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Given that the core flux variability of a sample of sources is reported as $1.36\pm0.15$ mJy and the jet flux variability as $0.51\pm0.12$ mJy, what is the maximum percentage variability that can be observed in the core flux compared to its average value? | |
What is the maximum value of the density function of the 2D Normal distribution $N(0, \left[ \begin{array}{cc} 1 & -1/4 \\ -1/4 & 1 \end{array} \right])$ over the region defined by $\{ x \geq 10, y \geq 10 \}$? | |
In the category of finite sets and bijections, what are the properties of group objects, and specifically, can you provide examples or descriptions of any group objects found within this category? Additionally, considering functor categories derived from finite sets and bijections, what would be the implications of def... | |
Consider a family of classification problems \( f_i \) where for any fixed depth \( d \) of a rectified linear unit (ReLU) neural network, there exists an index \( m \) such that problems with index \( n > m \) require exponential width to represent the function \( f_m \). If the width of the network is denoted by \( W... | |
What is the predicted day-night asymmetry in the solar neutrino flux for maximal mixing angles in the context of the MSW solution, given that the flux suppression is stated to be 1/2? | |
Determine the coordinates of the vertices and the asymptotes of the hyperbola represented by the equation \( x^2 - y^2 = 9 \). | |
Given the relationship for the Nusselt number in a heated horizontal plate subjected to free convection in a rapidly rotating system, expressed as \(\Nu \sim \frac{1}{2} \Ra^{1/2} \Ek^{3/4} [\ln (\Ra^{-1} \Ek^{-5/2})]^{1/2}\), where \(\Ra\) is the Rayleigh number and \(\Ek\) is the Ekman number, calculate the Nusselt n... | |
Prove that for every $\infty$-category $\mathcal{E}$, the mapping from $X$ to $\operatorname{id}_{X}$ induces an equivalence of $\infty$-categories between $\mathcal{E}$ and $\operatorname{Fun}(\Delta^0, \mathcal{E})$ and between $\operatorname{Fun}(\Delta^1[W^{-1}], \mathcal{E})$ and $\operatorname{Isom}(\mathcal{E})$... | |
If the Player Hands object in the Unity project applies three different forces with magnitudes of \( F_1 \), \( F_2 \), and \( F_3 \) representing the camera follow strength, spring force, and sway drag respectively, and if the resultant force acting on the Player Hands object is \( F_{\text{resultant}} = F_1 + F_2 - F... | |
In a cooperative game with 2001 players, where 1000 players own 1 left shoe each and 1001 players own 1 right shoe each, what is the unique imputation in the core, and why can't any coalition block this outcome? | |
If K is the midpoint of segment JL, where segment JK is represented by the expression \( 8x - 8 \) and segment KL is represented by \( 7x - 6 \), what is the length of segment JL when \( x = 2 \)? Please provide the exact numerical value for the length of JL. | |
What is the maximum number of different environments you can define in a LaTeX document if you use the command `\newenvironment{name}{begin}{end}` and you define 5 different environments? Include the meanings of each environment. | |
Given a finite vector space \( V \) of dimension \( d \) and a subset \( A \subseteq V \) such that the \( 2m \)-fold additive energy \( E_{2m}(A) > |A|^{2m}d^{-m}(2^{4m+3}m^{6m+4}C^{2m} + 2m^{2m+1}) \) for some constant \( C > 0 \), what is the minimum size of a subset \( S \subseteq A \) such that the dimension of th... | |
What is the correct expression for the density of states for a hole, \( D_h(\epsilon) \), assuming that the energy of a hole is given by the relation \( \epsilon = \frac{\hbar^2 k^2}{2m_h} - E_F \), and how can I ensure the result is positive? | |
What is the minimum number of shares required in Shamir's Simple Sharing Scheme to reconstruct the original secret if the scheme is designed such that a single share can be erroneous and still allow for successful secret recovery? | |
What is the value of the expression \( 4^{\frac{3}{2}} \cdot 2^{-\frac{1}{2}} \)? | |
In a time series analysis using the ResamplerModel interface, if the fitted values based on a given set of OLS residuals are represented by the matrix \( F \) and the conditional variances over time are represented by the matrix \( \sigma^2 \), what is the mathematical relationship between the standardized innovations ... | |
Given the need to configure a compilation environment for Qt with Phonon using MSVC 2005 on a Windows 7 operating system, what are the specific SDK and Compiler versions required, and what command line script should be used to set the environment for a 32-bit architecture? | |
Tomas saved 230 CZK at the beginning of the year, and by the end of six months, his total savings reached 590 CZK. How much did he save each month if he saved the same amount every month? | |
Consider the sequence of polygons defined by the vertices given by the points \((0, 0)\), \((0, i)\), \((1, i)\), and \((1, 0)\) for values of \(i\) in the set \([0.1, 1, 10, 100]\).
1. Calculate the area of each polygon described for the respective values of \(i\).
2. Determine the ratio of the areas of the polygons... | |
Given the parametric equations \( x = \sec(t) \) and \( y = \tan(t) \) for \( -\frac{\pi}{2} \leq t \leq \frac{\pi}{2} \), derive an explicit relationship between \( y \) and \( x \) and determine the value of \( y \) when \( x = 2 \). | |
Given a smooth projective variety \( V \) over a finitely generated field \( k \) and a prime \( l \) different from the characteristic of \( k \), if the Tate conjecture holds for that variety, can you determine how many homology classes of algebraic cycles of codimension \( i \) exist on \( V \otimes_k \overline{k} \... | |
Given the angle \( t = -\frac{5\pi}{6} \), find the values of \( \sin t, \cos t, \tan t, \sec t, \csc t, \) and \( \cot t \). | |
Given a wave function at time \( t = 0 \) for a spherically propagating shell containing \( N \) neutrons, expressed as
\[
\psi(r,0) = 4\pi i j_{1}(kr) \left( \frac{3}{\sqrt{34}} Y^{0}_{1} + \frac{5}{\sqrt{34}} Y^{-1}_{1} \right),
\]
how do you determine the wave function \( \psi(r,t) \) at a later time \( t \)? Con... | |
Given the vectors \(\vec{A} = 4 \hat{i} + 3 \hat{j}\) and \(\vec{B} = 5 \hat{i} - 2 \hat{j}\), what is the vector difference \(\vec{A} - \vec{B}\) expressed in terms of unit vectors \(\hat{i}\) and \(\hat{j}\)? | |
Given the Hamming distance formula \( D_{Hamming} = S_{A} + S_{B} - 2S_{AB} \), where \( S_{A} \) is the number of nonzero OTUs in community A, \( S_{B} \) is the number of nonzero OTUs in community B, and \( S_{AB} \) is the number of OTUs shared between communities A and B, calculate the Hamming distance between the ... | |
What would be the predicted miles per gallon (mpg) for a car with 6 cylinders, a displacement of 200, horsepower of 150, weight of 3200, acceleration of 12.5, year of 75, and of European origin, using the linear regression model derived from the provided dataset? | |
What is the period of oscillation for a hemispherical shell rocking back and forth on a flat surface, given that the moment of inertia about the equilibrium point of contact is \( I = \frac{2}{5} m r^2 \) and the center of mass is at a distance \( \frac{2}{3} r \) from the point of contact, where \( m \) is the mass of... | |
Given an image with dimensions \(W \times H\) (width by height) and its pixel intensity values represented as a function of two variables \(I(x,y)\), where \(x\) and \(y\) are the pixel coordinates, the Fourier transform \(F(u,v)\) of the image is defined as:
\[
F(u,v) = \int_0^W \int_0^H I(x,y) e^{-2\pi i (ux/W + vy/... | |
Prove that if a vector space \( V \) is defined over a field \( F \) that is not a scalar field (such as the integers), then \( V \) cannot have scalar multiples that satisfy the properties of a vector space, specifically the properties of scalar multiplication. What is the significance of having scalars from a field i... | |
What is the relationship between the applied torque and the angular acceleration of a thick ring when measured experimentally, and how does this relate to the moment of inertia calculated for the ring? | |
Let \( a \in (0,1) \) be a real number such that \( a \) is not of the form \( \frac{1}{n} \) for any natural number \( n \). Construct a continuous function \( f: [0,1] \to \mathbb{R} \) such that \( f(0) = f(1) \) and \( f(x) \neq f(x+a) \) for all \( x \) satisfying \( x, x+a \in [0, 1] \).
What is the form of suc... | |
What is the relationship between the number density of phonons \( n \), the frequency \( v \), and the total energy density \( E_{\text{Total}} \) of a sound wave, and how does the energy change affect the amplitude of the wave without altering its frequency? Additionally, how can this understanding reconcile the appar... | |
What is the percentage increase in the value of the worldwide additive manufacturing market from 2012 to 2021, given that it grew from $2.2 billion to more than $10.8 billion? | |
Given a point particle of mass \( m = 2 \, \text{kg} \) and charge \( q = 1 \, \text{C} \) moving in a constant magnetic field \( \vec{B} = (0, 0, 1) \, \text{T} \) with an initial velocity \( \vec{v}(0) = (3, 4, 0) \, \text{m/s} \), determine the trajectory \( \vec{r}(t) \) of the particle as a function of time. What ... | |
What is the maximum number of unique monster configurations that can be generated using a single monster class object, given that each monster has three attributes (name, max health, and ability) and that there are 5 possible names, 10 possible max health values (from 1 to 100), and 4 possible abilities? Moreover, if p... | |
In designing integrated circuits, if a resistor has a sheet resistance of 1 Ω/square, what would be the resistance of a rectangular track that is 50 µm wide and 200 µm long? | |
In a deformable isosceles triangle, the vertex \( C \) moves along a certain path, affecting the length of diagonal \( AC \). Given that the movement can be represented by the expressions \( a \) and \( d \) for the horizontal change, and \( b \) and \( c \) for the vertical change, can you derive and explain why the i... | |
Consider a scenario where you want to create a session management feature in C++ that mimics the behavior of PHP's `$_SESSION` associative array for storing user data. You decide to implement this using a `std::map<std::string, boost::any>` to allow dynamic types.
1. If you store the string `"John Doe"` under the key ... | |
Given a subset \( A \subseteq \mathbb{F}_q \) with size \( |A| > q^{\frac{2}{3} - c} \) for some constant \( c > 0 \), what is the minimum possible size of the set of products of differences \( (A - A) \cdot (A - A) \)? | |
Given the analysis of the Australian unemployment rate, what is the Akaike Information Criterion (AIC) value for the model that includes only interest rates as an explanatory variable when applied to the entire dataset from the late 1970s? | |
Given the following sentences, determine the correct preposition to complete each statement. Each sentence is missing either "since" or "for." How many sentences correctly utilize "since"?
1. Mary has been waiting for a reply _____ weeks.
2. Mr. Smith has been in that meeting _____ hours!
3. I have worked for Internat... | |
How many distinct five-card poker hands contain exactly two pairs and one other card, and what is the reasoning behind the counting method used to arrive at this number? | |
Consider the following regular expression designed to validate an IP address:
\[ \text{Pattern: } ^([01]?\d\d?|2[0-4]\d|25[0-5])\.([01]?\d\d?|2[0-4]\d|25[0-5])\.([01]?\d\d?|2[0-4]\d|25[0-5])\.([01]?\d\d?|2[0-4]\d|25[0-5])\$ \]
How many valid IP addresses exist that consist of four segments, where each segment is an i... | |
Given a Python dictionary `myDict = {'learnshareit': 'LearnShareIT', 'author': None}`, if you convert `myDict.items()` into a list and access the first element, what will be the output? Provide the value that is returned. | |
Given a directory containing 100 files of various types (30 JPEG images, 25 PDF documents, 20 Word documents, and 25 other miscellaneous files), determine the number of commands needed to categorize these files into three separate directories: one for images, one for PDFs, and one for Word documents. Each command is as... | |
Given the S-parameters \( S_{11}, S_{21}, S_{22} \) for a metamaterial, calculate the effective refractive index \( n_{eff} \) using the formula:
\[
n_{eff} = \frac{1}{k d} \cos^{-1}\left(\frac{1 - S_{11}^{2} + S_{21}^{2}}{2 S_{21}}\right)
\]
Assuming \( S_{11} = 0.5 \), \( S_{21} = 0.8 \), the wave vector \( k = 2\p... | |
In a company of 25 employees, 8 are graduates. If 3 employees are selected at random, what is the probability that at most 1 of the selected employees is a graduate? | |
Given a pricing experiment where 1,000 items are assigned to a positive control group and another 1,000 items to a negative control group, and the sales volumes for the positive group increase by 25% while the negative group’s sales volumes decrease by 10%, calculate the ratio of sales volumes in the positive group to ... | |
Given a power series \(\sum_{n=0}^{\infty} a_n x^n\) that converges for all \(x \in (x_0, x_1)\), what can you conclude about its radius of convergence \(R\), and how can you determine its value using the formula \( \frac{1}{R} = \limsup_{n \to \infty} |a_n|^{1/n} \)? | |
Let \( A \) be a \( 4 \times 4 \) matrix with three distinct eigenvalues, where one eigenspace is one-dimensional, and one of the other eigenspaces is two-dimensional. Is it possible for \( A \) to be diagonalizable? Justify your answer based on the properties of eigenvalues and eigenspaces. | |
Let \( \{s_n\} \) be the sequence defined by \( s_n = (-1)^n \cdot n \). Determine the values of \( \limsup_{n \to \infty} s_n \) and \( \liminf_{n \to \infty} s_n \). Provide a rigorous proof for your answer. | |
If I want to save for a vacation that costs $5,000, and I currently have $1,000 saved, plan to save $200 each month, and can invest my savings with a monthly interest rate of 0.5%, how many months will it take for me to afford the vacation? | |
What is the order of reactivity for the following reducing agents when reacting with carbonyl compounds, from least reactive to most reactive: $$\ce{NaBH4}$$, $$\ce{LiBH4}$$, $$\ce{LiAlH4}$$, and $$\ce{BH3}$$? | |
Let \( G \) be a group and \( H \) be a core-free permutable subnormal subgroup of \( G \) with subnormal depth \( k \). What is the maximum possible solvable length of \( H \)? | |
If the total price \( P \) of \( n \) commodities, each priced at \( c \), is represented by the equation \( P = c \cdot n \), what is the relationship between the total price and the quantity of commodities when the price per commodity remains constant?
A) directly proportional to
B) inversely proportional to
C)... | |
If the sum to infinity of an infinite geometric series is \( S_{\infty} = \frac{81}{2} \) and the ratio of the sum of the first \( n \) terms to the \( n \)-th term is given by \( \frac{S_n}{T_n} = \frac{3^n - 1}{2} \), what is the common ratio \( r \) of the geometric series? | |
Consider a column of fluid with uniform density \( \rho \) that extends to a depth \( h \). Determine a formula for the hydrostatic pressure \( P \) at a depth \( h \) in terms of the density \( \rho \), the gravitational acceleration \( g \), and the depth \( h \). How does the pressure change with an increase in dept... | |
A regular hexagon has a side length of \( s \) cm. If \( s = 10 \), what is the area of the hexagon in square centimeters? | |
Given the transactions between you and Ms. X, where you sold her a product for $229,214 and then sold her another product for $429,529, while Ms. X sold you a product for $456,193, can you calculate how much each of you owes the other to settle the accounts to a zero balance without either party profiting? | |
What is the maximum number of unit squares that can be shaded blue in a \(5 \times 5\) grid if at most 5 out of 9 unit squares can be shaded blue in any \(3 \times 3\) sub-grid? | |
What are the universal accumulation points in the spectrum of local operators at large transverse spin \( s \) in a defect conformal field theory, and how do they relate to the bulk to defect operator product expansion (OPE)? | |
Consider a string \( t \) of length \( m \) and an array of substrings \( s \) of size \( n \). Suppose we aim to cover the entire string \( t \) using the substrings from \( s \), such that each character in \( t \) is covered exactly once. What is the minimum number of substrings needed to cover \( t \)? Additionally... | |
Given the velocity \( v(t) \) of an object moving along the x-axis is represented by the graph between \( t=0 \) to \( t=2 \) seconds, calculate the total displacement of the object over this time interval in meters. Also, at what points in time is the object's acceleration equal to zero? Show your work. | |
In a quantum system described by a time-independent Hamiltonian with an initial energy eigenstate of \( E = 1 \), if a measurement is made of a non-commuting observable and subsequently the energy is re-measured, what is the expected value of the change in energy after the second measurement, assuming the system can tr... | |
How can I modify the LaTeX `nomencl` package to display a nomenclature list where each nomenclature entry is followed by an equals sign and the description, and how can I ensure that the widths of the entries are adjusted automatically? | |
Let \( f(x) = x^2 + \sqrt{1 + 2x} \) for \( 4 \leq x \leq 6 \). Use the definition of the definite integral to find a limit expression for the area under the graph of \( f \) from \( x = 4 \) to \( x = 6 \). Do not evaluate the limit. What is the expression for the area in terms of \( n \)? | |
Consider a uniformly charged surface whose perimeter is given by the infinite iteration of the triadic Koch curve. If the total charge distributed across the surface is \( Q \) and the area enclosed by the Koch curve at the \( n \)-th iteration is \( A_n \), derive the expression for the electric field \( E \) at a per... | |
What is the expected value of the sum of squares of random variables \( X_{n,1}, X_{n,2}, \ldots, X_{n,n} \) distributed according to the probability measure \( P_{\beta,n} \), specifically given by
\[
\mathbb{E}\left(X_{n,1}^2 + X_{n,2}^2 + \cdots + X_{n,n}^2\right)?
\]
for \( \beta > 0 \) and \( n \geq 2 \)? | |
What is the maximum number of characters that need to be added or removed in a single edit of a post for the edit reason to still be considered as "minor" according to typical standards in online forums? | |
A circular ripple created by a pebble dropped in a pond increases in radius at a rate of \(6\) inches per second. What is the rate of change of the area of the circle at \(t = 2\) seconds? | |
Let \( V \) be the vector space of all real-valued functions defined on \( \mathbb{R} \). Define the set \( W = \{ f \in V \mid f(x^2) = f(x)^2 \text{ for all } x \in \mathbb{R} \} \).
Is \( W \) a subspace of \( V \)? Justify your answer by determining whether \( W \) is closed under addition and scalar multiplicati... | |
Given the electric potential \( V = x^3yz + 2y^2z + xz^4 \), calculate the divergence of the electric field \( E \) at the point \( (2, -3, 1) \) and determine the corresponding charge density \( \rho \) at that point. Assume \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/\text{N} \cdot \text{m}^2 \). What is the v... | |
Given the dataset of mushrooms classified into two groups—edible and non-edible—using the provided disjunctive rules for poisonous mushrooms, calculate the accuracy of the model after removing the features associated with the first three disjunctive rules (odor, spore-print-color, and habitat). If the model's accuracy ... | |
Determine the open intervals in which the function $$f(x) = 2\,x^3 - 15\,x^2 + 36\,x + 1$$ is increasing and decreasing. Provide your answer in the format of intervals and select the correct option from the choices below:
A. $$f$$ is increasing in $$(-\infty, 2) \cup (3, \infty)$$ and decreasing in $$(2, 3)$$
B. $$f$... | |
Given that there are fifteen switches available to control the visibility of components of exercises, if an author uses 6 switches with the value "yes" and the rest with the value "no," how many different combinations of visibility settings are possible for this selection of switches? | |
What is the maximum number of round-trips required to ensure that the counter of a document in MongoDB is correctly incremented exactly once when designing a resilient update operation, and how can the update operation be made idempotent? | |
What is the difference in the periods of a simple pendulum of length 1 meter when measured on Earth and on Venus, given that the acceleration due to gravity on Earth is approximately \( 9.80 \, \text{m/s}^2 \) and on Venus is \( 8.87 \, \text{m/s}^2 \)? | |
What is the maximum achievable velocity change for a rocket given an exhaust velocity \( V_e \) and a structural fraction \( f \), using the equation \( \frac{\Delta V}{V_e} = \log{\frac{1+f}{f}} \)? Calculate this maximum velocity change assuming \( V_e = 4500 \) m/s and \( f = 0.1 \). | |
Given a hyperbolic isomorphism \( A: E \to E \) with skewness \( \lambda < 1 \), and assuming \( \varphi \) and \( \psi \) are Lipschitz continuous mappings satisfying certain conditions, what is the maximum Lipschitz constant \( K \) such that for any \( \varepsilon \)-Lipschitz map \( \varphi \) defined on a neighbor... | |
Let \( a, b, c \) be positive real numbers such that \( a^2 + b^2 + c^2 = 1 \). Determine the minimum value of the expression
\[
E = \frac{a^2}{1 + 2bc} + \frac{b^2}{1 + 2ca} + \frac{c^2}{1 + 2ab}.
\]
What is this minimum value? | |
What are the eigenvalues of the operator \( L_x \) when \( x \) is an idempotent element in an Euclidean Jordan algebra, and how does this relate to the Peirce decomposition of the space? Additionally, could you clarify how the concept of Jordan frames connects to the orthonormal basis of the underlying vector space, p... | |
Determine the ascending order of the numbers \( 2^2, 2^e, e^2, e^e \). Which of these numbers is the second largest? | |
If a jsonfilter.py command takes 3 seconds to extract 10,341 objects from a 41.6MB file containing 131,515 objects, how long would it take to extract all 131,515 objects from the same file? | |
What is the refrigerating effect, in BTUs per cubic foot, of saturated ammonia gas, and how many cubic feet of ammonia gas are required to achieve one ton of refrigeration? | |
What is the result of evaluating DiscreteLimit[f[x], x -> Infinity] if f is defined as f[x_Integer] := 0? Provide a brief explanation for your answer. | |
How many times will the for loop execute if it is structured to iterate with the conditions `i = 0`, `i < 10`, and `i++`? | |
Given a continuous function \( f(x, y) = 0.74894\, e^{0.5xy} \cos \left( \frac{3 \pi x y}{2} \right) \) defined over the square domain \(-1 \leq x \leq 1, -1 \leq y \leq 1\), calculate the integral
\[
I = \int_{-1}^{1} \int_{-1}^{1} f(x, y) \, dy \, dx
\]
using Gaussian quadrature with an order of 3 for both dimens... | |
In the context of training deep neural networks, if we initialize the weights of a layer using a variance of \( \frac{1}{n} \), where \( n \) is the number of input features, what is the expected variance of the outputs from that layer if the input features have a mean of 0 and a standard deviation of 1? Additionally, ... | |
What is the minimum communication complexity required for a protocol to ensure both computational security and statistical security under common security definitions such as Reactive Simulatability and Universal Composability? | |
What are the coordinates of a point that is 5 units in front of an entity located at (0, 0) and facing north-east (45 degrees) on a 2D plane? | |
Given a symmetric matrix \( A \) of order \( n = 4 \) defined as:
\[
A = \begin{pmatrix}
1 & 2 & 3 & 4 \\
2 & 2 & 3 & 4 \\
3 & 3 & 3 & 4 \\
4 & 4 & 4 & 4
\end{pmatrix}
\]
Compute the eigenvalues of matrix \( A \) that fall within the half-open interval \((-1, 1]\). What are the eigenvalues found in this interval? | |
What is the normalization factor for the function \( f(\mathbf{x}) = \exp(-\mathbf{x}' A \mathbf{x}) \) to make it a valid density function over the set \( S_d = \{ (x_1, \ldots, x_d) \in \mathbb{R}^d \mid \sum_{i=1}^d x_i = 0 \} \) when \( A \) is the \( d \times d \) identity matrix? | |
If an original research paper submitted to the journal exceeds the maximum recommended length of 25 double-spaced pages and contains more than 15 figures or tables, by how many pages or figures/tables would the submission be considered non-compliant with the journal's guidelines? | |
Given the logic function \( F = A'C'D + B'C D + AC'D + BCD \), find a minimal sum-of-products expression using Karnaugh map minimization techniques. | |
What is the maximum possible mass of a neutron star given that its radius is typically around 10 kilometers and the average density of a neutron star is approximately \(4 \times 10^{17} \, \text{kg/m}^3\)? Assume a spherical shape for the neutron star. | |
What is the 10th centered square prime number? | |
What are the total number of unique anchors used in the Region Proposal Network (RPN) if there are 3 different scales and 3 different aspect ratios, and how many bounding boxes does this imply RPN is predicting per location in the feature map? | |
Given the Dirichlet and Neumann boundary value problems for the heat equation in a wedge with an edge of arbitrary codimension, what specific conditions must the wedge satisfy to guarantee the existence and uniqueness of solutions under \( L^p \) estimates? Additionally, provide a brief explanation of how \( L^p \) est... | |
Given a positive integer \( n \), how many times does the loop run when printing the numbers from \( n \) down to \( 1 \) in reverse order using a for loop that decrements by \( 1 \) each iteration? |
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