The big (-1) -Jacobi, Bannai–Ito, and (-1) -Meixner–Pollaczek polynomials have appeared in the mathematical physics literature through differential- or difference-reflection eigenvalue problems. We show that these three families are governed by a single alternating mechanism, formulated at the level of orthogonality functionals, transpose operators, and structural identities. Starting from an ordinary orthogonal polynomial sequence in the quadratic variable, the resulting algorithm constructs the direct alternating family and transfers its moments, squared norms, and recurrence coefficients. One regular Geronimus inversion then produces the big (-1) -Jacobi and Bannai–Ito families, whereas the (-1) -Meixner–Pollaczek family is obtained directly.
In this paper, we define θ -orthotomicoids of spherical curves in spherical space. The relationship between θ -orthotomicoids and θ -evolutoids of spherical curves is established. We generalize the notion to the category of spherical frontals and find that θ -orthotomicoids of spherical frontals may have singularities, while θ -orthotomicoids of spherical unit speed curves are regular. Then, we investigate properties of θ -orthotomicoids of spherical frontals by using the singularity theory. From the viewpoint of the contact geometry, we prove the singularities of θ -orthotomicoids are deeply related to the order of contact between θ -orthotomicoids and specific circles on the unit sphere. Finally, we provide some examples to demonstrate main results.
Graph generation has attracted considerable attention in recent years. Most existing work focuses on generating graphs from collections, such as proteins, while the problem of generating a proxy from a single graph with comparable properties remains underexplored. Prior approaches often emphasize modifying neural architectures or training on edges with fixed node sets, which tends to produce high edge overlap and limited diversity. In addition, many neural graph generation methods overlook the mathematical structure of the input graph. For instance, key metrics such as wedge counts, which are determined by degree distributions, are often neglected. To address these limitations, we introduce a graph generation method that incorporates graph-theoretic principles into the learning process. Our approach preserves both global and local characteristics of the input graph while correcting the degree distribution to avoid duplicating the original topology. Unlike other single-graph methods that fix the node set, we generate new node embeddings and sample fresh node sets, yielding new connection topologies with substantially lower edge overlap.
Using recently developed algorithms, we compute and compare best L^2 and L^∞ rational approximations of analytic functions on the unit disk. Although there is some theory for these problems going back decades, this may be the first computational study. To compute the L^2 best approximations, we employ a new formulation of TF-IRKA in barycentric form.
