Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)
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Global solvability of the Laplace equation in weighted Sobolev spaces
No full textWe consider a non-local boundary value problem for the Laplace equation in an unbounded strip, studying the weak and strong solvability of the problem within the framework of the weighted Sobolev space W1,pν with a Muckenhoupt weight. Utilising tools from non-harmonic analysis, we prove that any weak solution belonging to W2,pν is also a strong solution and satisfies the corresponding boundary conditions. It is worth noting that such problems do not fall within the scope of the general theory of elliptic equations and therefore require a specialized approach
The two-loop amplituhedron
Full text linkThe loop-Amplituhedron A(L)n is a semialgebraic set in the product of Grassmannians GrR(2,4)L. Recently, many aspects of this geometry for the case of L=1 have been elucidated, such as its algebraic and face stratification, its residual arrangement and the existence and uniqueness of the adjoint. This paper extends this analysis to the simplest higher loop case given by the two-loop four-point Amplituhedron A(2)4$
Chow-Lam recovery
Full text linkWe study the conditions under which a subvariety of the Grassmannian may be recovered from certain of its linear projections. In the special case that our Grassmannian is projective space, this is equivalent to asking when a variety can be recovered from its Chow form; the answer is "always" by work of Chow in 1937. In the general Grassmannian setting, the analogous question is when a variety can be recovered from its Chow-Lam form. We give both necessary conditions for recovery and families of examples where, in contrast with the projective case, recovery is not possible
Uniqueness of MHV gravity amplitudes
Full text linkWe investigate MHV tree-level gravity amplitudes as defined on the spinor-helicity variety. Unlike their gluon counterparts, the gravity amplitudes do not have logarithmic singularities and do not admit Amplituhedron-like construction. Importantly, they are not determined just by their singularities, but rather their numerators have interesting zeroes. We make a conjecture about the uniqueness of the numerator and explore this feature from a more mathematical perspective. This leads us to a new approach for examining adjoints. We outline steps of our proposed proof and provide computational evidence for its validity in specific cases
Comparing Hilbert depth of I with Hilbert depth of S/I
No full textLet S = K[x1,...,xn] be the ring of polynomials over a field K and let I be a monomial ideal of S. We prove that the following are equivalent: (i) I is principal, (ii) hdepth(I) = n, (iii) hdepth(S/I) = n − 1.
If I is squarefree, we prove that if hdepth(S/I) ≤ 3 or n ≤ 5, then hdepth(I) ≥ hdepth(S/I) + 1. Also, we prove that if hdepth(S/I) ≤ 5 or n ≤ 7, then hdepth(I) ≥ hdepth(S/I)
Nonnil-Noetherian pairs of the form (R, R[X]) and some related results
No full textThe rings considered in this paper are commutative with identity and are nonzero. Let R be a ring. An ideal I of R is said to be nonnil if it is not contained in the nilradical of R. We say that R is nonnil-Noetherian (resp., nonnil-Laskerian) if each proper nonnil ideal of R is finitely generated (resp., admits a primary decomposition). Whenever T is an extension ring of R, we assume that R contains the identity element of T. Let T be an extension ring of R. We say that (R, T) is a Nonnil-Noetherian pair (resp., Nonnil-Laskerian pair) if f A is nonnil-Noetherian (resp., nonnil-Laskerian) for any intermediate ring A between R and T. This paper aims to characterize R such that (R, R[X]) is a nonnil-Noetherian pair (resp., nonnil-Laskerian pair), where R[X] is the polynomial ring in one variable X over R. Also, this paper aims to characterize R such that each intermediate ring A between R and R[X] posses a property which is related to being nonnil-Noetherian (resp., nonnil-Laskerian)
Binary geometries from pellytopes
Full text linkBinary geometries have recently been introduced in particle physics in connection with stringy integrals. In this work, we study a class of simple polytopes, called \emph{pellytopes}, whose number of vertices are given by Pell\u27s numbers. We provide a new family of binary geometries determined by pellytopes as conjectured by He--Li--Raman--Zhang. We relate this family to the moduli space of curves by comparing the pellytope to the ABHY associahedron
What is Positive Geometry?
Full text linkThis article serves as an introduction to the special volume on Positive Geometry in the journal Le Matematiche. We attempt to answer the question in the title by describing the origins and objects of positive geometry at this early stage of its development. We discuss the problems addressed in the volume and report on the progress. We also list some open challenges.
On almost commutative unital complex normed Q-algebras
No full textWe show that a unital complex normed Q-algebra (A,∥.∥) in which the spectral radius satisfies:
ρA(x) = inf{p(x) : p ∈ Eun(A), p ≤ ∥.∥},
where Eun(A) denotes the set of all algebra-norms p on A equivalent to the given algebra-norm ∥.∥ such that p(e) = 1, is commutative modulo its Jacobson radical. The same conclusion is obtained if (A, ∥.∥) satisfies:
ρAb(xy) ≤ ρAb(x) ∥y∥ for every x ∈ Ab, y ∈ A, where Ab is the completion of (A, ∥.∥)
Differential equations for moving hyperplane arrangements
Full text linkWe investigate Mellin integrals of products of hyperplanes, raised to an individual power each. We refer to the resulting functions as combinatorial correlators. We investigate their behavior when moving the hyperplanes individually. To encode these functions as holonomic functions in the constant terms of the hyperplanes, we aim to construct a holonomic annihilating D-ideal purely in terms of the hyperplane arrangement