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.

Modelo predictivo del fracaso de la terapia eléctrica en la fibrilación auricular paroxística / Predictive model of electrical therapy failure in paroxysmal atrial fibrillation

Introducción: La fibrilación auricular es la arritmia recurrente más habitual en la práctica clínica. Su prevalencia se multiplica en la población actual y tiene diferentes causas fisiopatológicas que la convierten en una pandemia mundial. Objetivos: Diseñar un modelo predictivo de fracaso de la terapia eléctrica en pacientes con fibrilación auricular paroxística. Métodos: Se realizó un estudio de casos y controles, con 33 casos y 66 controles. Variables predictoras: edad, fracción de eyección ≤ 40 por ciento, volumen de aurícula izquierda ≥ 34 mL/m2. A partir de la regresión logística se obtuvo un modelo en el que fueron incluidos el valor predictivo positivo, valor predictivo negativo, la sensibilidad y especificidad. Resultados: Los factores de riesgo predictores fueron: edad ≥ 55 años (p= 0,013; odds ratio (OR)= 3,58; intervalo de confianza -IC- 95 por ciento: 1,33-9,67); la fracción de eyección del ventrículo izquierdo (FEVI) ≤ 40 por ciento se observó en 20 pacientes (22,7 por ciento) (p= 0,004; OR= 4,45; IC95 por ciento: 1,54-12,8); presión de aurícula izquierda elevada, volumen de aurícula izquierda elevado (p= 0,004; OR= 3,11; IC95 por ciento: 1,24-8,77), según el modelo de regresión logística. Se realizó la validación interna por división de datos; se confirmó que el modelo pronostica bien los que van a tener éxito en el resultado terapéutico. Conclusiones: El modelo predictivo elaborado está compuesto por los predictores edad > 55 años, FEVI; volumen de aurícula izquierda; presenta un buen ajuste y poder discriminante, sobre todo valor predictivo positivo(AU)

Introduction: Atrial fibrillation is the most common recurrent arrhythmia in clinical practice. Its prevalence is multiplying in the current population and has different pathophysiological causes that make it a global pandemic. Objectives: To design a predictive model for failure of electrical therapy in patients with paroxysmal atrial fibrillation. Methods: A case-control study was carried out with 33 cases, and 66 controls. Predictor variables: age, ejection fraction ≤ 40 percent, left atrial volume ≥ 34 mL/m2. From logistic regression, a model was obtained in which the positive predictive value, negative predictive value, sensitivity and specificity were included. Results: The predictive risk factors were: age ≥ 55 years (p= 0.013; odds ratio (OR)= 3.58; 95 percent confidence interval -CI-: 1.33-9.67); left ventricular ejection fraction (LVEF) ≤ 40 percent was observed in 20 patients (22.7 percent) (p= 0.004; OR= 4.45; 95 percent CI: 1.54-12.8); elevated left atrial pressure, elevated left atrial volume (p= 0.004; OR= 3.11; 95 percent CI: 1.24-8.77), according to the logistic regression model. Internal validation was carried out by data division; It was confirmed that the model predicts very well those who will be successful in the therapeutic result. Conclusions: The predictive model developed is composed of the predictors age > 55 years, LVEF; left atrial volume; It presents a good fit and discriminating power, especially positive predictive value(AU)

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.

Contribution of Attentional Networks to Basic Arithmetic Achievement in School-age Children / Contribución de las redes atencionales al rendimiento en aritmética básica en niños de edad escolar

Nowadays, there is still debate about the origin of arithmetic disabilities. Two predominant hypotheses have been developed in this regard: 1) difficulties in arithmetic appear due to a deficit in domain-specific skills (numerical skills) and 2) there is a strong relationship between the development of the domain-general processes and arithmetical skills. In this study we explore the specific contribution of orienting and executive control attentional networks to individual variability in basic mental arithmetic. Participants were elementary school children, from second to third grade, with and without difficulties in basic mental arithmetic. Results provide support for the two proposed hypotheses. Children with difficulties in arithmetic showed difficulties both in numerical skills and in attentional networks. These findings suggest that arithmetic difficulties may be associated with a heterogeneous combination of deficits, including difficulties in attentional functioning.

En la actualidad existe aún debate sobre el origen de las dificultades en aritmética. Se han desarrollado dos hipótesis predominantes al respecto: 1) las dificultades en aritmética aparecen por un déficit en las capacidades de dominio específico (capacidades numéricas) y 2) hay una estrecha relación entre el desarrollo de los procesos de dominio general y las habilidades aritméticas. En este estudio se explora la contribución específica de las redes de atención ejecutiva y orientación atencional a la explicación de la varianza en aritmética básica. Participaron niños de segundo y tercer grado de primaria, con y sin dificultades en aritmética básica. Nuestros resultados ofrecen sustento a las dos hipótesis propuestas, ya que los niños con dificultades en aritmética mostraron dificultades tanto en las capacidades numéricas como en las redes atencionales. Estos hallazgos sugieren que las dificultades en aritmética pudieran estar asociadas a una combinación heterogénea de déficits, incluyendo dificultades en el funcionamiento atencional.

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)

The Moderating Role of Mathematical Skill Level when Using Curricular Methods to Learn Multiplication Tables / El papel moderador del nivel de habilidad matemática al emplear métodos curriculares para la enseñanza de las tablas de multiplicación

The present study explores the effect of two instructional methods for children with different levels of mathematical skills. One of these methods uses a conventional approach to learning multiplication and emphasizes the memorization of all arithmetic facts, whereas the other method is based on psychological principles and combines: a) the memorization of a small subset of problems aided by color cues and a portable time-table, with b) the use of single-step rules. One hundred and sixty second-grade children (aged 7-8) received instruction in one of these approaches – either the conventional method or the memory and rules method (M&R) – over the course of 6 months as part of their normal school education. Moderation analysis revealed that children with poor mathematical skills in the conventional group scored significantly better than their counterparts in the M&R group, whereas a significant advantage was observed in the M&R group for those children with strong mathematical skills. (AU)

El presente estudio explora el efecto de dos métodos de enseñanza de la multiplicación simple en alumnos de primaria con diferentes niveles de habilidades matemáticas. Un método se basa en el enfoque convencional para el aprendizaje de las multiplicaciones que enfatiza la memorización de todas ellas, mientras que el otro se basa en principios psicológicos y combina: a) la memorización de un pequeño subconjunto de multiplicaciones auxiliadas con claves de color y una tabla portátil con las multiplicaciones con b) el uso de reglas de un solo paso. Ciento sesenta niños y niñas de segundo de primaria (de 7 a 8 años) recibieron instrucción en uno de estos métodos, ya fuera el convencional o el método de memorización y reglas (M&R), durante 6 meses como parte de su educación escolar normal. El análisis de moderación reveló que los niños con habilidades matemáticas bajas en el grupo convencional obtuvieron puntuaciones significativamente mejores que sus pares en el grupo M&R, mientras que se observó una ventaja significativa en el grupo M&R para aquellos niños con altas habilidades matemáticas. (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.

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.

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.