an inhomogeneous semilinear equation in entire space abstract necessary conditions and a priori estimates are obtained for the inhomogeneous semilinear equation u up f x 0 in r n also several existence theorems are proved given explicit conditions on the inhomogeneous termf

generic mean curvature flow i generic singularities it has long been conjectured that starting at a generic smooth closed embedded surface in r 3 the mean curvature ow remains smooth until it arrives at a singularity in a neighborhood of which the ow looks like concentric spheres or cylinders that is the only singularities of a generic ow are spherical or cylindrical we will address this conjecture here and in a sequel the higher dimensional case will be addressed elsewhere the key to showing this conjecture is to show that shrinking spheres cylinders and planes are the only stable self shrinkers under the mean curvature ow we prove this here in all dimensions an easy consequence of this is that every singularity other than spheres and cylinders can be perturbed away

minimal cones and self expanding solutions for mean curvature flows in this paper we study self expanding solutions for mean curvature flows and their relationship to minimal cones in euclidean space ilmanen lectures on mean curvature flow and related equations trieste notes 1995 proved the existence of self expanding hypersurfaces with prescribed tangent cones at infinity if the cone is c 3 alpha regular and mean convex but not area minimizing we can prove that the corresponding self expanding hypersurfaces are smooth embedded and have positive mean curvature everywhere see theorem 1 1 as a result for regular minimal but not area minimizing cones we can give an affirmative answer to a problem arisen by lawson brothers in proc sympos pure math 44 441 464 1986

existence theorems for certain elliptic and parabolic semilinear equations existence theorems for the nonlinear parabolic differential equation u t u u p f x t 0 in rn 0 with zero initial value are established given explicit conditions on the nonhomogeneous termf x t an existence theorem is also demonstrated for the corresponding elliptic equation

semisimple hopf algebras via geometric invariant theory abstract we study hopf algebras via tools from geometric invariant theory we show that all the invariants we get can be constructed using the integrals of the hopf algebra and its dual together with the multiplication and the comultiplication and that these invariants determine the isomorphism class of the hopf algebra we then define certain canonical subspaces i n v i j of tensor powers of h and h and use the invariant theory to prove that these subspaces satisfy a certain non degeneracy condition using this non degeneracy condition together with results on symmetric monoidal categories we prove that the spaces i n v i j can also be described as h i h j a where a is the group of hopf automorphisms of h as a result we prove that the number of possible hopf orders of any semisimple hopf algebra over a given number ring is finite we give some examples of these invariants arising from the theory of frobenius schur indicators and from reshetikhin turaev invariants of three manifolds we give a complete description of the invariants for a group algebra proving that they all encode the number of homomorphisms from some finitely presented group to the group we also show that if all the invariants are algebraic integers then the hopf algebra satisfies kaplansky s sixth conjecture the dimensions of the irreducible representations of h divide the dimension of h

on twisting of finite dimensional hopf algebras in this paper we study the properties of drinfeld s twisting for finite dimensional hopf algebras we determine how the integral of the dual to a unimodular hopf algebra h changes under twisting of h we show that the classes of cosemisimple unimodular cosemisimple involutive cosemisimple quasitriangular finite dimensional hopf algebras are stable under twisting we also prove the cosemisimplicity of a coalgebra obtained by twisting of a cosemisimple unimodular hopf algebra by two different twists on two sides such twists are closely related to bi galois extensions and describe the representation theory of its dual next we define the notion of a non degenerate twist for a hopf algebra h and set up a bijection between such twists for h and h this bijection is based on miyashita ulbrich actions of hopf algebras on simple algebras it generalizes to the non commutative case the procedure of inverting a non degenerate skew symmetric bilinear form on a vector space finally we apply these results to classification of twists in group algebras and of cosemisimple triangular finite dimensional hopf algebras in positive characteristic generalizing the previously known classification in characteristic zero

existence of solutions for an inhomogeneous fractional semilinear heat equation abstract we obtain necessary conditions and sufficient conditions on the existence of solutions to the cauchy problem for a fractional semilinear heat equation with an inhomogeneous term we identify the strongest spatial singularity of the inhomogeneous term for the solvability of the cauchy problem

a sharp lower bound for the entropy of closed hypersurfaces up to dimension six the entropy is a natural geometric quantity which measures the complexity of a hypersurface in mathbb r n 1 it is non increasing along the mean curvature flow and so plays a significant role in analyzing the dynamics of this flow in colding et al j differ geom 95 1 53 69 2013 colding ilmanen minicozzi white showed that within the class of closed smooth self shrinking solutions of the mean curvature flow in mathbb r n 1 the entropy is uniquely minimized at the round sphere they conjectured that for 2 le n le 6 the round sphere minimizes the entropy among all closed hypersurfaces using an appropriate weak mean curvature flow we prove their conjecture for these dimensions our approach also gives a new proof of the main result of colding et al j differ geom 95 1 53 69 2013 and extends its conclusions to compact singular self shrinking solutions

on the existence of orders in semisimple hopf algebras the first author was supported by the projects mtm2011 27090 from micinn and feder and by the research group fqm0211 from junta de andalucia the second author was supported by the danish national research foundation dnrf through the centre for symmetry and deformation the authors would like to thank dmitriy rumynin and martin lorenz for bring ing to their attention the notion of weak order cesar galindo for pointing out that these examples were already discussed in 8 and yevgenia kashina and sonia na tale for useful discussions on the classification of low dimensional semisimple hopf algebras

classification of semisimple hopf algebras of dimension 16 abstract in this paper we completely classify nontrivial semisimple hopf algebras of dimension 16 we also compute all the possible structures of the grothendieck ring of semisimple non commutative hopf algebras of dimension 16 moreover we prove that non commutative semisimple hopf algebras of dimension pn p prime cannot have a cyclic group of grouplikes

semisimple hopf actions on weyl algebras abstract we study actions of semisimple hopf algebras h on weyl algebras a over an algebraically closed field of characteristic zero we show that the action of h on a must factor through a group action in other words if h acts inner faithfully on a then h is cocommutative the techniques used include reduction modulo a prime number and the study of semisimple cosemisimple hopf actions on division algebras

asymptotic behavior and stability of mean curvature flow with a conical end abstract if the initial hypersurface of an immortal mean curvature flow is asymptotic to a regular cone whose entropy is small the flow will become asymptotically self expanding moreover the expander that gives rise to the limiting flow is asymptotically stable as an equilibrium solution of the normalized mean curvature flow

a local regularity theorem for mean curvature flow this paper proves curvature bounds for mean curvature flows and other related flows in regions of spacetime where the gaussian densities are close to 1

a new critical phenomenon for semilinear parabolic problems abstract we study the cauchy problem of the inhomogeneous semilinear parabolic equations u up ut w 0 onmn 0 with initial valueu0 0 wheremnis a riemannian manifold with possibly nonnegative ricci curvature there is an exponentp which is critical in the following sense when 1 p the problem has a global positive solution for somew w x 0 and someu0 0

on semisimple hopf algebras of dimension pq2 let p and q be distinct prime numbers we prove a result on the existence of nontrivial group like elements in a certain class of semisimple hopf algebras of dimension pqr we conclude the classification of semisimple hopf algebras a of dimension pq2 over an algebraically closed field k of characteristic zero such that both a and a are of frobenius type we also complete the classification of semisimple hopf algebras of dimension pq2 100