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1. Parabolic Regularity of Spectral Functions

   https://doi.org/10.1287/moor.2023.0010  

   Mohammadi, Ashkan; Sarabi, Ebrahim (August 2024, Mathematics of Operations Research)

   This paper is devoted to the study of the second-order variational analysis of spectral functions. It is well-known that spectral functions can be expressed as a composite function of symmetric functions and eigenvalue functions. We establish several second-order properties of spectral functions when their associated symmetric functions enjoy these properties. Our main attention is given to characterize parabolic regularity for this class of functions. It was observed recently that parabolic regularity can play a central rule in ensuring the validity of important second-order variational properties, such as twice epi-differentiability. We demonstrates that for convex spectral functions, their parabolic regularity amounts to that of their symmetric functions. As an important consequence, we calculate the second subderivative of convex spectral functions, which allows us to establish second-order optimality conditions for a class of matrix optimization problems. 

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2. Distribution of values of Gaussian hypergeometric functions

   https://doi.org/10.4310/PAMQ.2023.v19.n1.a14  

   Ono, Ken; Saad, Hasan; Saikia, Neelam (January 2023, Pure and Applied Mathematics Quarterly)

   In the 1980’s, Greene defined hypergeometric functions over finite fields using Jacobi sums. The framework of his theory establishes that these functions possess many properties that are analogous to those of the classical hypergeometric series studied by Gauss and Kummer. These functions have played important roles in the study of Ap ́ery-style supercongruences, the Eichler-Selberg trace formula, Galois representations, and zeta-functions of arithmetic varieties. We study the value distribution (over large finite fields) of natural families of these functions. For the 2F1 functions, the limiting distribution is semicircular (i.e. SU(2)), whereas the distribution for the 3F2 functions is the Batman distribution for the traces of the real orthogonal group O3. 

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3. Targeted Lossy Functions and Applications

   https://doi.org/10.1007/978-3-030-84259-8\15  

   Quach, Willy; Waters, Brent; Wichs, Daniel (August 2021, CRYPTO 2021)

   Lossy trapdoor functions, introduced by Peikert and Waters (STOC ’08), can be initialized in one of two indistinguishable modes: in injective mode, the function preserves all information about its input, and can be efficiently inverted given a trapdoor, while in lossy mode, the function loses some information about its input. Such functions have found countless applications in cryptography, and can be constructed from a variety of Cryptomania assumptions. In this work, we introduce targeted lossy functions (TLFs), which relax lossy trapdoor functions along two orthogonal dimensions. Firstly, they do not require an inversion trapdoor in injective mode. Secondly, the lossy mode of the function is initialized with some target input, and the function is only required to lose information about this particular target. The injective and lossy modes should be indistinguishable even given the target. We construct TLFs from Minicrypt assumptions, namely, injective pseudorandom generators, or even one-way functions under a natural relaxation of injectivity. We then generalize TLFs to incorporate branches, and construct all-injective-but-one and all-lossy-but-one variants. We show a wide variety of applications of targeted lossy functions. In several cases, we get the first Minicrypt constructions of primitives that were previously only known under Cryptomania assumptions. Our applications include: Pseudo-entropy functions from one-way functions. Deterministic leakage-resilient message-authentication codes and improved leakage-resilient symmetric-key encryption from one-way functions. Extractors for extractor-dependent sources from one-way functions. Selective-opening secure symmetric-key encryption from one-way functions. A new construction of CCA PKE from (exponentially secure) trapdoor functions and injective pseudorandom generators. We also discuss a fascinating connection to distributed point functions. 

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4. A Collective Colony Migration Model with Hill Functions in Recruitment

   https://doi.org/10.1142/S0218127422502133  

   Wang, Lisha; Qiu, Zhipeng; Kang, Yun (November 2022, International Journal of Bifurcation and Chaos)

   Social insect colonies’ robust and efficient collective behaviors without any central control contribute greatly to their ecological success. Colony migration is a leading subject for studying collective decision-making in migration. In this paper, a general colony migration model with Hill functions in recruitment is proposed to investigate the underlying decision making mechanism and the related dynamical behaviors. Our analysis provides the existence and stability of equilibrium, and the global dynamical behavior of the system. To understand how piecewise functions and Hill functions in recruitment impact colony migration dynamics, the comparisons are performed in both analytic results and bifurcation analysis. Our theoretical results show that the dynamics of the migration system with Hill functions in recruitment differs from that of the migration system with piecewise functions in the following three aspects: (1) all population components in our colony migration model with Hill functions in recruitment are persistent; (2) the colony migration model with Hill functions in recruitment has saddle and saddle-node bifurcations, while the migration system with piecewise functions does not; (3) the system with Hill functions has only equilibrium dynamics, i.e. either has a global stability at one interior equilibrium or has bistablity among two locally stable interior equilibria. Bifurcation analysis shows that the geometrical shape of the Hill functions greatly impacts the dynamics: (1) the system with flatter Hill functions is less likely to exhibit bistability; (2) the system with steeper functions is prone to exhibit bistability, and the steady state of total active workers is closer to that of active workers in the system with piecewise function. 

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5. Log orthogonal functions: approximation properties and applications

   https://doi.org/10.1093/imanum/draa087  

   Chen, Sheng; Shen, Jie (December 2020, IMA Journal of Numerical Analysis)

   Abstract We present two new classes of orthogonal functions, log orthogonal functions and generalized log orthogonal functions, which are constructed by applying a $$\log $$ mapping to Laguerre polynomials. We develop basic approximation theory for these new orthogonal functions, and apply them to solve several typical fractional differential equations whose solutions exhibit weak singularities. Our error analysis and numerical results show that our methods based on the new orthogonal functions are particularly suitable for functions that have weak singularities at one endpoint and can lead to exponential convergence rate, as opposed to low algebraic rates if usual orthogonal polynomials are used. 

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