Open problems in billiards and optics from a workshop.
problem Open problems in billiards and geometric optics.
method Collection of open problems from a workshop.
result No specific key result mentioned; collection of problems.
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48 results for “open problems”
Open problems on surfaces with boundary and constant mean curvature.
problem Open problems on compact constant mean curvature surfaces with boundary.
method Survey of current status of open problems.
result Collection of open problems in the theory of surfaces.
Recalls intrinsically harmonic forms and open problems.
problem Open problems related to intrinsically harmonic forms.
method Recalling definitions and known results.
result Clarification of intrinsically harmonic forms and open questions.
Abstract collects open problems in billiards and symplectic geometry.
problem Open problems in billiards and symplectic geometry.
method Compilation of open problems from discussions.
result Compilation of open problems.
Open geometry puzzles keep the author engaged.
problem Open problems in geometry that challenge the author.
method Collection of open problems based on puzzle-charm.
result No spare hands in solving the problems.
Survey on broken ray transforms and open problems.
problem Understanding broken ray transforms and their applications.
method Survey of recent developments and open problems.
result Discussion of open problems in broken ray transforms.
Solves Brezis' first open problem on ball solutions.
problem Existence of solutions to Brezis-Nirenberg problem on a 3D ball.
method Building on sign-changing solutions to the Yamabe problem.
result Infinitely many sign-changing, nonradial solutions found.
Quadratic Hessian equationmath.AP
Abstract: Survey on quadratic Hessian equations, their properties, and open problems.
problem Understanding quadratic Hessian equations and their solutions.
method Survey and review of existing research.
result Survey of entire solutions, viscosity solutions, and Hessian estimates.
Some open problemsmath.OC
We discuss some challenging open problems in the geometric control theory and sub-Riemannian geometry.
The paper surveys open problems and questions related to different aspects of integrable systems with finitely many degrees of freedom. Many of the open problems were suggested by the participants of the conference "Finite-dimensional Integrable Systems, FDIS 2017" held at CRM, Barcelona in July 2017.
Solves open problems on curved projective varieties.
problem Structure theorems for curved projective varieties.
method Supplements and proposes open problems.
result Provides new insights into structure of curved projective varieties.
The paper surveys open problems and questions related to geodesics defined by Riemannian, Finsler, semi Riemannian and magnetic structures on manifolds.
Open problem: fixed-budget best arm identification complexity.
problem Understanding the complexity of identifying the best arm in a fixed budget setting.
method Analyzing existing results and conjectures in the fixed-confidence setting.
result Open questions remain about the fixed-budget setting.
Hyperkähler manifoldsmath.AG
Explains hyperkähler manifolds and their properties.
problem None explicitly stated; focuses on introduction and properties.
method Elementary introduction and survey of properties.
result Properties of hyperkähler manifolds discussed.
We review some applications of noncommutative geometry to the study of transverse geometry of Riemannian foliations and discuss open problems.
Survey of open problems linking integrable systems and Nijenhuis geometry.
problem Interplay between integrable systems and Nijenhuis geometry.
method Discussion of open problems and questions.
result Challenges and simple questions in the field.
On Brezis Open Problem 3.1math.AP
Brezis' open problem on harmonic maps resolved
problem Existence of explicit solutions to the harmonic map equation with Dirichlet boundary condition
method Boundary rigidity argument
result Proving the uniqueness of the explicit maps as weak harmonic maps
Fatal accidents are a major issue hindering the wide acceptance of safety-critical systems using machine-learning and deep-learning models, such as automated-driving vehicles. Quality assurance frameworks are required for such machine learning systems, but there are no widely accepted and established quality-assurance …
Complete open Kahler manifolds with nonnegative bisectional curvature and non-maximal volume growthmath.DG
It is still an open problem that a complete open Kahler manifold with positive bisectional curvature is Stein. This paper partially resolve the problem by putting a restriction to volume growth condition. The partial solution here improves the observation in ([8], page 341). The improvement is based on assuming a weake…
We propose a list of open problems in pluripotential theory partially motivated by their applications to complex differential geometry. The list includes both local questions as well as issues related to the compact complex manifold setting.
End-to-end PGL framework tackles open-set domain shift.
problem Real-world domain shift with unknown additional classes.
method Episodic training in graph neural network with adversarial learning.
result Guarantees tighter upper bound of target error.
This is a list of open problems on invariants of knots and 3-manifolds with expositions of their history, background, significance, or importance. This list was made by editing open problems given in problem sessions in the workshop and seminars on `Invariants of Knots and 3-Manifolds' held at Kyoto in 2001.
The paper compares spectral geometry in hyperbolic and spherical manifolds.
problem Understanding spectral geometry in spherical manifolds.
method Survey of known results and open problems.
result Analogous results hold in hyperbolic manifolds but not necessarily in spherical manifolds.
This paper presents a natural extension to foliated spaces of the following result due to Gromov : the h-principle for open, invariant differential relations is valid on open manifolds. The definition of openness for foliated spaces adopted here involves a certain type of Morse functions. Consequences concerning the pr…
The Chern-Ricci flowmath.DG
Survey on Chern-Ricci flow for complex manifolds.
problem Understanding and solving open problems in Chern-Ricci flow.
method Parabolic flow of Hermitian metrics on complex manifolds.
result Open problems and new directions in Chern-Ricci flow highlighted.
Geometric Hydrodynamics tackles open problems in fluid dynamics.
problem Open problems in fluid dynamics and invariant metrics.
method Variational settings, models for invariant metrics, Cauchy and boundary value problems.
result New constructions and recent developments in fluid dynamics.
This paper tackles open problem of tight bounds for KBs with Bernoulli rewards.
problem Open problem of tight bounds for Kernelized Bandits with Bernoulli rewards.
method Focus on Bernoulli model, not subgaussian noise, and optimize function in RKHS.
result Open problem remains unsolved in this context.
Integrates multiple datasets to solve open set crowdsourcing problems.
problem Crowdsourcing with unknown label space and unfamiliar tasks.
method Integrates multiple crowdsourced datasets, weights them based on category correlation, and uses open set transfer learning.
result Proves OSCrowd solves open set crowdsourcing problems and outperforms related solutions.
Paper solves open problem in complex Finsler geometry.
problem Existence of non-Kähler weakly Kähler Finsler metrics.
method Constructs a family of weakly Kähler Finsler metrics.
result Proves uniformization theorem for unitary invariant complex Randers metrics.
This study reviews decentralized prediction markets, identifying key design variants and open problems.
problem Designing and implementing decentralized prediction markets with desirable properties.
method Modular workflow comprising eight stages: infrastructure, market topic, share structure, pricing, market initialization, trading, resolution, settlement, and archiving. Analysis of design variants and trade-offs.
result Identification of open problems for researchers in the field of decentralized prediction markets.
This paper has been submitted to the Proceedings of the Australian-German Workshop on Differential Geometry in the Large held at the mathematical research institute MATRIX in Creswick, Victoria, Australia, Feb.2-Feb.14, 2019. We describe and discuss 2 important open problems in Sasaki geometry.
We give a short solution to one of the main open problems in subriemannian geometry. Namely, we prove that length minimizers do not have corner-type singularities. With this result we solve Problem II of Agrachev's list, and provide the first general result toward the 30-year-old open problem of regularity of subrieman…
Study finds open manifolds without complete metrics with positive scalar curvature.
problem Topological obstruction to positive scalar curvature on open manifolds.
method Defined Schoen-Yau-Schick and weak Schoen-Yau-Schick manifolds to prove the absence of complete metrics with positive scalar curvature.
result Proved no complete metric with positive scalar curvature on open Schoen-Yau-Schick manifolds.
Survey of recent metric geometry in Kähler metrics space.
problem Understanding the metric geometry of Kähler metrics space.
method Survey and highlighting of recent results.
result Highlighting of open problems in the field.
Open problem seeks an online learning algorithm for binary classification.
problem Existence of an online learning algorithm for binary classification with sublinear mistakes.
method Assumption of sequence allowing learning algorithm's existence.
result Specific condition determines sequence's learnability.
This paper proposes a method to use deep neural networks as end-to-end open-set classifiers. It is based on intra-class data splitting. In open-set recognition, only samples from a limited number of known classes are available for training. During inference, an open-set classifier must reject samples from unknown class…
Open set recognition problems exist in many domains. For example in security, new malware classes emerge regularly; therefore malware classification systems need to identify instances from unknown classes in addition to discriminating between known classes. In this paper we present a neural network based representation…
Continuous LWEcs.CC
Introduces a continuous version of LWE problem.
problem Hardness of learning mixtures of Gaussians.
method Polynomial-time quantum reduction from CLWE to lattice problems.
result CLWE shares hardness with LWE.
Characterizes quasiconformally homogeneous ladder surfaces.
problem Characterize quasiconformally homogeneous Riemann surfaces.
method Breaks the problem into four cases and proves one case.
result Every quasiconformally homogeneous ladder surface is quasiconformally equivalent to a regular cover of a closed surface.
We discuss recent results and open questions on the broad theme of (Nielsen) realization problems. Beyond realizing subgroups of mapping class groups, there are many other natural instances where one can ask if a surjection from a group of diffeomorphisms of a manifold to another group admits a section over particular …
Camera model identification refers to the problem of linking a picture to the camera model used to shoot it. As this might be an enabling factor in different forensic applications to single out possible suspects (e.g., detecting the author of child abuse or terrorist propaganda material), many accurate camera model att…
We survey the known existence and non-existence results for -instantons on non-compact cohomogeneity-1 -manifolds and their consequences, including an explicit example of a family of -instantons where bubbling, removable singularities and conservation of energy phenomena occur. We also describe several o…
In this note we briefly survey and propose some open problems related to isoparametric theory.
Uniform Closure Method and Bayes classifier perform similarly in classifying open knots.
problem Classifying knots in open macromolecular chains.
method Used the Bayes MAP classifier and compared it to the Uniform Closure Method.
result Both methods have comparable accuracy and positive predictive value.
We introduce a new class of possibly noncompact n-dimensional manifolds without boundary associated to finite data which we call topological automata. This class is large enough to contain many interesting examples of open 2-dimensional and 3-dimensional manifolds of interest to low-dimensional topologists. Our main re…
The study examines elastic curves with self-intersections and their properties.
problem Understanding self-intersecting elastic curves in Euclidean space.
method Review of classical elastica theory and discussion of open problems.
result Proof of Li--Yau type inequality for self-intersecting curves.
In this paper, we study a time-inconsistent consumption-investment problem with random endowments in a possibly incomplete market under general discount functions. We provide a necessary condition and a verification theorem for an open-loop equilibrium consumption-investment pair in terms of a coupled forward-backward …
Unified approach to equity markets with open and hybrid Jacobi models.
problem Stochastic Portfolio Theory problems in equity markets.
method Combining open markets and hybrid Jacobi processes.
result Stability of capital distribution curve and growth optimal strategies.