On chirality groups and regular coverings of regular oriented hypermaps

We prove that if the Walsh bipartite map M = W (ℋ) of a regular oriented hypermap ℋ is also orientably regular then both M and ℋ have the same chirality group, the covering core of M (the smallest regular map covering M) is the Walsh bipartite map of the covering core of ℋ and the closure cover of M (the greatest regular map covered by M) is the Walsh bipartite map of the closure cover of ℋ. We apply these results to the family of toroidal chiral hypermaps (3, 3, 3) _b,c = W ⁻¹{6, 3} _b,c induced by the family of toroidal bipartite maps {6, 3} _b,c .     

Bounds on Supremum Norms for Hecke Eigenfunctions of Quantized Cat Maps

We study extreme values of desymmetrized eigenfunctions (so called Hecke eigenfunctions) for the quantized cat map, a quantization of a hyperbolic linear map of the torus. In a previous paper it was shown that for prime values of the inverse Planck’s constant N = 1/h, such that the map is diagonalizable (but not upper triangular) modulo N, the Hecke eigenfunctions are uniformly bounded. The purpose of this paper is to show that the same holds for any prime N provided that the map is not upper triangular modulo N. We also find that the supremum norms of Hecke eigenfunctions are ≪_ε N^ε for all ε > 0 in the case of N square free. Submitted: March 6, 2006; Accepted: April 30, 2006     

Valuation-like maps and the congruence subgroup property

Let D be a finite dimensional division algebra and N a subgroup of finite index in D ^×. A valuation-like map on N is a homomorphism ϕ:N?Γ from N to a (not necessarily abelian) linearly ordered group Γ satisfying N _<-α+1⊆N _<-α for some nonnegative α∈Γ such that N _<-α≠=?, where N _<-α={x∈N|ϕ(x)<-α}. We show that this implies the existence of a nontrivial valuation v of D with respect to which N is (v-adically) open. We then show that if N is normal in D ^× and the diameter of the commuting graph of D ^×/N is ≥4, then N admits a valuation-like map. This has various implication; in particular it restricts the structure of finite quotients of D ^×. The notion of a valuation-like map is inspired by [27], and in fact is closely related to part (U3) of the U-Hypothesis in [27]. Oblatum 14-VII-2000 & 22-XI-2000?Published online: 5 March 2001     

Convective instability of a rotating layer of ferromagnetic fluid

The instability of a hot horizontal layer of ferromagnetic fluid rotating about a vertical axis has been investigated when the Prandtl numberP < 1. Earlier it was shown that forP > 1 the overstability cannot occur. In this paper the convective and overstable marginal states have been investigated separately forP < 1 and it is found that though convective marginal state is possible for alla, the non-dimensional wave number, and N the Taylor number, the overstability is possible only ifN > (1 +P)π ⁴/(1 −P) and in case the condition is satisfied, overstability is possible for all those values ofa which satisfya ² < [N(1 −P)π ²/(1 +P)] ^1/3 − π². IfR _c ^(con) andR _c ^(o.s) are the critical values of the convective and the overstable marginal states respectively, then it is also found thatR _c ^(con) <R _c ^(o.s) providedN is not sufficiently large.     

Minimal cocycles with the scaling property and substitutions

‘Fractal’ functions are formulated as a minimal cocycle on a topological dynamics which admits nontrivial scaling transformations. In this paper, it is proved that if in addition it admits a continuous family of scaling transformations, then itscapacity is not ino(N ²). We define minimal cocycles with nontrivial scaling transformations coming from substitutions on a finite alphabet which are proved to have capacityO(N), so that they admit only a discrete family of scaling transformations. We also construct one which has capacityO(N ²) and admit a continuous family of scaling transformations. Supported by C.N.R.S.(U.R.A 225); Partially supported by the Japan Society for the Promotion of Science.     

Categorified Quantum sl(2) and Equivariant Cohomology of Iterated Flag Varieties

A 2-category was introduced in that categorifies Lusztig’s integral version of quantum sl(2). Here we construct for each positive integer N a representation of this 2-category using the equivariant cohomology of iterated flag varieties. This representation categorifies the irreducible (N + 1)-dimensional representation of quantum sl(2).     

Commutative rings whose zero-divisor graph is a proper refinement of a star graph

A graph is called a proper refinement of a star graph if it is a refinement of a star graph, but it is neither a star graph nor a complete graph. For a refinement of a star graph G with center c, let G _c ^* be the subgraph of G induced on the vertex set V (G)\ {c or end vertices adjacent to c}. In this paper, we study the isomorphic classification of some finite commutative local rings R by investigating their zero-divisor graphs G = Γ(R), which is a proper refinement of a star graph with exactly one center c. We determine all finite commutative local rings R such that G _c ^* has at least two connected components. We prove that the diameter of the induced graph G _c ^* is two if Z(R)² ≠ {0}, Z(R)³ = {0} and G _c ^* is connected. We determine the structure of R which has two distinct nonadjacent vertices α, β ∈ Z(R)* \ {c} such that the ideal [N(α) ∩ N(β)]∪ {0} is generated by only one element of Z(R)*\{c}. We also completely determine the correspondence between commutative rings and finite complete graphs K _n with some end vertices adjacent to a single vertex of K _n .     

Gumbel statistics for the longest interval of identical spins in a one-dimensional Gibbs measure

We consider one-dimensional Gibbs measures on spin configurations σ ∈ {–1,+1}^ℤ. For N ∈ ℕ let l _N denote the length of the longest interval of consecutive spins of the same kind in the interval [0,N]. We show that the distribution of a suitable continuous modification l _c (N) of l _N converges to the Gumbel distribution, i.e., for some α, β ∈ (0, ∞) and γ ∈ ℝ, lim _{N →∞} ℙ(l _c (N) ≤ α log N + βx + γ) = e ^{–e –x} . Received: 2 September 2002     

A primary obstruction to topological embeddings¶and its applications

For a proper continuous map f:M→N between topological manifolds M and N with m≡ dimM < dimN≡m+k, a primary obstruction to topological embeddings θ(f) ∈H ^c _{m − k} (M; Z ₂) has been defined and studied by the authors in {9, 8, 2, 3], where H ^c _* denotes the singular homology with closed support. In this paper, we study the obstruction from the viewpoint of differential topology and give various applications. We first give some characterizations of embeddings among generic differentiable maps, which are refinements of the results in [9, 10]. Then we give a result concerning the number of connected components of the complement of the image of a codimension-1 continuous map with a normal crossing point, which generalizes the results in [6, 4, 5, 9]. Finally we give a simple proof of a theorem of Li and Peterson [20] about immersions of m-manifolds into (2m-1)-manifolds. Received: 3 December 1999 / Revised version: 10 October 2000     

A superrank of varieties of Lie algebras

A variety of Lie algebras over a field of characteristic 0 has a finite superrank if it is generated by the Grassmann envelope of a finitely generated Lie superalgebra. We prove that every commutator variety not in N_cA has infinite superrank. Consequently, infinite are superranks of all polynilpotent varieties of Lie algebras except N_c and N_cA. Supported by RFFR grant No. 96-01-00146. Translated fromAlgebra i Logika, Vol. 37, No. 4, pp. 394–412, July–August. 1998.     

Iteration of closed geodesics in stationary Lorentzian manifolds

Following the lines of Bott in (Commun Pure Appl Math 9:171–206, 1956), we study the Morse index of the iterates of a closed geodesic in stationary Lorentzian manifolds, or, more generally, of a closed Lorentzian geodesic that admits a timelike periodic Jacobi field. Given one such closed geodesic γ, we prove the existence of a locally constant integer valued map Λ_γ on the unit circle with the property that the Morse index of the iterated γ ^N is equal, up to a correction term ε_γ∈{0,1}, to the sum of the values of Λ_γ at the N-th roots of unity. The discontinuities of Λ_γ occur at a finite number of points of the unit circle, that are special eigenvalues of the linearized Poincaré map of γ. We discuss some applications of the theory.     

The number of elliptic curves over ℚ with conductorNwith conductorN

We prove that the number of elliptic curves E/ℚ with conductorN isO(N ^1/2+ε). More generally, we prove that the number of elliptic curves E/ℚ with good reduction outsideS isO(M ^1/2+ε), whereM is the product of the primes inS. Assuming various standard conjectures, we show that this bound can be improved toO(M ^c/loglogM ). Research partially supported by NSF DMS-9424642.     

Codeterminants and <Emphasis Type="Italic">q</Emphasis>-Schur Algebras

We study codeterminants in the q-Schur algebra S _q (n,r) and prove that the standard ones form a basis of S _q (n,r), using a quantized version of the Désarménien matrix. We find elements of the form F _S 1_λ E _T in Lusztig’s modified enveloping algebra of gl(n), which, up to powers of q, map to the basis of standard codeterminants, where F _S ∈U ^– and E _T ∈U ⁺ are explicitly given products of root vectors, depending on Young tableaux S and T.     

Complete classification of the positive solutions of −Δ<Emphasis Type="Italic">u</Emphasis> + <Emphasis Type="Italic">u</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>

We study the equation −Δu + u ^q = 0, q > 1, in a bounded C ² domain Ω ⊂ ℝ ^N . A positive solution of the equation is moderate if it is dominated by a harmonic function and σ-moderate if it is the limit of an increasing sequence of moderate solutions. It is known that in the subcritical case, 1 < q <, q _c = (N + 1)/(N − 1), every positive solution is σ-moderate [32]. More recently, Dynkin proved, by probabilistic methods, that this remains valid in the supercritical case for q ≤ 2, [15]. The question remained open for q > 2. In this paper, we prove that for all q ≥ q _c , every positive solution is σ-moderate. We use purely analytic techniques, which apply to the full supercritical range. The main tools come from linear and non-linear potential theory. Combined with previous results, our result establishes a one-to-one correspondence between positive solutions and their boundary traces in the sense of [36].     

Configurations combinatoires des polynômes fibrés de degré 2

In this Note, we are interested in quadratic fibered polynomials which are skew products of the kind P_c : X × C → X × C, P_c(x, z) = (ƒ(x),z² + c(x)), where X is a compact set, ƒ is a continuous map from X into X, and c is a continuous complex valued function on X considered as a parameter. We construct a compact connected configuration space which gives a combinatorial model of a subset of the parameters space. Then, we explain in which way an abstract configuration can be realized by a quadratic fibered polynomial.     

Baryon resonances in the relativistic mean field approach

We suggest a new approach according to which baryon resonances can be viewed as collective excitations about “intrinsic” one-quark excitations in a mean field of definite symmetry. This standpoint is justified in the limit of a large number N_c of colors. Although N_c = 3 in the real world, we obtain a good agreement with the observed resonance spectrum up to 2 GeV. A possible consequence of the scheme is the existence of new exotic charmed (and bottom) baryons that may be stable against strong decays.     

Generalised Kostka–Foulkes polynomials and cohomology of line bundles on homogeneous vector bundles

Let G be a simple algebraic group and B a Borel subgroup. We consider generalisations of Lusztig’s q-analogues of weight multiplicity, where the set of positive roots is replaced with the multiset of weights of a B-submodule N of an arbitrary finite-dimensional G-module V. The corresponding polynomials in q are called generalised Kostka–Foulkes polynomials (gKF). We prove vanishing theorems for the cohomology of line bundles on G ×  _B N and derive from this a sufficient condition for the non-negativity of the coefficients of gKF. We also consider in detail the case in which V is the simple G-module whose highest weight is the short dominant root and N is the B-submodule whose weights are all short positive roots.     

The Blow-up Locus of Heat Flows for Harmonic Maps

Abstract Let M and N be two compact Riemannian manifolds. Let u _k (x, t) be a sequence of strong stationary weak heat flows from M×R ⁺ to N with bounded energies. Assume that u _k→u weakly in H ^{1, 2}(M×R ⁺, N) and that Σ^t is the blow-up set for a fixed t > 0. In this paper we first prove Σ^t is an H ^m−2-rectifiable set for almost all t∈R ⁺. And then we prove two blow-up formulas for the blow-up set and the limiting map. From the formulas, we can see that if the limiting map u is also a strong stationary weak heat flow, Σ^t is a distance solution of the (m− 2)-dimensional mean curvature flow [1]. If a smooth heat flow blows-up at a finite time, we derive a tangent map or a weakly quasi-harmonic sphere and a blow-up set ∪_t<0Σ^t× {t}. We prove the blow-up map is stationary if and only if the blow-up locus is a Brakke motion. This work is supported by NSF grant     

Linear series computing the Clifford index of a projective curve

For a (smooth irreducible) curveC of genus g and Clifford indexc>2 with a linear seriesg _d ^r computing c (so ) it is well known thatc + 2 ≤d ≤2 (c + 2), and if then 2c + 1 ≤g ≤ 2c + 4 unlessd = 2c + 4 in which caseg = 2c + 5. Let c ≥ 0 andg be integers. If 2c + 1 ≤g ≤2c + 4 we prove that for any integerd <g such thatd ≡c mod 2 andc + 2 ≤d < 2(c + 2) there exists a curve of genus g and Clifford index c with a g_d ^r computing c. Ford ≥c + 6 (i.e.r ≥ 3) we construct this curve on a surface of degree 2r-2 in ℙ^r, and ford ≥c + 8 (i.e.r ≥ 4) we show that such a curve cannot be found on a surface in ℙ^r of smaller degree. In fact, if g_d ^r computes the Clifford index c of C such thatc + 8 ≤d ≤ 2c + 3 then the birational morphism defined by this series cannot map C onto a (maybe, singular) curve contained in a surface of degree at most 2r-3 in ℙ^r.