Questions tagged [unimodality]
The unimodality tag has no usage guidance.
13
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Peakedness of conditioned distributions
I'm struggling to prove the following:
Let $X,Y,Z$ be iid random variables (with pdf $f$) that are unimodal and symmetric around 0. Then $X \mid (X = Z)$ is more peaked than $X \mid \left(\tfrac12 X +...
- 291
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20
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Unimodality of distribution from Lévy symbol
Also posted in MSE.
Assume that one want to study a distribution $f$ on $\mathbb{R}$ for which the Lévy symboln, i.e.:
$$
\forall u\in\mathbb{R},\quad\psi(u) := \log \mathbb{E}\left[e^{iuX}\right]
$$
...
- 393
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88
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Pre-images of the critical point of $3.83 x(1-x)$
This question may be easy; however, I have been unable to locate any references regarding the specific scenario described below.
Let $T:[0,1]\to [0,1]$ be the quadratic map $T(x) = 3.83 x (1-x)$. It ...
- 243
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1
answer
271
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Operation preserving log-concavity of sequences
Here a log-concave sequence $(a_0,a_1,a_2,\ldots)$ is a sequence of positive real numbers such that $a_i^2 \geq a_{i-1}a_{i+1}$ for each $i\geq 1$. These are pervasive within mathematics.
A polynomial ...
- 1,959
1
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1
answer
142
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Question about the inverse operator on PSL(2,R) with respect to Liouville measure
In GTM 259 chapter 9 and Katok Hasselbaltt Introduction to Modern Theory of Dynamical System chapter 5 (using the Iwasawa KAN decomposition)
we see the Unit Tangent bundle of Hyperbolic half plane is ...
- 39
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130
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Exact classifications of numerical sequences in combinatorics
Recently there has been tremendous progress in showing that certain sequences of numbers $(a_0,a_1,\ldots,a_n)$ attached to a combinatorial object (such as the coefficients of the characteristic ...
- 23.1k
3
votes
1
answer
246
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Unimodality of $f$-vectors of $0/1$-polytopes
It is known that the face vectors (aka $f$-vectors) of general polytopes need not be unimodal. This even fails for simple or simplicial polytopes, as was shown first by Björner.
My question is if ...
- 1,959
15
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3
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831
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Log-concavity of matroids: characterization of equality?
Let $M$ be a (loopless) matroid of rank $r$.
The characteristic polynomial $\chi_M(x)$ is defined by $\chi_M(x)=\sum_{F \in \mathcal{L}(M)}\mu(\hat{0},F) \cdot x^{\mathrm{rk}(F)}$, where $ \mathcal{L}(...
- 23.1k
3
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1
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179
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Strict unimodality of bipartite partitions
For non-negative integers $k$ and $l$ let $p(k,l)$ denote the number of vector partitions of $(k,l)$. In other words, $p(k,l)$ is the number of ways of writing
$$
(k,l) = (k_1,l_1)+\dotsb + (k_r,l_r),
...
- 5,614
2
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1
answer
97
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De-Convolution of Distributions
Under what conditions a continuous unimodal distribution G(x) can be represented as a convolution of N of the same F(x) distributions?
I.e. G(x)= F(x) * F(x) * F(x) *......
Also does F(x) also ...
- 21
3
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239
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Are all mixtures of these unimodal functions unimodal?
Let us say that a function $F\colon(0,\infty)\to\mathbb{R}$ is increasing-decreasing if, for some $c\in[0,\infty]$, $F$ is non-decreasing on $(0,c]$ and non-increasing on $[c,\infty)$. Is it true that ...
- 121k
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168
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Unimodality of a certain parametric integral
Suppose $f: [0,1] \to [0,\infty)$ is a smooth, concave and strictly increasing function satisfying $f(0)=0$.
Is it true that the map
$$
F(y) = \int_0^1 \frac{y^{3/2}}{(y+f(x))^2} dx
$$
has exactly one ...
- 191
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490
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Unimodality of length of longest increasing subsequence
For $w \in S_n$, the symmetric group on $n$ letters, let $\mathrm{is}(w)$ denote the length of the longest increasing subsequence of $w$. Define, $g_n(p) := |\{w \in S_n \colon \mathrm{is}(w) = p\}|$. ...
- 23.1k