Long zero-free sequences in finite cyclic groups

A sequence in an additively written abelian group is called zero-free if each of its nonempty subsequences has sum different from the zero element of the group. The article determines the structure of the zero-free sequences with lengths greater than $n/2$ in the additive group \Zn/ of integers modulo $n$. The main result states that for each zero-free sequence $(a_i)_{i=1}^\ell$ of length $\ell>n/2$ in \Zn/ there is an integer $g$ coprime to $n$ such that if $\bar{ga_i}$ denotes the least positive integer in the congruence class $ga_i$ (modulo $n$), then $\Sigma_{i=1}^\ell\bar{ga_i}<n$. The answers to a number of frequently asked zero-sum questions for cyclic groups follow as immediate consequences. Among other applications, best possible lower bounds are established for the maximum multiplicity of a term in a zero-free sequence with length greater than $n/2$, as well as for the maximum multiplicity of a generator. The approach is combinatorial and does not appeal to previously known nontrivial facts.Comment: 13 page

Topological Semimetals with Triply Degenerate Nodal Points in \theta-phase Tantalum Nitride

Using first-principles calculation and symmetry analysis, we propose that \theta-TaN is a topological semimetal having a new type of point nodes, i.e., triply degenerate nodal points. Each node is a band crossing between degenerate and non-degenerate bands along the high-symmetry line in the Brillouin zone, and is protected by crystalline symmetries. Such new type of nodes will always generate singular touching points between different Fermi surfaces and 3D spin texture around them. Breaking the crystalline symmetry by external magnetic field or strain leads to various of topological phases. By studying the Landau levels under a small field along $c$-axis, we demonstrate that the system has a new quantum anomaly that we call "helical anomaly".Comment: 21 pages, 5 figures with supplemental material

Long nn-zero-free sequences in finite cyclic groups

A sequence in the additive group ${\mathbb Z}_n$ of integers modulo $n$ is called $n$-zero-free if it does not contain subsequences with length $n$ and sum zero. The article characterizes the $n$-zero-free sequences in ${\mathbb Z}_n$ of length greater than $3n/2-1$. The structure of these sequences is completely determined, which generalizes a number of previously known facts. The characterization cannot be extended in the same form to shorter sequence lengths. Consequences of the main result are best possible lower bounds for the maximum multiplicity of a term in an $n$-zero-free sequence of any given length greater than $3n/2-1$ in ${\mathbb Z}_n$, and also for the combined multiplicity of the two most repeated terms. Yet another application is finding the values in a certain range of a function related to the classic theorem of Erd\H{o}s, Ginzburg and Ziv.Comment: 11 page

Sampled-Data and Harmonic Balance Analyses of Average Current-Mode Controlled Buck Converter

Dynamics and stability of average current-mode control of buck converters are analyzed by sampled-data and harmonic balance analyses. An exact sampled-data model is derived. A new continuous-time model "lifted" from the sampled-data model is also derived, and has frequency response matched with experimental data reported previously. Orbital stability is studied and it is found unrelated to the ripple size of the current-loop compensator output. An unstable window of the current-loop compensator pole is found by simulations, and it can be accurately predicted by sampled-data and harmonic balance analyses. A new S plot accurately predicting the subharmonic oscillation is proposed. The S plot assists pole assignment and shows the required ramp slope to avoid instability.Comment: Submitted to International Journal of Circuit Theory and Applications on August 9, 2011; Manuscript ID: CTA-11-016