Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics

Nonlocal models are ubiquitous in all branches of science and engineering, with a rapidly expanding range of mathematical and computational applications due to the ability of such models to capture effects and phenomena that traditional models cannot. While spatial nonlocalities have received considerable attention in the research community, the same cannot be said about nonlocality in time, in particular when nonlocal initial conditions are present. This paper aims at filling this gap, providing an overview of the current status of nonlocal models and focusing on the mathematical treatment of such models when nonlocal initial conditions are at the heart of the problem. Specifically, our representative example is given for a nonlocal-in-time problem for the abstract Schrödinger equation. By exploiting the linear nature of nonlocal conditions, we derive an exact representation of the solution operator under assumptions that the spectrum of Hamiltonian is contained in the horizontal strip of the complex plane. The derived representation permits us to establish the necessary and sufficient conditions for the problem’s well-posedness and the existence of its solution under different regularities. Furthermore, we present new sufficient conditions for the existence of the solution that extend the existing results in this field to the case when some nonlocal parameters are unbounded. Two further examples demonstrate the developed methodology and highlight the importance of its computer algebra component in the reduction procedures and parameter estimations for nonlocal models. Finally, a connection of the considered models and developed analysis is discussed in the context of other reduction techniques, concentrating on the most promising from the viewpoint of data-driven modelling environments, and providing directions for further generalizations.

A simple and compact broadband circularly polarized circular slot antenna for WLAN/WiMAX/DBS applications

A simple and compact circularly polarized broadband circular slot antenna is proposed for WLAN/WiMAX/DBS applications. The main objective of this work is to design a microstrip line fed broadband circularly polarized antenna that is achieved by introducing an asymmetric perturbation over the circular slot onto the ground plane. A broad axial ratio bandwidth is achieved by using a small circular segment cut into the circular slotted ground and by adding a short stub on the feedline. To achieve a broad impedance bandwidth, a horizontal strip onto the slotted ground is placed just above the feed line on the opposite side. The proposed antenna is fabricated with an overall dimension of 20 × 20 × 1.6 mm3 (0.34λ o × 0.34λ o × 0.03λ o, where λ o represents the free-space wavelength at the center operating frequency). It is found that the impedance bandwidth of 75.82% ranges from 3.21 to 7.13 GHz and the 3 dB axial ratio (AR) bandwidth reaches 54.27% from 3.8 to 6.6 GHz. Throughout this paper, the improvement and validation process of the proposed antenna outcomes to accomplish desired characteristics are discussed.

Resistance of ETICS with Fire Barriers to Cyclic Hygrothermal Impact

The article presents the results of a set of hygrothermal experiments of an external wall insulated with an ETICS. As an add-on to previous studies, thermal insulation in the form of polystyrene with an additional horizontal strip of mineral wool was used. Laboratory tests were carried out in accordance with ETAG 004. The ETICS test rig was composed of combustible expanded polystyrene foam (EPS) and horizontal strips of noncombustible mineral wool (MW) fire barriers over windows. The physical and mechanical properties of four types of finishing renders (without an additional reinforcement mesh in base coat of the fire barriers) were analyzed across full hygrothermal cycles in a climate chamber. Temperature sensors were mounted onto different ETICS layers to collect thermal data during the weathering. The testing of ETICS regarding their hygrothermal performance revealed that there were no visible defects on any renderings and over the junctions depending on the type of used insulation materials. Results also showed that the joints of EPS and MW have approximately half of their bond strength from polystyrene strength.

Influence of Tillage Practices and Nitrogen Sources on Nitrogen Availability and Uptake of Rice

Aim: Evaluation of tillage practices and nitrogen sources on nitrogen availability in soil and nitrogen uptake of rice Study design: The experiment was laid out in strip-plot design with different tillage practices in horizontal strip and different nitrogen sources in vertical strip and was replicated thrice. Place and duration of study: The field experiment was conducted during the kharif season of 2019 and 2020 at the Agricultural College Farm, Bapatla, ANGRAU, Lam, Guntur, Andhra Pradesh. Methodology: The experiment was performed with twenty treatments in strip-plot design. The horizontal strip comprised four different tillage practices and vertical strip with five different nitrogen sources. Rice variety “BPT-5204” was taken as the test variety. Observations of the crop and soil during the experimentation were recorded at regular intervals. The significance of the treatment impact was examined by the test. Results: Among tillage practices, higher nitrogen uptake by rice grain (85.30 and 86.96 Kg ha-1) and straw (41.21 and 41.91 Kg ha-1) during 2019-20 and 2020-21, respectively was recorded in conventional tillage (T4) which was found statistically on a par with dry seeding on puddled soil (T1) (81.15 and 82.79 Kg ha-1, 38.59 and 39.20 Kg ha-1 during 2019-20 and 2020-21, respectively) and significantly superior over reduced tillage (T2) and minimum tillage (T3). Among nitrogen source, application of 50% STBN through fertilizer + 50% N through cured poultry manure (N5) registered the highest uptake of nitrogen in grain and straw (85.38 and 87.02 Kg ha-1. 42.09 and 42.76 Kg ha-1 during 2019-20 and 2020-21, respectively) and it was found significantly on a par with 75% STBN through fertilizer + 25% N through cured poultry manure (N4) and significantly superior over 50% STBN through fertilizer + 50% N through FYM (N3), 75% STBN through fertilizer + 25% N through FYM (N2), and 100% STBN through fertilizer (N1) treatment.

AN UNBOUNDED OPERATOR WITH SPECTRUM IN A STRIP AND MATRIX DIFFERENTIAL OPERATORS

Let A and $\tilde A$ be unbounded linear operators on a Hilbert space. We consider the following problem. Let the spectrum of A lie in some horizontal strip. In which strip does the spectrum of $\tilde A$ lie, if A and $\tilde A$ are sufficiently ‘close’? We derive a sharp bound for the strip containing the spectrum of $\tilde A$ , assuming that $\tilde A-A$ is a bounded operator and A has a bounded Hermitian component. We also discuss applications of our results to regular matrix differential operators.

Numerical study of the turbulent natural convection of nanofluids in a partially heated cubic cavity

In this work we study numerically the three-dimensional turbulent natural convection in a partially heated cubic cavity filled with water containing metallic nanoparticles, metallic oxides and others based on carbon.The objective is to study and compare the effect of the addition of nanoparticles studied in water and also the effect of the position of the heated partition on the heat exchange by turbulent natural convection in this type of geometry, which can significantly improve the design of heat exchange systems for better space optimization. For this we have treated numerically for different volume fractions the turbulent natural convection in the two cases where the cavity is heated respectively by a vertical and horizontal strip in the middle of one of the vertical walls. To take into account the effects of turbulence, we used the standard turbulence model ? - ?. The governing equations are discretized by the finite volume method using the power law scheme which offers a good stability characteristic in this type of flow. The results are presented in the form of isothermal lines and current lines. The variation of the mean Nusselt number is calculated for the two positions of the heated partition as a function of the volume fraction of the nanoparticles studied in water for different Rayleigh numbers.The results show that carbon-based nanoparticles intensify heat exchange by convection better and that the position of the heated partition significantly influences heat exchange by natural convection. In fact, an improvement in the average Nusselt number of more than 20% is observed for the case where the heated partition is horizontal.

A boundary value problem with a displacement for the generalized equation of Trikomi with a spectral parameter in an unbounded domain

В данной работе для обобщенного уравнения Трикоми со спектральным параметром в неограниченной области эллиптическая часть которой является горизонтальной полосой исследуется задача со смещением на характеристиках разных семейств. Единственность решения задачи доказывается методом интегралов энергии, а существование решения задачи методом функций Грина и методом интегральных уравнений. In this paper, for the generalized Tricomi equation with a spectral parameter in an unbounded domain, the elliptical part of which is a horizontal strip, we study a problem with a shift on the characteristics of different families. The uniqueness of the solution to the problem is proved by the method of energy integrals, and the existence of the solution to the problem by the method of Green’s functions and the method of integral equations