A natural problem in the context of the coupon collector's problem is the behavior of the maximum of independent geometrically distributed random variables (with distinct parameters). This question has been addressed by Brennan et al. (British J. of Math. & CS. 8 (2015), 330-336). Here we provide explicit asymptotic expressions for the moments of that maximum, as well as of the maximum of exponential random variables with corresponding parameters. We also deal with the probability of each of the variables being the maximal one.

The calculations lead to expressions involving Hurwitz's zeta function at certain special points. We find here explicitly the values of the function at these points. Also, the distribution function of the maximum we deal with is closely related to the generating function of the partition function. Thus, our results (and proofs) rely on classical results pertaining to the partition function.

We study a problem concerning parabolic induction in certain p-adic unitary groups. More precisely, for $E/F$ a quadratic extension of p-adic fields the associated unitary group $G=mathrm{U}(n,n)$ contains a parabolic subgroup $P$ with Levi component $L$ isomorphic to $mathrm{GL}_n(E)$. Let $pi$ be an irreducible supercuspidal representation of $L$ of depth zero. We use Hecke algebra methods to determine when the parabolically induced representation $iota_P^G pi$ is reducible.

We construct a smooth moduli stack of tuples consisting of genus two nodal curves, line bundles, and twisted fields. It leads to a desingularization of the moduli of genus two stable maps to projective spaces. The construction of this new moduli is based on systematical application of the theory of stacks with twisted fields (STF), which has its prototype appeared in arXiv:1906.10527 and arXiv:1201.2427 and is fully developed in this article. The results of this article are the second step of a series of works toward the resolutions of the moduli of stable maps of higher genera.

The main goal of this paper is to achieve a parametrization of the solution set of the truncated matricial Hausdorff moment problem in the non-degenerate and degenerate situation. We treat the even and the odd cases simultaneously. Our approach is based on Schur analysis methods. More precisely, we use two interrelated versions of Schur-type algorithms, namely an algebraic one and a function-theoretic one. The algebraic version, worked out in our former paper arXiv:1908.05115, is an algorithm which is applied to finite or infinite sequences of complex matrices. The construction and discussion of the function-theoretic version is a central theme of this paper. This leads us to a complete description via Stieltjes transform of the solution set of the moment problem under consideration. Furthermore, we discuss special solutions in detail.

This paper develops a method for obtaining guaranteed outer approximations for global attractors of continuous and discrete time nonlinear dynamical systems. The method is based on a hierarchy of semidefinite programming problems of increasing size with guaranteed convergence to the global attractor. The approach taken follows an established line of reasoning, where we first characterize the global attractor via an infinite dimensional linear programming problem (LP) in the space of Borel measures. The dual to this LP is in the space of continuous functions and its feasible solutions provide guaranteed outer approximations to the global attractor. For systems with polynomial dynamics, a hierarchy of finite-dimensional sum-of-squares tightenings of the dual LP provides a sequence of outer approximations to the global attractor with guaranteed convergence in the sense of volume discrepancy tending to zero. The method is very simple to use and based purely on convex optimization. Numerical examples with the code available online demonstrate the method.

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.

A subset of vertices in a graph $G$ is called a maximum dissociation set if it induces a subgraph with vertex degree at most 1 and the subset has maximum cardinality. The dissociation number of $G$, denoted by $psi(G)$, is the cardinality of a maximum dissociation set. A subcubic tree is a tree of maximum degree at most 3. In this paper, we give the lower and upper bounds on the dissociation number in a subcubic tree of order $n$ and show that the number of maximum dissociation sets of a subcubic tree of order $n$ and dissociation number $psi$ is at most $1.466^{4n-5psi+2}$.

We apply the paracontrolled calculus to study the asymptotic behavior of a certain quasilinear PDE with smeared mild noise, which originally appears as the space-time scaling limit of a particle system in random environment on one dimensional discrete lattice. We establish the convergence result and show a local in time well-posedness of the limit stochastic PDE with spatial white noise. It turns out that our limit stochastic PDE does not require any renormalization. We also show a comparison theorem for the limit equation.

The main aim of this paper is to detect dynamical properties of the Gy"orgyi-Field model of the Belousov-Zhabotinsky chemical reaction. The corresponding three-variable model given as a set of nonlinear ordinary differential equations depends on one parameter, the flow rate. As certain values of this parameter can give rise to chaos, the analysis was performed in order to identify different dynamics regimes. Dynamical properties were qualified and quantified using classical and also new techniques. Namely, phase portraits, bifurcation diagrams, the Fourier spectra analysis, the 0-1 test for chaos, and approximate entropy. The correlation between approximate entropy and the 0-1 test for chaos was observed and described in detail. Moreover, the three-stage system of nested subintervals of flow rates, for which in every level the 0-1 test for chaos and approximate entropy was computed, is showing the same pattern. The study leads to an open problem whether the set of flow rate parameters has Cantor like structure.

In this work, the generalized $N$-component Fokas-Lenells(FL) equations, which have been studied by Guo and Ling (2012 J. Math. Phys. 53 (7) 073506) for $N=2$, are first investigated via Riemann-Hilbert(RH) approach. The main purpose of this is to study the soliton solutions of the coupled Fokas-Lenells(FL) equations for any positive integer $N$, which have more complex linear relationship than the analogues reported before. We first analyze the spectral analysis of the Lax pair associated with a $(N+1) imes (N+1)$ matrix spectral problem for the $N$-component FL equations. Then, a kind of RH problem is successfully formulated. By introducing the special conditions of irregularity and reflectionless case, the $N$-soliton solution formula of the equations are derived through solving the corresponding RH problem. Furthermore, take $N=2,3$ and $4$ for examples, the localized structures and dynamic propagation behavior of their soliton solutions and their interactions are discussed by some graphical analysis.

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor. We consider the system $mathcal L (H)$ of all sets of lengths of $H$ and study when $mathcal L (H)$ contains or is contained in a system $mathcal L (H')$ of a Krull monoid $H'$ with finite class group $G'$, prime divisors in all classes and Davenport constant $mathsf D (G')=mathsf D (G)$. Among others, we show that if $G$ is either cyclic of order $m ge 7$ or an elementary $2$-group of rank $m-1 ge 6$, and $G'$ is any group which is non-isomorphic to $G$ but with Davenport constant $mathsf D (G')=mathsf D (G)$, then the systems $mathcal L (H)$ and $mathcal L (H')$ are incomparable.

In the present paper by the Fourier transform we show that every linear differential equations of $n$-th order has a solution in $L^1(Bbb{R})$ which is infinitely differentiable in $Bbb{R} setminus {0}$. Moreover the Hyers-Ulam stability of such equations on $L^1(Bbb{R})$ is investigated.

Our goal in this article is to study the global Lorentz estimates for gradient of weak solutions to $p$-Laplace double obstacle problems involving the Schr"odinger term: $-Delta_p u + mathbb{V}|u|^{p-2}u$ with bound constraints $psi_1 le u le psi_2$ in non-smooth domains. This problem has its own interest in mathematics, engineering, physics and other branches of science. Our approach makes a novel connection between the study of Calder'on-Zygmund theory for nonlinear Schr"odinger type equations and variational inequalities for double obstacle problems.

We investigate the quantum twistor bundle constructed as a $U(1)$-quotient of the quantum instanton bundle of Bonechi, Ciccoli and Tarlini. It is an example of a locally trivial noncommutative bundle fulfilling conditions of the framework recently proposed by Brzezi'nski and Szyma'nski. In particular, we give a detailed description of the corresponding $C^*$-algebra of 'continuous functions' on its noncommutative total space. Furthermore, we analyse a different construction of a quantum instanton bundle due to Landi, Pagani and Reina, find a basis of its polynomial algebra and discover an intriguing and unexpected feature of its enveloping $C^*$-algebra.

The congruence subgroup problem for a finitely generated group $Gamma$ and $Gleq Aut(Gamma)$ asks whether the map $hat{G} o Aut(hat{Gamma})$ is injective, or more generally, what is its kernel $Cleft(G,Gamma ight)$? Here $hat{X}$ denotes the profinite completion of $X$. In the case $G=Aut(Gamma)$ we denote $Cleft(Gamma ight)=Cleft(Aut(Gamma),Gamma ight)$.

Let $Gamma$ be a finitely generated group, $ar{Gamma}=Gamma/[Gamma,Gamma]$, and $Gamma^{*}=ar{Gamma}/tor(ar{Gamma})congmathbb{Z}^{(d)}$. Denote $Aut^{*}(Gamma)= extrm{Im}(Aut(Gamma) o Aut(Gamma^{*}))leq GL_{d}(mathbb{Z})$. In this paper we show that when $Gamma$ is nilpotent, there is a canonical isomorphism $Cleft(Gamma ight)simeq C(Aut^{*}(Gamma),Gamma^{*})$. In other words, $Cleft(Gamma ight)$ is completely determined by the solution to the classical congruence subgroup problem for the arithmetic group $Aut^{*}(Gamma)$.

In particular, in the case where $Gamma=Psi_{n,c}$ is a finitely generated free nilpotent group of class $c$ on $n$ elements, we get that $C(Psi_{n,c})=C(mathbb{Z}^{(n)})={e}$ whenever $ngeq3$, and $C(Psi_{2,c})=C(mathbb{Z}^{(2)})=hat{F}_{omega}$ = the free profinite group on countable number of generators.

In this paper, we give a criterion of the Gorenstein property of the Ehrhart ring of the stable set polytope of an h-perfect graph: the Ehrhart ring of the stable set polytope of an h-perfect graph $G$ is Gorenstein if and only if (1) sizes of maximal cliques are constant (say $n$) and (2) (a) $n=1$, (b) $n=2$ and there is no odd cycle without chord and length at least 7 or (c) $ngeq 3$ and there is no odd cycle without chord and length at least 5.

In 1978, Dwight Duffus---editor-in-chief of the journal "Order" from 2010 to 2018 and chair of the Mathematics Department at Emory University from 1991 to 2005---wrote that "it is not obvious that $P$ is connected and $P^P$ isomorphic to $Q^Q$ implies that $Q$ is connected," where $P$ and $Q$ are finite non-empty posets. We show that, indeed, under these hypotheses $Q$ is connected and $Pcong Q$.

Given a standard graded polynomial ring over a commutative Noetherian ring $A$, we prove that the cohomological dimension and the height of the ideals defining any of its Veronese subrings are equal. This result is due to Ogus when $A$ is a field of characteristic zero, and follows from a result of Peskine and Szpiro when $A$ is a field of positive characteristic; our result applies, for example, when $A$ is the ring of integers.

In this paper, we characterize a mixed graph $F$ which contains $k$ edge and arc disjoint spanning mixed arborescences $F_{1}, ldots, F_{k}$, such that for each $v in V(F)$, the cardinality of ${i in [k]: v ext{ is the root of } F_{i}}$ lies in some prescribed interval. This generalizes both Nash-Williams and Tutte's theorem on spanning tree packing for undirected graphs and the previous characterization on digraphs which was given by Cai [in: Arc-disjoint arborescences of digraphs, J. Graph Theory 7(2) (1983), 235-240] and Frank [in: On disjoint trees and arborescences, Algebraic Methods in Graph Theory, Colloquia Mathematica Soc. J. Bolyai, Vol. 25 (North-Holland, Amsterdam) (1978), 159-169].

In recent years, a large class of nuclear $C^ast$-algebras have been classified, modulo an assumption on the Universal Coefficient Theorem (UCT). We think this assumption is redundant and propose a strategy for proving it. Indeed, following the original proof of the classification theorem, we propose bridging the gap between reduction theorems and examples. While many such bridges are possible, various approximate ideal structures appear quite promising.

In this paper, we give new constructions of strongly regular Cayley graphs on abelian groups as generalizations of a series of known constructions: the construction of covering extended building sets in finite fields by Xia (1992), the product construction of Menon-Hadamard difference sets by Turyn (1984), and the construction of Paley type partial difference sets by Polhill (2010). Then, we obtain new large families of strongly regular Cayley graphs of Latin square type or negative Latin square type.

We investigate the evolution in time of the position of a fixed number inthe insertion tableau when the Robinson-Schensted-Knuth algorithm is applied to asequence of random numbers. When the length of the sequence tends to infinity, a typical trajectory after scaling converges uniformly in probability to some deterministiccurve.

Let $G=(V,E)$ be a finite graph and $M_G$ be the centered Hardy-Littlewood maximal operator defined there. We found the optimal value $C_{G,p}$ such that the inequality $$Var_{p}(M_{G}f)le C_{G,p}Var_{p}(f)$$ holds for every every $f:V o mathbb{R},$ where $Var_p$ stands for the $p$-variation, when: (i)$G=K_n$ (complete graph) and $pin [frac{ln(4)}{ln(6)},infty)$ or $G=K_4$ and $pin (0,infty)$;(ii) $G=S_n$(star graph) and $1ge pge frac{1}{2}$; $pin (0,frac{1}{2})$ and $nge C(p)<infty$ or $G=S_3$ and $pin (1,infty).$ We also found the optimal value $L_{G,2}$ such that the inequality $$|M_{G}f|_2le L_{G,2}|f|_2$$ holds for every $f:V o mathbb{R}$, when: (i)$G=K_n$ and $nge 3$;(ii)$G=S_n$ and $nge 3.$

We derive sufficient conditions for exponential decay of solutions of the delay negative feedback equation with distributed delay. The conditions are written in terms of exponential moments of the distribution. Our method only uses elementary tools of calculus and is robust towards possible extensions to more complex settings, in particular, systems of delay differential equations. We illustrate the applicability of the method to particular distributions - Dirac delta, Gamma distribution, uniform and truncated normal distributions.

We equip the type A diagrammatic Hecke category with a special derivation, so that after specialization to characteristic p it becomes a p-dg category. We prove that the defining relations of the Hecke algebra are satisfied in the p-dg Grothendieck group. We conjecture that the $p$-dg Grothendieck group is isomorphic to the Iwahori-Hecke algebra, equipping it with a basis which may differ from both the Kazhdan-Lusztig basis and the p-canonical basis. More precise conjectures will be found in the sequel.

Here are some other results contained in this paper. We provide an incomplete proof of the classification of all degree +2 derivations on the diagrammatic Hecke category, and a complete proof of the classification of those derivations for which the defining relations of the Hecke algebra are satisfied in the p-dg Grothendieck group. In particular, our special derivation is unique up to duality and equivalence. We prove that no such derivation exists in simply-laced types outside of finite and affine type A. We also examine a particular Bott-Samelson bimodule in type A_7, which is indecomposable in characteristic 2 but decomposable in all other characteristics. We prove that this Bott-Samelson bimodule admits no nontrivial fantastic filtrations in any characteristic, which is the analogue in the p-dg setting of being indecomposable.

We prove that all non-degenerate relative equilibria of the planar Newtonian $n$--body problem can be continued to spaces of constant curvature $kappa$, positive or negative, for small enough values of this parameter. We also compute the extension of some classical relative equilibria to curved spaces using numerical continuation. In particular, we extend Lagrange's triangle configuration with different masses to both positive and negative curvature spaces.

In this work we study the dynamical behavior Tonelli Lagrangian systems defined on the tangent bundle of the torus $mathbb{T}^2=mathbb{R}^2 / mathbb{Z}^2$. We prove that the Lagrangian flow restricted to a high energy level $ E_L^{-1}(c)$ (i.e $ c> c_0(L)$) has positive topological entropy if the flow satisfies the Kupka-Smale propriety in $ E_L^{-1}(c)$ (i.e, all closed orbit with energy $c$ are hyperbolic or elliptic and all heteroclinic intersections are transverse on $E_L^{-1}(c)$). The proof requires the use of well-known results in Aubry-Mather's Theory.

For an odd prime $p$, we determined the Brown-Peterson cohomology of $BPU_n$ in dimensions $-(2p-2)leq ileq 2p+2$, where $BPU_n$ is the classifying space of the projective unitary group $PU_n$. We construct a family of $p$-torsion classes $eta_{p,k}in BP^{2p^{k+1}+2}(BPU_n)$ for $p|n$ and $kgeq 0$ and identify their images under the Thom map with well understood cohomology classes in $H^*(BPU_n;mathbb{Z}_{(p)})$.

We prove that there are no irreducible representations of $B_n$ of dimension $n+1$ for $ngeq 10.$

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".

Let $H$ be an infinite dimensional, reflexive, separable Hilbert space and $NA(H)$ the class of all norm-attainble operators on $H.$ In this note, we study an implicit scheme for a canonical representation of nonexpansive contractions in norm-attainable classes.

We present the correspondence between Lax integrable systems with spectral parameter on a Riemann surface, and Conformal Field Theories, in quite general set-up suggested earlier by the author. This correspondence turns out to give a prequantization of the integrable systems in question.

The main purpose of this paper is to study and characterize the existing of general asymptotic regional gradient observer which observe the current gradient state of the original system in connection with gradient strategic sensors. Thus, we give an approach based to Luenberger observer theory of linear distributed parameter systems which is enabled to determinate asymptotically regional gradient estimator of current gradient system state. More precisely, under which condition the notion of asymptotic regional gradient observability can be achieved. Furthermore, we show that the measurement structures allows the existence of general asymptotic regional gradient observer and we give a sufficient condition for such asymptotic regional gradient observer in general case. We also show that, there exists a dynamical system for the considered system is not general asymptotic gradient observer in the usual sense, but it may be general asymptotic regional gradient observer. Then, for this purpose we present various results related to different types of sensor structures, domains and boundary conditions in two dimensional distributed diffusion systems

High-performance implementations of graph algorithms are challenging to implement on new parallel hardware such as GPUs, because of three challenges: (1) difficulty of coming up with graph building blocks, (2) load imbalance on parallel hardware, and (3) graph problems having low arithmetic intensity. To address these challenges, GraphBLAS is an innovative, on-going effort by the graph analytics community to propose building blocks based in sparse linear algebra, which will allow graph algorithms to be expressed in a performant, succinct, composable and portable manner. In this paper, we examine the performance challenges of a linear algebra-based approach to building graph frameworks and describe new design principles for overcoming these bottlenecks. Among the new design principles is exploiting input sparsity, which allows users to write graph algorithms without specifying push and pull direction. Exploiting output sparsity allows users to tell the backend which values of the output in a single vectorized computation they do not want computed. Load-balancing is an important feature for balancing work amongst parallel workers. We describe the important load-balancing features for handling graphs with different characteristics. The design principles described in this paper have been implemented in "GraphBLAST", the first open-source linear algebra-based graph framework on GPU targeting high-performance computing. The results show that on a single GPU, GraphBLAST has on average at least an order of magnitude speedup over previous GraphBLAS implementations SuiteSparse and GBTL, comparable performance to the fastest GPU hardwired primitives and shared-memory graph frameworks Ligra and Gunrock, and better performance than any other GPU graph framework, while offering a simpler and more concise programming model.

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.

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.

Large-scale pretrained language models are the major driving force behind recent improvements in performance on the Winograd Schema Challenge, a widely employed test of common sense reasoning ability. We show, however, with a new diagnostic dataset, that these models are sensitive to linguistic perturbations of the Winograd examples that minimally affect human understanding. Our results highlight interesting differences between humans and language models: language models are more sensitive to number or gender alternations and synonym replacements than humans, and humans are more stable and consistent in their predictions, maintain a much higher absolute performance, and perform better on non-associative instances than associative ones. Overall, humans are correct more often than out-of-the-box models, and the models are sometimes right for the wrong reasons. Finally, we show that fine-tuning on a large, task-specific dataset can offer a solution to these issues.

The quantum Fourier transform brings efficiency in many respects, especially usage of resource, for most operations on quantum computers. In this study, the existing QFT-based and non-QFT-based quantum arithmetic operations are examined. The capabilities of QFT-based addition and multiplication are improved with some modifications. The proposed operations are compared with the nearest quantum arithmetic operations. Furthermore, novel QFT-based subtraction and division operations are presented. The proposed arithmetic operations can perform non-modular operations on all signed numbers without any limitation by using less resources. In addition, novel quantum circuits of two's complement, absolute value and comparison operations are also presented by using the proposed QFT based addition and subtraction operations.

With the increase in use of Unmanned Aerial Vehicles (UAVs)/drones, it is important to detect and identify causes of failure in real time for proper recovery from a potential crash-like scenario or post incident forensics analysis. The cause of crash could be either a fault in the sensor/actuator system, a physical damage/attack, or a cyber attack on the drone's software. In this paper, we propose novel architectures based on deep Convolutional and Long Short-Term Memory Neural Networks (CNNs and LSTMs) to detect (via Autoencoder) and classify drone mis-operations based on sensor data. The proposed architectures are able to learn high-level features automatically from the raw sensor data and learn the spatial and temporal dynamics in the sensor data. We validate the proposed deep-learning architectures via simulations and experiments on a real drone. Empirical results show that our solution is able to detect with over 90% accuracy and classify various types of drone mis-operations (with about 99% accuracy (simulation data) and upto 88% accuracy (experimental data)).

Embodiment is an important characteristic for all intelligent agents (creatures and robots), while existing scene description tasks mainly focus on analyzing images passively and the semantic understanding of the scenario is separated from the interaction between the agent and the environment. In this work, we propose the Embodied Scene Description, which exploits the embodiment ability of the agent to find an optimal viewpoint in its environment for scene description tasks. A learning framework with the paradigms of imitation learning and reinforcement learning is established to teach the intelligent agent to generate corresponding sensorimotor activities. The proposed framework is tested on both the AI2Thor dataset and a real world robotic platform demonstrating the effectiveness and extendability of the developed method.

Hearing is often believed to be more sensitive than touch. This assertion is based on a comparison of sensitivities to weak stimuli. The respective stimuli, however, are not easily comparable since hearing is gauged using acoustic pressure and touch using skin displacement. We show that under reasonable assumptions the auditory and tactile detection thresholds can be reconciled on a level playing field. The results indicate that the capacity of touch and hearing to detect weak stimuli varies according to the size of a sensed object as well as to the frequency of its oscillations. In particular, touch is found to be more effective than hearing at detecting small and slow objects.

In this paper we prove that any odd multigrid with non-zero rational offsets is regular, which means that its dual is a rhombic tiling. To prove this result we use a result on trigonometric diophantine equations.

Monadic second order logic can be used to express many classical notions of sets of vertices of a graph as for instance: dominating sets, induced matchings, perfect codes, independent sets or irredundant sets. Bounds on the number of sets of any such family of sets are interesting from a combinatorial point of view and have algorithmic applications. Many such bounds on different families of sets over different classes of graphs are already provided in the literature. In particular, Rote recently showed that the number of minimal dominating sets in trees of order $n$ is at most $95^{frac{n}{13}}$ and that this bound is asymptotically sharp up to a multiplicative constant. We build on his work to show that what he did for minimal dominating sets can be done for any family of sets definable by a monadic second order formula.

We first show that, for any monadic second order formula over graphs that characterizes a given kind of subset of its vertices, the maximal number of such sets in a tree can be expressed as the extit{growth rate of a bilinear system}. This mostly relies on well known links between monadic second order logic over trees and tree automata and basic tree automata manipulations. Then we show that this "growth rate" of a bilinear system can be approximated from above.We then use our implementation of this result to provide bounds on the number of independent dominating sets, total perfect dominating sets, induced matchings, maximal induced matchings, minimal perfect dominating sets, perfect codes and maximal irredundant sets on trees. We also solve a question from D. Y. Kang et al. regarding $r$-matchings and improve a bound from G'orska and Skupie'n on the number of maximal matchings on trees. Remark that this approach is easily generalizable to graphs of bounded tree width or clique width (or any similar class of graphs where tree automata are meaningful).

Many efforts of research are devoted to semantic role labeling (SRL) which is crucial for natural language understanding. Supervised approaches have achieved impressing performances when large-scale corpora are available for resource-rich languages such as English. While for the low-resource languages with no annotated SRL dataset, it is still challenging to obtain competitive performances. Cross-lingual SRL is one promising way to address the problem, which has achieved great advances with the help of model transferring and annotation projection. In this paper, we propose a novel alternative based on corpus translation, constructing high-quality training datasets for the target languages from the source gold-standard SRL annotations. Experimental results on Universal Proposition Bank show that the translation-based method is highly effective, and the automatic pseudo datasets can improve the target-language SRL performances significantly.

A brain-computer interface (BCI) enables a user to communicate with a computer directly using brain signals. Electroencephalograms (EEGs) used in BCIs are weak, easily contaminated by interference and noise, non-stationary for the same subject, and varying across different subjects and sessions. Therefore, it is difficult to build a generic pattern recognition model in an EEG-based BCI system that is optimal for different subjects, during different sessions, for different devices and tasks. Usually, a calibration session is needed to collect some training data for a new subject, which is time consuming and user unfriendly. Transfer learning (TL), which utilizes data or knowledge from similar or relevant subjects/sessions/devices/tasks to facilitate learning for a new subject/session/device/task, is frequently used to reduce the amount of calibration effort. This paper reviews journal publications on TL approaches in EEG-based BCIs in the last few years, i.e., since 2016. Six paradigms and applications -- motor imagery, event-related potentials, steady-state visual evoked potentials, affective BCIs, regression problems, and adversarial attacks -- are considered. For each paradigm/application, we group the TL approaches into cross-subject/session, cross-device, and cross-task settings and review them separately. Observations and conclusions are made at the end of the paper, which may point to future research directions.

Event-related desynchronization and synchronization (ERD/S) and movement-related cortical potential (MRCP) play an important role in brain-computer interfaces (BCI) for lower limb rehabilitation, particularly in standing and sitting. However, little is known about the differences in the cortical activation between standing and sitting, especially how the brain's intention modulates the pre-movement sensorimotor rhythm as they do for switching movements. In this study, we aim to investigate the decoding of continuous EEG rhythms during action observation (AO), motor imagery (MI), and motor execution (ME) for standing and sitting. We developed a behavioral task in which participants were instructed to perform both AO and MI/ME in regard to the actions of sit-to-stand and stand-to-sit. Our results demonstrated that the ERD was prominent during AO, whereas ERS was typical during MI at the alpha band across the sensorimotor area. A combination of the filter bank common spatial pattern (FBCSP) and support vector machine (SVM) for classification was used for both offline and pseudo-online analyses. The offline analysis indicated the classification of AO and MI providing the highest mean accuracy at 82.73$pm$2.38\% in stand-to-sit transition. By applying the pseudo-online analysis, we demonstrated the higher performance of decoding neural intentions from the MI paradigm in comparison to the ME paradigm. These observations led us to the promising aspect of using our developed tasks based on the integration of both AO and MI to build future exoskeleton-based rehabilitation systems.

The global health threat from COVID-19 has been controlled in a number of instances by large-scale testing and contact tracing efforts. We created this document to suggest three functionalities on how we might best harness computing technologies to supporting the goals of public health organizations in minimizing morbidity and mortality associated with the spread of COVID-19, while protecting the civil liberties of individuals. In particular, this work advocates for a third-party free approach to assisted mobile contact tracing, because such an approach mitigates the security and privacy risks of requiring a trusted third party. We also explicitly consider the inferential risks involved in any contract tracing system, where any alert to a user could itself give rise to de-anonymizing information.

More generally, we hope to participate in bringing together colleagues in industry, academia, and civil society to discuss and converge on ideas around a critical issue rising with attempts to mitigate the COVID-19 pandemic.

Existing deep learning methods for image deblurring typically train models using pairs of sharp images and their blurred counterparts. However, synthetically blurring images do not necessarily model the genuine blurring process in real-world scenarios with sufficient accuracy. To address this problem, we propose a new method which combines two GAN models, i.e., a learning-to-Blur GAN (BGAN) and learning-to-DeBlur GAN (DBGAN), in order to learn a better model for image deblurring by primarily learning how to blur images. The first model, BGAN, learns how to blur sharp images with unpaired sharp and blurry image sets, and then guides the second model, DBGAN, to learn how to correctly deblur such images. In order to reduce the discrepancy between real blur and synthesized blur, a relativistic blur loss is leveraged. As an additional contribution, this paper also introduces a Real-World Blurred Image (RWBI) dataset including diverse blurry images. Our experiments show that the proposed method achieves consistently superior quantitative performance as well as higher perceptual quality on both the newly proposed dataset and the public GOPRO dataset.

Given a fixed graph $H$ that embeds in a surface $Sigma$, what is the maximum number of copies of $H$ in an $n$-vertex graph $G$ that embeds in $Sigma$? We show that the answer is $Theta(n^{f(H)})$, where $f(H)$ is a graph invariant called the `flap-number' of $H$, which is independent of $Sigma$. This simultaneously answers two open problems posed by Eppstein (1993). When $H$ is a complete graph we give more precise answers.