Square | EPFL Graph Search

In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ABCD would be denoted . Characterizations A quadrilateral is a square if and only if it is any one of the following:

A rectangle with two adjacent equal sides
A rhombus with a right vertex angle
A rhombus with all angles equal
A parallelogram with one right vertex angle and two adjacent equal sides
A quadrilateral with four equal sides and four right angles
A quadrilateral where the diagonals are equal, and are the perpendicular bisectors of each other (i.e., a rhombus with equal diagonals)
A convex quadrilateral with successive sides a,

A rectangle with two adjacent equal sides
A rhombus with a right vertex angle
A rhombus with all angles equal
A parallelogram with one right vertex angle and two adjacent equal sides
A quadrilateral with four equal sides and four right angles
A quadrilateral where the diagonals are equal, and are the perpendicular bisectors of each other (i.e., a rhombus with equal diagonals)
A convex quadrilateral with successive sides a,

Construction Of Self-Dual Integral Normal Bases In Abelian Extensions Of Finite And Local Fields

Erik Jarl Pickett

Let

F/E

be a finite Galois extension of fields with abelian Galois group

\Gamma

. A self-dual normal basis for

F/E

is a normal basis with the additional property that

Tr_{F/E}(g(x),h(x))=\delta_{g,h}

for

g,h\in\Gamma

. Bayer-Fluckiger and Lenstra have shown that when

char(E)\neq 2

, then

F

admits a self-dual normal basis if and only if

[F:E]

is odd. If

F/E

is an extension of finite fields and

char(E)=2

, then

F

admits a self-dual normal basis if and only if the exponent of

\Gamma

is not divisible by

4

. In this paper we construct self-dual normal basis generators for finite extensions of finite fields whenever they exist. Now let

K

be a finite extension of

\Q_p

, let

L/K

be a finite abelian Galois extension of odd degree and let

\bo_L

be the valuation ring of

L

. We define

A_{L/K}

to be the unique fractional

\bo_L

-ideal with square equal to the inverse different of

L/K

. It is known that a self-dual integral normal basis exists for

A_{L/K}

if and only if

L/K

is weakly ramified. Assuming

p\neq 2

, we construct such bases whenever they exist.

2010

Explicit construction of self-dual integral normal bases for the square-root of the inverse different

Erik Jarl Pickett

Let K be a finite extension of Q(p), let L/K be a finite abelian Galois extension of odd degree and let D-L be the valuation ring of L. We define A(L/K) to be the unique fractional D-L-ideal with square equal to the inverse different of L/K. For p an odd prime and L/Q(p) contained in certain cyclotomic extensions, Erez has described integral normal bases for A(L)/Q(p) that are self-dual with respect to the trace form. Assuming K/Q(p) to be unramified we generate odd abelian weakly ramified extensions of K using Lubin-Tate formal groups. We then use Dwork's exponential power series to explicitly construct self-dual integral normal bases for the square-root of the inverse different in these extensions. (C) 2009 Elsevier Inc. All rights reserved.

2009

Properties of Ir4 clusters in the gas phase and on oxide substrates

Vladan Stevanovic

Results of a theoretical study on the properties of Ir4 clusters in the gas–phase and on oxide surfaces are presented. The work is based on density functional theory (DFT) within the generalized gradient approximation (GGA) and ultrasoft pseudopotentials. Properties of a small particle such as Ir4 cluster are entirely determined by its geometry. The already known result that the most stable form of Ir4 in the gas–phase is the square structure which is significantly more stable than the butterfly and tetrahedron is confirmed. This result is in contradiction with experiments which indicate that the oxide supported Ir4 adopts a tetrahedral configuration. It is shown in this thesis that the chemical environment has a strong influence on the relative stability of Ir4 clusters. On MgO(100) surface, the square isomer remains the most stable Ir4 structure, well separated in energy from the other two. Moreover, the tetrahedron is heavily distorted by the interaction with the surface oxygen. Presence of point defects (neutral and charged O vacancies) affects the energy ordering making tetrahedron and square very close in energy, but the structural distortion of the tetrahedron is even bigger and the predicted data do not correspond to experiments. On TiO2(110) the tetrahedron and square structures become degenerate and the butterfly becomes the least stable isomer. Moreover, structural distortions are very small, in agreement with experimental data. It is shown that the TiO2 surface influences the relative stability of the three isomers through a particularly strong electrostatic field. Interactions of Ir4 with H, C and O atoms as well as with CO molecules have been studied. Adsorption of a single C atom strongly influences the relative stability of the three isomers. Upon C adsorption, the butterfly becomes the most stable gas–phase isomer while on both surfaces the tetrahedron is the most probable structure. Adsorption of a single H or O atom does not produce the same effect. The interaction with CO molecules is also important given the experimental procedure used for producing supported Ir4 clusters. It is shown that on MgO(100), CO dissociation is as probable as the competing process CO desorption justifying the presence of carbon adatoms on Ir4 clusters which brings theoretical predictions in better agreement with experimental data.

EPFL2009

Construction Of Self-Dual Integral Normal Bases In Abelian Extensions Of Finite And Local Fields

Erik Jarl Pickett

Let

F/E

be a finite Galois extension of fields with abelian Galois group

\Gamma

. A self-dual normal basis for

F/E

is a normal basis with the additional property that

Tr_{F/E}(g(x),h(x))=\delta_{g,h}

for

g,h\in\Gamma

. Bayer-Fluckiger and Lenstra have shown that when

char(E)\neq 2

, then

F

admits a self-dual normal basis if and only if

[F:E]

is odd. If

F/E

is an extension of finite fields and

char(E)=2

, then

F

admits a self-dual normal basis if and only if the exponent of

\Gamma

is not divisible by

4

. In this paper we construct self-dual normal basis generators for finite extensions of finite fields whenever they exist. Now let

K

be a finite extension of

\Q_p

, let

L/K

be a finite abelian Galois extension of odd degree and let

\bo_L

be the valuation ring of

L

. We define

A_{L/K}

to be the unique fractional

\bo_L

-ideal with square equal to the inverse different of

L/K

. It is known that a self-dual integral normal basis exists for

A_{L/K}

if and only if

L/K

is weakly ramified. Assuming

p\neq 2

, we construct such bases whenever they exist.

2010