arXiv research

Mathematical method based on a direct or indirect analysis of growth rates is described. It is shown how simple assumptions and a relatively easy analysis can be used to describe mathematically complicated trends and to predict growth. Only rudimentary knowledge of calculus is required. Projected trajectories based on …

The concept of $\Zn$-supermanifold has been recently proposed as a natural generalization of classical ($\Zs$-graded) supergeometry, allowing for more complicated commutativity constraints. Here we continue the study of $\Zn$-supergeometry by developing the foundations of differential calculus on $\Zn$-supermanifolds.

Abstract: Extends geometric concepts to generalized tangent bundle and describes flows.

In this paper, a direct continuation of math.DG/0411165, we generalize S. Lie's linearization criterion of an ordinary second order differential equation to the case of several independent variables (x^1, x^2 ..., x^n), n >1, and a single dependent variable y. Strikingly, as in math.DG/0411165, the (complicated) charac…

We outline in detail the general caloron correspondence for the group of automorphisms of an arbitrary principal $G$-bundle $Q$ over a manifold $X$, including the case of the gauge group of $Q$. These results are used to define characteristic classes of gauge group bundles. Explicit but complicated differential form re…

Teaches matrix calculus for machine learning and optimization.

EI-MTD defends edge intelligence against adversarial attacks with dynamic scheduling.

The Killing operator on a Riemannian manifold is a linear differential operator on vector fields whose kernel provides the infinitesimal Riemannian symmetries. The Killing operator is best understood in terms of its prolongation, which entails some simple tensor identities. These simple identities can be viewed as aris…

We define a differential graded algebra associated to Legendrian knots in Seifert fibered spaces with transverse contact structures. This construction is distinguished from other combinatorial realizations of contact homology invariants by the existence of orbifold points in the Reeb orbit space of the contact manifold…

Based on the Lie theoretical methods of algebraic Fourier transformation, we classify in the case of generic values of inducing parameters the scalar singular vectors corresponding to the diagonal branching rules for scalar generalized Verma modules in the case of orthogonal Lie algebra and its conformal parabolic suba…

VSDN models sporadic time series with neural SDEs.

We investigate the computational aspects of the basket CDS pricing with counterparty risk under a credit contagion model of multinames. This model enables us to capture the systematic volatility increases in the market triggered by a particular bankruptcy. The drawback of this problem is its analytical complication due…

This paper introduces a spline-based method for nonparametric ADVI that handles complex posterior distributions.

The fundamental tool in the classification of orthogonal coordinate systems in which the Hamilton-Jacobi and other prominent equations can be solved by a separation of variables are second order Killing tensors which satisfy the Nijenhuis integrability conditions. The latter are a system of three non-linear partial dif…

Data that is gathered adaptively --- via bandit algorithms, for example --- exhibits bias. This is true both when gathering simple numeric valued data --- the empirical means kept track of by stochastic bandit algorithms are biased downwards --- and when gathering more complicated data --- running hypothesis tests on c…

MissNODAG learns cyclic causal graphs from incomplete data.

Neural system optimizes glucose levels in diabetics.

Study describes severe dengue ICU patients in Brazil, 2012-2024.

Type 2 diabetes mellitus (T2DM) is a chronic disease that often results in multiple complications. Risk prediction and profiling of T2DM complications is critical for healthcare professionals to design personalized treatment plans for patients in diabetes care for improved outcomes. In this paper, we study the risk of …

Deep learning models predict postoperative complications more accurately than random forests.

The change of numeraire gives very important computational simplification in option pricing. This technique reduces the number of sources of risks that need to be accounted for and so it is useful in pricing complicated derivatives that have several sources of risks. In this article, we considered the underlying mathem…

A Poincaré-Hopf Theorem for line fields with point singularities on orientable surfaces can be found Hopf's 1956 Lecture Notes on Differential Geometry. In 1955 Markus presented such a theorem in all dimensions, but Markus' statement only holds in even dimensions $2k \geq 4$. In 1984 Jänich presented a Poincaré-Hopf th…

Hamiltonian Monte Carlo (HMC) is arguably the dominant statistical inference algorithm used in most popular "first-order differentiable" Probabilistic Programming Languages (PPLs). However, the fact that HMC uses derivative information causes complications when the target distribution is non-differentiable with respect…

Epileptic seizure activity shows complicated dynamics in both space and time. To understand the evolution and propagation of seizures spatially extended sets of data need to be analysed. We have previously described an efficient filtering scheme using variational Laplace that can be used in the Dynamic Causal Modelling…

Most known examples of doubly periodic minimal surfaces in $\mathbb{R}^3$ with parallel ends limit as a foliation of $\mathbb{R}^3$ by horizontal noded planes, with the location of the nodes satisfying a set of balance equations. Conversely, for each set of points providing a balanced configuration, there is a correspo…

Paper proposes a differentially private test for joint dependence among random vectors.

New method calibrates complex ODEs from noisy data using neural networks.

There has been much recent, exciting work on combining the complementary strengths of latent variable models and deep learning. Latent variable modeling makes it easy to explicitly specify model constraints through conditional independence properties, while deep learning makes it possible to parameterize these conditio…

Deep learning models are often trained on datasets that contain sensitive information such as individuals' shopping transactions, personal contacts, and medical records. An increasingly important line of work therefore has sought to train neural networks subject to privacy constraints that are specified by differential…

Compared with artificial neural networks (ANNs), spiking neural networks (SNNs) are promising to explore the brain-like behaviors since the spikes could encode more spatio-temporal information. Although pre-training from ANN or direct training based on backpropagation (BP) makes the supervised training of SNNs possible…

Parametric quantile regressions are a useful tool for creating probabilistic energy forecasts. Nonetheless, since classical quantile regressions are trained using a non-differentiable cost function, their creation using complex data mining techniques (e.g., artificial neural networks) may be complicated. This article p…

We solve the mean parametrization of von Mises-Fisher distribution.

Paper improves privacy bounds for shuffle model using novel numerical techniques.

The physics informed neural network (PINN) is evolving as a viable method to solve partial differential equations. In the recent past PINNs have been successfully tested and validated to find solutions to both linear and non-linear partial differential equations (PDEs). However, the literature lacks detailed investigat…

A spiral unibike track emerges from a mathematical construction.

Reliable training of generative adversarial networks (GANs) typically require massive datasets in order to model complicated distributions. However, in several applications, training samples obey invariances that are \textit{a priori} known; for example, in complex physics simulations, the training data obey universal …

We construct families of manifolds that have pairs of genus $g$ Heegaard splittings that must be stabilized roughly $g$ times to become equivalent. We also show that when two unstabilized, boundary-unstabilized Heegaard splittings are amalgamated by a "sufficiently complicated" map, the resulting splitting is unstabili…

FuDGE estimates differences between functional graphs in high-dimensional settings.

Stratifying patients at risk for postoperative complications may facilitate timely and accurate workups and reduce the burden of adverse events on patients and the health system. Currently, a widely-used surgical risk calculator created by the American College of Surgeons, NSQIP, uses 21 preoperative covariates to asse…

Iteratively reweighted least squares (IRLS) is a widely-used method in machine learning to estimate the parameters in the generalised linear models. In particular, IRLS for L1 minimisation under the linear model provides a closed-form solution in each step, which is a simple multiplication between the inverse of the we…

Asynchronous stochastic gradient descent (ASGD) is a popular parallel optimization algorithm in machine learning. Most theoretical analysis on ASGD take a discrete view and prove upper bounds for their convergence rates. However, the discrete view has its intrinsic limitations: there is no characterization of the optim…

Diabetes is a major public health problem in the United States, affecting roughly 30 million people. Diabetes complications, along with the mental health comorbidities that often co-occur with them, are major drivers of high healthcare costs, poor outcomes, and reduced treatment adherence in diabetes. Here, we evaluate…

Partial connections are (singular) differential systems generalizing classical connections on principal bundles, yielding analogous decompositions for manifolds with nonfree group actions. Connection forms are interpreted as maps determining projections of the tangent bundle onto the partial connection; this approach e…

Let $M$ be a 3-manifold with torus boundary components $T_1$ and $T_2$. Let $φ\colon T_1 \to T_2$ be a homeomorphism, $M_φ$ the manifold obtained from $M$ by gluing $T_1$ to $T_2$ via the map $φ$, and $T$ the image of $T_1$ in $M_φ$. We show that if $φ$ is "sufficiently complicated" then any incompressible or strongly …

LEM efficiently models long-term sequences with gradients.

Solar improves variable selection in high-dimensional data with complicated dependence structures.

The theory of differential forms began with a discovery of Poincare who found conservation laws of a new type for Hamiltonian systems - The Integral Invariants. Even in the absence of non-trivial integrals of motion, there exist invariant differential forms: a symplectic two-form, or a contact one-form for geodesic flo…

How can prior knowledge on the transformation invariances of a domain be incorporated into the architecture of a neural network? We propose Equivariant Transformers (ETs), a family of differentiable image-to-image mappings that improve the robustness of models towards pre-defined continuous transformation groups. Throu…