Symbolic Algebras Research Articles

Bridging Cognitive Architectures and Generative Models with Vector Symbolic Algebras

Recent developments in generative models have demonstrated that with the right data set, techniques, computational infrastructure, and network architectures, it is possible to generate seemingly intelligent outputs, without explicitly reckoning with underlying cognitive processes. The ability to generate novel, plausible behaviour could be a boon to cognitive modellers. However, insights for cognition are limited, given that generative models' blackbox nature does not provide readily interpretable hypotheses about underlying cognitive mechanisms. On the other hand, cognitive architectures make very strong hypotheses about the nature of cognition, explicitly describing the subjects and processes of reasoning. Unfortunately, the formal framings of cognitive architectures can make it difficult to generate novel or creative outputs. We propose to show that cognitive architectures that rely on certain Vector Symbolic Algebras (VSAs) are, in fact, naturally understood as generative models. We discuss how memories of VSA representations of data form distributions, which are necessary for constructing distributions used in generative models. Finally, we discuss the strengths, challenges, and future directions for this line of work.

Subrings of polynomial rings and the conjectures of Eisenbud and Evans

Let R be a commutative Noetherian ring of dimension d. In 1973, Eisenbud and Evans proposed three conjectures on the polynomial ring R[T]. These conjectures were settled in the affirmative by Sathaye, Mohan Kumar and Plumstead. One of the primary objectives of this article is to investigate the validity of these conjectures over Noetherian subrings of R[T] of dimension d+1, containing R. We formulate a class of such rings, which includes polynomial rings, Rees algebras, Rees-like algebras and Noetherian symbolic Rees algebras, and exhibit that all three conjectures hold for rings belonging to this class.

Symbolic Rees algebras and set-theoretic complete intersections

In this paper we extend a result of Cowsik on set-theoretic complete intersection and a result Huneke, Morales and Goto and Nishida about Noetherian symbolic Rees algebras of ideals. As applications, we show that the symbolic Rees algebras of the following ideals are Noetherian and the ideals are set-theoretic complete intersections: (a) the edge ideal of a complete graph, (b) the Fermat ideal and (c) the Jacobian ideal of a certain hyperplane arrangement.

Analytic spread of filtrations and symbolic algebras

In this paper we define and explore the analytic spread ℓ ( I ) $\ell (\mathcal {I})$ of a filtration in a local ring. We show that, especially for divisorial and symbolic filtrations, some basic properties of the analytic spread of an ideal extend to filtrations, even when the filtration is non-Noetherian. We also illustrate some significant differences between the analytic spread of a filtration and the analytic spread of an ideal with examples. In the case of an ideal I $I$ , we have the classical bounds ht ( I ) ⩽ ℓ ( I ) ⩽ dim R $\mbox{ht}(I)\leqslant \ell (I)\leqslant \dim R$ . The upper bound ℓ ( I ) ⩽ dim R $\ell (\mathcal {I})\leqslant \dim R$ is true for filtrations I $\mathcal {I}$ , but the lower bound is not true for all filtrations. We show that for the filtration I $\mathcal {I}$ of symbolic powers of a height two prime ideal p $\mathfrak {p}$ in a regular local ring of dimension three (a space curve singularity), so that ht ( I ) = 2 $\mbox{ht}(\mathcal {I}) =2$ and dim R = 3 $\dim R=3$ , we have that 0 ⩽ ℓ ( I ) ⩽ 2 $0\leqslant \ell (\mathcal {I})\leqslant 2$ and all values of 0, 1 and 2 can occur. In the cases of analytic spread 0 and 1 the symbolic algebra is necessarily non-Noetherian. The symbolic algebra is non-Noetherian if and only if ℓ ( p ( n ) ) = 3 $\ell (\mathfrak {p}^{(n)})=3$ for all symbolic powers of p $\mathfrak {p}$ and if and only if ℓ ( I a ) = 3 $\ell (\mathcal {I}_a)=3$ for all truncations I a $\mathcal {I}_a$ of I $\mathcal {I}$ .

On regularity of symbolic Rees algebras and symbolic powers of vertex cover ideals of graphs

In this work, we study the bigraded regularities of the symbolic Rees algebras R s ( J ( G ) ) , R s ( I ( G ) ) R_s(J(G)), R_s(I(G)) , of the vertex cover ideal J ( G ) J(G) and the edge ideal I ( G ) I(G) , of a graph G G respectively. We give combinatorial upper bounds for the ( 1 , 0 ) (1,0) -regularities of R s ( J ( G ) ) R_s(J(G)) and R s ( I ( G ) ) R_s(I(G)) . By using this upper bounds, we give general linear upper bounds for r e g ( J ( G ) ( k ) ) , r e g ( I ( G ) ( k ) ) reg(J(G)^{(k)}), reg(I(G)^{(k)}) for any k ≥ 1 k\geq 1 . Let G G be a graph on n n vertices and deg ⁡ ( J ( G ) ) \deg (J(G)) be the maximum degree of minimal generators of J ( G ) J(G) . We show that if G G is a non-bipartite graph, then k deg ⁡ ( J ( G ) ) ≤ r e g ( J ( G ) ( k ) ) ≤ k deg ⁡ ( J ( G ) ) + α 0 ( G ) − 1 + | A 0 ∪ { x i 1 , … , x i r } | − r , \begin{equation*} k \deg (J(G)) \!\leq \! reg(J(G)^{(k)})\!\leq \! k \deg (J(G))+ \alpha _0(G)-1+|A_0\cup \{x_{i_1}, \ldots , x_{i_r}\}|-r, \end{equation*} for all k ≥ 1 k \geq 1 , where α 0 ( G ) \alpha _0(G) denotes the vertex cover number of G G , A 0 A_0 is a maximal independent set in G G of maximal cardinality, and r r is the number of 0 0 -covers that are present in an irreducible representation of the affine cone associated with the irreducible covers of G G . Also if G G is a non-bipartite perfect graph, then 2 k ≤ r e g ( I ( G ) ( k ) ) ≤ 2 k + n − r + 1 , \begin{equation*} 2k \leq reg(I(G)^{(k)})\leq 2k+n-r+1, \end{equation*} for all k ≥ 1 k \geq 1 , where r r is the number of 0 0 -covers of Γ ( G ) \Gamma (G) that are present in an irreducible representation of the affine cone associated with the irreducible covers of Γ ( G ) \Gamma (G) .

Symbolic blowup algebras and invariants associated to certain monomial curves in

In this paper we explicitly describe the symbolic powers of the ideal defining the curve in parametrized by where q, m are positive integers, and We show that the symbolic blowup algebra is Noetherian and Gorenstein. An explicit formula for the resurgence and the Waldschmidt constant of the prime ideal defining the curve is computed. We also give a formula for the Castelnuovo-Mumford regularity of the symbolic powers for all

Symbolic blowup algebras and invariants of certain monomial curves in an affine space

Let and be integers such that Let be the defining ideal of the monomial curve in parametrized by where for all In this paper, we describe the symbolic powers for all As a consequence, we show that the symbolic blowup algebras and are Cohen–Macaulay. This gives a positive answer to a question posed by S. Goto (1994). We also discuss when these blowup algebras are Gorenstein. Moreover, for d = 3, considering as a weighted homogeneous ideal, we compute the resurgence, the Waldschmidt constant and the Castelnuovo–Mumford regularity of for all The techniques of this paper for computing are new, and we hope that these will be useful to study the symbolic powers of other prime ideals.

Splittings and Symbolic Powers of Square-free Monomial Ideals

Abstract We study the symbolic powers of square-free monomial ideals via symbolic Rees algebras and methods in prime characteristic. In particular, we prove that the symbolic Rees algebra and the symbolic associated graded algebra are split with respect to a morphism that resembles the Frobenius map and that exists in all characteristics. Using these methods, we recover a result by Hoa and Trung that states that the normalized $a$-invariants and the Castelnuovo–Mumford regularity of the symbolic powers converge. In addition, we give a sufficient condition for the equality of the ordinary and symbolic powers of this family of ideals and relate it to Conforti–Cornuéjols conjecture. Finally, we interpret this condition in the context of linear optimization.

Analytical solution to heat transfer in compressible laminar flow in a flat minichannel

Heat transfer in compressible laminar flow in mini-/micro-channels, a classical and general topic in fields of fuel cells, electronics, micro heat exchanger, etc., is revisited. Based on a two-dimensional continuum flow model, analytical solutions of the dimensionless model are achieved in closed-form symbolic algebras of Whittaker eigenfunctions, corresponding to two kinds of boundary conditions with arbitrarily prescribed wall temperature or wall heat flux. As the eigenvalues and eigenfunctions are independent on the dimensionless quantities, which influence the along-the-channel behaviors, the algorithm reveals the common features of compressible laminar thermal flows. The algorithms do not require the assumption of a linear pressure distribution, which is proved to be untenable in some cases (e.g. constant wall heat flux). The algorithms are validated well by the exact (numerical) computations in exemplary cases of both small and moderate Reynolds number, Mach number and Eckert number of air. Although expressed in a series of eigenfunctions, only several terms (sometimes one or two terms) of solutions are required for a practical computation.

Ordinary and symbolic Rees algebras for ideals of Fermat point configurations

Fermat ideals define planar point configurations that are closely related to the intersection locus of the members of a specific pencil of curves. These ideals have gained recent popularity as counterexamples to some proposed containments between symbolic and ordinary powers [6]. We give a systematic treatment of the family of Fermat ideals, describing explicitly the minimal generators and the minimal free resolutions of all their ordinary powers as well as many symbolic powers. We use these to study the ordinary and the symbolic Rees algebra of Fermat ideals. Specifically, we show that the symbolic Rees algebras of Fermat ideals are Noetherian. Along the way, we give formulas for the Castelnuovo–Mumford regularity of powers of Fermat ideals and we determine their reduction ideals.

Implosive graphs: Square-free monomials on symbolic Rees algebras

Let [Formula: see text] be the edge monomial ideal of a graph [Formula: see text], whose vertex set is [Formula: see text]. [Formula: see text] is implosive if the symbolic Rees algebra [Formula: see text] of [Formula: see text] has a minimal system of generators [Formula: see text] where [Formula: see text] are square-free monomials. We give some structural properties of implosive graphs and we prove that they are closed under clique-sums and odd subdivisions. Furthermore, we prove that universally signable graphs are implosive. We show that odd holes, odd antiholes and some Truemper configurations (prisms, thetas and even wheels) are implosive. Moreover, we study excluded families of subgraphs for the class of implosive graphs. In particular, we characterize which Truemper configurations and extensions of odd holes and antiholes are minimal nonimplosive.

M ¯ 0 , n is not a Mori dream space

Building on the work of Goto, Nishida, and Watanabe on symbolic Rees algebras of monomial primes, we prove that the moduli space of stable rational curves with n punctures is not a Mori dream space for n > 133 . This answers a question posed by Hu and Keel.

A theorem about Cremona maps and symbolic Rees algebras

This work is about the structure of the symbolic Rees algebra of the base ideal of a Cremona map. We give sufficient conditions under which this algebra has the "expected form" in some sense. The main theorem in this regard seemingly covers all previous results on the subject so far. The proof relies heavily on a criterion of birationality and the use of the so-called inversion factor of a Cremona map. A pretty long selection of examples of plane and space Cremona maps has been given to test against the conditions of the theorem, with special emphasis on Cohen–Macaulay base ideals.

On the prime ideal structure of symbolic Rees algebras

This paper contributes to the study of the prime spectrum and dimension theory of symbolic Rees algebra over Noetherian domains. We first establish some general results on the prime ideal structure of subalgebras of affine domains, which actually arise, in the Noetherian context, as domains between a domain $A$ and $A[a^{-1}]$. We then examine closely the special context of symbolic Rees algebras (which yielded the first counter-example to the Zariski-Hilbert problem). One of the results states that if $A$ is a Noetherian domain and $p$ a maximal ideal of $A$, then the Rees algebra of $p$ inherits the Noetherian-like behavior of being a stably strong S-domain. We also investigate graded rings associated with symbolic Rees algebras of prime ideals $p$ such that $A_{p}$ is a rank-one DVR and close with an application related to Hochster's result on the coincidence of the ordinary and symbolic powers of a prime ideal.

Symbolic powers of monomial ideals and vertex cover algebras

We introduce and study vertex cover algebras of weighted simplicial complexes. These algebras are special classes of symbolic Rees algebras. We show that symbolic Rees algebras of monomial ideals are finitely generated and that such an algebra is normal and Cohen–Macaulay if the monomial ideal is squarefree. For a simple graph, the vertex cover algebra is generated by elements of degree 2, and it is standard graded if and only if the graph is bipartite. We also give a general upper bound for the maximal degree of the generators of vertex cover algebras.

Straightening Law and Powers of Determinantal Ideals of Hankel Matrices

In this paper we establish a standard monomial theory for generic Hankel matrices. By a generic Hankel matrix we mean a matrix Y=( yij) with yij=x i+ j&1 where the x i are indeterminates over a field K. We use this structure to determine the symbolic powers and the primary decomposition of the powers of the determinantal ideals It of Y. Further we prove that the symbolic and ordinary Rees algebras of It are Cohen Macaulay normal domains. The first standard monomial theory was developed by Hodge [H] to study the homogeneous coordinate ring of the Grassmannian variety. Later standard monomial theories were established for generic matrices by Doubilet, Rota and Stein [DRS], and for generic symmetric and generic skew symmetric matrices by De Concini and Procesi [DP]. These are all examples of algebras with straightening law (ASL for short) over a poset or over a doset. The abstract notion of ASL was introduced and developed by Eisenbud [E1], and by De Concini, Eisenbud and Procesi [DEP2], see also [BV]. These structures turned out to be an extremely powerful tool in studying determinantal rings and ideals arising from the above mentioned generic matrices. Let now x1 , ..., xn be indeterminates over an arbitrary field K. For j=1, ..., n we denote by Xj the j_(n+1& j) Hankel matrix with entries x1 , ..., xn , that is,

Gorenstein algebras, symmetric matrices, self-linked ideals, and symbolic powers

Inspired by recent work in the theory of central projections onto hypersurfaces, we characterize self-linked perfect ideals of grade 2 2 as those with a Hilbert–Burch matrix that has a maximal symmetric subblock. We also prove that every Gorenstein perfect algebra of grade 1 1 can be presented, as a module, by a symmetric matrix. Both results are derived from the same elementary lemma about symmetrizing a matrix that has, modulo a nonzerodivisor, a symmetric syzygy matrix. In addition, we establish a correspondence, roughly speaking, between Gorenstein perfect algebras of grade 1 1 that are birational onto their image, on the one hand, and self-linked perfect ideals of grade 2 2 that have one of the self-linking elements contained in the second symbolic power, on the other hand. Finally, we provide another characterization of these ideals in terms of their symbolic Rees algebras, and we prove a criterion for these algebras to be normal.

The Cohen-Macaulay symbolic Rees algebras for curve singularities

The Cohen-Macaulay symbolic Rees algebras for curve singularities

Positive Characteristic Finite Generation of Symbolic Rees Algebras and Roberts' Counterexamples to the Fourteenth Problem of Hilbert

Positive Characteristic Finite Generation of Symbolic Rees Algebras and Roberts' Counterexamples to the Fourteenth Problem of Hilbert

The Gorensteinness of the symbolic blow-ups for certain space monomial curves

Let p = p ( n 1 , n 2 , n 3 ) {\mathbf {p}} = {\mathbf {p}}({n_1},{n_2},{n_3}) denote the prime ideal in the formal power series ring A = k [ [ X , Y , Z ] ] A = k[[X,Y,Z]] over a field k k defining the space monomial curve X = T n 1 X = {T^{{n_1}}} , Y = T n 2 Y = {T^{{n_2}}} , and Z = T n 3 Z = {T^{{n_3}}} with GCD ( n 1 , n 2 , n 3 ) = 1 {\text {GCD}}({n_1},{n_2},{n_3}) = 1 . Then the symbolic Rees algebras R s ( p ) = ⊕ n ≥ 0 p ( n ) {R_s}({\mathbf {p}}) = { \oplus _{n \geq 0}}{{\mathbf {p}}^{(n)}} are Gorenstein rings for the prime ideals p = p ( n 1 , n 2 , n 3 ) {\mathbf {p}} = {\mathbf {p}}({n_1},{n_2},{n_3}) with min { n 1 , n 2 , n 3 } = 4 \min \{ {n_1},{n_2},{n_3}\} = 4 and p = p ( m , m + 1 , m + 4 ) {\mathbf {p}} = {\mathbf {p}}(m,m + 1,m + 4) with m ≠ 9 , 13 m \ne 9,13 . The rings R s ( p ) {R_s}({\mathbf {p}}) for p = p ( 9 , 10 , 13 ) {\mathbf {p}} = {\mathbf {p}}(9,10,13) and p = p ( 13 , 14 , 17 ) {\mathbf {p}} = {\mathbf {p}}(13,14,17) are Noetherian but non-Cohen-Macaulay, if ch k = 3 \operatorname {ch}\,k = 3 .