Field Analogue Research Articles

A geometric approach to counting norms in cyclic extensions of function fields

In this paper we prove an explicit version of a function field analogue of a classical result of Odoni about norms in number fields in the case of a cyclic Galois extensions. In the particular case of a quadratic extension, we recover the result of Bary-Soroker, Smilanski, and Wolf which deals with finding asymptotics for a function field version on sums of two squares, improved upon by Gorodetsky , and reproved by the author in his Ph.D thesis using the method of this paper. The main tool is a twisted Grothendieck Lefschetz trace formula, inspired by the work of Church, Farb and Ellenberg on representation stability and asymptotic for point counts on varieties. Using a combinatorial description of the cohomology we obtain a precise quantitative result which works in the $q^n\rightarrow \infty$ regime, and a new type of homological stability phenomena, which arises from the computation of certain inner products of representations.

Codes from unit groups of division algebras over number fields

Lenstra and Guruswami described number field analogues of the algebraic geometry codes of Goppa. Recently, the first author and Oggier generalised these constructions to other arithmetic groups: unit groups in number fields and orders in division algebras; they suggested to use unit groups in quaternion algebras but could not completely analyse the resulting codes. We prove that the noncommutative unit group construction yields asymptotically good families of codes for the sum-rank metric from division algebras of any degree, and we estimate the size of the alphabet in terms of the degree.

Product formulas for periods of CM abelian varieties and the function field analog

We survey Colmez's theory and conjecture about the Faltings height and a product formula for the periods of abelian varieties with complex multiplication, along with the function field analog developed by the authors. In this analog, abelian varieties are replaced by Drinfeld modules and A-motives. We also explain the necessary background on abelian varieties, Drinfeld modules and A-motives, including their cohomology theories and comparison isomorphisms and their theory of complex multiplication.

V1460 Her: a fast spinning white dwarf accreting from an evolved donor star

ABSTRACT We present time-resolved optical and ultraviolet (UV) spectroscopy and photometry of V1460 Her, an eclipsing cataclysmic variable with a 4.99-h orbital period and an overluminous K5-type donor star. The optical spectra show emission lines from an accretion disc along with absorption lines from the donor. We use these to measure radial velocities, which, together with constraints upon the orbital inclination from photometry, imply masses of $M_1=0.869\pm 0.006\, \mathrm{M}_\odot$ and $M_2=0.295\pm 0.004\, \mathrm{M}_\odot$ for the white dwarf and the donor. The radius of the donor, $R_2=0.43\pm 0.002\, \mathrm{\it R}_\odot$, is ≈50 per cent larger than expected given its mass, while its spectral type is much earlier than the M3.5 type that would be expected from a main-sequence star with a similar mass. Hubble Space Telescope (HST) spectra show strong N v 1240-Å emission but no C iv 1550-Å emission, evidence for CNO-processed material. The donor is therefore a bloated, overluminous remnant of a thermal time-scale stage of high mass transfer and has yet to reestablish thermal equilibrium. Remarkably, the HST UV data also show a strong 30 per cent peak-to-peak, $38.9\,$s pulsation that we explain as being due to the spin of the white dwarf, potentially putting V1460 Her in a similar category to the propeller system AE Aqr in terms of its spin frequency and evolutionary path. AE Aqr also features a post-thermal time-scale mass donor, and V1460 Her may therefore be its weak magnetic field analogue since the accretion disc is still present, with the white dwarf spin-up a result of a recent high accretion rate.

7-order enhancement of the Stern-Gerlach effect of neutrons diffracting in a crystal

We measured the spatial splitting of a non-polarized neutron beam passed through a crystal under diffraction conditions in heterogeneous magnetic field (analog to the Stern-Gerlach effect) into two polarized components with opposite polarization. The measurements were carried out using Laue diffraction scheme, small gradients of the magnetic field and Bragg angles close to orthogonality θB=(78−82)∘. After a flight path in crystal of 21.6 cm a splitting of 4.1±0.1 cm was achieved (using a field gradient of ∼3 G/cm and a diffraction angle of 82∘). In the absence of a diffraction (crystal) but otherwise the same flight path and field gradient the spatial splitting would be ∼4⋅10−7 cm. From those we deduce an experimental amplification factor in the order of about ∼2⋅105tan2⁡θB due to the use of diffraction in crystals, which agrees with theory.

Coherent representation of fields and deformation quantization

Motivated by some well-known results in the phase space description of quantum optics and quantum information theory, we aim to describe the formalism of quantum field theory by explicitly considering the holomorphic representation for a scalar field within the deformation quantization program. Notably, the symbol of a symmetric ordered operator in terms of holomorphic variables may be straightforwardly obtained by the quantum field analogue of the Husimi distribution associated with a normal ordered operator. This relation also allows to establish a [Formula: see text]-equivalence between the Moyal and the normal star-products. In addition, by writing the density operator in terms of coherent states we are able to directly introduce a series representation of the Wigner functional distribution, which may be convenient in order to calculate probability distributions of quantum field observables without performing formal phase space integrals at all.

Effect of impact shock on extremophilic Halomonas gomseoemensis EP-3 isolated from hypersaline sulphated lake Laguna de Peña Hueca, Spain

The geologic histories of planetary surfaces reveal that Earth and Mars have been pummeled by cataclysmic impact events. The surface of Mars has been perused to have an impact origin for its hemispheric dichotomy. The spallation during impact events causes the interplanetary transfer of material from Mars to Earth or Mars to Phobos/Deimos. Assessing the survival of micro-organisms in impact conditions is critical for the development of planetary protection strategies for future missions. Shock waves are generated during such major impact events. The objective of the present investigation was to explore the microbial diversity of the hypersaline sulphated Laguna de Peña Hueca, Spain and to study the effect of shock waves on extremophilic bacteria isolated from the lake. Peña Hueca is a hypersaline sulphated lagoon rich in Mg–Na–SO4–Cl, epsomite and hexahydrate and it potentially serves as Planetary field analogue site for Martian chloride deposits and salt-rich subsurface brines of Ocean worlds like Enceladus and Europa. The microbial community structure of the lagoon was studied by 16S rRNA metagenomic sequencing. The phylogenetic studies indicated the presence of phyla Euryarchaeota, Proteobacteria, and Bacteroides in the hypersaline brines of the lagoon. The anoxic sediments of Peña Hueca showed the presence of Haloanaerobiaeta and Hadesarchaeota including the anoxic genus of Haloanaerobium, Desulfosalsimonas and Desulfovermiculum. The effect of impact shock on the halophilic bacterium Halomonas gomseomensis EP-3 isolated from Laguna de Peña Hueca was studied in a Reddy shock tube. The halophilic bacterium was exposed to shock waves at a peak shock pressure of 300 ​kPa and a temperature of 400 ​K. The results of shock recovery experiments of halophilic bacteria reveal 97% killing at 300 ​kPa and Mach number of 1.47 in comparison with Bacillus sp. This study indicates that gram-positive spore-forming Bacillus sp. are better adapted to survival in impact shock waves in comparison to non-sporulating halophiles. In the current study, we present the first report on response of halophiles in impact shock. Furthermore, we demonstrate a novel application of the simple handheld Reddy shock tube in astrobiology. The survival studies of halophiles isolated from terrestrial analogue sites in impact shock can provide valuable insights in astrobiology and microbial physiology in impact shock stress.

On a conjecture of Furusho over function fields

In the classical theory of multiple zeta values (MZV’s), Furusho proposed a conjecture asserting that the p-adic MZV’s satisfy the same \({\mathbb {Q}}\)-linear relations that their corresponding real-valued MZV counterparts satisfy. In this paper, we verify a stronger version of a function field analogue of Furusho’s conjecture in the sense that we are able to deal with all linear relations over an algebraic closure of the given rational function field, not just the rational linear relations. To each tuple of positive integers \({\mathfrak {s}}=(s_1, \ldots , s_r)\), we construct a corresponding t-module together with a specific rational point. The fine resolution (via fiber coproduct) of this construction actually allows us to obtain nice logarithmic interpretations for both the \(\infty \)-adic MZV and v-adic MZV at \({\mathfrak {s}}\), completely generalizing the work of Anderson–Thakur (Ann Math (2) 132(1):159–191, 1990) in the case of \(r=1\). Furthermore it enables us to apply Yu’s sub-t-module theorem (Yu in Ann Math (2) 145(2):215–233, 1997), connecting any \(\infty \)-adic linear relation on MZV’s with a sub-t-module of a corresponding giant t-module. This makes it possible to arrive at the same linear relation for v-adic MZV’s.

Dynamic of a lacustrine sedimentary system during late rifting at the Cretaceous–Palaeocene transition: Example of the Yacoraite Formation, Salta Basin, Argentina

AbstractThe architecture of lacustrine systems is the result of the complex interaction between tectonics, climate and environmental parameters, and constitute the main forcing parameters on the lake dynamics. Field analogue studies have been performed to better assess such interactions, and their impact on the facies distribution and the stratigraphic architecture of lacustrine systems. The Yacoraite Formation (Late Cretaceous/Early Palaeocene), deposited during the sag phase of the Salta rift basin in Argentina, is exposed in world‐class outcrops that allowed the dynamics of this lacustrine system to be studied through facies analysis and stratigraphic evolution. On the scale of the Alemania‐Metán‐El Rey Basin, the Yacoraite Formation is organized with a siliciclastic‐dominated margin to the west, and a carbonate‐dominated margin to the east. The Yacoraite can be subdivided into four main ‘mid‐term’ sequences and further subdivided into ‘short‐term’ sequences recording high frequency climate fluctuations. Furthermore, the depositional profiles and identified system tracts have been grouped into two end‐members at basin scale: (a) a balanced ‘perennial’ depositional system for the lower part of the Yacoraite Formation and (b) a highly alternating ‘ephemeral’ depositional system for the upper part of the Yacoraite Formation. The transition from a perennial system to an ephemeral system indicates a change in the sedimentary dynamics of the basin, which was probably linked with the Cretaceous/Tertiary boundary that induced a temporary shutdown of carbonate production and an increase in siliciclastic supply.

A representation theory approach to integral moments of L-functions over function fields

We propose a new heuristic approach to integral moments of L-functions over function fields, which we demonstrate in the case of Dirichlet characters ramified at one place (the function field analogue of the moments of the Riemann zeta function, where we think of the character n^{it} as ramified at the infinite place). We represent the moment as a sum of traces of Frobenius on cohomology groups associated to irreducible representations. Conditional on a hypothesis on the vanishing of some of these cohomology groups, we calculate the moments of the L-function and they match the predictions of the Conrey-Farmer-Keating-Rubinstein-Snaith recipe. In this case, the decomposition into irreducible representations seems to separate the main term and error term, which are mixed together in the long sums obtained from the approximate functional equation, even when it is dyadically decomposed. This makes our heuristic statement relatively simple, once the geometric background is set up. We hope that this will clarify the situation in more difficult cases like the L-functions of quadratic Dirichlet characters to squarefree modulus. There is also some hope for a geometric proof of this cohomological hypothesis, which would resolve the moment problem for these L-functions in the large degree limit over function fields.

Certain product formulas and values of Gaussian hypergeometric series

In this article we find finite field analogues of certain product formulas satisfied by the classical hypergeometric series. We express product of two ${_2}F_1$-Gaussian hypergeometric series as ${_4}F_3$- and ${_3}F_2$-Gaussian hypergeometric series. We use properties of Gauss and Jacobi sums and our earlier works on finite field Appell series to deduce these product formulas satisfied by the Gaussian hypergeometric series. We then use these transformations to evaluate explicitly some special values of ${_4}F_3$- and ${_3}F_2$-Gaussian hypergeometric series. By counting points on CM elliptic curves over finite fields, Ono found certain special values of ${_2}F_1$- and ${_3}F_2$-Gaussian hypergeometric series containing trivial and quadratic characters as parameters. Later, Evans and Greene found special values of certain ${_3}F_2$-Gaussian hypergeometric series containing arbitrary characters as parameters from where some of the values obtained by Ono follow as special cases. We show that some of the results of Evans and Greene follow from our product formulas including a finite field analogue of the classical Clausen's identity.

Quantifying uncertainty for remote spectroscopy of surface composition

Remote surface measurements by imaging spectrometers play an important role in planetary and Earth science. To make these measurements, investigators calibrate instrument data to absolute units, invert physical models to estimate atmospheric effects, and then determine surface properties from the spectral reflectance. This study quantifies the uncertainty in this process. Global missions demand predictive uncertainty models that can estimate future errors for varied environments and observing conditions. Here we validate uncertainty predictions with remote surface composition retrievals and in situ measurements in a field analogue of Earth and planetary exploration. We consider rover transects at Cuprite, Nevada, and remote observations by NASA's Next-Generation Airborne Visible Infrared Imaging Spectrometer (AVIRIS-NG). We show that accounting for input uncertainties can benefit mineral detection methods such as constrained spectrum fitting. This suggests that operational uncertainty estimates could improve future NASA missions like the Earth Mineral dust source InvesTigation (EMIT) and the Lunar Trailblazer mission, as well as NASA's Decadal Surface Biology and Geology (SBG) Investigation.

Evolution of salt structures of the Pyrenean rift (Chaînons Béarnais, France): From hyper-extension to tectonic inversion

The Chaînons Béarnais is a salt-detached fold belt in the northern Pyrenees that formerly occupied the axis of the Cretaceous Pyrenean rift. Geological map revision and cross-section construction from surface geology and industrial well and seismic reflection data emphasize the role of salt diapirism in the folding of the belt during the Cretaceous extension and the subsequent Pyrenean orogeny. Pre-rift Triassic evaporites played a fundamental role during rifting, allowing the sedimentary basin lying above to detach and slide down the hyper-extended margins onto a central exhumed mantle tract. Since the Early Cretaceous (and locally probably since the Jurassic) a system of low-amplitude salt walls evolved in shallow marine environments punctuated by episodic emersion. During the main stage of crustal extension, in late Aptian to early Cenomanian times, carbonate shelves rapidly drowned giving rise to deeper marine sedimentation. This was a period of major rise of salt walls, progressively detached from their substratum. These salt walls enclosed minibasins that accumulated thick flysch deposits arranged in growth stratal patterns. Depocenter migration and foundering of previous diapiric highs controlled further flysch deposition during the Late Cretaceous, while moderate extension probably persisted until the onset of the Pyrenean compression. During the late Santonian to Paleogene Pyrenean orogeny, the sedimentary lid of the Chaînons Béarnais basin climbed back along the Triassic detachment onto the colliding continental margins, leading to salt wall squeezing and further rising. Based on the Cretaceous timing and style of growth folding, we suggest that salt wall squeezing was not solely related to the Pyrenean compression, but shortening affected the diapiric ridges during the syn-rift sliding by gravity and crowding in the basin center, as the rifted margins were pulled apart from beneath. This makes the Chaînons Béarnais belt a unique field analog for contractional salt-tectonic systems in distal continental margins.

Distinguished representations, Shintani base change and a finite field analogue of a conjecture of Prasad

Let E/F be a quadratic extension of fields, and G a connected quasi-split reductive group over F. Let Gop be the opposition group obtained by twisting G by the duality involution considered by Prasad. Assume that the field F is finite. Let π be an irreducible generic representation of G(E). When π is a Shintani base change lift of some representation of Gop(F), we give an explicit nonzero G(F)-invariant vector in terms of the Whittaker vector of π. This shows particularly that π is G(F)-distinguished.When the field F is p-adic, the paper also proves that the duality involution takes an irreducible admissible generic representation of G(F) to its contragredient. As a special case of this result, all generic representations of G2,F4 or E8 are self-dual.

Gonality of dynatomic curves and strong uniform boundedness of preperiodic points

Fix$d\geqslant 2$and a field$k$such that$\operatorname{char}k\nmid d$. Assume that$k$contains the$d$th roots of$1$. Then the irreducible components of the curves over$k$parameterizing preperiodic points of polynomials of the form$z^{d}+c$are geometrically irreducible and have gonality tending to$\infty$. This implies the function field analogue of the strong uniform boundedness conjecture for preperiodic points of$z^{d}+c$. It also has consequences over number fields: it implies strong uniform boundedness for preperiodic points of bounded eventual period, which in turn reduces the full conjecture for preperiodic points to the conjecture for periodic points. Our proofs involve a novel argument specific to finite fields, in addition to more standard tools such as the Castelnuovo–Severi inequality.

The fourth power mean of Dirichlet L-functions in {mathbb {F}}_q [T

We prove results on moments of L-functions in the function field setting, where the moment averages are taken over primitive characters of modulus R, where R is a polynomial in {mathbb {F}}_{q}[T]. We consider the behaviour as {{,mathrm{deg},}}R rightarrow infty and the cardinality of the finite field is fixed. Specifically, we obtain an exact formula for the second moment provided that R is square-full, an asymptotic formula for the second moment for any R, and an asymptotic formula for the fourth moment for any R. The fourth moment result is a function field analogue of Soundararajan’s result in the number field setting that improved upon a previous result by Heath-Brown. Both the second and fourth moment results extend work done by Tamam in the function field setting who focused on the case where R is prime. As a prerequisite for the fourth moment result, we obtain, for the special case of the divisor function, the function field analogue of Shiu’s generalised Brun–Titchmarsh theorem.

Equidistribution on homogeneous spaces and the distribution of approximates in Diophantine approximation

The present paper is concerned with equidistribution results for certain flows on homogeneous spaces and related questions in Diophantine approximation. Firstly, we answer in the affirmative, a question raised by Kleinbock, Shi and Weiss regarding equidistribution of orbits of arbitrary lattices under diagonal flows and with respect to unbounded functions. We then consider the problem of Diophantine approximation with respect to rationals in a fixed number field. We prove a number field analogue of a famous result of W. M.Schmidt which counts the number of approximates to Diophantine inequalities for a certain class of approximating functions. Further we prove "spiraling" results for the distribution of approximates of Diophantine inequalities in number fields. This generalizes the work of Athreya, Ghosh and Tseng as well as Kleinbock, Shi and Weiss.

Finite field analogue of restriction theorem for general measures

We study restriction problem in vector spaces over finite fields. We obtain finite field analogue of Mockenhaupt-Mitsis-Bak-Seenger restriction theorem, and we show that the range of the exponentials is sharp.

Limiting Properties of the Distribution of Primes in an Arbitrarily Large Number of Residue Classes

AbstractWe generalize current known distribution results on Shanks–Rényi prime number races to the case where arbitrarily many residue classes are involved. Our method handles both the classical case that goes back to Chebyshev and function field analogues developed in the recent years. More precisely, let $\unicode[STIX]{x1D70B}(x;q,a)$ be the number of primes up to $x$ that are congruent to $a$ modulo $q$. For a fixed integer $q$ and distinct invertible congruence classes $a_{0},a_{1},\ldots ,a_{D}$, assuming the generalized Riemann Hypothesis and a weak version of the linear independence hypothesis, we show that the set of real $x$ for which the inequalities $\unicode[STIX]{x1D70B}(x;q,a_{0})>\unicode[STIX]{x1D70B}(x;q,a_{1})>\cdots >\unicode[STIX]{x1D70B}(x;q,a_{D})$ are simultaneously satisfied admits a logarithmic density.

Divergent neuronal activity patterns in the avian hippocampus and nidopallium.

Sleep-related brain activity occurring during non-rapid eye-movement (NREM) sleep is proposed to play a role in processing information acquired during wakefulness. During mammalian NREM sleep, the transfer of information from the hippocampus to the neocortex is thought to be mediated by neocortical slow-waves and their interaction with thalamocortical spindles and hippocampal sharp-wave ripples (SWRs). In birds, brain regions composed of pallial neurons homologous to neocortical (pallial) neurons also generate slow-waves during NREM sleep, but little is known about sleep-related activity in the hippocampus and its possible relationship to activity in other pallial regions. We recorded local field potentials (LFP) and analogue multiunit activity (AMUA) using a 64-channel silicon multi-electrode probe simultaneously inserted into the hippocampus and medial part of the nidopallium (i.e., caudal medial nidopallium; NCM) or separately into the caudolateral nidopallium (NCL) of adult female zebra finches (Taeniopygia guttata) anesthetized with isoflurane, an anesthetic known to induce NREM sleep-like slow-waves. We show that slow-waves in NCM and NCL propagate as waves of neuronal activity. In contrast, the hippocampus does not show slow-waves, nor sharp-wave ripples, but instead displays localized gamma activity. In conclusion, neuronal activity in the avian hippocampus differs from that described in mammals during NREM sleep, suggesting that hippocampal memories are processed differently during sleep in birds and mammals.