Scattered linear sets generated by collineations between pencils of lines

Let \(A\) and \(B\) be two points of \(\mathrm{{PG}}(2,q^n)\) , and let \(\Phi \) be a collineation between the pencils of lines with vertices \(A\) and \(B\) . In this paper, we prove that the set of points of intersection of corresponding lines under \(\Phi \) is either the union of a scattered \(\mathrm{{GF}}(q)\) -linear set of rank \(n+1\) with the line \(AB\) or the union of \(q-1\) scattered \(\mathrm{{GF}}(q)\) -linear sets of rank \(n\) with \(A\) and \(B\) . We also determine the intersection configurations of two scattered \(\mathrm{{GF}}(q)\) -linear sets of rank \(n+1\) of \(\mathrm{{PG}}(2,q^n)\) both meeting the line \(AB\) in a \(\mathrm{{GF}}(q)\) -linear set of pseudoregulus type with transversal points \(A\) and \(B\) .     

Generalized derivations with annihilating and centralizing Engel conditions on Lie ideals

Let $R$ be a non-commutative prime ring, with center $Z(R)$ , extended centroid $C$ and let $F$ be a non-zero generalized derivation of $R$ . Denote by $L$ a non-central Lie ideal of $R$ . If there exists $0\ne a\in R$ such that $a[F(x),x]_k\in Z(R)$ for all $x\in L$ , where $k$ is a fixed integer, then one of the followings holds: (1) either there exists $\lambda \in C$ such that $F(x)=\lambda x$ for all $x\in R$ , (2) or $R$ satisfies $s_4$ , the standard identity in $4$ variables, and $char(R)=2$ ; (3) or $R$ satisfies $s_4$ and there exist $q\in U, \gamma \in C$ such that $F(x)=qx+xq+\gamma x$ .     

Integer-valued polynomials over matrices and divided differences

Let $D$ be an integrally closed domain with quotient field $K$ and $n$ a positive integer. We give a characterization of the polynomials in $K[X]$ which are integer-valued over the set of matrices $M_n(D)$ in terms of their divided differences. A necessary and sufficient condition on $f\in K[X]$ to be integer-valued over $M_n(D)$ is that, for each $k$ less than $n$ , the $k$ th divided difference of $f$ is integral-valued on every subset of the roots of any monic polynomial over $D$ of degree $n$ . If in addition $D$ has zero Jacobson radical then it is sufficient to check the above conditions on subsets of the roots of monic irreducible polynomials of degree $n$ , that is, conjugate integral elements of degree $n$ over $D$ .     

Some totally geodesic submanifolds of the nonlinear Grassmannian of a compact symmetric space

Let \(M\) and \(N\) be two connected smooth manifolds, where \(M\) is compact and oriented and \(N\) is Riemannian. Let \(\mathcal {E}\) be the Fréchet manifold of all embeddings of \(M\) in \(N\) , endowed with the canonical weak Riemannian metric. Let \(\sim \) be the equivalence relation on \(\mathcal {E}\) defined by \(f\sim g\) if and only if \(f=g\circ \phi \) for some orientation preserving diffeomorphism \(\phi \) of \(M\) . The Fréchet manifold \(\mathcal {S}= \mathcal {E}/_{\sim }\) of equivalence classes, which may be thought of as the set of submanifolds of \(N\) diffeomorphic to \(M\) and is called the nonlinear Grassmannian (or Chow manifold) of \(N\) of type \(M\) , inherits from \( \mathcal {E}\) a weak Riemannian structure. We consider the following particular case: \(N\) is a compact irreducible symmetric space and \(M\) is a reflective submanifold of \(N\) (that is, a connected component of the set of fixed points of an involutive isometry of \( N\) ). Let \(\mathcal {C}\) be the set of submanifolds of \(N\) which are congruent to \(M\) . We prove that the natural inclusion of \(\mathcal {C}\) in \(\mathcal {S}\) is totally geodesic.     

A Question from a Famous Paper of Erdős

Given a convex body $K$ K , consider the smallest number $N$ N so that there is a point $P\in \partial K$ P ∈ ? K such that every circle centred at $P$ P intersects $\partial K$ ? K in at most $N$ N points. In 1946 Erd?s conjectured that $N=2$ N = 2 for all $K$ K , but there are convex bodies for which this is not the case. As far as we know there is no known global upper bound. We show that no convex body has $N=\infty $ N = ∞ and that there are convex bodies for which $N = 6$ N = 6 .     

Open Gromov–Witten invariants in dimension six

Let $L$ be a closed orientable Lagrangian submanifold of a closed symplectic six-manifold $(X , \omega )$ . We assume that the first homology group $H_1 (L ; A)$ with coefficients in a commutative ring $A$ injects into the group $H_1 (X ; A)$ and that $X$ contains no Maslov zero pseudo-holomorphic disc with boundary on $L$ . Then, we prove that for every generic choice of a tame almost-complex structure $J$ on $X$ , every relative homology class $d \in H_2 (X , L ; \mathbb{Z })$ and adequate number of incidence conditions in $L$ or $X$ , the weighted number of $J$ -holomorphic discs with boundary on $L$ , homologous to $d$ , and either irreducible or reducible disconnected, which satisfy the conditions, does not depend on the generic choice of $J$ , provided that at least one incidence condition lies in $L$ . These numbers thus define open Gromov–Witten invariants in dimension six, taking values in the ring $A$ .     

The Saito–Kurokawa lifting and Darmon points

Let $E_{/_\mathbb{Q }}$ be an elliptic curve of conductor $Np$ with $p\not \mid N$ and let $f$ be its associated newform of weight $2$ . Denote by $f_\infty $ the $p$ -adic Hida family passing though $f$ , and by $F_\infty $ its $\varLambda $ -adic Saito–Kurokawa lift. The $p$ -adic family $F_\infty $ of Siegel modular forms admits a formal Fourier expansion, from which we can define a family of normalized Fourier coefficients $\{\widetilde{A}_T(k)\}_T$ indexed by positive definite symmetric half-integral matrices $T$ of size $2\times 2$ . We relate explicitly certain global points on $E$ (coming from the theory of Darmon points) with the values of these Fourier coefficients and of their $p$ -adic derivatives, evaluated at weight $k=2$ .     

Neighbour-transitive codes in Johnson graphs

The Johnson graph \(J(v,k)\) has, as vertices, the \(k\) -subsets of a \(v\) -set \(\mathcal {V}\) and as edges the pairs of \(k\) -subsets with intersection of size \(k-1\) . We introduce the notion of a neighbour-transitive code in \(J(v,k)\) . This is a proper vertex subset \(\Gamma \) such that the subgroup \(G\) of graph automorphisms leaving \(\Gamma \) invariant is transitive on both the set \(\Gamma \) of ‘codewords’ and also the set of ‘neighbours’ of \(\Gamma \) , which are the non-codewords joined by an edge to some codeword. We classify all examples where the group \(G\) is a subgroup of the symmetric group \(\mathrm{Sym}\,(\mathcal {V})\) and is intransitive or imprimitive on the underlying \(v\) -set \(\mathcal {V}\) . In the remaining case where \(G\le \mathrm{Sym}\,(\mathcal {V})\) and \(G\) is primitive on \(\mathcal {V}\) , we prove that, provided distinct codewords are at distance at least \(3\) , then \(G\) is \(2\) -transitive on \(\mathcal {V}\) . We examine many of the infinite families of finite \(2\) -transitive permutation groups and construct surprisingly rich families of examples of neighbour-transitive codes. A major unresolved case remains.     

Uncertainty principles and sum complexes

Let \(p\) be a prime and let \(A\) be a nonempty subset of the cyclic group \(C_p\) . For a field \({\mathbb F}\) and an element \(f\) in the group algebra \({\mathbb F}[C_p]\) let \(T_f\) be the endomorphism of \({\mathbb F}[C_p]\) given by \(T_f(g)=fg\) . The uncertainty number \(u_{{\mathbb F}}(A)\) is the minimal rank of \(T_f\) over all nonzero \(f \in {\mathbb F}[C_p]\) such that \(\mathrm{supp}(f) \subset A\) . The following topological characterization of uncertainty numbers is established. For \(1 \le k \le p\) define the sum complex \(X_{A,k}\) as the \((k-1)\) -dimensional complex on the vertex set \(C_p\) with a full \((k-2)\) -skeleton whose \((k-1)\) -faces are all \(\sigma \subset C_p\) such that \(|\sigma |=k\) and \(\prod _{x \in \sigma }x \in A\) . It is shown that if \({\mathbb F}\) is algebraically closed then $$\begin{aligned} u_{{\mathbb F}}(A)=p-\max \{k :\tilde{H}_{k-1}(X_{A,k};{\mathbb F}) \ne 0\}. \end{aligned}$$ The main ingredient in the proof is the determination of the homology groups of \(X_{A,k}\) with field coefficients. In particular it is shown that if \(|A| \le k\) then \(\tilde{H}_{k-1}(X_{A,k};{\mathbb F}_p)\!=\!0.\)      

Characterization of ellipses as uniformly dense sets with respect to a family of convex bodies

Let \(K\subset \mathbb R ^N\) be a convex body containing the origin. A measurable set \(G\subset \mathbb R ^N\) with positive Lebesgue measure is said to be uniformly \(K\) -dense if, for any fixed \(r>0\) , the measure of \(G\cap (x+r K)\) is constant when \(x\) varies on the boundary of \(G\) (here, \(x+r K\) denotes a translation of a dilation of \(K\) ). We first prove that \(G\) must always be strictly convex and at least \(C^{1,1}\) -regular; also, if \(K\) is centrally symmetric, \(K\) must be strictly convex, \(C^{1,1}\) -regular and such that \(K=G-G\) up to homotheties; this implies in turn that \(G\) must be \(C^{2,1}\) -regular. Then for \(N=2\) , we prove that \(G\) is uniformly \(K\) -dense if and only if \(K\) and \(G\) are homothetic to the same ellipse. This result was already proven by Amar et al. in 2008 . However, our proof removes their regularity assumptions on \(K\) and \(G\) , and more importantly, it is susceptible to be generalized to higher dimension since, by the use of Minkowski’s inequality and an affine inequality, avoids the delicate computations of the higher-order terms in the Taylor expansion near \(r=0\) for the measure of \(G\cap (x+r\,K)\) (needed in 2008).