THE DESIGN AND ANALYSIS OF EXPERIMENTS. By Oscar Kempthorne. New York, John Wiley and Sons, Inc., 1952. 631 pp. $8.50. This book by a teacher of statistics (as well as a consultant for \"experimenters\") is a comprehensive study of the philosophical background for the statistical design of experiment. It is necessary to have some facility with algebraic notation and manipulation to be able to use the volume intelligently. The problems are presented from the theoretical point of view, without such practical examples as would be helpful for those not acquainted with mathematics. The mathematical justification for the techniques is given. As a somewhat advanced treatment of the design and analysis of experiments, this volume will be interesting and helpful for many who approach statistics theoretically as well as practically. With emphasis on the \"why,\" and with description given broadly, the author relates the subject matter to the general theory of statistics and to the general problem of experimental inference. MARGARET J. ROBERTSON

The Design and Analysis of Experiments

Cluster algebras were conceived by Fomin and Zelevinsky (1) in the spring of 2000 as a tool for studying dual canonical bases and total positivity in semisimple Lie groups. However, the theory of cluster algebras has since taken on a life of its own, as connections and applications have been discovered in diverse areas of mathematics, including representation theory of quivers and finite dimensional algebras, cf., for example, refs. 2⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–15; Poisson geometry (16⇓⇓–19); Teichmuller theory (20⇓⇓⇓–24); string theory (25⇓⇓⇓⇓⇓–31); discrete dynamical systems and integrability (6, 32⇓⇓⇓⇓⇓–38); and combinatorics (39⇓⇓⇓⇓⇓⇓⇓–47).

Quite remarkably, cluster algebras provide a unifying algebraic and combinatorial framework for a wide variety of phenomena in these and other settings. We refer the reader to the survey papers (36, 48⇓⇓⇓⇓–53) and to the cluster algebras portal (www.math.lsa.umich.edu/~fomin/cluster.html) for various introductions to cluster algebras and their links with other subjects in mathematics (and physics).

In brief, a cluster algebra A of rank k is a subring of an ambient field ℱ of rational functions in k variables, say x 1, …, x k . Unlike most commutative rings, a cluster algebra is not presented at the outset via a complete set of generators and relations. Instead, from the data of the initial seed — which includes the k initial cluster variables x 1, …, x k , plus an exchange matrix — one uses an iterative procedure called “mutation” to produce the rest of the cluster variables. In particular, each new cluster … 

[↵][1]1To whom correspondence should be addressed. Email: williams{at}math.berkeley.edu.

 [1]: #xref-corresp-1-1

https://www.pnas.org/content/pnas/111/27/9676.full.pdf

Cluster algebras

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Reflection Groups And Coxeter Groups

In [GHK11], Conjecture 0.6, the first three authors conjectured that the ring of regular functions on a natural class of affine log Calabi-Yau varieties (those with maximal boundary) has a canonical vector space basis parameterized by the integral tropical points of the mirror. Further, the structure constants for the multiplication rule in this basis should be given by counting broken lines (certain combinatorial objects, morally the tropicalisations of holomorphic discs). Here we prove the conjecture in the case of cluster varieties, where the statement is a more precise form of the Fock-Goncharov dual basis conjecture, [FG06], Conjecture 4.3. In particular, under suitable hypotheses, for each Y the partial compactification of an affine cluster variety U given by allowing some frozen variables to vanish, we obtain canonical bases for H0(Y,OY ) extending to a basis of H0(U,OU ). Each choice of seed canonically identifies the parameterizing sets of these bases with integral points in a polyhedral cone. These results specialize to basis results of combinatorial representation theory. For example, by considering the open double Bruhat cell U in the basic affine space Y we obtain a canonical basis of each irreducible representation of SLr, parameterized by a set which each choice of seed identifies with integral points of a lattice polytope. These bases and polytopes are all constructed essentially without representation theoretic considerations. Along the way, our methods prove a number of conjectures in cluster theory, including positivity of the Laurent phenomenon for cluster algebras of geometric type.

/pdf/canonical-bases-for-cluster-algebras-f4gz0rr2up.pdf

Canonical bases for cluster algebras

In previous work, the first three authors conjectured that the ring of regular functions on a natural class of affine log Calabi-Yau varieties (those with maximal boundary) has a canonical vector space basis parameterized by the integral tropical points of the mirror. Further, the structure constants for the multiplication rule in this basis should be given by counting broken lines (certain combinatorial objects, morally the tropicalisations of holomorphic discs). Here we prove the conjecture in the case of cluster varieties, where the statement is a more precise form of the Fock-Goncharov dual basis conjecture. In particular, under suitable hypotheses, for each Y the partial compactification of an affine cluster variety U given by allowing some frozen variables to vanish, we obtain canonical bases for the ring of functions on Y extending to a basis for functions on U. Each choice of seed canonically identifies the parameterizing sets of these bases with integral points in a polyhedral cone. These results specialize to basis results of combinatorial representation theory. For example, by considering the open double Bruhat cell U in the basic affine space Y we obtain a canonical basis of each irreducible representation of SL_r, parameterized by a set which each choice of seed identifies with integral points of a lattice polytope. These bases and polytopes are all constructed essentially without representation theoretic considerations. Along the way, our methods prove a number of conjectures in cluster theory, including positivity of the Laurent phenomenon for cluster algebras of geometric type.

We prove the positivity conjecture for all skew-symmetric cluster algebras.

https://annals.math.princeton.edu/wp-content/uploads/Lee_Schiffler.pdf

Positivity for cluster algebras

Serverless computing is attracting considerable attention recently, but many published papers use micro-benchmarks for evaluation that might result in impracticality. To address this, we present FunctionBench, a suite of practical function workloads for public services. It contains realistic data-oriented applications that utilize various resources during execution. The source codes customized for various cloud service providers are publicly available. We are positive that it suggests opportunities for new function applications with lessen experiment setup overheads.

FunctionBench: A Suite of Workloads for Serverless Cloud Function Service

A lot of recent activity in the theory of cluster algebras has been directed toward various constructions of “natural” bases in them. One of the approaches to this problem was developed several years ago by Sherman and Zelevinsky who have shown that the indecomposable positive elements form an integer basis in any rank 2 cluster algebra of finite or affine type. It is strongly suspected (but not proved) that this property does not extend beyond affine types. Here, we go around this difficulty by constructing a new basis in any rank 2 cluster algebra that we call the greedy basis. It consists of a special family of indecomposable positive elements that we call greedy elements. Inspired by a recent work of Lee and Schiffler; Rupel, we give explicit combinatorial expressions for greedy elements using the language of Dyck paths.

Greedy elements in rank 2 cluster algebras

Cloud computing resources that are equipped with GPU devices are widely used for applications that require extensive parallelism, such as deep learning. When the demand of cloud computing instance is low, the surplus of resources is provided at a lower price in the form of spot instance by AWS EC2. This paper proposes DeepSpotCloud that utilizes GPU-equipped spot instances to run deep learning tasks in a cost efficient and fault-tolerant way. Thorough analysis about spot instance price history logs reveals that GPU spot instances show more dynamic price change pattern than other general types of cloud computing resources. To deal with the price dynamicity of the GPU spot instance, DeepSpotCloud utilizes instances in different regions across continents as a single resource pool. This paper also proposes a task migration heuristic by utilizing a checkpointing mechanism of existing deep learning analysis platform to conduct fast task migration when a running spot instance is interrupted. Extensive experiments using real AWS services prove that the proposed task migration method is effective even in a WAN environment with limited network bandwidth. Comprehensive simulations by replaying AWS EC2 price history logs reveal that DeepSpotCloud can achieve 13% more cost gain than a state-of-the-art interrupt-driven scheduling policy. The prototype of DeepSpotCloud is implemented using various cloud computing services provided by AWS to serve real deep learning tasks.

https://www.computer.org/csdl/pds/api/csdl/proceedings/download-article/12OmNvjgWqP/pdf

DeepSpotCloud: Leveraging Cross-Region GPU Spot Instances for Deep Learning

Let r be any positive integer, and let x 1,x 2 be indeterminates. We consider the sequence {x n } defined by the recursive relation $$x_{n+1} =\bigl(x_n^r +1\bigr)/{x_{n-1}}$$ for any integer n. Finding a combinatorial expression for x n as a rational function of x 1 and x 2 has been an open problem since 2001. We give a direct elementary formula for x n in terms of subpaths of a specific lattice path in the plane. The formula is manifestly positive, providing a new proof of a result by Nakajima and Qin.

/pdf/a-combinatorial-formula-for-rank-2-cluster-variables-33virotbvu.pdf

Kyungyong Lee

Papers

Positivity for cluster algebras

FunctionBench: A Suite of Workloads for Serverless Cloud Function Service

Greedy elements in rank 2 cluster algebras

DeepSpotCloud: Leveraging Cross-Region GPU Spot Instances for Deep Learning

A combinatorial formula for rank 2 cluster variables