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1. Hypergeometric Integrals Modulo p and Hasse–Witt Matrices

   https://doi.org/10.1007/s40598-020-00168-2  

   Slinkin, Alexey; Varchenko, Alexander (January 2020, Arnold Mathematical Journal)

   null (Ed.)

   We consider the KZ differential equations over C in the case, when the hypergeometric solutions are one-dimensional integrals.We also consider the same differential equations over a finite field F_p. We study the space of polynomial solutions of these differential equations over F_p, constructed in a previous work by Schechtman and the second author. Using Hasse–Witt matrices, we identify the space of these polynomial solutions over F_p with the space dual to a certain subspace of regular differentials on an associated curve. We also relate these polynomial solutions over F_p and the hypergeometric solutions over C. 

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2. Lump solutions to nonlinear partial differential equations via Hirota bilinear forms

   Ma, Wen-Xiu; Zhou, Yuan (February 2018, Journal of differential equations)

   Lump solutions are analytical rational function solutions localized in all directions in space. We analyze a class of lump solutions, generated from quadratic functions, to nonlinear partial differential equations. The basis of success is the Hirota bilinear formulation and the primary object is the class of positive multivariate quadratic functions. A complete determination of quadratic functions positive in space and time is given, and positive quadratic functions are characterized as sums of squares of linear functions. Necessary and sufficient conditions for positive quadratic functions to solve Hirota bilinear equations are presented, and such polynomial solutions yield lump solutions to nonlinear partial differential equations under the dependent variable logarithmic derivative transformations. Applications are made for a few generalized KP and BKP equations. 

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3. Obtaining Approximately Optimal and Diverse Solutions via Dispersion

   https://doi.org/10.1007/978-3-031-20624-5_14  

   Gao, Jie; Goswami, Mayank; C.S., Karthik; Tsia, Meng-Tsung; Tsai, Shih-Yu; Yang Hao-Tsung (November 2022, Latin American Symposium on Theoretical Informatics)

   There has been a long-standing interest in computing diverse solutions to optimization problems. In 1995 J. Krarup [28] posed the problem of finding k-edge disjoint Hamiltonian Circuits of minimum total weight, called the peripatetic salesman problem (PSP). Since then researchers have investigated the complexity of finding diverse solutions to spanning trees, paths, vertex covers, matchings, and more. Unlike the PSP that has a constraint on the total weight of the solutions, recent work has involved finding diverse solutions that are all optimal. However, sometimes the space of exact solutions may be too small to achieve sufficient diversity. Motivated by this, we initiate the study of obtaining sufficiently-diverse, yet approximately-optimal solutions to optimization problems. Formally, given an integer k, an approximation factor c, and an instance I of an optimization problem, we aim to obtain a set of k solutions to I that a) are all c approximately-optimal for I and b) maximize the diversity of the k solutions. Finding such solutions, therefore, requires a better understanding of the global landscape of the optimization function. Given a metric on the space of solutions, and the diversity measure as the sum of pairwise distances between solutions, we first provide a general reduction to an associated budget-constrained optimization (BCO) problem, where one objective function is to optimized subject to a bound on the second objective function. We then prove that bi-approximations to the BCO can be used to give bi-approximations to the diverse approximately optimal solutions problem. As applications of our result, we present polynomial time approximation algorithms for several problems such as diverse c-approximate maximum matchings, shortest paths, global min-cut, and minimum weight bases of a matroid. The last result gives us diverse c-approximate minimum spanning trees, advancing a step towards achieving diverse c-approximate TSP tours. We also explore the connection to the field of multiobjective optimization and show that the class of problems to which our result applies includes those for which the associated DUALRESTRICT problem defined by Papadimitriou and Yannakakis [35], and recently explored by Herzel et al. [26] can be solved in polynomial ti 

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4. Obtaining Approximately Optimal and Diverse Solutions via Dispersion

   https://doi.org/10.1007/978-3-031-20624-5_14  

   Gao, Jie; Goswami, Mayank; C. S., Karthik; Tsai, Meng-Tsung; Tsai, Shih-Yu; Yang, Hao-Tsung (November 2022, Latin American Symposium on Theoretical Informatics)

   There has been a long-standing interest in computing diverse solutions to optimization problems. In 1995 J. Krarup [28] posed the problem of finding k-edge disjoint Hamiltonian Circuits of minimum total weight, called the peripatetic salesman problem (PSP). Since then researchers have investigated the complexity of finding diverse solutions to spanning trees, paths, vertex covers, matchings, and more. Unlike the PSP that has a constraint on the total weight of the solutions, recent work has involved finding diverse solutions that are all optimal. However, sometimes the space of exact solutions may be too small to achieve sufficient diversity. Motivated by this, we initiate the study of obtaining sufficiently-diverse, yet approximately-optimal solutions to optimization problems. Formally, given an integer k, an approximation factor c, and an instance I of an optimization problem, we aim to obtain a set of k solutions to I that a) are all c approximately-optimal for I and b) maximize the diversity of the k solutions. Finding such solutions, therefore, requires a better understanding of the global landscape of the optimization function. Given a metric on the space of solutions, and the diversity measure as the sum of pairwise distances between solutions, we first provide a general reduction to an associated budget-constrained optimization (BCO) problem, where one objective function is to optimized subject to a bound on the second objective function. We then prove that bi-approximations to the BCO can be used to give bi-approximations to the diverse approximately optimal solutions problem. As applications of our result, we present polynomial time approximation algorithms for several problems such as diverse c-approximate maximum matchings, shortest paths, global min-cut, and minimum weight bases of a matroid. The last result gives us diversec-approximate minimum spanning trees, advancing a step towards achieving diverse c-approximate TSP tours. We also explore the connection to the field of multiobjective optimization and show that the class of problems to which our result applies includes those for which the associated DUALRESTRICT problem defined by Papadimitriou and Yannakakis [35], and recently explored by Herzel et al. [26] can be solved in polynomial time. 

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5. Differential operators and families of automorphic forms on Unitary groups of arbitrary signature

   Eischen, Ellen; Fintzen, Jessica; Mantovan, Elena; Varma, Ila (June 2018, Documenta mathematica)

   In the 1970's, Serre exploited congruences between q-expansion coefficients of Eisenstein series to produce p-adic families of Eisenstein series and, in turn, p-adic zeta functions. Partly through integration with more recent machinery, including Katz's approach to p-adic differential operators, his strategy has influenced four decades of developments. Prior papers employing Katz's and Serre's ideas exploiting differential operators and congruences to produce families of automorphic forms rely crucially on q-expansions of automorphic forms. The overarching goal of the present paper is to adapt the strategy to automorphic forms on unitary groups, which lack q-expansions when the signature is of the form (a,b), a≠b. In particular, this paper completely removes the restrictions on the signature present in prior work. As intermediate steps, we achieve two key objectives. First, partly by carefully analyzing the action of the Young symmetrizer on Serre-Tate expansions, we explicitly describe the action of differential operators on the Serre-Tate expansions of automorphic forms on unitary groups of arbitrary signature. As a direct consequence, for each unitary group, we obtain congruences and families analogous to those studied by Katz and Serre. Second, via a novel lifting argument, we construct a p-adic measure taking values in the space of p-adic automorphic forms on unitary groups of any prescribed signature. We relate the values of this measure to an explicit p-adic family of Eisenstein series. One application of our results is to the recently completed construction of p-adic L-functions for unitary groups by the first named author, Harris, Li, and Skinner. 

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