scholarly journals Orthogonal Tits Quadrangles

10.53733/105 ◽  
2021 ◽  
Vol 52 ◽  
pp. 427-452
Author(s):  
Bernhard Mühlherr ◽  
Richard Weiss
Keyword(s):  
Quadratic Space ◽  
Witt Index ◽  
Standard Construction

We show that every 4-plump razor-sharp normal Tits quadrangle X is uniquely determined by a non-degenerate quadratic space whose Witt index m is at least 2. If this Witt index is finite, then X is the Tits quadrangle arising from the corresponding building of type B_m or D_m by a standard construction.

Orthogonal Buildings

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Vector Space ◽  
Quadratic Space ◽  
Positive Dimension ◽  
Witt Index ◽  
Coxeter Diagram ◽  
Vertex Set ◽  
Dimension 2

This chapter presents a few results about certain forms of orthogonal buildings. It begins with notations stating that V is a K-vector space of positive dimension, (K, V, q) is a quadratic space of positive dimension, (K, V, q) is a regular quadratic space of positive Witt index, S is the vertex set of the Coxeter diagram, (K, V, q) is a hyperbolic quadratic space of dimension 2n for some n greater than or equal to 3, S is the vertex set of the Coxeter diagram for some n greater than or equal to 3, and Dn.l,script small l is the Tits index of absolute type Dn for n greater than or equal to 3. The chapter also considers propositions dealing with regular quadratic spaces and hyperbolic quadratic spaces.

Existence

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Structure Theorem ◽  
Quadratic Extension ◽  
Quadratic Space ◽  
Division Algebras ◽  
Separable Extension ◽  
Valued Fields ◽  
Quaternion Division Algebra

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.

Totally Wild Quadratic Forms of Type E7

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Quadratic Space ◽  
Trace Map ◽  
Quaternion Division Algebra

This chapter assumes that (K, L, q) is a totally wild quadratic space of type E₇. The goal is to prove the proposition that takes into account Λ‎ of type E₇, D as the quaternion division algebra over K whose image in Br(K) is the Clifford invariant of q, and the trace and trace map. The chapter also considers two other propositions: the first states that if the trace map is not equal to zero, then the Moufang residues R₀ and R₁ are not indifferent; the second states that if the trace map is equal to zero, then the Moufang residues R₀ and R₁ are both indifferent.

Linked Tori, II

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Integral Domain ◽  
Quadratic Space ◽  
Small Observation ◽  
Quaternion Division Algebra

This chapter proves several more results about weak isomorphisms between Moufang sets arising from quadratic forms and involutory sets. It first fixes a non-trivial anisotropic quadratic space Λ‎ = (K, L, q) before considering two proper anisotropic pseudo-quadratic spaces. It then describes a quaternion division algebra and its standard involution, a second quaternion division algebra and its standard involution, and an involutory set with a quaternion division algebra and its standard involution. It concludes with one more small observation regarding a pointed anisotropic quadratic space and shows that there is a unique multiplication on L that turns L into an integral domain with a multiplicative identity.

Alignment of three sequences in quadratic space

1993 ◽  
Vol 1 (2) ◽  
pp. 7-11 ◽  
Author(s):  
Xiaoqiu Huang
Keyword(s):  
Quadratic Space

scholarly journals Chirality in Geometric Algebra

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1521
Author(s):  
Michel Petitjean
Keyword(s):  
Geometric Algebra ◽  
Orthogonal Group ◽  
Matrix Representation ◽  
Quadratic Space ◽  
Complex Numbers ◽  
Chiral Algebra

We define chirality in the context of chiral algebra. We show that it coincides with the more general chirality definition that appears in the literature, which does not require the existence of a quadratic space. Neither matrix representation of the orthogonal group nor complex numbers are used.

scholarly journals $$(O(V \oplus F), O(V))$$ is a Gelfand pair for any quadratic space V over a local field F

2008 ◽  
Vol 261 (2) ◽  
pp. 239-244 ◽  
Author(s):  
Avraham Aizenbud ◽  
Dmitry Gourevitch ◽  
Eitan Sayag
Keyword(s):  
Local Field ◽  
Quadratic Space ◽  
Gelfand Pair

The Integral Extension of Isometries of Quadratic Forms Over Local Fields

1966 ◽  
Vol 18 ◽  
pp. 920-942 ◽  
Author(s):  
Allan Trojan
Keyword(s):  
Vector Space ◽  
Quadratic Forms ◽  
Sufficient Conditions ◽  
Local Fields ◽  
The Other ◽  
Quadratic Space ◽  
Integral Extension ◽  
Ring Of Integers ◽  
Extension Of Isometries ◽  
Necessary And Sufficient

Let F be a local field with ring of integers 0 and prime ideal π0. If V is a vector space over F, a lattice L in F is defined as an 0-module in the vector space V with the property that the elements of L have bounded denominators in the basis for V. If V is, in addition, a quadratic space, the lattice L then has a quadratic structure superimposed on it. Two lattices on V are then said to be isometric if there is an isometry of V that maps one onto the other.In this paper, we consider the following problem: given two elements, v and w, of the lattice L over the regular quadratic space V, find necessary and sufficient conditions for the existence of an isometry on L that maps v onto w.

scholarly journals Technical requirements for translucent structures of schools and kindergartens

2019 ◽  
Vol 91 ◽  
pp. 05019
Author(s):  
Aleksandr Konstantinov ◽  
Elena Romanerikova ◽  
Margarita Borisova
Keyword(s):  
Operating Conditions ◽  
Normal Operation ◽  
Future Research ◽  
Natural Lighting ◽  
Technical Requirements ◽  
Standard Construction ◽  
Temperature And Humidity ◽  
Typical Composition ◽  
Room Lighting ◽  
Construction Practice

The article presents some features of the translucent structures design of schools and kindergartens. The analysis of technical requirements for translucent structures of schools and kindergartens was represented. Based on a review of the typical composition of the premises of schools and kindergartens, it was found that different technical requirements should be established for translucent structures installed in rooms of various functional purposes and operating conditions. It was determined that the existing construction practice doesn’t take into account the differentiation of requirements for translucent structures of various premises of schools and kindergartens, and their design, as a rule, is taken to be the same for the entire construction objects. In future, it leads to disruption of the normal operation of the facility (especially in terms of ensuring normal temperature and humidity conditions and natural lighting of the premises). The features of the replacement of translucent structures in reconstructed schools and kindergartens were considered. It has been revealed that the use of modern types of translucent structures of standard construction (first of all, PVC window units) in the reconstruction leads to a significant decrease in the indicators of natural room lighting. Ways to solve this problem were given. Moreover, perspective directions for future research of the issue under consideration were considered.

scholarly journals Representations of quadratic forms

1978 ◽  
Vol 69 ◽  
pp. 117-120
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Direct Summand ◽  
Quadratic Forms ◽  
Quadratic Space ◽  
List Type

We have shown in [1]Theorem A. Let L be a lattice in a regular quadratic space U over Q; then L has a submodule M satisfying the following conditions 1),2): 1)dM ≠ 0, rank M = rank L — 1, and M is a direct summand of L as a module.2)Let L′ be a lattice in some regular quadratic space U′ over Q satisfying dL′ = dL, rank L′ — rank L, tp(L′) ≥ tp(L) for any prime p. If there is an isometry α from M into L′ such that α(M) is a direct summand of L′ as a module, then L′ is isometric to L.

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