arXiv subjects

Feihu Liu

Publications and source records attributed to Feihu Liu.

43 records · Page 3Linked to original sources

arXiv2023

A Note on Generalized Repunit Numerical Semigroups

Let $A=(a_1, a_2, ..., a_n)$ be relative prime positive integers with $a_i\geq 2$. The Frobenius number $F(A)$ is the largest integer not belonging to the numerical semigroup $\langle A\rangle$ generated by $A$. The genus $g(A)$ is the number of positive integer elements that are not in $\langle A\rangle$. The Frobenius problem is to find $F(A)$ and $g(A)$ for a given sequence $A$. In this note, we study the Frobenius problem of $A=\left(a,ba+d,b^2a+\frac{b^2-1}{b-1}d,...,b^ka+\frac{b^k-1}{b-1}d\right)$ and obtain formulas for $F(A)$ and $g(A)$ when $a\geq k-1$. Our formulas simplifies further for some special cases, such as repunit, Mersenne and Thabit numerical semigroups. The idea is similar to that in [\cite{LiuXin23},arXiv:2306.03459].

math.NT↗

arXiv2023

The Frobenius Formula for $A=(a,ha+d,ha+b_2d,...,ha+b_kd)$

Given relative prime positive integers $A=(a_1, a_2, ..., a_n)$, the Frobenius number $g(A)$ is the largest integer not representable as a linear combination of the $a_i$'s with nonnegative integer coefficients. We find the ``Stable" property introduced for the square sequence $A=(a,a+1,a+2^2,\dots, a+k^2)$ naturally extends for $A(a)=(a,ha+dB)=(a,ha+d,ha+b_2d,...,ha+b_kd)$. This gives a parallel characterization of $g(A(a))$ as a ``congruence class function" modulo $b_k$ when $a$ is large enough. For orderly sequence $B=(1,b_2,\dots,b_k)$, we find good bound for $a$. In particular we calculate $g(a,ha+dB)$ for $B=(1,2,b,b+1)$, $B=(1,2,b,b+1,2b)$, $B=(1,b,2b-1)$ and $B=(1,2,...,k,K)$. Our idea also applies to the case $B=(b_1,b_2,...,b_k)$, $b_1> 1$.

math.CO↗

arXiv2023

A Combinatorial Approach to Frobenius Numbers of Some Special Sequences (Complete Version)

Let $A=(a_1, a_2, ..., a_n)$ be relative prime positive integers with $a_i\geq 2$. The Frobenius number $g(A)$ is the greatest integer not belonging to the set $\big\{ \sum_{i=1}^na_ix_i\ |x_i\in \mathbb{N}\big\}$. The general Frobenius problem includes the determination of $g(A)$ and the related Sylvester number $n(A)$ and Sylvester sum $s(A)$. We present a new approach to the Frobenius problem. Basically, we transform the problem into an easier optimization problem. If the new problem can be solved explicitly, then we will be able to obtain a formula of $g(A)$. We illustrate the idea by giving concise proof of some existing formulas and finding some interesting new formulas of $g(A), n(A), s(A)$. Moreover, we find that MacMahon's partition analysis applies to give a new way of calculating $n(A), s(A)$ by using a rational function representation of a polynomial determined by $A$.

math.CO↗

arXiv2019

Pressure induced electronic structure transformation of topological semimetal

We study the electronic structure change of yttrium trihydride $\mathrm{YH_3}$ by applying a hydrostatic pressure. At zero pressure, $\mathrm{YH_3}$ has the structure with energy favored symmetry group $P\bar{3}c1$ (165). From first principle calculation, we argue that the band crossings are caused by overlapping of an electron- and hole-like bands. Besides the space inversion symmetry ($\mathcal{I}$) and the time reversal symmetry, the band structure also exhibits an approximate particle-hole symmetry. Thus, $\mathrm{YH_3}$ can be viewed as a pseudo nodal surface semimetal belongs to class BDI of the ten-fold AZ+ $\mathcal{I}$ classifications of gapless topological matter. As pressure increases, the approximate particle-hole symmetry is gradually broken and the pseudo nodal surface turns into a nodal ring belonging to the class AI with fewer non-spatial symmetries. Also, the nodal ring is shrinking in the process. At about $31$ GPa, which is higher than the reported structure phase transition pressure $21$ GPa, the nodal ring shrinks to a nodal point. When above $31$ GPa, all band crossings are gapped out and $\mathrm{YH_3}$ becomes a trivial insulator eventually.

cond-mat.mtrl-sci↗

arXiv2017

No-go for tree-level R-symmetry breaking

We show that in gauge mediation models with tree-level R-symmetry breaking where supersymmetry and R-symmetries are broken by different fields, the gaugino mass either vanishes at one loop or finds a contribution from loop-level R-symmetry breaking. Thus tree-level R-symmetry breaking for phenomenology is either no-go or redundant in the simplest type of models. Including explicit messenger mass terms in the superpotential with a particular R-charge arrangement is helpful to bypass the no-go theorem, and the resulting gaugino mass is suppressed by the messenger mass scale.

hep-ph↗

arXiv2017

Free compact boson on branched coverings of the torus

We have studied the free compact boson on a $n$-sheeted covering of the torus gluing alone $m$ branch cuts. It is interesting because when the branched cuts are chosen to be real, the partition function is related to the $n$-th Rényi entanglement entropy of $m$ disjoint intervals in a finite system at finite temperature. After proposing a canonical homology basis and its dual basis of the covering surface, we find that the partition function can be written in terms of theta functions.

hep-th↗

arXiv2015

Two intervals Rényi entanglement entropy of compact free boson on torus

We compute the $N=2$ Rényi entanglement entropy of two intervals at equal time in a circle, for the theory of a 2d compact complex free scalar at finite temperature. This is carried out by performing functional integral on a genus 3 ramified cover of the torus, wherein the quantum part of the integral is captured by the four point function of twist fields on the worldsheet torus, and the classical piece is given by summing over winding modes of the genus 3 surface onto the target space torus. The final result is given in terms of a product of theta function and certain multi-dimensional theta function. We demonstrate the T-duality invariance of the result. We also study its low temperature limit. In the case in which the size of the intervals and of their separation are much smaller than the whole system, our result is in exact agreement with the known result for two intervals on an infinite system at zero temperature \cite{eeoftwo}. In the case in which the separation between the two intervals is much smaller than the interval length, the leading thermal corrections take the same universal form as proposed in \cite{Cardy:2014jwa,Chen:2015cna} for Rényi entanglement entropy of a single interval.

hep-th↗