5: Advanced Theories of Covalent Bonding
We have examined the basic ideas of bonding, showing that atoms share electrons to form molecules with stable Lewis structures and that we can predict the shapes of those molecules by valence shell electron pair repulsion (VSEPR) theory. These ideas provide an important starting point for understanding chemical bonding. But these models sometimes fall short in their abilities to predict the behavior of real substances. How can we reconcile the geometries of s, p, and d atomic orbitals with molecular shapes that show angles like 120° and 109.5°? Furthermore, we know that electrons and magnetic behavior are related through electromagnetic fields.
Contributors and Attributions
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Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>).


Chapter 6: Exponential and Logarithmic Functions
In this chapter, we will explore exponential functions, which can be used for, among other things, modeling growth patterns such as those found in bacteria. We will also investigate logarithmic functions, which are closely related to exponential functions. Both types of functions have numerous real-world applications when it comes to modeling and interpreting data.
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- 6.1: Exponential Functions
- When populations grow rapidly, we often say that the growth is “exponential,” meaning that something is growing very rapidly. To a mathematician, however, the term exponential growth has a very specific meaning. In this section, we will take a look at exponential functions, which model this kind of rapid growth.
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- 6.2: Graphs of Exponential Functions
- Working with an equation that describes a real-world situation gives us a method for making predictions. Most of the time, however, the equation itself is not enough. We learn a lot about things by seeing their pictorial representations, and that is exactly why graphing exponential equations is a powerful tool. It gives us another layer of insight for predicting future events.
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- 6.6: Exponential and Logarithmic Equations
- Uncontrolled population growth can be modeled with exponential functions. Equations resulting from those exponential functions can be solved to analyze and make predictions about exponential growth. In this section, we will learn techniques for solving exponential functions.
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- 6.7: Applications of Exponential Functions
- We will concentrate on two basic types of models in this section: exponential growth and exponential decay. Having already worked with each of these functions gives us an advantage. Knowing their formal definitions, the behavior of their graphs, and some of their real-world applications gives us the opportunity to deepen our understanding. As each model is presented, key features and definitions of its associated function are included for review.
Thumbnail: The functions \(y=e^x\) and \(y=\ln(x)\) are inverses of each other, so their graphs are symmetric about the line \(y=x\).
Contributors
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Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/precalculus .


13.20: Product Layout
- Explain how a retailer classifies its products into layout groupings
An excellent cut of bacon and some fresh eggs off the farm . . . each of those is an example of good grocery store merchandise. Displaying some of that bacon near the eggs in the refrigerator section of the grocery store . . . that’s an example of good grocery store merchandising.
You can have all kinds of great product in your store, but if you don’t organize it logically and attractively, it won’t sell.
Retailers use layout groupings to help the shopper find the product he wants – and maybe a few products he doesn’t know he wants. You don’t see Lowe’s or Home Depot selling paint on one side of the store, and then selling paint brushes on the other side of the store. That would confuse the customer and waste his time. Good merchandising means displaying your paint near your paint brushes, rollers, tape and trays. The customer might come in for paint, and then decide to pick up a new brush while he’s there. He might have forgotten the tape if he hadn’t seen it. And he has a paint tray, but that new one looks pretty slick—so he picks that up, too.
Before we take a look at some common merchandise groupings, let’s take a moment to learn the three rules of visual merchandising
- Make merchandise visible—what isn’t seen, isn’t purchased.
- Make merchandise tangible and accessible—customers want to see and touch before they purchase.
- Give customers good choices—but not too many choices. Shoppers purchase more when they have fewer products to choose from but a nice selection.
Considering these rules, we’ll look at a few different ways merchandise can be displayed to its best advantage, giving consideration to those three rules above and helping increase sales conversion.
Bundled Grouping
Bundled presentations of merchandise allow for a group of like items to be presented together, because they can be purchased together. IKEA does this magnificently.  When a customer walks through, they can see all the products working together.
This is how it works: You, the customer, come upon the display. Is that room about the size of your room? It is! Would that coffee table work well in your living room, given it’s the right size for the room and the right color? Why, yes! And you weren’t looking for end tables, but because you see them there, it seems like a good idea to pick them up while you’re there.
Bundled presentations don’t always come in the form of fully furnished rooms. You also see them at the grocery store, when they’re showing off their wine in a basket with bread and cheese. You’re not buying that basket, of course, but you’re picking those items off a shelf nearby.
Complementary groupings
Complementary groupings (often referred to as “cross-merchandising”) are similar to bundled presentations in that they sell different items that go together. But they’re sold adjacent to one another, not necessarily as part of a separate display. They can be grouped right on the shelves.
This is how it works: You, the shopper, head into the grocery store to grab some pancake mix. While you’re picking it off the shelf, you see the bottles of Vermont maple syrup placed right next to it. You pick up one of those, too.
The eggs and bacon, and the painting equipment, are examples of complementary groupings. Birthday cards and wrapping paper, mobile phones and chargers, flashlights and batteries, all of them are complementary groupings. Without the merchandising grouping, one might run in and buy the birthday card and forget the wrapping paper, or buy a new mobile phone and pick up a charger for the car, even though it wasn’t the shopper’s original intention.
Prop groupings
Mannequins are perhaps the most commonly used “prop” in merchandising. Outfits are created on the mannequin and then tables or racks of that merchandise are grouped around it so shoppers can buy what they see. Mannequins can attract shoppers from a distance away, standing like a beacon over other fixtures in the store. Used wisely, they’ll pull shoppers into areas they might otherwise skip in the normal flow of traffic around the store.
This is how it works: You, the shopper, are headed through the department store to purchase a new pair of running shoes. On your way, you see a mannequin wearing a pair of fabulous biking pants. You stop and check out the displays around the mannequin and find your size.  It goes home with you as well.
Mannequins aren’t the only kind of props to be found in a store. Wine stores frequently use barrels to display their merchandise. Natural baskets might be used to display vegetables in a grocery store. An inflatable palm tree might be used to draw attention to a table of suntan lotions in the middle of a northern winter.
Groupings of products positively impacts sales conversion. Putting like products together helps shoppers buy everything they need—it takes the hassle out of their shopping experience and tempts them with items they weren’t necessarily looking for. Bundling them and using props helps the shopper see the product “in action.” They can envision the product on their person, in their home, and see how good it’s going to be.
Contributors and Attributions
- Product Layout. Authored by : Freedom Learning Group. Provided by : Lumen Learning. License : CC BY: Attribution


4.2: Minerals
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The technical definition of a mineral is elegant: a naturally occurring crystalline solid . That definition calls for a bit of commentary, though. By “crystalline” is meant that the atoms of the mineral are arranged in a regular three-dimensional array, called its crystal structure . The qualification “naturally occurring” excludes the multitude of crystalline solids that can be synthesized in the laboratory but are not found in the natural environment. Also, there are naturally occurring solids, like glass or amber, that do not have a crystal structure and therefore technically are not minerals.
There are thousands of named minerals—but, fortunately for us in this course, only a few are common rock-forming minerals, and even fewer figure prominently in the unconsolidated material that mantles the continents.
Most of the common minerals are of a class called silicate minerals . Silicate minerals have as their basic building blocks a polyatomic (five-atom) unit, called a silica tetrahedron , that consists of one atom of silicon, relatively small, surrounded by four atoms of oxygen, relatively large, to give the shape of a tetrahedron (Figure 2-3). The five atoms are bonded very strongly together. The reason why silicate minerals are so common is that, in terms of abundances of the chemical elements in the crust, oxygen is the big number one and silicon is second.
The atoms in mineral crystals are held together by strong forces called bonds . An understanding of the nature of bonds necessitates some familiarity with atomic structure. All atoms consist of a positively charged nucleus around which a number of negatively charged electrons are in orbit. The fundamental nature of the electron orbits is complex, but for the purposes of understanding bonds in minerals it’s sufficient to know that electron orbits exist in distinctive groups known as shells .
Chemical elements like helium, neon, or argon whose atoms have just enough electrons to fill all of the shells are chemically almost entirely inert. Elements whose atoms have their outermost electron shell almost filled have a strong tendency to take on one or a few electrons to fill the shell, and in so doing they acquire a negative electric charge. Similarly, elements whose atoms have only one or a few electrons in their outermost shell lose those electrons to revert to a filled-shell configuration, and in so doing they acquire a positive electric charge. Such atoms with electric charges are called ions .
According to what is known in physics as Coulomb’s law, unlike electric charges attract one another and like charges repel one another. Certain combinations of positively charged and negatively charged ions can become packed together in a regular three-dimensional array in such a way that the sum of all the forces, attractive and repulsive, among all of the ions in the structure is attractive, meaning that the structure is a stable one and can exist as a mineral. Almost all minerals are of this nature, and are called ionic crystals ; the bonds in such a crystal are called ionic bonds . Ions can also be formed from two or more atoms that share their electrons in such a way that each atoms effectively has a filled-shell configuration. The bonds among such atoms are called covalent bonds . Covalent bonds are much stronger than ionic bonds, and the covalently bonded atoms form a single ion that is available for ionic bonding with other ions.
It turns out that in terms of electric charge the silica tetrahedron has a net negative charge. The silica tetrahedra are bonded to various positively charged atoms, mainly of the elements aluminum, iron, magnesium, calcium, potassium, and sodium. These ionic bonds are weaker than the strong covalent bonds within the silica tetrahedra. When a mineral is broken, it breaks along these ionic bonds rather than through the bonds within the silica tetrahedra.
There is an important further complication to silicate minerals, which makes them highly diverse (and almost unique in the universe!): individual tetrahedra can be joined together by sharing one or more (two, three, or even all four) of their oxygens. The phenomenon is called polymerization . By polymerization, zillions of tetrahedra can be joined into chains or sheets that stretch all the way through a macroscopic mineral grain. In some minerals, all of the oxygens of the tetrahedra are shared, resulting in a complex three-dimensional network of shared tetrahedra.
There are other important minerals besides silicate minerals. Oxide minerals are perhaps the most important kind, in the context of this course. Oxide minerals are those in which negatively charged oxygen atoms are bonded with various positively charged ions. The most important oxides are iron oxides and aluminum oxides . These oxides of iron and aluminum are generally unspectacular in their appearance, but they are voluminous and important in Earth-surface environments. Carbonate minerals are also important near-surface non-silicate minerals. In carbonates, a negatively charged carbonate ion, consisting of one carbon atom covalently bonded with three oxygen atoms, is ionically bonded with calcium, magnesium, and/or iron ions; calcite and dolomite are common carbonate minerals.
Conclusion
To conclude this section, I’ll list just the very most important kinds of minerals in the Earth’s upper crust. It’s important to know something about these minerals, because the mineralogy of solid Earth materials is of great importance in determining how those Earth materials are affected by their physical and chemical environment.
- ferromagnesian silicates (“femags”): This collection of minerals (olivines, pyroxenes, amphiboles, biotite, and certain metamorphic minerals) contain iron and/or magnesium, in various proportions, as their main positive ions. They tend to be dark in color. They are abundant in igneous and metamorphic rocks—but not in sedimentary rocks, because they tend to weather rapidly.
- micas: Micas consist of sheets of polymerized silica tetrahedrons, each with three of their four oxygens shared, with various positive ions between the sheets. The main micas are muscovite (a potassium mica), which is common in igneous, metamorphic, and sedimentary rocks, and biotite (a ferromagnesian mineral, with iron and magnesium rather than potassium). Biotite is common in both metamorphic and igneous rocks, but it weathers rapidly, so it’s uncommon in sediments and sedimentary rocks.
- feldspars: Feldspars are the most abundant kind of mineral in the Earth’s crust. They are network silicates with either potassium (these are called potassium feldspar, or just Kspar for short) or varying proportions of sodium and calcium (these are called plagioclase) in their structure. Feldspars are the major minerals of igneous rocks and are common in many metamorphic rocks. Potassium feldspar is common in sediments and sedimentary rocks, but plagioclase is not, because it weathers readily.
- quartz: Quartz, with the simple formula SiO2, is a network silicate mineral with a three-dimensional network of silica tetrahedra, with all of the oxygens shared with adjacent tetrahedra. It is common but not abundant in igneous rocks, is variably abundant in metamorphic rocks, and is very abundant in sediments and sedimentary rocks.
- iron oxides: Iron oxides, of which there are several forms, are common only in sediments and sedimentary rocks. In soils, there are several poorly crystallized hydrous iron oxide minerals.
- aluminum oxides: In poorly crystallized form, these are characteristic of soils formed by deep weathering of aluminum-bearing rocks under intense weathering conditions.
- carbonates: The calcium carbonate minerals calcite and aragonite, along with the calcium–magnesium mineral dolomite, are the main minerals in chemically precipitated carbonate sedimentary rocks, and they are common as well in sediments in the warm and shallow oceans.
- clay minerals: Several minerals with sheet-silicate structures are common as very fine-grained plate-shaped particles, some even smaller than a micrometer (one-thousandth of a millimeter; too small even to be seen with a powerful optical microscope). Clay minerals are the main constituent of most fine-grained muds.


1.14: Reading- Scientific Models
Scientists use models to help them understand and explain ideas. Models explain objects or systems in a more simple way. Models often only show only a part of a system. The real situation is more complicated. Models help scientists to make predictions about complex systems. Some models are something that you can see or touch. Other types of models use an idea or numbers. Each type is useful in certain ways.
Scientists create models with computers. Computers can handle enormous amounts of data. This can more accurately represent the real situation. For example, Earth’s climate depends on an enormous number of factors. Climate models can predict how climate will change as certain gases are added to the atmosphere. To test how good a model is, scientists might start a test run at a time in the past. If the model can predict the present it is probably a good model. It is more likely to be accurate when predicting the future.
Physical Models
A physical model is a representation of something using objects. It can be three-dimensional, like a globe. It can also be a two-dimensional drawing or diagram. Models are usually smaller and simpler than the real object. They most likely leave out some parts, but contain the important parts. In a good model the parts are made or drawn to scale. Physical models allow us to see, feel and move their parts. This allows us to better understand the real system.
An example of a physical model is a drawing of the layers of Earth (figure 1). A drawing helps us to understand the structure of the planet. Yet there are many differences between a drawing and the real thing. The size of a model is much smaller, for example. A drawing also doesn’t give good idea of how substances move. Arrows showing the direction the material moves can help. A physical model is very useful but it can’t explain the real Earth perfectly.
Ideas as Models
Some models are based on an idea that helps scientists explain something. A good idea explains all the known facts. An example is how Earth got its Moon. A Mars-sized planet hit Earth and rocky material broke off of both bodies (figure 2). This material orbited Earth and then came together to form the Moon. This is a model of something that happened billions of years ago. It brings together many facts known from our studies of the Moon’s surface. It accounts for the chemical makeup of rocks from the Moon, Earth, and meteorites. The physical properties of Earth and Moon figure in as well. Not all known data fits this model, but much does. There is also more information that we simply don’t yet know.
Models that Use Numbers
Models may use formulas or equations to describe something. Sometimes math may be the only way to describe it. For example, equations help scientists to explain what happened in the early days of the universe. The universe formed so long ago that math is the only way to describe it. A climate model includes lots of numbers, including temperature readings, ice density, snowfall levels, and humidity. These numbers are put into equations to make a model. The results are used to predict future climate. For example, if there are more clouds, does global temperature go up or down? Models are not perfect because they are simple versions of the real situation. Even so, these models are very useful to scientists. These days, models of complex things are made on computers.
Contributors and Attributions
- 1.1: The Nature of Science. Provided by : CK-12. Located at : http://www.ck12.org/book/CK-12-Earth-Science-For-Middle-School/section/1.1/ . License : CC BY-NC: Attribution-NonCommercial


46.1: Ecology of Ecosystems
Skills to Develop
- Describe the basic types of ecosystems on Earth
- Explain the methods that ecologists use to study ecosystem structure and dynamics
- Identify the different methods of ecosystem modeling
- Differentiate between food chains and food webs and recognize the importance of each
Life in an ecosystem is often about competition for limited resources, a characteristic of the theory of natural selection. Competition in communities (all living things within specific habitats) is observed both within species and among different species. The resources for which organisms compete include organic material from living or previously living organisms, sunlight, and mineral nutrients, which provide the energy for living processes and the matter to make up organisms’ physical structures. Other critical factors influencing community dynamics are the components of its physical and geographic environment: a habitat’s latitude, amount of rainfall, topography (elevation), and available species. These are all important environmental variables that determine which organisms can exist within a particular area.
An ecosystem is a community of living organisms and their interactions with their abiotic (non-living) environment. Ecosystems can be small, such as the tide pools found near the rocky shores of many oceans, or large, such as the Amazon Rainforest in Brazil (Figure \(\PageIndex{1}\)).
There are three broad categories of ecosystems based on their general environment: freshwater, ocean water, and terrestrial. Within these broad categories are individual ecosystem types based on the organisms present and the type of environmental habitat.
Ocean ecosystems are the most common, comprising 75 percent of the Earth's surface and consisting of three basic types: shallow ocean, deep ocean water, and deep ocean surfaces (the low depth areas of the deep oceans). The shallow ocean ecosystems include extremely biodiverse coral reef ecosystems, and the deep ocean surface is known for its large numbers of plankton and krill (small crustaceans) that support it. These two environments are especially important to aerobic respirators worldwide as the phytoplankton perform 40 percent of all photosynthesis on Earth. Although not as diverse as the other two, deep ocean ecosystems contain a wide variety of marine organisms. Such ecosystems exist even at the bottom of the ocean where light is unable to penetrate through the water.
Freshwater ecosystems are the rarest, occurring on only 1.8 percent of the Earth's surface. Lakes, rivers, streams, and springs comprise these systems; they are quite diverse, and they support a variety of fish, amphibians, reptiles, insects, phytoplankton, fungi, and bacteria.
Terrestrial ecosystems, also known for their diversity, are grouped into large categories called biomes, such as tropical rain forests, savannas, deserts, coniferous forests, deciduous forests, and tundra. Grouping these ecosystems into just a few biome categories obscures the great diversity of the individual ecosystems within them. For example, there is great variation in desert vegetation: the saguaro cacti and other plant life in the Sonoran Desert, in the United States, are relatively abundant compared to the desolate rocky desert of Boa Vista, an island off the coast of Western Africa (Figure \(\PageIndex{2}\)).
Ecosystems are complex with many interacting parts. They are routinely exposed to various disturbances, or changes in the environment that effect their compositions: yearly variations in rainfall and temperature and the slower processes of plant growth, which may take several years. Many of these disturbances are a result of natural processes. For example, when lightning causes a forest fire and destroys part of a forest ecosystem, the ground is eventually populated by grasses, then by bushes and shrubs, and later by mature trees, restoring the forest to its former state. The impact of environmental disturbances caused by human activities is as important as the changes wrought by natural processes. Human agricultural practices, air pollution, acid rain, global deforestation, overfishing, eutrophication, oil spills, and illegal dumping on land and into the ocean are all issues of concern to conservationists.
Equilibrium is the steady state of an ecosystem where all organisms are in balance with their environment and with each other. In ecology, two parameters are used to measure changes in ecosystems: resistance and resilience. The ability of an ecosystem to remain at equilibrium in spite of disturbances is called resistance . The speed at which an ecosystem recovers equilibrium after being disturbed, called its resilience . Ecosystem resistance and resilience are especially important when considering human impact. The nature of an ecosystem may change to such a degree that it can lose its resilience entirely. This process can lead to the complete destruction or irreversible altering of the ecosystem.
Food Chains and Food Webs
The term “food chain” is sometimes used metaphorically to describe human social situations. In this sense, food chains are thought of as a competition for survival, such as “who eats whom?” Someone eats and someone is eaten. Therefore, it is not surprising that in our competitive “dog-eat-dog” society, individuals who are considered successful are seen as being at the top of the food chain, consuming all others for their benefit, whereas the less successful are seen as being at the bottom.
The scientific understanding of a food chain is more precise than in its everyday usage. In ecology, a food chain is a linear sequence of organisms through which nutrients and energy pass: primary producers, primary consumers, and higher-level consumers are used to describe ecosystem structure and dynamics. There is a single path through the chain. Each organism in a food chain occupies what is called a trophic level . Depending on their role as producers or consumers, species or groups of species can be assigned to various trophic levels.
In many ecosystems, the bottom of the food chain consists of photosynthetic organisms (plants and/or phytoplankton), which are called primary producers . The organisms that consume the primary producers are herbivores: the primary consumers . Secondary consumers are usually carnivores that eat the primary consumers. Tertiary consumers are carnivores that eat other carnivores. Higher-level consumers feed on the next lower tropic levels, and so on, up to the organisms at the top of the food chain: the apex consumers . In the Lake Ontario food chain shown in Figure \(\PageIndex{3}\), the Chinook salmon is the apex consumer at the top of this food chain.
One major factor that limits the length of food chains is energy. Energy is lost as heat between each trophic level due to the second law of thermodynamics. Thus, after a limited number of trophic energy transfers, the amount of energy remaining in the food chain may not be great enough to support viable populations at yet a higher trophic level.
The loss of energy between trophic levels is illustrated by the pioneering studies of Howard T. Odum in the Silver Springs, Florida, ecosystem in the 1940s (Figure \(\PageIndex{4}\)). The primary producers generated 20,819 kcal/m 2 /yr (kilocalories per square meter per year), the primary consumers generated 3368 kcal/m 2 /yr, the secondary consumers generated 383 kcal/m 2 /yr, and the tertiary consumers only generated 21 kcal/m 2 /yr. Thus, there is little energy remaining for another level of consumers in this ecosystem.
There is a one problem when using food chains to accurately describe most ecosystems. Even when all organisms are grouped into appropriate trophic levels, some of these organisms can feed on species from more than one trophic level; likewise, some of these organisms can be eaten by species from multiple trophic levels. In other words, the linear model of ecosystems, the food chain, is not completely descriptive of ecosystem structure. A holistic model—which accounts for all the interactions between different species and their complex interconnected relationships with each other and with the environment—is a more accurate and descriptive model for ecosystems. A food web is a graphic representation of a holistic, non-linear web of primary producers, primary consumers, and higher-level consumers used to describe ecosystem structure and dynamics (Figure \(\PageIndex{5}\)).
A comparison of the two types of structural ecosystem models shows strength in both. Food chains are more flexible for analytical modeling, are easier to follow, and are easier to experiment with, whereas food web models more accurately represent ecosystem structure and dynamics, and data can be directly used as input for simulation modeling.
Link to Learning
Head to this online interactive simulator to investigate food web function. In the Interactive Labs box, under Food Web , click Step 1 . Read the instructions first, and then click Step 2 for additional instructions. When you are ready to create a simulation, in the upper-right corner of the Interactive Labs box, click OPEN SIMULATOR .
Two general types of food webs are often shown interacting within a single ecosystem. A grazing food web (such as the Lake Ontario food web in Figure \(\PageIndex{5}\)) has plants or other photosynthetic organisms at its base, followed by herbivores and various carnivores. A detrital food web consists of a base of organisms that feed on decaying organic matter (dead organisms), called decomposers or detritivores. These organisms are usually bacteria or fungi that recycle organic material back into the biotic part of the ecosystem as they themselves are consumed by other organisms. As all ecosystems require a method to recycle material from dead organisms, most grazing food webs have an associated detrital food web. For example, in a meadow ecosystem, plants may support a grazing food web of different organisms, primary and other levels of consumers, while at the same time supporting a detrital food web of bacteria, fungi, and detrivorous invertebrates feeding off dead plants and animals.
Evolution Connection: Three-spined Stickleback
It is well established by the theory of natural selection that changes in the environment play a major role in the evolution of species within an ecosystem. However, little is known about how the evolution of species within an ecosystem can alter the ecosystem environment. In 2009, Dr. Luke Harmon, from the University of Idaho in Moscow, published a paper that for the first time showed that the evolution of organisms into subspecies can have direct effects on their ecosystem environment. 1
The three-spines stickleback ( Gasterosteus aculeatus ) is a freshwater fish that evolved from a saltwater fish to live in freshwater lakes about 10,000 years ago, which is considered a recent development in evolutionary time (Figure \(\PageIndex{6}\)). Over the last 10,000 years, these freshwater fish then became isolated from each other in different lakes. Depending on which lake population was studied, findings showed that these sticklebacks then either remained as one species or evolved into two species. The divergence of species was made possible by their use of different areas of the pond for feeding called micro niches.
Dr. Harmon and his team created artificial pond microcosms in 250-gallon tanks and added muck from freshwater ponds as a source of zooplankton and other invertebrates to sustain the fish. In different experimental tanks they introduced one species of stickleback from either a single-species or double-species lake.
Over time, the team observed that some of the tanks bloomed with algae while others did not. This puzzled the scientists, and they decided to measure the water's dissolved organic carbon (DOC), which consists of mostly large molecules of decaying organic matter that give pond-water its slightly brownish color. It turned out that the water from the tanks with two-species fish contained larger particles of DOC (and hence darker water) than water with single-species fish. This increase in DOC blocked the sunlight and prevented algal blooming. Conversely, the water from the single-species tank contained smaller DOC particles, allowing more sunlight penetration to fuel the algal blooms.
This change in the environment, which is due to the different feeding habits of the stickleback species in each lake type, probably has a great impact on the survival of other species in these ecosystems, especially other photosynthetic organisms. Thus, the study shows that, at least in these ecosystems, the environment and the evolution of populations have reciprocal effects that may now be factored into simulation models.
Research into Ecosystem Dynamics: Ecosystem Experimentation and Modeling
The study of the changes in ecosystem structure caused by changes in the environment (disturbances) or by internal forces is called ecosystem dynamics . Ecosystems are characterized using a variety of research methodologies. Some ecologists study ecosystems using controlled experimental systems, while some study entire ecosystems in their natural state, and others use both approaches.
A holistic ecosystem model attempts to quantify the composition, interaction, and dynamics of entire ecosystems; it is the most representative of the ecosystem in its natural state. A food web is an example of a holistic ecosystem model. However, this type of study is limited by time and expense, as well as the fact that it is neither feasible nor ethical to do experiments on large natural ecosystems. To quantify all different species in an ecosystem and the dynamics in their habitat is difficult, especially when studying large habitats such as the Amazon Rainforest, which covers 1.4 billion acres (5.5 million km 2 ) of the Earth’s surface.
For these reasons, scientists study ecosystems under more controlled conditions. Experimental systems usually involve either partitioning a part of a natural ecosystem that can be used for experiments, termed a mesocosm , or by re-creating an ecosystem entirely in an indoor or outdoor laboratory environment, which is referred to as a microcosm . A major limitation to these approaches is that removing individual organisms from their natural ecosystem or altering a natural ecosystem through partitioning may change the dynamics of the ecosystem. These changes are often due to differences in species numbers and diversity and also to environment alterations caused by partitioning (mesocosm) or re-creating (microcosm) the natural habitat. Thus, these types of experiments are not totally predictive of changes that would occur in the ecosystem from which they were gathered.
As both of these approaches have their limitations, some ecologists suggest that results from these experimental systems should be used only in conjunction with holistic ecosystem studies to obtain the most representative data about ecosystem structure, function, and dynamics.
Scientists use the data generated by these experimental studies to develop ecosystem models that demonstrate the structure and dynamics of ecosystems. Three basic types of ecosystem modeling are routinely used in research and ecosystem management: a conceptual model, an analytical model, and a simulation model. A conceptual model is an ecosystem model that consists of flow charts to show interactions of different compartments of the living and nonliving components of the ecosystem. A conceptual model describes ecosystem structure and dynamics and shows how environmental disturbances affect the ecosystem; however, its ability to predict the effects of these disturbances is limited. Analytical and simulation models, in contrast, are mathematical methods of describing ecosystems that are indeed capable of predicting the effects of potential environmental changes without direct experimentation, although with some limitations as to accuracy. An analytical model is an ecosystem model that is created using simple mathematical formulas to predict the effects of environmental disturbances on ecosystem structure and dynamics. A simulation model is an ecosystem model that is created using complex computer algorithms to holistically model ecosystems and to predict the effects of environmental disturbances on ecosystem structure and dynamics. Ideally, these models are accurate enough to determine which components of the ecosystem are particularly sensitive to disturbances, and they can serve as a guide to ecosystem managers (such as conservation ecologists or fisheries biologists) in the practical maintenance of ecosystem health.
Conceptual Models
Conceptual models are useful for describing ecosystem structure and dynamics and for demonstrating the relationships between different organisms in a community and their environment. Conceptual models are usually depicted graphically as flow charts. The organisms and their resources are grouped into specific compartments with arrows showing the relationship and transfer of energy or nutrients between them. Thus, these diagrams are sometimes called compartment models.
To model the cycling of mineral nutrients, organic and inorganic nutrients are subdivided into those that are bioavailable (ready to be incorporated into biological macromolecules) and those that are not. For example, in a terrestrial ecosystem near a deposit of coal, carbon will be available to the plants of this ecosystem as carbon dioxide gas in a short-term period, not from the carbon-rich coal itself. However, over a longer period, microorganisms capable of digesting coal will incorporate its carbon or release it as natural gas (methane, CH 4 ), changing this unavailable organic source into an available one.  This is thought to be a major factor in the rise of the atmospheric carbon dioxide levels in the industrial age. The carbon dioxide released from burning fossil fuels is produced faster than photosynthetic organisms can use it. This process is intensified by the reduction of photosynthetic trees because of worldwide deforestation. Most scientists agree that high atmospheric carbon dioxide is a major cause of global climate change.
Conceptual models are also used to show the flow of energy through particular ecosystems. Figure \(\PageIndex{7}\) is based on Howard T. Odum’s classical study of the Silver Springs, Florida, holistic ecosystem in the mid-twentieth century. 2 This study shows the energy content and transfer between various ecosystem compartments.
Exercise
Why do you think the value for gross productivity of the primary producers is the same as the value for total heat and respiration (20,810 kcal/m 2 /yr)?
- Answer
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According to the first law of thermodynamics, energy can neither be created nor destroyed. Eventually, all energy consumed by living systems is lost as heat or used for respiration, and the total energy output of the system must equal the energy that went into it.
Analytical and Simulation Models
The major limitation of conceptual models is their inability to predict the consequences of changes in ecosystem species and/or environment. Ecosystems are dynamic entities and subject to a variety of abiotic and biotic disturbances caused by natural forces and/or human activity. Ecosystems altered from their initial equilibrium state can often recover from such disturbances and return to a state of equilibrium. As most ecosystems are subject to periodic disturbances and are often in a state of change, they are usually either moving toward or away from their equilibrium state. There are many of these equilibrium states among the various components of an ecosystem, which affects the ecosystem overall. Furthermore, as humans have the ability to greatly and rapidly alter the species content and habitat of an ecosystem, the need for predictive models that enable understanding of how ecosystems respond to these changes becomes more crucial.
Analytical models often use simple, linear components of ecosystems, such as food chains, and are known to be complex mathematically; therefore, they require a significant amount of mathematical knowledge and expertise. Although analytical models have great potential, their simplification of complex ecosystems is thought to limit their accuracy. Simulation models that use computer programs are better able to deal with the complexities of ecosystem structure.
A recent development in simulation modeling uses supercomputers to create and run individual-based simulations, which accounts for the behavior of individual organisms and their effects on the ecosystem as a whole. These simulations are considered to be the most accurate and predictive of the complex responses of ecosystems to disturbances.
Summary
Ecosystems exist on land, at sea, in the air, and underground. Different ways of modeling ecosystems are necessary to understand how environmental disturbances will affect ecosystem structure and dynamics. Conceptual models are useful to show the general relationships between organisms and the flow of materials or energy between them. Analytical models are used to describe linear food chains, and simulation models work best with holistic food webs.
Footnotes
- 1 Nature (Vol. 458, April 1, 2009)
- 2 Howard T. Odum, “Trophic Structure and Productivity of Silver Springs, Florida,” Ecological Monographs 27, no. 1 (1957): 47–112.
Glossary
- analytical model
- ecosystem model that is created with mathematical formulas to predict the effects of environmental disturbances on ecosystem structure and dynamics
- apex consumer
- organism at the top of the food chain
- conceptual model
- (also, compartment models) ecosystem model that consists of flow charts that show the interactions of different compartments of the living and non-living components of the ecosystem
- detrital food web
- type of food web in which the primary consumers consist of decomposers; these are often associated with grazing food webs within the same ecosystem
- ecosystem
- community of living organisms and their interactions with their abiotic environment
- ecosystem dynamics
- study of the changes in ecosystem structure caused by changes in the environment or internal forces
- equilibrium
- steady state of an ecosystem where all organisms are in balance with their environment and each other
- food chain
- linear representation of a chain of primary producers, primary consumers, and higher-level consumers used to describe ecosystem structure and dynamics
- food web
- graphic representation of a holistic, non-linear web of primary producers, primary consumers, and higher-level consumers used to describe ecosystem structure and dynamics
- grazing food web
- type of food web in which the primary producers are either plants on land or phytoplankton in the water; often associated with a detrital food web within the same ecosystem
- holistic ecosystem model
- study that attempts to quantify the composition, interactions, and dynamics of entire ecosystems; often limited by economic and logistical difficulties, depending on the ecosystem
- mesocosm
- portion of a natural ecosystem to be used for experiments
- microcosm
- re-creation of natural ecosystems entirely in a laboratory environment to be used for experiments
- primary consumer
- trophic level that obtains its energy from the primary producers of an ecosystem
- primary producer
- trophic level that obtains its energy from sunlight, inorganic chemicals, or dead and/or decaying organic material
- resilience (ecological)
- speed at which an ecosystem recovers equilibrium after being disturbed
- resistance (ecological)
- ability of an ecosystem to remain at equilibrium in spite of disturbances
- secondary consumer
- usually a carnivore that eat primary consumers
- simulation model
- ecosystem model that is created with computer programs to holistically model ecosystems and to predict the effects of environmental disturbances on ecosystem structure and dynamics
- tertiary consumer
- carnivore that eat other carnivores
- trophic level
- position of a species or group of species in a food chain or a food web


2.2: Concepts, Constructs, and Variables
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We discussed in Chapter 1 that although research can be exploratory, descriptive, or explanatory, most scientific research tend to be of the explanatory type in that they search for potential explanations of observed natural or social phenomena. Explanations require development of concepts or generalizable properties or characteristics associated with objects, events, or people. While objects such as a person, a firm, or a car are not concepts, their specific characteristics or behavior such as a person’s attitude toward immigrants, a firm’s capacity for innovation, and a car’s weight can be viewed as concepts.
Knowingly or unknowingly, we use different kinds of concepts in our everyday conversations. Some of these concepts have been developed over time through our shared language. Sometimes, we borrow concepts from other disciplines or languages to explain a phenomenon of interest. For instance, the idea of gravitation borrowed from physics can be used in business to describe why people tend to “gravitate” to their preferred shopping destinations. Likewise, the concept of distance can be used to explain the degree of social separation between two otherwise collocated individuals. Sometimes, we create our own concepts to describe a unique characteristic not described in prior research. For instance, technostress is a new concept referring to the mental stress one may face when asked to learn a new technology.
Concepts may also have progressive levels of abstraction. Some concepts such as a person’s weight are precise and objective, while other concepts such as a person’s personality may be more abstract and difficult to visualize. A construct is an abstract concept that is specifically chosen (or “created”) to explain a given phenomenon. A construct may be a simple concept, such as a person’s weight , or a combination of a set of related concepts such as a person’s communication skill , which may consist of several underlying concepts such as the person’s vocabulary , syntax , and spelling . The former instance (weight) is a unidimensional construct , while the latter (communication skill) is a multi-dimensional construct (i.e., it consists of multiple underlying concepts). The distinction between constructs and concepts are clearer in multi-dimensional constructs, where the higher order abstraction is called a construct and the lower order abstractions are called concepts. However, this distinction tends to blur in the case of unidimensional constructs.
Constructs used for scientific research must have precise and clear definitions that others can use to understand exactly what it means and what it does not mean. For instance, a seemingly simple construct such as income may refer to monthly or annual income, before-tax or after-tax income, and personal or family income, and is therefore neither precise nor clear. There are two types of definitions: dictionary definitions and operational definitions. In the more familiar dictionary definition, a construct is often defined in terms of a synonym. For instance, attitude may be defined as a disposition, a feeling, or an affect, and affect in turn is defined as an attitude. Such definitions of a circular nature are not particularly useful in scientific research for elaborating the meaning and content of that construct. Scientific research requires operational definitions that define constructs in terms of how they will be empirically measured. For instance, the operational definition of a construct such as temperature must specify whether we plan to measure temperature in Celsius, Fahrenheit, or Kelvin scale. A construct such as income should be defined in terms of whether we are interested in monthly or annual income, before-tax or after-tax income, and personal or family income. One can imagine that constructs such as learning , personality , and intelligence can be quite hard to define operationally.
A term frequently associated with, and sometimes used interchangeably with, a construct is a variable. Etymologically speaking, a variable is a quantity that can vary (e.g., from low to high, negative to positive, etc.), in contrast to constants that do not vary (i.e., remain constant). However, in scientific research, a variable is a measurable representation of an abstract construct. As abstract entities, constructs are not directly measurable, and hence, we look for proxy measures called variables. For instance, a person’s intelligence is often measured as his or her IQ ( intelligence quotient ) score , which is an index generated from an analytical and pattern-matching test administered to people. In this case, intelligence is a construct, and IQ score is a variable that measures the intelligence construct. Whether IQ scores truly measures one’s intelligence is anyone’s guess (though many believe that they do), and depending on whether how well it measures intelligence, the IQ score may be a good or a poor measure of the intelligence construct. As shown in Figure 2.1, scientific research proceeds along two planes: a theoretical plane and an empirical plane. Constructs are conceptualized at the theoretical (abstract) plane, while variables are operationalized and measured at the empirical (observational) plane. Thinking like a researcher implies the ability to move back and forth between these two planes.
Depending on their intended use, variables may be classified as independent, dependent, moderating, mediating, or control variables. Variables that explain other variables are called independent variables , those that are explained by other variables are dependent variables , those that are explained by independent variables while also explaining dependent variables are mediating variables (or intermediate variables), and those that influence the relationship between independent and dependent variables are called moderating variables . As an example, if we state that higher intelligence causes improved learning among students, then intelligence is an independent variable and learning is a dependent variable. There may be other extraneous variables that are not pertinent to explaining a given dependent variable, but may have some impact on the dependent variable. These variables must be controlled for in a scientific study, and are therefore called control variables .
To understand the differences between these different variable types, consider the example shown in Figure 2.2. If we believe that intelligence influences (or explains) students’ academic achievement, then a measure of intelligence such as an IQ score is an independent variable, while a measure of academic success such as grade point average is a dependent variable. If we believe that the effect of intelligence on academic achievement also depends on the effort invested by the student in the learning process (i.e., between two equally intelligent students, the student who puts is more effort achieves higher academic achievement than one who puts in less effort), then effort becomes a moderating variable. Incidentally, one may also view effort as an independent variable and intelligence as a moderating variable. If academic achievement is viewed as an intermediate step to higher earning potential, then earning potential becomes the dependent variable for the independent variable academic achievement , and academic achievement becomes the mediating variable in the relationship between intelligence and earning potential. Hence, variable are defined as an independent, dependent, moderating, or mediating variable based on their nature of association with each other. The overall network of relationships between a set of related constructs is called a nomological network (see Figure 2.2). Thinking like a researcher requires not only being able to abstract constructs from observations, but also being able to mentally visualize a nomological network linking these abstract constructs.


15.15: Assignment- Create Mail Merge Letters
For this assignment, you will create a Word document that is mail merged with an Access customer contact table. Mail Merge can be used to create bulk labels, letters, envelopes, and emails. You will be creating letters to send with the newsletter you previously edited. Follow the directions below, then submit your assignment. If you get stuck on a step, review this module and ask your classmates for help in the discussion forum.
You have been tasked with mailing the newsletter you previously created to customers of Rowan Retail. In order to accomplish this task, you need to merge the customer contact information in the Access file with a Word document in order to make mailing labels. Follow these steps to accomplish the task.
- The first step is to download the customer contact Access file and save it to the Rowan folder on your desktop.
- Open Word and create a new document by clicking on the Blank document option in the window.
- Save the document by using Save As and rename the document to BA132_LastName_MailMerge.docx , replacing “LastName” with your own last name. (Example: BA132_Hywater_MailMerge)
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Begin Mail Merge: Select the Mailings tab and use the Step-by-Step Mail Merge Wizard for this mail merge letter.
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Wizard: Walk through the wizard and click the Next: Starting document link leaving the selection of document type to ‘Letters’.
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Create letter: Leave the document type to ‘
Use the current document
‘ and click the
Next: Select recipients
link.
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Contact List: You already have a file with a contact list. Leave the selection ‘Use an existing list’ selected and click on the Browse link.
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Insert File: Find where you saved the Rowan Retail Customer Contact Info.accdb file and use this for the mail merge contacts file.
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Prepare contact file: Find Richard Castle and Britney Spears and uncheck their boxes. Go to the Next: Write your letter link step.
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Compose the letter: Decide where to place your cursor in the document for adding in an address and use the Address Block option from the wizard or the ribbon button, to select the naming style.
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Decide Greeting: Decide where you would like the greeting to be in the letter. Select the
Greeting line
from the wizard or the ribbon button and decide the style of greeting.
- Paste the following text to serve as the body of your letter. Enclosed is the quarterly newsletter for Rowan Retail. Please enjoy the information therein and pay special attention to the dates for the special sales this summer. If you have any questions or concerns, please contact us as soon as possible.
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Finish Letter: Close the letter with ‘Sincerely’ and then your name. Save your work and click Next: Preview your letters.
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Check Letters: Scroll through the letters by clicking on the arrows. Britney Spears and Richard Castle should be missing in the recipients.
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Complete the Merge: Once the merge is completed, click on
Edit individual letters…
if you need to make any changes to just a few letters.
- For this assignment don’t make any edits to individual letters and continue to the final step.
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Optional Task: To double check the letters you can print them out. Make sure ‘All’ records are selected to make sure all print or select specific ones if you want to print a few.
- Your merge mail file is now complete! Save your file once more, this time as a PDF (use the .pdf file extension), and submit the PDF in your course online.
Contributors and Attributions
- Assignment: Use Mail Merge. Authored by : Sherri Pendleton. Provided by : Lumen Learning. License : CC BY: Attribution


17.6: Standards Regulatory Bodies
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“Standards are industry’s way of codifying obsolescence”. – Anonymous
Technology changes rapidly. Accordingly, the development of standards for e-learning is like a moving target. Many institutions and organizations first laid claim to “the standard” for online content and delivery. Several organizations have gained prominence in developing e-learning standards including:
- Aviation Industry CBT Committee (AICC)
- Sharable Courseware Object Reference Model (SCORM)
- IMS Global Learning Consortium (IMS)
- Institute of Electrical and Electronic Engineers Learning Technology Standards Committee (IEEE— LTSC)
- Canadian Core Learning Resource Metadata Application Profile (CanCore).
While compliance to standards and membership in any organization is voluntary, most major content developers and technology providers conform to some or all of the standards recommended by these organizations. In many cases regulatory bodies reference a set or sub-set of each other’s standards. Others list only specifications and guidelines rather than standards, as the development and/or adoption of what will become a standard will continue. The following provides a brief background on each organization. The References section at the end of the chapter lists additional organizations and websites that may be of interest. AVIATION
Aviation Industry CBT Committee (AICC)
The Aviation Industry CBT Committee (AICC) is an international association of technology-based training professionals. The AICC develops guidelines for the aviation industry in the development, delivery, and evaluation of CBT and related training technologies. The AICC has developed methods that allow learning management systems to exchange information and track the results of contents.
Although AICC primarily attends to the aviation industry, their focus has led to very well developed specifications for learning, and particularly for computer-managed instruction. As a result, a wide range of learning consortiums and accredited standards groups adapt the AICC guidelines to their suit their own industries. The main link for the AICC is http://www.aicc.org /index.html.
Shareable Courseware Object Reference Model (SCORM)
The Department of Defense and the White House Office of Science and Technology Policy launched the Advanced Distributed Learning (ADL) initiative in 1997 to develop an open architecture for online learning. Its purpose was to support access to quality education and training resources tailored to individual learner needs and available as required.
The ADL Shareable Courseware Object Reference Model (SCORM) specification provides a common technical framework for computer and web-based learning that attempts to foster the creation of reusable learning content as "instructional objects". SCORM is based on AICC and the IMS Global Learning Consortium specifications. The ADL provides interoperability testing laboratories and intends to establish a certification program. The main website for SCORM is: http://www.adlnet.org/ .
IMS Global Learning Consortium
The IMS Global Learning Consortium represents a number of large and small educational institutions, training organizations, government and software vendors interested in incorporating learning resource metadata into their software products. IMS is developing and promoting open specifications for facilitating online distributed learning activities such as locating and using educational content, tracking learner progress, reporting learner performance, and exchanging learner records between administrative systems. The IMS Project is funded solely by membership (the highest level of participation is the contributing member, with an annual fee of $50,000). The main link for IMS is www.ims project.org/.
Institute of Electrical and Electronic Engineers Learning Technology Standards Committee (IEEE-LTSC)
The Learning Technology Standards Committee (LTSC), part of IEEE, is a formal standards body that produces standards with legal standing. The formal standardization process is generally based on existing process; in the case of the LTSC, the other organizations listed here provide input. The LTSC itself comprises several working groups that are developing technical standards, recommended practices, and guidelines for software components, tools, and technologies. They also design methods that facilitate the development, deployment, maintenance, and inter-operation of computer implementations of education and training components and systems. The URL for the LTSC is http://ieeeltsc.org/ .
Canadian Core Learning Resource Metadata Application Resource (CanCore)
CanCore interprets and simplifies the IMS metadata specification, which is a subset of the SCORM specifications. SCORM has been developed in the context of military and training applications, whereas CanCore’s authors and audience have been the public and educators. CanCore enhances the ability of educators, researchers, and students in Canada and around the world to search and locate material from online collections of educational resources. CanCore is based on, and fully compatible with, the IEEE Learning Object Metadata standard and the IMS Learning Resource Meta-data specification. However, the IMS and IEEE are global consortia of educational, industry, and government bodies and the standards they produce are cumbersome and complicated. (Some standards require support of a set of metadata with more than 80 single elements). CanCore was developed to identify a minimum baseline of elements that end-users and institutions could agree were essential, simplifying complexity and providing guidance on general details related to the use of content..
Successful implementation of e-learning requires consistent interpretation of a standard’s purpose and CanCore was devised to realize economies of scale in this process. Since its inception, CanCore has:
- conducted research into the field of learning object metadata;
- devised a workable, consensual sub-set of the IMS learning Object Meta-data Information Model, known as the CanCore Element Set ( http://www.cancore.ca /guidelines/drd/);
- become a participant in IMS through the sponsorship of Industry Canada;
- developed informal ties with Dublin Core;
- written and presented numerous papers in the field of learning object metadata;
- created an XML-record bank showcasing sample CanCore records; and
- written the CanCore Learning Resource Metadata Profile Guidelines.
The key documents on which the CanCore Guidelines were based are:
- IMS Learning Resource Metadata Information Model (www.imsproject.org/metadata/imsmdv1p2p1 /imsmd_infov1p2p1.html);
- IMS Learning Resource Metadata Binding Specification (www.imsproject.org/metadata/imsmdv1 p2p1/imsmd_bindv1p2p1.html);
- IMS Learning Resource Metadata Best Practices and Implementation Guide ( http://www.imsproject.org /metadata/imsmdv1p2p1/imsmd_bestv1p2p1.html).
- The IMS Learning Resource Information Model is itself based on the IEEE LOM.
The guidelines were also developed with consideration of the Dublin Core Initiative (www.dublincore .org/), particularly its element descriptions (www .dublincore.org/documents/dces/), qualifier descriptions ( http://www.dublincore.org/documents/dcmes-qualifiers/ ), and usage guide ( http://www.dublincore.org/documents /usageguide/).
E-Learning Standards Advisory Council of Canada
The E-learning Standards Advisory Council of Canada enables different provinces to work together to identify common requirements of their respective educational systems and to communicate requirements to those who develop standards. As there are multiple standards in development, eLSACC was intended to ensure standards being developed meet Canadian needs. eLSAAC was initially supported by the Minister of Education of Quebec and Council of Ministers of Education of Canada. Five provinces, including British Columbia, have agreed to fund eLSAAC for a five-year period. The eLSAAC can be found at elsacc.ca.


1.8: Geometry of Numbers
We have already seen that geometrical concepts are sometimes useful in illuminating number theoretic considerations. With the introduction by Minkowski of geometry of numbers a real welding of important parts of number theory and geometry was achieved. This branch of mathematics has been in considerable vogue in the last 20 years, particularly in England where it was and is being developed vigorously by Mordell, Davenport, Mahler and their students.
We shall consider a very brief introduction to this subject. First, we shall examine a proof of the fundamental theorem of Minkowski due to Hajos (1934), then we shall discuss some generalizations and applications of this theorem, and finally we shall investigate some new results and conjectures that are closely related.
In its simplest form the fundamental theorem of Minkowski is the following.
Let \(R\) be a region in the \(x-y\) plane of area \(A > 4\), symmetric about the origin and convex. Then \(R\) contains a lattice point other than the origin.
First, some preliminary remarks. In the condition \(A > 4\), the 4 cannot be replaced by any smaller number. This may be seen by considering the square of side \(2 − \epsilon\), centered at the origin. Indeed this example might at first suggest that the theorem is quite intuitive, as it might seem that squeezing this region in any direction and keeping its area fixed would necessarily force the region to cover some lattice point. However the matter is not quite so simple, as other examples reveal that neither central symmetry nor convexity are indispensable. As far as convexity is concerned what is really needed is that with the vectors \(\vec{V_1}\) and \(\vec{V_2}\) the region should also contain \(\dfrac{1}{2} (\vec{V_1} + \vec{V_2})\). The symmetry means that with \(\vec{V_1}\) the vector \(-\vec{V_1}\) should also be in \(R\). Thus the symmetry and convexity together imply that, if \(\vec{V_1}\) and \(\vec{V_2}\) are in \(R\), so is \(\dfrac{1}{2} (\vec{V_1} - \vec{V_2})\). This last condition is really sufficient for our purpose and may replace the conditions of symmetry and convexity. It is implied by symmetry and convexity but does not imply either of these conditions.
Another example that perhaps illuminates the significance of Minkowski’s theorem is the following. Consider a line through \(O\) having irrational slope \(\tan \theta\); see Figure 4. This line passes through no lattice point other than the origin. If we take a long segment of this line, say extending length \(R\) on either side of \(O\), then there will be a lattice point closest to, and a distance \(r\) from,
this segment. Hence, no matter how large \(R\) is, we can construct a rectangle containing this line segment, which contains no lattice point other than \(O\). By the fundamental theorem of Minkowski the area \(4rR\) of this rectangle does not exceed 4. Thus \(r \le \dfrac{1}{R}\). Note that if \((p, q)\) is a lattice point on the border of the rectangle then \(\dfrac{p}{q} \approx \tan \theta\), so that the fundamental theorem of Minkowski will give some information about how closely an irrational number can be approximated by rationals.
Let us now return to Hajos proof of the fundamental theorem of Minkowski. Consider the \(x-y\) plane cut up into an infinite chessboard with the basic square of area 4 determined by \(|x| \le 1\), \(|y| \le 1\). We now cut up the chessboard along the edges of the squares and superimpose all the squares that contain parts of the region \(R\). We have now compressed an area > 4 into a region of area 4. This implies that there will be some overlapping, i.e., one can stick a pin through the square so as to pierce \(R\) into two points say \(V_1\) and \(V_2\). Now reassemble the region and let the points \(V_1\) and \(V_2\) be the vectors \(\vec{V_1}\) and \(\vec{V_2}\). Consider the fact that the \(x\) and \(y\) coordinates of \(V_1\) and \(V_2\) differ by a multiple of 2. We write \(V_1 \equiv V_2\) (mod 2), which implies \(\dfrac{1}{2} (V_1 - V2) \equiv 0\) (mod 1). Thus \(\dfrac{1}{2} (V_1 - V_2)\) is a lattice point different from O (since \(V_1 \ne V_2\)) in \(R\).
The fundamental theorem of Minkowski can easily be generalized to \(n\)-dimensional space. Indeed we need only replace 4 in the fundamental theorem of Minkowski by 2n and Hajos’ proof goes through. Many extensions and re- finements of the fundamental theorem of Minkowski have been given. I shall return to some of them later.
One of Polya’s earliest papers has the long and curious title “Zahlhlentheoretisches und Wahrscheinlichkeitstheoretisches \(\ddot{u}\)ber die Sichtweite in Walde und durch Schneefall”. A proof of Polya’s main result in this paper can be greatly simplified and somewhat refined using the fundamental theorem of Minkowski. The problem is this.
Suppose every lattice point other than \(O\) is surrounded by a circle of radius \(r \le \dfrac{1}{2}\) (a tree in a forest). A man stands at \(O\). In direction \(\theta\) he can see a distance \(f(r, \theta)\). distance f(r,θ). What is the furthest he can see in any direction? That is, determine
\(F(r) = \text{max}_{\theta} f(\theta, r)\)
By looking past the circle centered at (1, 0) (Figure 5), we can see almost a distance \(\dfrac{1}{r}\). On the other hand we can prove that \(F(r) \le \dfrac{1}{r}\). For suppose that we can see a distance \(F(r)\) in direction θ. About this line of vision construct a rectangle with side \(2r\). This rectangle contains no lattice point, for otherwise the tree centered at such a lattice point would obstruct our line of vision; see Figure 6.
Hence, by the fundamental theorem of Minkowski \(4F(r) r \le 4\) and \(F(r) \le \dfrac{1}{r}\) as required. Note that no lattice point can be in either semicircle in the diagram. This enables us to improve slightly on Polya’s result. I shall leave the details as an exercise.
A more significant application of the fundamental theorem of Minkowski concerns the possibility of solving in integers a set of linear inequalities.
Consider the inequalities
\(|a_{11} x_{1} + a_{12}x_{2} + \cdot\cdot\cdot + a_{1n}x_{n}| \le \lambda_1,\)
\(|a_{21} x_{1} + a_{22}x_{2} + \cdot\cdot\cdot + a_{2n}x_{n}| \le \lambda_2,\)
.
.
.
\(|a_{n1} x_{1} + a_{n2}x_{2} + \cdot\cdot\cdot + a_{nn}x_{n}| \le \lambda_n,\)
where the \(a_{ij}\) are real numbers and the \(\lambda_1, \lambda_2, ..., \lambda_n\) are positive numbers. The problem is to find sufficient conditions for the existence of integers \(x_1, ..., x_n\), not all 0 satisfying the system. The fundamental theorem of Minkowski can be used to prove that a solution will exist provided the determinant det(aij) of the coefficients is, in absolute value, less than the product \(\lambda_1 \cdot \lambda_2 \cdot\cdot\cdot\cdot\cdot \lambda_n\). This comes about in the following way. Geometrically, the inequalities determine an \(n\)−dimensional parallelepiped whose volume (or content) is
\(\dfrac{1}{\text{det} (a_{ij})} \cdot 2^n \cdot \lambda_1 \cdot \lambda_2 \cdot\cdot\cdot\cdot\cdot \lambda_n.\)
If \(\lambda_1 \cdot \lambda_2 \cdot\cdot\cdot\cdot\cdot \lambda_n > \text{det} (a_{ij})\) then the content exceeds \(2^n\) and so contains a lattice point different from \(O\).
A very recent analogue of the fundamental theorem of Minkowski is the following. Let \(R\) be a convex region, not necessarily symmetric about O, but having its centroid at \(O\). If its area exceeds \(\dfrac{9}{2}\), then it contains a lattice point not \(O\). The constant \(\dfrac{9}{2}\) is again best possible, but an n-dimensional analogue of this result is unknown.
The following is a conjectured generalization of the fundamental theorem of Minkowski, which we have unfortunately been unable to prove. Perhaps you will be able to prove or disprove it. Let \(R\) be a convex region containing the origin and defined by \(r = f(\theta)\), \(0 \le \theta < 2 \pi\). If
\(\int_0^{\pi} f(\theta) f(\theta + \pi) d \theta > 4\)
then \(R\) contains a nontrivial lattice point. For symmetrical regions \(f(\theta) = f(\theta + \pi)\), and the conjecture reduces to the fundamental theorem of Minkowski.
Here is a somewhat related and only partially solved problem. Let \(M(n)\) be defined as the smallest number such that any convex region of area \(M(n)\) can be placed so as to cover \(n\) lattice points. Clearly \(M(1) = 0\). It is not difficult to show that \(M(2) = \dfrac{\pi}{4}\), i.e., any convex region whose area exceeds that of a circle of diameter 1 can be used to cover 2 lattice points. To determine \(M(3)\) already seems difficult. What one can easily prove is that \(M(n) \le n -1\) and we conjecture the existence of a positive constant \(c\) such that \(M(n) < n - c \sqrt{n}\).