DEVELOPMENT OF A METHODOLOGYFOR DETERMINING THE BEST OPTION FOR THE ROUTE OF THE HEAT NETWORK AT THE INITIAL DESIGN STAGE

Statement of the problem. Choosing the best option for the route of the thermal network at the initial stage of design is a complex multifactorial task, in addition, due to the lack of a number of necessary design calculations, its solution is accompanied by a limited set of initial data. Thus, it becomes relevant to develop a new methodology for designing the optimal route of the heat supply system considering the qualitative and quantitative characteristics of the discussed object.Results. A mathematical model of a generalized additive vector optimality criterion has been developed, taking into account the material consumption of the heat network, its reliability, construction time, annual thermal losses, heat turnover and temperature dispersion at the consumer. A method is proposed for determining the best option for the route of a thermal network at the initial design stage by jointly solving the optimization problem using vector optimization and matrix generalization methods. The expediency of the joint application of the methods of pairwise comparison and vector optimization in solving the problem under consideration is noted.Conclusions. An important characteristic of the developed mathematical model of the generalized criterion is the possibility of obtaining a more accurate solution to the optimization problem under consideration with an uneven distribution of the heat load by means of a biased estimate of the temperature variance among consumers. A combination of the methods of matrix generalization, pairwise comparison and vector optimization can improve the accuracy of the calculation while solving the optimization problem of choosing the best route of the thermal network.

On the Question of the Bäcklund Transformations and Jordan Generalizations of the Second Painlevé Equation

We demonstrate the way to derive the second Painlevé equation P2 and its Bäcklund transformations from the deformations of the Nonlinear Schrödinger equation (NLS), all the while preserving the strict invariance with respect to the Schlesinger transformations. The proposed algorithm allows for a construction of Jordan algebra-based completely integrable multiple-field generalizations of P2 while also producing the corresponding Bäcklund transformations. We suggest calling such models the JP-systems. For example, a Jordan algebra JMat(N,N) with the Jordan product in the form of a semi-anticommutator is shown to generate an integrable matrix generalization of P2, whereas the VN algebra produces a different JP-system that serves as a generalization of the Sokolov’s form of a vectorial NLS.

A new factor theorem on absolute matrix summability method

In this paper, we have a new matrix generalization with absolute matrix summability factor of an infinite series by using quasi-β-power increasing sequences. That theorem also includes some new and known results dealing with some basic summability methods

Development of a Methodology for Determining the Best Option for the Route of the Heat Network at the Initial Design Stage

Постановка задачи. Выбор наилучшего варианта трассы тепловой сети на начальном этапе проектирования является сложной многофакторной задачей, кроме того, ввиду отсутствия ряда необходимых конструктивных расчетов ее решение сопровождается ограниченностью набора исходных данных. Таким образом, становится актуальной разработка новой методики проектирования оптимальной трассы системы теплоснабжения, учитывающей качественные и количественные характеристики рассматриваемого объекта. Результаты. Разработана математическая модель обобщенного аддитивного векторного критерия оптимальности, учитывающая материалоемкость тепловой сети, ее надежность, время строительства, годовые тепловые потери, оборот теплоты и дисперсию температуры у потребителя. Предложен способ определения наилучшего варианта трассы тепловой сети на начальном этапе проектирования путем совместного решения задачи оптимизации методами векторной оптимизации и матричного обобщения. Отмечена целесообразность совместного применения методов попарного сравнения и векторной оптимизации при решении рассматриваемой задачи. Выводы. Важной характеристикой разработанной математической модели обобщенного критерия является возможность получения более точного решения рассматриваемой оптимизационной задачи при неравномерным распределении тепловой нагрузки посредством смещенной оценки дисперсии температуры у потребителей. Совместное применение методов матричного обобщения, попарного сравнения и векторной оптимизации позволяет повысить точность расчета при решении оптимизационной задачи выбора наилучшей трассы тепловой сети. Statement of the problem. Choosing the best option for the route of the thermal network at the initial stage of design is a complex multifactorial task, in addition, due to the lack of a number of necessary design calculations, its solution is accompanied by a limited set of initial data. Thus, it becomes relevant to develop a new methodology for designing the optimal route of the heat supply system, taking into account the qualitative and quantitative characteristics of the object under consideration. Results. A mathematical model of a generalized additive vector optimality criterion has been developed, taking into account the material consumption of the heat network, its reliability, construction time, annual thermal losses, heat turnover and temperature dispersion at the consumer. A method is proposed for determining the best option for the route of a thermal network at the initial design stage by jointly solving the optimization problem using vector optimization and matrix generalization methods. The expediency of the joint application of the methods of pairwise comparison and vector optimization in solving the problem under consideration is noted. Conclusions. An important characteristic of the developed mathematical model of the generalized criterion is the possibility of obtaining a more accurate solution to the optimization problem under consideration with an uneven distribution of the heat load by means of a biased estimate of the temperature variance among consumers. The combined application of the methods of matrix generalization, pairwise comparison and vector optimization can improve the accuracy of the calculation when solving the optimization problem of choosing the best route of the thermal network.

Matrix-Based RSA Encryption of Streaming Data

This paper describes a new method for performing secure encryption of blocks of streaming data. This algorithm is an extension of the RSA encryption algorithm. Instead of using a public key (e,n) where n is the product of two large primes and e is relatively prime to the Euler Totient function, φ(n), one uses a public key (n,m,E), where m is the rank of the matrix E and E is an invertible matrix in GL(m,φ(n)). When m is 1, this last condition is equivalent to saying that E is relatively prime to φ(n), which is a requirement for standard RSA encryption. Rather than a secret private key (d,φ(n)) where d is the inverse of e (mod φ(n)), the private key is (D,φ(n)), where D is the inverse of E (mod (φ(n)). The key to making this generalization work is a matrix generalization of the scalar exponentiation operator that maps the set of m-dimensional vectors with integer coefficients modulo n, onto itself.

The Grassmannian-like coset model and the higher spin currents

In the Grassmannian-like coset model, $$ \frac{\mathrm{SU}{\left(N+M\right)}_k}{\mathrm{SU}{(N)}_k\times \mathrm{U}{(1)}_{kNM\left(N+M\right)}} $$ SU N + M k SU N k × U 1 kNM N + M , Creutzig and Hikida have found the charged spin-2, 3 currents and the neutral spin-2, 3 currents previously. In this paper, as an extension of Gaberdiel-Gopakumar conjecture found ten years ago, we calculate the operator product expansion (OPE) between the charged spin-2 current and itself, the OPE between the charged spin-2 current and the charged spin-3 current and the OPE between the neutral spin-3 current and itself for generic N, M and k. From the second OPE, we obtain the new charged quasi primary spin-4 current while from the last one, the new neutral primary spin-4 current is found implicitly. The infinity limit of k in the structure constants of the OPEs is described in the context of asymptotic symmetry of MM matrix generalization of AdS3 higher spin theory. Moreover, the OPE between the charged spin-3 current and itself is determined for fixed (N, M) = (5, 4) with arbitrary k up to the third order pole. We also obtain the OPEs between charged spin-1, 2, 3 currents and neutral spin-3 current. From the last OPE, we realize that there exists the presence of the above charged quasi primary spin-4 current in the second order pole for fixed (N, M) = (5, 4). We comment on the complex free fermion realization.

The $$ \mathcal{N} $$ = 4 higher spin algebra for generic μ parameter

The $$ \mathcal{N} $$ N = 4 higher spin generators for general superspin s in terms of oscillators in the matrix generalization of AdS3 Vasiliev higher spin theory at nonzero μ (which is equivalent to the ’t Hooft-like coupling constant λ) were found previously. In this paper, by computing the (anti)commutators between these $$ \mathcal{N} $$ N = 4 higher spin generators for low spins s1 and s2 (s1 + s2 ≤ 11) explicitly, we determine the complete $$ \mathcal{N} $$ N = 4 higher spin algebra for generic μ. The three kinds of structure constants contain the linear combination of two different generalized hypergeometric functions. These structure constants remain the same under the transformation μ ↔ (1 − μ) up to signs. We have checked that the above $$ \mathcal{N} $$ N = 4 higher spin algebra contains the $$ \mathcal{N} $$ N = 2 higher spin algebra, as a subalgebra, found by Fradkin and Linetsky some time ago.

The 𝒩 = 4 coset model and the higher spin algebra

By computing the operator product expansions between the first two [Formula: see text] higher spin multiplets in the unitary coset model, the (anti-)commutators of higher spin currents are obtained under the large [Formula: see text] ’t Hooft-like limit. The free field realization with complex bosons and fermions is presented. The (anti-)commutators for generic spins [Formula: see text] and [Formula: see text] with manifest [Formula: see text] symmetry at vanishing ’t Hooft-like coupling constant are completely determined. The structure constants can be written in terms of the ones in the [Formula: see text] [Formula: see text] algebra found by Bergshoeff, Pope, Romans, Sezgin and Shen previously, in addition to the spin-dependent fractional coefficients and two [Formula: see text] invariant tensors. We also describe the [Formula: see text] higher spin generators, by using the above coset construction results, for general superspin [Formula: see text] in terms of oscillators in the matrix generalization of [Formula: see text] Vasiliev higher spin theory at nonzero ’t Hooft-like coupling constant. We obtain the [Formula: see text] higher spin algebra for low spins and present how to determine the structure constants, which depend on the higher spin algebra parameter, in general, for fixed spins [Formula: see text] and [Formula: see text].