Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2607.11242

4x4 Matrix Representation of Hybrid Numbers, Gram Matrix and Golden Ratio

Abdullah Mağden
View original source ↗Read source document ↗Export metadata ↓

Abstract

This study investigates the Lie algebra structure and geometric properties of hybrid numbers, under a specific non-associative multiplication table. Within the scope of the study, the structure constants of the system were derived through commutator relations, and 4x4 matrix representation was constructed. By defining the Gram matrix, complex and hybrid conjugates, the connection of the eigenvalues of Gram matrices to the golden ratio has been obtained. Theoretical analysis reveal that this hybrid Lie algebra possesses a semi-simple structure and that the characteristic spectrum of its Killing form is directly related to the Golden Ratio ({\phi}). Furthermore, it is proven that the Complex and Hyperbolic conjugation operations are symmetry transformations that leave the Killing form invariant.

Explore related subjects

math.AG

Keep this discovery

BibTeXRIS

Abdullah Mağden. 2026-07-13. 4x4 Matrix Representation of Hybrid Numbers, Gram Matrix and Golden Ratio. https://arxiv.org/abs/2607.11242

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Perverse Euler Characteristics of Hermitian Locally Symmetric Spaces

We prove that finite-volume locally Hermitian symmetric spaces of noncompact type have nonnegative perverse Euler characteristics. To show this, we obtain a nefness result for the logarithmic cotangent bundle of a smooth toroidal compactification. Combining this with a positivity criterion for Euler characteristics of perverse sheaves, we deduce the nonnegativity result. We further prove that the inequality is strict for perverse sheaves with full support. As applications, we get nonnegativity results for perverse Euler characteristics on various moduli spaces.

math.AG↗
arXiv2026

Coupled Pklt Tuples and Varieties of Pklt Type

We introduce asymptotic multiplier ideal sheaves and log canonical thresholds associated with tuples of pseudoeffective divisors on a projective klt pair. We prove that the threshold of a coupled potentially klt tuple is computed by a quasi-monomial valuation. For varieties of potentially klt type, we prove that every big divisor admits a birational Zariski decomposition with semiample positive part. We also prove finite generation of multisection rings of big divisors and give a criterion for a variety of potentially klt type to be a Mori dream space.

math.AG↗
arXiv2026

Graded Betti numbers of general curves of large degree

Let $C$ be a smooth projective complex curve of genus $g$ and gonality $k$, and $L$ be a very ample line bundle on $C$. When $L$ has sufficiently large degree, the vanishing and nonvanishing of the Koszul cohomology groups $K_{p,q}(C,L)$ have been determined previously, but the exact values of the graded Betti numbers $\kappa_{p,q}(C, L)$ remain largely unknown. In this paper, we give explicit closed formulas for all graded Betti numbers $\kappa_{p,q}(C, L)$ when the Brill--Noether locus $W_k^1(C)$ has the expected dimension and $H^1(C, L \otimes \omega_C^{-1})=0$. Consequently, we determine the complete Betti table for a general curve when $\deg L \geq 4g-3$ or when $\deg L \geq 3g-3$ and $L$ is general. We also explicitly compute the Boij--S\"{o}derberg coefficient of the section ring $R(C, L)$ governing asymptotic purity, and show eventual monotonicity of the remaining coefficients: they decrease for hyperelliptic curves and increase under a natural generic reducedness assumption on the relevant Brill--Noether loci.

math.AG↗