Proposition Research Articles

Dynamic spatial price equilibrium, dynamic user equilibrium, and freight transportation in continuous time: A differential variational inequality perspective

In this paper we provide a statement of dynamic spatial price equilibrium (DSPE) in continuous time as a basis for modeling freight flows in a network economy. The model presented describes a spatial price equilibrium due to its reliance on the notion that freight movements occur in response to differences between the local and distant prices of goods for which there is excess demand; moreover, local and distant delivered prices are equated at equilibrium. We propose and analyze a differential variational inequality (DVI) associated with dynamic spatial price equilibrium to study the Nash-like aggregate game at the heart of DSPE using the calculus of variations and optimal control theory. Our formulation explicitly considers inventory and the time lag between shipping and demand fulfillment. We stress that such a time lag cannot be readily accommodated in a discrete-time formulation. We provide an in-depth analysis of the DVI's necessary conditions that reveals the dynamic user equilibrium nature of freight flows obtained from the DVI, alongside the role played by freight transport in maintaining equilibrium commodity prices and the delivered-price-equals-local-price property of spatial price equilibrium. By intent, our contribution is wholly theoretical in nature, focusing on a mathematical statement of the defining equations and inequalities for dynamic spatial price equilibrium (DSPE), while also showing there is an associated differential variational inequality (DVI), any solution of which is a DSPE. The model of spatial price equilibrium we present integrates the theory of spatial price equilibrium in a dynamic setting with the path delay operator notion used in the theory of dynamic user equilibrium. It should be noted that the path delay operator used herein is based on LWR theory and fully vetted in the published dynamic user equilibrium literature. This integration is new and constitutes a significant addition to the spatial price equilibrium and freight network equilibrium modeling literatures. Among other things, it points the way for researchers interested in dynamic traffic assignment to become involved in dynamic freight modeling using the technical knowledge they already possess. In particular, it suggests that algorithms developed for dynamic user equilibrium may be adapted to the study of urban freight modelled as a dynamic spatial price equilibrium. As such, our work provides direction for future DSPE algorithmic research and application. However, no computational experiments are reported herein; instead, the computing of dynamic spatial price equilibria is the subject of a separate manuscript.

Implications for galaxy property estimation revealed by CO luminosity-FWHM relations in local star-forming galaxies

ABSTRACT This study explores a relationship between the $\rm {CO}$ luminosity-full width at half-maximum linewidth linear relation (i.e. the $\rm {CO}$ LFR) and mean galaxy property of the local star-forming galaxy sample in the xCOLDGASS data base, via a mathematical statement. The whole data base galaxies were separated into two subsamples based on their stellar masses and redshifts, being a help to examine the dependence issue of the $\rm {CO}$ LFR. Selecting the galaxy data with a stringent requirement was also implemented in order to assure the validly of the $\rm {CO}$ LFR. An algorithm of the linear regression was conducted with the data of the subsample. An assessment about the linear correlation manifested a valid $\rm {CO}$ LFR occurs in the selected galaxy of the subsample, and the intercept of the $\rm {CO}$ LFR may be related with the mean galaxy properties such as the molecular gas fraction and galaxy size. For the finding on the intercept of the $\rm {CO}$ LFR, we aligned that intercept with those galaxy properties via the involvement of a $\psi$ parameter. On evaluating the $\psi$ value with our local star-forming galaxy sample, we numerically determined a relationship between the statistical result and the galaxy property in a different stellar mass range. It also shows a possible method on estimating galaxy property.

Sharkovskii’s theorem and the limits of digital computers for the simulation of chaotic dynamical systems

Chaos is a unique paradigm in classical physics within which systems exhibit extreme sensitivity to the initial conditions. Thus, they need to be handled using probabilistic methods commonly based on ensembles. However, initial conditions generated by digital computers fall within the sparse set of discrete IEEE floating point numbers which have non-uniform distributions along the real axis. Therefore, there are many missing initial conditions whose absence might be expected to degrade the computed statistical properties of chaotic systems. The universality of this problem is enshrined in Sharkovskii’s theorem which is the simplest mathematical statement of the fact that no finite number representation of a chaotic dynamical system can account for all of its properties and shows that the precision of the representation limits the accuracy of the resulting digital behaviour.

Vitamin D-Do Diet Recommendations for Health Remain Strong?

How will the scientific community and authoritative bodies define future nutritional requirements for vitamin D? At the International Symposium on Nutritional Aspects of Musculoskeletal Health, the authors debated the strength of current evidence for setting vitamin D intake recommendations from diet: the positive side of the strength of the evidence (PRO) suggests there is a physiological requirement for vitamin D and the opposing view (CON) that in light of negative results from large, recent trials, particularly those with fractures and bone health outcomes, we are left rudderless. Should we provide recommendations based on empiric treatment of vitamin D for most groups and conditions? It is becoming increasingly evident that vitamin D plays a role in many physiological functions and processes associated with long-term human health; however, to what extent are these benefits apparent beyond what is needed for adequate nutritional status, measured as serum 25-hydroxyvitamin D levels, for active calcium absorption? The meeting attendees voted for the PRO vs. CON position at the end of the session.

OPTIMIZATION PROBLEM FOR PROPYLENE GLYCOL PRODUCTION

Currently, in the rapid development era of new technologies and in order to increase the economic efficiency of production, more and more attention is paid to the current automation, control, and optimization problems of technological processes in the oil refining, petrochemical, and chemical industries [1-11]. It is known that the production of commercial propylene glycol is one of the most important industries in modern chemical technology. Propylene glycol can act as one of the components in the obtaining drug's process. In addition, it is used for lubrication and preservation of food packaging machines, is used as a plasticizer in the production of cellophane and polyvinyl chloride films, as a solvent for natural and synthetic substances in the preparation of ointments, pastes, creams, shampoos, etc., in the pharmaceutical and cosmetic industries, in the food industry as a solvent for food additives and a tobacco humectant, etc. Based on the relevance of the above, this article, based on a comprehensive study of the features of the chemical-technological process for obtaining propylene glycol, in particular, the chemical reaction of obtaining commercial propylene glycol in one of the main apparatuses for the studied block - the hydrator, a physically substantiated mathematical statement of the optimal control problem of this apparatus is formulated. The optimal control problem of the studied technological apparatus is reduced to obtaining the maximum amount of the target product, considering the restrictions imposed on the control parameters. It is established that by its nature and specificity, it is a nonlinear programming problem, for the solution of which in the presented article an algorithm for optimizing the modes of the technological process is proposed, based on the use of the Lagrange multiplier method. An analysis of the results for solving the optimal control problem of the obtaining propylene glycol process showed that when maintaining the optimal values of the control parameters obtained by us, the amount of propylene glycol obtained increases by 3.5% compared to the actual amount. Keywords: technological process, commercial propylene glycol, optimal control problem, control parameters, Lagrange multiplier method, technological apparatus.

Fate and transport of ten plant protection products of emerging concern in a coastal lagoon: Application and evaluation of a multimedia level III fugacity model

Multimedia fugacity models are effective tools for studying the environmental behaviour and occurrence of contaminants of emerging concern (CECs) and assessing associated risks, especially when experimental data is limited. These models describe processes controlling chemical partitioning, transport, and reactions in environmental media using mathematical statements based on the concept of fugacity.To aid in identifying and prioritizing CECs for future local monitoring, we present here the application of a level III multimedia fugacity model assuming non-equilibrium between compartments and steady-state conditions. This model estimated predicted environmental concentrations (PECs), persistence, distribution, and transport of ten plant protection products (PPPs) in the Venice Lagoon, a complex coastal environment under high anthropogenic pressure.The model was evaluated through uncertainty and sensitivity analysis using the Monte Carlo approach and by comparing PECs with PPP concentrations measured during four sampling campaigns. Results showed good agreement with field data, with the highest concentrations in water and sediments estimated for glyphosate, followed by imidacloprid, metaflumizone, and triallate. The model indicated accumulation of all investigated PPPs in sediments. For most chemicals, advection outflow and degradation in the water column were the main removal mechanisms, while volatilization was significant only for oxadiazon and triallate.Sensitivity and uncertainty analysis revealed that degradation rates, organic carbon/water partitioning coefficients (KOC), and parameters describing air-water interactions had the strongest influence on the model's results, followed by inputs accounting for sediment sinking and resuspension. The lack of data on PPP degradation in brackish waters accounted for most of the uncertainty in model results.This work shows how a relatively simple multimedia model can offer new insights into the environmental behaviour of PPPs in a complex transitional waterbody such as the Venice lagoon, providing useful data for the identification of the CECs to be prioritised in future local monitoring efforts.

Students’ use of generative artificial intelligence for proving mathematical statements

AbstractIn this exploratory study, we investigate undergraduate students’ engagement with generative Artificial Intelligence (genAI) in proving mathematical statements. We selected six mathematical statements to conduct interviews with three students. We present the emergent framework, Students’ Interactive Proving Experience with AI (SIPE-AI), which explains the processes of students’ use of genAI in their proving and the factors influencing these processes. Our findings identify three factors that shape students’ use of genAI: conceptions of proof, conceptions of genAI, and ethical considerations. The results suggest a need to guide undergraduate students in critically engaging with genAI tools, rather than passively accepting their outputs. We also discuss the implications of these findings for enhancing undergraduate mathematics education by fostering informed and critical use of genAI in mathematical proving.

Undergraduate Students’ Logical Consistency in Mathematical Thinking: Implications for Teaching and Learning

This paper introduces the concept of logical consistency in students’ thinking in mathematical contexts. We present the Logical in-Consistency (LinC) instrument as a valuable assessment tool designed to examine the prevalence and types of logical inconsistencies among undergraduate students’ evaluation of mathematical statements and accompanying arguments. Beyond its utility as an assessment tool, this paper also underscores the potential of the LinC instrument as an instructional aid, effective for identifying students who may inadvertently overlook logical contradictions within their mathematical assertions. Furthermore, this paper outlines a practical classroom activity based on the LinC instrument format. This activity is tailored to enhance students’ capacity for reasoning with logical consistency, specifically in proof-oriented mathematics courses. We hope this activity helps instructors address and mitigate logical inconsistencies in students’ thinking throughout their classroom activities.

Uni/multi variate polynomial embeddings for zkSNARKs

AbstractA zero-knowledge proof is a cryptographic primitive that enables a prover to convince a verifier the validity of a mathematical statement (an NP statement) without revealing any secret inputs to the verifier. A special case, called zero-knowledge Succinct Non-interactive ARgument of Knowledge (zkSNARK) is particularly designed for arithmetic circuit proof systems which have important applications in blockchain privacy. The major computations in this type of zkSNARK proofs with post-quantum security are polynomial evaluations and Lagrange interpolations over finite fields. Given a sequence over a finite field, in the field of coding and sequences research, we understand that there are two representations of the sequence, one is a univariate polynomial and the other, a multivariate polynomial. This is exactly what is done in those zero-knowledge proof systems to transform the proof of a R1CS relation to evaluate uni/multi variate polynomials at some random points in the finite field. In this paper, we present a comparative analysis on how to convert a rank 1 constrained satisfiability (R1CS) system (more general than a circuit system) into a polynomial equality and provide analysis on the concrete complexities of provers, proof sizes and verifiers. We use two concrete zkSNARK schemes, i.e., Polaris, univariate polynomial encodings and Spartan, multivariate polynomial encodings, as examples to show our analysis. Secondly, we propose to select interpolating sets as subfields instead of affine spaces of a large field for Lagrange interpolation. This new method has improved the performance of R1CS encodings largely. We comment that post-quantum secure zkSNARKs yield post-quantum digital signatures with security only depending on symmetric-key schemes. Some open problems are proposed at the end of the paper.

Methodology for modeling and optimizing liquid cooling plates in the design of electronic equipment

Problem formulation. When solving the problem of excessive heat generation, which occurs when operating radio-electronic devices with a high density of electronic components and increased power, it is necessary to use cooling. Otherwise, this problem can lead to overheating of the electronics, which in turn can cause malfunctions of the devices or even their failure. Purpose. Development of a technique that will improve the cooling process of radioelectronic devices due to op-thymization of the parameters of liquid cooling plates. Results. A mathematical statement was formed, a modeling technique was proposed and procedures for optimizing liquid cooling plates were performed. Based on the developed 3D model of the liquid cooling plate, multivariate analysis procedures have been implemented using modern automated design tools. The liquid cooling system was optimized using modern modeling software, in particular the Solidworks Simulation program. Practical significance. The presented simulation results can be used in designing and creating cooling systems for various devices and devices.

The Early Determination Method of Reservoir Drive of Oil Deposits Based on Jamalbayli Indexes

Summary Historically, the concept of “reservoir drive” aimed to simplify the mathematical modeling in reservoir engineering. Within this framework, the energy of the reservoir, particularly its aquifer, was idealized, leading to classifications such as “partial waterdrive,” “full waterdrive,” “gas cap drive,” and so forth. However, in reality, all existing energy sources interact simultaneously within reservoirs. Accordingly, this study aims to develop a new concept for a more realistic description of reservoir drive mechanisms and evaluation of reservoir energy performance. Numerous computer simulations have revealed a strong correlation between the ratio of relative changes in pore volume to relative changes in reservoir pressure and the reservoir’s energy nature and activity level. Moreover, the noted ratio did not depend on production technology, pressure/volume/temperature properties of hydrocarbon systems, rheological properties of reservoir rocks, or other factors. Based on this correlation, specific parameters termed as Jamalbayli Indexes (JI) have been identified to quantitatively describe reservoir energetic performance. JI consist of two parameters. One of them describes the relative change in pore volume per unit of relative change in reservoir pressure, and the second is the relative change in pore volume per unit of relative change in formation porosity. Here, “relative change” means a change in a parameter relative to its original value. These parameters are dimensionless and can have values around or equal to unity. A new conceptual framework for describing reservoir drive mechanisms based on JI has been formulated. According to this framework, reservoir drive mechanism is determined by comparing the computed JI values with unity rather than relying on subjective assessments of the trend of some functional dependencies. For the first time, it has become possible to express the reservoir drive performance quantitatively and determine the level of energy activity of the reservoirs with the help of JI. Additionally, a technique has been developed to evaluate the numerical values of JI for specific oil (including volatile oil) deposits based on the production data at any stage of production. The proposed methodology was tested using data from the eighth horizon of the Russkiy Khutor field in Russia. The test results not only confirmed the reliability of the obtained model but also demonstrated the adequacy of the proposed concept as a whole. Summarizing the results of other works by the authors, the adequacy of the proposed concept for both oil, gas, and gas condensate deposits has been confirmed. The research findings are expected to contribute to updating the traditional principles used for the mathematical problem statements in fluid flow in porous formations. Additional Key Terms and Phrases Reservoir Drive Mechanism; Early Determination Methods; Reservoir Performance Analysis; Production Data Interpretation; Enhanced Oil Recovery; Reservoir Management; Drive Mechanism Identification; Reservoir Fluid Dynamics; Production Rate Analysis; Reservoir Engineering Techniques; Field Development Planning; Reservoir Simulation; Pressure/Volume/Temperature (PVT) Analysis; Production Monitoring and Evaluation drive mechanisms; production data analysis; volatile oil; Jamalbayli indexes; drive performance indexes

Modified projection generalized two-point two-stage extragradient quasinewton method for saddle point problems

Abstract. The purpose of this work is to investigate a new method mentioned in the article’s name. This method is designed for solving saddle problems with convexo-concave differentiable function that is defined on a convex closed subset of some finite-dimensional euclidean space and has «ravine» level hypersurfaces. The paper contains a brief survey of native publications devoted to new projection gradient methods for solving saddle problems. A mathematical statement of a saddle problem, information about solution method, some auxiliary inequalities, and method’s convergence are discussed in the article as well. Moreover, iterative formulas are exemplified for another perspective saddle method for convexo-concave differentiable saddle functions, which may be validated as well as formulas proved in this work. New auxiliary inequalities complete mathematical apparatus of convex analysis for justification of convergence and rate of convergence and have value also for justification of another methods of operations research. By using obtained inequalities, convex analysis and numerical mathematics, convergence of the saddle method for convexo-concave smooth saddle functions with Lipschitz partial gradients is proved. Under supplementary conditions, for twice continuously differentiable saddle functions, superlinear and quadratic rate of convergence of saddle method are proved, too.

Transitioning to proof via writing scripts on the rules of a new discourse

Coming from the commognitive standpoint, we consider proof-based mathematics as a distinct discourse, the transition to which requires special rules for endorsement and rejection of mathematical statements. In this study, we investigate newcomers’ learning of these rules when being taught them explicitly. Our data come from academically motivated high-school students who took a special course in undergraduate mathematics. The course teacher dedicated three academic hours to introducing and explaining selected rules of proof to support students’ shift to the new discourse. The homework assignment consisted of typical proof-requiring problems and a scriptwriting task, asking students to compose a dialogue between fictional characters about a proof-related mistake of their choice. We analyzed the differences and similarities between the rules discussed in the classroom and those that students addressed and implemented in their proofs. The analysis showed that while students’ solutions to proof-requiring problems required rule implementation, fictitious dialogues opened the space for rule formulation and substantiation. In many cases, the students discussed the rules presented in the classroom, extending, elaborating, and specifying the teacher’s formulations. Furthermore, while the students’ proofs were mainly consistent with the teacher’s expectations, some of their rule formulations were more radical and overgeneralized than expected. These findings suggest that newcomers’ communication about the rules of proof may lag behind their capability to implement those rules to prove mathematical statements.

Interplay of superconductivity and magnetism in the Fe1+yTe1‐xSex iron‐based superconductor: A theoretical study

AbstractThe competition between superconductivity and magnetism has been studied for the ongoing investigation of mechanisms in unconventional superconductors. Fe1+yTe1−xSex is distinct among iron‐based superconductors due to its structural simplicity, comprising solely FeTe/Se layers that are favorable for probing the superconductivity mechanism. The main goal of this article is to focus on the computational investigation of the interplay of magnetism and superconductivity in the Fe1+yTe1−xSex multiband superconductor. From the electronic structure of the Fe1+yTe1−xSex superconductor, a two‐band Hamiltonian model was considered to examine the interplay of magnetism and superconductivity in the Fe1+yTe1‐xSex superconductor. Using Green's function formalism with the Hamiltonian model, the mathematical statements for the superconducting (Sc) gap parameter, spin density wave (SDW) ordering parameter, Sc and magnetic transitional temperatures have been obtained. Using these mathematical expressions, the phase diagram of transition temperatures versus the gap parameters for the Fe1+yTe1‐xSex superconductor has been plotted. The intersection region in the phase diagram of transition temperatures as a function of the SDW order parameter has been plotted to show the possible interplay of superconductivity and magnetism in a Fe1+yTe1‐xSex iron‐based superconductor.

On Constructing Magnetic and Gravity Images of Mercury from Satellite Data

A new technique for simultaneous reconstruction of “gravity” and “magnetic” images of Mercury from satellite data based on the regional version of S-approximations is proposed. The mathematical statement of the inverse problem on finding the images of a planet from the data on the potential fields recorded at different times with different accuracy is reduced to solving ill-conditioned systems of linear algebraic equations (LAES) with approximate right-hand sides. Based on the analytical approximations of the Mercury’s magnetic and gravity fields determined from the solution of the ill-conditioned SLAE, the distributions of the equivalen sources on the spheres are determined. The results of the mathematical experiment on constructing the magnetic image of Mercury from the radial component of the magnetic induction vector analytically continued towards the field sources are presented.

Machine Learning and Information Theory Concepts towards an AI Mathematician

The current state of the art in artificial intelligence is impressive, especially in terms of mastery of language, but not so much in terms of mathematical reasoning. What could be missing? Can we learn something useful about that gap from how the brains of mathematicians go about their craft? This essay builds on the idea that current deep learning mostly succeeds at system 1 abilities—which correspond to our intuition and habitual behaviors—but still lacks something important regarding system 2 abilities—which include reasoning and robust uncertainty estimation. It takes an information-theoretical posture to ask questions about what constitutes an interesting mathematical statement, which could guide future work in crafting an AI mathematician. The focus is not on proving a given theorem but on discovering new and interesting conjectures. The central hypothesis is that a desirable body of theorems better summarizes the set of all provable statements, for example, by having a small description length while at the same time being close (in terms of number of derivation steps) to many provable statements.

Students' Errors in Constructing Mathematical Proofs by Direct Method

In mathematics, there are several methods for proving mathematical statements, two of them are direct method and indirect method. Both the methods were included in the mathematics curriculum at senior high school level as a learning topic. It implied that the teachers or pre-service teachers must master these methods. In addition, learning advanced mathematics courses also needed the ability of mathematical proving. The direct method was often used in proving mathematical statements. This descriptive qualitative study was conducted to examine what errors were written by the students in constructing mathematical proofs by direct method. Two basic problems (mathematical statements) were administered to 13 college students. They were pre-service mathematics teachers in one of the public universities in Aceh, Indonesia, and were asked to prove the statements by direct method. The data were analyzed by using Miles and Huberman step. The results showed that there were three main errors, (1) errors in using definition correctly, (2) errors in algebraic process, and (3) proving by examples. It concluded that the students must be accustomed to mathematical proving activities to avoid errors in constructing mathematical proofs. Keywords: constructing mathematical proofs, direct method, students’ errors

Метод чисельного визначення динамічних характеристик лопаті повітряного гвинта з композиційного матеріалу

The subject matter of this article is the dynamic characteristics of a composite propeller blade. Determination of the modes and frequencies of natural vibrations is necessary to predict the dangerous resonance modes of aircraft engines and to identify the most stressed local zones of the blade surface. The goal of this study is to develop a verified method for determining the dynamic characteristics of composite rotor parts of aircraft engines based on the known properties of the structural components of the composite material. A general mathematical statement of the problem of elasticity theory for the analysis of the dynamic characteristics of composite structures is described. A complete system of equations that describes the mechanical state of the body within the framework of the continuum mechanics approach was developed. Geometric modeling of the propeller blades was performed. Modeling and numerical investigation of the natural frequencies and mode shapes of the propeller blade were performed using the finite element method using the ANSYS software package. Based on the results of numerical research of the stressed and deformed state of the propeller blade, the first five natural vibration frequencies and the distribution of the local stress field were determined. The most stressed local zones on the blade surface were determined for each form of natural vibration. The propeller blade model was verified using an experimental study of the first five forms and frequencies of natural blade vibrations. The eigenfrequencies of the blade vibration were experimentally determined using the method of free (natural) vibrations. The resonance method was used to experimentally determine the resonant frequencies and vibration modes of the blade. The distribution of the blade deformation field was investigated using the strain gauge method. The highest error in the verification of numerical and experimental research is 4.11% for the fourth vibration frequency. Conclusions. The scientific novelty of the results obtained is that the effective elastic properties of the composite material for calculations should be determined using the procedure of numerical homogenisation of composite materials of different reinforcement structures by the properties of the matrix and fibers. The method does not require experimental determination of the effective elastic constants for the layers of blade components of different weave architecture patterns.

Pro-Con Debate: Universal Versus Selective Continuous Monitoring of Postoperative Patients.

In this Pro-Con commentary article, we discuss use of continuous physiologic monitoring for clinical deterioration, specifically respiratory depression in the postoperative population. The Pro position advocates for 24/7 continuous surveillance monitoring of all patients starting in the postanesthesia care unit until discharge from the hospital. The strongest arguments for universal monitoring relate to inadequate assessment and algorithms for patient risk. We argue that the need for hospitalization in and of itself is a sufficient predictor of an individual's risk for unexpected respiratory deterioration. In addition, general care units carry the added risk that even the most severe respiratory events will not be recognized in a timely fashion, largely due to higher patient to nurse staffing ratios and limited intermittent vital signs assessments (e.g., every 4 hours). Continuous monitoring configured properly using a "surveillance model" can adequately detect patients' respiratory deterioration while minimizing alarm fatigue and the costs of the surveillance systems. The Con position advocates for a mixed approach of time-limited continuous pulse oximetry monitoring for all patients receiving opioids, with additional remote pulse oximetry monitoring for patients identified as having a high risk of respiratory depression. Alarm fatigue, clinical resource limitations, and cost are the strongest arguments for selective monitoring, which is a more targeted approach. The proponents of the con position acknowledge that postoperative respiratory monitoring is certainly indicated for all patients, but not all patients need the same level of monitoring. The analysis and discussion of each point of view describes who, when, where, and how continuous monitoring should be implemented. Consideration of various system-level factors are addressed, including clinical resource availability, alarm design, system costs, patient and staff acceptance, risk-assessment algorithms, and respiratory event detection. Literature is reviewed, findings are described, and recommendations for design of monitoring systems and implementation of monitoring are described for the pro and con positions.

Unitizing Predicates and Reasoning About the Logic of Proofs

This article offers the construct unitizing predicates to name mental actions important for students’ reasoning about logic. To unitize a predicate is to conceptualize (possibly complex or multipart) conditions as a single property that every example has or does not have, thereby partitioning a universal set into examples and nonexamples. This explains the cognitive work that supports students to unify various statements with the same logical form, which is conventionally represented by replacing parts of statements with logical variables p or P(x). Using data from a constructivist teaching experiment with two undergraduate students, we document barriers to unitizing predicates and demonstrate how this activity influences students’ ability to render mathematical statements and proofs as having the same logical structure.