Vector-valued interval functions and the Dedekind completion of <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">X</Emphasis>, <Emphasis Type="Italic">E</Emphasis>)

We consider interval valued functions with values in a Banach lattice E. Certain notions of continuity introduced earlier for real interval valued functions are generalised to the more general case considered here. As an application, we characterise the Dedekind completion of the space of continuous, E-valued functions on a paracompact \(T_{1}\)-space, extending a result of Anguelov.     

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 \(B_p(s)\).     

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.     

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.     

On (<Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">C</Emphasis>)-Isometric Operators

In this paper we study the spectral properties of (m, C)-isometric operators. In particular, if \(T\in \mathcal{{L(H)}}\) is (m, C)-isometric operators, then the power of (m, C)-isometric operators is also (m, C)-isometric operators. Moreover, if \(T^{*}\) has the single-valued extension property, then T has the single-valued extension property. Finally, we investigate conditions for (m, C)-isometric operators to be (1, C)-isometric operators.     

On the equation Δ<Emphasis Type="Italic">u</Emphasis> + <Emphasis Type="Italic">q</Emphasis>(<Emphasis Type="Italic">x</Emphasis>)<Emphasis Type="Italic">u</Emphasis> = 0

Sufficient conditions for the blow-up of nontrivial generalized solutions of the interior Dirichlet problem with homogeneous boundary condition for the homogeneous elliptic-type equation Δu + q(x)u = 0, where either q(x) ≠ const or q(x) = const= λ > 0, are obtained. A priori upper bounds (Theorem 4 and Remark 6) for the exact constants in the well-known Sobolev and Steklov inequalities are established.     

A note on Green’s relations in the semigroups <Emphasis Type="Italic">T</Emphasis>(<Emphasis Type="Italic">X</Emphasis>,<Emphasis Type="Italic">ρ</Emphasis>)

Let X be a set and the full transformation semigroup on X. Let ρ be an equivalence relation on X and

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].     

Bishop’s theorem and differentiability of a subspace of <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">b</Emphasis></Subscript>(<Emphasis Type="Italic">K</Emphasis>)

Let K be a Hausdorff space and C _b (K) be the Banach algebra of all complex bounded continuous functions on K. We study the Gateaux and Fréchet differentiability of subspaces of C _b (K). Using this, we show that the set of all strong peak functions in a nontrivial separating separable subspace H of C _b (K) is a dense G _δ subset of H, if K is compact. This gives a generalized Bishop’s theorem, which says that the closure of the set of all strong peak points for H is the smallest closed norming subset of H. The classical Bishop’s theorem was proved for a separating subalgebra H and a metrizable compact space K.     

The <Emphasis Type="Italic">ψ</Emphasis>-associate <Emphasis Type="Italic">X</Emphasis><Superscript>#</Superscript> of a flat <Emphasis Type="Italic">X</Emphasis> in PG(<Emphasis Type="Italic">n</Emphasis>, 2)

For a given hypersurface ψ in PG(n, 2), with equation Q(x) = 0, where Q is a polynomial of (reduced) degree d > 1, a definition is given of the ψ-associate X ^# of a flat X in PG(n, 2). The definition involves the fully polarized form of the polynomial Q; in the cubic case d = 3 it reads: X ^# = {z ∈ PG(n, 2) | T(x, y, z) = 0 for all x, y ∈ X}, where T(x, y, z) denotes the alternating trilinear form obtained by completely polarizing Q. Some general results, valid for any degree d and projective dimension n, are first expounded. Thereafter several choices of ψ are visited, but for each choice just a few aspects are highlighted. Despite the partial nature of the survey quite a variety of behaviours of the ψ-associate are uncovered. Many of the choices of ψ which are considered are of cubic hypersurfaces in PG(5, 2). If ψ is the Segre variety it is shown that the 48 planes external to fall into eight pairs of ordered triplets {(P ₁, R ₁, S ₁), (P ₂, R ₂, S ₂)} such that and . Further those lines L of PG(5, 2) which are singular, satisfying that is L ^# = PG(5.2), are in this case shown to form a complete spread of 21 lines. Another result of note arises in the case where ψ is the underlying 35-set of a non-maximal partial spread Σ₅ of five planes in PG(5, 2), where it is shown that one plane is singled out by the property that every line is singular.      

On Lie’s problem and differential invariants of ODEs <Emphasis Type="Italic">y</Emphasis>″ = <Emphasis Type="Italic">F</Emphasis>(<Emphasis Type="Italic">x</Emphasis>, <Emphasis Type="Italic">y</Emphasis>)

A point classification of ordinary differential equations of the form y″ = F(x, y) is considered. The algebra of differential invariants of the action of the point symmetry pseudogroup on the right-hand sides of equations of the form y″ = F(x, y) is calculated, and Lie’s problem on the point equivalence of such equations is solved.     

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.     

A closed-form expression for the Drinfeld modular polynomial Φ<Subscript><Emphasis Type="Italic">T</Emphasis></Subscript>(<Emphasis Type="Italic">X</Emphasis>, <Emphasis Type="Italic">Y</Emphasis>)

In this paper we give a closed-form expression for the Drinfeld modular polynomial \({\Phi_T(X,Y) \in \mathbb{F}_q(T)[X,Y]}\) for arbitrary q and prove a conjecture of Schweizer. A new identity involving the Catalan numbers plays a central role.     

On Lagrange interpolation for |<Emphasis Type="Italic">X</Emphasis>|<Superscript><Emphasis Type="Italic">α</Emphasis></Superscript> (0 < <Emphasis Type="Italic">α</Emphasis> < 1)

We study the optimal order of approximation for |x| ^α (0 < α < 1) by Lagrange interpolation polynomials based on Chebyshev nodes of the first kind. It is proved that the Jackson order of approximation is attained. Supported by the NSFC, 10601065.     

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 the Spectrum of the Sizes of Maximal Partial Line Spreads in <Emphasis Type="Italic">PG</Emphasis>(2<Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">q</Emphasis>), <Emphasis Type="Italic">n</Emphasis> ≥ 3

A lot of research has been done on the spectrum of the sizes of maximal partial spreads in PG(3,q) [P. Govaerts and L. Storme, Designs Codes and Cryptography, Vol. 28 (2003) pp. 51–63; O. Heden, Discrete Mathematics, Vol. 120 (1993) pp. 75–91; O. Heden, Discrete Mathematics, Vol. 142 (1995) pp. 97–106; O. Heden, Discrete Mathematics, Vol. 243 (2002) pp. 135–150]. In [A. Gács and T. Sznyi, Designs Codes and Cryptography, Vol. 29 (2003) pp. 123–129], results on the spectrum of the sizes of maximal partial line spreads in PG(N,q), N 5, are given. In PG(2n,q), n 3, the largest possible size for a partial line spread is q^2n-1+q^2n-3+...+q³+1. The largest size for the maximal partial line spreads constructed in [A. Gács and T. Sznyi, Designs Codes and Cryptography, Vol. 29 (2003) pp. 123–129] is (q²ⁿ⁺¹–q)/(q²–1)–q³+q²–2q+2. This shows that there is a non-empty interval of values of k for which it is still not known whether there exists a maximal partial line spread of size k in PG(2n,q). We now show that there indeed exists a maximal partial line spread of size k for every value of k in that interval when q 9.J. Eisfeld: Supported by the FWO Research Network WO.011.96NP. Sziklai: The research of this author was partially supported by OTKA D32817, F030737, F043772, FKFP 0063/2001 and Magyary Zoltan grants. The third author is grateful for the hospitality of Ghent University.     

On the divergence of trigonometric Fourier series in classes <Emphasis Type="Italic">ϕ</Emphasis>(<Emphasis Type="Italic">L</Emphasis>) close to <Emphasis Type="Italic">L</Emphasis>

We show the unimprovability of a theorem on sufficient convergence conditions for the trigonometric Fourier series of a function in classes ?(L) in the case when the class ?(L) is “close” to L.     

Convex Solid Subsets of <Emphasis Type="Italic">L</Emphasis><Subscript>0</Subscript>(<Emphasis Type="Italic">X</Emphasis>, μ)

If A⊂ L₀(X, μ) is a convex solid subset of L₀(X, μ), then there exist disjoint X₀ and X₁ with X = X₀∪ X₁ such that A_{| X_0} is dense in L₀(X₀, μ) and A_{|X_1} is bounded in measure in L₀(X₁, μ).     

A characteristic property of the algebra <Emphasis Type="Italic">C</Emphasis>(Ω)<Subscript><Emphasis Type="Italic">ß</Emphasis></Subscript>

We study some properties of the algebras of continuous functions on a locally compact space whose topology is defined by the family of all multiplication operators (β-uniform algebras). We introduce the notion of a β-amenable algebra and show that a β-uniform algebra is β-amenable if and only if it coincides with the algebra of bounded functions on a locally compact space (an analog of M. V. She?nberg’s theorem for uniform algebras).     

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).