Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)
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The quest for Diophantine finite-fold-ness
Full text linkThe Davis-Putnam-Robinson theorem showed that every partially computable -ary function f(a1, ..., am) = c on the natural numbers can be specified by means of an exponential Diophantine formula involving, along with parameters a1, ... am, c, some number k of existentially quantified variables. Yuri Matiyasevich improved this theorem in two ways: on the one hand, he proved that the same goal can be achieved with no recourse to exponentiation and, thereby, he provided a negative answer to Hilbert\u27s 10th problem; on the other hand, he showed how to construct an exponential Diophantine equation specifying f which, once a1, ... am have been fixed, is solved by at most one tuple < v0, ..., vk > of values for the remaining variables. This latter property is called single-foldness. Whether there exists a single- (or, at worst, finite-) fold polynomial Diophantine representation of any partially computable function on the natural numbers is as yet an open problem. This work surveys relevant results on this subject and tries to draw a route towards a hoped-for positive answer to the finite-fold-ness issue
Spectra of generalized corona of graphs constrained by vertex subsets
Full text linkIn this paper, we introduce a generalization of corona of graphs. This construction generalizes the generalized corona of graphs (consequently, the corona of graphs), the cluster of graphs, the corona-vertex subdivision graph of graphs and the corona-edge subdivision graph of graphs. Further, it enables to get some more variants of corona of graphs as its particular cases. To determine the spectra of the adjacency, Laplacian and the signless Laplacian matrices of the above mentioned graphs, we define a notion namely, the coronal of a matrix constrained by an index set, which generalizes the coronal of a graph matrix. Then we prove several results pertain to the determination of this value. Then we determine the characteristic polynomials of the adjacency and the Laplacian matrices of this graph in terms of the characteristic polynomials of the adjacency and the Laplacian matrices of the constituent graphs and the coronal of some matrices related to the constituent graphs. Using these, we derive the characteristic polynomials of the adjacency and the Laplacian matrices of the above mentioned existing variants of corona of graphs, and some more variants of corona of graphs with some special constraints
Linear spaces of symmetric matrices with non-maximal maximum likelihood degree
Full text linkWe study the maximum likelihood degree of linear concentration models in algebraic statistics. We relate the geometry of the reciprocal variety to that of semidefinite programming. We show that the Zariski closure in the Grassmanian of the set of linear spaces that do not attain their maximal possible maximum likelihood degree coincides with the Zariski closure of the set of linear spaces defining a projection with non-closed image of the positive semidefinite cone. In particular, this shows that this closure is a union of coisotropic hypersurfaces
Equations and multidegrees for inverse symmetric matrix pairs
Full text linkWe compute the equations and multidegrees of the biprojective variety that parametrizes pairs of symmetric matrices that are inverse to each other.
As a consequence of our work, we provide an alternative proof for a result of Manivel, Micha\l{}ek, Monin, Seynnaeve and Vodi\v{c}ka that settles a previous conjecture of Sturmfels and Uhler regarding the polynomiality of maximum likelihood degree
On solvability of pseudodifferential operators with constant coefficients
Full text linkThe paper presents a result on local solvability, in classical Sobolev spaces, of pseudodifferential operators with constant coefficients. The proof of this result is based on a new Ho ̈rmander’s type inequality for such pseudodifferential operators
Reciprocal maximum likelihood degrees of brownian motion tree models
Full text linkWe give an explicit formula for the reciprocal maximum likelihood degree of Brownian motion tree models. To achieve this, we connect them to certain toric (or log-linear) models, and express the Brownian motion tree model of an arbitrary tree as a toric fiber product of star tree models
Coloured graphical models and their symmetries
Full text linkColoured graphical models are Gaussian statistical models determined by an undirected coloured graph. These models can be described by linear spaces of symmetric matrices. We outline a relationship between the symmetries of the graph and the linear forms that vanish on the reciprocal variety of the model. In particular, we give four families for which such linear forms are completely described by symmetries
About perfection of circular mixed hypergraphs
Full text linkA mixed hypergraph is a triple H = (X,C,D), where X is the vertex set and each of C and D is a family of subsets of X, the C-edges and D-edges, respectively. A proper k-coloring of H is a mapping c : X → {1,...,k} such that each C-edge has two vertices with a common color and each D-edge has two vertices with different colors. Maximum number of colors in a coloring using all the colors is called upper chromatic number χ ̄(H). Maximum cardinality of subset of vertices which contains no C-edge is C-stability number αC (H). A mixed hypergraph is called C-perfect if χ ̄ (H\u27) = αC (H\u27) for any induced subhypergraph H\u27. A mixed hyper- graph H is called circular if there exists a host cycle on the vertex set X such that every edge (C- or D-) induces a connected subgraph on the host cycle. We give a characterization of C-perfect circular mixed hypergraphs
A generalization of the space of complete quadrics
Full text linkTo any homogeneous polynomial h we naturally associate a variety Ωh which maps birationally onto the graph Γh of the gradient map ∇h and which agrees with the space of complete quadrics when h is the determinant of the generic symmetric matrix. We give a sufficient criterion for Ωh being smooth which applies for example when h is an elementary symmetric polynomial. In this case Ωh is a smooth toric variety associated to a certain generalized permutohedron. We also give examples when Ωh is not smooth
Inverting catalecticants of ternary quartics
Full text linkWe study the reciprocal variety to the LSSM of catalecticant matrices associated with ternary quartics.
With numerical tools, we obtain 85 to be its degree and 36 to be the ML-degree of the LSSM.
We provide a geometric explanation to why equality between these two invariants is not reached,
as opposed to the case of binary forms,
by describing the intersection of the reciprocal variety and the orthogonal of the LSSM in the rank loci.
Moreover, we prove that only the rank- locus, namely the Veronese surface ν4(P2), contributes to the degree of the reciprocal variety