We introduce a new class of affine connections on Hessian manifolds, called hybrid connections, characterized by the compatibility between their projective geometry with the underlying affine structure on the one hand and their infinitesimal holonomy with the Hessian metric on the other hand. In this paper, we investigate the properties of hybrid connections and prove that, on a given Hessian manifold, they are completely determined by the choice of a Hessian potential for the metric. In the special case of pseudo-Euclidean manifolds, we identify canonical models and construct, in particular, a natural connection on the open unit ball that combines features of the Cayley–Klein and Poincaré models of hyperbolic geometry. We also prove the existence and uniqueness (up to scaling) of a pseudo-Riemannian metric h such that the geodesics of ∇ admit parameterizations of constant speed with respect to h, which we call the isochrone metric.
In this paper, we study n-dimensional (n≥ 3) conformally flat minimal Lagrangian submanifolds in complex space forms. Our main result is a classification of such submanifolds under the assumption that the Ricci tensor is semi-parallel. This generalizes the recent classification result of Song–Wang–Xing [1] on conformally flat minimal Lagrangian submanifolds in complex space forms with parallel Ricci tensor.
Let S⊂ℝ^3 be a smooth embedded sphere whose normal curvatures have absolute value at most 1. We prove that if S is contained in an open ball of radius 2, then the body bounded by S contains a unit ball.
It is well-known that the Euclidean plane has a standard 6-regular triangulation by equilateral triangles and the hyperbolic plane has a k-regular geometric triangulation for every k>6 . In this note, we construct 6-regular geometric triangulations of the hyperbolic plane with a uniform bound on edge lengths, resolving a conjecture by Feng Luo [6]. We also show that there is no uniform bound on stretch factors of Delaunay triangulations in the hyperbolic plane, in contrast to the existence of such a bound in the Euclidean plane.
The girth of a graph is the length of a shortest cycle in it. If the graph does not contain any cycles (that is, it is a forest), its girth is defined to be infinity. For a finitely generated group, we can define its girth as the supremum of girths of Cayley graphs of it with respect to all finite generating sets. A given class of finitely generated groups is said to satisfy Girth Alternative if any group from this class is either virtually solvable or has infinite girth. We prove the Girth Alternative for a sub-class of HNN extensions as well as for a sub-class of amalgamated free products of finitely generated groups, and indicate counterexamples to show that beyond our class, the alternative fails in general. We also prove the Girth Alternative for HNN extensions of non-elementary word hyperbolic groups.
We study the numerical accuracy of some specific computer programs performing numerical computations. Such a numerical accuracy is expressed in terms of a bound on the difference between the floating-point computations and the corresponding rounding-free computations using mathematical real numbers. We do not only seek to discover such bounds “on paper” but we aim at obtaining computer-assisted formal proofs that these bounds are correct for any possible inputs. The functions we study come from the domain of machine learning: a function computing the logarithm of the sum of exponentials of a sequence, and a similar function related to the computation of mutual information. The bounds obtained are original results, because they are parameterized by assumed error bounds of the underlying implementations of the logarithm and exponential functions. The methodology we follow to conduct our formal proofs is also original, using a combination of the Why3 environment for deductive verification, an original modeling of floating-point computations using unbounded numbers, and the J ^3 environment for proving properties on C source code.
In this paper we prove that all the homology groups of the real nerve of the moduli space of Riemann surfaces of even genus g>4 are nontrivial. The crucial role in our constructions comes from the information known about the Riemann surfaces admitting the maximal number of symmetries and the ones admitting the maximal numbers of ovals.
In this paper, we prove the dimension of the space of weighted caloric functions with polynomial growth is bounded by the degree of growth times the dimension of weighted harmonic functions with the same growth on any weighted Riemannian manifold with polynomial volume growth. This generalizes the results of Colding-Minicozzi[5].
Sketch-based program synthesis allows users to guide the synthesizer by writing partial programs (sketches). Traditionally, the specifications of these sketches are safety properties expressed either as assertions or semantic equivalences. These specifications, however, lack expressiveness when the user wants to establish how the execution of the desired program evolves over time. This is especially important when synthesizing reactive programs, where the whole specification is about how the computation evolves over time. We explore an alternative method letting the user specify desired program executions as linear temporal logic (LTL) formulae. We define a method for transforming sketches with LTL assertions into sketches with only standard assertions. Specifically, for terminating programs, our method transforms and implements, within the sketch, the LTL formulae as runtime monitors. For non-terminating programs, our procedure defines and implements, into the sketch, fairness conditions (analyzing when an acceptance state of the Büchi automata equivalent to these LTL formulae occurs infinitely often). We prove the correctness of both constructions. Evaluation of our implementation in Sketch shows that our method enables the system to synthesize non-terminating programs, such as a round-robin arbiter for a variable number of devices and a lift controller for a variable number of floors. For terminating programs, our approach improves the synthesizer performance by restricting the search of program candidates without modifying the sketches structures.
For any positive integers h and n, we show that a knot surgered elliptic surface E(n)_T(2,2h+1) for a (2,2h+1) -torus knot T(2,2h+1) admits a handle decomposition without 1- and 3-handles using a Kirby diagram derived from a Lefschetz fibration on it. As a corollary, an elliptic surface E(1)_2,2h+1 has such a handle decomposition.
In [1], Chow and Glickenstein considered a second order linear semi-discrete analogue to the curve shortening flow for closed curves formed by joining an ordered set of points in the plane or in higher dimensional Euclidean space. In this article we consider similar flows with boundary conditions. Given distinct endpoints, results include characterisation of self-similar scaling solutions, and existence and uniqueness for solutions that evolve by pure translation in any given fixed direction and solutions that evolve by pure rotation with any fixed angular velocity. More generally, given any initial piecewise-linear curve with distinct boundary points, we provide a representation formula for the unique solution of the flow. Next we consider a semi-discrete analogue to the Yau problem of flowing one curve to another by curvature flow, here with boundaries. We show that any piecewise linear curve with distinct boundary points may flow to any other under the corresponding nonhomogeneous curvature flow; the solution is not unique unless the precise boundary point trajectories are prescribed. In the last section we consider two further variants of our semi-discrete curvature flow, where we allow anisotropy and vertices of varying mobility.
Using Symbolic Computer Algebra (SCA) enabled a huge progress in formal verification of arithmetic circuits in recent years. Several different approaches have been proposed showing great success especially for the verification of multipliers. Some of them are based on precomputing and simplifying polynomials for specific circuit structures like converging cones while others take advantage of known or detected hierarchy information to replace and simplify particular subcircuits of the design. In this paper we propose a new method that avoids the use of such methods and applies only two dynamic approaches: (1) choosing a good substitution order for the backward rewriting process and (2) adjusting the phases of signals occurring in the intermediate polynomials during the verification process. Both methods are simply based on a greedy local search taking the sizes of intermediate polynomials into account. Our experimental results show that this method is very competitive with already existing tools and it improves their robustness, e.g. against optimizations of the verified circuits using logic synthesis.
Hybrid automata networks (HANs) provide a powerful formalism to represent interactions within distributed hybrid systems, where each subsystem exhibits both continuous and discrete dynamics. Although the properties of HANs can be verified by model checking exhaustively, the effectiveness of the verification process is challenged by the state explosion problem inherent in large-scale HANs. This research proposes shan, a novel verification framework specifically designed for HANs. shan employs task scenarios in the form of sequence diagrams to facilitate the verification of HANs. It systematically navigates the state spaces that developers consider important. The task scenarios and the HAN under verification are encoded as SMT formulas to simulate the execution of the HAN to accelerate the verification. The framework accelerates the verification process further using parallelization and minimal unsatisfiable core techniques for efficient instruction pruning. Moreover, we introduce metrics designed to evaluate the extent of HAN verification coverage, providing developers with a comprehensive tool for assessing the completeness of HAN verification.
We define a new notion of translations in the hyperbolic plane and explicitly solve the equation of the curve shortening flow. Next, we consider the class of ancient convex solutions and solve the equation of the curve shortening flow when the curvature function is given by separation of variables. Lastly, we prove some area estimates for closed ancient solutions of the curve shortening flow.
Apisa-Wright conjectured that all branched covers of quadratic differentials in 𝒬(-1^4) with at most one cylinder in each direction are cyclic covers. We provide infinitely many counterexamples to this conjecture.
We prove that any dual leaf L^# of a simply connected, complete nonnegatively curved polar manifold M is totally geodesic and closed in M, and L^# is itself a complete nonnegatively curved polar manifold. Furthermore, the dual foliation on M induces a Riemannian submersion with totally geodesic fibers from M to a homogeneous space.
In the 1980s, M. J. Markowitz introduced a conformally invariant pseudodistance on pseudo-Riemannian manifolds, inspired by the Kobayashi metric in projective geometry. This construction relies on a distinguished class of parametrized lightlike geodesics, called projectively parametrized. We begin by reviewing the fundamental properties of this pseudodistance and provide several families of examples where it is non-degenerate and, in some cases, complete. In particular, we investigate three classes of manifolds: closed manifolds, conformally convex domains of the Einstein universe, and globally hyperbolic, conformally flat, C-maximal spacetimes. For the first two classes, we obtain results analogous to those of Brody and Barth concerning the complex Kobayashi metric. Finally, we apply Markowitz’s pseudodistance to classify all quasi-homogeneous domains of the Einstein-de Sitter space, that is, a half-space of the Minkowski space bounded by a spacelike hyperplane. Up to conformal transformations, only finitely many such domains exist, and all of them turn out to be homogeneous.
We show that the winding of low-lying closed geodesics on the modular surface has a Gaussian limiting distribution when normalized by any standard notion of length, in contrast to the Cauchy distribution arising when allowing arbitrarily deep excursions into the cusp. In addition, we prove a Berry-Esseen bound and a local limit theorem.
In this paper, we examine the combinatorial properties of conic arrangements in the complex projective plane that possess certain quasi–homogeneous singularities. First, we introduce a new tool that enables us to characterize the property of being plus–one generated within the class of conic arrangements with some naturally chosen quasi–homogeneous singularities. Next, we present a classification result on plus–one generated conic arrangements admitting only nodes and tacnodes as singularities. Building on results regarding conic arrangements with nodes and tacnodes, we present new examples of strong Ziegler pairs of conic-line arrangements – that is, arrangements having the same strong combinatorics but distinct derivation modules.
We construct novel families of exact immersed and embedded Lagrangian translating solitons and special Lagrangian submanifolds in ℂ^m that are invariant under the action of various admissible compact subgroups G ≤ SU (m-1) with cohomogeneity-two. These examples are obtained via an Ansatz generalising a construction of Castro–Lerma in ℂ^2 . We give explicit examples of admissible group actions, including a full classification for G simple. We also describe novel Lagrangian translators symmetric with respect to non-compact subgroups of the affine special unitary group SU (m)⋉ℂ^m , including cohomogeneity-one examples.
