brain

Let D(a,b;c)=\sum_{j=0}^{c-1} \left(\!

Hmm.

The problem asks for the maximum possible value of $d$ in the notation of Theorem P, given that $m = 10^{10}$ and the potency of the generator is 10.

From Eq.

**Exercise 3.

Let ${Y_n}$ be a binary sequence generated by the linear recurrence over $\mathbb{F}_2$ Y_n = (a_1 Y_{n-1} + \cdots + a_k Y_{n-k}) \bmod 2, with period $2^e - 1$, and initial state not the all-zero st...

The previous implementation fails because it blindly alternates the column in a zigzag without checking **preexisting cacti** in adjacent cells.

We are asked to determine the asymptotic value of the probability that $k+1$ consecutive bits generated by Y_n = (Y_{n-1} + Y_{n-2}) \bmod 2 contain more 1s than 0s, under the conditions that $k > 2l$...

Let the sequence $(Y_n)$ satisfy the recurrence Y_n = (Y_{n-21} + Y_{n-55}) \bmod 2 over $\mathbb{F}_2$.

The statement is **false**.

Let $b_{n,r,s}(m)$ be defined as in Exercise 28: it counts the number of $n$-tuples $(y_1, \ldots, y_n)$ with $0 \le y_j < m$ that have exactly $r$ equal spacings and $s$ zero spacings.

The exercise explicitly depends on results developed across Exercises 28 and 29, especially the generating functions for $b_{n,r,0}(m)$, and it asks for a fairly deep asymptotic expansion whose deriva...

Let $U_1,\ldots,U_n$ be independent uniform $(0,1)$ deviates and let $S_1,\ldots,S_n$ denote their spacings in increasing order, so that $0 \le S_{(1)} \le \cdots \le S_{(n)}, \qquad \sum_{i=1}^n S_{(...

Let the linear congruential sequence be X_{n+1} \equiv aX_n + c \pmod m, \qquad b=a-1, \qquad d=\gcd(m,c),

Let $y_1,\dots,y_n$ be i.

**Exercise 3.

Let $Y_1,\dots,Y_n$ be a cyclic sequence over $\{0,1,\dots,d-1\}$.

Algorithm P (as defined earlier in Section 3.

Let $(Y_n)$ and $(Y'_n)$ be integer sequences with period lengths $\lambda$ and $\lambda'$, respectively, and values in ${0,1,\ldots,d-1}$.

**Exercise 3.

Let $U_0,\ldots,U_{n-1}$ be independent identically distributed random variables.

Let the serial correlation coefficient (23) be C=\frac{N}{D}, where

Let the means of the sequences be $\bar{u} = \frac{1}{n} \sum_{0 \le k < n} U_k, \qquad \bar{v} = \frac{1}{n} \sum_{0 \le k < n} V_k,$ and define the centered sequences $U_k' = U_k - \bar{u}, \qquad V...

**Exercise 3.

**a)** Let Z_{jt} = \max(U_j, U_{j+1}, \ldots, U_{j+t-1}).

In the maximum-of-$t$ test, the $j$th observation is V_j=\max(U_{tj},U_{tj+1},\ldots,U_{tj+t-1}).

Let $\langle X_i \rangle = X_0, X_1, X_2, \ldots$ be a sequence of distinct numbers.

Pattern (15) is the unimodal pattern x_0 < x_1 < \cdots < x_p > x_{p+1} > \cdots > x_{p+q}, on $p+q+1$ distinct elements.

An ascending run is a maximal consecutive subsequence U_i,U_{i+1},\ldots,U_j such that

We are given a square grid of size $n times n$, where each cell represents the annual revenue generated by a table in a restaurant. The restaurant layout is a perfect square, so the grid has exactly four corners: top-left, top-right, bottom-left, and bottom-right.

Let $R$ denote the length of a single segment in the generalized coupon collector's test of exercise 9.

\pi = (1, 3, 5, 4, 6, 2, 7) since (9) in that section is usually this permutation.

Let $L$ denote the length of one coupon-collector segment produced by Algorithm C.

Let $Y_0, Y_1, \dots$ be independent and uniformly distributed integers between $0$ and $d-1$, with $d \ge 2$.

Let $e = 2.71828\ldots$ and consider its expansion in an integer base $b \ge 2$, giving digits $e = \sum_{k=-1}^{\infty} e_k b^{-k}, \quad e_k \in \{0,1,\dots,b-1\},$ where $e_{-1} = 2$ for the intege...

**Exercise 3.

Let $\langle U_n \rangle = U_0, U_1, U_2, \ldots$ be a sequence of independent uniform random variables on $[0,1)$, and let $0 \le \alpha < \beta \le 1$.

Let I_j = \begin{cases} 1,& \alpha \le U_j < \beta,\\ 0,& \text{otherwise},\end{cases} and define

Let ${U_j}$ be a sequence of independent and uniformly distributed random variables on $[0,1)$, and let $p = \beta - \alpha$ denote the probability that $U_j$ lies in the interval $[\alpha, \beta)$.

The serial test is defined in terms of $n$ observations of pairs that are intended to behave like independent draws from the $d^2$ equally likely categories.

For triples, quadruples, or generally $k$ successive values, the serial test is formed by grouping the sequence $\langle Y_n \rangle$ into disjoint blocks of length $k$.

Let Y_i=\sum_{j=1}^{n}a_{ij}X_j+\mu_i,\qquad 1\le i\le m, where $X_1,\ldots,X_n$ are independent random variables with

Let $n$ be a fixed positive integer, and let each of $n$ independent trials result in one of three categories with probabilities $p$, $q$, and $r$, satisfying $p + q + r = 1,\quad p,q,r \ge 0.$ Let $Y...

We are given a sequence of positive integers representing strengths of participants arranged in a line. The task is to choose a single split position such that the array is divided into a left prefix and a right suffix.

Let $X_1,\ldots,X_n$ be independent observations from a distribution function $F$, and let F_n(x)=\frac1n\#\{j:X_j\le x\}.

Let the empirical distribution function be F_n(x) = \frac{1}{n} \sum_{i=1}^n \mathbf{1}_{\{X_i \le x\}}, and define the Kolmogorov-Smirnov statistics as in formula (13):

Investigate the "improved" KS test suggested in the answer to exercise 6.

The previous solution fails because it assumes, without justification, that the finite-$n$ Kolmogorov–Smirnov distribution admits a power series expansion in $n^{-1/2}$ obtained by Euler–Maclaurin app...

A natural multivariate analogue of the Kolmogorov-Smirnov test is obtained by comparing the empirical distribution function F_n(x_1,\ldots,x_s) = \frac1n \#\{\,j: X_{j1}\le x_1,\ldots,X_{js}\le x_s\,\...

We are asked to generalize Theorem 1.

Let $t$ be a fixed real number and, for $0 \le k \le n$, define P_{nk}(x) = \int_{-t}^{t} dx_n \int_{-t}^{t} dx_{n-1} \cdots \int_{-t}^{t} dx_{k+1} \int_0^x dx_k \int_0^{x_k} dx_{k-1} \cdots \int_0^{x...

Let each observation in the experiment be an outcome in a finite set $\Omega$, and let $P$ be the probability measure assigning probability $p_s$ to category $s$, with independent observations.

Let Y_i=np_i+\sqrt{np_i}\,Z_i , where $Z_i$ is defined by Eq.

We compute the Jacobian of the transformation x_k = r\sin\theta_1\cdots\sin\theta_{k-1}\cos\theta_k \quad (1\le k<n), \qquad x_n = r\sin\theta_1\cdots\sin\theta_{n-1}.

Let the original KS test be based on $n$ observations $X_1,\ldots,X_n$, with empirical distribution function $F_n(x)$.

Equations (11) and (13) in Section 3.

**Solution to Exercise 3.

Let the original chi-square test be based on a partition of outcomes into categories $1,2,\dots,k$.

Let the 20 values of $K_{10}^+$ be X_1,\dots,X_{20}, and let the corresponding 20 values of $K_{10}^-$ be

Let the underlying distribution function be $F(x)$.

In Section 3.

The statistic $K_{10}^{+}$ is computed from blocks of length $10$, but the Kolmogorov-Smirnov test in this exercise is not being applied to the original observations within those blocks.

Let the first die be fair, and let the second die be loaded so that it can show only $1$ or $6$, each with probability $\tfrac12$.

Let the first die be biased toward the value $1$, and let the second die be biased toward the value $6$.

**Corrected Solution for Exercise 3.

The value $V = 7\frac{1}{16}$ corresponds to the chi-square statistic computed from $k = 11$ categories, as in Eq.

Let f(x)=a x^{-1}+c \pmod{2^e}, with

**Solution to Exercise 3.

Let f(x)=x^2-cx-a over the field $\mathbf F_p$, where $p$ is prime.

Let X_n=(X_{n-2}+X_{n-55})\pmod m .

Stopped thinking

**Exercise 3.

**Corrected Solution for Exercise 3.

Let $(X_n)$ be the sequence defined modulo $p^\lambda$ by X_n=x_n \pmod{p^\lambda}, \qquad 0\le n<k, and

Let $(X_n)$ be a sequence of integers modulo $m$, with period length $\lambda \gg k$, and let Algorithm B act on $(X_n)$ as described in Section 3.

**Exercise 3.

In Program A of Section 3.

Let Y_n=(Y_{n-l}+Y_{n-k}) \pmod 2, \qquad 0<l<k, and suppose that every nonzero sequence satisfying this recurrence has period

The recurrence is X_n=(X_{n-31}-X_{n-24})\pmod m.

Let S=(\mathbb Z_m)^k and write a state as

Let $m = p_1 p_2 \cdots p_s$, where $p_1,\ldots,p_s$ are distinct primes.

Method (10) of Section 3.

We restart from the correct criterion and remove the unsupported construction.

Let $X_n$ be the binary sequence generated by method (10) with $k=35$ and CONTENTS$(A)=(a_1a_2\ldots a_{35})_2$, where $a_{35}=1,\quad a_{31}=a_{33}=a_{35}=1,\quad a_i=0 \text{ otherwise in the final...

Let $m, k \in \mathbb{Z}^+$, and define the sequence $(X_n)$ by X_1 = X_2 = \cdots = X_k = 0, and, for $n \ge 1$,

Let the binary representation of $\mathrm{CONTENTS}(A_n)$ be \mathrm{CONTENTS}(A_n) = (c_{n,1} c_{n,2} \ldots c_{n,k})_2, where $c_{n,i} \in {0,1}$ for $1 \le i \le k$, and $c_{n,1}$ is the most signi...

Let $(X_n)$ and $(Y_n)$ be integer sequences modulo $m$, with periods $\lambda_1$ and $\lambda_2$.

The previous solution fails because it never constructs a valid global structure linking the return-time function $q_n$ with the indexing of the base period of $X_n$, and it incorrectly treats periodi...

Let $(X_n)$ and $(Y_n)$ be sequences of integers modulo $m$ with periods $\lambda_1$ and $\lambda_2$, respectively.

The sequence is defined modulo $2^e$ by $X_{n+1} = aX_n + bX_{n-1} + c \pmod{2^e}, \qquad n \ge 1.$ The goal is to choose integers $a,b,c,X_0,X_1$ so that the resulting sequence has maximal possible p...

Let X_{n+1}=X_n+X_{n-1}\pmod{2^e} and write the state vector

Let $m = 2^e$ and consider the modified middle-square sequence defined by Coveyou: $X_0 \text{ given}, \qquad X_{n+1} = \operatorname{middle}(X_n^2 + 2^{e-1} X_n), \eqno(4)$ where the function $\opera...

Let $R_{p^r} = (\mathbb{Z}/p^r\mathbb{Z})[z]/(f(z))$ with $f(0)=1$, and denote by $\overline{z}$ the residue class of $z$ in $R_{p^r}$.

Let the MIX machine have accumulator $A$, index register $X$, and overflow toggle $O$.

Let the binary method (10) be the scheme in which a word $X$ is updated by shifting and inserting a random bit, so that each step effectively appends a new random least significant bit while discardin...

Work modulo $8$ throughout.

**Corrected Solution to Exercise 3.

We consider the linear congruential generator (LCG) in its standard integer form: X_{n+1} = (a X_n + c) \bmod m, \quad X_0 \in \{0,1,\dots,m-1\}.

From the Fibonacci generator, X_{n+1} = (X_n + X_{n-1}) \bmod m, there exists an integer $t \in {0,1}$ such that

By Exercise 5, if m=p_1^{e_1}\cdots p_r^{e_r}, \qquad a=1+k\,p_1^{f_1}\cdots p_r^{f_r},