quantum analogues of richardson varieties in the grassmannian and their toric degeneration abstract in the present paper we are interested in natural quantum analogues of richardson varieties in the type a grassmannians to be more precise the objects that we investigate are quantum analogues of the homogeneous coordinate rings of richardson varieties which appear naturally in the theory of quantum groups our point of view here is geometric we are interested in the regularity properties of these non commutative varieties such as their irreducibility normality cohen macaulayness in the spirit of non commutative algebraic geometry a major step in our approach is to show that these algebras have the structure of an algebra with a straightening law from this it follows that they degenerate to some quantum analogues of toric varieties

blowup rate for heat equation in lipschitz domains with nonlinear heat source terms on the boundary abstract this paper establishes the blowup rate estimate near the blowup time for the heat equation ut u with the nonlinear boundary condition un up on the boundary 0 t for a bounded lipschitz domain and for the nonlinear boundary condition un exp u on the boundary for a two dimensional disc with nonradially symmetric data

canonical bases for the quantum group of type a_r and piecewise linear combinatorics this work was motivated by the following two problems from the classical representation theory both problems make sense for an arbitrary complex semisimple lie algebra but since we shall deal only with the ar case we formulate them in this generality 1 construct a good basis in every irreducible finite dimensional slr 1 module v which materializes the littlewood richardson rule a precise formulation of this problem was given in 3 we shall explain it in more detail a bit later 2 construct a basis in every polynomial representation of glr 1 such that the maximal element w0 of the weyl group sr 1 considered as an element of glr 1 acts on this basis by a permutation up to a sign and explicitly compute this permutation this problem is motivated by recent work by john stembridge 10 and was brought to our attention by his talk at the jerusalem combinatorics conference may 1993

nonexistence of positive solutions to a semilinear elliptic system and blow up estimates for a reaction diffusion system abstract we get simple conditions under which the elliptic system u u p 1 v q 1 v u p 2 v q 2 in r n n 3 with p i q i 1 i 1 2 has no positive radially symmetric solutions then by using this nonexistence result we establish blow up estimates for reaction diffusion system of fujita type u t u u p 1 v q 1 v t v u p 2 v q 2 with the homogeneous dirichlet boundary value condition

non simultaneous blow up for a multi coupled reaction diffusion system abstract this paper deals with a reaction diffusion system multi coupled via inner sources and nonlinear boundary flux the necessary and sufficient condition for the existence of non simultaneous blow up is proposed some regions of parameters are determined where both simultaneous and non simultaneous blow up can occur in addition sufficient conditions for non simultaneous blow up under any positive initial data are obtained

a short proof of tutte s characterization of totally unimodular matrices we give in terms of totally unimodular matrices a short and easy proof of tutte s characterization of regular matroids

blowup estimates for a semilinear reaction diffusion system this paper deals with the blowup estimates of positive solutions for a semilinear reaction diffusion system ut u u vp vt v uqv with null dirichlet boundary conditions the upper and lower bounds of blowup rates are obtained

geometric lifting of the canonical basis and semitoric degenerations of richardson varieties in the sl n case a berenstein and a zelevinsky studied the schutzenberger involution in terms of lusztig s canonical basis 3 we generalize their construction and formulas for any semisimple lie algebra we use for this the geometric lifting of the canonical basis on which an analogue of the schutzenberger involution can be given as an application we construct semitoric degenerations of richardson varieties following a method of p caldero 6

critical fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux we establish the critical fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux and then determine the blow up rates and the blow up sets for the nonglobal solutions

okounkov bodies and toric degenerations let vardelta be the okounkov body of a divisor d on a projective variety x we describe a geometric criterion for vardelta to be a lattice polytope and show that in this situation x admits a flat degeneration to the corresponding toric variety this degeneration is functorial in an appropriate sense

on theq commutations inuq n at roots of one abstract let e be a root of one and g a semisimple lie algebra with triangular decomposition g n h n letu e resp ures e be the nonrestricted resp restricted quantum enveloping algebra of n we prove that fract u eis a quantum weyl field we then give a description of the e center ofu e letufin ebe the finite part ofures e via the drinfeld correspondence theufin e covariant space of a weyl module is e central in case g s l n this enables us to describe this space in terms of semistandard young tableaux

the blow up rate for a strongly coupled system of semilinear heat equations with nonlinear boundary conditions abstract the paper deals with the blow up rate of positive solutions to the system u t u xx u l 11 v l 12 v t v xx u l 21 v l 22 with boundary conditions u x 1 t u p 11 v p 12 1 t and v x 1 t u p 21 v p 22 1 t under some assumptions on the matrices l l ij and p p ij and on the initial data u 0 v 0 the solution u v blows up at finite time t and we prove that max x 0 1 u x t resp max x 0 1 v x t goes to infinity as t t 1 2 resp t t 2 2 where i p id 1 2 t 1 1 t

blow up rate estimates for semilinear parabolic systems abstract this paper deals with the blow up rate estimates of positive solutions for semilinear parabolic systems with null dirichlet boundary conditions the upper and lower bounds of blow up rate were obtained

blow up for a semilinear reaction diffusion system coupled in both equations and boundary conditions we study the blow up behavior for a semilinear reaction diffusion system coupled in both equations and boundary conditions the main purpose is to understand how the reaction terms and the absorption terms affect the blow up properties we obtain a necessary and sufficient condition for blow up derive the upper bound and lower bound for the blow up rate and find the blow up set under certain assumptions

critical non simultaneous blow up exponents for a reaction diffusion system the paper deals with the critical non simultaneous blow up exponents for the coupled reaction diffusion equations ut u u m v p and vt v u q v n subject to homo geneous dirichlet boundary conditions for the radial blow up solutions we propose precise criteria to completely identify the four cases for non simultaneous and simultaneous blow up situations under admissible initial data there exist initial data such that non simultaneous or simultaneous blow up of solutions occurs or any blow up must be non simultaneous or simultaneous the necessary sufficient conditions are established for each case with a complete classification for all the parameters there moreover all possible simultaneous blow up rates are obtained these extend the results in the previous works for the same model

blow up rate estimates for a doubly coupled reaction diffusion system abstract this paper deals with a reaction diffusion system with coupled nonlinear inner sources and a nonlinear boundary flux blow up rates are determined for four different blow up situations the so called characteristic algebraic system is introduced to get a very simple and clear description for the desired blow up rate estimates it is pointed out that one cannot directly use super and sub solutions to establish blow up rate estimates since they do not share the same blow up time in general

blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition abstract in this paper we study the large time behavior of positive solutions of the heat equation under the nonlinear boundary condition u f u where is the outward normal and f is nondecreasing with f u 0 for u 0 we show that if b r and 1 f is integrable at infinity there is finite time blow up for any initial datum in the two dimensional case we show that this is true for any smooth simply connected domain in the radially symmetric case if f c2 is convex and satisfies the properties above we show that blow up occurs only at the boundary

the blow up estimate for heat equations with non linear boundary conditions this paper deals with the blow up rate estimates of positive solutions for a system of heat equations with non linear boundary conditions the upper and lower bounds of blow up rate are obtained

non simultaneous blow up of n components for nonlinear parabolic systems this paper deals with non simultaneous and simultaneous blow up for radially symmetric solution u1 u2 un to heat equations coupled via nonlinear boundary ui uipiui 1qi 1 i 1 2 n it is proved that there exist suitable initial data such that ui i 1 2 n blows up alone if and only if qi 1 pi all of the classifications on the existence of only two components blowing up simultaneously are obtained we find that different positions different values of k i n of ui k and ui leads to quite different blow up rates it is interesting that different initial data lead to different blow up phenomena even with the same requirements on exponent parameters we also propose that ui k ui k 1 ui i 1 2 n k 0 1 2 n 1 blow up simultaneously while the other ones remain bounded in different exponent regions moreover the blow up rates and blow up sets are obtained

recognizing binet matrices binet matrices generalize network matrices and play an important role in combinatorial optimization a first polynomial time algorithm for recognizing binet matrices appeared in the author s doctoral thesis in this paper we present some key ideas and results involved in the design of this algorithm we show how we can find a camion basis of the input matrix whenever this one is binet and then reduce the recognition problem to that of special binet matrices called bicyclic and cyclic

blow up estimates of the positive solution of a parabolic system this paper establishes the blow up estimates for the systems ut u 0 vt v 0 in br 0 t br rn with the nonlinear boundary conditions u um1vn1 and v um2vn2 on sr 0 t where 0 m1 1 m2 and 0 n2 1 n1 we prove that c t t 2 max u x t c t t 2 and c t t 2 max v x t c t t 2 under some monotonicity assumptions on the initial values where n1 n2 1 m2 m1 1 and n1m2 1 m1 1 n2

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

blow up analysis for a quasilinear parabolic system with multi coupled nonlinearities this paper deals with a quasilinear parabolic system coupled via both nonlinear reaction terms and nonlinear boundary flux as the results of the interaction among the multi coupled nonlinearities in the system some appropriate conditions for global existence and global nonexistence of solutions are determined respectively