Ground Field Research Articles

On cohomology of filiform Lie superalgebras

Suppose the ground field F is an algebraically closed field of characteristic different from 2, 3. We determine the Betti numbers and make a decomposition of the associative superalgebra of the cohomology for the model filiform Lie superalgebra. We also describe the associative superalgebra structures of the (divided power) cohomology for some low-dimensional filiform Lie superalgebras.

The nilpotence degree of quantum Lie nilpotent algebras

We consider the quantum analog of the Lie commutator [Formula: see text] for an invertible element [Formula: see text] of the ground field and prove lower and upper bounds for the nilpotence degree of an associative algebra satisfying an identity of the form [Formula: see text].

Minimum ranks of sign patterns and zero–nonzero patterns and point–hyperplane configurations

A sign pattern (matrix) is a matrix whose entries are from the set {+,−,0}. The minimum rank (respectively, rational minimum rank) of a sign pattern matrix A is the minimum of the ranks of the real (respectively, rational) matrices whose entries have signs equal to the corresponding entries of A. A sign pattern A is said to be condensed if A has no zero row or column and no two rows or columns are identical or negatives of each other. A zero–nonzero pattern (matrix) is a matrix whose entries are from the set {0,⋆}, where ⋆ indicates a nonzero entry. Many of the sign pattern notions carry over to zero–nonzero patterns, assuming that the ground field is R. In this paper, a direct connection between condensed m×n sign patterns and zero–nonzero patterns with minimum rank r and m point–n hyperplane configurations in Rr−1 is established. In particular, condensed sign patterns (and zero–nonzero patterns) with minimum rank 3 are closely related to point–line configurations on the plane. Using this connection, we construct the smallest known sign pattern whose minimum rank is 3 but whose rational minimum rank is greater than 3. It is proved that for any sign pattern or zero–nonzero pattern A, if the number of zero entries on each column of A is at most 2, then the rational and real minimum ranks of A are equal. Further, it is shown that for any zero–nonzero pattern A with minimum rank r≥3, if the number of zero entries on each column of A is at most r−1, then the rational minimum rank of A is also r. A few related conjectures and open problems are raised.

Investigation the influence of soil’s moisture regime on their degradation using the remote sensing and ground field verification

In the article were used the field and distance methods of determination of moisture content in the upper layer of soil cover. Comparison of field measurements of soil moisture and Sentinel-2 image processing results revealed their correlation. On the basis of the moisture content data in the soil layer and the vegetation state analysis the soil fertility prediction was made. The research were done near Berezan, the Baryshivskyi raion in east-central Kiev Oblast of Ukraine.

A reduction theorem for tau -rigid modules

We prove a theorem which gives a bijection between the support tau -tilting modules over a given finite-dimensional algebra A and the support tau -tilting modules over A / I, where I is the ideal generated by the intersection of the center of A and the radical of A. This bijection is both explicit and well-behaved. We give various corollaries of this, with a particular focus on blocks of group rings of finite groups. In particular we show that there are tau -tilting-finite wild blocks with more than one simple module. We then go on to classify all support tau -tilting modules for all algebras of dihedral, semidihedral and quaternion type, as defined by Erdmann, which include all tame blocks of group rings. Note that since these algebras are symmetric, this is the same as classifying all basic two-term tilting complexes, and it turns out that a tame block has at most 32 different basic two-term tilting complexes. We do this by using the aforementioned reduction theorem, which reduces the problem to ten different algebras only depending on the ground field k, all of which happen to be string algebras. To deal with these ten algebras we give a combinatorial classification of all tau -rigid modules over (not necessarily symmetric) string algebras.

Systems with Parameters, or Efficiently Solving Systems of Polynomial Equations: 33 Years Later. I

Consider a system of polynomial equations with parametric coefficients over an arbitrary ground field. We show that the variety of parameters can be represented as a union of strata. For values of the parameters from each stratum, the solutions of the system are given by algebraic formulas depending only on this stratum. Each stratum is a quasiprojective algebraic variety with degree bounded from above by a subexponential function in the size of the input data. The number of strata is also subexponential in the size of the input data. Thus, here we avoid double exponential upper bounds on the degrees and solve a long-standing problem

Bounded linear endomorphisms of rigid analytic functions

Let $K$ be a field of characteristic zero complete with respect to a non-trivial, non-Archimedean valuation. We relate the sheaf $\widehat{\mathcal{D}}$ of infinite order differential operators on smooth rigid $K$-analytic spaces to the algebra $\mathcal{E}$ of bounded $K$-linear endomorphisms of the structure sheaf. In the case of complex manifolds, Ishimura proved that the analogous sheaves are isomorphic. In the rigid analytic situation, we prove that the natural map $\widehat{\mathcal{D}} \to \mathcal{E}$ is an isomorphism if and only if the ground field $K$ is algebraically closed and its residue field is uncountable.

On derived Hall numbers for tame quivers

In the present paper we study the derived Hall algebra for the bounded derived category of the finite dimensional nilpotent representations of a Dynkin or tame quiver over a finite field. We show that for any three given objects in the bounded derived category, the associated derived Hall numbers are given by a rational function in the cardinalities of ground fields.

Rational points of bounded height on general conic bundle surfaces

A conjecture of Manin predicts the asymptotic distribution of rational points of bounded height on Fano varieties. In this paper we use conic bundles to obtain correct lower bounds or a wide class of surfaces over number fields for which the conjecture is still far from being proved. For example, we obtain the conjectured lower bound of Manin's conjecture for any del Pezzo surface whose Picard rank is sufficiently large, or for arbitrary del Pezzo surfaces after possibly an extension of the ground field of small degree.

Varieties over finite fields: quantitative theory

Algebraic varieties over finite fields are considered from the point of view of their invariants such as the number of points of a variety that are defined over the ground field and its extensions. The case of curves has been actively studied over the last thirty-five years, and hundreds of papers have been devoted to the subject. In dimension two or higher, the situation becomes much more difficult and has been little explored. This survey presents the main approaches to the problem and describes a major part of the known results in this direction. Bibliography: 102 titles.

Cohomology of model filiform Lie superalgebras

Suppose the ground field [Formula: see text] is an algebraically closed field of characteristic zero. By means of spectral sequences, the computation of the first cohomology group of the model filiform Lie superalgebra [Formula: see text] with coefficients in the adjoint module is reduced to the computation of the first cohomology group of an Abel ideal and a one-dimensional subalgebra. Then, by calculating the outer derivations, the algebra structure of the first cohomology group of [Formula: see text] is completely characterized.

The Milnor Number of Plane Branches with Tame Semigroups of Values

The Milnor number of an isolated hypersurface singularity, defined as the codimension $$\mu (f)$$ of the ideal generated by the partial derivatives of a power series f that represents locally the hypersurface, is an important topological invariant of the singularity over the complex numbers. However it may loose its significance when the base field is arbitrary. It turns out that if the ground field is of positive characteristic, this number depends upon the equation f representing the hypersurface, hence it is not an invariant of the hypersurface. For a plane branch represented by an irreducible convergent power series f in two indeterminates over the complex numbers, it was shown by Milnor that $$\mu (f)$$ always coincides with the conductor c(f) of the semigroup of values S(f) of the branch. This is not true anymore if the characteristic of the ground field is positive. In this paper we show that, over algebraically closed fields of arbitrary characteristic, this is true, provided that the semigroup S(f) is tame, that is, the characteristic of the field does not divide any of its minimal generators.

Leading terms of anticyclotomic Stickelberger elements and 𝑝-adic periods

Let E E be a quadratic extension of a totally real number field. We construct Stickelberger elements for Hilbert modular forms of parallel weight 2 in anticyclotomic extensions of E E . Extending methods developed by Dasgupta and Spieß from the multiplicative group to an arbitrary one-dimensional torus we bound the order of vanishing of these Stickelberger elements from below and, in the analytic rank zero situation, we give a description of their leading terms via automorphic L \mathcal {L} -invariants. If the field E E is totally imaginary, we use the p p -adic uniformization of Shimura curves to show the equality between automorphic and arithmetic L \mathcal {L} -invariants. This generalizes a result of Bertolini and Darmon from the case that the ground field is the field of rationals to arbitrary totally real number fields.

Cylindrical harmonic analysis of the magnetic field in the aperture of the superconducting winding of an electromagnet

It is of practical interest to create such models of the electromagnetic field of an electromagnet that help correct the mean integral coefficients of the harmonics of magnetic induction by the geometric parameters of the magnet. The aim of the study was to obtain analytical expressions for the coefficients of the mean integral lengths of the cylindrical harmonics of magnetic induction created by the current in the superconducting winding inside the aperture of a dipole or quadrupole direct electromagnet on the basis of the geometric parameters of the winding. The analytical solution found is based on the sectorial spherical harmonics of the internal solution of the Laplace equation for the scalar potential of the magnetic field and allows associating them with a series of polar harmonics. The study shows the proportionality between the mean integral contributions to the magnetic induction, produced by uncharacteristic harmonics, and the mean integral value of the ground field. It has been determined that the contribution to the mean integral values of the harmonics coefficients from the end elements of the winding appears only in the presence of the iron framework as a result of its saturation. The received analytical representations make it possible to calculate necessary correction of geometrical parameters of a current winding when optimizing the magnetic field inside the aperture of dipole and quadrupole electromagnets with given mean integral factors.

CANONICAL FACTORIZATION OF THE QUOTIENT MORPHISM FOR AN AFFINE G $$ \mathbb{G} $$ a-VARIETY

Working over a ground field of characteristic zero, this paper studies the quotient morphism π : X → Y for an affine $$ \mathbb{G} $$ a-variety X with affine quotient Y. It is shown that the degree modules associated to the Ga-action give a uniquely determined sequence of dominant $$ \mathbb{G} $$ a-equivariant morphisms, $$ X={X}_r\to {X}_{r-1}\to \cdots \to {X}_1\to {X}_0=Y, $$ where Xi is an affine $$ \mathbb{G} $$ a-variety and Xi + 1 → Xi is birational for each i ≥ 1. This is the canonical factorization of 𝜋. We give an algorithm for finding the degree modules associated to the $$ \mathbb{G} $$ a-action, and this yields the canonical factorization of 𝜋. The algorithm is applied to compute the canonical factorization for several examples, including the homogeneous (2, 5) action on $$ {\mathbb{A}}^3 $$ The Freeness Conjecture, introduced in the paper's last section, asserts that, for any $$ \mathbb{G} $$ a-action on $$ X={\mathbb{A}}^3 $$ the polynomial ring k[X] is a free module over $$ k{\left[X\right]}^{{\mathbb{G}}_a} $$ .

Mapping the Spatial Distribution of Rice Fields in Southern Coast of Caspian Sea Using Landsat 8 Time-series Images

Since rice is one of the important crops not only in Iran but also in the world, it is vital to determine the paddy rice field as accurately as possible using fast and economical methods such as remote sensing and GIS. Accurate and timely rice paddy field maps with a fine spatial resolution would greatly improve our understanding of the effects of paddy rice agriculture on greenhouse gases missions, food and water security, and human health. In the past, the accuracy and efficiency for rice paddy field mapping at fine spatial resolutions were limited by the poor data availability and image-based algorithms. The unique physical features of rice paddy fields during the flooding/open-canopy period were captured with the dynamics of vegetation indices, which are then used to identify rice paddy fields. In order to prepare information about spatial distribution of rice fields, Multi-Spectral and Multi-Temporal data can be helpful because in addition to the rice, all fields could be covered by mixture of water and soil regarding the time of crop calendar. The Landsat 8 data with 11 bands has visible, near infrared, shortwave infrared as well as thermal bands; and therefore, different vegetation indices can be calculated, including Normalized Difference Vegetation Index (NDVI) and Land Surface Water Index (LSWI2105) which is sensitive to leaf water and soil moisture. In this study, special mapping algorithm that uses NDVI and LSWI2105 time series data derived from LANDSAT8 16-days 30-meter vegetation indices product of LANDSAT imagery developed to identify paddy rice fields. This algorithm works based on the sensitivity of LSWI2105 to the surface moisture and NDVI to the vegetation chlorophyll content. In this research method has been developed to define the relationship between the NDVI and LSWI2105 to detect the location of paddy rice fields in North part of Iran in 2014-15. The results, a validated with ground field works data at 56 well-distributed sample points. The overall accuracy of the method was 69.0909%.

Supporting Capacity-Building Project Characteristics Test of Zhoushan Multi-Terminal HVDC Flexible Demonstration Project

In this paper, a flexible Zhoushan multiterminal HVDC test demonstration projects supporting capacity-building project for the study, the DC cables through the flexible testing ground field measurements of its surrounding environment, the DC cables of a flexible environment affect the characteristics of the test site. The results show: the project on the surrounding environment has little effect of electromagnetic radiation, which affect the field strength generated by the project on the environment is negligible, the project generated frequency electric field, power frequency magnetic fields, DC magnetic field, radio interference strength, noise are smaller than the corresponding standard limits.

Hermitian Forms and Systems of Quadratic Forms

We associate to every symmetric (antisymmetric) Hermitian form a system of quadratic forms over the base field which determines its isotropy and metabolicity behaviour. It is shown that two even Hermitian forms are isometric if and only if their associated systems are equivalent. As an application, it is also shown that an anisotropic symmetric Hermitian form over a quaternion division algebra in characteristic two remains anisotropic over all odd degree extensions of the ground field.

Cellular structures using TEXTBACKSLASHmathboldTEXTBACKSLASHUq-tilting modules

We use the theory of $\textbf{U}_q$-tilting modules to construct cellular bases for centralizer algebras. Our methods are quite general and work for any quantum group $\textbf{U}_q$ attached to a Cartan matrix and include the non-semisimple cases for $q$ being a root of unity and ground fields of positive characteristic. Our approach also generalizes to certain categories containing infinite-dimensional modules. As applications, we give a new semisimplicty criterion for centralizer algebras, and recover the cellularity of several known algebras (with partially new cellular bases) which all fit into our general setup.

Separable extensions of orthogonal involutions in characteristic two

It is shown that an anisotropic orthogonal involution in characteristic two is totally decomposable if it is totally decomposable over a separable extension of the ground field. In particular, this settles a characteristic two analogue of a conjecture formulated by Bayer-Fluckiger et al.