325-338页

作者： Imran Ahmed Shahid Muhmood Department of Mathematics COMSATS University Islamabad Lahore CampusLahorePakistan

We introduce first the spanning simplicial complex(SSC)of a multigraph g,which gives a generalization of the SSC associated with a simple graph *** properties are discussed for the SSC of a family of uni-cyclic multigraphs U_(n)^(r),m with n edges including r multiple edges within and outside the cycle of length m,which are then used to compute the f-vector and Hilbert series of face ring k[△s(U_(n)^(r),m)]for the SSC △s(U_(n)^(r),m)(un,m).Moreover,we find the associated primes of the facet ideal I_(F)(△s(U_(n)^(r),m).Finally,we device a formula for homology groups of △s(U_(n)^(r),m) prove that the SsC of a family of uni-cyclic multigraphs is Cohen-Macaulay.

关键词： simplicial complexes spanning trees Hilbert series vertex covers betti numbers

Biggs Theorem for Directed Cycles and Topological Invariants of Digraphs

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Advances in Pure Mathematics 2021年第6期11卷 573-594页

作者： Michael Hecht Ivo F. Sbalzarini Technische Universitä t Dresden Faculty of Computer Science Dresden Germany Max Planck Institute of Molecular Cell Biology and Genetics Dresden Germany Center for Systems Biology Dresden Dresden Germany

We generalize Biggs Theorem to the case of directed cycles of multi-digraphs allowing to compute the dimension of the directed cycle space independently of the graph representation with linear runtime complexity. By considering two-dimensional CW complex of elementary cycles and deriving formulas for the betti numbers of the associated cellular homology groups, we extend the list of representation independent topological inavariants measuring the graph structure. We prove the computation of the 2nd betti number to be sharp #P hard in general and present specific representation invariant sub-fillings yielding efficiently computable homology groups. Finally, we suggest how to use the provided structural measures to shed new light on graph theoretical problems as graph embeddings, discrete Morse theory and graph clustering.

关键词： Biggs Theorem Elementary and Simple Cycles CW Complexes of Graphs Cellular and Singular Homology betti numbers

A Unified Newtonian-Relativistic Quantum Resolution of the Supposedly Missing Dark Energy of the Cosmos and the Constancy of the Speed of Light

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International Journal of Modern Nonlinear Theory and Application 2013年第1期2卷 43-54页

作者： M.S.El Naschie Department of Physics University of AlexandriaAlexandriaEgypt

Time dilation, space contraction and relativistic mass are combined in a novel fashion using Newtonian dynamics. In this way we can surprisingly retrieve an effective quantum gravity energy-mass equation which gives the accurate experimental value of vacuum density. Furthermore Einstein’s equation of special relativity E = mc2, where m is the mass and c is the velocity of light developed assuming smooth 4D space time is transferred to a rugged Calabi-Yau and K3 fuzzy Kahler manifolds and revised to become E=(mc2)/(22), where the division factor 22 maybe interpreted as the compactified bosonic dimensions of Veneziano-Nambu strings. The result is again an accurate effective quantum gravity energy-mass relation akin to the one found using Newtonian dynamics which correctly predicts that 95.4915028% of the energy in the cosmos is the hypothetical missing dark energy. The agreement with WMAP and supernova measurements is in that respect astounding. In addition different theories are used to check the calculations and all lead to the same quantitative result. Thus the theories of varying speed of light, scale relativity, E-infinity theory, M-theory, Heterotic super strings, quantum field in curved space time, Veneziano’s dual resonance model, Nash Euclidean embedding and super gravity all reinforce, without any reservation, the above mentioned theoretical result which in turn is in total agreement with the most sophisticated cosmological measurements which was deservingly awarded the 2011 Nobel Prize in Physics. Finally and more importantly from certain viewpoints, we reason that the speed of light is constant because it is a definite probabilistic expectation value of a variable velocity in a hierarchical fractal clopen, i.e. closed and open micro space time.

关键词： Dark Energy Homology of Fuzzy Kahler betti numbers Heterotic Strings Revised Special Relativity Speed of Light as a Probabilistic Expectation Value

Mohamed El Naschie’s Revision of Albert Einstein’s E=m_(0)c^(2):A Definite Resolution of the Mystery of the Missing Dark Energy of the Cosmos

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International Journal of Modern Nonlinear Theory and Application 2013年第1期2卷 55-59页

作者： J.H.He L.Marek-Crnjac Nantong Textile Institute National Engineering Laboratory for Modern SilkCollege of Textile and ClothingSoochow UniversitySuzhouChina Technical School Center of Maribor MariborSlovenia

The Egyptian engineering scientist and theoretical physicist Mohamed El Naschie has found a definite resolution to the missing dark energy of the cosmos based on a revision of the theory of Relativity. Einstein’s equation of special relativity E = m0c2, where m0 is the controversial rest mass and c is the velocity of light developed in smooth 4D space-time was transferred by El Naschie to a rugged Calabi-Yau and K3 fuzzy Kahler manifold. The result is an accurate, effective quantum gravity energy-mass relation which correctly predicts that 95.4915028% of the energy in the cosmos is the missing hypothetical dark energy. The agreement with WMAP and supernova measurements is astounding. Different theories are used by El Naschie to check the calculations and all lead to the same quantitative result. Thus the theories of varying speed of light, scale relativity, E-infinity theory, M-theory, Heterotic super strings, quantum field in curved space-time, Veneziano’s dual resonance model and Nash’s Euclidean embedding all reinforce, without any reservation, the above mentioned theoretical result of El Naschie which in turn is in total agreement with the most sophisticated cosmological measurement. Incidentally these experimental measurements and analysis were awarded the 2011 Nobel Prize in Physics to Adam Riess, Brian Schmidt, and Saul Perlmutter.

关键词： Dark Matter Homology of Fuzzy Kahler betti numbers Heterotic Strings New Special Relativity Theory