We look for bounded periodic solutions for a parametric fractional problem involving a continuous nonlinearity with subcritical growth. By using a variant of Caffarelli and Silvestre extension method adapted to the periodic case and variational tools we prove the existence of at least three bounded periodic solutions when the parameter varies in an appropriate range.

We define tautological relations for the moduli space of stable maps to a target variety. Using the double ramification cycle formula for target varieties of Janda–Pandharipande–Pixton–Zvonkine [20], we construct non-trivial tautological relations parallel to Pixton’s double ramification cycle relations for the moduli of curves. Examples and applications are discussed.

Let $G = \lbrace h_t \; \vert \; t \in \mathbb{R} \rbrace$ be a continuous flow on a connected $n$-manifold $M$. The flow $G$ is said to be strongly reversible by an involution $\tau$ if $h_{-t} = \tau h_t \tau$ for all $t \in \mathbb{R}$, and it is said to be periodic if $h_s = $ identity for some $s \in \mathbb{R}^\ast$. A closed subset $K$ of $M$ is called a global section for $G$ if every orbit $G(x)$ intersects $K$ in exactly one point. In this paper, we study how the two properties “strongly reversible” and “has a global section” are related. In particular, we show that if $G$ is periodic and strongly reversible by a reflection, then $G$ has a global section.

We study $L^p$ inequalities that sharpen the triangle inequality for sums of $N$ functions in $L^p$.

We prove that the locus of Prym curves $(C, \eta)$ of genus $g \geq 5$ for which the Prym-canonical system $\lvert \omega_C (\eta) \rvert$ is base point free but the Prym-canonical map is not an embedding is irreducible and unirational of dimension $2g + 1$.

We obtain new classes of singular equivalences which are constructed from Gorenstein-projective modules.

For each real $\alpha , 0 \leq \alpha \lt 1$, we give examples of endomorphisms in dimension one with infinite topological entropy which are $\alpha$‑Hölder; and for each real $p , 1 \leq p\lt \infty$, we also give examples of endomorphisms in dimension one with infinite topological entropy which are $(1, p)$-Sobolev. These examples are constructed within a family of endomorphisms with infinite topological entropy and which traverse all $\alpha$-Hölder and $(1, p)$-Sobolev classes. Finally, we also give examples of endomorphisms, also in dimension one, which lie in the big and little Zygmund classes, answering a question of M. Benedicks.

We prove that $\operatorname{log} n$ is an almost everywhere convergence Weyl multiplier for the orthonormal systems of non-overlapping Haar polynomials. Moreover, it is done for the general systems of martingale difference polynomials.

We show that the inner distance inside a bounded planar domain is at most the one-dimensional Hausdorff measure of the boundary of the domain. We prove this sharp result by establishing an improved Painlevé length estimate for connected sets and by using the metric removability of totally disconnected sets, proven by Kalmykov, Kovalev, and Rajala. We also give a totally disconnected example showing that for general sets the Painlevé length bound $\varkappa (E) \leq \pi \mathcal{H}^1 (E)$ is sharp.

In [9] Anderson’s conjecture was proven by comparing values of Bloch functions with the variation of the function. We extend that result on Bloch functions from two to arbitrary dimension and prove that

\[ \int \limits_{[0, x]} \lvert \nabla b(\zeta) \rvert e^{b(\zeta)} \: d \lvert \zeta \rvert \lt \infty \; \textrm{.} \]

In the second part of the paper, we show that the area or volume integral

\[ \int \limits_{B^d} \lvert \nabla u(w) \rvert p(w,\theta) \: dA(w) \]

for positive harmonic functions $u$ is bounded by the value $cu(0)$ for at least one $\theta$. The integral is also transferred to simply connected domains and interpreted from the point of view of stochastics. Several emerging open problems are presented.

The Separatrix Theorem of C. Camacho and P. Sad says that there exists at least one invariant curve (separatrix) passing through the singularity of a germ of holomorphic foliation on complex surface, when the surface underlying the foliation is smooth or when it is singular and the dual graph of resolution surface singularity is a tree. Under some assumptions, we obtain existence of separatrix even when the resolution dual graph of the surface singular point is not a tree. It will be necessary to require an extra condition of the foliation, namely, absence of saddle-node in its reduction of singularities.

In this paper, we investigate the relationship between the index of reducibility and Chern coefficients for primary ideals. As an application, we give characterizations of a Cohen–Macaulay ring in terms of its type, irreducible multiplicity, and Chern coefficients with respect to certain parameter ideals in Noetherian local rings.