Chaitanya's Random Pages

December 15, 2014

Types of -saurs that are not dinosaurs

Filed under: nature — ckrao @ 12:06 pm

Below is a reference for myself of types of (mostly) prehistoric animals that are not dinosaurs but have names ending in -saur (sauria means lizard but most of these are not that closely related to lizards).

Group Prefix meaning When it lived Brief description
Aetosaur eagle late Triassic heavily armoured
Anteosaur Antaeus (son of Poseidon and Gaia) 272-260 Ma large Dinocephalians (therapsid)
Cotylosaur cup late Carboniferous-Permian basal reptile (also known as Captorhinids)
Ichthyosaur fish 245-90 Ma dolphin-like marine reptile
Ictidosaur ferret late Triassic – mid Jurassic mammal-like cynodonts, also known as tritheledontids
Mesosaur middle 299-280 Ma like a small aligator
Mosasaur Meuse River late Cretaceous marine reptile similar to monitor lizards
Nothosaur false/hybrid Triassic slender marine reptile
Pachypleurosaur thick-ribbed Triassic like an aquatic lizard
Pareiasaur shield 270-250 Ma large anapsid
Pelycosaur axe or bowl 320-251 Ma non-therapsid synapsids (e.g. Dimetrodon)
Phytosaur plant 228-200 Ma long-snouted archosauriforms
Plesiosaur close to/near 210-65 Ma marine reptile with broad flat body and short tail
Pliosaur closely 200-89 Ma short-necked plesiosaur
Poposaur discovered on Popo Agie River (ref) late Triassic carnivorous paracrocodylomorphs
Protorosaur early Permian-Triassic long-necked archosauromorphs
Pterosaur winged 228-65 Ma closest relatives to dinosauromorphs
Rhynchosaur beaked Triassic beaked archosauromorphs
Teleosaur end/last early Jurassic – early Cretaceous marine crocodyliforms
Thalattosaur ocean Triassic marine reptile with long flat tail
Trilophosaur three ridged late Triassic lizard-like archosauromorphs
Xenosaur strange present (Cenozoic) knob-scaled lizards

December 14, 2014

Basic combinatorics results

Filed under: mathematics — ckrao @ 8:11 pm

The following lists most of the introductory combinatorics formulas one might see in a first course expressed in terms of the number of arrangements of letters in which repetition or order matters.

number of letters alphabet size letters repeated? order matters? formula comments
k n yes yes n^k samples with replacement
k  n no yes \begin{aligned} & n(n-1)\ldots (n-k+1)\\ &= \frac{n!}{(n-k)!} = P(n,k) = (n)_k\end{aligned} samples without replacement (permutations)
n  n no  yes n! if letters in a line

(n-1)! if letters in a ring and rotations are considered equivalent

(n-1)!/2 if letters in a ring and rotations & reflections are considered equivalent

 k n no no \frac{n!}{k!(n-k)!} = \binom{n}{k} = C(n,k) binomial coefficient

(combinations)

n 2 yes: k of type 1, n-k of type 2 yes \binom{n}{k}
n m yes: k_1 of type i for i = 1,\ldots, m yes \frac{n!}{k_1! k_2! \ldots k_m!} = \frac{(k_1 + k_2 + \ldots + k_m)!}{k_1! k_2! \ldots k_m!} = \binom{n}{k_1, k_2, \ldots, k_m} multinomial coefficient (multiset permutations)

e.g. number of arrangements of “BANANA” is \frac{6!}{3!2!1!}

n k yes no \binom{n+k-1}{k-1} = \binom{n+k-1}{n} = \left(\!\!{n \choose k}\!\!\right) multiset coefficient (combinations with repetition):

number of non-negative integer solutions to n_1 + n_2 + \ldots + n_k = n

n 2 no two consecutive letters of type 1 yes F_{n+2} Fibonacci number where F_{n+2} = F_{n+1} + F_n and F_1 = F_2 = 1

e.g. number of sequences of 6 coin tosses with no two consecutive heads is F_8 = 21

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