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### $A^{*}A=B^{*}B$ [closed]

Let $A,B$ be two invertible $n\times n$ matrices of complex numbers. If $$AA^{*}=BB^{*},$$ does it follow that $$A^{*}A=B^{*}B$$ if not, then under what conditions this would be true? ($A^{*}$ is the ...
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### Integrability condition for flat connections

I am reading Kobayashi's book "Differential geometry of complex vector bundles". More precisely, I am on section 2 of chapter 1, page 5. Kobayashi is trying to prove that if $E$ is a vector ...
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### Going from a class of path functions to a topology

Someone asked a version of this 10 years ago, but no satisfactory answer was given, so I want to try again. I have a class of functions F from the reals to a set S (the specifics are complicated and ...
52 views

### If (a+b):(b+c):(c+a)=2:3:4 and a+b+c=27, then how much is c? [closed]

I found this question in an old mathematics book and I'm stuck. Does anyone know the answer?
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### Discrete control matrix - Upper limit

I have several questions regarding control theory which might be of interest on the mathematics community. One might recall a continuous-time linear system has the form $\dot{x} = A x + B \, u$. For ...
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### Cellular chain complex of $G$-CW-complexes & their differentials

I not completly understand EXAMPLE 2.31 (page 19) dealing with homology of $G$-CW-complexes. Source: http://www.ltcc.ac.uk/media/london-taught-course-centre/documents/LTCC-notes-Lecture3-2019.pdf ...
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Let $S$ be an infinite set of positive integers. Let us define the following quantities: $N_S(z)$ is the number of elements of $S$, less or equal to $z$ $r_S(z)$ if the number of positive integer ...
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### Finding a non-negative similar matrix

Let $B$ be some matrix. Is there a way to decide whether there exists an invertible matrix $P$, such that the matrix: $$A = P^{-1}BP$$ Is non-negative? (That is - all the entries in the matrix are ...
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### Holomorphic map from a punctured affine line to $M_g$ with Zariski-dense image

Denote by $M_g$ the coarse moduli space of connected smooth projective complex curves of genus $g$. Is there a holomorphic map $U\to M_g$ with Zariski-dense image where $U$ is the affine line with ...
237 views

### Cusp forms with integer Fourier-coefficients

Let $S_k(\Gamma_1(N))(\mathbb{Z})$ be the set of modular forms of weight $k$ and level $N$ with integer Fourier coefficients. Then is true that any cusp form can be written as $\mathbb{Q}$ linear ...
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### Let $T$ be a maximal torus of $SU(k+1)$. Who is the normalizer $N(T)$ of $T$ in $O(2k+2)$?

I'm reading an article which I cannot understand a paragraph very well. $T$ is a maximal torus of $SU(k+1)$ acting linearly on $\mathbb{C}^{k+1}$. And here is what is written that I cannot fully ...
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### Difference between semilinear and fully nonlinear

I'm confused why the Hamilton Jacobi Bellman equation: $$\frac{\partial u}{\partial t}(t,x)+\Delta u(t,x) -\lambda||\nabla u(t,x) ||^{2}=0$$ is considered fully nonlinear, but not semilinear. By ...
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### Formula to calculate 500 million base with an increase of 5% a year for X amount of years [closed]

I'm starting with 500 million user base with an average increase of 5% a year. I can get the result by doing: 500 million + 5% = A + 5% = B + 5% = C + 5% = D + 5% = E E will be the result of 5 years ...
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### Algebras Morita equivalent with the Weyl Algebra and its smash products with a finite group

My question os motivated, naturally, by the problem of classifying symplectic reflection algebras up to Morita equivalence (a classical reference for rational Cherednik algebras is Y. Berest, P. ...
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### On the Dirichlet divisor problem. Proof that $\Delta(n) = O(n^{\frac{1}{4} + \epsilon})$?

Hello dear mathematicians! I have a few questions regarding my current work (paper) on counting the number of lattice points under a hyperbola $\frac{n}{xy},\; 1 \leq x \leq n,\; 1 \leq y \leq n$. ...
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### Is the cohomology of Hilbert modular surfaces spanned by special cycles?

We consider the Hilbert modular surface $X$ that parametrize abelian surface with real multiplication by $\mathcal{O}_F$ and $\mathfrak{a}$-polarization, where $F$ is a real quadratic field with ...
Let $X$ be a CW complex. If $E$ is a vector bundle over $X$, then it's well-known that the Stiefel-Whitney classes $w_j(E) \in H^j(X,\mathbb F_2)$ of $E$ are determined from the classes $w_{2^k}(E)$ (...