text:
Now showing items 64724-64743 of 70386

• Transport properties of a Bose-Einstein condensate with tunable interactions in the presence of a disordered or single defect potential

(2010)
Bose-Einstein condensates (BECs) have proven to be remarkable systems with which to study some of the foundational models of condensed matter physics. The observation of a critical velocity for the breakdown of superfluidity ...
• Transport Properties of One-dimensional Quantum Systems

(2018-08-08)
We study quantum dynamical properties of discrete one-dimensional models of Schrodinger operators. The first goal of this thesis is to understand the time evolution of initial states supported on more than one site. We ...
• Transport Properties of Topological Phases in Broken Gap Indium Arsenide/Gallium Antimonide Based Quantum Wells

(2012)
The quantum Spin Hall Insulator (QSHI) is a two-dimensional variant of a novel class of materials characterized by topological order, whose unique properties have recently triggered much interest and excitement in the ...
• Transportation technology as a determinant of urban growth

(1969)
THESIS: that changing transportation technology helped determine the limit of maximum expansion of the city and that the physical characteristics of transportation have determined the pattern of growth to that limit. ...
• Transpositions CU

(2006)
The demarcations of territorial urban isms have not embedded within themselves with the potential to negotiate in the city-to-come. Transpositions CU considers the UNAM's Ciudad Universitaria (Mexico City) as a point of ...

(1999)
• Transvaluing immaturity: Hellenism, primitivism, and a reverse discourse of male homosexuality in late-Victorian and Edwardian narrative

(1998)
Late-Victorian and Edwardian British ideology represented male homosexuals as psychically and somatically arrested. In response, many middle-class, male homosexual writers drew on versions of Hellenic pederasty and primitivism ...

(1968)
• Transverse momentum spectra of inclusive b jets in pPb collisions at sNN=5.02 TeV

(2016)
We present a measurement of b jet transverse momentum (pT) spectra in proton-lead (pPb) collisions using a dataset corresponding to about 35 nb−1 collected with the CMS detector at the LHC. Jets from b quark fragmentation ...
• Transverse relaxation in sandstones due to the effect of internal field gradients and characterizing the pore structure of vuggy carbonates using NMR and Tracer analysis

(2013-09-16)
Nuclear magnetic resonance (NMR) has become an indispensable tool in petroleum industry for formation evaluation. This dissertation addresses two problems. • We aim at developing a theory to better understand the phenomena ...
• Transverse single-spin asymmetry and cross section for π0 and η mesons at large Feynman x in p↑+p collisions at √s=200  GeV

(2012)
Measurements of the differential cross section and the transverse single-spin asymmetry, AN, vs xF for π0 and η mesons are reported for 0.4<xF<0.75 at an average pseudorapidity of 3.68. A data sample of approximately ...
• TRANSVERSE SPIN DEPENDENCE OF THE PROTON-PROTON TOTAL CROSS-SECTION FROM 300 MEV TO 800 MEV

(1984)
The transverse spin dependence of the proton-proton total cross-section, (DELTA)(sigma)(,T), the difference between the proton-proton total cross-sections for protons in anti-parallel and parallel transverse spin states, ...
• Transverse spin-dependent azimuthal correlations of charged pion pairs measured in p↑ + p collisions at s=500 GeV

(2018)
The transversity distribution, which describes transversely polarized quarks in transversely polarized nucleons, is a fundamental component of the spin structure of the nucleon, and is only loosely constrained by global ...
• Transverse thermal conductivity of thin dielectric films

(1998)
Earlier work revealed that the transverse thermal conductivity of thin films of amorphous SiO$\sb2$ and $\rm Si\sb3N\sb4$ deposited on monocrystalline silicon decreased substantially when the film thickness was less than ...

(1938)
• Trap losses induced by near-resonant Rydberg dressing of cold atomic gases

(2016)
The near-resonant dressing of cold strontium gases and Bose-Einstein condensates contained in an optical dipole trap (ODT) with the 5s30s3S1 Rydberg state is investigated as a function of the effective two-photon Rabi ...
• Trap-loss collisions of ultracold lithium atoms

(1995)
Accurate measurements are presented of the rate of trap-loss-producing collisions between ultracold magneto-optically trapped Li7 atoms for a range of trap laser intensities and frequencies. Intensities from near the atomic ...
• Trapping and evaporation of strontium-87 and strontium-88 mixtures

(2010)
This thesis describes trapping and evaporative cooling of ultracold 87Sr and 88Sr mixtures in an optical dipole trap to produce the first Bose-Einstein condensate (BEC) of 88Sr. Furthermore, this work presents thermalization ...

(2002)
• Trauermusik

(1980-02-21)


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ForK Library

# krigingfuncvariance.f90 File Reference

Subroutine to Predict variance of function values from Kriging Surface (requires buildkriging to be called first) More...

Go to the source code of this file.

## Functions/Subroutines

subroutine krigingfuncvariance (ndim, ntot, X, stot, H, hyper, mtot, Xm, Hm, S, covarflagi)
Because of the stochastic nature of a Kriging model, the variance associated with mean function values can be predicted throughout the domain. The variance for function predictions are given as:

$V\left[y_*\right]= cov(\vec{x}_{*},\vec{x}_{*})-k_*^T K^{-1} k_*+ R(\vec{x}_*) A^{-1} R(\vec{x}_*)^{T}$

## Detailed Description

Subroutine to Predict variance of function values from Kriging Surface (requires buildkriging to be called first)

Definition in file krigingfuncvariance.f90.

## Function Documentation

 subroutine krigingfuncvariance ( integer,intent(in) ndim, integer,intent(in) ntot, real(8),dimension(ndim,ntot),intent(in) X, integer,intent(in) stot, real(8),dimension(stot,ntot),intent(in) H, real(8),dimension(ndim+2),intent(in) hyper, integer,intent(in) mtot, real(8),dimension(ndim,mtot),intent(in) Xm, real(8),dimension(stot,mtot),intent(in) Hm, real(8),dimension(mtot),intent(out) S, integer,intent(in) covarflagi )

Because of the stochastic nature of a Kriging model, the variance associated with mean function values can be predicted throughout the domain. The variance for function predictions are given as:

$V\left[y_*\right]= cov(\vec{x}_{*},\vec{x}_{*})-k_*^T K^{-1} k_*+ R(\vec{x}_*) A^{-1} R(\vec{x}_*)^{T}$

The first two terms, $$cov(\vec{x}_{*},\vec{x}_{*})-k_*^T K^{-1} k_*$$, account for the variance of associated with a zero mean gaussian process while the last term $$R(\vec{x}_*) A^{-1} R(\vec{x}_*)^{T}$$ accounts for the variance associated with the uncertainty associated with the regression parameters, where $$R(\vec{x}_*)$$ is given as:

$R(\vec{x}_{*}) = h^{T} - k_*^T K^{-1} H^{T}$

To compute the function variance, the hyperparameters within the covariance function are required (supplied by buildkriging). Hence, these subroutines must be used in conjunction with buildkriging called first.
Using the variance, a confidence interval can be established on the function predictions. For example, for a well training model, $$95 \%$$ of the true function values should lie within 2 standard deviations from the mean prediction, represented functionally as:

\begin{eqnarray*} y_{lower} &=& y_{*} - 2 \sqrt{V\left[y_*\right]} \\ y_{upper} &=& y_{*} + 2 \sqrt{V\left[y_*\right]} \end{eqnarray*}

Using this variance prediction, predictions from the model can be bounded. Additionally, the quality of the Kriging model can be tested using cross-validation procedures.

Date:
May 2, 2012
Parameters:
 in) ndim : The dimension of the problem in) ntot : The number of Training points in) X : The location of the training points (size=[ndimxntot]) in) stot : Number of Terms in the regression in) H: The collocation matrix for the regression (size=[stotxntot]) in) hyper: Hyperparameters for the Kriging Model (size=[ndim+2]) Supplied by buildkriging subroutine in) mtot : The number of test points, the places where function prediction are desired in) Xm : The location of the test points (size=[ndimxmtot]) in) Hm : The collocation matrix evaluated at the test points (size=[stotxmtot]) out) S: The variance of the function values (size=[mtot]) Predicting the variance values requires the construction of the covariance matrix for the training data so it is more expensive than function predictions Best to predict the variance associated with multiple points using a single function call if possible. in) covarflagi: Flag to govern which covariance function is used covarflag==0 Uses Matern function with $$\nu=1/2$$ covarflag==1 Uses Matern function with $$\nu=3/2$$ covarflag==2 Uses Matern function with $$\nu=5/2$$ The parameter $$\nu$$ governs the smoothness and differentiability of the covariance function. For function only Kriging, all three options are available. Must supply the same covariance flag as used in buildkriging.

Definition at line 52 of file krigingfuncvariance.f90.

Referenced by funcmod::func_kriging(), and krigingwrapper().

format: plain


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Sidebar

seminars:stat:141023

Statistics Seminar
Department of Mathematical Sciences

 DATE: Thursday, October 23, 2014 TIME: 1:15pm to 2:15pm PLACE: OW 100E SPEAKER: Wenyu Du (Binghamton University) TITLE: On Robustness of the Shiryaev–Roberts Change-Point Detection Procedure under Parameter Misspecification in the Post-Change Distribution

Abstract

The gist of the quickest change-point detection problem is to detect the presence of a change in the statistical behavior of a series of sequentially made observations, and do so in an optimal way. When optimality is understood either in the generalized Bayesian sense or as defined in Shiryaev’s multi-cyclic setup, the so-called Shiryaev–Roberts (SR) detection procedure has been shown to be the “best one can do”, provided, however, that both the pre- and post-change distributions of the observations are fully specified. As the latter condition is practically restrictive, we consider a more realistic setup, viz. one where the post-change distribution is assumed known only up to a parameter, so that the “anticipated” value of the parameter need not match the true (unknown) one. The question of interest is the sensitivity (or robustness) of the otherwise “best” SR procedure with respect to a possible misspecification of the post-change parameter. To answer this question, we develop a numerical framework to compute the performance of the SR procedure that may be “out of tune” in the way of the post-change parameter. We then employ the proposed framework in a specific Gaussian scenario and quantify the effect that a misspecification of the post-change parameter has on the SR procedure’s performance. The overall qualitative conclusion is that the less (more) contrast the change, the less (more) robust the SR procedure devised to detect it.


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# On a particularly boring transatlantic flight, one of the authors amused himself by counting the heads...

###### Question:

On a particularly boring transatlantic flight, one of the authors amused himself by counting the heads of the people in the seats in front of him. He noticed that all 28 of them either had black hair or had a whole row to themselves (or both). Of this total, 25 had black hair and 7 were fortunate enough to have a whole row of seats to themselves. How many of the black-haired people had whole rows to themselves? people

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Colloquia and Seminars

To join the email distribution list of the cs colloquia, please visit the list subscription page.

Computer Science events calendar in HTTP ICS format for of Google calendars, and for Outlook.

Upcoming Colloquia & Seminars

• Interplays between Machine Learning and Optimization

Speaker:
Tomer Koren - CS-Lecture
Date:
Monday, 18.12.2017, 10:30
Place:
Room 601 Taub Bld.
Affiliation:

Over the past two decades, machine learning has rapidly evolved and emerged as a highly influential discipline of computer science and engineering. One of the pillars of machine learning is mathematical optimization, and the connection between the two fields has been a primary focus of research. In this talk, I will present two recent works that contribute to this study, focusing on online learning---a central model in machine learning for sequential decision making and learning under uncertainty. In the first part of the talk, I will describe a foundational result concerned with the power of optimization in online learning, and give answer to the question: does there exist a generic and efficient reduction from online learning to black-box optimization? In the second part, I will discuss a recent work that employs online learning techniques to design a new efficient and adaptive preconditioned algorithm for large-scale optimization. Despite employing preconditioning, the algorithm is practical even in modern optimization scenarios such as those arising in training state-of-the-art deep neural networks. I will present the new algorithm along with its theoretical guarantees and demonstrate its performance empirically. Short Bio: ========== Tomer Koren is a Research Scientist at Google, Mountain View. His research focuses on machine learning and optimization, with an emphasis on online and statistical learning, sequential decision making, and stochastic optimization. Tomer joined Google in 2016 after receiving his Ph.D. from the Technion---Israel Institute of Technology, under the guidance of Prof. Elad Hazan. During his doctoral studies, he was also a research intern with Microsoft Research Herzliya, Microsoft Research Redmond, and Yahoo Research Labs in Haifa.

• Faster and Simpler Distributed CONGEST-Algorithms for Testing and Correcting Graph Properties

Speaker:
Guy Even - COLLOQUIUM LECTURE
Date:
Tuesday, 19.12.2017, 14:30
Place:
Room 337 Taub Bld.
Affiliation:
Dept. of Electrical Engineering-Systems, Tel-Aviv University
Host:
Yuval Filmus

We consider the following problem introduced by [Censor-Hillel et al., DISC 2016]. Design a distributed algorithm (called an $\epsilon$-tester) that tests whether the network over which the algorithm is running satisfies a given property (e.g., acyclic, bipartite) or is $\epsilon$-far from satisfying the property. If the network satisfies the property, then all processors must accept. If the network is $\epsilon$-far from satisfying the property, then (with probability at least $2/3$) at least one processor must reject. Being $\epsilon$-far from a property means that at least $\epsilon\cdot |E|$ edges need to be deleted or inserted to satisfy the property. Suppose we have an $\epsilon$-tester that runs in $O(Diameter)$ rounds. We show how to transform this tester to an $\epsilon$-tester that runs in $O((\log |V|)/\epsilon))$ rounds. Since cycle-freeness and bipartiteness are easily tested in $O(Diameter)$ rounds, we obtain $\epsilon$-testers for these properties with a logarithmic number of rounds. Moreover, for cycle-freeness, we obtain a \emph{corrector} of the graph that locally corrects the graph so that the corrected graph is acyclic. The corrector deletes at most $\epsilon \cdot |E| + distance(G,P)$ edges and requires $O((\log |V|)/\epsilon))$ rounds. Joint work with Reut Levy and Moti Medina. Short Bio: Guy Even received his B.Sc. degree in 1988 in Mathematics and Computer Science from the Hebrew University. Received his M.Sc. and D.Sc. in Computer Science from the Technion in 1991 and 1994, respectively. Dr. Even spent his post-doctorate in the University of the Saarland during 1995–1997 with Prof. Wolfgang Paul. Since 1997, he is a faculty member in the School of Electrical Engineering in Tel-Aviv University. Dr. Even is interested in algorithms and their applications in various fields. He has published papers on the theory of VLSI, approximation algorithms, computer arithmetic, online algorithms, frequency assignment, scheduling, packet-routing, linear-programming decoding of LDPC codes, and rateless codes. He is on the editorial board of "Theory of Computing Systems". Together with Moti Medina, Dr. Even wrote a digital hardware textbook titled "Digital Logic Design: A Rigorous Approach" published in 2012 by Cambridge University Press. ============================================= Refreshments will be served from 14:15 Lecture starts at 14:30

• Overcoming Intractability in Learning

Speaker:
Roi Livni - CS-Lecture
Date:
Thursday, 21.12.2017, 10:30
Place:
Room 601 Taub Bld.
Affiliation:
Princeton University, CS Dept.

Machine learning has recently been revolutionized by the introduction of Deep Neural Networks. However, from a theoretical viewpoint these methods are still poorly understood. Indeed the key challenge in Machine Learning today is to derive rigorous results for optimization and generalization in deep learning. In this talk I will present several tractable approaches to training neural networks. At the second part I will discuss a new sequential algorithm for decision making that can take into account the structure in the action space and is more tuned with realistic decision making scenarios. I will present our work that provides some of the first positive results and yield new, provably efficient, and practical algorithms for training certain types of neural networks. In a second work I will present a new online algorithm that learns by sequentially sampling random networks and asymptotically converges, in performance, to the optimal network. Our approach improves on previous random features based learning in terms of sample/computational complexity, and expressiveness. In a more recent work we take a different perspective on this problem. I will provide sufficient conditions that guarantee tractable learning, using the notion of refutation complexity. I will then discuss how this new idea can lead to new interesting generalization bounds that can potentially explain generalization in settings that are not always captured by classical theory. In the setting of reinforcement learning I will present a recently developed new algorithm for decision making in a metrical action space. As an application, we consider a dynamic pricing problem in which a seller is faced with a stream of patient buyers. Each buyer buy at the lowest price in a certain time window. We use our algorithm to achieve an optimal regret, improving on previously known regret bound. Short Bio: ========== Roi Livni is a research instructor at the computer science department in Princeton University. He is a recipient of the Eric and Wendy Schmidt fellowship for strategic innovation, and a Yad Hanadiv fellow for postdoctoral position. He completed his PhD at the Hebrew University, during which he was a recipient of a Google Europe fellowship in learning theory. He also served, during his graduate studies, as a long term intern at Microsoft Research. His graduate work won a Best paper award in ICML'13, and a best student paper award in COLT'13.

• On the Expressive Power of ConvNets and RNNs as a Function of their Architecture

Speaker:
Amnon Shashua - COLLOQUIUM LECTURE
Date:
Tuesday, 2.1.2018, 14:30
Place:
Room 337 Taub Bld.
Affiliation:
Hebrew University; CEO & CTO, Mobileye; Senior Vice President, Intel Corporation
Host:
Yuval Filmus
• Bridging the Gap between End-users and Knowledge Sources: Discovery, Selection and Utilization

Speaker:
Yael Amsterdamer - COLLOQUIUM LECTURE
Date:
Tuesday, 9.1.2018, 14:30
Place:
Room 337 Taub Bld.
Affiliation:
Department of Computer Science, Bar Ilan University
Host:
Yuval Filmus

TBA

• Personalization is a Two-Way Street

Speaker:
Ronny Lempel - COLLOQUIUM LECTURE
Date:
Tuesday, 16.1.2018, 14:30
Place:
Room 337 Taub Bld.
Affiliation:
Outbrain
Host:
Yuval Filmus

Recommender systems are first and foremost about matching users with items the systems believe will delight them. The "main street" of personalization is thus about modeling users and items, and matching per user the items predicted to best satisfy the user. This holds for both collaborative filtering and content-based methods. In content discovery engines, difficulties arise from the fact that the content users natively consume on publisher sites does not necessarily match the sponsored content that drives the monetization and sustains those engines. The first part of this talk addresses this gap by sharing lessons learned and by discussing how the gap may be bridged at scale with proper techniques. The second part of the talk focuses on personalization of audiences on behalf of content marketing campaigns. From the marketers' side, optimizing audiences was traditionally done by refining targeting criteria, basically limiting the set of users to be exposed to their campaigns. Marketers then began sharing conversion data with systems, and the systems began optimizing campaign conversions by serving the campaign to users likely to transact with the marketer. Today, a hybrid approach of lookalike modeling combines marketers' targeting criteria with recommendation systems' models to personalize audiences for campaigns, with marketer ROI as the target. This talk was given as a keynote in ACM RecSys 2017. Short bio: ========== Ronny Lempel joined Outbrain in May 2014 as VP of Outbrain's Recommendations Group, where he oversees the computation, delivery and auction mechanisms of the company's recommendations. Prior to joining Outbrain, Ronny spent 6.5 years as a Senior Director at Yahoo Labs. Ronny joined Yahoo in October 2007 to open and establish its Research Lab in Haifa, Israel. During his tenure at Yahoo, Ronny led R&D activities in diverse areas, including Web Search, Web Page Optimization, Recommender Systems and Ad Targeting. In January 2013 Ronny was appointed Yahoo Labs' Chief Data Scientist in addition to his managerial duties. Prior to joining Yahoo, Ronny spent 4.5 years at IBM Research, where his duties included research and development in the area of enterprise search systems. During his tenure at IBM, Ronny managed the Information Retrieval Group at IBM's Haifa Research Lab for two years. Ronny received his PhD, which focused on search engine technology, from the Faculty of Computer Science at Technion, Israel Institute of Technology in early 2003. Ronny has authored over 40 research papers in leading conferences and journals, and holds 18 granted US patents. He regularly serves on program and organization committees of Web-focused conferences, and has taught advanced courses on Search Engine Technologies and Big Data Technologies at Technion. ========================================= Refreshments will be served from 14:15 Lecture starts at 14:30

• Network Measurement meets Virtual Switching

Speaker:
Ran Ben-Basat, Ph.D. Thesis Seminar
Date:
Wednesday, 14.2.2018, 14:30
Place:
Taub 601


text:
I was solving this problem that needed to guess some expected value identity. I tried to prove it but couldn't.

There are $n$ pairs of numbers $(a,b)$. In one operation, you will choose one pair uniformly randomly. Let $(X_i,Y_i)$ denote the chosen pair in the $i^{th}$ operation.

Prove that $\large{E(\frac{\sum_{i=1}^{k}X_i}{\sum_{i=1}^{k}Y_i}) == \frac{E(\sum_{i=1}^{k}X_i)}{E(\sum_{i=1}^{k}Y_i)} == \frac{\sum_{j=1}^{n}a_j}{\sum_{j=1}^{n}b_j}}$ as $k$ tends to infinity.


text:
# At what acid concentration does potassium metabisulfite react?

I'm in the process of developing a well preserved food-stuff containing potassium metabisulfite and a number of acids (acetic, ascorbic, malic, tartaric, citric, lactic). Wine-making literature suggests that K-meta is added for its antibacterial and antioxidative properties.

However, other literature claims that K-meta reacts with acids. In what concentration does this acid reaction occur? How violent is the reaction, and are any very harmful (if ingested) substances formed?

A simple test shows that the pH of my food-stuff is lower than 4.0, but greater than 3.0 before the addition of K-meta.

Potassium metabisulfite ($$\ce{K2S2O5}$$) is a white crystalline powder, which also known as potassium pyrosulfite. It is mainly used as an antiseptic for winemaking, process engraving, and as a source for sulfurous acid. It also acts as a potent antioxidant, and hence use as a food preservative (known as E224). For example, potassium metabisulfite is sometimes added to lemon juice as a preservative (Wikipedia). Lemon juice contains citric acid ($$\mathrm{p}K_{\mathrm{a1}}= 3.13$$) and ascorbic acid (Vitamin C; $$\mathrm{p}K_{\mathrm{a1}}= 4.17$$) as ingredients, and hence, has $$\mathrm{pH}$$ ranging from 2 to 3. For that reason alone, your concern about reaction of $$\ce{K2S2O5}$$ with citric acid or a mixture of acids within $$\mathrm{pH}$$ 3-4 has no base (a good news).

However, a major reason for addition of potassium metabisulfite ($$\ce{K2S2O5}$$) as a food preservative is to have sulfur dioxide ($$\ce{SO2}$$) as an active ingredient. Its melting point is listed as $$\pu{190 ^{\circ}C}$$, but it decomposes to two compounds: $$\ce{K2S2O5(s) -> K2SO3(s) + SO2(g)}$$ It is also in equilibrium with $$\ce{K2O}$$ and $$\ce{SO2}$$ when contacted with water (Ref.1): $$\ce{K2S2O5(aq) -> K2O(aq) + 2SO2(aq)}$$ In the aqueous media, there is an equilibrium between molecular sulfur dioxide ($$\ce{SO2}$$; E220) and the anions, bisulfite ($$\ce{HSO3-}$$) and sulfite ($$\ce{SO3^2-}$$), which are collectively and conventionally defined as ‘free $$\ce{SO2}$$’ (Ref.1). The reference also states that:

The active substance is sulfur dioxide (($$\ce{SO2}$$, E 220) which in the present application is generated in situ from potassium metabisulfite ($$\ce{K2S2O5}$$, E 224).

During their investigation:

The ANS Panel noted that sulfur dioxide, bisulfite and sulfite ions coexist in solution in a series of equilibria and that a read across between the different sulfites used as food additives is possible.

In general, following equilibria exist in aqueous solution of $$\ce{S2O5^2-}$$ (potassium or sodium): $$\ce{S2O5^2- + H2O <=> 2 HSO3-}$$ $$\ce{HSO3- + H2O <=> 2 SO3^2- + H3O+}$$ $$\ce{HSO3- + H3O+ <=> SO2(H2O) + H2O}$$ Where $$\ce{SO2(H2O)}$$ is called hydrated $$\ce{SO2}$$.

A final note: Iowa State University has listed following values as desired 'free $$\ce{SO2}$$’ by $$\mathrm{pH}$$ of wine:

$$\begin{array}{ccc} \hline\ \text{pH of wine} & '\text{free }\ce{SO2}', \pu{ppm} & & \text{pH of wine} & '\text{free }\ce{SO2}', \pu{ppm} \\\hline 2.9 & 11 & & 3.5 & 40 \\ 3.0 & 13 & & 3.6 & 50 \\ 3.1 & 16 & & 3.7 & 63 \\ 3.2 & 21 & & 3.8 & 79 \\ 3.3 & 26 & & 3.9 & 99 \\ 3.4 & 32 & & 4.0 & 125 \\\hline \end{array}$$

References:

1. Claudia Bolognesi, Laurence Castle, et al. (Panel members), EFSA CEF Panel (EFSA Panel on Food Contact Materials, Enzymes, Flavourings and Processing Aids), “Safety assessment of the active substance potassium metabisulfite, for use in active food contact materials,” EFSA Journal 2016, 14(5), 4465, 8 pp. ( doi: 10.2903/j.efsa.2016.4465).
• The formation of K2O in water is impossible. How can they (ref 1) come up with this equilibrium? Apr 28 '19 at 5:31
• Thank you for the additional insights. I was not aware that the effectiveness of the preservitave was inhibited by a more acidic environment. Those insights are invaluable. Apr 28 '19 at 20:10


text:
# What should the null hypothesis be?

Tags:
1. Apr 4, 2017

### Tyto alba

1. The problem statement, all variables and given/known data

In a study of effecftiveness of an antipsychotic drug, patients are treated with the antipsychotic drug and were compared to those receiving a placebo.
In terms of the number relapsing, 698 of 1068 patients were released after taking the placebo while 639 out of 2,127 relapsed after taking the antipsychotic drug,
Test the prediction that the antipsychotic drug is significantly more ffective in preventing relapse than placebo.(Have to apply Chi-square)

2. Relevant equations

What should the Ho and H1 be?

3. The attempt at a solution

As I know, the Ho should be that the observed data follows some theoretical model and any deviation(which will almost always be) is due to chance. But here since we have no theoretical model to compare our data with I have thought in this way:

Since H1 is often the hypothesis that the scientist is trying to prove and the Null could be set as what might be needed to disprove1?

So,

Ho= The effect of antipsychotic drug and the placebo is the same, any deviation/difference is due to chance.
Ha= The effect of antipsychotic drug and the placebo is not the same and the drug is significantly more effective.

1 This is what I presume, tell me if it isn't right (ideally with citation).

2. Apr 4, 2017

### vbrasic

Note that these two possibilities do not encompass all possible outcomes. The null hypothesis, $H_0$, you've proposed is $E_{drug}=E_{placebo}$, where $E$ is simply the efficacy of the drug/placebo, while your alternative hypothesis, $H_a$ states, $E_{drug}>E_{placebo}$. What about if the drug is less effective? You haven't accounted for that possibility within your hypotheses.

3. Apr 4, 2017

### Tyto alba

Should this be a two-tailed test? Is there an authentic source online to learn it?

4. Apr 4, 2017

### vbrasic

A two-tailed test is of the form $\mu\neq\mu_0$. That is, you want to test whether or not the mean efficacy of the placebo is greater than or less than the mean efficacy of the placebo. In your case however, you only care if the drug performs better than the placebo (not worse).

I recommend reading this. It's a webpage from a chemistry course taught at the University of Toronto.

5. Apr 4, 2017

### Staff: Mentor

I disagree with vrbasic's recommendation. Based on what I quoted, the null hypothesis should be H0: Effdrug ≤ Effplacebo and the alternate hypothesis, Ha should be Ha: Effdrug > Effplacebo. This would be a one-tailed test.

6. Apr 4, 2017

### vbrasic

That is precisely what I was suggesting... I don't think it should be two-tailed. Sorry for any miscommunication.

7. Apr 4, 2017

### Staff: Mentor

Sorry, what you wrote was clear enough -- I simply misunderstood it.


text:
# How to graph h(x)=x-4+1?

Jul 5, 2017

?

Solve for what?

See below

#### Explanation:

For

$h \left(x\right) = x - 4 + 1$

we can simplify this to:

$h \left(x\right) = x - 3$

And we can graph the solution to this (with the vertical axis being the $h$-axis):

graph{x-3}


text:
Job: Security

Our only real economic security lies in our power to meet human needs.

So the worse conditions become, the more evident human needs become.

Our security does not lie in our organizations or our jobs; disruptive technologies may simply make them irrelevant.

— The 8th Habit

— Stephen Covey

.

.

2008.03.25 Tuesday $CHK_2$


text: Logistic regression is a variation of ordinary regression that is used when the dependent (response) variable is dichotomous (i. e., takes two values). The dichotomous variable represents the occurrence or non-occurrence of some outcome event, usually coded as 0 or 1, and the independent (input) variables are continuous, categorical, or both (i.e., in a medical study, the patient survives or does not survive).  Unlike ordinary linear regression, logistic regression does not assume that the relationship between the independent and dependent variables are linear. Nor does it assume that the dependent variable or the error terms are distributed normally. In logistic regression, a binary logistic model is used to estimate the probability of a binary response based on one or more predictor or independent variables. The binary logistic model is displayed as in the following   where, p is the probability that Y=1 and X1, X2,.. .,Xk are the independent variables (predictors). b0 , b1, b2, .... bk are known as the regression coefficients, which are estimated from the data. Logistic regression estimates the probability of a certain event occurring. Logistic regression thus forms a predictor variable (log (p/(1-p)) that is a linear combination of the explanatory variables. The values of this predictor variable are then transformed into probabilities by a logistic function. Such a function has the shape of an S. The values of the predictor variable are displayed on the horizontal axis, and the probabilities are on the vertical axis.      Logistic regression produces Odds Ratios (OR) associated with each predictor value. The odds of an event are defined as the probability of the event occurring divided by the probability of the event not occurring. In general, the OR is one set of odds divided by another. The odds ratio for a predictor is defined as the relative amount by which the odds of the outcome increase (OR greater than 1.0) or decrease (OR less than 1.0) when the value of the predictor variable is increased by 1.0 units. In other words, (odds for PV+1)/(odds for PV) where PV is the value of the predictor variable.


text:
Acta Metall Sin  2018, Vol. 54 Issue (2): 174-192    DOI: 10.11900/0412.1961.2017.00418
 Orginal Article Current Issue | Archive | Adv Search |
Characterization and Modeling Study on Interfacial Heat Transfer Behavior and Solidified Microstructure of Die Cast Magnesium Alloys
Shoumei XIONG1,2(), Jinglian DU1, Zhipeng GUO1, Manhong YANG1, Mengwu WU1, Cheng BI1, Yongyou CAO1
1 School of Materials Science and Engineering, Tsinghua University, Beijing 100084, China
2 Key Laboratory for Advanced Materials Processing Technology, Ministry of Education, Tsinghua University, Beijing 100084, China
Abstract

Magnesium alloys are widely used in various fields because of their outstanding properties. High-pressure die casting (HPDC) is one of the primary manufacturing methods of magnesium alloys. During the HPDC process, the solidification manner of casting is highly dependent on the heat transfer behavior at metal-die interface, which directly affects the solidified microstructure evolution, defect distribution and mechanical properties of the cast products. As common solidified microstructures of die cast magnesium alloys, the externally solidified crystals (ESCs), divorced eutectics and primary dendrites have important influences on the final performance of castings. Therefore, investigations on the interfacial heat transfer behavior and the solidified microstructures of magnesium alloys have considerable significance on the optimization of die-casting process and the prediction of casting quality. In this paper, recent research progress on theoretical simulation and experimental characterization of the heat transfer behaviors and the solidified microstructures of die cast magnesium alloys was systematically presented. The contents include：(1) A boundary-condition model developed based on the interfacial heat transfer coefficients (IHTCs), which could precisely simulate the boundary condition at the metal-die interface during solidification process. Accordingly, the IHTCs can be divided into four stages, namely the initial increasing stage, the high value maintaining stage, the fast decreasing stage and the low value maintaining stage. (2) A numerical model developed to simulate and predict the flow patterns of the externally solidified crystals (ESCs) in the shot sleeve during mold filling process, together with discussion on the influence of the ESCs distribution on the defect bands of die cast magnesium alloys. (3) Nucleation and growth models of the primary α-Mg phases developed by considering the ESCs in the shot sleeve. (4) Nucleation and growth models of the divorced eutectic phase, which can be used to simulate the microstructure evolution of die cast magnesium alloys. (5) The 3D morphology and orientation selection of magnesium alloy dendrite. It was found that magnesium alloy dendrite exhibits an eighteen-primary branch pattern in 3D, with six growing along $<112?0>$ in the basal plane and the other twelve along $<112?3>$ in non-basal planes. Accordingly, an anisotropy growth function was developed and coupled into the phase field model to achieve the 3D simulation of magnesium alloy dendrite.

 Fig.1  The size, geometry and actual casting of step shape (a, b) and finger shape (c, d) castings (unit: mm)[12] Fig.2  Configuration and boundary conditions of shot sleeve (P—symmetrical plane, q—heat density, F—symmetric line, Δx—spatial step along the x direction, x and z—coordinate axis, respectively)[57](a) cross section (b) longitudinal section (c) heat transfer model Table 1  Thermal physical parameters of the materials[16,17,18,19,20] Fig.3  Variation correlation between solid phase fraction (fs) and h [20] (hmax is the peak value of IHTC, h0 is the initial value of IHTC; while, kh-f, fs0, fsc, fde, hde, hsolid and ε are the fitting parameters; Pc, Ps and P0 are the demarcation points of the four stages, which correspond to the points with the maximum IHTC, the spacing IHTC and the final stable IHTC) Fig.4  The interfacial heat transfer coefficient (h) of A380 alloy (a) and AZ91D alloy (b) in single cycle at the different locations of chamber under normal die casting condition[1] (t—time) Fig.5  Optical micrograph of typical AM50 microstructure at the surface and central regions of the 'step shape' die-casting (a), area fraction distributions of externally solidified crystals (ESCs) over cross section of the five steps (b), and the step 4 (c) on the 'step shape’ die casting [44] (Tp is the casting temperature and Vs is the low speed velocity) Fig.6  Particle distributions on the cross sections of the tensile bars with semi-circle ingate at filling percentage of 37% (a), 45% (b) and 55% (c), and for the tensile bars with circle ingate at filling percentage of 44% (d), 63% (e) and 81% (f) [45] (n—content of ESCs) Table 2  Thermal physical parameters of Mg-Al eutectic alloy[50,53] Fig.7  Comparison of the the microstructure evolution at the central of magnesium alloy die castings (AM60B)[50] (a, b) the modeling results (c) the experimental result Fig.8  Dendrites of AZ91 alloy casting (a, b), microstructures of Mg-30%Gd alloys in as casting state (c, d)(The dendrites are with different growth patterns, such as six-branches, five-branches, four-branches and butterfly shape symbolled as No.1 to No.5, respectively) Fig.9  3D reconstructed dendrites extracted for the Mg-30%Sn (a, b) and Mg-30%Gd (c, d) alloys, and a dendrite of Mg-30%Sn alloy viewed from different perspectives (e, f) (A1~A6 and B1~B4 are used to indicate the preferred growth directions of the dendrite)[33] Fig.10  EBSD analyses on the preferred growth directions of the Mg-30%Gd alloy dendrite[33]: dendritic morphology and the according (0001) orientation (a, b) while those for $(033?1)$ (c, d), and pole distributions with respect to the $<112?3>$ for thirty dendrite (e) Fig.11  Comparison of the dendritic morphologies between phase field simulation (a) and X-ray tomography experiment (b). Dendritic morphologies with respect to four different view directions including $<0001>$, $<112?0>$, $<101?0>$ and $<112?3>$ are shown in Figs.11c~f for simulation and Figs.11g~j for experiment, respectively[29] Fig.12  Atomic densities (a) and inter-planar distances (b) of Mg low index crystallographic planes, and the pyramidal planes for the $<112?0>$ and $<112?3>$ growth directions (c)[33] (The correpsonding a and c, donote the lattice parameters of Mg) Fig.13  The surface energies of low index planes (a) and high index planes (b) of pure Mg by four different potentials[39], the correlation for $<112?x>{112?e}$ (c), the according morphologies in the basal and non-basal plane (d), and the surface energies of low index planes (e) and high index planes (f) of Mg and its alloys by PAW-LDA potentials (PAW, USPP, GGA and LDA are abbreviations of the projector-augmented wave, the ultra-soft pseudo-potential, the generalized gradient approximation and the local density approximation, respectively)[39,40]


text:
Stefano Zacchiroli <leader@debian.org> writes:
> \subsection{Guidelines for Using Logos}
>
> \begin{itemize}
>
> \item Any scaling must retain the original proportions of the logo.
>
> \item Logo should only use official'' logo colors.

What's the reason for this? While the policy as I understand it is not
binding for Debian (we can use our own trademark as we like), it may
create some confusion that we as a project did not use the "official"
color on several occasions in our "official" artwork (eg. current wheezy
default desktop background). I don't see a reason why this should not be
allowed for others.

Gaudenz

--
Ever tried. Ever failed. No matter.
Try again. Fail again. Fail better.
~ Samuel Beckett ~


text:
Fusion for Kdb+ / PyQ

# What’s new

## Version 4.1

Released 2017.08.30

### Highlights

• Improved calling Python functions from q:
• Objects of any type supported by K() can be returned
• Python functions can be called from multiple q threads
• Calls from Python to q will now release GIL allowing concurrent execution of q code from multiple threads
• Added a backtrace mode (requires kdb+ V3.5+) in which q stack trace is printed on q errors
• Experimental Windows support
• Performance improvements

### Return arbitrary objects from Python functions called from q

When a Python function is called from q, the returned Python objects are now automatically converted to q. Any type accepted by the K() constructor can be successfully converted. For example, the numpy.eye function returns a 2-D array with 1s on the diagonal and 0s elsewhere. It can be called from q as follows:

  q)p)import numpy
q)p)q.eye = numpy.eye
q)eye 3 4 1
0 1 0 0
0 0 1 0
0 0 0 1

### Calling Python from peach

Python functions can now be safely called from multiple q threads. For example, the following session illustrates how peach distributes work between four threads:

$q -s 4  q)p)import threading q)p)q.tid = threading.get_ident q){group (distinct x)?x} {tid()} peach til 19 0| 0 4 8 12 16 1| 1 5 9 13 17 2| 2 6 10 14 18 3| 3 7 11 15  ### Backtrace mode Backtrace mode is available when PyQ is running under kdb+ V3.5+. To activate the backtrace mode, set the environment variable PYQ_BACKTRACE. For example: $ PYQ_BACKTRACE=yes pyq

>>> q('f:{g[]};g:{1+};f[]')
Traceback (most recent call last):
..
_k.error: ('type', k('((({1+};2);("..g";"";-1;"{1+}");2;5);(({g[]};2);("..f";"";-1;"{g[]}");1;4);..'))
kdb+ backtrace:
[5]  g:{1+}
^
[4]  f:{g[]}
^
[3]  f:{g[]};g:{1+};f[]
^
..

### Experimental Windows support

PyQ can now be installed on Windows. We tested PyQ 4.1.0 with Python 3.6 build using VS 2015 and 2017 and 32-bit kdb+ 3.5.

PyQ 4.1.1 adds support for Python 2.7.x with Microsoft Visual C++ Compiler for Python 2.7.

Latest updates on Windows support, as well as latest patches can be found at Github.

## Version 4.0

Released 2017.03.02

### Highlights

• Enhanced q) prompt with syntax highlighting
• New operators: <<, >> and @
• Improved means for constructing K objects of arbitrary types
• Type casts using attribute syntax
• Improved numpy interoperability
• Restored support for KDB+ 2.x
• Better documentation
• More k.h functions are exposed to Python internally
• Added convenience scripts for starting different interactive sessions
• Additional conversions between ~pyq.K and native Python objects

### Enhanced q) prompt

The q) prompt will now use the prompt toolkit when available to provide a separate command history, q syntax highlighting and a status bar displaying system information.

### New operators

Three new operators are defined for K objects: <<, >> and @.

#### Shift operators

Shift operators << and >> can now be used to shift elements in K lists:

>>> q.til(10) << 3
k('3 4 5 6 7 8 9 0N 0N 0N')
>>> q.til(10) >> 3
k('0N 0N 0N 0 1 2 3 4 5 6')

### The @ operator

Users of Python 3.5 or later can now use the new binary operator @ to call q functions without using parentheses:

>>> q.til @ 5
k('0 1 2 3 4')

The same operator between two functions creates a function composition. For example, the dot product can be defined succinctly as

>>> dot = q.sum @ q('*')
>>> dot([1, 2, 3], [3, 2, 1])
k('10')

### Typed constructors and casts

Atoms and lists of like atoms can now be constructed from Python objects using typed constructors. For example, by default, a list of strings passed to the default K constructor becomes a symbol list:

>>> colors = K(['white', 'blue', 'red'])
>>> colors k('whitebluered')

If you want to create a list of strings, you can use a typed constructor:

>>> K.string(["Donald E. Knuth", "Edsger W. Dijkstra"])
k('("Donald E. Knuth";"Edsger W. Dijkstra")')

If you already have a symbol list and want to convert it to strings, you can use the attribute-access notation to perform the cast:

>>> colors.string
k('("white";"blue";"red")')

Similar operations can be performed with numeric data. For example, to create a matrix of single-precision floats (real), call

>>> m = K.real([[1, 0, 0],
...             [0, 1, 0],
...             [0, 0, 1]])
>>> m
k('(1 0 0e;0 1 0e;0 0 1e)')

To cast the result to booleans — access the K.boolean attribute:

>>> m.boolean.show()
100b 010b 001b

Unlike q, Python does not have special syntax for missing values and infinities. Those values can now be created in PyQ by accessing na and inf attributes on the typed constructors:

>>> for x in [K.int, K.float, K.date, K.timespan]:
...     print(x.na, x.inf)
0Ni 0Wi
0n 0w
0Nd 0Wd
0Nn 0Wn

### Interoperability with NumPy

#### Matrices and arrays of higher dimensions

Arrays with ndim > 1 can now be passed to q and they become nested lists. For example:

>>> q.x = numpy.arange(12, dtype=float).reshape((2, 3, 2))
>>> q.x
k('((0 1f;2 3f;4 5f);(6 7f;8 9f;10 11f))')

Similarly, ndim > 1 arrays can be constructed from lists of regular shape:

>>> numpy.array(q.x)
array([[[ 0., 1.],
[ 2., 3.],
[ 4., 5.]],

[[  6., 7.],
[  8., 9.],
[ 10., 11.]]])

### Times, dates and timedeltas

Prior to 4.0, conversion of temporal data to NumPy arrays would expose internal integer values. For example, a list of months

>>>months = q('2001.01m + til 3')

would become an integer array when converted to NumPy:

>>> numpy.array(months).tolist()
[12, 13, 14]

Now, an array of type datetime64 is returned:

>>> numpy.array(months)
array(['2001-01', '2001-02', '2001-03'], dtype='datetime64[M]')

Note that the resulting array has different numeric values and cannot share the data with the K object. To share the data and/or to get an array as in older versions, one should use the new data attribute:

>>> a = numpy.asarray(months.data)
>>> a.tolist()
[12, 13, 14]

An array constructed from the data attribute will use the same underlying storage. This means that changing the array will change the K object.

>>> a[:] += 998*12
>>> months
k('2999.01 2999.02 2999.03m')

#### Complex numbers

Complex numbers can now be passed to and obtained from kdb+. When passed to kdb+, complex numbers are automatically converted to dictionaries with keys re and im and lists of complex numbers are converted to tables with columns re and im.

>>> q.z = [1 + 2j, 3 + 4j, 5 + 6j]
>>> q.z.show()
re im
-----
1 2
3 4
5 6
>>> [complex(x) for x in q.z]
[(1+2j), (3+4j), (5+6j)]

#### Path objects

Path objects can now be used where q path handle symbols are expected

>>> import pathlib
>>> path = pathlib.Path('xyz')
>>> q.set(path, 42)
k(':xyz')
>>> q.get(path)
k('42')

#### Named tuples

Named tuples are now converted to dictionaries:

>>> from collections import namedtuple
>>> Point = namedtuple('Point', 'x,y')
>>> q.point = Point(1, 2)
>>> q.point
k('xy!1 2')

As a consequence, a uniform list of named tuples is converted to a table:

>>> q.points = [Point(1, 2), Point(3, 4), Point(5, 6)]
>>> q.points.show()
x y
---
1 2
3 4
5 6

### Redesigned iterators

Iterators (formerly adverbs) can now be used on functions with different ranks. For example, scan and over can be used with unary functions. To illustrate, the following code generates a Pascal triangle:

>>> f = q('{(0,x)+x,0}')
>>> f.scan(6, 1).show()
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1

If only the last row is of interest – use over:

>>> f.over(6, 1)
k('1 6 15 20 15 6 1')


text:
# You're reading: Posts Tagged: fractions

### Do you use mixed fractions?

I’m at the MATRIX conference in Leeds, where I’ve just been talking to Adam Atkinson. He told me that he’s trying to compile a definitive list of countries that don’t use mixed fractions.

Here’s a mixed fraction: $2 \frac{2}{3}$
And here’s a non-mixed fraction: $\frac{8}{3}$
Actually, here’s an interesting fact about that number: $2 \sqrt{ \frac{2}{3} } = \sqrt{ 2 \frac{2}{3} }$
This only makes sense if you believe in mixed fractions (and unicode character U+2062, “invisible times”)

This is going to be one of those wipe-your-bum-standing-up situations: it’s entirely possible that you can be on either side of this divide and not know the other exists. Apparently, in some countries mixed fractions just don’t exist: an integer written next to a fraction is incorrect.

So, to help Adam on his way, I thought I’d start another in our long-running series of Aperiodical Surveys. Please tell us where you live, and if mixed fractions are OK in your book.

### Ability with fractions and division aged 10 predicts ability with algebra aged 16

Last week we reported that the UK Government have released a draft primary school Programme of Study for mathematics for consultation. A report from the Telegraph quoted in that article mentioned that “the use and multiplication of fractions” was “a vital precursor to studying algebra”. A piece of research published in the journal Psychological Science, ‘Early Predictors of High School Mathematics Achievement‘, investigates this area. The findings indicate the importance of learning about fractions and division by showing that these “uniquely predict” students’ knowledge of algebra and overall mathematics achievement 5 or 6 years later.


text:
Back to Course

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• ## Do you like this content?

Lesson 4, Topic 3
In Progress

# Electrical Potential

Lesson Progress
0% Complete

Electric potential (V) at a point P is the work done in bringing a unit positive charge from infinity to that point in the electric field of another charge.

It is a scalar quantity. Its unit is Joules per Coulomb (J/c)

Electric potential = $$\frac{Work \; done}{charge}$$

V = $$\frac{w}{q}$$

If r = distance from infinity to the point charge

F = force of attraction

Q1 = Unit charges

Q2 = attracting charge

Work done = electric potential (V) x  r = F x r

= $$\frac{Q_1 Q_2}{4 \pi \varepsilon_0 r^2} \scriptsize \; \times \; r$$

Q2 = 1

V = $$\frac{Q}{4 \pi \varepsilon_0 r^2}$$

Electrical potential between two points (distance between a and b)

= $$\frac{Q}{4 \pi \varepsilon_0 r^2} \left[ \frac{1}{a} \; – \; \frac{1}{b} \right]$$

Types of electric potential

Zero potential

All potentials around charges are measured using a reference point. Two reference points for charges are

• Earth
• A point at infinity

Equipotential surface

This is an imaginary surface drawn in an electric field such that all point on the surface have the same potential. Their characteristics are as follows:

• It is the same distance from the source of field
• The electric lines of force are always perpendicular to the surface
• The work done in moving a positive charge along the surface is zero

error:


text:
D. Horse Races
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

Petya likes horse racing very much. Horses numbered from l to r take part in the races. Petya wants to evaluate the probability of victory; for some reason, to do that he needs to know the amount of nearly lucky horses' numbers. A nearly lucky number is an integer number that has at least two lucky digits the distance between which does not exceed k. Petya learned from some of his mates from Lviv that lucky digits are digits 4 and 7. The distance between the digits is the absolute difference between their positions in the number of a horse. For example, if k = 2, then numbers 412395497, 404, 4070400000070004007 are nearly lucky and numbers 4, 4123954997, 4007000040070004007 are not.

Petya prepared t intervals [li, ri] and invented number k, common for all of them. Your task is to find how many nearly happy numbers there are in each of these segments. Since the answers can be quite large, output them modulo 1000000007 (109 + 7).

Input

The first line contains two integers t and k (1 ≤ t, k ≤ 1000) — the number of segments and the distance between the numbers correspondingly. Next t lines contain pairs of integers li and ri (1 ≤ l ≤ r ≤ 101000). All numbers are given without the leading zeroes. Numbers in each line are separated by exactly one space character.

Output

Output t lines. In each line print one integer — the answer for the corresponding segment modulo 1000000007 (109 + 7).

Examples
Input
1 21 100
Output
4
Input
1 270 77
Output
2
Input
2 11 2080 100
Output
00
Note

In the first sample, the four nearly lucky numbers are 44, 47, 74, 77.

In the second sample, only 74 and 77 are in the given segment.


text:
# Mcrypt PHP extension required.IN Laravel With Ubuntu 14

I am trying to migrate laravel app into ubuntu and getting error like Mcrypt PHP extension required. when i check php details with phpinfo() function it showing below screen.

PLlease find the Terminal Screenshot below,

php.ini

How to solve this issue ?

• This may helps you askubuntu.com/questions/507796/… – Vinod VT Jan 30 '15 at 6:30
• Install mcrypt and Uncomment or add the mcrypt extension in php.ini file – Sougata Bose Jan 30 '15 at 6:30
• Your php sapi may not be the same when serving scripts for your http server from when invoking in command line, so the php.ini file maybe a different one. Try php --ini in your bash to see which is the php.ini file for cli. – demental Jan 30 '15 at 6:41
• @demental, how can i solve this. – Jishad Jan 30 '15 at 6:43
• @demental, Configuration File (php.ini) Path: /etc/php5/cli Loaded Configuration File: /etc/php5/cli/php.ini Scan for additional .ini files in: /etc/php5/cli/conf.d Additional .ini files parsed: /etc/php5/cli/conf.d/05-opcache.ini, /etc/php5/cli/conf.d/10-pdo.ini, /etc/php5/cli/conf.d/20-json.ini, /etc/php5/cli/conf.d/20-readline.ini – Jishad Jan 30 '15 at 6:46

## 2 Answers

If you have installed PHP through apt-get then find the mcrypt extension you are using either for apache by using

\$ locate *mcrypt.ini
/etc/php5/apache2/conf.d/20-mcrypt.ini
/etc/php5/cli/conf.d/20-mcrypt.ini
/etc/php5/fpm/conf.d/20-mcrypt.ini
/etc/php5/mods-available/mcrypt.ini
/usr/share/php5/mcrypt/mcrypt.ini
/var/lib/ucf/cache/:etc:php5:mods-available:mcrypt.ini

go into the directory that you have your extension in

cd /etc/php5/apache2/conf.d/20-mcrypt.ini

and you will probably find there is a symlink to the mcrypt extension in the mods-available directory. If that's the case all you have to do is cd into the /etc/php5/cli/conf.d directory and create a similar symlink

cd /etc/php5/cli/conf.d
sudo ln -s ../../mods-available/mcrypt.ini ./

then restart php maybe with sudo service php5 restart

That should work

• getting error Warning: require(/opt/lampp/htdocs/timegear/bootstrap/../vendor/autoload.php): failed to open stream: Permission denied in /opt/lampp/htdocs/timegear/bootstrap/autoload.php on line 17 Fatal error: require(): Failed opening required '/opt/lampp/htdocs/timegear/bootstrap/../vendor/autoload.php' (include_path='.:/opt/lampp/lib/php') in /opt/lampp/htdocs/timegear/bootstrap/autoload.php on line 17 – Jishad Jan 30 '15 at 7:16
• Unless any one else has any more ideas it sounds to me like Laravel is unable to read the Composer Autoload file. As you are getting permission denied I am guessing that you have run composer update. I would run sudo chmod 0755 -R /opt/lampp/htdocs/timegear && chmod 0777 -R /opt/lampp/htdocs/timegear/app/storage if I have your file structure figured out correctly. Again: other people may have other ideas. – John Evans Jan 30 '15 at 7:35
• when i run composer update terminal showing No command 'composer' found, did you mean: Command 'compose' from package 'mime-support' (main) composer: command not found – Jishad Jan 30 '15 at 8:15
• Try installing Composer globally as described in the Composer Getting started page curl -sS https://getcomposer.org/installer | php && sudo mv composer.phar /usr/local/bin/composer – John Evans Jan 30 '15 at 11:47
• HI John, i tried the above command and showed it successfully installed.Then moved to the /usr/local/bin/composer directory. I am create d the composer directory before moving. and after i tried to run composer it will showing above mentioned error. – Jishad Feb 2 '15 at 3:59

Find mcrypt.ini file in /etc/php5/mods-available/mcrypt.ini ,if this file does not exist,means php5-mcrypt could create the file.

• create the file mcrypt.ini in dir /etc/php5/mods-available/
• And add the following config to it:

extension=mcrypt.so

• Run this commands

sudo php5enmod mcrypt sudo service php5-fpm restart

This worked for me.Cheers


text:
# Find cdf, pdf and expextation value of a random variable

1. Nov 7, 2007

### strangequark

1. The problem statement, all variables and given/known data

Let X represent the random choice of a real number on the interval [-1,1] which has a uniform distribution such that the probability density function is$$f_{X}(x)=\frac{1}{2}$$ when $$-1\leqx\leq1$$. Let$$Y=X^{2}$$ a. Find the cumulative distribution $$F_{Y}(y)$$ b. the density function $$f_{Y}(y)$$ and c. the expected value $$E(Y)$$.

2. Relevant equations

my book gives a great explanation on how to change variables for joint distributions, but little is said about functions of one random variable, so I'm kind of at a loss here.

3. The attempt at a solution

first, if $$Y=X^{2}$$, then I want to say we need to find Y over the interval
[0,1]. And integrating I have that:
$$F_{X}(x)=\frac{x+1}{2}$$ which is $$P(X\leqx)$$...
now I want to say $$P(X\leqx)=P(\sqrt{Y}\leqx)$$...

i'm not sure where to go from here...
can I just substitute $$\sqrt{y}$$ for x so I have:

$$F_{Y}(y)=\frac{\sqrt{y}+1}{2}$$ ??

2. Nov 8, 2007

### D H

Staff Emeritus
One way to look at the cumulative distribution function F(X) for some random process is F(x) = p(X<x). In other words, the value of the F(x) is the probability that a random number drawn from the distribution is smaller than x. If the random process has some lower bound and upper bound, the CDF F(x) must be identically zero below the lower bound and identically one above the upper bound. Your conjectured function does not exhibit this behavior. (Look at FY(0).)

Using the above definition of the CDF, FY(y) = p(Y<y). Translate this expression to the original random variable and evaluate the resultant probability expression.

3. Nov 8, 2007

### sprint

part a:

1. graph and draw a picture of y = x^2

2. note that y is only valid from 0 to 1 ( since y = x^2 can only produce zero and positive numbers)

3. to get the CDF formula, find the limits of the integral. y goes from 0 to x^2 and x goes from 0 to infinity. once the limits are found, plug the pdf into the CDF formula and then do the integration

part b:

part c:

for part c there are two ways of doing this. use the expectation formula. or find the characteristic function and then evaluate -j times the first derivative of the characteristic function when the characteristic function's variable equals to zero.

Last edited: Nov 8, 2007


text:
# Flag & close as duplicate not working with Opera

I am trying to flag this post as a duplicate of this one, but I cannot seem to get it to work as the button is greyed out.

I tried to enter:

• https://codereview.stackexchange.com/questions/62864/are-these-the-correct-functions-for-cleaning-data-to-be-sent-to-database-and-ins
• 62864
• etc.

But none of them work. Voting to close as a duplicate has the same problem.

Here is a screenshot:

I'm using Opera (on Linux Mint).

If I'm using Firefox it works. Can someone confirm if this is just my version of Opera, or if this is a general issue?

• Is it still buggy ? – Marc-Andre Sep 22 '14 at 13:09
• @Marc-Andre yes, for me it still looks like in the screenshot. – tim Sep 22 '14 at 13:50
• My gut reaction here is "Opera + Linux Mint is not a super-common setup." Technically, we support only current and previous versions of Opera. What version are you using? – Pops Oct 10 '14 at 19:17
• @Pops good point. 12.16, and I'm just realizing that yes, that's pretty old (there doesn't seem to be a new version for linux, so I guess I'll have to switch to firefox). – tim Oct 10 '14 at 19:26


text:
# An argument involving well-orders

• Non Existence
Few days ago, I posted an answer for a mathoverflow question. The question was of elementary nature.

The question had a lot of views, and my answer didn't get any upvote. Now since I got a downvote, perhaps there was a (major) mistake somewhere in the argument I posted? Tbh I was reasonably confident about my answer [since its not that unusual for accurate answers to go by with 0 upvotes], but the downvote def. confused me a bit.

I left a comment asking the reason for downvote, but it is not very likely that will get a response (would be good if it gets one). It is hard to tell whether the downvote is just impulse or based upon identification of an actual (major) mistake in answer. I would definitely like to know if its the latter.

I didn't think about the answer quite thoroughly (since I posted it after few hours of thinking), so maybe there is some aspect I missed? Nevertheless, I should say that the argument is of basic nature and it looked OK to me on few cursory/initial checks. But sometimes one can keep thinking thinking about along the same (wrong) lines when re-checking (especially when re-checks are not thorough enough). So a "second opinion"/"further input" would be quite helpful.

Link: https://mathoverflow.net/questions/381527/the-sum-of-two-well-ordered-subsets-is-well-ordered

Here is the question (not mine):
if ##A,B## are two well-ordered subsets of ##\mathbb{R}## (or any ordered group — with the induced order of course), the subset ##A+B:=\{a+b\,|\,a\in A,b\in B\}## is well-ordered. How does one see that?

Here is the answer I posted:
Let's suppose the well-order of ##A## and ##B## have order-types ##p## and ##q## respectively. So we use the notation ##a_i## (##i<p##) to denote the ##i##-th element of ##A##. We have ##a_i<a_j## whenever ##i<j<p##. Similarly, we use ##b_i## (##i<q##) to denote the ##i##-th element of ##B##.

Since ##A+B## is already linearly-ordered, we want to prove that ##A+B## has no infinite descent. In other words, we want to disprove the existence of a function ##f:\mathbb{N} \rightarrow \mathbb{R}## such that (i) For all ##x## in domain of ##f## we have ##f(x)=a_i+b_j## (for some ##i<p## and ##j<q##) (ii) ##f## is a 1-1 function (iii) For all ##i,j \in \mathbb{N}## (with ##j>i##) we must have ##f(j)<f(i)##.

So if we have ##f(x)=a_i+b_j## then we can write ##\mathrm{first}(f(x))## and ##\mathrm{second}(f(x))## to denote ##a_i## and ##b_j## respectively. Now we can define ##\alpha_0=min\{\alpha \in Ord:first(f(x))=a_{\alpha} \wedge x \in \mathbb{N} \}##. Let's denote ##n_0 \in \mathbb{N}## as the "last" value for which ##first(f(n_0))=a_{\alpha_0}##.

Now we define ##\alpha_1=min\{\alpha \in Ord:first(f(x))=a_{\alpha} \wedge x \in \mathbb{N} \wedge x>n_0 \}##. Let's denote ##n_1 \in \mathbb{N}## as the "last" value for which ##first(f(n_1))=a_{\alpha_1}##.

Now because of ##\alpha_1>\alpha_0##, we should get ##a_{\alpha_1}>a_{\alpha_0}## and hence ##second(f(n_1))<second(f(n_0))##. So it seems to me that when we define ##\alpha_2=min\{\alpha \in Ord:first(f(x))=a_{\alpha} \wedge x \in \mathbb{N} \wedge x>n_1 \}## and ##n_2## as the "last" value for which ##first(f(n_2))=a_{\alpha_2}##, then we should have similarly ##second(f(n_2))<second(f(n_1))##. The last inequality is supposed to follow from ##a_{\alpha_2}>a_{\alpha_1}## (because ##\alpha_2>\alpha_1##).

The previous paragraph seems to be suggestive of defining ##\alpha_i##, ##n_i## generally for all natural numbers ##i## and then creating an infinite descent for the second components:##......<second(f(n_3))<second(f(n_2))<second(f(n_1))<second(f(n_0))##.

And a comment regarding notation that I wrote:
It seems that instead of writing ##\alpha_1=min\{\alpha \in Ord:first(f(x))=a_{\alpha} \wedge x \in \mathbb{N} \wedge x>n_0 \}## it would probably be better to write something like: ##\alpha_1=min\{\alpha \in Ord: \exists x \in \mathbb{N}(first(f(x))=a_\alpha \wedge x>n_0)\}##

## Answers and Replies

andrewkirk
Science Advisor
Homework Helper
Gold Member
Now we can define ##\alpha_0=min\{\alpha \in Ord:first(f(x))=a_{\alpha} \wedge x \in \mathbb{N} \}##
What do you mean by Ord? If you mean the collection of all ordinals, that is not a set, so you cannot simply use the axiom schema of specification as you have done here to define a set being all members of Ord that obey a certain criterion.

Secondly, you need an existential quantifier for ##x##, so that you would write:
$$\alpha_0=\min\{\alpha \in Ord\ |\ \exists x\in\mathbb N: first(f(x))=a_{\alpha} \}$$

Thirdly, what do you mean by "last" in
Let's denote ##n_0 \in \mathbb{N}## as the "last" value for which ##first(f(n_0))=a_{\alpha_0}##.
If you mean 'smallest' then 'last' seems a strange word to use rather than 'first'. If you mean 'largest' then you need to prove that the collection of all such values has a maximum.

I didn't read past there. Perhaps you can correct these problems. I suggest you try to do so and then see if you can continue the proof.

In general, you need to be more wary of unseen assumptions buried in your statements - eg that something is a set, or that a given set of integers has a maximum.

I don't think that justifies a downvote. I consider downvoting on places like stackexchange or overflow as unnecessarily aggressive. But I can understand how come prickly character may have gotten irritated at the uncareful presentation and taken the lazy way out of just downvoting rather than pointing out problems.

Also, I would not say the question has an elementary nature. It looks difficult to me.

Few points w.r.t. post#1:
(1) In post#1, at some points comparison relation ##<## is used for real numbers and at some points it is used for ordinals.

(2) We can think of ##first## as a function ##first:A+B \rightarrow A##. Similarly, we can think of ##second## as a function ##second:A+B \rightarrow B##.

(3) If we have ##first(f(i))=a_\alpha## for some specific natural number ##i## and some specific ordinal ##\alpha##, then there can't exist arbitrarily large natural numbers ##j>i## such that ##first(f(j))=a_\alpha##. This justifies the use of the phrase/word "last" in post#1.

(4) When we write: ##\alpha_0=min\{\alpha \in Ord: \exists x \in \mathbb{N}(first(f(x))=a_\alpha)\}##, the existence of minimum is justified due to a lack of infinite descent (in ordinals). If we want to be completely thorough, we could write a bit more working.

Last edited:
What do you mean by Ord? If you mean the collection of all ordinals, that is not a set, so you cannot simply use the axiom schema of specification as you have done here to define a set being all members of Ord that obey a certain criterion.

Secondly, you need an existential quantifier for ##x##, so that you would write:
$$\alpha_0=min\{\alpha \in Ord\ |\ \exists x\in\mathbb N: first(f(x))=a_{\alpha} \}$$
For the second comment, see what I wrote towards the end of post#1. I think writing something like:
##\alpha_0=min\{\alpha \in Ord\ |\ \exists x\in\mathbb N (first(f(x))=a_{\alpha}) \}##
is definitely better than:
##\alpha_0=min\{\alpha \in Ord:first(f(x))=a_{\alpha} \wedge x \in \mathbb{N} \}##
or
##\alpha_0=min\{\alpha \in Ord: x \in \mathbb{N} \wedge first(f(x))=a_{\alpha} \}##

With regards to first comment, if we are being really strict you are correct. But it is quite clear that ##\alpha_0, \alpha_1, \alpha_2,....## etc. will be well-defined since for all ##\alpha_i## we have ##\alpha_i<p## (with ##p## being an ordinal). So what I wrote for ##\alpha_0## correctly identifies an ordinal [rather than something too big to be a set]. I suppose we could also write:
##\alpha_0=\min\{\alpha \in Ord\ |\ \alpha<p \wedge \exists x\in\mathbb N (first(f(x))=a_{\alpha}) \}##
or better (formally):
##\alpha_0=\min\{\alpha \in p\ |\ \exists x\in\mathbb N (first(f(x))=a_{\alpha}) \}##
Some further comments towards the end [1]. At any rate, this doesn't affect the main point of post#1. In case this is a point of confusion, the last definition [which strictly identifies a minimum of a set (in strict sense) of ordinals] can be taken for ##\alpha_0## (and similar definitions for ##\alpha_1,\alpha_2,....## etc.).

Thirdly, what do you mean by "last" in

If you mean 'smallest' then 'last' seems a strange word to use rather than 'first'. If you mean 'largest' then you need to prove that the collection of all such values has a maximum.
By "last" I did not mean smallest. I meant last occurrence. Because there are finite number of natural numbers ##i## such that ##first(f(i))=a_{\alpha_0}##, the natural number ##n_0## is "last value" for which ##first(f(n_0))=a_{\alpha_0}##

[1]
When I wrote the original def. of ##\alpha_0## with ##Ord##, I had in mind something like a "class set". For example, strictly speaking, of course we can't write ##f:Ord \rightarrow Ord## and say ##f## is a function (from perspective of set-theory). However, it is common to call such functions "class functions" (and they can be formalized).

So when I said "class set" above, I meant a function like ##f:Ord \rightarrow \{0,1\}## identifying the elements of ##Ord##. For example, we could say something like "all limit ordinals" and it would be a "class set". Same for "all successor ordinals". For example, if we denote the class set for all successor ordinals as ##S##, formally we would write it as:
##S=\{\alpha \in Ord \,|\, \exists x \in Ord(x+1=\alpha)\}##
Similarly the class set denoting all limit ordinals as ##L##, we can write:
##L=\{\alpha \in Ord \,|\, \alpha \notin S\}##

So when we write ##\alpha \in Ord## instead of writing ##\alpha \in p##, what I had in mind was:
(i) The initial specification for ##\alpha_0## identifies a "class set".
(ii) However, what follows after [the part ##\exists x\in\mathbb N (first(f(x))=a_{\alpha})##] clearly makes it evaluate to minimum of a set (in strict sense) of ordinals.

So this was my perspective in writing ##\alpha \in Ord## instead of writing ##\alpha \in p##. At any rate, this is a tangent to main point in post#1. Since it came up, I thought I would describe what I was thinking. I do agree that writing ##\alpha \in p## seems to be more well-suited since it identifies from the outset that we are talking about ##\alpha_0## as minimum of a set (in strict sense) of ordinals.

Last edited:
So I think I will edit the overflow post with two changes. I was thinking about making these changes before, but didn't make them (to avoid small edits, since every edit is bumped). But these seem reasonably sufficient reasons for edit and do seem to make it easier to understand what I wrote.

(1) Changing:
##\alpha_0=min\{\alpha \in Ord:first(f(x))=a_{\alpha} \wedge x \in \mathbb{N} \}##
to:
##\alpha_0=min\{\alpha \in Ord\ |\ \exists x\in\mathbb N (first(f(x))=a_{\alpha}) \}##
or better yet [if the upper-bound was a complicated, non-obvious ordinal (or worse it didn't exist), then there would be a point to use the second definition instead of third. In this case, the below definition seems better than second]:
##\alpha_0=min\{\alpha \in p\ |\ \exists x\in\mathbb N (first(f(x))=a_{\alpha}) \}##

EDIT:
One issue I see with the third definition is that for non-limit the "intended definition" would need to be (even though the previous one would work too .... but we will need to add some extra justifications):
##\alpha_0=min\{\alpha \in p+1\ |\ \exists x\in\mathbb N (first(f(x))=a_{\alpha}) \}##

Of course we might just define (for both limit and non-limit ##p##):
##\alpha_0=min\{\alpha \in p+1\ |\ \exists x\in\mathbb N (first(f(x))=a_{\alpha}) \}##
END

(2)
Mention the points I wrote in post#3. They seemed reasonably clear, but it seems it wouldn't hurt to add them.

=============================

Anyway, in case it seemed that I made a thread because of downvote. Well perhaps sort of, but that's a tertiary reason. Net negative score for answers don't matter as much for questions [where net negative leads to deletion]. I wanted to confirm/double-check whether I had a mistake in reasoning or not. In-case of a mistake somewhere in reasoning, please do let me know.

Last edited:
Here is the (slightly) modified version of post#1 with the changes described in previous post incorporated. The main argument (and the text) is unchanged, but it should be a bit easier to read (better notation, some indentation).

BEGIN:
"
Let's suppose the well-order of ##A## and ##B## have order-types ##p## (assume limit for simplicity) and ##q## respectively. So we use the notation ##a_i## (##i<p##) to denote the ##i##-th element of ##A##. We have ##a_i<a_j## whenever ##i<j<p##. Similarly, we use ##b_i## (##i<q##) to denote the ##i##-th element of ##B##.

Since ##A+B## is already linearly-ordered, we want to prove that ##A+B## has no infinite descent. In other words, we want to disprove the existence of a function ##f:\mathbb{N} \rightarrow \mathbb{R}## such that (i) For all ##x## in domain of ##f## we have ##f(x)=a_i+b_j## (for some ##i<p## and ##j<q##) (ii) ##f## is a 1-1 function (iii) For all ##i,j \in \mathbb{N}## (with ##j>i##) we must have ##f(j)<f(i)##.

So if we have ##f(x)=a_i+b_j## then we can write ##first(f(x))## and ##second(f(x))## to denote ##a_i## and ##b_j## respectively. Now we can define ##\alpha_0=\min\{\alpha \in p \,|\, \exists x \in \mathbb{N}(first(f(x))=a_\alpha)\}##. Let's denote ##n_0 \in \mathbb{N}## as the "last" value for which ##first(f(n_0))=a_{\alpha_0}##.

Now we define ##\alpha_1=\min\{\alpha \in p \,|\, \exists x \in \mathbb{N}(first(f(x))=a_\alpha \, \wedge \,x>n_0)\}##. Let's denote ##n_1 \in \mathbb{N}## as the "last" value for which ##first(f(n_1))=a_{\alpha_1}##. Now because of ##\alpha_1>\alpha_0##, we should get ##a_{\alpha_1}>a_{\alpha_0}## and hence ##second(f(n_1))<second(f(n_0))##.

So it seems to me that when we define ##\alpha_2=\min\{\alpha \in p \,|\, \exists x \in \mathbb{N}(first(f(x))=a_\alpha \, \wedge \,x>n_1)\}## and ##n_2## as the "last" value for which ##first(f(n_2))=a_{\alpha_2}##, then we should have similarly ##second(f(n_2))<second(f(n_1))##. The last inequality is supposed to follow from ##a_{\alpha_2}>a_{\alpha_1}## (because ##\alpha_2>\alpha_1##).

The previous paragraph seems to be suggestive of defining ##\alpha_i##, ##n_i## generally for an arbitrary natural number ##i## and then creating an infinite descent for the second components: ##......<second(f(n_3))<second(f(n_2))<second(f(n_1))<second(f(n_0))##. However, this descent goes against the assumption of ##B## being a well-ordered set.

Few clarification points:
(1) At some points comparison relation ##<## is used for real numbers and at some points it is used for ordinals.
(2) We can think of ##first## as a function ##first:A+B \rightarrow A##. Similarly, we can think of ##second## as a function ##second:A+B \rightarrow B##.
(3) If we have ##first(f(i))=a_\alpha## for some specific natural number ##i## and some specific ordinal ##\alpha##, then there can't exist arbitrarily large natural numbers ##j>i## such that ##first(f(j))=a_\alpha##."
END

============================================================================================================================================================================================================================================================

P.S.

One miscellaneous point that I think is worth a brief mention (since I can't edit post#4). When talking about "class sets" towards the end of post#4, one thing to note is that the "minimum" of a "subset of ##Ord##" [informal sense] should always be a well-defined ordinal just like the minimum of a "subset of an ordinal".

Post#4 was a fairly scattered/unfocused. The brief point here is that whenever (for defining an ordinal) we write something like:
##min\{ \alpha \in Ord \,|\, ## "insert some condition" ##\}##
it is as good as:
##min\{ \alpha \in \beta \,|\,## "insert some condition" ##\}## (##\beta## being a fixed ordinal)

Even though the second notion is formal and the first one is (initially) informal, the first notion can be made formal [and hence importantly, it isn't ambiguous or ill-defined]. So I don't think there is an issue with using the first notion.

Last edited:
Office_Shredder
Staff Emeritus
Science Advisor
Gold Member
first and second aren't even well defined as functions, are they?

SSequence
first and second aren't even well defined as functions, are they?
Eh, good point. Thanks for pointing this out. It is quite helpful and I think you are right. That was silly on my part indeed.

It seems one fix is to do something like this:
We define:
##first(r)=min\{x\in A\,|\, \exists y \in B (x+y=r) \}##
##second(r)=r-first(r)##

But I need to think over it a bit probably. It seems that in general a definition such as one given for ##first(r)## wouldn't necessarily be well-defined [because the set over which we are taking the min has to be non-empty and needs to have a minimum element or better yet .... well-ordered].

However, because ##A## is assumed to be well-ordered, this definition seems to become justified [e.g. any subset of a well-ordered set is well-ordered]. Because here, assuming ##r \in A+B##, the set ##\{x\in A\,|\, \exists y \in B (x+y=r) \}## is a (non-empty) subset of ##A##.

Edit:
I guess another alternative definition for ##first(r)## (with ##r \in A+B##) could go like (this should be equivalent to def. just above it seems):
"Find the smallest value ##\alpha<p## such that ##a_\alpha+y=r## (where ##y \in B##). Then ##first(r)=a_\alpha##."

Note that some such value ##\alpha## would exist simply because of the virtue of ##r \in A+B## and ##A=\{ a_i\, |\, i\in p \}## (assuming limit ##p##).

Last edited:
Just in case it was a bit unclear, at least as I understand currently, the definition(s) of the function ##first## and ##second## I wrote in last post (post#8) together with what I wrote in post#6 [under the begin and end headings] or in OP [but see comment towards end of OP] should be fine. If there is still something that is missing/unclear (or incorrect), then please do let me know.

========================================

As mentioned in post#7, the original description of ##first## and ##second## was indeed ill-defined. As an example suppose that:
##\{4,6\} \subseteq A##
##\{4,6\} \subseteq B##
Then we have:
##10 \in A+B##

However, now if this value ##10## belongs to ##range(f)##, then it is ill-defined as to what value of ##first(10)## would be since:
##4+6=6+4=10##

Last edited:
fresh_42
Mentor
I think you use a lot of concepts without actually knowing how. E.g. the problem with the definition of ##A+B## where you used the projections 'first' and 'last', but you do not have a direct product, so what is it? An example with numbers cannot be generalized. You can always use the lexicographic ordering on the pairs of ##A## and ##B##, however, you used a sort of addition.

Moreover, it is never a good idea to discuss the same subject at two places at the time, which has no clear answer since the problem is ill-defined. Mathoverflow requires a certain level of knowledge compared to the ordinary Stackexchange math page. I did not get the impression that you have this knowledge on set theory and logic, i.e. a professional level, since otherwise you would not use the language you used (begin, end, first,second etc.). This might explain the downvote.

However, your basic questions were: a) Am I right?, and the answer is 'No', simply because you are far too vague in your wording. Set theory and logic requires an especially precise usage of terms, definitions and language in general. If you do not use this strict corset, you are almost automatically wrong. b) Why I got the downvote?' I think this thread offered more than one reason why. Nonetheless we could only guess and guesswork is outside our mission.

This thread is closed now.

jim mcnamara


text:
Approximation Theory / Théorie d'approximation
Org: Richard Fournier and/et Paul Gauthier (Montreal)

THOMAS BLOOM, University of Toronto, 100 St. George Street, Toronto, ON M1M 3W5
Random Polynomials and Green Functions

Let K be a regular (in the sense of pluripotential theory) compact set in Cn and let VK(z) denote its pluricomplex Green function with a logarithemic singularity at ¥. Then, with probability 1, a sequence of random polynomials {fa} gives the pluricomplex Green function via the formula

 æè \varlimsupa 1 deg(fa) log|fa(z)| öø * = VK (z)     for all z Î Cn
In the one-dimensional case, this result may be used to generalize a result of Shiffman-Zelditch on the limiting normalized distributon of zeroes of random polynomials.

ANDRÉ BOIVIN, Univerity of Western Ontario
Approximation on Riemann surfaces: recent results

Some recent results concerning approximation on Riemann surfaces will be presented. These will include generalizations to Riemann surfaces of a theorem of A. G. Vitushkin on uniform approximation by rational (meromorphic) functions, and of results by T. W. Gamelin and J. B. Garnett on bounded pointwise approximation.

This is joint work with Jiang B.

MAXIM BURKE, University of Prince Edward Island, Charlottetown, PE C1A 4P3
Entire functions mapping uncountable dense sets of reals onto each other monotonically

When A and B are countable dense subsets of R, it is a well-known result of Cantor that A and B are order-isomorphic. A theorem of K. F. Barth and W. J. Schneider states that the order-isomorphism can be taken to be very smooth, in fact the restriction to R of an entire function. J. E. Baumgartner showed that consistently 2À0 > À1 and any two subsets of R having À1 points in every interval are order-isomorphic. However, U. Abraham, M. Rubin and S. Shelah produced a ZFC example of two such sets for which the order-isomorphism cannot be taken to be smooth. A useful variant of Baumgartner's result for second category sets was established by S. Shelah. He showed that it is consistent that 2À0 > À1 and second category sets of cardinality À1 exist while any two sets of cardinality À1 which have second category intersection with every interval are order-isomorphic. In this paper, we show that the order-isomorphism in Shelah's theorem can be taken to be the restriction to R of an entire function. Moreover, using an approximation theorem of L. Hoischen, we show that given a nonnegative integer n, a nondecreasing surjection g :R ® R of class Cn and a positive continuous function e: R ® R, we may choose the order-isomorphism f so that for all i = 0,1,...,n and for all x Î R, |Di f(x)-Di g(x)| < e(x).

DIMITER DRYANOV, Concordia University, 7141 Sherbrooke Street West, Montreal, QC H4B 1R6
A Refinement of an Inequality of R. J. Duffin and A. C. Schaeffer

Let p be a polynomial of degree n. Let ms : = n-1 sin-2( sp/(2n) ), s odd, 1 £ s < n, and mn = (1-(-1)n ) / (4n). By using ms and xj = cos(jp/n), j integer, we define the linear functionals

 Lj (p) = nå s=1, s odd ms æè p(xj) - p(xj+s) + p(xj-s) 2 öø
for j = 0,1,...,n.

The following inequality is established:

 max x Î [-1,1] |p¢(x) | £ n max 0 £ j £ n |Lj(p) |.

Remark 1 In view of

 nå s=1, s odd ms = n 2
and by making use of the sets
 To : = {xj : j odd, 0 £ j £ n},     Te : = {xj : j even, 0 £ j £ n}
we have:
 max x Î [-1,1] | p¢(x) |
 £ n max 0 £ j £ n | Lj(p) |
(1)
 £ n2 2 max x Î Co, y Î Ce |p(x)-p(y)| £ n2 max 0 £ j £ n |p(xj)|.
(2)

Remark 2 The inequality obtained is a refinement of the well known inequality:

 max x Î [-1,1] |p¢(x) | £ n2 max 0 £ j £ n |p(xj)|
which is due to Duffin and Schaeffer [1].

## References

[1]
R. J. Duffin and A. C. Schaeffer, A refinement of an inequality of the brothers Markoff. Trans. Amer. Math. Soc. 50(1941), 517-528.

SERGE DUBUC, Université de Montréal, C.P. 6128, Succ. Centre-ville, Montréal, Québec H3C 3J7
The joint spectral radius of a family of matrices

Let A,B be the two 2×2 matrices

æ
ç
è
 1+2l
 2l
 -1/2-2l+m/2
 1/2-2l-m/2
ö
÷
ø
,     æ
ç
è
 1/2-2l-m/2
 -1/2-2l+m/2
 2l
 1+2l
ö
÷
ø
,
we define the subset of the plane W = {(l,m) :r(A,B) < 1} where r(A,B) is the joint spectral radius of {A,B}. We discuss the approximation of W.

RICHARD FOURNIER, CRM, Université de Montréal, C.P. 6128, Succ.  Centre-ville, Montréal, Québec H3C 3J7
A New Inequality for Polynomials

Let D be the unit disc of the complex plane C. We prove that for any polynomial p of degree at most n

 max q Î R êê p(eiq) - p(e-iq) eiq - e-iq êê £ n max 0 £ j £ n êê p(eijp/n) + p(e-ijp/n) 2 êê .
We shall also discuss how this inequality is related to classical results or Bernstein and Markov and to more recent ones due to Duffin and Schaeffer, Frappier, Rahman and Ruscheweyh.

This is joint work with Dimiter Dryanov.

PAUL GAUTHIER, Université de Montréal, Centreville, Montréal, QC H3C 3J7
Approximation of and by the Riemann zeta function

Firstly, we approximate the Riemann zeta function by meromorphic functions for which the Riemann hypothesis fails. Secondly, we approximate arbitrary holomorphic functions by linear combinations of translates of the Riemann zeta function. The first result is joint work with E. S. Zeron. The second work is joint with N. N. Tarkhanov.

DAN KUCEROVSKY, University of New Brunswick
Quasi-monotone sequences

Functions from finite sets to Rm occur very frequently in applied problems. If m=1, then there is a standard definition of monotonicity, and it is often useful to break up the function into (approximately) monotone segments. For the case of higher dimensional range spaces, there is no order structure, so we instead consider breaking up the functions into segments of bounded curvature. This leads to the problem of determining a bound on the curvature of a segment in a computationally efficient way, which we do by a recursive formula. We compare with quasi-linear fitting, obtained by a least-squares method, and find that the curvature method is more robust, in particular, with respect to deletion of data points.

Joint work with Daniel Lemire, UQAM.

Zeros of functions in the Dirichlet space

There are several uniqueness theorems for functions in the classical Dirichlet space. But, a necessary and sufficient condition (like the Blaschke condition for Hardy spaces) for this space is not yet available. We will discuss a new uniqueness theorem.

This is a joint work with Thomas Ransford and Abdellatif Bourhim.

THOMAS RANSFORD, Université Laval, Dép. de mathématiques, Québec (QC), G1K 7P4
A Denjoy-Carleman maximum principle

We prove a quantitative form of the classical Denjoy-Carleman theorem on quasi-analytic classes. As an application, we derive an extension of Carleman's theorem on the unique determination of probability measures by their moments.

Joint work with Isabelle Chalendar, Laurent Habsieger and Jonathan Partington.

GERALD SCHMIEDER, Universitaet Oldenburg, Fak. V, Inst. f. Math., 26111 Oldenburg, Germany
Extension of the Fusion Lemma

Let K1, K2 be compact sets in the complex plane C. We say that K1, K2 is a fusion pair if there exists some constant a = a(K1,K2) with the following property: for all rational functions r1, r2 and for each compact set K Ì C there is a rational function r which fulfills ||rj-r||KjÈK £ a·||r1-r2||K simultaneously for j=1,2.

Alice Roth proved in 1976 that K1,K2 is a fusion pair if the sets K1, K2 are disjoint. If K1ÇK2 = Æ is not required we have of course to replace ||r1-r2||K above by ||r1-r2||KÈ(K1ÇK2). But in general K1, K2 is no fusion pair (examples are due to Gauthier and Gaier).

Under rather natural topological restrictions (especially that K1,K2 have no common interior points) we can characterize the fusion pairs by the simple condition that K1 ÇK2 = (K1ÈK2).

JIE XIAO, Memorial University of Newfoundland, St. John's, NL A1C 5S7
The Heat Equation II: Regularity and Approximation

We use the Carleson-Sobolev estimates for an operator valued solution of the heat equation to give the regularity of the solution and its application to the Weierstrass approximation theorem.

EDUARDO ZERON, Cinvestav (Math), Apartado postal 14-740, Mexico DF, 07000, Mexico
Homotopical obstructions to rational approximation

There are several criteria to decide whether a continuous function F(z) defined from a compact set K Ì Cn into C can be approximated by holomorphic rational functions (P/Q)(z). Now, any rational function (P/Q)(z) can be seen as a holomorphic function defined from an open set of Cn into the Riemann sphere S2. And we can even generalise the concept of rational function to consider holomorphic functions with range into the complex projective space CPn.

We may then ask about the rational approximation of continuous functions G(z) defined from a compact set K Ì Cn into CPn. We want to show, in this talk, that there is essentially one extra necessary condition on G(z) to be approximated by rational functions: Function G must be homotopically trivial.

PING ZHOU, St. Francis Xavier University, Antigonish, NS B2G 2W5
Divided differences in construction of multivariate Padé approximants

We explicitly construct multivariate Padé approximants to some functions of the form

 F(x1,x2,...,xm) = ¥å n=0 f(n) å j1+j2+¼ +jm=n x1j1 x2j2 ¼xmjm,
where f(n) is a certain function of n, by using divided differences. Examples include the multivariate exponential function
 E(x1,x2,...,xm) : = ¥å j1,j2,...,jm=0 x1j1 x2j2 ¼xmjm (j1+j2+¼ +jm+1)! ,
the multivariate logarithm function
 L(x1,x2,...,xm) : = ¥å j1,j2,...,jm=0 x1j1 x2j2 ¼xmjm j1+j2+¼+jm+1 ,
and others.