If $J \neq 0$ is an ideal in $R=\mathbb{Z}[i] .$ Then I need to show that $J \cap \mathbb{Z}$ is an ideal in $\mathbb{Z}$.  My approach (Or the way I wanted to approach the problem): In order for $J \cap \mathbb{Z}$ to be an ideal, I need to show it's an additive subgroup of $\mathbb{Z}$, and that $\lambda x\in J \cap \mathbb{Z}$; for every $\lambda \in \mathbb{Z}$ and $x \in J \cap \mathbb{Z}$. But I really don't know how to show it in such a general case, when I don't know anything about the elements in J. I hope you can help me.
Let $\phi : R \to S$ be a homomorphism. If $N$ is an ideal of $S$, then $\phi ^{-1} (N)$ is an ideal of $R$.
Rules when using a Joker I had on my rack among other tiles, a black 9, 1 x blue 10, and a black 12.  On the table was a set of 3 x 10s including a Joker(no blue 10), and other sets of tiles including an available black 11.  Can I replace the joker with my blue 10 and then set out :  my black 9, the joker(representing a black 10), a black 11 from the table, and my black 12? $(\mathbb{R}^n,\|\cdot\|_{p})$ is isometrically isomorphic to $(\mathbb{R}^n,\|\cdot\|_{q})$ iff $p=q,$ Prove that $g$ is tangent to the graph of $f$ I live in Hong Kong, and the Play Store displays everything in the local Cantonese language. I have set everything I can (Android language, Account language, even the app locale via Xposed Module "App Settings") to English, but I still see all the categories in the store in Chinese. Can someone help me change those to English?  (Click image to enlarge) How to combine multiple conditions to subset a data-frame using "OR"? mysql_fetch_array()/mysql_fetch_assoc()/mysql_fetch_row()/mysql_num_rows etc... expects parameter 1 to be resource Constructing an explicit isomorphism between finite extensions of finite fields Suppose $K$ is a finite field, $K = \mathbb F_{p^s}$. If we take an irreducible polynomial $f$ of degree $d$ over $K$, then the splitting field $L$ of $f$ is $K(\alpha)$ where $f$ is the minimal polynomial of $\alpha$. But then $L = \mathbb F_{p^{sd}} $. Since $\mathbb F_{p^{sd}}$ is unique, we see that this is the splitting field of every irreducible polynomial of degree $d$ over $K$.   Take $K = \mathbb F_2$ and let $P(X) = X^3 + X + 1$, $Q(X) = X^3 + X^2 + 1$. Let $L$ be the splitting field of $P$ and $L&#39;$ be the splitting field of $Q$. The above tells us that $L$ and $L&#39;$ are isomorphic. I would like to construct an explicit isomorphism between $L$ and $L&#39;$.   I know that $L \cong \mathbb F_2[X] /(X^3 +X + 1)$ and $L&#39; \cong \mathbb F_2[X] / (X^3 + X^2 + 1)$. Intuitively, I want to find an isomorphism $\phi : \mathbb F_2[X] \to \mathbb F_2[X]$ such that $\phi((X^3 + X + 1)) = (X^3 + X^2 + 1)$. A little playing around gives me $\phi(X) = X+1$. It now feels like I'm falling at the last hurdle: how do I finish the construction of an isomorphism between $L$ and $L&#39;$? I don't think $\phi$ makes sense as a map from $L$ to $L&#39;$, yet it seems the map I want. Did I correctly derive this recurrence equation formula Does the derivative of a continuous function goes to zero if the function converges to it? Physicist here. I am puzzled by a question: looking at a continuous function $g :\mathbb{R} \rightarrow \mathbb{R}$ that goes to zero at infinity, I am interested in the behavior of its derivative $\beta = g'$. Precisely, does it go to zero too? By writing on paper it looks like: $$ \beta(+ \infty)=\text{lim}_{x\rightarrow \infty}\text{lim}_{h\rightarrow 0} \frac{g(x+h)-g(x)}{h}$$ And if I can invert the two limits, I get what I expect: $\beta(+\infty)=0$, so I was curious about the hypothesis behind this permutation. Is the requirement of continuity enough? Thanks. Asymptotic expansion for Fresnel Integrals If you take the fresnel integrals to be $$S(x) = \int_{0}^{x}\sin \left(\frac { \pi \cdot  t^2}{2} \right) dt$$ How do you find the asymptotic expansion? I know it begins with a $1/2$ but how? Zooming in to inspect texture details in 3D view clips some closer faces like shown in the image. It's not the camera view so changing camera settings doesn't help.   Where can the distance to the clipping plane be adjusted? Subgradients of non-convex functions In  (section 2.3), it is stated that:     A point $x^*$ is a minimizer of a function $f$ (not necessarily convex) if and only if $f$ is subdifferentiable at $x^*$ and $0 \in\partial f(x^*).$   Could anybody provide me with references for a proof of the above statement?   Is there a reference where we can learn more about subgradients of non-convex functions?  In Section 3. Calculus of subgradients of the above notes,many properties of subgradients are presented for convex functions. I would like to know which ones among these properties still hold for non-convex functions.  Thank you in advance! dpkg-divert: error: rename involves overwriting `/usr/lib/xorg/protocol.txt' with different file `/usr/lib/xorg/protocol-precise.txt', not allowed Problem: We have matrix $A\in M_{n-1\times n}(\mathbb Z)$ so that the sum of elements in each row is zero. Prove that $\det(AA^T)=nk^2$, where $k\in \mathbb Z$.  What have I considered so far:   First I thought, since sum of all elements in each row is zero, zero is eigenvalue of $A$ but $A\in M_{(n-1)\times n}(\mathbb Z)$ confused me. I see that $AA^T$ will be $(n-1)$ by $(n-1)$, so I tried calculating $AA^T$ but I failed to see any connection.   I did some research and found  question but we had no mention of it, which makes me believe I am on the wrong track.  Thank you all for your help. I am sure this question has been asked in some capacity on this site. I am a first year graduate student. I was contracted for a summer program that would have looked very good on my CV about 6 weeks ago.   However, due to the spread of COVID-19 the program was cancelled. Is there any way I can express this on my CV (possibly have an asterisk next to this activity)? If not, how should I explain the situation when applying for positions next year (because, now it is unlikely I will have anything noteworthy to put on my CV for this summer).  Thank you Show $\inf_f\int_0^1|f'(x)-f(x)|dx=1/e$ for continuously differentiable functions with $f(0)=0$, $f(1)=1$. How did Jaqen H'ghar get caught? Following a recent 9gag post:    Did Jaqen H'ghar end up in that cage because he knew that he would meet and "recruit" Arya, or was it some strange coincidence? Is it described somewhere in the books, or left unexplained? How do I *really* justify a horizontal menu in HTML+CSS? You find plenty of tutorials on menu bars in HTML, but for this specific (though IMHO generic) case, I haven't found any decent solution:  #  THE MENU ITEMS    SHOULD BE    JUSTIFIED     JUST AS    PLAIN TEXT     WOULD BE  # #  ^                                                                             ^  #    There's an varying number of text-only menu items and the page layout is fluid. The first menu item should be left-aligned, the last menu item should be right-aligned. The remaining items should be spread optimally on the menu bar. The number is varying,so there's no chance to pre-calculate the optimal widths.   Note that a TABLE won't work here as well:   If you center all TDs, the first and the last item aren’t aligned correctly. If you left-align and right-align the first resp. the last items, the spacing will be sub-optimal.   Isn’t it strange that there is no obvious way to implement this in a clean way by using HTML and CSS? I am looping over the features of a shapefile containing multiple polygons. If  certain criteria are met, I want to save the current feature in a new shapefile.  I am doing this using GDAL 1.10.0 with Python 2.7.3  The code runs, creates the files I want, calculates the area correctly. In the shapefile that is created, the attribute table contains the propoer fields ('name' and FID). Two things are wrong:   No polygons are visible (in ArcMap). This seems to me particularly weird, because the polygons seem to me proper, in the sense that they are closed (and the area is calculated properly).  The coordinate system seems off (in Arc Catalog there appears in the bottom -20,9667 0,0000 (are these degrees??)), despite the fact that I am creating a projection file (they are in German national grid 3)   Any hints as to what the problems could be? I tried to put the relevant excerpt of the code below:  from osgeo import gdal, gdalnumeric, ogr, osr, gdalconst  ### prepare shapefile into which to save driver = ogr.GetDriverByName('ESRI Shapefile') path_matched = r'.../.../path' datasource = driver.CreateDataSource(path_matched) lyr_matched = 'matched_%s' % cur_id  # from above # Remove output shapefile if it already exists name_shpfile = lyr_matched + '.shp' if os.path.exists(os.path.join(path_matched, name_shpfile)):     driver.DeleteDataSource(os.path.join(path_matched, name_shpfile)) layer = datasource.CreateLayer(lyr_matched, geom_type=ogr.wkbPolygon) #layer.ResetReading() field_name = ogr.FieldDefn("name", ogr.OFTString) layer.CreateField(field_name)  ### prepare file in which I want to write the areas of matched polygons path_area = path_matched file_area = 'matched_area_%s.txt' % (cur_id) fobj_area = open(os.path.join(path_area, file_area), 'w')  ### original shape file containing all polygons shp = ogr.Open(shp_all)  # os fobj from above lyr = shp.GetLayer()  # loop over polygons in original shameful for FID in featList:     feat = lyr.GetFeature(FID)      # this is from the shapefile      cur_name = feat.GetField(1)      # ... work on conditions ...      if condition = true:         myPoly = ogr.Geometry(type=ogr.wkbPolygon)         cur_poly = feat.GetGeometryRef()         myPoly.AddGeometry(cur_poly)         cur_spat_ref = cur_poly.GetSpatialReference()          cur_area =  cur_poly.GetArea()         cur_area_string = "%s \t %f \n" % (gw_name_EZG, cur_area)         fobj_area.write(cur_area_string)          featur_dfn = layer.GetLayerDefn()         feature_abc = ogr.Feature(featur_dfn)         feature_abc.SetGeometry(myPoly)         feature_abc.SetField("name", cur_name)         feature_abc.SetFID(FID)         layer.CreateFeature(feature_abc)         myPoly.Destroy()         feature.Destroy()   # generate projection file prj_filename = lyr_matched + '.prj' file = open(os.path.join(path_matched, prj_filename), 'w') targetSR = cur_spat_ref targetSR.MorphToESRI() file.write(targetSR.ExportToWkt()) file.close()   datasource.Destroy() Why doesn't 'find' show this file?