Period Map Research Articles

Return values of temperature and snow loadings for 50, 100 and 120-year return periods to support building design standards in Ireland

Abstract. This research produced gridded datasets and maps for use in building design standards to enhance resilience in support of climate change adaptation in Ireland. The new isothermal maps of return values of maximum and minimum air temperatures at mean sea level for 50, 100 and 120-year return periods based on the generalised extreme value distribution will be crucial to inform the design of buildings and bridges. The warming of the maximum and minimum air temperatures due to climate change has increased the intensity of the highest maximum air temperature while decreasing the intensity of the lowest extreme minimum air temperature of the new isothermal maps compared to previously published maps for a 50-year return period. Specifically, the new extreme isotherms are 32 ∘C for the maximum air temperature and −14 ∘C for the minimum air temperature, whereas the processor maps presented 30 and −16 ∘C, respectively. The geographical distribution of the isotherms for the 120-year return period range from 28 to 34 ∘C for the maximum air temperature and from −6 to −18 ∘C for the minimum air temperature. For the first time, isothermal maps of return values of the lowest 10 cm soil temperature for 50, 100 and 120-year return periods based on the generalised extreme value distribution have been produced for Ireland. The results presented here will be paramount to supporting the design of building structures. The values of the 120-year return period range from 0 to −2 ∘C. The produced maps represent the worst-case scenario in the current context of climate warming. The new maps of return values of snow loading at 100 m above mean sea level for 50, 100 and 120-year return periods based on the generalised Pareto distribution will be indispensable to support the design of buildings and civil engineering works such as roof patterns or bridges. The values of the 50-year return period map present four classes spread North-East to South-West: < 0.3, 0.3–0.4, 0.4–0.5 and 0.5–0.6 kN m−2, which is more accurate than the previously published map. It is expected that the comprehensive explanation of the methods and the rationale for the new maps presented here as being more accurate than the preceding maps will assist regulators in adopting these new maps in their own jurisdictions. Furthermore, these new maps will be of interest to a diversity of sectors, planners and policymakers to make long, lasting and climate-based sensitive decisions.

Overconvergent Hilbert modular forms via perfectoid modular varieties

We give a new construction of p-adic overconvergent Hilbert modular forms by using Scholze’s perfectoid Shimura varieties at infinite level and the Hodge–Tate period map. The definition is analytic, closely resembling that of complex Hilbert modular forms as holomorphic functions satisfying a transformation property under congruence subgroups. As a special case, we first revisit the case of elliptic modular forms, extending recent work of Chojecki, Hansen and Johansson. We then construct sheaves of geometric Hilbert modular forms, as well as subsheaves of integral modular forms, and vary our definitions in p-adic families. We show that the resulting spaces are isomorphic as Hecke modules to earlier constructions of Andreatta, Iovita and Pilloni. Finally, we give a new direct construction of sheaves of arithmetic Hilbert modular forms, and compare this to the construction via descent from the geometric case.

Erratum to “Tame topology of arithmetic quotients and algebraicity of Hodge loci”

We correct an error in the functoriality of the R a l g \mathbb {R}_\mathrm {alg} -definable structures on arithmetic quotients Γ ∖ G / M \Gamma \backslash G/M constructed in [B. Bakker, B. Klingler, and J. Tsimerman, Tame topology of arithmetic quotients and algebraicity of Hodge loci, J. Amer. Math. Soc. 33 (2020), no. 4, 917-939]. The statements for Hodge manifolds and period maps are unaffected.

Infinitesimal categorical Torelli theorems for Fano threefolds

Let X be a smooth Fano variety and Ku(X) its Kuznetsov component. A Torelli theorem for Ku(X) states that Ku(X) is uniquely determined by a certain polarized abelian variety associated to it. An infinitesimal Torelli theorem for X states that the differential of the period map is injective. A categorical variant of the infinitesimal Torelli theorem for X states that the morphism η:H1(X,TX)→HH2(Ku(X)) is injective. In the present article, we use the machinery of Hochschild (co)homology to relate the aforementioned three Torelli-type theorems for smooth Fano varieties via a commutative diagram. As an application, we prove infinitesimal categorical Torelli theorems for a class of prime Fano threefolds. We then prove, infinitesimally, a restatement of the Debarre–Iliev–Manivel conjecture regarding the general fibre of the period map for ordinary Gushel–Mukai threefolds.

Canopy gap patterns in Mediterranean forests: a spatio-temporal characterization using airborne LiDAR data

ContextIn the last century European forests are experiencing tree damage and mortality rise and it is expected to continue due to increased disturbances under global change. Disturbances generally creates canopy gaps, which leads to secondary succession, compositional changes and landscape mosaic transformations. Forest gap characterization has traditionally been performed in light-limited tropical and boreal forests, but no studies have been found on water-limited Mediterranean forests. Characterising canopy gaps and their dynamics in Mediterranean forests will help to better understand their dynamics across landscapes under ongoing global change.ObjectivesWe aimed to characterize canopy gaps and quantify their dynamics identifying hotspots of openings and closings in Mediterranean forests.MethodsWe used low density multitemporal airborne LiDAR data between 2010 and 2016, over a large region (Madrid, Spain, 1732.7 km2) with forests ranging from monospecific conifer and broadleaved to mixed forests, to delineate canopy gaps. The characterization was made through its Gap Size Frequency Distribution (GSFD) by forest type and year. We analysed canopy gap dynamics and identified statistically significant hotspots of gap openings and closings in each forest type.ResultsThere were major differences between conifers and broadleaved forest in terms of gap characteristics and GSFD. In general, we found a great dynamism in Mediterranean forests with high rates of forest openings and closings, but a net closing trend. A high spatial heterogeneity was observed finding hotspots of gap openings and closings across the entire study area.ConclusionsWe characterised for the first-time large-scale structure and dynamics of canopy gaps in Mediterranean forests. Our results represents the characterisation of the GSFD of Mediterranean forests and could be considered a benchmark for future studies. The provision of up-to-date periodic maps of hotspots of gap opening, closing and net change help to understand landscape mosaic changes as well as to prioritise forest management and restoration strategies.

Big Picard theorems and algebraic hyperbolicity for varieties admitting a variation of Hodge structures

In this paper, we study various hyperbolicity properties for a quasi-compact K\"ahler manifold $U$ which admits a complex polarized variation of Hodge structures so that each fiber of the period map is zero-dimensional. In the first part, we prove that $U$ is algebraically hyperbolic and that the generalized big Picard theorem holds for $U$. In the second part, we prove that there is a finite \'etale cover $\tilde{U}$ of $U$ from a quasi-projective manifold $\tilde{U}$ such that any projective compactification $X$ of $\tilde{U}$ is Picard hyperbolic modulo the boundary $X-\tilde{U}$, and any irreducible subvariety of $X$ not contained in $X-\tilde{U}$ is of general type. This result coarsely incorporates previous works by Nadel, Rousseau, Brunebarbe and Cadorel on the hyperbolicity of compactifications of quotients of bounded symmetric domains by torsion-free lattices.

Odd periodic waves for some Klein–Gordon type equations: Existence and stability

In this paper, we establish the existence and stability properties of odd periodic waves related to the Klein–Gordon type equations, which include the well known and models. Existence of periodic waves is determined by using a general planar theory of ODE. The spectral analysis for the corresponding linearized operator is established using the monotonicity of the period map combined with an improvement of the standard Floquet theory. Orbital stability in the odd sector of the energy space is proved using exclusively the monotonicity of the period map. The orbital instability of explicit solutions for the and models is presented using the abstract approach of instability for general abstract Hamiltonian systems.

Analytical study of Bragg resonances by a finite periodic array of congruent trapezoidal bars or trenches on a sloping seabed

For water waves propagating over a finite periodic array of congruent trapezoidal bars on a sloping seabed or trenches dug on a sloping seabed, an analytical solution in terms of Taylor series to the modified mild-slope equation (MMSE) is constructed. Based on the analytical solution, influences of various topographic parameters (e.g., the slope of seabed, bar/trench number, bar height or trench depth, and bar/trench width) on the peak value, peak phase and zero reflection of Bragg resonances are investigated. Especially, starting from the original definition of Bragg resonances, a modified Bragg's law for characterizing the excitation condition of the nth order Bragg resonance is derived, which shows that the excitation of Bragg resonances depends entirely on the average phase velocity of the water wave within the bar/trench field. When the influence of the bar/trench field on the average phase velocity is quite small, the modified Bragg's law degenerates into the traditional Bragg’s law borrowed from X-ray crystallography, which also reflects the essential difference between the water wave as a mechanical wave and the non-mechanical X-ray wave. In addition, for N bars/trenches on a flat seabed, there are at least N−1 zero reflections between any two adjacent Bragg resonance peaks. When the bars are very low or the trenches are very shallow, the phase of the N−1 zero reflections in those zero reflections is exactly the position of the N−1 zero points of the Chebyshev polynomial of the second kind, UN−1(cosk0d), where k0 is the wavenumber on the flat seabed and d is the bar/trench spacing. However, once a sloping seabed is added, all these N−1 zero reflections disappear possibly due to the destruction of the symmetry of the background bar/trench field. Finally, regardless of the number of bars/trenches, when the bar/trench width varies, the peak value of the nth order Bragg resonance oscillates periodically, leading to n local maxima. A comparative study shows that these characteristics are consistent with the band structure shown in the Bloch periodic gap map (PGM).

Future Projections of EURO‐CORDEX Raw and Bias‐Corrected Daily Maximum Wind Speeds Over Scandinavia

AbstractTwenty historical and future Coordinated Regional Climate Downscaling Experiment ensemble for Europe simulations are bias corrected to investigate the future changes in the daily maximum wind speed over Scandinavia. We use quantile mapping to adjust the wind for the historical period (1985–2014) and quantile delta mapping for two future periods (2041–2070, 2071–2100, RCP8.5) with the 3‐km spatial resolution NORA3 hindcast as the reference data set. Decomposing the variance, we find that most of the inter‐model spread in the bias and the response to climate change is due to the regional climate model over land and mainly to the general climate model over sea. On average, the mean daily maximum wind speed is projected to increase everywhere except along the western coast of Norway and Denmark. In summer, we see an opposite sign over Sweden and Finland. The Norwegian Sea experiences weaker mean and high wind whereas the Baltic Sea experiences a strengthening. Bias correction influences the amplitude of the response, not the response pattern. Wind speed distributions can have different shapes in the future, with, for example, a flattening of the distribution off the coast of Norway with more frequent weak and very strong winds. Apart from a few locations in Norway, we find an increase in the number of days in the local highest historical wind category. Overall, summer exhibits opposite signals compared to the three other seasons. At country scale, the sign of the change in the mean and 98th percentile varies greatly depending on the region and among the simulations, especially for Norway.

THE EFFECT OF ANCIENT GEOLOGICAL PERIODS IN BIOLOGY

At the end of the 19th century and the beginning of the 20th century, extensive works began in the world for the construction of railways, and for these works, the previous biography of land is necessary. William Smith was the first scientist. The rocks found in the Somerset region from the lower layers of the earth and belong to the Jurassic period (20–144 million years ago). He compared the rocks of the coastal areas of the northern seas and using these rocks, he drew a map of the past periods of the earth in England. Even so far, the stones that came from the lower layers of the earth represented its place, that is, by using those stones, William Smith not only made a map of the past of the earth, but also a map of the lower layers of the earth. This map has a special place in the knowledge of modern geology it helps us to create a table of the past periods. From the information contained in this map, we can see the underground layers and their contents, iron, coal, etc. even if it is in the middle of the ocean. Fossils are very valuable to us. By using fossils, humans can identify the living creatures of the past times and get to know their ways of life. It can also be done by using fossils to divide the age table of the earth into cycles first and then the cycles are divided into periods. For this reason, the previous age table is first divided into periods and then the periods are divided into periods. 1. Upper Carboniferous Period; 2. Lower Carboniferous Period; 3. Middle Jurassic Period; 4. Jurassic Period and 5. Triassic Period. Keywords: Period, fossils, Evolutionary modeling, extinction, Biodiversity and organisms.

Reconstructing bird trajectories from pressure and wind data using a highly optimized hidden Markov model

Abstract Tracking technologies have widely expanded our understanding of bird migration routes, destinations and underlying strategies. However, determining the entire trajectory of small birds equipped with lightweight geolocators remains a challenge. We develop a highly optimized hidden Markov model (HMM) for reconstructing bird trajectories. The observation model is defined by pressure and, optionally, light measurements, while the movement model incorporates wind data to constrain consecutive positions based on realistic airspeeds. To reduce the computational costs associated with a large state space, we prune the HMM states and transitions based on flight and observation constraints to efficiently model the entire trajectory. The approach presented is based on a mathematically exact procedure and is fast to compute. We demonstrate how to compute (1) the most likely trajectory, (2) the marginal probability map of each stationary period, (3) simulated trajectories and (4) the wind conditions (wind support/drift) encountered by the bird during each migratory flight. We construct a version of an HMM optimized for reconstructing a bird's migration trajectory based on lightweight geolocator data. To render this approach easily accessible to researchers, we designed a dedicated R package GeoPressureR (https://raphaelnussbaumer.com/GeoPressureR/).

Nearly Periodic Maps and Geometric Integration of Noncanonical Hamiltonian Systems

M. Kruskal showed that each continuous-time nearly periodic dynamical system admits a formal U(1)-symmetry, generated by the so-called roto-rate. When the nearly periodic system is also Hamiltonian, Noether’s theorem implies the existence of a corresponding adiabatic invariant. We develop a discrete-time analog of Kruskal’s theory. Nearly periodic maps are defined as parameter-dependent diffeomorphisms that limit to rotations along a U(1)-action. When the limiting rotation is non-resonant, these maps admit formal U(1)-symmetries to all orders in perturbation theory. For Hamiltonian nearly periodic maps on exact presymplectic manifolds, we prove that the formal U(1)-symmetry gives rise to a discrete-time adiabatic invariant using a discrete-time extension of Noether’s theorem. When the unperturbed U(1)-orbits are contractible, we also find a discrete-time adiabatic invariant for mappings that are merely presymplectic, rather than Hamiltonian. As an application of the theory, we use it to develop a novel technique for geometric integration of non-canonical Hamiltonian systems on exact symplectic manifolds.

A positive factorization for the balanced superelliptic rotation

The balanced superelliptic rotation is a periodic map on an oriented closed surface of order k≥3. We give a positive factorization for the balanced superelliptic rotation. By this result, we also give an infinite family of Lefschetz fibrations over a 2-sphere corresponding to the relators that the k-th power of this positive factorization. As an application, we observe their topological properties.

Harder-Narasimhan strata and 𝑝-adic period domains

We revisit the Harder-Narasimhan stratification on a minuscule p p -adic flag variety, by the theory of modifications of G G -bundles on the Fargues-Fontaine curve. We compare the Harder-Narasimhan strata with the Newton strata introduced by Caraiani-Scholze. As a consequence, we get further equivalent conditions in terms of p p -adic Hodge-Tate period domains for fully Hodge-Newton decomposable pairs. Moreover, we generalize these results to arbitrary cocharacters case by considering the associated B d R + B_{dR}^+ -affine Schubert varieties. Applying Hodge-Tate period maps, our constructions give applications to p p -adic geometry of Shimura varieties and their local analogues.

Integral points on algebraic subvarieties of period domains: from number fields to finitely generated fields

We show that for a variety which admits a quasi-finite period map, finiteness (resp. non-Zariski-density) of S-integral points implies finiteness (resp. non-Zariski-density) of points over all mathbb {Z}-finitely generated integral domains of characteristic zero. Our proofs rely on foundational results in Hodge theory due to Deligne, Griffiths, and Schmid, and Bakker-Brunebarbe-Tsimerman. We give straightforward applications to arithmetic locally symmetric varieties, the moduli space of smooth hypersurfaces in projective space, and the moduli of smooth divisors in an abelian variety.

Perfectoid spaces arising from arithmetic jet spaces

Using arithmetic jet spaces, we attach perfectoid spaces to smooth schemes and to $\delta$-morphisms of smooth schemes. We also study perfectoid spaces attached to arithmetic differential equations defined by some of the remarkable $\delta$-morphisms appearing in the theory such as the $\delta$-characters of elliptic curves and the $\delta$-period map on modular curves.

A transfer principle: from periods to isoperiodic foliations

We classify the possible closures of leaves of the isoperiodic foliation defined on the Hodge bundle over the moduli space of genus $$g\ge 2$$ curves and prove that the foliation is ergodic on those sets. The results derive from the connectedness properties of the fibers of the period map defined on the Torelli cover of the moduli space. Some consequences on the topology of Hurwitz spaces of primitive branched coverings over elliptic curves are also obtained. To prove the results we develop the theory of augmented Torelli space, the branched Torelli cover of the Deligne–Mumford compactification of the moduli space of curves.

Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes

These notes aim at providing a complete and systematic account of some foundational aspects of algebraic supergeometry, namely, the extension to the geometry of superschemes of many classical notions, techniques and results that make up the general backbone of algebraic geometry, most of them originating from Grothendieck's work. In particular, we extend to algebraic supergeometry such notions as projective and proper morphisms, finiteness of the cohomology, vector and projective bundles, cohomology base change, semicontinuity theorems, relative duality, Castelnuovo-Mumford regularity, flattening, Hilbert and Quot schemes, faithfully flat descent, quotient étale relations (notably, Picard schemes), among others. Some results may be found elsewhere, and, in particular, there is some overlap with [51]. However, many techniques and constructions are presented here for the first time, notably, a first development of Grothendieck relative duality for proper morphisms of superschemes, the construction of the Hilbert superscheme in a more general situation than the one already known (which in particular allows one to treat the case of sub-superschemes of supergrassmannians), and a rigorous construction of the Picard superscheme for a locally superprojective morphism of noetherian superschemes with geometrically integral fibres. Moreover, some of the proofs given here are new as well, even when restricted to ordinary schemes. In a final section we construct a period map from an open substack of the moduli of proper and smooth supercurves to the moduli stack of principally polarized abelian schemes.

The bad locus in the moduli of super Riemann surfaces with Ramond punctures

The bad locus in the moduli of super Riemann surfaces with Ramond punctures parametrizes those super Riemann surfaces that have more than the expected number of independent closed holomorphic 1-forms. There is a super period map that depends on certain discrete choices. For each such choice, the period map blows up along a divisor that contains the bad locus. Our main result is that away from the bad locus, at least one of these period maps remains finite. In other words, we identify the bad locus as the intersection of the blowup divisors. The proof abstracts the situation into a question in linear algebra, which we then solve. We also give some bounds on the dimension of the bad locus.

Monodromy and period map of the Winger pencil

AbstractThe sextic plane curves that are invariant under the standard action of the icosahedral group on the projective plane make up a pencil of genus ten curves (spanned by a sum of six lines and three times a conic). This pencil was first considered in a note by R. M. Winger in 1925 and is nowadays named after him. The second author recently gave this a modern treatment and proved among other things that it contains essentially every smooth genus ten curve with icosahedral symmetry. We here show that the Jacobian of such a curve contains the tensor product of an elliptic curve with a certain integral representation of the icosahedral group. We find that the elliptic curve comes with a distinguished point of order 3, which proves that the monodromy on this part of the homology is the full congruence subgroup and subsequently identify the base of the pencil with the associated modular curve. We also observe that the Winger pencil “accounts” for the deformation of the Jacobian of Bring's curve as a principal abelian fourfold with an action of the icosahedral group.