scholarly journals A Bloch-sphere-type model for two qubits in the geometric algebra of a 6D Euclidean vector space

2004 ◽  
Author(s):  
Timothy F. Havel ◽  
Chris J. L. Doran
Keyword(s):  
Vector Space ◽  
Geometric Algebra ◽  
Type Model ◽  
Bloch Sphere ◽  
Euclidean Vector Space

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 Product rule for vector fractional derivatives

2012 ◽  
Vol 15 (3) ◽  
Author(s):  
Diogo Bolster ◽  
Mark Meerschaert ◽  
Alla Sikorskii
Keyword(s):  
Vector Space ◽  
Fractional Derivatives ◽  
Fourier Transforms ◽  
Product Rule ◽  
Finite Dimensional ◽  
Euclidean Vector Space ◽  
Derivatives Of

AbstractThis paper establishes a product rule for fractional derivatives of a realvalued function defined on a finite dimensional Euclidean vector space. The proof uses Fourier transforms.

scholarly journals Geometric Algebra Framework Applied to Circuits with Non-sinusoidal Voltages and Currents

2021 ◽  
Author(s):  
Jan Cieśliński ◽  
Cezary Walczyk
Keyword(s):  
Vector Space ◽  
Clifford Algebra ◽  
Geometric Algebra ◽  
First Principles ◽  
Theoretical Physics ◽  
Electrical Circuits ◽  
Algebra Approach ◽  
Physical Components

We apply a well known technique of theoretical physics, known as Geometric Algebra or Clifford algebra, to linear electrical circuits with non-sinusoidal voltages and currents. We rederive from the first principles the Geometric Algebra approach to the apparent power decomposition. The important new point consists in a choice of a natural convenient basis in the Clifford vector space which simplifies considerably the presentation. Thus we are able to derive a number of general results which are missing in the former papers. In particular, a natural correspondence with the Current Physical Components approach is shown.

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.

Planimetry of Economic States

2015 ◽  
Vol 3 (2) ◽  
pp. 16-24
Author(s):  
S. Melnyk ◽  
I. Tuluzov ◽  
A. Melnyk
Keyword(s):  
Vector Space ◽  
Economic System ◽  
Scalar Product ◽  
Physical Method ◽  
Physical Methods ◽  
New Information ◽  
Euclidean Vector Space

The new information physical method of constructing the space of economic states is proposed. Unlike the existing theories of consumption, its properties are completely determined axiomatically by the operation of measurement and do not require phenomenological assumptions. The authors consider a transaction of exchange of valuables between two proprietors as such operation. The result of measurement is a dimensional number equal to the proportion of exchange. The constructed space appears to be Euclidean vector space with ordinary operators of composition of vectors, their scalar product, etc. The task of determining the parameters of equilibrium of a complex economic system can be formulated as a task of statics in the constructed space and can be solved by one of the physical methods.

scholarly journals SPECIAL VINBERG CONES

2021 ◽  
Author(s):  
D. V. ALEKSEEVSKY ◽  
V. CORTÉS
Keyword(s):  
Vector Space ◽  
Convex Cone ◽  
Euclidean Jordan Algebra ◽  
Jordan Algebras ◽  
Euclidean Case ◽  
Euclidean Metric ◽  
Euclidean Vector Space ◽  
Lorentz Cone ◽  
Homogeneous Convex ◽  
Vinberg Algebras

AbstractThe paper is devoted to the generalization of the Vinberg theory of homogeneous convex cones. Such a cone is described as the set of “positive definite matrices” in the Vinberg commutative algebra ℋn of Hermitian T-matrices. These algebras are a generalization of Euclidean Jordan algebras and consist of n × n matrices A = (aij), where aii ∈ ℝ, the entry aij for i < j belongs to some Euclidean vector space (Vij ; 𝔤) and $$ {a}_{ji}={a}_{ij}^{\ast }=\mathfrak{g}\left({a}_{ij},\cdot \right)\in {V}_{ij}^{\ast } $$ a ji = a ij ∗ = g a ij ⋅ ∈ V ij ∗ belongs to the dual space $$ {V}_{ij}^{\ast }. $$ V ij ∗ . The multiplication of T-Hermitian matrices is defined by a system of “isometric” bilinear maps Vij × Vjk → Vij ; i < j < k, such that |aij ⋅ ajk| = |aij| ⋅ |aik|, alm ∈ Vlm. For n = 2, the Hermitian T-algebra ℋn= ℋ2 (V) is determined by a Euclidean vector space V and is isomorphic to a Euclidean Jordan algebra called the spin factor algebra and the associated homogeneous convex cone is the Lorentz cone of timelike future directed vectors in the Minkowski vector space ℝ1,1⊕ V . A special Vinberg Hermitian T-algebra is a rank 3 matrix algebra ℋ3(V; S) associated to a Clifford Cl(V )-module S together with an “admissible” Euclidean metric 𝔤S.We generalize the construction of rank 2 Vinberg algebras ℋ2(V ) and special Vinberg algebras ℋ3(V; S) to the pseudo-Euclidean case, when V is a pseudo-Euclidean vector space and S = S0 ⊕ S1 is a ℤ2-graded Clifford Cl(V )-module with an admissible pseudo-Euclidean metric. The associated cone 𝒱 is a homogeneous, but not convex cone in ℋm; m = 2; 3. We calculate the characteristic function of Koszul-Vinberg for this cone and write down the associated cubic polynomial. We extend Baez’ quantum-mechanical interpretation of the Vinberg cone 𝒱2 ⊂ ℋ2(V ) to the special rank 3 case.

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