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1. On Φ-variation for 1-d scalar conservation laws

   https://doi.org/10.1142/S0219891620500277  

   Jenssen, Helge Kristian; Ridder, Johanna (December 2020, Journal of Hyperbolic Differential Equations)

   null (Ed.)

   Let [Formula: see text] be a convex function satisfying [Formula: see text], [Formula: see text] for [Formula: see text], and [Formula: see text]. Consider the unique entropy admissible (i.e. Kružkov) solution [Formula: see text] of the scalar, 1-d Cauchy problem [Formula: see text], [Formula: see text]. For compactly supported data [Formula: see text] with bounded [Formula: see text]-variation, we realize the solution [Formula: see text] as a limit of front-tracking approximations and show that the [Formula: see text]-variation of (the right continuous version of) [Formula: see text] is non-increasing in time. We also establish the natural time-continuity estimate [Formula: see text] for [Formula: see text], where [Formula: see text] depends on [Formula: see text]. Finally, according to a theorem of Goffman–Moran–Waterman, any regulated function of compact support has bounded [Formula: see text]-variation for some [Formula: see text]. As a corollary we thus have: if [Formula: see text] is a regulated function, so is [Formula: see text] for all [Formula: see text]. 

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2. Some remarks on the Erdős Distinct subset sums problem

   https://doi.org/10.1142/S1793042123500860  

   Steinerberger, Stefan (September 2023, International Journal of Number Theory)

   Let [Formula: see text] be a set of positive integers, [Formula: see text] denoting the largest element, so that for any two of the [Formula: see text] subsets the sum of all elements is distinct. Erdős asked whether this implies [Formula: see text] for some universal [Formula: see text]. We prove, slightly extending a result of Elkies, that for any [Formula: see text], [Formula: see text] with equality if and only if all subset sums are [Formula: see text]-separated. This leads to a new proof of the currently best lower bound [Formula: see text]. The main new insight is that having distinct subset sums and [Formula: see text] small requires the random variable [Formula: see text] to be close to Gaussian in a precise sense. 

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3. Residually finite rationally p groups

   https://doi.org/10.1142/S0219199719500160  

   Koberda, Thomas; Suciu, Alexander I. (May 2020, Communications in Contemporary Mathematics)

   In this paper, we develop the theory of residually finite rationally [Formula: see text] (RFR[Formula: see text]) groups, where [Formula: see text] is a prime. We first prove a series of results about the structure of finitely generated RFR[Formula: see text] groups (either for a single prime [Formula: see text], or for infinitely many primes), including torsion-freeness, a Tits alternative, and a restriction on the BNS invariant. Furthermore, we show that many groups which occur naturally in group theory, algebraic geometry, and in 3-manifold topology enjoy this residual property. We then prove a combination theorem for RFR[Formula: see text] groups, which we use to study the boundary manifolds of algebraic curves [Formula: see text] and in [Formula: see text]. We show that boundary manifolds of a large class of curves in [Formula: see text] (which includes all line arrangements) have RFR[Formula: see text] fundamental groups, whereas boundary manifolds of curves in [Formula: see text] may fail to do so. 

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4. Smoothing and growth bound of periodic generalized Korteweg–De Vries equation

   https://doi.org/10.1142/S0219891621500260  

   Oh, Seungly; Stefanov, Atanas G. (December 2021, Journal of Hyperbolic Differential Equations)

   For generalized Korteweg–De Vries (KdV) models with polynomial nonlinearity, we establish a local smoothing property in [Formula: see text] for [Formula: see text]. Such smoothing effect persists globally, provided that the [Formula: see text] norm does not blow up in finite time. More specifically, we show that a translate of the nonlinear part of the solution gains [Formula: see text] derivatives for [Formula: see text]. Following a new simple method, which is of independent interest, we establish that, for [Formula: see text], [Formula: see text] norm of a solution grows at most by [Formula: see text] if [Formula: see text] norm is a priori controlled. 

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5. For interpolating kernel machines, minimizing the norm of the ERM solution maximizes stability

   https://doi.org/10.1142/S0219530522400115  

   Rangamani, Akshay; Rosasco, Lorenzo; Poggio, Tomaso (January 2023, Analysis and Applications)

   In this paper, we study kernel ridge-less regression, including the case of interpolating solutions. We prove that maximizing the leave-one-out ([Formula: see text]) stability minimizes the expected error. Further, we also prove that the minimum norm solution — to which gradient algorithms are known to converge — is the most stable solution. More precisely, we show that the minimum norm interpolating solution minimizes a bound on [Formula: see text] stability, which in turn is controlled by the smallest singular value, hence the condition number, of the empirical kernel matrix. These quantities can be characterized in the asymptotic regime where both the dimension ([Formula: see text]) and cardinality ([Formula: see text]) of the data go to infinity (with [Formula: see text] as [Formula: see text]). Our results suggest that the property of [Formula: see text] stability of the learning algorithm with respect to perturbations of the training set may provide a more general framework than the classical theory of Empirical Risk Minimization (ERM). While ERM was developed to deal with the classical regime in which the architecture of the learning network is fixed and [Formula: see text], the modern regime focuses on interpolating regressors and overparameterized models, when both [Formula: see text] and [Formula: see text] go to infinity. Since the stability framework is known to be equivalent to the classical theory in the classical regime, our results here suggest that it may be interesting to extend it beyond kernel regression to other overparameterized algorithms such as deep networks. 

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