On Existence of Primitive Normal Elements of Cubic Form over Finite Fields

For a prime [Formula: see text]and a positive integer[Formula: see text], let [Formula: see text] and [Formula: see text] be the extension field of [Formula: see text]. We derive a sufficient condition for the existence of a primitive element [Formula: see text] in[Formula: see text] such that [Formula: see text] is also a primitive element of [Formula: see text], a sufficient condition for the existence of a primitive normal element [Formula: see text] in [Formula: see text] over [Formula: see text] such that [Formula: see text] is a primitive element of [Formula: see text], and a sufficient condition for the existence of a primitive normal element [Formula: see text] in [Formula: see text] over [Formula: see text] such that [Formula: see text] is also a primitive normal element of [Formula: see text] over [Formula: see text].

The existence of primitive normal elements of quadratic forms over finite fields

For [Formula: see text] ([Formula: see text]), denote by [Formula: see text] the finite field of order [Formula: see text] and for a positive integer [Formula: see text], let [Formula: see text] be its extension field of degree [Formula: see text]. We establish a sufficient condition for existence of a primitive normal element [Formula: see text] such that [Formula: see text] is a primitive element, where [Formula: see text], with [Formula: see text] satisfying [Formula: see text] in [Formula: see text].

Normal elements in the mod-𝑝 Iwasawa algebra over SL𝑛(ℤ𝑝): A computational approach

This is the last in a series of articles where we are concerned with normal elements of noncommutative Iwasawa algebras over {\mathrm{SL}_{n}(\mathbb{Z}_{p})}. Our goal in this portion is to give a positive answer to an open question in [D. Han and F. Wei, Normal elements of noncommutative Iwasawa algebras over \mathrm{SL}_{3}(\mathbb{Z}_{p}), Forum Math. 31 2019, 1, 111–147] and make up for an earlier mistake in [F. Wei and D. Bian, Normal elements of completed group algebras over \mathrm{SL}_{n}(\mathbb{Z}_{p}), Internat. J. Algebra Comput. 20 2010, 8, 1021–1039] simultaneously. Let n ({n\geq 2}) be a positive integer. Let p ({p>2}) be a prime integer, {\mathbb{Z}_{p}} the ring of p-adic integers and {\mathbb{F}_{p}} the finite filed of p elements. Let {G=\Gamma_{1}(\mathrm{SL}_{n}(\mathbb{Z}_{p}))} be the first congruence subgroup of the special linear group {\mathrm{SL}_{n}(\mathbb{Z}_{p})} and {\Omega_{G}} the mod-p Iwasawa algebra of G defined over {\mathbb{F}_{p}}. By a purely computational approach, for each nonzero element {W\in\Omega_{G}}, we prove that W is a normal element if and only if W contains constant terms. In this case, W is a unit. Also, the main result has been already proved under “nice prime” condition by Ardakov, Wei and Zhang [Non-existence of reflexive ideals in Iwasawa algebras of Chevalley type, J. Algebra 320 2008, 1, 259–275; Reflexive ideals in Iwasawa algebras, Adv. Math. 218 2008, 3, 865–901]. This paper currently provides a new proof without the “nice prime” condition. As a consequence of the above-mentioned main result, we observe that the center of {\Omega_{G}} is trivial.

Normal elements of noncommutative Iwasawa algebras over SL3(ℤ_ p )

Let p be a prime integer and let {\mathbb{Z}_{p}} be the ring of p-adic integers. By a purely computational approach we prove that each nonzero normal element of a noncommutative Iwasawa algebra over the special linear group {\mathrm{SL}_{3}(\mathbb{Z}_{p})} is a unit. This gives a positive answer to an open question in [F. Wei and D. Bian, Erratum: Normal elements of completed group algebras over \mathrm{SL}_{n}(\mathbb{Z}_{p}) [mr2747414], Internat. J. Algebra Comput. 23 2013, 1, 215] and makes up for an earlier mistake in [F. Wei and D. Bian, Normal elements of completed group algebras over \mathrm{SL}_{n}(\mathbb{Z}_{p}) , Internat. J. Algebra Comput. 20 2010, 8, 1021–1039] simultaneously.

Simulating of Bird Strike on Aircraft Laminated Glazing

A bird strike is a critical problem in the context of safety in the aviation industry. All modern aircraft structures are designed with account of likely collision with birds. Thus, aviation standards in force require that the aircraft construction would allow the crew to conclude the flight safely after collision with a 1.81-kg bird. A method for analysing the stress-strained state of laminated airplane glazing at different operational factors is presented. The method includes a technique for strength analysis of the laminated airplane glazing at bird impact, and a technique for analysis of excess pressure. The model of laminated glazing is based on the refined first-order theory accounting for transverse shear strains, thickness reduction and normal element rotation inertia in each layer. The mathematical model of the pressure impulse authentically reproducing the bird impact is based on experimental research. Theoretical results are in good agreement with experimental data, thus allowing to recommend the method to develop new airplane glazing elements.

NORMAL ELEMENT ON IDENTIFY PROPERTIES

One type of element in the ring with involution is normal element. Their main properties is commutative with their image by involution in ring. Group invers of element in ring is always commutative with element which is commutative with itself. In this paper, properties of normal element in ring with involution which also have generalized Moore Penrose invers are constructed by using commutative property of group invers in ring. Keywords: Normal, Moore Penrose, group, involution

On primitive normal elements over finite fields

Let [Formula: see text] be an extension of the field [Formula: see text] of degree [Formula: see text] where [Formula: see text] for some positive integer [Formula: see text] and prime [Formula: see text] In this paper, we establish a sufficient condition for the existence of a primitive element [Formula: see text] such that [Formula: see text] is also primitive as well as a primitive normal element [Formula: see text] of [Formula: see text] over [Formula: see text] such that [Formula: see text] is primitive.

Some notes on the k-normal elements and k-normal polynomials over finite fields

Recently, the [Formula: see text]-normal elements over finite fields are defined and characterized by Huczynska et al. In this paper, we give a new characterization of [Formula: see text]-normal elements and define [Formula: see text]-normal polynomials over finite fields. In what follows, we show that the problem of existence of a primitive 1-normal element in [Formula: see text] over [Formula: see text], for all [Formula: see text] and [Formula: see text], which has been stated by Huczynska et al., is not satisfied. Furthermore, we extend a recursive method given by Kyuregyan for constructing an infinite family of [Formula: see text]-polynomials, to constructing an infinite sequence of [Formula: see text]-normal polynomials over [Formula: see text].

On the bound of the complexity of the normal basis generated by the trace of the dual element of a Type I optimal normal element

Let 𝔽q be a finite field and 𝔽qn be an extension, where a Type I optimal normal basis exists. Suppose n = mk. We provide a bound for the complexity of the normal basis of 𝔽qm over 𝔽q generated by the trace of the dual element of the generator of a Type I optimal normal basis of 𝔽qn over 𝔽q. Further, we show that our bounds are better than the known bounds under certain conditions on m and k.