global regularity for the 2d mhd equations with mixed partial dissipation and magnetic diffusion abstract whether or not classical solutions of the 2d incompressible mhd equations without full dissipation and magnetic diffusion can develop finite time singularities is a difficult issue a major result of this paper establishes the global regularity of classical solutions for the mhd equations with mixed partial dissipation and magnetic diffusion in addition the global existence conditional regularity and uniqueness of a weak solution is obtained for the 2d mhd equations with only magnetic diffusion

decomposition algebras and axial algebras abstract we introduce decomposition algebras as a natural generalization of axial algebras majorana algebras and the griess algebra they remedy three limitations of axial algebras 1 they separate fusion laws from specific values in a field thereby allowing repetition of eigenvalues 2 they allow for decompositions that do not arise from multiplication by idempotents 3 they admit a natural notion of homomorphisms making them into a nice category we exploit these facts to strengthen the connection between axial algebras and groups in particular we provide a definition of a universal miyamoto group which makes this connection functorial under some mild assumptions we illustrate our theory by explaining how representation theory and association schemes can help to build a decomposition algebra for a given permutation group this construction leads to a large number of examples we also take the opportunity to fix some terminology in this rapidly expanding subject

global existence and decay of smooth solution for the 2 d mhd equations without magnetic diffusion abstract we prove the global existence and the decay estimates of small smooth solution for the 2 d mhd equations without magnetic diffusion this confirms the numerical observation that the energy of the mhd equations is dissipated at a rate independent of the ohmic resistivity

small scale creation for solutions of the incompressible two dimensional euler equation we construct an initial data for the two dimensional euler equation in a disk for which the gradient of vorticity exhibits double exponential growth in time for all times this estimate is known to be sharp the double exponential growth is the fastest possible growth rate

tannaka krein duality for compact matrix pseudogroups twisted su n groups the notion of concrete monoidalw category is introduced and investigated a generalization of the tannaka krein duality theorem is proved it leads to new examples of compact matrix pseudogroups among them we have twistedsu n groups denoted bys u n it is shown that the representation theory fors u n is similar to that ofsu n irreducible representations are labeled by young diagrams and formulae for dimensions and multiplicity are the same as in the classical case

anomalous dissipation for 1 5 holder euler flows recently the second and fourth authors developed an iterative scheme for obtaining rough solutions of the 3d incompressible euler equations in holder spaces the motivation comes from onsager s conjecture the construction involves a superposition of weakly interacting perturbed beltrami flows on infinitely many scales an obstruction to better regularity arises from the errors in the linear transport of a fast periodic flow by a slow velocity field in a recent paper the third author has improved upon the methods introducing some novel ideas on how to deal with this obstruction thereby reaching a better holder exponent albeit weaker than the one conjectured by onsager in this paper we give a shorter proof of this final result adhering more to the original scheme of the second and fourth authors and introducing some new devices more precisely we show that for any positive e there exist periodic solutions of the 3d incompressible euler equations that dissipate the total kinetic energy and belong to the holder class c 1 5 epsilon

on slightly degenerate fusion categories abstract in this paper we first show for a slightly degenerate pre modular fusion category c that squares of dimensions of simple objects divide half of the dimension of c and that slightly degenerate fusion categories of fp dimensions 2 p n d and 4 p n d are nilpotent where p is an odd prime and d is an odd square free integer then we classify weakly integral slightly degenerate fusion categories of particular dimensions

a jordan holder theorem for weakly group theoretical fusion categories we prove a version of the jordan holder theorem in the context of weakly group theoretical fusion categories this allows us to introduce the composition factors and the length of such a fusion category mathcal c which are in fact morita invariants of mathcal c

on the grothendieck rings of equivariant fusion categories in this paper we describe a mackey type decomposition for group actions on abelian categories this allows us to define new mackey functors which associates to any subgroup the k theory of the corresponding equivariantized abelian category in the case of an action by tensor autoequivalences the mackey functor at the level of grothendieck rings has a green functor structure as an application we give a description of the grothendieck rings of equivariantized fusion categories under group actions by tensor autoequivalences on graded fusion categories in this settings a new formula for the tensor product of any two simple objects of an equivariantized fusion category is given simplifying the fusion formula from burciu and natale j math phys 54 013511 2013

a proof of onsager s conjecture for any 1 3 we construct weak solutions to the 3d incompressible euler equations in the class c tc x that have nonempty compact support in time on r t 3 and therefore fail to conserve the total kinetic energy this result together with the proof of energy conservation for 1 3 due to eyink and constantin e titi solves onsager s conjecture that the exponent 1 3 marks the threshold for conservation of energy for weak solutions in the class l t c x the previous best results were solutions in the classc tc x for 1 5 due to isett and in the class l t 1c x for 1 3 due to buckmaster de lellis szekelyhidi both based on the method of convex integration developed for the incompressible euler equations by de lellis szekelyhidi the present proof combines the method of convex integration and a new gluing approximation technique the convex integration part of the proof relies on the mikado flows introduced by daneri szekelyhidi and the framework of estimates developed in the author s previous work

energy and cross helicity conservation for the three dimensional ideal mhd equations in a bounded domain abstract in this paper we prove the energy and cross helicity conservation of weak solutions to the three dimensional ideal mhd equations in a bounded domain under the interior besov regularity conditions which are exactly same as the three dimensional periodic domain case in 3 and the boundedness and the besov type continuity conditions for both the velocity and magnetic fields near the boundary which seem crucial for the bounded domain case due to the boundary effect note that our besov type continuity conditions near the boundary are consistent with the interior besov regularity conditions

regularity and energy conservation for the compressible euler equations we give sufficient conditions on the regularity of solutions to the inhomogeneous incompressible euler and the compressible isentropic euler systems in order for the energy to be conserved our strategy relies on commutator estimates similar to those employed by constantin et al for the homogeneous incompressible euler equations

onsager s conjecture on the energy conservation for solutions of euler s equation we give a simple proof of a result conjectured by onsager 1 on energy conservation for weak solutions of euler s equation

global well posedness of the incompressible magnetohydrodynamics this paper studies the cauchy problem of the incompressible magnetohydro dynamic systems with or without viscosity under the assumption that the initial velocity field and the displacement of the initialmagnetic field froma non zero constant are sufficiently small in certain weighted sobolev spaces the cauchy problem is shown to be globally well posed for all 0 and all spaces with dimension n 2 such a result holds true uniformly in nonnegative viscosity parameters the proof is based on the inherent strong null structure of the systems introduced by lei commun pure appl math 69 11 2072 2106 2016 and the ghost weight technique introduced by alinhac invent math 145 3 597 618 2001

semisimple hopf algebras of dimension 60 abstract we determine the isomorphism classes of semisimple hopf algebras of dimension 60 which are simple as hopf algebras

dissipative continuous euler flows we show the existence of continuous periodic solutions of the 3d incompressible euler equations which dissipate the total kinetic energy

the lazy homology of a hopf algebra to any hopf algebra h we associate two commutative hopf algebras h1l h and h2l h which we call the lazy homology hopf algebras of h these hopf algebras are built in such a way that they are predual to the lazy cohomology groups based on the so called lazy cocycles we establish two universal coefficient theorems relating the lazy cohomology to the lazy homology and we compute the lazy homology of the sweedler algebra

global small solutions to the compressible 2d magnetohydrodynamic system without magnetic diffusion abstract this paper establishes the global existence and uniqueness of smooth solutions to the two dimensional compressible magnetohydrodynamic system when the initial data is close to an equilibrium state in addition explicit large time decay rates for various sobolev norms of the solutions are also obtained these results are achieved through a new approach of diagonalizing a system of coupled linearized equations the standard method of diagonalization via the eigenvalues and eigenvectors of the matrix symbol is very difficult to implement here this new process allows us to obtain an integral representation of the full system through explicit kernels in addition in order to overcome various difficulties such as the anisotropicity and criticality we fully exploit the structure of the integral representation and employ extremely delicate fourier analysis and associated estimates

on a symmetry of muger s centralizer for the drinfeld double of a semisimple hopf algebra in this paper we prove a formula that relates muger s centralizer in the category of representations of a factorizable hopf algebra to the notion of hopf kernel of a representation of the dual hopf algebra using this relation we obtain a complete description for muger s centralizer of some fusion subcategories of the fusion category of finite dimensional representations of a drinfeld double of a semisimple hopf algebra

on the brauer picard groups of fusion categories we develop methods of computation of the brauer picard groups of fusion categories and apply them to compute such groups for several classes of fusion categories of prime power dimension representation categories of elementary abelian groups with twisted associativity extra special p groups and the kac paljutkin hopf algebra we conclude that many finite groups of lie type occur as composition factors of the brauer picard groups of pointed fusion categories

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

simple compact quantum groups i the notion of simple compact quantum group is introduced as non trivial noncommutative and noncocommutative examples the following families of compact quantum groups are shown to be simple a the universal quantum groups bu q for q gl n c satisfying qq in n 2 b the quantum automorphism groups aaut b of finite dimensional c algebras b endowed with the canonical trace when dim b 4 including the quantum permutation groups aaut xn on n points n 4 c the standard deformations kq of simple compact lie groups k and their twists kqu as well as rieffel s deformation kj

quantum symmetry groups of finite spaces we determine the quantum automorphism groups of finite spaces these are compact matrix quantum groups in the sense of woronowicz

longtime dynamics of a conductive fluid in the presence of a strong magnetic field we prove existence in the large of localized solutions to the mhd equations for an ideal conducting fluid subject to a strong magnetic field we show that for large time the dynamics may reduce to linear alfven waves

classification of rank 5 premodular categories we survey a number of classification tools developed in recent years and employ them to classify pseudo unitary rank 5 premodular categories up to grothendieck equivalence

onsager s conjecture with physical boundaries and an application to the vanishing viscosity limit author s bardos claude titi edriss s wiedemann emil abstract we consider the incompressible euler equations in a bounded domain in three space dimensions recently the first two authors proved onsager s conjecture for bounded domains i e that the energy of a solution to these equations is conserved provided the solution is h quot older continuous with exponent greater than 1 3 uniformly up to the boundary in this contribution we relax this assumption requiring only interior h quot older regularity and continuity of the normal component of the energy flux near the boundary the significance of this improvement is given by the fact that our new condition is consistent with the possible formation of a prandtl type boundary layer in the vanishing viscosity limit

the energy balance relation for weak solutions of the density dependent navier stokes equations abstract we consider the incompressible inhomogeneous navier stokes equations with constant viscosity coefficient and density which is bounded and bounded away from zero we show that the energy balance relation for this system holds for weak solutions if the velocity density and pressure belong to a range of besov spaces of smoothness 1 3 a density dependent version of the classical karman howarth monin relation is derived

weakly group theoretical and solvable fusion categories we introduce two new classes of fusion categories which are obtained by a certain procedure from finite groups weakly group theoretical categories and solvable categories these are fusion categories that are morita equivalent to iterated extensions in the world of fusion categories of arbitrary respectively solvable finite groups weakly group theoretical categories have integer dimension and all known fusion categories of integer dimension are weakly group theoretical our main results are that a weakly group theoretical category c has the strong frobenius property i e the dimension of any simple object in an indecomposable c module category divides the dimension of c and that any fusion category whose dimension has at most two prime divisors is solvable a categorical analog of burnside s theorem for finite groups this has powerful applications to classification of fusion categories and semsisimple hopf algebras of a given dimension in particular we show that any fusion category of integer dimension 84 is weakly group theoretical i e comes from finite group theory and give a full classification of semisimple hopf algebras of dimensions pqr and pq2 where p q r are distinct primes

nilpotent fusion categories abstract in this paper we extend categorically the notion of a finite nilpotent group to fusion categories to this end we first analyze the trivial component of the universal grading of a fusion category c and then introduce the upper central series of c for fusion categories with commutative grothendieck rings e g braided fusion categories we also introduce the lower central series we study arithmetic and structural properties of nilpotent fusion categories and apply our theory to modular categories and to semisimple hopf algebras in particular we show that in the modular case the two central series are centralizers of each other in the sense of m muger

equivalent notions of normal quantum subgroups compact quantum groups with properties f and fd and other applications abstract the notion of normal quantum subgroup introduced in algebraic context by parshall and wang when applied to compact quantum groups is shown to be equivalent to the notion of normal quantum subgroup introduced by the author as applications a quantum analog of the third fundamental isomorphism theorem for groups is obtained which is used along with the equivalence theorem to obtain results on structure of quantum groups with property f and quantum groups with property fd other results on normal quantum subgroups for tensor products free products and crossed products are also proved

on gauging symmetry of modular categories topological order of a topological phase of matter in two spacial dimensions is encoded by a unitary modular tensor category umc a group symmetry of the topological phase induces a group symmetry of its corresponding umc gauging is a well known theoretical tool to promote a global symmetry to a local gauge symmetry we give a mathematical formulation of gauging in terms of higher category formalism roughly given a umc with a symmetry group g gauging is a 2 step process first extend the umc to a g crossed braided fusion category and then take the equivariantization of the resulting category gauging can tell whether or not two enriched topological phases of matter are different and also provides a way to construct new umcs out of old ones we derive a formula for the h 4 obstruction prove some properties of gauging and carry out gauging for two concrete examples

exponential growth of the vorticity gradient for the euler equation on the torus abstract we prove that there are solutions to the euler equation on the torus with c 1 vorticity and smooth except at one point such that the vorticity gradient grows in l at least exponentially as t the same result is shown to hold for the vorticity hessian and smooth solutions our proofs use a version of a recent result by kiselev and sverak 6

decomposition of products in nilpotent table algebras abstract two theorems on the decomposition of products in nilpotent commutative table algebras are established this generalizes several results of adan bante on the decomposition of products of irreducible characters and of conjugacy classes of finite p groups and extends these results to fusion rule algebras and association schemes

global solutions to the 2d viscous non resistive mhd system with large background magnetic field abstract in this paper we consider the 2d viscous non resistive mhd system if the background magnetic field is sufficiently large then we can obtain the global existence of strong solutions

on thin residues and basis digraphs of nilpotent table algebras and applications to nilpotent groups abstract in this paper we study characterizations of nilpotent table algebras in terms of thin residues thin residue matrices thin residue digraphs and basis digraphs we will also discuss applications to nilpotent groups in particular we give a conceptual explanation of the results in takegahara and yoshida 2008 ty

on the nonuniqueness of weak solution of the euler equation weak solution of the euler equations is defined as an l2 vector field satisfying the integral relations expressing the mass and momentum balance their general nature has been quite unclear in this work an example of a weak solution on a 2 dimensional torus is constructed that is identically zero outside a finite time interval this example is simpler and more transparent than the previous example of v scheffer j geom anal 3 4 1993 pp 343 401 1997 john wiley sons inc

on the grothendieck rings of generalized drinfeld doubles abstract in this paper it is shown that any irreducible representation of a drinfeld double d a of a semisimple hopf algebra a can be obtained as an induced representation from a certain hopf subalgebra of d a this generalizes a well known result concerning the irreducible representations of drinfeld doubles of finite groups 11 using this description we also give a formula for the fusion rules of semisimple drinfeld doubles this shows that the grothendieck rings of these drinfeld doubles have a ring structure similar to the grothendieck rings of drinfeld doubles of finite groups

on fusion categories using a variety of methods developed in the literature in particular the theory of weak hopf algebras we prove a number of general results about fusion categories in characteristic zero we show that the global dimension of a fusion category is always positive and that the s matrix of any not necessarily hermitian modular category is unitary we also show that the category of module functors between two module categories over a fusion category is semisimple and that fusion categories and tensor functors between them are undeformable generalized ocneanu rigidity in particular the number of such categories functors realizing a given fusion datum is finite finally we develop the theory of frobenius perron dimensions in an arbitrary fusion category at the end of the paper we generalize some of these results to positive characteristic

energy conservation for inhomogeneous incompressible and compressible euler equations abstract energy conservations are studied for inhomogeneous incompressible and compressible euler equations with general pressure law in a torus or a bounded domain we provide sufficient conditions for a weak solution to conserve the energy by exploiting a suitable test function the spatial regularity for the density is only required to be of order 2 3 in the incompressible case and of order 1 3 in the compressible case when the density is constant we recover the existing results for classical incompressible euler equation

fusion rings with few degrees abstract the main result characterizes fusion rings whose degree set consists of 1 and a fixed prime p a well known theorem of isaacs and passman on finite groups whose irreducible character degrees are 1 and p follows as a corollary numerically compatible nc fusion rings are defined as a potential generalization of the grothendieck rings of fusion categories and the nc fusion rings of order p p 2 p 3 pq and p q 2 where p q are primes are classified in terms of certain standard table algebras as another application of the main theorem

stability of perturbations near a background magnetic field of the 2d incompressible mhd equations with mixed partial dissipation abstract the stability and large time behavior problem on some partially dissipated systems is not well understood the vorticity gradient of the 2d incompressible euler equation can grow double exponentially in time while the same quantity to the 2d navier stokes equation decays algebraically in time however the stability and large time behavior of the vorticity gradients of the 2d navier stokes equation with only vertical or horizontal dissipation appears to be unknown this paper presents a global stability result on perturbations near a background magnetic field to the 2d incompressible magnetohydrodynamic mhd equations with vertical dissipation and horizontal magnetic diffusion this stability result provides a significant example for the stabilizing effects of the magnetic field on electrically conducting fluids in addition we obtain an explicit decay rate for the solution of this nonlinear system

fermionic modular categories and the 16 fold way we study spin and super modular categories systematically as inspired by fermionic topological phases of matter which are always fermion parity enriched and modelled by spin topological quantum field theories at low energy we formulate a 16 fold way conjecture for the minimal modular extensions of super modular categories to spin modular categories which is a categorical formulation of gauging the fermion parity we investigate general properties of super modular categories such as fermions in twisted drinfeld doubles verlinde formulas for naive quotients and explicit extensions of psu 2 4m 2 with an eye towards a classification of the low rank cases

global small solutions of 2 d incompressible mhd system abstract in this paper we consider the global wellposedness of 2 d incompressible magneto hydrodynamical system with smooth initial data which is close to some non trivial steady state it is a coupled system between the navier stokes equations and a free transport equation with a universal nonlinear coupling structure the main difficulty of the proof lies in exploring the dissipative mechanism of the system to achieve this and to avoid the difficulty of propagating anisotropic regularity for the free transport equation we first reformulate our system 1 1 in the lagrangian coordinates 2 19 then we employ anisotropic littlewood paley analysis to establish the key a priori l 1 r lip r 2 estimate for the lagrangian velocity field y t with this estimate we can prove the global wellposedness of 2 19 with smooth and small initial data by using the energy method we emphasize that the algebraic structure of 2 19 is crucial for the proofs to work the global wellposedness of the original system 1 1 then follows by a suitable change of variables

some properties of group theoretical categories abstract we first show that every group theoretical category is graded by a certain double coset ring as a consequence we obtain a necessary and sufficient condition for a group theoretical category to be nilpotent we then give an explicit description of the simple objects in a group theoretical category following o2 and of the group of invertible objects of a group theoretical category in group theoretical terms finally under certain restrictive conditions we describe the universal grading group of a group theoretical category

free unitary groups are almost simple we show that the quotients of wang and van daele s universal quantum groups by their centers are simple in the sense that they have no normal quantum subgroups thus providing the first examples of simple compact quantum groups with non commutative fusion rings

tensor categories with fusion rules of self duality for finite abelian groups abstract semisimple tensor categories with fusion rules of self duality for finite abelian groups are classified as an application we prove that the tannaka duals of the dihedral and the quaternion groups of order 8 and the eight dimensional hopf algebra of kac and paljutkin are not isomorphic as abstract tensor categories whereas they have the same fusion rule

some mathematical questions related to the mhd equations some questions relating to the large time behavior of the solutions of mhd equations for a viscous incompressible resistive fluid are investigated the physical system is briefly described and the functional setting of the equations a flow in a bounded domain or in whole space with a space periodicity property in all directions the main existence and uniqueness results for weak and strong solutions of the mhd equations are recalled regularity properties and bounds on the solutions to the equations which are valid for all time are established and the concept of functional invariant sets is introduced which is contained in the space of smooth functions if the data are sufficiently regular the squeezing property of the trajectories are stated and it is shown that any functional invariant set for the mhd equations and in particular any attractor has a finite haussdorf dimension the flow is found to be totally determined for large dimensions by a finite number of parameters 26 references