arXiv subjects

Jonathan Fine

Publications and source records attributed to Jonathan Fine.

At least 19 recordsLinked to original sources

arXiv2024

Paddy: Evolutionary Optimization Algorithm for Chemical Systems and Spaces

Optimization of chemical systems and processes have been enhanced and enabled by the guidance of algorithms and analytical approaches. While many methods will systematically investigate how underlying variables govern a given outcome, there is often a substantial number of experiments needed to accurately model these relations. As chemical systems increase in complexity, inexhaustive processes must propose experiments that efficiently optimize the underlying objective, while ideally avoiding convergence on unsatisfactory local minima. We have developed the Paddy software package around the Paddy Field Algorithm, a biologically inspired evolutionary optimization algorithm that propagates parameters without direct inference of the underlying objective function. Benchmarked against the Tree of Parzen Estimator, a Bayesian algorithm implemented in the Hyperopt software Library, Paddy displays efficient optimization with lower runtime, and avoidance of early convergence. Herein we report these findings for the cases of: global optimization of a two-dimensional bimodal distribution, interpolation of an irregular sinusoidal function, hyperparameter optimization of an artificial neural network tasked with classification of solvent for reaction components, and targeted molecule generation via optimization of input vectors for a decoder network. We anticipate that the facile nature of Paddy will serve to aid in automated experimentation, where minimization of investigative trials and or diversity of suitable solutions is of high priority.

math.OC↗

arXiv2019

Some stumbling first steps towards linear homology in a nutshell

In 1985 Bayer and Billera defined a flag vector $f(X)$ for every convex polytope $X$, and proved some fundamental properties. The flag vectors $f(X)$ span a graded ring $\mathcal{R}=\bigoplus_{d\geq0}\mathcal{R}_d$. Here $\mathcal{R}_d$ is the span of the $f(X)$ with $\dim X=d$. It has dimension the Fibonacci number $F_{d+1}$. This paper introduces and explores the conjecture, that $\mathcal{R}$ has a counting basis $\{e_i\}$. If true then the equation $f(X) = \sum g_i(X)e_i$ conjecturally provides a formula for the Betti numbers $g_i(X)$ of a new homology theory. As the $g_i(X)$ are linear functions of $f(X)$, we call the new theory linear homology. Further, assuming the conjecture each $g_i$ will have a rank $r\geq0$. The rank zero part of linear homology will be (middle perversity) intersection homology. The higher rank $g_i$ measure successively more complicated singularities. In dimension $d$ we will have $\dim\mathcal{R}_d$ linearly independent Betti numbers. This paper produces a basis $\{e_i\}$ for $\mathcal{R}$, that is conjecturally a counting basis. Warning: Conjecture withdrawn in version 2.

math.CO↗

arXiv2018

The algebra of balanced dessins

This paper gives a key definition, for a new approach to dessins and algebraic numbers. The distant goal is to construct from each dessin $D$ an algebraic number $\eta_D$, in a systematic and useful way. The algebra of balanced dessins is generated by formal sums $\psi_D$ of dessins, intended to be intermediate between $D$ and $\eta_D$.

math.CO↗

arXiv2015

Bias and dessins

Grothendieck's theory of dessins provides a bridge between algebraic numbers and combinatorics. This paper adds a new concept, called 'bias', to the bridge. This produces: (i) from a biased plane tree the construction of a sequence of algebraic numbers, and (ii) a Galois invariant lattice structure on the set of biased dessins. Bias brings these benefits by (i) using individual polynomials instead of equivalence classes of polynomials, and (ii) applying properties of covering spaces and the fundamental group. The new features give new opportunities. At the 2014 SIGMAP conference the author spoke [1] on 'The decorated lattice of biased dessins'. This decorated lattice $\mathcal{L}$ is combinatorially defined, and its automorphism group contains the absolute Galois group $\Gamma$, perhaps as an index 6 subgroup. This paper defines new families of invariants of dessins, although they require further work to be understood and useful. For this, $\mathcal{L}$ is vital. This paper relies on the the existing, unbiased, theory. Also, it only sketches the construction of $\mathcal{L}$. In [2], [3] the author will remove this dependency, develop the biased theory further, with a focus on $\Gamma$, and make the theory more accessible.

math.CO↗

arXiv2010

Axioms for the g-vector of general convex polytopes

McMullen's g-vector is important for simple convex polytopes. This paper postulates axioms for its extension to general convex polytopes. It also conjectures that, for each dimension d, a stated finite calculation gives the formula for the extended g-vector. This calculation is done by computer for d=5 and the results analysed. The conjectures imply new linear inequalities on convex polytope flag vectors. Underlying the axioms is a hypothesised higher-order homology extension to middle perversity intersection homology (order-zero homology), which measures the failure of lower-order homology to have a ring structure.

math.CO↗

arXiv2010

A complete $g$-vector for convex polytopes

We define an extension of the toric (middle perversity intersection homology) $g$-vector of a convex polytope $X$. The extended $g(X)$ encodes the whole of the flag vector $f(X)$ of $X$, and so is called complete. We find that for many examples that $g_k(X)\geq 0$ for most $k$ (independent of $X$).

math.CO↗

arXiv2009

A complete h-vector for convex polytopes

This note defines a complete h-vector for convex polytopes, which extends the already known toric (or mpih) h-vector and has many similar properties. Complete means that it encodes the whole of the flag vector. First we define the concept of a generalised h-vector and state some properties that follow. The toric h-vector is given as an example. We then define a complete generalised h-vector, and again state properties. Finally, we show that this complete h-vector and all with similar properties will sometimes have negative coefficients. Most of the proofs, and further investigations, will appear elsewhere.

math.CO↗

arXiv2009

A note on braids and Parseval's theorem

In 1988 Falk and Randell, based on Arnol'd's 1969 paper on braids, proved that the pure braid groups are residually nilpotent. They also proved that the quotients in the lower central series are free abelian groups. This brief note uses an example to provide evidence for a much stronger statement: that each braid $b$ can be written as an infinite sum $b =\sum_0^\infty b_i$, where each $b_i$ is a linear function of the $i$-th Vassiliev-Kontsevich $Z_i(b)$ invariant of $b$. The example is pure braids on two strands. This leads to solving $e^\tau=q$ for $\tau$ a Laurent series in $q$. We set $\tau = \sum_1^\infty (-1)^{n+1} (q^n - q^{-n})/n$ and use Fourier series and Parseval's theorem to prove $e^\tau=q$. For more than two strands the stronger statement seems to rely on an as yet unstated Plancherel theorem for braid groups, which is likely both to be both and to have deep consequences

math.QA↗

arXiv2009

Vassiliev-Kontsevich invariants and Parseval's theorem

We use an example to provide evidence for the statement: the Vassiliev-Kontsevich invariants $k_n$ of a knot (or braid) $k$ can be redefined so that $k = \sum_0^\infty k_n$. This constructs a knot from its Vassiliev-Kontsevich invariants, like a power series expansion. The example is pure braids on two strands $P_2\cong \mathbb{Z}$, which leads to solving $e^τ=q$ for $τ$ a Laurent series in $q$. We set $τ= \sum_1^\infty (-1)^{n+1} (q^n - q^{-n})/n$ and use Parseval's theorem for Fourier series to prove $e^τ=q$. Finally we describe some problems, particularly a Plancherel theorem for braid groups, whose solution would take us towards a proof of $k=\sum_0^\infty k_n$.

math.QA↗

arXiv2009

A filtration question on Belyi pairs and dessins

A Bely\uı pair is a holomorphic map from a Riemann surface to $S^2$ with additional properties. A dessin d'enfants is a bipartite graph with additional structure. It is well know that there is a bijection between Belyĭ pairs and dessins d'enfants. Vassiliev has defined a filtration on formal sums of isotopy classes of knots. Motivated by this, we define a filtration on formal sums of Bely\uı pairs, and another on dessin d'enfants. We ask if the two definitions give the same filtration.

math.AG↗

arXiv2006

Some notes on the inverse problem for braids

The Kontsevich integral $Z$ associates to each braid $b$ (or more generally knot $k$) invariants $Z_i(b)$ lying in finite dimensional vector spaces, for $i = 0, 1, 2, ...$. These values are not yet known, except in special cases. The inverse problem is that of determining $b$ from its invariants $Z_i(b)$. In this paper we study the case of braids on two strands, which is already sufficient to produce interesting and unexpected mathematics. In particular, we find connections with number theory, numerical analysis and field theory in physics. However, we will carry this study out with an eye to the more general case of braids on $n$ strands. We expect that solving the inverse problem even for $n=3$ will present real difficulties. Most of the concepts in this paper also apply to knots, but to simplify the exposition we will rarely mention this. The organisation and bulk of the writing of this paper predates its most significant results. We hope later to present better and develop further these results.

math.QA↗

arXiv1998

Flag vectors

This paper defines for each object $X$ that can be constructed out of a finite number of vertices and cells a vector $fX$ lying in a finite dimensional vector space. This is the flag vector of $X$. It is hoped that the quantum topological invariants of a manifold $M$ can be expressed as linear functions of the flag vector of the $i$-graph that arises from any suitable triangulation $T$ of $M$. Flag vectors are also defined for finite groups and more generally for $n$-ary relations. Some problems, and suggested connections with other constructions, particularly that of the associahedron and so on, conclude the presentation.

math.CO↗

arXiv1998

Stratified simplices and intersection homology

Intersection homology is obtained from ordinary homology by imposing conditions on how the embedded simplices meet the strata of a space $X$. In this way, for the middle perversity, properties such as strong Lefschetz are preserved. This paper defines local-global intersection homology groups, that record global information about the singularities of $X$. They differ from intersection homology in that stratified rather than ordinary simplices are used. An example of such is $σ_j\times Cσ_i$, where $σ_i$ and $σ_j$ are ordinary simplices, and $C$ is the coning operator. The paper concludes with a sketch of the relationship between local-global homology and the geometry of convex polytopes. This paper is a more formal exposition of part of the author's `Local-global intersection homology', alg-geom/9709011.

math.AT↗

arXiv1998

Graphs, flags and partitions

This paper defines, for each graph $G$, a flag vector $fG$. The flag vectors of the graphs on $n$ vertices span a space whose dimension is $p(n)$, the number of partitions on $n$. The analogy with convex polytopes indicates that the linear inequalities satisfied by $fG$ may be both interesting and accessible. Such would provide inequalities both sharp and subtle on the combinatorial structure of $G$. These may be related to Ramsey theory.

math.CO↗

arXiv1998

Ring structure, uniform expressions and intersection homology

Although intersection homology lacks a ring structure, certain expressions (called uniform) in the intersection homology of an irreducible projective variety $X$ always give the same value, when computed via the decomposition theorem on any resolution $X_r\to X$. This paper uses uniform (and non-uniform) expressions to define what is believed to be the usual intersection homology (and its local-global variant) of a convex polytope (or a projective toric variety). Such expressions are generated by the facets, and so may lead to necessary numerical conditions on the flag vector. Most of the concepts, however, apply to more general algebraic varieties, and perhaps some other situations also.

math.AG↗

arXiv1998

Vassiliev theory and regional change

The purpose of this note is to state some definitions that may be useful in the study of knots, manifolds and the like. They apply to anything for which the concept of a regional change can be defined, such as a product of elements in a group.

math.QA↗

arXiv1998

Hodge's harmonic $p$-sets and Pontrjagin classes

This paper shows how Hodge's theory of harmonic $p$-sets (a discrete version of his theory of harmonic forms) allows a new approach to be taken to the problem of providing a combinatorial definition of the Pontrjagin classes of a compact manifold. This approach is then related to the author's definition of flag vectors for hypergraphs, and other objects constructed out of vertices and cells.

math.GT↗

arXiv1997

Convex polytopes and linear algebra

This paper defines, for each convex polytope $Δ$, a family $H_wΔ$ of vector spaces. The definition uses a combination of linear algebra and combinatorics. When what is called exact calculation holds, the dimension $h_wΔ$ of $H_wΔ$ is a linear function of the flag vector $fΔ$. It is expected that the $H_wΔ$ are examples, for toric varieties, of the new topological invariants introduced by the author in "Local-global intersection homolog" (preprint alg-geom/9709011).

alg-geom↗