The study measures similarity in introductory programming items, offering a method and evaluation.
problem Measuring similarity in a diverse pool of programming items for personalized learning.
method General approach to measuring similarity, specific measures for introductory programming, three levels of abstraction evaluation.
result Evaluation of similarity measures using diverse programming environments.
Graph-based approach repairs programs from diagnostic feedback.
problem Learning to repair programs from limited labeled data and compiler error messages.
method Introduces program-feedback graph and graph neural network for reasoning, and self-supervised learning with unlabeled programs.
result DrRepair significantly outperforms prior work, achieving high repair rates.
An article based on a four-lecture introductory minicourse on minimal surface theory given at the 2013 summer program of the Institute for Advanced Study and the Park City Mathematics Institute.
An introductory course on hyperbolic geometry for advanced students.
problem None explicitly stated; focuses on teaching hyperbolic geometry.
method Textbook format, intended for advanced undergraduate and early graduate students.
result Teaches advanced students about hyperbolic geometry.
These notes are an expanded version of an introductory lecture on contact geometry given at the 2001 Georgia Topology Conference. They are intended to present some of the "topological" aspects of three dimensional contact geometry.
This is an introductory text to differential geometry (written in Polish) aimed for high-school students.
This is a detailed introductory survey of the cohomological dimension theory of compact metric spaces.
Course on arithmetic lattices at EPFL.
problem Understanding arithmetic lattices.
method Introductory course on arithmetic lattices.
result Introduction to arithmetic lattices.
This book is a textbook for the basic course of differential geometry. It is recommended as an introductory material for this subject.
Paper tackles zero-shot code feedback using rubric sampling with deep learning.
problem Lack of historical data for supervised learning in introductory programming assignments.
method Human-in-the-loop rubric sampling with deep learning inference.
result Autonomous feedback for first students is more accurate than data-hungry algorithms and approaches human level fidelity.
We present some plausible definitions for the tangent grupoid of a manifold M, as well as some of the known applications of the structure. This is a kind of introductory note.
These notes provide an introductory exposition of the Seiberg-Witten gauge theory. They collect the material presented in a series of seminars given by the author at the University of Milano.
This is an expository and introductory note on some results obtained in "Coisotropic embeddings in Poisson manifolds" (ArXiv math/0611480). Some original material is contained in the last two sections, where we consider linear Poisson structures.
We propose an extensive analysis of the behavior of majority votes in binary classification. In particular, we introduce a risk bound for majority votes, called the C-bound, that takes into account the average quality of the voters and their average disagreement. We also propose an extensive PAC-Bayesian analysis that …
Attention mechanism improves various NLP tasks.
problem Improving performance in natural language processing tasks.
method Assigning importance scores to sequence elements for encoding.
result Significant improvement in various NLP tasks.
Algebras of generalized functions offer possibilities beyond the purely distributional approach in modelling singular quantities in non-smooth differential geometry. This article presents an introductory survey of recent developments in this field and highlights some applications in mathematical physics.
The homology and cohomology of quandles and racks are used in knot theory: given a finite quandle and a cocycle, we can construct a knot invariant. This is a quick introductory survey to the invariants of knots derived from quandles and racks.
Geometric flows help solve the swampland problem by preserving Einstein equations.
problem Addressing the swampland conjecture in string theory.
method Analyzing scalar and metric bubble solutions under Perelman's flow, deriving geometric flow equations, and introducing an additional energy-momentum tensor term.
result A supplementary energy-momentum tensor term precisely reproduces the infinite tower of states with exponentially dropping masses.
This paper has two purposes. First it partially extends the result in the author's previous work concerning the asymptotic expansion of the Tian-Yau metrics, by considering a slightly larger class of quasi-projective manifolds. This text is also intended to provide a quick introductory reference to the study of Ricci-f…
The Ricci flow is a parabolic evolution equation in the space of Riemannian metrics of a smooth manifold. To some extent, Einstein equations give rise to a similar hyperbolic evolution. The present text is an introductory exposition to Bianchi-Ricci and Bianchi-Einstein flows, that is, the restricted finitely dimension…
We give an introductory account of functional determinants of elliptic operators on manifolds and Polyakov-type formulas for their infinitesimal and finite conformal variations. We relate this to extremal problems and to the Q-curvature on even-dimensional conformal manifolds. The exposition is self-contained, in the s…
This is an introductory text on the more topological aspects of contact geometry, written for the Handbook of Differential Geometry vol. 2. After discussing (and proving) some of the fundamental results of contact topology (neighbourhood theorems, isotopy extension theorems, approximation theorems), I move on to a deta…
These lecture notes are a systematic and self-contained exposition of the cohomological theories naturally related to partial differential equations: the Vinogradov C-spectral sequence and the C-cohomology, including the formulation in terms of the horizontal (characteristic) cohomology. Applications to computing invar…
Dissertation tackles geodesic ray transform on Riemannian manifolds.
problem Determining functions from line integrals along geodesics.
method Establishes conditions for unique and stable determination of functions.
result New numerical model for computed tomography imaging created.
Paper defines nearly Frobenius algebras and explores their properties.
problem Defining and studying algebras without traces.
method Survey of foundational results and applications.
result Characterization of nearly Frobenius algebras and their properties.
We survey work on the topology of the space AH(M) of all (marked) hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold M with boundary. The interior of AH(M) is quite well-understood, but the topology of the entire space can be quite complicated. However, the topology is well-behaved at many points …
We give an introductory survey on the universal Vassiliev invariant called the perturbative series expansion of the Chern-Simons theory of links in euclidean space, and on its relation with the Kontsevich integral. We also prove an original geometric property of the anomaly of Bott, Taubes, Altschuler, Freidel and D. T…
The paper explains how importance sampling can be used for optimization of rare events.
problem Minimizing tail risks in stochastic optimization formulations.
method Importance sampling for reducing sample requirements in estimating rare events.
result Effective importance sampling techniques for optimization of rare events.
Notes on linear classifiers and outliers, improving decision boundaries.
problem Negative impact of outliers on linear classifier decision boundaries.
method Use of least-squares cost function, scaling to reduce outlier significance.
result Improved decision boundaries through scaling outliers.
Intro to Poisson geometry, focusing on basics and recent tools.
problem Understanding Poisson structures and their applications.
method Exercises leading to conclusions on foundational and recent topics.
result Introduction to Poisson geometry basics and recent developments.
This paper introduces accurate coresets for basic problems.
problem Solving problems on a small set that approximates the full set.
method Simple and accurate coreset constructions for basic problems.
result Unified notation and comparison table for modern results.
Explains Arnold's J+ invariant for curves, using basic math.
problem Understanding Arnold's J+ invariant for planar curves.
method Explains computation methods like Viro's sum.
result Basic undergraduate math suffices to grasp the invariant.
A tutorial on variational inference for high-dimensional models.
problem Approximating marginal likelihood and posterior in Bayesian models.
method Parametric approach to variational inference.
result Variational inference is now preferred for high-dimensional models and large datasets.
This is an introductory article on high dimensional knots for the beginners. High dimensional knot theory is an exciting field. It is a field of knot theory, which is one of topology and is connected with many ones. In this article we use few literal expressions, equations, functions, etc. We barely suppose that the re…
Expanding submanifold theory in products of multiple space forms.
problem Generalizing tensors and the Bonnet theorem for products of many space forms.
method Generalizing tensors and the Bonnet theorem from products of 2 space forms to products of many space forms.
result Generalization of tensors and the Bonnet theorem to products of many space forms.
Lecture notes on crystallography and discrete surfaces.
problem Mathematical modeling of crystal structures.
method Variational principle and discrete surface theory.
result Most symmetric crystal structures identified.
Introduces machine learning basics and algorithms.
problem Developing and analyzing machine learning algorithms.
method Mathematical foundations, optimization, statistical prediction, reproducing kernel theory, Hilbert space techniques, sampling methods, Markov chains, graphical models, variational methods, deep learning, clustering, factor analysis, manifold learning.
result Theoretical support and practical algorithms for machine learning.
Research on unique continuation principles in medical and seismic imaging.
problem Understanding unique continuation principles for inverse problems.
method Integral geometry and fractional calculus methods applied to various imaging problems.
result Developed new techniques for solving inverse problems with partial data.
This article is a standalone introduction to sutured Floer homology for graduate students in geometry and topology. It is divided into three parts. The first part is an introductory level exposition of Lagrangian Floer homology. The second part is a construction of Heegaard Floer homology as a special, and slightly mod…
This is an introductory article about the Boy surface. Boy found 1901 that RP2 can be immersed into R3, and published it. (The image of) the immersion is called the Boy surface after Boy's discovery. We have created a way to construct the Boy surface by using a pair of scissors, a piece of paper…
Notes on Whitney towers in 4-manifolds, focusing on local surface manipulations and invariants.
problem Classifying and understanding Whitney towers in 4-manifolds.
method Local manipulations of surfaces, definitions of Whitney towers and trees, geometric Jacobi identities, classification of twisted Whitney towers.
result Classification of order n twisted Whitney towers in the 4-ball and related invariants.
Survey on minimal surface equation on Cartan-Hadamard manifolds.
problem Asymptotic Dirichlet problem for minimal surface equation.
method Introduction of Cartan-Hadamard manifolds, discussion of results and methods used in proofs.
result Discussion of results and methods used in proofs.
This is an introductory review of topological field theories (TFTs) called AKSZ sigma models. The AKSZ construction is a mathematical formulation for the construction and analyses of a large class of TFTs, inspired by the Batalin-Vilkovisky formalism of gauge theories. We begin by considering a simple two-dimensional t…
In 2008, Lomonaco and Kauffman introduced a knot mosaic system to define a quantum knot system. A quantum knot is used to describe a physical quantum system such as the topology or status of vortexing that occurs on a small scale can not see. Kuriya and Shehab proved that knot mosaic type is a complete invariant of tam…
This monograph describes a Riemannian geometric reduction approach to the three-body problem. The fundamental theorems are presented in the introductory part, whereas their proofs are provided in later chapters where specific topics are analyzed in more detail. The basic idea is to reduce the kinematic and dynamics of …
Study cliquet options pricing using geometric Meixner model.
problem Pricing cliquet options in a geometric Meixner model.
method Inference of semi-analytic expressions using Meixner distribution and Fourier transform techniques.
result Inferred semi-analytic expressions for cliquet option price.
SED integrates synthesis, execution, and debugging for neural program synthesis.
problem Challenges in synthesizing complex programs that match specifications.
method SED combines synthesis, execution, and debugging to improve neural program generation.
result SED reduces error rates and outperforms standard decoding methods.
Stochastic programs simplify complex models with noise and nondeterminism.
problem Handling models with nuisance parameters, noise, and nondeterminism.
method Developed a reference implementation for stochastic probabilistic programs and inference.
result Efficient inference in models with noise and nondeterminism is possible.