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Russell Miller

Publications and source records attributed to Russell Miller.

At least 19 recordsLinked to original sources
arXiv2026

Stochastic Cluster Expansion for Excited State Energies

Excited-state electronic structure in strongly correlated systems remains challenging due to the exponential scaling of the many-body Hilbert space and the difficulty of constructing systematically controlled active spaces. Building on the stochastic cluster expansion (SCE) framework previously developed for ground-state correlation energies, we extend the formalism to excitation gaps by expressing energy differences directly as a hierarchy of orbital-space cluster contributions. In this formulation, excitation energies are reconstructed from reduced-rank calculations involving a minimal frontier chemical subspace (FCS), treated exactly, together with stochastic sampling of the remaining orbital environment. This approach eliminates the need for large or chemically preselected active spaces. We demonstrate the method on charge-transfer complexes and polyacenes, where accurate singlet-triplet gaps are obtained that agree with full-system results. The method converges with low-order cluster terms and provides a systematically improvable framework for excited states in correlated systems.

physics.chem-ph↗
arXiv2025

Computability for tree presentations of continuum-size structures

We formalize an existing computability-theoretic method of presenting first-order structures whose domains have the cardinality of the continuum. Work using these methods until now has emphasized their topological properties. We shift the focus to first-order properties, using computable structure theory (on countable structures) as a guide. We present three basic questions to be asked when a structure is presented as the set of paths through a computable tree, as in our definition, and also propose the concept of tree-decidability as an analogue to the notion of decidability for a countable structure. As examples, we prove decidability results for certain additive and multiplicative groups of $p$-adic integers, products of these (such as the profinite completion of $\mathbb Z$), and the field of real numbers.

math.LO↗
arXiv2025

Computable $K$-theory for $\mathrm{C}^*$-algebras: UHF algebras

We initiate the study of the effective content of $K$-theory for $\mathrm{C}^*$-algebras. We prove that there are computable functors which associate, to a computably enumerable presentation of a $\mathrm{C}^*$-algebra $\boldA$, computably enumerable presentations of the abelian groups $K_0(\boldA)$ and $K_1(\boldA)$. When $\boldA$ is stably finite, we show that the positive cone of $K_0(\boldA)$ is computably enumerable. We strengthen the results in the case that $\boldA$ is a UHF algebra by showing that the aforementioned presentation of $K_0(\boldA)$ is actually computable. In the UHF case, we also show that $\boldA$ has a computable presentation precisely when $K_0(\boldA)$ has a computable presentation, which in turn is equivalent to the supernatural number of $\boldA$ being lower semicomputable; we give an example that shows that this latter equivalence cannot be improved to requiring that the supernatural number of $\boldA$ is computable. Finally, we prove that every UHF algebra is computably categorical.

math.LO↗
arXiv2024

Torsion-free abelian groups of finite rank and fields of finite transcendence degree

Let $\operatorname{TFAb}_r$ be the class of torsion-free abelian groups of rank $r$, and let $\operatorname{FD}_r$ be the class of fields of characteristic $0$ and transcendence degree~$r$. We compare these classes using various notions. Considering Scott complexity of the structures in the classes and the complexity of the isomorphism relations on the classes, the classes seem very similar. Hjorth and Thomas showed that the $\operatorname{TFAb}_r$ are strictly increasing under Borel reducibility. This is not so for the classes $\operatorname{FD}_r$. Thomas and Velickovic showed that for sufficiently large $r$, the classes $\operatorname{FD}_r$ are equivalent under Borel reducibility. We try to compare the groups with the fields, using Borel reducibility, and also using some effective variants. We give functorial Turing computable embeddings of $\operatorname{TFAb}_r$ in $\operatorname{FD}_r$, and of $\operatorname{FD}_r$ in $\operatorname{FD}_{r+1}$. We show that under computable countable reducibility, $\operatorname{TFAb}_1$ lies on top among the classes we are considering. In fact, under computable countable reducibility, isomorphism on $\operatorname{TFAb}_1$ lies on top among equivalence relations that are effective $\Sigma_3$, along with the Vitali equivalence relation on $2^\omega$.

math.LO↗
arXiv2023

Computability for the absolute Galois group of $\mathbb{Q}$

The absolute Galois group Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$ of the field $\mathbb{Q}$ of rational numbers can be presented as a highly computable object, under the notion of type-2 Turing computation. We formalize such a presentation and use it to address several effectiveness questions about Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$: the difficulty of computing Skolem functions for this group, the arithmetical complexity of various definable subsets of the group, and the extent to which countable subgroups defined by complexity (such as the group of all computable automorphisms of the algebraic closure $\overline{\mathbb{Q}}$) may be elementary subgroups of the overall group.

math.LO↗
arXiv2023

Differentially closed fields and universality on a cone

The class of all countable differentially closed differential fields $K$ of characteristic $0$ was shown by Marker and the author to be "one jump away" from universal for spectra of structures: for every nontrivial countable structure $\mathfrak M$, there is some $K$ whose spectrum is the preimage under jump of the spectrum of $\mathfrak M$, and conversely, for every $K$, there is such an $\mathfrak M$. We show that the missing jump can be accounted for by adding to the signature of differential fields a predicate describing a certain algebraic transcendence property. The ensuing universality results for differentially closed fields in the new signature include not only spectra of structures, but also many properties related to computable categoricity. However, these latter universality results hold only on the cone above a specific $\Sigma^0_1$ oracle set, whose decidability status remains unknown. Moreover, differentially closed fields simply fail flat-out to be universal for automorphism groups, even non-effectively. We also include a small erratum to an earlier work.

math.LO↗
arXiv2020

A topological approach to undefinability in algebraic extensions of $\mathbb{Q}$

For any subset $Z \subseteq \mathbb{Q}$, consider the set $S_Z$ of subfields $L\subseteq \overline{\mathbb{Q}}$ which contain a co-infinite subset $C \subseteq L$ that is universally definable in $L$ such that $C \cap \mathbb{Q}=Z$. Placing a natural topology on the set $\text{Sub}(\overline{\mathbb{Q}})$ of subfields of $\overline{\mathbb{Q}}$, we show that if $Z$ is not thin in $\mathbb{Q}$, then $S_Z$ is meager in $\text{Sub}(\overline{\mathbb{Q}})$. Here, thin and meager both mean "small", in terms of arithmetic geometry and topology, respectively. For example, this implies that only a meager set of fields $L$ have the property that the ring of algebraic integers $\mathcal{O}_L$ is universally definable in $L$. The main tools are Hilbert's Irreducibility Theorem and a new normal form theorem for existential definitions. The normal form theorem, which may be of independent interest, says roughly that every $\exists$-definable subset of an algebraic extension of $\mathbb Q$ is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible polynomials.

math.NT↗
arXiv2020

Forcing as a computational process

We investigate how set-theoretic forcing can be seen as a computational process on the models of set theory. Given an oracle for information about a model of set theory $\langle M,\in^M\rangle$, we explain senses in which one may compute $M$-generic filters $G\subseteq\mathbb{P}\in M$ and the corresponding forcing extensions $M[G]$. Specifically, from the atomic diagram one may compute $G$, from the $\Delta_0$-diagram one may compute $M[G]$ and its $\Delta_0$-diagram, and from the elementary diagram one may compute the elementary diagram of $M[G]$. We also examine the information necessary to make the process functorial, and conclude that in the general case, no such computational process will be functorial. For any such process, it will always be possible to have different isomorphic presentations of a model of set theory $M$ that lead to different non-isomorphic forcing extensions $M[G]$. Indeed, there is no Borel function providing generic filters that is functorial in this sense.

math.LO↗
arXiv2020

Interpreting a field in its Heisenberg group

We improve on and generalize a 1960 result of Maltsev. For a field $F$, we denote by $H(F)$ the Heisenberg group with entries in $F$. Maltsev showed that there is a copy of $F$ defined in $H(F)$, using existential formulas with an arbitrary non-commuting pair $(u,v)$ as parameters. We show that $F$ is interpreted in $H(F)$ using computable $\Sigma_1$ formulas with no parameters. We give two proofs. The first is an existence proof, relying on a result of Harrison-Trainor, Melnikov, R. Miller, and Montalb\'an. This proof allows the possibility that the elements of $F$ are represented by tuples in $H(F)$ of no fixed arity. The second proof is direct, giving explicit finitary existential formulas that define the interpretation, with elements of $F$ represented by triples in $H(F)$. Looking at what was used to arrive at this parameter-free interpretation of $F$ in $H(F)$, we give general conditions sufficient to eliminate parameters from interpretations.

math.LO↗
arXiv2019

Effectivizing Lusin's Theorem

Lusin's Theorem states that, for every Borel-measurable function $\bf{f}$ on $\mathbb R$ and every $\epsilon>0$, there exists a continuous function $\bf{g}$ on $\mathbb R$ which is equal to $\bf{f}$ except on a set of measure $<\epsilon$. We give a proof of this result using computability theory, relating it to the near-uniformity of the Turing jump operator, and use this proof to derive several uniform computable versions. Easier results, which we prove by the same methods, include versions of Lusin's Theorem with Baire category in place of Lebesgue measure and also with Cantor space $2^{\mathbb N}$ in place of $\mathbb R$. The distinct processes showing generalized lowness for generic sets and for a set of full measure are seen to explain the differences between versions of Lusin's Theorem.

math.LO↗
arXiv2019

HTP-complete rings of rational numbers

For a ring $R$, Hilbert's Tenth Problem $HTP(R)$ is the set of polynomial equations over $R$, in several variables, with solutions in $R$. We view $HTP$ as an enumeration operator, mapping each set $W$ of prime numbers to $HTP(\mathbb Z[W^{-1}])$, which is naturally viewed as a set of polynomials in $\mathbb Z[X_1,X_2,\ldots]$. It is known that for almost all $W$, the jump $W'$ does not $1$-reduce to $HTP(R_W)$. In contrast, we show that every Turing degree contains a set $W$ for which such a $1$-reduction does hold: these $W$ are said to be "HTP-complete." Continuing, we derive additional results regarding the impossibility that a decision procedure for $W'$ from $HTP(\mathbb Z[W^{-1}])$ can succeed uniformly on a set of measure $1$, and regarding the consequences for the boundary sets of the $HTP$ operator in case $\mathbb Z$ has an existential definition in $\mathbb Q$.

math.LO↗
arXiv2019

Degree spectra for transcendence in fields

We show that for both the unary relation of transcendence and the finitary relation of algebraic independence on a field, the degree spectra of these relations may consist of any single computably enumerable Turing degree, or of those c.e. degrees above an arbitrary fixed $\Delta^0_2$ degree. In other cases, these spectra may be characterized by the ability to enumerate an arbitrary $\Sigma^0_2$ set. This is the first proof that a computable field can fail to have a computable copy with a computable transcendence basis.

math.LO↗
arXiv2019

Model completeness and relative decidability

We study the implications of model completeness of a theory for the effectiveness of presentations of models of that theory. It is immediate that for a computable model $\mathcal A$ of a computably enumerable, model complete theory, the entire elementary diagram $E(\mathcal A)$ must be decidable. We prove that indeed a c.e. theory $T$ is model complete if and only if there is a uniform procedure that succeeds in deciding $E(\mathcal A)$ from the atomic diagram $\Delta(\mathcal A)$ for all countable models $\mathcal A$ of $T$. Moreover, if every presentation of a single isomorphism type $\mathcal A$ has this property of relative decidability, then there must be a procedure with succeeds uniformly for all presentations of an expansion $(\mathcal A,\vec{a})$ by finitely many new constants. We end with a conjecture about the situation when all models of a theory are relatively decidable.

math.LO↗
arXiv2018

Degree Spectra of Real Closed Fields

Several researchers have recently established that for every Turing degree $\boldsymbol{c}$, the real closed field of all $\boldsymbol{c}$-computable real numbers has spectrum $\{\boldsymbol{d}~:~\boldsymbol{d}'\geq\boldsymbol{c}"\}$. We investigate the spectra of real closed fields further, focusing first on subfields of the field $\mathbb{R}_{\boldsymbol{0}}$ of computable real numbers, then on archimedean real closed fields more generally, and finally on non-archimedean real closed fields. For each noncomputable, computably enumerable set $C$, we produce a real closed $C$-computable subfield of $\mathbb{R}_{\boldsymbol{0}}$ with no computable copy. Then we build an archimedean real closed field with no computable copy but with a computable enumeration of the Dedekind cuts it realizes, and a computably presentable nonarchimedean real closed field whose residue field has no computable presentation.

math.LO↗
arXiv2017

The Hilbert's-Tenth-Problem Operator

For a ring $R$, Hilbert's Tenth Problem $HTP(R)$ is the set of polynomial equations over $R$, in several variables, with solutions in $R$. We view $HTP$ as an operator, mapping each set $W$ of prime numbers to $HTP(\mathbb Z[W^{-1}])$, which is naturally viewed as a set of polynomials in $\mathbb Z[X_1,X_2,\ldots]$. For $W=\emptyset$, it is a famous result of Matiyasevich, Davis, Putnam, and Robinson that the jump $\emptyset~\!'$ is Turing-equivalent to $HTP(\mathbb Z)$. More generally, $HTP(\mathbb Z[W^{-1}])$ is always Turing-reducible to $W'$, but not necessarily equivalent. We show here that the situation with $W=\emptyset$ is anomalous: for almost all $W$, the jump $W'$ is not diophantine in $\mathbb Z[W^{-1}]$. We also show that the $HTP$ operator does not preserve Turing equivalence: even for complementary sets $U$ and $\overline{U}$, $HTP(\mathbb Z[U^{-1}])$ and $HTP(\mathbb Z[\overline{U}^{-1}])$ can differ by a full jump. Strikingly, reversals are also possible, with $V<_T W$ but $HTP(\mathbb Z[W^{-1}]) <_T HTP(\mathbb Z[V^{-1}])$.

math.LO↗
arXiv2017

Isomorphism and classification for countable structures

We introduce a topology on the space of all isomorphism types represented in a given class of countable models, and use this topology as an aid in classifying the isomorphism types. This mixes ideas from effective descriptive set theory and computable structure theory, extending concepts from the latter beyond computable structures to examine the isomorphism problem on arbitrary countable structures. We give examples using specific classes of fields and of trees, illustrating how the new concepts can yield classifications that reveal differences between seemingly similar classes. Finally, we use a computable homeomorphism to define a measure on the space of isomorphism types of algebraic fields, and examine the prevalence of relative computable categoricity under this measure.

math.LO↗
arXiv2017

On existential definitions of C.E. subsets of rings of functions of characteristic 0

We extend results of Denef, Zahidi, Demeyer and the second author to show the following. (1) Rational integers have a single-fold Diophantine definition over the ring of integral functions of any function field of characteristic 0. (2) Every c.e. set of integers has a finite-fold Diophantine definition over the ring of integral functions of any function field of characteristic $0$. (3) All c.e. subsets of polynomial rings over totally real number fields have finite-fold Diophantine definitions. (These are the first examples of infinite rings with this property.) (4) If $k$ is algebraic over $\Q$ and is embeddable into a finite extension of $\Q_p$ for odd $p$, and $K$ is a one-variable function field over $k$, then the valuation ring of any function field valuation of $K$ has a Diophantine definition over $K$. (5) If $k$ is algebraic over $\Q$ and is embeddable into $\R$, and $K$ is a function field over $k$, then "almost" all function field valuations of $K$ have a valuation ring Diophantine over $K$. (6) Let $K$ be a one-variable function field over a number field and let $S$ be a finite set of its primes. Then all c.e. subsets of $O_{K,S}$ are existentially definable. (Here $O_{K,S}$ is the ring of $S$-integers or a ring of integral functions.)

math.NT↗