I would like to think of "state of an object" as, classically, the simultaneous knowledge of its position, momentum, rotational angular momentum, orbital angular momentum, kinetic energy, etc. In quantum mechanics, "state of an object" would be simultaneous knowledge of the probability distributions of all the observables.  But we are typically taught that "state" is a vector, sometimes is represented by a wave function if we are talking about its position and momentum in space sometimes is just an abstract vector if we are talking about spin.  This definition of state does not seem to agree with what I want a state to be. But is the following true?     Suppose a state induces a probability distribution for each observables, be it position, momentum, kinetic energy, spin, etc. There is no other state that induces the same probability distributions of observables.   (if two states differ by a scalar multiple, they are considered the same)  In particular, if two wave functions induces the same probability distributions of position and momentum, are they the same wave function?  If two wave functions $f,g$ have such properties, then their probability amplitude are equal that $\lvert f(x)\rvert^2=\lvert g(x)\rvert^2$, meaning $f(x)=e^{i\theta(x)}g(x)$ for some real-valued function $\theta$. I believe $\theta$ may have to be a constant function if $f,g$ induces the same probability distributions of momentum.  By the same arguments, their Fourier transform are related by $\hat f(k)=e^{i\phi(k)}\hat g(k)$. This should lead to $$\int_{-\infty}^{\infty}e^{i\theta(x)}g(x)e^{-ikx}dx=\int_{-\infty}^{\infty}e^{i\phi(k)}g(x)e^{-ikx}dx.$$Not sure how that is helpful.
Consider a "wavefunction" $\psi(x)$, which has a Fourier transform $\tilde \psi(p)$  Suppose that we know, for each $x$, $|\psi(x)|^2$, and that we know, for each $p$, $|\tilde \psi(p)|^2$.  Have we enough information to reconstruct the "wavefunction" $\psi(x)$, that is, obtain the phase of the "wavefunction" for all $x$ (up to a global phase, we are only interesting to the relative phases between the $x$)?
Using CSS to affect div style inside iframe Schema for a multilanguage database I'm developing a multilanguage software. As far as the application code goes, localizability is not an issue. We can use language specific resources and have all kinds of tools that work well with them.  But what is the best approach in defining a multilanguage database schema? Let's say we have a lot of tables (100 or more), and each table can have multiple columns that can be localized (most of nvarchar columns should be localizable). For instance one of the tables might hold product information:  CREATE TABLE T_PRODUCT (   NAME        NVARCHAR(50),   DESCRIPTION NTEXT,   PRICE       NUMBER(18, 2) )   I can think of three approaches to support multilingual text in NAME and DESCRIPTION columns:   Separate column for each language  When we add a new language to the system, we must create additional columns to store the translated text, like this:  CREATE TABLE T_PRODUCT (   NAME_EN        NVARCHAR(50),   NAME_DE        NVARCHAR(50),   NAME_SP        NVARCHAR(50),   DESCRIPTION_EN NTEXT,   DESCRIPTION_DE NTEXT,   DESCRIPTION_SP NTEXT,   PRICE          NUMBER(18,2) )  Translation table with columns for each language  Instead of storing translated text, only a foreign key to the translations table is stored. The translations table contains a column for each language.  CREATE TABLE T_PRODUCT (   NAME_FK        int,   DESCRIPTION_FK int,   PRICE          NUMBER(18, 2) )  CREATE TABLE T_TRANSLATION (   TRANSLATION_ID,   TEXT_EN NTEXT,   TEXT_DE NTEXT,   TEXT_SP NTEXT )  Translation tables with rows for each language  Instead of storing translated text, only a foreign key to the translations table is stored. The translations table contains only a key, and a separate table contains a row for each translation to a language.  CREATE TABLE T_PRODUCT (   NAME_FK        int,   DESCRIPTION_FK int,   PRICE          NUMBER(18, 2) )  CREATE TABLE T_TRANSLATION (   TRANSLATION_ID )  CREATE TABLE T_TRANSLATION_ENTRY (   TRANSLATION_FK,   LANGUAGE_FK,   TRANSLATED_TEXT NTEXT )  CREATE TABLE T_TRANSLATION_LANGUAGE (   LANGUAGE_ID,   LANGUAGE_CODE CHAR(2) )    There are pros and cons to each solution, and I would like to know what are your experiences with these approaches, what do you recommend and how would you go about designing a multilanguage database schema. WCF - Faults / Exceptions versus Messages We're currently having a debate whether it's better to throw faults over a WCF channel, versus passing a message indicating the status or the response from a service.  Faults come with built-in support from WCF where by you can use the built-in error handlers and react accordingly. This, however, carries overhead as throwing exceptions in .NET can be quite costly.  Messages can contain the necessary information to determine what happened with your service call without the overhead of throwing an exception. It does however need several lines of repetitive code to analyze the message and determine actions following its contents.  We took a stab at creating a generic message object we could utilize in our services, and this is what we came up with:  public class ReturnItemDTO&lt;T&gt; {     [DataMember]     public bool Success { get; set; }      [DataMember]     public string ErrorMessage { get; set; }      [DataMember]     public T Item { get; set; } }   If all my service calls return this item, I can consistently check the "Success" property to determine if all went well. I then have an error message string in the event indicating something went wrong, and a generic item containing a Dto if needed.  The exception information will have to be logged away to a central logging service and not passed back from the service.  Thoughts? Comments? Ideas? Suggestions?  Some further clarification on my question  An issue I'm having with fault contracts is communicating business rules.  Like, if someone logs in, and their account is locked, how do I communicate that? Their login obviously fails, but it fails due to the reason "Account Locked".  So do I:  A) use a boolean, throw Fault with message account locked  B) return AuthenticatedDTO with relevant information Ubuntu 12.04 on Amilo LA1703 after istalling graphics doesn't start? Control 220V AC line using MCU Starting iPhone app development in Linux? I've heard that you need to get a Mac if you want to develop iPhone apps. Is this true?   Is it possible to develop iPhone apps using Linux? If yes, what do I need and where do I download the necessary tools? Is there an example of a ring $S$ with identity $1_S$ containing a non-trivial subring $R$ which itself has an identity $1_R$, but $1_R\neq 1_S$ (or equivalently $1_S\notin R$). I'd also like to know under what conditions the identities have to be equal. I know they must be equal if $S$ has no zero divisors, since for every $r\in R$ we have $(1_S-1_R)r=0$  In other words: Is the category of unital rings with identity-preserving morphisms a full subcategory of the category of rings (where morphism are only required to respect addition and multiplication)? Difference between generalized linear models & generalized linear mixed models Finding Radii of Convergence for $\sum a_n z^{2n}$ and $\sum a_n^2 z^n$ Setting: Let $\sum a_n z^n$ have radius of convergence $R$.  We have that  $$ R = \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n \right|^{1/n}} $$  via Hadamard's formula for the radius of convergence.  Question: What are the radii of convergence for (i) $\sum a_n z^{2n}$ and (ii) $\sum a_n^2 z^n$?  Attempt for $\sum a_n^2 z^n$:   For ease of notation, let $R_1$ and $R_2$ denote the radii of convergence for power series $\sum a_n z^{2n}$ and $\sum a_n^2 z^n$ respectively. From Hadamard's formula, we then have  $$ R_2 = \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n^2 \right|^{1/n}} = \left( \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n \right|^{1/n}} \right)^2 = R^2 $$   But what about $R_1$? $n$th derivative of $e^x \sin x$ I have to compute this integral $$\int_{0}^{\infty} \frac{dt}{1+t^4}$$ to solve a problem in a homework. I have tried in many ways, but I'm stuck. A search in the web reveals me that it can be do it by methods of complex analysis. But I have not taken this course yet. Thanks for any help. UBUNTU 14.04 cheese booth not detect webcam Cheese Booth shows an error:     There was an error playing video from the webcam     I test if webcam works fine or not by opening google hangout, its works fine.    Please tell, what is the issue in cheese booth.  Note: Sometimes the cheese booth works fine but most of time not Preview on the right please, optionally? I don't know how many of you have experienced the  principle when posting questions on SO, but I know I have on a few occasions. And then I just discard my question. So here's the question:  Should we go ahead and post the question?  Of course if it's something stupid like forgetting a semicolon, etc., I would say no... but there are several occasions, like if a data structure doesn't make sense, that at least someone would eventually benefit from the question/answers.  And if you should ask the question, how would you recommend choosing an answer for it? What should I do when my neural network doesn't learn? There are $25$ horses. You can take $5$ horses at a time and race them. Each horse always finishes the race in the same amount of time, and there are no ties. The only information you get from each race is the order that the $5$ horses finished in. What is the smallest number of races you need to find the $3$ fastest horses, in order?   I found this problem on  and cannot figure out the best answer. Please help. What algorithm is used by computers to calculate logarithms? Showing that if a subset of a complete metric space is closed, it is also complete Let $(X, d(x,y))$ be a complete metric space. Prove that if $A\subseteq X$ is a closed set, then $A$ is also complete.  My attempt: I tried to prove that every Cauchy sequence $(b_n)$ of points of $A$ converges to a point $b\in A$. However could not figure out the exit way. Maybe I am on the wrong track. Could you please help me?  edit: More from my attempt:  Suppose $A$ is a closed set and let $(x_n)$ be a sequence of points $A$ such that $\lim_n x_n\to b$.  Suppose now that $A$ has the property that $b\in A$, whenever $x_n$ converges to $b$. We know that every element of $x_n$ which is convergent in $X$ also converges to a point in $A$. Since $x_n$ is a Cauchy sequence in $A$, it must converge to a point $y\in A$. But the limit of a convergent sequence is unique.   Take $x\in A$ and select an appropriate $n$, which enables $x_n$ to converge to a point $x$ in $X$. Since the limit is unique, it must follow that $x=y$. Thus $x\in A$ and $A$ is closed.   If $A$ is a closed subset of $X$, then any Cauchy sequence of a point in $A$ is convergent in $X$ and hence converges to a point in $A$. Thus $A$ is complete. The ability to undo reviews I recently had somebody tell me that a mutual friend of ours who is  prefers that people refer to him/her using the gender-indefinite pronoun they.   In some cases, this almost seems okay:   Kris left their umbrella at our house.  On the other hand, if Kris is sitting right next to you, it feels very odd to say       They (meaning just Kris) would like more cake.   Or even odder,      Kris would like some more cake, can you please pass it to they/them?"   Are these usages grammatically correct? Are they in the process of becoming grammatically correct? Are there more correct alternatives?