Structure of the Period Space

Permanent groundwater storage in basaltic dyke fractures and termite mound viability

We introduce a new method to approximate integrals∫Rdf(x)dx\int _{\mathbb {R}^d} f(\boldsymbol {x}) \,\mathrm {d}\boldsymbol {x}which simply scales lattice rules from the unit cube[0,1]d[0,1]^dto properly sized boxes onRd\mathbb {R}^d, hereby achieving higher-order convergence that matches the smoothness of the integrand functionffin a certain Sobolev space of dominating mixed smoothness. Our method only assumes that we can evaluate the integrand functionffand does not assume a particular density nor the ability to sample from it. In particular, for the theoretical analysis we show a new result that the method of adding Bernoulli polynomials to a function to make it “periodic” on a box without changing its integral value over the box is equivalent to an orthogonal projection from a well chosen Sobolev space of dominating mixed smoothness to an associated periodic Sobolev space of the same dominating mixed smoothness, which we call a Korobov space. We note that the Bernoulli polynomial method is often not used because of its excessive computational complexity and also here we only make use of it in our theoretical analysis. We show that our new method of applying scaled lattice rules to increasing boxes can be interpreted as orthogonal projections with decreasing projection error. Such a method would not work on the unit cube since then the committed error caused by non-periodicity of the integrand would be constant, but for integration on the Euclidean space we can use the certain decay towards zero when the boxes grow. Hence we can bound the truncation error as well as the projection error and show higher-order convergence in applying scaled lattice rules for integration on Euclidean space. We illustrate our theoretical analysis by numerical experiments which confirm our findings.

In this paper it is shown that every periodic space of interior type with a real simple group of motions is completely determined by the set of simple roots of the reduced and irreducible root system. A relationship is established between periodic spaces with flag manifolds and Sabinin trisymmetric spaces. A certain generalization of trisymmetric spaces is considered.

Aims. We analyze systematics in the asteroseismological mass determination methods in pulsating PG 1159 stars. Methods. We compare the seismic masses resulting from the comparison of the observed mean period spacings with the usually adopted asymptotic period spacings, �� a, and the average of the computed period spacings, ��l. Computations are based on full PG1159 evolutionary models with stellar masses ranging from 0.530 to 0.741M⊙ that take into account the complete evolution of progenitor stars. Results. We conclude that asteroseismology is a precise and powerful technique that determines the masses to a high internal accuracy, but it depends on the adopted mass determination method. In particular, we find that in the case of pulsating PG 1159 stars characterized by short pulsation periods, like PG 2131+066 and PG 0122+200, the employment of the asymptotic period spacings overestimates the stellar mass by about 0.06 M⊙ as compared with inferences from the average of the period spacings. In this case, the discrepancy between asteroseismological and spectroscopical masses is markedly reduced when use is made of the mean period spacing ��l instead of the asymptotic period spacing �� a.

We develop the analog in equal positive characteristic of Fontaine’s theory for crystalline Galois representations of a p-adic eld. In particular we describe the analog of Fontaine’s functor which assigns to a crystalline Galois representation an isocrystal with a Hodge ltration. In equal characteristic the role of isocrystals and Hodge ltrations is played by z-isocrystals and Hodge-Pink structures. The latter were invented by Pink. Our rst main result in this article is the analog of the Colmez-Fontaine Theorem that \weakly implies admissible. Next we construct period spaces for Hodge-Pink structures on a xed z-isocrystal. These period spaces are analogs of the Rapoport-Zink period spaces for Fontaine’s ltered isocrystals in mixed characteristic and likewise are rigid analytic spaces. For our period spaces we prove the analog of a conjecture of Rapoport and Zink stating the existence of a \universal local system on a Berkovich open subspace of the period space. As a consequence of \weakly implies admissible this Berkovich open subspace contains every classical rigid analytic point of the period space. As the principal tool to demonstrate these results we use the analog of Kedlaya’s Slope Filtration Theorem which we also formulate and prove here.

KIC 10553698A is a hot pulsating subdwarf B (sdB) star observed by the Kepler satellite. It exhibits dipole (l = 1) and quadrupole (l = 2) gravity modes with a clear period spacing structure. The seismic properties of the KIC 10553698A provide a test of stellar evolution models, and offer a unique opportunity to determine mixing processes. We consider mixing due to convective overshooting beyond the boundary of the helium burning core. Very small overshooting ( f = 10^{-6} ) results in a progressive increase in the size of convective core. However, moderate ( f = 10^{-2} ) and small ( f = 10^{-5} ) overshooting both lead to the occurrence of inert outer convective shells in the near-core region. We illustrate that the chemical stratifications induced by convective shells are able to change the g-mode period spacing pattern of a sdB star appreciably. The mean period spacing and trapping of the gravity modes in the model with moderate and small core overshooting are fully consistent with the period spacing trends observed in KIC 10553698A. Atomic diffusion driven by gravitational settling as well as thermal and chemical gradients is applied to reach a better match with the observed period spacings. Models that include small or very small overshooting with atomic diffusion have a decreased lifetime of the extreme horizontal branch phase and produce chemical stratification induced by convective shells during helium burning phase. In addition of being consistent with asteroseismology, their calculated values of the R2 parameter are more compatible with the observed R2 values.

Three groups of DAV star models are evolved with time-dependent element\ndiffusion by \\texttt{WDEC}. The core compositions of these models are directly\nfrom white dwarf models evolved by \\texttt{MESA}, which are results of\nthermonuclear burning. Based on these DAV star models, we try to study the\ndispersion of period spacing. The thickness of hydrogen atmosphere can\nseriously affect the deviation degree of minimal period spacings. The minimal\nperiod spacings dominate the dispersion of period spacing. The thinner the\nhydrogen atmosphere, basically, the more dispersive the period spacing.\nStandard deviations are used to analyze the dispersion of period spacing.\nStudying the dispersion of period spacing on a DAV star KUV03442+0719\npreliminarily, we suggest that log($M_{\\rm H}/M_{*}$) is from -8.5 to -5.5. In\naddition, modes partly trapped in C/O core are found based on those DAV star\nmodels. The identified modes and average period spacings indicate that\nKUV03442+0719 may be the first star to 'observe' modes partly trapped in C/O\ncore.\n

We investigate the possibility of nearly-equally spaced periods in 13 hot subdwarf B (sdB) stars observed with the Kepler spacecraft and one observed with CoRoT. Asymptotic limits for gravity (g-)mode pulsations provide relationships between equal period spacings of modes with differing degrees and relationships between periods of the same radial order but differing degrees. Period transforms, Kolmogorov-Smirnov tests, and linear least-squares fits have been used to detect and determine the significance of equal period spacings. We have also used Monte Carlo simulations to estimate the likelihood that the detected spacings could be produced randomly. Period transforms for nine of the Kepler stars indicate ell=1 period spacings, with five also showing peaks for ell=2 modes. 12 stars indicate ell=1 modes using the Kolmogorov-Smirnov test while another shows solely ell=2 modes. Monte Carlo results indicate that equal period spacings are significant in 10 stars above 99% confidence and 13 of the 14 are above 94% confidence. For 12 stars, the various methods find consistent regular period spacing values to within the errors, two others show some inconsistencies, likely caused by binarity, and the last has significant detections but the mode assignment disagrees between methods. We find a common ell=1 period spacing spanning a range from 231 to 272 s allowing us to correlate pulsation modes with 222 periodicities and that the ell=2 period spacings are related to the ell=1 spacings by the asymptotic relationship $1/\sqrt{3}$. We briefly discuss the impact of equal period spacings which indicate low-degree modes with a lack of significant mode trappings.

BackgroundCorrect temporal and spatial gene expression during metazoan development relies on combinatorial interactions between different transcription factors. As a consequence, cis-regulatory elements often colocalize in clusters termed cis-regulatory modules. These may have requirements on organizational features such as spacing, order and helical phasing (periodic spacing) between binding sites. Due to the turning of the DNA helix, a small modification of the distance between a pair of sites may sometimes drastically disrupt function, while insertion of a full helical turn of DNA (10–11 bp) between cis elements may cause functionality to be restored. Recently, de novo motif discovery methods which incorporate organizational properties such as colocalization and order preferences have been developed, but there are no tools which incorporate periodic spacing into the model.ResultsWe have developed a web based motif discovery tool, HeliCis, which features a flexible model which allows de novo detection of motifs with periodic spacing. Depending on the parameter settings it may also be used for discovering colocalized motifs without periodicity or motifs separated by a fixed gap of known or unknown length. We show on simulated data that it can efficiently capture the synergistic effects of colocalization and periodic spacing to improve detection of weak DNA motifs. It provides a simple to use web interface which interactively visualizes the current settings and thereby makes it easy to understand the parameters and the model structure.ConclusionHeliCis provides simple and efficient de novo discovery of colocalized DNA motif pairs, with or without periodic spacing. Our evaluations show that it can detect weak periodic patterns which are not easily discovered using a sequential approach, i.e. first finding the binding sites and second analyzing the properties of their pairwise distances.

We develop the analog in equal positive characteristic of Fontaine's theory for crystalline Galois representations of a p-adic field. In particular we describe the analog of Fontaine's functor which assigns to a crystalline Galois representation an isocrystal with a Hodge filtration. In equal characteristic the role of isocrystals and Hodge filtrations is played by z-isocrystals and Hodge-Pink structures. The latter were invented by Pink. Our first main result in this article is the analog of the Colmez-Fontaine Theorem that "weakly admissible implies admissible". Next we construct period spaces for Hodge-Pink structures on a fixed z-isocrystal. These period spaces are analogs of the Rapoport-Zink period spaces for Fontaine's filtered isocrystals in mixed characteristic and likewise are rigid analytic spaces. For our period spaces we prove the analog of a conjecture of Rapoport and Zink stating the existence of a "universal local system" on a Berkovich open subspace of the period space. As a consequence of "weakly admissible implies admissible" this Berkovich open subspace contains every classical rigid analytic point of the period space. As the principal tool to demonstrate these results we use the analog of Kedlaya's Slope Filtration Theorem which we also formulate and prove here.

Context. The Kepler and TESS space missions have revealed the rich gravity (g-)mode pulsation spectra of many hot subdwarf B (sdB) stars in detail. These spectra exhibit complex behaviors, with some stars exhibiting trapped modes interposing in the asymptotic period sequences of regular period spacing, while others do not. Aims. We aim to thoroughly compute theoretical g-mode pulsation spectra, using our current sdB models, useful for future reference when comparing to observations. This also enables us to explore relationships with features of the internal structure of these stars. Such studies provide guidance in conducting future asteroseismic analyses of these pulsators and insights on how to interpret their outcomes. Methods. We used our STELlar modeling from the Université de Montréal (STELUM) code to compute static (parametric) and evolutionary models of sdB stars, with different prescriptions for their chemical and thermal structures. We used our adiabatic PULSE code to compute the theoretical spectra of g-mode pulsations for degrees of ℓ = 1 to 4 and for periods between 1000 s and 15 000 s, amply covering the range of observed g-modes in these stars. Results. We show that g-mode pulsation spectra and, in particular, the appearance of trapped modes are highly dependent on the chemical and thermal structures in the models as the star evolves, particularly in the region just above the He-burning core. Depending on the prescriptions and specific evolutionary stage, we observe mainly three types of spectra for mid to high radial-order g-modes (the ones observed in sdB stars): “flat” spectra of nearly constant period spacing; spectra with deep minima of the period spacing interposing between modes with more regular spacing (which correspond to trapped modes); and spectra showing a “wavy” pattern in period spacing. For the two latter cases, we have identified the region where the modes are trapped in the star. Conclusions. Detailed comparisons with observed g-mode spectra ought to be carried out next to progress on this issue and constrain the internal structure of core-He burning stars via asteroseismology, in particular, for the region above the He-burning core.

Context. Red clump (RC) stars still pose open questions regarding several physical processes, such as the mixing around the core or the nuclear reactions, which are ill-constrained by theory and experiments. The oscillations of RC stars, which are of a mixed gravito-acoustic nature, allow us to directly investigate the interior of these stars and thereby better understand their physics. In particular, the measurement of their period spacing is a good probe of the structure around the core. Aims. We aim to explain the distribution of period spacings in RC stars observed by Kepler by testing different prescriptions of core-boundary mixing and the nuclear reaction rate. Methods. Using the MESA stellar evolution code, we computed several grids of core-helium-burning tracks, with varying masses and metallicities. Each of these grids has been computed assuming a certain core boundary mixing scheme, or 12 C( α , γ ) 16 O reaction rate. We then sampled these grids, in a Monte-Carlo fashion, using observational spectroscopic metallicity and seismic mass priors, in order to retrieve a period spacing distribution, which we compared to the observations. Results. We find that the best-fitting distribution is obtained when using a “maximal overshoot” core-boundary scheme, which has similar seismic properties as a model whose modes are trapped outside a semi-convective region, and which does not exhibit core-breathing pulses at the end of the core-helium-burning phase. If no mode trapping is assumed, then no core boundary mixing scheme is compatible with the observations. Moreover, we find that extending the core with overshoot worsens the fit. Additionally, reducing the 12 C( α , γ ) 16 O reaction rate (by around 15%) improves the fit to the observed distribution. Finally, we note that an overpopulation of early RC stars with period spacing values around 250 s is predicted by the models but not found in the observations. Conclusions. Assuming a semi-convective region and mode trapping, along with a slightly lower than nominal 12 C( α , γ ) 16 O rate, allowed us to reproduce most of the features of the observed period spacing distribution, except for those of early RC stars.

Light absorption enhancement in a CdSe-quantum dot (QD)-sensitized TiO2 nanorod periodic array-based photoanode is calculated by the finite differential time domain (FDTD) model through optimizing the height, diameter, and periodic space between TiO2 nanorods. Results show that light absorption in the CdSe-QD-sensitized TiO2 nanorods is enhanced at the visible wavelength range and the absorption edge is shifted toward higher wavelengths as a result of decreasing the height and increasing the diameter of the TiO2 nanorods. Consequently, the equivalent bandgap of the CdSe-QD-sensitized TiO2 nanorod photoanode was decreased. Also, absorption spectra indicate that the position of the absorption peak related to TiO2 has a redshift with increasing periodic space between the TiO2 nanorods, suggesting a decrease in the bandgap of TiO2. In addition, results show that for TiO2 nanorod heights lower than 300 nm, light absorption in the visible region is considerably enhanced as the periodic space is reduced from 400 nm to 100 nm. In comparison, for TiO2 nanorod heights larger than 300 nm, light absorption is considerably enhanced. Finally, it is found from absorption spectra that coupling optical modes of TiO2 nanorods and CdSe QDs are enhanced by increasing the diameter and decreasing the height and periodic space between the TiO2 nanorods. As a result, the light absorption is increased in the photoanodes at the visible region, leading to an improvement in the performance of the photoanode at visible light.

Context. The recent arrival of continuous photometric observations of unprecedented quality from space missions has strongly promoted the study of pulsating stars and caused great interest in the stellar astrophysics community. In the particular case of pulsating white dwarfs, the TESS mission is taking asteroseismology of these compact stars to a higher level, emulating or even surpassing the performance of its predecessor, the Kepler mission. Aims. We present a detailed asteroseismological analysis of six GW Vir stars that includes the observations collected by the TESS mission. Methods. We processed and analyzed TESS observations of RX J2117+3412 (TIC 117070953), HS 2324+3944 (TIC 352444061), NGC 6905 (TIC 402913811), NGC 1501 (TIC 084306468), NGC 2371 (TIC 446005482), and K 1−16 (TIC 233689607). We carried out a detailed asteroseismological analysis of these stars on the basis of PG 1159 evolutionary models that take into account the complete evolution of the progenitor stars. We constrained the stellar mass of these stars by comparing the observed period spacing with the average of the computed period spacings, and we employed the individual observed periods to search for a representative seismological model when possible. Results. In total, we extracted 58 periodicities from the TESS light curves of these GW Vir stars using a standard prewhitening procedure to derive the potential pulsation frequencies. All the oscillation frequencies that we found are associated with g-mode pulsations, with periods spanning from ∼817 s to ∼2682 s. We find constant period spacings for all but one star (K 1−16), which allowed us to infer their stellar masses and constrain the harmonic degree ℓ of the modes. Based on rotational frequency splittings, we derive the rotation period of RX J2117+3412, obtaining a value in agreement with previous determinations. We performed period-to-period fit analyses on five of the six analyzed stars. For four stars (RX J2117+3412, HS 2324+3944, NGC 1501, and NGC 2371), we were able to find an asteroseismological model with masses that agree with the stellar mass values inferred from the period spacings and are generally compatible with the spectroscopic masses. Obtaining seismological models allowed us to estimate the seismological distance and compare it with the precise astrometric distance measured with Gaia. Finally, we find that the period spectrum of K 1−16 exhibits dramatic changes in frequency and amplitude that together with the scarcity of modes prevented us from meaningful seismological modeling of this star. Conclusions. The high-quality data collected by the TESS space mission, considered simultaneously with ground-based observations, provide very valuable input to the asteroseismology of GW Vir stars, similar to the case of other classes of pulsating white dwarf stars. The TESS mission, in conjunction with future space missions and upcoming surveys, will make impressive progress in white dwarf asteroseismology.

We use photometry from the Kepler Mission to study oscillations in gamma Doradus stars. Some stars show remarkably clear sequences of g modes and we use period echelle diagrams to measure period spacings and identify rotationally split multiplets with l=1 and l=2. We find small deviations from regular period spacings that arise from the gradient in the chemical composition just outside the convective core. We also find stars for which the period spacing shows a strong linear trend as a function of period, consistent with relatively rapid rotation. Overall, the results indicate it will be possible to apply asteroseismology to a range of gamma Dor stars.