These are notes for a class on double affine Hecke algebras and Macdonald polynomials given in the Fall of 2007 at the University of Minnesota. Here we introduce the (single and double) affine braid groups and Hecke algebras. We are following chapters 3,4, and 5 of Macdonald’s book [Mac]. There is also an (optional) aside giving applications to the T -equivariant K-theory of flag varieties G/B....

Let G be a graph. Denote by L(G) its i-iterated line graph and denote by W (G) its Wiener index. Dobrynin, Entringer and Gutman stated the following problem: Does there exist a non-trivial tree T and i ≥ 3 such that W (L(T )) = W (T )? In a series of five papers we solve this problem. In a previous paper we proved that W (L(T )) > W (T ) for every tree T that is not homeomorphic to a path, claw...

a p p r e h e n d s b u t d o e s n o t a p p r e h e n d t h i s s h a p e i n a w a y t h a t i s c o m p l e t e l y i n d e p e n d e n t f r o m m a t e r i a l c o n d i t i o n s . T h e p r o c e s s o f a b s t r a c t i o n i n i n c o m p l e t e s i n c e t h e c a t o n l y k n o w s t h i s s h a p e a s i t c h a r a c t e r i z e s a p a r t i c u l a r o b j e c t a n d n o t i...

Let L s L u be a linear differential operator acting on functions u: V a R that are defined on a fixed manifold V. These functions will belong to a Ž . certain family B ; F V, R , and the definition of B may include some boundary conditions or other requirements that must be satisfied by any function u g B. It is said that L satisfies the maximum principle if the differential inequality L u G 0...

a . t r u e . b . F a l s e . P e o p l e w i t h i m p a i r m e n t s d o n o t l i k e t o b e p i t i e d . I t i s b e t t e r t o t r e a t t h e m w i t h t h e s a m e r e s p e c t a n d d i g n i t y y o u w o u l d s h o w t o p e o p l e w i t h o u t i m p a i r m e n t s . c . F a l s e . I m p a i r m e n t s r e f e r t o t h e p h y s i c a...

Rhode R i v e r s y s t e m have shown t h a t a d i v e r s e w e t l a n d ( M i l l Swamp) h a s a s i g n i f i c a n t i m p a c t on n u t r i e n t r u n o f f f r o m u p l a n d w a t e r s h e d s . I n t h i s p a p e r we p r e s e n t d a t a t o d e m o n s t r a t e how w a t e r q u a l i t y p a r a m e t e r s change as s u r f a c e and s u b s u r f a c e w a t e r moves t h...

for simultaneously diagonalizable matrices A,B ∈ C . The unbounded drift term is defined by a skew-symmetric matrix S ∈ R. Differential operators of this form appear when investigating rotating waves in time-dependent reaction diffusion systems. As shown in a companion paper, one key assumption to prove resolvent estimates of L∞ in L (R,C ), 1 < p < ∞, is the following L-dissipativity condition...

V 3 + W 0 3 # 0 6X W Y 6060(* Z / -,[ / # W 0 56 / -,[ / ' ) 0 \ / ". 0 [6 ( Q@ Z @ [". @6 ], A<>D ^_ @4 @ ` a>D W ` b6c =6 ], 3 de 0 3 ,\Q0 8^_ 0 f / #" (?6 8 2". @6 ], g # W J @ 0 #"9 N 6 6 0 2 W L 6 (* h 86% : %(* @ i W + % / D6 8 #Tkj)>? b6f^_ Wlnm@( @6 L D6C ^_ 9 : C a>D o6 + #"J W"` 5606 (* ' ) 8 D6 : o . 0 ."J @ Z / 3L @ W 0 # 0 ' @ / W (* / 86@MZI0pF /> Z(* @ Z / 7c : o W + C6 Nqr @6 )"...

The m a j o r i t y o f n o r m a l h u m a n se ra h e m o l y z e sheep e r y t h r o c y t e s in the p r e s e n c e o f c o m p l e m e n t ( "hemoly zers") , w h e r e a s s e ra o f the r e m a i n i n g populat ion do not lyze sheep e r y t h r o e y t e s C~nonhemolyzers") . The antibodies respons ib le for this lyt ic ac t iv i ty a r e n o w identified as be ing specific to F o r s s...

Let g be a finite dimensional complex semi-simple Lie algebra with Weyl group W and simple reflections S. For I ⊆ S let g I be the corresponding semi-simple subalgebra of g. Denote by W I the Weyl group of g I and let w • and w I • be the longest elements of W and W I , respectively. In this paper we show that the answer to Kostant's problem, i.e. whether the universal enveloping algebra surjec...