Exceptional real Lucas sequences
... An index n, greater than 2, is exceptional in (U) if every prime dividing Un also divides UXU2U^ ••• CT^. In the study of exceptional indices it suffices to take I > 0. For if (U) and (Uf) are generated by #2 — Iz + m and z2 + Iz + m, respectively, then Z7W = (—ly^Ul. In all that follows we therefor ...
... An index n, greater than 2, is exceptional in (U) if every prime dividing Un also divides UXU2U^ ••• CT^. In the study of exceptional indices it suffices to take I > 0. For if (U) and (Uf) are generated by #2 — Iz + m and z2 + Iz + m, respectively, then Z7W = (—ly^Ul. In all that follows we therefor ...
Cantor`s Legacy Outline Let`s review this argument Cantor`s Definition
... f(x) = x/2 if x is odd -x/2 if x is even The odd numbers in N map to the positive integers in Z. The even numbers in N map to negative integers in Z. ...
... f(x) = x/2 if x is odd -x/2 if x is even The odd numbers in N map to the positive integers in Z. The even numbers in N map to negative integers in Z. ...
DYNAMIC PROCESSES ASSOCIATED WITH NATURAL NUMBERS
... means of the function ψ. If we call x̂ŷ plane the set (ψ(R+ )) , the hyperbolas xy = k (k > 0) of the xy plane with x > 0 and y > 0 are transformed by means of the function ψ × ψ at the x̂ ⊗ ŷ = k̂ “hyperbolas” of the x̂ŷ plane. We will restrict our attention to the points in the x̂ŷ plane that ...
... means of the function ψ. If we call x̂ŷ plane the set (ψ(R+ )) , the hyperbolas xy = k (k > 0) of the xy plane with x > 0 and y > 0 are transformed by means of the function ψ × ψ at the x̂ ⊗ ŷ = k̂ “hyperbolas” of the x̂ŷ plane. We will restrict our attention to the points in the x̂ŷ plane that ...
"Restricted Partition Function as Bernoulli and Eulerian Polynomials of Higher Order"
... The summation is made over all primitive roots of unity ρ j = exp(2πin/j) for n relatively prime to j (including unity) and smaller than j. This result is just a recipe for calculation of the partition function and it does not provide any explicit formula. Using the Sylvester recipe we find an expli ...
... The summation is made over all primitive roots of unity ρ j = exp(2πin/j) for n relatively prime to j (including unity) and smaller than j. This result is just a recipe for calculation of the partition function and it does not provide any explicit formula. Using the Sylvester recipe we find an expli ...
Week 3. Functions: Piecewise, Even and Odd.
... 1. Find the largest possible domain and range for the following functions and sketch them: (i) f(x) = x |x|, ...
... 1. Find the largest possible domain and range for the following functions and sketch them: (i) f(x) = x |x|, ...
Random geometric complexes in the thermodynamic regime
... The first, and perhaps most natural topological question to ask about these sets is how connected are they. This is more a graph theoretic question than a topological one, and has been well studied in this setting, with [30] being the standard text in the area. There are various ‘regimes’ in which i ...
... The first, and perhaps most natural topological question to ask about these sets is how connected are they. This is more a graph theoretic question than a topological one, and has been well studied in this setting, with [30] being the standard text in the area. There are various ‘regimes’ in which i ...
Fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the function's integral.The first part of the theorem, sometimes called the first fundamental theorem of calculus, is that the definite integration of a function is related to its antiderivative, and can be reversed by differentiation. This part of the theorem is also important because it guarantees the existence of antiderivatives for continuous functions.The second part of the theorem, sometimes called the second fundamental theorem of calculus, is that the definite integral of a function can be computed by using any one of its infinitely-many antiderivatives. This part of the theorem has key practical applications because it markedly simplifies the computation of definite integrals.