Number Learning |VERIFIED|

I have read some elementary number theory from David Burton's text and I know groups and rings from Herstein's book Topics in Algebra and some field theory from different sources online. I am currently learning commutative algebra from Atiyah and Macdonald's engaging book on commutative algebra.

I would really like to learn some algebraic number theory and I was wondering if someone could provide me a sequence of books/ steps so that I can understand enough number theory to be a strong graduate student about to begin research. I apologize sincerely if my question is off-topic here or partly vague.

Number Learning

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Stewart and Tall's Algebraic Number Theory (mentioned in answers to linked questions): treats number fields, with a little bit on elliptic curves and modular forms. gentle, but doesn't cover some important topics.

I'm not saying you should read all of these right now, or in this order, or that reading these will give you deep enough knowledge to do serious research on these topics. I am suggesting you learn the basics of some of the major topics in (algebraic-ish) number theory. (The above list of books still don't cover all the "basics"--e.g., I'm worried Gauss' theory of binary quadratic forms will be missing and Dirichlet's theorem on arithmetic progressions maybe missing, though if you get through the above you can find this stuff in my number theory ii notes, for instance, which also take the approach of a smattering of topics). Then you can try to go deeper in what you're most interested in, or maybe you'll be in grad school with more guidance at that point.

I don't want to have more than 10-15 cards in learning, but when I have 20+ cards in "review" and 30+ cards in "new cards" Anki just fills up the learning deck to such an amount that I totally blank when a card come up again.

Explore all of our learning numbers worksheets (recognizing and printing numbers), counting worksheets (counting objects, skip counting, counting backwards) and comparing numbers worksheets ("more than", "less than", ordering numbers).

As a group, children from disadvantaged, low-income families perform substantially worse in mathematics than their counterparts from higher-income families. Minority children are disproportionately represented in low-income populations, resulting in significant racial and social-class disparities in mathematics learning linked to diminished learning opportunities. The consequences of poor mathematics achievement are serious for daily functioning and for career advancement. This article provides an overview of children's mathematics difficulties in relation to socioeconomic status (SES). We review foundations for early mathematics learning and key characteristics of mathematics learning difficulties. A particular focus is the delays or deficiencies in number competencies exhibited by low-income children entering school. Weaknesses in number competence can be reliably identified in early childhood, and there is good evidence that most children have the capacity to develop number competence that lays the foundation for later learning.

The learning process is a multidimensional function with a wide intra- and interindividual scattering. To determine the learning process in anesthesia, we evaluated 11 first-year residents according to their rate of success or failure when applying manual anesthesiological skills, such as performance of spinal, epidural, or brachial plexus anesthesia and tracheal intubation or insertion of an arterial line. Epidural anesthesia was the most difficult procedure (P < 0.05). Significant differences were found between epidural anesthesia and tracheal intubation (P < 0.05), insertion of an arterial line (P < 0.05), and brachial plexus block (P < 0.05), as well as between spinal anesthesia and orotracheal intubation (P < 0.05). Learning curves are a valid tool for monitoring institutional and individual success.

Implications: To investigate the learning process in anesthesia, typical anesthetic procedures were performed by inexperienced residents during their first year. Learning curves were generated for each procedure performed. Epidural anesthesia was the most difficult procedure to perform (P < 0.05).

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Counting includes 1) the ability to say number words in correspondence with objects (enumerate objects), 2) understanding that the last number word said when counting refers to how many items have been counted (cardinality), and 3) using counting strategies to solve problems.

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Count on these learning cards for a unique twist on number fun! Poke the irresistible buttons to hear satisfying clicks and pops while developing early learning skills. Each side of the 13 sturdy cards features a number or a shape and a color, with corresponding dots to poke. Poke-a-Dot learning cards help encourage counting, color and shape recognition, and fine motor skills. Patented Pop-a-Tronic technology lets kids discover different popping sounds by poking the dots from the front or the back, with cards raised or flat!

The turnover number kcat, a measure of enzyme efficiency, is central to understanding cellular physiology and resource allocation. As experimental kcat estimates are unavailable for the vast majority of enzymatic reactions, the development of accurate computational prediction methods is highly desirable. However, existing machine learning models are limited to a single, well-studied organism, or they provide inaccurate predictions except for enzymes that are highly similar to proteins in the training set. Here, we present TurNuP, a general and organism-independent model that successfully predicts turnover numbers for natural reactions of wild-type enzymes. We constructed model inputs by representing complete chemical reactions through differential reaction fingerprints and by representing enzymes through a modified and re-trained Transformer Network model for protein sequences. TurNuP outperforms previous models and generalizes well even to enzymes that are not similar to proteins in the training set. Parameterizing metabolic models with TurNuP-predicted kcat values leads to improved proteome allocation predictions. To provide a powerful and convenient tool for the study of molecular biochemistry and physiology, we implemented a TurNuP web server.

The turnover number kcat is the maximal rate at which one active site of an enzyme converts molecular substrates into products. kcat is a central parameter for quantitative studies of enzymatic activities, and is of key importance for understanding cellular metabolism, physiology, and resource allocation. In particular, comprehensive sets of kcat values are essential for metabolic models that consider the cost of producing or maintaining enzymes1,2,3,4,5,6,7,8,9, a prerequisite for accurate simulations of cellular physiology and growth10. Currently, no high-throughput experimental assays exist for kcat, and experiments are both time consuming and expensive. Thus, kcat estimates are unavailable for most reactions; even for Escherichia coli, arguably the biochemically best-characterized organism, in vitro kcat is known for only ~10% of all enzyme-catalyzed reactions11. In genome-scale kinetic models of cellular metabolism, this issue is typically addressed by either sampling missing kcat values or fitting them to large datasets7,8,12,13. However, these techniques typically result in inaccurate results, and fitted kcat values bear little relationship to known in vitro estimates7,12,13.

Here, we present a general machine and deep learning approach for predicting in vitro kcat values for natural reactions of wild-type enzymes. In contrast to previous approaches, we represent chemical reactions through numerical fingerprints that consider the complete set of substrates and products of a reaction. To capture the enzyme properties, we use fine-tuned state-of-the-art protein representations as additional model inputs (Fig. 1). We created these enzyme representations using Transformer Networks, deep neural networks for sequence processing, which were trained with millions of protein sequences17. It has been shown for various prediction tasks that Transformer Networks outperform protein representations created with convolutional neural networks (CNNs)18,19, which were used in previous models for predicting enzyme turnover numbers16.

We compiled a dataset that connects kcat measurements with the corresponding enzyme sequences, reactant IDs, and reaction equations. The underlying data is derived from the three databases BRENDA20, UniProt21, and Sabio-RK22. Our aim was to build a turnover number prediction model for natural reactions of wild-type enzymes. We hypothesized that we do not have enough data to train a model to predict the catalytic effect of enzyme mutations or to predict the kcat value of non-natural enzyme-reaction pairs, which have not been shaped by natural selection. Hence, we removed all data points with non-wild-type enzymes and all non-natural reactions (see Methods, Data preprocessing"). We removed redundancy by deleting data that was identical to other data points in the set, and we excluded data points with incomplete reaction or enzyme information. We also removed 55 outliers with unrealistically low or high measurements, i.e., reported kcat values that are either very close to zero (105/s)23. If multiple different kcat values existed for the same enzyme-reaction pair, we took the geometric mean across these values. e24fc04721