scholarly journals On the local pairing behavior of critical points and roots of random polynomials

2020 ◽  
Vol 25 (0) ◽  
Author(s):  
Sean O’Rourke ◽  
Noah Williams
Keyword(s):  
Critical Points ◽  
Random Polynomials ◽  
Pairing Behavior ◽  
Local Pairing

scholarly journals Pairing of zeros and critical points for random polynomials

2017 ◽  
Vol 53 (3) ◽  
pp. 1498-1511 ◽  
Author(s):  
Boris Hanin
Keyword(s):  
Critical Points ◽  
Random Polynomials

scholarly journals Pairing between zeros and critical points of random polynomials with independent roots

2018 ◽  
Vol 371 (4) ◽  
pp. 2343-2381 ◽  
Author(s):  
Sean O’Rourke ◽  
Noah Williams
Keyword(s):  
Critical Points ◽  
Random Polynomials

scholarly journals Correlations and Pairing between Zeros and Critical Points of Gaussian Random Polynomials

2013 ◽  
Vol 2015 (2) ◽  
pp. 381-421 ◽  
Author(s):  
Boris Hanin
Keyword(s):  
Critical Points ◽  
Random Polynomials

scholarly journals Distances between zeroes and critical points for random polynomials with i.i.d. zeroes

2019 ◽  
Vol 24 (0) ◽  
Author(s):  
Zakhar Kabluchko ◽  
Hauke Seidel
Keyword(s):  
Critical Points ◽  
Random Polynomials

The relationship of the scleractinian corals to the rugose corals

Paleobiology ◽  
1980 ◽  
Vol 6 (02) ◽  
pp. 146-160 ◽  
Author(s):  
William A. Oliver
Keyword(s):  
Critical Points ◽  
Scleractinian Corals ◽  
Convincing Evidence ◽  
Early Triassic ◽  
Rugose Corals ◽  
History Of ◽  
The Subject ◽  
Relationship Of ◽  
The Relationship ◽  
Biologic Factors

The Mesozoic-Cenozoic coral Order Scleractinia has been suggested to have originated or evolved (1) by direct descent from the Paleozoic Order Rugosa or (2) by the development of a skeleton in members of one of the anemone groups that probably have existed throughout Phanerozoic time. In spite of much work on the subject, advocates of the direct descent hypothesis have failed to find convincing evidence of this relationship. Critical points are:(1) Rugosan septal insertion is serial; Scleractinian insertion is cyclic; no intermediate stages have been demonstrated. Apparent intermediates are Scleractinia having bilateral cyclic insertion or teratological Rugosa.(2) There is convincing evidence that the skeletons of many Rugosa were calcitic and none are known to be or to have been aragonitic. In contrast, the skeletons of all living Scleractinia are aragonitic and there is evidence that fossil Scleractinia were aragonitic also. The mineralogic difference is almost certainly due to intrinsic biologic factors.(3) No early Triassic corals of either group are known. This fact is not compelling (by itself) but is important in connection with points 1 and 2, because, given direct descent, both changes took place during this only stage in the history of the two groups in which there are no known corals.

On Computing Critical Points for a Bivariate Chi-Square Random Variable

1973 ◽  
Vol 2 (3) ◽  
pp. 221-224
Author(s):  
Richard Gunst
Keyword(s):  
Critical Points ◽  
Random Variable ◽  
Chi Square

MULTIPLE CRITICAL POINTS OF PERTURBED SYMMETRIC FUNCTIONALS INW01,p

1987 ◽  
Vol 7 (4) ◽  
pp. 431-445
Author(s):  
Gongbao Li
Keyword(s):  
Critical Points
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