scholarly journals Lefschetz property and powers of linear forms in 𝕂[x,y,z]

2018 ◽  
Vol 30 (4) ◽  
pp. 857-865 ◽  
Author(s):  
Charles Almeida ◽  
Aline V. Andrade
Keyword(s):  
Linear Form ◽  
Characteristic Zero ◽  
Maximal Rank ◽  
General Linear ◽  
Linear Forms ◽  
Lefschetz Property ◽  
Powers Of Linear Forms

Abstract In [9], Migliore, Miró-Roig and Nagel proved that if {R=\mathbb{K}[x,y,z]} , where {\mathbb{K}} is a field of characteristic zero, and {I=(L_{1}^{a_{1}},\dots,L_{4}^{a_{4}})} is an ideal generated by powers of four general linear forms, then the multiplication by the square {L^{2}} of a general linear form L induces a homomorphism of maximal rank in any graded component of {R/I} . More recently, Migliore and Miró-Roig proved in [7] that the same is true for any number of general linear forms, as long the powers are uniform. In addition, they conjectured that the same holds for arbitrary powers. In this paper, we will prove that this conjecture is true, that is, we will show that if {I=(L_{1}^{a_{1}},\dots,L_{r}^{a_{r}})} is an ideal of R generated by arbitrary powers of any set of general linear forms, then the multiplication by the square {L^{2}} of a general linear form L induces a homomorphism of maximal rank in any graded component of {R/I} .

scholarly journals The weak Lefschetz property for quotients by quadratic monomials

2020 ◽  
Vol 126 (1) ◽  
pp. 41-60
Author(s):  
Juan Migliore ◽  
Uwe Nagel ◽  
Hal Schenck
Keyword(s):  
Linear Form ◽  
Monomial Ideals ◽  
Geometric Characterization ◽  
General Linear ◽  
Laplace Equations ◽  
Weak Lefschetz Property ◽  
Lefschetz Property

Michałek and Miró-Roig, in J. Combin. Theory Ser. A 143 (2016), 66–87, give a beautiful geometric characterization of Artinian quotients by ideals generated by quadratic or cubic monomials, such that the multiplication map by a general linear form fails to be injective in the first nontrivial degree. Their work was motivated by conjectures of Ilardi and Mezzetti, Miró-Roig and Ottaviani, connecting the failure to Laplace equations and classical results of Togliatti on osculating planes. We study quotients by quadratic monomial ideals, explaining failure of the Weak Lefschetz Property for some cases not covered by Michałek and Miró-Roig.

scholarly journals On the weak Lefschetz property for powers of linear forms

2012 ◽  
Vol 6 (3) ◽  
pp. 487-526 ◽  
Author(s):  
Juan C. Migliore ◽  
Rosa M. Miró-Roig ◽  
Uwe Nagel
Keyword(s):  
Linear Forms ◽  
Weak Lefschetz Property ◽  
Lefschetz Property ◽  
Powers Of Linear Forms

scholarly journals On a characterization theorem on a-adic solenoids

2019 ◽  
Vol 489 (3) ◽  
pp. 227-231
Author(s):  
G. M. Feldman
Keyword(s):  
Gaussian Distribution ◽  
Real Line ◽  
Linear Form ◽  
Conditional Distribution ◽  
Random Variables ◽  
Characterization Theorem ◽  
The Other ◽  
Independent Random Variables ◽  
Linear Forms ◽  
The Real

According to the Heyde theorem the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given the other. We prove an analogue of this theorem for linear forms of two independent random variables taking values in an -adic solenoid containing no elements of order 2. Coefficients of the linear forms are topological automorphisms of the -adic solenoid.

scholarly journals ELEMENTARY INVARIANTS FOR CENTRALIZERS OF NILPOTENT MATRICES

2009 ◽  
Vol 86 (1) ◽  
pp. 1-15 ◽  
Author(s):  
JONATHAN BROWN ◽  
JONATHAN BRUNDAN
Keyword(s):  
Lie Algebra ◽  
Characteristic Zero ◽  
Universal Enveloping Algebra ◽  
General Linear ◽  
Enveloping Algebra ◽  
Nilpotent Matrix ◽  
Nilpotent Matrices ◽  
Algebra Over A Field

AbstractWe construct an explicit set of algebraically independent generators for the center of the universal enveloping algebra of the centralizer of a nilpotent matrix in the general linear Lie algebra over a field of characteristic zero. In particular, this gives a new proof of the freeness of the center, a result first proved by Panyushev, Premet and Yakimova.

Lifting n-Dimensional Galois Representations

2008 ◽  
Vol 60 (5) ◽  
pp. 1028-1049 ◽  
Author(s):  
Spencer Hamblen
Keyword(s):  
Characteristic Zero ◽  
Galois Representations ◽  
Galois Groups ◽  
Geometric Shape ◽  
General Linear ◽  
Linear Groups ◽  
Selmer Group ◽  
General Linear Groups

AbstractWe investigate the problem of deforming n-dimensional mod p Galois representations to characteristic zero. The existence of 2-dimensional deformations has been proven under certain conditions by allowing ramification at additional primes in order to annihilate a dual Selmer group. We use the same general methods to prove the existence of n-dimensional deformations.We then examine under which conditions we may place restrictions on the shape of our deformations at p, with the goal of showing that under the correct conditions, the deformations may have locally geometric shape. We also use the existence of these deformations to prove the existence as Galois groups over ℚ of certain infinite subgroups of p-adic general linear groups.

Reciprocal velocity-porosity general linear form provides consistent and systematic industry-wide applications

2019 ◽  
Vol 7 (4) ◽  
pp. T751-T759
Author(s):  
Killian Ikwuakor
Keyword(s):  
Linear Form ◽  
Oil And Gas ◽  
Rock Physics ◽  
Time Lapse ◽  
General Linear ◽  
S Wave ◽  
Rock Property ◽  
Formation Evaluation ◽  
Clastic Rocks ◽  
Universal Applicability

Velocity is an important rock property that is required and used in different applications in petrophysics, rock physics, and seismic. The published literature shows a plethora of equations and models that relate velocity and porosity, a critical reservoir property. Attempts to account for the presence of shale in the formation invariably lead to more complicated relations. The inability of the industry to streamline these relations handicaps advancements in rock physics and formation evaluation, complicates the application of best practices in time-lapse seismic and fluid substitutions, and jeopardizes the integration of petrophysical, geologic, and seismic characteristics of oil and gas reservoirs. I have considered the following criteria to grade some of the different velocity-porosity relations in use today: (1) the significance of effective stress, (2) usefulness for interpreting geology, (3) predictive capability, and (4) universal applicability. Judging by these criteria, the general linear form, first prescribed by the late George R. Pickett, is the clear winner. The general linear form is a linear relationship between the reciprocal velocity and porosity. It passes theoretical and empirical justification. It is also valid for P- and S-wave velocities, yields easily to mathematical manipulation, and satisfies carbonate as well as clastic rocks for porosities encountered in everyday subsurface investigations. I evaluate practical examples in which the general linear form is the basis for multiple rock-typing criteria, comparative formation evaluation, and interpretive use of the [Formula: see text] ratio. Appropriate integration of the general linear form with other rock property relations provides avenues to redefine the [Formula: see text] ratio and acoustic impedance, and it expands the understanding and applications of reservoir elastic properties, as well as it constrains and streamlines rock physics models and applications.

scholarly journals Real and Complex Rank for Real Symmetric Tensors with Low Ranks

Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Edoardo Ballico ◽  
Alessandra Bernardi
Keyword(s):  
Homogeneous Polynomial ◽  
Symmetric Tensors ◽  
Linear Forms ◽  
The Real ◽  
The Difference ◽  
Powers Of Linear Forms

We study the case of a real homogeneous polynomial whose minimal real and complex decompositions in terms of powers of linear forms are different. We prove that if the sum of the complex and the real ranks of is at most , then the difference of the two decompositions is completely determined either on a line or on a conic or two disjoint lines.

A general canonical expression

1933 ◽  
Vol 29 (4) ◽  
pp. 465-469 ◽  
Author(s):  
J. Bronowski
Keyword(s):  
Linear Space ◽  
Dimensional Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
General Point ◽  
Linear Forms ◽  
Powers Of Linear Forms ◽  
Necessary And Sufficient

1. In a recent paper I established new conditions for a form φ of order n, homogeneous in r + 1 variables, to be expressible as the sum of nth powers of linear forms in these variables; and for this expression, if it exists, to be unique. These conditions, I further showed, may be stated as general theorems regarding the secant spaces of manifolds Mr in higher space, namely:Necessary and sufficient conditions that through a general point of a space N, of h (r + 1) − 1 dimensions, there passes (i) no, (ii) a unique (h − 1)-dimensional space containing h points of a manifold Mr lying in N are that(i) the space projecting a general point of Mr from the join of h − 1 general r-dimensional tangent spaces of Mr meets Mr in a curve, so that Mr cannot be so projected upon a linear space of r dimensions;(ii) the space projecting a general point of Mr from the join of h − 1 general r-dimensional tangent spaces of Mr does not meet Mr again, so that Mr can be so projected, birationally, upon a linear space of r dimensions..

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