An algebraic invariant for Jordan automorphisms on В(H): The set of idempotents

Let H be an infinite dimensional complex Hilbert space. Denote by B(H) the algebra of all bounded linear operators on H, and by I(H) the set of all idempo-tents in B(H). Suppose that Φ is a surjective map from B(H) onto itself. If for every λ ∈ {-1,1,2,3,1/2,1/3} and A, B ∈ B(H), A - λB ∈ I(H) (?) Φ(A) - λΦ(B) ∈ I(H), then Φ is a Jordan ring automorphism, i.e. there exists a continuous invertible linear or conjugate linear operator T on H such that Φ(A) = TAT-1 for all A ∈ B(H), or Φ(A) = TA*T-1 for all A ∈ B(H); if, in addition, A-iB ∈ I(H) (?) Φ(A) -ιΦ(B) ∈ I(H), here ι is the imaginary unit, then Φ is either an automorphism or an anti-automorphism.     

A note on invariant submanifolds of (<Emphasis Type="Italic">k</Emphasis>, <Emphasis Type="Italic">μ</Emphasis>)-contact manifolds

The object of the present paper is to study invariant submanifolds of a (k, μ)-contact manifold and to find the necessary and sufficient conditions for an invariant submanifold of a (k, μ)-contact manifold to be totally geodesic.     

Bilocal automorphisms of <Emphasis Type="Italic">T</Emphasis><Subscript>∞</Subscript>(<Emphasis Type="Italic">F</Emphasis>)

We prove that if F is a field such that |F| > 2, then every bilocal automorphism of T _∞(F) - the algebra of ? × ? upper triangular matrices over F, is an automorphism.     

The crossing number of <Emphasis Type="Italic">G</Emphasis> ▭ <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> for the graph <Emphasis Type="Italic">G</Emphasis> on six vertices

The crossing numbers of Cartesian products of paths, cycles or stars with all graphs of order at most four are known. The crossing numbers of G□C _n for some graphs G on five and six vertices and the cycle C _n are also given. In this paper, we extend these results by determining the crossing number of the Cartesian product G □ C _n , where G is a specific graph on six vertices.     

Normal Subgroups of Hecke Groups <Emphasis Type="Italic">H</Emphasis>(<Emphasis Type="Italic">λ</Emphasis>)

Let λ ≥ 2 and let H(λ) be the Hecke group associated to λ. Also let H(λ)\U be the Riemann surface associated to the Hecke group H(λ). In this article, we study the even subgroup H _e (λ) and the power subgroups H ^m (λ) of the Hecke groups H(λ). We also study some genus 0 normal subgroups of finite index of H(λ). Finally, we discuss free normal subgroups of H(λ).     

The smooth structure set of <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> × <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>

We calculate \({\mathcal{S}^{{\it Diff}}(S^p \times S^q)}\), the smooth structure set of S ^p × S ^q , for p, q ≥ 2 and p + q ≥ 5. As a consequence we show that in general \({\mathcal{S}^{Diff}(S^{4j-1}\times S^{4k})}\) cannot admit a group structure such that the smooth surgery exact sequence is a long exact sequence of groups. We also show that the image of the forgetful map \({\mathcal{S}^{Diff}(S^{4j}\times S^{4k}) \rightarrow \mathcal{S}^{Top}(S^{4j}\times S^{4k})}\) is not in general a subgroup of the topological structure set.     

An <Emphasis Type="Italic">L</Emphasis>-approach for packing (<Emphasis Type="Italic">ℓ</Emphasis>, <Emphasis Type="Italic">w</Emphasis>)-rectangles into rectangular and <Emphasis Type="Italic">L</Emphasis>-shaped pieces

This paper presents an approach using a recursive algorithm for packing (?, w)-rectangles into larger rectangular and L-shaped pieces. Such a problem has actual applications for non-guillotine cutting and pallet/container loading. Our motivation for developing the L-approach is based on the fact that it can solve difficult pallet loading instances. Indeed, it is able to solve all testing problems (more than 20 000 representatives of infinite equivalence classes of the literature), including the 18 hard instances unresolved by other heuristics. We conjecture that the L-approach always finds optimum packings of (?, w)-rectangles into rectangular pieces. Moreover, the approach may also be useful when dealing with cutting and packing problems involving L-shaped pieces.     

Period of the adelic Ikeda lift for <Emphasis Type="Italic">U</Emphasis>(<Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">m</Emphasis>)

We give a period formula for the adelic Ikeda lift of an elliptic modular form f for U(m, m) in terms of special values of the adjoint L-functions of f. This is an adelic version of Ikeda’s conjecture on the period of the classical Ikeda lift for U(m, m).     

On Analytic Campanato and <Emphasis Type="Italic">F</Emphasis>(<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>, <Emphasis Type="Italic">s</Emphasis>) Spaces

We investigate the relation between analytic Campanato spaces \(\mathcal {AL}_{p,s}\) and the spaces F(p, q, s), characterize the bounded and compact Riemann–Stieltjes operators from \(\mathcal {AL}_{p,s}\) to \(F(p,p-s-1,s)\). We also describe the corona theorem and the interpolating sequences for the class \(F(p,p-2,s)\), which is the Möbius invariant subspace of the analytic Besov type spaces \(B_p(s)\).     

Systematic Construction of <Emphasis Type="Italic">q</Emphasis>-Analogs of <Emphasis Type="Italic">t</Emphasis>−(<Emphasis Type="Italic">v</Emphasis>,<Emphasis Type="Italic">k</Emphasis>,λ)-Designs

In the present paper we consider a q-analog of t–(v,k,)-designs. It is canonic since it arises by replacing sets by vector spaces over GF(q), and their orders by dimensions. These generalizations were introduced by Thomas [Geom.Dedicata vol. 63, pp. 247–253 (1996)] they are called t –(v,k,;q)- designs. A few of such q-analogs are known today, they were constructed using sophisticated geometric arguments and case-by-case methods. It is our aim now to present a general method that allows systematically to construct such designs, and to give complete catalogs (for small parameters, of course) using an implemented software package. &nbsp; In order to attack the (highly complex) construction, we prepare them for an enormous data reduction by embedding their definition into the theory of group actions on posets, so that we can derive and use a generalization of the Kramer-Mesner matrix for their definition, together with an improved version of the LLL-algorithm. By doing so we generalize the methods developed in a research project on t –(v,k,)-designs on sets, obtaining this way new results on the existence of t–(v,k,;q)-designs on spaces for further quintuples (t,v,k,;q) of parameters. We present several 2–(6,3,;2)-designs, 2–(7,3,;2)-designs and, as far as we know, the very first 3-designs over GF(q).classification 05B05     

Derivatives for smooth representations of <Emphasis Type="Italic">GL</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, ℝ) and <Emphasis Type="Italic">GL</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, ℂ)

The notion of derivatives for smooth representations of GL(n, ? _p ) was defined in [BZ77]. In the archimedean case, an analog of the highest derivative was defined for irreducible unitary representations in [Sah89] and called the “adduced” representation. In this paper we define derivatives of all orders for smooth admissible Fréchet representations of moderate growth. The real case is more problematic than the p-adic case; for example, arbitrary derivatives need not be admissible. However, the highest derivative continues being admissible, and for irreducible unitarizable representations coincides with the space of smooth vectors of the adduced representation.In the companion paper [AGS] we prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations.We apply those results to finish the computation of adduced representations for all irreducible unitary representations and to prove uniqueness of degenerate Whittaker models for unitary representations, thus completing the results of [Sah89, Sah90, SaSt90, GS13a].     

The size of the set of <Emphasis Type="Italic">μ</Emphasis>-irregular points of a measure <Emphasis Type="Italic">μ</Emphasis>

Let μ be a compactly suppported positive measure on the real line. A point x∈supp?[μ] is said to be μ-regular, if, as n→∞,

$\sup_{\deg\, (P) \le n}\left(\frac{|P(x)|}{{\|P\|}_{L_{2}(d\mu)}}\right )^{1/n}\to1.$

Otherwise it is a μ-irregular point. We show that for any such measure, the set of μ-irregular points in {μ′>0} (with a suitable definition of this set) has Hausdorff \(m_{h_{\beta}}\) measure 0, for \(h_{\beta}(t) =\left(\log \frac{1}{t}\right)^{-\beta}\), any β>1.

     

A theorem of the alternatives for the equation |<Emphasis Type="Italic">Ax</Emphasis>| − |<Emphasis Type="Italic">B</Emphasis>||<Emphasis Type="Italic">x</Emphasis>| = <Emphasis Type="Italic">b</Emphasis>

A theorem of the alternatives for the equation \({|Ax|-|B||x|=b\ (A,B\in{\mathbb{R}}^{n\times n},\, b\in{\mathbb{R}}^n)}\) is proved and several consequences are drawn. In particular, a class of matrices A, B is identified for which the equation has exactly 2 ⁿ solutions for each positive right-hand side b.     

An <Emphasis Type="Italic">R</Emphasis> = <Emphasis Type="Italic">T</Emphasis> theorem for imaginary quadratic fields

We prove the modularity of certain residually reducible p-adic Galois representations of an imaginary quadratic field assuming the uniqueness of the residual representation. We obtain an R = T theorem using a new commutative algebra criterion that might be of independent interest. To apply the criterion, one needs to show that the quotient of the universal deformation ring R by its ideal of reducibility is cyclic Artinian of order no greater than the order of the congruence module T/J, where J is an Eisenstein ideal in the local Hecke algebra T. The inequality is proven by applying the Main conjecture of Iwasawa Theory for Hecke characters and using a result of Berger [Compos Math 145(3):603–632, 2009]. This strengthens our previous result [Berger and Klosin, J Inst Math Jussieu 8(4):669–692, 2009] to include the cases of an arbitrary p-adic valuation of the L-value, in particular, cases when R is not a discrete valuation ring. As a consequence we show that the Eisenstein ideal is principal and that T is a complete intersection.     

Relative <Emphasis Type="Italic">t</Emphasis>-designs in binary Hamming association scheme <Emphasis Type="Italic">H</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, 2)

A relative t-design in the binary Hamming association schemes H(n, 2) is equivalent to a weighted regular t-wise balanced design, i.e., certain combinatorial t-design which allows different sizes of blocks and a weight function on blocks. In this paper, we study relative t-designs in H(n, 2), putting emphasis on Fisher type inequalities and the existence of tight relative t-designs. We mostly consider relative t-designs on two shells. We prove that if the weight function is constant on each shell of a relative t-design on two shells then the subset in each shell must be a combinatorial \((t-1)\)-design. This is a generalization of the result of Kageyama who proved this under the stronger assumption that the weight function is constant on the whole block set. Using this, we define tight relative t-designs for odd t, and a strong restriction on the possible parameters of tight relative t-designs in H(n, 2). We obtain a new family of such tight relative t-designs, which were unnoticed before. We will give a list of feasible parameters of such relative 3-designs with \(n \le 100\), and then we discuss the existence and/or the non-existence of such tight relative 3-designs. We also discuss feasible parameters of tight relative 4-designs on two shells in H(n, 2) with \(n \le 50\). In this study we come up with the connection on the topics of classical design theory, such as symmetric 2-designs (in particular 2-\((4u-1,2u-1,u-1)\) Hadamard designs) and Driessen’s result on the non-existence of certain 3-designs. We believe Problems 1 and 2 presented in Sect. 5.2 open a new way to study relative t-designs in H(n, 2). We conclude our paper listing several open problems.     

On common values of <Emphasis Type="Italic">φ</Emphasis>(<Emphasis Type="Italic">n</Emphasis>) and <Emphasis Type="Italic">σ</Emphasis>(<Emphasis Type="Italic">m</Emphasis>). I

We show, conditional on a uniform version of the prime k-tuples conjecture, that there are x/(log x)^1+o(1) numbers not exceeding x common to the ranges of φ and σ. Here φ is Euler’s totient function and σ is the sum-of-divisors function.     

The commutation relation <Emphasis Type="Italic">xy</Emphasis>=<Emphasis Type="Italic">qyx</Emphasis>+<Emphasis Type="Italic">hf</Emphasis>(<Emphasis Type="Italic">y</Emphasis>) and Newton’s binomial formula

Let x and y be two variables satisfying the commutation relation xy=qyx+hf(y), where f(y) is a polynomial. In this paper, using Young diagrams and generating functions techniques, we study the binomial formula (x+y) ⁿ and we present an identity for x ^m y. The connection to Operator Calculus is discussed and several special cases are treated explicitly.     

The fractional Hausdorff operators on the Hardy spaces <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>(ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

In this paper, we study the high-dimensional fractional Hausdorff operators and establish their boundedness on the real Hardy spaces H ^p (? ⁿ ) for 0 < p < 1.     

The critical group of <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">m</Emphasis></Subscript> × <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

In this paper, the structure of the critical group of the graph K _m × C _n is determined, where m, n ≥ 3.     

On blocks with <Emphasis Type="Italic">K</Emphasis>(<Emphasis Type="Italic">B</Emphasis>)−<Emphasis Type="Italic">L</Emphasis>(<Emphasis Type="Italic">B</Emphasis>)=1

We use the method of local representation and original method of Brauer to study the block with K(B)−L(B)=1, and get some properties on the defect group and the structure of this kind of blocks. Then, we show that K(B) conjecture holds for this kind of blocks.