The axiom of real Blackwell determinacy

The theory of infinite games with slightly imperfect information has been developed for games with finitely and countably many moves. In this paper, we shift the discussion to games with uncountably many possible moves, introducing the axiom of real Blackwell determinacy ${\mathsf{Bl-AD}_\mathbb{R}}$ (as an analogue of the axiom of real determinacy ${\mathsf{AD}_\mathbb{R}}$ ). We prove that the consistency strength of ${\mathsf{Bl-AD}_\mathbb{R}}$ is strictly greater than that of AD.     

Subgame perfect equilibria in stopping games

Stopping games (without simultaneous stopping) are multi-player sequential games in which at every stage one of the players is chosen according to a stochastic process, and that player decides whether to continue the interaction or to stop it, whereby the terminal payoff vector is obtained by another stochastic process. We prove that if the payoff process is integrable, a $\delta $ -approximate subgame perfect ${\epsilon }$ -equilibrium exists for every $\delta ,\epsilon >0$ ; that is, there exists a strategy profile that is an ${\epsilon }$ -equilibrium in all subgames, except possibly in a set of subgames that occurs with probability at most $\delta $ (even after deviation by some of the players).     

First-order algorithm with {\mathcal{O}({\rm ln}(1{/}\epsilon))} convergence for {\epsilon}-equilibrium in two-person zero-sum games

We propose an iterated version of Nesterov’s first-order smoothing method for the two-person zero-sum game equilibrium problem $$\min_{x \in Q_1}\max_{y \in Q_2} {x}^{\rm T}{Ay} = \max_{y \in Q_2} \min_{x \in Q_1} {x}^{\rm T}{Ay}.$$ This formulation applies to matrix games as well as sequential games. Our new algorithmic scheme computes an ${\epsilon}$ -equilibrium to this min-max problem in ${\mathcal {O}\left(\frac{\|A\|}{\delta(A)} \, {\rm ln}(1{/}\epsilon)\right)}$ first-order iterations, where δ(A) is a certain condition measure of the matrix A. This improves upon the previous first-order methods which required ${\mathcal {O}(1{/}\epsilon)}$ iterations, and it matches the iteration complexity bound of interior-point methods in terms of the algorithm’s dependence on ${\epsilon}$ . Unlike interior-point methods that are inapplicable to large games due to their memory requirements, our algorithm retains the small memory requirements of prior first-order methods. Our scheme supplements Nesterov’s method with an outer loop that lowers the target ${\epsilon}$ between iterations (this target affects the amount of smoothing in the inner loop). Computational experiments both in matrix games and sequential games show that a significant speed improvement is obtained in practice as well, and the relative speed improvement increases with the desired accuracy (as suggested by the complexity bounds).     

Smooth Stable Planes

This paper deals with smooth stable planes which generalize the notion of differentiable (affine or projective) planes [7]. It is intended to be the first one of a series of papers on smooth incidence geometry based on the Habilitationsschrift of the author. It contains the basic definitions and results which are needed to build up a foundation for a systematic study of smooth planes. We define smooth stable planes, and we prove that point rows and line pencils are closed submanifolds of the point set and line set, respectively (Theorem (1.6)). Moreover, the flag space is a closed submanifold of the product manifold $P\times {\cal L}$ (Theorem (1.14)), and the smooth structure on the set P of points and on the set ${\cal L}$ of lines is uniquely determined by the smooth structure of one single line pencil. In the second section it is shown that for any point p \te P the tangent space T_pP carries the structure of a locally compact affine translation plane ${\cal A}_p$ , see Theorem (2.5). Dually, we prove in Section 3 that for any line $L \in {\cal L}$ the tangent space ${\rm T}_L{\cal L}$ together with the set ${\cal \rm S}_L=\lbrace {\rm T}_{L}{\cal L}_p\mid p \in L\rbrace$ gives rise to some shear plane. It turned out that the translation planes ${\cal A}_p$ are one of the most important tools in the investigation of smooth incidence geometries. The linearization theorems (3.9), (3.11), and (4.4) can be viewed as the main results of this paper. In the closing section we investigate some homogeneity properties of smooth projective planes.     

Homotopy Groups of Spheres and Lipschitz Homotopy Groups of Heisenberg Groups

We provide a sufficient condition for the nontriviality of the Lipschitz homotopy group of the Heisenberg group, ${\pi_m^{\rm Lip}(\mathbb{H}_n)}$ , in terms of properties of the classical homotopy group of the sphere, ${\pi_m(\mathbb{S}^n)}$ . As an application we provide a new simplified proof of the fact that ${\pi_n^{\rm Lip}(\mathbb{H}_n)\neq \{0\}, n=1,2,\ldots}$ , and we prove a new result that ${\pi_{4n-1}^{\rm Lip}(\mathbb{H}_{2n})\neq \{0\}}$ for n = 1,2,… The last result is based on a new generalization of the Hopf invariant. We also prove that Lipschitz mappings are not dense in the Sobolev space ${W^{1,p}(\mathcal{M},\mathbb{H}_{2n})}$ when ${\dim \mathcal{M} \geq 4n}$ and 4n?1 ≤  p < 4n.     

Rational preimages in families of dynamical systems

Let ${\phi}$ be a rational function of degree at least two defined over a number field k. Let ${a \in \mathbb{P}^1(k)}$ and let K be a number field containing k. We study the cardinality of the set of rational iterated preimages Preim ${(\phi, a, K) = \{x_{0} \in \mathbb{P}^1(K) | \phi^{N} (x_0) = a {\rm for some} N \geq 1\}}$ . We prove two new results (Theorems 2 and 4) bounding ${|{\rm Preim}(\phi, a, K)|}$ as ${\phi}$ varies in certain families of rational functions. Our proofs are based on unit equations and a method of Runge for effectively determining integral points on certain affine curves. We also formulate and state a uniform boundedness conjecture for Preim ${(\phi, a, K)}$ and prove that a version of this conjecture is implied by other well-known conjectures in arithmetic dynamics.     

A Born–Oppenheimer Expansion in a Neighborhood of a Renner–Teller Intersection

We perform a rigorous mathematical analysis of the bending modes of a linear triatomic molecule that exhibits the Renner–Teller effect. Assuming the potentials are smooth, we prove that the wave functions and energy levels have asymptotic expansions in powers of ${\epsilon}$ , where ${\epsilon^4}$ is the ratio of an electron mass to the mass of a nucleus. To prove the validity of the expansion, we must prove various properties of the leading order equations and their solutions. The leading order eigenvalue problem is analyzed in terms of a parameter ${\tilde{b}}$ , which is equivalent to the parameter originally used by Renner. For ${0< \tilde{b}< 1}$ , we prove self-adjointness of the leading order Hamiltonian, that it has purely discrete spectrum, and that its eigenfunctions and their derivatives decay exponentially. Perturbation theory and finite difference calculations suggest that the ground bending vibrational state is involved in a level crossing near ${\tilde{b}=0.925}$ . We also discuss the degeneracy of the eigenvalues. Because of the crossing, the ground state is degenerate for ${0< \tilde{b}< 0.925}$ and non-degenerate for ${0.925< \tilde{b}< }$ .     

Computing α-Invariants of Singular del Pezzo Surfaces

We prove a new local inequality for divisors on surfaces and utilize it to compute α-invariants of singular del Pezzo surfaces, which implies that del Pezzo surfaces of degree one whose singular points are of type $\mathbb{A}_{1}$ , $\mathbb{A}_{2}$ , $\mathbb{A}_{3}$ , $\mathbb{A}_{4}$ , $\mathbb{A}_{5}$ , or $\mathbb{A}_{6}$ are Kähler-Einstein.     

Optimization of expected shortfall on convex sets

In this article, we prove that the minimization problem of the expected shortfall over a convex but not necessarily closed set of financial positions $ \mathcal{X}\subseteq {L^1} $ has a solution. We provide both minimax and variational approaches on this problem. In the case where the optimization conclusions arise from the application of subgradient arguments, we need the assumption that the set of financial positions $ \mathcal{X} $ is closed.     

Eigenvalue estimates for generalized Dirac operators on Sasakian manifolds

We consider a two-parameter generalization $D_{ab}$ of the Riemann Dirac operator $D$ on a closed Sasakian spin manifold, focusing attention on eigenvalue estimates for $D_{ab}$ . We investigate a Sasakian version of twistor spinors and Killing spinors, applying it to establish a new connection deformation technique that is adapted to fit with the Sasakian structure. Using the technique and the fact that there are two types of eigenspinors of $D_{ab}$ , we prove several eigenvalue estimates for $D_{ab}$ which improve Friedrich’s estimate (Friedrich, Math Nachr 97, 117–146, 1980).     

Global injectivity conditions for planar maps

We prove that a planar $C^1$ -smooth map $f:D\longrightarrow \mathbb{R }^{2n}$ , where $D\subseteq \mathbb{R }^{2n}$ is a convex open set, is injective if $\mathbb{R }\cap \mathrm{Spec}(df)_z=\emptyset $ for all $z\in D$ . We continue by showing that the triangulability of the differentials $(df)_z$ , $z\in D$ , ensure the global injectivity as well.     

A note on admissible rules and the disjunction property in intermediate logics

With any structural inference rule A/B, we associate the rule ${(A \lor p)/(B \lor p)}$ , providing that formulas A and B do not contain the variable p. We call the latter rule a join-extension ( ${\lor}$ -extension, for short) of the former. Obviously, for any intermediate logic with disjunction property, a ${\lor}$ -extension of any admissible rule is also admissible in this logic. We investigate intermediate logics, in which the ${\lor}$ -extension of each admissible rule is admissible. We prove that any structural finitary consequence operator (for intermediate logic) can be defined by a set of ${\lor}$ -extended rules if and only if it can be defined through a set of well-connected Heyting algebras of a corresponding quasivariety. As we exemplify, the latter condition is satisfied for a broad class of algebraizable logics.     

The \mathfrak {sl}_{3}-web algebra

In this paper we use Kuperberg’s $\mathfrak {sl}_3$ -webs and Khovanov’s $\mathfrak {sl}_3$ -foams to define a new algebra $K^S$ , which we call the $\mathfrak {sl}_3$ -web algebra. It is the $\mathfrak {sl}_3$ analogue of Khovanov’s arc algebra. We prove that $K^S$ is a graded symmetric Frobenius algebra. Furthermore, we categorify an instance of $q$ -skew Howe duality, which allows us to prove that $K^S$ is Morita equivalent to a certain cyclotomic KLR-algebra of level 3. This allows us to determine the split Grothendieck group $K^{\oplus }_0(\mathcal {W}^S)_{\mathbb {Q}(q)}$ , to show that its center is isomorphic to the cohomology ring of a certain Spaltenstein variety, and to prove that $K^S$ is a graded cellular algebra.     

Locally homogeneous rigid geometric structures on surfaces

We study locally homogeneous rigid geometric structures on surfaces. We show that a locally homogeneous projective connection on a compact surface is flat. We also show that a locally homogeneous unimodular affine connection ${\nabla}$ on a two dimensional torus is complete and, up to a finite cover, homogeneous. Let ${\nabla}$ be a unimodular real analytic affine connection on a real analytic compact connected surface M. If ${\nabla}$ is locally homogeneous on a nontrivial open set in M, we prove that ${\nabla}$ is locally homogeneous on all of M.     

Estimates of three classical summations on the spaces F_p~(α,q)(R~n)(0 < p ≤ 1)

We obtain the boundedness on ˙Fα,qp(Rn) for the Poisson summation and Gauss summation. Their maximal operators are proved to be bounded from˙Fα,qp(Rn) to L∞(Rn).For the maximal operator of the Bochner-Riesz summation, we prove that it is bounded from˙Fα,qp(Rn) to Lpnn-pα,∞(Rn).     

Biharmonic PNMC submanifolds in spheres

We obtain several rigidity results for biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector fields. We classify biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector fields and with at most two distinct principal curvatures. In particular, we determine all biharmonic surfaces with parallel normalized mean curvature vector fields in $\mathbb{S}^{n}$ . Then we investigate, for (not necessarily compact) proper-biharmonic submanifolds in $\mathbb{S}^{n}$ , their type in the sense of B.-Y. Chen. We prove that (i) a proper-biharmonic submanifold in $\mathbb{S}^{n}$ is of 1-type or 2-type if and only if it has constant mean curvature f=1 or f∈(0,1), respectively; and (ii) there are no proper-biharmonic 3-type submanifolds with parallel normalized mean curvature vector fields in $\mathbb{S}^{n}$ .     

Gamma-convergence of nonlocal perimeter functionals

Given ${\Omega\subset\mathbb{R}^{n}}$ open, connected and with Lipschitz boundary, and ${s\in (0, 1)}$ , we consider the functional $$\mathcal{J}_s(E,\Omega)\,=\, \int_{E\cap \Omega}\int_{E^c\cap\Omega}\frac{dxdy}{|x-y|^{n+s}}+\int_{E\cap \Omega}\int_{E^c\cap \Omega^c}\frac{dxdy}{|x-y|^{n+s}}\,+ \int_{E\cap \Omega^c}\int_{E^c\cap \Omega}\frac{dxdy}{|x-y|^{n+s}},$$ where ${E\subset\mathbb{R}^{n}}$ is an arbitrary measurable set. We prove that the functionals ${(1-s)\mathcal{J}_s(\cdot, \Omega)}$ are equi-coercive in ${L^1_{\rm loc}(\Omega)}$ as ${s\uparrow 1}$ and that $$\Gamma-\lim_{s\uparrow 1}(1-s)\mathcal{J}_s(E,\Omega)=\omega_{n-1}P(E,\Omega),\quad \text{for every }E\subset\mathbb{R}^{n}\,{\rm measurable}$$ where P(E, ??) denotes the perimeter of E in ?? in the sense of De Giorgi. We also prove that as ${s\uparrow 1}$ limit points of local minimizers of ${(1-s)\mathcal{J}_s(\cdot,\Omega)}$ are local minimizers of P(·, ??).