# INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem of distinguishing between absolutely continuous and purely singular probability distributions on the real line. In fact, there are no consistent decision rules for distinguishing between absolutely continuous distributions and distributions supported by Borel sets of Hausdorff dimension 0. It follows that there is no consistent sequence of estimators of the Hausdorff dimension of a probability distribution. 1. Introduction Let X 1, X 2,... be independent, identically distributed random variables with common distribution µ. A decision rule for choosing between the null hypothesis that µ A and the alternative hypothesis that µ B, where A and B are mutually exclusive sets of probability distributions, is a sequence (1.1) φ n = φ n (X 1, X 2,..., X n ) of Borel measurable functions taking values in the two-element set {0, 1}. A decision rule is said to be consistent for a probability distribution µ if (1.2) (1.3) lim n(x 1, X 2,..., X n ) = 0 n a.s.(µ) if µ A and lim n(x 1, X 2,..., X n ) = 1 n a.s.(µ) if µ B, and it is said to be consistent if it is consistent for every µ A B. Theorem 1. There is no consistent decision rule for choosing between the null hypothesis that µ is absolutely continuous and the alternative hypothesis that µ is singular. The Hausdorff dimension of a probability distribution µ on the real line is defined to be the infimal Hausdorff dimension of a measurable set with µ probability one. A probability distribution of Hausdorff dimension less than one is necessarily singular, as sets of Hausdorff dimension less than one have Lebesgue measure 0, and so the set of probability distributions with Hausdorff dimension less than one is contained in the set of singular probability distributions. In fact the containment is strict, as there are singular probability distributions with Hausdorff dimension 1. Date: July 26, Mathematics Subject Classification. Primary 62F03 Secondary 62G10. Key words and phrases. consistent decision rule, singular measure, Hausdorff dimension. Supported by NSF grant DMS

2 2 STEVEN P. LALLEY AND ANDREW NOBEL Theorem 2. There is no consistent decision rule for choosing between the null hypothesis that µ is absolutely continuous and the alternative hypothesis that µ has Hausdorff dimension 0. Theorem 2 implies that it is impossible to discriminate between probability distributions of Hausdorff dimension α and probability distributions of Hausdorff dimension β, for any 0 α < β 1. It also implies that it is impossible to consistently estimate the Hausdorff dimension of a probability distribution µ: Corollary 1.1. There is no sequence of estimators θ n = θ n (X 1, X 2,..., X n ) such that (1.4) lim n θ n(x 1, X 2,..., X n ) = dim H (µ) a.s. (µ) Proof. If there were such estimators, then the decision rule φ n := 1{θ n > 1/2} would be consistent for choosing between the null hypothesis that the Hausdorff dimension of µ is 0 and the alternative hypothesis that the Hausdorff dimension of µ is Construction of Singular Measures To prove Theorem 1, we shall prove that, for any decision rule {φ n } n 1 that is universally consistent for absolutely continuous distributions, there is at least one singular distribution for which {φ n } n 1 is not consistent. In fact, we shall construct such a singular distribution; more precisely, we shall construct a random singular distribution µ Γ in such a way that, almost surely, the decision rule {φ n } n 1 is not consistent for µ Γ. Furthermore, the random singular distribution µ Γ will, almost surely, be supported by a set of Hausdorff dimension 0, and therefore it will follow that there is no consistent decision rule for distinguishing between absolutely continuous distributions and singular distributions with Hausdorff dimension α, for any value of α < 1. This will prove Theorem 2. Note: Professor Vladimir Vapnik has informed us that a similar construction to ours was used by N. N. Chentsov [1] in a somewhat different context. We have not been able to decipher Chentsov s arguments. In this section we outline a general procedure for randomly choosing a singular probability distribution of Hausdorff dimension 0. In the following section, we show that the parameters of this construction can be adapted to particular decision rules so as to produce, almost surely, singular distributions for which the decision rules fail to be consistent Condensates of Uniform Distributions. Let F = N J i be a finite union of N nonoverlapping subintervals J i of the unit interval I = [0, 1], each of positive length. For any pair (m, n) of integers both greater than 1, define an (m, n) combing of F to be one of the n mn subsets F of F that can be obtained in the following manner: First, partition each constituent interval J i of F into m nonoverlapping subintervals J i,j of equal lengths J i /m; then partition each subinterval J i,j into n nonoverlapping subintervals J i,j,k of equal lengths J i /mn; choose exactly one interval J i,j,k in each interval J i,j, and let F be the union of these. Note that any

3 INDISTINGUISHABILITY OF DISTRIBUTIONS 3 subset F constructed in this manner must itself be a finite union of mn intervals, and that the Lebesgue measure of F must be 1/n times that of F. Now let µ = µ F be the uniform probability distribution on the set F. For integers m, n 2, define the (m, n) condensates of µ to be the uniform distributions µ F on the (m, n) combings of F, and set (2.1) U m,n (µ) = {(m, n) condensates of µ}. The following simple lemma will be of fundamental importance in the arguments to follow. Its proof is entirely routine, and is therefore omitted. Lemma 2.1. For any integers m, n 2, the uniform distribution µ on a finite union of nonoverlapping intervals F = J i is the average of its (m, n) condensates: 1 (2.2) µ = µ. #U m,n (µ) µ U m,n(µ) Here # denotes cardinality. Notice that Lemma 2.1 has the following interpretation: If one chooses an m, n condensate µ of µ at random, then chooses X at random from the distribution µ, the unconditional distribution of X will be µ Trees in the Space of Absolutely Continuous Distributions. A tree is a connected graph with no nontrivial cycles, equivalently, a graph in which any two vertices are connected by a unique self-avoiding path. If a vertex is designated as the root of the tree, then the vertices may be arranged in layers, according to their distances from the root. Thus, the root is the unique vertex at depth 0, and for any other vertex v the depth of v is the length of the unique self-avoiding path γ from the root to v. The penultimate vertex v on this path is called the parent of v, and v is said to be an offspring of v. Any vertex w through which the path γ passes on its way to its terminus v is designated an ancestor of v, and v is said to be a descendant of w. For any sequence of integer pairs (m n, m n) satisfying m n 2 and m n 2, there is a unique infinite rooted tree T = T ({(m n, m n)}) in the space A of absolutely continuous probability distributions (that is, a tree whose vertices are elements of A) satisfying the following properties: (a) The root node is the uniform distribution on the unit interval. (b) The offspring of any vertex µ at depth n 0 are its (m n, m n) condensates. Observe that each vertex of T is the uniform distribution on a finite union of nonoverlapping intervals of equal lengths, and its support is contained in that of its parent, and, therefore, in the supports of all its ancestors. For each vertex µ at depth n, support(µ) is the union of n m i intervals, each of length 1/ n m im i, and so has Lebesgue measure n (2.3) support(µ) = 1/ m i Ends of the Trees T. Fix sequences of integers m n, m n 2, and let T be the infinite rooted tree in A described above. The ends of T are defined to be the infinite, self-avoiding paths in T beginning at the root. Thus, an end of T is an infinite sequence γ = (µ 0, µ 1, µ 2,... ) of vertices with µ 0 = root and such that each vertex µ n+1 is an offspring of µ n. The boundary (at infinity) of T is the set T of all ends, endowed with the topology of coordinatewise convergence.

4 4 STEVEN P. LALLEY AND ANDREW NOBEL Proposition 2.2. Let γ = (µ 0, µ 1, µ 2,... ) be any end of the tree T. Then (2.4) lim µ D n = µ γ n exists and is a singular distribution. Proof. To show that the sequence µ n converges weakly, it suffices to show that on some probability space are defined random variables X n with distributions µ n such that lim n X n exists almost surely. Such random variables X n may be constructed on any probability space supporting a random variable U with the uniform distribution on the unit interval, by setting (2.5) X n = G 1 n (U), where G n is the cumulative distribution function of the measure µ n. Since γ is a self-avoiding path in T beginning at the root, every step of γ is from a vertex to one of its offspring; consequently, each element of the sequence µ n must be an m n, m n condensate of its predecessor µ n 1. Now if µ is an m, m condensate of µ, where µ is the uniform distribution on the finite union J i of nonoverlapping intervals J i, then µ must assign the same total mass to each of the intervals J i as does µ, and so the cumulative distribution functions of µ and µ have the same values at all endpoints of the intervals J i. Thus, for each n, n X n+1 X n 1/ m i m i 1/4 n. This implies that the random variables X n converge almost surely. Because each µ n has support F n contained in that of its predecessor µ n 1, the support of the weak limit µ γ is the intersection of the sets F n. Since these form a nested sequence of nonempty compact sets, their intersection is a nonempty compact set whose Lebesgue measure is lim n F n. This limit is zero, by equation (2.3). Therefore, µ γ is a singular measure. Proposition 2.3. Assume that n (2.6) lim log m i n log m = 0. n Then for every end γ of T, the support of µ γ has Hausdorff dimension 0. Proof. Since the Hausdorff dimension of any compact set is dominated by its lower Minkowski dimension, it suffices to show that the lower Minkowski dimension of support(µ γ ) is 0. Recall ([2], section 5.3) that the lower Minkowski (box-counting) dimension of a compact set K is defined to be (2.7) dim M (K) = lim inf log N(K; ε)/ log ε 1 ε 0 where N(K; ε) is the minimum number of intervals of length ε needed to cover K. Consider the set K =support(µ γ ). If γ = (µ 0, µ 1,... ), then for each n 0 the support of µ n contains K. But support(µ n ) is the union of n m i intervals, each of length ε n := 1/ n m im i. Consequently, the hypothesis (2.6) implies that lim log N(K; ε n)/ log ε 1 n = 0. n i

5 INDISTINGUISHABILITY OF DISTRIBUTIONS Random Singular Distributions. By Proposition 2.3, every end of the tree T is a singular measure, and by Proposition 2.2, if relation (2.6) holds then every end is supported by a compact set of Hausdorff dimension 0. Thus, if µ Γ is a randomly chosen end of T then it too must have these properties. The simplest and most natural way to choose an end of T at random is to follow the random path (2.8) Γ := (Γ 0, Γ 1, Γ 2,... ), where Γ 0 is the root and Γ n+1 is obtained by choosing randomly among the offspring of Γ n. We shall refer to the sequence Γ n as simple self-avoiding random walk on the tree, and the distribution of the random end µ Γ as the Liouville distribution on the boundary. Proposition 2.4. Assume that µ Γ has the Liouville distribution on T, and that X is a random variable whose (regular) conditional distribution given µ Γ is µ Γ. Then the unconditional distribution of X is the uniform distribution on the unit interval. More generally, the conditional distribution of X given the first k steps Γ 1, Γ 2,..., Γ k of the path Γ is Γ k. Proof. For each n let X n = G 1 n (U), where U is uniformly distributed on the unit interval and G n is the cumulative distribution function of Γ n. As was shown in the proof of Proposition 2.2, the random variables X n converge almost surely to a random variable X whose conditional distribution, given the complete random path Γ, is µ Γ. Therefore, for any integer k 0, the conditional distribution of X given the first k steps Γ 1, Γ 2,..., Γ k is the limit as n of the conditional distribution of X n. By construction, the conditional distribution of X n given the first n steps of the random walk Γ is Γ n. Since Γ n is equidistributed among the offspring of Γ n 1, it follows that the conditional distribution of X n given the first n 1 steps of the path is Γ n 1, by Lemma 2.1. Hence, by induction, the conditional distribution of X n given the first k steps of the path, for any positive integer k < n, is Γ k Recursive Structure of the Liouville Distribution. Each individual step of the random walk Γ consists of randomly choosing an (m n, m n) condensate Γ n of the distribution Γ n 1. This random choice comprises m n N independent random choices of subintervals, m n in each of the N constituent intervals in the support of Γ n 1. Not only are the subchoices in the different intervals independent, but they are of the same type (choosing among subintervals of equal lengths) and use the same distribution (the uniform distribution on a set of cardinality m n). This independence and identity in law extends to the choices made at subsequent depths n + 2, n + 3,... in different intervals of support(γ n 1 ). Thus, the restrictions of the random measure µ Γ to the distinct intervals in support(γ n ) are themselves independent random measures, with the same distribution (up to translations), conditional on Γ n. Consider the very first step Γ 1 of the random walk: this consists of randomly choosing m 1 intervals J i,j(i) of length 1/m 1 m 1, one in each of the intervals J i = (i/m 1, (i+1)/m 1 ], and letting Γ 1 be the uniform distribution on the union of these. The restriction of µ Γ to any of these intervals J i,j(i) is now constructed in essentially the same manner, but using the tree T = T ({(m n+1, m n+1)}) specified by the

6 6 STEVEN P. LALLEY AND ANDREW NOBEL shifted sequence (m n+1, m n+1). We have already argued that the restrictions of µ Γ to different intervals J i,j(i) are independent. Thus, we have proved the following structure theorem for the Liouville distribution. Proposition 2.5. Let (J i,j(i) ) 1 i m1 be a random (m 1, m 1) combing of the unit interval, and for each i let T i be the affine transformation that maps the unit interval onto J i,j(i). Let µ i, for 1 i m 1, be independent, identically distributed random measures, each with the Liouville distribution for the tree T specified by the shifted sequence (m n+1, m n+1). Then (2.9) µ Γ D = 1 m 1 m 1 µ i T 1 i The decomposition (2.9) exhibits µ Γ as a mixture of random probability measures µ i with nonoverlapping supports. This implies that a random variable Y with distribution µ Γ (conditional on Γ) may be obtained by first choosing an interval J i at random from among the m 1 intervals in the initial decomposition of the unit interval; then choosing at random a subinterval J i,j(i) ; then choosing a random measure µ i at random from the Liouville distribution on T ; and then choosing Y at random from µ i T 1 i, where T i is the increasing affine transformation mapping [0, 1] onto J i,j(i). Observe that if interval J i is chosen in the initial step, then the other measures µ i in the decomposition (2.9) play no role in the selection of Y. As the measures µ j are independent, this proves the following: Corollary 2.6. Let µ Γ be a random measure with the Liouville distribution on T, and let Y be a random variable whose conditional distribution, given µ Γ, is µ Γ. Then conditional on the event {Y J i }, the random variable Y is independent of the restriction of µ Γ to [0, 1] \ J i. Because the independent random measures µ i in the decomposition (2.9) are themselves chosen from the Liouville distribution on T, they admit similar decompositions; and the random measures that appear in their decompositions admit similar decompositions; and so on. For each of these decompositions, the argument that led to Corollary 2.6 again applies: Thus, conditional on the event {Y J}, for any interval J of the form n 1 J = (k/m n j=1 n 1 m j m j, (k + 1)/m n j=1 m j m j], the random variable Y is independent of the restriction of µ Γ to [0, 1] \ J Sampling from the Random Singular Distribution µ Γ. Proposition 2.4 implies that if µ Γ is chosen randomly from the Liouville distribution on T, and if X is then chosen randomly from µ Γ, then X is unconditionally distributed uniformly on the unit interval. This property only holds for random samples of size one: If µ Γ has the Liouville distribution, and if X 1, X 2,..., X n are conditionally i.i.d. with distribution µ Γ then the unconditional distribution of the random vector (X 1, X 2,..., X n ) is not that of a random sample of size n from the uniform distribution. Nevertheless, if the integer m 1 in the specification of the tree T is sufficiently large compared to the sample size n, then the unconditional distribution of (X 1, X 2,..., X n ) is close, in the total variation distance, to that of a uniform random sample. More generally, if the sample size is small compared to m k+1, then

7 INDISTINGUISHABILITY OF DISTRIBUTIONS 7 the conditional distribution of (X 1, X 2,..., X n ) given Γ 1, Γ 2,..., Γ k will be close to that of a random sample of size n from the distribution Γ k : Proposition 2.7. Assume that X 1, X 2,..., X n are conditionally i.i.d. with distribution µ Γ, where µ Γ has the Liouville distribution on T. Let Q n k denote the conditional joint distribution of X 1, X 2,..., X n given Γ 1, Γ 2,..., Γ k. Then ( ) k+1 n (2.10) Q n k Γ n k T V / m i 2 Here T V denotes the total variation norm, and Γ n k the product of n copies of Γ k, that is, the joint distribution of a random sample of size n, with replacement, from the distribution Γ k. Proof. It suffices to show that, on a suitable probability space, there are random vectors X = (X 1, X 2,..., X n ) and Y = (Y 1, Y 2,..., Y n ) whose conditional distributions, given Γ 1, Γ 2,..., Γ k, are Q n k and satisfy ( ) k+1 n (2.11) P {X Y} / m i. 2 and Γ n k, respectively, For ease of exposition, we shall consider the case k = 0, so that Γ 0 is the uniform distribution on the unit interval; the general case k 0 is similar. Let Y be a random sample of size n from the uniform distribution. For each 1 i n, define J i to be the interval I j := (j/m 1, (j+1)/m 1 ] containing Y i. Since there are precisely m 1 intervals from which to choose, the vector J = (J 1, J 2,..., J n ) constitutes a random sample (with replacement!) of size n from a population of size m 1. The probability that two elements of this random sample coincide is bounded by the expected number of coincident pairs; thus, ( ) n P {#{J 1, J 2,..., J n } < n} /m 1. 2 To complete the proof of inequality (2.11), we will show that random vectors X and Y with the desired conditional distributions may be constructed in such a way that X = Y on the event that the entries J 1, J 2,..., J n are distinct. Consider the following sampling procedures: Procedure A: Choose intervals J 1, J 2,..., J n by sampling with replacement from the m 1 element set I 1, I 2,..., I m1. For each index i n, choose an interval J i,j(i) at random from among the m 1 nonoverlapping subintervals of J i of length 1/m 1 m 1, and independently choose µ i at random from the Liouville distribution on T. Finally, choose Y i at random from µ i T 1 i, where T i is the increasing affine transformation mapping the unit interval onto J i,j(i). Procedure B: Choose intervals J 1, J 2,..., J n by sampling with replacement from the m 1 element set I 1, I 2,..., I m1. For each index i n, choose an interval J i,j(i) at random from among the m 1 nonoverlapping subintervals of J i of length 1/m 1 m 1, and independently choose µ i at random from the Liouville distribution on T,

8 8 STEVEN P. LALLEY AND ANDREW NOBEL provided that the interval J i has not occurred earlier in the sequence J i ; otherwise, set µ i = µ i and J i,j(i) = J i,j(i ), where i is the smallest index where J i = J i. Finally, choose X i at random from the distribution µ i T 1 i. By Proposition 2.4 and Corollary 2.6, Procedure A produces a random sample Y from the uniform distribution, and Procedure B produces a random sample X with distribution Q n 0. Clearly, the random choices in Procedures A and B can be made in such a way that the two samples coincide if the sample J 1, J 2,..., J n contains no duplicates. The case k 1 is similar; the only difference is that the intervals J i are chosen from a population of size k+1 m i. 3. Indistinguishability of Absolutely Continuous and Singular Measures Let φ n = φ n (X 1, X 2,..., X n ) be a decision rule for choosing between the null hypothesis that µ is absolutely continuous and the alternative hypothesis that µ has Hausdorff dimension 0. Assume that this decision rule is consistent for all absolutely continuous probability distributions, that is, (3.1) lim n φ n(x 1, X 2,..., X n ) = 0 a.s.(µ). We will show that sequences m n, m n of integers greater than 1 can be constructed in such a way that if µ Γ is chosen from the Liouville distribution on T, where T = T ({(m n, m n)}), then with probability one µ Γ has Hausdorff dimension 0, and has the property that the decision rule φ n is inconsistent for µ Γ. The tree is constructed one layer at a time, starting from the root (the uniform distribution on [0, 1]). Specification of the first k entries of the sequences m n, m n determines the vertices V k and edges of T to depth k. The sequences m n, m n, along with a third sequence ν n are chosen so that, for each n 1, n (3.2) log m n n log m n (3.3) (3.4) ( ) n+1 νn / m i e n ; 2 and µ{φ k (X 1, X 2,..., X k ) = 1 for some k ν n } e n µ V n 1. The consistency hypothesis (3.1) and the fact that each set V k is finite ensure that, for each n 1, a positive integer ν n > ν n can be chosen so that inequality (3.4) holds. Once ν n is determined, m n+1 is chosen so that inequality (3.3) holds, and then m n+1 may be taken so large that (3.2) holds. Inequality (3.2) guarantees that for any end γ of the tree T, the distribution µ γ will have Hausdorff dimension 0, by Proposition 2.3. Proposition 3.1. Let µ Γ be a random probability measure chosen from the Liouville distribution on T, where T is the tree specified by sequences m n and m n satisfying relations (3.2), (3.3), and (3.4). Let X 1, X 2,... be random variables

9 INDISTINGUISHABILITY OF DISTRIBUTIONS 9 that are conditionally i.i.d. with common conditional distribution µ Γ. Then with probability one, (3.5) lim inf n φ n(x 1, X 2,..., X n ) = 0. Proof. The conditional distribution of the random vector (X 1, X 2,..., X νn+1 ), given the first n steps of the random walk Γ, differs in total variation norm from the product measure Γ νn+1 n by less than e n, by inequality (3.3) and Proposition 2.7. Consequently, by inequality (3.4), P {φ νn+1 (X 1, X 2,..., X νn+1 ) = 1} 2e n. By the Borel-Cantelli Lemma, it must be that, with probability one, for all sufficiently large n, φ νn+1 (X 1, X 2,..., X νn+1 ) = 0. Corollary 3.2. With probability one, the singular random measure µ Γ property that the decision rule φ n is inconsistent for µ Γ. has the References [1] N. N. Chentsov (1981) Correctness of the problem of statistical point estimation. (Russian) Teor. Veroyatnost. i Primenen [2] P. Mattila (1995) Geometry of Sets and Measures in Euclidean Spaces. Cambridge U. Press. University of Chicago, Department of Statistics, 5734 University Avenue, Chicago IL address: University of North Carolina, Department of Statistics, Chapel Hill NC address:

### The Ideal Class Group

Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned

### Random graphs with a given degree sequence

Sourav Chatterjee (NYU) Persi Diaconis (Stanford) Allan Sly (Microsoft) Let G be an undirected simple graph on n vertices. Let d 1,..., d n be the degrees of the vertices of G arranged in descending order.

### MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

### Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely

### The Henstock-Kurzweil-Stieltjes type integral for real functions on a fractal subset of the real line

The Henstock-Kurzweil-Stieltjes type integral for real functions on a fractal subset of the real line D. Bongiorno, G. Corrao Dipartimento di Ingegneria lettrica, lettronica e delle Telecomunicazioni,

### Classification of Cartan matrices

Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms

### Ideal Class Group and Units

Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals

### Mathematics for Econometrics, Fourth Edition

Mathematics for Econometrics, Fourth Edition Phoebus J. Dhrymes 1 July 2012 1 c Phoebus J. Dhrymes, 2012. Preliminary material; not to be cited or disseminated without the author s permission. 2 Contents

### THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear

### SMALL SKEW FIELDS CÉDRIC MILLIET

SMALL SKEW FIELDS CÉDRIC MILLIET Abstract A division ring of positive characteristic with countably many pure types is a field Wedderburn showed in 1905 that finite fields are commutative As for infinite

### 3. INNER PRODUCT SPACES

. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

### CHAPTER 5. Product Measures

54 CHAPTER 5 Product Measures Given two measure spaces, we may construct a natural measure on their Cartesian product; the prototype is the construction of Lebesgue measure on R 2 as the product of Lebesgue

### An example of a computable

An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept

### IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible

### On an anti-ramsey type result

On an anti-ramsey type result Noga Alon, Hanno Lefmann and Vojtĕch Rödl Abstract We consider anti-ramsey type results. For a given coloring of the k-element subsets of an n-element set X, where two k-element

### The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

### About the inverse football pool problem for 9 games 1

Seventh International Workshop on Optimal Codes and Related Topics September 6-1, 013, Albena, Bulgaria pp. 15-133 About the inverse football pool problem for 9 games 1 Emil Kolev Tsonka Baicheva Institute

### SOLUTIONS TO ASSIGNMENT 1 MATH 576

SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts

### Some Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.

Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,

### Extension of measure

1 Extension of measure Sayan Mukherjee Dynkin s π λ theorem We will soon need to define probability measures on infinite and possible uncountable sets, like the power set of the naturals. This is hard.

### Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs

CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like

### Subsets of Euclidean domains possessing a unique division algorithm

Subsets of Euclidean domains possessing a unique division algorithm Andrew D. Lewis 2009/03/16 Abstract Subsets of a Euclidean domain are characterised with the following objectives: (1) ensuring uniqueness

### THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING

THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING 1. Introduction The Black-Scholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages

Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650

### CHAPTER 1 BASIC TOPOLOGY

CHAPTER 1 BASIC TOPOLOGY Topology, sometimes referred to as the mathematics of continuity, or rubber sheet geometry, or the theory of abstract topological spaces, is all of these, but, above all, it is

### A 2-factor in which each cycle has long length in claw-free graphs

A -factor in which each cycle has long length in claw-free graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science

### INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES

INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES FERNANDO BOMBAL AND IGNACIO VILLANUEVA Abstract. We study and characterize the integral multilinear operators on a product of C(K) spaces in terms of the

### A Sublinear Bipartiteness Tester for Bounded Degree Graphs

A Sublinear Bipartiteness Tester for Bounded Degree Graphs Oded Goldreich Dana Ron February 5, 1998 Abstract We present a sublinear-time algorithm for testing whether a bounded degree graph is bipartite

### Dedekind s forgotten axiom and why we should teach it (and why we shouldn t teach mathematical induction in our calculus classes)

Dedekind s forgotten axiom and why we should teach it (and why we shouldn t teach mathematical induction in our calculus classes) by Jim Propp (UMass Lowell) March 14, 2010 1 / 29 Completeness Three common

### The Union-Find Problem Kruskal s algorithm for finding an MST presented us with a problem in data-structure design. As we looked at each edge,

The Union-Find Problem Kruskal s algorithm for finding an MST presented us with a problem in data-structure design. As we looked at each edge, cheapest first, we had to determine whether its two endpoints

### o-minimality and Uniformity in n 1 Graphs

o-minimality and Uniformity in n 1 Graphs Reid Dale July 10, 2013 Contents 1 Introduction 2 2 Languages and Structures 2 3 Definability and Tame Geometry 4 4 Applications to n 1 Graphs 6 5 Further Directions

### CONTINUED FRACTIONS AND FACTORING. Niels Lauritzen

CONTINUED FRACTIONS AND FACTORING Niels Lauritzen ii NIELS LAURITZEN DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF AARHUS, DENMARK EMAIL: niels@imf.au.dk URL: http://home.imf.au.dk/niels/ Contents

### 1. Introduction. PROPER HOLOMORPHIC MAPPINGS BETWEEN RIGID POLYNOMIAL DOMAINS IN C n+1

Publ. Mat. 45 (2001), 69 77 PROPER HOLOMORPHIC MAPPINGS BETWEEN RIGID POLYNOMIAL DOMAINS IN C n+1 Bernard Coupet and Nabil Ourimi Abstract We describe the branch locus of proper holomorphic mappings between

### PSEUDOARCS, PSEUDOCIRCLES, LAKES OF WADA AND GENERIC MAPS ON S 2

PSEUDOARCS, PSEUDOCIRCLES, LAKES OF WADA AND GENERIC MAPS ON S 2 Abstract. We prove a Bruckner-Garg type theorem for the fiber structure of a generic map from a continuum X into the unit interval I. We

### 11 Ideals. 11.1 Revisiting Z

11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(

### The degree, size and chromatic index of a uniform hypergraph

The degree, size and chromatic index of a uniform hypergraph Noga Alon Jeong Han Kim Abstract Let H be a k-uniform hypergraph in which no two edges share more than t common vertices, and let D denote the

### Classification/Decision Trees (II)

Classification/Decision Trees (II) Department of Statistics The Pennsylvania State University Email: jiali@stat.psu.edu Right Sized Trees Let the expected misclassification rate of a tree T be R (T ).

### 1 Sets and Set Notation.

LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most

### ABSTRACT. For example, circle orders are the containment orders of circles (actually disks) in the plane (see [8,9]).

Degrees of Freedom Versus Dimension for Containment Orders Noga Alon 1 Department of Mathematics Tel Aviv University Ramat Aviv 69978, Israel Edward R. Scheinerman 2 Department of Mathematical Sciences

### ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE. 1. Introduction

ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE J.M. CALABUIG, J. RODRÍGUEZ, AND E.A. SÁNCHEZ-PÉREZ Abstract. Let m be a vector measure taking values in a Banach space X. We prove that

### ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD. 1. Introduction

ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD A.D. SCOTT Abstract. Gallai proved that the vertex set of any graph can be partitioned into two sets, each inducing a subgraph with all degrees even. We prove

### ON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION. E. I. Pancheva, A. Gacovska-Barandovska

Pliska Stud. Math. Bulgar. 22 (2015), STUDIA MATHEMATICA BULGARICA ON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION E. I. Pancheva, A. Gacovska-Barandovska Smirnov (1949) derived four

### DEGREES OF CATEGORICITY AND THE HYPERARITHMETIC HIERARCHY

DEGREES OF CATEGORICITY AND THE HYPERARITHMETIC HIERARCHY BARBARA F. CSIMA, JOHANNA N. Y. FRANKLIN, AND RICHARD A. SHORE Abstract. We study arithmetic and hyperarithmetic degrees of categoricity. We extend

### Mean Ramsey-Turán numbers

Mean Ramsey-Turán numbers Raphael Yuster Department of Mathematics University of Haifa at Oranim Tivon 36006, Israel Abstract A ρ-mean coloring of a graph is a coloring of the edges such that the average

### RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES

RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES GAUTAM BHARALI AND INDRANIL BISWAS Abstract. In the study of holomorphic maps, the term rigidity refers to certain types of results that give us very specific

### Universal Algorithm for Trading in Stock Market Based on the Method of Calibration

Universal Algorithm for Trading in Stock Market Based on the Method of Calibration Vladimir V yugin Institute for Information Transmission Problems, Russian Academy of Sciences, Bol shoi Karetnyi per.

### Notes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand

Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such

### WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?

WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly

### A Turán Type Problem Concerning the Powers of the Degrees of a Graph

A Turán Type Problem Concerning the Powers of the Degrees of a Graph Yair Caro and Raphael Yuster Department of Mathematics University of Haifa-ORANIM, Tivon 36006, Israel. AMS Subject Classification:

### Discernibility Thresholds and Approximate Dependency in Analysis of Decision Tables

Discernibility Thresholds and Approximate Dependency in Analysis of Decision Tables Yu-Ru Syau¹, En-Bing Lin²*, Lixing Jia³ ¹Department of Information Management, National Formosa University, Yunlin, 63201,

### High degree graphs contain large-star factors

High degree graphs contain large-star factors Dedicated to László Lovász, for his 60th birthday Noga Alon Nicholas Wormald Abstract We show that any finite simple graph with minimum degree d contains a

University of Oslo MAT2 Project The Banach-Tarski Paradox Author: Fredrik Meyer Supervisor: Nadia S. Larsen Abstract In its weak form, the Banach-Tarski paradox states that for any ball in R, it is possible

### Wald s Identity. by Jeffery Hein. Dartmouth College, Math 100

Wald s Identity by Jeffery Hein Dartmouth College, Math 100 1. Introduction Given random variables X 1, X 2, X 3,... with common finite mean and a stopping rule τ which may depend upon the given sequence,

### MINITAB ASSISTANT WHITE PAPER

MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. One-Way

### Cycles in a Graph Whose Lengths Differ by One or Two

Cycles in a Graph Whose Lengths Differ by One or Two J. A. Bondy 1 and A. Vince 2 1 LABORATOIRE DE MATHÉMATIQUES DISCRÉTES UNIVERSITÉ CLAUDE-BERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS

### CONTRIBUTIONS TO ZERO SUM PROBLEMS

CONTRIBUTIONS TO ZERO SUM PROBLEMS S. D. ADHIKARI, Y. G. CHEN, J. B. FRIEDLANDER, S. V. KONYAGIN AND F. PAPPALARDI Abstract. A prototype of zero sum theorems, the well known theorem of Erdős, Ginzburg

### Low upper bound of ideals, coding into rich Π 0 1 classes

Low upper bound of ideals, coding into rich Π 0 1 classes Antonín Kučera the main part is a joint project with T. Slaman Charles University, Prague September 2007, Chicago The main result There is a low

CS787: Advanced Algorithms Topic: Online Learning Presenters: David He, Chris Hopman 17.3.1 Follow the Perturbed Leader 17.3.1.1 Prediction Problem Recall the prediction problem that we discussed in class.

### THE DEGREES OF BI-HYPERHYPERIMMUNE SETS

THE DEGREES OF BI-HYPERHYPERIMMUNE SETS URI ANDREWS, PETER GERDES, AND JOSEPH S. MILLER Abstract. We study the degrees of bi-hyperhyperimmune (bi-hhi) sets. Our main result characterizes these degrees

### Inner Product Spaces and Orthogonality

Inner Product Spaces and Orthogonality week 3-4 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,

### Catalan Numbers. Thomas A. Dowling, Department of Mathematics, Ohio State Uni- versity.

7 Catalan Numbers Thomas A. Dowling, Department of Mathematics, Ohio State Uni- Author: versity. Prerequisites: The prerequisites for this chapter are recursive definitions, basic counting principles,

### ON FIBER DIAMETERS OF CONTINUOUS MAPS

ON FIBER DIAMETERS OF CONTINUOUS MAPS PETER S. LANDWEBER, EMANUEL A. LAZAR, AND NEEL PATEL Abstract. We present a surprisingly short proof that for any continuous map f : R n R m, if n > m, then there

### 3. Linear Programming and Polyhedral Combinatorics

Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the

### CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian. Pasquale Candito and Giuseppina D Aguí

Opuscula Math. 34 no. 4 2014 683 690 http://dx.doi.org/10.7494/opmath.2014.34.4.683 Opuscula Mathematica CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian Pasquale

### Large induced subgraphs with all degrees odd

Large induced subgraphs with all degrees odd A.D. Scott Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England Abstract: We prove that every connected graph of order

### Adaptive Search with Stochastic Acceptance Probabilities for Global Optimization

Adaptive Search with Stochastic Acceptance Probabilities for Global Optimization Archis Ghate a and Robert L. Smith b a Industrial Engineering, University of Washington, Box 352650, Seattle, Washington,

### Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay

Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture - 17 Shannon-Fano-Elias Coding and Introduction to Arithmetic Coding

### Solutions of Equations in One Variable. Fixed-Point Iteration II

Solutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011

### From Binomial Trees to the Black-Scholes Option Pricing Formulas

Lecture 4 From Binomial Trees to the Black-Scholes Option Pricing Formulas In this lecture, we will extend the example in Lecture 2 to a general setting of binomial trees, as an important model for a single

### OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN. E.V. Grigorieva. E.N. Khailov

DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Supplement Volume 2005 pp. 345 354 OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN E.V. Grigorieva Department of Mathematics

### SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by

SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples

### A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS. In memory of Rou-Huai Wang

A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS XAVIER CABRÉ, MANEL SANCHÓN, AND JOEL SPRUCK In memory of Rou-Huai Wang 1. Introduction In this note we consider semistable

### Section 1.1. Introduction to R n

The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

### 4: SINGLE-PERIOD MARKET MODELS

4: SINGLE-PERIOD MARKET MODELS Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2015 B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market

### TAKE-AWAY GAMES. ALLEN J. SCHWENK California Institute of Technology, Pasadena, California INTRODUCTION

TAKE-AWAY GAMES ALLEN J. SCHWENK California Institute of Technology, Pasadena, California L INTRODUCTION Several games of Tf take-away?f have become popular. The purpose of this paper is to determine the

### Corollary. (f є C n+1 [a,b]). Proof: This follows directly from the preceding theorem using the inequality

Corollary For equidistant knots, i.e., u i = a + i (b-a)/n, we obtain with (f є C n+1 [a,b]). Proof: This follows directly from the preceding theorem using the inequality 120202: ESM4A - Numerical Methods

### Turing Degrees and Definability of the Jump. Theodore A. Slaman. University of California, Berkeley. CJuly, 2005

Turing Degrees and Definability of the Jump Theodore A. Slaman University of California, Berkeley CJuly, 2005 Outline Lecture 1 Forcing in arithmetic Coding and decoding theorems Automorphisms of countable

### A CONSTRUCTION OF THE UNIVERSAL COVER AS A FIBER BUNDLE

A CONSTRUCTION OF THE UNIVERSAL COVER AS A FIBER BUNDLE DANIEL A. RAMRAS In these notes we present a construction of the universal cover of a path connected, locally path connected, and semi-locally simply

### PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

### MATH1231 Algebra, 2015 Chapter 7: Linear maps

MATH1231 Algebra, 2015 Chapter 7: Linear maps A/Prof. Daniel Chan School of Mathematics and Statistics University of New South Wales danielc@unsw.edu.au Daniel Chan (UNSW) MATH1231 Algebra 1 / 43 Chapter

### Ph.D. Thesis. Judit Nagy-György. Supervisor: Péter Hajnal Associate Professor

Online algorithms for combinatorial problems Ph.D. Thesis by Judit Nagy-György Supervisor: Péter Hajnal Associate Professor Doctoral School in Mathematics and Computer Science University of Szeged Bolyai

### Degrees that are not degrees of categoricity

Degrees that are not degrees of categoricity Bernard A. Anderson Department of Mathematics and Physical Sciences Gordon State College banderson@gordonstate.edu www.gordonstate.edu/faculty/banderson Barbara

### COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS

COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS Alexander Burstein Department of Mathematics Howard University Washington, DC 259, USA aburstein@howard.edu Sergey Kitaev Mathematics

### On The Existence Of Flips

On The Existence Of Flips Hacon and McKernan s paper, arxiv alg-geom/0507597 Brian Lehmann, February 2007 1 Introduction References: Hacon and McKernan s paper, as above. Kollár and Mori, Birational Geometry

### Unique Factorization

Unique Factorization Waffle Mathcamp 2010 Throughout these notes, all rings will be assumed to be commutative. 1 Factorization in domains: definitions and examples In this class, we will study the phenomenon

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### Lecture 6 Online and streaming algorithms for clustering

CSE 291: Unsupervised learning Spring 2008 Lecture 6 Online and streaming algorithms for clustering 6.1 On-line k-clustering To the extent that clustering takes place in the brain, it happens in an on-line

### All trees contain a large induced subgraph having all degrees 1 (mod k)

All trees contain a large induced subgraph having all degrees 1 (mod k) David M. Berman, A.J. Radcliffe, A.D. Scott, Hong Wang, and Larry Wargo *Department of Mathematics University of New Orleans New

### On a conjecture by Palis

1179 2000 103-108 103 On a conjecture by Palis (Shuhei Hayashi) Introduction Let \$M\$ be a smooth compact manifold without boundary and let Diff1 \$(M)\$ be the set of \$C^{1}\$ diffeomorphisms with the \$C^{1}\$

### FIXED POINT SETS OF FIBER-PRESERVING MAPS

FIXED POINT SETS OF FIBER-PRESERVING MAPS Robert F. Brown Department of Mathematics University of California Los Angeles, CA 90095 e-mail: rfb@math.ucla.edu Christina L. Soderlund Department of Mathematics

### Reading material on the limit set of a Fuchsian group

Reading material on the limit set of a Fuchsian group Recommended texts Many books on hyperbolic geometry and Kleinian and Fuchsian groups contain material about limit sets. The presentation given here

### This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.

IEEE/ACM TRANSACTIONS ON NETWORKING 1 A Greedy Link Scheduler for Wireless Networks With Gaussian Multiple-Access and Broadcast Channels Arun Sridharan, Student Member, IEEE, C Emre Koksal, Member, IEEE,

### ON DEGREES IN THE HASSE DIAGRAM OF THE STRONG BRUHAT ORDER

Séminaire Lotharingien de Combinatoire 53 (2006), Article B53g ON DEGREES IN THE HASSE DIAGRAM OF THE STRONG BRUHAT ORDER RON M. ADIN AND YUVAL ROICHMAN Abstract. For a permutation π in the symmetric group

### Shape Optimization Problems over Classes of Convex Domains

Shape Optimization Problems over Classes of Convex Domains Giuseppe BUTTAZZO Dipartimento di Matematica Via Buonarroti, 2 56127 PISA ITALY e-mail: buttazzo@sab.sns.it Paolo GUASONI Scuola Normale Superiore

### Chapter ML:IV. IV. Statistical Learning. Probability Basics Bayes Classification Maximum a-posteriori Hypotheses

Chapter ML:IV IV. Statistical Learning Probability Basics Bayes Classification Maximum a-posteriori Hypotheses ML:IV-1 Statistical Learning STEIN 2005-2015 Area Overview Mathematics Statistics...... Stochastics

### TOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS

TOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS VIPUL NAIK Abstract. In this journey, we are going to explore the so called separation axioms in greater detail. We shall try to understand how these axioms

### Tail inequalities for order statistics of log-concave vectors and applications

Tail inequalities for order statistics of log-concave vectors and applications Rafał Latała Based in part on a joint work with R.Adamczak, A.E.Litvak, A.Pajor and N.Tomczak-Jaegermann Banff, May 2011 Basic

### This chapter is all about cardinality of sets. At first this looks like a

CHAPTER Cardinality of Sets This chapter is all about cardinality of sets At first this looks like a very simple concept To find the cardinality of a set, just count its elements If A = { a, b, c, d },