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This paper studies the geometric structure of the locus $��{\Phi }_3��\left(�X�\right)��$ of rank 3 quadratic equations of the Veronese variety $��X�=�{\nu }_d��\left(�P^n�\right)��$. Specifically, we investigate the minimal irreducible decomposition of $��{\Phi }_3��\left(�X�\right)��$ of rank 3 quadratic equations and analyze the geometric properties of the irreducible components of $��{\Phi }_3��\left(�X�\right)��$ such as their desingularizations. Additionally, we explore the non-singularity and singularity of these irreducible components of $��{\Phi }_3��\left(�X�\right)��$.

Fraser–Sargent surfaces are free boundary minimal surfaces in the four-dimensional unit Euclidean ball. Extended infinitely they define immersed minimal surfaces in the Euclidean space. The parts of these surfaces outside the ball are exterior-free boundary minimal surfaces. We prove that they are stable. Independently of it, we find an upper bound on the index of Fraser–Sargent surfaces inside the ball. Also, we provide computational experiments and state a conjecture about an improved index lower bound of the orientable cover of Fraser–Sargent surfaces inside the ball. Finally, based on a similar computational experiment for infinitely extended Fraser–Sargent surfaces, we state a conjecture about their index.

On real hypersurafces of the Kähler surface $��S^2�\times �S^2�$ and $��H^2�\times �H^2�$, there are three typical operators: the shape operator, the structure Lie operator, and the contact Lie operator. In this paper, we study real hypersurfaces in $��S^2�\times �S^2�$ and $��H^2�\times �H^2�$ related to these operators. As the main results, we classify real hypersurfaces of both $��S^2�\times �S^2�$ and $��H^2�\times �H^2�$

Cordes' characterization of Heisenberg-smooth operators bridges a gap between the theory of pseudo-differential operators and quantum harmonic analysis (QHA). We give a new proof of the result by using the phase-space formalism of QHA. Our argument is flexible enough to generalize Cordes' result in several directions: (1) we can admit general quantization schemes, (2) allow for other phase-space geometries, (3) obtain Schatten-class analogs of the result, and (4) are able to characterize precisely ‘Heisenberg-analytic’ operators. For (3), we use QHA to derive Schatten versions of the Calderón–Vaillancourt theorem, which might be of independent interest.

In this paper, the approximation of Dirac operators with general $�\delta$-shell potentials supported on $�C^2$-curves in $�R^2$ or $�C^2$-surfaces in $�R^3$, which may be bounded or unbounded, is studied. It is shown under suitable conditions on the weight of the $�\delta$-interaction that a family of Dirac operators with regular, squeezed potentials converges in the norm resolvent sense to the Dirac operator with the $�\delta$-shell interaction.

This paper is dedicated to the classification of uniform vector bundles of rank $��d�+�1�$ over the Grassmannian $��G�(�d�,�n�)�$ ($��2�\le �d�\le �n�-�d�$) over an algebraically closed field in positive characteristics. Specifically, we show that all uniform vector bundles with rank $��d�+�1�$ over $��G�(�d�,�n�)�$ are homogeneous.