Formal definition of fraction
Is $\frac{1}{\sqrt{2}}$ a fraction or not?
While looking at online code samples, I have sometimes come across an assignment of a String constant to a String object via the use of the new operator.  For example:  String s; ... s = new String("Hello World");   This, of course, compared to  s = "Hello World";   I'm not familiar with this syntax and have no idea what the purpose or effect would be.  Since String constants typically get stored in the constant pool and then in whatever representation the JVM has for dealing with String constants, would anything even be allocated on the heap? Continuity and existence of directional derivatives at all directions of a $f:\mathbb{R}^{2} \to \mathbb{R}$ given by $f(x,y)=\frac{2x^2y}{x^2+y^2}$ Let $f(x,y)=     \begin{cases}       \frac{2x^2y}{x^2+y^2} &amp; if &amp; (x,y) \neq (0,0) \\       0       &amp; if &amp; (x,y)=(0,0) .  \end{cases}$  (i) Prove the directional derivatives of $f$ exist in any direction at the point $(0,0)$.  (ii)¿Is $f$ continuous on (0,0)?  For (i) I took $u=(u_{1},u_{2}) \in \mathbb{R^{2}}$ such $\|u\|=1$ and $0=(0,0)$. So $$\lim_{t \to 0} \frac{f(0+ t u)-f(0)}{t}=\lim_{t \to 0}\frac{2(tu_{1})^2 (tu_{2})}{(tu_{1})^4+(tu_{2})^2}=\lim_{t \to 0}\frac{(t^2 u_{1}^2)(t^2 u_{1}^{2}) tu_{2}}{t^{4}u_{1}^{4}+t^{2} u_{2}^{2}}.$$ But I cannot find the limit when $t$ aproaches to $0$. Which by finding them I would prove the directional derivative exist at every direction right?  For (ii) I got the intuition $f$ is not continuous at $(0,0)$ so I took $\lbrace (\frac{1}{k},\frac{1}{k}) \rbrace_{k \in \mathbb{N}}$ and $\lbrace (\frac{1}{k},0) \rbrace_{k \in \mathbb{N}}$ which are two different sequences in $\mathbb{R}^{2}$ converging to $(0,0)$. However, $\lbrace f(\frac{1}{k},\frac{1}{k}) \rbrace_{k \in \mathbb{N}}=\lbrace \frac{1}{k^{2}} \rbrace_{n \in \mathbb{N}} \to 0$ and $\lbrace f(\frac{1}{k},0) \rbrace_{k \in \mathbb{N}} \to 0$. So maybe my intuition was not right? Can anyone help me end the proof of continuity or not continuity , please? What is the equivalent of Persian idiom "When the reed blooms"? In Persian, we say "When the  blooms" when we want to  express that an event:   Never happens. (This is only the opinion of the speaker so it's not a fact) It's very unlikely to happen. It's going to take so long to actually happen. (It's so long that you'd be better off giving up expecting it)  3.1. We also use it when someone is always postponing something they have to do so you never actually see it happen.   For example:      Kyle: When are you going to pay my money back?      Cartman: When the  blooms.   Now I don't know if I got the plant's name right because there are some other versions like:        In any event, they have mentioned this plant in the idiom because apparently some versions of it actually never bloom. Others believe it's because farmers harvest them before they get the chance to bloom (e.g to make music instruments), so it never happens. I'd love to know the equivalent idioms or expressions for this.  PS: I only know "When pigs fly" but I'm not a fan of it because the word pig sounds a bit rude to me and partly because it implies a strong impossibility which doesn't work for the #3 use. I’m writing a series of blog posts and I’m finding a little bit of a conundrum. I want the audience that can be manufactured on Medium but I also want the SEO that comes from long form text on my website.  Is there any type of SEO conflict that would arise from putting the exact same posts on both? How to prove the Fibonacci sum $\sum \limits_{n=0}^{\infty}\frac{F_n}{p^n} = \frac{p}{p^2-p-1}$ Alignment of numbers on the individual bars how to get the selected date in jquery Prove that $\left | \frac{a-b}{a+b}+\frac{b-c}{b+c}+\frac{c-a}{c+a} \right | < \frac{1}{8}.$ Let $a,b,$ and $c$ be the lengths of the sides of a triangle. Prove that $$\left | \dfrac{a-b}{a+b}+\dfrac{b-c}{b+c}+\dfrac{c-a}{c+a} \right | &lt; \dfrac{1}{8}.$$   The best idea I had was to expand the fractions to get something nicer. So we get $\dfrac{a-b}{a+b}+\dfrac{b-c}{b+c}+\dfrac{c-a}{c+a} = \dfrac{a^2 b-a^2 c-a b^2+a c^2+b^2 c-b c^2}{(a+b) (a+c) (b+c)}.$ Then I would try to relate this to side lengths of a triangle to prove the desired inequality but that is the step where I am unsure. Is there a more efficient way to clamp real numbers than using if statements or ternary operators? I want to do this both for doubles and for a 32-bit fixpoint implementation (16.16). I'm not asking for code that can handle both cases; they will be handled in separate functions.  Obviously, I can do something like:  double clampedA; double a = calculate(); clampedA = a &gt; MY_MAX ? MY_MAX : a; clampedA = a &lt; MY_MIN ? MY_MIN : a;   or  double a = calculate(); double clampedA = a; if(clampedA &gt; MY_MAX)     clampedA = MY_MAX; else if(clampedA &lt; MY_MIN)     clampedA = MY_MIN;   The fixpoint version would use functions/macros for comparisons.  This is done in a performance-critical part of the code, so I'm looking for an as efficient way to do it as possible (which I suspect would involve bit-manipulation)   EDIT: It has to be standard/portable C, platform-specific functionality is not of any interest here. Also, MY_MIN and MY_MAX are the same type as the value I want clamped (doubles in the examples above). What is time, does it flow, and if so what defines its direction? If I click to open achievements from a question when I click the X to close the achievements view I end up on the answers tab of the question, I assume because the X is over the tab and it's tapping through.    It appears if I hit the X it's ok, but if I tap anywhere else in the top row (where the X is) then I can switch to the question or answer tab. I'm a student who uses the laptop all the time and I have a problem with Wifi on the laptop. I don't really know what it is. I can describe the problem tho. For example, i closed my laptop last night, so it goes in suspend mode. I turn it on this morning and my wifi doesn't want to connect until I restart the laptop. It says that it's connecting and in Wifi settings the circle animation is spinning, turning Wifi off and on doesn't do anything. Here's what happens when I check to see my network info: sudo lshw -C network   *-network                         description: Wireless interface        product: Wireless 3160        vendor: Intel Corporation        physical id: 0        bus info: pci@0000:02:00.0        logical name: wlp2s0        version: 83        serial: 00:1e:64:e6:1c:93        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress bus_master cap_list ethernet physical wireless        configuration: broadcast=yes driver=iwlwifi driverversion=5.4.0-52-generic firmware=17.3216344376.0 ip=192.168.0.14 latency=0 link=yes multicast=yes wireless=IEEE 802.11        resources: irq:129 memory:df100000-df101fff   *-network        description: Ethernet interface        product: RTL810xE PCI Express Fast Ethernet controller        vendor: Realtek Semiconductor Co., Ltd.        physical id: 0        bus info: pci@0000:03:00.0        logical name: enp3s0        version: 07        serial: f8:ca:b8:52:72:49        capacity: 100Mbit/s        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress msix vpd bus_master cap_list ethernet physical tp mii 10bt 10bt-fd 100bt 100bt-fd autonegotiation        configuration: autonegotiation=on broadcast=yes driver=r8169 firmware=rtl8106e-1_0.0.1 06/29/12 latency=0 link=no multicast=yes port=MII        resources: irq:17 ioport:d000(size=256) memory:df000000-df000fff memory:d0000000-d0003fff  Also I have to mention that my Linux is doublebooted alongside Windows 10, but I don't think that makes any difference. Help would be much appreciated, I wouldn't want to miss my classes because of this hahah Thanks! Does a polymorphed creature remember who his friends are? Signed to unsigned conversion in C - is it always safe? Suppose I have the following C code.  unsigned int u = 1234; int i = -5678;  unsigned int result = u + i;   What implicit conversions are going on here, and is this code safe for all values of u and i? (Safe, in the sense that even though result in this example will overflow to some huge positive number, I could cast it back to an int and get the real result.) How to make a field NULL using MySQL GUI Tools? I'm using the MySQL Query Browser (part of the ) and need to change a field to NULL, but I can't figure out how to do it - if I delete the value it tries to update it to ''. Typing "NULL" makes it try to update to 'NULL' (a string).  I know I could just write a query to do it, but that defeats the entire purpose of the tool, no? I installed the WSL Bash on Windows 10 and set my user to e.g. 'abc' but now I want to change it to something else.  How can I change the user name of the user I'm logged in as?  When I open bash it automatically logs me in with my current user. Windows 10 update broke my MBR / GRUB pgfmathanglebetweenpoints always returns 90 Importing 3D DXF file into QGIS with z information as attribute value? Check all dependencies of a Record Type in the Organisation before deactivating it