Solving olympiad geometry without human demonstrations.

Proving mathematical theorems at the olympiad level represents a notable milestone in human-level automated reasoning1-4, owing to their reputed difficulty among the world's best talents in pre-university mathematics. Current machine-learning approaches, however, are not applicable to most mathematical domains owing to the high cost of translating human proofs into machine-verifiable format. The problem is even worse for geometry because of its unique translation challenges1,5, resulting in severe scarcity of training data. We propose AlphaGeometry, a theorem prover for Euclidean plane geometry that sidesteps the need for human demonstrations by synthesizing millions of theorems and proofs across different levels of complexity. AlphaGeometry is a neuro-symbolic system that uses a neural language model, trained from scratch on our large-scale synthetic data, to guide a symbolic deduction engine through infinite branching points in challenging problems. On a test set of 30 latest olympiad-level problems, AlphaGeometry solves 25, outperforming the previous best method that only solves ten problems and approaching the performance of an average International Mathematical Olympiad (IMO) gold medallist. Notably, AlphaGeometry produces human-readable proofs, solves all geometry problems in the IMO 2000 and 2015 under human expert evaluation and discovers a generalized version of a translated IMO theorem in 2004.

The sums are larger than their natural number addends: Relation to operands understanding predicts growth in arithmetic/algebraic problem solving.

The relation to operands (RO) principles describe the relation between operands and answers in arithmetic problems (e.g., the sum is always larger than its positive addends). Despite being a fundamental property of arithmetic, its empirical relation with arithmetic/algebraic problem solving has seldom been investigated. The current longitudinal study aimed to address this issue. Two-hundred-and-two Chinese fifth graders (57% boys) were assessed on their RO understanding. Their arithmetic/algebraic problem solving were assessed multiple times over 2 years. Results from latent growth curve modeling showed that RO understanding predicted the growth in arithmetic/algebraic problem solving when other known predictors of arithmetic/algebraic problem solving were controlled. The findings highlight the role of RO understanding in children's mathematical development. Interventions should be developed to enhance children's RO understanding. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Inertial and geometrical effects of self-propelled elliptical Brownian particles.

Active particles that self-propel by transforming energy into mechanical motion represent a growing area of research in mathematics, physics, and chemistry. Here we investigate the dynamics of nonspherical inertial active particles moving in a harmonic potential, introducing geometric parameters which take into account the role of eccentricity for nonspherical particles. A comparison between the overdamped and underdamped models for elliptical particles is performed. The model of overdamped active Brownian motion has been used to describe most of the basic aspects of micrometer-sized particles moving in a liquid ("microswimmers"). We consider active particles by extending the active Brownian motion model to incorporate translation and rotation inertia and account for the role of eccentricity. We show how the overdamped and the underdamped models behave in the same way for small values of activity (Brownian case) if eccentricity is equal to zero, but increasing eccentricity leads the two dynamics to substantially depart from each other-in particular, the action of a torque induced by external forces, induced a marked difference close to the walls of the domain if eccentricity is high. Effects induced by inertia include an inertial delay time of the self-propulsion direction from the particle velocity, and the differences between the overdamped and underdamped systems are particularly evident in the first and second moments of the particle velocities. Comparison with the experimental results of vibrated granular particles shows good agreement and corroborates the notion that self-propelling massive particles moving in gaseous media are dominated by inertial effects.

Effects of Perinatal Stroke on Executive Functioning and Mathematics Performance in Children.

The goal of this study was to examine executive functioning, math performance, and visuospatial processing skills of children with perinatal stroke, which have not been well explored in this population. Participants included 18 children with perinatal stroke (aged 6-16 years old) and their primary caregiver. Each child completed standardized tests of executive function and visuospatial processing skills, Intelligence Quotient (IQ), and math achievement. Performance on executive function, IQ, math, and visuospatial processing tests was significantly lower in children with perinatal stroke when compared to normative means. Poorer inhibitory control was associated with worse math performance. Increased age at testing was associated with better performance on visuospatial ability (using standardized scores), and females performed better than males on a test of inhibitory control. Children with perinatal stroke displayed a range of neuropsychological impairments, and difficulties with executive function (inhibition) may contribute to math difficulties in this population.

Active math and grammar learning engages overlapping brain networks.

We here demonstrate common neurocognitive long-term memory effects of active learning that generalize over course subjects (mathematics and vocabulary) by the use of fMRI. One week after active learning, relative to more passive learning, performance and fronto-parietal brain activity was significantly higher during retesting, possibly related to the formation and reactivation of semantic representations. These observations indicate that active learning conditions stimulate common processes that become part of the representations and can be reactivated during retrieval to support performance. Our findings are of broad interest and educational significance related to the emerging consensus of active learning as critical in promoting good long-term retention.

Diagnóstico matemático del infarto agudo de miocardio y de la falla cardiaca mediante la entropía proporcional / Mathematical Diagnosis of Acute Myocardial Infarction and Failure Using Proportional Entropy

Introducción: Las teorías físicas y matemáticas han permitido el desarrollo de nuevas metodologías diagnósticas de la dinámica cardiaca. Entre estas se encuentra la evaluación de las proporciones de la entropía proporcional para diferenciar la normalidad de la enfermedad cardiaca, aunque su capacidad diagnóstica debe comprobarse en escenarios clínicos críticos específicos, como en la falla cardiaca y el infarto agudo de miocardio. Objetivo: Describir evaluaciones diagnósticas de la dinámica cardiaca en pacientes con infarto agudo de miocardio o falla cardiaca aguda. Métodos: En un estudio a doble ciegos con 20 Holter, 5 normales, 8 con falla cardiaca aguda y 7 con infarto agudo de miocardio, se aplicó un método fundamentado en las proporciones de la entropía tomando los valores máximos y mínimos de la frecuencia cardiaca y el número total de latidos por hora, en un mínimo de 18 horas, generando un atractor numérico. Se evaluó cada dinámica con base en la entropía y sus proporciones. Finalmente, se comparó la precisión diagnóstica del método matemático con respecto al diagnóstico clínico convencional. Resultados: Se diferenciaron matemáticamente los casos normales y patológicos mediante la evaluación en 18 horas con el método descrito, encontrando valores de sensibilidad y especificidad del 100 por ciento y un coeficiente Kappa de uno, indicando una concordancia diagnóstica perfecta del método matemático con respecto al diagnóstico clínico. Conclusiones: Las proporciones de la entropía permiten establecer diagnósticos objetivos de la dinámica cardiaca, diferenciando matemáticamente dinámicas normales de aquellas que presentan infarto agudo de miocardio y falla cardiaca aguda(CU)

Introduction: Physical and mathematical theories have allowed the development of new diagnostic methodologies of cardiac dynamics, as one based on the evaluation of entropy proportions to differentiate normality from cardiac disease, although its diagnostic capacity must be yet determined in specific critical scenarios as acute heart failure and acute myocardial infarction Objective: To describe diagnostic evaluations of cardiac dynamics in patients diagnosed with acute myocardial infarction or acute heart failure. Methods: A blind study was developed with 20 Holter registries; 5 normal, 8 with acute cardiac failure and 7 with acute myocardial infarction. Then, a method based on the proportions of the entropy of the numerical attractors was applied. The maximum and minimum values of the heart rate and the total number of beats per hour were taken for at least 18 hours, with which numerical attractors were generated, which measure the probability of consecutive heart rate pairs. An evaluation of all dynamics was made based on the entropy and its proportions. Finally, a comparison between the diagnostic precision of the mathematical method with respect to the conventional clinical diagnosis was performed. Results: Normal cases were mathematically differentiated from the pathological ones through the evaluation of Holter registries for 18 hours, achieving values of sensitivity and specificity of 100 percent as well as a Kappa coefficient of 1, indicating a perfect diagnostic concordance between the mathematical method to diagnose the cardiac dynamics with respect to the clinical diagnosis. Conclusions: The proportions of entropy allow to establish objective diagnoses of cardiac dynamics, mathematically differentiating normal dynamics from those with acute myocardial infarction and with acute cardiac failure(AU)

Examining the effect of perceived performance-contingent gains, losses and errors on arithmetic.

Gains and losses have previously been found to differentially modulate Executive Functions and cognitive performance depending on performance contingency. Following recent findings suggesting that random gains and losses modulate arithmetic performance, the current study aimed to investigate the effect of perceived performance-contingent gains and losses on arithmetic performance. In the current study, an arithmetic equation judgment task was administered, with perceived performance-contingent gain, loss, and error feedback presented upon each trial. The results from two experiments suggest that when perceiving gain and loss as performance-contingent, the modulation of arithmetic performance, seen previously under random contingency conditions was entirely eliminated. In addition, another type of feedback was examined in the context of an arithmetic task: post-error adjustments. When performance after error feedback was compared to performance after other aversive performance feedback such as loss signals, only errors, but not other aversive feedback, modulated performance in the subsequent trial. These findings further extend the knowledge regarding the influence of gain and loss situations, as well as errors, on arithmetic performance.

A mathematician's view of the unreasonable ineffectiveness of mathematics in biology.

This paper discusses, from a mathematician's point of view, the thesis formulated by Israel Gelfand, one of the greatest mathematicians of the 20th century, and one of the pioneers of mathematical biology, about the unreasonable ineffectiveness of mathematics in biology as compared with the obvious success of mathematics in physics. The author discusses the limitations of the mainstream mathematics of today when it is used in biology. He suggests that some emerging directions in mathematics have potential to enhance the role of mathematics in biology.

Estimation of the marbling development pattern in Japanese Black cattle by using serial ultrasound measurement data.

The objective of this study was to develop mathematical equations for describing the change in marbling in Japanese Black steers using longitudinal measurements. Serial ultrasound measurements were taken at 14, 16, 20, and 26 months of age and analyzed using an image analysis software. The longitudinal marbling measurements from the ultrasound images and carcasses were fitted into a nonlinear logistic curve. Data used for the analysis consisted of 749 steers that converged in nonlinear curve fitting and showed reasonable estimated parameters of the logistic curves. The average predicted mature beef marbling score (BMS) and maturation rate were 6.26 and 0.353, respectively, and the average maturity levels at 24 months of age were 83.9%. The heritability estimates for the predicted maturity traits were moderate, indicating that these traits may have potential for genetic improvement. There was a negative relationship between the expected progeny differences between carcass BMS and maturity traits, suggesting that genetic improvement by carcass BMS may lead to the selection of bulls with late maturity for marbling. The results indicate that ultrasound and model building for marbling can be useful tools to correctly select candidate bulls with high marbling in the early fattening period.

A Fast Algorithm for Computing the Fourier Spectrum of a Fractional Period.

Directly computing Fourier power spectra at fractional periods of real sequences can be beneficial in many digital signal processing applications. In this article, we present a fast algorithm to compute the fractional Fourier power spectra of real sequences. For a real sequence of length of m = n l , we may deduce its congruence derivative sequence with a length of l. The discrete Fourier transform of the original sequence can be calculated by the discrete Fourier transform of the congruence derivative sequence. The relation of discrete Fourier transforms between the two sequences may derive the special features of Fourier power spectra of the integer and fractional periods for a real sequence. It has been proved mathematically that after calculating the Fourier power spectrum (FPS) at an integer period, the Fourier power spectra of the fractional periods related this integer period can be easily represented by the computational result of the FPS at the integer period for the sequence. Computational experiments using a simulated sinusoidal data and protein sequence show that the computed results are a kind of Fourier power spectra corresponding to new frequencies that cannot be obtained from the traditional discrete Fourier transform. Therefore, the algorithm would be a new realization method for discrete Fourier transform of the real sequence.

Effect of Mental Calculation and Number Comparison on a Manual-Pointing Movement.

The present study aimed at examining the effect of mental calculation and number comparison on motor performance measured as the movement time of a fast manual-pointing movement. Three experiments, involving a total number of 65 undergraduate subjects, examined the effect of mental subtraction (complex) and, respectively, of (a) mental addition (simple or complex), (b) mental multiplication (simple or complex), and (c) the comparison of dot sets and number comparison. Each number was written in Arabic. The movement times were analyzed by using a multilevel linear mixed-effect model. The results showed significant improvement of manual-pointing movement performance only after the complex calculations and after number comparison. Possible implication of attentional mechanisms specific to this arithmetical activity is further discussed.

Meaning to multiply: Electrophysiological evidence that children and adults treat multiplication facts differently.

Multiplication tables are typically memorized verbally, with fluent retrieval leading to better performance in advanced math. Arithmetic development is characterized by strategy shifts from procedural operations to direct fact retrieval, which would not necessitate access to the facts' conceptual meaning. This study tested this hypothesis using a combination of event related brain potentials (ERP) and behavioral measures with 3rd-5th grade children and young adults. Participants verified the solutions to simple multiplication problems (2 × 3 = 6 or = 7) and the semantic fit of word-picture pairs, separately. Children showed an N400 effect to multiplication solutions with larger (more negative) amplitude for incorrect than correct solutions, reflecting meaning-level processing. A similar ERP response was observed in the word-picture verification task, with larger negative amplitude for word-picture pairs that were semantically mismatched compared to matched. In contrast, adults showed a P300 response for correct solutions, suggesting that they treated these solutions as potential targets in over-rehearsed mathematical expressions. This P300 response was specific to math fact processing, as the word-picture verification task elicited a classic N400 in adults. These ERP findings reveal an overlooked developmental transition that occurs after fifth grade, and speak to theories of arithmetic that have been based primarily on adult data.

Mathematics, executive functioning, and visual-spatial skills in Chinese kindergarten children: Examining the bidirectionality.

This study examined the bidirectional relationships among Chinese children's mathematics, executive functioning, and visual-spatial skills during their transition from kindergarten to primary school. Participants were 172 Cantonese-speaking Hong Kong children (mean age at Time 1 = 62.75 months; 88 male) and their parents. At Time 1 (kindergarten K3), children were administered the measures of mathematics (calculation and applied problems), executive functioning (working memory and inhibitory control), and visual-spatial skills. They were reassessed on these measures at Time 2 (primary 1) 1 year later. Results from the cross-lagged panel model showed that, controlling for child age, gender, and family socioeconomic status, children's visual-spatial skills at Time 1 were significantly predictive of their mathematics at Time 2 and children's executive functioning and visual-spatial skills reciprocally predicted each other across times. However, children's mathematics at Time 1 were not predictive of their executive functioning or visual-spatial skills at Time 2. The findings highlight the desirability of improving children's executive functioning and visual-spatial skills to promote their mathematical performance during the formal school transition.

Children's discrete proportional reasoning is related to inhibitory control and enhanced by priming continuous representations.

Children can successfully compare continuous proportions as early as 4 years of age, yet they struggle to compare discrete proportions at least to 10 years of age, especially when the discrete information is misleading. This study examined whether inhibitory control contributes to individual differences in discrete proportional reasoning and whether reasoning could be enhanced by priming continuous information. A total of 49 second-graders completed two tasks. In the Hearts and Flowers (H&F) task, a measure of inhibition, children pressed on either the corresponding or opposite side, depending on the identity of the displayed figure. In the Spinners task, a measure of proportional reasoning, children chose the spinner with the proportionally larger red area across continuous and two discrete formats. In the discrete adjacent format, the continuous stimuli were segmented into sections, which could be compatible with the proportional information or misleading; the discrete mixed format interspersed the colored sections from the discrete adjacent conditions. Finally, two priming groups were formed. Children who saw the continuous format immediately before the discrete adjacent format formed the continuous priming group (n = 26). Children who saw the discrete mixed format immediately before the discrete adjacent format formed the discrete priming group (n = 23). Our results showed that children who performed better on the H&F task also had better performance on the discrete counting misleading trials. Furthermore, children in the continuous priming group outperformed children in the discrete priming group, specifically in contexts where discrete information was misleading. These results suggest that children's proportional reasoning may be improved by fostering continuous representations of discrete stimuli and by enhancing inhibitory control skills.

Examining the mutual relations between language and mathematics: A meta-analysis.

This study presents a meta-analysis of the relation between language and mathematics. A moderate relation between language and mathematics was found in 344 studies with 393 independent samples and more than 360,000 participants, r = .42, 95% CI [.40, .44]. Moderation and partial correlation analyses revealed the following: (a) more complicated language and mathematics skills were associated with stronger relations between language and mathematics; after partialing out working memory and intelligence, rapid automatized naming showed the strongest relation to numerical knowledge; (b) the relation between language and mathematics was stronger among native language speakers than among second-language learners, but this difference was not found after partialing out working memory and intelligence; (c) working memory and intelligence together explained over 50% of the variance in the relation between language and mathematics and explained more variance in such relations involving complex mathematics skills; (d) language and mathematics predicted the development of one another even after controlling for initial performance. These findings suggest that we may use language as a medium to communicate, represent, and retrieve mathematics knowledge as well as to facilitate working memory and reasoning during mathematics performance and learning. With development, the use of language to retrieve mathematics knowledge may be more important for foundational mathematics skills, which in turn further strengthens linguistic thought processes for performing more advanced mathematics tasks. Such use of language may boost the mutual effects of cognition and mathematics across development. (PsycInfo Database Record (c) 2020 APA, all rights reserved).

Symbolic fractions elicit an analog magnitude representation in school-age children.

A fundamental question about fractions is whether they are grounded in an abstract nonsymbolic magnitude code similar to that postulated for whole numbers. Mounting evidence suggests that symbolic fractions could be grounded in mechanisms for perceiving nonsymbolic ratio magnitudes. However, systematic examination of such mechanisms in children has been lacking. We asked second- and fifth-grade children (prior to and after formal instructions with fractions, respectively) to compare pairs of symbolic fractions, nonsymbolic ratios, and mixed symbolic-nonsymbolic pairs. This paradigm allowed us to test three key questions: (a) whether children show an analog magnitude code for rational numbers, (b) whether that code is compatible with mental representations of symbolic fractions, and (c) how formal education with fractions affects the symbolic-nonsymbolic relation. We examined distance effects as a marker of analog ratio magnitude processing and notation effects as a marker of converting across numerical codes. Second and fifth graders' reaction times and error rates showed classic distance and notation effects. Nonsymbolic ratios were processed most efficiently, with mixed and symbolic notations being relatively slower. Children with more formal instruction in symbolic fractions had a significant advantage in comparing symbolic fractions but had a smaller advantage for nonsymbolic ratio stimuli. Supplemental analyses showed that second graders relied on numerator distance more than holistic distance and that fifth graders relied on holistic fraction magnitude distance more than numerator distance. These results suggest that children have a nonsymbolic ratio magnitude code and that symbolic fractions can be translated into that magnitude code.

Insights from Microbial Transition State Theory on Monod's Affinity Constant.

Microbial transition state theory (MTS) offers a theoretically explicit mathematical model for substrate limited microbial growth. By considering a first order approximation of the MTS equation one recovers the well-known Monod's expression for growth, which was regarded as a purely empirical function. The harvest volume of a cell as defined in MTS theory can then be related to the affinity concept, giving a new physical interpretation to it, and a new way to determine its value. Consequences of such a relationship are discussed.