Automation of Diagrammatic Proofs in Mathematics

Theorems in automated theorem proving are usually proved by logical formal proofs. However, there is a subset of problems which can also be proved in a more informal way by the use of geometric operations on diagrams, so called diagrammatic proofs. Insight is more clearly perceived in these than in the corresponding logical proofs: they capture an intuitive notion of truthfulness that humans find easy to see and understand. The proposed research project is to identify and ultimately automate this diagrammatic reasoning on mathematical theorems. The system that we are in the process of implementing will be given a theorem and will (initially) interactively prove it by the use of geometric manipulations on the diagram that the user chooses to be the appropriate ones. These operations will be the inference steps of the proof. The constructive !-rule will be used as a tool to capture the generality of diagrammatic proofs. In this way, we hope to verify and to show that the diagra..

Searching for a Solution to Program Verification=Equation Solving in CCS

International audienceUnder non-exponential discounting, we develop a dynamic theory for stopping problems in continuous time. Our framework covers discount functions that induce decreasing impatience. Due to the inherent time inconsistency, we look for equilibrium stopping policies, formulated as fixed points of an operator. Under appropriate conditions, fixed-point iterations converge to equilibrium stopping policies. This iterative approach corresponds to the hierarchy of strategic reasoning in game theory and provides “agent-specific” results: it assigns one specific equilibrium stopping policy to each agent according to her initial behavior. In particular, it leads to a precise mathematical connection between the naive behavior and the sophisticated one. Our theory is illustrated in a real options model

On Automating Diagrammatic Proofs of Arithmetic Arguments

. Theorems in automated theorem proving are usually proved by formal logical proofs. However, there is a subset of problems which humans can prove by the use of geometric operations on diagrams, so called diagrammatic proofs. Insight is often more clearly perceived in these proofs than in the corresponding algebraic proofs; they capture an intuitive notion of truthfulness that humans find easy to see and understand. We are investigating and automating such diagrammatic reasoning about mathematical theorems. Concrete, rather than general diagrams are used to prove particular concrete instances of the universally quantified theorem. The diagrammatic proof is captured by the use of geometric operations on the diagram. These operations are the "inference steps" of the proof. An abstracted schematic proof of the universally quantified theorem is induced from these proof instances. The constructive !-rule provides the mathematical basis for this step from schematic proofs to theoremhood. In ..

Dietary patterns and non-communicable disease risk in Indian adults : secondary analysis of Indian Migration Study data

Acknowledgements The authors thank the IMS study team members and field staff involved in the generation and processing of IMS data. Financial support: This study forms part of the Sustainable and Healthy Diets in India (SAHDI) project supported by the Wellcome Trust ‘Our Planet, Our Health’ programme (grant number 103932). The Wellcome Trust had no role in the design, analysis or writing of this article. The IMS was funded by Wellcome Trust (grant number GR070797MF). L.A.’s PhD studentship is funded by the Leverhulme Centre for Integrative Research on Agriculture and Health (LCIRAH).Peer reviewedPublisher PD

Glutamine and GABA alterations in cingulate cortex may underlie alcohol drinking in a rat model of co-occurring alcohol use disorder and schizophrenia: an 1H-MRS study

Alcohol use disorder commonly occurs in patients with schizophrenia and significantly worsens the clinical course of the disorder. The neurobiological underpinnings of alcohol drinking are not well understood. Magnetic resonance spectroscopy (MRS) has been used to assess the neurochemical substrates that may be associated with alcohol drinking in patients; however, the causal impact of these findings remains elusive, highlighting the need for studies in animal models. This study performed MRS in the neonatal ventral hippocampal lesioned (NVHL) rat model, a model of co-occurring schizophrenia and substance use disorders. NVHL lesions (or sham surgeries) were performed on post-natal day 7 and animals were given brief exposure to alcohol during adolescence (10% v/v in a 2-bottle choice design). Animals were re-exposed to alcohol during adulthood (20% v/v) until a stable drinking baseline was established, and then forced into abstinence to control for the effects of differential alcohol drinking. Animals were scanned for MRS after one month of abstinence. NVHL rats consumed significantly more alcohol than sham rats and in the cingulate cortex showed significantly higher levels of GABA and glutamine. Significantly lower GABA levels were observed in the nucleus accumbens. No differences between the NVHL and sham animals were observed in the hippocampus. Correlation analysis revealed that GABA and glutamine concentrations in the cingulate cortex significantly correlated with the rats’ alcohol drinking prior to 30 days of forced abstinence. These findings suggest that a potential dysfunction in the glutamate/GABA–glutamine cycle may contribute to alcohol drinking in a rat model of schizophrenia, and this dysfunction could be targeted in future treatment-focused studies

Machine Learning Based Classification of Deep Brain Stimulation Outcomes in a Rat Model of Binge Eating Using Ventral Striatal Oscillations

Neuromodulation-based interventions continue to be evaluated across an array of appetitive disorders but broader implementation of these approaches remains limited due to variable treatment outcomes. We hypothesize that individual variation in treatment outcomes may be linked to differences in the networks underlying these disorders. Here, Sprague-Dawley rats received deep brain stimulation separately within each nucleus accumbens (NAc) sub-region (core and shell) using a within-animal crossover design in a rat model of binge eating. Significant reductions in binge size were observed with stimulation of either target but with significant variation in effectiveness across individuals. When features of local field potentials (LFPs) recorded from the NAc were used to classify the pre-defined stimulation outcomes (response or non-response) from each rat using a machine-learning approach (lasso), stimulation outcomes could be classified with greater accuracy than expected by chance (effect sizes: core = 1.13, shell = 1.05). Further, these LFP features could be used to identify the best stimulation target for each animal (core vs. shell) with an effect size = 0.96. These data suggest that individual differences in underlying network activity may relate to the variable outcomes of circuit based interventions, and measures of network activity could have the potential to individually guide the selection of an optimal stimulation target to improve overall treatment response rates

Machine Learning Based Classification of Deep Brain Stimulation Outcomes in a Rat Model of Binge Eating Using Ventral Striatal Oscillations

Neuromodulation-based interventions continue to be evaluated across an array of appetitive disorders but broader implementation of these approaches remains limited due to variable treatment outcomes. We hypothesize that individual variation in treatment outcomes may be linked to differences in the networks underlying these disorders. Here, Sprague-Dawley rats received deep brain stimulation separately within each nucleus accumbens (NAc) sub-region (core and shell) using a within-animal crossover design in a rat model of binge eating. Significant reductions in binge size were observed with stimulation of either target but with significant variation in effectiveness across individuals. When features of local field potentials (LFPs) recorded from the NAc were used as predictors of the pre-defined stimulation outcomes (response or non-response) from each rat using a machine-learning approach (lasso), stimulation outcomes could be predicted with greater accuracy than expected by chance (effect sizes: core = 1.13, shell = 1.05). Further, these LFP features could be used to identify the best stimulation target for each animal (core vs. shell) with an effect size = 0.96. These data suggest that individual differences in underlying network activity may contribute to the variable outcomes of circuit based interventions and that measures of network activity have the potential to individually guide the selection of an optimal stimulation target and improve overall treatment response rates

EVALUATING CLINICAL MASTITIS IN DAIRY CATTLE FED MONENSIN

The effect of Monensin on clinical mastitis in dairy cattle was evaluated from data collected at nine geographical clinical field trials using 966 Holstein cows and heifers in the United States and Canada. At each site, a randomized complete block design was conducted. Monensin (Rumensin®) was fed at concentrations of 0, 8, 16, or 24 ppm in a total mixed ration beginning 21 days before first calving for all nine sites, up to 7 days after second calving for six sites, and 203 days after second calving for three sites. Quarter milk samples were taken and cultured to determine the causative pathogen for each mastitis case and if clinical signs were observed the disease data were grouped according to etiology and analyses conducted. Analyses were conducted for all clinical mastitis cases as well as for a breakdown of the clinical mastitis cases into microorganism group levels. A generalized linear mixed model and a linear mixed model were used to determine if there were significant differences in clinical mastitis between the non-zero concentrations of Monensin and controls. Response variables for the clinical mastitis cases that were analyzed using a generalized linear mixed model were Animal rate, Quarter rate, Observation rate, and Incident rate. An additional response variable, Average case duration, was analyzed using a linear mixed model. Inferences from the analyses indicate that Monensin does not influence the susceptibility of dairy cattle to clinical mastitis