Do there exist any 3-regular graphs with an odd number of vertices? I'm starting a delve into graph theory and can prove the existence of a 3-regular graph for any even number of vertices 4 or greater, but can't find any odd ones.
I'm having a bit of a trouble with the below question     Given $G$ is an undirected graph, the degree of a vertex $v$, denoted by $\mathrm{deg}(v)$, in graph $G$ is the number of neighbors of $v$.   Prove that the number of vertices of odd degree in any graph $G$ is even.
I am doing some forecasting in R using Rob Hyndman's forecast package. The paper belonging to the package can be found .  In the paper, after explaining the automatic forecasting algorithms, the authors implement the algorithms on the same data set. However, after estimating both an exponential smoothing and ARIMA model they make a statement I do no understand (on page 17):     Note that the information criteria are not comparable.    I thought that an advantage of using AIC for model selection is that we can compare AIC values from different models, as long as they are estimated using the same data set. Is this incorrect?  This matter is of particular interest to me, as I was planning on combining forecasts from different model classes (e.g. exponential smoothing and ARIMA) using so called Akaike weights (see Burnham and Anderson, 2002, for discussion on Akaike weights)  References   Burnham, K. P., &amp; Anderson, D. R. (2002). Model selection and multi-model inference: a practical information-theoretic approach. Springer Verlag. Recently I needed to do weighted random selection of elements from a list, both with and without replacement. While there are well known and good algorithms for unweighted selection, and some for weighted selection without replacement (such as modifications of the resevoir algorithm), I couldn't find any good algorithms for weighted selection with replacement. I also wanted to avoid the resevoir method, as I was selecting a significant fraction of the list, which is small enough to hold in memory.  Does anyone have any suggestions on the best approach in this situation? I have my own solutions, but I'm hoping to find something more efficient, simpler, or both. How can I make realistic looking potions as handouts Each  has a compiled cost that contributes towards the  compiled character formula limit.  What are the current compiled costs associated with each formula function? Samsung netbook NP300E5A dimming on battery My Samsung netbook NP300E5A screen is continually dimming.  The screen is forty percent dim on battery, but if I plug in the charger, the screen will be normal. I have tried all I know and I could not fix it. What should I do? what am trying to explain  is that the system is dimming like flooricent light  the screen resolution systems are all ok. When I pluging my charge the dimming we stop but when is on battery it look like flashing someone  light when driving Clone Method not working I am trying to Clone a lead when certain condition is arrived using before trigger but i am not able to see cloned record.   for( Lead ld : Trigger.new ) {                          if(ld.Integration_Source__c == 'Hubspot' &amp;&amp; Trigger.isUpdate){         Lead LeadBeforeUpd = Trigger.oldMap.get( ld.Id );         if(ld.Program__c != LeadBeforeUpd.Program__c ){            Lead ClonedLead = new Lead();            ClonedLead = LeadBeforeUpd.clone(false, false, false, false);         }     }  } How to pass an ArrayList to a varargs method parameter? If $\frac{\cos x}{\cos y}+\frac{\sin x}{\sin y}=-1$, then what's the value of $4\left(\frac{\cos^3y}{\cos x}+\frac{\sin^3y}{\sin x}\right)$? Prevent Windows 10 installer from using the preinstalled serial key without disabling UEFI My laptop (a Lenovo Flex 2 15) came preinstalled with Windows 8 Standard (with the serial key preinstalled into the BIOS), and I've since installed an SSD (a Samsung Evo 840 120GB) &amp; upgraded to Windows 10.  When the laptop still had its hard drive, I upgraded it to Professional so I could use the resources provided by the domain on my homeserver; because of this, I can only use the Professional editions of Windows (Home can't join domains).  I bought Windows 10 Professional &amp; completed a fresh-install (complete reinstallation) of Windows, only that it installed Windows 10 Home instead of Professional. I'm guessing this is due to the Windows 8 Standard Edition serial key preinstalled in the BIOS.  I then downloaded an All-In-One ISO image of Windows 10 from MSDN (of which I have limited access to), transferred the installer onto a USB stick, &amp; reinstalled (a clean install again). However it's still installing Windows 10 Home.  If I try to install using the AIO image from within Windows, the installer does not present me with a list of options for the edition I want to install; instead, it assumes that I want to install Home Edition:    I am able to get Windows 10 Professional to install if I disable UEFI &amp; use Legacy instead, but I feel that this is stepping around the issue rather than resolving it.  Is it possible to have the Windows 10 installer ignore any serial keys preinstalled in the BIOS/UEFI? If so, how? If not, is using Legacy mode really the only way around this? Change audio output depending on which one is on I have a PC that is hooked up to my HDTV (via a long hdmi cable) and to a monitor, which is in another room. I have speakers directly plugged to the PC audio out. The PC is next to the monitor and speakers.   I am not sure but I think Windows can play sound on only 1 audio device at a time. And I can only set 1 device as the default output. I can get sound on the TV or speaker depending on which device I set to default. But I would have to do this every time I switch between using my TV or my monitor!   Is there some way to configure such that sound plays through the TV if the TV is on, else it plays in the speakers? If this is not possible, then the next best alternative would be to get the sound to play on both the devices at the same time. Annoying space before the first item in an enumeration which starts a theorem Consider the following example:  \documentclass{article} \newtheorem{Theorem}{Theorem} \begin{document} \begin{Theorem} On the other hand, this one has only one claim. \end{Theorem} \begin{Theorem} \begin{enumerate} \item First claim \item Second claim \end{enumerate} \end{Theorem} \end{document}   Typesetting it results in:    Notice there is extra horizontal space before the first item in the list of the second theorem.     Is there a sensible way to get rid of that?   Ideally, the solution would work with amsthm, thmtools, enumitem.  N.B.: this is related to  but not quite the same. I'm reading through a proof in my lecture notes:     Every convergent sequence of a metric space has a unique limit.   Suppose $\lim_{n \to \infty} x_n = l$ and $\lim_{n \to \infty} x_n = m.$ Then for every $\epsilon &gt; 0,$ there exists $N$ such that for every $n \geqslant N,$  $$d(l,m) \leqslant d(l,x_n) + d(x_n,m) &lt; \epsilon + \epsilon = 2 \epsilon. $$  So $(\star) \hspace{1mm} 0 \leqslant \frac{d(l,m)}{2} &lt; \epsilon \Longrightarrow \hspace{1mm} (\star^\prime) \hspace{1mm} \frac{d(l,m)}{2} =0 \Longrightarrow l = m.$  I can follow the proof up until the jump from $(\star)$ to $(\star^\prime)$. Why does $\frac{d(l,m)}{2}=0$? What does "cpu MHz" field mean in the /proc/cpuinfo of a hyper-threading cpu? The CPU is a i7-3770@3.40GHz. It has 4 cores and each core has 2 threads. Here is the dmidecode output:  # dmidecode -t 4 # dmidecode 2.9 SMBIOS 2.7 present.  Handle 0x0042, DMI type 4, 42 bytes Processor Information     Socket Designation: SOCKET 0     Type: Central Processor     Family: &lt;OUT OF SPEC&gt;     Manufacturer: Intel(R) Corporation     ID: A9 06 03 00 FF FB EB BF     Version: Intel(R) Core(TM) i7-3770 CPU @ 3.40GHz     Voltage: 1.1 V     External Clock: 100 MHz     Max Speed: 3800 MHz     Current Speed: 3400 MHz     Status: Populated, Enabled     Upgrade: &lt;OUT OF SPEC&gt;     L1 Cache Handle: 0x003F     L2 Cache Handle: 0x003E     L3 Cache Handle: 0x0040     Serial Number: Not Specified     Asset Tag: Fill By OEM     Part Number: Fill By OEM     Core Count: 4     Core Enabled: 4     Thread Count: 8     Characteristics:         64-bit capable   It will be 8 logic core in a system, like what shows in /proc/cpuinfo. But can any one tell why the cpu MHz of a core is 1600MHz? I guess there is 2 threads in a core, so a hw thread freq may be about the half of the core's? How this number is calculated?  processor   : 7 vendor_id   : GenuineIntel cpu family  : 6 model       : 58 model name  : Intel(R) Core(TM) i7-3770 CPU @ 3.40GHz stepping    : 9 cpu MHz     : 1600.000 cache size  : 8192 KB physical id : 0 siblings    : 8 core id     : 3 cpu cores   : 4 apicid      : 7 initial apicid  : 7 fpu     : yes fpu_exception   : yes cpuid level : 13 wp      : yes flags       : fpu vme de pse tsc msr pae mce cx8 apic sep mtrr pge mca cmov pat pse36 clflush dts acpi mmx fxsr sse sse2 ss ht tm pbe syscall nx rdtscp lm constant_tsc arch_perfmon pebs bts rep_good xtopology nonstop_tsc aperfmperf pni pclmulqdq dtes64 monitor ds_cpl vmx smx est tm2 ssse3 cx16 xtpr pdcm sse4_1 sse4_2 x2apic popcnt aes xsave avx lahf_lm ida arat tpr_shadow vnmi flexpriority ept vpid bogomips    : 7013.49 clflush size    : 64 cache_alignment : 64 address sizes   : 36 bits physical, 48 bits virtual power management:   Also, here is the output of lshw and lscpu command. There are also 1600MHz mentioned.   lshw info:  #lshw -class processor   *-cpu                           description: CPU        product: Intel(R) Core(TM) i7-3770 CPU @ 3.40GHz        vendor: Intel Corp.        physical id: 42        bus info: cpu@0        version: Intel(R) Core(TM) i7-3770 CPU @ 3.40GHz        slot: SOCKET 0        size: 1600MHz        capacity: 3800MHz        width: 64 bits        clock: 100MHz        capabilities: fpu fpu_exception wp vme de pse tsc msr pae mce cx8 apic sep mtrr pge mca cmov pat pse36 clflush dts acpi mmx fxsr sse sse2 ss ht tm pbe syscall nx rdtscp x86-64 constant_tsc arch_perfmon pebs bts rep_good xtopology nonstop_tsc aperfmperf pni pclmulqdq dtes64 monitor ds_cpl vmx smx est tm2 ssse3 cx16 xtpr pdcm sse4_1 sse4_2 x2apic popcnt aes xsave avx lahf_lm ida arat tpr_shadow vnmi flexpriority ept vpid cpufreq   lscpu info:  #lscpu Architecture:          x86_64 CPU op-mode(s):        32-bit, 64-bit CPU(s):                8 Thread(s) per core:    2 Core(s) per socket:    4 CPU socket(s):         1 NUMA node(s):          1 Vendor ID:             GenuineIntel CPU family:            6 Model:                 58 Stepping:              9 CPU MHz:               1600.000 Virtualization:        VT-x L1d cache:             32K L1i cache:             32K L2 cache:              256K L3 cache:              8192K Where can questions about our community-run moderation tools be asked? The question  was recently closed due to not being "about the software that powers the Stack Exchange network, within the scope defined in the help center." MEE is Johann Gambolputty made a comment that     SmokeDetector is a community run project to help to fight spam. Therefore your question is unfortunately off-topic for this site. You can find help in  chatroom.   If questions about community run moderation bots (e.g. SmokeDetector, FireAlarm, Natty, Queen, etc.) are off-topic here on Meta Stack Exchange, is there anywhere on the Stack Exchange network where one can post a Question and Answer-style question on them, or are they only on-topic in appropriate chat rooms? Uk visa refused as self employed Using url package without hyperref makes link hit boxes erratic I've currently got a url link in a footnote in my document. With the hyperref package, everything works correctly. Because the IEEE PDF Express tool doesn't like having the hyperref package, I've removed it and only left in the url package.  However, some strange behavior occurs as a result of this. Instead of just the link being clickable, entire paragraphs of text before/after end up being clickable and going to the same link. This occurs for both Skim and Preview in Mac OSX.  Does anyone have an inkling of what might be causing this? Solving a Diophantine equation with a large solution Divisibility by 7 rule, and Congruence Arithmetic Laws Sometimes when adding an object to the scene I see that all options (on toolbar or in window after pressing F6) are disabled. Can anyone tell me when can this happen? What is causing it and how to revert it - make it enabled again. It worked for me when I restarted Blender and reloaded the scene. However I would really like to know what can cause it? A characterisation of quadratic extensions contained in cyclic extensions of degree 4 Let $D$ be a squarefree integer. I am trying to prove that $\mathbb Q[\sqrt D]$ is contained in a Galois extension of $\mathbb Q$ with Galois group $\mathbb Z/4$ if and only if $D$ is the sum of two squares, $D = a^2 + b^2$ with $a,b\in \mathbb Q$.   The hint in the exercise in Dummit and Foote suggests considering the extension $\mathbb Q[\sqrt{s + s\sqrt D}]$ for the forward direction. However, I am unsure how to show that this extension is Galois, much less that its Galois group is cyclic of order 4.   For the reverse direction, I expressed the given Galois extension as $\mathbb Q[\sqrt{a+b\sqrt D}]$ for some $a,b\in \mathbb Q$. I showed that this extension has $\mathbb Q[\sqrt{c^2 - d^2 D}]$ as a subfield for some $c,d \in \mathbb Q$. From this, I deduced that the $\sqrt D$ and $\sqrt{c^2 - d^2 D}$ generate the same extension, so $\sqrt D = x \sqrt{c^2 - d^2 D}$ for some $x$, and hence that $D = \frac{x^2 c^2}{1 + x^2 d^2} $. It seems that there should be a way to decompose this as a sum of two squares, but I did not see how.   I did not tag this as homework as I am studying it independently.