TheoremOne is an innovation and engineering company that advises clients on product strategy, engineering, design, and culture, then partners with them to build and launch technology-driven solutions to their most complex problems

TheoremOne

theoremone

The way we used to do it. TeX (/ˈtɛx/ tekh as in Greek) • MiniLatex Live Demo App II Recent news Disappointed in the appearance and esthetics of the proofs for for the second volume of his , the famed computer scientist Donald Knuth set out to build the right tool for setting mathematical text, a job he felt would take about six months. This was 1977. By 1978, he had working program, baptized and in it was released in final form. It has been the foundation of mathematical typesetting ever since. Thus, to produce the formula Art of Computer Programming TeX, 1989 one writes the incantation $$\int_{-\infty}^\infty e^{-x^2/2} dx = \sqrt{2\pi}$$ Knuth’s program plays the role of master magician, turning the incantation into the beautiful formula displayed above. It may seem retrograde, or maybe just retro, to write such an arcane sequence of symbols to obtain the desired artistry. Surely in the age of the mouse and the touch screen, there is something better! Perhaps, but those who do mathematics day in and day out do it this way, not out of ignorance or stubbornness, but because it is the BETTER WAY: faster, once you know what you are doing, guaranteed to give you a beautiful result without messing around, and able to handle the weirdest that you may discover or invent. mathematics MathJax TeX, LaTeX, and Postscript taken together solve the problem of composing mathematical text for print. Add PDF to the mix and the problem of distributing whole mathematical texts electronically is also solved. But what about the worldwide web? Can one edit and display LaTeX in the browser? MathJax (see ) solves half of the browser problem — the display of mathematical text per se, that which is fenced off by dollar signs or double dollar signs, as in the example above. The results are beautiful, a tribute to the skill and craft of the MathJax team. MathJax is how the blogs you find on the internet deal with mathematical text. mathjax.org Nonetheless, there is a gap, because LaTeX does more than display formulas— there are automatically numbered sections and subsections, equations which are cross-referenced by a convenient system of labels, special “environments” for theorems, itemized lists, verbatim text, etc., etc. So the question is: what can be done for the part of LaTeX outside the dollar signs? MiniLaTeX To reproduce all of LaTeX in the browser is too much to ask, but what is not too much is to define a reasonable subset which can be processed with the usual tools or with some new browser-based tool. To do this is the aim of the . While still in an experimental stage, the project shows promise — look, for example, at these documents and drafts: , , , and . They were written using the editing tools at using MiniLatex, and so provide a primitive proof-of-concept. MiniLaTeX project MiniLaTeX Infinity Quantum Field Theory Notes Elm by Example www.knode.io You can experiment with MiniLaTeX using the . To write documents using MiniLaTeX, try As this project is very much in a fluid stage of development, I welcome (and need) comments, criticism, and bug reports (jxxcarlson at gmail). MiniLaTeX Demo App . www.knode.io Nuts and bolts The MiniLaTeX project employs a divide-and-conquer strategy, using MathJax to render all of the text fenced off by dollar signs. To handle the part of LaTeX outside of the fences, one uses a — a specialized program — that reads the LaTeX source text and converts it to an intermediate form called an Abstract Syntax Tree (AST). The AST “knows” the grammar of LaTeX and encodes this knowledge in such a way that it easy for a second program, the , to convert it into HTML, which is the language in which web pages are written. To put it in a few words, it is all a matter of translating from one language to another. parser renderer Getting Technical Let’s now get technical, or at least a little bit technical. For the real deal, see the . MiniLaTeX Technical Description and Progress Report The first step in setting up a language-translation toolchain is to define the language to be translated. For this, it is a good idea to start out with a language that is expressive enough to make writing a parser easy (not to mention possible). The language I chose is a recent one, , created and developed by Evan Czaplicki, who conceived it as part of his 2012 senior thesis at Harvard University. Elm is a purely functional language for building front-end web apps. It compiles to and has a number of remarkable properties: (1) there are no runtime exceptions, (2) the compiler gives the best error messages on the planet, transforming it from an annoying nag to an incredibly friendly and helpful assistant, (3) refactoring, even of the most drastic kind, is easy; as a result, so is maintaining code over the long run. Oh — and one more thing. It is fast. The deciding factor for this project, however, is the fact that there are , akin to the parser combinators in Haskell’s parsec. programming Elm Javascript excellent parser-writing tools The MiniLaTeX language definition as it stands now is quite small, just nine lines of code: type LatexExpression= LXString String| Comment String| Item Int LatexExpression| InlineMath String| DisplayMath String| Macro String (List LatexExpression)| Environment String LatexExpression| LatexList (List LatexExpression) The parser is also quite small — at present 304 lines. (See the ). Of course, there is more, because once a passage of MiniLaTeX source text has been parsed into an AST, it has to be rendered into HTML. The renderer is larger (548 lines). However, it is much easier, indeed almost routine to write. For more details, please see the . GitHub repository MiniLaTeX Technical Description and Progress Report Peeking under the Hood For those who would like to peek under the hood just a little more, consider this fanciful piece of LaTeX text: This is MiniLaTeX:\begin{theorem}This is a test: $\alpha^2 = 7$ \foo{1}\begin{a}la di dah\end{a}\end{theorem} The experimental parser, version 1.0, transforms it into the following AST: Ok (LatexList (LXString "This is MiniLaTeX:",[Environment "theorem" (LatexList ([LXString "This is a test:",InlineMath "\alpha^2 = 7",Macro "foo" ["1"],Environment "a" (LatexList ([LXString "la di dah"]))]))])) And here is how the AST is rendered: This is MiniLaTeX:<div class="environment"><strong>Theorem</strong><div class="italic">This is a test: $\alpha^2 = 7$ \foo{1}<div class="environment"><strong>A</strong><div class="italic">la di dah</div></div></div></div> Comments on Rendering Your might ask, “What’s the deal with ?” It was passed on as-is. Well, that turns out to be a good strategy. If MiniLaTeX understands a TeX macro, it acts accordingly. For example, in the case , the text “Wow!” is italicized. But if MiniLaTeX doesn’t know about the macro, it just passes it on. If the macro appears inside something like an equation environment, then MathJax may know how to handle it, so passing it on as-is the right thing to do. If neither MiniLaTeX nor MathJax knows how to handle it, the author can see that something is wrong, and he or she can take corrective action. \foo{1} \emph{Wow!} You might also ask: “What’s the deal the ?” There is no environment named “a” in LaTeX. True, of course. The idea here is to have a default way of handling environments that MiniLaTeX does not know about, so that they will “just work.” When running a MiniLaTeX document through a conventional tool such as , the user can customize these environments, or use a macro package the defines them. For example, a trial lawyer may wish to use an “objection” environment, or a psychic may wish to have “prediction” environment. Whatever. In the meantime, both the lawyer and the psychic can use these environments in MiniLaTeX. Thus, even though the prediction environment is not wired into MiniLaTeX, the text \begin{a} pdflatex \begin{prediction} MiniLaTeX will be a big success. \end{prediction}\begin{prediction} You will win the lottery. \end{prediction} renders as _MiniLaTeX will be a big success._ Prediction 1 Prediction 2 You will win the lottery. Yes! Notes We are exploring ways for users to define non-default environment behaviors in the browser. The same goes for macros used outside the dollar and double-dollar fences. Authors using can use whatever macro definitions they need for the MathJax part of MiniLaTeX documents. One way is to put their definitions inside double dollar signs in the text of the document. Another way is to set up a document of macro definitions that is included in other MiniLatex documents. www.knode.io — a draft, but gives an idea of what is going on. The grammar is LL(*) because it has to look arbitrarily far ahead to recognize a begin — end block. Since the largest unit parsed is a logical paragraph, one should think of it as LL(N), where N is the number of characters in the paragraph to be parsed. A logical paragraph is either an ordinary paragraph or an outer begin — end block. MiniLatex Grammar seems to work much better in Firefox than in Chrome — at least for me. knode.io Credits I wish to acknowledge the generous help that I have received throughout the work on this project from the community at , with special thanks to Ilias van Peer. elmlang.slack.com — Jim Carlson

Towards LaTeX in the Browser

This paper is available on arxiv under CC 4.0 license. Authors: (1) Md Masud Rana, Department of Mathematics, University of Kentucky; (2) Duc Duy Nguyen, Department of Mathematics, University of Kentucky & ducnguyen@uky.edu. Table of Links Abstract & Introduction Datasets and Results Methods Conclusion, Data and Software Availability, Competing interests, Acknowledgments & References 2 Datasets and Results In this section, we perform validation and evaluation of our proposed models on several benchmark datasets. We develop two types of GGL-PPI models: GGL-PPI1 and GGL-PPI2. The first model, GGL-PPI1, is built solely on geometric graph features discussed in Section 3. On the other hand, GGL-PPI2 incorporates both geometric graph features and auxiliary features, as detailed by Wang et al. [41]. The electrostatic potential calculations for the auxiliary components are conducted using the MIBPB software [42]. 2.1 Validation To validate our models, we primarily consider the AB-Bind dataset [25], SKEMPI 1.0 dataset [23], and SKEMPI 2.0 dataset [24]. We employ a rigorous evaluation methodology by conducting a 10-times 10-fold cross-validation (CV) on each datasets. The mean Pearson correlation coefficient (R ) and root-mean-square error (RMSE) serve as our evaluation metrics. p In comparing the CV performance of our proposed models with other existing methods, we specifically assess TopNetTree [41], Hom-ML-V2 [43], and Hom-ML-V1 [43]. Both TopNetTree and Hom-ML-V2 incorporate auxiliary features in conjunction with their topology-based and Hom-complex-based features, respectively. On the other hand, Hom-ML-V1 solely relies on Hom-complex-based features without utilizing any auxiliary features. Validation on AB-Bind S645 Data Set The AB-Bind dataset contains 1,101 mutational data points for 32 antibody-antigen complexes, providing experimentally determined binding affinity changes upon mutations. Pires et al. curated a subset known as AB-Bind S645 [44], consisting of 645 single-point mutations observed in 29 antibody-antigen complexes. The dataset comprises a mix of stabilizing (20%) and destabilizing (80%) mutations. Additionally, the dataset includes 27 non-binders that do not show any binding within the assay’s sensitivity range. For these non-binders, the binding free energy changes have been uniformly set to a value of 8 kcal/mol. It is crucial to consider these non-binders as outliers during model development and evaluation to ensure model accuracy and robustness. Our GGL-PPI2 achieved an Rp of 0.58 on the AB-Bind S645 dataset, as shown in Figure 2a. The comparison results in Table 1 indicate that our model tied for second place with Hom-ML-V2 43, while TopNetTree [41] claimed the top position. However, when we exclude the 27 nonbinders from the dataset, our model outperforms all other existing models. Specifically, the Rp value increases to 0.74 from 0.58 after removing the nonbinders (Figure 2b). Furthermore, GGL-PI1, our purely geometric graph-based features model, demonstrated competitive performance with an Rp of 0.57 on the AB-Bind S645 dataset. Intriguingly, when excluding the nonbinders, GGL-PPI1 surpassed all other models with an improved Rp of 0.73. These performances reveal that our multiscale weighted colored geometric graphs can effectively characterize the wide range of interactions in biomolecular complexes. Validation on SKEMPI 1.0 S1131 Data Set The SKEMPI 1.0 dataset consists of a collection of 3,047 mutations of 158 complexes obtained from literature sources, where the complexes have experimentally determined structures [23]. The dataset includes both single-point mutations and multi-point mutations. Specifically, there are 2,317 entries in the dataset that represent single-point mutations, which are collectively known as the SKEMPI S2317 set. Additionally, a subset of 1,131 non-redundant interface single-point mutations has been selected from the SKEMPI S2317 set and labeled as the SKEMPI S1131 set [45]. This subset focuses on studying the impact of single-point mutations on protein-protein interactions. Table 1: Performance comparison of different methods in terms of Pearson correlation coefficients (Rp) for the AB-Bind (S645) dataset. Figure 2c shows that our model GGL-PPI2 achieves an Rp of 0.873 and an RMSE of 1.21 kcal/mol in 10-fold CV on the S1131 dataset. Table 2 presents the performance comparison of various methods on the S1131 dataset, including our proposed models, GGL-PPI1 and GGL-PPI2. Among them, our model, GGL-PPI2, achieved the highest performance, underscoring its superiority in predicting binding affinity changes due to mutation. Notably, even without auxiliary features, our GGL-PPI1 outperformed both TopNetTree and Hom-ML-V2 methods that do leverage auxiliary features. This again highlights the efficacy of our geometric graph-based molecular representation. Table 2: Performance comparison of different methods in terms of Pearson correlation coefficients (Rp) for the single-point mutations in the SKEMPI 1.0 (S1131) dataset. Validation on SKEMPI 2.0 S4169 and S8338 Data Sets The SKEMPI 2.0 dataset is an updated and expanded version of the original SKEMPI dataset, incorporating new mutations collected from various sources [24]. Released in 2018, it significantly increased in size, now containing a total of 7,085 entries, including both single-point and multi-point mutations. The data was obtained by merging several databases, including SKEMPI 1.0 [23], AB-Bind [25], PROXiMATE27, and dbMPIKT46. Additionally, new data from the literature were manually curated and added to the dataset. The mutations cover a wide range of protein complexes, such as protease-inhibitor, antibody-antigen, and TRCpMHC complexes. Among the mutations, approximately 3,000 are single-point alanine mutations, 2,000 are single-point non-alanine mutations, and another 2,000 involve multiple mutations. Notably, the authors of the mCSM-PPI2 [8] method filtered the single-point mutations, yielding S4169 set, comprising 4,169 variants in 139 different complexes/ The S8338 set, derived from S4169, represents hypothetical reverse mutation energy changes with negative values. This comprehensive dataset serves as a valuable resource for studying protein interactions and their thermodynamic properties. Perforamnce-wise, Our GGL-PPI2 model posts an Rp of 0.81 with an RMSE of 1.03 kcal/mol for the S4169 dataset as shown in Figure 2d, outstripping all existing models (Table 3). It is noteworthy that our GGL-PPI1 model, which solely relies on geometric graph-based features, demonstrated comparable performance to GGL-PPI2, outperforming TopNetTree and mCSM-PPI2 with an Rp of 0.80 and an RMSE of 1.06 kcal/mol. In the case of the S8338 dataset, we applied a stratified cross-validation approach similar to mCSM-PPI2. We ensured that hypothetical reverse mutations were consistently placed either in the training or test sets during the dataset splits, maintaining their relationship to the corresponding original mutations intact throughout the cross-validation process. GGL-PPI2 achieved an Rp of 0.85 with an RMSE of 1.07 kcal/mol as depicted in Figure 2e, and GGL-PPI1 closely followed, attaining an Rp of 0.84 with the same RMSE value. As Table 3 attests, our GGL-PPI2 is on par with TopNetTree and outperforms mCSM-PPI2 on the S8338 dataset. Table 3: Performance comparison of different methods in terms of Pearson correlation coefficients (Rp) for the single-point mutations in the SKEMPI 2.0 (S4169 and S8338) dataset. 2.2 Evaluation To evaluate our proposed model for predicting binding free energy (BFE) changes of protein-protein interactions, we consider two datasets sourced from the ProTherm database [22]. The first dataset, carefully selected by Pucci et al. [36], named S[sym] dataset. This data assembles 684 mutations from the ProTherm, comprising 342 direct mutations and their corresponding reverse mutations, resulting in a balanced dataset. The dataset specifically focuses on mutations in fifteen protein chains with solved 3D structures, ensuring highresolution data with a resolution of at least 2.5˚A. By providing experimentally measured ∆∆G values and a balanced representation of stabilizing and destabilizing mutations, the S[sym] dataset serves as a valuable resource for evaluating prediction biases in the context of predicting mutation-induced binding affinity changes. To address the issue of data leakage and enhance the generalization capability of our method, we employed the Q1744 dataset [47]. Quan et al. [48] compiled the Q3421 dataset from ProTherm, consisting of 3421 single-point mutations across 150 proteins with available PDB structures. However, the presence of homologous proteins in both the training and test set can lead to interdependent effects of mutations, compromising the model’s performance. To mitigate this, Li et al. [47] created the Q1744 dataset, derived by excluding overlapping data points and refining protein-level homology between Q3421 and S[sym] datasets, resulting in 1744 distinct mutations. Furthermore, the Q3488 dataset was created by augmenting reverse mutations in the Q1744 set. We utilized the Q3488 dataset as our training set, thereby enhancing our ∆∆G predictor’s capability to accurately predict BFE changes in PPIs. We conduct an evaluation of our model on the blind test set S[sym], with a distinct focus on both direct and reverse mutations. To assess the performance, we utilize the Pearson correlation coefficient and root-mean-square error as our primary metrics. Additionally, to discern any prediction bias, we incorporated two statistical measures: Rpdir−rev and δ. The former calculates the Pearson correlation between predictions for direct and reverse mutations, while the latter represents the sum of predicted ∆∆G values for both types of mutations. The hypothesis is that an unbiased predictor would yield Rpdir−rev = −1 and an average δ ( ¯δ) of 0 kcal/mol. Our main focus is to highlight the effectiveness of our model, GGL-PPI2, particularly emphasizing its robust geometric graph-based molecular featurization. GGL-PPI2 has demonstrated exceptional prediction accuracy, maintaining consistency for both direct and reverse mutations. As depicted in Figure 3a and 3b, our model achieves consistent Rp values of 0.57 and an RMSE of 1.28 kcal/mol, indicating its efficiency against overfitting to direct mutations. Additionally, the analysis reveals that a significant proportion of mutations fall within a prediction error of 0.5 kcal/mol and 1.0 kcal/mol, with 34.6% and 65.8% for direct mutations and 35.1% and 66.0% for reverse mutations, as depicted in Figure 3d and 3e. Furthermore, Figure 3c demonstrates that GGL-PPI2 effectively addresses prediction bias by achieving a nearly perfect Rpdir−rev value of -0.999 and an extremely low average ¯δ of 0.006 kcal/mol. Finally, the distribution plot in Figure 3f illustrates that 99.4% of mutations exhibit a prediction bias under 0.05 kcal/mol. In Table 4, we present the prediction results of our models and conduct a comprehensive comparison with other ∆∆G predictors. We observe that our GGL-PPI2 model outperforms ThermoNet [47], which was also trained on the homologyreduced set Q3488, across all evaluation measures. It outperforms ThermoNet by 21.3% for direct mutations and 18.7% for reverse mutations. Furthermore, the GGL-PPI1 model, which only uses geometric graph-based features, also performs better than ThermoNet in both direct and reverse prediction tasks. This further emphasizes the effectiveness of our geometric-graph approach. For a broader comparison against other ∆∆G predictors, we introduce the GGL-PPI2∗ model, trained on the Q6428 set constructed before the homology reduction of the set Q3421 [47]. As illustrated in Table 4, GGL-PPI2∗ excels over other methods in reverse mutation predictions. It is noteworthy that while some methods surpass GGL-PPI2∗ for direct mutations, they frequently exhibit significant bias towards reverse mutations. This paper is available on under CC 4.0 license. Arxiv

Our Datasets and Results From Our Study: GGL-PPI Models

আমাদের গবেষণা থেকে আমাদের ডেটাসেট এবং ফলাফল: GGL-PPI মডেল

Unsere Datensätze und Ergebnisse unserer Studie: GGL-PPI-Modelle

Nuestros conjuntos de datos y resultados de nuestro estudio: modelos GGL-PPI

Nos ensembles de données et résultats de notre étude : modèles GGL-PPI

हमारे डेटासेट और हमारे अध्ययन के परिणाम: जीजीएल-पीपीआई मॉडल

私たちのデータセットと研究結果: GGL-PPI モデル

연구 데이터 세트 및 결과: GGL-PPI 모델

Nossos conjuntos de dados e resultados de nosso estudo: modelos GGL-PPI

Наши наборы данных и результаты нашего исследования: модели GGL-PPI

Veri Setlerimiz ve Çalışmamızın Sonuçları: GGL-PPI Modelleri

Bộ dữ liệu và kết quả từ nghiên cứu của chúng tôi: Mô hình GGL-PPI

我们的数据集和研究结果：GGL-PPI 模型

This paper is available on arxiv under CC 4.0 license. Authors: (1) Jeffrey Ouyang-Zhang, UT Austin (2) Daniel J. Diaz, UT Austin (3) Adam R. Klivans, UT Austin (4) Philipp Krähenbühl, UT Austin Table of Links Abstract & Introduction Related Works Preliminary Methodology Results Limitations Conclusion, Acknowledgements, & References Experimental Setting & Additional Results 2 Related Works 2.1 Protein Engineering Protein engineering is the process of mutating a natural protein to improve particular phenotypes, such as thermodynamic stability and function. The field has historically relied on rational design and stochastic methods, such as error-prone PCR [1], DNA shuffling [71], and directed evolution (DE) [4, 27], to identify gain-of-function mutations. Rational design is limited to proteins with solved structures and requires an extensive understanding of biochemistry and specific knowledge of the particular protein to select mutations highly likely to improve the target phenotypes [33, 79]. Directed evolution requires little to no knowledge of the protein and instead generates a library of protein variants that are then screened for the target phenotypes (e.g. fluorescence brightness, antibiotic resistance, stability, activity) [4, 27]. The library can be generated via site-saturated mutagenesis of a handful of positions in the sequence [12] or DNA shuffling, in which the gene is digested into random fragments and reassembled into full-length sequences [71]. After screening the library, the most "fit" variant is then selected as the initial sequence for the next round of directed evolution. This iterative process repeats until it obtains a protein variant with the desired phenotypes. Machine learning has demonstrated its ability to augment rational design and accelerate the stabilization of a variety of proteins [20, 28, 40, 52, 68]. Separately, machine learning-guided directed evolution (MLDE)[75] has been shown to improve the likelihood of obtaining the global fitness maximum by 81-fold compared to traditional DE [76]. MLDE has accelerated the engineering of several proteins, such as the enantioselectivity of enzymes for kinetic resolution of epoxides [23] and the activity and expression of a glutathione transferase [46]. Mutate Everything empowers the experimentalist to accelerate the stabilization of a protein for both rational design and MLDE. 2.2 Machine Learning for Protein Structure Prediction Recent advances in machine learning have led to remarkable progress in protein structure prediction. AlphaFold [31] has demonstrated that deep learning is highly effective at predicting protein structures from a sequence by using evolutionary history via a multiple sequence alignment (MSA). AlphaFold passes the MSA and a pairwise representation of the sequence into Evoformer to capture the coevolutionary patterns between residues. The Evoformer output is then processed by the Structure Module, which predicts the protein’s structure. We challenge prior works postulating that AlphaFold cannot be used for stability prediction and show that fine-tuning these rich co-evolutionary and structural features yield highly performant stability predictors [53]. Evolutionary Scale Modeling (ESM) [38, 62] has shown that protein structure prediction can be performed without MSAs and specialized architectures by leveraging large transformers pre-trained on masked token prediction. Other works extended this masked pre-training framework, including MSA-Transformer [60] which incorporates a sequence’s MSA as input, and Tranception [49], which develops a hybrid convolution and attention based autoregressive architecture. We show that finetuning these evolutionary representations excels at protein stability assessment without MSAs. 2.3 Protein Stability Assessment Traditional approaches to protein stability assessment relied on a combination of physics-based methods, statistical analysis, and traditional machine learning techniques. Physics-based tools, such as FoldX, Rosetta, and SDM, utilize energy functions and statistical patterns to assess how mutations affect a protein’s stability [32, 67, 77]. DDGun [44] directly inferred ∆∆G from heuristics, including Blosum substitution scores, differences in interaction energy with neighboring residues, and changes in hydrophobicity. Many traditional machine learning approaches used support vector machines and decision trees with physics-based feature engineering [11, 14, 15, 17, 37, 66]. Others ensemble existing machine learning methods [35, 56, 59, 63, 64]. Recently, deep learning-based approaches have recently begun to outperform existing physics and traditional machine learning approaches [13, 19, 34]. ACDC-NN [6] trains an asymmetric convolutional neural network for predicting forward and reverse mutation effects. Thermonet [36] voxelizes and feeds both the wild-type and mutant protein structures into a 3D-CNN to regress a ∆∆G value. PROSTATA feeds the wild-type and mutant protein sequence into a pre-trained ESM2 model and then regresses a ∆∆G value [74]. Stability Oracle takes the native and mutant amino acid type along with the local structure as input and regresses ∆∆G [18]. Several deep learning frameworks [64, 80, 81] model multiple mutations in addition to single mutations. In this paper, we develop a framework that models the protein to enable efficient enumeration of all mutation candidates.

Predicting a Protein’s Stability
under a Million Mutations: Related Works

I have worked with two different kinds of leaders and managers over the years. The one who believes in fixed abilities and promotes a fixed mindset “those who don’t perform well can never do better” and the other who believes in growing those abilities thereby promoting a growth mindset “people can be coached into improving their skills.” The first kind engages with the workplace to boost their self-esteem, establish their superiority and prove their smartness. They focus on their employees' weaknesses, refuse to coach them, blame others for failure, do not seek feedback, divide people into competent or incompetent and hire for natural talent instead of looking for ways to grow talent within their organization. They praise people for their intelligence while failing to take notice of those who show improvement through effort and hard work. After all, their fixed mindset leads them with the belief that people can’t get any better and that makes them stick with their initial impression about them. How can you spot them? You will often see them shutting down people and their ideas. They refuse to take responsibility for failures, discourage dissent, treat mistakes as a personal fault and give feedback that puts talent above effort. The second kind engages with the workplace to create a learning environment providing development opportunities with focus on a high learning culture. They focus on their employees' strengths, coach them into developing critical thinking skills, take responsibility for failures, engage with the conflict with the desire to gain knowledge, seek critical feedback to learn from criticism and hire people who show the desire to learn and grow. They invest in people and their growth since they believe that anyone can get better with the right kind of strategies, determination and persistence. How can you spot them? You will often see their eyes light up with curiosity while listening to new ideas. They embrace challenges, offer advice to experiment, try harder and implement better strategies. They encourage difference of opinion, using failures and mistakes as valuable lessons to get better and give feedback that involves personal mastery. In other words, they help you become the best version of yourself. What’s the impact of this mindset on employees? Employees who work for growth mindset organizations where learning and growth is highly valued have far more trust in their company and a greater sense of empowerment and ownership. They are more committed to their companies and ready to go the extra mile for it. They welcome challenges, work harder, and persevere in the face of struggle which makes them more successful learners and better contributors to their organizations. Stanford University psychologist Carol Dweck’s has found that employees in a “growth mindset” company are: research 47% likelier to say that their colleagues are trustworthy, 34% likelier to feel a strong sense of ownership and commitment to the company, 65% likelier to say that the company supports risk taking, and 49% likelier to say that the company fosters innovation. Employees in the fixed-mindset companies not only say that their companies are less likely to support them in risk-taking and innovation, they are also far more likely to agree that their organizations are rife with cutthroat or unethical behavior. When organizations put the premium on natural talent, then everyone wants to be the superstar, everyone wants to shine brighter than the others, and people may be more likely to cheat or cut corners to do so. Teamwork can take a nosedive - Carol Dweck She suggests “Our best bet is not simply to hire the most talented managers we can find and turn them loose, but to look for managers who also embody a growth mindset: a zest for teaching and learning, an openness to giving and receiving feedback, and an ability to confront and surmount obstacles. It also means we need to train leaders, managers, and employees to believe in growth, in addition to training them in the specifics of effective communication and mentoring.” Main attributes of a growth mindset workplace A growth mindset environment in which people can thrive: Value hard work, determination and perseverance over talent Help employees learn new skills Coach them to develop critical thinking skills to navigate challenging problems Encourage experimentation to try new ideas Use failures as valuable learning opportunities Empower to take responsibility and own up to mistakes Encourage people to speak up and respect diverse viewpoints Give feedback that promotes learning and desire to get better Growth mindset environment promotes the idea that anyone can succeed if they apply the right strategies, work hard and ask for help to improve a little each day. People may have plenty of talent and still achieve nothing. It’s the effort that builds skills and it’s the effort that turns those skills into accomplishment. Talent is not all there is to achievement. Talent is how quickly your skills improve when you invest effort. Achievement is what happens when you take your acquired skills and use them. Skill is not the same thing as achievement. Without effort, your talent is nothing more than your unmet potential. Without effort, your skill is nothing more than what you could have done but didn’t. With effort, talent becomes skill and, at the very same time, effort makes skill productive - Angela Duckworth Questions to examine mindset in the workplace Examine how your own mindset is getting in the way of your growth and those of people around you. If you are a manager or a leader, your mindset can have a profound impact on others. Are you in a fixed-mindset or growth-mindset workplace? How do people around you behave? Do you feel being judged all the time or others help you grow? How are mistakes and failures handled in the workplace? Does your company promote talent or effort? Are you encouraged to build new skills? How has been your learning in the past? Does your manager encourage you to do better? Does your manager coach you into solving problems on your own? Do you feel a greater sense of empowerment and ownership? Does your company encourage risk-taking and innovation? Do you find your colleagues trustworthy? Have you considered the possibility that you can be a problem? How do you react to mistakes? Are you defensive about them or use them as a learning opportunity to get better? Do you apply the feedback that you get or simply ignore it? Do you seek criticism as a tool to get better? What’s your behaviour towards others in the workplace? Do you try to take advantage of others? Do you demean them? Do you worry about high performing employees in your organization? Do they threaten you or challenge you to do better? Do you embrace new opportunities as a learning experience or avoid them with fear of failure? Do you spend time in building new skills or spend the same time blaming others for everything not going so well for you? Strategies to promote growth mindset as a manager If you are a manager, consider ways in which you can promote a growth mindset on the job. What are the ways in which you can help employees build new skills - coaching? How can you hire people who value learning over talent? What can you do to encourage collaboration and effective team communication? How can you promote teamwork, knowledge sharing, open communication and learning over the desire to look smart? What strategies can you employ to maximize your team’s potential and grow their knowledge? Apply these 3 strategies to promote growth mindset in the workplace as a manager: 1. Give growth oriented feedback While giving constructive criticism, ensure your employees feel in control of their situation. Even when they feel vulnerable, they shouldn’t engage in self-doubt. Because feeling that they can’t really do much to fix the situation can lead to pessimism and fixed mindset thinking about their abilities – considering the situation as permanent and pervasive as opposed to attaching a more specific and temporary explanation. They can spiral into hopelessness with the fear that nothing they do can fix their situation – helpless, hopeless, not good enough, not worthy enough. If you want them to take responsibility for their own failures, don’t undermine their sense of control over their own performance. Rather, encourage them to see how a change in their behaviour and actions can be useful to get over this temporary bump in the road. Without a sense of how they can make things better, they might either consider themselves as a failure or get defensive in order to protect their self-esteem, leading to a fixed mindset. Telling people how smart they are when they achieve a big milestone or do otherwise great work also promotes a belief in the fixed mindset – talent and success is all that matters. It can make them feel good in the moment, but it interferes with their ability to handle setbacks later. They refuse to acknowledge mistakes. They consider failure as a reflection on their incompetence. They are reluctant to take up new challenges with the fear they might fail. They try to say and do things to showcase their smartness and prove their brilliance as opposed to embracing opportunities that will make them learn. The right kind of praise isn’t based on applauding people for their talent. It’s given with the purpose to promote a growth mindset – skills and abilities can be built with the right kind of effort. It requires praising people for their effort, persevering through a difficult task, trying new strategies, seeking help and staying determined to achieve their goals. 2. Encourage experimentation Show curiosity to experiment with new ideas. Promote the belief that every new initiative won’t lead to success, but it will lead to a lot of learning. When someone voices their opinion, show enthusiasm to explore it further “tell me more about this idea.” Even if it does not seem right at first or does not match your opinion, do not reject without hearing out first. Rather, help them evaluate their own ideas using a framework to make better decisions: Why is this idea useful? What problem does it solve and why is it important? What are the risks associated with not solving it? Why do they think this is the right time? How much time/effort will it take? Where does it stack against other priorities on our list? Managers can promote a growth mindset in the workplace by making people feel safe to raise their hands and share their viewpoint, to experiment, to explore, and most importantly to value their contribution without judging it based on specific outcomes. When they try something new - a new strategy, a challenging project, recognize it, regardless of whether it leads to success or not. 3. Accept mistakes and share learnings Look at every failed pitch as a learning lesson. Whenever someone makes a mistake or fails at something, instead of pointing fingers, encourage them to adopt a solution mindset by asking these questions: What caused this to not work? Use the five-whys technique designed by Sakichi Toyoda who used it within Toyota Motor Corporation during the evolution of its manufacturing methodologies to get to the underlying root-cause. Encourage employees to shift from self-justification to self-awareness. What strategies can you employ to fix this situation? What has been your most valuable learning from this mistake or failure? Another important strategy to promote a growth mindset in the workplace is to share your own mistakes and failures. When employees see their managers open up about their own vulnerabilities, they feel safe to accept their own shortcomings instead of hiding behind them. Vulnerability is not weakness, and the uncertainty, risk, and emotional exposure we face every day are not optional. Our only choice is a question of engagement. Our willingness to own and engage with our vulnerability determines the depth of our courage and the clarity of our purpose; the level to which we protect ourselves from being vulnerable is a measure of our fear and disconnection - Brene Brown Managers can promote a growth mindset by choosing courage and clarity over fear and disconnection. Strategies to promote growth mindset as a leader While a manager’s impact is typically limited to their team, it’s the leader who sets the tone for the entire organization. They need to take initiatives to promote a growth mindset in the workplace. The idea that effort, hard work, learning and growth is valued more than talent shouldn’t be communicated only through speech, it should be demonstrated in their action. Apply these 3 strategies to promote growth mindset in the workplace as a leader: 1. Create learning opportunities Launch programs where people can learn new skills. A leader of a technology department shouldn’t focus only on their people’s technical skills, but create opportunities for employees to build other skills that are also necessary for their growth. Communication, collaboration, conflict management, decision making, risk-analysis, estimation, strategic planning and prioritization are equally important to have a successful career. Encourage cross-domain learning by enabling people to interact with others outside their department. E.g. people in technology to interact with marketing and design. It not only builds better relationships at work, but also leads to better problem solving. Cross-domain thinking can be extremely powerful to recombine ideas from multiple disciplines and solve problems. People who have adopted cross-domain thinking have revolutionized industries. Charles Babbage’s invention of computational machines powered by punch cards was inspired by his knowledge of the silk-weaving industry, which used cards with holes to create patterns in the silk fabric. Henry Ford’s idea of the car manufacturing assembly line was inspired by Singer sewing machines and meat-packing plants. Claude Shannon combined telephone call-routing technology with his knowledge of boolean algebra to encode and transmit information electronically. David Epstein wrote about how generalists triumph in a specialized world in Range “Increasing specialization has created a system of parallel trenches in the quest for innovation. Everyone is digging deeper into their own trench and rarely standing up to look in the next trench over, even though the solution to their problem happens to reside there...Modern life requires range, making connections across far-flung domains and ideas...Our greatest strength is the exact opposite of narrow specialization. It is the ability to integrate broadly.” 2. Message it right When looking to hire people, don’t talk about hiring the best and the brightest, talk about opportunities to grow, develop and learn. Don’t tolerate toxic behaviour even if it comes from your most valuable employee. Set the right example for everyone by helping the person correct their behaviour or letting them go if they refuse to change. Don’t reward people for their exemplary performance, reward them for their hard work, their strategies, their ability to tackle challenges and for their perseverance. Use language that communicates a growth mindset message “we value learning and growth over talent and success.” Celebrate failures with the desire to learn from them - actively kill ideas that don’t really matter, host idea funerals to create a safe space for people to learn from their failures, change KPIs to reward risks. These are just a few ways to promote the growth mindset in the organization. 3. Overcome groupthink Fixed mindset leads to groupthink, a tendency to conform to ideas and beliefs with conservative thinking, ignoring potential signs of failure, and making decisions with incomplete and biased information. When such behavior becomes part of an organization’s culture, it gives rise to collective blindness to unethical ways, regressive thinking that ignores the future demands of business and a propensity to ignore truth especially if it requires taking a hard stance. Carol Dweck writes in Mindset “There are so many ways the fixed mindset creates groupthink. Leaders are seen as gods who never err. A group invests itself with special talents and powers. Leaders, to bolster their ego, suppress dissent. Or workers, seeking validation from leaders, fall into line behind them. That’s why it’s critical to be in a growth mindset when important decisions are made. By relieving people of the illusions or the burdens of fixed ability—leads to a full and open discussion of the information and to enhanced decision making.” What can leaders do to avoid groupthink and promote growth mindset? She writes “Create ways to foster alternative views and constructive criticism. Assign people to play the devil’s advocate, taking opposing viewpoints so you can see the holes in your position. Get people to wage debates that argue different sides of the issue. Have an anonymous suggestion box that employees must contribute to as part of the decision-making process. Remember, people can be independent thinkers and team players at the same time. Help them fill both roles.” Summary Organizations that promote fixed mindset: Value talent over effort, determination and hard work. They are less concerned about learning and more about achieving success. Managers and leaders in such organizations are less likely to support risk-taking and innovation. They are more focused on establishing their power in the organization than taking care of employees’ well-being. People learn to value smartness over the ability to do smart work. Often leads to a competitive environment with unethical behaviour where success at all costs is valued. Mistakes are considered as personal failings and a sign of incompetence. Organizations that promote growth mindset: Value the ability to learn and grow over innate-talent. Understand that hard work, determination and experimentation are the only way to succeed in the long run. Managers and leaders in such organizations encourage risk-taking and innovation. They empower their people to develop critical thinking skills for better decision-making. People learn to value challenges and opportunities to practice their skills over proving their smartness. They aren’t afraid to voice their opinions and enjoy constructive disagreements. Leads to effective collaboration and teamwork. Mistakes and failures are considered as necessary tools to grow. Previously published here .

How to Cultivate Growth Mindset in the Workplace

Definition, Significance, and Applications TLDR: You can find the whole code . here on GitHub The Central Limit Theorem captures the following phenomenon: Take any distribution! (say a distribution of the number of passes in a football match) Start taking n samples from that distribution (say n = 5) multiple times [say m = 1000] times. Take the mean of each sample set (so we would have m = 1000 means) The distribution of means would be (more or less) . (You will get that famous bell curve if you plot the means on the x-axis and their frequency on the y-axis.) normally distributed Increase n to get a smaller standard deviation and increase m to get a better approximation to normal distribution. But Why Should I Care? Are you unable to load the whole data for processing? No problem, take out multiple samples from the data and use the central limit theorem to estimate the data parameters like mean, standard deviation, sum, etc. It can save you resources in terms of time and money. Because now, we can work on samples significantly smaller than the population and draw inferences for the entire population! Does a certain sample belong to a certain population (or a data set)? Let's check that using the sample mean, the population mean, the sample standard deviation, and the population standard deviation. Definition Given a dataset with an unknown distribution (it could be uniform, binomial, or completely random), the sample means will approximate the normal distribution. Explanation If we take any dataset or a population and we start taking samples from the population, let's say we take 10 samples and take the mean of those samples, and we keep on doing this, a few times, say 1000 times, after doing this, we get 1000 means and when we plot it, we get a distribution called a sampling distribution of sample means. This sampling distribution (more or less) follows a normal distribution! This is the Central Limit theorem. A normal distribution has a number of properties which are useful for analysis. Fig.1 Sampling distribution of sample means (following a normal distribution) Properties of a normal distribution: The mean, mode, and median are all equal. 68% of the data falls within one standard deviation of the mean. 95% of the data falls within two standard deviations of the mean. The curve is symmetric at the center (i.e., around the mean, μ). Moreover, the mean of the sampling distribution of sample means is equal to the population mean. If μ is the population mean and μX̅ is the mean of the sample, means then: Fig.2 population mean = sample mean And the standard deviation of the population(σ) has the following relation to the standard deviation sampling distribution (σX̅): If σ is the standard deviation of population and σX̅ is the standard deviation of sample means, and n is the sample size, then we have Fig.3 Relation between population standard deviation and sampling distribution standard deviation Intuition Since we are taking multiple samples from the population, the means would be equal (or close) to the actual population mean more often than not. Hence, we can expect a peak (mode) in the sampling distribution of sample means equal to the actual population mean. Multiple random samples and their means would lie around the actual population mean. Hence, we can assume 50% of the means would be greater than the population mean and 50% would be less than that (median). If we increase the sample size (from 10 to 20 to 30), more and more of the sample means would fall closer to the population mean. Hence, the average(mean) of those means should be more or less similar to the population mean. Consider the extreme case where the sample size is equal to the population size. So, for each sample, the mean would be the same as the population mean. This is the narrowest distribution (standard deviation of sample means, here is 0). Hence, as we increase the sample size (from 10 to 20 to 30) the standard deviation would tend to decrease (because the spread in the sampling distribution would be limited and more of the sample means would be focused toward the population mean). This phenomenon is captured in the formula in "Fig. 3" where the standard deviation of the sample distribution is inversely proportional to the square root of sample size. If we take more and more samples (from 1,000 to 5,000 to 10,000), then the sampling distribution would be a more smoother curve, because more of the samples would behave according to the central limit theorem, and the pattern would be cleaner. "Talk Is Cheap, Show Me the Code!" - Linus Torvalds So, let’s simulate the central limit theorem via code: Some Imports: import random
from typing import List
import matplotlib.pyplot as plt
import matplotlib
import statistics
import pandas as pd
import math Create a population using . You can try different distributions to generate data. The following code generates a (sort of) monotonically decreasing distribution: random.randint() def create_population(sample_size: int) -> List[int]:
    """Generate a population of sample_size

    Args:
        sample_size (int): The size of the population

    Returns:
        List[int]: a list of randomly generated integers
    """
    population = []
    for _ in range(sample_size):
        random_number = (random.randint(0, random.randint(1, 1000)))
        population.append(random_number)
    return population Create samples, and take their mean number of times: sample_count def generate_sample_mean_list(population: List[int],
                              sample_size: int,
                              sample_count: int) -> List[int]:
    """From the population generate samples of sample_size, sample_count times

    Args:
        population (List[int]): List of random numbers
        sample_size (int): Number of elements in each sample
        sample_count (int): Number of sample means in sample_mean_list

    Returns:
        List[int]: a list of sample means
    """
    sample_mean_list = []
    for _ in range(sample_count):
        sample = random.sample(population, sample_size)
        sample_mean = statistics.mean(sample)
        sample_mean_list.append(sample_mean)
    return sample_mean_list Function to plot the distribution of data along with some labels. def plot_hist(data: List[int],
              ax: matplotlib.axes.Axes,
              xlabel: str,
              ylabel: str,
              title: str,
              texts: List[str]) -> None:
    """Plot a histogram with labels and additional texts

    Args:
        data (List[int]): the list of data points to be plotted
        ax (matplotlib.axes.Axes): Axes object for text plotting
        xlabel (str): label on x axis
        ylabel (str): label on y axis
        title (str): title of the plot
        texts (List[str]): Additional texts to be plotted
    """
    plt.hist(data, 100)
    plt.xlabel(xlabel)
    plt.ylabel(ylabel)
    plt.title(title)

    i = 0.0
    for text in texts:
        plt.text(0.8,
                 0.8 - i,
                 text,
                 horizontalalignment="center",
                 verticalalignment="center",
                 transform=ax.transAxes)
        i += 0.05
    plt.grid(True)
    plt.show() The main function to run the code: def main(plot=True):
    """Driver Function

    Args:
        plot (bool, optional): Decide whether to plot or not. Defaults to True.
    """
    fig, ax = plt.subplots()
    population_size = int(1E5)
    population = create_population(population_size)
    if plot:
        plot_hist(population,
                  ax,
                  "Value",
                  "Frequency",
                  "Histogram of Population of Random Numbers",
                  [f"population_size={population_size}"])

    population_mean = statistics.mean(population)
    population_stdev = statistics.stdev(population)

    sample_size_list = [50, 500]
    sample_count_list = [500, 5000]

    records = []
    for sample_size in sample_size_list:
        for sample_count in sample_count_list:
            sample_mean_list = generate_sample_mean_list(
                population, sample_size, sample_count)

            # also called as mean of sample distribution of sample means
            mean_of_sample_means = round(statistics.mean(sample_mean_list), 2)

            # also called standard dev of sample distribution of sample means
            std_error = round(statistics.stdev(sample_mean_list), 2)
            if plot:
                plot_hist(sample_mean_list,
                          ax,
                          "Mean Value",
                          "Frequency",
                          "Sampling Distribution of Sample Means",
                          [
                              f"sample_count={sample_count}",
                              f"sample_size={sample_size}",
                              f"mean_of_sample_means={mean_of_sample_means}",
                              f"std_error={std_error}"])

            record = {
                "sample_size": sample_size,
                "sample_count": sample_count,
                "population_mean": population_mean,
                "sample_mean": mean_of_sample_means,
                "population_stdev": population_stdev,
                "population_stdev_using_formula": std_error*math.sqrt(sample_size),
                "sample_stdev": std_error,
            }

            records.append(record)

    df = pd.DataFrame(records)
    print(df)


if __name__ == "__main__":
    main(plot=True) You can find the whole code . here on GitHub References: Central Limit Theorem In Action Central Limit Theorem: a real-life application Introduction to the Central Limit Theorem A Gentle Introduction to the Central Limit Theorem for Machine Learning Central Limit Theorem Cover Image Credits: Casey Dunn & Creature Cast on Vimeo Suggested Reading (Suggested Videos): khanacademy/central-limit-theorem Machine Learning Data Science statistics Also published here

Data Science: The Cental Limit Theorem Explained

তথ্য বিজ্ঞান: কেন্দ্রীয় সীমা উপপাদ্য ব্যাখ্যা করা হয়েছে

Data Science: Der zentrale Grenzwertsatz erklärt

Ciencia de datos: explicación del teorema del límite central

Science des données : le théorème central limite expliqué

डेटा साइंस: सेंट्रल लिमिट प्रमेय की व्याख्या

データ サイエンス: 中心極限定理の説明

데이터 과학: 중심 극한 정리 설명

Ciência de dados: o teorema do limite central explicado

Наука о данных: объяснение центральной предельной теоремы

Veri Bilimi: Merkezi Limit Teoreminin Açıklaması

Khoa học dữ liệu: Giải thích định lý giới hạn trung tâm

数据科学：中心极限定理解释

Stories, News & Mentions about THEOREMONE

Theoremone’s Stories on HackerNoon