A Criterion for Decomposabilty in QYBE

In this paper, set theoretic solutions of the Quantum Yang–Baxter Equations are considered. Etingof et al. [ 8] defined the structure group for non-degenerate solutions and gave some properties of this group. In particular, they provided a criterion for decomposability of involutive solutions based on the transitivity of the structure group. In that paper, the diagonal permutation $T$ is also introduced. It is known that this permutation is trivial exactly when the solution is square free. Rump [ 12] proved that these solutions are decomposable except in the trivial case. Later, Ramirez and Vendramin [ 11] gave some criteria for decomposability related with the diagonal permutation $T$. In this paper it was proven that an involutive solution is decomposable when the number of symbols of the solution and the order of the diagonal permutation $T$ are coprime.

Quadrature protection of squeezed states in a one-dimensional photonic topological insulator

What is the role of topology in the propagation of quantum light in photonic lattices? We address this question by studying the propagation of squeezed states in a topological one-dimensional waveguide array, benchmarking our results with those for a topologically trivial localized state, and studying their robustness against disorder. Specifically, we study photon statistics, one-mode and two-mode squeezing, and entanglement generation when the localized state is excited with squeezed light. These quantum properties inherit the shape of the localized state but, more interestingly, and unlike in the topologically trivial case, we find that propagation of squeezed light in a topologically protected state robustly preserves the phase of the squeezed quadrature as the system evolves. We show how this latter topological advantage can be harnessed for quantum information protocols.

Spanning trees in random regular uniform hypergraphs

Let $${{\mathcal G}_{n,r,s}}$$ denote a uniformly random r-regular s-uniform hypergraph on the vertex set {1, 2, … , n}. We establish a threshold result for the existence of a spanning tree in $${{\mathcal G}_{n,r,s}}$$ , restricting to n satisfying the necessary divisibility conditions. Specifically, we show that when s ≥ 5, there is a positive constant ρ(s) such that for any r ≥ 2, the probability that $${{\mathcal G}_{n,r,s}}$$ contains a spanning tree tends to 1 if r > ρ(s), and otherwise this probability tends to zero. The threshold value ρ(s) grows exponentially with s. As $${{\mathcal G}_{n,r,s}}$$ is connected with probability that tends to 1, this implies that when r ≤ ρ(s), most r-regular s-uniform hypergraphs are connected but have no spanning tree. When s = 3, 4 we prove that $${{\mathcal G}_{n,r,s}}$$ contains a spanning tree with probability that tends to 1, for any r ≥ 2. Our proof also provides the asymptotic distribution of the number of spanning trees in $${{\mathcal G}_{n,r,s}}$$ for all fixed integers r, s ≥ 2. Previously, this asymptotic distribution was only known in the trivial case of 2-regular graphs, or for cubic graphs.

Polynomial and power series ring extensions from sequences

Let [Formula: see text] be a commutative ring with identity. Let [Formula: see text] and [Formula: see text] be the collection of polynomials and, respectively, of power series with coefficients in [Formula: see text]. There are a lot of multiplications in [Formula: see text] and [Formula: see text] such that together with the usual addition, [Formula: see text] and [Formula: see text] become rings that contain [Formula: see text] as a subring. These multiplications are from a class of sequences [Formula: see text] of positive integers. The trivial case of [Formula: see text], i.e. [Formula: see text] for all [Formula: see text], gives the usual polynomial and power series ring. The case [Formula: see text] for all [Formula: see text] gives the well-known Hurwitz polynomial and Hurwitz power series ring. In this paper, we study divisibility properties of these polynomial and power series ring extensions for general sequences [Formula: see text] including UFDs and GCD-domains. We characterize when these polynomial and power series ring extensions are isomorphic to each other. The relation between them and the usual polynomial and power series ring is also presented.

Verification of a resetting protocol for an uncontrolled superconducting qubit

Quantum resetting protocols allow a quantum system to be sent to a state in the past by making it interact with quantum probes when neither the free evolution of the system nor the interaction is controlled. We experimentally verify the simplest non-trivial case of a quantum resetting protocol, known as the $${{\mathcal{W}}}_{4}$$ W 4 protocol, with five superconducting qubits, testing it with different types of free evolutions and target–probe interactions. After projection, we obtained a reset state fidelity as high as 0.951, and the process fidelity was found to be 0.792. We also implemented 100 randomly chosen interactions and demonstrated an average success probability of 0.323 for $$\left|1\right\rangle$$ 1 and 0.292 for $$\left|-\right\rangle$$ − , and experimentally confirmed the nonzero probability of success for unknown interactions; the numerical simulated values are about 0.3. Our experiment shows that the simplest quantum resetting protocol can be implemented with current technologies, making such protocols a valuable tool in the eternal fight against unwanted evolution in quantum systems.

Alternative methods to derive the Black-Scholes-Merton equation

We investigate the derivation of option pricing involving several assets following the Geometric Brownian Motion (GBM). First, we propose some derivations based on the basic ideas of the assets. Next, we consider the trivial case where we have n assets. Finally, we consider different drifts, volatilities and Wiener processes but now from n stochastic assets taking into account a fixed-income.

Effective writing style transfer via combinatorial paraphrasing

Stylometry can be used to profile or deanonymize authors against their will based on writing style. Style transfer provides a defence. Current techniques typically use either encoder-decoder architectures or rule-based algorithms. Crucially, style transfer must reliably retain original semantic content to be actually deployable. We conduct a multifaceted evaluation of three state-of-the-art encoder-decoder style transfer techniques, and show that all fail at semantic retainment. In particular, they do not produce appropriate paraphrases, but only retain original content in the trivial case of exactly reproducing the text. To mitigate this problem we propose ParChoice: a technique based on the combinatorial application of multiple paraphrasing algorithms. ParChoice strongly outperforms the encoder-decoder baselines in semantic retainment. Additionally, compared to baselines that achieve nonnegligible semantic retainment, ParChoice has superior style transfer performance. We also apply ParChoice to multi-author style imitation (not considered by prior work), where we achieve up to 75% imitation success among five authors. Furthermore, when compared to two state-of-the-art rule-based style transfer techniques, ParChoice has markedly better semantic retainment. Combining ParChoice with the best performing rulebased baseline (Mutant-X [34]) also reaches the highest style transfer success on the Brennan-Greenstadt and Extended-Brennan-Greenstadt corpora, with much less impact on original meaning than when using the rulebased baseline techniques alone. Finally, we highlight a critical problem that afflicts all current style transfer techniques: the adversary can use the same technique for thwarting style transfer via adversarial training. We show that adding randomness to style transfer helps to mitigate the effectiveness of adversarial training.

Biomimicry of iridescent, patterned insect cuticles: comparison of biological and synthetic, cholesteric microcells using hyperspectral imaging

Biological systems inspire the design of multifunctional materials and devices. However, current synthetic replicas rarely capture the range of structural complexity observed in natural materials. Prior to the definition of a biomimetic design, a dual investigation with a common set of criteria for comparing the biological material and the replica is required. Here, we deal with this issue by addressing the non-trivial case of insect cuticles tessellated with polygonal microcells with iridescent colours due to the twisted cholesteric organization of chitin fibres. By using hyperspectral imaging within a common methodology, we compare, at several length scales, the textural, structural and spectral properties of the microcells found in the two-band cuticle of the scarab beetle Chrysina gloriosa with those of the polygonal texture formed in flat films of cholesteric liquid crystal oligomers. The hyperspectral imaging technique offers a unique opportunity to reveal the common features and differences in the spectral-spatial signatures of biological and synthetic samples at a 6-nm spectral resolution over 400 nm–1000 nm and a spatial resolution of 150 nm. The biomimetic design of chiral tessellations is relevant to the field of non-specular properties such as deflection and lensing in geometric phase planar optics.

REGULARITY OF THE WEIGHTED BERGMAN PROJECTION ON THE FOCK–BARGMANN–HARTOGS DOMAIN

The Fock–Bargmann–Hartogs domain $D_{n,m}(\,\unicode[STIX]{x1D707}):=\{(z,w)\in \mathbb{C}^{n}\times \mathbb{C}^{m}:\Vert w\Vert ^{2}<e^{-\unicode[STIX]{x1D707}\Vert z\Vert ^{2}}\}$, where $\unicode[STIX]{x1D707}>0$, is an unbounded strongly pseudoconvex domain with smooth real-analytic boundary. We compute the weighted Bergman kernel of $D_{n,m}(\,\unicode[STIX]{x1D707})$ with respect to the weight $(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}}$, where $\unicode[STIX]{x1D70C}(z,w):=\Vert w\Vert ^{2}-e^{-\unicode[STIX]{x1D707}\Vert z\Vert ^{2}}$ and $\unicode[STIX]{x1D6FC}>-1$. Then, for $p\in [1,\infty ),$ we show that the corresponding weighted Bergman projection $P_{D_{n,m}(\,\unicode[STIX]{x1D707}),(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}}}$ is unbounded on $L^{p}(D_{n,m}(\,\unicode[STIX]{x1D707}),(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}})$, except for the trivial case $p=2$. This gives an example of an unbounded strongly pseudoconvex domain whose ordinary Bergman projection is $L^{p}$ irregular when $p\in [1,\infty )\setminus \{2\}$, in contrast to the well-known positive $L^{p}$ regularity result on a bounded strongly pseudoconvex domain.