In recent years, the computer graghics have improved highly. In this paper,  we present a special kind  of  the computer  graghics named  vectorial  graphics and point out  to  their  mathematical backgrounds  by using  the NURBS curves  and surfaces which are often unknown. Then, we propose the use of the presented mathematical formulas in the vectorial computer ghraghics.

This study aimed to estimate the genetic parameters of body weight traits in Markhoz goats, using B-SPLINE random regression models. The data used in this study included 19549 records collected during 29 years (1992-2021) in Markhoz goat Breeding Research Station, located in Sanandaj, Iran. The model used to analyze data included fixed effects (year of birth, sex, type of birth and age of dam) and random effects including direct additive genetic, maternal additive genetic, permanent environmental and maternal permanent environmental assuming homogeneous and heterogeneous residual variance during the time. Akaike (BIC) and Bayesian (BIC) information criteria were used to compare the models and bspq.4.4.4.4 was selected as the best model. The direct heritability values for birth, 3-month, 6-month, 9-month and 12-month weights were estimated to be 0.14, 0.16, 0.08, 0.28 and 0.26, respectively. Genetic correlation between body weights at birth and 3-month, birth and 6-month, birth and 9-month, birth and 12-month, 3-month and 6-month, 3-month and 9-month, 3-month and 12-month, 6-months and 9-month and 9-month and 12-month were 0.22, 0.38, 0.21, 0.56, -0.26, 0.30, 0.62, 0.86 and 0.77, respectively. The highest phenotypic correlation was between the weight of 9-month and 12-month (0.82) and the lowest correlation was between birth weight and 3-month and 6-month (0.12). The results showed that the 9-month weight is a good criterion for selection in Markhoz goats.

The collocation methods based on cubic B-SPLINE, are developed to approximate solution of the second and first kind Fredholm integral equations. First we collocate the solution by B-SPLINE and the Newton-Cotes formula is used to approximate integral. Convergence analysis has been investigated and proved that the quadratur rule is fourth order convergent. The presented methods are tested to the problem, and the absolute error in the solution are compared with existing methods [1, 2, 7, 10] to verify the accuracy and convergent nature of proposed methods.

Purpose: The aims of the paper are to obtain necessary and sufficient conditions of existence and smoothness for non-polynomial SPLINE spaces of fifth order, to establish the uniqueness of the Bφ-SPLINE spaces in the class C4 among mentioned spaces (under condition of fixed grid), and to prove the embedding of the Bφ-SPLINE spaces corresponding to embedded grids.Methods: In the paper, the approximation relations with initial grid and with complete chain of vectors are applied to obtain the minimal SPLINE spaces. Usage of locally orthogonal chain of vectors gives opportunity to construct special approximation relations from which the initial space of Bφ SPLINEs is constructed.Results: Deletion of a knot from initial grid gives a new grid, and as result, a new space of Bφ SPLINEs is embedded in the initial space mentioned above.Conclusions: Consequent deletion of the knots (one by one) generates the sequence of the embedded spaces of Bφ SPLINEs. Obtained results are successfully proved. They may be applied to SPLINE-wavelet decompositions.

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In this paper we develop a non polynomial cubic SPLINE function which we called "TS SPLINE", based on trigonometric functions. The convergence analysis of this SPLINE is investigated in details. The de, nition of B-SPLINE basis function for TS SPLINE is extended and "TS B-SPLINE" is introduced. This paper attempts to develop collocation method based on this B-SPLINE for the numerical solution of the nonlinear Klein-Gordon equation. The convergence analysis of this approach is discussed, the second order of convergence is proved consequently. The proposed method is applied on some test examples and the numerical results are compared with those already available in literature. Observed errors in the solutions show the e, ciency and numerical applicability of the proposed method.