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Workshop Speakers

Prof. Krzysztof Apt (University of Amsterdam, Netherlands)

Title: Social Network Games

Abstract: In this lecture we survey our recent works on social network games. They are tailored to study a model of social networks introduced by Apt and Markakis in 2011 in which the nodes influenced by their neighbours can adopt one out of several products. In these games the payoff of each player weakly increases when more players choose his strategy, which is exactly opposite to the congestion games.

We show that such games may have no Nash equilibrium and that determining an existence of a Nash equilibrium is NP-complete. We also clarify the status and the complexity of the finite best response property (FBRP), the finite improvement property (FIP).

Further, we exhibit in this framework some paradoxes. One of them allows us to explain 'bubbles' in a financial market, in which a decision of a trader to switch to some new financial product triggers a sequence of transactions, as a result of which all traders involved become worse off.

Based on joint works with Evangelos Markakis and Sunil Simon.

Prof. Ken Badcock (University of Liverpool, Pro-Vice Chancellor)

Prof. Themis Bowcock (University of Liverpool, UK)

Prof. Artur Czumaj (University of Warwick, UK)

Title: Testing Cluster Structure of Graphs

Abstract: Cluster analysis is a fundamental task in data analysis that aims at partitioning a set of objects into a disjoint collection of objects with similar characteristics. In this talk, we will use the concept of conductance to measure the quality of cluster structure and will focus on a question of approximately recognizing cluster structure of a graph in sublinear time in the framework of property testing in the bounded degree model. We show how to test in O*(\sqrt{n}) time whether a graph with nodes can be partitioned into no more than k parts (clusters) such that the outer-conductance of each cluster is \phi_o at most and the inner-conductance of the induced subgraph on each cluster is at least phi_i, for a large spectrum of parameters k, phi_o, phi_i. By the lower bound of \Omega(\sqrt{n}) for testing graph expansion, which corresponds to the case when k=1 in our problem, our algorithm is asymptotically optimal up to polylogarithmic factors. This is joint work with Pan Peng and Christian Sohler.

Prof. Francesco Falciani (University of Liverpool, UK)

Title: Discovering Biological mechanisms by integrating multi-level, large scale biological datasets

Abstract: With the advent of Functional genomics, any competent Biological laboratory can generate genome-wide measurements representing different levels of biological complexity. A consequence of this has been the rapid growth of public databases containing genetics information as well as the results of thousands of gene expression, proteomics and metabolomics experiments. This has generated the expectation that such vast amount of information would result in a rapid increase in our understanding of biology and that this in turn would contribute a whole spectrum of clinically relevant biomarkers, novel drug targets, and ultimately improved personalized therapies.

However, understanding this complexity has proven to be more challenging than expected. The main difficulty has been integrating such large datasets within a computational framework designed to infer the underlying structure of biological pathways from observational data.

In this talk, I will present several case studies demonstrating the successful identification of novel biological mechanisms using a set of network inference techniques. I will also explore the issue of biological interpretability in using computational models for the generation of testable hypothesis.

Prof. Antonio Fernández Anta (IMDEA, Spain)

Title: Saving Energy by Powering Down Links

Abstract: In this talk I will be presenting two related papers that have to do with energy savings in wired networks. In the first paper I present a model that allows to estimate the energy savings of links that implement the new standards IEEE 802.3az Energy Efficient Ethernet, as a function of simple traffic parameters. Under this standard, the link can go to a low power mode when there is not traffic over the link. Then, in a network of (idealized) EEE links, I present algorithms to schedule and route packets in order to reduce energy consumption while maintaining the latency low.

Prof. Paul Goldberg (University of Oxford, UK)

Title: Computing Nash equilibrium with low communication

Abstract: Various recent papers in the CS/game theory literature have attempted to model the computation of game-theoretic equilibria in a decentralised style. That is, for a solution concept to be realistic, it should be obtainable using distributed computation by the agents, as opposed to, say, passing all information about the game to a central controller, who in turn tells everyone how to play. In the talk I will present some results about communication-bounded computation, and how this helps to capture the notion of decentralisation.

Prof. David Hutchison (Lancaster University, UK)

Title: A strategy for network resilience

Abstract: Resilience needs to be one of the most important properties of future computer networks, because of the increasing use of networks as critical infrastructures and their support for mission-critical services. This talk introduces network resilience and a strategy for its realisation, alongside some relevant research issues.

Prof. Evangelos Kranakis (Carleton University, Canada)

Title: Displacing Random Sensors: Coverage and Interference Problems

Abstract:  When placing sensors at random on a line we want to ensure that interference is minimized and coverage is guaranteed. Proximity between neighbouring sensors affects their transmission and reception signals and degrades network performance In fact, the closer their distance the higher the resulting interference and hence performance degradation. Similarly, monitoring a line domain requires that every point in the line is within the range of a sensor.

In general we are interested in the following questions: What is the expected min total sensor displacement required to avoid interference and/or ensure coverage? We study these two problems separately. For the interference problem we develop tradeoffs between theinter-sensor distance and the expected min total sensor displacement and prove the existence of a critical regime. Similarly, for the coverage problem we develop tradeoffs between the sensor range and the expected min total sensor displacement.

Prof. Mirek Kutylowski (Wroclaw University of Technology, Poland)

Abstract: One of the major challenges of radio communication in ad hoc networks is to synchronize the terminals attempting to send via a shared radio channel. As the prior knowledge of the network state may be unreliable and the state may change dynamically, it is hard to coordinate the terminals and avoid collisions of the transmissions. Our aim is to avoid such collisions and increase channel utilization via self organization strategies. We present some efficient solutions that lead to practical improvements over current industrial practice in terms of reduction of the communication overhead.

Dr. Paul Laycock (University of Liverpool, UK and CERN, Switzerland)

Title: How the ATLAS Collaboration used networks to discover the Higgs

Abstract: The ATLAS collaboration is a team of ~3,000 scientists working towards a better understanding of nature, using one of the most complicated experimental apparatus ever built, and analysing datasets measured in Petabytes. This talk will give a brief overview of the many ways in which ATLAS used networks, from computing networks to information networks, to discover the Higgs boson and produce ~300 publications that address fundamental questions in particle physics. The knowledge required to understand both the enormous experimental apparatus, and the diverse physical processes involved in the experimental measurements, is communicated by various means among the collaboration members. The unique detector-level dataset is processed in diverse ways to satisfy the various research interests of the collaboration's members, using a "Grid" of computing resources distributed worldwide. Maintaining individual flexibility for innovation while working efficiently as a collaboration is a complicated problem, both sociologically and technically. At the end of the first phase of the LHC, the ATLAS computing resources were stretched to their limit, provoking a review and subsequent update of the data analysis model and highlighting its importance.

Prof. Alan Marshall (University of LiverpoolDepartment of Electronics and Electrical Engineering, UK)

Title: Public Access Wireless Networks: Vulnerabilities, Detection and Mitigation Strategies

Abstract: This talk identifies a range of vulnerabilities and threats that can be mounted against domestic and public wireless access networks that employ 802.11 (or Wi-Fi) standards. Compared to enterprise or corporate networks that permit wireless access in a controlled manner, such networks are necessarily more open in nature, in order to provide access to a greater range of users and their wireless devices. Consequently the rapid increase in the use of these networks has led to an associated increase in the number and types of attacks that can be mounted against them, by malicious or criminal users. The talk will outline the types of attacks that can be launched against these types of networks, and identify detection and mitigation strategies.

Dr. George Mertzios (Durham University, UK)

Title: Determining Majority in Networks with Local Interactions and very Small Local Memory

Abstract: In this work we study the problem of determining the majority type in an arbitrary connected network, each vertex of which has initially two possible types (states). The vertices may have a few additional possible states and can interact in pairs only if they share an edge. We first provide a population protocol with 4 states per vertex that always computes the initial majority value, under any fair scheduler. Under the uniform probabilistic scheduler of pairwise interactions, we prove that our protocol stabilizes in expected polynomial time for any connected network and is quite fast on the clique. It turns out that this protocol is optimal, in the sense that there does not exist any population protocol that always computes majority with fewer than 4 states per vertex. Furthermore we analyze an elegant and very natural majority protocol with 3 states per vertex, introduced in (Angluin, Aspnes, Eisenstat, Distributed Computing, 21(2):87-102, 2008). As it is known that this protocol determines the correct initial majority type in the clique graph very fast and whp under the uniform probabilistic scheduler, we study the performance of this protocol in arbitrary networks. We prove that, when the two initial states are put uniformly at random on the vertices, the protocol of Angluin et al. converges to the initial majority with probability higher than the probability of converging to the initial minority. In contrast, we present an infinite family of graphs, on which the protocol of Angluin et al. can fail, i.e. it can converge to the initial minority type whp, even when the difference between the initial majority and the initial minority is $n - \Theta(\ln{n})$. Finally we present another infinite family of graphs in which the protocol of Angluin et al. takes an expected exponential time to converge. Surprisingly, the resistance of the clique to failure causes the failure in general graphs.

Dr. Kieran Sharkey (University of Liverpool, UK)

Title: Epidemic dynamics on networks

Abstract: I will briefly discuss the application of networks in epidemic modelling. I then hope to convey a flavour of two representations of Susceptible-Infectious-Removed (SIR) epidemic dynamics. These are the moment-closure representation as well as the message-passing formalism of Karrer and Newman. Particular focuses are the circumstances under which these two representations become equivalent and the circumstances when they become exact. I will discuss the relative merits of both representations. Although applied in an epidemic context, the underlying principles should be readily transferable to other domains.


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# Schwarzschild radius in black hole density

The textbook from which I teach physics at the end of secondary school has a question about the density of a non-rotating black hole. Because the density at the singularity is perhaps infinite or beyond the scope of GR, this can only be some "mean density" mass/volume within a sphere of a certain radius. Some authors use the Schwarzschild radius.

Why this value of the radius? How can I explain that this choice of radius makes more sense than another radius?

• The Schwarzschild radius is the distance fro the center at which the escape velocity of the black hole surpasses the speed of light. – Brionius Apr 19 '15 at 19:41
• @Brionius: As far as I remember the physics lecture (I'm engineer, not physicist) this definition is based on non-relativistic calculations. You get the correct radius using this calculation however the calculation has (as far as I remember) nothing to do with the physical truth. – Martin Rosenau Apr 19 '15 at 19:50
• Very related: Do black holes have infinite areas and volumes?. Basically, the volume of a black hole is not frame-independent, and so is not unambiguously defined. However, for a simple (non-rotating, uncharged, isolated) Schwarzschild black hole, one fairly 'obvious' choice of frame gives the same volume as the usual Euclidean formula with Schwarzschild radius. – Stan Liou Apr 19 '15 at 19:58
• @MartinRosenau while the common Newtonian derivation of the Schwartzschild radius is a happy accident, it still is the dividing line between the point when light (or anything else) could escape. It's a convenient, accurate shorthand, but it shouldn't be taken as an excuse to apply Newtonian gravity to a non-Newtonian system. – Brionius Apr 19 '15 at 20:10

Therefore, it makes more sense to try to calculate volume (and hence indirectly, density) of a black hole according to the extent of the event horizon rather than the singularity. It so happens that for the simplest of black hole solutions, the spherically symmetric Schwarzschild black hole, the Schwarzschild radius characterizes the size of the horizon: the Schwarzschild radius is actually $\sqrt{A/4\pi}$, where $A$ is the area of the horizon. This is actually (implicitly) by definition of the Schwarzschild radial coordinate, which directly corresponds to the area of a sphere.

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# A question about physical chemistry

1. Oct 14, 2015

### hgnk708113

• Please post this type of questions in HW section using the template.
For the problem of particle in a 1D box
V(x)=0 for 0≦x≦a
∞ for anywhere outside the box
I know that Hamiltonian operator commutes with momentum operator
so they should have smae eigenfunction but it's obvious that the wavefunction
(2/a)^1/2sin(nπx/a) is not a eigenfunction of linear momentum operator
what is the reason for the contradiction ?

Last edited: Oct 14, 2015
2. Oct 14, 2015

### blue_leaf77

The momentum operator in infinite well potential is not well defined to begin with, this is because momentum operator is supposed to act as a generator of translation, while such a translation in infinite well potential cannot take place. Moreover, even after you define a "momentum-like" operator $\hat{p}$, it does not commute with the global Hamiltonian.

3. Oct 14, 2015


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## An Edgeworth expansion for symmetric finite population statistics.(English)Zbl 1010.62017

From the paper: Given a set $${\mathcal X}=\{x_1, \dots,x_N)$$, let $$(X_1, \dots,X_N)$$ be a random permutation of the ordered set $$(x_2,\dots, x_N)$$. We assume that the random permutation is uniformly distributed over the class of permutations. Let $$T=t(X_1, \dots,X_n)$$ denote a symmetric statistic of the first $$n$$ observations $$X_1,\dots, X_n$$, where $$n<N$$. That is, $$t$$ is a real function defined on the class of subsets $$\{x_{i_1}, \dots, x_{i_n}\} \subset {\mathcal X}$$ of size $$n$$ and we assume that $$t(x_{i_1}, \dots, x_{i_n})$$ is invariant under permutations of its arguments. Since $$X_1,\dots, X_n$$ represents a sample drawn without replacement from the population $${\mathcal X}$$, we call $$T$$ a symmetric finite population statistic.
Assuming that the linear part of Hoeffding’s decomposition of $$T$$ is nondegenerate we construct a one term Edgeworth expansion for the distribution function of $$T$$ and prove the validity of the expansion with the remainder $$O(1/n^*)$$ as $$n^*\to \infty$$, where $$n^*=\min \{n,N-n\}$$.

### MSC:

 62E20 Asymptotic distribution theory in statistics 62D05 Sampling theory, sample surveys 60F05 Central limit and other weak theorems
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### References:

 [1] ALBERS, W., BICKEL, P. J. and VAN ZWET, W. R. (1976). Asy mptotic expansions for the power of distribution free tests in the one-sample problem. Ann. Statist. 4 108-156. · Zbl 0321.62049 [2] BABU, GUTTI JOGESH, BAI, Z. D. (1996). Mixtures of global and local Edgeworth expansions and their applications. J. Multivariate Anal. 59 282-307. · Zbl 0864.62006 [3] BABU, G. J. and SINGH, K. (1985). Edgeworth expansions for sampling without replacement from finite populations. J. Multivariate Anal. 17 261-278. · Zbl 0589.62009 [4] BENTKUS, V., GÖTZE, F. and VAN ZWET, W. R. (1997). An Edgeworth expansion for sy mmetric statistics. Ann. Statist. 25 851-896. · Zbl 0920.62016 [5] BERTAIL, P. (1997). Second-order properties of an extrapolated bootstrap without replacement under weak assumptions. Bernoulli 3 149-179. · Zbl 0919.62035 [6] BHATTACHARy A, R. N. and GHOSH, J. K. (1978). On the validity of the formal Edgeworth expansion. Ann. Statist. 6 434-451. · Zbl 0396.62010 [7] BICKEL, P. J. (1974). Edgeworth expansions in non parametric statistics. Ann. Statist. 2 1-20. · Zbl 0284.62018 [8] BICKEL, P. J., GÖTZE, F. and VAN ZWET, W. R. (1986). The Edgeworth expansion for U-statistics of degree two. Ann. Statist. 14 1463-1484. · Zbl 0614.62015 [9] BICKEL, P. J., GÖTZE, F. and VAN ZWET, W. R. (1997). Resampling fewer than n observations: gains, losses, and remedies for losses. Statist. Sinica 7 1-31. · Zbl 0927.62043 [10] BICKEL, P. J. and ROBINSON, J. (1982). Edgeworth expansions and smoothness. Ann. Probab. 10 500-503. · Zbl 0485.60018 [11] BICKEL, P. J. and VAN ZWET, W. R. (1978). Asy mptotic expansions for the power of distribution free tests in the two-sample problem. Ann. Statist. 6 937-1004. · Zbl 0378.62047 [12] BIKELIS, A. (1972). On the estimation of the remainder term in the central limit theorem for samples from finite populations. Studia Sci. Math. Hungar. 4 345-354 (in Russian). [13] BLOZNELIS, M. (1999). A Berry-Esseen bound for finite population Student’s statistic. Ann. Probab. 27 2089-2108. · Zbl 0984.62010 [14] BLOZNELIS, M. and GÖTZE, F. (2000). An Edgeworth expansion for finite population U-statistics. Bernoulli 6 729-760. · Zbl 0982.62009 [15] BLOZNELIS, M. and GÖTZE, F. (1999). One term Edgeworth expansion for finite population U-statistics of degree two. Acta Appl. Math. 58 75-90. · Zbl 0939.62021 [16] BLOZNELIS, M. and GÖTZE, F. (2001). Orthogonal decomposition of finite population statistic and its applications to distributional asy mptotics. Ann. Statist. 29 899-917. · Zbl 1012.62009 [17] BOLTHAUSEN, E. and GÖTZE, F. (1993). The rate of convergence for multivariate sampling statistics. Ann. Statist. 21 1692-1710. · Zbl 0798.62023 [18] BOOTH, J. G. and HALL, P. (1993). An improvement of the jackknife distribution function estimator. Ann. Statist. 21 1476-1485. · Zbl 0792.62036 [19] CALLAERT, H. and JANSSEN, P. (1978). A Berry-Esseen theorm for U-statistics. Ann. Statist. 6 417-421. · Zbl 0393.60022 [20] CALLAERT, H., JANSSEN, P. and VERAVERBEKE, N. (1980). An Edgeworth expansion for U-statistics. Ann. Statist. 8 299-312. · Zbl 0428.62016 [21] CHIBISOV, D. M. (1972). Asy mptotic expansion for the distribution of a statistic admitting a stochastic expansion. Theory Probab. Appl. 17 620-630. · Zbl 0279.60018 [22] ERD OS, P. and RÉNy I, A. (1959). On the central limit theorem for samples from a finite population. Publ. Math. Inst. Hungar. Acad. Sci. 4 49-61. · Zbl 0086.34001 [23] FELLER, W. (1968). An Introduction to Probability Theory and Its Applications. I. Wiley, New York. · Zbl 0155.23101 [24] GÖTZE, F. (1979). Asy mptotic expansions for bivariate von Mises functionals. Z. Wahrsch. Verw. Gebiete 50 333-355. · Zbl 0405.60009 [25] GÖTZE, F. and VAN ZWET, W. R. (1991). Edgeworth expansions for asy mptotically linear statistics. Preprint 91-034, SFB 343 Univ. Bielefeld. [26] HELMERS, R. (1982). Edgeworth expansions for linear combinations of order statistics. Math. Centre Tracts 105. Math. Centrum, Amsterdam. · Zbl 0485.62017 [27] HELMERS, R. and VAN ZWET, W. R. (1982). The Berry-Esseen bound for U-statistics. In Statistical Decision Theory and Related Topics III (S. S. Gupta and J. O. Berger, eds.) 1 497-512. Academic Press, New York. · Zbl 0598.62016 [28] HÖGLUND, T. (1978). Sampling from a finite population: a remainder term estimate. Scand. J. Statist. 5 69-71. · Zbl 0382.60028 [29] KOKIC, P. N. and WEBER, N. C. (1990). An Edgeworth expansion for U-statistics based on samples from finite populations. Ann. Probab. 18 390-404. · Zbl 0703.60016 [30] LAI, T. L. and WANG, J. Q. (1993). Edgeworth expansions for sy mmetric statistics with applications to bootstrap methods. Statist. Sinica 3 517-542. · Zbl 0822.62010 [31] NANDI, H. K. and SEN, P. K. (1963). On the properties of U-statistics when the observations are not independent. II. Unbiased estimation of the parameters of a finite population. Calcutta Statist. Assoc. Bull. 12 124-148. · Zbl 0123.37002 [32] PETROV, V. V. (1975). Sums of Independent Random Variables. Springer, New York. · Zbl 0322.60043 [33] PFANZAGL, J. (1973). Asy mptotic expansions related to minimum contrast estimators. Ann. Statist. 1 993-1026. · Zbl 0273.62015 [34] POLITIS, D. N. and ROMANO, J. P. (1994). Large sample confidence regions based on subsamples under minimal assumptions. Ann. Statist. 22 2031-2050. · Zbl 0828.62044 [35] PRAWITZ, H. (1972). Limits for a distribution, if the characteristic function is given in a finite domain. Scand. Aktuar Tidskr. 138-154. · Zbl 0275.60025 [36] RAO, C. R. and ZHAO, L. C. (1994). Berry-Esseen bounds for finite-population t-statistics. Statist. Probab. Lett. 21 409-416. · Zbl 0806.62014 [37] ROBINSON, J. (1978). An asy mptotic expansion for samples from a finite population. Ann. Statist. 6 1005-1011. · Zbl 0387.60030 [38] SCHNELLER, W. (1989). Edgeworth expansions for linear rank statistics. Ann. Statist. 17 1103-1123. · Zbl 0689.62016 [39] SERFLING, R. J. (1980). Approximation Theorems of Mathematical Statistics. Wiley, New York. · Zbl 0538.62002 [40] SHAO, J. (1989). The efficiency and consistency of approximations to the jackknife variance estimators. J. Amer. Statist. Assoc. 84 114-119. JSTOR: · Zbl 0677.62042 [41] VAN ZWET, W. R. (1984). A Berry-Esseen bound for sy mmetric statistics. Z. Wahrsch. Verw. Gebiete 66 425-440. · Zbl 0525.62023 [42] WU, C. F. J. (1990). On the asy mptotic properties of the jackknife histogram. Ann. Statist. 18 1438- 1452. · Zbl 0705.62044 [43] ZHAO, L. C. and CHEN, X. R. (1987). Berry-Esseen bounds for finite-population U-statistics. Sci. Sinica Ser. A 30 113-127. · Zbl 0665.60029 [44] ZHAO, L. C. and CHEN, X. R. (1990). Normal approximation for finite-population U-statistics. Acta Math. Appl. Sinica 6 263-272. · Zbl 0724.60024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


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# Where is the __ command?

One tricky thing is understanding that there are a certain number of commands that come with PowerShell, _while _most commands do not.
Every new version of PowerShell includes at least a few new commands. For example, Start-Job appeared for the first time in PowerShell v2, while Invoke-AsWorkflow was introduced in PowerShell v3.
What confuses people is that a new version of PowerShell also tends to correspond with a new version of the Windows operating system? and the OS itself comes with hundreds of commands. For example, you may have used Get-SmbShare for the first time in Windows Server 2012, which included PowerShell v3. But Get-SmbShare is part of the operating system, not part of PowerShell. That is, you won't have Get-SmbShare on every system that has PowerShell v3 or later, because the command isn't a "feature of PowerShell," it's a "feature of Windows."
So? where do you get commands?
Usually, with whatever product those commands are a part of. Want the Exchange Server commands? Install the Exchange Server admin tools. Want the Windows Server 2012 commands? Install the Remote Server Administration Toolkit (RSAT), which contains the server admin tools.


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### Top 10 Arxiv Papers Today in Systems And Control

##### #1. Investigating Flight Envelope Variation Predictability of Impaired Aircraft using Least-Squares Regression Analysis
###### Ramin Norouzi, Amirreza Kosari, Mohammad Hossein Sabour
Aircraft failures alter the aircraft dynamics and cause maneuvering flight envelope to change. Such envelope variations are nonlinear and generally unpredictable by the pilot as they are governed by the aircraft complex dynamics. Hence, in order to prevent in-flight Loss of Control it is crucial to practically predict the impaired aircraft's flight envelope variation due to any a-priori unknown failure degree. This paper investigates the predictability of the number of trim points within the maneuvering flight envelope and its centroid using both linear and nonlinear least-squares estimation methods. To do so, various polynomial models and nonlinear models based on hyperbolic tangent function are developed and compared which incorporate the influencing factors on the envelope variations as the inputs and estimate the centroid and the number of trim points of the maneuvering flight envelope at any intended failure degree. Results indicate that both the polynomial and hyperbolic tangent function-based models are capable of...
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###### Tweets
arxivml: "Investigating Flight Envelope Variation Predictability of Impaired Aircraft using Least-Squares Regression Analysi… https://t.co/lCvSS5E81k
arxiv_cs_LG: Investigating Flight Envelope Variation Predictability of Impaired Aircraft using Least-Squares Regression Analysis. Ramin Norouzi, Amirreza Kosari, and Mohammad Hossein Sabour https://t.co/hGejfq18jk
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Total Words: 16156
Unqiue Words: 3622

##### #2. Randomized Algorithms for Data-Driven Stabilization of Stochastic Linear Systems
Data-driven control strategies for dynamical systems with unknown parameters are popular in theory and applications. An essential problem is to prevent stochastic linear systems becoming destabilized, due to the uncertainty of the decision-maker about the dynamical parameter. Two randomized algorithms are proposed for this problem, but the performance is not sufficiently investigated. Further, the effect of key parameters of the algorithms such as the magnitude and the frequency of applying the randomizations is not currently available. This work studies the stabilization speed and the failure probability of data-driven procedures. We provide numerical analyses for the performance of two methods: stochastic feedback, and stochastic parameter. The presented results imply that as long as the number of statistically independent randomizations is not too small, fast stabilization is guaranteed.
more | pdf | html
###### Tweets
arxiv_org: Randomized Algorithms for Data-Driven Stabilization of Stochastic Linear Systems. https://t.co/skBYsMjg4B https://t.co/rlyYRSky5j
arxivml: "Randomized Algorithms for Data-Driven Stabilization of Stochastic Linear Systems", Mohamad Kazem Shirani Faradonbe… https://t.co/x3ICbJlVrz
StatsPapers: Randomized Algorithms for Data-Driven Stabilization of Stochastic Linear Systems. https://t.co/ugznyLlktM
DrPjenFI: RT @arxiv_org: Randomized Algorithms for Data-Driven Stabilization of Stochastic Linear Systems. https://t.co/skBYsMjg4B https://t.co/rlyYR…
qiming82: RT @arxiv_org: Randomized Algorithms for Data-Driven Stabilization of Stochastic Linear Systems. https://t.co/skBYsMjg4B https://t.co/rlyYR…
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Total Words: 2939
Unqiue Words: 1093

##### #3. Global analysis of synchronization performance for power systems: bridging the theory-practice gap
The issue of synchronization in the power grid is receiving renewed attention, as new energy sources with different dynamics enter the picture. Global metrics have been proposed to evaluate performance and analyzed under highly simplified assumptions. In this paper, we extend this approach to more realistic network scenarios and more closely connect it with metrics used in power engineering practice. In particular, our analysis covers networks with generators of heterogeneous ratings and richer dynamic models of machines. Under a suitable proportionality assumption in the parameters, we show that the step response of bus frequencies can be decomposed in two components. The first component is a {system-wide frequency} that captures the aggregate grid behavior, and the residual component represents the individual bus frequency deviations from the aggregate. Using this decomposition, we define --and compute in closed form-- several metrics that capture dynamic behaviors that are of relevance for power engineers. In particular, using...
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Authors: 2
Total Words: 12802
Unqiue Words: 3076

##### #4. Sparsity-Promoting Optimal Control of Cyber-Physical Systems over Shared Communication Networks
###### Nandini Negi, Aranya Chakrabortty
Recent years have seen several new directions in the design of sparse control of cyber-physical systems (CPSs) driven by the objective of reducing communication cost. One common assumption made in these designs is that the communication happens over a dedicated network. For many practical applications, however, communication must occur over shared networks, leading to two critical design challenges, namely - time-delays in the feedback and fair sharing of bandwidth among users. In this paper, we present a set of sparse H2 control designs under these two design constraints. An important aspect of our design is that the delay itself can be a function of sparsity, which leads to an interesting pattern of trade-offs in the H2 performance. We present three distinct algorithms. The first algorithm preconditions the assignable bandwidth to the network and produces an initial guess for a stabilizing controller. This is followed by our second algorithm, which sparsifies this controller while simultaneously adapting the feedback delay...
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Sample Sizes : [2, 2, 3, 3, 2]
Authors: 2
Total Words: 13065
Unqiue Words: 2962

##### #5. Non-signaling Approximations of Stochastic Team Problems
###### Naci Saldi, Can Deha Karıksız, Maxim Raginsky, Eric Chitambar
In this paper, we consider non-signaling approximation of finite stochastic teams. We first introduce a hierarchy of team decision rules that can be classified in an increasing order as randomized policies, quantum-correlated policies, and non-signaling policies. Then, we establish an approximation of team-optimal policies for sequential teams via extendible non-signaling policies. We prove that the distance between extendible non-signaling policies and decentralized policies is small if the extension is sufficiently large. Using this result, we establish a linear programming (LP) approximation of sequential teams. Finally, we state an open problem regarding computation of optimal value of quantum-correlated policies.
more | pdf | html
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###### Tweets
DO: Non-signaling Approximations of Stochastic Team Problems. https://t.co/lSW6UbpyZ5
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Sample Sizes : None.
Authors: 4
Total Words: 10798
Unqiue Words: 1967

##### #6. A Structure-Tensor Approach to Integer Matrix Completion with Applications to Differentiated Energy Services
###### Yanfang Mo, Wei Chen, Sei Zhen Khong, Li Qiu
Efficient resource allocation is one of the main driving forces of human civilizations. Of the many existing approaches to resource allocation, matrix completion is one that is frequently applied. In this paper, we investigate a special type of matrix completion problem concerning the class of $(0,1)$-matrices with given row/column sums and certain zeros prespecified. We provide a necessary and sufficient condition under which such a class is nonempty. The condition is stated in the form of the nonnegativity of a structure tensor constructed from the information regarding the given row/column sums and fixed zeros. Moreover, we show that a more general matrix completion problem can be studied in a similar manner, namely that involving the class of nonnegative integer matrices with prescribed row/column sums, predetermined zeros, and different bounds across the rows. To illustrate the utility of our results, we apply them to demand response applications in smart grids. Specifically, we address two adequacy problems in differentiated...
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##### #7. Distribution Electricity Pricing under Uncertainty
###### Robert Mieth, Yury Dvorkin
Distribution locational marginal prices (DLMPs) facilitate the efficient operation of low-voltage electric power distribution systems. We propose an approach to internalize the stochasticity of renewable distributed energy resources (DERs) and risk tolerance of the distribution system operator in DLMP computations. This is achieved by means of applying conic duality to a chance-constrained AC optimal power flow. We show that the resulting DLMPs consist of the terms that allow to itemize the prices for the active and reactive power production, balancing regulation, and voltage support provided. Finally, we prove the proposed DLMP constitute a competitive equilibrium, which can be leveraged for designing a distribution electricity market, and show that imposing chance constraints on voltage limits distort the equilibrium.
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##### #8. Control Theory Meets POMDPs: A Hybrid Systems Approach
Partially observable Markov decision processes (POMDPs) provide a modeling framework for a variety of sequential decision making under uncertainty scenarios in artificial intelligence (AI). Since the states are not directly observable in a POMDP, decision making has to be performed based on the output of a Bayesian filter (continuous beliefs). Hence, POMDPs are often computationally intractable to solve exactly and researchers resort to approximate methods often using discretizations of the continuous belief space. These approximate solutions are, however, prone to discretization errors, which has made POMDPs ineffective in applications, wherein guarantees for safety, optimality, or performance are required. To overcome the complexity challenge of POMDPs, we apply notions from control theory. The goal is to determine the reachable belief space of a POMDP, that is, the set of all possible evolutions given an initial belief distribution over the states and a set of actions and observations. We begin by casting the problem of...
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##### #9. Multi-Armed Bandits for Boolean Connectives in Hybrid System Falsification (Extended Version)
###### Zhenya Zhang, Ichiro Hasuo, Paolo Arcaini
Hybrid system falsification is an actively studied topic, as a scalable quality assurance methodology for real-world cyber-physical systems. In falsification, one employs stochastic hill-climbing optimization to quickly find a counterexample input to a black-box system model. Quantitative robust semantics is the technical key that enables use of such optimization. In this paper, we tackle the so-called scale problem regarding Boolean connectives that is widely recognized in the community: quantities of different scales (such as speed [km/h] vs. RPM, or worse, RPH) can mask each other's contribution to robustness. Our solution consists of integration of the multi-armed bandit algorithms in hill climbing-guided falsification frameworks, with a technical novelty of a new reward notion that we call hill-climbing gain. Our experiments show our approach's robustness under the change of scales, and that it outperforms a state-of-the-art falsification tool.
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##### #10. A simple but energy-efficient HVAC control synthesis for data centers
###### Michel Fliess, Cédric Join, Maria Bekcheva, Alireza Moradi, Hugues Mounier
The air conditioning management of data centers, a key question with respect to energy saving, is here tackled via the recent model-free control synthesis. Mathematical modeling becomes useless in this approach. The tuning of the corresponding intelligent proportional controller is straightforward. Computer simulations show excellent tracking performances in various realistic situations, like CPU load or temperature changes.
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text:
# How do you solve -5x \geq 7( x + 7) - 13?

Jul 16, 2017

Treat the larger than or equal sign as an equals sign.

#### Explanation:

$- 5 x = 7 \left(x + 7\right) - 13$

Expand the bracket

$- 5 x = 7 x + 49 - 13$

Simplify

$- 5 x = 7 x + 36$

Minus the $36$ and add the $5 x$, you want the $x$ to be positive

$- 36 = 12 x$

Divide by $12$ on both sides

$- 3 = x$

Replace equals with larger than or equals sign

$- 3 \ge x$


text:
## Electronic Journal of Statistics

### Sufficient dimension reduction via principal L$q$ support vector machine

#### Abstract

Principal support vector machine was proposed recently by Li, Artemiou and Li (2011) to combine L1 support vector machine and sufficient dimension reduction. We introduce the principal L$q$ support vector machine as a unified framework for linear and nonlinear sufficient dimension reduction. By noticing that the solution of L1 support vector machine may not be unique, we set $q>1$ to ensure the uniqueness of the solution. The asymptotic distribution of the proposed estimators are derived for $q>1$. We demonstrate through numerical studies that the proposed L2 support vector machine estimators improve existing methods in accuracy, and are less sensitive to the tuning parameter selection.

#### Article information

Source
Electron. J. Statist., Volume 10, Number 1 (2016), 783-805.

Dates
First available in Project Euclid: 6 April 2016

https://projecteuclid.org/euclid.ejs/1459967423

Digital Object Identifier
doi:10.1214/16-EJS1122

Mathematical Reviews number (MathSciNet)
MR3486417

Zentralblatt MATH identifier
06576607

#### Citation

Artemiou, Andreas; Dong, Yuexiao. Sufficient dimension reduction via principal L$q$ support vector machine. Electron. J. Statist. 10 (2016), no. 1, 783--805. doi:10.1214/16-EJS1122. https://projecteuclid.org/euclid.ejs/1459967423

#### References

• Abe, S. (2002). Analysis of support vector machines. In, Neural Networks for Signal Processing XII – Proceedings of the 2002 IEEE Signal Processing Society Workshops, 89–98.
• Abe, S. (2010)., Support Vector Machines for Pattern Classification. Second Edition. Springer.
• Artemiou, A. and Shu, M. (2014). A cost based reweighed scheme of principal support vector machine. In, Topics in Nonparametric Statistics, Springer Proceedings in Mathematics & Statistics, 74, 1–12.
• Bura, E. and Pfeiffer, R. (2008). On the distribution of the left singular vectors of a random matrix and its applications., Statistics and Probability Letters, 78, 2275–2280.
• Burges, C. J. C. and Crisp, S. J. (1999). Uniqueness of the SVM solution. In, Proceedings of Neural Information Processing Systems, 12, 223–229.
• Cook, R. D. (1998a)., Regression Graphics: Ideas for Studying Regressions through Graphics. New York: Wiley.
• Cook, R. D. (1998b). Principal Hessian directions revisited (with discussion)., Journal of the American Statistical Association, 93, 84–100.
• Cook, R. D. (2004). Testing predictors contributions in sufficient dimension reduction., The Annals of Statistics, 32, 1062–1092.
• Cook, R. D. (2007). Fisher lecture: dimension reduction in regression., Statistical Science, 22, 1–40.
• Cook, R. D. and Weisberg, S. (1991). Discussion of “Sliced inverse regression for dimension reduction”., Journal of the American Statistical Association. 86, 316–342.
• Cortes, C. and Vapnik, V. (1995). Support vector networks., Machine Learning, 20, 1–25.
• Fukumizu, K., Bach, F. R., and Jordan, M. I. (2009). Kernel dimension reduction in regression., The Annals of Statistics, 37, 1871–1905.
• Jiang, B., Zhang, X., and Cai, T. (2008). Estimating the confidence interval for prediction errors of support vector machine classifiers., Journal of Machine Learning Research, 9, 521–540.
• Koo, J.-Y., Lee, Y., Kim, Y. and Park, C. (2008). A Bahadur representation of the linear support vector machine., Journal of Machine Learning Research, 9, 1343–1368.
• Li, B., Artemiou, A. and Li, L. (2011). Principal support vector machine for linear and nonlinear sufficient dimension reduction., The Annals of Statistics, 39, 3182–3210
• Li, B. and Wang, S. (2007). On directional regression for dimension reduction., Journal of the American Statistical Association, 102, 997–1008.
• Li, B., Zha, H., and Chiaromonte, F. (2005). Contour regression: a general approach to dimension reduction., The Annals of Statistics, 33, 1580–1616.
• Li, K. C. (1991). Sliced inverse regression for dimension reduction (with discussion)., Journal of the American Statistical Association, 86, 316–342.
• Shin, S. J., Wu, Y., Zhang, H. H. and Liu, Y. (2014). Probability-enhanced sufficient dimension reduction for binary classification., Biometrics, 70, 546–555.
• Suykens, J. A. K., Gestel, T. V., Brabanter, J. D., Moor, B. D. and Vandewalle, J. (2002). Least Squares Support Vector Machines. World Scientific Pub. Co., Singapore.
• Vapnik, N. V. (1998)., Statistical Learning Theory. John Wiley & Sons, Inc.
• Wang, Q. and Yin, X. (2008). Sufficient dimension reduction and variable selection for regression mean function with categorical predictors., Statistics and Probability Letters, 78, 2798–2803.
• Wu, H. M. (2008). Kernel sliced inverse regression with applications on classification., Journal of Computational and Graphical Statistics, 17, 590–610.
• Xia, Y., Tong, H., Li, W. K. and Zhu, L. X. (2002). An adaptive estimation of optimal regression subspace., Journal of the Royal Statistical Society, Series B., 64, 363–410.
• Ye, Z. and Weiss, R. (2003). Using the bootstrap to select one of a new class of dimension reduction methods., Journal of the American Statistical Association, 98, 968–979.
• Yeh, I. C. (2007). Modeling slump flow of concrete using second-order regressions and artificial neural networks., Cement and Concrete Composites, 29, 474–480.
• Yeh, Y. R., Huang, S. Y. and Lee, Y. Y. (2009). Nonlinear dimension reduction with kernel sliced inverse regression., IEEE Transactions on Knowledge and Data Engineering, 21, 1590–1603.
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• Zhu, L. X., Miao, B. and Peng, H. (2006). On sliced inverse regression with large dimensional covariates., Journal of the American Statistical Association, 101, 630–643.


text:
# Energy of a cylinder rolling down a path

Problem statement:

A cylinder rolls without slipping down a hill. It is released from height h. What is its speed when it come down? The cylinder mass may be completely concentrated on the radius R, which is the radius of the cylinder.

My thoughts:

At the top(hight h) the potential energy of the cylinder is E=mgh and at the bottom(h=0) all energy has become kinetic energy since friction and air drag is neglected in this context.(I assumed this). Thus:

$mgh=\frac{1}{2}mv^2 \Leftrightarrow v=\sqrt{2gh}$

Correct answer is however $v=\sqrt{gh}$ which means that energy must have been lost,correct?

What have i missed?

-

You're missing the rotational kinetic energy at the bottom, $\frac{1}{2}I \omega ^2$. The key word in the problem is, 'rolling without slipping'. Also remember the equation, $$v=r\omega$$ The cylinder's moment of inertia should be looked up in a table or given.
On wikipedia, the moment of inertia of a thin cylindrical shell is given as: $$I=mr^2$$
"insection": n., the intersection between inspection, inception, perception, and introspection. ;-) –  rubenvb Jun 27 '14 at 9:25


text:
# inequalities

• April 3rd 2009, 08:27 PM
ose90
inequalities
given that x+y=1, prove that 1/x + 1/y >= 4

Appreciate any help/hints!
• April 3rd 2009, 09:08 PM
NonCommAlg
Quote:

Originally Posted by ose90

given that x+y=1, prove that 1/x + 1/y >= 4

Appreciate any help/hints!

you also need to have x > 0 and y > 0. first note that from $(t-1)^2 \geq 0,$ we get: $t + \frac{1}{t} \geq 2,$ for any $t > 0.$ thus: $\frac{1}{x} + \frac{1}{y} = (x+y) \left(\frac{1}{x} + \frac{1}{y} \right)=2 + \frac{x}{y} + \frac{y}{x} \geq 4.$
• April 5th 2009, 01:51 AM
ose90
Quote:

Originally Posted by NonCommAlg
you also need to have x > 0 and y > 0. first note that from $(t-1)^2 \geq 0,$ we get: $t + \frac{1}{t} \geq 2,$ for any $t > 0.$ thus: $\frac{1}{x} + \frac{1}{y} = (x+y) \left(\frac{1}{x} + \frac{1}{y} \right)=2 + \frac{x}{y} + \frac{y}{x} \geq 4.$

Thanks alot!


text:
# Fourier Series (Even and Odd Functions)

Let $f \in E$ (where $E$ is a linear space of complex-valued piecewise continuous functions defined on the interval $[-\pi,\pi]$) and $$f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n}\cos{nx}+b_{n}\sin{nx}\right]$$ denote its Fourier series. Define $$g(x)=\frac{f(x)+f(-x)}{2}, \quad \quad h(x)=\frac{f(x)-f(-x)}{2}$$ Find the Fourier series of $g$ and of $h$.

Is it just as simple as to write out the definition of $f$ and then use the properties of the trigonometric functions and series, as done below? $$h(x)=\frac{\frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n}\cos{nx}+b_{n}\sin{nx}\right]-\left(\frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n}\cos{-nx}+b_{n}\sin{-nx}\right]\right)}{2}$$ $$=\frac{\frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n}\cos{nx}+b_{n}\sin{nx}\right]-\left(\frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n}\cos{nx}-b_{n}\sin{nx}\right]\right)}{2}$$

$$=\frac{\sum_{n=1}^{\infty}\left[\left(a_{n}\cos{nx}-a_{n}\cos{nx} \right)+\left(b_{n}\sin{nx}+b_{n}\sin{nx} \right) \right]}{2}=\sum_{n=1}^{\infty}b_{n}\sin{nx}$$

If so, then one easily see that $g(x)=\sum_{n=1}^{\infty}a_{n}\cos{nx}$, which is to say it is an even function, whereas $h(x)$ is odd.

What you should do instead is to consider the definition of Fourier series. Thus, you should try to calculate $a_n(g)$ and $b_n(g)$ (and $a_n(h)$ and $b_n(h)$).


text:
# Equation describing the shape of a concave mirror

What is the equation describing the shape of a mirror with the property that light rays from a point source placed at the origin will be reflected in a beam of rays parallel to the x-axis?

From the Wikipedia page I can quote that the relation is $4 f y = x^2$ where $f$ is the distance of the focus to the origin of the parabola.
Your focus is at $(a, 0)$, so the equation would be $4 a x = y^2$. You can solve that equation for $y$ and have a function $y = f(x) = \sqrt{4 a x}$.


text:
# How do you find the number of complex zeros for the function f(x)=27x^6+208x^3-64?

Oct 17, 2016

$f \left(x\right)$ has $2$ Real zeros and $4$ non-Real Complex zeros.

#### Explanation:

$f \left(x\right) = 27 {x}^{6} + 208 {x}^{3} - 64$

By the Fundamental Theorem of Algebra, $f \left(x\right)$ has exactly $6$ Complex (possibly Real) zeros counting multiplicity, since it is of degree $6$.

$\textcolor{w h i t e}{}$
Descartes' Rule of Signs

The signs of the coefficients of $f \left(x\right)$ follow the pattern $+ + -$

With one change of sign, that means that $f \left(x\right)$ has exactly one positive Real zero.

The signs of the coefficients of $f \left(- x\right)$ follow the pattern $+ - -$

With one change of sign, that means that $f \left(x\right)$ has exactly one negative Real zero.

The other $4$ zeros must be non-Real Complex.

$\textcolor{w h i t e}{}$
Bonus - Direct solution

$27 f \left(x\right) = 27 \left(27 {x}^{6} + 208 {x}^{3} - 64\right)$

$\textcolor{w h i t e}{27 f \left(x\right)} = 729 {x}^{6} + 5616 {x}^{3} - 1728$

$\textcolor{w h i t e}{27 f \left(x\right)} = {\left(27 {x}^{3}\right)}^{2} + 2 \left(104\right) \left(27 {x}^{3}\right) + {\left(104\right)}^{2} - {112}^{2}$

$\textcolor{w h i t e}{27 f \left(x\right)} = {\left(27 {x}^{3} + 104\right)}^{2} - {112}^{2}$

$\textcolor{w h i t e}{27 f \left(x\right)} = \left(\left(27 {x}^{3} + 104\right) - 112\right) \left(\left(27 {x}^{3} + 104\right) + 112\right)$

$\textcolor{w h i t e}{27 f \left(x\right)} = \left(27 {x}^{3} - 8\right) \left(27 {x}^{3} + 216\right)$

$\textcolor{w h i t e}{27 f \left(x\right)} = 27 \left({\left(3 x\right)}^{3} - {2}^{3}\right) \left({x}^{3} + {2}^{3}\right)$

$\textcolor{w h i t e}{27 f \left(x\right)} = 27 \left(3 x - 2\right) \left(9 {x}^{2} + 6 x + 4\right) \left(x + 2\right) \left({x}^{2} - 2 x + 4\right)$

$\textcolor{w h i t e}{27 f \left(x\right)} = 27 \left(3 x - 2\right) \left(3 x - 2 \omega\right) \left(3 x - 2 {\omega}^{2}\right) \left(x + 2\right) \left(x + 2 \omega\right) \left(2 + 2 {\omega}^{2}\right)$

where $\omega = - \frac{1}{2} + \frac{\sqrt{3}}{2} i$ is the primitive Complex cube root of $1$

So the Real zeros are:

$x = \frac{2}{3}$

$x = - 2$

and the non-Real Complex zeros are:

$x = \frac{2}{3} \omega$

$x = \frac{2}{3} {\omega}^{2}$

$x = - 2 \omega$

$x = - 2 {\omega}^{2}$


text:
# Make vim yankings global via a shared text file

buffervimwindowyank

In a nutshell, I want the following:

• y(ank) actually writes the yanked text to a line in a text file, maybe in your .vim directory
• that line is prefixed with the name of the register, so "qyy yields q>my line of text
• p(ut) gets the expected result from that file

The reasoning is to get more of a 'system wide' vim buffer to solve this problem:
Sharing Vim Yank Register

• I had the same question. I found following simple solution:

map <C-y> <Esc>:'<,'>! cat \| tee ~/tmp/.myvimbuf<CR>

map <C-p> o<Esc>:.!cat ~/tmp/.myvimbuf<CR>

map <C-P> O<Esc>:.!cat ~/tmp/.myvimbuf<CR>

and make sure the directory ~/tmp is in place.

Open two instances of vim. If you now highlight a portion of text in one instance and hit CTRL+y and then hit CTRL+p or CTRL+SHIFT+p in the other instance, you should get what you want.

Obviously, the key combinations could be modified to whatever you like.

This needs cat and tee; it should work on any Mac/Linux machine.


text:
# Mass flow problem

1. Sep 22, 2016

### Mr Davis 97

1. The problem statement, all variables and given/known data
A sand-spraying locomotive sprays sand horizontally into a freight car. The locomotive and the freight car are not attached. The engineer in the locomotive maintains his speed so that the distance to the freight car is constant. The sand is transferred at a rate $\displaystyle \frac{dm}{dt}$, with a velocity $\vec{u}$ relative to the locomotive. The freight car starts from rest with an initial mass $M_0$. Find the speed of the freight car for all time t.

2. Relevant equations
Change in momentum

3. The attempt at a solution
Since this is a flow of mass problem, we will use the concept of mass transfer and momentum, rather than F = ma.

First, we isolate the system such that we initially have a stationary freight car and some sand traveling towards the car.

Taking this system as it is, we can find the change of momentum.

$P(t) = \Delta m u$, where $u$ is the relative velocity of the sand with respect to the freight car, and $\Delta m$ is the mass of the little portion of sand we are analyzing.

$P(t + \Delta t) = M \Delta v + \Delta m \Delta v$

$P(t + \Delta t) - P(t) = M \Delta v + \Delta m \Delta v -\Delta m u$

$\displaystyle \frac{\Delta P}{\Delta t} = M \frac{\Delta v}{\Delta t} + \frac{\Delta m \Delta v}{\Delta t}- \frac{\Delta m}{\Delta t} u$

$\displaystyle \frac{dP}{dt} = M \frac{dv}{dt} - \frac{dm}{dt} u$

Is this correct so far? If so, how do I proceed? Is $\displaystyle \frac{dP}{dt} = 0$?

2. Sep 24, 2016

### Mr Davis 97

Bump. Some help would be nice...

3. Sep 24, 2016

### Staff: Mentor

I might approach the problem in a slightly different way. Let $r = \frac{dm}{dt}$ be the rate at which sand is delivered to the freight car. The relative velocity between the arriving sand and the freight car is fixed at $u$ thanks to the matching speed of the locomotive.

What can you say about the mass of the freight car + sand as a function of time? Can you write an expression for it in terms of $M_o , r, \text{ and } t$?

What is the rate of delivery of momentum to the freight car (that's $\frac{dp}{dt}$)? Can you write an expression for it in terms of $r, u, \text{ and }t$?

What's another name for $\frac{dp}{dt}$?


text:
## Thursday, February 6, 2020

### Selected 2019 Particle Data Group Physical Constant Values

There are 30 experimentally determined fundamental physical constants in the core theory made up of the Standard Model and General Relativity. This is close to, but not quite, a minimal set of physical constants for these theories, as a small number of them can be derived in principle from some of the others (e.g. the Z boson mass could be, in principle derived from the W boson mass and two of the three coupling constants).  As the data below indicate, these are known to varying degrees of precision. Lighter particle masses and smaller quantities are known to greater absolute precision, while relative precision varies considerably.

One of the great unsolved problems of physics is how to derive more of these experimentally measured constants from a smaller number of fundamental quantities through a "deeper" or "within the Standard Model" theory, as they naively appear to have some sort of pattern and functional relationship to each other, but no widely accepted physics theories has done so successfully in the half a century since the Standard Model was devised.

There are also integer or simple fraction constants in the Standard Model that are set by theory like the quantum numbers of the various standard model fundamental particles, the masses of the photon and the gluon (zero), and the CP violation parameter of the strong force (zero).

The global average of the best available measurements of these measurements, compiled by the Particle Data Group (link in sidebar) which is the "default" source for these values by physicists, with a handful of additional notable data points, are set forth below. These are updated at least once a year, and sometimes more often (for physical constants for which there have been new experimental measurements) which is why I am posting the current values for easy reference.

In addition to the Standard Model constants, the core equations of the Standard Model are summed up in a Standard Model Lagrangian and some related renormalization beta functions, along with definitions of what the terms in it mean and conventions regarding how they are operationalized, together with some foundational concepts of quantum mechanics generally. The Lagrangian take a full t-shirt to set forth in full detail in fairly small print, and each of the beta functions (one for each of the 26 constants particular to the Standard Model) is similar in size.

The electromagnetic and weak force components of the Standard Model can be applied in a practical manner basically directly from the Lagrangian with high precision.

The QCD component is, in practice, to difficult to calculate with analytically from first principles in all but the most simplified, stylized and symmetric circumstances, and so a variety of approximations with limited ranges of applicability including Lattice QCD for most non-perturbative QCD calculations, and various perturbative QCD methods for higher energy scale applications, are used rather than the exact QCD Lagrangian terms, in most cases. There are roughly half a dozen to a dozen perturbative QCD approximations used to do actual calculations in wide use.

Core theory is known to be an incomplete description of the fundamental laws of nature because it does not explain the phenomena attributed to dark matter (although it does explain "dark energy").

Standard Model fundamental boson masses (3)

Higgs boson mass 125.10 ± 0.14 GeV

W boson mass 80.379 ± 0.012 GeV (the W boson mass/Z boson mass ratio is 0.088147 ± 0.00013)

Z boson mass 91.1876 ± 0.0021 GeV

Standard Model fundamental charged fermion masses (9)

The heavy quarks (c, b ad t) and the charged lepton masses are pole masses (i.e. evaluated at the energy scale where the rest mass of the particle and the energy scale are the same) and at a 2 GeV energy scale for the light quarks (u, d and s).

Electron mass 0.5109989461 ± 0.0000000031 MeV

Muon mass 105.6583745 ± 0.0000024 MeV

Tau lepton mass 1,776.86 ± 0.12 MeV

Up quark mass 2.16 + 0.49 - 0.26 MeV

Down quark mass 4.57 + 0.48 - 0.17 MeV

Strange quark mass 93 + 11 -5 MeV

Charm quark mass 1.27 ± 0.02 GeV

Bottom quark mass 4.18 + 0.03 - 0.02 GeV

Top quark mass 172.9 ± 0.4 GeV (direct measurements)

The charged lepton masses and top quark masses are direct measurements. The up, down, strange, charm and bottom quarks are only observed indirectly since they are always confined in hadrons, and in practice, are determined via Lattice QCD methods (which sometimes purport to measure these quantities with relative precisions greater than shown above from the Particle Data Group).

It is not feasible to measure the pole masses of the up, down and strange quarks, which are ill defined, because these quarks are always confined in hadrons (i.e. composite particles with components bound by the strong force) which characteristic mass-energies far in excess of their masses a regime in which perturbative QCD in which pole mass is well defined, is not applicable. The lightest hadrons containing only up and down valence quarks are the proton, the neutron and the pions. The lightest hadrons containing only strange quarks are kaons.

Strictly speaking, the "Yukawas" of these particles, rather than their masses, are the fundamental quantities in the Standard Model. A Yukawa is basically the "Higgs field charge" of a particle, while the mass of each of these particles is a function of the Higgs vev (discussed below) and its Yukawa.

Standard Model Coupling Constants (3)

Strong force, aka SU(3), coupling constant 0.1179 ± 0.0010 (this is a downward revision as of 2019) evaluated at the Z boson mass-energy scale.

Fermi coupling constant 1.166 378 7 ± 0.000 0006 × 10^−5 GeV^−2  (functionally related to the weak force, aka SU(2), coupling constant)

Fine structure constant 7.297 352 5693 ± 0.000 000 0011 ×10^−3 = 1/137.035 999 084 ± 0.000 000 021 (functionally related to the electromagnetic force, aka U(1), coupling constant)

Standard Model CKM Matrix Physical Constants (4)

The CKM matrix in the Standard Model sets forth the probability of each kind of quark producing a different flavor of quark via a W boson mediated interaction (a flavor changing charged current). This has nine components, the square of which is a probability, and can be summarized with four parameters that can be defined in multiple ways, all of which produce the same matrix elements.

Standard Model Neutrino Physical Constants (7)

Neutrino masses and mixings (7)

In principle, the Standard Model has seven parameters related to neutrinos: three neutrino mass eigenvalues and four parameters that describe the PMNS matrix which describes the probability of a neutrino of a particular flavor oscillating into a neutrino of a different flavor in a manner closely analogous to that of the CKM matrix (as in the case of the CKM matrix there are multiple ways to parameterize the PMNS matrix which produce identical values for its nine entires).

In practice, some of the PMNS matrix parameters (particularly the charge parity violation phase) are known only very imprecisely, and we know the magnitude of the differences between the mass eigenvalues and have certain floors and ceilings to the sum of the three neutrino eigenvalues (from very different methodologies) and to the maximum masses of each of the three neutrino mass eigenvalues, but have only very imprecise ranges of values for the absolute masses of the neutrino mass eigenstate. In particular, it is not yet certain if the neutrino masses observe the same lightest, medium and heaviest mass hierarchy with generation seen in the other fundamental fermions of the Standard Model or not, although cosmology and mixing parameter values taken together increasingly favor a normal mass hierarchy.

The Standard Model and the PMNS matrix, implicitly assume that neutrinos have what is known as "Dirac mass" rather than Majorana mass (in which case there would be additional CP violating phases), but without associating that with the Higgs mechanism use for the fundamental charged particles of the Standard Model and without truly explaining this with the neat consistency of the electroweak model explanation of other fundamental particle masses in the Standard Model at all.

The data below are in the form in which the actual values are directly measured (sine squared values of real valued mixing angles and squared values of mass differences) rather than the underlying parameter values which are easily derived from them with a scientific calculator.

in^2(theta23):

The full data for the parameters shown as ". . . " in the chart above are as follows:

sin^2(theta23) theta23 could be either side of a 45 degree angle based upon existing measurements and assuming a "normal" mass hierarchy for the neutrino masses. But existing experiments, while capable of determining that theta23 is not 45 degrees, but can't determine if it is greater or smaller than those values, which is why there is both an octant I and an octant II value.

 $\mathbf{0.512}{}_{\mathbf{-}\mathbf{0.022}}^{\mathbf{+}\mathbf{0.019}}$ OUR FIT  Normal ordering, octant I $\mathbf{0.542}{}_{\mathbf{-}\mathbf{0.022}}^{\mathbf{+}\mathbf{0.019}}$ OUR FIT  Normal ordering, octant II −0.028$\mathbf{0.536}{}_{\mathbf{-}\mathbf{0.028}}^{\mathbf{+}\mathbf{0.023}}$ OUR FIT  Inverted ordering

Delta m32^2 with normal ordering is 2.444 ± 0.034 (the number shown in chart is inverse mass hierarchy).

Sum of neutrino masses Σmν < 0.12 eV (95%, CMB + BAO); ≥ 0.06 eV (mixing).

Directly measured neutrino mass limits:

Neutrino Flavors

The number of neutrino flavors in the Standard Model is a theoretically determined, rather than experimentally measured value, but the experimental measurements are consistent at the two sigma level with the Standard Model value of 3:

Effective number of neutrino flavors Neff 2.99 ± 0.17 (cosmology measurements) (the expected value of this measured physical constant with exactly three types of neutrinos is 3.045 rather than zero for technical reasons related to the way that radiation impacts the relevant observables). This measurement includes all light neutrinos (up to the order of roughly 1-10 eV in mass) that oscillate with each other, and is independent of whether or not the interact via the weak force.

Number of light (i.e. less than 45 GeV) neutrino flavors from Z boson decays Nν = 2.984 ± 0.008. The Standard Model theoretical value is 3.

General Relativity Physical Constants (2)

Newton's constant G 6.708 83 ± 0.000 15 × 10^−39 hc (GeV/c^2)^−2

Cosmological constant Λ 1.088 ± 0.030 × 10^−56 cm^−2

General relativity can be boiled down to a single tensor valued equation with definitions of the symbols and terms used in it and the conventions about how this equation is applied. It is difficult, although not entirely intractable to compute predictions with analytically outside highly stylized and symmetric circumstances, and in other circumstances, approximations motivated by the core equation are used.

The cosmological constant is small enough that it can be ignored even in quite large scale general relativity calculations, such as those on the scale of whole galaxies. The cosmological constant is one of multiple ways of describing what is also known as "dark energy".

Note also that the Standard Model and General Relativity are inconsistent at a fundamental level. The Standard Model is a quantum field theory, while General Relativity is a deterministic classical physics theory. Efforts to formulate a theory of quantum gravity to bridge this gap have been elusive. The Standard Model is fully consistent, however, with Special Relativity. Fortunately, because there are few overlaps between the domains of applicability of the Standard Model an those of General Relativity, and there are simplifications and approximations that can be used in the places where there are overlaps (like the internal properties of neutron stars), the inconsistencies between the Standard Model and General Relativity doesn't greatly impair practical applications of these core theories. Usually, general relativity can be safely ignored in Standard Model applications, and the Standard Model's subatomic scale laws of nature are irrelevant to applications of General Relativity.

Other fundamental constants and unit conversions (2)

In the SI system (the most up to date version of the metric system) to physical constants, the speed of light and Planck's constant, have been used to define the meter per second, and the Joule*second, respectively, at values as close as experimental measurements as of 2019 allowed to the values of the more arbitrary values of the meter, second and Joule prior to being defined in terms of fundamental constants, in order to insure back compatibility.

Both of these constants are used in the Standard Model. The speed of light is a physical constant also used in General Relativity.

Speed of light in vacuum c 299 792 458 m s−1  (exact due to revised definition of units used)

Planck constant h 6.626 070 15×10−34 J s (exact due to revised definition of units used)

The reduced Planck constant a.k.a. "h bar" h (h divided by two pi) is 6.582 119 569. . . × 10−22 MeV s. This value is exact, but it a transcendental number since it includes a factor of pi (i.e. it has an infinite number of base ten digits).

The units of electromagnetic charge in the SI system, in contrast, is not defined directly in terms of fundamental electron charge. One electron charge in SI units is 1.602 176 634×10^−19 C

Footnote Regarding Derived Physical Constants

Many other physical constants, in principle, are possible to derive from the fundamental constants in the Standard Model. In practice, however, direct measurements of these quantities are actually used. Among them are:

* The decay width of every fundamental particle and hadron (this is inversely proportional to the half-life and mean lifetime of a an unstable particle). In theory, electrons, photons and gluons are stable and do not decay or oscillate in the Standard Model, as are protons and as are neutrons that are bound in stable atomic nuclei. The decay width of a free quark other than a top quark, however, isn't terribly meaningful or well defined in the Standard Model, as other flavors of quarks are always confined outside quark-gluon plasma.

* The electric and magnetic dipole moments of every particle.

* The decay width and branching fraction of every possible decay path of fundamental particles and hadrons that are not stable. The hypothetical lepton number violating processes like neutrinoless double beta decay and flavor changing neutral currents are not possible at the tree level in the Standard Model and are profoundly suppressed beyond the tree level in the Standard Model.

* The nuclear binding force between protons and neutrons in an atom.

* The half-life of every unstable atomic isotype.

* The rest mass of every possible hadron.

* The chemical properties of every atomic isotype.

* All of the Standard Model constants other than c and Planck's constant, at energy scales other than those at which they are conventionally quoted per renormalization beta functions, all coefficients of which can be derived from theoretical first principles. This may be ill defined at extremely high Planck energy scales that have never existed anywhere in the universe at any time more than tiny factions of a nanosecond after the Big Bang.

* The characteristic QCD energy scale called Λ.

* The Higgs vacuum expectation value (vev), which is also derived from the Fermi coupling constant, per a non-PDG source using PDG values, is 246,227.9579 MeV with an uncertainty on the order of 0.0010 MeV.

Some of the derived physical constants, such as the masses of the proton and neutron, are known to far greater precision than the fundamental constants from which they are derived.

This is particularly common for physical constants that require QCD to determine. This is because doing calculations with QCD is computationally extremely difficult to do to any given level of precision, and for the closely related reason that as a result of this computational difficulty, it is very difficult to measure the quark masses and the strong force coupling constant to high relative precision. The roughly 1% relative precision in the strong force coupling constant places a ceiling on the precision of all QCD calculations to the extent that they depend upon this fundamental physical constant as all QCD calculations do to some degree or another.

Footnote regarding non-fundamental, non-derived physical constants

Fundamental physical constants are those necessary to apply the "laws of nature" set forth in the Standard Model and General Relativity and any beyond the Standard Model theory that adds additional such constants. Derived physical constants are those that can, in principle, be determined from the fundamental physical constants.

There are also a variety of observationally measured physical constants which are neither fundamental, nor capable of being derived from fundamental physical constants. These are mostly descriptive constants of particular physical systems including the universe.

These include: the mass of the Earth, the Sun, the Milky Way galaxies, the universe and various other physical objections or systems in space; the age of the universe; Hubble's constant (although this is closely related to the cosmological constant); the aggregate baryon number of the universe, the aggregate lepton number of the universe; the number of bosons in existence at any one time in the universe; the mix of atomic elements in the universe or other smaller physical systems (e.g. the planet Earth or the Sun); the distances between particular places on Earth and/or in the Universe; the proportionate mix of ordinary matter other than neutrinos, neutrinos, radiation, dark matter,  and dark energy in the total mass-energy of the universe.

neo said...

i'm unclear how gravity in GR as 4D spacetime can be represnted as a spin-2 boson of QFT

2 seemingly different categories

andrew said...

This is not a short or simple question, although the leading textbooks in GR contain derivations of their analysis. Worthy of a full post, at a minimum.

andrew said...

FWIW, in my humble opinion, the two are not, in fact, exactly equivalent, although a spin-2 boson of QFT in the classical limit, is approximately equivalent to GR in 4D.

For example, generically, in a QFT where a force is transmitted via a carrier boson, the energy associated with the force can be localized subject to the limitation of the Heisenberg uncertainty principle, which is not true in classical GR. Similarly, the QFT has both local and global conservation of mass-energy which is inconsistent with the cosmological constant. GR conserves mass-energy only locally, and not globally.


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# Show $a_1 \mathbb Z$ ∩ $a_2 \mathbb Z$ = $N \mathbb Z$ [duplicate]

Let $$a_1$$, $$a_2$$$$\mathbb Z$$ and let $$N$$ = lcm($$a_1$$, $$a_2$$).

Show

$$a_1 \mathbb Z$$$$a_2 \mathbb Z$$ = $$N \mathbb Z$$

where $$c \mathbb Z$$ = {$$c · n : n$$$$\mathbb Z$$}.

My attempt:

let $$x$$$$a_1 \mathbb Z$$$$a_2 \mathbb Z$$, thus $$x$$$$a_1 \mathbb Z$$ and $$x$$$$a_2 \mathbb Z$$, so we can write $$x$$ = $$a_1.n$$ and $$x$$ = $$a_2.m$$, where $$n,m$$$$\mathbb Z$$.

Since $$N$$ = lcm($$a_1$$, $$a_2$$), then $$a_1$$/$$N$$ and $$a_2$$/$$N$$, thus there exists $$l,s$$$$\mathbb Z$$ such that $$l.a_1 = N$$ and $$s.a_2 = N$$.

Let $$y$$$$N \mathbb Z$$, thus we can write $$y$$ = $$N.b$$ for some $$b$$$$\mathbb Z$$

But $$x$$ = $$a_1.n$$ = $$\frac Nl . n$$, and $$x$$ = $$a_2.m$$ = $$\frac Ns . m$$.

But $$x=x$$, thus $$\frac Nl . n$$ = $$\frac Ns . m$$ and hence $$\frac nl$$ = $$\frac ms$$, and since $$b$$$$\mathbb Z$$, it can be written as $$\frac nl$$.

Therefore $$y = N.b = N. \frac nl$$ = $$x$$.

Hence for all $$x$$$$a_1 \mathbb Z$$$$a_2 \mathbb Z$$, $$x$$$$N \mathbb Z$$ and so, $$a_1 \mathbb Z$$$$a_2 \mathbb Z$$ = $$N \mathbb Z$$

## marked as duplicate by Javi, ThorWittich, Yanior Weg, Matthew Daly, nmasantaSep 16 at 2:17

• $\operatorname{lcm}(a,b)$ is the unique positive integer $k$ such that $a \Bbb Z \cap b \Bbb Z = k \Bbb Z$. This is the abstract way to define the lcm. – Henno Brandsma Sep 15 at 7:21


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In this installment of our series on type coercions, we’re going to introduce a way to support type widening in a language interpreter. We’ll present a general approach based on semilattices of types and a particular implementation of this approach that uses a straightforward encoding in Scala’s type system. We’ll work from a simple interpreter to allow for a clear exposition, but our general techniques will be applicable to more involved languages as well.

### Widening functions

A widening function maps values from some type T to a wider type U. We can implement a trivial but generic type-widening method based on the partial ordering of types encoded in Scala’s type system:

Invoking widen[A,B] on an argument of type A will succeed if there is a witness object for A => B. By default, we’ll be able to see the predefined widenings in Scala, including reflexive widenings. Note that there are no implicitly defined narrowings, though:

It’s important to note that we could declare other witnesses to A => B for other A and B types and have them in scope; we aren’t constrained by Scala’s predefined definitions or by static relationships between implementation types. (We’ll come back to this point later, when we’re thinking about how to model types in the interpreted language.)

### A simply-typed interpreter

We’ll start with a simple interpreter with four kinds of values (integers, doubles, strings, and nulls) and one kind of expression representing numeric addition or string concatenation, depending on the types of its operands. It is a stretch to call the language embodied in this interpreter “typed” (since it has only literal values and expressions but no variables). However, because of the way we’ve encoded the interpreter in Scala, it is impossible to express programs with runtime errors. In particular, it is only possible to create an AddExpression with two arguments that evaluate to values of the same type.

If we have an expression of the form t1 • t2, where the left-hand side is of type T1 and the right-hand side is of type T2, we will be able to convert this to an expression in which both operands have the same type if the following conditions are met:

1. There must exist some type U such that T1UT2U, and
2. There must exist widening functions with the signatures T1 ⇒ U and T2 ⇒ U.

Finding U is simply finding the least upper bound of T1 and T2 on a semilattice of types. Once we have this least upper bound, if we also have appropriate widening functions, we can convert both t1 and t2 to values of the same type. In the following code, we extend our simple interpreter by modeling interpreter types in Scala, making types properties of values, and adding widening conversions to AddExpression by explicitly encoding the partial ordering of types – in this case, based on a relationship between the nativeType type member of each Type class. (We’re doing widening statically through Scala’s type system in this case, but there’s no reason why we couldn’t take a similar approach dynamically, handling errors by raising exceptions at runtime.)

In WideningInterpreter we extend AddExpression to allow it to have two potentially distinct argument types (Value[T] and Value[U]) and by requiring evidence of an implicit conversion from a pair of values with distinct types to a pair of values with the same type.1 We define two witness functions, leftWiden for the case in which the left element of the pair is narrower than the right, and rightWiden for the case in which the right element of the pair is narrower than the left. In both cases, we determine that a type T is narrower than another type U if Scala knows how to widen values of the representation type (Type#nativeType) of T to the representation type of U; this is the case if an implicit resolution exists for the conv argument.

The problem we might encounter is that, because our partial ordering is reflexive, if T and U are the same type, then there will be witnesses both for T#nativeType => U#nativeType and U#nativeType => T#nativeType. So if we were to have naive implementations of leftWiden and rightWiden that only depended on evidence of such a conversion, Scala would be unable to unambiguously resolve which would apply in the case of monomorphic AddExpressions. We resolve this problem by adding a test for type inequality (due to Miles Sabin) to the implicit argument list of rightWiden, so that it will not apply if the arguments are of the same type.2

Note that the partial ordering among interpreter types (IntType, DoubleType, and StringType) does not depend on Scala-level subtyping relationships between interpreter types. This is important because in a more realistic language we will want the flexibility to model data types independently of properties of our object-language implementation. Instead, we have a generic partial ordering in this example based on predefined relationships between representation types, and we could extend the partial ordering to other types by adding other instances of =>[A,B] for other types of interest.

For a small number of interpreter types, we could also explicitly encode the partial ordering, as in the example below:

Since in this example we have a total ordering among types, we can also easily widen one argument at a time by adding a witness object for the least upper bound of T and U,3 as in the example below:

Again, a similar approach would be applicable in an untyped Scala representation of interpreter-language types: we could represent types as terms, implement the least-upper-bound relation as a partial function mapping from a pair of terms to the least upper bound of the pair, and implement widenings as functions taking a value and a term representing the type to widen to.

1. See the signature for the widen implicit argument to AddExpression: (Value[T], Value[U]) => (Value[V], Value[V])

2. This type-inequality test will fail if it is given two identical types because both neqAmbig1 and neqAmbig2 will both be applicable.

3. It occurred to me while developing these examples that using witness objects in this way is a lot like forward-chaining logic programming. (Note that we use negation-as-failure in implicit resolution when testing for type inequality and we use implicit arguments to guide further implicit resolution with the WiderThan witness.) Unsurprisingly, it turns out that other people have had the same idea! See the discussion on this post or this talk for two examples.

type systemssqlhivescalatype coercions • You may reply to this post on Twitter or


text: How to compute the first eigenvalue of $M = R \times {}_{\cosh t}N$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T09:27:08Z http://mathoverflow.net/feeds/question/119831 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119831/how-to-compute-the-first-eigenvalue-of-m-r-times-cosh-tn How to compute the first eigenvalue of $M = R \times {}_{\cosh t}N$ jiangsaiyin 2013-01-25T13:40:09Z 2013-01-27T06:03:12Z <p>$$M = R \times N$$with the warped product metric$$d{s^2} = d{t^2} + {\cosh ^2}\left( t \right)ds_N^2$$where N(dimN=n-1) is a compact manifold with $$Ric \ge - \left( {n - 2} \right)$$It should be mentioned that M may not be a Riemannian manifold but an Alexandrov space.So how to compute the first eigenvalue of M?If we restrict to the case $$N = {S^{n - 1}}\left( {\frac{1}{2}} \right)$$an n-1 dim sphere with radius 1/2,then the result?</p>


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# Cotangent bundle tensor product tangent bundle

What is the meaning of Cotangent bundle tensor product tangent bundle:

$T^*M\otimes TM$? what will an element of this space be?

This is isomorphic to $Hom(TM,TM)$. Namely it is the endomorphism bundle of the tangent bundle. In general given two vector spaces(or bundles) we have an isomorphism between $W \otimes V^*$ and $Hom(V,W)$.
• You need $V$ to be finite-dimensional here. – Qiaochu Yuan Jan 20 '16 at 7:28
• @QiaochuYuan Or for $W$ to be finite, no? Either way every element of $\mathrm{Hom}(V,W)$ has finite rank. – Oscar Cunningham Jan 21 '16 at 11:29


text:
I am trying to find local extrema for multi variable functions. Solution for . The difference is that the solution to the inequality is not the drawn line but the area of the coordinate plane that satisfies the inequality. Required fields are marked *, How to find the boundary line of an inequality. To illustrate this point, we first turn to the minimization of a function F of n real variables over a convex set C; the minimizer x is characterized by the condition Then, starting at (say) the point with the highest Y value, trace a route around the outside following the connected line with the smallest exterior angle/bearing. You must be logged into your Facebook account in order to share via Facebook. boundaries := [[-1<=x],[ x<=1], [-1<=y], [y<=1]]; We are interested in variational problems involving weights that are singular at a point of the boundary of the domain. Example 1: Graph the linear inequality y > 2x − 1. Is it a solution to the inequality? Inequality solver that solves an inequality with the details of the calculation: linear inequality, quadratic inequality. How many times has meghan markle been married www cbs young and the restless com video, Graphing linear inequalities (Pre-Algebra, Graphing and functions) � Mathplanet, Your email address will not be published. Please log-in to your MaplePrimes account. The solutions for a linear inequality are in a region of the coordinate plane. Linear inequalities can be graphed on a coordinate plane.The solutions for a linear inequality are in a region of the coordinate plane. I am trying to find local extrema for multi variable functions. In today's blog, I define boundary points and show their relationship to open and closed sets. Let’s take another point on the left side of the boundary line and test whether or not it is a solution to the inequality . The easiest solution method for polynomial inequalities is to use what you know about polynomial shapes, but the shape isn't always enough to give you the answer. Every point in that region is a solution of the inequality. The boundary line for the linear inequality goes through the points (-6,-4) and (3,-1). The boundary line for the inequality is drawn as a solid line if the points on the line itself do satisfy the inequality, as in the cases of ≤ and ≥. In this tutorial, you'll learn about this kind of boundary! A linear inequality describes an area of the coordinate plane that has a boundary line. 5. If points on the boundary line are solutions, then use a solid line for drawing the boundary line. The point clearly looks to be to the left of the boundary line, doesn’t it? Using Hessian matrix and eigen values I am able to find the global extrema. The resulting values of x are called boundary points or critical points. All points on the left are solutions. The test-point method from your book will give you the answer eventually, but it can be a lot of work. This boundary is either included in the solution or not, depending on the given inequality. By … Be sure to show your boundary point, number line, and test number work. Many free boundary problems can profitably be viewed as variational inequalities for the sake of analysis. 62/87,21 The boundary of the graph is the graph of . Combine multiple words with dashes(-), and seperate tags with spaces. A point is in the form \color{blue}\left( {x,y} \right). You must be logged in to your Twitter account in order to share. Since this is an "or equal to" inequality, the boundary points of the intervals (the intercepts themselves) are included in the solution. Lance Taylor with Özlem Ömer, Macroeconomic Inequality from Reagan to Trump: Market Power, Wage Repression, Asset Price Inflation, and Industrial Decline, Cambridge University Press, 2020. Please refresh the page and try again. boundary point means. Click the button below to share this on Google+. 1. Select a point not on the boundary line and substitute its x and y values into the original inequality. Let’s take another point on the left side of the boundary line and test whether or not it is a solution to the inequality . c. Substitute 50 for x and 50 for y in the inequality . Linear inequalities can be graphed on a coordinate plane. We show that by making the line dashed, not solid. To find this region, we will graph each inequality separately and then locate the region where they are both true. Once your linear equation is graphed, you then must focus on the inequality symbol and perform two more steps. This indicates that any ordered pair in the shaded region, including the boundary line, will satisfy the inequality. The region that does not contain (0, 0) is shaded. Posted: Rohith 60. optimization extrema inequality + Manage Tags. 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Math Help - Please Help Find Real Solutions for X

1. Please Help Find Real Solutions for X

$\frac{5}{x+2} - \frac{2x-1}{5} = 0$

2. Originally Posted by BadMaterial
$\frac{5}{x+2} - \frac{2x-1}{5} = 0$

Multiply both sides by the common denominator and then make some order and then multiply both sides by $-1$:

$25-2x^2-3x+2=0\,\Longrightarrow\,2x^2+3x-27=0$

Now just solve this easy quadratic equation using the well known formula:

$x_{1,2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ ....I think the solutions are $-4.5\,,\,\,3$ , but check this carefully.

Tonio