If $f\in C^2(\mathbb R)$ then $M_1^2 \le 2M_0 M_2$, where $M_k = \text {sup}_x |(d/dx)^k f(x)|$ for $k=0,1,2.$
Inequality involving the sup of a function and its first and second derivatives
Prove there's a simple path of length $k$ in a simple graph $G$ where all the vertices have degree of at least $k$.   Relevant definitions:  $G$ is a simple graph that consists of a vertex set $V(G) = \{v_1, v_2, ..., v_n\}$ and an edge set $E(G) = \{e_1, e_2, ..., e_m\}$ where each edge is an ordered pair of vertices. The edge $\{u,v\}$ is denoted $uv$.  A walk of length $k$ is a sequence $v_0,e_1,v_1,e_2,...,e_k,v_k$ of vertices and edges such that $e_i=v_{i-1}v_i$ for all $i$. A path is a walk with no repeated vertex.   My attempt:  Induction, for $k=1$ it's obvious.  Suppose for $k-1$ and we'll prove for $k$.  Let $G$ be a simple graph such that $\forall v \in V : d(v)\ge k$.  We can assume that there are at least $k+1$ vertices since there are $k$ neighbours to every vertex.  Let $v_0$ be some vertex, it has $k$ neighbours, we'll move to one of its neighbours say $v_1$ and remove $v_0$. So now there are $k$ vertices with a degree of at least $k-1$ and from the induction hypothesis we'll have a path of length $k$.  Is using only $k$ vertices from $n$ right? Does it keep the generality when using only all the neighbours of $v_0$? Ball swung in a horizontal circle, can the string ever by exactly horizontal to the ground? You swing a ball on the end of a lightweight string in a horizontal circle at constant speed.  Can the string ever by truly horizontal?  If not, would it slope above the horizontal or below the horizontal?  Why? Any software for windows Writing custom (intelligent) microsoft windows firewall extension (C#/C++/C) Writing custom (intelligent) firewall for windows Installing nginx etc. on the same computer and use it as proxy with rules.   "You could not solve all Ddos problems with software" is not answer for this question.I want to reduce Ddos vulnarebility of my server.(It will stop low-level Ddos attacks)  Hosted solutions or cloud services are not related with this question.  Closing some of the ports and protocols via Microsot Windows Firewall is not related with this question.  Any idea?  EDIT After Questions  Low level: sample scenerio: 500 zombie clients and each of them sending 100 Custom Get/Post requests per second)   Web Application is on IIS. I want to stop attacks before they reach to IIS.  EDIT After Helpful Comment  How can the DDOS attack achieve its aim?   Exhausting bandwidth ( Not related with my question) Exhausting network hardware of server (NIC) ( Not related with my question) Exhausting server resources (RAM,CPU)  (Yep, I mean this)   If you stop attacks late, then you lost more resources! (Is this false?)   You can stop attacks on Web Application Level (in Asp.net MVc Code or in Handler) You can stop attacks on IIS (Dynamic Ip Restrictions)  You can stop attacks on lower levels (Firewall ..)   EXTRA INFO ABOUT THE LAST ATTACK (7 Hours ago)  When I connected server during the attack IIS Cpu Utilization was %92-%99. And when I try to connect homepage I got this error:  A connection was successfully established with the server, but then an error occurred during the pre-login handshake. (provider: SSL Provider, error: 0 - The wait operation timed out.) I'm still making my way along in Niven's Intro to Number Theory, and the title problem is giving me a little trouble near the end, and I was hoping someone could help get me through it.  Now $x^8\equiv 16\pmod{2}$ is solvable with $x\equiv 0\pmod{2}$, so I assume $p$ is an odd prime. From a theorem earlier in the text,      If $p$ is a prime and $(a,p)=1$, then the congruence $x^n\equiv a\pmod{p}$ has $(n,p-1)$ solutions or no solution according as $a^{(p-1)/(n,p-1)}\equiv 1\pmod{p}$ or not.   So since $(16,p)=1$, the problem reduces to showing that $16^{(p-1)/(8,p-1)}\equiv 1\pmod{p}$ holds for all $p$. I note that $(8,p-1)$ can only take values $2,4,8$. For $2$, the above equivalence is then $4^{p-1}\equiv 1\pmod{p}$, which is true by Fermat's little Theorem. For $4$, it is then $2^{p-1}\equiv 1\pmod{p}$, which again holds by FlT. However, the case where $(8,p-1)=8$ is throwing me off. At best I see that $16^{(p-1)/8}\equiv 2^{(p-1)/2}\pmod{p}$, but I'm not sure how to show this is congruent to $1$ modulo $p$. Maybe there's a more elegant way to do it without looking at cases. Thanks for any insight. understanding TTNavigator I have noticed that the smackpacks say on the package that the package is meant for a 5 gal. batch and that you should pour the pack directly into the wort.   On the otherhand - nearly everyone tells me to always make a starter.   Is this just because making a starter never hurts or does the package contain wrong instructions?   I would have guessed that the yeast producers would tell you on the package if you should need some extra steps to get a good product. How to increase Scrollback lines in Ubuntu(14.04.2) server edition? I can see only limited lines on terminal by using shift+ pageup. in Ubuntu desktop version, there is option to make scroll-back lines unlimited. how to do it in Ubuntu server version. please suggest. Two logos in beamer after certain frame Is it possible to transfer Minecraft PE worlds to Minecraft PC? Can't run "dpkg --add-architecture i386" on 12.04 64-bit to run eclipse ADT Escape velocity from a rotating black hole some \newcommand instructions not working I am a total newbie to Ubuntu, and I'm not sure what to do here. I have a Toshiba Chromebook. I had a friend install the latest version of Ubuntu on it for me. For a while, the audio in Ubuntu was working fine, but now I can't hear any audio whatsoever.  How can I fix this? The problem of instant velocity The concept of velocity is by definition the movement divided by the interval of time between initial position and final position.  If $f(t)$ is the position of a particle at time $t$; the velocity in the interval $[t_0;t_1]$ is $\dfrac{f(t_1)-f(t_0)}{t_1-t_0}$  The problem is that in a single instant there is no movement and the time is not changed; so no velocity.  I can consider $\lim_{t_1 \to t_0} \dfrac{f(t_1)-f(t_0)}{t_1-t_0}$, but mathematically it is only the limit of average velocity function and doesn't represent velocity at instant $t_0$  What are your views about this problem ? How to count the number of occurrences of an element in a List I have an ArrayList, a Collection class of Java, as follows:  ArrayList&lt;String&gt; animals = new ArrayList&lt;String&gt;(); animals.add("bat"); animals.add("owl"); animals.add("bat"); animals.add("bat");   As you can see, the animals ArrayList consists of 3 bat elements and one owl element. I was wondering if there is any API in the Collection framework that returns the number of bat occurrences or if there is another way to determine the number of occurrences.  I found that Google's Collection Multiset does have an API that returns the total number of occurrences of an element. But that is compatible only with JDK 1.5. Our product is currently in JDK 1.6, so I cannot use it. Why is "can I get" replacing "could I have"? I noticed the other day when serving the public that when asking for something, people were saying "Can I get an xyz, please". The previous time I had such a job it was "Could I have an xyz", or "May I have..." if they were being very polite.  Does anyone know when this trend started and what caused it? Can I check iPhone, iPad, Watch battery on Mac? My goal is to have battery status of other Apple device on my Mac (preferably on menubar). I'd like to have a glimpse of the battery status of my iPhone, iPad, Watch, and Airpods.  I'm a user of Bitbar and iStat menu, but I haven't found any widgets can wirelessly check other Apple device's battery status.  What I know is that I can click the Wi-Fi icon on menu bar and it will show my iPhone battery status. To show the percentage, I have to option + click the Wi-Fi icon and hover over my mouse on the iPhone hot spot section. It's pretty clunky, but I guess at least the API is there.  I am also open to third party apps which can let me see battery status across devices. Ordered N-Tuples as Cartesian Products of Indexed Families In Introduction to Topology, Mendelson defines the Cartesian Product of an Indexed Family as follows:     DEFINITION  Let $\{X_\alpha\}_{\alpha \in I}$ be an indexed family of sets. The product  of the sets $\{X_\alpha\}_{\alpha \in I}$,    written  $\prod_{x\in I}X_\alpha$ consists of all functions $x$ with domain the indexing set $I$ having the property that for each $\alpha \in I$, $x(\alpha)\in X_\alpha$.    Halmos, in Naive Set Theory, makes the following observation about this sort of product:     Ordered triples, ordered quadruples, etc., may be defined as families whose index sets are unordered triples, quadruples, etc.   However, it is not immediately apparent to me why this is true. It makes sense if we understand the prior assertions in Halmos that $\prod_{i \in I} X_i = X_i$ and that $\prod_{i \in I} X_i = X_a \times X_b$ when $I = \{a, b\}$, but these already pose difficulty for me.  Let $I = \{0, 1\}$, let $X_0 = \{A, B\}$, and let $X_1 = \{C, D\}$. Then by Mendelson's (and Halmos's) definitions, it would seem to me that . . .  $$\prod_{i \in I} X_ i = \{\{(0, A), (1, C)\}, \{(0, A), (1, D)\}, \{(0, B), (1, C)\}, \{(0, B), (1, D)\}\}$$  This would appear to me to be the set of all functions matching the definition given, but it is not equal to $X_0 \times X_1$. Moreover, extending this to a case where the index set $I$ is a triple, we would not end up with ordered triples, but with each subset of the product containing three ordered pairs.  When Halmos posits the equalities he does, the idea seems easy to grasp, but the peculiar wording of Mendelson's definition suggests something different to me. Why doesn't 2.__add__(3) work in Python? I have a MacBook Pro with Mini DisplayPort and I want to use target display mode to use a Thunderbolt iMac as an external display. If this works, would I need a Mini DisplayPort cable or a Thunderbolt cable?