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MUTAFCHIEV, LJUBEN. "The Size of the Largest Part of Random Weighted Partitions of Large Integers." Combinatorics, Probability and Computing 22, no. 3 (February 21, 2013): 433–54. http://dx.doi.org/10.1017/s0963548313000047.
Повний текст джерелаАнотація:
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.
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Molteni, Giuseppe. "Recent results about the prime ideal theorem." Bollettino dell'Unione Matematica Italiana 10, no. 1 (May 31, 2016): 19–28. http://dx.doi.org/10.1007/s40574-016-0084-y.
Повний текст джерела
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Riznyk, V. V. "FORMALIZATION CODING METHODS OF INFORMATION UNDER TOROIDAL COORDINATE SYSTEMS." Radio Electronics, Computer Science, Control, no. 2 (July 7, 2021): 144–53. http://dx.doi.org/10.15588/1607-3274-2021-2-15.
Повний текст джерелаАнотація:
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.
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Kanai, Yasuo. "On a generalization of distributivity." Journal of Symbolic Logic 59, no. 3 (September 1994): 1055–67. http://dx.doi.org/10.2307/2275928.
Повний текст джерелаАнотація:
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.
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ANDREWS, GEORGE E., SYLVIE CORTEEL, and CARLA D. SAVAGE. "ON q-SERIES IDENTITIES ARISING FROM LECTURE HALL PARTITIONS." International Journal of Number Theory 05, no. 02 (March 2009): 327–37. http://dx.doi.org/10.1142/s1793042109002134.
Повний текст джерелаАнотація:
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.
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Johnson, C. A. "Distributive ideals and partition relations." Journal of Symbolic Logic 51, no. 3 (September 1986): 617–25. http://dx.doi.org/10.2307/2274018.
Повний текст джерелаАнотація:
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).
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Schmerl, James H. "Partitioning large vector spaces." Journal of Symbolic Logic 68, no. 4 (December 2003): 1171–80. http://dx.doi.org/10.2178/jsl/1067620179.
Повний текст джерелаАнотація:
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.
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Mhammed. E, Showq, and . "On JD-Fuzzy Ideal of BH-Algebra." International Journal of Engineering & Technology 7, no. 3.27 (August 15, 2018): 310. http://dx.doi.org/10.14419/ijet.v7i3.27.17896.
Повний текст джерелаАнотація:
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.
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Costa, Barbara, Aron Simis, and Zaqueu Ramos. "A theorem about Cremona maps and symbolic Rees algebras." International Journal of Algebra and Computation 24, no. 08 (December 2014): 1191–212. http://dx.doi.org/10.1142/s0218196714500532.
Повний текст джерелаАнотація:
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.
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ADJI, SRIWULAN, IAIN RAEBURN, and RIZKY ROSJANUARDI. "GROUP EXTENSIONS AND THE PRIMITIVE IDEAL SPACES OF TOEPLITZ ALGEBRAS." Glasgow Mathematical Journal 49, no. 1 (January 2007): 81–92. http://dx.doi.org/10.1017/s0017089507003436.
Повний текст джерелаАнотація:
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.
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Klausner, L. D., and T. Weinert. "THE POLARISED PARTITION RELATION FOR ORDER TYPES." Quarterly Journal of Mathematics 71, no. 3 (June 4, 2020): 823–42. http://dx.doi.org/10.1093/qmathj/haaa003.
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Abstract We analyse partitions of products with two ordered factors in two classes where both factors are countable or well-ordered and at least one of them is countable. This relates the partition properties of these products to cardinal characteristics of the continuum. We build on work by Erd̋s, Garti, Jones, Orr, Rado, Shelah and Szemerédi. In particular, we show that a theorem of Jones extends from the natural numbers to the rational ones, but consistently extends only to three further equimorphism classes of countable orderings. This is made possible by applying a 13-year-old theorem of Orr about embedding a given order into a sum of finite orders indexed over the given order.
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Widiastuti, Ratna Sari. "Radikal Prima Bi-Ideal Dalam Semiring Ternari." Jurnal Matematika 9, no. 2 (December 31, 2019): 78. http://dx.doi.org/10.24843/jmat.2019.v09.i02.p113.
Повний текст джерелаАнотація:
A ternary semiring is an additive commutative semigroup with a ternary multiplication which satisfying some condition. This paper will be discuss about prime bi-ideal radikal in semiring ternary with definition and some theorem. If B is a bi-ideal in semiring ternary T, then radical prime bi-ideal of B in semiring ternary T is an intersection of all prime bi-ideal in semiring ternary T which containing B.
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Raghavan, K. "Uniform annihilation of local cohomology and of Koszul homology." Mathematical Proceedings of the Cambridge Philosophical Society 112, no. 3 (November 1992): 487–94. http://dx.doi.org/10.1017/s0305004100071164.
Повний текст джерелаАнотація:
Let R be a ring (all rings considered here are commutative with identity and Noetherian), M a finitely generated R-module, and I an ideal of R. The jth local cohomology module of M with support in I is defined byIn this paper, we prove a uniform version of a theorem of Brodmann about annihilation of local cohomology modules. As a corollary of this, we deduce a generalization of a theorem of Hochster and Huneke about uniform annihilation of Koszul homology.
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Bigatti, Anna, Elisa Palezzato, and Michele Torielli. "Extremal behavior in sectional matrices." Journal of Algebra and Its Applications 18, no. 03 (March 2019): 1950041. http://dx.doi.org/10.1142/s0219498819500415.
Повний текст джерелаАнотація:
In this paper, we recall the object sectional matrix which encodes the Hilbert functions of successive hyperplane sections of a homogeneous ideal. We translate and/or reprove recent results in this language. Moreover, some new results are shown about their maximal growth, in particular, a new generalization of Gotzmann’s Persistence Theorem, the presence of a GCD for a truncation of the ideal, and applications to saturated ideals.
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PINTER, CHARLES C. "STONE SPACE OF CYLINDRIC ALGEBRAS AND TOPOLOGICAL MODEL SPACES." Journal of Symbolic Logic 81, no. 3 (September 2016): 1069–86. http://dx.doi.org/10.1017/jsl.2016.11.
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AbstractThe Stone representation theorem was a milestone for the understanding of Boolean algebras. From Stone’s theorem, every Boolean algebra is representable as a field of sets with a topological structure. By means of this, the structural elements of any Boolean algebra, as well as the relations between them, are represented geometrically and can be clearly visualized. It is no different for cylindric algebras: Suppose that ${\frak A}$ is a cylindric algebra and ${\cal S}$ is the Stone space of its Boolean part. (Among the elements of the Boolean part are the diagonal elements.) It is known that with nothing more than a family of equivalence relations on ${\cal S}$ to represent quantifiers, ${\cal S}$ represents the full cylindric structure just as the Stone space alone represents the Boolean structure. ${\cal S}$ with this structure is called a cylindric space.Many assertions about cylindric algebras can be stated in terms of elementary topological properties of ${\cal S}$. Moreover, points of ${\cal S}$ may be construed as models, and on that construal ${\cal S}$ is called a model space. Certain relations between points on this space turn out to be morphisms between models, and the space of models with these relations hints at the possibility of an “abstract” model theory. With these ideas, a point-set version of model theory is proposed, in the spirit of pointless topology or category theory, in which the central insight is to treat the semantic objects (models) homologously with the corresponding syntactic objects so they reside together in the same space.It is shown that there is a new, purely algebraic way of introducing constants in cylindric algebras, leading to a simplified proof of the representation theorem for locally finite cylindric algebras. Simple rich algebras emerge as homomorphic images of cylindric algebras. The topological version of this theorem is especially interesting: The Stone space of every locally finite cylindric algebra ${\frak A}$ can be partitioned into subspaces which are the Stone spaces of all the simple rich homomorphic images of ${\frak A}$. Each of these images completely determines a model of ${\frak A}$, and all denumerable models of ${\frak A}$ appear in this representation.The Stone space ${\cal S}$ of every cylindric algebra can likewise be partitioned into closed sets which are duals of all the types in ${\frak A}$. This fact yields new insights into miscellaneous results in the model theory of saturated models.
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Viktorovich, Pivnev Vitaliy, and Basan Sergey Nikolaevich. "Functionally Complete Elemental Basis of Mathematical Schemes of Electrical Circuits." Applied Mechanics and Materials 448-453 (October 2013): 2120–24. http://dx.doi.org/10.4028/www.scientific.net/amm.448-453.2120.
Повний текст джерелаАнотація:
The paper gives the definition of the mathematical electrical circuit scheme, the extended element basis, the minimum and full functional element basis. The theorem about functionality of the full minimum element basis for the synthesis of mathematical diagrams of electrical circuits, which consists of: ideal conductor, ideal switch, one independent and one dependent source of electrical energy, capacitance or inductance, was convinced. There are a few variants of functionally complete minimal element bases.
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Budd, Samuel, and Adam Van Tuyl. "Newton Complementary Duals of -Ideals." Canadian Mathematical Bulletin 62, no. 02 (October 15, 2018): 231–41. http://dx.doi.org/10.4153/s0008439518000024.
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AbstractA square-free monomial ideal $I$ of $k[x_{1},\ldots ,x_{n}]$ is said to be an $f$ -ideal if the facet complex and non-face complex associated with $I$ have the same $f$ -vector. We show that $I$ is an $f$ -ideal if and only if its Newton complementary dual $\widehat{I}$ is also an $f$ -ideal. Because of this duality, previous results about some classes of $f$ -ideals can be extended to a much larger class of $f$ -ideals. An interesting by-product of our work is an alternative formulation of the Kruskal–Katona theorem for $f$ -vectors of simplicial complexes.
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Hemmo, Meir, and Orly Shenker. "The Multiple-Computations Theorem and the Physics of Singling Out a Computation." Monist 105, no. 2 (March 9, 2022): 175–93. http://dx.doi.org/10.1093/monist/onab030.
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Abstract The problem of multiple-computations discovered by Hilary Putnam presents a deep difficulty for functionalism (of all sorts, computational and causal). We describe in outline why Putnam’s result, and likewise the more restricted result we call the Multiple-Computations Theorem, are in fact theorems of statistical mechanics. We show why the mere interaction of a computing system with its environment cannot single out a computation as the preferred one amongst the many computations implemented by the system. We explain why nonreductive approaches to solving the multiple-computations problem, and in particular why computational externalism, are dualistic in the sense that they imply that nonphysical facts in the environment of a computing system single out the computation. We discuss certain attempts to dissolve Putnam’s unrestricted result by appealing to systems with certain kinds of input and output states as a special case of computational externalism, and show why this approach is not workable without collapsing to behaviorism. We conclude with some remarks about the nonphysical nature of mainstream approaches to both statistical mechanics and the quantum theory of measurement with respect to the singling out of partitions and observables.
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Senashov, S. I., and A. M. Vinogradov. "Symmetries and conservation laws of 2-dimensional ideal plasticity." Proceedings of the Edinburgh Mathematical Society 31, no. 3 (October 1988): 415–39. http://dx.doi.org/10.1017/s0013091500006817.
Повний текст джерелаАнотація:
Symmetry theory is of fundamental importance in studying systems of partial differential equations. At present algebras of classical infinitesimal symmetry transformations are known for many equations of continuum mechanics [1, 2, 4]. Methods foi finding these algebras go back to S. Lie's works written about 100 years ago. Ir particular, knowledge of symmetry algebras makes it possible to construct effectively wide classes of exact solutions for equations under consideration and via Noether's theorem to find conservation laws for Euler–Lagrange equations. The natural development of Lie's theory is the theory of “higher” symmetries and conservation laws [5].
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Ramakrishnan, T. S., and D. J. Wilkinson. "Water-Cut and Fractional-Flow Logs From Array-Induction Measurements." SPE Reservoir Evaluation & Engineering 2, no. 01 (February 1, 1999): 85–94. http://dx.doi.org/10.2118/54673-pa.
Повний текст джерелаАнотація:
Summary Despite the importance of relative permeabilities in reservoir simulation, no information regarding them is available from current logs. In this paper, for the first time, we demonstrate a continuous log of multiphase flow properties. Mud filtrate invasion is usually regarded as a process that corrupts the true logs. In reality, the multiphase flow characteristics that influence filtrate flow also determine the subsequent reservoir performance. We propose the notion that invasion is an experiment, albeit uncontrolled, that may be used to invert for multiphase flow properties. Thus, in principle, inversion of array induction measurements in terms of the fractional flow curve is possible. The forward model for filtrate invasion is based on two-phase (aqueous and oleic), three-component (oil, water and salt) transport. Hysteretic behavior of relative permeability functions is included. The radial conductivity profiles calculated from the flow model are converted to induction logs using radial response functions. An algorithm for rapid calculations of the forward logs by combining the electromagnetic and flow models is developed. A nonlinear least squares method is used for parameter inversion from measurements. Additional data of near-wellbore resistivity, or logs obtained during drilling, may be included. Presentations for several output logs have been developed: a reserves estimate that partitions porosity into residual and movable saturations, initial water cut in the production stream, the fractional flow curve as a function of saturation, filtrate loss per unit depth, and a quality indicator. A field example of the processing, and its comparison with production data is also discussed. Introduction Drilling mud is usually weighted to maintain the wellbore hydrostatic pressure above that of the formation. This prevents the well from blowing out, but leads to invasion of borehole fluids into the formation, during which a mudcake is deposited on the borehole surface. The invasion process may consist of beneath-the-bit loss, dynamic filtration during mud circulation and finally static mud loss.1 While filtration beneath the bit may be important at the time of drilling, at the time of wireline logging most of the invasion is due to radial loss from the borehole wall. Except in tight formations, this loss is largely controlled by the mudcake, owing to its low permeability of about 1 nm2 [1 µD].2 One of the main objectives of logging is to determine the native formation resistivity in order to estimate oil reserves accurately. But the presence of an invaded region around the borehole distorts the electromagnetic logs and can make interpretation difficult. For understanding logs in the presence of invasion, a model based on a step resistivity change has been widely used, beginning with the work of Dumanoir et al.3 The step model consists of two zones of resistivity Rxo and Rt with the zone boundary at some distance ri Charts have been developed based on this model for various shoulder and mud resistivities to help the analyst deduce Rt For economic viability, in addition to knowing the reserves, it is important to know the recoverable amount. Here invasion has been regarded as representative of a waterflood. Thus, Rxo is a direct measure of the residual oil saturation Sor and tools to measure shallow resistivity have been built. Another unanticipated benefit of invasion has been discussed by Campbell and Martin 4 where a resistivity annulus is used as a pay zone indicator. The depth of invasion has also been believed to be related to permeability, although given the ultralow mudcake permeability, the correlation is probably weak. The motivation for the present work is provided by Ramakrishnan and Wilkinson,5 who developed the notion of interpreting conductivity profiles around the borehole by using fluid-flow physics. Based on these profiles, a rigorous and useful inversion result was proved. It was shown that with an ideal logging tool that could measure radial conductivity variation, the fractional flow curve could be exactly inverted provided the assumptions of the invasion model are met. This was true with just a single snapshot of the profile. The filtrate loss volume at every depth is also determined. A resistivity contrast between the mud filtrate and the connate water is required. Thus, for the first time in the history of logging, the possibility of obtaining multiphase flow properties was demonstrated. Although there is no ideal logging tool that measures conductivity profiles, tools that have multiple depths of investigation are becoming available. With the array induction imaging (AIT**) 28 raw measurements (not all independent), or more appropriately, five resolution matched channels are available. These may be combined with a shallow log and one which measures resistivity such as a log while drilling, e.g., MicroSFL** and compensated dual resistivity (CDR**). Then seven channels are obtained. The main purpose of this paper is to utilize such measurements that have different depths of investigation and demonstrate the practical utility of the inversion theorem 5,6 for obtaining fractional flow. From this, one is also able to obtain the initial water cut upon production, at any depth of interest. Rather than simply obtaining a resistivity profile based on one or two steps,7 the present work computes profiles that are constrained by the laws of fluid transport. Since the inverted flow parameters have restricted physical ranges, quality checks may be imposed. All of the familiar logs, such as Rt and Rxo can also be computed with little extra effort. Here we note that the work of Semmelbeck et al.8 done in parallel with ours, is an attempt to estimate single phase permeability (for low permeability gas sands) from array logs, quite different from the aim of this paper. Finally, it is important to point out that the principles behind the work presented here are applicable to any set of array logs that have multiple depths of investigation and are not restricted to the logging tools discussed in this paper.
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Zamani, Naser. "Rees Modules and Reduction of an Ideal Relative to a Noetherian Module." Algebra Colloquium 18, spec01 (December 2011): 739–48. http://dx.doi.org/10.1142/s1005386711000617.
Повний текст джерелаАнотація:
Let (A, 𝔪) be a Noetherian local ring, E a non-zero finitely generated A-module, and 𝔟 a proper ideal of A. The aim of this paper is, under some assumptions on𝔟 and E, to clarify a necessary and sufficient condition for the Rees module of E associated to 𝔟 to obtain Cohen-Macaulayness. Also, among other independent results about the reduction number of 𝔟 relative to E, we expand a theorem of Marley.
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Krueger, John. "Fat sets and saturated ideals." Journal of Symbolic Logic 68, no. 3 (September 2003): 837–45. http://dx.doi.org/10.2178/jsl/1058448442.
Повний текст джерелаАнотація:
AbstractWe strengthen a theorem of Gitik and Shelah [6] by showing that if κ is either weakly inaccessible or the successor of a singular cardinal and S is a stationary subset of κ such that NSκ↾S is saturated then κ ∖ S is fat. Using this theorem we derive some results about the existence of fat stationary sets. We then strengthen some results due to Baumgartner and Taylor [2], showing in particular that if I is a λ+++-saturated normal ideal on Pκλ then the conditions of being λ+-preserving, weakly presaturated, and presaturated are equivalent for I.
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Burke, Douglas, and Yo Matsubara. "Ideals and combinatorial principles." Journal of Symbolic Logic 62, no. 1 (March 1997): 117–22. http://dx.doi.org/10.2307/2275734.
Повний текст джерелаАнотація:
It is well known that if σ is a strongly compact cardinal and λ a regular cardinal ≥ σ, then for every stationary subset X of {α < λ: cof (α) = ω} there is some β < λ such that X ⋂ β is stationary in β. In fact the existence of a uniform, countably complete ultrafilter over λ is sufficient to prove the same conclusion about stationary subsets of {α < λ: cof (α) = ω}. See [13] or [10]. By analyzing the proof of this theorem as presented in [10], we realized the same conclusion will follow from the existence of a certain ideal, not necessarily prime, on . Throughout we will assume that σ is a regular uncountable cardinal and use the word “ideal” to mean fine ideal.
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Ionescu, Marius, Alex Kumjian, Aidan Sims, and Dana P. Williams. "A stabilization theorem for Fell bundles over groupoids." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 148, no. 1 (October 17, 2017): 79–100. http://dx.doi.org/10.1017/s0308210517000129.
Повний текст джерелаАнотація:
We study the C*-algebras associated with upper semi-continuous Fell bundles over second-countable Hausdorff groupoids. Based on ideas going back to the Packer–Raeburn ‘stabilization trick’, we construct from each such bundle a groupoid dynamical system whose associated Fell bundle is equivalent to the original bundle. The upshot is that the full and reduced C*-algebras of any saturated upper semi-continuous Fell bundle are stably isomorphic to the full and reduced crossed products of an associated dynamical system. We apply our results to describe the lattice of ideals of the C*-algebra of a continuous Fell bundle by applying Renault's results about the ideals of the C*-algebras of groupoid crossed products. In particular, we discuss simplicity of the Fell-bundle C*-algebra of a bundle over G in terms of an action, described by Ionescu and Williams, of G on the primitive-ideal space of the C*-algebra of the part of the bundle sitting over the unit space. We finish with some applications to twisted k-graph algebras, where the components of our results become more concrete.
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Simpson, Stephen G. "Ordinal numbers and the Hilbert basis theorem." Journal of Symbolic Logic 53, no. 3 (September 1988): 961–74. http://dx.doi.org/10.2307/2274585.
Повний текст джерелаАнотація:
In [5] and [21] we studied countable algebra in the context of “reverse mathematics”. We considered set existence axioms formulated in the language of second order arithmetic. We showed that many well-known theorems about countable fields, countable rings, countable abelian groups, etc. are equivalent to the respective set existence axioms which are needed to prove them.One classical algebraic theorem which we did not consider in [5] and [21] is the Hilbert basis theorem. Let K be a field. For any natural number m, let K[x1,…,xm] be the ring of polynomials over K in m commuting indeterminates x1,…,xm. The Hilbert basis theorem asserts that for all K and m, every ideal in the ring K[x1,…,xm] is finitely generated. This theorem is of fundamental importance for invariant theory and for algebraic geometry. There is also a generalization, the Robson basis theorem [11], which makes a similar but more restrictive assertion about the ring K〈x1,…,xm〉 of polynomials over K in mnoncommuting indeterminates.In this paper we study a certain formal version of the Hilbert basis theorem within the language of second order arithmetic. Our main result is that, for any or all countable fields K, our version of the Hilbert basis theorem is equivalent to the assertion that the ordinal number ωω is well ordered. (The equivalence is provable in the weak base theory RCA0.) Thus the ordinal number ωω is a measure of the “intrinsic logical strength” of the Hilbert basis theorem. Such a measure is of interest in reference to the historic controversy surrounding the Hilbert basis theorem's apparent lack of constructive or computational content.
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Johnson, C. A. "Seminormal λ-generated ideals on Pκλ". Journal of Symbolic Logic 53, № 1 (березень 1988): 92–102. http://dx.doi.org/10.1017/s0022481200028942.
Повний текст джерелаАнотація:
In this paper we consider the problem of lifting properties of the Fréchet ideal Iκ = {X ⊆ κ: ∣X∣ < κ} on a regular uncountable cardinal κ, to an analogue about Iκλ, the ideal of not unbounded subsets of Pκλ. With this in mind, in §1 we introduce and study the class of seminormal λ-generated ideals on Pκλ. We shall see that ideals belonging to this class display properties which are clearly analogous to those of the Fréchet ideal on κ (for instance, with regard to saturation, normality and weak selectivity) and yet are closely related to Iκλ. Our results here show that if λ<λ = λ, then many restrictions of Iκλ are weakly selective, nowhere precipitous and, quite suprisingly, seminormal (but nowhere normal). These latter two results suggest the question of whether any restriction of Iκλ can ever be normal. In §2 we prove that if κ is strongly inaccessible, λ<κ = 2λ and NSκλ, the ideal of nonstationary subsets of Pκλ, has a mild selective property, then NSκλ∣A = Iκλ∣A for some stationary A ⊆ Pκλ.In [1] Baumgartner showed that if κ is weakly compact and P is the collection of indescribable subsets of κ, then P → (P, κ)2. As a Pκλ analogue of indescribability, Carr (see [3]–[5]) introduced the λ-Shelah property, but was unable to derive the natural Pκλ analogue of Baumgartner's result, (where NShκλ is the normal ideal on Pκλ induced by the λ-Shelah property). In §3 we show that the problem lies in the fact that, as far as we know, NShκλ is not sufficiently distributive, and derive conditions which are sufficient and, in a sense, necessary to yield partitions related to .
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Kadykov, V. Yu, and A. B. Levina. "HOMOMORPHIC OPERATIONS WITHIN IDEAL LATTICE BASED ENCRYPTION SYSTEMS." Vestnik komp'iuternykh i informatsionnykh tekhnologii, no. 198 (December 2020): 40–46. http://dx.doi.org/10.14489/vkit.2020.12.pp.040-046.
Повний текст джерелаАнотація:
By 2009 the first system of fully homomorphic encryption had been constructed, and it was thought-provoking for many future works based on it. Instead of legacy encryption systems which depend on sharing a key (public or private) among endpoints involved in exchanging en encrypted message the fully homomorphic encryption can keep service without depending on shared keys and does not necessarily need to access the content. Such property allows any third party to operate on the encrypted data without decrypting it in advance. In this work, the possibility of using the ideal lattices for the construction of homomorphic operations is researched with a detailed level of math.The paper represents the analysis method based on the primitive of a union of ideals in lattice space. A segregated analysis between homomorphic and security properties is the advantage of this method. The work will be based on the analysis of generalized operations over ciphertext using the concept of the base reducing element which shares all about the method above. It will be shown how some non-homomorphic encryption systems can be supplemented by homomorphic operations which invoke different parameters choosing. Thus such systems can be decomposed from ciphertext structure to decryption process which will be affected by separately analyzed base reduction elements. Distinct from the encryption scheme the underlying math can be used to analyze only the homomorphic part, particularly under some simplifications. The building of such ideal-based ciphertext is laying on the assumption that ideals can be extracted further. It will be shown that the “remainder theorem” can be one of the principal ways to do this providing a simple estimate of an upper bound security strength of ciphertext structure.
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Ellerman, David. "Logical Entropy: Introduction to Classical and Quantum Logical Information Theory." Entropy 20, no. 9 (September 6, 2018): 679. http://dx.doi.org/10.3390/e20090679.
Повний текст джерелаАнотація:
Logical information theory is the quantitative version of the logic of partitions just as logical probability theory is the quantitative version of the dual Boolean logic of subsets. The resulting notion of information is about distinctions, differences and distinguishability and is formalized using the distinctions (“dits”) of a partition (a pair of points distinguished by the partition). All the definitions of simple, joint, conditional and mutual entropy of Shannon information theory are derived by a uniform transformation from the corresponding definitions at the logical level. The purpose of this paper is to give the direct generalization to quantum logical information theory that similarly focuses on the pairs of eigenstates distinguished by an observable, i.e., qudits of an observable. The fundamental theorem for quantum logical entropy and measurement establishes a direct quantitative connection between the increase in quantum logical entropy due to a projective measurement and the eigenstates (cohered together in the pure superposition state being measured) that are distinguished by the measurement (decohered in the post-measurement mixed state). Both the classical and quantum versions of logical entropy have simple interpretations as “two-draw” probabilities for distinctions. The conclusion is that quantum logical entropy is the simple and natural notion of information for quantum information theory focusing on the distinguishing of quantum states.
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Ellerman, David. "Introduction to Quantum Logical Information Theory: Talk." EPJ Web of Conferences 182 (2018): 02039. http://dx.doi.org/10.1051/epjconf/201818202039.
Повний текст джерелаАнотація:
Logical information theory is the quantitative version of the logic of partitions just as logical probability theory is the quantitative version of the dual Boolean logic of subsets. The resulting notion of information is about distinctions, differences, and distinguishability, and is formalized using the distinctions (“dits”) of a partition (a pair of points distinguished by the partition). All the definitions of simple, joint, conditional, and mutual entropy of Shannon information theory are derived by a uniform transformation from the corresponding definitions at the logical level. The purpose of this talk is to outline the direct generalization to quantum logical information theory that similarly focuses on the pairs of eigenstates distinguished by an observable, i.e., “qudits” of an observable. The fundamental theorem for quantum logical entropy and measurement establishes a direct quantitative connection between the increase in quantum logical entropy due to a projective measurement and the eigenstates (cohered together in the pure superposition state being measured) that are distinguished by the measurement (decohered in the postmeasurement mixed state). Both the classical and quantum versions of logical entropy have simple interpretations as “two-draw” probabilities for distinctions. The conclusion is that quantum logical entropy is the simple and natural notion of information for quantum information theory focusing on the distinguishing of quantum states.
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SHIMOMURA, TAKASHI. "Graph covers and ergodicity for zero-dimensional systems." Ergodic Theory and Dynamical Systems 36, no. 2 (October 7, 2014): 608–31. http://dx.doi.org/10.1017/etds.2014.72.
Повний текст джерелаАнотація:
Bratteli–Vershik systems have been widely studied. In the context of general zero-dimensional systems, Bratteli–Vershik systems are homeomorphisms that have Kakutani–Rohlin refinements. Bratteli diagrams are well suited to analyzing such systems. Besides this approach, general graph covers can be used to represent any zero-dimensional system. Indeed, all zero-dimensional systems can be described as certain kinds of sequences of graph covers that may not be brought about by Kakutani–Rohlin partitions. In this paper, we follow the context of general graph covers to analyze the relations between ergodic measures and circuits of graph covers. First, we formalize the condition for a sequence of graph covers to represent minimal Cantor systems. In constructing invariant measures, we deal with general compact metrizable zero-dimensional systems. In the context of Bratteli diagrams with finite rank, it has previously been mentioned that all ergodic measures should be limits of some combinations of towers of Kakutani–Rohlin refinements. We demonstrate this for the general zero-dimensional case, and develop a theorem that expresses the coincidence of the time average and the space average for ergodic measures. Additionally, we formulate a theorem that signifies the old relation between uniform convergence and unique ergodicity in the context of graph circuits for general zero-dimensional systems. Unlike previous studies, in our case of general graph covers there arises the possibility of the linear dependence of circuits. We give a condition for a full circuit system to be linearly independent. Previous research also showed that the bounded combinatorics imply unique ergodicity. We present a lemma that enables us to consider unbounded ranks of winding matrices. Finally, we present examples that are linked with a set of simple Bratteli diagrams having the equal path number property.
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BACKES, MICHAEL, and BORIS KÖPF. "Quantifying information flow in cryptographic systems." Mathematical Structures in Computer Science 25, no. 2 (November 10, 2014): 457–79. http://dx.doi.org/10.1017/s0960129513000662.
Повний текст джерелаАнотація:
We provide a novel definition of quantitative information flow, called transmissible information, that is suitable for reasoning about informational-theoretically secure (or non-cryptographic) systems, as well as about cryptographic systems with their polynomially bounded adversaries, error probabilities, etc. Transmissible information captures deliberate communication between two processes, and it safely over-approximates the quantity of information that a process unintentionally leaks to another process.We show that transmissible information is preserved under universal composability, which constitutes the prevalent cryptographic notion of a secure implementation. This result enables us to lift quantitative bounds of transmissible information from simple ideal functionalities of cryptographic tasks to actual cryptographic systems.We furthermore prove a connection between transmissible information in the unconditional setting and channel capacity, based on the weak converse of Shannon's coding theorem. This connection enables us to compute an upper bound on the transmissible information for a restricted class of protocols, using existing techniques from quantitative information flow.
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CAGLIERO, LEANDRO, and NADINA ROJAS. "FAITHFUL REPRESENTATIONS OF MINIMAL DIMENSION OF CURRENT HEISENBERG LIE ALGEBRAS." International Journal of Mathematics 20, no. 11 (November 2009): 1347–62. http://dx.doi.org/10.1142/s0129167x09005790.
Повний текст джерелаАнотація:
Given a Lie algebra 𝔤 over a field of characteristic zero k, let μ(𝔤) = min{dim π : π is a faithful representation of 𝔤}. Let 𝔥m be the Heisenberg Lie algebra of dimension 2m + 1 over k and let k [t] be the polynomial algebra in one variable. Given m ∈ ℕ and p ∈ k [t], let 𝔥m, p = 𝔥m ⊗ k [t]/(p) be the current Lie algebra associated to 𝔥m and k [t]/(p), where (p) is the principal ideal in k [t] generated by p. In this paper we prove that [Formula: see text]. We also prove a result that gives information about the structure of a commuting family of operators on a finite dimensional vector space. From it is derived the well-known theorem of Schur on maximal abelian subalgebras of 𝔤𝔩(n, k ).
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Henrion, Claudia. "Properties of subtle cardinals." Journal of Symbolic Logic 52, no. 4 (December 1987): 1005–19. http://dx.doi.org/10.2307/2273834.
Повний текст джерелаАнотація:
Subtle cardinals were first introduced in a paper by Jensen and Kunen [JK]. They show that ifκis subtle then ◇κholds. Subtle cardinals also play an important role in [B1], where Baumgartner proposed that certain large cardinal properties should be considered as properties of their associated normal ideals. He shows that in the case of ineffables, the ideals are particularly useful, as can be seen by the following theorem,κis ineffable if and only ifκis subtle andΠ½-indescribableandthe subtle andΠ½-indescribable ideals cohere, i.e. they generate a proper, normal ideal (which in fact turns out to be the ineffable ideal).In this paper we examine properties of subtle cardinals and consider methods of forcing that destroy the property of subtlety while maintaining other properties. The following is a list of results.1) We relativize the following two facts about subtle cardinals:i) ifκisn-subtle then {α<κ:αis notn-subtle} isn-subtle, andii) ifκis (n+ 1)-subtle then {α<κ:αisn-subtle} is in the (n+ 1)-subtle filter to subsets ofκ:i′) ifAis ann-subtle subset ofκthen {α ϵ A:A∩αis notn-subtle} isn-subtle, andii′) ifAis an (n+ 1)-subtle subset ofκthen {α ϵ A:A∩αisn-subtle} is (n+ 1)-subtle.2) We show that although a stationary limit of subtles is subtle, a subtle limit of subtles is not necessarily 2-subtle.3) In §3 we use the technique of forcing to turn a subtle cardinal into aκ-Mahlo cardinal that is no longer subtle.4) In §4 we extend the results of §3 by showing how to turn an (n+ 1)-subtle cardinal into ann-subtle cardinal that is no longer (n+ 1)-subtle.
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Diesl, Alexander J., Samuel J. Dittmer, and Pace P. Nielsen. "Idempotent lifting and ring extensions." Journal of Algebra and Its Applications 15, no. 06 (March 30, 2016): 1650112. http://dx.doi.org/10.1142/s0219498816501127.
Повний текст джерелаАнотація:
We answer multiple open questions concerning lifting of idempotents that appear in the literature. Most of the results are obtained by constructing explicit counter-examples. For instance, we provide a ring [Formula: see text] for which idempotents lift modulo the Jacobson radical [Formula: see text], but idempotents do not lift modulo [Formula: see text]. Thus, the property “idempotents lift modulo the Jacobson radical” is not a Morita invariant. We also prove that if [Formula: see text] and [Formula: see text] are ideals of [Formula: see text] for which idempotents lift (even strongly), then it can be the case that idempotents do not lift over [Formula: see text]. On the positive side, if [Formula: see text] and [Formula: see text] are enabling ideals in [Formula: see text], then [Formula: see text] is also an enabling ideal. We show that if [Formula: see text] is (weakly) enabling in [Formula: see text], then [Formula: see text] is not necessarily (weakly) enabling in [Formula: see text] while [Formula: see text] is (weakly) enabling in [Formula: see text]. The latter result is a special case of a more general theorem about completions. Finally, we give examples showing that conjugate idempotents are not necessarily related by a string of perspectivities.
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BEZHANISHVILI, NICK, and WESLEY H. HOLLIDAY. "CHOICE-FREE STONE DUALITY." Journal of Symbolic Logic 85, no. 1 (August 29, 2019): 109–48. http://dx.doi.org/10.1017/jsl.2019.11.
Повний текст джерелаАнотація:
AbstractThe standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras.
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Scheepers, Marion. "Concerning n-tactics in the countable-finite game." Journal of Symbolic Logic 56, no. 3 (September 1991): 786–94. http://dx.doi.org/10.2178/jsl/1183743727.
Повний текст джерелаАнотація:
In the paper [S1] I introduced a game, denoted by MG(J) (where J is a free ideal on some infinite set S) and called “the meager nowhere dense game for J”. The special case when J is the collection of finite subsets of the set S is called the countable-finite game on S. It proceeds as follows.First player ONE picks a countable set C1, then player TWO picks a finite set F1. Then in the second inning ONE picks a countable set C2 with C1 ⊂ C2 (unless explicitly indicated otherwise, “⊂” means “is a proper subset of”) and TWO responds with a finite set F2, and so on. The players construct a sequence (C1,F1,C2,F2,…,Ck,Fk,…) where for each positive integer k(i) Ck denotes ONE's countable set picked during the kth inning,(ii) Fk denotes TWO's finite set picked during the kth inning, and(iii) Ck ⊂ Ck + 1.Such a sequence is a play of the countable-finite game on S, and TWO wins this play if is contained in . The notion of a winning perfect information strategy is defined as usual (see, for example, [S1]). Zermelo-Fraenkel set theory together with the axiom of choice (denoted by ZFC; for a statement of the axioms see pp. xv–xvi of [K]) is a strong enough theory to build a winning perfect information strategy for player TWO in this game.Does TWO have a winning strategy requiring less than perfect information? Fix a positive integer k. A strategy of TWO which requires knowledge of only at the most the k most recent moves of ONE is said to be a k-tactic. For the countable-finite game on an infinite set S the following facts about the existence of winning k-tactics for TWO are proved in [S1]:1) TWO does not have a winning 1-tactic (Theorem 1 of [S1]).2) If the cardinality of S is less than ℵ2 then TWO has a winning 2-tactic (Corollary 4 of [S1]).3) If TWO has a winning k-tactic in the countable-finite game on an infinite set S, then TWO has a winning 3-tactic (Proposition 15 of [SI]).
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Kuperberg, Greg. "Kasteleyn Cokernels." Electronic Journal of Combinatorics 9, no. 1 (June 24, 2002). http://dx.doi.org/10.37236/1645.
Повний текст джерелаАнотація:
We consider Kasteleyn and Kasteleyn-Percus matrices, which arise in enumerating matchings of planar graphs, up to matrix operations on their rows and columns. If such a matrix is defined over a principal ideal domain, this is equivalent to considering its Smith normal form or its cokernel. Many variations of the enumeration methods result in equivalent matrices. In particular, Gessel-Viennot matrices are equivalent to Kasteleyn-Percus matrices. We apply these ideas to plane partitions and related planar of tilings. We list a number of conjectures, supported by experiments in Maple, about the forms of matrices associated to enumerations of plane partitions and other lozenge tilings of planar regions and their symmetry classes. We focus on the case where the enumerations are round or $q$-round, and we conjecture that cokernels remain round or $q$-round for related "impossible enumerations" in which there are no tilings. Our conjectures provide a new view of the topic of enumerating symmetry classes of plane partitions and their generalizations. In particular we conjecture that a $q$-specialization of a Jacobi-Trudi matrix has a Smith normal form. If so it could be an interesting structure associated to the corresponding irreducible representation of SL$(n,C)$. Finally we find, with proof, the normal form of the matrix that appears in the enumeration of domino tilings of an Aztec diamond.
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Einstein, David, Miriam Farber, Emily Gunawan, Michael Joseph, Matthew Macauley, James Propp, and Simon Rubinstein-Salzedo. "Noncrossing Partitions, Toggles, and Homomesies." Electronic Journal of Combinatorics 23, no. 3 (September 30, 2016). http://dx.doi.org/10.37236/5648.
Повний текст джерелаАнотація:
We introduce $n(n-1)/2$ natural involutions ("toggles") on the set $S$ of noncrossing partitions $\pi$ of size $n$, along with certain composite operations obtained by composing these involutions. We show that for many operations $T$ of this kind, a surprisingly large family of functions $f$ on $S$ (including the function that sends $\pi$ to the number of blocks of $\pi$) exhibits the homomesy phenomenon: the average of $f$ over the elements of a $T$-orbit is the same for all $T$-orbits. We can apply our method of proof more broadly to toggle operations back on the collection of independent sets of certain graphs. We utilize this generalization to prove a theorem about toggling on a family of graphs called "$2$-cliquish." More generally, the philosophy of this "toggle-action", proposed by Striker, is a popular topic of current and future research in dynamic algebraic combinatorics.
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Klazar, Martin. "On Growth Rates of Permutations, Set Partitions, Ordered Graphs and Other Objects." Electronic Journal of Combinatorics 15, no. 1 (May 31, 2008). http://dx.doi.org/10.37236/799.
Повний текст джерелаАнотація:
For classes ${\cal O}$ of structures on finite linear orders (permutations, ordered graphs etc.) endowed with containment order $\preceq$ (containment of permutations, subgraph relation etc.), we investigate restrictions on the function $f(n)$ counting objects with size $n$ in a lower ideal in $({\cal O},\preceq)$. We present a framework of edge $P$-colored complete graphs $({\cal C}(P),\preceq)$ which includes many of these situations, and we prove for it two such restrictions (jumps in growth): $f(n)$ is eventually constant or $f(n)\ge n$ for all $n\ge 1$; $f(n)\le n^c$ for all $n\ge 1$ for a constant $c>0$ or $f(n)\ge F_n$ for all $n\ge 1$, $F_n$ being the Fibonacci numbers. This generalizes a fragment of a more detailed theorem of Balogh, Bollobás and Morris on hereditary properties of ordered graphs.
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Kehayopulu, Niovi. "Comment on: “Bi-interior ideals of semigroups”." Asian-European Journal of Mathematics, June 3, 2021, 2250056. http://dx.doi.org/10.1142/s1793557122500565.
Повний текст джерелаАнотація:
This is about the paper “Bi-interior ideals of semigroups” by M. Murali Krishna Rao in Discuss. Math. Gen. Algebra Appl. 38 (2018) 69–78. According to Theorem 3.11 (also Theorem 3.3(8)) of the paper, the intersection of a bi-interior ideal [Formula: see text] of a semigroup [Formula: see text] and a subsemigroup [Formula: see text] of [Formula: see text] is a bi-interior ideal of [Formula: see text]. Regarding to Theorem 3.6, every bi-interior ideal of a regular semigroup is an ideal of [Formula: see text]. We give an example that the above two results are not true for semigroups. According to the same paper, if [Formula: see text] is a regular semigroup then, for every bi-interior ideal [Formula: see text], every ideal [Formula: see text] and every left ideal [Formula: see text] of [Formula: see text], we have [Formula: see text]. The proof is wrong, we provide the corrected proof. In most of the results of the paper the assumption of unity is not necessary. Care should be taken about the proofs in the paper.
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Andrews, George E. "Concave Compositions." Electronic Journal of Combinatorics 18, no. 2 (April 23, 2011). http://dx.doi.org/10.37236/2002.
Повний текст джерелаАнотація:
Concave compositions are compositions (i.e. ordered partitions) of a number in which the parts decrease up to the middle summand(s) and increase thereafter. Perhaps the most surprising result is for even length, concave compositions where the generating function turns out to be the quotient of two instances of the pentagonal number theorem with variations of sign. The false theta function discoveries also lead to new facts about concatenatable, spiral, self-avoiding walks.
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Onar, Serkan, Erdogan Mehmet Özkan, Bayram Ali Ersoy та Kostaq Hila. "2-Absorbing δ-Primary Intuitionistic Fuzzy Ideals of Commutative Rings". New Mathematics and Natural Computation, 1 березня 2022, 1–18. http://dx.doi.org/10.1142/s1793005723500011.
Повний текст джерелаАнотація:
In this paper, we study the primary intuitionistic fuzzy ideal, the intuitionistic fuzzy ideal expansion and [Formula: see text]-primary intuitionistic fuzzy ideal which assemble prime intuitionistic fuzzy ideals and primary intuitionistic fuzzy ideals. Some properties of them are investigated. Also, we scrutinize the relationships of [Formula: see text]-primary intuitionistic fuzzy ideal and [Formula: see text]-primary ideal of a commutative ring [Formula: see text]. Moreover, we give a fundamental result about correspondence theorem for [Formula: see text]-primary intuitionistic fuzzy ideals. Further, we introduce 2-absorbing [Formula: see text]-primary intuitionistic fuzzy ideals which are the generalization of 2-absorbing intuitionistic fuzzy ideals and 2-absorbing primary intuitionistic fuzzy ideals. Some properties of them are obtained.
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Kolos, O. "CONTRADICTIONS WITHIN ZERMELO–FRAENKEL SET THEORY WITH AXIOM OF CHOICE." ΛΌГOΣ МИСТЕЦТВО НАУКОВОЇ ДУМКИ, June 14, 2020. http://dx.doi.org/10.36074/2663-4139.10.05.
Повний текст джерелаАнотація:
By using a counterexample to the known equivalent of the axiom of choice (Krull’s theorem about maximal ideal existence) contradictions within Zermelo–Fraenkel set theory with axiom of choice was shown. Ring ideals set which satisfies the Zorn’s lemma conditions, but with no maximal ele-ment was built.
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Kaučikas, Algirdas. "On the left strongly prime modules and their radicals." Lietuvos matematikos rinkinys 51 (December 21, 2010). http://dx.doi.org/10.15388/lmr.2011.05.
Повний текст джерелаАнотація:
We give the new results on the theory of the one-sided (left) strongly prime modules and their strongly prime radicals. Particularly, the conceptually new and short proof of the A.L.Rosenberg’s theorem about one-sided strongly prime radical of the ring is given. Main results of the paper are: presentation of each left stongly prime ideal p of a ring R as p = R ∩ M, where M is a maximal left ideal in a ring of polynomials over the ring R; characterization of the primeless modules and characterization of the left strongly prime radical of a finitely generated module M in terms of the Jacobson radicals of modules of polynomes M(X1, . . . , Xni) .
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"On a characterization of finitely presented functors." Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 425, no. 1869 (October 9, 1989): 329–39. http://dx.doi.org/10.1098/rspa.1989.0109.
Повний текст джерелаАнотація:
The concept of finitely presented functor was introduced by Auslander. Proposition 3.1 of Auslander & Reiten provides a way of dealing with the category of finitely presented functors, that seems concrete and easy to use, at least in some examples. The study of this category, using this particular line of thought, is the main purpose of this work. In §1 I recall some basic definitions and give the required notation. In §2 I state the theorem of Auslander & Reiten referred to above and give a new proof of this result. The first part of this proof is an immediate consequence of the theory developed by Green. In §3 I state and prove an unpublished theorem by J. A. Green and I introduce a new category I such that the category of finitely presented functors. mmod A , is equivalent to a quotient category I / J , where J is an ideal of I . In §4 I give some examples of properties of mmod A , stated and proved in terms of the category I , by using the equivalence of categories referred to in §3. In §5 I consider the particular case where A = A q = k -alg < z : z q = 0>, apply the results of previous sections to study mmod A q and make conclusions about the representation type of the Auslander algebra of A q .