## Real exponential field with restricted analytic functions: $\mathbb R_{an, exp, log}$ has quantifier elimination, but $\mathbb R_{an, exp}$ does not.

At a talk sometime ago a result was presented, which I believe originates from:

van den Dries, Lou; Miller, Chris, On the real exponential field with restricted analytic functions, Isr. J. Math. 85, No. 1-3, 19-56 (1994). ZBL0823.03017.

At some point it was mentioned that $$\mathbb R_{an,exp,log}$$ admits quantifier elimination while $$\mathbb R_{an,exp}$$ does not. Here $$\mathbb R_{an,exp}$$ is the theory of the (ordered) real exponential field with function symbols for all restricted analytic functions. Then of course $$\mathbb R_{an,exp,log}$$ is just adding a function symbol for logarithms.

Someone in the audience remarked that $$log(x)$$ (or more precisely, its graph) is quantifier-free definable by $$x = exp(y)$$. Then a fairly simple formula was presented to show why you really need $$log$$ as a function symbol for quantifier elimination, and there is my question: I just cannot remember or reconstruct that formula. So what would be a simple example of some formula in this setting that is not equivalent to a quantifier-free formula in $$\mathbb R_{an,exp}$$?

I am probably missing something obvious here, but now it’s haunting me.

## Determine all idempotents, nilpotents, and units in $F[x]/\langle h\rangle$, where $F$ is a field, $h=x^2-x$.

Problem: Determine all idempotents, nilpotents, and units in $$F[x]/\langle h \rangle$$, where $$F$$ is a field, $$h=x^2-x$$.

I know $$F[x]/\langle h \rangle = \{a_0 + a_1t \mid a_i \in F, t^2-t=0 \}$$, So first starting with idempotents, if $$z \in F[x]/\langle h \rangle$$ is an idempotent, then $$z=a_0+a_1t$$ satisfies $$z^2=z \rightarrow a_0^2+2a_0a_1t+a_1^2t^2=a_0+a_1t$$ which implies that $$a_0=0$$ or $$1$$.

For the nilpotents, if there is some $$k \leq 0$$ such that $$z^k=0$$, I’m not sure what can be said about $$z$$. Hints appreciated.

## Locality of a tensor product with a fixed field extension

Given a strict (not necessarily finite) field extension $$F \subset K$$, does there always exist a field extension $$F \subset L$$ such that $$K \otimes_F L$$ is not local?

## Is the bits field a unique representation of the target?

The bits field is the compact representation of the target.

Example:

bits: 1d00ffff target: 00ffff0000000000000000000000000000000000000000000000000000  bits: 1cfff00 target: ffff0000000000000000000000000000000000000000000000000000 

But these two actually represent the same number.

int(target) -> 26959535291011309493156476344723991336010898738574164086137773096960 for both of the above targets.

What (if anything) makes bits a unique representation of the target?

## Is there a functor which is equivalent to discriminant of number field?

Let $$K$$ be a number field, i.e. a finite extension of $$\mathbb{Q}$$. The ring of integer $$O_K$$ is a free $$\mathbb{Z}$$-module. Let $$\{ a_1, \cdots , a_n\}$$ be a integral basis of $$O_K$$. Then, $$\Delta_{K/ \mathbb{Q}} = \det (\mathrm{Tr}(a_ia_j)_{i,j})$$ is independent of choice of integral basis. We call $$\Delta_{K/ \mathbb{Q}}$$ a discriminant of number field $$K$$ over $$\mathbb{Q}$$.

My question is: Is there some categories $$C,D$$ and a functor $$F \colon C \to D$$ such that you have a simple way to get the discriminant $$\Delta_{K/ \mathbb{Q}}$$ from an object $$F(K)$$? I want $$F$$ to be a canonical one.

Thanks.

## Reading text from MetaInfo field

For research purposes, I’m trying to read the text in a Sharepoint database MetaInfo field which is stored as tCompressedBinary(varbinary(max)).

In order to read this, I tried the following solution which I found online:

select top 4 cast(cast(MetaInfo as varbinary(2048)) as varchar(2048)) from AllDocs where Extension = 'pdf' 

But this returned “¨©01\f for all four selected fields. If I run the following query:

select top 4 MetaInfo from AllDocs where Extension = 'pdf' 

It returns binary fields which start with : a8a930310c000000 (and continue like this).

Do you know how to turn the info in the MetaInfo table to a readable string?

## A duplicate field name “StartDate” was found in SharePoint 2013

You cannot vote on your own post 0

After completion of Migration from SharePoint 2010 to SP2013, didn’t see any timer jobs under Job Definition. So we have deactivated the feature and activated at SiteCollection Level then able to see the Custom Timer Job Under Timer Job Definition, Except One Custom Timer Job for that we have deactivated and while activating the feature at web level getting the error A duplicate field name “StartDate” was found in SharePoint 2013.

In SharePoint 2010 there is no issue, but after migration to SharePoint 2013 getting A duplicate field name “StartDate” was found.

We have two Custom List Definitions (Activity Taks and Workflow Task) created the StartDate Column using below field attributes in Schema.xml

And also have many custom ContentTypes referred the StartDate columns like below mentioned in the Elments.xml file

This StartDate Field had used in the code in many places. Can you please help on this issue how to resolve this issue.

## How do I remove the title field from a node’s custom view mode

In my Drupal 8 website I created a custom view mode through the UI. I use it to output nodes using a view.

In the view result I do not want the titles of the nodes to appear. The UI (content type > manage display) does not seem to give the option to hide the title.

How can I hide the title in a specific view mode for an entity via the UI or via custom code?

## Why central isogeny of reductive group over genereal field F map maximal F split torus onto a maximal split F torus

let $$f$$ be an central isogeny of reductive groups over a field F, why $$f$$ map a maximal F split torus onto a maximal split F torus.

## I have three models HomePage, Callout, FeatureContent like below:

class FeatureTip(models.Model):    feature_tip_title = models.CharField(max_length=120, null=True, blank=False)    feature_tip_description = models.TextField()    def __str__(self):       return self.feature_tip_title   class Citie(models.Model):    name = models.CharField(max_length=120, null=True, blank=False)    description = models.TextField()     def __str__(self):       return self.name   class HomePage(models.Model):    header = models.CharField(max_length=120, null=True, blank=False)    cities = models.ManyToManyField(Citie)    featured_tips = models.ManyToManyField(FeatureTip)     def __str__(self):       return 'Home Page'   class Callout(models.Model):    header = models.CharField(max_length=120, null=True, blank=False)    home_page = models.ForeignKey(HomePage,on_delete=models.CASCADE, null=True, blank=False)    def __str__(self):       return self.header     def get_city(self):       return self.cities.all()  class FeatureContent(models.Model):    title = models.CharField(max_length=120, null=True, blank=False)    home_page = models.ForeignKey(HomePage, on_delete=models.CASCADE, null=True)    def __str__(self):       return self.feature_article_title_en 

class CalloutInline(admin.StackedInline):     model = Callout     fields = ['header']     extra = 4     max_num = 4     def get_queryset(self, request):         HomePage.objects.filter(name="Eminem")  class FeatureContentInline(admin.StackedInline):     model = FeatureContent     fields = ['title']     extra = 1     max_num = 1  class HomePageAdmin(admin.ModelAdmin):     filter_horizontal = ['cities', 'featured_tips']      inlines = [CalloutInline, FeatureContentInline]