Data-driven model reduction captures non-Markovian dynamics using Koopman and Mori-Zwanzig formalisms.
problem Modeling complex, non-Markovian dynamics efficiently and understanding their underlying mechanisms.
method Formulates data-driven model reduction within Koopman and Mori-Zwanzig formalisms, deriving NARMAX models from dynamical systems.
result Shows how data-driven methods can represent non-Markovian dynamics using Koopman and Mori-Zwanzig formalisms.
Regression learns Mori-Zwanzig operators for dynamical systems.
problem Learning Mori-Zwanzig operators for complex dynamical systems.
method Statistical regression to extract Markov and memory operators.
result Regression models improve learning of memory-dependent corrections.
Develops a new deep learning formulation using Mori-Zwanzig formalism.
problem Improves deep learning by introducing a new concept of memory.
method Uses Mori-Zwanzig formalism to propagate quantities of interest through neural networks.
result Rigorously transforms deep networks into shallow ones using decay property of memory operator.
The well-known Mori-Zwanzig theory tells us that model reduction leads to memory effect. For a long time, modeling the memory effect accurately and efficiently has been an important but nearly impossible task in developing a good reduced model. In this work, we explore a natural analogy between recurrent neural network…
Novel autoencoder method approximates Koopman operator in low dimensions.
problem Challenges in approximating finite Koopman operators using data-driven methods.
method Mori-Zwanzig autoencoder (MZ-AE) for robust Koopman operator approximation.
result Improved predictive capability and robust long-term statistical performance.
A new method predicts non-Markovian closure terms for complex systems.
problem Predicting the effect of unresolved variables on resolved dynamics in high-dimensional systems.
method Mamba-Assisted Closure (MAC) framework: sequence model trained to predict closure from resolved trajectory, coupled with reduced-order equations.
result Substantially outperforms existing methods in predictive accuracy and long-time stability.
iLED framework offers interpretable dynamics for multiscale systems.
problem Modeling high-dimensional multiscale systems is challenging.
method Interpretable Learning Effective Dynamics (iLED) framework based on Mori-Zwanzig and Koopman operator theory.
result Comparable accuracy to state-of-the-art approaches with added interpretability.
A new neural network approach for diffusion on networks.
problem Inference and estimation of diffusion on network structures.
method Neural mean-field dynamics derived from Mori-Zwanzig formalism, approximated by learnable time convolution operators.
result Significantly outperforms existing approaches in accuracy and efficiency.
Deep neural networks with memory learn reduced equations from partial data.
problem Constructing governing equations for unknown dynamical systems from limited data.
method Formulate a discrete approximation of memory integrals, use deep neural networks to incorporate history terms.
result Deep neural networks can learn reduced equations with memory from partial data.
The paper defines projective structures for Lie bialgebras and Poisson-Lie groups.
problem Defining projective analogues of Lie bialgebras and Poisson-Lie groups.
method Introducing projective tensor products and adapting classical notions to these structures.
result Every quasi-triangular projective r-matrix gives rise to a projective Banach Lie bialgebra.
Characterizes minor-minimal separating projective planar graphs and their generalizations.
problem Understanding projective planar graphs and their properties.
method Analyzing minors, embeddings, and specific link types.
result Partial characterization of minor-minimal separating projective planar graphs and their generalizations.
Legendrian knots can be represented by projections with multi-crossings.
problem Representing Legendrian knots with multi-crossings.
method Investigating übercrossing and petal projections in front and Lagrangian projections.
result Legendrian knots with übercrossing projections in front are smoothly isotopic to the unknot.
The study classifies holomorphic projective connections on complex threefolds.
problem Characterizing holomorphic projective connections on complex threefolds.
method Analyzing properties of holomorphic projective connections on complex projective threefolds.
result Holomorphic projective connections on complex threefolds are either flat or translation invariant on abelian threefolds.
Defines projective Ricci curvature and proves rigidity for sprays.
problem Defining and studying projective Ricci curvature.
method Introduced projective Ricci-flat sprays and studied Randers metrics.
result Global rigidity result for projectively Ricci-flat sprays with nonnegative Ricci curvature.
The paper calculates delta invariants for specific geometric structures.
problem Computing delta invariants for projective bundles and cones of Fano type.
method Provides a precise formula for delta invariants.
result A formula to compute delta invariants for projective bundles and cones of Fano type.
This paper proposes a method to select project schedules with the lowest risk.
problem Selecting schedules that meet project deadlines while minimizing risk.
method Integrating aleatory uncertainty into project scheduling to quantify and compare risks.
result Proposes a method to select schedules with the lowest risk.
Projective rigidity of circle packings on complex surfaces proved.
problem Proving rigidity of circle packings on complex projective surfaces.
method Proved projective rigidity through triangulations and complex projective structures.
result Space of circle packings is projectively rigid on complex projective surfaces.
A small projective 4-manifold created via Dehn filling.
problem Creating a small positive Euler characteristic closed convex projective 4-manifold.
method Explicit construction through continuous path of projective cone-manifolds and Dehn filling of a cusped hyperbolic 4-manifold.
result Obtained a closed orientable convex projective four-manifold with small positive Euler characteristic.
Classifies flat projective structures with specific symmetries.
problem Classifying local projective structures with non-trivial Lie symmetries.
method Analyzes flat projective structures with positive-dimensional Lie algebra of projective vector fields.
result Obtained a classification of flat projective structures.
Affine links in projective space have a specific group property.
problem Characterizing links in projective space.
method Examined the fundamental group of link complements.
result Affine links have a non-trivial element of order two in their fundamental group.
The authors give a complete classification of projective threefolds admitting a holomorphic normal projective connection. Moreover, they prove a general structure theorem on complex projective manifolds admitting a holomorphic normal projective connection, saying in particular, that any such manifold is either the proj…
Report examines Muskrat Falls Project's cost and schedule overruns.
problem Analyzing Muskrat Falls Project's cost and schedule overruns.
method Examines national and international context, causes, and recommendations.
result Provides insights into preventing cost and schedule overruns in hydroelectric dam projects.
In the asymmetric setting, Hilbert's fourth problem asks to construct and study all (non-reversible) projective Finsler metrics: Finsler metrics defined on open, convex subsets of real projective n-space for which geodesics lie on projective lines. While asymmetric norms and Funk metrics provide many examples of esse…
In 2D, Finsler metrics with 3+ projective fields are projectively equivalent to Randers.
problem Characterizing Finsler metrics with multiple projective vector fields.
method Analyzing the Lie algebra of projective vector fields and showing equivalence to Randers metrics.
result A complete list of 2D Finsler metrics with at least 3 projective fields up to equivalence.
Paper studies pseudo-projective tensors on warped products.
problem Characterizing pseudo-projectively flat warped products.
method Analyzes sequential warped products and pseudo-projective tensors.
result Necessary and sufficient conditions for pseudo-projectively flat sequential warped products.
Study projective KLT varieties with projectively flat cotangent sheaves.
problem Uniformisation problems on projective varieties with klt singularities.
method Generalising Jahnke-Radloff's work, study torus quotients and varieties with semistable cotangent sheaves and extremal Chern classes.
result Torus quotients are the only klt varieties with semistable cotangent sheaves and extremal Chern classes.
New framework uses elliptic operators to study projective maps.
problem Understanding projective structures on Riemannian manifolds.
method Develops two elliptic operators of second and fourth order.
result Establishes a natural correspondence between analytical and geometric properties.
Projective manifolds with specific bundles are isomorphic to simpler spaces.
problem Characterizing projective manifolds with tangent bundles containing strictly nef subsheaves.
method Analyzing the structure of the tangent bundle and using properties of strictly nef subsheaves.
result Projective manifolds with the described bundles are isomorphic to projective bundles over hyperbolic manifolds or projective spaces.
An axis of a link projection is a closed curve which lies symmetrically on each region of the link projection. In this paper we define axis systems of link projections and characterize axis systems of the standard projections of twist knots.
This paper combines two classical theories, namely metric projective differential geometry and superintegrability. We study superintegrable systems on 2-dimensional geometries that share the same geodesics, viewed as unparametrized curves. We give a definition of projective equivalence of such systems, which may be con…
Paper classifies Randers metrics based on Ricci curvature properties.
problem Investigating isotropic projective Ricci curvature in Randers metrics.
method Classification of Randers metrics based on isotropic projective Ricci curvature properties.
result Randers metric of isotropic projective Ricci curvature is reversible if and only if it is of square projective Ricci curvature.
Paper defines conditions for projective links in projective 3-space.
problem Characterizing links in projective 3-space.
method Combinatorial conditions and antipodal symmetry.
result Easy condition to prevent alternating projective links.
The paper studies new curvature properties in Finsler geometry.
problem Properties of projectively equivalent Finsler metrics and their curvature structures.
method Introducing new characterizations of quadratic curvature properties in Finsler manifolds.
result Novel insights into curvature behavior under generalized projective sprays.
We present an algorithm for converting an indoor spherical panorama into a photograph with a simulated overhead view. The resulting image will have an extremely wide field of view covering up to 4π steradians of the spherical panorama. We argue that our method complements the stereographic projection commonly used in t…
Characterizes projective special complex manifolds using c-projective structures.
problem Characterizing projective special complex manifolds.
method Defining S1-bundles and constructing conical special complex manifolds. result Intrinsic characterization of projective special complex manifolds.
A new indicator measures project risk from activity durations.
problem Managing project risks throughout the lifecycle.
method Activity Risk Index (ARI) based on Schedule Risk Baseline.
result Identifies activities contributing most to project uncertainty.
This paper surveys various methods for dimensionality reduction and nearest neighbor search.
problem Efficiently reducing high-dimensional data to lower dimensions while preserving essential information.
method Linear and nonlinear random projections, including sparse random projections, random Fourier Features, and Random Kitchen Sinks.
result Various methods for dimensionality reduction and nearest neighbor search are explained and compared.
Study on projective structures on a hyperbolic 3-orbifold using tetrahedra.
problem Analyzing projective structures on hyperbolic 3-orbifolds.
method Parameterization using classical invariants, traces, and geometric cross ratios.
result Computed and analyzed the moduli space of projective structures.
Paper studies binary random projections with controllable sparsity patterns for computational and accuracy advantages.
problem Improving computational efficiency and accuracy in random projections.
method Proposes two sparse binary projection models with controllable sparsity patterns.
result Significant computational advantages and improved accuracies in empirical evaluations.
Two projective structures on Riemann surfaces are described and shown not to be identical.
problem Characterizing and distinguishing projective structures on Riemann surfaces.
method Construction and analysis of projective structures pu and ph using the period map and moduli space. result The Hodge projective structure ph is not the same as the uniformization projective structure pu. A new method prioritizes project risks using Monte Carlo Simulation.
problem Determining the relative importance of project risks.
method Monte Carlo Simulation (MCS) for quantitative prioritization.
result Differentiates critical risks based on their impact on project duration and cost.
The article constructs strong Carrollian geometries at infinity for Ricci flat Einstein manifolds.
problem Understanding projective and Carrollian geometries at infinity for Ricci flat Einstein manifolds.
method Developed a new type of Cartan geometry based on non-effective homogeneous models for projective geometry.
result Carrollian geometries are determined by the projective compactification data of Ricci flat Einstein manifolds.
Study calculates Kulkarni limit sets for quaternionic projective groups.
problem Computing Kulkarni limit sets for quaternionic projective groups.
method Natural action of quaternionic projective linear group on quaternionic projective space.
result Computed Kulkarni limit sets for cyclic subgroups.
Study projective connections on surfaces using osculating spaces.
problem Understanding projective connections on surfaces.
method Analyzing the second neighborhood and osculating behavior of integral curves.
result Geometrical interpretation of joint invariants of a related group.
Introduces PIT-plot for prioritizing projects based on their impact.
problem Optimizing R&D investments in project portfolios.
method Develops a new tool (PIT-plot) focusing on project impact rather than project properties.
result Identifies projects with the largest impact for risk mitigation or value-adding.
Defines super projective modules and explores their properties.
problem Exploring the geometric-algebraic link in super geometry.
method Defined and explored super projective modules over supersmooth functions.
result Module of vector fields over a supersphere is a super projective module.
This paper tabulates prime knot projections up to eight double points.
problem Tabulating prime knot projections and their mirror images up to a certain number of double points.
method Systematic flypes and enumeration of tangles with at most four double points, using arrow diagrams.
result Complete table of prime knot projections with their mirror images up to eight double points.
New projection techniques reduce the frequency of projections in solving LCPs.
problem Solving linearly constrained problems efficiently with reduced projection frequency.
method Delayed projection technique to call a projection less frequently.
result Theoretical and practical improvements in convergence rates and efficiency.