bifurcations with o 3 symmetry including applications to the benard problem this paper has two purposes to study from the singularity theory point of view certain bifurcation problems which commute with the five dimensional irreducible representation of the orthogonal group o 3 and to give some implications of this study for the benard problem in spherical geometry we shall now describe in general terms our results and compare them with the work of chossat 3 whose paper motivated our own interest in the problem the benard problem is concerned with convection in a viscous fluid when it is heated from below the fluid is assumed to be confined in a spherical shell of outer radius r and inner radius qro where q is near 0 3 this choice of q is partially motivated by considering convection within the molten layer of the core of the earth see busse l for further discussion we consider the binard problem in the boussinesq approximation in this model there is a trivial solution representing pure heat conduction radially outward as the temperature on the inner sphere that is the rayleigh number r is increased this trivial solution loses stability say at r r the benard problem is the study of the resulting bifurcation chossat studies this problem using the lyapunov schmidt reduction in the boussinesq approximation the fluid is driven from the pure conduction state by a term in the momentum equation involving the gravity vector g r and a term in the temperature equation involving the gradient of the conduction temperature vt assuming that the production of heat and the density are uniform throughout the shell these vectors have the form

a lattice boltzmann approach for the non newtonian effect in the blood flow abstract a new lattice boltzmann approach within the framework of d2q9 lattice for simulating shear thinning non newtonian blood flows described by the power law carreau yasuda and casson rheology models is proposed in this study the essence of this method lies in splitting the complete non newtonian effect up into two portions one as the newtonian result and the other as an effective external source this arrangement takes the advantage in remaining fixed relaxation time during the whole course of numerical simulation that can avoid the potential numerical instability caused by the relaxation time approaches to 1 2 an inherent difficulty in the conventional lattice boltzmann methods using varying relaxation times for the non newtonian effect macroscopically consistency of the proposed model with the equations of motion for the three target non newtonian models is demonstrated through the technique of chapman enskog multi scale expansion the feasibility and accuracy of the method are examined by comparing with the analytical solutions of the two dimensional poiseuille flows based on the power law and casson models the results show that the velocity profiles agree very well with those of analytical solutions and the error analyses demonstrate that the proposed scheme is with second order accuracy the present approach also demonstrates its superiority over the conventional lattice boltzmann method in the extent of numerical stability for simulating the power law based shear thinning flows the straightforwardness in scheme derivation and implementation renders the present approach as a potential method for the complex non newtonian flows

partial stability of catastrophe sections and its application to the cuspoids and conic umbilics the control space of a system described by an elementary catastrophe frequently has locally at least a cartesian product structure caused for example by physical distinctions among control variables or by observational necessity the effect of this structure like that of symmetries can be to stabilise singular catastrophe sections partial unfoldings the authors define partial stability to be the structural stability of a stratum of a catastrophe section under perturbations respecting the product structure these ideas are illustrated by reference to a recent observation of double cusp diffraction and then applied to previous studies of singular catastrophe sections

classification and unfoldings of degenerate hopf bifurcations abstract this paper initiates the classification up to symmetry covariant contact equivalence of perturbations of local hopf bifurcation problems which do not satisfy the classical non degeneracy conditions the only remaining hypothesis is that i should be simple eigenvalues of the linearized right hand side at criticality then the lyapunov schmidt method allows a reduction to a scalar equation g x 0 where g x g x a definition is given of the codimension of g and a complete classification is obtained for all problems with codimension 3 together with the corresponding universal unfoldings the perturbed bifurcation diagrams are given for the cases with codimension 2 and for one case with codimension 3 for this last case one of the unfolding parameters is a modal parameter such that the topological codimension equals in fact 2 formulas are given for the calculation of the taylor coefficients needed for the application of the results and finally the results are applied to two simple problems a model of glycolytic oscillations and the fitzhugh nerve equations

free entropy dimension in amalgamated free products we calculate the microstates free entropy dimension of natural generators in an amalgamated free product of certain von neumann algebras with amalgamation over a hyperfinite subalgebra in particular some exotic popa algebra generators of free group factors are shown to have the expected free entropy dimension we also show that microstates and non microstates free entropy dimension agree for generating sets of many groups in the appendix the first l 2 betti number for certain amalgamated free products of groups is calculated

a method for the numerical evaluation of the oscillatory integrals associated with the cuspoid catastrophes application to pearcey s integral and its derivatives a numerical method for the evaluation of the cuspoid canonical integrals and their derivatives is described the method exploits cauchy s theorem and jordan s lemma to write the infinite integration path along different contours in the complex plane the method is straightforward to implement on a computer and in many cases results of high accuracy can be obtained using standard quadrature techniques application is made to pearcey s integral p x y and its two partial derivatives and the method is shown to have some significant advantages over other techniques that have been applied to this problem tables of p x y delta p x y delta x delta p x y delta y and the real zeros of p x y are presented for the grid 8 0 or x or 8 0 and 0 or y or 8 0

on sofic actions and equivalence relations abstract the notion of sofic equivalence relation was introduced by gabor elek and gabor lippner their technics employ some graph theory here we define this notion in a more operator algebraic context starting from connes embedding problem and prove the equivalence of these two definitions we introduce a notion of sofic action for an arbitrary group and prove that an amalgamated product of sofic actions over amenable groups is again sofic we also prove that an amalgamated product of sofic groups over an amenable subgroup is again sofic

applications of catastrophe theory to the physical sciences abstract catastrophe theory was introduced by rene thom in the late 1960 s as an attempt to model morphogentic changes in nature using ideas from topological dynamics and the theory of singularities of mappings thom envisaged a very general approach to topological changes in the solutions to parametrized systems of equations such as differential and difference equations and in particular discussed the special case of elementary catastrophe theory singularities of smooth real valued functions popular expositions have tended to overemphasize this special case but it remains the major source of ideas and methods here we survey the applications of catastrophe theory to the physical sciences physics chemistry engineering fluid mechanics etc for brevity we confine attention to an area lying between elementary and general catastrophe theory usually known as singularity theory this is the theory of singularities of smooth vector valued functions which mathematically is a straightforward though non trivial generalization of the real valued case in the last few years it has developed into a powerful and useful technique in several areas of theoretical physics notably optics and bifurcation theory equivariant catastrophe theory taking account of symmetry is likely to prove especially interesting

discrete lattice effects on the forcing term in the lattice boltzmann method we show that discrete lattice effects must be considered in the introduction of a force into the lattice boltzmann equation a representation of the forcing term is then proposed with the representation the navier stokes equation is derived from the lattice boltzmann equation through the chapman enskog expansion several other existing force treatments are also examined

on the second painleve equation the connection formula via a riemann hilbert problem and other results abstract we prove the existence of asymptotic expansions to all orders for the solutions of the second painleve equation p ii using the linearization obtained via a riemann hilbert problem we then derive the amplitude part of the connection formula for this equation and give some other properties of the expansion

entropy and the variational principle for actions of sofic groups recently lewis bowen introduced a notion of entropy for measure preserving actions of a countable sofic group on a standard probability space admitting a generating partition with finite entropy by applying an operator algebra perspective we develop a more general approach to sofic entropy which produces both measure and topological dynamical invariants and we establish the variational principle in this context in the case of residually finite groups we use the variational principle to compute the topological entropy of principal algebraic actions whose defining group ring element is invertible in the full group c algebra

accuracy of non newtonian lattice boltzmann simulations this work deals with the accuracy of non newtonian lattice boltzmann simulations previous work for newtonian fluids indicate that depending on the numerical value of the dimensionless collision frequency additional artificial viscosity is introduced which negatively influences the accuracy since the non newtonian fluid behavior is incorporated through appropriate modeling of the dimensionless collision frequency a dependent error e is introduced and its influence on the overall error is investigated here simulations with the srt and the mrt model are carried out for power law fluids in order to numerically investigate the accuracy of non newtonian lattice boltzmann simulations a goal of this accuracy analysis is to derive a recommendation for an optimal choice of the time step size and the simulation mach number respectively for the non newtonian case an error estimate for e in the form of a functional is derived on the basis of a series expansion of the lattice boltzmann equation this functional can be solved analytically for the case of the hagen poiseuille channel flow of non newtonian fluids with the help of the error functional the prediction of the global error minimum of the velocity field is excellent in regions where the e error is the dominant source of error with an optimal simulation mach number the simulation is about one order of magnitude more accurate additionally for both collision models a detailed study of the convergence behavior of the method in the non newtonian case is conducted the results show that the simulation mach number has a major impact on the convergence rate and second order accuracy is not preserved for every choice of the simulation mach number

the free entropy dimension of hyperfinite von neumann algebras suppose m is a hyperfinite von neumann algebra with a normal tracial state and 1 n is a set of selfadjoint generators for m we calculate 0 1 n the modified free entropy dimension of 1 n moreover we show that 0 1 n depends only on m and consequently 0 1 n is independent of the choice of generators for m in the course of the argument we show that if b 1 b n is a set of selfadjoint generators for a von neumann algebra r with a normal tracial state and b 1 b n has finite dimensional approximants then 0 n 0 b 1 b n for any hyperfinite von neumann subalgebra n of r combined with a result by voiculescu this implies that if r has a regular diffuse hyperfinite von neumann subalgebra then 0 b 1 b n 1

cout des relations d equivalence et des groupes we study a new dynamical invariant for dicrete groups the cost it is a real number in 1 1 n 1 bounded by the number of generators of the group and it is well behaved with respect to finite index subgroups namely the quantities 1 minus the cost are related by multiplying by the index the cost of every infinite amenable group equals 1 we compute it in some other situations including free products free products with amalgamation and hnn extensions over amenable groups and for direct product situations for instance the cost of the free group on n generators equals n we prove that each possible finite value of the cost is achieved by a finitely generated group it is dynamical because it relies on measure preserving free actions on probability borel spaces in most cases groups have fixed price which implies that two freely acting groups which define the same orbit partition must have the same cost it enables us to distinguish the orbit partitions of probability preserving free actions of free groups of different ranks at the end of the paper we give a mercuriale i e a list of costs of different groups the cost is in fact an invariant of ergodic measure preserving equivalence relations and is defined using graphings a treeing is a measurable way to provide every equivalence class orbit with the structure of a simplicial tree this an example of graphing not every relation admits a treeing we prove that every free action of a cost 1 non amenable group is not treeable but we prove that subrelations of treeable relations are treeable we give examples of relations which cannot be produced by an action of any finitely generated group the cost of a relation which can be decomposed as a direct product is shown to be 1 we define the notion for a relation to be a free product or an hnn extension and compute the cost for the resulting relation from the costs of the building blocks the cost is also an invariant of the pairs von neumann algebra cartan subalgebra

a comparison of non newtonian models for lattice boltzmann blood flow simulations in the present paper three non newtonian models for blood are used in a lattice boltzmann flow solver to simulate non newtonian blood flows exact analytical solutions for two of these models have been derived and presented for a fully developed 2d channel flow original results for the use of the k l model in addition to the casson and carreau yasuda models are reported for non newtonian flow simulations using a lattice boltzmann lb flow solver numerical simulations of non newtonian flow in a 2d channel show that these models predict different mass flux and velocity profiles even for the same channel geometry and flow boundary conditions which in turn suggests a more careful model selection for more realistic blood flow simulations the agreement between predicted velocity profiles and those of exact solutions is excellent indicating the capability of the lb flow solver for such complex fluid flows

an lp version of von neumann dimension for banach space representations of sofic groups abstract in 5 a gournay defined a notion of l p dimension for invariant subspaces of l q n with amenable the number dim l q l p v is dim v when p q and is preserved by a certain class of equivariant bounded linear isomorphisms we develop a notion of dim l p y where y is a banach space with a uniformly bounded action of a sofic group and is a sofic approximation in particular our definition makes sense for a large class of non amenable groups we also develop a notion of dim s p y with an r embeddable group and s p the space of finite dimensional schatten p class operators these numbers are invariant under bounded equivariant linear isomorphisms and under the natural translation action of dim l p l p v dim v and dim s p l p v dim v for 1 p 2 in particular this shows that l p v is not isomorphic to l p w as a representation of if dim v dim w and is r embeddable we discuss other concrete computations in a follow up paper including proving that our dimension agrees with von neumann dimension for representations contained in a multiple of the left regular representation

power law fluid flow in driven enclosures with undulation using mrt lattice boltzmann method abstract in this article the flow of non newtonian fluid which is represented by the power law model in two dimensional 2d driven enclosures is studied the enclosure consists of regular rectangular shaped undulations on the bottom wall multiple relaxation time mrt collision model for the lattice boltzmann equation method lbm is employed first numerical validation is performed by comparing the mrt lbm results of power law fluid flow inside the wall driven square enclosure no undulation and flow inside diagonally flipped l shaped enclosure with the literature the strain rate profiles for square enclosure without undulations are compared using various equations available for strain rate calculation in the lbm literature further the effect of different values of the non hydrodynamic relaxation parameters on the flow is examined then for the undulated enclosures flow features and eddy dynamics are analyzed and discussed for the variations in the power law index n to represent shear thinning and shear thickening fluids the effects of various parameters such as reynolds numbers wall undulation heights and wavelength of undulations on the power law fluid flow are analyzed also the variation of viscosity with spatial location for steady state flow and total kinetic energy within the computational domain are presented for various values of power law index

lattice boltzmann simulations of blood flow non newtonian rheology and clotting processes the numerical simulation of thrombosis in stented aneurysms is an important issue to estimate the efficiency of a stent in this paper we consider a lattice boltzmann lb approach to bloodflow modeling and we implement a non newtonian correction in order to reproduce more realistic flow profiles we obtain a good agreement between simulations and casson s model of blood rheology in a simple geometry finally we discuss how by using a passive scalar suspension model with aggregation on top of the lb dynamics we can describe the clotting processes in the aneurysm

a second order accurate lattice boltzmann non newtonian flow model a second order accurate lattice boltzmann model is presented for non newtonian flow the non newtonian nature of the flow is implemented using a power law model this is used to enable the accuracy of the model to be assessed and is not a limitation of the model the second order accuracy is demonstrated for a range of power law model parameter values representing shear thinning and shear thickening fluids these results are compared with those of gabbanelli et al 2006 phys rev e 72 046312 and it is noted that a higher order of accuracy and greater computational efficiency are achieved these results demonstrate the suitability of the lbm for shear dependent non newtonian flow simulations

analytic singularities in dispersive wave phenomena based on topological singularities of dispersion relations the connection between topological singularities of dispersion relations and analytic singularities with respect to frequency of spectral densities and wavefunctions in dispersive media due to time harmonic sources is demonstrated the topological singularities are discussed within the framework of imperfect bifurcation theory regarding the frequency as a distinguished bifurcation parameter and the wavenumbers as bifurcation variables and using a recently obtained classification of topological singularities up to codimension four which is given in terms of a list of normal forms to each normal form corresponds an analytic singularity governed by characteristic exponents which are tabulated

lattice boltzmann method for non newtonian power law fluids we study an ad hoc extension of the lattice boltzmann method that allows the simulation of non newtonian fluids described by generalized newtonian models we extensively test the accuracy of the method for the case of shear thinning and shear thickening truncated power law fluids in the parallel plate geometry and show that the relative error compared to analytical solutions decays approximately linear with the lattice resolution finally we also tested the method in the reentrant flow geometry in which the shear rate is no longer a scalar and the presence of two singular points requires high accuracy in order to obtain satisfactory resolution in the local stress near these points in this geometry we also found excellent agreement with the solutions obtained by standard finite element methods and the agreement improves with higher lattice resolution