Artykuły w czasopismach: „Iterated forcing”

1

Friedman, Sy D. "Iterated Class Forcing". Mathematical Research Letters 1, nr 4 (1994): 427–36. http://dx.doi.org/10.4310/mrl.1994.v1.n4.a3.

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Groszek, Marcia J. "Applications of iterated perfect set forcing". Annals of Pure and Applied Logic 39, nr 1 (lipiec 1988): 19–53. http://dx.doi.org/10.1016/0168-0072(88)90044-9.

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Ferrero, Daniela, Thomas Kalinowski i Sudeep Stephen. "Zero forcing in iterated line digraphs". Discrete Applied Mathematics 255 (luty 2019): 198–208. http://dx.doi.org/10.1016/j.dam.2018.08.019.

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4

Spinas, O. "Iterated forcing in quadratic form theory". Israel Journal of Mathematics 79, nr 2-3 (październik 1992): 297–315. http://dx.doi.org/10.1007/bf02808222.

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Ihoda, Jaime I., i Saharon Shelah. "Souslin forcing". Journal of Symbolic Logic 53, nr 4 (grudzień 1988): 1188–207. http://dx.doi.org/10.1017/s0022481200028012.

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AbstractWe define the notion of Souslin forcing, and we prove that some properties are preserved under iteration. We define a weaker form of Martin's axiom, namely , and using the results on Souslin forcing we show that is consistent with the existence of a Souslin tree and with the splitting number s = ℵ1. We prove that proves the additivity of measure. Also we introduce the notion of proper Souslin forcing, and we prove that this property is preserved under countable support iterated forcing. We use these results to show that ZFC + there is an inaccessible cardinal is equiconsistent with ZFC + the Borel conjecture + -measurability.

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Audrito, Giorgio, i Matteo Viale. "Absoluteness via resurrection". Journal of Mathematical Logic 17, nr 02 (27.11.2017): 1750005. http://dx.doi.org/10.1142/s0219061317500052.

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The resurrection axioms are forcing axioms introduced recently by Hamkins and Johnstone, developing on ideas of Chalons and Veličković. We introduce a stronger form of resurrection axioms (the iterated resurrection axioms [Formula: see text] for a class of forcings [Formula: see text] and a given ordinal [Formula: see text]), and show that [Formula: see text] implies generic absoluteness for the first-order theory of [Formula: see text] with respect to forcings in [Formula: see text] preserving the axiom, where [Formula: see text] is a cardinal which depends on [Formula: see text] ([Formula: see text] if [Formula: see text] is any among the classes of countably closed, proper, semiproper, stationary set preserving forcings). We also prove that the consistency strength of these axioms is below that of a Mahlo cardinal for most forcing classes, and below that of a stationary limit of supercompact cardinals for the class of stationary set preserving posets. Moreover, we outline that simultaneous generic absoluteness for [Formula: see text] with respect to [Formula: see text] and for [Formula: see text] with respect to [Formula: see text] with [Formula: see text] is in principle possible, and we present several natural models of the Morse–Kelley set theory where this phenomenon occurs (even for all [Formula: see text] simultaneously). Finally, we compare the iterated resurrection axioms (and the generic absoluteness results we can draw from them) with a variety of other forcing axioms, and also with the generic absoluteness results by Woodin and the second author.

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7

Ishiu, Tetsuya, i Paul B. Larson. "Some results about (+) proved by iterated forcing". Journal of Symbolic Logic 77, nr 2 (czerwiec 2012): 515–31. http://dx.doi.org/10.2178/jsl/1333566635.

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AbstractWe shall show the consistency of CH+⌝(+) and CH+(+)+there are no club guessing sequences on ω1. We shall also prove that ◊+ does not imply the existence of a strong club guessing sequence on ω1.

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8

Shelah, Saharon. "Iterated forcing and normal ideals onω 1". Israel Journal of Mathematics 60, nr 3 (grudzień 1987): 345–80. http://dx.doi.org/10.1007/bf02780398.

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9

Mitchell, William. "Prikry forcing at κ+ and beyond". Journal of Symbolic Logic 52, nr 1 (marzec 1987): 44–50. http://dx.doi.org/10.2307/2273859.

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If U is a normal measure on κ then we can add indiscernibles for U either by Prikry forcing [P] or by taking an iterated ultrapower which will add a sequence of indiscernibles for over M. These constructions are equivalent: the set C of indiscernibles for added by the iterated ultrapower is Prikry generic for [Mat]. Prikry forcing has been extended for sequences of measures of length by Magidor [Mag], and his method readily extends to . In this case the measure U is replaced by a sequence of measures and the set C of indiscernibles is replaced by a system of indiscernibles for : is a function such that (κ, β) is a set of indiscernibles for (κ, β) for each . The equivalence between forcing and iterated ultra-powers still holds true for such sequences: there is an interated ultrapower j: V → M (which is defined in detail later in this paper) such that the system of indiscernibles for j() constructed by j is Magidor generic over M.The construction of the system of indiscernibles works equally well for o(κ) ≧ κ+. Radin has defined a variant of Prikry forcing which also works for o(κ) > κ+ ([R]; see also [Mi82] where Radin forcing is applied specifically to sequences of measures, rather than to hypermeasures as in Radin's paper), but Radin's forcing is weaker than Magidor's extension of Prikry forcing in the sense that the system of indiscernibles generated by the interated ultrapower is not Radin generic for j(), but only the set . That is, an indiscernible does not belong to a specific measure, but only to the whole sequence of measures on the cardinal κ.

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10

Kanovei, Vladimir. "On non-wellfounded iterations of the perfect set forcing". Journal of Symbolic Logic 64, nr 2 (czerwiec 1999): 551–74. http://dx.doi.org/10.2307/2586484.

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AbstractWe prove that if I is a partially ordered set in a countable transitive model of ZFC then can be extended by a generic sequence of reals ai, i ∈ I, such that is preserved and every ai is Sacks generic over [〈aj: j < i〉]. The structure of the degrees of -constructibility of reals in the extension is investigated.As applications of the methods involved, we define a cardinal invariant to distinguish product and iterated Sacks extensions, and give a short proof of a theorem (by Budinas) that in ω2-iterated Sacks extension of L the Burgess selection principle for analytic equivalence relations holds.

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11

Eisworth, Todd. "On iterated forcing for successors of regular cardinals". Fundamenta Mathematicae 179, nr 3 (2003): 249–66. http://dx.doi.org/10.4064/fm179-3-4.

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Tohmé, Fernando, Gianluca Caterina i Jonathan Gangle. "Iterated Admissibility Through Forcing in Strategic Belief Models". Journal of Logic, Language and Information 29, nr 4 (29.05.2020): 491–509. http://dx.doi.org/10.1007/s10849-020-09317-4.

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Shelah, Saharon. "Some notes on iterated forcing with $2^{\aleph_0}>\aleph_2$." Notre Dame Journal of Formal Logic 29, nr 1 (grudzień 1987): 1–17. http://dx.doi.org/10.1305/ndjfl/1093637766.

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Mohammadpour, Rahman. "New methods in forcing iteration and applications". Bulletin of Symbolic Logic 29, nr 2 (czerwiec 2023): 300–302. http://dx.doi.org/10.1017/bsl.2023.7.

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AbstractThe Theme. Strong forcing axioms like Martin’s Maximum give a reasonably satisfactory structural analysis of $H(\omega _2)$ . A broad program in modern Set Theory is searching for strong forcing axioms beyond $\omega _1$ . In other words, one would like to figure out the structural properties of taller initial segments of the universe. However, the classical techniques of forcing iterations seem unable to bypass the obstacles, as the resulting forcings axioms beyond $\omega _1$ have not thus far been strong enough! However, with his celebrated work on generalised side conditions, I. Neeman introduced us to a novel paradigm to iterate forcings. In particular, he could, among other things, reprove the consistency of the Proper Forcing Axiom using an iterated forcing with finite supports. In 2015, using his technology of virtual models, Veličković built up an iteration of semi-proper forcings with finite supports, hence reproving the consistency of Martin’s Maximum, an achievement leading to the notion of a virtual model.In this thesis, we are interested in constructing forcing notions with finitely many virtual models as side conditions to preserve three uncountable cardinals. The thesis constitutes six chapters and three appendices that amount to 118 pages, where Section 1 is devoted to preliminaries, and Section 2 is a warm-up about the scaffolding poset of a proper forcing. In Section 3, we present the general theory of virtual models in the context of forcing with sets of models of two types, where we, e.g., define the “meet” between two virtual models and prove its properties.The main results are joint with Boban Veličković, and partly appeared in Guessing models and the approachability ideal, J. Math. Log. 21 (2021).Pure Side Conditions. In Section 4, we use two types of virtual models (countable and large non-transitive ones induced by a supercompact cardinal, which we call Magidor models) to construct our forcing with pure side conditions. The forcing covertly uses a third type of models that are transitive. We also add decorations to the conditions to add many clubs in the generic $\omega _2$ . In contrast to Neeman’s method, we do not have a single chain, but $\alpha $ -chains, for an ordinal $\alpha $ with $V_\alpha \prec V_\lambda $ . Thus, starting from suitable large cardinals $\kappa <\lambda $ , we construct a forcing notion whose conditions are finite sets of virtual models described earlier. The forcing is strongly proper, preserves $\kappa $ , and has the $\lambda $ -Knaster property. The relevant quotients of the forcing are strongly proper, which helps us prove strong guessing model principles. The construction is generalisable to a ${<}\mu $ -closed forcing, for any given cardinal $\mu $ with $\mu ^{<\mu }=\mu <\kappa $ .The Iteration Theorem. In Section 5, we use the forcing with pure side conditions to iterate a subclass of proper and $\aleph _2$ -c.c. forcings and obtain a forcing axiom at the level of $\aleph _2$ . The iterable class is closely related to Asperó–Mota’s forcing axiom for finitely proper forcings.Guessing Model Principles. Section 6 encompasses the main parts of the thesis. We prove the consistency of the guessing principle $\mathrm {GMP}^+(\omega _3,\omega _1)$ that states for any cardinal ${\theta \geq \omega _3}$ , the set of $\aleph _2$ -sized elementary submodels M of $H(\theta )$ , which are the union of an $\omega _1$ -continuous $\in $ -chain of $\omega _1$ -guessing, I.C. models is stationary in $\mathcal P_{\omega _3}(H(\theta ))$ . The consistency and consequences of this principle are demonstrated in the following diagram. We also prove that one can obtain the above guessing models in a way that the $\omega _1$ -sized $\omega _1$ -guessing models remain $\omega _1$ -guessing model in any outer transitive model with the same $\omega _1$ , and we denote this principle by $\rm{SGMP}^+(\omega_3,\omega_1)$ .In the following diagram, $\mathrm{TP}$ stands for the tree property; $w\mathrm{KH}$ stands for the weak Kurepa Hypothesis; $\mathrm{MP}$ stands for Mitchell property, i.e., the approachability ideal is trivial modulo the nonstationary ideal; $\mathrm{AP}$ stands for the approachability property; $\mathrm {AMTP}(\kappa ^+)$ states that if $2^\kappa <\aleph _{\kappa ^+}$ , then every forcing which adds a new subset of $\kappa ^+$ whose initial segments are in the ground model, collapses some cardinal $\leq 2^{\kappa }$ . The dotted arrow denotes the relative consistency, while others are logical implications.Appendices. Appendix A includes merely the above diagram. Appendix B presents a proof of the Mapping Reflection Principle with finite conditions under $\mathrm {PFA}$ . Appendix C contains open problems. Finally, the thesis’s bibliography consists of 42 items.Abstract prepared by Rahman MohammadpourE-mail: rahmanmohammadpour@gmail.comURL: https://theses.hal.science/tel-03209264

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15

Apter, Arthur W. "Some structural results concerning supercompact cardinals". Journal of Symbolic Logic 66, nr 4 (grudzień 2001): 1919–27. http://dx.doi.org/10.2307/2694985.

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Abstract.We show how the forcing of [5] can be iterated so as to get a model containing supercompact cardinals in which every measurable cardinal δ is δ+ supercompact. We then apply this iteration to prove three additional theorems concerning the structure of the class of supercompact cardinals.

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Schlindwein, Chaz. "Shelah's work on non-semi-proper iterations, II". Journal of Symbolic Logic 66, nr 4 (grudzień 2001): 1865–83. http://dx.doi.org/10.2307/2694981.

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One of the main goals in the theory of forcing iteration is to formulate preservation theorems for not collapsing ω1 which are as general as possible. This line leads from c.c.c. forcings using finite support iterations to Axiom A forcings and proper forcings using countable support iterations to semi-proper forcings using revised countable support iterations, and more recently, in work of Shelah, to yet more general classes of posets. In this paper we concentrate on a special case of the very general iteration theorem of Shelah from [5, chapter XV]. The class of posets handled by this theorem includes all semi-proper posets and also includes, among others, Namba forcing.In [5, chapter XV] Shelah shows that, roughly, revised countable support forcing iterations in which the constituent posets are either semi-proper or Namba forcing or P[W] (the forcing for collapsing a stationary co-stationary subset ofwith countable conditions) do not collapse ℵ1. The iteration must contain sufficiently many cardinal collapses, for example, Levy collapses. The most easily quotable combinatorial application is the consistency (relative to a Mahlo cardinal) of ZFC + CH fails + whenever A ∪ B = ω2 then one of A or B contains an uncountable sequentially closed subset. The iteration Shelah uses to construct this model is built using P[W] to “attack” potential counterexamples, Levy collapses to ensure that the cardinals collapsed by the various P[W]'s are sufficiently well separated, and Cohen forcings to ensure the failure of CH in the final model.In this paper we give details of the iteration theorem, but we do not address the combinatorial applications such as the one quoted above.These theorems from [5, chapter XV] are closely related to earlier work of Shelah [5, chapter XI], which dealt with iterated Namba and P[W] without allowing arbitrary semi-proper forcings to be included in the iteration. By allowing the inclusion of semi-proper forcings, [5, chapter XV] generalizes the conjunction of [5, Theorem XI.3.6] with [5, Conclusion XI.6.7].

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17

Metzler, Wolfgang. "Iterated Differentiable Maps with Nowhere Differentiable Basin Boundaries". Zeitschrift für Naturforschung A 48, nr 5-6 (1.06.1993): 669–72. http://dx.doi.org/10.1515/zna-1993-5-616.

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Abstract The fractal basin boundary of a two-dimensional discrete dynamical system modelling a chaotic forcing applied to bistability is shown to be identical to the graph of an infinite series F(x,t)= of weighted iterates of an ergodic unimodal interval function f. In the special case, when f is the logistic map in "full chaos", i.e. ƒ: x ↦ 4x(1 - x), F is a nowhere differentiable function of x for each t > exp(-λf) (even equal to the Weierstrass function), where λf >0 is denoting the Lyapunov exponent of f. For further chaotic functions f, nowhere-differentiability is shown to be obvious from computer simulations.

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Mitchell, William J. "A Gitik iteration with nearly Easton factoring". Journal of Symbolic Logic 68, nr 2 (czerwiec 2003): 481–502. http://dx.doi.org/10.2178/jsl/1052669060.

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AbstractWe reprove Gitik's theorem that if the GCH holds and o(κ) = κ + 1 then there is a generic extension in which κ is still measurable and there is a closed unbounded subset C of κ such that every ν ∈ C is inaccessible in the ground model.Unlike the forcing used by Gitik, the iterated forcing ℛλ+1 used in this paper has the property that if λ is a cardinal less then κ then ℛλ+1 can be factored in V as ℛκ+1 = ℛλ+1 × ℛλ+1,κ where ∣ℛλ+1∣ ≤ λ+ and ℛλ+1,κ does not add any new subsets of λ.

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Dordal, Peter Lars. "A model in which the base-matrix tree cannot have cofinal branches". Journal of Symbolic Logic 52, nr 3 (wrzesień 1987): 651–64. http://dx.doi.org/10.1017/s0022481200029662.

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AbstractA model of ZFC is constructed in which the distributivity cardinal h is , and in which there are no ω2-towers in [ω]ω. As an immediate corollary, it follows that any base-matrix tree in this model has no cofinal branches. The model is constructed via a form of iterated Mathias forcing, in which a mixture of finite and countable supports is used.

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Wathan, F., R. Hoshyar i R. Tafazolli. "Dynamic Grouped Chip-Level Iterated Multiuser Detection Based on Gaussian Forcing Technique". IEEE Communications Letters 12, nr 3 (marzec 2008): 167–69. http://dx.doi.org/10.1109/lcomm.2008.071931.

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Kanovei, Vladimir, i Vassily Lyubetsky. "On the Significance of Parameters in the Choice and Collection Schemata in the 2nd Order Peano Arithmetic". Mathematics 11, nr 3 (1.02.2023): 726. http://dx.doi.org/10.3390/math11030726.

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We make use of generalized iterations of the Sacks forcing to define cardinal-preserving generic extensions of the constructible universe L in which the axioms of ZF hold and in addition either (1) the parameter-free countable axiom of choice ACω* fails, or (2) ACω* holds but the full countable axiom of choice ACω fails in the domain of reals. In another generic extension of L, we define a set X⊆P(ω), which is a model of the parameter-free part PA2* of the 2nd order Peano arithmetic PA2, in which CA(Σ21) (Comprehension for Σ21 formulas with parameters) holds, yet an instance of Comprehension CA for a more complex formula fails. Treating the iterated Sacks forcing as a class forcing over Lω1, we infer the following consistency results as corollaries. If the 2nd order Peano arithmetic PA2 is formally consistent then so are the theories: (1) PA2+¬ACω*, (2) PA2+ACω*+¬ACω, (3) PA2*+CA(Σ21)+¬CA.

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DILÃO, RUI, i JOÃO GRACIANO. "EVALUATING DETERMINISTIC POLICIES IN TWO-PLAYER ITERATED GAMES". International Journal of Bifurcation and Chaos 19, nr 12 (grudzień 2009): 4039–53. http://dx.doi.org/10.1142/s0218127409025213.

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We construct a statistical ensemble of games, wherein each independent subensemble we have two players playing the same game. We derive the mean payoffs per move of the representative players of the game, and we evaluate all the deterministic policies with finite memory. In particular, we show that if one of the players has a generalized tit-for-tat policy, the mean payoff per move of both players is the same, forcing the equalization of the mean payoffs per move. In the case of symmetric, noncooperative and dilemmatic games, we show that generalized tit-for-tat or imitation policies together with the condition of not being the first to defect, leads to the highest mean payoffs per move for the players. Within this approach, it can be decided which policies perform better than others. In particular, it shows that reciprocity in noncooperative iterated games forces equality of mean payoffs. We prove a simple ergodic theorem for symmetric and noncooperative games. The Prisoner's Dilemma and the Hawk–Dove games have been analyzed, and the equilibrium states of the infinitely iterated games have been determined. In infinitely iterated games with the player choosing their moves with equal probabilities, strict Nash solutions are not necessarily reachable equilibrium solutions of games.

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Shelah, Saharon. "Two cardinal invariants of the continuum (∂<α) and FS linearly ordered iterated forcing". Acta Mathematica 192, nr 2 (2004): 187–223. http://dx.doi.org/10.1007/bf02392740.

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Jockusch, Carl G., i Robert I. Soare. "Boolean algebras, Stone spaces, and the iterated Turing jump". Journal of Symbolic Logic 59, nr 4 (grudzień 1994): 1121–38. http://dx.doi.org/10.2307/2275695.

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AbstractWe show, roughly speaking, that it requires ω iterations of the Turing jump to decode nontrivial information from Boolean algebras in an isomorphism invariant fashion. More precisely, if α is a recursive ordinal, is a countable structure with finite signature, and d is a degree, we say that has αth-jump degreed if d is the least degree which is the αth jump of some degree c such there is an isomorphic copy of with universe ω in which the functions and relations have degree at most c. We show that every degree d ≥ 0(ω) is the ωth jump degree of a Boolean algebra, but that for n < ω no Boolean algebra has nth-jump degree d < 0(n). The former result follows easily from work of L. Feiner. The proof of the latter result uses the forcing methods of J. Knight together with an analysis of various equivalences between Boolean algebras based on a study of their Stone spaces. A byproduct of the proof is a method for constructing Stone spaces with various prescribed properties.

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Gitik, Moti. "On generic elementary embeddings". Journal of Symbolic Logic 54, nr 3 (wrzesień 1989): 700–707. http://dx.doi.org/10.2307/2274734.

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Suppose that I is a precipitous ideal over a cardinal κ and j is a generic embedding of I. What is the nature of j? If we assume the existence of a supercompact cardinal then, by Foreman, Magidor and Shelah [FMS], it is quite unclear where some of such j's are coming from. On the other hand, if ¬∃κ0(κ) = κ++, then, by Mitchell [Mi], the restriction of j to the core model is its iterated ultrapower by measures of it. A natural question arising here is if each iterated ultrapower of can be obtained as the restriction of a generic embedding of a precipitous ideal. Notice that there are obvious limitations. Thus the ultrapower of by a measure over λ cannot be obtained as a generic embedding by a precipitous ideal over κ ≠ λ. But if we fix κ and use iterated ultrapowers of which are based on κ, then the answer is positive. Namely a stronger statement is true:Theorem. Let τ be an ordinal and κ a measurable cardinal. There exists a generic extension V* of V so that NSℵ1 (the nonstationary ideal on ℵ1) is precipitous and, for every iterated ultrapower i of V of length ≤ τ by measures of V based on κ, there exists a stationary set forcing “the generic ultrapower restricted to V is i”.Our aim will be to prove this theorem. We assume that the reader is familiar with the paper [JMMiP] by Jech, Magidor, Mitchell and Prikry. We shall use the method of that paper for constructing precipitous ideals. Ideas of Levinski [L] for blowing up 2ℵ1 preserving precipitousness and of our own earlier paper [Gi] for linking together indiscernibles will be used also.

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Lee, Chia-Ying, Michael K. Tippett, Adam H. Sobel i Suzana J. Camargo. "Autoregressive Modeling for Tropical Cyclone Intensity Climatology". Journal of Climate 29, nr 21 (12.10.2016): 7815–30. http://dx.doi.org/10.1175/jcli-d-15-0909.1.

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Abstract An autoregressive model is developed to simulate the climatological distribution of global tropical cyclone (TC) intensity. The model consists of two components: a regression-based deterministic component that advances the TC intensity in time and depends on the storm state and surrounding large-scale environment and a stochastic forcing. Potential intensity, deep-layer mean vertical shear, and midlevel relative humidity are the environmental variables included in the deterministic component. Given a storm track and its environment, the model is initialized and then iterated along the track. Model performance is evaluated by its ability to represent the observed global and basin distributions of TC intensity as well as lifetime maximum intensity (LMI). The deterministic model alone captures the spatial features of the climatological TC intensity distribution but with intensities that remain below 100 kt (1 kt ≈ 0.51 m s−1). Addition of white (uncorrelated in time) stochastic forcing reduces this bias by improving the simulated intensification rates and the frequency of major storms. The model simulates a realistic range of intensities, but the frequency of major storms remains too low in some basins.

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Rashidinia, Jalil, Mehri Sajjadian, Jorge Duarte, Cristina Januário i Nuno Martins. "On the Dynamical Complexity of a Seasonally Forced Discrete SIR Epidemic Model with a Constant Vaccination Strategy". Complexity 2018 (2.12.2018): 1–11. http://dx.doi.org/10.1155/2018/7191487.

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In this article, we consider the discretized classical Susceptible-Infected-Recovered (SIR) forced epidemic model to investigate the consequences of the introduction of different transmission rates and the effect of a constant vaccination strategy, providing new numerical and topological insights into the complex dynamics of recurrent diseases. Starting with a constant contact (or transmission) rate, the computation of the spectrum of Lyapunov exponents allows us to identify different chaotic regimes. Studying the evolution of the dynamical variables, a family of unimodal-type iterated maps with a striking biological meaning is detected among those dynamical regimes of the densities of the susceptibles. Using the theory of symbolic dynamics, these iterated maps are characterized based on the computation of an important numerical invariant, the topological entropy. The introduction of a degree (or amplitude) of seasonality, ε, is responsible for inducing complexity into the population dynamics. The resulting dynamical behaviors are studied using some of the previous tools for particular values of the strength of the seasonality forcing, ε. Finally, we carry out a study of the discrete SIR epidemic model under a planned constant vaccination strategy. We examine what effect this vaccination regime will have on the periodic and chaotic dynamics originated by seasonally forced epidemics.

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Nord-Larsen, Thomas, Henrik Meilby i Jens Peter Skovsgaard. "Simultaneous estimation of biomass models for 13 tree species: effects of compatible additivity requirements". Canadian Journal of Forest Research 47, nr 6 (czerwiec 2017): 765–76. http://dx.doi.org/10.1139/cjfr-2016-0430.

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A desirable feature of biomass models distinguishing different tree components is compatible additivity of the component functions. Due to forcing of parameter estimates, such additivity is achieved at an expense of precision of the component functions. This study aimed to analyse the loss of precision incurred by forcing of parameters in tree biomass models due to (i) additivity constraints, (ii) combining global and species-specific parameters, and (iii) estimating component functions simultaneously as a system instead of as individual equations. Based on biomass data from 697 trees including 13 different species, we estimated a set of compatibly additive, nonlinear biomass models using simultaneous estimation and compared these with less restricted model systems. In line with other similar studies, the overall model system explained 88%–99% of the variation in individual biomass components. Compared with the unrestricted model, restricting parameters to obtain compatible additivity resulted in a change in RMSE of –0.6% to 5.4%. When restricting parameter estimates using both species-specific and global parameters, RMSE increased by 1.7%–13.1%. Estimating model parameters using simultaneous estimation (nonlinear iterated seemingly unrelated regression, NSUR) increased model bias compared with ordinary least squares estimation (OLS) for most biomass components. Contrary to expectations, NSUR estimation did not lead to a reduction in the standard error of estimates.

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BAIYA, SUPARAT, i KASAMSUK UNGCHITTRAKOOL. "Modified inertial Mann’s algorithm and inertial hybrid". Carpathian Journal of Mathematics 39, nr 1 (30.07.2022): 27–43. http://dx.doi.org/10.37193/cjm.2023.01.02.

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"In this work, we introduce and study the modified inertial Mann’s algorithm and inertial hybrid algorithm for approximating some fixed points of a k-strict pseudo-contractive mapping in Hilbert spaces. Weak convergence to a solution of fixed-point problems for a k-strict pseudo-contractive mapping is obtained by using the modified inertial Mann’s algorithm. In order to obtain strong convergence, we introduce an inertial hybrid algorithm by using the inertial extrapolation method mixed with the convex combination of three iterated vectors and forcing for strong convergence by the hybrid projection method for a k-strict pseudo-contractive mapping in Hilbert spaces. The strong convergence theorem of the proposed method is proved under mild assumptions on the scalars. For illustrating the performance of the proposed algorithms, we provide some new nonlinear k-strict pseudo-contractive mappings which are not nonexpansive to create some numerical experiments to show the advantage of the two new inertial algorithms for a k-strict pseudo-contractive mapping."

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30

Blass, Andreas. "Marcia J. Groszek. Applications of iterated perfect set forcing. Annals of pure and applied logic, vol. 39 (1988), pp. 19– 53." Journal of Symbolic Logic 55, nr 1 (marzec 1990): 360–61. http://dx.doi.org/10.2307/2274996.

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Beal, Aubrey N., i Robert N. Dean. "A Random Stimulation Source for Evaluating MEMS Devices using an Exact Solvable Chaotic Oscillator". Additional Conferences (Device Packaging, HiTEC, HiTEN, and CICMT) 2015, DPC (1.01.2015): 001594–625. http://dx.doi.org/10.4071/2015dpc-wp32.

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MEMS devices are nearly ubiquitous, with applications ranging from automobiles to toys, medical equipment to missiles, and cell phones to industrial equipment. At the microscale, fabrication tolerances are significantly less precise than at the scale of traditional machining techniques. This can result in significant differences in the operating characteristics between otherwise identical MEMS devices. A wide bandwidth random excitation source is ideal for evaluating these components, whether used as the forcing function for an electromechanical shaker employed to measure transmissibility, or as a voltage source to evaluate actuator structure resonances and instabilities. An electronic chaotic oscillator provides an ideal wide bandwidth voltage source which is provably random from first principles and may be simply integrated for the aforementioned MEMS testing. This type of system is easily integrated through standard Si MEMS processes and readily lends itself to application as a built-in-self test (BIST) component. These systems guarantee uniform frequency content from D.C. up to 100kHz due to their characteristically random behavior and serve as a strong candidate for providing uniform spectral density to a device under test. The proposed system is a simple, electronic circuit that creates a random, wideband excitation voltage for observing characteristics of MEMS devices. This functionality is achieved by the analog, digital or mixed signal computation of nonlinear differential equations that describe various exactly solvable chaotic systems. By creating Si microsystems which perform these computations, these test sources may be readily fabricated as integrated BIST components for MEMS devices or fabricated separately and integrated by flip chip assembly techniques. Furthermore, by considering the iterated map of this particular category of stimulation source, a direct and easy measurement of the stimulation entropy may be monitored and corrected. This work begins as a theoretical treatment involving the Nonlinear Dynamics of these types of systems including chaotic systems which permit closed form solutions. These systems are described classically through nonlinear differential equations and intuitively through iterated maps. These techniques reveal inherent methods for entropy measurement in these sources which may be implemented and controlled easily using electronic circuits. Subsequently, the simulation, circuit design methodology, circuit simulation, fabrication, testing and hardware verification of these wideband chaotic sources is presented. The development of this work delineates simple, wideband electronic testing circuits which may be fully integrated with MEMS devices using standard Si MEMS processes. The resulting microsystem may be used as the forcing function when measuring transmissibility of MEMS devices, or as a BIST element to evaluate MEMS microstructure characteristics through direct microelectronic fabrication or flip chip integration.

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32

LÜCKE, PHILIPP, RALF SCHINDLER i PHILIPP SCHLICHT. "Σ1(κ)-DEFINABLE SUBSETS OF H(κ+)". Journal of Symbolic Logic 82, nr 3 (wrzesień 2017): 1106–31. http://dx.doi.org/10.1017/jsl.2017.36.

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AbstractWe study Σ1(ω1)-definable sets (i.e., sets that are equal to the collection of all sets satisfying a certain Σ1-formula with parameter ω1 ) in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is Σ1(ω1)-definable, the set of all stationary subsets of ω1 is not Σ1(ω1)-definable and the complement of every Σ1(ω1)-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$ is not Σ1(ω1)-definable. In contrast, we show that the existence of a Woodin cardinal is compatible with the existence of a Σ1(ω1)-definable well-ordering of H(ω2) and the existence of a Δ1(ω1)-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$. We also show that, if there are infinitely many Woodin cardinals and a measurable cardinal above them, then there is no Σ1(ω1)-definable uniformization of the club filter on ω1. Moreover, we prove a perfect set theorem for Σ1(ω1)-definable subsets of ${}_{}^{{\omega _1}}\omega _1^{}$, assuming that there is a measurable cardinal and the nonstationary ideal on ω1 is saturated. The proofs of these results use iterated generic ultrapowers and Woodin’s ℙmax-forcing. Finally, we also prove variants of some of these results for Σ1(κ)-definable subsets of κκ, in the case where κ itself has certain large cardinal properties.

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33

Argyros, I. K. "Forcing sequences and inexact Newton iterates in Banach space". Applied Mathematics Letters 13, nr 1 (styczeń 2000): 77–80. http://dx.doi.org/10.1016/s0893-9659(99)00148-2.

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34

Poveda, Alejandro. "Contributions to the Theory of Large Cardinals through the Method of Forcing". Bulletin of Symbolic Logic 27, nr 2 (czerwiec 2021): 221–22. http://dx.doi.org/10.1017/bsl.2021.22.

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AbstractThe dissertation under comment is a contribution to the area of Set Theory concerned with the interactions between the method of Forcing and the so-called Large Cardinal axioms.The dissertation is divided into two thematic blocks. In Block I we analyze the large-cardinal hierarchy between the first supercompact cardinal and Vopěnka’s Principle (Part I). In turn, Block II is devoted to the investigation of some problems arising from Singular Cardinal Combinatorics (Part II and Part III).We commence Part I by investigating the Identity Crisis phenomenon in the region comprised between the first supercompact cardinal and Vopěnka’s Principle. As a result, we generalize Magidor’s classical theorems [2] to this higher region of the large-cardinal hierarchy. Also, our analysis allows to settle all the questions that were left open in [1]. Finally, we conclude Part I by presenting a general theory of preservation of $C^{(n)}$ -extendible cardinals under class forcing iterations. From this analysis we derive several applications. For instance, our arguments are used to show that an extendible cardinal is consistent with “ $(\lambda ^{+\omega })^{\mathrm {HOD}}<\lambda ^+$ , for every regular cardinal $\lambda $ .” In particular, if Woodin’s HOD Conjecture holds, and therefore it is provable in ZFC + “There exists an extendible cardinal” that above the first extendible cardinal every singular cardinal $\lambda $ is singular in HOD and $(\lambda ^+)^{\textrm {{HOD}}}=\lambda ^+$ , there may still be no agreement at all between V and HOD about successors of regular cardinals.In Part II and Part III we analyse the relationship between the Singular Cardinal Hypothesis (SCH) with other relevant combinatorial principles at the level of successors of singular cardinals. Two of these are the Tree Property and the Reflection of Stationary sets, which are central in Infinite Combinatorics.Specifically, Part II is devoted to prove the consistency of the Tree Property at both $\kappa ^+$ and $\kappa ^{++}$ , whenever $\kappa $ is a strong limit singular cardinal witnessing an arbitrary failure of the SCH. This generalizes the main result of [3] in two senses: it allows arbitrary cofinalities for $\kappa $ and arbitrary failures for the SCH.In the last part of the dissertation (Part III) we introduce the notion of $\Sigma $ -Prikry forcing. This new concept allows an abstract and uniform approach to the theory of Prikry-type forcings and encompasses several classical examples of Prikry-type forcing notions, such as the classical Prikry forcing, the Gitik-Sharon poset, or the Extender Based Prikry forcing, among many others.Our motivation in this part of the dissertation is to prove an iteration theorem at the level of the successor of a singular cardinal. Specifically, we aim for a theorem asserting that every $\kappa ^{++}$ -length iteration with support of size $\leq \kappa $ has the $\kappa ^{++}$ -cc, provided the iterates belong to a relevant class of $\kappa ^{++}$ -cc forcings. While there are a myriad of works on this vein for regular cardinals, this contrasts with the dearth of investigations in the parallel context of singular cardinals. Our main contribution is the proof that such a result is available whenever the class of forcings under consideration is the family of $\Sigma $ -Prikry forcings. Finally, and as an application, we prove that it is consistent—modulo large cardinals—the existence of a strong limit cardinal $\kappa $ with countable cofinality such that $\mathrm {SCH}_\kappa $ fails and every finite family of stationary subsets of $\kappa ^+$ reflects simultaneously.

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35

Claverie, Benjamin, i Ralf Schindler. "Increasing u2 by a stationary set preserving forcing". Journal of Symbolic Logic 74, nr 1 (marzec 2009): 187–200. http://dx.doi.org/10.2178/jsl/1231082308.

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AbstractWe show that if I is a precipitous ideal on ω1 and if θ > ω1 is a regular cardinal, then there is a forcing ℙ = ℙ(I, θ) which preserves the stationarity of all I-positive sets such that in Vℙ, ⟨Hθ; ∈, I⟩ is a generic iterate of a countable structure ⟨M; ∈, Ī⟩. This shows that if the nonstationary ideal on ω1 is precipitous and exists, then there is a stationary set preserving forcing which increases . Moreover, if Bounded Martin's Maximum holds and the nonstationary ideal on ω1 is precipitous, then .

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36

Argyros, I. K. "Relations Between Forcing Sequences and Inexact Newton Iterates in Banach Space". Computing 63, nr 2 (1.09.1999): 131–44. http://dx.doi.org/10.1007/s006070050055.

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37

Argyros, Ioannis K. "Relations between forcing sequences and inexact newton-like iterates in banach space". International Journal of Computer Mathematics 71, nr 2 (styczeń 1999): 235–46. http://dx.doi.org/10.1080/00207169908804804.

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38

Si, Jian-Guo, i Sui Sun Cheng. "Smooth solutions of a nonhomogeneous iterative functional differential equation". Proceedings of the Royal Society of Edinburgh: Section A Mathematics 128, nr 4 (1998): 821–31. http://dx.doi.org/10.1017/s0308210500021806.

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This paper is concerned with an iterative functional differential equation x(t) = c1x(t) + c2x[2](t) + … cmχ[m](t) + F(t), where x[i](t) is the i-th iterate of the function x(t). By means of Schauder's Fixed Point Theorem, we establish a local existence theorem for smooth solutions which also depend continuously on the forcing function F(t).

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39

Kellner, Jakob, i Saharon Shelah. "Saccharinity". Journal of Symbolic Logic 76, nr 4 (grudzień 2011): 1153–83. http://dx.doi.org/10.2178/jsl/1318338844.

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AbstractWe present a method to iterate finitely splitting lim-sup tree forcings along non-wellfounded linear orders. As an application, we introduce a new method to force (weak) measurability of all definable sets with respect to a certain (non-ccc) ideal.

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40

Wang, Peiguang, i Xiang Liu. "Rapid Convergence for Telegraph Systems with Periodic Boundary Conditions". Journal of Function Spaces 2017 (2017): 1–10. http://dx.doi.org/10.1155/2017/1982568.

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The generalized quasilinearization method is applied in this paper to a telegraph system with periodic boundary conditions. We consider the case in which the forcing function F(t,x,U) satisfies the following condition: ∂n-1F(t,x,U)/∂Un-1 exists and is quasimonotone nondecreasing or nonincreasing. We develop nonlinear iterates of order n-1 which will be different with n being even or odd. Finally, we develop two sequences which converge to the solution of the telegraph system and the convergence is of order n.

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41

Andres, Jan. "Randomized Sharkovsky-type theorems and their application to random impulsive differential equations and inclusions on tori". Stochastics and Dynamics 19, nr 05 (19.08.2019): 1950036. http://dx.doi.org/10.1142/s0219493719500369.

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Our randomized versions of the Sharkovsky-type cycle coexistence theorems on tori and, in particular, on the circle are applied to random impulsive differential equations and inclusions. The obtained effective coexistence criteria for random subharmonics with various periods are formulated in terms of the Lefschetz numbers (in dimension one, in terms of degrees) of the impulsive maps and their iterates w.r.t. the (deterministic) state variables. Otherwise, the forcing properties of certain periods of the given random subharmonics are employed, provided there exists a random harmonic solution. In the single-valued case, the exhibition of deterministic chaos in the sense of Devaney is detected for random impulsive differential equations on the factor space [Formula: see text]. Several simple illustrative examples are supplied.

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42

Chang, Edmund K. M. "An Idealized Nonlinear Model of the Northern Hemisphere Winter Storm Tracks". Journal of the Atmospheric Sciences 63, nr 7 (1.07.2006): 1818–39. http://dx.doi.org/10.1175/jas3726.1.

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Abstract In this paper, a nonlinear dry model, forced by fixed radiative forcing alone, has been constructed to simulate the Northern Hemisphere winter storm tracks. A procedure has been devised to iterate the radiative equilibrium temperature profile such that at the end of the iterations the model climate closely resembles the desired target climate. This iterative approach is applied to simulate the climatological storm tracks in January. It is found that, when the three-dimensional temperature distribution in the model resembles the observed distribution, the model storm tracks are much too weak. It is hypothesized that this is due to the fact that eddy development is suppressed in a dry atmosphere, owing to the lack of latent heat release in the ascending warm air. To obtain storm tracks with realistic amplitudes, the static stability of the target climate is reduced to simulate the enhancement in baroclinic energy conversion due to latent heat release. With this modification, the storm tracks in the model simulation closely resemble those observed except that the strength of the Atlantic storm track is slightly weaker than observed. The model, when used as a forecast model, also gives high-quality forecasts of the evolution of observed eddies. The iterative approach is applied to force the model to simulate climate anomalies associated with ENSO and the interannual variations of the winter Pacific jet stream/storm tracks. The results show that the model not only succeeds in simulating the climatology of storm tracks, but also produces realistic simulations of storm track anomalies when the model climate is forced to resemble observed climate anomalies. An extended run of the control experiment is conducted to generate monthly mean flow and storm track statistics. These statistics are used to build a linear statistical model relating storm track anomalies to mean flow anomalies. This model performs well when used to hindcast observed storm track anomalies based on observed mean flow anomalies, showing that the storm track/mean flow covariability in the model is realistic and that storm track distribution is not sensitive to the exact form of the applied forcings.

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43

Britto, Abraham Benjamin, i Sathesh Mariappan. "Lock-in phenomenon of vortex shedding in oscillatory flows: an analytical investigation pertaining to combustors". Journal of Fluid Mechanics 872 (7.06.2019): 115–46. http://dx.doi.org/10.1017/jfm.2019.353.

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An analytical investigation is performed to understand the lock-in phenomenon, observed in vortex shedding combustors. Several aeroengine afterburners and ramjets use a bluff body to stabilize the flame. The bluff body sheds vortices. During the occurrence of high-amplitude combustion instability, the frequency of vortex shedding locks in to the frequency of the chamber acoustic field. This phenomenon is termed vortex-acoustic lock-in. In general, there is a two-way coupling between the vortex shedding process and the acoustic field, making analytical investigation difficult. Since the frequency of the latter remains largely unaltered, performing an investigation to study the response of vortex shedding to external excitation not only allows one to gain insights, but also make the problem analytically tractable. We begin with a lower-order model available in the literature to describe the vortex shedding process in non-reacting flows, arising from sharp corners in the presence of upstream velocity excitation. The continuous time domain model is transformed to a discrete map, which connects the time instances of two successive vortex shedding events. The frequency and amplitude of excitation are varied to study the instantaneous vortex shedding time period, as the response of the system. In the absence of forcing, the iterates of the map form a period-1 solution with the frequency equalling the natural vortex shedding frequency. On increasing the amplitude of excitation, quasi-periodic behaviour of the iterates is observed, followed by a period-1 lock-in solution, where vortex shedding occurs at the excitation frequency. On further increasing the amplitude, de-lock-in occurs. From the map, an analytical solution is extracted, which represents the lock-in state. The condition and thereby the region in the frequency–amplitude parameter space where a general$p:1$lock-in occurs is then identified. Several important analytical expressions, such as for (1) critical threshold frequency above which lock-in occurs, (2) boundary of lock-in region in the parameter space, that are of direct importance to the design of quieter combustors are obtained. The study also identifies the transition of higher-order$p:1$to$1:1$lock-in state, through a series of lock-in and de-lock-in steps, whose occurrence could be verified from future experiments.

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44

Lewis, Christian. "Anthony Trollope's Formal Experiment: Repetition and the Anti-romantic Marriage Plot". Novel: A Forum on Fiction 57, nr 1 (1.05.2024): 67–85. http://dx.doi.org/10.1215/00295132-11052367.

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Abstract In Miss Mackenzie, Anthony Trollope attempted to write a marriage plot without love or romance but admitted that he was unable to do so. This essay argues that Trollope's formal experiment developing the “anti-romantic” marriage plot did not end with Miss Mackenzie but continued across eighteen novels written from 1864 to 1876. The essay performs a midrange reading of this set of novels to argue that across this experiment Trollope iterates his anti-romantic marriage plot, replicating his central conflict and establishing three stock characters (moral maidens, hesitating heroines, and husband hunters). Trollope's experiment clashed against his own gender and class politics, forcing him to confront and reevaluate his own ideas and gradually empathize with women who choose to marry not for romance but for financial security. While Trollope's novels have, since their publication, been criticized as repetitious, the essay argues that his use of repetition is not a weakness but a strength, a formal tactic with unique affordances. This reframing offers a way to reconsider Trollope's literary technique and prodigious output as well as appreciate the impact that his experiment had on the trajectory of the Victorian realist novel. While the 1860s and 1870s are often considered the peak of Victorian high realism, Trollope's novels (and the influence they have on other major novelists) reveal how during these decades the marriage plot and realism itself was in crisis.

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45

Dobrinen, Natasha. "James Cummings and Ernest Schimmerling, editors. Lecture Note Series of the London Mathematical Society, vol. 406. Cambridge University Press, New York, xi + 419 pp. - Paul B. Larson, Peter Lumsdaine, and Yimu Yin. An introduction to Pmax forcing. pp. 5–23. - Simon Thomas and Scott Schneider. Countable Borel equivalence relations. pp. 25–62. - Ilijas Farah and Eric Wofsey. Set theory and operator algebras. pp. 63–119. - Justin Moore and David Milovich. A tutorial on set mapping reflection. pp. 121–144. - Vladimir G. Pestov and Aleksandra Kwiatkowska. An introduction to hyperlinear and sofic groups. pp. 145–185. - Itay Neeman and Spencer Unger. Aronszajn trees and the SCH. pp. 187–206. - Todd Eisworth, Justin Tatch Moore, and David Milovich. Iterated forcing and the Continuum Hypothesis. pp. 207–244. - Moti Gitik and Spencer Unger. Short extender forcing. pp. 245–263. - Alexander S. Kechris and Robin D. Tucker-Drob. The complexity of classification problems in ergodic theory. pp. 265–299. - Menachem Magidor and Chris Lambie-Hanson. On the strengths and weaknesses of weak squares. pp. 301–330. - Boban Veličković and Giorgio Venturi. Proper forcing remastered. pp. 331–362. - Asger ToÖrnquist and Martino Lupini. Set theory and von Neumann algebras. pp. 363–396. - W. Hugh Woodin, Jacob Davis, and Daniel RodrÍguez. The HOD dichotomy. pp. 397–419." Bulletin of Symbolic Logic 20, nr 1 (marzec 2014): 94–97. http://dx.doi.org/10.1017/bsl.2014.1.

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Eklof, Paul C. "Fred Appenzeller. An independence result in quadratic form theory: infinitary combinatorics applied to ε-Hermitian spaces. The journal of symbolic logic, vol. 54 (1989), pp. 689–699. - Otmar Spinas. Linear topologies on sesquilinear spaces of uncountable dimension. Fundamenta mathematicae, vol. 139 (1991), pp. 119–132. - James E. Baumgartner, Matthew Foreman, and Otmar Spinas. The spectrum of the Γ-invariant of a bilinear space. Journal of algebra, vol. 189 (1997), pp. 406–418. - James E. Baumgartner and Otmar Spinas. Independence and consistency proofs in quadratic form theory. The journal of symbolic logic, vol. 56 (1991), pp. 1195–1211. - Otmar Spinas. Iterated forcing in quadratic form theory. Israel journal of mathematics, vol. 79 (1992), pp. 297–315. - Otmar Spinas. Cardinal invariants and quadratic forms. Set theory of the reals, edited by Haim Judah, Israel mathematical conference proceedings, vol. 6, Gelbart Research Institute for Mathematical Sciences, Bar-Ilan University, Ramat-Gan 1993, distributed by the American Mathematical Society, Providence, pp. 563–581. - Saharon Shelah and Otmar Spinas. Gross spaces. Transactions of the American Mathematical Society, vol. 348 (1996), pp. 4257–4277." Bulletin of Symbolic Logic 7, nr 2 (czerwiec 2001): 285–86. http://dx.doi.org/10.2307/2687785.

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47

Corazza, Paul. "Forcing with Non-wellfounded Models". Australasian Journal of Logic 5 (30.11.2007). http://dx.doi.org/10.26686/ajl.v5i0.1784.

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We develop the machinery for performing forcing over an arbitrary (possibly non-wellfounded) model of set theory. For consistency results, this machinery is unnecessary since such results can always be legitimately obtained by assuming that the ground model is (countable) transitive. However, for establishing properties of a given (possibly non-wellfounded) model, the fully developed machinery of forcing as a means to produce new related models can be useful. We develop forcing through iterated forcing, paralleling the standard steps of presentation found in [19] and [14].

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48

Bakke Haga, Karen, David Schrittesser i Asger Törnquist. "Maximal almost disjoint families, determinacy, and forcing". Journal of Mathematical Logic, 10.05.2021, 2150026. http://dx.doi.org/10.1142/s0219061321500264.

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We study the notion of [Formula: see text]-MAD families where [Formula: see text] is a Borel ideal on [Formula: see text]. We show that if [Formula: see text] is any finite or countably iterated Fubini product of the ideal of finite sets [Formula: see text], then there are no analytic infinite [Formula: see text]-MAD families, and assuming Projective Determinacy and Dependent Choice there are no infinite projective [Formula: see text]-MAD families; and under the full Axiom of Determinacy + [Formula: see text] or under [Formula: see text] there are no infinite [Formula: see text]-mad families. Similar results are obtained in Solovay’s model. These results apply in particular to the ideal [Formula: see text], which corresponds to the classical notion of MAD families, as well as to the ideal [Formula: see text]. The proofs combine ideas from invariant descriptive set theory and forcing.

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49

Krishnamurthy, Deepak, i Manu Prakash. "Emergent programmable behavior and chaos in dynamically driven active filaments". Proceedings of the National Academy of Sciences 120, nr 28 (5.07.2023). http://dx.doi.org/10.1073/pnas.2304981120.

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How the behavior of cells emerges from their constituent subcellular biochemical and physical parts is an outstanding challenge at the intersection of biology and physics. A remarkable example of single-cell behavior occurs in the ciliate Lacrymaria olor , which hunts for its prey via rapid movements and protrusions of a slender neck, many times the size of the original cell body. The dynamics of this cell neck is powered by a coat of cilia across its length and tip. How a cell can program this active filamentous structure to produce desirable behaviors like search and homing to a target remains unknown. Here, we present an active filament model that allows us to uncover how a “program” (time sequence of active forcing) leads to “behavior” (filament shape dynamics). Our model captures two key features of this system—time-varying activity patterns (extension and compression cycles) and active stresses that are uniquely aligned with the filament geometry—a “follower force” constraint. We show that active filaments under deterministic, time-varying follower forces display rich behaviors including periodic and aperiodic dynamics over long times. We further show that aperiodicity occurs due to a transition to chaos in regions of a biologically accessible parameter space. We also identify a simple nonlinear iterated map of filament shape that approximately predicts long-term behavior suggesting simple, artificial “programs” for filament functions such as homing and searching space. Last, we directly measure the statistical properties of biological programs in L. olor , enabling comparisons between model predictions and experiments.

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Kaplan, Eyal. "The Magidor iteration and restrictions of ultrapowers to the ground model". Israel Journal of Mathematics, 28.10.2024. http://dx.doi.org/10.1007/s11856-024-2674-1.

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AbstractWe study the Magidor iteration of Prikry forcings below a measurable limit of measurables κ. We first characterize all the normal measures κ carries in the generic extension, building on and extending the main result of [1]. Then, for every such normal measure, we prove that the restriction of its ultrapower, from the generic extension to the ground model, is an iterated ultrapower of V by normal measures. This is done without core model theoretic assumptions; GCH≤κ in the ground model suffices.

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