Related papers: Euclidean Quadratic Forms and ADC Forms I

We determine the shapes of pure cubic fields and show that they fall into two families based on whether the field is wildly or tamely ramified (of Type I or Type II in the sense of Dedekind). We show that the shapes of Type I fields are…

A strong consequence of quadratic forms becoming hyperbolic over the function field of a form is established. This result is invoked to obtain a new characterisation of hyperbolicity over function fields, and to recover a number of…

Austere submanifolds in Euclidean space were introduced by Harvey and Lawson in connection with their study of calibrated geometries. The algebraic possibilities for second fundamental forms of 4-dimensional austere submanifolds were…

The model of kappa-deformed space is an interesting example of a noncommutative space, since it allows a deformed symmetry. In this paper we present new results concerning different sets of derivatives on the coordinate algebra of…

We observe structure in the sequences of quotients and remainders of the Euclidean algorithm with two families of inputs. Analyzing the remainders, we obtain new algorithms for computing modular inverses and representating prime numbers by…

We determine the border rank of each power of any quadratic form in three variables. Since the problem for rank $1$ and rank $2$ quadratic forms can be reduced to determining the rank of powers of binary forms, we primarily focus on…

There are (at least) two different approaches to define equivariant analogue of the Euler charateristic for a space with a finite group action. The first one defines it as an element of the Burnside ring of the group. The second approach…

Quadratic algebras are generalizations of Lie algebras which include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical…

We extend to characteristic two recent results about isotropy of quadratic forms over function fields. In particular, we provide a characterization of function fields not only of quadratic forms but also more generally of polynomials in…

We verify a conjecture of Emil Artin, for the case of a Cubic and Quadratic form over any $p$-adic field, provided the cardinality of the residue class field exceeds 293. That is any Cubic and Quadratic form with at least 14 variables has a…

A (positive definite and integral) quadratic form $f$ is said to be $\textit{universal}$ if it represents all positive integers, and is said to be $\textit{primitively universal}$ if it represents all positive integers primitively. We also…

For a discrete valuation ring $R$ with quotient field $K$ and residue field $F$ both of characteristic not 2, we study low-dimensional quadratic forms with Witt class in the $n$-th power of the fundamental ideal of $F$ resp. $K$ and point…

For a given triangle $\triangle ABC$, we define two sequences of points on line $BC$ and provide their generalizations to real functions such that centers of circumscribed circles around $A$ and adjacent points in subsequences generate a…

The Hasse principle and weak approximation is established for equations of the shape P(t)=N(x_1,x_2,x_3,x_4), where P is an irreducible quadratic polynomial in one variable and N is a norm form associated to a quartic extension of the…

Four classes of three dimensional quadratic algebras of the type $\lsb Q_0 , Q_\pm \rsb$ $=$ $\pm Q_\pm$, $\lsb Q_+ , Q_- \rsb$ $=$ $aQ_0^2 + bQ_0 + c$, where $(a,b,c)$ are constants or central elements of the algebra, are constructed using…

We consider generalized quadratic forms over real quadratic number fields and prove, under a natural positive-definiteness condition, that a generalized quadratic form can only be universal if it contains a quadratic subform that is…

Finite frames can be viewed as mass points distributed in $N$-dimensional Euclidean space. As such they form a subclass of a larger and rich class of probability measures that we call probabilistic frames. We derive the basic properties of…

A general form of the octupole collective Hamiltonian is introduced and analyzed based on fundamental tensors in the seven-dimensional tensor space. Possible definitions of intrinsic frames of reference possessing cubic symmetry for the…

We find general solutions of some field equations (systems of equations) in pseudo-Euclidian spaces (so-called primitive field equations). These equations are used in the study of the Dirac equation and Yang-Mills equations. These equations…

Positive and negative quadratic forms are well known and widely used. They are multivariate homogeneous polynomials of degree two taking positive or negative values respectively for any values of their arguments not all zero. In the present…