W. charged into the Compositae and Umbelliferae like a hero, and demolished ever so many in grand style.
"The Life and Letters of Charles Darwin, Volume I (of II)" by Charles Darwin
I was struck with your remark about the Compositae, etc.
"More Letters of Charles Darwin" by Charles Darwin
I am surprised at what you say about Compositae and Gramineae.
"More Letters of Charles Darwin Volume II" by Charles Darwin
He is even able to leave the family of the Compositae and to go gleaning more or less everywhere.
"Bramble-bees and Others" by J. Henri Fabre
The largest bush in the island (belonging to the family of Compositae) is scarcely so tall as our gorse.
"A Naturalist's Voyage Round the World" by Charles Darwin
Such composita are possible in many different ways.
"The Critique of Pure Reason" by Immanuel Kant
Cum simplicia non possunt neccessitas cogit ad composita.
"The Anatomy of Melancholy" by Democritus Junior
Compositae, Cruciferae, and Gramineae form the bulk of the vegetation.
"Journals of Travels in Assam, Burma, Bhootan, Afghanistan and TheNeighbouring Countries" by William Griffith
In the labiatae none has been discovered, nor in the compositae among the highest plants.
"Scientific American Supplement, No. 623, December 10, 1887" by Various
As in all the Compositae, the anthers are here united in a tube, the pollen being discharged within.
"My Studio Neighbors" by William Hamilton Gibson
Cassini on flowers of compositae, 145.
"On the Origin of Species by Means of Natural Selection" by Charles Darwin
It was late in March, and the ground about the cabin was yellow with low-growing compositae.
"The Wizard's Daughter and Other Stories" by Margaret Collier Graham
Siegesbeckia, one of the Compositae, consists of two species, one inhabiting the Mascarene islands, the other Peru.
"Island Life" by Alfred Russel Wallace
Leaves opposite or in whorls of four 119a, in =COMPOSITAE=, p. 123.
"The Plants of Michigan" by Henry Allan Gleason
Compositae, comprising mugwort, southernwood, and wormwood.
"The New Gresham Encyclopedia. Vol. 1 Part 2" by Various
Compositae, found in Britain and other parts of Europe, and in Asia.
"The New Gresham Encyclopedia" by Various
Compositae formed a minor part of the cover in most of the habitats studied.
"Natural History of the Prairie Vole (Mammalian Genus Microtus)" by E. W. Jameson
COMPOSITAE, gradation of species among the, i.
"The Descent of Man and Selection in Relation to Sex" by Charles Darwin
In Compositae, the tubular flowers of the head as distinct from the ray.
"The Manual of the Botany of the Northern United States" by Asa Gray
From Epilobium on through Umbelliferae and Compositae.
"The Life of Johannes Brahms (Vol 2 of 2)" by Florence May
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Because f (n) is an integral function, the values of compositae F ∆(n, k) are integers.
On a property of superposition of the generating functions ln(1/(1-F(x)))
Key words: exponential generating function, composition of generating functions, composita, primality, Touchard’s Congruence, Bell numbers, Euler numbers.
Integer properties of a composition of exponential generating functions
Kruchinin introduced the notion of the composita of a given generating function F (x) = Pn>0 f (n)xn .
Integer properties of a composition of exponential generating functions
The expression F (n, k) is the composita and it’s denoted by F ∆(n, k).
Integer properties of a composition of exponential generating functions
Calculation of the composita is essential for obtaining the coeﬃcients function of a composition of generating functions.
Integer properties of a composition of exponential generating functions
Let us establish the following theorem on calculation of compositae of reciprocal generating functions.
The method for obtaining expressions for coefficients of reverse generating functions
Suppose we have the generating function B (x), b(0) 6= 0 and the composita B∆ (n, k) of the generating function xB (x).
The method for obtaining expressions for coefficients of reverse generating functions
Let us obtain the composita of the function 1 (B (x) − b0 ).
The method for obtaining expressions for coefficients of reverse generating functions
Now, let us establish the theorem on obtaining a explicit formula for the composita of reverse generating function.
The method for obtaining expressions for coefficients of reverse generating functions
R∆ (n, m) is a reciprocal composita for the generating function xH (x) (see the formula 7).
The method for obtaining expressions for coefficients of reverse generating functions
Suppose we have the generating function F (x) = x − x2 − x3 , the composita of reverse function needs to be found.
The method for obtaining expressions for coefficients of reverse generating functions
To do this we need to ﬁnd the composita of reciprocal function of R(x) = x 1−x−x2 .
The method for obtaining expressions for coefficients of reverse generating functions
To do this we need to ﬁnd the composita of the generating function F (x) = 2 ln(1 + x) − x.
The method for obtaining expressions for coefficients of reverse generating functions
The composita of generating function (−x) has the expression (−1)k δ (n, k).
The method for obtaining expressions for coefficients of reverse generating functions
To do this we ﬁnd the composita of the generating function R(x) = 2x − ex + 1.
The method for obtaining expressions for coefficients of reverse generating functions
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