We prove a range of new sum-product type growth estimates over a general field $\mathbb{F}$, in particular the special case $\mathbb{F}=\mathbb{F}_{p}$. They are unified by the theme of “breaking the $3/2$ threshold”, epitomizing the previous state of the art. This concerns two questions pivotal for the sum-product theory, which are lower bounds for the number of distinct cross-ratios determined by a finite subset of $\mathbb{F}$, as well as the number of values of the symplectic form determined by a finite subset of $\mathbb{F}^{2}$. We establish the estimate $|R[A]|\gtrsim |A|^{8/5}$ for cardinality of the set $R[A]$ of distinct cross-ratios, defined by triples of elements of a set $A\subset \mathbb{F}$ (sufficiently small if $\mathbb{F}$ has positive characteristic, similarly for the rest of the estimates), pinned at infinity. The cross-ratio bound enables us to break the threshold in the second question: for a non-collinear point set $P\subset \mathbb{F}^{2}$, the number of distinct values of the symplectic form $\unicode[STIX]{x1D714}$ on pairs of distinct points $u,u^{\prime }$ of $P$ is $|\unicode[STIX]{x1D714}(P)|\gtrsim |P|^{2/3+c}$, with an explicit $c$. Symmetries of the cross-ratio underlie its local growth properties under both addition and multiplication, yielding an onset of growth of products of difference sets, which is another main result herein. Our proofs make use of specially suited applications of new incidence bounds over $\mathbb{F}$, which apply to higher moments of representation functions. The technical thrust of the paper uses additive combinatorics to relate and adapt these higher moment bounds to growth estimates. A particular instance of this is breaking the threshold in the few sums, many products question over any $\mathbb{F}$, by showing that if $A$ is sufficiently small and has additive doubling constant $M$, then $|AA|\gtrsim M^{-2}|A|^{14/9}$. This result has a second moment version, which allows for new upper bounds for the number of collinear point triples in the set $A\times A\subset \mathbb{F}^{2}$, the quantity often arising in applications of geometric incidence estimates.

We study Piatetski-Shapiro sequences $(\lfloor n^{c}\rfloor )_{n}$ modulo $m$, for non-integer $c>1$ and positive $m$, and we are particularly interested in subword occurrences in those sequences. We prove that each block $\in \{0,1\}^{k}$ of length $k<c+1$ occurs as a subword with the frequency $2^{-k}$, while there are always blocks that do not occur. In particular, those sequences are not normal. For $1<c<2$, we estimate the number of subwords from above and below, yielding the fact that our sequences are deterministic and not morphic. Finally, using the Daboussi–Kátai criterion, we prove that the sequence $\lfloor n^{c}\rfloor$ modulo $m$ is asymptotically orthogonal to multiplicative functions bounded by 1 and with mean value 0.

A flag area measure on an $n$-dimensional euclidean vector space is a continuous translation-invariant valuation with values in the space of signed measures on the flag manifold consisting of a unit vector $v$ and a $(p+1)$-dimensional linear subspace containing $v$ with $0\leqslant p\leqslant n-1$. Using local parallel sets, Hinderer constructed examples of $\text{SO}(n)$-covariant flag area measures. There is an explicit formula for his flag area measures evaluated on polytopes, which involves the squared cosine of the angle between two subspaces. We construct a more general sequence of smooth $\text{SO}(n)$-covariant flag area measures via integration over the normal cycle of appropriate differential forms. We provide an explicit description of our measures on polytopes, which involves an arbitrary elementary symmetric polynomial in the squared cosines of the principal angles between two subspaces. Moreover, we show that these flag area measures span the space of all smooth $\text{SO}(n)$-covariant flag area measures, which gives a classification result in the spirit of Hadwiger’s theorem.

We prove that, for any finite set $A\subset \mathbb{Q}$ with $|AA|\leqslant K|A|$ and any positive integer $k$, the $k$-fold product set of the shift $A+1$ satisfies the bound

$$\begin{eqnarray}|\{(a_{1}+1)(a_{2}+1)\cdots (a_{k}+1):a_{i}\in A\}|\geqslant \frac{|A|^{k}}{(8k^{4})^{kK}}.\end{eqnarray}$$

$K$

$c\log |A|$

$c=c(k)$

$\unicode[STIX]{x1D6EC}$

We apply Newton’s method to stochastic functional evolution equations in Hilbert spaces using semigroup methods. The first-order convergence is based on our generalization of the Gronwall-type inequality. We also establish a second-order convergence in a probabilistic sense.

The lonely runner conjecture, now over fifty years old, concerns the following problem. On a unit-length circular track, consider $m$ runners starting at the same time and place, each runner having a different constant speed. The conjecture asserts that each runner is lonely at some point in time, meaning at a distance at least $1/m$ from the others. We formulate a function field analogue, and give a positive answer in some cases in the new setting.

We demonstrate that Radon measures which arise as the limit of the Modica-Mortola measures associated with phase-fields with uniformly bounded diffuse area and Willmore energy may be singular at the boundary of a domain and discuss implications for practical applications. We furthermore give partial regularity results for the phase-fields uε at the boundary in terms of boundary conditions and counterexamples without boundary conditions.

We use bifurcation and topological methods to investigate the existence/nonexistence and the multiplicity of positive solutions of the following quasilinear Schrödinger equation

$$\left\{ {\matrix{ {-\Delta u-\kappa \Delta \left( {u^2} \right)u = \beta u-\lambda \Phi \left( {u^2} \right)u{\mkern 1mu} {\mkern 1mu} } \hfill & {{\rm in}\;\Omega ,} \hfill \cr {u = 0} \hfill & {{\rm on}\;\partial \Omega } \hfill \cr } } \right.$$

In this series of papers, we explore moments of derivatives of L-functions in function fields using classical analytic techniques such as character sums and approximate functional equation. The present paper is concerned with the study of mean values of derivatives of quadratic Dirichlet L-functions over function fields when the average is taken over monic and irreducible polynomials P in 𝔽q[T]. When the cardinality q of the ground field is fixed and the degree of P gets large, we obtain asymptotic formulas for the first moment of the first and the second derivative of this family of L-functions at the critical point. We also compute the full polynomial expansion in the asymptotic formulas for both mean values.

In this paper, we analyse the semilinear fourth-order problem ( − Δ)2u = g(u) in exterior domains of ℝN. Assuming the function g is nondecreasing and continuous in [0, + ∞) and positive in (0, + ∞), we show that positive classical supersolutions u of the problem which additionally verify − Δu > 0 exist if and only if N ≥ 5 and

$$\int_0^\delta \displaystyle{{g(s)}\over{s^{(({2(N-2)})/({N-4}))}}} {\rm d}s \lt + \infty$$

In this paper, we are concerned with the following bi-harmonic equation with Hartree type nonlinearity

$$\Delta ^2u = \left( {\displaystyle{1 \over { \vert x \vert ^8}}* \vert u \vert ^2} \right)u^\gamma ,\quad x\in {\open R}^d,$$

Let Q be the open unit square in ℝ2. We prove that in a natural complete metric space of BV homeomorphisms f : Q → Q with f|∂Q = Id, residually many homeomorphisms (in the sense of Baire categories) map a null set onto a set of full measure, and vice versa. Moreover, we observe that for 1 ⩽ p < 2, the family of W1,p homemomorphisms satisfying the above property is of the first category.

We consider an elliptic problem of the type

$$\left\{ {\matrix{ {-\Delta u = f(x,u)\quad } \hfill & {{\rm in}\,\Omega } \hfill \cr {u = 0} \hfill & {{\rm on}\,\Gamma _1} \hfill \cr {\displaystyle{{\partial u} \over {\partial \nu }} = g(x,u)} \hfill & {{\rm on}\,\Gamma _2} \hfill \cr } } \right.$$

$\partial \Omega = \overline {\Gamma _1} \cup \overline {\Gamma _2} $

Under a Morse index condition, we prove cylindrical symmetry results for solutions of the above problem.

As an intermediate step, we relate the Morse index of a solution of the nonlinear problem to the eigenvalues of the following linear eigenvalue problem

$$\left\{ {\matrix{ {-\Delta w_j + c(x)w_j = \lambda _jw_j} \hfill & {{\rm in }\Omega } \hfill \cr {w_j = 0} \hfill & {{\rm on }\Gamma _1} \hfill \cr {\displaystyle{{\partial w_j} \over {\partial \nu }} + d(x)w_j = \lambda _jw_j} \hfill & {{\rm on }\Gamma _2} \hfill \cr } } \right.$$

The Ginzburg–Landau functional is a phase transition model which is suitable for classification type problems. We study the asymptotics of a sequence of Ginzburg–Landau functionals with anisotropic interaction potentials on point clouds Ψn where n denotes the number data points. In particular, we show the limiting problem, in the sense of Γ-convergence, is related to the total variation norm restricted to functions taking binary values, which can be understood as a surface energy. We generalize the result known for isotropic interaction potentials to the anisotropic case and add a result concerning the rate of convergence.

In this paper, we study under what boundary conditions the inequality

$${\rm \Vert }\nabla \omega {\rm \Vert }_{L^2(\Omega )}^2 \les C({\rm \Vert }{\rm curl}\omega {\rm \Vert }_{L^2(\Omega )}^2 + {\rm \Vert }{\rm div}\omega {\rm \Vert }_{L^2(\Omega )}^2 + {\rm \Vert }\omega {\rm \Vert }_{L^2(\Omega )}^2 )$$

We present some fundamental properties of quasi-Reinhardt domains, in connection with Kobayashi hyperbolicity, minimal domains and representative domains. We also study proper holomorphic correspondences between quasi-Reinhardt domains.

Let n ⩾ 3, 0 ⩽ m < n − 2/n, ρ1 > 0, $\beta>\beta_{0}^{(m)}=(({m\rho_{1}})/({n-2-nm}))$, αm = ((2β + ρ1)/(1 − m)) and α = 2β+ρ1. For any λ > 0, we prove the uniqueness of radially symmetric solution υ(m) of Δ(υm/m) + αmυ + βx · ∇υ = 0, υ > 0, in ℝn∖{0} which satisfies $\lim\nolimits_{|x|\to 0|}|x|^{\alpha _m/\beta }v^{(m)}(x) = \lambda ^{-((\rho _1)/((1-m)\beta ))}$ and obtain higher order estimates of υ(m) near the blow-up point x = 0. We prove that as m → 0+, υ(m) converges uniformly in C2(K) for any compact subset K of ℝn∖{0} to the solution υ of Δlog υ + α υ + β x·∇ υ = 0, υ > 0, in ℝn\{0}, which satisfies $\lim\nolimits_{ \vert x \vert \to 0} \vert x \vert ^{{\alpha}/{\beta}}v(x)=\lambda^{-{\rho_{1}}/{\beta}}$. We also prove that if the solution u(m) of ut = Δ (um/m), u > 0, in (ℝn∖{0}) × (0, T) which blows up near {0} × (0, T) at the rate $ \vert x \vert ^{-{\alpha_{m}}/{\beta}}$ satisfies some mild growth condition on (ℝn∖{0}) × (0, T), then as m → 0+, u(m) converges uniformly in C2 + θ, 1 + θ/2(K) for some constant θ ∈ (0, 1) and any compact subset K of (ℝn∖{0}) × (0, T) to the solution of ut = Δlog u, u > 0, in (ℝn∖{0}) × (0, T). As a consequence of the proof, we obtain existence of a unique radially symmetric solution υ(0) of Δ log υ + α υ + β x·∇ υ = 0, υ > 0, in ℝn∖{0}, which satisfies $\lim\nolimits_{ \vert x \vert \to 0} \vert x \vert ^{{\alpha}/{\beta}}v(x)=\lambda^{-{\rho_{1}}/{\beta}}$.