Can anyone show me concrete examples of unital MF algebra and non-unital MF algebra resepctively? Thanks!

Let $R$ be a U.F.D. and \begin{align*} T \,\colon= R[[X_1,\ldots,X_d]]. \end{align*} Suppose that we have $d$ elements $f_1,\ldots,f_d \in T$ and let us consider an ideal $J$ of $T$ such that $(f_1,\ldots,f_d) \subset J$ and the following three conditions$\colon$ \begin{align*} & 1. \quad \overline{f_i} \,\colon \overset{{\mathrm{def}}}{=} f_i~{\mathrm{mod}}(X_1,\ldots,X_d)~{\mathrm{is~ a~ prime~ element~ of~}} R ~{\mathrm{for~each}}~ 1 \leq i \leq d\\ & 2. \quad T/(f_1,\ldots,f_d) \phantom{a} {\mathrm{is}}~not~{\mathrm{finite~over}}~ R \\ & 3. \quad T/J \phantom{a} {\mathrm{is~finite~over}}~R. \end{align*}

Q. Does the following equality hold$\colon$$\phantom{A}$${\mathrm{ht}}(J) > d$$\,$?I'm studying the structural stability of vector fields and I'm interested in learning about this phenomenon on compact $2$-manifolds with boundary.

Let $M^2$ be a compact connected 2-manifold and $\mathfrak{B}$ the space of vector fields with the $\mathcal{C}^1$ topology. The paper "M. M. Peixoto - Structural stability on two-dimensional manifolds" defines structural stability as

(By an $\varepsilon$-homeomorphism of $M^2$ onto itself we understand a homeomorphism which moves each point by less than $\varepsilon$, using a Riemannian metric. However, I'm not interested in this condition, we can change the word $\varepsilon$- homomorphism by just homeomorphism)

Using this definition Peixoto was able to demonstrate the following theorem:

**My Question:** I would like to know if someone knows a result, like the above, about
Structural Stability in a compact $2$-manifold with boundary.

Searching online I found the paper Clark Robison - Structural stability on manifolds with boundary, however, the author defines a strange topology using the flows of the vector field, and I don't know if it is the same topology as in the definition 1.

It is also worth to mention that on the paper "M.C. Peixoto and M.M. Peixoto - Structural Stability in the Plane with Enlarged Boundary Condition", the following result was proved. Let $G\subset \mathbb{R}^2$ be a compact region, such that the boundary $L$ of $G$ is a $\mathcal{C}^1$ simple curve, then $X$ (a vector field in $G$) is structurally stable $\Leftrightarrow$ $X$ satisfies conditions $A$ and $B$.

Can anyone help me?

I have asked Question this reptaly but not use knowone one had not gave me satsifed answer I have born on 18.03.2000 on 7.30 pm my question is How to clacaulte my age? I have read one website in that website in that they have put who were born on march 18.3.2000 their age will be 16 going ti complete 17will start this is true in information please answer my question

This is related to the proposition 5.1. of Mumford's GIT. It states that:

There is a unique subscheme $H$ in the Hilbert scheme $Hilb_{\mathbb{P}^n}^{P(x)}$ such that, for any morphism $f : S \to Hilb$, $f$ factors through $H$ iff:

i) the induced subscheme $\Gamma$ in $\mathbb{P}^n_S$ is a curve of genus $g$ over $S$,

ii) the invertible sheaf $\mathscr{O}_{\mathbb{P}}(1)|_{\Gamma}$ is isomorphic to $\Omega_{\Gamma/S}^{\otimes \nu} \otimes \pi^* \mathscr{L}$ for some invertible sheaf $\mathscr{L}$ on $S$,

iii) for every geometric point $s$ of $S$, the fibre $\Gamma_s$ spans $\mathbb{P}_s^n$.

(Now the induced scheme is $\Gamma = W \times_{Hilb} S$, for the universal object $W$ of the functor $Hilb(-)$, and $\nu \ge 3$, $P(x) = (2\nu x - 1)(g-1)$, $n = P(1) -1 $, $\pi : \Gamma \to S$. And a curve of genus $g$ is a proper smooth morphism whose geometric fibres are connected curves and of genus $g$.)

And in Deligne-Mumford's "the irreducibility of the space of curves of given genus", the authors says:

Following standard arguments (the proof of 5.1. of GIT), it is easy to prove that there is a subscheme $H \subset Hilb_{\mathbb{P}^{5g - 6}}^{P(x)}$ of "all" tri-canonically embedded stable curves. To be precise, there is an isomorphism of functors: $$Hom(S, H) \cong \{ \text{stable curve } \pi : C \to S, \text{plus isomorphisms } \mathbb{P}(\pi_*(\omega_{C/S}^{\otimes 3})) \cong \mathbb{P}_S^{5g-6} \}.$$

This is the analogy of the first statement for $\nu = 3$.

But for me, these are a priori slightly different. I think that, if (ii) and (iii) is equivalent to $\mathbb{P}(\pi_*(\Omega_{\Gamma/S}^{\otimes \nu})) \cong \mathbb{P}_S^{n}$, these two statements are the same. But I can't show this. So my question is:

Let $f : S \to Hilb_{\mathbb{P^n}}^{P(x)}$ be a morphism and $\pi : \Gamma \to S$ the corresponding scheme. Assume this $\pi$ is an $S$-curve. Then (ii) and (iii) of 5.1. iff $\mathbb{P}(\pi_*(\Omega_{\Gamma/S}^{\otimes \nu})) \cong \mathbb{P}_S^{n}$ over $S$?

Here is what I tried so far: Let $i : \Gamma \to \mathbb{P}$ be the immersion, $p : \mathbb{P} \to S$ the projection. Then (iii) is quivalent to the canonical map $p_*\mathscr{O}_{\mathbb{P}}(1) \to \pi_* i^* \mathscr{O}_{\mathbb{P}}(1)$ is an isomorphism. So for "only if", it's sufficient to show that $\mathbb{P}(\pi_*(\Omega_{\Gamma/S}^{\otimes \nu} \otimes \pi^* \mathscr{L})) \cong \mathbb{P}(\pi_*(\Omega_{\Gamma/S}^{\otimes \nu})) $. And for the converse, I showed that (iii) holds.

Thank you very much!

The Church encoding of the sum type $A + B$ goes like that:

$$\prod_{X:\mathsf{Set}_{\mathcal{U}}} (A\to X)\to (B\to X) \to X$$

But it lacks an induction principle.

According to this blog article by Mike Shulman, one can recover the induction principle by extending the above encoding with a naturality condition:

$ \sum_{\alpha:\prod_{X:\mathsf{Set}_{\mathcal{U}}} (A\to X)\to (B\to X) \to X} \prod_{X,Y:\mathsf{Set}_{\mathcal{U}}} \prod_{f:X\to Y} \prod_{h:A\to X} \prod_{k:B\to X} f(\alpha_X(h,k)) = \alpha_Y(f\circ h, f\circ k) $

In Coq syntax (with the option -impredicative-set), it gives:

Variables A B : Set. Record sum : Set := { type :> forall (X : Set), (A -> X) -> (B -> X) -> X; natural : forall (X Y : Set) (f : X -> Y) (inl : A -> X) (inr : B -> X), f (sum_carrier X inl inr) = sum_carrier Y (fun a => f (inl a)) (fun b => f (inr b)) }.One can the define the constructors inl and inr:

Definition inl (a : A) : sum := {| type := fun X l r => l a; natural := ltac:(reflexivity) |}. Definition inr (b : B) : sum := {| type := fun X l r => r b; natural := ltac:(reflexivity) |}.Then the induction principle goes like that:

Definition sum_ind (P : sum -> Set) (Hleft : forall a, P (inl a)) (Hright : forall b, P (inr b)) (s : sum) : P s.However I cannot figure out how to use the naturality of s to conclude P s, even by daring using the infamous option -type-in-type allowing me to instantiate the morphism f with P.

How does the added naturality condition allow for proving the induction principle?

This Hausdorff dimension of the graph of an increasing function shows that:

Let $f$ be a continuous, strictly increasing function from $[0,1]$ to itself with $f(0)=0, f(1)=1$. Then $dim_H \; G = 1$ where $G$ is the graph of $f$.

I have at hand the Casino function, described as follows in Massopoust's *Interpolation and Approximation with Splines and Fractals*:

Let $X = [0,1] \times \mathbb{R}$, $N = 4$ and $Y = \{(x_v,y_v):0 = x_0 < \ldots x_N = 1, 0 = y_0 < \ldots < y_N = 1\}$. Define an IFS by $f_i(x,y) = \begin{pmatrix} x_i-x_{i-1} & 0 \\ 0 & y_i - y_{i-1} \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} x_{i-1} \\ y_{i-1} \end{pmatrix} $ for $i = 1, \ldots, N$.

The associated RB operator $T$ is contractive and its unique fixed point is called a Casino function $c:[0,1] \to [0,1]$. These functions are monotone increasing and therfore $dim_H \; graph(c) = \dim_B \; graph(c) = 1$.

I was wondering how can I show that $dim_B \; graph(c) = 1$ and whether there is a general argument establishing:

Let $f$ be a continuous, strictly increasing function from $[0,1]$ to itself with $f(0)=0, f(1)=1$. Then $dim_B \; G = 1$ where $G$ is the graph of $f$.

I don't find an argument stablishing $dim_B \; G \le 1$.

We make subsequent throws of a fake cubic cube for which the probability of falling out six is 1/6 - epsilon, the probability of falling out of one is 1/6 + epsilon and the others eyes drop out with the same probability 1/6. Provide a consistent and unbalanced estimator epsilon parameter and calculate its variance. I need help i start my trip with this and it is so hard for me :/

Let $f: \mathbb{R}^m\rightarrow \mathbb{R}^{m-k}$ be a Lipschitz map.

Can we get a uniform estimate on the Hausdorff dimension of the boundaries of fibres of $f$? I.e. do we have an upper bound for $$ \sup_{y\in \mathbb{R}^{n-k}} \dim_H(\partial f^{-1}(\{y\})) ?$$

Theorem 2.5 in [1] tells us, that for almost every $y\in \mathbb{R}^{n-k}$ we have that $\dim_H(f^{-1}(y))\leq k$. This tells us $$ \text{essup}_{y\in \mathbb{R}^{n-k}} \dim_H(\partial f^{-1}(\{y\})) \leq k.$$ Can we pass to the supremum? And are there even better bounds? I mean, I used $\partial f^{-1}(\{y\})\subseteq f^{-1}(\{y\})$ as $f$ is continuous and the monotonicity of the Hausdorff dimension, but I guess that one can do better than this.

**[1] G. Alberti, S. Bianchini, G. Crippa,** Structure of level sets and Sard-type properties of Lipschitz maps: results and counterexamples.
*Ann. Sc. Norm. Super. Pisa Cl. Sci.* (5) 12 (2013), no. 4, 863–902.

Machin-like formulas for $\pi$ have the following general form: $$c_{0} \frac{\pi}{4}=\sum_{n=1}^{N} c_{n} \arctan \frac{a_{n}}{b_{n}}$$ Recently browsing through this question here, I really became very curious in finding out Machin Type formula's for logarithms,of course using the inverse hyperbolic tangent function.

Now, its relatively easy and well known to find out Machin Type formulas for $\pi$ using complex numbers the original equation can be written as $$(1+i)^{c_{0}}=\prod_{n=1}^{N}\left(b_{n}+a_{n} i\right)^{c_{n}}$$ and then suitable $a_n$ and $b_n$ could be found using algorithms like branch and bound search, with arbitrary small $\frac{a_n}{b_n}$ for efficient computation.

Now what I am really curious about is the find whether there are methods for obtaining such a series for $\log k$ which could be in the form of $$c_{0} \log k=\sum_{n=1}^{N} c_{n} \tanh ^{-1}\frac{a_{n}}{b_{n}}$$ For arbitary $k$?

Let $f$ is Borel function from $x_1, ..., x_n, ...$ and $\forall i \in \mathbb N : f(x_1, ..., x_{i-1}, x_i, x_{i+1}, x_{i+2}, ...)=f(x_1, ..., x_{i-1}, x_{i+1}, x_i, x_{i+2}, ...)$, $X_1, ..., X_n, ...$ are mutually independent random variables with the same distribution. How to show, that random variable $f(X_1, ..., X_n, ...)$ with probability 1 is a constant?

Suppose $k_{\varepsilon}$ is a family of solution which are uniformly bounded \begin{equation} \ \ \left\{\begin{aligned} -\Delta k_{\varepsilon} & = g _{\varepsilon} \hbox{ in } B _{\varepsilon} \\ k _{\varepsilon} & = 0 \hbox{ in } \partial B _{\varepsilon} \\ k _{\varepsilon}(0) & = 1 \end{aligned} \right. \end{equation} where $B_{\varepsilon}$ is a ball which increases to $\mathbb R^N$ as $\varepsilon\to 0.$ Here $g _{\varepsilon}$ is a uniform bounded sequence and $g _{\varepsilon} \to 0$ and $k _{\varepsilon} \to k$ locally uniformly in $C^2(\mathbb R^N)$. Then $k$ satisfies $$\ \ \left\{\begin{aligned} -\Delta k & = 0 \hbox{ in } \mathbb R^N \\ k & \in L^{\infty}, k(0)=1 . \end{aligned} \right.$$ But by the Liouville theorem, $k\equiv 1$. Is it possible to arrive at a contradiction.

Question: suppose that $H_{n_1}, H_{n_2}, H_{n_3} \in L^{2}(\mathbb{S}^2)$ are Spherical Harmonics of degrees $n_j$ $(j = 1, 2, 3)$ with $n_1 > n_2 + n_ 3$. Then, it is true that $$ \int_{\mathbb{S}^2} H_{n_1} H_{n_2} H_{n_3} dS = 0 \; ? $$

Obs.: My failed attempt to solve this question was to show that $H_{n_2} H_{n_3}$ is a sum of Spherical Harmonics $H_m$ of degrees at most $m < n_1$ and to use the orthogonality property in $L^2({S}^2)$.

For every $\epsilon \in [0, 1)$ and $n$ in $\mathbb N$, let $g(n, \epsilon)$ be a function $\mathbb R \to \mathbb R$.

Suppose for all $\epsilon \in [0, 1)$, $g(n, \epsilon)$ is $C^n$ smooth and that for each $n$, $g(n, \epsilon)$ converges uniformly in $C^n$ norm to $0$ as $e \to 1-$. Suppose further that for every $\epsilon$, $\lim_{N \to \infty}$ $ \sum_{n = 1}^N g(n, \epsilon) \epsilon^n$ converges uniformly.

If $F := \lim_{\epsilon \to 1-} \sum_{n = 1}^\infty g(n, \epsilon) \epsilon^n$ exists in the uniform sense, is $F$ necessarily $C^\infty$ smooth?

The dual Steenrod algebra ($p=2$) has generators $\xi_n$ and these have conjugates that are often labeled $\zeta_n$. I am curious about the left and right actions of the Steenrod algebra on its dual, and in particular, what the total square is. I have seen in papers that $(\xi_n)Sq = \xi_n + \xi_{n-1}$ and $Sq(\xi_n) = \xi_n + \xi_{n-1}^2$ [1]. On the other hand, I have seen that $(\zeta_n)Sq = \zeta_n + \zeta_{n-1}^2 + \dots + \zeta_1^{2^{n-1}} + 1$ [2]. I can't find a reference anywhere for the left total square on $\zeta_n$. I am not sure how to prove these actions, although it seems to me that it should follow from fairly elementary Kronecker product arithmetic along with duality knowledge.

I am interested in either a reference for the left total square, or a way to prove it.

[1] See, for example, Mahowald -- bo-resolutions, page 369.

[2] Bruner, May, McClure, Steinberger -- $H_\infty$ Ring Spectra and their Applications, page 78. (There is a typo: 1 should be $i$.)

My question maybe a bit stupid, but I can't quite grasp what the Weil restriction actually does... In particular:

Given a projective variety $V$ defined over $L$ algebraically closed, of characteristic 0, given a finite extension $K$ of $\mathbb{Q}$, is the Weil restriction of $V$ to $K$ a model of $V$ over K?

I am looking for examples of constructions for transfinite towers $(X_{\alpha})_{\alpha}$ generated by structures $X$ where the problem of determining whether the tower $(X_{\alpha})_{\alpha}$ stops growing is a non-trivial problem or the problem of determining the ordinal in which $X_{\alpha}$ stops growing is a non-trivial problem.

In particular, I want the tower $(X_{\alpha})_{\alpha}$ to be generated by the following construction. Suppose that for each object $X$, there is a new object $C(X)$ and a morphism $e:X\rightarrow C(X)$. Then define the tower generated by $X$ by letting $X_{0}=X$, $X_{\alpha+1}=C(X_{\alpha})$ and $X_{\gamma}=\varinjlim_{\alpha<\lambda}X_{\alpha}$ for limit ordinals $\gamma$ where the direct limit is taken in the category that $X$ belongs to.

I want all of the objects $X$ and each $X_{\alpha}$ to be set sized.

**Non-Example:** The hierarchy of sets $(V_{\alpha}[X])_{\alpha}$ where
$V_{0}[X]=X,V_{\alpha+1}[X]=P(V_{\alpha}[X])$ and $V_{\gamma}[X]=\bigcup_{\alpha<\gamma}V_{\alpha}[X]$ does not count as an example of what I am looking for since the tower $(V_{\alpha}[X])_{\alpha}$ never stops growing and therefore whether $(V_{\alpha}[X])_{\alpha}$ terminates is now a trivial mathematics problem.

**Example 1:** Suppose that $G$ is a group. Let $G_{0}=G$, and let
$G_{\alpha+1}=\mathrm{Aut}(G_{\alpha})$ and let $G_{\gamma}=\varinjlim_{\alpha<\gamma}G_{\alpha}$. The transition mapping from $G_{\alpha}$ to $G_{\alpha+1}$ is the mapping $e$ where $e(g)(h)=ghg^{-1}$. Then $(G_{\alpha})_{\alpha}$ is the automorphism group tower generated by $G$. The automorphism group tower always terminates. In this case, the mapping $G\mapsto G_{\alpha}$ is not functorial.

**Example 2:** Frames are the objects that people study in point-free topology. If $L$ is a frame, then let $\mathfrak{C}(L)$ denote the lattice of congruences of the frame $L$. Then $\mathfrak{C}(L)$ is always a frame. Define a mapping $e:L\rightarrow\mathfrak{C}(L)$ by letting $(x,y)\in e(a)$ if and only if $x\vee a=y\vee a$. Then the function $e$ is a frame homomorphism.

There are frames $L$ where the congruence tower generated by $L$ never terminates. However, for ordinals $\alpha$, it is a difficult open problem to determine whether there is a frame $L$ where $e:L_{\beta}\rightarrow\mathfrak{C}(L_{\beta})$ is a surjection if and only if $\beta\geq\alpha$ since in all known examples, the congruence tower either terminates before the fourth step or so or it never terminates.

If $L$ is a frame and $e:L\rightarrow\mathfrak{C}(L)$ is an isomorphism, then $L$ is a complete Boolean algebra.

Unlike Example 1, Example 2 is functorial.

** I simplified the question: **

On bounded domains, the maximum principle implies that the solution to the heat equation is (strictly) positive, if the initial and boundary data is positive.

I would like to know whether this result is also true if applied on an unbounded domain:

Let $a>0$ and $g$ a continuous function with $g(a,t)>0$ for all $t \in [0,T]$ and $u(x,0) = u_0(x) >0$ for all $x \in (a,\infty)$ with $u_0 \in C^{\infty} \cap L^{\infty}.$

Let $u(x,t)$ be the solution to the heat equation $\left(\partial_t - \partial_{x}^2 \right)u=0$ with the above boundary data.

Does there exist then a version of the maximum principle saying that $u(x,t)>0$ for $(x,t) \in [a,\infty) \times [0,T]$?

Oftentimes open problems will have some evidence which leads to a prevailing opinion that a certain proposition, $P$, is true. However, more evidence is discovered, which might lead to a consensus that $\neg P$ is true. In both cases the evidence is not simply a "gut" feeling but is grounded in some heuristic justification.

Some examples that come to mind:

Because many decision problems, such as graph non-isomorphism, have nice

*probabilistic*protocols, i.e. they are in $\mathsf{AM}$, but are not known to have certificates in $\mathsf{NP}$, a reasonable conjecture was that $\mathsf{NP}\subset\mathsf{AM}$. However, based on the conjectured existence of strong-enough*pseudorandom*number generators, a reasonable statement nowadays is that $\mathsf{NP}=\mathsf{AM}$, etc.I learned from Andrew Booker that opinions of the number of solutions of $x^3+y^3+z^3=k$ with $(x,y,z)\in \mathbb{Z}^3$ have varied, especially after some heuristics from Heath-Brown. It is reasonable to state that

**most**$k$ have an**infinite**number of solutions.Numerical evidence suggests that for all $x$, $y$, we have $\pi(x+y)\leq \pi(x)+\pi(y)$. This is commonly known as the "second Hardy-Littlewood Conjecture". See also this MSF question. However, a 1974 paper showed that this conjecture is incompatible with the other, more likely first conjecture of Hardy and Littlewood.

- Number theory may also be littered with other such examples.

I'm interested if it has ever happened whether the process has ever *repeated* itself. That is:

Have there ever been situations wherein it is reasonable to suppose $P$, then, after some heuristic analysis, it is reasonable to supposed $\neg P$, then, after *further* consideration, it is reasonable to suppose $P$?

I have read that Cantor thought the Continuum Hypothesis is true, then he thought it was false, then he gave up.

Given $k$ girls, they are given $kn$ balls so that each girl has $n$ balls. Balls are coloured with $n$ colours so that there are $k$ balls of each colour. Two girls may exchange the balls (1 ball for 1 ball, so each girl still has $n$ balls), but no ball may participate in more than one exchange. They want to achieve the situation when each girl has balls of all $n$ colours. Is it always possible?

On other language. Given is a bipartite multigraph $G=(V_1,V_2,E)$, $|V_1|=k$, $|V_2|=n$, each vertex in $V_1$ has degree $n$ and each vertex in $V_2$ has degree $k$. We may replace two edges $ab,cd$ ($a,c\in V_1, b,d \in V_2$) to $ad,cb$, but new edges can not be used in exchanges anymore. Is it possible to get a usual $K_{k,n}$ without multiplicities?

If yes, this implies the positive answer to this question, which I find quite interesting itself.

I think I may prove it when $\min(n,k)\leqslant 3$, but already for $3$ there are many cases to consider.