SPITZER`S FORMULA: A SHORT PROOF
... random variables with the same distribution, and let {Sn} he the usual sequence of partial sums. In studying problems of first passage, ruin and the like, it is often useful to consider random variables 7?n = max (0, Si, S2, ■ ■ • , Sn). Spitzer [l] has obtained a beautiful formula from which one ca ...
... random variables with the same distribution, and let {Sn} he the usual sequence of partial sums. In studying problems of first passage, ruin and the like, it is often useful to consider random variables 7?n = max (0, Si, S2, ■ ■ • , Sn). Spitzer [l] has obtained a beautiful formula from which one ca ...
RATIONAL FUNCTIONS AND REAL SCHUBERT CALCULUS 1
... 1. Let q = 2, a1 = a2 = d − 1. Then the problem always has one solution, this solution is real, and the condition that Aj are separated is redundant. 2. Let us consider the limiting situation when all points in each Aj collide. Here aj = card Aj − 1 are arbitrary integers satisfying (1). Then Wj are ...
... 1. Let q = 2, a1 = a2 = d − 1. Then the problem always has one solution, this solution is real, and the condition that Aj are separated is redundant. 2. Let us consider the limiting situation when all points in each Aj collide. Here aj = card Aj − 1 are arbitrary integers satisfying (1). Then Wj are ...
On the representation of integers as sums of triangular number
... n as a sum of 24 triangular numbers. It turns out that for odd n we obtain an interesting relation between δ24 (n − 3) and N2n , the number of lattice points on the Leech lattice with norm 2n [3, p, 131]. As a corollary to the formula for δ24 (n), we get interesting congruences for the Ramanujan τ ( ...
... n as a sum of 24 triangular numbers. It turns out that for odd n we obtain an interesting relation between δ24 (n − 3) and N2n , the number of lattice points on the Leech lattice with norm 2n [3, p, 131]. As a corollary to the formula for δ24 (n), we get interesting congruences for the Ramanujan τ ( ...
3.4.2 Direct Proof 3.4.3 Contrapositive Proof 3.4.4 Deductive Proof
... This theorem provides the logical basis for the method of proof by deduction. It states that in order to establish the validity of an implication a ⇒ b, we may introduce a number (n ∈ Z+ ) of intermediate statements c1 , c2 , · · · , cn and prove the implications a ⇒ c1 , c1 ⇒ c2 , · · · , cn−1 ⇒ cn ...
... This theorem provides the logical basis for the method of proof by deduction. It states that in order to establish the validity of an implication a ⇒ b, we may introduce a number (n ∈ Z+ ) of intermediate statements c1 , c2 , · · · , cn and prove the implications a ⇒ c1 , c1 ⇒ c2 , · · · , cn−1 ⇒ cn ...
Continuity of Local Time: An applied perspective
... Equation (1) is often referred to as a continuity equation for the conserved quantity η(y)u(t, y); Fourier’s flux law for heat conduction and the corresponding Fick’s law for diffusion being among the most notable such occurrences. The right-side of the pde is the divergence of the diffusive flux 12 ...
... Equation (1) is often referred to as a continuity equation for the conserved quantity η(y)u(t, y); Fourier’s flux law for heat conduction and the corresponding Fick’s law for diffusion being among the most notable such occurrences. The right-side of the pde is the divergence of the diffusive flux 12 ...
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.