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In this very personal hunt for the meaning of the bacterial cell cycle, the snark, I briefly revisit and update some of the mechanisms we and many others have proposed to regulate the bacterial cell cycle. These mechanisms, which include the dynamics of calcium, membranes, hyperstructures, and networks, are based on physical and physico-chemical concepts such as ion condensation, phase transition, crowding, liquid crystal immiscibility, collective vibrational modes, reptation, and water availability. I draw on ideas from subjects such as the 'prebiotic ecology' and phenotypic diversity to help with the hunt. Given the fundamental nature of the snark, I would expect that its capture would make sense of other parts of biology. The route, therefore, followed by the hunt has involved trying to answer questions like "why do cells replicate their DNA?", "why is DNA replication semi-conservative?", "why is DNA a double helix?", "why do cells divide?", "is cell division a spandrel?", and "how are catabolism and anabolism balanced?". Here, I propose some relatively unexplored, experimental approaches to testing snark-related hypotheses and, finally, I propose some possibly original ideas about DNA packing, about phase separations, and about computing with populations of virtual bacteria. [ABSTRACT FROM AUTHOR]

In this paper we extend the SuperHyperAlgebra, SuperHyperGraph, SuperHyperTopology, SuperHyperSoft Set, endowed with SuperHyperOperations, SuperHyperAxioms, and SuperHyperFunctions, to the most general form of structure, from our real world, called SuperHyperStructure in any field of knowledge. A practical application of the SuperHyperStructure is presented at the end. The prefix “Hyper” [Marty, 1934] stand for the codomain of the functions and operations to be P(H), or the powerset of the set H. While the prefix “Super” [Smarandache, 2016] stands for using the Pn (H), n ≥ 2, or the n-th PowerSet of the Set H {because the set (or system) 𝐻 (that may be a set of items, a company, institution, country, region, etc.) is organized in sub-systems, which in their turn are organized in sub-sub-systems, and so on} in the domain and/or codomain of the functions and operations.

In this very personal hunt for the meaning of the bacterial cell cycle, the snark, I briefly revisit and update some of the mechanisms we and many others have proposed to regulate the bacterial cell cycle. These mechanisms, which include the dynamics of calcium, membranes, hyperstructures, and networks, are based on physical and physico-chemical concepts such as ion condensation, phase transition, crowding, liquid crystal immiscibility, collective vibrational modes, reptation, and water availability. I draw on ideas from subjects such as the ‘prebiotic ecology’ and phenotypic diversity to help with the hunt. Given the fundamental nature of the snark, I would expect that its capture would make sense of other parts of biology. The route, therefore, followed by the hunt has involved trying to answer questions like “why do cells replicate their DNA?”, “why is DNA replication semi-conservative?”, “why is DNA a double helix?”, “why do cells divide?”, “is cell division a spandrel?”, and “how are catabolism and anabolism balanced?”. Here, I propose some relatively unexplored, experimental approaches to testing snark-related hypotheses and, finally, I propose some possibly original ideas about DNA packing, about phase separations, and about computing with populations of virtual bacteria.

In this paper we extend the SuperHyperAlgebra, SuperHyperGraph, SuperHyperTopology, SuperHyperSoft Set, endowed with SuperHyperOperations, SuperHyperAxioms, and SuperHyperFunctions, to the most general form of structure, from our real world, called SuperHyperStructure in any field of knowledge. A practical application of the SuperHyperStructure is presented at the end. The prefix "Hyper" [Marty, 1934] stand for the codomain of the functions and operations to be P(H), or the powerset of the set H. While the prefix "Super" [Smarandache, 2016] stands for using the Pn(H), n = 2, or the n-th PowerSet of the Set H {because the set (or system) H (that may be a set of items, a company, institution, country, region, etc.) is organized in sub-systems, which in their turn are organized in sub-subsystems, and so on} in the domain and/or codomain of the functions and operations. [ABSTRACT FROM AUTHOR]

The Hv-structures defined on a set admits a partial order and has a numerous of applications in mathematics and other sciences. The weak axioms, which the Hv-structures satisfy, is the main reason that they can be used even as organized device. On the other side, in applications one needs small results, thus, the researchers ask for Hv-structures which are near to the classical structures. For this reason, we focus on the raised Hv-fields. [ABSTRACT FROM AUTHOR]

Fluid-induced vibration is a common phenomenon in fluid–structure interaction. A flow-induced vibrational energy harvester based on a corrugated hyperstructure bluff body which can improve energy collection efficiency under low wind speeds is proposed in this paper. CFD simulation of the proposed energy harvester was carried out with COMSOL Multiphysics. The flow field around the harvester and the output voltage in different flow velocities is discussed and validated with experiments. Simulation results show that the proposed harvester has an improved harvesting efficiency and higher output voltage. Experimental results show that the output voltage amplitude of the harvester increased by 189% under 2 m/s wind speed. [ABSTRACT FROM AUTHOR]

The hyperstructures have applications in mathematics and other sciences such as‎ ‎biology‎, ‎physics‎, ‎linguistics‎, ‎sociology‎, ‎to mention but a few‎. ‎For this‎, ‎mainly‎, ‎the‎ ‎largest class of the hyperstructures‎, ‎the $H_v$-structures‎, ‎is used‎, ‎which satisfy the‎ weak axioms where the non-empty intersection replaces the equality and they are‎ ‎straightly related to fuzzy set theory‎. ‎The fundamental relations connect the $H_v$-structures with the classical ones‎, ‎moreover‎, ‎they reveal new concepts as the $H_{v}$-fields‎. $H_{v}$-numbers are called the elements of an $H_v$-field and they are used in‎ ‎representation theory‎. ‎We introduce the raised finite $H_{v}$-fields‎, ‎and present some‎ ‎results and examples on $2\times 2$ representations on them‎.

In the last few decennium, the acquaintance between algebraic hyperstructures and fuzzy sets have been considered for its theoretical as well as applications in various fields. Fuzzy hyperstructures are fascinating research topic and substantial amount of researches have been undergoing as of now. As a consequence, fuzzy hyperlattice was introduced by Pengfei He, Xiaolong Xin. But the applications on fuzzy hyperlattice are not defined so far. By scrutinizing the fuzzy hyperlattice, our objective is to present some applications of fuzzy hyperlattice in Biological and Physical Sciences. We find the first and second applications in dihybrid cross of Drosophila melanogaster (housefly) and Pisum sativum (Peas), respectively. Third application is in particle Physics (Elementary particles interaction).

The purpose of this paper is to investigate a new connection between fuzzy sets and population genetics. More precisely, we consider the fuzzy set of alleles in a population, determine sequences of join spaces associated with each of its types, and investigate their fuzzy grades. [ABSTRACT FROM AUTHOR]

Fluid-induced vibration is a common phenomenon in fluid–structure interaction. A flow-induced vibrational energy harvester based on a corrugated hyperstructure bluff body which can improve energy collection efficiency under low wind speeds is proposed in this paper. CFD simulation of the proposed energy harvester was carried out with COMSOL Multiphysics. The flow field around the harvester and the output voltage in different flow velocities is discussed and validated with experiments. Simulation results show that the proposed harvester has an improved harvesting efficiency and higher output voltage. Experimental results show that the output voltage amplitude of the harvester increased by 189% under 2 m/s wind speed.

The primary aim of this paper is to investigate useful generalizations of the classical concept of action of a hyperstructure on a non-empty set. The main goal is to develop the theory of dynamical system to the theory of n-ary dynamical hypersystem. We also give some principal properties of an n-ary dynamical hypersystem. [ABSTRACT FROM AUTHOR]

The generation in a bacterial population of a diversity that is coherent with present and future environments is a fundamental problem. Here, we use modeling to investigate growth rate diversity. We show that the combination of (1) association of extended assemblies of macromolecules with the DNA strands and (2) the segregation of DNA strands during cell division allows cells to generate different patterns of growth rate diversity with little effect on the overall growth rate of the population and thereby constitutes an example of “order for free” on which evolution can act.

The generation in a bacterial population of a diversity that is coherent with present and future environments is a fundamental problem. Here, we use modeling to investigate growth rate diversity. We show that the combination of (1) association of extended assemblies of macromolecules with the DNA strands and (2) the segregation of DNA strands during cell division allows cells to generate different patterns of growth rate diversity with little effect on the overall growth rate of the population and thereby constitutes an example of "order for free" on which evolution can act. [ABSTRACT FROM AUTHOR]

The generation of a phenotypic diversity that is coherent across a bacterial population is a fundamental problem. We propose here that the DNA strand-specific segregation of certain nucleoid-associated proteins or NAPs results in these proteins being asymmetrically distributed to the daughter cells. We invoke a variety of mechanisms as responsible for this asymmetrical segregation including those based on differences between the leading and lagging strands, post-translational modifications, oligomerisation and association with membrane domains. [ABSTRACT FROM AUTHOR]

A brief excursus on the last results on Hyperstructures and their connections with Fuzzy Sets. At the end a calculation of the Fuzzy Grade of Hv-structures of order two.

The largest class of hyperstructures is the class of H v -structures. This is the class of hyperstructures where the equality is replaced by the non-empty intersection. This extremely large class can used to define several objects that they are not possible to be defined in the classical hypergroup theory. It is convenient, in applications, to use more axioms and conditions to restrict the research in smaller classes. In this direction, in the present paper we continue our study on H v -structures which have strong-inverse elements. More precisely we study the small finite cases.

In a series of papers, we have discussed higher structures in science in general, and developed a framework called hyperstructures for describing and working with higher structures. We discussed the philosophy behind higher structures and formulated a principle in six stages – the Hyperstructure Principle – for forming higher structures. In this paper, we will relate hyperstructures and the general principle to known mathematical structures. We also discuss how they may give rise to new mathematical structures and prepare a framework for a mathematical theory. [ABSTRACT FROM AUTHOR]

Higher structures occur and play an important role in all sciences and their applications. In a series of papers, we have developed a framework called Hyperstructures for describing and working with higher structures. The purpose of this paper is to describe and elaborate the philosophical ideas behind hyperstructures and structure formation in general and emphasize the key ideas of the Hyperstructure Program. [ABSTRACT FROM AUTHOR]

Hyperstructure is the generalized concept of algebraic structure. The study of it and its applications is extremely interesting, offering interesting results to one who is willing to look for structure. This paper deals with hyperstructures associated to the symmetric group S n . First, we define a new hyperoperation ⋆ on S n and study its properties. Next, we prove that (S n , ⋆) is a single-power cyclic hypergroup with finite period. Then, we determine the set of all automorphisms of (S n , ⋆) and prove that it is a finite group under the operation of function composition. Finally, we construct a sequence of join spaces on S 3 and find its fuzzy grade. [ABSTRACT FROM AUTHOR]

In this paper, first we introduce the categories of "n-ary hyperstructures" (HSn) and "n-ary hyperstructures with particular carriers and relations" (HSPn), and we prove that the categories of HSn and HSPn-1 are isomorphic. Thenwe prove that the category of "(n - 1)-ary hypergroupoids" (HGn-1) is a full subcategory of the category HSn and the category of "unary binding n-ary algebra with particular properties" (UBAn) is a full subcategory of the category HSPn. Next, we define two functors F and G, and show that the restriction of the functor F to the category HGn is an isomorphism of HGn onto UBAn, and the restriction of the functor G to the category UBAn is an isomorphism of UBAn onto HGn. [ABSTRACT FROM AUTHOR]

A paradigm shift in one field can trigger paradigm shifts in other fields. This is illustrated by the paradigm shifts that have occurred in bacterial physiology following the discoveries that bacteria are not unstructured, that the bacterial cell cycle is not controlled by the dynamics of peptidoglycan, and that the growth rates of bacteria in the same steady-state population are not at all the same. These paradigm shifts are having an effect on longstanding hypotheses about the regulation of the bacterial cell cycle, which appear increasingly to be inadequate. I argue that, just as one earthquake can trigger others, an imminent paradigm shift in the regulation of the bacterial cell cycle will have repercussions or “paradigm quakes” on hypotheses about the origins of life and about the regulation of the eukaryotic cell cycle.

Hyperstructure theory has many applications to several areas of pure and applied sciences. This paper investigates some properties of hyper B -algebras which is a generalization of B -algebras. This paper also introduces the notion of hyper B -ideals, weak hyper B -ideals and strong hyper B -ideals in hyper B - algebras and gives some relations among these hyper B -ideals. Relations between hyper B -ideals and subhyper B -algebras of H is also discussed. Moreover, homomorphism on hyper B -algebra is defined and some related properties are given. Finally, proof of the relationship between the hyper B -algebras and hypergroups is provided. [ABSTRACT FROM AUTHOR]

After introducing the definition of hyperring by Krasner, the study of hyperrings has been of great importance. In this paper, we define a new hyperoperation associated to the set G of all arithmetic functions and give a complete analysis of its properties. Also, we combine our hyperoperation (+) with an already defined hyperoperation (⋅) and define ( G , + , ⋅ ) in which we study its properties and the possibility of containing hyperring. First, we prove that ( G , + , ⋅ ) is a weak hyperring. Then we find the largest hyperring contained in ( G , + , ⋅ ) . [ABSTRACT FROM AUTHOR]

After introducing the definition of hypergroups by Marty, the study of hyperstructures and its applications has been of great importance. In this paper, we find a link between hyperstructures and the infinite non-abelian group, braid group . This is the first connection to be done between these two different domains. First, we define a new hyperoperation ⋆ associated to and study its properties. Next, we prove that is a single-power cyclic hypergroup with infinite period. Then, we define an onto homomorphism from to another hypergroup. Finally, we determine the set of all automorphisms of and prove that it is a group under the operation of functions composition. [ABSTRACT FROM AUTHOR]

There are several ways that our species might try to send a message to another species separated from us by space and/or time. Synthetic biology might be used to write an epitaph to our species, or simply “Kilroy was here”, in the genome of a bacterium via the patterns of either (1) the codons to exploit Life's non-equilibrium character or (2) the bases themselves to exploit Life's quasi-equilibrium character. We suggest here how DNA movies might be designed using such patterns. We also suggest that a search for mechanisms to create and preserve such patterns might lead to a better understanding of modern cells. Finally, we argue that the cutting-edge microbiology and synthetic biology needed for the Kilroy project would put origin-of-life studies in the vanguard of research.

In the “ecosystems-first” approach to the origins of life, networks of non-covalent assemblies of molecules (composomes), rather than individual protocells, evolved under the constraints of molecular complementarity. Composomes evolved into the hyperstructures of modern bacteria. We extend the ecosystems-first approach to explain the origin of eukaryotic cells through the integration of mixed populations of bacteria. We suggest that mutualism and symbiosis resulted in cellular mergers entailing the loss of redundant hyperstructures, the uncoupling of transcription and translation, and the emergence of introns and multiple chromosomes. Molecular complementarity also facilitated integration of bacterial hyperstructures to perform cytoskeletal and movement functions.

The problem of not only how but also why cells divide can be tackled using recent ideas. One idea from the origins of life – Life as independent of its constituents – is that a living entity like a cell is a particular pattern of connectivity between its constituents. This means that if the growing cell were just to get bigger the average connectivity between its constituents per unit mass – its cellular connectivity – would decrease and the cell would lose its identity. The solution is division which restores connectivity. The corollary is that the cell senses decreasing cellular connectivity and uses this information to trigger division. A second idea from phenotypic diversity – Life on the Scales of Equilibria – is that a bacterium must find strategies that allow it to both survive and grow. This means that it has learnt to reconcile the opposing constraints that these strategies impose. The solution is that the cell cycle generates daughter cells with different phenotypes based on sufficiently complex equilibrium and non-equilibrium cellular compounds and structures appropriate for survival and growth, respectively, alias `hyperstructures'. The corollary is that the cell senses both the quantity of equilibrium material and the intensity of use of non-equilibrium material and then uses this information to trigger the cell cycle. A third idea from artificial intelligence – Competitive Coherence – is that a cell selects the active subset of elements that actively determine its phenotype from a much larger set of available elements. This means that the selection of an active subset of a specific size and composition must be done so as to generate both a coherent cell state, in which the cell’s contents work together harmoniously, and a coherent sequence of cell states, each coherent with respect to itself and to an unpredictable environment. The solution is the use of a range of mechanisms ranging from hyperstructure dynamics to the cell cycle itself.

Hv-structures are hyperstructures where the equality is replaced by the nonempty intersection. Therefore, this class of the hyperstructures is very large, one can use it in order to define several objects that they are not possible to be defined in the classical hypergroup theory. In the present paper we focus on the large classes of P-hyperstructures and ∂-hyperstructures. We introduce and study a property on inverse elements. [ABSTRACT FROM AUTHOR]

Hypergroups first were introduced by Marty in 1934. Up to now many researchers have been working on this field of modern algebra and developed it. It is purpose of this paper to provide examples of hypergroups associated with chemistry. The examples presented are connected to construction from chain reactions.

We show that on an arbitrary collection of objects there is a wide variety of higher order architectures governed by hyperstructures. Higher order gluing, local to global processes, fusion of collections, bridges and higher order types are discussed. We think that these types of architectures may have interesting applications in many areas of science. [ABSTRACT FROM AUTHOR]

In this paper, the notion of fuzzy Krasner(m, n)-hyperrings (F(m,n)-hyperrings) by using the notion of Fm-hyperoperations and F(m,n)-operations is introduced and some related properties are investigated. In this regards, relationships between Krasner F(m,n)-hyperrings and Krasner F(m,n)-hyperrings are considered. We shall prove that every Krasner F(m,n)-hyperring is extended by a Krasner F(2,n)-hyperring. The concepts of normal F-hyperideals and homomorphisms of Krasner F(m,n)-hyperrings are adopted. Also, the quotient of Krasner F(m,n)-hyperrings by defining regular relations are studied. Finally, the classical isomorphism theorems of groups are generalized to Krasner F(m,n)-hyperrings provided the F-hyperideals considered in them are normal. [ABSTRACT FROM AUTHOR]

Background: Fundamental problems faced by the protocells and their modern descendants include how to go from one phenotypic state to another; escape from a basin of attraction in the space of phenotypes; reconcile conflicting growth and survival strategies (and thereby live on ‘the scales of equilibria’); and create a coherent, reproducible phenotype from a multitude of constituents. Presentation of the hypothesis: The solutions to these problems are likely to be found with the organic and inorganic molecules and inorganic ions that constituted protocells, which we term SUMIs for Simple Universal Molecules and Ions. These SUMIs probably included polyphosphate (PolyP) as a source of energy and of phosphate; poly-(R)-3-hydroxybutyrate (PHB) as a source of carbon and as a transporter in association with PolyP; polyamines as a source of nitrogen; lipids as precursors of membranes; as well as peptides, nucleic acids, and calcium. Here, we explore the hypothesis that the direct interactions between PHB, PolyP, polyamines and lipids – modulated by calcium – played a central role in solving the fundamental problems faced by early and modern cells. Testing the hypothesis: We review evidence that SUMIs (1) were abundant and available to protocells; (2) are widespread in modern cells; (3) interact with one another and other cellular constituents to create structures with new functions surprisingly similar to those of proteins and RNA; (4) are essential to creating coherent phenotypes in modern bacteria. SUMIs are therefore natural candidates for reducing the immensity of phenotype space and making the transition from a “primordial soup” to living cells. Implications of the hypothesis: We discuss the relevance of the SUMIs and their interactions to the ideas of molecular complementarity, composomes (molecular aggregates with hereditary properties based on molecular complementarity), and a prebiotic ecology of co-evolving populations of composomes. In particular, we propose that SUMIs might limit the initial phenotype space of composomes in a coherent way. As examples, we propose that acidocalcisomes arose from interactions and self-selection among SUMIs and that the phosphorylation of proteins in modern cells had its origin in the covalent modification of proteins by PHB. [ABSTRACT FROM AUTHOR]

One approach to answering the question of what properties of life are universal is to try to answer the question of what are the essential properties of biology's best understood model organism, Escherichia coli. One of these properties is competitive coherence whereby E. coli reconciles the generation of a coherent cell state with the generation of a coherent sequence of cell states. The second property is differentiation which occurs ineluctably when E. coli divides. The third property is dualism which is how E. coli navigates between the two main attractors of phenotypes - survival and growth - which are based on quasi-equilibrium and non-equilibrium structures, respectively. The fourth property is complementarity: the interactions between the molecules and macromolecules that constitute E. coli protect them from degradation and confer new properties. The fifth property is multi-scale existence: E. coli exists at levels extending from the bacterium to the global super-organism. The sixth property is maintenance of connectivity; growth alters connectivity and, in the case of E. coli, alters the phenotype. The seventh property is the combination of intensity sensing (the constituents can work no harder) and quantity sensing (too much unused material has been made); this combination is used by E. coli to drive its cell cycle and thereby generate an environmentally adapted population of cells. The eighth property is subjective experience which exists even at the level of a single E. coli but which only becomes important at higher levels of organisation. I propose that the search for life at other times and in other places be based on the above eight universal properties and be independent of both particular substances and spatio-temporal scales. [ABSTRACT FROM AUTHOR]

In this paper, the notion of fuzzy n-hypergroups (Fn-hypergroups) by using the notion of Fn-hyperoperations is introduced and some related properties are investigated. In this regards relationships between Fn-hypergroups and n-hypergroups are considered. Next, some properties of direct product of Fn-hypergroups are presented. We then study the quotient of Fn-hypergroups by a fundamental relation and describe some of their properties. [ABSTRACT FROM AUTHOR]

The notion of intuitionistic fuzzy sets was introduced by Atanassov as a generalization of the notion of fuzzy sets. In this paper, we apply this concept to Γ - semihypergroups and we introduce the notion of an intuitionistic fuzzy Γ - hyperideal of a Γ - semihypergroup. Then, we investigate some properties connected with intuitionistic fuzzy Γ - hyperideals in a Γ - semihypergroup. [ABSTRACT FROM AUTHOR]

We discuss the nature of structure and organization and the process of making new things. Hyperstructures are introduced as binding and organizing principles, and we show how they can transfer from one situation to another. A guiding example is the hyperstructure of higher order Brunnian rings and similarly structured many-body systems. [ABSTRACT FROM AUTHOR]