Proof of Eckart–Young–Mirsky theorem (for spectral norm)

Proof of Eckart–Young–Mirsky theorem (for Frobenius norm)

Weighted low-rank approximation problems

Entry-wise Lp low-rank approximation problems

Distributed/Streaming low-rank approximation problem

Image and kernel representations of the rank constraints

A Variant: convex-restricted low rank approximation


Method of moments for polynomial distributions

Positive Hankel matrices and the Hamburger moment problems


Allowing observation errors in all variables


Rank-reduced singular value decomposition

Querying and augmenting LSI vector spaces


Categorical by categorical variable interactions

Categorical by continuous variable interactions


Determinantal rank – size of largest non-vanishing minor

Tensor rank – minimum number of simple tensors


Classification of critical points and extrema


Finite-dimensional complex normed spaces

Maximum norm (special case of: infinity norm, uniform norm, or supremum norm)

Classification of seminorms: absolutely convex absorbing sets


Computing PCA using the covariance method

Derivation of PCA using the covariance method

Discriminant analysis of principal components


Constructing loss and objective functions


Approximate non-negative matrix factorization

Convex non-negative matrix factorization

Different cost functions and regularizations


Criteria for determining the number of factors

Exploratory factor analysis (EFA) versus principal components analysis (PCA)


Reinforcement learning for recommender systems


History and relationships to other fields

Proprietary software with free and open-source editions


Lower bound on number of observed entries

Algorithms for Low-Rank Matrix Completion


Methods: Rules, statistics, neural networks

Lexical semantics (of individual words in context)

Relational semantics (semantics of individual sentences)

Discourse (semantics beyond individual sentences)

General tendencies and (possible) future directions


Theoretical and algorithmic advancements

Synthetic example of a traveling pattern


Matrix norms induced by vector α- and β- norms


Rotation, coordinate scaling, and reflection

Singular values as semiaxes of an ellipse or ellipsoid

The columns of U and V are orthonormal bases

Singular values, singular vectors, and their relation to the SVD


Graphics and graphical user interface programming


Differences between linear and nonlinear least squares


Estimation of the noise covariances Qk and Rk

Deriving the posteriori estimate covariance matrix

Simplification of the posteriori error covariance formula

Relationship to recursive Bayesian estimation

Variants for the recovery of sparse signals


Comparison with other identifier schemes


Deriving the Solution of Linear Time-Invariant Differential Equations

Discrete-time systems from continuous-time systems


Wars with the Huns, Vandals, and Persians


FBI career and first espionage activities (1976–1981)

FBI counterintelligence unit, further espionage activities (1985–1991)

Later FBI career, continued espionage activities (1992–2001)


Songs written by Barry Mann and Cynthia Weil


British logistics in the Normandy campaign


Valentinus's revolt and Martinus's death


Blending Christianity with classical culture


Before the 2017 constitutional amendment

After the 2017 constitutional referendum

Before the 2017 constitutional referendum


Complete IMSA SportsCar Championship results


Suite of musical instruments, utensils, and furniture


Ukrainian unrest, Russian activity in Crimea and Donbas

Initial invasion of Ukraine (24 February – 7 April)

Southeastern front (8 April – 5 September)

Russian annexations and Ukrainian counterattacks (6 September – 11 November 2022)

Russian annexation of Donetsk, Kherson, Luhansk and Zaporizhzhia oblasts

Winter stalemate, attrition campaign and military surge (12 November 2022 – present)


Wikimedia Foundation and affiliate movements

Internal research and operational development


Differences between encyclopedias and dictionaries


Interaction with the mid-latitude westerlies

Natural phenomena caused or worsened by tropical cyclones


Three circles of English-speaking countries

English loanwords and calques in other languages


Regions, districts and major communities


Independence and the later nineteenth century


Sheikh-ul-Jamia, Jamia Millia Islamia (1926–1948)

Basic National Education Committee (1937)

Vice-Chancellor, Aligarh Muslim University (1948–1956)


Important presidential interventions in the past


Overall results in the Champions League T20

Rivalry with Royal Challengers Bangalore


Azerbaijani and Russian interference allegations


2006: House Armed Services Committee Hearing


Decomposition into symmetric and skew-symmetric


Choice of basis as a choice of isomorphism


Fundamental operators on scalar and vector fields


Motivating example: Euclidean vector space

Bessel's inequality and Parseval's formula


Properties of the determinant in relation to other notions

Eigenvalues and characteristic polynomial

Determinants for finite-dimensional algebras

Related notions for non-commutative rings


Eigenvalues and eigenvectors of matrices

Eigenvalues and the characteristic polynomial

Eigenspaces, geometric multiplicity, and the eigenbasis for matrices

Diagonalization and the eigendecomposition

Three-dimensional matrix example with complex eigenvalues

Matrix with repeated eigenvalues example

Eigenvalues and eigenfunctions of differential operators

Associative algebras and representation theory

Eigenvalues of geometric transformations


The Plancherel theorem and Parseval's theorem

Relation to the multidimensional Fourier transform

Fourier transform in d dimensions (radially symmetric case)

Relation to the Fourier and Abel transforms


Multiplication and division in polar form

Matrix representation of complex numbers

Complex exponential and related functions

Electromagnetism and electrical engineering


Matrix inverses in real-time simulations

Matrix inverses in MIMO wireless communication


EAN format used in barcodes, and upgrading


Society for Industrial and Applied Mathematics


Addition, scalar multiplication, and transposition

Abstract algebraic aspects and generalizations

Symmetries and transformations in physics

Other historical usages of the word "matrix" in mathematics


Nobel prize winners published by Cambridge University Press

Organisational governance and operational structure

Relationship with the University of Cambridge

Cambridge University Press partnerships and acquisitions


Decomposition into Hermitian and skew-Hermitian


Associative product of Hermitian matrices

Complex Hermitian forms vector space over R

Decomposition into Hermitian and skew-Hermitian matrices


Compressed sparse row (CSR, CRS or Yale format)


Similarity to symmetric tridiagonal matrix


Variants and definitions of component matrices

Singularity and relations to the Mertens function and special series

Factorizations of sums encoded by these matrices

Bounds and properties of non-trivial eigenvalues

Matrix products expanding Dirichlet convolutions and Dirichlet inverses

Generalizations of the Redheffer matrix forms: GCD sums and other matrices whose entries denote inclusion in special sets


Commutation and anti-commutation relations


Determinants of the Cartan matrices of the simple Lie algebras

Representations of finite-dimensional algebras


Confusion matrices with more than two categories


Distance Matricies Using Gaussian Mixture distance

Evaluation of the similarity or dissimilarity of Cosine similarity and Distance matrices

Neighborhood Retrieval Visualizer (NeRV)

Interconversion mechanisms between two permutational isomers

Distance Polynomials and Distance Spectra


Example 2: polar-Cartesian transformation

Example 3: spherical-Cartesian transformation


Computational and theoretical relationship to other unitary transformations


Fundamental matrix (linear differential equation)


Non-standard orientation of the coordinate system

Conversion from rotation matrix to axis–angle


Mathematical statistics and numerical analysis

Convergence to Wigner semicircular distribution

The variational problem for the unitary ensembles


Use of different similarity measure formulas


One-way ANOVA (offset from reference group)


Covariance matrix as a parameter of a distribution


Informal derivation of the Cramér–Rao bound


Terminology and theory of inferential statistics

Applied statistics, theoretical statistics and mathematical statistics

Statistics applied to mathematics or the arts


Other measures of dependence among random variables

Uncorrelatedness and independence of stochastic processes


Subgraphs, induced subgraphs, and minors


Proof of the Birkhoff–von Neumann theorem


Transmission coefficient and reflection coefficient


Specialized substitution matrices and their extensions


Estimation of the state-transition matrix


Linear dependence and independence of two vectors


The determinant of the matrix exponential

Inequalities for exponentials of Hermitian matrices

Directional derivatives when restricted to Hermitian matrices

Using the Jordan–Chevalley decomposition

Evaluation by implementation of Sylvester's formula

Inhomogeneous case generalization:  variation of parameters


Application to linear differential equations


Prediction (interpolation and extrapolation)


Least-squares estimation and related techniques

Maximum-likelihood estimation and related techniques


Matrices with conditions on eigenvalues or eigenvectors

Matrices used in science and engineering

Other matrix-related terms and definitions


Matrix form and calculation of estimates


Simple linear regression without the intercept term (single regressor)


Probit link function as popular choice of inverse cumulative distribution function


Definition of the inverse of the logistic function

Many explanatory variables, two categories

Multinomial logistic regression: Many explanatory variables and many categories

Iteratively reweighted least squares (IRLS)

Deviance and likelihood ratio test ─ a simple case


Latent variable interpretation / derivation


As a set of independent binary regressions

Application in natural language processing


Correlation in unobserved factors over time


Common features of discrete choice models

Properties of discrete choice models implied by utility theory

Prominent types of discrete choice models

A. Logit with attributes of the person but no attributes of the alternatives

B. Probit with attributes of the person but no attributes of the alternatives

C. Logit with variables that vary over alternatives

D. Probit with variables that vary over alternatives

Multinomial choice without correlation among alternatives

E. Logit with attributes of the person but no attributes of the alternatives

F. Logit with variables that vary over alternatives (also called conditional logit)

Multinomial choice with correlation among alternatives

G. Nested Logit and Generalized Extreme Value (GEV) models


Equality of fixed effects and first difference estimators when T=2


Applications to longitudinal (repeated measures) data

Alternative ways of analyzing hierarchical data


Maximum likelihood-based parameter estimation


Smooth coefficient/varying coefficient models


Fundamental characteristics and applications of the PCR estimator

Optimality of PCR among a class of regularized estimators


List of general-purpose nonparametric regression algorithms


History and unpopularity of robust regression


Example: Modeling cognitive decline in Alzheimer's disease

Example: Population Pharmacokinetic/pharmacodynamic modeling

Example: COVID-19 epidemiological modeling

Example: Prediction of oil production curve of shale oil wells at a new location with latent kriging


Conditional quantile and quantile regression

Computation of estimates for regression parameters

Equivariance to reparameterization of design

Bayesian methods for quantile regression

Machine learning methods for quantile regression


Segmented linear regression, two segments


Calculation of the Jacobian by numerical approximation

Parameter errors, confidence limits, residuals etc.


Independent and identically distributed (iid)


Reduced-rank vector generalized linear models

Quadratic reduced-rank vector generalized linear models


Differences between linear and nonlinear least squares


The model and the proportional odds assumption


Rotation, coordinate scaling, and reflection

Singular values as semiaxes of an ellipse or ellipsoid

The columns of U and V are orthonormal bases

Singular values, singular vectors, and their relation to the SVD


Scale-invariant curves and self-similarity

Scale invariance in stochastic processes

Scale invariance in classical field theory

Scale invariance of field configurations

Scale invariance in quantum field theory

Newtonian fluid mechanics with no applied forces

Hidden scale invariance in liquids and solids


Other uses of the word "error" in statistics


Gauss–Markov theorem as stated in econometrics


Constructing loss and objective functions


Other forms of multivariate location and scatter


Textbook analysis using a normal distribution

Statistical models for observational data


Algebraic fitting of functions to data points

Fitting lines and polynomial functions to data points

Geometric fitting of plane curves to data points


Modern formulation via differentiable manifolds

Interpretation of the Lagrange multipliers


Covariance matrix as a parameter of a distribution


Second-order efficiency after correction for bias

Application of maximum-likelihood estimation in Bayes decision theory

Relation to minimizing Kullback–Leibler divergence and cross entropy

Discrete distribution, finite parameter space

Discrete distribution, continuous parameter space

Continuous distribution, continuous parameter space


Terminology and theory of inferential statistics

Applied statistics, theoretical statistics and mathematical statistics

Statistics applied to mathematics or the arts


Other measures of dependence among random variables

Uncorrelatedness and independence of stochastic processes


Publication Type: Clinical queries/systematic reviews


Interpretation of the size of a correlation


Semipartial correlation (part correlation)


Decreasing the potential for confounding


Assumption 2: homogeneity of error variances

Assumption 3: independence of error terms

Assumption 5: homogeneity of regression slopes

Test the homogeneity of variance assumption

Test the homogeneity of regression slopes assumption


Partitioning the sum of squares in linear regression


Correspondence analysis based on Spearman's ρ

Approximating Spearman's ρ from a stream


Methods to assist in choosing the set of candidate models


Comparisons with other model selection methods


Linear MMSE estimator for linear observation process

Special Case: vector observation with uncorrelated noise


Choosing an optimality criterion and robustness

Flexible optimality criteria and convex analysis

Model uncertainty and Bayesian approaches

System identification and stochastic approximation

Specifying the number of experimental runs


Discretization and numerical integration

Numerical stability and well-posed problems

Interpolation, extrapolation, and regression

Solving equations and systems of equations

Solving eigenvalue or singular value problems


Quadrature rules based on interpolating functions

Conservative (a priori) error estimation


Example of two-point Gauss quadrature rule

Computation of Gaussian quadrature rules


Relations between the two kinds of Chebyshev polynomials

Chebyshev polynomials as special cases of more general polynomial families

Families of polynomials related to Chebyshev polynomials


Algebraic fitting of functions to data points

Fitting lines and polynomial functions to data points

Geometric fitting of plane curves to data points


Persons influential in the field of statistics


Disciplines and paradigms in linguistic semantics


Consistency between real and complex definitions

Uniqueness up to unitary transformations

Extension for non-Hermitian square matrices


Hurwitz matrix and the Hurwitz stability criterion


A matrix representation formula for a nonzero projection operator

Finding projection with an inner product


Relationships between idempotent matrices


Matrix inverses in real-time simulations

Matrix inverses in MIMO wireless communication


Dot product, bilinear form and sesquilinear form

Computational complexity depends on parenthezation


Classification of unipotent groups over characteristic 0


Definition for 1-variable case for a real measure


Frequency characteristics of convolution filters

Comparison with other filters and alternatives


Relations to more specialized optimal design theory


Comparison to multiple linear regression


Relationship with Generalized Method of Moments


Symmetric Laplacian via the incidence matrix

Symmetric Laplacian for a directed graph

Left (random-walk) and right normalized Laplacians

Definitions for graphs with weighted edges

Interpretation as the discrete Laplace operator approximating the continuous Laplacian

Generalizations and extensions of the Laplacian matrix


Derivation of the fundamental matrix using coplanarity condition


Metaphor as an organizational principle for lexis


Variance of the mean and predicted responses


Lexical relations: how meanings relate to each other

Syntactic basis of event structure: a brief history

Micro-syntactic theories: 1990s to the present

Intransitive verbs: unaccusative versus unergative

Transitivity alternations: the inchoative/causative alternation

Beck & Johnson's 2004 double object construction


Example: equal frequencies of men and women


Other uses of the word "error" in statistics


Ridge regression (or Tikhonov regularization)


Basic approach of response surface methodology

Simplex geometry and mixture experiments


Relation to singular-value decomposition and Wiener filter


List of unsolved problems in linguistics

Historical linguistics and the evolution of language


The generalized Lomb–Scargle periodogram

Korenberg's "fast orthogonal search" method


Research activities on corporate applications


The von Neumann equation for time evolution

Wigner functions and classical analogies


Statistical experiments, following Charles S. Peirce

Discussion topics when setting up an experimental design


Denotational semantics of non-deterministic programs

Denotational semantics for programs of restricted complexity

Denotational semantics as source-to-source translation

Connections to other areas of computer science


Term frequency-inverse document frequency weights

Models based on and extending the vector space model

Software that implements the vector space model


Triangle inequality for cosine similarity


Set-theoretic interpretation of signature


Intuitionistic logic, denotational semantics, linear logic, logical pluralism


Weakest-preconditions and strongest-postconditions of imperative expressions


Other measures of dependence among random variables

Uncorrelatedness and independence of stochastic processes


Distributional semantic modeling in vector spaces


Second dimension: properties of the model


Performance and professionalism supervision


Eigenvalues and eigenvectors of matrices

Eigenvalues and the characteristic polynomial

Eigenspaces, geometric multiplicity, and the eigenbasis for matrices

Diagonalization and the eigendecomposition

Three-dimensional matrix example with complex eigenvalues

Matrix with repeated eigenvalues example

Eigenvalues and eigenfunctions of differential operators

Associative algebras and representation theory

Eigenvalues of geometric transformations


Abstract interpretation of computer programs


"Content-based" versus "request-based" classification


Term frequency–inverse document frequency


Methodology used in measuring performance


Connectivity-based clustering (hierarchical clustering)

Biology, computational biology and bioinformatics


Effective date of patents and patent applications as prior art

Public participation in patent examination


Simultaneous approximation of extreme eigenvalues


Non-national treatment in the application procedure


Sense inventory and algorithms' task-dependency


Rotation, coordinate scaling, and reflection

Singular values as semiaxes of an ellipse or ellipsoid

The columns of U and V are orthonormal bases

Singular values, singular vectors, and their relation to the SVD


Physiological and neural architecture of language and speech


As slices of generalized Pascal's triangle

Equivalence tests for multinomial distributions


Standard contents of a contingency table

Cramér's V and the contingency coefficient C


Science Applications International Corporation


Guidelines for evaluating a concept search engine


Recursive computation with Taylor Series expansion

Using the Taylor series and Newton's method for the inverse function

Fourier transform and characteristic function

Moment and cumulant generating functions

Operations and functions of normal variables

Operations on multiple independent normal variables

Operations on multiple correlated normal variables

Infinite divisibility and Cramér's theorem

Bayesian analysis of the normal distribution

Generating values from normal distribution

Numerical approximations for the normal CDF and normal quantile function


Examples of probability for Poisson distributions

Once in an interval events: The special case of λ = 1 and k = 0

Examples that violate the Poisson assumptions

Sums of Poisson-distributed random variables

As a Binomial distribution with infinitesimal time-steps

Simultaneous estimation of multiple Poisson means

Poisson regression and negative binomial regression


Evaluation measures (information retrieval)


Computing PCA using the covariance method

Derivation of PCA using the covariance method

Discriminant analysis of principal components


Applications and systems for summarization

Submodular functions as generic tools for summarization

Domain-specific versus domain-independent summarization


Topic models for quantitative biomedicine


Types of electronically stored information


Second dimension: properties of the model


"Content-based" versus "request-based" classification


Methods: Rules, statistics, neural networks

Lexical semantics (of individual words in context)

Relational semantics (semantics of individual sentences)

Discourse (semantics beyond individual sentences)

General tendencies and (possible) future directions


Relation to human cognitive and brain development


Bigram frequency in the English language


Detecting and indexing similar documents in large corpora


Free or open source software and services


Contrast to "shallow linguistic processing"


Notable theories on the structure of concepts


Scientific literature mining and academic applications

Methods for scientific literature mining

Digital humanities and computational sociology


Sense inventory and algorithms' task-dependency


Potential AI-based techniques to improve translations

Using machine translation as a teaching tool

Machine translation and signed languages


Early approaches and some sample heuristics


Challenges with statistical machine translation

Systems implementing statistical machine translation


Computing PCA using the covariance method

Derivation of PCA using the covariance method

Discriminant analysis of principal components


Language resource publication, dissemination and creation


Radiology and intelligent word embeddings (IWE)

Preservation of semantic and syntactic relationships


Machine-readable dictionary vs. NLP dictionary


Linguistic Data: Scope and Classification


Development as W3C Recommendation (2006–2009)


Automatic identification and data capture

Overview of automatic identification methods


Speech segmentation in infants and non-natives


Creative writing and computational humor


Birth of smart virtual assistants: 1990s–2010s

Artificial intelligence and language models: 2020s-present

Privacy policy of prominent virtual assistants

Presumed and observed interest for the consumer

Presumed added value as allowing a new way of interactions


Dynamic time warping (DTW)-based speech recognition

Deep feedforward and recurrent neural networks


Voice command software products on computing devices


Spell-checking for languages other than English


Of a tetrahedron and n-dimensional simplex


Controllers of gene expression heterogeneity


Alternative definition using relational algebra

Well-definedness under an equivalence relation

Fundamental theorem of equivalence relations

Equivalence relations and mathematical logic


Relationships between the mean, median and mode


Central tendency and statistical dispersion

Scale types and Stevens's "operational theory of measurement"

Same variable may be different scale type depending on context


Categorical by categorical variable interactions

Categorical by continuous variable interactions


Definition in terms of limits of functions

Definition in terms of limits of sequences

Weierstrass and Jordan definitions (epsilon–delta) of continuous functions

Definition in terms of control of the remainder

Relation to differentiability and integrability

Continuous functions between metric spaces

Uniform, Hölder and Lipschitz continuity

Continuous functions between topological spaces

Closure operator and interior operator definitions

Defining topologies via continuous functions


Bayesian inference using conjugate prior


Universal donors and universal recipients


Pascal and syntactically similar languages

Algebraic data type in functional programming


Discretization of linear state space models


Standard contents of a contingency table

Cramér's V and the contingency coefficient C


Mathematical and combinatoric foundations


The unit circle and common trigonometric values

Trigonometric functions of real or complex variables


Absolutely continuous probability distributions

Common probability distributions and their applications

Exponential growth (e.g. prices, incomes, populations)

Bernoulli trials (yes/no events, with a given probability)

Categorical outcomes (events with K possible outcomes)

Poisson process (events that occur independently with a given rate)

Absolute values of vectors with normally distributed components

Normally distributed quantities operated with sum of squares

As conjugate prior distributions in Bayesian inference

Some specialized applications of probability distributions


Second-order efficiency after correction for bias

Application of maximum-likelihood estimation in Bayes decision theory

Relation to minimizing Kullback–Leibler divergence and cross entropy

Discrete distribution, finite parameter space

Discrete distribution, continuous parameter space

Continuous distribution, continuous parameter space


Theorem of Apollonios on conjugate diameters


Law of large numbers, binomial distribution and central limit theorem


Points, lines, and circles associated with a triangle

Further formulas for general Euclidean triangles

Medians, angle bisectors, perpendicular side bisectors, and altitudes

Specifying the location of a point in a triangle


Efficient computation of the sample median


Software for power and sample size calculations


Nonparametric bootstrap hypothesis testing


Relation to the necessary truths and contingent truths


Sample distributions of mean and variance


Proof of

lim

p
→
0

M

p

=

M

0

 (geometric mean)

Equivalence of inequalities between means of opposite signs


Comparison of scores measured on different scales: ACT and SAT

Percentage of observations below a z-score

Cluster analysis and multidimensional scaling

Relative importance of variables in multiple regression: Standardized regression coefficients

Standardizing in mathematical statistics


Textbook analysis using a normal distribution

Statistical models for observational data


Central tendency and statistical dispersion

Scale types and Stevens's "operational theory of measurement"

Same variable may be different scale type depending on context


Qualitative and quantitative interactions

Example: Interaction of species and air temperature and their effect on body temperature

Example: effect of stroke severity and treatment on recovery

Example: Interaction of temperature and time in cookie baking

ANOVA model 1: no interaction term; yield ~ temperature + time

ANOVA model 2: include interaction term; yield ~ temperature * time


Multiplication and division in polar form

Matrix representation of complex numbers

Complex exponential and related functions

Electromagnetism and electrical engineering


Categorical manifest variables as latent variable


Bronze Age: parallel script forms and gradual evolution

Unification: seal script, vulgar writing, and proto-clerical

Dominance and maturation of regular script

Comparisons of traditional Chinese, simplified Chinese, and Japanese


Absolutely continuous probability distributions

Common probability distributions and their applications

Exponential growth (e.g. prices, incomes, populations)

Bernoulli trials (yes/no events, with a given probability)

Categorical outcomes (events with K possible outcomes)

Poisson process (events that occur independently with a given rate)

Absolute values of vectors with normally distributed components

Normally distributed quantities operated with sum of squares

As conjugate prior distributions in Bayesian inference

Some specialized applications of probability distributions


Interquartile range test for normality of distribution


Laboratory measures of intra-assay and inter-assay CVs

As a measure of standardisation of archaeological artefacts


Ring chart, sunburst chart, and multilevel pie chart


Independent and identically distributed random variables with random sample size

Student approximation when σ value is unknown

Standard error of mean versus standard deviation

Correction for correlation in the sample


Required sample sizes for hypothesis tests


Reliability and validity of self-report measures

Guidelines for the effective wording of questions


Software for power and sample size calculations


Involuntary facades and false correlations

Social media as a source of opinion on candidates


Inference when the number of clusters is small


Minimizing cross-validation estimated squared error


Population standard deviation of grades of eight students

Standard deviation of average height for adult men

Confidence interval of a sampled standard deviation

Experiment, industrial and hypothesis testing

Relationship between standard deviation and mean


The linear interpolation between closest ranks method


Mean absolute deviation around a central point

Mean absolute deviation around the median

Median absolute deviation around a central point

Median absolute deviation around the median


Addition and multiplication by a constant

Matrix notation for the variance of a linear combination

Product of statistically dependent variables

Relations with the harmonic and arithmetic means


Basic rules for questionnaire item construction


Pearson's first skewness coefficient (mode skewness)

Pearson's second skewness coefficient (median skewness)


General equation to compute empirical quantiles


Interpretation of the size of a correlation


Standardized and unstandardized effect sizes

Correlation family: Effect sizes based on "variance explained"

Difference family: Effect sizes based on differences between means

Distribution of effect sizes based on means

Categorical family: Effect sizes for associations among categorical variables

Confidence intervals by means of noncentrality parameters

t-test for mean difference of single group or two related groups

t-test for mean difference between two independent groups

One-way ANOVA test for mean difference across multiple independent groups


Expected value of the order statistic for a uniform distribution

Expected value of the order statistic for a standard normal distribution


Blocking used for nuisance factors that can be controlled

Block a few of the most important nuisance factors


By outcome of interest (efficacy vs. effectiveness)

By hypothesis (superiority vs. noninferiority vs. equivalence)

Relative importance and observational studies


Subsequent developments and Polyak–Ruppert averaging

Example (where the stochastic gradient method is appropriate)

Subsequent developments and important issues


Examples of exponential family distributions

Normal distribution: unknown mean, known variance

Normal distribution: unknown mean and unknown variance

Moments and cumulants of the sufficient statistic

Moment-generating function of the sufficient statistic

Bayesian estimation: conjugate distributions

Hypothesis testing: uniformly most powerful tests


Uniform distribution (with two parameters)


Estimating the distribution of sample mean

Methods for improving computational efficiency

Deriving confidence intervals from the bootstrap distribution

Bias, asymmetry, and confidence intervals

Methods for bootstrap confidence intervals

Relation to other approaches to inference

Relationship to other resampling methods


Converting a single distribution to a location–scale family


Modes of convergence of random variables


Families of distributions satisfying MLR

Relation to other statistical properties

Uniformly most powerful tests: The Karlin–Rubin theorem

Lifetime analysis: Survival analysis and reliability


Estimators of the mean of u.i.d. variables


Fisherian reduction and Neyman-Pearson operational criteria

The statistical philosophy of frequentism


Influence function and sensitivity curve


Relationship between the likelihood and probability density functions

Likelihoods for mixed continuous–discrete distributions

Likelihood ratio and relative likelihood

Likelihoods that eliminate nuisance parameters

Interpretations under different foundations


Minimum-variance mean-unbiased estimator (MVUE)

Point estimate v.s. confidence interval estimate


Maximum of a discrete uniform distribution

Bias with respect to other loss functions

Example: Estimation of population variance


Stationary distribution of dose allocations

Durham and Flournoy's biased coin design

k

-in-a-row (or "transformed" or "geometric") UDD


Example of advantages of factorial experiments

Components of interaction and confounding with blocks


The p-norm in infinite dimensions and ℓp spaces


Online randomized controlled experiments

Empirical evidence that randomization makes a difference


Cumulative distribution function of order statistics

Probability distributions of order statistics

Order statistics sampled from a uniform distribution

The joint distribution of the order statistics of the uniform distribution

Order statistics sampled from an exponential distribution

Order statistics sampled from an Erlang distribution

The joint distribution of the order statistics of an absolutely continuous distribution

Application: confidence intervals for quantiles

Application: Non-parametric density estimation


Model-based analysis of randomized experiments

Frequentist inference, objectivity, and decision theory

Bayesian inference, subjectivity and decision theory


Example for when using a non-complete minimal sufficient statistic


Example of two-sided sign test for matched pairs

Example of one-sided sign test for matched pairs

Example of sign test for median of a single sample


Kruskal–Wallis one-way analysis of variance

Test for differences in ozone levels by month


Setting confidence limits for the shape of a distribution function

The Kolmogorov–Smirnov statistic in more than one dimension


Nonparametric bootstrap hypothesis testing


Completeness and Lehmann–Scheffé minimum variance


Confidence procedure for uniform location


Prior probability in statistical mechanics

Priori probability and distribution functions


Example: estimating p in a binomial distribution


Inference over exclusive and exhaustive possibilities

Parametric formulation: motivating the formal description

Formal description of Bayesian inference

In frequentist statistics and decision theory

Bioinformatics and healthcare applications


Efficient computation of the sample median


Classification of multiple hypothesis tests

Assessing whether any alternative hypotheses are true


Equal or unequal sample sizes, similar variances (1/2 < sX1/sX2 < 2)

Equal or unequal sample sizes, unequal variances (sX1 > 2sX2 or sX2 > 2sX1)

Exact method for unequal variances and sample sizes

Alternatives to the t-test for location problems

A design which includes both paired observations and independent observations

The two-sample t-test is a special case of simple linear regression


As most powerful test for small deviations

Relationship with other hypothesis tests


Computing a standard normal approximation


Estimating the population median of a symmetric population


Chi-squared test for variance in a normal population

Example chi-squared test for categorical data


Assumptions and formal statement of hypotheses

Area-under-curve (AUC) statistic for ROC curves

Proportion of concordance out of all pairs


k-fold cross-validation with validation and test set


Objective and subjective Bayesian probabilities

Personal probabilities and objective methods for constructing priors


Hypothesis testing and confidence interval


Statistics: Generalized multivariate analysis


Using cross-equation restrictions to achieve identification

Limited information maximum likelihood (LIML)


Double exponential smoothing (Holt linear)

Triple exponential smoothing (Holt Winters)


A single break in mean with a known breakpoint

Structural breaks in cointegration models


Complementary cumulative distribution function (tail distribution)

Two normally distributed random variables need not be jointly bivariate normal

Classification into multivariate normal classes


Fourier transforms on arbitrary locally compact abelian topological groups


Canonical discriminant analysis for k classes


Cross-correlation of deterministic signals

Cross-correlation of stochastic processes

Definition for wide-sense stationary stochastic process


Whitening and probabilistic canonical correlation analysis


Autocorrelation and partial autocorrelation plots

Assumptions for a stable univariate process


Estimation of the covariance matrix of the errors

Estimation of the estimator's covariance matrix

Forecasting using an estimated VAR model


Definition for complex stochastic process


Real data may need more complicated models


Continuous wavelet transforms (continuous shift and scale parameters)

Discrete wavelet transforms (discrete shift and scale parameters, continuous in time)

Multiresolution based discrete wavelet transforms (continuous in time)

Comparisons with Fourier transform (continuous-time)


Auto-correlation of stochastic processes

Properties of the autocorrelation matrix

Auto-correlation of deterministic signals

Auto-correlation of continuous-time signal

Auto-correlation of discrete-time signal


Time-varying predictors and coefficients


Dealing with uncertainty about including the intercept and deterministic time trend terms


Computing and interpreting the Durbin–Watson statistic


Autoregressive–moving-average model with exogenous inputs model (ARMAX model)


Connectivity-based clustering (hierarchical clustering)

Biology, computational biology and bioinformatics


Definitions of common terms in survival analysis

Example: Acute myelogenous leukemia survival data

Log-rank test: Testing for differences in survival in the aml data

Cox proportional hazards (PH) regression analysis

Example: Cox proportional hazards regression analysis for melanoma

Cox model using a covariate in the melanoma data

Lifetime distribution function and event density

Hazard function and cumulative hazard function

Quantities derived from the survival distribution


Joint distribution for independent variables

Joint distribution for conditionally dependent variables


Moves to standardise seasonal adjustment processes

Use of seasonally adjusted data in regressions

Shortcomings of using seasonally adjusted data


Derivation of the Kaplan–Meier estimator

Derivation as a maximum likelihood estimator


Bioinformatics advances in databases, data mining, and biological interpretation

Use of computationally intensive methods


First passage time of a 1D Brownian particle

First-hitting-time applications in many families of stochastic processes

Threshold regression: first-hitting-time regression


Reliability and availability program plan

Reliability culture / human errors / human factors

Quantitative system reliability parameters—theory

Basic reliability and mission reliability


High-throughput single cell data analysis

Bioinformatics workflow management systems


'Common' and 'special' sources of variation

Application to non-manufacturing processes


Applications to other forms of insurance

Actuarial science related to modern financial economics


Classification, pattern recognition, clustering


Conflicts of interest and unfavorable studies

Aggregation of safety data during clinical development

Participant recruitment and participation


Definition of measurement in the social sciences


Screening of drug users, mass shooters, and terrorists


Reported behavior versus actual behavior


Scientific organisations related to geostatistics


Obtaining and organizing a sampling frame


Projections, coordinate systems, and registration


Outline of types of designs for clinical studies


Design and analysis of computer experiments


The second dimension of spatial association

Mobile geospatial and hydrospatial Computing

Geographic information science and spatial analysis


Most common indicators used in official statistics


Characteristics of qualitative research interviews

Participant in qualitative research interviews


Possible characteristics of a less structured interview

Readability and validity of the unstructured interview

Perceived difficulties in comparing data

Types of use in feminist research methods


Key methodological issues of online surveys


Confirmatory versus exploratory research


Relationship between simple random sample and other methods

Distinction between a systematic random sample and a simple random sample


List of fields of application of statistics


Statistical corrections for a biased sample


International influence from the Internet


Techniques for analyzing quantitative data

Did the implementation of the study fulfill the intentions of the research design?

Final stage of the initial data analysis


World Values Paper Series: World Values Research


Basic equation regarding development of a population

Demographic analysis in institutions and organizations


Obtaining and organizing a sampling frame


The Planning Committee, collaborators and the CSES Plenary


Suffolk University Political Research Center


National Health and Nutrition Examination Survey


List of household surveys in the United States


World Association for Public Opinion Research


Household, Income and Labour Dynamics in Australia Survey


Religion and the Christchurch earthquakes

The Multi-dimensional Model of Māori and Cultural Engagement

The Pacific Identity and Wellbeing Scale


American Association for Public Opinion Research


Parallel feed-forward compensation with derivative (PFCD)


Gray-box testing assumptions for object-oriented software


Unknown constant in additive white Gaussian noise


Open-loop and closed-loop (feedback) control

Analysis techniques - frequency domain and time domain

Deterministic and stochastic systems control


Choosing an optimality criterion and robustness

Flexible optimality criteria and convex analysis

Model uncertainty and Bayesian approaches

System identification and stochastic approximation

Specifying the number of experimental runs


Sociology, cognitive science and management research


Participation by income and wealth strata

Determinants and possible explanations of stock market participation

According to the negotiation phase of financial assets

According to the geographical perspective


Direct derivation from differential equations


Barbalat's lemma and stability of time-varying systems


Local existence and uniqueness theorem simplified

Global uniqueness and maximum domain of solution


Frequentist or Bayesian approach to pattern recognition

Classification methods (methods predicting categorical labels)

Clustering methods (methods for classifying and predicting categorical labels)

Ensemble learning algorithms (supervised meta-algorithms for combining multiple learning algorithms together)

General methods for predicting arbitrarily-structured (sets of) labels

Multilinear subspace learning algorithms (predicting labels of multidimensional data using tensor representations)

Real-valued sequence labeling methods (predicting sequences of real-valued labels)

Regression methods (predicting real-valued labels)

Sequence labeling methods (predicting sequences of categorical labels)


Relation to other formulations of classical physics


Dirty, dangerous, dull, or inaccessible tasks

Home automation for the elderly and disabled


Nonlinear autoregressive exogenous model


Reciprocal gain, a.k.a. proportional band


General-purpose computing on graphics processing unit (GPGPU)


Generative (state-space) models in generalized coordinates

Generalized filtering and model parameters

Relationship to Bayesian filtering and predictive coding

Generalized filtering and Kalman filtering

Generalized filtering and particle filtering

Generalized filtering and variational Bayes

Generalized filtering and predictive coding


Terminology and theory of inferential statistics

Applied statistics, theoretical statistics and mathematical statistics

Statistics applied to mathematics or the arts


Estimators of the mean of u.i.d. variables


Kernel, image and the rank–nullity theorem

Algebraic classifications of linear transformations


Examples of systems that require pivoting


Size of spanning set is at least size of linearly independent set

Closed linear span (functional analysis)


Code assignment, maintenance and look-up


Matrix definitions involving transposition

Implementation of matrix transposition on computers

Transposes of linear maps and bilinear forms


Definition via tensor products of vector spaces


Linear dependence and independence of two vectors


General systems research and systems inquiry


Second example: ordered pairs of numbers

Complex numbers and other field extensions

Normed vector spaces and inner product spaces


Proof that every vector space has a basis


Example: Growth and decline of companies


Connection to a determinant or a product of eigenvalues

Using for solution to linear inverse problems


Affine, conical, and convex combinations


Rotation, coordinate scaling, and reflection

Singular values as semiaxes of an ellipse or ellipsoid

The columns of U and V are orthonormal bases

Singular values, singular vectors, and their relation to the SVD


Ill-posedness of the standard approximation problem


The additive and the multiplicative group of a field

Constructing fields within a bigger field

Finite fields: cryptography and coding theory


IEEE 754: floating point in modern computers

Representable numbers, conversion and rounding

Binary-to-decimal conversion with minimal number of digits

Machine precision and backward error analysis

Minimizing the effect of accuracy problems


Kernel, image and the rank–nullity theorem

Algebraic classifications of linear transformations


Rank factorization from reduced row echelon forms


Background, Present Appointments and Scientific Contributions


Other methods to determine passive elements


Static systems and general topological spaces


Multiplicative identity and the term "ring"

Function field of an irreducible algebraic variety


Nonhomogeneous systems of linear equations


Example: Continuous (C0) But Not Differentiable

Example: Finitely-times Differentiable (Ck)

Example: Differentiable But Not Continuously Differentiable (not C1)

Example: Differentiable But Not Lipschitz Continuous

Example: Smooth (C∞) but not Analytic (Cω)

Smooth functions on and between manifolds

Smooth functions between subsets of manifolds


Continuous linear time-invariant (LTI) systems

Discrete linear time-invariant (LTI) systems

Controllability in the behavioral framework


Relation of its properties to properties of the equation system


Surjections as right invertible functions

Cardinality of the domain of a surjection

Induced surjection and induced bijection


Pushforward from multiplication on Lie groups


Patching together Euclidean pieces to form a manifold

Definition of tangent space and differentiation in local coordinates

Differentiability of mappings between manifolds


Definition for unbounded operators between Banach spaces

Definition for bounded operators between Hilbert spaces

Adjoint of densely defined unbounded operators between Hilbert spaces

Counterexample where the adjoint is not densely defined


Definition via tensor products of vector spaces


Matroid rank functions as utility functions


Opposite, parallel, and antiparallel vectors

Conversion between multiple Cartesian bases

Vectors, pseudovectors, and transformations


Interest rates for different terms to maturity


Dot product, bilinear form and sesquilinear form

Computational complexity depends on parenthezation


Decompositions related to solving systems of linear equations

Decompositions based on eigenvalues and related concepts


Geometric algebra and the geometric product


Basic results, terminology, and definitions

Real and complex parts of inner products


Similar libraries (not compatible with BLAS)


Stability in numerical differential equations


Extensions of the inner and exterior products

Hypervolume of a parallelotope spanned by vectors


Quotient of a Banach space by a subspace


Tensor product of modules over a non-commutative ring


Construction for vector spaces and abelian groups

Construction for an arbitrary family of modules

Direct sum of modules with additional structure

Direct sum of modules with bilinear forms


Basis for the sum and intersection of two subspaces


IEEE 754: floating point in modern computers

Representable numbers, conversion and rounding

Binary-to-decimal conversion with minimal number of digits

Machine precision and backward error analysis

Minimizing the effect of accuracy problems


Formal definitions and algebraic properties

Axiomatic characterization and properties


Test functions for single-objective optimization

Test functions for constrained optimization

Test functions for multi-objective optimization


Minimal model of set theory is countable


Banach and Hilbert spaces and quantum mechanics

Complex-valued function of a real variable

Cardinality of sets of functions of a real variable


Relation to constraint-satisfaction problems


Programming language theory and formal methods

Computational science, finance and engineering

Social computing and human–computer interaction

Concurrent, parallel and distributed computing


Engineering and technological engineering

Operations research and management science


Relationship with astrology and esotericism

Psychology (aesthetic, creativity and intuition)


Methods based on real algebraic geometry

Response surface methodology-based approaches


Optimal vertices (and rays) of polyhedra

Ellipsoid algorithm, following Khachiyan

Current matrix multiplication time algorithm

Comparison of interior-point methods and simplex algorithms

Solvers and scripting (programming) languages


Injective, surjective and bijective functions

In the foundations of mathematics and set theory


Conservation of energy and mass in transformation

Reversible and non-reversible transformations


Horizontal alliances between logistics service providers


Differential calculus and calculus of variations


Controversy and anti-semitism allegations


Boundedness in topological vector spaces


Non-probabilistic robust optimization models

Probabilistically robust optimization models


Exact algorithms for a small number of variables


Relation with other optimization problems

Stochastic second-order cone programming


Discrete analogues of continuous mathematics

Calculus of finite differences, discrete analysis, and discrete calculus


Example 3 (Goemans–Williamson max cut approximation algorithm)


Deterministic equivalent of a stochastic problem

Monte Carlo sampling and Sample Average Approximation (SAA) Method

Example: multistage portfolio optimization


Newell and Simon: heuristic search hypothesis


Nature-inspired and metaphor-based metaheuristics


Generalization to other boundary value problems

Eigenvalue problems in several dimensions

Variations and sufficient condition for a minimum


Example from economics: Ramsey's problem of optimal saving

Dijkstra's algorithm for the shortest path problem

Faster DP solution using a different parametrization


The problem of finding the minimal value of a real-valued function of real variable

The general finite-dimensional variational inequality

The variational inequality for the Signorini problem


Analytical concepts in dynamic programming


Relation to multi-objective optimization


Mathematical programming with equilibrium constraints


Propositional satisfiability for classical logic


Other population-based metaheuristic methods


Generalizations and extensions for convexity


Solvers and scripting (programming) languages


Visualization in bi-objective problems: tradeoff curve

Visualization in high-order multi-objective optimization problems


Civil War and Reconstruction (1849–1877)

Gilded Age, Progressive Era, and World War I (1877–1929)

Great Depression, New Deal, and World War II (1929–1945)


Probabilistic methods for uncertain reasoning

Classifiers and statistical learning methods

Machine consciousness, sentience and mind


Cellular automata, DNA and the universal constructor

Scientific computing and numerical analysis


Precise statement of monotonicity properties

Precise statement of first-derivative test

Second-derivative test (single variable)


Interpretation: KKT conditions as balancing constraint-forces in state space

Regularity conditions (or constraint qualifications)


Motivation by comparison to linear programming


Iterating towards a solution of the original problem


Functions to which the theorem does not apply

Generalization to metric and topological spaces

Alternative proof of the extreme value theorem


Applications to mechanism design and auction theory

Applications to multidimensional parameter spaces

Applications to parameterized constraints


Description of the problem addressed by conjugate gradients

The preconditioned conjugate gradient method

The flexible preconditioned conjugate gradient method

Vs. the locally optimal steepest descent method

Conjugate gradient method as optimal feedback controller for double integrator

Conjugate gradient on the normal equations

Conjugate gradient method for complex Hermitian matrices


Modern formulation via differentiable manifolds

Interpretation of the Lagrange multipliers


Subgradient-projection and bundle methods


Critical point of a single variable function


Economics and partial differential equations: Minimax theorems

Operations not preserving quasiconvexity


Quantum approximate optimization algorithm


Proof 1: Non-vanishing derivatives implies not extremum

Proof 2: Extremum implies derivative vanishes


Newell and Simon: heuristic search hypothesis


Primal-dual interior-point method for nonlinear optimization


Computer language for Euclid's algorithm

An inelegant program for Euclid's algorithm

An elegant program for Euclid's algorithm

Measuring and improving the Euclid algorithms

History: Development of the notion of "algorithm"

Manipulation of symbols as "place holders" for numbers: algebra

Mechanical contrivances with discrete states

Mathematics during the 19th century up to the mid-20th century

Emil Post (1936) and Alan Turing (1936–37, 1939)

J. B. Rosser (1939) and S. C. Kleene (1943)


An example of the role of the stability assumption


Subdifferential set and the Fenchel-Young inequality


Upper and lower bounds on the complexity of problems

Problems in NP not known to be in P or NP-complete

Separations between other complexity classes


Local variables, recursion and reentrancy


Rules for calculus of finite difference operators


An analogy for understanding gradient descent

Choosing the step size and descent direction


Consistency between real and complex definitions

Uniqueness up to unitary transformations

Extension for non-Hermitian square matrices


Conservative vector fields and the gradient theorem


Machine learning and statistical classification

Information theory and signal processing

Memory allocation and deallocation algorithms


Simultaneous perturbation stochastic approximation

Extension to second-order (Newton) methods


Virtual work of forces acting on a rigid body

D'Alembert's form of the principle of virtual work


Lower bounds on the solution value, and primal-dual analysis


Selection of an individual learning method or meme to be used for a particular problem or individual

Determination of the individual learning frequency

Selection of the individuals to which individual learning is applied

Specification of the intensity of individual learning

Choice of Lamarckian or Baldwinian learning


Solving for the competitive equilibrium price


B. History of Economic Thought, Methodology, and Heterodox Approaches

C. Mathematical and Quantitative Methods

E. Macroeconomics and Monetary Economics

M. Business Administration and Business Economics • Marketing • Accounting • Personnel Economics

O. Economic Development, Innovation, Technological Change, and Growth

Q. Agricultural and Natural Resource Economics • Environmental and Ecological Economics

R. Urban, Rural, Regional, Real Estate, and Transportation Economics


One possible variation of the NM algorithm


Relationship between the primal problem and the dual problem

The strong Lagrangian principle: Lagrange duality


Other population-based metaheuristic methods


Dynamically adaptive stochastic tunneling


Convex quadratic-minimization: Minimum conditions


Search from known distributions and heterogeneous costs


Macro and micro analysis of labour markets

Personnel economics: hiring and incentives


Measures of risk aversion under expected utility theory

Implications of increasing/decreasing absolute and relative risk aversion

Limitations of expected utility treatment of risk aversion

Public understanding and risk in social activities

The behavioural approach to employment status


Empirical Evidence for the Heckscher–Ohlin model

Unemployment in international trade situations


First microwave communication experiments


Utility maximisation of perfect complements

Utility maximisation of perfect substitutes


Examples of problems solved with hydrological optimization


Humanity's economic system as a subsystem of the global environment


Discoveries of specific stochastic processes


Perfect information and imperfect information

Stochastic outcomes (and relation to other fields)

Use of game theory in project management

Artificial Intelligence and Machine Learning


An Essay on the Nature and Significance of Economic Science (1932)


Free energy and equilibria of conformational isomers

Factors contributing to the free energy of conformers

Isolation or observation of conformational isomers


Statistical ranking and selection methods (R/S)

Dynamic programming and neuro-dynamic programming


The New Palgrave Dictionary of Economics

The New Palgrave: A Dictionary of Economics

Criticisms of undue weight for heterodox approaches


Global positioning systems (GPS) and geographical information systems (GIS)


Solving overdetermined systems of equations


An Essay on the Nature and Significance of Economic Science


Classes of deterministic global optimization problems

Mixed-integer linear programming problems (MILP)

Mixed-integer non-linear programming problems (MINLP)

Deterministic global optimization solvers


History and relationships to other fields

Proprietary software with free and open-source editions


List of important publications in mathematics

The Nine Chapters on the Mathematical Art

Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa'l-muqābala

Līlāvatī, Siddhānta Shiromani and Bijaganita

Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse

Articles Publiés par Galois dans les Annales de Mathématiques

Traité des substitutions et des équations algébriques

"Sur Quelques Points d'Algèbre Homologique"

Géométrie Algébrique et Géométrie Analytique

Le théorème de Riemann–Roch, d'après A. Grothendieck

"Beweis des Satzes, daß jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält"

"Über die Anzahl der Primzahlen unter einer gegebenen Grösse"

Fourier Analysis in Number Fields and Hecke's Zeta-Functions

"Endlichkeitssätze für abelsche Varietäten über Zahlkörpern"

"Modular Elliptic Curves and Fermat's Last Theorem"

The geometry and cohomology of some simple Shimura varieties

"Le lemme fondamental pour les algèbres de Lie"

Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illi calculi genus

Philosophiae Naturalis Principia Mathematica

Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum

Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe

Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse

Produits Tensoriels Topologiques et Espaces Nucléaires

Sur certains espaces vectoriels topologiques

Mémoire sur la propagation de la chaleur dans les corps solides

Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données

On convergence and growth of partial sums of Fourier series

Disquisitiones generales circa superficies curvas

Über die Hypothesen, welche der Geometrie zu Grunde Liegen

Leçons sur la théorie génerale des surfaces et les applications géométriques du calcul infinitésimal

L'anneau d'homologie d'une représentation, Structure de l'anneau d'homologie d'une représentation

Quelques propriétés globales des variétés differentiables

"General Theory of Natural Equivalences"

Categories for the Working Mathematician

"Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"

"The consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory"

"The Independence of the Continuum Hypothesis"

"Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I"

"On sets of integers containing no k elements in arithmetic progression"

Solutio problematis ad geometriam situs pertinentis

Winning Ways for your Mathematical Plays

How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension

Essai d'une nouvelle méthode pour déterminer les maxima et les minima des formules intégrales indéfinies

"Математические методы организации и планирования производства"

"Decomposition Principle for Linear Programs"

"Полиномиальный алгоритм в линейном программировании"

The Schoolmaster's Assistant, Being a Compendium of Arithmetic both Practical and Theoretical

Set Theory: An Introduction to Independence Proofs

Topology from the Differentiable Viewpoint

Number Theory, An approach through history from Hammurapi to Legendre

An Introduction to the Theory of Numbers

Hodge Theory and Complex Algebraic Geometry II


Classification of critical points and extrema


Generalization to other boundary value problems

Eigenvalue problems in several dimensions

Variations and sufficient condition for a minimum


Optimal vertices (and rays) of polyhedra

Ellipsoid algorithm, following Khachiyan

Current matrix multiplication time algorithm

Comparison of interior-point methods and simplex algorithms

Solvers and scripting (programming) languages


MST on complete graphs with random weights


Machine learning and statistical classification

Information theory and signal processing

Memory allocation and deallocation algorithms


Comparison with the simplex algorithm for linear optimization

Computational complexity: Worst and average cases

Other optimization problems with linear constraints


Types are propositions; terms are proofs


Comparison with other shortest path algorithms


Mathematics during the Scientific Revolution


"Current-arc" data structure and discharge operation


Alternating direction method of multipliers


Setting 1 (Periodic Descent Direction Setting)


Python implementation of Edmonds–Karp algorithm


Kullback–Leibler divergence (information gain)

Channels with memory and directed information

Intelligence uses and secrecy applications

Integrated process organization of neural information


Directed graphs with nonnegative weights

Directed graphs with arbitrary weights without negative cycles

Directed graphs with arbitrary weights with negative cycles

General algebraic framework on semirings: the algebraic path problem

Shortest path in stochastic time-dependent networks


Popper's two senses of number statements


Dynamical systems and differential equations

Information theory and signal processing


Platonism as a philosophy of mathematics

Philosophical consequences of Gödel's completeness theorem


Practical optimizations and infinite graphs


Proof theory and constructive mathematics


Universal constructions, limits, and colimits


Patent controversy – can mathematics be patented?


Topological properties of the real numbers

Uniform and pointwise convergence for sequences of functions

Generalizations and related areas of mathematics


Areas of mathematics with the word algebra in their name


Combinatorial and geometric group theory


Cartesian coordinates (in a plane or space)

Finding intersections of geometric objects

Spherical and nonlinear planes and their tangents


Approaches and subfields of combinatorics


Rational function and birational equivalence

Cylindrical algebraic decomposition (CAD)

Asymptotic complexity vs. practical efficiency


Classifying spaces and homotopy operations

Obstruction theory and characteristic class


Early-to-mid 20th century: algebraic developments and the Weil conjectures

Mid-to-late 20th century: developments in modularity, p-adic methods, and beyond


Measure of countable unions and intersections


Topologies on the real and complex numbers

Defining topologies via continuous functions


As a description of the structure of space


Differential topology versus differential geometry


Deterministic processes (dynamical systems)

Stochastic processes (random dynamical systems)


Operations research and management science


Risk and portfolio management: the P world


Marginalists and the roots of neoclassical economics

Variational calculus and optimal control

Example: The effect of a corporate tax cut on wages

Adequacy of mathematics for qualitative and complicated economics

Testing predictions of mathematical economics

Mathematical economics as a form of pure mathematics


Relativity and quantum relativistic theories

List of prominent contributors to mathematical physics in the 20th century


Foundations of mathematics considerations


Classical Greece and the early Hellenistic period


Classical antiquity until the Renaissance (300 BC – 1600 AD)

Intrinsic geometry and non-Euclidean geometry (1800–1900)

Modern differential geometry (1900–2000)


Statistics, mathematics, and mathematical statistics


Other types of systems of linear equations


Important theorems in algebraic topology


Classification of finite projective spaces by geometric dimension


Differences between low-dimensional and high-dimensional topology


Mathematics and psychology before the 19th century

Mathematics and psychology in the 19th century

Sensation, perception, and psychophysics


Lists by nationality, ethnicity or religion


Modeling formalisms and graphical representations


Oil and gas industry offshore (API 14C; ISO 10418)


Systems analysis, requirements definition


Origins: from ancient Greece to the Renaissance


Academic programs in agricultural and biological systems engineering


Online blogs, podcasts, and YouTube channels


Information technology as an enabler for business process management

Reporting as an essential base for execution


Types of software development specifications


Operating system configuration management


Information Enterprise Systems Engineering (IESE)


Programming paradigms for robust collective behavior

Understanding robust collective behavior in biological systems


History of computer engineering education

Computer engineering licensing and practice

Coding, cryptography, and information protection

Computer networks, mobile computing, and distributed systems

Computer systems: architecture, parallel processing, and dependability

Integrated circuits, VLSI design, testing and CAD


Earth systems engineering and management


Structured Analysis and Design Technique


Popular concepts for a human presence in space


Transitive extensions and transitive closure

Relation types that require transitivity


Basic results, terminology, and definitions

Real and complex parts of inner products


Iterative and incremental project management

Project control and project control systems

Positive, appropriate (requisite), and negative complexity

Multilevel success framework and criteria - project success vs. project performance

Work breakdown structure and other breakdown structures


Einstein's proof by dissection without rearrangement

Trigonometric proof using Einstein's construction

Euclidean distance in other coordinate systems

Sets of m-dimensional objects in n-dimensional space


Uniformly at random on the (n − 1)-sphere


Memory institutions (museums, libraries and archives)


Multi-valued relationships and branch points

Restricting the domain of meromorphic functions


Comparison to other engineering disciplines


Exponential, logarithm, and power functions

Three-dimensional and four-dimensional rotation groups

Quaternions as the even part of Cl3,0(R)


Normed spaces as quotient spaces of seminormed spaces


Relationship to other norm-like concepts

Stronger, weaker, and equivalent seminorms


Mathematical expression of the constraint on the sides of a triangle

Example of the generalized polygon inequality for a quadrilateral

Triangle inequality for cosine similarity


Square roots of negative and complex numbers

Principal square root of a complex number

Geometric construction of the square root


Relation to Minkowski functions and open convex sets


Relationship to asymmetric and antisymmetric relations


Hurwitz's theorem (composition algebras)


Proof that every vector space has a basis


Europe and its colonies (12th-17th centuries)

Early United States (17th-19th centuries)

Ecological features, rain water absorption, and pollutant generation


Metrics valued in structures other than the real numbers


Hinduism and other religious affiliations


Defining topologies using neighborhoods of the origin

Non-Hausdorff spaces and the closure of the origin


A construction using compactly supported functions

A construction using mean values of functions


Area in terms of Cartesian coordinates of vertices

Parallelograms arising from other figures


Continuous and bounded linear functions and seminorms

Tensor products and the approximation property

Characterizations of Hilbert space among Banach spaces


Generalization with more than two functions

Hölder's inequality for increasing seminorms


Involution throughout the fields of mathematics


Sedrakyan's lemma - Positive real numbers


Classification of low-dimensional unital associative algebras over the complex numbers


Series of elements in a topological abelian group

Convergence of sequence of functions on a topological space

Series of functions on a topological abelian group


Examples of Hausdorff and non-Hausdorff spaces


Trigonometric functions on the unit circle


Construction through three noncollinear points

Inscription in or circumscription about other figures


Relation to Minkowski functions and open convex sets


Relation with classification of quadratic forms


Homogeneous polynomials of higher degree


The multiplicativity formula for degrees

Proof of the multiplicativity formula in the finite case

Proof of the formula in the infinite case


Common conditions guaranteeing gauges are seminorms

Positive homogeneity characterizes Minkowski functionals

Characterizing Minkowski functionals on star sets

Characterizing Minkowski functionals that are seminorms

Positive sublinear functions and Minkowski functionals


Metrics valued in structures other than the real numbers


Matrix norms induced by vector α- and β- norms


Separable elements and separable extensions

Separable extensions within algebraic extensions

Separability of transcendental extensions


American Revolution and the early United States


Ways of defining a locally convex topology

Finest and coarsest locally convex topology

Examples of spaces lacking local convexity


Weak topology induced by the continuous dual space

Weak topology induced by the algebraic dual


A canonical tensor product as a subspace of the dual of Bi(X, Y)

Grothendieck's representation of elements of

X

⊗
^

π

Y

Nuclear operators between locally convex spaces

Nuclear operators between Hilbert spaces


Continuous bilinear maps as a tensor product

Canonical identification of separately continuous bilinear maps with linear maps

Relation to projective tensor product and nuclear spaces

Canonical identifications of bilinear and linear maps

Dual spaces of the injective tensor product and its completion

Injective tensor product of Hilbert spaces

Space of continuously differentiable vector-valued functions

Spaces of continuous maps from a compact space


Representation as a disjoint sum of positive elements


Uniform spaces and translation-invariant uniformities

TVS completeness vs completeness of (pseudo)metrics

Preliminaries: Complete pseudometric spaces

Complete pseudometrics and complete TVSs

Properties of maps preserved by extensions to a completion

Examples and sufficient conditions for a complete TVS


Countable family of seminorms definition


Relation to maximum and minimum elements


Banach spaces in other areas of analysis

Banach spaces serving as counterexamples


Cross norms and tensor products of Banach spaces

Tensor products of locally convex topological vector spaces


Corresponce between asymmetric seminorms and convex subsets of the dual space


Pseudometrics and values on topological groups

An invariant pseudometric that doesn't induce a vector topology

Topology induced by a family of F-seminorms

Of (pseudo)metrics induced by (semi)norms

Metrically bounded sets and bounded sets


Finite-dimensional complex normed spaces

Maximum norm (special case of: infinity norm, uniform norm, or supremum norm)

Classification of seminorms: absolutely convex absorbing sets


Unit spheres and balls in Euclidean space


Symmetric, skew-symmetric and alternating forms


A linear map from a finite-dimensional space is always continuous


Relationship with other topologies on B(H)

SOT and WOT on B(X,Y) when X and Y are normed spaces

Relationships between different topologies on B(X,Y)


Kernel, image and the rank–nullity theorem

Algebraic classifications of linear transformations


Symmetric operators and self-adjoint operators


Relationships between multiple linear functionals

Equicontinuity of families of linear functionals


Some history: B*-algebras and C*-algebras


A canonical tensor product as a subspace of the dual of Bi(X, Y)

Nuclear operators between Hilbert spaces

Nuclear operators between locally convex spaces


Sufficient conditions for continuity of all positive linear functionals

Positive linear functionals (C*-algebras)


The double dual of a normed linear space


Equivalence of boundedness and continuity

Characterizations of bounded linear operators

Properties of the space of bounded linear operators


Weak topology induced by the continuous dual space

Weak topology induced by the algebraic dual


Properties of bounded self-adjoint operators

Schrödinger operators with singular potentials

Self-adjoint extensions in quantum mechanics

A symmetric operator that is not essentially self-adjoint


Relationship between common classes of operators

Trace class as the dual of compact operators


Spectral theory of ordinary differential equations

Solutions of ordinary differential equations

Limit circle and limit point for singular equations

Spectral theorem and Titchmarsh–Kodaira formula

Application to the hypergeometric equation

Generalisations and alternative approaches


Upper bounds on the spectral radius of a matrix

Upper bounds for spectral radius of a graph


Classification of points in the spectrum

Spectra of particular classes of operators


Normed spaces as quotient spaces of seminormed spaces


Spectral subspaces and projection-valued measures


Relation with the weak (operator) topology


Bounded on a neighborhood implies continuous implies bounded

Continuous and bounded but not bounded on a neighborhood

Characterizing continuous linear functionals

Properties of continuous linear functionals


Sufficient conditions for a closed graph

Closed graph theorems: When a closed graph implies continuity


Relation with the strong (operator) topology


Pseudometrics and values on topological groups

An invariant pseudometric that doesn't induce a vector topology

Topology induced by a family of F-seminorms

Of (pseudo)metrics induced by (semi)norms

Metrically bounded sets and bounded sets


Topologies of uniform convergence on arbitrary spaces of maps

𝒢-topologies on spaces of continuous linear maps

𝒢-ℋ topologies on spaces of bilinear maps


Numerical determination of the matrix polar decomposition


Relation to the axiom of choice and other statements


Generalization of the Pythagorean theorem


Mathematics vs. physics notations and definitions of inner product

Canonical norm and inner product on the dual space and anti-dual space

Constructions of the representing vector

Example in finite dimensions using matrix transformations

Relationship with the associated real Hilbert space

Canonical injections into the dual and anti-dual

Extending the bra–ket notation to bras and kets

Descriptions of self-adjoint, normal, and unitary operators


A function with infinitely many fixed points

A function that does not satisfy convexity

A function that does not satisfy closed graph

Relation to Brouwer's fixed-point theorem


Generalization to topological vector spaces


Total derivative, total differential and Jacobian matrix


Using the Fundamental Theorem of Calculus


Law of large numbers for Pettis-integrable random variables


Extension to functions defined on the whole real line


Functional derivative of iterated function

Using the delta function as a test function


Open mapping theorem (functional analysis)


Example: pointwise convergence of Fourier series

Uniform boundedness in topological vector spaces

Generalizations involving nonmeager subsets


Proof of the continuous extension theorem

Geometric Hahn–Banach (the Hahn–Banach separation theorems)

Relation to axiom of choice and other theorems


Weak convergence of orthonormal sequences


Vector-valued holomorphic functions defined in the complex plane

Holomorphic functions between Banach spaces

Holomorphic functions between topological vector spaces


Functional calculus for a bounded operator

On the assumption that f be holomorphic over an open neighborhood of σ(T)

Continuity with respect to compact convergence


Examples of continuous maps that do not have a closed graph

Closed graph theorem in point-set topology


Extensions of projection-valued measures, integrals and the spectral theorem


Blunt, pointed, flat, salient, and proper cones


Relationship between barycentric and affine coordinates


Bornology and fundamental systems of bounded sets


Relation between the two notions of absolute continuity


BV functions have only jump-type or removable discontinuities

V(·, Ω) is lower semi-continuous on L1(Ω)


Connection to real Hardy spaces on the circle

Factorization into inner and outer functions (Beurling)

Real-variable techniques on the unit circle


Continuous functions on a compact Hausdorff space


Properties of ℓp spaces and the space c0

ℓ2 is isomorphic to all separable, infinite dimensional Hilbert spaces


The p-norm in infinite dimensions and ℓp spaces


Absolutely continuous on lines (ACL) characterization of Sobolev functions


Examples of functions in the Schwartz space

Relation of Schwartz spaces with other topological vector spaces


Specialization for the canonical duality


Motivating example: Euclidean vector space

Bessel's inequality and Parseval's formula


Ring of polynomial differential operators

Ring of univariate polynomial differential operators

Ring of multivariate polynomial differential operators

A differential operator of infinite order


Mathematical formulation of quantum mechanics

The "old quantum theory" and the need for new mathematics

Mathematical structure of quantum mechanics


Extensions of the Atiyah–Singer index theorem


A proof outline of existence and uniqueness of the solution

General form of the finite element method

Scaled boundary finite element method (SBFEM)

Crystal plasticity finite element method (CPFEM)

Link with the gradient discretization method

Comparison to the finite difference method


Non-commutative algebra of generalized functions


Lindelöf hypothesis and growth of the zeta function

Analytic criteria equivalent to the Riemann hypothesis

Consequences of the generalized Riemann hypothesis

Dirichlet L-series and other number fields

Function fields and zeta functions of varieties over finite fields

Arithmetic zeta functions of arithmetic schemes and their L-factors

Montgomery's pair correlation conjecture

Arithmetic zeta functions of models of elliptic curves over number fields

Theorem of Hadamard and de la Vallée-Poussin

Arguments for and against the Riemann hypothesis


Original motivation: The Schwartz kernel theorem


Induced by a bounded disk – Banach disks

Properties of disk induced seminormed spaces


Theorems involving prevalence and shyness


Properties; examples of uncomplemented subspaces


Application: Manifolds of mappings between finite dimensional manifolds


Noncommutative C*-algebras, von Neumann algebras

Noncommutative affine and projective schemes


Counterexample in the non-Hermitian case


Group algebra of a locally compact group

The algebra Cc(G) of continuous functions with compact support

von Neumann algebras associated to groups


Definition for unbounded operators between Banach spaces

Definition for bounded operators between Hilbert spaces

Adjoint of densely defined unbounded operators between Hilbert spaces

Counterexample where the adjoint is not densely defined


Closed graph theorem (functional analysis)

Characterizations of closed graphs (general topology)

Basic properties of maps with closed graphs

Codomain not complete or (pseudo) metrizable


Definitions of test functions and distributions

Trivial extensions and independence of Ck(K)'s topology from U

Topology on the space of distributions and its relation to the weak-* topology

Extensions and restrictions to an open subset

Gluing and distributions that vanish in a set

Support in a point set and Dirac measures

Distributions of finite order with support in an open subset

Decomposition of distributions as sums of derivatives of continuous functions

Preliminaries: Transpose of a linear operator

Differential operators acting on smooth functions

Multiplication of distributions by smooth functions

Convolution of a test function with a distribution

Convolution of a smooth function with a distribution

Locally integrable functions as distributions

Structure of distributions of finite order

Tempered distributions and Fourier transform

Expressing tempered distributions as sums of derivatives

Restriction of distributions to compact sets

Using holomorphic functions as test functions


Joint distribution for independent variables

Joint distribution for conditionally dependent variables


Inverse Probability Weighted Estimator (IPWE)

Augmented Inverse Probability Weighted Estimator (AIPWE)


Publication of "Paper Mill" generated papers


As a maximization–maximization procedure


Relations between the interpolation methods

Interpolation of Sobolev and Besov spaces


Why Most Published Research Findings Are False (2005)

Male Circumcision for Reduction of HIV Infection Risk (2005)

Tamiflu clinical trial data sharing (2012)


Free open-source data mining software and applications

Proprietary data-mining software and applications


London School of Hygiene and Tropical Medicine

Faculty of Epidemiology and Population Health

Faculty of Infectious and Tropical Diseases


Stationary distribution relation to eigenvectors and simplices

Time-homogeneous Markov chain with a finite state space

Convergence speed to the stationary distribution


Function complexity and amount of training data


Behavioural revolution and new institutionalism


Reporting bias and social desirability bias


Space of smooth compactly supported functions

Direct limit of finite-dimensional spaces


Sequentially complete topological vector spaces

Properties of sequentially complete topological vector spaces


Characterizations of semi-reflexive spaces


Criminality, deviance, law and punishment

Communications and information technologies


Online learning: recursive least squares


Comparison with other forms of machine learning


Criticism of existing tools and processes


Grammatical inference by trial-and-error

Grammatical inference by genetic algorithms

Grammatical inference by greedy algorithms


Multi-relational decision tree learning (MRDTL)


The weighted nearest neighbour classifier

Condensed Nearest Neighbor for data reduction


Unsupervised learning with random forests

Relation between infinite KeRF and infinite random forest


Object categorization in computer vision

Convex vs. non-convex boosting algorithms


Constructing a classifier from the probability model


Calculations with the clustering features

Numerical issues in BIRCH clustering features


Contrast with discriminative classifiers


t-distributed stochastic neighbor embedding


Approximate non-negative matrix factorization

Convex non-negative matrix factorization

Different cost functions and regularizations


Linear algebra simulation with quantum amplitudes

Quantum machine learning algorithms based on Grover search

Classical learning applied to quantum problems


Comparison of reinforcement learning algorithms


Online dictionary learning (LASSO approach)


Inference complexity and approximation algorithms


Neutral theory and origin-fixation dynamics


Cylindrical and spherical coordinate systems

Coordinate lines/curves and planes/surfaces


Neocognitron, origin of the CNN architecture

Image recognition with CNNs trained by gradient descent


Gaussian states of interacting quantum harmonic oscillators

Reservoir computing on gate-based near-term superconducting quantum computers


Complementary cumulative distribution function (tail distribution)

Two normally distributed random variables need not be jointly bivariate normal

Classification into multivariate normal classes


Relation to other statistical machine learning methods

GANs with particularly large or small scales


The relation between Occam and PAC learning

Theorem (Occam learning implies PAC learning)

Theorem (Occam learning implies PAC learning, cardinality version)

Proof that Occam learning implies PAC learning

Improving sample complexity for common problems


Overview of VC theory in empirical processes


International Conference on Learning Representations


Comparison of different self-play techniques


Cross-disciplinary fields involving machine learning

Other machine learning methods and problems

Machine learning conferences and workshops


Comparison with Convolutional Neural Networks


Bias–variance decomposition of mean squared error


Impact of crowdsourcing on product quality

Entrepreneurs contribute less capital themselves


Conference on Neural Information Processing Systems


International Conference on Machine Learning


List of datasets for machine-learning research

List of portals suitable for multiple types of ML applications

List of portals suitable for a specific subtype of ML application


The domain of a function of several variables


Differential operators in three dimensions


Cross-covariance of stochastic processes

Cross-covariance of deterministic signals


Estimating the distribution of sample mean

Methods for improving computational efficiency

Deriving confidence intervals from the bootstrap distribution

Bias, asymmetry, and confidence intervals

Methods for bootstrap confidence intervals

Relation to other approaches to inference

Relationship to other resampling methods


Possible fundamental limitations of predictive models based on data fitting


Properties of the Karhunen–Loève transform

The Karhunen–Loève transform decorrelates the process

The Karhunen–Loève expansion minimizes the total mean square error

The Karhunen–Loève expansion has the minimum representation entropy property

Applications in signal estimation and detection

Detection of a known continuous signal S(t)

Detection of a Gaussian random signal in Additive white Gaussian noise (AWGN)


Determining the ellipse of a plane section


Hessian locally-linear embedding (Hessian LLE)

Modified Locally-Linear Embedding (MLLE)

t-distributed stochastic neighbor embedding

Topologically constrained isometric embedding

Uniform manifold approximation and projection


Y-DNA haplotypes from genealogical DNA tests


Comparison of scores measured on different scales: ACT and SAT

Percentage of observations below a z-score

Cluster analysis and multidimensional scaling

Relative importance of variables in multiple regression: Standardized regression coefficients

Standardizing in mathematical statistics


Addition and multiplication by a constant

Matrix notation for the variance of a linear combination

Product of statistically dependent variables

Relations with the harmonic and arithmetic means


Consistency between real and complex definitions

Uniqueness up to unitary transformations

Extension for non-Hermitian square matrices


Computing PCA using the covariance method

Derivation of PCA using the covariance method

Discriminant analysis of principal components


Duality formula for variational inference

Relation to other quantities of information theory

Principle of minimum discrimination information

Relationship to other probability-distance measures


In terms of PMFs for discrete distributions

In terms of PDFs for continuous distributions

Relation to conditional and joint entropy

Bayesian estimation of mutual information


Relation with equivalence, congruence, and isomorphism

Set equality based on first-order logic with equality

Set equality based on first-order logic without equality


Linear MMSE estimator for linear observation process

Special Case: vector observation with uncorrelated noise


Covariance matrix as a parameter of a distribution


Eigenvalues and eigenvectors of matrices

Eigenvalues and the characteristic polynomial

Eigenspaces, geometric multiplicity, and the eigenbasis for matrices

Diagonalization and the eigendecomposition

Three-dimensional matrix example with complex eigenvalues

Matrix with repeated eigenvalues example

Eigenvalues and eigenfunctions of differential operators

Associative algebras and representation theory

Eigenvalues of geometric transformations


Auto-covariance matrix of real random vectors

Cross-covariance matrix of real random vectors

Cross-covariance sesquilinear form of random vectors in a real or complex Hilbert space

In meteorological and oceanographic data assimilation


Independent and identically distributed random variables

Definition for more than two random variables


Other measures of dependence among random variables

Uncorrelatedness and independence of stochastic processes


Fundamental theory of matrix eigenvectors and eigenvalues

Useful facts regarding eigendecomposition


Producers with increasing returns to scale: marginal cost pricing


Factor analysis and work on intelligence


Short-term loans to good government bodies

Mid- and long-term loans to good government bodies

Short-term loans to blue-chip corporations

Mid- and long-term loans to blue-chip corporations


Capital structure substitution - asset pricing formula


Allegations of UNDP resources used by Hamas

Russia UNDP GEF Project Corruption Scandal


Maturation of the electrical properties of the action potential

Taxonomic distribution and evolutionary advantages


Mechanisms for propagating action potentials


Notation of floating-point number system

Roundoff error under different rounding rules

Calculating roundoff error in IEEE standard

Measuring roundoff error by using machine epsilon

Roundoff error caused by floating-point arithmetic


Engineering applications of neuroscience


2021 Human Development Index (2022 report)


Graphics and graphical user interface programming


Similar libraries (not compatible with BLAS)


Spearman's two-factor theory of intelligence


Computing PCA using the covariance method

Derivation of PCA using the covariance method

Discriminant analysis of principal components


Preliminaries: Gradient descent for eigenvalue problems

Locking of previously converged eigenvectors

Partial Principal component analysis (PCA) and Singular Value Decomposition (SVD)


Simultaneous approximation of extreme eigenvalues


United States military selection in World War I

IQ testing and the eugenics movement in the United States

Test bias or differential item functioning

Non-shared family environment and environment outside the family


Connections to other applications, programming languages, and services


Fabrication and operation of microarrays


Approximate non-negative matrix factorization

Convex non-negative matrix factorization

Different cost functions and regularizations


Critical exponents and universality classes


Standard contents of a contingency table

Cramér's V and the contingency coefficient C


Chi-squared test for variance in a normal population

Example chi-squared test for categorical data


Notes on Semidefinite Programming Relaxation


Examples of software produced with Free Pascal


Ill-posedness of the standard approximation problem


Exact solution of exponential complexity


Multilinear principal component analysis


As an extension of correspondence analysis

Multiple correspondence analysis and principal component analysis


Multilinear independent component analysis

Multilinear linear discriminant analysis

Multilinear canonical correlation analysis


Application to Ensemble Weather Forecasts

Application to Ensemble Climate Model Projections


Journal of the American Statistical Association


Canterbury College, University of New Zealand, 1873–1960

Independence of the University of Canterbury, 1961–2010

Staff reductions and academic freedom issues


Open letter for the people of Gaza (2014)

February 2020 letter dismissing lab-leak theory

Retracted papers and scientific controversies

Andrew Wakefield and the MMR vaccine (1998)

Metastudy on the use of hydroxychloroquine and chloroquine (2020)

Covid Commission head pushed US lab origin conspiracy theory (2022)


Aristotelian natural philosophy (400 BC–1100 AD)

Newton and the scientific revolution (1600–1800)


Definition and the Four Essential Freedoms of Free Software

Binary blobs and other proprietary software


People associated with the social sciences


Philosophy of mind and cognitive science


General principles of the physical sciences


Description of units and physical quantities


Society for Industrial and Applied Mathematics


Constitutive equations in electromagnetism and related areas

Without magnetic or dielectric materials

Thermoelectric and electromagnetic properties of matter


Fundamental notions of first-order model theory

Compactness and the Löwenheim-Skolem theorem

Connections to related branches of mathematical logic


Susan Wojcicki and going mainstream (2014–2018)

YouTube as a tool to promote conspiracy theories and far-right content

Handling of COVID-19 pandemic and other misinformation


Concrete and abstract thought in psychology


Journal of Computational and Graphical Statistics


Connection with principal component analysis


Contemporary issues in methodology and practice


Linearization of a multivariable function


Energy efficiency relating to aircraft jet engines


A popular but inaccurate analogy for chaos


Slope–intercept form or Gradient-intercept form


Difficulty in calculating the derivative of a function

Failure of the method to converge to the root

Slow convergence for roots of multiplicity greater than 1

Proof of quadratic convergence for Newton's iterative method

Multiplicative inverses of numbers and power series

Numerical verification for solutions of nonlinear equations


Difference operator and difference equations

Solving linear recurrence relations with constant coefficients

Solving first-order non-homogeneous recurrence relations with variable coefficients

Solving general homogeneous linear recurrence relations

Solving first-order rational difference equations

Stability of linear higher-order recurrences

Stability of linear first-order matrix recurrences

Stability of nonlinear first-order recurrences


Absolute versus statistical reversibility


Rationalizability and iterated dominance


Relation to the necessary truths and contingent truths


Kinematics of a particle trajectory in a non-rotating frame of reference

Particle trajectories in cylindrical-polar coordinates

Point trajectories in a body moving in the plane

Point trajectories in body moving in three dimensions


Proof using the Kakutani fixed-point theorem

Alternate proof using the Brouwer fixed-point theorem


Potential functions of one active variable

Potential functions of two active variables


Relation to randomness and probability in quantum mechanics


Prior probability in statistical mechanics

Priori probability and distribution functions