Prove that $\prod_{i=1}^n a_i^{\lambda_i}\le \sum_{i=1}^n\lambda_i a_i.$ Question: Let $a_1,a_2,\cdots,a_n$ be positive real numbers and $\lambda_1,\lambda_2,\cdots,\lambda_n$ be non-negative real numbers such that $$\sum_{k=1}^n\lambda_k=1.$$ Prove that $$\prod_{i=1}^n a_i^{\lambda_i}\le \sum_{i=1}^n\lambda_i a_i.$$    Solution: Observe that for each $a_i$, $1\le i\le n$, we have an unique $y_i\in\mathbb{R}$, such that $a_i=e^{y_i}$. Therefore, we have $$a_1=e^{y_1},a_2=e^{y_2},\cdots, a_n=e^{y_n}.$$  Discrete Jensen Inequality: Let $f$ be a convex function defined on an interval $I\in\mathbb{R}$. Assume that $x_1,x_2,\cdots,x_n\in I$ and let $\lambda_1,\lambda_2,\cdots, \lambda_n$ be non-negative real numbers such that $\sum_{k=1}^n\lambda_k=1$. Then, $$f\left(\sum_{i=1}^n \lambda_i x_i\right)\le \sum_{i=1}^n\lambda_if(x_i).$$   Now $f(x):=e^x, \forall x\in\mathbb{R}$ is a convex function. Also we have $a_i=e^{y_i}$, $1\le i\le n$, where each $y_i\in\mathbb{R}$ and $\lambda_1,\lambda_2,\cdots,\lambda_n$ are given non-negative real numbers such that $\sum_{k=1}^n\lambda_k=1$. Thus, applying Discrete Jensen Inequality we have $$e^{\sum_{i=1}^n\lambda_i y_i}\le \sum_{i=1}^n\lambda_i e^{y_i}\\\implies \prod_{i=1}^n (e^{y_i})^{\lambda_i}\le \sum_{i=1}^n\lambda_i e^{y_i}\\\implies \prod_{i=1}^n a_i^{\lambda_i}\le \sum_{i=1}^n\lambda_i a_i.$$ Hence, we are done.   Is this solution correct and rigorous enough? And, is there any other way to solve the same?
$\sum_{1}^{n}\left ( a_{i}b_{i} \right )\geq \prod_{1}^{n}\left ( a_{i}^{b_{i}} \right )$ Let $a,b,c,x,y,z&gt;0$ and $x+ y+ z= 1$. Show that $$ax+ by+ cz\geq  a^{x}.b^{y}.c^{z}$$ I have found the generalization of this inequality: For 2 sets of numbers $\left \{ a_{i} \right \}, \left \{ b_{i} \right \}&gt; 0$ and $\sum_{1}^{n}b_{i}= 1$, we have: $$\sum_{1}^{n}\left ( a_{i}b_{i} \right )\geq  \prod_{1}^{n}\left ( a_{i}^{b_{i}} \right )$$ First solution Simply compare the logs: as the natural logarithm is a concave function, one has $$\ln\bigl(a^x b^y c^z)=x\ln a+y\ln b+z\ln c\le \ln(ax+by+cz)$$ for all non-negative $x,y,z$ such that $x+y+z=1$. How about another solution? I really hope to see that. Thanks!
Basically, I want to have a single operator which can do multiple things, based on some attributes values.  class MyPanel(bpy.types.Panel):     # (...)     layout = self.layout     row = layout.row()     row.operator("my.button", text="Button text")  class MY_BUTTON_OT_Button(bpy.types.Operator):     bl_idname = "my.button"     bl_description = "Button description"     bl_label = "Button"     foo = bpy.props.IntProperty()     bar = bpy.props.BoolProperty()      if bar:         # Do something with foo     else:         # Do something else :P   I know how to set one attribute, like so:      row.operator("my.button", text="Button text").foo=5   But what about multiple arguments? I tried several things that didn't worked, like (e.g.):   With setattr(): row.operator("my.button", text="Button text").setattr(foo=5,bar=True)   With a custom method setVal() in the operator class: row.operator("my.button", text="Button text").setVal(foo=5,bar=True)   Maybe I must use a single EnumProperty? I hope to avoid this, so my question is: Is it possible to pass multiple custom arguments to set attributes in an operator class? And if yes, how ?  Thanks for your help. Should the eigenkets be weighted in $|P\rangle = \sum\limits_{r}|\xi^r\rangle$? Page 37 of Dirac's book The Principles of Quantum Mechanics, states  The condition for the eigenstates of $\xi$ to form a complete set must thus be formulated, that any ket $|P\rangle$ can be expressed as an integral plus a sum of eigenkets of $\xi$, i.e. $$|P\rangle = \int |\xi'c\rangle\; d\xi' + \sum\limits_{r}|\xi^rd\rangle$$ where the $|\xi'c\rangle$, $|\xi^rd\rangle$ are all eigenkets of $\xi$, the labels c and d being inserted to distinguish them when the eigenvalues $\xi'$ and $\xi^r$ are equal, and where the integral is taken over the whole range of eigenvalues and the sum is taken over any selection of them.  I find it inconsistent that the eigenkets under the integral are weighted by what appears to be a differential eigenvalue $d\xi'$, yet all the eigenkets under the summation are weighted by the value 1. Is this correct? How to generate XML file dynamically using PHP? If the Standard Normal PDF is $$f(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}$$  and the CDF is  $$F(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-x^2/2}\mathrm{d}x\,,$$  how does this turn into an error function of $z$? Can Compound Topographic Index (CTI) / Topographic Wetness Index (TWI) produce negative values? I am calculating Topographic Wetness Index for a study area using SRTM DEM (90 and 30m).   I've seen several versions of TWI (CTI).  I have been using: TWI = ln(As +1 / tanB)       where As = catchment area, B = slope (degrees/radians??)  When I run this function, the output contains negative values (using slope in degrees and radians).  Are negative values acceptable? Or does this indicate an error in the analysis??  I understand the output should be scaled to 1-10, but negative values seems counter intuitive. Could someone explain what negative values could mean (due to flat areas, no data values?)  On a side not, the calculations are being run on on a very large area, which overaps several watersheds. Is it appropriate to run TWI over such a large extent? Replaced wrong car in garage in GTA Online I was asked to replace a car in order to put another car in my garage. However I thought I would be able to swap the car I wanted in and take out the other car that was already in there so I could bring it to my other garage. Can I get my car back? How to run a background task in a servlet based web application? Robust method to extract "pure" part of a variable I am trying to perform a multi linear regression model:  $$y_i = β_0 + β_1x_{i1} + β_2x_{i2} +... + β_px_{ip} + ε_i$$  where $$x_{i1}, x_{i2}, ..., x_{ip}$$ are highly correlated with each other (VIFs can be as low as 5 and high as 30).  I am just wondering if there exists a procedure with the following properties:  1) extracts the pure part of each independent variable, where pure is defined as a part of the variable which is uncorrelated with other variables.  2) the pure part and the original variable should be similar to each other, whether similar is defined as moments such as mean and variance and correlation coefficient (ideally >0.90) What language is the Ubuntu Software Center written in? Just been wondering, since elementary AppCenter is written with Vala and Gtk+3.  Also, what GUI wrapper does it use? If $\Gamma$ is satisfiable then $\Gamma$ is consistent We say that $\Gamma$ is consistent if for no $\phi$, then both $\Gamma\vdash \neg\phi$ and $\Gamma\vdash\phi$. We say that $\Gamma$ is satisfiable if there is a structure $M$ s.t. $\models_M\gamma[s]$ for alle $\gamma\in\Gamma$.  I want to show that if $\Gamma$ is satisfiable, then $\Gamma$ is consistent.  I'm thinking we assume that $\Gamma$ is not consistent, and then show that $\Gamma$ is not satisfiable. So if $\Gamma$ is not consistent then both $\Gamma\vdash \neg\phi$ and $\Gamma\vdash\phi$ for some $\phi$. But by soundness theorem then $\Gamma\models\neg\phi$ and $\Gamma\models\phi$, meaning for structure $M$, we have $\models_M\neg\phi[s]$ (whereas $\not\models_M\phi[s]$) and $\models_M\phi[s]$. So would this mean that $\Gamma$ is not satisfiable? I'm a little confused, because wouldn't this instead be a contradiction? Why is charge of the electron negative? 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Reverse of number in PHP not showing correct output Is it possible to get total render time via Python code when render’s done? I have seen some comments and speculations about the robot having an off switch or whatever. But I want to know if there is any source material or comment from the author on how they managed to pass the exam.  I have only seen the actual author's comment on Mineta:     But I was wondering if there are author notes for the other characters. How do you prove that there are infinitely many primes of the form $5 + 6n$?