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.

In this paper, a general framework for surface modeling using geometric partial differential equations (PDEs) is presented. Starting with a general integral functional, we derive an Euler–Lagrange equation and then a geometric evolution equation (also known as geometric flow). This evolution equation is universal, containing several well-known geometric partial differential equations as its spe...

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...

In this paper, stability conditions for distributed control problems are derived under general integral quadratic constraints to achieve quadratic performance. These results take the form of coupled LMIs, and the multipliers are specified by the underlying integral quadratic constraints to model interconnections between the subsystems. It is further shown that these stability results can be exp...