Considering
the ring of integers modulo n, the classical Fermat-Euler theorem establishes the existence of a specific natural number
satisfying the following property:
for all x belonging to the ...group of units of
In this manuscript, this result is extended to a class of rings that satisfies some mild conditions.
The modified Totient function of Carmichael λ(.) is revisited, where important properties have been highlighted. Particularly, an iterative scheme is given for calculating the λ(.) function. A ...comparison between the Euler φ and the reduced totient λ(.) functions aiming to quantify the reduction between is given.
The (i) reciprocity relations for the relative Fisher information (RFI, hereafter) and (ii) a generalized RFI–Euler theorem are self-consistently derived from the Hellmann–Feynman theorem. These new ...reciprocity relations generalize the RFI–Euler theorem and constitute the basis for building up a mathematical Legendre transform structure (LTS, hereafter), akin to that of thermodynamics, that underlies the RFI scenario. This demonstrates the possibility of translating the entire mathematical structure of thermodynamics into a RFI-based theoretical framework. Virial theorems play a prominent role in this endeavor, as a Schrödinger-like equation can be associated to the RFI. Lagrange multipliers are determined invoking the RFI–LTS link and the quantum mechanical virial theorem. An appropriate ansatz allows for the inference of probability density functions (pdf’s, hereafter) and energy-eigenvalues of the above mentioned Schrödinger-like equation. The energy-eigenvalues obtained here via inference are benchmarked against established theoretical and numerical results. A principled theoretical basis to reconstruct the RFI-framework from the FIM framework is established. Numerical examples for exemplary cases are provided.
•Legendre transform structure for the RFI is obtained with the Hellmann–Feynman theorem.•Inference of the energy-eigenvalues of the SWE-like equation for the RFI is accomplished.•Basis for reconstruction of the RFI framework from the FIM-case is established.•Substantial qualitative and quantitative distinctions with prior studies are discussed.
Outsourcing computation allows an outsourcer with limited resource to delegate the computation load to a powerful server without the exposure of true inputs and outputs. It is well known that modular ...exponentiation is one of the most expensive operations in public key cryptosystems. Currently, most of outsourcing algorithms for modular exponentiation are based on two untrusted servers or have small checkability with single server. In this paper, we first propose an efficient outsourcing algorithm of modular exponentiation based on two untrusted servers, where the outsourcer can detect the error based on Euler theorem with a probability of 1 if one of the servers misbehaves. We then present an outsourcing algorithm of modular exponentiation with single server, and the outsourcer can also check the failure with a probability of 1. Therefore, the proposed algorithm with single server improves efficiency and checkability simultaneously compare with the previous ones. Finally, we provide the experimental evaluations to demonstrate that the proposed two algorithms are the most efficient ones in all of the outsourcing algorithms for an outsourcer.
Variational extremization of the relative Fisher information (RFI, hereafter) is performed. Reciprocity relations, akin to those of thermodynamics are derived, employing the extremal results of the ...RFI expressed in terms of probability amplitudes. A time independent Schrödinger-like equation (Schrödinger-like link) for the RFI is derived. The concomitant Legendre transform structure (LTS, hereafter) is developed by utilizing a generalized RFI-Euler theorem, which shows that the entire mathematical structure of thermodynamics translates into the RFI framework, both for equilibrium and non-equilibrium cases. The qualitatively distinct nature of the present results vis-á-vis those of prior studies utilizing the Shannon entropy and/or the Fisher information measure (FIM, hereafter) is discussed. A principled relationship between the RFI and the FIM frameworks is derived. The utility of this relationship is demonstrated by an example wherein the energy eigenvalues of the Schrödinger-like link for the RFI are inferred solely using the quantum mechanical virial theorem and the LTS of the RFI.
•Generalized RFI-Euler theorem is formulated.•Legendre transform structure for the RFI is obtained.•Inference model for energy eigenvalues is formulated.
We consider commutative regular and context-free grammars, or, in other
words, Parikh images of regular and context-free languages. By using linear
algebra and a branching analog of the classic Euler ...theorem, we show that,
under an assumption that the terminal alphabet is fixed, the membership problem
for regular grammars (given v in binary and a regular commutative grammar G,
does G generate v?) is P, and that the equivalence problem for context free
grammars (do G_1 and G_2 generate the same language?) is in $\mathrm{\Pi_2^P}$.
The Euler Hadamard/DCT polynomial is defined in this paper. This polynomial is similar to the Euler theorem in that it calculates the unit operation. However, the Euler Hadamard/DCT polynomial is ...computed by using matrix operations and angle information. The computations of Euler Hadamard/DCT polynomial can be used to construct higher order of Hadamard/DCT matrices and other real orthogonal matrices.
Specially, the inverses of these Euler Hadamard polynomials are simply from the element inverse and the basic idea is corresponding to the polynomial function
X
N
·(
X
N
)
T
=
NI
N
with Hadamard computations
H
N
·(
H
N
)
T
=
N
I
N
, which is the unit operation of orthogonal matrix.
Otherwise, from the geometric view, we give a briefly description to the Euler Hadamard/DCT polynomial. The geometric structure shows that there possibly exist some other orthogonal or element-wise inverse matrices (or polynomials) by using the generalized Euler Hadamard/DCT polynomial.
Until the seventeenth century, rational numbers were represented as fractions. It was starting from that century, thanks to Simon Stevin, that for all practical purposes the use of decimal notation ...became widespread. Although decimal numbers are widely used, their organic development is lacking, while a vast literature propagates misconceptions about them, among both students and teachers. In particular, the case of the period 9 is especially interesting. In this paper, changing the point of view, we propose substituting the usual definitions of the periodic decimal representation of a rational number - the one obtained through long division introduced in the secondary school and the one obtained through the series concept for undergraduate level - with one which, instead of infinite progressions, uses a known property of fractions deducible from Euler's Totient Theorem. The long division becomes a convenient algorithm for obtaining the decimal expansion of a rational number. The new definition overcomes the difficulties noted in the literature. An elementary proof of Euler's Totient Theorem will be given depending only on Euclidean Division Theorem.
Rigid Body Kinematics
Fundamental Spacecraft Dynamics and Control,
09/2015
Book Chapter
This chapter introduces satellite attitude notions and their representations by different methods and also describes a few attitude determination algorithms and their realizations.
PDF
