For an ideal I of a Noetherian local ring （R, m, k） one hasβR1（I） -β0R（I） ≥-1. It is demonstrated that some residual intersections of an ideal I for whichβ1R（I） -β0R（I） = -1 or 0 are perfect.
The authors compute the(rational) Betti number of real toric varieties associated to Weyl chambers of type B, and furthermore show that their integral cohomology is p-torsion free for all odd primes p.
In this paper, we introduce the notion of bounded Betti numbers, and show that the bounded Betti numbers of a closed Riemannian n-manifold （M, g） with Ric （M） ≥ -（n - 1） and Diam （M） ≤D are bounded by a numb...In this paper, we introduce the notion of bounded Betti numbers, and show that the bounded Betti numbers of a closed Riemannian n-manifold （M, g） with Ric （M） ≥ -（n - 1） and Diam （M） ≤D are bounded by a number depending on D and n. We also show that there are only finitely many isometric isomorphism types of bounded cohomology groups （H（M）,Ⅱ·Ⅱ∞） among closed Riemannian manifold （M, g） with K（M） ≥ -1 and Diam （M） ≤ D.展开更多
We show that if the fiber of a closed 4-dimensional mapping torus X is reducible and not S2× S1 or RP3#P3, then the virtual first Betti number of X is infinite and X is not virtually symplectic. This confirms two...We show that if the fiber of a closed 4-dimensional mapping torus X is reducible and not S2× S1 or RP3#P3, then the virtual first Betti number of X is infinite and X is not virtually symplectic. This confirms two conjectures made by Li and Ni （2014） in an earlier paper.展开更多
Let Mn be a smooth closed n-manifold with a locally standard （Z2）n-action. This paper deals with the relationship among the rood 2 Betti numbers of Mn, the mod 2 Betti numbers and the h-vector of the orbit space of ...Let Mn be a smooth closed n-manifold with a locally standard （Z2）n-action. This paper deals with the relationship among the rood 2 Betti numbers of Mn, the mod 2 Betti numbers and the h-vector of the orbit space of the action.展开更多
We provide the regularity and the Cohen-Macaulay type of binomial edge ideals of Cohen-Macaulay cones,and we show the extremal Betti numbers of some classes of Cohen-Macaulay binomial edge ideals:Cohen-Macaulay bipart...We provide the regularity and the Cohen-Macaulay type of binomial edge ideals of Cohen-Macaulay cones,and we show the extremal Betti numbers of some classes of Cohen-Macaulay binomial edge ideals:Cohen-Macaulay bipartite and fan graphs.In addition,we compute the Hilbert-Poincare series of the binomial edge ideals of some Cohen-Macaulaybipartitegraphs.展开更多
In this paper, Betti numbers are evaluated for several classes of graphs whose complements are bipartite graphs. Relations are established for the general case, and counting formulae are given in several particular ca...In this paper, Betti numbers are evaluated for several classes of graphs whose complements are bipartite graphs. Relations are established for the general case, and counting formulae are given in several particular cases, including the union of several mutually disjoint complete graphs.展开更多
We show that the torsion module Tor_(j)^(R)(R/a,H_(a)^(i)(X))is in a Serre subcategory for the bounded below R-complex X.In addition,we prove the isomorphism Tor_(s-t)^(R)(R/a,X)≅Tor_(s)^(R)(R/a,H_(a)^(t)(X))in some c...We show that the torsion module Tor_(j)^(R)(R/a,H_(a)^(i)(X))is in a Serre subcategory for the bounded below R-complex X.In addition,we prove the isomorphism Tor_(s-t)^(R)(R/a,X)≅Tor_(s)^(R)(R/a,H_(a)^(t)(X))in some case.As an application,the Betti number of a complex X in a prime ideal p can be computed by the Betti number of the local cohomology modules of X in p.展开更多
Microstructure topology evolution during severe plastic deformation(SPD)is crucial for understanding and optimising the mechanical properties of metallic materials,though its prediction remains challenging.Herein,we c...Microstructure topology evolution during severe plastic deformation(SPD)is crucial for understanding and optimising the mechanical properties of metallic materials,though its prediction remains challenging.Herein,we combine discrete cell complexes(DCC),a fully discrete algebraic topology model-with finite element analysis(FEA)to simulate and analyse the microstructure topology of pure copper under SPD.Using DCC,we model the evolution of microstructure topology characterised by Betti numbers(β_(0),β_(1),β_(2))and Euler characteristic(χ).This captures key changes in GBNs and topological features within representative volume elements(RVEs)containing several hundred grains during SPD-induced recrystallisation.As SPD cycles increase,high-angle grain boundaries(HAGBs)progressively form.Topological analysis reveals an overall decrease in β_(0)values,indicating fewer isolated HAGB substructures,while β_(2) values show a steady upward trend,highlighting new grain formation.Leveraging DCC-derived RVE topology and FEA-generated plastic strain data,we directly simulate the evolution and spatial distribution of microstructure topology and HAGB fraction in a copper tube undergoing cyclic parallel tube channel angular pressing(PTCAP),a representative SPD technique.Within the tube,the HAGB fraction continuously increases with PTCAP cycles,reflecting the microstructure’s gradual transition from subgrains to fully-formed grains.Analysis of Betti number distribution and evolution reveals the microstructural reconstruction mechanism underpinning this subgrain to grain transition during PTCAP.We further demonstrate the significant influence of spatially non-uniform plastic strain distribution on microstructure reconstruction kinetics.This study demonstrates a feasible approach for simulating microstructure topology evolution of metals processed by cyclic SPD via the integration of DCC and FEA.展开更多
Particle suspension and deposition dynamics are significant factors affecting the level of mixing quality in solidliquid two-phase stirring processes. In general, the ability to increase the suspension rate and minimi...Particle suspension and deposition dynamics are significant factors affecting the level of mixing quality in solidliquid two-phase stirring processes. In general, the ability to increase the suspension rate and minimize depositioneffects is instrumental in improving the uniformity of particle mixing, accelerating the reaction of involved solidliquid two-phase, and improving the efficiency of production operations. In this work, suspension and depositionindicator based on the Betti number and a uniformity indicator are introduced and obtained by means of imageanalysis. The influence of the blade type, rotation speed, blade diameter and blade bottom height on the particlesuspension/deposition characteristics and mixing uniformity are carefully investigated. The experimental resultsshow that the two-phase motion region can be divided into three local regions, including a bottom motion alongthe wall, a low-degree suspension region under the blade and a high suspension region above the blade. The bestdegree of particle suspension is attained by the double-inclined blade paddle at a speed of 270 r/min, a paddlediameter ratio of 0.414, and a height-diameter ratio of 0.086. The double-inclined blade paddle has a better effecton promoting particle suspension and solid-liquid two-phase mixing uniformity.展开更多
文摘For an ideal I of a Noetherian local ring （R, m, k） one hasβR1（I） -β0R（I） ≥-1. It is demonstrated that some residual intersections of an ideal I for whichβ1R（I） -β0R（I） = -1 or 0 are perfect.
基金supported by the Basic Science Research Program through the National Research Foundation of Korea(Nos.NRF-2016R1D1A1A09917654,NRF-2015R1C1A1A01053495)
文摘The authors compute the(rational) Betti number of real toric varieties associated to Weyl chambers of type B, and furthermore show that their integral cohomology is p-torsion free for all odd primes p.
文摘In this paper, we introduce the notion of bounded Betti numbers, and show that the bounded Betti numbers of a closed Riemannian n-manifold （M, g） with Ric （M） ≥ -（n - 1） and Diam （M） ≤D are bounded by a number depending on D and n. We also show that there are only finitely many isometric isomorphism types of bounded cohomology groups （H（M）,Ⅱ·Ⅱ∞） among closed Riemannian manifold （M, g） with K（M） ≥ -1 and Diam （M） ≤ D.
基金supported by National Science Foundation of USA(Grant No.DMS1252992)an Alfred P.Sloan Research Fellowship
文摘We show that if the fiber of a closed 4-dimensional mapping torus X is reducible and not S2× S1 or RP3#P3, then the virtual first Betti number of X is infinite and X is not virtually symplectic. This confirms two conjectures made by Li and Ni （2014） in an earlier paper.
基金supported by the National Natural Science Foundation of China(No.10931005)the Research Fund for the Doctoral Program of Higher Education of China(No.20100071110001)
文摘Let Mn be a smooth closed n-manifold with a locally standard （Z2）n-action. This paper deals with the relationship among the rood 2 Betti numbers of Mn, the mod 2 Betti numbers and the h-vector of the orbit space of the action.
基金The first author was supported by the“National Group for Algebraic and Geometric Structures,and Their Applications”(GNSAGA-INdAM).
文摘We provide the regularity and the Cohen-Macaulay type of binomial edge ideals of Cohen-Macaulay cones,and we show the extremal Betti numbers of some classes of Cohen-Macaulay binomial edge ideals:Cohen-Macaulay bipartite and fan graphs.In addition,we compute the Hilbert-Poincare series of the binomial edge ideals of some Cohen-Macaulaybipartitegraphs.
基金This research was supported by the National Natural Science Foundation of China （Grant No. 11271250）.
文摘In this paper, Betti numbers are evaluated for several classes of graphs whose complements are bipartite graphs. Relations are established for the general case, and counting formulae are given in several particular cases, including the union of several mutually disjoint complete graphs.
基金Natural Science Foundation of Gansu Province(23JRRA866)Higher Education Innovation Fund of Gansu Provincial Department of Education(2025A-132)+1 种基金University-level Scientific Research and Innovation Project of Gansu University of Political Science and Law(GZF2024XQN16)Youth Foundation of Lanzhou Jiaotong University(2023023)。
文摘We show that the torsion module Tor_(j)^(R)(R/a,H_(a)^(i)(X))is in a Serre subcategory for the bounded below R-complex X.In addition,we prove the isomorphism Tor_(s-t)^(R)(R/a,X)≅Tor_(s)^(R)(R/a,H_(a)^(t)(X))in some case.As an application,the Betti number of a complex X in a prime ideal p can be computed by the Betti number of the local cohomology modules of X in p.
基金support from Outstanding Youth Fund of Jiangsu Province(BK20240077)Key Project(Provincial-Municipal Joint)of Jiangsu Province(BK20243044)+2 种基金Fundamental Research Funds for the Central Universities(NE2024001)National Youth Talents Programof Chinaa project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
文摘Microstructure topology evolution during severe plastic deformation(SPD)is crucial for understanding and optimising the mechanical properties of metallic materials,though its prediction remains challenging.Herein,we combine discrete cell complexes(DCC),a fully discrete algebraic topology model-with finite element analysis(FEA)to simulate and analyse the microstructure topology of pure copper under SPD.Using DCC,we model the evolution of microstructure topology characterised by Betti numbers(β_(0),β_(1),β_(2))and Euler characteristic(χ).This captures key changes in GBNs and topological features within representative volume elements(RVEs)containing several hundred grains during SPD-induced recrystallisation.As SPD cycles increase,high-angle grain boundaries(HAGBs)progressively form.Topological analysis reveals an overall decrease in β_(0)values,indicating fewer isolated HAGB substructures,while β_(2) values show a steady upward trend,highlighting new grain formation.Leveraging DCC-derived RVE topology and FEA-generated plastic strain data,we directly simulate the evolution and spatial distribution of microstructure topology and HAGB fraction in a copper tube undergoing cyclic parallel tube channel angular pressing(PTCAP),a representative SPD technique.Within the tube,the HAGB fraction continuously increases with PTCAP cycles,reflecting the microstructure’s gradual transition from subgrains to fully-formed grains.Analysis of Betti number distribution and evolution reveals the microstructural reconstruction mechanism underpinning this subgrain to grain transition during PTCAP.We further demonstrate the significant influence of spatially non-uniform plastic strain distribution on microstructure reconstruction kinetics.This study demonstrates a feasible approach for simulating microstructure topology evolution of metals processed by cyclic SPD via the integration of DCC and FEA.
基金support from the Yunnan Fundamental Research Project,China(No.202201BE070001-026)Interdisciplinary Research Project of Kunming University of Science and Technology(No.KUST-xk2022001)+2 种基金Yunnan Major Scientific and Technological Projects(No.202302AQ370001-4)Open Foundation of State Environmental Protection Key Laboratory of Mineral Metallurgical Resources Utilization and Pollution Control(No.HB202204)Young Elite Scientist Sponsorship Program by China Association for Science and Technology,China(No.YESS20210106).
文摘Particle suspension and deposition dynamics are significant factors affecting the level of mixing quality in solidliquid two-phase stirring processes. In general, the ability to increase the suspension rate and minimize depositioneffects is instrumental in improving the uniformity of particle mixing, accelerating the reaction of involved solidliquid two-phase, and improving the efficiency of production operations. In this work, suspension and depositionindicator based on the Betti number and a uniformity indicator are introduced and obtained by means of imageanalysis. The influence of the blade type, rotation speed, blade diameter and blade bottom height on the particlesuspension/deposition characteristics and mixing uniformity are carefully investigated. The experimental resultsshow that the two-phase motion region can be divided into three local regions, including a bottom motion alongthe wall, a low-degree suspension region under the blade and a high suspension region above the blade. The bestdegree of particle suspension is attained by the double-inclined blade paddle at a speed of 270 r/min, a paddlediameter ratio of 0.414, and a height-diameter ratio of 0.086. The double-inclined blade paddle has a better effecton promoting particle suspension and solid-liquid two-phase mixing uniformity.
