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The uniqueness conjecture for Markov numbers [159] that every Markov number is the largest number in exactly one normalized solution to the Markov Diophantine equation. Pillai's conjecture : for any A , B , C {\displaystyle A,B,C} , the equation A x m − B y n = C {\displaystyle Ax^{m}-By^{n}=C} has finitely many solutions when m , n ...
This notation represents (i) the number of vertices, (ii) the number of polygons around each vertex (arranged clockwise) and (iii) the number of sides to each of those polygons. For example: 3 6 ; 3 6 ; 3 4 .6, tells us there are 3 vertices with 2 different vertex types, so this tiling would be classed as a ‘3-uniform (2-vertex types)’ tiling.
Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The n th triangular number is the number of dots in the triangular arrangement with n dots on each side, and is equal to the sum of the n natural numbers from 1 to n. The sequence of triangular numbers, starting with the 0th triangular number, is
The transformation of a linear program to one in standard form may be accomplished as follows. [16] First, for each variable with a lower bound other than 0, a new variable is introduced representing the difference between the variable and bound. The original variable can then be eliminated by substitution. For example, given the constraint
The Fraction class in the fractions module provides arbitrary precision for rational numbers. [ 129 ] Due to Python's extensive mathematics library, and the third-party library NumPy that further extends the native capabilities, it is frequently used as a scientific scripting language to aid in problems such as numerical data processing and ...
The combinatorial meaning of the expression of mixed moments in terms of cumulants is easier to understand than that of cumulants in terms of mixed moments, see Equation (3.2.6) in: [11] = (:).
[39] [40] The factorial number system is a mixed radix notation for numbers in which the place values of each digit are factorials. [41] Factorials are used extensively in probability theory, for instance in the Poisson distribution [42] and in the probabilities of random permutations. [43]
The β form can be crystallized from supercooled solutions at −26 °C, and melts partially at −1.83 °C. [19] The "sodium hydroxide" of commerce is often the monohydrate (density 1.829 g/cm 3). Physical data in technical literature may refer to this form, rather than the anhydrous compound.