Show that the set of all polynomials of degree at most m is a (m + 1) -dimensional linear manifold in C[a,b].

I asked this question in two parts on Math Exchange so I did not get an answer. Link

From the material/books/etc. that I have read - an Ito diffusion always has a fixed initial condition

$$ dX(t)= \mu dt + \sigma dW(t) \hspace{10mm} (1) \\ X(0)=x_0 $$

Does it make sense to model a diffusion with a normally distributed initial condition as an example

$$ dX(t)= \mu dt + \sigma dW(t) \hspace{10mm} (2)\\ X(0)= N(0,1) $$

Additional information:

Monte Carlo in probability theory:

Let $g(x)$ be a function of a random variable $X$, then $$ E[g(x)]= \intop\nolimits_{-\infty }^{\infty} g(x)f(x)dx $$

Where $f(x)$ is the probability distribution of $X$.

Now, assume $g(x,t)$ is the solution of the Faynman-Kac formula

$$ \frac{\partial g(x,t)}{\partial t} = -v(x)g(x,t) + \mu \frac{\partial g(x,t)}{\partial t} + \frac{ \sigma^2 }{2} \frac{\partial^2 g(x,t) }{\partial x^2} \hspace{10mm} (3) $$ Example graph: (See other post for picture Link)

- The blue line (or any parallel line to it) in the graph gives us the conditional solution $[g(x,t)|X(0)=x]$ - specifically $X(0)=0$ in this case.

Ok, great now imagine that I am modeling a system of (100 particles or items).

- Each particle follows the diffusion model defined for $X(t)$ in (1).
- Also, suppose that I have solved the Faynman-Kac for the function $g(x,t)$ but I do not know exactly the initial condition of each particle.
- But, I do know that their initial condition follows a distribution i.e.: $f(x_0) \approx N(0,1)$.

So, if I want to find an average solution $E[g(x,t)]$, I suppose i can write the following:

$$ E[g(x,t)]= \intop\nolimits_{-\infty }^{\infty} g(x,t|X(0)=x)f(x_0)dx \approx \frac{1}{N} \sum_{i=i}^{N}g(x^i,t) \hspace{10mm} (4) $$

Thus, if I were to simulate $10,000$ diffusion paths $(N=10000)$ with $X(0)=N(0,1)$ - would approximate the integral.

So, in this context how wrong am I?

Show that C[a,b] is infinite-dimensional. Consider the sequence of functions 1, t, t^2, ..., t^n,... and show that 1, t, t^2, ..., t^n are linearly independent for any natural n. Show that C^k[a,b] also infinite-dimensional.

Let $A$ be a UFD, $M$ a free module, and $N$ a finitely generated rank 1 submodule. I have a relatively quick proof (about half a page) that if $A^{-1}N \cap M = N$, then $N$ is also free. This is done by induction, reducing the number of generators to 1.

However, I feel like my proof is superficial, and that there should be a "deeper" proof in some sense. So my question is:

Is there some deep reason that the above statement should be obvious?

As we know,if $u$ and $v$ are tempered distribution, and $\dot{\Delta}$ is the Little-wood Paley operator,so $u=\sum_{j\in \mathbb{Z}} \dot{\Delta}_ju$ , $v=\sum_{j\in \mathbb{Z}} \dot{\Delta}_jv$ , so at least formally , $uv=\sum{j,j^{\prime}} \dot{\Delta}_ju\dot{\Delta}_{j^\prime}v$ ,and we defined $T_uv=\sum_{j\in \mathbb{Z}}\dot{S}_{j-1}u\dot{\Delta}_jv$ ,$R(u,v)=\sum_{|k-j|\leq 1}\dot{\Delta}_ku\dot{\Delta}_jv$ , the following is the Bony decomposition$$uv=T_uv+T_vu+R(u,v)$$ , my question is if $u$ and $v$ is the function ,why the the Bony decomposition $$uv=T_uv+T_vu+R(u,v)$$ satisfy pointwise? I think the Bony decomposition is just one of the order of the product of two series.

If det(A) = 12, then det($A^3$) = det(A)$^3$ = 12$^3$.

What happens when you add a scalar, though?

det(A) = 12, det(2*$A^3$) = ?

I'm getting det(2*$A^3$) = det$(A)^3$ * 2$^3$ = 12$^3$ * 2$^3$

but the answer is apparently 2$^4$ * 12$^3$

I am currently reading a paper which, somewhat indirectly, asserts the following result:

**Lemma**: Let $\Delta \subset \mathbb{R}^d$ denote the simplex $\{(x_1,\ldots,x_d):\sum_{i=1}^d x_i=1\}$, let $A_1,\ldots,A_N$ be positive $d \times d$ matrices and let $\overline{A}_\ell \colon \Delta \to \Delta$ be the projective transformation induced by $\overline{A}_\ell$, that is if $A=[a_{ij}]_{i,j=1}^d$ then
$$\overline{A}\left((x_i)_{i=1}^d\right):=\left(\frac{\sum_{k=1}^d a_{ik}x_k}{\sum_{j,k=1}^d a_{jk}x_k}\right)_{i=1}^d.$$
Then (up to identifying $\Delta$ with a subset of $\mathbb{R}^{d-1}$ by an affine change of co-ordinates) there exists a complex neighbourhood $D\subset \mathbb{C}^{d-1}$ of $\Delta$ such that each $\overline{A}_\ell$ extends to a holomorphic map $\overline{A}_\ell \colon D \to D$ with the property that $\overline{A}_\ell D$ is precompact in $D$.

The case $d=2$ is given in the paper as an example: in a natural way we can identify $\Delta$ with $[0,1]$ and each induced map $\overline{A}_\ell$ with a linear fractional transformation. If we let $D$ be a complex disc centred at $1/2$ with diameter slightly larger than $1$ then each $\overline{A}_i$ maps the interval $[0,1]$ to a proper subinterval and therefore maps $D$ to a disc which is centred somewhere in $(0,1)$, has diameter smaller than one, and is precompact in $D$. However, I do not understand how to generalise this argument to higher dimensions. Is it perhaps more clear to someone better-versed in multivariate complex analysis?

The paper goes on to assert this result for the more general situation where $\mathcal{K}\subseteq \mathbb{R}^d$ is a cone which is mapped strictly inside itself by each $A_\ell$ and where $\Delta$ is the intersection of $\mathcal{K}$ with a suitable hyperplane. This seems even more challenging to me, since $\Delta$ is then not merely a simplex but an arbitrary compact convex set with interior. How might the lemma be proved in this case?

The theory of $q$-characters for quantum affine algebras are studied in The q-characters of representations of quantum affine algebras and deformations of W-algebras, Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras, and Spectra of Tensor Products of Finite Dimensional Representations of Yangians.

I think that the following results are true. But I didn't find their proofs in the above references. Do the results follow from the general theory of $q$-characters? Thank you very much.

Let $m_1, \ldots, m_k$ be dominant monomials in $\chi_q(M_1) \chi_q(M_2)$, where $M_1, M_2$ are two simple $U_q(\hat{g})$-modules and $\chi_q(M)$ is the $q$-character of $M$. Then we have the following two results.

Suppose that $m_i$ is not contained in any $\chi_q(L(m_j))$, $j \neq i$. Then $L(m_i)$ is an irreducible subfactor of $M_1 \otimes M_2$. Here $L(m_i)$ is the simple $U_q(\hat{g})$-module with highest $l$-weight $m_i$.

Suppose that $L(m_i)$ is a subfactor of $M_1 \otimes M_2$. Suppose that $m_j$ appears in $\chi_q(M_1) \chi_q(M_2)$ with multiplicity $p$ and $m_j$ appears in $\chi_q(L(m_i))$ with multiplicity $p$. Then $m_j$ is not a subfactor of $M_1 \otimes M_2$.

Suppose that we have a Markov process $\{Z_t\}_{t=0}^\infty$, where $Z_t \geq 0$ for any $t$. Assume that, conditioning on $Z_t = z_t$, we have $ \mathbb{E}\{Z_{t+1}|Z_t = z_t\} \leq \kappa z_t^2 $. Here $\kappa > 0$ is a constant.

Question: Conditioning on that the realization of $Z_0$ is sufficiently small, can we prove that $\mathbb{E}\{Z_{t}\} \leq c\exp(-t^2)$, where $c$ is an constant, or something like $\mathbb{E}\{Z_{t}^2\} \leq c\exp(-t^2)$? If not, what additional conditions on $\kappa$ or the value of $Z_0$ do we need? Or is there any counter example for this claim?

As Philip Scott says about Densi Higgs:

In category theory, he wrote an influential and beautiful long paper, "A category approach to Boolean valued set theory", which initiated many early students in topos theory in the 1970s to Omega-valued sets.

I wonder to know if there is a way to get access to the following paper by Higgs:

``**A category approach to Boolean valued set theory**''

**Remarks.** I am aware of some similar works, for example A category-theoretic approach to boolean-valued models of set theory or the book Sheaves in geometry and logic by Mac Lane and Moerdijk.

Let $f$ be a compactly supported function in $\Omega \subset \mathbb{R}^3$ and

$\Delta u=f$ in $\Omega$

such that $D^{\alpha}u=0$ on $\partial \Omega$ for every multi-index $\alpha$ with $|\alpha| \geq 0$. Is $f$ necessarily zero?

For varieties $X,Y$ over an algebraically closed field, and a surjective morphism $f:X\rightarrow Y$, $\dim f^{-1}(y)\geq\dim X-\dim Y$ for all closed $y\in Y$, and $\dim f^{-1}(y)=\dim X-\dim Y$ for all closed $y$ in a nonempty open subset of $Y$. If the requirement that $X$ and $Y$ are of finite type is dropped, and we just require them to be integral, separated, Noetherian schemes over an algebraically closed field, there are two ways I can see to interpret this claim:

(1) $\dim f^{-1}(y)\#\dim Y\geq\dim X$ for all closed $y\in Y$, and $\dim f^{-1}(y)\#\dim Y=\dim X$ for all closed $y$ in a nonempty open subset of $Y$, where $\dim$ refers to ordinal Krull dimension, and $\#$ is natural sum ($\alpha\#\beta$ is the largest ordinal that can be reached by interleaving $\alpha$ and $\beta$).

(2) $\text{codim}(f^{-1}(y)\text{ in }X)\geq\dim Y$ for all closed $y\in Y$, and $\text{codim}(f^{-1}(y)\text{ in }X)=\dim Y$ for all closed $y$ in a nonempty open subset of $Y$, where $\text{codim}(Z\text{ in }X)$ refers to the supremum of order types of chains of irreducible closed subsets of $X$ containing $Z$.

These two claims are not clearly equivalent in the infinite-dimensional setting. Are either or both of them true?

Consider the $n$-dimensional unit ball $B$ centered at the origin and a hyperplane $H$ that intersects $B$. Suppose that there is a simplex $S$ inscribed in $B\cap H$, so that the vertices of $S$ lie in the boundary of $B\cap H$. Let $|S|$ denote the $(n-1)$ volume of $S$. Is there a formula for the volume of the region of the (smaller) cap that lies above $S$, in terms of $|S|$? e.g. volume of $S$ times an integral, etc.

The picture looks like this in three dimensions, except instead of a facet on the top of the object, we have a spherical polytope, and instead of going directly up, we go radially out to the cap from the origin (although straight up suffices too for the purposes of my question).: https://math.stackexchange.com/questions/435060/volume-of-n-dimensional-solid-w-n-1-dimensional-simplex-as-a-base

If one cannot come up with such a formula, can one obtain a lower bound for the volume of this region, again in the form "$|S|\times$ something", that does not involve the factor $1/n$? In the link below, someone asked a similar question about inscribing an $n$-simplex in a cap, but the volume of this set involves a $1/n$ factor, whereas, e.g. that of a prism does not.

Thank you in advance.

Related: Maximal volume of a simplex inscribed in a spherical cap

I have quite a practical question motivated by physics.

Consider the Riccati equation whose solution gives a quantum-mechanical (QM) analogue of the classical momentum: $$ (p(x))^2 + \dfrac{\hbar}{i}p'(x)= 2 (E-V(x)) \quad. $$

Clearly, in the $\hbar\to0$ limit one obtains the definition of the classical momentum: $$ p_{c}(x)= \sqrt{ E-V(x)}\quad. $$ It's easy to determine what's the Riemann surface on which $p_c(x)$ is defined $-$ a two-dimensional oriented manifold with, typically, a finite number of punctured points (let's limit ourselves with polynomial potentials).

Now, the standard approach in QM is to consider an expansion of $p(x)$ in powers of $\hbar$, which leads to the (Generalised) Bohr-Sommerfeld qantisation: $$ \begin{gathered} p(x) = \sum \limits_{k=0}^\infty \left(\dfrac{\hbar}{i}\right)^k p_k(x)\quad,\\ p_0(x) \equiv p_c(x) \quad. \end{gathered} $$ Here $p_k(x)$ are, of course, assumed to be $\hbar$-independent.

It is now a simple exercise to show that all the $p_k(x)$ live on the same Riemann surface as $p_c(x)$: they all are obtained from $p_c(x)$ recursively by means of algebraic operations and taking derivatives $-$ none of these can drag us out of the Riemann surface.

This result suggests me to make a way stronger statement: namely, that the solutions of the original Riccati equation have to live on the same Riemann surface as $p_c(x)$. Basically, I'm saying that $\hbar\to0$ limit does not change the Riemann surface (well, that's a tricky limit since it turns a differential equation into a trivial equality).

It may be temptingly to say that the last equation (the infinite sum) is exactly what I need. However, those who are familiar with things like asymptotic series know that it's not like that. The reason for this is that $p(x)$ may depend on $\hbar$ in a non-polynomial way, like $\exp(-1/\hbar)$ or $\log (1/\hbar)$ (non-perturbatively, in physical jargon).

So my question is:

**Given the Riccati equation (the top one in the question), is it possible to prove that its solutions live on the same Riemann surface as the function $p_c(x)$ defined by the $\hbar\to 0$ limit of the equation?**

I would prefer to get the answer which will not rely on employing any (trans-)series expansions of $p(x)$, but would rather be based on a global analysis of the differential equation's Riemann surface (or smth like that).

P.S. If I'm asking something trivial, any references to the relevant textbooks are greatly appreciated.

**UPDATE**

I'm still very much interested to hear any useful comments/references on the topic, however I've just realised that the answer to my question is **negative**. Different solutions may live on different Riemann surfaces. Unfortunately, this conclusion relies not on strict mathematical statements, but rather on my knowledge from physics. The quantum momentum $p(x)$ has a first-order pole wherever the wave function $\psi(x)$ has a zero:
$$
p(x) =\dfrac{\hbar}{i} \dfrac{1}{\psi(x)} \dfrac{\operatorname{d} \psi(x)}{\operatorname{d} x}
$$
These wave function may have arbitrary number of poles on the branch cut of the classical momentum. In the classical limit this sequence of first-order poles coalesces into a branch cut, just like $\int \dfrac{\operatorname{d}x}{x}=\log x$. Which tells us that for non-zero $\hbar$ the Riemann surface is very different from the $\hbar=0$ case.

Is it possible to give an example of $n$ dimensional manifold with the property that the tangent bundle $TM$ cannot be expressed as Whitney sum of two subbundles? It is certain true for two sphere; it is certainly not true for three dimensional manifold since every three manifold is parallelizable. You can always split $TM$ as $\gamma \oplus Q$ where $\gamma$ is one dimensional if you can find non vanishing vector field (i.e. where the Euler class is zero which is the case for odd dimension) but what about examples with $n$ even, larger than $2$?

Consider an $n\times n$ matrix $M_n$ where the sequence $1,2,3,\dots,n^2$ forms a clock-wise spiral, in that given order. For example, $$M_4=\begin{bmatrix} 1&2&3&4\\ 12&13&14&5\\ 11&16&15&6 \\ 10&9&8&7 \end{bmatrix} \qquad \text{and} \qquad M_5=\begin{bmatrix} 1&2&3&4&5\\ 16&17&18&19&6 \\ 15&24&25&20&7 \\ 14&23&22&21&8 \\ 13&12&11&10&9 \end{bmatrix}.$$

**Question.** What are the diagonal entries in the Smith normal form of the matrix $M_n$?

I'm looking for software that can compute symmetric powers of medium-size square (say rational, 100 by 100) matrices, and ideally can do so efficiently if the matrix is sparse enough. I haven't found any function for symmetric power in Sage or sympy, and a google search hasn't turned up anything. (An aside: Do people studying matrix algorithms have a different name for symmetric power, or they just aren't interested?) Any suggestions?

Edit: Macaulay2 only does symmetric powers of one-rowed matrices, it seems. Singular seems hopelessly inefficient on medium-sized sparse matrices (20 by 20).

In the chapter "A Mathematician's Gossip" of his renowned *Indiscrete Thoughts*, Rota launches into a diatribe concerning the "replete injustice" of misplaced credit and "forgetful hero-worshiping" of the mathematical community. He argues that a particularly egregious symptom of this tendency is the cyclical rediscovery of forgotten mathematics by young mathematicians who are unlikely to realize that their work is fundamentally unoriginal. My question is about his example of this phenomenon.

In all mathematics, it would be hard to find a more blatant instance of this regrettable state of affairs than the theory of symmetric functions. Each generation rediscovers them and presents them in the latest jargon. Today it is K-theory yesterday it was categories and functors, and the day before, group representations. Behind these and several other attractive theories stands one immutable source: the ordinary, crude definition of the symmetric functions and the identities they satisfy.

I don't see how K-theory, category theory, and representation theory all fundamentally have at their core "the ordinary, crude definition of the symmetric functions and the identities they satisfy." I would appreciate if anyone could give me some insight into these alleged connections and, if possible, how they exemplify Rota's broader point.

I am interested in the simplicial approximation of Serre or Hurewicz fibrations (or even fibre bundles). Let's assume $E$ and $B$ are finite simplicial complexes (or their associated geometric realizations) (therefore compact spaces) and $p\colon E \to B$ is a fibration (in the topological sense, I mean, the homotopy lifting property is satisfied).

Are there any sufficient conditions which guarantee that a simplicial approximation of $p$, $\widetilde{p}\colon E\to B$, is also a fibration? or at least that all the fibres of $\widetilde{p}$ have the same homotopy type and the same homotopy type of the fibers of $p$ (in the case of Hurewicz fibrations)? And in the case of fiber bundles?

I am both interested in arguments or references where you think some information about these topics could be provided. I am also interested in counterexamples which show under which conditions what I ask is not possible.

Thanks in advance and any help would be appreciated.

Let $P$ be an analytic linear differential operator defined on some open interval $X=(a,b)$ and $\mathcal{M}=\mathcal{D}_X/P\bullet \mathcal{D}_X$ the corresponding $\mathcal{D}$-module. I'm trying to understand **what's the correct definition of a Green's function for $P$ in this context?**

For starters i'm not sure what kind of algebraic object it is. Should it be a...

- A function on $X \times X - \triangle$?
- A $1$-form on $X \times X - \triangle$?
- A class in local cohomology $H^1_{\triangle}(X \times X, \Omega^1_{X \times X})$?
- A $\mathcal{D}_{X \times X}$- module?
- None of the above... :(

Obviously i'm phrasing everything in the most simple terms but ideally I'm looking for the answer that will generalize easily and immediately to general case.

If there is a standard reliable reference for these kind of questions i'd be happy to know about it.