On the subalgebra of a Fourier–Stieltjes algebra generated by pure positive definite functions

For a locally compact group $G$ , the first-named author considered the closed subspace $a_0(G)$ which is generated by the pure positive definite functions. In many cases $a_0(G)$ is itself an algebra. We illustrate using Heisenberg groups and the $2\times 2$ real special linear group, that this is not the case in general. We examine the structures of the algebras thereby created and examine properties related to amenability.     

Inequalities for the expectation of Δ-monotone functions

Summary For some subsets of the set of all -monotone functions on [0,1] ⁿ we characterize distribution functions F, G such that E _F fE_G f for all f within these subsets. Furthermore, we determine sharp upper and lower bounds of integrals of functions in these subsets w.r.t. all distributions with fixed marginals and give some applications of these results.     

On condition numbers in hp-FEM with Gauss–Lobatto-based shape functions

Sharp bounds on the condition number of stiffness matrices arising in hp/spectral discretizations for two-dimensional problems elliptic problems are given. Two types of shape functions that are based on Lagrange interpolation polynomials in the Gauss–Lobatto points are considered. These shape functions result in condition numbers O(p) and $\text{O(p}\text{ln}\text{p)}$ for the condensed stiffness matrices, where p is the polynomial degree employed. Locally refined meshes are analyzed. For the discretization of Dirichlet problems on meshes that are refined geometrically toward singularities, the conditioning of the stiffness matrix is shown to be independent of the number of layers of geometric refinement.     

On the Picard principle for Δ?+?μ

Given a (local) Kato measure?μ on ${{\mathbb{R}^d} \setminus \{0\},\,d \ge 2}$ , let ${{\mathcal H}_0^{\Delta+\mu}(U)}$ be the convex cone of all continuous real solutions u?≥ 0 to the equation Δu?+?u μ?=?0 on the punctured unit ball U satisfying ${\lim_{|x|\to 1} u(x)=0}$ . It is shown that ${{\mathcal H}_0^{\Delta+\mu}(U)\ne \{0\}}$ if and only if the operator ${f\mapsto \int_U G(\cdot,y)f(y)\,d\mu(y)}$ , where G denotes the Green function on U, is bounded on ${\mathcal L^2(U,\mu)}$ and has a norm which is at most one. Moreover, extremal rays in ${{\mathcal H}_0^{\Delta+\mu}(U)}$ are characterized and it is proven that Δ?+?μ satisfies the Picard principle on U, that is, that ${{\mathcal H}_0^{\Delta+\mu}(U)}$ consists of one ray, provided there exists a suitable sequence of shells in U such that, on these shells,?μ is either small or not too far from being radial. Further, it is shown that the verification of the Picard principle can be localized. Several results on L ²-(sub)eigenfunctions and 3G-inequalities which are used in the paper, but may be of independent interest, are proved at the end of the paper.     

On algebraic relations for Ramanujan’s functions

Let P,Q, and R denote the Ramanujan Eisenstein series. We compute algebraic relations in terms of P(q ⁱ ) (i=1,2,3,4), Q(q ⁱ ) (i=1,2,3), and R(q ⁱ ) (i=1,2,3). For complex algebraic numbers q with 0<|q|<1 we prove the algebraic independence over ? of any three-element subset of {P(q),P(q ²),P(q ³),P(q ⁴)} and of any two-element subset of {Q(q),Q(q ²),Q(q ³)} and {R(q),R(q ²),R(q ³)}, respectively. For all the results we use some expressions of $P(q^{i_{1}}), Q(q^{i_{2}}) $ , and $R(q^{i_{3}}) $ in terms of theta constants. Computer-assisted computations of functional determinants and resultants are essential parts of our proofs.     

On a condition for α-starlikeness

In this paper we give a sufficient condition for function to be α-starlike function and some of its applications. We use the techniques of convolution and differential subordinations.     

On the Shapiro–Lopatinkii condition for elliptic problems

This paper considers elliptic problems with high-order derivatives multiplied by a small parameter. We found the algebraic conditions for an operator and the boundary conditions that guarantee the Fredholm property. An a priori estimate for the solution with a constant independent of the small parameter is proved. These results are known for elliptic boundary-value problems with small parameter in the half-space R _n ⁺. We extend them to the case of bounded domains with smooth boundary. The small-parameter coercive conditions are formulated, and a two-sided estimate is proved.     

On some Turán-type inequalities for q-special functions

Intrinsic inequalities involving Turán-type inequalities for some q-special functions are established. A special interest is granted to q-Dunkl kernel. The results presented here would provide extensions of those given in the classical case.     

On estimating road distances by mathematical functions—A rejoinder

The results presented by Love and Morris are discussed from three more points of view.     

On α-close-to-convex functions

LetP(α) denote the class of functionsf analytic in the unit discE, withf(0)=0,f(z)≠0 (0<|z|<1) andf′(z)≠0 inE, satisfying the condition $$\int\limits_{\theta _1 }^{\theta _2 } {\operatorname{Re} } \left\{ {a\left( {1 + \frac{{zf''\left( z \right)}}{{f'\left( z \right)}}} \right) + \left( {1 - a} \right)\frac{{zf'\left( z \right)}}{{f\left( z \right)}}} \right\}d\theta > - \pi $$ whenever 0≤θ₁≤θ₂≤θ₁+2π,z=re^iθ r<1 and α is any positive real number. The functions inP(α) unify the classes of close-to-starlike (α=0) and close-to-convex (α=1) functions. We callf∈P(α) and α-close-to-convex function. In this paper we investigate certain properties of the classP(α).     

Commutators generated by multilinear Calderón-Zygmund type singular integral and Lipschitz functions

In this paper, we establish the boundedness of commutators generated by the multilinear Calderon- Zygmud type singular integrals and Lipschitz functions on the Triebel-Lizorkin space and Lipschitz spaces.     

On the representation of localized functions in ℝ2 by the Maslov canonical operator

We prove that localized functions can be represented in the form of an integral over a parameter, the integrand being the Maslov canonical operator applied to an amplitude obtained from the Fourier transform of the function to be represented. This representation generalizes an earlier one obtained by Dobrokhotov, Tirozzi, and Shafarevich and permits representing localized initial data for wave type equations with the use of an invariant Lagrangian manifold, which simplifies the asymptotic solution formulas dramatically in many cases.     

On α-quasi-irresolute functions

Joseph and Kwack [9] introduced the notion of (θ,s)-continuous functions in order to investigateS-closed spaces due to Thompson [32]. In [26], the present authors investigated further properties of (θ,s)-continuous functions. In this paper, we introduce a new class of functions called α-quasi-irresolute functions which is weaker than (θ,s)-continuous and improve some results established in [26].     

A pure smoothness condition for radó’s theorem for α-analytic functions

Let \(\Omega \subset {{\Bbb C}^n}\) be a bounded, simply connected ?-convex domain. Let α ∈ ?₊ⁿ and let f be a function on Ω which is separately \({C^{2{\alpha _j} - 1}}\)-smooth with respect to z_j (by which we mean jointly \({C^{2{\alpha _j} - 1}}\)-smooth with respect to Rez_j, Imz_j). If f is α-analytic on Ω\f^?1(0), then f is α-analytic on Ω. The result is well-known for the case α_i = 1, 1 ? i ? n, even when f a priori is only known to be continuous.     

The Collet–Eckmann condition for rational functions on the Riemann sphere

We show that the set of Collet–Eckmann maps has positive Lebesgue measure in the space of rational maps on the Riemann sphere for any fixed degree d ≥ 2.     

On the Bose-Barlotti Δ-planes

Connections with flocks of quadratic cones in PG(3,q) to translation planes are used to produce Bose-Barlotti -planes that contain unitals and are of Lenz-Barlotti class II-1 (exactly one incident point-line transitivity).Dedicato ad Adriano Barlotti in occasione del suo settantesimo compleannoPartially supported by M.U.R.S.T.     

On Taylor’s formula for functions of several variables

Elementary courses in mathematical analysis often mention some trick that is used to construct the remainder of Taylor’s formula in integral form. The trick is based on the fact that, differentiating the difference $f(x) - f(t) - f'(t)\frac{{(x - t)}} {{1!}} - \cdots - f^{(r - 1)} (t)\frac{{(x - t)^{r - 1} }} {{(r - 1)!}} $ between the function and its degree r ? 1 Taylor polynomial at t with respect to t, we obtain $ - f^{(r)} (t)\frac{{(x - t)^{r - 1} }} {{(r - 1)!}} $ , so that all derivatives of orders below r disappear. The author observed previously a similar effect for functions of several variables. Differentiating the difference between the function and its degree r ? 1 Taylor polynomial at t with respect to its components, we are left with terms involving only order r derivatives. We apply this fact here to estimate the remainder of Taylor’s formula for functions of several variables along a rectifiable curve.