In this paper, we completely solve the simultaneous Diophantine equations x − az = 1, y − bz = 1 provided the positive integers a and b satisfy b−a ∈ {1, 2, 4}. Further, we show that these equations possess at most one solutions in positive integers (x, y, z) if b − a is a prime or prime power, under mild conditions. Our approach is (relatively) elementary in nature and relies upon classical re...

The manuscript demonstrates that the solution of algebraic quadratic equations on the set of real numbers sometimes gives the results which contradict the transitional processes in oscillation systems. The solution of algebraic quadratic equations on the set of complex numbers, on the contrary, always agrees with the transitional processes in oscillation systems.

We give asymptotic formulas for the number of biquadratic extensions of Q that admit a quadratic extension which is a Galois extension of Q with a prescribed Galois group, for example, with a Galois group isomorphic to the quaternionic group. Our approach is based on a combination of the theory of quadratic equations with some analytic tools such as the Siegel–Walfisz theorem and the double osc...

Introduction Perceived waiting time (PWT) at signalized intersections differs from the real value of waiting time (WT), and varies with signal design. The onerousness of delay depends on the conditions under which it is experienced. Using weighted travel time may contribute to optimal signal control if its use can improve upon assuming that all time is weighted equally by users. This research e...

In this paper, an aggregate production-planning problem is formulated as a stochastic linear quadratic problem with chance constraints on state and control variables. The stochastic model proposed extends the classical aggregate model developed by Holt, Modigliani, Muth and Simon (HMMS). As an example of application, an equivalent deterministic problem is developed from the stochastic model. Th...

This paper introduces the emph{interval unilateral quadratic matrix equation}, $IUQe$ and attempts to find various analytical results on its AE-solution sets in which $A,B$ and $CCC$ are known real interval matrices, while $X$ is an unknown matrix. These results are derived from a generalization of some results of Shary. We also give sufficient conditions for non-emptiness of some quasi-solutio...

An analytical equation of state was applied to calculate the thermodynamic properties for argon.The equation of state is that of Song and Mason. It is based on a statistical-mechanical perturbationtheory of the hard convex bodies and can be written as fifth-order polynomial in the density. Thereexist three temperature-dependent parameters: the second virial coefficient, an effective molecularvo...

The paper proposes a new method, called the Fast Quadratic Transform (FQT), to solve the general indefinite two-variable quadratic equation in integers. The paper presents the new approach, discusses its properties, and provides a comparative evaluation with the classical technique. The FQT is demonstrated to be markedly superior for all cases in which it applies, including examples for more th...

Let E be an elliptic curve over Q without complex multiplication, and for each prime p of good reduction, let nE(p) = |E(Fp)|. For any integer b, we are studying in this paper elliptic pseudoprimes to the base b. More precisely, let QE,b(x) be the number of primes p 6 x such that bE ≡ b (modnE(p)), and π E,b (x) be the number of compositive nE(p) such that b nE(p) ≡ b (modnE(p)) (also called el...