isogeometric analysis using t splines we explore t splines a generalization of nurbs enabling local refinement as a basis for isogeometric analysis we review t splines as a surface design methodology and then develop it for engineering analysis applications we test t splines on some elementary two dimensional and three dimensional fluid and structural analysis problems and attain good results in all cases we summarize the current status of t splines their limitations and future possibilities

local refinement of analysis suitable t splines we develop a local refinement algorithm for analysis suitable t splines which does not produce excessive propagation of control points we then demonstrate its use as an adaptive framework for isogeometric analysis analysis suitable t splines are a class of t splines which are linearly independent and form a partition of unity these properties coupled with local refinement make this class of t splines appealing as a basis for isogeometric analysis

global existence and asymptotic properties of the solution to a two species chemotaxis system abstract this paper deals with the cauchy problem for a two species chemotactic keller segel system u t u 1 u w v t v 2 v w w t w w 1 u 2 v in r 2 0 where 0 1 2 and 1 2 are real numbers we obtain the global existence of solutions if u 0 1 v 0 1 and w 0 2 are small and the asymptotic behavior of the small data solution as follows if 0 the solution is asymptotic to a self similar solution for large time if 0 the solution behaves like a multiple of the heat kernel as t

characterization of t splines with reduced continuity order on t meshes abstract the use of t splines 30 in isogeometric analysis 24 has been proposed in 5 as a tool to enhance the flexibility of isogeometric methods if t splines are a very general concept their success in isogeometric analysis relies upon some basic properties that needs to be true as e g i linear independence of blending functions and ii polynomial reproducibility at element level in this paper we study these properties for t splines of a reduced regularity order namely for t splines of degree p and regularity p 1 p 2 our results are both for odd and even degree under mild assumptions on the underlying t mesh t splines are shown to be linearly independent and the space they span is characterized in terms of piecewise polynomials on a topological extension of the t mesh also as p is odd we construct a new topological local refinement algorithm and demonstrate its locality properties through numerical examples

selective and reduced numerical integrations for nurbs based isogeometric analysis we propose a new approach to construct selective and reduced integration rules for isogeometric analysis based on nurbs elements the notion of an approximation space that approximates the target space is introduced we explore the use of various approximation spaces associated with optimal patch wise numerical quadratures that exactly integrate the polynomials in approximation spaces with the minimum number of quadrature points patch rules exploit the higher continuity of spline basis functions the tendency of smooth spline functions to exhibit numerical locking in nearly incompressible problems when using a full gauss legendre quadrature is alleviated with selective or reduced integration stability and accuracy of the schemes are examined analyzing the discrete spectrum in a generalized eigenvalue problem we propose a local algorithm which is robust and computationally efficient to compute element by element the quadrature points and weights in patch rules the performance of the methods is assessed on several numerical examples in two dimensional elasticity and reissner mindlin shell structures

minimal forbidden factors of circular words minimal forbidden factors are a useful tool for investigating properties of words and languages two factorial languages are distinct if and only if they have different antifactorial sets of minimal forbidden factors there exist algorithms for computing the minimal forbidden factors of a word as well as of a regular factorial language conversely crochemore et al ipl 1998 gave an algorithm that given the trie recognizing a finite antifactorial language m computes a dfa of the language having m as set of minimal forbidden factors in the same paper they showed that the obtained dfa is minimal if the input trie recognizes the minimal forbidden factors of a single word we generalize this result to the case of a circular word we also discuss combinatorial properties of the minimal forbidden factors of a circular word finally we characterize the minimal forbidden factors of the circular fibonacci words

existence and uniqueness theorem on weak solutions to the parabolic elliptic keller segel system abstract in r n n 3 we first define a notion of weak solutions to the keller segel system of parabolic elliptic type in the scaling invariant class l s 0 t l r r n for 2 s n r 2 with n 2 r n any condition on derivatives of solutions is not required at all the local existence theorem of weak solutions is established for every initial data in l n 2 r n we prove also their uniqueness as for the marginal case when r n 2 we show that if n 4 then the class c 0 t l n 2 r n enables us to obtain the only weak solution

two way string matching a new string matching algorithm is presented which can be viewed as an intermediate between the classical algorithms of knuth morris and pratt on the one hand and boyer and moore on the other hand the algorithm is linear in time and uses constant space as the algorithm of galil and seiferas it presents the advantage of being remarkably simple which consequently makes its analysis possible the algorithm relies on a previously known result in combinatorics on words called the critical factorization theorem which relates the global period of a word to its local repetitions of blocks

isogeometric analysis using lr b splines abstract the recently proposed locally refined b splines denoted lr b splines by dokken et al 2013 6 may have the potential to be a framework for isogeometric analysis to enable future interoperable computer aided design and finite element analysis in this paper we propose local refinement strategies for adaptive isogeometric analysis using lr b splines and investigate its performance by doing numerical tests on well known benchmark cases the theory behind lr b spline is not presented in full details but the main conceptual ingredients are explained and illustrated by a number of examples

symmetrization techniques on unbounded domains application to a chemotaxis system on rn the authors study the parabolic elliptic system on rn u t u u v 0 v v u u 0 u0 a version of the mathematical model of chemotaxis proposed by keller and segel a differential inequality for the quantity s0u t d where u is the decreasing rearrangement of the solution u t with respect to the spatial variable is obtained as a consequence they obtain e g lp bounds of the solution u v on r2 and global in time existence of solutions under the condition r2u0 8 then the solution u v blows up in a finite time compared to the previous work of diaz diaz and nagai adv math sci appl 5 1995 no 2 659 680 mr1361010 96j 35246 where this problem has been considered on bounded domains of rn there are some additional technical difficulties connected with the regularity of the derivative u t

global strong solution to the semi linear keller segel system of parabolic parabolic type with small data in scale invariant spaces abstract we shall show existence of global strong solution to the semi linear keller segel system in r n n 3 of parabolic parabolic type with small initial data u 0 h n r 2 r r n and v 0 h n r r r n for max 1 n 4 r n 2 our method is based on the perturbation of linealization together with the l p l q estimates of the heat semigroup and the fractional powers of the laplace operator as a by product of our method we shall prove the decay property of solutions as the time goes to infinity

isogeometric spline forests in this paper we present isogeometric spline forests an isogeometric spline forest is a hierarchical spline representation capable of representing surfaces or volumes of arbitrarily complex geometry and topological genus spline forests can accomodate arbitrary degree and smoothness in the underlying hierarchical basis as well as non uniform knot interval configurations we describe adaptive h refinement and coarsening algorithms for isogeometric spline forests and develop a bezier extraction framework which provides a simple and efficient single level finite element description of the complex multi level unstructured hierarchical spline basis we then demonstrate the potential of spline forests as a basis for analysis in the context of transient advection diffusion problems where fully integrated adaptivity is demonstrated for the first time in an isogeometric simulation in all cases the adaptive process remains local even in the case of moving fronts and preserves exact geometry at the coarsest level of the discretization the accuracy and robustness of the approach is demonstrated in all cases

adaptive finite element methods for elliptic equations over hierarchical t meshes isogeometric analysis using nurbs non uniform rational b splines as basis functions gives accurate representation of the geometry and the solution but it is not well suited for local refinement in this paper we use the polynomial splines over hierarchical t meshes pht splines to construct basis functions which not only share the nice smoothness properties as the b splines but also allow us to effectively refine meshes locally we develop a residual based a posteriori error estimate for the finite element discretization of elliptic equations using pht splines basis functions and study their approximation properties in addition we conduct numerical experiments to verify the theory and to demonstrate the effectiveness of the error estimate and the high order approximations provided by the numerical solution

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

a posteriori error estimation for variational problems with uniformly convex functionals the objective of this paper is to introduce a general scheme for deriving a posteriori error estimates by using duality theory of the calculus of variations we consider variational problems of the form inf f v g v where f v r is a convex lower semicontinuous functional g y r is a uniformly convex functional v and y are reflexive banach spaces and v y is a bounded linear operator we show that the main classes of a posteriori error estimates known in the literature follow from the duality error estimate obtained and thus can be justified via the duality theory

the structure of subword graphs and suffix trees of fibonacci words we use automata theoretic approach to analyze properties of fibonacci words the directed acyclic subword graph dawg is a useful deterministic automaton accepting all suffixes of the word we show that dawg s of fibonacci words have particularly simple structure our main result is a unifying framework for a large collection of relatively simple properties of fibonacci words the simple structure of dawgs of fibonacci words gives in many cases simplified alternative proofs and new interpretation of several well known properties of fibonacci words in particular the structure of lengths of paths corresponds to a number theoretic characterization of occurrences of any subword using the structural properties of dawg s it can be easily shown that for a string we can check if is a subword of a fibonacci word in time o and o 1 space compact dawg s of fibonacci words show a very regular structure of their suffix trees and show how the suffix tree for the fibonacci word grows extending the leaves in a very simple way into the suffix tree for the next fibonacci word

on uniqueness theorem on weak solutions to the parabolic parabolic keller segel system of degenerate and singular types abstract the uniqueness of weak solutions to the parabolic parabolic keller segel systems ks m below with m max 1 2 1 n 0 is proved in the class of holder continuous functions for any space dimension n since holder continuity is an optimal regularity for weak solutions of the porous medium equation it seems to be reasonable to investigate its uniqueness in such a class of solutions our proof is based on the standard duality argument coupled with vanishing viscosity method which recovers degeneracy for m 1 and which removes singularities for 0 m 1 in the energy class of solutions

a characterization of the squares in a fibonacci string a finite fibonacci stringfn is defined as follows f0 b f1 a for every integer n 2 fn fn 1fn 2 for n 1 the length of fn is denoted by the infinite fibonacci stringf is the string which contains every fn n 1 as a prefix apart from their general theoretical importance fibonacci strings are often cited as worst case examples for algorithms which compute all the repetitions or all the abelian squares in a given string in this paper we provide a characterization of all the squares in f hence in every prefix fn this characterization naturally gives rise to a algorithm which specifies all the squares of fn in an appropriate encoding this encoding is made possible by the fact that the squares of fn occur consecutively in runs the number of which is by contrast the known general algorithms for the computation of the repetitions in an arbitrary string require time and produce outputs when applied to a fibonacci string fn

ill posedness of the two dimensional keller segel model in triebel lizorkin spaces abstract this article proves the ill posedness of the cauchy problem for the two dimensional keller segel model in triebel lizorkin spaces f 2 1 r r 2 for 2 r in particular it is shown that solutions can develop norm inflation under certain settings in that the solution can become arbitrarily large after an arbitrarily short time even for small initial data

some a posteriori error estimators for elliptic partial differential equations we present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations the estimators are based on solving local neumann problems in each element the estimators differ in how they enforce consistency of the neumann problems we prove that as the mesh size decreases under suitable assumptions two of the error estimators approach upper bounds on the norm of the true error and all three error estimators are within multiplicative constants of the norm of the true error we present numerical results in which one of the error estimators appears to converge to the norm of the true error

a posteriori error estimates based on hierarchical bases the authors present an analysis of an a posteriors error estimator based on the use of hierarchical basis functions the authors analyze nonlinear nonselfadjoint and indefinite problems as well as the selfadjoint positive definite case because both the analysis and the estimator itself are quite simple it is easy to see how various approximations affect the quality of the estimator as examples the authors apply the theory to some scalar elliptic equations and the stokes system of equations

existence and uniqueness theorem on mild solutions to the keller segel system in the scaling invariant space we prove the existence and uniqueness of solutions u v to the keller segel system of parabolic parabolic type in rn for n 3 in the scaling invariant class u lq 0 t lr rn v lr 0 t h r rn where 2 q n r 2 2 q n r 2 provided the initial data u0 v0 is chosen as u0 ln 2 rn v0 h 2 n 2 rn for n 2 n 2 1 2 in particular our uniqueness result holds for all n 2 even though we impose an assumption only on u such as lq 0 t lr rn for 2 q n r 2 with n 2 r as for the marginal case when r n 2 we show that if n 3 and if u c 0 t ln 2 rn v c 0 t ln rn then u v is the only solution

variationally consistent domain integration for isogeometric analysis abstract spline type approximations for solving partial differential equations are the basis of isogeometric analysis while the common approach of using integration cells defined by single knot spans using standard e g gaussian quadrature rules is sufficient for accuracy more efficient domain integration is still in high demand the recently introduced concept of variational consistency provides a guideline for constructing accurate and convergent methods requiring fewer quadrature points than standard integration techniques in this work variationally consistent domain integration is proposed for isogeometric analysis test function gradients are constructed to meet the consistency conditions which only requires solving small linear systems of equations the proposed approach allows for significant reduction in the number of quadrature points employed while maintaining the stability accuracy and optimal convergence properties of higher order quadrature rules several numerical examples are provided to illustrate the performance of the proposed domain integration technique

anisotropic nurbs approximation in isogeometric analysis abstract we investigate the approximation error of mapped nurbs spaces such as those used in isogeometric analysis in accordance to the isoparametric paradigm we extend the results of 6 to the case of anisotropic meshes such as those needed in the presence of boundary layers the theoretical results cover anisotropic h refinement and are supported by numerical tests involving different geometries

singularity patterns in a chemotaxis model the authors study a chemotactic model under certain assumptions and obtain the existence of a class of solutions which blow up at the center of an open disc in finite time such a finite time blow up of solutions implies chemotactic collapse namely concentration of species to form sporae the model studied is the limiting case of a basic chemotactic model when diffusion of the chemical approaches infinity which has the form ut u u v 0 v u 1 on r2 where is an open disc with no flux homogeneous neumann boundary conditions the initial conditions are continuous functions u x 0 u0 x 0 v x 0 v0 x 0 for x under these conditions the authors prove there exists a radially symmetric solution u r t which blows up at r 0 t t a specific description of such a solution is presented the authors also discuss the strong similarity between the chemotactic model they study and the classical stefan problem

simple a posteriori error estimators in adaptive isogeometric analysis in this article we propose two simple a posteriori error estimators for solving second order elliptic problems using adaptive isogeometric analysis the idea is based on a serendipity11according to wikipedia serendipity means a fortunate happenstance or pleasant surprise it was coined by horace walpole in 1754 one aspect of walpole s original definition of serendipity is the need for an individual to be sagacious enough to link together apparently innocuous facts in order to come to a valuable conclusion we feel that this applies for the present discovery but it is of course up to the readers to judge pairing of discrete approximation spaces s h p k m s h p 1 k 1 m where the space s h p 1 k 1 m is considered as an enrichment of the original basis of s h p k m by means of the k refinement a typical unique feature available in isogeometric analysis the space s h p 1 k 1 m is used to obtain a higher order accurate isogeometric finite element approximation and using this approximation we propose two simple a posteriori error estimators the proposed a posteriori error based adaptive h refinement methodology using lr b splines is tested on classical elliptic benchmark problems the numerical tests illustrate the optimal convergence rates obtained for the unknown as well as the effectiveness of the proposed error estimators