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Keçeci Numbers (first defined: July 27, 2022) constitute a unique numerical system that generates sequences based on specific initial conditions and a set of iterative rules. This system relies on a complex interplay of fundamental arithmetic operations (addition), special divisibility rules (periodically dividing by 2 or 3), and the primality properties of numbers. One of the most notable aspects of Keçeci Numbers is their definability across various number sets, such as integers, rational numbers, negative numbers, complex numbers, and quaternions, exhibiting distinct behaviours in each. This offers a rich ground for investigating how fundamental number theoretic principles manifest within different algebraic structures. The "Keçeci Prime Number" refers to the number (or its positive integer representative, based on the absolute value of its real or scalar part, according to the number type) that is prime and appears most frequently within the generated Keçeci Number sequence. This value can be interpreted as a kind of "characteristic signature" or "condensation point" regarding the sequence's structure. The potential significance of Keçeci Numbers lies not in being an established sequence, but in providing a "computational laboratory" for exploring how fundamental concepts in number theory (divisibility, primality, arithmetic progression) can generate new and unpredictable patterns under specific constraints and conditions. The application of such rules, particularly in higher-dimensional number systems like complex numbers and quaternions, might produce sequences exhibiting fractal-like structures or chaotic behaviour. Such sequences could potentially inspire models for certain dynamic systems in theoretical physics, or new designs for pseudo-random number generators in signal processing engineering or cryptography. As Keçeci Numbers are still in an exploratory phase, their true value will emerge with deeper mathematical analysis in the future (e.g., concerning the long-term behaviour of the sequences, their limits, periodicity, or the distribution of Keçeci Prime Numbers). Such original numerical systems demonstrate that mathematics is not only about solving existing problems but is also a self-enriching discipline that generates new structures and questions. This can serve as a source of motivation, reminding students and researchers of the vast, unexplored territories within the infinite world of numbers.

The visualization of scientific data is a fundamental step in uncovering hidden patterns and relationships within complex systems. While conventional force-directed graph layout algorithms (e.g., spring layout) are effective at displaying the general topology and clustering tendencies of systems, they possess a critical shortcoming: they are inherently non-deterministic and obscure the natural, sequential, or index-based identities of nodes. This poses a significant limitation when analysing systems that contain an intrinsic sequential logic, such as quantum circuits, metabolic pathways, crystal structures, or chronological data structures. This paper introduces the "Keçeci Layout," a deterministic and order-preserving algorithm designed to fill this analytical gap. The algorithm operates by arranging nodes sequentially along a primary axis according to their IDs, employing a predictable zigzag pattern on a secondary axis to prevent node overlaps. This approach establishes a direct and intuitive bridge between a system's abstract data structure (e.g., qubit indices, reaction steps) and its visual representation. To demonstrate the versatility and efficacy of the Keçeci Layout, four case studies from distinct scientific disciplines are presented: Chemistry, Physics, Biology, Quantum Computing, Materials Science, Molecular Biology, Computer Science, etc. In conclusion, the Keçeci Layout is proven to be a powerful scientific tool that offers analytical depth by remaining faithful to the underlying data structure, moving beyond purely aesthetic presentation. The algorithm provides a significant advantage over conventional methods, particularly for the visual analysis of ordered, hierarchical, or index-dependent systems.

Nodal-line semimetals constitute a fascinating subclass within the family of topological materials in condensed matter physics. These unique materials are characterized by the intersection of their electronic energy bands along one or more closed or open lines within the Brillouin Zone, rather than at isolated points. These lines of intersection are termed "nodal lines," and along these lines, electrons can behave as if they were massless, leading to extraordinary electronic properties. The significance of nodal-line semimetals stems from the fundamental physical phenomena offered by these unique linear crossings, as well as the promising potential they hold for future technologies, particularly quantum computers. One of the most striking features of these materials is that their nodal lines are often topologically protected by specific crystal symmetries (e.g., mirror symmetry). This protection renders the nodal lines and their associated electronic states highly resilient to minor imperfections and disorder within the material. Considering that decoherence (the loss of quantum states by qubits) is the primary adversary of quantum computers, such topological stability is of vital importance. Nodal-line semimetals, through this inherent protection mechanism, could offer new pathways for developing more long-lived and fault-tolerant qubits. Another distinguishing characteristic of nodal-line semimetals is their ability to host exotic surface states with nearly flat energy bands, known as "drumhead surface states." These surface states are localized within the region bounded by the projection of the nodal line onto the surface and can possess a high density of states. The manipulation of these special surface states might enable novel quantum information processing schemes or topologically protected information carriers. Furthermore, the geometry and number of nodal lines can significantly influence the material's response to electromagnetic fields or mechanical strain. This makes nodal-line semimetals potential candidates for sensitive sensors or tuneable electronic/optical components. When a magnetic field is applied or symmetries are broken, a nodal line can split into Weyl points or a band gap can open, offering the possibility to dynamically control the material's properties. Such control mechanisms could be exploited for designing quantum switches or complex quantum circuit elements. In conclusion, nodal-line semimetals, with the rich physics offered by their linear band crossings and the advantages conferred by topological protection, carry immense potential for applications in quantum computing and beyond. Their drumhead surface states and tuneable electronic structures make them exciting systems for fundamental scientific research and prime candidates to become building blocks for future quantum technologies. Ongoing research in this area is anticipated to pave the way for more robust and powerful quantum computers.

Weyl semimetals (WSMs) have emerged as one of the most groundbreaking discoveries in condensed matter physics in recent years, deepening our fundamental scientific understanding and opening new horizons for future technologies. These materials are named after Hermann Weyl, who in 1929 predicted Weyl fermions-massless, chiral relativistic particles. WSMs, as low-energy excitations of these fermions in solid-state systems, possess special points in their electronic band structure where bands cross linearly, known as "Weyl points." These points act as sources or sinks (monopoles) of Berry curvature in momentum space and carry a definite chirality (right-or left-handed). For a material to host a WSM phase, either timereversal symmetry or inversion symmetry must be broken. One of the most distinctive features of Weyl semimetals is the existence of unique topological surface states known as "Fermi arcs" on their surfaces. These arcs connect Weyl points of opposite chirality within the bulk and form open Fermi surfaces, which are absent in ordinary metals. The presence of Fermi arcs is strong evidence of the topological nature of WSMs and has been experimentally observed using techniques like angle-resolved photoemission spectroscopy (ARPES). Another crucial phenomenon is the "chiral anomaly." This is a quantum field theory effect where chiral charge is not conserved in the presence of parallel electric and magnetic fields, resulting in the pumping of Weyl fermions from one chirality to the other. In solid-state WSMs, the chiral anomaly leads to unusual transport properties, such as a characteristic negative longitudinal magnetoresistance and a giant anomalous Hall effect. These features are important experimental signatures for confirming the existence of Weyl fermions. WSM phases have been theoretically predicted and experimentally realized in various materials, including the TaAs family (TaAs, TaP, NbAs, NbP), WTe₂, MoTe₂, and magnetic WSMs. These materials exhibit a wide range of exotic physical properties, such as anomalous Hall conductivity, giant nonlinear optical responses, thermoelectric effects, and potential catalytic activities. The unique electronic and topological properties of Weyl semimetals make them highly promising candidates for potential applications such as low-energy-consumption electronics, spintronic devices, topological quantum computing, and even high-efficiency catalysts. Intensive research in material synthesis, characterization, and theoretical modeling is ongoing to unlock the full potential of these materials. The discovery of Weyl semimetals has opened up a new and exciting research area in condensed matter physics, highlighting the richness and complexity of topological matter.

This study presents a comparative analysis of static and dynamic number sequences, using the classical Oresme numbers and the novel Keçeci numbers, developed by Mehmet Keçeci, as primary case studies. Static sequences are characterized by a fixed, predictable recurrence relation. The Oresme numbers—the partial sums of the harmonic series (Η_n=∑_(k=1)^n 1/k)—exemplify this category. Their generation follows a simple, deterministic rule (Η_n= Η_(n-1)+1/n), and their predictable divergence, proven by Nicole Oresme, serves as a foundational concept in mathematical analysis and pedagogy. In stark contrast, Keçeci numbers are defined as a dynamic sequence generated by a state-dependent algorithm. Their progression is not linear but determined by the properties of the terms themselves. The algorithm initiates with a value and an increment, but each subsequent term is derived through a conditional pathway involving division by an alternating divisor (2 or 3). If division fails, a primality check is performed on the term's principal component (e.g., the real part of a complex number). A prime result triggers the unique "Augment/Shrink & Check (ASK)" rule, modifying the term before re-attempting division. This process, implemented in Python for number sets including integers, rationals, complex numbers, and quaternions, generates a complex, path-dependent behavior. The comparison reveals a fundamental dichotomy. Oresme numbers provide a robust, transparent framework ideal for theoretical exploration and teaching mathematical series. Conversely, the dynamic and adaptive structure of Keçeci numbers offers significant flexibility, suggesting potential applications in modern computational fields such as algorithm design, cryptographic systems, and procedural generation in simulations. While the predictable nature of static sequences like Oresme's provides a solid theoretical bedrock for analysis, the computationally intensive and pseudo-random characteristics of dynamic sequences like Keçeci numbers open new research avenues in computer science and complex systems modeling.