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1. On the Equations of Poizat and Liénard

   https://doi.org/10.1093/imrn/rnac322  

   Freitag, James; Jaoui, Rémi; Marker, David; Nagloo, Joel (November 2022, International Mathematics Research Notices)

   Abstract We study the structure of the solution sets in universal differential fields of certain differential equations of order two, the Poizat equations, which are particular cases of Liénard equations. We give a necessary and sufficient condition for strong minimality for equations in this class and a complete classification of the algebraic relations for solutions of strongly minimal Poizat equations. We also give an analysis of the non-strongly minimal cases as well as applications concerning the Liouvillian and Pfaffian solutions of some Liénard equations. 

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2. Trimmed serendipity finite element differential forms

   https://doi.org/10.1090/mcom/3354  

   Gillette, Andrew; Kloefkorn, Tyler (May 2018, Mathematics of Computation)

   We introduce the family of trimmed serendipity finite element differential form spaces, defined on cubical meshes in any number of dimensions, for any polynomial degree, and for any form order. The relation between the trimmed serendipity family and the (non-trimmed) serendipity family developed by Arnold and Awanou [Math. Comp. 83(288) 2014] is analogous to the relation between the trimmed and (non-trimmed) polynomial finite element differential form families on simplicial meshes from finite element exterior calculus. We provide degrees of freedom in the general setting and prove that they are unisolvent for the trimmed serendipity spaces. The sequence of trimmed serendipity spaces with a fixed polynomial order r provides an explicit example of a system described by Christiansen and Gillette [ESAIM:M2AN 50(3) 2016], namely, a minimal compatible finite element system on squares or cubes containing order r-1 polynomial differential forms. 

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3. Minimal free resolutions of differential modules

   https://doi.org/10.1090/tran/8754  

   Brown, Michael; Erman, Daniel (January 2022, Transactions of the American Mathematical Society)

   We propose a notion of minimal free resolutions for differential modules, and we prove existence and uniqueness results for such resolutions. We also take the first steps toward studying the structure of minimal free resolutions of differential modules. Our main result in this direction explains a sense in which the minimal free resolution of a differential module is a deformation of the minimal free resolution of its homology; this leads to structural results that mirror classical theorems about minimal free resolutions of modules. 

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4. Impact of Static Pressure Differential Between Supply Air and Return Exhaust on Server Level Performance

   https://doi.org/10.1109/ITHERM.2018.8419536  

   Siddarth, Ashwin; Eiland, Richard; Fernandes, John; Agonafer, Dereje (July 2018, 2018 17th IEEE Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems (ITherm))

   Modern Information Technology (IT) servers are typically assumed to operate in quiescent conditions with almost zero static pressure differentials between inlet and exhaust. However, when operating in a data center containment system the IT equipment thermal status is a strong function of the non- homogenous environment of the air space, IT utilization workloads and the overall facility cooling system design. To implement a dynamic and interfaced cooling solution, the interdependencies of variabilities between the chassis, rack and room level must be determined. In this paper, the effect of positive as well as negative static pressure differential between inlet and outlet of servers on thermal performance, fan control schemes, the direction of air flow through the servers as well as fan energy consumption within a server is observed at the chassis level. In this study, a web server with internal air-flow paths segregated into two separate streams, each having dedicated fan/group of fans within the chassis, is operated over a range of static pressure differential across the server. Experiments were conducted to observe the steady-state temperatures of CPUs and fan power consumption. Furthermore, the server fan speed control scheme’s transient response to a typical peak in IT computational workload while operating at negative pressure differentials across the server is reported. The effects of the internal air flow paths within the chassis is studied through experimental testing and simulations for flow visualization. The results indicate that at higher positive differential pressures across the server, increasing server fans speeds will have minimal impact on the cooling of the system. On the contrary, at lower, negative differential pressure server fan power becomes strongly dependent on operating pressure differential. More importantly, it is shown that an imbalance of flow impedances in internal airflow paths and fan control logic can onset recirculation of exhaust air within the server. For accurate prediction of airflow in cases where negative pressure differential exists, this study proposes an extended fan performance curve instead of a regular fan performance curve to be applied as a fan boundary condition for Computational Fluid Dynamics simulations. 

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5. Deciding accuracy of differential privacy schemes

   https://doi.org/10.1145/3434289  

   Barthe, Gilles; Chadha, Rohit; Krogmeier, Paul; Sistla, A. Prasad; Viswanathan, Mahesh (January 2021, Proceedings of the ACM on Programming Languages)

   null (Ed.)

   Differential privacy is a mathematical framework for developing statistical computations with provable guarantees of privacy and accuracy. In contrast to the privacy component of differential privacy, which has a clear mathematical and intuitive meaning, the accuracy component of differential privacy does not have a generally accepted definition; accuracy claims of differential privacy algorithms vary from algorithm to algorithm and are not instantiations of a general definition. We identify program discontinuity as a common theme in existing ad hoc definitions and introduce an alternative notion of accuracy parametrized by, what we call, — the of an input x w.r.t.  a deterministic computation f and a distance d , is the minimal distance d ( x , y ) over all y such that f ( y )≠ f ( x ). We show that our notion of accuracy subsumes the definition used in theoretical computer science, and captures known accuracy claims for differential privacy algorithms. In fact, our general notion of accuracy helps us prove better claims in some cases. Next, we study the decidability of accuracy. We first show that accuracy is in general undecidable. Then, we define a non-trivial class of probabilistic computations for which accuracy is decidable (unconditionally, or assuming Schanuel’s conjecture). We implement our decision procedure and experimentally evaluate the effectiveness of our approach for generating proofs or counterexamples of accuracy for common algorithms from the literature. 

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