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    Number of independent subcubes of a hypercube

    Number of "independent" subcubes of a hypercube Hello, I am trying to solve this problem: I have an n-dimensional hypercube and m of its vertices. Now I want to compute the maximum number of subcubes of the entire hypercube such that: - each subcube from the set may contain only those m...
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    Reproduction issue

    Hello, let's have this scenario. We have a parameter n and a exponentially large (in n) universe of organisms. We are given a subset of this universe, which is only polynomially large. The only action which the organisms are able to do is that two compatible organisms join together and a new...
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    Is this sum exponential?

    Hello, I've been solving a problem which forces me to answer the question: "Is there a boolean function with exponential number (in variable count) of prime implicants of the length n - 1?" Anyway, during solving this problem I came to this point: Is the following sum exponential in 2n...
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    What is canonical and prime DNF?

    Hello, I'm little confused about canonical and prime DNF. I found on web that prime DNF is DNF consisting of exactly the set of the prime implicants. In school we've been told that canonical DNF is set of all prime implicants, so it gives me that prime DNF = canonical DNF. Then we...
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    Linear regression

    Homework Statement Consider model of linear regression: Y_i = \beta_0 + x_i \beta_1 + \epsilon_i i = 1, ..., 5, where \epsilon_i \sim \mathcal{N}(0, \sigma^2) are independent. Find expected value and variance of predicted values \widehat{Y}_i considering that observations are...
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    Confidence interval

    Let's say we know this: \sqrt{n}\left(\widehat{\theta} - \theta\right) \sim \mathcal{N}\left(0, \frac{1}{F(\theta)}\right) How do we get from this information to this expression of confidence interval for \theta? \left( \widehat{\theta} \pm...
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    Estimation of parameters using maximum likelihood method

    Homework Statement Let's have random value X defined by its density function: f(x; \beta) = \beta^2x \mbox{e}^{-\beta x} where \beta > 0 for x > 0 and f(x) = 0 otherwise. Expected value of X is EX = \frac{2}{\beta} and variance is \mbox{var } X = \frac{2}{\beta^2}. Next...
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    Recursively enumerable predicate

    Hi all, could someone please explain to me, why, having recursively enumerable predicate Q(x,y) which is not recursive, the function defined as f(x) \simeq \mu_{y} Q(x,y) doesn't define partially recursive function? Ok, here's the argument for it: the program will "try" if Q(x,y)...
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    Equivalent characterization of uniform convergence

    Hi all, I'm learning some calculus theory and I found one point I don't fully understand: \mbox{Let M} \subset \mathbb{R} \mbox{ be non-empty set and let } f, f_{n}, n \in \mathbb{N} \mbox{ be functions defined on M. Then the following is true:} f_n \rightrightarrows f \mbox{ on...
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    Computing axis forces of a construction

    Hi all, I'm trying to compute the axis forces in the construction I put to attachment. I already have computed the R forces, but I don't know where should I start computing the axis forces from, from which point. Could please someone give me some advice? Thank you very much, best regards.
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    Little bit of statistics

    Homework Statement The bank is opened from 9:00 to 17:00. From 9:00 to 10:30 the average client count who come into bank is 32/hour, from 10:30 to 15:30 it is 26/hour and from 15:30 to 17:00 it is 50 clients per hour. I have to test the hypothesis that the average count of clients who...
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    Is this inequality correct?

    I've encountered this nice-looking inequality: \left(A+B\right)^{p} \le p\left(A^{p}+B^{p}\right) (p can irrational as well) but I can't find a way to prove or disprove its correctness. I've tried using the binomial theorem, but it didn't seem it would lead me to the finish...
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    Expected value of number of carps pool

    Homework Statement There is k carps in the pool, m of them are marked. I randomly fish out n carps and see that x of them are marked. What is the expected value of number of carps in the pool? (ie. expected value of number of the carps in the pool in the beginning) How will the...
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    Women and men around the table probability question

    Homework Statement Let's have 2n persons, n men and n women. Suppose they sit randomly around a table with 2n chairs. What is the probability that no two persons of the same sex will sit next to each other? The Attempt at a Solution Here's my idea: I will model this situation with...
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    Probability problem

    Homework Statement Bowman shoots into a dartboard, with possible gain ranging from 0 to 10 points. Probability that he shoots 30 points in 3 shots is 0.008. Probability that he shoots < 8 in one shot is 0.4. Probability that he shoots exactly 8 in one shot is 0.15. What is the...
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    Probability question in telephone network failures

    Hi, I have this problem Homework Statement In telephone network, an average number of failures during a month for one telephone owner is 8. What is the probability that some telephone owner will experience more than 4 failures? The Attempt at a Solution I have no idea what...
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    Logic formula

    Homework Statement p \leftrightarrow \left( \forall x \right)\left( x \wedge \neg y \right) Homework Equations The Attempt at a Solution
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    Why aren't these lattices isomorphic?

    Hi, I can't see why these lattices aren't isomorphic: [tex] (\mathbb{N} \mbox{ u } \left\{-\infty, +\infty\right\}, \le) \mbox{ and } (\mathbb{N} \mbox{ u } \left\{-\infty, +\infty\right\}, \le)^{inverse} [/itex] I thought that this isomorphism would straightforwardly map an element...
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    Is there a isomorphism between N and Q?

    Hi all, I wonder if there is an isomorphism between the group of \mathbb{N} and the group of \mathbb{Q} (or \mathbb{Q}+). I know there is a proof that there is a bijection between these sets, but I didn't find a way how to construct the isomorphism. What confuses me a little is that (I...
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    Lattice of subgroups

    Homework Statement How do the atoms and coatoms of the lattice of all subgroups of the group \mathbb{Z}(+,-,0) look like? Homework Equations Let (L,\le) be a lattice and e, f \in L is the minimum (maximum) elements of L. Then we say that a, b \in L is the atom (coatom) of L if a covers...
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    Central limit theorem problem

    Hi all, I just wrote a test from probability and had troubles doing this problem: Homework Statement The assurance company makes an insurance for 1000 people of the same age. The probability of death during the year is 0.01 for each of them. Each insured person pays 1.200 dollars a year. In...
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    What is the probability that the temperature will drop

    Homework Statement Meteorologists expect rapid temperature drop in the following 48 hours. What is the probability that the temperature drop will occur between 15:45 and 24:00? The Attempt at a Solution I can't find out, what distribution is suitable for this kind of problem...Would...
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    Probability question

    Hi all! I have a problem solving this exercise: Homework Statement Peter and Paul play a card game and after each round the winner gets (and the defeated pays) one dollar. Both of them have equal chance to win in the round, the round can't end in a tie. Homework Equations a) Find out...
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    Reaction of a solid desk

    Hi all! Homework Statement What I'm trying to figure out is a reaction of a solid desk (don't know if this is the proper term) given this picture: [Broken] I need to compute the reactions A, B and C (they corresponding to points denoted by 'a'...
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    Solving equation in finite field

    Hi all, I encountered a problem I don't know how to solve: In \mathbb{Z}_{21}, I want to find numbers x such that x^2 = x. I don't know what to do with that, I just wrote that (since 21 divides x^2 - x) 21y = x^2 - x for some integer y, but it didn't seem to help me much...
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    Find all subgroups of the given group

    Hi all, I've been practising some algebra excercises and don't know how to solve this one: Given the group (\mathbb{Z}_{12}, +, 0), find all its subgroups. How many elements will have these subgroups? The only idea which came to my mind is to see the subgroups generated by each...
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    Proving a propositional logic formula

    Hello all, first I hope there's no problem putting this question here, since I didn't find any special forum dedicated to propositional logic. I really have very basic question, I'm trying to prove \vdash (A \rightarrow (B \rightarrow C)) \rightarrow (B \rightarrow (A \rightarrow C))...
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    Proof: Sufficient condition for local extreme existence

    Let G \subset \mathbb{R}^{n}\mbox{ open } a \in G f : G \rightarrow \mathbb{R} f \in C^{1}(G) Df(a) = \overrightarrow{0} Then: (i) if D^2f(a) is positively definite, then f has local minimum in a (ii) if D^2f(a) is negatively definite, then f has local maximum in a (iii) if...
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    Proof question - extremes of continous function on compact

    Hi all, you all sure know this theorem. We've it as follows: Let (P,\rho) be metric space, let K \subset P be compact and let f:\ K \rightarrow \mathbb{R} is continuous with respect to K. Then f has its maximum and minimum on K. Proof: f(K) is compact (we know from previous...
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    Transcendental equation

    Hi, can somebody see if there is a way to solve this analytically? x = \mbox{cotg } x I know it could be solved numerically, but I'm interested in analytical solution only (if it exists, of course). Thank you very much.