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One-dimensional potentials defined by $V^{(S)}(x) =S(S+1) \hbar^2 \pi^2 /[2ma^2\sin^2(\pi x/a)]$ (for integer $S$) arise in the repeated supersymmetrization of the infinite square well, here defined over the region $(0,a)$. We review the derivation of this hierarchy of potentials and then use the methods of supersymmetric quantum mechanics, as well as more familiar textbook techniques, to derive compact closed-form expressions for the normalized solutions, $\psi_n^{(S)}(x)$, for all $V^{(S)}(x)$ in terms of well-known special functions in a pedagogically accessible manner. We also note how the solutions can be obtained as a special case of a family of shape-invariant potentials, the trigonometric P\"oschl-Teller potentials, which can be used to confirm our results. We then suggest additional avenues for research questions related to, and pedagogical applications of, these solutions, including the behavior of the corresponding momentum-space wave functions $\phi_n^{(S)}(p)$ for large $|
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The purpose of this work is to rigorously prove the existence of traveling waves in neural field models with lateral inhibition synaptic coupling types and sigmoidal firing rate functions. In the case of traveling fronts, we utilize theory of linear operators and the implicit function theorem on Banach spaces, providing a variation of the homotopy approach originally proposed by Ermentrout and McLeod (1992) in their seminal study of monotone fronts in neural field models. After establishing the existence of traveling fronts, we move to a well-studied singularly perturbed system with linear feedback. For the special case where the synaptic coupling kernel is a difference of exponential functions, we are able to combine our results for the front with theory of invariant manifolds in autonomous dynamical systems to prove the existence of fast traveling pulses that are comparable to singular homoclinical orbits. Finally, using a numerical approximation scheme, we derive the ubiquitous Evan
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We investigate the algebraic structure of complex Lie groups equipped with left-invariant metrics which are expanding semi-algebraic solitons to the Hermitian curvature flow (HCF). We show that the Lie algebras of such Lie groups decompose in the semidirect product of a reductive Lie subalgebra with their nilradicals. Furthermore, we give a structural result concerning expanding semi-algebraic solitons on complex Lie groups. It turns out that the restriction of the soliton metric to the nilradical is also an expanding algebraic soliton and we explain how to construct expanding solitons on complex Lie groups starting from expanding algebraic solitons on their nirladicals.
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We study the Lie bialgebra structures that can be built on the one-dimensional central extension of the Poincar\'e and (A)dS algebras in (1+1) dimensions. These central extensions admit more than one interpretation, but the simplest one is that they describe the symmetries of (the noncommutative deformation of) an Abelian gauge theory, $U(1)$ or $SO(2)$ on the (1+1) dimensional Minkowski or (A)dS spacetime. We show that this highlights the possibility that the algebra of functions on the gauge bundle becomes noncommutative. This is a new way in which the Coleman-Mandula theorem could be circumvented by noncommutative structures, and it is related to a mixing of spacetime and gauge symmetry generators when they act on tensor-product states. We obtain all Lie bialgebra structures on centrally-extended Poincar\'e and (A)dS which are coisotropic w.r.t. the Lorentz algebra, and therefore all of them admit the construction of a noncommutative principal gauge bundle on a quantum homogeneous M
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Over an algebraically closed field, various finiteness results are known regarding the automorphism group of a K3 surface and the action of the automorphisms on the Picard lattice. We formulate and prove versions of these results over arbitrary base fields, and give examples illustrating how behaviour can differ from the algebraically closed case.
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In this article we define intersection Floer homology for exact Lagrangian cobordisms between Legendrian submanifolds in the contactisation of a Liouville manifold, provided that the Chekanov-Eliashberg algebras of the negative ends of the cobordisms admit augmentations. From this theory we derive several exact sequences relating the Morse homology of an exact Lagrangian cobordism with the bilinearised contact homologies of its ends. These are then used to investigate the topological properties of exact Lagrangian cobordisms.
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Error bound condition has recently gained revived interest in optimization. It has been leveraged to derive faster convergence for many popular algorithms, including subgradient methods, proximal gradient method and accelerated proximal gradient method. However, it is still unclear whether the Frank-Wolfe (FW) method can enjoy faster convergence under error bound condition. In this short note, we give an affirmative answer to this question. We show that the FW method (with a line search for the step size) for optimization over a strongly convex set is automatically adaptive to the error bound condition of the problem. In particular, the iteration complexity of FW can be characterized by $O(\max(1/\epsilon^{1-\theta}, \log(1/\epsilon)))$ where $\theta\in[0,1]$ is a constant that characterizes the error bound condition. Our results imply that if the constrained set is characterized by a strongly convex function and the objective function can achieve a smaller value outside the considered
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We consider stochastic (partial) differential equations appearing as Markovian lifts of affine Volterra processes with jumps from the point of view of the generalized Feller property which was introduced in, e.g., D\"orsek-Teichmann (2010). In particular we provide new existence, uniqueness and approximation results for Markovian lifts of affine rough volatility models of general jump diffusion type. We demonstrate that in this Markovian light the theory of stochastic Volterra processes becomes almost classical.
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We study the one-parameter family of twisted Kahler Taub-NUT metrics (discovered by Donaldson), along with two exceptional Taub-NUT-like instantons, and understand them to the extend that should be sufficient for blow-up and gluing arguments. In particular we parametrize their geodesics from the origin, determine curvature fall-off rates, volume growth rates for metric balls, and find blow-down limits.
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At least two U.S. states are investigating a breach at Alphabet's Google that may have exposed private profile data of at least 500,000 users to hundreds of external developers. From a report: The investigation follows Google's announcement on Monday that it would shut down the consumer version of its social network Google+ and tighten its data-sharing policies after a "bug" potentially exposed user data that included names, email addresses, occupations, genders and ages. "We are aware of public reporting on this matter and are currently undertaking efforts to gain an understanding of the nature and cause of the intrusion, whether sensitive information was exposed, and what steps are being taken or called for to prevent similar intrusions in the future," Jaclyn Severance, a spokeswoman for Connecticut Attorney General George Jepsen, told Reuters in an email. The New York Attorney General's office also said it was looking into the breach.
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Fully self-driving cars may be on the fast lane to U.S. roads under a pilot program the Trump administration said on Tuesday it was considering, which would allow real-world road testing for a limited number of the vehicles. Reuters: Self-driving cars used in the program would potentially need to have technology disabling the vehicle if a sensor fails or barring vehicles from traveling above safe speeds, the National Highway Traffic Safety Administration (NHTSA) said in a document made public Tuesday. NHTSA said it was considering whether it would have to be notified of any accident within 24 hours and was seeking public input on what other data should be disclosed including near misses. The U.S. House of Representatives passed legislation in 2017 to speed the adoption of self-driving cars, but the Senate has not approved it. Several safety groups oppose the bill, which is backed by carmakers. It has only a slender chance of being approved in 2018, congressional aides said.
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An anonymous reader shares a report: Last month, an activist from the German art collective Peng! walked into her local government office in Berlin and applied for a new passport. "I probably have broken the law," the woman, a chemist living in the Western Saxony region, told Motherboard, "but our lawyers don't know which one." The woman applied for a passport using a photo of two separate people. Using specialized software created by Peng!, the collective merged the facial vectors from two different faces from two different images into one. Billie Hoffman (a pseudonym used by everyone in the Peng! Collective when talking to journalists), she told me how easy the whole process was: "Officials didn't mention fraud at any point." Hoffman's passport application was approved, and now she has an official German passport using the digitally altered photo. The photo is half her, half Federica Mogherini, an Italian politician who is the High Representative of the European Union for Foreign Aff
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At its Pixel 3 launch event, Google announced a smart speaker called the Google Home Hub, featuring a 7-inch display to give you visual information, making it easier to control smart home devices and view photos and the weather. Interestingly, Google decided not to include a camera in this device for privacy reasons, as they want you to feel comfortable placing it in an intimate location, such as a bedroom. PhoneDog reports: Google explains that Home Hub will be able to recognize who is speaking to it using Voice Match to provide info for that specific person, which should help to make the device more useful in homes with multiple people. And when you're not using Home Hub, a feature called Live Albums will let you select certain people and have Google Photos create albums with images of these people. Another feature of Google's Home Hub is the Home View. With it, you can easily see and control your smart home devices. And then there's Ambient EQ, which uses a sensor that'll adjust the
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Not everything got leaked before Google's event today. One surprise announcement that wowed was Call Screen, a new feature that lets the Google Assistant answer your incoming calls and politely ask what the caller wants. A real-time transcript will appear on your screen, allowing you to decide whether or not you want to pick up. When your Pixel rings, a "Screen call" button shows up alongside the usual controls. Tapping it will prompt the Google Assistant to tell your caller that the call is being screened and ask what it's about. Their explanation is transcribed on your screen, and you have options to mark the call as spam or tell the caller you'll get back to them, among others. This is an amazing feature that will save a lot of people a lot of frustration. I want this feature on my phone now. On a related note, Google Duplex, the feature whereby the Google Assistant will call restaurants and such on your behalf, will be rolled out to Pixel phones next month.
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Alongside the new Pixel smartphones, and the Pixel Slate laptop-tablet hybrid, Google on Tuesday also announced a new version of its Chromecast streaming adapter, the third generation of the company's streaming device, which supports playback video at higher frame rates and can also stream multiroom audio. From a report: The new device goes on sale Tuesday in the U.S., Australia, Canada, Denmark, Finland, Great Britain, Japan, Netherlands, New Zealand, Norway, Singapore and Sweden. Stateside, the new Chromecast once again costs $35 -- the same as its predecessor. [...] The bigger changes are on the inside: The new Chromecast is 15% faster than the previous model, which allows it to stream 1080p HD video with a rate of up to 60 frames per second (fps). "Everything becomes much smoother," said Google Home product manager Chris Chan during a recent interview with Variety. He specifically cited the growth of 60fps content on YouTube as one of the reasons Google added the new feature.
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Google on Tuesday unveiled the Pixel 3 and Pixel 3 XL, its latest flagship Android smartphones. "For life on the go, we designed the world's best camera and put it in the world's most helpful phone," said Google's hardware chief Rick Osterloh. From a report: The Pixel 3 starts at $799 for 64GB, with the 3 XL costing $899. Add $100 to either for the 128GB storage option. Core specs for both include a Snapdragon 845, 4GB RAM (there's no option for more), Bluetooth 5.0, and front-facing stereo speakers. Also inside is a new Titan M security chip, which Google says provides "on-device protection for login credentials, disk encryption, app data, and the integrity of the operating system." Preorders for both phones begin today, and buyers will get six months of free YouTube Music service. The Pixel 3 and 3 XL both feature larger screens than last year's models thanks to slimmed down bezels -- and the controversial notch in the case of the bigger phone. The 3 XL has a 6.3-inch display (up fro
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Google intends to launch a censored version of its Search app for China sometime in the next six to nine months, according to a leaked transcript from a private employee meeting held last month. The Intercept's Ryan Gallagher today reported the company's Search engine chief, Ben Gomes, held a meeting to congratulate a room full of employees working on the platform, dubbed Project Dragonfly. From a report: According to The Intercept, Gomes talked about the launch timeline: "While we are saying it's going to be six and nine months [to launch], the world is a very dynamic place." He goes on to point out that the current political climate makes it difficult to pinpoint a definite timeline, but indicates employees should be ready to launch whenever a "window opens." These comments come in stark contrast to public statements given recently by both Gomes and Google's chief privacy officer, Kieth Enright. Speaking to members of Congress last month, Enright tried to skirt the issue of the Drago
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In addition to announcing new flagship phones today, Google took the wraps off a new premium tablet called the Pixel Slate. It's a Chrome OS-powered slate with a 12.3-inch display that's supposed to be the sharpest in its class. Google claims this isn't just a laptop pretending to be a tablet or a phone pretending to be a computer. From a report: It has a resolution of 3,000 x 2,000 -- i.e., a pixel density of 293 ppi, which Google says is the highest for a premium 12-inch tablet. For reference, the Surface Pro 6 and iPad Pro (12.9 inch) come in at 267 ppi and 264 ppi, respectively. Google was able to make the screen so sharp because of an energy-efficient LCD technology called Low Temperature PolySilicon (LTPS), which let the company pack in more pixels without sacrificing size or battery. In fact, the Pixel Slate is supposed to last up to 12 hours on a charge, which is impressive for its skinny 7mm profile. [...] What stands out about the Pixel Slate is the version of Chrome OS it ru
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Google unveiled its new Pixel phones today, as well as the Pixel Slate, a ChromeOS tablet/laptop device that's basically a cross between an iPad Pro and a Surface Pro. Virtually everything from the event was leaked over the past few weeks, so there were few - if any - surprises. The new devices are certainly interesting, but Google continues its policy of not making these products available in most of the world, so there's little for me to say about them - I have never seen them, let alone used them. One thing that stood out to me about the Pixel Slate are its specifications - it runs on Intel processors, and in order to get a processor that isn't a slow Celeron or m3, you need to shell out some big bucks. I don't have particularly good experiences with Celeron or m3 processors, and even Intel's mobile i5 chips have never really managed to impress me - hence why I opted for the i7 version of the latest Dell XPS 13 when I bought a new laptop a few weeks ago. In The Verge's video, yo
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Google's human-sounding AI software that makes calls for you is coming to Pixel smartphones next month in select markets, like New York, Atlanta, Phoenix, and the San Francisco Bay Area. Google Duplex, as it is called, will be a feature of Google Assistant and, for now, will only be able to call restaurants without online booking systems, which are already supported by the assistant. Wired reports: A Google spokesperson told WIRED that the company now has a policy to always have the bot disclose its true nature when making calls. Duplex still retains the human-like voice and "ums," "ahs," and "umm-hmms" that struck some as spooky, though. Nick Fox, the executive who leads product and design for Google search and the company's assistant, says those interjections are necessary to make Duplex calls shorter and smoother. "The person on the other end shouldn't be thinking about how do I adjust my behavior, I should be able to do what I normally do and the system adapts to that," he says. Fo
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A relation between the precanonical quantization of pure Yang-Mills fields and the functional Schr\"odinger representation in the temporal gauge is discussed. It is shown that the latter can be obtained from the former when the ultraviolet parameter $\varkappa$ introduced in precanonical quantization goes to infinity. In this limiting case, the Schr\"odinger wave functional can be expressed as the trace of the Volterra product integral of Clifford-algebra-valued precanonical wave functions restricted to a certain field configuration, and the canonical functional derivative Schr\"odinger equation together with the quantum Gau\ss\ constraint are derived from the Dirac-like precanonical Schr\"odinger equation.
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So I learned something new today. Back in the early and mid-90s, IBM tried to build a PC-like platform and ecosystem around its PowerPC processor. They called it the PowerPC Reference Platform, or PReP, and with it, you could build what were effectively PC clones with PowerPC processors, ready to run a number of operating systems, including AIX, Windows NT, OS/2, and Apple's failed Taligent project. None of this is news to me. What is news to me, however, is that aside from a number of desktop PReP machines, IBM also developed and sold a number of PReP laptops under the ThinkPad brand. Sometime in 1994, IBM started working on a prototype mobile system named Woodfield and designated as type 6020. Very little is known about this system; it was never officially announced or sold. On June 19, 1995, IBM announced the ThinkPad 850 and 820 (announcement letters 195-178 and 195-179, respectively) with a planned availability date of July 24, 1995. The ThinkPad 820 designation was typ
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The problems of partially observed major minor LQG and nonlinear mean field game (PO MM LQG MFG) problems where it is assumed the major agent's state is partially observed by each minor agent, and the major agent completely observes its own state have been analysed in the literature. In this paper, PO MM LQG MFG problems with general information patterns are studied where (i) the major agent has partial observations of its own state, and (ii) each minor agent has partial observations of its own state and the major agent's state. The assumption of partial observations by all agents leads to a new situation involving the recursive estimation by each minor agent of the major agent's estimate of its own state. For the general case of indefinite LQG MFG systems, the existence of $\epsilon$-Nash equilibria together with the individual agents' control laws yielding the equilibria are established via the Separation Principle.
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The connection matrix is a powerful algebraic topological tool from Conley index theory that captures relationships between isolated invariant sets. Conley index theory is a topological generalization of Morse theory in which the connection matrix subsumes the role of the Morse boundary operator. Over the last few decades, the ideas of Conley have been cast into a purely computational form. In this paper we introduce a computational, categorical framework for the connection matrix theory. This contribution transforms the computational Conley theory into a computational homological theory for dynamical systems. More specifically, within this paper we have two goals: 1) We cast the connection matrix theory into appropriate categorical, homotopy-theoretic language. We identify objects of the appropriate categories which correspond to connection matrices and may be computed within the computational Conley theory paradigm by using the technique of reductions. 2) We describe an algorithm for
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Traditionally, there are several polynomial algorithms for linear programming including the ellipsoid method, the interior point method and other variants. Recently, Chubanov [Chubanov, 2015] proposed a projection and rescaling algorithm, which has become a potentially \emph{practical} class of polynomial algorithms for linear feasibility problems and also for the general linear programming. However, the Chubanov-type algorithms usually perform much better on the infeasible instances than on the feasible instances in practice. To explain this phenomenon, we derive a new theoretical complexity bound for the infeasible instances based on the condition number, which shows that algorithms can indeed run much faster on infeasible instances in certain situations. In order to speed up the feasible instances, we propose a \emph{Polynomial-time Primal-Dual Projection} algorithm (called PPDP) by explicitly developing the dual algorithm. The numerical results validate that our PPDP algorithm achi
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We consider solving the surface Helmholtz equation on a smooth two dimensional surface embedded into a three dimensional space meshed with tetrahedra. The mesh does not respect the surface and thus the surface cuts through the elements. We consider a Galerkin method based on using the restrictions of continuous piecewise linears defined on the tetrahedra to the surface as trial and test functions.Using a stabilized method combining Galerkin least squares stabilization and a penalty on the gradient jumps we obtain stability of the discrete formulation under the condition $h k < C$, where $h$ denotes the mesh size, $k$ the wave number and $C$ a constant depending mainly on the surface curvature $\kappa$, but not on the surface/mesh intersection. Optimal error estimates in the $H^1$ and $L^2$-norms follow.
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The stable center conjecture asserts that the space of stable distributions in the Bernstein center of a reductive p-adic is closed under convolution. It is closely related to the notion of an L-packet and endoscopy theory. We describe a categorical approach to the depth zero part of the conjecture. As an illustration of our method, we show that the Bernstein projector to the depth zero spectrum is stable.
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We study a categorical construction called the cobordism category, which associates to each Waldhausen category a simplicial category of cospans. We prove that this construction is homotopy equivalent to Waldhausen's $S_{\bullet}$-construction and therefore it defines a model for Waldhausen $K$-theory. As an example, we discuss this model for $A$-theory and show that the cobordism category of homotopy finite spaces has the homotopy type of Waldhausen's $A(*)$. We also review the canonical map from the cobordism category of manifolds to $A$-theory from this viewpoint.
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It has been discovered that linear codes may be described by binomial ideals. This makes it possible to study linear codes by commutative algebra and algebraic geometry methods. In this paper, we give a decoding algorithm for binary linear codes by utilizing the Groebner bases of the associated ideals.
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In this note, we establish an expression of the Loewner energy of a Jordan curve on the Riemann sphere in terms of Werner's measure on simple loops of SLE$_{8/3}$ type. The proof is based on a formula for the change of the Loewner energy under a conformal map that is reminiscent of the restriction properties derived for SLE processes.
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We design and analyze solution techniques for a linear-quadratic optimal control problem involving the integral fractional Laplacian. We derive existence and uniqueness results, first order optimality conditions, and regularity estimates for the optimal variables. We propose two strategies to discretize the fractional optimal control problem: a semidiscrete approach where the control is not discretized - the so-called variational discretization approach - and a fully discrete approach where the control variable is discretized with piecewise constant functions. Both schemes rely on the discretization of the state equation with the finite element space of continuous piecewise polynomials of degree one. We derive a priori error estimates for both solution techniques. We illustrate the theory with two-dimensional numerical tests.
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We analyse features of the patterns formed from a simple model for a martensitic phase transition. This is a fragmentation model that can be encoded by a general branching random walk. An important quantity is the distribution of the lengths of the interfaces in the pattern and we establish limit theorems for some of the asymptotics of the interface profile. We are also able to use a general branching process to show almost sure power law decay of the number of interfaces of at least a certain size. We discuss the numerical aspects of determining the behaviour of the density profile and power laws from simulations of the model.
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The Weinstein operator has several applications in pure and applied Mathematics especially in Fluid Mechanics and satisfies some uncertainty principles similar to the Euclidean Fourier transform. The aim of this paper is establish a generalization of uncertainty principles for Weinstein transform in $L_\alpha^p$-norm. Firstly, we extend the Heisenberg-Pauli-Weyl uncertainty principle to more general case. Then we establish three continuous uncertainty principles of concentration type. The first and the second uncertainty principles are $L_\alpha^p$ versions and depend on the sets of concentration $\Omega$ and $\Sigma$, and on the time function $\varphi$. However, the third uncertainty principle is also $L_\alpha^p$ version depends on the sets of concentration and he is independent on the band limited function $\varphi$. These $L_\alpha^p$-Donoho-Stark-type inequalities generalize the results obtained in the case $p=q=2$.
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We describe main issues and design principles of an efficient implementation, tailored to recent generations of Nvidia Graphics Processing Units (GPUs), of an Algebraic Multigrid (AMG) preconditioner previously proposed by one of the authors and already available in the open-source package BootCMatch: Bootstrap algebraic multigrid based on Compatible weighted Matching for standard CPU. The AMG method relies on a new approach for coarsening sparse symmetric positive definite (spd) matrices, named "coarsening based on compatible weighted matching". It exploits maximum weight matching in the adjacency graph of the sparse matrix, driven by the principle of compatible relaxation, providing a suitable aggregation of unknowns which goes beyond the limits of the usual heuristics applied in the current methods. We adopt an approximate solution of the maximum weight matching problem, based on a recently proposed parallel algorithm, referred as the Suitor algorithm, and show that it allow us to o
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A path (cycle) in a $c$-edge-colored multigraph is alternating if no two consecutive edges have the same color. The problem of determining the existence of alternating Hamiltonian paths and cycles in $2$-edge-colored multigraphs is an $\mathcal{NP}$-complete problem and it has been studied by several authors. In Bang-Jensen and Gutin's book "Digraphs: Theory, Algorithms and Applications", it is devoted one chapter to survey the last results on this topic. Most results on the existence of alternating Hamiltonian paths and cycles concern on complete and bipartite complete multigraphs and a few ones on multigraphs with high monochromatic degrees or regular monochromatic subgraphs. In this work, we use a different approach imposing local conditions on the multigraphs and it is worthwhile to notice that the class of multigraphs we deal with is much larger than, and includes, complete multigraphs, and we provide a full characterization of this class. Given a $2$-edge-colored multigraph $G$,
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We use Brascamp-Lieb's inequality to obtain new decoupling inequalities for general Gaussian vectors, and for stationary cyclic Gaussian processes. In the second case, we use a version by Bump and Diaconis of the strong Szego limit theorem. This extends results of Klein, Landau and Shucker.
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Let $(X,{\mathcal A},\mu)$ be a probability space and let $S\colon X\to X$ be a measurable transformation. Motivated by the paper of K. Nikodem [Czechoslovak Math. J. 41(116) (4) (1991) 565--569], we concentrate on a functional equation generating measures that are absolutely continuous with respect to $\mu$ and $\varepsilon$-invariant under $S$. As a consequence of the investigation, we obtain a result on the existence and uniqueness of solutions $\varphi\in L^1([0,1])$ of the functional equation $$ \varphi(x)=\sum_{n=1}^{N}|f_n'(x)|\varphi(f_n(x))+g(x), $$ where $g\in L^1([0,1])$ and $f_1,\ldots,f_N\colon[0,1]\to[0,1]$ are functions satisfying some extra conditions.
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Let $p\equiv 4,7\mod 9$ be a rational prime number such that $3\mod p$ is not a cubic residue. In this paper we prove the 3-part of the product of the full BSD conjectures for $E_p$ and $E_{3p^3}$ is true using an explicit Gross-Zagier formula, where $E_p: x^3+y^3=p$ and $E_{3p^2}: x^3+y^3=3p^2$ are the elliptic curves related to the Sylvester conjecture and cube sum problems.
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In this paper we obtain a very general Gauss-Green formula for weakly differentiable functions and sets of finite perimeter. This result is obtained by revisiting Anzellotti's pairing theory and by characterizing the measure pairing $(\boldsymbol{A}, Du)$ when $\boldsymbol{A}$ is a bounded divergence measure vector field and $u$ is a bounded function of bounded variation.
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Given an open subset $\Omega$ of a Banach space and a Lipschitz function $u_0: \overline{\Omega} \to \mathbb{R},$ we study whether it is possible to approximate $u_0$ uniformly on $\Omega$ by $C^k$-smooth Lipschitz functions which coincide with $u_0$ on the boundary $\partial \Omega$ of $\Omega$ and have the same Lipschitz constant as $u_0.$ As a consequence, we show that every $1$-Lipschitz function $u_0: \overline{\Omega} \to \mathbb{R},$ defined on the closure $\overline{\Omega}$ of an open subset $\Omega$ of a finite dimensional normed space of dimension $n \geq 2$, and such that the Lipschitz constant of the restriction of $u_0$ to the boundary of $\Omega$ is less than $1$, can be uniformly approximated by differentiable $1$-Lipschitz functions $w$ which coincide with $u_0$ on $\partial \Omega$ and satisfy the equation $\| D w\|_* =1$ almost everywhere on $\Omega.$ This result does not hold in general without assumption on the restriction of $u_0$ to the boundary of $\Omega$.
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We prove, using variational methods, the existence in dimension two of positive vector ground states solutions for the Bose-Einstein type systems \begin{equation} \begin{cases} -\Delta u+\lambda_1u=\mu_1u(e^{u^2}-1)+\beta v\left(e^{uv}-1\right) \text{ in } \Omega, &\\ -\Delta v+\lambda_2v=\mu_2v(e^{v^2}-1)+\beta u\left(e^{uv}-1\right)\text{ in } \Omega, &\\ u,v\in H^1_0(\Omega) \end{cases} \end{equation} where $\Omega$ is a bounded smooth domain, $\lambda_1,\lambda_2>-\Lambda_1$ (the first eigenvalue of $(-\Delta,H^1_0(\Omega))$, $\mu_1,\mu_2>0$ and $\beta$ is either positive (small or large) or negative (small). The nonlinear interaction between two Bose fluids is assumed to be of critical exponential type in the sense of J. Moser. For `small' solutions the system is asymptotically equivalent to the corresponding one in higher dimensions with power-like nonlinearities.
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We consider the problem of private linear computation (PLC) in a distributed storage system. In PLC, a user wishes to compute a linear combination of $f$ messages stored in noncolluding databases while revealing no information about the coefficients of the desired linear combination to the databases. In extension of our previous work we employ linear codes to encode the information on the databases. We show that the PLC capacity, which is the ratio of the desired linear function size and the total amount of downloaded information, matches the maximum distance separable (MDS) coded capacity of private information retrieval for a large class of linear codes that includes MDS codes. In particular, the proposed converse is valid for any number of messages and linear combinations, and the capacity expression depends on the rank of the coefficient matrix obtained from all linear combinations.
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The performance enhancements observed in various models of continuous quantum thermal machines have been linked to the buildup of coherences in a preferred basis. But, is this connection always an evidence of 'quantum-thermodynamic supremacy'? By force of example, we show that this is not the case. In particular, we compare a power-driven three-level quantum refrigerator with a four-level combined cycle, partly driven by power and partly by heat. We focus on the weak driving regime and find the four-level model to be superior since it can operate in parameter regimes in which the three-level model cannot, it may exhibit a larger cooling rate, and, simultaneously, a better coefficient of performance. Furthermore, we find that the improvement in the cooling rate matches the increase in the stationary quantum coherences exactly. Crucially, though, we also show that the thermodynamic variables for both models follow from a classical representation based on graph theory. This implies that w
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A successive cancellation list (SCL) decoder with limited list size for polar codes can not be analyzed as a successive cancellation (SC) decoder, nor as a maximum likelihood (ML) decoder, due to the complicated decoding errors caused by path elimination. To address this issue, an analytical tool, named as cluster pairwise error probability (CPEP), is proposed in this paper to measure the competitiveness of the correct path against the error paths in an SCL decoder. It is shown that the sum of CPEPs over error paths could be used as an indicator of the probability of correct path being eliminated from the decoder list. Then, we use CPEP to explain the error performance gain of parity-check-concatenated (PCC) polar code, and apply CPEP as the optimization criterion in the construction of PCC polar codes, aiming to reduce the elimination probability of the correct path in an SCL decoder with limited list size. Simulation results show that the constructed CRC-PCC polar codes outperform th
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We study numerical invariants of identities of finite-dimensional solvable Lie superalgebras. We define new series of finite-dimensional solvable Lie superalgebras $L$ with non-nilpotent derived subalgebra $L'$ and discuss their codimension growth. For the first algebra of this series we prove the existence and integrality of $exp(L)$.
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We study a natural growth process with competition, which was recently introduced to analyze MDLA, a challenging model for the growth of an aggregate by diffusing particles. The growth process consists of two first-passage percolation processes $\text{FPP}_1$ and $\text{FPP}_\lambda$, spreading with rates $1$ and $\lambda>0$ respectively, on a graph $G$. $\text{FPP}_1$ starts from a single vertex at the origin $o$, while the initial configuration of $\text{FPP}_\lambda$ consists of infinitely many \emph{seeds} distributed according to a product of Bernoulli measures of parameter $\mu>0$ on $V(G)\setminus \{o\}$. $\text{FPP}_1$ starts spreading from time 0, while each seed of $\text{FPP}_\lambda$ only starts spreading after it has been reached by either $\text{FPP}_1$ or $\text{FPP}_\lambda$. A fundamental question in this model, and in growth processes with competition in general, is whether the two processes coexist (i.e., both produce infinite clusters) with positive probabilit
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Electrical impedance tomography (EIT) is highly affected by modeling errors regarding electrode positions and the shape of the imaging domain. In this work, we propose a new inclusion detection technique that is completely independent of such errors. Our new approach is based on a combination of frequency-difference and ultrasound modulated EIT measurements.
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Lie group methods are applied to the time-dependent, monoenergetic neutron diffusion equation in materials with spatial and time dependence. To accomplish this objective, the underlying 2nd order partial differential equation (PDE) is recast as an exterior differential system so as to leverage the isovector symmetry analysis approach. Some of the advantages of this method as compared to traditional symmetry analysis approaches are revealed through its use in the context of a 2nd order PDE. In this context, various material properties appearing in the mathematical model (e.g., a diffusion coefficient and macroscopic cross section data) are left as arbitrary functions of space and time. The symmetry analysis that follows is restricted to a search for translation and scaling symmetries; consequently the Lie derivative yields specific material conditions that must be satisfied in order to maintain the presence of these important similarity transformations. The principal outcome of this wor
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Graded bundles are a particularly nice class of graded manifolds and represent a natural generalisation of vector bundles. By exploiting the formalism of supermanifolds to describe Lie algebroids we define the notion of a weighted $A$-connection on a graded bundle. In a natural sense weighted $A$-connections are adapted to the basic geometric structure of a graded bundle in the same way as linear $A$-connections are adapted to the structure of a vector bundle. This notion generalises directly to multi-graded bundles and in particular we present the notion of a bi-weighted $A$-connection on a double vector bundle. We prove the existence of such adapted connections and use them to define (quasi-)actions of Lie algebroids on graded bundles.
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We give a constructive proof of the existence of the almost revlex ideal $J\subset K[x_1,\dots,x_n]$ with the same Hilbert function of a complete intersection defined by $n$ forms of degrees $d_1\leq \dots \leq d_n$, when for every $i\geq 4$ the degrees satisfy the condition $d_i\geq \bar u_{i-1}+1=\min\Bigl\{\Big\lfloor\frac{\sum_{j=1}^{i-1}d_j-i+1}{2}\Big\rfloor, \sum_{j=1}^{i-2} d_j-i+2\Bigr\}+1$. The further property that, for every $t\geq \bar u_n+1$, all terms of degree $t$ outside $J$ are divisible by the last variable has an important role in our inductive and constructive proof, which is different from the more general construction given by Pardue in 2010.
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We prove the existence of a subsonic weak solution $({\bf u}, \rho, p)$ to steady Euler system in a two-dimensional infinitely long nozzle when prescribing the value of the entropy $(= \frac{p}{\rho^{\gamma}})$ at the entrance by a piecewise $C^2$ function with a discontinuity at a point. Due to the variable entropy condition with a discontinuity at the entrance, the corresponding solution has a nonzero vorticity and contains a contact discontinuity $x_2=g_D(x_1)$. We construct such a solution via Helmholtz decomposition. The key step is to decompose the Rankine-Hugoniot conditions on the contact discontinuity via Helmholtz decomposition so that the compactness of approximated solutions can be achieved. Then we apply the method of iteration to obtain a piecewise smooth subsonic flow with a contact discontinuity and nonzero vorticity. We also analyze the asymptotic behavior of the solution at far field.
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This work serves as a primer to our efforts in arriving at convergence estimates for the fixed stress split iterative scheme for single phase flow coupled with small strain anisotropic poroelastoplasticity. The fixed stress split iterative scheme solves the flow subproblem with stress tensor fixed using a mixed finite element method, followed by the poromechanics subproblem using a conforming Galerkin method in every coupling iteration at each time step. The coupling iterations are repeated until convergence and Backward Euler is employed for time marching. The convergence analysis is based on studying the equations satisfied by the difference of iterates to show that the iterative scheme is contractive.
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We compute the asymptotic growth rate of the number N(C, R) of closed geodesics of length less than R in a connected component C of a stratum of quadratic differentials. We prove that for any 0 < \theta < 1, the number of closed geodesics of length at most R that spend at least \theta-fraction of time outside of a compact subset of C is exponentially smaller than N(C, R). The theorem follows from a lattice counting statement. For points x, y in the moduli space M of Riemann surfaces, and for 0 < \theta < 1, we find an upper-bound for the number of geodesic paths of length less than R in C which connect a point near x to a point near y and spend a \theta-fraction of the time outside of a compact subset of C.
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We prove that under certain conditions, phase separation is enough to sustain a regime in which current flows along the concentration gradient, a phenomenon which is known in the literature as \textit{uphill diffusion}. The model we consider here is a version of that proposed in [G. B. Giacomin, J. L. Lebowitz, Phase segregation dynamics in particle system with long range interactions, Journal of Statistical Physics 87(1) (1997): 37-61], which is the continuous mesoscopic limit of a $1d$ discrete Ising chain with a Kac potential. The magnetization profile lies in the interval $\left[-\varepsilon^{-1},\varepsilon^{-1}\right]$, $\varepsilon>0$, staying in contact at the boundaries with infinite reservoirs of fixed magnetization $\pm\mu$, $\mu\in(m^*\left(\beta\right),1)$, where $m^*\left(\beta\right)=\sqrt{1-1/\beta}$, $\beta>1$ representing the inverse temperature. At last, an external field of Heaviside-type of intensity $\kappa>0$ is introduced. According to the axiomatic non
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We introduce the cycle intersection graph of a graph, an adaptation of the cycle graph of a graph, and use the structure of these graphs to prove an upper bound for the decycling number of all even graphs. This bound is shown to be significantly better when an even graph admits a cycle decomposition in which any two cycles intersect in at most one vertex. Links between the cycle rank of the cycle intersection graph of an even graph and the decycling number of the even graph itself are found. The problem of choosing an ideal cycle decomposition is addressed and is presented as an optimization problem over the space of cycle decompositions of even graphs.
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This paper is devoted for classifying all capable Heisenberg Lie superalgebras where we have shown that there exists at least one capable Lie superalgebra of corank $\geq 4$. This paper can be thought up as a super symmetric extension of a recent result by Peyman Niroomand, Mohesen Parvizi, Francesco G. Russo [20] who classify all capable Heisenberg Lie algebras where they have shown that there exists atleast one capable Lie algebra of arbitrary corank.
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Let $U^-_q = U^-_q(\mathfrak g)$ be the negative part of the quantum group associated to a finite dimensional simple Lie algebra $\mathfrak g$, and $\sigma : \mathfrak g \to \mathfrak g$ be the automorphism obtained from the diagram automorphism. Let $\mathfrak g^{\sigma}$ be the fixed point subalgebra of $\mathfrak g$, and put $\underline U^-_q = U^-_q(\mathfrak g^{\sigma})$. Let $B$ be the canonical basis of $U_q^-$ and $\underline B$ the canonical basis of $\underline U_q^-$. $\sigma$ induces a natural action on $B$, and we denote by $B^{\sigma}$ the set of $\sigma$-fixed elements in $B$. Lusztig proved that there exists a canonical bijection $B^{\sigma} \simeq \underline B$ by using geometric considerations. In this paper, we construct such a bijection in an elementary way. We also consider such a bijection in the case of certain affine quantum groups, by making use of PBW-bases constructed by Beck and Nakajima.
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We consider a second order differential operator $\mathscr{A}$ on an (typically unbounded) open and Dirichlet regular set $\Omega\subset \mathbb{R}^d$ and subject to nonlocal Dirichlet boundary conditions of the form \[ u(z) = \int_\Omega u(x)\mu (z, dx) \quad \mbox{ for } z\in \partial \Omega. \] Here, $\mu : \partial\Omega \to \mathscr{M}(\Omega)$ is a $\sigma (\mathscr{M}(\Omega), C_b(\Omega))$-continuous map taking values in the probability measures on $\Omega$. Under suitable assumptions on the coefficients in $\mathscr{A}$, which may be unbounded, we prove that a realization $A_\mu$ of $\mathscr{A}$ subject to the nonlocal boundary condition, generates a (not strongly continuous) semigroup on $L^\infty(\Omega)$. We also establish a sufficient condition for this semigroup to be Markovian and prove that in this case, it enjoys the strong Feller property. We also study the asymptotic behavior of the semigroup.
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We study the distinguishability of a particular type of maximally entangled states -- the "ququad-ququad" states which are tensor products of Bell states in $\mathbb{C}^4\otimes\mathbb{C}^4$. We first prove that any three orthogonal ququad-ququad maximally entangled states can be distinguished with LOCC. Then we use a new approach of semidefinite program to construct all sets of four ququad-ququad orthogonal maximally entangled states that are PPT-indistinguishable and we find some interesting sets of six states having interesting property of distinguishability. Also, we show that our approach of the optimization problem can make some computational complex problem more tractable.
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It is the purpose of this article to outline a course that can be given to engineers looking for an understandable mathematical description of the foundations of distribution theory and the necessary functional analytic methods. Arguably, these are needed for a deeper understanding of basic questions in signal analysis. Objects such as the Dirac delta and Dirac comb require a proper definition, and it should be possible to explain how one can reconstruct a band-limited function from its samples by means of simple series expansions. It should also be useful for graduate students who want to see how functional analysis can help to understand fairly practical problems, or teachers who want to offer a course related to the "Mathematical Foundations of Signal Processing". The course requires only an understanding of the basic terms from linear functional analysis, namely Banach spaces and their duals, bounded linear operators and a simple version of weak$^{*}$-convergence. As a matter of fa
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This paper addresses the trajectory tracking control problem for underactuated VTOL UAVs. According to the different actuation mechanisms, the most common UAV platforms can achieve only a partial decoupling of attitude and position tasks. Since position tracking is of utmost importance for applications involving aerial vehicles, we propose a control scheme in which position tracking is the primary objective. To this end, this work introduces the concept of attitude planner, a dynamical system through which the desired attitude reference is processed to guarantee the satisfaction of the primary objective: the attitude tracking task is considered as a secondary objective which can be realized as long as the desired trajectory satisfies specific trackability conditions. Two numerical simulations are performed by applying the proposed control law to a hexacopter with and without tilted propellers, which accounts for unmodeled dynamics and external disturbances not included in the control d
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We give formulae for the multiplicities of eigenvalues of generalized rotation operators in terms of generalized Frobenius-Schur indicators in a semisimple spherical tensor category $\mathcal{C}$. In particular, this implies that the entire collection of rotation eigenvalues for a fusion category can be computed from the fusion rules and the traces of rotation at finitely many tensor powers. We also establish a rigidity property for FS indicators of fusion categories with a given fusion ring via Jones's theory of planar algebras. If $\mathcal{C}$ is also braided, these formulae yield the multiplicities of eigenvalues for a large class of braids in the associated braid group representations. When $\mathcal{C}$ is modular, this allows one to determine the eigenvalues and multiplicities of braids in terms of just the $S$ and $T$ matrices.
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We consider two models of a compressible inviscid isentropic two-fluid flow. The first one describes the liquid-gas two-phase flow. The second one can describe the mixture of two fluids of different densities or the mixture of fluid and particles. Introducing an entropy-like function, we reduce the equations of both models to a symmetric form which looks like the compressible Euler equations written in the nonconservative form in terms of the pressure, the velocity and the entropy. Basing on existing results for the Euler equations, this gives a number of instant results for both models. In particular, we conclude that all compressive shock waves in these models exist locally in time. For the 2D case, we make the conclusion about the local-in-time existence of vortex sheets under a "supersonic" stability condition. In the sense of a much lower regularity requirement for the initial data, our result for 2D vortex sheets essentially improves a recent result for vortex sheets in the liqui
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Let U(N) be the quasi-split unitary group in N variables for a quadratic unramified extension of p-adic fields. We compute the characters of simple supercuspidal representations of twisted GL(N) and U(N). Comparing them by the endoscopic character relation, we determine the liftings of simple supercuspidal representations of U(N) to GL(N), under the assumption that p is not equal to 2.
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