global existence and blow up of solutions to a nonlocal quasilinear degenerate parabolic system abstract this paper investigates the properties of nonnegative solutions of a quasilinear degenerate parabolic system u t div u p 2 u a v x t d x v t div v q 2 v b u x t d x with zero dirichlet boundary conditions in a smooth bounded domain r n n 1 where p q 2 1 and a b 0 are positive constants under appropriate hypotheses we first establish the local existence and uniqueness of solutions then we show that whether or not the solution blows up in finite time depends on the initial data and the relations between and p 1 q 1 in the special case of q 1 and p 1 we conclude that the solution exists globally if p 1 d x q 1 d x 1 a b while if p 1 d x q 1 d x 1 a b then the solution blows up in finite time here x and x denote the unique solution of the following elliptic problem div p 2 1 in x 0 and div q 2 1 in x 0 respectively

an explicit description of the second cohomology group of a quandle we use the inflation restriction sequence and a result of etingof and grana on the rack cohomology to give a explicit description of 2 cocycles of finite indecomposable quandles with values in an abelian group several applications are given

from racks to pointed hopf algebras abstract a fundamental step in the classification of finite dimensional complex pointed hopf algebras is the determination of all finite dimensional nichols algebras of braided vector spaces arising from groups the most important class of braided vector spaces arising from groups is the class of braided vector spaces c x c q where x is a rack and q is a 2 cocycle on x with values in c racks and cohomology of racks appeared also in the work of topologists this leads us to the study of the structure of racks their cohomology groups and the corresponding nichols algebras we will show advances in these three directions we classify simple racks in group theoretical terms we describe projections of racks in terms of general cocycles we introduce a general cohomology theory of racks containing properly the existing ones we introduce a fourier transform on racks of certain type finally we compute some new examples of finite dimensional nichols algebras

new large rank nichols algebras over nonabelian groups with commutator subgroup z2 abstract in this article we explicitly construct new finite dimensional indecomposable nichols algebras with dynkin diagrams of type a n c n d n e 6 7 8 f 4 over any group g of order 2 l with commutator subgroup isomorphic to z 2 the construction is generic in the sense that the type just depends on the rank and center of g and thus positively answers for all groups of this class a question raised by susan montgomery in 1995 22 4 our construction uses the new notion of a covering nichols algebra as a special case of a covering hopf algebra 18 and produces non faithful nichols algebras we give faithful examples of doi twists for type a 3 c 3 d 4 c 4 f 4 over several nonabelian groups of order 16 and 32 these are hence the first known examples of nondiagonal finite dimensional indecomposable nichols algebras of rank 2 over nonabelian groups

a note on blow up of solutions of a quasilinear heat equation with vanishing initial energy abstract in this work we consider an initial boundary value problem related to the equation u t div u m 2 u f u and prove under suitable conditions on f a blow up result for solutions with vanishing or negative initial energy

diffusivity versus absorption through the boundary for all 0 1 then we do expect to find that any solution becomes unbounded in finite time see 13 15 63 it is our goal to prove an analogous result for m f 1 for a 1 one can find that if v is a positive solution of

blow up and global solutions for a class of nonlinear parabolic equations with different kinds of boundary conditions abstract by constructing different auxiliary functions and using hopf s maximum principle the sufficient conditions for the blow up and global solutions are presented for nonlinear parabolic equation u t a u b x c t u f x u q t with different kinds of boundary conditions the upper bounds of the blow up time and the upper estimates of global solutions are provided finally some examples are presented as the application of the obtained results

nichols algebras over groups with finite root system of rank two iii abstract we study nichols algebras of direct sums of two absolutely simple yetter drinfeld modules over non abelian groups and arbitrary fields we construct families of new examples where the nichols algebra is finite dimensional and not of diagonal type

a new condition for the concavity method of blow up solutions to p laplacian parabolic equations abstract in this paper we consider an initial boundary value problem of the p laplacian parabolic equation u t x t div u x t p 2 u x t f u x t x t 0 u x t 0 x t 0 u x 0 u 0 0 x where p 2 and is a bounded domain of r n n 1 with smooth boundary the main contribution of this work is to introduce a new condition c p 0 u f s d s u f u u p u 0 for some 0 with 0 p 1 p p where 1 p is the first eigenvalue of p laplacian p and we use the concavity method to obtain the blow up solutions to the above equations in fact it will be seen that the condition c p improves the conditions ever known so far

a quasilinear reaction diffusion system coupled via nonlocal sources in this paper we establish the critical exponent and the blow up rate for a quasilinear reaction diffusion system with nonlocal nonlinear sources and inner absorptions subject to homogeneous dirichlet conditions and nonnegative initial data it is found that the critical exponent is determined by the interaction among all the six nonlinear exponents from all the three kinds of the nonlinearities while the blow up rate is independent of the nonlinear diffusion exponents due to the effects of the coupled nonlocal sources two kinds of characteristic algebraic systems are introduced to get simple descriptions for the critical exponent and the blow up rate respectively we prove moreover that the blow up could be global due to the nonlocality of the nonlinear sources

a freeness theorem for nichols algebras abstract we prove a freeness result for nichols algebras over certain nichols subalgebras this result allows us in particular to classify pointed hopf algebras of prime and prime squared index

the problem of dirichlet for evolution one dimensional p laplacian with nonlinear source in the present paper we consider the dirichlet problem for one dimensional p laplacian with nonlinear source we obtain new a priori estimates of a solution and of the gradient of a solution and formulate conditions guaranteeing the global solvability of this problem our consideration includes singular case as well

global existence and blow up of solutions to an evolution p laplace system coupled via nonlocal sources abstract the aim of this paper is to investigate the behavior of positive solutions to the following system of evolution p laplace equations coupled via nonlocal sources u t u x p 1 1 u x x 0 a v m 1 t d x t in 0 a 0 t v t v x p 2 1 v x x 0 a u m 2 t d x t in 0 a 0 t with nonlinear boundary conditions u x x 0 0 u x x a u q 11 v q 12 x a v x x 0 0 v x x a u q 21 v q 22 x a and the initial data u 0 v 0 where p 1 p 2 1 m 1 m 2 q 11 q 12 q 21 q 22 0 under appropriate hypotheses the authors first prove a local existence result by a regularization method then the authors discuss the global existence and blow up of positive weak solutions by using a comparison principle

quandle cohomology and state sum invariants of knotted curves and surfaces the 2 twist spun trefoil is an example of a sphere that is knotted in 4 dimensional space a proof is given in this paper that this sphere is distinct from the same sphere with its orientation reversed our proof is based on a state sum invariant for knotted surfaces developed via a cohomology theory of racks and quandles also known as distributive groupoids a quandle is a set with a binary operation the axioms of which model the reidemeister moves in classical knot theory colorings of diagrams of knotted curves and surfaces by quandle elements together with cocycles of quandles are used to define state sum invariants for knotted circles in 3 space and knotted surfaces in 4 space cohomology groups of various quandles are computed herein and applied to the study of the state sum invariants non triviality of the invariants is proved for a variety of knots and links and conversely knot invariants are used to prove non triviality of cohomology for a variety of quandles

necessary and sufficient conditions for complete blow up and extinction for one dimensional quasilinear heat equations we characterize the occurrence of complete or incomplete blow up and extinction for a general quasilinear heat equation of the form he text u t phi u xx pm f u text in r times 0 t in terms of the constitutive functions and f we assume that u 0 for u 0 and that f u 0 for the positive sign before f u in he with f u superlinear as u blow up occurs in finite time supxu x t as t t t i e after the singularity occurs if such continuation exists we say that the blow up extinction is incomplete otherwise it is called complete our characterization is based on the qualitative behaviour of the family of travelling wave solutions and a proper use of the intersection comparison argument the analysis applies to other nonlinear models like the equations with gradient dependent diffusivity

lifting of quantum linear spaces and pointed hopf algebras of orderp3 abstract we propose the following principle to study pointed hopf algebras or more generally hopf algebras whose coradical is a hopf subalgebra given such a hopf algebraa consider its coradical filtration and the associated graded coalgebra gr a then gr ais a graded hopf algebra since the coradicala0ofais a hopf subalgebra in addition there is a projection gr a a0 letrbe the algebra of coinvariants of then by a result of radford and majid ris a braided hopf algebra and gr ais the bosonization or biproduct ofranda0 gr a r a0 the principle we propose to studyais first to studyr then to transfer the information to gr avia bosonization and finally to lift toa in this article we apply this principle to the situation whenris the simplest braided hopf algebra a quantum linear space as consequences of our technique we obtain the classification of pointed hopf algebras of orderp3 pan odd prime over an algebraically closed field of characteristic zero with the same hypothesis the characterization of the pointed hopf algebras whose coradical is abelian and has indexporp2 and an infinite family of pointed nonisomorphic hopf algebras of the same dimension this last result gives a negative answer to a conjecture of i kaplansky

finite dimensional pointed or copointed hopf algebras over affine racks abstract we study the pointed or copointed liftings of nichols algebras associated to affine racks and constant cocycles for any finite group admitting a principal yd realization of these racks in the copointed case we complete the classification for the six affine racks whose nichols algebra is known to be of finite dimension in the pointed case our method allows us to finish four of them in all of the cases the hopf algebras obtained turn out to be cocycle deformations of their associated graded hopf algebras all of them are new examples of finite dimensional copointed or pointed hopf algebras over non abelian groups

on the classification of finite dimensional pointed hopf algebras we classify finite dimensional complex hopf algebras a which are pointed that is all of whose irreducible comodules are one dimensional and whose group of group like elementsg a is abelian such that all prime divisors of the order of g a are 7 since these hopf algebras turn out to be deformations of a natural class of generalized small quantum groups our result can be read as an axiomatic description of generalized small quantum groups

doubly nonlinear degenerate parabolic systems with coupled nonlinear boundary conditions in this paper we study the global existence and the global nonexistence of doubly nonlinear degenerate parabolic systems with nonlinear boundary conditions we first prove a local existence result by the regularization method next we construct a weak comparison principle then we discuss the large time behavior of solutions by using a modified upper and lower solution methods and constructing various upper and lower solutions necessary and sufficient conditions on the global existence of all positive weak solutions are obtained

classification of arithmetic root systems arithmetic root systems are invariants of nichols algebras of diagonal type with a certain finiteness property they can also be considered as generalizations of ordinary root systems with rich structure and many new examples on the other hand nichols algebras are fundamental objects in the construction of quantized enveloping algebras in the noncommutative differential geometry of quantum groups and in the classification of pointed hopf algebras by the lifting method of andruskiewitsch and schneider in the present paper arithmetic root systems are classified in full generality as a byproduct many new finite dimensional pointed hopf algebras are obtained

a necessary and sufficient condition for global existence for a degenerate parabolic boundary value problem consider the degenerate parabolic boundary value problemut u f u on 0 in which is a bounded domain inrnand thec 0 functionsfand are nonnegative and nondecreasing with s f s 0 ifs 0 and 0 0 assume homogeneous neumann boundary conditions and an initial condition that is nonnegative nontrivial and continuous on because the function is not sufficiently nice to allow this problem to have a classical solution we consider generalized solutions in a manner similar to that of benilan crandall and sacks appl math optim 17 1988 203 224 we show that this initial boundary value problem has such a nonnegative generalized solution if and only ifi 0 ds 1 f s

blow up and global existence for nonlinear parabolic equations with neumann boundary conditions in this paper we study the global and blow up solutions of the following problem h u t a u t b x u g t f u in dx 0 t u n 0on dx 0 t u x 0 u 0 x 0in d where d r n is a bounded domain with smooth boundary d by constructing auxiliary functions and using maximum principles and comparison principles the sufficient conditions for the existence of global solution an upper estimate of the global solution the sufficient conditions for the existence of the blow up solution an upper bound for the blow up time and an upper estimate of the blow up rate are specified under some appropriate assumptions on the functions a b f g and h

divisibility relations for the dimensions and hilbert series of nichols algebras of non abelian group type abstract we present a divisibility relation for the dimensions and hilbert series of certain classes of nichols algebras of non abelian group type which generalizes nichols algebras over coxeter groups with constant cocycle 1 for this we introduce three groups of isomorphisms acting on nichols algebras which generalizes the exchange operator introduced by milinski and schneider 2000 18 for coxeter groups

robust semi supervised least squares classification by implicit constraints abstract we introduce the implicitly constrained least squares icls classifier a novel semi supervised version of the least squares classifier this classifier minimizes the squared loss on the labeled data among the set of parameters implied by all possible labelings of the unlabeled data unlike other discriminative semi supervised methods this approach does not introduce explicit additional assumptions into the objective function but leverages implicit assumptions already present in the choice of the supervised least squares classifier this method can be formulated as a quadratic programming problem and its solution can be found using a simple gradient descent procedure we prove that in a limited 1 dimensional setting this approach never leads to performance worse than the supervised classifier experimental results show that also in the general multidimensional case performance improvements can be expected both in terms of the squared loss that is intrinsic to the classifier and in terms of the expected classification error

liouville type results for a class of quasilinear parabolic problems the aim of this paper is to study the properties of positive solutions to the nonlinear diffusion equation with a nonlinear bistable source we show that there exists a universal a priori upper bound for all positive solutions moreover under some assumptions we prove the convergence of these solutions to a constant if t

coverings of graded pointed hopf algebras we introduce the concept of a covering of a graded pointed hopf algebra the theory developed shows that coverings of a bosonized nichols algebra can be concretely expressed by biproducts using a quotient of the universal coalgebra covering group of the nichols algebra if there are enough quadratic relations the universal coalgebra covering is given by the bosonization by the enveloping group of the underlying rack

large margin semi supervised learning in classification semi supervised learning occurs when a large amount of unlabeled data is available with only a small number of labeled data in such a situation how to enhance predictability of classification through unlabeled data is the focus in this article we introduce a novel large margin semi supervised learning methodology using grouping information from unlabeled data together with the concept of margins in a form of regularization controlling the interplay between labeled and unlabeled data based on this methodology we develop two specific machines involving support vector machines and learning denoted as ssvm and spsi through difference convex programming in addition we estimate the generalization error using both labeled and unlabeled data for tuning regularizers finally our theoretical and numerical analyses indicate that the proposed methodology achieves the desired objective of delivering high performance in generalization particularly against some strong performers

porous medium equation with absorption and a nonlinear boundary condition where is a bounded domain with smooth boundary is the outer normal derivative m 1 p and q are positive parameters and u0 is in l problems of this form arise in mathematical models in a number of areas of science for instance in models for gas or uid ow in porous media 3 and for the spread of certain biological populations 13 in the semilinear case that is for m 1 there is an extensive literature about global existence and blow up results for this type of problems see among others 5 9 16 and the literature therein for the degenerate case that is for m 1 with a nonlinear boundary condition local existence and uniqueness of weak solutions which are limit of solutions of nondegenerate problems has been established in 1 also in 2 existence and uniqueness of global weak solutions for a similar

the weyl groupoid of a nichols algebra of diagonal type the theory of nichols algebras of diagonal type is known to be closely related to that of semi simple lie algebras in this paper the connection between both theories is made closer for any nichols algebra of diagonal type invertible transformations are introduced which remind one of the action of the weyl group on the root system associated to a semi simple lie algebra they give rise to the definition of a groupoid as an application an alternative proof of classification results of rosso andruskiewitsch and schneider is obtained without using any technical assumptions on the braiding

stability and instability for solutions of the convective porous medium equation with a nonlinear forcing at the boundary ii abstract we continue our study of the long time behavior of nonnegative solutions for the degenerate parabolic equation u t u m xx n u n x 0 x t 0 subject to nonlinear boundary conditions here we consider solutions of this equation subject to the boundary conditions here we consider solutions of this equation subject to the boundary conditions u m x 0 t au p 0 t u 1 t 0 t 0 as in part i j r anderson j differential equations to appear we give the bifurcation diagrams for the stationary solutions due to the existence of singular stationary states some of these diagrams are three dimensional we also examine the stability properties of these states when n p m 1

semi supervised nearest mean classification through a constrained log likelihood we cast a semi supervised nearest mean classifier previously introduced by the first author in a more principled log likelihood formulation that is subject to constraints this in turn leads us to make the important suggestion to not only investigate error rates of semi supervised learners but also consider the risk they originally aim to optimize we demonstrate empirically that in terms of classification error mixed results are obtained when comparing supervised to semi supervised nearest mean classification while in terms of log likelihood on the test set the semi supervised method consistently outperforms its supervised counterpart comparisons to self learning a standard approach in semi supervised learning are included to further clarify the way in which our constrained nearest mean classifier improves over regular supervised nearest mean classification

global solutions and blow up problems for a nonlinear degenerate parabolic system coupled via nonlocal sources this paper concerns with a nonlinear degenerate parabolic system coupled via nonlocal sources subjecting to homogeneous dirichlet boundary condition the main aim of this paper is to study conditions on the global existence and or blow up in finite time of solutions and give the estimates of blow up rates of blow up solutions

nonlinear degenerate parabolic equations with neumann boundary condition abstract we study neumann problem for a class of nonlinear degenerate parabolic pde a typical nonlinearity we have in mind is for instance u 1 u u 0 we establish a necessary and sufficient condition on given data for existence of solution

nichols algebras of group type with many quadratic relations abstract we classify nichols algebras of irreducible yetter drinfeld modules over nonabelian groups satisfying an inequality for the dimension of the homogeneous subspace of degree two all such nichols algebras are finite dimensional and all known finite dimensional nichols algebras of nonabelian group type appear in the result of our classification we find a new finite dimensional nichols algebra over fields of characteristic two

an overdetermined initial and boundary value problem for a reaction diffusion equation with m 1 p 1 and p 0 for p 1 2 m the minimum speed is 0 here we are interested in understanding the phenomenon of nonuniqueness therefore we restrict ourselves to the exponent range 2 m i p 1 we remind the reader that negative exponents p i 0 are also allowed with the convention that up 0 wherever u 0 we will consider initial data u0 such that problem 1 1 admits more than one solution and we will try to