# All Questions

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### Adjusting matrix in generalized eigenvalue problem for the design of eigenfunctions

Take two matrices $A,B \in \mathbb{R}^{n\times n}$. I am considering the generalized eigenvalue problem ($\gamma \in \mathbb{R}, \phi \in \mathbb{R}^n$) $$A\phi = \gamma B\phi.$$ Is there a ...
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### dual (p,q)-property

If the set system $(X,S)$ has the $(p,q)$-property does its dual system also have the property? (Possibly, for different $p$ and $q$.) Explicitly, I am asking about the equivalence of the following ...
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### Maximum sets of lattice points such that only a few points collinear

Consider all the integer points $\in [0,n]\times[0,n]$, I want to find the maximum subset $S$ of which such that there are at most $n^\varepsilon(0<\varepsilon<1)$ points in $S$ collinear. So, ...
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### Degree of a Segre variety (without Hilbert polynomial)

Algebraic Geometry is not my topic of research and I am having troubles in understanding the following: In Harris book, Algebraic Geometry, a firs course, Example 18.15, there is a proof of the ...
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### Minimum number of transitive paths in tournament

Let $T$ be a tournament with $n$ vertices (i.e., between every pair of vertices there exists an edge in exactly one direction.) For any $k$, the vertices $A_1,A_2,...,A_k$ form a transitive path if ...
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### Global sections in negative degree line bundles over singular curves

Let $C$ be a local complete intersection projective curve in $\mathbb{P}^3$. Assume that $C$ is integral. Let $\mathcal{L}$ be a line bundle on $C$ of negative degree. We know that if $C$ is smooth ...
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### An inequality for moments of a random variable

I'm interested in a class C of $R^1$-valued random variables $\xi$ which satisfy an inequality of the type $$(1) \qquad E|\xi|^p \leq F(E|\xi|^2),$$ where $p>2$, $F$ is a certain ...
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### Hello, everyone, I want to ask you a question about a proof in the Terence Tao's Real Analysis notes [migrated]

everyone. I am using Terence Tao's Real Analysis notes to self learn Analysis 1. There is one thing in the proof of Theorem 27 (Least Upper Bound Property) in “week 2 note” that I don’t understand ...
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### Why does the Fourier transform of bounded domain, e.g., [-1 1], differs from that of complete domain, e.g., real line. [closed]

Recently, I asked a question about proving the positive semidefiniteness of following kernel function. $$k(x, x') = 1 - 2|x-x'|$$ When using the Bochner's theorem, showing psd of the kernel ...
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### Composition of a transcendental function with a rational function [closed]

The problem is: let $f: \mathbb{R}\to \mathbb{R}$ be an analytic transcendental function and let $\psi(x)=\frac{x}{2(1+x^2)}$. Is the function $f(\psi(x))$ transcendental?
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### Representations of parabolic subgroups of the general linear group over the complex numbers

In all that follows, we are working over $\mathbb{C}$. Let $B \subseteq P \subseteq {\rm GL}(n)$ be a parabolic subgroup. Can you say anything in general about the representations of $P$? I suspect ...
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### biproduct and tensorial product [closed]

Let $\mathcal{C}$ be a monoidal abelian category. Let A,B, C $\in$ $\mathcal{C}$.There is an isomorphisms between this objects? (A$\bigoplus$B)$\otimes$C; A$\otimes$(B $\bigoplus$C)
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### Sufficient Conditions For Monotonic Decreasing of Multivariate Function

I found the following theorem on sufficient conditions for decreasing monotonicity of an absolutely continuous function $\phi : \mathbb{R} \rightarrow \mathbb{R}$, I would like to know if it is ...
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### Standard name for a Monoid/Semigroup with a+b <= a, b?

I have seen suplattice and inflattice being used when dealing with a lattice. What about when you don't have a lattice? For instance, for positive reals a, b define a |+| b === 1/((1/a) + (1/b)), you ...
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### What is the arithmetic Nullstellensatz?

The only precise statement (coming from a reliable source) of the "arithmetic Nullstellensatz" I can find is in Gowers's book, stating that two polynomials with integral coefficients have the same ...
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### “embedding” various matrix equivalences into the equivalence of particular linear map

Consider the square matrices over a (local) ring $R$, up to conjugation, $A\rightarrow UAU^{-1}$, where $U$ is an invertible matrix over $R$. Such an equivalence embeds into the "left-right" ...
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### Absolute irreducibility of a symmetric square?

This is a question I received today by email, which somebody more experienced with finite group representations can probably answer directly. Take $F:=\mathbb{F}_q$ for some prime power $q$, so ...
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### Minimal density hitting set for k-length arithmetic progressions

This is problem which came up in the process of designing a game. Thus, I don't know any previous work relevant to the problem. Fix a small set $D$ of a natural numbers. For example, $D=\{1,2,3\}$. ...
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### What arrangement of unit cubes minimizes surface area?

For each of these two questions, one can assume that the arrangements are polycubes (for which a definition can be found in the excerpt-image below). Question A. How does one arrange $n$ unit cubes ...
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### Which forcing types preserve the axiom of determinacy?

Do we have some rudimentary understanding of some properties that a forcing can have in order to guarantee that it doesn't violate the axiom of determinacy? To be more specific, in Which forcings ...
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### “Is it possible to give a restricted set-theoretical definition of addition of natural numbers in terms of successor?” [Tarski]

In his paper "Restricted set-theoretical defintions in arithmetic" Raphael Robinson cites a problem posed by Tarski: Is it possible to give a restricted set-theoretical definition of addition of ...
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### Results for resolution of equations in polynomial ring

Is there any reference for resoluttion of equations in a polynomial ring, such as $x^2+y^2=z^2$ in $\mathbb{C}[t]$? Thanks!
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### Which axioms of ZF are used for finite choice?

Apologies if this is a silly question, not an expert in set theory but just wondering about it. ZF implies finite choice. But let's suppose one wanted to work without it. The thinking here is being ...
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### $f(x)$ is irreducible but $f(x^n)$ is reducible

Does there exist an irreducible polynomial $f(x)\in \mathbb{Z}[x]$ with degree greater than one such that for each $n>1$, $f(x^n)$ is reducible (over $\mathbb{Z}[x]$)?
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### questions on steenrod algebra

I plan to give a talk on Steenrod algebra in a student seminar. but there are some questions that I didn't find an answer to, and it seems to me that I'm missing something: if the algebra of ...
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### if V(f) is irreducible, then how to show that the polynomial f itself is irreducible? [closed]

V(f) is the zero locus of the polynomial f in the polynomial ring k[x1, x2, ..., xn] with k an algebraically closed field. If V(f) is irreducible, then how to show that 'f' is irreducible?
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### prime number sequence and the tendency [closed]

before I ask.. I'm not good at english. (because I'm not an English.) so you may..not be able to understand what I want to say. 1> definite new sequences prime number sequence {Pn} = 2 3 5 7 11 13 ...
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### Weil restriction

I've already asked a similar question in SE, without success, so I've decided to post here a more general version of my question. Let $f: Y \to X$ be a finite etale morphism of smooth proper ...
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### What are TQFTs that are multiplicative under connected sums? Do bordisms with connected sum as monoidal product exist?

In general, one extracts a manifold invariant from a TQFT by interpreting the closed manifold as a bordism from the empty set to the empty set. The TQFT sends this bordism to a homomorphism of the ...
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### Counterexample to Pólya's conjecture

It is known that Polya's conjecture is false and the smallest counter-example is about $10^9$. Assuming that we are searching for a counter-example not knowing that it exists. What useful information ...
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### Kempe chain color swaps in a partially colored map

Crossposted from math.stackexchange.com: http://math.stackexchange.com/questions/904932/kempe-chain-color-swaps-in-a-partially-colored-map Question: In this partially Tait's colored map, using ...
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### Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space?

Preliminary Definition We define the Zygmund spaces $C^r_{*}$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are allowed to have values in $\mathbb{R}^m$, via working with ...
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### Asymptotics on prime divisors

Somewhat inspired from the Zsigmondy's theorem is my question. Suppose Let $a_{1}> a_{2}> \cdots > a_{k}$ be nonzero integers, with $k \geq 2$. Let $a(n) : = a_{1}^{n}+\cdots+a_{k}^{n}$ for ...
What are the complex rational homogenous spaces $G/P$ ($G$ a semi-simple complex Lie group, $P$ a parabolic subgroup) such that the set of real points $(G/P)(\mathbb R)$ is a (compact) riemannian ...