In this paper, we mainly establish the existence and uniqueness theorem for solutions of the exterior Dirichlet problem for a class of fully nonlinear second-order elliptic equations related to the eigenvalues of the Hessian, with prescribed generalized symmetric asymptotic behavior at infinity. Moreover, we give some new results for the Hessian equations, Hessian quotient equations and the special Lagrangian equations, which have been studied previously.

In this paper we collect some open set-theoretic problems that appear in the large-scale topology (called also Asymptology). In particular we ask problems about critical cardinalities of some special (large, indiscrete, inseparated) coarse structures on $omega$, about the interplay between properties of a coarse space and its Higson corona, about some special ultrafilters ($T$-points and cellular $T$-points) related to finitary coarse structures on $omega$, about partitions of coarse spaces into thin pieces, and also about coarse groups having some extremal properties.

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

Let $w$ be a word in $k$ variables. For a finite nilpotent group $G$, a conjecture of Amit states that $N_w(1) ge |G|^{k-1}$, where $N_w(1)$ is the number of $k$-tuples $(g_1,...,g_k)in G^{(k)}$ such that $w(g_1,...,g_k)=1$. Currently, this conjecture is known to be true for groups of nilpotency class 2. Here we consider a generalized version of Amit's conjecture, and prove that $N_w(g) ge |G|^{k-2}$, where $g$ is a $w$-value in $G$, for finite groups $G$ of odd order and nilpotency class 2. If $w$ is a word in two variables, we further show that $N_w(g) ge |G|$, where $g$ is a $w$-value in $G$ for finite groups $G$ of nilpotency class 2. In addition, for $p$ a prime, we show that finite $p$-groups $G$, with two distinct irreducible complex character degrees, satisfy the generalized Amit conjecture for words $w_k =[x_1,y_1]...[x_k,y_k]$ with $k$ a natural number; that is, for $g$ a $w_k$-value in $G$ we have $N_{w_k}(g) ge |G|^{2k-1}$.

Finally, we discuss the related group properties of being rational and chiral, and show that every finite group of nilpotency class 2 is rational.

In cite{CC} the authors, investigating a model of population dynamics, find the following result. Let $Omegasubset mathbb{R}^N$, $Ngeq 1$, be a bounded smooth domain. The weighted eigenvalue problem $-Delta u =lambda m u $ in $Omega$ under homogeneous Dirichlet boundary conditions, where $lambda in mathbb{R}$ and $min L^infty(Omega)$, is considered. The authors prove the existence of minimizers $check m$ of the principal positive eigenvalue $lambda_1(m)$ when $m$ varies in a class $mathcal{M}$ of functions where average, maximum, and minimum values are given. A similar result is obtained in cite{CCP} when $m$ is in the class $mathcal{G}(m_0)$ of rearrangements of a fixed $m_0in L^infty(Omega)$. In our work we establish that, if $Omega$ is Steiner symmetric, then every minimizer in cite{CC,CCP} inherits the same kind of symmetry.

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

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

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.

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.

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.

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 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.

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.

We study the language universality problem for One-Counter Nets, also known as 1-dimensional Vector Addition Systems with States (1-VASS), parameterized either with an initial counter value, or with an upper bound on the allowed counter value during runs. The language accepted by an OCN (defined by reaching a final control state) is monotone in both parameters. This yields two natural questions: 1) Does there exist an initial counter value that makes the language universal? 2) Does there exist a sufficiently high ceiling so that the bounded language is universal? Despite the fact that unparameterized universality is Ackermann-complete and that these problems seem to reduce to checking basic structural properties of the underlying automaton, we show that in fact both problems are undecidable. We also look into the complexities of the problems for several decidable subclasses, namely for unambiguous, and deterministic systems, and for those over a single-letter alphabet.

In this paper we analyze a continuous version of the maximal covering location problem, in which the facilities are required to be interconnected by means of a graph structure in which two facilities are allowed to be linked if a given distance is not exceed. We provide a mathematical programming framework for the problem and different resolution strategies. First, we propose a Mixed Integer Non Linear Programming formulation, and derive properties of the problem that allow us to project the continuous variables out avoiding the nonlinear constraints, resulting in an equivalent pure integer programming formulation. Since the number of constraints in the integer programming formulation is large and the constraints are, in general, difficult to handle, we propose two branch-&-cut approaches that avoid the complete enumeration of the constraints resulting in more efficient procedures. We report the results of an extensive battery of computational experiments comparing the performance of the different approaches.

We show that the traveling salesman problem (TSP) and its many variants may be modeled as functional optimization problems over a graph. In this formulation, all vertices and arcs of the graph are functionals; i.e., a mapping from a space of measurable functions to the field of real numbers. Many variants of the TSP, such as those with neighborhoods, with forbidden neighborhoods, with time-windows and with profits, can all be framed under this construct. In sharp contrast to their discrete-optimization counterparts, the modeling constructs presented in this paper represent a fundamentally new domain of analysis and computation for TSPs and their variants. Beyond its apparent mathematical unification of a class of problems in graph theory, the main advantage of the new approach is that it facilitates the modeling of certain application-specific problems in their home space of measurable functions. Consequently, certain elements of economic system theory such as dynamical models and continuous-time cost/profit functionals can be directly incorporated in the new optimization problem formulation. Furthermore, subtour elimination constraints, prevalent in discrete optimization formulations, are naturally enforced through continuity requirements. The price for the new modeling framework is nonsmooth functionals. Although a number of theoretical issues remain open in the proposed mathematical framework, we demonstrate the computational viability of the new modeling constructs over a sample set of problems to illustrate the rapid production of end-to-end TSP solutions to extensively-constrained practical problems.

The Cannabis Issue As we look ahead to sunnier days, few things are as satisfying as a scoop of nice, cold ice cream.…

Solving computational problems may include generating a logic circuit representation of the computational problem, encoding the logic circuit representation as a discrete optimization problem, and solving the discrete optimization problem using a quantum processor. Output(s) of the logic circuit representation may be clamped such that the solving involves effectively executing the logic circuit representation in reverse to determine input(s) that corresponds to the clamped output(s). The representation may be of a multiplication circuit. The discrete optimization problem may be composed of a set of miniature optimization problems, where each miniature optimization problem encodes a respective logic gate from the logic circuit representation. A multiplication circuit may employ binary representations of factors, and these binary representations may be decomposed to reduce the total number of variables required to represent the multiplication circuit.

A computing method includes accepting a definition of a computing task, which includes multiple Processing Elements (PEs) having execution dependencies. The computing task is compiled for concurrent execution on a multiprocessor device, by arranging the PEs in a series of two or more invocations of the multiprocessor device, including assigning the PEs to the invocations depending on the execution dependencies. The multiprocessor device is invoked to run software code that executes the series of the invocations, so as to produce a result of the computing task.

This invention relates to a method, computer readable medium, and apparatus for identifying one or more problematic loops in an application. This invention provides a Directed Acyclic Graph or DAG representation of structure of one or more loops in the application by performing a static and a dynamic analysis of the application source code and depicts the loop information as LoopID, loop weight, total loop iteration, average loop iteration, total loop iteration time, average loop iteration time and embedded vector size. This aids a programmer to concentrate on problematic loops in the application and analyze them further for potential parallelism.

A method for designing a transformer in an integrated circuit includes receiving one or more desired characteristics of the transformer from user input and iteratively determining a design solution for the transformer through one or more simulations and modifications using a rule-set. The method combines the one or more desired characteristics with other preset characteristics of the transformer or the integrated circuit. A first model of the transformer is defined with typical load impedances and simulated having the combined characteristics to determine performance. Results of the simulation are processed to calculate performance with the load impedances specified by the user. The results are further processed to obtain a mathematical model that includes tuning capacitors. The first and subsequent models are modified by drawing on a rule-set of expert knowledge relating to general dependency of at least one design criterion, such as a physical, geometrical or performance characteristic, with another design criterion.

Plus, I have no interest in my husband now that he is smoking again. I'm a cis bi woman, and I mainly have sex with people with penises. I have a really gross problem, sorry.…

Gov. Roy Cooper declared Tuesday, May 5 a “Day of Awareness for Murdered and Missing Indigenous Women.” The advocacy group Shatter the Silence reports that 31 native women have gone missing or been murdered in eastern North Carolina since 1998. The state tracked at least 90 cases of murdered or missing indigenous women in North Carolina since 1994. But advocates say the real numbers are likely much higher.

Ohio is among the top states for several agricultural crops and for food production and processing. But while farming is considered an essential business under the various shutdown orders, it's a tough time for those who run the state's 76,000 farms. Statehouse correspondent Jo Ingles has more.

Talking about the issue four-year-old Remington once had, the 'American Idol' alum admits before finding out the problem, she used to worry that the boy was hearing impaired.

Conservatives demand council and police act on growing menace.

SFA vice-president Mike Mulraney believes it would be “foolhardy” to dismiss the chances of any of Scotland’s major clubs going out of business due to the Covid-19 pandemic.

Tech companies in Silicon Valley are under the microscope for not living up to their idealistic pledges to save the world. On this week’s episode of Reveal, we investigate companies on the cutting edge that are struggling to solve some old-fashioned problems: Worker safety at Tesla, and diversity at Google and beyond.

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Emily Ligawiec has to sign in visitors to the recovery program she attends in a grand Victorian house in Holyoke, Mass. She can't bring people to her room. She only recently earned phone and car privileges. "We get 24, 48, 72-hour passes every weekend," she said. But Ligawiec doesn't mind the restrictions. The 29-year-old is grateful she's alive to follow them, after a decade of addiction — first to prescription painkillers, then pills she bought in the street, then heroin. "I had gone down a pretty dark path," she said. What finally turned her around was a 911 call last winter. She had gotten high on heroin and stolen her mother's car. When she returned it a few hours later, Officer John Cacela of Ware, Mass., was waiting in the driveway. In the past, Cacela might have immediately read Ligawiec her rights, "because for the longest time, the whole idea was — arrest, arrest," he said. Instead, he tapped on the car window and assured her she wasn't in trouble. "I closed the window on him

Finanzhilfen für besonders von der Krise getroffene Länder wie Italien? Die Erfahrung mit der deutschen Währungsunion zeigt, dass das allein nicht reichen wird. Es gilt, die Wirtschaftsstruktur anzupassen. Dazu braucht es relative Preisänderungen.

In den kommenden Monaten könnten viele Kreditnehmer in Zahlungsschwierigkeiten kommen. Schon nach kurzer Zeit können Banken dann das Darlehen kündigen. Doch es gibt Hoffnung. Das sollten Verbraucher jetzt wissen.

Pedro Vacca, director de la Fundación para la Libertad de Prensa, dijo que esas “listas negras”, pueden generar consecuencias fatales.

Q: I had such a terrible allergy attack that I couldn’t get my head off my desk to drive myself home. It was 1987, and I was very reluctant to take any medication. My boss gave me a pill she said was safe because it was plant-based. It was quercetin. When she checked on me […]

In Germany, unions are pushing Amazon.com to prevent warehouse workers from congregating like bunches of “grapes” before their shifts. In Italy, where a COVID-19 outbreak hit Amazon’s main logistics depot, unions staged an 11-day strike that ended after the company granted employees an additional five-minute break to practice better personal hygiene. And in what amounts […]

Let’s all remember that COVID-19 is a global pandemic. As bad as it is in the U.S., low-income countries are suffering in nightmarish ways. The gains made in child immunization are lost because of a critical shortage of health workers. We have clean water and sanitation not available in many countries. I’m asking our representatives […]

Recent protests against stay-at-home orders are political theater and a distraction from the real problem facing us — which is that government is failing to ramp up enough testing. The virus isn’t much chastened by guns or bellicose threats, but it can be hunted down relentlessly and isolated, by science. Why aren't we doing it?

A Saskatchewan laboratory is working on a coronavirus vaccine, Michael Pollan on how caffeine rules our world, how to retrofit an '80s shopping mall, why the NFL hires so few black head coaches, the impeachment drama skids towards acquittal and more.

How has the outbreak affected your life? What problems or struggles are you facing right now? Our expert panel takes your questions about money, medicine, mental health and kids today on a special Q&A edition of Cross Country Checkup.

Building an artificial reef could be the key to stabilising a popular beach at Queensland's Inskip Point that suffers from repeated landslips, a geotechnical expert says.

The passionfruit vine is a typical grievance for those with a green thumb, a temperamental plant that just won't do what it's told.

Panic immediately set in at the sight of an injured Ellyse Perry limping from the field, but her limited role in the Australian T20 team might mean the side can cope better than expected, writes Geoff Lemon.

Victoria's one-stop shop for all things to do with fines has been crippled by serious IT issues. So what's behind the multi-million-dollar problem, and what does it mean for people facing a payment deadline?

Whips will likely be banned from horseracing in the next 10 to 15 years, the CEO of Racing Victoria says, as rain and animal activists dampen the annual Melbourne Cup parade.

An innovative baiting method being trialled on Queensland's Cape York Peninsula could be used on a larger scale to reduce the impact feral pigs have on Australia's marine turtle populations.

Labor's Home Affairs spokesperson Kristina Keneally has sparked controversy after declaring the COVID-19 crisis should force a revamp of the temporary migration program.

Research into a mysterious substance that makes up 85 per cent of the universe, along with good old-fashioned gold and a new hydroponic farm, are driving a jobs boom in the small Victorian town of Stawell, but there's a problem.

We all know the value of planning, but in a complex, complicated and often confounding world it can be difficult knowing how to start. Scenario Planning is planning tool for uncertain situations - find out what it entails and how it might benefit organisations and businesses.

When an outback town in WA was faced with the unusual problem of cows leaving their paddocks and roaming their streets by night in search of food, two men were tasked with doing something about it.

Tasmania is known for being clean and green, but Eva Pagett can't drink her tank water because it's contaminated with lead.

A man who assaulted his then-pregnant girlfriend on a weekly basis, destroyed her possessions by driving over them and smeared faeces on her clothes is jailed and described by the presiding judge as "domineering" and "cowardly".