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We show that in many extender models, e.g., the minimal one with infinitely many Woodin cardinals or the minimal with a Woodin cardinal that is a limit of Woodin cardinals, there are no generic embeddings with critical point $\omega _1$ that resemble the stationary tower at the second Woodin cardinal. The meaning of “resemble” is made precise in the paper (see Definition 0.3).

We show that if the universe is self-iterable and κ \kappa is an inaccessible limit of Woodin cardinal then A D R + “ Θ \mathsf {AD}_{\mathbb {R}}+“\Theta is regular” holds in the derived model at κ \kappa . The proof is fine-structure free, and only assumes basic knowledge of iteration trees and iteration strategies. Our proof can be viewed as the fine-structure free version of the well-known fact that A D R + “ Θ \mathsf {AD}_{\mathbb {R}}+“\Theta is regular” is true in the derived models of hod mice that have inaccessible limit of Woodin cardinals (see for example Sargsyan [Mem. Amer. Math. Soc. 236 (2015), p. viii+172]). However, the proof uses a different set of ideas and is more general.

A set of reals is universally Baire if all of its continuous preimages in topological spaces have the Baire property. ${\sf Sealing}$ is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by set forcings. The ${\sf Largest\ Suslin\ Axiom}$ ( ${\sf LSA}$ ) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable surjections. Let ${\sf LSA}$ - ${\sf over}$ - ${\sf uB}$ be the statement that in all (set) generic extensions there is a model of $\sf {LSA}$ whose Suslin, co-Suslin sets are the universally Baire sets. We outline the proof that over some mild large cardinal theory, $\sf {Sealing}$ is equiconsistent with $\sf {LSA}$ - $\sf {over}$ - $\sf {uB}$ . In fact, we isolate an exact theory (in the hierarchy of strategy mice) that is equiconsistent with both (see Definition 3.1). As a consequence, we obtain that $\sf {Sealing}$ is weaker than the theory “ $\sf {ZFC}$ + there is a Woodin cardinal which is a limit of Woodin cardinals.” This significantly improves upon the earlier consistency proof of $\sf {Sealing}$ by Woodin. A variation of $\sf {Sealing}$ , called $\sf {Tower \ Sealing}$ , is also shown to be equiconsistent with $\sf {Sealing}$ over the same large cardinal theory. We also outline the proof that if V has a proper class of Woodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy, then $\sf {Sealing}$ holds after collapsing the successor of the least strong cardinal to be countable. This result is complementary to the aforementioned equiconsistency result, where it is shown that $\sf {Sealing}$ holds in a generic extension of a certain minimal universe. This theorem is more general in that no minimal assumption is needed. A corollary of this is that $\sf {LSA}$ - $\sf {over}$ - $\sf {uB}$ is not equivalent to $\sf {Sealing}$ .

We show that if V V has a proper class of Woodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy (as defined in the paper) then S e a l i n g \mathsf {Sealing} holds after collapsing the successor of the least strong cardinal to be countable. This result is complementary to other work by the authors where it is shown that S e a l i n g \mathsf {Sealing} holds in a generic extension of a certain minimal universe. The current theorem is more general in that no minimality assumption is needed. A corollary of the main theorem is that S e a l i n g \mathsf {Sealing} is consistent relative to the existence of a Woodin cardinal which is a limit of Woodin cardinals. This improves significantly on the first consistency of S e a l i n g \mathsf {Sealing} obtained by W.H. Woodin. The L a r g e s t S u s l i n A x i o m \mathsf {Largest\ Suslin\ Axiom} ( L S A \mathsf {LSA} ) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable bijections. Let L S A - o v e r - u B \mathsf {LSA\text {-}over\text {-}uB} be the statement that in all (set) generic extensions there is a model of L S A \mathsf {LSA} whose Suslin, co-Suslin sets are the universally Baire sets. The other main result of the paper shows that assuming V V has a proper class of inaccessible cardinals which are limit of Woodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy, in the universe V [ g ] V[g] , where g g is V V -generic for the collapse of the successor of the least strong cardinal to be countable, the theory L S A \mathsf {LSA} - o v e r \mathsf {over} - U B \mathsf {UB} fails; this implies that L S A - o v e r - U B \mathsf {LSA\text {-}over\text {-}UB} is not equivalent to S e a l i n g \mathsf {Sealing} (over the base theory of V [ g ] V[g] ). This is interesting and somewhat unexpected, in light of other work by the authors. Compare this result with Steel’s well-known theorem that “ A D L ( R ) \mathsf {AD}^{L(\mathbb {R})} holds in all generic extensions” is equivalent to “the theory of L ( R ) L(\mathbb {R}) is sealed” in the presence of a proper class of measurable cardinals.

We show that assuming the determinacy of all games on reals, every set of reals is $$\Theta $$ universally baire.

The main goal of bridge health monitoring is to show the condition of structure in real time mode with using special sensors, which can measure different characteristics, which can indicate about the changes in structure. One of the easier measured indicators what can be used is frequency of structure vibration. The diagnostic of the health condition of structure is dependent of correct evaluation of changes range of measured indicators. This paper presents the theoretical calculations to study the effect of cracking on the frequency response of typical reinforced concrete T type beams which has widespread in road bridges construction.

We re-prove Becker’s theorem from Becker (Isr J Math 40(3–4):229–234, 1981) by showing that $$AD^{L({\mathbb {R}})}$$ implies that $$L({\mathbb {R}})\vDash ``\omega _2$$ is -supercompact”. Our proof uses inner model theoretic tools instead of Baire category. We also show that $$\omega _2$$ is $$

We introduce a mouse whose derived model satisfies $AD_ + {\rm{\Theta }} \ge \theta _{\aleph _2 } $. More generally, we will introduce a class of large cardinal properties yielding mice whose derived models can satisfy properties as strong as $AD_ + {\rm{\Theta }} = \theta _{\rm{\Theta }} $.

Suppose M is a transitive class size model of A D R + “Θ is regular” . M is a minimal model of A D R + “Θ is measurable” if (i) R , O r d ⊆ M (ii) there is μ ∈ M such that M ⊨ “ μ is a normal R -complete measure on Θ” and (iii) for any transitive class size N ⊊ M such that R ⊆ N , N ⊨ “there is no R -complete measure on Θ” . Continuing Trang's work in [8] , we compute HOD of a minimal model of A D R + “Θ is measurable” .

We examine nucleon-nucleon realistic interactions, based on their SU(3) decomposition to SU(3)-symmetric components. We find that many of these interaction components are negligible, which, in turn, allows us to identify a subset of physically relevant components that are sufficient to describe the structure of low-lying states in $^{12}$C and related observables, such as excitation energies, electric quadrupole transitions and rms radii. We find that paring the interaction down to half of the SU(3)-symmetric components or more yields results that practically coincide with the corresponding ab initio calculations with the full interaction. In addition, we show that while various realistic interactions differ in their SU(3) decomposition, their renormalized effective counterparts exhibit a striking similarity and composition that can be linked to dominant nuclear features such as deformation, pairing, clustering, and spin-orbit effect., 9 pages, 7 figures

We analyze the hereditarily ordinal definable sets $\operatorname{HOD}$ in $M_n(x)[g]$ for a Turing cone of reals $x$, where $M_n(x)$ is the canonical inner model with $n$ Woodin cardinals build over $x$ and $g$ is generic over $M_n(x)$ for the L\'evy collapse up to its bottom inaccessible cardinal. We prove that assuming $\boldsymbol\Pi^1_{n+2}$-determinacy, for a Turing cone of reals $x$, $\operatorname{HOD}^{M_n(x)[g]} = M_n(\mathcal{M}_{\infty} | \kappa_\infty, \Lambda),$ where $\mathcal{M}_\infty$ is a direct limit of iterates of $M_{n+1}$, $\delta_\infty$ is the least Woodin cardinal in $\mathcal{M}_\infty$, $\kappa_\infty$ is the least inaccessible cardinal in $\mathcal{M}_\infty$ above $\delta_\infty$, and $\Lambda$ is a partial iteration strategy for $\mathcal{M}_{\infty}$. It will also be shown that under the same hypothesis $\operatorname{HOD}^{M_n(x)[g]}$ satisfies $\operatorname{GCH}$., Comment: 30 pages

We develop a basic translation procedure that translates a hod mouse to an equivalent mouse. Unlike the translation procedure used by Steel and Zhu, our procedure works in a coarse setting without assuming AD. Nevertheless, the procedure resembles the one developed by Steel. We use the translation procedures to answer a question of Trevor Wilson. Namely, we show that if there is a stationary class of lambda such that lambda is a limit of Woodin cardinals and the derived model at lambda satisfies AD the first member of the Solovay sequence is is less than Theta then there is a transitive model M such that M contains the ordinals and satisfies there is a proper class of Woodin cardinals and a strong cardinal.

We introduce a new framework for studying clustering and for calculating alpha partial widths using ab initio wave functions. We demonstrate the formalism for $^{20}$Ne, by calculating the overlap between the $^{16}$O$+\alpha$ cluster configuration and states in $^{20}$Ne computed in the ab initio symmetry-adapted no-core shell model. We present spectroscopic amplitudes and spectroscopic factors, and compare those to no-core symplectic shell-model results in larger model spaces, to gain insight into the underlying physics that drives alpha-clustering. Specifically, we report on the alpha partial width of the lowest $1^-$ resonance in $^{20}$Ne, which is found to be in good agreement with experiment. We also present first no-core shell-model estimates for asymptotic normalization coefficients for the ground state, as well as for the first excited $4^{+}$ state in $^{20}$Ne that lies in a close proximity to the $^{16}$O$+\alpha$ threshold. This outcome highlights the importance of correlations for developing cluster structures and for describing alpha widths. The widths can then be used to calculate alpha-capture reaction rates for narrow resonances of interest to astrophysics. We explore the reaction rate for the alpha-capture reaction $^{16}$O$(\alpha,\gamma)^{20}$Ne at astrophysically relevant temperatures and determine its impact on simulated X-ray burst abundances.

Let Msw denote the least iterable inner model with a strong cardinal above a Woodin cardinal. By [11], Msw has a fully iterable core model, ${K^{{M_{{\rm{sw}}}}}}$, and Msw is thus the least iterable extender model which has an iterable core model with a Woodin cardinal. In V, ${K^{{M_{{\rm{sw}}}}}}$ is an iterate of Msw via its iteration strategy Σ.We here show that Msw has a bedrock which arises from ${K^{{M_{{\rm{sw}}}}}}$ by telling ${K^{{M_{{\rm{sw}}}}}}$ a specific fragment ${\rm{\bar{\Sigma }}}$ of its own iteration strategy, which in turn is a tail of Σ. Hence Msw is a generic extension of $L[{K^{{M_{{\rm{sw}}}}}},{\rm{\bar{\Sigma }}}]$, but the latter model is not a generic extension of any inner model properly contained in it.These results generalize to models of the form Ms (x) for a cone of reals x, where Ms (x) denotes the least iterable inner model with a strong cardinal containing x. In particular, the least iterable inner model with a strong cardinal above two (or seven, or boundedly many) Woodin cardinals has a 2-small core model K with a Woodin cardinal and its bedrock is again of the form $L[K,{\rm{\bar{\Sigma }}}]$.

We present a covering conjecture that we expect to be true below superstrong cardinals. We then show that the conjecture is true in hod mice. This work is a continuation of the work that started in Covering with Universally Baire Functions Advances in Mathematics, and the main conjecture of the current paper is a revision of the UB Covering Conjecture of the aforementioned paper.

By forcing with $\mathbb{P}_{\rm max}$ over strong models of determinacy, we obtain models where different square principles at $\omega_2$ and $\omega_3$ fail. In particular, we obtain a model of $2^{\aleph_0}=2^{\aleph_1}=\aleph_2 + \lnot\square(\omega_2) + \lnot\square(\omega_3)$., Comment: Revised version, incorporating the referee's suggestions

This article discusses the emergence – from first principles – of collectivity and clustering in light and intermediate-mass nuclei, up to the calcium region, with implications for reproducing enhanced E2 transitions in deformed nuclei, and for further understanding the formation of alpha clustering. Structure calculations use nucleon-nucleon chiral potentials and are based on the ab initio symmetry-adapted no-core shell model (SA-NCSM) that expands the reach of the no-core shell-model theory to heavier nuclei and enables the description of spatially enhanced nuclear configurations. This is achieved by using a physically relevant symmetry-adapted basis, which exploits approximate symmetries that, we find, dominate the nuclear dynamics. Important implications that have deepened our understanding of emergent phenomena in nuclei, such as enhanced collectivity, giant resonances, pairing, and clustering, are discussed for Be, C, O, Ne, and Ti isotopes.This article discusses the emergence – from first principles – of collectivity and clustering in light and intermediate-mass nuclei, up to the calcium region, with implications for reproducing enhanced E2 transitions in deformed nuclei, and for further understanding the formation of alpha clustering. Structure calculations use nucleon-nucleon chiral potentials and are based on the ab initio symmetry-adapted no-core shell model (SA-NCSM) that expands the reach of the no-core shell-model theory to heavier nuclei and enables the description of spatially enhanced nuclear configurations. This is achieved by using a physically relevant symmetry-adapted basis, which exploits approximate symmetries that, we find, dominate the nuclear dynamics. Important implications that have deepened our understanding of emergent phenomena in nuclei, such as enhanced collectivity, giant resonances, pairing, and clustering, are discussed for Be, C, O, Ne, and Ti isotopes.

We show that the Mouse Set Conjecture for sets of reals is true in the minimal model of ADℝ + “Θ is regular”. As a consequence, we get that below ADℝ + “Θ is regular”, models of AD++¬ADℝ are hybrid mice over ℝ. Such a representation of models of AD+ is important in core model induction applications.

We introduce a covering conjecture and show that it holds below A D R + “ Θ is regular” . We then use it to show that in the presence of mild large cardinal axioms, PFA implies that there is a transitive model containing the reals and ordinals and satisfying A D R + “ Θ is regular” . The method used to prove the Main Theorem of this paper is the core model induction. The paper contains the first application of the core model induction that goes significantly beyond the region of A D + + θ 0 Θ .

The competition of isovector and isoscalar pairing in A=18 and 20 even-even N≈Z nuclei is analyzed in the framework of the mean-field plus the dynamic quadurpole-quadurpole, pairing and particle-hole interactions, whose Hamiltonian is diagonalized in the basis in the L = 0 configuration subspace. Besides the pairing interaction, it is observed that the quadurpole-quadurpole and particle-hole interactions also play a significant role in determining the relative positions of low-lying excited 0+ and 1+ levels and their energy gaps, which can result in the ground state first-order quantum phase transition from J = 0 to J = 1. The strengths of the isovector and isoscalar pairing interactions in these even-even nuclei are estimated with respect to the energy gap and the total contribution to the binding energy. Most importantly, it is shown that although the mechanism of the particle-hole contribution to the binding energy is different, it is indirectly related to the Wigner term in the binding energy.

Exploiting exact and special symmetries to unmask simplicity within complexity, which remains the “holy grail” of nuclear physics, will be considered within its historical context and as evolving through 21st century ab initio methods, including emerging results linked to the internal structure of nucleons. Some exemplar results for very light to medium mass nuclei will be presented, and what these may portend for heavier systems, including species beyond known lines of stability, will be proffered.

Working under AD, we investigate the length of prewellorderings given by the iterates of ℳ2k+1, which is the minimal proper class mouse with 2k + 1 many Woodin cardinals. In particular, we answer some questions from [4] (the discussion of the questions appears in the last section of [2]).

Starting from a supercompact cardinal κ , we force and construct a model in which κ is both the least strongly compact and least supercompact cardinal and κ ’s strong compactness, but not its supercompactness, is indestructible under arbitrary κ -directed closed forcing.

We obtain an equiconsistency for a weak form of universal indestructibility for strongness. The equiconsistency is relative to a cardinal weaker in consistency strength than a Woodin cardinal, Stewart Baldwin's notion of hyperstrong cardinal. We also briefly indicate how our methods are applicable to universal indestructibility for supercompactness and strong compactness.

We investigate the indestructibility properties of strongly compact cardinals in universes where strong compactness suffers from identity crisis. We construct an iterative poset that can be used to establish Kimchi–Magidor theorem from (in The independence between the concepts of compactness and supercompactness, circulated manuscript), i.e., that the first n strongly compact cardinals can be the first n measurable cardinals. As an application, we show that the first n strongly compact cardinals can be the first n measurable cardinals while the strong compactness of each strongly compact cardinal is indestructible under Levy collapses (our theorem is actually more general, see Sect. 3). A further application is that the class of strong cardinals can be nonempty yet coincide with the class of strongly compact cardinals while strong compactness of any strongly compact cardinal κ is indestructible under κ-directed closed posets that force GCH at κ.

As the monetary and financial systems in transition economies have evolved so have monetary policies. Many central banks have followed a course from exchange rate targeting, to targeting monetary aggregates to targeting inflation. We examine the development of monetary policy in Armenia, a small, open transition economy that adopted inflation targeting in January 2006. First, we review monetary policy in Armenia. Then we evaluate the capacity of the Central Bank of Armenia to implement inflation targeting. Finally, we estimate and simulate reaction functions to evaluate the effectiveness of monetary policy. We find that the Central Bank may have been overly restrictive. Comparative Economic Studies (2008) 50, 421–437. doi:10.1057/ces.2008.22

During his Fall 2005 set theory seminar, Woodin asked whether V-supercompactness implies HOD-supercompactness. We show, as he predicted, that that the answer is no.

We establish two theorems concerning strongly compact cardinals and universal indestructibility for degrees of supercompactness. In the first theorem, we show that universal indestructibility for degrees of supercompactness in the presence of a strongly compact cardinal is consistent with the existence of a proper class of measurable cardinals. In the second theorem, we show that universal indestructibility for degrees of supercompactness is consistent in the presence of two non-supercompact strongly compact cardinals, each of which exhibits a significant amount of indestructibility for its strong compactness.

We show that it is consistent, relative to n ∈ ω supercompact cardinals, for the strongly compact and measurable Woodin cardinals to coincide precisely. In particular, it is consistent for the first n strongly compact cardinals to be the first n measurable Woodin cardinals, with no cardinal above the n th strongly compact cardinal being measurable. In addition, we show that it is consistent, relative to a proper class of supercompact cardinals, for the strongly compact cardinals and the cardinals which are both strong cardinals and Woodin cardinals to coincide precisely. We also show how the techniques employed can be used to prove additional theorems about possible relationships between Woodin cardinals and strongly compact cardinals.

In this note, we provide an affirmative answer to the title question by giving two examples of cardinals satisfying conditions implying they are non-Rowbottom which can be turned into Rowbottom cardinals via forcing. In our second example, our cardinal is also non-Jonsson.

Working in the theory ”ZF + There is a nontrivial elementary embedding j : V → V“, we show that a final segment of cardinals satisfies certain square bracket finite and infinite exponent partition relations. As a corollary to this, we show that this final segment is composed of Jonsson cardinals. We then show how to force and bring this situation down to small alephs. A prototypical result is the construction of a model for ZF in which every cardinal μ ≥ ℵ2 satisfies the square bracket infinite exponent partition relation . We conclude with a discussion of some consistency questions concerning different versions of the axiom asserting the existence of a nontrivial elementary embedding j: V → V. By virtue of Kunen's celebrated inconsistency result, we use only a restricted amount of the Axiom of Choice.

Assuming V=L(R)+AD, using methods from inner model theory, we give a new proof of the strong partition property for δ∼12. The result was originally proved by Kechris et al.

In this paper, we show that the failure of the unique branch hypothesis ([Formula: see text]) for tame iteration trees implies that in some homogenous generic extension of [Formula: see text] there is a transitive model [Formula: see text] containing [Formula: see text] such that [Formula: see text] is regular. The results of this paper significantly extend earlier works from [Non-tame mice from tame failures of the unique branch bypothesis, Canadian J. Math. 66(4) (2014) 903–923; Core models with more Woodin cardinals, J. Symbolic Logic 67(3) (2002) 1197–1226] for tame trees.

In this paper, we show that the failure of the unique branch hypothesis (UBH) for tame trees (see Definition 5.1) implies that in some homogenous generic extension of V there is a transitive model M containing Ord∪R such that M AD + Θ > θ0. In particular, this implies the existence (in V ) of a non-tame mouse. The results of this paper significantly extend Steel’s earlier results from [16] for tame trees. In this paper, we establish, using the core model induction, a lower bound for certain failures of the Unique Branch Hypothesis, (UBH), which is the statement that every iteration ∗2000 Mathematics Subject Classifications: 03E15, 03E45, 03E60. †

A paper for general audience about descriptive inner model theory., To appear in BSL

Building on the work of Schimmerling [Coherent sequences and threads, Adv. Math.216(1) (2007) 89–117] and Steel [PFA implies AD L(ℝ), J. Symbolic Logic70(4) (2005) 1255–1296], we show that the failure of square principle at a singular strong limit cardinal implies that there is a nontame mouse. The proof presented is the first inductive step beyond L(ℝ) of the core model induction that is aimed at getting a model of ADℝ + "Θ is regular" from the failure of square at a singular strong limit cardinal or PFA.

This paper is the first attempt to look at inflation dynamics and monetary transmission mechanisms in Armenia in the context of a full information model containing three interrelated markets: foreign exchange, money, and labor. Using the vector error correction model (VECM) approach, we find that the exchange rate pass-through to prices is very strong relative to credit, wage, and interest rate channels. The analysis suggests a relatively fast adjustment of prices to long-run disequilibria in the exchange rate market, albeit with initial overshooting of the price level. In addition, we find no evidence of prices responding to changes in money and wages in a statistically significant manner.