In this article, we have proposed an efficient Bee Colony Optimization method, namely Phased Bee Colony Optimization (PBCO) technique for solving **mathematical** functions in a multidimensional space. The search process of the optimization approach is directed towards a region or a hypercube in a multidimensional space to find a global optimum or near global optimum after a predefined number of iterations. The process in the entire search area to another region (new search area) surrounding the optimum value found so far after a few iterations and restarts the search process in the new region. However, the search area of the new region is reduced compared to previous search area. Thus, the search process finding advances and jumps to a new search space (with reduced area space) in several phases until the algorithm is terminated. The PBCO technique has tested on a set of **mathematical** benchmark functions with number of dimensions up to 100 and compared with several relevant optimizing approaches to evaluate the performance of the algorithm. It has observed that the proposed technique performs either better or similar to the performance of other optimization methods.

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Encryption is the concept spread all over in today’s scenario. In the perspicacious computing and perspicacity scenario information must be kept secret and only be shared with the trusted parties. Encryption techniques integrate delays incremented the overhead of the astute computing contrivances. Digital logic circuits works more expeditious than any other techniques so as it can be utilized for more expeditious encryption minimizing the delays. **Mathematical** equations are the intricate equations which makes encryption secure and intricate than any other techniques. Avalanche effect is the property of encryption in which various keys engenders unique cipher-text of the information. In this, an intricate mechanism of encryption is defined by **mathematical** equation utilizing variant compliments with Gray code has been proposed, which utilizes the **mathematical** functions to engender a pseudo number with avails in generation of variants keys with which desultory and independent cipher text is obtained. Performing crypt- analysis of the method gives the security and vigor of the algorithm and keys.

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The suitable chosen **function** for the transition curve description or for the entry in the algorithm of a suitable **function** from many offered gives the possibility to define the measured values evalua- tion with more precision. In our opinion the values entry in MathLab program and the curve plotting inclusive the transfer in the text editors is more simply and the results are more accurate than when using the accessible tabulator processor excel. In addition the searched results can be automated, as the transition temperature determination using the minimum energy method, which is used for the making of the constructional steels truncated sym- bols, or if need be, for the transition temperature as the inflexion point searching. In MathLab cal- culation the inflexion point of demanded param- eters.

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This paper describes the method of derivation of **mathematical** description of a cam orbit in rotary rakes (hay aggregators). At ﬁ rst, the authors describe basic construction elements of the rotary hay rake mechanism and their mutual links and relationships. Therea er, they deﬁ ne the origin and the orientation of the system of coordinates, in which all calculations are carried out. In the next step they deﬁ ne basic requirements concerning the assurance of an optimum functioning of cam mechanisms as well as their transformation into **mathematical** equations. These requirements represent a base for the **mathematical** formulation of an optimum transition curve and it is emphasized that an optimum formulation of parameters of this curve is very important. In the course of calculation, they use also a normalized transition curve, which is used for the optimizations of the total number of parameters of the transition curve. Therea er, they take into account mechanical and operational parameters of the hay aggregators and convert the optimum transition curve to that part of the space curve, which agrees at best with these parameters. Finally, the whole cam orbit is constructed using individual segments and presented as a sequentially deﬁ ned space curve. Its individual parts concur sequentially to the level of the second derivation and are described as explicit **mathematical** functions of mechanical and operational parameters of the hay aggregator. The deﬁ nition of the system of coordinates, the execution phase of calculations and the ﬁ nal shape of the cam orbit are illustrated in graphs.

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HELO follows the theory of probability [15], [16] and defines the class probability as a subclass of the class mathematical function to enable mathematical operations with probabilities [r]

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The cell mechanism **function** presented by previous studies does contain logarithmic and exponential **function** forms [19-24]. In view of the above functions, they are the objective reflection of nature law phenomenon in mathematics. When the **mathematical** functions are constructed, from the perspective of mathematics, it should not change the symbols and exponential form of the **function** to meet the data obtained from such experiments. Only in this way, it can be ensured that the physical phenomenon of the functions will not be distorted by the change of symbol or the adjustment of index. In addition, the selected **mathematical** functions should be proportioned according to the dimensional and physical requirements.

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In order to verify the effectiveness of the proposed **mathematical** approach, a six-generating units sample test system including transmission loss is considered[13]. The operating ranges of all committed generating units are restricted by their minimum and maximum power generation limits. The sample system contains of six thermal generating units. The total load demand of the system is 1263 MW. The cost coefficients, generation limits of each generating unit and transmission loss coefficient are given in Appendix. The proposed approach is applied to sample test system using Matlab 6.5 programming language. The optimal generation schedule obtained through the proposed method minimizes the total fuel cost and satisfies power balance constraint i.e. total output minus loss should be equal to total load demand of the system and other operating constraints. The number of iterations taken by the method for the convergence is four. The optimal solution obtained through the proposed **mathematical** method is Table 1.

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Upreti and Deb [18] present an optimal procedure of an ammonia synthesis reactor using genetic algorithm. They used correct objective **function** and correct stoichiometric expression of the partial pressures of N 2 , H 2 , NH 3 . They used simple GA in combination with Gear package of NAG library's subroutine, D02EBF, for the optimization of ammonia synthesis reactor. They obtained mass flow rate of nitrogen, feed gas temperature and reaction gas temperature at every 0.01 m of 10 m reactor length. Moreover there is a contradiction in the temperatures and gas flow rate profiles obtained. They reported the profiles that were not smooth as in earlier literature. Also, they reported reverse reaction condition at the top temperature of 664K which was not found in the literature earlier. Babu and Angira [2] [3] used a correct objective **function** and correct stoichometeric expression of pressures of N 2 , H 2 , NH 3 but there is a contradiction in the temperatures and gas flow rate profiles obtained. Hence, our research is carried out in order to take care of bad models or bad computations.

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With regard to the strategies used by students to solve the problems as a **function** of the degree of abstraction, we shall briefly analyze some aspects of the most notable strategies: direct modeling, counting, and numerical facts (see Bermejo et al., 2002). Direct modeling is used significantly more often in concrete problems and with drawings than in verbal and numerical problems, F(3, 276) = 5.06, p < .05, as reported by other authors (Bermejo et al., 1998; Carpenter, Hiebert, & Moser, 1983; Fuson & Briars, 1990; Fuson & Burghardt, 2003). Counting strategies appear mainly at the concrete, pictorial, and verbal levels in second grade, whereas the numerical level is especially noted in fourth grade. More specifically, in second grade, counting in was significantly higher drawing and verbal problems than in numerical problems, F(3, 90) = 3.71, p < .05, and, in third grade, it was significantly higher in concrete problems than in verbal ones, F(3, 90) = 2.76, p < .05. It is interesting to note that counting strategies in general are much more frequent when the unknown is located in the result than when it is located in the first term, as can be seen in Figure 3. The significance of the interaction Type of problem × Location of unknown, F(3, 276) = 4.85, p < .05, indicates that counting is used in a different way depending on the type of problem and the location of the unknown. In effect, verbal problems lead to a higher number of counting strategies when the unknown is in the result, and its frequency is also the lowest in these problems when the unknown is located in the first term, F(3, 276) = 3.64, p < .05.

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as the objective **function** [8]. Cross-sectional area was chosen as the design variable. The main aim was to find the minimum cost and minimum displacement. Analysis were performed and results shows that there was a saving by 4-16% in cost. A new method of optimization of tower design was done for extra high-voltage transmission lines taking tower weight and geometry as the desired **function** [9]. Nonlinear optimization method was incorporated taking into consideration the configuration and design of the tower as the prime interest. Fuzzy optimization technique was used for the double circuit transmission tower under multiple loading condition. Structural optimization of transmission tower was done for obtaining minimum weight using an optimization methodology that allows to obtain more efficient solution than conventional designs of towers [10]. Studies were done on design of a 33KV double circuit transmission line tower with rectangular base self- supporting lattice tower that optimizes the present geometry. The structural behavior of existing tower was studied and Excel programs are developed for calculation of load in STAAD-Pro. Analysis was performed in STAAD- Pro, and the axial stress, compressive stress of the tower member were obtained. While weight optimization, the height of the tower, base width and basic outline of the tower were design variables. Optimization of truss tower for least weight was studied for tower subjected to multiple combination of dead load, wind load and seismic load [11]. Optimization problem is divided into two design spaces. Hooke and Jeeves method were used for changing the coordinate variables. Design variables chosen are the area

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Participant D used inductive reasoning to arrive at her answer. Naming the asymptote as -2 is mathematically incorrect, since -2 cannot be located on the Cartesian plane. There is a tendency by some learners to think that p or q could replace x and y respectively, or that they were synonymous such that one could replace the other interchangeably. Participant D first wrote a general equation , and had to be reminded of the asymptotes of the **function**. She then substituted the number three for a. She further equated q to the asymptote. Finally, she wrote that asymptote = -2. The object, which Participant D wrote, did not exist on the Cartesian plane because the word asymptote did not show a clear representation on the coordinate system. She did not do anything about the vertical asymptote. She also did not mention the vertical asymptote. The participant’s routines were ritualised because she followed- up the routines of others as her own routines, which did not lead to mathematically endorsed narratives. Participant D’s equations were not **mathematical** as they could not be located on the Cartesian plane. Her level of communication is at level one because she failed to identify the asymptote from an algebraic representation showing some disjointed relationships.

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Figure 3. Relative density of the argument A question remains open with defined in such a manner density of the **function** and density of the argument: why the density of the corresponding arrays {x} and {y} changes – whether the distance between their values is changed (which contradicts to their continuity with power of continuum) or the values themselves increase their size (which contradicts the fact that they are presented on the number axis as points without any size)?

It is interesting to note that though both the objective **function** formulations are practically of the same general nature, i.e. to minimize the difference between the simulated response of the aquifer and the actual response of the aquifer, there is contrasting difference in the accuracy achieved by both the formulations while using iden- tical optimization algorithm for solution. Although this difference may not be so pronounced when using error free concentration measurements, it is more apparent when erroneous measurements are utilized. This can be explained by analysing the two objective formulations. If we imagine the solution response space representing the objective **function** values for corresponding decision variable values as a rough plane, such that roughness of the surface creates multiple local optima, the objective is to find the global optimum amongst these local optima. The second objective formulation which uses the absolute difference between the observed and the estimated pollutant concentration value do not alter the roughness of the search space at all. However, when the difference between the observed and the simulated pollutant concentration is divided by the observed pollutant concentra- tion measurement, the difference gets normalised. This gives proportional weight age to pollutant concentration measurements for all the observation locations at all times. As a result there is greater distinction between dif- ferent local optima and it is easier for the optimization algorithm to get out of any local optima. The biggest challenge of any optimization algorithm is to get out of local optima and find a global optimum.

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According to some influential accounts of anthropocentric epistemology and even metaphysics, what counts as a (“real”) pattern depends on the human being’s sensory and epistemic capabilities, as well as human motives and interests. But since scientists make use of various kinds of technical auxiliaries, human sensory and epistemic capabilities cannot serve to restrict the class of patterns in science. To make our notion of patterns in science adaptable to past, contemporary and future developments in science in principle, we cannot claim much more than that patterns are (in an ontological and epistemological sense) **mathematical** properties. This claim is justified by the historical fact that computers are more and more able to precisely explicate various patterns in data, including very non-trivial cases, such as human facial recognition. However, if I accept such an abstract notion of patterns, which I do, I can still claim that a pattern must be constructive for epistemic reasons. Even if we use our fastest computers or some help from an extraterrestrial colleague, there must still be a **mathematical** way to test or explicate the pattern feasibly. This cannot be achieved with the general framework from the information theory of Shannon, Kolmogorov and Chaitin. Grenander’s general pattern theory provides a well-suited approach, due to its constructiveness. To clarify the connection between patterns and phenomena, which is not an identity, I distinguish between general and concrete patterns. Further scientific assumptions that are additional to the observations and data in question, deter- mine what class of concrete patterns make a general pattern that corresponds to a phenomenon. In principle, these assumptions can be mathematically explicated, but the significant structural dissimilarities between many different concrete pat- terns of the same general pattern justify my differentiation between concrete and general patterns.

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If X is a compact Hausdorﬀ space, then C(X) is a C ∗ -algebra and every commutative uni- tal C ∗ -algebra is isometrically isomorphic to a C(K)-space for some compact Hausdor ﬀ space K . Every linear surjective isometry between C(K )-spaces is a weighted substitu- tion operator as we have seen earlier. The concept of a disjointness preserving operator between noncommutative algebras can be discussed in terms of zero-product preserv- ing map which by definition is a linear map T between two algebras A and B such that T(a)T (b) = 0 whenever ab = 0. The C(K)-spaces carry diﬀerent **mathematical** structures on them like commutative ring structure, Banach algebra structure, Banach lattice struc- ture, C ∗ -algebra structure, and so forth. It is easy to see in light of results in Section 1 and Section 2 that surjective isometries and lattice isomorphisms between C(K)-spaces turn out to be zero product preserving maps. In fact, T preserves zero product between C(K)-spaces if and only if T is a weighted substitution operator.

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This article aims at achieving desirable response properties at variable settings, namely low bias and variance, using a compromise programming-based criterion along with optimization performance measures as alternative to other methods mathematically sound but less appealing to practitioners, namely to those who have limited **mathematical** or statistical background. The feasibility and effectiveness of the suggested approach is illustrated through examples from the literature and its results compared with those of several methods.

In this paper, we propose a new region-based Active Contour Model (ACM) that employs signed pressure force (SPF) as a level set **function**. Further, a flood fill algorithm is incorporated along with SPF **function** for robust object extraction. Signed pressure force (SPF) parameters, is able to control the direction of evolution of the region. The proposed system shares all advantages of the C–V and GAC models. The proposed ACM has an additional advantage i.e. of selective local or global segmentation. Flood Fill framework is employed for retrieving the object upon successful detection in the image. In addition, the computer simulation results show that the proposed system could address object detection within an image and its extraction with highest order of efficiency.

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[5] F. Chaspeau-Blondeau and A. Monir, “Numerical Evalua- tion of the Lambert W **Function** and Application to Gen- eration of Generalized Gaussian Noise with Exponent ½,” IEEE Transactions on Signal Processing, Vol. 50, No. 1, 2002, pp. 2160-2165. doi:10.1109/TSP.2002.801912 [6] R. M. Corless, G. H. Gonnet, D. E. Hare and D. J. Jeffrey,

II. REVIEWS OF PREVIOUS WORKS An insight on how to model a simple nonlinear conveyor system was provided by[3]. The paper did an overview of the conveyor belt system with fair consideration of non-linear friction. The work later derived the **mathematical** model of the conveyor system considering the non-linear friction on the system. The work by[4] designed a conveyor system which includes; belt speed, belt width, motor selection, belt specification, shaft diameter, and pulley and gear box selection using standard model calculation.

Abstract. Some obstacles create vulnerable situations in financial market. Overcome this unexpected situation, it is essential to reform the financial market by measuring the risk of share market. This project investigates the sensitivity of radial basis functions to construct different volatility surface by radial basis **function** approaches to understand the risk of share market. Different types of radial basis functions on the basis of different error measurement such as average error as well as relative average error of Dhaka Stock Exchange (DSE) are measured and multiquadratic **function** gives the best result with compare to other functions especially Gaussian and Thin plate spline **function**.

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