On the reduction of the multidimensional Schroedinger equation to a first order equation and its relation to the pseudoanalytic function theory

Given a particular solution of a one-dimensional stationary Schroedinger equation (SE) this equation of second order can be reduced to a first order linear differential equation. This is done with the aid of an auxiliary Riccati equation. We show that a similar fact is true in a multidimensional situation also. We consider the case of two or three independent variables. One particular solution of (SE) allows us to reduce this second order equation to a linear first order quaternionic differential equation. As in one-dimensional case this is done with the aid of an auxiliary Riccati equation. The resulting first order quaternionic equation is equivalent to the static Maxwell system. In the case of two independent variables it is the Vekua equation from theory of generalized analytic functions. We show that even in this case it is necessary to consider not complex valued functions only, solutions of the Vekua equation but complete quaternionic functions. Then the first order quaternionic equation represents two separate Vekua equations, one of which gives us solutions of (SE) and the other can be considered as an auxiliary equation of a simpler structure. For the auxiliary equation we always have the corresponding Bers generating pair, the base of the Bers theory of pseudoanalytic functions, and what is very important, the Bers derivatives of solutions of the auxiliary equation give us solutions of the main Vekua equation and as a consequence of (SE). We obtain an analogue of the Cauchy integral theorem for solutions of (SE). For an ample class of potentials (which includes for instance all radial potentials), this new approach gives us a simple procedure allowing to obtain an infinite sequence of solutions of (SE) from one known particular solution

On a factorization of second order elliptic operators and applications

We show that given a nonvanishing particular solution of the equation (divpgrad+q)u=0 (1) the corresponding differential operator can be factorized into a product of two first order operators. The factorization allows us to reduce the equation (1) to a first order equation which in a two-dimensional case is the Vekua equation of a special form. Under quite general conditions on the coefficients p and q we obtain an algorithm which allows us to construct in explicit form the positive formal powers (solutions of the Vekua equation generalizing the usual powers of the variable z). This result means that under quite general conditions one can construct an infinite system of exact solutions of (1) explicitly, and moreover, at least when p and q are real valued this system will be complete in ker(divpgrad+q) in the sense that any solution of (1) in a simply connected domain can be represented as an infinite series of obtained exact solutions which converges uniformly on any compact subset of . Finally we give a similar factorization of the operator (divpgrad+q) in a multidimensional case and obtain a natural generalization of the Vekua equation which is related to second order operators in a similar way as its two-dimensional prototype does

On a complex differential Riccati equation

We consider a nonlinear partial differential equation for complex-valued functions which is related to the two-dimensional stationary Schrodinger equation and enjoys many properties similar to those of the ordinary differential Riccati equation as, e.g., the famous Euler theorems, the Picard theorem and others. Besides these generalizations of the classical "one-dimensional" results we discuss new features of the considered equation like, e.g., an analogue of the Cauchy integral theorem

On the Klein-Gordon equation and hyperbolic pseudoanalytic function theory

Elliptic pseudoanalytic function theory was considered independently by Bers and Vekua decades ago. In this paper we develop a hyperbolic analogue of pseudoanalytic function theory using the algebra of hyperbolic numbers. We consider the Klein-Gordon equation with a potential. With the aid of one particular solution we factorize the Klein-Gordon operator in terms of two Vekua-type operators. We show that real parts of the solutions of one of these Vekua-type operators are solutions of the considered Klein-Gordon equation. Using hyperbolic pseudoanalytic function theory, we then obtain explicit construction of infinite systems of solutions of the Klein-Gordon equation with potential. Finally, we give some examples of application of the proposed procedure

Algorithmic tools for optimizing the temperature regime of evaporator at absorption refrigeration units of ammonia production

Проведено аналіз випарників абсорбційно-холодильних установок блоку вторинної конденсації виробництва аміаку як об’єктів керування. Визначені координати векторів стану, керування та зовнішніх збурень. Обґрунтована необхідність розв'язання задачi мiнiмiзацiї температури охолодження циркуляційного газу у випарниках для пiдвищення енергоефективностi виробництва. За результатами аналізу промислового апаратурно-технологічного оформлення блоків первинної i вторинної конденсації з'ясовані особливостi умов роботи випарника, що зумовлюють параметричну невизначенiсть у функцiонуваннi об’єктiв керування. Основна з таких невизначеностей пов’язана з керуючою дією витрати флегми. Методом математичного моделювання за розробленим алгоритмом визначені закономiрностi керуючої дiї витрати флегми на ефективнiсть процесiв теплообміну у випарниках абсорбцiйно холодильних установок. Встановлено екстремальний характер залежностi тепловогопотоку (холодопродуктивностi) та температури охолодження циркуляцiйного газу вiд витрати флегми. Максимальна холодопродуктивнiсть, а отже i мiнiмальна температура охолодження циркуляцiйного газу за певного температурного напору, обумовленi досягненням критичного режиму бульбашкового кипiння холодоагенту. Подальше збiльшення температурного напору з пiдвищенням витрати флегми сприяє встановленню перехiдного режиму i зниженню ефективностi поверхнi теплообмiну. Визначенi показники енергоефективностi виробництва амiаку, а саме витрати природного газу в умовах змiни керуючої дiї витрати флегми та значень координат вектора збурень. Розроблене алгоритмiчне забезпечення дозволяє здiйснити розв’язання задачi мiнiмiзацiї температури охолодження циркуляцiйного газу безградiєнтним способом крокового типу з використанням методiв одномiрного пошуку екстремуму. Показано, що за рахунок мiнiмiзацiї температури охолодження циркуляцiйного газу рiчна витрата природного газу може бути знижена в середньому на 500 тис. нм3

Pricing double barrier options on homogeneous diffusions: a Neumann series of Bessel functions representation

This paper develops a novel analytically tractable Neumann series of Bessel functions representation for pricing (and hedging) European-style double barrier knock-out options, which can be applied to the whole class of one-dimensional time-homogeneous diffusions, even for the cases where the corresponding transition density is not known. The proposed numerical method is shown to be efficient and simple to implement. To illustrate the flexibility and computational power of the algorithm, we develop an extended jump to default model that is able to capture several empirical regularities commonly observed in the literature.info:eu-repo/semantics/submittedVersio

Study of Macro- and Microelement Status in Patients with Nodular Goiter Residing in Kyiv Region

Sixty-one residents of Kiev region (16 individuals with nodular goiter and 45 individuals without thyroid pathology – the control group) were examined. When studying urinary iodine excretion, median urinary iodine concentration in the control group was 65.0 μg/l, while in patients with nodular goiter, it was 72.15 μg/l indicating mild iodine deficiency. In patients with nodular goiter, there were observed decreased serum levels of calcium - 74.17 mg/l (p<0.05), magnesium - 17.67 mg/l, zinc - 0.73 mg/l (p<0.05) and selenium - 0.03 mg/l (p<0.05) as compared to those in the control group. The relative risk of developing nodular goiter in decreased serum calcium concentration was 1.66 (95% confidence interval 1.07-2.09), (p<0.05); in decreased serum concentration of both calcium and selenium, it was 2.30 (95% confidence interval 1.147–4.085), (p<0.05); in low serum magnesium concentration, the relative risk was 2.6 (95% confidence interval 1.11-6.09) (p<0.05).

R-Functions and WA-Systems of Functions in Modern Information Technologies

The review report consists of five parts. It describes the main physical applications of atomic, WA-systems and R-functions