We consider the broadest concept of the gradient Yamabe soliton, the conformal gradient soliton. In this paper, we elucidate the structure of complete gradient conformal solitons under some assumption, and provide some applications to gradient Yamabe solitons. These results enhance the understanding gained from previous research. Furthermore, we give an affirmative partial answer to the Yamabe soliton version of Perelman’s conjecture.

The present paper is concerned with the infimum of the norm of potentials for Sturm–Liouville eigenvalue problems with Dirichlet boundary condition such that the first two eigenvalues are known. The explicit quantity of the infimum is given by the two eigenvalues.

For a real number $0<\epsilon <1/3$, we show that the anti-canonical volume of an $\epsilon $-klt Fano $3$-fold is at most $3,200/\epsilon ^4$, and the order $O(1/\epsilon ^4)$ is sharp.

We show that if an open set in $\mathbb{R}^d$ can be fibered by unit n-spheres, then $d \geq 2n+1$, and if $d = 2n+1$, then the spheres must be pairwise linked, and $n \in \left\{0, 1, 3, 7 \right\}$. For these values of n, we construct unit n-sphere fibrations in $\mathbb{R}^{2n+1}$.

In this paper, we mainly prove the following conjectures of Sun [16]: Let p > 3 be a prime. Then

\begin{align*}&A_{2p}\equiv A_2-\frac{1648}3p^3B_{p-3}\ ({\rm{mod}}\ p^4),\\&A_{2p-1}\equiv A_1+\frac{16p^3}3B_{p-3}\ ({\rm{mod}}\ p^4),\\&A_{3p}\equiv A_3-36738p^3B_{p-3}\ ({\rm{mod}}\ p^4),\end{align*}

where $A_n=\sum_{k=0}^n\binom{n}k^2\binom{n+k}{k}^2$ is the nth Apéry number, and Bn is the nth Bernoulli number.

In this article, we investigate the spectra of the stability and Hodge–Laplacian operators on a compact manifold immersed as a hypersurface in a smooth metric measure space, possibly with singularities. Using ideas developed by A. Ros and A. Savo, along with an ingenious computation, we have obtained a comparison between the spectra of these operators. As a byproduct of this technique, we have deduced an estimate of the Morse index of such hypersurfaces.

We show that for every $\eta \gt 0$ every sufficiently large $n$-vertex oriented graph $D$ of minimum semidegree exceeding $(1+\eta )\frac k2$ contains every balanced antidirected tree with $k$ edges and bounded maximum degree, if $k\ge \eta n$. In particular, this asymptotically confirms a conjecture of the first author for long antidirected paths and dense digraphs.

Further, we show that in the same setting, $D$ contains every $k$-edge antidirected subdivision of a sufficiently small complete graph, if the paths of the subdivision that have length $1$ or $2$ span a forest. As a special case, we can find all antidirected cycles of length at most $k$.

Finally, we address a conjecture of Addario-Berry, Havet, Linhares Sales, Reed, and Thomassé for antidirected trees in digraphs. We show that this conjecture is asymptotically true in $n$-vertex oriented graphs for all balanced antidirected trees of bounded maximum degree and of size linear in $n$.

A result of Gyárfás [12] exactly determines the size of a largest monochromatic component in an arbitrary $r$-colouring of the complete $k$-uniform hypergraph $K_n^k$ when $k\geq 2$ and $k\in \{r-1,r\}$. We prove a result which says that if one replaces $K_n^k$ in Gyárfás’ theorem by any ‘expansive’ $k$-uniform hypergraph on $n$ vertices (that is, a $k$-uniform hypergraph $G$ on $n$ vertices in which $e(V_1, \ldots, V_k)\gt 0$ for all disjoint sets $V_1, \ldots, V_k\subseteq V(G)$ with $|V_i|\gt \alpha$ for all $i\in [k]$), then one gets a largest monochromatic component of essentially the same size (within a small error term depending on $r$ and $\alpha$). As corollaries we recover a number of known results about large monochromatic components in random hypergraphs and random Steiner triple systems, often with drastically improved bounds on the error terms.

Gyárfás’ result is equivalent to the dual problem of determining the smallest possible maximum degree of an arbitrary $r$-partite $r$-uniform hypergraph $H$ with $n$ edges in which every set of $k$ edges has a common intersection. In this language, our result says that if one replaces the condition that every set of $k$ edges has a common intersection with the condition that for every collection of $k$ disjoint sets $E_1, \ldots, E_k\subseteq E(H)$ with $|E_i|\gt \alpha$, there exists $(e_1, \ldots, e_k)\in E_1\times \cdots \times E_k$ such that $e_1\cap \cdots \cap e_k\neq \emptyset$, then the smallest possible maximum degree of $H$ is essentially the same (within a small error term depending on $r$ and $\alpha$). We prove our results in this dual setting.

For a graph $H$ and a hypercube $Q_n$, $\textrm{ex}(Q_n, H)$ is the largest number of edges in an $H$-free subgraph of $Q_n$. If $\lim _{n \rightarrow \infty } \textrm{ex}(Q_n, H)/|E(Q_n)| \gt 0$, $H$ is said to have a positive Turán density in a hypercube or simply a positive Turán density; otherwise, it has zero Turán density. Determining $\textrm{ex}(Q_n, H)$ and even identifying whether $H$ has a positive or zero Turán density remains a widely open question for general $H$. By relating extremal numbers in a hypercube and certain corresponding hypergraphs, Conlon found a large class of graphs, ones having so-called partite representation, that have zero Turán density. He asked whether this gives a characterisation, that is, whether a graph has zero Turán density if and only if it has partite representation. Here, we show that, as suspected by Conlon, this is not the case. We give an example of a class of graphs which have no partite representation, but on the other hand, have zero Turán density. In addition, we show that any graph whose every block has partite representation has zero Turán density in a hypercube.

Tao and Vu showed that every centrally symmetric convex progression $C\subset \mathbb{Z}^d$ is contained in a generalized arithmetic progression of size $d^{O(d^2)} \# C$. Berg and Henk improved the size bound to $d^{O(d\log d)} \# C$. We obtain the bound $d^{O(d)} \# C$, which is sharp up to the implied constant and is of the same form as the bound in the continuous setting given by John’s theorem.

This paper is concerned with stochastic Schrödinger delay lattice systems with both locally Lipschitz drift and diffusion terms. Based on the uniform estimates and the equicontinuity of the segment of the solution in probability, we show the tightness of a family of probability distributions of the solution and its segment process, and hence the existence of invariant measures on $l^2\times L^2((-\rho,\,0);l^2)$ with $\rho >0$. We also establish a large deviation principle for the solutions with small noise by the weak convergence method.

In this paper, we analyse Turing instability and bifurcations in a host–parasitoid model with nonlocal effect. For a ordinary differential equation model, we provide some preliminary analysis on Hopf bifurcation. For a reaction–diffusion model with local intraspecific prey competition, we first explore the Turing instability of spatially homogeneous steady states. Next, we show that the model can undergo Hopf bifurcation and Turing–Hopf bifurcation, and find that a pair of spatially nonhomogeneous periodic solutions is stable for a (8,0)-mode Turing–Hopf bifurcation and unstable for a (3,0)-mode Turing–Hopf bifurcation. For a reaction–diffusion model with nonlocal intraspecific prey competition, we study the existence of the Hopf bifurcation, double-Hopf bifurcation, Turing bifurcation, and Turing–Hopf bifurcation successively, and find that a spatially nonhomogeneous quasi-periodic solution is unstable for a (0,1)-mode double-Hopf bifurcation. Our results indicate that the model exhibits complex pattern formations, including transient states, monostability, bistability, and tristability. Finally, numerical simulations are provided to illustrate complex dynamics and verify our theoretical results.

We generalise and improve some recent bounds for additive energies of modular roots. Our arguments use a variety of techniques, including those from additive combinatorics, algebraic number theory and the geometry of numbers. We give applications of these results to new bounds on correlations between Salié sums and to a new equidistribution estimate for the set of modular roots of primes.

In this paper, we study the existence of solutions for a critical time–harmonic Maxwell equation in nonlocal media

\[ \begin{cases} \nabla\times(\nabla\times u)+\lambda u=\left(I_{\alpha}\ast|u|^{2^{{\ast}}_{\alpha}}\right)|u|^{2^{{\ast}}_{\alpha}-2}u & \mathrm{in}\ \Omega,\\ \nu\times u=0 & \mathrm{on}\ \partial\Omega, \end{cases} \]

$\Omega \subset \mathbb {R}^{3}$

$\mathcal {C}^{1,1}$

$\nu$

$\lambda <0$

$2^{\ast }_{\alpha }=3+\alpha$

$0<\alpha <3$

$W^{\alpha,2^{\ast }_{\alpha }}_{0}(\mathrm {curl};\Omega )$