Differentiability of projective transformations in dimension 2

2020 ◽  
Vol 20 (4) ◽  
pp. 553-557
Author(s):  
Julius Lang

AbstractIt is proven by elementary methods that in dimension 2, every locally injective continuous map, sending the curves of a Ck-spray to curves of another Ck-spray as oriented point sets, is a Ck-diffeomorphism. This extends the result [1] for dimension three and higher from 1965.

1962 ◽  
Vol 40 (8) ◽  
pp. 992-999
Author(s):  
R. O. Robinson

The derivation of thermodynamic and magnetic properties of a crystal, on the basis of the Ising model, concerns essentially two non-commuting matrices which are here called A and B. In particular, one must evaluate the traces of Am, Bm, AmBm, and (AB)m for general integer m. The first two traces relate to the one-dimensional case; the third and fourth to the two-dimensional case. A and B are of dimension 2n, m and n being respectively the numbers of rows and columns.Previous methods of evaluating these traces have been elaborate, involving concepts such as spinor rotations in a space of 2n dimensions. In this paper elementary methods are used to derive closed expressions for the first three and the fourth is evaluated in an interesting special case.To date the Ising model has been treated exactly in two cases only. For this reason the search for simple calculational methods is of particular importance.


2003 ◽  
Vol 40 (3) ◽  
pp. 269-286 ◽  
Author(s):  
H. Nyklová

In this paper we study a problem related to the classical Erdos--Szekeres Theorem on finding points in convex position in planar point sets. We study for which n and k there exists a number h(n,k) such that in every planar point set X of size h(n,k) or larger, no three points on a line, we can find n points forming a vertex set of a convex n-gon with at most k points of X in its interior. Recall that h(n,0) does not exist for n = 7 by a result of Horton. In this paper we prove the following results. First, using Horton's construction with no empty 7-gon we obtain that h(n,k) does not exist for k = 2(n+6)/4-n-3. Then we give some exact results for convex hexagons: every point set containing a convex hexagon contains a convex hexagon with at most seven points inside it, and any such set of at least 19 points contains a convex hexagon with at most five points inside it.


2010 ◽  
Vol 36 (8) ◽  
pp. 1073-1083 ◽  
Author(s):  
Xu-Fang PANG ◽  
Ming-Yong PANG ◽  
Chun-Xia XIAO
Keyword(s):  

1992 ◽  
Author(s):  
L. V. Meisel ◽  
M. A. Johnson

1986 ◽  
Vol 12 (1) ◽  
pp. 377 ◽  
Author(s):  
Morgan
Keyword(s):  

2021 ◽  
Vol 186 (Supplement_1) ◽  
pp. 445-451
Author(s):  
Yifei Sun ◽  
Navid Rashedi ◽  
Vikrant Vaze ◽  
Parikshit Shah ◽  
Ryan Halter ◽  
...  

ABSTRACT Introduction Early prediction of the acute hypotensive episode (AHE) in critically ill patients has the potential to improve outcomes. In this study, we apply different machine learning algorithms to the MIMIC III Physionet dataset, containing more than 60,000 real-world intensive care unit records, to test commonly used machine learning technologies and compare their performances. Materials and Methods Five classification methods including K-nearest neighbor, logistic regression, support vector machine, random forest, and a deep learning method called long short-term memory are applied to predict an AHE 30 minutes in advance. An analysis comparing model performance when including versus excluding invasive features was conducted. To further study the pattern of the underlying mean arterial pressure (MAP), we apply a regression method to predict the continuous MAP values using linear regression over the next 60 minutes. Results Support vector machine yields the best performance in terms of recall (84%). Including the invasive features in the classification improves the performance significantly with both recall and precision increasing by more than 20 percentage points. We were able to predict the MAP with a root mean square error (a frequently used measure of the differences between the predicted values and the observed values) of 10 mmHg 60 minutes in the future. After converting continuous MAP predictions into AHE binary predictions, we achieve a 91% recall and 68% precision. In addition to predicting AHE, the MAP predictions provide clinically useful information regarding the timing and severity of the AHE occurrence. Conclusion We were able to predict AHE with precision and recall above 80% 30 minutes in advance with the large real-world dataset. The prediction of regression model can provide a more fine-grained, interpretable signal to practitioners. Model performance is improved by the inclusion of invasive features in predicting AHE, when compared to predicting the AHE based on only the available, restricted set of noninvasive technologies. This demonstrates the importance of exploring more noninvasive technologies for AHE prediction.


Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 318
Author(s):  
Jiafan Zhang ◽  
Yuanyuan Meng

In this paper, we use the elementary methods and properties of classical Gauss sums to study the calculation problems of some mean values of character sums of special polynomials, and obtained several interesting calculation formulae for them. As an application, we give a criterion for determining that 2 is the cubic residue for any odd prime p.


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