A Study On Dual Generalized Edouard Numbers

In this paper, we introduce and develop the concept of generalized dual hyperbolic Pierre numbers, a novel class of number sequences that extends the structural framework of classical Pierre-type sequences through duality and hyperbolic transformations. This generalization offers a unified approach that encompasses both established and newly constructed numerical models. As distinguished special cases, we examine the dual hyperbolic Pierre numbers and their Lucas-type counterparts, emphasizing their algebraic relationships and unique structural features. Our study presents a comprehensive set of mathematical results, including closed-form identities, matrix representations, and recurrence relations that define the behavior of these sequences. We further derive Binet-type formulas for explicit term computation and construct generating functions that capture their combinatorial and analytical properties. Additionally, we explore Simson’s formulas and establish various summation identities that reveal deeper interconnections among sequence elements. This investigation contributes to the broader theory of Pierre-type sequences and offers new tools for research in discrete mathematics, algebraic structures, and computational number theory.

In this research, we introduce the generalized dual hyperbolic Edouard numbers, a novel class of number sequences that extends existing recurrence relations into a new mathematical framework. Several special cases of these numbers are examined in detail, including the dual hyperbolic Edouard numbers and the dual hyperbolic Edouard-Lucas numbers, each revealing intriguing combinatorial and algebraic properties. Explicit expressions for these sequences are derived, such as Binet-type formulas, generating functions, and summation identities, which offer analytical insight into their behavior and structural patterns. In addition, we explore matrix representations associated with these sequences, providing an elegant algebraic tool for further theoretical development and potential applications.

This paper is a tutorial introduction to a general methodology, consisting of categorical constructions, for the definition of new algebraic semantics for transition-based formalisms. The methodology individuates three levels of semantic description: programs, structured transition systems, and models. The various formalisms differ for the structure on states and transitions, with respect to which all the categorical constructions of the methodology are parametric. These constructions generate in an automatic way the induced transition system and the free model of a program. One characterizing issue of the methodology is that structured transition systems have a (usually algebraic) structure on both the states and the transitions, while for example in Plotkin's SOS approach just the structure of states is relevant. The advantage of considering transition systems with an algebraic structure on transitions, too, resides in the fact that often the same structure can be automatically extended to the computations of the system. This yields to categories whose arrows are not simple sequences of elementary transitions, but are instead abstract computations, equipped with a rich algebraic structure. The methodology generalizes the algebraic treatment of Petri nets proposed in [MM88], and includes the main ideas of the algebraic semantics for Horn Clause Logic presented in [Co90, CM90], and of the algebraic treatment of Milner's CCS, reported in [Fe90, FM90].

This work introduces complex hybrid numbers, an extension of hybrid numbers, which are defined as Z = z0 + z1i + z2ɛ + z3ℎ where zi are complex numbers and iℎ = −ℎi = i + ɛ. We investigate their algebraic properties, classify them into ℂ1, ℂ2, D, and ℙ-types, and analyze their matrix representations. Numerical examples illustrate key findings, highlighting both their theoretical structure and potential areas of application. Key findings are demonstrated with numerical examples, emphasizing their structural importance and their uses.

It is well-known that sums and products of holomorphic functions are holomorphic, and the holomorphic functions on a complex manifold form a commutative algebra over C . The study of complex manifolds using algebras of holomorphic functions upon them is called complex algebraic geometry. The purpose of this paper is to develop an analogue of complex algebraic geometry, in which the complex numbers C are replaced by the quaternions H . The natural quaternionic analogue of a complex manifold is called a hypercomplex manifold. A class of H -valued functions on hypercomplex manifolds will be defined, called q-holomorphic functions, that are analogues of holomorphic functions on complex manifolds. Now, the set of holomorphic functions on a complex manifold is a commutative algebra over C . Therefore one asks: does the set of q-holomorphic functions on a hypercomplex manifold have an analogous algebraic structure, and if so, what is it? We shall show that the q-holomorphic functions on a (noncompact) hypercomplex manifold do indeed possess a rich algebraic structure. To describe it, we shall introduce a theory of quaternionic algebra, which is a quaternionic analogue of real linear algebra. This theory is built on three building blocks: AH-modules, the analogues of vector spaces, AH-morphisms, the analogues of linear maps, and the quaternionic tensor product, the analogue of tensor product of real vector spaces. As far as the author can tell, these ideas seem to be new. They enable us to construct algebraic structures over H as though H were a commutative field. Quaternionic algebra describes the algebraic structure of hypercomplex manifolds in a remarkable way, and it seems to be the natural language of hypercomplex algebraic geometry, the algebraic geometry of hypercomplex manifolds. We believe that quaternionic algebra is worth studying for its own sake. It has many similarities with linear algebra over R or C , which is why the analogies between complex and quaternionic theories work so well, but there are also deep differences, which give quaternionic algebra a flavour all of its own. Quillen [12] has given a sheaf-theoretic interpretation of the ideas of quaternionic algebra, based on a previous version of this paper. He finds a contravariant equivalence between a class of AH-modules and regular sheaves on a real form of CP. Regular sheaves on CP are equivalent to representations of the Kronecker quiver, and out of such a representation Quillen constructs an AH-module. Under Quillen’s equivalence stable AH-modules correspond to regular vector bundles over the real form of CP.

We analyze the mathcal {N}=2 superconformal field theories that arise when a pair of D3-branes probe an F-theory singularity from the perspective of the associated vertex operator algebra. We identify these vertex operator algebras for all cases; we find that they have a completely uniform description, parameterized by the dual Coxeter number of the corresponding global symmetry group. We further present free field realizations for these algebras in the style of recent work by three of the authors. These realizations transparently reflect the algebraic structure of the Higgs branches of these theories. We find fourth-order linear modular differential equations for the vacuum characters/Schur indices of these theories, which are again uniform across the full family of theories and parameterized by the dual Coxeter number. We comment briefly on expectations for the still higher-rank cases.

This paper presents a comprehensive investigation into the structure and properties of the standard Cayley-Dickson algebras $A_x$ (dimension $2^x$). Our research, driven by computational exploration and pattern recognition, yields both significant quantitative and qualitative results. We establish an explicit formula, $N(x) = \frac{(2^x - 2)(2^x - 4)(2^x - 8)}{16}$, which accurately counts the number of unique zero divisor pairs specifically of the form $(e_i + e_u) * (e_j \pm e_v) = 0$ for $x \geq 4$, as identified and computationally validated by our exploratory methodology. This formula provides a precise quantitative measure for this structurally significant class of zero divisors. Independently, we unveil a detailed structural analysis of the Cayley-Dickson multiplication tables, demonstrating a recursive decomposition into 8x8 blocks (reflecting octonion substructures). We introduce a novel "block type" classification ('x' or 'y') based on the sign of a single indicator element $e_{(8k, 8k+1)}$, which dictates the location (Upper/Lower Triangular Matrix relative to the anti-diagonal) of associated zero divisors within those blocks. This analysis reveals hidden recursive patterns and self-similarity across dimensions. Furthermore, we discuss the conceptual frameworks of an "Observed Pattern Multiplication Table" (OPMT) construction and a sign compression scheme, which served as valuable tools during our investigation. A Python script implementing the standard construction, featuring an exploratory zero divisor finder with computational validation and deduplication, is provided as supplementary material. Our combined findings offer a substantially deeper understanding of Cayley-Dickson algebras and suggest potential applications in related fields. This paper is based on empirical results and lacks most of the time rigorous proofs, so what isn't proven is a conjecture.

Investigations of adequate mathematical model of digital computer have lasted for a few years. But when a digital computer was well represented by some suggested model, this model had extremely complicated algebraic structure and it couldnrt be studied by modern algebraic methods. On the other hand if investigated model has a simple algebraic structure (e.g. a unary algebra) then lot of notions associated with digital computers had to be excluded from such a model (e.g. programs). The starting-point of this pa~er is the notion of stored program computer presented by PAWIAK,Z. in ~4J. The stored program computers have a very rich algebraic structure and because of difficulties mentioned above to each stored program computer a new kind of object is associated, more useful in algebraic studies; it's called a M-groupoid. Such associations were done very often in mathematics. To investigate properties of some objects we associate a new kind of objects. And we find the properties of our investigated objects from studies of objects associated with them (e.g. associating to first order predicate calculus the Tarski-Lindenbaum algebra (see ~J), to given Boolean algebra the Stone space (see E6J), to given topological space the fundamental group (see F7J ), to given finite automaton the input monoid (see [1~)). Such an approach often makes possible simple proofs of complex results. On the other hand it#s good to be aware of differences between an associated object and the investigated one (it means how much information about the invastigated object is contained in the associated object). Theory of categories will be useful for it (it will be shown, that the assotiation between stored program computers and M-groupoids is a functor from the category of stored program computers to the category of M-groupoids).It will be shown that stored program computers and M-groupoids are closely related, namely theirs categories are equiva-

Porous tantalum represents a relatively new solution for primary and revision total knee arthroplasty, offering several unmatched properties. Tantalum is a transition metal, with excellent biocompatibility and bioactivity due to its intrinsic physical and structural characteristics. A widespread clinical use of porous tantalum tibial components for primary total knee arthroplasty has been partly hindered by the difficulty in removing this type of implant after bone in growth, often leading to a significant bone defect. On the contrary, in the case here reported, removal of the trabecular metal tibial component was unexpectedly easy at a 7-month follow-up due to the absence of bone ingrowth but with a complete preservation of the tibial plate bone stock. Causes for the lack of bone ingrowth are discussed.

Using as example an incomplete information system with support a set of objects X, we discuss a possible algebraization of the concrete algebra of the power set of X through quasi BZ lattices. This structure enables us to define two rough approximations based on a similarity and on a preclusive relation, with the second one always better that the former. Then, we turn our attention to Pawlak rough sets and consider some of their possible algebraic structures. Finally, we will see that also Fuzzy Sets are a model of the same algebras. Particular attention is given to HW algebra which is a strong and rich structure able to characterize both rough sets and fuzzy sets.

Gadolinium (Gd[III])-based nanoaggregates are potential noninvasive magnetic resonance imaging (MRI) probes with excellent spatial and temporal resolution for cancer diagnosis. Peptides conjugated with Gd3+ can aid in supramolecular scaffolding for MRI nanoagents because of their inherent biocompatibility and degradability. We report here a strategy to tune the MR relaxivity of tumor cell-targeted nanoagents and enhance the antimicrobial and anticancer activities of nanoagents based on rationally designed antimicrobial peptide (AMP) assembly. A tripeptide with glycyl-l-histidyl-l-lysine (GHK) capable of Gd3+ chelation was attached to short AMPs containing pyrazole amino acids that spontaneously assembled as a function of the number of hydrophobic amino acid residues and the peptide length of AMPs. Aqueous coassembly of GHK with tumor-targeting, cyclic arginine-glycine-aspartic acid (cRGD)-tagged AMPs resulted in the formation of micelles, fibrils, vesicles, sheets, and planar networks. Interestingly, the two-dimensional planar network nanostructure showed less antibacterial activity and tumor cell cytotoxicity but greater drug loading/delivery and magnetic resonance signaling than micelles because of its intrinsic structural characteristics. This study can provide a rational approach for the design and fabrication of clinically useful nanoagents.

Flexible conductive materials with intrinsic structural characteristics are currently in the spotlight of both fundamental science and advanced technological applications due to their functional preponderances such as the remarkable conductivity, excellent mechanical properties, and tunable physical and chemical properties, and so on. Typically, conductive hydrogel fibers (CHFs) are promising candidates owing to their unique characteristics including light weight, high length‐to‐diameter ratio, high deformability, and so on. Herein, a comprehensive overview of the cutting‐edge advances the CHFs involving the architectural features, function characteristics, fabrication strategies, applications, and perspectives in flexible electronics are provided. The fundamental design principles and fabrication strategies are systematically introduced including the discontinuous fabrication (the capillary polymerization and the draw spinning) and the continuous fabrication (the wet spinning, the microfluidic spinning, 3D printing, and the electrospinning). In addition, their potential applications are crucially emphasized such as flexible energy harvesting devices, flexible energy storage devices, flexible smart sensors, and flexible biomedical electronics. This review concludes with a perspective on the challenges and opportunities of such attractive CHFs, allowing for better understanding of the fundamentals and the development of advanced conductive hydrogel materials.

Two indium–organic framework isomers, namely [Me2NH2][In(BPDC)2] (InOF-3) and [MeNH3][In(BPDC)2] (InOF-4), have been solvothermally prepared from a biphenyl-3,3′-dicarboxylate ligand (H2BPDC) with changeable conformation, and feature skeletal frameworks with equivalent chemical formulae. For the 2D close-packing InOF-3, each 3-connected [In(O2CR)4] node connects to 3 others through syn-BPDC2− ligands, leading to a 3-connected uninodal layer with a point symbol of {63}. For the 3D microporous InOF-4, the framework is assembled from [In(O2CR)4] nodes bridged by dicarboxylate ligands in the anti- fashion. Careful examination of the structure shows that InOF-4 is a 4-fold interpenetrating network and topologically adopts a diamond-like 4-connected uninodal net with a point symbol of {66}. More importantly, the theoretical calculations and the experimental single component gas uptake measurements reveal that the 3D porous InOF-4 has a better low pressure gas storage capacity than its 2D framework isomer InOF-3, with a saturated N2 uptake of 168.2 cm3 g−1 at 77 K and H2 uptake capacities of up to 126.5 cm3 g−1 (1.13 wt%) at 77 K and 1.0 bar, and 98.1 cm3 g−1 (0.88 wt%) at 87 K and 1.0 bar. Finally, the IAST calculations show us that both microporous materials have an outstanding low-pressure selectivity between CO2 and N2, probably due to their intrinsic structural characteristics.

Since the discovery of carbon nanotubes, many scientists have developed a numbers of nanomaterial (nanotubes, nanorods, nanowires) in order to minimizing the size of the devices. A large series of devices have been created on the basis of nanomaterials[1]. In addition, many properties of nanomaterials will be dramatically changed due to extremely small size or unique shape. For instance, electrical and optical properties, etc. That's why one-dimensional (1D) materials are important to nano-level photoelectric and electronic component. In the past decades, the In-Se system[2] has attracted to a great deal of attention in high-efficiency thermoelectric conversion due to its extraordinary adjustable electrical and thermal properties. The physical properties of indium selenide mainly depend on its intrinsic structural characteristics, including composition, phase, crystal structure. Structurally, indium selenide belongs to a composite system of different stoichiometry, including In4Se3 [3], In2Se3 [4], InSe[5], which is also used in electrochemical devices due to its low-dimensional structure. In general, nanowires are made by chemical vapor deposition (CVD) process. However, using CVD to obtain heterostructure is relatively complicated. Thus, injection molding process is a simple way to let nanowire's phase separated. The topic of this research was divided into two main processes, the anodic treatment of aluminum sheet and In-Se system materials synthesized by vacuum melting, annealing and torch sintering, sequentially. The effects of melting and annealing process on the phase, composition, microstructure, of polycrystalline In4Se3/In were investigated. First, the high purity aluminum (99.999%) was carried out electrolytic- polishing, and then form nano-porous array of anodic aluminum oxide (AAO) by anodic oxidation process. After that, AAO was soaked in acid to get alumina template having aperture of 70~100 nm and thickness of 30 μm. It would be used for filling in functional In-Se one-dimensional (1D) materials. In the second part, the powder with 70 at% indium and 30 at% selenium were uniformly mixed in the quartz tube, and then melted into In0.7Se0.3 bulk in vacuum at high temperature. Then, the melting In0.7Se0.3 bulk was cast into AAO auxiliary template, a special high-temperature oven including In0.7Se0.3 bulk and AAO was heated up to melting point. Afterwards, press into the AAO template to form In + In4Se3/AAO composite by vacuum injection molding process. As-obtained samples were immersed in chromic acid to get heterostructure nanowires. Afterward these nanowires can be transformed into segmental structure of In4Se3/In by annealing. Moreover, the length ratio of In4Se3/In segmentation would be 4:1 via theoretical calculating. The morphology of the nanowires was observed by optical microscopy (OM). After annealing, the segmentations of In/In4Se3 nanowires were analyzed by field-emission scanning electron microscope (FE-SEM). It can be observed that the nanowires were obviously separated into In and In4Se3 parts. Using energy dispersive spectrum (EDS) to confirm the composition of the segment structure, the dark area of the nanowires was In4Se3 and the bright one was In. That is corresponding to the bulk composition of In4Se3 identified by Raman analysis. It has great potential that nanowire product will be applied in photovoltaic semiconductors.

Structural and environmental characteristics of multiple topographies of stereotypic behaviors shown by 8 individuals with developmental disabilities are described. Analysis of structural characteristics show that the parameters of stereotyped behavior (percentage of time, bout length, and bout length variability) were highly correlated. Analysis of the influence of environmental variables indicated that topographies of stereotyped behaviors were more likely to occur under conditions of low stimulation and less likely during conditions involving social contact. Individuals whose stereotyped behaviors were less influenced by environmental factors were more likely to engage in them for greater periods of time. Results suggest that an examination of individual topographies of stereotyped behaviors and their structural and environmental characteristics may provide useful insights for understanding the origins of these behaviors.