Holomorphic spectral geometry of magnetic Schrödinger operators on Riemann surfaces

We study magnetic Schrödinger operators on line bundles over Riemann surfaces endowed with metrics of constant curvature. We show that for harmonic magnetic fields the spectral geometry of these operators is completely determined by the Bochner Laplacians of the line bundles. Therefore we are led to examine the spectral problem for the Bochner Laplacian ∇^∗∇ of a Hermitian line bundle L with connection ∇ over a Riemann surface S. This spectral problem is analyzed in terms of the natural holomorphic structure on L defined by the Cauchy-Riemann operator associated with ∇. By means of an elliptic chain of line bundles obtained by twisting L with the powers of the canonical bundle we prove that there exists a certain subset of the spectrum σhol(∇^∗∇) such that the eigensections associated with λ∈σhol(∇^∗∇) are given by the holomorphic sections of a certain line bundle of the elliptic chain. For genus p=0,1 we prove that σhol(∇^∗∇) is the whole spectrum, whereas for genus p>1 we get a finite number of eigenvalues.     

Skew linear recurring sequences of maximal period over galois rings

Let p be a prime number, and R = GR(q ^d , p ^d ) be a Galois ring of q ^d = p ^rd elements and of characteristic p ^d . Denote by S = GR(q ^nd , p ^d ) a Galois extension of the ring R of dimension n and by ? the ring of all linear transformations of the module _R S. We call a sequence v over the ring S with the law of recursion $$ {\mathrm{for}\ \mathrm{all}\ }i \in {\mathbb{N}_0}:v\left( {i + m} \right) = {\psi_{m - 1}}\left( {v\left( {i + m - 1} \right)} \right) + \cdots + {\psi_0}\left( {v(i)} \right),\quad {\psi_0}, \ldots, {\psi_{m - 1}} \in \textit{\v{S}} $$ (i.e., a linear recurring sequence of order m over the module _? S) a skew LRS over S. It is known that the period T(v) of such a sequence satisfies the inequality T(v) ?? ?? = (q ^nm ?1)p ^d?1. If T(v) = ?? , then we call v a skew LRS of maximal period (a skew MP LRS) over S. A new general characterization of skew MP LRS in terms of coordinate sequences corresponding to some basis of a free module _R S is given. A simple constructive method of building a big enough class of skew MP LRS is stated, and it is proved that the linear complexity of some of them (the rank of the linear recurring sequence) over the module _S S is equal to mn, i.e., to the linear complexity over the module _R S.     

Two-parameter estimates for joint spectral projections on complex spheres

We prove sharp two-parameter estimates for the L ^p -L ² norm, 1 ≤ p ≤ 2, of the joint spectral projectors associated to the Laplace–Beltrami operator and to the Kohn Laplacian on the unit sphere S ^2n-1 in . Then, by using the notion of contraction of Lie groups, we deduce the estimates recently obtained by H. Koch and F. Ricci for joint spectral projections on the reduced Heisenberg group h ¹.      

Isomorphism of circulant graphs and digraphs

Let S? {1, …, n?1} satisfy ?S = S mod n. The circulant graph G(n, S) with vertex set {v₀, v₁,…, v_n?1} and edge set E satisfies v_iv_j?E if and only if j ? i ∈ S, where all arithmetic is done mod n. The circulant digraph G(n, S) is defined similarly without the restriction S = ? S. Ádám conjectured that $\text{G(n, S) ? G(n, S′)}$ if and only if S = uS′ for some unit u mod n. In this paper we prove the conjecture true if n = pq where p and q are distinct primes. We also show that it is not generally true when n = p², and determine exact conditions on S that it be true in this case. We then show as a simple consequence that the conjecture is false in most cases when n is divisible by p² where p is an odd prime, or n is divisible by 2⁴.     

On spherically convex sets and Q-matrices

Let K be a closed spherically convex subset of S^n?1 that is contained in a hemisphere, and x?(K) the radial projection onto S^n?1 of the centroid of K. Then p^Tx?(K)>0 for all p ? K. A specialization of this result to spherical simplices is used to derive a necessary condition for Q-matrices, i.e., matrices for which every corresponding linear complementarity problem has at least one solution.     

On some finite ramified coverings of Pn

Let S be a hypersurface in Pⁿ (n≧3) with only normal crossings and let ƒ : XPⁿ be a finite ramified covering which is unramified over Pⁿ − S. Then S. Kawai has shown that there are neither regular 1-forms nor regular 2-forms on X. The aim of this article is to derive a stronger conclusion: H⁰(X,Ω_X^p)= 0 for 1≦p<n , and moreover H⁰(X,Ω_X^p)= 0 if deg S≦n+1.     

Conical Embeddings of Steiner systems

Any classicalS(3,2 ^a +1;2 ^ab +1) is embedded intoPG(2,2 ^ab ) as point set one may use any conic, the blocks being determined by subplanes of order 2 ^a . Consequently, every classicalS(3,2 ^a +1;2 ^ab +1) is naturally embedded intoPG(2,K) whereK is the algebraic closure ofGF(2).     

Clarkson–McCarthy Inequalities for l p -Spaces of Operators in Schatten Ideals

In this paper we obtain generalized Clarkson–McCarthy inequalities for spaces l _q (S ^p ) of operators from Schatten ideals S ^p . We show that all Clarkson–McCarthy type inequalities are, in fact, some estimates on the norms of operators acting on the spaces l _q (S ^p ) or from one such space into another. We also extend some inequalities for partitioned operators and for Cartesian decomposition of operators.     

Weighted fractional Bernstein’s inequalities and their applications

This paper studies the weighted, fractional Bernstein inequality for spherical polynomials on S^d-1\(\left( {0.1} \right)\;{\left\| {{{\left( { - {\Delta _0}} \right)}^{{\raise0.7ex\hbox{$r$} \!\mathord{\left/ {\vphantom {r 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}}}f} \right\|_{p,w}} \leqslant {C_w}{n^r}{\left\| f \right\|_{p,w}}\;for\;all\;f \in \Pi _n^d\), where Π_n^d denotes the space of all spherical polynomials of degree at most n on S^d-1 and (-Δ₀)^r/2 is the fractional Laplacian-Beltrami operator on S^d-1. A new class of doubling weights with conditions weaker than the A_p condition is introduced and used to characterize completely those doubling weights w on S^d-1 for which the weighted Bernstein inequality (0.1) holds for some 1 ≤ p ≤ 8 and all r > t. It is shown that in the unweighted case, if 0 < p < 8 and r > 0 is not an even integer, (0.1) with w = 1 holds if and only if r > (d - 1)((1/p) - 1). As applications, we show that every function f ∈ L_p(S^d-1) with 0 < p < 1 can be approximated by the de la Vallée Poussin means of a Fourier-Laplace series and establish a sharp Sobolev type embedding theorem for the weighted Besov spaces with respect to general doubling weights.     

Large dimension homomorphism spaces between Specht modules for symmetric groups

Let F be a field of characteristic p. We show that Hom_FΣn(S^λ,S^μ) can have arbitrarily large dimension as n and p grow, where S^λ and S^μ are Specht modules for the symmetric group Σ_n. Similar results hold for the Weyl modules of the general linear group. Every previously computed example has been at most one-dimensional, with the exception of Specht modules over a field of characteristic two. The proof uses the work of Chuang and Tan, providing detailed information about the radical series of Weyl modules in Rouquier blocks.     

A limit theorem for order preserving nonexpansive operators inL 1

We prove the following theorem:Let T be an order preserving nonexpansive operator on L ₁ (μ) (or L ₁ ⁺ ) of a σ-finite measure, which also decreases theL _∞-norm, and let S=tI+(1?t)T for 0<t<1. Then for everyf ∈ L_p (1<p<∞),the sequence S ⁿf converges weakly in L_p. (The assumptions do not imply thatT is nonexpansive inL _p for anyp>1, even ifμ is finite.) For the proof we show that ∥S ⁿ⁺¹ f?S ⁿf∥ _p → 0 for everyf ∈L _p, 1<p<∞, and apply toS the following theorem:Let T be order preserving and nonexpansive in L ₁ ⁺ , and assume that T decreases theL _∞-norm. Then forg ∈L _p (1<p<∞) Tⁿg is weakly almost convergent. If forf ∈ L_p we have T ⁿ⁺¹ f?T ⁿ f → 0weakly, then T ⁿf converges weakly in L_p (1<p<∞).     

On interpolation of interpolating sequences

Let A be a uniform algebra on the compact space X and σ a probability measure on X. We define the Hardy spaces H^P(σ) and the H^P(σ) interpolating sequences S in the p-spectrum M_p of σ. Under some structural hypotheses on (A, σ), we prove that if a sequence S ⊂ M_p is H^P(σ) interpolating, then it is H^s(σ) interpolating for s < p. In the special case of the unit ball B of ?ⁿ this answers a natural question asked in [8].     

Directional operators and radial functions on the plane

LetEçS ¹ be a set with Minkowski dimensiond(E)1. We consider the Hardy-Littlewood maximal function, the Hilbert transform and the maximal Hilbert transform along the directions ofE. The main result of this paper shows that these operators are bounded onL _rad ^p (R²) forp>1+d(E) and unbounded whenp<1+d(E). We also give some end-point results.     

Bubbles with prescribed mean curvature: The variational approach

Let H:R³→R be a C¹ mapping such that H(p)→H_∞>0 as ∣p∣→∞. We show that when H satisfies some global conditions then there exists an H-bubble, namely a sphere S in R³ such that the mean curvature of S at any regular point p∈S equals H(p).     

The complex Frobenius Theorem and uniqueness of solutions to the tangential Cauchy-Riemann equations

Suppose M is a C^∞ real k-dimensional CR-submanifold of $\text{C}$ⁿ, n > 1, and suppose that $\text{?}\text{?}\text{t6}{}_{\mathrm{M}}$ is the tangential Cauchy-Riemann operator on M. Let S be a C¹ real (k ? 1)-dimensional submanifold of M which is noncharacteristic for ??t6_M at p?S. Conditions are found so that a C^∞ solution f of $\text{?}\text{?}\text{t6}{}_{\mathrm{M}}\text{f = 0}$ which vanishes on one side of S in M must vanish in a neighborhood of p in M. If M is a real hypersurface, it is known that such unique continuation always exists. If the codimension of M in $\text{C}$ⁿ is greater than 1, and if the excess dimension of the Levi algebra on M is constant, then it is proved that CR-functions on M which vanish on one side of S must vanish in a full neighborhood of p. The assumption on the dimension of the Levi algebra allows us to use the Complex Frobenius Theorem. Other methods to prove such unique continuation results are also developed.     

Modulus of stability for vector fields on 3-manifolds

Let X be a vector field on M³ which exhibits a saddle connection between a singularity p₁ and a periodic orbit σ₁. We give necessary conditions and also sufficient ones in order to have the finite modulus of stability. They rely heavily upon restrictions on the behaviour of p₁ and σ₁.     

L p -L q Estimates for Bergman Projections in Bounded Symmetric Domains of Tube Type

Let D be an irreducible bounded symmetric domain of tube type in ? ⁿ . The class of Bloch functions is well known in this context, in connection with Hankel operators or duality of Bergman spaces. Contrary to what happens in the unit ball, Bloch functions do not belong to all Lebesgue spaces L ^p (D) for p<∞ in higher rank. We give here both necessary and sufficient conditions on p for such an embedding. This question is equivalent to local boundedness properties of the Bergman projection in the tube domain over a symmetric cone that is conformally equivalent to D. We are linked to consider L ^∞–L ^q inequalities on symmetric cones, which may be of independent interest, and study more systematically estimates with loss for the Bergman projection. The proofs are based on a very precise estimate on an integral related to the Gamma function of a symmetric cone.     

A conjecture of recski in combinatorial set theory

A p-cover of $\text{n = {1, 2,…,}\text{n}\text{}}$ is a family of subsets $\text{S}{}_{\mathrm{i}}\text{≠ ?}$ such that ∪ S_i = n and |S_i ∩ S_i| ? p for i ≠ j. We prove that for fixed p, the number of p-cover of n is O(n^p+1logn).     

A series of Siamese twin designs

A {0,±1}-matrix S is called a Siamese twin design sharing the entries of I if S=I+K−L, where I,K,L are nonzero {0,1}-matrices and both I+K and I+L are incidence matrices of symmetric designs with the same parameters. Let p and 2p+3 be prime powers and . We construct a Siamese twin design with parameters (4(p+1)²,2p²+3p+1,p²+p).     

Plemelj Formulas for Rarita-Schwinger Type Operators

We introduce Plemelj formulas for Rarita-Schwinger operators defined over Lipschitz graphs in \({\mathbb{R}^{n}}\) and their corresponding surfaces on the sphere, S ⁿ and real projective spaces. We introduce the corresponding Hardy p-spaces for \({1 < p < \infty}\) . We also introduce Rarita-Schwinger analogues of the classical Szegö projection operators and Kerzman-Stein formulas.