Courant Institute of Mathematical Sciences

About: Courant Institute of Mathematical Sciences is a education organization based out in New York, New York, United States. It is known for research contribution in the topics: Nonlinear system & Boundary value problem. The organization has 2414 authors who have published 7759 publications receiving 439773 citations. The organization is also known as: CIMS & New York University Department of Mathematics.

Creating an automated trigger for sepsis clinical decision support at emergency department triage using machine learning

Abstract: Objective To demonstrate the incremental benefit of using free text data in addition to vital sign and demographic data to identify patients with suspected infection in the emergency department. Methods This was a retrospective, observational cohort study performed at a tertiary academic teaching hospital. All consecutive ED patient visits between 12/17/08 and 2/17/13 were included. No patients were excluded. The primary outcome measure was infection diagnosed in the emergency department defined as a patient having an infection related ED ICD-9-CM discharge diagnosis. Patients were randomly allocated to train (64%), validate (20%), and test (16%) data sets. After preprocessing the free text using bigram and negation detection, we built four models to predict infection, incrementally adding vital signs, chief complaint, and free text nursing assessment. We used two different methods to represent free text: a bag of words model and a topic model. We then used a support vector machine to build the prediction model. We calculated the area under the receiver operating characteristic curve to compare the discriminatory power of each model. Results A total of 230,936 patient visits were included in the study. Approximately 14% of patients had the primary outcome of diagnosed infection. The area under the ROC curve (AUC) for the vitals model, which used only vital signs and demographic data, was 0.67 for the training data set, 0.67 for the validation data set, and 0.67 (95% CI 0.65–0.69) for the test data set. The AUC for the chief complaint model which also included demographic and vital sign data was 0.84 for the training data set, 0.83 for the validation data set, and 0.83 (95% CI 0.81–0.84) for the test data set. The best performing methods made use of all of the free text. In particular, the AUC for the bag-of-words model was 0.89 for training data set, 0.86 for the validation data set, and 0.86 (95% CI 0.85–0.87) for the test data set. The AUC for the topic model was 0.86 for the training data set, 0.86 for the validation data set, and 0.85 (95% CI 0.84–0.86) for the test data set. Conclusion Compared to previous work that only used structured data such as vital signs and demographic information, utilizing free text drastically improves the discriminatory ability (increase in AUC from 0.67 to 0.86) of identifying infection.

Kähler-Einstein metrics and the generalized Futaki invariant

Abstract: In 1983, Futaki introduced his famous invariant. This invariant generalizes the obstruction of Kazdan-Warner to prescribing Gauss curvature on S 2 (cf. [Ful l ) . The Futaki invariant is defined for any compact K/ihler manifold with positive first Chern class that has nontrivial holomorphic vector fields. It is a Lie algebraic character from the Lie algebra of holomorphic vector fields into •, and its vanishing is a necessary condition for the existence of a K/ihler-Einstein metric on the underlying manifold. Therefore, it can be used to test the existence of K/ihlerEinstein metrics on a given compact K/ihler manifold with positive first Chern class. An excellent reference on the Futaki invariant is Futaki 's book [Fu2]. Until now, all known nontrivial obstructions to K/ihler-Einstein metrics come from holomorphic vector fields. This suggests the following conjecture.

Towards exact geometric computation

Abstract: Exact computation is assumed in most algorithms in computational geometry. In practice, implementors perform computation in some fixed-precision model, usually the machine floating-point arithmetic. Such implementations have many well-known problems, here informally called “robustness issues”. To reconcile theory and practice, authors have suggested that theoretical algorithms ought to be redesigned to become robust under fixed-precision arithmetic. We suggest that in many cases, implementors should make robustness a non-issue by computing exactly. The advantages of exact computation are too many to ignore. Many of the presumed difficulties of exact computation are partly surmountable and partly inherent with the robustness goal. This paper formulates the theoretical framework for exact computation based on algebraic numbers. We then examine the practical support needed to make the exact approach a viable alternative. It turns out that the exact computation paradigm encompasses a rich set of computational tactics. Our fundamental premise is that the traditional “BigNumber” package that forms the work-horse for exact computation must be reinvented to take advantage of many features found in geometric algorithms. Beyond this, we postulate several other packages to be built on top of the BigNumber package.

The large deviation principle for the Erdős-Rényi random graph

Abstract: What does an Erdos-Renyi graph look like when a rare event happens? This paper answers this question when p is fixed and n tends to infinity by establishing a large deviation principle under an appropriate topology. The formulation and proof of the main result uses the recent development of the theory of graph limits by Lovasz and coauthors and Szemeredi's regularity lemma from graph theory. As a basic application of the general principle, we work out large deviations for the number of triangles in G(n,p). Surprisingly, even this simple example yields an interesting double phase transition.

Differentiation of discrete multidimensional signals

Abstract: We describe the design of finite-size linear-phase separable kernels for differentiation of discrete multidimensional signals. The problem is formulated as an optimization of the rotation-invariance of the gradient operator, which results in a simultaneous constraint on a set of one-dimensional low-pass prefilter and differentiator filters up to the desired order. We also develop extensions of this formulation to both higher dimensions and higher order directional derivatives. We develop a numerical procedure for optimizing the constraint, and demonstrate its use in constructing a set of example filters. The resulting filters are significantly more accurate than those commonly used in the image and multidimensional signal processing literature.