arXiv subjects

Joseph Johnson

Publications and source records attributed to Joseph Johnson.

13 recordsLinked to original sources

arXiv2025

The influence of boundary conditions and interfacial slip on the time taken to achieve a nonequilibrium steady-state for highly confined flows

We investigate the equilibration time to attain steady-state for a system of liquid molecules under boundary-driven planar Couette flow via nonequilibrium molecular dynamics (NEMD) simulation. In particular, we examine the equilibration time for the two common types of boundary driven flow: one in which both walls slide with equal and opposite velocity, and the other in which one wall is fixed and the other moves with twice the velocity. Both flows give identical steady-state strain rates, and hence flow properties, but the transient behaviour is completely different. We find that in the case of no-slip boundary conditions, the equilibration times for the counter-sliding walls flow are exactly 4 times faster than those of the single sliding wall system, and this is independent of the atomistic nature of the fluid, i.e., it is an entirely hydrodynamic feature. We also find that systems that exhibit slip have longer equilibration times in general and the ratio of equilibration times for the two types of boundary-driven flow is even more pronounced. We analyse the problem by decomposing a generic planar Couette flow into a linear sum of purely symmetric and antisymmetric flows. We find that the no-slip equilibration time is dominated by the slowest decaying eigenvalue of the solution to the Navier-Stokes equation. In the case of slip, the longest relaxation time is now dominated by the transient slip velocity response, which is longer than the no-slip response time. In the case of a high-slip system of water confined to graphene channels, the enhancement is over two orders of magnitude.

physics.flu-dyn↗

arXiv2025

Modeling Social Systems: Transparency, Reproducibility, and Responsibility

Mathematical models of complex social systems can enrich social scientific theory, inform interventions, and shape policy. From voting behavior to economic inequality and urban development, such models influence decisions that affect millions of lives. Thus, it is especially important to formulate and present them with transparency, reproducibility, and humility. Modeling in social domains, however, is often uniquely challenging. Unlike in physics or engineering, researchers often lack controlled experiments or abundant, clean data. Observational data is sparse, noisy, partial, and missing in systematic ways. In such an environment, how can we build models that can inform science and decision-making in transparent and responsible ways?

math.HO↗

arXiv2025

Characteristic Imsets for Cyclic Linear Causal Models and the Chickering Ideal

Two directed graphs are called covariance equivalent if they induce the same set of covariance matrices, up to a Lebesgue measure zero set, on the random variables of their associated linear structural equation models. For acyclic graphs, covariance equivalence is characterized both structurally, via essential graphs and characteristic imsets, and transformationally, through sequences of covered edge flips. However, when cycles are allowed, only a transformational characterization of covariance equivalence has been discovered. We consider a linear map whose fibers correspond to the sets of graphs with identical characteristic imset vectors, and study the toric ideal associated to its integer matrix. Using properties of this ideal we show that directed graphs with the same characteristic imset vectors are covariance equivalent. In applications, imsets form a smaller search space for solving causal discovery via greedy search.

math.ST↗

arXiv2024

Hyperplane Representations of Interventional Characteristic Imset Polytopes

Characteristic imsets are 0/1-vectors representing directed acyclic graphs whose edges represent direct cause-effect relations between jointly distributed random variables. A characteristic imset (CIM) polytope is the convex hull of a collection of characteristic imsets. CIM polytopes arise as feasible regions of a linear programming approach to the problem of causal disovery, which aims to infer a cause-effect structure from data. Linear optimization methods typically require a hyperplane representation of the feasible region, which has proven difficult to compute for CIM polytopes despite continued efforts. We solve this problem for CIM polytopes that are the convex hull of imsets associated to DAGs whose underlying graph of adjacencies is a tree. Our methods use the theory of toric fiber products as well as the novel notion of interventional CIM polytopes. Our solution is obtained as a corollary of a more general result for interventional CIM polytopes. The identified hyperplanes are applied to yield a linear optimization-based causal discovery algorithm for learning polytree causal networks from a combination of observational and interventional data.

math.CO↗

arXiv2024

New directions in algebraic statistics: Three challenges from 2023

In the last quarter of a century, algebraic statistics has established itself as an expanding field which uses multilinear algebra, commutative algebra, computational algebra, geometry, and combinatorics to tackle problems in mathematical statistics. These developments have found applications in a growing number of areas, including biology, neuroscience, economics, and social sciences. Naturally, new connections continue to be made with other areas of mathematics and statistics. This paper outlines three such connections: to statistical models used in educational testing, to a classification problem for a family of nonparametric regression models, and to phase transition phenomena under uniform sampling of contingency tables. We illustrate the motivating problems, each of which is for algebraic statistics a new direction, and demonstrate an enhancement of related methodologies.

math.ST↗

arXiv2023

Plane partitions and rowmotion on rectangular and trapezoidal posets

We define a birational map between labelings of a rectangular poset and its associated trapezoidal poset. This map tropicalizes to a bijection between the plane partitions of these posets of fixed height, giving a new bijective proof of a result by Proctor. We also show that this map is equivariant with respect to birational rowmotion, resolving a conjecture of Williams and implying that birational rowmotion on trapezoidal posets has finite order.

math.CO↗

arXiv2023

The Codegree, Weak Maximum Likelihood Threshold, and the Gorenstein Property of Hierarchical Models

The codegree of a lattice polytope is the smallest integer dilate that contains a lattice point in the relative interior. The weak maximum likelihood threshold of a statistical model is the smallest number of data points for which there is a non-zero probability that the maximum likelihood estimate exists. The codegree of a marginal polytope is a lower bound on the maximum likelihood threshold of the associated log-linear model, and they are equal when the marginal polytope is normal. We prove a lower bound on the codegree in the case of hierarchical log-linear models and provide a conjectural formula for the codegree in general. As an application, we study when the marginal polytopes of hierarchical models are Gorenstein, including a classification of Gorenstein decomposable models, and a conjectural classification of Gorenstein binary hierarchical models.

math.CO↗

arXiv2022

Piecewise-linear promotion and RSK in rectangles and moon polyominoes

We study piecewise-linear and birational lifts of Sch\"utzenberger promotion, evacuation, and the RSK correspondence defined in terms of toggles. Using this perspective, we prove that certain chain statistics in rectangles shift predictably under the action of these maps. We then use this to construct piecewise-linear and birational versions of Rubey's bijections between fillings of equivalent moon polyominoes that preserve these chain statistics, and we show that these maps form a commuting diagram. We also discuss how these results imply Ehrhart equivalence and Ehrhart quasi-polynomial period collapse of certain analogues of chain polytopes for moon polyominoes.

math.CO↗

arXiv2022

Toric Ideals of Characteristic Imsets via Quasi-Independence Gluing

Characteristic imsets are 0-1 vectors which correspond to Markov equivalence classes of directed acyclic graphs. The study of their convex hull, named the characteristic imset polytope, has led to new and interesting geometric perspectives on the important problem of causal discovery. In this paper we begin the study of the associated toric ideal. We develop a new generalization of the toric fiber product, which we call a quasi-independence gluing, and show that under certain combinatorial homogeneity conditions, one can iteratively compute a Gr\"obner basis via lifting. For faces of the characteristic imset polytope associated to trees, we apply this technique to compute a Gr\"obner basis for the associated toric ideal. We end with a study of the characteristic ideal of the cycle and propose directions for future work.

math.ST↗

arXiv2022

Birational Rowmotion and the Octahedron Recurrence

We use the octahedron recurrence to give a simplified statement and proof of a formula for iterated birational rowmotion on a product of two chains, first described by Musiker and Roby. Using this, we show that weights of certain chains in rectangles shift in a predictable way under the action of rowmotion. We then define generalized Stanley-Thomas words whose cyclic rotation uniquely determines birational rowmotion on the product of two chains. We also discuss the relationship between rowmotion and birational RSK and give a birational analogue of Greene's theorem in this setting.

math.CO↗

arXiv2021

Determinantal Formulas for SEM Expansions of Schubert Polynomials

We show that for any permutation $w$ that avoids a certain set of 13 patterns of lengths 5 and 6, the Schubert polynomial $\mathfrak S_w$ can be expressed as the determinant of a matrix of elementary symmetric polynomials in a manner similar to the Jacobi-Trudi identity. For such $w$, this determinantal formula is equivalent to a (signed) subtraction-free expansion of $\mathfrak S_w$ in the basis of standard elementary monomials.

math.CO↗

arXiv2017

Multimodal Content Analysis for Effective Advertisements on YouTube

The rapid advances in e-commerce and Web 2.0 technologies have greatly increased the impact of commercial advertisements on the general public. As a key enabling technology, a multitude of recommender systems exists which analyzes user features and browsing patterns to recommend appealing advertisements to users. In this work, we seek to study the characteristics or attributes that characterize an effective advertisement and recommend a useful set of features to aid the designing and production processes of commercial advertisements. We analyze the temporal patterns from multimedia content of advertisement videos including auditory, visual and textual components, and study their individual roles and synergies in the success of an advertisement. The objective of this work is then to measure the effectiveness of an advertisement, and to recommend a useful set of features to advertisement designers to make it more successful and approachable to users. Our proposed framework employs the signal processing technique of cross modality feature learning where data streams from different components are employed to train separate neural network models and are then fused together to learn a shared representation. Subsequently, a neural network model trained on this joint feature embedding representation is utilized as a classifier to predict advertisement effectiveness. We validate our approach using subjective ratings from a dedicated user study, the sentiment strength of online viewer comments, and a viewer opinion metric of the ratio of the Likes and Views received by each advertisement from an online platform.

cs.AI↗

arXiv2005

Statistical Mechanics of Amplifying Apparatus

We implement Feynman's suggestion that the only missing notion needed for the puzzle of Quantum Measurement is the statistical mechanics of amplifying apparatus. We define a thermodynamic limit of quantum amplifiers which is a classically describable system in the sense of Bohr, and define macroscopic pointer variables for the limit system. Then we derive the probabilities of Quantum Measurement from the deterministic Schroedinger equation by the usual techniques of Classical Statistical Mechanics.

quant-ph↗