Related papers: On ternary positive-definite quadratic forms with …

We present two new canonical forms for real congruence of a real square matrix $A$. The first one is a direct sum of canonical matrices of four different types and is obtained from the canonical form under $^*$congruence of complex matrices…

We call a positive definite Hermitian lattice regular if it represents all integers which can be represented locally by the lattice. We investigate binary regular Hermitian lattices over imaginary quadratic fields $\mathbb{Q}(\sqrt{-m})$…

A positive definite and integral quadratic form $f$ is called irrecoverable if there is a quadratic form $F$ such that it represents all proper subforms of $f$, whereas it does not represent $f$ itself. In this case, $F$ is called an…

A polynomial transformation of the real plane $\Bbb R^2$ is a mapping $\Bbb R^2\to\Bbb R^2$ given by two polynomials of two variables. Such a transformation is called cubic if the degrees of its polynomials are not greater than three. It…

If $D$ is the definite quaternion algebra over $\qu$ of discriminant $p$, we compute, for any prime $p>3$, the number of infinite dimensional cusp forms on $D^*$ which are trivial at infinity, tamely ramified at $p$, and have given…

Motivated by a problem in local differential geometry of Cauchy--Riemann (CR) structures of hypersurface type, we find a canonical form for pairs consisting of a nondegenerate Hermitian form and a self-adjoint antilinear operator, or,…

Genus Theory is a classical feature of integral binary quadratic forms. Using the author's generalization of the well-known correspondence between quadratic form classes and ideal classes of quadratic algebras, we extend it to the case when…

Orthogonal spaces are vector spaces together with a quadratic form whose associated bilinear form is non-degenerate. Over fields of characteristic two, there are many quadratic forms associated to a given bilinear form and quadratic…

The roots of -1 in the set of biquaternions (quaternions with complex components, or complex numbers with quaternion real and imaginary parts) are studied and it is shown that there is an infinite number of non-trivial complexified…

We present a new and simple proof of a theorem due to Kaplansky which unifies theorems of Kolchin and Levitzki on triangularizability of semigroups of matrices. We also give two different extensions of the theorem. As a consequence, we…

For any given positive definite binary quadratic form $Q$ with integer coefficients, we establish two results on Diophantine approximation with integers represented by $Q$. Firstly, we show that for every irrational number $\alpha$, there…

Let $T:E\rightarrow F$ be a non-necessarily continuous triple homomorphism from a (complex) JB$^*$-triple (respectively, a (real) J$^*$B-triple) to a normed Jordan triple. The following statements hold: (1) $T$ has closed range whenever $T$…

For quadratic forms over fields of characteristic different from two, there is a so-called Vishik criterion, giving a purely algebraic characterization of when two quadratic forms are motivically equivalent. In analogy to that, we define…

For a differential form on a manifold, having constant components in suitable local coordinates trivially implies being parallel relative to a torsion-free connection, and the converse implication is known to be true for $p$-forms in…

We study quantum field theory in six dimensions with two of them compactified on a square. A simple boundary condition is the identification of two pairs of adjacent sides of the square such that the values of a field at two identified…

This paper gives explicit formulas for the formal total mass Dirichlet series for integer-valued ternary quadratic lattices of varying determinant and fixed signature over number fields F where p = 2 splits completely. We prove this by…

We prove a local-global principle for primitive representations of binary quadratic forms by quaternary quadratic forms. Our method is a variant of Linnik's ergodic method showing density for certain homogenous toral sets. The central…

We present a survey of recent results, scattered in a series of papers that appeared during past five years, whose common denominator is the use of cubic relations in various algebraic structures. Cubic (or ternary) relations can represent…

In this article we establish two new results on quantitative Diophantine approximation for one-parameter families of diagonal ternary indefinite forms. In the first result, we consider quadratic forms taking values at prime points. In the…

We prove a local-global principle for representations of binary by quaternary quadratic forms. One of the main ingredients is a recent measure rigidity result of Einsiedler and Lindenstrauss for diagonalizable actions on quotients of…