Existence of a function between two functions $f$ and $g$ where $f in o(g)$

Is this statement true?

For each two functions $f$ and $g$, where $f in o(g)$, there exists a function $h$ where $f in o(h)$ and $h in o(g)$

Please note that I am using small $o$ notation.

ag.algebraic geometry – Existence of solutions to quadratic vector equation

Given
$$A_{i,j,k} X_j^* X_k + C_i = 0$$
where $𝐴_{i,j,k}$ and $𝐶_i$ are arbitrary complex numbers for all $𝑗, 𝑘$ which are $𝑁$-dimensional indices and $i$ which is an $m$-dimensional index where $m<N$. Note $*$ is complex conjugation and there is a summation for repeated indices. Are there conditions on $A$ and $C$ to guarantee the existence of a solution to this set of equations?

algorithms – Suffix -Tree checks existence of P pattern before k position in T string

I need to design an algorithm that given a T string of n length, after a process O(n), for every string P of m length and a k value between 1 to n, to checks in O(m) time, if P appears on T before k position, only using Suffix Tree.

Unfortunately there are not any good bioinformatics books with fair examples and practical methodologies. Dan Gusfield book does not offer a solution manual.

logic – Analogue of disjunction and existence properties for a Turing-complete programming language?

Quoting from Wikipedia:

In mathematical logic, the disjunction and existence properties are
the “hallmarks” of constructive theories such as Heyting arithmetic
and constructive set theories (Rathjen 2005).

The disjunction property is satisfied by a theory if, whenever a
sentence A ∨ B is a theorem, then either A is a theorem, or B is a
theorem.

The existence property or witness property is satisfied by a theory
if, whenever a sentence (∃x)A(x) is a theorem, where A(x) has no other
free variables, then there is some term t such that the theory proves
A(t).

In the realm of logic (to the best of my understanding), these properties are violated when a logic is “classical”, which, through the Curry-Howard lens, corresponds to it having support for capturing undelimited continuations.

For Turing-complete programming languages, the presence or absence of first-class control and undelimited continuations is also a meaningful and important distinction. However, being inconsistent as logics, the disjunction and existence properties in their above form hold trivially: every ‘sentence’ is a ‘theorem’.

So: Is there any formal property that can distinguish Turing-complete programming languages based on whether or not they have undelimited continuations, which is similar in spirit to the disjunction and existence properties for logics? (Let’s assume the languages are statically typed and do possess sum types and/or existential types.)

fa.functional analysis – Existence of periodic solution to ODE

We shall consider the matrix-valued differential operator

$$(L u)(x) :=u'(x) – begin{pmatrix} 0 & 1+2sin(2pi x-frac{pi}{6})\ 1 – 2sin(2pi x+frac{pi}{6}) & 0 end{pmatrix} u(x).$$

This is a $1$-periodic operator. Thus, does there exist a $lambda in mathbb C$ and a $1$-periodic solution to this ODE such that

$$ (L – lambda)u = 0.$$

Probably there is no explicit solution, but can we show the existence of such a solution?

ap.analysis of pdes – Existence of divergence-free unit vector field in conformally flat metric

Question. Let $Omega subset mathbf{R}^2$ be a convex polygonal domain, equipped with a Riemannian metric $g$. Under which conditions on $g$ is there a vector field $X$ in $Omega$ with $mathrm{div}_g X = 0$, $g(X,X) = 1$, and orthogonal to the boundary?

  • Here an open domain $Omega subset mathbf{R}^2$ is called polygonal if its boundary is a union of line segments. This is allowed to be unbounded: the special case I would be most interested is when $Omega = { (x_1,x_2) in mathbf{R}^2 mid x_1,x_2 > 0 }$.

  • If that helps, the metric may be assumed to be conformally flat, of the form $g = e^{2varphi} (mathrm{d} x_1^2 + mathrm{d} x_2^2)$ for some function $varphi: Omega to mathbf{R}$. However $varphi$ may go to $-infty$ approaching the boundary.

  • The orthogonality to the boundary is meant in the Euclidean sense: $X cdot nu = X^1 nu^1 + X^2 nu^2 = 0$ along $partial Omega$. This orthogonality need only be satisfied at the smooth portions of the boundary. (The vector field cannot be defined continuously at the corners.)

The assumption that $X$ be a unit vector field is crucial. Without it, for example the gradient $X = nabla u$ of a harmonic function with $frac{partial u}{partial nu} = 0$ on $partial Omega$ is divergence-free.

Expressed in coordinates it looks like an overdetermined first-order PDE:
begin{equation}
begin{cases}
partial_i(sqrt{det g} X^i) = 0 text{ in $Omega$} \
g_{ij} X^i X^j = 1 text{ in $Omega$} \
delta_{ij} X^i nu^j = 0 text{ on $partial Omega$}.
end{cases}
end{equation}

Some special cases admit solutions: if e.g. $Omega = { (x_1,x_2) in mathbf{R}^2 mid x_1 > 0 }$ and $g = mathrm{d} x_1^2 + mathrm{d} x_2^2$, then $X = partial_2$ is a solution. What properties of the metric $g$ play a role in the solvability of the problem? I have little experience with this kind of problem, so any suggestions or references would be welcome!

How to prove that existence of one-way functions implies P!=NP?

Wikipedia:

The existence of such one-way function… would prove that the complexity classes P and NP are not equal.

How is this proved?

general topology – Existence of a Jordan path in a path-connected set.

A path in set $E$ is a continuous function
$$gamma:(a,b)longrightarrow E, $$
with $A:=gamma(a)$ its starting point and $B:=gamma(b)$ its ending point.
Any two points $A, B$ in a path-connected set can be connected by a path, with $A, B$ as its starting and ending points.
A Jordan path $gamma$ is a injective path.
Any two points $P, Q$ in a Jordan-path connected set can be connected by a Jordan path.


Question: For any path-connected set $E$, is $E$ always Jordan-path connected?


When there are at most countably many self-intersecting points, it’s possible to ‘cut’ the loop and make the path a Jordan one. But I got problems when there are uncountably many self-intersecting points on the path.

general topology – Checking Existence of a Surjective Continuous Map between Spaces

I am studying topology and recently tried to solve these two questions concerning the existence of a surjective continuous map between spaces. But I am not sure how to construct one, or to claim that there is no such surjective continuous map. The questions(in the form of T/F question) are as follows:

  1. There exists a surjective continuous map from $S^2$ to $mathbb{R^2}$ where $S^2$ is the unit sphere $S^2 := {(x,y,z) in mathbb{R^3}: x^2+y^2+z^2=1}.$
  1. There exists a surjective continuous map from $S^1$ to $S^1 times S^1$, where $S^1$ is the unit circle $S^1 := {(x,y) in mathbb{R^2}: x^2+y^2=1}.$

From what I think, I am guessing question 2 is False. (One of my friend told me it seems false due to topological group… But as I did not learn the concept of topological group, I felt there should be some other way to solve this.) That is, there is no surjective continuous map between spaces. But overall, I am not sure whether these statements are true or not. Also, I am not sure of a way to construct a surjective continuous map for them.

Is there some method or way of constructing surjective continuous map in this kind of situations? (Or some criteria that can tell whether these kinds of map exist or not.)

Thank you.

graphs – How to prove the existence of the spectral expander with the given parameteres?

I need to prove the existence of the $(1944, 144, 0.5)$ spectral expander. I tried to construct it using tensor product of the following graphs:
$$
(1944, 144, 0.5) = (9^2, 9, 1/3) otimes (24, 16, 0.5)
$$

I already know that expander with parameters $(9^2, 9, 1/3)$ exists and corresponds to the affine expander, but I am not sure about $(24, 16, 0.5)$ one.