How to show that there are infinitely many prime numbers $p$ such that the polynomial f has a zero in $\mathbb Z_p$? Let $f\in \mathbb Z[X]$ be a polynomial of positive degree.How to show that there are infinitely many prime numbers $p$ such that the polynomial $f$ has a zero in $\mathbb Z/p \mathbb Z$ ?  I have no idea of how to start even.  Need help
Primes dividing a polynomial Let $g(x)\in \mathbb{Z}[x]$, a nonconstant polynomial. Show that the set of primes $p$ such that $p\mid g(n)$ for some $n\in \mathbb{Z}$ is infinite.  I don't know how to start. I have tried asuming that the set is finite but i haven't reached any contradiction.
Shortest path from points in one layer to points in another layer via roads in QGIS I'm looking for a way to calculate the shortest path from addresses to subway stations via a network of roads.  I have the , however, the addresses and subway stations are point layers that do not overlap with vertices in the roads layer.  In QGIS, is there a way to calculate the shortest distance from the address/subway station points to the roads, create vertices at those locations, then calculate the shortest distance via the roads network?   black squares - addresses white circles - subway stops gray lines - roads $f:\mathbb R \to \mathbb R$ be a continuous such that $f(f(f(x)))=x,\forall x \in \mathbb R$ , is $f(x)=x , \forall x \in \mathbb R$? Let $f:\mathbb R \to \mathbb R$ be a continuous function such that $f(f(f(x)))=x,\forall x \in \mathbb R$ , then is it true that $f(x)=x , \forall x \in \mathbb R$ ? I can see that $f$ is bijective , hence strictly monotone . But I cannot progress further . Please help. Thanks in advance What is the angle at which water is splashed when a vehicle tyre rides in water on the road? I would like to know the following:   What is the angle at which water gets splashed when I ride my vehicle through a water on the road? How does angle of water varies with speed? What is the relation between the distance the water goes with the speed of the vehicle?   Thanks! Replacing the nth instance of a regex match in Javascript I'm trying to write a regex function that will identify and replace a single instance of a match within a string without affecting the other instances.  For example, I have this string:  12||34||56   I want to replace the second set of pipes with ampersands to get this string:  12||34&amp;&amp;56   The regex function needs to be able to handle x amount of pipes and allow me to replace the nth set of pipes, so I could use the same function to make these replacements:  23||45||45||56||67 -&gt; 23&amp;&amp;45||45||56||67  23||34||98||87 -&gt; 23||34||98&amp;&amp;87   I know that I could just split/replace/concat the string at the pipes, and I also know that I can match on /\|\|/ and iterate through the resulting array, but I'm interested to know if it's possible to write a single expression that can do this.  Note that this would be for Javascript, so it's possible to generate a regex at runtime using eval(), but it's not possible to use any Perl-specific regex instructions. How soon can I re-apply for a UK visa after getting a refusal? I have removed the mentioned libraries. Then my system disconnected from NIS service and many packages are being removed. The problem is I am not even able to login in my system. When i restarted it is showing "low graphics mode error". I do not know what to do Please help me. I am a beginner to Ubuntu. Define functions in structs Can hard drives fully utilize SATA III? My motherboard currently supports both SATA II and III (3 Gb/s and 6Gb/s respectively), but after reading some online, it seems as though connecting a HDD to the 6 Gb/s is pretty much pointless. If this is the case I'm going to leave my two SATA III ports open for something that can actually use it (which at this point I'm not really sure what can).  This is part of an article I read:     In terms of everything else, we saw basically no difference from the same single drive being plugged into a SATA 6Gb/s or SATA 3Gb/s port. All of this is obviously due to the hard drive not being able to actually take advantage of the 6Gb/s bus.   from . Natural examples of deformation retracts that are not strong deformation retracts The question  asks for examples of deformation retracts which are not strong deformation retracts. A comment is given refering the asker to Exercise 0.6 in Hatcher. (The description of this space also appears in , which is about this very same exercise.)  This space, however, seems to be constructed specifically for this purpose.     Are there any naturally occurring (by which I mean not constructed specifically for this purpose, arising in some other way) spaces that are deformation retracts but not strong deformation retracts?   The main reason for asking is that it doesn't seem very likely to me that one would stumble upon such a space at random, so I don't know what to watch out for. One can never be too careful. Is there a rule to know when to pronounce letter c as a "soft c" (s) or "hard c" (k)? Why is the upper Riemann integral the infimum of all upper sums? I was reading the theory of Riemann integration when I cam across the following ,   If $f$ is bounded on $[a,b]$, and $P = \{x_0,x_1,x_2.......x_n\}$ is a partition of $[a,b]$, let $$M_j = \sup_{x_{j-1}\leq x\leq x_j}f(x)$$ The upper sum of f over P is $$S(P) = \sum_{j=1}^{n} M_j(x_j-x_{j-1})$$ and the upper integral of $f$ over $[a,b]$, denoted by $$\int_{a}^{b^-} f(x)dx$$ is the infimum of all upper sums.  The theorem similarly goes on to state the result for lower sums.  My doubt is : I do not understand how is $$\int_{a}^{b^-} f(x)dx$$ the infimum of all upper sums. I understand that if we refine the partition P, then the upper sum would decrease, so it may be a lower limit for all the upper sums computed on the refinements of P (but still being the lower limit does not prove that it is the infimum) and what about those partitions for which P itself is the refinement of?  How do I know that it will be a lower limit for those, let alone a infimum? OS is openSUSE 12.1, PostgreSQL - 9.1, PostGIS - 2.0. Postgre/PostGIS are installed from Application:Geo repository.  After recent updates when I try to connect to my database I get the following error:      ERROR:  could not load library "/usr/lib/postgresql91/lib64/postgis-2.0.so":  /usr/lib/postgresql91/lib64/postgis-2.0.so: undefined symbol: pj_get_spheroid_defn   I searched the web for this error (, ) and seems that it may have something to do with ld.so.config (or ldconfig command). But I can't figure out what I have to do in my case. An $n \times n$ matrix $M$ of integers has the property that $M_{ab} = M_{xy}$ iff $a+b=x+y$. How many distinct elements are there in $M$? Using iOS 5 rich text editor What am I doing when I separate the variables of a differential equation? I see an equation like this:  $$y\frac{\textrm{d}y}{\textrm{d}x} = e^x$$  and solve it by "separating variables" like this:  $$y\textrm{d}y = e^x\textrm{d}x$$ $$\int y\textrm{d}y = \int e^x\textrm{d}x$$ $$y^2/2  = e^x + c$$  What am I doing when I solve an equation this way?  Because $\textrm{d}y/\textrm{d}x$ actually means  $$\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$$  they are not really separate entities I can multiply around algebraically.    I can check the solution when I'm done this procedure, and I've never run into problems with it.  Nonetheless, what is the justification behind it?  What I thought of to do in this particular case is write  $$\int y \frac{\textrm{d}y}{\textrm{d}x}\textrm{d}x = \int e^x\textrm{d}x$$ $$\int \frac{\textrm{d}}{\textrm{d}x}(y^2/2)\textrm{d}x = e^x + c$$  then by the fundamental theorem of calculus  $$y^2/2 = e^x + c$$  Is this correct?  Will such a procedure work every time I can find a way to separate variables? This error message is being presented, any suggestions?     Allowed memory size of 33554432 bytes exhausted (tried to allocate   43148176 bytes) in php Change in "under review" date I have passed the calc sequence, and thought that I understood things.  But I have trouble with limits now and again.  From the way I understand it, the 'best' way to evaluate a limit, is to evaluate the expression at the number in the limit.  In that way it's not only a limit, but an algebraic expression (so, I think, the need for limits goes away entirely, superseded by accuracy.)  But then I see something like $\lim_{x\to 2} \frac{x^2-4}{x-2}$.  Now, I think in my mind 'the best way to evaluate this is just to plug in $2$'  But it doesn't work that way, as the limit is $4$, and if I plug in, the answer is $0/0$.  I think that people here, with their natural talent in math, can perceive where my train of thought is going wrong, and correct me.  Please! Convergence of $\sum_{n=1}^\infty\frac{n}{(n+1)!}$ Can someone give an explanation using the definition of convergence in partial sum to show how the above infinite sum converges to 1?  Thanks Do the 46 human chromosomes form a single unbroken DNA helix?  Or is it rather that a human's genome consists of 46 disconnected helices?    If it is the former, does the common numbering scheme for the chromosomes have any correlation to their actual ordering in the one large strand?  If is the latter, is there a convention on how the chromosomes are ordered in genomic datasets?  Also, is there a clear understanding of how sister chromosomes "find" each other in Meiosis I?  Generally, during periods when Mitosis/Meiosis are not occurring,  what's a good physical picture for how the chromosomes are physically arranged (e.g. a bowl of 46 spaghetti noodles, or maybe the sister chromosomes always stay close together, etc)  thanks!