Grid peeling and ACSF show similar behavior on convex curves.
problem Understanding the connection between computational and differential geometry processes.
method Empirical evidence and theoretical arguments.
result Grid peeling of N2 behaves similarly to ACSF on convex curves. A grid layout method for graph classification using CNNs.
problem How to project graphs onto grids for CNNs to work effectively.
method Proposes a novel graph-preserving grid layout (GPGL) using integer programming to minimize topological loss, and solves it approximately with a regularized Kamada-Kawai algorithm.
result Demonstrates the success of the method for graph classification using multi-scale maxout CNNs.
We give combinatorial descriptions of the Heegaard Floer homology groups for arbitrary three-manifolds (with coefficients in Z/2). The descriptions are based on presenting the three-manifold as an integer surgery on a link in the three-sphere, and then using a grid diagram for the link. We also give combinatorial descr…
Study grid homology of diagonal knots, finding key terms related to prime factors and decompositions.
problem Determine grid homology of diagonal knots and compare them to other knot types.
method Use grid diagrams and combinatorial knot Floer homology to analyze diagonal knots.
result Grid homology detects the number of prime factors and decompositions of the knot into non-integer tangles.
GraPhyR uses GNNs to optimize power grid reconfiguration in real-time.
problem Optimizing power grid reconfiguration for reliability and efficiency.
method Physics-informed Graph Neural Network (GNN) framework.
result GraPhyR learns to optimize DyR tasks efficiently.
New method characterizes surface quadrilateral layouts as special immersions.
problem Characterize surface quadrilateral layouts mathematically.
method Characterizes quadrilateral layouts as special immersions of a cut representation of the surface into the Euclidean plane.
result Mathematically describes and generalizes integer grid maps.
Link Floer homology is an invariant for links which has recently been described entirely in a combinatorial way. Originally constructed with mod 2 coefficients, it was generalized to integer coefficients thanks to a sign refinement. In this paper, thanks to the spin extension of the permutation group we give an alterna…
New proof confirms petal number for torus knots without modular condition.
problem Proving the petal number of torus knots without modular condition.
method Constructing petal grid diagrams for any torus knots.
result The conjecture that $p(T_{n,s})\le 2s-2\left\lfloor \frac sn
ight
floor+1$ holds for any 2≤n<s is confirmed. This paper recovers smooth functions from noisy modulo samples using a three-stage strategy.
problem Recovering Hölder smooth functions from noisy modulo samples.
method Three-stage strategy: denoising with local polynomial estimators, unwrapping, and spline-based quasi-interpolant.
result Uniform error rates for Hölder class functions with high probability.
Grid-scale batteries' bid patterns in price uncertainty markets
problem Interpreting bids from grid-scale batteries in wholesale electricity markets under price uncertainty
method Developing an asset-level model of a price-taking battery
result Empirical results deliver insights into withholding behavior, uncertainty effects, and risk management reshaping bid curves
Exact and scalable algorithm for Gaussian process regression with Matérn correlations.
problem Efficient Gaussian process regression with Matérn correlations.
method Novel kernel packet theory and sparse representation of covariance matrix.
result Significantly superior to existing alternatives in computational time and predictive accuracy.
Pipeline-aware hyperparameter tuning speeds up machine learning pipelines by reusing intermediate computations.
problem High computational burden in hyperparameter tuning of multi-stage pipelines.
method Proposes a hybrid hyperparameter tuning method and a caching problem formulated as an ILP to maximize reuse.
result Pipeline-aware approach offers over an order-of-magnitude speedup over independent evaluations.
We present a grid diagram analogue of Carter, Rieger and Saito's smooth movie theorem. Specifically, we give definitions for grid movies, grid movie isotopies and present a definition of grid planar isotopy as a particular subset of the grid diagram moves: stabilization, destabilization and commutation. We show that gr…
Paper proposes a network framework for prosumers to manage peak loads in Iran.
problem Balancing renewable prosumers' self-sufficiency with grid integration under uncertainty.
method Distributed contextual stochastic optimization (DCSO) framework with consensus-based sharing.
result Integration of prediction and optimization reduces peak loads and costs.
The paper introduces triple grid diagrams to construct Lagrangian surfaces in complex projective space.
problem Constructing Lagrangian surfaces in complex projective space.
method Defining and analyzing triple grid diagrams to determine Lagrangian caps and surfaces.
result Triple grid diagrams can determine closed Lagrangian surfaces in CP2 under certain conditions. Half grid diagrams prove every link can be represented by a special type of grid diagram.
problem Representing links using grid diagrams and related invariants.
method Defining half grid diagrams and constructing canonical pairs, proving equivalence to Jones' construction, relating to classical link invariants.
result Established a new method to relate the oriented Thompson index to classical link invariants and provided bounds for knot invariants.
Grid homology confirms the Upsilon invariant in knot theory.
problem Verifying the equivalence of Upsilon invariants in knot theory.
method Reconstructed Upsilon invariant using grid homology and proved equivalence.
result Upsilon invariants in knot Floer and grid homology are equivalent.
GridPyM handles grid diagrams for knot theory.
problem Handling grid diagrams for knot theory.
method Generates and simplifies grids, models local transformations.
result Models local transformations between grid diagrams.
Grid homology theory for spatial graphs extends skein sequence.
problem No specific problem stated; focuses on extending a sequence.
method Defined grid homology theory for spatial graphs and extended skein sequence.
result Skein exact sequence extended to grid homology for spatial graphs.
New method finds grid diagrams for many fibered knots.
problem Detecting fibered knots using grid diagrams.
method Developed an efficient method to identify grid diagrams with unique maximal Alexander grading states.
result Found suitable grid diagrams for 5385 of 5397 fibered prime knots with crossing number ≤ 13.
Extends knot invariant to filtered grid complexes.
problem Knot invariants and grid complexes.
method Combining Ozsváth-Szabó-Stipsicz crossing-change maps with Alishahi-Eftekhary l(K) invariant.
result Combinatorial formulation of knot invariant.
Grid homology properties for MOY graphs studied.
problem Defining and studying properties of grid homology for MOY graphs.
method Defined grid homology from Harvey and O'Donnol's work. Studied properties using oriented skein relation, edge contraction, and parallel edge unification.
result Properties of grid homology for MOY graphs were studied and defined.
Grid homology invariant proved for lens space links.
problem Proving combinatorial invariance of grid homology for lens space links.
method Combining combinatorial methods with sign assignments to prove invariance.
result Grid homology is a link invariant for lens space links.
New trading strategy beats traditional grid in crypto markets.
problem Low expected return of traditional grid trading strategy.
method Dynamic Grid Trading (DGT) strategy that adapts to market conditions.
result DGT strategy outperforms traditional grid and buy-and-hold strategies.
New method constructs moduli spaces of Lagrangian surfaces in CP^2 from grid diagrams.
problem Constructing explicit examples of triple grid diagrams for Lagrangian surfaces in CP^2.
method Elegant geometric construction reducing to linear algebra.
result Explicit construction of moduli space of triple grid diagrams.
Develops equivariant grid homology for strongly invertible knots.
problem Invariants of strongly invertible knots.
method Equivariant grid diagrams and mapping cones.
result Equivariant unknotting numbers and genus bounds.
Optimal Volt/VAR control rules designed using deep learning.
problem Designing optimal Volt/VAR control rules for DERs to regulate voltage fluctuations.
method Formulated as a deep learning problem, where a DNN emulates Volt/VAR dynamics and optimizes rule parameters.
result DNN-based optimization outperforms MINLP in efficiency and accuracy.
Computes homology of an obstruction chain complex in grid homology.
problem Computing the homology of an obstruction chain complex in grid homology.
method Defined and computed the homology of the obstruction chain complex of the full grid.
result Results about the existence of sign assignments in grid homology.
Grid homology shows knot unknotting lower bound.
problem Knot unknotting number determination
method Grid homology analysis
result Torsion homology classes order bounds unknotting number
SKI accelerates GP inference with sparse grids to handle higher dimensions.
problem SKI scales poorly in high dimensions due to dense grid size.
method Sparse grids within SKI framework, novel matrix-vector multiplication algorithm.
result SKI can be scaled to higher dimensions while maintaining accuracy.
The paper develops algorithms to accurately identify power grid topology.
problem Identifying the exact topology of a power grid in real-time.
method Graphical model framework for topology estimation using voltage measurements.
result Exact extraction of operational topology is theoretically guaranteed for grid topologies without three-node cycles.
System detects power grid health using AI and machine learning.
problem Detecting and preventing power grid malfunctions.
method Artificial intelligence, machine learning, recurrent neural networks, SVM, LSTM.
result High accuracy in detecting grid health, scalable for complex architectures.
This paper improves image generation models using a multi-grid method.
problem Improving image generation models.
method A multi-grid method for learning energy-based generative ConvNet models.
result The multi-grid method outperforms traditional models.
Combinatorial proof shows knot invariant in Lipshitz's grid homology.
problem Proving knot invariance in Lipshitz's grid homology.
method Purely combinatorial proof.
result Proves 'minus' version of Lipshitz's double-point enhanced grid homology is a knot invariant.
The paper studies grid homology for spatial graphs and proves a Künneth formula for connected sums.
problem Understanding grid homology for spatial graphs with various types of edges.
method Developed grid homology for spatial graphs with cut edges and applied it to prove a Künneth formula for connected sums.
result A Künneth formula for knot Floer homology of connected sums is proven using grid homology.
We present a method for proving upper bounds on the eigenvalues of the graph Laplacian. A main step involves choosing an appropriate "Riemannian" metric to uniformize the geometry of the graph. In many interesting cases, the existence of such a metric is shown by examining the combinatorics of special types of flows. T…
Hexagon grid patterns emerge from conformal isometry in grid cell neural networks.
problem Understanding the algebraic, geometric, and topological properties of grid cells.
method Investigating recurrent neural network models of grid cells, focusing on Lie group and Lie algebra representations, conformal isometry, and hexagon periodic patterns.
result Conformal isometry leads to hexagon periodic patterns in grid cell responses and accurate path integration.
Minimal grid diagrams for 12-crossing prime knots identified.
problem Identifying minimal grid diagrams for prime knots.
method Listed minimal grid diagrams for 12-crossing prime knots.
result Provided a list of minimal grid diagrams for 12-crossing prime knots.
The paper analyzes how grid cells perform path integration and learns hexagon grid patterns.
problem Understanding how grid cells perform path integration calculations.
method Theoretical analysis of a general representation model of path integration by grid cells, identifying group representation and isotropic scaling conditions.
result The learned model of hexagon grid patterns is capable of accurate long distance path integration.
Let L be a link in an integral homology three-sphere. We give a description of the Heegaard Floer homology of integral surgeries on L in terms of some data associated to L, which we call a complete system of hyperboxes for L. Roughly, a complete systems of hyperboxes consists of chain complexes for (some versions of) t…
It will be shown that according to theorems of K. Menger, every neuron grid if identified with a curve is able to preserve the adopted qualitative structure of a data space. Furthermore, if this identification is made, the neuron grid structure can always be mapped to a subset of a universal neuron grid which is constr…
The increasing penetration of distributed energy resources poses numerous reliability issues to the urban distribution grid. The topology estimation is a critical step to ensure the robustness of distribution grid operation. However, the bus connectivity and grid topology estimation are usually hard in distribution gri…
Grid diagrams define invariants for knots in lens spaces.
problem Characterizing knots in lens spaces using grid diagrams.
method Using combinatorial grid diagrams to define invariants.
result Invariants are equivalent to those defined in previous works.
The paper studies how grid cell patterns emerge in neural networks.
problem Understanding how grid cells in the brain form hexagonal firing patterns.
method Training recurrent neural networks with conformal normalization of velocity inputs.
result Conformal normalization is crucial for the emergence of hexagonal grid patterns in neural networks.
Combinatorial proof of grid homology properties.
problem Properties of double-point enhanced grid homology.
method Purely combinatorial proof, extended to Z coefficients. result Skein exact sequence obeyed by grid homology.
System identifies power grid location from media recordings.
problem Identifying the origin of power distribution grid from media recordings.
method Cascaded SVM and pole-matching classifiers for grid identification.
result Cascaded system improves accuracy by 15.57%.
DeepGDL models create realistic power grids from confidential data.
problem Creating realistic power grids from confidential data.
method Graph distribution learning (GDL) with a deep nonlinear recurrent structure.
result DeepGDL models accurately create synthetic power grids.
Study finds cherry-picking load shaping strategies outperforms others in reducing grid CO2 emissions.
problem Lack of detailed counterfactual data makes it hard to assess load shaping strategies' effectiveness.
method Calibrated granular ERCOT simulations for counterfactual analysis of load shaping strategies.
result LMP-based load shaping outperforms other strategies in reducing grid CO2 emissions.