This paper continues the exploration of a tachyonic, bosonic string theory in which fermions are not strings, but rather occupy a curved D-brane boundary of open strings, and the open-string tachyon supplies the Higgs sector. The fermionic D-brane occupies the 11 + 1 dimensions of the Clifford algebra Cl(11,1) , while open strings fill out the additional 14 dimensions of a maximal torus of the rank 14 subgroup SU(8)_R ×SU(8)_L of the group G^2(10) = U(16) generated by multivectors of grade 2 (mod 4) in Cl(10) , for a total of 12+14 = 26 dimensions. An important result is that there exist open-string solutions whose D-brane boundary is curved, but those solutions are purely holomorphic (right-moving) or purely antiholomorphic (left-moving) without an associated closed string solution. The only closed-string modes are completely neutral under all internal gauge transformations; for massless modes, these comprise just 4 modes, the spin ± 2 graviton, and two spin 0 modes, the scalar dilaton and the pseudoscalar 2-form.
We establish refined Eneström–Kakeya type bounds for the zeros of right quaternionic polynomials using a scaled-difference approach. The main result provides an optimizable upper bound in terms of weighted coefficient variation, while coefficient reversal gives corresponding lower bounds. The finite-defect formulas extend the classical monotonicity conditions, and angular restrictions give further estimates based on coefficient directions. We also develop orthogonal-frame and projection bounds, with Haar averaging providing the universal factor 16/(3π ) . The numerical examples illustrate the effectiveness of scaling and compare the resulting bounds with the actual zero moduli.
It is shown that a careful study of the simplest family of generalized Clifford algebras (GCAs) associated with the Nth root of unity in d-dimensions leads to the following generalized anti-commutator with N entries { e_i_1, e_i_2, e_ i_3, … , e_ i_N} = e_i_1 e_i_2 e_ i_3… e_ i_N + permutations = N! η _ i_1 i_2 … i_N e , where e is the unit element and all the N! terms of the permutations appear with the same positive sign. The components of the rank-N metric are η _ i_1 i_2 … i_N = 1, iff i_1 = i_2 = i_3 … = i_ N , and 0 otherwise. The range of the indices i_1, i_2, … i_N is 1,2, … , d . We proceed to explore the Nth norm extensions of the quadratic norm and write down a generalized Finsler-like arc-length based on the rank-N metric g_μ _1 μ _2 …μ _N . We continue by constructing the different expressions of the Dirac operators associated with the (Generalized) Clifford Spaces corresponding to these GCA’s. Dirac operators are essential in the study of Spectral Geometry in Noncommutative Geometry after imposing the correspondence between the geodesic distance and the inverse of the Dirac operator (a fermion propagator). These generalized anti-commutators above are special types of an N-ary algebraic structure. We finalize by displaying the relation among GCAs and the algebras underlying the noncommutative fuzzy torus and discuss applications in condensed matter and quantum groups. We conclude with some remarks on N-ary algebras and their applications in Mathematics and Physics.
This paper revisits α -hyperholomorphic functions in Clifford analysis, that is, functions satisfying the equation Du - uα = 0, where D denotes the Dirac operator in ℝ^n and α∈ℂ_0,n . A transmutation operator for α -hyperholomorphic functions has previously been constructed, mapping each monogenic function in a star-like domain to an α -hyperholomorphic one. In the present work, we derive explicit integral kernels for the inverse of this operator.
The aim of this paper is to establish several versions of Paley–Wiener-type theorems for the two-sided quaternion Dunkl transform (QDT) in ℝ^2 . This transform combines the non-commutative structure of quaternions with the reflection symmetry and weighted analysis inherent in Dunkl theory.
