# All Questions

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### smoothness of free boundary [closed]

For in $\mathbb{R}^n$, there is a an positive obstacle it has the expression $\phi(x) = \chi_{BR(0)} \cdot \max\{0,a(x)\}$, where $a(x)$ is an analytic function or smooth function on the whole of ...
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### Some regularity results of free boundary quesions for a special case [on hold]

In $R^n$, there is an obstacle $\phi \in C_0(R^n)$, and $\phi$ is analytic on its support and has analytic continuation in an open set containing its support. Then I just want to solve the following ...
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### Approximate rank of the set formed by all delayed replicas of a bandlimited signals between 0 and T

My question is given a complex-valued signal with a certain delay $s(t-\tau)$ for which we sample $N$ isntants: $$\mathbf{s(\tau)}=\left[s(0-\tau),\ldots,s\left(\frac{N-1}{f_s}-\tau\right)\right]^T$$ ...
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### Panning for gold nuggets: a type of isoperimetric problem

Let $C$ be a unit-radius circle in the plane. Suppose you have a total length $L$ of string available, and your task is to connect chords of $C$ using no more than $L$ of string to minimize the ...
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### Gaussian Curvature of Exponentiated 2-Planes

Consider a Riemannian manifold $M$ with sectional curvatures $K\ge 0$ and let $\Pi$ be a 2-plane in the tangent space of $M$ at a point $p$. In a small enough neighborhood $U$ of 0 the exponential map ...
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### A question on the size of an admissible ordinal

Let $\mathbf{L}_{\varsigma}$ be the level of ordinal $\varsigma$ of Gödel's constructible universe $\mathbf{L}$. Let $\Sigma_{3}$-KP be Kripke-Platek set theory with infinity and ...
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### About the reduceness of the commuting scheme associated with a symmetric pair

my question is the following one: Let $G$ be a connected reductive algebraic group over the field of complex numbers, and let $V$ be a linear representation of $G$ obtained as the isotropy ...
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### Orthogonal projection

Let $G$ be an operator with compact resolvent on a Hilbert space $H$ such that $\ker G \neq \{0\}$. Further let $P$ be the orthogonal projection onto $\ker G$, and let $G_{0} := G+P$. My question is: ...
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### Condition Number and CFL Condition in Finite difference Methods

when applying a Finite Difference scheme for an IVP, two factors come to mind when considering stability: One factor would be the condition number of the approximation operator. The other factor ...
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### What is the best reference for Spectral theory?

I'm studying Bernard Aupetit: A Primer on Spectral Theory but the textbook we are using is a little bit heavy going for me. Is there a best book to learn about these things? Thank you.
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### Does this condition imply a polynomial is a product of linear factors

Let $\Lambda$ be a lattice (i.e. $\Lambda \simeq \mathbb{Z}^n$) with a positive subcone $\Lambda^+$. Let $H: \Lambda^+ \rightarrow \mathbb{C}$ be a function such that $\forall\mu \in \Lambda^+$, ...
Understanding moments and subconvexity bounds for $L$-functions is a big topic with a lot of activity. I'm currently looking at a related problem, bounding $$\int_0^T L\left(\tfrac{1}{2} + it, f ... 0answers 55 views ### Lower periodic subsets of groups and semigroups Suppose that A and B are subsets of a group or semigroup. We call A left upper [resp. lower] B-periodic if BA\subseteq A [resp. A\subseteq BA]. If A is both left upper and lower ... 0answers 57 views ### Game theory question Folk Theorem [closed] I wonder what is the strategy is here.. I have calculated the potential result start from p1 choose C and p2 choose D at first time, and then they both confess. However someone points this is not the ... 0answers 56 views ### Verification of Gauss Bonnet Theorem on Beltrami pseudosphere and bent sphere patches [closed] Given that boundary geodesic curvature k_g and Gauss curvature K are constant, patch area = A and perimeter length = p.  K\, A + k_g\, p = 2 \pi  For a flat circle patch  k_g= 1/R,   ... 1answer 127 views ### Graph automorphism that swaps two pairs of nodes Suppose we have two automorphisms on a graph G such that each one swaps a separate pairs of vertices. Is it possible to construct (or prove the existence of) a third automorphism that swaps both ... 1answer 195 views ### Subquotients in the Verma filtration on Verma modules Let \lambda be a dominant integral weight of \mathfrak g, a finite-dimensional reductive Lie algebra over \mathbb C. Let M(w\cdot \lambda) denote the Verma module with high weight w\cdot ... 1answer 68 views ### Standard names and methods for this type of fitting minimization In material science research, we have come across the following type of problem. Given a m by n matrix A, a m vector b, and error tolerance \varepsilon, we want to do this minimization$$\eqalign{ ...
I'm interested in a definition of cocommutative Hopf-algebra objects in the $\infty$-category of associative (read: $A_\infty$) ring spectra. One thought I had was to think of cocommutative ...