Volume 37, Issue 4

Let r ⩾ 3 r\geqslant 3 be an integer and 𝑄 any positive definite quadratic form in 𝑟 variables. We establish asymptotic formulae with power-saving error terms for the number of rational points of bounded height on singular hypersurfaces S Q S_{Q} defined by x 3 = Q ⁢ ( y 1 , … , y r ) ⁢ z x^{3}=Q(y_{1},\dots,y_{r})z . This confirms Manin’s conjecture for any S Q S_{Q} . Our proof is based on analytic methods, and uses some estimates for character sums and moments of 𝐿-functions. In particular, one of the ingredients is Siegel’s mass formula in the argument for the case r = 3 r=3 .

In this paper, we prove the fractional Hardy inequality on polarisable metric measure spaces. The integral Hardy inequality for 1 < p ≤ q < ∞ 1<p\leq q<\infty is playing a key role in the proof. Moreover, we also prove the fractional Hardy–Sobolev type inequality on metric measure spaces. In addition, logarithmic Hardy–Sobolev and fractional Nash type inequalities on metric measure spaces are presented. In addition, we present applications on homogeneous groups and on the Heisenberg group.

In this paper, the weighted estimates for multilinear pseudo-differential operators were systematically studied in rearrangement invariant Banach and quasi-Banach spaces. These spaces contain the Lebesgue space, the classical Lorentz space and Marcinkiewicz space as typical examples. More precisely, the weighted boundedness and weighted modular estimates, including the weak endpoint case, were established for multilinear pseudo-differential operators and their commutators. As applications, we show that the above results also hold for the multilinear Fourier multipliers, multilinear square functions, and a class of multilinear Calderón–Zygmund operators.

Let { u λ } \{u_{\lambda}\} be a sequence of L 2 L^{2} -normalized Laplacian eigenfunctions on a compact two-dimensional smooth Riemanniann manifold ( M , g ) (M,g) . We seek to get an L p L^{p} restriction bound of the Neumann data λ − 1 ⁢ ∂ ν u λ | γ \lambda^{-1}\partial_{\nu}u_{\lambda}|_{\gamma} along a unit geodesic 𝛾. Using the 𝑇- T ∗ T^{*} argument, one can transfer the problem to an estimate of the norm of a Fourier integral operator and show that such bound is O ⁢ ( λ − 1 p + 3 2 ) O(\lambda^{-\frac{1}{p}+\frac{3}{2}}) . The stationary phase theorem plays the crucial role in our proof. Moreover, this upper bound is shown to be optimal.

We introduce the notion of quasi-triangular Leibniz bialgebras, which can be constructed from solutions of the classical Leibniz Yang–Baxter equation (CLYBE) whose skew-symmetric parts are invariant. In addition to triangular Leibniz bialgebras, quasi-triangular Leibniz bialgebras contain factorizable Leibniz bialgebras as another subclass, which lead to a factorization of the underlying Leibniz algebras. Relative Rota–Baxter operators with weights on Leibniz algebras are used to characterize solutions of the CLYBE whose skew-symmetric parts are invariant. On skew-symmetric quadratic Leibniz algebras, such operators correspond to Rota–Baxter type operators. Consequently, we introduce the notion of skew-symmetric quadratic Rota–Baxter Leibniz algebras, such that they give rise to triangular Leibniz bialgebras in the case of weight 0, while they are in one-to-one correspondence with factorizable Leibniz bialgebras in the case of nonzero weights.

A module over a ring R is pure projective provided it is isomorphic to a direct summand of a direct sum of finitely presented modules. We develop tools for the classification of pure projective modules over commutative noetherian rings. In particular, for a fixed finitely presented module M , we consider Add ⁡ ( M ) {\operatorname{Add}(M)} , which consists of direct summands of direct sums of copies of M . We are primarily interested in the case where R is a one-dimensional, local domain, and in torsion-free (or Cohen–Macaulay) modules. We show that, even in this case, Add ⁡ ( M ) {\operatorname{Add}(M)} can have an abundance of modules that are not direct sums of finitely generated ones. Our work is based on the fact that such infinitely generated direct summands are all determined by finitely generated data. Namely, idempotent/trace ideals of the endomorphism ring of M and finitely generated projective modules modulo such idempotent ideals. This allows us to extend the classical theory developed to study the behaviour of direct sum decomposition of finitely generated modules comparing with their completion to the infinitely generated case. We study the structure of the monoid V * ⁢ ( M ) {V^{*}(M)} , of isomorphism classes of countably generated modules in Add ⁡ ( M ) {\operatorname{Add}(M)} with the addition induced by the direct sum. We show that V * ⁢ ( M ) {V^{*}(M)} is a submonoid of V * ⁢ ( M ⊗ R R ^ ) {V^{*}(M\otimes_{R}\widehat{R})} , this allows us to make computations with examples and to prove some realization results.

We show that, for any given q ≥ 0 {q\geq 0} , any Sasakian structure on a closed manifold M is approximated in the C q {C^{{q}}} -norm by structures induced by CR embeddings into weighted Sasakian spheres. In order to obtain this result, we also strengthen the approximation of an orbifold Kähler form by projectively induced ones given in [J. Ross and R. Thomas, Weighted projective embeddings, stability of orbifolds, and constant scalar curvature Kähler metrics, J. Differential Geom. 88 2011, 1, 109–159] in the C 0 {C^{0}} -norm to a C q {C^{{q}}} -approximation.

We study the quotient variety of the space of foliations on ℂ ⁢ ℙ 2 {\mathbb{CP}^{2}} of degree 2 up to change of coordinates. We find the intersection Betti numbers of this variety. As a corollary, we have that these intersection Betti numbers coincide with the intersection Betti numbers of the quotient variety of quartic plane curves. Finally, we give an explicit isomorphism between the space of foliations of degree 2 with different singular points, without invariant lines and the space of smooth quartic plane curves.

Let ( S , 𝔫 ) {(S,{\mathfrak{n}})} be a commutative noetherian local ring and let ω ∈ 𝔫 {\omega\in{\mathfrak{n}}} be non-zerodivisor. This paper is concerned with the two categories of monomorphisms between finitely generated (Gorenstein) projective S -modules, such that their cokernels are annihilated by ω. It is shown that these categories, which will be denoted by 𝖬𝗈𝗇 ⁢ ( ω , 𝒫 ) {{\mathsf{Mon}}(\omega,\mathcal{P})} and 𝖬𝗈𝗇 ⁢ ( ω , 𝒢 ) {{\mathsf{Mon}}(\omega,\mathcal{G})} , are both Frobenius categories with the same projective objects. It is also proved that the stable category 𝖬𝗈𝗇 ¯ ⁢ ( ω , 𝒫 ) {\underline{\mathsf{Mon}}(\omega,\mathcal{P})} is triangle equivalent to the category of D-branes of type B, 𝖣𝖡 ⁢ ( ω ) {\mathsf{DB}(\omega)} , which has been introduced by Kontsevich and studied by Orlov. Moreover, it will be observed that the stable categories 𝖬𝗈𝗇 ¯ ⁢ ( ω , 𝒫 ) {\underline{\mathsf{Mon}}(\omega,\mathcal{P})} and 𝖬𝗈𝗇 ¯ ⁢ ( ω , 𝒢 ) {\underline{\mathsf{Mon}}(\omega,\mathcal{G})} are closely related to the singularity category of the factor ring R = S / ( ω ) {R=S/({\omega)}} . Precisely, there is a fully faithful triangle functor from the stable category 𝖬𝗈𝗇 ¯ ⁢ ( ω , 𝒢 ) {\underline{\mathsf{Mon}}(\omega,\mathcal{G})} to 𝖣 𝗌𝗀 ⁡ ( R ) {\operatorname{\mathsf{D_{sg}}}(R)} , which is dense if and only if R (and so S ) are Gorenstein rings. Particularly, it is proved that the density of the restriction of this functor to 𝖬𝗈𝗇 ¯ ⁢ ( ω , 𝒫 ) {\underline{\mathsf{Mon}}(\omega,\mathcal{P})} , guarantees the regularity of the ring S .

Let A be a superalgebra over a field F of characteristic zero. We prove tight relations between graded automorphisms, pseudoautomorphisms, superautomorphisms and K -gradings on A , where K is the Klein group. Moreover, we investigate the consequences of such connections within the theory of polynomial identities. In the second part we focus on the superalgebra U ⁢ T n ⁢ ( F ) {UT_{n}(F)} of n × n {n\times n} upper triangular matrices by completely classifying the graded-pseudo-super automorphism that one can define on it. Finally, we compute the ideals of identities of U ⁢ T n ⁢ ( F ) {UT_{n}(F)} endowed with a graded or a pseudo automorphism, for any n , and the ideals of identities with superautomorphism in the cases n = 2 {n=2} and n = 3 {n=3} .

The aim of this short note is to prove that if G is a (homomorphic images of a) soluble periodic linear group and N is a locally nilpotent normal subgroup of G such that N and G / N {G/N} have no isomorphic G -chief factors, then two supplements to N in G are conjugate provided that they have the same intersection with N . This result follows from well-known theorems in the theory of Schunck classes (see [A. Ballester-Bolinches and L. M. Ezquerro, On conjugacy of supplements of normal subgroups of finite groups, Bull. Aust. Math. Soc. 89 2014, 2, 293–299]), and it appeared as the main theorem of [C. Parker and P. Rowley, A note on conjugacy of supplements in finite soluble groups, Bull. Lond. Math. Soc. 42 2010, 3, 417–419].

We study lower bounds for the Riemann zeta function ζ ⁢ ( s ) {\zeta(s)} along vertical arithmetic progressions in the right-half of the critical strip. We show that the lower bounds obtained in the discrete case coincide, up to the constants in the exponential, with the ones known for the continuous case, that is when the imaginary part of s ranges on a given interval. Our methods are based on a discretization of the resonance method for estimating extremal values of ζ ⁢ ( s ) {\zeta(s)} .

A map f : R → S {f\colon R\to S} between (associative, unital, but not necessarily commutative) rings is a brachymorphism if f ⁢ ( 1 + x ) = 1 + f ⁢ ( x ) {f(1+x)=1+f(x)} and f ⁢ ( x ⁢ y ) = f ⁢ ( x ) ⁢ f ⁢ ( y ) {f(xy)=f(x)f(y)} whenever x , y ∈ R {x,y\in R} . We tackle the problem whether every brachymorphism is additive (i.e., f ⁢ ( x + y ) = f ⁢ ( x ) + f ⁢ ( y ) {f(x+y)=f(x)+f(y)} ), showing that in many contexts, including the following, the answer is positive: • R is finite (or, more generally, R is left or right Artinian); • R is any ring of 2 × 2 {2\times 2} matrices over a commutative ring; • R is Engelian; • every element of R is a sum of π-regular and central elements (this applies to π-regular rings, Banach algebras, and power series rings); • R is the full matrix ring of order greater than 1 over any ring; • R is the monoid ring K ⁢ [ M ] {K[M]} for a commutative ring K and a π-regular monoid M ; • R is the Weyl algebra 𝖠 1 ⁢ ( K ) {\mathsf{A}_{1}(K)} over a commutative ring K with positive characteristic; • f is the power function x ↦ x n {x\mapsto x^{n}} over any ring; • f is the determinant function over any ring R of n × n {n\times n} matrices, with n ≥ 3 {n\geq 3} , over a commutative ring, such that if n > 3 {n>3} , then R contains n scalar matrices with non zero divisor differences.

In this note, we give an upper bound on the Hausdorff dimension of removable sets for elliptic and canceling homogeneous differential operators with constant coefficients in the class of bounded functions, using a simple extension of Frostman’s lemma in Euclidean space with an additional power decay.

We consider the set of m × n {m\times n} matrices with rational entries having numerator and denominator of size at most H and obtain various upper bounds on the number of such matrices of a given rank, or with a given determinant, or a given characteristic polynomial. We also consider similar questions for matrices whose entries are Egyptian fractions.

In this article we consider diffeomorphism groups of manifolds with smooth boundary. We show that the diffeomorphism groups of the manifold and its boundary fit into a short exact sequence which admits local sections. In other words, they form an infinite-dimensional fibre bundle. Manifolds with boundary are of interest in numerical analysis and with a view towards applications in machine learning we establish controllability results for families of vector fields. This generalises older results due to Agrachev and Caponigro in the boundary-less case. Our results show in particular that the diffeomorphism group of a manifold with smooth boundary is generated by the image of the exponential map.