Matrix Calculus and Euclidean Vector Space

2014 ◽  
pp. 239-248
Author(s):  
Jérôme Pagès
Keyword(s):  
Vector Space ◽  
Euclidean Vector Space ◽  
Matrix Calculus

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.

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.

Groups Preserving a Class of Bilinear Functions

1985 ◽  
Vol 28 (3) ◽  
pp. 267-271
Author(s):  
W. H. Greub ◽  
J. Malzan ◽  
J. R. Vanstone
Keyword(s):  
Vector Space ◽  
Bilinear Function ◽  
Finite Dimensional ◽  
Euclidean Vector Space ◽  
Bilinear Functions

AbstractGiven a finite dimensional Euclidean vector space V, ( , ) and an involution τ of V, one can form the bilinear function ( , )τ defined by (x, y)τ = (τ(x), y), x,y ∊ V.Let O(τ) = {ϕ ∊ GL(V)|(ϕx, ϕy)τ = (x, y)τ}.If t is self-adjoint the structure of O(t) is well known. The purpose of this paper is to detemine the structure of O(t) in the general case. This structure is also determined in the complex and quaternionic case, as well as the case when the condition on t is replaced by τ2 = ∊ι, ∊ ∈ ℝ.

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