We introduce a framework for Riemannian diffeology. To this end, we use the tangent functor in the sense of Blohmann and one of the options of a metric on a diffeological space in the sense of Iglesias-Zemmour. As a consequence, the category consisting of weak Riemannian diffeological spaces and isometries is established. With a technical condition for a definite weak Riemannian metric, we show that the pseudodistance induced by the metric is indeed a distance. As examples of weak Riemannian diffeological spaces, an adjunction space of manifolds, a space of smooth maps and the mixed one are considered.

In this paper, we consider a reaction-diffusion equation that models the time-almost periodic response to climate change within a straight, infinite cylindrical domain. The shifting edge of the habitat is characterised by a time-almost periodic function, reflecting the varying pace of environmental changes. Note that the principal spectral theory is an important role to study the dynamics of reaction-diffusion equations in time heterogeneous environment. Initially, for time-almost periodic parabolic equations in finite cylindrical domains, we develop the principal spectral theory of such equations with mixed Dirichlet–Neumann boundary conditions. Subsequently, we demonstrate that the approximate principal Lyapunov exponent serves as a definitive threshold for species persistence versus extinction. Then, the existence, exponential decay and stability of the forced wave solutions $U(t,x_{1},y)=V\left (t,x_{1}-\int ^{t}_{0}c(s)ds,y\right )$ are established. Additionally, we analyse how fluctuations in the shifting speed affect the approximate top Lyapunov exponent.

Recently, Donoso, Le, Moreira, and Sun studied the asymptotic behaviour of the averages of completely multiplicative functions over the Gaussian integers. They derived Wirsing’s theorem for Gaussian integers, answered a question of Frantzikinakis and Host for the sum of two squares, and obtained a variant of a theorem of Bergelson and Richter on ergodic averages along the number of prime factors of integers. In this paper, we will show the analogue of these results for co-prime integer pairs. Moreover, building on Frantzikinakis and Host’s results, we obtain some convergences on the multilinear averages of multiplicative functions over primitive lattice points.

We consider the following problem

\begin{equation*}\varepsilon^{2 s} P_{\hat{h}}^s u+u=u^p, \quad u \gt 0 \quad \text {on } \,(M, \hat{h}),\end{equation*}

$(X, g^{+})$

$(M,[\hat{h}])$

$s\in (0,1)$

$P_{\hat{h}}^s$

$1 \lt p \lt \frac{n+2s}{n-2s}$

$\varepsilon$

$C^1$

$H$

$0 \lt s \lt {1}/{2}$

$C^1$

${1}/{2}\le s \lt 1$

We introduce a natural weighted enumeration of lattice points in a polytope, and give a Brion-type formula for the corresponding generating function. The weighting has combinatorial significance, and its generating function may be viewed as a generalization of the Rogers–Szegő polynomials. It also arises from the geometry of the toric arc scheme associated to the normal fan of the polytope. We show that the asymptotic behaviour of thecoefficients at $q=1$ is Gaussian.

In this paper, we study the cohomology of the unitary unramified PEL Rapoport-Zink space of signature $(1,n-1)$ at hyperspecial level. Our method revolves around the spectral sequence associated to the open cover by the analytical tubes of the closed Bruhat-Tits strata in the special fiber, which were constructed by Vollaard and Wedhorn. The cohomology of these strata, which are isomorphic to generalized Deligne-Lusztig varieties, has been computed in an earlier work. This spectral sequence allows us to prove the semisimplicity of the Frobenius action and the non-admissibility of the cohomology in general. Via p-adic uniformization, we relate the cohomology of the Rapoport-Zink space to the cohomology of the supersingular locus of a Shimura variety with no level at p. In the case $n=3$ or $4$, we give a complete description of the cohomology of the supersingular locus in terms of automorphic representations.

In the 1980s, Erdős and Sós initiated the study of Turán problems with a uniformity condition on the distribution of edges: the uniform Turán density of a hypergraph $H$ is the infimum over all $d$ for which any sufficiently large hypergraph with the property that all its linear-size subhypergraphs have density at least $d$ contains $H$. In particular, they asked to determine the uniform Turán densities of $K_4^{(3)-}$ and $K_4^{(3)}$. After more than 30 years, the former was solved in [Israel J. Math. 211 (2016), 349 – 366] and [J. Eur. Math. Soc. 20 (2018), 1139 – 1159], while the latter still remains open. Till today, there are known constructions of $3$-uniform hypergraphs with uniform Turán density equal to $0$, $1/27$, $4/27$, and $1/4$ only. We extend this list by a fifth value: we prove an easy to verify sufficient condition for the uniform Turán density to be equal to $8/27$ and identify hypergraphs satisfying this condition.

We introduce the $\ell ^1$-ideal intersection property for crossed product ${\mathrm {C}}^*$-algebras. It is implied by ${\mathrm {C}}^*$-simplicity as well as ${\mathrm {C}}^*$-uniqueness. We show that topological dynamical systems of arbitrary lattices in connected Lie groups, arbitrary linear groups over the integers in a number field and arbitrary virtually polycyclic groups have the $\ell ^1$-ideal intersection property. On the way, we extend previous results on ${\mathrm {C}}^*$-uniqueness of -groupoid algebras to the general twisted setting.

In this paper, we study the existence of $k$-$11$-representations of graphs. Inspired by work on permutation patterns, these representations are ways of representing graphs by words where adjacencies between vertices are captured by patterns in the corresponding letters. Our main result is that all graphs are $1$-$11$-representable, answering a question originally raised by Cheon et al. in 2018 and repeated in several follow-up papers – including a very recent paper, where it was shown that all graphs on at most $8$ vertices are $1$-$11$-representable. Moreover, we prove that all graphs are permutationally $1$-$11$-representable – that is representable as the concatenation of permutations of the vertices – answering the existence question in extremely strong fashion. Our construction leads to nearly optimal bounds on the length of the words, as well. It can, moreover, be adapted to represent all acyclic orientations of graphs; this generalizes the fact that word-representations capture semi-transitive orientations of graphs. Our construction also adapts easily to other $k \geq 2$ as well, giving representations using a linear number of permutations when the best known previous bounds used a quadratic number. Finally, we also consider the (non-)existence of ‘even–odd’-representations of graphs. This answers a question raised by Wanless after a conference talk in 2018.

By extending some basic results about cohomological dimension of tensor products to non-positive DG-rings, the Intersection Theorem for DG-modules is examined over commutative noetherian local DG-rings with bounded cohomology. Some applications are provided. The first is to improve the DG-setting of the amplitude inequality in [Forum Math. 22 (2010) 941–948]. The second is to show Minamoto’s conjecture in [Israel J. Math. 242 (2021) 1–36]. The third is to obtain the DG-version of the Vasconcelos conjecture about Gorenstein rings.

A general way to represent stochastic differential equations (SDEs) on smooth manifolds is based on the Schwartz morphism. In this manuscript, we are interested in SDEs on a smooth manifold $M$ that are driven by p-dimensional Wiener process $W_t \in \mathbb{R}^p$ and time $t$. In terms of the Schwartz morphism, such an SDE is represented by a Schwartz morphism that morphs the semimartingale $(t,W_t)\in\mathbb{R}^{p+1}$ into a semimartingale on the manifold $M$. We show that it is possible to construct such Schwartz morphisms using special maps that we call diffusion generators. We show that one of the ways to construct a diffusion generator is by considering the flow of differential equations. One particular case is the construction of diffusion generators using Lagrangian vector fields. Using the diffusion generator approach, we also give the extended Itô formula (also known as generalized Itô formula or Itô–Wentzell formula) for SDEs on manifolds.

We consider the existence of normalized solutions to non-linear Schrödinger equations on non-compact metric graphs in the L2-supercritical regime. For sufficiently small prescribed mass (L2 norm), we prove existence of positive solutions on two classes of graphs: periodic graphs and non-compact graphs with finitely many edges and suitable topological assumptions. Our approach is based on mountain pass techniques. A key point to overcome the serious lack of compactness is to show that all solutions with small mass have positive energy. To complement our analysis, we prove that this is no longer true, in general, for large masses. To the best of our knowledge, these are the first results with an L2-supercritical non-linearity extended on the whole graph and unravelling the role of topology in the existence of solutions.

A reflection mapping is a singular holomorphic mapping obtained by restricting the quotient mapping of a complex reflection group. We study the analytic structure of double point spaces of reflection mappings. In the case where the image is a hypersurface, we obtain explicit equations for the double point space and for the image as well. In the case of surfaces in ${\mathbb C}^3$, this gives a very efficient method to compute the Milnor number and delta invariant of the double point curve.

Fix integers $r \ge 2$ and $1\le s_1\le \cdots \le s_{r-1}\le t$ and set $s=\prod _{i=1}^{r-1}s_i$. Let $K=K(s_1, \ldots , s_{r-1}, t)$ denote the complete $r$-partite $r$-uniform hypergraph with parts of size $s_1, \ldots , s_{r-1}, t$. We prove that the Zarankiewicz number $z(n, K)= n^{r-1/s-o(1)}$ provided $t\gt 3^{s+o(s)}$. Previously this was known only for $t \gt ((r-1)(s-1))!$ due to Pohoata and Zakharov. Our novel approach, which uses Behrend’s construction of sets with no 3-term arithmetic progression, also applies for small values of $s_i$, for example, it gives $z(n, K(2,2,7))=n^{11/4-o(1)}$ where the exponent 11/4 is optimal, whereas previously this was only known with 7 replaced by 721.

The Generalised Baker–Schmidt Problem (1970) concerns the Hausdorff measure of the set of $\psi$-approximable points on a non-degenerate manifold. Beresnevich-Dickinson-Velani (in 2006, for the homogeneous setting) and Badziahin-Beresnevich-Velani (in 2013, for the inhomogeneous setting) proved the divergence part of this problem for dual approximation on arbitrary non-degenerate manifolds. The divergence part has also been resolved for the $p$-adic setting by Datta-Ghosh in 2022, for the inhomogeneous setting. The corresponding convergence counterpart represents a challenging open question. In this paper, we prove the homogeneous $p$-adic convergence result for hypersurfaces of dimension at least three with some mild regularity condition, as well as for some other classes of manifolds satisfying certain conditions. We provide similar, slightly weaker results for the inhomogeneous setting. We do not restrict to monotonic approximation functions.