predict the window of opportunity for screening) National Cancer Institute – What is Cancer 3A computational model for cancer initiation 4 A single cell • Consider a bunch of ‘pseudo’ cells. • Cells have a stochastic process (Markov process). Birth event Mutation event Random Time 𝑢12 𝜆1A computational model for cancer initiation 4 A single cell • Consider a bunch of ‘pseudo’ cells. • Cells have rnd_generator; std::exponential_distribution<> exp{rate}; double time = exp(rnd_generator); 𝑢12 𝜆1A computational model for cancer initiation 5 Healthy cell Cell that harbors a neutral mutation Cancerous

accelerates widely-used deep learning frameworks such as PyTorch. PyTorch is a GPU-accelerated tensor computational framework with a Python front end. Functionality can be easily extended with common Python libraries GEMMs and convolutions with FP16 inputs can run on Tensor Cores, which provide an 8X increase in computational throughput over FP32 arithmetic. APEX AMP is included to support models that currently rely on GEMMs and convolutions with FP16 inputs can run on Tensor Cores, which provide an 8X increase in computational throughput over FP32 arithmetic. APEX AMP is included to support models that currently rely on

INTRODUCTION 1.1 Computers and Theorem Proving Formal verification involves the use of logical and computational methods to establish claims that are expressed in precise mathematical terms. These can include reasoning about complex systems, and to verify claims in both domains. Lean’s underlying logic has a computational interpretation, and Lean can be viewed equally well as a programming lan- guage. More to the point aspects of Lean are explored in a companion tutorial to this one, Programming in Lean, though computational aspects of the system will make an appearance here. 1.2 About Lean The Lean project was launched

device-level token-dropping strategy during training. This approach first computes the average computational budget for each device, which means that the capacity factor for each device is equivalent to al. (2021), we drop tokens with the lowest affinity scores on each device until reaching the computational budget. In addition, we ensure that the tokens belonging to approximately 10% of the training for Computational Linguistics: Human Language Technologies, NAACL-HLT 2019, Minneapolis, MN, USA, June 2-7, 2019, Volume 1 (Long and Short Papers), pages 2368– 2378. Association for Computational Linguistics

which one can define mathematical objects and reason about them. But expressions in Lean have a computational interpretation, which is to say, they can be evaluated. Any closed term of type nat – that is of type nat without free variables – evaluates to a numeral, as long as it is defined in the computational fragment of Lean’s foundational framework. Similarly, any closed term of type list nat evaluates on Lean: instead of thinking of it as a theorem prover whose language just happens to have a computational interpretation, think of it as a programming language that just happens to come equipped with

The evaluator is not very efficient, however, and is not intended to be used for substantial computational tasks. For that purpose, Lean also generates bytecode for every definition of a computable object expression. • #eval can be used to run the bytecode evaluator on any closed term that has a computational interpretation. Lean’s standard library defines a number of data types, such as nat, int, list type. They are used in the standard library to define, for example, the rational numbers, and a computational representation of finite sets (as lists, without duplicates, up to permutation). Inductive types

Complexity @ CppCon 2021 Computational Complexity Computational Complexity or simply Complexity of an algorithm is the amount of resources required to run it. - from Computational complexity in wikipedia The amount of time, storage, or any other resource. 15 Algorithmic Complexity @ CppCon 2021 Computational Complexity n → f(n) n - is size of input f(n) - is the amount of resources required to run Amount of memory required Often denoted by S(n) or s(n) 16Algorithmic Complexity @ CppCon 2021 Computational Complexity Worst-Case Complexity Maximum amount of resources needed over all inputs of size

details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 8 Computational tools 101 8.1 Statistical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . analysis toolkit, Release 0.7.3 100 Chapter 7. Indexing and selecting data CHAPTER EIGHT COMPUTATIONAL TOOLS 8.1 Statistical functions 8.1.1 Covariance The Series object has a method cov to compute s[’d’] = s[’b’] # so there’s a tie In [175]: s.rank() Out[175]: a 2.0 b 3.5 102 Chapter 8. Computational tools pandas: powerful Python data analysis toolkit, Release 0.7.3 c 1.0 d 3.5 e 5.0 rank

arithmetic calculation involving decimal numbers. In AI, total FLOPs are often used to estimate the computational cost of training or running a model. Note: Only language models shown (per Epoch AI, includes development – one that builds on recent exponential gains in model scale, training data, and computational efficiency. Timelines for AGI remain uncertain, but expert expectations have shifted forward scale and sophistication of artificial intelligence is demanding an extraordinary amount of computational horsepower, primarily from AI-focused data centers. These facilities – purpose-built to train

details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 8 Computational tools 95 8.1 Statistical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . data analysis toolkit, Release 0.7.1 94 Chapter 7. Indexing and selecting data CHAPTER EIGHT COMPUTATIONAL TOOLS 8.1 Statistical functions 8.1.1 Covariance The Series object has a method cov to compute s[’d’] = s[’b’] # so there’s a tie In [175]: s.rank() Out[175]: a 2.0 b 3.5 96 Chapter 8. Computational tools pandas: powerful Python data analysis toolkit, Release 0.7.1 c 1.0 d 3.5 e 5.0 rank