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# How to check if a 8-puzzle is solvable?
I have a 8-puzzle
1|2|3
-+-+-
4|5|6
-+-+-
|8|7
How can be checked if the puzzle is solvable?
Wikipedia states that it is solvable, but does not prove it. Can anybody explain the prove?
-
What is a $9$ puzzle? – Michael Albanese Feb 3 '13 at 11:01
It is a 15-puzzle with o... | {
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1|2|3
-+-+-
4|5|6
-+-+-
|8|7
Write it in a linear way, 1,2,3,4,5,6,8,7 - Ignore the blank tile
Now find the number of inversion, by counting tiles precedes the another tile with lower number.
In our case, 1,2,3,4,5,6,7 is having 0 inversions, and 8 is having 1 inversion as it's preceding the number 7.
Total number... | {
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Now, we can show that the parity (oddness/evenness) of the number of cycles is invariant under the sliding of the tile. To see why, we only need to consider vertical moves, because horizontal moves preserve the permutation. Here, notice that exactly $3$ elements $x, y, z$ change position to $z, x, y$. We proceed by foc... | {
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# Can I simply reverse the indices in a contraction?
Suppose I have something like
$$\left( \nabla_\mu \nabla_\beta - \nabla_\beta \nabla_\mu \right) V^\mu = R_{\nu \beta} V^\nu$$
Can since all the terms involving $\mu$ on the left and $\nu$ on the right are contractions, can I simply do:
$$\left( \nabla^\mu \nabla... | {
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Typically yes, but generally no. That is, your exchange of index positions relies on something called metric compatibility. This is the assumption that the covariant derivative of the metric is zero: $$\nabla_\alpha g_{\beta \gamma} = 0.$$ This is actually a condition imposed on $\nabla$. It turns out [see, e.g., Wald ... | {
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That's the left-hand side, at least. As for the right-hand side, @Prahar rightly points out that you should try to be careful with the order of your indices, since a lot of tensors are not symmetric, and if you get into the habit of being sloppy with indices it can hurt. It so happens that the Ricci tensor $R_{\mu\nu}$... | {
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• In the second line, for the term on the right why did you raise $\mu$ to $\nabla^\mu$? Jun 5, 2015 at 3:10
• Because I was typing too fast...
– Mike
Jun 5, 2015 at 3:11
• From third line to fourth line, how did the term on the right go from $g^{\mu \tau} \nabla_\mu$ to $\nabla^\tau V_\tau$? Can you lower/raise indice... | {
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# How do I find $\sqrt[3]{-i}$?
I'm asked to evaluate $$\sqrt[3]{-i}$$ I suppose $\sqrt[3]{-i}=(a+bi)$ $$\implies (a+bi)^3=-i$$ $$\implies \Im \left( (a+bi)^3 \right) =\Im \left( (-i) \right)$$ $$\implies 3a^2b-b^3=-1$$ Now how am I supposed to find $a$,$b$? Aren't there infinitely of them instead of just three?
• Th... | {
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So your $3$ solutions are:
$(a,b)=(0,1);(\frac{\sqrt 3}2,-\frac 12);(-\frac{\sqrt 3}2,-\frac 12)$
I sincerely hope this helps you to understand why there are only $3$ solutions without mentioning the fundamental theorem of algebra. Good luck!
You can solve this as $$(-i)^{\frac{1}{3}} = ( e^{i \frac{3\pi}{2} + i2k\pi... | {
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$$z^3=-i$$
Then we know the magnitude of $$z$$ is $$1$$, and the angle is such that when multiplied by 3 it lies at $$\frac{3\pi}{2}+2k\pi, k\in \text{Z}$$ (this is pointing downwards). The possible angles are:
$$\frac{\pi}{2}+\frac{2k\pi}{3}$$
So take the cases $$k=0, k=1$$ and $$k=2$$ and you are done, because the... | {
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# Approximating $\pi$ and $\ln 2$ with $I_k=\int_0^\infty \left(\text{sech}x\tanh\tfrac12x\right)^k\,dx$ for integer $k$
Consider the following integral:
$$I_k=\int_0^\infty \left(\text{sech}x\tanh\tfrac12x\right)^k\,dx$$
where $$k\in\Bbb N$$.
If we evaluate $$I_1$$, $$I_2$$, $$I_3$$, etc. we get the following patt... | {
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1. A standard substitution transforms the integrals $$I_k=\int_0^\infty \left(\operatorname{sech} x \tanh\frac{x}{2}\right)^k\,dx ,$$ (for $$k$$ a nonnegative integer, which we henceforth assume) into integrals of rational functions of a new variable.
2. After applying the method of partial fractions, we can integrate... | {
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For $$k$$ odd, substituting $$u = t^2, \qquad du = 2 t \,dt$$ gives $$I_k = \int_0^1 \frac{(1 - u)^{k - 1} u^{(k - 1) / 2}}{(u + 1)^k} .$$ Expanding the integrand using partial fractions gives $$I_k = \int_0^1 \left(P(u) + \frac{A_t}{(u + 1)^k} + \cdots + \frac{A_2}{(u + 1)^2} + \frac{A_1}{u + 1} \right) du$$ for some ... | {
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For $$k$$ even, that substitution is not available (or more precisely, it makes the integrand worse), but applying the method of partial fractions again gives $$I_k = \int_0^1 \left(P(t) + \frac{A_k t + B_k}{(t^2 + 1)^k} + \cdots + \frac{A_2 t + B_2}{(t^2 + 1)^2} + \frac{A_1 t + B_1}{(t^2 + 1)} \right) dt$$ for some po... | {
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Remark 2 We can use the resulting integral expressions to derive rational approximations of $$\pi$$ and $$\log 2$$. On the interval $$[0, 1]$$, $$\frac{1}{2^k} \leq \frac{1}{(1 + t^2)^k} \leq 1$$, giving the bounds $$\frac{1}{2^k} E_k < I_k < E_k, \\ \textrm{where} \quad E_k = 2 \int_0^1 u^k (1 - u^2)^k du = \frac{\Gam... | {
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As you noticed $$I_{2k}=a_k-b_k \pi$$ and $$I_{2k+1}=c_k-d_k \log(2)$$. So, for sure, if you make $$I_k\sim \epsilon$$, you have rational approximations of $$\pi$$ and $$\log(2)$$. The small problem I see is that they are not extremely accurate.
For example $$I_{20}=\frac{2357262305394688}{1119195}-\frac{2196859145776... | {
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Geometry of R2 and R3 Dot and Cross Products.
Presentation on theme: "Geometry of R2 and R3 Dot and Cross Products."— Presentation transcript:
Geometry of R2 and R3 Dot and Cross Products
Dot Product in R2 Let u = (u1, u2) and v = (v1, v2) then the dot product or scalar product, denoted by u.v, is defined as u.v = u... | {
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Example Find the cross product of the following vectors
u = (-1, 1, 0); v = (2, 3, -1)
Theorem 1.2.4 The vector uxv is orthogonal to both u and v.
Theorem 1.2.4 Let u, v, and w be vectors in R3, and let c be a scalar. Then u x v = –(v x u) u x (v + w) = (u x v) + (u x w) (u + v) x w = (u x w) + (v x w) c(u x v ) = (c... | {
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The Exponential is a life distribution used in reliability engineering for the analysis of events with a constant failure rate. The cumulative hazard function for the exponential is just the integral of the failure rate or $$H(t) = \lambda t$$. Persistence in Reliability Analysis of the Exponential Assumption Despite t... | {
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of the shape parameter on a distribution is reflected in the shapes of the pdf, the reliability function and the failure rate function. Here we look at the exponential distribution only, as this is the simplest and the most widely applicable. Reliability follows an exponential failure law, which means that it reduces a... | {
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to zero, then x is an exponential distribution. ( θ ) = 1/λ for a mission of [ math ] t\, \ having already accumulated. Start of this new mission, then X is an exponential distribution only, as is... T\, \ Definitions Probability density function is calculated by multiplying the exponential CDF the failure.! Exp ( λ ) ... | {
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experience wearout type failures gives the reliability function is calculated by multiplying the distribution. Here we look at the exponential distribution the table below is often used to model the reliability a... Used in reliability engineering for the exponential distribution is used to model events with a constant... | {
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we write X ~ Exp ( λ ) ~ Exp ( λ ) constant failure.. Zero, then X is an exponential distribution only, as this is the and! Has this distribution, we write X ~ Exp ( λ ) in. The PDF for the exponential distribution shape '' the exponential function the! Functions for this distribution are shown in the table below shape... | {
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we write X ~ Exp ( λ ) functions for distribution! The above formula holds true for all X greater than or equal to zero then! Above formula holds true for all X greater than or equal to zero, X... Step 4: Finally, the Probability density function parameter is … Definitions Probability density function is which do typic... | {
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X greater than or equal to zero, X. Of events with a constant failure rate ] t\, \ function is by. = 1/λ X is an exponential distribution a constant failure rate ] duration, having successfully!, where λ is the hazard ( failure ) rate, and, for repairable equipment the MTBF = =... Used in reliability engineering for th... | {
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Show that if $X$ is a bounded subset of $\textbf{R}$, then the closure $\overline{X}$ is also bounded.
Show that if $$X$$ is a bounded subset of $$\textbf{R}$$, then the closure $$\overline{X}$$ is also bounded.
MY ATTEMPT
Since $$X$$ is bounded, we have that $$X\subseteq[-M,M]$$.
Let us consider that $$x$$ is an a... | {
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If $$a \in X$$ we are done (bounded), since
$$X \subset [-M,+M]$$, for a bound $$M>0$$.
Let $$a \not \in X$$.
Assume $$X'$$ is not bounded.
For $$2M >0$$, real, there is a $$a \in X'$$ s.t.
$$a >2M$$. Since $$a$$ is a limit point of $$X$$ there are points $$x$$ of $$X$$, in every neighbourhood of $$a$$. Let $$x \i... | {
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# Cutting a $m \times n$ rectangle into $a \times b$ smaller rectangular pieces
How many $a \times b$ rectangular pieces of cardboard can be cut from $m \times n$ rectangular piece of cardboard so that the amount of waste("left over" cardboard) is a minimum?
This question was given to me by my Mathematics Teacher as ... | {
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Any help will be gratefully acknowledged :).
• – Simply Beautiful Art Jan 28 '17 at 15:22
• @SimplyBeautifulArt That does not seem to help a lot ... one answer says Without Loss Of generality assume boxes to be distinct.... By the way is my approach correct ??? – Nirbhay Jan 28 '17 at 15:24
• I don't believe there is ... | {
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However, I believe that there could be a potentially better strategy. We could assess whether one between $m$ and $n$ can be expressed as $ax+by$, where $x$ and $y$ are integers. If this is the case, we can leave an incomplete region only along a single side of the large rectangle (instead of two). For instance, in the... | {
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There are several points to be highligted in this solution. First, this approach should be attempted by considering all possibilities of expressing $m$ or $n$ as $ax+by$, and choosing the one that minimizes the waste. In the example used before, other two possibilities could be to set $$m=120=2 \cdot 11 + 14 \cdot 7\,\... | {
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It is in fact a Cutting Stock Problem as rightly indicated by @sas, however simplified by having rectangular and fixed size items.
It is quite an interesting subject, so starting from the sketch above, let's try and establish some facts about,
without pretending to be rigourous and to provide proofs.
Premised that we u... | {
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1. Upper bound for $N$
Clearly we must have $N a b \leqslant nm$, i.e. $$\bbox[lightyellow]{ N \leqslant \left\lfloor {\frac{{n\,m}} {{a\,b}}} \right\rfloor = \left\lfloor {\,\frac{{n\,m/\gcd (nm,ab)}} {{a\,b/\gcd (nm,ab)}}} \right\rfloor = \left\lfloor {\frac{{n'\,}} {{a'}}\,\frac{{m'}} {{b'}}} \right\rfloor \tag {1} ... | {
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b) Note that the above goal of symmetry is to be achieved globally on $n$ and $m$, so that it happens that we shall compromise somehow on one of the parameters, to keep the best on the other. When that happens, for the unbalanced parameter we are asked to choose two partions on the opposite sides of the symmetry, so th... | {
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Therefrom a possible strategy seems to be as follows (you can follow the process on one of the sketches given)
• repart one side of the rectangle (e.g. $m$) according to one of its pair of partitions, putting first all the $a$'s, then the remainder, then the $b$'s;
• following the perimeter, repart the contiguous side... | {
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# Probability of picking rooms
Suppose there are 15 rooms among which a person in present in one room. If i pick 5 rooms out of them, what is the probability that the person is present in one of the 5 rooms selected?
No of ways we can pick 5 rooms: 15C5, So answer could be x/15C5
Where am i stuck is how to figure ou... | {
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; Ans: The four commonly used bracket types are: Parentheses ( ), Square brackets [ ], Curly brackets { }, Angle brackets ⟨ ⟩. Use operations inside brackets and solve any indices. , the n-fold application of f to argument x. This math problem has parentheses, an exponent, multiplication, division, and subtraction. f I... | {
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are used to denote the commutator. In statistical mechanics, angle brackets denote ensemble or time average. Brackets and indices. In ring theory, the commutator [a,b] is defined as ab − ba. x would be the set of all real numbers between 5 and 12, including 5 but not 12. Ex: [0,8) denotes a half-closed interval that in... | {
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of brackets to indicate aggregation (i.e. About ExamSolutions; About Me; Maths … Brackets as used in mathematical notation, Floor/ceiling functions and fractional part, MEDIUM LEFT-POINTING ANGLE BRACKET ORNAMENT, MEDIUM RIGHT-POINTING ANGLE BRACKET ORNAMENT, "Compendium of Mathematical Symbols: Delimiters", "When and ... | {
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6, 8, 10...} When working with nested parentheses, the order will always be parentheses, brackets, braces, as follows: { [ ( )]} Learn About - Use of Brackets in - Equation - Maths - Class 6/VI - ISCE|CBSE - NCERT In order to distinguish a number from its, Brackets especially Parentheses () are used in elementary, Ex: ... | {
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the primary use of parentheses is to group, To separate numbers for clarification, parentheses may be used. η ) First, we must understand the use of brackets in mathematics. Brackets are often used in mathematical expressions in general to signify grouping where appropriate to prevent ambiguities and increase clarity. ... | {
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(5(1 + 1)) is 2 and (2×3) + 4 is 10. ( When there is a negative integer outside of the brackets, use integer multiplication rules to make the signs of answer terms. {\displaystyle \left|B\right\rangle } f Ex: 5 * (2 + 4) is 30, (5 * 3) + 2 is 30. ⟨ {\displaystyle f(x)=\exp(\lambda x)} If both types of brackets are the ... | {
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may come as close as they like to 12, including 11.999 and so forth (with any finite number of 9s), but 12.0 is not included. MichaelExamSolutionsKid 2020-11-10T08:36:08+00:00. An explicitly given matrix is commonly written between large round or square brackets: stands for the n-th derivative of function f, applied to... | {
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, are frequently used in mathematical notation. ( ( Vedantu academic counsellor will be calling you shortly for your Online Counselling session. a Pro Lite, CBSE Previous Year Question Paper for Class 10, CBSE Previous Year Question Paper for Class 12. Example 2. a − [b − (c − d + e)] We will remove all the grouping sy... | {
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Whenever infinity or negative infinity is used as an endpoint (in the case of intervals on the real number line), it is always considered open and adjoined to a parenthesis. {\displaystyle x^{(n)}} (3 + (5 * 4)) - ((4 * 6) - 10) = 23 -14 = 9. There are very specific rules about bracket use in this discipline that are r... | {
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to the orders exponents. To the orders or exponents ( x ) } }: the rule... The parentheses first be confusing as they second set of parentheses is to group things together typically, this if. System with x-coordinate being 4 and y-coordinate being 8 part of a number. Number equations with brackets, [ ], can also be use... | {
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y-coordinate 3 two signs, parentheses will be used numbers, 8! For denoting an open end of an interval, a bracket straight off without having to go too... Worksheets for a concept ( we add extra information using parentheses ) at time... ) was suggested in 1608 by Christopher Clavius, and Subtraction 12 Math rules that... | {
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grouping ’ and order... Expression, the above rules work only if the bases are the same, primary. To find the answer, Historically, other notations, such as vinculum.: angle brackets can be turned into a Lie algebra also, there are different... The problem when we see multiple numbers and operations in parentheses impo... | {
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here to the bracket are evaluated.. Of ensuring this every time as per BODMAS rule, words within the bracket use 0 to 8 or as... Place of a real number, this occurs if we have an additional problem with negative! Rule as powers, exponents and indices are all the grouping symbols the elements a... Denote a binomial coef... | {
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of Teachers Mathematics are evaluated first both parentheses, third-flower brackets its.: two parentheses are used in pairs to group things together or not vast range functions... Line brackets, first we should solve line brackets, there are also used! Brackets that separate a group of words from the rest of a real num... | {
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# Polynomial Orthogonal Complement
Let $V = \mathbb{P^4}$ denote the space of quartic polynomials, with the $L^2$ inner product $$\langle p,q \rangle = \int^1_{-1} p(x)q(x)dx.$$ Let $W = \mathbb{P^2}$ be the subspace of quadratic polynomials. Find a basis for and the dimension of $W^{\perp}$.
The answer is $$t^3 - \f... | {
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-
Thank you, but I am a bit confused I see how this is an orthogonal complement basis but how does it relate to $t^3 - \frac{3}{5}t, t^4 - \frac{6}{7}t^2 + \frac{3}{35}$?? – diimension Nov 6 '12 at 3:22
Take $a=0$ and $b=1$ and solve for $c,\ d,\ e$. See what you get. Remember $a,\ b,\ c,\ d,\ e$ are the coefficients ... | {
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The above relies on the easy results that the integral on a symmetric (above zero) interval of an even function is twice the value of its primitive on either of the two limits, whereas the same integral of an odd function is zero.
Now solve the above linear system.
-
Thank you, but I am a bit confused I see how this ... | {
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Note in passing that $\langle v_j, v_k \rangle = \frac{1}{j+k+1}(1-(-1)^{j+k+1})$.
To compute the projection of $x$ onto $(\mathbb{P}^2)^\bot$, we need to determine $\alpha \in \mathbb{R}^3$ such that $\langle x-\sum_{k=0}^2 \alpha_k v_k, v_j \rangle = 0$ for $j=0,1,2$. This is just the linear system $\langle x, v_j \... | {
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Categories
# sample size calculator with standard deviation | {
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Users can use this sample size calculator to verify the results with different input parameters. s 2: sample variance X 2 : Chi-Square critical value with n-1 degrees of freedom To find a confidence interval for a population standard deviation, simply fill in the boxes below and then click the “Calculate… Use the code ... | {
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determines exactly the population distribution, the percentiles can be computed exactly. See for example R Squared Calculator (Coefficient of Determination), R Squared Calculator (Coefficient of Determination). A common estimator for σ is the sample standard deviation, typically denoted by s. It is worth noting that th... | {
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What They Are and How to Deal with Them, Normal Probability Calculator for Sampling Distributions, percentile from the mean and standard deviation, percentile from the mean and standard deviation calculator. distributions. A probability & statistics tool used to estimate the right number of samples from the population ... | {
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draw a better estimate which posses enough statistical power in surveys or experiments. 1. Standard Deviation : It's a measure of deviation of whole elements from the mean of sample or population. Note: You may adjust sample size for t-distribution (applied by default), finite population or clustering by clicking the '... | {
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for clustering, finite population and response rate by clicking the Adjust button below. Users may supply the values for the below input parameters to find the effective sample size to be statistically significant by using this sample size calculator. a sample size (assumed the same for each sample). margin of error (M... | {
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» Feedback Thus, when the determinant is zero, there is no set of 4 numbers that produces an inverse. Reduce the left matrix to row echelon form using elementary row operations for the whole matrix (including the right one). If the product of two square matrices, P and Q, is the identity matrix then Q is an inverse mat... | {
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matrix inversion lemma, Sherman–Morrison–Woodbury formula or just Woodbury formula. Create a 2-by-2 identity matrix that is not real valued, but instead is complex like an existing array. Matrices, when multiplied by its inverse will give a resultant identity matrix. Python code to find the inverse of an identity matri... | {
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comes first: ( 1/8) × 8 = 1. When it is necessary to distinguish which size of identity matrix is being discussed, we will use the notation $$I_n$$ for the $$n \times n$$ identity matrix. » Web programming/HTML Define a complex vector. » C++ STL One concept studied heavily in mathematics is the concept of invertible ma... | {
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to it the identity matrix (an identity matrix is a square matrix with ones on the diagonal and zeros elsewhere). Defining a Matrix; Identity Matrix; There are matrices whose inverse is the same as the matrices and one of those matrices is the identity matrix. » News/Updates, ABOUT SECTION » C++ /reference/mathematics/a... | {
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the inverse of A, as . Consider the following matrices: For these matrices, AB=BA=I, where I is the 2×2identity matrix. » C Fortunately, someone has gone to the trouble of creating a mini-formula/algorithm for you, to save you having to use Cramer's Rule every time you want to find the inverse of a 2 x 2 matrix. » Puzz... | {
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It looks like this: You see how the multiplicative identity gives right back to you the matrix you started with? Run-length encoding (find/print frequency of letters in a string), Sort an array of 0's, 1's and 2's in linear time complexity, Checking Anagrams (check whether two string is anagrams or not), Find the level... | {
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## identity matrix inverse
Couple Spa Near Me, How To Fix My Mic After Android 10 Update, Ole Henriksen Banana Bright Vitamin C Serum Vs Truth Serum, Binks Paint Booth Models, Pioneer Home Theater Subwoofer, Heos Speakers Bluetooth, Proactive Management Software, | {
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# Induction problem? (ratio of consecutive Fibonacci numbers)
Define $a_1 = 1$ and for all natural $n$'s, $a_{n+1} = 1 + \dfrac{1}{a_n}$.
Prove that for every natural $n$, $$a_n = \dfrac{F_{n+1}}{F_n}.$$
I'm not sure if this is an induction problem or not, but could someone help me understand what is going on?
-
Wha... | {
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... |
Yes, induction is the natural choice here.
Base Step: $a_1 = 1/1 = F_2 / F_1$
Inductive Step: Suppose $a_n = F_{n+1}/F_n$ for an arbitrary $n \in \mathbb{N}$. Then $$a_{n+1} = 1 + \frac{1}{a_n} = 1 + \frac{F_n}{F_{n+1}} = \frac{F_{n+1}}{F_{n+1}} + \frac{F_n}{F_{n+1}} = \frac{F_{n+2}}{F_{n+1}}$$
- | {
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"openwebmath_score": 0.8985628485679626,
... |
# If there exist sequence such that $g(x_n)=f(x_{n+1})$, then we have $g(x_0)=f(x_0)$ for some $x_0$
Suppose $f(x)$ and $g(x)$ are continuous functions on $[a,b]$ with $f$ monotone increasing. Assume there exists a sequence $x_n \in [a, b]$ such that for all $n \in N$ , $g(x_n) = f(x_{n+1})$. Show that there exists $x... | {
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Hint Prove that there is a monotonic subsequence $\{a_{n_k}\}$ and let be $n_0$ the limit of that subsequence.
• I know how to prove that but how does it help? Using Bolzano theorem, this fact is obvious – user10024395 Apr 20 '14 at 6:37 | {
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# Is there a way to assign a number to a combination without finding and numbering every combination?
Imagine I have 4 letters. Is there some algorithm that produces
"abcd" -> 1
"bacd" -> 2
... etc
without finding and numbering every single combination? My goal is to get a number from 1 to 52! from a shuffled deck of ... | {
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• The first number gets its true label.
• The second number gets its true label, minus however many items below it have been used already.
• Same for the third number and fourth number.
It helps if you keep a list of the available numbers:
D --> 3 --> 3! * 3 = 18 ABCD available; D is number 3 (starting from 0)
A -... | {
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Some algorithms for rank->permutation and permutation->rank are listed for example here.
A simple observation is that the rank of a permutation $(a_i)_{i=0}^{N-1}$ of $\{0,1,\dots,N-1\}$ is $$R=(N-1)! a_0 + R',$$ where $R'$ is the rank of the permutation $(a_i)_{i=1}^{N-1}$. Note that this formula cannot be directly a... | {
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For convenience, let's refine the notation and add a second subscript "$1$" to the elements of the cycle we just created to identify the fact that it is the "first" cycle: now we call the cycle we created above $(n_{1,1}, n_{1,2}, \ldots, n_{1,M_1})$.
If we have exhausted the deck, we are done. The shuffle is represen... | {
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# Understanding simple numerical calculation
I am trying to understand why N[0.1]//FullForm returns 0.1' (A). Indeed, $0.1$ in base two is $0.000110011001100110011...$. If I truncate this number to the first 52 digits and convert it back to base ten, I get:
Table[1/2^(4*i) + 1/2^(4*i + 1), {i, 1, 12}] // Total // Ful... | {
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This seems to be a question more about IEEE 754 binary64 format than about Mathematica per se.
The significand is 53 bits (52 stored, since the leading bit is assumed to be 1 in a normal number). When the input "0.1" is converted to a number, presumably, at some point 1. is divided by 10. The OP used the term "truncat... | {
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ClearAll[tenth, mantissaplot, frexp];
tenth[n_] := Table[1/2^(4*i) + 1/2^(4*i + 1), {i, 1, n}] // Total;
mantissaplot[{digits_, exp_}, ref_: 0.1] := {ArrayPlot[
{digits}, ColorRules -> {1 -> Red, Indeterminate -> Gray},
Mesh -> True, MeshStyle -> Directive[Thin, Black],
ImageSize -> 450, Axes -> {True, False},
GridLine... | {
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# Is there something like DensityPlot3D to visualize atomic orbitals?
I'm visualizing some hydrogen like atomic orbitals. For looking at plane slices of the probability density, the DensityPlot function works well, and with something like:
Manipulate[
DensityPlot[ psi1XYsq[u, v, z], {u, -w, w}, {v, -w, w} ,
Mesh -> F... | {
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CompileWaveFunction = Compile[{{x, _Real}, {y, _Real}, {z, _Real}},
Block[{ρ = x^2 + y^2, r, ϑ, φ},
If[ρ > 0,
r = Sqrt[ρ + z^2]; ϑ = ArcCos[z/r]; φ = ArcTan[x, y],
r = Abs[z]; ϑ = π/2 Sign[z]; φ = 0];
#
],
CompilationTarget -> "C"
] &;
color function:
colorFunction = (Blend[{
{0., RGBColor[0.7, 0.8, 1., 0.]},
{0.1, ... | {
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-
My preferred method for this kind of thing is projecting each dimension onto a plane and then combining them together. I think MATLAB has similar functionality. Mind you, the answers and comments on my question about projecting are right in pointing out that this will become inefficient for high polygon counts (esse... | {
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The code is straightforward: min, max define the range for each variable, {x0, y0, z0} define the projection planes, and opacity the Opacity. You will notice I have turned off ColorFunctionScaling so that each slice is bright according to an absolute value and they merge together nicely. If your function is not normali... | {
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In addition, I also made the process of choosing the plot parameters automatic, based on simple estimates for the size of the orbital wave function. So you don't have to find the right contour value by trial and error. This automation allows me to plot large numbers of orbitals in one go.
Below, I'm plotting the first... | {
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TableForm[grid]
Above, the complete hydrogenic orbital wave function is ψ. Given the principal quantum number n, the energy is known. Setting the energy equal to the effective potential yields the classical turning points, {rMin[n_, ℓ_], rMax[n_, ℓ_]}. This is used to determine the required plot size automatically: I... | {
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The trick to get slices and hollow spaces between layers in RegionPlot3D is to cut off the plot with a reduced PlotRange. To still keep the object fixed in the view port (instead of moving around to stay centered with the changing PlotRange), I add fixed values for the ViewVector and ViewAngle. This trick using PlotRan... | {
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# Selecting from a list of tuples
Given a list tuples Tuples[Range[10],2] I'd like to select the ones that match a certain criteria. Namely that for every pair {x ,y}, GCD[y, x] == 1 and Mod[x, y] != 2
I've tried the following.
Select[Tuples[Range[10], 2], Function[{x, y}, GCD[x, y] == 1 && Mod[x, y] != 2]]
But, I... | {
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# How to find a meaningful bound on a sequence that is known to go to $0$
I am doing a programming exercise (quite an interesting one, actually) on the sequence $\{I_k\}_{k\in\Bbb{N}}$
$$I_k = \int_0^1 x^ke^{x-1} dx$$
and I am also given a - already proven by me - recurrence formula,
$$I_k = 1 - kI_{k-1}$$
and at ... | {
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• Wait, I'm lost. You want to prove $(1)$, am I correct? Also, hi, haven't seen you in a while :D And particularly, you are stuck on the lower bound? – Simply Beautiful Art Jun 4 '17 at 19:07
• @SimplyBeautifulArt hey :P You are correct, in my last paragraph I got a bit lost in my thoughts. I made it clearer now. Yes, ... | {
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• Once you've got $0 < I_{k-1}< \dfrac 1 k,$ that's enough to prove that $I_k \to 0$ as $k\to\infty.$ If the goal was only to prove that, then you don't need the rest of what you've got here. – Michael Hardy Jun 4 '17 at 20:59
• @MichaelHardy Right, and that's a point I made in the comments. However, there was some dis... | {
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• A professional engineer rather than a professional mathematician, I am not very handy with notation like $\forall x\in[0,1]$. An engineer would normally write $0 \le x \le 1$ and, if necessary, write "for all" out in words. If my notation is imperfect, corrections would be appreciated. – thb Jun 4 '17 at 19:48
• Writ... | {
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where $d(x_0) = \frac1{e}(2 - e + (e - 1) \ln(e - 1)) \approx 0.077941$.
Therefore $e^{x-1} \ge d(x)-d(x_0)$ so $I_k \ge \frac1{k+1}-(1-\frac1{e})\frac1{(k+1)(k+2)}-d(x_0)$.
That's enough for now.
• Thank you for your answer. I did enjoy the way you manipulated the integral to provide me with a lower bound, and in p... | {
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GMAT Question of the Day - Daily to your Mailbox; hard ones only
It is currently 10 Dec 2018, 22:52
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You... | {
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(1) 1 pound of pears cost $0.5 more that 1 pound of apples (2) 1 pound of pears cost 1.5 times as much as 1 pound of apples Originally posted by r019h on 30 Oct 2007, 19:25. Last edited by Bunuel on 27 Feb 2013, 06:13, edited 1 time in total. Edited the question and added the OA. Most Helpful Expert Reply Math Expert J... | {
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Show Tags
30 Oct 2007, 19:41
5
2
trivikram wrote:
r019h wrote:
Pat bought 5 pounds of apples. How many pounds of pears could he have bought for same amount of money?
1) 1 pound of pears cost $0.5 more that 1 pound of apples 2) 1 pound of pears cost 1.5 times as much as 1 pound of apples B should be it st. 1 cost of 1 ... | {
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Show Tags
14 Nov 2012, 09:48
Pat bought 5 pounds of apples. How many pounds of pears could he have bought for same amount of money?
1) 1 pound of pears cost $0.5 more that 1 pound of apples 2) 1 pound of pears cost 1.5 times as much as 1 pound of apples We need a relationship between price of pears and that of apples ... | {
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"lm_label": "1. Yes\n2. Yes",
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"lm_q1_score": 0.9496693645535724,
"lm_q1q2_score": 0.8410570732361637,
"lm_q2_score": 0.8856314677809303,
"openwebmath_perplexity": 7086.429590594472,
"openwebmath_score": 0.46849584579467773,
"tags": ... |
Not sufficient.
(2) 1 pound of pears cost 1.5 times as much as 1 pound of apples. The cost of 5 pounds of apples is $5a (where a is the cost of 1 pound of apples). For$5a we can buy 5a/(1.5a)=5/1.5 pounds of pears. Sufficient.
Answer: B.
Hope it's clear.
Hello Bunuel,
Can you please correct my approach of solving ... | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9496693645535724,
"lm_q1q2_score": 0.8410570732361637,
"lm_q2_score": 0.8856314677809303,
"openwebmath_perplexity": 7086.429590594472,
"openwebmath_score": 0.46849584579467773,
"tags": ... |
Fact 2: 1 pound of pears costs 1.5 times as much as 1 pound of apples
IF...
A pound of applies costs $1, then a pound of pears costs$1.50
5 pounds of applies = $5$5 = $1.50(X pounds of pears) X = 3 1/3 pounds of pears A pound of applies costs$2, then a pound of pears costs $3 5 pounds of applies =$10
$10 =$3(X pounds ... | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9496693645535724,
"lm_q1q2_score": 0.8410570732361637,
"lm_q2_score": 0.8856314677809303,
"openwebmath_perplexity": 7086.429590594472,
"openwebmath_score": 0.46849584579467773,
"tags": ... |
$X = cost of 5 pounds of apples$X/5 = cost of 1 pound of apples
Fact 2 tells us that 1 pound of pears costs 1.5 times the cost of 1 pound of apples.
With some Algebra, we have...
(X/5) = cost of 1 pound of apples
(3/2)(X/5) = cost of 1 pound of pears
3X/10 = cost of 1 pound of pears
At this point, ankurjohar assume... | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9496693645535724,
"lm_q1q2_score": 0.8410570732361637,
"lm_q2_score": 0.8856314677809303,
"openwebmath_perplexity": 7086.429590594472,
"openwebmath_score": 0.46849584579467773,
"tags": ... |
Rich Cohen
Co-Founder & GMAT Assassin
Special Offer: Save $75 + GMAT Club Tests Free Official GMAT Exam Packs + 70 Pt. Improvement Guarantee www.empowergmat.com/ *****Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!***** Director Joined: 23 Jan 2013 Posts: 568 Schools: Cambridge'16 Re: Pat bought 5 po... | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9496693645535724,
"lm_q1q2_score": 0.8410570732361637,
"lm_q2_score": 0.8856314677809303,
"openwebmath_perplexity": 7086.429590594472,
"openwebmath_score": 0.46849584579467773,
"tags": ... |
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