Let $$T\left(q\right)=\sum _{k=1}^{\infty }d\left(k\right)q^k,\left|q\right|<1,$$ where $d\left(k\right)$ denotes the number of positive divisors of the natural number $k$. We present monotonicity properties of functions defined in terms of $T$. More specifically, we prove that $$H\left(q\right)=T\left(q\right)-\frac{log(1-q)}{log\left(q\right)}$$ is strictly increasing on $(0,1)$, while $$F\left(q\right)=\frac{1-q}{q}H\left(q\right)$$ is strictly decreasing on $(0,1)$. These results are then applied to obtain various inequalities, one of which states that the double inequality $$\alpha \frac{q}{1-q}+\frac{log(1-q)}{log\left(q\right)}<T\left(q\right)<\beta \frac{q}{1-q}+\frac{log(1-q)}{log\left(q\right)},0<q<1,$$ holds with the best possible constant factors $\alpha =\gamma$ and $\beta =1$. Here, $\gamma$ denotes Euler’s constant. This refines a result of Salem, who...

We show that if $\alpha >1$, then the logarithmically weighted Bergman space $A_{{log}^{\alpha }}^2$ is mapped by the Libera operator $ℒ$ into the space $A_{{log}^{\alpha -1}}^2$, while if $\alpha >2$ and $0<\epsilon \le \alpha -2$, then the Hilbert matrix operator $H$ maps $A_{{log}^{\alpha }}^2$ into $A_{{log}^{\alpha -2-\epsilon }}^2$.We show that the Libera operator $ℒ$ maps the logarithmically weighted Bloch space ${ℬ}_{{log}^{\alpha }}$, $\alpha \in ℝ$, into itself, while $H$ maps ${ℬ}_{{log}^{\alpha }}$ into ${ℬ}_{{log}^{\alpha +1}}$.In Pavlović’s paper (2016) it is shown that $ℒ$ maps the logarithmically weighted Hardy-Bloch space ${ℬ}_{{log}^{\alpha }}^1$, $\alpha >0$, into ${ℬ}_{{log}^{\alpha -1}}^1$. We show that this result is sharp. We also show that $H$ maps ${ℬ}_{{log}^{\alpha }}^1$, $\alpha \ge 0$,...

Dati due elementi $f$ e $g$ in un'algebra uniforme $A$, sia $G=f(M_A/f({\partial }_A)$. Nella presente Nota si danno, fra l’altro, due nuove dimostrazioni elementari del fatto che la funzione $\lambda \to \log \max g(f^{-1}(\lambda ))$ è subarmonica su $G$ e che l’applicazione $\lambda \to g(f^{-1}(\lambda ))$ è analitica nel senso di Oka.

For the real quadratic map $P_a\left(x\right)=x^2+a$ and a given $ϵ\&gt;0$ a point $x$ has good expansion properties if any interval containing $x$ also contains a neighborhood $J$ of $x$ with $P_a^n{|}_J$ univalent, with bounded distortion and $B(0,ϵ)\subseteq P_a^n\left(J\right)$ for some $n\in \mathbb{N}$. The $ϵ$-weakly expanding set is the set of points which do not have good expansion properties. Let $\alpha$ denote the negative fixed point and $M$ the first return time of the critical orbit to $[\alpha ,-\alpha ]$. We show there is a set $ℛ$ of parameters with positive Lebesgue measure for which the Hausdorff...

The purpose of this paper is to investigate efficient representations of the residue classes modulo $q$, by performing sum and product set operations starting from a given subset $A$ of ${\mathbb{Z}}_q$. We consider the case of very small sets $A$ and composite $q$ for which not much seemed known (nontrivial results were recently obtained when $q$ is prime or when log $\left|A\right|\sim logq$). Roughly speaking we show that all residue classes are obtained from a $k$-fold sum of an $r$-fold product set of $A$, where $r\ll logq$ and $logk\ll logq$, provided the...

Let $\left[x\right]$ be an integer part of $x$ and $d\left(n\right)$ be the number of positive divisor of $n$. Inspired by some results of M. Jutila (1987), we prove that for $1<c<\frac{6}{5}$, $$\sum _{n\le x}d\left(\left[n^c\right]\right)=cxlogx+(2\gamma -c)x+O\left(\frac{x}{logx}\right),$$ where $\gamma$ is the Euler constant and $\left[n^c\right]$ is the Piatetski-Shapiro sequence. This gives an improvement upon the classical result of this problem.

Dati due elementi $f$ e $g$ in un'algebra uniforme $A$, sia $G=f(M_A/f({\partial }_A)$. Nella presente Nota si danno, fra l’altro, due nuove dimostrazioni elementari del fatto che la funzione $\lambda \to \log \max g(f^{-1}(\lambda ))$ è subarmonica su $G$ e che l’applicazione $\lambda \to g(f^{-1}(\lambda ))$ è analitica nel senso di Oka.

In this paper we investigate Ramanujan’s inequality concerning the prime counting function, asserting that $\pi {\left(x\right)}^2<\frac{\mathrm{e}x}{logx}\pi \left(\frac{x}{\mathrm{e}}\right)$ for $x$ sufficiently large. First, we study its sharpness by giving full asymptotic expansions of its left and right hand sides expressions. Then, we discuss the structure of Ramanujan’s inequality, by replacing the factor $\frac{x}{logx}$ on its right hand side by the factor $\frac{x}{logx-h}$ for a given $h$, and by replacing the numerical factor $\mathrm{e}$ by a given positive $\alpha$. Finally, we introduce and study inequalities...

Let $f:I\to I$ be a $C^2$ multimodal interval map satisfying polynomial growth of the derivatives along critical orbits. We prove the existence and uniqueness of equilibrium states for the potential ${\phi }_t:x\mapsto -tlog\left|Df\left(x\right)\right|$ for $t$ close to $1$, and also that the pressure function $t\mapsto P\left({\phi }_t\right)$ is analytic on an appropriate interval near $t=1$.

Let K be an algebraic number field with non-trivial class group G and ${}_K$ be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let $F_k\left(x\right)$ denote the number of non-zero principal ideals $a_K$ with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that $F_k\left(x\right)$ behaves for x → ∞ asymptotically like $x{\left(logx\right)}^{1-1/\left|G\right|}{\left(loglogx\right)}^{{}_k\left(G\right)}$. We prove, among other results, that $₁\left(C_{n₁}\oplus C_{n₂}\right)=n₁+n₂$ for all integers n₁,n₂ with 1 < n₁|n₂.

For any given positive integer k, and any set A of nonnegative integers, let $r_{1,k}(A,n)$ denote the number of solutions of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. We prove that if k,l are multiplicatively independent integers, i.e., log k/log l is irrational, then there does not exist any set A ⊆ ℕ such that both $r_{1,k}(A,n)=r_{1,k}(\mathbb{N}\setminus A,n)$ and $r_{1,l}(A,n)=r_{1,l}(\mathbb{N}\setminus A,n)$ hold for all n ≥ n₀. We also pose a conjecture and two problems for further research.

Let ${𝒯}_n$ denote the set of log canonical thresholds of pairs $(X,Y)$, with $X$ a nonsingular variety of dimension $n$, and $Y$ a nonempty closed subscheme of $X$. Using non-standard methods, we show that every limit of a decreasing sequence in ${𝒯}_n$ lies in ${𝒯}_{n-1}$, proving in this setting a conjecture of Kollár. We also show that ${𝒯}_n$ is closed in $𝐑$; in particular, every limit of log canonical thresholds on smooth varieties of fixed dimension is a rational number. As a consequence of this property, we see that in...

The inequality $ʃlog|\sum a_n{e}^{2\pi i{\phi }_n}|d{\phi }_1\dots d{\phi }_n\ge Clog\left(\sum \right|a_n{{|}^2)}^{1/2}$ with an absolute constant C, and similar ones, are extended to the case of $a_n$ belonging to an arbitrary normed space X and an arbitrary compact group of unitary operators on X instead of the operators of multiplication by $e^{2\pi i\phi }$.

For any positive integer k and any set A of nonnegative integers, let $r_{1,k}(A,n)$ denote the number of solutions (a₁,a₂) of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. Let k,l ≥ 2 be two distinct integers. We prove that there exists a set A ⊆ ℕ such that both $r_{1,k}(A,n)=r_{1,k}(\mathbb{N}\setminus A,n)$ and $r_{1,l}(A,n)=r_{1,l}(\mathbb{N}\setminus A,n)$ hold for all n ≥ n₀ if and only if log k/log l = a/b for some odd positive integers a,b, disproving a conjecture of Yang. We also show that for any set A ⊆ ℕ satisfying $r_{1,k}(A,n)=r_{1,k}(\mathbb{N}\setminus A,n)$ for all n ≥ n₀, we have $r_{1,k}(A,n)\to \infty$ as n → ∞.

We give some characterizations of the class ${}_m\left(\Omega \right)$ and use them to establish a lower estimate for the log canonical threshold of plurisubharmonic functions in this class.

We present a method that in certain sense stores the inverse of the stiffness matrix in $O(NlogN)$ memory places, where $N$ is the number of degrees of freedom and hence the matrix size. The setup of this storage format requires $O\left(N^{3/2}\right)$ arithmetic operations. However, once the setup is done, the multiplication of the inverse matrix and a vector can be performed with $O(NlogN)$ operations. This approach applies to the first order finite element discretization of linear elliptic and parabolic problems in triangular...

An inequality of Brascamp and Lieb provides a bound on the covariance of two functions with respect to log-concave measures. The bound estimates the covariance by the product of the $L^2$ norms of the gradients of the functions, where the magnitude of the gradient is computed using an inner product given by the inverse Hessian matrix of the potential of the log-concave measure. Menz and Otto [Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site...

In this paper we study the degree of integrability of quasiharmonic fields. These fields are connected with the study of the equation $div(A(x)\nabla u(x))=0$, where the symmetric matrix $A(x)$ satisfies the condition ${|\xi |}^2+{|A(x)\xi |}^2\le K(x)⟨A(x)\xi ,\xi ⟩$.The nonnegative function $K(x)$ belongs to the exponential class, i.e. $\exp (\beta K(x))$ is integrable for some $\beta >0$. We prove that the gradient of a local solution of the equation belongs to the Zygmund spaces $L_{\text{loc}}^2{\log }^{\alpha -1}L$, . Moreover we show exactly how the degree of improved regularity depends on $\beta$.

We give a new proof using the classic Calderón-Zygmund decomposition that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space $L^{p(\cdot )}$ whenever the exponent function $p(\cdot )$ satisfies log-Hölder continuity conditions. We include the case where $p(\cdot )$ assumes the value infinity. The same proof also shows that the fractional maximal operator $M_a$, , maps $L^{p(\cdot )}$ into $L^{q(\cdot )}$, where $1/p(\cdot )-1/q(\cdot )=a/n$.