In this paper, we provide some topological criteria for the Poisson Dixmier-Moeglin equivalence for $A$ in terms of the poset $({ m P. spec A}, subseteq)$ and the symplectic leaf or core stratification on its maximal spectrum. In particular, we prove that the Zariski topology of the Poisson prime spectrum and of each symplectic leaf or core can detect the Poisson Dixmier-Moeglin equivalence for any complex affine Poisson algebra. Moreover, we generalize the weaker version of the Poisson Dixmier-Moeglin equivalence for a complex affine Poisson algebra proved in [J. Bell, S. Launois, O.L. S'anchez, and B. Moosa, Poisson algebras via model theory and differential algebraic geometry, J. Eur. Math. Soc. (JEMS), 19(2017), no. 7, 2019-2049] to the general context of a commutative differential algebra.

The goal of this article is to investigate infinite dimensional affine diffusion processes on the canonical state space. This includes a derivation of the corresponding system of Riccati differential equations and an existence proof for such processes, which has been missing in the literature so far. For the existence proof, we will regard affine processes as solutions to infinite dimensional stochastic differential equations with values in Hilbert spaces. This requires a suitable version of the Yamada-Watanabe theorem, which we will provide in this paper. Several examples of infinite dimensional affine processes accompany our results.

We study an equivariant extension of the Batalin-Vilkovisky formalism for quantizing gauge theories. Namely, we introduce a general framework to encompass failures of the quantum master equation, and we apply it to the natural equivariant extension of AKSZ solutions of the classical master equation (CME). As examples of the construction, we recover the equivariant extension of supersymmetric Yang-Mills in 2d and of Donaldson-Witten theory.

In this paper we consider the "quasidensity" of a subset of the product of a Banach space and its dual, and give a connection between quasidense sets and sets of "type (NI)". We discuss "coincidence sets" of certain convex functions and prove two sum theorems for coincidence sets. We obtain new results on the Fitzpatrick extension of a closed quasidense monotone multifunction. The analysis in this paper is self-contained, and independent of previous work on "Banach SN spaces". This version differs from the previous version because it is shown that the (well known) equivalence of quasidensity and "type (NI)" for maximally monotone sets is not true without the monotonicity assumption and that the appendix has been moved to the end of Section 10, where it rightfully belongs.

This paper is devoted to the study of the singularity phenomenon of timelike extremal hypersurfaces in Minkowski spacetime $mathbb{R}^{1+3}$. We find that there are two explicit lightlike self-similar solutions to a graph representation of timelike extremal hypersurfaces in Minkowski spacetime $mathbb{R}^{1+3}$, the geometry of them are two spheres. The linear mode unstable of those lightlike self-similar solutions for the radially symmetric membranes equation is given. After that, we show those self-similar solutions of the radially symmetric membranes equation are nonlinearly stable inside a strictly proper subset of the backward lightcone. This means that the dynamical behavior of those two spheres is as attractors. Meanwhile, we overcome the double roots case (the theorem of Poincar'{e} can't be used) in solving the difference equation by construction of a Newton's polygon when we carry out the analysis of spectrum for the linear operator.

We introduce an infinite-dimensional $p$-adic affine group and construct its irreducible unitary representation. Our approach follows the one used by Vershik, Gelfand and Graev for the diffeomorphism group, but with modifications made necessary by the fact that the group does not act on the phase space. However it is possible to define its action on some classes of functions.

We introduce primal and dual stochastic gradient oracle methods for decentralized convex optimization problems. Both for primal and dual oracles the proposed methods are optimal in terms of the number of communication steps. However, for all classes of the objective, the optimality in terms of the number of oracle calls per node in the class of methods with optimal number of communication steps takes place only up to a logarithmic factor and the notion of smoothness. By using mini-batching technique we show that all proposed methods with stochastic oracle can be additionally parallelized at each node.

The central aim of this paper is to study (regional) fractional Poincar'e type inequalities on unbounded domains satisfying the finite ball condition. Both existence and non existence type results are established depending on various conditions on domains and on the range of $s in (0,1)$. The best constant in both regional fractional and fractional Poincar'e inequality is characterized for strip like domains $(omega imes mathbb{R}^{n-1})$, and the results obtained in this direction are analogous to those of the local case. This settles one of the natural questions raised by K. Yeressian in [ extit{Asymptotic behavior of elliptic nonlocal equations set in cylinders, Asymptot. Anal. 89, (2014), no 1-2}].

We prove that, given a constant $K> 2$ and a bounded linear operator $T$ from a JB$^*$-triple $E$ into a complex Hilbert space $H$, there exists a norm-one functional $psiin E^*$ satisfying $$|T(x)| leq K , |T| , |x|_{psi},$$ for all $xin E$. Applying this result we show that, given $G > 8 (1+2sqrt{3})$ and a bounded bilinear form $V$ on the Cartesian product of two JB$^*$-triples $E$ and $B$, there exist norm-one functionals $varphiin E^{*}$ and $psiin B^{*}$ satisfying $$|V(x,y)| leq G |V| , |x|_{varphi} , |y|_{psi}$$ for all $(x,y)in E imes B$. These results prove a conjecture pursued during almost twenty years.

The Gabriel-Roiter measure is used to give an alternative proof of the finiteness of the representation dimension for Artin algebras, a result established by Iyama in 2002. The concept of Gabriel-Roiter measure can be extended to abelian length categories and every such category has multiple Gabriel-Roiter measures. Using this notion, we prove the following broader statement: given any object $X$ and any Gabriel-Roiter measure $mu$ in an abelian length category $mathcal{A}$, there exists an object $X'$ which depends on $X$ and $mu$, such that $Gamma = operatorname{End}_{mathcal{A}}(X oplus X')$ has finite global dimension. Analogously to Iyama's original results, our construction yields quasihereditary rings and fits into the theory of rejective chains.

We deal with various Diophantine equations involving the Euler totient function and various sequences of numbers, including factorials, powers, and Fibonacci sequences.

In this paper we first review the setting for the geometric Langlands functoriality and establish a result for the `backward' functoriality functor. We illustrate this by known examples of the geometric theta-lifting. We then apply the above result to obtain new Hecke eigen-sheaves. The most important application is a construction of the automorphic sheaf for G=GSp_4 attached to a G^L-local system on a curve X such that its standard representation is an irreducible local system of rank 4 on X.

We consider Landau-Ginzburg models stemming from groups comprised of non-diagonal symmetries, and we describe a rule for the mirror LG model. In particular, we present the non-abelian dual group, which serves as the appropriate choice of group for the mirror LG model. We also describe an explicit mirror map between the A-model and the B-model state spaces for two examples. Further, we prove that this mirror map is an isomorphism between the untwisted broad sectors and the narrow diagonal sectors for Fermat type polynomials.

In this paper we study the following set[A={p(n)+2^nd mod 1: ngeq 1}subset [0.1],] where $p$ is a polynomial with at least one irrational coefficient on non constant terms, $d$ is any real number and for $ain [0,infty)$, $a mod 1$ is the fractional part of $a$. By a Bernoulli decomposition method, we show that the closure of $A$ must have full Hausdorff dimension.

This work presents a novel data-driven framework for constructing eigenfunctions of the Koopman operator geared toward prediction and control. The method leverages the richness of the spectrum of the Koopman operator away from attractors to construct a rich set of eigenfunctions such that the state (or any other observable quantity of interest) is in the span of these eigenfunctions and hence predictable in a linear fashion. The eigenfunction construction is optimization-based with no dictionary selection required. Once a predictor for the uncontrolled part of the system is obtained in this way, the incorporation of control is done through a multi-step prediction error minimization, carried out by a simple linear least-squares regression. The predictor so obtained is in the form of a linear controlled dynamical system and can be readily applied within the Koopman model predictive control framework of [12] to control nonlinear dynamical systems using linear model predictive control tools. The method is entirely data-driven and based purely on convex optimization, with no reliance on neural networks or other non-convex machine learning tools. The novel eigenfunction construction method is also analyzed theoretically, proving rigorously that the family of eigenfunctions obtained is rich enough to span the space of all continuous functions. In addition, the method is extended to construct generalized eigenfunctions that also give rise Koopman invariant subspaces and hence can be used for linear prediction. Detailed numerical examples with code available online demonstrate the approach, both for prediction and feedback control.

Let $hat G$ be the finite simply connected version of an exceptional Chevalley group, and let $V$ be a nontrivial irreducible module, of minimal dimension, for $hat G$ over its field of definition. We explore the overgroup structure of $hat G$ in $mathrm{GL}(V)$, and the submodule structure of the exterior square (and sometimes the third Lie power) of $V$. When $hat G$ is defined over a field of odd prime order $p$, this allows us to construct the smallest (with respect to certain properties) $p$-groups $P$ such that the group induced by $mathrm{Aut}(P)$ on $P/Phi(P)$ is either $hat G$ or its normaliser in $mathrm{GL}(V)$.

We study the geometry and topology of exotic Springer fibers for orbits corresponding to one-row bipartitions from an explicit, combinatorial point of view. This includes a detailed analysis of the structure of the irreducible components and their intersections as well as the construction of an explicit affine paving. Moreover, we compute the ring structure of cohomology by constructing a CW-complex homotopy equivalent to the exotic Springer fiber. This homotopy equivalent space admits an action of the type C Weyl group inducing Kato's original exotic Springer representation on cohomology. Our results are described in terms of the diagrammatics of the one-boundary Temperley-Lieb algebra (also known as the blob algebra). This provides a first step in generalizing the geometric versions of Khovanov's arc algebra to the exotic setting.

We derive finite rational formulas for the traces of cycle integrals of certain meromorphic modular forms. Moreover, we prove the modularity of a completion of the generating function of such traces. The theoretical framework for these results is an extension of the Shintani theta lift to meromorphic modular forms of positive even weight.

Gabber and Joseph introduced a ladder diagram between two natural sequences of extensions. Their diagram is used to produce a 'twisted' sequence that is applied to old and new results on extension groups in category $mathcal{O}$.

In this work, we propose a simple modification of the forward-backward splitting method for finding a zero in the sum of two monotone operators. Our method converges under the same assumptions as Tseng's forward-backward-forward method, namely, it does not require cocoercivity of the single-valued operator. Moreover, each iteration only requires one forward evaluation rather than two as is the case for Tseng's method. Variants of the method incorporating a linesearch, relaxation and inertia, or a structured three operator inclusion are also discussed.

We prove an inequality between the sum of the Betti numbers of a complex projective manifold and its total curvature, and we characterize the complex projective manifolds whose total curvature is minimal. These results extend the classical theorems of Chern and Lashof to complex projective space.

The two-dimensional directed spanning forest (DSF) introduced by Baccelli and Bordenave is a planar directed forest whose vertex set is given by a homogeneous Poisson point process $mathcal{N}$ on $mathbb{R}^2$. If the DSF has direction $-e_y$, the ancestor $h(u)$ of a vertex $u in mathcal{N}$ is the nearest Poisson point (in the $L_2$ distance) having strictly larger $y$-coordinate. This construction induces complex geometrical dependencies. In this paper we show that the collection of DSF paths, properly scaled, converges in distribution to the Brownian web (BW). This verifies a conjecture made by Baccelli and Bordenave in 2007.

We prove the effectiveness of the canonical bundle of several Hurwitz spaces of degree k covers of the projective line from curves of genus 13<g<20.

We consider the stochastic 2-dimensional Cahn-Hilliard equation which is driven by the derivative in space of a space-time white noise. We use two different approaches to study this equation. First we prove that there exists a unique solution $Y$ to the shifted equation (see (1.4) below), then $X:=Y+{Z}$ is the unique solution to stochastic Cahn-Hilliard equaiton, where ${Z}$ is the corresponding O-U process. Moreover, we use Dirichlet form approach in cite{Albeverio:1991hk} to construct the probabilistically weak solution the the original equation (1.1) below. By clarifying the precise relation between the solutions obtained by the Dirichlet forms aprroach and $X$, we can also get the restricted Markov uniquness of the generator and the uniqueness of martingale solutions to the equation (1.1).

Tree balance plays an important role in different research areas like theoretical computer science and mathematical phylogenetics. For example, it has long been known that under the Yule model, a pure birth process, imbalanced trees are more likely than balanced ones. Therefore, different methods to measure the balance of trees were introduced. The Sackin index is one of the most frequently used measures for this purpose. In many contexts, statements about the minimal and maximal values of this index have been discussed, but formal proofs have never been provided. Moreover, while the number of trees with maximal Sackin index as well as the number of trees with minimal Sackin index when the number of leaves is a power of 2 are relatively easy to understand, the number of trees with minimal Sackin index for all other numbers of leaves was completely unknown. In this manuscript, we fully characterize trees with minimal and maximal Sackin index and also provide formulas to explicitly calculate the number of such trees.

The article is devoted to the expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity $k$ $(kinmathbb{N})$ based on the generalized iterated Fourier series. The case of Fourier-Legendre series as well as the case of trigonotemric Fourier series are considered in details. The obtained expansion provides a possibility to represent the iterated Stratonovich stochastic integral in the form of iterated series of products of standard Gaussian random variables. Convergence in the mean of degree $2n$ $(nin mathbb{N})$ of the expansion is proved. Some modifications of the mentioned expansion were derived for the case $k=2$. One of them is based of multiple trigonomentric Fourier series converging almost everywhere in the square $[t, T]^2$. The results of the article can be applied to the numerical solution of Ito stochastic differential equations.

We extend the analytic theory of Frobenius manifolds to semisimple points with coalescing eigenvalues of the operator of multiplication by the Euler vector field. We clarify which freedoms, ambiguities and mutual constraints are allowed in the definition of monodromy data, in view of their importance for conjectural relationships between Frobenius manifolds and derived categories. Detailed examples and applications are taken from singularity and quantum cohomology theories. We explicitly compute the monodromy data at points of the Maxwell Stratum of the A3-Frobenius manifold, as well as at the small quantum cohomology of the Grassmannian G(2,4). In the latter case, we analyse in details the action of the braid group on the monodromy data. This proves that these data can be expressed in terms of characteristic classes of mutations of Kapranov's exceptional 5-block collection, as conjectured by one of the authors.

High dimensional expanders is a vibrant emerging field of study. Nevertheless, the only known construction of bounded degree high dimensional expanders is based on Ramanujan complexes, whereas one dimensional bounded degree expanders are abundant.

In this work, we construct new families of bounded degree high dimensional expanders obeying the local spectral expansion property. This property has a number of important consequences, including geometric overlapping, fast mixing of high dimensional random walks, agreement testing and agreement expansion. Our construction also yields new families of expander graphs which are close to the Ramanujan bound, i.e., their spectral gap is close to optimal.

The construction is quite elementary and it is presented in a self contained manner; This is in contrary to the highly involved previously known construction of the Ramanujan complexes. The construction is also very symmetric (such symmetry properties are not known for Ramanujan complexes) ; The symmetry of the construction could be used, for example, in order to obtain good symmetric LDPC codes that were previously based on Ramanujan graphs.

The main tool that we use for is the theory of coset geometries. Coset geometries arose as a tool for studying finite simple groups. Here, we show that coset geometries arise in a very natural manner for groups of elementary matrices over any finitely generated algebra over a commutative unital ring. In other words, we show that such groups act simply transitively on the top dimensional face of a pure, partite, clique complex.

In this paper, we are concerned with the numerical solution of one type integro-differential equation by a probability method based on the fundamental martingale of mixed Gaussian processes. As an application, we will try to simulate the estimation of ruin probability with an unknown parameter driven not by the classical L'evy process but by the mixed fractional Brownian motion.

We use Ricci flow to obtain a local bi-Holder correspondence between Ricci limit spaces in three dimensions and smooth manifolds. This is more than a complete resolution of the three-dimensional case of the conjecture of Anderson-Cheeger-Colding-Tian, describing how Ricci limit spaces in three dimensions must be homeomorphic to manifolds, and we obtain this in the most general, locally non-collapsed case. The proofs build on results and ideas from recent papers of Hochard and the current authors.

We study flows on C*-algebras with the Rokhlin property. We show that every Kirchberg algebra carries a unique Rokhlin flow up to cocycle conjugacy, which confirms a long-standing conjecture of Kishimoto. We moreover present a classification theory for Rokhlin flows on C*-algebras satisfying certain technical properties, which hold for many C*-algebras covered by the Elliott program. As a consequence, we obtain the following further classification theorems for Rokhlin flows. Firstly, we extend the statement of Kishimoto's conjecture to the non-simple case: Up to cocycle conjugacy, a Rokhlin flow on a separable, nuclear, strongly purely infinite C*-algebra is uniquely determined by its induced action on the prime ideal space. Secondly, we give a complete classification of Rokhlin flows on simple classifiable $KK$-contractible C*-algebras: Two Rokhlin flows on such a C*-algebra are cocycle conjugate if and only if their induced actions on the cone of lower-semicontinuous traces are affinely conjugate.

Bernstein, Frenkel, and Khovanov have constructed a categorification of tensor products of the standard representation of $mathfrak{sl}_2$, where they use singular blocks of category $mathcal{O}$ for $mathfrak{sl}_n$ and translation functors. Here we construct a positive characteristic analogue using blocks of representations of $mathfrak{sl}_n$ over a field $ extbf{k}$ of characteristic $p$ with zero Frobenius character, and singular Harish-Chandra character. We show that the aforementioned categorification admits a Koszul graded lift, which is equivalent to a geometric categorification constructed by Cautis, Kamnitzer, and Licata using coherent sheaves on cotangent bundles to Grassmanians. In particular, the latter admits an abelian refinement. With respect to this abelian refinement, the stratified Mukai flop induces a perverse equivalence on the derived categories for complementary Grassmanians. This is part of a larger project to give a combinatorial approach to Lusztig's conjectures for representations of Lie algebras in positive characteristic.

We study a class of fourth order nonlinear parabolic equations which include the thin-film equation and the quantum drift-diffusion model as special cases. We investigate these equations by first developing functional inequalities of the type $ int_Omega u^{2gamma-alpha-eta}Delta u^alphaDelta u^eta dx geq cint_Omega|Delta u^gamma |^2dx $, which seem to be of interest on their own right.

The paper proves the Riemann Hypothesis.

We use the classical Fourier analysis to introduce analytic families of weighted differential operators on the unit sphere. These operators are polynomial functions of the usual Beltrami-Laplace operator. New inversion formulas are obtained for totally geodesic Funk transforms on the sphere and the correpsonding lambda-cosine transforms.

We consider the most general class of linear inhomogeneous boundary-value problems for systems of ordinary differential equations of an arbitrary order whose solutions and right-hand sides belong to appropriate Sobolev spaces. For parameter-dependent problems from this class, we prove a constructive criterion for their solutions to be continuous in the Sobolev space with respect to the parameter. We also prove a two-sided estimate for the degree of convergence of these solutions to the solution of the nonperturbed problem.

We define local indices for projective umbilics and godrons (also called cusps of Gauss) on generic smooth surfaces in projective 3-space. By means of these indices, we provide formulas that relate the algebraic numbers of those characteristic points on a surface (and on domains of the surface) with the Euler characteristic of that surface (resp. of those domains). These relations determine the possible coexistences of projective umbilics and godrons on the surface. Our study is based on a "fundamental cubic form" for which we provide a closed simple expression.

We consider two special cases of the connection problem for the second Painlev'e equation (PII) using the method of uniform asymptotics proposed by Bassom et al.. We give a classification of the real solutions of PII on the negative (positive) real axis with respect to their initial data. By product, a rigorous proof of a property associate with the nonlinear eigenvalue problem of PII on the real axis, recently revealed by Bender and Komijani, is given by deriving the asymptotic behavior of the Stokes multipliers.

The characterization of phase dynamics in coupled oscillators offers insights into fundamental phenomena in complex systems. To describe the collective dynamics in the oscillatory system, order parameters are often used but are insufficient for identifying more specific behaviors. We therefore propose a topological approach that constructs quantitative features describing the phase evolution of oscillators. Here, the phase data are mapped into a high-dimensional space at each time point, and topological features describing the shape of the data are subsequently extracted from the mapped points. We extend these features to time-variant topological features by considering the evolution time, which serves as an additional dimension in the topological-feature space. The resulting time-variant features provide crucial insights into the time evolution of phase dynamics. We combine these features with the machine learning kernel method to characterize the multicluster synchronized dynamics at a very early stage of the evolution. Furthermore, we demonstrate the usefulness of our method for qualitatively explaining chimera states, which are states of stably coexisting coherent and incoherent groups in systems of identical phase oscillators. The experimental results show that our method is generally better than those using order parameters, especially if only data on the early-stage dynamics are available.

We give a simple proof of the strong maximum principle for viscosity subsolutions of fully nonlinear elliptic PDEs on the form $$ F(x,u,Du,D^2u) = 0 $$ under suitable structure conditions on the equation allowing for non-Lipschitz growth in the gradient terms. In case of smooth boundaries, we also prove the Hopf lemma, the boundary Harnack inequality and that positive viscosity solutions vanishing on a portion of the boundary are comparable with the distance function near the boundary. Our results apply to weak solutions of an eigenvalue problem for the variable exponent $p$-Laplacian.

We investigate the Boundary Harnack Principle in H"older domains of exponent $alpha>0$ by the analytical method developed in our previous work "A short proof of Boundary Harnack Principle".

Input transformation based defense strategies fall short in defending against strong adversarial attacks. Some successful defenses adopt approaches that either increase the randomness within the applied transformations, or make the defense computationally intensive, making it substantially more challenging for the attacker. However, it limits the applicability of such defenses as a pre-processing step, similar to computationally heavy approaches that use retraining and network modifications to achieve robustness to perturbations. In this work, we propose a defense strategy that applies random image corruptions to the input image alone, constructs a self-correlation based subspace followed by a projection operation to suppress the adversarial perturbation. Due to its simplicity, the proposed defense is computationally efficient as compared to the state-of-the-art, and yet can withstand huge perturbations. Further, we develop proximity relationships between the projection operator of a clean image and of its adversarially perturbed version, via bounds relating geodesic distance on the Grassmannian to matrix Frobenius norms. We empirically show that our strategy is complementary to other weak defenses like JPEG compression and can be seamlessly integrated with them to create a stronger defense. We present extensive experiments on the ImageNet dataset across four different models namely InceptionV3, ResNet50, VGG16 and MobileNet models with perturbation magnitude set to {epsilon} = 16. Unlike state-of-the-art approaches, even without any retraining, the proposed strategy achieves an absolute improvement of ~ 4.5% in defense accuracy on ImageNet.

Quantum computing has evolved quickly in recent years and is showing significant benefits in a variety of fields. Malware analysis is one of those fields that could also take advantage of quantum computing. The combination of software used to locate the most frequent hashes and $n$-grams between benign and malicious software (KiloGram) and a quantum search algorithm could be beneficial, by loading the table of hashes and $n$-grams into a quantum computer, and thereby speeding up the process of mapping $n$-grams to their hashes. The first phase will be to use KiloGram to find the top-$k$ hashes and $n$-grams for a large malware corpus. From here, the resulting hash table is then loaded into a quantum machine. A quantum search algorithm is then used search among every permutation of the entangled key and value pairs to find the desired hash value. This prevents one from having to re-compute hashes for a set of $n$-grams, which can take on average $O(MN)$ time, whereas the quantum algorithm could take $O(sqrt{N})$ in the number of table lookups to find the desired hash values.

Robots operating in household environments must find objects on shelves, under tables, and in cupboards. Previous work often formulate the object search problem as a POMDP (Partially Observable Markov Decision Process), yet constrain the search space in 2D. We propose a new approach that enables the robot to efficiently search for objects in 3D, taking occlusions into account. We model the problem as an object-oriented POMDP, where the robot receives a volumetric observation from a viewing frustum and must produce a policy to efficiently search for objects. To address the challenge of large state and observation spaces, we first propose a per-voxel observation model which drastically reduces the observation size necessary for planning. Then, we present a novel octree-based belief representation which captures beliefs at different resolutions and supports efficient exact belief update. Finally, we design an online multi-resolution planning algorithm that leverages the resolution layers in the octree structure as levels of abstractions to the original POMDP problem. Our evaluation in a simulated 3D domain shows that, as the problem scales, our approach significantly outperforms baselines without resolution hierarchy by 25%-35% in cumulative reward. We demonstrate the practicality of our approach on a torso-actuated mobile robot searching for objects in areas of a cluttered lab environment where objects appear on surfaces at different heights.

Predicting the spatial configuration of gas molecules in nanopores of shale formations is crucial for fluid flow forecasting and hydrocarbon reserves estimation. The key challenge in these tight formations is that the majority of the pore sizes are less than 50 nm. At this scale, the fluid properties are affected by nanoconfinement effects due to the increased fluid-solid interactions. For instance, gas adsorption to the pore walls could account for up to 85% of the total hydrocarbon volume in a tight reservoir. Although there are analytical solutions that describe this phenomenon for simple geometries, they are not suitable for describing realistic pores, where surface roughness and geometric anisotropy play important roles. To describe these, molecular dynamics (MD) simulations are used since they consider fluid-solid and fluid-fluid interactions at the molecular level. However, MD simulations are computationally expensive, and are not able to simulate scales larger than a few connected nanopores. We present a method for building and training physics-based deep learning surrogate models to carry out fast and accurate predictions of molecular configurations of gas inside nanopores. Since training deep learning models requires extensive databases that are computationally expensive to create, we employ active learning (AL). AL reduces the overhead of creating comprehensive sets of high-fidelity data by determining where the model uncertainty is greatest, and running simulations on the fly to minimize it. The proposed workflow enables nanoconfinement effects to be rigorously considered at the mesoscale where complex connected sets of nanopores control key applications such as hydrocarbon recovery and CO2 sequestration.

In this work, we investigate multi-task learning as a way of pre-training models for classification tasks in digital pathology. It is motivated by the fact that many small and medium-size datasets have been released by the community over the years whereas there is no large scale dataset similar to ImageNet in the domain. We first assemble and transform many digital pathology datasets into a pool of 22 classification tasks and almost 900k images. Then, we propose a simple architecture and training scheme for creating a transferable model and a robust evaluation and selection protocol in order to evaluate our method. Depending on the target task, we show that our models used as feature extractors either improve significantly over ImageNet pre-trained models or provide comparable performance. Fine-tuning improves performance over feature extraction and is able to recover the lack of specificity of ImageNet features, as both pre-training sources yield comparable performance.

Large transformer-based language models have been shown to be very effective in many classification tasks. However, their computational complexity prevents their use in applications requiring the classification of a large set of candidates. While previous works have investigated approaches to reduce model size, relatively little attention has been paid to techniques to improve batch throughput during inference. In this paper, we introduce the Cascade Transformer, a simple yet effective technique to adapt transformer-based models into a cascade of rankers. Each ranker is used to prune a subset of candidates in a batch, thus dramatically increasing throughput at inference time. Partial encodings from the transformer model are shared among rerankers, providing further speed-up. When compared to a state-of-the-art transformer model, our approach reduces computation by 37% with almost no impact on accuracy, as measured on two English Question Answering datasets.

We show that a random puncturing of a code with good distance is list recoverable beyond the Johnson bound. In particular, this implies that there are Reed-Solomon codes that are list recoverable beyond the Johnson bound. It was previously known that there are Reed-Solomon codes that do not have this property. As an immediate corollary to our main theorem, we obtain better degree bounds on unbalanced expanders that come from Reed-Solomon codes.

Temporal event segmentation of a long video into coherent events requires a high level understanding of activities' temporal features. The event segmentation problem has been tackled by researchers in an offline training scheme, either by providing full, or weak, supervision through manually annotated labels or by self-supervised epoch based training. In this work, we present a continual learning perceptual prediction framework (influenced by cognitive psychology) capable of temporal event segmentation through understanding of the underlying representation of objects within individual frames. Our framework also outputs attention maps which effectively localize and track events-causing objects in each frame. The model is tested on a wildlife monitoring dataset in a continual training manner resulting in $80\%$ recall rate at $20\%$ false positive rate for frame level segmentation. Activity level testing has yielded $80\%$ activity recall rate for one false activity detection every 50 minutes.

Differential machine learning (ML) extends supervised learning, with models trained on examples of not only inputs and labels, but also differentials of labels to inputs.

Differential ML is applicable in all situations where high quality first order derivatives wrt training inputs are available. In the context of financial Derivatives risk management, pathwise differentials are efficiently computed with automatic adjoint differentiation (AAD). Differential ML, combined with AAD, provides extremely effective pricing and risk approximations. We can produce fast pricing analytics in models too complex for closed form solutions, extract the risk factors of complex transactions and trading books, and effectively compute risk management metrics like reports across a large number of scenarios, backtesting and simulation of hedge strategies, or capital regulations.

The article focuses on differential deep learning (DL), arguably the strongest application. Standard DL trains neural networks (NN) on punctual examples, whereas differential DL teaches them the shape of the target function, resulting in vastly improved performance, illustrated with a number of numerical examples, both idealized and real world. In the online appendices, we apply differential learning to other ML models, like classic regression or principal component analysis (PCA), with equally remarkable results.

This paper is meant to be read in conjunction with its companion GitHub repo https://github.com/differential-machine-learning, where we posted a TensorFlow implementation, tested on Google Colab, along with examples from the article and additional ones. We also posted appendices covering many practical implementation details not covered in the paper, mathematical proofs, application to ML models besides neural networks and extensions necessary for a reliable implementation in production.