on some results in fuzzy metric spaces abstract we define a hausdorff topology on a fuzzy metric space introduced by kramosil and michalek kybernetica 11 1975 326 334 and prove some known results of metric spaces including baire s theorem for fuzzy metric spaces

optimum design of flextensional piezoelectric actuators into two spatial dimensions piezoelectric actuators are in common use for control of distributed parameter systems we consider the topology optimization of a multiphysic model in piezoelectricity into two spatial dimensions the topological derivative of a tracking type shape functional is derived in its closed form for the purpose of shape optimization of piezoelectric actuators the optimum design procedure is applied to a micromechanism which transforms the electrical energy supplemented via its piezoceramic part into the elastic energy of an actuator the domain decomposition technique and the steklov poincare pseudodifferential boundary operator are employed in the asymptotic analysis of the shape functional defined on a part of the boundary of the elastic body under consideration the new method of sensitivity analysis is general and can be used for shape topological optimization in a broad class of multiphysics models our numerical results confirm the efficiency of the proposed approach to optimum design in multiphysics

complete metrizability of topologies of strong uniform convergence on bornologies abstract we continue the study of topologies of strong uniform convergence on bornologies initiated by beer and levi 2009 4 in beer and levi 2009 4 the metrizability of such topologies restricted to continuous functions was characterized and a compatible metric was displayed we find necessary and sufficient conditions for completeness of the metric for complete metrizability and for polishness of topologies of strong uniform convergence on bornologies on continuous functions polishness of such topologies forces their coincidence with the compact open topology

on some results of analysis for fuzzy metric spaces abstract a necessary and sufficient condition for a fuzzy metric space to be complete is given we prove that a subspace of a separable fuzzy metric space is separable and every separable fuzzy metric space is second countable uniform limit theorem is generalized to fuzzy metric spaces

s topologies and bounded convergences abstract recently lechicki levi and spakowski have studied set convergence of the attouch wets type expressed in terms of truncations by members of a prescribed family of sets when this family is either the family of metrically bounded sets or the family of all nonempty subsets it is well known that convergence means uniform convergence of distance functions on members of the family in this paper we give necessary and sufficient conditions for coincidence to occur when the family is a general bornology of subsets

active contours without edges we propose a new model for active contours to detect objects in a given image based on techniques of curve evolution mumford shah 1989 functional for segmentation and level sets our model can detect objects whose boundaries are not necessarily defined by the gradient we minimize an energy which can be seen as a particular case of the minimal partition problem in the level set formulation the problem becomes a mean curvature flow like evolving the active contour which will stop on the desired boundary however the stopping term does not depend on the gradient of the image as in the classical active contour models but is instead related to a particular segmentation of the image we give a numerical algorithm using finite differences finally we present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable also the initial curve can be anywhere in the image and interior contours are automatically detected

local and subquotient inheritance of simplicity in jordan systems abstract in this paper we prove that the local algebras of a simple jordan pair are simple jordan pairs all of which local algebras are simple are also studied showing that they have a nonzero simple heart which is described in terms of powers of the original pair similar results are given for jordan triple systems and algebras finally we characterize the inner ideals of a simple pair which determine simple subquotients answering the question posed by o loos and e neher 1994 j algebra166 255 295

when is the space of minimal usco cusco maps a topological vector space abstract we will define a vector structure on the space of minimal usco resp cusco maps from a locally compact space to a stegall locally convex space we will prove that these spaces equipped with the topology of uniform convergence on compacta are locally convex spaces

shape identification for navier stokes problem using shape sensitivity analysis and level set method the shape identification for incompressible steady state navier stokes problem is considered we propose an algorithm based upon a combination of the classical shape sensitivity analysis and the level set method the adjoint method is used to derive the optimal system for the shape gradient of the cost functional the promising features of this algorithm are illustrated by two benchmark numerical examples

local orders in jordan algebras abstract making use of results on general algebras of quotients of jordan algebras we study a notion of local order based on the version for linear jordan algebras of the ideas of fountain and gould 14 as adapted to the jordan context by fernandez lopez and garcia rus in 7 in particular we characterize the set of lesieur croisot elements of a nondegenerate jordan algebra as those elements of the jordan algebra lying in the socle of its maximal algebra of quotients and apply this relationship to extend to quadratic jordan algebras the results of fernandez lopez and garcia rus on local orders in nondegenerate jordan algebras satisfying the descending chain condition on principal inner ideals and not containing ideals which are nonartinian quadratic factors

epigraphical and uniform convergence of convex functions we examine when a sequence of lsc convex functions on a banach space converges uniformly on bounded sets resp compact sets provided it converges attouch wets resp painlev e kuratowski we also obtain related results for pointwise convergence and uniform convergence on weakly compact sets some known results concerning the convergence of sequences of linear functionals are shown to also hold for lsc convex functions for example a sequence of lsc convex functions converges uniformly on bounded sets to a continuous affine function provided that the convergence is uniform on weakly compact sets and the space does not contain an isomorphic copy of ell 1

snakes active contour models a snake is an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges snakes are active contour models they lock onto nearby edges localizing them accurately scale space continuation can be used to enlarge the capture region surrounding a feature snakes provide a unified account of a number of visual problems including detection of edges lines and subjective contours motion tracking and stereo matching we have used snakes successfully for interactive interpretation in which user imposed constraint forces guide the snake near features of interest

shape topology optimization of stokes flow via variational level set method in this paper the conventional level set method is extended as an efficient approach for shape optimization the motion of the dynamic interface is governed by stokes equations although this method is not specifically designed for topology optimization it can handle topology changes easily furthermore a relatively smooth evolution can be maintained without re initialization the cost of this method is moderate the accuracy and efficiency of the proposed algorithm are illustrated by several numerical examples

grids and the arithmetics of jordan pairs abstract we prove that every 2 finitedimensional covering standard division grid of a jordan pair v over a henselian field canonically determines a norm on v this is used to classify maximal orders which contain a covering division grid of v we show that weakly separable orders always satisfy this condition and link their existence to ramification properties of v

a level set approach to domain wall kinetics and domain patterning in elastic ferroelectrics the level set method is a powerful tool for tracking a moving interface in ferroelectric materials the boundaries between domains of uniform polarization referred to as domain walls are viewed as interfaces in this work a variational level set method is used to formulate a model for the behavior of elastic ferroelectrics and more particularly to describe the domain formation and the domain wall kinetics a regularized version of the sharp interface theory is introduced by substitution of the singular surfaces with thin transition layers inserting so inhomogeneities into the energy density function it is proved that the forces driving domain wall motion are in fact inhomogeneity forces with this description the level set method becomes an appropriate framework to insert and exploit material forces the proposed model may interpret typical microstructures in 2 d elastic ferroelectrics with tetragonal crystal symmetry comparisons with phase field models and experiments show good agreement

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

a new algorithm for topology optimization using a level set method the level set method has been recently introduced in the field of shape optimization enabling a smooth representation of the boundaries on a fixed mesh and therefore leading to fast numerical algorithms however most of these algorithms use a hamilton jacobi equation to connect the evolution of the level set function with the deformation of the contours and consequently they can hardly create new holes in the domain at least in 2d in this work we propose an evolution equation for the level set function based on a generalization of the concept of topological gradient this results in a new algorithm allowing for all kinds of topology changes

compatible subtriples of jordan triples abstract the purpose of this paper is to extend the notion of compatibility of tripotents in an anisotropic jordan triple a to that of subtriples of a for a subtriple b of a define the kernel ker b and the annihilator b of b to be the subsets a a b a b 0 and a a b a a 0 respectively the peirce spaces b i i 0 1 2 of b are defined in terms of the kernel and the annihilator by b 0 b b 1 ker b ker b and b 2 b a pair b c of subtriples of a is said to be compatible if a coincides with i j 0 1 2 b i c j the subtriple b is said to be self compatible if it is compatible with itself a self compatible subtriple b is complemented in that b ker b coincides with a which implies that b is an inner ideal in a it follows that a pair u v of tripotents in a is compatible if and only if the ranges im p 2 u im p 2 v of the peirce projections p 2 u and p 2 v form a compatible pair of subtriples moreover for any tripotent u im p 2 u is a self compatible subtriple of a for the special case of a jbw triple a it is shown that a subtriple i is compatible with every self compatible subtriple j if and only if i is a complemented ideal in a these results are then applied to abelian jbw triples w algebras and hilbert spaces

the samuel realcompactification of a metric space abstract in this paper we introduce a realcompactification for any metric space x d defined by means of the family of all its real valued uniformly continuous functions we call it the samuel realcompactification according to the well known samuel compactification associated to the family of all the bounded real valued uniformly continuous functions among many other things we study the corresponding problem of the samuel realcompactness for metric spaces at this respect we prove that a result of kat tov shirota type occurs in this context where the completeness property is replaced by bourbaki completeness a notion recently introduced by the authors and the closed discrete subspaces are replaced by the uniformly discrete ones more precisely we see that a metric space x d is samuel realcompact iff it is bourbaki complete and every uniformly discrete subspace of x has non measurable cardinal as a consequence we derive that a normed space is samuel realcompact iff it has finite dimension and this means in particular that realcompactness and samuel realcompactness can be very far apart the paper also contains results relating this realcompactification with the so called lipschitz realcompactification also studied here with the classical hewitt nachbin realcompactification and with the completion of the initial metric space

discrete adjoint sensitivity analysis for fluid flow topology optimization based on the generalized lattice boltzmann method abstract a discrete adjoint sensitivity analysis for fluid flow topology optimization based on the lattice boltzmann method lbm with multiple relaxation times mrt is developed the lattice boltzmann fluid solver is supplemented by a porosity model using a darcy force the continuous transition from fluid to solid facilitates a gradient based optimization process of the design topology of fluidic channels the adjoint lbm equation which is used to compute the gradient of the optimization objective with respect to the design variables is derived in moment space and found to be as simple as the original lbm the moment based spatial momentum derivatives used to express the discrete objective functional cost function have the advantage that the local stress tensor is a local quantity avoiding the numerical computation of gradients of the discrete velocity field this is particularly useful if dissipation is a design criterion as demonstrated in this paper the method is validated by a detailed comparison with results obtained by borrvall et al for stokes flow while their approach is only valid for stokes flow i e very low reynolds numbers our approach in its present form can in principle be applied for flows of different reynolds numbers just like the navier stokes equation based approaches this point is demonstrated with a bending pipe example for various reynolds numbers

the topological asymptotic for pde systems the elasticity case the aim of the topological sensitivity analysis is to obtain an asymptotic expansion of a design functional with respect to the creation of a small hole in this paper such an expansion is obtained and analyzed in the context of linear elasticity for general functionals and arbitrary shaped holes by using an adaptation of the adjoint method and a domain truncation technique the method is general and can be easily adapted to other linear pdes and other types of boundary conditions

shape topology optimization for navier stokes problem using variational level set method we consider the shape topology optimization of the navier stokes problem a new algorithm is proposed based on the variational level set method by this algorithm a relatively smooth evolution can be maintained without re initialization and drastic topology change can be handled easily finally the promising features of the proposed method are illustrated by two benchmark examples

shape optimization of an acoustic horn the shape of an acoustic horn is optimized with respect to its ability to provide impedance matching to the surrounding air the resulting minimization problem is solved using a bfgs quasi newton algorithm the wave propagation is modeled by a finite element approximation of the helmholtz equation and the necessary objective function gradient is calculated from the solution of the associated adjoint equation local solutions to the optimization problem corresponding to irregular horn contours are avoided by using a filtering technique that applies smoothing to the design updates and the gradient the method converges fast to a smooth solution and the resulting horn has good impedance properties throughout the entire frequency band of interest

an improved implicit re initialization method for the level set function applied to shape and topology optimization of fluid this paper presents an accurate implicit re initialization approach in the framework of a level set method the improved method includes two schemes for keeping the zero level set unperturbed the first scheme is to derive and use a new formula for the smoothing parameter in the conventional re initialization equation based on the principle of the interface not moving the second scheme is to reduce the local time step to avoid the interface moving across grid points when the sign of the level set function near the interface is changed the example presented suggests that the new algorithm has a better approximation to the signed distance function and obtains a more accurate interface than the algorithm presented by sussman et al 1994 for shape and topology optimization of fluid the effectiveness of the developed method in this paper was demonstrated by the internal and external flow examples

a level set method for structural topology optimization this paper presents a new approach to structural topology optimization we represent the structural boundary by a level set model that is embedded in a scalar function of a higher dimension such level set models are flexible in handling complex topological changes and are concise in describing the boundary shape of the structure furthermore a well founded mathematical procedure leads to a numerical algorithm that describes a structural optimization as a sequence of motions of the implicit boundaries converging to an optimum solution and satisfying specified constraints the result is a 3d topology optimization technique that demonstrates outstanding flexibility of handling topological changes fidelity of boundary representation and degree of automation we have implemented the algorithm with the use of several robust and efficient numerical techniques of level set methods the benefit and the advantages of the proposed method are illustrated with several 2d examples that are widely used in the recent literature of topology optimization especially in the homogenization based methods

incorporating topological derivatives into shape derivatives based level set methods shape derivatives and topological derivatives have been incorporated into level set methods to investigate shape optimization problems the shape derivative measures the sensitivity of boundary perturbations while the topological derivative measures the sensitivity of creating a small hole in the interior domain the combination of these two derivatives yields an efficient algorithm which has more flexibility in shape changing and may escape from a local optimal examples on finding the optimal shapes for maximal band gaps in photonic crystal and acoustic drum problems are demonstrated

convergence of dynamic programming models weak conditions are presented for approximating dynamic programming models for a sequence of these models continuous convergence of the sequence of associated optimal value functions is obtained under the condition that state and action space converge in the sense of kuratowski and that the mappings of admissible actions as well as the transition law the discount factors and the reward functions converge continuously further a relation for the associated sets of optimal actions is given the analysis is based on results about convergence preserving properties of supremum value functions and integrals the approximation results are extended to so called upper semi continuous convergent sequences and are related to discretization procedures by using projections

a new approach for fuzzy topology ii abstract we point out a new approach for fuzzy topology with fuzzy logic and discuss the neighborhood structure of a point and the convergence of nets and filters in this new framework

locally lipschitz functions cofinal completeness and uc spaces abstract let x d be a metric space we find necessary and sufficient conditions on the space for the locally lipschitz functions to coincide with each of two more restrictive classes of locally lipschitz functions studied by several authors the uniformly locally lipschitz functions and the lipschitz in the small functions in the first case we get the cofinally complete spaces and in the second the uc spaces we address this question to which family of subsets of x d is the restriction of each function in each class actually lipschitz finally we determine exactly when the class of uniformly locally lipschitz functions is uniformly dense in the cauchy continuous real valued functions a class that naturally contains them in fact our theorem is valid when the target space is any banach space our density theorem complements the uniform approximation results of garrido and jaramillo 12 13

level set topology optimization of fluids in stokes flow we propose the level set method of topology optimization as a viable robust and efficient alternative to density based approaches in the setting of fluid flow the proposed algorithm maintains the discrete nature of the optimization problem throughout the optimization process leading to significant advantages over density based topology optimization algorithms specifically the no slip boundary condition is implemented directly this is accurate removes the need for interpolation schemes and continuation methods and gives significant computational savings by only requiring flow to be modeled in fluid regions topological sensitivity information is utilized to give a robust algorithm in two dimensions and familiar two dimensional power dissipation minimization problems are solved successfully computational efficiency of the algorithm is also clearly demonstrated on large scale three dimensional problems

the lipschitz metric for real valued continuous functions abstract for a continuous real function f defined on a metric space x let f denote its minimal lipschitz constant if f is lipschitz and put f otherwise we study the extended real valued metric on the continuous real functions defined by d f g max f x 0 g x 0 f g when x a b this metric provides new insight into a classical result regarding the derivative of a limit of a sequence of real valued functions defined on the interval

the range of a structural projection abstract let a be a jbw triple a linear subspace j of a is called an inner ideal in a provided that the subspace j a j is contained in j a subtriple b in a is said to be complemented if a b ker b where ker b a a b a b 0 a complemented subtriple in a is a weak closed inner ideal a linear projection on a is said to be structural if for all elements a b and c in a formula the range of a structural projection is a complemented subtriple and conversely a complemented subtriple is the range of a unique structural projection we analyze the structure of the weak closed inner ideal generated by two arbitrary tripotents in a jbw triple in terms of the simultaneous peirce spaces of three suitably chosen pairwise compatible tripotents this result is then used to show that every weak closed inner ideal j in a jbw triple a is a complemented subtriple in a and therefore the range of a unique structural projection on a as an application structural projections on w algebras are considered

strong uniform continuity abstract let b be an ideal of subsets of a metric space x d this paper considers a strengthening of the notion of uniform continuity of a function restricted to members of b which reduces to ordinary continuity when b consists of the finite subsets of x and agrees with uniform continuity on members of b when b is either the power set of x or the family of compact subsets of x the paper also presents new function space topologies that are well suited to this strengthening as a consequence of the general theory we display necessary and sufficient conditions for continuity of the pointwise limit of a net of continuous functions

l valued bornologies on powersets in m abel and a ostak 2011 1 the concept of an l fuzzy bornology was introduced actually an l fuzzy bornology on a set x is a certain ideal in the family lx of l fuzzy subsets of a set x here we propose an alternative approach to fuzzification of the concept of bornology we define an l valued bornology on a set x as an l fuzzy subset b of the powerset 2x satisfying l valued analogues of the axioms of a bornology basic properties of l valued bornological spaces are studied our special interest concerns l valued bornologies induced by fuzzy metrics and relative compactness type l valued bornologies in changgoguen l topological spaces

local pi theory of jordan systems ii abstract in this paper we study jordan systems having nonzero local algebras that satisfy a polynomial identity we prove that in nondegenerate jordan systems the set of elements at which the local algebra is pi is an ideal and that if it is nonzero it coincides with the socle when the system is primitive

incorporating topological derivatives into level set methods the aim of this paper is to investigate the use of topological derivatives in combination with the level set method for shape reconstruction and optimization problems we propose a new approach generalizing the standard speed method which is obtained by using a source term in the level set equation that depends on the topological derivative of the objective functional the resulting approach can be interpreted as a generalized fixed point iteration for the optimality system with respect to topological and shape variations moreover we apply the new approach for a simple model problem in shape reconstruction where the topological derivative can be computed without additional effort finally we present numerical tests related to this model problem which demonstrate that the new method based on shape and topological derivative successfully reconstructs obstacles in situations where the standard level set approach fails

an artinian theory for lie algebras abstract complemented lie algebras are introduced in this paper a notion similar to that studied by o loos and e neher in jordan pairs we prove that a lie algebra is complemented if and only if it is a direct sum of simple nondegenerate artinian lie algebras moreover we classify simple nondegenerate artinian lie algebras over a field of characteristic 0 or greater than 7 and describe the lie inner ideal structure of simple lie algebras arising from simple associative algebras with nonzero socle

essentially smooth lipschitz functions in this paper we address some of the most fundamental questions regarding the differentiability structure of locally lipschitz functions defined on separable banach spaces for example we examine the relationship between integrability d representability and strict differentiability in addition to this we show that on any separable banach space there is a significant family of locally lipschitz functions that are integrable d representable and possess desirable differentiability properties we also present some striking applications of our results to distance functions

a semismooth newton method for topology optimization abstract this paper deals with elliptic optimal control problems for which the control function is constrained to assume values in 0 1 based on an appropriate formulation of the optimality system a semismooth newton method is proposed for the solution convergence results are proved and some numerical tests illustrate the efficiency of the method

minimal usco and minimal cusco maps and compactness abstract we prove a nice generalization of the arzela ascoli theorem from continuous functions to minimal usco cusco maps into metric spaces let x be a locally compact space y d be a metric space k y be the space of all nonempty compact subsets of y and mu x y be the space of minimal usco maps from x to y the family e mu x y is compact in k y x equipped with the topology uc of uniform convergence on compact sets if and only if e is closed in mu x y uc pointwise bounded and densely equicontinuous the same result holds also for compact subsets of mc x y uc the space of minimal cusco maps from x to a banach space y

involutions on rectangular jordan pairs abstract jordan triple systems are equivalent to jordan pairs with involution in recent work with d amour on triples of clifford type we described involutions on pairs m 1 q generalizing these results in this paper we describe all involutions on nondegenerate pairs of rectangular type a r m f j having a simple artinian coordinate algebra r or more generally a simple unital coordinate algebra such that the form f is unital valued f u v 1 for some u m v m the involutions are of hermitian type determined by an involution anti automorphism with 2 1 on the coordinate ring automorphism type determined by an automorphism on the coordinate ring with 2 inner or of isomorphism type determined by an isomorphism of the necessarily non artinian coordinate ring onto a proper subring with 2 somewhat inner

topology optimization for stationary fluid structure interaction problems using a new monolithic formulation this paper outlines a new procedure for topology optimization in the steady state fluid structure interaction fsi problem a review of current topology optimization methods highlights the difficulties in alternating between the two distinct sets of governing equations for fluid and structure dynamics hereafter the fluid and structural equations respectively and in imposing coupling boundary conditions between the separated fluid and solid domains to overcome these difficulties we propose an alternative monolithic procedure employing a unified domain rather than separated domains which is not computationally efficient in the proposed analysis procedure the spatial differential operator of the fluid and structural equations for a deformed configuration is transformed into that for an undeformed configuration with the help of the deformation gradient tensor for the coupling boundary conditions the divergence of the pressure and the darcy damping force are inserted into the solid and fluid equations respectively the proposed method is validated in several benchmark analysis problems topology optimization in the fsi problem is then made possible by interpolating young s modulus the fluid pressure of the modified solid equation and the inverse permeability from the damping force with respect to the design variables

convergence of partial maps abstract given metric spaces x d and y a partial map between x and y is a pair d u where d is a closed subset of x and u d y is a function we introduce a general convergence notion for nets of such partial functions while our initial description is variational in nature we show that this description amounts to bornological convergence of the associated net of graphs as defined by lechicki levi and spakowski 26 with respect to a natural bornology on x y and which places the work on continuous partial functions of brandi ceppitelli and hola 12 13 20 21 in a general framework

a consistent relaxation of optimal design problems for coupling shape and topological derivatives in this article we introduce and analyze a general procedure for approximating a black and white shape and topology optimization problem with a density optimization problem allowing for the presence of grayscale regions our construction relies on a regularizing operator for smearing the characteristic functions involved in the exact optimization problem and on an interpolation scheme which endows the intermediate density regions with fictitious material properties under mild hypotheses on the smoothing operator and on the interpolation scheme we prove that the features of the approximate density optimization problem material properties objective function etc converge to their exact counterparts as the smoothing parameter vanishes in particular the gradient of the approximate objective functional with respect to the density function converges to either the shape or the topological derivative of the exact objective these results shed new light on the connections between these two different notions of sensitivities for functions of the domain and they give rise to different numerical algorithms which are illustrated by several experiments

topological derivative based topology optimization of structures subject to drucker prager stress constraints abstract an algorithm for topology optimization of elastic structures under plane stress subject to the drucker prager stress constraint is presented the algorithm is based on the use of the topological derivative of the associated objective functional in conjunction with a level set representation of the structure domain in this context a penalty functional is proposed to enforce the point wise stress constraint and a closed formula for its topological derivative is derived the resulting algorithm is of remarkably simple computational implementation it does not require post processing procedures of any kind and features only a minimal number of user defined algorithmic parameters this is in sharp contrast with current procedures for topological structural optimization with local stress constraints the effectiveness and efficiency of the algorithm presented here are demonstrated by means of numerical examples the examples show in particular that it can easily handle structural optimization problems with underlying materials featuring strong asymmetry in their tensile and compressive yield strengths

level set methods for optimization problems involving geometry and constraints i frequencies of a two density inhomogeneous drum many problems in engineering design involve optimizing the geometry to maximize a certain design objective geometrical constraints are often imposed in this paper we use the level set method devised in osher and sethian j comput phys 79 12 1988 the variational level set calculus presented in zhao et al j comput phys 127 179 1996 and the projected gradient method as in rudin et al physica d 60 259 1992 to construct a simple numerical approach for problems of this type we apply this technique to a model problem involving a vibrating system whose resonant frequency or whose spectral gap is to be optimized subject to constraints on geometry our numerical results are quite promising we expect to use this approach to deal with a wide class of optimal design problems in the future

on the topological derivative in shape optimization in this paper the topological derivative for an arbitrary shape functional is defined examples are provided for elliptic equations and the elasticity system in the plane the topological derivative can be used for solving shape optimization problems in structural mechanics

structural optimization using sensitivity analysis and a level set method in the context of structural optimization we propose a new numerical method based on a combination of the classical shape derivative and of the level set method for front propagation we implement this method in two and three space dimensions for a model of linear or nonlinear elasticity we consider various objective functions with weight and perimeter constraints the shape derivative is computed by an adjoint method the cost of our numerical algorithm is moderate since the shape is captured on a fixed eulerian mesh although this method is not specifically designed for topology optimization it can easily handle topology changes however the resulting optimal shape is strongly dependent on the initial guess

goldie theory for jordan algebras it is shown that zelmanov s version of goldie s conditions still characterizes quadratic jordan algebras having an artinian algebra of quotients which is nondegenerate at the same time jordan versions of the main notions of the associative theory such as those of the uniform ideal uniform element singular ideal and uniform dimension are studied moreover it is proved that the nondegenerate unital jordan algebras of finite capacity are precisely the algebras of quotients of nondegenerate jordan algebras having the property that an inner ideal is essential if and only if it contains an injective element 2002 elsevier science usa

an immersed boundary element method for level set based topology optimization summary in this paper we propose a new bem for level set based topology optimization in the proposed bem the nodal coordinates of the boundary element are replaced with the nodal level set function and the nodal coordinates of the eulerian mesh that maintains the level set function because this replacement causes the nodal coordinates of the boundary element to disappear the boundary element mesh appears to be immersed in the eulerian mesh therefore we call the proposed bem an immersed bem the relationship between the nodal coordinates of the boundary element and the nodal level set function of the eulerian mesh is clearly represented and therefore the sensitivities with respect to the nodal level set function are strictly derived in the immersed bem furthermore the immersed bem completely eliminates grayscale elements that are known to cause numerical difficulties in topology optimization by using the immersed bem we construct a concrete topology optimization method for solving the minimum compliance problem we provide some numerical examples and discuss the usefulness of the constructed optimization method on the basis of the obtained results copyright 2012 john wiley sons ltd

the shape and topological optimizations connection abstract in 1973 jean cea introduced a topological shape optimization algorithm based on a fixed point method the recent topological gradient concept provides a mathematical justification of cea s powerful method the aim of the topological gradient is to compute the sensitivity of a cost function when we drill a little hole in the domain some optimality conditions are derived when classical shape gradient and topological gradient are used simultaneously and numerical results are shown both in mechanics and electromagnetics

complementary inner ideals in jbw triples abstract the annihilator l of a subspace l of a jbw triple a consists of the elements a in a for which l a a is equal to 0 the kernel ker l of l consists of those elements a in a for which l a l is equal to 0 and the inner ideal inid l in a associated with l consists of the elements a in a for which a l a is equal to 0 and l a a is contained in l a weak closed subspace j is said to be an inner ideal in a if j a j is contained in j in which case a j j 1 j where j 1 is the intersection of the kernels of j and j the inner ideal inid j in a associated with a weak closed inner ideal j in a forms a complementary weak closed inner ideal to j it turns out that inid j is compatible with j and coincides with inid j k j m j in the case where j is a peirce inner ideal in a by completely identifying inid j it is shown that inid j is a peirce inner ideal in a and the inner ideal inid inid j in a associated with inid j is equal to j

real valued lipschitz functions and metric properties of functions abstract the purpose of this article is to explore the very general phenomenon that a function between metric spaces has a particular metric property if and only if whenever it is followed in a composition by an arbitrary real valued lipschitz function the composition has this property the key tools we use are the efremovic lemma 21 and a theorem of garrido and jaramillo 22 that says that a function h between metric spaces is lipschitz if and only if whenever it is followed by a lipschitz real valued function in a composition the composition is lipschitz we also present a streamlined proof of the garrido jaramillo result itself but one that still relies on their natural continuous linear operator from the lipschitz space for the target space to the lipschitz space for the domain separately we include a highly applicable uniform closure theorem that yields the most important uniform density theorems for lipschitz type functions as special cases

primitivity in jordan systems is ubiquitous abstract the main result of the paper asserts that if a jordan pair v v v is primitive at some 0 b v then it is primitive at any 0 b v also if a jordan triple system t is primitive at some 0 b t then it is primitive at any 0 b t as a tool similar results concerning one sided primitivity and primitivity of associative pairs and triple systems are established equivalences between our definitions of primitivity and some other appearing in the literature are also obtained as a consequence

regularized perimeter for topology optimization the perimeter functional is known to oppose serious difficulties when it has to be handled within a topology optimization procedure in this paper a regularized perimeter functional mbox per varepsilon defined for two and three dimensional domains is introduced on one hand the convergence of mbox per varepsilon to the exact perimeter when varepsilon tends to zero is proved on the other hand the topological differentiability of mbox per varepsilon for varepsilon 0 is analyzed these features lead to the design of a topology optimization algorithm suitable for perimeter dependent objective functionals several numerical results illustrate the method

arzela s theorem and strong uniform convergence on bornologies abstract in 1883 arzela 1983 1984 2 gave a necessary and sufficient condition via quasi uniform convergence for the pointwise limit of a sequence of real valued continuous functions on a compact interval to be continuous arzela s work paved the way for several outstanding papers a milestone was the p s alexandroff convergence introduced in 1948 to tackle the question for a sequence of continuous functions from a topological space not necessarily compact to a metric space in 2009 in the realm of metric spaces beer and levi 2009 10 found another necessary and sufficient condition through the novel notion of strong uniform convergence on finite sets we offer a direct proof of the equivalence of arzela alexandroff and beer levi conditions the proof reveals the internal gear of these important convergences and sheds more light on the problem we also study the main properties of the topology of strong uniform convergence of functions on bornologies initiated in beer and levi 2009 10

pointwise convergence of partial functions the gerlits nagy problem abstract for a set x r let b x r x denote the space of borel real valued functions on x with the topology inherited from the tychonoff product r x assume that for each countable a b x each f in the closure of a is in the closure of a under pointwise limits of sequences of partial functions we show that in this case b x is countably frechet urysohn that is each point in the closure of a countable set is a limit of a sequence of elements of that set this solves a problem of arnold miller the continuous version of this problem is equivalent to a notorious open problem of gerlits and nagy answering a question of salvador hernandez we show that the same result holds for the space of all baire class 1 functions on x we conjecture that in the general context the answer to the continuous version of this problem is negative but we identify a nontrivial context where the problem has a positive solution the proofs establish new local to global correspondences and use methods of infinite combinatorial topology including a new fusion result of francis jordan

a variational level set approach to multiphase motion a coupled level set method for the motion of multiple junctions of e g solid liquid and grain boundaries which follows the gradient flow for an energy functional consisting of surface tension proportional to length and bulk energies proportional to area is developed the approach combines the level set method of s osher and j a sethian with a theoretical variational formulation of the motion by f reitich and h m soner the resulting method uses as many level set functions as there are regions and the energy functional is evaluated entirely in terms of level set functions the gradient projection method leads to a coupled system of perturbed by curvature terms hamilton jacobi equations the coupling is enforced using a single lagrange multiplier associated with a constraint which essentially prevents a regions from overlapping and b the development of a vacuum the numerical implementation is relatively simple and the results agree with and go beyond the theory as given in 12 other applications of this methodology including the decomposition of a domain into subregions with minimal interface length are discussed finally some new techniques and results in level set methodology are presented

idempotents and peirce gradings of jordan algebras abstract in this paper we find a necessary and sufficient condition for a peirce grading of a jordan algebra j to come from a peirce decomposition with respect to an idempotent of a jordan algebra j containing j as a subalgebra we also show that the above condition holds automatically when j is nondegenerate

structural boundary design via level set and immersed interface methods we develop and test an algorithmic approach to the boundary design of elastic structures the goal of our approach is two fold first to develop a method which allows one to rapidly solve the two dimensional lam equations in arbitrary domains and compute for example the stresses and second to develop a systematic way of modifying the design to optimize chosen properties at the core our approach relies on two distinct steps given a design we first apply an explicit jump immersed interface method to compute the stresses for a given design shape we then use a narrow band level set method to perturb this shape and progress towards an improved design the equations of 2d linear elastostatics in the displacement formulation on arbitrary domains are solved quickly by domain embedding and the use of fast elastostatic solvers this effectively reduces the dimensionality of the problem by one once the stresses are found the level set method which represents the design structure through an embedded implicit function is used in the second step to alter the shape with velocities depending on the stresses in the current design criteria are provided for advancing the shape in an appropriate direction and for correcting the evolving shape when given constraints are violated

semicontinuous limits of nets of continuous functions in this paper we present a topology on the space of real valued functions defined on a functionally hausdorff space x that is finer than the topology of pointwise convergence and for which 1 the closure of the set of continuous functions mathcal c x is the set of upper semicontinuous functions on x and 2 the pointwise convergence of a net in mathcal c x to an upper semicontinuous limit automatically ensures convergence in this finer topology

a level set method for vibration and multiple loads structural optimization abstract we extend the level set method for shape and topology optimization to new objective functions such as eigenfrequencies and multiple loads this method is based on a combination of the classical shape derivative and of the osher sethian level set algorithm for front propagation in two and three space dimensions we maximize the first eigenfrequency or we minimize a weighted sum of compliances associated to different loading configurations the shape derivative is used as an advection velocity in a hamilton jacobi equation for changing the shape this level set method is a low cost shape capturing algorithm working on a fixed eulerian mesh and it can easily handle topology changes

lipschitz type functions on metric spaces in order to find metric spaces x for which the algebra lip x of bounded lipschitz functions on x determines the lipschitz structure of x we introduce the class of small determined spaces we show that this class includes precompact and quasi convex metric spaces we obtain several metric characterizations of this property as well as some other characterizations given in terms of the uniform approximation and the extension of uniformly continuous functions in particular we show that x is small determined if and only if every uniformly continuous real function on x can be uniformly approximated by lipschitz functions