We consider a 10-dimensional family of Lehn–Lehn–Sorger–van Straten hyperkähler eightfolds, which have a non-symplectic automorphism of order 3. Using the theory of finite-dimensional motives, we show that the action of this automorphism on the Chow group of 0-cycles is as predicted by the Bloch–Beilinson conjectures. We prove a similar statement for the anti-symplectic involution on varieties in this family. This has interesting consequences for the intersection product of the Chow ring of these varieties.

There are well-known identities involving the Ext bifunctor, coproducts, and products in AB4 abelian categories with enough projectives. Namely, for every such category \[\mathcal{A}\], given an object X and a set of objects \[{\{ {{\text{A}}_{\text{i}}}\} _{{\text{i}} \in {\text{I}}}}\], an isomorphism \[Ext_\mathcal{A}^{\text{n}}({ \oplus _{{\text{i}} \in {\text{I}}}}{{\text{A}}_{\text{i}}},{\text{X}}) \cong \prod\nolimits_{{\text{i}} \in {\text{I}}} {Ext_\mathcal{A}^{\text{n}}({{\text{A}}_{\text{i}}},{\text{X}})} \] can be built, where \[Ex{t^{\text{n}}}\] is the nth derived functor of the Hom functor. The goal of this paper is to show a similar isomorphism for the nth Yoneda Ext, which is a functor equivalent to \[Ex{t^{\text{n}}}\] that can be defined in more general contexts. The desired isomorphism is constructed explicitly by using colimits in AB4 abelian categories with not necessarily enough projectives nor injectives, extending a result by Colpi and Fuller in [8]. Furthermore, the isomorphisms constructed are used to characterize AB4 categories. A dual result is also stated.

Galego and Samuel showed that if K, L are metrizable, compact, Hausdorff spaces, then $C(K)\widehat{\otimes}_\pi C(L)$ is c0-saturated if and only if it is subprojective if and only if K and L are both scattered. We remove the hypothesis of metrizability from their result and extend it from the case of the twofold projective tensor product to the general n-fold projective tensor product to show that for any $n\in\mathbb{N}$ and compact, Hausdorff spaces K1, …, Kn, $\widehat{\otimes}_{\pi, i=1}^n C(K_i)$ is c0-saturated if and only if it is subprojective if and only if each Ki is scattered.

Let G be a group and A a set equipped with a collection of finitary operations. We study cellular automata $$\tau :{A^G} \to {A^G}$$ that preserve the operations AG of induced componentwise from the operations of A. We show τ that is an endomorphism of AG if and only if its local function is a homomorphism. When A is entropic (i.e. all finitary operations are homomorphisms), we establish that the set EndCA(G;A), consisting of all such endomorphic cellular automata, is isomorphic to the direct limit of Hom(AS, A), where S runs among all finite subsets of G. In particular, when A is an R-module, we show that EndCA(G;A) is isomorphic to the group algebra $${\rm{End}}(A)[G]$$ . Moreover, when A is a finite Boolean algebra, we establish that the number of endomorphic cellular automata over AG admitting a memory set S is precisely $${(k|S|)^k}$$ , where k is the number of atoms of A.

The problem of finding the number of ordered commuting tuples of elements in a finite group is equivalent to finding the size of the solution set of the system of equations determined by the commutator relations that impose commutativity among any pair of elements from an ordered tuple. We consider this type of systems for the case of ordered triples and express the size of the solution set in terms of the irreducible characters of the group. The obtained formulas are natural extensions of Frobenius’ character formula that calculates the number of ways a group element is a commutator of an ordered pair of elements in a finite group. We discuss how our formulas can be used to study the probability distributions afforded by these systems of equations, and we show explicit calculations for dihedral groups.

The aim of this paper is to study complete (noncompact) m-quasi-Einstein manifolds with λ=0 satisfying a fourth-order vanishing condition on the Weyl tensor and zero radial Weyl curvature. In this case, we are able to prove that an m-quasi-Einstein manifold (m>1) with λ=0 on a simply connected n-dimensional manifold(Mn, g), (n ≥ 4), of nonnegative Ricci curvature and zero radial Weyl curvature must be a warped product with (n–1)–dimensional Einstein fiber, provided that M has fourth-order divergence-free Weyl tensor (i.e. div4W =0).

Let K be a field of arbitrary characteristic, $${\cal A}$$ be a commutative K-algebra which is a domain of essentially finite type (e.g., the algebra of functions on an irreducible affine algebraic variety), $${a_r}$$ be its Jacobian ideal, and $${\cal D}\left( {\cal A} \right)$$ be the algebra of differential operators on the algebra $${\cal A}$$ . The aim of the paper is to give a simplicity criterion for the algebra $${\cal D}\left( {\cal A} \right)$$ : the algebra $${\cal D}\left( {\cal A} \right)$$ is simple iff $${\cal D}\left( {\cal A} \right)a_r^i{\cal D}\left( {\cal A} \right) = {\cal D}\left( {\cal A} \right)$$ for all i ≥ 1 provided the field K is a perfect field. Furthermore, a simplicity criterion is given for the algebra $${\cal D}\left( R \right)$$ of differential operators on an arbitrary commutative algebra R over an arbitrary field. This gives an answer to an old question to find a simplicity criterion for algebras of differential operators.

For any odd prime p, we construct an infinite family of imaginary quadratic fields whose class numbers are divisible by p. We give a corollary that settles Iizuka’s conjecture for the case n=1 and p>2.

Weight criteria for embedding of the weighted Sobolev–Lorentz spaces to the weighted Besov–Lorentz spaces built upon certain mixed norms and iterated rearrangement are investigated. This gives an improvement of some known Sobolev embedding. We achieve the result based on different norm inequalities for the weighted Besov–Lorentz spaces defined in some mixed norms.

We continue our investigation of the real space H of Hermitian matrices in $${M_n}(\mathbb{C})$$ with respect to norms on $${\mathbb{C}^n}$$. We complete the commutative case by showing that any proper real subspace of the real diagonal matrices on $${\mathbb{C}^n}$$ can appear as H. For the non-commutative case, we give a complete solution when n=3 and we provide various illustrative examples for n ≥ 4. We end with a short list of problems.

Let π : X → C be a fibration with integral fibers over a curve C and consider a polarization H on the surface X. Let E be a stable vector bundle of rank 2 on C. We prove that the pullback π*(E) is a H-stable bundle over X. This result allows us to relate the corresponding moduli spaces of stable bundles $${{\mathcal M}_C}(2,d)$$ and $${{\mathcal M}_{X,H}}(2,df,0)$$ through an injective morphism. We study the induced morphism at the level of Brill–Noether loci to construct examples of Brill–Noether loci on fibered surfaces. Results concerning the emptiness of Brill–Noether loci follow as a consequence of a generalization of Clifford’s Theorem for rank two bundles on surfaces.

An effective estimate for the measure of the set of real numbers for which the inequality |P(x)|<Q-w for $w > {3 \over 2}n + 1$ has a solution in integral polynomials P of degree n and of height H(P) at most $Q \in {\rm{\mathbb N}}$ is obtained.

The article focuses on the evaluation of convolution sums $${W_k}(n): = \mathop \sum \nolimits_{_{m < {n \over k}}} \sigma (m)\sigma (n - km)$$ involving the sum of divisor function $$\sigma (n)$$ for k =21, 33, and 35. In this article, our aim is to obtain certain Eisenstein series of level 21 and use them to evaluate the convolution sums for level 21. We also make use of the existing Eisenstein series identities for level 33 and 35 in evaluating the convolution sums for level 33 and 35. Most of the convolution sums were evaluated using the theory of modular forms, whereas we have devised a technique which is free from the theory of modular forms. As an application, we determine a formula for the number of representations of a positive integer n by the octonary quadratic form $$(x_1^2 + {x_1}{x_2} + ax_2^2 + x_3^2 + {x_3}{x_4} + ax_4^2) + b(x_5^2 + {x_5}{x_6} + ax_6^2 + x_7^2 + {x_7}{x_8} + ax_8^2)$$, for (a, b)=(1, 7), (1, 11), (2, 3), and (2, 5).

Let $\pi$ be a group satisfying the Farrell–Jones conjecture and assume that $B\pi$ is a 4-dimensional Poincaré duality space. We consider topological, closed, connected manifolds with fundamental group $\pi$ whose canonical map to $B\pi$ has degree 1, and show that two such manifolds are s-cobordant if and only if their equivariant intersection forms are isometric and they have the same Kirby–Siebenmann invariant. If $\pi$ is good in the sense of Freedman, it follows that two such manifolds are homeomorphic if and only if they are homotopy equivalent and have the same Kirby–Siebenmann invariant. This shows rigidity in many cases that lie between aspherical 4-manifolds, where rigidity is expected by Borel’s conjecture, and simply connected manifolds where rigidity is a consequence of Freedman’s classification results.

In this paper, we characterize surjective isometries on certain classes of noncommutative spaces associated with semi-finite von Neumann algebras: the Lorentz spaces $L^{w,1}$ , as well as the spaces $L^1+L^\infty$ and $L^1\cap L^\infty$ . The technique used in all three cases relies on characterizations of the extreme points of the unit balls of these spaces. Of particular interest is that the representations of isometries obtained in this paper are global representations.

Let $f\,{:}\,(\mathbb R^n,0)\to (\mathbb R,0)$ be an analytic function germ with non-isolated singularities and let $F\,{:}\, (\mathbb{R}^{1+n},0) \to (\mathbb{R},0)$ be a 1-parameter deformation of f. Let $ f_t ^{-1}(0) \cap B_\epsilon^n$ , $0 < \vert t \vert \ll \epsilon$ , be the “generalized” Milnor fiber of the deformation F. Under some conditions on F, we give a topological degree formula for the Euler characteristic of this fiber. This generalizes a result of Fukui.

We prove that the profinite completion of a profinite projective group is projective.

Following Ryan and Tornaría, we prove that moduli of congruences of Hecke eigenvalues, between Saito–Kurokawa lifts and non-lifts (certain Siegel modular forms of genus 2), occur (squared) in denominators of central spinor L-values (divided by twists) for the non-lifts. This is conditional on Böcherer’s conjecture and its analogues and is viewed in the context of recent work of Furusawa, Morimoto and others. It requires a congruence of Fourier coefficients, which follows from a uniqueness assumption or can be proved in examples. We explain these factors in denominators via a close examination of the Bloch–Kato conjecture.