t is known that a good choice of notations is essentially important for mathematics in general (as a language of science) and for calculus in particular. There is no uniform notation for them. Therefore, several different notations for derivatives and differential equations have been introduced by...
According to backward induction, the first-order and second-order partial derivatives for \(\tau\) from Eq. (1) are \(\frac{{\partial \Pi_{M} }}{\partial \tau } = \left( {1 - \Phi } \right)\left[ \alpha w - \left( {c\alpha + \beta \tau } \right)\left( {1 - \Phi...
Detecting the impact of the most important variables on the market outcome is straightforward: We obtain the following results (Appendix A shows the concrete derivatives):(28)∂ϝ*∂cP>0;∂ϝ*∂φ>0;∂ϝ*∂μ>0;∂ϝ*∂cA<0;∂ϝ*∂ψ<0∂ϝ*∂γ>0;∂ϝ...
The first and second order partial derivatives of C11(t1, T) and C12(t1, T) with respect to t1 and T are given below∂C11(T,t1)∂T=1T2A+Dt12h2+πD(T-t1)22+12Ip(cDT-DMp1+12MI-D(T-t1)p(1+MI))2pD-12pDt1-M-cDT-DMp1+12MI-D(T-t1)p(1+MI)pD2Ie-pDt1-Mi-cDT-...
Lizarraga-Celaya, CarlosThe Mathematical Intelligencerhingareva, I. K., & i a'rraga-Celaya, C. (2015). On different symbolic notations for derivatives. The Mathematical Intelligencer, 37(3), 33-38. https://doi.org/10.1007/s00283-014-9526-5...
All notations applied in the article can be found in App. 1. 3.2 Set of preferential rights We structure the allocation mechanisms of exit proceeds subject to exit relevant cash flow rights in two consecutive phases.Footnote 6 The first phase contains the allocation due to explicit preferential ...
Evaluating the first- and second-order derivatives of J E T C t ( Q , n , L , G ) 𝐽𝐸𝑇𝐶𝑡(𝑄,𝑛,𝐿,𝐺) with respect to (w.r.t) Q 𝑄, we obtain ∂ J E T C t ( Q , n , L , G ) ∂ Q = − D n Q 2 ( 1 − Y e ) [ n A + ...
Here, the time derivatives are solved by applying the third-order accurate Total Variation Diminishing (TVD) Runge–Kutta scheme [5]. The same time discretisation is also used in (6) and (7). Turbulence time advancement is solved using implicit methods due to its source term driven character...
Then, computing the second-order partial derivatives of w, t for Equation (2), we can obtain the Hessian matrix: 𝐻=⎡⎣⎢⎢−𝑏𝛽2𝛽2−𝑘⎤⎦⎥⎥H=[−bβ2β2−k]. When 4𝑏𝑘−𝛽2>04bk−β2>0, it is a negative definite matrix, and the profi...
Hydroxycinnamic acids and their derivatives have been reported to possess properties of antioxidant, anti-inflammatory, antimicrobial, and anti-tyrosinase activities [77] which might suggest that phaseolic acid can be exploited as an anti-aging agent. Glycinamide ribonucleotide is an intermediate in de ...