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1. A new theory of fractional differential calculus

   https://doi.org/10.1142/s0219530521500019  

   Feng, Xiaobing; Sutton, Mitchell (July 2021, Analysis and Applications)

   null (Ed.)

   This paper presents a self-contained new theory of weak fractional differential calculus in one-dimension. The crux of this new theory is the introduction of a weak fractional derivative notion which is a natural generalization of integer order weak derivatives; it also helps to unify multiple existing fractional derivative definitions and characterize what functions are fractionally differentiable. Various calculus rules including a fundamental theorem calculus, product and chain rules, and integration by parts formulas are established for weak fractional derivatives. Additionally, relationships with classical fractional derivatives and detailed characterizations of weakly fractional differentiable functions are also established. Furthermore, the notion of weak fractional derivatives is also systematically extended to general distributions instead of only to some special distributions. This new theory lays down a solid theoretical foundation for systematically and rigorously developing new theories of fractional Sobolev spaces, fractional calculus of variations, and fractional PDEs as well as their numerical solutions in subsequent works. 

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2. Synthesis and characterization of 3- O -esters of N -acetyl- d -glucosamine derivatives as organogelators

   https://doi.org/10.1039/c9nj00630c  

   Chen, Anji; Samankumara, Lalith P.; Garcia, Consuelo; Bashaw, Kristen; Wang, Guijun (May 2019, New Journal of Chemistry)

   Carbohydrate derived low molecular weight organogelators are interesting compounds with many potential applications. Selective functionalization of the different hydroxyl substituents on d -glucose and d -glucosamine resulted in small molecular gelators. Previously we have found that the C-2 acylated derivatives including esters and carbamates of 4,6- O -benzylidene protected glucose and glucosamine derivatives have shown remarkable applications as molecular gelators. In this research, in order to probe the structural influence of sugar derivatives on molecular self-assembly, we introduced acylation functional groups to the 3-hydroxyl group of 4,6- O -benzylidene acetal protected N -acetyl glucosamine derivatives. A library of fourteen ester derivatives was synthesized and characterized. The ester derivatives typically formed gels in pump oil and aqueous mixtures of dimethyl sulfoxide or ethanol. The resulting gels were characterized using optical microscopy, and rheology, etc. All alkyl ester derivatives were gelators for pump oil. A short chain ester derivative was able to form gels in a few different oils and the corresponding oil water mixtures phase selectively. The compound was also used to trap naproxen sodium and formed a stable co-gel. The controlled release of the drug from the gel to the aqueous phase was analyzed using UV-vis spectroscopy. These results show that the functionalization at the 3-OH position of the N -acetyl glucosamine derivative is a feasible strategy in designing new classes of organogelators. 

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3. Linear-Quadratic Stochastic Differential Games on Random Directed Networks

   Yichen Feng, Jean-Pierre Fouque (January 2020, Journal of mathematics and statistical science)

   The study of linear-quadratic stochastic differential games on directed networks was initiated in Feng, Fouque & Ichiba [7]. In that work, the game on a directed chain with finite or infinite players was defined as well as the game on a deterministic directed tree, and their Nash equilibria were computed. The current work continues the analysis by first developing a random directed chain structure by assuming the interaction between every two neighbors is random. We solve explicitly for an open-loop Nash equilibrium for the system and we find that the dynamics under equilibrium is an infinite-dimensional Gaussian process described by a Catalan Markov chain introduced in [7]. The discussion about stochastic differential games is extended to a random two-sided directed chain and a random directed tree structure. 

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4. Linear-Quadratic Stochastic Differential Games on Random Directed Networks

   Feng, Yichen; Fouque, Jean-Pierre; Ichiba, Tomoyuki (January 2021, Journal of mathematics and statistical science)

   null (Ed.)

   The study of linear-quadratic stochastic differential games on directed networks was initiated in Feng, Fouque & Ichiba [7]. In that work, the game on a directed chain with finite or infinite players was defined as well as the game on a deterministic directed tree, and their Nash equilibria were computed. The current work continues the analysis by first developing a random directed chain structure by assuming the interaction between every two neighbors is random. We solve explicitly for an open-loop Nash equilibrium for the system and we find that the dynamics under equilibrium is an infinite-dimensional Gaussian process described by a Catalan Markov chain introduced in [7]. The discussion about stochastic differential games is extended to a random two-sided directed chain and a random directed tree structure. 

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5. Perturbation confusion in forward automatic differentiation of higher-order functions

   https://doi.org/10.1017/S095679681900008X  

   MANZYUK, OLEKSANDR; PEARLMUTTER, BARAK A.; RADUL, ALEXEY ANDREYEVICH; RUSH, DAVID R.; SISKIND, JEFFREY MARK (January 2019, Journal of Functional Programming)

   Abstract Automatic differentiation (AD) is a technique for augmenting computer programs to compute derivatives. The essence of AD in its forward accumulation mode is to attach perturbations to each number, and propagate these through the computation by overloading the arithmetic operators. When derivatives are nested, the distinct derivative calculations, and their associated perturbations, must be distinguished. This is typically accomplished by creating a unique tag for each derivative calculation and tagging the perturbations. We exhibit a subtle bug, present in fielded implementations which support derivatives of higher-order functions, in which perturbations are confused despite the tagging machinery, leading to incorrect results. The essence of the bug is as follows: a unique tag is needed for each derivative calculation, but in existing implementations unique tags are created when taking the derivative of a function at a point. When taking derivatives of higher-order functions, these need not correspond! We exhibit a simple example: a higher-order function f whose derivative at a point x , namely f ′( x ), is itself a function which calculates a derivative. This situation arises naturally when taking derivatives of curried functions. Two potential solutions are presented, and their deficiencies discussed. One uses eta expansion to delay the creation of fresh tags in order to put them into one-to-one correspondence with derivative calculations. The other wraps outputs of derivative operators with tag substitution machinery. Both solutions seem very difficult to implement without violating the desirable complexity guarantees of forward AD. 

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