Decomposing knots and links into tangles is a useful technique for understanding their properties. The notion of prime tangles was introduced by Kirby and Lickorish; Lickorish proved that by summing prime tangles one obtains a prime link. In a similar spirit, summing two prime alternating tangles will produce a prime alternating link if summed correctly with respect to the alternating property. Given a prime alternating link, we seek to understand whether it can be decomposed into two prime tangles, each of which is alternating. We refine results of Menasco and Thistlethwaite to show that if such a decomposition exists, either it is visible in an alternating link diagram or the link is of a particular form, which we call a pseudo-Montesinos link.

The hitting and mixing times are two often-studied quantities associated with Markov chains. Yuval Peres, Perla Sousi and Roberto Oliveira showed that the mixing times and “worst-case” hitting times of reversible Markov chains on finite state spaces are “equivalent”—that is, equal up to some universal multiplicative constant. We have extended this strong connection between mixing and hitting times to Markov chains satisfying the strong Feller property in an earlier work. In the present paper, we further extend the results to include Metropolis–Hastings chains, the popular Gibbs sampler (from statistics), and Glauber dynamics (from statistical physics), which make “one-dimensional” updates and thus do not satisfy the strong Feller property. We also apply this result to obtain decomposition bounds for such Markov chains. Our main tools come from nonstandard analysis.

In a minimal flow, the hitting time is the exponent of the power law, as r goes to zero, for the time needed by orbits to become r-dense. We show that on the so-called Ornithorynque origami, the hitting time of the flow in an irrational slope equals the diophantine type of the slope. We give a general criterion for such equality. In general, for genus at least two, hitting time is strictly bigger than diophantine type.

Let R be a commutative Noetherian ring. Denote by $\mathrm{mod}R$ the category of finitely generated R-modules. In this paper, we study dominant resolving subcategories of $\mathrm{mod}R$. We introduce a new $\mathbb{N}$-valued function on $\mathrm{Spec}R$ which we call a moderate function. Under an acceptable assumption, we construct explicit bijections between the set of dominant resolving subcategories of $\mathrm{mod}R$ and the set of moderate functions on $\mathrm{Spec}R$.

The v-function is a key ingredient in the wild McKay correspondence. In this paper, we give a formula to compute it in terms of valuations of Witt vectors, when the given group is a cyclic group of prime power order. We apply it to study singularities of a quotient variety by a cyclic group of prime square order. We give a criterion whether the stringy motive of the quotient variety converges or not. Furthermore, if the given representation is indecomposable, then we also give a simple criterion for the quotient variety being terminal, canonical, log canonical, and not log canonical. With this criterion, we obtain more examples of quotient varieties which are Kawamata log terminal (klt) but not Cohen–Macaulay.

We explicitly construct a strongly aperiodic subshift of finite type for the discrete Heisenberg group. Our example builds on the classical aperiodic tilings of the plane due to Raphael Robinson. Extending those tilings to the Heisenberg group by exploiting the group’s structure and posing additional local rules to prune out remaining periodic behavior, we maintain a rich projective subdynamics on $\mathbb{Z}2$ cosets. In addition, the obtained subshift factors onto a strongly aperiodic, minimal sofic shift via a map that is invertible on a dense set of configurations.

In this paper, we calculate the relative multifractal Hausdorff and packing dimensions of measures in a probability space. Also, we obtain the analogue of Frostman’s lemma in a probability space for a relative multifractal Hausdorff measure. In the same way, there is a valid result for the relative multifractal packing pre-measure. Furthermore, we obtain the representations of the functions b and B by means of the analogue of Frostman’s lemma, and we provide a technique for showing that E is a $(q,\mathrm{\mu })$-fractal with respect to ν. In addition, we suggest new proofs of theorems on the relative multifractal formalism in a probability space. They yield results even at a point q for which the multifractal functions $b(q)$ and $B(q)$ differ.

In this paper, we are concerned about the local-in-time well-posedness of the Vlasov–Poisson–Fokker–Planck equation in $E_{2,1}^s({\mathbb{R}}^{2n})$ which is a hybrid modulation Lebesgue space and related to the Gevery class only with respect to x variable. The difficulty lies in the estimates of the electronic term ${\nabla }_x\mathit{\varphi }$. To handle this, we establish a product formula and $L^2$-$L^q$ estimate.

The purpose of this paper is to establish convergence of random walks on the moduli space of abelian differentials on compact Riemann surfaces in two different modes: convergence of the n-step distributions from almost every starting point in an affine invariant submanifold toward the associated affine invariant measure, and almost sure pathwise equidistribution toward the affine invariant measure on the ${\mathrm{SL}}_2(\mathbb{R})$-orbit closure of an arbitrary starting point. These are analogues to previous results for random walks on homogeneous spaces.