ON A FINITENESS THEOREM ABOUT PROBLEMS INVOLVING INEQUALITIES

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Finiteness results for equifocal hypersurfaces in compact symmetric spaces

We prove that there are only finitely many diffeomorphism types of curvature-adapted equifocal hypersurfaces in a simply connected compact symmetric space.Moreover,if the symmetric space is of rank one,the result can be strengthened by dropping the condition curvature-adapted.

Finiteness of Mapping Degree Sets for 3-Manifolds

By constructing certain maps, this note completes the answer of the question： For which closed orientable 3-manifold N, is the set of mapping degrees D（M, N） finite for any closed orientable 3-manifold M？

On the Residual Finiteness of a Class of Infinite Soluble Groups

Let A be a completely decomposable homogeneous torsion-free abelian group of rank n(n≥2).Let m(n)=A×(a)be the split extension of A by an automorphismαwhich is a cyclic permutation of the direct components twisted by a rational integer m.Then Om(n)is an infinite soluble group.In this paper,the residual finiteness of Om(n)is investigated.

On the Finiteness Dimension of Local Cohomology Modules

Let R be a commutative Noetherian ring, α an ideal of R, and M a non-zero finitely generated R-module. Let t be a non-negative integer. In this paper, it is shown that dim Supp Hi a（M） ≤ 1 for all i 〈 t if and only if there exists an ideal b of R such that dimR/b ≤ 1 and Hia（M） ≌ Hi b（M） for all i 〈 t. Moreover, we prove that dimSuppHia（M） 〈≤dim M - i for all i.

A generalized π2-diffeomorphism finiteness theorem

Theπ2-diffeomorphism finiteness result of F.Fang-X.Rong and A.Petrunin-W.Tuschmann(independently)asserts that the diffeomorphic types of compact n-manifolds M with vanishing first and second homotopy groups can be bounded above in terms of n,and upper bounds on the absolute value of sectional curvature and diameter of M.In this paper,we will generalize thisπ2-diffeomorphism finiteness by removing the condition thatπ1(M)-0 and asserting the diffeomorphism finiteness on the Riemannian universal cover of M.

Fatigue Strength Analysis of Dissimilar Aluminum Alloy TIG Welds

The constant amplitude loading fatigue tests were carried out on the 6061/7075 aluminum alloy TIG fillet welded lap specimens in this study,and the weld seam cross-section hardness was measured.The experimental results show that most specimens mainly failed at the 7075 side weld toes even though the base material tensile strength of 7075 is higher than that of 6061.The maximum stress-strain concentration in the two finite element models is located at the 7075 side weld toe,which is basically consistent with the actual fracture location.The weld zone on the 7075 side experiences severe material softening,with a large gradient.However,the Vickers hardness value on the 6061 side negligibly changes and fluctuates around 70 HV.No obvious defects are found on the fatigue fracture,but a large number of secondary cracks appear.Cracks germinate from the weld toe and propagate in the direction of the plate thickness.Weld reinforcement has a serious impact on fatigue life.Fatigue life will decrease exponentially as the weld reinforcement increases under low stress.It is found that the notch stress method can give a better fatigue life prediction for TIG weldments,and the errors of the predicted results are within the range of two factors,while the prediction accuracy decreases under low stress.The equivalent structural stress method can also be used for fatigue life prediction of TIG weldments,but the errors of prediction results are within the range of three factors,and the accuracy decreases under high stress.

Cofiniteness of Local Cohomology Modules

Let M be a non-zero finitely generated module over a commutative Noetherian local ring （R, m）. In this paper we consider when the local cohomology modules are finitely generated. It is shown that if t≥ 0 is an integer and p C Supp H^t_p （M）, then Hm^t＋dim R/p （M） is not p-cofinite. Then we obtain a partial answer to a question raised by Huneke. Namely, if R is a complete local ring, then H^n_m （M） is finitely generated if and only if 0 ≤ n ￠ W, where W ---- {t ＋ dimR/p丨p ∈ SuppH^t_p（M）／V（m）}. Also, we show that if J C I are 1-dimensional ideals of R, then H^t_I（M） is J-cominimax, and H^t_I（M） is finitely generated （resp., minimax） if and only if H}R, （Mp） is finitely generated for all p C Spec R （resp., p ∈ SpecR／MaxR）. Moreover, the concept of the J-cofiniteness dimension cJ（M） of M relative to I is introduced, and we explore an interrelation between c^I_m（M） and the filter depth of M in I. Finally, we show that if R is complete and dim M/IM ≠ 0, then c^I_m （R） ---- inf{depth Mp ＋ dim R/p 丨 P ∈ Supp M/IM／V（m）}.

FINITENESS CONDITIONS FOR GENERALIZED EXPONENTS OF DIGRAPHS

Necessary and sufficient conditions are given for the finiteness of the generalized exponents expD(k), f(D, k) and F(D, k) for digraphs which are not necessarily primitive. Also the largest finite value of the generalized exponent expD(k) for digraphs of order n is determined and the complete characterizations of the extreme digraphs are given.

Residual finiteness of outer automorphism groups of certain 1-relator groups

We prove that certain 1-relator groups have Property E. Using this fact, we characterize all conjugacy separable 1-relator groups of the form a,b;(a-αbβaαbγ)t , t 1, having residually finite outer automorphism groups.

Positive Definiteness of a Class of Analytic Functions

Some singular characteristics of analytic functions with positive definiteness are considered. To avoid these singular cases, the finite truncate condition is proposed. When the finite truncate condition is satisfied, the positive definiteness of an analytic function can be judged by the partial sum of its Taylor series, i e , by a polynomial. This discussion will be useful in the construction of Lyapunov functions for nonlinear systems.

Rapid rupture characterization for the 2023 M_(S)6.2 Jishishan earthquake

On December 18, 2023, the M_(S)6.2 Jishishan earthquake occurred in the northeastern region of the QinghaiXizang Plateau, causing heavy casualties and property damage in Gansu and Qinghai Provinces. In this study,we integrate space imaging geodesy, finite fault inversion, and back-projection methods to decipher its rupture property, including fault geometry, coseismic slip distribution, rupture direction, and propagation speed. The results reveal that the seismogenic fault dips to the southwest at an angle of 29°. The major slip asperity is dominated by reverse slip and is concentrated within a depth range of 7–16 km, which explains the significant uplift near the epicenter observed by both the Sentinel-1 ascending and descending In SAR data. Moreover, the teleseismic array waveforms indicate a northwest propagating rupture with an overall slow rupture velocity of~1.91 km/s(AK array) or 1.01 km/s(AU array).

Finite Element Analysis for Magneto-Convection Heat Transfer Performance in Vertical Wavy Surface Enclosure:Fin Size Impact

The goal of this paper is to represent a numerical study of magnetohydrodynamic mixed convection heat transfer in a lid-driven vertical wavy enclosure with a fin attached to the bottomwall.We use a finite elementmethod based on Galerkin weighted residual(GWR)techniques to set up the appropriate governing equations for the present flow model.We have conducted a parametric investigation to examine the impact of Hartmann and Richardson numbers on the flow pattern and heat transmission features inside a wavy cavity.We graphically represent the numerical results,such as isotherms,streamlines,velocity profiles,local and mean Nusselt numbers,and average surface temperature.Comparisons between the results of this work and previously published work in a literature review have been produced to examine the reliability and consistency of the data.The different sizes of the fin surface significantly impact flow creation and temperature fields.Additionally,the long fin size is necessary to enhance the heat transfer rate on the right surface at large Richardson numbers and low Hartmann numbers.Fin surfaces can significantly increase the mixing of fluid inside the enclosure,which can mean reductions in reaction times and operating costs,along with increases in heat transfer and efficiency.

Numerical modeling of fracture propagation of supercritical CO_(2)compound fracturing

The exploitation of shale gas is promising due to depletion of the conventional energy and intensification of the greenhouse effect.In this paper,we proposed a heat-fluid-solid coupling damage model of supercritical CO_(2)(SC-CO_(2))compound fracturing which is expected to be an efficient and environmentally friendly way to develop shale gas.The coupling model is solved by the finite element method,and the results are in good agreement with the analytical solutions and fracturing experiments.Based on this model,the fracture propagation characteristics at the two stages of compound fracturing are studied and the influence of pressurization rate,in situ stress,bedding angle,and other factors are considered.The results show that at the SC-CO_(2)fracturing stage,a lower pressurization rate is conducive to formation of the branches around main fractures,while a higher pressurization rate inhibits formation of the branches around main fractures and promotes formation of the main fractures.Both bedding and in situ stress play a dominant role in the fracture propagation.When the in situ stress ratio(δ_(x)/δ_(y))is 1,the presence of bedding can reduce the initiation pressure and failure pressure.Nevertheless,it will cause the fracture to propagate along the bedding direction,reducing the fracture complexity.In rocks without bedding,hydraulic fracturing has the lengthening and widening effects for SC-CO_(2)induced fracture.In shale,fractures induced at the hydraulic fracturing stage are more likely to be dominated by in situ stresses and have a shorter reorientation radius.Therefore,fracture branches propagating along the maximum principal stress direction may be generated around the main fractures induced by SC-CO_(2)at the hydraulic fracturing stage.When the branches converge with the main fractures,fracture zones are easily formed,and thus the fracture complexity and damage area can be significantly increased.The results are instructive for the design and application of SC-CO_(2)compound fracturing.

Numerical analysis of moving train induced vibrations on tunnel,surrounding ground and structure

This study is focused on the effect of vibration induced by moving trains in tunnels on the surrounding ground and structures.A three-dimensional finite element model is established for a one-track railway tunnel and an adjacent twelve-storey building frame by using commercial software Midas GTS-NX(2019)and Midas Gen.This study considered the moving load effect of a complete train,which varies with space as well as with time.The effect of factors such as train speed,overburden pressure on the tunnel and variation in soil properties are studied in the time domain.As a result,the variations in horizontal and vertical acceleration for two different sites,i.e.,the free ground surface(without structure)and the area containing the structure,are compared.Also,the displacement pattern of the raft foundation is plotted for different train velocities.At lower speeds,the heaving phenomenon is negligible,but as the speed increases,both the heaving and differential settlement increase in the foundation.This study demonstrates that the effect of moving train vibrations should be considered in the design of new nearby structures and proper ground improvement should be considered for existing structures.

Numerical Study of the Biomechanical Behavior of a 3D Printed Polymer Esophageal Stent in the Esophagus by BP Neural Network Algorithm

Esophageal disease is a common disorder of the digestive system that can severely affect the quality of life andprognosis of patients. Esophageal stenting is an effective treatment that has been widely used in clinical practice.However, esophageal stents of different types and parameters have varying adaptability and effectiveness forpatients, and they need to be individually selected according to the patient’s specific situation. The purposeof this study was to provide a reference for clinical doctors to choose suitable esophageal stents. We used 3Dprinting technology to fabricate esophageal stents with different ratios of thermoplastic polyurethane (TPU)/(Poly-ε-caprolactone) PCL polymer, and established an artificial neural network model that could predict the radial forceof esophageal stents based on the content of TPU, PCL and print parameter. We selected three optimal ratios formechanical performance tests and evaluated the biomechanical effects of different ratios of stents on esophagealimplantation, swallowing, and stent migration processes through finite element numerical simulation and in vitrosimulation tests. The results showed that different ratios of polymer stents had different mechanical properties,affecting the effectiveness of stent expansion treatment and the possibility of postoperative complications of stentimplantation.

Experimental and simulation research on hollow AZ31 magnesium alloy three-channel joint by hot extrusion forming with sand mandrel

Magnesium alloy is one of the lightest metal structural materials.The weight is further reduced through the hollow structure.However,the hollow structure is easily damaged during processing.In order to maintain the hollow structure and to transfer the stresses during the high temperature deformation,the sand mandrel is proposed.In this paper,the hollow AZ31 magnesium alloy three-channel joint is studied by hot extrusion forming.Sand as one of solid granule medium is used to fill the hollow magnesium alloy.The extrusion temperatures are 230℃ and 300℃,respectively.The process parameters(die angle,temperature,bottom thickness,sidewall thickness,edge-to-middle ratio in bottom,bottom shape)of the hollow magnesium alloy are analyzed based on the results of experiments and the finite element method.The results are shown that the formability of the hollow magnesium alloy will be much better when the ratio of sidewall thickness to the bottom thickness is 1:1.5.Also when edge-to-middle ratio in bottom is about 1:1.5,a better forming product can be received.The best bottom shape in these experiments will be convex based on the forming results.The grain will be refined obviously after the extrusion.Also the microstructures will be shown as streamlines.And these lines will be well agreement with the mold in the corner.

A stable implicit nodal integration-based particle finite element method(N-PFEM)for modelling saturated soil dynamics

In this study,we present a novel nodal integration-based particle finite element method(N-PFEM)designed for the dynamic analysis of saturated soils.Our approach incorporates the nodal integration technique into a generalised Hellinger-Reissner(HR)variational principle,creating an implicit PFEM formulation.To mitigate the volumetric locking issue in low-order elements,we employ a node-based strain smoothing technique.By discretising field variables at the centre of smoothing cells,we achieve nodal integration over cells,eliminating the need for sophisticated mapping operations after re-meshing in the PFEM.We express the discretised governing equations as a min-max optimisation problem,which is further reformulated as a standard second-order cone programming(SOCP)problem.Stresses,pore water pressure,and displacements are simultaneously determined using the advanced primal-dual interior point method.Consequently,our numerical model offers improved accuracy for stresses and pore water pressure compared to the displacement-based PFEM formulation.Numerical experiments demonstrate that the N-PFEM efficiently captures both transient and long-term hydro-mechanical behaviour of saturated soils with high accuracy,obviating the need for stabilisation or regularisation techniques commonly employed in other nodal integration-based PFEM approaches.This work holds significant implications for the development of robust and accurate numerical tools for studying saturated soil dynamics.

Numerical analysis of downward progressive landslides in long natural slopes with sensitive clay

Landslides occurring in sensitive clay often result in widespread destruction,posing a significant risk to human lives and property due to the substantial decrease in undrained shear strength during deformation.Assessing the consequences of these landslides is challenging and necessitates robust numerical methods to comprehensively investigate their failure mechanisms.While studies have extensively explored upward progressive landslides in sensitive clays,understanding downward progressive cases remains limited.In this study,we utilised the nodal integration-based particle finite element method(NPFEM)with a nonlinear strain-softening model to analyse downward progressive landslides in sensitive clay on elongated slopes,induced by surcharge loads near the crest.We focused on elucidating the underlying failure mechanisms and evaluating the effects of different soil parameters and strainsoftening characteristics.The simulation results revealed the typical pattern for downward landslides,which typically start with a localised failure in proximity to the surcharge loads,followed by a combination of different types of failure mechanisms,including single flow slides,translational progressive landslides,progressive flow slides,and spread failures.Additionally,inclined shear bands occur within spread failures,often adopting distinctive ploughing patterns characterised by triangular shapes.The sensitive clay thickness at the base,the clay strength gradient,the sensitivity,and the softening rate significantly influence the failure mechanisms and the extent of diffused displacement.Remarkably,some of these effects mirror those observed in upward progressive landslides,underscoring the interconnectedness of these phenomena.This study contributes valuable insights into the complex dynamics of sensitive clay landslides,shedding light on the intricate interplay of factors governing their behaviour and progression.

Extended finite element-based cohesive zone method for modeling simultaneous hydraulic fracture height growth in layered reservoirs

In this study,a fully coupled hydromechanical model within the extended finite element method(XFEM)-based cohesive zone method(CZM)is employed to investigate the simultaneous height growth behavior of multi-cluster hydraulic fractures in layered porous reservoirs with modulus contrast.The coupled hydromechanical model is first verified against an analytical solution and a laboratory experiment.Then,the fracture geometry(e.g.height,aperture,and area)and fluid pressure evolutions of multiple hydraulic fractures placed in a porous reservoir interbedded with alternating stiff and soft layers are investigated using the model.The stress and pore pressure distributions within the layered reservoir during fluid injection are also presented.The simulation results reveal that stress umbrellas are easily to form among multiple hydraulic fractures’tips when propagating in soft layers,which impedes the simultaneous height growth.It is also observed that the impediment effect of soft layer is much more significant in the fractures suppressed by the preferential growth of adjoining fractures.After that,the combined effect of in situ stress ratio and fracturing spacing on the multi-fracture height growth is presented,and the results elucidate the influence of in situ stress ratio on the height growth behavior depending on the fracture spacing.Finally,it is found that the inclusion of soft layers changes the aperture distribution of outmost and interior hydraulic fractures.The results obtained from this study may provide some insights on the understanding of hydraulic fracture height containment observed in filed.