Random geometric complexes in the thermodynamic regime
... ergodic theorems for χ(CB (Φ, r)) when the underlying point process Φ is itself ergodic. More recently, a slew of results have been established for χ(CB (P, r)) (i.e. the Poisson case) in the preprint [17]. The arguments in this paper replace more classic integral geometric arguments, and are based ...
... ergodic theorems for χ(CB (Φ, r)) when the underlying point process Φ is itself ergodic. More recently, a slew of results have been established for χ(CB (P, r)) (i.e. the Poisson case) in the preprint [17]. The arguments in this paper replace more classic integral geometric arguments, and are based ...
Introduction to HyperReals
... infinitesimal if b is either positive infinitesimal, negative infinitesimal, or 0. ...
... infinitesimal if b is either positive infinitesimal, negative infinitesimal, or 0. ...
Bayesian Belief Net: Tutorial
... in a train strike (A) then this in turn will increase our belief in both Martin being late (B) and Norman being late (C). Of more interest is whether information about B can be transmitted to C (and vice versa). Suppose we have no hard evidence about A (that is, we do not know for certain whether or ...
... in a train strike (A) then this in turn will increase our belief in both Martin being late (B) and Norman being late (C). Of more interest is whether information about B can be transmitted to C (and vice versa). Suppose we have no hard evidence about A (that is, we do not know for certain whether or ...
Full text
... and so on. It is clear that when n ∈ N is large, the probability that a number chosen at random from {0, 1, 2, . . . , n} will end in 1 when written in binary is approximately onehalf. In fact, by taking n sufficiently large, the probability that a randomly-chosen number from {0, 1, 2, . . . , n} wi ...
... and so on. It is clear that when n ∈ N is large, the probability that a number chosen at random from {0, 1, 2, . . . , n} will end in 1 when written in binary is approximately onehalf. In fact, by taking n sufficiently large, the probability that a randomly-chosen number from {0, 1, 2, . . . , n} wi ...
Full text
... The next theorem indicates precisely which real numbers have an alpha expansion whose defining sequence k(i) does not include any two consecutive integers. Theorem 2.3. The real number θ has an alpha expansion whose defining sequence {k(i)} does not include any two consecutive integers if and only i ...
... The next theorem indicates precisely which real numbers have an alpha expansion whose defining sequence k(i) does not include any two consecutive integers. Theorem 2.3. The real number θ has an alpha expansion whose defining sequence {k(i)} does not include any two consecutive integers if and only i ...
CENTRAL LIMIT THEOREM FOR THE EXCITED RANDOM WALK
... (ii) (Central limit theorem). There exists σ = σ(p, d), 0 < σ < +∞, such that t 7→ n−1/2 (Xbntc · e1 − vbntc), converges in law as n → +∞ to a Brownian motion with variance σ 2 . Our proof is based on the well-known construction of regeneration times for the random walk, the key issue being to obtai ...
... (ii) (Central limit theorem). There exists σ = σ(p, d), 0 < σ < +∞, such that t 7→ n−1/2 (Xbntc · e1 − vbntc), converges in law as n → +∞ to a Brownian motion with variance σ 2 . Our proof is based on the well-known construction of regeneration times for the random walk, the key issue being to obtai ...
1 - WordPress.com
... 8. Imagine a toy spinner with the numbers with the numbers 1, 2, 3, 3, and 8. Printed on it. When the arrow is spun, it is equally likely to land on any of the five numbers. (The spinner is pictured below.) a. Suppose you spin it twice. What is the polynomial multiplication associated with this pro ...
... 8. Imagine a toy spinner with the numbers with the numbers 1, 2, 3, 3, and 8. Printed on it. When the arrow is spun, it is equally likely to land on any of the five numbers. (The spinner is pictured below.) a. Suppose you spin it twice. What is the polynomial multiplication associated with this pro ...
Stochastic Processes - Institut Camille Jordan
... More generally, when then are defined, the quantities E[X k ], k ∈ N, are called the moments of X. Definition 4.19. We have defined random variables as being real-valued only, but in Quantum Probability Theory one often considers complex-valued functions of random variables. If X is a real-valued ra ...
... More generally, when then are defined, the quantities E[X k ], k ∈ N, are called the moments of X. Definition 4.19. We have defined random variables as being real-valued only, but in Quantum Probability Theory one often considers complex-valued functions of random variables. If X is a real-valued ra ...
1991-Analyses of Instance-Based Learning Algorithms
... bounded by E. Also, A is not required to always generate sufficiently accurake concept descriptions (i.e., witkn c), but must do so only wit11 probability at least (1-S). The polynomial time bound is automatic for IBl since the amount of time it takes to genera.te a prediction is polynomia.1 in the ...
... bounded by E. Also, A is not required to always generate sufficiently accurake concept descriptions (i.e., witkn c), but must do so only wit11 probability at least (1-S). The polynomial time bound is automatic for IBl since the amount of time it takes to genera.te a prediction is polynomia.1 in the ...
Signal Detection Theory handout
... of response, due to thermal isomerizations of photopigment molecules. Barlow called this the “dark light” because a spontaneous isomerization will lead to the same neural signal as if a photon was actually absorbed. The subject will not be able to tell the difference between real light and dark ligh ...
... of response, due to thermal isomerizations of photopigment molecules. Barlow called this the “dark light” because a spontaneous isomerization will lead to the same neural signal as if a photon was actually absorbed. The subject will not be able to tell the difference between real light and dark ligh ...
A Characterization of Entropy in Terms of Information Loss
... Then H(p) = ln 2, while H(q) = 0. The information loss associated with the map f is defined to be H(p) − H(q), which in this case equals ln 2. In other words, the measure-preserving map f loses one bit of information. On the other hand, f is also measure-preserving if we replace p by the probability ...
... Then H(p) = ln 2, while H(q) = 0. The information loss associated with the map f is defined to be H(p) − H(q), which in this case equals ln 2. In other words, the measure-preserving map f loses one bit of information. On the other hand, f is also measure-preserving if we replace p by the probability ...
Infinite monkey theorem
The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare.In this context, ""almost surely"" is a mathematical term with a precise meaning, and the ""monkey"" is not an actual monkey, but a metaphor for an abstract device that produces an endless random sequence of letters and symbols. One of the earliest instances of the use of the ""monkey metaphor"" is that of French mathematician Émile Borel in 1913, but the first instance may be even earlier. The relevance of the theorem is questionable—the probability of a universe full of monkeys typing a complete work such as Shakespeare's Hamlet is so tiny that the chance of it occurring during a period of time hundreds of thousands of orders of magnitude longer than the age of the universe is extremely low (but technically not zero). It should also be noted that real monkeys don't produce uniformly random output, which means that an actual monkey hitting keys for an infinite amount of time has no statistical certainty of ever producing any given text.Variants of the theorem include multiple and even infinitely many typists, and the target text varies between an entire library and a single sentence. The history of these statements can be traced back to Aristotle's On Generation and Corruption and Cicero's De natura deorum (On the Nature of the Gods), through Blaise Pascal and Jonathan Swift, and finally to modern statements with their iconic simians and typewriters. In the early 20th century, Émile Borel and Arthur Eddington used the theorem to illustrate the timescales implicit in the foundations of statistical mechanics.