cssi/SciSciGPT-SciSciNet huggingface dataset:
Critical heights on the moduli space of polynomials

Abstract Let M d be the moduli space of one-dimensional, degree d ⩾ 2 , complex polynomial dynamical systems. The escape rates of the critical points determine a critical heights map G : M d → R d − 1 . For generic values of G , we show that each connected component of a fiber of G is the deformation space for twist deformations on the basin of infinity. We analyze the quotient space T d ⁎ obtained by collapsing each connected component of a fiber of G to a point. The space T d ⁎ is a parameter-space analog of the polynomial tree T ( f ) associated to a polynomial f : C → C , studied in DeMarco and McMullen (2008) [6] , and there is a natural projection from T d ⁎ to the space of trees T d . We show that the projectivization P T d ⁎ is compact and contractible; further, the shift locus in P T d ⁎ has a canonical locally finite simplicial structure. The top-dimensional simplices are in one-to-one correspondence with topological conjugacy classes of structurally stable polynomials in the shift locus.
An Online Plug-and-Play Algorithm for Regularized Image Reconstruction

Plug-and-play priors (PnP) is a powerful framework for regularizing imaging inverse problems by using advanced denoisers within an iterative algorithm. Recent experimental evidence suggests that PnP algorithms achieve the state-of-the-art performance in a range of imaging applications. In this paper, we introduce a new online PnP algorithm based on the proximal gradient method (PGM). The proposed algorithm uses only a subset of measurements at every iteration, which makes it scalable to very large datasets. We present a new theoretical convergence analysis, for both batch and online variants of PnP-PGM, for denoisers that do not necessarily correspond to proximal operators. We also present simulations illustrating the applicability of the algorithm to image reconstruction in diffraction tomography. The results in this paper have the potential to expand the applicability of the PnP framework to very large datasets.
Renormalized oscillation theory for Hamiltonian systems

Abstract We extend a result on renormalized oscillation theory, originally derived for Sturm–Liouville and Dirac-type operators on arbitrary intervals in the context of scalar coefficients, to the case of general Hamiltonian systems with block matrix coefficients. In particular, this contains the cases of general Sturm–Liouville and Dirac-type operators with block matrix-valued coefficients as special cases. The principal feature of these renormalized oscillation theory results consists in the fact that by replacing solutions by appropriate Wronskians of solutions, oscillation theory now applies to intervals in essential spectral gaps where traditional oscillation theory typically fails.
Conversion of HOL Light proofs into Metamath

We present an algorithm for converting proofs from the OpenTheory interchange format, which can be translated to and from any of the HOL family of proof languages (HOL4, HOL Light, ProofPower, and Isabelle), into the ZFC-based Metamath language. This task is divided into two steps: the translation of an OpenTheory proof into a Metamath HOL formalization, hol.mm , followed by the embedding of the HOL formalization into the main ZFC foundations of the main Metamath library, set.mm . This process provides a means to link the simplicity of the Metamath foundations to the intense automation efforts which have borne fruit in HOL Light, allowing the production of complete Metamath proofs of theorems in HOL Light, while also proving that HOL Light is consistent, relative to Metamath's ZFC axiomatization.
Computing weight q-multiplicities for the representations of the simple Lie algebras

The multiplicity of a weight $$\mu $$ in an irreducible representation of a simple Lie algebra $$\mathfrak {g}$$ with highest weight $$\lambda $$ can be computed via the use of Kostant’s weight multiplicity formula. This formula is an alternating sum over the Weyl group and involves the computation of a partition function. In this paper we consider a q-analog of Kostant’s weight multiplicity and present a SageMath program to compute q-multiplicities for the simple Lie algebras.
Direct Learning to Rank And Rerank

Learning-to-rank techniques have proven to be extremely useful for prioritization problems, where we rank items in order of their estimated probabilities, and dedicate our limited resources to the top-ranked items. This work exposes a serious problem with the state of learning-to-rank algorithms, which is that they are based on convex proxies that lead to poor approximations. We then discuss the possibility of "exact" reranking algorithms based on mathematical programming. We prove that a relaxed version of the "exact" problem has the same optimal solution, and provide an empirical analysis.
A Push-Pull Gradient Method for Distributed Optimization in Networks

In this paper, we focus on solving a distributed convex optimization problem in a network, where each agent has its own convex cost function and the goal is to minimize the sum of the agents' cost functions while obeying the network connectivity structure. In order to minimize the sum of the cost functions, we consider a new distributed gradient-based method where each node maintains two estimates, namely, an estimate of the optimal decision variable and an estimate of the gradient for the average of the agents' objective functions. From the viewpoint of an agent, the information about the decision variable is pushed to the neighbors, while the information about the gradients is pulled from the neighbors (hence giving the name “push-pull gradient method”). The method unifies the algorithms with different types of distributed architecture, including decentralized (peer-to-peer), centralized (master-slave), and semi-centralized (leader-follower) architecture. We show that the algorithm converges linearly for strongly convex and smooth objective functions over a directed static network. In our numerical test, the algorithm performs well even for time-varying directed networks.
Quantum knot invariants

This is a survey talk on one of the best known quantum knot invariants, the colored Jones polynomial of a knot, and its relation to the algebraic/geometric topology and hyperbolic geometry of the knot complement. We review several aspects of the colored Jones polynomial, emphasizing modularity, stability and effective computations. The talk was given in the Mathematische Arbeitstagung June 24–July 1, 2011.
Mean Field Residual Networks: On the Edge of Chaos

We study randomly initialized residual networks using mean field theory and the theory of difference equations. Classical feedforward neural networks, such as those with tanh activations, exhibit exponential behavior on the average when propagating inputs forward or gradients backward. The exponential forward dynamics causes rapid collapsing of the input space geometry, while the exponential backward dynamics causes drastic vanishing or exploding gradients. We show, in contrast, that by adding skip connections, the network will, depending on the nonlinearity, adopt subexponential forward and backward dynamics, and in many cases in fact polynomial. The exponents of these polynomials are obtained through analytic methods and proved and verified empirically to be correct. In terms of the "edge of chaos" hypothesis, these subexponential and polynomial laws allow residual networks to "hover over the boundary between stability and chaos," thus preserving the geometry of the input space and the gradient information flow. In our experiments, for each activation function we study here, we initialize residual networks with different hyperparameters and train them on MNIST. Remarkably, our initialization time theory can accurately predict test time performance of these networks, by tracking either the expected amount of gradient explosion or the expected squared distance between the images of two input vectors. Importantly, we show, theoretically as well as empirically, that common initializations such as the Xavier or the He schemes are not optimal for residual networks, because the optimal initialization variances depend on the depth. Finally, we have made mathematical contributions by deriving several new identities for the kernels of powers of ReLU functions by relating them to the zeroth Bessel function of the second kind.
Domain Wall Six-Vertex Model with Half-Turn Symmetry

We obtain asymptotic formulas for the partition function of the six-vertex model with domain wall boundary conditions and half-turn symmetry in each of the phase regions. The proof is based on the Izergin–Korepin–Kuperberg determinantal formula for the partition function and its reduction to orthogonal polynomials, and on an asymptotic analysis of the orthogonal polynomials under consideration in the framework of the Riemann–Hilbert approach.
A note on density estimation for binary sequences

Abstract A histogram estimate of the Radon–Nikodym derivative of a probability measure with respect to a dominating measure is developed for binary sequences in { 0 , 1 } N . A necessary and sufficient condition for the consistency of the estimate in the mean-square sense is given. It is noted that the product topology on { 0 , 1 } N and the corresponding dominating product measure pose considerable restrictions on the rate of sampling required for the requisite convergence.
Moduli spaces of critical Riemannian metrics with L^n/2 norm curvature bounds

Abstract We consider the moduli space of a class of critical metrics on compact manifolds that includes extremal Kahler metrics. We show that under the conditions of volume and diameter bounds, L n 2 curvature bounds, and Sobolev constant bounds, this Moduli space can be compactified by including orbifolds with finitely many orbifold points. In dimension >4 these orbifold points are C ∞ Riemannian orbifold points, and in the 4 dimensional extremal Kahler case, they are also of class C ∞ .
Optimizing the Number of Fog Nodes for Cloud-Fog-Thing Networks

Fog networking has recently received considerable attention from a theoretical perspective, but in order for such networks to be practical several open areas need to be addressed. This paper determines the optimum number of nodes that should be upgraded to fog nodes with additional computing capabilities in order to maximize the average data rate and minimize the transmission delay. The optimization is performed for a given set of wireless channel conditions and a fixed total number of network nodes. It will be shown that, having more or less fog nodes than the optimum degrades the data rate. The numerical results indicate that the average data rate can increase nearly an order of magnitude for an optimized number of fog nodes in case of shadowing and fading. It is further shown that the optimum number of fog nodes does not increase in direct proportion to the increase in the total number of nodes. Furthermore, the optimum number of fog nodes decreases when channels have high path loss exponents. These findings suggest that the fog nodes must be selected among those that have the highest computation capability for densely deployed networks and high path loss exponent channels.
Infinite product exponents for modular forms

Recently, Choi obtained a description of the coefficients of the infinite product expansions of meromorphic modular forms over $$\Gamma _0(N)$$ Γ 0 ( N ) . Using this result, we provide some bounds on these infinite product coefficients for holomorphic modular forms. We give an exponential upper bound for the growth of these coefficients. We show that this bound is also a lower bound in the case that the genus of the associated modular curve $$X_0(N)$$ X 0 ( N ) is 0 or 1.
Another observation about operator compressions

Let T be a self-adjoint operator on a finite dimensional Hilbert space. It is shown that the distribution of the eigenvalues of a compression of T to a subspace of a given dimension is almost the same for almost all subspaces. This is a coordinate-free analogue of a recent result of Chatterjee and Ledoux on principal submatrices. The proof is based on measure concentration and entropy techniques, and the result improves on some aspects of the result of Chatterjee and Ledoux.
Enhanced Word Representations for Bridging Anaphora Resolution

Most current models of word representations (e.g., GloVe) have successfully captured fine-grained semantics. However, semantic similarity exhibited in these word embeddings is not suitable for resolving bridging anaphora, which requires the knowledge of associative similarity (i.e., relatedness) instead of semantic similarity information between synonyms or hypernyms. We create word embeddings (embeddings_PP) to capture such relatedness by exploring the syntactic structure of noun phrases. We demonstrate that using embeddings _PP alone achieves around 30% of accuracy for bridging anaphora resolution on the ISNotes corpus. Furthermore, we achieve a substantial gain over the state-of-the-art system (Hou et al., 2013b) for bridging antecedent selection.
The Peterson variety and the wonderful compactification

We look at the centralizer in a semisimple algebraic group $G$ of a regular nilpotent element, and show that its closure in the wonderful compactification is isomorphic to the Peterson variety. It follows that the closure in the wonderful compactification of the centralizer $G^x$ of any regular element $x$ is isomorphic to the closure of a general $G^x$-orbit in the flag variety. We also give a description of the $G^e$-orbit structure of the Peterson variety.
Ensemble-LES analysis of perturbation response of turbulent partially-premixed flames

Abstract The response to small perturbations of the Sandia CH4/air flame series is studied using a dynamical systems formulation. Here, the Sandia D and E turbulent partially-premixed flames are computed using large eddy simulation (LES) with a flamelet/progress variable approach. Using 300 simultaneous LES computations of each flame, the partial Lyapunov spectrum of the flow is obtained, which provides a set of Lyapunov exponents (LEs) and Lyapunov vectors (LVs). Special numerical procedures to handle such large data sets and perform matrix manipulations on-the-fly are discussed. From the LEs, the dimension of the attractor for the flames was estimated, for the first time, to be at least 5000 (flame D) and 10500 (flame E). This is a significant result indicating that the chaoticity of the turbulent flame is confined to a volume of phase-space that is quantified. The growth of perturbations occurs at time-scales that are much larger than the smallest strain-rate based times scales, but smaller than the integral scales. Further, the Lyapunov vectors that determine the perturbation response show that the strongest perturbations are aligned with the jet breakdown region, while other LVs are aligned with the shear layer. It is found that both the extinction/re-ignition process and the turbulence-induced jet motion strongly contribute to the flame chaoticity but are not equally important for all the LVs. These results provide new insights into the perturbation response of turbulent flames. More broadly, this Lyapunov approach allows a full characterization of the chaotic dynamics associated with turbulent flames, which have applications for both modeling and control of such complex flows.
Exponential random graph models with big networks: Maximum pseudolikelihood estimation and the parametric bootstrap

With the growth of interest in network data across fields, the Exponential Random Graph Model (ERGM) has emerged as the leading approach to the statistical analysis of network data. ERGM parameter estimation requires the approximation of an intractable normalizing constant. Simulation methods represent the state-of-the-art approach to approximating the normalizing constant, leading to estimation by Monte Carlo maximum likelihood (MCMLE). MCMLE is accurate when a large sample of networks is used to approximate the normalizing constant. However, MCMLE is computationally expensive, and may be prohibitively so if the size of the network is on the order of 1,000 nodes (i.e., one million potential ties) or greater. When the network is large, one option is maximum pseudolikelihood estimation (MPLE). The standard MPLE is simple and fast, but generally underestimates standard errors. We show that a resampling method — the parametric bootstrap — results in accurate coverage probabilities for confidence intervals. We find that bootstrapped MPLE can be run in 1/5th the time of MCMLE. We study the relative performance of MCMLE and MPLE with simulation studies, and illustrate the two different approaches by applying them to a network of bills introduced in the United State Senate.
Simultaneous semi-stable reduction for curves with ADE singularities

A key tool in the study of algebraic surfaces and their moduli is Brieskorn's simultaneous resolution for families of algebraic surfaces with simple (du Val or ADE) singularities. In this paper we show that a similar statement holds for families of curves with at worst simple (ADE) singularities. For a family X ! B of ADE curves, we give an explicit and natural resolution of the rational map B 99K Mg. Moreover, we discuss a lifting of this map to the moduli stack Mg, i.e. a simultaneous semi-stable reduction for the family X /B. In particular, we note that in contrast to what might be expected from the case of surfaces, the natural Weyl cover of B is not a sufficient base change for a lifting of the map B 99K Mg to Mg.
Detecting localized eigenstates of linear operators

We describe a way of detecting the location of localized eigenvectors of the eigenvalue problem $$Ax = \lambda x$$ for eigenvalues $$\lambda $$ with $$|\lambda |$$ comparatively large. We define the family of functions $$f_{\alpha }: \left\{ 1,2, \dots , n\right\} \rightarrow {\mathbb {R}}_{}$$ $$\begin{aligned} f_{\alpha }(k) = \log \left( \Vert A^{\alpha } e_k \Vert _{\ell ^2} \right) , \end{aligned}$$ where $$\alpha \ge 0$$ is a parameter and $$e_k = (0,0,\ldots , 0,1,0, \ldots , 0)$$ is the kth standard basis vector. We prove that eigenvectors associated with eigenvalues with large absolute value localize around local maxima of $$f_{\alpha }$$ : the metastable states in the power iteration method (slowing down its convergence) can be used to predict localization. We present a fast randomized algorithm and discuss different examples: a random band matrix, discretizations of the local operator $$-\Delta + V$$ , and the nonlocal operator $$(-\Delta )^{3/4} + V$$ .
Optimal estimation of low rank density matrices

The density matrices are positively semi-definite Hermitian matrices of unit trace that describe the state of a quantum system. The goal of the paper is to develop minimax lower bounds on error rates of estimation of low rank density matrices in trace regression models used in quantum state tomography (in particular, in the case of Pauli measurements) with explicit dependence of the bounds on the rank and other complexity parameters. Such bounds are established for several statistically relevant distances, including quantum versions of Kullback-Leibler divergence (relative entropy distance) and of Hellinger distance (so called Bures distance), and Schatten p-norm distances. Sharp upper bounds and oracle inequalities for least squares estimator with von Neumann entropy penalization are obtained showing that minimax lower bounds are attained (up to logarithmic factors) for these distances.
Towards a Calculus of Echo State Networks

Abstract Reservoir computing is a recent trend in neural networks which uses dynamical perturbations in the phase space of a system to compute a desired target function. We show one can for- mulate an expectation of system performance in a simple model of reservoir computing called echo state networks. In contrast with previous theoretical frameworks, which uses annealed approximation, we calculate the exact optimal output weights as a function of the structure of the system and the properties of the input and the target function. Our calculation agrees with numerical simulations. To the best of our knowledge this work presents the first exact analytical solution to optimal output weights in echo state networks.
Learning Optimal Linear Regularizers

We present algorithms for efficiently learning regularizers that improve generalization. Our approach is based on the insight that regularizers can be viewed as upper bounds on the generalization gap, and that reducing the slack in the bound can improve performance on test data. For a broad class of regularizers, the hyperparameters that give the best upper bound can be computed using linear programming. Under certain Bayesian assumptions, solving the LP lets us "jump" to the optimal hyperparameters given very limited data. This suggests a natural algorithm for tuning regularization hyperparameters, which we show to be effective on both real and synthetic data.
The univalence axiom for elegant Reedy presheaves

We show that Voevodsky's univalence axiom for intensional type theory is valid in categories of simplicial presheaves on elegant Reedy categories. In addition to diagrams on inverse categories, as considered in previous work of the author, this includes bisimplicial sets and $\Theta_n$-spaces. This has potential applications to the study of homotopical models for higher categories.
Some results on asymptotic regularity of ideal sheaves

Abstract Let I be an ideal sheaf on P n defining a subscheme X . Associated to I there are two elementary invariants: the invariant s which measures the positivity of I , and the minimal number d such that I ( d ) is generated by its global sections. It is now clear that the asymptotic behavior of reg I t is governed by s but usually not linear. In this paper, we first describe the linear behavior of the asymptotic regularity by showing that if s = d , i.e., s reaches its maximal value, then for t large enough reg I t = d t + e for some positive constant e . We then turn to concrete geometric settings to study the asymptotic regularity of I in the case that X is a nonsingular variety embedded by a very ample adjoint line bundle. Our approach also gives regularity bounds for I t once we know reg I and assume that X is a local complete intersection.
Quantifying the accuracy of approximate diffusions and Markov chains

Markov chains and diffusion processes are indispensable tools in machine learning and statistics that are used for inference, sampling, and modeling. With the growth of large-scale datasets, the computational cost associated with simulating these stochastic processes can be considerable, and many algorithms have been proposed to approximate the underlying Markov chain or diffusion. A fundamental question is how the computational savings trade off against the statistical error incurred due to approximations. This paper develops general results that address this question. We bound the Wasserstein distance between the equilibrium distributions of two diffusions as a function of their mixing rates and the deviation in their drifts. We show that this error bound is tight in simple Gaussian settings. Our general result on continuous diffusions can be discretized to provide insights into the computational-statistical trade-off of Markov chains. As an illustration, we apply our framework to derive finite-sample error bounds of approximate unadjusted Langevin dynamics. We characterize computation-constrained settings where, by using fast-to-compute approximate gradients in the Langevin dynamics, we obtain more accurate samples compared to using the exact gradients. Finally, as an additional application of our approach, we quantify the accuracy of approximate zig-zag sampling. Our theoretical analyses are supported by simulation experiments.
Conjugacy search problem and the Andrews–Curtis conjecture

We develop new computational methods for studying potential counterexamples to the Andrews-Curtis conjecture, in particular, Akbulut-Kurby examples AK(n). We devise a number of algorithms in an attempt to disprove the most interesting counterexample AK(3). To improve metric properties of the search space (which is a set of balanced presentations of the trivial group) we introduce a new transformation (called an ACM-move here) that generalizes the original Andrews-Curtis transformations and discuss details of a practical implementation. To reduce growth of the search space we introduce a strong equivalence relation on balanced presentations and study the space modulo automorphisms of the underlying free group. Finally, we prove that automorphism-moves can be applied to AK(n)-presentations. Unfortunately, despite a lot of effort we were unable to trivialize any of AK(n)-presentations, for n>2.
Towards Formal Fault Tree Analysis Using Theorem Proving

Fault Tree Analysis FTA is a dependability analysis technique that has been widely used to predict reliability, availability and safety of many complex engineering systems. Traditionally, these FTA-based analyses are done using paper-and-pencil proof methods or computer simulations, which cannot ascertain absolute correctness due to their inherent limitations. As a complementary approach, we propose to use the higher-order-logic theorem prover HOL4 to conduct the FTA-based analysis of safety-critical systems where accuracy of failure analysis is a dire need. In particular, the paper presents a higher-order-logic formalization of generic Fault Tree gates, i.e., AND, OR, NAND, NOR, XOR and NOT and the formal verification of their failure probability expressions. Moreover, we have formally verified the generic probabilistic inclusion-exclusion principle, which is one of the foremost requirements for conducting the FTA-based failure analysis of any given system. For illustration purposes, we conduct the FTA-based failure analysis of a solar array that is used as the main source of power for the Dong Fang Hong-3 DFH-3 satellite.
Cosmic String Loops and Gravitational Radiation

Understanding of the signatures of cosmic string networks is limited by a large uncertainty in the sizes at which cosmic string loops form. We review cosmic string network evolution, and the gravitational signatures, with emphasis on this uncertainty. We then review a recent analytic model of cosmic string networks. In combination with recent simulations, this suggests that 90% of the string goes into very small loops, at the gravitational radiation scale, and 10% into loops near the Hubble scale. We discuss cosmic string signatures in such a scenario, and the ‘inverse problem’ of determining the microscopic cosmic string properties from observations. To appear in the Proceedings of the 11th Marcel Grossman meeting.
Coefficients of McKay-Thompson series and distributions of the moonshine module

In a recent paper, Duncan, Griffin and Ono provide exact formulas for the coefficients of McKay-Thompson series and use them to find asymptotic expressions for the distribution of irreducible representations in the moonshine module $V^\natural = \bigoplus_n V_n^\natural$. Their results show that as $n$ tends to infinity, $V_n^\natural$ is dominated by direct sums of copies of the regular representation. That is, if we view $V_n^\natural$ as a module over the group ring $\mathbb{Z}[\mathbb{M}]$, the free-part dominates. A natural problem, posed at the end of the aforementioned paper, is to characterize the distribution of irreducible representations in the non-free part. Here, we study asymptotic formulas for the coefficients of McKay-Thompson series to answer this question. We arrive at an ordering of the series by the magnitude of their coefficients, which corresponds to various contributions to the distribution. In particular, we show how the asymptotic distribution of the non-free part is dictated by the column for conjugacy class 2A in the monster's character table. We find analogous results for the other monster modules $V^{(-m)}$ and $W^\natural$ studied by Duncan, Griffin, and Ono.
Effective bounds on the dimensions of Jacobians covering abelian varieties

We show that any polarized abelian variety over a finite field is covered by a Jacobian whose dimension is bounded by an explicit constant. We do this by first proving an effective version of Poonen's Bertini theorem over finite fields, which allows us to show the existence of smooth curves arising as hypersurface sections of bounded degree and genus. Additionally, we show that for simple abelian varieties a better bound is possible. As an application of these results we show that if $E$ is an elliptic curve over a finite field then for any $n\in \mathbb{N}$ there exist smooth curves of bounded genus whose Jacobians have a factor isogenous to $E^n$.
Quantifying Reticulation in Phylogenetic Complexes Using Homology

Reticulate evolutionary processes result in phylogenetic histories that cannot be modeled using a tree topology. Here, we apply methods from topological data analysis to molecular sequence data with reticulations. Using a simple example, we demonstrate the correspondence between nontrivial higher homology and reticulate evolution. We discuss the sensitivity of the standard filtration and show cases where reticulate evolution can fail to be detected. We introduce an extension of the standard framework and define the median complex as a construction to recover signal of the frequency and scale of reticulate evolution by inferring and imputing putative ancestral states. Finally, we apply our methods to two datasets from phylogenetics. Our work expands on earlier ideas of using topology to extract important evolutionary features from genomic data.
Popular Progression Differences in Vector Spaces

Green used an arithmetic analogue of Szemeredi's celebrated regularity lemma to prove the following strengthening of Roth's theorem in vector spaces. For every $\alpha>0$, $\beta<\alpha^3$, and prime number $p$, there is a least positive integer $n_p(\alpha,\beta)$ such that if $n \geq n_p(\alpha,\beta)$, then for every subset of $\mathbb{F}_p^n$ of density at least $\alpha$ there is a nonzero $d$ for which the density of three-term arithmetic progressions with common difference $d$ is at least $\beta$. We determine for $p \geq 19$ the tower height of $n_p(\alpha,\beta)$ up to an absolute constant factor and an additive term depending only on $p$. In particular, if we want half the random bound (so $\beta=\alpha^3/2$), then the dimension $n$ required is a tower of twos of height $\Theta \left((\log p) \log \log (1/\alpha)\right)$. It turns out that the tower height in general takes on a different form in several different regions of $\alpha$ and $\beta$, and different arguments are used both in the upper and lower bounds to handle these cases.
Primitive Deformations of Quantum p -Groups

In this paper, working over an algebraically closed field k of prime characteristic p, we introduce a concept, called Primitive Deformation, to provide a structured technique to classify certain finite-dimensional Hopf algebras which are Hopf deformations of restricted universal enveloping algebras. We illustrate this technique for the case when the restricted Lie algebra has dimension 3. Together with our previous classification results, we provide a complete classification of p3-dimensional connected Hopf algebras over k of characteristic p > 2.
Some results on superpatterns for preferential arrangements

Abstract A superpattern is a string of characters of length n over [ k ] = { 1 , 2 , … , k } that contains as a subsequence, and in a sense that depends on the context, all the smaller strings of length k in a certain class. We prove structural and probabilistic results on superpatterns for preferential arrangements, including (i) a theorem that demonstrates that a string is a superpattern for all preferential arrangements if and only if it is a superpattern for all permutations; and (ii) a result that is reminiscent of a still unresolved conjecture of Alon on the smallest permutation on [ n ] that contains all k-permutations with high probability.
Counting with Borel’s triangle

Abstract Borel’s triangle is an array of integers closely related to the classical Catalan numbers. In this paper we study combinatorial statistics counted by Borel’s triangle. We present various combinatorial interpretations of Borel’s triangle in terms of lattice paths, binary trees, and pattern avoiding permutations and matchings, and derive a functional equation that is useful in analyzing the involved structures.
Lower bound for ranks of invariant forms

We give a lower bound for the Waring rank and cactus rank of forms that are invariant under an action of a connected algebraic group. We use this to improve the Ranestad–Schreyer–Shafiei lower bounds for the Waring ranks and cactus ranks of determinants of generic matrices, Pfaffians of generic skew-symmetric matrices, and determinants of generic symmetric matrices.
Amazon in the White Space: Social Recommendation Aided Distributed Spectrum Access

Distributed spectrum access (DSA) is challenging, since an individual secondary user often has limited sensing capabilities only. One key insight is that channel recommendation among secondary users can help to take advantage of the inherent correlation structure of spectrum availability in both time and space, and enable users to obtain more informed spectrum opportunities. With this insight, we advocate to leverage the wisdom of crowds, and devise social recommendation aided DSA mechanisms to orient secondary users to make more intelligent spectrum access decisions, for both strong and weak network information cases. We start with the strong network information case where secondary users have the statistical information. To mitigate the difficulty due to the curse of dimensionality in the stochastic game approach, we take the one-step Nash approach and cast the social recommendation aided DSA decision making problem at each time slot as a strategic game. We show that it is a potential game, and then devise an algorithm to achieve the Nash equilibrium by exploiting its finite improvement property. For the weak information case where secondary users do not have the statistical information, we develop a distributed reinforcement learning mechanism for social recommendation aided DSA based on the local observations of secondary users only. Appealing to the maximum-norm contraction mapping, we also derive the conditions under which the distributed mechanism converges and characterize the equilibrium therein. Numerical results reveal that the proposed social recommendation aided DSA mechanisms can achieve a superior performance using real social data traces and its performance loss in the weak network information case is insignificant, compared with the strong network information case.
An elementary, illustrative proof of the Rado–Horn theorem

Abstract The Rado–Horn theorem provides necessary and sufficient conditions for when a family of vectors can be partitioned into a fixed number of linearly independent sets. Such partitions exist if and only if every subfamily of the vectors satisfies the so-called Rado–Horn inequality. In this paper we provide an elementary proof of the Rado–Horn theorem as well as results for the redundant case. Previous proofs give no information about how to actually partition the vectors; we use ideas present in our proof to find subfamilies of vectors which may be used to construct a kind of “optimal” partition.
Why capillary flows in slender triangular grooves are so stable against disturbances

Lyapunov and non-modal stability analysis reveal why stationary and self-similar capillary flows in slender open triangular channels are so stable to disturbances. This robust feature allows superior flow management for applications ranging from space micropropulsion to microfluidic diagnostics.
Closed hierarchies and non-equilibrium steady states of driven systems

Abstract We present a class of tractable non-equilibrium dynamical quantum systems which includes combinations of injection, detection and extraction of particles interspersed by unitary evolution. We show how such operations generate a hierarchy of equations tying lower correlation functions with higher order ones. The hierarchy closes for particular choices of measurements and leads to a rich class of evolutions whose long time behavior can be simulated efficiently. In particular, we use the method to describe the dynamics of current generation through a generalized quantum exclusion process, and exhibit an explicit formula for the long time energy distribution in the limit of weak driving.
Soliton Solutions of the KP Equation and Application to Shallow Water Waves

The main purpose of this paper is to give a survey of recent developments on a classification of soliton solutions of the Kadomtsev–Petviashvili equation. The paper is self-contained, and we give complete proofs of theorems needed for the classification. The classification is based on the totally nonnegative cells in the Schubert decomposition of the real Grassmann manifold, Gr(N, M), the set of N-dimensional subspaces in . Each soliton solution defined on Gr(N, M) asymptotically consists of the N number of line-solitons for y ≫ 0 and the M − N number of line-solitons for y ≪ 0. In particular, we give detailed description of the soliton solutions associated with Gr(2, 4), which play a fundamental role in the study of multisoliton solutions. We then consider a physical application of some of those solutions related to the Mach reflection discussed by J. Miles in 1977.
Pareto optimal multi-robot motion planning

This paper studies a class of multi-robot coordination problems where a team of robots aim to reach their goal regions with minimum time and avoid collisions with obstacles and other robots. A novel numerical algorithm is proposed to identify the Pareto optimal solutions where no robot can unilaterally reduce its traveling time without extending others'. The consistent approximation of the algorithm in the epigraphical profile sense is guaranteed using set-valued numerical analysis. Simulations show the anytime property and increasing optimality of the proposed algorithm.
Valuation: Accounting for Risk and the Expected Return

Under accounting principles, the recognition of earnings is path-dependent and the path depends on risk and its resolution: under the so-called realization principle, earnings are not booked until uncertainty is resolved. In asset pricing terms, the principle means that earnings cannot be recognized until the firm can book a low-beta asset such as cash or a near-cash discounted receivable. If the risk to which this accounting responds is priced risk, the accounting indicates the expected return. This paper connects accounting under this principle to risk and return, summarizes the supporting empirical evidence, and examines the implications for research on the implied cost of capital, cash-flow betas, asset pricing models that imbed accounting numbers, and papers that assume an autoregressive model for the earnings path to infer the expected return. The accounting that captures risk and its resolution also has implications for the unsolved issue of specifying the appropriate accounting for accounting-based valuation models and, indeed, for financial accounting standards.
On facility location with general lower bounds

In this paper, we give the first constant approximation algorithm for the lower bounded facility location (LBFL) problem with general lower bounds. Prior to our work, such algorithms were only known for the special case where all facilities have the same lower bound: Svitkina [27] gave a 448-approximation for the special case, and subsequently Ahmadian and Swamy [2] improved the approximation factor to 82.6.As in [27] and [2], our algorithm for LBFL with general lower bounds works by reducing the problem to the capacitated facility location (CFL) problem. To handle the challenges raised by the general lower bounds, it involves more reduction steps. One main complication is that after aggregating the clients and facilities at a few locations, each of these locations may contain many facilities with different opening costs and lower bounds. To address this issue, we introduce and reduce the LBFL problem to two intermediate problems called the LBFL with penalty (LBFL-P) and the transportation with configurable supplies and demands (TCSD) problems, which in turn can be reduced to the CFL problem.
Xcel-RAM: Accelerating Binary Neural Networks in High-Throughput SRAM Compute Arrays

Deep neural networks are biologically inspired class of algorithms that have recently demonstrated the state-of-the-art accuracy in large-scale classification and recognition tasks. Hardware acceleration of deep networks is of paramount importance to ensure their ubiquitous presence in future computing platforms. Indeed, a major landmark that enables efficient hardware accelerators for deep networks is the recent advances from the machine learning community that have demonstrated the viability of aggressively scaled deep binary networks. In this paper, we demonstrate how deep binary networks can be accelerated in modified von Neumann machines by enabling binary convolutions within the static random access memory (SRAM) arrays. In general, binary convolutions consist of bit-wise exclusive-NOR (XNOR) operations followed by a population count (popcount). We present two proposals: one based on charge sharing approach to perform vector XNOR and approximate popcount and another based on bit-wise XNOR followed by a digital bit-tree adder for accurate popcount. We highlight the various tradeoffs in terms of circuit complexity, speed-up, and classification accuracy for both the approaches. Few key techniques presented as a part of the manuscript are the use of low-precision, low-overhead analog-to-digital converter (ADC), to achieve a fairly accurate popcount for the charge-sharing scheme and proposal for sectioning of the SRAM array by adding switches onto the read-bitlines, thereby achieving improved parallelism. Our results on benchmark image classification datasets for CIFAR-10 and SVHN on a binarized neural network architecture show energy improvements of up to $6.1\times $ and $2.3\times $ for the two proposals, compared to conventional SRAM banks. In terms of latency, improvements of up to $15.8\times $ and $8.1\times $ were achieved for the two respective proposals.
On the dimension of matrix embeddings of torsion-free nilpotent groups

Abstract Since the work of Jennings (1955), it is well-known that any finitely generated torsion-free nilpotent group can be embedded into unitriangular integer matrices U T N ( Z ) for some N. In 2006, Nickel proposed an algorithm to calculate such embeddings. In this work, we show that if U T n ( Z ) is embedded into U T N ( Z ) using Nickel's algorithm, then N ≥ 2 n / 2 − 2 if the standard ordering of the Mal'cev basis (as in Nickel's original paper) is used. In particular, we establish an exponential worst-case running time of Nickel's algorithm. On the other hand, we also prove a general exponential upper bound on the dimension of the embedding by showing that for any torsion free, finitely generated nilpotent group the matrix representation produced by Nickel's algorithm has never larger dimension than Jennings' embedding. Moreover, when starting with a special Mal'cev basis, Nickel's embedding for U T n ( Z ) has only quadratic size. Finally, we consider some special cases like free nilpotent groups and Heisenberg groups and compare the sizes of the embeddings.
Hardening Quantum Machine Learning Against Adversaries

Security for machine learning has begun to become a serious issue for present day applications. An important question remaining is whether emerging quantum technologies will help or hinder the security of machine learning. Here we discuss a number of ways that quantum information can be used to help make quantum classifiers more secure or private. In particular, we demonstrate a form of robust principal component analysis that, under some circumstances, can provide an exponential speedup relative to robust methods used at present. To demonstrate this approach we introduce a linear combinations of unitaries Hamiltonian simulation method that we show functions when given an imprecise Hamiltonian oracle, which may be of independent interest. We also introduce a new quantum approach for bagging and boosting that can use quantum superposition over the classifiers or splits of the training set to aggregate over many more models than would be possible classically. Finally, we provide a private form of $k$--means clustering that can be used to prevent an all powerful adversary from learning more than a small fraction of a bit from any user. These examples show the role that quantum technologies can play in the security of ML and vice versa. This illustrates that quantum computing can provide useful advantages to machine learning apart from speedups.
Matrix Convex Sets Without Absolute Extreme Points

Abstract This article shows the existence of a class of closed bounded matrix convex sets which do not have absolute extreme points. The sets we consider are noncommutative sets, K X , formed by taking matrix convex combinations of a single tuple X . In the case that X is a tuple of compact operators with no nontrivial finite dimensional reducing subspaces, K X is a closed bounded matrix convex set with no absolute extreme points. A central goal in the theory of matrix convexity is to find a natural notion of an extreme point in the dimension free setting which is minimal with respect to spanning. Matrix extreme points are the strongest type of extreme point known to span matrix convex sets; however, they are not necessarily the smallest set which does so. Absolute extreme points, a more restricted type of extreme points that are closely related to Arveson's boundary, enjoy a strong notion of minimality should they span. This result shows that matrix convex sets may fail to be spanned by their absolute extreme points.
Synthesis of separable controlled invariant sets for modular local control design

Many correct-by-construction control synthesis methods suffer from the curse of dimensionality. Motivated by this challenge, we seek to reduce a correct-by-construction control synthesis problem to subproblems of more modest dimension. As a step towards this goal, in this paper we consider the problem of synthesizing decoupled robustly controlled invariant sets for dynamically coupled linear subsystems with state and input constraints. Our approach, which gives sufficient conditions for decoupled invariance, is based on optimization over linear matrix inequalities which are obtained using slack variable identities. We illustrate the applicability of our method on several examples, including one where we solve local control synthesis problems in a compositional manner.
A Second-Order Time-Stepping Scheme for Simulating Ensembles of Parameterized Flow Problems

We consider settings for which one needs to perform multiple flow simulations based on the Navier-Stokes equations, each having different values for the physical parameters and/or different initial condition data, boundary conditions data, and/or forcing functions. For such settings, we propose a second-order time accurate ensemble-based method that to simulate the whole set of solutions, requires, at each time step, the solution of only a single linear system with multiple right-hand-side vectors. Rigorous analyses are given proving the conditional stability and error estimates for the proposed algorithm. Numerical experiments are provided that illustrate the analyses.