Добірка наукової літератури з теми "Theorem about the ideal partitions"

We consider partitions of the positive integernwhose parts satisfy the following condition. For a given sequence of non-negative numbers {bk}k≥1, a part of sizekappears in exactlybkpossible types. Assuming that a weighted partition is selected uniformly at random from the set of all such partitions, we study the asymptotic behaviour of the largest partXn. LetD(s)=∑k=1∞bkk−s,s=σ+iy, be the Dirichlet generating series of the weightsbk. Under certain fairly general assumptions, Meinardus (1954) obtained the asymptotic of the total number of such partitions asn→∞. Using the Meinardus scheme of conditions, we prove thatXn, appropriately normalized, converges weakly to a random variable having Gumbel distribution (i.e., its distribution function equalse−e−t, −∞<t<∞). This limit theorem extends some known results on particular types of partitions and on the Bose–Einstein model of ideal gas.

Contents. Coding and processing large information content actualizes the problem of formalization of interdependence between information parameters of vector data coding systems on a single mathematical platform. Objective. The formalization of relationships between information parameters of vector data coding systems in the optimized basis of toroidal coordinate systems with the achievement of a favorable compromise between contradictory goals. Method. The method involves the establishing harmonious mutual penetration of symmetry and asymmetry as the remarkable property of real space, which allows use decoded information for forming the mathematical principle relating to the optimal placement of structural elements in spatially or temporally distributed systems, using novel designs based on the concept of Ideal Ring Bundles (IRB)s. IRBs are cyclic sequences of positive integers which dividing a symmetric sphere about center of the symmetry. The sums of connected sub-sequences of an IRB enumerate the set of partitions of a sphere exactly R times. Two-and multidimensional IRBs, namely the “Glory to Ukraine Stars”, are sets of t-dimensional vectors, each of them as well as all modular sums of them enumerate the set node points grid of toroid coordinate system with the corresponding sizes and dimensionality exactly R times. Moreover, we require each indexed vector data “category-attribute” mutually uniquely corresponds to the point with the eponymous set of the coordinate system. Besides, a combination of binary code with vector weight discharges of the database is allowed, and the set of all values of indexed vector data sets are the same that a set of numerical values. The underlying mathematical principle relates to the optimal placement of structural elements in spatially and/or temporally distributed systems, using novel designs based on tdimensional “star” combinatorial configurations, including the appropriate algebraic theory of cyclic groups, number theory, modular arithmetic, and IRB geometric transformations. Results. The relationship of vector code information parameters (capacity, code size, dimensionality, number of encodingvectors) with geometric parameters of the coordinate system (dimension, dimensionality, and grid sizes), and vector data characteristic (number of attributes and number of categories, entity-attribute-value size list) have been formalized. The formula system is derived as a functional dependency between the above parameters, which allows achieving a favorable compromise between the contradictory goals (for example, the performance and reliability of the coding method). Theorem with corresponding corollaries about the maximum vector code size of conversion methods for t-dimensional indexed data sets “category-attribute” proved. Theoretically, the existence of an infinitely large number of minimized basis, which give rise to numerous varieties of multidimensional “star” coordinate systems, which can find practical application in modern and future multidimensional information technologies, substantiated. Conclusions. The formalization provides, essentially, a new conceptual model of information systems for optimal coding and processing of big vector data, using novel design based on the remarkable properties and structural perfection of the “Glory to Ukraine Stars” combinatorial configurations. Moreover, the optimization has been embedded in the underlying combinatorial models. The favorable qualities of the combinatorial structures can be applied to vector data coded design of multidimensional signals, signal compression and reconstruction for communications and radar, and other areas to which the GUS-model can be useful. There are many opportunities to apply them to numerous branches of sciences and advanced systems engineering, including information technologies under the toroidal coordinate systems. A perfection, harmony and beauty exists not only in the abstract models but in the real world also.

In this paper, we generalize the notion of distributivity and consider some properties of distributive ideals, that is, ideals I such that the algebra P(κ)/I is distributive in our sense.Our notation and terminology is explained in §1, while the main results of this paper begin in §2. We shall show here some relations of the distributivity and the ideal theoretic partitions. In §3, we shall study the class of distributive ideals over κ whose existence is equivalent to the ineffability of κ, and other classes. Finally, in §4, we shall consider the equivalence of the Boolean prime ideal theorem and show that the existence of certain distributive ideals characterizes several large cardinals. As a byproduct, we can give a simple proof of Ketonen's theorem that κ is strongly compact if and only if for any regular cardinal λ ≥ κ there exists a nontrivial κ-complete prime ideal over λ.

In this paper, we highlight two q-series identities arising from the "five guidelines" approach to enumerating lecture hall partitions and give direct, q-series proofs. This requires two new finite corollaries of a q-analog of Gauss's second theorem. In fact, the method reveals stronger results about lecture hall partitions and anti-lecture hall compositions that are only partially explained combinatorially.

It is a theorem of Rowbottom [12] that ifκis measurable andIis a normal prime ideal onκ, then for eachλ<κ,In this paper a natural structural property of ideals, distributivity, is considered and shown to be related to this and other ideal theoretic partition relations.The set theoretical terminology is standard (see [7]) and background results on the theory of ideals may be found in [5] and [8]. Throughoutκwill denote an uncountable regular cardinal, andIa proper, nonprincipal,κ-complete ideal onκ.NSκis the ideal of nonstationary subsets ofκ, andIκ= {X⊆κ∣∣X∣<κ}. IfA∈I+(=P(κ) −I), then anI-partitionofAis a maximal collectionW⊆,P(A) ∩I+so thatX∩ Y ∈IwheneverX, Y∈W, X≠Y. TheI-partitionWis said to be disjoint if distinct members ofWare disjoint, and in this case, fordenotes the unique member ofWcontainingξ. A sequence 〈Wα∣α<η} ofI-partitions ofAis said to be decreasing if wheneverα<β<ηandX∈Wβthere is aY∈Wαsuch thatX⊆Y. (i.e.,WβrefinesWα).

The theme of this paper is the generalization of theorems about partitions of the sets of points and lines of finite-dimensional Euclidean spaces ℝd to vector spaces over ℝ of arbitrary dimension and, more generally still, to arbitrary vector spaces over other fields so long as these fields are not too big. These theorems have their origins in the following striking theorem of Sierpiński [12] which appeared a half century ago.Sierpiński's Theorem. The Continuum Hypothesis is equivalent to: There is a partition {X, Y, Z} of ℝ3such that if ℓ is a line parallel to the x-axis [respectively: y-axis, z-axis] then X ∩ ℓ [respectively: Y ∩ ℓ, Z ∩ ℓ] is finite.The history of this theorem and some of its subsequent developments are discussed in the very interesting article by Simms [13]. Sierpiński's Theorem was generalized by Kuratowski [9] to partitions of ℝn+2 into n + 2 sets obtaining an equivalence with . The geometric character that Sierpiński's Theorem and its generalization by Kuratowski appear to have is bogus, since the lines parallel to coordinate axes are essentially combinatorial, rather than geometric, objects. The following version of Kuratowski's theorem emphasizes its combinatorial character.Kuratowski's Theorem. Let n < ω and A be any set. Then ∣A∣ ≤ ℵnif and only if there is a partition P: An+2 → n + 2 such that if i ≤ n + 1 and ℓ is a line parallel to the i-th coordinate axis, then {x ∈ ℓ: P(x) = i} is finite.

In this analysis, you think about the construct JD-fuzzy ideal of BH-Algebra and we can study a number of properties, theorem and that we can provide some examples then we gave the construct JD –fuzzy extension ideal.

This work is about the structure of the symbolic Rees algebra of the base ideal of a Cremona map. We give sufficient conditions under which this algebra has the "expected form" in some sense. The main theorem in this regard seemingly covers all previous results on the subject so far. The proof relies heavily on a criterion of birationality and the use of the so-called inversion factor of a Cremona map. A pretty long selection of examples of plane and space Cremona maps has been given to test against the conditions of the theorem, with special emphasis on Cohen–Macaulay base ideals.

Abstract.Let Γ be a totally ordered abelian group andIan order ideal in Γ. We prove a theorem which relates the structure of the Toeplitz algebraT(Γ) to the structure of the Toeplitz algebrasT(I) andT(Γ/I). We then describe the primitive ideal space of the Toeplitz algebraT(Γ) when the set Σ(Γ) of order ideals in Γ is well-ordered, and use this together with our structure theorem to deduce information about the ideal structure ofT(Γ) when 0→I→ Γ→ Γ/I→ 0 is a non-trivial group extension.

Метою роботи було дослідження способів доведення та узагальнення Теореми про ідеальні розбиття множин на підмножини, кількість елементів в якій кратна пяти та степеням п'ятірки. Розглянуто та описано деякі частинні випадки теореми, розроблено явні схеми ідеальних розбиттів, які утворюють повне покриття множини та не перетинаються між собою. Підраховано кількість етапів розбиттів та кількості ітерацій на кожному етапі. Виведено формул для пошуку загальної кількості етапів методом комбінування елементів в множинах. Підраховано кількість ітерацій користуючись методами комбінаторики. Створено схемі з вершинами та графами, щоб проілюструвати часткові випадки. За основу взято спосіб доведення для часткового випадку кратності 3 був показаний та використаний у моїй бакалаврській роботі. Об’єктом дослідження є теорема про ідеальні розбиття підмножин та її доведення в явному вигляді для випадку, коли кількість елементів множини є будь-яким числом. Предметом дослідження є доведення теореми в явному вигляді для множини з кількістю елементів різної кратності та виведення узагальнених формул.
The aim of the work was to study the methods of proving and generalizing the theorem on ideal divisions of subsets into sets, the number of elements in which is a multiple of five and powers of five, consideration and description of some partial cases of the theorem, development of explicit schemes of ideal partitions. among ourselves. Counting the number of stages of partitions and the number of iterations at each stage. Derivation of formulas for finding the total number of stages by combining elements in sets. Counting the number of iterations using combinatorics methods. Create diagrams with vertices and graphs to illustrate partial cases. The method of proof for the partial case of multiplicity 3 was shown and used in my bachelor's thesis. The object of research is the theorem on ideal partitioning of subsets and its explicit proof for the case when the number of elements of the set is any number. The subject of the research is to prove the theorem explicitly for a set with the number of elements of different multiplicity and to derive generalized formulas.

Relativity is the phenomenon that things relate to things: parents to their offspring; doubles to halves; larger things to smaller things. This book is about how ancient philosophers, particularly Plato, Aristotle, the Stoics, and Sextus Empiricus, understood this phenomenon and how their theories of relativity affected, and were affected by, their broader philosophical outlooks. Many scholars have thought that ancient thinkers were either fundamentally confused about the phenomenon of relativity, or held a view that is a trivial variation on a modern view. This book argues that neither is the case. In fact, ancient philosophers shared a close-knit family of views, referred to as ‘constitutive relativity’: a relative is not simply linked by a relation, but is constituted by it. The book shows that this view is present in Plato, and is exploited by him in some key arguments concerning the Forms and the partition of the soul. Aristotle adopts the constitutive view in his discussions of relativity in Categories 7 and the Topics, and retains the constitutive view in his later discussion in Metaphysics 5.15. The Relatives Argument of Aristotle’s lost work On Ideas also involves constitutive relativity. The book moves on to examine a complex report of Stoic relativity and the role relativity played in Stoic philosophy. Finally, the book discusses Sextus Empiricus’ way of thinking about relativity, which does not appeal to the nature of relatives, but rather to how we conceive of things as correlative.

The renormalization-group (RG) method discussed in this chapter has assumed a pivotal role in the modern theory of critical phenomena. It attempts to relate the partition function of a given system to that of a “similar system” with decreased degrees of freedom through a process referred to as renormalization. Exactly how these degrees of freedom are removed from the system, what we mean by a “similar” system, and how successive systems are coupled to one another are essentially the questions we take up in the introductory treatment given in this chapter. The RG method is a topic with large scope and found widely disseminated in an extensive physics literature on the topic; however, it is seldom found in engineering journals. Our purpose here is to try and make sense of some basic ideas with the RG approach so that it is more accessible to this wider community. For this we often rely upon some prior exposes of the subject in more specialized settings [1, 2, 3, 5]. In its complete sense, the RG method has only been made to work, at least analytically, for a few simple statistical-mechanical models. But aside from these numerical results, many important and quite general insights about critical phenomena can be developed from studying this approach to the problem, especially the central role played by length scale as a factor in describing the phenomenology. These ideas have significantly enhanced our understanding of ideas like scale invariance, universality classes, relevant scaling fields (as opposed to irrelevant ones), Hamiltonian renormalization, and so on; these and related concepts lie at the center of modern discourse on the subject. The essential concepts of the approach can be well illustrated using the Ising system since, with this model, lattice spins are fixed in space, which makes the analytical work quite transparent. This approach, called real space renormalization, is the RG method studied in this chapter.

TORA is a highly adaptive routing protocol that provides loop free routing of data packets in ad hoc network at every instant. Each node of network has the capability to contain information about its local topology changes and regular coordination with their neighbour nodes in order to detect partitions occurs in network. In Ad-Hoc networks, multicast communication is not ideal due to the frequent nodes mobility and failures. Such problems in transmission of data become reasons for increased delays and decreased throughput, and also sudden changes in delay, traffic, and link breakage occur due to nodes mobility. The network parameters like network size, number of nodes, number of hops per route, traffic patterns (bits/packets), link capacity, mobility trajectory, and frequency of sleeping nodes do have great results on the performance of routing protocols in the mobile ad hoc networks. TORA has three basic route functions: establishment, maintenance, and erasing. In TORA routing protocol, each node has capable to contain a structure of network describing node’s height and status of all connected links. The importance of these heights in TORA routing protocol are that a node in network may only forward data packets downstream instead of upstream. TORA are more appropriate for QOS in ad hoc networks with a higher mobility and data transfer rate.

This chapter develops the normative ideal of acknowledged freedom. I begin with an investigation of strategic interaction, coordination, and cooperation, using insights from game theory on common knowledge. Something is common knowledge among us if we know it, we know that we know it, and so on—that is, whenever it is entirely transparent among us. Using consumer behaviour as an example, I show how common knowledge facilitates coordination and collaboration, and offers protection to freedom. In that context, I discuss the no-trade theorem from economics, the idea of perspective change from the literature on privacy, as well as the role of common knowledge in Philip Pettit’s republicanism. I apply these insights to questions about stakeholder informed consent, the ethics of communication, as well as recent legislation offering consumer protection in finance such as the 2008 EU Consumer Credit Directive.