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## 信息流

• A function $U:\left[ \omega_{1}\right] ^{2}\longrightarrow\omega$ is called $\left( 1,\omega_{1}\right)$\emph{-weakly universal }if for every function $F:\left[ \omega_{1}\right] ^{2}\longrightarrow\omega$ there is an injective function $h:\omega_{1}\longrightarrow\omega_{1}$ and a function $e:\omega \longrightarrow\omega$ such that $F\left( \alpha,\beta\right) =e\left( U\left( h\left( \alpha\right) ,h\left( \beta\right) \right) \right)$ for every $\alpha,\beta\in\omega_{1}$. We will prove that it is consistent that there are no $\left( 1,\omega_{1}\right)$\emph{-}weakly universal functions, this answers a question of Shelah and Stepr\={a}ns. In fact, we will prove that there are no $\left( 1,\omega_{1}\right)$\emph{-}weakly universal functions in the Cohen model and after adding $\omega_{2}$ Sacks reals side-by-side. However, we show that there are $\left( 1,\omega _{1}\right)$\emph{-}weakly universal functions in the Sacks model. In particular, the existence of such graphs is co

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• The aim of this paper is to present the global bounds for renormalized solutions to the following quasilinear elliptic problem: \begin{align*} \begin{cases} -\div(A(x,\nabla u)) &= \mu \quad \text{in} \ \ \Omega, \\ u &=0 \quad \text{on} \ \ \partial \Omega, \end{cases} \end{align*} in Lorentz-Morrey spaces, where $\Omega \subset \mathbb{R}^n$ ($n \ge 2$), $\mu$ is a finite Radon measure, $A$ is a monotone Carath\'eodory vector valued function defined on $W^{1,p}_0(\Omega)$ and the $p$-capacity uniform thickness condition is imposed on our domain. There have been research activities on the gradient estimates in Lorentz-Morrey spaces with various hypotheses. For instance, in \cite{55Ph1} Nguyen Cong Phuc proposed the Morrey global bounds of solution to this equation, but for the regular case $2-\frac{1}{n}<p\le n$, in \cite{MP2018}, our first result provides us with the good-$\lambda$ bounds of solution in Lorentz space for $\frac{3n-2}{2n-1}<p \le 2 - \frac{1}{n}$; and in

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• This paper derives non-central asymptotic results for non-linear integral functionals of homogeneous isotropic Gaussian random fields defined on hypersurfaces in $\mathbb{R}^d$. We obtain the rate of convergence for these functionals. The results extend recent findings for solid figures. We apply the obtained results to the case of sojourn measures and demonstrate different limit situations.

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• In this article we lay out the details of Fukaya's A $\infty$-structure of the Morse com-plexe of a manifold possibily with boundary. We show that this A $\infty$-structure is homotopically independent of the made choices. We emphasize the transversality arguments that some fiber product constructions make valid.

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• We prove that if the edges of a graph G can be colored blue or red in such a way that every vertex belongs to a monochromatic k-clique of each color, then G has at least 4(k-1) vertices. This confirms a conjecture of Bucic, Lidicky, Long, and Wagner (arXiv:1805.11278[math.CO]) and thereby solves the 2-dimensional case of their problem about partitions of discrete boxes with the k-piercing property. We also characterize the case of equality in our result.

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• In this paper, we present new applications of our general minimax theorems. In particular, one of them concerns the multiplicity of global minima for the integral functional of the Calculus of Variations.

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• Comon's conjecture on the equality of the rank and the symmetric rank of a symmetric tensor, and Strassen's conjecture on the additivity of the rank of tensors are two of the most challenging and guiding problems in the area of tensor decomposition. We survey the main known results on these conjectures, and, under suitable bounds on the rank, we prove them, building on classical techniques used in the case of symmetric tensors, for mixed tensors. Finally, we improve the bound for Comon's conjecture given by flattenings by producing new equations for secant varieties of Veronese and Segre varieties.

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• We obtain an improved version of a recent result concerning the existence of nonnegative nonradial solutions $u\in D^{1,2}(\mathbb{R}^{N})\cap L^{2}(\mathbb{R}^{N},\left| x\right| ^{-\alpha }dx)$ to the equation $-\triangle u+\displaystyle\frac{A}{\left| x\right| ^{\alpha }}u=f\left( u\right) \quad \text{in }\mathbb{R}^{N},\quad N\geq 3,\quad A,\alpha >0,$ where $f$ is a continuous nonlinearity satisfying $f\left( 0\right) =0$.

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• In this paper, we develop a new computational approach which is based on minimizing the difference of two convex functionals (DC) to solve a broader class of phase retrieval problems. The approach splits a standard nonlinear least squares minimizing function associated with the phase retrieval problem into the difference of two convex functions and then solves a sequence of convex minimization sub-problems. For each subproblem, the Nesterov's accelerated gradient descent algorithm or the Barzilai-Borwein (BB) algorithm is used. In the setting of sparse phase retrieval, a standard $\ell_1$ norm term is added into the minimization mentioned above. The subproblem is approximated by a proximal gradient method which is solved by the shrinkage-threshold technique directly without iterations. In addition, a modified Attouch-Peypouquet technique is used to accelerate the iterative computation. These lead to more effective algorithms than the Wirtinger flow (WF) algorithm and the Gauss-Newton (

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• In this paper, we prove an isoperimetric inequality for lower order eigenvalues of the Dirichlet Laplacian in bounded domains of a Euclidean space which strengthens the well-known Ashbaugh-Beguria inequality about the ratio of the first two Dirichlet eigenvalues of the same domains and supports strongly a conjecture of Ashbaugh-Benguria.

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• Consider an i.i.d. sample from an unknown density function supported on an unknown manifold embedded in a high dimensional Euclidean space. We tackle the problem of learning a distance between points, able to capture both the geometry of the manifold and the underlying density. We prove the convergence of this microscopic distance, as the sample size goes to infinity, to a macroscopic one that we call Fermat distance as it minimizes a path functional, resembling Fermat principle in optics. The proof boils down to the study of geodesics in Euclidean first-passage percolation for nonhomogeneous Poisson point processes.

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• We introduce first the large-cardinal notion of $\Sigma_n$-supercompactness as a higher-level analog of the well-known Magidor's characterization of supercompact cardinals, and show that a cardinal is $C^{(n)}$-extendible if and only if it is $\Sigma_{n+1}$-supercompact. This yields a new characterization of $C^{(n)}$-extendible cardinals which underlines their role as natural milestones in the region of the large-cardinal hierarchy between the first supercompact cardinal and Vop\v{e}nka's Principle ($\rm{VP}$). We then develop a general setting for the preservation of $\Sigma_n$-supercompact cardinals under class forcing iterations. As a result we obtain new proofs of the consistency of the GCH with $C^{(n)}$-extendible cardinals (cf.~\cite{Tsa13}) and the consistency of $\rm{VP}$ with the GCH (cf.~\cite{Broo}). Further, we show that $C^{(n)}$-extendible cardinals are preserved after forcing with standard Easton class forcing iterations for any $\Pi_1$-definable possible behaviour of

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• The dichromatic number of a digraph $D$, denoted by $\chi_A(D)$, is the minimum $k$ such that $D$ admits a $k$-coloring of its vertex set in such a way that each color class is acyclic. In 1976, Bondy proved that the chromatic number of a digraph $D$ is at most its circumference, the length of a longest cycle. In this paper we will construct three graphs from $D$ whose chromatic numbers will bound $\chi_A(D)$. Moreover, we prove: i) for integers $k\geq 2$, $s\geq 1$ and $r_1, \ldots, r_s$ with $k\geq r_i\geq 0$ and $r_i\neq 1$ for each $i\in[s]$, that if all cycles in $D$ have length $r$ modulo $k$ for some $r\in\{r_1,\ldots,r_s\}$, then $\chi_A(D)\leq 2s+1;$ ii) if $D$ has girth $g$, the length of a shortest cycle, and circumference $c$, then $\chi_A(D)\leq \lceil \frac{c-1}{g-1} \rceil +1$, which improves, substantially, the bound proposed by Bondy; iii) if $D$ has girth $g$ and there are integers $k$ and $p,$ with $k\geq g-1\geq p\geq 1$ such that $D$ contains no cycle of length $r$

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• Non-transversal intersection of the free and fixed boundary is shown to hold and a classification of blow-up solutions is given for obstacle problems generated by fully nonlinear uniformly elliptic operators in two dimensions which appear in the mean-field theory of superconducting vortices.

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• For a noetherian scheme that has an ample family of invertible sheaves, we prove that direct products in the category of quasi-coherent sheaves are not exact unless the scheme is affine. This result can especially be applied to all quasi-projective schemes over commutative noetherian rings. The main tools of the proof are the Gabriel-Popescu embedding and Roos' characterization of Grothendieck categories satisfying Ab6 and Ab4*.

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•   10-18 MIT Technology 251

This paper is the first step in the project of categorifying the bialgebra structure on the half of quantum group $U_{q}(\mathfrak{g})$ by using geometry and Hall algebras. We equip the category of D-modules on the moduli stack of objects of the category $Rep_{\mathbb{C}}(Q)$ of representations of a quiver with the structure of an algebra object in the category of stable $\infty$-categories. The data for this construction is provided by an extension of the Waldhausen construction for the category $Rep_{\mathbb{C}}(Q)$. We discuss the connection to the Khovanov-Lauda-Rouquier categorification of half of the quantum group $U_{q}(\mathfrak{g})$ associated to the quiver $Q$ and outline our approach to the categorification of the bialgebra structure.

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•   10-18 Hacker News 121

Good sleep, good learning, good life (2012)

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•   10-18 Hacker News 251

Android Creator’s Startup Essential Products Cuts About 30% of Staff

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•   10-18 Hacker News 160

Building a Titan: Better security through a tiny chip

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•   10-18 Hacker News 201

Building a Cross-Browser, Cross-Platform, Real-Time Game Streaming Protocol

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• In the early days of what ultimately became Waymo, Google's self-driving car division (known at the time as "Project Chauffeur"), there were "more than a dozen accidents, at least three of which were serious," according to a new article in The New Yorker . From a report: The magazine profiled Anthony Levandowski, the former Google engineer who was at the center of the Waymo v. Uber trade secrets lawsuit. According to the article, back in 2011, Levandowski also modified the autonomous software to take the prototype Priuses on "otherwise forbidden routes." Citing an anonymous source, The New Yorker reports that Levandowski sat behind the wheel as the safety driver, along with Isaac Taylor, a Google executive. But while they were in the car, the Prius "accidentally boxed in another vehicle," a Camry. As The New Yorker wrote: "A human driver could easily have handled the situation by slowing down and letting the Camry merge into traffic, but Google's software wasn't prepared for this scena

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•   10-18 Hacker News 201

CodeStream Master Plan: addressing the messaging gap for developers with markers

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•   10-18 THE NEW YORKER 104

Daniel Penny writes about the rapper Gucci Mane’s appearance at the Hypefest convention, at the Brooklyn Navy Yard, to promote a collaboration with the Italian denim brand Diesel, called “Hate Couture.”

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• The future of technology will be brought to you by the number five, say speakers at ARM TechCon

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•   10-18 THE NEW YORKER 189

Helen Rosner writes about how restaurants like the Spotted Pig have responded to sexual-misconduct scandals by promoting women to leadership roles to clean up the messes men have made.

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•   10-18 Slashdot 144

An anonymous reader quotes a report from Bloomberg: Apple on Wednesday began allowing users in the U.S. to download a copy of all of the data that they have stored with the company from a single online portal. U.S. users will be able to download data such as all of their address book contacts, calendar appointments, music streaming preferences and details about past Apple product repairs. Previously, customers could get their data by contacting Apple directly. In May, when Apple first launched the online privacy portal, it only allowed U.S. users to either correct their data or delete their Apple accounts.

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•   10-18 Hacker News 190

Dandelion Seeds Fly Using ‘Impossible’ Method Never Before Seen in Nature

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•   10-18 Hacker News 201

Cambly (YC W14) is hiring engineers who are passionate about language education

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•   10-18 Ars Technica 224

Part two of a chat with Sam Harris takes us on an MDMA-fueled journey to enlightenment.

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•   10-18 Hacker News 211

A university IT guy plans to bid in two government airwave auctions

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•   10-18 Hacker News 113

Are Black Holes Actually Dark Energy Stars?

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•   10-18 MIT Technology 211

As climate dangers rise, researchers are testing new ways of communicating clearly about uncertainty.

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•   10-18 Slashdot 107

People's Facebook posts might predict whether they are suffering from depression, researchers reported this week. From a report: The researchers found that the words people used seemed to indicate whether they would later be diagnosed with depression. The findings offer a way to flag people who may be in need of help, but they also raise important questions about people's health privacy, the team reported in the Proceedings of the National Academy of Sciences. People who were later clinically diagnosed with depression used more "I" language, according to Johannes Eichstaedt of the University of Pennsylvania and his colleagues. They also used more words reflecting loneliness, sadness and hostility. "We observed that users who ultimately had a diagnosis of depression used more first-person singular pronouns, suggesting a preoccupation with the self," they wrote. That is an indicator of depression in some people. The team recruited 683 people who visited an emergency room for their study

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• Three state treasurers and a top official from New York have joined a shareholders' motion to install an independent chairman at Facebook, claiming the move would improve governance and accountability. [...] The move comes as Facebook was presented with a new legal challenge. The technology company has been accused of misleading advertisers by inflating the viewing figures for videos on its site. A group of US advertisers launched a fraud claim against the social media giant on Tuesday, stating that it had overstated the average viewing time of advertising videos on the site by between 100 and 900pc before reporting them in 2016. All tech companies are pretty terrible as far as companies go, but Facebook really seems to be going out of its way to lead the pack. As far as I'm concerned, we shut it down. Would anyone really miss it?

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•   10-18 Slashdot 137

An anonymous reader shares a report: Amazon cracked down on fake reviews two years ago by prohibiting shoppers from getting free products directly from merchants in exchange for writing reviews. It was a major turning point for the world's largest online retailer, which had previously seen "incentivized reviews" as a key way for consumers to discover new products. Amazon changed course because it realized some merchants were using such reviews to game its search algorithm, undermining faith in the customer feedback that helps drive e-commerce. Amazon instead used its "Vine" program, in which Amazon serves as a middleman between prolific Amazon reviewers and vendors eager for exposure. Amazon would still allow freebies in exchange for feedback so long as there was no direct contact between its retail partners and reviewers, theoretically lessening the chance of quid-pro-quo. Amazon would select shoppers eligible for the program, and Amazon vendors would pay a fee and provide free produc

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•   10-18 Ars Technica 177

Google announces a number of Android licensing changes in response to the EC's ruling.

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•   10-18 Hacker News 259

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• OpenSourceAllTheWay writes: There are many fantastic open-source tools out there for everything from scanning documents to making interactive music to creating 3D assets for games. Many of these tools have an Achilles heel though -- while the code quality is great and the tool is fully functional, the user interface (UI) and user experience (UX) are typically significantly inferior to what you get in competing commercial tools. In an nutshell, with open source, the code is great, the tool is free, there is no DRM/activation/telemetry bullshit involved in using the tool, but you very often get a weak UI/UX with the tool that -- unfortunately -- ultimately makes the tool far less of a joy to use daily than should be the case. A prime example would be the FOSS 3D tool Blender, which is great technically, but ultimately flops on its face because of a poorly designed UI that is a decade behind commercial 3D software. So here is the question: should open-source developer teams for larger FOS

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• Essential Products, a startup founded in 2015 by Android creator Andy Rubin, was started to create a smartphone with high-end design features that wasn't associated with a particular operating-system maker. Unfortunately, reaching that goal has been harder than anticipated as the company has laid off about 30 percent of its staff. Fortune reports: Cuts were particularly deep in hardware and marketing. The company's website indicates it has about 120 employees. A company spokesperson didn't confirm the extent of layoffs, but said that the decision was difficult for the firm to make and, "We are confident that our sharpened product focus will help us deliver a truly game changing consumer product." The firm was Rubin's first startup after leaving Google in 2014, which had acquired his co-founded firm, Android, in 2005. Essential's first phone came out in August 2017, a few weeks later than initially promised. It received mixed reviews, with most critics citing its lower quality and missi

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•   10-18 Slashdot 117

An anonymous reader shares a report: Sellers have been modifying lower end NVIDIA graphics cards and selling them more powerful cards online. In a recent version of the GPU-Z graphics card information utility, TechPowerUp has added the ability to now detect these fake NVIDIA cards. This new feature allows buyers of cards to detect if the card is actually a relabled NVIDIA G84, G86, G92, G94, G96, GT215, GT216, GT218, GF108, GF106, GF114, GF116, GF119, or GK106 GPU by displaying an exclamation point where the NVIDIA logo would normally appear and also prepends the string "[FAKE]" before the card's name.

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• Millions of dollars in funding and billions of dollars in valuations have made scooters the next big thing since the last big thing. From a report: When Michael Ramsey, an analyst for technology research firm Gartner, started in February to put together his 2018 "hype cycle" report for the future of transportation, he had plenty of topics to choose from: electric vehicles, flying cars, 5G, blockchain, and, of course, autonomous vehicles. But one type of transportation is conspicuously absent from the results of the report: electric scooters. "At the time, outside of California, these scooters were really not that common," Ramsey said. "That's how much has happened." As for autonomous vehicles, which have enjoyed years of hype as the next big thing, Ramsey labeled them sliding into "the trough of disillusionment," which Ramsey described as "when expectations don't meet the truth." In a matter of months, electric scooter startups have gone from tech oddity to global phenomenon. In some c

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• At its annual Adobe Max conference, Adobe announced plans to bring a complete version of Photoshop to the iPad in 2019. Photoshop CC for iPad will feature a revamped interface designed specifically for a touch experience, but it will bring the power and functionality people are accustomed to on the desktop. This is the real, full photoshop - the same codebase as the regular Photoshop, but running on the iPad with a touch UI. The Verge's Dami Lee and artist colleagues at The Verge got to test this new version of Photoshop, and they are very clear to stress that the biggest news here isn't even having the "real" Photoshop on the iPad, but the plans Adobe has for the PSD file format. But the biggest change of all is a total rethinking of the classic .psd file for the cloud, which will turn using Photoshop into something much more like Google Docs. Photoshop for the iPad is a big deal, but Cloud PSD is the change that will let Adobe bring Photoshop everywhere. This does se

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• In this work, we show that along a particular choice of Hermitian curvature flow, the non-positivity of Chern-Ricci curvature will be preserved if the initial metric has non-positive bisectional curvature. As a corollary, we show that the canonical line bundle of a compact Hermitian manifold with nonpositive bisectional curvature and quasi-negative Chern-Ricci curvature is ample.

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• A tetravalent $2$-arc-transitive graph of order $728$ is either the known $7$-arc-transitive incidence graph of the classical generalized hexagon $GH(3,3)$ or a normal cover of a $2$-transitive graph of order $182$ denoted $A[182,1]$ or $A[182,2]$ in the $2009$ list of Poto\v{c}nik.

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• Let $R=K[x_1,...,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I$ be a monomial ideal generated in degree $d$. Bandari and Herzog conjectured that a monomial ideal $I$ is polymatroidal if and only if all its monomial localizations have a linear resolution. In this paper we give an affirmative answer to the conjecture in the following cases: $(i)$ ${\rm height}(I)=n-1$; $(ii)$ $I$ contains at least $n-3$ pure powers of the variables $x_1^d,...,x_{n-3}^d$; $(iii)$ $I$ is a monomial ideal in at most four variables.

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• A sequence is called $r$-sparse if every contiguous subsequence of length $r$ has no repeated letters. A $DS(n, s)$-sequence is a $2$-sparse sequence with $n$ distinct letters that avoids alternations of length $s+2$. Pettie and Wellman (2018) asked whether there exist $r$-sparse $DS(n, s)$-sequences of length $\Omega(s n^{2})$ for $s \geq n$ and $r > 2$, which would generalize a result of Roselle and Stanton (1971) for the case $r = 2$. We construct $r$-sparse $DS(n, s)$-sequences of length $\Omega(s n^{2})$ for $s \geq n$ and $r > 2$. Our construction uses linear hypergraph edge-coloring bounds. We also use the construction to generalize a result of Pettie and Wellman by proving that if $s = \Omega(n^{1/t} (t-1)!)$, then there are $r$-sparse $DS(n, s)$-sequences of length $\Omega(n^{2} s / (t-1)!)$ for all $r \geq 2$. In addition, we find related results about the lengths of sequences avoiding $(r, s)$-formations.

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• We study the continuous-time evolution of the recombination equation of population genetics. This evolution is given by a differential equation that acts on a product probability space, and its solution can be described by a Markov chain on a set of partitions that converges to the finest partition. We study an explicit form of the law of this process by using a family of trees. We also describe the geometric decay rate to the finest partition and the quasi-stationary behavior of the Markov chain when conditioned on the event that the chain does not hit the limit.

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• We propose an index for gapped quantum lattice systems that conserve a $\mathrm{U}(1)$-charge. This index takes integer values and it is therefore stable under perturbations. Our formulation is general, but we show that the index reduces to (i) an index of projections in the non-interacting case, (ii) the filling factor for translational invariant systems, (iii) the quantum Hall conductance in the two-dimensional setting without any additional symmetry. Example (ii) recovers the Lieb-Schultz-Mattis theorem, (iii) provides a new and short proof of quantization of Hall conductance in interacting many-body systems. Additionally, we provide a new proof of Bloch's theorem on the vanishing of ground state currents.

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•   10-17 Ars Technica 220

DOJ says Colton Grubbs "has no respect for the law."

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• In this work we present an algorithm to construct sparse-paving matroids over finite set $S$. From this algorithm we derive some useful bounds on the cardinality of the set of circuits of any Sparse-Paving matroids which allow us to prove in a simple way an asymptotic relation between the class of Sparse-paving matroids and the whole class of matroids. Additionally we introduce a matrix based method which render an explicit partition of the $r$-subsets of $S$, $\binom{S}{r}=\sqcup_{i=1}^{\gamma }\mathcal{U}_{i}$ such that each $\mathcal{U}_{i}$ defines a sparse-paving matroid of rank $r$.

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• In this paper we study the contribution of monopole bubbling to the expectation value of supersymmetric 't Hooft defects in Lagrangian theories of class $\mathcal{S}$ on $\mathbb{R}^3\times S^1$. This can be understood as the Witten index of an SQM living on the world volume of the 't Hooft defect that couples to the bulk 4D theory. The computation of this Witten index has many subtleties originating from a continuous spectrum of scattering states along the non-compact vacuum branches. We find that even after properly dealing with the spectral asymmetry, the standard localization result for the 't Hooft defect does not agree with the result obtained from the AGT correspondence. In this paper we will explicitly show that one must correct the localization result by adding an extra term to the standard Jeffrey-Kirwan residue formula. This extra term accounts for the contribution of ground states localized along the non-compact branches. This extra term restores both the expected symmetry

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•   10-17 Ars Technica 243

Almost everyone has now migrated to TLS 1.2, and a few have moved to TLS 1.3.

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• For families of smooth complex projective varieties we show that normal functions arising from algebraically trivial cycle classes are algebraic, and defined over the field of definition of the family. As a consequence, we prove a conjecture of Charles and Kerr-Pearlstein, that zero loci of normal functions arising from algebraically trivial cycle classes are algebraic, and defined over the field of definition of the family. In particular, this gives a short proof of a special, algebraically motivated case of a result of Saito, Brosnan-Pearlstein, and Schnell, conjectured by Green-Griffiths, on zero loci of admissible normal functions.

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• Recently, Misanantenaina and Wagner characterized the set of induced $N$-free and bowtie-free posets as a certain class of recursively defined subposets which they term "$\mathcal{V}$-posets". Here we offer a new characterization of $\mathcal{V}$-posets by introducing a property we refer to as \emph{autonomy}. A poset $\mathcal{P}$ is said to be \emph{autonomous} if there exists a directed acyclic graph $D$ (with adjacency matrix $U$) whose transitive closure is $\mathcal{P}$, with the property that any total ordering of the vertices of $D$ so that Gaussian elimination of $U^TU$ proceeds without row swaps is a linear extension of $\mathcal{P}$. Autonomous posets arise from the theory of pressing sequences in graphs, a problem with origins in phylogenetics. The pressing sequences of a graph can be partitioned into families corresponding to posets; because of the interest in enumerating pressing sequences, we investigate when this partition has only one block, that is, when the pressing

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• In this paper, we prove a new cohomology theory that is an invariant of a planar trivalent graph with a given perfect matching. This bigraded cohomology theory appears to be very powerful: the graded Euler characteristic of the cohomology is a one variable polynomial (called the 2-factor polynomial) that, if nonzero when evaluated at one, implies that the perfect matching is even. This polynomial can be used to construct a polynomial invariant of the graph called the even matching polynomial. We conjecture that the even matching polynomial is positive when evaluated at one for all bridgeless planar trivalent graphs. This conjecture, if true, implies the existence of an even perfect matching for the graph, and thus the trivalent planar graph is 3-edge-colorable. This is equivalent to the four color theorem---a famous conjecture in mathematics that was proven using a computer program in 1970s. While these polynomial invariants may not have enough strength as invariants to prove such a co

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• Under the assumption that sequences of graphs equipped with resistances, associated measures, walks and local times converge in a suitable Gromov-Hausdorff topology, we establish asymptotic bounds on the distribution of the $\varepsilon$-blanket times of the random walks in the sequence. The precise nature of these bounds ensures convergence of the $\varepsilon$-blanket times of the random walks if the $\varepsilon$-blanket time of the limiting diffusion is continuous with probability one at $\varepsilon$. This result enables us to prove annealed convergence in various examples of critical random graphs, including critical Galton-Watson trees, the Erd\H{o}s-R\'enyi random graph in the critical window and the configuration model in the scaling critical window. We highlight that proving continuity of the $\varepsilon$-blanket time of the limiting diffusion relies on the scale invariance of a finite measure that gives rise to realizations of the limiting compact random metric space, and t

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• Let $\{T(t)\}_{t\ge 0}$ be a $C_0$-semigroup on a separable Hilbert space $H$. We characterize that $T(t)$ is an $m$-isometry for every $t$ in terms that the mapping $t\in \Bbb R^+ \rightarrow \|T(t)x\|^2$ is a polynomial of degree less than $m$ for each $x\in H$. This fact is used to study $m$-isometric right translation semigroup on weighted $L^p$-spaces. We characterize the above property in terms of conditions on the infinitesimal generator operator or in terms of the cogenerator operator of $\{ T(t)\}_{t\geq 0}$. Moreover, we prove that a non-unitary $2$-isometry on a Hilbert space satisfying the kernel condition, that is, $$T^*T(KerT^*)\subset KerT^*\;,$$ then $T$ can be embedded into a $C_0$-semigroup if and only if $dim (KerT^*)=\infty$.

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•   10-17 Hacker News 109

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•   10-17 Hacker News 207

American Mercenaries Were Hired to Assassinate Politicians in the Middle East

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•   10-17 Ars Technica 201

Verizon network still in bad shape—FCC demands refunds and will investigate.

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• It is known that fuzzy set theory can be viewed as taking place within a topos. There are several equivalent ways to construct this topos, one is as the topos of \'{e}tal\'{e} spaces over the topological space $Y=[0,1)$ with lower topology. In this topos, the fuzzy subsets of a set $X$ are the subobjects of the constant \'{e}tal\'{e} $X\times Y$ where $X$ has the discrete topology. Here we show that the type-2 fuzzy truth value algebra is isomorphic to the complex algebra formed from the subobjects of the constant relational \'{e}tal\'{e} given by the type-1 fuzzy truth value algebra $\mathfrak{I}=([0,1],\wedge,\vee,\neg,0,1)$. More generally, we show that if $L$ is the lattice of open sets of a topological space $Y$ and $\mathfrak{X}$ is a relational structure, then the convolution algebra $L^\mathfrak{X}$ is isomorphic to the complex algebra formed from the subobjects of the constant relational \'{e}tal\'{e} given by $\mathfrak{X}$ in the topos of \'{e}tal\'{e} spaces over $Y$.

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•   10-17 Slashdot 158

Facebook is developing hardware for the TV, news outlet Cheddar reported Tuesday. From the report: The world's largest social network is building a camera-equipped device that sits atop a TV and allows video calling along with entertainment services like Facebook's YouTube competitor, according to people familiar with the matter. The project, internally codenamed "Ripley," uses the same core technology as Facebook's recently announced Portal video chat device for the home. Portal begins shipping next month and uses A.I. to automatically detect and follow people as they move throughout the frame during a video call. Facebook currently plans to announce project Ripley in the spring of 2019, according to a person with direct knowledge of the project. But the device is still in development and the date could be changed.

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• This paper presents a modified quasi-reversibility method for computing the exponentially unstable solution of a nonlocal terminal-boundary value parabolic problem with noisy data. Based on data measurements, we perturb the problem by the so-called filter regularized operator to design an approximate problem. Different from recently developed approaches that consist in the conventional spectral methods, we analyze this new approximation in a variational framework, where the finite element method can be applied. To see the whole skeleton of this method, our main results lie in the analysis of a semi-linear case and we discuss some generalizations where this analysis can be adapted. As is omnipresent in many physical processes, there is likely a myriad of models derived from this simpler case, such as source localization problems for brain tumors and heat conduction problems with nonlinear sinks in nuclear science. With respect to each noise level, we benefit from the Faedo-Galerkin meth

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• This article is based on a talk at the RIEMain in Contact conference in Cagliari, Italy in honor of the 78th birthday of David Blair one of the founders of modern Riemannian contact geometry. The present article is a survey of a special type of Riemannian contact structure known as Sasakian geometry. An ultimate goal of this survey is to understand the moduli of classes of Sasakian structures as well as the moduli of extremal and constant scalar curvature Sasaki metrics, and in particular the moduli of Sasaki-Einstein metrics.

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•   10-17 Hacker News 316

A new python HTTP service framework from the maker of requests and pipenv

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•   10-17 Ars Technica 256

Neuroscientist Sam Harris discusses intellectualism, politics, and Ben Affleck.

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• The spectral properties of two special classes of Jacobi operators are studied. For the first class represented by the $2M$-dimensional real Jacobi matrices whose entries are symmetric with respect to the secondary diagonal, a new polynomial identity relating the eigenvalues of such matrices with their matrix { entries} is obtained. In the limit $M\to\infty$ this identity induces some requirements, which should satisfy the scattering data of the resulting infinite-dimensional Jacobi operator in the half-line, which super- and sub-diagonal matrix elements are equal to -1. We obtain such requirements in the simplest case of the discrete Schr\"odinger operator acting in ${l}^2( \mathbb{N})$, which does not have bound and semi-bound states, and which potential has a compact support.

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• The performance of all-optical dual-hop relayed free-space optical communication systems is analytically studied and evaluated. We consider the case when the total received signal undergoes turbulence-induced channel fading, modeled by the versatile mixture-Gamma distribution. Also, the misalignment-induced fading due to the presence of pointing errors is jointly considered in the enclosed analysis. The performance of both amplify-and-forward and decode-and-forward relaying transmission is studied, when heterodyne detection is applied. New closed-form expressions are derived regarding some key performance metrics of the considered system; namely, the system outage probability and average bit-error rate.

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• In this paper, we extend a class of globally convergent evolution strategies to handle general constrained optimization problems. The proposed framework handles relaxable constraints using a merit function approach combined with a specific restoration procedure. The unrelaxable constraints in our framework, when present, are treated either by using the extreme barrier function or through a projection approach. The introduced extension guaranties to the regarded class of evolution strategies global convergence properties for first order stationary constraints. Preliminary numerical experiments are carried out on a set of known test problems as well as on a multidisciplinary design optimization problem

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• In many contemporary optimization problems, such as hyperparameter tuning for deep learning architectures, it is computationally challenging or even infeasible to evaluate an entire function or its derivatives. This necessitates the use of stochastic algorithms that sample problem data, which can jeopardize the guarantees classically obtained through globalization techniques via a trust region or a line search. Using subsampled function values is particularly challenging for the latter strategy, that relies upon multiple evaluations. On top of that all, there has been an increasing interest for nonconvex formulations of data-related problems. For such instances, one aims at developing methods that converge to second-order stationary points, which is particularly delicate to ensure when one only accesses subsampled approximations of the objective and its derivatives. This paper contributes to this rapidly expanding field by presenting a stochastic algorithm based on negative curvature a

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• We prove existence, uniqueness and regularity of solutions of nonlocal heat equations associated to anisotropic stable diffusion operators. The main features are that the right-hand side has very few regularity and that the spectral measure can be singular in some directions. The proofs require having good enough estimates for the corresponding heat kernels and their derivatives.

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• We exhibit a new construction of edge-regular graphs with regular cliques that are not strongly regular. The infinite family of graphs resulting from this construction includes an edge-regular graph with parameters $(24,8,2)$. We also show that edge-regular graphs with $1$-regular cliques that are not strongly regular must have at least $24$ vertices.

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• We look at the number $L(n)$ of $O$-sequences of length $n$. This interesting and naturally-defined sequence $L(n)$ was first investigated in a recent paper by commutative algebraists Enkosky and Stone, inspired by Huneke. In this note, we significantly improve both of their upper and lower bounds, by means of a very short partition-theoretic argument. In particular, it turns out that, for suitable positive constants $c_1$ and $c_2$ and all $n\ge 1$, $$e^{c_1\sqrt{n}}\le L(n)\le e^{c_2\sqrt{n}\log n}.$$ It remains an open problem to determine an exact asymptotic estimate for $L(n)$.

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• Emerging applications of sensor networks for detection sometimes suggest that classical problems ought be revisited under new assumptions. This is the case of binary hypothesis testing with independent - but not necessarily identically distributed - observations under the two hypotheses, a formalism so orthodox that it is used as an opening example in many detection classes. However, let us insert a new element, and address an issue perhaps with impact on strategies to deal with "big data" applications: What would happen if the structure were streamlined such that data flowed freely throughout the system without provenance? How much information (for detection) is contained in the sample values, and how much in their labels? How should decision-making proceed in this case? The theoretical contribution of this work is to answer these questions by establishing the fundamental limits, in terms of error exponents, of the aforementioned binary hypothesis test with unlabeled observations draw

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• The set of all perfect matchings of a plane (weakly) elementary bipartite graph equipped with a partial order is a poset, moreover the poset is a finite distributive lattice and its Hasse diagram is isomorphic to $Z$-transformation directed graph of the graph. A finite distributive lattice is matchable if its Hasse diagram is isomorphic to a $Z$-transformation directed graph of a plane weakly elementary bipartite graph, otherwise non-matchable. We introduce the meet-irreducible cell with respect to a perfect matching of a plane (weakly) elementary bipartite graph and give its equivalent characterizations. Using these, we extend a result on non-matchable distributive lattices, and obtain a class of new non-matchable distributive lattices.

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• The characterization of local regularity is a fundamental issue in signal and image processing, since it contains relevant information about the underlying systems. The 2-microlocal frontier, a monotone concave downward curve in $\mathbb {R}^2$, provides a complete and profound classification of pointwise singularity. In \cite{Meyer1998}, \cite{GuiJaffardLevy1998} and \cite{LevySeuret2004} the authors show the following: given a monotone concave downward curve in the plane it is possible to exhibit one function (or distribution) such that its 2-microlocal frontier al $x_0$ is the given curve. In this work we are able to unify the previous results, by obtaining a large class of functions (or distributions), that includes the three examples mentioned above, for which the 2-microlocal frontier is the given curve. The three examples above are in this class. Further, if the curve is a line, we characterize all the functions whose 2-microlocal frontier at $x_0$ is the given line.

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• The Newton strata of a reductive $p$-adic group are introduced in \cite{Newton} and play some role in the representation theory of $p$-adic groups. In this paper, we give a geometric interpretation of the Newton strata.

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• We prove a blow-up criterion in terms of an $L_2$-bound of the curvature for solutions to the curve diffusion flow if the maximal time of existence is finite. In our setting, we consider an evolving family of curves driven by curve diffusion flow, which has free boundary points supported on a line. The evolving curve has fixed contact angle $\alpha \in (0, \pi)$ with that line and satisfies a no-flux condition. The proof is led by contradiction: A compactness argument combined with the short time existence result enables us to extend the flow, which contradicts the maximality of the solution.

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• # question_answer The r.m.s. velocity of a gas at a certain temperature is $\sqrt{2}$ times than that of the oxygen molecules at that temperature. The gas can be:- A)  ${{H}_{2}}$                B)  HeC)  $C{{H}_{4}}$               D)  $S{{O}_{2}}$

${{V}_{rms}}\propto \sqrt{\frac{T}{{{M}_{\infty }}}}$ ${{V}_{rms}}=\sqrt{2}\,{{({{V}_{rms}})}_{{{O}_{2}}}}$ $\sqrt{\frac{T}{{{M}_{\infty }}}}=\sqrt{2}\,\sqrt{\frac{T}{32}}$ ${{M}_{w}}=16\,gm$ so gas is $C{{H}_{4}}$

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# Careers

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# Confusion about the usage of points vs. vectors

As far as definitions go, understand the difference between a vector and a point. A vector can be translated and still be the same vector, whereas a point is fixed. But I would like some clarification on the usage of vectors and the usage of points, because it seems like in many cases they are used interchangeably. For example:

1. It has always been my understanding that addition is not defined for two points. But in this question, two points are being added together in this equation:

$\overline{C}= \lbrace tx+(1-t)x': x\in C, x' \in C', t\in [0,1]\rbrace$

2. Sometimes $\mathbb{R}^n$ is used to denote the set of $n$ dimensional vectors, and sometimes it denotes the set of points in $n$ dimensional space.
3. In vector calculus, it is often said that a function with multiple inputs takes a vector as an input, but I have rarely seen a function written as $f(\vec{v})$. Even though I understand that what it means to say is, the vector that originates at the origin, to me, it doesn't seem entirely correct to say that the input is a vector without explicitly saying that the vector's tail is at the origin.
Can anyone clarify these points of confusion? Are point and vector interchangeable in these cases? Also, is there a notation for "converting" one to the other? E.g. how to "convert" $\langle{x,y,z}\rangle$ to $(x,y,z)$ or vice versa?

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I'll try to convince you that they are geometrically quite obviously different, but when it comes to naming them, the begin to look alike :)

Geometrically you probably already have a good picture of what a point is: it's just the primitive notion of a point you have in geometry. That is, a single dimensionless location in space.

A vector should be thought of as having two qualities: a ray that has direction and magnitude. In basic vector algebra in $\Bbb R^n$, we learn that such a ray can slide all around $\Bbb R^n$, and as long as you aren't changing the direction or the length of the ray, then it is still the same vector.

Now when it comes to naming these two things, they start to look alike! With Cartesian coordinates, points in $\Bbb R^n$ are labeled by their projections to the axes, and that creates a list of real numbers. Similarly, when we go about naming vectors, we have this convention of sliding the vector so that it is being emitted from the origin, and then we check to see what point is on its arrowhead. The vector is named after this point.

So in both cases, a similar list of real numbers is used to identify the object. Since this is the case, it's common to just start referring to any ordered $n$-tuple of things from a field (like $\Bbb R$) as a "vector," even if we aren't thinking of it as a ray in that application.

One example is that of vector fields. Since these are functions of position, the inputs they take are points of $\Bbb R^n$ (which look like an ordered $n$-tuple). The outputs are vectors (which again look like an ordered $n$-tuple), but we are interpreting these as the vectors they represent, slid over from the origin to the point we're at.

You can, of course, really have vector inputs! For instance, the length of a vector in $\Bbb R^n$ creates a function from vectors into $\Bbb R$. Of course, the same function could be reinterpreted as the distance-to-zero function on points of $\Bbb R^n$.

So, the difference is all in how you are interpreting that particular list of numbers.

For #1 in your post, you are probably thinking of it as the line segment between points $x$ and $x'$. The addition that's going on is vector addition though. Drawing the vectors that $x$ and $x'$ represent, you see you have two vectors extending from the orign to these two points. For any two vectors $v,w$, $v-w$ yields the vector which fits between the two tips of $w$ and $v$, and points to the tip of $v$. So, you can see that $x-x'$ has the point $x$ on its tip.

What does the $t$ contribute? If you multiply out $xt+(1-t)x'=x'+t(x-x')$, you can see that the vector $x-x'$ is being scaled by $t$ to something shorter, and then is being concatenated onto the tip of $x'$. The tip of this arrow gives another point on the segment. Ranging over all $t$ between 0 and 1, you get vectors pointing to all points on that segment.

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1. It is indeed true that addition is not defined for points. But subtraction can be defined (for flat spaces, at least), and then gives a vector: If $A$ and $B$ are points, then $v=B-A$ is the vector pointing from $A$ to $B$. Also you then can add a vector to a point, giving another point (namely the point where the vector ends if you start it at the given point). So for the vector $v$ from above, $A+v=B$.

Now, the points in the straight line segment from $A$ to $B$ have the form $A+t(B-A)$ with $t\in[0,1]$. Now if you ignore that you're dealing with different types and apply the normal arithmetic laws, you can expand $t(B-A)=tB-tA$ and then factor $A-tA=(1-t)A$. As written, it doesn't make sense with points and vectors as defined above.

However there's also another way to see it: You can identify each point with the vector fro the origin to that point. As soon as you do that, you've got only vectors, and thus can apply all vector operations. The expression you've seen does make immediate sense in that case, because you've basically dispensed with points.

Finally, there's yet another way to think about it, which combines the advantages of both: Add an additional number to the tuples, which has the following meaning:

• If it is 1, you've got a point.
• If it is 0, you've got a vector.
• Any other value is invalid.

Then you can allow any operation where you get a valid (i.e. vector or point) result. For example, in 2 dimensions, you'd have $A=(a_1,a_2,1)$ (the final $1$ marking it as point), and $B=(b_1,b_2,1)$ (again, a point). Now $A+B$ would not be allowed (last component would be $2$), but $A-B$ is allowed and gives a vector (last component $0$). And the formula in question gives as last component $(1-t)+t=1$, so it is valid and a vector.

Note however that all those things only work on flat spaces with linear coordinates. On non-flat spaces or general coordinates you'll not be able to do such an identification (or even to define the vector from one point to another).

2. $\mathbb{R}^n$ is the set of n-tuples of real numbers, not more and not less. Now together with an operation $+:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}^n$ and an operation $\cdot:\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}^n$ which fulfil certain relations, you get the vector space $(\mathbb{R}^n,+,\cdot)$. If you supply other operations, you get something different. Now it is quite common to only write the set when you mean the complete algebraic structure, meaning that the rest of the structure is implied. So if you speak about the vector space $\mathbb{R}^n$ you actually mean "the vector space you get if you use $\mathbb{R}^n$ together with the canonical choice of component-wise addition and scalar multiplication". If you speak about the points as $\mathbb{R}^n$, you of course normally don't consider the vector space structure included.

3. A function may operate on vectors, on points, or simply on $n-tuples$. All those concepts make sense. A function on points could e.g. be a temperature map: Each point is mapped to the temperature at that point. Here, it would not make sense to consider it as function of a vector (except if identifying points with vectors, see above). A function on vectors could e.g. be the function which tells you the length of the vector. That function makes sense only on vectors, not on points. And of course any function on $n$ arguments operates on $n-tuples$. If you describe your points or vectors with $n-tuples$, you'll give the function on those argument as $n-tuples$ as well. On the reverse, since derivations are linear, and linear spaces are vector spaces (indeed, both are just two names for the same thing), whenever you do derivations on a multi-argument function, you can interpret that argument tuple as a vector.

You can "convert" a point to a vector by subtracting the origin (which is a point), and you can "convert" a vector to a point by adding it to the origin.

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The most important difference between a vector and a point is that vectors can be added, while points cannot. Instead, you can take two points and "subtract" them to get the vector between them, and add a point to a vector and get a new point. There's really no notion of a vector's "tail": in some sense, a vector just represents the difference between it's head and tail.

When we talk about a (real) vector, we usually mean an element of $\Bbb{R}^n$, which we give componentwise addition and multiplication. A point is an element of an affine space over $\Bbb{R}^n$. However, if we impose coordinates on that affine space, which is equivalent to picking a single point of the affine space and calling it the origin, we get a correspondence between points and vectors. So in your question 1, those points can be added, because they are in a coordinate system, and. Similarly, in question 2, since affine spaces are so closely related to vector spaces, we can use the same notation for both.

Regarding your question three, the notation $\vec{v}$ is really just helpful to avoid confusion, but, in most cases where confusion between vectors and scalars is uncommon, is unnecessary.

Note both that a vector can be defined more generally as an element of a vector space, which is a much more general notion than just $\Bbb{R}^n$, and that "point" can refer to elements of a much broader variety of spaces, like topological spaces or affine varieties.

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Evaluate where $f(x) = 3 x^2 + 5$. Enter I for $\infty$, -I for $-\infty$, and DNE if the limit does not exist.

Limit =


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# The Phase Rule

 Consider the general case: C components and P phases. At equilibrium all pressures, temperatures, and each chemical potential is constant: \begin{align}  & p_{a}=p_{b}=\ldots =p_{P} \\ & T_{a}=T_{b}=\ldots =T_{P} \\ & \mu _{ia}=\mu _{ib}=\ldots =\mu _{iP};\text{ }i=1,2,\ldots C \\  \end{align} The number of unknowns is the number of intensive variables in each phase: (C+1) intensive variables in each phase: p, T, and (C-1) mole fractions times the number of phases P or $=\left( C+1 \right)P$] The number of equations: $=\left( P-1 \right)\left( C+2 \right)$ Therefore the degrees of freedom are: $F=\left( C+1 \right)P-\left( P-1 \right)\left( C+2 \right)=C+2-P$ $F=C+2-P$]


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# Henry's Law

Written by Jerry Ratzlaff on . Posted in Thermodynamics

## Henry's Law

Henry's Law (one of the Gas Laws), at a constant temperature, the volume of a gas which will dissolve into a solution is directly porportional to the partial pressure of that gas above the solution in equilibrium with the liquid.

### Henry's Law Constant Formila

$$\large{ k_H = \frac{ p_{gas} }{ c_{gas} } }$$

Where:

$$\large{ c_{gas} }$$ = the concentration of the solubility (gas) at a fixed temperature in a partial solution

$$\large{ k_H }$$ = Henry's law constant

$$\large{ p_{gas} }$$ = the partial pressure of the gas


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Energy Change Equation

Tags:
1. Nov 12, 2016

Lil Uzi Vert

1. The problem statement, all variables and given/known data
A person performs 6.1 J of work to lift an object without acceleration to a particular height on Earth. Write an equation describing that energy change and analyze it to determine how different the energy would be required to lift the same object to the same height on the moon where the value of g is 1.622 m/s^(2).

No height is given.
No mass is given.
No displacement is given.
2. Relevant equations
W= F*D
Eg=mgh
P=W/T

3. The attempt at a solution
I tried using the power equation but we only have work as energy (6.1 J) and no time or power.
Using W=FD, i was able to make two equations
1. 6.1= m*9.8*h*D
2. E= m*1.622*h*D
After that i did not know where to go from there.
I'm stuck on this question and have no clue how to create this equation describing the energy change to find the energy required to lift the object to the same height on the moon.

2. Nov 12, 2016

PeroK

You've nearly done this problem without realising it. That said, it isn't clear to me exactly what they are looking for. But, perhaps, any correct answer will do.

One mistake, however, is why do you have $h*D$ in your equations?

Also, if you are going to use any numbers like $9.8$ you are going to have to put units on everything. Alternatively, use $g$ or $g_e$ for the Earth's surface gravity and leave the numbers out of the equations.

3. Nov 12, 2016

Staff: Mentor

The h is the D in these equations, so just use D for the height change.

Oops, beaten by PeroK again!

4. Nov 12, 2016

Lil Uzi Vert

I am having trouble with these two equations because I do not know how to create them into what the question is looking for. Can I have a hint?

5. Nov 12, 2016

PeroK

Here's what I would do:

1) I would describe the sort of energy we are talking about.

2 Because we have two scenarios (Earth and moon), I would use $E_1$ for the Earth and $E_2$ for the moon.

3) I would do the equation all in letters and then, separately, state what quantities we know.

So, for the first one I would do:

Let $E_1$ be (you have to describe what energy we are talking about here)

Then:

$E_1 =$ (your equation involving $m, g, h$ etc.)

Where $E_1 = 6.1J$

6. Nov 12, 2016

Staff: Mentor

Hi Lil Uzi Vert, Welcome to Physics Forums!

A handy way to proceed in these sorts of problems is to form ratios from your formula. Best to start with symbols and plug in numbers later. So let:

$M$ be the mass of the object (same for both cases)
$h$ be the height (same for both cases)
$g_e$ be the gravity on Earth
$g_m$ be the gravity on the Moon (that's your 1.622 m/s^2)
$w_e$ be the work done on Earth (that's your 6.1 J)
$w_m$ be the work done on the Moon

Write the expressions (symbolically) for the work done in each case, then form ratios from the two sides of the equations. Solve for your unknown value.

For example, suppose you had a general expression for some quantity: $f = a*x$ and you had one case where you know all the values: $f_1 = a_1*x_1$, but you have another case where the constant $a$ is different and you don't know what $f_2$ will be. Then you first write out the two cases:

$f_1 = a_1 * x_1$
$f_2 = a_2*x_2$

Form ratios:

$\frac{f_1}{f_2} = \frac{a_1 * x_1}{a_2 * x_2}$

If you can fill in values for all but one of the variables then you can solve for that variable.

7. Nov 12, 2016

Lil Uzi Vert

Thanks, this really clarified it for me.


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115 CFA
File:
115.030.10.06 Economics - Reading 12 - 6. Marginal Revenue, Marginal Cost, and Profit Maximization

# 6. Marginal Revenue, Marginal Cost, and Profit Maximization

## d. describe the phenomenon of diminishing marginal returns;

What is Total Revenue and what is the formula?
- Total revenue (TR) is the sum of individual units sold multiplied by their respective prices
- $$\sum P_i \times Q_i$$

What is the formula for Average Revenue (AR)?
$$\frac{total revenue}{quantity} = \frac{TR}{Q}$$

What is marginal revenue and what is the formula?
- Marginal revenue (MR) is the change in revenue from selling one extra unit of output
- $$MR = \frac{\Delta T R}{\Delta Q}$$

Why and how does reducing price (and thus increasing sales) impact a firm's marginal revenue?
The lower cost will cause more sales, but since each unit is at a lower price, the marginal revenue (the change in revenue from selling one extra unit of output) will always be less than the price of the good.

What is Total Fixed Cost?
The sum of the costs that do not vary with output. They will be incurred as long as a firm continues in business and the assets have alternative uses. Examples of fixed costs include rent, property taxes and insurance premiums.

What is Average Fixed Cost?
Total fixed cost divided by the number of units produced. It always declines as output increases.

What is Total Variable Cost?
The sum of those costs that rise as output increases. Total variable costs are zero if output is zero. Examples are wages paid to workers and payments for raw materials.

What is Average Variable Cost?
The total variable cost divided by the number of units produced.

What is Average Total Cost?
- Total cost divided by the number of units produced. It is sometimes called per unit cost.
- ATC is high at low levels of output, decreases as output increases (since fixed costs are spread across more units), and then increases as the firm's maximum capacity is approached (since marginal costs increase).

What is Marginal Cost?
The change in total cost required to produce an additional unit of output.

• CFA


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# Generating unit-test data with Julia

## TDD the Easy Way!

Most of the development I do uses TDD and at present I’m writing some new Python classes to do basic 3D Maths which include Mat3, Mat4 and Vec3 and Vec4 classes.

These classes are very similar to the ones in my c++ NGL library and I started to test them using the same Unit test data.

For example the ngl::Mat4 inverse() method has the following test.

TEST(Mat4,inverse)
{
// test verified from wolfram alpha
// 1,0,0,0,0,0.4, -0.4, 0 ,0 , 0.1 , 0.4 0 ,0,0,0,1
ngl::Mat4 test(1,0,0,0,0,2,2,0,0,-0.5,2,0,0,0,0,1);
test=test.inverse();
ngl::Mat4 result(1,0,0,0,0,0.4f,-0.4f,0,0,0.1f,0.4f,0,0,0,0,1);

EXPECT_TRUE(test == result);
}

As you can see from the comments I used wolfram alpha to generate a simple test result then verify by running my test and comparing.

Whilst this works ok, it is only 1 simple case and generating more is time consuming using the web interface, and doesn’t scale well to multiple tests with different data.

I decided I needed another tool to help auto generate the tests for me.

## What to use?

I’m lucky that I have access to quite a bit of software (being an academic), however I wanted to use something FOSS if possible. So I discounted MATLAB and Mathematica immediately.

I considered Octave which I have used before for projects as well as just using Numpy or GLM but in the end I decided I would try Julia as I quite fancied learning something new.

## Julia

Julia is a dynamically typed programming language with a REPL. It’s syntax is similar to C like languages but has the feel of a scripting language. It’s is simple to install and works across Mac, Linux and Windows which are my target operating systems.

At it’s simplest level we can just run the REPL and do basic maths.

julia> 2*3*5
30
julia>

However for me the most useful thing is the built in Matrix types.

julia> zeros((4,4))
4×4 Matrix{Float64}:
0.0  0.0  0.0  0.0
0.0  0.0  0.0  0.0
0.0  0.0  0.0  0.0
0.0  0.0  0.0  0.0
julia>

We can also build a matrix by passing in values using newline to start a new row

julia> a=[1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16]
4×4 Matrix{Int64}:
1   2   3   4
5   6   7   8
9  10  11  12
13  14  15  16

julia> transpose(a)
4×4 transpose(::Matrix{Int64}) with eltype Int64:
1  5   9  13
2  6  10  14
3  7  11  15
4  8  12  16

This works well but what I really need is a more complex matrix of floating point values to do maths with. I investigated the random number generation routines here and found it was easy to generate a random 4x4 matrix using the following code

julia> rand(Float64,(4,4))
4×4 Matrix{Float64}:
0.604374   0.0912643  0.372395  0.203154
0.0834194  0.851173   0.754508  0.835495
0.336754   0.436165   0.255739  0.433446
0.333849   0.0981714  0.439347  0.331527

So using this basic process I can create some random 4x4 matrices and do some maths.

julia> a=rand(Float64,(4,4))
4×4 Matrix{Float64}:
0.155732   0.206663  0.0922702  0.827322
0.338653   0.399044  0.188655   0.168776
0.0293418  0.444229  0.816516   0.0736119
0.22251    0.255415  0.0859794  0.63177

julia> b=rand(Float64,(4,4))
4×4 Matrix{Float64}:
0.96249    0.462971  0.376175  0.908213
0.0793394  0.989267  0.154777  0.873103
0.931258   0.281352  0.293946  0.187418
0.518119   0.144784  0.585945  0.860775

julia> c=a*b
4×4 Matrix{Float64}:
0.680866  0.422288  0.602457  1.05131
0.620743  0.629062  0.343504  0.836611
0.862013  0.693432  0.362938  0.6309
0.64183   0.471351  0.518691  0.985017

## Packages

One of the first issues I noticed was the random matrix generation is within the range 0-1, whilst this is fine for my tests, I felt it would be better to have negative values as well (as these are very common in 3D). This is something that is not supported by the default random generator and needs to use the Distributions package.

Julia has a built in package manager which can be accessed via the ] key.

(@v1.7) pkg> add Distributions

This will install the package and dependencies We can then exit the pkg took by pressing backspace. We can then use it as follows

julia> using Distributions

julia> dist=Uniform(-100,100)
Uniform{Float64}(a=-100.0, b=100.0)

julia> rand(dist,(4,4))
4×4 Matrix{Float64}:
-48.6763  -63.7468    46.7309   45.7332
-64.3383   53.3985    27.7707  -77.1511
11.4589   -2.73745  -59.6267  -51.2212
10.7504  -16.7751   -94.7999   77.6484

## Test Generator

Now I can do the basics I need to design my program. I decided that I needed to generate the following tests using 3x3 and 4x4 matrix values

1. a*b
2. a+b
3. a-b
4. inv(a) the inverse
5. det(a) the determinant

I want to write this data out to a python file as a series of lists of float values that I can then use to construct my tests, something like this

a=[ [.......],[......]]
b=[ [.......],[......]]
a_times_b[ [.......],[......]]

The basic structure is a function called generateTests, this is passed in a few parameters which are generated from the command line of the script. size is the matrix size (3x3 or 4x4 for now), file is the name of the file to write and loopSize is the number of test to generate (the size of the list).

The overall structure of this function is documented below but a few things to mention :-

1. Scope is usually delimited with keyword then end
2. Julia indexes from 1 not 0 (gets me a lot!)
3. ! function names that end with an exclamation mark modify one or more of their arguments by convention
#=
size = Matrix size will be square usually 4 or 3
file = name of file to output to (.py)
loopSize = the number of list elements to generate
=#
function generateTests(size,file,loopSize)
# generate some empty lists for our data
a=Matrix{Float64}[]
b=Matrix{Float64}[]
a_times_b=Matrix{Float64}[]
a_plus_b=Matrix{Float64}[]
a_minus_b=Matrix{Float64}[]
a_inv=Matrix{Float64}[]
a_det=[]
# uniform distribution for rng
dist=Uniform(-100,100)
# generate our data (don't forget we index from 1!)
for i = 1:loopSize
# create our base matrix values a and b
push!(a,rand(dist, size, size))
push!(b,rand(dist, size, size))
# do our calculations and store
push!(a_times_b,a[i]*b[i])
push!(a_plus_b,a[i]+b[i])
push!(a_minus_b,a[i]-b[i])
push!(a_inv,inv(a[i]))
push!(a_det,det(a[i]))
end
# open our file for io
open(file,"w") do io
# println writes a line and \n
println(io,"# file generated by gen_mat4_tests.jl")
# dump our results to the file
writePythonVar(io,"a",a)
writePythonVar(io,"b",b)
writePythonVar(io,"a_times_b",a_times_b)
writePythonVar(io,"a_plus_b",a_plus_b)
writePythonVar(io,"a_minus_b",a_minus_b)
writePythonVar(io,"a_inv",a_inv)
writePythonVar(io,"a_det",a_det)
end # end io
end # end function

The main work is done by the function writePythonVar this will write the data to python lists.

#=
io = file pointer for the python file to write to
array = the array to write to file
=#
function printMatrix(io,array)
print(io,"[")
for i = 1:length(array)
print(io,array[i],",")
end
print(io,"],\n")
end

#=
io = file pointer for the python file to write to
var = the name of the variable to write to the file
data = the array of data to write
=#
function writePythonVar(io,var,data)
println(io,var,"=[")
for i = 1:length(data)
printMatrix(io,data[i])
end
println(io,"]")
end

I have also added command line parsing using the ArgParse module the full code can be seen here with the final data generated in the file mat4Data.py

## Using the files

The files are placed in the same folder as my unit tests and imported as a python module.

import tests.mat4Data as mat4Data  # this is generated from the julia file gen_mat4_tests.jl

A sample test will take the data values and perform a calculation on them and then compare to the result Julia created. The following test if for the mat4*mat4 (note the use of the new python __matmult__ operator)

    def compare_matrix(self, a, b, places=6):
for r, v in zip(a, b):
self.assertAlmostEqual(r, v, places=places)

def test_mat4_times_mat4(self):
for a, b, result in zip(mat4Data.a, mat4Data.b, mat4Data.a_times_b):
m1 = Mat4.from_list(a)
m2 = Mat4.from_list(b)
value = m1 @ m2
self.compare_matrix(value.get_matrix(), result)

## Conclusions

I’m happy with the results of this, it was about a days coding and learning and it’s highlighted a few issues I had with my implementation for the Mat classes. I’m going to add a few more generated values (for example Mat * Vec) and I can now increase and decrease the test amounts when I need to try stress testing etc.

I have considered generating the test data fresh each time by using the unitest fixture and running the julia script each time. This could make the tests more interesting as they could be different each time you run the tests, but could also be problematic when trying to identify issues.

At present I hard code the seed by using Random.seed!(12345) as this helps to get predictable test values, but again this can be removed easily.

Now I have this working for python I may integrate the same data / setups into my NGL library for the C++ and PyNGL tests as well.

Previous


text:
### Elliptic Curves

#### Definition

Let $$K$$ be a field. An elliptic curve $$E$$ is a plane curve defined over the field $$K$$ as follows: $$E(K)=\{y^2=x^3+ax+b : (x,y) \in K^2\} \cup \{\mathcal{O}\}$$ where $$(a,b) \in K^2$$. The point $$\mathcal{O}$$ is called the infinity point of the curve. The set $$E(K)$$ forms an abelian group with identity element $$\mathcal{O}$$.

In addition, we need the curve to have no cusps, self-intersections, or isolated points. Algebraically, this can be defined by the condition $$4a^3+27b^2 \neq 0$$ in the field $$K$$.

The $$j-$$ invariant of an elliptic curve is defined to be $$-1728\frac{4a^3}{4a^3+27b^2}$$. Two elliptic curves are isomorphic to each other if and only if they have the same $$j-$$ invariant value.

The endomorphism ring of $$E$$ is denoted $$End(E)$$. The structure of $$End(E)$$ can be found in Chapter 3.9 of Silverman's book.

For an integer $$n$$, we define $$E[n]=\{(x,y) \in E(K) | n*(x,y)=\mathcal{O}\}$$

Over a field $$\mathbb{F}_p$$, there are two types of curves: Ordinary and Supersingular, based on the set $$E[p]$$. We are interested in studying Supersingular curves, since the isogeny graph on these curves has nice structure and properties.


text:
# A right triangle has sides A, B, and C. Side A is the hypotenuse and side B is also a side of a rectangle. Sides A, C, and the side of the rectangle adjacent to side B have lengths of 10 , 6 , and 3 , respectively. What is the rectangle's area?

Area of the rectangle $= 24 \text{ }$square units

#### Explanation:

compute by Pythagorean Theorem the side b of the right triangle

$b = \sqrt{{a}^{2} - {c}^{2}} = \sqrt{{10}^{2} - {6}^{2}} = 8$

One side of the rectangle is 3 which is adjancent to b=8

So the sides of the rectangle are 3 and 8

Area $= 3 \cdot 8 = 24$


text:
Author Topic: Boot Code Transfer to OS  (Read 8144 times)

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ralphdspam

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Boot Code Transfer to OS
« on: December 23, 2011, 04:23:13 pm »
How does boot code give control to the OS?  Where does it jump to?  Are there any integrity checks?  If so, what are they?
ld a, 0
ld a, a

thepenguin77

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Re: Boot Code Transfer to OS
« Reply #1 on: December 23, 2011, 10:04:01 pm »
A fun question

Well, firstly, I recommend you actually take a look at the boot code yourself. In all honesty, this will allow you to understand it far better than I can explain it. Just start at $4000 and figure out what it's doing. The route from boot to OS is only like 300 instructions. But here's what happens (I'm reading from a disassembly): The boot code... 1. sets up all of the calculator hardware through ports (port (07) gets 80h except for an OS installation where it gets 00h). 2. checks for Stat and Del and does its usual if they are pressed 3. checks (0038h) to make sure it's not FFh 4. checks (0056h) to make sure it's A55Ah, (5Ah, A5h in memory) 5. jumps to 0053h Also, if you are going to send your own OS, (0056h) should be$A5FF when sending. Something gets angry if it's not, but I don't remember what it is.
zStart v1.3.013 9-20-2013
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ralphdspam

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Re: Boot Code Transfer to OS
« Reply #2 on: December 23, 2011, 11:59:18 pm »
1. sets up all of the calculator hardware through ports (port (07) gets 80h except for an OS installation where it gets 00h).
2. checks for Stat and Del and does its usual if they are pressed
3. checks (0038h) to make sure it's not FFh
4. checks (0056h) to make sure it's A55Ah, (5Ah, A5h in memory)
5. jumps to 0053h

Also, if you are going to send your own OS, (0056h) should be $A5FF when sending. Something gets angry if it's not, but I don't remember what it is. Thanks! I'll try that. Ok, so I took a look at the code and made some notes. I only got so far until I got incredibly annoyed at the code. Spoiler For Spoiler: 0002 Check if calc is 83+BE (I have an 84+SE.) 029A Set up hardware timers and memory banking. 8130 waste time... 813D waste moar time... 814C Set up hardware timers 4151 MUST WASTE ALL THE TIME!!!!! 4165 Unlock flash 4173 Abuses SP in a very weird way () 4193 The coding on this is so bad... Please put me out of this misery! 60E1 Set up link port 41D3 Set up LCD 4204 Set up flash and ram execution limits 421A What are port$39 and $4A? 4225 No comment about the programming... 628F AARRGGH!!! You used CPL before!!! DX 423D Check for keypresses (Such as DEL) 4248 Check if 00:0053 is$A55A
FFFF FFUUUU Ok, I give up on this.  The coding on this is the worst I've ever seen... Since worse TI stuff.  uugghh...
« Last Edit: December 24, 2011, 12:01:16 am by ralphdspam »
ld a, 0
ld a, a

thepenguin77

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Re: Boot Code Transfer to OS
« Reply #3 on: December 24, 2011, 09:02:08 am »
Your analysis entertained me, so here is a commented version I just whipped up.

A few notes:
• The boot code starts on page 7F at 4000, not 0000 on page 00 of the OS. (Technically the boot code starts on page 7F at 0000, but I'm not going into that)
• A lot of the useless code you saw was so people can't trick the boot code into unlocking flash for them. When we unlock flash, we have to trick those things into returning to our control with flash unlocked
• And as much as you'd like to think otherwise, TI's code is actually pretty well written
zStart v1.3.013 9-20-2013
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ralphdspam

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Re: Boot Code Transfer to OS
« Reply #4 on: December 25, 2011, 01:47:53 am »
• The boot code starts on page 7F at 4000, not 0000 on page 00 of the OS. (Technically the boot code starts on page 7F at 0000, but I'm not going into that)
I noticed the way the code switches out the pages and jumps up to the \$4000 bank.  It seemed useless at first, but it's actually kind of clever.
Quote
• A lot of the useless code you saw was so people can't trick the boot code into unlocking flash for them. When we unlock flash, we have to trick those things into returning to our control with flash unlocked
• And as much as you'd like to think otherwise, TI's code is actually pretty well written :)
Yeah, maybe I'm just nitpicking.
« Last Edit: December 25, 2011, 01:48:40 am by ralphdspam »
ld a, 0
ld a, a


text:
# [texhax] Resetting \labelsep

Philip Taylor P.Taylor at Rhul.Ac.Uk
Wed Jul 29 10:56:44 CEST 2015

Professor John Simmie wrote:

> I have a very large document in which the table of contents messes up
> after the first 99 \sections, with the section number crashing into
> the section heading
>
> Redefining via \setlength{\labelsep}{36pt} in the preamble does not
> work

It may be reset by \@BeginDocument; try setting it as the first line after

\begin {document}

Philip Taylor

More information about the texhax mailing list


text:
Existence of non-abelian split metacyclic extensions.

Is there any necessary conditions that must be held in order to guarantee the existence of a non-abelian split metacyclic extension? i.e. for which $m,n\in\mathbb{Z}$ there exist a non abelian split extension G $$1\to\mathbb{Z}_m\to G\to\mathbb{Z}_n\to1$$ G must obviously have the form $G=\mathbb{Z}_n\rtimes_\phi\mathbb{Z}_m$, so the question can be formulated as: "For which $m,n\in\mathbb{Z}$ there exist $\phi:\mathbb{Z}_n\to Aut(\mathbb{Z}_m)$ such that $G=\mathbb{Z}_n\rtimes_\phi\mathbb{Z}_m$ is non-abelian?".

• Judging by exact sequence you would write $G=\mathbb Z_n\ltimes_\phi\mathbb Z_m$. – Alex W May 10 '15 at 1:55

1 Answer

Nontrivial homomorphism $\phi:\mathbb Z_n\to Aut(\mathbb Z_m)$ there exist if and only if $\gcd(n,\varphi(m))\neq 1$, where $\varphi$ - Euler function. Indeed, let $p$ be a prime divisor of $\gcd(n,\varphi(m))$, then $p\shortmid|\mathbb Z_n|$ and $p\shortmid|Aut(\mathbb Z_m)|$, since $|Aut(\mathbb Z_m)|=\varphi(m)$. From $p\shortmid|\mathbb Z_n|$ follow, that there exist $A\leq \mathbb Z_n$ such,that $\mathbb Z_n/A\cong\mathbb Z_p$. From $p\shortmid|Aut(\mathbb Z_m)|$ follow, that there exist $B\leq Aut(\mathbb Z_m)$ of order $p$. Thus $\mathbb Z_n/A\cong B\leq Aut(\mathbb Z_m)$, therefore there exist nontrivial homomorphism $\phi$.

Conversely, suppose that there exist nontrivial homomorphism $\phi:\mathbb{Z}_n\to Aut(\mathbb Z_m)$. Then $1\neq \phi(\mathbb Z_n)\leq Aut(\mathbb Z_m)$, hence $1\neq|\phi(\mathbb Z_n)|\shortmid|Aut(\mathbb Z_m)|=\varphi(m)$. From the other side, $|\phi(\mathbb Z_n)|\shortmid|\mathbb Z_n|=n$. So $1\neq |\phi(\mathbb Z_n)|\shortmid\gcd(n,\varphi(m))$.

• Thanks. Do you have any reference for this result? Know about any paper of book dealing with these type of non-abelian extensions? – Jose Paternina May 10 '15 at 1:49
• @JosePaternina Unfortunately, I do not know anything special about extensions, except what is in the standard textbooks on group theory. I do not know what to advise. – Alex W May 10 '15 at 2:09


text:
## Algebra: A Combined Approach (4th Edition)

$10y^{2}-19y-15$
Step 1: A Special Product cannot be applied here. In its place, we will use the FOIL method. Step 2: $(5y+3)(2y-5)=5y(2y)+5y(-5)+3(2y)+3(-5)$ Step 3: $5y(2y)+5y(-5)+3(2y)+3(-5)=10y^{2}-25y+6y-15$ Step 4: $10y^{2}-25y+6y-15=10y^{2}-19y-15$ Step 5: Therefore, the required expression is $10y^{2}-19y-15$


text:
Modulo operation

Jump to navigation Jump to search

In mathematics, modulo operation, represented by the symbol ${\displaystyle {\text{mod}}}$, returns the remainder of an integer division (also known as Euclidean division), where the division of two integers produces a quotient and a remainder.

More specifically, given an integer ${\displaystyle n}$ and a non-zero integer ${\displaystyle d}$, the expression ${\displaystyle n{\text{ mod }}d}$ refers to the remainder of the division ${\displaystyle n\div d}$. Here, ${\displaystyle n}$ is also known as the dividend; ${\displaystyle d}$ is also known as the divisor (or the modulus), and the remainder can be also called the least non-negative residue.[1][2][3]

However, other conventions and definitions of the modulo operation are also possible, since computers and calculators have various ways of storing and representing numbers. Their definition of the modulo operation depends on the programming language and/or the underlying hardware.

References

1. "The Definitive Glossary of Higher Mathematical Jargon: Modulo". Math Vault. 2019-08-01. Retrieved 2020-08-27.
2. Weisstein, Eric W. "Congruence". mathworld.wolfram.com. Retrieved 2020-08-27.
3. Caldwell, Chris. "residue". Prime Glossary. Retrieved August 27, 2020.


text:
# First Map of Coherent Low Frequency Continuum Radiation in the Sky

Martin Fullekrug, Kuang Liang Koh, Zhongjian Liu, Andrey Mezentsev

Research output: Contribution to journalArticle

### Abstract

Lightning discharges and radio transmitters emit low-frequency (\mbox{$\sim$3-300 kHz}) electromagnetic waves with large electric field strengths and stable phases. This phase stability makes it possible to map the source locations of lightning and transmitters in the sky. Electromagnetic waves with smaller electric field strengths generally exhibit a reduced phase stability, caused by numerous simultaneous physical processes that blend into an underlying continuum radiation trapped inside the Earth-ionosphere cavity. It is therefore currently not known whether the source locations of continuum radiation can be determined. Here we show the first map of coherent continuum radiation in the sky above an array of high precision radio receivers. The source locations of the coherent continuum radiation are found at elevation angles \mbox{$\sim30^{\rm o}-60^{\rm o}$} above the horizon. The identified source locations are attributed to intermittent radio transmitters which emit electromagnetic waves with electric field strengths $\sim2$ orders of magnitude below the instrumental noise floor. The results demonstrate that it is possible to simultaneously map the signals from coherent continuum radiation, lightning discharges, and radio transmitters in the sky. This work thereby lays the foundation toward the discovery of many more coherent source locations of low frequency electromagnetic waves in the sky. It is expected that the identified source locations vary with time as a result of the impact of solar variability on the D-region ionosphere. Future studies have therefore the potential to contribute to a novel remote sensing and an improved understanding of the D-region ionosphere, influenced by the near-Earth space environment.
Original language English Radio Science https://doi.org/10.1029/2018RS006705 Published - 21 Dec 2018

### Cite this

First Map of Coherent Low Frequency Continuum Radiation in the Sky. / Fullekrug, Martin; Koh, Kuang Liang; Liu, Zhongjian; Mezentsev, Andrey.

Research output: Contribution to journalArticle

Fullekrug, Martin ; Koh, Kuang Liang ; Liu, Zhongjian ; Mezentsev, Andrey. / First Map of Coherent Low Frequency Continuum Radiation in the Sky. In: Radio Science. 2018.
@article{ff6d1596080c4501a9b862a3f15a90c4,
title = "First Map of Coherent Low Frequency Continuum Radiation in the Sky",
abstract = "Lightning discharges and radio transmitters emit low-frequency (\mbox{$\sim$3-300 kHz}) electromagnetic waves with large electric field strengths and stable phases. This phase stability makes it possible to map the source locations of lightning and transmitters in the sky. Electromagnetic waves with smaller electric field strengths generally exhibit a reduced phase stability, caused by numerous simultaneous physical processes that blend into an underlying continuum radiation trapped inside the Earth-ionosphere cavity. It is therefore currently not known whether the source locations of continuum radiation can be determined. Here we show the first map of coherent continuum radiation in the sky above an array of high precision radio receivers. The source locations of the coherent continuum radiation are found at elevation angles \mbox{$\sim30^{\rm o}-60^{\rm o}$} above the horizon. The identified source locations are attributed to intermittent radio transmitters which emit electromagnetic waves with electric field strengths $\sim2$ orders of magnitude below the instrumental noise floor. The results demonstrate that it is possible to simultaneously map the signals from coherent continuum radiation, lightning discharges, and radio transmitters in the sky. This work thereby lays the foundation toward the discovery of many more coherent source locations of low frequency electromagnetic waves in the sky. It is expected that the identified source locations vary with time as a result of the impact of solar variability on the D-region ionosphere. Future studies have therefore the potential to contribute to a novel remote sensing and an improved understanding of the D-region ionosphere, influenced by the near-Earth space environment.",
author = "Martin Fullekrug and Koh, {Kuang Liang} and Zhongjian Liu and Andrey Mezentsev",
year = "2018",
month = "12",
day = "21",
doi = "10.1029/2018RS006705",
language = "English",
issn = "0048-6604",
publisher = "American Geophysical Union",

}

TY - JOUR

T1 - First Map of Coherent Low Frequency Continuum Radiation in the Sky

AU - Fullekrug, Martin

AU - Koh, Kuang Liang

AU - Liu, Zhongjian

AU - Mezentsev, Andrey

PY - 2018/12/21

Y1 - 2018/12/21

N2 - Lightning discharges and radio transmitters emit low-frequency (\mbox{$\sim$3-300 kHz}) electromagnetic waves with large electric field strengths and stable phases. This phase stability makes it possible to map the source locations of lightning and transmitters in the sky. Electromagnetic waves with smaller electric field strengths generally exhibit a reduced phase stability, caused by numerous simultaneous physical processes that blend into an underlying continuum radiation trapped inside the Earth-ionosphere cavity. It is therefore currently not known whether the source locations of continuum radiation can be determined. Here we show the first map of coherent continuum radiation in the sky above an array of high precision radio receivers. The source locations of the coherent continuum radiation are found at elevation angles \mbox{$\sim30^{\rm o}-60^{\rm o}$} above the horizon. The identified source locations are attributed to intermittent radio transmitters which emit electromagnetic waves with electric field strengths $\sim2$ orders of magnitude below the instrumental noise floor. The results demonstrate that it is possible to simultaneously map the signals from coherent continuum radiation, lightning discharges, and radio transmitters in the sky. This work thereby lays the foundation toward the discovery of many more coherent source locations of low frequency electromagnetic waves in the sky. It is expected that the identified source locations vary with time as a result of the impact of solar variability on the D-region ionosphere. Future studies have therefore the potential to contribute to a novel remote sensing and an improved understanding of the D-region ionosphere, influenced by the near-Earth space environment.

AB - Lightning discharges and radio transmitters emit low-frequency (\mbox{$\sim$3-300 kHz}) electromagnetic waves with large electric field strengths and stable phases. This phase stability makes it possible to map the source locations of lightning and transmitters in the sky. Electromagnetic waves with smaller electric field strengths generally exhibit a reduced phase stability, caused by numerous simultaneous physical processes that blend into an underlying continuum radiation trapped inside the Earth-ionosphere cavity. It is therefore currently not known whether the source locations of continuum radiation can be determined. Here we show the first map of coherent continuum radiation in the sky above an array of high precision radio receivers. The source locations of the coherent continuum radiation are found at elevation angles \mbox{$\sim30^{\rm o}-60^{\rm o}$} above the horizon. The identified source locations are attributed to intermittent radio transmitters which emit electromagnetic waves with electric field strengths $\sim2$ orders of magnitude below the instrumental noise floor. The results demonstrate that it is possible to simultaneously map the signals from coherent continuum radiation, lightning discharges, and radio transmitters in the sky. This work thereby lays the foundation toward the discovery of many more coherent source locations of low frequency electromagnetic waves in the sky. It is expected that the identified source locations vary with time as a result of the impact of solar variability on the D-region ionosphere. Future studies have therefore the potential to contribute to a novel remote sensing and an improved understanding of the D-region ionosphere, influenced by the near-Earth space environment.

U2 - 10.1029/2018RS006705

DO - 10.1029/2018RS006705

M3 - Article


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# Finite-Time Error Bounds For Linear Stochastic Approximation and TD Learning

3 Feb 2019R. SrikantLei Ying

We consider the dynamics of a linear stochastic approximation algorithm driven by Markovian noise, and derive finite-time bounds on the moments of the error, i.e., deviation of the output of the algorithm from the equilibrium point of an associated ordinary differential equation (ODE). We obtain finite-time bounds on the mean-square error in the case of constant step-size algorithms by considering the drift of an appropriately chosen Lyapunov function... (read more)

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# Given $n\times n$ matrix $A$ with integer entries, find #$k$SAT formula that yields $\mathrm{perm}(A)>0$

For each #$$k$$SAT instance one can build a matrix $$A$$ such that $$\mathrm{perm}(A) = F(\Sigma)$$, where $$\Sigma$$ is the solution count of the $$k$$SAT formula and $$F$$ an easy to invert function.

My question is whether there is a similar reduction in the opposite direction: given an integer valued matrix $$A$$ with $$\mathrm{perm}(A)>0$$, is there a systematic way to build a #$$k$$SAT formula such that $$\Sigma=F(\mathrm{perm}(A))$$, where $$F$$ an easy to invert function?

• If the entries of $A$ are nonnegative, then yes, just because #kSAT is #P-complete and perm of nonnegative integer matrices is in #P. If you allow negative entries, this seems closely related to GapP vs #P. – Joshua Grochow May 18 at 3:16
• @JoshuaGrochow What if I have negative entries but I am somehow guaranteed that $\mathrm{perm}(A)>0$? Also, how does the reduction in the case of only nonnegative entries? – delete000 May 18 at 12:36


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# Instead of axiomatizing ordered fields, can we axiomatize just the right half?

Since ordered fields can always be split into two halves whose only common element is $0$, namely $(\leftarrow,0]$ and $[0,\rightarrow),$ I was wondering if we can axiomatize just the right half. Note that the left half fails to be closed under products, so its not a good candidate for axiomatization.

Anyway, has anyone done this?

If not, here's a not overly-serious proposal. Start with the ordered field axioms, drop the requirement that every number have an additive inverse, and add weaker the requirement that if $x \leq y$, then there exists $k$ such that $x+k=y$. This new system of axioms is still satisfied by every ordered field, so in order to exclude them, we also assume that $0 \leq x$ for all $x$. We can also drop the axiom that if $x \geq 0$ and $y \geq 0$, then $xy \geq 0$, since this is immediate.

I don't know if this really works - we want a natural bijection between the class of all ordered fields and the class of all 'right halves.'

A bit of motivation.

Firstly, I like the idea that there are many ways to obtain $\mathbb{R}$. The 'usual' approach is to begin with $\mathbb{N}$, adjoin negatives to obtain $\mathbb{Z}$, adjoin products and reciprocals to obtain $\mathbb{Q}$, and finally adjoin suprema to obtain $\mathbb{R}.$

Another approach would be to begin with $\mathbb{N}$, adjoin reciprocals and products to obtain$[0,\rightarrow)_\mathbb{Q}$, adjoin suprema to obtain $[0,\rightarrow)_\mathbb{R}$, and finally adjoin negatives to obtain $\mathbb{R}.$

Now the step where suprema are adjoined is basically just completion via Dedekind cuts. But, in the usual approach, where negative numbers are adjoined right at the outset, products of Dedekind cuts are messy to define. This problem does not arise, though, in the latter approach.

A similar issue seems to appear in Conway's 'On Numbers And Games,' where the product of Surreal numbers is rather messy (I only just received this book, hence the question). So I'm wondering if Conway's definition of the product of Surreal numbers can be simplified by restricting ourselves to the non-negative numbers, and only adjoining additive inverses afterwards.

• Why would you want to do that? – tomasz Jun 16 '13 at 12:38
• @tomasz, are you genuinely asking, or are you suggesting this isn't the right thing to do? – goblin Jun 16 '13 at 12:42
• I'm genuinely asking. Is there any motivation behind this question, or are you asking just because? – tomasz Jun 16 '13 at 12:44
• This is a en.wikipedia.org/wiki/Semiring in particular a semiring with the natural order. Note that these don't generally have multiplicative inverses, but they can be added, e.g. non-negative rationals. – András Salamon Jun 16 '13 at 12:46
• @tomasz, I added some motivation to the question. – goblin Jun 16 '13 at 13:09

This is because for any positive part of the ordered field, we can define an extension of a structure $(X,\leq,+,\cdot,0,1)$ to an ordered field in the usual manner, by identifying the ordered pair $(x,0)$ with $x$ and $(x,1)$ with $-x$.
It shouldn't be hard to check that this extension satisfies the axioms of field whenever the $X$ satisfies the theory of the positive part of an ordered field.