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Let R be a ring with no torsion. In the subsequent considerations, the universal formal group will play the role of a very general composition law admissible for the construction of the entropies of the Z -family. In this section, we shall first define the notion of composability, in the strict and weak sense, by following and slightly generalizing [ 23 ]. This axiom is necessary to ensure that a given entropy may be suitable for thermodynamic purposes.

Indeed, it should be obviously symmetric with respect to the exchange of the labels A and B. At the same time, if we compose a given system with another system in a state with zero entropy, the total entropy should coincide with that of the given system. Finally, the composability of more than two independent systems in an associative way is crucial to define a zeroth law.

The weak formulation essentially requires the composability of the generalized logarithm associated with a given entropy. The weak property is satisfied by almost all of the generalized entropies proposed in the literature. However, it is not obvious: indeed, there are entropic forms that satisfy the first three SK axioms, but are not weakly composable.

There is a simple construction allowing a generalized logarithm from a given group law to be defined. Needless to say, many other definitions of a generalized logarithm can be proposed. Observe that the eq. One of our key results is the following simple proposition, which is a restatement of the previous observation.

Let G be a continuous strictly increasing function, vanishing at zero.

The function F G x defined by. This result can also be formulated in a field of characteristic zero for G in the class of formal power series. Let G be a strictly increasing real analytic function of the form eq. Many other choices are allowed. Thus, given a group law, under mild hypotheses we may determine a generalized group logarithm, for instance by means of relation 4. The inverse of a generalized group logarithm will be called the associated generalized group exponential. We get the formal power series.

Let us denote by N the number of particles of a complex system. The equation for the multiplicative formal group allows us to give a two-parametric presentation of the logarithm 4. Indeed, we observe that the functional equation. This is coherent with the general theory of functional equations [ 29 ]. From this observation, we can also formulate a two-parametric presentation of the Tsallis entropy S q , physically equivalent to it, whose main interest in this context is that it is still strictly composable. Note that the entropies S a , q and S q are related by means of the simple formulae.

The entropy 5.

Let A and B be two independent systems. The entropy S a , q is strictly composable, with composition law given by. The entropy S a , q represents the general solution of the previous functional equation in the trace-form class. This entropy, therefore, belongs to the family of composable entropies. However, it is not of trace-form type. The Tsallis entropy with the Boltzmann entropy as its reduction is the only known entropy of the trace-form class which is strictly composable.

If we restrict ourselves to the trace-form class, composability is realized essentially in the case of the Tsallis entropy, with the Boltzmann entropy as its fundamental reduction. However, if we relax the trace-form requirement, new possibilities arise.

The following theorem establishes one of the most relevant properties of the Z -class of entropies: its strict composability.

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References

We have. The symmetry property is trivial. The null-composability comes from the relation. It is obtained by identifying the generalized logarithm with the standard one. A crucial property of the Z -class is that it is compatible with the SK requirements, as made clear by the following statement. We conclude that the Z -entropies are group entropies. We shall discuss the extensivity properties of the Z -family of entropies over the uniform probability distribution i.

In other words, we wish to determine the conditions ensuring that a Z -entropy is proportional to the number N of particles of a given system, in the large N limit, when all states are equiprobable. We have that, for N large and focusing on the leading constribution to the asymptotic behaviour. However, the function W N must be interpretable as a phase space growth rate. These requirements usually restrict the space of allowed parameters of the considered entropy.

Consequently, provided the previous condition is satisfied, the entropies of the Z -family are extensive on a specific regime, given by a growth rate W N. The previous properties altogether indicate the potential relevance of the notion of Z -entropy in thermodynamical contexts and, more generally, in the theory of complex systems. We recall here some basic facts about majorization [ 31 ]. Precisely, we have to prove that, for any probability distribution p ,.

The previous construction allows us to establish a correspondence among trace-form and non-trace-form entropies. Precisely, given a well-defined trace-form entropy, weakly composable , we can associate with it a new, non-trace-form, but strictly composable , generalized entropy. In this way, we can generate a tower of new examples of group entropies that parallel the well-known entropies, with infinitely many more in addition.

Types of Groups

Let us consider the first example of non-additive entropic function of the Z -class 1. The composition law for the entropy 8. The entropy 8. The associated logarithm is the Kaniadakis logarithm [ 34 ], defined by. A new element of the Z -family is therefore the following entropy. The strict composability of the entropy 8. This procedure can be easily extended to infinitely many group laws; correspondingly, it yields a sequence of entropic forms. Another particularly interesting case will be discussed below. In this section, we present a first example of a composable, three-parametric entropy.

Perhaps the most appealing feature of this entropy is that it generalizes some of the most important entropies known in the literature, which have played a prominent role in information theory, thermodynamics and generally speaking in applied mathematics. The Z a , b -entropy is defined to be the function. We also have the following notable properties. The Z a , b -entropy is related to the Borges—Roditi logarithm:. The Abel exponential 9. The composition laws associated are remarkable: the Abel formal group laws , defined by.

The ring over which the universal Abel formal group law can be defined has been studied in [ 37 , 36 ]; the cohomology theory associated with this formal group law has been studied in [ 37 ] and in [ 38 ]. The class of Z -entropies can be quantized by means of a standard procedure. For each specific entropy, the set of parameters appearing in equation We shall discuss now the example of the quantum version of the Z a , b -entropy. Similar considerations apply to other cases. One of the most interesting aspects of the family of quantum entropies Concerning the quantum Z a , b -entropy, it possesses other interesting limits.

For instance, it can be related to the quantum version of the Sharma—Mittal entropy. A priori , due to the multi-parametric nature of these entropies, they can be particularly useful in the study of entangled systems. An analysis of this aspect is outside the scope of this paper and will be discussed elsewhere.

As an example of application of the entropic forms presented in the previous sections, we shall consider the generalized isotropic Lipkin—Meshkov—Glick model of N interacting particles introduced in [ 27 ]. Its Hamiltonian reads. The ground state entanglement entropies for this model can be easily computed in the case of our group entropies, starting from the results of [ 27 ].

There, the reduced density matrix for a block of L spins was computed, when the system is in its ground state. In the large L limit, we obtain the formulae. The relevance of formula Observe that, instead, the entropy Z a ,0 can be made linear in L. The present work represents a first exploration of the mathematical properties of a new, large family of non-trace-form group entropies coming from formal group theory via the composability axiom.

The underlying group-theoretical structure is responsible for essentially all the relevant properties of this family. The results obtained above suggest that the Z -class can be a flexible tool, offering in a future perspective new insight into different contexts of the theory of classical and quantum complex systems, in ecology and social sciences. For instance, we plan to define generalized diversity indices related to the class. We wish to point out that some of the definitions of the theory can be reformulated in different ways, and some conditions can be relaxed.

The present approach has the advantage of providing a large class of group entropies in a simple way; however, the problem of determining the most general form for group entropies is open. We shall discuss in detail progress on these issues elsewhere. In our opinion, the crucial role played by the group-theoretical approach in the description of compound systems paves the way towards a re-foundation of the theory of generalized entropies in terms of group entropies.

An open problem is to establish whether the Z -entropies are Lesche stable. We conjecture that, in this respect, the entropies of the class 1. This conjecture will be thoroughly analysed in a future work. We also wish to mention that a generalization of the correspondence among entropies and zeta functions [ 22 , 25 , 26 ] to the case of multi-parametric entropies and multiple zeta values and polylogarithms is an interesting open problem. I thank J. Carrasco, A. Sicuro for useful discussions. As customary in the literature on generalized entropies, here we re-state the original requirements of Shannon and Khinchin for an entropy to be admissible, in terms of four requirements.

Europe PMC requires Javascript to function effectively. Recent Activity. A crucial aspect is that every Z-entropy is composable Tempesta Ann. The composability axiom, introduced as a generalization of the fourth Shannon-Khinchin axiom postulating additivity , is a highly non-trivial requirement.

However, in the non-trace form class, the Z-entropies arise as new entropic functions possessing the mathematical properties necessary for information-theoretical applications, in both classical and quantum contexts. The snippet could not be located in the article text.

This may be because the snippet appears in a figure legend, contains special characters or spans different sections of the article. Proc Math Phys Eng Sci. PMID: Piergiulio Tempesta 1, 2. Received Feb 24; Accepted Sep 8. Published by the Royal Society. All rights reserved. Keywords: generalized entropies, group theory, information theory. Introduction Since the pioneering work by Boltzmann, Clausius and Gibbs, the notion of entropy has been widely investigated for its prominence in thermodynamics and statistical mechanics, in both classical and quantum contexts [ 1 — 3 ].

Groups and entropies: a general approach In the subsequent sections, we shall present a comprehensive theory of generalized entropies based on formal group laws. Definition 2. The composability axiom In this section, we shall first define the notion of composability, in the strict and weak sense, by following and slightly generalizing [ 23 ]. Definition 3. Remark 3. Generalized logarithms and exponentials from group laws There is a simple construction allowing a generalized logarithm from a given group law to be defined. Definition 4.

Proposition 4. Remark 4.

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On the trace-form class of entropies a Basic properties Let us denote by N the number of particles of a complex system. Definition 5.

Formal groups 2 (Algebraic Number Theory IV)

Proposition 5. Remark 5. New strictly composable entropies a The Z -family If we restrict ourselves to the trace-form class, composability is realized essentially in the case of the Tsallis entropy, with the Boltzmann entropy as its fundamental reduction. Remark 6. Definition 6. Theorem 6.

Proposition 7. SK1 By construction, the function 1. SK2 The entropy 1. Definition 7. Theorem 7. A tower of new entropic forms The previous construction allows us to establish a correspondence among trace-form and non-trace-form entropies. Definition 8.

Definition 9. Proposition 9. Quantum Z -entropies The class of Z -entropies can be quantized by means of a standard procedure. Definition Open problems and future perspectives The present work represents a first exploration of the mathematical properties of a new, large family of non-trace-form group entropies coming from formal group theory via the composability axiom. Acknowledgements I thank J. Appendix A.


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The Shannon—Khinchin axioms As customary in the literature on generalized entropies, here we re-state the original requirements of Shannon and Khinchin for an entropy to be admissible, in terms of four requirements. SK1 Continuity. The function S p 1 ,…, p W is continuous with respect to all its arguments. SK2 Maximum principle. SK3 Expansibility. SK4 Additivity.

Footnotes 1 The torsion-free hypothesis was implicit also in [ 23 , 26 ]. Competing interests I have no competing interests. References 1. Thermodynamics of chaotic systems: an introduction. Callen HB.

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Formal groups are knowingly and systematically created. Spontaneously created. Informal groups are not really created, they are naturally formed. Systematic structure.

Visual Group Theory, Lecture 1.1: What is a group?

So there is a system in place. Loose structure. In informal groups, there is no structure at all. They just come together for some time. There is no junior and senior…everybody is equal. Importance to position. In a formal group, importance is always given to the position. Ex: the group leader, the head of department, etc. The position gets importance in a formal way because there is a system in place. Importance to the person. The beauty of being in an informal group is that the position does not exist at all because there is no structure.

So importance is always given to the person. Relationship is official. The relationship is very formal and official in a formal group. So the relationship and behavior is almost prescribed in a formal group. Relationship is personal. In an informal group, the interaction and the attachment becomes very personal and not official. Communication is restricted and slow.

Communication is very slow because a certain procedure has to be followed if we want to pass down the information. Communication is free and fast. Everybody wants to tell everyone all the information. With the gossip network in an informal group, information is passed very quickly. As human being, we tend to tilt towards informal groups. For example: If we remain informal all the time, the organizational goals cannot be met. When we interact with people in an informal scale, we develop good relationships with them. So this develops better working relationships and therefore creates better efficiency in the organization.

So, good organizations today are promoting informal activities and formation of informal groups. Ex: Parties in companies, which promotes informal interaction. Human dignity is something which is very important for every human being. As human beings, we want to be respected and treated in a particular way. So, self respect is very important for a human being. In a formal group setup, very often, our dignity is destroyed.

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So our dignity is destroyed. But what supports us?? The grapevine is the plant on which the grape fruit grows. The characteristic of a grapevine is that it grows in all directions. So in an organization, the grapevine is the informal communication network that has grown into every group and corner of the organization. If we want information to be passed on very fast to everybody in the organization, then the grapevine is the best way. Example 2: When director makes an announcement for a holiday, information is passed very quickly.

So the grapevine is active. And if a formal notice is made, then it takes time for the information to reach everybody because the formal network is very slow. You are commenting using your WordPress.