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An anonymous reader quotes a report from Ars Technica: Swiss company Climeworks has announced the opening of a new plant in Italy that will collect carbon dioxide (CO2) from ambient air and pair it with renewably-made hydrogen (H2) to make methane fuel that would add little or no CO2 to the atmosphere. The plant in Troia, Italy, was completed in July and went into operation this week as part of a research program funded by the European Union. The new Italian plant will be run for more than 4,000 hours over the next 17 months (that's just under eight hours a day) in order to demonstrate the viability of fuel production as a potential revenue source for carbon capture. Gebald said that pure, captured CO2 could even be processed into jet fuel. When that fuel is burned, he said, it would again create CO2 that could be captured at an arbitrary Direct Air Capture plant and turned back into jet fuel. The plant consists of three air collectors that are more energy efficient than Climeworks' fi
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talonyx writes: D-Wave Systems, the contentious but scrappy maker of quantum annealing processors, has launched a cloud-based platform where developers can sign up for free and run problems on their quantum processor unit (QPU). There's an in-depth set of demos, documentation, and an open-source Python SDK to look at. "Leap is the latest addition to the quantum cloud -- services that virtualize quantum computing for almost anyone with a computer and a broadband connection to use," reports IEEE Spectrum. "Leap allows anyone to sign up, giving them one minute of time on a cloud-connected 2000Q each month. That might not sound like much, but a key advantage of quantum computing is to be able to solve in milliseconds problems like factoring large numbers, optimizing routes, or calculating molecular structures that could take traditional computers days or weeks." "D-Wave estimates that each user's free minute of quantum computing time should be enough to run between 400 and 4,000 jobs each
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About 8,000 light-years away, a giant planet circles an aging star, marching once around its sun in a single Earth-year. But that planet, called Kepler 1625b, might not be traveling completely alone. From a report: Scientists now suspect the planet's skies are filled by an orbiting mega-moon, a stunningly large world the size of Neptune that may be the first moon spotted outside our solar system. Early hints of its existence surfaced in July 2017, when scientists tentatively announced that they'd found some evidence of an orbital companion for Kepler 1625b. But it wasn't until the Hubble Space Telescope aimed its eye at the faraway star a year ago that scientists were able to gather enough data to build the case for the so-called exomoon's presence. Now, the two scientists behind the discovery are hoping for independent confirmation of their finding to really shore up the extraordinary claim. "I'm confident that we've done a thorough job vetting this thing, but I also anticipate there
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Emil Protalinski, reporting for VentureBeat: Until just a few days ago, some Facebook users could not delete their accounts -- the option to do so simply didn't work. After VentureBeat reached out to Facebook regarding the issue, an engineer was able to squash the bug. Two weeks ago, I got an email from a VentureBeat reader who couldn't delete his Facebook account. He claimed there were others also having issues -- no matter what they tried, they simply could not delete Facebook. I didn't believe him at first. [...] I did my due diligence. The least I could do was help him delete his account. Upon request, the reader was gracious enough to let me log into his Facebook account so I could see for myself. No matter what I tried, and regardless of which browser I used, the Facebook help page for deleting your account would not load when logged into his account. The reporter contacted a Facebook spokesperson, who after looking into the matter concluded that a bug prevented some people with
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A Hell-Raising Mezzo-Soprano Cuts Through a Dreary First Week at the Metrop
10-05 THE NEW YORKER 2466 -
The $\mathbb{Z}_{2}$-equivariant Heegaard Floer cohomlogy $\widehat{HF}_{\mathbb{Z}_{2}}(\Sigma(K))$ of a knot $K$ in $S^{3}$, constructed by Hendricks, Lipshitz, and Sarkar, is an isotopy invariant which is defined using bridge diagrams of $K$ drawn on a sphere. We prove that $\widehat{HF}_{\mathbb{Z}_{2}}(\Sigma(K))$ can be computed from knot Heegaard diagrams of $K$ and show that it is a strong Heegaard invariant. As a topolocial application, we construct a transverse knot invariant $\hat{\mathcal{T}}_{\mathbb{Z}_{2}}(K)$ as an element of $\widehat{HFK}_{\mathbb{Z}_{2}}(\Sigma(K),K)$, which is a refinement of $\widehat{HF}_{\mathbb{Z}_{2}}(\Sigma(K))$, and show that it is a refinement of both the LOSS invariant $\hat{\mathcal{T}}(K)$ and the $\mathbb{Z}_{2}$-equivariant contact class $c_{\mathbb{Z}_{2}}(\xi_{K})$.
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We show that for any $P= 6^{m+1}.N -1 $ is a prime number for any $1 < N \le 13$ , $N \ne 8$ and $N \ne i^{m+1}Mod(6i+1) $ where $ i \in Z^+ $ and $ m \in $ $odd$ $Z^+ $ for $1 < N \le 13$ and $N \ne 8$ and also we further discussed that $P= 6^{m+1}.N -1 $ is a prime number for $ N >13 $ if and only if , $N \ne i^{m+1}Mod(6i+1) +(6i +1)a $ $ ; i,a \le Z^+ $
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This paper reviews minimax best equivariant estimation in these invariant estimation problems: a location parameter, a scale parameter and a (Wishart) covariance matrix. We briefly review development of the best equivariant estimator as a generalized Bayes estimator relative to right invariant Haar measure in each case. Then we prove minimaxity of the best equivariant procedure by giving a least favorable prior sequence based on non-truncated Gaussian distributions. The results in this paper are all known, but we bring a fresh and somewhat unified approach by using, in contrast to most proofs in the literature, a smooth sequence of non truncated priors. This approach leads to some simplifications in the minimaxity proofs.
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We construct a spectral sequence converging to the homology of the ordered configuration spaces of a product of parallelizable manifolds. To identify the second page of this spectral sequence, we introduce a version of the Boardman--Vogt tensor product for linear operadic modules, a purely algebraic operation. Using the rational formality of the little cubes operads, we show that our spectral sequence collapses in characteristic zero.
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This paper analyzes an interface-unfitted numerical method for distributed optimal control problems governed by elliptic interface equations. We follow the variational discretization concept to discretize the optimal control problems, and apply a Nitsche-eXtended finite element method to discretize the corresponding state and adjoint equations, where piecewise cut basis functions around the interface are enriched into the standard linear element space. Optimal error estimates of the state, co-state and control in a mesh-dependent norm and the $L^2$ norm are derived. Numerical results are provided to verify the theoretical results.
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This work develops a proximal primal-dual distributed strategy for multi-agent optimization problems that involve multiple coupled affine constraints, and where each constraint may involve only a subset of the agents. The constraints are generally sparse, meaning that only a small subset of the agents are involved in them. This scenario arises in many applications including distributed control formulations, resource allocation problems, and smart grids. Traditional distributed solutions tend to ignore the structure of the constraints and lead to degraded performance. We instead develop a distributed solution that exploits the sparsity structure. Under constant step-size learning, we establish the asymptotic convergence of the distributed algorithm in the presence of non-smooth terms, and further show that convergence occurs at a linear rate in the smooth case. We also examine how the performance of the algorithm is influenced by the sparsity of the constraints. Simulations illustrate t
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Consider the trilinear form for twisted convolution on $\mathbb{R}^{2d}$: \begin{equation*} \mathcal{T}_t(\mathbf{f}):=\iint f_1(x)f_2(y)f_3(x+y)e^{it\sigma(x,y)}dxdy,\end{equation*} where $\sigma$ is a symplectic form and $t$ is a real-valued parameter. It is known that in the case $t\neq0$ the optimal constant for twisted convolution is the same as that for convolution, though no extremizers exist. Expanding about the manifold of triples of maximizers and $t=0$ we prove a sharpened inequality for twisted convolution with an arbitrary antisymmetric form in place of $\sigma$.
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We give a combinatorial formula for the Ehrhart $h^*$-vector of the hypersimplex. In particular, we show that $h^{*}_{d}(\Delta_{k,n})$ is the number of hypersimplicial decorated ordered set partitions of type $(k,n)$ with winding number $d$, thereby proving a conjecture of Nick Early. We do this by proving a more general conjecture of Nick Early on the Ehrhart $h^*$-vector of a generic cross-section of a hypercube.
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We prove the equivalence between the categories of motives of rigid analytic varieties over a perfectoid field $K$ of mixed characteristic and over the associated (tilted) perfectoid field $K^{\flat}$ of equal characteristic. This can be considered as a motivic generalization of a theorem of Fontaine and Wintenberger, claiming that the Galois groups of $K$ and $K^\flat$ are isomorphic. A main tool for constructing the equivalence is Scholze's theory of perfectoid spaces.
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Picard-Lefschetz theory is applied to solutions of the Helmholtz equation, formulated in terms of sums of integrals of a proper-time, or `einbein', wave function $\Psi(\Lambda) = \exp(i\mathbb S(\Lambda))$ along complex contours bounded by essential singularities of $\Psi$, or poles of $\mathbb S$. There is a one to one map between steepest descent paths connecting poles, also known as Lefschetz thimbles, and both real and complex eigenrays. Residues of poles at finite $\Lambda$ always vanish at some spatial points, which correspond to the location of a source if only one Lefschetz thimble is bounded at the pole. If there are two oppositely oriented contours ending at a pole, points of vanishing residue are not source locations, but are argued to be the locus on which caustic curves may have singularities such as cusps. The map between $\mathbb S$ and the generating function in the Thom--Arnold classification of catastrophes is discussed. Monodromies of the solution set with respect to
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We study the kink-antikink scattering within the double sine-Gordon model. In the numerical simulations we found a critical value $v_{cr}$ of the initial velocity $v_{in}$, which separates two different scenarios: at $v_{in}<v_{cr}$ the kinks capture each other and form a bound state, while at $v_{in}>v_{cr}$ the kinks pass through each other and escape to infinities. We obtain non-monotonous dependence of $v_{cr}$ on the model parameter $R$. Besides that, at some initial velocities below $v_{cr}$ we observe formation and interaction of the so-called oscillons (new phenomenon), as well as escape windows (well-known phenomenon).
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The aim of this paper is to provide new perspectives on relative finite element accuracy which is usually based on the asymptotic speed of convergence comparison when the mesh size $h$ goes to zero. Starting from a geometrical reading of the error estimate due to Bramble-Hilbert lemma, we derive two probability distributions that estimate the relative accuracy, considered as a random variable, between two Lagrange finite elements $P_k$ and $P_m$, ($k < m$). We establish mathematical properties of these probabilistic distributions and we get new insights which, among others, show that $P_k$ or $P_m$ is more likely accurate than the other, depending on the value of the mesh size $h$.
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For $k\ge 2$ even, let $d_{k,N}$ denote the dimension of the largest simple Hecke submodule of $S_{k}(\Gamma_0(N); \mathbb{Q})$. We show, using a simple analytic method, that $d_{k,N} \gg_k \log\log N / \log(2p)$ with $p$ the smallest prime co-prime to $N$. Previously, bounds of this quality were only known for prime $N \equiv 7 \pmod{8}$ and $k=2$, using an algebraic method based on congruences. We also establish similar (and sometimes stronger) results concerning $S_{k}(\Gamma_0(N), \chi)$, with $k \geq 2$ an integer and $\chi$ an arbitrary nebentypus.
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Nearly 60 years ago, Erd\H{o}s and Szekeres raised the question of whether $$\liminf_{N\to \infty} \prod_{r=1}^N \left| 2\sin \pi r \alpha \right| =0$$ for all irrationals $\alpha$. Despite its simple formulation, the question has remained unanswered. It was shown by Lubinsky in 1999 that the answer is yes if $\alpha$ has unbounded continued fraction coefficients, and it was suggested that the answer is yes in general. However, we show in this paper that for the golden ratio $\varphi=(\sqrt{5}-1)/2$, $$\liminf_{N\to \infty} \prod_{r=1}^N \left| 2\sin \pi r \varphi \right| >0 ,$$ providing a negative answer to this long-standing open problem.
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Connections via Baecklund transformations among different non-linear evolution equations are investigated aiming to compare corresponding Abelian and non Abelian results. Specifically, links, via Baecklund transformations, connecting Burgers and KdV-type hierarchies of nonlinear evolution equations are studied. Crucial differences as well as notable similarities between Baecklund charts in the case of the Burgers - heat equation, on one side and KdV -type equations are considered. The Baecklund charts constructed in [16] and [17], respectively, to connect Burgers and KdV-type hierarchies of operator nonlinear evolution equations show that the structures, in the non-commutative cases, are richer than the corresponding commutative ones.
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We characterize the accuracy of analyzing the performance of a NOMA system where users are ranked according to their distances instead of instantaneous channel gains, i.e., product of distance-based path-loss and fading channel gains. Distance-based ranking is analytically tractable and can lead to important insights. However, it may not be appropriate in a multipath fading environment where a near user suffers from severe fading while a far user experiences weak fading. Since the ranking of users in a NOMA system has a direct impact on coverage probability analysis, impact of the traditional distance-based ranking, as opposed to instantaneous signal power-based ranking, needs to be understood. This will enable us to identify scenarios where distance-based ranking, which is easier to implement compared to instantaneous signal power-based ranking, is acceptable for system performance analysis. To this end, in this paper, we derive the probability of the event when distance-based ranking
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We obtain the first positive results for bounded sample compression in the agnostic regression setting. We show that for p in {1,infinity}, agnostic linear regression with $\ell_p$ loss admits a bounded sample compression scheme. Specifically, we exhibit efficient sample compression schemes for agnostic linear regression in $R^d$ of size $d+1$ under the $\ell_1$ loss and size $d+2$ under the $\ell_\infty$ loss. We further show that for every other $\ell_p$ loss (1 < p < infinity), there does not exist an agnostic compression scheme of bounded size. This refines and generalizes a negative result of David, Moran, and Yehudayoff (2016) for the $\ell_2$ loss. We close by posing a general open question: for agnostic regression with $\ell_1$ loss, does every function class admit a compression scheme of size equal to its pseudo-dimension? This question generalizes Warmuth's classic sample compression conjecture for realizable-case classification (Warmuth, 2003).
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We study limit models in the class of abelian groups with the subgroup relation and in the class of torsion-free abelian groups with the pure subgroup relation. We show: $\textbf{Theorem}$ (1) If $G$ is a limit model of cardinality $\lambda$ in the class of abelian groups with the subgroup relation, then $G \cong \mathbb{Q}^{(\lambda)} \oplus (\oplus_{p} \mathbb{Z}(p^\infty)^{(\lambda)})$. (2) If $G$ is a limit model of cardinality $\lambda$ in the class of torsion-free abelian groups with the pure subgroup relation, then: * If the length of the chain has uncountable cofinality, then $G \cong \mathbb{Q}^{(\lambda)} \oplus \Pi_{p} \overline{\mathbb{Z}_{(p)}^{(\lambda)}}$. * If the length of the chain has countable cofinality, then $G$ is not algebraically compact. We also study the class of finitely Butler groups with the pure subgroup relation, we show that it is an AEC, Galois-stable and $(<\aleph_0)$-tame and short.
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Using a mild variant of polar codes we design linear compression schemes compressing Hidden Markov sources (where the source is a Markov chain, but whose state is not necessarily observable from its output), and to decode from Hidden Markov channels (where the channel has a state and the error introduced depends on the state). We give the first polynomial time algorithms that manage to compress and decompress (or encode and decode) at input lengths that are polynomial $\it{both}$ in the gap to capacity and the mixing time of the Markov chain. Prior work achieved capacity only asymptotically in the limit of large lengths, and polynomial bounds were not available with respect to either the gap to capacity or mixing time. Our results operate in the setting where the source (or the channel) is $\it{known}$. If the source is $\it{unknown}$ then compression at such short lengths would lead to effective algorithms for learning parity with noise -- thus our results are the first to suggest a s
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We give a short proof of some sparse bounds for the spherical maximal operator of Magyar, Stein and Wainger. This proof also includes certain endpoint estimates. The new method of proof is inspired by ones by Bourgain and Ionescu, is very efficient, and not been used in the proof of sparse bounds before. The Hardy-Littlewood Circle method is used to decompose the multiplier into major and minor arc components. The efficiency arises as one only needs a single estimate on each element of the decomposition.
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In this paper, elliptic control problems with integral constraint on the gradient of the state and box constraints on the control are considered. The optimal conditions of the problem are proved. To numerically solve the problem, we use the 'First discretize, then optimize' approach. Specifically, we discretize both the state and the control by piecewise linear functions. To solve the discretized problem efficiently, we first transform it into a multi-block unconstrained convex optimization problem via its dual, then we extend the inexact majorized accelerating block coordinate descent (imABCD) algorithm to solve it. The entire algorithm framework is called finite element duality-based inexact majorized accelerating block coordinate descent (FE-dABCD) algorithm. Thanks to the inexactness of the FE-dABCD algorithm, each subproblems are allowed to be solved inexactly. For the smooth subproblem, we use the generalized minimal residual (GMRES) method with preconditioner to slove it. For th
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In the present paper, we consider a non self adjoint hyperbolic operator with a vector field and an electric potential that depend not only on the space variable but also on the time variable. More precisely, we attempt to stably and simultaneously retrieve the real valued velocity field and the real valued potential from the knowledge of Neumann measurements performed on the whole boundary of the domain. We establish in dimension n greater than two, stability estimates for the problem under consideration. Thereafter, by enlarging the set of data we show that the unknown terms can be stably retrieved in larger regions including the whole domain. The proof of the main results are mainly based on the reduction of the inverse problem under investigation to an equivalent and classic inverse problem for an electro-magnetic wave equation.
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In this paper, we develop approximate dynamic programming methods for stochastic systems modeled as Markov Decision Processes, given both soft performance criteria and hard constraints in a class of probabilistic temporal logic called Probabilistic Computation Tree Logic (PCTL). Our approach consists of two steps: First, we show how to transform a class of PCTL formulas into chance constraints that can be enforced during planning in stochastic systems. Second, by integrating randomized optimization and entropy-regulated dynamic programming, we devise a novel trajectory sampling-based approximate value iteration method to iteratively solve for an upper bound on the value function while ensuring the constraints that PCTL specifications are satisfied. Particularly, we show that by the on-policy sampling of the trajectories, a tight bound can be achieved between the upper bound given by the approximation and the true value function. The correctness and efficiency of the method are demonstr
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In practical conjugate gradient (CG) computations it is important to monitor the quality of the approximate solution to $Ax=b$ so that the CG algorithm can be stopped when the required accuracy is reached. The relevant convergence characteristics, like the $A$-norm of the error or the normwise backward error, cannot be easily computed. However, they can be estimated. Such estimates often depend on approximations of the smallest or largest eigenvalue of~$A$. In the paper we introduce a new upper bound for the $A$-norm of the error, which is closely related to the Gauss-Radau upper bound, and discuss the problem of choosing the parameter $\mu$ which should represent a lower bound for the smallest eigenvalue of $A$.The new bound has several practical advantages, the most important one is that it can be used as an approximation to the $A$-norm of the error even if $\mu$ is not exactly a lower bound for the smallest eigenvalue of $A$. In this case, $\mu$ can be chosen, e.g., as the smallest
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In this work we establish functional asymmetric versions of the celebrated Blaschke-Santal\'o inequality. As consequences of these inequalities we recover their geometric counterparts with equality cases, as well as, another inequality with strong probabilistic flavour that was firstly obtained by Lutwak, Yang and Zhang. We present a brief study on an $L_p$ functional analogue to the center of mass that is necessary for our arguments and that might be of independent interest.
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Let $G$ be a (multi)graph of order $n$ and let $u,v$ be vertices of $G$. The maximum number of internally disjoint $u$-$v$ paths in $G$ is denoted by $\kappa_G(u,v)$, and the maximum number of edge-disjoint $u$-$v$ paths in $G$ is denoted by $\lambda_G (u,v)$. The average connectivity of $G$ is defined by $\overline{\kappa}(G)=\sum_{\{u,v\}\subseteq V(G)} \kappa_G(u,v)/\tbinom{n}{2},$ and the average edge-connectivity of $G$ is defined by $\overline{\lambda}(G)=\sum_{\{u,v\}\subseteq V(G)} \lambda_G(u,v)/\tbinom{n}{2}$. A graph is called ideally connected if $\kappa(u,v)=\min\{\mbox{deg}(u),\mbox{deg}(v)\}$ for all pairs of vertices $\{u,v\}$. We prove that every minimally $2$-connected graph of order $n$ with largest average connectivity is bipartite, with the set of vertices of degree $2$ and the set of vertices of degree at least $3$ being the partite sets. We use this structure to prove that $\overline{\kappa}(G)<\tfrac{9}{4}$ for any minimally $2$-connected graph $G$ (this boun
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The averaging principle is established for the slow component and the fast component being two dimensional stochastic Navier-Stokes equations and stochastic reaction-diffusion equations, respectively. The classical Khasminskii approach based on time discretization is used for the proof of the slow component strong convergence to the solution of the corresponding averaged equation under some suitable conditions. Meanwhile, some powerful techniques are used to overcome the difficulties caused by the nonlinear term and to release the regularity of the initial value.
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This paper examines the long-run behavior of learning with bandit feedback in non-cooperative concave games. The bandit framework accounts for extremely low-information environments where the agents may not even know they are playing a game; as such, the agents' most sensible choice in this setting would be to employ a no-regret learning algorithm. In general, this does not mean that the players' behavior stabilizes in the long run: no-regret learning may lead to cycles, even with perfect gradient information. However, if a standard monotonicity condition is satisfied, our analysis shows that no-regret learning based on mirror descent with bandit feedback converges to Nash equilibrium with probability $1$. We also derive an upper bound for the convergence rate of the process that nearly matches the best attainable rate for single-agent bandit stochastic optimization.
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For an Ornstein-Uhlenbeck process driven by a fractional Brownian motion with Hurst parameter 0<H<1/2, one shows the Berry-Ess\'{e}en bound of the least squares estimator of the drift parameter. Thus, a problem left in the previous paper (Chen, Kuang and Li in Stochastics and Dynamics, 2019+) is solved, where the Berry-Ess\'{e}en bound of the least squares estimator is proved for 1/2<=H<=3/4. An approach based on Malliavin calculus given by Kim and Park \cite{kim 3} is used
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Using wall-crossing for K3 surfaces, we establish birational equivalence of moduli spaces of stable objects on generic Enriques surfaces for different stability conditions. As an application, we prove in the case of a Mukai vector of odd rank that they are birational to Hilbert schemes and that under an extra assumption every minimal model can be described as a moduli space. The argument makes use of a new Chow-theoretic result, showing that moduli spaces on an Enriques surface give rise to constant cycle subvarieties of the moduli spaces of the covering K3.
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In this paper we study the moduli spaces $Simp^m_n$ of degree $n+1$ morphisms $ \mathbb{A}^1_{K} \to \mathbb{A}^1_{K}$ with "ramification length $<m$" over an algebraically closed field $K$. For each $m$, the moduli space $Simp^m_n$ is a Zariski open subset of the space of degree $n+1$ polynomials over $K$ up to $Aut (\mathbb{A}^1_{K})$. It is, in a way, orthogonal to the many papers about polynomials with prescribed zeroes- here we are prescribing, instead, the ramification data. Exploiting the topological properties of the poset that encodes the ramification behaviour, we use a sheaf-theoretic argument to compute $H^*(Simp^m_n(\mathbb{C});\mathbb{Q})$ as well as the \'etale cohomology $H^*_{\'et}({Simp^m_n}_{/K};\mathbb{Q}_{\ell})$ for $char K=0$ or $char K> n+1$. As a by-product we obtain that $H^*(Simp^m_n(\mathbb{C}); \mathbb{Q})$ is independent of $n$, thus implying rational cohomological stability. When $char K>0$ our methods compute $H^*_{\'et}(Simp^m_n;\mathbb{Q}_{\el
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We consider the graphs involved in the theoretical physics model known as the colored Sachdev-Ye-Kitaev (SYK) model. We study in detail their combinatorial properties at any order in the so-called $1/N$ expansion, and we enumerate these graphs asymptotically. Because of the duality between colored graphs involving $q+1$ colors and colored triangulations in dimension $q$, our results apply to the asymptotic enumeration of spaces that generalize unicellular maps - in the sense that they are obtained from a single building block - for which a higher-dimensional generalization of the genus is kept fixed.
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The formalism of multiple context-free grammars (MCFG) is a non-trivial generalization of context-free grammars (CFG), where basic constituents on which rules operate are discontinuous tuples of words rather than single words. Just as context-free ones, multiple context-free grammars have polynomial parsing algorithms, but their expressive power is strictly stronger. It is well known that CFG generate the same class of languages as type logical grammars based on Lambek calculus, which is, basically, a variant of noncommutative linear logic. We construct a system of type logical grammars based on ordinary commutative linear logic and show that these grammars are in the same relationship with MCFG as Lambek grammars with CFG. It turns out that tuples of words on which MCFG operate can be organized into a symmetric monoidal category, very similar to the category of topological cobordisms; we call it the category of word cobordisms. In particular, this category is compact closed and, thus,
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We consider unions of interior disjoint discs in the plane such that the graph whose vertices are disc centers and edges connect centers of mutually tangent discs is triangulated, called compact packings. There is only one compact packing by discs all of the same size, called the hexagonal compact packing. It has been proven that there are exactly $9$ values of $r$ which allow a compact packing with discs of radius $1$ and $r$. It has also been proven that at most $11462$ pairs $(r,s)$ allow a compact packing with discs of radius $1$, $r$ and $s$. This paper shows that there are exactly $164$ such pairs $(r,s)$.
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While fundamental physically realistic Hamiltonians should be invariant under time reversal, time asymmetric Hamiltonians can occur as mathematical possibilities or effective Hamiltonians. Here, we study conditions under which non-relativistic Hamiltonians involving particle creation and annihilation, as come up in quantum field theory (QFT), are time asymmetric. It turns out that the time reversal operator T can be more complicated than just complex conjugation, which leads to the question which criteria determine the correct action of time reversal. We use Bohmian trajectories for this purpose and show that time reversal symmetry can be broken when charges are permitted to be complex numbers, where `charge' means the coupling constant in a QFT that governs the strength with which a fermion emits and absorbs bosons. We pay particular attention to the technique for defining Hamiltonians with particle creation based on interior-boundary conditions, and we find them to generically be tim
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Each sub-Riemannian geometry with bracket generating distribution enjoys a background structure determined by the distribution itself. At the same time, those geometries with constant sub-Riemannian symbols determine a unique Cartan connection leading to their principal invariants. We provide cohomological description of the structure of these curvature invariants in the cases where the background structure is one of the parabolic geometries. As an illustration, constant curvature models are discussed for certain sub-Riemannian geometries.
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Recently, Branco da Silva and Silva described an efficient encoding and decoding algorithm for Construction D$^\prime$ lattices. Using their algorithm, we propose a Construction D$^\prime$ lattice based on binary quasi-cyclic low-density parity-check (QC-LPDC) codes and single parity-check product codes. The underlying codes designed by the balanced-distances rule contribute in a balanced manner to the squared minimum distance of the constructed lattice, which results in a high lattice coding gain. The proposed lattice based on IEEE 802.16e QC-LDPC codes is shown to provide competitive error-rate performance on the power-unconstrained additive white Gaussian noise channel.
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It is proved that, for an indefinite quadratic programming problem under linear constraints, any iterative sequence generated by the Proximal DC decomposition algorithm $R$-linearly converges to a Karush-Kuhn-Tucker point, provided that the problem has a solution. Another major result of this paper says that DCA sequences generated by the algorithm converge to a locally unique solution of the problem if the initial points are taken from a suitably-chosen neighborhood of it. To deal with the implicitly defined iterative sequences, a local error bound for affine variational inequalities and novel techniques are used. Numerical results together with an analysis of the influence of the decomposition parameter, as well as a comparison between the Proximal DC decomposition algorithm and the Projection DC decomposition algorithm, are given in this paper. Our results complement a recent and important paper of Le Thi, Huynh, and Pham Dinh (J. Optim. Theory Appl. 179 (2018), 103-126).
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Adam is a popular variant of the stochastic gradient descent for finding a local minimizer of a function. The objective function is unknown but a random estimate of the current gradient vector is observed at each round of the algorithm. This paper investigates the dynamical behavior of Adam when the objective function is non-convex and differentiable. We introduce a continuous-time version of Adam, under the form of a non-autonomous ordinary differential equation (ODE). The existence and the uniqueness of the solution are established, as well as the convergence of the solution towards the stationary points of the objective function. It is also proved that the continuous-time system is a relevant approximation of the Adam iterates, in the sense that the interpolated Adam process converges weakly to the solution to the ODE.
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In this paper, we propose a dynamical systems perspective of the Expectation-Maximization (EM) algorithm. More precisely, we can analyze the EM algorithm as a nonlinear state-space dynamical system. The EM algorithm is widely adopted for data clustering and density estimation in statistics, control systems, and machine learning. This algorithm belongs to a large class of iterative algorithms known as proximal point methods. In particular, we re-interpret limit points of the EM algorithm and other local maximizers of the likelihood function it seeks to optimize as equilibria in its dynamical system representation. Furthermore, we propose to assess its convergence as asymptotic stability in the sense of Lyapunov. As a consequence, we proceed by leveraging recent results regarding discrete-time Lyapunov stability theory in order to establish asymptotic stability (and thus, convergence) in the dynamical system representation of the EM algorithm.
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We consider the problem of finding an approximate second-order stationary point of a constrained non-convex optimization problem. We first show that, unlike the unconstrained scenario, the vanilla projected gradient descent algorithm may converge to a strict saddle point even when there is only a single linear constraint. We then provide a hardness result by showing that checking (\epsilon_g,\epsilon_H)-second order stationarity is NP-hard even in the presence of linear constraints. Despite our hardness result, we identify instances of the problem for which checking second order stationarity can be done efficiently. For such instances, we propose a dynamic second order Frank--Wolfe algorithm which converges to (\epsilon_g, \epsilon_H)-second order stationary points in {\mathcal{O}}(\max\{\epsilon_g^{-2}, \epsilon_H^{-3}\}) iterations. The proposed algorithm can be used in general constrained non-convex optimization as long as the constrained quadratic sub-problem can be solved efficien
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We compute the radiative quantum corrections to the critical exponents and amplitude ratios for O($N$) $\lambda\phi^{4}$ scalar high energy nonextensive $q$-field theories. We employ the field theoretic renormalization group approach through six methods for evaluating the high energy nonextensive critical exponents up to next-to-leading order while the high energy nonextensive amplitude ratios are computed up to leading level by applying three methods. Later we generalize these high energy nonextensive finite loop order results for any loop level. We find that the high energy nonextensive critical exponents are the same when obtained through all the methods employed. The same fact occurs for the high energy nonextensive amplitude ratios. Furthermore, we show that these high energy nonextensive universal quantities are equal to their low energy extensive counterparts, thus showing that the nonextensivity is broken down at high energies.
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A complete theory of linearly ordered structures $T$ satisfies condition (LB), called linear binarity, if every complete type of an increasing sequence of elements of a model of $T$ is uniquely determined by all the 2-types of pairs of consecutive elements; (LB) is shared by all complete theories of coloured orders as shown by Rubin. We introduce a strong form of linear binarity (SLB) and a weak form, called linear finiteness (LF). We prove that convex, parametrically definable subsets of models of theories satisfying (LF) are Boolean combinations of intervals and classes of definable convex equivalence relations; if we allow also unary definable sets in these combinations and assume (SLB), then we get description of all definable sets. Remarkably, we prove that (SLB), up to inter-definability, characterizes theories of coloured orders expanded by arbitrary convex equivalence relations.
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In this work, we consider an advanced noise reduction differential chaotic shift keying (NR-DCSK) system in which a single antenna source communicates with a single antenna destination under the attack of a single antenna jammer. We devote our efforts to design a novel anti-jamming (AJ) receiver for the considered system. Particularly, we propose a variational mode decomposition-independent component analysis-wavelet packet decomposition-based (VMD-ICA-WPD-based) structure, in which the VMD method is firstly exploited to generate multiple signals from the single received one. Secondly, the ICA method is applied to coarsely separate chaotic and jamming signals. After that, the WPD method is used to finely estimate and mitigate jamming signals that exist on all outputs of the ICA method. Finally, an inverse ICA procedure is carried out, followed by a summation, and the outcome is passed through the conventional correlation-based receiver for recovering the transmitted information. Simula
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The distinguishing number $D(G)$ of a graph $G$ is the smallest number of colors that is needed to color the vertices of $G$ such that the only color preserving automorphism is the identity. For infinite graphs $D(G)$ is bounded by the supremum of the valences, and for finite graphs by $\Delta(G)+1$, where $\Delta(G)$ is the maximum valence. Given a finite or infinite tree $T$ of bounded finite valence $k$ and an integer $c$, where $2 \leq c \leq k$, we are interested in coloring the vertices of $T$ by $c$ colors, such that every color preserving automorphism fixes as many vertices as possible. In this sense we show that there always exists a $c$-coloring for which all vertices whose distance from the next leaf is at least $\lceil\log_ck\rceil$ are fixed by any color preserving automorphism, and that one can do much better in many cases.
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Let d be a positive integer. There are several versions of d-gonality for tropical curves, stable d-gonality and divisorial d-gonality, which are both inspired by d-gonality for compact Riemann surfaces. However, that conditions are not equivalent. We have a condition of divisors equivalent to stable d-gonality for tropical curves.
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This paper rigorously analyses preconditioners for the time-harmonic Maxwell equations with absorption, where the PDE is discretised using curl-conforming finite-element methods of fixed, arbitrary order and the preconditioner is constructed using Additive Schwarz domain decomposition methods. The theory developed here shows that if the absorption is large enough, and if the subdomain and coarse mesh diameters and overlap are chosen appropriately, then the classical two-level overlapping Additive Schwarz preconditioner (with PEC boundary conditions on the subdomains) performs optimally -- in the sense that GMRES converges in a wavenumber-independent number of iterations -- for the problem with absorption. An important feature of the theory is that it allows the coarse space to be built from low-order elements even if the PDE is discretised using high-order elements. It also shows that additive methods with minimal overlap can be robust. Numerical experiments are given that illustrate t
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This paper is Part III of the series of work by the first named author on duality theories for p-primary etale cohomology, whose Parts I and II were published in 1986 and 1987, respectively. In this Part III, we study a duality for p-primary etale nearby cycles on smooth schemes over henselian discrete valuation rings of mixed characteristic (0, p) whose residue field is not necessary perfect.
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For a commutative non-autonomous dynamical system we show that topological transitivity of the non-autonomous system induced on probability measures (hyperspaces) is equivalent to the weak mixing of the induced systems. Several counter examples are given for the results which are true in autonomous but need not be true in non-autonomous systems. Wherever possible sufficient conditions are obtained for the results to hold true. For a commutative periodic non-autonomous system on intervals, it is proved that weakly mixing implies Devaney chaos. Given a periodic non-autonomous system, it is shown that sensitivity is equivalent to some stronger forms of sensitivity on a closed unit interval.
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Massive machine-type communication (mMTC) is a newly introduced service category in 5G wireless communication systems to support a variety of Internet-of-Things (IoT) applications. In recovering sparsely represented multi-user vectors, compressed sensing based multi-user detection (CS-MUD) can be used. CS-MUD is a feasible solution to the grant-free uplink non-orthogonal multiple access (NOMA) environments. In CS-MUD, active user detection (AUD) and channel estimation (CE) should be performed before data detection. In this paper, we propose the expectation propagation based joint AUD and CE (EP-AUD/CE) technique for mMTC networks. The expectation propagation (EP) algorithm is a Bayesian framework that approximates a computationally intractable probability distribution to an easily tractable distribution. The proposed technique finds the best approximation of the posterior distribution of the sparse channel vector. Using the approximate distribution, AUD and CE are jointly performed. We
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We present an analysis of edge domain walls in exchange-biased ferromagnetic films appearing as a result of a competition between the stray field at the film edges and the exchange bias field in the bulk. We introduce an effective two-dimensional micromagnetic energy that governs the magnetization behavior in exchange-biased materials and investigate its energy minimizers in the strip geometry. In a periodic setting, we provide a complete characterization of global energy minimizers corresponding to edge domain walls. In particular, we show that energy minimizers are one-dimensional and do not exhibit winding. We then consider a particular thin film regime for large samples and relatively strong exchange bias and derive a simple and comprehensive algebraic model describing the limiting magnetization behavior in the interior and at the boundary of the sample. Finally, we demonstrate that the asymptotic results obtained in the periodic setting remain true in the case of finite rectangula
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We show that an enlargeable Riemannian metric on a (possibly nonspin) manifold cannot have uniformly positive scalar curvature. This extends a well-known result of Gromov and Lawson to the nonspin setting. We also prove that every noncompact manifold admits a nonenlargeable metric. In proving the first result, we use the main result of the recent paper by Schoen and Yau on minimal hypersurfaces to obstruct positive scalar curvature in arbitrary dimensions. More concretely, we use this to study nonzero degree maps f from a manifold X to the product of the k-sphere with the n-k dimensional torus, with k=1,2,3. When X is a closed oriented manifold endowed with a metric g of positive scalar curvature and the map f is (possibly area) contracting, we prove inequalities relating the lower bound of the scalar curvature of g and the contracting factor of the map f.
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In this paper, we propose a comprehensive research over triple hop all-optical relaying free-space optical (FSO) systems in the presence of all main noise sources including background, thermal and amplified spontaneous emission (ASE) noise and by considering the effect of the optical degree-of- freedom (DoF). Using full CSI relaying, we derive the exact expressions for the noise variance at the destination. Then, in order to simplify the analytical expressions of full CSI relaying, we also propose and investigate the validity of different approximations over noise variance at the destination. Finally, we evaluate the the performance of considered triple-hop all-optical relaying FSO system in term of ergodic capacity.
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For topologically mixing locally expanding semigroup actions generated by a finite collection of $C^{1+\alpha}$ conformal local diffeomorphisms, we provide a countable Markov partition satisfying the finite images and the finite cycle properties. We show that they admit inducing schemes and describe the tower constructions associated with them. An important feature of these towers is that their induced maps are equivalent to a subshift of countable type. Through the investigating the ergodic properties of induced map, we prove the existence of a liftable absolutely continuous stationary measure for the original action. We then establish a thermodynamic formalism for induced map and deduce the unicity of eqiliberium state.
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For a stationary and axisymmetric spacetime, the vacuum Einstein field equations reduce to a single nonlinear PDE in two dimensions called the Ernst equation. By solving this equation with a {\it Dirichlet} boundary condition imposed along the disk, Neugebauer and Meinel in the 1990s famously derived an explicit expression for the spacetime metric corresponding to the Bardeen-Wagoner uniformly rotating disk of dust. In this paper, we consider a similar boundary value problem for a rotating disk in which a {\it Neumann} boundary condition is imposed along the disk instead of a Dirichlet condition. Using the integrable structure of the Ernst equation, we are able to reduce the problem to a Riemann-Hilbert problem on a genus one Riemann surface. By solving this Riemann-Hilbert problem in terms of theta functions, we obtain an explicit expression for the Ernst potential. Finally, a Riemann surface degeneration argument leads to an expression for the associated spacetime metric.
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The traveling waves for surface diffusion of plane curves are studied. We consider an evolving plane curve with two endpoints, which can move freely on the x-axis with generating constant contact angles. For the evolution of this plane curve governed by surface diffusion, we discuss the existence, the uniqueness and the convexity of traveling waves. The main results show that the uniqueness and the convexity can be lost in depending on the conditions of the contact angles, although the existence holds for any contact angles in the interval $(0,\pi/2)$.
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