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# Determination of statistically relevant quantitative/qualitative variables

I have a database containing records of several parameters: one is a quantitative parameter $$D$$ (e.g. average fuel consumption), others are qualtitative parameters $$X_1, ..., X_n$$ (e.g. make, color, engine type, ...) and quantitative parameters $$Y_1, ... Y_m$$ (e.g. average speed, driver's age, ...).

I am trying to assess whether there is any statistical relationship between $$D$$ and the other parameters $$X_i, Y_j$$. The ultimate goal would be to identify which of the $$X_i, Y_j$$ are relevant to "predict" $$D$$, and which are the ones that should be dismissed (because they are actually unrelated to $$D$$).

Please note that at this stage, I am just trying to identify what are the relevant parameters. The question of determining a quantitative relationship between $$D$$ and those parameters would come after this analysis.

I was thinking about performing chi-squared tests of independance between $$D$$ and the various $$Y_j$$, but I am rather confused about what to do with the qualtitative $$X_i$$. My questions are therefore:

1. What tests/analysis can be used for the qualtitative parameters $$X_i$$ ?
2. Is there any additional analysis that could be performed on top of the chi-squared tests for the $$Y_i$$?

I'd be inclined to use a multiple regression model with $$D$$ as dependent/target variable and the rest as predictors (all $$X_i$$ and $$Y_i$$). Note that you can use qualitative/categorical variables or continuous variables in this approach. And by including all predictors together in one model instead of doing separate chi-squared tests, you can account for the covariance between the predictors, and also model any interactions between them that you may expect.

Alternately, depending on how much data you have, I'd consider using a random forest (again $$D$$ as dependent/target variable and the rest as predictors). Random forests allow you to model complex interactions and nonlinear responses without specifying them a priori, which is why they may be of use. With this approach too you can include both categorical and continuous variables together.

In either case, you can get estimates of how important each predictor is, as well as quantitative descriptions of how they relate to $$D$$ (although this is easier with multiple regression than with random forests).

• The problem I see with multiple regression is that I cannot assume a specific relationship (e.g. linear) between the parameters and D a priori. I will have a look at random forests as I am not familiar with this concept. For my specific case, I have around 3000 records with less than 7 parameters for each. – G. Vaurs Aug 2 '19 at 13:11
• @G.Vaurs Random forests can handle complex nonlinear relationships without your having to specify them. I've edited my answer to reflect that. They should work fine with 3000 data points. – mkt - Reinstate Monica Aug 2 '19 at 14:05
• I tried the random forest approach -which seems a very useful tool- on my dataset. What I feared happened: it seems that the 7 parameters (predictors) are not exhaustive enough to grasp an actual understanding of D, the OOB error estimate being around 75%. Should I conclude that modeling D based on my parameters is a vain attempt? – G. Vaurs Aug 7 '19 at 15:40
• @G.Vaurs Hard to say, we don't know that much about your data or area. You could try the multiple regression approach, possibly with some regularization and see how well it does. – mkt - Reinstate Monica Aug 7 '19 at 16:56

If you want an alternative (simpler and to some extent parametric) to the solutions mentioned above, without making a clear hypothesis about the shape of the relationship between dependent variable and predictors and the interaction between predictors (as in a regression), just look at how each single predictor is associated to the dependent variable regardless the other candidate predictors or shape of the relationship (is each predictor significantly associated to the dependent variable?). This way you can start grabbing an idea of what predictors are significantly associated/correlated to the dependent variable without forcing too many assumptions on the shape of the relationship ( as well as the relationship between all predictors when crowded into a multivariate regression). Typically this goes under the label of univariate ANOVA (t-tests for numerical variables and and tests of association for categoriacal variables).

To some extent this is similar to checking the classification power of a predictor on the dependent variable without making the shape of the relationship explicit. And it has the advantage that it is parametric for a sufficiently large sample size (due to the central limit theorem) and easy to carry out and well documented.

Then you can use the reduced set of significant predictors found above to specify a multivariate model, by comparing different model structures (linear, non-linear alternatives, ecc and choosing the predictors for each one) based on goodness of fit (you can use iterative procedures as well to accomplish this)

• The problem I am facing with ANOVA is that the residuals do not seem normally distributed, even after applying a transform, thus violating one one the key assumptions of ANOVA. – G. Vaurs Aug 7 '19 at 6:44


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# Structure of foliations of codimension 2 on three dimensional torus

Is it possible to have a one-dimensional foliation on three dimensional torus such that the foliation has a trefoil knot as its leaf? Moreover, does a one dimensional foliation on three dimensional torus have both compact and non-compact leaves?

Thank you very much!

[Apologies: the answer I wrote below is for $S^2 \times S^1$, not the 3-torus. Corrections added. The case of a 3-torus, or any 3-manifold, is included at the end.]

1. First, there are different kinds of "trefoil knots" in $M = S^1 \times S^2$. Let's say we are interested in the "split trefoil knot", which lies in a 3-ball in $M$. This does in fact occur as a leaf of a foliation and it is easy to see.

2. To build it, first recall the standard foliation of $S^3$ by circles where the one leaf is the trefoil knot $T$. This can be obtained by taking the orbits of the circle action $(z,w) \mapsto (t^2 z, t^3 w)$, $|t|=1$, on the 3-sphere in $\mathbb C^2$.

3. Next we observe that the trefoil knot $T$ and a "disjoint" (split) unknot $U$ in $S^3$ can arise as the leaves of a foliation of $S^3$. (Split means there is an $S^2$ separating $U$ from $T$). To see this, consider the unknot $U'$ in (2) that arises as the orbit of $(z,w) = (1,0)$. This would almost do, except that $U'$ links $T$, i.e. $U'$ and $T$ are not split.

4. To fix this, we use the fact (mentioned by Zare) that the space of constant slope foliations of a 2-torus is connected (it is just a circle). First adjust the foliation in (2) so it contains a solid torus $D^2 \times S^1$ foliated by $p\times S^1$'s with $U' = 0 \times S^1$. Here $D^2 = D^2(1)$ is the unit disk. By rotating the foliations on the tori $N_r = S^1(r) \times S^1$ as $r$ varies from 0 to 1, we can arrange that for some $r>0$ the torus $N_r$ is foliated by leaves of the form $U = S^1(r)\times p$, $p$ in $S^1$. Each of these leaves is an unknot split from $T$, so we have shown 3.

5. To complete the picture, we now observe that $S^2 \times S^1$ can be obtained from $S^3$ by Dehn surgery along $U$, i.e. by cutting out a solid torus (disjoint from a ball containing $T$) with core curve $U$ and gluing it in again differently. Using the rotating foliation idea again, it is easy to extend the foliation from (3) to a foliation of $M$ with $T$ as a leaf. By construction, $T$ is a split trefoil in $M$.

Case of the 3-torus: It is also true that for any closed 3-manifold $M$ and any link $L$ in $M$, there exists a nowhere zero vector field on $M$ such that $L$ is invariant under the flow. One way to see this is to use Zare's construction, which depends on two facts: (a) $M$ can be obtained by Dehn surgery on a link $L'$ in $S^3$ (Lickorish) and (b) any link $L''$ in $S^3$ can be presented as a closed braid (Alexander). This construction will be a little less explicit than the one above, since it uses (a) and (b).

The main point is to prove the statement when $M=S^3$. For this one can start with any flow (e.g. the Hopf flow) such that there exists a closed trajectory. Thickening this flow line, we obtain an unknotted torus $S^1\times D^2$ with the product foliation. The foliation is obtained by suspending the identity map $D^2 \rightarrow D^2$. Now let us present $L$ as the closure of a braid $B$ with $n$ strands. Then $B$ can be obtained by suspending a diffeomorphism of $D^2$ to itself, fixing the boundary, and permuting $n$ points. Plug this new foliation into the original $D^2 \times S^1$. We then have a flow on $S^3$ such that $L$ is periodic.

For the general case, let $W = L' \cup L''$ such that $M$ is obtained by Dehn surgery on $L'$, and $L''$ becomes the desired link $L$ in $M$ after surgery. Then apply the construction above to $(S^3,W)$, and rotate the foliation near $L'$ as in step (5) above so it is compatible with Dehn surgery.

• $T^3$ cannot be obtained by Dehn surgery on the unknot in $S^3$, since $T^3$ has Heegaard genus 3, which is not a lens space. Jul 30 '18 at 5:41

The $1$-dimensional foliations of $3$-manifolds are very flexible. You can have any tame knot $K$ as a closed leaf of a $1$-dimensional foliation of a $3$-torus.

Step 1: Foliate the torus by parallel circles, $S^1 \times \mathbb{T}^2$.

Step 2: Change the foliation near one circle $C_0$ to include a meridional circle $C_1$ with a product foliation around that. You can maintain cylindrical symmetry and that the leaves are tangent to tori around $C_0$. Change the angle with a longitude smoothly from $0$ to $\pi/2$, hold constant for an interval, then change the angle back to a multiple of $\pi$.

Step 3: Perturb the foliation near $C_1$ to include $K$ as a leaf with any representation of $K$ as a closed braid you want.

• The knot leaf (trefoil knot) on 3-torus, I mean is that the bounded ball in 3-torus intersects with the leaf is the knotted 1-tangle. Hence, I think the braid you construct is not the one I wanted. Thank you! Aug 5 '16 at 6:29
• What I constructed was a foliation with a trefoil knot as a leaf, which is what you asked. The trefoil is contained in a ball. If you mean that you want the knot to be nontrivial in homology, then I don't think it is normally called a trefoil knot, but it is easy to modify the construction to produce such a knot. Aug 6 '16 at 8:28
• Would you please give some more details in step 2 and how is the knot K constructed in step 2? Some references and some pictures about the construction in step 2 are preferred. (I am working on dynamical systems and not quite familiar with your sentences of knot theory in step 2 and so I think I have some misunderstanding. I think that C_0 is a leaf, C_1 is another generator of the 2-torus other than C_0, the neighborhood you consider is the thickened torus C_0*C_1* I where I is the interval you mentioned above. Am I right? What is the cylindrical symmetry, the angle and "hold constant"?) Aug 7 '16 at 6:56
• $C_0$ is a leaf of the foliation from step 1. It is not homologically trivial. $C_1$ is a meridional curve on a solid torus whose core is $C_0$. $C_1$ is homologically trivial. In step 2, I replace the foliation in a larger solid torus around $C_0$ so that it agrees with the old foliation on the boundary. Inside, all of the leaves are tangent to the tori of fixed distances from $C_0$. The induced foliations of the tori change in slope from the longitude to the meridian, $C_1$, then back to the longitude next to $C_0$. Aug 7 '16 at 12:05

We can foliate a torus (product of circles) by circle factors, with all leaves compact. If the torus has dimension 2, we can foliate it by lines with a given irrational slope quotiented down from the plane, and similarly in any dimension 2 or more, with no leaves compact. I don't know about knots.

Suppose that the three dimensional torus is the quotient of $R^3$ by the group $G$ generated by the translations $t_{e_i},=1,2,3$ of respective directions $e_i, i=1,2,3$. We denote by $p:R^3\rightarrow T^3$ the covering map.

The vector field $X$ of $R^3$ defined by $X_(x,y,z)=e_1+sin(2\pi y)e_2$ is invariant by $G$ and defines a vector field $Y$ on $T^3$, the orbit of $p(0,0,0)$ is compact, but the orbit through $p(0,1/8,0)$ is not compact.

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Project data
Authors Scott Gennings Designed Prototyped 2010 Yes No CAD 8.67 LED lightingPhotovoltaic device Download Upload your project too!body.poncho-dark-mode .mw-parser-output .mw-ui-button{background:#303134;border-color:#3c4043;color:#bdc1c6}@media screen and (max-width:500px){.mw-parser-output .button .mw-ui-button{display:block;margin:.5em 0}}

Partially prototyped -- needs full test with photovoltaics.. The main goal of this Mech425 project is to design a simple, cost effective indoor light source that can be used in areas of the world that do not have access to electricity or safe lighting at night. If this design proves viable, the technical specifications as well as methodology and costs will be available to anyone who has access to the internet or to programs such as the One Laptop Per Child (OLPC) Program.

The proposed design utilizes a recycled bottle and cap (with a cap diameter equal to or greater than that of a 2L pop bottle) as a portable, durable platform for a high efficiency LED lighting system that is charged using a small solar panel. The bottle cap will house the lighting equipment and act as the bulb ballast and the pop bottle will act as the diffuser and stand.

## Background

### LED Technology

LED is an acronym for a lighting system known as a light-emitting diode. LED's are semiconductors that emit light when a junction is exposed to a forward biased electrical charge (electro luminescence). This application will use white LED's, which can be used as a much more efficient replacement for incandescent light bulbs. The main advantages of LED use are the efficiency and lifespan compared to standard incandescent bulbs. First of all, LED's produce many more lumens/watt of light than incandescent bulbs, resulting in a reduced power requirement as well as increased running time when powered by batteries. They also have extremely long lifetimes (anywhere from 10 000 to 100 000 hours) compared to incandescent bulbs (1000-2000 hours).[1][2][3]

These qualities have made LED lighting ideal for grass roots, low power applications such as this project, and have been used in similar endeavors. Some examples include:

### Lighting Issues Around the World

Approximately two billion people, almost 30% of the world population have no access to electricity, thus relying on fuel-based lighting, such as kerosene wick lamps and candles. This is a very dangerous alternative that is unsafe, expensive, and provides very inefficient lighting. Being limited by the amount of sunlight during the day makes it very difficult to perform tasks efficiently at night. The main problem is that these tasks include homework or studies by adults and children, and without proper lighting, reading books or chalkboards is impossible. This presents a barrier to education and economic development, not to mention literacy improvement. Efficient, self sufficient lighting can therefore impact the well being of people who cannot function efficiently after the sun has set.[7]

A major push to switch to LED lighting is the widespread use of dangerous, expensive fuels such as kerosene. Kerosene is very expensive for people living in poverty. In places such as rural India, purchasing enough kerosene to light a small household can represent more than 4% of a household budget. The light that these kerosene lamps produce is also very inefficient; LED's can produce more than 200 times the useful light. For example, an entire village can be lit using less energy than a single 100W normal bulb.[8]

Making the switch to solar powered LED lighting systems not only presents a more cost effective, efficient alternative, but also a much safer one. According to the World Health Organization, indoor air pollution from kerosene and other burned fuels used for indoor lighting and cooking is the cause of more than 1.5 million deaths to date [8]. There is also additional risk of fire when using kerosene, providing yet another reason to switch to LED lighting.

## LED Light Source Design Requirements

The design must fulfill the following functional requirements in order to be an appropriate solution to the lighting problem:

## Bottle Cap LED Fixture Design

There are two main methodologies that have been explored in order to develop a suitable appropriate technology designation for this design. The first of which relies on mass production (or at least semi-mass production) to justify the housing design as well as the parts purchases. The methodology relies on recycling all parts, which would be relatively difficult where there is little technology available, and designing based on parts available. The first methodology has been applied to develop a single part and assembly methodology that can be mass produced using injection molding and labor in China. The second methodology has been applied through the development of a prototype that utilizes the most recycled components as possible.

### Custom Designed Fixture

#### Required Parts

The mass production design requires a larger investment on the part of a sponsor or company in the short term, but will provide people with a reliable, water resistant, efficient lighting source that requires minimal maintenance. This design involves the production of a central component housing with the purchase of multiple electrical components. The parts required for the design are as follows:

 Part Name Information CAD Representation 1 x Housing To be produced 1 x LED 5mm, White, 3V 1 x Transistor BC549, 30V, 100mA, NPN 1 x Diode IN5711, 70V, 15mA, DO-35 1 x Resistor 271-1321, 1/4W, 1kohm 1 x Transformer Ferrous Ring 1 x AAA Rechargeable Battery NIMH AAA, 1.2V, 1000mAH 1 x Plastic Washer 15.5mm Plastic Washer 1 x Switch DPDT Slide Switch 2 x 3mm Screw 3mm Threaded Flat Screw 1 x Solar Panel 2V, 70mA, 0.14W

#### Housing

The part that must be manufactured is the housing for all of the components, and it can be seen in Figure 3, with it's dimensioned engineering specifications in Figure 4. It can be made in China using injection molding in order to minimize the overall cost of production. The bottom concave surface around the LED must be sprayed with a reflective paint in order to maximize light projection into the diffuser (bottle).

Figure 3: Housing Design

Ortho View Front View Bottom View

The engineering drawing is shown in Figure 4.

Figure 4: Housing Dimensions

#### Assembly

The assembly process is performed as follows:

1. Construct the circuit shown in Figure 3, leaving the solar panel and switch wires disconnected
2. Insert the connected LED,transistor, diode, resistor, transformer and battery into the housing in that order
3. Feed the wires for the solar panel and switch through the hole of the plastic housing cap
4. Apply glue to the edges of the cap, and press it into place, making sure to line up the flat edge with the switch location
5. Slide the switch into the side opening
6. Thread the 3mm screws into the switch screw holes to secure it in place
7. Connect the switch wires to the leads on the back side of the switch
8. Connect the solar panel wires to the leads on the reverse side of the solar panel
9. Apply glue to the top lip of the housing (square section)
10. Line up and set the solar panel on the top lip of the housing (bottom side down, making sure to tuck the wires into the opening
11. Hold the panel in place until the glue has set, and the panel is firmly set in place

The assembled final product can be seen in Figure 5 and the dimensioned engineering specs can be seen in Figure 6.

Figure 5: Final LED Fixture Design

Ortho View Front View Bottom View

Figure 6: Assembly Dimensions

#### Installation Into Bottle

Installing the light fixture is very simple.

1. Cut a 175 mm hole in the center of the recycled pop bottle cap using a sharp knife
2. Insert the end of the assembled fixture through the hole (through the top of the cap)
3. Push it in until it reaches the lip approximately 3/4 of the way up the housing
4. Force the lip through the hole and stop when the housing is completely inserted into the cap
5. Thread the cap with the fixture installed onto the top of the bottle to complete the luminaire
6. If the bottle is top heavy, which may happen in tall or small diameter bottles, it can be inverted to rest on the solar cell

Figure 7: Cap and Bottle Assembly

Cut 175mm Hole Push Fixture Through Cap Screw Assembly onto Bottle

### Prototype

A prototype was constructed in order to demonstrate a simplistic methodology utilizing minimal capital and maximizing the use of recycled materials. This ideology presents some major problems in that recycled materials are not always the correct size and do not always come together into an optimal design. It is a fair price to pay in order to minimize the overall cost and maximize the practicality of the design.

In this case, the parts attained were not exactly to specification, but worked anyway. For example, the recycled ferrite toroid ring is much larger than required, but no other parts were available. The leads for the recycled resistor and LED were very short, making soldering difficult. New parts will make the construction process much more efficient.

#### Circuit Construction

The first step to constructing the prototype is to make the internal circuit. To help simplify the process, a helpful walk through for the construction of the voltage booster portion of the circuit can be seen HERE. The visual steps can be seen in Figure 8 and is followed by a step by step walk through.

Figure 8: Prototype Circuit Construction

1. Required Parts 2. Ferrite Ring 3. Inductor
4. Transistor, Resistor, LED 5. Battery Test 6. Diode

Construction Walk through:

Tools Required:

• Soldering Iron
• Solder
• Needle Nose Pliers
• Side Cutters

Walk Through:

1. Twist together one of the pair of ends of the wires selected for the inductor portion of the circuit (see picture 2)
2. Pull the twisted end through the center of the ring and continue to feed the wires through, wrapping the ring in the wire pairs. (see picture 2 for the partially completed inductor and picture 3 for the final product)
3. Solder one of the blue wires to one of the red wires, leaving the solder and wire ends exposed
4. Solder on of the remaining wires to the 1K resistor and clip the remaining lead off
5. Spread the two exterior leads on the transistor apart and bend the middle lead backwards
6. Solder the middle lead to the lead on the resistor and clip the remaining lead off
7. Bend the leads on the LED outward
8. Solder the LED leads to the remaining transistor leads (one LED lead to one transistor lead)
9. Solder the remaining wire from the inductor to the lead opposite to the flat side of the LED (see picture 4)
10. Solder a small section of wire to the remaining lead of the LED, stripping both ends (this will act as the battery contact)
11. Connect the battery by connecting the small section of wire from the LED to the positive end of the battery (do not solder, simply test the circuit by touching the lead), and the soldered ends of the inductor wire to the negative end of the battery (this end can be soldered)(see picture 5) (hint: I utilized a recycled battery housing in order to be able to easily connect to the battery electrodes)
12. Solder the diode (white striped end closest) to the positive battery lead (see picture 6)
13. Solder a wire extension to the remaining diode lead (this will act as the positive solar panel connection)
14. Solder another wire extension to the negative lead of the battery (this will act as the negative solar panel connection)

The circuit assembly can now be installed into a suitable bottle cap and bottle.

#### Cap Assembly Construction

This portion of the prototype is open to customization because each bottle and circuit housing situation will be very different. The visual steps to constructing the bottle fixture can be seen in Figure 9 and is followed by a step by step walk through.

Figure 9: Cap and Bottle Prototype Assembly

1. Cap Size Test 2. Housing Installation 3. Cap Installation
6. Cap Wiring 5. Lighting Test

Construction Walk through:

Tools Required:

• Sharp knife
• Cut proof gloves
• Glue (can be substituted by cutting larger hole)
• Tape
• Circular housing for the circuit (I utilized a tape core, but the part can easily be substituted)

Walk Through:

1. Inspect the cap and neck opening of the bottle to ensure that the circuit assembly can be easily fit through (see picture 1)
2. Cut a small hole in the center of the cap for the wires if you have glue, or cut a large hole to suspend the housing if you do not have glue
3. Insert the wiring assembly into the selected housing, making sure to feed the two solar panel wires and connection leads through (the connection leads will be connected to a switch) (see picture 2)
4. If you have glue: Apply glue to the top side of the housing and install it on the inside of the cap. If you do not have glue: Slide the round housing through the hole in the cap (it should be a tight fit)
5. The next steps are not shown in the pictures because I was not able to obtain a functional switch or solar panel in time for the end of this project
6. Solder the switch to the two switch leads protruding through the bottle cap
7. Glue the switch to the side of the cap if you have glue, or pressure fit it if possible
8. Solder the positive and negative solar panel wires to the corresponding leads on the solar panel
9. Glue or attach the solar panel to the top of the cap
10. Screw the assembly onto the bottle (see pictures 3 and 4)
11. Turn off the lights in the room and test the light fixture (see picture 5)

The final product (unlike the final design in Figure 9 which is missing the solar panel and switch) is a self sustaining indoor and outdoor solar LED light fixture. It is important to note here that the design outlined here has not been tested in a completed form with the PV. It can be recharged daily by leaving it in the sun during peak daylight hours. The total battery capacity is 1000 mAh when new, which at a draw of 15 mA, will last up to 67 hours, so the light will not need to be charged every day. Make sure not to leave the assembly in the rain as it is not waterproof.

## Economic Considerations

The custom designed fixture will require very high investment in the short term, but if mass produced, costs could be reduced drastically, making it a viable solution. The prototype solution has lower initial costs, but buying individual parts for each reproduction could be more expensive than mass producing a design. An economic feasibility analysis will be performed in order to determine which is more feasible over the long term.

### Custom Designed Fixture

This portion of the economic analysis has been based on the paper [Early Cost Estimation for Injection Molded Parts] [21] produced by Dr. O. Kazmer. The paper presents a valid model for predicting the overall production costs involved in tooling and producing injection molded parts, such as the housing portion of this design. He also generated a Java Applet that calculates individual unit costs based on the number of parts produced,[22] which I used as a general model to predict costs over several production stages. The plot of the model can be seen in Figure 10.

Figure 10: Cost Model Plot

The same assumptions and part costs used in the initial circuit analysis have been applied to this analysis, with the addition of the Kazmer cost model and the results are as follows:

 Number of Units Produced 1 100 1000 10000 100000 967.65 102.73 33.47 10.91 3.55 1.20 1.2 1.2 1.2 1.2 Total Parts Cost / Unit ($) 6.72 3.36 1.68 0.84 0.42 Total Cost / Unit ($) 975.15 107.29 36.35 12.95 5.17

The economic analysis spreadsheet for the ideal design can be seen here: File:Mass Production Economic Analysis.xls

The resultant costs in the last row of Table 5 indicate that the final product will become extremely affordable when production exceeds 10000. The closest comparable product on the market sells for almost $25 [4], so the costs of$12.95 / unit at 10000 produced, and $5.17 / unit at 100000 produced are very reasonable. If mass production is undergone, and up to 100 000 are made, the final cost of$5.17 plus any additional profit margin markups will be much cheaper than the prototype cost which includes the initial circuit cost of $8.67 (labor included) + any additional part costs associated with construction. ### Prototype This portion of the design requires a much more simple cost analysis that is almost identical to the wiring cost analysis with the addition of the assembly parts cost. Most of the parts that I used were recycled, but I am going to consider a worst case scenario where all parts must be purchased (excluding the bottle and bottle cap which must be recycled). The entire construction and assembly process required 30 minutes but I was not being paid (and neither will someone else if they are making it themselves) so labor costs will not be included in the analysis.  Component Cost ($) Circuit 6.57 (labor subtracted from quoted cost) Wiring 0.15 Solder 0.05 Housing 0.50 Bottle + Cap 0.00 Total Cost / Unit 7.27 (without labor)

The cost analysis spreadsheet for the prototype can be found here: File:Prototype Cost Analysis.xls

The total cost per unit is higher than the mass production final cost (at 100 000), but will be much more affordable in the short term. The prototype solution requires very little overhead investment and will provide a suitable, reliable lighting solution. The fixture will have a shorter life cycle than the mass produced option due to construction inconsistencies and potential quality shortcomings of poorly soldered connections. The reduced overhead cost does however justify this chance of quality reduction.

## Discussion

The main goal of this Mech425 project was to design a simple, cost effective indoor light source that can be used in areas of the world that do not have access to electricity or safe lighting at night. The design utilizes a recycled bottle and cap (with a cap diameter equal to or greater than that of a 2L pop bottle) as a portable, durable platform for a high efficiency LED lighting system that is charged using a small solar panel. The final design had to comply with the functional requirements of minimal cost, maximum simplicity, minimal maintenance, maximum self sufficiency, minimal environmental impact and maximum ease of use.

These requirements led to the choice of a voltage boosted, single AAA, solar charged circuit design that would be applied in either a mass produced, or individually produced housing assembly. Both options come with distinct advantages and disadvantages. Mass production offers very high quality control of parts and assembly, along with minimal parts and labor costs, but only comes with a reduced unit price when high numbers are made. Individual, prototype style production comes with higher parts cost with reduced quality but offers a simple design that can be replicated at a relatively low cost by individuals; thus maximizing simplicity of production. In order to fulfill the ideology of developing a simple, cost effective light source that can be used in areas of the world that function without electricity; the 'prototype' style design would be most appropriate. This option allows the builder to customize the light fixture depending on available parts and tools. This final prototyped design totals a cost of $7.27 for parts, assuming that the builder will not be paid to assemble the unit and all required tools are available. If funding is provided, the mass production option is viable with a final unit cost of$5.17. This is a much more economically suitable solution for developing nations compared to similar products on the market right now which can cost up to $55.00. The main conclusion that can be deduced from this project analysis is that appropriate solutions, with minimal cost, for problems in developing countries can be easily solved by switching our mindsets from profitable designs to minimal cost designs. It has been proven in this design that a serious problem can be solved with an investment of less than$10.00. These lessons should be applied to engineers and designers everywhere in order to shift mindsets and make a difference in the world.

If there are any questions or concerns, feel free to contact me --S. Gennings 01:53, 31 March 2010 (UTC)

## Future Work

• Discussion of charge controller
• Build full prototype with PV
• Optimize sizing of PV and LED components

## References

1. Appropedia, "LED Lighting", https://www.appropedia.org/LED, Accessed March 23, 2010
2. Marktech Optoelectronics, White LED's, http://www.marktechopto.com/Engineering-Services/white-leds.cfm, Accessed March 24, 2010
3. Theledlight.com, "Technical Information", http://www.theledlight.com/LED101.html, Accessed March 23, 2010
4. Barefoot Power, "Barefoot Power", http://www.barefootpower.com/, Accessed March 20, 2010
5. Simply Brilliant, LLC, "Sollight",http://web.archive.org/web/20130320113706/http://www.sollight.com/products/lc200.cfm, accessed March 20, 2010
6. Cosmos Ignite Innovations, "Mighty Light", http://web.archive.org/web/20170806125717/http://www.cosmosignite.com:80/product-brief.htm, Accessed March 20, 2010
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## Related projects

Page data
Type Project led, bottle cap lighting, micro solar, solar, photovolatics, plastic SDG07 Affordable and clean energy, SDG09 Industry innovation and infrastructure, SDG10 Reduced inequalities Scott Gennings, Kathy Nativi 2010 CC-BY-SA-3.0 Category:Queens University, MECH425 Engineering for Sustainable Development English (en) 4,028 Scott Gennings, Kathy Nativi (2010). "Bottle Cap LED Lighting". Appropedia. Retrieved August 18, 2022.
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text:
Definition:Population

Definition

The population of a statistical study is everything in the universe of discourse that is relevant to the study.

It includes all objects, measures, and observations under discussion.

Finite Population

A finite population is a population which is finite.

Infinite Population

An infinite population is a population which is infinite.

Also known as

The population (of a particular study) can also be referred to as the universe.

See universal set.


text:
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## Swipe to navigate through the articles of this issue

01-06-2015 | Issue 3/2015

# The largest Erdős–Ko–Rado sets in $$2-(v,k,1)$$ designs

Journal:
Designs, Codes and Cryptography > Issue 3/2015
Author:
Maarten De Boeck
Important notes
Communicated by L. Teirlinck.

## Abstract

An Erdős–Ko–Rado set in a block design is a set of pairwise intersecting blocks. In this article we study Erdős–Ko–Rado sets in $$2\,-\,(v,k,1)$$ designs, Steiner systems. The Steiner triple systems and other special classes are treated separately. For $$k\ge 4$$, we prove that the largest Erdős–Ko–Rado sets cannot be larger than a point-pencil if $$r\ge k^{2}-3k+\frac{3}{4}\sqrt{k}+2$$ and that the largest Erdős–Ko–Rado sets are point-pencils if also $$r\ne k^{2}-k+1$$ and $$(r,k)\ne (8,4)$$. For unitals we also determine an upper bound on the size of the second-largest maximal Erdős–Ko–Rado sets.