We establish Ohno-type identities for multiple harmonic ($q$-)sums which generalize Hoffman's identity and Bradley's identity. Our result leads to a new proof of the Ohno-type relation for $\mathcal{A}$-finite multiple zeta values recently proved by Hirose, Imatomi, Murahara and Saito. As a further application, we give certain sum formulas for $\mathcal{A}_2$- or $\mathcal{A}_3$-finite multiple zeta values.

We investigate the simple resonances of a 2 by 2 matrix of $n$-dimensional semiclassical Schrödinger operators that interact through a first order differential operator. We assume that one of the two (analytic) potentials admits a well with non empty interior, while the other one is non trapping and creates a barrier between the well and infinity. Under a condition on the resonant state inside the well, we find an optimal lower bound on the width of the resonance. The method of proof relies on Carleman estimates, microlocal propagation of the microsupport, and a refined study of a non involutive double characteristic problem in the framework of Sjöstrand's analytic microlocal theory.

We build a bijection between the set ${\rm s}\tau\mbox{-tilt}\hspace{.01in} \Lambda$ of isomorphism classes of basic support $\tau$-tilting modules over the Auslander algebra $\Lambda$ of $K[x]/(x^n)$ and the symmetric group $\mathfrak{S}_{n+1}$, which is an anti-isomorphism of partially ordered sets with respect to the generation order on ${\rm s}\tau\mbox{-tilt}\hspace{.01in} \Lambda$ and the left order on $\mathfrak{S}_{n+1}$. This restricts to the bijection between the set ${\rm tilt}\hspace{.01in} \Lambda$ of isomorphism classes of basic tilting $\Lambda$-modules and the symmetric group $\mathfrak{S}_n$ due to Brüstle, Hille, Ringel and Röhrle. Regarding the preprojective algebra $\Gamma$ of Dynkin type $A_n$ as a factor algebra of $\Lambda$, we show that the tensor functor $-\otimes_{\Lambda} \Gamma$ induces a bijection between ${\rm s}\tau\mbox{-tilt}\hspace{.01in} \Lambda\to {\rm s}\tau\mbox{-tilt}\hspace{.01in} \Gamma$. This recover Mizuno's anti-isomorphism $\mathfrak{S}_{n+1} \to {\rm s}\tau\mbox{-tilt}\hspace{.01in} \Gamma$ of posets for type $A_n$.

In the present paper we construct two new explicit complex bordisms between any two projective bundles over $\mathbb{C} P^1$ of the same complex dimension, including the Milnor hypersurface $H_{1,n}$ and $\mathbb{C} P^1 \times \mathbb{C} P^{n-1}$. These constructions reduce the bordism problem to the null-bordism of some projective bundle over $\mathbb{C} P^1$ with the non-standard stably complex structure.

Given an arbitrary non-zero simplicial cycle and a generic vector coloring of its vertices, there is a way to produce a graded Poincare duality algebra associated with these data. The procedure relies on the theory of volume polynomials and multi-fans. The algebras constructed this way include many important examples: cohomology algebras of toric varieties and quasitoric manifolds, and Gorenstein algebras of triangulated homology manifolds, introduced and studied by Novik and Swartz. In all these examples the dimensions of graded components of such duality algebras do not depend on the vector coloring. It was conjectured that the same holds for any simplicial cycle. We disprove this conjecture by showing that the colors of singular points of the cycle may affect the dimensions. However, the colors of nonsingular points are irrelevant. By using bistellar moves we show that the number of distinct dimension vectors arising on a given 3-dimensional pseudomanifold with isolated singularities is a topological invariant. This invariant is trivial on manifolds, but nontrivial on general pseudomanifolds.

We characterize strong type and weak type inequalities with two weights for positive operators on filtered measure spaces. These estimates are probabilistic analogues of two-weight inequalities for positive operators associated to the dyadic cubes in $\mathbb R^n$ due to Lacey, Sawyer and Uriarte-Tuero [30]. Several mixed bounds for the Doob maximal operator on filtered measure spaces are also obtained. In fact, Hytönen–Pérez type and Lerner–Moen type norm estimates for Doob maximal operator are established. Our approaches are mainly based on the construction of principal sets.

In this paper, we study the existence of finite-to-one zero-dimensional covers of dynamical systems. Kulesza showed that any homeomorphism $f:X \to X$ on an $n$-dimensional compactum $X$ with zero-dimensional set $P(f)$ of periodic points can be covered by a homeomorphism on a zero-dimensional compactum via an at most $(n + 1)^n$-to-one map. Moreover, Ikegami, Kato and Ueda showed that in the theorem of Kulesza, the condition of at most $(n + 1)^n$-to-one map can be strengthened to the condition of at most $2^n$-to-one map. In this paper, we will show that the theorem is also true for more general maps except for homeomorphisms. In fact we prove that the theorem is true for a class of maps containing two-sided zero-dimensional maps. For the special case, we give a theorem of symbolic extensions of positively expansive maps. Finally, we study some dynamical zero-dimensional decomposition theorems of spaces related to such maps.

We reformulate Milgram's model of a double loop suspension in terms of a preoperad of posets, each stage of which is the poset of all ordered partitions of a finite set. Using this model, we give a combinatorial model for the evaluation map and use it to study the Cohen representation for the group of homotopy classes of maps between double loop suspensions. Demonstrating the general theory, we recover Wu's shuffle relations and further provide a type of secondary relations in Cohen groups by using Toda brackets. In particular, we prove certain maps are null-homotopic by combining our relations and the classical James–Hopf invariants.

For knots in $S^3$, it is well-known that the Alexander polynomial of a ribbon knot factorizes as $f(t)f(t^{-1})$ for some polynomial $f(t)$. By contrast, the Alexander polynomial of a ribbon 2-knot in $S^4$ is not even symmetric in general. Via an alternative notion of ribbon 2-knots, we give a topological condition on a 2-knot that implies the factorization of the Alexander polynomial.

Relying on the main results of [GT], we classify all unitary $t$-groups for $t \geq 2$ in any dimension $d \geq 2$. We also show that there is essentially a unique unitary 4-group, which is also a unitary 5-group, but not a unitary $t$-group for any $t \geq 6$.

The aim of this paper is to study two local moves $V(n)$ and $V^{n}$ on welded links for a positive integer $n$, which are generalizations of the crossing virtualization. We show that the $V(n)$-move is an unknotting operation on welded knots for any $n$, and give a classification of welded links up to $V(n)$-moves. On the other hand, we give a necessary condition for two welded links to be equivalent up to $V^{n}$-moves. This leads us to show that the $V^{n}$-move is not an unknotting operation on welded knots except for $n = 1$. We also discuss relations among $V^{n}$-moves, associated core groups and the multiplexing of crossings.

We introduce the sphere fibration for real map germs with radial discriminant and we address the problem of its equivalence with the Milnor–Hamm tube fibration. Under natural conditions, we prove the existence of open book structures with singularities and solve the equivalence problem.

The Cuntz–Toeplitz algebra $E_{n+1}$ for $n \geq 1 $ is the universal C*-algebra generated by $n + 1$ isometries with mutually orthogonal ranges. In this paper, we determine the homotopy groups of the automorphism group of $E_{n+1}$.

Let ${\mathcal R}_\mathbb{K}[H]$ be the Hibi ring over a field $\mathbb{K}$ on a finite distributive lattice $H$, $P$ the set of join-irreducible elements of $H$ and $\omega$ the canonical ideal of ${\mathcal R}_\mathbb{K}$[H]. We show the powers $\omega^{(n)}$ of $\omega$ in the group of divisors $\mathrm{Div}({\mathcal R}_\mathbb{K}[H])$ is identical with the ordinary powers of $\omega$, describe the $\mathbb{K}$-vector space basis of $\omega^{(n)}$ for $n \in \mathbb{Z}$. Further, we show that the fiber cones $\bigoplus_{n \geq 0} \omega^n/\mathfrak{m} \omega^n$ and $\bigoplus_{n \geq 0} (\omega^{(-1)})^n/\mathfrak{m} (\omega^{(-1)})^n$ of $\omega$ and $\omega^{(-1)}$ are sum of the Ehrhart rings, defined by sequences of elements of $P$ with a certain condition, which are polytopal complex version of Stanley–Reisner rings. Moreover, we show that the analytic spread of $\omega$ and $\omega^{(-1)}$ are maximum of the dimensions of these Ehrhart rings. Using these facts, we show that the question of Page about Frobenius complexity is affirmative: $\lim_{p \to \infty} \mathrm{cx}_F ({\mathcal R}_\mathbb{K}[H]) = \dim(\bigoplus_{n \geq 0} \omega^{(-n)}/\mathfrak{m} \omega^{(-n)}) - 1$, where $p$ is the characteristic of the field $\mathbb{K}$.