The classical Bohr inequality says that |a+b|2 £ p½a½2+q½b½2 for all scalars a, b and p, q > 0 with 1/p + 1/q = 1. The equality holds if and only if (p-1)a = b. Several authors discussed operator version of Bohr inequality. In this paper, we give a unified proof to operator generalizations of Bohr inequality. One viewpoint of ours is a MATRIX inequality, and the other is a generalized parallelogram law for absolute value of operators, i.e., for operators A and B on a Hilbert space and t¹0, ½A-B½2+ 1/t½T A+B½2= (1+t) A½2 +(1+1/t) B½2.

This paper uses a new framework for solving a class of linear MATRIX di erential equations. For doing so, the operational MATRIX of the derivative based on the shifted Bernstein polynomials together with the collocation method are exploited to decrease the principal problem to system of linear MATRIX equations. An error estimation of this method is provided. Numerical experiments are reported to show the applicably and e ciency of the propounded method.

In The current paper presents an idea for solving a class of linear MATRIX di erential equations of second ORDER. To perform so, the operational MATRIX of the integration based on the Bernstein multiscaling polynomials are used to reduce the main problem to system of MATRIX equations. Numerical experiments illustrate the applicably and e ciency of the propounded technique.

Intelligent agents are considered as significant means towards realizing the semantic web vision. On the Semantic Web, integrating ontologies and rules enables software agents to interoperate between them, however, this leads to a problem, that no studies have focused on effective distributed reasoning for integrating ontologies and rules in multiple knowledge-bases. The methods that have been presented for distributed reasoning not only get a lot of times and memory, but also do not lead to a complete and sound reasoning. In this paper, to solve this problem, we present a distributed reasoning system that deals with the representation of the knowledge-base of ORDER sorted logic. This logic is able to describe the hierarchy of predicates and inheritance of expressions that there are in our natural language. To have a distributed reasoning, our proposed method uses the expansion of rigid and valid-non-rigid properties between knowledge-bases. Furthermore, with considering time and the situation of properties for reasoning, the non-rigid properties have not been ignored, in fact, in their valid time and situation, they are used. With this method, we achieve a complete reasoning and, moreover, the extracted knowledge is completely considered in the knowledge-bases and we have a distributed reasoning with high efficiency and sound without missing any information.

Background and Objectives: This paper proposes a new Model ORDER ‎Reduction (MOR) method based on the Bilinear Balanced Truncation ‎‎(BBT) approach. In the BBT method, solving the generalized Lyapunov ‎equations is necessary to determine the bilinear system's controllability ‎and observability Gramians. Since the bilinear systems are generally of ‎high ORDER, the computation of the Gramians of controllability and ‎observability have huge computational volumes. In addition, the accuracy ‎of reduced-ORDER model obtained by BT is relatively low. In fact, the ‎balanced truncation method is only available for local energy bands due ‎to the use of type I Gramians. In this paper, BBT based on type II ‎controllability and observability Gramians would be considered to fix ‎these drawbacks.‎Methods: At first, a new iterative method is proposed for determining ‎the proper ORDER for the reduced-ORDER bilinear model, which is related to ‎the number of Hankel singular values of the bilinear system whose real ‎parts are closest to origin and have the most significant amount of ‎energy. Then, the problem of determining of type II controllability and ‎observability Gramians of the high-ORDER bilinear system have been ‎formulated as a constrained optimization problem with some Linear ‎MATRIX Inequality (LMI) constraints for an intermediate middle-ORDER ‎system. Then, the achieved Gramians are applied to the BBT method to ‎determine the reduced-ORDER model of the bilinear system. Next, the ‎steady state accuracy of the reduced model would be improved via ‎employing a tuning factor.‎Results: Using the concept of type II Gramians and via the proposed ‎method, the accuracy of the proposed bilinear BT method is increased. ‎For validation of the proposed method, three high-ORDER bilinear models ‎are approximated. The achieved results are compared with some well-‎known MOR approaches such as bilinear BT, bilinear Proper Orthogonal ‎Decomposition (POD) and Bilinear Iterative Rational Krylov subspace ‎Algorithm (BIRKA) methods.‎Conclusion: According to the obtained results, the proposed MOR ‎method is superior to classical bilinear MOR methods, but is almost ‎equivalent to BIRKA. It is out-performance respecting to BIRKA is its ‎guaranteed stability and convergence. ‎

The aim of this article is to find an effective method for solving variable-ORDER fractional integro-differential equations. This method transforms the problem into a system of algebraic equations. For this purpose, we first express Vieta-Lucas orthogonal polynomials, then, we express the operational matrices of these polynomials. At this stage, all components of the equation will be expressed in terms of the new shifted Vieta-Lucas operational matrices. After that, by placing these operational matrices in the main equation and using the spectral collocation method, the variable-ORDER fractional integro-differential equation will become an algebraic system. By solving this algebraic system, we will find an approximate solution to the original equation. In the following, an analysis of the error is also presented by preparing some theorems. In the end, in ORDER to express the efficiency and capability of the method, some numerical examples are given. Additionally, for the numerical examples, the condition number, numerical convergence ORDER, and the computed CPU time are evaluated. Based on the obtained results, it was concluded that the proposed method is relatively stable, highly accurate and efficient, and has an appropriate convergence rate.

In this paper, two sets of sufficient conditions are obtained to ensure the existence and stability of a unique equilibrium point of unforced first ORDER fuzzy relational dynamical systems by using two different approaches which are both based on the fuzzy relational MATRIX of the model. In the first approach, the equilibrium point of the system is one of the centers of the related membership functions. In the second approach, the equilibrium point of the system is the origin (the center of the middle membership function) and the behavior of the system, though can be nonlinear, is symmetric around the origin. The results are approved by numerical examples.

In recent years, using power electronic equipment in residential and office buildings, commercial complexes, manufacturing unit's loads such as heating, lighting, domestic appliances have significantly increased. Since, most of them are single-phase with variable load pattern, LV power systems are imbalanced and harmonic polluted. The presence of harmonics in imbalance power systems can generate vital problems such as excessive neutral current losses and costs. Therefore, a detailed study into these systems is essential. In this paper, instantaneous neutral power is used as an indication of identifying imbalances in low voltage systems. By applying MATRIX Pencil Method to instantaneous neutral power, and based on the rules introduced, the harmonics that have contributions in unbalancing of the system are found. According to this result and according to another set of proposed rules, the loads that affect power losses are identified in ORDER to be switched between phases and, further, to optimize the loss in neutral conductor.

Introduction: The watershed is a geomorphologic unit suitable for managing natural resources and achieving sustainable development. The drainage network of the catchment area is a series of streams and rivers that guide and drain the surface water flow. These spatial data play an important role in physiographic studies, hydrology, erosion and sedimentation of a catchment area. The network of canals can be considered as a system that is dynamic in a state of equilibrium, in which dependent variables such as form; the slope and horizontal projection of the river or the amount of data and the coincidence that whenever there is a change in the data, This system reacts quickly with the coordination of their morphological characteristics with the new situation. The drainage network pattern is one of the most prominent landforms of the earth's surface. The ORDER and distribution of hydrographic networks are different. The purpose of this study is to identify the factors of balance and imbalance of the right and left side of the research area and which part of the research area has severe asthma and chaos. The basis of this research is the method analysis ORDER MATRIX of basin Ramhormoz at the beginning of the study. The exact coordinates of basin Ramhormoz boundary from Global Mapper. 20 software were taken in Arc GIS 10. 5 software using the Dem area of the study; for Draw curves, lane lines and line basin s, slope direction, fracture of streams and formations. Arc GIS software has been used to evaluate the Ramhormoz river basins using the ORDER MATRIX method. To set the categories in MATRIX form. ORDERs 1 to 1, 2 to 2, 3 to 3, and to the last. ORDER that is the main. ORDER, are placed in tables in the two views (right and left) in the array view. In the following, the. ORDER 1, which joins the large. ORDERs r, is calculated separately and the rest of the ORDERs are calculated in this way (ORDER 2 to 3, 4, 5, . . . 3rd to 4th, 5th And other ORDERs). In the study, the split density and drainage density are used. In this study, the number of waterways in the upper reaches of the two sides of the study area with a very significant difference indicates the imbalance in the research area. Erosion, slope changes, channel displacement, channel redirection, tectonic activity of these factors in causing significant differences and effective imbalances on both sides of the research area as well as the manner, extent and stages of changes in the form of the canals affected by time as a The main variable is to create the threshold, and the reason that the canals tend to trigger the chaos process, and even show a slight change, causing them to balance and imbalance. on both sides of the research area, the most balanced average of the lengths of the waterways, the ORDER of waterway 1 falls to the 7th (right 819. 2 km and the left is 819. 1km). Due to changes in the geomorphological parameters of the basin, due to changes in the parameters of the basin, resulting in no change in their form during the period. The correlation coefficient between the longitudinal lengths of the two sides of the research with the coefficient of determination (0. 9932), due to the slope and elevation variations, shortage of the path and length of the branches of the canals, which are located on both sides of the research area, in the right side of the study, correlation relationship Between the length and length of the waterways with the coefficient of determination (0. 929), the left side of the study area, the length and length of the waterways with a coefficient of determination (0. 8821) indicating the presence of the threshold and the chaos on both sides of the research area and these factors Due to the effects of erosion that indicate tectonic activity, side erosion, floor, grooves, and fault systems, especially in the left-hand region of the research, these factors (other than erosion) are more severe. The imbalance and the threshold on the left of the research area are more intense. The right side of the study area, with a branching ratio of 3. 34 and drainage density of 7. 87, the left side of the research area with a split ratio of 3. 50 and a drainage density of 4. 06, due to the split ratio of the two river research areas, has a moderate construction turbulence. The drainage density is correlated with the amount of erosion. In the right zone of the study, the moderate drainage density is equal to the rate of high erosion, which is due to the high percentage of Aghajari Formation in this research area and the greater proportion of branching in the left side of the research area to the right of the research is due to the Bakhtiari Formation. Conclusion: The most evolution of the waterways of both parts of the research, especially the left, is the tectonic research, erosion, and formations, Bakhtiari, Gachsaran and Aghajari formations are more than the Mishan Formation and the terraces and Low level alluvial fan and valley terrace deposits in the creation of chaos, Threshold and erosion. Lateral erosion in the right side of the research is more and less erosion of the floor, while the left side erosion of the research is less and the erosion of the floor is greater, these factors are due to imbalance in the two areas of research. Tectonic activity, albeit moderately, interfered with changing the state of the trenches, eroding, shortening the length of the waterways, and varying the number of waterways. Keywords: Ramhormoz River, Fluvial Geomorphology, MATRIX, ORDER MATRIX Model, Threshold.