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In the context of an A* search, a heuristic function is said to be admissible if it does not overestimate the cost to reach the goal. Such functions can also be viewed as being "optimistic".

When using an admissible heuristic, A* is guaranteed to return a cost-optimal solution, i.e. the best path. Let's prove it by contradiction:

Assume that the algorithm returned a path, the cost of which is greater than that of the optimal path $P$. Let's call $C$ the cost of the path that was followed, and $C^{\ast }$ the cost of $P$, noting that $C^{\ast }<C$. First, we can safely assume that at least one node in $P$ was not expanded during the algorithm's execution (if all nodes of $P$ were expanded, then $P$ would have been chosen instead since that would lead to a lower path cost${}^{[1]}$). Without loss of generallity, let's take the first occurance of such unexpanded node and name it "n". Let's call the actual cost from n to the destination $h^{\ast }(n)$ and define $g^{\ast }(n)$ as the cost of the optimal path starting from the origin all the way to n. Remember that A* is a best-first search on the value of $f$, so $f(x)\ge C$ for all unexpanded nodes. Here's what we've got:

Now note that $g^{\ast }(n)+h^{\ast }(n)$ is equal to "the cost to reach n following the optimal path" + "the cost to reach the goal starting from n, following the optimal path". That's just equal to the total cost of the optimal path! So, both $f(n)>C^{\ast }$ and $f(n)\le C^{\ast }$ hold simultaneously which obviously constitutes a contradiction. $\square$

[1]: Remember than $h(n)$ (our heuristic) is just a hint to prioritize certain expansions over others. When everything is expanded however, $g(n)$ is the sole metric that will be considered, which will always lead to the optimal path being selected, that being $P$.

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When we boil rice for example, we observe the grains swelling and becoming much more soft. This manifests itself in the process of starch gelatinization.

At high temperatures, the intermolecular bonds of the starch molecules (for example the double helices formed by amylopectin) are broken down (new water-starch hydrogen bonds are formed) and thus the structure of the starch granules is altered. First the amorphous regions are disrupted and then the granule's whole structure gets effected. The granules lose their integrity and burst. Amylose (and a smaller amount of amylopectin) molecules leave the granule and contribute to the increased viscosity of the liquid.

When the temperature drops, recrystallization occurs. This is referred to as starch retrogradation and is responsible for bread staling.

Glass Chemistry by Werner Vogel (1994)

The Sun's rays are not monochromatic and represent photons of various wavelength that may or may not be part of the visible light. In glass, some ions absorb particular wavelengths in the visible light and emit only the remaining wavelength. This is what contitutes the colour of an object. If an ion absorbs photons of red light, it appears bluish. These ions are normally transition metals as they can absorb photons of wavelengths that we can see.

The colour that we perceive by the glass is not only related to the ions contained in the glass but it also depends on the network-former/modifiers present. This is because the ions interact with them, which influences each ion's orbital energy levels and consequently the light that they can absorb. Lastly the colour also depends on the concentration of the ion, its nature (valency) and the heterogeneity of the glass. A portion of the light that arrives in the surface of the glass gets reflected and doesn't interact further.

A list summarizing some generally perceived colours and their associated ions:

Glasses can also be coloured by metal colloids or to coloured particles.

This type of glass accounts for 90% of commercially used glass. Its uses range from windows to even jars. It is typically made of (w/w) 70% silica ($Si{O}_2$), 10% lime ($CaO$) and 15% soda ($N{a}_{2}O$).

Silica is the network-former of the glass. Silica, is found in nature as quartz and its pure form is the ideal glass. Nevertheless, soda is added in order to reduce the temperature at which the glass is softened, thus making its production cheaper and better to work with. Since sodium cations are very soluble in water, lime is added in order to decrease the solubility of the glass.

Silica is obtained by sand and mixed with soda ash ($N{a}_{2}C{O}_3$) and limestone ($CaC{O}_3$). These materials are mixed together to get a powder called batch which is later combined with cullet (recycled glass pieces) in order for the glass to once again soften at even lower temperatures. The mix is heated into an oven at temperatures around $1000^{o}C$. Impurities that arise are removed and the melted mass is put into moulds to take its final shape. Annealing is also taking place at the end to ensure glass durability, which is the process of reheated the bottle and then slowly cooling it down in order to further stabilize the structure of the glass.

Soda lime glass is recyclable.

The works of Zachariasen and Warren (Network Theory and its verification by X-ray diffraction) resulted in big advancements in the study of glass structure. This theory can predict a large number properties of conventional glasses. His theory was later extended by Dietzel.

Zachariasen presented three types of cations that form a glass:

In order for glass to form, the network-former should (although there are exceptions) form polyhedral groups in its simplest form. (For example silica, $Si{O}_2$ forms $Si{O}_4^{4-}$ tetrahedrals). Each polyhedron should only connect with its neighbouring once via bonding bridges formed by the anions (in this example an Si-O-Si).

The network-former of silicate glass is $Si{O}_2$ which is

A common misconsception suggests that glass is a liquid of high viscosity. This not the case. Glass is its own distinct state of matter that doesn't coincide with any other classical one. Every liquid (except Helium) can be turned into glass, if a sufficiently rapid cooling takes place. This process is called vitrification. When a liquid is cooled (water for example) it normally goes through the process of crystallization. We say that the water is frozen as ice forms which has a very specific structure that is characterized by its stability (crystalline solid). If the cooling happens quickly enough, the water molecules don't have the opportunity to occupy the lowest energy sites and this is how the amorphous solids forms.

Annealing illustrates this phenomenon. When glass is cooled down in order to solidify, this process must be done during a specific time interval in order for the molecules to have enough time to position themselves in a more stable manner. If during the glass making process, the produced glass has been solidified too rapidly, then the stress present in the solid makes it too fragile to the point where it can rupture/shatter even during handling. By reheating the glass and slowly droping the temperature again, we ensure that the glass object has gained a much more stable structure.

As crystalline solids have a melting point, amorphous ones have a glass transition temperature which surprisingly depends on thermal history (how rapidly was the former liquid made into glass?). Around this temperature point, the viscosity of the glass increases rapidly and can be classified as a solid (under classical terms). It should be noted that the viscosity as well as other properties of the substance made into glass, present a continuous change as the temperature changes. This is not the case with "ordinary" freezing as liquid water turns spontaneously into a solid in a discontinuous manner (during the freezing process the temperature stays constant. Immediately below $0^{o}C$ all the water has turned into ice).

The glass state is metastable and its transition to a crystalline solid is thermodynamically favoured although kinetically inert.

The most commonly used component of glass is silica, an element that is abundant in the Earth's crust (30% by mass). During the formation of our planet, rocks containing a high percentage of silica were melted and cooled rapidly under the hostile conditions of the environment (volcanic eruptions, lighting strikes), thus natural glass was made such as obsidian. Man-made glass was first produced about 4000 years ago and it was used mainly for jewellery. Then, glass couldn't be obtained in a pure/homogenous form and lacked transparency.

Glass continued to spread around the world and new techniques were invented but still, transparent glass could not be achieved until the middle ages.

This important step took place in the island of Murano. Cristallo was the first clear, colourless, transparent glass and was made by the Venetian glass blower Cesaro Barovier in the 15th century.

Another marking point in the glassmaking industry was the invention of crown glass, which was a simpler process that prevailed in the middle ages. The glass that was manufactured by this method is sometimes thicker in the bottom than at the top. That is not attributed to the viscosity of the glass (a common misconception) but rather to the glass making process itself. It was slowly replaced in the 19th century by cylinder blown sheet glass, another hand-blown technique.

It took many years for man to study the relation of glass composition and properties. The industry only saw very steep scientific progress in the 19th century.

After we became aware of the underlying science of the craft, the glass making process was automated and improved to accomondate all our needs. The mass production of glass objects is done with the help of machines that create the desired shapes of the products using moulds.

White is often seen as a symbol of purity and virtue (even the ancient Egyptians and Greeks shared this view ), thus there was a need for effective fabric whitening methods.

Since the ancient times, people used a widely available "bleaching product", the sun itself (the small amounts of UV radiation that reach the surface of the earth, break up bonds of stain molecules that are responsible for the colours that we perceive).

Later in ancient rome, before the invention of soap, other alkaline products were used as cleaning agents, such as urine (which breaks down to ammonia). These only removed the grease stains from the cloth, before it was left to dry and whiten in the sun.

Ultimately, from wood ashes a better cleaning product was made which was lye. Lye is used as the main ingredient is soap making.
In the 17th century, the Dutch were leading the bleaching market. They invented a new method of bleaching which involved additions of bases and acid to the cloth. Firstly, the fabrics were soaked in lye in order to remove grease stains and then washed clean. Then they were spread out in specific places, called bleachfields. There, on the grass, they got whitened under the sun. This process was repeated several times. Lastly, sour milk was placed on the fabric in order to neutralize the bases added prior and once again a wash took place. This process was not only physically demanding, but also took months to finish.

Thankfully, after the discovery of elemental Chlorine by Karl Wilhelm Scheele, another scientist came up with a faster way to bleach clothes. He was Claude Berthollet, who made in 1789, what is known as "Eau de Javel" (commonly known as liquid bleach), which is a widely used bleach in laundry, cleaning and water treatment. That bleach consists of a solution of 3-6% NaClO.

Later, in 1798 Charles Tennant made a solution of $Ca(OCl{)}_2$ known as bleaching powder. Peroxide-based bleach only became popular in the 20th century, after being made in a century ago.

Glass is a state of matter, characterizing amorphous solids. The structure of glass neither exactly resembles that of a crystalline solid nor that of a liquid.

The properties of glass vary depending on its composition (over 400.000 types of glasses have been manufactuered) and thus it has big range of uses.

As starch is just a chain of glucose, $C_6{H}_{12}{0}_6$, molecules (and is broken down by the enzyme amylase found in our saliva) it can provide an organism with its main source of energy through aerobic or anaerobic respitation. This process uses glycose with (aerobic) or without oxygen (anaerobic) to create ATP, the energy "currency" of organisms.