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... Let Cw (E) and Bw (E) denote the space of all real valued continuous and Borel measurable functions on E with finite w-norm, respectively [15]. Let P(E) denote the set of all probability measures on E. A sequence R {µn } of measures R on E is said to converge weakly (resp., setwise) [16] to a measur ...
... Let Cw (E) and Bw (E) denote the space of all real valued continuous and Borel measurable functions on E with finite w-norm, respectively [15]. Let P(E) denote the set of all probability measures on E. A sequence R {µn } of measures R on E is said to converge weakly (resp., setwise) [16] to a measur ...
ON FINITE SUMS OF RECIPROCALS OF DISTINCT
... Theorem A is an immediate consequence of a result of the author [2, Theorem 4] together with the fact that every sufficiently large integer is the sum of distinct nth. powers of positive integers (cf., [8], [7] or [3]). The main results* We begin with several definitions. Let S = (su s2, •••) denote ...
... Theorem A is an immediate consequence of a result of the author [2, Theorem 4] together with the fact that every sufficiently large integer is the sum of distinct nth. powers of positive integers (cf., [8], [7] or [3]). The main results* We begin with several definitions. Let S = (su s2, •••) denote ...
Strong Theorems on Coin Tossing - International Mathematical Union
... 1.2. The length of blocks containing at most T tails. Theorems 1.1-1.4 are characterizing the length of the longest run containing no tails at all. One can also ask similar questions about the length of the longest run containing at most T tails. ...
... 1.2. The length of blocks containing at most T tails. Theorems 1.1-1.4 are characterizing the length of the longest run containing no tails at all. One can also ask similar questions about the length of the longest run containing at most T tails. ...
Stats random variables
... some value X? For instance, between 68 and 70 inches P(68 < X < 70)? Because the woman is selected at random, X is a random variable. ...
... some value X? For instance, between 68 and 70 inches P(68 < X < 70)? Because the woman is selected at random, X is a random variable. ...
Handout11B - Harvard Mathematics
... So far, we have only considered probability functions on finite sets and on the set of non-negative integers. The task for this handout is to introduce probability functions on the whole real line, R, or on some subinterval in R such as the interval where 0 ≤ x ≤ 1. Let me start with an example to m ...
... So far, we have only considered probability functions on finite sets and on the set of non-negative integers. The task for this handout is to introduce probability functions on the whole real line, R, or on some subinterval in R such as the interval where 0 ≤ x ≤ 1. Let me start with an example to m ...
Full text
... where b is an integer _> 2. Then we may consider Am to be the base b representation of a number between 0 and 1. Then Theorem 4 implies that this number is irrational if and only if B - BR. As the last result of this paper, we state a theorem giving a description similar to that in Theorem 3 for ano ...
... where b is an integer _> 2. Then we may consider Am to be the base b representation of a number between 0 and 1. Then Theorem 4 implies that this number is irrational if and only if B - BR. As the last result of this paper, we state a theorem giving a description similar to that in Theorem 3 for ano ...
Communication Systems and Networks
... Suppose that the process is non-homogeneous i.e., the rate varies with time, i.e., λ(t) ≤λ, for all t
... Suppose that the process is non-homogeneous i.e., the rate varies with time, i.e., λ(t) ≤λ, for all t
Outline Recall: For integers Euclidean algorithm for finding gcd’s
... Using the physical analogy that Hamming distance is really like distance: the set of all length n codewords with an alphabet with q letters (maybe q = 2) is like a container codewords with specified minimum distance d = 2e + 1 between are like balls of radius e and the question of how many codewords ...
... Using the physical analogy that Hamming distance is really like distance: the set of all length n codewords with an alphabet with q letters (maybe q = 2) is like a container codewords with specified minimum distance d = 2e + 1 between are like balls of radius e and the question of how many codewords ...
A note on the random greedy triangle-packing algorithm
... in which every edge is in exactly one triangle; in other words, the remaining graph is an approximate partial Steiner triple system. If Q drops significantly below this bound then the process will soon terminate. ...
... in which every edge is in exactly one triangle; in other words, the remaining graph is an approximate partial Steiner triple system. If Q drops significantly below this bound then the process will soon terminate. ...
A 1 ∪A 2 ∪…∪A n |=|A 1 |+|A 2 |+…+|A n
... Theorem 4.6: The number of circular rpermutations of a set of n elements is given by p(n,r)/r=n!/r(n-r)! . In particular,the number of circular permutations of n elements is (n-1)! . Proof: The set of linear r-permutations can be partitioned into parts in such a way that two linear r-permutations ar ...
... Theorem 4.6: The number of circular rpermutations of a set of n elements is given by p(n,r)/r=n!/r(n-r)! . In particular,the number of circular permutations of n elements is (n-1)! . Proof: The set of linear r-permutations can be partitioned into parts in such a way that two linear r-permutations ar ...
Arithmetic Sequences
... Mix and Match Activity: Each student receives a card. When prompted, students are to seek the person who has a card that matches their card. Students should be asking appropriate questions using ...
... Mix and Match Activity: Each student receives a card. When prompted, students are to seek the person who has a card that matches their card. Students should be asking appropriate questions using ...
FREE PROBABILITY THEORY Lecture 3 Freeness and Random
... well-known paper of Harer and Zagier, 1986.) ...
... well-known paper of Harer and Zagier, 1986.) ...
Power Spectrum Estimation
... each sample would be represented by a different value of the ensemble of possible sample functions. Hence, it is not a frequency representation of the process but only of one member of the process. It might still be possible, however, to use this function by finding its mean or expected value over t ...
... each sample would be represented by a different value of the ensemble of possible sample functions. Hence, it is not a frequency representation of the process but only of one member of the process. It might still be possible, however, to use this function by finding its mean or expected value over t ...
Average Running Time of the Fast Fourier Transform
... THEOREM 5. As x ten& to infinity, there is a c > 0 such that the second moment of A(n), A*(n), G(n), or P,(n) equals s+l ...
... THEOREM 5. As x ten& to infinity, there is a c > 0 such that the second moment of A(n), A*(n), G(n), or P,(n) equals s+l ...
Ballot theorems for random walks with finite variance
... a simple random walk with step size X. Before stating our ballot theorem, we introduce a small amount of terminology. We say X is a lattice random variable with period d > 0 if there is a constant z such that dX − z is an integer random variable and d is the smallest positive real number for which t ...
... a simple random walk with step size X. Before stating our ballot theorem, we introduce a small amount of terminology. We say X is a lattice random variable with period d > 0 if there is a constant z such that dX − z is an integer random variable and d is the smallest positive real number for which t ...
