Suppose each $A_i,i \in I$ is a finite dimensionl $C^*$ algebra,then $\bigoplus A_i$ is nuclear,but $\prod A_i$ is not nuclear,how many nuclear $C^*$ algebras between $\bigoplus A_i$ and $\prod A_i$?(I is infinite)

There appears to be a dearth of resources and references for the question of 'locality' in Floer theory. In particular, I cannot seem to find any complete statement of what people refer to as 'Kontsevich's cosheaf conjecture' or why the work of Ganatra-Pardon-Shende ('18) resolves this conjecture (partially or wholly). I would appreciate a direction to a reference of Kontsevich's conjecture (if one exists) or a sketch of its formulation. Furthermore, I would appreciate if someone could link this statement directly to the work of Ganatra-Pardon-Shende.

I calculated the quantity of something to be $\sum^x_{k=0} \binom{n}{k} 2^k$. When I enter this in Wolfram Alpha, it appears the sum can be simplified to $3^n$.

In fact, if the exponential part is $a^k$ for any $a$, the result simplifies to $(a+1)^n$.

I have tried deriving this by hand, even trying to prove it by induction, but haven't had any luck. Is there a way to derive this simplification by hand?

Suppose that $X$ is the continuous-time simple symmetric random walk on the lattice $\mathbb Z^d$ (i.e., a simple symmetric random walk with i.i.d. exponential jump times), and let $$u(t,x):=\mathbf E\left[\exp\left(\int_0^tV(X_s)~ds\right)f(X_t)\bigg|X_0=x\right] \tag{1}$$ for $(t,x)\in[0,\infty)\times\mathbb Z^d$, where $V,f:\mathbb Z^d\to\mathbb R$ are functions.

According to the Feynman-Kac formula, we know that $u(t,x)$ solves the lattice/matrix heat equation $$\partial_tu=\tfrac12\Delta u+Vu,\qquad u(0,x)=f(x),\tag{2}$$ where $$\Delta:=\left[ \begin{array}{ccccc} &\ddots&\ddots&\\ &\ddots&-2&1&\\ &&1&-2&1&\\ &&&1&-2&\ddots\\ &&&&\ddots&\ddots& \end{array}\right]$$ is the discrete Laplacian, and we think of $V$ as the diagonal matrix $$V=\left[ \begin{array}{ccccc} &\ddots&&\\ &&V(-1)&&\\ &&&V(0)&&\\ &&&&V(1)&&\\ &&&&&\ddots&& \end{array}\right].$$

As an alternative to $(2)$, a common model for a lattice heat equation is to consider $$\partial_tu=\tfrac12\Delta u+\tilde Vu,\qquad u(0,x)=f(x),\tag{3}$$ where the potential $\tilde V$ is instead of the form $$\tilde V=\left[ \begin{array}{ccccc} &\ddots&\ddots&\\ &\ddots&0&V(-1)&\\ &&V(-1)&0&V(0)&\\ &&&V(0)&0&V(1)&\\ &&&&V(1)&0&\ddots\\ &&&&&\ddots&\ddots& \end{array}\right],$$ or a more general tridiagonal matrix $$\tilde V=\left[ \begin{array}{ccccc} &\ddots&\ddots&\\ &\ddots&U(-1)&V(-1)&\\ &&V(-1)&U(0)&V(0)&\\ &&&V(0)&U(1)&V(1)&\\ &&&&V(1)&U(2)&\ddots\\ &&&&&\ddots&\ddots& \end{array}\right].$$

**Question.** Does there exist a Feynman-Kac formula *similar to* $(1)$ for lattice operators with non-diagonal potential such as $(3)$?

To clarify a bit what I mean by *similar to* $(1)$: It's easy enough to come up with *some* probabilistic representation of the solution of $(3)$ (for example by using the Trotter-Kato theorem: $e^{\Delta/2+V}\approx(e^{\Delta/2n}e^{V/n})^n$ for large $n$), but I can't get anything *nice* like $(1)$, and It's not clear to me if we should/shouldn't expect such a nice representation in those cases.

By performing some computations using the Singular software, I've noticed the following pattern: if $\mu$ is the Milnor Number of a holomorphic germ $f\in \mathcal{O}_n$ at the origin, then the Milnor Number of its real and imaginary parts are equal and are $\mu^2$, that is, $$ \mu_{(\mathcal{R}e(f),0)} = \mu_{f,0}^2 $$

By Milnor number of the real part of $f$, I mean $u=\mathcal{R}e(f)$ as a germ of real analytic function of $2n$ real variables (the real and imaginary parts of each complex variable). If $\mathcal{A_{2n}}$ is the ring of germs of such real analytic functions, then $$ \mu_{(\mathcal{R}e(f),0)}=\text{dim}_{\mathbb{R}} \dfrac{\mathcal{A}_{2n}}{\left<\frac{\partial u}{\partial x_1},..., \frac{\partial u}{\partial x_n}, \frac{\partial u}{\partial y_1},..., \frac{\partial u}{\partial y_n}\right> } $$

I would like to know if there's any generalisation to this. I've tried using some direct sum properties on ideals but got nowhere. I suspect there might be some tensor products involved, but also got nowhere.

I am interested in the $\ell^1$ analogue of direct sums for Operator spaces, e.g. Operator Space Dictionary. Briefly, and operator space is either a concrete subspace of $B(H)$, the operators on a Hilbert space, or equivalently a normed space $E$ with norms on the matrix spaces $M_n(E)$, satisfying Ruan's axioms. The analogue of the $\ell^\infty$ direct sum can be stated as $M_n(E\oplus F) = M_n(E)\oplus M_n(F)$, which is compatible with the diagonal embedding $B(H)\oplus B(K)\rightarrow B(H\oplus K)$.

As a Banach space, the dual of $E\oplus F$ is $E^*\oplus_1 F^*$ the $\ell^1$ sum. We use this idea to define an operator space structure on $E\oplus_1 F$ be embedding it into $(E^*\oplus F^*)^*$. One can then show, in increasing order of difficulty (IMHO):

- $E\oplus_1 F$ has the universal property that if $u:E\rightarrow X, v:F\rightarrow X$ are complete contractions, then $u\oplus v: E\oplus_1 F\rightarrow X$ is a complete contraction.
- $E\rightarrow E\oplus_1 F$ (and for $F$) is a complete isometry, and $E\oplus_1 F\rightarrow E$ (and for $F$) is a complete quotient map.
- $(E\oplus_1 F)^* = E^* \oplus F^*$
- $(E\oplus F)^* = E^*\oplus_1 F^*$. This in particular had me stumped for a bit; I needed to use the fact that $M_n(E)^{**} = M_n(E^{**})$.

I am struggling to find references. Points (2) and (3) above are covered in these notes of Blecher. The books by Paulsen and Effros & Ruan seem not to consider $\oplus_1$. The book of Blecher & Le Merdy leaves all the proofs to the reader. The original paper of Blecher also does not give details for points (3) and (4). Pisier's book instead defines $\oplus_1$ using point (1) (the universal property) but leaves (3) and (4) as exercises.

I would like a reference to a clear proof of (3) and (4).

Alternatively, am I missing some genuinely "easy" argument?

In particular, just using the universal properties, I can show that $(E\oplus_1 F)^* = E^*\oplus F^*$. How can one give an analogous proof that $(E\oplus F)^* = E^*\oplus_1 F^*$?

Let $F$ be a finite field, and $T$ be a torus over $F$. Assume that $T_1,T_2$ are two $F$-subtori of $T$, such that $T_1 \times T_2 \to T,(t_1,t_2) \mapsto t_1 t_2$ is surjective with finite kernel $K$. I wonder whether $T_1(F)$ and $T_2(F)$ would generate $T(F)$. Of course the case that $T$ is $F$-split is trivial. For general case, one could use Galois cohomology to describe it, namely that $1 \to K(F) \to T_1(F) \times T_2(F) \to T(F) \to H^1(\langle Frob \rangle,K) \to 1$. But I do not think the last cohomology group would vanish in general. Is the claim wrong or did I miss something?

Consider $X$ be a topological space and $f:X\rightarrow\mathbb{R}$ a continuous function. If $\mu$ and $\nu$ are two different measures defined over $X$ with $\operatorname{supp}\mu\subseteq \operatorname{supp}\nu$ and $\mu(f)=-\infty$ then can we affirm that $\nu(f)=-\infty$?

The definition of the metric has the following: a) $d(x,y)=0 \leftrightarrow x=y $

b) $d(x,y) \leq d(x,z)+d(z,y) $

c) $d(x,y)=d(y,x)$

How to prove that the axiom of symmetry(c) is corollary to the first two axioms?

Let $d$ be an integer. It is a well-known theorem, attributed to Hermite and Minkowski, which asserts that the number of number fields $K$, allowed to have *any* degree over $\mathbb{Q}$, having discriminant $\Delta_K = d$ is finite.

Let $S(d)$ be the number of isomorphism classes of number fields of discriminant $d$. The Hermite-Minkowski theorem is the assertion that $S(d) < \infty$ for all $d \in \mathbb{Z}$. Is $S(d)$ uniformly bounded? That is, does there exist a positive integer $N$ such that $S(d) \leq N$ for all $d \in \mathbb{Z}$?

We can refine the question and instead only count (isomorphism classes of) number fields of fixed degree over $\mathbb{Q}$. Let $S_n(d)$ be the number of isomorphism classes of number fields of discriminant equal to $d$ and degree equal to $n$. Does there exist a number $N(n)$ such that $S_n(d) \leq N(n)$ for all $d \in \mathbb{Z}$?

Observe that the answer is yes for $n = 2$. Indeed we find that $S_2(d) \leq 1$ for all $d \in \mathbb{Z}$, with equality if and only if $d$ is a fundamental discriminant.

Suppose we are given the coordinates of three points $f=(f_x,f_y)$, $g=(g_x,g_y)$, $p=(p_x,p_y)$. Which is the most direct and smart way to compute the quadratic equation of the ellipse with foci $f$, $g$ and passing through $p$? Something like

$$ \left\{ \begin{array}{rcl} aux_1 & = & F_1(f_x,f_y,g_x,g_y,p_x,p_y)\\ aux_2 & = & F_2(f_x,f_y,g_x,g_y,p_x,p_y,aux_1)\\ & \vdots & \\ Q & = & \left( \begin{array}{ccc} A & B & D \\ B & C & E \\ D & E & F \\ \end{array}\right)\\ \end{array}\right. $$

Analogously for a hyperbola, given two foci and a point, and a parabola, given its focus and directrix

Let $v\in \mathbb{R}^n$ be uniformly distributed on the unit sphere. Let $\lambda_1,...,\lambda_n$ be given real numbers. What is the distribution of $$X=\sum_{i=1}^n\lambda_iv_i^2\;?$$ Does it happen to belong to any known family of distributions? I think this is a very flexible way to model the distribution with compact support. When $n=2$, $X$ is just the celebrated arcsine distribution supported on $(\lambda_{\min},\lambda_{\max})$. What about for general $n$? I also think $X$ can capture the ''spreadness'' of the sequence $\lambda_1,..,\lambda_n$.

The Hermite-Lindemann-Weierstrass theorem is the following statement regarding the exponential function $\exp : \mathbb{C} \rightarrow \mathbb{C}$:

Theorem (Hermite-Lindemann-Weierstrass): Let $\beta_1, \cdots, \beta_n$ be algebraic numbers which are linearly independent over $\mathbb{Q}$. Then $\exp(\beta_1), \cdots, \exp(\beta_n)$ are algebraically independent over $\mathbb{Q}$.

This theorem is obviously false for algebraic functions $f: \mathbb{C} \rightarrow \mathbb{C}$, and one can easily construct transcendental functions for which the statement is false for some particular algebraic numbers $\beta_1, \cdots, \beta_n$.

How does one classify the set of transcendental functions for which Hermite-Lindemann-Weierstrass holds, perhaps up to finitely many exceptions?

I have this question stuck a bit and Google is not really a good help.

The question is about standardized cumulative average abnormal returns. What is the advantage of this method that we should use it? It's like what is the advantage of using returns standardized by their standard deviation in event studies? And when we use standardized cumulative abnormal returns?

Let $(M^{n},g)$ be a Riemannian manifold, we say that $M$ is *parabolic* if the constant functions over $M$ are the only subharmonic functions which are bounded above, i.e, for a function $u \in C^{2}(M)$, if we have $\Delta u \geq 0$ and $u\leq u^{*}<\infty$, then $u$ is constant. Liouville's theorem for subharmonic functions asserts that $\mathbb{R}^{2}$ is a parabolic manifold.

I would like to know two things about this definition:

What is the motivation for the study of parabolic manifolds?

If $N$ is a complete Riemannian manifold, then $\mathbb{R}^{2} \times N$ is parabolic. Why?

Let $\mathcal{G}$ be compact quantum group in the sense of S. L. Woronowicz. As is well-known, every compact quantum group contains a dense Hopf algebra, called the **polynomial Hopf algebra** Pol$(\mathcal{G})$.
For example, consider the famous $C(SU_q(2))$, the $q$-deformation of $SU(2)$, with generators $\alpha$ and $\beta$ :

https://en.wikipedia.org/wiki/Compact_quantum_group

The dense Hopf algebra is now the polynomial $*$-algebra generated by $\alpha$ and $\beta$. A well-known fact about Pol($SU_q(2)$) is that it has no zero-divisors, that is, it is a domain. What is a good example of a compact quantum group $\mathcal{G}$ such that Pol$(\mathcal{G})$ **has** zero divisors? On the other hand, is there any abstract characterization of the compact quantum groups such that the polynomial Hopf algebra is a domain?

Apologies if this is a simple question, but I've left PDE as a field and a friend recently asked me the following question regarding solutions $u$ and $v$ of a system of PDE. Consider $$ \nabla \cdot (a(u,v)\nabla u + \lambda u a(u,v)\nabla v)=0,$$ $$ \nabla \cdot (b(u,v)\nabla u - \lambda u b(u,v)\nabla v)=0,$$ where $a$ and $b$ are arbitrary smooth functions, $\lambda > 0$ is a real constant, and the spatial domain is a sufficiently smooth bounded domain $\Omega\subset \mathbb{R}^2$ where $u$ and $v$ satisfy Neumann or periodic conditions on the boundary. The question is: do nonconstant (smooth) solutions exist? If so, what conditions on $a$ and $b$ could preclude nonconstant solutions? I imagine that some positivity assumptions on $a$ and $b$ are necessary to use any tools from Elliptic PDE, such as Maximum Principles etc, but my memory of these things is quite shallow (and I never went much beyond what is in Evans' book). As an example, if $a = \exp(c_1u+c_2v)$ and $b = \exp(c_3u+c_4v)$, where $c_1, c_2, c_3,$ and $c_4$ are real constants, can nonconstant smooth solutions exist?

As a small note, my friend is interested specifically in the case of a rectangle with Neumann conditions on the vertical boundaries and periodic on the horizontal, but any simpler geometric or boundary condition choices should be sufficient to see the key ideas.

Consider two black holes with masses $m_1,m_2$ and zero angular momenta merging to form a single one with the mass $m$ and the rotation parameter $a=J/m$. Hawking, in "Black Holes in General Relativity" Commun. math. Phys. 25 (1972), 152—166 proposed an inequality $$m^2+m\sqrt{m^2-a^2}>2(m_1^2+m_2^2)$$ for this process (in fact, for a more general one, see p. 14 of the paper). I learned about this bound ages ago from the Lightman-Press-Price-Teukolsky relativity problem book and had no doubt about it. But now I think that the proof given in this paper is total rubbish despite being published in a supposedly mathematical journal.

The inequality is derived from what is now called an area theorem which sates that the area of the event horizon never decreases. There is nothing wrong with the theorem itself except the way it is formulated makes it completely useless for obtaining an inequality of this sort. (And probably for any other meaningful conclusion.) The fishy point here is the assumption that the area of a black hole event horizon is given by the formula (in geometric units $c=G=1$) $$A=8\pi m(m+\sqrt{m^2-a^2}).$$ No doubt, this assumption is true for a Kerr black hole but there is a big problem. The event horizon as it is defined in the formulation of the area theorem depends on the (arbitrarily distant) future evolution of a black hole, so even it the thing looks exactly as a standard Kerr black hole now its event horizon may still well be very different from what one of a Kerr hole is supposed to be, with very different area. There is no formula for the actual area of this event horizon in terms of the mass and the angular momentum.

To see where the problem really lies it is convenient to consider a scattering of two black holes instead of their merger. This process has an *inverse* which is also perfectly physical even if not likely to ever happen in reality. (Because general relativity dynamics is,
of course, time-symmetric.) Then exactly the same argument as in the paper when applied to both processes gives two inequalities which contradict each other.

Admittedly, from reading more recent physical literature I have the impression that the problem is more or less known. However, it is never mentioned explicitly. Apparently, physicists believe that the inequality is true anyway and do not care much about gaps in its proof. A mathematician like myself would rather like to see an actual proof though. Is such a proof already known or, at the very least, was the problem ever considered seriously? This is my question.

The du Bois-Reymond lemma reads as follows:

Let $ f \in L^1 (a,b) $ satisfies \begin{equation*} \int^b_a f(t) \varphi'(t) dt =0, \ \ \forall \varphi \in C^{\infty}_0(a,b), \end{equation*} then $ f(t) = c, \ a.e. \ t \in (a,b)$.

Now we consider to generalize it into case of dimension $ 2 $ with mixed partial derivatives:

Assume that $ g \in L^1((a,b)\times(a,b))$ and satifies \begin{equation*} \int^b_a\int^b_a g \frac{\partial^2 \varphi}{\partial x \partial y} dxdy =0, \ \ \forall \varphi \in C^{\infty}_0((a,b)\times(a,b)). \end{equation*} Can we claim that $ g = c_0 p (x) +c_1 q(y) +c_2, \ a.e. \ (x,y) \in (a,b) \times (a,b) $?

Let $\tau>0$, and let $T\in \mathcal{D}'(\mathbb{R})$ be a $\tau$-periodic distribution (that is, $ \langle T, \varphi(\cdot+\tau)\rangle= \langle T,\varphi\rangle $ for all $\varphi \in \mathcal{D}(\mathbb{R})$). Then $$ T=\sum_{n\in \mathbb{Z}} c_n e^{i 2\pi t/\tau}, $$ for some $c_n\in \mathbb{C}$, and where the equality means that the symmetric partial sums of the series on the right hand side converge in $\mathcal{D}'(\mathbb{R})$ to $T$. What are the $c_n$s in terms of $T$? One would think that they are given by $c_n=\langle T, e^{-in2\pi /\tau}\rangle/\tau$, but $e^{-in2\pi/\tau}$ is not a test function in $\mathcal{D}(\mathbb{R})$.