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In this paper, using thermodynamic formalism for the sub-additive potential, upper bounds for the Hausdorff dimension and the box dimension of non-conformal repellers are obtained as the sub-additive Bowen equation. The map f only needs to be C 1 , without additional conditions. We also prove that all the upper bounds for the Hausdorff dimension obtained in earlier papers coincide. This unifies their results. Furthermore we define an average conformal repeller and prove that the dimension of an average conformal repeller equals the unique root of the sub-additive Bowen equation.

Let a 0 ( n ) a_0(n) and b 0 ( n ) b_0(n) be the normalized Fourier coefficients of the two holomorphic Hecke eigenforms f ( z ) ∈ S 2 k ( Γ ) f(z)\in S_{2k}(\Gamma ) and φ ( z ) ∈ S 2 l ( Γ ) \varphi (z)\in S_{2l}(\Gamma ) respectively. In 1999, Fomenko studied the following average sums of a 0 ( n ) a_0(n) and b 0 ( n ) b_0(n) : \[ ∑ n ≤ x a 0 ( n ) 3 , ∑ n ≤ x a 0 ( n ) 2 b 0 ( n ) , ∑ n ≤ x a 0 ( n ) 2 b 0 ( n ) 2 , ∑ n ≤ x a 0 ( n ) 4 . \sum _{n \leq x}a_0(n)^3, \quad \sum _{n \leq x}a_0(n)^2b_0(n), \quad \sum _{n \leq x}a_0(n)^2b_0(n)^2, \quad \sum _{n \leq x}a_0(n)^4. \] In this paper, we are able to improve on Fomenko’s results.

We obtain average results on the Sato-Tate conjecture for elliptic curves for small angles.

We obtain an asymptotic formula for the 2k-th power mean of odd primitive character sums over the interval [1, q/4).

We consider the classical autonomous constrained variational problem of minimization of ∫ a b ∫(v(t),v'(t))dt in the class Ω:={v ∈ W 1,1 (a,b): v(a) α,v(b) β,v'(t) > 0 a.e. in (a,b)}, where ∫: [a,β] x [0,+∞)→ R is a lower semicontinuous, nonnegative integrand, which can be nonsmooth, nonconvex and noncoercive. We prove a necessary and sufficient condition for the optimality of a trajectory v 0 ∈ Ω in the form of a DuBois-Reymond inclusion involving the subdifferential of Convex Analysis. Moreover, we also provide a relaxation result and necessary and sufficient conditions for the existence of the minimum expressed in terms of an upper limitation for the assigned mean slope ξ0 = (β-α)/(b-a). Applications to various noncoercive variational problems are also included.

We prove that the set of Farey fractions of order T T , that is, the set { α / β ∈ Q : gcd ⁡ ( α , β ) = 1 , 1 ≤ α , β ≤ T } \{\alpha /\beta \in \mathbb {Q}\ : \ \operatorname {gcd}(\alpha , \beta ) = 1, \ 1 \le \alpha , \beta \le T\} , is uniformly distributed in residue classes modulo a prime p p provided T ≥ p 1 / 2 + ε T \ge p^{1/2 +\varepsilon } for any fixed ε > 0 \varepsilon >0 . We apply this to obtain upper bounds for the Lang–Trotter conjectures on Frobenius traces and Frobenius fields “on average” over a one-parametric family of elliptic curves.

In this paper, we show that nonempty closed convex subsets of a metric tree enjoy many properties shared by convex subsets of Hilbert spaces and admissible subsets of hyperconvex spaces. Furthermore, we prove that a set-valued mapping T ∗ T^* of a metric tree M M with convex values has a selection T : M → M T: M\rightarrow M for which d ( T ( x ) , T ( y ) ) ≤ d H ( T ∗ ( x ) , T ∗ ( y ) ) d(T(x),T(y))\leq d_H(T^*(x),T^*(y)) for each x , y ∈ M x,y \in M . Here by d H d_H we mean the Hausdroff distance. Many applications of this result are given.

For A ⊂ X = { x 1 , . . . , x n | x i ≥ 0 , i = 1 , 2 , . . . , n } , A\subset X=\{x_1,...,x_n ~|~x_i\geq 0, ~i=1,2,...,n\}, let a A a_A and g A g_A denote the arithmetic mean and geometric mean of elements of A A , respectively. It is proved that if k k is an integer in ( n 2 , n ] (\frac {n}{2}, n] , then \[ ( ∏ | A | = k A ⊂ X a A ) 1 C n k ≥ 1 C n k ( ∑ | A | = k A ⊂ X g A ) , \Big (\prod _{|A|=k\atop A\subset X}a_{A}\Big )^{\frac {1}{C_n^k}} \geq \frac {1}{C_n^k}\Big (\sum _{|A|=k\atop A\subset X}g_{A}\Big ), \] with equality if and only if x 1 = . . . = x n x_1=...=x_n . Furthermore, as a generalization of this inequality, a mixed power-mean inequality for subsets is established.

In this paper we are concerned with the estimation of integrals on the unit circle of the form ∫02π f(eiθ)ω(θ)dθ by means of the so-called Szego quadrature formulas, i.e., formulas of the type Σj=1n λjf(xj) with distinct nodes on the unit circle, exactly integrating Laurent polynomials in subspaces of dimension as high as possible. When considering certain weight functions ω(θ) related to the Jacobi functions for the interval [-1, 1], nodes {xj}j=1n and weights {λj}j=1n in Szego quadrature formulas are explicitly deduced. Illustrative numerical examples are also given.

In this paper we consider a non-commutative analogue of the arithmetic-geometric mean inequality a r b 1-r + (r - 1)b ≥ ra for two positive numbers a, b and for r > 1. We show that under certain assumptions the non-commutative analogue of a r b 1-r which satisfies this inequality is unique and equal to the r-mean. The case 0 < r < 1 is also considered. In particular, we give a new characterization of the geometric mean.