3.8
... Such matrices are called stochastic matrices. The above formulation has in effect assumed that the substitution process is a Markov chain, the essence of which is as follows. Consider three time points in evolution, t1 t 2 t 3 . The Markovian model assumes that the status of the nucleotide site ...
... Such matrices are called stochastic matrices. The above formulation has in effect assumed that the substitution process is a Markov chain, the essence of which is as follows. Consider three time points in evolution, t1 t 2 t 3 . The Markovian model assumes that the status of the nucleotide site ...
Some applications of the theory of distributions
... (Note that at least one derivative of P is a constant =1= 0. Therefore P(Ç) is greater than some > 0 constant; and the inequality (34a) only involves the large values of |{|.) There are different proofs of this theorem. An indispensable step in the proof of the sufficiency is the proof that, for eve ...
... (Note that at least one derivative of P is a constant =1= 0. Therefore P(Ç) is greater than some > 0 constant; and the inequality (34a) only involves the large values of |{|.) There are different proofs of this theorem. An indispensable step in the proof of the sufficiency is the proof that, for eve ...
The algebra of essential relations on a finite set
... is inessential. Proof : Suppose that a and b are equivalent and a 6= b. Then the rows Ra and Rb are equal and Corollary 2.2 applies. We need a few basic facts about reflexive relations. Recall that a relation S on X is reflexive if S contains ∆ = {(x, x) | x ∈ X}. Moreover, a preorder is a relation ...
... is inessential. Proof : Suppose that a and b are equivalent and a 6= b. Then the rows Ra and Rb are equal and Corollary 2.2 applies. We need a few basic facts about reflexive relations. Recall that a relation S on X is reflexive if S contains ∆ = {(x, x) | x ∈ X}. Moreover, a preorder is a relation ...
Full text
... of rational numbers. It will be supposed that each such rational number is written as a quotient of relatively prime integers. A rational number so written is said to be in standard form. It is immaterial for this discussion whether the denominator be positive or negative. The purpose of this paper ...
... of rational numbers. It will be supposed that each such rational number is written as a quotient of relatively prime integers. A rational number so written is said to be in standard form. It is immaterial for this discussion whether the denominator be positive or negative. The purpose of this paper ...
![arXiv:math/9911224v2 [math.GT] 9 Dec 1999](http://s1.studyres.com/store/data/016370695_1-31d422643fefeca7faefc96120ceb1d7-300x300.png)
2 and
... To multiply two binomials, the distributive property is used so that every term in one polynomial is multiplied by every term in the other polynomial. Example: Multiply. (7x + 3)(2x + 4) (7x + 3)(2x + 4) = (7x + 3)(2x) + (7x + 3)(4) ...
... To multiply two binomials, the distributive property is used so that every term in one polynomial is multiplied by every term in the other polynomial. Example: Multiply. (7x + 3)(2x + 4) (7x + 3)(2x + 4) = (7x + 3)(2x) + (7x + 3)(4) ...
Stringy Hodge numbers and Virasoro algebra
... Remark 1.4 We that if X is a K3-surface, then the relation 1.3 is equivalent to the equality c2 (X) = 24. For smooth Calabi-Yau 4-folds X the relation 1.3 has been observed by Sethi, Vafa, and Witten [11] (it is equivalent to the equality c4 (X) = 6(8 − h1,1 (X) + h2,1 (X) − h3,1 (X)), if h1,0 (X) ...
... Remark 1.4 We that if X is a K3-surface, then the relation 1.3 is equivalent to the equality c2 (X) = 24. For smooth Calabi-Yau 4-folds X the relation 1.3 has been observed by Sethi, Vafa, and Witten [11] (it is equivalent to the equality c4 (X) = 6(8 − h1,1 (X) + h2,1 (X) − h3,1 (X)), if h1,0 (X) ...