We discuss the representation theory of the non-linear chiral algebra W1+∞ of Gaberdiel and Gopakumar and its connection to the Yangian of û(1) whose presentation was given by Tsymbaliuk. The characters of completely degenerate representations of W1+∞ are given by the topological vertex. The Yangian picture provides an infinite number of commuting charges which can be explicitly diagonalized in...

for various pairs of weight functions w1 : X → R + and w2 : Y → R . Roughly speaking, these weighted estimates are a variant of the usual L estimates which assert that if f avoids the regions where w1 is large, then Tf avoids the regions where w2 is large. If X = Y and one can establish the above type of estimate for many pairs w1, w2, with w1 “looking similar to” w2, then these estimates begin...

Let p(n) denote the ordinary partition function. Subbarao conjectured that in every arithmetic progression r (mod t) there are infinitely many integers N ≡ r (mod t) for which p(N) is even, and infinitely many integers M ≡ r (mod t) for which p(M) is odd. In the even case the conjecture was settled by Ken Ono. In this paper we prove the odd part of the conjecture which together with Ono’s resul...

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The prevalence of minor physical anomalies was examined in a sample of 109 children with idiopathic mental retardation (65 boys and 44 girls). Control group consisted of 246 healthy schoolchildren (123 boys and 123 girls) aged 8 to 12 years. A comparison was made between number of found minor anomalies per child (W1) and their Waldrop weight scores (W2) in healthy and mentally retarded (MR) chi...

In a recent paper Folsom and Ono constructed a grid of Poincaré series of weights 3/2 and 1/2. They conjectured that the coefficients of the holomorphic parts of these series are rational integers. We prove that these coefficients are indeed rational numbers with bounded denominators.

In this paper we study the asymptotic distribution of the moments of (nonnormalized) traces Tr(w1),Tr(w2), . . . ,Tr(wr), where w1, w2, . . . , wr are reduced words in unitaries in the group U(N). We prove that as N → ∞ these variables are distributed as normal gaussian variables √ j1Z1, . . . , √ Zr , where j1, . . . , jr are the number of cyclic rotations of the words w1, . . . , ws leaving t...

Let us consider an edge-to-edge and strongly balanced tiling of plane by pentagons. A node of valence s (≥3) in an edge-to-edge tiling is a point that is the common vertex of s tiles. Let W1 be a finite closed disk satisfying the property that the average valence of nodes in W1 is nearly equal to 10/3. Then, let T denote the union of the set of pentagons meeting the boundary of W1 but not conta...

Background and Aim: Orthodontic patients are at an increased risk of caries due to difficulty in plaque control. Antimicrobial agents, such as fluoride and chlorhexidine, can be useful in preventing tooth demineralization. The present study aimed to determine the effect of fluoride and chlorhexidine gels and their mixture on the shear bond strength (SBS) of orthodontic brackets. Materials and ...

The main result is the following. Let be a bounded Lipschitz domain in Rd , d 2. Then for every f ∈ Ld( ) with ∫ f = 0, there exists a solution u ∈ C0( ) ∩ W1,d ( ) of the equation divu= f in , satisfying in addition u= 0 on ∂ and the estimate ‖u‖L∞ + ‖u‖W1,d C‖f ‖Ld , where C depends only on . However one cannot choose u depending linearly on f . To cite this article: J. Bourgain, H. Brezis, C...