Period Map Research Articles

Bergman kernel and period map for curves

As for any symmetric space the tangent space to Siegel upper-half space is endowed with an operation coming from the Lie bracket on the Lie algebra. We consider the pull-back of this operation to the moduli space of curves via the Torelli map. We characterize it in terms of the geometry of the curve, using the Bergman kernel form associated to the curve. It is known that the second fundamental form of the Torelli map outside the hyperelliptic locus can be seen as the multiplication by a certain meromorphic form. Our second result says that the Bergman kernel form is the harmonic representative—in a suitable sense—of this meromorphic form.

Geometric Interpretation of Hermitian Modular Forms via Burkhardt Invariants

We give an exact theta expression of a tuple of Hermitian modular forms as an inverse period mapping of lattice polarized K3 surfaces. Our result gives a non-trivial relation among moduli of K3 surfaces, theta functions and the finite complex reflection group of rank 5.

On two mod p period maps: Ekedahl–Oort and fine Deligne–Lusztig stratifications

Consider a Shimura variety of Hodge type admitting a smooth integral model S at an odd prime pge 5. Consider its perfectoid cover S^{text {ad}}(p^infty ) and the Hodge–Tate period map introduced by Caraiani and Scholze. We compare the pull-back to S^{text {ad}}(p^infty ) of the Ekedahl–Oort stratification on the mod p special fiber of a toroidal compactification of S and the pull back to S^text {ad}(p^infty ) of the fine Deligne–Lusztig stratification on the mod p special fiber of the flag variety which is the target of the Hodge–Tate period map. An application to the non-emptiness of Ekedhal–Oort strata is provided.

Reducibility for a class of almost periodic Hamiltonian systems which are degenerate

<abstract><p>This paper studies the reducibility for a class of Hamiltonian almost periodic systems that are degenerate in a small perturbation parameter. We prove for most of the sufficiently small parameter, the Hamiltonian system is reducible by a symplectic almost periodic mapping.</p></abstract>

Embedding and Global Stability in Periodic 2-Dimensional Maps of Mixed Monotonicity

Embedding and Global Stability in Periodic 2-Dimensional Maps of Mixed Monotonicity

Differential of a period mapping at a singularity

Differential of a period mapping at a singularity

Pointwise periodic maps with quantized first integrals

We describe the global dynamics of some pointwise periodic piecewise linear maps in the plane that exhibit interesting dynamic features. For each of these maps we find a first integral. For these integrals the set of values are discrete, thus quantized. Furthermore, the level sets are bounded sets whose interior is formed by a finite number of open tiles of certain regular or uniform tessellations. The action of the maps on each invariant set of tiles is described geometrically.

The evolution dynamics of an impulsive hybrid population model with spatial heterogeneity

This paper is devoted to the study of evolution dynamics for an impulsive hybrid population model with nonlocal dispersal/random diffusion operators in a heterogeneous landscape. Since this hybrid model generates a time-periodic semiflow on the space of continuous functions, the problem reduces to the investigation of a discrete-time recursion system defined by the Poincaré period map. We obtain a threshold-type result on the global dynamics of the population in a single-patch landscape. Moreover, the recursion admits a unique positive steady state which is globally asymptotically stable. For the population spread in a periodic-patchy landscape, we establish the existence of spreading speed and its coincidence with the minimal wave speed for spatially periodic traveling waves. Simulations are presented to illustrate our analytic results.

Hydraulic Model for Flood Forecasting of Tajabad Khwar in Hayatabad Phase III Peshawar, Pakistan: A Case Study

Among all other natural disasters occurring throughout the world, floods are considered to be the worst and most devastating catastrophe as it causes loss of billions of lives. Flooding is caused due to inundation of water over the areas which are in close proximity of river or natural waterways resulting in severe damages to commercial and residential areas in the surrounding. Thus, an efficient flood forecasting system through the development of a combined hydrological and hydraulic model for the prediction of future flood events through marking the potential high-risk zone is required to minimize the damages. Due to large number of encroachments made in the waterway of Tajabad khwar located near Deans Residential Apartment of Hayatabad Phase III, a hydraulic model is developed for its flood forecasting as the floods in this khwar may cause severe damages to the inhabitants of the adjacent areas. In this research work, Flood zone maps are developed for 10 years, 20 years, 50 years, and 100 years flood return periods in order for deterring extent of the inundation as a result of these encroachments and to identify the areas at potential risk. Flood discharge for each return period was estimated using HEC-HMS software and was found to be 772, 1036, 1392 and 1666 m^3/sec for 10 years, 20 years, 50 years, and 100 years flood return periods respectively. The corresponding water surface elevation determine using HEC-RAS and was found to be 196 m, 197m, 201m, 202m. This model provides a basic idea for developing flood zone maps of a given period of return for the assessment of areas that can get adversely pretentious by floods.

A sinusoidal transformation of the visual field is the basis for periodic maps in area V2

Retinotopic maps of many visual areas are thought to follow the fundamental principles described for the primary visual cortex (V1), where nearby points on the retina map to nearby points on the surface of V1, and orthogonal axes of the retinal surface are represented along orthogonal axes of the cortical surface. Here we demonstrate a striking departure from this mapping in the secondary visual area (V2) of the tree shrew best described as a sinusoidal transformation of the visual field. This sinusoidal topography is ideal for achieving uniform coverage in an elongated area like V2, as predicted by mathematical models designed for wiring minimization, and provides a novel explanation for periodic banded patterns of intra-cortical connections and functional response properties in V2 of tree shrews as well as several other species. Our findings suggest that cortical circuits flexibly implement solutions to sensory surface representation, with dramatic consequences for large-scale cortical organization.

Dynamics of tax revenues in the budget of Ukraine and their forecast during the crisis period

It is extremely important for the budget process to obtain accurate forecasts of potential tax revenues, especially in periods of disruption and crisis. The paper is devoted to the study of dynamics of tax revenues’ volumes in the budget of Ukraine and the forecast of their values during the crisis.The dynamics of tax revenues in the Consolidated Budget of Ukraine, studied by using randomized R|S-analysis, fractal and probabilistic analyses as well as entropy calculation based on the data on monthly tax revenues for the period 2011–2021, is anti-persistent, fractal-like and unpredictable based on parametric dependencies, simple and complex trends. The topological dimension of the lines of dynamics for tax revenues of all types of taxes is much higher than 1, and the Hirst index indicates either fractal similarity of dynamics or its chaos. The map of dissipation periods of tax revenues in Ukraine, determined on the basis of entropy calculation and periods of negative entropy production according to the dynamics of tax revenues, coincided with the periods of maximum reduction in their volumes. The most crisis periods in the formation of tax revenues are 2019–2020, for certain types of taxes – 2016–2020, but the dissipation of tax revenues is projected for 2021–2022.The comparison of the level of fractal similarity in dynamics of the volume of tax revenues and peculiarities of the dynamics of entropy and entropy production, allowed to substantiate the division of taxes into nine types, of which five were found in Ukraine.

Periods of cubic surfaces with the automorphism group of order 54

To any cubic surface, one can associate a cubic threefold given by a triple cover of $\mathbb P^3$ branched in this cubic surface. D. Allcock, J. Carlson, and D. Toledo used this construction to define the period map for cubic surfaces. It is interesting to calculate this map for some specific cubic surfaces. In this paper, we have calculated it in the case when the cubic surface is given by a triple covering of $\mathbb P^2$ branched in a smooth elliptic curve. In this case, the periods can be expressed through periods of the corresponding elliptic curve.

Geometry and Global Stability of 2D Periodic Monotone Maps

Geometry and Global Stability of 2D Periodic Monotone Maps

Unobstructedness of hyperk\xe4hler twistor spaces

A family of irreducible holomorphic symplectic (ihs) manifolds over the complex projective line has unobstructed deformations if its period map is an embedding. This applies in particular to twistor spaces of ihs manifolds. Moreover, a family of ihs manifolds over a subspace of the period domain extends to a universal family over an open neighborhood in the period domain.

On non-hyperbolic algebraic automorphisms of a two-dimensional torus

Abstract. This paper contains a complete classification of algebraic non-hyperbolic automorphisms of a two-dimensional torus, announced by S. Batterson in 1979. Such automorphisms include all periodic automorphisms. Their classification is directly related to the topological classification of gradient-like diffeomorphisms of surfaces, since according to the results of V. Z. Grines and A.N. Bezdenezhykh, any gradient like orientation-preserving diffeomorphism of an orientable surface is represented as a superposition of the time-1 map of a gradient-like flow and some periodic homeomorphism. J. Nielsen found necessary and sufficient conditions for the topological conjugacy of orientation-preserving periodic homeomorphisms of orientable surfaces by means of orientation-preserving homeomorphisms. The results of this work allow us to completely solve the problem of realization all classes of topological conjugacy of periodic maps that are not homotopic to the identity in the case of a torus. Particularly, it follows from the present paper and the work of that if the surface is a two-dimensional torus, then there are exactly seven such classes, each of which is represented by algebraic automorphism of a two-dimensional torus induced by some periodic matrix.

Geometry of the Wiman–Edge monodromy

The Wiman–Edge pencil is a pencil of genus 6 curves for which the generic member has automorphism group the alternating group [Formula: see text]. There is a unique smooth member, the Wiman sextic, with automorphism group the symmetric group [Formula: see text]. Farb and Looijenga proved that the monodromy of the Wiman–Edge pencil is commensurable with the Hilbert modular group [Formula: see text]. In this note, we give a complete description of the monodromy by congruence conditions modulo 4 and 5. The congruence condition modulo 4 is new, and this answers a question of Farb–Looijenga. We also show that the smooth resolution of the Baily–Borel compactification of the locally symmetric manifold associated with the monodromy is a projective surface of general type. Lastly, we give new information about the image of the period map for the pencil.

P-adic Dynamics of Hecke Operators on Modular Curves

In this paper we study the p-adic dynamics of prime-to-p Hecke operators on the set of points of modular curves in both cases of good ordinary and supersingular reduction. We pay special attention to the dynamics on the set of CM points. In the case of ordinary reduction we employ the Serre–Tate coordinates, while in the case of supersingular reduction we use a parameter on the deformation space of the unique formal group of height 2 over 𝔽 ¯ p , and take advantage of the Gross–Hopkins period map.

In vitro detection of structural changes in chondrogenic mesenchymal stem cell pellets at the nano-scale using nano-sensitive optical coherance tomography

In vitro detection of structural changes in chondrogenic mesenchymal stem cell pellets at the nano-scale using nano-sensitive optical coherance tomography

Three-dimensional imaging of electron spin resonance-magnetic resonance force microscopy at room temperature

In this study, we demonstrated the three-dimensional (3D) imaging by magnetic resonance force microscopy (MRFM) based on electron spin resonance (ESR) measurements at room temperature. For a microsample containing radicals, the 3D force distribution was obtained using a custom-made Si nanowire and a permanent magnetic particle. Calculation using precise values of the magnetic field distribution is required to define an accurate response function for the 3D deconvolution processing of the spin density distribution. A symmetric resonance magnetic field produces good periodic force maps using a spherical micromagnet, which simplifies the deconvolution processing with resonant slice systems. In addition, the 3D imaging method was processed in the wavenumber space by a Fourier transform that used a simple convolution with noise parameters in the response function. After the reconstruction of the distribution of electron spins (radicals), the shape of the sample agreed with that of the optical image; thus, the accuracy of the internal density structure was verified. We believe that the combination of a Si nanowire and a spherical magnetic particle used for magnetic resonance detection is a good candidate for Fourier transform 3D deconvolution in multiple MRFM applications.

Mapping the seismic noise field in Europe: spatio-temporal variations in wavefield composition and noise source contributions

SUMMARYWe study spatial and temporal characteristics of the microseismic noise field across Europe. Rather than focusing on the areas of noise generation, the scope of this work is to characterize, at the scale of Europe, the spatio-temporal evolution of the noise wavefield that results from the interplay of the seismic noise sources and the propagation effect. To that end, we perform single station analysis in three period bands (PB1: 2.5−5 s; PB2: 5−10 s and PB3: 10−20 s) using three-component seismic data recorded by ∼1000 broad-band stations in the time period 2011–2019. We calculate, for each period band, station and day, a set of parameters that are practically possible to apply to a large data set, yet yields insight into the spatio-temporal evolution of the wavefield. These parameters are: the total energy level, the dominant period of the Primary and Secondary microseismic peaks, the horizontal direction with the most energy, the horizontal direction of the dominant Rayleigh waves and the square root of the energy ratio between the horizontal and vertical components. The analysis of these parameters shows that the noise field in Europe is dominated by surface waves from the North Atlantic Ocean with, in PB1 and PB2, an additional and significant contribution from the eastern part of the Mediterranean Sea. The relative contribution of these two source regions depends on the season, the influence of the eastern Mediterranean Sea being strongest in summer. The map of the peak period of the Primary and Secondary microseismic peaks indicates that the relative contribution of these two source regions is frequency dependent: the period of the Primary microseismic peak exhibits an overall increase with distance to the North Atlantic Sources, because of stronger attenuation of high-frequency wave contents. By contrast, the period of the Secondary microseismic peak is simultaneously influenced by sources in both the North Atlantic Ocean and eastern Mediterranean Sea. We show that in both microseismic peaks (PB2 and PB3), the wavefield is dominated by Love waves, as the horizontal components have the highest energy at approximately 90° angle to the direction of elliptical polarization. Moreover, our results show that lateral heterogeneities in the crust have a major influence on the noise field. In particular, the propagation directions of Love and Rayleigh waves show strong dependency on location (but not on time of year), with very sharp boundaries for example at the edge of the Alps. Thus, the scattering that takes place in the heterogeneous Alpine crust partly randomizes the directions of the microseismic wavefield in particular in PB1 and PB2. Finally, we show that the temporal evolution of the amplitude ratio between the horizontal and vertical components reflects the relative amounts of surface waves from the North Atlantic Ocean with respect to body waves from sources in the Southern Hemisphere. Thus, this ratio can be used as a proxy to identify time periods where body waves are significant in the noise wavefield.