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We obtain a sharp bound on the number of self-intersections of a closed planar curve with trigonometric parameterization. Moreover, we show that a generic curve of this form is normal in the sense of Whitney.

The remarkable properties of toroidal metasurfaces, featuring ultrahigh-Q bound states in the continuum (BIC) resonances and nonradiating anapole modes, have garnered significant attention. The active manipulation of quasi-BIC resonance characteristics offers substantial potential for advancing tunable metasurfaces. Our study explores explicitly the application of vanadium dioxide, a phase change material widely used in active photonics and room-temperature bolometric detectors, to control quasi-BIC resonances in toroidal metasurfaces. The phase change transition of vanadium dioxide occurs in a narrow temperature range, providing a large variation in material resistivity. Through heating thin film patches of vanadium dioxide integrated into a metasurface comprising gold split-ring resonators on a sapphire substrate, we achieve remarkable control over the amplitude and frequency of quasi-BIC resonances due to their high sensitivity to losses present in the system. Breaking the symmetry of meta-atoms reveals enhanced tunability. The predicted maximum change in the quasi-BIC resonance amplitude reaches 14 dB with a temperature variation of approximately 10 {\deg}C., Comment: Submitted to Advanced Photonics Research

The existence of a median-type ternary operation on a metric space is known to have a number of implications for the geometry of the space. For such operations, if two of the three arguments coincide, they also coincide with the output of the operation. We consider ternary operations with the opposite property: if two of the arguments coincide, the output is equal to the third one. The existence of such an operation is a necessary condition for the space to be an absolute retract.

We study parametric amplification of electromagnetic waves using metasurfaces. We design a variable capacitor-loaded metasurface that can amplify incident electromagnetic waves. We analyze various regimes of operation of the system and find that we can achieve a significant gain (over 10 dB) in just one layer of such a structure, and this gain can be controlled by parametric modulation. We study the instability threshold for this system and show that a simple theoretical model agrees well with the results of full numerical simulations., Comment: This article was published in Optical Materials Express (https://opg.optica.org/ome)

Single-section quantum dash semiconductor lasers have attracted much attention as an integrated and simple platform for the generation of THz-wide and flat optical frequency combs in the telecom C-band. In this work, we present an experimental method allowing to increase the spectral width of the laser comb by the injection of a broadband optical noise from an external semiconductor optical amplifier that is spectrally overlapped with the quantum dash laser comb. The noise injection induces an amplification of the side modes of the laser comb which acquire a fixed phase relationship with the central modes of the comb. We demonstrate a broadening of the laser comb by a factor of two via this technique., Comment: 4 pages, 4 figures

The standard arithmetic measures of center, the mean and median, have natural topological counterparts which have been widely used in continuum theory. In the context of metric spaces it is natural to consider the Lipschitz continuous versions of the mean and median. We show that they are related to familiar concepts of the geometry of metric spaces: the bounded turning property, the existence of quasisymmetric parameterization, and others.

In this paper, the Lipschitz clustering property of a metric space refers to the existence of Lipschitz retractions between its finite subset spaces. Obstructions to this property can be either topological or geometric features of the space. We prove that uniformly disconnected spaces have the Lipschitz clustering property, while for some connected spaces, the lack of sufficiently short connecting curves turns out to be an obstruction. This property is shown to be invariant under quasihomogeneous maps, but not under quasisymmetric ones., Comment: added Corollary 4.7, corrected typos

We prove the continuity of logarithmic capacity under Hausdorff convergence of uniformly perfect planar sets. The continuity holds when the Hausdorff distance to the limit set tends to zero at sufficiently rapid rate, compared to the decay of the parameters involved in the uniformly perfect condition. The continuity may fail otherwise., Comment: 13 pages

A finite Blaschke product, restricted to the unit circle, is a smooth covering map. The maximum and minimum values of the derivative of this map reflect the geometry of the Blaschke product. We identify two classes of extremal Blaschke products: those that maximize the difference between the maximum and minimum of the derivative, and those that minimize it. Both classes turn out to have the same algebraic structure, being the quotient of two hypergeometric polynomials., Comment: Corrected typos, added references

For any metric space $X$, finite subset spaces of $X$ provide a sequence of isometric embeddings $X=X(1)\subset X(2)\subset\cdots$. The existence of Lipschitz retractions $r_n\colon X(n)\to X(n-1)$ depends on the geometry of $X$ in a subtle way. Such retractions are known to exist when $X$ is an Hadamard space or a finite-dimensional normed space. But even in these cases it was unknown whether the sequence $\{r_n\}$ can be uniformly Lipschitz. We give a negative answer by proving that $\operatorname{Lip}(r_n)$ must grow with $n$ when $X$ is a normed space or an Hadamard space.

This paper continues the investigation of the relation between the geometry of a circle embedding and the values of its Fourier coefficients. First, we answer a question of Kovalev and Yang concerning the support of the Fourier transform of a starlike embedding. An important special case of circle embeddings are homeomorphisms of the circle onto itself. Under a one-sided bound on the Fourier support, such homeomorphisms are rational functions related to Blaschke products. We study the structure of rational circle homeomorphisms and show that they form a connected set in the uniform topology.

Hardy spaces in the complex plane and in higher dimensions have natural finite-dimensional subspaces formed by polynomials or by linear maps. We use the restriction of Hardy norms to such subspaces to describe the set of possible derivatives of harmonic self-maps of a ball, providing a version of the Schwarz lemma for harmonic maps. These restricted Hardy norms display unexpected near-isometric duality between the exponents 1 and 4, which we use to give an explicit form of harmonic Schwarz lemma.

We numerically analyze a delay differential equation model of a short-cavity semiconductor laser with an intracavity frequency swept filter and reveal a complex bifurcation structure responsible for the asymmetry of the output characteristics of this laser. We show that depending on the direction of the frequency sweep of a narrowband filter, there exist two bursting cycles determined by different parts of a continuous-wave solutions branch., Comment: 5 pages, 5 figures