Data from a comparative judgement survey consisting of 62 working mathematics educators (ME) at Norwegian universities or city colleges, and 57 working mathematicians at Norwegian universities. A total of 3607 comparisons of which 1780 comparisons by the ME and 1827 ME. The comparative judgement survey consisted of respondents comparing pairs of statements on mathematical definitions compiled from a literature review on mathematical definitions in the mathematics education literature. Each WM was asked to judge 40 pairs of statements with the following question: “As a researcher in mathematics, where your target group is other mathematicians, what is more important about mathematical definitions?” Each ME was asked to judge 41 pairs of statements with the following question: “For a mathematical definition in the context of teaching and learning, what is more important?” The comparative judgement was done with No More Marking software (nomoremarking.com) The data set consists of the following data: comparisons made by ME (ME.csv) comparisons made by WM (WM.csv) Look up table of codes of statements and statement formulations (key.csv) Each line in the comparison represents a comparison, where the "winner" column represents the winner and the "loser" column the loser of the comparison.

This dataset is about book series. It has 1 row and is filtered where the books is A mathematical and philosophical dictionary : containing an explanation of the terms, and an account of the several subjects, comprized under the heads mathematics, astronomy, and philosophy, both natural and experimental. It features 10 columns including number of authors, number of books, earliest publication date, and latest publication date.

A new approach to the validation of surface texture form removal methods is introduced. A linear algebra technique is presented that obtains total least squares (TLS) model fits for a continuous mathematical surface definition. This model is applicable to both profile and areal form removal, and can be used for a range of form removal models including polynomial and spherical fits. The continuous TLS method enables the creation of mathematically traceable reference pairs suitable for the assessment of form removal algorithms in surface texture analysis software. Multiple example reference pairs are presented and used to assess the performance of four tested surface texture analysis software packages. The results of each software are compared against the mathematical reference, highlighting their strengths and weaknesses.

Protein-Protein, Genetic, and Chemical Interactions for MATH-34 (Caenorhabditis elegans) curated by BioGRID (https://thebiogrid.org); DEFINITION: Protein MATH-34

Protein-Protein, Genetic, and Chemical Interactions for MATH-50 (Caenorhabditis elegans) curated by BioGRID (https://thebiogrid.org); DEFINITION: math-50 encodes a protein which has a meprin-associated Traf homology (MATH) domain and may be involved in apoptosis.

ABSTRACT Defining the optimum points for installing of a cable logging system is a problem faced by forestry planners. This study evaluated the application of a mathematical programming model for optimal location of cable logging in wood extraction. The study was conducted in a forestry company located in Parana State, Brazil. We collected data during timber harvesting and developed mathematical models to define the optimal location of the cable logging considering the variables “cycle time” and “extraction distance”. The variable “cycle time” affected the definition of the optimal location of equipment resulted in a reduced number of installation points with the largest coverage area. The variable “distance extraction” negatively influenced the location, with an increased number of installation points with smaller coverage. The developed model was efficient, but needs to be improved in order to ensure greater accuracy in wood extraction over long distances.

The HWRT database of handwritten symbols contains on-line data of handwritten symbols such as all alphanumeric characters, arrows, greek characters and mathematical symbols like the integral symbol.

The database can be downloaded in form of bzip2-compressed tar files. Each tar file contains:

• symbols.csv: A CSV file with the rows symbol_id, latex, training_samples, test_samples. The symbol id is an integer, the row latex contains the latex code of the symbol, the rows training_samples and test_samples contain integers with the number of labeled data.
• train-data.csv: A CSV file with the rows symbol_id, user_id, user_agent and data.
• test-data.csv: A CSV file with the rows symbol_id, user_id, user_agent and data.

All CSV files use ";" as delimiter and "'" as quotechar. The data is given in YAML format as a list of lists of dictinaries. Each dictionary has the keys "x", "y" and "time". (x,y) are coordinates and time is the UNIX time.

About 90% of the data was made available by Daniel Kirsch via github.com/kirel/detexify-data. Thank you very much, Daniel!

The XSD defines a data structure for an unambiguous, easy-to-use, safe and uniform exchange of fundamental physical constants and mathematical constants in a machine-readable data format. The data elements for constants are defined in the Digital System of Units (D-SI) metadata model from the EMPIR project 17IND02 SmartCom. The development of the data structure was based on an own analysis of minimum requirement for transfering the data of fundamental physical constants "as are" provided by CODATA into a machine-readable form. Furthermore, the considerations for the transformation comprise traceability to the original CODATA values.

The availability of machine-readable data of fundamental physical constants that can easy be accessed by software and that are traceable to a international accepted definition is essential for various areas of metrology where the International System of Units (SI) is of key-importance.

This work studies the intersection of chromatics, mathematical algorithms, and innovative concepts such as grammatical geometry. Analyzes the impact of colors on language perception and introduces original mathematical formulas, extracted from the definitions of grammatical concepts, providing a deep and authentic approach to the study of language, which allows for a more precise interpretation of grammatical rules, thus facilitating systematic learning of the language. The concept of grammatical geometry, proposed in the pages of this book, opens new horizons in understanding the relationships between the elements of a sentence and how they interact. This geometry is not limited to visual representations but offers a solid theoretical framework for analyzing linguistic structures from various and multidimensional perspectives. Another remarkable aspect of this work is the presentation of the first chromatic map of grammar, which illustrates how different shades of colors can correspond to different grammatical functions and concepts. This map not only enriches the perspective on learning grammar but also serves as an innovative visual tool that helps associate colors with certain linguistic structures, thus facilitating the process of memorization and understanding. Due to the correlation of these four fields – algorithms, grammar, mathematics, and chromatics – the formula of metamorphosis has been identified, a new formula with potential applicability in other fields such as psychology, art, design, medicine, geology, biology, etc. This formula opens new perspectives for interdisciplinary explorations, contributing to a deeper understanding of the interaction between language and perception in various contexts, as well as within any discipline that generates the idea of metamorphosis.

Protein-Protein, Genetic, and Chemical Interactions for BATH-31 (Caenorhabditis elegans) curated by BioGRID (https://thebiogrid.org); DEFINITION: BTB and MATH domain containing

Supplementary Materials for Chapter 2 of the Doctoral Dissertation "How we think about numbers - Early counting and mathematical abstraction". Contains Preregistrations, open data and open materials for study 1 and study 2.As children learn to count, they make one of their first mathematical abstractions. They initially learn how numbers in the count sequence correspond to quantities of physical things if the rules of counting are followed (i.e., if you say the numbers in order “one two three four …” as you tag each thing with a number). Around the age of four-years-old, children discover that these rules also define numbers in relation to each other, such that numbers contain meaning in themselves and without reference to the physical world (e.g., “five” is “one” more than “four”). It is through learning to count, that children discover the natural numbers as mathematical symbols defined by abstract rules.In this dissertation, I explored the developmental trajectory and the cognitive mechanisms of how we gain an understanding of the natural numbers as children. I present new methodological, empirical, and theoretical insights on how and when in the process of learning to count, children discover that numbers represent cardinalities, that numbers can be defined in relation to each other by the successor function and that numbers refer to units. Lastly, I explore this mathematical abstraction as the foundation of how we think about numbers as adults.My work critically tested prominent theories on how learning to count gives meaning to numbers through analogical mapping and conceptual bootstrapping. Findings across five empirical studies suggest that the process is more gradual and continuous than previous theories have proposed. Children begin to understand numbers as cardinalities defined in relation to other numbers by the successor function before they fully grasp the rules of counting. With learning the rules of counting this understanding continuously expands and matures. I further suggest that children may only fully understand numbers as abstract mathematical symbols once they understand how counting and numbers refer to the abstract notion of units rather than to physical things.The central finding of this dissertation is that learning to count does not change children’s understanding of numbers altogether and all at once. Nonetheless, when learning to count, children accomplish a fascinating mathematical abstraction, which builds the foundation for lifelong mathematical learning.© Theresa Elise Wege, CC BY-NC 4.0

The NaturalProofs Dataset is a large-scale dataset for studying mathematical reasoning in natural language. NaturalProofs consists of roughly 20,000 theorem statements and proofs, 12,500 definitions, and 1,000 additional pages (e.g. axioms, corollaries) derived from ProofWiki, an online compendium of mathematical proofs written by a community of contributors.

Thematic meanings of numerical definitions of subject data in various fields of science lead to manipulation of digital codes of known physical, chemical, biological, genetic and other quantities. In principle, each scientific justification contains, to one degree or another, a quantitative, qualitative characteristic of comparison or content. Thus, the language of natural numbers, like mathematical operations, can be accompanied by any definition in any terminology. In this text, the author does not use well-known terms related to the main scientific areas. In this text, the numbers speak for themselves. Any combination of orders or compositions of complex numerical structures presented in this text has its own logical meaning. Any paradox of numerical combinations is an algorithm of real values of numbers.

Protein-Protein, Genetic, and Chemical Interactions for BATH-38 (Caenorhabditis elegans) curated by BioGRID (https://thebiogrid.org); DEFINITION: BTB and MATH domain containing

Reference data for the Perram and Wertheim (1985) contact function of ellipsoids

This dataset provides reference values of the contact function of two ellipsoids, as defined by Perram and Wertheim (Perram, J. W., & Wertheim, M. S. (1985). Statistical mechanics of hard ellipsoids. I. Overlap algorithm and the contact function. Journal of Computational Physics, 58(3), 409–416. DOI:10.1016/0021-9991(85)90171-8). This paper will be referred to as PW85 in what follows.

Reference values of the F function

The data is shared as a HDF5 file pw85_ref_data-YYYYMMDD.h5, which contains the following datasets (to be described below)

directions: a 12×3 array,

F: a 108×108×12×9 array,

lambdas: a length-9 array,

radii: a length-3 array,

spheroids: a 108×6 array.

The attached Python script pw85_gen_ref_data.py was used to generate the data; it uses the mpmath library.

Mathematical definition of the contact function

The contact function is defined in PW85 as the maximum over (0, 1) of the F function which is defined as follows [see Eq. (3.7) in PW85, with slightly different notations]

F(λ) = λ(1-λ)r₁₂ᵀ⋅Q⁻¹⋅r₁₂,

where 0 ≤ λ ≤ 1 is a scalar, r₁₂ is the center-to-center vector. Q is the matrix defined as follows

Q = (1-λ)Q₁ + λQ₂,

where Qᵢ is the symmetric, positive definite matrix that defines ellipsoid Ωᵢ through

m ∈ Ωᵢ iff (m-cᵢ)ᵀ⋅Qᵢ⁻¹⋅(m-cᵢ) ≤ 1,

where cᵢ is the center of Ωᵢ. Then, the contact function F₁₂ is defined as the maximum of F [see Eq. (3.8) in PW85]

F₁₂(r₁₂, Q₁, Q₂) = max{ F(λ), 0 ≤ λ ≤ 1 }.

Parametrization

The reference data is restricted to spheroids (equatorial radius: aᵢ; polar radius: cᵢ; direction of axis of revolution: nᵢ)

Qᵢ = aᵢ²I + (cᵢ²-aᵢ²)nᵢᵀ⋅nᵢ,

(I: identity matrix). The radii take the following values

aᵢ, cᵢ ∈ {0.01999, 1.999, 9.999}.

These values of the radii are stored in the radii dataset of the HDF5 file. The orientations nᵢ coincide with the vertices of an icosahedron

nᵢ = [0, ±u, ±v]ᵀ or nᵢ = [±v, 0, ±u]ᵀ or nᵢ = [±u, ±v, 0]ᵀ,

where

  1        φ         1+√5

u = ───────, v = ─────── and φ = ────. √(1+φ²) √(1+φ²) 2

The orientations are stored in the directions dataset as a 12×3 array. The matrices Qᵢ are precomputed and stored in the spheroids dataset as a 108×6 array (note: 108 = 12 orientations × 3 equatorial radii × 3 polar radii). spheroids[i, :] stores the upper triangular part of the corresponding matrix in row-major order

⎡ spheroids[i, 0] spheroids[i, 1] spheroids[i, 2] ⎤ ⎢ spheroids[i, 3] spheroids[i, 4] ⎥. ⎣ sym. spheroids[i, 5] ⎦

The scalar λ takes tabulated values (see the lambdas dataset)

λ ∈ {0.1, 0.2, …, 0.9}.

Note that λ = 0.0 and λ = 1.0 are excluded, since F is uniformly 0 in that case.

Reference values of the F function

The reference values of the function F are stored in the F dataset, which is a 108×108×12×9, such that F[i, j, h, k] is the value of F for

Q₁ = spheroids[i], Q₂ = spheroids[j], r₁₂ = directions[h] and λ = lambdas[k].

Note that the r₁₂ vector takes values in the directions dataset. In other words, only unit-length center-to-center vectors are considered here. Indeed, F trivially depends on the norm of r₁₂, which is therefore not considered here in order to reduce the size of the dataset.

Reference values of the contact function

Note: the following is not implemented yet, as reference values of the contact function were not deemed useful. Indeed, once F is validated, it is straightforward to check that the implementation of F₁₂ to be tested indeed maximizes F.

The reference values of the contact function F₁₂ are stored in the contact_function dataset, which is a 108×108×12×3 array, such that contact_function[i, j, h, k] is the value of F₁₂ for

Q₁ = spheroids[i], Q₂ = spheroids[j] and r₁₂ = radii[h] * directions[k].

Note that the r₁₂ vector is not normed, here.

Protein-Protein, Genetic, and Chemical Interactions for BPM3 (Arabidopsis thaliana (Columbia)) curated by BioGRID (https://thebiogrid.org); DEFINITION: BTB/POZ and MATH domain-containing protein

This tutorial presents an introduction to Electrochemical Impedance Spectroscopy (EIS) theory and has been kept as free from mathematics and electrical theory as possible. If you still find the material presented here difficult to understand, don't stop reading. You will get useful information from this application note, even if you don't follow all of the discussions.

Four major topics are covered in this Application Note.

AC Circuit Theory and Representation of Complex Impedance Values

Physical Electrochemistry and Circuit Elements

Common Equivalent Circuit Models

Extracting Model Parameters from Impedance Data

No prior knowledge of electrical circuit theory or electrochemistry is assumed. Each topic starts out at a quite elementary level, then proceeds to cover more advanced material.

GPQA stands for Graduate-Level Google-Proof Q&A Benchmark. It's a challenging dataset designed to evaluate the capabilities of Large Language Models (LLMs) and scalable oversight mechanisms. Let me provide more details about it:

Description: GPQA consists of 448 multiple-choice questions meticulously crafted by domain experts in biology, physics, and chemistry. These questions are intentionally designed to be high-quality and extremely difficult. Expert Accuracy: Even experts who hold or are pursuing PhDs in the corresponding domains achieve only 65% accuracy on these questions (or 74% when excluding clear mistakes identified in retrospect). Google-Proof: The questions are "Google-proof," meaning that even with unrestricted access to the web, highly skilled non-expert validators only reach an accuracy of 34% despite spending over 30 minutes searching for answers. AI Systems Difficulty: State-of-the-art AI systems, including our strongest GPT-4 based baseline, achieve only 39% accuracy on this challenging dataset.

The difficulty of GPQA for both skilled non-experts and cutting-edge AI systems makes it an excellent resource for conducting realistic scalable oversight experiments. These experiments aim to explore ways for human experts to reliably obtain truthful information from AI systems that surpass human capabilities¹³.

In summary, GPQA serves as a valuable benchmark for assessing the robustness and limitations of language models, especially when faced with complex and nuanced questions. Its difficulty level encourages research into effective oversight methods, bridging the gap between AI and human expertise.

(1) [2311.12022] GPQA: A Graduate-Level Google-Proof Q&A Benchmark - arXiv.org. https://arxiv.org/abs/2311.12022. (2) GPQA: A Graduate-Level Google-Proof Q&A Benchmark — Klu. https://klu.ai/glossary/gpqa-eval. (3) GPA Dataset (Spring 2010 through Spring 2020) - Data Science Discovery. https://discovery.cs.illinois.edu/dataset/gpa/. (4) GPQA: A Graduate-Level Google-Proof Q&A Benchmark - GitHub. https://github.com/idavidrein/gpqa. (5) Data Sets - OpenIntro. https://www.openintro.org/data/index.php?data=satgpa. (6) undefined. https://doi.org/10.48550/arXiv.2311.12022. (7) undefined. https://arxiv.org/abs/2311.12022%29.

Protein-Protein, Genetic, and Chemical Interactions for BATH-9 (Caenorhabditis elegans) curated by BioGRID (https://thebiogrid.org); DEFINITION: BTB and MATH domain containing

MMLU (Massive Multitask Language Understanding) is a new benchmark designed to measure knowledge acquired during pretraining by evaluating models exclusively in zero-shot and few-shot settings. This makes the benchmark more challenging and more similar to how we evaluate humans. The benchmark covers 57 subjects across STEM, the humanities, the social sciences, and more. It ranges in difficulty from an elementary level to an advanced professional level, and it tests both world knowledge and problem solving ability. Subjects range from traditional areas, such as mathematics and history, to more specialized areas like law and ethics. The granularity and breadth of the subjects makes the benchmark ideal for identifying a model’s blind spots.