Modern mathematical sciences are hard to imagine without appeal to efficient computational algorithms. We address several conceptual problems arising from this interaction by outlining rival but complementary perspectives on mathematical tractability. More specifically, we articulate three alternative characterizations of the complexity hierarchy of mathematical problems that are themselves based on different understandings of computational constraints. These distinctions resolve the tension between epistemic contexts in which exact solutions can be found and the ones in which they cannot; however, contrary to a (...) persistent myth, we conclude that having an exact solution is not generally more epistemologically beneficial than lacking one. (shrink)

The present collection is the very first contribution of this type in the field of sparse recovery. Compressed sensing is one of the important facets of the broader concept presented in the book, which by now has made connections with other branches such as mathematical imaging, inverse problems, numerical analysis and simulation. The book consists of four lecture notes of courses given at the Summer School on "Theoretical Foundations and Numerical Methods for Sparse Recovery" held at the (...) Johann Radon Institute for Computational and Applied Mathematics in Linz, Austria, in September 2009. This unique collection will be of value for a broad community and may serve as a textbook for graduate courses. (shrink)

In this paper, DSEK model with fractional derivatives of the Atangana-Baleanu Caputo is proposed. This paper gives a brief overview of the ABC fractional derivative and its attributes. Fixed point theory has been used to establish the uniqueness and existence of solutions for the fractional DSEK model. According to this theory, we will define two operators based on Lipschitzian and prove that they are contraction mapping and relatively compact. Ulam-Hyers stability theorem is implemented to prove the fractional DSEK model’s stability (...) in Banach space. Also, fractional Euler’s numerical method is derived for initial value problems with ABC fractional derivative and implemented on fractional DSEK model. The symmetric properties contribute to determining the appropriate method for finding the correct solution to fractional differential equations. The numerical solutions generated using fractional Euler’s method have been plotted for different values of α where α ∈ 0,1 and different step sizes h. Result discussion will be given, describing the changes that occur due to the step size h. (shrink)

An important task in mathematical sciences is to make quantitative predictions, which is often done via the solution of differential equations. In this paper, we investigate why, to perform this task, scientists sometimes choose to use numerical methods instead of analytical solutions. Via several examples, we argue that the choice for numerical methods can be explained by the fact that, while making quantitative predictions seems at first glance to be facilitated by analytical solutions, this is actually (...) often much easier with numerical methods. Thus we challenge the widely presumed superiority of analytical solutions over numerical methods. (shrink)

In this study, we implemented a new numerical method known as the Chebyshev Pseudospectral method for solving nonlinear delay differential equations having fractional order. The fractional derivative is defined in Caputo manner. The proposed method is simple, effective, and straightforward as compared to other numerical techniques. To check the validity and accuracy of the proposed method, some illustrative examples are solved by using the present scenario. The obtained results have confirmed the greater accuracy than the modified Laguerre wavelet (...) method, the Chebyshev wavelet method, and the modified wavelet-based algorithm. Moreover, based on the novelty and scientific importance, the present method can be extended to solve other nonlinear fractional-order delay differential equations. (shrink)

In this work we report a simple way to assign a single numeric value in a three-dimensional space to a given nucleotide sequence. The method reported allows for theoretical comparisons of naturally occurring nucleotide sequences.

To solve fractional delay differential equation systems, the Laguerre Wavelets Method is presented and coupled with the steps method in this article. Caputo fractional derivative is used in the proposed technique. The results show that the current procedure is accurate and reliable. Different nonlinear systems have been solved, and the results have been compared to the exact solution and different methods. Furthermore, it is clear from the figures that the LWM error converges quickly when compared to other approaches. When (...) compared with the exact solution to other approaches, it is clear that LWM is more accurate and gets closer to the exact solution faster. Moreover, on the basis of the novelty and scientific importance, the present method can be extended to solve other nonlinear fractional-order delay differential equations. (shrink)

In this paper is examined the acceleration of convergence of alternative and non-alternative numerical and power series by means of an Euler-Abel type operator, that is defined earlier by G.A.Sorokin and I.Z.Milovanovic. Through this type of linear operator of generalized difference of sequence is achieved that alternative and non-alternative numerical and power series to be transformed into series with higher speed of convergence than the initial series. At the end of this paper is given the implementation of this (...) method through a numeric example. (shrink)

Phenomenological psychology has emphasized that experience as it is immediately "given" to the experiencing individual is an appropriate subject matter for psychological investigation. Consideration of the methodological implications of this stance suggests that certain text analytic and cluster analytic methods could be used to discern the identifying properties of different types of experience. We present results of a study in which textual analysis was used to identify recurrent properties of participants' verbal accounts of their experience, cluster analysis was used (...) to classify participants' accounts according to the similarity of their profiles of properties, and the resulting clusters were examined for their more or less characteristic prpoerties. Using these methods, three distinct types of experience of a Renaissance painting were identified and described. This demonstration of numerically aided phenomenological methods indicates the compatibility of rigorous and sensitive descriptions of experiential accounts. (shrink)

In this paper, we present an intensive investigation of the finite volume method compared to the finite difference methods. In order to show the main difference in the way of approaching the solution, we take the Burgers equation and the Buckley–Leverett equation as examples to simulate the previously mentioned methods. On the one hand, we simulate the results of the finite difference methods using the schemes of Lax–Friedrichs and Lax–Wendroff. On the other hand, we apply Godunov’s scheme (...) to simulate the results of the finite volume method. Moreover, we show how starting with a variational formulation of the problem, the finite element technique provides piecewise formulations of functions defined by a collection of grid data points, while the finite difference technique begins with a differential formulation of the problem and continues to discretize the derivatives. Finally, some graphical and numerical comparisons are provided to illustrate and corroborate the differences between these two main methods. (shrink)

ArgumentJohannes Kepler published hisAstronomia novain 1609, based upon a huge amount of computations. The aim of this paper is to show that Kepler's new astronomy was grounded on methods from numerical analysis. In his research he applied and improved methods that required iterative calculations, and he developed precompiled mathematical tables to solve the problems, including a transcendental equation. Kepler was aware of the shortcomings of his novel methods, and called for a new Apollonius to offer a (...) formal mathematical deduction. He was also in great need of computational power, and his friend and colleague, Wilhelm Schickard, constructed the first prototype of a true mechanical calculator, although it never came into regular use. The article concludes that Kepler's new astronomy was clearly backed up by numerical methods and embedded concepts and challenges of great importance for the future development of numerical analysis. (shrink)

Given an argumentation network with initial values to the arguments, we look for algorithms which can yield extensions compatible with such initial values. We find that the best way of tackling this problem is to offer an iteration formula that takes the initial values and the attack relation and iterates a sequence of intermediate values that eventually converges leading to an extension. The properties surrounding the application of the iteration formula and its connection with other numerical and non-numerical (...) techniques proposed by others are thoroughly investigated in this paper. (shrink)

In philosophical studies regarding mathematical models of dynamical systems, instability due to sensitive dependence on initial conditions, on the one side, and instability due to sensitive dependence on model structure, on the other, have by now been extensively discussed. Yet there is a third kind of instability, which by contrast has thus far been rather overlooked, that is also a challenge for model predictions about dynamical systems. This is the numerical instability due to the employment of numerical (...) class='Hi'>methods involving a discretization process, where discretization is required to solve the differential equations of dynamical systems on a computer. We argue that the criteria for numerical stability, as usually provided by numerical analysis textbooks, are insufficient, and, after mentioning the promising development of backward analysis, we discuss to what extent, in practice, numerical instability can be controlled or avoided. (shrink)

It has been known for long time that most of the existing cosmology models have singularity problem. Cosmological singularity has been a consequence of excessive symmetry of flow, such as “Hubble’s law”. More realistic one is suggested, based on Newtonian cosmology model but here we include the vertical-rotational effect of the whole Universe. We review a Riccati-type equation obtained by Nurgaliev, and solve the equation numerically with Mathematica. It is our hope that the new proposed method can be verified with (...) observation data. (shrink)

The first aim of the paper is to show that under certain conditions generative syntax can be made suitable for Montague semantics, based on his type logic. One of the conditions is to make branching in the so-called X-bar syntax strictly binary, This makes it possible to provide an adequate semantics for Noun Phrases by taking them as referring to sets of collections of sets of entities ( type <ett,t>) rather than to sets of sets of entities (ett).

It has been long known that a year after Schrödinger published his equation, Madelung also published a hydrodynamics version of Schrödinger equation. Quantum diffusion is studied via dissipative Madelung hydrodynamics. Initially the wave packet spreads ballistically, than passes for an instant through normal diffusion and later tends asymptotically to a sub‐diffusive law. In this paper we will review two different approaches, including Madelung hydrodynamics and also Bohm potential. Madelung formulation leads to diffusion interpretation, which after a generalization yields to Ermakov (...) equation. Since Ermakov equation cannot be solved analytically, then we try to find out its solution with Mathematica package. It is our hope that these methods can be verified and compared with experimental data. But we admit that more researches are needed to fill all the missing details. (shrink)

A new class of polynomials investigates the numerical solution of the fractional pantograph delay ordinary differential equations. These polynomials are equipped with an auxiliary unknown parameter a, which is obtained using the collocation and least-squares methods. In this study, the numerical solution of the fractional pantograph delay differential equation is displayed in the truncated series form. The upper bound of the solution as well as the error analysis and the rate of convergence theorem are also investigated in (...) this study. In five examples, the numerical results of the present method have been compared with other methods. For the first time, a -polynomials are used in this study to numerically solve delay equations, and accurate approximations have been displayed. (shrink)

An interpretive system for the IBM 704 computer permitting interpretation of the genetic pattern of a numeric symbioorganism as a game strategy has been developed. Selection for best performance in a simple game has been applied in a preliminary experiment. An objective method to measure the quality of a game played is described. The results presented in the article show a small but significant improvement of game quality during a period of 2300 generations.The general characteristics of the phenomena observed are (...) similar to those of preceding evolution experiments . However, a startling infection process caused by a parasite whose behaviour was influenced by the game competition is described. The parasite never developed an independent game strategy to any degree of efficiency and was entirely dependent on its host organism for game competitions with, and transmission of the infection to, uninfected hosts.The consequences of substituting the reproduction and mutation rules used in the preceding evolution experiments are discussed. Particularly, the use of rules applying the reproduction pattern ofDNA-molecules is considered, and the theoretic aspects of symbiogenetic evolution experiments based onDNA-norm are discussed. A few inferences concerning the origin and history of terrestrial life suggested by the results of the symbiogenesis experiments are presented.Ein interpretierendes System für die IBM 704 Rechenmaschine, dass Interpretation des genetischen Modells von einem numerischen Symbio-organismus wie eine Spiel Strategie, erlaubt, wurde entwickelt. Eine Forführung für Fähigkeit in einem einfachen Spiel wurde angewandt in einem preliminarisches Experiment. Eine objektive Methode zur Messung der Qualität des betreffenden Spiels wurde beschrieben. Die in dem Artikel angegebenen Resultate zeigen eine kleine, aber bedeutende Verbesserung der Spiel-Qualität während einr Periode von 2300 Generationen.Die allgemeinen Kennziechen von den beobachteten Phänomenen sind denen von vorhergehenden Evolutions-Experimenten ähnlich . Jedoch, ein überraschender Infektions-Prozess, verursacht durch einen Parasit, wessen Verhalten beinflusst wurde durch die Spiel-Konkurrenz, wurde beschrieben. Der Parasit entwickelte niemals eine unabhängige Spiel-Strategie zu irgendeinem Masze von Fähigkeit und war ganz abhängig von seinem Gastherr-Organismus für Spiel-Konkurrenz mit, und Überbringung von der Infektion auf, uninfektierte Gäste.Die Folgerungen von dem Ersetzen der Fortpflanzung und Mutations-Regeln , angewandt in vorhergehenden Evolutions-Experimente wurden besprochen. Besonders der Gebrauch von Regeln welche das FortpflanzungsModell von DNA-Molekülen anwenden wurde betrachtet, und die theorethischen Aspekte von Symbiogenetischen Evolutions-Experimenten basiert auf DNA-Norm wurden besprochen. Einige Schlussfolgerungen in Bezug auf die Entstehung und Geschichte von dem Erdenleben suggeriert durch die Resultate von den Symbiogenese-Experimenten wurden gezeigt. (shrink)

Learning programs that generalize from real‐world examples will have to deal with many different kinds of data. Continuous numeric data can cause problems for algorithms that search for examples with identical property values. These problems can be surmounted by categorizing the numeric data. However, this process has problems of its own. In this paper, we look at the need for categorizing numeric data and several methods for doing so. We concentrate on the use of generalization‐based memory, a memory organization (...) where actual examples are stored along with generalizations, which leads to a generalization‐based categorization algorithm. We also consider how to use a number heuristic, looking for gaps. These methods have been implemented in the UNIMEM computer system. Examples are presented of these algorithms categorizing data about the states of the United States. (shrink)

Pantographs are important devices on high-speed trains. When a train runs at a high speed, concave and convex parts of the train cause serious airflow disturbances and result in flow separation, eddy shedding, and breakdown. A strong fluctuation pressure field will be caused and transformed into aerodynamic noises. When high-speed trains reach 300 km/h, aerodynamic noises become the main noise source. Aerodynamic noises of pantographs occupy a large proportion in far-field aerodynamic noises of the whole train. Therefore, the problem of (...) aerodynamic noises for pantographs is outstanding among many aerodynamics problems. This paper applies Detached Eddy Simulation to conducting numerical simulations of flow fields around pantographs of high-speed trains which run in the open air. Time-domain characteristics, frequency-domain characteristics, and unsteady flow fields of aerodynamic noises for pantographs are obtained. The acoustic boundary element method is used to study noise radiation characteristics of pantographs. Results indicate that eddies with different rotation directions and different scales are in regions such as pantograph heads, hinge joints, bottom frames, and insulators, while larger eddies are on pantograph heads and bottom frames. These eddies affect fluctuation pressures of pantographs to form aerodynamic noise sources. Slide plates, pantograph heads, balance rods, insulators, bottom frames, and push rods are the main aerodynamic noise source of pantographs. Radiated energies of pantographs are mainly in mid-frequency and high-frequency bands. In high-frequency bands, the far-field aerodynamic noise of pantographs is mainly contributed by the pantograph head. Single-frequency noises are in the far-field aerodynamic noise of pantographs, where main frequencies are 293 Hz, 586 Hz, 880 Hz, and 1173 Hz. The farther the observed point is from the noise source, the faster the sound pressure attenuation will be. When the distance of two adjacent observed points is increased by double, the attenuation amplitude of sound pressure levels for pantographs is around 6.6 dB. (shrink)

The progress of Mathematics during the nineteenth century was characterised both by an enormous acquisition of new knowledge and by the attempts to introduce rigour in reasoning patterns and mathematical writing. Cauchy’s presentation of Mathematical Analysis was not immediately accepted, and many writers, though aware of that new style, did not use it in their own mathematical production. This paper is devoted to an episode of this sort that took place in Spain during the first half of the century: It (...) deals with the presentation of a method for numerically solving algebraic equations by José Mariano Vallejo, a late Spanish follower of the Enlightenment ideas, politician, writer, and mathematician who published it in the fourth (1840) edition of his book Compendio de Matemáticas Puras y Mistas, claiming to have discovered it on his own. Vallejo’s main achievement was to write down the whole procedure in a very careful way taking into account the different types of roots, although he paid little attention to questions such as convergence checks and the fulfilment of the hypotheses of Rolle’s Theorem. For sure this lack of mathematical care prevented Vallejo to occupy a place among the forerunners of Computational Algebra. (shrink)

In this paper, we investigate the numerical solution of the coupled fractional massive Thirring equation with the aid of He’s fractional complex transform. This study plays a significant aspect in the field of quantum physics, weakly nonlinear thrilling waves, and nonlinear optics. The main advantage of FCT is that it converts the fractional differential equation into its traditional parts and is also capable to handle the fractional order, whereas the homotopy perturbation method is employed to tackle the nonlinear terms (...) in the coupled fractional massive Thirring equation. An example is illustrated to present the efficiency and validity of the two-scale theory. The solutions are obtained in the form of series with simple and easy computations which confirm that the present approach is good in agreement and is easy to implement for such type of complex systems in science and engineering. (shrink)

What does it look like when a group of scientists set out to re-envision an entire field of biology in symbolic and formal terms? I analyze the founding and articulation of Numerical Taxonomy between 1950 and 1970, the period when it set out a radical new approach to classification and founded a tradition of mathematics in systematic biology. I argue that introducing mathematics in a comprehensive way also requires re-organizing the daily work of scientists in the field. Numerical (...) taxonomists sought to establish a mathematical method for classification that was universal to every type of organism, and I argue this intrinsically implicated them in a qualitative re-organization of the work of all systematists. I also discuss how Numerical Taxonomy’s re-organization of practice became entrenched across systematic biology even as opposing schools produced their own competing mathematical methods. In this way, the structure of the work process became more fundamental than the methodological theories that motivated it. (shrink)

This paper suggests a method to calculate a degree of validity for the proof of a statement which is derived from empirical statements by means of logic conclusions. The empirical statements are assumed not to be completely valid or their validity to be doubtful. The suggested rules are consistent with two-valued logic, yield decreasing validities with increasing number of applications of modus ponens and obey the law of the excluded middle. The actual calculation of validity values, the relation of the (...) suggested method to some truth tables of multi-valued logic and to fuzzy logic are discussed. (shrink)

Solid-state processing of metal material is a very complex physical and chemical process, which is coupled by a series of variations including heat transfer, momentum transfer, mass transfer, and phase change. Applying three-dimensional finite element numerical method to the simulation of solid-state processing can perform analysis of metal material’s forging processes before production trial production, can obtain their relevant information such as material flow law, temperature field, and strain field under the minimum physical test conditions, thereby predicting metal material’s (...) forming defects and improving their forging quality. On the basis of summarizing and analyzing previous research works, this paper expounded the current status and significance of solid-state processing of metal materials, elaborated the development background, current status, and future challenges of 3D finite element numerical simulation, introduced the discrimination method and free surface solution method of numerical simulation calculation, conducted finite element model’s geometric assumptions, material selection, element division, model establishment, parameter selection, and initial and boundary condition determination, and simulated and analyzed rheological casting, remelting heating, thixoforming, and rotary piercing processes of metal materials. The results show that the 3D finite element numerical method can not only simulate various processes of flow field, temperature field, stress field, and microstructure in solid-state processing but also can provide a reliable basis for effectively obtaining a reasonable description and finding a more optimized design plan for metal material processing in a short time, which plays an important role in understanding and analyzing solid metal forming process, controlling and optimizing process parameters, guiding and mastering rheological casting, and secondary heating and rotary piercing of metal materials. (shrink)

Much research effort has been directed in different physiological contexts towards describing realistic behaviors with differential equations. One observes obviously that more state-variables give the model more accuracy. Unfortunately, the computational cost involved is higher. A new algorithm is presented for simulating a model described by a system of differential equations in which efficiency may not be altered by its size. In order to do this, the method is based on a polynomial description of the state-variables' evolution and on a (...) computation distributed control. Evaluations and results performed with classical models like Fitzhugh Nagumo or Hodgkin Huxley, allow validation of the method and exhibits its potential to decrease the computational costs. (shrink)

We consider an approach to some philosophical problems that I call the Method of Conceptual Articulation: to recognize that a question may lack any determinate answer, and to re-engineer concepts so that the question acquires a definite answer in such a way as to serve the epistemic motivations behind the question. As a case study we examine “Galileo’s Paradox”, that the perfect square numbers seem to be at once as numerous as the whole numbers, by one-to-one correspondence, and yet less (...) numerous, being a proper subset. I argue that Cantor resolved this paradox by a method at least close to that proposed—not by discovering the true nature of cardinal number, but by articulating several useful and appealing extensions of number to the infinite. Galileo was right to suggest that the concept of relative size did not apply to the infinite, for the concept he possessed did not. Nor was Bolzano simply wrong to reject Hume’s Principle (that one-to-one correspondence implies equal number) in the infinitary case, in favor of Euclid’s Common Notion 5 (that the whole is greater than the part), for the concept of cardinal number (in the sense of “number of elements”) was not clearly defined for infinite collections. Order extension theorems now suggest that a theory of cardinality upholding Euclid’s principle instead of Hume’s is possible. Cantor’s refinements of number are not the only ones possible, and they appear to have been shaped by motivations and fruitfulness, for they evolved in discernible stages correlated with emerging applications and results. Galileo, Bolzano, and Cantor shared interests in the particulate analysis of the continuum and in physical applications. Cantor’s concepts proved fruitful for those pursuits. Finally, Gödel was mistaken to claim that Cantor’s concept of cardinality is forced on us; though Gödel gives an intuitively compelling argument, he ignores the fact that Euclid’s Common Notion is also intuitively compelling, and we are therefore forced to make a choice. The success of Cantor’s concept of cardinality lies not in its truth (for concepts are not true or false), nor its uniqueness (for it is not the only extension of number possible), but in its intuitive appeal, and most of all, its usefulness to the understanding. (shrink)

In the first half of this paper, we present a fragment of relational syllogisms named RELSYLL consisting of quantified statements with a special set of numerical quantifiers, and introduce a number of concepts that are useful for the later sections, including indirect reduction, quantifier transformations and equivalence of syllogisms. After determining the valid and invalid syllogisms in RELSYLL, we then introduce two Derivation Methods which can be used to derive valid relational syllogisms based on known valid simple syllogisms. (...) We also show that the two Methods are sound and complete for RELSYLL. In the second half of this paper, we discuss ways to extend the Derivation Methods, including the use of more valid syllogisms and the use of existential assumptions. In this way, we are able to derive more relational syllogisms that contain other types of non-classical quantifiers, including “only” and proportional quantifiers. Finally, we state and prove a proposition concerning the relationship between the two Methods. (shrink)

The focus of this paper is two-fold. First, similarities generated from six semantic models were compared to human ratings of paragraph similarity on two datasets—23 World Entertainment News Network paragraphs and 50 ABC newswire paragraphs. Contrary to findings on smaller textual units such as word associations (Griffiths, Tenenbaum, & Steyvers, 2007), our results suggest that when single paragraphs are compared, simple nonreductive models (word overlap and vector space) can provide better similarity estimates than more complex models (LSA, Topic Model, SpNMF, (...) and CSM). Second, various methods of corpus creation were explored to facilitate the semantic models’ similarity estimates. Removing numeric and single characters, and also truncating document length improved performance. Automated construction of smaller Wikipedia-based corpora proved to be very effective, even improving upon the performance of corpora that had been chosen for the domain. Model performance was further improved by augmenting corpora with dataset paragraphs. (shrink)

Building a meaningful model of biological regulatory network is usually done by specifying the components and their interactions, by guessing the values of parameters, by comparing the predicted behaviors to the observed ones, and by modifying in a trial-error process both architecture and parameters in order to reach an optimal fitness. We propose here a different approach to construct and analyze biological models avoiding the trial-error part, where structure and dynamics are represented as formal constraints. We apply the method to (...) Hopfield-like networks, a formalism often used in both neural and regulatory networks modeling. The aim is to characterize automatically the set of all models consistent with all the available knowledge. The available knowledge is formalized into formal constraints. The latter are compiled into Boolean formula in conjunctive normal form and then submitted to a Boolean satisfiability solver. This approach allows to formulate a wide range of queries, expressed in a high level language, and possibly integrating formalized intuitions. In order to explore its potential, we use it to find cycles for 3-nodes networks and to determine the flower morphogenesis regulatory network of Arabidopsis thaliana. Applications of this technique are numerous and concern the building of models from data as well as the design of biological networks possessing specified behaviors. (shrink)

This paper describes the investigations performed to better understand two-stage rotor speed matching in a contrarotating fan. In addition, this study develops a comprehensive measuring and communication system for a contrarotating fan using ZigBee network. The investigation method is based on three-dimensional RANS simulations; the RANS equations are solved by the numerical method in conjunction with a SST turbulence model. A wireless measurement system using big data method is first designed, and then a comparison is done with experimental measurements (...) to outline the capacity of the numerical method. The results show that when contrarotating fan worked under designed speed, performance of two-stages rotors could not be matched as the designed working condition was deviated. Rotor 1 had huge influences on flow rate characteristics of a contrarotating fan. Rotor 2 was influenced by flow rates significantly. Under large flow rate condition, the power capability of rotor 2 became very weak; under working small flow rate condition, overloading would take place to class II motor. In order to solve the performance mismatch between two stages of CRF under nondesigned working conditions, under small flow rate condition, the priority shall be given to increase of the speed of rotor 1, while the speed of rotor 2 shall be reduced appropriately; under large flow rate condition, the speed of rotor 1 shall be reduced and the speed of rotor 2 shall be increased at the same time. (shrink)

As a generalization of both single-valued neutrosophic element and hesitant fuzzy element, single-valued neutrosophic hesitant fuzzy element is an efficient tool for describing uncertain and imprecise information. Thus, it is of great significance to deal with single-valued neutrosophic hesitant fuzzy information for many practical problems. In this paper, we study the aggregation of SVNHFEs based on some normalized operations from geometric viewpoint. Firstly, two normalized operations are defined for processing SVNHFEs. Then, a series of normalized aggregation operators which fulfill some (...) basic conditions of a valid aggregation operator are proposed. Additionally, a decision-making method is developed for resolving multiattribute decision-making problems based on the proposed operators. Finally, a numerical example is provided to illustrate the feasibility and effectiveness of the method. (shrink)

Organic evolution, or change, among the diatoms has proceeded with considerable regularity. The origins, the extinctions, and the increases in numbers of species and genera, on the whole, have submitted to law and order, as rules-within-limits . The changes of evolution have followed from two sorts of phenomena — 1) the origin and extinction of shortlived or unstable species, in a proportion which has been more or less constant, but different in the two groups of diatoms, Centricae and Pennatae; these (...) short-lived species died out in the subperiod of their origin even under favourable enough environment or surrounding conditions, because of the innate instability of these particular species: and 2) the origin of new species which demonstrated their innate capacity for permanence by remaining in the living condition from their origins mostly up to the present time. The continued or continual additions of new and long-living species, in proportion to the number available as possible and parental species, has given the geometrical progressions.Consequently evolution in the diatoms has been monopodial, or one-stemmed, and not sympodial or not with many substitutions of one species after another in successions. The two kinds of species-durations are well-known in other classes of plants and animals. Exactly the same processes of additions, subtractions, and multiplications, particularly as found in the centric diatoms, are very probable in these classes of plants and animals which have given the same answer or arithmetical result for the whole of their previous histories, in the form of the frequency distribution of the present sizes of their genera.The rules-within-limits of this basic evolution are described, defined, and discussed in their relations to various theories of evolution which perhaps only concern the working of factors of superimposed modifications that reduce the results of the pattern or plan of basic evolution.Die phylogenetische Entwicklung wurde im Bereich der Kieselalgen von bestimmten Gesetzmässigkeiten beherrscht. Die Entstehung neuer Arten und Gattungen, ihr Wiederverschwinden und die Zunahme des Formenreichtums, resultierend aus der Zahl der neuentstandenen, vermindert um die der wieder verschwindenen Arten und Gattungen und vollzog sich nach Regeln , die durch folgende Phänomene gekennzeichnet sind: Zwischen der Zahl der neuentstehenden und der Zahl der in der gleichen geologischen Subperiode aus inneren Gründen wieder aussterbenden Arten besteht eine mehr oder wenig feste Beziehung. Diese ist für die Centricae und Pennatae verschieden. Ein bestimmter Anteil der neuentstandenen Arten und Gattungen erwies sich aus inneren Gründen als „ausdauernd” und blieb bis auf den heutigen Tag erhalten. Die Zunahme der Arten- Zahl durch das Hinzukommen langlebiger Formen war stetig und vollzog sich nach Massgabe geometrischer Progressionen.Diese Tatsachen führen zu dem Schluss, dass die phylogenetische Entwicklung der Kieselalgen nach monopodialem, und nicht nach sympodialem Muster vorsichging. Auch in andern Pflanzengruppen und im Tierreich durften die gleichen Gesetzmässigkeiten herrschend sein. Am klarsten gelangen sie bei den Centricae zum, Ausdruck und werden vom Autor als das Fazit von Additions-, Subtraktions-, und Multiplikationsprozessen aufgefasst. Die heutige Situation bezüglich Häufigkeit der Grössen der Gattungen, in Zahl der Arten, ist das Endergebnis der summativen Wirkung dieser Einzelprozesse.Diese als ‚rules-within-limits’ bezeichneten Gesetzmässigkeiten werden im einzelnen beschrieben, erklärt und zusammenfassend dargestellt. Zu andern Evolutionstheorien in Beziehung gebracht, werden sie schliesslich für Erörterung des Problems benutzt, inwieweit noch andere evolutionsgeschichtliche Prozesse als übergeordnete Faktoren in den vom Verfasser dargestellten Grundplan des Evolutionsgeschehens eingreifen können.L'évolution organique ou le changement parmi les Diatomées s'est effectué avec une régularité considérable. Les origines et les extinctions, et les accroissements des nombres des espèces et des genres, se sont soumis en tout aux lois et aux ordres, commes des régularités à l'intérieur de légères déviations. Le changement de l'évolution fut la conséquence de deux sortes de phénomène: l'origine et l'extinction des espèces éphémères dans une proportion plus ou moins déterminée mais différente pour les deux classes des Diatomées, Centricae et Pennatae; ces espèces éphémères moururent dans la subpériode de leur origines malgré le caractère assez favorable des circonstances environnantes, à cause de l'évanescence inhérente aux espèces particulières, et l'origine des espèces nouvelles qui démontrèrent leur qualité innée de permanence tout en continuant de vivre depuis leurs origines, pour la plupart, jusqu'à présent. Les additions continuelles d'espèces nouvelles et permanentes, en nombres proportionnés aux nombres disponibles comme espèces parentes et possibles, donnèrent des accroissements par progressions géométriques.Par conséquent, l'évolution parmi les Diatomées fut monopodiale et non pas sympodiale. Les deux sortes d'espèces-durées sont bien connues parmi beaucoup d'autres classes de plantes et d'animaux. Les mêmes procédés d'additions, de soustractions et de multiplications, particulièrement tels qu'on les trouve dans les Centricae, sont très probables pour ces classes de plantes et d'animaux qui ont donné un résultat le plus semblable pour la somme totale de leur histoire précédente, sous la forme de la répartition des fréquences de leurs grandeurs générique actuelles.Les régularités à l'intérieur de légères déviations de cette évolution fondamentale sont détaillées, définies et discutées dans leurs relations aux diverses théories touchant les autres méthodes de l'évolution qui jouent le rôle de facteurs modifiants superposés au dessin général ou au dessein de l'évolution fondamentale. (shrink)