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Also in NHS Grampian Health and Care Framework: Health Services Fit For the Future:
Background to the Health and Care Framework
NHS Grampian aims to ensure that care is provided at the right time, by the right person and in the right place which is high quality, effective, efficient, sustainable and affordable.
- Helping people to maintain good health, and protect against and prevent ill-health.
- Care being available closer to people's homes and away from specialist hospitals where possible.
- Care being delivered by staff with the appropriate level of skills, training and experience across a range of professions.
- Radical changes and difficult choices will need to be made.
In order to ensure our services and staff are all focussed and working to the same objectives, NHS Grampian produces the Grampian Health Plan.
As part of the development of the Grampian Health Plan for 2010-13, staff, the public and partners told us that:
- They supported NHS Grampian's aims, values and priorities.
- They want more detail about what this will mean in practice.
- The way services are currently provided is unlikely to be sustainable in the future because of the many challenges ahead.
NHS Grampian is facing a number of challenges. The main challenges include:
- A growing elderly population who are more likely to have more than one long term condition.
- New technology and treatments which will continue to be developed.
- The global financial situation which will mean there is less money available than in previous years.
- High expectations about what should be delivered.
- The requirement to deliver national policy and targets.
Why do we need a Health and Care Framework?
The purpose of the Health and Care Framework is to provide greater detail and direction, based on the strategy set out in the Grampian Health Plan 2010-13.
What will the Health and Care Framework look like?
The Health and Care Framework will have three main parts which will:
- Describe how health and care in the future will be safe, effective and sustainable, which meets future population needs and is delivered within available resources - the Health and Care Model.
- Set out detailed plans to deliver the agreed model.
- Guide services and the organisation in allocating resources and making decisions based on the model of care.
What difference will it make?
- Services will collectively focus on the key priorities for health and care.
- We will be clearer about where we need to focus our resources such as staff, property and equipment.
- We will set out clear improvement areas (outcomes) which will focus on health and healthcare needs, evidence of effectiveness and available resources.
- We will better manage current and future challenges.
- Care will be delivered at the right time, by the right person and in the right place.
When will the Health and Care Framework be developed?
The proposed Health and Care Model and an update on the resource allocation and decision making framework will be submitted to the Grampian NHS Board in April 2011.
Following approval by the Board, we will develop and consult on an action plan to support the implementation of the agreed model. The final action plan will then be submitted to the Board for approval in October 2011.
How will it be developed?
We understand that it is important to develop the Health and Care Framework jointly with staff, members of the public and our partners.
A number of existing groups have been identified to inform the development of the different parts of the Health and Care Framework. These groups involve staff, the public and partner organisations.
In addition, a number of pathfinder projects have been identified. The pathfinders
have been chosen as part of the Health and Care Framework to look at different services and communities in detail, and to identify the health and care needs for these specific groups, for example, older people, children and young families, and local communities.
All of the information from the pathfinders will be collated with all of the other feedback in order to build the Health and Care Framework.
How can I get involved?
There are a number of ways in which you can get involved:
- Become involved in one of the pathfinder projects
- Join the virtual forum, where we will e-mail questions and documents to get your views.
- Give your views via the formal consultation process.
We will also keep staff, members of the public and partners informed and up-to-date on developments by:
- Distributing newsletters at key stages.
- Placing regular updates on this website.
If you would like to be involved, please e-mail _EMAIL_ or call 01224 558584.


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Mango Kush Full Spectrum Terpene Blend
- Reg. Certificates-FMK
A top seller in the category, this light yet fruity blend is an incredible representation of the strain.
How to Use Terpenes
Use 1%-5% by weight
Warm up your concentrate as you normally would. Calculate how much terpenes you will need to use, by weight. Always measure by weight, and never by the drop. We will use an example of a 100g batch, and our average add back rate of 4%. Using a calculator, input 100 x 0.04 (4%). You should get an answer of 4 grams.
1. WARM MATERIAL AND CALCULATE ADD BACK RATE
Remove your concentrate from heat and allow it to cool down enough so that you can still manipulate it. Concentrates should at the very most be 100°F at the time of adding terpenes.
2. ALLOW MATERIAL TO COOL
Take a pyrex beaker and a scale that is accurate to at least 0.00 decimal places. Place the beaker on the scale, and using the same example of 4% usage in a 100g batch, carefully weigh out 4g of terpenes, using a disposable plastic pipette. Set aside in preparation for mixing.
3. WEIGH TERPENES
Once your material is at 100°F or less, it is time to add back the 4g of terpenes. Pour them into the concentrate, and using either a magnetic stir bar or a hand tool, mix the solution for up to 20 minutes.
4. ADD AND MIX
Many people make the mistake of tasting the product right after mixing. It is always better to let the solution sit for 24 hours. Once it has steeped for a day, and it has passed your visual test, you may apply it to whichever end product the material is going into and give it an official taste test.
5. WAIT AND TASTE
Once you are satisfied with your end product, it is ready to go into production. If this is the first time you have developed this product, be sure to document your add back rate for each terpene product for future use. Always store your finished product between 60° and 75°F. This will help prevent separation, crystallization, and oxidation.
Does NOT contain THC or CBD as all of our products are botanically derived from plants and NOT cannabis or hemp.

source: openbmb/Ultra-FineWeb


"This yeast is tightly associated with the flowers of the sweet potatoes. This probably has some relation with human migration, because we know that sweet potatoes come from the Andes. We are using yeasts to track human migration and contacts. That is part of what we call'microbiological archaeology.'"
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Drawing flowers is fun but making them truly unique with your own fingerprints is even better! The middle of each of these flowers is one of a kind!
1. Gather your materials. You will need a stamp pad, some markers and a piece of paper.
2. Press your finger tip into the stamp pad.
3. Print onto the paper.
4. Add petals.
5. Add stems and leaves!


Why assertiveness matters in business
In tough times, organisations look for well-balanced assertive leaders. How do you measure up?
Assertiveness by itself will not get you a promotion, nor the respect of your peers and direct reports. But failing to stand your ground effectively, and - conversely – being too controlling and aggressive, can halt your advance up the corporate ladder. So why are organisations seeking out assertive leaders and where do you fit on the assertiveness barometer?
Assertiveness is a way of describing how people defend their interests. Assertive leaders can create a compelling vision, communicate strategy and clearly define objectives and service quality standards. They inspire their people, gather support, and create alignment within a team so that everyone moves in the same direction. They avoid the confusion and disorientation caused when team members are trying to second-guess what they're supposed to be doing and why.
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Let $G$ be an algebraic variety over an algebraically closed field $k$ (any characteristic). Suppose that: (1) the set of $k$-points has the structure of a group. (2) for any $g\in G$ the right-multiplication by $g$ is a morphism of algebraic varieties $G\to G$. (3) the inverse map is a morphism $G\to G$.
Does it imply that $G$ is an algebraic group? (i.e. is the multiplication $G\times G\to G$ a morphism?)
share|improve this question
you might specify if you have in mind characteristic zero or arbitrary characteristic. – YCor Nov 17 '12 at 18:01
What set are you using to be the elements of the group? – Will Sawin Nov 17 '12 at 18:03
The multplicative group of quaternions (presented as $\mathbb{C}\times \mathbb{C} \setminus \{0\}$ yields a non-algebraic example of your (1)+(2); yet it fails (3). – Mikhail Bondarko Nov 17 '12 at 19:02
_USER_: In char. $p$ such near-misses (losing (3)) abound. Let $H$ be a linear algebraic group and $H'$ an infinitesimal non-normal closed subgroup scheme of $H$. The quotient scheme $G = H'\backslash H$ is a smooth affine variety and the natural map $H(k) \rightarrow G(k)$ is bijective. If $G(k)$ is thereby equipped with the group structure of $H(k)$ then this is non-algebraic (as otherwise the quotient map $H \rightarrow G$ would be a homomorphism of algebraic groups and so its schematic kernel $H'$ would be normal), and (1) and (2) hold but (3) fails. – user28172 Nov 17 '12 at 19:25
Since there was some discussion about the extent to which the group structure is needed, the following might be relevant. If $k$ is a countably infinite field, then there is a function $f:k\times k\to k$ that is not a polynomial even though, for each fixed value of $x$ in $k$, $f(x,y)$ is a polynomial function of $y$ and, for each fixed $y\in k$, $f(x,y)$ is a polynomial function of $x$. (There is no such $f$ when $k$ is uncountable.) – Andreas Blass Nov 18 '12 at 2:21
1 Answer 1
up vote 12 down vote accepted
I predict that in whatever is the situation of motivating interest, you know more: for any algebraically closed field $K/k$ you likewise have a group structure on $G(K)$ functorially in $K/k$ making the translations and inversions by $G(K)$ also be morphisms on $G_K$. Under this additional condition the answer is always affirmative, by using a trick that I believe goes back to Hasse, namely base change to the function field of $G$ (or rather, in our situation, an algebraic closure thereof). This is a Weil-style way of getting at Yoneda's Lemma.
The inversion hypothesis implies that left-translations are also morphisms, and we will use this condition instead of the inversion hypothesis (and deduce at the end that inversion is a morphism too). In fact, in char. 0 we won't even need this condition with the left-translations (or inversion) at all, but we are using a stronger assumption on the existence of group laws on $K$-points for lots of $K/k$. This stronger initial assumption across many $K/k$ bypasses Bondarko's examples in char. 0. The near-miss examples avoiding the inversion condition in char. $p$ in the comments are not ruled out by our stronger hypothesis across all $K/k$, but they are ruled out by the left-translations being morphisms (!), so matters will be more delicate in positive characteristic. We need the left-translation condition in positive characteristic to circumvent some inseparability issues with function fields of $k$-varieties.
Before we begin the actual arguments, note that since $k$ is algebraically closed, so $G(k)$ is Zariski-dense in $G$, for any field extension $F/k$ the subset $G(k)$ inside $G(F) = G_F(F)$ is Zariski-dense $G_F$. Hence, $F$-morphisms among $G_F$, $G_F \times G_F$, etc. are determined by their effect on $k$-points promoted to $F$-points. I mention this at the outset for peace of mind in some later discussions.
The "translation is a morphism" hypothesis ensures that $G$ is smooth, so its connected components are irreducible. I will assume that the variety $G$ is connected, a harmless assumption since (i) for $g \in G^0(k)$ the right-translation morphism by $g$ preserves $G^0$ because it carries $e$ to $g$, (ii) inversion preserves $G^0$ since it fixes $e$, (iii) the morphism property on $G^0$ clearly implies the general case by using the "morphism" property for a few translations by points in the various connected components of $G$. Thus, now $G$ is irreducible and so has a "function field" $k(G)$.
By applying the functoriality in $K/k$ to $k$-automorphisms of $K$, we see via Galois descent that the initial hypothesis for algebraically closed extensions is actually valid for all perfect extensions of $k$ as well. Let $K$ be the perfect closure of $k(G)$, and let $\eta \in G(k(G)) \subset G(K)$ correspond to the generic point. This induces a right-translation morphism $\rho_K:G_K \simeq G_K$ (computing right-translation by $\eta_K$). Since $G$ is finite type over $k$, by expressing $K$ as a directed union of finite purely inseparable extensions of $k(G)$ we obtain a finite purely inseparable extension $F$ of $k(G)$ and an $F$-morphism $\rho_F:G_F \simeq G_F$ that descends $\rho_K$ and clearly computes right-translation by $\eta_F$. The normalization $X$ of $G$ in the finite extension $F/k(G)$ is a $k$-variety finite radiciel over $G$, and $X_{\eta} = {\rm{Spec}}(F)$ over $k(G)$. Thus, by expressing $k(G)$ as the directed union of the coordinate rings of affine opens around $\eta$ in $G$, we find such an open $U$ so that over its preimage $U'$ in $X$ (which is finite radiciel over $U$) there is a $U'$-morphism $\rho_{U'}:G_{U'} \simeq G_{U'}$ extending $\rho_F$.
The map $U'(k) \rightarrow U(k)$ is bijective, so for each $g \in U(k)$ let $g' \in U'(k)$ be the unique point over $g$. Consider the specialization $\rho_{g'}:G \simeq G$ of $\rho_{U'}$ over $g'$. I claim that this is the right-translation by $g$. To see this, pick $h \in G(k)$ and consider the morphism of left-translation $\ell_h:G \simeq G$ by $h$. This carries $\eta$ to another $k(G)$-point of $G$ (over $k$) that "spreads out" to a $U$-point of $G$ whose specialization at $g \in U(k)$ is $\ell_h(g) = hg$ (so if we work instead with the $U'$-point over this via the canonical $U' \rightarrow U$ then its specialization at $g' \in U'(k)$ is also $\ell_h(g) = hg$). Now inside the group $G(K)$ we have $\ell_h(\eta_K) = \rho_K(h_K)$, so $\rho_{U'}(h_{U'})$ is the $U'$-point obtained from applying $\ell_h$ to the canonical $U'$-point of $G$ (spreading out $\eta_F$) because this comparison of $U'$-points may be checked by working at the generic point of $U'$ (and this generic point is dominated by the canonical $K$-point, over which we have the equality $\ell_h(\eta_K) = \rho_K(h_K)$). Specializing this equality of $U'$-points at the point $g' \in U'(k)$ over $g \in U(k) \subset G(k)$ gives the equality of $k$-points $\rho_{g'}(h) = \ell_h(g) = hg$, as desired.
To summarize, we have constructed a $U'$-morphism $\rho:G_{


Lightstep accepts Prometheus Counters, Gauges, and Histograms through the OpenTelemetry Collector or the Prometheus Sidecar. However, there may be differences in how these metric instruments and measurements are displayed in Lightstep as opposed to Prometheus. This document collects these differences for reference.
Prometheus Cumulative Counters
Lightstep accepts a Counter, but for technical and philosophical reasons, we do not provide the underlying cumulative count to our graphs and instead plot them as deltas. The resolution of these deltas varies based on the age of the data - for example, 45 second buckets for data recorded in the past hour, 2 minute buckets for data recorded between one and four hours ago, and so forth.
For an example of how this might manifest, consider the promhttp_metric_handler_requests_total counter that Prometheus exposes for itself (at the /metrics endpoint). Assuming a regular scape cadence of 1 scrape per 10 seconds, graphing this metric in Prometheus would result in a plot of values increasing steadily, up and to the right. In Lightstep, the same metric represented as a count would result in a straight line over time assuming a consistent rate of scrape requests.
Default Grouping for Rate Calculations
When taking the rate of a counter in Prometheus, the default plot will implicitly group the plot by labels. In Lightstep, you must explicitly specify the label to group by.


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As we have an increasing number of roofing clients, C&C Consulting has pledged its support behind the National Federation of Roofing Contractors (NFRC) Safe2Torch campaign.
This simple but effective campaign seeks to significantly reduce the risk of roof fires when using gas torches, either to dry out roofs or when used to install torch-on membranes. Put simply, Safe2Torch is a positive message which puts safety first at the design stage.
Roof fires caused by gas torches, no matter how minor, pose a serious threat to life, property, the image of the industry and possibly even the long-term future of torch-on as an accepted method of covering a roof.
By completing a safe to torch survey during the planning stage of this type of roof work, surveyors and contractors can identify high risk areas in the building process where naked flames should be avoided. Listed below are some areas that should be considered high risk and where naked flames should be avoided:
Expansion joints: Expansion joints which are filled with combustible fillers, such as foam or fibreboard, should not come into contact with high heat or naked flame
Adjoining a pitched roof: If a membrane roof is adjoining a pitched roof with details to be completed under abutments to roof tiles and slates, the slates or tiles should be removed and a compatible torch-free product should be selected.
Abutments with open cavities or open perpends: A compatible torch-free product should be selected for any abutments with open cavities or open perpends.
Fixed timber, plastic fascia or soffits: Naked flames should be avoided where there is fixed timber, plastic fascia or soffits
Here are some key points to reduce the risk of starting a fire in roofing work:
- Survey the roof area, highlight on working drawings high risk areas and mark them clearly as No Flame Zones
- Ensure a hot works permit is prepared for the works and controls are effective in managing the risks
- Provide flameless alternative tools for use in the areas designated as No Flame Zones
- Ensure fire extinguishers are provided and deployed at the point of work. Ensure this equipment is fit for purpose, inspected and maintained
- Employ competent supervision on site, that is following the Safe Systems of Work and Safe2Torch guidance at all times
- Have effective emergency arrangements in place that ensure the site workers and potential occupants of the premises can take appropriate emergency action upon hearing an alarm etc.
Many of the items above are pre-requisites for a Safe System of Work relating to hot works on roofs; the Safe2Torch Campaign seeks to reduce the risk potential by identifying and removing naked flames from the areas where flames are more likely to take hold.
The full Safe2Torch guide can be downloaded from here: _URL_


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Type your tag names separated by a space and hit enter
Knowledge of breast cancer and practice of breast self examination among female senior secondary school students in Abuja, Nigeria.
Breast cancer is a public health problem that is increasing throughout the world especially in developing countries. The study was aimed at assessing the knowledge of breast cancer and practice of breast self examination (BSE) among female senior secondary school students in the municipal council area of Abuja, Nigeria.
This descriptive cross sectional study was carried out among female senior secondary school students from selected schools in the municipal area council of Abuja. The tool for data collection was a structured self administered questionnaire. Data were analysed using SPSS version 16.0.
Two hundred and eighty-seven students participated in the study. Their mean age was 16.5 +/- 1.4 years. A greater proportion of respondents 163 (56.8%) had poor knowledge of breast cancer while 217 (75.6%) had poor knowledge of BSE. Only 114 (39.7%) of the respondents knew that being a female was a risk factor for breast cancer and the least known risk factors were obesity and aging. The major source of information for breast cancer and BSE among the respondents was the mass media. Only 29 (10.1%) of respondents had practiced BSE. Knowledge of BSE was significantly associated with BSE practice.
This study revealed that female secondary school students have poor knowledge of breast cancer. A good proportion of them knew that BSE could be used as a screening method for breast cancer but only few had practiced BSE. There is need for adequate health education on breast cancer and BSE among adolescent females in Nigeria.
Health Knowledge, Attitudes, Practice
Pub Type(s)Journal Article


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It makes sense conceptually to me I would just like this verified.
share|improve this question
The dot product can be distributed over addition: $\mathbf n\cdot(p\mathbf a+q\mathbf b)=p(\mathbf n\cdot\mathbf a)+q(\mathbf n\cdot\mathbf b)$... – Guess who it is. Oct 18 '11 at 1:12
1 Answer 1
up vote 4 down vote accepted
Yes! One way to check that two vectors are orthogonal is that their dot product is zero. So now you just need to algebraically check that if $\vec{v}\cdot\vec{a}=0$ and $\vec{v}\cdot\vec{b}=0$ then for any scalars $r,s$ we have $$\vec{v}\cdot(r\vec{a}+s\vec{b})=0.$$
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Thanks! Seems blindingly obvious now. :D – Jon Martin Oct 18 '11 at 2:10
Your Answer


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�2-inch thickness do not have to be turned.
Cooking Time Critical
The most important point to remember when cooking seafood is not to overcook it. Perfectly grilled seafood is moist and flavorful. Overcooked seafood becomes dry and tasteless. To estimate cooking time, measure the seafood at its thickest part (including stuffing). Grill 10 minutes (at approximately 400 degrees F) per inch of thickness. Turn the seafood halfway through the cooking time.
Fish is cooked but still moist when it turns opaque or just starts to flake when tested with a fork. To prevent foodborne illness, grill a large whole or stuffed fish with indirect heat 10 to 12 minutes per inch of thickness to reach an internal temperature of 145 degrees F. (Use a calibrated food thermometer to determine doneness in the thickest part.)
Shrimp and soft-shell crab turn opaque (milky white) when done. Mollusks in the shell, such as oysters, clams, and mussels, open when cooked.
Once seafood has reached its safe internal temperature, remove it from the grill and place it on a clean platter. Do not put cooked food on the same platter that held raw seafood. Clean the food thermometer with hot, soapy water.
Keep food at a safe temperature during the entire cookout. Refrigerate leftovers within 2 hours or 1 hour if the air temperature is above 90 degrees F.
Raw Seafood and Foodborne Illness Risk
If you are in a high-risk population group, which includes younger children (age 5 and under), older adults (age 65 and older), pregnant women, and immune-compromised individuals, you are at a higher risk of getting a foodborne illness. In addition, certain health conditions put you at risk for serious illness or death. Individuals with the following conditions are at higher risk:
- Liver disease, either from excessive alcohol intake, viral hepatitis, or other causes
- Hemochromatosis, an iron disorder
- Stomach problems, including recent stomach surgery and low stomach acid (from antacid use)
- Immune disorders, including HIV infection
- Long-term steroid use (as for asthma and arthritis)
Check with your doctor if you are unsure of your risk. Eating raw or undercooked seafood, especially raw oysters, can put you at risk of infections such as vibriosis. Vibriosis is caused by some types of Vibrio bacteria. An oyster that contains Vibrio does not look, smell, or taste any different from any other oyster. When eating shellfish be sure they are properly and thoroughly cooked to kill harmful bacteria. Shellfish safe to eat will open during cooking. Throw out any shellfish that do not open fully after cooking.
- Specialty woods, such as oak, hickory, pecan, mesquite, fruit woods, and grapevine cuttings lend a subtle flavor to grilled seafood. Soak wood shavings or chips for 30 minutes before cooking. Place a handful of drained, damp chips on briquettes (or in a separate aluminum or cast iron smoke pan just above the heat source in electric or gas grills) right before cooking.
- Fresh or dried herbs such as thyme, rosemary, dill, basil, marjoram, paprika, and oregano, enhance the flavor of seafood. Soak herbs in water for several minutes, drain, and pat dry. Toss damp herbs directly onto briquettes (or lava rocks) just before cooking.
- Marinades can be as easy as a bottled salad dressing or a homemade combination of oil with vinegar or fresh lemon or lime juice and your choice of seasonings. Save extra, unused marinade to brush on seafood as it cooks.
- Sauces should enhance—not mask—the taste of seafood. Match the flavor level of the sauce to the seafood being grilled. Baste with a mixture of equal parts lemon juice and butter or margarine. Alternately, mayonnaise can be spread on fish fillets to keep them moist.
Other General Pointers
- Be sure fish and crustaceans are completely cleaned (scaled, eviscerated, peeled) before grilling.
- Make sure mollusks are alive. Remove mud from shells by scrubbing with a brush under cold, running water.
- Pat seafood dry with paper towels before applying oil.
- Use square-sided, not round, skewers for seafood kabobs.
Revised by Alice Moore, Regional Extension Agent, Food Safety and Quality, Auburn University. Originally written by Brian E. Perkins, former Extension Seafood Technologist.
Appreciation is expressed to P. J. Waters and Dani Reams for their review of this material and to Eat Alabama Seafood, National Fisheries Institute, UF/IFAS Extension, Delaware Sea Grant, Clemson Cooperative Extension, CDC, FDA and the Florida Department of Natural Resources, Bureau of Seafood Marketing, for information used. This work is partly a result of research sponsored by NOAA, Office of Sea Grant, Department of Commerce, under Grant No. NA18OAR4170080. MASGP-21-063.
New January 2022, Grilling Alabama Seafood, ANR-0758


Reference > Brewer's Dictionary > Chrysippus.
 Chrysa'or [ch = k].Chubb (Thomas).
NisiChrysippus fuisset, Porticus non esset. Chrysippus of Soli was a disciple of Zeno the Stoic, and Cleanthes his successor. He did for the Stoics what St. Paul did for Christianity—that is, he explained the system, showed by plausible reasoning its truth, and how it was based on a solid foundation. Stoicism was founded by Zeno, it is true; but if Chrysippus had not advocated it, the system would never have taken root. 1
 Chrysa'or [ch = k].Chubb (Thomas).


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These two plates reproduce works from the Petit Palais belonging to the Dutuit collection. The images were obtained using the photochromy process.
In the recent catalogue Photochromie. Voyage en couleur, _PHONE_ [Photochromy. A Journey in Colour, _PHONE_], Sabine Arqué describes this complex process: " […] photochromes are mass-produced colorized lithographic proofs obtained from a black and white photographic negative […].
Léon Vidal filed a patent for the invention of the photochromy technique in 1872. Photochromies were successful, especially for reproductions of works of art. The effect is quite characteristic: the way in which the image is printed gives the impression of relief to the photograph, as well as bright colours which appealed to enthusiasts. In 1872, Léon Vidal embarked on the huge project of using photochromy to reproduce exhaustively the major works of art in French collections. The luxurious book was called Trésor artistique de la France, but never progressed beyond volume II.
Auguste Dutuit, an occasional photographer, paid close attention to the dissemination of his work and was enthusiastic about the most up-to-date techniques such as chromolithography. The works reproduced in these two plates were probably part of a much larger collection. Could they have been plates from the Trésor artistique de la France? Or plates commissioned directly by Auguste Dutuit to enhance his collection?

source: HuggingFaceFW/fineweb-edu


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$$p(x,y) = x^N + a_{N-1}(y)x^{N-1} + \ldots + a_0(y), \quad x,y \in \mathbf{C}$$
is a monic polynomial in $x$, and the coefficients $a_j$ are analytic functions of $y$, then the roots of $p$ have expansions in Puiseux series (in powers of $y^{1/m}$ for some $m$) which are convergent for $y$ sufficiently close to 0.
Is this true in an asymptotic sense when the $a_j$ are only assumed to be smooth? i.e. Do the roots have asymptotic expansions which are formal Puiseux series (not necessarily convergent)?
For $A$ to have an asymptotic expansion $A \sim B_1 + B_2 + \ldots$ with respect to some grading $\mathcal{O}(n)$ means that $B_n \in \mathcal{O}(n)$, and for each $N$, $A - \sum_{n=1}^N B_n \in O(N+1)$. Here $\mathcal{O}(n)$ means $\mathcal{O}(|y|^{n/m})$ in the usual big-O notation, as $y\to 0$.
share|improve this question
How would you expand the two roots of $x^2=exp(-1/y^2)$? – Piero D'Ancona Aug 13 '10 at 7:10
I would have thought that you get an asymptotic expansion by replacing the $a_i$ by their asymptotic expansions, i.e., their Taylor series (which are formal power series), and then do the usual formal Puiseux series expansion of $x$. – Torsten Ekedahl Aug 13 '10 at 8:29
_USER_ I believe the expansions would simply be 0 for both roots. _USER_ Ah, I guess I've never actually computed a Puiseux series, but I suppose the basic algorithm is just to backsolve for the coefficients. If the algorithm works, I guess it has to work asymptotically. I'm a little surprised at this outcome as, combined with the Malgrange preparation theorem, this should imply that the germ of zeros of any smooth function has such an expansion. – Mike Hall Aug 13 '10 at 16:36
1 Answer 1
up vote 3 down vote accepted
The constructive proof of the Newton-Puiseux theorem works formally, and a posteriori one can show that if the original coefficients are convergent, then the Puiseux series are convergent (see the details in Casas-Alvero's book, "Singularities of Plane Curves"). So Torsten Ekedahl's comment is right, of course. However, I wonder how useful this would be when the power series don't converge to the original smooth functions.
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That was exactly my point. You can not hope to represent the solutions by a Puiseux series, even in the simplest examples. And you can use $C^\infty$ functions to produce much weirder examples (e.g., infinite number of oscillations near many points) – Piero D'Ancona Aug 14 '10 at 17:10
The Taylor series gives the asymptotic expansion of a smooth function in terms of powers of $y$ and as such is uniquely determined and does give a lot of information about the behaviour of the function around the point. Of course the flat functions (those which are $O(y^n)$ for all $n$) get ignored so that $\exp(-1/y^2) \sim 0$ and then also so is any solution of $x^2=\exp(-1/y^2)$. Whether or not an asymptotic expansion of this type is useful or not will of course depend on your needs but the OP seems to have found it so as the question is about asymptotic expansions. – Torsten Ekedahl Aug 14 '10 at 17:44
_USER_: sure. I am questioning that the Puiseux expansion may be given any reasonable meaning. I'm thinking of examples like Glaeser's non negative, smooth function, flat at 0, whose square root fails to be $C^2$. In that case what is the meaning of the expansion? which would be a 'Taylor' expansion of any order of a non smooth function? I'm just perplexed and slightly curious about the usefulness of Mike's efforts. – Piero D'Ancona Aug 14 '10 at 19:18
_USER_: Interesting, I didn't know that you had that kind of complications. I agree that Puiseux expansions as asymptic expansions for functions on the real line look a little fishy. For one thing it seems safer to have them defined only for positive reals as otherwise $y^{m/n}$ may not even be well defined. – Torsten Ekedahl Aug 15 '10 at 22:12
I think the expansions certainly tell you something meaningful about where the zeros are, namely that they are within $\mathcal{O}(y^\infty)$ of several curves with the given Puiseux expansions. It's true that the $\mathcal{O}(y^\infty)$ error means you have essentially no control over derivatives. Still, I think this gives a nice picture of what the zero set looks like near a point. – Mike Hall Aug 16 '10 at 17:51
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Definition of Long-wool
1. Adjective. (of sheep) having relatively long wool.
Click the following link to bring up a new window with an automated collection of images related to the term: Long-wool Images
Lexicographical Neighbors of Long-wool
Literary usage of Long-wool
Below you will find example usage of this term as found in modern and/or classical literature:
1. Report of the Secretary for Agriculture by United States Dept. of Agriculture (1866)
"The demand for long wool is still unsatisfied, and the movement of... The breeding of long-wool sheep, especially of Cotswolds, Shropshire, and other Downs..."
2. Report of the Secretary of Agriculture by United States Dept. of Agriculture (1865)
"The old Lincoln sheep produced a long wool, making a fabric of lustrous... 4 Mr. Unwin observed that there was a greater demand for long wool and a wider..."
3. The Monthly Review by Ralph Griffiths (1782)
"... long wool; for tbe duty on exportation mutt be laid in a determinate fum on every pound of wool exported...."
4. The Encyclopaedia Britannica: A Dictionary of Arts, Sciences, and General by Thomas Spencer Baynes (1890)
"It will be obvious that the gill-boxes through which long wool passes, while possessing the power of bringing the fibres into s smooth, equal,..."
5. English Farming Past & Present by Rowland Edmund Prothero Ernle (1917)
"In England, long wool was employed mainly for worsted fabrics,... In long wool, or combing-wool, England had practically a monopoly of the markets,..."
6. The Parliamentary Debatesby Thomas Curson Hansard, Great Britain Parliament by Thomas Curson Hansard, Great Britain Parliament (1825)
"Every pound of long wool enabled the manufacturer to work up a certain quantity of inferior fine wool ; and, if the long wool were exported, the inferior..."


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Become an Interior Architect: Step-by-Step Career Guide
Research the requirements to become an interior architect. Learn about the job description and duties, and read the step-by-step process to start a career as an interior architect.
Requirements for a Career as an Interior Architect
Interior architects might work for architectural firms or other companies that design hospitals, offices, homes or schools; they also might be self-employed. Interior architects prepare drawings of interior spaces and might even design the furniture that will be used in those spaces. Although the work of interior architects sometimes overlaps with that of interior designers, the fields are not the same. In all states, interior architects must be licensed as architects, while certification or licensure for interior designers is only required in some states. The chart below covers the basic skills, training and licensure necessary to work as an interior architect:
|Degree Level||Professional architecture degree from an accredited school, commonly a 5-year bachelor's degree or a master's degree*|
|Licensure/Certification||All states require architects to be licensed; certification is voluntary*|
|Experience||A 1-3 year internship is usually required prior to licensure*; some employers prefer candidates with 5-10 years of experience***|
|Key Skills||Visualization, analytical, critical-thinking, communication and organizational skills*; knowledge of engineering, technology, public safety, building and construction**|
|Computer Skills||Able to use computer-aided design, query, photo editing, graphics, document management and project management software programs**|
Sources: *U.S. Bureau of Labor Statistics (BLS), **O*NET OnLine, ***CareerBuilder.com job postings (November 2012).
Step 1: Earn a Bachelor's or Master's Degree
All states require architects to be licensed. To become licensed, an individual must graduate from a program accredited by the National Architectural Accrediting Board (NAAB). Bachelor's and master's degree programs in interior architecture are available; however, these programs are usually not accredited by NAAB, and therefore can't prepare an individual for licensure.
Accredited bachelor's degree programs in architecture are commonly offered as a Bachelor of Architecture (B.Arch.) and last five years. Topics of study might include urban design, technology, interior architecture, physics, architectural math and the history of architecture. The curriculum of these programs often includes both lectures and design studios.
Master of Architecture (M.Arch.) programs require 1-2 years of study and are usually designed for students who possess a bachelor's degree in a field other than architecture. These programs might address environmental systems, computer-aided design and drafting, urban determinants and methods of construction. Instruction is often delivered through lectures and design studios. Some M.Arch. programs are combined with B.Arch. programs; these generally last 5-6 years.
- Complete courses related to interior architecture. Students who plan to work as interior architects often can choose electives related to this specialty. These courses might explore topics like lighting or building materials.
Step 2: Complete an Internship
Most states require that applicants for architect licensure have completed the National Council of Architectural Registration Boards (NCARB) Intern Development Program (IDP). This program mandates completion of a 5,600-hour paid internship.
- Prepare for the Architect Registration Examination (ARE). The NCARB offers study guides and practice exams to individuals intending to take the ARE. These materials might help an individual pass the ARE on his or her first try.
Step 3: Pass the Architect Registration Exam
Most states require that aspiring architects pass NCARB's ARE to become licensed. This exam can only be taken after an individual has completed the internship requirement. It consists of seven areas, including site planning and design, building systems and schematic design. Continuing education is required to renew architecture licensure. Depending on the state, renewal might be annual or biennial.
Step 4: Earn Professional Certification
Architects can apply for voluntary NCARB certification once they've earned state licensure. In addition to holding a license, candidates must have completed an NAAB-accredited bachelor's or master's degree program and the IDP and passed the ARE. NCARB certification must be renewed annually. The BLS notes that certification may ease the transition of licensing across states.
Architects with 'green' construction or design experience also might pursue one of the multiple Leadership in Energy and Environmental Design (LEED) certifications available through the U.S. Green Building Council. LEED credentials must be renewed every two years by completing 15 hours of continuing education.
Related to Become an Interior Architect: Step-by-Step Career Guide
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Explore more education and human sciences grant opportunities or find fellowships.
American Speech-Language-Hearing Foundation: Supports the advancement of knowledge in this field and seeks to improve the lives of people with speech, language, or hearing disorders. Programs include:
- Clinical Research Grant: Seeks to advance knowledge of the efficacy of treatment and assessment practices in speech-language pathology and audiology. Priority will be given to proposals investigating approaches to improve the everyday functioning of individuals with, or at-risk for, communication and related impairments. Awards are up to $75,000 for two years.
- New Century Scholars Research Grant: Supports studies that are innovative, groundbreaking, or that can meet research needs not yet addressed. Awards are up to $10,000 for up to two years.
- New Investigators Research Grant: Seeks to help further research activities of new investigators and should have particular clinical relevance to audiology and/or speech-language pathology. Awards are $5,000.
- Speech Science Research Grant: Supports research on the acoustic properties of speech production and perception. Awards are $5,000.
Autism Speaks: Supports research into the diagnosis, causes, prevention, and treatment of autism or its disabling symptoms. Awards range from $60,000 to $150,000.
Deafness Research Foundation: Seeks novel investigations of auditory and vestibular function and dysfunction. All research proposals in hearing and balance science, including those for basic, translational and applied clinical research, will be considered for funding. Awards are up to $25,000 for one year.
Helen Bader Foundation: Seeks to make Wisconsin a leader, not just in how we address the challenges posed by Alzheimer's disease, but in how we view growing older and the later stages of life. Through an emphasis on program development, applied research, and public policy, the Foundation aims to offer hope to families grappling with Alzheimer's. In the broader interest of healthy aging, the Foundation supports education and training efforts that raise the capacity of professionals who serve older adults.


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Well-trained comparative medical scientists are needed to meet the research needs of the 21st Century. Veterinarians currently are underrepresented in biomedical research but can make a unique contribution because of their expertise in clinical practice and fundamental biology, as well as their knowledge of spontaneous animal models of human disease.
The goal of the Tufts REVEAL Program for residents is to attract veterinarians to academic and research careers. The program is designed to first create the desire to pursue research and then to nurture these candidates with strong mentoring and programs, as well as by providing readily accessible research opportunities and removing common barriers to research.
The program includes: 1) Symposia on spontaneous animal models of human disease to provide greater interaction with researchers from other disciplines and to increase opportunities for collaborative research; 2) A multi-function website to facilitate research, including a web-based course on applied statistical methods; 3) An academic seminar series that includes topics to foster an interest in research and to facilitate research training; 4) Short-term introductory research electives; and 5) An active mentoring program.
For more information, contact Dr. Lisa Freeman


KnowledgeWorks and Prolog User Guide > 4 Objects > 4.2 Relational Database Objects
4.2.1 Example
(sql:def-view-class vehicle
 (standard-db-object standard-kb-object)
 ((vehicle_no :db-kind :key)
 (owner :db-kind :join
 :db-info (:home-key :keeper
 :foreign-key person_id
 :retrieval :deferred
 :join-class person))))
defines a database class vehicle where the person object in the keeper slot is retrieved from the person table in the database using the value of the keeper slot as key, only when queried. In the list of superclasses, standard-kb-object should appear after sql:standard-db-object.


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Let $X,Y$ be complex manifolds and let $f: X \rightarrow Y$ be a continuous map. When exactly do we say that $f$ is "holomorphic"? I am interested in the basic definition (possibly using charts), not an equivalent characterization.
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1 Answer 1
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If $p$ is a point of $X$, then we say $f$ is holomorphic at $p$ if there is a chart $(U, \phi)$ on $X$ with $p \in U$, and a chart $(V, \psi)$ on $Y$ with $\phi(p) \in V$, such that $f(U) \subseteq V$ and the induced map
$$ \phi(U) \stackrel{\phi^{-1}}\longrightarrow U \stackrel{f}{\to} V \stackrel{\psi}{\to}\mathbb{C}^n $$
is holomorphic in the familiar sense (using $\phi(U) \subseteq \mathbb{C}^n$).
We say $f$ is holomorphic if it is holomorphic at all points.
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Do we say that if $f$ satisfies the property you described using charts for a point $p$, then it is "locally holomorphic"? – Manos Sep 18 '12 at 18:04
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How to Open a Jammed Freezer Drawer?
To open a jammed freezer drawer, start by wiggling the handle to check for any obstructions that may be causing it to stick.
If that doesn't work, unplug the freezer and wait for about an hour to allow it to thaw.
Be cautious to not damage the freezer while attempting to open the drawer.
If the door or drawer is still stuck after thawing, there may be a problem with the freezer itself and it's advisable to call a repairman.
Possible issues with the freezer door or drawer include a faulty gasket, hinges, latch, closure cams, or gears.
A faulty gasket can cause frost to build up, and it can be easily replaced by ordering a new one and installing it correctly.
It's essential to avoid overfilling the freezer to maintain proper air circulation and leave at least half of it full.
Check for old or spoiled food and dispose of it to create more space.
If unsure about the safety of the food, it's best to discard it.
Lastly, if the freezer door opens on its own, it may be due to a faulty gasket or latch, issues with hinges or closure cams, overfilling, or the freezer not being level.
Freezer drawers can be replaced, but they must be the same model and size.
- Wiggle the handle to check for obstructions causing the freezer drawer to stick
- If the handle wiggling doesn't work, unplug the freezer and let it thaw for about an hour
- Be careful not to damage the freezer while trying to open the drawer
- If the door or drawer is still stuck after thawing, call a repairman
- Possible issues with the freezer door or drawer include faulty gasket, hinges, latch, closure cams, or gears
- Faulty gaskets can be easily replaced by ordering and installing a new one correctly.
Did You Know?
1. The world's largest popsicle was made in 2005, weighing a whopping 17.5 tons! It was a strawberry-flavored delight measuring 21 feet in length, which is more than twice the size of an average freezer drawer!
2. Did you know that before the invention of freezers, people used to store food in iceboxes? These early refrigerators relied on large blocks of ice to keep food cool, and the ice would need to be replenished regularly.
3. The concept of freezing food for long-term storage dates back centuries. Some ancient civilizations, like the Chinese and the Romans, would use mountainside ice caves to store food during harsh winters. This frozen food would then be enjoyed throughout the year.
4. Have you ever wondered why ice cubes sometimes stick together in the freezer? This phenomenon is called "bridging" and occurs when the cubes melt slightly and then refreeze together. It's caused by the water molecules bonding together when they come into contact.
5. When a freezer drawer becomes jammed, one simple trick to open it is to use a hairdryer. By directing hot air onto the edges of the drawer, the heat can cause the metal or plastic to expand slightly, making it easier to slide open. However, make sure to be cautious and avoid overheating any surrounding materials!
Wiggling The Handle To Check For Obstructions
When you encounter a jammed freezer drawer or door, the first step is to try wiggling the handle gently. Sometimes, an obstruction such as a stray item or ice buildup may be blocking the door's movement. By applying slight pressure while moving the handle back and forth, you can check for any hindrances that might be causing the jam. If you feel any resistance, do not force the handle as it may lead to further damage. Instead, move on to the next troubleshooting step.
If the handle wiggles freely without any obstructions but the freezer door or drawer remains stuck, there may be a different underlying issue. In such cases, it is important to proceed with caution to avoid damaging the freezer.
Freezer Door Frozen Shut? Try Thawing It Out
One common reason behind a jammed freezer door is the formation of frost, which freezes the door shut. If the handle wiggling technique does not resolve the problem, it is likely due to ice accumulation. In this case, you can unplug the freezer and allow it to thaw for approximately an hour. This will give enough time for the ice to melt, restoring the door's functionality.
Bear in mind that opening the freezer door frequently or leaving it open for extended periods of time can lead to frost buildup. It is essential to maintain proper freezer habits to prevent such issues in the future. Regularly defrosting the freezer and keeping the door closed tightly will help avoid freezing problems.
Avoid Damaging The Freezer – Open With Care
While attempting to open a jammed freezer door or drawer, it is crucial to exercise caution and prevent any potential damage to the appliance. Forcing the door open or using excessive strength may result in broken hinges, damaged gaskets, or other mechanical problems. These situations can be expensive to repair and even render the freezer unusable.
If you are unable to open the door or drawer after trying the aforementioned methods, the best course of action is to contact a professional repair service. They possess the necessary expertise to handle such situations without causing further harm to the appliance.
- Exercise caution when opening a jammed freezer door or drawer.
- Using excessive force can lead to broken hinges, damaged gaskets, or other mechanical problems.
- Contact a professional repair service if unable to open the door or drawer after attempting these methods.
Calling A Repairman For A Stuck Door
If you have exhausted all the quick-fix options and the freezer door or drawer remains stubbornly jammed, it is time to reach out to a qualified repairman. Persistent door jams may indicate a more serious problem with the freezer itself. Attempting to fix the issue without proper knowledge and experience can lead to further damage or void any warranty that may be in place.
When contacting a repairman, be prepared to provide them with detailed information about the jammed freezer drawer or door. They will be able to diagnose the problem accurately and provide you with the necessary solutions. Remember that safety should always come first, and it is better to seek professional help than risk causing irreversible damage.
Possible Issues With Freezer Door Mechanisms
Several factors can contribute to a freezer door or drawer getting stuck. Common culprits include faulty gaskets, hinges, latches, closure cams, or gears. These components ensure smooth operation and airtight seals, which are essential for preserving the freshness and quality of your frozen goods.
A faulty gasket, for instance, can allow warm air to infiltrate the freezer, causing frost to accumulate over time. Luckily, replacing a gasket is relatively easy. By ordering a new one and installing it according to the manufacturer's instructions, you can restore the integrity of the freezer's seal and prevent future jams from occurring.
It is important to note that freezer drawers can also be replaced, but it is crucial to ensure that the new drawer matches the model and size of the existing one. Mismatched accessories may not fit properly, leading to functionality issues and potential damage to the freezer.
Easy Fix For A Faulty Gasket
A faulty gasket is a common cause of freezer door jams and compromised insulation. Replacing a faulty gasket is a simple fix that can be easily accomplished. Here are the steps to follow:
Order a new gasket that is specifically designed for your freezer model.
Unplug the freezer to avoid any electrical hazards.
Gently remove the damaged gasket by detaching it from the anchor points around the door. Take note of how the old gasket was installed for guidance during the installation of the new gasket.
Carefully align the new gasket along the door, ensuring that it fits snugly into place.
Reattach the new gasket to the anchor points using the appropriate fasteners or adhesive, following the manufacturer's instructions.
Perform a quick test by closing and opening the door. You should notice a notable improvement in the ease of use and the freezer's overall performance.
Encountering a jammed freezer door can be frustrating, but with the right troubleshooting steps, it can be resolved without causing further damage. By checking for obstructions and trying to thaw out the freezer door, if necessary, you can often alleviate the issue. However, if the door remains stuck or you suspect a more significant problem, it is best to contact a professional repairman.
Remember to maintain proper freezer habits, such as avoiding overfilling and regularly defrosting, to prevent future jams. With these tips and a little patience, you can keep your freezer accessible and functioning smoothly for years to come.
Check this out:
Frequently Asked Questions
How do you open a stuck freezer drawer?
To open a stuck freezer drawer, start by unplugging the refrigerator and allowing the freezer to defrost completely. This typically takes around 24-48 hours. Once defrosted, carefully open the freezer and ensure that any moisture is wiped dry, paying close attention to the rails that the drawer sits on. Afterward, it is important to let the freezer air dry for another 24 hours before plugging the refrigerator back in. This will help prevent any future sticking issues and ensure smooth operation of the drawer.
How do you open a jammed freezer door?
To open a jammed freezer door, try pressing your finger into the inside corner of the door seal to alleviate some of the suction created. This can help release the door and make it easier to open after closing. Additionally, if the door has just been closed and is difficult to open, it is advisable to wait for approximately one minute. This allows the pressure to equalize and makes it easier to open the door without excessive force.
Why is


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For those times when Windows completely freezes and you can't even open the start menu, you can also restart your computer from Task Manager instead of just powering the machine off.
Doing a hard reset isn't as healthy for your computer's internal components, this method would be preferable if possible.
There are two ways you can bring up Task Manager:
- Hit Ctrl+Alt+Del and click on Task Manager.
- Hit Ctrl+Shift+Esc to bring up Task Manager
Once you are in Task Manager, you can restart a hung process in the processes list, or you can click on Shutdown \ Restart to restart your computer.
Pretty simple tip, but very useful


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Gravity Irrigation with Pump and Flow Control [Success Story]
Counter Nodes with flow meters are used to monitor the water flowing to orchards to allow for precise water metering and accurate billing for water use. For installations without access to wired power, a solar panel or battery pack may be used to power the FlexPower Nodes and sensors.
The Gateway, or wireless network controller, monitors the water level inside the water tank. When the water level drops, a Node installed with the reservoir pumps uses its outputs to cycle the pumps on or off. A series of Nodes monitors the amount of water flowing into each orchard. A 1 Watt Data Radio connected to the Gateway transmits the data to a mobile asset or a centralized control center for data processing and billing.
Because of the difficult terrain, using Data Radios allowed us to reliably transmit data over great distances and around obstacles. SureCross FlexPower devices using efficient power management technology allow users to choose from solar, battery, or dc power without compromising the performance or response time of the network.


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Question: "Why did God sometimes change a person's name in the Bible?"
Answer: When God changed a person's name and gave him a new name, it was usually to establish a new identity. God changed Abram's "high father" name to "Abraham," "father of a multitude" (Genesis 17:5) and his wife's name from "Sarai," "my princess," to "Sarah," "mother of nations" (Genesis 17:15). We know from history that the descendants of Abraham and Sarah formed many nations, including the Jews' and Muslims'.
God changed Jacob's "supplanter" name to "Israel," "having power with God" (Genesis 32:28). He changed Simon's "God has heard" name to "Peter," "rock" (John 1:42). Why did Jesus occasionally call Peter "Simon" after He had changed His name to "Peter"? Probably because Simon sometimes acted like his old self instead of the rock God called him to be. The same is true for Jacob. God continued to call him "Jacob" to remind him of his past and to remind him to depend on God's strength.
Why did God choose new names for some people? The Bible doesn't give us His reasons, but perhaps it was to let them know they were destined for a new mission in life. The new name was a way to let them in on the divine plan and also to assure them that God's plan would be fulfilled in them.
© Copyright 2002-2013 Got Questions Ministries.


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Measurement in today's natural gas pipeline systems encompasses many challenges. Measurement departments are faced with tighter system measurement balances than they were ten short years ago. Ultrasonic meters are the meter of choice to help with these challenges. Pipeline companies have pushed USM manufacturers to advance the USM technology to help with the challenges of measurement. Manufacturers have improved the ultrasonic technology greatly over the past 15 years. Many of these enhancements have been in the form of diagnostics and processing speed and power. This has aided measurement departments in reaching their goals of increased system balances and improved overall measurement uncertainty. Flow calibrating the ultrasonic meter is a vital part of the process. Most users require flow calibrations to improve meter performance and overall measurement uncertainty. The latest revision of AGA Report No. 9, Measurement of Gas by Multipath Ultrasonic Meters, Second Addition [Ref 1], requires flow calibration for ultrasonic flow meters when being used for custody transfer applications.
ULTRASONIC METER FLOW CALIBRATIONS CONSIDERATIONS AND BENIFITS
Explore the Latest Papers at ASGMT


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"Fuming" is a method of coloring wood by exposing it to ammonia fumes. In short, the wood and ammonia are sealed into a fuming chamber where the concentrated fumes react with the tannin in the wood and cause it to turn anywhere from a rich brown to almost black, depending on the length of exposure. The method became popular around the turn of the century, especially with Arts and Crafts furniture makers, who liked it because it evenly colored the quarter sawn white oak they favored. The process lives on, and is still considered the method of choice for Mission style furniture. And it's not all that difficult, although there a few important safety considerations. Below, Michael Dresdner explains the process – including how to protect yourself – in answer to a Woodworkers Journal eZine reader's question.
Q. "Fuming wood. It's a finishing technique used in Mission style furniture. What strength ammonia is required and where to obtain it? Does it require heat and how much heat? How long does the process take, and how much ammonia is required? Does anyone have any ideas for the "Fuming Chamber"?
A. Michael Dresdner: "Fuming is a dangerous process, so gear up with eye protection, respirator, and gloves. Suit up BEFORE you open the jug, or open the chamber. Use 35% solution ammonium hydroxide, which you can get from a chemical supply house or blueprint supplier (this is the ammonia used in blueprint machines — something becoming increasingly rare thanks to computer plotters.) The ammonia generates fumes that are trapped in a chamber — hence, very little is required, since a small amount will generate a good bit of fumes. One bowl will do a tented table or chair, and a gallon will do an entire chamber full of furniture. Naturally, it can be used again and again."
"The longer the furniture is in the chamber, the darker it gets. Typical schedules go from 12 hours (a very light fuming) to 72 hours (rather dark). Test scraps first.
"The temperature in the chamber not only affects speed of fuming, but also color. The hotter the chamber, the more reds it brings out. Colder chambers result in a more greenish brown. Make one section of the chamber of clear plastic, and shine a heat lamp into it. Remember, ammonia corrodes aluminum, so no electrical connections should be inside the chamber. You'll notice that when the air inside gets up to 80 or 82 degrees, the reaction goes faster and the colors get slightly redder."
"A simple chamber can be nothing more than a frame draped with clear or black 2 mil plastic. Make the frame of wood scraps or plastic pipe — anything that will hold up plastic and surround the piece you are fuming. Seal the plastic at the bottom with duct tape. (Obviously, don't fume on a wood floor — fuming works even through most finishes.) For a more permanent, reusable one, try one of those ubiquitous Rubbermaid storage sheds. Close it and seal up the edges and seams with duct tape."
From the Woodworker's Journal eZine Archives
You'll also find some great tips on fuming white oak in the Woodweb Knowledge Base. As a bonus, the discussion includes a glimpse of how they did things in the old days:
"There was this old story about fuming my grandfather told about oak and some other woods done in France. He said the work was interior installations of small stores. When the work was complete, everything inside the space was removed, and all windows and doors were sealed except one. This would be sealed from the outside. When it was sealed really tight, the finisher would set a big flat pan of ammonia on a metal grate, and light a real thick candle under it. Next morning he'd come by, check the color, make sure the candle was still lit, and wait for the right tint to the wood. (Ahh… the old tricks–who would care to hand stain an entire architectural molded interior space?)"
(We assume that wood fumers of years past used appropriate, government approved safety equipment as well – of course they did. – ed.)
If you have access to the Fine Woodworking magazine archives (a modest subscription fee is required) make sure you read Kevin Rodel's "Fuming with Ammonia" (FW # 126). You'll find a thorough explanation of the process and a look at how fuming works with species other than the traditional white oak. Marc Spagnuolo (aka the Wood Whisperer) offered his wood fuming method recently (see May 14). Note the emphasis, again, on taking the proper safety precautions.
Finally, if you want to learn Mission furniture making from start to finish, pick up a copy of The Furniture of Gustav Stickley. It covers the entire process, including how to fume your finished projects to the point of perfect authenticity.


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Animal Species:Cape York Sheathtail Bat
Very little is known about Cape York Sheathtail Bats in Australia. They are also found in Papua New Guinea.
Cape York Sheathtail Bats seem to roost in tree hollows in Australia, but in New Guinea they are known from limestone caves.
Feeding and Diet
Cape York Sheathtail Bats eat beetles, flying ants, grasshoppers and moths. They fly high and straight over the top of trees in open forests hunting for prey.
Cape York Sheathtail Bats are vulnerable to destruction of roost sites in hollows trees by clearing for agriculture and changes to feeding areas by agricultural activities.


text: - While breast milk is considered the best nourishment for newborns, they have a lowered ability to digest the fat rich liquid due to decreased lipase secretion for the first few months after birth. To offset this, breast milk itself contains Bile Salt-Stimulated Lipase (BSSL) which significantly heightens fat digestion ability. Pre-term infants (<37 weeks gestation age) are often fed donor breast milk, which is treated through Holder pasteurization by nonprofit milk banks to ensure microbiological safety. Due to the fact that Holder inactivates up to 95% of BSSL, two alternative techniques also proven to eliminate harmful breast milk pathogens were assessed for their ability to retain BSSL: high pressure processing (HPP) and gamma cell irradiation (IR). After treatment of breast milk samples with either Holder (62.5°C, 30 min), HPP (550 MPa, 5 mins), or IR (2.5 Mrads), BSSL activity was determined using a spectrometric assay and compared to raw untreated milk. IR treatment was shown to retain 32% of BSSL activity while HPP was shown to retain 84%, suggesting HPP could be an effective alternative to Holder pasteurization.


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8. to misunderstand: di maintindihan, di maunawaan, di mawatasan, di maulinigan
9. to fail to keep or be present at: di pumasok, magpalya, pumalya, lumiban
1. a form of address given to a girl or to a woman who is not married: binibini, Mis, Miss
2. a failure to reach: di pag-abot, pagkahuli
3. a failure to hit: sala, mintis
1. to catch fire: magningas
2. to set on fire: magsindi, sindihan, magpaningas, paningasan
3. to become stirred up or aroused: mag-alab, magsiklab
1. the evolution of heat and light: apoy
2. a destructive burning as of a building: sunog
3. to build a fire: magparikit, magparikit ng apoy
4. to catch fire: masunog, masilab, magliyab
5. to set on fire: sunugin, silaban, susuhan
1. to dismiss or be dismissed from a job (coll.): magalis (alisin) sa trabaho, masisante, isisante, magtiwalag
2. to play with fire: maglaro ng apoy
3. to be on fire means (a) burning: nasusunog, nasisilab,
lasso, a long rope with a noose for catching horses, cattle, etc.: silo, panilo, lubid na panilo
to catch a horse, a cow, etc. with a lariat: sumilo, manilo, siluin
a strong, thick line or cord made by twisting smaller cords together: lubid
1. to tie, to fasten, or to bind with a rope: gumapos, gapusin, igapos, magtali, talian, itali (ng lubid)
2. to catch with a lasso: sumilo, siluin
humabol, habulin (-um:-in) to catch up with, to appeal, to run after manghabol, habulin (mang-:-in)
to run after somebody. Hinabol ni Pedro si Juan. Peter ran after John.
late, tardy, last mahuli (ma-)
to be late or tardy. Dali ka at baka ka mahuli! Hurry up or you might be late.
humuli (-um-) to catch, to arrest, to apprehend manghuli, hulihin (mang-:-in)
to catch someone, to chase after someone. Nanghuli siya ng masasamang tao. He caught bad people.
sumalo (-um-) to catch. Sino ang sumalo ng bola? Who caught the ball?
lice comb, fine toothed comb, farmer's harrow
magsuyod, suyurin (mag-:-in) to comb one's hair with a fine nit comb, to catch lice, to harrow the soil, to make an intensive search in a place. Magsuyod ka at baka may kuto ka. Comb your hair with a fine tooth comb because you might have lice.
ipitin (-in) to crush. Huwag mong ipitin ang kamay mo. Don't crush your hand (e.g. on the car door). maipit (ma)
to get caught in between, to be clipped. Naipit siya sa pinto. She got pinched in the door [She (her hand, foot, etc.) got caught in the door].
mag-ipit (mag-) to clip, pin (hair). Mag-ipit ka ng buhok. Pin your hair in place. mang-ipit, ipitin (mang-:-in, i-)
to clip, to clasp, to press, to catch in between
salary sumahod (-um)
to catch the flow of something with a receptacle, to receive a salary or wage. Sumahod ako kahapon. I got/received my pay yesterday.
magsahod, sahurin (mag-:-in) to catch the dripping. Sahurin mo ang tumutulong tubig sa gripo. Catch the dripping water from the faucet (in a receptacle).


text:
Illustrated by Eric Gurney
"I can read it all by myself" is the Beginner Books motto, and behind it is an understanding of how important it is for children to take pride and pleasure in their early reading. These books have been designed to appeal directly to children through the use of humour, rhyme and bright pictures that can be "read" even by the non-reading child. Some Beginner Books are simple stories, others are hilarious nonsense: both types have been designed to give children confidence and make them want to go on reading.
When Sam's new dog learns to dig holes, he can't stop. He digs up gardens, roads, and buildings, and finally digs himself into a hole he can't get out of. The adventures of 'The Digging-est Dog' are certain to appeal to the Beginner reader age group.
- Publication Date:
- 17 / 10 / 1983
- 163 x 225mm


October 2020
Neuromuscular Control of Ankle-stabilizing Muscles-specific Effects of Sex and Menstrual Cycle
Authors: Iman Akef Khowailed 1, Haneul Lee 2
Author information:
 1. University of St Augustine for Health Sciences, College of Rehabilitative Sciences San Marcos California Campus
 2. Department of Physical Therapy, Gachon University – Medical Campus, Incheon, Korea
Journal: International Journal of Sports Medicine - September 2020, Volume 42, Issue 3, Pages 270-276 (DOI: 10.1055/a-_PHONE_)
The purpose of this study was to examine the differences in neuromuscular control and mechanical properties of the ankle-stabilizing muscles between men and women, and during different phases of menstrual cycle in women. Fifteen women with regular menstrual cycles and 17 male counterparts were included in this study. Electromyographic signals were recorded from the peroneus longus (PL) and tibialis anterior (TA) muscles while performing three balance tasks.
Muscle tone, stiffness, and elasticity of muscles were measured using a MyotonPRO in the resting position. Outcomes were measured twice (ovulation and early follicular phases) for women, while measurements were acquired only once for men.
Significantly higher tibialis anterior-peroneus longus co-contraction (TA/PL ratio) was observed in all balance tasks in women than in men (p< 0.05); however, significant differences between phases of the menstrual cycle were noted only in the 2 most difficult tasks (p< 0.05). A similar pattern was observed in the postural sway.
Keywords: estrogen, menstrual cycle, mechanical properties, sex-difference, lower limbs, neuromuscular activation
These results highlight the importance of sex-specific hormonal effects on neuromuscular control and mechanical properties, and as well as the differences during phases of the menstrual cycle. These insights assume significance in the context of developing neuromuscular strategies for the purpose of preventing lower extremity injuries during sports activities.


text:
Awareness of oncologic emergencies is raised through education program
In an oncologic emergency, nurses must intervene quickly to prevent life-threatening complications and save lives. This is especially important given that the incidence of oncologic emergencies increases along with the number of people diagnosed with cancer.
According to Suzanne Brady, RN, BSN, CCRN, the Nursing Staff Executive Council at Roswell Park Cancer Institute, Buffalo, New York, determined the need existed to provide nurses with the most current information available regarding emergencies in cancer patients. To address that need, a day-long oncologic emergencies educational program was held to provide registered nurses in a comprehensive cancer center and the surrounding community with an overview of the signs, symptoms, and nursing management of the following oncologic emergencies: malignant pleural effusions, tumor lysis syndrome, sepsis, increased intracranial pressure, and superior vena cava syndrome, Brady noted in a presentation at the Oncology Nursing Society 36th Annual Congress.
Speakers at the oncologic emergencies educational program included three doctors, a staff development instructor, and a nurse practitioner. The program offered 5.25 continuing nursing education credits. Members of the Nursing Staff Executive Council were involved in planning, advertising, and hosting the event, which was attended by 59 nurses.
Attendee evaluation of the program revealed the goal of increasing awareness of oncologic emergencies had been met. Comments about the program included, "good presentation," and "very informative."
"By providing education to nurses and the community we can recognize and intervene in these emergencies earlier and may even save a life," the researchers concluded.


text:
Examples of scheduled monuments
Below are examples of scheduled monuments, ranging from early prehistoric to modern monuments.
Early prehistoric monuments
Despite recycling of stone for later uses, early prehistoric burial
sites such as this chambered cairn still contain important and unique information about burial practices
around 5000 years ago.
Later prehistoric monuments
Around 2000 years ago people in northern and western Scotland lived in substantial stone houses, often with defensive qualities. Here a submerged causeway leads from the shore to an island dun.
For a time in the 2nd century AD, the Antonine Wall was the north-west frontier of the Roman empire.
Early medieval monuments
The Picts of the 4th to 7th centuries AD built forts in many parts of Scotland, here reusing masonry from a nearby Roman fort.
Later medieval monuments
Later medieval churches were at the heart of their communities. Along with their graveyards they can tell us about changing religious beliefs and how this was reflected in the church architecture and burial practices.
It can be difficult to identify the remains of medieval settlements, but deserted settlements are characteristic of much of post-medieval Scotland. This fermtoun appears to contain at least eight separate units or farmsteads, pre-dating the transition to the dispersed farmsteads of the post-Improvement era.
We now recognise the cultural significance of the military remains of the 20th century.
In Scotland, seven wrecks dating from the First World War are scheduled in Scapa Flow, Orkney. Scottish Ministers can also designate wrecks under the Protection of Wrecks Act 1973.


Chanting Nam Myoho Renge Kyo
Nichiren (1222-82) established the chanting of Nam-myoho-renge-kyo as the way to awaken one's Buddha nature and tap into the deepest level of our existence, where our own lives and that of the universe are one. He first taught the invocation of the phrase to a small group at Seicho-ji temple in Awa province, Japan, on April 28, 1253.
Nam derives from the Sanskrit word namu , meaning "to devote oneself." Nichiren established the practice of chantingNam-myoho-renge-kyo as a means to enable all people to put their lives in harmony or rhythm with the Buddhist law, or Dharma. In the original Sanskrit, namu indicates the elements of action and attitude, and refers therefore to the correct action one needs to take and the attitude one needs to develop in order to attain Buddhahood in this lifetime.
The lotus flower grows and blooms in a muddy pond, and yet remains pristine and free from any defilement,symbolising the emergence of Buddhahood from within the life of an ordinary person in the midst of the struggles of day-to-day existence.
Kyo literally means sutra, the voice or teaching of a Buddha. In this sense, it also means sound, rhythm or vibration. In a broad sense kyo conveys the concept that all things in the universe are a manifestation of the Mystic Law.
SGI President Daisaku Ikeda on chanting daimoku:
"Being human, it's quite natural for our minds to wander, for all sorts of thoughts and memories to surface. [...] There is no set form or pattern for how we should pray. Buddhism emphasises being natural. Therefore, simply chant earnestly without pretence or artifice, just as you are. In time, as your faith develops, you'll naturally find it easier to focus your mind when you chant.
"It's natural for prayers to centre on your own desires and dreams. [...] By chanting very naturally, without affectation or reservation, for what you seek most of all you'll gradually come to develop a higher and more expansive life-condition. Of course, it's perfectly fine as well to chant with the resolve to become a bigger-hearted person or for the welfare of your friends and for kosen-rufu - the happiness and flourishing of all humankind."


TCS Papers: Sample Questions 265 - 265 of 502
Question number: 265
» Basic CS » Operating System
Essay Question▾
Describe in Detail
What is process migration?
 - It is the transfer of sufficient amount of the state of process from one machine to the target machine.
 - Process migration is the relocation of a process from its source node to another destination node.
 - A process can either be migrated before it starts execution on its source node which is called as non-preemptive process or during the course of its execution that is known as preemptive process migration.
 - Preemptive process migration is more costly compared to non-preemptive because the process environment must accompany the process to its new node.
 - Steps involved in process migration:
 1. Process is selected for migration.
 2. Select destination node for the process to be migrated.
 3. Transfer of selected process to the destination node.
 - Migration policy is responsible for first two steps while third step is handles by migration mechanism.
 - Migration of a process is complex that involves handling various activities to meet the requirements for a good process migration.
 - The sub activities involved are:
 1. Freezing the process on its source node and restarting it on destination node
 2. Transferring the processes address space from restarting from its source to destination node.
 3. Forwarding messages meant for the migrant process.
 4. Handling communication between processes that have been placed at different nodes.
 - A preemptive process migration facility allows the transfer of an executing process from one node to another.
 - On the other hand, in system supporting only non-preemptive migration facility, a process can only be transferred prior to beginning its execution.
 - Preemptive process migration is costlier than non-preemptive process migration since the process state, which must accompany the process to its new node, becomes much more complex after execution begins
 Image of Process migration
 Image of Process Migration
 Given the image is define the way a process migrates from one node to another is shown in the figure.
 Advantages of process migration:
 1. Reducing average response time of processes
 To reduce the average response time of the processes, processes of a heavily loaded node are migrated to idle or underutilized nodes.
 2. Speeding up individual jobs
 - A migration of job to different node is done and executes them concurrently.
 - Migrate a job to a node having a faster CPU or to a node at which it has minimum turnaround time.
 - More speed up more migration cost involved.


Mini Schedules
Are you looking for special education resources to help your student or child gain more independence as they navigate their day-to-day routines?
Do you create a daily schedule for your student or child with the tasks you expect them to complete that day? If you need additional reinforcement for a particular task or portion of the schedule, you may choose to create a mini-schedule. A mini-schedule is an abbreviated list of tasks within a larger schedule.
For example, if your goal is to help your child get ready for school independently, you may use a mini-schedule in which each step in the morning routine is outlined using either visual representations of each task or written prompts as appropriate for your child.
In this example, the mini-schedule may contain steps such as: brush your teeth, get dressed, eat breakfast, get your backpack, put on your coat, etc.
Mini-schedules can be fully customizable and made to fit any situation!
Learn how to use mini-schedules to increase independence and promote on-task behavior.
This resource was authored by Watson Institute Special Education Consultant, Teresa O'Brien, M.Ed.


Lemuel Shaw
Lemuel Shaw.
Lemuel Shaw, (born January 9, 1781Barnstable, Massachusetts, U.S.—died March 30, 1861Boston), chief justice of the Supreme Judicial Court of Massachusetts (1830–60), who left an indelible mark on the law of that state and significantly contributed to the structure of American law.
Shaw was educated at Harvard, studied law privately, was admitted to the bar in 1804 in New Hampshire, and entered private practice. Extremely successful, he became prominent in the public life of the state. He drafted the first municipal charter for Boston in 1822 virtually without precedents to guide him and in 1830 accepted appointment as chief justice of Massachusetts. His decisions on that bench were formative in the development of both Massachusetts and national jurisprudence. He is particularly remembered for two opinions. In Commonwealth v. Hunt (1842), his ruling in favour of a striking labour union provided the first major precedent for removing labour unions from the province of the law of conspiracy. In Roberts v. City of Boston (1849), his ruling upholding the city's segregated schools was later reflected in the U.S. Supreme Court's "separate but equal" doctrine in Plessy v. Ferguson (1896), but its more immediate effect was to stimulate the passage in Massachusetts of the only 19th-century American desegregation law.
Judge Shaw was a fellow and an overseer of Harvard, and he was the father-in-law and major financial supporter of novelist Herman Melville.


The Southern Baptist Convention (SBC) was organized in 1845 as like-minded churches joined forces to accomplish more together for Christ's Kingdom than what they could do by themselves. A distinctive of the denomination is that each church, association, and state convention or fellowship is autonomous, but agrees with basic beliefs (see the Baptist Faith and Message) and gives as led to the Cooperative Program which supports missionaries in North America and missionaries serving in countries around the world.
According to its Bylaws and in practice, the State Convention of Baptists maintains a cooperative relationship with the Southern Baptist Convention in all its work and agencies.
Find out more by visiting the following SBC organizations' websites:


text:
How many dekanewton in 1 sthene?
The answer is 100.
We assume you are converting between dekanewton and sthene.
You can view more details on each measurement unit:
dekanewton or sthene
The SI derived unit for force is the newton.
1 newton is equal to 0.1 dekanewton, or 0.001 sthene.
Note that rounding errors may occur, so always check the results.
Use this page to learn how to convert between dekanewtons and sthenes.
Type in your own numbers in the form to convert the units!
1 dekanewton to sthene = 0.01 sthene
10 dekanewton to sthene = 0.1 sthene
50 dekanewton to sthene = 0.5 sthene
100 dekanewton to sthene = 1 sthene
200 dekanewton to sthene = 2 sthene
500 dekanewton to sthene = 5 sthene
1000 dekanewton to sthene = 10 sthene
You can do the reverse unit conversion from sthene to dekanewton, or enter any two units below:
dekanewton to yottanewton
dekanewton to dyne
dekanewton to gram
dekanewton to nanonewton
dekanewton to kip
dekanewton to zeptonewton
dekanewton to meganewton
dekanewton to micronewton
dekanewton to pound
dekanewton to dekagram
The SI prefix "deka" represents a factor of 101, or in exponential notation, 1E1.
So 1 dekanewton = 101 newtons.
The definition of a newton is as follows:
In physics, the newton (symbol: N) is the SI unit of force, named after Sir Isaac Newton in recognition of his work on classical mechanics. It was first used around 1904, but not until 1948 was it officially adopted by the General Conference on Weights and Measures (CGPM) as the name for the mks unit of force.
The sthene is the unit of force in the former Soviet mts system, _PHONE_. The symbol is sn.
ConvertUnits.com provides an online conversion calculator for all types of measurement units. You can find metric conversion tables for SI units, as well as English units, currency, and other data. Type in unit symbols, abbreviations, or full names for units of length, area, mass, pressure, and other types. Examples include mm, inch, 100 kg, US fluid ounce, 6'3", 10 stone 4, cubic cm, metres squared, grams, moles, feet per second, and many more!


New Research Points to the Benefits of Being all Fingers and Thumbs
Posted on Thursday 3rd February 2011
Gesturing with our hands helps us to solve spatial visualisation problems, a skill that is necessary in many professions and in our daily lives, according to research by University of Birmingham psychologists published this week in the Journal of Experimental Psychology: General by Dr Mingyuan Chu and Dr Sotaro Kita.
Spatial visualisation is the visual technique that a medical doctor would use to look at an x-ray and imagine a patient's bone structure from different angles or an architect uses to imagine how a building might look from a plan. It is the tool that we use, for example, when we are deciding how to re-arrange our living room without actually moving the furniture.
In the study the researchers asked groups of students to perform tasks in which objects presented on a computer screen had to be mentally rotated. Firstly 132 students were asked to solve the mental rotation task while the researchers observed spontaneous hand gestures using a hidden camera. They found that the number of hand gestures increased as the problems became harder. Secondly 66 students were divided into three groups. One group was encouraged to use hand gestures, the second was given no instructions, and the third group was asked to complete the task while sitting on their hands. The gesture-encouraged group were more successful in completing the task than the other two groups.
In the third experiment 32 students were asked to solve mental rotation problems. Half of them were encouraged to gesture and the other half were told to sit on their hand while solving the problems. Then both groups were asked to solve different spatial visualisation problems while sitting on their hands. The gesture-encouraged group were more successful than the gesture-prohibited group not only in solving the first mental rotation problems but also in the second different visualisation problems when they did not gesture. Gesture therefore has a lasting beneficial effect and the benefit spreads to a subsequent different visualisation task performed without gestures.
Dr Sotaro Kita, from the University's School of Psychology, said, 'We found that when people were encouraged to spontaneously produce 'co-thought gestures' with their hands while trying to complete a spatial visualisation task they found the problem easier to solve. And the gestural benefit lingers on and applies to a subsequent different visualisation task even if you don't gesture in the new task."
He continued: 'We are endowed with a wonderful cognitive tool - hand gesture - that helps us to visualise objects from different angles. Our research shows that we are more competent at solving a problem when we are able to move our hands while thinking through a solution.
'Computers can also help us with visualisation problems, but gestures are more convenient because they are readily available wherever you are as long as your hands are free.
'Spatial visualisation is important in many scientific fields, including mathematics, physics and engineering, but it also helps in any occupation that requires the use of images or diagrams. The research could have practical implications for education.'
Notes to Editors
The Nature of Gestures: Beneficial Role in Spatial Problem Solving. Journal of Experimental Psychology: General. Vol X, No. X.
For further information


How many helicopters crashed in Vietnam?
According to the Vietnam Helicopter Pilots Association, a total of 11,846 helicopters were shot down or crashed during the war, resulting in nearly 5,000 American pilots and crew killed. Of those servicepeople, 2,382 were killed while serving aboard UH-1 Iroquois, better known as the ubiquitous "Huey."
How many Huey helicopters were lost in Vietnam?
During the war 7,013 UH-1s served in Vietnam and of these 3,305 were destroyed. In total 1,074 Huey pilots were killed, along with 1,103 other crew members.
What was the life expectancy of a 2nd Lieutenant in Vietnam?
For the average life expectancy to be 16 minutes, it would mean for every 2nd Lieutenant that survived a year (525,000 minutes,) 30,000 or so would have to die in 1 minute. Did hundreds of millions of 2nd Lieutenants die in Vietnam?
How many tanks were lost in Vietnam?
On average 11,000 small arms, 68 tanks, and 30 aircraft were lost each day. Entries are listed below in alphanumeric order. During the Vietnam War, thousands of U.S. aircraft were lost to antiaircraft artillery (AAA), surface-to-air missiles (SAMs), and fighter interceptors ()s.
FASCINATINGLY: Question: When was letter K used in the Philippines?
Keep Calm and Travel


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This question is the outcome of a few naive thoughts, without reading the proof of Gelfand-Neumark theorem.
Given a compact Hausdorff space $X$, the algebra of complex continuous functions on it is enough to capture everything on its space. In fact, by the Gelfand-Neumark theorem, it is enough to consider the commutative C*-algebras instead of considering compact Hausdorff spaces.
The important thing here is that $C*$-algebras have a complex structure. The real structure is not enough. Given the algebra $C(X, \mathbb R) \oplus C(X, \mathbb R)$ of real continuous functions on $X$, the algebra $C(X, \mathbb{C})$ is simply the direct sum $C(X, \mathbb R) \oplus C(X, \mathbb R)$, as a Banach algebra(and this can be given a complex structure, (seeing it as the complexification...)). But to obtain a C*-algebra, we need an additional C*-algebra, and the obvious way, ie, defining $(f + ig)$* $= (f - ig)$ does not work out. More precisely, the C* identity does not hold.
So one cannot weaken (as it stands) the condition in the Gelfand-Neumark theorem that we need the algebra of complex continuous functions on the space $X$, since we do need the C* structure. Of course, this is without an explicit counterexample. Which brings us to:
Qn 1. Please given an example of two non-homeomorphic compact Hausdorff spaces $X$ and $Y$ such that the function algebras $C(X, \mathbb R)$ and $C(Y, \mathbb R)$ are isomorphic(as real Banach algebras)?
(Here I am hoping that such an example exists).
Then again,
Qn 2. From the above it appears that the structure of complex numbers is involved when the algebra of complex functions captures the topology on the space. So how exactly is this happening?
(The vague notions concerning this are something like: the complex plane minus a point contains nontrivial $1$-cycles, so perhaps the continuous maps to the complex plane might perhaps capture all the information in the first homology, etc..)..
Note : Edited in response to the answers. Fixed the concerns of Andrew Stacey, and changed Gelfand-Naimark to Gelfand-Neumark, as suggested by Dmitri Pavlov.
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If you know a little algebraic geometry, an exercise toward the end of chapter 1 in Atiyah-Macdonald walks you through an attractive proof that a CH space is determined by its ring of real valued functions. What the C* algebra formalism gets you in this context is the "image" of the functor $X \mapsto C(X, \mathbb{C})$ from CH spaces to $\mathbb{C}$-algebras. It is possible to formulate an analogous condition for $\mathbb{R}$-algebras, but the entire theory is more awkward (though more powerful in some ways!). – Paul Siegel May 25 '10 at 23:48
Let me try to justify my last remark. Aside from the awkwardness of the definition, the Bott periodicity theorem for the K-theory of real C* algebras has period 8 rather than period 2 in the complex case, and thus many arguments are much more arduous. But there is also room for more subtlety. For example, the complex structure of C* algebras is at some level responsible for the fact that the Atiyah Singer index theorem is most naturally formulated for even dimensional manifolds. One way to formulate an odd dimensional analogue is to use real C* algebras. This is not without applications. – Paul Siegel May 25 '10 at 23:57
_USER_: That is a nice exercise, but I am confused by "If you know a little algebraic geometry". I guess you mean because of the topology placed on the maximal ideal space? Also, thanks for addressing some differences between the real and complex case. _USER_: In spite of Paul's comments, I must say I'm a little confused by the noncommutative-geometry tag in light of the fact that there is nothing noncommutative in your question. – Jonas Meyer May 26 '10 at 2:40
_USER_: In AM, $X$ is recovered from $C(X)$ as the subspace of $Spec(C(X))$ (with the Zariski topology) consisting of maximal ideals. So the exercise requires a casual familiarity with $Spec$, which one usually accumulates via AG. For what it's worth, I support the NCG tag for this question. I don't always work directly with NC C*-algebras in a substantial way, but I usually tell people I work in NCG because I often use $C(X)$ as a proxy for $X$. Maybe I am "mistagging" myself. :) – Paul Siegel May 26 '10 at 14:03
Makes sense. This isn't the first time I've heard of someone working in commutative NCG, but I still appreciate the novelty. – Jonas Meyer May 26 '10 at 19:34
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3 Answers
up vote 4 down vote accepted
Here is a slightly different, perhaps simpler take on showing that $C(X,\mathbb{R})$ determines $X$ if $X$ is compact Hausdorff. For each closed subset $K$ of $X$, define $\mathcal{I}_K$ to be the set of elements of $C(X,\mathbb{R})$ that vanish on $K$. The map $K\mapsto\mathcal{I}_K$ is a bijection from the set of closed subsets of $X$ to the set of closed ideals of $C(X,\mathbb{R})$. Urysohn's lemma and partitions of unity are enough to see this, with no complexification, Gelfand-Neumark, or (explicitly) topologized ideal spaces required. I remember doing this as an exercise in Douglas's Banach algebra techniques in operator theory in the complex setting, but the same proof works in the real setting.
Here are some details in response to a prompt in the comments. (Added later: See Theorem 3.4.1 in Kadison and Ringrose for another proof. Again, the functions are assumed complex-valued there, but you can just ignore that, read $\overline z$ as $z$ and $|z|^2$ as $z^2$, to get the real case.)
I will take it for granted that each $\mathcal{I}_K$ is a closed ideal. This doesn't require that the space is Hausdorff (nor that $K$ is closed). Suppose that $K_1$ and $K_2$ are unequal closed subsets of $X$, and without loss of generality let $x\in K_2\setminus K_1$. Because $X$ is compact Hausdorff and thus normal, Urysohn's lemma yields an $f\in C(X,\mathbb{R})$ such that $f$ vanishes on $K_1$ but $f(x)=1.$ Thus, $f$ is in $\mathcal{I}_{K_1}\setminus\mathcal{I}_{K_2}$, and this shows that $K\mapsto \mathcal{I}_K$ is injective. The work is in showing that it is surjective.
Let $\mathcal{I}$ be a closed ideal in $C(X,\mathbb{R})$, and define $K_\mathcal{I}=\cap_{f\in\mathcal{I}}f^{-1}(0)$, so that $K_\mathcal{I}$ is a closed subset of $X$. Claim: $\mathcal{I}=\mathcal{I}_{K_\mathcal{I}}$.
It is immediate from the definition of $K_\mathcal{I}$ that each element of $\mathcal{I}$ vanishes on $K_\mathcal{I}$, so that $\mathcal{I}\subseteq\mathcal{I}_{K_\mathcal{I}}.$ Let $f$ be an element of $\mathcal{I}_{K_\mathcal{I}}$. Because $\mathcal{I}$ is closed, to show that $f$ is in $\mathcal{I}$ it will suffice to find for each $\epsilon>0$ a $g\in\mathcal{I}$ with $\|f-g\|_\infty<3\epsilon$. Define $U_0=f^{-1}(-\epsilon,\epsilon)$, so $U_0$ is an open set containing $K_\mathcal{I}$. For each $y\in X\setminus U_0$, because $y\notin K_\mathcal{I}$ there is an $f_y\in \mathcal{I}$ such that $f_y(y)\neq0$. Define $$g_y=\frac{f(y)}{f_y(y)}f_y$$ and $U_y=\{x\in X:|g_y(x)-