This paper provides a convenient and practical method to compute the homology and intersection pairing of a branched double cover of the 4-ball.

To projections of links in the 3-ball, and to projections of surfaces in the 4-ball into the boundary sphere, we associate a sequence of homology groups, called the disoriented homology. We show that the disoriented homology is isomorphic to the homology of the double branched cover of the link or surface. We define a pairing on the first disoriented homology group of a surface and show that this is equal to the intersection pairing of the branched cover. These results generalize work of Gordon and Litherland, for embedded surfaces in the 3-sphere, to arbitrary surfaces in the 4-ball. We also give a generalization of the signature formula of Gordon–Litherland to the general setting.

Our results are underpinned by a theorem describing a handle decomposition of the branched double cover of a codimension-2 submanifold in the n-ball, which generalizes previous results of Akbulut–Kirby and others.

Using probabilistic tools, we prove that any weak* continuous semigroup $(T_t)_{t \geqslant 0}$ of self-adjoint unital completely positive measurable Schur multipliers acting on the space $\mathrm {B}({\mathrm {L}}^2(X))$ of bounded operators on the Hilbert space ${\mathrm {L}}^2(X)$, where X is a suitable measure space, can be dilated by a weak* continuous group of Markov $*$-automorphisms on a bigger von Neumann algebra. We also construct a Markov dilation of these semigroups. Our results imply the boundedness of the McIntosh’s ${\mathrm {H}}^\infty $ functional calculus of the generators of these semigroups on the associated Schatten spaces and some interpolation results connected to ${\mathrm {BMO}}$-spaces. We also give an answer to a question of Steen, Todorov, and Turowska on completely positive continuous Schur multipliers.

We present a unified theory for the almost periodicity of functions with values in an arbitrary Banach space, measures and distributions via almost periodic elements for the action of a locally compact abelian group on a uniform topological space. We discuss the relation between Bohr- and Bochner-type almost periodicity, and similar conditions, and how the equivalence among such conditions relates to properties of the group action and the uniformity. We complete the paper by demonstrating how various examples considered earlier all fit in our framework.

We provide a correspondence between one-sided coideal subrings and one-sided ideal two-sided coideals in an arbitrary bialgebroid. We prove that, under some expected additional conditions, this correspondence becomes bijective for Hopf algebroids. As an application, we investigate normal Hopf ideals in commutative Hopf algebroids (affine groupoid schemes) in connection with the study of normal affine subgroupoids.

We provide a criterion for a coherent sheaf to be an Ulrich sheaf in terms of a certain bilinear form on its global sections. When working over the real numbers, we call it a positive Ulrich sheaf if this bilinear form is symmetric or Hermitian and positive-definite. In that case, our result provides a common theoretical framework for several results in real algebraic geometry concerning the existence of algebraic certificates for certain geometric properties. For instance, it implies Hilbert’s theorem on nonnegative ternary quartics, via the geometry of del Pezzo surfaces, and the solution of the Lax conjecture on plane hyperbolic curves due to Helton and Vinnikov.

We consider direct and inverse Jacobi transforms with measures

$$\begin{align*}d\mu(t)=2^{2\rho}(\operatorname{sinh} t)^{2\alpha+1}(\operatorname{cosh} t)^{2\beta+1}\,dt\end{align*}$$

$$\begin{align*}d\sigma(\lambda)=(2\pi)^{-1}\Bigl|\frac{2^{\rho-i\lambda}\Gamma(\alpha+1)\Gamma(i\lambda)} {\Gamma((\rho+i\lambda)/2)\Gamma((\rho+i\lambda)/2-\beta)}\Bigr|^{-2}\,d\lambda,\end{align*}$$

$$\begin{align*}\inf\Lambda((-1)^{m-1}f), \quad m\in \mathbb{N}, \end{align*}$$

$\Lambda (f)=\sup \,\{\lambda>0\colon f(\lambda )>0\}$

$m\ge 2$

$\int _{0}^{\infty }\lambda ^{2k}f(\lambda )\,d\sigma (\lambda )=0$

$k=0,\dots ,m-2$

We prove that admissible functions for this problem are positive-definite with respect to the inverse Jacobi transform. The solution of Logan’s problem was known only when $\alpha =\beta =-1/2$. We find a unique (up to multiplication by a positive constant) extremizer $f_m$. The corresponding Logan problem for the Fourier transform on the hyperboloid $\mathbb {H}^{d}$ is also solved. Using the properties of the extremizer $f_m$ allows us to give an upper estimate of the length of a minimal interval containing not less than n zeros of positive definite functions. Finally, we show that the Jacobi functions form the Chebyshev systems.

We first provide a detailed proof of Kato’s classification theorem of log p-divisible groups over a Noetherian Henselian local ring. Exploring Kato’s idea further, we then define the notion of a standard extension of a classical finite étale group scheme (resp. classical étale p-divisible group) by a classical finite flat group scheme (resp. classical p-divisible group) in the category of finite Kummer flat group log schemes (resp. log p-divisible groups), with respect to a given chart on the base. These results are then used to prove that log p-divisible groups are formally log smooth. We then study the finite Kummer flat group log schemes $T_n(\mathbf {M}):=H^{-1}(\mathbf {M}\otimes _{{\mathbb Z}}^L{\mathbb Z}/n{\mathbb Z})$ (resp. the log p-divisible group $\mathbf {M}[p^{\infty }]$) of a log 1-motive $\mathbf {M}$ over an fs log scheme and show that they are étale locally standard extensions. Lastly, we give a proof of the Serre–Tate theorem for log abelian varieties with constant degeneration.

For a closed, connected, and oriented 3-manifold with a non-singular flow, we construct its virtual knot diagram via a “flow-spine” of the manifold. Then, we introduce a coloring invariant of 3-manifolds through their virtual knot diagrams, and classify some 3-manifolds by using the invariant.

In this work, we obtain the pointwise almost everywhere convergence for two families of multilinear operators: (a) the doubly truncated homogeneous singular integral operators associated with $L^q$ functions on the sphere and (b) lacunary multiplier operators of limited smoothness. The a.e. convergence is deduced from the $L^2\times \cdots \times L^2\to L^{2/m}$ boundedness of the associated maximal multilinear operators.

Let G be a compact connected Lie group, and let $\operatorname {Hom}({\mathbb {Z}}^m,G)$ be the space of pairwise commuting m-tuples in G. We study the problem of which primes $p \operatorname {Hom}({\mathbb {Z}}^m,G)_1$, the connected component of $\operatorname {Hom}({\mathbb {Z}}^m,G)$ containing the element $(1,\ldots ,1)$, has p-torsion in homology. We will prove that $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$ for $m\ge 2$ has p-torsion in homology if and only if p divides the order of the Weyl group of G for $G=SU(n)$ and some exceptional groups. We will also compute the top homology of $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$ and show that $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$ always has 2-torsion in homology whenever G is simply-connected and simple. Our computation is based on a new homotopy decomposition of $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$, which is of independent interest and enables us to connect torsion in homology to the combinatorics of the Weyl group.

We formulate a generalization of Riesz-type criteria in the setting of L-functions belonging to the Selberg class. We obtain a criterion which is sufficient for the grand Riemann hypothesis (GRH) for L-functions satisfying axioms of the Selberg class without imposing the Ramanujan hypothesis on their coefficients. We also construct a subclass of the Selberg class and prove a necessary criterion for GRH for L-functions in this subclass. Identities of Ramanujan–Hardy–Littlewood type are also established in this setting, specific cases of which yield new transformation formulas involving special values of the Meijer G-function of the type ${G^{n , 0}_{0 , n}}$.