Let {{\left( {{n}_{k}} \right)}_{k\ge 1}}${{\left( {{n}_{k}} \right)}_{k\ge 1}}$ be an increasing sequence of positive integers, and let f\left( x \right)$f\left( x \right)$ be a real function satisfying

1

f\left( x+1 \right)=f\left( x \right),\int\limits_{0}^{1}{f\left( x \right)}dx=0,\text{Va}{{\text{r}}_{\left[ 0,1 \right]}}f<\infty $$f\left( x+1 \right)=f\left( x \right),\int\limits_{0}^{1}{f\left( x \right)}dx=0,\text{Va}{{\text{r}}_{\left[ 0,1 \right]}}f<\infty$$

If {{\lim }_{k\to \infty }}\frac{{{n}_{k+1}}}{{{n}_{k}}}=\infty ${{\lim }_{k\to \infty }}\frac{{{n}_{k+1}}}{{{n}_{k}}}=\infty$ the distribution of

2

\frac{\sum\nolimits_{k=1}^{N}{f\left( {{n}_{k}}x \right)}}{\sqrt{N}} $$\frac{\sum\nolimits_{k=1}^{N}{f\left( {{n}_{k}}x \right)}}{\sqrt{N}}$$

converges to a Gaussian distribution. In the case

1<\underset{k\to \infty }{\mathop{\lim \inf }}\,\,\frac{{{n}_{k+1}}}{{{n}_{k}}},\,\,\underset{k\to \infty }{\mathop{\text{lim}\,\text{sup}}}\,\,\frac{{{n}_{k+1}}}{{{n}_{k}}}<\infty $$1<\underset{k\to \infty }{\mathop{\lim \inf }}\,\,\frac{{{n}_{k+1}}}{{{n}_{k}}},\,\,\underset{k\to \infty }{\mathop{\text{lim}\,\text{sup}}}\,\,\frac{{{n}_{k+1}}}{{{n}_{k}}}<\infty$$

there is a complex interplay between the analytic properties of the function f$f$, the number-theoretic properties of {{\left( {{n}_{k}} \right)}_{k\ge 1}}${{\left( {{n}_{k}} \right)}_{k\ge 1}}$, and the limit distribution of (2).

In this paper we prove that any sequence {{\left( {{n}_{k}} \right)}_{k\ge 1}}${{\left( {{n}_{k}} \right)}_{k\ge 1}}$ satisfying \lim {{\sup }_{k\to \infty }}{{n}_{k+1}}/{{n}_{k}}=1$\lim {{\sup }_{k\to \infty }}{{n}_{k+1}}/{{n}_{k}}=1$ contains a nontrivial subsequence {{\left( {{m}_{k}} \right)}_{k\ge 1}}${{\left( {{m}_{k}} \right)}_{k\ge 1}}$ such that for any function satisfying (1) the distribution of

\frac{\sum\nolimits_{k=1}^{N}{f\left( {{m}_{k}}x \right)}}{\sqrt{N}} $$\frac{\sum\nolimits_{k=1}^{N}{f\left( {{m}_{k}}x \right)}}{\sqrt{N}}$$

converges to a Gaussian distribution. This result is best possible: for any \varepsilon >0$\varepsilon >0$ there exists a sequence {{\left( {{n}_{k}} \right)}_{k\ge 1}}${{\left( {{n}_{k}} \right)}_{k\ge 1}}$ satisfying lim \underset{k\to \infty }{\mathop{\sup }}\,\frac{{{n}_{k+1}}}{{{n}_{k}}}\le 1+\varepsilon$\underset{k\to \infty }{\mathop{\sup }}\,\frac{{{n}_{k+1}}}{{{n}_{k}}}\le 1+\varepsilon$ such that for every nontrivial subsequence {{\left( {{m}_{k}} \right)}_{k\ge 1}}${{\left( {{m}_{k}} \right)}_{k\ge 1}}$ of {{\left( {{n}_{k}} \right)}_{k\ge 1}}${{\left( {{n}_{k}} \right)}_{k\ge 1}}$ the distribution of (2) does not converge to a Gaussian distribution for some f$f$.

Our result can be viewed as a Ramsey type result: a sufficiently dense increasing integer sequence contains a subsequence having a certain requested number-theoretic property.

Let \Delta $\Delta$ be a Euclidean quiver. We prove that the closures of the maximal orbits in the varieties of representations of \Delta $\Delta$ are normal and Cohen–Macaulay (even complete intersections). Moreover, we give a generalization of this result for the tame concealed-canonical algebras.

In this paper, we generalize a conjecture due to Darmon and Logan in an adelic setting. We study the relation between our construction and Kudla's works on cycles on orthogonal Shimura varieties. This relation allows us to conjecture a Gross-Kohnen-Zagier theorem for Darmon's points.

In this paper we study systems of weakly coupled Hamilton-Jacobi equations with implicit obstacles that arise in optimal switching problems. Inspired by methods from the theory of viscosity solutions and weak \text{KAM}$\text{KAM}$ theory, we extend the notion of Aubry set for these systems. This enables us to prove a new result on existence and uniqueness of solutions for the Dirichlet problem, answering a question of F. Camilli, P. Loreti, and N. Yamada.

Let C$C$ and D$D$ be digraphs. A mapping f:V\left( D \right)\to V\left( C \right)$f:V\left( D \right)\to V\left( C \right)$ is a C$C$-colouring if for every arc uv$uv$ of D$D$, either f\left( u \right)f\left( v \right)$f\left( u \right)f\left( v \right)$ is an arc of C$C$ or f\left( u \right)=f\left( v \right)$f\left( u \right)=f\left( v \right)$, and the preimage of every vertex of C$C$ induces an acyclic subdigraph in D$D$. We say that D$D$ is C$C$-colourable if it admits a C$C$-colouring and that D$D$ is uniquely C$C$-colourable if it is surjectively C$C$-colourable and any two C$C$-colourings of D$D$ differ by an automorphism of C$C$. We prove that if a digraph D$D$ is not C$C$-colourable, then there exist digraphs of arbitrarily large girth that are D$D$-colourable but not C$C$-colourable. Moreover, for every digraph D$D$ that is uniquely D$D$-colourable, there exists a uniquely D$D$-colourable digraph of arbitrarily large girth. In particular, this implies that for every rational number r\ge 1$r\ge 1$, there are uniquely circularly r$r$-colourable digraphs with arbitrarily large girth.

We study properties of composition operators induced by symbols acting from the unit disk to the polydisk. This result will be involved in the investigation of weighted composition operators on the Hardy space on the unit disk and, moreover, be concerned with composition operators acting from the Bergman space to the Hardy space on the unit disk.

We present a new construction of the entropy-maximizing, invariant probability measure on a Smale space (the Bowen measure). Our construction is based on points that are unstably equivalent to one given point, and stably equivalent to another, i.e., heteroclinic points. The spirit of the construction is similar to Bowen's construction from periodic points, though the techniques are very different. We also prove results about the growth rate of certain sets of heteroclinic points, and about the stable and unstable components of the Bowen measure. The approach we take is to prove results through direct computation for the case of a Shift of Finite type, and then use resolving factor maps to extend the results to more general Smale spaces.

In 1960, Sobolev proved that for a finite reflection group G$G$, a G$G$-invariant cubature formula is of degree t$t$ if and only if it is exact for all G$G$-invariant polynomials of degree at most t$t$. In this paper, we make some observations on invariant cubature formulas and Euclidean designs in connection with the Sobolev theorem. First, we give an alternative proof of theorems by Xu (1998) on necessary and sufficient conditions for the existence of cubature formulas with some strong symmetry. The new proof is shorter and simpler compared to the original one by Xu, and, moreover, gives a general interpretation of the analytically-written conditions of Xu's theorems. Second, we extend a theorem by Neumaier and Seidel (1988) on Euclidean designs to invariant Euclidean designs, and thereby classify tight Euclidean designs obtained from unions of the orbits of the corner vectors. This result generalizes a theorem of Bajnok (2007), which classifies tight Euclidean designs invariant under the Weyl group of type B$B$, to other finite reflection groups.

Using ideas from Shelah's recent proof that a completely separable maximal almost disjoint family exists when c<{{\aleph }_{\omega }}$c<{{\aleph }_{\omega }}$, we construct a weakly tight family under the hypothesis \mathfrak{s}\le \mathfrak{b}<{{\aleph }_{\omega }}$\mathfrak{s}\le \mathfrak{b}<{{\aleph }_{\omega }}$. The case when \mathfrak{s}<\mathfrak{b}$\mathfrak{s}<\mathfrak{b}$ is handled in ZFC and does not require \mathfrak{b}<{{\aleph }_{\omega }}$\mathfrak{b}<{{\aleph }_{\omega }}$, while an additional PCF type hypothesis, which holds when \mathfrak{b}<{{\aleph }_{\omega }}$\mathfrak{b}<{{\aleph }_{\omega }}$ is used to treat the case \mathfrak{s}=\mathfrak{b}$\mathfrak{s}=\mathfrak{b}$. The notion of a weakly tight family is a natural weakening of the well-studied notion of a Cohen indestructible maximal almost disjoint family. It was introduced by Hrušák and García Ferreira [8], who applied it to the Katétov order on almost disjoint families.

This article gives an existence theory for weak solutions of second order non-elliptic linear Dirichlet problems of the form

{\nabla }'P\left( x \right)\nabla u+\text{HR}u+\text{{S}'G}u\text{+ F}u\text{=}\text{f}\,\text{+}\,\text{{T}'g}\,\text{in}\,\Theta $${\nabla }'P\left( x \right)\nabla u+\text{HR}u+\text{{S}'G}u\text{+ F}u\text{=}\text{f}\,\text{+}\,\text{{T}'g}\,\text{in}\,\Theta$$

u=\varphi \,\,on\,\,\partial \Theta $$u=\varphi \,\,on\,\,\partial \Theta$$

The principal part {\xi }'P\left( x \right)\xi${\xi }'P\left( x \right)\xi$ of the above equation is assumed to be comparable to a quadratic form \mathcal{Q}\left( x,\xi \right)={\xi }'Q\left( x \right)\xi$\mathcal{Q}\left( x,\xi \right)={\xi }'Q\left( x \right)\xi$ that may vanish for non-zero \xi \in {{\mathbb{R}}^{n}}$\xi \in {{\mathbb{R}}^{n}}$. This is achieved using techniques of functional analysis applied to the degenerate Sobolev spaces Q{{H}^{1}}\left( \Theta \right)={{W}^{1,2}}\left( \Theta ,Q \right)$Q{{H}^{1}}\left( \Theta \right)={{W}^{1,2}}\left( \Theta ,Q \right)$ and QH_{0}^{1}\left( \Theta \right)=W_{0}^{1,2}\left( \Theta ,Q \right)$QH_{0}^{1}\left( \Theta \right)=W_{0}^{1,2}\left( \Theta ,Q \right)$ as defined in previous works. E.T. Sawyer and R.L. Wheeden (2010) have given a regularity theory for a subset of the class of equations dealt with here.

Analytical study of the regularization of the Boussinesq system is performed in frequency space using Fourier theory. Existence and uniqueness of weak solutions with minimum regularity requirement are proved. Convergence results of the unique weak solution of the regularized Boussinesq system to a weak Leray–Hopf solution of the Boussinesq system are established as the regularizing parameter \alpha$\alpha$ vanishes. The proofs are done in the frequency space and use energy methods, the Arselà-Ascoli compactness theorem and a Friedrichs-like approximation scheme.