random walks on the hypercube

... Distributions and Conditional Probability The Uniform Distribution 8.2.1 Additional Properties of the Uniform Distribution The Exponential Distribution 8.3.1 Additional Properties of the Exponential ... tvhere the summation is o ver all y in the range The definition states that the conditional expectation of a random variabl e is, like the expectation , a weighted sum of the values it assumes The ... Visually, a convex function f has the property that , if you connect two points on the graph of the function by a straight line, thi s line lies on or above the graph of the func­ tion The following...

Ngày tải lên: 14/07/2016, 15:42

... t;  t  T , is a Brownian motion P Caveat: This theorem requires a technical condition on the size of If IE exp  ZT 2 u du  1; everything is OK We make the following remarks: Z t is ... have not proved Girsanov’s Theorem We will not prove it completely, but here is the beginning of the proof Lemma 1.55 Using the notation of Girsanov’s Theorem, we have the martingale property fe ... tell us anything useful about I Thus, P P we consider subsets of , rather than individual elements of e Distribution of B T  If is constant, then n o Z T  = exp ,B T  , T e T  = T +...

Ngày tải lên: 18/10/2013, 03:20

... deg g(x) Here the notation deg f (x) means the degree of polynomial f (x) From the properties of polynomial, we have Proposition 2.1 If ϕ1 and ϕ2 are the rational functions on F, then 1) deg(ϕ1 ... Generalied Mason’s Theorem 35 Proof of the main theorem Let ϕ(x) = f (x) ≡ be a rational function, where f (x), g(x) are non - zero and g(x) relatively prime polynomials on F The degree of ϕ(x), ... be a the rational function on F and let the derivatives order k, ϕ(k) ≡ Then ϕ(k) −k ordα ϕ Proof Let ϕ(x) = (x − α)m f (x) , where f (x), g(x) are relatively prime and f (α)g(α) = g(x) Then,...

Ngày tải lên: 13/02/2014, 04:20

... literature The information concerning distribution along arithmetic progressions to small moduli comes from the prime number theorem of Siegel and Walfisz 1617 ROTH’S THEOREM IN THE PRIMES Proposition ... collection using the notation M= Ma,q q (log N ) (a,q)=1 B (Q)∧ For these θ, the Fourier transforms λb,m,N and λ∧ b,m,N depend on the distribution of the almost-primes and primes along arithmetic ... zero Secondly, we must formulate and prove a result of the form (6.4) For this we need Theorem 2.1, the restriction theorem for primes The idea of constructing a1 , and the technique for constructing...

Ngày tải lên: 06/03/2014, 08:21

... Figure which shows the original chain and the domain Ωn on the left, three iterations of the reflections in the center and the corresponding domain Dn on the right Ωn Figure 4: Reflections in a circle ... covered by the components of U1 there is a finite collection of these components which also covers Let the closure of the union of these components be F , which clearly has capacity ≤ ε The complement ... completes the initial step of the induction Now suppose we have verified the induction hypotheses up to step n We describe the construction of fn+1 We will not give the details for gn+1 , but the construction...

Ngày tải lên: 22/03/2014, 20:21

... orientations of the tetrahedron FIGURE 64 Is the orientation of the tetrahedron preserved by the following permutations: (rotation by 120° around the alti- The intersection of many sets is the set ... permutation of tetrahedra When we carry out two rotations of the cube one after the other, the permutation of the tetrahedra so obtained is the product of the permutations of the tetrahedra corresponding ... new position (i.e., in the original position of the axis and after this, consider the transformation as the reflection with respect to the new position of the axis (i.e., with respect to the original...

Ngày tải lên: 31/03/2014, 14:57

... another billion electrons are measured, then the polarizers are set at 90 degrees/zero degrees for another billion electrons The result of the experiment is that the inequality is violated The ... mechanism More recent tests set the orientation of the the polarizers randomly after the electrons have left the source The results of these tests are the same as the previous experiments: Bell's ... shall close this section by briefly considering the consequences of discarding the assumption of the validity of logic and then the consequences of discarding the assumption of a reality separate...

Ngày tải lên: 05/06/2014, 10:58

... Dhompongsa Fixed Point Theory and Applications 2011, 2011:34 http://www.fixedpointtheoryandapplications.com/content/2011/1/34 Page of 14 Further, for each x Î C, the limit point of {Tμα x}is the ... leads to a contradiction Step x0 = Tμx We suppose on the contrary and let h := d(x0, Tμx) >0 Let I = [Tμx, x0] and πI : C ® I be the nearest point projection onto I Since {Tsx} is bounded, there exists ... is easy to see that if {x a } Δ-converges to x, then every subnet of {x a } also Δ-converges to x On the other hand, suppose {xa} does not Δ-converge to x Thus, there exists a subnet {xb} of {xa}...

Ngày tải lên: 21/06/2014, 01:20

... that the point z, obtained in the Theorem 5, satisfies the statement of the theorem Following the same notation in the proof of Theorem 5, it is needed to show that x ∉ S(z) for x ≠ z Assume the ... conclude our claim □ The following theorem is an extension of the result of Caristi ([9]; Theorem 2.1) Theorem Let (X, p) be a complete PMS,  : X ® ℝ+ a lower semi-continuous (l s.c) function ... Point Theory and Applications 2011, 2011:4 http://www.fixedpointtheoryandapplications.com/content/2011/1/4 Page of Theorem Let (X, p) be a complete PMS Assume for each x Î X, the function px...

Ngày tải lên: 21/06/2014, 02:20

... convergence, convergence in probability, or almost sure convergence is called the Shannon-McMillan theorem or the entropy theorem or the asymptotic equipartition property AEP in information theory ... theory The Shannon-McMillan theorem on the Markov chain has been studied extensively see 2, In the recent years, with the development of the information theory scholars get to study the Shannon-McMillan ... between the arbitrary random fields and the Markov chain fields on the generalized Bethe tree Main Results Theorem 2.1 Let X {Xt , t ∈ T} be an arbitrary random field on the generalized Bethe tree...

Ngày tải lên: 21/06/2014, 05:20

... that the point y, obtained in Theorem 2.4, satisfies the statement of the theorem Following the same notation in the proof of Theorem 2.4, it is needed to show that x / S y for x / y Assume the contrary ... positive constants Then X, d is a CMS Note that the cone P is normal with the normal constant K Definition 1.5 Let X, d be a CMS, x ∈ X, and let {xn }n≥1 be a sequence in X Then i {xn }n≥1 converges ... xm 0); → iii the sequence {xn } converges to x and the sequence {yn } converges to y then d xn , yn → d x, y Lemma 1.7 see 1, Let X, d be a CMS over a cone P in E Thenone has the following...

Ngày tải lên: 21/06/2014, 20:20

Ngày tải lên: 05/08/2014, 09:46

Ngày tải lên: 05/08/2014, 10:20

Ngày tải lên: 05/08/2014, 15:20

Ngày tải lên: 05/08/2014, 15:20

... obtaining the minimal disjunctive forms of a Boolean function using prime implicants of the function or for minimizing the number of states of an incompletely specified Mealy type automaton A by ... if xi , xj ∈ XA then xi xj ∈ E(G) since otherwise A ∪ {xi , xj } induces a / complete subgraph in G whose vertex set contains strictly ci and cj , which contradicts the definition of a clique Similarly, ... specified sequential functions, IRE Trans Electronic Computers, Vol EC-8 (1959), 356-367 [5] S R Petrick, A direct determination of the irredundant forms of a Boolean function from the set of prime implicants,...

Ngày tải lên: 07/08/2014, 06:23

... G there is only one unknown on the RHS of Equation (29) All G(v(G), l, (ni, mi )l ) i=1 other terms on the RHS are known by the induction hypothesis We can compute the unknown term to obtain the ... The diagonal entries are Except N11 , all the other entries in the first row are We construct the remaining entries of N (G) by induction on the rank The base case is rank It corresponds to the ... from the matrix entries Therefore, we conclude that the computation of the chromatic polynomial can be performed on the matrix entries alone, even if the induced subgraphs indexing the rows and the...

Ngày tải lên: 07/08/2014, 13:21

... quadrilateralization graph GQ whose vertices correspond to the vertices of the polygon and whose edges correspond to the edges of polygon P , to the diagonals of quadrilateralization Q, and to the added ... times The same argument can be applied to any orthogonal polygon with h holes, but only if the dual graph consists of h cycles connected by trees: cut the polygon along h − diagonals of the quadrilateralization ... first consider the dual graph of the quadrilateralization of the polygon from Fig It consists of four cycles connected by single edges, as illustrated in Fig Cutting along three diagonals of the...

Ngày tải lên: 07/08/2014, 13:21

... k(n − k), (5) i∈K From the definition of M we see that ρ(∪i∈K Ai ) = where k = |K| By (5), the rank of ∪i∈K Ai depends only on k = |K| By the monotonicity assumption on the ri , i∈K ri is largest ... observed, because of the presence of the quantity γ(K), whether or not the inequalities (10) in Theorem 3.1 are satisfied depends on the initial labeling of the vertices of T These conditions may not ... , r2 , , rn be a sequence of nonnegative integers with n ri = n (Now we make i=1 no monotone assumption on the ri or the si ) An obvious necessary condtion for T to be completed to a tournament...

Ngày tải lên: 07/08/2014, 21:21

... satisfying the Newton inequalities (2) of order m I.e we dropped the condition of nonnegativity from the definition of ultra-logconcavity It is not true that c = a a satisfies the Newton inequalities ... (but powerful) representation of the convolution in terms of the product of the corresponding polynomials The original Liggett’s proof does not rely on this representation Let a = (a0 , , am ) be ... ab) and the number c2 a2 =2 c4 c6 b (1 − ab) converges to zero if the positive numbers a, b, a converge to zero b The reader can find further implications (and their generalizations) of Theorem(2.3)...

Ngày tải lên: 07/08/2014, 21:21