Novel Research Aspects in Mathematical and Computer Science Vol. 2

In the present work we have gone a step forward towards integration by part of higher order Malliavin derivatives by formulating and extending some formulas and results on Malliavin calculus and ordinary stochastic differential equations to include delay stochastic differential equations as well as ordinary SDE’s (see [1–11]). Here we have also stated clearly what we mean by the Malliavin derivatives and densities of distributions of the solutions process for delay stochastic differential equations which we are considering. Generally speaking we can say that our work extends the first three chapters of the work by Norris to include delay SDE’s as well as ordinary SDE’s; see Theorems 2.3, 3.1 and 3.2 in [12]. We will also show in a Sequal paper to this work that the distribution of the solution process has smooth density. Also we will establish an integration by parts formula involving Malliavin derivatives of higher order. Observe that the delay SDE ([E: V 3]) is an extension of the SDE (3.3) in Norris to include delay SDE’s as well as ordinary SDE’s. We can see this by considering only the terms in ([E: V 3]) which include derivatives of the coefficients with respect to the space variable and in the same time it contain no derivative with respect to the delay variable. If we do this then we are automatically in the Norris case of SDE’s. Observe that the SDE’s (2.31), (2.32) and (2.33) are equivalent to the SDE’s (3.1), (3.2) and (3.3) in Norris [12] respectively. Thus we can see that our delay stochastic differential equations (2.28), (2.15) and (2.30) in fact extend the SDE’s (3.1), (3.2) and (3.3) in Norris; they include the case of delay as well as ordinary SDE’s.

The fixed point theorem for extending onto self-mappings in full cone metric spaces is proved in this study. Some analogous results in the literature are improved and extended by our findings.

A fuzzy analogue of the binomial option pricing tree model introduced by Cox et.al [1] was proposed by Yoshida [2] to study fuzzy American put option model in an uncertain environment using discrete time fuzzy stochastic process. Later, Muzzioli [3] used non overlapping triangular and trapezoidal fuzzy numbers to model the jump factors in American put option pricing model admitting impreciseness only in volatility while Xcaojian Yu [4] persumed impreciseness in both risk-free interest rate and volatility of the underlying stock involving non-overlapping trapezoidal fuzzy numbers. K. Meenakshi et al. [5] defined new fuzzy risk-neutral probability measures using general trapezoidal fuzzy numbers to study "Problem of Pricing American Fuzzy Put Option Buyer’s Model". The two up and down jump factors and the risk-free interest rate fluctuates oftenly and hence are uncertain in nature. When financial investors come across a high volatile or low volatile (up and down jump factors) fuzzy stocks, non-overlapping type of fuzzy numbers will not be sufficient to predict the underlying fuzzy stock prices as the fuzzy stock prices would go up only in the up state and would go down only in the down state of the fuzzy binomial tree. The uncertainty associated in the above stated fuzzy option pricing parameters could not be captured completely here. Such a scenario needs attention. To handle such a situation, we need to consider fuzzy numbers which are not only non - overlapping but partially or fully overlapping and/ or contained in too. Also not much consideration had been given to American fuzzy put option model involving fuzzy martingales. This paper will consider the study of the fuzzy analogue of martingale pricig theory in the context of fuzzy option pricing theory.

In this study, we discuss American Fuzzy Put Option Seller’s Model (AFPOSM) based on fuzzy future contract involving general linear octagonal fuzzy numbers (GLOFN); general in the sense that they could be either overlapping or non-overlapping partially or completely and/or contained in using the fuzzy risk-neutral probability measures introduced by us. We record a computational procedure to obtain the fuzzy profit and loss (PL) values of sellers using two-period fuzzy binomial tree model wherein the fuzzy stock price and the fuzzy future price following discrete time fuzzy stochastic process. The same is performed by deploying the real stock market data obtained from the website [6] that includes American style Microsoft Corporation (MSFT) shares of the year 2018 under the bullish (the constant initial fuzzy stock price goes above the constant fuzzy strike price) scenario. Utilizing the PL values, we estimate an optimal exercise time and an optimal exercise price of sellers. Also the discounted fuzzy intrinsic values of AFPOSM on fuzzy future contract are validated. The computations carried out in AFPOSM are performed using MATLAB 2016a software.

The Artificial Neural Network is one of the popular machine learning method, which has recently been applied in many areas of medical and medically related fields. Diabetes is one of the chronic diseases that occur when the blood sugar is too high. Early prediction of the disease may reduce the complications. Artificial Neural Network is known as an excellent classifier for nonlinear data. Some major issues are to be considered while constructing the network model.  The network structure, learning rate parameter and normalization of input vectors. The proposed research focuses on various normalization methods applied in back propagation neural networks to enhance the reliability of the trained network. The experimental results show that the performance of the classifier model can be increased based on the selection of the normalization method.

This paper aims to find the distance, distance Laplacian and distance signless Laplacian of \(\Gamma\left(Z_{n}\right)\),for some values of n. For a commutative ring  with non-zero identity, \(Z^{*}(R)\) denote the set of nonzero zero-divisors of R . The zero-divisor graph of R denoted by \(\Gamma\) (R), is a simple undirected graph with all non-zero zero-divisors as vertices and two distinct vertices \(x, y \in Z^{*}(R)\) are adjacent if and only if xy = 0 . In this paper, the distance, distance Laplacian and the distance singless Laplacian spectrum of \(\Gamma\left(Z_{n}\right)\) for n = p³ , pq are investigated. The specific combinatorial structure as well as the typical block structure of the distance related matrices of the graphs mentioned in this study both contributed to our decision to use matrices in the spectrum computation.

Five basic mistakes are found in the Riemann’s original paper proposed in 1859. The Riemann hypothesis becomes meaningless. 1. It is proved that on the real axis of complex plane and in the domain of the function, when the left-hand side of equation is finite, the right-hand side may be infinite, and vice versa. The Riemann Zeta function equation holds only at the point Re(s) = 1/2(s= a+ \(i b\)) . However, at this point, the Zeta function is infinite, rather than zero, to make Riemann hypothesis untenable. 2. An integral item around the original point of coordinate system was ignored when Riemann deduced the integral form of Zeta function. The item was convergent when Re(s) > 1but divergent when Re(s) < 1 .The integral form of Zeta function does not change the divergence of its series form. 3. A summation formula was used in the deduction of the integral form of Zeta function. The applicative condition of this formula is x > 0 . At point  x = 0 , the formula is meaningless. However, the lower limit of Zeta function’s integral is x = 0 , so the formula can not be used. 4. Because the integral lower limit of Zeta function is zero, the integrand function is not uniformly convergent, so integral sign and sum sign can not be exchanged. But Riemann made them interchangeable, resulting in that the integral form of Zeta function is untenable. 5.The formula \(\theta(x)=\sqrt{x} \theta(1 / x)\) of Jacobi function was used to prove the symmetry of Zeta function equation. The applicable condition of this formula is also  x > 0 . Because the lower limit of integral in the deduction was  x = 0 , this formula can not be used too. The zero calculation of Riemann Zeta function is discussed at last. It is pointed out that because approximate methods are used, the analytic property of the original function is destroyed and the Cauchy-Riemann equation can not be satisfied. So they are not the real zeros of strict Riemann Zeta function.

A standard method is proposed to prove strictly that the Riemann Zeta function equation has no non-trivial zeros on the whole complex plain. The real part and imaginary part of the Riemann Zeta function equation are separated completely. Let \(\zeta(a+i b)=\zeta_{1}(a, b)+i \zeta_{2}(a, b)\) and comparing the real part and the imaginary part of Zeta function equation individually, a set of equation about  a and b is obtained. It is proved that this equation set only has the solutions of trivial zeros. In order to obtain possible non-trivial zeros, the only way is that \(\zeta_{1}(a, b)\) and \(\zeta_{2}(a, b)\) are equal to zero simultaneously. However, by using the compassion method of infinite series, it is proved that \(\zeta_{1}(a, b)\) and \(\zeta_{2}(a, b)\) can not be equal to zero simultaneously. So the Riemann Zeta function equation has no non-trivial zeros. The Riemann hypothesis does not hold.

Cable-based parallel robots are a special kind of parallel robots, where rigid links are replaced by cables. The robots show several promising advantages over their rigid-link counterparts. Using the cables with unilateral driving properties, however, introduces many new challenges, and one most critical challenge focused herein is the stability of the robot. The minimum cable tension distributions in the workspace for cable-based parallel robots are investigated in this research to determine the relationship between stability and minimum cable tensions. The wrench matrix is used to create the kinematic model of a cable-based parallel robot. Then, using convex optimization theory, a non-iterative polynomial-based optimization algorithm with the suitable optimal objective function is developed, in which the least cable tension at every position is determined. Furthermore, three performance indices are presented to depict the distributions of minimum cable tensions in a given workspace region. The three performance indicators can be used to assess the stability of cable-based parallel robots, which is crucial. A new workspace, the Preset Minimum Cable Tension Workspace (SMCTW), is also added, in which all minimum tensions exceed a given amount, achieving the specified stability criteria. Finally, to show the distributions of the minimum cable tensions in the workspace and the relationship between the three performance indices and the stability, a camera robot parallel driven by four cables for aerial panorama photography is chosen.  

The ill-posedness of derivative interpolation is examined in this chapter, as well as a regularised derivative interpolation for band-limited signals. The Shannon Sampling Theorem is used to examine the ill-posedness. The convergence of the regularized derivative interpolation is studied by the combination of a regularized Fourier transform and the Shannon Sampling Theorem. The error estimation is given, and high order derivatives are also considered. The regularised derivative interpolation algorithm is compared to derivative interpolation with other algorithms. Some examples are used to prove and test the convergence property. The numerical findings reveal that for band-limited signals, the regularised sampling algorithm's derivative interpolation is more effective in decreasing noise.

In this paper, we consider the general quasi-differential expressions \(\tau_{1}, \tau_{2}, \ldots, \tau_{n}\) each of order  with complex coefficients and their formal adjoints on the interval ( a, b ) . It is shown in direct sum spaces \(L_{w}^{2}\left(I_{p}\right), p=\) 1 , 2,..., N of functions defined on each of the separate intervals with the cases of one and two singular end-points and when all solutions of the product equation \(\left[\prod_{j=1}^{n} \tau_{j}-\lambda w\right] u=0\) and its adjoint  \(\left[\prod_{j=1}^{n} \tau_{j}^{+}-\bar{\lambda} w\right] v=\)  0 are in \(L_{w}^{2}(a, b)\)  (the limit circle case) that all well-posed extensions of the minimal operator T₀ (\(\tau_{1}, \tau_{2}, \ldots, \tau_{n}\) ) have resolvents which are Hilbert-Schmidt integral operators and consequently have a wholly discrete spectrum. This implies that all the regularly solvable operators have all the standard essential spectra to be empty. These results are extension of those of formally symmetric expressions and those of general quasi-differential expressions.