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Every polygon with n vertices in the complex projective plane is naturally associated with its adjoint curve of degree n-3. Hence the adjoint of a heptagon is a plane quartic. We prove that a general plane quartic is the adjoint of exactly 864 distinct complex heptagons. This number had been numerically computed by Kohn et al. We use intersection theory and the Scorza correspondence for quartics to show that 864 is an upper bound, complemented by a lower bound obtained through explicit analysis of the famous Klein quartic., Comment: 18 pages, 1 figure

Nuclear spin hyperpolarization enables real-time observation of metabolism and intermolecular interactions in vivo. 1-13C-Pyruvate is the leading hyperpolarized tracer currently under evaluation in several clinical trials as a promising molecular imaging agent. Still, the quest for a simple, fast, and efficient hyperpolarization technique is ongoing. Here, we describe that continuous, weak irradiation in the audio-frequency range of the 13C spin at 121 {\mu}T magnetic field (\sim twiceEarth\apos s field) enables spin order transfer from parahydrogen to 13C magnetization of 1-13C-pyruvate. These so-called LIGHT-SABRE pulses couple nuclear spin states of parahydrogen and pyruvate via the J-coupling network of reversibly exchanging Ir-complexes. Using \sim 100% parahydrogen at ambient pressure, we polarized 51 mM of 1-13C-pyruvate in the presence of 5.1 mM Ir-complex continuously and repeatedly to a polarization of 1.1% averaged over free and catalyst-bound pyruvate. The experiments were conducted at -8{\deg}C), where almost exclusively bound pyruvate was observed, corresponding to an estimated 11% polarization on bound pyruvate. The obtained hyperpolarization levels closely match those obtained via SABRE-SHEATH under otherwise identical conditions. The creation of three different types of spin orders was observed: transverse 13C magnetization along the applied magnetic field, 13C z-magnetization along the main field B_0, and 13C-1H zz-spin-order. With a superconducting quantum interference device (SQUID) for detection, we found that the generated spin orders result from tiny 1H-13C J-coupling interactions, which are not visible even with our narrow linewidth below 0.3 Hz.

Although 19F has high potential to serve as a background-free molecular marker in bioimaging, the molar amount of marker substance is often too small to enable 19F MR imaging or 19F NMR spectroscopy with a sufficiently high signal-to-noise ratio (SNR). Hyperpolarization methods such as parahydrogen-based hyperpolarization or dynamic nuclear polarization (DNP) can significantly improve the SNR, but require expensive and complex sample preparation and the removal of toxic catalysts and solvents. Therefore, we used the biologically compatible model of the fluorinated amino acid 3-fluoro-DL-tyrosine with riboflavin 5'-monophosphate (FMN) as a chromophore dissolved in D2O with 3.4% H2Odest. allowing to transform light energy into hyperpolarization of the 19F nucleus via photo-chemically induced dynamic nuclear polarization (photo-CIDNP). We used a low-cost high-power blue LED to illuminate the sample replacing traditionally used laser excitation, which is both potentially harmful and costly. For the first time, we present results of hyperpolarized 19F MRI and 19F NMR performed with a low-cost 0.6 T benchtop MRI system. The device allowed simultaneous dual channel 1H/19F NMR. 19F imaging was performed with a (0.94 mm)2 in-plane resolution. This enabled the spatial resolution of different degrees of hyperpolarization within the sample. We estimated the photo-CIDNP-based 19F signal enhancement at 0.6 T to be approximately 465. FMN did not bleach out even after multiple excitations, so that the signal-to-noise ratio could be further improved by averaging hyperpolarized signals. The results show that the easy-to-use experimental setup has a high potential to serve as an efficient preclinical tool for hyperpolarization studies in bioimaging., Comment: 16 pages, 5 figures, supplementary information (4pages, 4 figures)

Parahydrogen (pH2) is a convenient and cost efficient source for magnetic resonance signal enhancement. Transient interaction of pH2 with a metal organic complex in a signal amplification by reversible exchange (SABRE) experiment enabled more than 10% polarization for some 15N molecules. Here, we analyzed a variant of SABRE, consisting of an outer magnetic field alternating between a low field of ~1 \muT, where a polarization transfer takes place, and a higher field >50 \muT (alt-SABRE). We found effects of both of these fields on amplitude and the frequency of polarization transfer. Deviation of a lower magnetic field from a "perfect" condition of level anti-crossing increases the frequency of polarization transfer that can be exploited for polarization of short-lived transient SABRE complexes i.e. some substrates. Moreover, the coherences responsible for polarization transfer at a lower field persisted during magnetic field variation and continued their spin evolution at higher field with a frequency of 2.5 kHz at 54 \muT. The latter should be taken into consideration for an efficient alt-SABRE.

Given a real algebraic curve, embedded in projective space, we study the computational problem of deciding whether there exists a hyperplane meeting the curve in real points only. More generally, given any divisor on such a curve, we may ask whether the corresponding linear series contains an effective divisor with totally real support. This translates into a particular type of parametrized real root counting problem that we wish to solve exactly. On the other hand, it is known that for a given genus and number of real connected components, any linear series of sufficiently large degree contains a totally real effective divisor. Using the algorithms described in this paper, we solve a number of examples, which we can compare to the best known bounds for the required degree.

We study families of faces for convex semi-algebraic sets via the normal cycle which is a semi-algebraic set similar to the conormal variety in projective duality theory. We propose a convex algebraic notion of a "patch" -- a term recently coined by Ciripoi, Kaihnsa, L\"ohne, and Sturmfels as a tool for approximating the convex hull of a semi-algebraic set. We discuss geometric consequences, both for the semi-algebraic and convex geometry of the families of faces, as well as variations of our definition and their consequences., Comment: 22 pages

Kippenhahn's Theorem asserts that the numerical range of a matrix is the convex hull of a certain algebraic curve. Here, we show that the joint numerical range of finitely many Hermitian matrices is similarly the convex hull of a semi-algebraic set. We discuss an analogous statement regarding the dual convex cone to a hyperbolicity cone and prove that the class of bases of these dual cones is closed under linear operations. The result offers a new geometric method to analyze quantum states., Comment: 28 pages, 5 figures; version v2 extends version v1 in applications and examples and has more detailed proofs. Any comments are welcomed

We present a method for the simultaneous observation of heteronuclear multi-quantum coherences (up to the 3rd order), which give an additional degree of freedom for ultralow magnetic field (ULF) MR experiments, where the chemical shift is negligible. The nonequilibrium spin state is generated by Signal Amplification By Reversible Exchange (SABRE) and detected at ULF with SQUID-based NMR. We compare the results obtained by the heteronuclei Correlated SpectroscopY (COSY) with a Flip Angle FOurier Series (FAFOS) method. COSY allows a quantitative analysis of homo- and heteronuclei quantum coherences.

Hyperbolic polynomials are real multivariate polynomials with only real roots along a fixed pencil of lines. Testing whether a given polynomial is hyperbolic is a difficult task in general. We examine different ways of translating hyperbolicity into nonnegativity conditions, which can then be tested via sum-of-squares relaxations., Comment: 12 pages, 1 figure; minor changes in Section 2 and a few corrections

We describe a new method for constructing a spectrahedral representation of the hyperbolicity region of a hyperbolic curve in the real projective plane. As a consequence, we show that if the curve is smooth and defined over the rational numbers, then there is a spectrahedral representation with rational matrices. This generalizes a classical construction for determinantal representations of plane curves due to Dixon and relies on the special properties of real hyperbolic curves that interlace the given curve., Comment: 18 pages, 8 figures

We present a computational study of smooth curves of degree six in the real projective plane. In the Rokhlin-Nikulin classification, there are 56 topological types, refined into 64 rigid isotopy classes. We developed software that determines the topological type of a given sextic and used it to compute empirical probability distributions on the various types. We list 64 explicit representatives with integer coefficients, and we perturb these to draw many samples from each class. This allows us to explore how many of the bitangents, inflection points and tensor eigenvectors are real. We also study the real tensor rank, the construction of quartic surfaces with prescribed topology, and the avoidance locus, which is the locus of all real lines that do not meet a given sextic. This is a union of up to 46 convex regions, bounded by the dual curve., Comment: 28 pages, 8 figues; corrections and improved exposition

Hyperbolic programming is the problem of computing the infimum of a linear function when restricted to the hyperbolicity cone of a hyperbolic polynomial, a generalization of semidefinite programming. We propose an approach based on symbolic computation, relying on the multiplicity structure of the algebraic boundary of the cone, without the assumption of determinantal representability. This allows us to design exact algorithms able to certify the multiplicity of the solution and the optimal value of the linear function., Comment: Final version, published in the Journal of Algebra and its Applications (2017). 16 pages, 1 figure