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Fix $t\in [1,\infty]$. Let $S$ be an atomic commutative semigroup and, for all $x\in S$, let $\mathscr{L}_t(S):=\{\|f\|_t:f\in Z(x)\}$ be the "$t$-length set" of $x$ (using the standard $l_p$-space definition of $\|\cdot\|_t$). The $t$-Delta set of $x$ (denoted $\Delta_t(S)$) is the set of gaps between consecutive elements of $\mathscr{L}_t(S)$; the Delta set of $S$ is then defined by $\bigcup\limits_{x\in S} \Delta_t(S)$. Though all existing literature on this topic considers the $1$-Delta set, recent results on the $t$-elasticity of Numerical Semigroups (Behera et. al.) for $t\neq 1$ have brought attention to other invariants, such as the $t$-Delta set for $t\neq 1$, as well. Here we characterize $\Delta_t(S)$ for all numerical semigroups $\langle a_1,a_2\rangle$ and all $t\in(1,\infty)$ outside a small family of extremal examples. We also determine the cardinality and describe the distribution of that aberrant family.

Understanding topological matter in the fractional quantum Hall (FQH) effect requires identifying the nature of edge state quasiparticles. FQH edge state at the filling factor $\nu=2/3$ in the spin-polarized and non-polarized phases is represented by the two modes of composite fermions (CF) with the parallel or opposite spins described by the chiral Luttinger liquids. Tunneling through a quantum point contact (QPC) in such systems between different or similar spin phases is solved exactly. With the increase of the applied voltage, the QPC conductance grows from zero and saturates at $e^2/2h$ while a weak electron tunneling between the edge modes with the same spin transforms into a backscattering carried by the charge $q=e/2$ quasiparticles. These unusual quasiparticles and conductance plateau emerge in the QPC with one or two CF modes scattering into a single mode, as occurs in these systems. We propose experiments on the applied voltage and temperature dependence of the QPC conductance and noise that can shed light on the nature of edge states and FQH transport., Comment: 6 pages, 1 figure + 4-page supplementary material in a single file

Length density is a recently introduced factorization invariant, assigned to each element $n$ of a cancellative commutative atomic semigroup $S$, that measures how far the set of factorization lengths of $n$ is from being a full interval. We examine length density of elements of numerical semigroups (that is, additive subsemigroups of the non-negative integers).

Domain walls in fractional quantum Hall ferromagnets are gapless helical one-dimensional channels formed at the boundaries of topologically distinct quantum Hall (QH) liquids. Na\"{i}vely, these helical domain walls (hDWs) constitute two counter-propagating chiral states with opposite spins. Coupled to an s-wave superconductor, helical channels are expected to lead to topological superconductivity with high order non-Abelian excitations. Here we investigate transport properties of hDWs in the $\nu=2/3$ fractional QH regime. Experimentally we found that current carried by hDWs is substantially smaller than the prediction of the na\"{i}ve model. Luttinger liquid theory of the system reveals redistribution of currents between quasiparticle charge, spin and neutral modes, and predicts the reduction of the hDW current. Inclusion of spin-non-conserving tunneling processes reconciles theory with experiment. The theory confirms emergence of spin modes required for the formation of fractional topological superconductivity., Comment: 26 pages, 8 figures

A numerical set $S$ is a cofinite subset of $\mathbb{N}$ which contains $0$. We use the natural bijection between numerical sets and Young diagrams to define a numerical set $\widetilde{S}$, such that their Young diagrams are complements. We determine various properties of $\widetilde{S}$, particularly with an eye to closure under addition (for both $S$ and $\widetilde{S}$), which promotes a numerical set to become a numerical semigroup., Comment: 12 pages

For a commutative cancellative monoid $M$, we introduce the notion of the length density of both a nonunit $x\in M$, denoted $\mathrm{LD}(x)$, and the entire monoid $M$, denoted $\mathrm{LD}(M)$. This invariant is related to three widely studied invariants in the theory of non-unit factorizations, $L(x)$, $\ell(x)$, and $\rho(x)$. We consider some general properties of $\mathrm{LD}(x)$ and $\mathrm{LD}(M)$ and give a wide variety of examples using numerical semigroups, Puiseux monoids, and Krull monoids. While we give an example of a monoid $M$ with irrational length density, we show that if $M$ is finitely generated, then $\mathrm{LD}(M)$ is rational and there is a nonunit element $x\in M$ with $\mathrm{LD}(M)=\mathrm{LD}(x)$ (such a monoid is said to have accepted length density). While it is well-known that the much studied asymptotic versions of $L(x)$, $\ell (x)$ and $\rho (x)$ (denoted $\overline{L}(x)$, $\overline{\ell}(x)$, and $\overline{\rho} (x)$) always exist, we show the somewhat surprising result that $\overline{\mathrm{LD}}(x) = \lim_{n\rightarrow \infty} \mathrm{LD}(x^n)$ may not exist. We also give some finiteness conditions on $M$ that force the existence of $\overline{\mathrm{LD}}(x)$.

A numerical semigroup $S$ is an additive subsemigroup of the non-negative integers, containing zero, with finite complement. Its multiplicity $m$ is its smallest nonzero element. The Apery set of $S$ is the set $\text{Ap}(S) = \{n \in S : n-m \notin S\}$. Fixing a numerical semigroup, we ask how many elements of its Apery set have nonunique factorization, and define several new invariants.

For affine monoids of dimension 2 with embedding dimension 2 and 3, we study the problem of determining when a vector is an element of the monoid, and the problem of determining the elasticity of a monoid element.

A numerical semigroup $S$ is a subset of the non-negative integers containing $0$ that is closed under addition. The Hilbert series of $S$ (a formal power series equal to the sum of terms $t^n$ over all $n \in S$) can be expressed as a rational function in $t$ whose numerator is characterized in terms of the topology of a simplicial complex determined by membership in $S$. In this paper, we obtain analogous rational expressions for the related power series whose coefficient of $t^n$ equals $f(n)$ for one of several semigroup-theoretic invariants $f:S \to \mathbb R$ known to be eventually quasipolynomial.

A numerical semigroup $S$ is an additive subsemigroup of the non-negative integers with finite complement, and the squarefree divisor complex of an element $m \in S$ is a simplicial complex $\Delta_m$ that arises in the study of multigraded Betti numbers. We compute squarefree divisor complexes for certain classes numerical semigroups, and exhibit a new family of simplicial complexes that are occur as the squarefree divisor complex of some numerical semigroup element.

The catenary degree of an element $n$ of a cancellative commutative monoid $S$ is a nonnegative integer measuring the distance between the irreducible factorizations of $n$. The catenary degree of the monoid $S$, defined as the supremum over all catenary degrees occurring in $S$, has been heavily studied as an invariant of nonunique factorization. In this paper, we investigate the set $\mathsf C(S)$ of catenary degrees achieved by elements of $S$ as a factorization invariant, focusing on the case where $S$ in finitely generated (where $\mathsf C(S)$ is known to be finite). Answering an open question posed by Garc\'ia-S\'anchez, we provide a method to compute the smallest nonzero element of $\mathsf C(S)$ that parallels a well-known method of computing the maximum value. We also give several examples demonstrating certain extremal behavior for $\mathsf C(S)$, and present some open questions for further study.

We generalize the geometric sequence $\{a^p, a^{p-1}b, a^{p-2}b^2,...,b^p\}$ to allow the $p$ copies of $a$ (resp. $b$) to all be different. We call the sequence $\{a_1a_2a_3\cdots a_p, b_1a_2a_3\cdots a_p, b_1b_2a_3\cdots a_p,\ldots, b_1b_2b_3\cdots b_p\}$ a \emph{compound sequence}. We consider numerical semigroups whose minimal set of generators form a compound sequence, and compute various semigroup and arithmetical invariants, including the Frobenius number, Ap\'ery sets, Betti elements, and catenary degree. We compute bounds on the delta set and the tame degree.

Let $G$ be an abelian group, let $S$ be a sequence of terms $s_1,s_2,...,s_{n}\in G$ not all contained in a coset of a proper subgroup of $G$, and let $W$ be a sequence of $n$ consecutive integers. Let $$W\odot S=\{w_1s_1+...+w_ns_n:\;w_i {a term of} W,\, w_i\neq w_j{for} i\neq j\},$$ which is a particular kind of weighted restricted sumset. We show that $|W\odot S|\geq \min\{|G|-1,\,n\}$, that $W\odot S=G$ if $n\geq |G|+1$, and also characterize all sequences $S$ of length $|G|$ with $W\odot S\neq G$. This result then allows us to characterize when a linear equation $$a_1x_1+...+a_rx_r\equiv \alpha\mod n,$$ where $\alpha,a_1,..., a_r\in \Z$ are given, has a solution $(x_1,...,x_r)\in \Z^r$ modulo $n$ with all $x_i$ distinct modulo $n$. As a second simple corollary, we also show that there are maximal length minimal zero-sum sequences over a rank 2 finite abelian group $G\cong C_{n_1}\oplus C_{n_2}$ (where $n_1\mid n_2$ and $n_2\geq 3$) having $k$ distinct terms, for any $k\in [3,\min\{n_1+1,\,\exp(G)\}]$. Indeed, apart from a few simple restrictions, any pattern of multiplicities is realizable for such a maximal length minimal zero-sum sequence.

We have studied the zero-temperature statistics of charge transfer between the two edges of Quantum Hall liquids with filling factors $\nu_{0,1}=1/(2 m_{0,1}+1)$ forming Mach-Zehnder interferometer. The known Bethe ansatz solution for symmetric interferometer is used to obtain the cumulant-generating function of charge at constant voltage $V$ between the edges. Its low-$V$ behavior can be interpreted in terms of electron tunneling, while its large-$V$ asymptotics reproduces the $m$-state dynamics ($m\equiv 1+m_{0}+m_{1}$) of quasiparticles with fractional (for $m>1$) charge and statistics. We also analyze the transition region between electrons and quasiparticles., Comment: 5 two-column pages in RevTex4, 2 eps figures, final version with corrected misprints, references, and introduction as appeared in PRB Rapid Comm

We have studied theoretically the tunneling between two edges of Quantum Hall liquids (QHL) of different filling factors, $\nu_{0,1}=1/(2 m_{0,1}+1)$, with $m_0 \geq m_1\geq 0$, through two separate point contacts in the geometry of Mach-Zehnder interferometer [Y. Ji et al., Nature {\bf 422}, 415 (2003); I. Neder et al., Phys.\ Rev.\ Lett. {\bf 96}, 016804 (2006)]. The quasi-particle formulation of the interferometer model is derived as a dual to the initial electron model, in the limit of strong electron tunneling reached at large voltages or temperatures. For $m\equiv 1+m_{0}+m_{1}>1$, the tunneling of quasiparticles of fractional charge $e/m$ leads to non-trivial $m$-state dynamics of effective flux through the interferometer, which restores the regular "electron" periodicity of the current in flux despite the fractional charge and statistics of quasiparticles. The exact solution available for equal times of propagation between the contacts along the two edges demonstrates that the interference pattern of modulation of the tunneling current by flux depends on voltage and temperature only through a common amplitude., Comment: fourteen two-column pages in RevTex4, 4 eps figure, extended final verson as appeared in PRB

We consider tunneling between two edges of Quantum Hall liquids (QHL) of filling factors $\nu_{0,1}=1/(2 m_{0,1}+1)$, with $m_0 \geq m_1\geq 0$, through two point contacts forming Mach-Zehnder interferometer. Quasi-particle description of the interferometer is derived explicitly through the instanton duality transformation of the initial electron model. For $m_{0}+m_{1}+1\equiv m>1$, tunneling of quasiparticles of charge $e/m$ leads to non-trivial $m$-state dynamics of effective flux through the interferometer, which restores the regular ``electron'' periodicity of the current in flux. The exact solution available for equal propagation times between the contacts along the two edges demonstrates that the interference pattern in the tunneling current depends on voltage and temperature only through a common amplitude., Comment: five two-column pages in RevTex4, 1 eps figure

Physical Review Letter {\bf 90}, 026802 (2003) by U. Z\"{u}licke and E. Shimshoni and subsequent extended paper proposed an ``exact solution'' of the problem of a tunnel junction of length $L$ between two single-mode edges of the Fractional Quantum Hall Liquids with different filling factors $\nu_{j}=1/(2m_j+1)$, $j=1,2$; $m_j=0,1,...$. The purpose of this comment is to point out that the solution found in these papers is not correct and to present correct solution of a particular instance of this model., Comment: two two-column pages in RevTex4, no figures

We consider tunneling through two point contacts between two edges of Quantum Hall liquids of different filling factors $\nu_{0,1}=1/ (2m_{0,1}+1)$ with $m_0-m_1\equiv m>0$. Properties of the antidot formed between the point contacts in the strong-tunneling limit are shown to be very different from the $\nu_0 =\nu_1$ case, and include vanishing average total current in the two contacts and quasiparticles of charge $e/m$. For $m>1$, quasiparticle tunneling leads to non-trivial $m$-state dynamics of effective flux through the antidot which restores the regular ``electron'' periodicity of the current in flux despite the fractional charge and statistics of quasiparticles., Comment: 5 two-column pages, 2 figures

The $q$-round Renyi-Ulam pathological liar game with $k$ lies on the set $[n]:=\{1,...,n\}$ is a 2-player perfect information zero sum game. In each round Paul chooses a subset $A\subseteq [n]$ and Carole either assigns 1 lie to each element of $A$ or to each element of $[n]\setminus A$. Paul wins if after $q$ rounds there is at least one element with $k$ or fewer lies. The game is dual to the original Renyi-Ulam liar game for which the winning condition is that at most one element has $k$ or fewer lies. We prove the existence of a winning strategy for Paul to the existence of a covering of the discrete hypercube with certain relaxed Hamming balls. Defining $F^*_k(q)$ to be the minimum $n$ such that Paul can win the $q$-round pathological liar game with $k$ lies and initial set $[n]$, we find $F^*_1(q)$ and $F^*_2(q)$ exactly. For fixed $k$ we prove that $F_k^*(q)$ is within an absolute constant (depending only on $k$) of the sphere bound, $2^q/\binom{q}{\leq k}$; this is already known to hold for the original Renyi-Ulam liar game due to a result of J. Spencer., Comment: 18 pages, 1 figure * Typo corrected in Definition 17. * Reference information completed for reference [4]

We have found solution to a model of tunneling between a multi-channel Fermi liquid reservoir and an edge of the principal fractional quantum Hall liquid (FQHL) in the strong coupling limit. The solution explains how the absence of the time-reversal symmetry at high energies due to chiral edge propagation makes the universal two-terminal conductance of the FQHL fractionally quantized and different from that of a 1D Tomonaga-Luttinger liquid wire, where a similar model but preserving the time-reversal symmetry predicts unsuppressed free-electron conductance., Comment: 5 twocolumn pages in RevTex, no figures, more explanations added, a short version was published in JETP Letters, vol.74, 87 (2001)

We have analyzed Coulomb drag between currents of interacting electrons in two parallel one-dimensional conductors of finite length $L$ attached to external reservoirs. For strong coupling, the relative fluctuations of electron density in the conductors acquire energy gap $M$. At energies larger than $\Gamma = const \times v_- \exp (-LM/v_-)/L + \Gamma_{+}$, where $\Gamma_{+}$ is the impurity scattering rate, and for $L>v_-/M$, where $v_-$ is the fluctuation velocity, the gap leads to an ``ideal'' drag with almost equal currents in the conductors. At low energies the drag is suppressed by coherent instanton tunneling, and the zero-temperature transconductance vanishes, indicating the Fermi liquid behavior., Comment: 5 twocolumn pages in RevTex, added 1 eps-Figure and calculation of trans-resistance

A low energy crossover (see cond-mat/9711167) induced by Fermi liquid reservoirs in transport through a 1D Mott-Hubbard insulator of finite length $L$ is examined in the presence of impurity pinning. Under the assumption that the Hubbard gap 2M is large enough: $M > T_L \equiv v_c/L$ ($v_c$: charge velocity in the wire) and the impurity backscattering rate $\Gamma_1 \ll T_L$, the conductance vs. voltage/temperature displays a zero-energy resonance. Transport through a spin gapped 1D system is also described availing of duality between the backscattered current of this system and the direct current of the Mott-Hubbard insulator., Comment: 5 twocolumn pages in RevTex, no figures

Suppression of electron current $ \Delta I$ through a 1D channel of length $L$ connecting two Fermi liquid reservoirs is studied taking into account the Umklapp interaction induced by a periodic potential. This interaction opens band gaps at the integer fillings and Hubbard gaps $2m$ at some rational fillings in the infinite wire: $L \to \infty$. In the perturbative regime where $m \ll v_c/L$ ($v_c:$ charge velocity), and for small deviations $\delta n$ of the electron density from its commensurate values $- \Delta I/V$ can diverge with some exponent as voltage or temperature $V,T$ decreases above $E_c=max(v_c/L,v_c \delta n)$, while it goes to zero below $E_c$. This results in a non-monotonous behavior of the conductance. In the case when the Umklapp interaction creates a large Mott-Hubbard gap $2m \gg T_L $ inside the wire, the transport is suppressed near half-filling everywhere inside the gap except for an exponentially small region of $V,T < T_L exp(-2m/T_L)$., Comment: to appear in "Highlights in Condensed Matter Physics", proceedings of the APCTP/ICTP Joint International Conference, June 1998, Seoul, Y.M. Cho and M. Virasoro eds, World Scientific (Singapore); 10 twocolumn pages in RevTex, 4 figures included

The ground state of a phase-coherent mesoscopic system is sensitive to its environment. We investigate the persistent current of a ring with a quantum dot which is capacitively coupled to an external circuit with a dissipative impedance. At zero temperature, zero-point quantum fluctuations lead to a strong suppression of the persistent current with decreasing external impedance. We emphasize the role of displacement currents in the dynamical fluctuations of the persistent current and show that with decreasing external impedance the fluctuations exceed the average persistent current., Comment: 4 pages, 2 eps figures

Transport through a one channel wire of length $L$ confined between two leads is examined when the 1D electron system has an energy gap $2M$: $M > T_L \equiv v_c/L$ induced by the interaction in charge mode ($v_c$: charge velocity in the wire). In spinless case the transformation of the leads electrons into the charge density wave solitons of fractional charge $q$ entails a non-trivial low energy crossover from the Fermi liquid behavior below the crossover energy $T_x \propto \sqrt{T_L M} e^{-M /[T_L(1-q^2)]}$ to the insulator one with the fractional charge in current vs. voltage, conductance vs. temperature, and in shot noise. Similar behavior is predicted for the Mott insulator of filling factor $\nu = integer/(2 m')$., Comment: 5 twocolumn pages in RevTex, no figures

Transport through a 1D Mott-Hubbard insulator of a finite length $L$ is studied beyond perturbative approach. At special value of the low energy constant of the interaction we have mapped the problem onto the exactly solvable models and found current vs. voltage $V$ at high temperature $T>max(m,T_L)$ and at low energy $T,V

One-dimensional interacting electrons in a quantum wire connected to reservoirs are studied theoretically. The difference in the Tomonaga-Luttinger interaction constants between the wire (g) and reservoirs $(g_{\infty})$ produces the cross-correlation between the right- and left-going chiral components of the charge density wave field. The low energy asymptotics of this field correlator, which is determined by (g) and $(g_{\infty})$, specifies renormalization of physical quantities. We have found that charge of the carriers in the shot noise is determined by $g_\infty$ (no renormalization for the Fermi liquid reservoirs) at any energy, meanwhile the factor g renormalizing the charge and spin susceptibilities emerges in the threshold structures at some rational fillings., Comment: Extended version will appear in PRB

Finite length of a one channel wire results in crossover from a Tomonaga-Luttinger to Fermi liquid behavior with lowering energy scale. In condition that voltage drop $(V)$ mostly occurs across a tunnel barrier inside the wire we found coefficients of temperature/voltage expansion of low energy conductance as a function of constant of interaction, right and left traversal times. At higher voltage the finite length contribution exhibits oscillations related to both traversal times and becomes a slowly decaying correction to the scale-invariant $V^{1/g-1}$ dependence of the conductance., Comment: 12 pages of RevTex file and 1 PS file figure

Suppression of electron current $ \Delta I$ through a 1D channel of length $L$ connecting two Fermi liquid reservoirs is studied taking into account the Umklapp electron-electron interaction induced by a periodic potential. This interaction causes Hubbard gaps $E_H$ for $L \to \infty$. In the perturbative regime where $E_H \ll v_c/L$ ($v_c:$ charge velocity), and for small deviations $\delta n$ of the electron density from its commensurate values $- \Delta I/V$ can diverge with some exponent as voltage or temperature $V,T$ decreases above $E_c=max(v_c/L,v_c \delta n)$, while it goes to zero below $E_c$. This results in a nonmonotonous behavior of the conductance., Comment: Final variant published in PRL, 79, 1714; minor corrections

The effect of Coulomb screening on the magnetic impurity behavior is analysed. Two types of the behavior corresponding to either integer or fractional occupation numbers of the low lying magnetic level are described. The features in the dependence of the resistivity on the magnetic field specific for these regimes are predicted., Comment: 4 pages, Latex, 1 Postcript figure in a separate file, to appear in Proceedings of the ISQM-Tokyo '95, Quantum Coherence and Decoherence. Ed. Y.A.Ono, Elsevier