scholarly journals Interpretation of Geometry on Manifolds as a Geometry in a Space with Projective Metric

2019 ◽  
Vol 65 (1) ◽  
pp. 1-10
Author(s):  
A Artikbaev ◽  
S S Saitova
Keyword(s):  
Vector Space ◽  
Three Dimensional ◽  
Projective Metric

In this paper, we give essential concepts of geometry of three-dimensional spaces in vector formulation in an affine-vector space An.

scholarly journals Identities Generalizing the Theorems of Pappus and Desargues

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1382
Author(s):  
Roger D. Maddux
Keyword(s):  
Projective Plane ◽  
Vector Space ◽  
Invariant Theory ◽  
Lattice Theory ◽  
Three Dimensional ◽  
Dimensional Vector ◽  
Dimensional Vector Space

The Theorems of Pappus and Desargues (for the projective plane over a field) are generalized here by two identities involving determinants and cross products. These identities are proved to hold in the three-dimensional vector space over a field. They are closely related to the Arguesian identity in lattice theory and to Cayley-Grassmann identities in invariant theory.

Constitutive relations in classical optics in terms of geometric algebra

2015 ◽  
Vol 55 (2) ◽  
Author(s):  
Adolfas Dargys
Keyword(s):  
Wave Propagation ◽  
Electromagnetic Wave ◽  
Vector Space ◽  
Closed System ◽  
Geometric Algebra ◽  
Electromagnetic Wave Propagation ◽  
Constitutive Relations ◽  
Three Dimensional ◽  
Basis Vectors

To have a closed system, the Maxwell electromagnetic equations should be supplemented by constitutive relations which describe medium properties and connect primary fields (E, B) with secondary ones (D, H). J.W. Gibbs and O. Heaviside introduced the basis vectors {i, j, k} to represent the fields and constitutive relations in the three-dimensional vectorial space. In this paper the constitutive relations are presented in a form of Cl3,0 algebra which describes the vector space by three basis vectors {σ1, σ2, σ3} that satisfy Pauli commutation relations. It is shown that the classification of electromagnetic wave propagation phenomena with the help of constitutive relations in this case comes from the structure of Cl3,0 itself. Concrete expressions for classical constitutive relations are presented including electromagnetic wave propagation in a moving dielectric.

scholarly journals On Rotor calculus, I

1966 ◽  
Vol 6 (4) ◽  
pp. 402-423 ◽  
Author(s):  
H. A. Buchdahl
Keyword(s):  
Vector Space ◽  
Lorentz Transformation ◽  
Real World ◽  
Three Dimensional ◽  
Linear Vector ◽  
Complex Rotation ◽  
Starting Point ◽  
Complex Linear ◽  
Definition Of ◽  
Linear Vector Space

SummaryIt is known that to every proper homogeneous Lorentz transformation there corresponds a unique proper complex rotation in a three-dimensional complex linear vector space, the elements of which are here called “rotors”. Equivalently one has a one-one correspondence between rotors and self- dual bi-vectors in space-time (w-space). Rotor calculus fully exploits this correspondence, just as spinor calculus exploits the correspondence between real world vectors and hermitian spinors; and its formal starting point is the definition of certain covariant connecting quantities τAkl which transform as vectors under transformations in rotor space (r-space) and as tensors of valence 2 under transformations in w-space.

VERSAL DEFORMATIONS OF THREE-DIMENSIONAL LIE ALGEBRAS AS L∞ ALGEBRAS

2005 ◽  
Vol 07 (02) ◽  
pp. 145-165 ◽  
Author(s):  
ALICE FIALOWSKI ◽  
MICHAEL PENKAVA
Keyword(s):  
Vector Space ◽  
Lie Algebras ◽  
Moduli Space ◽  
Deformation Theory ◽  
Three Dimensional ◽  
3 Dimensional ◽  
Exterior Degree ◽  
Versal Deformations

We consider versal deformations of 0|3-dimensional L∞ algebras, also called strongly homotopy Lie algebras, which correspond precisely to ordinary (non-graded) three-dimensional Lie algebras. The classification of such algebras is well-known, although we shall give a derivation of this classification using an approach of treating them as L∞ algebras. Because the symmetric algebra of a three-dimensional odd vector space contains terms only of exterior degree less than or equal to three, the construction of versal deformations can be carried out completely. We give a characterization of the moduli space of Lie algebras using deformation theory as a guide to understanding the picture.

scholarly journals A computational theory of visual surface interpolation

1982 ◽  
Vol 298 (1092) ◽  
pp. 395-427 ◽  
Keyword(s):  
Vector Space ◽  
Visual Information ◽  
Null Space ◽  
Rotational Symmetry ◽  
Three Dimensional ◽  
Quadratic Variation ◽  
Inner Product ◽  
Functional Surface ◽  
Computational Theory ◽  
Consistency Constraint

Computational theories of structure-from-motion and stereo vision only specify the computation of three-dimensional surface information at special points in the image. Yet the visual perception is clearly of complete surfaces. To account for this a computational theory of the interpolation of surfaces from visual information is presented. The problem is constrained by the fact that the surface must agree with the information from stereo or motion correspondence, and not vary radically between these points. Using the image irradiance equation, an explicit form of this surface consistency constraint can be derived. To determine which of two possible surfaces is more consistent with the surface consistency constraint, one must be able to compare the two surfaces. To do this, a functional from the space of possible functions to the real numbers is required. In this way, the surface most consistent with the visual information will be that which minimizes the functional. To ensure that the functional has a unique minimal surface, conditions on the form of the functional are derived. In particular, if the functional is a complete semi-norm that satisfies the parallelogram law, or the space of functions is a semi-Hilbert space and the functional is a semi-inner product, then there is a unique (to within possibly an element of the null space of the functional) surface that is most consistent with the visual information. It can be shown, based on the above conditions plus a condition of rotational symmetry, that there is a vector space of possible functionals that measure surface consistency, this vector space being spanned by the functional of quadratic variation and the functional of square Laplacian. Arguments based on the null spaces of the respective functionals are used to justify the choice of the quadratic variation as the optimal functional. Possible refinements to the theory, concerning the role of discontinuities in depth and the effects of applying the interpolation process to scenes containing more than one object, are discussed.

scholarly journals ON RIGID AND INFINITESIMAL RIGID DISPLACEMENTS IN THREE-DIMENSIONAL ELASTICITY

2003 ◽  
Vol 13 (11) ◽  
pp. 1589-1598 ◽  
Author(s):  
PHILIPPE G. CIARLET ◽  
CRISTINEL MARDARE
Keyword(s):  
Vector Space ◽  
Tangent Space ◽  
Three Dimensional ◽  
Connected Subset ◽  
Rigid Displacement ◽  
Open Set ◽  
Open Connected Subset ◽  
Linearized Elasticity ◽  
Proper Perspective ◽  
Three Dimensional Elasticity

Let Ω be an open connected subset of ℝ3 and let Θ be an immersion from Ω into ℝ3. It is first established that the set formed by all rigid displacements, i.e. that preserve the metric, of the open set Θ(Ω) is a submanifold of dimension 6 and of class [Formula: see text] of the space H1(Ω). It is then shown that the vector space formed by all the infinitesimal rigid displacements of the same set Θ(Ω) is nothing but the tangent space at the origin to this submanifold. In this fashion, the familiar "infinitesimal rigid displacement lemma" of three-dimensional linearized elasticity is put in its proper perspective.

scholarly journals Projective Metric Geometry and Clifford Algebras

2021 ◽  
Vol 76 (4) ◽  
Author(s):  
Hans Havlicek
Keyword(s):  
Quadratic Form ◽  
Projective Space ◽  
Vector Space ◽  
Clifford Algebra ◽  
Clifford Algebras ◽  
Metric Geometry ◽  
Algebraic Description ◽  
Projective Metric ◽  
Point Set ◽  
Distinguished Group

AbstractEach vector space that is endowed with a quadratic form determines its Clifford algebra. This algebra, in turn, contains a distinguished group, known as the Lipschitz group. We show that only a quotient of this group remains meaningful in the context of projective metric geometry. This quotient of the Lipschitz group can be viewed as a point set in the projective space on the Clifford algebra and, under certain restrictions, leads to an algebraic description of so-called kinematic mappings.

Bistability theorem in a cancer model

2018 ◽  
Vol 11 (01) ◽  
pp. 1850004 ◽  
Author(s):  
Jens Christian Larsen
Keyword(s):  
Vector Space ◽  
Cancer Metastasis ◽  
Metastatic Cancer ◽  
Three Dimensional ◽  
Rate Of Change ◽  
Mass Action ◽  
Cancer Growth ◽  
Ode Model ◽  
Euclidean Vector Space ◽  
Appl Math

In this paper, I continue the study of the mathematical models presented in [J. C. Larsen, Models of cancer growth, J. Appl. Math. Comput. 53(1–2) (2015) 613–645] and [J. C. Larsen, The bistability theorem in a model of metastatic cancer, to appear in Appl. Math.]. I shall prove the bistability theorem for the ODE model from [Larsen, 2015]. It is a mass action kinetic system in the variables [Formula: see text] cancer, GF growth factor and GI growth inhibitor. This theorem says that for some values of the parameters, there exist two positive singular points [Formula: see text], [Formula: see text] of the vector field. Here [Formula: see text] and [Formula: see text] is stable and [Formula: see text] is unstable, see Sec. 2. There is also a discrete model in [Larsen, 2015], it is a linear map ([Formula: see text]) on three-dimensional Euclidean vector space with variables [Formula: see text] where these variables have the same meaning as in the ODE model above. In [Larsen, 2015], I showed that one can sometimes find affine vector fields on three-dimensional Euclidean vector space whose time one map is [Formula: see text]. I shall also show this in the present paper in a more general setting than in [Larsen, 2015]. This enables me to find an expression for the rate of change of cancer growth on the coordinate hyperplane [Formula: see text] in Euclidean vector space. I also present an ODE model of cancer metastasis with variables [Formula: see text] where [Formula: see text] is primary cancer and [Formula: see text] is metastatic cancer and GF, GI are growth factors and growth inhibitors, respectively.

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