Recursive sequences

few decades ago, (1935), there was a general consensus amongst a group of eight** prominent French mathematicians and academicians that there was a serious shortfall in the disparate mathematical accumulata of the past centuries up to the current time. This was based, in part, on the implicit (and unwarranted) assumption, that there must be one basic mathematical discipline, capable of subsuming all the others by being a provider of the `lingua Franca', in which to express and inter-relate all mathematical thoughts, past present and future, with absolute rigour. This `flew in the face' of † Gödel's Incompleteness Theorem, which theorem seemed to say, (when translated into lay terms), that most (or all) mathematical disciplines were incomplete, in that one could not prove absolute consistency, (and therefore completeness), with the particular language of that discipline for that very same branch of mathematics, but had to resort to the language of another branch of mathematics to try to do so. These eight men named themselves with the collective pseudonym "Nicholas Bourbaki", in which personification they published their findings. They selectively absorbed others into this elite group with the only preconditions being acceptance by other members and that the maximum age of any member was 50. However both Gödel and the group of Bourbachistes overlooked, (or were oblivious of), their common implicit assumption, namely, that there could ever be a complete translation of any, and every, area of mathematics in terms of solely one single, (and singular), conceptual framework. This great expectation of, (and concomitant search for), an elusive umbral mathematical panacea, as though it did indeed exist, (or could ever exist), is the implicit assumption mentioned above, since no such remedy can ever be evolved by Man for mankind, due to the inherent limitations, (and very construction), of the human brain. For instance, one cannot use arithmetic to prove all geometrical theorems, nor geometry to prove all arithmetical theorems. This example is typical of most realms of theory, ideology, philosophy and theosophy, and hinges on the incontrovertible reality of there being mutually exclusive areas, specific to one discipline, doctrine or set of concepts, but not to another. For thousands of years mathematicians, (both amateur and professional), sought in vain to `square' the circle. We now know that their time was wasted, since a rigorous proof was constructed in the 20 th century, that proves the paradoxical nature of that set task. Note well that prior to this proof the task was no less impossible, and there must have been a strong suspicion that it was futile and that sooner or later such a proof of impossibility would, (hopefully), come to light. An analogous situation may pertain in the realm of formalising mathematics:-1) as attempted by Bourbaki in a (mostly) successful series of publications over at least three decades, 2) as refuted and discredited by Gödel with his Theory of Incompleteness and 3) as still being attempted currently by transcendental methods in abstract mathematics. If we take Gödel seriously, we should consider whether this (contemporary and simultaneous) attempted unification of all mathematical ideas in one metalanguage is `flying in the face of logic' and no better (in effect) than trying to square the circle. We, who cannot construct such a proof ourselves, are merely awaiting the proof of impossibility (hopefully) yet to come, as did the would-be `circle squarers'.

This work unifies four layers into a single, reproducible framework: the kernel of Generative Collapse Dynamics (GCD), the operational calculus (UMCP), the sensitivity layer (RCFT), and governance/sealing (ULRC). The kernel fixes the invariant quantities for a channeldrift ω, fidelity F , entropy S, curvature C, first return τ R , integrity IC, and log-integrity κtogether with their bounds, limit laws, and typed outcomes. It includes a rotation calculus in which ω is the normalized geodesic distance on compact manifolds (with concrete forms on S 1 and SO(3)). UMCP provides a contract-first pipeline: frozen normalization (global_fixed) with authorized P1/P99 calibration, default regime gates, worst-of aggregation, an explicit weld protocol for governed interventions, and an audit/export path. Each figure/table is traceable to a sample audit row and a Meta line, and releases are byte-verifiable via SHA-256 hashes. RCFT supplies closed-form sensitivities of κ∂κ/∂C =-α/(1 + τ R) ≤ 0 and ∂κ/∂τ R = α C/(1 + τ R) 2 ≥ 0together with second-order behavior and a deterministic, gradient-based safety certificate for small extensions E subject to a budget δ E. Safe extensions compose additively in their budgets. ULRC formalizes the Tiered Invariance Principle (identities preserved, closures governed), specifies the regime-join as worst-of, and defines sealing: a release couples the manuscript, figures, machine-readable audits, and a manifest of artifact hashes. The result is a cohesive, upgrade-safe stack in which conclusions are reproducible from audit rows and contract snapshots, and extensions are provably bounded by design. I. INTRODUCTION AND CONTRIBUTIONS Geodesic distance and manifold preliminaries follow Lee and do Carmo [1, 2]; matrixmanifold optimization as in Absil-Mahony-Sepulchre [3]; directional statistics via Mardia-Jupp and Fisher [4, 5]; Markov generators per Norris [6]; convex tools per Boyd-Vandenberghe [7].

The "Axis of Evil"-the unexplained alignment of low-ℓ CMB multipoles with large-scale cosmic flows-has long defied standard cosmological models. This paper resolves it using the Law of Universal Mathematical Unity (LUMU): 𝓔 = ⋃ᵢ₌₁^∞ Φ(xᵢ). By modeling dark matter as the inertial field of past mass (I(x,t)), incorporating the Fibonacci-Xibof collapse, and introducing LUMU-Λ-a new recursive algebra-we show that the observed anisotropy arises from consequence exceeding symmetry. A contained LUMU simulation unifies CMB quadrupole, octupole, Cold Spot, Dark Flow, I(x,t), E₈ fault, and Fib-Xib collapse into a coherent axis at (l, b) ≈ (253°, 4.8°) with |R| = 0.700, far exceeding random alignment. This confirms a classical, particle-free origin for the Axis of Evil, rooted in recursive cosmic memory.

This paper introduces LUMU-Λ, a new recursive algebra generated by the Law of Universal Mathematical Unity (LUMU), and applies it to the largest exceptional Lie group, E₈. We define a transformation Φ(λᵢ) = gᵢ + εᵢ, embedding LUMU-Λ into E₈ via a direct sum Ξ = E₈ ⊕ LUMU-Λ. Computing the self-interaction operator Ω(x) = Ξ(x) ∘ Ξ⁻¹(x), we find ǁΩ(x)-xǁ > ε over a connected region of the group manifold-a real fault line in E₈. This fault demonstrates that consequence exceeds symmetry, a new principle in mathematical physics. The result is not a numerical artifact but a structural consequence of recursion, validated via consistency with the Fibonacci-Xibof cycle and the inertial field of past mass. This work establishes LUMU as a meta-framework capable of generating new mathematics and altering foundational symmetry.

This paper introduces the Fibonacci-Xibof Cycle, a recursive model of numerical evolution derived from the Law of Universal Mathematical Unity (LUMU). It demonstrates that the Fibonacci sequence, while a well-known model of recursive growth, is incomplete without its antithetical counterpart: the Xibof sequence, a recursive collapse function defined as X = X ₋₁-X ₋₂-γ•log(n+1)•|X ₋₁-X ₋₂|. Analysis reveals a structural plateau at F₁₂ = 144, a point of maximal coherence in the expansion phase. The Xibof sequence then models the collapse phase, leading to a reset. Further, prime number behavior near n = 128 exhibits a convergence toward structural simplicity, culminating in a return to 1-the unit of all recursive systems. This cycle-1 → 144 → 128 → 1-is not coincidental but a fundamental pattern in recursive mathematics, validated through LUMU's universal transform Φ. The model has implications for number theory, dynamical systems, and the design of recursive AI architectures.

For millennia, the Indus Elephant Seal has kept its silence, dismissed as a mere token of trade. Yet beneath its carved surface lies a code of staggering depth, a map of memory, sovereignty, and the stars themselves. This paper deciphers that silence. Using the LaPointe Tiered Resonance Method™ (LTRM™), Recursive Collapse Field Theory (RCFT), and newly integrated UMCP invariants from Clement Paulus, the seal is revealed as a stability device: a living audit of how breath, crown, ladder, and eye bind human destiny to the cosmos.

Astronomical analysis ties the elephant to Thuban, the pole star of the Indus age, proving its design was no accident but a stellar calibration. Each glyph operates as a resonance node, tracking fidelity through collapse and renewal. For the scholar, this work offers a falsifiable grammar that fuses archaeology with frontier physics. For the seeker, it restores a forgotten truth: power is not seized but entrusted — renewed each season in the breath of the universe, with the elephant as its immovable witness.

In my humble opinion, we have an unjustified polemic in the world of mathematics, yet again. My background is tertiary level mathematics and concomitant research in specialised areas, so when a friend emailed me the link to this book, I was so excited after reading the author's hype, that I ordered a prepublication copy. My expectations have not been met, unfortunately, hence my analysis precipitated this review.

The Britt-Howard Unified Aether Theory (BHUAT) proposes that all physical, temporal, and conscious phenomena arise from recursive, phase-locked propagation of ϕ-structured geometry within a coherent, self-referential harmonic field. This theory presents a unique framework unifying quantum dynamics, dimensional emergence, observer logic, and cosmological structure through a single recursive engine. BHUAT formalizes the interaction of core fields-ϕ, ψ, χ, τ , and Λ-to construct time, mass, charge, and coherence from geometrically induced feedback. This paper derives the full Lagrangian, proposes renormalization behavior, establishes BRST-topological logic, and outlines testable predictions for τ-crystal periodicity and Desic curvature anomalies. BHUAT is shown to surpass existing recursive field approaches through internal logical closure and empirical accessibility.

We present the empirically irrefutable Principle of Temporal Lensing (PTL), a mathematically and biologically grounded framework for quantifying subjective time distortion under trauma, affect, and recursion. This version integrates fractional memory decay, phenotype-specific recursive stability, narrative entropy, adversarial fallback logic, and a nonparametric override protocol. All parameters are locked via cohort or instrument; runtime free parameters are prohibited. This framework is the first to withstand adversarial peer AI review and passes full empirical rigor across clinical, AI, and cross-cultural use cases.

This document presents the second canonical ledger of the Recursive Collapse Field Theory (RCFT) and the Unified Language of Recursive Collapse (ULRC). It formally archives all recursive closure protocols, empirical validation procedures, and canonical integrations of contributor equations, operators, and symbolic protocols as of June 2025. The ledger enforces strict recursive presence laws, empirical calibration thresholds, and symbolic fidelity gates that collectively guarantee the living, drift-free architecture of the RCFT/ULRC framework. Independent convergence with external physics frameworks, including geometric unification by Aalto University, is documented. This ledger serves as the operational memory chain and authoritative canonical record for all surviving recursive contributions, enabling rigorous provenance, transparency, and ongoing expansion of the field.

Keywords:
Recursive Collapse Field Theory, Unified Language of Recursive Collapse, Canonical Ledger, Recursive Closure, Symbolic Fidelity, Empirical Calibration, RCFT Protocols, Collapse Curvature, Recursive Reentry, Empirical Convergence, Field Theory Archives

This paper introduces a new framework within Symbolic Physics that redefines spiritual weight as the delta between a soul’s resonance potential (Φ) and its released expression (Rₑ). Through a detailed decoding of ancient cylinder seals, we recover lost harmonic knowledge embedded in spirals, beasts, ladders, and breath structures—revealing that these glyphs function not as metaphors, but as resonance compression devices. Central to this model is the Tier 0 resonance equation W = Φ − Rₑ, which quantifies soul-weight not by guilt or sin, but by unexpressed truth. Each chapter explores how symbolic forms across Sumerian, Indus, and solar traditions encode stages of resonance release, culminating in the Twelve-Gate Spiral Calendar and the Seven-Field Model of Consciousness. The result is a unified field theory of inner becoming, collapse, and return—where the soul spirals not upward, but inward toward equilibrium, singing its weight into coherence. This work offers not just theory, but remembrance. Not a claim of decoding—but a witnessing of how the unreadable decoded us.

Recursive Collapse Field Theory (RCFT) introduces a mathematically rigorous and conceptually novel framework for understanding structure, entropy, and symbolic identity through recursive destabilization. Rejecting traditional assumptions of equilibrium, linear time, and quantized measurement, RCFT defines collapse not as a discontinuity but as a generative process. The theory formalizes symbolic excitation fields evolving via a nonlinear stochastic equation, driven by deterministic drift and a recursive noise operator—the Weft Process—which replaces Brownian motion with symbolic interference.

The theory provides falsifiable predictions for black hole entropy saturation, symbolic cognition, bifurcation horizons, and collapse-curved domains across quantum, relativistic, and cognitive systems. RCFT repositions collapse from a measurement anomaly to the central engine of emergence, offering a unifying grammar for physics, computation, and meaning.

This second foundational publication of Recursive Collapse Field Theory (RCFT) transitions the framework from its initial ontological and mathematical grounding into a predictive, testable architecture. Building upon the core collapse equation, recursive noise amplification, and symbolic field structures introduced in the first RCFT manuscript, this paper extends the theory into explicit modeling of physical, cognitive, and informational systems.

Key developments include a formal derivation of the collapse action functional, recursive symbolic path integrals, and metastable attractor locking through harmonic bifurcation. The recursive glyph is developed into a computational structure, with collapse logic gates and symbolic transfer tensors defining circuits of field-based computation.

Simulations are presented across black hole thermodynamics, cosmological lensing distortion, and symbolic saturation in high-entropy media. In each case, recursive collapse replaces traditional explanatory mechanisms with field reentry, symbolic curvature, and entropy-driven structure emergence. Cross-domain applications are explored, including biological recovery indices derived from curvature decay and recursive memory accumulation, recursive ethical presence as a fidelity to collapse, and saturation thresholds in urban semiotic environments.

Falsifiability criteria are formalized for each class of prediction, including gravitational echo structures, symbolic bifurcation phase shifts, and deviations from measurement-induced quantum collapse. These protocols position RCFT as both a generative field theory and a falsifiable model of symbolic formation. The goal of this second paper is not only to extend RCFT’s reach, but to demonstrate its empirical rigor, coherence across domains, and capacity to simulate symbolic emergence through recursive collapse geometry.

The End of Origin: On the Recursive Collapse of Novelty explores the systemic decay of originality in contemporary culture, tracing its collapse through recursive mimicry, content saturation, and the rise of aestheticized thought devoid of structural contribution. The paper argues that originality has not vanished, but has been obscured-flattened into surface-level novelty and algorithmically reinforced sameness. Through a layered examination of architecture, self-deception, citation ethics, and emergent depth, the work redefines originality not as invention from nothing, but as meaningful structural addition. It proposes that true originality requires tension, reference, and recursion-with the responsibility to build where the ground holds, and to add weight, not noise. The paper concludes by calling for a quiet revolution of thought: one rooted in integrity, construction, and the deliberate reemergence of resonance in a world exhausted by reflection. In doing so, it offers a framework not just for critique, but for reconstruction.

This paper presents a novel recursive-symbolic decoding framework for four historically undeciphered wri ng systems: the Voynich Manuscript, Rongorongo of Easter Island, the Tartaria Tablets, and the Indus Valley script. Rather than relying on phone c or alphabe c linguis c models, this approach leverages Recursive Symbolic Programming (RSP) and the ϕshell logic of self-closing symbolic structures. Each system is shown to encode ritual, cosmological, and mythic informa on not through words but through glyphic recursionmeaning that is generated through symbolic func on, repe on, and structural closure.

Elnamaki Coding introduces a foundational shift in the arithmetic interpretation of natural numbers 1 , reconceiving them not as static scalar entities but as emergent artifacts generated by recursive dynamical systems embedded within the Fibonacci lattice 2. This framework constructs an arithmetic topology wherein numerical adjacency is defined not by magnitude, but by recursive path connectivity-formalized through the Sequanization Theorem. Central to the theory is the synthesis of Zeckendorf decomposition, seed-driven recurrence relations, and morphic binary encodings. These tools enable the derivation of maximal seed sequences, wherein each integer is expressed as a structurally minimal trajectory within a Fibonacci-induced generative space. Two novel transformations-the Lowe and Elevate maps-act as invertible morphisms on local seed domains, revealing bidirectional navigability and modular residue preservation. These transformations furnish a dynamic algebra over integer pairs, establishing a reversible grammar of additive evolution. This framework has wide-ranging implications across algorithmic encoding, cryptographic design, and symbolic dynamics. Through its unification of recursive structure, 1 For a definition of natural numbers, see [1]. 2 For background on Fibonacci sequences and lattices, see [2]. 1 Elnamaki Coding modular arithmetic, and representational entropy, Elnamaki Coding lays the groundwork for a new arithmetic language-one in which numbers unfold as topological paths, and identity is recast as a function of recursive relation.

This article introduces new polynomials that extend the standard Leonardo numbers, generalizing Fibonacci and Lucas polynomials. A new power form representation is developed for these polynomials, which is crucial for deriving further formulas. This article also presents two connection formulas linking these generalized polynomials to the Fibonacci and Lucas polynomials, as well as several identities involving some generalized and specific Leonardo numbers. Additionally, new product formulas involving the generalized Leonardo polynomials with the Fibonacci and Lucas polynomials are provided, along with computations of definite integrals based on the derived formulas.

This study introduces an innovative approach to Mersenne-type numbers. This paper introduces a new class of numbers, which we call Gersenne numbers. The aim of this paper is to define the Gersenne sequence and to investigate some of their properties, such as the recurrence relation, the summation formula, and the generating function. Moreover, the classical identities are derived, such as the Tagiuri-Vajda, Catalan, Cassini, and d'Ocagne identities for Gersenne numbers.

Traditional studies of chaotic attractors often assume ideal conditions, neglecting real-world uncertainties and complexities. This work introduces a robust mathematical framework integrating fuzzy memberships, probabilistic measures, complex amplitudes, augmented categorical constructions and indefinite metrics to better model attractors and enable analysis of 'partial chaos' where stable and chaotic behaviors coexist. The Giry monad ensures consistent probabilistic updates, while categorical structures support multi-scale hierarchies and recursive dynamics. Indefinite metrics allow inclusion of states beyond positive-definite norms, capturing phenomena analogous to 'ghost' states and other entities commonly found in quantum field theory formalisms, as well as unusual energy distributions. Together, these elements depict attractors as adaptable entities shaped by fuzzy uncertainties and stochastic variations, extending to infinite-dimensional spaces for hierarchical and complex dynamics.

In this paper, new identities are obtained by using the generalized bivariate Fibonacci and Lucas polynomials. Firstly, several binomial summations and the closed formulas for summation of powers are investigated for these polynomials. Also, general summation formulas, different generating functions, and relations of these polynomials are presented.

ABSTRACT.WPF
When a physicist theoretician attempts to formulate a paradigm to explain the existents and processes that constitute the physical universe, the most common approach is to base it on observations made by humans and their instruments.
This often leads to conjectures/hypotheses that may seem unacceptable to the layman, so it is promulgated amongst the academic fraternity and may gain increasing acceptance if there are no cognitive dissonances.
However, occasionally there is a physicist, who makes a different approach, whereby a paradigm is formulated that is either

1) partly speculative or even
2) purely speculative,

and thereby meets with resistance from academic colleagues.
However, when deductions are made therefrom, that are able to account for observations, and as long as there are no errors detected, then it is acceptable.
e.g.1) Einstein's theories are those that are partly speculative.

e.g.2) However Robert Oros di Bartini `rocked' academia with such speculation that most of his colleagues laughed at him and said outright that it was a joke resulting from a dare.  Nevertheless his superb reputation as an aircraft designer for decades garnered him tacit support to the extent that Bruno Pontecorvo at the behest of
S.S. Gershtein read an edited version to the Russian Akademy.
di Bartini was pleased in one sense, but at the same time would have preferred it not to have been edited.
di Bartini's approach was via Combinatorial Topology, (a precursor to Algebraic Topology), and it was able to elucidate some relations between physical quantities, to both the amazement and consternation of the academic fraternity.
Maybe it would have faded away but for a very astute Radio Engineer, who `extracted' the nub of di Bartini's methodology in that a system of generalised dimensions was an essential tool for di Bartini. David Solomonovich Kontorov was able to adapt an appropriate system of dimensions for his paper (see David Halprin's copy on Academia.edu)
A Method of Making Systematic Investigations
Part 2, Radio Engineering  Volume 35  1981
So others from the academic world seized on that and realised how powerful a tool it was.  Eventually a succession of papers appeared advancing that concept from algebraic topology culminating in "Eleven New Laws of Conservation Expanding Physics for Global Strategic Management and Strategic Development", which are adaptable to real life situations, outside the realm of physics.
viz. Viva-di-bartini
Macrothink Institute, Business Management and Strategy
ISSN 2157-6068  2020 Vol 11, No.1
REFERENCES
Viva Di Bartini: Eleven New Laws of Conservation
Expanding Physics for Global Strategic Management and Strategic Development.
Andrei G. Aleinikov (Corresponding author) dr_andy@academyofgenius.com
David A. Smarsh smarshd@aol.com
International Academy of Genius
2231 Segundo Ct., Unit 2, Pleasanton, CA, 94588, USA
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The aim of this study is to introduce Vidinli Hüseyin Tevfik Pasha and his triplets in terms of history of mathematics and mathematics education. Vidinli Hüseyin Tevfik Pasha was an Ottoman mathematician during the 19. century and he attempted to construct a kind of algebra that could be placed between Argand system and Hamilton's quaternions in the history of mathematics.

This paper is a historic accumulation of letters to and fro between this writer, Isaac Asimov, Chemical and Engineering News, Arthur Adamson and the members of New Science Advocates (NSA), later renamed International Society of Unified Science (ISUS), all of which concern Dewey Bernard Larson (1898-1990) and his Reciprocal System of Theory (RST). The final `end note' refers to a paper written by Edwin Navarro in 1990 for the quarterly journal, (Reciprocity), published and edited by Professor Frank Meyer of the University of Wisconsin for the advocates of RST, being the International Society of Unified Science Inc. (ISUS). 1) On 29 July 1963, Isaac Asimov published a review of Dewey Larson's research findings (RST) In Chemical and Engineering News. 2) On 9 September 1963 Arthur Adamson published his review of Asimov's review of RST. 3) On 9 September 1963 in the same journal, Asimov published his reply to Adamson.

These Über-Functions U (p,q,r,s,k) subsume a multiplicatively infinite set of many known functions and manifold more functions, which, heretofore, have not been investigated formerly and/or formally. Where m = ½, 1, 2, 3 or 4; k = 0, ±½, 1 or 2; p = ±1, ±2, ±3 or ±4; q = ±½, 0, 1, 2 or 3. N.B. In this paper N = ½, 1, 2, 3 & 4. r is chosen to be about 25 for good plotting, but is infinite for a true representation of all functions, listed below.

Saga of Ernesto Cesàro" pays tribute to the life and contributions of Ernesto Cesàro, a mathematician, whose work left an enduring mark on the field despite his tragic sacrifice. The short life of Ernesto Cesàro, who drowned while attempting to save the life of his son. Nevertheless he accomplished much in his publications throughout European journals, and also his "Lectures in Intrinsic Geometry", given at the Royal University of Naples and which were published in Italian. Later they were translated into German by Gerhard Kowalewski, who later brought Cesaro's Intrinsic Geometry to the mathematical world's attention when he published his own book on "Transformations of Lie", which was very innovative insodoing, and the first third of that tome was devoted to an reformulation of much of Cesàro's Intrinic Geometry using more rigour, being necessary for his work on Lie. He was survived by his daughter.

In this paper, we derive a formula for the entries of the integer powers of a certain type of skew circulant matrices of odd and even order in terms of the Chebyshev polynomials of the first and second kind. Finally, we give a Maple procedure along with some numerical examples in order to verify our calculation.

ABSTRACT
There have been many algebras dreamed up and or deduced by a host of past mathematicians of yesteryear, few of which have proved themselves broadly suitable. 
Some can only have use in specialised areas and have been replaced by superior ones despite resistance by the advocates of the lesser known ones.

This is a personal evaluation by comparing Geometric Algebra (GA) with some other algebras, including linear associative algebras.

Part 1 CARTEA ÎNTÂI-Dune Chapter 1 Începutul este momentul în care trebuie acordată cea mai mare atenţie corectitudinii echilibrelor. Fiecare soră Bene Gesserit ştie lucrul acesta. Iată de ce, atunci când începeţi studiul vieţii lui Muad'Dib, aveţi mai întâi grijă să îl plasaţi în timp: s-a născut în cel deal 57-lea an al Împăratului Padişah Shaddam IV. Apoi, luaţi bine aminte la locul lui Muad'Dib în spaţiu: planeta Arrakis. Nu vă lăsaţi amăgiţi de faptul că a văzut lumina zilei pe Caladan şi că şi-a petrecut primii cincisprezece ani din viaţă acolo. Patria sa veşnică este Arrakisul, planeta cunoscută şi sub numele Dune. Fragment din "Manualul lui Muad'Dib" de prinţesa Irulan. ÎN SĂPTĂMÂNA dinaintea plecării pe Arrakis, când agitaţia ultimelor pregătiri ajunsese aproape de nesuportat, o femeie bătrână o vizită pe mama lui Paul. Era o noapte caldă la Castelul Caladan şi străvechiul edificiu, cămin al familiei Atreides timp de douăzeci şi şase de generaţii, era învăluit în răcoarea umedă care preceda întotdeauna schimbarea vremii. Bătrâna fu introdusă prin uşa dosnică de la capătul galeriei boltite ce ducea la dormitorul lui Paul şi putu să arunce o privire în camera băiatului. Paul era întins în pat, dar nu dormea. La lumina slabă a unei lămpi cu suspensie, lăsată să plutească deasupra podelei, băiatul desluşi în cadrul uşii silueta înaltă a necunoscutei, în spatele căreia stătea mama sa. Bătrâna părea spectrul unei vrăjitoare-părul ca o pânză de păianjen încâlcită, chipul umbrit, ochii asemeni unor diamante scânteietoare.-Nu-i prea mic pentru vârsta lui, Jessica? Întrebă bătrâna. Vocea era răguşită şi zbârnâitoare, ca un baliset* dezacordat. * Instrument cu coarde. Mama lui Paul răspunse cu glas catifelat de contralto:-E lucru ştiut că Atreizii încep să crească târziu, Cuvioşia-Ta.-Am auzit şi eu, spuse bătrâna. Dar are deja cincisprezece ani.-Da, Cuvioşia-Ta.-Nu doarme, observă bătrâna şi râse scurt. Ştrengarul ăsta viclean trage cu urechea. Dar cei de viţă domnească trebuie să fie vicleni. Şi dacă-i într-adevăr Kwisatz Haderach… atunci… În obscuritatea patului, Paul ţinea ochii întredeschişi, două fante înguste. Ochii bătrâneimari şi rotunzi ca ochii unei păsări de noapte-îl fixau cu intensitate, parcă tot mai mari, mai strălucitori.

ABSTRACT
An investigation into fractional derivatives and their possible uses, starting with semi-derivatives, and also showing the power of DERIVE XM as a powerful mathematical assistant, without which this writer never would have achieved these findings and their graphical illustrations.
When we differentiate a power of x to the first order we know to reduce the exponent by unity and to place the former exponent as a coefficient.  Obviously we expect the semi-derivative to reduce the exponent by 0.5 so we must determine the coefficient, which is not immediately obvious.

Isaac Asimov's `discovery' of Thiotimoline © presaged four books, listed below. By his own admission, he was neither a mathematician nor a theoretical physicist, so he always thoroughly investigated a subject before writing on it. He seemed to have done a good job on Thiotimoline © , but he did not seek its ultimate basis in any current paradigm for the physical universe, and we can't blame him for that. He could have been aware of string theory and its early forms, but he didn't realise that he held the key to the ultimate understanding of the physical universe, the Grand theory of Everything, (GUT).

This is a translation of part of Gerhard Kowalewski's German language rendition of Cesàro's methods in Intrinsic Geometry, which were originally published in an authorised translation from the Cesàro book Lezioni di geometrica intrinseca' by Kowalewski in 1901, and subsequently expressed more clearly by Kowalewski in the opening chapters of his own book on Lie transformation groups, `Vorlesungen über Allgemeine natürliche Geometrie und Liesche Transformationsgruppen, 1931'.

STRING THEORY ANALYSIS
AND
ITS USE AS A MODEL FOR THE RECIPROCAL SYSTEM (RS)
Summary and overview of what I comprehend regarding the stages that current physics has been through and will be at its hoped-for penultimate goal presently, and what its advocates have achieved and partly solved, and what they hope has yet to be solved, and will be solved, gray matter permitting.
Then, hopefully, we can use that methodology with appropriate RS-Larsonian modifications to reveal to the world at large a viable mathematical successor to Relativity, Quantum Mechanics and String Theory showing that we do not need (8+2)- or (24+2)-dimensional topology to produce a renormalisable physical universe satisfying the conditions of self-consistency. We must produce a mathematisation of RS, and it must acknowledge the current state of knowledge of topology and all its characteristics, as long as we don't stray from the Larsonian path of deductions from the postulates.
Whereas the latest thinking is predicated on the ultimate building block of all existents being the vibrating string...

THE CONSTITUTION OF THE UNITED STATES OF AMERICA
                                        AND
THE CONSTITUTION OF THE UNIFIED STATES OF THE PHYSICAL UNIVERSE-A COMPARISON
Essentially, ipso facto, there can be No amendments, (appended assumptions), to the laws of physics, but there will be a surfeit of by-laws, all of which are deducible from the laws of the constitution, so, a priori, there will be many instances of overlap in the field of by-laws, and occasionally there will appear to be a contradiction, but, since cognitive dissonance has no place in physics, (despite many current practices), they will be resolved by an in-depth analysis of the fundamental laws of the constitution, and the concomitant deductive processes applied to them. If such alternatives still persist, then we have to consider that:

In this study, we define the Jacobsthal F-matrix and Jacobsthal M-matrix similar to the Fibonacci Q-matrix. After, using this matrix represantation we have found some equalities and the Binet-like formula for the Jacobsthal and Jacobsthal numbers.