An object has a location (or center) (x, y, z) and a size (or dimensions) (width, height, length). What is a single word for both? I thought of &quot;measurements&quot; but it's not so accurate.
I am developing an application and can not figure out a proper abstract term that would fit an interface that defines an object as Movable and Resizable (In essence, something that has a defined size and defined position). Calling it Movable and resizable is kind of silly. Layout was a term that came to my mind but it is not good as layout is something that is responsible for defining position and size, not containing it. Thanks a lot.
How to use manual partitioning during installation? Let $h$ the irreducible polynomial over $\Bbb F_2 = \{0, 1\}$  $$h  =  x^4 +x +1$$  Let $E=\Bbb F_2(\alpha)$, i.e. the field $\Bbb F_2$ extended with $\alpha$, where $\alpha$ is a root of $h$.   How do we determine the numbers of elements in the extension $E$? Can every non-zero element in the extension be expressed in the form $(\alpha)^n$ for some $n\in\Bbb N$ ? How do we find all the roots of $h$ over the extension $E$? I know it's possible to match a word and then reverse the matches using other tools (e.g. grep -v). However, is it possible to match lines that do not contain a specific word, e.g. hede, using a regular expression?   Input:  hoho hihi haha hede   Code:  grep "&lt;Regex for 'doesn't contain hede'&gt;" input   Desired output:  hoho hihi haha If $R$ is a local ring and $M$ and $N$ are finitely generated $R$-modules such that $M\otimes N=0$ then how does it follow from Nakayama's lemma that either $M=0$ or $N=0$?    This is an exercise in Atiyah and Macdonald. The part I could not show in the hints is $({M{\otimes}_R N)}_{k}=0$ implies $M_{k}{\otimes }_{k} N_{k}=0$, where $k=R/\mathfrak m$ and $\mathfrak m$ is the maximal ideal of $R$. The Megahertz Myth became a promotional tactic due to differences between the PC's INTEL 8086 processor and Apple's Rockwell 6502 processor.  The 8086 ran at 4.77MHz while the 6502 ran at 1MHz.  However, instructions on the 6502 needed fewer cycles; so many fewer, in fact, that it ran faster than the 8086.  Why do some instructions need fewer cycles?  And why can't the instructions of the 6502, needing fewer cycles, be combined with a fast cycling processor of the 8086?  Wikipedia's article for instructions per cycle (IPC) says     Factors governing IPC   A given level of instructions per second can be achieved with a high IPC and a low clock speed...or from a low IPC and high clock speed.   Why can't you have both high instructions per cycle and high clock speed?  Maybe this has to do with what a clock cycle is?  Wikipedia mentions synchronization of circuits?  Not sure what that means.  Or maybe this has to do with how a pipeline works?  I'm not sure why instructions in a short pipeline are different from instructions in a long pipeline.  Any insight would be great!  Just trying to understand the architecture behind the myth.  Thanks!  References: Can an inanimate thing lift Mjolnir? Can an inanimate thing can lift Mjolnir without the help of a living being (unworthy)  Scenario 1  Mjolnir in a very light car with a sail attached to it, then a very strong wind blew and moved the car uphill.  It is possible or not?  Scenario 2  Can upward wind alone lift the Mjolnir? How can you check for a #hash in a URL using JavaScript? When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$? I have a raster aerial photo.  I have county wide parcel data.  I want to clip/screen shot the raster at each individual parcel.  Think of this application - a post card with an image of your house --here is your house--  I want to have each parcel attribute to contain information + hyperlink of the aerial photo.  Any other solutions welcome, but this is the only method I could think of. Although I am unsure of how to achieve it efficiently.  Using ArcInfo 10. Rounded caps on a Bezier Curve pipe This may be a simple question.  Essentially I am procedurally generating random 3D figures using a bezier curve and bezier circle bevel object, however, one of the only things I can't figure out how to do is how to round the end caps of the object.   I have tried using a taper object like suggested  but have had only limited success.  This is about as close as I've gotten:   I should note that all of this is done through Python and I'm not modeling any of this by hand. So any suggestions for how I can make the rounded caps programmatically would be greatly appreciated. Characterization of short exact sequences The following is the first part of Proposition 2.9 in "Introduction to Commutative Algebra" by Atiyah &amp; Macdonald.     Let $A$ be a commutative ring with $1$. Let $$M' \overset{u}{\longrightarrow}M\overset{v}{\longrightarrow}M''\longrightarrow  0\tag{1} $$ be sequence of $A$-modules and homomorphisms. Then the sequence (1) is exact if and only if for all $A$-modules $N$, the   sequence  $$0\longrightarrow \operatorname{Hom}(M'', N)  \overset{\overline{v}}{\longrightarrow}\operatorname{Hom}(M,  N)\overset{\overline{u}}{\longrightarrow}\operatorname{Hom}(M', N)  \tag{2} $$ is exact.   Here, $\overline{v}$ is the map defined by $\overline{v}(f)=f\circ v$ for every $f\in\operatorname{Hom}(M'', N)$ and $\overline{u}$ is defined likewise.   The proof one of direction, namely $(2)\Rightarrow (1)$ is given in the book, which I am having some trouble understanding. So assuming (2) is exact sequence, the authors remark that "since $\overline{v}$ is injective for all $N$, it follows that $v$ is surjective". Could someone explain why this follows?  Given that $\overline{v}$ is injective, we know that whenever $f(v(x))=g(v(x))$ for all $x\in M$, we have $f=g$. I am not sure how we conclude from this surjectivity of $v$.  Thanks! Review of a proof for De Moivre's theorem using mathematical induction The following is a rigorous proof of De Moivre's theorem by means of mathematical induction. The theorem put simply is that:  Any complex number, $z = a+bi$, on a cartesian plane can be expressed in polar form, where $a=r\cos\theta$ and $b=r\sin\theta$ and $r$ is the absolute distance from the origin to the point $z$. In light of this, the expression for the complex number $z$ in polar form is $z=r(\cos\theta+i\sin\theta)$ In order to find the $n^{th}$ power of $z$, the following rule applies: $$\displaystyle z^n=(r(\cos\theta+i\sin\theta))^n=r^n(\cos n\theta+i\sin n\theta)$$ where $n\in\mathbb N$ (I have confined $n$ to the natural numbers given that I intend on using mathematical induction to prove the theorem.  Base step: We must prove that what is stated is true for $n=1$, hence $$z^1=r^1(\cos 1\cdot\theta+i\sin1\cdot \theta)$$ $$=r(\cos\theta+i\sin\theta)$$ which is true. Next we assume it is true for $n=k$, and therefore proceed to the inductive step. Inductive step: When calculating $z^{n+1}$, it's the same as calculating $z^nz$, hence $$z^{n+1}=(r^n(\cos n\theta + i\sin n\theta))(r(\cos \theta+i\sin \theta))$$ $$=r^nr(\cos n\theta+i\sin n\theta)(\cos\theta +i\sin \theta)$$ $$=r^{n+1}(\cos n\theta\cdot\cos\theta+\cos n\theta \cdot i\sin\theta+i\sin n\theta\cdot\cos\theta+i^2\sin n\theta\cdot\sin\theta)$$ from here, $i^2=-1$, and therefore $+i^2\sin n\theta\cdot\sin\theta$ becomes $-\sin n\theta\cdot\sin\theta$. In addition to this, the following trigonometric identities will be used: $$\sin(n\theta + \theta) = \sin n\theta\cdot\cos\theta+\cos n\theta\cdot\sin\theta$$ $$or$$ $$k\sin(n \theta+\theta)=k\sin n \theta\cdot\cos \theta + \cos n\theta\cdot k\sin\theta$$ $$and$$ $$\cos(n \theta + \theta)=\cos n\theta\cdot\cos\theta-\sin n\theta\cdot\sin\theta$$ by using these, the original equation has now become $$=r^{n+1}(\cos(n \theta + \theta) + i\sin(n \theta + \theta))$$ then, after factoring out $\theta$, the theorem becomes $$=r^{n+1}(\cos((n+1)\theta)+i\sin((n+1)\theta))$$ which completes the proof. Question: is this proof correct all the way through? Or have I missed any conditions? If all is well in the case of this proof, could somebody aid me in proving it for $\{n|n\in\mathbb R,n\geq1\}$ When I am writing I often come across the words "around" and "round." I was wondering what is different between them, and how they would be used in different contexts. Custom question lists: finding questions you can answer For example I want to use \iint from ams package  and also mathabx package in one file together. Actually I want a command that neutralizes ams package effect and replace it with mathabx package effect from here on in my file. Thanks for your responsibility.  This is my example minimal file:  \documentclass[12pt,a4paper]{report} \usepackage{amsmath}  \usepackage{mathabx}  \begin{document} i want this one in ams style:        \[ \iiint\,\iiint\, \iiint\,\oint\, \] text text text text text text text text text text text text   and this one in mathabx style:   \[ \iiint\,\iiint\, \iiint\,\oint\, \]   but both of them are the same and i don't like it ..... \end{document} I have a data set for a predictive model(predicting survival rate with certain acute medical condition on some animals) with 25 predictors where around 30% of the predictors are complete, 3 predictors are sitting at 30%,25% and 20% missing, and the others are missing at around 5% level. Around 50% of my data are complete cases. I'm new to dealing with missing values so I have a couple of questions regarding how to deal with it:   What can I do with the variable with 30% missing assuming it's MAR? Is 30% too high for imputation? What kinds of metrics can I use to make the decision between removal of the predictor, listwise removal, imputation or some other options? How should I deal with a predictor with 20% - 25% missing when I have reason to believe that it's MNAR? I'm thinking of using imputation on the remaining predictors with around 5% missingness. How do I make the decision on which imputation methods to use? Are they chosen on a case by case basis based on the individual predictors? How does one impute categorical values?  How does(should?) the imputation carry out in practice? Should it use complete cases or compute it iteratively somehow?  Does feature selection comes in before or after dealing with missing values? if at all? deploying and retrieving Address settings (State & Country picklists) Is the universe really "fine-tuned for life"? I understand what is meant when we say , but I've seen it claimed it's is fine tuned "for life". Is it?  When I look at known or even speculated life in our universe, I see life clinging to a tiny crumbs separated by vast voids. Our life needs baryonic matter, only 5% of the universe. Its density is $10^{-30}  g/cm^3$, but it clumps together leaving vast voids between the clumps. 99% is in stars and black holes and nebula utterly inhospitable to known life.  That leaves the parts "fine tuned for life" to be 0.05% of the universe. The planets, asteroids, and comets seem like just debris separated by  than the life-sustaining clumps preventing life from spreading.  Even that seems to be used very inefficiently. Life requires the energy of a star, but it's spewed out in all directions. We only receive a tiny fraction of it. Similarly, life requires heavier elements created in supernova, but these are also spewed out in all directions. And it took billions of years to develop a being capable of asking "is the universe fine tuned for life?"  It seems like we live on the crumbs of creation warmed by a rounding error of the heat from our star. Taking this view, the universe appears extremely inhospitable to life. It could be a lot better tuned.  Is it that the universe is fine-tuned for life? Or is it that the universe is so large and life requires such an infinitesimal fraction of the universe that it's a near guarantee were we to roll the dice again that some crumb, maybe not even baryonic matter, could develop something complex enough to ask "why is the universe fine tuned for life"? Am I just restating the weak anthropic principle?  I understand this is straying into philosophy. If that's off topic, are my  physical assumptions correct? This question has troubled me for ages despite my several attempts of looking it up in dictionaries or usage books. Do we say, "Do you have any ideas" or "Do you have any idea"? I do see an example where "any" means "it doesn't matter who/which/what", therefore "You can borrow any book you like." Also, does it matter between using it in questions and negatives? Suppose we want to define a first-order language to do set theory (so we can formalize mathematics). One such construction can be found . What makes me uneasy about this definition is that words such as "set", "countable", "function", and "number" are used in somewhat non-trivial manners. For instance, behind the word "countable" rests an immense amount of mathematical knowledge: one needs the notion of a bijection, which requires functions and sets. One also needs the set of natural numbers (or something with equal cardinality), in order to say that countable sets have a bijection with the set of natural numbers.  Also, in set theory one uses the relation of belonging "$\in$". But relation seems to require the notion an ordered pair, which requires sets, whose properties are described using belonging...  I found the following in Kevin Klement's,  (pages 2-3).  "You have to use logic to study logic. There’s no getting away from it. However, I’m not going to bother stating all the logical rules that are valid in the metalanguage, since I’d need to do that in the metametalanguage, and that would just get me started on an infinite regress. The rule of thumb is: if it’s OK in the object language, it’s OK in the metalanguage too."  So it seems that, if one proves a fact about the object language, then one can also use it in the metalanguage. In the case of set theory, one may not start out knowing what sets really are, but after one proves some fact about them (e.g., that there are uncountable sets) then one implicitly "adds" this fact also to the metalanguage.  This seems like cheating: one is using the object language to conduct proofs regarding the metalanguage, when it should strictly be the other way round.  To give an example of avoiding circularity, consider the definition of the integers. We can define a binary relation $R\subseteq(\mathbf{N}\times\mathbf{N})\times(\mathbf{N}\times\mathbf{N})$, where for any $a,b,c,d\in\mathbf{N}$, $((a,b),(c,d))\in R$ iff $a+d=b+c$, and then defining $\mathbf{Z}:= \{[(a,b)]:a,b\in\mathbf{N}\}$, where $[a,b]=\{x\in \mathbf{N}\times\mathbf{N}: xR(a,b)\}$, as in  question or  on Wikipedia. In this definition if set theory and natural numbers are assumed, then there is no circularity because one did not depend on the notion of "subtraction" in defining the integers.  So my question is:     Question Is the definition of first-order logic circular?   If not, please explain why.   If the definitions are circular, is there an alternative definition which avoids the circularity?   Some thoughts:   Perhaps there is the distinction between what sets are (anything that obeys the axioms) and how sets are expressed (using a formal language). In other words, the notion of a set may not be circular, but to talk of sets using a formal language requires the notion of a set in a metalanguage. In foundational mathematics there also seems to be the idea of first defining something, and then coming back with better machinery to analyse that thing. For instance, one can define the natural numbers using the Peano axioms, then later come back to say that all structures satisfying the axioms are isomorphic. (I don't know any algebra, but that seems right.) Maybe sets, functions, etc., are too basic? Is it possible to avoid these terms when defining a formal language?