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A rigorous formulation of Vessiot's vector field approach to the analysis of general systems of partial differential equations is provided. It is shown that this approach is equivalent to the formal theory of differential equations and that…
One aim of this paper is to develop some aspects of the theory of monoidal derivators. The passages from categories and model categories to derivators both respect monoidal objects and hence give rise to natural examples. We also introduce…
Data are often represented as graphs. Many common tasks in data science are based on distances between entities. While some data science methodologies natively take graphs as their input, there are many more that take their input in…
The transformation of the partial fractional derivatives under spatial rotation in $R^2$ are derived for the Riemann-Liouville and Caputo definitions. These transformation properties link the observation of physical quantities, expressed…
A covariant scalar-tensor-vector gravity theory is developed which allows the gravitational constant $G$, a vector field coupling $\omega$ and the vector field mass $\mu$ to vary with space and time. The equations of motion for a test…
We use the classical Fourier analysis to introduce analytic families of weighted differential operators on the unit sphere. These operators are polynomial functions of the usual Beltrami-Laplace operator. New inversion formulas are obtained…
In this paper, a function on any pair of graphs is defined whose properties are similar to the properties of dot product in vector space. This function enables us to define graph orthogonality and, also, a new metric on isomorphism classes…
We seek the inverse formulas for the cosmological unifying relation between gluons and conformally coupled scalars. We demonstrate that the weight-shifting operators derived from the conformal symmetry at the dS late-time boundary can serve…
A simple theory of the covariant derivatives, deformed derivatives and relative covariant derivatives of multivector and multiform fields is presented using algebraic and analytical tools developed in previous papers.
The paper is devoted to differential geometric invariants determining a Frenet curve in up to a direct similarity These invariants can be presented by the Euclidean curvatures in terms of an arc lengths of the spherical indicatrices. Then,…
We discuss different generalizations of the classical notion of the index of a singular point of a vector field to the case of vector fields or 1-forms on singular varieties, describe relations between them and formulae for their…
It is shown that a separated sequence of points in the unit disc of the complex plane is in fact uniformly separated, if there exists a certain intermediate sequence whose separated subsequences are uniformly separated. This property is…
In this article, we demonstrate the common fixed point theorems for three transformations on vector S-metric space by utilizing weakly compatible and point of coincidence. Moreover, some of our results generalize the existing results in the…
We set out an elementary approach to derive Visible Point Identities summed on lattice points of inverted triangle (2D), pyramid (3D), hyperpyramid (4D, 5D and so on) utilizing the greatest common divisor for the nD Visible Point Vectors.…
We present a complete suite of algorithms for finding isotropic vectors of quadratic forms (of any dimension) over an arbitrary global field of characteristic different from 2. This is a new version with numerous changes and improvements.
We explain the orthogonal planes split (OPS) of quaternions based on the arbitrary choice of one or two linearly independent pure unit quaternions $f,g$. Next we systematically generalize the quaternionic Fourier transform (QFT) applied to…
I adapt a recently introduced method for integrating over the unitary group (S. Aubert and C.S. Lam, J.Math.Phys. 44, 6112-6131 (2003)) to the orthogonal group. I derive explicit formulas for a number of one, two and three-vector integrals,…
Conformal fields are a recently discovered class of representations of the algebra of vector fields in $N$ dimensions. Invariant first-order differential operators (exterior derivatives) for conformal fields are constructed.
A set of integral relations for rotational and translational zero modes in the vicinity of the soliton solution are derived from the particle-like properties of the latter and verified for a number of models (solitons in 1+1-dimensions,…
It is sometimes desirable to produce for a nonlinear system of ODEs a new representation of simpler structural form, but it is well known that this goal may imply an increase in the dimension of the system. This is what happens if in this…