Let M be a short extender mouse. We prove that if
$E\in M$
and
$M\models $
“E is a countably complete short extender whose support is a cardinal
$\theta $
and
$\mathcal {H}_\theta \subseteq \mathrm ...{Ult}(V,E)$
”, then E is in the extender sequence
$\mathbb {E}^M$
of M. We also prove other related facts, and use them to establish that if
$\kappa $
is an uncountable cardinal of M and
$\kappa ^{+M}$
exists in M then
$(\mathcal {H}_{\kappa ^+})^M$
satisfies the Axiom of Global Choice. We prove that if M satisfies the Power Set Axiom then
$\mathbb {E}^M$
is definable over the universe of M from the parameter
$X=\mathbb {E}^M\!\upharpoonright \!\aleph _1^M$
, and M satisfies “Every set is
$\mathrm {OD}_{\{X\}}$
”. We also prove various local versions of this fact in which M has a largest cardinal, and a version for generic extensions of M. As a consequence, for example, the minimal proper class mouse with a Woodin limit of Woodin cardinals models “
$V=\mathrm {HOD}$
”. This adapts to many other similar examples. We also describe a simplified approach to Mitchell–Steel fine structure, which does away with the parameters
$u_n$
.
In this paper, we study the stability of the ring solution of the N-body problem in the entire sphere
$\mathbb {S}^2$
by using the logarithmic potential proposed in Boatto et al. (2016, Proceedings ...of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 472, 20160020) and Dritschel (2019, Philosophical Transactions of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 377, 20180349), derived through a definition of central force and Hodge decomposition theorem for 1-forms in manifolds. First, we characterize the ring solution and study its spectral stability, obtaining regions (spherical caps) where the ring solution is spectrally stable for
$2\leq N\leq 6$
, while, for
$N\geq 7$
, the ring is spectrally unstable. The nonlinear stability is studied by reducing the system to the homographic regular polygonal solutions, obtaining a 2-d.o.f. Hamiltonian system, and therefore some classic results on stability for 2-d.o.f. Hamiltonian systems are applied to prove that the ring solution is unstable at any parallel where it is placed. Additionally, this system can be reduced to 1-d.o.f. by using the angular momentum integral, which enables us to describe the phase portraits and use them to find periodic ring solutions to the full system. Some of those solutions are numerically approximated.
Multiple factors operating across different spatial and temporal scales affect β-diversity, the variation in community composition among sites. Disentangling the relative influence of co-occurring ...ecological drivers over broad biogeographic gradients and time is critical to developing mechanistic understanding of community responses to natural environmental heterogeneity as well as predicting the effects of anthropogenic change. We partitioned taxonomic β-diversity in phytoplankton communities across 75 north-temperate lakes and reservoirs in Alberta, Canada, using data-driven, spatially constrained null models to differentiate between spatially structured, spatially independent, and spuriously correlated associations with a suite of biologically relevant environmental variables. Phytoplankton β-diversity was largely independent of space, indicating spatial processes (e.g., dispersal limitation) likely play a minor role in structuring communities at the regional scale. Our analysis also identified seasonal differences in the importance of environmental factors, suggesting a general shift toward greater relevance of local, in-lake (e.g., nutrients and Secchi depth) over regional, atmospheric and catchment-level (e.g., monthly solar radiation and grassland coverage) drivers as the open-water growing season progressed. Several local and regional variables explained taxonomic variation jointly, reflecting climatic and land-use linkages (e.g., air temperature and water column stability or pastureland and nutrient enrichment) that underscore the importance of understanding how phytoplankton communities integrate, and may serve as sentinels of, broader anthropogenic changes. We also discovered similar community composition in natural and constructed water bodies, demonstrating rapid filtering of regional species to match local environmental conditions in reservoirs comparable to those in natural habitats. Finally, certain factors related to human footprint (e.g., cropland development) explained the composition of bloom-forming and/or toxic cyanobacteria more than the overall phytoplankton community, suggesting their heightened importance to integrated watershed management.
We prove Fermat’s Last Theorem over
$\mathbb {Q}(\sqrt {5})$
and
$\mathbb {Q}(\sqrt {17})$
for prime exponents
$p \ge 5$
in certain congruence classes modulo
$48$
by using a combination of the ...modular method and Brauer–Manin obstructions explicitly given by quadratic reciprocity constraints. The reciprocity constraint used to treat the case of
$\mathbb {Q}(\sqrt {5})$
is a generalization to a real quadratic base field of the one used by Chen and Siksek. For the case of
$\mathbb {Q}(\sqrt {17})$
, this is insufficient, and we generalize a reciprocity constraint of Bennett, Chen, Dahmen, and Yazdani using Hilbert symbols from the rational field to certain real quadratic fields.
Abstract
The stereotypic and oversimplified relationship between female sex hormones and
undesirable behavior dates to the earliest days of human society, as already the
ancient Greek word for the ...uterus, “hystera” indicated an
aversive connection. Remaining and evolving throughout the centuries,
transcending across cultures and various aspects of everyday life, its
perception was only recently reframed. Contemporarily, the complex interaction
of hormonal phases (i. e., the menstrual cycle), hormonal medication
(i. e., oral contraceptives), women’s psychological well-being,
and behavior is the subject of multifaceted and more reflected discussions. A
driving force of this ongoing paradigm shift was the introduction of this highly
interesting and important topic into the realm of scientific research. This
refers to neuroscientific research as it enables a multimodal approach combining
aspects of physiology, medicine, and psychology. Here a growing body of
literature points towards significant alterations of both brain function, such
as lateralization of cognitive functions, and structure, such as gray matter
concentrations, due to fluctuations and changes in hormonal levels. This
especially concerns female sex hormones. However, the more research is conducted
within this field, the less reliable these observations and derived insights
appear. This may be due to two particular factors: measurement inconsistencies
and diverse hormonal phases accompanied by interindividual differences. The
first factor refers to the prominent unreliability of one of the primarily
utilized neuroscientific research instruments: functional magnetic resonance
imaging (fMRI). This unreliability is seemingly present in paradigms and
analyses, and their interplay, and is additionally affected by the second
factor. In more detail, hormonal phases and levels further influence
neuroscientific results obtained through fMRI as outcomes vary drastically
across different cycle phases and medication. This resulting vast uncertainty
thus tremendously hinders the further advancement of our understanding of how
female sex hormones might alter brain structure and function and, ultimately,
behavior.
This review summarizes parts of the current state of research and outlines the
essential requirements to further investigate and understand the female
brain’s underlying physiological and anatomical features.
On the basis of Poincaré and Weyl’s view of predicativity as invariance, we develop an extensive framework for predicative, type-free first-order set theory in which
$\Gamma _0$
and much bigger ...ordinals can be defined as von Neumann ordinals. This refutes the accepted view of
$\Gamma _0$
as the “limit of predicativity”.
Let G be a locally compact unimodular group, and let
$\phi $
be some function of n variables on G. To such a
$\phi $
, one can associate a multilinear Fourier multiplier, which acts on some n-fold ...product of the noncommutative
$L_p$
-spaces of the group von Neumann algebra. One may also define an associated Schur multiplier, which acts on an n-fold product of Schatten classes
$S_p(L_2(G))$
. We generalize well-known transference results from the linear case to the multilinear case. In particular, we show that the so-called “multiplicatively bounded
$(p_1,\ldots ,p_n)$
-norm” of a multilinear Schur multiplier is bounded above by the corresponding multiplicatively bounded norm of the Fourier multiplier, with equality whenever the group is amenable. Furthermore, we prove that the bilinear Hilbert transform is not bounded as a vector-valued map
$L_{p_1}(\mathbb {R}, S_{p_1}) \times L_{p_2}(\mathbb {R}, S_{p_2}) \rightarrow L_{1}(\mathbb {R}, S_{1})$
, whenever
$p_1$
and
$p_2$
are such that
$\frac {1}{p_1} + \frac {1}{p_2} = 1$
. A similar result holds for certain Calderón–Zygmund-type operators. This is in contrast to the nonvector-valued Euclidean case.
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 investigate Maker–Breaker games on graphs of size
$\aleph _1$
in which Maker’s goal is to build a copy of the host graph. We establish a firm dependence of the outcome of the game on the axiomatic ...framework. Relating to this, we prove that there is a winning strategy for Maker in the
$K_{\omega ,\omega _1}$
-game under ZFC+MA+
$\neg $
CH and a winning strategy for Breaker under ZFC+CH. We prove a similar result for the
$K_{\omega _1}$
-game. Here, Maker has a winning strategy under ZF+DC+AD, while Breaker has one under ZFC+CH again.
In this note, we revisit Ramanujan-type series for
$\frac {1}{\pi }$
and show how they arise from genus zero subgroups of
$\mathrm {SL}_{2}(\mathbb {R})$
that are commensurable with
$\mathrm ...{SL}_{2}(\mathbb {Z})$
. As illustrations, we reproduce a striking formula of Ramanujan for
$\frac {1}{\pi }$
and a recent result of Cooper et al., as well as derive a new rational Ramanujan-type series for
$\frac {1}{\pi }$
. As a byproduct, we obtain a Clausen-type formula in some general sense and reproduce a Clausen-type quadratic transformation formula closely related to the aforementioned formula of Ramanujan.
