The paper is inspired by a spectral decomposition and fractal eigenvectors for a class of piecewise linear maps due to Tasaki et al. [1994] and by an ad hoc explicit derivation of the Heisenberg uncertainty relation based on a Peano–Hilbert planar curve, due to El Nashie [1994]. It is also inspired by an elegant generalization by Zhang [2008] of the exact solution by Onsager [1944] to the problem of description of the Ising lattices [Ising, 1925]. This generalization involves, in particular, opening the knots by a rotation in a higher dimensional space and studying important commutators in the corresponding algebra. The investigations of Onsager and Zhang, involving quaternion matrices of order being a power of two, can be reformulated with the use of the "quaternionic" sequence of Jordan algebras implied by the fundamental paper of Jordan et al. [1934]. It is closely related to Heisenberg's approach to quantum theories, as summarized by him in his essay dedicated to Bohr on the occasion of Bohr's seventieth birthday (1955). We show that the Jordan structures are closely related to some types of fractals, in particular, fractals of the algebraic structure. Our study includes fractal renormalization and the renormalized Dirac operator, meromorphic Schauder basis and hyperfunctions on fractal boundaries, and a final discussion.

It is well known that starting with real structure, the Cayley–Dickson process gives complex, quaternionic, and octonionic (Cayley) structures related to the Adolf Hurwitz composition formula for dimensions p = 2, 4 and 8, respectively, but the procedure fails for p = 16 in the sense that the composition formula involves no more a triple of quadratic forms of the same dimension; the other two dimensions are n = 27. Instead, Ławrynowicz and Suzuki (2001) have considered graded fractal bundles of the flower type related to complex and Pauli structures and, in relation to the iteraton process p → p + 2 → p + 4 → ⋯, they have constructed 24-dimensional "bipetals" for p = 9 and 27-dimensional "bisepals" for p = 13. The objects constructed appear to have an interesting property of periodicity related to the gradating function on the fractal diagonal interpreted as the "pistil" and a family of pairs of segments parallel to the diagonal and equidistant from it, interpreted as the "stamens." The present paper aims at an effective, explicit determination of the periods and expressing them in terms of complex and quaternionic structures, thus showing the quaternionic background of that periodicity. The proof of the Periodicity Theorem is given in the case where the index of the generator of the algebra in question exceeds the order of the initial algebra. In contrast to earlier results, the fractal bundle flower structure, in particular sepals, bisepals, perianths, and calyces are not introduced ab initio; they are quoted a posteriori, when they are fully motivated. The same concerns canonical two-layer pairing of prianth sepals and the relationship of fractal bundles of algebraic structure with the Hurwitz problem.

No abstract

No abstract

About one-tenth to one-third of patients with severe aortic stenosis (AS) do not develop left ventricular hypertrophy (LVH). Intriguingly, the absence of LVH despite severe AS is associated with lower prevalence of heart failure (HF), which challenges the classical notion of LVH as a beneficial compensatory response. Notably, the few studies that have attempted to characterize AS subjects with inadequately low left ventricular (LV) mass relative to LV afterload (i-lowLVM) described better prognosis and enhanced LV performance in AS associated with i-lowLVM, but those reports were limited to severe AS. Our aim was to compare myocardial function between moderate and severe AS with i-lowLVM. We retrospectively analyzed in-hospital records of 225 clinically stable nondiabetic patients with isolated moderate or severe degenerative AS in sinus rhythm, free of coexistent diseases. Subjects with i-lowLVM were compared to those with appropriate or excessive LVM (a/e-LVM), defined on the basis of the ratio of a measured LVM to the LVM predicted from an individual hemodynamic load. Patients with i-lowLVM and a/e-LVM did not differ in aortic valve area, LV end-diastolic diameter (LVd, a measure of LV preload), and circumferential end-systolic LV wall stress (cESS), an estimate of LV afterload. Compared to a/e-LVM, patients with i-lowLVM had increased LV ejection fraction (EF) and especially higher LV midwall fractional shortening (a better index of LV myocardial function than EF in concentric LV geometry) (p < 0.001–0.01), in both moderate and severe AS. LVd and cESS were similar in the four subgroups of the study subjects, i.e., moderate AS with i-lowLVM, moderate AS with a/e-LVM, severe AS with i-lowLVM, and severe AS with a/e-LVM (p > 0.6). Among patients with i-lowLVM, LVM did not differ significantly between moderate and severe AS (p > 0.4), while in those with a/e-LVM, LVM was increased in severe versus moderate AS (p < 0.001). In conclusion, the association of the low-LVM phenotype with better myocardial contractility may already develop in moderate AS. Additionally, cESS appears to be a controlled variable, which is kept constant over AS progression irrespective of LVM category, but even when controlled (by increasing LVM), is not able to prevent deterioration of LV function. Whether improved myocardial performance contributes to favorable prognosis and the preventive effect against HF in AS without LVH, remains to be studied.