Abstract: Since the heart of the hybrid system is the fuel cell and it has vital impact on efficiency and performance of cycle, in this study, the major modeling of electrochemical reaction within the fuel cell is analyzed. Also, solid oxide fuel cell is integrated with the gas turbine and thermodynamic analysis on different elements of hybrid system is applied. Next, in predefined operational points of hybrid cycle, the simulation results are obtained. Then, different source of irreversibility in fuel cell is modeled and influence of different major parameters on different irreversibility is computed and applied. Then, the effect of important parameters such as thickness and surface of electrolyte fuel cell are simulated in fuel cell and its dependency to these parameters is explained. At the end of the paper, different impact of parameters on fuel cell with a gas turbine and current density and voltage of fuel cell are simulated.
Abstract: In this work, we consider the rational points on elliptic curves over finite fields Fp where p ≡ 5 (mod 6). We obtain results on the number of points on an elliptic curve y2 ≡ x3 + a3(mod p), where p ≡ 5 (mod 6) is prime. We give some results concerning the sum of the abscissae of these points. A similar case where p ≡ 1 (mod 6) is considered in [5]. The main difference between two cases is that when p ≡ 5 (mod 6), all elements of Fp are cubic residues.
Abstract: In elliptic curve theory, number of rational points on
elliptic curves and determination of these points is a fairly important
problem. Let p be a prime and Fp be a finite field and k ∈ Fp. It
is well known that which points the curve y2 = x3 + kx has and
the number of rational points of on Fp. Consider the circle family
x2 + y2 = r2. It can be interesting to determine common points of
these two curve families and to find the number of these common
points. In this work we study this problem.
Abstract: In this work, we consider the rational points on elliptic
curves over finite fields Fp. We give results concerning the number
of points Np,a on the elliptic curve y2 ≡ x3 +a3(mod p) according
to whether a and x are quadratic residues or non-residues. We use
two lemmas to prove the main results first of which gives the list of
primes for which -1 is a quadratic residue, and the second is a result
from [1]. We get the results in the case where p is a prime congruent
to 5 modulo 6, while when p is a prime congruent to 1 modulo 6,
there seems to be no regularity for Np,a.
Abstract: In this work, we consider the number of integer solutions
of Diophantine equation D : y2 - 2yx - 3 = 0 over Z and
also over finite fields Fp for primes p ≥ 5. Later we determine the
number of rational points on curves Ep : y2 = Pp(x) = yp
1 + yp
2
over Fp, where y1 and y2 are the roots of D. Also we give a formula
for the sum of x- and y-coordinates of all rational points (x, y) on
Ep over Fp.
Abstract: Let p be a prime number, Fp be a finite field and t ∈ F*p= Fp- {0}. In this paper we obtain some properties of ellipticcurves Ep,t: y2= y2= x3- t2x over Fp. In the first sectionwe give some notations and preliminaries from elliptic curves. In the second section we consider the rational points (x, y) on Ep,t. Wegive a formula for the number of rational points on Ep,t over Fnp for an integer n ≥ 1. We also give some formulas for the sum of x?andy?coordinates of the points (x, y) on Ep,t. In the third section weconsider the rank of Et: y2= x3- t2x and its 2-isogenous curve Et over Q. We proved that the rank of Etand Etis 2 over Q. In the last section we obtain some formulas for the sums Σt∈F?panp,t for an integer n ≥ 1, where ap,t denote the trace of Frobenius.
Abstract: Let p be a prime number, Fpbe a finite field and let Qpdenote the set of quadratic residues in Fp. In the first section we givesome notations and preliminaries from elliptic curves. In the secondsection, we consider some properties of rational points on ellipticcurves Ep,b: y2= x3+ b2 over Fp, where b ∈ F*p. Recall that theorder of Ep,bover Fpis p + 1 if p ≡ 5(mod 6). We generalize thisresult to any field Fnp for an integer n≥ 2. Further we obtain someresults concerning the sum Σ[x]Ep,b(Fp) and Σ[y]Ep,b(Fp), thesum of x- and y- coordinates of all points (x, y) on Ep,b, and alsothe the sum Σ(x,0)Ep,b(Fp), the sum of points (x, 0) on Ep,b.
Abstract: Let p be a prime number, Fp be a finite field, and let k ∈ F*p. In this paper, we consider the number of rational points onconics Cp,k: x2 − ky2 = 1 over Fp. We proved that the order of Cp,k over Fp is p-1 if k is a quadratic residue mod p and is p + 1 if k is not a quadratic residue mod p. Later we derive some resultsconcerning the sums ΣC[x]p,k(Fp) and ΣC[y]p,k(Fp), the sum of x- and y-coordinates of all points (x, y) on Cp,k, respectively.
Abstract: Let p ≥ 5 be a prime number and let Fp be a finite
field. In this work, we determine the number of rational points on
singular curves Ea : y2 = x(x - a)2 over Fp for some specific
values of a.