We study p-Wasserstein spaces over the branching spaces $\mathbb {R}^2$ and $[-1,1]^2$ equipped with the maximum norm metric. We show that these spaces are isometrically rigid for all $p\geq 1,$ meaning that all isometries of these spaces are induced by isometries of the underlying space via the push-forward operation. This is in contrast to the case of the Euclidean metric since with that distance the $2$-Wasserstein space over $\mathbb {R}^2$ is not rigid. Also, we highlight that the $1$-Wasserstein space is not rigid over the closed interval $[-1,1]$, while according to our result, its two-dimensional analog, the closed unit ball $[-1,1]^2$ with the more complicated geodesic structure is rigid.

A real number is simply normal to base b if its base-b expansion has each digit appearing with average frequency tending to $1/b$. In this article, we discover a relation between the frequency at which the digit $1$ appears in the binary expansion of $2^{p/q}$ and a mean value of the Riemann zeta function on vertical arithmetic progressions. In particular, we show that

$$\begin{align*}\lim_{l\to \infty} \frac{1}{l}\sum_{0<|n|\leq 2^l } \zeta\left(\frac{2 n\pi i}{\log 2}\right) \frac{e^{2n\pi i p/q} }{n} =0 \end{align*}$$

$2^{p/q}$

$2$

Let $\Omega $ be a compact subset of $\mathbb {C}$ and let A be a unital simple, separable $C^*$-algebra with stable rank one, real rank zero, and strict comparison. We show that, given a Cu-morphism ${\alpha :\mathrm { Cu}(C(\Omega ))\to \mathrm {Cu}(A)}$ with , there exists a homomorphism $\phi : C(\Omega )\to A$ such that $\mathrm {Cu}(\phi )=\alpha $. Moreover, if $K_1(A)$ is trivial, then $\phi $ is unique up to approximate unitary equivalence. We also give classification results for maps from a large class of $C^*$-algebras to A in terms of the Cuntz semigroup.

We prove a conjecture of Lê and Sikora by providing a comparison between various existing $SL_n$ skein theories. While doing so, we show that the full subcategory of the spider category, $\mathcal {S}p(SL_n)$, defined by Cautis–Kamnitzer–Morrison, whose objects are monoidally generated by the standard representation and its dual, is equivalent as a spherical braided category to Sikora’s quotient category. This also answers a question from Morrison’s Ph.D. thesis. Finally, we show that the skein modules associated with the CKM and Sikora’s webs are isomorphic.

As is well-known, the conformal class of a surface M with boundary is determined by its Diriclet-to-Neumann (DN) map $\Lambda $. We propose an algorithm for determination of the b-period matrix $\mathbb {B}$ of the (Schottky) double of M via $\Lambda $.

Let F be a non-archimedean local field of characteristic not equal to 2. In this article, we prove the local converse theorem for quasi-split $\mathrm {O}_{2n}(F)$ and $\mathrm {SO}_{2n}(F)$, via the description of the local theta correspondence between $\mathrm {O}_{2n}(F)$ and $\mathrm {Sp}_{2n}(F)$. More precisely, as a main step, we explicitly describe the precise behavior of the $\gamma $-factors under the correspondence. Furthermore, we apply our results to prove the weak rigidity theorems for irreducible generic cuspidal automorphic representations of $\mathrm {O}_{2n}(\mathbb {A})$ and $\mathrm {SO}_{2n}(\mathbb {A})$, respectively, where $\mathbb {A}$ is a ring of adele of a global number field L.

This article focuses on two kinds of generalized special Lagrangian type equations. We investigate the Dirichlet problem for these equations with supercritical phase and critical phase in $\mathbb {R}^n$, deriving the a priori estimates and establishing the existence under the assumption of a subsolution. Furthermore, we also consider the corresponding special Lagrangian curvature type equations with supercritical phase and critical phase.

We investigate the notion of tracial ${\mathcal {Z}}$-stability beyond unital $\mathrm {C}^*$-algebras, and we prove that this notion is equivalent to ${\mathcal {Z}}$-stability in the class of separable simple nuclear $\mathrm {C}^*$-algebras.