In this note, we mainly show the analogue of one Alladi's formulas over Q with respect to Dirichlet convolutions involving Möbius function μ(n), which is related natural densities sets primes by recent work Dawsey, Sweeting and Woo, Kural et al. This would give us several new analogues. particular, get that if (k,ℓ)=1, then−∑n⩾2p(n)≡ℓ(modk)μ(n)φ(n)=1φ(k), where p(n) smallest prime divisor n, φ(...

Enumerating formulae are constructed which count the number of partitions of a positive integer into positive summands in arithmetic progression with common difference D. These enumerating formulae (denoted pD(n)) which are given in terms of elementary divisor functions together with auxiliary arithmetic functions (to be defined) are then used to establish a known characterisation for an intege...

Soy-based infant formulas have been consumed in the United States since 1909, and currently constitute a significant portion of the infant formula market. There are efforts underway to generate genetically modified soybeans that produce therapeutic agents of interest with the intent to deliver those agents in a soy-based infant formula platform. The threefold purpose of this review article is t...

Steven Romans Logarithmic Binomial Formula analogue has been found and is presented here also for the case of fibonomial coefficients-which recently have been given a combinatorial interpretation by the present author.

We have seen that irreducible representations of a compact Lie group G can be constructed starting from a highest weight space and applying negative roots to a highest weight vector. One crucial thing that this construction does not easily tell us is what the character of this irreducible representation will be. The character would tell us not just which weights occur in the representation, but...

Orbital acceleration in polar: a = (r̈ − rθ̇2)r̂ + (rθ̈ + 2ṙθ̇)θ̂ Legendre polynomials: P1(x) = 1, P2(x) = x, P3(x) = (3x 2 − 1)/2; orthogonality: ∫ 1 −1 Pm(x)Pn(x)dx = 2 2n+1δmn Eccentricity: e > 1 hyperbola; e = 1 parabola; 0 < e < 1 ellipse; e = 0 circle Stirling’s Approximation: lnN ! = N lnN −N Gaussian Integral: ∫∞ −∞ e −ax2dx = √ π/a ∫∞ −∞ x 2e−ax 2 dx = √ π/2a3/2 ∫∞ 0 x 3e−αx 2 = 1/2α2 Delta ...

Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain. e-mail: [email protected] Abstract We apply the formal W.K.B. method in the complex plane to the quantum field theory to obtain the Schwinger formula for spin and spinless particles; i.e., we obtain the probability that the vacuum state remains unchanged in presence of a constant el...

where V , E, and F are, respectively, the number of vertices, edges, and faces of p. The formula was first stated in print by Euler in 1758 [11]. The proof given here is based on Poincaré’s linear algebraic proof, stated in [17] (with a corrected proof in [18]), as adapted by Imre Lakatos in the latter’s Proofs and Refutations [15]. As is well known, Euler’s formula is not true for all polyhedr...