I have a probability mass function of some experimental data who's log looks like the following: (please ignore the fact that it is not normalized)

(meaning if p(x) is the pmf, this is log(p(x)) ) Does anyone know what parametric family it might belong to? (note that this is a discrete distribution)

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This came up today while thinking about topoi in seminar, as the title suggests my question is;

Is there a finitely complete category with terminal object but NO subobject classifier?

Hopefully if the answer is yes, you can give an interesting example that is useful in some way. One of my professors has a saying that

for every definition you should have an interesting example and an interesting non-example

So I am trying to stick to this.

Thanks!

Suppose $G$ is a simple (linear) algebraic group over an algebraically closed field of characteristic zero, that $n$ is a positive natural number, and that $g\in G$. Can we always find an $h\in G$ such that $h^n=g$?

(It appears to be possible to check this for the classical algebraic groups by direct computations in each case, but covering the exceptional Lie Algebras this way seems like it might be tricky, and anyhow, I'm inclined to think that a case analysis is probably not the optimal way of approaching this problem!)

Note added: Kovalev made a comment showing that the answer is `no' in general. The counterexamples appear to revolve around non-semisimple elements. I wonder whether the answer becomes positive if one restricts oneself to $g$ of finite order?

Let $\mathbf{T}$ be the reduced nearly ordinary Hecke algebra of level $N$ of Hida theory for $\operatorname{GL}_{2}$ over $\mathbb{Q}$ (or more generally over a totally real field $F$). Then $\mathbf{T}$ is finitely generated over a regular ring $\Lambda$ of dimension 3. Let $\mathfrak{m}$ be a maximal non-Eisentein ideal of $\mathbf{T}$.

By patching pseudo-representations attached to algebraic modular forms, Wiles (and Hida) have constructed a two-dimensional $G_{\mathbb{Q}}$-representation $(V,\rho)$ with coefficients in $\mathbf{T}_{\mathfrak{m}}\otimes_{\Lambda}\operatorname{Frac}(\Lambda)$. This representation admits a free 1-dimensional quotient $V^{+}$ and a free 1-dimensional quotient $V^{-}$ both stable under the action of $G_{\mathbb{Q}_{p}}$. Because $\mathfrak{m}$ is non-Eisenstein, there exists a choice of basis of $V$ such that $\rho$ has values in $\operatorname{GL}_{2}(\mathbf{T}_{\mathfrak{m}})$. The lattice $L\subset V$ corresponding to this choice of basis admits a free sub-module $L^{+}=L\cap V^{+}$ of rank 1 stable under $G_{\mathbb{Q}_{p}}$. However, it is unclear to me whether $L$ admits a free rank 1 quotient stable under $G_{\mathbb{Q}_{p}}$. This is true if $\rho$ modulo $\mathfrak{m}$ is of the form $$\rho\sim\begin{pmatrix}\chi_{1}&*\\ 0&\chi_{2}\end{pmatrix}$$ with $\chi_{1}\neq\chi_{2}$ because then $L/L^{+}$ is generated by a single element according to Nakayama lemma. However, without this hypothesis, I don`t see an obvious proof of this fact, nor have I good reasons to believe it should be true. Does anyone know for sure?

Given a symmetric monoidal category and a monoid object A in it, one can form the category of modules over this monoid object, i.e. objects are $A \otimes M \rightarrow M$ satisfying the natural properties analogous to modules over a ring and morphisms respecting this. The following seems to be true and I would like to know why:

If the category of modules has a closed symmetric monoidal structure with A as unit object, then A is a commutative monoid.

This is how I read the statement right after Proposition 2.3.4 in Hovey/Shipley/Smith's paper "Symmetric Spectra" and it would give an excellent motivation for introducing symmetric spectra...

Let $X$, $Y$ and $Z$ be Noetherian schemes.

If $f: Y \to X$ and $g: Z \to X$ are morphisms of finite type, such that at each point of $X$, at least one of the two morphisms is smooth/étale/unramified (at all points of its inverse image), can we conclude that the induced morphism $Y \times_X Z \to X$ is smooth/étale/unramified everywhere?

If not, which results can we obtain?

(In his textbook on Algebraic Geometry, Liu asks to prove that the answer is always "yes"...)

EDIT. So, indeed, the problem statement in the book is wrong...

There are nonequivalent geometries, nonequivalent groups finite and infinite, nonequivalent logics ( fregean and nofregean http://www.formalontology.it/suszkor.htm), even nonequivalent logicians;-)

Are there nonequivalent randomnesses?

The main two theories we know dealing with randomness and probability is Kolmogorov Probability Theory ( by means of measure theory and Borel $\sigma$-algebras), and Bayesian a priori approach. Are they equivalent enough to say that they are the same in some deeper meaning?

@ Johannes Hahn - no, I am not asking about non isomorphic probability spaces as it would be trivial. rather I as about possible probability theories as different as different are geometries euclidean and noneuclidean. The obvious generalization is to change linearity in the second axiom of probability ($P(A U B) = P(A) + P(B)$ when A,B are independent.

In fact I mention about it after reading probability overview by Terence Tao http://terrytao.wordpress.com/2010/01/01/254a-notes-0-a-review-of-probability-theory/ He wrotes:

probability theory is only “allowed” to study concepts and perform operations which are preserved with respect to extension of the underlying sample space.

Which in my opinion is something very deep ( but I am only a hobbyist;-). So probably if You have initial probability space, and You need to extent it to describe some additional phenomena, You have have done some kind of morphisms between structures of first space and another wider one. Is there a unique, canonical or any defined way of doing this? May we perform this kind of extension always in the same way or there are different ways of doing that? Gives it any predictable and interesting structure?

@sheldon-cooper Bayesian approach to probability is sometimes seen as alternative ( not very well defined) to Kolmogorov axiomatic probability system, because it do not require given a priori probability space. For example in this approach we may say that probability that tomorrow will be could day ( say temp<-10) is defined, whilst in Kolmogorov approach You probably cannot define proper space ( because You cannot have equivalent population of Wednesdays which are tomorrow days, with different temperatures). Agree - when You have possibility to use properly defined probability space this two approach coincides. From Wikipedia: http://en.wikipedia.org/wiki/Bayesian_probability

Bayesian probability interprets the concept of probability as "a measure of a state of knowledge",[1] in contrast to interpreting it as a frequency or a physical property of a system.

@Qiaochu Yuan - of course randomness here is colloquialism. Yes You have right: maybe I just should ask about different probability theories, but note that non-euclidean geometries in analogy are just geometries but in different spaces, with some special properties. So in fact they share the same meaning of geometric set, figure, space, even so complicated objects as coordinate systems, and angle. But they have different relations between them. So I ask about something similar: different kinds of randomness which are in scope of probability theory but describes different relations between for example different classes of ways of extensions of probability spaces. If the last procedure changes anything in resultants;-) Agree - maybe this is not very interesting question. Maybe it would be more interesting in scope of algorithmic information theory and its randomness concept?

Suppose that $X$ is a variety (in char 0) with an action of an affine algebraic group $G$. Let $Y \subset X$ be a subvariety fixed by $G$--the action map agrees with projection upon restriction to $Y$. Let $\widehat{Y}$ be the formal completion of $X$ along $Y$. Furthermore let $\widehat{G}$ be the the completion of $G$ at the identity, viewed as a formal group. There is a restriction functor $j^*$ from the $Qcoh^G(X)$, the category of $G$-equivariant quasicoherent sheaves on $X$, to $Qcoh^{\widehat{G}}(\widehat{Y})$, the category of $\widehat{G}$-equivariant quasicoherent sheaves on $\widehat{Y}$.

1) Is this situation considered in the literature? Where?
2) What tools are available to control this functor? How might one describe the essential image?

Although curious about this general package, I specifically care about the case $G =\mathbb{G}_m$.

Are supersingular primes and supersingular elliptic curves related?

(this was essentially a subquestion in my earlier question, but still looks sufficiently different to me to deserve a separate post)