Let $d\in \mathbb{N}$, and let $f$ be a function in the Orlicz class $L({log}^{+}L{)}^{d-1}$ defined on the unit cube $[0,1{]}^d$ in ${\mathbb{R}}^d$. Given knot sequences ${\Delta }_1,\dots ,{\Delta }_d$ on $[0,1]$, we first prove that the orthogonal projection $P_{({\Delta }_1,\dots ,{\Delta }_d)}\left(f\right)$ onto the space of tensor product splines with arbitrary orders $(k_1,\dots ,k_d)$ and knots ${\Delta }_1,\dots ,{\Delta }_d$ converges to $f$ almost everywhere as the mesh diameters $\left|{\Delta }_1\right|,\dots ,\left|{\Delta }_d\right|$ tend to zero. This extends the one-dimensional result in [9] to arbitrary dimensions.

In the second step, we show that this result is optimal, that is, given any “bigger” Orlicz class $X=\sigma \left(L\right)L({log}^{+}L{)}^{d-1}$ with an arbitrary function $\sigma$ tending to zero at infinity, there exist a function $\phi \in X$ and partitions of the unit cube such that the orthogonal projections of $\phi$ do not converge almost everywhere.

We show that, for every principal $G$-bundle over a complete Riemannian manifold of nonpositive Ricci curvature, if the projection of the $G$-bundle is biharmonic, then it is harmonic.

A conjecture of Sokal [24], regarding the domain of nonvanishing for independence polynomials of graphs, states that given any natural number $\Delta \ge 3$, there exists a neighborhood in $\mathbb{C}$ of the interval $[0,(\Delta -1{)}^{\Delta -1}/(\Delta -2{)}^{\Delta })$ on which the independence polynomial of any graph with maximum degree at most $\Delta$ does not vanish. We show here that Sokal’s conjecture holds, as well as a multivariate version, and prove the optimality for the domain of nonvanishing. An important step is to translate the setting to the language of complex dynamical systems.

Based on a maximal inequality-type result of Cuculescu, we establish some noncommutative maximal inequalities such as the Hajék–Penyi and Etemadi inequalities. In addition, we present a noncommutative Kolmogorov-type inequality by showing that if $x_1,x_2,\dots ,x_n$ are successively independent self-adjoint random variables in a noncommutative probability space $(\mathfrak{M},\tau )$ such that $\tau \left(x_k\right)=0$ and $s_k{s}_{k-1}=s_{k-1}{s}_k$, where $s_k={\sum }_{j=1}^k{x}_j$, then, for any $\lambda >0$, there exists a projection $e$ such that

$$1-\frac{(\lambda +{max\hspace{0.17em}}_{1\le k\le n}\Vert x_k\Vert {)}^2}{{\sum }_{k=1}^{n}var\left(x_k\right)}\le \tau (e)\le \frac{\tau \left(s_n^2\right)}{{\lambda }^2}.$$ As a result, we investigate the relation between the convergence of a series of independent random variables and the corresponding series of their variances.

We consider the 3-point blowup of the manifold $(S^2\times S^2,\sigma \oplus \sigma )$, where $\sigma$ is the standard symplectic form that gives area 1 to the sphere $S^2$, and study its group of symplectomorphisms $Symp(S^2\times S^2\#3{\overline{\mathbb{C}\mathbb{P}}}^2,\omega )$. So far, the monotone case was studied by Evans [6], who proved that this group is contractible. Moreover, Li, Li, and Wu [13] showed that the group ${Symp}_h(S^2\times S^2\#3{\overline{\mathbb{C}\mathbb{P}}}^2,\omega )$ of symplectomorphisms that act trivially on homology is always connected, and recently, in [14], they also computed its fundamental group. We describe, in full detail, the rational homotopy Lie algebra of this group.

We show that some particular circle actions contained in the symplectomorphism group generate its full topology. More precisely, they give the generators of the homotopy graded Lie algebra of $Symp(S^2\times S^2\#3{\overline{\mathbb{C}\mathbb{P}}}^2,\omega )$. Our study depends on Karshon’s classification of Hamiltonian circle actions and the inflation technique introduced by Lalonde and McDuff. As an application, we deduce the rank of the homotopy groups of $Symp(\mathbb{C}{\mathbb{P}}^2\#5{\overline{\mathbb{C}\mathbb{P}}}^2,\tilde{\omega })$ in the case of small blowups.

We give a classification of three-dimensional partially hyperbolic systems that have at least one torus tangent to the center-stable bundle.

Consider two continuous linear operators $T:X_1(\mu )\to Y_1(\nu )$ and $S:X_2(\mu )\to Y_2(\nu )$ between Banach function spaces related to different $\sigma$-finite measures $\mu$ and $\nu$. By means of weighted norm inequalities we characterize when $T$ can be strongly factored through $S$, that is, when there exist functions $g$ and $h$ such that $T(f)=gS(hf)$ for all $f\in X_1(\mu )$. For the case of spaces with Schauder basis, our characterization can be improved, as we show when $S$ is, for instance, the Fourier or Cesàro operator. Our aim is to study the case where the map $T$ is besides injective. Then we say that it is a representing operator—in the sense that it allows us to represent each element of the Banach function space $X(\mu )$ by a sequence of generalized Fourier coefficients—providing a complete characterization of these maps in terms of weighted norm inequalities. We also provide some examples and applications involving recent results on the Hausdorff–Young and the Hardy–Littlewood inequalities for operators on weighted Banach function spaces.

It was proved by Nill that for any lattice simplex of dimension $d$ with degree $s$ that is not a lattice pyramid, we have $d+1\le 4s-1$. In this paper, we give a complete characterization of lattice simplices satisfying the equality, that is, the lattice simplices of dimension $4s-2$ with degree $s$ that are not lattice pyramids. It turns out that such simplices arise from binary simplex codes. As an application of this characterization, we show that such simplices are counterexamples for the conjecture known as the Cayley conjecture. Moreover, by slightly modifying Nill’s inequality we also see the sharper bound $d+1\le f(2s)$, where $f(M)={\sum }_{n=0}^{\lfloor {log}_{2}M\rfloor }\lfloor M/2^n\rfloor$ for $M\in {\mathbb{Z}}_{\ge 0}$. We also observe that any lattice simplex attaining this sharper bound always comes from a binary code.

We give a short proof of the $L^1$ criterion for Beurling generalized integers to have a positive asymptotic density. We in fact prove the existence of density under a weaker hypothesis. We also discuss related sufficient conditions for the estimate $m(x)={\sum }_{n_k\le x}\mu (n_k)/n_k=o(1)$ with the Beurling analog $\mu$ of the Möbius function.