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Now showing items 1-20 of 93

• #### A cellular decomposition of the manifold of observable conditioned invariant subspaces

(2000)
Article
Open Access
Given an observable system defined by a pair of matrices $(C,A)$ we obtain a cellular decomposition of the manifold of $(C,A)$-conditioned invariant subspaces having the restricted system fixed observability indices and ...
• #### A chaotic secure communication scheme based on duffing oscillators and frequency estimation

(2013)
Conference report
Open Access
This work presents a new technique to securely transmit and retrieve a message signal via chaotic systems. In our system, a two-valued message signal modulates the frequency of a Duffing oscillator sinusoidal term. An ...
• #### A coordinate atlas of the manifold of observable conditioned invariant subspaces

(1999)
Article
Open Access
Given an observable system (C,A)€ K^(mxn)X K^(nxn)with K=R or C, the set of (C,A)-invariant subspaces having the restricted system fixed observability indices is a smooth manifold embedded on the corresponding ...
• #### A defuzzification based new algorithm for the design of Mamdani-type fuzzy controllers

(Universitat Politècnica de Catalunya. Secció de Matemàtiques i Informàtica, 2000)
Article
Open Access
This paper presents a new learning algorithm for the design of Mamdani-type or fully-linguistic fuzzy controllers based on available input-output data. It relies on the use of a previously introduced parametrized defuzzification ...
• #### A genetic algorithm for the multistage control of a fuzzy system in a fuzzy environment

(Universitat Politècnica de Catalunya. Secció de Matemàtiques i Informàtica, 1997)
Article
Open Access
We discuss a prescriptive approach to multistage optimal fuzzy control of a fuzzy system, given by a fuzzy state transition equation. Fuzzy constraints and fuzzy goals at consecutive control stages are given, and ...
• #### A geometrical aprroach to Cp-globalization of pointwise matricial relations

(1999)
External research report
Open Access
• #### A mixed-signal architecture for high complexity CMOS fuzzy controllers

(Universitat Politècnica de Catalunya. Secció de Matemàtiques i Informàtica, 1999)
Article
Open Access
Analog circuits provide better area/power efficiency than their digital counterparts for low-medium precision requirements [1]. This limit in precision, as well as the lack of design tools when compared to the digital ...
• #### A modified Chua chaotic oscillator and its application to secure communications

(2014)
Article
Open Access
In this paper, a new modified Chua oscillator is introduced. The original Chua oscillator is well known for its simple implementation and mathemati- cal modeling. A modification of the oscillator is proposed in order to ...
• #### Actuator fault-tolerant control based on set separation

(2008)
External research report
Open Access
In this report an actuator fault-tolerant control (FTC) strategy based on set separation is presented. The proposed scheme employs a bank of observers which match the different fault situations that can occur in the plant. ...
• #### Admissible Model Matching Fault Tolerant Control based on LPV Fault Representation

(2009-11-19)
Conference report
Open Access
• #### Algebraic PGD for tensor separation and compression: an algorithmic approach

(2018-07)
Article
Open Access
Proper Generalized Decomposition (PGD) is devised as a computational method to solve high-dimensional boundary value problems (where many dimensions are associated with the space of parameters defining the problem). The ...
• #### Alimentación de modelos cuantitativos con información subjetiva: aplicación Delphi en la elaboración de un modelo de imputación del gasto turístico individual en Catalunya

Article
Open Access
El presente artículo presenta un estudio realizado para el Institut d'Estadística de Catalunya, donde se ha aplicado una técnica que trabaja con información subjetiva (el Método Delphi) para obtener unos datos que puedan ...
• #### An LPV Fractional Model for Canal Control

(PERGAMON PRESS. IFAC SERIES., 2009)
Conference report
Restricted access - publisher's policy
In this paper an LPV non-integer order control model of an irrigation canal is derived from system identification experiments. This model is experimentally obtained by using the described LPV fractional identification ...
• #### Analysis of the control structures for an integrated ethanol processor for proton exchange membrane fuel cell systems

(2009)
Article
Open Access
The aim of this work is to investigate which would be a good preliminary plantwide control structure for the process of Hydrogen production from bioethanol to be used in a proton exchange membrane (PEM) accounting only ...
• #### Analyzing the performance of a miniature 3D wind sensor for Mars

(Multidisciplinary Digital Publishing Institute (MDPI), 2020-10-20)
Article
Open Access
This paper analyzes the behavior of a miniature 3D wind sensor designed for Mars atmosphere. The sensor is a spherical structure of 10 mm diameter divided in four sectors. By setting all the sectors to constant temperature, ...
• #### Análisis tensión - deformación de la construcción incremental de presas de tierra: comparación entre FLAC y PLAXIS

(Universitat Politècnica de Catalunya / Universitat de Barcelona, 2002-11)
Minor thesis
Open Access
L’objectiu del treball realitzat consisteix en la comparació de dos dels programae comercials de mètodes numérics més utilitzats fins el moment, FLAC i PLAXIS, per al cas particular de l’anàlisi de tensió-deformació de ...
• #### Automatic synthesis of fuzzy logic controlers

(Universitat Politècnica de Catalunya. Secció de Matemàtiques i Informàtica, 1996)
Article
Open Access
This paper describes a design environment for the hardware realizations of fuzzy controllers which includes a set of CAD tools to ease the description, verification and synthesis of this kind of systems. Special emphasis ...
• #### Automatized Measurement Setup for Performance Evaluation of Wi-Fi Mesh Networks (A.M.S.P.E.W.M.N.)

(Universitat Politècnica de Catalunya, 2017-10-20)
Bachelor thesis
Open Access
Covenantee:   Technische Universität Wien
Telecommunications is an everchanging world. Everyday we are faced with improvements in the way we communicate and so in the tools that we use, especially in the field of Wi-Fi connection. The 5G connection will soon start ...
• #### Barcelona Virtual Mobility Lab: the multimodal transport simulation testbed for emerging mobility concepts evaluation

(International Academy, Research, and Industry Association (IARIA), 2018)
Conference report
Open Access
New sustainable mobility concepts and smart resilient ideas are arising every day. However, there is not an easy way to bring these ideas into reality, or to test how good they are as mobility solutions. Virtual Mobility ...


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# Integrating $\int_{\sqrt{2}}^2 \frac{1}{t^3\sqrt{t^2-1}}\,dt$.

I am trying to compute $$\int_{\sqrt{2}}^2 \frac{1}{t^3\sqrt{t^2-1}}\,dt.$$

This is what I got so far:

$t=\sec(x)$ and $dt=\sec(x)\tan(x)x\,dx$

So plugging this in gives me $$\int \frac{1}{\sec^3(x)}\cdot\sqrt{\sec^2(x)-1}\sec(x)\tan(x)\,dx.$$ By my trig property $1+\tan^2(x)=\sec^2(x)$, I get $$\int \frac{1}{\sec^3(x)\tan(x)}\sec(x)\tan(x)\,dx.$$ Then I simplify and get $$\int \frac{1}{\sec^2(x)}\,dx.$$ By the trig property I get $$\int\cos^2(x)\,dx.$$ Then I get $(x/2)+(1/4)\sin(2x).$

So now I need to get back to my original variable so I solve for $x$:

$t=\sec(x)$ and $x=\mathrm{arcsec}(t)$.

It's this part I'm confused about. I don't know what step to take next. I know I am supposed to draw my triangle, I just don't know how to get there.

• Welcome to MSE! Please use LaTeX or Mathjax to format your question. – graydad Jan 23 '15 at 19:42
• @graydad Lighten up a bit when it's a first time user, please. At the very least, provide a link to the tutorial. Some folks have never heard of mathjax. – Namaste Jan 23 '15 at 19:43
• Are LaTeX and Mathjax websites – Jessica Garcia Tejeda Jan 23 '15 at 19:44
• It's a way to typeset equations and mathematics to display nicely. See mathjax tutorial – Namaste Jan 23 '15 at 19:44
• In the future I will use Mathjax. I'm currently on a tablet device and am a new user of MSE. Thank you for sharing that Aaron. – Jessica Garcia Tejeda Jan 23 '15 at 19:52

$t=\sec(x)$

You can build a right triangle for this to translate your answer in terms of $t$ to in terms of $x$.

## Also here's a hint: $\sin(2x)=2\sin(x)\cos(x)$

oops didn't realize it was a definite integral

Let $x=\cosh t$, and use the fact that $\cosh^2t-\sinh^2t=1$, along with $\cosh't=\sinh t$ and $\sinh't=\cosh t$. You'll get $$I=\int\frac{\sinh t}{\cosh^3t\cdot\sinh t}dt=\int\frac1{\cosh^3t}dt=\int\frac{\cosh t}{\cosh^4t}dt=\int\frac{d(\sinh t)}{(1+\sinh^2t)^2}=\int\frac{du}{(1+u^2)^2}$$ which is trivial. $($Of course, you'll have to pay attention to evaluating the new limits of integration at each substitution$)$.

there is no need to go back to the original variable. you could change the limits of integration at the same time like:

change of variable: $t = 1/\cos x, dt = \frac{\sin x}{\cos^2 x} dx$ and at $t = \sqrt 2, x = \pi/4$ and $t = 2, x = \pi/3$

$$\int_{\sqrt 2}^2 \dfrac{1}{t^3\sqrt{t^2 - 1}} \ dt = \int_{\pi/4}^{\pi/3} \cos^2 x \ dx = \frac{1}{2}\int_{\pi/4}^{\pi/3} (1 + \cos 2x) \ dx$$

By substituting $u=\sqrt{t^2-1}$ you get using $\newcommand{\dd}{\; \mathrm{d}}\dd u = \frac{t}{\sqrt{t^2-1}} \dd t$ and $t^2=u^2+1$ that $$\int \frac1{t^3\sqrt{t^2-1}} \dd t= \int \frac1{t^4} \cdot \frac{t}{\sqrt{t^2-1}} \dd t = \int \frac1{(u^2+1)^2} \dd u.$$ (So in the end this seems just as a variation of Lucian's answer.)

Probably the easiest thing to do at this point is to use that for $0<x<\frac{\pi}{2}$,

$\;\;\;\sec x=\sqrt{2}\implies \cos x=\frac{1}{\sqrt{2}}\implies x=\frac{\pi}{4}$ and

$\;\;\;\sec x=2\implies \cos x=\frac{1}{2}\implies x=\frac{\pi}{3}$.

Alternatively, you can use $t=\sec x\implies\cos x=\frac{1}{t}$ and $\sin x=\frac{\sqrt{t^2-1}}{t}$ $\;\;$if $0<x<\frac{\pi}{2}$,

and $\sin 2x=2\sin x\cos x$.

• I though that I couldn't just do that because I changed the variable. So because I changed the variable, I can't use the original bounds. – Jessica Garcia Tejeda Jan 23 '15 at 19:58
• You can either change the limits and then use the antiderivative you have (in terms of x), or you can write the antiderivative in terms of t and then use the original limits. – user84413 Jan 23 '15 at 20:03

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text:
GGP Home Agreements Purpose / Goals Mailing Address / Email List Links Newsletters Publications / Online Stations Glossary Maps

### Status Report 1994

David Crossley1 and Jacques Hinderer2

1Department of Earth and Planetary Sciences, McGill University, Montreal
2Institute de Physique du Globe, Strasbourg

in Proceedings of the Second IAG Workshop on Non-Tidal Gravity Changes: Intercomparison between absolute and superconducting gravimeters, Walferdange, Luxembourg, September 6-8, 1994

### Preface and Acknowledgments

This GGP Status Report records the activities of the various superconducting gravimeter groups since the last report (Aldridge et al., 1991). It is the outcome of discussions held at several scientific meetings held during 1994 which have helped to define how the GGP will operate. We emphasise the agreements that have been made with respect to Data Recording and Exchange and in the establishment of an Observation Period to begin in July 1995.

Although the concept of the GGP and the scientific objectives have not changed, to make this paper self contained we also include some background information on the scientific objectives of the GGP. Several concepts have evolved in connection with the Data Centre and Data Exchange and therefore this report considerably updates the previous one in these areas.

The GGP community is not large (there are at most 20 superconducting gravimeters (SGs) in operation worldwide), though the goals of the various groups running the instruments is extremely diverse. Due to the character of national research efforts, there is a natural tendency for each group to re-invent the wheel' either in terms of data acquisitions systems or data processing and data analysis techniques. Despite this diversity, the agreements that have been reached with respect to the GGP recording parameters and data exchange demonstrate that significant progress has been achieved.

It would be difficult, if not impossible, to properly reference each individual for all the comments that have been made. Instead we have summarised the various contributions in the body of the text. We would like to thank sincerely all those who not only have made the effort to be present at important scientific meetings at which GGP has been discussed, but also who submitted information concerning their stations and gave comments on the proposals.

### Brief Summary of the GGP

#### Objectives

The Global Geodynamics Project (GGP) is a proposal to monitor changes in the Earth's gravity field at periods of seconds and longer. The GGP is named to indicate the application of gravity data to the solution of a number of geodynamic problems; additionally GGP may become a source for absolute gravimeter data as well as other geodynamic data.

The measurements will be taken over a time span of 6 years at a small number of permanent observatories where there is a superconducting gravimeter (SG) currently installed. The 6-year period has been chosen as the minimum length of data required to separate annual and 14 month Chandler wobble components in the gravity record. The Observation Period will commence July 1995.

The SG has been, for the past two decades, the most sensitive, stable instrument for the measurement of the vertical component of the Earth's gravity field. Each of the currently operating SGs is the focus of a national effort to provide a continuous gravity record for geodetic and geophysical research. The GGP is an opportunity for the various SG groups to participate in a global campaign to monitor the gravity field and to exchange the raw data.

Precise global measurements of the Earth's gravity field are essential to answer a number of important questions in geophysics, which we outline in more detail in the next section (a) do internal gravity waves (inertial waves if the fluid is neutrally stratified) exist in the Earth's liquid core and are their gravitational effects at the Earth's surface detectable ? (b) what is the gravity effect of the global atmospheric loading and mass re-distribution on the solid Earth ? (c) through global tidal analysis, can we refine estimates of the nearly diurnal free wobble of the Earth and models of oceanic loading on the solid Earth ? (d) what changes in gravity are associated with slow and silent earthquakes, tectonic motions, sea-level changes and post-glacial rebound ? (e) can we monitor the location of the rotation pole of the Earth on a time scale of minutes ? (f) can SG recordings of the earth's normal modes enhance the global long period seismic and spring gravimeter networks ?

#### Benefits

The benefits of GGP are twofold :

1. To reassure users of SG data that extreme care has been taken in the sampling and pre-processing of the available data and that all pre-processing steps and other site-specific information such as atmospheric pressure, environmental data and a record of all site disturbances are available to users, and
2. To enable global signals to be extracted by various stacking procedures that would not be possible with single station recordings.

#### SG Groups

The GGP is open to all organisations with access to the appropriate instrumentation, i.e. a SG. Each independent organisation that manages a SG will be called a SG group; there may be several SG groups in any one country.

The GGP Agreements encourage SG groups to (a) upgrade existing SG facilities to a common standard of data acquisition, (b) participate in the 6-year observation period by maintaining the SG's in good operating conditions at fixed locations and (c) exchange raw gravity (and other important supplemental) data through high speed computer links.

Funding for GGP activities is the financial responsibility of the supporting government organisations for each of the national SG groups.

#### Data Centre and Exchange

It is not the intention of the GGP to restrict in any way the provision of SG data between individual SG groups and potential users, either individual or collective. Each SG group is free to provide its data as it wishes in response to normal requests. However GGP Data {\bf as a whole} is available on a restricted basis to protect the investment in time and effort that SG groups have made to acquire the data and pre-process it to GGP standards.

The GGP will include an International Data Centre from which all the SG data can be accessed according to the agreements reached between participating SG Groups. The Japanese SG Groups will establish a Japan GGP Subcentre to exchange data with the International Data Centre. SG data will be restricted to GGP participants for the first year following collection; thereafter the data will be placed in the public domain for general access.

Let the GGP participants be in no doubt that when GGP begins, some significant changes will occur in the pace with which SG data is processed. Each group will have a one year grace period in which to process its data and make it available to other GGP participants. Then there will be a further year for the various groups to collaborate in publishing the results of the data exchange. After two years the GGP data will be available to the scientific community in general. We are convinced this agenda will stimulate a more active dissemination of this important data and that the various SG groups can rise to the challenge of preparing the data.

#### Organisation

The activities of the GGP are by a Steering Committee with a Chairman (Crossley) and Secretary (Hinderer). The Steering Committee is responsible for all aspects of the GGP such as creation of sub-committees, setting the timetable for the project, setting standards for the data acquisition systems and data exchange protocols and coordinating activities of the Data Centre. The Steering Committee agrees to meet at least once a year, in person, preferably in conjunction with an appropriate scientific meeting.

The GGP has been endorsed by the IUGG Inter-Union Project SEDI (Study of the Earth's Deep Interior) and has been presented to the IAG. We will also present the proposal to IASPEI.

### Scientific Objectives

The scientific goals of the GGP include seismic, geodetic and geophysical components that range in frequency content from seismic normal modes, tides, core modes and wobble modes of the Earth to other long period variations in Earth's gravity field such as tectonic deformation (Fig. 1).

#### Global Data Acquisition and Distribution

Many of the Earth parameters of critical interest in global dynamics are recoverable from gravimetric signals at or below the ambient noise level. This includes internal gravity waves in the fluid core, post-glacial uplift and plate motions. The SG has a stated sensitivity at the nanogal level and most signals of interest are expected to be in this range. Since the ambient gravity noise is usually three orders of magnitude greater than this, signals identified on the record of an individual instrument at the nanogal level cannot be considered reliable unless and until confirmed with similar signals from other instruments. Furthermore, these instruments must be distributed widely around the Earth because many gravimetric signals are global in spatial extent and have theoretically predictable spatial and temporal global variations.

Rapid access to worldwide gravimetric data is essential for progress in global geodynamics for several reasons. First, high resolution gravimetric data makes it possible to measure free oscillations of the Earth with unprecedented precision. In real-time, an array of SG instruments would provide a means for global detection of silent earthquakes' (Beroza \& Jordan, 1990). Second, sub-milliarcsecond orientation can be obtained through measurement by an SG network for space-based measurements such as Satellite Laser Ranging and the US-proposed GLRS project to position points on the Earth's surface to the sub-centimetre level through the use of Earth-based retro-reflectors and satellite-based lasers. Third, Earth models which incorporate core resonances require access to gravimetric data at the nanogal level to successfully account for motion in the deep interior in all of the orientation calculations. The proposed network of instruments and communication makes access to gravimetric data sufficiently rapid that all the above tasks can be accomplished.

Lastly, a number of tectonics-related problems require global gravity field data for their resolution. In particular the problems of long-term secular changes in elevation, caused not only by post-glacial rebound and sea level changes but also by active plate-tectonic related deformation, need long-term gravity variations at continental scales. The long period behaviour of SG's is currently very variable, with the best instruments virtually drift-free at the level of about 1 microgal per year. If this level of stability can be maintained by even a small number of SG stations in the global network, particularly where calibrated with absolute gravimeters, then GGP will provide very important data for these tectonic problems. At shorter periods, the signatures of earthquake pre-cursor activity, the search for slow earthquakes and the precise monitoring of the normal modes following large earthquakes are all tasks for which the SG is particularly suited.

In the past an individual with access to a computing service could make major progress in the solution of both analytical and data analysis problems. However, the complexity of presently extant problems in global geodynamics is such that measurements on a global scale are needed to make even minimum progress. Concerted effort by cooperating scientists is needed to make any significant advance. Without uniform high precision global data it will be impossible to move toward the solution of the problems of the next geophysical frontier - the Earth's deep interior. High quality continuous vertical gravity values need to have global coverage to answer a number of important geophysical questions. Due to the extremely weak gravity signals (some at the nanogal level) associated with many of these phenomena, it is essential that very high sensitivity ground based observations using SGs be undertaken.

#### GGP Goals

The SG is capable of recording temporal gravity variations from seconds to years and thus the GGP has application to large number of scientific tasks. At long periods (months - years), we highly recommend the use of a SG supplemented by an Absolute Gravimeter to fully characterise secular trends in gravity.

• Earth tides and the nearly diurnal free wobble: the estimation of precise tidal parameters (e.g. gravitational delta factors) will allow the development of better models for correcting for ocean loading phenomena. In addition, the stacking of global delta factors provides important information on the diurnal free wobble of the Earth which is essential for theoretical work on the structure of the Earth's core.
• core modes: the search for internal gravity waves in the Earth's liquid core necessitates global, long-period, long-duration recordings to separate local gravity variations from a global coherent signal. The detection of these waves will give direct information on the mechanical equilibrium of the fluid in the core, and thus information on the operation of the geodynamo.
• atmospheric interactions: stacking global gravity and pressure data is essential to clarify the nature of the long period phenomena in the atmosphere and for evaluating the effects of global atmospheric surface pressure and mass redistribution on the Earth's gravity field.
• Earth rotation and polar motion: the measurement of the gravity effect of polar motion (orientation of the Earth's rotation axis) requires a global coverage of stations. It should be possible to continuously monitor the location of the rotation pole on the time scale of minutes and therefore provide an independent verification of the same measurement now made with space techniques; connections with the International Earth Rotation Service (IERS) service here will be valuable.
• gravity changes due to tectonic motions: the monitoring of long-term changes due to tectonic motions, sea-level changes affecting the survival of coastal cities, post-glacial uplift and the deformation associated with active tectonic events.
• enhancing absolute gravity measurements: SGs are a valuable aid to international programmes for the determination of absolute gravity values on a global scale as they provide a short-term, relative gravity reference level.
• general research tool: a high quality, continuous global data set will be a valuable resource for future geodetic and geophysical studies that involve the Earth's gravity.

There are important connections between the above goals and other scientific programmes of national concern. In particular, the geodetic community clearly recognises the importance of simultaneous geodetic (positional) information and gravity changes at fiducial stations which contain very high quality instrumentation. There are two primary connections :

• space techniques: Two space techniques which require detailed models of Earth deformation are satellite tracking and Very Long Baseline Interferometry (VLBI). At the proposed sub-centimeter level of accuracy, for projects in the 1990's such as the current Satellite Laser Ranging (SLR) and the proposed Geodynamics Laser Ranging System (GLRS) mission, precise knowledge of the Earth's dynamics, including resonances in the liquid core, are required. A global net of SG's will give the required information on dynamics of the liquid core.
• sea level changes: A satisfactory solution to the problem of defining the origins of sea-level changes requires input from different sources. The necessity of differentiating between the effects of height variations caused by post glacial rebound or plate tectonics and changes in sea level resulting from global warming demands the establishment of a global geodetic/geophysical observatory network, such as FLINN (Fiducial Laboratories for an International Natural Science Network), an IUGG-sponsored project initiated at the Coolfront Workshop in 1989. A central feature of such a network is the monitoring of the gravity field at a smaller group of fiducial stations equipped with SGs as well as precise positioning instrumentation (e.g. SLR, VLBI or GPS) and having accurate connections to the reference tide gauges.

Additionally, the SG's can be used for valuable local studies in the following areas:

• seasonal effects: long-period seasonal (annual, semi-annual) components have been observed in gravity variations at some single SG stations. These variations cannot be successfully modeled without comparisons with other SG stations.
• earthquakes: a SG with a bandwidth of 1 second to several years is the only instrument capable of monitoring both earthquake activity and tectonic motions. At intermediate time scales the SG is the ideal instrument for detecting slow and silent earthquakes.
• seismic normal modes: the SGs have excellent noise characteristics for the observations of the Earth's normal mode spectrum following a moderate to large earthquake.
• geodesy: as indicated above, single SGs, if located at strategic geodetic sites, can considerably enhance local models used to reduce VLBI and other precise measurements and if located near the coast, would provide data for estimating true local sea level changes.

### Recent Scientific Meetings

We summarise the discussions that took place at several meetings held in 1994, listed chronologically.

#### Walferdange, Luxmebourg, 14-16 March 1994

The following is a re-construction of the presentations and discussion during the GGP portion of the 76th Journées Luxembourgeoises de Geodynamique, Walferdange, Luxembourg, 14th-16th March, 1994. Those in attendance at the March Luxembourg meeting were :

T. Camelbeeck, V. Dehant, B. Ducarme, M. Everaerts, P. Melchior, F. Olivier, P. Paqet, C. Poitevin, F. Rosenbeek and H. Sun (Royal Observatory of Belgium, Brussels), D. Crossley (McGill U., Montreal), W. Schwahn and B. Richter (IFAG, Frankfurt), J. Hinderer (IPG, Strasbourg), J. Neumeyer, P. Schwintzer and W. Rongjiang (GeoForschungsZentrum, Potsdam), R. Warburton (GWR Instruments, San Diego), E. Smets (Lab. Hydraulics Research and Antwerp, U. Leuven, Belgium), B. Meurers (U. Vienna), H. Virtannen and J. Kääriänen (Finnish Geodetic Inst., Helsinki), G. Casula (Bologna U.), M. Bonatz (U. Bonn), L. Brinnick (Geophysical Inst., Bratislava, Slovakia), G. Jentzsch (Tech. U. Claustal), H. Hsu (Inst. Geodesy and Geophys., Wuhan), L. Mansinha (U. Western Ontario, Canada).

##### Status Reports of the Various GGP Groups

See Table 1 for station details.

Finnish SG - Kääriänen The instrument will be located in Mestahovi (30 km west of Helsinki), a site which includes a GGP receiver. The building is nearing completion (projected 1st April 1994) and the instrument will be installed during the summer of 1994. Calibration will be by ET and Absolute gravimeters.

Braunschweig - G. Jentzsch This SG was intended to measure the gravity changes associated with filling of a salt deposit with environmentally hazardous material. Due to difficulties with instrumentation (the SG took 6hr to stabilise following movement, the data was not reproducible and there were calibration problems), the experiment was not successful as intended. In addition the very big galleries show signs of instabilities. The possibility of using this SG for another experiment of several years was discussed.

Chinese SG (Wuhan) - G. Jentzsch The current operation uses a German designed digital recording system (since November 1988) with 6.5 digits, 20s sampling. Originally conceived as a test station, there are plans to move the station late this year to a new site. The recording suffers from a noisy cultural environment and gaps due to power failures. There are few offsets but many spikes which are often one-sided and correlate in occurrence with the working day. There is a problem with the air pressure correction; the admittance is high (-0.39 mgal/mbar) but there is a poor correlation coefficient and a strong annual signal. The residual gravity shows evidence of polar motion.

Chinese SG (Wuhan) - H. Hsu Hsu summarised the problems with the current site and discussed the planned move to a new geodetic site which has a SLR system and atomic time. He showed details of the new installation which he hoped could be occupied by the end of 1994.

German SG (Potsdam) - P. Schwintzer The Potsdam SG site is 3 km away from the town centre and located on sediments. It has been operational since July 1992 using a 33 channel data acquisition system from GWR Instruments. Calibration is via LaCoste tidal meters. Helium consumption is 0.4 liters/day with cool head maintenance every 2 yr. and compressor maintenance 1.5 yr. There is 1 download of data per week. The installation of custom designed feet for the dewar was found to reduce the noise level. Average drift is 3 mgal/year, water level admittance is 7 mgal/m.

Schwintzer expressed concern with the restricted distribution of European SG stations and raised the possibility of moving the Potsdam instrument to a geographically more useful site (Asia ?). Also discussed were the standardisation of data formats and the use of the IFAG calibration platform. This SG will participate in the GGP.

Various SG Installations - B. Richter Richter reviewed the status of several instruments with which he is involved. The original TT40 (GWR 1) is being upgraded and will be installed for a parallel recording with the new NOAA instrument to be installed at Boulder, Colorado in the autumn of 1994. After the IUGG (in the fall of 1995, the upgraded TT40 will be installed in Miami for a 3-4 year period. The other original Bad Homburg instrument, TT60, will continue to operate at Wettzell in Bavaria at the IFAG Fundamental Geodetic Station.

A new portable instrument, SG103, now at Bad Homburg is still being tested. It is not yet drift free (0.1 mgal/day) and is sensitive to room temperature, possibly through the foot screws. There are plans to install this instrument for a limited time period in Norway. A fourth instrument, SG101, has not yet been allocated to an experiment.

Richter emphasised the need for accurate rainfall and groundwater measurements and noted these were different, though correlated, parameters. This concern is particularly important at the Miami site where there is frequent heavy rainfall. He also discussed the benefits of parallel registration and showed that hourly differences between two closely sited instruments can be in the range ± 1 mgal. Additionally, the data processing is very important and the selection of offsets can be crucial to the overall record. GGP instruments need to be properly calibrated by Absolute meters, or preferably by the new calibration platforms.

French SG (Strasbourg) - J. Hinderer The Strasbourg instrument has been in operation since 1987, yielding 6.5 years of data, with the same levitation. Raw data, originally sampled every 2s, are stored every 5min through a digital filter; the same for atmospheric pressure. In the future, there are no planned changes for the site, but the data acquisition system should be replaced for the GGP. Calibration is by Absolute meter.

Canadian SG (Cantley) - D. Crossley The Canadian instrument operated in its first phase between November 1989 and October 1993 at a site provided by the GSC. It is currently being refurbished by GWR to correct the excessive drift (0.5 mgal/day), the highest of all operating SG's) and re-installation will occur in the second half of 1994. Despite the high drift, comparisons with the French instrument have shown low residual noise levels (due to the good site and data acquisition systems). The data shows four large offsets, two of which are connected with large electrical storms. One of these resulted in the necessity to re-center the ball in the summer of 1993. Other than that there have been almost no gaps, but the site does experience several earthquakes per week. Calibration has been done using ET and Absolute meters and by comparison with synthetic tides.

[Note Added: GWR T012 will be re-installed in December 94; we hope to test new installation and calibration procedures, as well as the dewar mounting and tilt compensation systems. It will be calibrated by the IFAG lift during 1995.]

Japanese SGs - D. Crossley In the absence of our Japanese colleagues, Crossley summarised some of the main characteristics of the SG instruments as reported in various publications.

Two SGs are located side-by-side at Kyoto University and run by Nakagawa and his colleagues at the Department of Geophysics. As reported by Nakagawa et. al. (paper presented at the 3rd International SEDI symposium, Mizusawa, 1992) the recordings from late 1988 - late 1989 are continuous. One has a decreasing drift with time, the other a steady upward trend (both of amplitude in the 100's of mgal over the year). Analysis of the recordings allows the FCN resonance to be modeled.

The SG at Kakioka is operated by J. Segawa and colleagues at the Ocean Research Institute in Tokyo. The data acquisition, processing methods and results are described in Seama et al. (1993). Data are recorded from late 1989. The raw data show a number of large offsets caused by large earthquakes and sphere re-levitations and spikes which appear to have a common signal shape. Calibration is based on a combined synthetic tides/ocean loading time series. The instrument drift appears to be well modeled by exponential functions. The current drift rate is of the order of 0.2 mgal/day. The residual gravity is correlated with polar motion.

The fourth Japanese instrument is at the Esashi Earth Tides Station near Mizusawa and operated by T. Sato, Y. Tamura and colleages. High rate (2s) gravity data is recorded and processed by the standard BAYTAP procedure (Tamura et al. (1993) used by all the Japanese groups.

Results have been reported concerning the Japanese SG installation at Syowa in Antarctica (Sato et al., 1993; Shibuya, 1993). Problems in transporting the original instrument in 1992 caused a leak in the dewar which was replaced by GWR. After re-installation, observations began on 19 March 1993 and initial results look promising. The helium for the dewar is liquefied on site from gaseous helium transported from Japan.

Belgian SGs - B. Ducarme The original Brussels instrument continues to function providing a twelve year record of data. The recording and analysis procedures have been reported several times in the literature. There are plans to install a second Belgian instrument at Membach which is an existing seismograph station located 100 km east of Brussels. The new site should be much quieter than Brussels but there is some concern over the interference between the cold head pump (2 Hz) and the existing long-period seismometers.

##### Instrument and Data Acquisition Issues

Instrument Design - R. Warburton Warburton (GWR Instruments) gave a review of the technical aspects of the gravimeter design and current issues concerning offsets and drift reduction. There are several possible causes of instrument drift behaviour, some resulting in positive drift, some negative. The characteristics of the instrument also result in offsets caused by the Pb shields used in some designs. There was some confidence that the latest design would answer the existing concerns.

He recommended a data acquisition calibration procedure to be applied by an electrostatic signal at the start of each SG record and a gravity calibration be done by a separate instrument. He was also concerned about the characteristics of the first tide anti-aliasing filter, that it be sufficient for the needs of the GGP. He described several upgrades that should be done, or considered, prior to the GGP. Of these, the upgrading to the HC2 compressor, attention to the hoses and water supply and the need for a consistent APD service regimen to be the easiest and most essential.

It is clearly going to be impossible to upgrade all existing instruments to identical performance before the start of GGP. GWR does not have the resources to do this and each group will have to make separate arrangements for specialised upgrades or modifications as appropriate. GWR would provide a shopping list' to help SG groups decide what was feasible.

[Note added:] Warburton presents a comprehensive review of recent instrumental issues and the preparation of instruments for the GGP in his article for this symposium (Warburton, 1994). We highly recommend that SG Groups follow the recommendations contained in that article for preparing their sites and data acquisition systems. GWR Instruments is presently discussing (with other SG groups) the specifications for an anti-aliasing filter to enable new data acquisition systems to sample at 10s, which is the appropriate rate for GGP Raw Data'.

Data Acquisition - J. Neumeyer Neumeyer described some problems with the GWR data acquisition system in the area of time-outs and spikes. He proposed a new data acquisition system based on the OS/2 operating system using a GPS clock and talked about the need for an agreed upon format for data exchange. It appears that the Potsdam group will continue with the development of their new system.

Data Acquisition - R. Warburton Warburton described the existing situation with respect to data acquisition systems available through GWR. There are four (a) the original GWR system (b) a hybrid system comprising the GWR system as modified by NOAA (c) a purely NOAA system being pursued for the installation at Boulder and (d) the Canadian (CSGI) system which GWR is providing to the new Finnish installation. Of these, only the last is currently fully operational, although it is being upgraded to include a GPS time clock system.

Data Acquisition - Discussion Considerable discussion of data acquisition systems followed. Jentzsch added that their group was considering upgrading the system they designed for the Wuhan installation. Thus in addition to existing installations and the four systems described by Warburton, there are 2-3 others being developed by other groups.

Crossley remarked that while in principle it was easy to conceive a data acquisition system on paper, the development of a robust and error-free system was an expensive and time consuming operation. In addition the Canadian system is the only one to date with a technical reference manual. Richter made the point that a multitasking system had to have precise task switching (time slicing) in order to preserve the timing requirements. Of all current mainstream possibilities (UNIX, OS/2, Windows), only the QNX system (used by Richter's installations and the Canadian system) has the requisite specifications.

[Note Added:] GWR Instruments has modified the Canadian Data Acquisition system (see Aldridge et al., 1991 for a complete description of the system) in a number of significant ways (a) the hardware and software were modified from WWVB to communicate with a GPS clock (Trimble ACUTIME PLUS or modified SVeeSix GPS receiver) receiver through a COM port using TSIP protocol (b) the software was modified to allow mass storage to be written in DOS format instead of QNX format and the mass storage was upgraded from a 20MB to 90MB Bernoulli (c) users can download files by modem in DOS, as well a QNX. This avoids the necessity for users to learn the QNX system locally.

Hinderer and others questioned the virtues of one high-accuracy channel versus the split signal/separate filter approach (MODE, TIDE) used by the current GWR system. Ducarme noted that long decimation filters spread data problems unnecessarily; for this reason Mansinha recommended all data repair be done at the original high-rate sampling.

It was clear that various philosophies existed with respect to the data acquisition systems and data processing methods and it is unlikely that a consensus can be established for the GGP to adopt any one data acquisition system or processing method.

[Note Added]: Options (a) and (b) above are now discontinued and NOAA is now committed to Option (c). There is some concern the NOAA system (as well as other systems) will not meet the timing requirement as it is not an interrupt driven system. QNX is a software interrupt driven system, a real time' operating system is by comparison hardware interrupt driven. Multitasking systems such as Windows/Unix/OS2 are not interrupt driven at all, though tasks are prioritised. Peters (NOAA) responds to the timing issue saying that practicalities preclude timing to better than 50ms or 0.1s, especially given the periods of the signals and the 1 minute samples required. While these are valid points, if the ultimate aim is to see 1 ngal signals (for example, seismic normal modes have amplitudes of several ngals at periods of several 100s), a timing jitter of 0.1s in recording the tide would yield a noise of 1ngal; admittedly atmospheric pressure probably contributes at least this amount and one only hopes these noise sources are uncorrelated.

##### Calibration

Italian Instrument (Bologna) - G. Casula Casula introduced the Italian instrument sited near Bologna. To date their main concern had been calibration. Experiments with a vertically movable ring surrounding the dewar yielded calibration constants that appear accurate to 0.2%. Though they had not yet made any long term gravimeter observations, there was the possibility that this instrument would remain in Bologna for some time and therefore may participate in GGP.

Vertical SG acceleration - B. Richter Direct vertical acceleration of the SG has been shown to be feasible by Richter for several years. Special instrument supports are added which can be computer coordinated to provide a precise vertical acceleration to the instrument. At periods of 300 and 2400s, the 5mm amplitude of motion yields accelerations of about 220 and 3 mgal respectively. It is anticipated that the accuracy of calibration can be increased to 0.01% by improvements to the system after June 1994.

Calibration - Discussion Ducarme reported that good results had been obtained by Van Ruymbek for the vertical acceleration of smaller LCR (D,G) meters. Warburton remarked that a new (small) GWR instrument would make a suitable test calibration instrument for the GGP.

In order to provide useful data for the determination of gravimetric tidal factors, GGP instruments should be calibrated to an accuracy of 10-3 or better. We anticipate the use of an oscillating platform (Richter et al., 1991; etc.).

The general subject of system calibration was also discussed. In particular, the SGs should be calibrated by a step response method (Richter & Wenzel, 1993) and the whole data acquisition system calibrated for frequency response and phase delays. Van Ruymbeck has pointed out the need for ground loops to be identified in the data acquisition electronics as a potential source of unwanted sugnals. Warburton (1994) also deals with SG calibration issues.

A GGP Data Centre - Discussion To date, there has been no establishment of a GGP Data Centre. In some respects a Data Centre is unnecessary if all sites are connected through INTERNET. In other respects a Data Centre brings a sense of cohesion to the project and provides facilities for the uniform processing of data and scientific visits between GGP participants.

It has generally been assumed that GGP Data would be made available to ICET for decimation to 1 hour for the global gravity data bank. It is left unresolved whether ICET should receive the GGP data (a) as a GGP Participant (b) after a suitable delay or (c) when it is made available to the general scientific community (should there be distinctions between the above times).

The Potsdam Group has offered to set up a GGP Data Centre and will present such a proposal at the High Precision Tidal Data processing meeting in Bonn (1994). A Data Centre was discussed in detail at the GGP meeting in Whistler (Aug 1994; attached).

The GGP secretariat (Crossley, Hinderer) could set up a GGP Information Centre at an INTERNET node that could act as the clearing house for GGP communications.

Funding The establishment of a Data Centre would require support. In other respects, funding would be useful to promote cooperative visits between GGP participants, to aid in establishing INTERNET facilities where none exist or to engage help in processing data. It is left unresolved whether funding should be pursued and, if so, through what channels.

##### Other Issues

Observation Period Based on the projections of the various groups (see Table 2, attached), the Observation Period should commence July 1995. This date is subject to ongoing agreement by the GGP participants.

New SG Installations Mansinha suggested that other countries should be invited to participate in GGP, such as India. This would be beneficial in view of fiducial geodetic stations already operating in Tibet and China. Australia has also been considered as a potential location for a SG. Clearly the very sparse GGP network should be enhanced in geographical locations far from existing installations. The possibility of siting an SG in Russia was also discussed but at the present time it was felt this might not be cost effective from the supplier's point of view.

GGP Organisation The meeting approved the nominations of D. Crossley (Canada) as Chair and J. Hinderer (France) as Secretary of GGP. Their terms should run four years from approval of these officers at the IUGG in Boulder Colorado (1995). These two officers should establish the membership of the GGP Steering Committee, which generally should include at least one representative from each SG Group. The GGP Steering Committee is the main body for establishing the terms of reference of the GGP. The GGP should be made known directly to the IASPEI and IAG Organisations of IUGG.

Other Presentations Mansinha discussed his modification of the wavelet transform, which he terms the S transform. The benefits of the S transform is that, unlike the wavelet transform, it is bi-directional which preserves phase information and allows transient signals to be filtered from a data set. He showed several synthetic examples.

GGP was discussed at two lunch time meetings during the International SEDI symposium at Whistler Mountain, Canada, 7-12 August 1994.

Those in attendance were :

K. Aldridge (York University, Toronto), D. Crossley (McGill U., Montr\'eal), J. Hinderer (IPG, Strasbourg), Y. Imanishi (Ocean Research Institute, University of Tokyo), A. Mukai (Kyoto University) J. Merriam (University of Saskatchewan, Saskatoon) B. Richter (IFAG, Frankfurt), M. Rieutord (Toulouse, France) T. Sato and Y. Tamura (National Astronomical Observatory, Mizusawa)

Data Recording Parameters The question of 10s data was discussed and it was emphasised that 10s data was to be provided on request following large seismic events. This data is in addition to the main GGP (1min) data. The use of the PRETERNA format (Wenzel, 1994a) was approved by the Japanese groups. Data format for 10s data needs to be discussed, as does the anti-aliasing filter.

Data Exchange Each 1 year of 1 minute data takes approximately 8 MB in CSGI (Canadian, unpacked, format). This would be reduced somewhat in ETERNA format if a packing algorithm is used.

Data Availability Sato reviewed the Japanese proposal for treatment and exchange of data. In particular he stated that Syowa station (Antarctica) data was in a special category as far as the Japanese were concerned because of the extreme effort to collect this data and the difficulty in transmitting the data back to Japan. After some discussion it was clear that the Syowa data would be made available to GGP 1 year after it is made available to the Japanese SG groups and this means approximately one year later than other GGP data.

Considerable discussion centred on the issue of data availability. All members were happy with the general statement that GGP Data should be available to other GGP participants within 1 year of its collection (original wording, see main text). Since the files are organised on a month by month basis, this means an additional 1 month delay is initially in effect. This was interpreted to mean that, for example, data collected during July 1995 would be available between 1 August 1995 (earliest) and 1 August 1996 (latest) to GGP.

Participants agreed that a further 1 year delay should be in effect before releasing GGP data to the general scientific community (open file). Thus, for example, all data collected through GGP for July 1995 would be available to the general community by 1 August 1997. Since ICET data falls into the completely open data category, ICET would receive the 1 min GGP Data no later than 2 years after its collection.

Data Centre Several participants endorsed the idea of a Data Centre. At a minimum, this could be an INTERNET node where GGP Information (site details, GGP News, publications etc) was stored, as well as file pointers to data files kept on individual sites and accessed through INTERNET (Type A Site or Network Centre). It was agreed that only one person per GGP group would be authorised to receive GGP data in the privileged (1 year after collection) time period. Thus the GGP could operate in a similar fashion to the current CSGI Gopher Site at McGill University.

A more elaborate Data Centre would have facilities for storing all the GGP Data Centre at a single site with manual release of the data to authorised persons associated with GGP (Type B Site). Hopefully a Type B Site might have the manpower to be able to process GGP data if required.

Aldridge proposed a Type A Site to be set up at York University. A proposal for a GGP Data Centre is scheduled to be presented at the Bonn meeting. It is difficult to ensure that a Type A Site could give adequate data protection as all INTERNET access is in principle subject to outside interference.

Funding The benefits of seeking funding were discussed. In particular it was felt that funds for scientific exchanges between individuals were worth pursuing as these may not be available elsewhere. Also the funding of GGP Workshops would be highly beneficial to the project. Members were asked to consider how possible funding might work within their own countries.

#### Bonn, Germany, August/September 1994

GGP has a natural connection with the activities of the Working Group High Precision Tidal Processing' which meets in Bonn every two years. This Working Group is established by the Earth Tides Commission of the IAG. At the 1994 meeting, there was no formal discussion of GGP proposals, but several presentations dealt with GGP-related matters. The central issue discussed was that of a GGP Data/Network Centre.

##### GGP Data Centre Proposals

Ritschel, Neumeyer and W\"achter (GFZ Postdam) presented a proposal for a GGP Data Centre to be set up in Potsdam. Ritschel outlined the elements of a modern computer-network based data centre, including the storage of data, the gathering of meta-data and messaging facilities. Possible access was through INTERNET/Mosaic or a RDBMS running at GFZ. The proposed GGP Data Centre would not be attached to the Potsdam SG Group, but would be at an independent geophysical data centre devoted to data of many different types.

The Japanese SG Group, led by Sato and Takemoto, proposed to set up a GGP Sub Centre in Japan to handle the GGP Data from all the Japanese instruments (their proposal is attached as Appendix B). In this proposal they suggested that a real (Type B, with all data physically at one location) Data Centre be established in a country and location where at least one active GGP member would be located.

After much discussion it became apparent that SG members wanted more than a Newtork Centre (Aldridge's proposal) from the point of view of data access and the possibility of data processing by experienced SG data handlers in future. Also a real Data Centre might be able to attract funding.

However, it was also clear that members were not convinced that it is appropriate to store the relatively small amount of GGP data (a projected 1-2 GB for the whole Observing Period) at a much large data centre (the GFZ proposal). Also the GFZ data centre does not, in itself, have the experience to process the data according to GGP needs. The idea of a separate European GGP Subcentre (say at Strasbourg or Potsdam) was considered unnecessary at the present time, though this should be reviewed in the future.

Thus it was finally suggested that the GGP Secretariat (Crossley, Hinderer) should set up a pilot GGP Data Centre in Canada, maybe at McGill University, based on INTERNET/Mosaic. Initially, the goal would be to store existing data in GGP format and experiment with the provision of GGP meta information and messages using the facilities (password protection, forms etc) of Mosaic. The option is open in future for moving this Data Centre to another site should the manpower and facilities required exceed those currently available.

#### Walferdange, Luxembourg, 6-8 September 1994

The second IAG Workshop on the Intercomparison of Superconducting and Absolute Gravimeters (which allows the presentation of this paper) dealt with many of the issues that will contribute to the GGP. We were scheduled to have a detailed discussion of GGP issues during this workshop, but unfortunately a lack of time prevented more than a cursory review of GGP issue such as the Data Centre, Recording Parameters etc. These issues were raised, and sometimes left unresolved, in previous meetings. Nonetheless as a result of the discussions that did take place, resolutions regarding the GGP was passed on to the International Gravity Commission Meeting in Graz, Austria. It is hoped that the IAG will respond to the GGP directly in the near future.

Finnish SG Update - H. Virtannen The Finnish SG Group has now received and installed GWR T020 and reported successful operation of the gravimeter and recording systems. There are still a few small bugs in the new GGP clock module that was an upgrade to the original Canadian design, but otherwise we are pleased to see this station now fully functional.

[Note Added:] KääriänenK\"a\"ari\"ainen notes two details concerning the station height reported in Table 1, first that the height reported is that of the sphere and second that due to the Fennoscandian postglacial uplift, and second that station ME has experienced a 0.14 m rise in elevation since 1960. Luckily even this rapid change has a minimal effect on theoretical tides!

GGP Dictionary and Information Sheets To avoid unnecessary confusion in discussion, the GGP community should adopt a common terminology with respect to such terms as drift, accuracy, nutation etc. In addition we will provide information sheets on a number of widely used topics (units, spectral analysis terms ...).

#### Tsukuba, Japan, 19 October 1994

Those Present: M. Ooe, T. Sato, Y. Tamura (representing NAOM), J. Segawa, Y. Imanishi (ORI, Kakioka), T. Higashi, A. Mukai, S. Takemoto, (representing Kyoto University) and K. Shibuya and others (representing Syowa, NIPR)

T. Sato reported the discussion on GGP in the meeting in Walferdange on September 7. S. Takemoto reported on the resolution of the IGC-ICG Joint Meeting in Graz, Austria, September 1994, in which a recommendation to support the activity of GGP was incorporated. All of the Japanese SG Groups agreed to promote the construction of the Japanese (Asian) Subcentre for GGP Data at ORI. Y. Imanishi will be the manager of this Centre. The Japanese GGP Data will be made available to the International Data Centre and other GGP Data Centres.


text:
# The Klein bottle and the Heawood Conjecture

Let $\Sigma_g$ be a surface of genus $g$. The Heawood Conjecture gives a closed formula in one variable, $\chi$ (the Euler characterstic of $\Sigma_g$), for the minimal number of of colors needed to color any graph drawn on $\Sigma_g$.

Ringel-Youngs, 1968: The Heawood Conjecture is true, except for $g=0$ (where we don't know the answer - this would be the Four-Color Conjecture) and the Klein bottle (where a graph was shown to be colorable with fewer than predicted by the formula).

Of course, the Four-Color Conjecture was eventually proven (not by Ringel or Youngs, although I think they were both involved to some extent), leaving the Klein bottle as the odd-man out.

(a) What is the "reason" for the Klein bottle's exceptionality here? (b) Does the answer to part (a) manifest in any other way - meaning, are there other theorems, patterns, etc to which the Klein bottle is an exception that can be traced back to (a)?

-
In my mind I classify this as another "low-dimensional obstruction"; in most theorems of this kind (e.g. A_n is simple, the classification of finite simple groups) one should expect finitely many counterexamples because there may not be enough room for generic behavior to happen initially. – Qiaochu Yuan Jan 12 '11 at 14:41

Denote by $N_c$ the non-orientable surface with $c$ pairwise disjoint crosscaps. So $N_1$ is the projective plane, $N_2$ is the Klein bottle, etc.

Then the "reason" of Klein bottle's exceptionality in Heawood conjecture is the following: for all $c \geq 3$, the complete graph $K_p$ embeds in $N_c$, where $c$ is the smallest integer which is at least $(p-3)(p-4)/6$. The case $c=2$ is exceptional, in fact $K_7$ does not embed in $N_2$. More details are here.

This result on Klein bottle was proven by Franklin, who also showed that there actually exists in $N_2$ a map with chromatic number $6$, obtained by embedding in the surface the so-called Franklin graph

-
The complete graph $K_6$ easily embeds in the real projective plane $N_1$ (as antipodal quotient of an icosahedron), hence in $N_2$ (and a Möbius band, by the way). – BS. Jan 12 '11 at 11:36
Ah, that's great, thanks! – Dr Shello Jan 13 '11 at 5:11

Even though there is always a preference towards uniformity in understanding of general results such as Heawood's theorem, this somehow feels like the wrong question to ask. While I am not familiar all the way with all the proofs involved (the full proof is massive, and involves the four color theorem as well) it is obvious that there are many special cases treated separately, with the most difficult cases being the planar case and $g\in \{59, 83,158,257\}$. So even if the resulting formula is uniform, the behavior of each case is not obviously so.

In the end, the answer to your question boils down to the reason that Francesco Polizzi mentions in his answer, and can perhaps be elaborated a bit more by mentioning the improvement of the Heawood result by Dirac, Albertson and Hutchinson that every graph in $\Sigma$ is actually $H(\Sigma)-1$ colorable unless it has a subgraph isomorphic to a complete graph on $H(\Sigma)$ vertices. You can see here that the Klein bottle is expected to behave differently as $K_7$ doesn't embed in it, but suddenly we get the planar case as another exception and the projective plane was dealt separately too.

-
Your second paragraph is very interesting - thanks for the info ;) – Dr Shello Jan 13 '11 at 5:12


text:
GENERALIZED STABILITY OF ISOMETRIES ON REAL BANACH SPACES

Title & Authors
GENERALIZED STABILITY OF ISOMETRIES ON REAL BANACH SPACES
Lee, Eun-Hwi; Park, Dal-Won;

Abstract
Let X and Y be real Banach spaces and ${\varepsilon}\;>\;0$, p > 1. Let f : $\small{X\;{\to}\;Y}$ be a bijective mapping with f(0) = 0 satisfying $\small{|\;{\parallel}f(x)-f(y){\parallel}-{\parallel}{x}-y{\parallel}\;|\;{\leq}{\varepsilon}{\parallel}{x}-y{\parallel}^p}$ for all $\small{x\;{\in}\;X}$ and, let $\small{f^{-1}\;:\;Y\;{\to}\;X}$ be uniformly continuous. Then there exist a constant ${\delta}\;>\;0$ and N($\small{{\varepsilon},p}$) such that lim N($\small{{\varepsilon},p}$)=0 and a unique surjective isometry I : X $\small{{\to}}$ Y satisfying $\small{{\parallel}f(x)-I(x){\parallel}{\leq}N({\varepsilon,p}){\parallel}x{\parallel}^p}$ for all $\small{x\;{\in}\;X\;with\;{\parallel}x{\parallel}{\leq}{\delta}}$.
Keywords
(${\varepsilon}$,p)-isometry;isometry;real Banach spaces;
Language
English
Cited by
References
1.
G. Dolinar, Generalized stability of isometries, J. Math. Anal. Appl. 242 (2000), no. 1, 39-56

2.
J. Gevirtz, Stability of isometries on Banach spaces, Proc. Amer. Math. Soc. 89 (1983), no. 4, 633-636

3.
P. M. Gruber, Stability of isometries, Trans. Amer. Math. Soc. 245 (1978), 263- 277

4.
J. Lindenstrauss and A. Szankowsik, Nonlinear perturbation of isometries, Asterisque 131 (1985), 357-371

5.
S. Mazur and S. Ulam, Sur les transformations isometriques despaces vectoriels normes, C. R. Acad. Sci. Paris Ser 194 (1932), 946-948

6.
M. Omladic and P. Semrl, On nonlinear pertubations of isometries, Math. Ann. 303 (1995), no. 4, 617-628


text:
# Recent questions and answers in Physics

How abounding years accept you acclimated to acquirement a Raymond Milling?

Raymond Mill is a frequently acclimated mineral milling equipment, in abounding areas of assembly will be seen, in fact, Raymond Mill is a abiding operation, the account activity is actual continued equipment, the apriorism is to buy the equipment, Because the bazaar does accept some bad manufacturers, the superior of accessories produced by austere unqualified, the user in the acquirement of equipment, to bouncer adjoin such manufacturers. The activity of Raymond Mill is afflicted by abounding factors.
natural factors
In fact, accustomed factors refers to the superior of the accessories itself, with the user's operation is not anon related. Raymond Mill on the bazaar cast is actual much, no one can agreement that anniversary architect will aftermath able products, Raymond Mill is a actual ample accessories in the assembly process, if the superior is not qualified, not alone not Will accompany the user a bigger assembly income, but aswell access the abrasion and breach of the accessories itself, abbreviate the account activity of equipment, amateur Raymond Mill achievement in the low superior of the plate, locations dislocation, accouterments on the bound And so on, the appropriate botheration is not austere will advance to added abrasion and breach in the assembly process.
Human Factors
Method of operation
Raymond Mill in the use of the process, the user to yield the actual adjustment of operation, if generally in accordance with the amiss way to operate, even if the accessories can run normally, will abate the accessories life.
Conservation methods
If the user can consistently according to the actual attention adjustment of Raymond Mill maintenance, can abundantly access the account activity of the equipment, but aswell consistently analysis the use of cutting parts, analysis whether the charge to alter the parts.
Maintenance method
Equipment in the accident of failure, it is all-important to yield the actual way to repair, do accepted aliment records, the charge for backup locations for appropriate replacement, so as to finer extend the account activity of the equipment.
Raymond Mill on the actual operation, maintenance, aliment methods, not one by one here, the user if you buy accessories on the actual operation and maintenance, Ultrafine Mill consumables use a three or 5 years is no botheration , If the antecedent acquirement of superior on the unqualified, able to use three or 5 months on the good, if you buy the accessories and the actual operation and maintenance, Raymond Mill apparatus use thirty or fifty years is no problem!

Application of Raymond mill in Atramentous Industrying

The province's atramentous industry durably authorize a people-oriented, safe development of the concept, to action the alteration of atramentous mines and gas administration of the two acute action to ensure that the atramentous industry, the Ministry of Finance, the Ministry of Science and Technology of the People's Republic of China, , To backpack out the development of "safe production," efforts to barrier atramentous accidents, abnormally the blow of above accidents, the advance of the industry's bread-and-butter operation superior and level, finer advance the province's atramentous industry safe and abiding development. The efforts of the government and amusing workers are the aftereffect of the achievements, the accurateness of the action and measures are adapted for the accelerated development of Henan Province.
In the 5 years, Henan Province, the best development of the atramentous industry, the fastest 5 years, the Raymond Mill to become safe assembly in Henan atramentous abundance the best situation, the a lot of abiding 5 years. During the period, the province's atramentous assembly with a almanac high, the accumulative assembly of about 140 actor bags of raw coal, than the "fifteen" aeon of about 50% increase; in 2010, the province's atramentous assembly accomplished 28 actor bags Around, not anarchy in the top level. During this aeon of time, safe assembly in Henan Arena with a almanac success in the aboriginal four years to accomplish cogent after-effects on the abject of the arena in 2010 the assurance of atramentous assembly almanac the best level. During the aeon of the Eleventh Five-Year Plan period, the amount of atramentous abundance blow deaths in the arena decreased by 50.8% compared with the aeon of 15%, of which the assurance of assembly is almost more, the plan burden is almost beyond city-limits and belt atramentous mines fell 44.6% ; The province's atramentous actor bags of bloodshed than the "fifteen" rsquo; aeon fell 67.2%, of which the canton and belt atramentous mines fell 67.9%.
"Eleventh Five-Year" period, the province's structural adjustment, agitated out a array of behavior issued, so that the clip of atramentous bread-and-butter development abundantly accelerated. The arena to adjust the cease and transformation of the capital agreeable of the advance of the atramentous abundance anatomy to decidedly advance the clip of the absolute cease of atramentous abundance 203, annihilate astern assembly accommodation of 3.89 actor tons, advanced of agenda and behind to complete the accompaniment issued 155 atramentous abundance task; From 803 at the end of 2005 to 601 at the end of 2010, atramentous abundance individual able-bodied assembly from the end of 2005 to 25,500 bags to the end of 2010 to 45,600 tons.
SBM Raymond apparatus business administrator concluded: Looking back, the Henan atramentous industry will added consolidate the assurance of assembly base, the atramentous industry in Henan Province, Speed up the clip of structural adjustment, the next footfall to change the way of development, and strive to bigger and faster development for the accurate development of Henan, backpack out bolt up, blooming acceleration to accommodate added reliable activity security.

Muscovite Raymond Mill

Mosson Raymond Comminute is mainly acclimated for the processing of muscovite, for those who wish to advance in the acreage of muscovite milling products, the amount of muscovite Raymond Comminute is the focus of its attention, abounding users will Accept to seek the amount advice on the Internet, which is futile, with the assembly of muscovite Raymond comminute manufacturers increased, the bazaar amount has become added diversified.
For online amount information, users do not accept too much, abounding manufacturers action advice is wrong, just a gimmick, in adjustment to allure the user's click. The assembly of muscovite Raymond Mill apparatus about will not accept to action online, first, because the arrangement loopholes, and second, because anniversary architect of beach authoritative apparatus specifications, altered prices, all models of beach Apparatus prices are apparent on the Internet is unrealistic.
Before the accommodation to acquirement muscovite Raymond Mill, the user should aboriginal accept the actual backdrop of muscovite, such as density, hardness, baptize content, accumulated with muscite cutting characteristics, the absolute assembly altitude and accepted assembly targets Select the adapted blueprint of the muscovite Raymond Mill. To actuate the acquirement of the muscovite Raymond Comminute blueprint afterwards the boutique around, by comparing the prices of assorted manufacturers, aftermarket, manufacturers and supply time and added factors to actuate one or added absorbed in the acquirement of the manufacturers, Factory visit, all-embracing compassionate of the manufacturers, to actuate the adapted acquirement manufacturers.
SBM aggregation specializes in the assembly of muscovite Raymond Mill, the company's assembly of muscovite Raymond Comminute vertical structure, simple structure, simple operation and maintenance, the device's arbor asset arbor optimized to abundantly access the accessories Operation of the adherence of my company's assembly of muscovite Raymond Comminute abortion rate, continued life, energy-saving aftereffect is good.

Sand making machine to become the best accessories for dry crumb

Dry adhering is mainly fabricated of Sand Making Machine, the sand of the bedrock about accept granite, accustomed river pebbles, andesite, rhyolite, diorite, sandstone, limestone, fabricated of sand by altered types of bedrock accept altered backbone The SBM machine is a able architect of abundance equipment, the capital varieties of its sand mining accessories are: appulse crusher, cone crusher, vertical shaft appulse crusher, Sand making machine, these accessories accept altered characteristics and altered uses, abnormally Is a sand authoritative machine in the dry crumb adhering processing has a cogent advantage.
Sand machine is the aboriginal bearing of appulse sand machine, SBM machine to blot the advantages of the additional bearing of VSI5X Sand Making Machine, and the use of a new blazon of hydraulic equipment, sand machine equipment, mainly acclimated in high-performance sand assembly band operations, abysmal atrium blazon Afterwards optimizing the architecture of the rotor, the actual throughput is added by about 30%. Afterwards the bounded abrasion of the bouncer console can be acclimated up and down, the actual appliance amount can be improved, and the account activity can be bigger by added than 48%. The accumulated bang architecture is alone bare to alter the abrasion Bang part, can abate the use of added than 30%. At the aforementioned time the new Sand Making Machine can aswell be in a array of ore, adverse corundum, bottle raw abstracts and added hard, harder abstracts in the acreage of accomplished sand is broadly used.

The abstruse characteristics of Raymond

Raymond cutting roller abstruse characteristics refers to the amount of cutting roller apparent teeth, cutting angle, cutting teeth and a brace of cutting caster cutting teeth and so on.
1) Raymond Mill Amount of Rows The amount of teeth is bidding as the amount of teeth in the circumferential breadth of the apparent of the roller, that is, the amount of teeth per centimeter or per inch.
2) Cutting bend The apparent of the roller is cut by the cartoon machine, and the bend amid the two bevels is alleged the tooth angle.
3) Raymond Cutting Tilt In adjustment to enhance the crushing and stripping ability, abate activity burning and advance the superior of flour, the apparent of the roller accept to accept a dip bend with the butt bus. If there is no abruptness of the cutting teeth, again the actual can not be continuously ground, the bran will be cut into strips, the plan will could cause the roller vibration, consistent in accident to cutting teeth. In general, the affection is bidding by the adjustment of the breadth L of the roller and the adjustment of the cutting distance, and the abruptness is s / L: l: 7 to 1: 10. You can aswell use the affection bend a, usually a = 10. ~ L6.
4) Ultrafine mill grinds The two rollers are abiding at altered speeds, and there are four altered means of adjustment because of the bend of the blaze and the birdbrained angle.

Speed of kangaroo
A kangaroo is capable jumping two a height of 2.62m ditermint the take off speed of the kangaroo
what is the push force F?

A 50 kg crate is being pushed on a horizontal floor at constant velocity. Given that the coefficient of kinetic friction between crate and floor is $\mu$k = 0.1, what is the push force F?

What is the minimum angle theta so that the ladder doesn't slip?

A uniform ladder of length L is leaning against the side of a building, as shown. A person of mass m = 75 kg is standing on it. The mass of the ladder is M = 10 kg. The coefficient of static friction between the ground and ladder is $\mu$s1 = 0.5, and the coefficient of static friction between the wall and ladder is $\mu$s2 = 0.3. What is the minimum angle $\theta$ so that the ladder doesn't slip?

Determine the acceleration of the blocks.

Two blocks of mass m and M are hanging off a single pulley. Determine the acceleration of the blocks. Ignore the mass of the pulley.

The spring has a stiffness k. How much is the spring compressed?

A block of mass M falls onto a spring. Before colliding with the spring the velocity of the block is V. The spring has a stiffness k. How much is the spring compressed?

A block of mass M is traveling on a surface inclined at angle $\theta$. The coefficient of kinetic friction is $\mu$ between block and surface. The velocity of the block is V before colliding with a spring. The spring has a stiffness k. How much is the spring compressed?


text:
# Limit from some russian book: lmt-0 [(1+mx)^n-(1+nx)^m]/x^2

1. Jul 13, 2010

### mateloco

limit from some russian book: lmt--0 [(1+mx)^n-(1+nx)^m]/x^2

Hey guys, i'm newbie in calculus, while doing a quick reading of my problem's book i find this interesting limit:
1. The problem statement, all variables and given/known data
solve the following limit when x tend to zero,
m and n belong to the set of naturals
$$\frac{(1+mx)^n-(1+nx)^m}{ x^2}$$

2. Relevant equations

3. The attempt at a solution
the anwser is (mn)(m-n)
i dont know how to solve the problem...
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jul 13, 2010

### Staff: Mentor

Re: limit from some russian book: lmt--0 [(1+mx)^n-(1+nx)^m]/x^2

The approach I would take would be to expand both binomials using the binomial theorem http://en.wikipedia.org/wiki/Binomial_theorem.

3. Jul 13, 2010

### qbert

Re: limit from some russian book: lmt--0 [(1+mx)^n-(1+nx)^m]/x^2

or use L'Hospital's rule (twice).

4. Jul 13, 2010

### Staff: Mentor

Re: limit from some russian book: lmt--0 [(1+mx)^n-(1+nx)^m]/x^2

5. Jul 13, 2010

### mateloco

Re: limit from some russian book: lmt--0 [(1+mx)^n-(1+nx)^m]/x^2

Exactly!
Before posting, i've tryed L'hopp, and i reached the following answer:
(mn)(m-n)(1/2)
Right?, but looking at the answer's section in the book, i realized that the answer given by the book is:
(mn)(m-n)

Warning! my silly idea:
expanding the binomial, and doing the substitution of all x, almost every term of the function will be zero, it doesn't matter the value of m and n, am i wrong? What do you think about it?
i appreciate all your comments and if you notice any mistakes in my writing (Sintax, grammar, etc.), let me know*(?), English is not my mother tongue, but I want to learn it.

6. Jul 13, 2010

### Tedjn

Re: limit from some russian book: lmt--0 [(1+mx)^n-(1+nx)^m]/x^2

When you expand with the binomial theorem, in the numerator, what is the coefficient of x? What happens when you divide that by x2 as x approaches zero?

EDIT: Also, your idea isn't silly. Most ideas aren't silly but some lead to silly conclusions when you realize that you didn't get anywhere. And some leave you feeling silly since you didn't follow them to completion. But the idea, well, that's just being a creative person.

Last edited: Jul 13, 2010
7. Jul 14, 2010

### mateloco

Re: limit from some russian book: lmt--0 [(1+mx)^n-(1+nx)^m]/x^2

Very nice comment Tedjn!!

8. Jul 15, 2010

### gomunkul51

Re: limit from some russian book: lmt--0 [(1+mx)^n-(1+nx)^m]/x^2

Alex Spivak ? :)