Let a cardinal $\kappa$ be $n$-shadow iff $\kappa$ does not have cofinality $\omega$ and for any $n$-th order sentence $\varphi$ in the language $\mathcal{L}_\in$, $\varphi\Leftrightarrow V_\kappa\models\varphi$.

The property of being $0$-shadow is (much) weaker than being Mahlo; in fact, much weaker than $\text{Ord is Mahlo}$.

The property of being $n$-shadow is also (much, much, much) weaker than being $n$-extendible. Furthermore, if there is an $n$-extendible cardinal, then $\text{Con}(\text{ZFC}^2+n\text{-shadow})$.

The reason I bring this up is because of the interesting properties of $1$-shadow cardinals when given an inner model $M$ of ZFC.

Assuming $M$ is an inner model and there is a $1$-shadow cardinal in $M$, then there is no nontrivial elementary embedding from $M$ into itself (in most cases this is equivalent to it's sharp not existing).

So, if $0^{\#}$ exists then there are no $1$-shadow cardinals in $L$. If $0^{\dagger}$ exists then there are no $1$-shadow cardinals in $L[U]$ for the standard $U$.

Specifically, if there is a nontrivial elementary embedding from $K$ into itself, then no cardinal is $1$-shadow in $K$, even though it should be that every uncountable cardinal has most large cardinal properties in $K$ if such an embedding exists, because of the tendencies of the core models.

**Questions: What is the consistency strength of $n$-shadow cardinals? What properties result from these properties of inner models?**

Russia's TASS News Agency a few days ago reported that a Russian scientist proposed a solution of two problems from Hilbert's list, including the Riemann hypothesis. The solution involves a numerical infinity denoted $\circledcirc$ or some variation thereof. I was unable to determine from the report whether the problem was solved affirmatively or negatively. My question is the following:

Is $\frac{1}{2}+i\circledcirc$ a zero of the Riemann zeta function?

To correct an erroneous impression that emerged in the *comments* below: the article in question is not a preprint and on the contrary has been published in a respectable European venue:

Sergeyev, Yaroslav; Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems. EMS Surv. Math. Sci. 4 (2017), no. 2, 219–320.

if the author wishes to exploit $\circledcirc$ to "propose a solution to RH" by "extending the precision of the traditional mathematical language", he would have to explain how to extend the zeta function, first.

In addition to the Seiberg-Witten invariant there has been further success with "finite dimensional approximations" of the Seiberg-Witten theory: Bauer-Furuta's stable (co)homotopy invariants, and Kronheimer's and Manolescu's homotopy invariants associated with Seiberg-Witten-Floer cohomology. Such approximations were then demonstrated by Frauenfelder on the *vortex equations* in 2 dimensions. So I naively ask:

**Is it impossible to achieve finite dimensional approximations of Donaldson's theory?** If it were possible, what are the obstacles? Is there an expected use of them that perhaps Seiberg-Witten theory wouldn't reach?

While there is an expected "equality" between the Seiberg-Witten invariants and the Donaldson invariants, there is at least one defining difference between them: lack of compactness of the moduli. I'm unsure how much this affects the finite-dimensional techniques (e.g. Kuranishi constructions) from being applied to the Yang-Mills equations. It probably makes it hard, but I wonder whether it leads to a dead-end. I'm unsure what (not) to expect, especially when there are compactifications of the moduli by Gieseker and Uhlenbeck. A glimpse at other obstacles would also be informative.

This is something that has come up in my research. Fix $n\geq 2$ and let $S_n$ be the symmetric group on $n$ letters with identity $e$. We consider elements of $S_n$ to be bijections $[n]\to [n]$. For $1\leq i<n$, let $s_i$ be the transposition exchanging $i$ and $i+1$. Consider the following algorithm that successively constructs subsets $A_N\subseteq S_n$.

- Let $A_0=\{e\}$.
- At step $K$, remove the least element from $A_K$ in lexicographical order, say $f\in A_K$, and for each subset $B\subseteq [n-1]$ such that if $B=\{b_1,\ldots,b_m\}$, then $|b_i-b_j|\geq 2$ for all $i\neq j$ and $f(b_i)<f(b_{i+1})$ for all $i$, add the permutation $fs_{b_1}s_{b_2}\cdots s_{b_m}$ to $A_K$. (In weak order, this will add all maximal elements of Boolean algebra intervals with minimal element $f$). After this one removal and all such additions we obtain the set $A_{K+1}$.

Step 2 is repeated until the ultimate set is empty. The penultimate set will contain only the longest element, the reversal permutation.

Now it seems to me anecdotally that the number $M_n$ defined by $$M_n=\max_{K}{|A_K|}$$ is $o(n!)$. We have $$M_{10}\leq 0.35\cdot 10!$$ $$M_{11}\leq 0.32\cdot 11!$$ $$M_{12}\leq 0.27\cdot 12!$$ I'm looking for a good upper bound (not aymptotic) for $M_n$. This will be the amount of space required in my algorithm for computing the sequence http://oeis.org/A006245. I'm mostly interested in $n=16$, since this is the first unknown value. If $M_{16}$ is around $0.27\cdot 16!$, then it's not feasible at the moment to do the computation, whereas if it's $0.2\cdot 16!$ it might be.

Let $A_1$, $A_2$ be $n\times n$ real matrices. Suppose that $A_1$ and $A_2$ are Schur stable (i.e., their eigenvalues are strictly inside the unit circle in the complex plane). Let $B_1$, $B_2$ be two $n\times m$ real matrices of full column rank and define $$ \mathcal{R}_1 :=\left[B_1\, |\, A_1B_1\, |\, A_1^2B_1\, |\, \cdots\, |\, A_1^{n-1}B_1\right] $$ $$ \mathcal{R}_2 :=\left[B_2\, |\, A_2B_2\, |\, A_2^2B_2\, |\, \cdots\, |\, A_2^{n-1}B_2\right] $$ (In control theory the above-defined matrices are called reachability or controllability matrices of the pairs $(A_1,B_1)$ and $(A_2,B_2)$, resp.)

**Question.** Under the assumption that $\mathrm{rank}\,\mathcal{R}_1=\mathrm{rank}\,\mathcal{R}_2=n$, can we conclude that the following series
$$
\sum_{k=0}^\infty A_1^k\, B_1\, B_2\, A_2^k
$$
is non-singular?
If not, do there exist some non-trivial conditions on the pairs $(A_1,B_1)$ and $(A_2,B_2)$ that guarantee that the above series is non-singular?

Consider a utility maximization problem

$\max_{\textbf{x}}~ U=\sum_i x_i^{\alpha_i}$

$s.t. ~~\textbf{C} \textbf{x}\leq B$

$~~~~~~~~~~~x_i\geq 0, \forall i,$

where $\alpha_i \in (0,1), \textbf{C}\in \{0,1\}^{N\times M}$ is a binary matrix.

It is intuitive that if we increase $B$, $U^*$ grows sub-linearly and $\frac{\partial U^*}{\partial B} \cdot B$ increases with $B$.

But I wonder if latter one can be proved?

Thanks!

Is it possible for an element to be a subset if the element is not null?

Eg in the set of natural numbers, can an arbitrary natural number be considered an element and subset?

This is my first post here, as someone from Mathsstack suggested this might me a more suitable forum for this specific question.

I have been reading some texts by Joaquim Lambek on formal languages, grammar and so on, and at some point he mentions that the kind of approach he advocates in the late 90s and 2000s has the advantage, as compared to some similar ones from the end 60s, that they allow for double adjoints, which, in turn, account for what other schools address with the use of traces.

Does that mean that double adjoints were ony developed in logic and mathematics in the 70s or later? Or how should I understand that claim? I am assuming that the authors in the 60s were not simply oblivious

I would be very grateful if you could me illustrate me on the history of double adjoints, with references, milestones, etc...

Thanks in advance.

I am looking for a reference of the proof of the above claim.

Basically it's the following claim:

Consider a Wigner matrix $X_N$ satisfying $r_k\le k^{Ck}$ for some constant $C$ and all positive integers $k$. Then, $\lambda_N^N$ converges to $2$ in $L^p$ norm.

where $r_k$ is defined as: $r_k := \max ( E|Z_{1,2}|^k,E|Y_1|^k )<\infty$; and $\{ Z_{i,j} \}_{1\le i <j}$ and $\{ Y_i \}_{i\ge 1}$ are real valued random variables iid with zero mean, s.t $E(Z_{1,2}^2)=1$.

Thanks!

Let $M,N$ be two differentiable manifolds, and let $V$ be a vector field on $M\times N$. What conditions on $V$ enable the existence of a fibration $\pi:M\times N\to M$ such that $V$ is tangent to the fibres?

Let $g_{\lambda}$ be a one parameter family of Riemannian metrics, which are complete and with bounded curvature, on the unit disk, depending smoothly on the parameter $\lambda$. Let $\Delta_{\lambda}$ be the corresponding family of Laplacian operators. Consider the following family of non linear parabolic equations $$\frac{\partial u}{\partial t}=e^{-u}\Delta u-e^{-u}R_{\lambda}+r,$$ where $R_{\lambda}$ is the curvature of $g_{\lambda}$ and $r$ is a constant. If we have a solution $u_{\lambda}(x,t)$ with $u_{\lambda}(\cdot,0)\equiv1$, is it true that $u_{\lambda}$ depends smoothly on $\lambda$?

In the book "Theorie der gewöhnlichen Differentialgleichungen" by Bieberbach, page 240, there is a solution to the hypergeometric differential equation $z(z-1)w^{\prime \prime}+(2z-1)w^{\prime}+\frac{w}{4}=0$. This equation corresponds to the hypergeometric function $F(\frac{1}{2},\frac{1}{2},1;z)$. In general, a basis for the solution of the hypergeometric differential equation $$z(z-1)w^{\prime \prime}+(z(1+\alpha+\beta)-\gamma )w^{\prime}+\alpha \beta w=0$$ is given by $F(\alpha, \beta, \gamma;z)$ and $z^{1-\gamma}F(\alpha-\gamma+1, \beta-\gamma+1,2-\gamma; z)$. However, when $\alpha = \beta=\frac{1}{2}$ and $\gamma=1$, these two solutions coincide. In order to remedy this, the author considers for the first basis element the function $w_1=F(\frac{1}{2},\frac{1}{2},1;z)$ and for the second $w_2=F_1(\frac{1}{2},\frac{1}{2},1;z)+F(\frac{1}{2},\frac{1}{2},1;z) \log z$, where $F_1(\frac{1}{2},\frac{1}{2},1;z)=(\frac{\partial F(\alpha, \beta, \gamma;z)}{\partial \alpha}+\frac{\partial F(\alpha, \beta, \gamma;z)}{\partial \beta}+2\frac{\partial F(\alpha, \beta, \gamma;z)}{\partial \gamma})_{\alpha=\frac{1}{2},\beta=\frac{1}{2}, \gamma=1}$. Of course one can check that this is really a solution, but my question is from where, exactly, comes the operator $\frac{\partial}{\partial \alpha}+\frac{\partial}{\partial \beta}+2\frac{\partial}{\partial \gamma}$? How one should get this idea?

A recent question Why do we need model categories? reminded me of this long-standing confusion of mine -- I mentioned it in an answer there, and then decided to ask a separate question about it. I even dare not to use the soft-question tag.

What confuses me is this. There is an obvious resemblance between model categories and factorization systems. Yet, one thing goes totally wrong: in factorization systems, "left halves" tend to be "like epimorphisms" and "right halves" - "like monomorphisms". At the same time, in model categories cofibrations have distinct flavor of monos to them, and fibrations - of epis.

You see, I cannot even formulate this rigorously, yet I hope you agree that what I now described contains undeniable truth.

So what is this truth? Does this phenomenon have any explanation? Living with this puzzle for many years, the only consideration that I've been able to come up with is this: take one step from sets to categories. In sets, monos are just inclusions. In categories, the "correct" notion of mono starts to involve some amount of "epiness", since "good" monos in categories are **full** and faithful functors, and fullness involves some surjectivity condition. I don't even know whether there is some sort of such "first-step-consideration" from the opposite end, relating "betterness" of epis with some sort of additional requirements which have to do with monomorphy.

I repeat - this is a strange question: although it is full of most vague handwaving, I hope you agree that it touches on something very rigorous, which I just fail to capture.