arXiv subjects

Slava Naprienko

Publications and source records attributed to Slava Naprienko.

9 recordsLinked to original sources

arXiv2026

Binomial coefficients with divisors avoiding an interval

We solve a fifty-year-old conjecture of Erd\H{o}s and Graham concerning whether the binomial coefficient ${n \choose k}$ with $1 \leq k \leq \frac{n}{2}$ must always have a divisor $\leq n$ that is ``close'' to $n$: that is, bigger than a constant times $n$. We show this is the case when $k$ is sufficiently large as a function of $n$. However, we show it is possible to find binomial coefficients ${n \choose k}$, where $k$ is small compared to $n$, such that ${n \choose k}$ does not have divisors $\leq n$ close to $n$. This latter, more substantial argument involves a restricted covering problem with residue classes, sieve methods, and various exponential sum estimates.

math.NT↗

arXiv2025

Data Leakage and Redundancy in the LIT-PCBA Benchmark

LIT-PCBA is widely used to benchmark virtual screening models, but our audit reveals that it is fundamentally compromised. We find extensive data leakage and molecular redundancy across its splits, including 2D-identical ligands within and across partitions, pervasive analog overlap, and low-diversity query sets. In ALDH1 alone, for instance, 323 active training -- validation analog pairs occur at ECFP4 Tanimoto similarity $\geq 0.6$; across all targets, 2,491 2D-identical inactives appear in both training and validation, with very few corresponding actives. These overlaps allow models to succeed through scaffold memorization rather than generalization, inflating enrichment factors and AUROC scores. These flaws are not incidental -- they are so severe that a trivial memorization-based baseline with no learnable parameters can exploit them to match or exceed the reported performance of state-of-the-art deep learning and 3D-similarity models. As a result, nearly all published results on LIT-PCBA are undermined. Even models evaluated in "zero-shot" mode are affected by analog leakage into the query set, weakening claims of generalization. In its current form, the benchmark does not measure a model's ability to recover novel chemotypes and should not be taken as evidence of methodological progress. All code, data, and baseline implementations are available at: https://github.com/sievestack/LIT-PCBA-audit

cs.LG↗

arXiv2025

The Six-Vertex Yang-Baxter Groupoid

A parametrized Yang-Baxter equation is usually defined to be a map from a group to a set of R-matrices, satisfying the Yang-Baxter commutation relation. These are a mainstay of solvable lattice models. We will show how the parameter space can sometimes be enlarged to a groupoid, and give two examples of such groupoid parametrized Yang-Baxter equations, within the six vertex model. A groupoid parametrized Yang-Baxter equation consists of a groupoid $\mathfrak{G}$ together with a map $\pi:\mathfrak{G}\to\operatorname{End}(V\otimes V)$ for some vector space $V$ such that the Yang-Baxter commutator $[[ \pi(u),\pi(w),\pi(v)]]=0$ if $u,v\in\mathfrak{G}$ are such that the groupoid composition $w=u\star v$ is defined. An important role is played by an object map $\Delta:\mathfrak{G}\to M$ for some set $M$ such that $\Delta(u)=\Delta(v')$, $\Delta(w)=\Delta(v)$ and $\Delta(w')=\Delta(u')$, where $v\mapsto v'$ is the groupoid inverse map. There are two main regimes of the six-vertex model: the free-fermionic point, and everything else. For the free-fermionic point, there exists a parametrized Yang-Baxter equation with a large parameter group $\operatorname{GL}(2)\times\operatorname{GL}(1)$. For non-free-fermionic six-vertex matrices, there are also well-known (group) parametrized Yang-Baxter equations, but these do not account for all possible interactions. Instead we will construct a groupoid parametrized Yang-Baxter equation that accounts for essentially all possible Yang-Baxter equations in the six-vertex model. We will also exhibit a separate groupoid for the five-vertex model. We will show how to construct solvable lattice models based on groupoid parametrized Yang-Baxter equations.

math.QA↗

arXiv2023

Switch Operators for the Six-Vertex Model

In this paper, we introduce and analyze a new switch operator for the six-vertex model. This operator, derived from the Yang-Baxter equation, allows us to express the partition function with arbitrary boundaries in terms of a base case with domain wall boundary conditions. As an application, we derive explicit formulas for the factorial Schur functions and their generalizations. Our results provide new insights into the relationship between boundary conditions and partition functions in the six-vertex model.

math.CO↗

arXiv2023

Free Fermionic Schur Functions

We introduce a new family of Schur functions $s_{\lambda/\mu;a,b}(x/y)$ that depend on two sets of variables and two sequences of parameters. These free fermionic Schur functions have a hidden symmetry between the two sets of parameters that allows us to generalize and unify factorial, supersymmetric, and dual Schur functions from literature. We then prove that these functions satisfy the supersymmetric Cauchy identity $$ \sum_{\lambda}s_{\lambda;a,b}(x/y)\widehat{s}_{\lambda;a,b}(z/w) = \prod_{i,j}\frac{1+y_iz_j}{1-x_iz_j}\frac{1+x_iw_j}{1-y_iw_j}, $$ where $\widehat{s}_{\lambda;a,b}(z/w) = s_{\lambda';b',a'}(w/z)$ are the dual functions. Our approach is based on the integrable six vertex model with free fermionic Boltzmann weights. We show that these weights satisfy the \textit{refined Yang-Baxter equation}, which allows us to prove well-known properties of Schur functions: supersymmetry, combinatorial descriptions, the Jacobi-Trudi identity, the N\"agelsbach-Kostka formula, the Giambelli formula, the Ribbon formula, the Weyl determinant formula, the Berele-Regev factorization, dual Cauchy identity, the flagged determinant formula, and many others. We emphasize that many of these results are novel even in special cases.

math.CO↗

arXiv2022

Colored Bosonic Models and Matrix Coefficients

We develop the theory of colored bosonic models (initiated by Borodin and Wheeler). We will show how a family of such models can be used to represent the values of Iwahori vectors in the "spherical model" of representations of $GL_r(F)$, where $F$ is a nonarchimedean local field. Among our results are a monochrome factorization, which is the realization of the Boltzmann weights by fusion of simpler weights, a local lifting property relating the colored models with uncolored models, and an action of the affine Hecke algebra on the partition functions of a particular family of models by Demazure-Lusztig operators. As an application of the local lifting property we reprove a theorem of Korff evaluating the partition functions of the uncolored models in terms of Hall-Littlewood plynomials. Our results are very closely parallel to the theory of fermionic models representing Iwahori Whittaker functions developed by Brubaker, Buciumas, Bump and Gustafsson, with many striking relationships between the two theories, confirming the philosophy that the spherical and Whittaker models of principal series representations are dual.

math.RT↗

arXiv2022

Integrability of the Six-Vertex model and the Yang-Baxter Groupoid

We study the Yang-Baxter equation for the $R$-matrices of the six-vertex model. We analyze the solutions and give new parametrizations of the Yang-Baxter equation. In particular, we find the maximal commutative families of parametrized solutions which generalize the $R$-matrices from the affine quantum (super)-groups. Then we give a new parametrization of the Yang-Baxter equation by a groupoid of non-free-fermionic matrices. In the appendix, we study the general algebraic structure of the solutions of the Yang-Baxter and formulate a conjecture that extends the conjecture by Brubaker, Bump, and Friedberg that the composition law on the Yang-Baxter solutions is always associative.

math.QA↗

arXiv2022

Enrichment Score: a better quantitative metric for evaluating the enrichment capacity of molecular docking models

The standard quantitative metric for evaluating enrichment capacity known as $\textit{LogAUC}$ depends on a cutoff parameter that controls what the minimum value of the log-scaled x-axis is. Unless this parameter is chosen carefully for a given ROC curve, one of the two following problems occurs: either (1) some fraction of the first inter-decoy intervals of the ROC curve are simply thrown away and do not contribute to the metric at all, or (2) the very first inter-decoy interval contributes too much to the metric at the expense of all following inter-decoy intervals. We fix this problem with LogAUC by showing a simple way to choose the cutoff parameter based on the number of decoys which forces the first inter-decoy interval to always have a stable, sensible contribution to the total value. Moreover, we introduce a normalized version of LogAUC known as $\textit{enrichment score}$, which (1) enforces stability by selecting the cutoff parameter in the manner described, (2) yields scores which are more intuitively meaningful, and (3) allows reliably accurate comparison of the enrichment capacities exhibited by different ROC curves, even those produced using different numbers of decoys. Finally, we demonstrate the advantage of enrichment score over unbalanced metrics using data from a real retrospective docking study performed using the program $\textit{DOCK 3.7}$ on the target receptor TRYB1 included in the $\textit{DUDE-Z}$ benchmark.

q-bio.QM↗

arXiv2021

Combinatorics of Iwahori Whittaker Functions

We give a combinatorial evaluation of Iwahori Whittaker functions for unramified genuine principal series representations on metaplectic covers of the general linear group over a non-archimedean local field. To describe the combinatorics, we introduce new combinatorial data that we call colored data: colored Lusztig data, colored Gelfand-Tsetlin patterns, and colored lattice models. We show that all three are equivalent. To achieve the result, we give an explicit Iwahori decomposition for the maximal unipotent subgroup of a split reductive group which gives the parametrization of the generalized Mirkovi\'c-Vilonen cycles in the affine flag varieties and is of interest in itself. Our result is based and naturally extends Peter McNamara's evaluation of the metaplectic spherical Whittaker function in terms of Lusztig data.

math.RT↗