We extend the extension by Ky Fan of the Krein–Milman theorem. The $\Phi$-extreme points of a $\Phi$-convex compact metrizable space are replaced by the $\Phi$-exposed points in the statement of Ky Fan theorem. Our main results are based on new variational principles which are of independent interest. Several applications will be given.

We give a mathematical exposition of the Page metric, and introduce an efficient coordinate system for it. We carefully examine the submanifolds of the underlying smooth manifold, and show that the Page metric does not have positive holomorphic bisectional curvature. We exhibit a holomorphic subsurface with flat normal bundle. We also give another proof of the fact that a compact complex surface together with an Einstein–Hermitian metric of positive orthogonal bisectional curvature is biholomorphically isometric to the complex projective plane with its Fubini–Study metric up to rescaling. This result relaxes the Kähler condition in Berger’s theorem, and the positivity condition on sectional curvature in a theorem proved by the second author.

Let $S$ be a subsemigroup of a second countable locally compact group $G$, such that $S^{-1}S=G$. We consider the $C^{*}$-algebra $C^{*}_{\delta }(S)$ generated by the operators of translation by all elements of $S$ in $L^{2}(S)$. We show that this algebra admits a comultiplication which turns it into a compact quantum semigroup. The same is proved for the von Neumann algebra $\operatorname{VN}(S)$ generated by $C^{*}_{\delta }(S)$.

We study the Donaldson–Thomas type invariants for the Calabi–Yau threefold Deligne–Mumford stacks under flops. A crepant birational morphism between two smooth Calabi–Yau threefold Deligne–Mumford stacks is called an orbifold flop if the flopping locus is the quotient of weighted projective lines by a cyclic group action. We prove that the Donaldson–Thomas invariants are preserved under orbifold flops.

We prove that every lens space contains a genus one homologically fibered knot, which is contrast to the fact that some lens spaces contain no genus one fibered knot. In the proof, the Chebotarev density theorem and binary quadratic forms in number theory play a key role. We also discuss the Alexander polynomial of homologically fibered knots.

In this article, we generalize to the case of measured quantum groupoids on a finite basis some important results concerning actions of locally compact quantum groups on C$^{*}$-algebras (Comm. Math. Phys. 235 (2003) 139–167). Let $\mathcal{G}$ be a measured quantum groupoid on a finite basis. We prove that if $\mathcal{G}$ is regular, then any weakly continuous action of $\mathcal{G}$ on a C$^{*}$-algebra is necessarily strongly continuous. Following (K-Theory 2 (1989) 683–721), we introduce and investigate a notion of $\mathcal{G}$-equivariant Hilbert C$^{*}$-modules. By applying the previous results and a version of the Takesaki–Takai duality theorem obtained in (Bull. Soc. Math. France 145 (2017) 711–802) for actions of $\mathcal{G}$, we obtain a canonical equivariant Morita equivalence between a given $\mathcal{G}$-C$^{*}$-algebra $A$ and the double crossed product $(A\rtimes\mathcal{G})\rtimes\widehat{\mathcal{G}}$.

Let $k$ be a field of characteristic zero. Let $f\in k[x_{0},y_{0}]$ be an irreducible polynomial. In this article, we study the space of polynomial partial differential equations of order one in the plane, which admit $f$ as a solution. We provide algebraic characterizations of the associated graded $k[x_{0},y_{0}]$-module (by degree) of this space. In particular, we show that it defines the general component of the tangent space of the curve $\{f=0\}$ and connect it to the $V$-filtration of the logarithmic differential operators of the plane along $\{f=0\}$.

For an integer $k\ge 2$, a tuple of $k$ positive integers $(M_{i})_{i=1}^{k}$ is called an amicable $k$-tuple if the equation \begin{equation*}\sigma (M_{1})=\cdots =\sigma (M_{k})=M_{1}+\cdots +M_{k}\end{equation*} holds. This is a generalization of amicable pairs. An amicable pair is a pair of distinct positive integers each of which is the sum of the proper divisors of the other. Gmelin (Über vollkommene und befreundete Zahlen (1917) Heidelberg University) conjectured that there is no relatively prime amicable pairs and Artjuhov (Acta Arith. 27 (1975) 281–291) and Borho (Math. Ann. 209 (1974) 183–193) proved that for any fixed positive integer $K$, there are only finitely many relatively prime amicable pairs $(M,N)$ with $\omega (MN)=K$. Recently, Pollack (Mosc. J. Comb. Number Theory 5 (2015), 36–51) obtained an upper bound \begin{equation*}MN<(2K)^{2^{K^{2}}}\end{equation*} for such amicable pairs. In this paper, we improve this upper bound to \begin{equation*}MN<\frac{\pi^{2}}{6}2^{4^{K}-2\cdot 2^{K}}\end{equation*} and generalize this bound to some class of general amicable tuples.

Let $(K,\nu)$ be a valued field and $K(x)$ a simple purely transcendental extension of $K$. In the nineteen thirties, in order to study the possible extensions of $\nu $ to $K(x)$, S. Mac Lane considered the special case when $\nu $ is discrete of rank $1$, and introduced the notion of key polynomials. M. Vaquié extended this definition to the case of arbitrary valuations.

In this paper we give a new definition of key polynomials (which we call abstract key polynomials) and study the relationship between them and key polynomials of Mac Lane–Vaquié.

The main result of this paper is that the toric degenerations of flag and Schubert varieties associated to string polytopes and certain Bott–Samelson resolutions of flag and Schubert varieties fit into a commutative diagram which gives a resolution of singularities of singular toric varieties corresponding to string polytopes. Our main tool is a result of Anderson which shows that the toric degenerations arising from Newton–Okounkov bodies are functorial in an appropriate sense. We also use results of Fujita which show that Newton–Okounkov bodies of Bott–Samelson varieties with respect to a certain valuation $\nu_{\mathrm{max}}$ coincide with generalized string polytopes, as well as previous results by the authors which explicitly describe the Newton–Okounkov bodies of Bott–Samelson varieties with respect to a different valuation $\nu_{\mathrm{min}}$ in terms of Grossberg–Karshon twisted cubes. A key step in our argument is that, under a technical condition, these Newton–Okounkov bodies coincide.

In this note, we prove the profile decomposition for hyperbolic Schrödinger (or mixed signature) equations on $\mathbb{R}^{2}$ in two cases, one mass-supercritical and one mass-critical. First, as a warm up, we show that the profile decomposition works for the ${\dot{H}}^{\frac{1}{2}}$ critical problem. Then, we give the derivation of the profile decomposition in the mass-critical case based on an estimate of Rogers-Vargas (J. Functional Anal. 241(2) (2006), 212–231).

We analyze a canonical construction of group-invariant CR Mappings between hyperquadrics due to D’Angelo. Given source hyperquadric of $Q(1,1)$, we determine the signature of the target hyperquadric for all finite subgroups of $SU(1,1)$. We also extend combinatorial results proven by Loehr, Warrington, and Wilf on determinants of sparse circulant determinants. We apply these results to study CR mappings invariant under finite subgroups of $U(1,1)$.

We define two families of expansions of $(\mathbb{Z},+)$ by unary predicates, and prove that their theories are superstable of $U$-rank $\omega $. The first family consists of expansions $(\mathbb{Z},+,A)$, where $A$ is an infinite subset of a finitely generated multiplicative submonoid of $\mathbb{Z}^{+}$. Using this result, we also prove stability for the expansion of $(\mathbb{Z},+)$ by all unary predicates of the form $\{q^{n}:n\in \mathbb{N}\}$ for some $q\in \mathbb{N}_{\geq 2}$. The second family consists of sets $A\subseteq \mathbb{N}$ which grow asymptotically close to a $\mathbb{Q}$-linearly independent increasing sequence $(\lambda_{n})_{n=0}^{\infty }\subseteq\mathbb{R}^{+}$ such that $\{\frac{\lambda_{n}}{\lambda_{m}}:m\leq n\}$ is closed and discrete.

Recently, Du, Guth and Li showed that the Schrödinger operator $e^{it\Delta }$ satisfies $\lim_{t\rightarrow 0}e^{it\Delta }f=f$ almost everywhere for all $f\in H^{s}(\mathbb{R}^{2})$, provided that $s>1/3$. In this paper, we discuss the rate of convergence on $e^{it\Delta }(f)$ by assuming more regularity on $f$. At $n=2$, our result can be viewed as an application of the Du–Guth–Li theorem. We also address the same issue on the cases $n=1$ and $n>2$.

For each ordinal $\xi$ and each $1\leqslant q<\infty$, we define the notion of $\xi$-$q$-summable Szlenk index. When $\xi=0$ and $q=1$, this recovers the usual notion of summable Szlenk index. We define for an arbitrary weak$^{*}$-compact set a transfinite, asymptotic analogue $\alpha_{\xi,p}$ of the martingale type norm of an operator. We prove that this quantity is determined by norming sets and determines $\xi$-Szlenk power type and $\xi$-$q$-summability of Szlenk index. This fact allows us to prove that the behavior of operators under the $\alpha_{\xi,p}$ seminorms passes in the strongest way to injective tensor products of Banach spaces. Furthermore, we combine this fact with a result of Schlumprecht to prove that a separable Banach space with good behavior with respect to the $\alpha_{\xi,p}$ seminorm can be embedded into a Banach space with a shrinking basis and the same behavior under $\alpha_{\xi,p}$, and in particular it can be embedded into a Banach space with a shrinking basis and the same $\xi$-Szlenk power type. Finally, we completely elucidate the behavior of the $\alpha_{\xi,p}$ seminorms under $\ell_{r}$ direct sums. This allows us to give an alternative proof of a result of Brooker regarding Szlenk indices of $\ell_{p}$ and $c_{0}$ direct sums of operators.

We derive explicit upper bounds for various counting functions for primes in arithmetic progressions. By way of example, if $q$ and $a$ are integers with $\mathop{\mathrm{gcd}}\nolimits (a,q)=1$ and $3\leq q\leq10^{5}$, and $\theta(x;q,a)$ denotes the sum of the logarithms of the primes $p\equiv a\ (\operatorname{mod}q)$ with $p\leq x$, we show that

\[\vert \theta(x;q,a)-{x}/{\varphi(q)}\vert <\frac{1}{160}\frac{x}{\log x}\] for all $x\geq8\cdot10^{9}$, with significantly sharper constants obtained for individual moduli $q$. We establish inequalities of the same shape for the other standard prime-counting functions $\pi(x;q,a)$ and $\psi(x;q,a)$, as well as inequalities for the $n$th prime congruent to $a\ (\operatorname{mod}q)$ when $q\le1200$. For moduli $q>10^{5}$, we find even stronger explicit inequalities, but only for much larger values of $x$. Along the way, we also derive an improved explicit lower bound for $L(1,\chi)$ for quadratic characters $\chi$, and an improved explicit upper bound for exceptional zeros.