We state that electrons are subatomic particles with no known subcomponents. We discover that these electrons behave as waves. We also discover that sometimes, these electrons behave as point-particles. We conclude that they're the both at the same time. This, to me, sounds nonsensical. If this happened in math, and one discovered that 1 = 2, one would not go around saying that we have one-two duality, would we? No, we would question the axioms that we used to perform our calculations. If one-two duality is an outcome of those axioms, we should not accept one-two duality, rather we should reject the axioms! Yet, in physics, this does not seem to be the case. Why? Why is wave-particle duality accepted, rather than the axioms rejected? And what are the axioms in this case? Well, it's the first line in this question. We assumed electrons have no known subcomponents. Well, clearly our experimens show we're wrong. They do have subcomponents, and when the wave-like behavior of electrons occurs, it occurs precisely because of interactions between those subcomponents that we are not familiar with and therefore cannot understand. So what am I missing? It seems like physicists have accepted this bizarre notion of wave-particle duality that even Einstein thought was an embarrassment of physics,  rather than consider that some premises may be wrong and it is those wrong premises that lead to the wave-particle duality.
I often hear about the wave-particle duality, and how particles exhibit properties of both particles and waves. However, I wonder, is this actually a duality? At the most fundamental level, we 'know' that everything is made up out of particles, whether those are photons, electrons, or maybe even strings. That light for example, also shows wave-like properties, why does that even matter? Don't we know that everything is made up of particles? In other words, wasn't Young wrong and Newton right, instead of them both being right?
I am reading Boyd &amp; Vandenberghe's Convex Optimization. The authors use curved greater than or equal to (\succcurlyeq)  $$f(x^*) \succcurlyeq \alpha$$  and curved less than or equal to (\preccurlyeq)  $$f(x^*) \preccurlyeq \alpha$$  Can someone explain what they mean? Is the collapse of the wave function inherently time asymmetric? This is Problem 6 of the 2007 Indian National Math Olympiad (INMO).  If $x, y, z$ are positive real numbers, prove that $(x+y+z)^2(yz+zx+xy)^2 \leq 3(y^2+yz+z^2)(z^2+zx+x^2)(x^2+xy+y^2).$  My best idea was to expand this and simplify. Although that doesn't look very feasible. Another idea is to see that $x^2+y^2+xy \geq x^2+y^2$. Then we just have to show that $(x+y+z)^2(yz+zx+xy)^2 \leq 3(x^2+y^2)(x^2+z^2)(y^2+z^2)$ if that is even true. Google is counting the domain home page ("/") as a duplicate of "/index.html" For only my index pages, I am receiving this warning in Google Search Console:     Duplicate, submitted URL not selected as canonical   I know that this occurs because Google is attempting to track both the default landing "page" (i.e. www.domain.example/) and the page to which to which DirectoryIndex points (i.e. www.domain.example/index.html). They are, in fact, the same page, and so the warning is the result of a false positive.  Two questions for our community:   Why does Google not conflate these and count them as a single page? Alternately, is there something wrong with my set up if this has happened? I can easily force both pages to point to a single canonical URL. But which is preferable, "/" or "index.html"? Intersecting transparent object causing artifacts in cycles When two objects -one of them transparent- intersects it creates some artifacts visible even in the alpha channel. Even with 1024 transparent bounces the artifacts remain.    In the example file there are two cubes in the same position, one transparent and one emission.    What is causing this? Any way to get rid of this artifacts? What is the use of anonymous classes in Java? Can we say that usage of anonymous class is one of the advantages of Java? Inserting the value from Datetime field form I have a form where the user can input datetime and then I want to store it in the database. Here's my schema :     'start' => array(           'type' => 'varchar', 'mysql_type' => 'datetime',           'not null' => FALSE,         ),   The form :     $form['add']['start'] = array(         '#type' => 'datetime',   '#default_value' => $date,       );   When I hit submit button, it shows me :     db_insert failed. Message = SQLSTATE[22007]: Invalid datetime format: 1292 Incorrect datetime value:...   What I got out of it was that the DateTime in Drupal is incompatible with MySQL's. Any suggestions or how should I process the value from the form so I can insert it into the db?  Alternatively, any way to convert DrupalDatetime into string?  (For what it's worth I also tried 'timestamp' as the MySQL type but it throws me the same error). A method for evaluating sums/discrete functions by assuming they can be made continuous and differentiable? Sum of indepedent random variables and a constant I'm reading . Then I'm not able to figure out the difference between hard voting and soft voting in context to ensemble based methods.  I quote descriptions of them from the book. The first two images from the top are description for hard voting, and last one is for soft voting.        In my view hard voting is majority decision, but I don't understand soft voting and the reason why soft voting is better than hard voting. Would anyone teach me these? Let's say I am building a logistic regression model where the dependent variable is binary and can take the values $0$ or $1$. Let the independent variables be $x_1, x_2, ..., x_m$ - there are $m$ independent variables. Let's say for the $k$th independent variable, the bivariate analysis shows a U-shaped trend - i.e., if I group $x_k$ into $20$ bins each containing roughly equal number of observations and calculate the 'bad rate' for each bin - # observations where y = 0 / total observations in each bin - then I get a U shaped curve.  My questions are:   Can I directly use $x_k$ as input while estimating the beta parameters? Are any statistical assumptions violated which might cause significant error in estimating the parameters? Is it necessary to 'linearize' this variable through a transformation (log, square, product with itself, etc.)? series of young adult science fiction books from early 1980s about last of a race of martial experts that sees the destruction of his home planet I remember that the main character is trying to find the group that destroyed his home planet while he was on a mission. His planet was very difficult to live on and the people became expert in martial arts and were the best soldiers in the galaxy. They were mercenaries and often helped against tyrants. There was a very memorable section in at least one book about an initiation task that the main character had to do in that he had to survive in their wilderness for days. He was at some point surrounded in a old building by thousands of poisonous worm/snake creatures that he managed to survive until rescued by the leaders. Proving that the Hopf map $S^3 \to S^2$ is a submersion I need to show that the Hopf map $H : S^3 \to S^2$ is a submersion. There are many ways to define it, but for now I've gone with the restriction of the function $F : \mathbb{R}^4 \to \mathbb{R}^3$ mapping $$(x,y,u,v)\mapsto (2(xu+yv), 2(xv-yu), x^2+y^2-u^2-v^2).$$ It wasn't too bad to show that $F$ is a submersion, but is there a way to go from here to show that $H$ is a submersion? I can write down $H$ as the composition $$S^3 \overset{\iota}{\hookrightarrow} \mathbb{R}^4 \overset{F}{\to} \mathbb{R}^3 \overset{\pi}{\to} S^2,$$ and define $\pi(p) = p/\lVert p\rVert$, but I don't really know where to go after computing the Jacobian of $\pi$, and I don't know if this will even work for my purposes (I don't see how to argue that the composition of the corresponding differentials here isn't rank $1$ — I know $d\iota$ has trivial kernel but things seem to break down once I get to $d\pi$). I'm really just trying to avoid having to compute the Jacobian of $H$ with respect to the stereographic projections and show that that beast of a thing is full rank. Am I on the right track, or is there a better way? How can I left-align a caption? I am using this for captions:  \usepackage[font=small,format=plain,labelfont=bf,up,textfont=normal,up]{caption}   But they all come out centred - I want them aligned with the left margin - how can I do this? In which US accent is “again” pronounced /əˈɡeɪn/ with the FACE vowel, not /əˈɡɛn/ with the DRESS vowel? Most dictionaries provide the US pronunciation of “again” as /əˈɡɛn/ (uh-gen) with the DRESS vowel. This is the most common pronunciation in the USA. However, I think I might have also come across /əˈɡeɪn/ (uh-gain) with the FACE vowel. This sounds like the word “gain” with an “a” at the beginning. Is there a US accent where “again” is pronounced this way? Why is inductance (L) proportional with turns-square (N²)? We start from the   $$ \mathbf{\nabla} \times \mathbf{B} = \mu \mathbf{J} + \overbrace{\mu \epsilon \dfrac{\partial \mathbf{E}}{\partial t}}^0. $$  We take surface integration of both sides, for the surface (\$s\$) inside the mean path (\$c\$) of the core.  $$ \int_s \left( \mathbf{\nabla} \times \mathbf{B} \right) \cdot d\mathbf{s} = \mu \int_s \mathbf{J} \cdot d\mathbf{s} $$  We use the  to rewrite the left hand side; where \$c\$ is in the same direction with the magnetic flux \$\Phi\$.  $$ \oint_c \mathbf{B} \cdot d \mathbf{\ell} = \mu N I $$  (The integral on the left hand side results \$NI\$, because there are \$N\$ different wires on the winding.)  The magnetic field density inside these kind of cores are considered to be uniform. So, we can write  $$ B \ell_c \overset\sim= \mu NI \implies B = \dfrac{\mu NI}{\ell_c}; $$  where \$\ell_c\$ is the mean path lenght of the core.  We can find magnetic flux from the magnetic flux density we have found by using the cross sectional area of the core \$A_c\$.  $$ \Phi = BA_c = \dfrac{\mu NIA_c}{\ell_c} $$  By definition, inductance is the amount of magnetic flux generated per applied current, that is  $$ L \overset\triangle= \dfrac{\Phi}{I}. $$  So, we find inductance of the system as  $$ \boxed{ L = \dfrac{\Phi}{I} = \dfrac{\dfrac{\mu NIA_c}{\ell_c}}{I} = \dfrac{\mu NA_c}{\ell_c} }. $$  But, all other sources () give inductance of an inductor like this as  $$ L = \dfrac{\mu N^2A_c}{\ell_c}. $$  What is the mistake I did in my derivation? Please explain in detail. Which Harry Potter works are considered canon? There are many different Harry Potter works including:   books short stories movies card games video games toys amusement parks web content   Which ones are considered to be canon? Textbook recommendations for self-studying high school math? I'm a high school student who is trying to figure out a complete course of self-study for each year of high school. How can I self-learn grades of math without devoting too much time? This is a complex issue for me, as other students at my competitive high school have tutors and the like. Please recommend textbooks that have detailed explanations and progressive practice problems, for self-study for each area such as:  Algebra  Geometry  Trigonometry and Analytic Algebra  Pre-calculus  BC Calculus  Other people have skipped grades of math due to help from tutors and parents. Can I cover all of geometry and trigonometry in 8 months without going insane and be able to skip a grade? /usr/sbin/grub never finishes I did apt-get upgrade it never finishes at this point:  Searching for GRUB installation directory ... found: /boot/grub Searching for default file ... found: /boot/grub/default Testing for an existing GRUB menu.lst file ... found: /boot/grub/menu.lst Searching for splash image ... none found, skipping ... Found kernel: /boot/vmlinuz-3.2.0-24-virtual Updating /boot/grub/menu.lst ... done  You already have a NoLOADER configuration in /etc/grub.conf Running boot loader as requested Running /usr/sbin/grub  ...    The system does not freeze (Ctrl-Z works, I get my console back) just that that /usr/sbin/grub seems to never complete.  What should I do? How to justify $\phi(n) \ge \sqrt{n}$ If $\phi(n)$ is the Euler-totient function, how can I show that $\phi(n) \ge \sqrt{n}$?