Abstract: Geometric modeling plays an important role in the
constructions and manufacturing of curve, surface and solid
modeling. Their algorithms are critically important not only in
the automobile, ship and aircraft manufacturing business, but are
also absolutely necessary in a wide variety of modern applications,
e.g., robotics, optimization, computer vision, data analytics and
visualization. The calculation and display of geometric objects
can be accomplished by these six techniques: Polynomial basis,
Recursive, Iterative, Coefficient matrix, Polar form approach and
Pyramidal algorithms. In this research, the coefficient matrix (simply
called monomial form approach) will be used to model polynomial
rectangular patches, i.e., Said-Ball, Wang-Ball, DP, Dejdumrong and
NB1 surfaces. Some examples of the monomial forms for these
surface modeling are illustrated in many aspects, e.g., construction,
derivatives, model transformation, degree elevation and degress
reduction.
Abstract: In this work, a five step continuous method for the solution of third order ordinary differential equations was developed in block form using collocation and interpolation techniques of the shifted Legendre polynomial basis function. The method was found to be zero-stable, consistent and convergent. The application of the method in solving third order initial value problem of ordinary differential equations revealed that the method compared favorably with existing methods.
Abstract: It is well known, that any interpolating polynomial
p (x, y) on the vector space Pn,m of two-variable polynomials with
degree less than n in terms of x and less than m in terms of y, has
various representations that depends on the basis of Pn,m that we
select i.e. monomial, Newton and Lagrange basis e.t.c.. The aim of
this short note is twofold : a) to present transformations between the
coordinates of the polynomial p (x, y) in the aforementioned basis
and b) to present transformations between these bases.
Abstract: Shifted polynomial basis (SPB) is a variation of
polynomial basis representation. SPB has potential for efficient
bit level and digi -level implementations of multiplication over
binary extension fields with subquadratic space complexity. For
efficient implementation of pairing computation with large finite
fields, this paper presents a new SPB multiplication algorithm based
on Karatsuba schemes, and used that to derive a novel scalable
multiplier architecture. Analytical results show that the proposed
multiplier provides a trade-off between space and time complexities.
Our proposed multiplier is modular, regular, and suitable for very
large scale integration (VLSI) implementations. It involves less
area complexity compared to the multipliers based on traditional
decomposition methods. It is therefore, more suitable for efficient
hardware implementation of pairing based cryptography and elliptic
curve cryptography (ECC) in constraint driven applications.
Abstract: This paper presents unified theory for local (Savitzky-
Golay) and global polynomial smoothing. The algebraic framework
can represent any polynomial approximation and is seamless from
low degree local, to high degree global approximations. The representation
of the smoothing operator as a projection onto orthonormal
basis functions enables the computation of: the covariance matrix
for noise propagation through the filter; the noise gain and; the
frequency response of the polynomial filters. A virtually perfect Gram
polynomial basis is synthesized, whereby polynomials of degree
d = 1000 can be synthesized without significant errors. The perfect
basis ensures that the filters are strictly polynomial preserving. Given
n points and a support length ls = 2m + 1 then the smoothing
operator is strictly linear phase for the points xi, i = m+1. . . n-m.
The method is demonstrated on geometric surfaces data lying on an
invariant 2D lattice.
Abstract: Polynomial bases and normal bases are both used for
elliptic curve cryptosystems, but field arithmetic operations such as
multiplication, inversion and doubling for each basis are implemented
by different methods. In general, it is said that normal bases, especially
optimal normal bases (ONB) which are special cases on normal bases,
are efficient for the implementation in hardware in comparison with
polynomial bases. However there seems to be more examined by
implementing and analyzing these systems under similar condition. In
this paper, we designed field arithmetic operators for each basis over
GF(2233), which field has a polynomial basis recommended by SEC2
and a type-II ONB both, and analyzed these implementation results.
And, in addition, we predicted the efficiency of two elliptic curve
cryptosystems using these field arithmetic operators.
Abstract: This paper presents a methodology towards the emulation of the electrical power consumption of the RF device during the cellular phone/handset transmission mode using the LTE technology. The emulation methodology takes the physical environmental variables and the logical interface between the baseband and the RF system as inputs to compute the emulated power dissipation of the RF device. The emulated power, in between the measured points corresponding to the discrete values of the logical interface parameters is computed as a polynomial interpolation using polynomial basis functions. The evaluation of polynomial and spline curve fitting models showed a respective divergence (test error) of 8% and 0.02% from the physically measured power consumption. The precisions of the instruments used for the physical measurements have been modeled as intervals. We have been able to model the power consumption of the RF device operating at 5MHz using homotopy between 2 continuous power consumptions of the RF device operating at the bandwidths 3MHz and 10MHz.