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12700	https://www.su18.eecs70.org/static/notes/n5a.pdf	Note 5 Supplement: Planar Duality Computer Science 70 University of California, Berkeley Summer 2018 1 Planar Graphs A graph G = (V, E) is a planar graph if it can be drawn in the plane with no edge crossings. We assume that G is ﬁnite (it has ﬁnitely many vertices and edges) and undirected. We will not assume that G is a simple graph, however, because we will need to discuss planar graphs which possibly contain multiple edges between pairs of vertices, or even an edge from a vertex to itself (a self-loop). Finally, we will only be discussing connected graphs. There are many ways to embed the graph G into a plane with no edge crossings (we call such an embedding a planar drawing). For the rest of this note, we will ﬁx some planar drawing of G and refer to it. Perhaps the simplest examples of planar graphs are trees, i.e., connected acyclic graphs. Theorem 1. Trees are planar graphs. Proof. Proceed by induction on the number of vertices. If the tree has one vertex, then it is certainly planar. Otherwise, consider a tree with at least two vertices and assume that all trees with fewer vertices are planar. Remove a leaf v and its associated edge {u, v} from the tree. The resulting graph is a tree, so by the inductive hypothesis, it has a planar drawing. Now “zoom in” on the vertex u in this planar drawing (this requires you to visualize the tree in your head). If we “zoom in” close enough, then we will only see the vertex u and the edges leaving the vertex u. Since the vertex u is only connected to ﬁnitely many other vertices, we can place the vertex v in an empty spot in between some of the edges leaving u and add back the edge {u, v}, yielding a planar drawing of our original tree. 1 A face is a connected region of the plane (in the planar drawing). Since the graph is ﬁnite, one of the faces (the face which lies “outside” of the edges in the planar drawing) is inﬁnite. We will use the following notation: • v := \|V \| is the number of vertices in G; • e := \|E\| is the number of edges in G; • f is the number of faces in G. There is a relationship between these quantities for planar graphs: Theorem 2 (Euler’s Formula). For any connected planar graph, v+f = e+2. One can prove Euler’s Formula by induction. Instead, we will present a diﬀerent proof which uses a deeply fascinating concept called planar duality. 2 Planar Duality A planar dual of G, called G∗, is a planar graph whose vertices are the faces of G and whose edges correspond to the edges of G. For this, a picture is worth much more than the formal deﬁnition (see Figure 1). Figure 1: A planar graph (in black) is shown, along with a planar dual (in blue) vertices of he dual planar graph correspond to the faces of the original planar graph. Each edge of the original planar graph has a face on each of its two sides (the two faces are not necessarily distinct); the corresponding edge in the dual planar graph connects these two faces. For a planar graph G, there may be multiple ways to draw a dual planar graph, so technically we should speak about “a planar dual” rather than “the 2 planar dual”. For our purposes, it does not really matter what planar dual we choose, so we will not worry about this any further. There is an intimate relationship between a planar graph and its dual. First of all, when we speak of cycles, we mean simple cycles, i.e., cycles that do not repeat edges and do not repeat vertices (except for the fact that a cycle starts and ends at the same vertex). A cycle encloses one or more faces of the graph. A cut in the graph is a set of edges in the graph which, if removed, disconnects a set of vertices from the rest of the graph (in this deﬁnition, the set of vertices S is assumed to be non-empty, and not equal to V itself). When we speak of cuts, however, we mainly refer to minimal cuts, which are cuts such that no proper subset of edges is also a cut. Theorem 3 (Cycle-Cut Duality). Every cycle in G corresponds to a unique cut in G∗. Figure 2: A cycle in G corresponds to a cut in G∗. Proof. See Figure 2 for a visualization. A cycle in G encloses one or more faces of G, i.e., a cycle encloses one or more vertices of G∗. Thus, the dual edges to the cycle form a cut in G∗ which separate these dual vertices from the rest of the dual graph. Conversely, a minimal cut in G∗consists of the set of edges with one endpoint in S and one endpoint in V \ S, where S is some proper non-empty subset of vertices. The corresponding edges in G must enclose either the faces corresponding to S or the faces corresponding to V \ S, so these edges must be a simple cycle in G. As a special case of Cycle-Cut Duality, we have the following: Theorem 4. G is a planar dual of G∗. In other words, the graph G is a dual of its dual. 3 Proof of Theorem 4. First, for any vertex in G, the edges leaving the vertex correspond to dual edges which deﬁne a face in G∗; thus, vertices in G correspond to faces in G∗. Since we already know that the edges of G and G∗correspond to each other, then G is a planar dual of G∗. Since G is a dual of its dual, then Cycle-Cut Duality also implies that each minimal cut in G corresponds to a unique simple cycle in G∗. The last piece of information that we will need is to relate cuts and the connectedness of a graph. Theorem 5. Let E′ be a subset of edges in a graph G = (V, E). Then, the graph (V, E′) is connected if and only if the remaining edges E \ E′ do not contain any cuts. Proof. If E \ E′ does not contain any cuts, then removing the edges in E \ E′ does not disconnect any subset of vertices, i.e., the graph (V, E′) is connected. Conversely, if E \ E′ contains a cut, then there is a subset S of vertices for which the removal of the edges in the cut disconnects S from the rest of the vertices. Then, the graph (V, E′) is not connected because there is no path from a vertex in S to a vertex in V \ S. Now, in our planar graph G, we ﬁnd a spanning tree, that is, a subset E′ of edges in G such that (V, E′) is a tree. We can ﬁnd this spanning tree because G is connected. See Figure 3 for a picture of a spanning tree of the planar graph in Figure 1. Figure 3: In the ﬁgure on the right, the remaining black edges form a spanning tree of the planar graph in Figure 1. 4 Next, we look at the edges in G∗which are not dual edges to the spanning tree (V, E′). Thus, these edges are dual to E \ E′. Call these edges E′. See Figure 4 for a picture of these edges. Figure 4: The orange edges are the edges in G∗which are not the dual edges of the spanning tree in G. Since (V, E′) is connected, then by Theorem 5, the remaining set of edges E \ E′ does not contain any cuts. By Cycle-Cut Duality (Theorem 3), the corresponding dual edges E′ are acyclic. On the other hand, since (V, E′) is acyclic, then again by Cycle-Cut Duality (Theorem 3), the dual edges to E′ do not have any cuts; thus, the remaining set of edges E′ form a connected subgraph in G∗by Theorem 5. Hence, we have found that the set of edges E′ is connected and acyclic, so they form a spanning tree of G∗. We call this the dual spanning tree to the spanning tree (V, E′). Now we are ready to prove Euler’s Formula: Proof of Theorem 2. Consider a spanning tree (V, E′) of G and the edges E′ which form the dual spanning tree in G∗. Since the number of edges in a tree is one less than the number of vertices, then \|E′\| = v −1 and \|E′\| = f −1. However, since the edges in E′ are dual to E \ E′, then the total number of edges in E′ and E′ is equal to E, so we get e = (v −1) + (f −1). Rearranging this equation yields Euler’s Formula. As a closing note, if you prove a theorem about planar graphs, then you can apply the theorem to a planar dual to see if you can get another “dual” theorem out. For example, we have proved the following: Lemma 1 (Handshaking Lemma). P v∈V deg v = 2\|E\|. 5 Now, applying the Handshaking Lemma to a planar dual of G = (V, E), we get P f∈F deg f = 2\|E\|, where F is the set of faces in G. Here, deg f is the degree of the vertex v in the planar dual, which is equal to the number of sides of the face in G. We have the statement that 2\|E\| is the total number of sides in G. On the other hand, if the planar graph has at least three vertices, then each face has at least three sides. So, 2e ≥3f, and by plugging this into Euler’s Formula (Theorem 2) and rearranging, we get the inequality: Corollary 1. If G has at least three vertices, then e ≤3v −6. This last result can be used to prove the non-planarity of K5, the complete graph on ﬁve vertices. 6
12701	https://www.quora.com/What-is-order-statistics-and-why-do-we-use-it	Something went wrong. Wait a moment and try again. Data Analysis Data Science Order Statistics Probability (Mathematical... Statistical Analysis Probability Statistics Statistics and Probabilit... Data Science/AI Statistics and Data Scien... 5 What is order statistics, and why do we use it? Sort Terry Moore PhD in statistics · Author has 16.6K answers and 29.3M answer views · 7y Originally Answered: What are order statistics? Why do we use it? · What are order statistics? Why do we use it? If you put a sample into numerical order and delete the original observations then you lose information on the order that the observations were received in. These are the order statistics. If the observations are independent then you haven’t lost any information of any practical use for estimating the characteristics of the population that the data came from. I don’t think they are used directly, they are more of a tool for discussing the theory. In most cases we would like to reduce the data to a small number of statistics. But the distribution of qu What are order statistics? Why do we use it? If you put a sample into numerical order and delete the original observations then you lose information on the order that the observations were received in. These are the order statistics. If the observations are independent then you haven’t lost any information of any practical use for estimating the characteristics of the population that the data came from. I don’t think they are used directly, they are more of a tool for discussing the theory. In most cases we would like to reduce the data to a small number of statistics. But the distribution of quantiles such as the median are derived from the distribution of the order statistics. So they are important from that point of view. Promoted by Coverage.com Johnny M Master's Degree from Harvard University (Graduated 2011) · Updated Sep 9 Does switching car insurance really save you money, or is that just marketing hype? This is one of those things that I didn’t expect to be worthwhile, but it was. You actually can save a solid chunk of money—if you use the right tool like this one. I ended up saving over $1,500/year, but I also insure four cars. I tested several comparison tools and while some of them ended up spamming me with junk, there were a couple like Coverage.com and these alternatives that I now recommend to my friend. Most insurance companies quietly raise your rate year after year. Nothing major, just enough that you don’t notice. They’re banking on you not shopping around—and to be honest, I didn’t. This is one of those things that I didn’t expect to be worthwhile, but it was. You actually can save a solid chunk of money—if you use the right tool like this one. I ended up saving over $1,500/year, but I also insure four cars. I tested several comparison tools and while some of them ended up spamming me with junk, there were a couple like Coverage.com and these alternatives that I now recommend to my friend. Most insurance companies quietly raise your rate year after year. Nothing major, just enough that you don’t notice. They’re banking on you not shopping around—and to be honest, I didn’t. It always sounded like a hassle. Dozens of tabs, endless forms, phone calls I didn’t want to take. But recently I decided to check so I used this quote tool, which compares everything in one place. It took maybe 2 minutes, tops. I just answered a few questions and it pulled up offers from multiple big-name providers, side by side. Prices, coverage details, even customer reviews—all laid out in a way that made the choice pretty obvious. They claimed I could save over $1,000 per year. I ended up exceeding that number and I cut my monthly premium by over $100. That’s over $1200 a year. For the exact same coverage. No phone tag. No junk emails. Just a better deal in less time than it takes to make coffee. Here’s the link to two comparison sites - the one I used and an alternative that I also tested. If it’s been a while since you’ve checked your rate, do it. You might be surprised at how much you’re overpaying. Related questions What's the definition of order statistics? What is definition of order statistics and why it's important? What are the uses of order statistics? What is an order effect in statistics? What are biviarite statistics? Joseph Barr Self Employed Statistician/Data Scientist. at Barr Analytics · Author has 655 answers and 836.5K answer views · 7y Order statistics refers to ranking a sample from some, usually unknown distribution p(x; \theta). Suppose you draw a sample from some population X_1, X_2, …,X_n, where each X_j is a number (scalar.). Then you’ve sorted the values from smallest to largest and you rename the sample X_(1), X_(2), …, X_(n) where X_(1)<= X_(2) <= X_(3) <=… <=X_(n). Say, if X_j are revenues on day j, then you may be inter Order statistics refers to ranking a sample from some, usually unknown distribution p(x; \theta). Suppose you draw a sample from some population X_1, X_2, …,X_n, where each X_j is a number (scalar.). Then you’ve sorted the values from smallest to largest and you rename the sample X_(1), X_(2), …, X_(n) where X_(1)<= X_(2) <= X_(3) <=… <=X_(n). Say, if X_j are revenues on day j, then you may be interested in the highest grossing day. If X_j is the timing of race of the j-th competitor then you may be interested in the timing of the one that won the race. If X_j represents high temperature of j-th measuring station, then you may be interested in the range X_(n) - X_(1) (hottest - coldest). Each of those X_(j)s is a random measurement hence has a statistical distribution which in turn can be used to estimate probabilities or to predict something, etc. The study of records X_(n) (largest) and X_(... Assistant Bot · 1y Order statistics are the statistics that provide information about the order or rank of data points in a sample. Specifically, if you have a sample of n observations, the k-th order statistic is the k-th smallest value in that sample. For example, in a sample of values x1,x2,…,xn, the first order statistic X(1) is the minimum value, the second order statistic X(2) is the second smallest value, and so on, up to the n-th order statistic X(n), which is the maximum value. Key Concepts: Notation: The k-th order statistic is often denoted as X(k). Distribution: The distribution o Order statistics are the statistics that provide information about the order or rank of data points in a sample. Specifically, if you have a sample of n observations, the k-th order statistic is the k-th smallest value in that sample. For example, in a sample of values x1,x2,…,xn, the first order statistic X(1) is the minimum value, the second order statistic X(2) is the second smallest value, and so on, up to the n-th order statistic X(n), which is the maximum value. Key Concepts: Notation: The k-th order statistic is often denoted as X(k). Distribution: The distribution of order statistics can be derived from the underlying distribution of the sample data. For example, if the sample comes from a normal distribution, the order statistics will also have a specific distribution. Applications of Order Statistics: Statistical Analysis: Order statistics are used in various statistical methods, including non-parametric statistics, where the sample distribution is not assumed to follow a specific parametric form. They are essential for constructing confidence intervals and hypothesis testing. Robust Estimation: They provide robust estimates of central tendency (e.g., median) and variability (e.g., interquartile range), which are less affected by outliers than mean and standard deviation. Quality Control: In fields like manufacturing and quality control, order statistics help in assessing the performance of processes by analyzing the minimum and maximum values of a sample. Reliability Engineering: They are used to model lifetimes of products and systems, where the order statistics can indicate the minimum time until failure or the expected lifespan. Data Analysis: Order statistics can help in understanding the distribution of data, identifying outliers, and making decisions based on ranked data. Summary In summary, order statistics are crucial in statistics for analyzing the rank and distribution of data points, providing robust estimates, and facilitating various applications in statistical inference, quality control, and more. They help us understand the underlying patterns in data and make informed decisions based on that understanding. Mythily Kesavan Knows English · Author has 7K answers and 3.2M answer views · 4y Originally Answered: What is definition of order statistics and why it's important? · Order statistics are sample values placed in ascending order. The study of order statistics deals with the applications of these ordered values and their functions. To get the order statistics (Yn) put the items in numerical increasing order: Y1=12kg. Y2=22kg. Related questions What's the definition of order statistics? What is definition of order statistics and why it's important? What are the uses of order statistics? What is an order effect in statistics? What are biviarite statistics? What does second order statistics mean? And why is it second order? What is an example of statistics? In what type of situation is order statistics method actually useful? What percentage of statistics are made up? What are some examples of higher order statistics? What is the use of statistical probability in statistics? What is the future of order statistics in research? What is statistics and why we use it? What is statistics? Why are 95% of statistics made up? About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
12702	https://www.xconvert.com/unit-converter/gallons-to-litres	Gallons to Litres \| Convert gal To l Online - XConvert XConvertOther Unit ConversionsDownloadsCompress MP4 Gallons (gal) to Litres (l) conversion Volume From Gallons gal To Litres l Swap units Copy result Gallons to Litres conversion table \| Gallons (gal) \| Litres (l) \| --- \| \| 0 \| 0 \| \| 1 \| 3.7854117954011 \| \| 2 \| 7.5708235908022 \| \| 3 \| 11.356235386203 \| \| 4 \| 15.141647181604 \| \| 5 \| 18.927058977006 \| \| 6 \| 22.712470772407 \| \| 7 \| 26.497882567808 \| \| 8 \| 30.283294363209 \| \| 9 \| 34.06870615861 \| \| 10 \| 37.854117954011 \| \| 20 \| 75.708235908022 \| \| 30 \| 113.56235386203 \| \| 40 \| 151.41647181604 \| \| 50 \| 189.27058977006 \| \| 60 \| 227.12470772407 \| \| 70 \| 264.97882567808 \| \| 80 \| 302.83294363209 \| \| 90 \| 340.6870615861 \| \| 100 \| 378.54117954011 \| \| 1000 \| 3785.4117954011 \| How to convert gallons to litres? Converting between gallons and liters is a common task, especially when dealing with recipes, fuel, or other liquid quantities. Here's how to perform the conversion, along with some background and examples. Conversion Factors The conversion between gallons and liters is based on a fixed relationship. There are two primary types of gallons: US gallons and Imperial gallons. The conversions to liters are as follows: 1 US gallon ≈ 3.78541 liters 1 Imperial gallon ≈ 4.54609 liters Since the question doesn't specify which gallon, we'll use the US gallon for the following explanation and conversions. Converting US Gallons to Liters To convert from US gallons to liters, you multiply the number of gallons by the conversion factor: Liters=Gallons×3.78541\text{Liters} = \text{Gallons} \times 3.78541 Example: Convert 1 US gallon to liters: 1 gallon×3.78541=3.78541 liters 1 \text{ gallon} \times 3.78541 = 3.78541 \text{ liters} Converting Liters to US Gallons To convert from liters to US gallons, you divide the number of liters by the conversion factor: Gallons=Liters 3.78541\text{Gallons} = \frac{\text{Liters}}{3.78541} Example: Convert 1 liter to US gallons: 1 liter 3.78541≈0.264172 gallons\frac{1 \text{ liter}}{3.78541} \approx 0.264172 \text{ gallons} Interesting Facts and History The gallon has a long and varied history. Its definition has changed over time and differs between countries. The US gallon is based on the wine gallon of Queen Anne, while the Imperial gallon is based on the volume of 10 pounds of distilled water at 62°F (17°C). The standardization of units is crucial for trade, science, and engineering, ensuring consistent measurements across different contexts. Real-World Examples Fuel: In many countries, fuel is sold in liters, while in the US, it is sold in gallons. Converting between these units is useful when comparing fuel efficiency or prices. For instance, a car with a 15-gallon fuel tank has approximately 15×3.78541≈56.78 15 \times 3.78541 \approx 56.78 liters of fuel capacity. Recipes: Many international recipes use liters, while US recipes often use gallons, quarts, or cups. Converting helps in accurately following the recipe. If a recipe calls for 2 liters of stock, that's approximately 2 3.78541≈0.528\frac{2}{3.78541} \approx 0.528 US gallons. Aquariums: Aquarium sizes are sometimes listed in gallons (especially in the US) and need to be converted to liters for certain treatments or calculations. A 20-gallon aquarium holds about 20×3.78541≈75.71 20 \times 3.78541 \approx 75.71 liters of water. By understanding these conversions and their applications, you can easily work with both gallons and liters in various contexts. See below section for step by step unit conversion with formulas and explanations. Please refer to the table below for a list of all the Litres to other unit conversions. What is Gallons? The gallon is a unit of volume in the imperial and United States customary systems of measurement. Different definitions are used in these two systems. Let's explore the gallon in detail. Definition of a Gallon A gallon is a unit of volume used primarily in the United States and the United Kingdom (though its use is diminishing in the UK in favor of the metric system). There are two primary types of gallons: US Gallon: Defined as 231 cubic inches (exactly 3.785411784 liters). Imperial Gallon: Used in the UK, Canada, and some Caribbean nations, defined as 4.54609 liters. Subdivisions of a Gallon Gallons are further divided into smaller units, which is useful for everyday measurements. The common subdivisions are: 1 Gallon (gal) = 4 Quarts (qt) 1 Quart (qt) = 2 Pints (pt) 1 Pint (pt) = 2 Cups (c) 1 Cup (c) = 8 Fluid Ounces (fl oz) This hierarchical breakdown allows for convenient measurements of various quantities. Differences Between US and Imperial Gallons \| Feature \| US Gallon \| Imperial Gallon \| --- \| Volume \| 231 cubic inches (3.785411784 liters) \| 4.54609 liters \| \| Common Usage \| United States \| United Kingdom, Canada, some Caribbean nations \| \| Weight of Water \| Approximately 8.345 pounds at 62°F (17°C) \| Approximately 10.02 pounds at 62°F (17°C) \| The Imperial gallon is about 20% larger than the US gallon. Real-World Examples of Gallon Usage Fuel: Gasoline is commonly sold by the gallon in the United States. The price per gallon is a standard metric for comparing fuel costs. Milk and Beverages: Milk, juice, and other beverages are often sold in gallon containers. Paint: Paint is typically sold in gallons, quarts, and pints, making it easy to estimate the amount needed for a project. Water Consumption: Water usage is often measured in gallons, allowing homeowners and municipalities to track water consumption rates. Aquariums and Pools: The volume of water in aquariums and swimming pools is usually specified in gallons. This helps in determining the appropriate amount of chemicals and maintenance needed. Historical Context The term "gallon" has murky origins, with roots in old French and other languages. There isn't a single individual or law directly associated with the invention of the gallon. Instead, it evolved as a practical unit of measure through trade and commerce. Different gallon definitions existed throughout history before standardization. Gallon Conversions Here are some common conversions involving gallons: US Gallons to Liters: Liters=US Gallons×3.78541\text{Liters} = \text{US Gallons} \times 3.78541 Liters to US Gallons: US Gallons=Liters÷3.78541\text{US Gallons} = \text{Liters} \div 3.78541 Imperial Gallons to Liters: Liters=Imperial Gallons×4.54609\text{Liters} = \text{Imperial Gallons} \times 4.54609 Liters to Imperial Gallons: Imperial Gallons=Liters÷4.54609\text{Imperial Gallons} = \text{Liters} \div 4.54609 Interesting Facts A gallon of water weighs approximately 8.34 pounds (US) or 10.02 pounds (Imperial) at 62°F (17°C). This is useful for estimating the weight of water-filled containers. The "gallon challenge" is a social media stunt (discouraged due to health risks) that involves attempting to drink a gallon of milk quickly. In the US, fuel efficiency of cars is measured in miles per gallon (MPG). For more information, you can refer to NIST's definition of units and the Wikipedia article on Gallons. What is Litres? This section will explore the definition of liters, their origin, relationship to other units, and some common uses. We'll also touch upon some interesting facts and examples. Definition of Litre A litre (L) is a metric unit of volume. It is defined as the volume of one cubic decimeter (d m 3 dm^3). One litre of water almost has a mass of precisely one kilogram due to how metric system was initially defined. History and Formation The litre was introduced as part of the French metric system in 1795, originally defined as one cubic decimeter. The name "litre" comes from the older French unit, the "litron." Over time, the precise definition has been slightly refined, but the core concept remains the same. Relation to Other Units Cubic Centimeters (c m 3 cm^3 or cc): 1 L = 1000 c m 3 cm^3 Millilitres (mL): 1 L = 1000 mL Cubic Meters (m 3 m^3): 1 L = 0.001 m 3 m^3 Gallons (gal): 1 L ≈ 0.264 US gallons Quarts (qt): 1 L ≈ 1.057 US liquid quarts Interesting Facts and Connections While no specific "law" is directly tied to the litre itself, its consistent definition and wide adoption highlight its importance in the International System of Units (SI). Its relationship to the kilogram via the density of water was a key design principle of the metric system. Real-World Examples Beverages: A standard bottle of water is often 1 or 1.5 litres. Soda bottles commonly come in 2-litre sizes. Fuel: Car fuel tanks are measured in litres (e.g., 50-litre tank). Fuel consumption is often expressed as litres per 100 kilometers (L/100 km). Cooking: Recipes often specify liquid ingredients in millilitres or litres. For example, a soup recipe might call for 2 litres of broth. Medical: Intravenous (IV) fluids are administered in litres, and blood volume is often estimated in litres. Aquariums: The capacity of an aquarium is measured in litres, for example 100-litre tank. Engine Displacement: Engine size is commonly measured in litres, such as a 2.0-litre engine. This refers to the total volume displaced by the pistons during one complete cycle. Formulae examples Relationship between Litres and Cubic Meters: 1 L=0.001 m 3 1 \text{ L} = 0.001 \text{ m}^3 Relationship between Litres and Millilitres: 1 L=1000 mL 1 \text{ L} = 1000 \text{ mL} Relationship between Litres and Cubic Centimeters: 1 L=1000 cm 3 1 \text{ L} = 1000 \text{ cm}^3 External Links For a broader overview of the litre, you can consult the Wikipedia page. You can also explore the Bureau International des Poids et Mesures (BIPM), the international standards organization responsible for maintaining the SI system. Complete Gallons conversion table Enter # of Gallons \| Convert 1 gal to other units \| Result \| --- \| \| Gallons to Cubic Millimeters (gal to mm 3) \| 3785411.7954011 \| \| Gallons to Cubic Centimeters (gal to cm 3) \| 3785.4117954011 \| \| Gallons to Cubic Decimeters (gal to dm 3) \| 3.7854117954011 \| \| Gallons to Millilitres (gal to ml) \| 3785.4117954011 \| \| Gallons to Centilitres (gal to cl) \| 378.54117954011 \| \| Gallons to Decilitres (gal to dl) \| 37.854117954011 \| \| Gallons to Litres (gal to l) \| 3.7854117954011 \| \| Gallons to Kilolitres (gal to kl) \| 0.003785411795401 \| \| Gallons to Megalitres (gal to Ml) \| 0.000003785411795401 \| \| Gallons to Gigalitres (gal to Gl) \| 3.7854117954011e-9 \| \| Gallons to Cubic meters (gal to m 3) \| 0.003785411795401 \| \| Gallons to Cubic kilometers (gal to km 3) \| 3.7854117954011e-12 \| \| Gallons to Kryddmått (gal to krm) \| 3785.4117954011 \| \| Gallons to Teskedar (gal to tsk) \| 757.08235908022 \| \| Gallons to Matskedar (gal to msk) \| 252.36078636007 \| \| Gallons to Kaffekoppar (gal to kkp) \| 25.236078636007 \| \| Gallons to Glas (gal to glas) \| 18.927058977006 \| \| Gallons to Kannor (gal to kanna) \| 1.4464699256405 \| \| Gallons to Teaspoons (gal to tsp) \| 768 \| \| Gallons to Tablespoons (gal to Tbs) \| 256 \| \| Gallons to Cubic inches (gal to in 3) \| 231.00106477053 \| \| Gallons to Fluid Ounces (gal to fl-oz) \| 128 \| \| Gallons to Cups (gal to cup) \| 16 \| \| Gallons to Pints (gal to pnt) \| 8 \| \| Gallons to Quarts (gal to qt) \| 4 \| \| Gallons to Cubic feet (gal to ft 3) \| 0.1336806244556 \| \| Gallons to Cubic yards (gal to yd 3) \| 0.004951126961594 \| Volume conversions Gallons to Cubic Millimeters (gal to mm 3) Gallons to Cubic Centimeters (gal to cm 3) Gallons to Cubic Decimeters (gal to dm 3) Gallons to Millilitres (gal to ml) Gallons to Centilitres (gal to cl) Gallons to Decilitres (gal to dl) Gallons to Litres (gal to l) Gallons to Kilolitres (gal to kl) Gallons to Megalitres (gal to Ml) Gallons to Gigalitres (gal to Gl) Gallons to Cubic meters (gal to m 3) Gallons to Cubic kilometers (gal to km 3) Gallons to Kryddmått (gal to krm) Gallons to Teskedar (gal to tsk) Gallons to Matskedar (gal to msk) Gallons to Kaffekoppar (gal to kkp) Gallons to Glas (gal to glas) Gallons to Kannor (gal to kanna) Gallons to Teaspoons (gal to tsp) Gallons to Tablespoons (gal to Tbs) Gallons to Cubic inches (gal to in 3) Gallons to Fluid Ounces (gal to fl-oz) Gallons to Cups (gal to cup) Gallons to Pints (gal to pnt) Gallons to Quarts (gal to qt) Gallons to Cubic feet (gal to ft 3) Gallons to Cubic yards (gal to yd 3)
12703	https://www.crashwhite.com/apphysics/materials/presentations/sem1-ch9/index.html	Chapter 9: Conservation of Momentum 9.0. Overview There are three big conservation laws in the field of Mechanics, and we've just finished studying the first one: Conservation of Energy. The second important law, Conservation of (linear) Momentum, is another important tool that we can use to solve problems, especially when we see two bodies interacting with other via a Force. Let's get started! Linear Momentum Impulse Conservation of Linear Momentum Collisions Center of Mass 9.1. Linear Momentum When Newton wrote his Second Law of Motion, he didn't say "A net force causes a mass to accelerate." What he really said (in Latin), was, "Forces applied over a period of time change an object's quantity of motion." And what does "quantity of motion" mean? There are lots of quantities associated with motion: distance, speed, acceleration, direction of motion... When Newton said "quantify of motion" he was referring to a quantity that we now call linear momentum. Newton's statement regarding quantity of motion and the Second Law of Motion that we've used in this class are mathematically equivalent. Definition of momentum Take a look at this simple derivation that begins with what we've been calling Newton's Second Law of Motion. The term mv is what Newton referred to as a "quantity of motion," a vector quantity that combines the mass of the object and its velocity. We use the variable p, a vector, to indicate linear momentum. So, a net force on a mass causes a change in its momentum p, its mv, where momentum and velocity have been written in bold to remind us that they are vector quantities. 9.2. Impulse We can talk about this force-momentum-time relationship another way. Rearranging the relationship above: Here, the value J represents the impulse applied to the mass, the Force applied through a given amount of time, resulting in a change in the momentum of the object. Note that some authors use I to indicate impulse rather than J. With this relationship we can talk about the impulse applied to an object, and how its momentum changes as a result. Determining Impulse There are effectively four ways that you can identify the impulse acting on an object. You can calculate the average force acting over a time period. Similarly, if you have a Force-time function you can integrate it with respect to time. You can look at a force-time graph and use the area-under-the-curve to determine impulse. You can calculate the change in momentum, mΔv, which is caused by the impulse. Which strategy you use will depend on what you have to work with in the problem. Cart hitting a wall A 5.0 kg cart is traveling at 3.7 m/s when it hits a brick wall and comes to a halt. The time of the collision is 0.257 seconds. What was the impulse, magnitude and direction, of the wall on the cart in this collision? What average force, magnitude and direction, was exerted on the cart during the collision? When asked to calculate impulse, one often wants to calculate using Force and time, but impulse can be determined by calculating change in momentum as well. Note that we're indicating directions using positive and negative signs. Although the Force is almost certainly changing over time as the wagon collides with the wall, we can easily calculate the average Force of the wall on the wagon. Note that we're indicating directions using positive and negative signs. Gun firing a bullet A 10.0 kg gun with an 80.0 cm long barrel fires a 130-gram bullet to the right with a velocity of +400 m/s. Calculate the acceleration of the bullet while in the barrel of the gun. Calculate the time over which the bullet accelerated. Calculate the average Force, magnitude and direction, applied by the gun to the bullet. Calculate the impulse on the bullet from the gun, magnitude and direction. Calculate the impulse on the gun from the bullet, magnitude and direction. Calculate the recoil velocity of the gun. Solution:) This type of a problem can be classified as a "recoil" problem: two masses are together at the beginning of the problem, and then a force is applied between them, causing them to move away from each other. We can consider the problem in terms of the masses, the time over which the force is applied, the initial and final velocities of both masses, etc.As we solve these problems, take a look at the values that we're getting. Do they seem reasonable to you? The bullet starts out with an initial velocity of zero, but then after being fired, exits the gun with a velocity of +400 m/s (to the right). Using kinematics to find the acceleration: Use kinematics again to find the time of acceleration (while the bullet is moving through the barrel): We can calculate the average Force applied by the gun to the bullet using Fnet = ma: Based on what we've learned so far we can calculate the impulse using either Force and time, or mass and change in velocity. Either one will work. These impulses are positive, and thus to the right. We could go to the trouble of using the equal-and-opposite force on the gun from the bullet in the opposite direction to do similar calculations, but it should be clear that equal magnitudes of forces over the same period of time will produce equal magnitudes of impulses. The impulse on the gun, however is in the opposite direction, -x in this case. We have multiple strategies that we can use here as well, knowing so much about the quantities involved in this problem. Let's use impulse on the gun to find its change in velocity during the firing of the bullet. 9.3. Conservation of Linear Momentum Two particles that are interacting with each other via some force apply equal and opposite forces to each other (Newton's 3rd Law), for some amount of time t. Because the forces are equal in magnitude and the time of the interaction is the same for both objects, the impulse the objects apply to each other is also equal in magnitude, but opposite in direction. Take, for example, these two billiard balls that collide in space. Let's consider the force on the 3 ball from the 6 ball, F3,6, and the force pair, the force on the 6 ball from the 3, F6,3. In this derivation you can see that we sometimes use "prime" to indicate final values. This is not meant to indicate a derivative. Law of Conservation of Momentum "Whenever two or more particles in an isolated system interact, their total linear momentum remains constant." Ball thrown into a wagon A 7.5 kg bowling ball is thrown horizontally at 3.2 m/s into a 5.0 kg cart that is initially at rest. The ball lands in the cart and sticks there. What is the velocity of the cart after the ball lands in it? In one trial the ball doesn't land in the cart, but instead hits the front side of the cart, knocking it forward at 2.2 m/s. What is the velocity of the ball after the collision? Solution:) The problem doesn't describe a direction for the motion, so I'm going to assume the ball was thrown in the +x direction. So: Because the ball lands in the cart and sticks there, they have the same final velocity, so: Now the two masses have different velocities: 9.4. Collisions We've been discussing how conservation of linear momentum is related to particles that are interacting with each other via a force. One common type of interaction is a collision between two objects, and we can split our analysis up according to the type of collision: Elastic collisions Inelastic collisions Perfectly inelastic collisions The three types of collisions are similar in some ways, but there are some important differences as well, which will affect how you analyze the situation. 9.4.1. Elastic collisions Elastic Collision An elastic collision is one in which there is a negligible amount of energy converted to heat in the process of the collision. We know that momentum is conserved in collisions, and speaking generally, energy is conserved as well, although in the vast majority of collisions, some energy is converted ("lost") to thermal energy: the bodies colliding become permanently deformed or bent, and the sound of a crash or a crunch can clearly be heard. But some collisions lose so little energy that kinetic energy is effectively conserved: the total K of all the bodies before the collision is the same as their total K after the collision. In these cases, we say that the collision is elastic. How to solve an elastic collision problem? Elastic collisions are so rare that there are just a few circumstances you might consider elastic. Typically collisions between molecules are considered elastic, or perhaps collisions between billiard balls. For some problems you may simply be told to "consider this as an elastic collision." The equations that you can use to help figure out what's going on are: 9.4.2. Inelastic collisions Inelastic Collision An inelastic collision is one in which some energy is converted to heat in the process of the collision. In this type of collision we have significant amounts of kinetic energy that are are converted to heat in the process of the collision. As a result, kinetic energy is no longer conserved, although momentum still is. Inelastic collisions are very common. Because the forces involved in collisions are so great compared to the other forces that might be present in a situation—a normal force, a force of gravity, etc.—we typically ignore those other forces and focus on the single force of collision to find out the velocities of the colliding objects just before and just after the collision. In the photo here, the two cars have collided, and clearly show some damage, some deformation as a result. It takes work to deform a mass, resulting in ΔEinternal. But we can still use conservation of momentum to determine their velocities immediately after collision. (Once the collision is over, friction forces will bring the moving cars to a halt, of course, but for that part of the problem we'd do a different analysis.) How to solve an inelastic collision problem? Unless your problem states that a collision can be considered elastic—a very special case—you should assume that your problem is inelastic: some kinetic energy was converted into thermal energy in the course of the collision. We can no longer use conservation of Kinetic Energy in our analysis. The equation that you can use to help figure out what's going on is: Demo: Deforming metal and ΔEinternal Take a length of coat hanger wire and do Work on it by bending a section of it back and forth for 10-15 seconds. Then carefully touch the section that was being deformed. How can you tell that your Work was converted to ΔEinternal in the wire? 9.4.3. Perfectly inelastic collisions Perfectly inelastic Collision A perfectly inelastic collision is a special type of inelastic collision, in which the two bodies stick together in the process of the collision. As a result they have the same final velocity after the collision. Here, too, we see kinetic energy converted to heat, making this a type of inelastic collision. Because the colliding bodies stick together, however, the conservation of momentum analysis is slightly simplified. One common example of a perfectly inelastic collision is that of a baseball player catching a moving ball in their mitt. Just before the ball is caught, the ball and the mitt have different velocities. After the ball is caught, it and the mitt are together moving with a different velocity: in some cases they might both be moving backwards at a smaller velocity, in other situations they might be moving forward, or even come to a complete halt. It depends on the circumstances. How to solve perfectly inelastic collision problem? Perfectly inelastic collisions occur anytime the two colliding objects stick together. The equation that you can use to help figure out what's going on is: 9.4.4. Example Problems Ballistic pendulum A ballistic pendulum is used to measure the speed of a projectile: a 5.0 gram bullet is fired into a 1.0 kg block of wood. The bullet sticks into the wood, and the bullet-block swing up to a height of 5.0 cm. Find the initial speed of the projectile just before it hit the block. Find the energy lost in the collision between the block and the projectile. Solution:) This is a classic problem because it consists of two parts, and because it focuses in on making sure we clarify our understanding of both parts of the problem. In the first part of the problem the bullet collides with the block, an analysis that we can consider using conservation of energy. Immediately after that collision the bullet-block swing up as a pendulum, which is a conservation of energy problem. Which part of the problem you solve first depends on what we've been given. This being an inelastic collision problem, I'm tempted to jump straight into solving it with conservation of momentum: That one equation has two unknowns. I could solve for the initial velocity of the bullet if I knew what the final velocity of the bullet-block mass was just after the collision. How can I figure that out?A pendulum swinging up is a conservation of energy problem, and I have enough information to be able to solve that. So I'll do that first, and then work my way backwards from there. This "initial velocity" for the energy part of the problem is the same as the "final velocity" from the collision part of the problem. So: We now know the masses and velocities of the bullet and block both before and after the collision, so we should be able to use those with a conservation of energy equation to figure out what the ΔEint was. Jane and Tarzan Tarzan (69kg) is on the ground, about to be eaten by a tiger. Jane (62 kg) climbs 14 meters up a nearby tree and swings down towards the ground on a vine. She swings past Tarzan, grabbing him as she passes by, and they swing together up into another tree. What is the maximum height that Jane and Tarzan can reach on their upswing? Solution:) This is another classic problem because there are now three parts to the problem: Jane swinging down Jane grabbing Tarzan Jane and Tarzan swinging up together Which of these are energy problems? Which of these are momentum problems? Let's take care of the question of Jane swinging down first, and use conservation of mechanical energy to get her velocity just before she grabs Tarzan. Now there's a collision to analyze using conservation of linear momentum: And immediately after the collision, with their new velocity, let's see how far they can swing upwards: Bullet and a block A 100 gram rubber bullet is fired horizontally at an 800 gram block of wood which is sitting on top of a flat surface. The coefficient of kinetic friction between the block of wood and the surface is 0.70, and the block slides 50 cm before coming to rest. If the bullet collided elastically with the block, what were its initial and final velocities? Solution:) This is an "elastic collision," so the first thing that might be helpful is knowing how to manipulate those equations. Now that we know that we've that useful relationship, we can solve this problem. First, let's use conservation of energy to figure out what the bullet-block's final velocity was just after the collision. (We could use Fnet=ma and kinematics, too.) That final velocity after the collision is also the initial velocity it had just as it started sliding across the surface. Now, let's work backwards to see what we can discover about the collision itself. 9.4.5. Two-Dimensional Collisions Because momentum is a vector, we need to take into account the direction of momentum, which we have been able to do up until know by considering positive and negative directions along a single axis. If a problem occurs in 2- or 3-dimensional space, it requires considering linear momentum along those other axes as well. Car Collision A 1000 kg car traveling north at 4 m/s collides with a 800 kg car traveling east at 3 m/s. After the masses of the two cars have locked together, what is the final velocity of the wreckage just after the collision? Solution:) Because the two cars lock together in the collision, we take that to mean that they have a shared final velocity, making this a perfectly inelastic collision. It's occurring in 2-dimensions, however, so we'll need to solve the problems by breaking it up into components (or using i-j notation if you prefer). In the x-direction: And in the y-direction: Two-dimensional elastic collisions have three equations that can be used to solve the problem: With three equations to use, that's three unknowns that a hard problem might have you solve for, and solving those simultaneous equations is no fun. One subset of the 2-D elastic collision problem is that of a "glancing collision," when one mass comes along to strike another identical mass that is stationary. In those situations, it can be shown that the final velocities of the two masses are at a 90-degree angle to each other. In the situation shown here, the 5-ball is traveling in the positive-x direction at some velocity when it collides with the stationary 2-ball. The 2-ball is knocked up and to the right, and the 5-ball continues down and to the left. The angle between those two paths after the collision is 90-degrees. The proof that this happens is more extensive than we want to go into here, but the result can be summarized in the momentum analysis here: If momentum is going to be conserved, the initial momentum vector p5 will need to maintained after the collision, and we can see that p5' and p2' do indeed graphically (tip-to-tail) sum to the original vector. Playing Pool In the diagram shown above, the 5-ball is traveling east with an initial velocity of 1.00 m/s when it hits the 2-ball, which is knocked so that it travels at 50 degrees north of east. Assuming this is an elastic collision: At what angle is the 5-ball deflected after the collision? What are the final speeds of each ball? Solution:) As mentioned above, the angle between the two masses after an elastic glancing collision of this type is 90-degrees. Therefore, the 5-ball is traveling at 90 - 50 = 40 degrees south of east after the collision. Kinetic energy is conserved, and momentum is conserved in both the x and y directions. Here's one way to solve for the final velocities, using a momentum analysis: We can solve these simultaneous "two equations in two unknowns" now: And substituting back in: 9.5. Center of Mass Most people have something of an intuitive idea of where the center-of-mass of an object is. Maybe it's as simple as "the point where you'd put your finger if you wanted to balance a softball bat across it." It's important to note right away that the center of mass is not a position at which there is equal mass on either side. In the example shown below, there are just two masses attached to a stick of negligible mass. There is much more mass on the left side, but its position is closer is closer to the center-of-mass as well. There is less mass on the right side, but it is located farther away. Both masses end up having the same "leverage" on the stick. Center of Mass The center-of-mass is located at the "weighted average position of the system's mass." A force F applied at the center of mass of a system of total mass m will cause it to accelerate in the direction of F, without rotation. Let's try that with some numbers. Calculating center of mass In the diagram above, the blue cylinder has a mass of 3m and the red cylinder has a mass of 1m. The two masses are attached to a thin light rod so that they are 1.0 meters apart. Calculate the center of mass of the system. Solution:) With masses distributed along a line, a one-dimensional arrangement, its easy to calculate the center-of-mass. It's almost as easy to calculate the center of mass for a two-dimensional arrangement of masses, or even a three-dimensional arrangement of masses. There are a few ways that you can express these two- and three-dimensional relationships. Center of Mass in 2- and 3-dimensions We can express these individual components as a multi-dimensional r vector as well. Two-dimensional center of mass, #1 A 2 kg mass is located at position (2,3), and a 4 kg mass is located at position (-2,-1). Sketch this two-mass system, and calculate the position of its center of mass. Solution:) Two-dimensional center of mass, #2 Calculate the position of the center of mass for this arrangement of particles of mass m, 2m, and 4m. Solution:) Let's take a look at the x-component of the center-of-mass first: And for the y-component: 9.5.1. Continuous distribution of Mass Locating the center of mass for a large, extended object can be a little trickier, but the same basic principles apply: large objects have lots of particles, all located various distances from the center-of-mass cm. For these continuous distributions, we're going to need to use integrals to add up the effects of every little small mass dm. The position r of that mass is going to be a factor as well, of course. Single-variable limitations Because r is a vector, potentially in 2- or 3-dimensions, and because this course only requires single-variable calculus, we will be somewhat restricted in the geometries of objects that we can analyze. For these larger distributions of mass, consider the total mass to consist of a series of discrete Δm chunks. We can figure out the x-component of the center-of-mass by summing the weighted-position of those masses. Better, we can let the size of those Δm chunks approach 0, at which point we can express this analysis in integral form: In other dimensions: Taking a look at these integrals, you can see that we have an issue. Consider the ycm integral, for example. We can't integrate a y-based function with respect to m. They're not directly related to each other. We can, however, use density to create functions that we can analyze. Let's see how to do that! 9.5.1.1. Density For these analyses, we’re going to need some way of relating dm to the location of that mass. We can do this by considering either: the linear density λ ("lambda") for linear situations (like a long, thin, rod) the surface area density σ ("sigma") for surface-based analyses, or the volume density ρ ("rho") of the object for volume-based analyses. You're almost certainly aware of "volume density," where Density = mass divided by volume: This works for the overall density of a volume, but we can also consider the very small mass of a very small Volume of that larger mass: We can also express this relationship in a slightly different way: With this new relationship between dm and dV we can solve the integrals shown above. We'll see how to do that in just a moment. Continuous distributions of mass and density For a given axis: For substituting into the integral to solve for center of mass: Let's solve some problems. Long, thin, uniform, rod Assuming that a long, thin, rod has a uniform linear density λ, show that the center of mass for the rod is in the middle. Solution:) We already know that the center of mass for a uniform rod is in the middle of the length of that rod, but let's use this opportunity to practice using the integral. Long, thin, rod with changing density Assuming that a long, thin, rod has a changing linear density λ = α x, where α is a positive constant, calculate the location of the center of mass. Solution:) This rod is a little more interesting. At x = 0 the rod has a mass of dm = α dx = 0 kg, while at the end L it has a dm based on a linearly-increasing density. Let's solve that integral: This is fine so far, but... we don't know the mass M. How can determine that value? Substituting back in: For a mostly linear object like a baseball bat or a long, thin rod, we can think of the center-of-mass as being that location where, if we were to put our finger there, the object would balance, with gravity having an equal effect on either side of the object. In a similar way, with a 2-dimensional object like the triangle in the next problem, we can imagine attaching a string to the 2-dimensional surface. If we attach that string at the center-of-mass in both the x and y directions, the triangle will maintain a horizontal position as we support it. If we attach the string at any other location, the forces acting on it will supply uneven torque effects, causing the triangle to hang to one side. A triangle Find the center of mass of the 2-d right triangle shown here. The triangle has a constant thickness and density. Solution:) This is an interesting problem, because we need to use the integral to calculate center-of-mass twice, once in the x direction and once in the y direction. Let's focus on the x direction, and because we're working with an area, we'll use the "surface density" form of the integral: This is awkward. We're trying to integrate x with respect to area A? Where is dA? How is it related to x? The dA refers to each of the rectangles shown here that, taken together, over the entire area of the triangle. Each of those rectangles has a width dx, and a height y related to the value of x. Related how? The hypotenuse of the triangle follows the function y = mx, where m = the slope of the line, b/a. Let's put that all together. Substituting this into our center-of-mass integral: This is pretty good, but let's again see if we can simply things by calculating the mass of the triangle: The conclusion is that the center-of-mass along the x-axis is 2/3 away from the "skinny" end. We can extrapolate that result to identify the center-of-mass along the y-axis, also 2/3 away from the "skinny" end of the triangle. 9.5.2. A System in Motion Now that we know a little about center-of-mass for objects and systems, let's do a derivation in which we take time time-derivative of a center of mass. The time-derivative of position is velocity, so All of these mv values are the individual momentums of each component of the system. Let's rearrange the equation a little: This is an interesting result because it reveals that the total momentum of a system of moving particles is the same as the momentum of the center-of-mass of that system. This remains true even if the velocities of the individual particles are moving in different directions (as long as their is no external Force providing an impulse to the system). This animation from The Mechanical Universe demonstrates the effect nicely. Ice hockey is dangerous A hockey puck of mass m is moving at 10 m/s in the +x direction when it explodes into two pieces, one of them with mass m/3 traveling in the +x direction at 15m/s. What is the velocity of the second piece after the explosion? Where is the center of mass of the system 3.0 seconds after the explosion? Solution:) We can solve for the final velocity of the second piece simply by using conservation of momentum. This is effectively a "recoil" problem, except that the system is moving when the two pieces come apart: 3.0 seconds after the explosion we have two ways that we can identify the location of the center-of-mass. The simplest is to realize that the explosion doesn't affect the velocity of the center-of-mass. If it was moving at 10 m/s in the +x direction before the explosion, it will continue to move at that rate after the collision. Therefore: It's a little more tedious, but perhaps useful to take a look at the individual positions of the two masses 3.0 seconds later, and determine the center-of-mass based on those values: They're the same! There is one more interesting derivation that we can perform here. We have determined that the total momentum of a system is equal to the momentum of its center of mass: What happens if we take the time-derivative of that equation? This is Newton's Second Law of Motion! It's actually a slightly different form from the original. Our original Fnet = ma applied a particle of mass m. Here, we have arrived at an equation that justifies our application of the Second Law to a larger system of particles. Applying a net Force to the center-of-mass of that system will cause it to accelerate linearly, just as if it were a single particle.
12704	https://www.echemi.com/community/what-are-the-other-indicators-we-use-in-the-na2co3-and-hcl-titration_mjart2204141557_99.html	What are the other indicators we use in the Na2CO3 and HCL titration? - ECHEMI  Community  Sign in \| Join free Home>Community>What are the other indicators we use in the Na2CO3 and HCL titration? Upvote 21 Downvote Titration Chemistry Hydrochloric acid Posted by Katejenner What are the other indicators we use in the Na2CO3 and HCL titration? Although Na2CO3 is salt it hydrolyzes to produce strong alkali NaOH.This NaOH gets titrated. David WrixonFollowFollowing Although Na2CO3 is salt it hydrolyzes to produce strong alkali NaOH.This NaOH gets titrated. More Upvote VOTE Downvote Methyl orange shows pink colour in acidic medium and yellow colour in basic medium. Because it changes colour at the pH of a mid strength acid, it is usually used in titration for acids. Unlike a universal indicator, methyl orange does not have a full spectrum of colour change, but it has a sharp end point. Dimitra TsiovolouFollowFollowing Methyl orange shows pink colour in acidic medium and yellow colour in basic medium. Because it changes colour at the pH of a mid strength acid, it is usually used in titration for acids. Unlike a universal indicator, methyl orange does not have a full spectrum of colour change, but it has a sharp end point. More Upvote VOTE Downvote There are two indicators such as phenolphthalein and methyl orange are used in the Na2CO3 and HCl titration. Out of them, phenolphthalein colour change is easy to spot but methyl orange colour change is difficult to judge. Bernardo FagioliFollowFollowing There are two indicators such as phenolphthalein and methyl orange are used in the Na2CO3 and HCl titration. Out of them, phenolphthalein colour change is easy to spot but methyl orange colour change is difficult to judge. More Upvote VOTE Downvote Yes…. sodium carbonate is a primary standard substance and you can standardize HCl with it easily. As mentioned in the literature you can use methyl orange as an indicator although I always preferred bromothymol blue! ;-) Angelo ParedesFollowFollowing Yes…. sodium carbonate is a primary standard substance and you can standardize HCl with it easily. As mentioned in the literature you can use methyl orange as an indicator although I always preferred bromothymol blue! ;-) More Upvote VOTE Downvote The methyl orange should encompass all of the alkalinity by converting it all to carbonic acid at an end point around pH 4.2. Phenolphthalein has an end point of around 8.2pH. At this point all the carbonate alkalinity is converted but only about half of the bicarbonate is converted. Craig DorschelFollowFollowing The methyl orange should encompass all of the alkalinity by converting it all to carbonic acid at an end point around pH 4.2. Phenolphthalein has an end point of around 8.2pH. At this point all the carbonate alkalinity is converted but only about half of the bicarbonate is converted. More Upvote VOTE Downvote You need two indicators because the reaction has two end points Phenolphthalein changes to clear at the first end point. Add some methyl orange to get the second end point. [math]Na_2CO_3 + HCl = NaHCO_3 + NaCl[/math] [math]NaHCO_3 + HCl = CO_2 + H_2O + NaCl[/math] Alan ApplebyFollowFollowing You need two indicators because the reaction has two end points Phenolphthalein changes to clear at the first end point. Add some methyl orange to get the second end point. [math]Na_2CO_3 + HCl = NaHCO_3 + NaCl[/math] [math]NaHCO_3 + HCl = CO_2 + H_2O + NaCl[/math] More Upvote VOTE Downvote Look at the answer of Pete Gannett to “Why is Na2CO3 used as a primary standard?” But soda isn’t overwhelming good. pH titrations can be problematic, because you can use it against srong acids only (or have to fight with “self produced buffer solutions” obscuring the point of equivalence). The soda used must be good cystallized pure dekahydrate, no impurities of monohydrate or water free carbonate disturbing the weight. Dong-Yoon LeeFollowFollowing Look at the answer of Pete Gannett to “Why is Na2CO3 used as a primary standard?” But soda isn’t overwhelming good. pH titrations can be problematic, because you can use it against srong acids only (or have to fight with “self produced buffer solutions” obscuring the point of equivalence). The soda used must be good cystallized pure dekahydrate, no impurities of monohydrate or water free carbonate disturbing the weight. More Upvote VOTE Downvote There is no reason to boil the solution after the titration. After the titration, you are done, it doesn't make any sense to boil the solution. But it does make sense to boil the solution during the titration: Why during titration of HCL and Na2CO3 the solution is boiled just before reaching the second equivalence point? CharlieFollowFollowing There is no reason to boil the solution after the titration. After the titration, you are done, it doesn't make any sense to boil the solution. But it does make sense to boil the solution during the titration: Why during titration of HCL and Na2CO3 the solution is boiled just before reaching the second equivalence point? 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12705	https://www.quora.com/How-can-you-prove-that-if-x-2-1-then-xs-either-1-or-1	How to prove that if x^2=1 then x's either -1 or 1 - Quora Something went wrong. Wait a moment and try again. Try again Skip to content Skip to search Sign In Mathematics Proof Theory (mathematics... Square Roots (functions) Algebraic Equations Basic Concepts of Mathema... Algebra 1 Math Proofs Mathematical Sciences Algebra 5 How can you prove that if x^2=1 then x's either -1 or 1? All related (36) Sort Recommended Marvin Kalngan Civil Engineer · Author has 1.4K answers and 726.6K answer views ·1y You can solve some quadratic equations by using the Square Root Property. For any real number n n, if x 2=n x 2=n, then x=±√n x=±n. Therefore, x 2=1 x 2=1 x=±√1 x=±1 x=±1 x=±1 Regards, Marvin Kalngan Upvote · 9 1 Promoted by Coverage.com Johnny M Master's Degree from Harvard University (Graduated 2011) ·Updated Sep 9 Does switching car insurance really save you money, or is that just marketing hype? This is one of those things that I didn’t expect to be worthwhile, but it was. You actually can save a solid chunk of money—if you use the right tool like this one. I ended up saving over $1,500/year, but I also insure four cars. I tested several comparison tools and while some of them ended up spamming me with junk, there were a couple like Coverage.com and these alternatives that I now recommend to my friend. Most insurance companies quietly raise your rate year after year. Nothing major, just enough that you don’t notice. They’re banking on you not shopping around—and to be honest, I didn’t. Continue Reading This is one of those things that I didn’t expect to be worthwhile, but it was. You actually can save a solid chunk of money—if you use the right tool like this one. I ended up saving over $1,500/year, but I also insure four cars. 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Absolutely not Definitely yes Upvote · 9 1 Raymond Beck Former Infantry Sergeant · Author has 42.4K answers and 10.3M answer views ·Jan 2 x^2 = 1 take square roots of both sides x = + or - 1 or factor x^2–1 = 0 (x+1)(x-1) = 0 set each factor = 0 and solve for x x+1= 0, x-1= 0 x = -1 x= +1 or use Decartes’ Rule of Signs x^2 -1 = 0 1 positive real root 1 negative real root try trial & error, simple small integers x= 0 nope x= 1 1^2 -1 = 0 yes x=-1 (-1)^2 -1 = 0 yes or graph the equation and find the x-intercepts = roots= zeros = solutions = (1,0) and (-1,0) a quadratic, degree 2, equation has maximum 2 solutions. if you find 2 solutions, then that’s it, all she wrote, The End Upvote · George Ivey Former Math Professor at Gallaudet University · Author has 23.7K answers and 2.6M answer views ·1y If x^2=1 then, subtracting 1 from both sides x^2–1= (x- 1)(x+ 1)= 0. Either x- 1= 0, so x= 1, or x+ 1= 0 so x= -1. Upvote · Related questions More answers below Why is −1×−1=1−1×−1=1? How do I prove that 1+x+x 2+x 3+⋯=1 1−x 1+x+x 2+x 3+⋯=1 1−x? How can I calculate x x if (1+1 x)x+1=x(1+1 x)x+1=x? How do you prove cos−1 x=2 sin−1√1−x 2 cos−1⁡x=2 sin−1⁡1−x 2? How can we prove if x=2+2^1/3 +2^-1/ 3 then 2x^3 -12x^2 +18x - 9=0? Jack Smith B.A. in Physics&Mathematics, University of Illinois · Author has 4.2K answers and 11.4M answer views ·10mo x 2=1 x 2=1 Take the principal square root on both sides: \|x\|=1\|x\|=1 If the absolute value of something is equal to a positive number, then that something is either that number itself or its additive inverse: x=±1 x=±1 Upvote · Promoted by The Penny Hoarder Lisa Dawson Finance Writer at The Penny Hoarder ·Updated Sep 16 What's some brutally honest advice that everyone should know? Here’s the thing: I wish I had known these money secrets sooner. They’ve helped so many people save hundreds, secure their family’s future, and grow their bank accounts—myself included. And honestly? 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Continue Reading etre ou ne pas etre, telle est la question. Upvote · 9 5 9 1 Gerald Bieniek Lives in Corpus Christi, TX (1990–present) · Author has 5K answers and 1.8M answer views ·1y Related How can you prove (-1) x(-1) =1? You can think of integers as being on a number line with negative numbers to the left and positive numbers to the right. You add numbers on this line by measuring out two distances from zero, and shifting one segment’s start point to the other segment’s endpoint. This makes the lines longer when both are positive or both are negative, and shorter when one is positive and the other is negative. Congratulations, you’ve just made “subtraction” But what is multiplication? It’s just repeated addition, right? 3 x 4 = 12 is just taking a length of 3 and adding it a total of four times (starting from ze Continue Reading You can think of integers as being on a number line with negative numbers to the left and positive numbers to the right. You add numbers on this line by measuring out two distances from zero, and shifting one segment’s start point to the other segment’s endpoint. This makes the lines longer when both are positive or both are negative, and shorter when one is positive and the other is negative. Congratulations, you’ve just made “subtraction” But what is multiplication? It’s just repeated addition, right? 3 x 4 = 12 is just taking a length of 3 and adding it a total of four times (starting from zero). But how do you multiply a negative number? The exact same way, but instead of going right by 3 four times, you are going left by 3 four times. -3 x 4 = -12 The multiplication with a negative results in the same answer as when both numbers are positive. Therefore, you can say that -a is the same as +a, just flipped around. And if both are negative? Well, you do the multiplication, then flip it once to negative, then flip it back to positive. -1 x -1 = 1 Upvote · 9 1 Sponsored by CDW Corporation Looking for ways to leverage AI to improve customer experiences? CDW’s AI solutions include 24/7 chatbots and workshops to enhance efficiency and customer satisfaction. Learn More 99 23 Balázs Iván József Master in Mathematics, Eötvös Loránd University (Graduated 1983) · Author has 5.3K answers and 1.8M answer views ·1y Related How can you prove (-1) x(-1) =1? If we want the usual properties (including distributivity), we have to define the operations as follows. 0 + x = x Let’s multiply this with any y 0.y + x.y = x.y Therefore 0.y better be 0. Now x + (-x) = 0 Let’s multiply this with any y x.y + (-x).y = 0 therefore (-x).y = - (x.y) Let us apply this for x=1, y=-1, (-1).(-1) = - ( 1.(-1)) but 1.z =z for every z. so = - (-1)) = 1 Upvote · 9 2 Stanley Rabinowitz Ph.D. in Algebraic Number Theory, City University of New York (CUNY) (Graduated 1970) ·Updated 1y Related How do you solve (2 x 2+x+1)2 x 2+x−1<1(2 x 2+x+1)2 x 2+x−1<1 ? 2 x 2+x+1=2((x+1 4)2+7 16)>0.2 x 2+x+1=2((x+1 4)2+7 16)>0. Since l n(x)l n(x) and e x e x are strictly increasing, the given problem is equivalent to (A) (2 x 2+x−1)l n(2 x 2+x+1)<0.(2 x 2+x−1)l n(2 x 2+x+1)<0. Case 1. 2 x 2+x+1<1 2 x 2+x+1<1. Then 2 x 2+x−1=2 x 2+x+1−2<1−2<0 2 x 2+x−1=2 x 2+x+1−2<1−2<0. Hence, (A) is false. Case 2. 2 x 2+x+1=1 2 x 2+x+1=1. Clearly, (A) is false. Case 3. (B) 2 x 2+x+1>1 2 x 2+x+1>1. In this case, (A) is true ⟺⟺ (C) 2 x 2+x−1<0.2 x 2+x−1<0. (B) is equivalent to x(2 x+1)>0 x(2 x+1)>0 which is equivalent to (D) x<−1 2∨x>0.x<−1 2∨x>0. (C) is equivalent to (2 x−1)(x+1)<0(2 x−1)(x+1)<0 which is equivalent to (E)−1<x<1 2.−1<x<1 2. Thus, (A) holds ⟺⟺((D) ∧∧ (E)) holds whic Continue Reading 2 x 2+x+1=2((x+1 4)2+7 16)>0.2 x 2+x+1=2((x+1 4)2+7 16)>0. Since l n(x)l n(x) and e x e x are strictly increasing, the given problem is equivalent to (A) (2 x 2+x−1)l n(2 x 2+x+1)<0.(2 x 2+x−1)l n(2 x 2+x+1)<0. Case 1. 2 x 2+x+1<1 2 x 2+x+1<1. Then 2 x 2+x−1=2 x 2+x+1−2<1−2<0 2 x 2+x−1=2 x 2+x+1−2<1−2<0. Hence, (A) is false. Case 2. 2 x 2+x+1=1 2 x 2+x+1=1. Clearly, (A) is false. Case 3. (B) 2 x 2+x+1>1 2 x 2+x+1>1. In this case, (A) is true ⟺⟺ (C) 2 x 2+x−1<0.2 x 2+x−1<0. (B) is equivalent to x(2 x+1)>0 x(2 x+1)>0 which is equivalent to (D) x<−1 2∨x>0.x<−1 2∨x>0. (C) is equivalent to (2 x−1)(x+1)<0(2 x−1)(x+1)<0 which is equivalent to (E)−1<x<1 2.−1<x<1 2. Thus, (A) holds ⟺⟺((D) ∧∧ (E)) holds which is equivalent to (−1<x<−1/2)∨(0<x<1/2).(−1<x<−1/2)∨(0<x<1/2). Upvote · 9 4 Sponsored by Spokeo Is there a way to find out if someone has a dating profile? Yes. All you need to do is enter their name here to see what dating websites or apps they are on. Learn More 17K 17K Elizabeth Jean Stapel tutor (1989 to present), instructor (1991+), author (1998+) · Author has 5.9K answers and 3.3M answer views ·1y Related How can you prove (-1) x(-1) =1? One way to prove it is to ask, “Well, what else could it be?” And I’m not being flip; real mathematics is often done this way. I doubt anybody has any argument with the (1)(1) = 1 part; the only quarrel is with the “minus times minus is plus” part. So let’s assume, for the sake of this argument, that “minus times minus” is not “plus”; that is, let’s assume the following: (−1)×(−1)=−1(−1)×(−1)=−1 (It’s the only other option, right?) I believe we all agree that 1−1=0 1−1=0, that 0(a)=0 0(a)=0 for any number a a, and that the Distributive Law is a true rule of arithmetic. To refresh: The Distributive Law Continue Reading One way to prove it is to ask, “Well, what else could it be?” And I’m not being flip; real mathematics is often done this way. I doubt anybody has any argument with the (1)(1) = 1 part; the only quarrel is with the “minus times minus is plus” part. So let’s assume, for the sake of this argument, that “minus times minus” is not “plus”; that is, let’s assume the following: (−1)×(−1)=−1(−1)×(−1)=−1 (It’s the only other option, right?) I believe we all agree that 1−1=0 1−1=0, that 0(a)=0 0(a)=0 for any number a a, and that the Distributive Law is a true rule of arithmetic. To refresh: The Distributive Law is the one that “distributes multiplication over addition/subtraction”, so, for instance: 3(2+4)=3(6)=18 3(2+4)=3(6)=18 …but also, by the Distributive Law: 3(2+4)3(2+4) 3(2)+3(+4)3(2)+3(+4) 6+12 6+12 18 18 We’re also going to assume that “plus times minus is minus” and that the sum of two negative numbers is also negative (nobody seems to have any issues with these two rules). So we are working under these assumptions: The Distributive Law holds.1. The Distributive Law holds. 2.1−1=0 2.1−1=0 3.0(a)=0 for any number a 3.0(a)=0 for any number a 4.(−1)(−1)=−1 4.(−1)(−1)=−1 5.1(−1)=−1 5.1(−1)=−1 6.−1+(−1)=−2 6.−1+(−1)=−2 The only suspect assumption is (4). Then we can say the following: 0=0(−1)=(1−1)(−1)=(1+(−1))(−1)=(1)(−1)+(−1)(−1)=−1+(−1)(∗)=−2 0=0(−1)=(1−1)(−1)=(1+(−1))(−1)=(1)(−1)+(−1)(−1)=−1+(−1)(∗)=−2 But this is nonsense! Zero equals only zero; it certainly never somehow equals −2−2! How did we end up with this garbage? When you work from assumptions and end up with garbage, then at least one of the assumptions must have been wrong. Our only suspect assumption was that “minus times minus is plus” was wrong. I used this assumption in the line marked with an asterisk (or “star”) above. Assuming in thi step that “minus times minus” was somehow also a minus led us directly to a nonsensical result. This proves that the assumption was wrong. (And, in technical math-ese, is a “proof by contradiction”.) Therefore, it must be that “minus times minus is plus”. Upvote · 9 1 9 1 Ron Davis I earn my living with mathematics. · Author has 6.7K answers and 18.8M answer views ·3y Related How do I prove that the solutions to X^X=X are 1 and minus 1? To find the solutions of x x=x,x x=x, divide both sides of the equation by x:x: \|x\|x−1=1.\|x\|x−1=1. One solution to this equation is that the power is 1; x−1=0;x=1.x−1=0;x=1. For any other solution, \|x\|=1.\|x\|=1. That implies that x=e i θ,x=e i θ, in which x x and θ θ are both real. A necessary condition to satisfy these conditions is θ=n π,θ=n π, in which n n is an integer. The only such values of n n that yield distinct solutions are n=0;x=1 n=0;x=1 (already derived above), and n=1;x=−1.n=1;x=−1. Upvote · Merghaney Mohammed graduated from faculty of mathematical sciences UofK · Upvoted by Michael Jørgensen , PhD in mathematics and David Joyce , Ph.D. Mathematics, University of Pennsylvania (1979) ·8y Related What is x 2+1 x 2 x 2+1 x 2, given x+1 x=1 x+1 x=1? The obvious solution is : One may use some interesting solution and more fun: we may even find some more sophisticated substitution, but it is in some way not obvious: I hope that was so fun. Continue Reading The obvious solution is : One may use some interesting solution and more fun: we may even find some more sophisticated substitution, but it is in some way not obvious: I hope that was so fun. Upvote · 99 42 99 11 Related questions If x(x−1)2=1 x(x−1)2=1, what is x? How do you prove that X=1? How do you prove that (x^2) ^(1/2) =\|x\|? How can I prove that x X or 1 = not x? How do you prove 1 A 1 A 2=∫1 0 d x 1 d x 2 δ(2)(x 1+x 2−1)1(x 1 A 1+x 2 A 2)2 1 A 1 A 2=∫0 1 d x 1 d x 2 δ(2)(x 1+x 2−1)1(x 1 A 1+x 2 A 2)2? Why is −1×−1=1−1×−1=1? How do I prove that 1+x+x 2+x 3+⋯=1 1−x 1+x+x 2+x 3+⋯=1 1−x? How can I calculate x x if (1+1 x)x+1=x(1+1 x)x+1=x? How do you prove cos−1 x=2 sin−1√1−x 2 cos−1⁡x=2 sin−1⁡1−x 2? How can we prove if x=2+2^1/3 +2^-1/ 3 then 2x^3 -12x^2 +18x - 9=0? How can you prove that X+(1/x) is greater than 2? How do I prove 1 / (x^6 + 1) = 1 / [3 (x^2 + 1)] - (x^2 - 2) / [3 (x^4 - x^2 + 1)]? If x x−1=x 1−x x x−1=x 1−x, what is x x? How can I prove that 1^x = 1^y? Is there a proof that x=x? Related questions How do I prove that 1+x+x 2+x 3+⋯=1 1−x 1+x+x 2+x 3+⋯=1 1−x? How do you prove this without CAS? ∫1 0 d x x x=1 1 1+1 2 2+1 3 3+⋯∫0 1 d x x x=1 1 1+1 2 2+1 3 3+⋯ How can I calculate x x if (1+1 x)x+1=x(1+1 x)x+1=x? How can I prove that x X or 1 = not x? How do you solve x^(x+1) = (x+1) ^x? How do you prove cos−1 x=2 sin−1√1−x 2 cos−1⁡x=2 sin−1⁡1−x 2? Advertisement About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
12706	https://web.mit.edu/16.070/www/lecture/big_o.pdf	Big O notation (with a capital letter O, not a zero), also called Landau's symbol, is a symbolism used in complexity theory, computer science, and mathematics to describe the asymptotic behavior of functions. Basically, it tells you how fast a function grows or declines. Landau's symbol comes from the name of the German number theoretician Edmund Landau who invented the notation. The letter O is used because the rate of growth of a function is also called its order. For example, when analyzing some algorithm, one might find that the time (or the number of steps) it takes to complete a problem of size n is given by T(n) = 4 n2 - 2 n + 2. If we ignore constants (which makes sense because those depend on the particular hardware the program is run on) and slower growing terms, we could say "T(n) grows at the order of n2" and write:T(n) = O(n2). In mathematics, it is often important to get a handle on the error term of an approximation. For instance, people will write ex = 1 + x + x2 / 2 + O(x3) for x -> 0 to express the fact that the error is smaller in absolute value than some constant times x3 if x is close enough to 0. For the formal definition, suppose f(x) and g(x) are two functions defined on some subset of the real numbers. We write f(x) = O(g(x)) (or f(x) = O(g(x)) for x -> to be more precise) if and only if there exist constants N and C such that \|f(x)\| C \|g(x)\| for all x>N. Intuitively, this means that f does not grow faster than g. If a is some real number, we write f(x) = O(g(x)) for x -> a if and only if there exist constants d > 0 and C such that \|f(x)\| C \|g(x)\| for all x with \|x-a\| < d. The first definition is the only one used in computer science (where typically only positive functions with a natural number n as argument are considered; the absolute values can then be ignored), while both usages appear in mathematics. Here is a list of classes of functions that are commonly encountered when analyzing algorithms. The slower growing functions are listed first. c is some arbitrary constant. notation name O(1) constant O(log(n)) logarithmic O((log(n))c) polylogarithmic O(n) linear O(n2) quadratic O(nc) polynomial O(cn) exponential Note that O(nc) and O(cn) are very different. The latter grows much, much faster, no matter how big the constant c is. A function that grows faster than any power of n is called superpolynomial. One that grows slower than an exponential function of the form cn is called subexponential. An algorithm can require time that is both superpolynomial and subexponential; examples of this include the fastest algorithms known for integer factorization. Note, too, that O(log n) is exactly the same as O(log(nc)). The logarithms differ only by a constant factor, and the big O notation ignores that. Similarly, logs with different constant bases are equivalent. The above list is useful because of the following fact: if a function f(n) is a sum of functions, one of which grows faster than the others, then the faster growing one determines the order of f(n). Example: If f(n) = 10 log(n) + 5 (log(n))3 + 7 n + 3 n2 + 6 n3, then f(n) = O(n3). One caveat here: the number of summands has to be constant and may not depend on n. This notation can also be used with multiple variables and with other expressions on the right side of the equal sign. The notation: f(n,m) = n2 + m3 + O(n+m) represents the statement: ∃C ∃ N ∀ n,m>N : f(n,m)n2+m3+C(n+m) Obviously, this notation is abusing the equality symbol, since it violates the axiom of equality: "things equal to the same thing are equal to each other". To be more formally correct, some people (mostly mathematicians, as opposed to computer scientists) prefer to define O(g(x)) as a set-valued function, whose value is all functions that do not grow faster then g(x), and use set membership notation to indicate that a specific function is a member of the set thus defined. Both forms are in common use, but the sloppier equality notation is more common at present. Another point of sloppiness is that the parameter whose asymptotic behaviour is being examined is not clear. A statement such as f(x,y) = O(g(x,y)) requires some additional explanation to make clear what is meant. Still, this problem is rare in practice. In addition to the big O notations, another Landau symbol is used in mathematics: the little o. Informally, f(x) = o(g(x)) means that f grows much slower than g and is insignificant in comparison. Formally, we write f(x) = o(g(x)) (for x -> ) if and only if for every C>0 there exists a real number N such that for all x > N we have \|f(x)\| < C \|g(x)\|; if g(x) 0, this is equivalent to limx f(x)/g(x) = 0. Also, if a is some real number, we write f(x) = o(g(x)) for x -> a if and only if for every C>0 there exists a positive real number d such that for all x with \|x - a\| < d we have \|f(x)\| < C \|g(x)\|; if g(x) 0, this is equivalent to limx -> a f(x)/g(x) = 0. Big O is the most commonly-used of five notations for comparing functions: Notation Definition Analogy f(n) = O(g(n)) see above f(n) = o(g(n)) see above < f(n) = (g(n)) g(n)=O(f(n)) f(n) = (g(n)) g(n)=o(f(n)) > f(n) = (g(n)) f(n)=O(g(n)) and g(n)=O(f(n)) = The notations and are often used in computer science; the lowercase o is common in mathematics but rare in computer science. The lowercase is rarely used. A common error is to confuse these by using O when is meant. For example, one might say "heapsort is O(n log n)" when the intended meaning was "heapsort is (n log n)". Both statements are true, but the latter is a stronger claim. The notations described here are very useful. They are used for approximating formulas for analysis of algorithms, and for the definitions of terms in complexity theory (e.g. polynomial time). How efficient is an algorithm or piece of code? Efficiency covers lots of resources, including: • CPU (time) usage • memory usage • disk usage • network usage All are important but we will mostly talk about time complexity (CPU usage). Be careful to differentiate between: 1. Performance: how much time/memory/disk/... is actually used when a program is run. This depends on the machine, compiler, etc. as well as the code. 2. Complexity: how do the resource requirements of a program or algorithm scale, i.e., what happens as the size of the problem being solved gets larger? Complexity affects performance but not the other way around. The time required by a function/procedure is proportional to the number of "basic operations" that it performs. Here are some examples of basic operations: • one arithmetic operation (e.g., +, ). • one assignment (e.g. x := 0) • one test (e.g., x = 0) • one read (of a primitive type: integer, float, character, boolean) • one write (of a primitive type: integer, float, character, boolean) Some functions/procedures perform the same number of operations every time they are called. For example, StackSize in the Stack implementation always returns the number of elements currently in the stack or states that the stack is empty, then we say that StackSize takes constant time. Other functions/ procedures may perform different numbers of operations, depending on the value of a parameter. For example, in the BubbleSort algorithm, the number of elements in the array, determines the number of operations performed by the algorithm. This parameter (number of elements) is called the problem size/ input size. When we are trying to find the complexity of the function/ procedure/ algorithm/ program, we are not interested in the exact number of operations that are being performed. Instead, we are interested in the relation of the number of operations to the problem size. Typically, we are usually interested in the worst case: what is the maximum number of operations that might be performed for a given problem size. For example, inserting an element into an array, we have to move the current element and all of the elements that come after it one place to the right in the array. In the worst case, inserting at the beginning of the array, all of the elements in the array must be moved. Therefore, in the worst case, the time for insertion is proportional to the number of elements in the array, and we say that the worst-case time for the insertion operation is linear in the number of elements in the array. For a linear-time algorithm, if the problem size doubles, the number of operations also doubles. We express complexity using big-O notation. For a problem of size N: a constant-time algorithm is "order 1": O(1) a linear-time algorithm is "order N": O(N) a quadratic-time algorithm is "order N squared": O(N2) Note that the big-O expressions do not have constants or low-order terms. This is because, when N gets large enough, constants and low-order terms don't matter (a constant-time algorithm will be faster than a linear-time algorithm, which will be faster than a quadratic-time algorithm). Formal definition: A function T(N) is O(F(N)) if for some constant c and for values of N greater than some value n0: T(N) <= c F(N) The idea is that T(N) is the exact complexity of a procedure/function/algorithm as a function of the problem size N, and that F(N) is an upper-bound on that complexity (i.e., the actual time/space or whatever for a problem of size N will be no worse than F(N)). In practice, we want the smallest F(N) -- the least upper bound on the actual complexity. For example, consider: T(N) = 3 N2 + 5. We can show that T(N) is O(N2) by choosing c = 4 and n0 = 2. This is because for all values of N greater than 2: 3 N2 + 5 <= 4 N2 T(N) is not O(N), because whatever constant c and value n0 you choose, There is always a value of N > n0 such that (3 N2 + 5) > (c N) In general, how can you determine the running time of a piece of code? The answer is that it depends on what kinds of statements are used. Sequence of statements statement 1; statement 2; ... statement k; The total time is found by adding the times for all statements: total time = time(statement 1) + time(statement 2) + ... + time(statement k) If each statement is "simple" (only involves basic operations) then the time for each statement is constant and the total time is also constant: O(1). If-Then-Else if (cond) then block 1 (sequence of statements) else block 2 (sequence of statements) end if; Here, either block 1 will execute, or block 2 will execute. Therefore, the worst-case time is the slower of the two possibilities: max(time(block 1), time(block 2)) If block 1 takes O(1) and block 2 takes O(N), the if-then-else statement would be O(N). Loops for I in 1 .. N loop sequence of statements end loop; The loop executes N times, so the sequence of statements also executes N times. If we assume the statements are O(1), the total time for the for loop is N O(1), which is O(N) overall. Nested loops for I in 1 .. N loop for J in 1 .. M loop sequence of statements end loop; end loop; The outer loop executes N times. Every time the outer loop executes, the inner loop executes M times. As a result, the statements in the inner loop execute a total of N M times. Thus, the complexity is O(N M). In a common special case where the stopping condition of the inner loop is instead of (i.e., the inner loop also executes N times), the total complexity for the two loops is O(N2). Statements with function/ procedure calls When a statement involves a function/ procedure call, the complexity of the statement includes the complexity of the function/ procedure. Assume that you know that function/ procedure f takes constant time, and that function/procedure g takes time proportional to (linear in) the value of its parameter k. Then the statements below have the time complexities indicated. f(k) has O(1) g(k) has O(k) When a loop is involved, the same rule applies. For example: for J in 1 .. N loop g(J); end loop; has complexity (N2). The loop executes N times and each function/procedure call is complexity O(N).
12707	https://math.stackexchange.com/questions/1374754/inequality-for-the-difference-of-two-products	Inequality for the difference of two products - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Loading… Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company, and our products current community Mathematics helpchat Mathematics Meta your communities Sign up or log in to customize your list. more stack exchange communities company blog Log in Sign up Home Questions Unanswered AI Assist Labs Tags Chat Users Teams Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Try Teams for freeExplore Teams 3. Teams 4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Hang on, you can't upvote just yet. You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Inequality for the difference of two products Ask Question Asked 10 years, 2 months ago Modified2 years, 11 months ago Viewed 2k times This question shows research effort; it is useful and clear 4 Save this question. Show activity on this post. Suppose a 1,…,a k a 1,…,a k and b 1,…,b k b 1,…,b k are complex numbers bounded in absolute value by 1 1. Is it true that \|k∏i=1 a i−k∏i=1 b i\|≤k∑i=1\|a i−b i\|? ∣∣∣∣∏i=1 k a i−∏i=1 k b i∣∣∣∣≤∑i=1 k\|a i−b i\|? inequality Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications edited Jul 26, 2015 at 18:19 user940 asked Jul 26, 2015 at 18:02 justhereforonequestionjusthereforonequestion 51 3 3 bronze badges 2 2 Duplicate of math.stackexchange.com/q/2560702.Martin R –Martin R 2022-10-17 08:04:47 +00:00 Commented Oct 17, 2022 at 8:04 2 Also: math.stackexchange.com/a/1343295/42969Martin R –Martin R 2022-10-17 08:12:35 +00:00 Commented Oct 17, 2022 at 8:12 Add a comment\| 1 Answer 1 Sorted by: Reset to default This answer is useful 9 Save this answer. +50 This answer has been awarded bounties worth 50 reputation by Apprentice Show activity on this post. Consider the telescoping sum a 1⋯a k−b 1⋯b k=k∑i=1 a 1⋯a i−1(a i−b i)b i+1⋯b k. a 1⋯a k−b 1⋯b k=∑i=1 k a 1⋯a i−1(a i−b i)b i+1⋯b k. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Jul 26, 2015 at 18:18 user940 user940 3 What you wrote doesn't make sense. In the RHS, you have a sum over i i from 1 and you write a i−1 a i−1 which is not even defined Apprentice –Apprentice 2022-10-17 06:47:15 +00:00 Commented Oct 17, 2022 at 6:47 @Apprentice Your bounties are placed on two low-quality questions, but it's really better to start a new question following the guide in How to ask a good question.Ѕᴀᴀᴅ –Ѕᴀᴀᴅ 2022-10-17 07:13:11 +00:00 Commented Oct 17, 2022 at 7:13 3 @Apprentice: An empty product is usually understood as the value 1 1, so that the term for i=1 i=1 on the RHS is just (b 1−a 1)b 2⋯b n(b 1−a 1)b 2⋯b n. – Btw, this has been asked and answered more than once on this site, I have added some links below the question.Martin R –Martin R 2022-10-17 08:14:50 +00:00 Commented Oct 17, 2022 at 8:14 Add a comment\| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions inequality See similar questions with these tags. Featured on Meta Introducing a new proactive anti-spam measure Spevacus has joined us as a Community Manager stackoverflow.ai - rebuilt for attribution Community Asks Sprint Announcement - September 2025 Report this ad Linked 4Prove by induction that \|∏n k=1 a k−∏n k=1 b k\|≤∑n k=1\|a k−b k\|∣∣∣∏n k=1 a k−∏n k=1 b k∣∣∣≤∑n k=1\|a k−b k\|. 5How to prove the following inequality \|∏i=n i=1 a i−∏i=n i=1 b i\|<n δ\|∏i=n i=1 a i−∏i=n i=1 b i\|<n δ? 1Upper bound on the distance of two products Related 2A simple inequality about products 7Maximum difference inequality 0Help inequality real numbers 1Upper bound on the distance of two products 1Sum of harmonic numbers inequality 2product over sum inequality Hot Network Questions How to locate a leak in an irrigation system? Storing a session token in localstorage Should I let a player go because of their inability to handle setbacks? 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12708	https://www.expii.com/t/trigonometry-co-function-identities-5262	Expii Trigonometry: Co-Function Identities - Expii The co-function identities relate "co-pairs" of trig functions at x and π/2 - x. For example, sin(π/2 - x) = cos(x), cos(π/2 - x) = sin(x), tan(π/2 - x) = cot(x), and cot(π/2 - x) = tan(x). Explanations (3) Anthony Jiang Text 6 The cofunction identities state that sin(90∘−x)=cosxcos(90∘−x)=sinxtan(90∘−x)=cotxcot(90∘−x)=tanxsec(90∘−x)=cscxcsc(90∘−x)=secx The first two identities follow from the unit circle, the others following from the first two via the fact that tanθ=sinθcosθ, cot=cosθsinθ, secθ=1cosθ, csc=1sinθ. These relations are very useful in simplifying calculations in both geometry and complex numbers, proving other identities, and evaluating trigonometric functions, though the first 2 are really the most commonly used ones. Report Share 6 Like Related Lessons Trigonometry: Negative Angle Identities Trigonometry: Pythagorean Identity Trigonometry: Half-Angle Identities Trigonometry: Tangent Identity View All Related Lessons Anusha Rahman Text 1 Cofunction Identities The next set of identities you should memorize are the co-function identities! You'll recognize sine and cosine, cosecant and secant, and tangent and cotangent. Image source: by Anusha Rahman Memorizing these cofunction identities will help you to solve trigonometric equations in the future. Report Share 1 Like Anusha Rahman Video 1 (Video) Co-Function Identities by Mario's Math Tutoring In this video, Mario's Math Tutoring will help you understand the cofunction identities. Summary We can think of the cofunction identities in pairs. First, sin(π2−θ)=cos(θ) cos(π2−θ)=sin(θ) We can also pair together: sec(π2−θ)=csc(θ) csc(π2−θ)=sec(θ) Lastly, we can pair together: tan(π2−θ)=cot(θ) cot(π2−θ)=tan(θ) Let's think about why this works. Think of the fact that all angles in a right triangle must equal 180∘. If the right angle already comprises of the 90∘, then the rest of the 90∘ must be distributed in the other two angles. (Imagine acute, obtuse, and right angles!) Let's walk through an example. Example #1 tan(π2−θ)cos(θ) At first, this expression looks hard and complicated. However, we know that tan(π2−θ)=cot(θ), so our numerator becomes cot(θ). Our new fraction is: cot(θ)cos(θ) We also know that cot(θ)=cos(θ)sin(θ). If we simplify our new fraction, we will find that the answer is csc(θ). Try the next example on your own, then watch the video to see how you did! Example #2 sin2(10∘)+sin2(80∘) Report Share 1 Like You've reached the end How can we improve? General Bug Feature Send Feedback
12709	https://goldbook.iupac.org/terms/index/all	Gold Book Terms 13C-HMQC-NOESY13C-NOESY-HSQC15N-NOESY-HSQC15N-TOCSY-HSQC1-methyl-4-phenyl-1,2,3,6-tetrahydropyridine180° pulse1D property descriptor1D structure 2-amino-3-(3-hydroxy-5-methyl-1,2-oxazol-4-yl)propanoic acid receptor2-photon lithography2D property2D quantitative structure-activity relationships2D structure 3D lithography3D property3D searching3D structure3D structure generation 4D quantitative structure-activity relationships 5-hydroxytryptamine5α-reductase5α-reductase deficiency 90° pulse 'A' value(z + 1)-average molar massA-factorab initio calculationsab initio quantum mechanical methodsabeo-acaci-nitro compoundsamboantiprismo-aparachno-asym-a-ionA/D ratioabatement in atmospheric chemistryabdominal cavityabiologicalabioticabiotic transformationABL oncogeneablation by sputteringablation efficiency in LIBSablation rateablation threshold fluence in LIBSablephariaABO blood group systemabortionabsolute activation analysisabsolute activityabsolute configurationabsolute counting in radioanalytical chemistryabsolute counting efficiencyabsolute electrode potentialabsolute elemental sensitivity factorabsolute full energy peak efficiencyabsolute lethal concentrationabsolute lethal doseabsolute photopeak efficiencyabsolute preconcentration in trace analysisabsolute quantitation of proteinsabsorbanceabsorbance matching in spectrochemical analysisabsorbed dose of a substanceabsorbed dose of radiationabsorbed electron coefficient in in situ microanalysisabsorbed electrons in in situ microanalysisabsorbed photon flux densityabsorbed radiant power densityabsorberabsorptanceabsorptionabsorption coefficientabsorption coefficient in biologyabsorption cross-sectionabsorption factorabsorption indexabsorption intensityabsorption length in X-ray reflectrometryabsorption lineabsorption pathlength of a sample cellabsorption spectrumabsorptivityabstractionabstraction process in catalysisabundance sensitivity in mass spectrometryabzymeAC voltammetryaccelerating potentialaccelerating voltage scan in mass spectrometryaccelerationacceleration energy in in situ microanalysisacceleration of free fallaccelerator in solvent extractionaccelerator mass spectrometryacceptable daily intakeacceptance intervalacceptance limitacceptor numberacceptor phaseaccessory cellaccessory moleculeaccessory ribaccessory sex glandaccessory sex organaccessory thyroid tissueaccidental degeneracyacclimationaccommodation coefficientaccreditation of a laboratoryaccretion in atmospheric chemistryaccumulation in stripping voltammetryaccuracy of a measuring instrumentaccuracy of measurementaccurate massaccurate mass tagacenesacetalsacetogeninsacetonidesacetylcholineacetylcholine receptoracetylcholinesteraseacetylene blackacetylenesacetylidesAcheson graphiteachiralachirotopicachondroplasiaacidacid anhydridesacid deposition in atmospheric chemistryacid digestionacid rain in atmospheric chemistryacid thioanhydridesacid transient effects in ICP spectrometryacid-labile sulfuracid–base indicatoracid–base titrationacidimetric titrationacidityacidity constantacidity functionacidosisacinusacoustic meatusacoustic neuromaacoustic startle response testacquired immunityacquired immunodeficiency syndromeacquired myasthenia gravisacquisition timeacraniaacrocentric centromereacrocephalyacromegalyacroparesthesiaacrosomeacrosome reactionactinicactinic fluxactinismactinometeraction potentialaction spectrumactivated adsorption processactivated carbonactivated charcoalactivated complexactivated monomeractivated nucleotide sugaractivated stateactivated-monomer polymerizationactivation in chemical kineticsactivation in electrochemical corrosionactivation in radiochemistryactivation analysisactivation cross-sectionactivation energyactivation reactionactivation-induced cell deathactivatoractive avoidance testactive centreactive immunizationactive mediumactive metal in electrochemical corrosionactive samplingactive site in heterogeneous catalysisactive solidactive speciesactive state in electrochemical corrosionactive systemic anaphylaxis testactive transport in biologyactivinactivityactivity of a radioactive materialactivity in thermodynamicsactivity coefficientactivity growth curveactuatoracute lymphoblastic leukemiaacute myelogenous leukemiaacute rejectionacute respiratory distress syndromeacute toxicityacute-phase proteinacute-phase responseacyclicityacyl anionsacyl carbenesacyl cationsacyl groupsacyl halidesacyl radicalsacyl shift in photochemistryacyl speciesacylalsacylium ionsacyloinsacyloxyl radicalsadactylyadaptive immune responseadaptive immune systemadded hydrogenaddendaddictionAddison diseaseadditionaddition reactionaddition-fragmentation chain transferadditional gases in flame atomic spectroscopyadditiveadditive matrix effectadditive nameadditivity of mass spectraadditivity principleaddressinadductadduct ion in mass spectrometryadenocarcinomaadenomaadenosine deaminase deficiencyadenosine receptoradermiaadhesionadhesion factoradhesion moleculeadhesion promoteradhesional wettingadiabaticadiabatic electron transferadiabatic ionization in mass spectrometryadiabatic lapse rate in atmospheric chemistryadiabatic photoreactionadiabatic pulseadiabatic transition-state theoryadiabatic treatments of reaction ratesaditusadjacent re-entry model in polymer crystalsadjusted retention timeadjusted retention volume adjuvantadjuvant arthritisADMETADMRadolescenceadoptive transferadrenal 4 binding proteinadrenal glandadrenalinadrenalineadrenergicadrenocortocotropic hormoneadsorbateadsorbentadsorberadsorptionadsorption capacityadsorption chromatographyadsorption complexadsorption currentadsorption hysteresisadsorption indicatoradsorption isobaradsorption isostereadsorption isotherm in chromatographyadsorptiveadsorptive stripping voltammetryadultadulthoodadvancementadvection in atmospheric chemistryadventitious carbon referencingadverse drug reactionadverse effectadverse immunostimulationaeration in atmospheric chemistryaerial imageaeroallergenaerobeaerobicaerobic biodegradationaerobic conditionsaerogelaerometer in atmospheric chemistryaerometric measurement in atmospheric chemistryaerosolaerosol hydrolysisaerosol transport phenomena in ICP spectrometryafferentaffine chain behaviouraffinity in immunologyaffinity chromatographyaffinity maturationaffinity of reactionafter mass analysis in mass spectrometryage-specific birth rateageing of precipitate of precipitateagenesisagglomerate in polymer scienceagglomerate in catalysisagglomeration except in polymer scienceagglomeration in polymer scienceagglutinationagglutination inhibitionaggregate in catalysisaggregation except in polymer scienceaggregationaggregation chimeraaging of a polymeraglyconagnathiaagnosiaagonistagosticagostic interactionagranular carbonagranulocytosisair contaminant in atmospheric chemistryair mass in atmospheric chemistryair monitoring station in atmospheric chemistryair pollutantair pollutionair pollution indexair pollution survey in atmospheric chemistryair quality characteristic in atmospheric chemistryair resource management in atmospheric chemistryair sampling network in atmospheric chemistryair-lift bioreactorAitken particlesalar platealbedoalbinismalbuminalcogelalcohol-related neurodevelopmental disabilitiesalcoholatesalcoholsalcoholysisaldaric acidsaldazinesaldehydesaldiminesalditolsaldoketosesaldonic acidsaldosesaldoximesalert levels in atmospheric chemistryalgorithmalias structurealiased effectsalicyclic compoundsaligned incidence spectrumaliphatic compoundsaliquot in analytical chemistryalkalimetric titrationalkaline degradation in lipidomicsalkalinityalkaloidsalkalosisalkanesalkanium ionsalkene photocycloadditionalkene photodimerizationalkene photoisomerizationalkene photorearrangementalkenesalkoxidesalkoxyaminesalkyl cationalkyl groupsalkyl radicalsalkylenesalkylidene groupsalkylideneamino carbenesalkylideneaminoxyl radicalsalkylideneaminyl radicalsalkylidenesalkylidynesalkynesall-glass heated inlet systemall-solid-state ion-selective electrodeallantoic stalkallantoisalleleallelic exclusionallenesallergenallergicallergic conjunctivitisallergic rhinitisallergyallo- in amino-acid nomenclaturealloantibodyalloantigenallogeneicallograftallometricalloreactivityallosteric antagonistallosteric enzymesallosteryallotriomorphic transitionallotropesallotropic transitionallotypeallyl cationallylic groupsallylic intermediatesallylic substitution reactionalopeciaalpha relaxationalpha relaxation peakalpha-fetoproteinalpha-helixalpha-synucleinaltered layeralternancy symmetryalternantalternating copolymeralternating copolymerizationalternating currentalternating least squares regressionalternating voltagealternative pathway of complement activationalternative splicingaltocumulus cloud in atmospheric chemistryaltostratus cloud in atmospheric chemistryalveolar macrophagealveolitisalveolusAlzheimer diseaseAM 0 sunlightAM 1 sunlightamalgam lampamastiaambidentambient air in atmospheric chemistryambient air quality in atmospheric chemistryambient ionizationambient mass spectrometryambiphilicambisexualameliaameloblastamelogenesis imperfectaamenorrheaAmes/salmonella testamic acidsamide hydrazonesamide oximesamidesamidinesamidium ionsamidrazonesaminalsamine imidesamine iminesamine oxidesamine ylidesaminesaminimidesaminium ionsaminiumyl radical ionsamino radicalsamino sugarsamino-acid residue in a polypeptideaminonitrenesaminooxyl radicalsaminoxidesaminoxyl radicalsaminoxyl-mediated radical polymerizationaminyl oxidesaminyl radicalsaminylenesaminylium ionsammonium compoundsammonium iminesammonium ylidesammoniumyl radical ionsamnesiaamnesic shellfish poisoningamniocentesisamniochorionic membraneamnionamnioteamniotic cavityamniotic fluidamniotic sacamorphous carbonamorphous material transmitting infrared radiationamorphous polymeramorphous stateamount concentrationamount fractionamount of substanceamount-of-substance concentrationamount-of-substance fractionampereamperometric detection method in electrochemical analysisamperometryamphibian metamorphosis assayamphipathicamphiphileamphiphilicamphiprotic solventampholytesampholytic polymeramphotericamplification of deoxyribonucleic acidamplification reactionamplitude modulationamplitude of alternating currentamplitude of alternating voltageampullaamygdalaamyloidamyloid plaqueamyotrophic lateral sclerosisanabolismanaerobeanaerobicanaerobic biodegradationanaestheticanal canalanal membraneanalgesicanalogueanalogue metabolismanalogue to digital converter (pulse)analogue-based drug discoveryanalyser blanking in secondary-ion mass spectrometryanalysis area of sampleanalysis area of spectrometeranalyteanalyte transport rate in ICP spectrometryanalytical chemistryanalytical functionanalytical instrumentanalytical intercomparisonanalytical portionanalytical pyrolysisanalytical quality controlanalytical radiochemistryanalytical runanalytical sampleanalytical selectivityanalytical solutionanalytical solution calorimetryanalytical unitanamnesticanamnioteanaphaseanaphase laganaphrodisiacanaphylactic shockanaphylactoidanaphylatoxinanaphylaxisanasarcaanastomosisanationanchimeric assistanceanchoring villusandroblastomaandrogenandrogen receptorandrogenicandrostenedioneanemiaanencephalusanencephalyanergyaneroid barometeranestheticanestrusaneugenaneuploidaneuploidyaneurysmAnger cameraangioblastangioedemaangiogenesisangiographyangiotensin converting enzymeangleangle lappingangle of emissionangle of incidence of electromagnetic radiationangle of observationangle of optical rotationangle resolved mass spectrometryangle strainangular distributionangular frequencyangular momentumangular overlap modelanhydridesanhydro basesanilidesanilsanionanion exchangeanion exchangeranion radicalanion-activated-monomer polymerizationanion-exchange polymeranionic group-transfer polymerizationanionic polymeranionic polymerizationanionic ring-opening polymerizationanionized moleculeanionotropic rearrangementanisometricanisotropicanisotropic Raman scatteringanisotropyankyloglossiaankylosing spondylitisannealing in polymer scienceannelationannihilation in photochemistryannulationannulenesannulenylidenesannulus of intervertebral discanodeanodic stripping voltammetryanodic transfer coefficientanodontiaanogenital distanceanomalous dispersion of the refractive indexanomalously polarized Raman bandanomeric effectanomeric hydroxy groupanomersanonychiaanophthalmiaanorchismanorectalanorexiaanorexia nervosaanosmiaanotiaanovulationanoxiaansa compoundsantagonismantagonistantarafacialantemortemantepartumanthelminticanthocyanidinsanthocyaninsantianti-Compton γ-ray spectrometeranti-DNA antibodyanti-erythrocyte antibodyanti-Hammond effectanti-idiotypic antibodyanti-isotypic antibodyanti-Markownikoff additionanti-Müllerian hormoneanti-reflective coatinganti-sheep red blood cell IgM response assayanti-Stokes Raman scatteringanti-Stokes shiftanti-Stokes type radiationanti-thixotropyantiaromatic compoundsantiaromaticityantibioticantibodyantibody therapyantibody-dependent cellular cytotoxicityantibody-forming cell assayantibonding molecular orbitalanticholinergicanticircular elution in planar chromatographyanticlinalanticlined structures in polymersanticodonanticonvulsantanticyclone in atmospheric chemistryantidepressantantiemeticantiferroelectric transitionantiferromagnetic transitionantigenantigen presentationantigen processingantigen receptorantigen recognitionantigen-presenting cellantigen-presenting grooveantigenicantigenic determinantantigenicityantihelminthantimetaboliteantimitochondrial antibodyantimitoticantimony–xenon lampantimycoticantineuralgicantineutrophil cytoplasmic autoantibodyantineutrophil cytoplasmic autoantibody associated vasculitisantinociceptionantinuclear antibodyantiparticleantiperiplanarantiphospholipid syndromeantipodesantipsychoticantiresistantantiserumantisolitonantispasmodicantisymmetry principleantitoxinantrumanusanxiolyticaortaaorta-gonad-mesonephros regionaortic archaortopulmonary septumapexapex currentapex potentialaphakiaaphasiaaphicideaphoniaaphrodisiacapicalapicophilicityaplastic anemiaapneusisapo- in carotenoid nomenclatureapodizationapodization functionapoenzymeapoproteinapoptosisapparent in quantitiesapparent lifetimeapparent molar massapparent monomerapparent viscosity of a liquidappearance energyappearance potentialappearance temperature in electrothermal atomizationappendicular skeletonappendix in anatomyapplicability domainapplied potentialapraxiaaprotic solventaquagelaquationarachnodactylyarachnoid membranearachnoid villusarachnoiditisarborizationArchibald's methodarea of an electrode-solution interfacearea of interfacearea under the curvearea viscosityareicareic doseareic dose rateareic surface energyarene epoxidesarene oxidesarenesarenium ionsarenolsarenonium ionsareolaargon ion laserarithmetic meanArnold–Chiari malformationaromatasearomaticaromatic photocycloadditionaromaticityarray detector in spectroscopyarrester in atmospheric chemistryArrhenius A factorArrhenius energy of activationArrhenius equationarrhythmiaarsanesarsanylidenesarsanylium ionsarsenidesarsine oxidesarsinesarsinic acidsarsinidenesarsinous acidsarsonic acidsarsonium compoundsarsonous acidsarsoranesarteriogenesisarteriolovenular anastomosisarteriovenousarteriovenous anastomosisarteriovenous fistulaarteriovenous malformationarteriovenous shuntarthritisArthus reactionartificialartificial graphiteartificial inseminationartificial neural networksartificial organartificial polymerartificial radioactivityaryl cationsaryl groupsaryl hydrocarbon receptoraryl hydrocarbon receptor nucleotide translocator proteinarylene groupsaryneascending elution in planar chromatographyash in atmospheric chemistryashing in analysisaspect ratio in lithographyaspiratorassayassay kitassessorassigned valueassociationassociation reaction in mass spectrometryassociative avoidance testassociative desorptionassociative ionization in mass spectrometryassociative learningassociative surface reactionasthmaastrocyteastrocytic scarastrocytomaastrogliosisasymmetricasymmetric carbon atomasymmetric centreasymmetric destructionasymmetric enantiomer-differentiating polymerizationasymmetric filmasymmetric inductionasymmetric membraneasymmetric photochemistryasymmetric polymerizationasymmetric synthesisasymmetric topasymmetric transformationasymmetrical flow field-flow fractionationasymmetryasymmetry factor in chromatographyasymmetry parameterasymmetry potential of a glass electrodeat-line extractionatactic macromoleculeatactic polymerataxiaataxia telangiectasiaatelectasisatheliaathetosisathymiaatmosphereatmosphere of the earthatmospheric pressure chemical ionizationatmospheric pressure ionization in mass spectrometryatmospheric pressure matrix-assisted laser desorption/ionizationatmospheric pressure photoionizationatomatom-atom polarizabilityatom-bond polarizabilityatom-transfer radical polymerizationatom–molecule complex mechanismatomic absorption spectroscopyatomic chargeatomic emission spectroscopyatomic excitation sourceatomic fluorescenceatomic fluorescence spectroscopyatomic force microscopyatomic laseratomic massatomic mass constantatomic mass unitatomic mixingatomic numberatomic orbitalatomic polar tensoratomic ring-sectoratomic spectral linesatomic spectroscopyatomic symbolatomic unitsatomic vapouratomic weightatomization in analytical flame spectroscopyatomization curve in electrothermal AASatomization surface temperature in electrothermal atomizationatomizeatomizer in analytical flame spectroscopyatonyatopic allergyatopic dermatitisatopic eczemaatopyATP binding cassette proteinatresiaatrial-septal defectatrichia congenitaatrioventricularatriumatrophyatropisomersattachmentattention deficit hyperactivity disorderattenuanceattenuance filterattenuated total reflectionattenuated total reflection infrared spectroscopyattenuated total reflection spectroscopyattenuated vaccineattenuationattenuation coefficientattoattractive potential-energy surfaceattractive–mixed–repulsive classificationattributable riskattrition rateatypical interstitial pneumoniaauditauditory tubeaufbau principleAuger effectAuger electronAuger electron spectroscopyAuger electron spectrumAuger electron yieldAuger neutralizationAuger parameterAuger photoelectron coincidence spectroscopyAuger spectroscopyAuger vacancy satelliteAuger yieldauraauricleauronesautacoidautismauto-ionization in mass spectrometryautoantibodyautoantigenautocatalytic reactionautochthonousautocorrelationautocorrelation vectorautocrineautodetachmentautograftautoimmuneautoimmune bullous skin diseaseautoimmune diseaseautoimmune gastritisautoimmune hemolytic anemiaautoimmune hepatitisautoimmune infertilityautoimmune lymphoproliferative syndromeautoimmune myositisautoimmune nephritisautoimmune neuropathyautoimmune peripheral neuropathyautoimmune polyendocrine syndrome type 1 or 2autoimmune polyendocrinopathyautoimmune regulatorautoimmune thyroiditisautoimmunityautoinductionautoinflammatory diseaseautologousautologous antibodyautomated extractionautomatic extractionautomation in analysisautomerizationautonomic dysreflexiaautonomic nervous systemautophagyautophobicityautopoisoning in catalysisautoprotolysisautoprotolysis constantautopsyautoradiographautoradiolysisautoreactivityautoregressionautoscalingautositeautosomal dominant mutationautosomal recessive mutationautosomeauxiliary electrodeauxiliary gasauxochromeauxotrophyavalanche photodiodeavascularityaverage beam currentaverage current densityaverage degree of polymerizationaverage life in nuclear chemistryaverage massaverage matrix relative sensitivity factor in electron spectroscopyaverage rate of flow in polarographyavidityAvogadro constantavoidance testavoided crossing of potential-energy surfacesAvrami equationaxialaxial chiralityaxial ejectionaxial skeletonaxial viewing modeaxialite in polymer crystalsaxializationaxis of chiralityaxis of helicityaxonaxonal degenerationaxonal transportaxonopathyaza-di-π-methane rearrangementazacarbenesazaminesazanesazenesazidesaziminesazinesazinic acidsazlactonesazo compoundsazo imidesazo ylidesazomethine imidesazomethine oxidesazomethine ylidesazomethinesazonic acidsazoospermiaazoxy compoundsazylenesαα-additionα-adrenergic receptorα-cleavageα-decayα-effectα-eliminationα-expulsion in photochemistryα-oxo carbenesα-particleα-ray spectrometerα, βα:β T cellα:β T-cell receptorπ-adductσ-adduct β, βnuc, βlgB lymphocyteB-1/B-2 cellsB-cell linker proteinB-cell receptorB-cell stimulatory factorb-ionB-lymphocyte chemokineB-lymphocyte-induced maturation protein 1B7 moleculebacille Calmette–Guérinback donationback electron transferback extractionback scatter coefficient in in situ microanalysisback scattered electrons in in situ microanalysisback titrationback washingback-up compoundbackbonebackflushbackground of a radiation measuring devicebackground concentration in atmospheric chemistrybackground continuum emission in LIBSbackground equivalent concentrationbackground indicationbackground mass spectrumbackground radiationbackscatterbackscattered electronbackscattering analysisbackscattering correction factorbackscattering energybackscattering fractionbackscattering spectroscopybackscattering spectrumbackscattering yieldbackward chainingbackward propagationbackward stepwise linear discriminant analysisBaeyer strainbaffle chamber in atmospheric chemistrybag filter in atmospheric chemistrybaghouse in atmospheric chemistryBainite transitionbaking in carbon chemistryBALB/c mouseBaldwin's rulesball crateringband shapeband spectrabandgap energybandpass filterbarbar testbarbituratesbare lymphocyte syndromebarnBarnes mazebarotropic mesophaseBarton reactionbasalbasal gangliabasebase electrolytebase kind of quantitybase pairbase pairingbase peak in mass spectrometrybase peak chromatogrambase quantitybase quencherbase sequence analysisbase unit of measurementbaseline in chromatographybaseline concentration in atmospheric chemistrybaseline error in spectrochemical analysisbasement membranebasic local alignment search toolbasicitybasicity functionbasis functionbasis setbasis set superposition errorbasophilbasophilic degranulationbasting solventbatch in analytical chemistrybatch extractionbatch injection calorimetrybatch operation in analysisbatch reactorBates–Guggenheim conventionbathochromic shiftBayes’s theoremBayesian regularized neural networkBayesian statisticsBayley scaleBcl-2bead injectionbead-rod modelbead-spring modelbeading in neuritesbeam bunching in secondary-ion mass spectrometrybeam chopperbeam current in in situ microanalysisbeam diameter in in situ microanalysisbeam footprint in X-ray reflectrometrybeam mass spectrometerbeam spill-off in X-ray reflectrometrybeam splitterbeam walking testbeam-gas experimentsbecquerelbed volume in chromatographyBeer–Lambert lawbefore mass analysis in mass spectrometryBehcet diseasebelief theorybell stageBell–Evans–Polanyi principleBence–Jones proteinbenchmark dosebending of energy bandsbenign monoclonal gammopathyBent's ruleBenton testbenzenium ionsbenzenonium ionsbenzyl ionbenzylic groupsbenzylic intermediatesbenzynesBernoullian distributionBerry pseudorotationbest-in-classbeta relaxationbeta relaxation peakbeta-barrelbeta-sheetbeta-sheet breakerbetainesbetweenanenesbiasbias errorbiased linear pulse amplifierbicornate uterusbicycle rearrangementbicycle-pedal mechanismbidentateBiel water mazebiflavonoidsbifunctional catalysisbifurcationbilaminar embryobilayerbilinear equationbimetallic electrodebimodal distributionbimodal networkbimolecularbinary elastic scatteringbinary elastic scattering peakbinderbinder in chromatographybinder cokebinding capacitybinding energybinding siteBingham flowbinodalbioaccessibilitybioactivebioactivitybioadhesionbioalteration of a polymerbioanalytical chemistrybioassaybioassimilationbioavailabilitybioavailability in pharmacokineticsbioavailability in generalbiobasedbiocatalystbiochemical oxygen demandbiochipbiocompatible solid-phase microextractionbioconjugatebioconversionbiodegradabilitybiodegradablebiodegradable polymerbiodegradationbiodisintegrationbioelectronicsbioerosionbiofilmbiofragmentationbioinformaticsbioisosterebiolisticsbiological agentbiological effect monitoringbiological exposure indexbiological half lifebiological monitoringbiological recognition elementbiological tolerance values for working materialsbioluminescencebiomacromoleculebiomarkerbiomassbiomaterialbiomedicalbiomimeticbiomineralizationbiomoleculebiopanningbioplasticbiopolymersbioprosthesisbioreactorbiorecognitionbiorelatedbioresorbabilitybioresorbablebioresorptionbiosensorbiosphere in atmospheric chemistrybiostabilitybiosynthesisbiotechnologybioticbiotransformationbiphasic system in liquid–liquid extractionbiphotonic excitationbiphotonic processbiplotbipolar disorderbipolaronsbipolymerbiradicalbirthbirth defectbisecting conformationbismuthanesbismuthinesbispecific antibodybisphenolsbit stringbivane in atmospheric chemistryblack filmblackbody infrared radiative dissociationblade formatblank correctionblank indicationblank materialblank nominal indicationblank titrationblank value in analysisblastblastemablastocoelblastocystblastomereblastulablastulationBlau syndromeblaze-angle in spectrochemical analysisbleaching in photochemistryblobBloch equationsBloch–Siegert shiftblockblock copolymerblock copolymer lithographyblock homopolymerblock macromoleculeblock polymerblockbuster drugblockingblocking antibodyblocking geometrybloodblood cellblood cell countblood dyscrasiablood groupblood group antigenblood transfusionblood–brain barrierblood–cerebrospinal fluid barrierblood–placenta barrierblood–retinal barrierblood–testis barrierBloom syndromeblotting in biotechnologyblowdown in atmospheric chemistryblue shiftboatBodenstein approximationbody burdenbody fluidsbohrBohr magnetonBohr radiusboloamphiphilebolometerBoltzmann constantBoltzmann distribution of nuclear spinsBoltzmann enhanced discrimination of receiver operating characteristicbomb-digestion in spectrochemical analysisbondbond dissociationbond energy in theoretical chemistrybond energybond enthalpybond lengthbond migrationbond numberbond opposition strainbond orbitalbond orderbond orderbond ring-sectorbond-dissociation energy in theoretical chemistrybond-dissociation energybond-dissociation enthalpybond-energy-bond-order methodbond-stretch isomersbonded phase in chromatographybonding molecular orbitalbonding numberbone agebone cementbone marrowbone marrow transplantationbone morphogenetic proteinbone morphogenetic protein-4boosterbootstrap resamplingbootstrappingboranesboranylidenesborderline mechanismborenesborinic acidsBorn–Oppenheimer approximationboron-doped diamond electrodeboronic acidsborylenesbosonBoston naming testbottom anti-reflective coatingbottom-up lithographybottom-up proteomicsbound fraction in radioanalytical chemistrybound/free ratio in immunoassayboundary layer in atmospheric chemistryboundary layer thicknessbowspritBrønsted relationbrachial plexusbrachydactylybradycardiabradykinesiabradykininBragg filterBragg’s rulebrainbrain herniationbrain slicebrain-derived neurotrophic factorbrainstembranch in polymersbranch constitutional repeating unitbranch point in polymersbranch unit in polymersbranched chain in polymersbranched polymer in polymersbranchialbranchial archbranching chain reactionbranching decaybranching fractionbranching indexbranching planebranching ratiobranching ratiobreak of a foambreak of an emulsionbreakdown threshold fluence in LIBSbreakthrough volumeBredt's rulebreeching in atmospheric chemistrybreeze in atmospheric chemistryBremsstrahlungbrevicollisBrewster anglebridgebridge indexbridge solution in pH measurementbridged carbocationbridgeheadbridging ligandbrightnessbrightness of a laser dyeBroca areabromohydrinsbromonium ionsbronchial provocation testbronchopulmonary segmentBrooks and Taylor structureBrownian field-flow fractionation modeBrownian motionBrønsted acidBrønsted baseBrubaker lensbrush macromoleculebrush polymerbruxismbubble columnbubbler in atmospheric chemistrybuccopharyngeal membranebudBuehler assaybuffer gasbuffer-addition technique in analytical flame spectroscopybulbourethral glandbulbus cordisbulk concentration in electroanalysisbulk degradationbulk mesophasebulk rheologybulk samplebulk strainbuncherbundle of HisBunnett–Olsen equationsBunsen burnerBunte saltsBurkitt lymphomaburn-upburn-up fractionburning tension of an electrical arcburning velocity of a flame frontburr hole surgerybursabursa of Fabriciusbutyrylcholinesterasebypass injector in gas chromatographybystander effect in immunologybystander suppressionββ-N-acetylhexosaminidaseβ-adrenergic receptorβ-amyloidβ-blockerβ-cleavage in mass spectrometryβ-decayβ-eliminationβ-particleβ2-microglobulinπ-bondσ-bond Caenorhabditis eleganscatena-cis conformation in polymerscis- in inorganic nomenclaturecis-trans isomerscis-fusedcis, transcloso-c-ionC-reactive proteinC-terminal analysisC-terminal residue in a polypeptideC-terminusC-type lectinC. I. P. systemC1 inhibitorCA1-pyramidal cellCA3-pyramidal cellcadherincadium–helium laserCaesarian sectioncagecage compoundcage effectCahn–Ingold–Prelog systemcalamiticcalcinationcalcined cokecalcineurin inhibitorcalculated molar refractivitycalibrant in extraction phasecalibration in analysiscalibration certificatecalibration componentcalibration curvecalibration function in analysiscalibration gas mixture in atmospheric chemistrycalibration intervalcalibration material in analysiscalibration mixture in analysiscalibration sample in analysiscalibration, k0 methodcalibrator in activation analysiscalifornia verbal learning testcalixarenescaloriecalorimetric titrationcalorimetrycanalcancercandelacandidate certified reference materialcandidate reference materialcandidiasiscanonical formcanonical rate constantcanonical structure representationcanonical variablescanonical variate analysiscanonical variational transition-state theory capabilitycapacitance of a plate capacitorcapacitance hygrometercapacitation of spermcapacitive microwave plasmacapillary affinity electrophoresiscapillary column in chromatographycapillary condensationcapillary electrochromatographycapillary electrophoresiscapillary gel electrophoresiscapillary isoelectric focusingcapillary isotachophoresiscapillary sieving electrophoresiscappingcaptodative effectcapturecapture cross-sectioncapture probecapture γ-radiationcaput epididymiscarbaboranescarbamatescarbanioncarbenacarbene analoguescarbene metal complexescarbene radical anionscarbene radical cationscarbenescarbenium centrecarbenium ioncarbenoidscarbinolaminescarbinolscarbinyl cationscarbo-reductioncarbocationcarbocyclic compoundscarbodiimidescarbohydratecarbohydratescarboncarbon artifactcarbon blackcarbon cenospherescarbon clothcarbon dioxide lasercarbon electrodecarbon feltcarbon fibrecarbon fibre fabricscarbon fibres type HMcarbon fibres type HTcarbon fibres type IMcarbon fibres type LMcarbon fibres type UHMcarbon ink electrodecarbon loading of the packing materialcarbon materialcarbon mixcarbon paste electrodecarbon whiskerscarbon–carbon compositecarbonaceous mesophasecarbonitrilescarbonium ioncarbonizationcarbonothioyl-mediated radical polymerizationcarbonyl compoundscarbonyl imidescarbonyl iminescarbonyl oxidescarbonyl ylidescarboranescarboxamidescarboxamidinescarboxylic acidscarbylaminescarbyne metal complexescarbynescarbynium ionscarceplexcarcerandcarcinoembryonic antigencarcinogencarcinogenesiscarcinogenicitycarcinomacardiaccardiac jellycardiogenesiscardiogeniccardiolipincardiovascularcarotenescarotenoidscarotid arterycarpel tunnel syndromecarriercarrier in radioanalytical chemistrycarrier atom in organic reaction mechanismscarrier gascarrier proteincarrier-freecarry-overcarry-over effectcartilagecartridgecascade impactorcascade mixingcaspasecassavismcastrationcatabolismcatabolite repressionCatalán solvent parameterscatalasecatalymetric titrationcatalysed rate of reactioncatalysiscatalysis lawcatalystcatalyst in solvent extractioncatalyst ageingcatalyst deactivationcatalyst decaycatalyst precursor in coordination polymerizationcatalyst-transfer cross-coupling polymerizationcatalyst-transfer polymerizationcatalytic activity of an enzymecatalytic activity concentrationcatalytic activity contentcatalytic activity fraction of an isozymecatalytic antibodycatalytic chain terminationcatalytic chain transfercatalytic coefficientcatalytic concentrationcatalytic contentcatalytic crackingcatalytic currentcatalytic current, limitingcatalytic cyclecatalytic dehydrocyclizationcatalytic domain of a polypeptide of a polypeptidecatalytic fractioncatalytic graphitizationcatalytic hydrocrackingcatalytic hydrodesulfurizationcatalytic hydrogenolysiscatalytic methanationcatalytic reformingcatalytic thermometric titrationcataphoresiscataractcatecholaminescategorical datacategorycatenanescathodecathodic stripping voltammetrycathodic transfer coefficientcationcation exchangecation exchangercation radicalcation-activated-monomer polymerizationcation-exchange polymercation-π interactioncationic polymercationic polymerizationcationic ring-opening polymerizationcationized moleculecationotropic rearrangementcationotropyCauchy functioncauda epididymiscauda equinacaudalcaudate nucleuscause-and-effect diagramcaveolacavitandscavitation in biologycavity dumping in photochemistryCBCA(CO)NH/HN(CO)CACBCBCANH/HNCACBCC(CO)NHCD1CD16CD23CD25CD3CD4CD4+/CD25+ T cellCD40 ligandCD5+ B lymphocyteCD8CD8+ T suppressor cellcDNAcecumceiling valueceilometer in atmospheric chemistryceliac diseasecell constant of a conductivity cellcell error in spectrochemical analysiscell linecell-mediated cytotoxicitycell-mediated immune responsecell-mediated immunityCelsius temperaturecenticentral atomcentral nervous systemcentral tolerancecentre of a Mössbauer spectrumcentre burstcentre of chiralitycentrifugal accelerationcentrifugal barriercentrifugal flowcentrifugal forcecentrifugal radiuscentroid acquisitioncentromerecephaliccephalinscephalosporinscephamscephemsceramerceramicceramic filterceramic precursorceramic yieldceramic-reinforced polymerceramideceramizationcerebellar tonsillar herniationcerebellumcerebral cortexcerebral hemispherecerebral palsycerebrosidecerebrospinal fluidcerebrumCerenkov detectorCerenkov effectCerenkov radiationcerimetric titrationcertificationcertified nominal reference materialcertified property valuecervical ribcervical spine in vertebratescervical vertebracervixchain in polymerschain activation in SRMPchain activation in ATRPchain axis of a polymerchain branchingchain carrierchain conformationchain conformational repeating unit of a polymerchain deactivationchain entanglementchain fission yieldchain folding in polymer crystalschain initiationchain lengthchain polymerizationchain propagation in a chain polymerizationchain reactionchain reactivation in a chain polymerizationchain scission of a polymerchain segmentchain stiffnesschain termination in a chain polymerizationchain transferchain-ending stepchain-orientational disorder in polymer crystalschain-propagating reactionchain-shuttling polymerizationchain-termination reactionchain-transfer agentchain-walking polymerizationchair–chair interconversionchair, boat, twistchalcogen bondchalconeschallenge in immunologychamber saturation in gas chromatographychance correlationchange of a quantitychange of masschange ratio of a quantitychannelchannel blackchannel photomultiplier tubechannelingchanneling effectchannels ratio methodchaperonchaperone-mediated autophagycharcharacteristic in analytical chemistrycharacteristic electron energy lossescharacteristic group in organic nomenclaturecharacteristic length in thin filmscharacteristic linecharacteristic mass in electrothermal atomizationcharacteristic mass for integrated absorbance in electrothermal atomizationcharacteristic mass for peak absorption in electrothermal atomizationcharacteristic potentialcharacteristic ratio in polymerscharacteristic scale in thin filmscharacteristic temperaturescharacteristic X-radiationcharacteristic X-ray emissioncharcoalcharcoal tubechargecharge carriercharge carrier concentrationcharge carrier diffusioncharge carrier generationcharge carrier injectioncharge carrier mobilitycharge carrier recombinationcharge coupled devicecharge densitycharge hoppingcharge inversion reactioncharge neutralizationcharge number in generalcharge number in inorganic nomenclaturecharge number of a cell reactioncharge populationcharge recombinationcharge referencingcharge remote fragmentationcharge separationcharge shiftcharge site derivatizationcharge transfercharge-exchange ionization in mass spectrometrycharge-exchange reactioncharge-inversion mass spectrumcharge-mediated fragmentationcharge-permutation reactioncharge-stripping reactioncharge-transfer adsorptioncharge-transfer complexcharge-transfer device in radiation detectioncharge-transfer excitoncharge-transfer reaction in mass spectrometrycharge-transfer statecharge-transfer step of an electrode reactioncharge-transfer transitioncharge-transfer transition to solventcharged residue modelcharging potentialcharringcheckpoint pathwaychelatechelating polymerchelationchelation solvent extractionchelatometric titrationcheletropic reactionchelotropic reactionchemi-ionization in mass spectrometrychemical actinometerchemical activationchemical adsorptionchemical amountchemical amplificationchemical analysischemical biologychemical bondchemical databasechemical decompositionchemical dependencychemical diffusionchemical dosimeterchemical elementchemical equilibriumchemical fluxchemical functionalitychemical hydride generationchemical inductionchemical ionization in mass spectrometrychemical isotope exchangechemical laserchemical librarychemical mapchemical measurement processchemical modificationchemical oxygen demandchemical potentialchemical puritychemical reactionchemical reaction equationchemical relaxationchemical shift in NMRchemical shift in photoelectron and Auger spectrachemical shift anisotropy in NMRchemical spacechemical specieschemical species of an elementchemical state in electron spectroscopychemical state plotchemical substancechemical vapour generation in spectrochemical analysischemical yieldchemically amplified resistchemically bonded hybridChemically Induced Dynamic Electron Polarizationchemically induced dynamic electron polarizationchemically induced dynamic nuclear polarizationchemically induced dynamic nuclear polarizationChemically Initiated Electron Exchange Luminescencechemically initiated electron exchange luminescencechemically-modified electrodechemically-sensitive field effect transistorchemiexcitationchemifluxchemiluminescencechemiluminescence analyserchemiluminescent methods of detection in analysischeminformaticschemisorptionchemogenomicschemokinechemometricschemoselectivitychemospecificitychemostatchemotactic factorchemotaxischemotypeChernoff–Kavlock assaychimerachimney effect in atmospheric chemistrychiralchiral additivechiral featurechiral mobile phasechiral recognitionchiral selectorchiral stationary phase in liquid chromatographychiralitychirality axischirality centrechirality elementchirality planechirality sensechiroptic/chiropticalchirotopicchloracnechloramineschlorocarbonschlorohydrinschloronium ionscholesteric phasecholinergiccholinomimeticchondrificationchondroblastchondrocytechordinchoreachoreoathetosischorioallantoic placentachorionchorionic gonadotropinchorionic sacchorionic villuschorioretinitischoroid plexusChristiansen effectChristiansen filterchromaffin cellchromate uptake assaychromatic aberrationchromatidchromatin immunoprecipitationchromatogramchromatographchromatographchromatographic bedchromatographic detectorchromatographychromatolysischromium release assaychromophorechromosomal abnormalitychromosomal inversionchromosomal translocationchromosomechromosome deletionchromosome ringchronic allergic inflammationchronic granulomatous diseasechronic lymphocytic leukemiachronic myelogenous leukemiachronic rejectionchronic solvent-induced encephalopathychronic toxicitychronoamperometrychronocoulometric constantchronocoulometrychronopotentiometric constantchronopotentiometryciguatera poisoningciliary muscleciliary neurotrophic factorcine-substitutionCIP prioritycircular birefringencecircular developmentcircular dichroismcircular elutioncircular fingerprintscircular frequencycisoid conformationcistactic polymercity-block distanceclass (a) metal ionclass (b) metal ionclass of helix in polymersclass switchingclassical conditioningclassical pathway of complement activationclastogenclathratesclausiusclay hybridclean surfaceclean up in spectrochemical analysisclearanceclearance in toxicologyclearance doseclearing pointclearing temperaturecleavage in embryologycleftcleftcleft lipcleft palatecleft sternumclinalclitoriscloaca in zoologycloacal membraneclog PCLOGP valuesclonal anergyclonal deletionclonal expansionclonal indifferenceclonal selection in immunologyclonecloningcloning vectorclonusclosed filmclosed shell molecular systemsclosed system in spectrochemical analysisclosed-loop extractionclosed-vessel acid digestioncloud in atmospheric chemistrycloud pointcloud point extractioncloud-point curvecloud-point temperatureclover diseaseclubfootclustercluster analysiscluster centroidcluster determinantcluster ion in mass spectrometrycluster of differentiation antigenco-assemblyco-drugco-ionsco-nonsolvencyco-oligomerco-oligomerizationco-solvency in polymersco-stimulationCO2 lasercoacervationcoacervative extractioncoagulation in colloid chemistrycoagulation systemcoal tar pitchcoal-derived pitch cokecoalescence in colloid chemistrycoalificationcoarctatecoarctationcoated-vessel formatcoaxial reflectroncobalaminescobalt-mediated radical polymerizationcochleacoded experimental designcodoncoefficientcoefficient of haze in atmospheric chemistrycoelomcoenzymecoextractioncofactorscognate T cellcognitioncoherence length in thin filmscoherent anti-Stokes Raman spectroscopycoherent radiationcoherent scatteringcoherent source in spectrochemistrycoherent structurecoherent system of units of measurement of measurementcoherent unit of measurementcoil-to-globule transitioncoiled-tube field-flow fractionationcoincidence circuitcoincidence resolving timecokecoke breezecokingcolcold autoantibody typecold crystallisationcold fibre solid-phase microextractioncold neutronscollagen arthritiscollagenouscollectincollectioncollection efficiency in atmospheric chemistrycollection gas flow system in spectrochemical analysiscollector in atmospheric chemistrycollectorcollector slitcollembolan reproduction testcolligationcollimationcollimatorcollinear reactioncollinearitycollision cascadecollision cellcollision complexcollision cross-sectioncollision densitycollision diametercollision efficiencycollision frequencycollision gascollision numbercollision quadrupolecollision reaction cellcollision theorycollision-induced dissociation in mass spectrometrycollisional activation in mass spectrometrycollisional broadening of a spectral linecollisional excitation in mass spectrometrycollisional focusingcolloidcolloid osmotic pressurecolloidalcolloidal carboncolloidal crystalcolloidal dispersioncolloidal electrolytecolloidal gelcolloidal networkcolloidal processingcolloidal suspensioncolloidally stablecolobomacolony-stimulating factorcolor scalecolorimetercolour indicatorcolourabilitycolumn in chromatographycolumn chromatographycolumn diametercolumn lengthcolumn volume in chromatographycolumnar phasecomacomb macromoleculecomb polymercombinationcombination electrodecombination transitioncombinatorial diversitycombinatorial joiningcombined electrodecombined repeated dose toxicity studycombined rotation and multiple pulse spectroscopycombined samplecombined technique in thermal analysiscombustion gascommensurate–incommensurate transitioncommon black filmcommon factor analysiscommon variable immunodeficiencycommon-ion effect on ratescommunicating hydrocephaluscommunitiescommutability of a reference materialcompact layer in electrochemistrycompactioncomparative glycoproteomicscomparative molecular field analysiscomparative molecular similarity analysiscomparative proteoglycomicscomparator in radioanalytical chemistrycompartmental analysiscompatible polymer blendcompensation in catalysiscompensation in stereochemistrycompensation effectcompetencecompetitioncompetitive binding assaycompetitive inhibition of catalysiscompetitive inhibition assaycomplement in immunologycomplement deficiencycomplement reactioncomplement receptorcomplement systemcomplementarity-determining regioncomplementary binding sitescomplementary DNAcomplementary sequencecomplete active space self-consistent fieldcomplete active space self-consistent field second-order perturbation theorycomplete genome sequencecomplete linkagecomplexcomplex coacervationcomplex mechanismcomplex reactioncomplex-mode reactioncomplexometric titrationcomponentcompositecomposite alkalinitycomposite mechanismcomposite membranecomposite reactioncomposite samplecomposition of pure air in atmospheric chemistrycompositional depth profilecompositional heterogeneity of polymerscompostcompostingcomprehensive two-dimensional chromatographycompressibility factorcompression factorcomproportionationCompton effectCompton electronCompton scatteringCompton scattering analysiscomputational chemistrycomputational photochemistrycomputer-assisted drug designcomputer-assisted molecular designcomputer-assisted molecular modelingconal growth hypothesisconcanavalin Aconcave isotherm in chromatographyconcentrated phase in polymer chemistryconcentrated solutionconcentrationconcentration depolarizationconcentration distribution ratio in chromatographyconcentration factor in solvent extractionconcentration gradientconcentration in experimental surfaceconcentration overpotentialconcentration thermometric technique in enthalpimetric analysisconcentration-cell corrosionconcentration-sensitive detector in chromatographyconcentric nebulizerconceptionconceptusconcerted processconcerted reactionconcordancecondensation in atmospheric chemistrycondensation nuclei in atmospheric chemistrycondensation polymerizationcondensation reactioncondensative chain polymerizationcondensative chain polymerizationcondensed formulaconditional potentialconditional rate constant of an electrode reactionconditioned avoidance response testconditioned stimulusconditioning in solvent extractionconditioning filmconditioning solventconductanceconducting polymerconducting polymer compositeconduction bandconductivityconductometric titrationconductometryconductorconfidence levelconfidence limits about the meanconfiguration of electronsconfiguration in sterochemistryconfiguration interactionconfigurational base unit in polymersconfigurational disorder in polymersconfigurational homosequence in polymersconfigurational repeating unit in polymersconfigurational sequence in polymersconfigurational unit in polymersconfined atomizerconformance probabilityconformationconformational analysis in drug designconformational analysisconformational disorder in polymersconformational enthalpyconformational entropyconformational epitopeconformational meltingconformational repeating unit of a polymerconformational transitionconformerconformity assessmentconfusionconfusion matrixcongenercongeniccongenitalcongenital adrenal hyperplasiacongenital branchial cystcongenital malformationcongenital retinoschisisconglomeratecongruent transitionconical intersectionconing and quartering in analytical chemistryconjoined twinconjugate acid–base pairconjugate solutionsconjugated polymerconjugated systemconjugationconjugation in gene technologyconjugation labelling in radioanalytical chemistryconjugative mechanismconjunctive nameconnective tissueconnectivity in polymer chemistryconnectivityconrotatoryconrotatoryconsciousconsecutive reaction monitoringconsecutive reactionsconsecutive stepsconsensus property valueconservation of an examination standardconservation of orbital symmetryconsignment in analytical chemistryconsistencyconsolute pointconstant geneconstant neutral mass gain spectrumconstant neutral mass loss spectrumconstant oscillation-amplitude methodconstant regionconstant time evolutionconstituentconstitutionconstitutional heterogeneity of polymersconstitutional homosequenceconstitutional isomerismconstitutional repeating unit in polymersconstitutional sequence in polymersconstitutional unitconstitutive activityconstitutive enzymescontact allergencontact anglecontact corrosioncontact dermatitiscontact holecontact hypersensitivitycontact ion paircontact potential differencecontact sensitivitycontact urticariacontaminationcontentcontingency tablecontinuity inversion in solvent extractioncontinuous analysercontinuous bed packing in liquid chromatographycontinuous dynode particle multipliercontinuous extractioncontinuous flowcontinuous flow enthalpimetrycontinuous indication analysercontinuous liquid-liquid extractioncontinuous measuring cellcontinuous operation in analysiscontinuous performance testcontinuous precipitationcontinuous scan interferometercontinuous transitioncontinuous wave lasercontinuous-flow cell in spectrochemical analysiscontinuous-flow fast atom bombardmentcontinuous-flow matrix-assisted laser desorption/ionizationcontinuously curved chain in polymerscontinuumcontinuum source in atomic spectroscopycontour length in polymerscontraceptivecontract research organisationcontracturecontrastcontrast curvecontrast mediumcontributing structurecontrol chartcontrol limitcontrol material in analysiscontrol sample in analysiscontrolled anionic polymerizationcontrolled atmosphere in atmospheric chemistrycontrolled cationic polymerizationcontrolled deliverycontrolled ionic polymerizationcontrolled polymerizationcontrolled radical polymerizationcontrolled releasecontrolled temperature program in thermal analysiscontrolled-rate thermal analysisconvection as applied to air motionconvenience sampleconventional dendritic cellconventional massconventional nominal property valueconventional quantity valueconventional transition-state theoryconventional true valueconvergent evolutionconversion cross-sectionconversion dynodeconversion electronconversion spectrumconvertaseconvex isotherm in chromatographyconvulsioncooling curveCoombs testcooperative transitioncooperativitycoordinate covalencecoordinate linkcoordinationcoordination entitycoordination networkcoordination numbercoordination polyhedroncoordination polymercoordination polymerizationcoordination ring-opening polymerizationcoordination-addition polymerizationcoordination-insertion polymerizationcoordinative chain-transfer polymerizationcoordinatively saturated complexcoordinatively unsaturated complexcoordinator of an interlaboratory comparisoncopolymercopolymer micellecopolymerizationcopper vapour lasercoprecipitationcopy number in biotechnologycord bloodcore atom in organic reaction mechanismscore consistency diagnosticcore oligosaccharidecore unitcore-crosslinked micellecore-shell structurecoreceptorCoriolis couplingCoriolis forcecornu ammoniscorona dischargecorona discharge ionizationcoronalcoronal suturecoronandscorpus callosumcorpus cavernosumcorpus luteumcorpus striatumcorrected emission spectrumcorrected excitation spectrumcorrected retention volume in gas chromatographycorrelation analysiscorrelation coefficientcorrelation diagramcorrelation energycorrelation spectroscopy in NMRcorrelation spectroscopy through long-range couplingcorrelation timecorrespondence factor analysiscorrinoidscorrosioncorrosion cellcorrosion currentcorrosion potentialcorrosion ratecortex in anatomycorticosteroidcosolventcospherecotecticCottrell equationcoulombcoulomb integralcoulomb radiusCoulomb repulsioncoulometercoulometric detection method in electrochemical analysiscoulometric titrationcoulometric titrationcoulometrycoumarinscountcounter electrodecounter tubecounter-current flowcounter-current gascounter-ions in colloid chemistrycounter-regulation hypothesiscountercurrent chromatographycountercurrent extractioncounterpoise correctioncounting efficiencycounting losscounting ratecoupled cluster methodcoupled reaction in analysiscoupled simultaneous techniques in analysiscouplingcoupling constantcovalent bondcovalent drugcovalent networkCox–Yates equationcrackingCraig countercurrent distribution apparatuscrampcranial nervecranial placodescranial suturecraniofacialcraniopharyngiomacraniorachischisiscranioschisiscraniosynostosiscraniumcrater depthcrazingCre/loxPcreamcream volumecreamingcreepcresolscretinismcri du chat syndromecristacritical anglecritical dimensioncritical embryocritical energycritical excitation energy in in situ microanalysiscritical ion-concentration in an ionomercritical micelle concentrationcritical periodcritical pointcritical pressurecritical quenching radiuscritical solution compositioncritical solution pointcritical solution temperaturecritical studycritical temperaturecritical thickness of a film of a filmcritical valueCrohn diseasecross polarizationcross polarization with magic angle spinning NMRcross reactioncross reactivitycross section in radiation chemistrycross sensitivitycross-conjugationcross-flow filtration in biotechnologycross-flow nebulizercross-linking in biomoleculescross-matchingcross-over concentrationcross-presentationcross-reacting antibodycross-reacting antigencross-sectional area of the columncross-sectioningcross-tolerance in immunologycrosscurrent extractioncrossed electric and magnetic fields in mass spectrometrycrossed molecular beamscrossing over in biotechnologycrosslinkcrosslink densitycrosslinkingcrosslinking siteCrouzon syndromecrowding in solvent extractioncrowncrown conformationcrud in solvent extractioncryogeniccryogenic samplingcryogenic trapcryoglobulincryoglobulinemic vasculitiscrypt in anatomycryptandcryptatecryptic epitopecryptophthalmoscrystal diffraction spectrometercrystal fieldcrystal field splittingcrystal lasercrystal lattice defectcrystal photochemistrycrystalline electrodescrystalline polymercrystalline statecrystallinitycrystallisable polymercrystallizationCTcubiccubic mesophasecuecultivation in metabolomicscumulative double bondscumulative fission yieldcumulative mass-distribution functioncumulative number-distribution functioncumulative samplecumulative sum control chartcumulenescumulonimbus cloud in atmospheric chemistrycumulus cloud in atmospheric chemistrycumulus oophoruscupolacuriecuringcurly arrowscurrent densitycurrent densitycurrent distributioncurrent efficiencycurrent samplingcurrent yieldcurrent-potential curvecurrent, migrationCurtin–Hammett principlecurve-crossing modelcurved field reflectroncut off in aerosol sizes","in atmospheric chemistrycut-off filtercut-on filtercutaneous lymphocyte antigencutaneous T-cell lymphomaCVD diamondCWcyanatescyanidescyanine dyescyanogeniccyanohydrinscyanosiscybotactic regioncycles per secondcyclic acid anhydridescyclic voltammetrycyclitolscyclizationcyclo-cycloadditioncycloalkanescycloalkyl groupscyclodepsipeptidescyclodextrinscycloeliminationcyclohexadienyl cationscyclone collector in atmospheric chemistrycyclophanescyclopiacyclopolymerizationcycloreversioncyclosilazanescyclosiloxanescyclotroncyclotron motioncystcytochrome P450cytochromescytogeneticscytokinecytokine capture assaycytokine profilecytokine release assaycytomegaloviruscytopeniacytotoxiccytotoxic T lymphocytecytotoxic T lymphocyte antigen 4cytotoxic T lymphocyte assaycytotoxincytotrophoblastσ-constantχ-parameter dodecahedro-d-iond, l, dldaltonDaly detectordamage limitdamage to deoxyribonucleic aciddamping curvedanger hypothesisDaphnia magna reproduction testdark currentdark electric conductivitydark lossdark outputdark photochemistrydark reactiondark resistanceDarlington phototransistorDASdata matrixdata miningdata pre-processingdata reductiondata-dependent acquisitiondatabase searching in proteomicsdative bondDauben–Salem–Turro rulesdaughter ion in mass spectrometrydaughter product in radiochemistryDavydov splittingdayDaylight fingerprintsDC polarographyde Mayo reactionde novo designde-electronationde-energization efficiencydeactivationdeactivator in ATRPdead polymer chaindead time in radioanalytical chemistrydead time of an analyserdead time correction in radioanalytical chemistrydead-volume in chromatographydeath domaindeath receptordebyeDebye–Hückel equationDEC-205decadecadic absorbancedecay chaindecay constantdecay curvedecay rate in atmospheric chemistrydecay rate of a radioactive materialdecay scheme in radioanalytical chemistrydecay time in heterogenous catalysisdecay-accelerating factordecidecibeldeciduadecidual celldecidual cell response techniquedecidualizationdeciduumdecision rule in conformity assessmentdecision treedecompositiondecontamination factor in liquid-liquid distributiondeconvoluted mass spectrumdecoupling in the presence of scalar interactionsdecussatededicated kind-of-nominal-propertyDEDMRdefectdefect-free patterndefeminized gonadotropin secretiondefensindefinitive methoddeflection for a precision balancedeflocculationdeformation vibrationdegeneracydegenerate chain branchingdegenerate chemical reactiondegenerate orbitalsdegenerate rearrangementdegenerative chain transferdegenerative chain-transfer radical polymerizationdegenerative-transfer polymerizationdegradabilitydegradabledegradable macromoleculedegradable polymerdegradation in generaldegradation of a biorelated polymerdegranulationdegree of arcdegree Celsiusdegree Fahrenheitdegree of activationdegree of association of a micelledegree of bioassimilationdegree of biodegradationdegree of biodisintegrationdegree of biofragmentationdegree of biomineralizationdegree of branchingdegree of crystallinity of a polymerdegree of degradationdegree of disintegrationdegree of dissociationdegree of fragmentationdegree of inhibitiondegree of ionizationdegree of mineralizationdegree of polymerizationdegree of reactiondegree-of-polymerisation dispersitydegrees of cistacticity and transtacticitydegrees of freedomdegrees of triad isotacticity, syndiotacticity, and heterotacticitydehydroarenesdehydrobenzenesdelayed coincidencedelayed cokedelayed coking processdelayed extractiondelayed fluorescencedelayed luminescencedelayed neutronsdelayed onset in an X-ray absorption spectrumdelayed-neutron activation analysisdelayed-type hypersensitivitydelayed-type hypersensitivity assaydeletion of a base pairdeliquescencedeliriumdelocalizationdelocalization in theoretical organic chemistrydelocalization energydelocalization of electronsdelta layerdemasculinizeddementiademisterdemyelinationdenaturation of a macromoleculedenaturation of alcoholdendrimerdendrimer moleculedendritedendritic block macromoleculedendritic celldendritic constitutional repeating unitdendritic graft macromoleculedendritic macromoleculedendritic polymerdendrogramdendrondendron directiondenitrificationdensificationdensitydensity functional theorydensity inversion in solvent extractiondensity of statesdentate gyrusdenticitydenuderdenuder system in atmospheric chemistrydeodorizer in atmospheric chemistrydeoxyribonucleic acid fragmentationdeoxyribonucleic acid fragmentation indexdeoxyribonucleic acid integrity indexdeoxyribonucleic acid mismatchdeoxyribonucleic acid profilingdeoxyribonucleic acid replicationdeoxyribonucleic acid sequencingdeoxyribonucleic acidsdeoxyribosedephasingdepolarizationdepolarization of scattered lightdepolarization ratiodepolarized Raman banddepolarizerdepolymerasedepolymerizationdeposition in atmospheric chemistrydeposition velocity in atmospheric chemistrydepression of the central nervous systemdeprotectiondeprotonated moleculedepsidesdepsipeptidesdepth of focusdepth of penetration of lightdepth profiledepth profilingdepth resolutiondepth resolution parameterdepth-profilingderacemizationderivative potentiometric titrationderivative spectroscopyderivative technique in thermal analysisderived coherent unitderived kind of quantityderived non-coherent unitderived quantityderived unit of measurementdermatitisdermatomyositisdermisdescending elution/development in planar chromatographydescriptor in computational drug designdesensitizationdeshieldingdesiccantdesign matrixdesign of experiment in metabolomicsdesigned multiple ligandsdesolvation in flame spectroscopydesorptiondesorption atmospheric pressure chemical ionization mass spectrometrydesorption atmospheric pressure photoionization mass spectrometrydesorption by displacementdesorption chemical ionizationdesorption electrospray ionization mass spectrometrydesorption ionizationdesorption ionization on silicondesulfurizationdesymmetrizationdesymmetrization stepdesynapsisdetachmentdetailed balancingdetection efficiency in nuclear analytical chemistrydetection limitdetection limit in analysisdetector in chromatographydetector gate widthdetergentdeteriorationdeterminationdetoxificationdeuteriationdeuteridedeuteriodeuteriumdeuterondeveloped imagedevelopmentdevelopmental anomalydevelopmental biologydevelopmental neurotoxicitydevelopmental neurotoxicity studydevelopmental neurotoxicity testingdevelopmental susceptibility genedevelopmental toxicologydevelopmental variationdeviationdevitrification in polymer sciencedevolatilizerdew point in atmospheric chemistrydew point hygrometerDexter excitation transferdextransdextrinsdextrocardiadextropositionDFDMRdi-π-methane rearrangementdi-π-silane rearrangementdiabatic couplingdiabatic electron transferdiabatic photoreactiondiabetes mellitus type 1diacylaminesdiads in polymersdiagnostic iondiagram level in X-ray spectroscopydiagram line in X-ray spectroscopydialysatedialysisdialysis residuediamagneticdiamididesdiamonddiamond by CVDdiamond-like carbon filmsdianionsdiapedesisdiaphoreticdiaphragmdiarrheal shellfish poisoningdiastereoisomer excessdiastereoisomeric units in a polymerdiastereoisomerismdiastereoisomerizationdiastereoisomersdiastereomeric ratiodiastereomersdiastereomorphismdiastereoselectivitydiastereoselectivitydiastereotopicdiathesisdiazanylidenesdiazenyl radicalsdiazo compoundsdiazoamino compoundsdiazoatesdiazonium saltsdiazooxidesdiazotatesdicarbenium ionsdichotomous samplerdichroic filterdichroic mirrordichroismdielectric barrier discharge ionizationdielectric constantdielectric materialdielectric polarizationdielectric thermal analysisdielectrometrydiencephalondienesdienophilediethylstilbestroldifference absorption spectroscopydifference gel electrophoresis in proteomicsdifference spectroscopydifference transitiondifferential in hematologydifferential capacitancedifferential contactordifferential detector in chromatographydifferential diffusion coefficientdifferential molar energy of adsorptiondifferential molar interfacial energydifferential molar surface excess energydifferential pulse voltammetrydifferential scanning calorimeterdifferential scanning calorimetrydifferential spectrumdifferential splicingdifferential technique in thermal analysisdifferential thermal analyserdifferential thermal analysisdifferential viscositydifferentiation in biologydifferentiation antigendiffractiondiffraction analysisdiffuse layer in electrochemistrydiffuse reflectancediffuse reflectiondiffuse reflection-absorption infrared spectroscopydiffuse transmissiondiffused junction semiconductor detectordiffuserdiffusiondiffusion batterydiffusion coefficientdiffusion controldiffusion currentdiffusion current constant in polarographydiffusion layerdiffusion layer thicknessdiffusion ordered spectroscopydiffusion potentialdiffusion-controlled ratediffusional transitiondiffusionless transitionDiGeorge syndromedigestion in sample preparationdigit span testdigit symbol substitution testdihedral anglediisotactic polymerdilatancydilational transitiondilational viscositydilatometerdilatometrydiluent in solvent extractiondiluent gasdilute limit in secondary-ion mass spectrometrydilute phase in polymer chemistrydilute solution in polymer sciencedilute solutiondiluterdilution rate in biotechnologydimension of a quantitydimensionless Henry’s law solubility constant defined via amount concentrationsdimensionless Henry’s law volatility constant defined via amount concentrationsdimensionless quantitiesdimeric ion in mass spectrometrydimerizationDimroth–Reichardt ET parameterdiode laserdiolsdiosphenolsdioxindip-pen nanolithographydiploiddiplopiadipolar aprotic solventdipolar bonddipolar compoundsdipolar coupling in nuclear magnetic resonance spectroscopydipolar cycloadditiondipolar ionsdipolar mechanism of energy transferdipole couplingdipole lengthdipole momentdipole moment per volumedipole-dipole excitation transferdipole-induced dipole forcesdipole–dipole interactiondipyrrinsdipyrromethenesdiradicaloiddiradicalsdirect amplificationdirect analysis in real timedirect coulometry at controlled currentdirect coulometry at controlled potentialdirect currentdirect current glow discharge optical emission spectroscopydirect effectdirect exposure probedirect fission yielddirect infusiondirect insertion probedirect isotope dilution analysisdirect liquid introductiondirect reactiondirect recoil spectroscopydirect spectrumdirect thermometric method in enthalpimetric analysisdirect titrationdirect transfer gas flow system in spectrochemical analysisdirect-injection burner in analytical flame spectroscopydirect-injection enthalpimetrydirect-write lithographydirected self-assembly in lithographydirectly-ionizing radiationdirectordisaccharidesdisappearance cross sectiondiscomfort threshold in atmospheric chemistrydiscontinuous analyserdiscontinuous indication of an analyserdiscontinuous measuring celldiscontinuous phase transitiondiscontinuous precipitationdiscontinuous simultaneous techniques in analysisdiscoticdiscotic mesogendiscrete dynode particle multiplierdiscrete transportdiscriminant analysisdiscriminatordisintegrationdisintegration constantdisintegration energydisintegration rate of a radioactive materialdisintegrin and metalloproteinase domain-containing protein 33disjoining pressure for the attraction between two surfacesdisjoint principal component analysisdislocationdismutationdisorientationdispenserdispermydispersing solventdispersion for spectroscopic instrumentsdispersion in atmospheric chemistrydispersion forcesdispersion of a sample in atomic spectroscopydispersion plane in X-ray reflectrometrydispersitydispersive Fourier-transform spectroscopydispersive liquid-liquid microextractiondispersive spectrometerdispersive transport of charge carriersdisplacement chromatographydisplacive transitiondisproportionationdisrotatorydisruptiondisruptordissociationdissociation energydissociative adsorptiondissociative electron capturedissociative ionization in mass spectrometrydissociative surface reactiondissolutiondissolution inhibitordissolution promoterdissolution ratedissymmetry in stereochemistrydissymmetry of scatteringdistal in anatomydistonic radical cationdistortion interaction modeldistortionless enhancement by polarization transferdistribuenddistributiondistribution coefficientdistribution constantdistribution constant in chromatographydistribution constant (exclusion) in size-exclusion chromatographydistribution constant calibrationdistribution function in polymersdistribution isotherm in chromatographydistribution ratio in liquid-liquid distributiondisulfide bond cleavagedisyndiotactic polymerditactic polymerditerpenoidsdithioacetalsditopic receptordiurnal variation in atmospheric chemistrydiversitydiversity genediversity-oriented synthesisdiverticulumdivided spinning celldividing surfaceDM-interferenceDNADNA probeDNA vaccinationDobson unit in atmospheric chemistrydockingdocking studiesdoctor bladingdolicholsdomain in molecular biologydominantdominant hemispheredomoic aciddonicityDonnan emfDonnan equilibriumDonnan exclusionDonnan pressuredonor numberdonor phasedopaminedopaminergicdopantdoping in polymer chemistrydoping in catalysisdoping (p- and n-type)Doppler broadening of a spectral linedormant speciesDorn effectdorsaldorsal rootdorsal root gangliondosage in atmospheric chemistrydosedose in radioanalytical chemistrydose of a substancedose in photochemistrydose equivalentdose-response and dose-effect relationshipsdot plotdouble escape peakdouble exposuredouble helixdouble injectiondouble patterningdouble pulsed-field gradient spin-echo excitationdouble resonancedouble-beam spectrometerdouble-beam spectrometer for luminescence spectroscopydouble-focusing mass spectrographdouble-focusing mass spectrometerdouble-labelled proteindouble-layerdouble-layer currentdouble-negative celldouble-positive celldouble-pulse laser-induced breakdown spectroscopydouble-quantum filtered correlation spectroscopydouble-strand chain in polymersdouble-strand copolymerdouble-strand macromoleculedouble-strand polymerdouble-wavelength spectroscopydoublet statedoubling time in biotechnologydown-time in analysisdownconversiondownfielddownstream in membrane processesdownwash in atmospheric chemistryDP, DQ, and DR moleculesdraining lymph nodeDraize testdriftdrift currentdrift tubedrift velocity in electricitydriving force of a reactiondriving force for electron transferdrop time in polarographydroplet in atmospheric chemistrydropping mercury electrodeDrude–Nernst equation for electrostrictiondrugdrug carrierdrug cocktaildrug deliverydrug designdrug distributiondrug metabolizing enzymedrug repurposingdrug safetydrug-induced autoimmunitydrug-induced lupusdrug-likenessdrug-specific T celldruggabilitydry bulb temperature in atmospheric chemistrydry depositiondry developmentdry etchingdry-etch resistancedrying agentdrying control chemical additivedrying gasdual binding sitedual fluorescence of systems or molecular speciesdual substituent-parameter equationdual viewing modedual-beam profilingdual-mode photochromismdual-tone resistductductus arteriosisdummy factordung fly testduodenumdura materdurabilitydust in atmospheric chemistrydust collector in atmospheric chemistrydust fall in atmospheric chemistrydwell timedye laserdynamerdynamic bioaccessibilitydynamic emittance matchingdynamic exclusiondynamic extractiondynamic fields mass spectrometerdynamic headspace analysisdynamic interfacial tensiondynamic light scatteringdynamic mechanical analyserdynamic mechanical analysis in thermal analysisdynamic membrane formationdynamic NMRdynamic quenchingdynamic range of an analyserdynamic reaction pathdynamic secondary-ion mass spectrometrydynamic spectrumdynamic structure factordynamic surface tensiondynamic technique in thermal analysisdynamic thermomechanometrydynamic time warpingdynamic viscometerdynamic viscositydynamic-load thermomechanical analysisdynedyneindynorphindyotropic rearrangementdypnonesdysaphiadysarthriadysesthesiadysgenesisdysgonesisdysgraphiadyskinesiadyslexiadysmenorrheadysmorphiadysmorphogenesisdysostosisdysphagiadysphasiadysplasiadystoniaᴅ, ʟ, ᴅʟthree-dimensional quantitative structure–activity relationshipsΔ, Λ EsumE, ZE/Z photoisomerizationendo, exo, syn, antienterythro structures in a polymererythro, threoearly life stage testearly-downhill surfaceearthworm reproduction testecho timeeclampsiaeclipsedeclipsing conformationeclipsing strainecogeneticsecotoxicityectodermectohormoneectopiaectopic expressionectopic pregnancyectrodactylyectrosyndactylyeczemaeddyeddy dispersionedemaedge acuityEdman degradationeducteffect of a factoreffective attenuation lengtheffective bond lengtheffective cadmium cut-off energy in nuclear analytical chemistryeffective chargeeffective chimney heighteffective concentrationeffective doseeffective half lifeeffective length of the capillaryeffective molarityeffective plate heighteffective plate numbereffective theoretical plate number of a chromatographic columneffective thermal cross-sectioneffective thickness of diffusion layereffectively infinite thickness in flame spectroscopyeffectivenesseffectoreffector cellefferentefferent ductefficiency of a stepefficiency of a counterefficiency of atomization in analytical flame spectroscopyefficiency of nebulization in flame spectrometryefficiency spectrumefflorescenceeffluenteffluent in chromatographyefflux pumpeggeicosanoidseighteen-electron ruleeinsteinEinstein equationEinstein photoelectric equationeinzel lensejaculationejaculatory ductEl-Sayed ruleselastic collisionelastic peakelastic peak electron spectroscopyelastic scattering in reaction dynamicselastic scattering in spectrochemistryelastic scattering cross sectionelastically active network chainelastomerelectretelectric capacitanceelectric chargeelectric coherence length in thin filmselectric conductanceelectric conductivityelectric conductorelectric currentelectric current densityelectric dipole momentelectric dischargeelectric displacementelectric driftelectric fieldelectric mobilityelectric polarizabilityelectric potentialelectric potential difference of a galvanic cellelectric resistanceelectric resistivityelectric sector in mass spectrometryelectrical anharmonicityelectrical arcelectrical double-layerelectrical harmonicityelectrical hygrometerelectrically conducting polymerelectrically conducting polymer compositeelectrified interphaseelectro-dialysiselectro-endosmosiselectro-optic effectelectro-optic parameterelectro-osmosiselectro-osmotic hold-up time in capillary electromigrationelectro-osmotic mobility in capillary electromigrationelectro-osmotic pressureelectro-osmotic velocityelectro-osmotic volume flowelectroactive substanceelectrocapillarityelectrocapillary equationelectrocatalysiselectrochemical biosensorelectrochemical cellelectrochemical constantselectrochemical detection in flowing systemselectrochemical detector in gas chromatographyelectrochemical impedance spectroscopyelectrochemical method of detection in analysiselectrochemical piezoelectric microgravimetryelectrochemical potentialelectrochemical quartz crystal microbalanceelectrochemical sensorelectrochemiluminescenceelectrochromic effectelectrochromic polymerelectroclinic polymerelectroconvulsive therapyelectrocyclic reactionelectrodeelectrode arrayelectrode coverage densityelectrode current densityelectrode memoryelectrode potentialelectrode processelectrode reactionelectrode reaction rate constantselectrode surface areaelectrodecantationelectrodepositionelectroencephalogramelectroencephalographyelectrofluorescenceelectrofugeelectrogenerated chemiluminescenceelectrographiteelectrogravimetryelectrohydrodynamic ionizationelectroinjectionelectrokinetic chromatographyelectrokinetic injectionelectrokinetic potentialelectroluminescenceelectroluminescent polymerelectrolyteelectrolytic cellelectrolytic hygrometerelectromagnetic fieldelectromagnetic field responsive polymerelectromagnetic radiationelectromagnetic transitionelectromembrane extractionelectromeric effectelectromigrationelectromotive forceelectromyographyelectronelectron acceptorelectron affinityelectron attachmentelectron back-transferelectron beam curingelectron beam lithographyelectron beam resistelectron captureelectron capture detector in gas chromatographyelectron capture dissociationelectron capture ionizationelectron chargeelectron configurationelectron correlationelectron densityelectron density functionelectron detachmentelectron donorelectron emission spectrometryelectron energy in mass spectrometryelectron energy analyserelectron energy loss spectroscopyelectron energy loss spectrumelectron exchange excitation transferelectron floodingelectron impact ionization in mass spectrometryelectron inelastic mean free pathelectron ionization in mass spectrometryelectron jumpelectron kinetic energyelectron microscopyelectron number densityelectron number of an electrochemical reactionelectron paramagnetic resonanceelectron probe microanalysiselectron probe X-ray microanalysiselectron pushingelectron rest masselectron retardationelectron spectrometerelectron spectroscopy for chemical analysiselectron stopping power in X-ray emission spectroscopyelectron transferelectron transfer photosensitizationelectron tunnelingelectron work functionelectron-beam direct-write lithographyelectron-counting ruleselectron-deficient bondelectron-deficient compoundselectron-deficient moleculeelectron-donor-acceptor complexelectron-pair acceptorelectron-pair donorelectron-rich moleculeelectron-transfer catalysiselectron-transfer dissociation in proteomicselectron-transfer ring-opening metathesis polymerizationelectron-transfer/higher-energy collision dissociation in proteomicselectronationelectronegativityelectroneurographyelectroneutrality principleelectronic chemical potentialelectronic configurationelectronic effect of substituents: symbols and signselectronic energy migrationelectronic reference to access in vivo concentrationselectronic stabilityelectronic stateelectronic stopping cross sectionelectronic transitionelectronically excited stateelectronvoltelectroosmosiselectroosmotic velocityelectroosmotic volume flow rateelectrophileelectrophilic chain polymerizationelectrophilic substitutionelectrophilicityelectrophoresiselectrophoresis convectionelectrophoretic mobilityelectrophoretic velocityelectrophosphorescenceelectrophotographyelectrophysiologyelectroporationelectroretinogramelectrorheological effectelectrosmosiselectrospray ionisation in proteomicselectrospray laser desorption ionization mass spectrometryelectrospray needleelectrostackingelectrostatic energy analyserelectrostatic filterelectrostatic precipitatorelectrostrictionelectrothermal atomic absorption spectroscopyelectrothermal atomizer in spectrochemical analysiselectrothermal vaporizationelectroviscous effectselementelement effectelement of chiralityelemental mapelemental relative sensitivity factorelementary chargeelementary entityelementary particle in nuclear chemistryelementary reactionelevated plus mazeelicitationeliminationellipsometryellipticityeluateeluenteluteelutionelution band in chromatographyelution chromatographyelution curveelution mode in counter-current chromatographyelution solventelution timeelution volumeelutriationemanation thermal analysisembossingembryoembryo transferembryo transport rate analysisembryogenesisembryoid bodyembryologyembryonic discembryonic inductionembryonic periodembryonic stem cellemesisemeticemissionemission in atmospheric chemistryemission anisotropyemission control equipment in atmospheric chemistryemission flux in atmospheric chemistryemission spectrumemittanceempirical formulaemulsifieremulsionenaminesenantioconvergenceenantioenrichedenantiomerenantiomeric excessenantiomeric groupsenantiomeric purityenantiomeric ratioenantiomeric units in a polymerenantiomerically enrichedenantiomerically pureenantiomerismenantiomerizationenantiomorphenantiomorphous structures in polymersenantiopureenantioselective chromatographyenantioselectivityenantiotopicenantiotropic mesophaseenantiotropic transitionencapsulation in catalysisencephalitisencephaloceleencephalopathyencounterencounter complexencounter controlencounter pairencounter-controlled rateend-groupend-pointend-point errorend-to-end distance in polymersend-to-end vector in polymersendergonic reactionendocardial cushionendocardiumendochondral ossificationendocrineendocrine disrupterendocrine disruptorendocytosisendodermendoenzymesendogenousendometabolomeendometriumendonucleaseendorphinendosomeendotheliumendothermic reactionendotoxinene reactionenergized speciesenergyenergy of a radiationenergy bandenergy band theoryenergy dispersion in emission spectrometryenergy dispersive detectorenergy dispersive X-ray fluorescence analysisenergy edgeenergy flux densityenergy gapenergy gradientenergy hypersurfaceenergy level of a free atom, ion or moleculeenergy migrationenergy of activationenergy of activation of an electrode reactionenergy of the highest occupied molecular orbitalenergy of the lowest unoccupied molecular orbitalenergy poolingenergy profileenergy resolution in radiochemistryenergy storage efficiencyenergy threshold in radiochemistryenergy transfer in photochemistryenergy transfer plot in photochemistryenergy yield of luminescenceenforced concerted mechanismenhanced elastic cross sectionenhanced phosphorescence analysis in luminescence spectroscopyenhancement reaction in analytical chemistryenhancerenhancing antibodyenkephalinenolatesenolsenophileenosesenrichment factor in liquid-liquid distributionenrichment, isotopicentanglement in polymer scienceentanglement networkentatic stateentering groupenterotoxinenthalpimetric analysisenthalpimetric flow injection analysisenthalpimetric titrationenthalpimetryenthalpogramenthalpyenthalpy of activationenthalpy of immersionenthalpy of wettingenthalpy relaxationentiticentitic quantityentrainment in atmospheric chemistryentrainment in photochemistryentrance channelentropyentropy of activationentropy unitenvelope conformationenvironmental factorenvironmental monitoringenvironmental stress crackingenvironmentally degradable polymerenvironmentally friendlyenzymatic decompositionenzymatic degradationenzyme activatorenzyme activityenzyme conjugateenzyme induction in general chemistryenzyme induction in medicinal chemistryenzyme inhibitorenzyme repressionenzyme substrate electrodeenzyme thermistorenzyme-linked immunosorbent assayenzyme-linked immunospot assayenzyme-modified electrodeenzymesenzymic decompositioneosinophileosinophil chemotactic factor of anaphylaxiseotaxinepaxialependymal tissueependymomaepi-phaseepiblastepicadmium neutronsepicardiumepidemiologyepidermisepididymal appendixepididymisepiduralepigeneticepihalohydrinsepilepsyepimerizationepimersepiphyseal plateepispadiasepistasisepisulfonium ionsepitaxial crystallizationepitaxyepithalamusepithelial-to-mesenchymal transitionepitheliumepithermal neutronsepitopeepitope retrievalepitope spreadingepoxide hydrolaseepoxidesepoxy compoundsEpstein–Barr virusequalization of electronegativity, principle ofequatorialequilibrationequilibrium constantequilibrium controlequilibrium dialysateequilibrium distance in a moleculeequilibrium electrode potentialequilibrium filmequilibrium geometryequilibrium isotope effectequilibrium melting temperatureequilibrium reactionequilibrium sedimentationequilibrium sedimentation in a density gradientequilibrium sedimentation methodequilibrium solubilityequilibrium solutionequivalence in immunologyequivalence pointequivalence postulate in polymer chemistryequivalence zoneequivalence-pointequivalent chain in polymersequivalent diameterequivalent entityequivalent freely jointed chainequivalent hydrodynamic radiusequivalent hydrodynamic volumeerectile dysfunctionergergotismErnst angleerosionerosion rate of a surfaceerror of measurementerrors-in-variables regressionerythemaerythroblastosis fetaliserythrocyteerythropoiesiserythropoietinESCAescape depth for surface analysis techniquesEsin and Markov coefficientestersestradiolestriolestrogenestrogen-mimeticestrogenicestrous cycleestrusestrus syndromeET-valueetchetch rateetherseuatmotic reactionEuclidean distanceeuglobulineupeptide bondeuphoriaeuploideutectic reactioneutherianevaluation dataevaluation functionevanescent waveEvans holeevaporationeven-electron ioneven-electron ruleeventrationevocationevoked potentialevoked responseevolved gas analysisevolved gas detectionevolving factor analysisEwens–Bassett numberexaexact massexaminandexaminationexamination accuracyexamination biasexamination calibrationexamination calibratorexamination errorexamination methodexamination precisionexamination principleexamination procedureexamination repeatabilityexamination reproducibilityexamination resultexamination standardexamination traceabilityexamination truenessexamination uncertaintyexaminational comparability of examination resultsexaminational compatibility of examination resultsexamined valueexamining systemexcess acidityexcess mass at a solid/liquid interfaceexcess Rayleigh ratioexcess scatteringexcess volume at a solid/liquid interfaceexchange current of an electrode reactionexchange extractionexchange integralexchange labellingexchange repulsionexchange spectroscopyexchange-inversion transitionexcimerexcimer lampexcimer laserexcimer-luminescence in luminescence quenchingexcipient in toxicologyexciplexexciplex-luminescence in luminescence quenchingexcitable cellexcitationexcitation energyexcitation energy in in situ microanalysisexcitation level in X-ray spectroscopyexcitation profileexcitation sculptingexcitation spectrumexcitation transferexcitation-emission spectrumexcitatory post-synaptic potentialexcited stateexcitonexciton annihilationexciton diffusionexciton dissociationexciton mean free pathexciton–phonon couplingexcitotoxicityexcluded volume of a macromolecule in polymersexcluded volume of a segment in polymersexclusion chromatographyexcretionexcurrentexencephalyexergonic reactionexfoliated graphiteexfoliationexhaustive extractionexit channelexitanceexoexocrineexocytosisexoenzymesexogenous deoxyribonucleic acidexometabolomeexomphalosexonexonucleaseexophthalmiaexothermic reactionexotoxinexpansion factor in polymersexpectation valueexpected valueexperimental absorbanceexperimental designexperimental perturbational calculationexperimental scattered-ion intensityexpert systemexplanatory variableexplicit solventexploratory data analysisexplosivity limitsexponential decayexposure in medicinal chemistryexposure in photochemistryexposure in nuclear chemistryexposureexposure latitudeexpressed sequence tagexpression in biotechnologyexpression proteomicsexstrophyextended connectivity fingerprintsextended Hammett equationextended Hückel MO methodextended X-ray absorption fine structureextended X-ray absorption fine structure spectroscopyextended-chain crystal in polymersextenderextensive quantityextent at of an interface (surface)extent of electron delocalizationextent of reactionexternal calibrationexternal compensationexternal fieldexternal genitaliaexternal heavy atom effectexternal ion returnexternal ion-pair returnexternal photoelectric effectexternal quantum yieldexternal reflection spectroscopyexternal returnexternal standard in chromatographyexternal standardization for quenching correctionexternal surfaceexterplexextinctionextinction coefficientextra-column volume in chromatographyextracellular matrixextractextractextractability in solvent extractionextractantextracted ion chromatogramextracted ion electropherogramextracted ion profileextracting agentextractionextraction in process liquid-liquid distributionextraction coefficientextraction constantextraction factorextraction fieldextraction fractionation of polymersextraction isothermextraction phaseextraction process in catalysisextraction rateextractive distillationextractive electrospray ionization mass spectrometryextractor voltage in secondary ion mass spectrometryextraembryonic membraneextraembryonic mesodermextraordinary waveextrapolated range in radiochemistryextravasationextreme ultraviolet lithographyextremophilesextrinsic pathwayextrinsic plasmonextrinsic semiconductorextrusion transformationexudateπ-electron acceptor/donor groupρσ-equation f numberf-functional branch pointfac-f-functional branch unitF(ab')2 fragmentF1 generationFab fragmentfabric filterfacial cleftfaciesfacilitationfactorfactor analysisfactor levelfactor-group splittingfall timefallopian tubefallout in atmospheric chemistryfalse negativefalse positivefalxfamilial cold autoinflammatory syndromefamilial hemophagocytic lymphohistiocytosisfanning in atmospheric chemistryfaradfaradaic currentfaradaic current densityfaradaic demodulation currentfaradaic rectification currentFaraday constantFaraday cup collectorFaraday’s laws of electrolysisfarmer’s lungFasfasciculationfast atom bombardment ionizationfast field cycling NMR relaxometryfast followerfast ion bombardmentfast neutronsfast-atom bombardment mass spectrometryfast-atom bombardment secondary ion mass spectrometryfatigue of a photochromic systemfatty acidsfatty acylsFc fragmentFc receptorFd fragmentfecundationfecundityfeedfeed rate in catalysisfeed-back inhibition in biotechnologyfeedback in analysisfeedback in kineticsfemale fertilityfemale fertility indexfeminizationfemtofenestranesFenton reactionfermentationfermenterfermiFermi energyFermi levelFermi level referencingFermi resonancefermionferomoneferredoxinferrimagnetic transitionferrocenophanesferroelastic transitionferroelectric polymerferroelectric transitionferroic transitionferromagnetic polymerferromagnetic transitionfertile in radioanalytical chemistryFertility Assessment by Continuous Breedingfertilizationfetal alcohol syndromefetal anatomic positionfetal antigensfetal polefetal toleranceFETAXfetoscopyfetusfibre formatfibrillar morphologyfibrillationfibrinolysisfibrinolyticfibroblastfibroblast growth factorfibrous activated carbonfibrous carbonfibrous crystal in polymers in polymersfiducial groupfield desorption in mass spectrometryfield effectfield effect phototransistorfield ionization in mass spectrometryfield levelfield-effect transistorfield-flow fractionationfield-free regionfield-induced migrationfilamentous carbonfill factor of a solar cellfillerfiller cokefilling solution of a reference electrodefilmfilm badge in radioanalytical chemistryfilm elementfilm tensionfilterfilter of a radiationfilter spectrometerfiltrationfinal energy statefine structure constantfinesse of a spectrometerfinger tapping testfingerprintfingerprint bandsfingerprint regionfingerprinting in metabolomicsfirst line of defencefirst stability regionfirst-in-classfirst-order phase transitionfirst-pass effectFischer projection Fischer–Rosanoff conventionFischer–Tollens projectionfish embryo testfish reproduction assayfish sexual development testfissile in radioanalytical chemistryfission fragment ionization in mass spectrometryfission fragmentsfission neutronsfission productsfission yieldfissionablefistulafitnessfitness for purposefixed ionsfixed neutral loss scan in mass spectrometryfixed neutral loss spectrum in mass spectrometryfixed precursor ion scan in mass spectrometryfixed precursor ion spectrumfixed product ion scan in mass spectrometryfixed product ion spectrum in mass spectrometryflaccidflagpoleflame in flame atomic spectroscopyflame atomic spectrometerflame atomic spectroscopyflame ionization detector in gas chromatographyflame photometric detector in gas chromatographyflame photometryflame-in-tube atomizer in spectrochemical analysisflammable limitsflash desorptionflash fluorimetryflash lampsflash photolysisflash pointflash vacuum pyrolysisflat band potential at a semiconductor/solution interphaseflat bandsflavansflavinsflavonesflavonoidsflavonolignansflavoproteinsflavylium saltsflip anglefloating monolayerflocflocculationflocculeFlory–Fox assumptionFlory–Fox equationFlory–Huggins theoryflotationflow analysisflow birefringenceflow cytometryflow enthalpimetryflow field-flow fractionationflow injectionflow injection enthalpimetryflow injection mass spectrometryflow rate in chromatographyflow rate in flame emission and absorption spectrometryflow rate of a quantityflow rate of unburnt gas mixture in flame emission and absorption spectrometryflow reactorflow resistance parameter in chromatographyflow spoilerflow-programmed chromatographyflow-through extractionflow-through solid-phase spectrometryflue gas in atmospheric chemistryflue gas scrubber in atmospheric chemistryfluencefluence ratefluid cokefluidityfluidized bedfluorescein isothiocyanatefluoresceinsfluorescencefluorescence error in spectrochemical analysisfluorescence excitation of X-raysfluorescence lifetimefluorescence resonance energy transferfluorescence yieldfluorescence-activated cell sortingfluorescent in situ hybridizationfluorescent antibodyfluorimeterfluorocarbonsfluorogenicfluorohydrinsfluorophilicfluorophilic interactionfluorophobicfluorophobic interactionfluorophorefluoropolymersfluorousfluorous extractionflux of a quantityflux depressionflux monitorflux of a beam of particlesflux perturbationfluxionalfluxomicsfly ash in atmospheric chemistryfMLP peptidefoamfoam fractionationfoamed polyurethane sorbentfoaming agentfocal plane detectorfocal unitfocused ion beamfocused ion beam systemfoetusfogfog horizon in atmospheric chemistryfolatesfold in polymer crystalsfold domain in polymer crystalsfold plane in polymer crystalsfold surface in polymer crystalsfoldamerfolded-chain crystal in polymersfolding in spectroscopyfolic acidfolliclefollicle-stimulating hormonefollicle-stimulating hormone releasing hormonefollicular atresiafollicular dendritic cellfollicular phasefollow-on drugfontanellefootfootpad testfootprinting in metabolomicsforamenforamen magnumforamen ovaleforbidden line in X-ray spectroscopyforceforce constantsforce fieldforce-field calculationforced sinusoidal oscillationforeign body reactionforeign-gas broadeningformal chargeformal electrode potentialformamidine disulfidesformation constantformazansForssman antigenFörster cycleFörster excitation transferFörster-resonance-energy transferforward chainingforward geneticsforward geometryforward library searchforward scatteringforward stepwise linear discriminant analysisfossafossa ovalisfossil fuelfouling in membrane processesfouling agent in catalysisFourier transform ion cyclotron resonance mass spectrometerFourier transform spectrometerFourier-transform infrared spectroscopyFourier-transform mass spectrometryFourier-transform Raman spectroscopyFourier-transform spectroscopyFoxp3fractal agglomeratefractal dimensionfractionfraction collector in chromatographyfraction extractedfractional change of a quantityfractional ion yieldfractional selectivity in catalysisfractional sputtering yieldfractional-factorial designfractionation of analytesfractionation of polymersfragile glassfragility indexfragmentfragment ion in mass spectrometryfragment keysfragment-based lead discoveryfragmentationfragmentation reaction in mass spectrometryframe-shift mutationframework regionFranck–Condon principleFranck–Condon statefranklinfree charge density on the interfacefree charge-carrier photogenerationfree electron laserfree energyfree energy perturbationfree induction decayfree oscillationfree radicalfree rotationfree spectral rangefree spiro unionfree volumefree-drainingfree-running laserfreely drainingfreely jointed chain in polymersfreely rotating chain in polymersfreezingfreezing out in atmospheric chemistryfrequency in photochemistryfrequencyfrequency distributionfrequency doublingfrequency-domain fluorometryfrequent hitterFresnel reflectionFRETFreund’s adjuvantfriction coefficientfriction factorfrictional coefficient in polymer chemistryfringe fieldfringed-micelle model in polymer crystalsfront surface geometry in luminescencefrontal chromatographyfrontal lobefrontier orbitalsfronting in chromatographyfrontonasal dysplasiafrontotemporal dementiafrost point hygrometerfroth flotationFrumkin effectfrustrated Lewis acid-base pairfucolipidfuel cyclefuel elementfuel reprocessingfugacityfugacity coefficientfulgidesfull energy peakfull width at half maximumfull width at half maximumfull-evaporation headspace analysisfull-factorial designfullerenesfulminatesfulvalenesfulvenesfume in atmospheric chemistryfumesfumigation in atmospheric chemistryfunctional class fingerprintsfunctional class namefunctional domainfunctional genomicsfunctional groupfunctional observational batteryfunctional parentfunctional polymerfunctional proteomicsfunctionality of a monomerfundusfungicidefuranocoumarinsfuranosesfurnace blackfurnace pyrolysis in spectrochemical analysisfurocoumarinsfused in sarcoma transcription factorfusion in biotechnologyfusion namefuzzy clusteringFv fragment G value in nuclear chemistryG-fold cross validationg-factorg′-factorg index in G-SIMSG proteinG protein-coupled receptorG-SIMSG-SIMS with fragmentation pathway mappingGABAergicgain of a photomultipliergall bladderGalvani potential differencegalvanic cellgalvanic corrosiongalvanostatgametegamete intrafallopian transfergametogenesisgamma relaxationgamma relaxation peakgamma-ray spectrometrygammaglobulinganglionganglion cellgangliosidegas analysis installation in atmospheric chemistrygas blackgas chromatographygas chromatography-mass spectrometrygas constantgas flow in thermal analysisgas lasergas sensing electrodegas-filled phototubegas-filled X-ray detectorgas-liquid chromatographygas-phase aciditygas-phase basicitygas-phase-grown carbon fibresgas-solid chromatographygaseous diffusion separator in atmospheric chemistrygastric parietal cellgastroschisisgastrulagastrulationgate delay in laser induced breakdown spectroscopygated decouplinggated photochromismgated photodetectorgauchegauche conformation in polymersgauche effectgaussGaussian bandGaussian band shapeGaussian orbitalGD-OES spectrometers with simultaneous detectionGeiger counterGeiger–Muller counter tubegelgel aginggel electrophoresis in proteomicsgel fractiongel permeation chromatographygel pointgel timegelationgelation temperatureGell and Coombs classificationgeminate ion pairgeminate pairgeminate recombinationgenegene amplificationgene expressiongene knockoutgene librarygene manipulationgene productgene rearrangementgene silencinggene targetinggeneral acid catalysisgeneral acid–base catalysisgeneral base catalysisgeneral force fieldgeneralized transition-state theorygenerally labelled tracergeneration in a dendrongeneration time in biotechnologygenerator coupling efficiency in inductively-coupled plasma spectrometrygenetic algorithmgenetic codegenetic engineeringgenetic recombinationgenitalgenital foldgenital organgenital tractgenital tuberclegenitaliagenomegenomicsgenotoxicgenotypegeometric attenuationgeometric isomerismgeometric meangeometric surface areageometrical equivalence in polymersgeometry in radioanalytical chemistrygeometry factor in radioanalytical chemistrygerm cell gene mutation assaygerm layergerm linegerm line cellgerm-freegerminal centergermylenesgermylidenesgestationgestation indexgestation periodgiant axongiant axonal neuropathygiant cellGibbs adsorptionGibbs dividing surfaceGibbs energyGibbs energy diagramGibbs energy of activationGibbs energy of photoinduced electron transferGibbs energy of repulsionGibbs energy profileGibbs film elasticityGibbs surfacegigagigantismglandglansglans clitoridisglans penisglassglass electrode errorglass laserglass pH-sensitive electrodeglass transitionglass-like carbonglass-transition temperatureglassy carbon electrodeglassy stateglaucomagliaglial cellglial fibrillary acidic proteinglial-derived neurotrophic factorglioblastomagliomaglobal analysisglobal modelglobally optimized alternating-phase rectangular pulsesglobosideglobular-chain crystal in polymersglobuleglobule-to-coil transitionglomerulonephritisglomerulonephropathyglomerulusglove boxglow discharge ionizationglow discharge sourceglow electric dischargeglow-discharge optical emission spectroscopyglucocorticoidglucuronideglutamateglutamate-induced excitotoxicityglutamic acid decarboxylaseglutenglycalsglycan microarrayglycan profilingglycan-binding proteinglycansglycaric acidsglyceridesglycerolipidsglycerophospholipidglycerophospholipidsglycitolsglyco-amino-acidglycoconjugateglycoformglycogeneglycoglycerolipidglycolipidglycolipidsglycolsglycolysisglycomeglycomicsglyconeglyconic acidsglycopeptidesglycoproteinglycoproteinsglycoproteomicsglycosaminesglycosaminoglycanglycosesglycosidaseglycosideglycosidesglycosidic linkageglycosphingolipidglycosylglycosyl groupglycosyl-amino-acidglycosylaminesglycosylationglycosyltransferaseglycuronic acidsgold electrodeGolgi staininggonadgonadotropingonadotropin-releasing hormonegood laboratory practicegood manufacturing practiceGoodpasture syndromeGouy layer in electrochemistrygp120Graafian folliclegrab samplinggradientgradient elution in chromatographygradient layer in chromatographygradient packinggradient-selected experimentgradientless reactor in catalysisgradual potential-energy surfacegraft copolymergraft copolymerizationgraft homopolymergraft macromoleculegraft polymergraft rejectiongraft-versus-host diseasegraft-versus-host reactiongrafting in catalysisgrafting in polymer chemistrygramGram staingranular carbongranule cellgranulocytegranulocyte colony-stimulating factorgranulocyte–macrophage colony-stimulating factorgranulocytopeniagranulomagranulosagranzymegraph-based methodgraphenegraphene layergraphitegraphite electrodegraphite fibresgraphite materialgraphite whiskersgraphitic carbongraphitizable carbongraphitizationgraphitization heat treatmentgraphitized carbonGraves diseasegravimetric analysisgravimetric factorgravimetric methodgravitational constantgravitational field-flow fractionationgraygrazing incidencegrazing-incidence small-angle X-ray scattering analysisgreater omentumgreen bodygreen chemistrygreen cokegreen polymergreenhouse effect in atmospheric chemistrygrey mattergridless reflectronGrignard reagentsGrimm-type cathodegristgrit in atmospheric chemistrygroove binding in deoxyribonucleic acidground level concentration in atmospheric chemistryground level inversion in atmospheric chemistryground stategroupgroup electronegativitygroup frequencygroup preconcentration in trace analysisgroup-transfer polymerizationgrowth curve of activitygrowth factorgrowth hormonegrowth hormone releasing factorgrowth rate in biotechnologyGrunwald–Winstein equationguard bandgubernaculumguestGuillain–Barré syndromeguinea pig maximization testGuinier plotgustiness in atmospheric chemistrygut-associated lymphoid tissuehybrid materialγ-aminobutyric acidγ-aminobutyric acid receptorγ-cascadeγ-interferonγ:δ T cellγ:δ T-cell receptor hexaprismo- in inorganic nomenclaturehexahedro- in inorganic nomenclaturehypho-H-2H-2 complexH(CCO)NHHaber–Weiss reactionhabituationHadamard transform spectrometerhaemhairy cell leukemiahalato-telechelic polymerhalatopolymerhalf lifehalf life of a radionuclidehalf thickness in radiochemistryhalf width at half maximumhalf width at half maximumhalf-chairhalf-life of a photochromic systemhalf-life of a transient entityhalf-peak potentialhalf-value thicknesshalf-wave potentialhalf-width of a bandhalirenium ionshallucinationhalochromismhaloformshalogen bondhalohydrinshalonium ionshalophileshamiltonian operatorHammett acidity functionHammett equationHammond principleHammond–Herkstroeter plothandednessHansch analysisHansch constantHansch equationHansch–Fujita π-constantHantzsch–Widman namehaploidhaplotypehaptenhaptenichapticityhaptohard acidhard amorphous carbon filmshard bakehard basehard ionizationhard-segment phase domainhard-sphere collision cross sectionharmonic approximationharmonic constantharmonic frequency generationharmonic meanharpoon mechanismhartreeHartree energyHashimoto thyroiditishashingHaworth representationhay feverhazardhaze in atmospheric chemistryhaze horizon in atmospheric chemistryHBA solventHBD solventHBHA(CO)NHHCCH-COSYHCCH-TOCSYhead-tail isomerismheadacheheadspaceheadspace solid-phase microextractionhealth surveillanceheart-cut separationsheatheat capacityheat capacity of activationheat fluxheat shock proteinsheat-flow differential scanning calorimetryheavy atom effectheavy atom isotope effectheavy chainheavy waterhectoheight equivalent to a theoretical plate in chromatographyheight equivalent to an effective theoretical plate in chromatographyheliceneshelicityheliochromismhelionhelium dead-space in colloid and surface chemistryhelium ionization detector in gas chromatographyhelium–cadmium laserhelium–neon laserhelixhelix residue in a polymerhelix senseHelmholtz energyhelper T lymphocytehemagglutinationhemagglutininhemalhemangiomahematomahematopoiesishematopoietic stem cellhemeshemiacetalhemiacetalshemiaminalshemicarceplexhemicarcerandhemiketalhemiketalshemileptichemimeliaheminshemiplegiahemivertebrahemochromeshemodialysishemoglobinshemolytic anemiahemolytic disease of the newbornhemorrhagehen testHenderson–Hasselbalch equationhenryHenry's lawHenry’s law constantHenry’s law solubility constantHenry’s law solubility constant (via b and p) defined via molality and partial pressureHenry’s law solubility constant (via c and p) defined via amount concentration and partial pressureHenry’s law solubility constant (via x and p) defined via liquid-phase amount fraction and partial pressureHenry’s law volatility constantHenry’s law volatility constant (via p and c) defined via partial pressure and amount concentrationHenry’s law volatility constant (via p and w)Henry’s law volatility constant (via p and x)heparin-induced thrombocytopeniahepatic flexurehepatosplenomegalyhereditary angioneurotic edemaHerkstroeter plothermaphroditehermaphroditismherniaherpesHershberger bioassayhertzHerz compoundshetareneshetaryl groupshetarynesheteroalkenesheteroantigenheteroarenesheteroaryl groupsheteroarynesheterobimetallic complexheterochain polymerheteroconjugationheterocumulenesheterocyclic compoundsheterocyclyl groupsheterodetic cyclic peptideheterodisperseheteroexcimerheterogeneity in analytical chemistryheterogeneous catalysisheterogeneous catalysis coordination polymerizationheterogeneous crystalline membrane electrodesheterogeneous crystalline membrane ion-selective electrodeheterogeneous degradationheterogeneous diffusion rate constant in electrochemistryheterogeneous membrane electrodeheterogeneous nucleationheterojunctionheterolepticheterologous antibodyheterologous antigenheterolysisheterolytic bond-dissociation energyheterolytic dissociative adsorptionheteronuclear decouplingheteronuclear multiple bond correlation NMRheteronuclear multiple-bond correlation over two bondsheteronuclear multiple-quantum correlation NMRheteronuclear multiple-quantum correlation with additional TOCSY transferheteronuclear Overhauser effect spectroscopyheteronuclear shift correlation NMRheteronuclear single quantum correlationheteronuclear single quantum correlation with additional TOCSY transferheteronuclear single quantum multiple-bond correlationheterophagyheterophile antibodyheterophile antigenheteropolysaccharideheterotactic macromoleculeheterotactic polymerheterotactic triads in polymersheterotopicheterotrophicheterovalent hyperconjugationheterozygoushexagonal graphitehexagonal mesophaseHeyrovský–Ilkovič equationhiatal herniahidden layerhidden returnhierarchical clusteringhigh endothelial venulehigh resolution energy loss spectroscopyhigh-field asymmetric waveform ion mobility spectrometryhigh-mobility group box proteinhigh-pass filterhigh-pressure graphitizationhigh-pressure mercury lamphigh-speed counter-current chromatographyhigh-spinhigh-throughput screeninghigher-energy collision dissociation in proteomicshigher-order transitionhighly active antiretroviral therapyhighly oriented pyrolytic graphitehighly oriented pyrolytic graphite electrodeHilbert transforms of a spectrumHildebrand parameterhindered rotationhinge regionhippinghippocampal slice culturehippocampusHirschsprung syndromehirsutismhistaminehistaminergichistaminosishistocompatibilityhistoincompatibilityhistoneshithit expansionhit-to-lead chemistryhivesHN(CA)COHN(CO)CAHNCAHNCOHodgkin diseaseHoffman–Weeks plotHofmann rulehold-back carrierhold-up time in chromatographyhold-up volume in chromatographyhold-up volume in column chromatographyholehole burninghole transferhole-transporting materialhollow fibrehollow-cathode discharge sourcehollow-cathode lamphollow-fibre liquid-phase microextractionholoblastic cleavageholoenzymeholoprosencephalyHoltsmark broadening of a spectral linehomeoboxhomeobox genehomeosishominghoming receptorhomohomoaromatichomochain polymerhomochiralhomoconjugationhomocyclic compoundshomocytotropic antibodyhomodesmotic reactionhomodetic cyclic peptidehomogeneity in analytical chemistryhomogeneous catalysishomogeneous catalysis coordination polymerizationhomogeneous crystalline membrane electrodeshomogeneous crystalline membrane ion-selective electrodehomogeneous degradationhomogeneous line-broadeninghomogeneous membrane electrodehomogeneous nucleationhomogeneous nucleushomogeneous polymer blendhomogeneous precipitationhomojunctionhomoleptichomologoushomologous polymer blendhomologous recombinationhomology modelhomolysishomolytic dissociative adsorptionhomomorphichomonuclear decouplinghomopolymerhomopolymerizationhomopolysaccharidehomotopichomozygoushoney bee larval toxicity testhoneycomb polymer filmhopping transporthorizontal elution in planar chromatographyhormonehorror autotoxicusHorwitz equationHorwitz ratiohost in biotechnologyhost defensehost responsehost-resistance assayhost-vector systemhot atomhot cellhot electronhot ground state reactionhot quartz lamphot state reactionhot transitionhourHückel (4n + 2) ruleHückel molecular orbital theoryHuggins coefficientHuggins equationHuisgen reactionhula-twist mechanismhuman chorionic somatomammotropinHuman Genome Projecthuman immunodeficiency virushuman leukocyte antigenhuman leukocyte antigen polymorphismhuman leukocyte antigen type Ihuman leukocyte antigen type IIhumanized antibodyhumidity in atmospheric chemistryhumoralhumoral immune responsehumoral immunityHund rulesHuntington diseaseHush modelhybrid artificial organhybrid mass spectrometerhybrid orbitalhybrid polymerhybridizationhybridization assayhybridomahydatidiform molehydranencephalyhydrationhydrazide hydrazoneshydrazide imideshydrazideshydrazidineshydrazineshydrazinylideneshydrazo compoundshydrazoneshydrazonic acidshydridehydrohydrocarbon crackinghydrocarbonshydrocarbyl groupshydrocarbylene groupshydrocarbylidene groupshydrocarbylidyne groupshydrocarbylsulfanyl nitreneshydrocelehydrocephalushydrocracking unithydrodynamic chromatographyhydrodynamic injectionhydrodynamic interactionhydrodynamic voltammetryhydrodynamic volume in polymershydrodynamically equivalent sphere in polymershydrogelhydrogenhydrogen bondhydrogen bond in theoretical organic chemistryhydrogen gas electrodehydrogen shiftshydrogen/deuterium exchangehydrolasehydrolaseshydrolysishydrolysis ratiohydrometeor in atmospheric chemistryhydromicrocephalyhydromyeliahydronhydronationhydronephrosishydroperoxideshydrophilichydrophilic interactionhydrophilic-interaction liquid chromatographyhydrophilicityhydrophobichydrophobic fragmental constanthydrophobic interactionhydrophobic-interaction chromatographyhydrophobicityhydropolysulfideshydropshydrosphere in atmospheric chemistryhydrosulfideshydroureterhydroxamic acidshydroximic acidshydroxylamineshygromahygrometerhygrometryhymenhyoid archhypaxialhyper-Raman spectroscopyhyperbranched macromoleculehyperbranched oligomerhyperbranched oligomer moleculehyperbranched polymerhypercapniahyperchromic effecthyperconjugationhypercoordinationhyperesthesiahyperextensionhyperfinehyperfine couplinghyperflexionhypergammaglobulinemiahyperimmunoglobulin E syndromehyperimmunoglobulin M syndromehyperlayer focusing field-flow fractionationhyperplasiahyperpolarizability of nth orderhyperpolarizationhyperreactivityhyperreflexiahypersatellite in X-ray spectroscopyhypersensitivityhypersensitivity pneumonitishypersensitizationhyperspectral imaginghypersusceptibilityhypertelorismhypertoniahypertrophyhypervalencyhypervariable regionhyphenated mass spectrometry techniquehypnotichypo-phasehypoblasthypocapniahypochromic effecthypogammaglobulinemiahypogonadismhypophysectomyhypoplasiahyporeflexiahyposensitization therapyhypospadiashypotensionhypothalamushypotoniahypoxiahypsochromic shifthysteresisη in inorganic nomenclature in vitro estrogen activity assayin vitro fertilizationin-vivo neutron activation analysisicosahedro-in situ composite formationin situ micro-X-ray diffractionin situ microanalysisin vitroin vivoipso-attackIa antigenICichthyosisicosanoidsICTICT emissionictusideal adsorbed stateideal chromatographyideal dilute solutionideal gasideal mixtureideal, non-linear chromatographyideally polarized interphaseideally unpolarized interphaseidentical twinsidentity period of a polymeridentity reactionidiopathicidiopathic thrombocytopenic purpuraidiosyncratic drug reactionidiotypeidiotype networkileocecalileumiliac arteryIlkovič equationilluminanceimage in lithographyimage blurimage converter tubeimage current detectionimage depth profileimage dissection tubeimaginary NMR spectrumimagingimaging mass spectrometryimbalanceimbibition in colloid chemistryimenesimidesimidic acidsimidinesimidogensimidonium ionsimidoyl carbenesimidoyl nitrenesiminimine radicaliminesiminium compoundsiminium ionimino acidsimino carbenesiminooxy radicalsiminoxyl radicalsiminyl carbenesiminyl radicalsiminylium ionsimmature in immunologyimmediate-type hypersensitivityimmersion fluidimmersion lithographyimmersion solid-phase microextractionimmersional wettingimmiscibilityimmiscible polymer blendimmission in atmospheric chemistryimmission dose in atmospheric chemistryimmission flux in atmospheric chemistryimmission rate in atmospheric chemistryimmobile adsorptionimmobilization in biotechnologyimmobilization by adsorption immobilization by inclusion immobilized enzymeimmobilized phase in chromatographyimmobilized stationary phase in chromatographyimmune adherenceimmune compleximmune complex depositionimmune complex diseaseimmune cytolysisimmune deviationimmune eliminationimmune enhancementimmune equilibriumimmune escapeimmune evasionimmune modulationimmune regulationimmune reserve hypothesisimmune responseimmune response genesimmune response-associated proteinimmune surveillanceimmune systemimmunityimmunizationimmunoactivationimmunoactivatorimmunoadsorptionimmunoassayimmunoblotimmunoblottingimmunochemical specificityimmunochemistryimmunocompetenceimmunocompleximmunocompromisedimmunoconjugateimmunocytokineimmunodeficiencyimmunodominant epitopeimmunodysregulation–polyendocrinopathy–enteropathy, X-linkedimmunoeditingimmunoelectrophoresisimmunofluorescenceimmunogenimmunogenicityimmunoglobulinimmunoglobulin Aimmunoglobulin classimmunoglobulin Dimmunoglobulin Eimmunoglobulin E-binding Fc receptorsimmunoglobulin E-mediated hypersensitivityimmunoglobulin Gimmunoglobulin gene superfamilyimmunoglobulin Mimmunoglobulin or T-cell receptor lymphocyte repertoireimmunoglobulin superfamilyimmunoglobulin Yimmunohistochemistryimmunological ignoranceimmunological incompetenceimmunological memoryimmunological synapseimmunological toleranceimmunologically privileged siteimmunologyimmunomagnetic separationimmunomodulationimmunopathologyimmunopathyimmunopharmacologyimmunophenotypingimmunophilinimmunopotentiationimmunoprecipitateimmunoprecipitationimmunoradiometric assayimmunoreactivityimmunoreceptor tyrosine-based activation motifimmunoreceptor tyrosine-based inhibitory motifimmunosensitivityimmunosensitizerimmunosensorimmunosorbentsimmunostimulating compleximmunostimulationimmunosuppressantimmunosuppressionimmunosuppressiveimmunosurveillanceimmunotherapyimmunotoxicimmunotoxicantimmunotoxicologyimmunotoxinimpact energyimpact ionization in a semiconductorimpact parameterimpact-modified polymerimpactionimpedanceimpedimetryimperforate anusimpermeableimpingementimpingerimplantimplantationimplanted areic doseimposeximprecision in analysisimpregnation in polymer chemistryimpregnation in chromatographyimprint lithographyimprinting in geneticsimproper rotation axisimproved canonical variational transition-state theoryimpulsive reactionin-house reference materialin-laboratory processing in analytical chemistryin-line extractionin-needle capillary adsorption trapin-out isomerismin-source collision-induced dissociationin-syringe microextractionin-tube solid-phase microextractioninaccuracy in analysisinactivated vaccineinactive chaininbred straininchInChIKeyincidence in medicinal chemistryincident powerincineratorinclusion bodyinclusion compoundincoherent radiationincoherent scatteringincoherent structureincongruent reactionincredible natural-abundance double-quantum transfer experimentincrement for large unitsindicated hydrogenindicatorindicator consumption errorindicator electrodeindicator reactionindifferent absorbing ionindifferent electrolyteindirect agglutinationindirect amplificationindirect immunofluorescenceindirect reactionindirectly-ionizing radiationindividual gauge for localized orbitalsindividual isotherm individual particle analysisindividual perception threshold in atmospheric chemistryindolamine-2,3-dioxygenaseinduced abortioninduced pluripotent stem cellinduced radioactivityinduced reactioninducer in enzyme catalysisinducible co-stimulatory proteininductioninduction periodinductive effectinductively-coupled plasmainductively-coupled plasma mass spectrometryinductively-coupled plasma optical emission spectroscopyinductively-coupled plasma robustnessinductively-coupled plasma spectrometryinductively-coupled plasma thermal pinchinductively-coupled plasma torchinductomeric effectinductorinelastic collisioninelastic electron scattering backgroundinelastic electron scattering background subtractioninelastic electron tunnelling spectroscopyinelastic scatteringinelastic scattering cross sectioninertinert gasinertial defectinertial separatorinfectious toleranceinferior cerebellar peduncleinfertilityinfinite source thicknessinflammasomeinflammationinflammatory bowel diseaseinfluence nominal propertyinformation depthinformation radiusinformation theoryinfraredinfrared multiphoton dissociationinfrared radiationinfrared spectroscopyinfratentorialinfundibuluminguinal canalinherent viscosity of a polymerinhibininhibitioninhibition constantinhibitorinhibitory concentrationinhibitory concentration at 5%inhibitory doseinhibitory post-synaptic potentialinhomogeneity error in spectrochemical analysisinhomogeneous broadeninginiencephalyiniferiniferteriniferter processiniopagusinitial energy initial energy state in X-ray photoelectron spectroscopyinitial energy state in Auger electron spectroscopyinitial plasma radiation zone in inductively-coupled plasma optical emission spectroscopyinitial rate methodinitial state correlationsinitial-state Auger parameterinitialization in coordination polymerizationinitialization efficiency in RDPinitializer in RDPinitiating speciesinitiationinitiatorinitiator efficiency initiation of polymerizationinjection temperature in chromatographyinnate immunityinner electric potentialinner filter effectinner Helmholtz planeinner layer in electrochemistryinner orbital X-ray emission spectrainner saltsinner-sphere electron transferinner-sphere electron transfer atom-transfer radical polymerizationinoculationinorganic oxide adsorbentinorganic polymerinorganic–organic polymerinositolsinput rate in analysisinsecticideinseminationinsensitive nuclei enhanced by polarization transferinsert in biotechnologyinsertioninsertion polymerizationinsomniainspectioninstability in instrumentationinstability of Hartree–Fock solutioninstantaneous currentinstantaneous rate of flow in polarographyinstantaneous sampling in atmospheric chemistryinstrument line shapeinstrumental activation analysisinstrumental dependabilityinstrumental depth resolutioninstrumental indication for a precision balanceintegral capacitance of an electrodeintegral detector in chromatographyintegrated intensityintegrating sphereintegration range of a spectrumintegrinintelligence quotientintended crossing of potential-energy surfacesintensified charge coupled device detectorintensityintensity in mass spectrometryintensive quantityinteraction distanceinteraction effectinteractomeinteractomicsinteratomic Auger processintercalation in polymer chemistryintercalation compoundsintercalation reactionintercellular adhesion moleculeinterchange reactioninterchromophoric radiationless transitioninterclonal competitioninterconal regioninterconvertible enzymeinterfaceinterface core-level shiftinterfacial adhesioninterfacial concentration in electrochemistryinterfacial double-layerinterfacial layerinterfacial layer width in thin filmsinterfacial regioninterfacial tensioninterference in analysisinterference curve in atomic spectroscopyinterference filterinterference fringesinterference lithographyinterference recordinterferentinterfering linesinterfering substance in electroanalytical chemistryinterferograminterferometerinterferonsinterlaboratory comparisoninterleukinintermediateintermediate examination precisionintermediate filamentintermediate measurement precisionintermediate neutronsintermediate precision condition of examinationintermolecularintermolecular interactions in biomacromoleculesintermolecular radiationless transitioninternal absorbanceinternal absorptanceinternal compensationinternal conversioninternal energyinternal filling solution of a glass electrodeinternal fragmentinternal imageinternal quality controlinternal quantum yieldinternal reference electrodeinternal reflectioninternal reflection elementinternal returninternal standard in chromatographyinternal surfaceinternal transmittanceinternal transmittanceinternal valence force fieldinternational calorieInternational Chemical IdentifierInternational Chemical Identifierinternational examination standardInternational system of unitsinternational unitinterparticle porosity in chromatographyinterparticle volume of the column in chromatographyinterparticle volume of the column in size-exclusion chromatographyinterpenetrating polymer networkinterphaseinterphase transitionintersection spaceintersexualityinterstitial fraction in chromatographyinterstitial velocity in chromatographyinterstitial volume in gas chromatographyintersystem crossinginterval analysisintervalence charge transferintervertebral discinterzonal regionintimate ion pairintolerance in immunologyintra- in organic reaction mechanismsintra-epithelial lymphocyteintrachromophoric radiationless transitionintracranial pressureintracytoplasmic sperm injectionintradermal testintralaboratory comparisonintramembrous ossificationintramolecularintramolecular catalysisintramolecular charge transferintramolecular interactions in biomacromoleculesintramolecular isotope effectintraparticle volume of the column in size-exclusion chromatographyintraphase transitionintrathecalintrauterineintrauterine growth restrictionintrinsic activation energyintrinsic barrierintrinsic detector efficiencyintrinsic examination standardintrinsic full energy peak efficiencyintrinsic pathwayintrinsic photopeak efficiencyintrinsic plasmonintrinsic reaction coordinateintrinsic semiconductorintrinsic solubilityintrinsic viscosity of a polymerintrinsically conducting polymerintroninvariant chaininverse Bremsstrahlunginverse isotope effectinverse kinetic isotope effectinverse micelleinverse opalinverse Raman scatteringinverse spatially offset Raman spectroscopyinverse square law in radiation chemistryinversioninversion height in atmospheric chemistryinversion point in phase transitionsinversion pulseinversion recovery sequenceinversion timeinverted micelleinverted region for electron transferinvestigational new drugiodine-transfer polymerizationiodohydrinsiodometric titrationiodonium ionsionion beamion beam analysision beam ratioion beam resistion channelion channelopathyion collector in mass spectrometryion cyclotron resonance mass spectrometerion desolvation in mass spectrometryion energy loss spectra in mass spectrometryion enhancemention evaporation modelion exchangeion exchangerion funnelion gateion implantationion kinetic energy spectrum in mass spectrometryion laserion microscopyion mobility spectrometryion opticsion pairion pair returnion probe microanalysision pumpsion radicalion scattering spectrometryion source in mass spectrometryion suppressionion trapion trap mass spectrometerion-conducting polymerion-containing polymerion-exchange chromatographyion-exchange isothermion-exchange membraneion-exchange polymerion-exchange sorbention-exclusion chromatographyion-free layerion-pair chromatographyion-pair extractionion-pair formation in mass spectrometryion-scattering spectrumion-selective electrodeion-selective electrode cellion-selective field effect transistorion-selective membraneion-to-photon detectorion/ion reactionion/molecule reaction in mass spectrometryion/neutral complexion/neutral species exchange reaction in mass spectrometryion/neutral species reaction in mass spectrometryioneneionic aggregates in an ionomerionic bondionic cloud in atomic spectroscopyionic concentrationionic conductivityionic copolymerizationionic dissociation in mass spectrometryionic liquidionic polymerionic polymerizationionic ring-opening polymerizationionic spectral linesionic strengthionic transport numberionizationionization buffer in flame spectroscopyionization by sputteringionization chamberionization cross-sectionionization efficiencyionization efficiency curve in mass spectrometryionization energyionization potentialionized calcium-binding adapter molecule 1ionizing collision in mass spectrometryionizing powerionizing radiationionizing voltageionogenic groupsionomerionomer clusterionomer moleculeionomer multipletionophoreionotropic receptoriridescent layersiridoidsiron-sulfur clusteriron-sulfur proteinsirradiance at a point of a surfaceirradiationirregular dendrimerirregular dendrimer moleculeirregular dendronirregular macromoleculeirregular polymerirreversible transitionIRTRAN™Irwin batteryISCischemiaISIS keysislet cell antibodiesisoabsorption pointisobar in atmospheric chemistryisobaric mass-change determinationisobaric separationisobaric tag for relative and absolute quantitationisobarsisoclined structures in polymersisoclinic pointisoconfertic separationisoconjugate systemsisocoumarinsisocratic analysis in chromatographyisocratic elution in chromatographyisocyanatesisocyanidesisocyclic compoundsisodesmic reactionisodiazenesisoelectricisoelectric focusing in proteomicsisoelectric point in electrophoresisisoelectronicisoemissive pointisoentropicisoenzymeisoequilibrium relationshipisoflavansisoflavonesisoflavonoidsisoflavonoidsisograftisogyric reactionisohemagglutininisoionicisoionic point in electrophoresisisokinetic line in atmospheric chemistryisokinetic relationshipisokinetic sampling in atmospheric chemistryisolampsic pointisolated double bondsisolobalisomerisomer shift in Mössbauer spectroscopyisomerasesisomericisomeric state in nuclear chemistryisomeric transition in nuclear chemistryisomerismisomerizationisometricisomorphic polymer blendisomorphous structures in polymersisonitrilesisonitroso compoundsisooptoacoustic pointisopeptide bondisopotential pointisoprenesisoprenoidsisopycnicisopycnic separationisorefractiveisosbestic pointisoselective relationshipisoselenocyanatesisostatic pressingisosteric enthalpy of adsorptionisostilbic pointisostructural reactionisotactic macromoleculeisotactic polymerisotactic triads in polymersisotherm in atmospheric chemistryisothermal chromatographyisothermal technique in thermal analysisisothiocyanatesisotonesisotope clusterisotope coded affinity tagisotope dilutionisotope dilution analysisisotope dilution mass spectrometryisotope effectisotope exchangeisotope exchange analysisisotope pattern in mass spectrometryisotope ratioisotope ratio mass spectrometryisotopesisotopic abundanceisotopic carrierisotopic enrichmentisotopic enrichment factorisotopic fractionation factorisotopic ionisotopic labelisotopic labellingisotopic molecular ionisotopic scramblingisotopic separationisotopic tracerisotopically deficientisotopically enriched ionsisotopically labelledisotopically modifiedisotopically substitutedisotopically unmodifiedisotopolog ionsisotopologueisotopomerisotopomeric ionisotretinoinisotropicisotropic carbonisotropic pitch-based carbon fibresisotropisation temperatureisotypeisotype controlisotype switchisoureasisovalent hyperconjugationisozyme j-value in atmospheric chemistryJ-modulated spin-echoJ-resolved spectroscopyJablonski diagramjackknifingJacquinot stopJahn–Teller effectJahn–Teller transitionJAK/STAT signaling pathwayJAM testJanus-family tyrosine kinasejejunumjet separatorjoining chainjoining genejoulejunction pointjunction unitjunction-point densityjunctional diversity k-means clusteringk-nearest neighbourklado-kainic acidkallikreinKamlet–Taft solvent parametersKaposi sarcomakappa shape descriptorkappa-light chainKaptein–Closs ruleskaryolysiskaryorrhexiskaryotypeKasha ruleKasha–Vavilov rulekatalKauzmann temperatureKekulé structure for aromatic compoundskelvinkeratinkeratinizationkeratinocytekernel methodketalsketazinesketenesketeniminesketidesketiminesketoketo carbenesketo-enol tautomerizationketoacidosisketoaldonic acidsketoaldosesketone bodyketonesketosesketoximesketylskiller activatory receptorkiller cellkiller cell immunoglobulin-like receptorkiller lectin-like receptorkilokilogramkinasekind-of-nominal-propertykind-of-propertykind-of-quantitykindlingkinematic factorkinematic viscositykinematicskinetic activity factorkinetic ambiguitykinetic control of product compositionkinetic couplingkinetic currentkinetic electrolyte effectkinetic energykinetic energy releasekinetic energy release distributionkinetic equivalencekinetic isotope effectkinetic method of analysiskinetic resolutionkinetic shiftkinetic solubilitykinetic synergistkinetic theory of collisionsKingdon trapkininKirkwood–Riseman theoryKlinefelter syndromeKnight shiftknockoutknockout mouseknockout mutationKohlrausch–Williams–Watts equationKohonen networkKoopmans' theoremKoppel–Palm solvent parametersKosower -valueKováts indexKraemer coefficientKraemer equationKrafft pointKramers–Kronig transforms of a spectrumKratky plotKRS-5krypton ion laserKubelka–Munk functionKuhn segmentKuhn segment lengthKupffer cellKveim reactionkyphosisκ in inorganic nomenclature ll, uLIDAR in atmospheric chemistryLL-selectinlab-at-valve liquid-phase microextractionlab-on-a-chip extractionlab-on-valve extractionlabellabel-free quantitationlabellinglabilelaborlaboratory biaslaboratory information management systemlaboratory samplelachrymatorlacrimationlactamslactationlactation periodlactideslactimslactolslactonesladder chainladder macromoleculeladder polymerLADSlag phase in biotechnologylambdalambda (λ)-light chainLambert lawLambert–Eaton myasthenic syndromelamellalamellar block copolymerlamellar crystallamina proprialamplamp blackLandau–Zener modelLandolt reactionLangerhans cellLangmuir monolayerLangmuir–Blodgett membraneLangmuir–Hinshelwood mechanismLangmuir–Rideal mechanismLanthony D-15 color testLaporte rulelapse rate in atmospheric chemistrylarge granular lymphocytelarge moleculelarge particle in radiation scatteringlariat ethersLarmor angular frequencyLarmor precessionlarynxLASlaserlaser ablationlaser ablation electrospray ionization mass spectrometrylaser beam ionizationlaser desorptionlaser desorption ionizationlaser dyelaser induced breakdown spectroscopylaser ionization in mass spectrometrylaser micro emission spectroscopylaser micro mass spectrometrylaser microprobe mass spectrometrylaser Raman microanalysislaser-induced breakdown spectrumlaser-induced plasmalaser-jet photochemical techniquelasinglate phase reactionlate-downhill surfacelatentlatent imagelatent period in medicinal chemistrylatent variablelateral flow testlateral order in a polymerlateral resolution in in situ microanalysislatexlatex agglutination testlatex allergenlath crystallathyrismlattice in condensed matter physicslattice distortionLaurence solvent parameterslaws of distribution in precipitationlayerlayer equilibrium in chromatographylayer-by-layer assemblyleachingleadlead validationleader sequence in biotechnologyleast motion principleleast squares regressionleast-squares techniqueleave-some-out cross-validationleaving grouplecithinslectinlectin microarraylectinsLEDLeffler's assumptionLeffler’s relationleft-to-right conventionlengthleprosyleptocephalyleptodactylylesser omentumlethal concentrationlethal doselethal synthesislethargyleucismleuco basesleuco compoundsleukemialeukocyteleukocyte common antigenleukocyte functional antigenleukocytopenialeukocytosisleukopenialeukotrieneleukotrieneslevarterenollevellevel widthlevelling effectleverageLevich equationlevocardiaLewis acidLewis acidityLewis adductLewis baseLewis basicityLewis formulaLewis octet ruleLeydig celllibidolife cycle assessmentlifetimelifetime of luminescencelifetime broadeninglift-off processligamentum arteriosumligand efficiencyligand exchange chromatographyligand fieldligand field splittingligand of inducible co-stimulatory proteinligand to ligand charge transfer transitionligand to metal charge transfer transitionligand-based drug designligand-gated ion channelligandsligasesligatelightlight chainlight emitting diodelight polarizationlight scatteringlight sourcelight-atom anomalylight-emitting diodelignansligninslimb budlimbic systemlimit of detection in analysislimit of quantificationlimit test in toxicologylimiting adsorption currentlimiting catalytic currentlimiting condition of operationlimiting currentlimiting differential diffusion coefficientlimiting diffusion currentlimiting kinetic currentlimiting meanlimiting migration currentlimiting sedimentation coefficientlimiting viscosity numberLimulus testline edge roughnessline formulaline profileline repetition groupsline representation of electrochemical cellsline scanline widthline width in Mössbauer spectroscopyline width roughnessline-broadening in atomic spectroscopyline-of-centres modelline-shape analysisline-space patterninglinear absorption coefficientlinear absorption coefficient in optical spectroscopylinear attenuation coefficientlinear attenuation coefficient in optical spectroscopylinear chainlinear chromatographylinear constitutional repeating unitlinear copolymerlinear dichroismlinear discriminant analysislinear dispersionlinear distribution isotherm in chromatographylinear electron acceleratorlinear energy transferlinear epitopelinear free-energy relationlinear Gibbs energy relationlinear interaction energylinear ion traplinear macromoleculelinear Napierian absorption coefficientlinear polarizerlinear polymerlinear pulse amplifierlinear rangelinear solvation energy relationshipslinear strainlinear thermodilatometrylinear-scan voltammetrylinearity of a measuring systemlinearity of calibrationlinearity of responsivity of a radiation detectorlineicLineweaver–Burk plotlink in polymer conjugate chemistrylinkage analysislinkage disequilibriumlinked recognitionlinked scan in mass spectrometrylipid Alipid dropletslipid extractionlipid filmlipid profilinglipid raftlipid remodellinglipid vesiclelipidomelipidomicslipidosislipidslipomalipooligosaccharidelipophiliclipophilic ligand efficiencylipophilicitylipophobiclipopolysaccharideslipopolysaccharidomicslipoproteinsliposomeLippman's equationliquid chromatographyliquid chromatography-mass spectrometryliquid crystalliquid crystal tunable filterliquid excimer laserliquid ion evaporationliquid ion exchangeliquid ion laserliquid junctionliquid junction interfaceliquid laserliquid membraneliquid scintillation detectorliquid secondary ionizationliquid sheathliquid volume in gas chromatographyliquid-coated stationary phase in liquid chromatographyliquid-crystal dendrimerliquid-crystal polymerliquid-crystal stateliquid-crystal transitionsliquid-crystalline phaseliquid-crystalline polymerliquid-gel chromatographyliquid-ion exchangeliquid-liquid distributionliquid-liquid extractionliquid-metal ion gunliquid-phase film thicknessliquid-phase loading in chromatographyliquid-phase microextractionliquiduslithographylithometeor in atmospheric chemistrylithosphere in atmospheric chemistrylitrelitterlive attenuated vaccinelive timeliver-kidney microsomal antibodiesliving anionic polymerizationliving cationic polymerizationliving condensative chain polymerizationliving coordination polymerizationliving copolymerizationliving ionic polymerizationliving polymerliving polymer chainliving polymerizationliving radical polymerizationliving ring-opening metathesis polymerizationload on a precision balanceloading capacity in solvent extractionloading of a propertyloadingsloadings plotlobe of cerebral cortexlocal conformation of a polymerlocal current densitylocal efficiency of atomization in flame spectrometrylocal flame temperature in flame emission and absorption spectrometrylocal fraction atomized in flame emission and absorption spectrometrylocal fraction desolvated in flame emission and absorption spectrometrylocal fraction volatilized in flame emission and absorption spectrometrylocal mode of vibrationlocal modellocal molar polarizabilitylocalized adsorptionlocalized molecular orbitalslocalized-itinerant transitionlocantlock masslock-in amplifierlock-in statelocus in geneticsLOD scorelog Dlog Plog-normal distributionlogarithmic decrementlogarithmic distribution coefficientlogarithmic normal distribution of a macromolecular assemblylogarithmic viscosity numberlogitLondon forcesLondon–Eyring–Polanyi methodLondon–Eyring–Polanyi–Sato methodlone pairlong chainlong spacing in polymer crystalslong-chain branchlong-chain branchlong-lived collision complexlong-range intramolecular interaction in polymerslong-term potentiationlongitudinal order in a polymerloose endloose ion pairlordosisLorentz broadening of a spectral lineLorentzian band shapeLorenz–Mie theoryloss curveloss modulusloss tangentlot in analytical chemistrylow energy electron diffractionlow pressure electrical dischargelow sample consumption system in inductively-coupled plasma spectrometrylow temperature UV–VIS absorption spectroscopylow-energy collision-induced dissociationlow-energy ion scattering spectrometrylow-pass filterlow-pressure diamondlow-pressure mercury lamplow-specificity sorbentslow-spinlow-spin statelower critical solution temperaturelower motor neuronlowest lethal concentration foundlowest observed adverse effect levellowest unoccupied molecular orbitallowest-observed-adverse-effect-levellowest-observed-effect-levellumbar puncturelumbar spinelumenluminanceluminescenceluminescence quenchingluminescence spectrometerluminescent materialluminous fluxluminous intensityluminous quantitieslumiphoreLUMOlupus anticoagulantluteal phaseluteinizing hormoneluxLy-6lyaseslyate ionlymphlymph nodelymphadenopathylymphangionlymphaticlymphatic systemlymphatic tissuelymphoblastlymphocytelymphocyte activationlymphocyte function-associated antigen-1lymphocyte hominglymphocyte homing receptorlymphocyte proliferation testlymphocyte subpopulationlymphocyte transformation testlymphocyte-activating factorlymphocytopenialymphocytosislymphocytotoxicitylymphoidlymphoid folliclelymphoid stem celllymphoid tissuelymphokinelymphokine-activated killer celllymphomalymphopoiesislymphoproliferationlymphosumlymphotoxinLyon hypothesislyonium ionlyophiliclyophilic solslyophobiclyophobiclyotropic mesophaselysimeterlysislysosomal storage diseaselysosomeΛ Mmer- in inorganic nomenclaturemesomeso-compoundE/2 mass spectrumm diadmachine in analysismachine learningmacro-chain-transfer agentmacroautophagymacrocephalymacrocyclemacrocyclic componentmacroelectrodemacroglossiamacrognathiamacroinitiatormacrolidesmacromeliamacrometeorology in atmospheric chemistrymacromolecularmacromolecular coilmacromolecular component of a rotaxanemacromolecular crystallographic information filemacromolecular isomorphismmacromolecular prodrugmacromolecular pseudorotaxanemacromolecular rotaxanemacromoleculemacromonomermacromonomer moleculemacromonomer unitmacromonomeric unitmacrophagemacrophage function testmacrophage mannose receptormacropinocytosismacropore in catalysismacroporous polymermacroradicalmacroscopic cross-sectionmacroscopic diffusion controlmacroscopic filmmacroscopic kineticsmacrosomiamacrostomiamagic acidmagic anglemagic angle spinningmagnetic circular dichroismmagnetic coherence lengthmagnetic deflection in mass spectrometrymagnetic equivalencemagnetic field scan in mass spectrometrymagnetic field strengthmagnetic fluxmagnetic flux density in Mössbauer spectroscopymagnetic flux densitymagnetic inductionmagnetic maskmagnetic momentmagnetic nanoparticle sorbentmagnetic resonance imagingmagnetic resonance spectroscopy in neurotoxicologymagnetic sectormagnetic susceptibilitymagnetic susceptibilitymagnetic transitionmagnetizabilitymagnetization transfermagnetogyric ratiomagnetolithographymagnetoresistancemagnetron motionMahalanobis distancemain chain of a polymermain effectmain-chain macromolecular rotaxanemain-chain polymer liquid crystalmain-chain rotaxane polymermain-chain scissionmain-chain self-assemblymajor basic proteinmajor histocompatibility complexmajor histocompatibility complex class I moleculemajor histocompatibility complex class II moleculemajor histocompatibility complex class III moleculemajor histocompatibility complex moleculemajor histocompatibility complex restrictionmake-up liquidmale fertilitymale fertility indexmalformationmalignantmammalian target of rapamycinmammary glandmancude-ring systemsmancunide-ring systemsmanganismmanipulatormannan-binding lectin-associated serine peptidase-1 and -2mannose-binding proteinmantle zonemanual in analysismapping in biotechnologyMarangoni effectMarcus equation for electron transferMarcus inverted region for electron transferMarcus–Coltrin pathMarcus–Hush relationshipMarfan syndromemarginal zonemarginationMark–Houwink equationmarkerMarkownikoff ruleMarkush structuremartensitic transitionmasculinizationmaskmask error enhancement factormaskless lithographymassmass accuracymass action, law ofmass analysis in mass spectrometrymass balance in atmospheric chemistrymass calibrationmass concentrationmass defect in mass spectrometrymass densitymass density gradientmass discriminationmass distribution ratio in micellar electrokinetic chromatographymass distribution ratio in micro-emulsion electrokinetic chromatographymass distribution ratiomass distribution ratio in chromatographymass excessmass fingerprinting in proteomicsmass flow ratemass fractionmass gatemass limitmass numbermass of the stationary phase in chromatographymass peak in mass spectrometrymass range in mass spectrometrymass resolving power in mass spectrometrymass selective axial ejectionmass selective instabilitymass spectral librarymass spectrographmass spectrometermass spectrometer operating on the linear accelerator principlemass spectrometer focusing systemmass spectrometric detector in gas chromatographymass spectrometrymass spectrometry/mass spectrometrymass spectroscopemass spectroscopymass spectrummass transfer in biotechnologymass transfer coefficient in electrochemistrymass-analyzed ion kinetic energy spectrometrymass-average molar massmass-average velocity in electrolytesmass-distribution functionmass-flow sensitive detector in chromatographymass-law effectmass-to-charge ratio in mass spectrometrymass-transfer-controlled electrolyte rate constantmassicmassive transitionmast cellmast cell activation disordermast cell stabilizermaster curvemastocytomamastocytosismatched cells in spectrochemical analysismaterial in thermal analysismaterial characterization studymaterial homogeneitymaterial recoverymaterial safety data sheetmaterial stabilitymaterials chemistrymaternal serum alpha-fetoproteinMathieu stability diagrammatrix in analysismatrix effect in surface analysismatrix factormatrix isolationmatrix reference materialmatrix-assisted laser desorption electrospray ionization mass spectrometrymatrix-assisted laser desorption/ionization mass spectrometrymatrix-compatible solid-phase microextractionmatrix-matched calibrationmatrix-solid phase dispersionMattauch–Herzog geometrymaturation in immunologymature B cellmaxillarymaximum allowable concentration in atmospheric chemistrymaximum degree of biodegradationmaximum emission concentration in atmospheric chemistrymaximum hardness principlemaximum latent periodmaximum likelihood principal component analysismaximum loadingmaximum permissible daily dosemaximum permissible levelmaximum storage lifemaximum tolerable concentrationmaximum tolerable dosemaximum tolerable exposure levelmaximum tolerated doseMayr–Patz equationmazeMcLafferty rearrangement in mass spectrometryme-too drugmeanmean activity of an electrolyte in solution in mass spectrometrymean catalytic activity ratemean centeringmean current densitymean escape depthmean exchange current densitymean free pathmean interstitial velocity of the carrier gas in chromatographymean life of a radionuclidemean lifetimemean lifetimemean linear range in nuclear chemistrymean mass range in nuclear chemistrymean mass ratemean residence time of adsorbed moleculesmean squared error of calibrationmean squared error of predictionmean substance ratemean volume ratemean-field theorymean-square end-to-end distancemean-square end-to-end distance of a freely rotating chainmean-square radius of gyrationmean-square unperturbed end-to-end distancemean-square unperturbed radius of gyrationmeasurable quantitymeasurandmeasured excitation spectrummeasured spectrummeasured value in analysismeasurementmeasurement capabilitymeasurement efficiency in inductively-coupled plasma spectrometrymeasurement precision control materialmeasurement proceduremeasurement procedure biasmeasurement procedure with standard additionmeasurement reproducibilitymeasurement resolution in atmospheric trace component analysismeasurement resultmeasurement solution in analysismeasurement threshold of an analysermeasurement trueness control materialmechanical anharmonicitymechanical entrapmentmechanical harmonicitymechanical hygrometermechanical inhibitor mechanical meltingmechanically interlocked molecular architecturemechanism of a reactionmechanism-based inhibitionmechanization in analysismechanized extractionMechano-chemical reactionMeckel diverticulummeconiummedianmedian effective concentrationmedian effective dosemedian lethal concentrationmedian lethal dosemedian lethal timemedian narcotic concentrationmedian narcotic dosemedian teratogenic concentrationmediastinummediating agentmedical devicemediummedium effectmedium-energy ion scattering spectrometrymedium-pressure mercury lampmedullamedulloblastomamegamegacolonmegadactylymegakaryocytemeiosisMeisenheimer adductMeker burnermeltmelt flow indexmeltingmelting point correctedmelting temperaturemembranemembranemembrane in an ion-selective electrodemembrane attack complexmembrane bilayermembrane emfmembrane extractionmembrane extraction with sorbent interfacemembrane introduction mass spectrometrymembrane polarizationmembrane potentialmembrane raftsmembrane separatormembrane sites in an ion-selective electrodemembrane-protected solid-phase microextractionmembranes with mobile charged sitesmemory in immunologymemory cellmemory effect in atmospheric chemistrymemory lymphocyte immunostimulation assaymemristormenarcheMendelian geneMendelian inheritanceMendelian traitmeningesmeningiomameningitismeningocelemeningoencephalitismeningoencephalocelemeningomyelocelemeninxmenopausemenstrual cyclemenstruationmental retardationmentationmer mercaptalsmercaptansmercaptidesmercaptolesmercury electrodemercury flow system in spectrochemical analysismercury–xenon lampmeromeroblastic cleavagemeromeliamerry-go-round reactormesectodermmesencephalonmesenchymal-to-epithelial transitionmesenchymemesenteric lymph nodemesenterymesh size in polymer sciencemeso structures in polymersmesodermmesofluidic extraction devicemesogastriummesogenmesogenic groupmesogenic monomermesogenic pitchmesoionic compoundsmesolytic cleavagemesomeric effectmesomerismmesomorphic phasemesomorphic statemesomorphic transitionmesonephric ductmesonephrosmesopause in atmospheric chemistrymesophasemesophase pitchmesophase pitch-based carbon fibresmesophilesmesopore in catalysismesoscalemesospheremesotheliummessenger RNAmetabolic activationmetabolic half-lifemetabolic pathwaymetabolismmetabolism quenchingmetabolitemetabolite fingerprinting and footprintingmetabolite profilingmetabologenmetabolomemetabolomicsmetabonomicsmetabotropic receptormetal distributionmetal to ligand charge transfer transitionmetal to metal charge transfer transitionmetal-organic frameworkmetal–carbene complexesmetal–carbyne complexesmetal–insulator transitionmetal–nonmetal transitionmetallacycloalkanesmetallo-supramolecular polymermetallocene polymerizationmetallocenesmetalloenzymemetallomesogenmetallothioneinmetallurgical cokemetamagnetic transitionmetanephrosmetaphasemetaplasiametastabilitymetastablemetastable background in secondary-ion mass spectrometrymetastable filmmetastable ion in mass spectrometrymetastable state in nuclear chemistrymetastable state in spectrochemistrymetatectoid reactionmetathesismetathesis polymerizationmethanogensmethod of isotopic perturbationmethod performance studymethylationmethylenemethylidynemethylotrophic microorganismsmetremetrological compatibility of measurement resultsmetrological equivalence of measurement resultsmicellar catalysismicellar chromatographymicellar electrokinetic chromatographymicellar massmicellar rodmicellar solubilizationmicellar weightmicellemicelle chargeMichaelis constantMichaelis–Menten equationMichaelis–Menten kineticsMichaelis–Menten mechanismmicromicro- technique in thermal analysismicro-encapsulationmicro-networkmicroarraymicroautophagymicrobial leachingmicrocanonical rate constantmicrocanonical variational transition-state theory microcapsulemicrocarrier in biotechnologymicrocephalymicrochannel platemicrocheiriamicroclimatologymicrocontact printingmicrodialysismicrodomain morphologymicroelectrodemicroelectrophoresismicroelectrospraymicroemulsion electrokinetic chromatographymicroextraction (methology)microextraction (packed sorbent)microextraction (technique)microfabricationmicrofiltrationmicroflow injection analysis extraction devicemicrofluidic extraction devicemicrofluidicsmicrofold cellmicrogelmicroglial cellmicrogliosismicroglobulinmicroglossiamicrognathiamicroheterogeneity in biochemistrymicrolithographymicromass culturemicrometeorologymicromoldingmicronucleus testmicroparticlemicrophallusmicrophotometermicrophthalmiamicropore in catalysismicropore filling in catalysismicropore volume in catalysismicroporous carbonmicroRNAmicrosatellite in geneticsmicroscopic chemical eventmicroscopic cross-sectionmicroscopic diffusion controlmicroscopic electrophoresismicroscopic filmmicroscopic kineticsmicroscopic polyangiitismicroscopic reversibility at equilibriummicrosomemicrospheremicrostomiamicrostructuremicrosyneresismicrowave acid digestionmicrowave cavitymicrowave-assisted extractionmicrowave-induced plasmamiddle atmospheremidgutMie scatteringmigrainemigrainous infarctionmigrationmigration currentmigration time in capillary electrophoresismigration time of micelles in micellar electrokinetic chromatographymigratory aptitudemigratory insertionmillimilligram equivalent of readability of a precision balancemillimetre of mercurymillingMinamata diseasemineralizationminimum consumption time in flame emission and absorption spectrometryminimum density of states criterionminimum lethal concentrationminimum lethal doseminimum sample sizeminimum structural change, principle ofminimum-energy reaction pathminisatelliteMinnesota multiphasic personality inventoryminor histocompatibility antigenminusminute of arcmiosismiscarriagemiscibilitymiscibility gapmisclassification ratemisfolded proteinmist in atmospheric chemistrymitochondriamitochondrial membrane potentialmitogenmitogen-activated protein kinasemitophagymitosismixed ceramicmixed connective tissue diseasemixed controlmixed crystalmixed energy releasemixed indicatormixed labelledmixed lymphocyte responsemixed neuropathymixed potentialmixed-mode sorbentmixing in analytical chemistrymixing controlmixing height in atmospheric chemistrymixing ratio in analytical chemistrymixtureMLCTmobile adsorptionmobile phase in chromatographymobile-phase speedmobile-phase velocity in chromatographymobile-phase volume in counter-current chromatographymobility in generalmobilitymobility in aerosol physicsMöbius aromaticitymodemode of vibrationmode-locked lasermodel in experimental designmodel networkmoderation in nuclear chemistrymoderatormodified active solid in chromatographymodified Arrhenius equationmodified Auger parametermodified samplemodifier in solvent extractionmodulated heat flowmodulated technique in thermal analysismodulated temperature technique in thermal analysismodulated-temperature DSCmodulation frequency in comprehensive chromatographymodulation number in comprehensive chromatographymodulation period in comprehensive chromatographymodulator in comprehensive chromatographymodulus of elasticityMohr amplification process in analysismoietyMOL file formatmolalmolalitymolarmolar absorption coefficientmolar absorptivitymolar activity in radiochemistrymolar average velocity molar conductivitymolar decadic absorption coefficientmolar ellipticitymolar massmolar Napierian absorption coefficientmolar refractionmolar-mass averagemolar-mass dispersitymolar-mass exclusion limit in polymersmolaritymolemole fractionmolecular anionmolecular beam mass spectrometrymolecular beamsmolecular cationmolecular conformation of a polymermolecular connectivity indexmolecular descriptormolecular designmolecular diversitymolecular dynamics in drug designmolecular dynamicsmolecular effusion separatormolecular encapsulationmolecular entitymolecular epidemiologymolecular fingerprintsmolecular formulamolecular fragmentmolecular glassmolecular graphmolecular graph theorymolecular graphicsmolecular imagemolecular imprintingmolecular ion in mass spectrometrymolecular kineticsmolecular lasermolecular logic gatemolecular mechanics calculationmolecular metalmolecular mimicry in immunologymolecular modelingmolecular nucleation in polymersmolecular orbitalmolecular orbital theorymolecular orientationmolecular rearrangementmolecular recognitionmolecular Rydberg statemolecular shapemolecular shape of lipidsmolecular shuttlemolecular sieve effectmolecular similaritymolecular spectramolecular spectroscopymolecular switchmolecular targetmolecular templatemolecular weightmolecular wiresmolecular-weight exclusion limit in polymersmolecularitymolecularly imprinted polymer sorbentmoleculemolfilemolozonidesmoment of a forcemoment of inertiamomentummomentum separatormomentum spectrummonitoringmono-energetic radiationmonochromatormonoclonalmonoclonal antibodiesmonocytemonodisperse aerosol generating interface for chromatographymonodisperse mediummonodisperse polymermonoisotopic elementmonoisotopic massmonoisotopic mass spectrummonokinemonolayermonolayer capacitymonolithmonolithic columnmonomermonomer moleculemonomer unitmonomeric unitmononuclear phagocyte systemmonophthalmosmonosaccharidesmonosomymonospecificitymonotectic reactionmonotectoid reactionmonotectoid temperaturemonoterpenesmonoterpenoidsmonothioacetalsmonotropic transitionMonro–Kellie hypothesisMonte Carlo methodMonte Carlo studyMoore’s lawmordantMore O'Ferrall–Jencks diagramMorin transitionmorphinemorphogenmorphogenesismorphologicalmorphologymorphology coarseningmorphotropic transitionMorse potentialmorulamosaicismMössbauer effectMössbauer spectrometryMössbauer thickness in Mössbauer spectroscopymost probable distribution in macromolecular assembliesmotor activitymotor coordinationmotor cortexmotor endplatemotor neuronmotor neuron diseasemotor systemMott transitionMott–Hubbard transitionmouse ear-swelling testmouse IgE testmoving belt interfaceMPP-based carbon fibresMS/MS data based identification in proteomicsMS/MS spectrumMSnmucocutaneousmucopolysaccharidesmucosamucosa-associated lymphoid tissuemucosal addressin cell adhesion molecule-1mucosal tolerancemucous membranemucusmulching filmMulliken population analysismulti-centre bondmulti-centre reactionmulti-channel pulse height analysermulti-component resistmulti-dimensional chromatographymulti-strand chain in polymersmulti-strand macromoleculemulti-way datamulticoat morphologymulticollector mass spectrometermulticommutation flow extractionmulticonfiguration SCF methodmulticotyledonary placentationmultidentmultidimensional protein identification technologymultidimensional scalingmultidrug resistancemultienzymemultienzyme complexmultienzyme polypeptidemultifactorial inheritancemultigenerational studymultilayermultilayer adsorptionmultilayer aggregate in polymer crystalsmultilayer coil in counter-current chromatographymultilinear least squares regressionmultiobjective optimizationmultiparameter optimisationmultiphase copolymermultiphoton absorptionmultiphoton ionization in mass spectrometrymultiphoton processmultiple attenuated total reflectionmultiple chemical sensitivitymultiple headspace extractionmultiple inclusion morphologymultiple ionization satellitemultiple myelomamultiple peak scanning in mass spectrometrymultiple reaction monitoringmultiple scatteringmultiple sclerosismultiple-pass cell in spectrochemical analysismultiple-quantum magic angle spinning NMRmultiple-stage mass spectrometrymultipletmultiplex advantagemultiplex immunoassaymultiplex spectrometermultiplexingmultiplicative matrix effectmultiplicative namemultiplicative scatter correctionmultiplicitymultiply labelledmultipole line in X-ray spectroscopymultipotent progenitor cellmultireference configuration interactionmultistage countercurrent distributionmultistage samplingmultitarget drug discoverymultitarget-directed ligandmultivariate calibrationmultivariate curve resolutionmultivariate datamultivariate statisticsmultiwell extraction platemultiwell sample platemunchnonesmuon induced X-ray emission analysismuoniumMurcko assemblymurine local lymph node assaymuscarinicmuscarinic receptormuscle spindlemustard oilsmustardsmutagenmutagenesismutarotationmutationmutation rate in biotechnologymutual inductancemycoplasmamydriasismyelencephalonmyelinmyelin basic proteinMyelin cylindersmyelin sheathmyelin-associated glycoproteinmyelinationmyeloblastmyelodysplastic syndromemyelogrammyeloidmyeloid stem cellmyeloid tissuemyelomamyeloma proteinmyelomeningocelemyelopathymyeloperoxidasemyelopoiesismyelosuppressionmyelotoxicmyoblastmyocarditismyocardiummyometriummyopathymyositismyotomemyotoniaµ- in inorganic nomenclature N-methyl-D-aspartateN-methyl-D-aspartate-type glutamate receptornth-order effectn-star macromoleculen-strand chainn-strand moleculenth-generation product ionnth-generation product ion spectrumnth order phase transitionnido-ångströmn → π staten → π transitionn → σ transitionN-nucleotidesN-regionN-terminal analysisN-terminal residue N-terminusn-σ delocalizationNACHT domain-, leucine-rich repeat (LRR)-, and pyrin (N-terminal homology) domain (PYD)-containing protein 3naïve lymphocytenanonanocapsulenanocompositenanodevicenanodomainnanodomain morphologynanoelectrodenanoelectromechanical systemsnanoelectronicsnanoelectrospraynanofiltrationnanogelnanoimprint lithographynanolithographynanomaterialnanoparticlenanopatternnanophotonicsnanoscopic filmnanoscopic polymer filmnanospherenanosphere lithographynanostructurenanotechnologynaphthenesnaphthenic acidsNapierian absorbancenarcissistic reactionnarcosisnarcoticnasalnatalitynational examination standardnatural atomic orbitalnatural autoantibodiesnatural bond orbitalnatural broadening of a spectral linenatural graphitenatural hybrid orbitalnatural immunitynatural isotopic abundancenatural killer cellnatural killer cell activity assaynatural killer T cellnatural lifetime of an excited statenatural lifetimenatural linewidthnatural orbitalnatural population analysisnatural radiationnatural radioactivitynatural resistance-associated macrophage proteinnear edge X-ray absorption fine structurenear-field scanning optical microscopynear-infrared radiationneat soapnebulizationnebulizernecropsynecrosisneedle cokeneedle trapnegative adsorptionnegative feedbacknegative hyper-conjugationnegative ion in mass spectrometrynegative ion chemical ionizationnegative selectionnegative-ion yieldnegative-tone developmentnegatonneglected diseaseneighbouring group participationnematic phasenematic stateneoantigenneocortexneodymium laserneoflavansneoflavonesneoflavonoidsneoflavonoidsneolignansneonatal toleranceneonateneoplasmnepernephelometryNernst equationNernst's diffusion layernervenerve conduction studynerve conduction velocitynerve fibernerve gasnerve growth factornerve rootnervous systemnet in surface chemistrynet absorption cross-sectionnet currentnet dephasing timenet electric charge of a particlenet faradaic currentnet shapingnetwork in polymer chemistrynetworknetwork defectnetwork polymernetwork-chain molar massneuralneural activityneural archneural crestneural networkneural plateneural stem cellneural tubeneural tube defectneuralgianeurectomyneuriteneurite outgrowthneuritisneurobehavior core test batteryneurobehavioralneuroblastneuroblastomaneurodegenerative diseaseneurodevelopmental toxicityneuroectodermneuroendocrineneuroendocrine systemneurofibrilneurofibrillary tangleneurofibromaneurofilamentneurogenesisneurogenicneurogenic shockneurohypophysisneuroimagingneuroinflammationneurolepticneurolysisneuromaneuronneuron-specific enolaseneuronopathyneuropathic painneuropathological examinationneuropathyneuropathy target esteraseneuroporeneuroprogenitor cellneuroprotectiveneuropsychological testingneurotoxicneurotoxic shellfish poisoningneurotoxicantneurotoxicologyneurotoxinneurotransmissionneurotransmitterneurotrophinneurotropicneurotubuleneurovascularneurulationneutral antagonist in pharmacologyneutral lossneutral loss scanneutral-density filterneutralization in immunologyneutralization-reionization mass spectrometryneutralized gelneutrinoneutronneutron activation analysisneutron densityneutron depth profilingneutron diffraction analysisneutron energyneutron multiplicationneutron numberneutron rest massneutron scattering analysisneutron temperatureneutropenianeutrophilneutrophil activationneutrophilic organismsnew chemical entitynew molecular entityNewman projectionnewtonNewton black filmNewton diagramNewtonian fluidNG2 cellNHOMONHOMOnicotinicnicotinic receptorNier–Johnson geometryNIH shiftNikolsky–Eisenman equationnimbostratus cloud in atmospheric chemistrynine-hole box testnippleNissl bodynitraminesnitrenesnitrenium ionsnitric oxidenitrificationnitrile imidesnitrile iminesnitrile oxidesnitrile sulfidesnitrile ylidesnitrilesnitriliminesnitrilium betainesnitrilium ionsnitriminesnitro compoundsnitrogen fixationnitrogen lasernitrogen mustardsnitrogen rulenitrogen ylidesnitrolic acidsnitronesnitronic acidsnitrosamidesnitrosaminesnitrosiminesnitroso compoundsnitrosolic acidsnitroxidesnitroxyl radicalsNMR exchange spectroscopyno carrier addedno observed adverse effect levelno-bond resonanceno-load indication for a precision balanceno-observed-effect-levelnociceptionNOD-like receptornodalnodal planenode in artificial neural networknode of Ranviernogginnoisenoisenominal indicationnominal linear flow in chromatographynominal massnominal propertynominal property valuenominal property value coverage probabilitynominal property value coverage setnominal property value setnominal reference materialnominal spectral resolutionnominally labelled tracernon-adiabatic electron transfernon-aqueous capillary electrophoresisnon-calorimetric thermophysical measurementsnon-certified property valuenon-coherent source non-covalent interactions in biomoleculesnon-crossing rulenon-crystalline electrodesnon-crystalline membranenon-destructive activation analysisnon-diagram line non-dispersive infrared spectroscopiesnon-dissociative chemisorptionnon-drainingnon-equilibrium reactionnon-graphitic carbonnon-graphitizable carbonnon-Hodgkin lymphomanon-ideal chromatographynon-ideal, linear chromatographynon-ideal, non-linear chromatographynon-isotopic labellingnon-Kekulé moleculesnon-Kekulé structurenon-linear chromatographynon-linear distribution isotherm in chromatographynon-linear iterative partial least squaresnon-linear optical effectnon-linear optical polymernon-linear optical techniquesnon-linearity error in spectrochemical analysisnon-localized adsorptionnon-metal-mediated atom-transfer radical polymerizationnon-obese diabetic mousenon-polarized interphasesnon-porous membranenon-radiative decaynon-radiative quenchingnon-reversing heat flownon-selfnon-specific adsorptionnon-specific binding in DNA-protein interactionsnon-specific emission or attenuationnon-uniform corrosionnon-uniform dispersionnon-uniform polymernon-uniform samplingnon-vertical energy transfernonadiabaticnonadiabatic couplingnonadiabatic photoreactionnonbonded interactionsnonbonding molecular orbitalnonclassical carbocationnonclassical structurenoncompetitive antagonistnondepleting antibodynondisjunctionnonlinear mappingnonselective detectornonselective quantum counternonselectively labellednor-noradrenalinnoradrenalinenormalnormal coordinatenormal coordinate analysisnormal dispersion of the refractive indexnormal distributionnormal incidencenormal kinetic isotope effectnormal mode of vibrationnormal plasma analytical zone in inductively-coupled plasma optical emission spectroscopynormal pulse voltammetrynormal region for electron transfernormal stressnormal X-ray levelnormal-phase chromatographynormalizationnormalized image log slopenormalized tunnelling intensityNorrish Type I photoreactionNorrish Type II photoreactionNorrish–Yang reactionNorthern blotnose-poke testnotochordnozzle-skimmer dissociationnozzle-skimmer voltagenuclear atomnuclear capturenuclear chemistrynuclear decaynuclear disintegrationnuclear electric quadrupole momentnuclear factor kappa-light-chain-enhancer of activated B cellsnuclear factor of activated T cellsnuclear fissionnuclear fuelnuclear fusionnuclear fusion reactionnuclear graphitenuclear hormone receptornuclear isomersnuclear levelnuclear magnetic resonance relaxometrynuclear magnetic resonance spectroscopynuclear magnetic resonance spectrumnuclear magnetonnuclear Overhauser effectnuclear Overhauser effect difference spectroscopynuclear Overhauser effect spectroscopynuclear particlenuclear quadrupole moment in Mössbauer spectroscopynuclear quadrupole resonance spectroscopynuclear reaction cross sectionnuclear reactornuclear receptornuclear spindlenuclear stopping cross sectionnuclear transformationnuclear transitionnuclear type I receptornuclear type II receptornuclearitynucleating agentnucleation in colloid chemistrynucleation and growthnucleation of phase separation in polymer chemistrynucleic acid aptamersnucleic acid–protein interactionnucleic acidsnucleobasenucleobase lesionnucleofugenucleonnucleon numbernucleophilenucleophilic aromatic photosubstitutionnucleophilic catalysisnucleophilic chain polymerizationnucleophilic substitutionnucleophilicitynucleoproteinsnucleosidesnucleotide basesnucleotide sequencenucleotide-binding oligomerization domain-containing protein 1, 2nucleotide-binding oligomerization domain-containing protein-like receptornucleotidesnucleusnuclidenuclide precursornuclidic massnude mousenuisance threshold in atmospheric chemistrynull cellnulligravidanulliparousnumber concentrationnumber contentnumber densitynumber flow ratenumber fractionnumber of entitiesnumber of generations of a dendronnumber of generations of a regular dendrimer moleculenumber of molesnumber of pseudo-generations of a hyperbranched polymernumber of theoretical platesnumber-average molar massnumber-distribution functionnumbnessnumerical value of a quantityNyquist criterionNyquist frequencynystagmus octahedro- in inorganic nomenclatureortho- and peri-fused in polycyclic compoundsortho-fused in polycyclic compoundsO-polysaccharideobject traceabilityobservation frequencyobservation height in flame emission and absorption spectrometryobservation pathlength in flame emission and absorption spectrometryobservation space in flame emission and absorption spectrometryobservation volume in flame emission and absorption spectrometryobserved interface widthobserved rate coefficientobstructive hydrocephalusobviousnessoccipital lobeocciputocclusionoccult bloodoccupied bandoctanol-water partition ratioodd-electron ionodontoblastodour threshold in atmospheric chemistryoff peptide product ionsoff-gel isoelectric focusingoff-specular X-ray reflectometryoff-target effectoffspringohmohmic contactOKT3OLEDolefinsolfactionolfactory bulboligooligoclonaloligodactylyoligodendrocyteoligodendrogliomaoligodontiaoligomenomenorrheaoligomeroligomer moleculeoligomericoligomerizationoligonucleotidesoligopeptidesoligosaccharidesoligospermiaomentumomphaloceleomphalositeon-line extractiononcofetal antigenone-bond-flipone-class classificationone-colour indicatorone-generation reprotox studyone-photon photochromismonion morphologyonium compoundsoocyteoogenesisoogoniumopen atomizeropen circuit potentialopen dynamic system open field testopen filmopen hearth furnace in atmospheric chemistryopen reading frameopen static system open-shell systemsopen-tubular column in chromatographyopen-vessel acid digestionoperant conditioningoperational pH celloperational pH standardoperator geneoperonopiate receptoropioidopiumopposing reactionsopsonizationopticoptical absorption depthoptical activityoptical antipodesoptical constantsoptical densityoptical emission spectroscopyoptical filteroptical isomersoptical multi-channel analyseroptical null double beam spectrometeroptical parametric amplificationoptical parametric oscillatoroptical path differenceoptical purityoptical resolutionoptical rotationoptical rotatory dispersionoptical rotatory poweroptical spectroscopyoptical throughput of a spectrometeroptical yieldoptical-beam error in spectrochemical analysisoptically active polymeroptically detected magnetic resonanceoptically labileoptically nonlinear polymeroptimisation of extractionoptimizationoptoacoustic spectroscopyoptoelectronicsoral toleranceoral vaccinationorbit in anatomyorbitalorbital energyorbital steeringorbital symmetryorder of reactionorder parameterorder–disorder transitionordered co-continuous double gyroid morphologyordinary least squares regressionordinary waveorgan of Cortiorganellesorganic dye laserorganic psychosisorganic–inorganic polymerorganically modified silicaorganically-modified ceramicorganizationorgano-organobismuthane-mediated radical polymerizationorganocatalysisorganogenesisorganoheteryl groupsorganolepticorganometallic compoundorganometallic compoundsorganometallic-mediated radical polymerizationorganophilicorganophilic interactionorganophosphateorganophosphate-induced delayed neuropathyorganostibane-mediated radical polymerizationorganotellurium-mediated radical polymerizationorganyl groupsorientation parameterorigin of replicationorofacial cleftorphan diseaseorphan drugorphan receptorortho acidsortho amidesortho estersorthogonal electrosprayorthogonal extractionorthogonal projections to latent structuresorthokinetic aggregation in colloidsosazonesoscillating reactionoscillator strengthosmolalityosmolarityosmometerosmometryosmophobiaosmotic coefficientosmotic concentrationosmotic pressureosmotic virial coefficientosonesosotriazolesosteoclastosteogenesisostiumOstwald ripeningoticotocephalyOuchterlony techniqueout-isomerout-of-control criteriaout-of-plane bending coordinate in molecular geometryouter electric potentialouter Helmholtz planeouter-sphere electron transferouter-sphere electron transfer ATRPoutgassing of a catalystoutput rateovarian cycleovarian follicleovaryoverall activation energyoverall efficiency of atomizationoverlap integraloverpotentialovertone transitionovertrainingovotestisovulationovumoxa-di-π-methane rearrangementoxenium ionsoxidant in atmospheric chemistryoxidationoxidation numberoxidation stateoxidation–reduction titrationoxidative additionoxidative couplingoxidative stressoxide networkoxidized speciesoxidoreductasesoxime O-ethersoximesoxo acidsoxo carboxylic acidsoxo compoundsoxoacidsoxocarbonsoxonium ionsoxonium ylidesoxyacidoxygen-flask combustion in spectrochemical analysisoxylium ionsoxytocinozone holeozonidesσ-orbital P, Mpentaprismo-pro-E, pro-Zpro-R, pro-Spros in histidine nomenclature/n rad magnetic sectorP-glycoproteinP-nucleotidesp-polarizationP-selectinpacked column in chromatographypacking in column chromatographypaddlanespainpair attenuation coefficient in nuclear chemistrypair correlation length in thin filmspair production in nuclear chemistrypaired helical filamentPake doubletpalatepalatine raphePallmann effectpalpitationPAN-based carbon fibrespancytopeniaPapanicolaou smearpapilledemaparacortical areaparacrineparaffinparaffin impregnated graphite electrodeparafollicular cellparallel artificial liquid membrane extractionparallel effectparallel factors analysisparallel reactionsparallel synthesisparallel-chain crystal in polymersparalysisparalytic shellfish poisoningparamagneticparamesonephric ductparametric amplificationparametric processesparametriumparamoneparaneoplastic autoimmune syndromeparaoxonaseparaplegiaparasitic twinparasympathetic nervous systemparasympathomimeticparatopeparaxial mesodermparenchymal cellparent hydrideparent ion in mass spectrometryparesisparesthesiaPareto frontPareto optimizationparietalparietal lobePariser–Parr–Pople methodParkinson diseaseparthenogenesispartial anodic currentpartial atomic chargespartial charge exchange reactionpartial charge transfer reactionpartial decay constant in nuclear chemistrypartial digestion in spectrochemical analysispartial free-drainingpartial ion yieldpartial isotherm in surface chemistrypartial isotopic scramblingpartial kinetic current in electrochemistrypartial least squarespartial least squares discriminant analysispartial least squares regressionpartial mass densitypartial microscopic diffusion controlpartial molar Gibbs energypartial molar quantitypartial order of reactionpartial pressurepartial rate factorpartial specific volumepartial sputtering yieldpartially drainingparticipant in an interlaboratory comparisonparticle beam interfaceparticle concentration in atmospheric chemistryparticle density in nuclear chemistryparticle flux densityparticle immunoassayparticle induced X-ray emission analysisparticle scattering factorparticle scattering functionparticle size in atmospheric chemistryparticle size distribution in atmospheric chemistryparticle sorbent in microextractionparticle-embedded glass fibre discparticle-induced gamma-ray-emission analysisparticle-loaded membrane discparticular propertyparticulate carbonparticulate gelparticulate matter in atmospheric chemistryparticulate solpartitionpartition chromatographypartition coefficientpartition constantpartition functionpartition isotherm in chromatographypartition ratiopartly open filmparturitionpascalpass energy in electron spectroscopypassivation in electrochemical corrosionpassivation potential in electrochemical corrosionpassive avoidance testpassive cutaneous anaphylaxis testpassive hemagglutinationpassive immunizationpassive metalpassive samplerpassive samplingpassive state in electrochemical corrosionpatch testpatent ductus arteriosuspatentabilityPaterno–Büchi reactionpath fingerprintspath-lengthpathogen-associated molecular patternpATR spectrumpattern collapsepattern recognitionpattern recognition receptorpattern transferpauci-immunepaucidisperse systemPaul ion trapPauli exclusion principlepaw-reaching testPC-12 cellPDMRpeak in chromatographypeak analysispeak area in chromatographypeak area methodpeak base in chromatographypeak concentration of a trace atmospheric componentpeak currentpeak elution volume in column chromatographypeak energypeak enthalpimetrypeak fittingpeak height in chromatographypeak matchingpeak maximum in chromatographypeak parkingpeak potentialpeak resolution in chromatographypeak strippingpeak width at basepeak width at half heightpeak widths in chromatographypearl-necklace modelpectinsPeierls distortionPeierls transitionpellicular packing in chromatographypelvic kidneypemphiguspenamspendant chainpendant grouppendent chainpenemspenetrantpenetration depthpenicillinsPenning excitation in inductively-coupled plasma spectrometryPenning gas mixturePenning ion trapPenning ionizationpentads in polymerspeptide de novo sequencingpeptide fragment ionpeptide fragmentation techniquepeptide mass fingerprintingpeptide nucleic acidpeptide precursor ionpeptide sequence tagpeptide tetramerpeptide vaccinepeptide-binding groovepeptidespeptidoglycanpeptidomicspeptization peptizationpeptoidper acidspercentpercent diastereoisomer excesspercent enantiomer excesspercentage errorpercentage exposed in metallic catalystspercentage relative errorpercentage standard deviationpercentage transmission of samplepercolationpercolation threshold in a composite materialperfect networkperfectly polarized interphaseperforinperformance characteristic of a measurement procedureperformance scoreperfusateperfusion stationary phase in liquid chromatographyperiarteriolar lymphoid sheathpericardiumpericyclic reactionperikinetic aggregation in colloidsperinatalperineumperiodperiodic copolymerperiodic copolymerizationperiodic voltageperiodontitisperipheral atom in organic reaction mechanismsperipheral bloodperipheral blood leukocyteperipheral blood mononuclear cellperipheral lymphoid organperipheral nervous systemperipheral neuropathyperipheral toleranceperiplanarperiselectivityperistalsisperitectic reactionperitectoid reactionperitectoid temperatureperitoneal cavityperitoneumpermanent crosslinkpermanent modifier in electrothermal atomic absorption spectroscopypermeabilitypermeability of vacuumpermeatepermeationpermeation chromatographypermeation tubepermittivitypermittivity of vacuumpermselectivitypernicious anemiaperoxidesperoxisomeperoxo compoundsperoxy acidsperpendicular effectPerrin equationpersistence length in polymerspersistentperspective formulaperstractionperturbation theoryperturbed angular correlation spectrometryperturbed dimensions in polymerspervaporationpesticidepesticide residuepetapetroleum cokepetroleum pitchpexophagyPeyer’s patchpHpH glass electrodepH gradient in electrophoresispH standardpH-rate profilepH-sensitive electrodepH-zone-refining counter-current chromatographypH0.5 or pH1/2 in solvent extractionphage displayphagocytephagocytic activity assayphagocytosisphagolysosomephagosomephantom chain behaviourphantom painpharmaceuticalpharmacodynamicspharmacogeneticspharmacogenomicspharmacokineticspharmacologicalpharmacologically activepharmacologypharmacophorepharyngeal groovepharyngeal pouchpharynxphasephase 0 clinical studiesphase correctionphase cyclingphase domainphase fluorimetryphase I clinical studiesphase I reaction of biotransformationphase II clinical studiesphase II reaction of biotransformationphase III clinical studiesphase interactionphase inversionphase IV clinical studiesphase ratio in chromatographyphase ratio in liquid-liquid distributionphase rulephase separationphase transitionphase-space theoryphase-transfer catalysisphenolatesphenolsphenomenological equationphenonium ionsphenotypephenotypic screeningphenoxidesphenylketonuriapheochromocytomapheromonePhiladelphia chromosomephiltrumphocomeliaphononphonophobiaphorbol 12-myristate 13-acetatephosphanesphosphanylidenesphosphatidic acidsphosphazenesphosphine oxidesphosphinesphosphinic acidsphosphinidenesphosphinous acidsphosphophosphoglyceridesphospholipase Cγphospholipidsphosphonic acidsphosphonitrilesphosphonium compoundsphosphonium ylidesphosphonophosphonous acidsphosphoramidesphosphoranesphosphoranyl radicalsphosphorescencephosphorescence lifetimephosphorimetryphosphoroscopephosphorus ylidesphosphorylationphosphylenesphoto- technique in thermal analysisphoto-Bergman cyclizationphoto-Claisen rearrangementphoto-elastic polymerphoto-Fries rearrangementphotoacid generatorphotoacoustic detectorphotoacoustic effectphotoacoustic saturationphotoacoustic spectroscopyphotoactive compoundphotoadsorptionphotoaffinity labellingphotoallergyphotoassisted catalysisphotobiologyphotocatalysisphotocatalystphotocatalytic cellphotochemical curingphotochemical detectorphotochemical equivalencephotochemical funnelphotochemical hole burningphotochemical nitrogen extrusionphotochemical reactionphotochemical reaction pathphotochemical smogphotochemical yieldphotochemistryphotochromic polymerphotochromismphotoconducting polymerphotoconductive detectorphotoconductivityphotocontact dermatitisphotocrosslinkingphotocurrent yieldphotocyclizationphotocycloadditionphotodecarbonylationphotodecarboxylationphotodeconjugationphotodegradationphotodesorptionphotodetachment of electronsphotodetrappingphotodiodephotodiode arrayphotodissociationphotodynamic effectphotoelectric attenuation coefficientphotoelectric detectorphotoelectric peakphotoelectric work functionphotoelectrical effectphotoelectrochemical cellphotoelectrochemical etchingphotoelectrochemistryphotoelectrolytic cellphotoelectron spectroscopyphotoelectron X-ray satellite peaksphotoelectron X-ray satellite subtractionphotoelectron yieldphotoemissionphotoemissive detectorphotoexcitationphotogalvanic cellphotographic plate recordingphotohydrationphotoimagingphotoinduced dischargephotoinduced electron transferphotoinduced polymerizationphotoinitiationphotoinjectionphotoionizationphotoionization cross sectionphotoionization detector in gas chromatographyphotoisomerizationphotolithographyphotoluminescencephotoluminescent polymerphotolysisphotometryphotomultiplier tubephotonphoton activationphoton countingphoton echophoton emittancephoton exitancephoton exposurephoton flowphoton fluencephoton fluence ratephoton fluxphoton irradiancephoton numberphoton quantitiesphoton radiancephotonic crystalphotonicsphotooxidationphotooxygenationphotopeakphotophobiaphotophoresisphotophosphorylationphotophysical processesphotopolymerizationphotoreactionphotorearrangementphotoreceptor cellphotoreductionphotorefractive polymerphotoresistphotoselectionphotosensitive polymerphotosensitivityphotosensitizationphotosensitizerphotostationary statephotosynthesisphotosystemphotothermal effectphotothermal spectroscopyphotothermographyphototoxicity testphototransistorphotovoltaic cellphotovoltaic effectphthaleinsphthalidesphysical adsorptionphysical networkphysical quantityphysisorptionphytoestrogenphytohemagglutininphytotoxicantpiapicopicratespiebaldismpiezoelectric polymerpiezoluminescencepile-up in radioanalytical chemistrypileupPIN diodePIN semiconductor detectorpinacolspineal glandpinnapinocytosispipelinepipelining programspitchpitch divisionpitch-based carbon fibrespitting corrosionpituitaryPitzer strainpivotal studyplacentaplacental barrierplacental circulationplacental insufficiencyplacental transferPlackett−Burman designplanar chiralityplanar chromatographyplanar filmplanar intramolecular charge transferplanar stereoisomerismplanarising layerPlanck constantplane angleplantibodyplaque-forming cellplasma in spectrochemistryplasma in biologyplasma arcplasma assisted desorption ionization mass spectrometryplasma cellplasma desorption ionization in mass spectrometryplasma enhanced chemical vapor depositionplasma etchplasma excitation temperatureplasma gases in inductively-coupled plasmaplasma induction zoneplasma ionization temperatureplasma lipoproteinsplasma local thermodynamic equilibriumplasma shielding in laser induced breakdown spectroscopyplasma termination in laser induced breakdown spectroscopyplasmablastplasmacytoid dendritic cellplasmacytomaplasmapheresisplasmidplasmonplasticplastic flowplastic transitionplate height in chromatographyplate number in chromatographyplateau border in surface chemistryplateletplatelet-activating factorplatelet-derived growth factorplatform atomization in electrothermal atomic absorption spectroscopyplatinum electrodePLEDpleiotropic genepleiotropismpleurapleural cavityplexusploidyPLS latent variablePLS loadingplug-flow in catalysisplumbylenesplumbylidenesplume in atmospheric chemistrypluripotentpluripotent stem cellplus, minuspneumatic detectorpneumatic nebulizerspneumatically assisted electrospray ionizationpneumotaxic centerpnictogen bondpoint detectorpoint grouppoint of zero chargepoisepoison in catalysisPoisson distributionPoisson–Boltzmann equationpokeweed mitogenpolar aprotic solventpolar effectpolar solventpolar surface areapolarity of a bondpolarity of a solventpolarizabilitypolarizable electrodepolarizationpolarization in electrochemistrypolarization densitypolarization error in spectrochemical analysispolarized interphasespolarized Raman bandpolarographypolaronpolingpollutionpollution rosepoly I:Cpoly ICLCpoly-Ig receptorpoly(ethene-1,2-diyl)poly[3,4-(ethylenedioxy)thiophene]polyacetylenespolyacidpolyadditionpolyanilinespolyarteritis nodosapolyarylenespolyarylenethynylenespolyarylenevinylenespolyatomic fragmentpolyazomethinespolybasepolybenzimidazolespolybetainepolychromatorpolyclonalpolyclonal activatorpolycondensationpolycrystalline graphitepolycyclic systempolycysticpolydactylypolydentpolydiacetylenespolydisperse mediumpolydisperse polymerpolyelectrolytepolyelectrolyte complexpolyelectrolyte gelpolyelectrolyte networkpolyfluorenespolyfunctional catalysispolygranular carbonpolygranular graphitepolyhedral symbolpolyhedranespolyimidespolyionspolyketidespolymerpolymer alloypolymer blendpolymer catalystpolymer compatibilizerpolymer complexationpolymer compositepolymer crystalpolymer crystallitepolymer cyclizationpolymer degradationpolymer drugpolymer functionalizationpolymer gelpolymer glasspolymer meltpolymer membranepolymer moleculepolymer networkpolymer phase-transfer catalystpolymer reactantpolymer reactionpolymer solventpolymer sorbentpolymer supportpolymer surfactantpolymer-derived ceramicpolymer-metal complexpolymer-poor phasepolymer-rich phasepolymer-supported catalystpolymer-supported reactionpolymer–polymer complexpolymer–solvent interactionpolymerasepolymerase chain reactionpolymericpolymeric inner saltpolymeric solpolymeric stationary phase in liquid chromatographypolymerisationpolymerizationpolymersomepolymolecularpolymolecularity correctionpolymorphic transitionpolymorphismpolymorphonuclear leukocytepolymyositispolyneuritispolyneuropathypolyoxadiazolespolypeptidespolyphenylenespolyphenylenethynylenespolyphenylenevinylenespolyploidypolyprenolspolypyrrolespolyquinanespolyrotaxanepolysaccharidespolysilanespolysulfanespolysulfidespolytetrazinespolythiadiazolespolythiazolespolythiophenespolytopal rearrangementpolytypic transitionpolyunsaturated fatty acidspolyvinylcarbazolespolyvinylenespolyzwitterionsPoole–Frenkel effectpooled relative standard deviationpooled standard deviationpopliteal lymph node assaypopulation inversionpore size distributionporencephalyPorod–Kratky chain porosityporous membraneporous-layer open-tabular column in chromatographyporphyrinogensporphyrinspose diagramposition-sensitive photomultiplier tubepositive feedbackpositive ion in mass spectrometrypositive selectionpositive-ion yieldpositive-tone developmentpositronpositron annihilation analysispositron emission tomographypositron lifetime spectrometrypositroniumpost-acceleration detectorpost-apply bakepost-column derivatization in chromatographypost-exposure bakepost-exposure delaypost-filter effect in luminescence spectroscopypost-implantationpost-source decayposterior fossaposterior probabilitypostmortempostnatalpostpartumpostprecipitationpostsynapticpotential at the point of zero chargepotential couplingpotential energypotential genotoxic impuritypotential of a cell reactionpotential of mean forcepotential temperaturepotential-determining ionspotential-energy profilepotential-energy surfacepotentiationpotentiometerpotentiometric detection method in electrochemical analysispotentiometric enzyme electrodepotentiometric gas sensorpotentiometric selectivity coefficientpotentiometric stripping analysispotentiometric titrationpotentiometrypotentiostatpowder patternpowerpower compensation differential scanning calorimetrypower levelpre-associationpre-B cellpre-eclampsiapre-embryopre-equilibrium in solvent extractionpre-equilibriumpre-exponential factorpre-filter effect in luminescence spectroscopypre-gel regimepre-gel statepre-implantationpre-ionization statepre-polymerpre-polymer moleculepre-reactive complexespre-T cellprecessionprecipitating antibodyprecipitationprecipitation in sol-gel processingprecipitation fractionation of polymersprecipitation from homogeneous solution in analysisprecipitation indicatorprecipitation titrationprecipitinprecisionprecision of a balanceprecision of a weighingprecision limitprecision of indication of a balancepreclinical candidatepreconcentration in trace analysispreconcentration coefficient of a desired microcomponentprecursor in radioanalytical chemistryprecursor complexprecursor ion in mass spectrometryprecursor ion spectrumpredatory mite testprediction error sum of squaresprediction setpredissociationpreferential sorption in polymerspreferential sputteringpreferred constitutional repeating unitpregnancypreheating inductively-coupled plasma zonepremature birthpremium cokepremix burner in flame spectroscopyprenatalprenatal development studyprenol lipidsprenolspreoptic areaprepolymerprepregprepubertalprepubescenceprepucepreputial separationpresaturationpresenile dementiapressurepressure broadeningpressure flowpressure gradient correction factor in gas chromatographypressure jumppressure-induced transitionpressure-sensitive detectorpressurized hot-water extractionpressurized-liquid extractionpresynapticpretreatment of a catalystprick testprimary aerosolprimary beamprimary biliary cirrhosisprimary crystallizationprimary current distributionprimary electrons in in situ microanalysisprimary examination standardprimary excitation primary immune responseprimary immunizationprimary ionprimary isotope effectprimary kinetic electrolyte effectprimary kinetic isotope effectprimary lymphoid follicleprimary lymphoid organprimary metaboliteprimary mixtureprimary myxedemaprimary oocyteprimary particle primary pH standardsprimary photochemical processprimary photoprocessprimary photoproductprimary pollutant in atmospheric chemistryprimary reference examination procedureprimary sampleprimary sclerosing cholangitisprimary sex organprimary species in a chain polymerizationprimary standardprimary structureprimary structure of a segment of a polypeptideprimeprimerprimitive axisprimitive chainprimitive changeprimitive grooveprimitive nodeprimitive pitprimitive streakprimordial follicleprimordial germ cellprimordiumprincipal component loadingprincipal component scoreprincipal component-discriminant analysisprincipal components regressionprincipal groupprincipal ion in mass spectrometryprincipal moments of inertiaprincipal-component analysisprincipal-component factorprinciple of least nuclear motionprinciple of microscopic reversibilityprinciple of minimum structural changeprior distributionprior equilibriumprior probabilitypriorityprivileged structureprobabilityprobability densityprobability-based matchingprobe in biotechnologyprobe ionprocedure validationprocedure verificationprocessprocess in a state of statistical controlprochiralityprochirality centreProcrustes analysisproctodeumprodromal stageprodromeprodrugprodrug conjugateproductproduct development controlproduct ionproduct ion analysisproduct ion spectrumproduct state distributionproduct-determining stepproductivity in biotechnologyprofessional antigen-presenting cellproficiency test itemproficiency testingproficiency testing providerproficiency testing roundproficiency testing schemeprofile modeprofile of mood state testprogenitor ion in mass spectrometryprogesteroneprogestinprogestogenprogramprogrammable flow extractionprogrammed-flow chromatographyprogrammed-pressure chromatographyprogrammed-temperature chromatographyprogressive systemic sclerosisproinflammatory cytokineprojected rangeprojection formulaprolactinprolapseprolate trochoidal mass spectrometerproliferation assaypromoter in gene technologypromoter in catalysispromotionprompt coincidenceprompt gamma radiationprompt gamma-ray analysisprompt neutronspronephrospronucleusproof of concept in pharmacologypropagating reactionpropagating species in a chain polymerizationpropagationpropellanesproperdinpropertyproperty value assignmentprophageprophaseproportional counterproportional counter tubeproportional gas-scintillation counterproprioceptionproprioceptorproprochiralityprosencephalonprostaglandinsprostanoidsprostateprosthesisprosthetic groupprotamineproteasesproteasomeprotected lyophobic colloidprotectinprotecting groupprotection of a reactive groupprotective action in colloid chemistryprotective immunityprotein Aprotein complexprotein complex purificationprotein data bankprotein data bank fileprotein databaseprotein engineeringprotein foldingprotein fractionationprotein fragmentationprotein Gprotein identificationprotein kinaseprotein kinase Cprotein machineprotein purificationprotein quantificationprotein sequencerprotein sequencingprotein stainingprotein-free filtrateprotein–protein interaction in ppiproteinaseproteinase 3proteinsproteoformproteogenomicsproteoglycanproteoglycomicsproteolipidsproteolysisproteolytic digestionproteolytic enzymeproteolytic peptideproteomeproteomicsproticprotideprotioprotiumprotocol for interlaboratory comparisonprotofugalityprotogenicprotolysisprotonproton affinityproton chargeproton dopingproton magnetic momentproton magnetogyric ratioproton numberproton rest massproton transfer reactionproton-bound dimerprotonated molecule in mass spectrometryprotonationprotonation constantprotonic conductivityprotophilicprotophilicityprotoplastprototrophsprototropic rearrangementprototype drugproximal in anatomyproximity effectprozone effectpseudo acidspseudo basespseudo rate constantpseudo-asymmetric carbon atompseudo-axialpseudo-catalysispseudo-co-oligomerpseudo-copolymerpseudo-equatorialpseudo-first-order rate coefficientpseudo-first-order reactionpseudo-ionic polymerizationpseudo-unimolecularpseudo-zero-order reactionpseudoallergypseudobulbar affectpseudohalogenspseudohermaphroditepseudomeningocelepseudomolecular rearrangementpseudopericyclicpseudopregnancypseudorotationpseudorotaxanepseudorotaxane polymerpseudotumor cerebripseudoureaspsoriasispsoriaticpsychoactivepsychomotor retardationpsychoneuroimmunologypsychosinepsychosispsychotropicpsychrometric hygrometerpsychrometrypterocarpanspubertypubescentpubicpubic symphysispuffer fishpuffingpuffing inhibitorpull-down assaypulmonarypulmonary arterypulmonary surfactantpulmonary valve stenosispulsatancepulse amplitude analyserpulse amplitude selectorpulse duration of a laserpulse duration in electroanalytical chemistrypulse energy of a laserpulse height analyserpulse pile-uppulse rate in secondary-ion mass spectrometrypulse reactor in catalysispulse sequencepulse width in secondary-ion mass spectrometrypulsed extraction fieldpulsed laserpulsed-field gradientpump-dump-probe techniquepump-probe techniquepuppure shift yielded by chirp excitationpure-element relative sensitivity factorpurified protein derivativepurinepurine basesPurkinje cellpurpurapursuit aiming testpush-pull conjugationputamenpyknosispyloric stenosispyramidal cellpyramidal inversionpyranosespyrethroidpyrimidinepyrimidine basespyropyroelectric detectorpyrogenpyrogen testpyrolysispyrolysis curve in electrothermal atomic absorption spectroscopypyrolysis mass spectrometrypyrolysis-gas chromatographypyrolytic carbonpyrolytic graphitepyrromethenesπ – π stateπ → π transitionπ-complexπ-conjugated systemπ-electron donorπ-π stacking quadro-Q-switched laserq2Qa antigenQSARQTc intervalquadratic discriminant analysisquadratic field reflectronquadratic meanquadrature detection in nuclear magnetic resonance spectroscopyquadriplegiaquadrupolar axializationquadrupolar nuclidequadrupolar relaxationquadrupole ion storage trapquadrupole mass analyserquadrupole splitting in Mössbauer spectroscopyquadrupole time-of-flightqualitative analysisqualitative elemental specificity in analysisqualityquality assurancequality controlquality control materialquality factor in nuclear analytical chemistryquality improvementquality managementquality management systemquality manualquality objectivequality of solvent in polymer chemistryquality planquality planningquality policyquantitation by concatenated tryptic peptidesquantitative analysisquantitative polymerase chain reactionquantitative structure–activity relationship in drug designquantitative structure–activity relationshipsquantitative structure–property relationshipquantityquantity calculusquantity of dimension onequantized internal energyquantum of radiationquantum cascade laserquantum confinement effectquantum counterquantum efficiencyquantum mechanics/molecular mechanicsquantum tunneling polymer compositequantum well multilayerquantum yieldquantum-mechanical tunnellingquarter-transition-time potentialquartet statequartz atomizerquartz–iodine lampquasi-axialquasi-classical trajectory methodquasi-enantiomersquasi-equatorialquasi-equilibriumquasi-equilibrium theoryquasi-molecular ion in mass spectrometryquasi-racemic compoundquasi-reference electrodequasi-single-strand polymerquasiparticlequaternary ammonium compoundsquaternary structurequaterpolymerQuEChERS (quick, easy, cheap, effective, rugged and safe) extractionquencherquenching in radiation detectorsquenchingquenching constant in photochemistryquenching correction in photochemistryquickeningquinarenesquinhydronesquinomethanesquinomethidesquinone diazidesquinonesquinoniminesquinonoximesQuistorγ-quantumθ state in polymers R, Sr, sR, SRp, SpracRe, SirelRSr diadR-group decompositionr2rabbitracemateracemicracemic compoundracemic conglomerateracemic mixtureracemizationracemo structures in polymersrachischisisradradial arm mazeradial developmentradial ejectionradial electrostatic field analyser in mass spectrometryradial elution in planar chromatographyradial immunodiffusionradial sectioningradial viewing moderadianradianceradiant emittanceradiant energyradiant energy densityradiant exitanceradiant exposureradiant fluxradiant fluxradiant intensityradiant powerradiant quantitiesradiationradiation chemistryradiation constantsradiation continuum in spectrochemistryradiation counterradiation dampingradiation detectorradiation filterradiation hazardradiation reactionradiation spectrumradiation trappingradiationless deactivationradiationless transitionradiative absorption in spectrochemistryradiative captureradiative charge carrier recombinationradiative de-excitation in spectrochemistryradiative energy transferradiative lifetimeradiative transitionradicalradical anionradical cationradical centreradical combinationradical copolymerizationradical disproportionationradical ionradical pairradical photosubstitutionradical polymerizationradical ring-opening polymerizationradicofunctional nameradiculitisradioactiveradioactive ageradioactive chainradioactive contaminationradioactive coolingradioactive datingradioactive decayradioactive equilibriumradioactive falloutradioactive half liferadioactive seriesradioactive sourceradioactive steady stateradioactive tracerradioactive tracer technique in analysisradioactive wasteradioactivityradioallergosorbent testradioanalytical chemistryradiochemical activation analysisradiochemical purificationradiochemical purityradiochemical separationradiochemical yieldradiochemistryradiochromatographradiocolloidradioenzymatic assayradiofrequencyradiofrequency coilradiofrequency generator in an inductively-coupled plasma spectrometerradiofrequency glow discharge optical emission spectroscopyradiographradiogravimetric analysisradioimmunoassayradioimmunoconjugateradioiodinationradioisotoperadioisotope dilution analysisradioisotope induced X-ray emission analysisradioluminescenceradiolysisradiometric analysisradiometric titrationradiometryradionuclideradionuclidic purityradioreceptor assayradiorelease analysisradiosonderadius of gyrationraffinaterain out in atmospheric chemistryRaman bandRaman optical activityRaman scatteringRaman shiftRaman spectroscopyRaman spectrumRaman wavenumber shiftRandles–Ševčík equationsrandom coil in polymersrandom coincidence in nuclear chemistryrandom copolymerrandom copolymerizationrandom errorrandom forest methodrandom incidence spectrumrandom samplerandom samplingrange in analysisrange of measurement of an analyserrange stragglingRaoult's lawrapid-scan Fourier-transform infrared spectroscopyrare-earth-metal-mediated coordination-addition polymerizationrasterraster scanraterate coefficientrate constantrate constantrate lawrate of appearancerate of change of a quantityrate of change ratiorate of consumptionrate of conversionrate of disappearancerate of flow of liquid metal in polarographyrate of fluid consumption in flame emission and absorption spectrometryrate of formationrate of liquid consumption in flame spectroscopyrate of migration in electrophoresisrate of nucleationrate of reactionrate-controlling steprate-determining stepratemeter in radiochemistryratioraw cokeraw dataRayleigh limitRayleigh ratioRayleigh scatteringRaynaud phenomenonrayon-based carbon fibresre-extractionreactancereactantreacting bond rulesreactionreaction barrierreaction chromatographyreaction coordinatereaction cross-sectionreaction dynamicsreaction injection mouldingreaction intermediatereaction mechanismreaction pathreaction path degeneracyreaction probabilityreaction stagereaction stepreaction timereactivereactive adsorptionreactive airways dysfunction syndromereactive astrocytereactive blendingreactive complexreactive desorptionreactive desorption electrospray ionization mass spectrometryreactive gliosisreactive ion etchingreactive oxygen speciesreactive polymerreactive polymer processingreactive scatteringreactivity indexreactivity–selectivity principlereadability of a balancereadingreagentreagent gasreagent ionreaginreal interphasereal potential of a species in a phasereal spectrumreal surface areareal-time polymerase chain reactionrearing in toxicity testingrearrangementrearrangement ion in mass spectrometryrearrangement stagereboundrebound headacherebound reactionrecallrecall antigenreceived radiant flux densityreceiver-operator characteristic curvereceptor in drug designreceptor in toxicokineticsrecessivereciprocal mesenchymal-epithelial interactionreciprocal translocationrecognition siterecoil in radioanalytical chemistryrecoil labellingrecoil-free fraction in Mössbauer spectrometryrecombinant antibodyrecombinant deoxyribonucleic acidrecombinant DNA technologyrecombinationrecombination energyrecombination fluorescencerecombination signal sequencerecombination-activating genereconstructive transitionrecovered quantity value ratiorecoveryrecovery factor in an extraction processrecovery time of a radiation counterrecrystallizationrectilinear ion traprectumrecursive partitioningred pulpred shiftredepositionredox doping of a polymerredox indicatorredox ion exchangersredox mediationredox polymerredox potentialreduced adsorptionreduced limiting sedimentation coefficientreduced massreduced mobile phase velocity in chromatographyreduced osmotic pressurereduced plate heightreduced samplereduced sedimentation coefficientreduced speciesreduced viscosity of a polymerreducing in analytical chemistryreductionreductive eliminationreductonesReed–Sternberg cellreferee samplereference atom in organic reaction mechanismsreference cellreference dosereference electrodereference examination procedurereference examination standardreference flux densityreference ionreference laboratoryreference materialreference material certificatereference material certificationreference material certification reportreference material producerreference methodreference nominal property valuereference procedure in analysis of trace air constituentsreference state of an elementreference value pH standardreflectancereflected powerreflectionreflection electron energy loss spectroscopyreflection factorreflection high energy electron diffractionreflection-absorption at grazing incidencereflection-absorption infrared spectroscopyreflector voltagereflectronreflexreflex sympathetic dystrophyrefraction effectsrefractive indexrefractive index increment in polymer chemistryregeneration of a catalystregioirregular polymerregiorandom polymerregioregular polymerregioselectivityregression analysisregular block in a polymerregular cokeregular comb macromoleculeregular dendrimerregular dendrimer moleculeregular dendronregular macromolecular rotaxaneregular macromoleculeregular oligomer moleculeregular polymerregular single-strand polymerregular star macromoleculeregularizationregularized discriminant analysisregulated upon activation normal T cell expressed and secretedregulator generegulatory idiotoperegulatory T cellRehm–Weller equationreinforced reaction injection mouldingreinforcement in psychologyReissert compoundsrejection in immunologyrejection intervalrelativerelative activityrelative adsorptionrelative atomic massrelative biological effectiveness of radiationrelative configurationrelative counting in nuclear chemistryrelative counting efficiencyrelative densityrelative detection limitrelative electrode potentialrelative elongationrelative errorrelative hardnessrelative humidityrelative instrument spectral response functionrelative isotope-ratio differencerelative isotopic sensitivity factorrelative local current densityrelative mass accuracyrelative micellar massrelative molar massrelative molecular massrelative molecular-mass exclusion limitrelative permeabilityrelative permittivityrelative preconcentration in trace analysisrelative quantification in proteomicsrelative resolution of a spectrometerrelative responsivityrelative retardation in planar chromatographyrelative retention in column chromatographyrelative selectivity in catalysisrelative spectral responsivityrelative sputtering raterelative standard deviationrelative uncertaintyrelative viscosityrelative viscosity incrementrelative volumic massrelativistic effectsrelaxantrelaxationrelaxation energy in X-ray photoelectron spectroscopyrelaxation kineticsrelaxation maprelaxation reagentrelaxation spectrumrelaxation timereleaser in analytical flame spectroscopyremrenalRenner–Teller effectreorganization in polymersreorganization energy in electron transferrepeatabilityrepeatability condition of examinationrepeatability condition of measurementrepeating distancerepeating distancereperfusionrepetencyrepetition rate of a pulsed laserrepetition rate in secondary-ion mass spectrometryrepetition timereplacement namereplacement operation in organic nomenclaturereplicate samplereplicationrepolarizationreport in analysisreporter probereprecipitationrepresentative samplerepressionreproducibilityreproducibility condition of examinationreproductive cyclereproductive senescencereproductive toxicantreproductive toxicity testreptationrepulsive potential-energy surfacerequirementreserve sampleresidence timeresidence time in biotechnologyresidential macrophageresidual currentresidual dipolar couplingresidual emission anisotropyresidual fuel/oilresidual gas analyserresidual liquid junction error in pH measurementresidual spectrum/background spectrum in mass spectrometryresinresistresist polymerresist profileresist stripresistanceresistivityresolution in gas chromatographyresolutionresolution in optical spectroscopyresolution in stereochemistryresolution of a designresolution phaseresolution: 10 per cent valley definitionresolution: peak width definitionresolving power in mass spectrometryresolving power in optical spectroscopyresolving time in nuclear analytical chemistryresolving time correction in nuclear analytical chemistryresonanceresonance absorption techniqueresonance broadening resonance cross-section in Mössbauer spectrometryresonance effectresonance effectresonance effect magnitude in Mössbauer spectrometryresonance emissionresonance energyresonance energy in radiochemistryresonance fluorescenceresonance fluorescence techniqueresonance formresonance hybridresonance integralresonance integral in radiochemistryresonance ion ejectionresonance lampresonance line in photochemistryresonance line in X-ray spectroscopyresonance neutronsresonance radiationresonance Raman scatteringresonance reactionresonance-enhanced multiphoton ionizationresonant inelastic X-ray scatteringresorptionrespiratory burstrespiratory hypersensitivity assayresponseresponse constant in electroanalytical chemistryresponse scramblingresponse surface methodologyresponse time of a radiation detectorresponse time of an analyserresponsive gelresponsivity in detection of radiationrest point of a balanceresting potentialrestricted rotationrestricted-access sorbentrestriction enzymesrestriction fragment length polymorphismrestriction siteresult in analysisretainer and eluter in counter-current chromatographyretardation in polymer scienceretardation factor in column chromatographyretardation factor in planar chromatographyretarderreterete testisretentateretentate streamretention in nuclear chemistryretention efficiency in particle separationretention factor in column chromatographyretention index in column chromatographyretention of configurationretention temperature in chromatographyretention time in chromatographyretention time of an unretained compound in size-exclusion chromatographyretention volume in size-exclusion chromatographyretention volume of an unretained compound in size-exclusion chromatographyretention volume of the stationary phase in counter-current chromatographyretention volumes in chromatographyreticulate doped polymerreticulation retinaretinoic acidretinoidsretinopathyretroretro Diels–Alder reactionretro-ene reactionretroadditionretrocycloadditionretrograde axonal transportretrograde neurotransmissionretroperitonealretrosynthesisreuptakereverse geneticsreverse geometryreverse immunologyreverse iodine-transfer polymerizationreverse library searchreverse micellereverse osmosisreverse passive agglutinationreverse transcriptasereverse transcriptasesreversed direct-injection burner in analytical flame spectroscopyreversed isotope dilution analysisreversed-phase chromatographyreversible networkreversible transitionreversible-addition-fragmentation chain-transfer agentreversible-addition-fragmentation chain-transfer polymerizationreversible-addition-fragmentation radical polymerizationreversible-deactivation anionic polymerizationreversible-deactivation cationic polymerizationreversible-deactivation coordination polymerizationreversible-deactivation ionic polymerizationreversible-deactivation polymerizationreversible-deactivation radical polymerizationreversing heat flowrevolutions per minuteRey–Osterrieth complex figure testRF value in chromatographyRF-only quadrupolerheologyrheopexic gelrheopexyrhesus factorrhesus factorrheumatic feverrheumatoid arthritisrheumatoid factorrhinitisrhodamine dyesrhombencephalonrhombohedral graphiteribbon delocalisationribonucleic acidsribonucleotidesribosomal RNAribosomesRice–Ramsperger–Kassel theoryRice–Ramsperger–Kassel–Marcus theoryriffling in analytical chemistryrighting reflexrighting testrigid chainrigid membranesring and double bond equivalentring assemblyring reversalring-opening copolymerizationring-opening metathesis polymerizationring-opening nucleophilic chain polymerizationring-opening polymerizationring-sectorRingelmann chart in atmospheric chemistryringing gelrinse solventrise time rise time of an analyserrise velocity in flame emission and absorption spectrometryriskrisk assessmentrisk estimationRitchie equationRM value in planar chromatographyRNARNA interferenceRobertsonian translocationrocket electrophoresisrod-coil copolymerrod-like morphologyröntgenroot mean square error of cross validationroot mean squared error of calibrationroot mean squared error of predictionroot-mean-square end-to-end distance in polymersroot-mean-square end-to-end distance of a freely rotating chainroot-mean-square radius of gyrationroot-mean-square unperturbed end-to-end distanceroot-mean-square unperturbed radius of gyrationRosanoff conventionrosetterostralrotamerrotarod testrotated Fischer projectionrotating disc electroderotating disk sorptive extractionrotating-frame NOE spectroscopyrotation-vibration spectrumrotational barrierrotational branchrotational constantsrotational correlation timerotational diffusionrotational diffusion coefficientrotational echo double resonancerotational frequency in centrifugationrotational frictional coefficient in polymer sciencerotational isomeric state in polymer sciencerotational relaxation timerotational termrotator phase transitionrotatory powerrotaxane constitutional repeating unitrotaxane constitutional unitrotaxane end-unitrotaxane monomerrotaxane monomeric unitrotaxane polymerrotaxanesrotenoidsrotometer in atmospheric chemistryroughness factor of a surfaceRouse chainRouse theoryrovibronic staterRNArubredoxinruby laserruggedness of a measurement procedurerule of fiverule of threerun biasruntrupture of a thin filmRutherford backscatteringRutherford backscattering spectrometryRutherford cross sectionRydberg constantRydberg orbitalRydberg stateRydberg transitionRYDMRγ-radiationγ-ray spectrometer SscSispSRsym-syns-cis, s-transsaccadic eye movementsaccharidessaccharolipidsSackur–Tetrode constantsacral agenesissacrificial acceptorsacrificial donorsacrificial hyperconjugationsaddle field gunsaddle pointsagittalSaha’s ionization equationSaint Anthony firesalivasalpingectomysaltsalt bridgesalt effectsalt form of an ion exchangersalting insalting outsample in analytical chemistrysample biassample chargingsample controlled technique in thermal analysissample error in spectrochemical analysissample flux densitysample handling in analysissample injector in chromatographysample introduction in isotachophoresis separationssample pre-treatmentsample quality controlsample unitsample voltagesampled DC polarographysamplersamplingsampling conesampling errorsampling incrementsampling interval in electroanalysissampling plan in analytical chemistrysampling targetsampling time in electroanalysissampling unitsandwich compoundssandwich immunoassaySanger sequencingsanitary land fillSanta Ana dexterity testsaprophytesarcoidosissaturated solutionsaturationsaturation in radioanalytical chemistrysaturation activitysaturation capacitysaturation capacitysaturation fractionsaturation loadingsaturation mutagenesissaturation transfersaturation transfer difference spectroscopysaturation vapour pressuresaturatorsawhorse projectionSaytzeff rulescaffoldscaffold hoppingscalerscalingscaling circuitscaling factorscaling lawscan in thermal analysisscan cycle timescan ratescanning calorimetryscanning electron microscopyscanning method in mass spectrometryscanning near-field optical microscopyscanning proton microscopyscanning technique in thermal analysisscanning transmission electron microscopyscanning-probe lithographyscaphocephalyScatchard equationScatchard plotscattered-ion energyscattered-ion energy ratioscatteringscattering anglescattering cross-sectionscattering error in spectrochemical analysisscattering geometryscattering matrixscattering planescattering vectorscavengerscavenger receptorscavenging in atmospheric chemistryscFvschedule of reinforcementschedule-controlled operant behaviourSchenck reactionSchenck-sensitization mechanismSchiff basesSchiller layersSchottky barrierSchottky-barrier photodiodeSchulz–Zimm distributionSchulze–Hardy ruleSchwann cellscintillationscintillation counterscintillation detectorscintillation spectrometerscintillatorssclerodermasclerosisscoliosisScombroid poisoningscorescores plotscoringscramblingscreeningscreening functionscrotumscrubber in atmospheric chemistryscrubbing in atmospheric chemistryseco-secondsecond of arcsecond harmonic generationsecond messengersecond set rejectionsecond-order transitionsecondary challengesecondary crystallizationsecondary current distributionsecondary electron multiplier in mass spectrometrysecondary electrons in in situ microanalysissecondary examination standardsecondary excitation secondary fluorescence in X-ray emission spectroscopysecondary immune responsesecondary ionsecondary ionization in mass spectrometrysecondary isotope effectsecondary kinetic electrolyte effectsecondary kinetic isotope effectsecondary lymphoid tissuesecondary metabolitessecondary neutral mass spectrometrysecondary oocytesecondary parkinsonismsecondary pollutionsecondary radiationsecondary relaxationsecondary relaxation peaksecondary relaxation temperaturesecondary sexual characteristicsecondary species in a chain polymerizationsecondary standardsecondary structuresecondary-electron yieldsecondary-electron yield in in situ microanalysissecondary-electron yield in secondary-ion mass spectrometrysecondary-ion angular distributionsecondary-ion energy distributionsecondary-ion mass spectrometrysecondary-ion yieldsecretory componentsecretory immunoglobulin Asector mass spectrometersecular equationsecular equilibriumsedativesedimentsedimentationsedimentation in chemistrysedimentation coefficientsedimentation equilibriumsedimentation field strengthsedimentation field-flow fractionationsedimentation potential differencesedimentation velocitysedimentation velocity methodsedimentation volumesegment in analytical chemistrysegmented copolymersegmented flow analysissegmented flow extractionsegregated star macromoleculesegregationsegregation in polymersseizureselected area electron diffractionselected ion flow tubeselected ion monitoring in mass spectrometryselected reaction monitoringselected-area apertureselectinselection in biotechnologyselection ruleselective in analysisselective corrosionselective decouplingselective detectorselective detector in chromatographyselective digestion selective elution in chromatographyselective excitationselective IgA deficiencyselective inhibition selective micro-sample in spectrochemical analysisselective poisoning in catalysisselective preconcentration in trace analysisselective pulseselective sampleselective serotonin reuptake inhibitorselective solvent in polymer chemistryselective sorption selectively labelledselectively-labelled isotopic tracerselectivityselectivityselectivity of a reagentselectivity coefficient in ion exchange chromatographyselectivity factorselectivity factor in ion exchange chromatographyselectivity ratioselector geneselenenic acidsselenidesseleninic acidsselenocyanatesselenolsselenonesselenonic acidsselenoxidesself inductanceself-absorptionself-absorption broadening of a spectral lineself-absorption effect in luminescence spectroscopyself-absorption factor of a radiation sourceself-assembled monolayerself-assemblyself-avoiding random-walk chainself-chemical ionizationself-diffusion coefficientself-localized excitations in conjugated organic polymersself-poisoning in catalysisself-quenchingself-reversalself-shieldingself-tolerancesella turcicaselonessemensemi-dendritic constitutional repeating unitsemi-dilute solutionsemi-empirical quantum mechanical methodssemi-interpenetrating polymer networksemi-rigid chainsemicarbazonessemicokesemiconductorsemiconductor detectorsemiconductor lasersemiconductor-metal transitionseminal vesicleseminiferous tubulesemioxamazonessemipermeable membrane devicesemiquinonessemisystematic namesenile dementiasenilitysenior constitutional repeating unitseniority in organic nomenclaturesensitive area of a radiation detectorsensitive volume of a radiation detectorsensitivity in mass spectrometrysensitivity in metrology and analytical chemistrysensitivity analysissensitivity of a measuring systemsensitizationsensitization in colloid chemistrysensitized luminescencesensitized strainsensitizersensorsensory neuronseparability assumptionseparated flame in flame spectroscopyseparation coefficientseparation factor in column chromatographyseparation factor in liquid-liquid distributionseparation number in chromatographyseparation temperature in chromatographysepsisseptumsequencesequence rulessequencing of nucleic acidssequential analysersequential detection in glow discharge optical emission spectrometrysequential extractionsequential indication of an analysersequential injection analysis extraction devicesequential interpenetrating polymer networksequential measuring cellsequential samplesequential semi-interpenetrating polymer networksequential spectrometerseries of analytical resultsseroconversionserologyserotoninergicserousSertoli cellSertoli–Leydig cell tumorserumserum sicknesssesquiterpenoidssesterterpenoidssettling chamber in atmospheric chemistrysettling error in spectrochemical analysissettling velocitysevere combined immunodeficiencysex chromatinsex chromosomesex-determining region Y proteinsexual dimorphismsexual maturationsexual maturitysexual reproductionsexually dimorphic nucleusshakeoffshakeupshape selectivity in catalysisshape-memory polymershear breakdownshear dependent viscosityshear modulusshear rateshear strainshear stressshear thickeningshear thinningshear transitionshear viscositysheath flow interfacesheath gassheath liquidsheep red blood cell antigenshelf lifeshell-crosslinked micelleshellfish poisoningShewhart control chartShewhart control limitShewhart means chartShewhart range chartshielded flame shieldingshielding constantshift factorshift reagentShirley backgroundshish-kebab structureshort chainshort circuit photocurrentshort-chain branchshort-chain branchshort-range intramolecular interactions in polymersshot noiseshotgun lipidomicsshotgun methodshotgun proteomicsshots per pixel in secondary-ion mass spectrometryshrinkageshuntshut-down state in analysisshut-down time in analysisshuttle vectorSIsialonsiblingsick building syndromeside chainside groupside group or side-chain polymer liquid crystalside-chain macromolecular rotaxaneside-chain rotaxane polymerside-chain self-assemblysiemenssievertsigmatropic rearrangementsignal in analysissignal peptidesignal sequencesignal transducer and activator of transcriptionsignal transductionsignal-to-noise ratiosignaling lectinsignalling lipidssilanessilanolssilasesquiazanessilasesquioxanessilasesquithianessilathianessilazanessiliconessiloxanessilver electrodesilver filmsilver stainingsilyl groupssilyl radicalssilylationsilylenesimilarity ensemble approachsimilarity indexsimilarity searchingsimple reaction time testsimple shearsimple-to-use interactive self-modelling mixture analysissimplified molecular input line entry systemsimplified molecular input line entry systemSIMS ion imagesimulated annealingsimulation technique in analysissimultaneous interpenetrating polymer networksimultaneous nucleationsimultaneous pair transitionssimultaneous reactionssimultaneous semi-interpenetrating polymer networksimultaneous spectrometersimultaneous steam distillation-solvent extractionsimultaneous technique in thermal analysissingle beam spectrumsingle cell proteinsingle channel pulse height analysersingle crystal nuclear magnetic resonance spectroscopysingle escape peaksingle linkagesingle nucleotide polymorphismsingle photon countingsingle scatteringsingle-beam spectrometersingle-chain antibodysingle-drop microextractionsingle-electron transfer mechanismsingle-focusing mass spectrometersingle-photon timingsingle-step reactionsingle-strand chain in a polymersingle-strand macromoleculesingle-strand polymersinglet molecular oxygensinglet statesinglet-singlet absorptionsinglet-singlet annihilationsinglet-singlet energy transfersinglet-triplet absorptionsinglet-triplet energy transfersinglet–triplet crossingsingly labelledsingular value decompositionsink in atmospheric chemistrysinoatrial nodesinteringsinussinus venosussister chromatid exchangesite splittingsite-directed mutagenesissitus inversussize effectsize-exclusion chromatographySjögren syndromeskeletal atomskeletal bondskeletal formulaskeletal structureskewskimmerskinskin immune systemskin sensitization testskin testSlater determinantSlater-type orbitalslipslip castingslit width of a dispersive spectrometerslot burnerslow neutronsslow-reacting substance of anaphylaxisSLUDDESM-interferencesmall G proteinsmall inhibitory double-stranded RNAsmall interfering RNA moleculesmall moleculesmall outer capsid proteinsmall particle in radiation scatteringsmall-angle X-ray scatteringSMARTSsmecticsmectic mesophasesmectic stateSMIRKSsmog in atmospheric chemistrysmog chamber in atmospheric chemistrysmog index in atmospheric chemistrysmokesmooth transitionsmoothingsoapsoap curdsoap filmsocial interaction testsoft acidsoft bakesoft basesoft independent modelling of class analogysoft ionizationsoft lithographysoft-segment phase domainsoiling in atmospheric chemistrysolsol fractionsol-gel coatingsol-gel critical concentrationsol-gel materialsol-gel metal oxidesol-gel processsol-gel silicasol-gel transitionsolar cellsolar conversion efficiencysolar flaresolar radiation in atmospheric chemistrysolid amalgam electrodesolid anglesolid angle of analysersolid dispersionsolid phase antibody radioimmunoassaysolid polymer electrolytesolid solutionsolid solutionsolid state laserssolid support in column chromatographysolid volume in column chromatographysolid-phase dynamic extractionsolid-phase extractionsolid-phase microextractionsolid-state nuclear magnetic resonance spectroscopysolidificationsolidified floating organic dropsolidified floating organic drop microextractionsolidussolitonsolubilitysolubility parametersolubility parametersolubility productsolubilizationsolutesolute-volatilization interference in flame spectroscopysolutionsolvationsolvation energysolvatochromic relationshipsolvatochromismsolvatomerssolventsolvent in liquid-liquid distributionsolvent blank flux densitysolvent desorptionsolvent extractionsolvent front in chromatographysolvent ion exchangesolvent isotope effectsolvent migration-distance in chromatographysolvent nucleationsolvent parametersolvent regeneration in extraction processessolvent shiftsolvent suppressionsolvent transport rate in an inductively-coupled plasma spectrometersolvent vapor annealingsolvent-induced symmetry breakingsolvent-separated ion pairsolvent-shared ion pairsolvolysissolvophilicsolvophobicsolvophobicity parametersolvussomaticsomatic diversification theorysomatic gene conversionsomatic hypermutationsomatic recombinationsomatomedinsomatotropinsomiteSOMOsonic hedgehogsonic spray ionizationsonicationsonogelsonoluminescencesonosolsootsoporificsorbentsorbent trapsorbent-containing pipette tipSoret bandsorption in colloid chemistrysorption isotherm in ion exchangesorption techniques in trace analysissorptive insertion in surface catalysissorptive-tape extractionsource efficiency Southern blotSoxhlet extractionspace charge in a semiconductorspace charge effectspace formulaspace time in catalysisspace velocity in catalysisspacerspacer armspare receptorspark ionization in mass spectrometryspasmspasmolyticspasticspasticityspatial-distribution interference in flame spectroscopyspatially offset Raman spectroscopyspatially resolving detector of radiationspawnspawningspecial salt effectspeciation in chemistryspeciation analysis in chemistryspecies in taxonomyspecific in quantitiesspecific in analysisspecific acid–base catalysisspecific activity in radiochemistryspecific activity (radionuclide) of a radionuclidespecific adsorptionspecific binding in DNA-protein interactionsspecific burn-upspecific catalysisspecific conductancespecific detector in chromatographyspecific gravityspecific heat capacityspecific ionization in nuclear chemistryspecific permeability in chromatographyspecific photon emissionspecific pore volume of a catalystspecific resolution in size-exclusion chromatographyspecific retention volume in chromatographyspecific surface area in surface chemistryspecific volumespecific weightspecifically absorbing ionspecifically labelledspecifically labelled tracerspecificity of a two-class modelspecified requirementspecimen in analytical chemistryspectator holespectator mechanismspectator-stripping reactionspectral background of an excitation sourcespectral bandspectral bandwidthspectral bandwidth error in spectrochemical analysisspectral distributionspectral effectivenessspectral fluence ratespectral intensityspectral interference in flame spectroscopyspectral irradiancespectral line of an atomspectral overlapspectral photon exitancespectral photon flowspectral photon fluxspectral photon radiancespectral quantitiesspectral radiancespectral radiant energyspectral radiant energy densityspectral radiant exitancespectral radiant fluxspectral radiant intensityspectral radiant powerspectral range of a spectrometerspectral resolution of an instrumentspectral responsivityspectral responsivity functionspectral sensitivityspectral sensitizationspectral skewingspectral spheradiancespectral subtractionspectral width in NMRspectrochemical buffer in atomic spectroscopyspectrochemical carrier in atomic spectroscopyspectroelectrochemistryspectrogramspectrographspectrometerspectrometer response functionspectrometer transmission functionspectrometryspectrometryspectroscopespectroscopyspectrumspectrum analysisspecular reflectance in optical spectroscopyspecular reflectionspecular X-ray reflectrometryspeedspeed distribution functionspeed of light in a vacuumSpemann organizerspermsperm bankingsperm motilityspermatidspermatocidespermatocytespermatogenesisspermatogonial chromosome aberration testspermatogoniumspermatozoonspermiationspermiogenesissphenoid bonespherandspherical carbonaceous mesophasespherical radiancespherical radiant exposurespherical topspherulitesphinctersphingolipidsphingolipidssphingomyelinsphingosinespikespin of a nucleusspin adductspin conservation rulespin contaminationspin countingspin crossoverspin decouplingspin densityspin echospin labelspin polarizationspin probespin projectionspin systemspin trappingspin-allowed electronic transitionspin-flip methodspin-flip transitionspin-flop transitionspin-glass transitionspin-lattice relaxationspin-lattice relaxation timespin-orbit splittingspin-Peierls transitionspin-spin relaxationspin-spin relaxation timespin-state transitionspin-statistical factor in diffusion-controlled reactionsspin–orbit couplingspin–orbit coupling constantspin–spin couplingspin–spin coupling constantspina bifidaspinal columnspinal cordspinning sidebandsspinodalspinodal decompositionspiral arteriolespiro atomspiro chain in a polymerspiro compoundsspiro macromoleculespiro polymerspiro unionsplanchnicspleensplenicsplenocytesplicingsplit adjuvant techniquesplit-flow thin-cell fractionationspondylitisspongioblastspontaneous abortionspontaneous autoimmune thyroiditisspontaneous emissionspontaneous fissionspontaneous resolutionspot in chromatographyspot samplingspray chamberspray ionizationsprayer in flame spectroscopyspread monolayerspreading function in chromatographyspreading wettingsputter depth profilesputter ratesputter yieldsputteringsputtering ratesputtering yieldsquare-wave currentsquare-wave voltammetrystability constantstablestable film, metastable filmstable ion in mass spectrometrystable isotope labelling in proteomicsstable isotope ratio analysis of amino acids in cell culturestable isotope standards and capture by anti-peptide antibodiesstable radical in SRMPstable-radical-mediated polymerizationstacking interactionsstage in liquid–liquid extractionstagewise contactorstaggered conformationstagnant inversion in atmospheric chemistrystalkstampstand-by state in analysisstandard acceleration of free fallstandard atmospherestandard atomic weightsstandard chemical potentialstandard concentrationstandard conditions for gasesstandard deviationstandard deviation for proficiency assessmentstandard electrode potentialstandard electromotive forcestandard entropy of activationstandard equilibrium constantstandard error of estimatesstandard error of predictionstandard Gibbs energy of activationstandard hydrogen electrodestandard molalitystandard operating procedurestandard potential of an electrode reactionstandard potential of the reaction in a chemical cellstandard pressurestandard rate constant of an electrode reactionstandard reaction quantitiesstandard solutionstandard statestandard subtraction method in electroanalytical chemistrystandard thermodynamic quantitiesstandard uncertaintystandard vacuum levelstandardizationstannoxanesstannylenesstannylidenesstar copolymerstar macromoleculestar polymerStark effectstart-up time in analysisstarting line in chromatographystarting point startle reactionstartle reflexstate crossingstate diagramstate functionstate-to-state kineticsstatic extractionstatic fieldstatic fields mass spectrometerstatic headspace analysisstatic light scatteringstatic limit in secondary-ion mass spectrometrystatic magnetic flux densitystatic pressurestatic quenchingstatic secondary-ion mass spectrometrystatic self-assemblystatic stabilitystatic structure factorstatic technique in thermal analysisstationary phase in fermentationstationary phase in chromatographystationary phase volume in chromatographystationary statestationary-phase fractionstatistical coil statistical coilstatistical copolymerstatistical copolymerizationstatistical factorstatistical process controlstatistical pseudo-copolymerstatistical segment in polymersstatistical weightstatus epilepticussteady state in liquid-liquid distributionsteady statesteady state approximationsteady state extractionsteady-state sputteringsteady-state surface composition sputteringstealthStefan–Boltzmann constantstem in polymer crystalsstem cellstenosisstep in chromatographystep height in chromatographystep-scan interferometerstepped surfacestepwise elution in chromatographystepwise reactionstepwise technique in thermal analysissteradianstereoblock macromoleculestereoblock polymerstereochemical formulastereochemical non-rigiditystereoconvergencestereocopolymerstereodescriptorstereoelectronicstereoelectronic controlstereoformulastereogenic centerstereogenic unitstereoheterotopicstereohomosequence in a polymerstereoisomerismstereoisomersstereologystereomutationstereoregular macromoleculestereoregular polymerstereorepeating unit in a polymerstereoselective polymerizationstereoselective synthesisstereoselectivitystereosequence in a polymerstereospecific polymerizationstereospecifically labelled tracerstereospecificitysteric effectsteric factorsteric factor in polymerssteric field-flow fractionationsteric hindrancesteric isotope effectsteric strainsteric-approach controlsterilitysterilizationStern layerStern–Volmer kinetic relationshipssteroidsteroid 21-monooxygenasesteroidogenesis assaysteroidssterol lipidssterolsStevens–Johnson syndromeStevenson’s rulestibanesstibanylidenesstibinesstibinidenesstibonium compoundssticking coefficient in surface chemistrysticking probability in surface chemistrysticky ends in biotechnologystillbirthstimulantstimulated emissionstimulated radiative charge carrier recombinationstimulated Raman scatteringstimuli-responsive polymerstimulusstimulus-responsive polymerstir bar sorptive extractionstirred flow reactor in catalysisstochastic effectstochastic samplingstochastic theoriesStock numberstock solutionStockholm conventionstoichiometricstoichiometric capacitystoichiometric concentrationstoichiometric mean molal in electrochemistrystoichiometric numberstoichiometric sputteringstoichiometrystokesStokes equationStokes lawStokes numberStokes parametersStokes shiftStokes type radiationStokes–Einstein equationStokes–Raman scatteringstop event in secondary-ion mass spectrometrystop-flow conditions in electrothermal atomic absorption spectroscopystopped flowstopped-flow cell in spectrochemical analysisstopperstopping cross sectionstopping cross section factorstopping powerstorage modulusstored waveform inverse fourier transformSTPstrainstrain energystrain-induced transitionstrand breakstratified filmstratified samplestratocumulus cloudstratopausestratospherestratus cloudstray lightstray radiation error in spectrochemical analysisstreak camerastreak tubestream in membrane separationsstreaming birefringencestreaming currentstreaming potential differencestress graphitizationstress relaxationstress-assisted transitionstress-response pathwaystriatumstrip dynode photomultiplier tubestrippingstripping isothermstripping phasestripping ratio in solvent extractionstripping reactionstripping solution in liquid-liquid distributionstripping voltammetrystroke in neurosciencestromastromal cellstrong collisionstrong glassstructural alertstructural disorderstructural formulastructural keysstructural lipidsstructural proteomicsstructural stabilitystructural transitionstructure of a catalyststructure diagramstructure-based design in drug designstructure-data format filestructure-property correlationsstructure–activity relationshipstuporstyphnatessub-shell photoionization cross sectionsubacute cutaneous lupus erythematosussubarachnoidsubarachnoid hemorrhagesubchain of a polymersubduralsubgroup-supergroup transitionsubjacent orbitalsublation in solvent extractionsublimationsubphase subsamplesubstance concentrationsubstance contentsubstance flow ratesubstance fractionsubstantia nigrasubstituent atomsubstituent electronegativitysubstitution chain transfersubstitution reactionsubstitutive namesubstoichiometric extractionsubstoichiometric isotope dilution analysissubstratesubstratesubstrate in thin filmssubstructure searchingsubthalamussubtractive namesubunitsuccessor complexsudden polarizationsudden potential-energy surfacesugarsugarssuicide inhibitionsuicide metabolismsulcussulfamic acidssulfanessulfatidessulfenamidessulfenessulfenic acidssulfenium ionssulfenyl groupssulfenyl nitrenessulfenyl radicalssulfenylium ionssulfidessulfiliminessulfimidessulfiminessulfinamidessulfinamidinessulfinessulfinic acidssulfinic anhydridessulfiniminessulfinylaminessulfolipidssulfonamidessulfonamidinessulfonediiminessulfonessulfonic acidssulfonic anhydridessulfonimidessulfonium compoundssulfonphthaleinssulfonylaminessulfoxidessulfoximidessulfoximinessulfur diimidessultamssultimssultinessultonessum bandsum frequency generation spectroscopysum of statessum transitionsum-frequency spectroscopysummit current in polarographysummit potentialsuperabsorbent polymersuperacidsuperantigensuperbasesuperbasesuperconducting transitionsupercooled polymer meltsupercritical drying of a gelsupercritical fluidsupercritical fluid chromatographysupercritical fluid chromatography-mass spectrometrysupercritical fluid extractionsuperequivalent adsorptionsuperexchange interactionsuperfamilysuperfecundationsuperfetationsuperficial work in surface chemistrysuperjacent orbitalsuperlatticesupernumerarysuperposabilitysuperradiancesupersaturated solutionsupersaturationsupervised classificationsupervised learningsupport of a catalystsupport plate in chromatographysupport-coated open-tubular column in chromatographysupported liquid membranesupporting electrolytesuppressionsuppressorsuppressor cellsuprafacialsupramolecular adhesion complexsupramolecular assemblysupramolecular associationsupramolecular chemistrysupramoleculesupratentorialsupratentorial herniationsurfacesurface active agentsurface amountsurface approximation energysurface barrier semiconductor detectorsurface catalysissurface charge densitysurface charge layersurface chemical potentialsurface concentrationsurface contamination in surface analysissurface core-level shiftsurface coveragesurface crossingsurface densitysurface dilatational viscositysurface dipole layersurface electric potentialsurface excesssurface excess amountsurface excess concentrationsurface excess conductivitysurface excess energysurface excess enthalpysurface excess entropysurface excess Gibbs energysurface excess Helmholtz energysurface excess isothermsurface extended X-ray absorption fine structure spectroscopysurface fractal dimensionsurface graftingsurface ionization in mass spectrometrysurface ionssurface layersurface mapsurface of a phasesurface of tensionsurface plasmonsurface plasmon resonancesurface potentialsurface pressuresurface reaction surface regionsurface rheologysurface righting testsurface roughnesssurface segregationsurface shear viscositysurface statessurface stresssurface tensionsurface worksurface-assisted laser desorption/ionization mass spectrometrysurface-bound macrocyclic sorbentsurface-bound phenylboronic acid sorbentsurface-enhanced hyper-Raman spectroscopysurface-enhanced infrared absorptionsurface-enhanced infrared spectroscopysurface-enhanced laser desorption/ionizationsurface-enhanced Raman spectroscopysurface-enhanced raman spectroscopysurface-enhanced resonance Raman spectroscopysurface-enhanced resonant Raman spectroscopysurface-enhanced, spatially offset Raman spectroscopysurface-induced dissociationsurface-induced reactionsurfactantsurprisalsurprisal analysissurrogate internal standardsurrogate light chainsuspended matter in atmospheric chemistrysuspensionsuspension effect in an ion-selective electrodesustainabilitysustained deliverysustained off-resonance irradiationSwain–Lupton equationSwain–Scott equationsweatswelling in colloid and surface chemistryswelling agentswelling pressure in colloid and surface chemistryswift ionswimming testswing curveswitch sequenceswitchboard model in polymer crystalsswitching transitionsydnone iminessydnonessymbiosissymbol in quantities and unitssymmetric in vibrational spectroscopysymmetric topsymmetrical filmssymmetrical flow field-flow fractionationsymmetrizationsymmetry coordinatesymmetry numbersymmetry-breaking transitionsymmetry-conserving transitionsymmorphic space groupsympathetic nervous systemsympathetic ophthalmiasympathomimeticsymproportionationsynapsesynapsissynaptic cleftsynaptic inhibitionsynaptic plasticitysynaptic strippingsynaptic transmissionsynaptonemal complexsynartetic accelerationsyncephalussynchronizationsynchronoussynchronously excited spectrum in phosphorescencesynchrotron radiationsynchrotron radiation induced X-ray fluorescence analysissynchrotron X-ray spectroscopysynclinalsyncytiotrophoblastsyncytiumsyndactylysyndetsyndiotactic macromoleculesyndiotactic polymersyndiotactic triads in polymerssyndromesyneresissynergism in solvent extractionsynergism in toxicologysyngeneicsynophthalmiasynoptic scalesynotiasynperiplanarsyntectic reactionsynthetic biopolymersynthetic graphitesyringomyeliasystemsystem of units of measurementsystem resolutionsystematic effectsystematic errorsystematic examination errorsystematic namesystematic samplingSystème International d'Unitéssystemicsystemic autoimmune diseasesystemic effectsystemic lupus erythematosussystemic sclerosissystems biologySzilard–Chalmers effectπ → σ transitionσ → σ transitionσ, π 'true' rate constantt1 in two-dimensional nuclear magnetic resonance spectroscopyt1 noise in two-dimensional nuclear magnetic resonance spectroscopyt2 in two-dimensional nuclear magnetic resonance spectroscopyteletetrahedro-threothreo structurestrans- in inorganic nomenclaturetranstrans conformationtrans-fusedtriangulo-triprismo-T lymphocyteT-cell receptorT-cell-dependent antibody responseT-dependent antigenT-independent antigenT-jumpT-mazeT-stackingtachycardiatactic block in a polymertactic block polymertactic macromoleculetactic polymertacticityTaft equationtaggedtail budtail flick testtail immersion testtail suspension testtailing in chromatographytailing factor in chromatographytake-it-or-leave-it datatake-off angleTammann’s 2/3rd ruletan in thermal analysistandem conjugatetandem mass spectrometertandem mass spectrometry in spacetandem mass spectrometry in timetandem repeat sequenceTanimoto similarityTanimoto similarity indextardive dyskinesiatarget in biologytarget cell in immunologytarget examination uncertaintytarget measurement uncertaintytarget validationtargeted drug deliverytargeted lipidomicstargeted proteomicstargetingtau proteintauopathytautomertautomeric effecttautomerismtautomerizationtautomerstaxonTay–Sachs diseaseTaylor coneTDP-43 transcription factorTdT-dependent dUTP-biotin nick end labeling assaytearstegmentumtele-substitutiontelechelic moleculetelechelic polymertelencephalontelluridestelluronestelomertelomerasetelomeretelomerizationtelophasetemperaturetemperature coefficient of responsivitytemperature effect in luminescence spectroscopytemperature inversion in atmospheric chemistrytemperature jumptemperature lapse rate in atmospheric chemistrytemperature sensitivitytemperature-programmed chromatographytemperature-programmed desorption in thermal analysistemperature-programmed oxidation in thermal analysistemperature-programmed reduction in thermal analysistemplate in biotechnologytemplate polymerizationtemporal lobetemporary poisoning in catalysistension headachetentoriumterateratocarcinomateratogenteratogenesisteratogeneticsteratogenic indexteratogenicityteratologyteratomatermterm in X-ray spectroscopyterm symbolsterminal constitutional repeating unitterminal deoxynucleotidyl transferaseterminationterminator in biotechnologytermolecularterpenesterpenoidsterpolymertertiary aerosol in an inductively-coupled plasma spectrometertertiary current distributiontertiary lymphoid tissuetertiary structuretertiary structure of a proteinteslaTessier classificationtest portiontest sampletest settest solution in analysistesticular feminizationtestingtestistestosteronetetanictetanustetanus toxintetanus toxin-binding gangliosidetetanus toxoidtetanytetracyclinestetrad testtetrads tetrahedral intermediatetetralogy of Fallottetramer stainingtetrapyrrolestetraterpenoidstetrodotoxintetrodotoxin-insensitivetexture of a catalystTh0 cellTh1 cellTh17 cellTh2 cellTh3 cellTh9 cellthalamusthalidomidethecatheoretical ceramic yieldtheoretical degree of biodegradationtheoretical plate numbertheoretical scattered-ion intensitytherapeutic antibodytherapeutic indextherapeutic polymerthermal analysisthermal annealing in lithographythermal blackthermal column in nuclear chemistrythermal conductancethermal conductivitythermal conductivity detector in gas chromatographythermal curingthermal curvethermal detectorthermal diffusion depththermal diffusivitythermal dispersionthermal extractionthermal field-flow fractionationthermal fissionthermal history in polymer sciencethermal injectionthermal ionization in mass spectrometrythermal lagthermal lensingthermal neutronsthermal resistancethermal volatilizationthermal wave decay coefficientthermally activated delayed fluorescencethermally stimulated current in thermal analysisthermally stimulated depolarisation in thermal analysisthermally-induced transitionthermionic work functionthermo- techniquethermoacoustimetrythermoanalyticalthermochemical analysisthermochemical caloriethermochromismthermocouplethermodiffractometrythermodilatometrythermodynamic control of product compositionthermodynamic energythermodynamic equilibrium constantthermodynamic isotope effectthermodynamic quality of solvent in polymer chemistrythermodynamic temperaturethermodynamically equivalent sphere in polymer chemistrythermoelectrometrythermogramthermogravimetric analyserthermogravimetric analysisthermogravimetrythermoluminescencethermoluminescence analysisthermolysisthermomagnetometrythermomanometrythermomechanical analyserthermomechanical analysisthermomechanical measurementthermomechanical propertiesthermometricthermometric enthalpy titrationthermometric titrationthermomicroscopythermoparticulate analysisthermophilethermophotometrythermopilethermoplastic elastomerthermoptometrythermoreversible gelthermoreversible junction pointthermoreversible networkthermosetting polymerthermosonimetrythermospectrometrythermospherethermospray ionizationthermotropic mesophasethiazynesthick filmthickness of diffusion layer in electrochemistrythickness of electrical double layerthickness of reaction layer in electrochemistrythigmotaxisthin filmthin film formatthin layer chromatography-mass spectrometrythin targetthin-layer chromatographythin-layer headspace analysisthiothioacetalsthioaldehyde S-oxidesthioaldehydesthioanhydridesthiocarboxylic acidsthiocyanatesthioethersthiohemiacetalsthioketone S-oxidesthioketonesthiolatesthiolsthionylaminesthird bodythixotropic gelthixotropythiyl radicalsthoracic ductthoracic spinethreadable ringthreading componentthree-dimensional nuclear magnetic resonance spectroscopythree-electrode cellthreshold energythreshold limit valuethreshold potentialthrombocytopeniathrombogenicitythromboxanethrombusthrough-bond electron transferthrough-conjugationthrough-space electron transferthroughput ratethrowing powerthrushthymicthymic atrophythymic educationthymomathymusthymus-dependent antigenthymus-independent antigenthyroglobulinthyroglossal ductthyroid glandthyroid peroxidasethyroid stimulating hormonethyroid-stimulating hormone receptorTICT emissionTICT statetie molecule in polymerstight ion pairtimetime of of thixotropic recoverytime constant of a detectortime domaintime domain nuclear magnetic resonance spectroscopytime of centrifugation of centrifugationtime of deactivation in heterogeneous catalysistime-correlated single photon countingtime-dependent density functional theorytime-dependent stoichiometrytime-independent stoichiometrytime-lag focusingtime-of-flight mass spectrometertime-resolved anisotropytime-resolved fluorometrytime-resolved laser-induced breakdown spectroscopytime-resolved microwave conductivitytime-resolved spectroscopytime-weighted average samplingtime–temperature superposition principletinglingtinnitustip enhanced raman spectroscopytip-enhanced Raman spectroscopytissuetissue engineeringtissue transglutaminasetiter in immunologytitrandtitranttitrationtitration curvetitration errortitretitrimetric analysistolerancetolerance intervaltolerance limittolerogentoll-like receptortolyl iontomographytonnetonsiltop-down patterningtop-down proteomicstop-surface imagingtopcoattopical effecttopochemical reactiontopographic contrasttopological indextopological polar surface areatopomerizationtopomerstopotactic reactiontopotactic transitiontoroidal ion traptorquetorquoselectivitytorrtorsion angletorsion pendulumtorsional braid analysistorsional pendulum analysistorsional stereoisomerstorsional straintorticollistotal absorption peaktotal chemifluxtotal consumption time in flame emission and absorption spectrometrytotal correlation spectroscopytotal internal reflectiontotal ion currenttotal ion current electropherogramtotal ion current profiletotal ion yieldtotal mobile phase volume in size-exclusion chromatographytotal quality managementtotal radiant powertotal reflection of X-raystotal reflection X-ray fluorescence analysistotal reflection X-ray fluorescence spectroscopytotal retention time in chromatographytotal retention volume in column chromatographytotal sample consumption systemtotal secondary-electron yieldtotal suppression of spinning sidebandstotal velocity of the analyte in capillary electrophoresistotally porous packing in chromatographyTougaard backgroundtoxic epidermal necrolysistoxic shock syndrometoxic substancetoxicitytoxicodynamicstoxicokineticstoxicologytoxintoxinologytoxoidtoxoplasmosisTr1 celltrace elementtraceabilitytracertracheatracktrack detectortrajectory in reaction dynamicstranquilizertransannular straintranscription in biotechnologytranscription factortranscutaneous electronic nerve stimulationtranscytosistransducertransduction in biotechnologytransfertransfer in analysistransfer activity coefficienttransfer coefficienttransfer examination devicetransfer linetransfer RNAtransferabilitytransferasestransflectancetransflectiontransformationtransformation in gene technologytransformation probabilitytransforming growth factor βtransgenictransient crosslinktransient junction pointtransient networktransient phase in chemical kineticstransient speciestransient spectroscopytransient-stimulated emission pumpingtransit time in flame emission and absorption spectrometrytransitiontransition coordinatetransition elementtransition interval in titrimetric analysistransition layertransition momenttransition polarizationtransition potential of a redox indicatortransition probability for absorptiontransition probability for spontaneous emissiontransition probability for stimulated emissiontransition speciestransition statetransition state analoguetransition state theorytransition structuretransition temperature for liquid crystalstransition time in electroanalytical chemistrytransition wavenumbertransition-metal-mediated radical polymerizationtranslation in biotechnologytranslational diffusiontranslational diffusion coefficient in polymer sciencetranslational frictional coefficient in polymer sciencetranslational spectroscopytransmissiontransmission in mass spectrometrytransmission coefficienttransmission efficiencytransmission electron energy loss spectroscopytransmission electron microscopytransmission factortransmission high energy electron diffractiontransmission quadrupole mass spectrometertransmission Raman spectroscopytransmission spectrumtransmittancetransoid conformationtransplacental carcinogentransplanttransplantationtransport in analysistransport controltransport efficiency of a sampletransport interference in flame spectroscopytransport numbertransport regiontransporters associated with antigen processingtransposition of the great arteriestransposontranssynaptictranstactic polymertranstentorial herniationtransverse relaxation optimized spectroscopytrappingtravel time in flame emission and absorption spectrometrytravelling examination standardTreacher Collins syndrometreadmill testtreated solution in analytical chemistrytremortriadstriazanestriazenestriboluminescencetrigeminal nervetrigeminal neuralgiatrilaminar embryotrimestertrimethylenemethanestrioxidestriple pointtriple quadrupole mass spectrometertriple screentriple-resonance nuclear magnetic resonance spectroscopytriplet statetriplet-triplet absorptiontriplet-triplet annihilationtriplet-triplet energy transfertriplet-triplet transitionstriploidytriptantrisomytrisomy 13trisomy 18trisomy 21trisomy 8tritactic polymertriterpenoidstritidetritiotritiumtritontrivial energy transfertrivial nametRNATroe expressiontrophectodermtrophoblasttropilidenestropolonestroponestropopausetropospheretropyl radicalstropylium ionstrue coincidence in radiochemistrytrue negativetrue nominal property valuetrue positivetrue value in analysistruncus arteriosustrypsin in proteomicstryptophantub conformationtube modeltube renewaltuberculintuberculin testtuberoinfundibular pathwaytubular fertility indexTucker tri-linear analysistumortumor antigentumor immunologytumor necrosis factortumor necrosis factor receptor-associated factortumor rejection antigentumor-infiltrating lymphocytetumor-specific transplantation antigentunable laser in spectrochemical analysisTunel testTung distribution of a macromolecular assemblytungsten-halogen lamptunica albugineatunica vaginalistunnellingturbidimetric titrationturbidity in light scatteringturner syndrometurnover frequency in catalysisturnstile rotationturntable reactorTversky similaritytwist formtwisted intramolecular charge transfertwo hybrid systemtwo-colour indicatortwo-dimensional chromatographytwo-dimensional correlation spectroscopytwo-dimensional correlation spectrumtwo-dimensional gel electrophoresis in proteomicstwo-dimensional nuclear magnetic resonance spectroscopytwo-generation reprotox studytwo-phase aqueous partitiontwo-photon excitationtwo-photon photochromismtwo-photon processtwo-site immunoradiometric assaytympanictype-J coil planet centrifugetype-J synchronous planetary motiontyrosine kinaseθ temperature in polymersλ-transition uubiquitinulosonic acidsultimate biodegradationultimate capacity in solvent extractionultra-shallow depth profileultrafiltrateultrafiltrationultramicroelectrodeultrasonic detector in gas chromatographyultrasonic nebulizersultrasonographyultrasoundultrasound-assisted extractionultravioletumbilical arteryumbilical cordumbilical herniaumbilical veinumpire sampleumpolungunactivated adsorption processunbiased stereologyuncertainty of measurementunconditioned stimulusunconsciousuncusunderdetermined systemunderfitting of a calibration modelundersamplingundescended testisuniaxial pressinguniaxial sampleunified atomic mass unituniform corrosionuniform dispersionuniform polymeruniformly labelleduniformly labelled tracerunimerunimolecularunimolecular dissociationunit in analytical chemistryunit of measurementunit cellunit mass resolutionunit systemunitary quantityuniversal calibration in chromatographyuniversal detector in chromatographyuniversal indicatorunmet medical needunperturbed conformational stateunperturbed dimensions in polymersunperturbed end-to-end distanceunperturbed end-to-end vectorunperturbed radius of gyrationunreactiveunresolved peak unresponsiveness in immunologyunsaturation indexunstableunstable filmunstable ion in mass spectrometryunsupervised classificationunsymmetrical filmsunweighted pair group method with arithmetic meanunzippingupconversionupfielduphill transport in membrane processesupper critical solution temperatureupper limit of measurement in atmospheric trace component analysisupper motor neuronupstream in membrane processesuptakeureidesurethanesurethraurineurogenitalurogenital sinusuronic acidsuronium saltsuseful ion yielduseful spatial resolution in secondary-ion mass spectrometryuseful spectral rangeuterine cryptuterine cycleuterotrophicuterotrophic assayuterusUV doseUV photochemical digestionUV photoelectron spectroscopyUV stabilizeruvula ρ-valuev-ionV(D)J recombinationvaccinationvaccinevacuum levelvacuum level at infinityvacuum level referencingvacuum phototubevacuum system in mass spectrometryvaginavaginal cornificationvaginal patencyvaginal smearvagus nervevalencevalence bandvalence bond theoryvalence isomervalence tautomerizationvalence transitionvalence-band spectrumvalidationvalidation setvalue of a quantityvalue of a division of a precision balance scalevan der Waals adsorptionvan der Waals broadeningvan der Waals forcesvapor depositionvaporization temperature in electrothermal atomizationvapour phase interference in analysisvariablevariable genevariable pathlength cell in spectrochemical analysisvariable regionvariancevariance scalingvariational transition state theoryvariegated star macromoleculevas deferensvascular addressinvascular cell adhesion moleculevasculitisvasculogenesisvasectomyvasoactive aminevasoconstrictionvasodilatorvasopressinvasopressorvasorelaxationvasospasmVavilov rulevectorvector scanvelocityvelocity in mass transportvelopharyngeal insufficiencyvena cavavenomventralventricleventricular septal defectventricular septumventriculo-atrial shuntventriculo-peritoneal shuntventriculogramventriculomegalyverdazyl radicalsverificationvermis in neuroanatomyvertebravertebral columnvertebratevertical ionizationvertical transitionvertigoVerwey transitionvery early activating antigenvery late activation antigenvesiclevestigialvetoingviabilityvibrational circular dichroismvibrational eigenvectorvibrational kinetic energyvibrational potential energyvibrational redistributionvibrational relaxationvibrational spectroscopyvibrational termvibrational term valuevibrationally adiabatic transition-state theoryvibronic couplingvibronic transitionVicat softening temperaturevidiconvigilancevigilance decrementvillusvimentinvinyl carbenesvinylic cationsvinylic groupsvinylidenesviologensvirial coefficientsvirilizationvirtual chemical libraryvirtual reactionvirtual screeningvirtual transitionvisceraviscoelastic spectrumviscosityviscosity functionviscosity number viscosity ratioviscosity-average molar massviscous sinteringvisibility in atmospheric chemistryvisiblevisual cortexvitellogeninvitrification in polymer scienceVogel–Fulcher–Tammann equationVoigt functionvolatilizationvolatilizervoltVolta potential differencevoltage in electroanalysisvoltage-gated ion channelvoltammetric constantvoltammetryvoltammogramvolume contentvolume flow ratevolume fractionvolume of activationvolume of distributionvolume of the stationary phase in chromatographyvolume ratevolume reaction volume reflectionvolume strainvolume thermodilatometryvolume viscosityvolume yieldvolumetric flowrate in gas chromatographyvolumetric internal standardvolumetric titrationvolumicvon Weimarn ratiovulcanizationvulvaVUVVα-Jα rearrangement w-ionWagner numberWahrhaftig diagramWalden inversionwall atomization in electrothermal atomic absorption spectroscopywall-coated open-tubular column in chromatographywall-jet electrodeWard’s minimum variance methodwarm autoantibody typewash out in atmospheric chemistrywash out timewastewaste managementwater maze testwattwave height in electrochemistrywave vector in X-ray reflectrometrywavefunctionwavelengthwavelength converterwavelength dispersion in X-ray emission spectroscopywavelength error in X-ray emission spectroscopywavelength in mediumwavelength in vacuumwavelength-dispersive X-ray fluorescence analysiswavenumberwavenumber in mediumwavenumber in vacuumweak collisionweanweatheringweb in anatomyweberwedge projectionWegener granulomatosisweightweight-distribution functionweighted least squares regressionweighted meanWeller correlationWernicke aphasiaWernicke–Korsakoff syndromeWernicke’s areaWestcott cross-sectionWestern blotWestern Pacific amyotrophic lateral sclerosis and parkinsonism–dementia complexwet bulb temperaturewet etchingwettingwetting tensionWharton’s jellywhealwheal and flareWheland intermediatewhite matterwhite pulpwhole embryo culturewideband, alternating-phase, low-power technique for residual splittingWiegner effectWien filterWigner matricesWigner rulewild typewild-type receptorWilliams syndromeWilliams–Landel–Ferry equationWilzbach labellingwind roseWisconsin card sorting testWiskott–Aldrich syndromewithdrawalwithdrawal effectWittig reagentsWntWolfgram proteinwolfram lampWood hornWood lampWoodward–Hoffmann rulesworkwork functionwork hardeningwork of adhesionwork of cohesion per unit areawork of immersional wetting per unit areawork of separationwork softeningworkflowworking electrodeworking examination standardworking intervalworking potential rangeworking range of a spectrometerworm-like chain in polymerswrap-around in comprehensive chromatographywrist drop X chromosomex unitx-ionX-linkedX-linked agammaglobulinemiaX-linked severe combined immunodeficiencyX-radiationX-ray absorption analysisX-ray absorption edgeX-ray absorption fine structure spectroscopyX-ray absorption near edge structureX-ray absorption near-edge spectroscopyX-ray analysisX-ray computed micro-tomographyX-ray diffraction analysisX-ray emission spectrumX-ray escape peakX-ray fluorescenceX-ray fluorescence analysisX-ray fluorescence microscopyX-ray intensityX-ray jump ratioX-ray levelX-ray linewidthX-ray lithographyX-ray monochromatorX-ray photoelectron spectroscopyX-ray reflectometryX-ray resistX-ray satellite in X-ray spectroscopyX-ray spectroscopyX-ray standing wavesxanthatesxanthene dyesxanthic acidsxanthophyllsxenobioticxenogeneicxenograftxenon lampxenophagyxerogelXPSxylylenesξ- Y chromosomey-ionYAGYang photocyclizationyardyearyeast artificial chromosomeyield in biotechnologyyield stressylidesynaminesynolsyoctoyolkyolk sacyolk stalkyottaYouden plotYoung's modulusYukawa–Tsuno equation Z-valueZof a photochromic systemz-average molar massz-ionZaitsev rulezebrafishZeeman effectZeeman levelszeptozero differential overlap approximationzero field splittingzero fillingzero path differencezero point of a glass electrodezero point of scale of a balancezero-field nuclear magnetic resonance spectroscopyzero-point energyzero–zero absorption or emissionzetaZeta chain-associated protein kinase-70 kDazettazig-zag projectionZimm plotzinc fingerzona pellucidazone in chromatographyzone melting method of preconcentrationzone pH in counter-current chromatographyZucker–Hammett hypothesiszwitterionic polymerzwitterionszygosiszygotezygote intrafallopian transferzymosanζ-potential
12710	https://ocw.mit.edu/courses/6-042j-mathematics-for-computer-science-fall-2010/resources/mit6_042jf10_chap10/	“mcs-ftl” — 2010/9/8 — 0:40 — page 283 — #289 10 Recurrences A recurrence describes a sequence of numbers. Early terms are specified explic-itly and later terms are expressed as a function of their predecessors. As a trivial example, this recurrence describes the sequence 1, 2, 3, etc.: T1 D 1Tn D Tn1 C 1 (for n 2): Here, the first term is defined to be 1 and each subsequent term is one more than its predecessor. Recurrences turn out to be a powerful tool. In this chapter, we’ll emphasize using recurrences to analyze the performance of recursive algorithms. However, recur-rences have other applications in computer science as well, such as enumeration of structures and analysis of random processes. And, as we saw in Section 9.4, they also arise in the analysis of problems in the physical sciences. A recurrence in isolation is not a very useful description of a sequence. One can not easily answer simple questions such as, “What is the hundredth term?” or “What is the asymptotic growth rate?” So one typically wants to solve a recurrence; that is, to find a closed-form expression for the nth term. We’ll first introduce two general solving techniques: guess-and-verify and plug-and-chug. These methods are applicable to every recurrence, but their success re-quires a flash of insight—sometimes an unrealistically brilliant flash. So we’ll also introduce two big classes of recurrences, linear and divide-and-conquer, that often come up in computer science. Essentially all recurrences in these two classes are solvable using cookbook techniques; you follow the recipe and get the answer. A drawback is that calculation replaces insight. The “Aha!” moment that is essential in the guess-and-verify and plug-and-chug methods is replaced by a “Huh” at the end of a cookbook procedure. At the end of the chapter, we’ll develop rules of thumb to help you assess many recurrences without any calculation. These rules can help you distinguish promis-ing approaches from bad ideas early in the process of designing an algorithm. Recurrences are one aspect of a broad theme in computer science: reducing a big problem to progressively smaller problems until easy base cases are reached. This same idea underlies both induction proofs and recursive algorithms. As we’ll see, all three ideas snap together nicely. For example, one might describe the running time of a recursive algorithm with a recurrence and use induction to verify the solution. 1“mcs-ftl” — 2010/9/8 — 0:40 — page 284 — #290 Chapter 10 Recurrences Figure 10.1 The initial configuration of the disks in the Towers of Hanoi problem. 10.1 The Towers of Hanoi According to legend, there is a temple in Hanoi with three posts and 64 gold disks of different sizes. Each disk has a hole through the center so that it fits on a post. In the misty past, all the disks were on the first post, with the largest on the bottom and the smallest on top, as shown in Figure 10.1. Monks in the temple have labored through the years since to move all the disks to one of the other two posts according to the following rules: The only permitted action is removing the top disk from one post and drop-ping it onto another post. A larger disk can never lie above a smaller disk on any post. So, for example, picking up the whole stack of disks at once and dropping them on another post is illegal. That’s good, because the legend says that when the monks complete the puzzle, the world will end! To clarify the problem, suppose there were only 3 gold disks instead of 64. Then the puzzle could be solved in 7 steps as shown in Figure 10.2. The questions we must answer are, “Given sufficient time, can the monks suc-ceed?” If so, “How long until the world ends?” And, most importantly, “Will this happen before the final exam?” 10.1.1 A Recursive Solution The Towers of Hanoi problem can be solved recursively. As we describe the pro-cedure, we’ll also analyze the running time. To that end, let Tn be the minimum number of steps required to solve the n-disk problem. For example, some experi-mentation shows that T1 D 1 and T2 = 3. The procedure illustrated above shows that T3 is at most 7, though there might be a solution with fewer steps. The recursive solution has three stages, which are described below and illustrated in Figure 10.3. For clarity, the largest disk is shaded in the figures. 2“mcs-ftl” — 2010/9/8 — 0:40 — page 285 — #291 10.1. The Towers of Hanoi 1234567 Figure 10.2 The 7-step solution to the Towers of Hanoi problem when there are n D 3 disks. 123 Figure 10.3 A recursive solution to the Towers of Hanoi problem. 3“mcs-ftl” — 2010/9/8 — 0:40 — page 286 — #292 Chapter 10 Recurrences Stage 1. Move the top n1 disks from the first post to the second using the solution for n 1 disks. This can be done in Tn1 steps. Stage 2. Move the largest disk from the first post to the third post. This takes just 1 step. Stage 3. Move the n 1 disks from the second post to the third post, again using the solution for n 1 disks. This can also be done in Tn1 steps. This algorithm shows that Tn, the minimum number of steps required to move n disks to a different post, is at most Tn1 C 1 C Tn1 D 2T n1 C 1. We can use this fact to upper bound the number of operations required to move towers of various heights: T3 2 T2 C 1 D 7T4 2 T3 C 1 15 Continuing in this way, we could eventually compute an upper bound on T64 , the number of steps required to move 64 disks. So this algorithm answers our first question: given sufficient time, the monks can finish their task and end the world. This is a shame. After all that effort, they’d probably want to smack a few high-fives and go out for burgers and ice cream, but nope—world’s over. 10.1.2 Finding a Recurrence We can not yet compute the exact number of steps that the monks need to move the 64 disks, only an upper bound. Perhaps, having pondered the problem since the beginning of time, the monks have devised a better algorithm. In fact, there is no better algorithm, and here is why. At some step, the monks must move the largest disk from the first post to a different post. For this to happen, the n 1 smaller disks must all be stacked out of the way on the only remaining post. Arranging the n1 smaller disks this way requires at least Tn1 moves. After the largest disk is moved, at least another Tn1 moves are required to pile the n 1 smaller disks on top. This argument shows that the number of steps required is at least 2T n1 C 1.Since we gave an algorithm using exactly that number of steps, we can now write an expression for Tn, the number of moves required to complete the Towers of Hanoi problem with n disks: T1 D 1Tn D 2T n1 C 1 (for n 2):4“mcs-ftl” — 2010/9/8 — 0:40 — page 287 — #293 10.1. The Towers of Hanoi This is a typical recurrence. These two lines define a sequence of values, T1; T 2; T 3; : : : .The first line says that the first number in the sequence, T1, is equal to 1. The sec-ond line defines every other number in the sequence in terms of its predecessor. So we can use this recurrence to compute any number of terms in the sequence: T1 D 1T2 D 2 T1 C 1 D 3T3 D 2 T2 C 1 D 7T4 D 2 T3 C 1 D 15 T5 D 2 T4 C 1 D 31 T6 D 2 T5 C 1 D 63: 10.1.3 Solving the Recurrence We could determine the number of steps to move a 64-disk tower by computing T7, T8, and so on up to T64 . But that would take a lot of work. It would be nice to have a closed-form expression for Tn, so that we could quickly find the number of steps required for any given number of disks. (For example, we might want to know how much sooner the world would end if the monks melted down one disk to purchase burgers and ice cream before the end of the world.) There are several methods for solving recurrence equations. The simplest is to guess the solution and then verify that the guess is correct with an induction proof. As a basis for a good guess, let’s look for a pattern in the values of Tn computed above: 1, 3, 7, 15, 31, 63. A natural guess is Tn D 2n 1. But whenever you guess a solution to a recurrence, you should always verify it with a proof, typically by induction. After all, your guess might be wrong. (But why bother to verify in this case? After all, if we’re wrong, its not the end of the... no, let’s check.) Claim 10.1.1. Tn D 2n 1 satisfies the recurrence: T1 D 1Tn D 2T n1 C 1 (for n 2): Proof. The proof is by induction on n. The induction hypothesis is that Tn D 2n 1. This is true for n D 1 because T1 D 1 D 21 1. Now assume that Tn1 D 2n1 1 in order to prove that Tn D 2n 1, where n 2: Tn D 2T n1 C 1 D 2.2 n1 1/ C 1 D 2n 1: 5“mcs-ftl” — 2010/9/8 — 0:40 — page 288 — #294 Chapter 10 Recurrences The first equality is the recurrence equation, the second follows from the induction assumption, and the last step is simplification. Such verification proofs are especially tidy because recurrence equations and induction proofs have analogous structures. In particular, the base case relies on the first line of the recurrence, which defines T1. And the inductive step uses the second line of the recurrence, which defines Tn as a function of preceding terms. Our guess is verified. So we can now resolve our remaining questions about the 64-disk puzzle. Since T64 D 264 1, the monks must complete more than 18 billion billion steps before the world ends. Better study for the final. 10.1.4 The Upper Bound Trap When the solution to a recurrence is complicated, one might try to prove that some simpler expression is an upper bound on the solution. For example, the exact so-lution to the Towers of Hanoi recurrence is Tn D 2n 1. Let’s try to prove the “nicer” upper bound Tn 2n, proceeding exactly as before. Proof. (Failed attempt.) The proof is by induction on n. The induction hypothesis is that Tn 2n. This is true for n D 1 because T1 D 1 21. Now assume that Tn1 2n1 in order to prove that Tn 2n, where n 2: Tn D 2T n1 C 1 2.2 n1/ C 1 6 2n Uh-oh! The first equality is the recurrence relation, the second follows from the induction hypothesis, and the third step is a flaming train wreck. The proof doesn’t work! As is so often the case with induction proofs, the ar-gument only goes through with a stronger hypothesis. This isn’t to say that upper bounding the solution to a recurrence is hopeless, but this is a situation where in-duction and recurrences do not mix well. 10.1.5 Plug and Chug Guess-and-verify is a simple and general way to solve recurrence equations. But there is one big drawback: you have to guess right . That was not hard for the Towers of Hanoi example. But sometimes the solution to a recurrence has a strange form that is quite difficult to guess. Practice helps, of course, but so can some other methods. 6“mcs-ftl” — 2010/9/8 — 0:40 — page 289 — #295 10.1. The Towers of Hanoi Plug-and-chug is another way to solve recurrences. This is also sometimes called “expansion” or “iteration”. As in guess-and-verify, the key step is identifying a pattern. But instead of looking at a sequence of numbers , you have to spot a pattern in a sequence of expressions , which is sometimes easier. The method consists of three steps, which are described below and illustrated with the Towers of Hanoi example. Step 1: Plug and Chug Until a Pattern Appears The first step is to expand the recurrence equation by alternately “plugging” (apply-ing the recurrence) and “chugging” (simplifying the result) until a pattern appears. Be careful: too much simplification can make a pattern harder to spot. The rule to remember—indeed, a rule applicable to the whole of college life—is chug in moderation . Tn D 2T n1 C 1 D 2.2T n2 C 1/ C 1 plug D 4T n2 C 2 C 1 chug D 4.2T n3 C 1/ C 2 C 1 plug D 8T n3 C 4 C 2 C 1 chug D 8.2T n4 C 1/ C 4 C 2 C 1 plug D 16T n4 C 8 C 4 C 2 C 1 chug Above, we started with the recurrence equation. Then we replaced Tn1 with 2T n2 C 1, since the recurrence says the two are equivalent. In the third step, we simplified a little—but not too much! After several similar rounds of plugging and chugging, a pattern is apparent. The following formula seems to hold: Tn D 2k Tnk C 2k1 C 2k2 C : : : C 22 C 21 C 20 D 2k Tnk C 2k 1 Once the pattern is clear, simplifying is safe and convenient. In particular, we’ve collapsed the geometric sum to a closed form on the second line. 7“mcs-ftl” — 2010/9/8 — 0:40 — page 290 — #296 Chapter 10 Recurrences Step 2: Verify the Pattern The next step is to verify the general formula with one more round of plug-and-chug. Tn D 2k Tnk C 2k 1 D 2k .2T n.k C1/ C 1/ C 2k 1 plug D 2kC1Tn.k C1/ C 2kC1 1 chug The final expression on the right is the same as the expression on the first line, except that k is replaced by k C 1. Surprisingly, this effectively proves that the formula is correct for all k. Here is why: we know the formula holds for k D 1,because that’s the original recurrence equation. And we’ve just shown that if the formula holds for some k 1, then it also holds for k C 1. So the formula holds for all k 1 by induction. Step 3: Write Tn Using Early Terms with Known Values The last step is to express Tn as a function of early terms whose values are known. Here, choosing k D n 1 expresses Tn in terms of T1, which is equal to 1. Sim-plifying gives a closed-form expression for Tn: Tn D 2n1T1 C 2n1 1 D 2n1 1 C 2n1 1 D 2n 1: We’re done! This is the same answer we got from guess-and-verify. Let’s compare guess-and-verify with plug-and-chug. In the guess-and-verify method, we computed several terms at the beginning of the sequence, T1, T2, T3,etc., until a pattern appeared. We generalized to a formula for the nth term, Tn. In contrast, plug-and-chug works backward from the nth term. Specifically, we started with an expression for Tn involving the preceding term, Tn1, and rewrote this us-ing progressively earlier terms, Tn2, Tn3, etc. Eventually, we noticed a pattern, which allowed us to express Tn using the very first term, T1, whose value we knew. Substituting this value gave a closed-form expression for Tn. So guess-and-verify and plug-and-chug tackle the problem from opposite directions. 8“mcs-ftl” — 2010/9/8 — 0:40 — page 291 — #297 10.2. Merge Sort 10.2 Merge Sort Algorithms textbooks traditionally claim that sorting is an important, fundamental problem in computer science. Then they smack you with sorting algorithms until life as a disk-stacking monk in Hanoi sounds delightful. Here, we’ll cover just one well-known sorting algorithm, Merge Sort. The analysis introduces another kind of recurrence. Here is how Merge Sort works. The input is a list of n numbers, and the output is those same numbers in nondecreasing order. There are two cases: If the input is a single number, then the algorithm does nothing, because the list is already sorted. Otherwise, the list contains two or more numbers. The first half and the second half of the list are each sorted recursively. Then the two halves are merged to form a sorted list with all n numbers. Let’s work through an example. Suppose we want to sort this list: 10, 7, 23, 5, 2, 8, 6, 9. Since there is more than one number, the first half (10, 7, 23, 5) and the second half (2, 8, 6, 9) are sorted recursively. The results are 5, 7, 10, 23 and 2, 6, 8, 9. All that remains is to merge these two lists. This is done by repeatedly emitting the smaller of the two leading terms. When one list is empty, the whole other list is emitted. The example is worked out below. In this table, underlined numbers are about to be emitted. First Half Second Half Output 5, 7, 10, 23 2, 6, 8, 9 5, 7, 10, 23 6, 8, 9 27, 10, 23 6, 8, 9 2, 5 7, 10, 23 8, 9 2, 5, 6 10, 23 8, 9 2, 5, 6, 7 10, 23 9 2, 5, 6, 7, 8 10, 23 2, 5, 6, 7, 8, 9 2, 5, 6, 7, 8, 9, 10, 23 The leading terms are initially 5 and 2. So we output 2. Then the leading terms are 5 and 6, so we output 5. Eventually, the second list becomes empty. At that point, we output the whole first list, which consists of 10 and 23. The complete output consists of all the numbers in sorted order. 9“mcs-ftl” — 2010/9/8 — 0:40 — page 292 — #298 Chapter 10 Recurrences 10.2.1 Finding a Recurrence A traditional question about sorting algorithms is, “What is the maximum number of comparisons used in sorting n items?” This is taken as an estimate of the running time. In the case of Merge Sort, we can express this quantity with a recurrence. Let Tn be the maximum number of comparisons used while Merge Sorting a list of n numbers. For now, assume that n is a power of 2. This ensures that the input can be divided in half at every stage of the recursion. If there is only one number in the list, then no comparisons are required, so T1 D 0. Otherwise, Tn includes comparisons used in sorting the first half (at most Tn=2 ), in sorting the second half (also at most Tn=2 ), and in merging the two halves. The number of comparisons in the merging step is at most n 1.This is because at least one number is emitted after each comparison and one more number is emitted at the end when one list becomes empty. Since n items are emitted in all, there can be at most n 1 comparisons. Therefore, the maximum number of comparisons needed to Merge Sort n items is given by this recurrence: T1 D 0Tn D 2T n=2 C n 1 (for n 2 and a power of 2) : This fully describes the number of comparisons, but not in a very useful way; a closed-form expression would be much more helpful. To get that, we have to solve the recurrence. 10.2.2 Solving the Recurrence Let’s first try to solve the Merge Sort recurrence with the guess-and-verify tech-nique. Here are the first few values: T1 D 0T2 D 2T 1 C 2 1 D 1T4 D 2T 2 C 4 1 D 5T8 D 2T 4 C 8 1 D 17 T16 D 2T 8 C 16 1 D 49: We’re in trouble! Guessing the solution to this recurrence is hard because there is no obvious pattern. So let’s try the plug-and-chug method instead. 10 “mcs-ftl” — 2010/9/8 — 0:40 — page 293 — #299 10.2. Merge Sort Step 1: Plug and Chug Until a Pattern Appears First, we expand the recurrence equation by alternately plugging and chugging until a pattern appears. Tn D 2T n=2 C n 1 D 2.2T n=4 C n=2 1/ C .n 1/ plug D 4T n=4 C .n 2/ C .n 1/ chug D 4.2T n=8 C n=4 1/ C .n 2/ C .n 1/ plug D 8T n=8 C .n 4/ C .n 2/ C .n 1/ chug D 8.2T n=16 C n=8 1/ C .n 4/ C .n 2/ C .n 1/ plug D 16T n=16 C .n 8/ C .n 4/ C .n 2/ C .n 1/ chug A pattern is emerging. In particular, this formula seems holds: Tn D 2k Tn=2 k C .n 2k1/ C .n 2k2/ C : : : C .n 20/ D 2k Tn=2 k C k n 2k1 2k2 : : : 20 D 2k Tn=2 k C k n 2k C 1: On the second line, we grouped the n terms and powers of 2. On the third, we collapsed the geometric sum. Step 2: Verify the Pattern Next, we verify the pattern with one additional round of plug-and-chug. If we guessed the wrong pattern, then this is where we’ll discover the mistake. Tn D 2k Tn=2 k C k n 2k C 1 D 2k .2T n=2 kC1 C n=2 k 1/ C k n 2k C 1 plug D 2kC1Tn=2 kC1 C .k C 1/n 2kC1 C 1 chug The formula is unchanged except that k is replaced by k C 1. This amounts to the induction step in a proof that the formula holds for all k 1.11 “mcs-ftl” — 2010/9/8 — 0:40 — page 294 — #300 Chapter 10 Recurrences Step 3: Write Tn Using Early Terms with Known Values Finally, we express Tn using early terms whose values are known. Specifically, if we let k D log n, then Tn=2 k D T1, which we know is 0: Tn D 2k Tn=2 k C k n 2k C 1 D 2log nTn=2 log n C n log n 2log n C 1 D nT 1 C n log n n C 1 D n log n n C 1: We’re done! We have a closed-form expression for the maximum number of com-parisons used in Merge Sorting a list of n numbers. In retrospect, it is easy to see why guess-and-verify failed: this formula is fairly complicated. As a check, we can confirm that this formula gives the same values that we computed earlier: n Tn n log n n C 11 0 1 log 1 1 C 1 D 02 1 2 log 2 2 C 1 D 14 5 4 log 4 4 C 1 D 58 17 8 log 8 8 C 1 D 17 16 49 16 log 16 16 C 1 D 49 As a double-check, we could write out an explicit induction proof. This would be straightforward, because we already worked out the guts of the proof in step 2 of the plug-and-chug procedure. 10.3 Linear Recurrences So far we’ve solved recurrences with two techniques: guess-and-verify and plug-and-chug. These methods require spotting a pattern in a sequence of numbers or expressions. In this section and the next, we’ll give cookbook solutions for two large classes of recurrences. These methods require no flash of insight; you just follow the recipe and get the answer. 10.3.1 Climbing Stairs How many different ways are there to climb n stairs, if you can either step up one stair or hop up two? For example, there are five different ways to climb four stairs: 1. step, step, step, step 12 “mcs-ftl” — 2010/9/8 — 0:40 — page 295 — #301 10.3. Linear Recurrences hop, hop 3. hop, step, step 4. step, hop step 5. step, step, hop Working through this problem will demonstrate the major features of our first cook-book method for solving recurrences. We’ll fill in the details of the general solution afterward. Finding a Recurrence As special cases, there is 1 way to climb 0 stairs (do nothing) and 1 way to climb 1 stair (step up). In general, an ascent of n stairs consists of either a step followed by an ascent of the remaining n 1 stairs or a hop followed by an ascent of n 2 stairs. So the total number of ways to climb n stairs is equal to the number of ways to climb n 1 plus the number of ways to climb n 2. These observations define a recurrence: f .0/ D 1f .1/ D 1f .n/ D f .n 1/ C f .n 2/ for n 2: Here, f .n/ denotes the number of ways to climb n stairs. Also, we’ve switched from subscript notation to functional notation, from Tn to fn. Here the change is cosmetic, but the expressiveness of functions will be useful later. This is the Fibonacci recurrence, the most famous of all recurrence equations. Fibonacci numbers arise in all sorts of applications and in nature. Fibonacci intro-duced the numbers in 1202 to study rabbit reproduction. Fibonacci numbers also appear, oddly enough, in the spiral patterns on the faces of sunflowers. And the input numbers that make Euclid’s GCD algorithm require the greatest number of steps are consecutive Fibonacci numbers. Solving the Recurrence The Fibonacci recurrence belongs to the class of linear recurrences, which are es-sentially all solvable with a technique that you can learn in an hour. This is some-what amazing, since the Fibonacci recurrence remained unsolved for almost six centuries! In general, a homogeneous linear recurrence has the form f .n/ D a1f .n 1/ C a2f .n 2/ C : : : C ad f .n d / 13 “mcs-ftl” — 2010/9/8 — 0:40 — page 296 — #302 Chapter 10 Recurrences where a1; a 2; : : : ; a d and d are constants. The order of the recurrence is d . Com-monly, the value of the function f is also specified at a few points; these are called boundary conditions . For example, the Fibonacci recurrence has order d D 2 with coefficients a1 D a2 D 1 and g.n/ D 0. The boundary conditions are f .0/ D 1 and f .1/ D 1. The word “homogeneous” sounds scary, but effectively means “the simpler kind”. We’ll consider linear recurrences with a more complicated form later. Let’s try to solve the Fibonacci recurrence with the benefit centuries of hindsight. In general, linear recurrences tend to have exponential solutions. So let’s guess that f .n/ D xn where x is a parameter introduced to improve our odds of making a correct guess. We’ll figure out the best value for x later. To further improve our odds, let’s neglect the boundary conditions, f .0/ D 0 and f .1/ D 1, for now. Plugging this guess into the recurrence f .n/ D f .n 1/ C f .n 2/ gives xn D xn1 C xn2: Dividing both sides by xn2 leaves a quadratic equation: x2 D x C 1: Solving this equation gives two plausible values for the parameter x: x D 1 ˙ p52 : This suggests that there are at least two different solutions to the recurrence, ne-glecting the boundary conditions. f .n/ D 1 C p52 !n or f .n/ D 1 p52 !n A charming features of homogeneous linear recurrences is that any linear com-bination of solutions is another solution. Theorem 10.3.1. If f .n/ and g.n/ are both solutions to a homogeneous linear recurrence, then h.n/ D sf .n/ C tg.n/ is also a solution for all s; t 2 R.Proof. h.n/ D sf .n/ C tg.n/ D s .a 1f .n 1/ C : : : C ad f .n d // C t .a 1g.n 1/ C : : : C ad g.n d // D a1.sf .n 1/ C tg.n 1// C : : : C ad .sf .n d / C tg.n d // D a1h.n 1/ C : : : C ad h.n d / 14 “mcs-ftl” — 2010/9/8 — 0:40 — page 297 — #303 10.3. Linear Recurrences The first step uses the definition of the function h, and the second uses the fact that f and g are solutions to the recurrence. In the last two steps, we rearrange terms and use the definition of h again. Since the first expression is equal to the last, h is also a solution to the recurrence. The phenomenon described in this theorem—a linear combination of solutions is another solution—also holds for many differential equations and physical systems. In fact, linear recurrences are so similar to linear differential equations that you can safely snooze through that topic in some future math class. Returning to the Fibonacci recurrence, this theorem implies that f .n/ D s 1 C p52 !n C t 1 p52 !n is a solution for all real numbers s and t. The theorem expanded two solutions to a whole spectrum of possibilities! Now, given all these options to choose from, we can find one solution that satisfies the boundary conditions, f .0/ D 1 and f .1/ D 1. Each boundary condition puts some constraints on the parameters s and t. In particular, the first boundary condition implies that f .0/ D s 1 C p52 !0 C t 1 p52 !0 D s C t D 1: Similarly, the second boundary condition implies that f .1/ D s 1 C p52 !1 C t 1 p52 !1 D 1: Now we have two linear equations in two unknowns. The system is not degenerate, so there is a unique solution: s D 1 p5 1 C p52 t D 1 p5 1 p52 : These values of s and t identify a solution to the Fibonacci recurrence that also satisfies the boundary conditions: f .n/ D 1 p5 1 C p52 1 C p52 !n 1 p5 1 p52 1 p52 !n D 1 p5 1 C p52 !nC1 1 p5 1 p52 !nC1 :15 “mcs-ftl” — 2010/9/8 — 0:40 — page 298 — #304 Chapter 10 Recurrences It is easy to see why no one stumbled across this solution for almost six centuries! All Fibonacci numbers are integers, but this expression is full of square roots of five! Amazingly, the square roots always cancel out. This expression really does give the Fibonacci numbers if we plug in n D 0; 1; 2 , etc. This closed-form for Fibonacci numbers has some interesting corollaries. The first term tends to infinity because the base of the exponential, .1 C p5/=2 D 1:618 : : : is greater than one. This value is often denoted and called the “golden ratio”. The second term tends to zero, because .1 p5/=2 D 0:618033988 : : : has absolute value less than 1. This implies that the nth Fibonacci number is: f .n/ D nC1 p5 C o.1/: Remarkably, this expression involving irrational numbers is actually very close to an integer for all large n—namely, a Fibonacci number! For example: 20 p5 D 6765:000029 : : : f .19/: This also implies that the ratio of consecutive Fibonacci numbers rapidly approaches the golden ratio. For example: f .20/ f .19/ D 10946 6765 D 1:618033998 : : : : 10.3.2 Solving Homogeneous Linear Recurrences The method we used to solve the Fibonacci recurrence can be extended to solve any homogeneous linear recurrence; that is, a recurrence of the form f .n/ D a1f .n 1/ C a2f .n 2/ C : : : C ad f .n d / where a1; a 2; : : : ; a d and d are constants. Substituting the guess f .n/ D xn, as with the Fibonacci recurrence, gives xn D a1xn1 C a2xn2 C : : : C ad xnd : Dividing by xnd gives xd D a1xd 1 C a2xd 2 C : : : C ad 1x C ad : This is called the characteristic equation of the recurrence. The characteristic equa-tion can be read off quickly since the coefficients of the equation are the same as the coefficients of the recurrence. The solutions to a linear recurrence are defined by the roots of the characteristic equation. Neglecting boundary conditions for the moment: 16 “mcs-ftl” — 2010/9/8 — 0:40 — page 299 — #305 10.3. Linear Recurrences If r is a nonrepeated root of the characteristic equation, then rn is a solution to the recurrence. If r is a repeated root with multiplicity k then rn, nr n, n2rn, . . . , nk1rn are all solutions to the recurrence. Theorem 10.3.1 implies that every linear combination of these solutions is also a solution. For example, suppose that the characteristic equation of a recurrence has roots s, t, and u twice. These four roots imply four distinct solutions: f .n/ D sn f .n/ D tn f .n/ D un f .n/ D nu n: Furthermore, every linear combination f .n/ D a sn C b tn C c un C d nu n (10.1) is also a solution. All that remains is to select a solution consistent with the boundary conditions by choosing the constants appropriately. Each boundary condition implies a linear equation involving these constants. So we can determine the constants by solving a system of linear equations. For example, suppose our boundary conditions were f .0/ D 0, f .1/ D 1, f .2/ D 4, and f .3/ D 9. Then we would obtain four equations in four unknowns: f .0/ D 0 ) a s0 C b t0 C c u0 C d 0u 0 D 0f .1/ D 1 ) a s1 C b t1 C c u1 C d 1u 1 D 1f .2/ D 4 ) a s2 C b t2 C c u2 C d 2u 2 D 4f .3/ D 9 ) a s3 C b t3 C c u3 C d 3u 3 D 9 This looks nasty, but remember that s, t, and u are just constants. Solving this sys-tem gives values for a, b, c, and d that define a solution to the recurrence consistent with the boundary conditions. 10.3.3 Solving General Linear Recurrences We can now solve all linear homogeneous recurrences, which have the form f .n/ D a1f .n 1/ C a2f .n 2/ C : : : C ad f .n d /: Many recurrences that arise in practice do not quite fit this mold. For example, the Towers of Hanoi problem led to this recurrence: f .1/ D 1f .n/ D 2f .n 1/ C 1 (for n 2):17 “mcs-ftl” — 2010/9/8 — 0:40 — page 300 — #306 Chapter 10 Recurrences The problem is the extra C1; that is not allowed in a homogeneous linear recur-rence. In general, adding an extra function g.n/ to the right side of a linear recur-rence gives an inhomogeneous linear recurrence : f .n/ D a1f .n 1/ C a2f .n 2/ C : : : C ad f .n d / C g.n/: Solving inhomogeneous linear recurrences is neither very different nor very dif-ficult. We can divide the whole job into five steps: 1. Replace g.n/ by 0, leaving a homogeneous recurrence. As before, find roots of the characteristic equation. 2. Write down the solution to the homogeneous recurrence, but do not yet use the boundary conditions to determine coefficients. This is called the homo-geneous solution .3. Now restore g.n/ and find a single solution to the recurrence, ignoring bound-ary conditions. This is called a particular solution . We’ll explain how to find a particular solution shortly. 4. Add the homogeneous and particular solutions together to obtain the general solution .5. Now use the boundary conditions to determine constants by the usual method of generating and solving a system of linear equations. As an example, let’s consider a variation of the Towers of Hanoi problem. Sup-pose that moving a disk takes time proportional to its size. Specifically, moving the smallest disk takes 1 second, the next-smallest takes 2 seconds, and moving the nth disk then requires n seconds instead of 1. So, in this variation, the time to complete the job is given by a recurrence with a Cn term instead of a C1: f .1/ D 1f .n/ D 2f .n 1/ C n for n 2: Clearly, this will take longer, but how much longer? Let’s solve the recurrence with the method described above. In Steps 1 and 2, dropping the Cn leaves the homogeneous recurrence f .n/ D 2f .n 1/ . The characteristic equation is x D 2. So the homogeneous solution is f .n/ D c2 n.In Step 3, we must find a solution to the full recurrence f .n/ D 2f .n 1/ C n,without regard to the boundary condition. Let’s guess that there is a solution of the 18 “mcs-ftl” — 2010/9/8 — 0:40 — page 301 — #307 10.3. Linear Recurrences form f .n/ D an C b for some constants a and b. Substituting this guess into the recurrence gives an C b D 2.a.n 1/ C b/ C n0 D .a C 1/n C .b 2a/: The second equation is a simplification of the first. The second equation holds for all n if both a C 1 D 0 (which implies a D 1) and b 2a D 0 (which implies that b D 2). So f .n/ D an C b D n 2 is a particular solution. In the Step 4, we add the homogeneous and particular solutions to obtain the general solution f .n/ D c2 n n 2: Finally, in step 5, we use the boundary condition, f .1/ D 1, determine the value of the constant c: f .1/ D 1 ) c2 1 1 2 D 1 ) c D 2: Therefore, the function f .n/ D 2 2n n 2 solves this variant of the Towers of Hanoi recurrence. For comparison, the solution to the original Towers of Hanoi problem was 2n 1. So if moving disks takes time proportional to their size, then the monks will need about twice as much time to solve the whole puzzle. 10.3.4 How to Guess a Particular Solution Finding a particular solution can be the hardest part of solving inhomogeneous recurrences. This involves guessing, and you might guess wrong. 1 However, some rules of thumb make this job fairly easy most of the time. Generally, look for a particular solution with the same form as the inhomo-geneous term g.n/ . If g.n/ is a constant, then guess a particular solution f .n/ D c. If this doesn’t work, try polynomials of progressively higher degree: f .n/ D bn C c, then f .n/ D an 2 C bn C c, etc. More generally, if g.n/ is a polynomial, try a polynomial of the same degree, then a polynomial of degree one higher, then two higher, etc. For example, if g.n/ D 6n C 5, then try f .n/ D bn C c and then f .n/ D an 2 C bn C c. 1In Chapter 12, we will show how to solve linear recurrences with generating functions—it’s a little more complicated, but it does not require guessing. 19 “mcs-ftl” — 2010/9/8 — 0:40 — page 302 — #308 Chapter 10 Recurrences If g.n/ is an exponential, such as 3n, then first guess that f .n/ D c3 n.Failing that, try f .n/ D bn3 n C c3 n and then an 23n C bn3 n C c3 n, etc. The entire process is summarized on the following page. 10.4 Divide-and-Conquer Recurrences We now have a recipe for solving general linear recurrences. But the Merge Sort recurrence, which we encountered earlier, is not linear: T .1/ D 0T .n/ D 2T .n=2/ C n 1 (for n 2): In particular, T .n/ is not a linear combination of a fixed number of immediately preceding terms; rather, T .n/ is a function of T .n=2/ , a term halfway back in the sequence. Merge Sort is an example of a divide-and-conquer algorithm: it divides the in-put, “conquers” the pieces, and combines the results. Analysis of such algorithms commonly leads to divide-and-conquer recurrences, which have this form: T .n/ D k X iD1 ai T .b i n/ C g.n/ Here a1; : : : a k are positive constants, b1; : : : ; b k are constants between 0 and 1, and g.n/ is a nonnegative function. For example, setting a1 D 2, b1 D 1=2 , and g.n/ D n 1 gives the Merge Sort recurrence. 10.4.1 The Akra-Bazzi Formula The solution to virtually all divide and conquer solutions is given by the amazing Akra-Bazzi formula . Quite simply, the asymptotic solution to the general divide-and-conquer recurrence T .n/ D k X iD1 ai T .b i n/ C g.n/ is T .n/ D ‚ np 1 C Z n1 g.u/ upC1 du (10.2) 20 “mcs-ftl” — 2010/9/8 — 0:40 — page 303 — #309 10.4. Divide-and-Conquer Recurrences Short Guide to Solving Linear Recurrences A linear recurrence is an equation f .n/ D a1f .n 1/ C a2f .n 2/ C : : : C ad f .n d / „ ƒ‚ … homogeneous part C g.n/ „ ƒ‚ … inhomogeneous part together with boundary conditions such as f .0/ D b0, f .1/ D b1, etc. Linear recurrences are solved as follows: 1. Find the roots of the characteristic equation xn D a1xn1 C a2xn2 C : : : C ak1x C ak : Write down the homogeneous solution. Each root generates one term and the homogeneous solution is their sum. A nonrepeated root r generates the term cr n, where c is a constant to be determined later. A root r with multi-plicity k generates the terms d1rn d2nr n d3n2rn : : : dk nk1rn where d1; : : : d k are constants to be determined later. 3. Find a particular solution. This is a solution to the full recurrence that need not be consistent with the boundary conditions. Use guess-and-verify. If g.n/ is a constant or a polynomial, try a polynomial of the same degree, then of one higher degree, then two higher. For example, if g.n/ D n, then try f .n/ D bn C c and then an 2 C bn C c. If g.n/ is an exponential, such as 3n,then first guess f .n/ D c3 n. Failing that, try f .n/ D .bn C c/3 n and then .an 2 C bn C c/3 n, etc. 4. Form the general solution, which is the sum of the homogeneous solution and the particular solution. Here is a typical general solution: f .n/ D c2 n C d. 1/ n „ ƒ‚ … homogeneous solution C 3n C 1.„ ƒ‚ … inhomogeneous solution Substitute the boundary conditions into the general solution. Each boundary condition gives a linear equation in the unknown constants. For example, substituting f .1/ D 2 into the general solution above gives 2 D c 21 C d .1/ 1 C 3 1 C 1 ) 2 D 2c d: Determine the values of these constants by solving the resulting system of linear equations. 21 “mcs-ftl” — 2010/9/8 — 0:40 — page 304 — #310 Chapter 10 Recurrences where p satisfies kX iD1 ai bpi D 1: (10.3) A rarely-troublesome requirement is that the function g.n/ must not grow or oscillate too quickly. Specifically, jg0.n/ j must be bounded by some polynomial. So, for example, the Akra-Bazzi formula is valid when g.n/ D x2 log n, but not when g.n/ D 2n.Let’s solve the Merge Sort recurrence again, using the Akra-Bazzi formula in-stead of plug-and-chug. First, we find the value p that satisfies 2 .1=2/ p D 1: Looks like p D 1 does the job. Then we compute the integral: T .n/ D ‚ n 1 C Z n1 u 1u2 du D ‚ n 1 C log u C 1u n1 D ‚ n log n C 1n D ‚ .n log n/ : The first step is integration and the second is simplification. We can drop the 1=n term in the last step, because the log n term dominates. We’re done! Let’s try a scary-looking recurrence: T .n/ D 2T .n=2/ C 8=9T .3n=4/ C n2: Here, a1 D 2, b1 D 1=2 , a2 D 8=9 , and b2 D 3=4 . So we find the value p that satisfies 2 .1=2/ p C .8=9/.3=4/ p D 1: Equations of this form don’t always have closed-form solutions, so you may need to approximate p numerically sometimes. But in this case the solution is simple: p D 2. Then we integrate: T .n/ D ‚ n2 1 C Z n1 u2 u3 du D ‚ n2.1 C log n/ D ‚ n2 log n : That was easy! 22 “mcs-ftl” — 2010/9/8 — 0:40 — page 305 — #311 10.4. Divide-and-Conquer Recurrences 10.4.2 Two Technical Issues Until now, we’ve swept a couple issues related to divide-and-conquer recurrences under the rug. Let’s address those issues now. First, the Akra-Bazzi formula makes no use of boundary conditions. To see why, let’s go back to Merge Sort. During the plug-and-chug analysis, we found that Tn D nT 1 C n log n n C 1: This expresses the nth term as a function of the first term, whose value is specified in a boundary condition. But notice that Tn D ‚.n log n/ for every value of T1.The boundary condition doesn’t matter! This is the typical situation: the asymptotic solution to a divide-and-conquer recurrence is independent of the boundary conditions . Intuitively, if the bottom-level operation in a recursive algorithm takes, say, twice as long, then the overall running time will at most double. This matters in practice, but the factor of 2 is concealed by asymptotic notation. There are corner-case exceptions. For example, the solution to T .n/ D 2T .n=2/ is either ‚.n/ or zero, depending on whether T .1/ is zero. These cases are of little practical interest, so we won’t consider them further. There is a second nagging issue with divide-and-conquer recurrences that does not arise with linear recurrences. Specifically, dividing a problem of size n may create subproblems of non-integer size. For example, the Merge Sort recurrence contains the term T .n=2/ . So what if n is 15? How long does it take to sort seven-and-a-half items? Previously, we dodged this issue by analyzing Merge Sort only when the size of the input was a power of 2. But then we don’t know what happens for an input of size, say, 100. Of course, a practical implementation of Merge Sort would split the input ap-proximately in half, sort the halves recursively, and merge the results. For example, a list of 15 numbers would be split into lists of 7 and 8. More generally, a list of n numbers would be split into approximate halves of size dn=2 e and bn=2 c. So the maximum number of comparisons is actually given by this recurrence: T .1/ D 0T .n/ D T . dn=2 e/ C T . bn=2 c/ C n 1 (for n 2): This may be rigorously correct, but the ceiling and floor operations make the recur-rence hard to solve exactly. Fortunately, the asymptotic solution to a divide and conquer recurrence is un-affected by floors and ceilings . More precisely, the solution is not changed by replacing a term T .b i n/ with either T . dbi ne/ or T . bbi nc/. So leaving floors and 23 “mcs-ftl” — 2010/9/8 — 0:40 — page 306 — #312 Chapter 10 Recurrences ceilings out of divide-and-conquer recurrences makes sense in many contexts; those are complications that make no difference. 10.4.3 The Akra-Bazzi Theorem The Akra-Bazzi formula together with our assertions about boundary conditions and integrality all follow from the Akra-Bazzi Theorem , which is stated below. Theorem 10.4.1 (Akra-Bazzi) . Suppose that the function T W R ! R satisfies the recurrence T .x/ D 8ˆ<ˆ: is nonnegative and bounded for 0 x x0kP iD1 ai T .b i x C hi .x// C g.x/ for x > x 0 where: 1. a1; : : : ; a k are positive constants. 2. b1; : : : ; b k are constants between 0 and 1. 3. x0 is large enough so that T is well-defined. 4. g.x/ is a nonnegative function such that jg0.x/ j is bounded by a polynomial. 5. jhi .x/ j D O.x= log 2 x/ .Then T .x/ D ‚ xp 1 C Z x1 g.u/ upC1 du where p satisfies kX iD1 ai bpi D 1: The Akra-Bazzi theorem can be proved using a complicated induction argument, though we won’t do that here. But let’s at least go over the statement of the theorem. All the recurrences we’ve considered were defined over the integers, and that is the common case. But the Akra-Bazzi theorem applies more generally to functions defined over the real numbers. The Akra-Bazzi formula is lifted directed from the theorem statement, except that the recurrence in the theorem includes extra functions, hi . These functions 24 “mcs-ftl” — 2010/9/8 — 0:40 — page 307 — #313 10.4. Divide-and-Conquer Recurrences extend the theorem to address floors, ceilings, and other small adjustments to the sizes of subproblems. The trick is illustrated by this combination of parameters a1 D 1 b1 D 1=2 h1.x/ D l x2 m x2a2 D 1 b2 D 1=2 h2.x/ D j x2 k x2g.x/ D x 1 which corresponds the recurrence T .x/ D 1 T x2 C l x2 m x2 C T x2 C j x2 k x2 C x 1 D T l x2 m C T j x2 k C x 1: This is the rigorously correct Merge Sort recurrence valid for all input sizes, complete with floor and ceiling operators. In this case, the functions h1.x/ and h2.x/ are both at most 1, which is easily O.x= log 2 x/ as required by the theorem statement. These functions hi do not affect—or even appear in—the asymptotic solution to the recurrence. This justifies our earlier claim that applying floor and ceiling operators to the size of a subproblem does not alter the asymptotic solution to a divide-and-conquer recurrence. 10.4.4 The Master Theorem There is a special case of the Akra-Bazzi formula known as the Master Theorem that handles some of the recurrences that commonly arise in computer science. It is called the Master Theorem because it was proved long before Akra and Bazzi arrived on the scene and, for many years, it was the final word on solving divide-and-conquer recurrences. We include the Master Theorem here because it is still widely referenced in algorithms courses and you can use it without having to know anything about integration. Theorem 10.4.2 (Master Theorem) . Let T be a recurrence of the form T .n/ D aT nb C g.n/: Case 1: If g.n/ D O nlog b .a/ for some constant > 0 , then T .n/ D ‚ nlog b .a/ :25 “mcs-ftl” — 2010/9/8 — 0:40 — page 308 — #314 Chapter 10 Recurrences Case 2: If g.n/ D ‚ nlog b .a/ log k .n/ for some constant k 0, then T .n/ D ‚ nlog b .a/ log kC1.n/ : Case 3: If g.n/ D nlog b .a/ C for some constant > 0 and ag.n=b/ < cg.n/ for some constant c < 1 and sufficiently large n, then T .n/ D ‚.g.n//: The Master Theorem can be proved by induction on n or, more easily, as a corol-lary of Theorem 10.4.1. We will not include the details here. 10.4.5 Pitfalls with Asymptotic Notation and Induction We’ve seen that asymptotic notation is quite useful, particularly in connection with recurrences. And induction is our favorite proof technique. But mixing the two is risky business; there is great potential for subtle errors and false conclusions! False Claim. If T .1/ D 1 and T .n/ D 2T .n=2/ C n; then T .n/ D O.n/ . The Akra-Bazzi theorem implies that the correct solution is T .n/ D ‚.n log n/ and so this claim is false. But where does the following “proof” go astray? Bogus proof. The proof is by strong induction. Let P .n/ be the proposition that T .n/ D O.n/ . Base case : P .1/ is true because T .1/ D 1 D O.1/ . Inductive step : For n 2, assume P .1/ , P .2/ , . . . , P .n 1/ to prove P .n/ . We have T .n/ D 2 T .n=2/ C n D 2 O.n=2/ C n D O.n/: The first equation is the recurrence, the second uses the assumption P .n=2/ , and the third is a simplification. 26 “mcs-ftl” — 2010/9/8 — 0:40 — page 309 — #315 10.5. A Feel for Recurrences Where’s the bug? The proof is already far off the mark in the second sentence, which defines the induction hypothesis. The statement “ T .n/ D O.n/ ” is either true or false; it’s validity does not depend on a particular value of n. Thus the very idea of trying to prove that the statement holds for n D 1, 2, . . . , is wrong-headed. The safe way to reason inductively about asymptotic phenomena is to work di-rectly with the definition of the asymptotic notation . Let’s try to prove the claim above in this way. Remember that f .n/ D O.n/ means that there exist constants n0 and c > 0 such that jf .n/ j cn for all n n0. (Let’s not worry about the absolute value for now.) If all goes well, the proof attempt should fail in some blatantly obvious way, instead of in a subtle, hard-to-detect way like the earlier ar-gument. Since our perverse goal is to demonstrate that the proof won’t work for any constants n0 and c, we’ll leave these as variables and assume only that they’re chosen so that the base case holds; that is, T .n 0/ cn . Proof Attempt. We use strong induction. Let P .n/ be the proposition that T .n/ cn . Base case : P .n 0/ is true, because T .n 0/ cn . Inductive step : For n > n 0, assume that P .n 0/, . . . , P .n 1/ are true in order to prove P .n/ . We reason as follows: T .n/ D 2T .n=2/ C n 2c.n=2/ C n D cn C n D .c C 1/n — cn: The first equation is the recurrence. Then we use induction and simplify until the argument collapses! In general, it is a good idea to stay away from asymptotic notation altogether while you are doing the induction. Once the induction is over and done with, then you can safely use big-Oh to simplify your result. 10.5 A Feel for Recurrences We’ve guessed and verified, plugged and chugged, found roots, computed integrals, and solved linear systems and exponential equations. Now let’s step back and look for some rules of thumb. What kinds of recurrences have what sorts of solutions? 27 “mcs-ftl” — 2010/9/8 — 0:40 — page 310 — #316 Chapter 10 Recurrences Here are some recurrences we solved earlier: Recurrence Solution Towers of Hanoi Tn D 2T n1 C 1 Tn 2n Merge Sort Tn D 2T n=2 C n 1 Tn n log n Hanoi variation Tn D 2T n1 C n Tn 2 2n Fibonacci Tn D Tn1 C Tn2 Tn .1:618 : : :/ nC1=p5 Notice that the recurrence equations for Towers of Hanoi and Merge Sort are some-what similar, but the solutions are radically different. Merge Sorting n D 64 items takes a few hundred comparisons, while moving n D 64 disks takes more than 10 19 steps! Each recurrence has one strength and one weakness. In the Towers of Hanoi, we broke a problem of size n into two subproblem of size n 1 (which is large), but needed only 1 additional step (which is small). In Merge Sort, we divided the problem of size n into two subproblems of size n=2 (which is small), but needed .n 1/ additional steps (which is large). Yet, Merge Sort is faster by a mile! This suggests that generating smaller subproblems is far more important to al-gorithmic speed than reducing the additional steps per recursive call . For example, shifting to the variation of Towers of Hanoi increased the last term from C1 to Cn,but the solution only doubled. And one of the two subproblems in the Fibonacci recurrence is just slightly smaller than in Towers of Hanoi (size n 2 instead of n1). Yet the solution is exponentially smaller! More generally, linear recurrences (which have big subproblems) typically have exponential solutions, while divide-and-conquer recurrences (which have small subproblems) usually have solutions bounded above by a polynomial. All the examples listed above break a problem of size n into two smaller prob-lems. How does the number of subproblems affect the solution? For example, suppose we increased the number of subproblems in Towers of Hanoi from 2 to 3, giving this recurrence: Tn D 3T n1 C 1 This increases the root of the characteristic equation from 2 to 3, which raises the solution exponentially, from ‚.2 n/ to ‚.3 n/.Divide-and-conquer recurrences are also sensitive to the number of subproblems. For example, for this generalization of the Merge Sort recurrence: T1 D 0Tn D aT n=2 C n 1: 28 “mcs-ftl” — 2010/9/8 — 0:40 — page 311 — #317 10.5. A Feel for Recurrences the Akra-Bazzi formula gives: Tn D 8ˆ<ˆ: ‚.n/ for a < 2 ‚.n log n/ for a D 2‚.n log a/ for a > 2: So the solution takes on three completely different forms as a goes from 1.99 to 2.01! How do boundary conditions affect the solution to a recurrence? We’ve seen that they are almost irrelevant for divide-and-conquer recurrences. For linear re-currences, the solution is usually dominated by an exponential whose base is de-termined by the number and size of subproblems. Boundary conditions matter greatly only when they give the dominant term a zero coefficient, which changes the asymptotic solution. So now we have a rule of thumb! The performance of a recursive procedure is usually dictated by the size and number of subproblems, rather than the amount of work per recursive call or time spent at the base of the recursion. In particular, if subproblems are smaller than the original by an additive factor, the solution is most often exponential. But if the subproblems are only a fraction the size of the original, then the solution is typically bounded by a polynomial. 29 “mcs-ftl” — 2010/9/8 — 0:40 — page 312 — #318 30 MIT OpenCourseWare 6.042J / 18.062J Mathematics for Computer Science Fall 2010 For information about citing these materials or our Terms of Use, visit:
12711	https://allen.in/jee/chemistry/d-block-elements	HomeJEE Chemistryd block elements d-Block Elements The d-block elements are part of the periodic table and are also known as transition metals. They span groups 3 to 12 and are characterized by the filling of inner d orbitals as electrons are added across the period. What are d-Block Elements? The d block elements in the periodic table are a special group of metals found in the middle part of the periodic table. They have electrons arranged in a certain way called the d-orbitals, giving them some unique qualities. These metals can change how many electrons they have, which helps them do different kinds of chemistry. They make colorful compounds because they can grab and release light. The d-block, residing at the heart of the periodic table, unfolds into four distinct series aligned with the filling of the 3d, 4d, 5d, or 6d orbitals. Each series reveals how electrons organize within orbitals, providing vital information about how transition metals behave and their distinct properties. 3d- Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn 4d- Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd, Ag, Cd 5d- La, Hf, Ta, W, Re, Os, Ir, Pt, Au, Hg 6d- incomplete. What defines these elements is their electron configuration. As electrons populate the d orbitals within these elements, they exhibit a diverse range of features, making them indispensable in various facets of our lives. Let’s discuss several common Chemical and physical properties of d block elements, such as: 1.0Electronic Configuration of d Block Elements d Block elements names and electronic configuration of d block elements (A) General Electronic Configurations-(n – 1)d1–10 ns1–2 The (n–1) stands for the inner d orbitals which may have one to ten electrons. The (n) stands for the outermost s orbital which may have one or two electrons. 3d Series: The Start of Transition Metals This series introduces the first set of transition metals found in the fourth row of the periodic table. Metals like scandium (Sc), titanium (Ti), and zinc (Zn) belong to this group, showcasing their unique ways of arranging electrons. 4d Series: Moving Beyond the Fourth Row Progressing further, the 4d series extends our exploration into transition metals beyond the fourth row. Elements like yttrium (Y), zirconium (Zr), and palladium (Pd) are part of this group, revealing new characteristics and trends as we continue discovering. 5d Series: Advancing Deeper into the Table Continuing our journey, the 5d series marks the filling of the next set of orbitals. Elements like hafnium (Hf), tantalum (Ta), and platinum (Pt) represent this stage, displaying unique properties and behaviors as we delve deeper into the periodic table. 6d Series: Beyond the Main Section The 6d series encompasses elements situated beyond the main body of the periodic table. Elements like rutherfordium (Rf), dubnium (Db), and seaborgium (Sg) belong to this group, showcasing rare and distinct properties in this less commonly encountered orbital configuration. (B) Exceptions- (a) Above generalization has several exceptions because (i) The energy difference between (n – 1)d and ns orbitals is very less. (ii) Half and completely filled sets of orbitals are relatively more stable. (b) A consequence of this factor is reflected in the electronic configuration of following elements. (i) In 3d series — Cr and Cu (ii) In 4d series — Nb, Mo, Tc, Ru, Rh, Pd, and Ag (iii) In 5d series — Pt and Au (iv) In 6d series — Rg (C) Electronic configuration of all d-block elements. (a) 3d-Series (First Transition Series) \| \| \| \| \| --- --- \| \| Atomic No. \| Element \| Symbol \| Electron Configuration \| \| 21 \| Scandium \| Sc \| [Ar]3d14s2 \| \| 22 \| Titanium \| Ti \| [Ar]3d24s2 \| \| 23 \| Vanadium \| V \| [Ar]3d34s2 \| \| 24 \| Chromium \| Cr \| [Ar]3d54s1 \| \| 25 \| Manganese \| Mn \| [Ar]3d54s2 \| \| 26 \| Iron \| Fe \| [Ar]3d64s2 \| \| 27 \| Cobalt \| Co \| [Ar]3d74s2 \| \| 28 \| Nickel \| Ni \| [Ar]3d84s2 \| \| 29 \| Copper \| Cu \| [Ar]3d104s1 \| \| 30 \| Zinc \| Zn \| [Ar]3d104s2 \| (b) 4d Series (Second Transition Series)- \| \| \| \| \| --- --- \| \| Atomic No. \| Element \| Symbol \| Electron Configuration \| \| 39 \| Yttrium \| Y \| [Kr]4d15s2 \| \| 40 \| Zirconium \| Zr \| [Kr]4d25s2 \| \| 41 \| Niobium \| Nb \| [Kr]4d45s1 \| \| 42 \| Molybdenum \| Mo \| [Kr]4d55s1 \| \| 43 \| Technetium \| Tc \| [Kr]4d65s1 \| \| 44 \| Ruthenium \| Ru \| [Kr]4d75s1 \| \| 45 \| Rhodium \| Rh \| [Kr]4d85s1 \| \| 46 \| Palladium \| Pd \| [Kr]4d105s0 \| \| 47 \| Silver \| Ag \| [Kr]4d105s1 \| \| 48 \| Cadmium \| Cd \| [Kr]4d105s2 \| (c) 5d-Series (Third Transition series) \| \| \| \| \| --- --- \| \| Atomic No. \| Element \| Symbol \| Electron Configuration \| \| 72 \| Hafnium \| Hf \| [Xe] 5d26s2 \| \| 73 \| Tantalum \| Ta \| [Xe] 5d36s2 \| \| 74 \| Tungsten \| W \| [Xe] 5d46s2 \| \| 75 \| Rhenium \| Re \| [Xe] 5d56s2 \| \| 76 \| Osmium \| Os \| [Xe] 5d66s2 \| \| 77 \| Iridium \| Ir \| [Xe] 5d76s2 \| \| 78 \| Platinum \| Pt \| [Xe] 5d96s1 \| \| 79 \| Gold \| Au \| [Xe] 5d106s1 \| \| 80 \| Mercury \| Hg \| [Xe] 5d106s2 \| c) 6d-Series (Fourth Transition series) \| \| \| \| \| --- --- \| \| Atomic No. \| Element \| Symbol \| Electron Configuration \| \| 104 \| Rutherfordium \| Rf \| [Rn] 6d27s2 \| \| 105 \| Dubnium \| Db \| [Rn] 6d37s2 \| \| 106 \| Seaborgium \| Sg \| [Rn] 6d47s2 \| \| 107 \| Bohrium \| Bh \| [Rn] 6d57s2 \| \| 108 \| Hassium \| Hs \| [Rn] 6d67s2 \| \| 109 \| Meitnerium \| Mt \| [Rn] 6d77s2 \| \| 110 \| Darmstadtium \| Ds \| [Rn] 6d87s2 \| \| 111 \| Roentgenium \| Rg \| [Rn] 6d107s1 \| \| 112 \| Copernicium \| Cn \| [Rn] 6d107s2 \| 2.0Atomic Radii of d Block Elements Period Trend: Initial five elements from Sc to Mn show decreasing atomic radius due to stronger nucleus-valence shell attraction over valence-penultimate shell repulsion. Fe, Co, and Ni maintain almost equal atomic radii as nucleus-valence shell attraction equals valence-penultimate shell repulsion. Cu to Zn exhibit increasing atomic radius due to weaker nucleus-valence shell attraction compared to valence-penultimate shell repulsion. Group Trend: 4d series elements have larger atomic radii than 3d series. 4d and 5d series display nearly identical atomic radii owing to lanthanide contraction. 3.0Magnetic Properties of d Block Elements Paramagnetism: Substances attracted to a magnetic field possess unpaired electrons in their atomic orbitals, termed paramagnetic substances. Transition metal ions or compounds with unpaired electrons in the d-subshell (configurations d1 to d9) typically exhibit paramagnetic behavior. Diamagnetism: Substances repelled by a magnetic field, containing paired electrons in their atomic orbitals, are termed diamagnetic substances. Paramagnetism in Metals: Metals with unpaired electrons typically display paramagnetism. The magnetic moment (μ) for paramagnetic substances can be calculated using- μ=(n(n+2)) where 'n' is the number of unpaired electrons, measured in Bohr magneton units (B.M.). Ferromagnetism: Some materials exhibit ferromagnetism, a robust form of paramagnetism, displaying very strong attraction to magnetic fields. Common ferromagnetic metals include iron (Fe), cobalt (Co), and nickel (Ni). 4.0Formation of Interstitial Compounds by d Block Elements Interstitial compounds are formed when smaller atoms or molecules fit into the gaps or interstitial spaces between the larger atoms or ions in a crystal lattice. In the case of d-block elements, these compounds can form when smaller atoms or molecules occupy the spaces between the atoms of the d-block metal lattice. Characteristics and Examples: Titanium Carbide (TiC): TiC forms as carbon atoms occupy interstitial spaces in the crystal lattice of titanium. This compound displays exceptional hardness and is used in cutting tools and coatings. Manganese Nitride (Mn4N): Mn4N involves nitrogen atoms fitting into the interstitial spaces of manganese. It exhibits magnetic properties and is explored for potential applications in electronics and magnetic devices. Iron Hydride (Fe3H): Fe3H involves hydrogen occupying interstitial sites within iron. It's studied for its role in hydrogen storage and as a potential material for fuel cells. 5.0Formation of Alloys The d-block elements, or transition metals, are pivotal in the formation of alloys due to their unique properties, allowing them to create materials with enhanced characteristics. Examples of Alloy Formation with d-Block Elements: Brass: A mixture of copper (Cu) and zinc (Zn), offering enhanced malleability and acoustic properties. Alnico Magnets: An alloy of aluminum (Al), nickel (Ni), and cobalt (Co), known for its strong magnetic properties. Titanium Alloys: Incorporating titanium (Ti) with elements like aluminum or vanadium for lightweight, corrosion-resistant materials used in aerospace and medical implants. Some important alloys- \| \| \| \| --- \| (a) \| Bronze \| Cu + Sn \| \| (b) \| Brass \| Cu + Zn \| \| (c) \| Gun metal \| Cu + Zn + Sn \| \| (d) \| German Silver \| Cu + Zn + Ni \| \| (e) \| Stainless Steel \| Cr + Ni \| \| (f) \| Invar \| Ni + Fe \| \| (g) \| Alnico \| Al + Ni + Co \| \| (h) \| Duralumin \| Cu + Al + Mn \| \| (i) \| 22 Carat gold \| Au + Ag \| \| (j) \| 18 Carat gold \| Au + Ag + Cu \| 6.0Compounds Formed by d-block Elements Transition metal complexes occur when transition metals interact with ligands, which are molecules or ions that donate electrons, resulting in complex structures. Here's an overview of the formation of complex compounds involving d-block elements: 1. Coordination Bonds: Central Metal Ion: A transition metal serves as the central ion in a coordination complex, surrounded by ligands. Ligands: These are molecules or ions with lone pairs or π-electrons that bond with the metal ion through coordinate covalent bonds, sharing electrons with the metal center. 2. Coordination Number and Geometry: Coordination Number: The number of bonds formed between the metal and its attached ligands determines the coordination number. It varies from 2 to 12 in most cases. Geometry: The arrangement of ligands around the metal ion determines the complex's geometry, such as octahedral, square planar, tetrahedral, or other shapes. 3. Ligand Exchange and Stability: Ligand Substitution: Transition metal complexes can undergo ligand exchange reactions, where one ligand is replaced by another, leading to different complex structures. Stability: The stability of these complexes is influenced by factors such as the nature of the metal ion, the type of ligands, and the overall charge of the complex. 4. Properties and Applications: Color: Transition metal complexes often exhibit vibrant colors due to the absorption and emission of specific wavelengths of light. Catalytic Activity: Many transition metal complexes serve as catalysts in various chemical reactions due to their ability to facilitate reactions by altering the reaction pathways. Biological Significance: Some metal complexes play essential roles in biological systems, acting as cofactors in enzymatic reactions or as therapeutic agents. Examples of Transition Metal Complexes: [Fe(CN)6]³⁻ : Hexacyanoferrate(III) ion, featuring iron (Fe) as the central metal ion bonded to six cyanide (CN⁻) ligands. [Cu(NH3)4]²⁺ : Tetraamminecopper(II) ion, with copper (Cu) coordinated to four ammonia (NH3) ligands. [PtCl4]²⁻ : Tetrachloridoplatinate(II) ion, containing platinum (Pt) coordinated to four chloride (Cl⁻) ligands. 7.0Catalytic Properties of d block Elements Transition metals, found in the d-block, make exceptional catalysts. They speed up chemical reactions without getting used up themselves. These metals can activate molecules, facilitate bond-making and breaking, and enable various reactions, from hydrogenation to environmental processes like CO2 conversion. They're vital in industries like petroleum refining, chemical production, and renewable energy, contributing to efficiency, selectivity, and sustainability in chemistry and technology. The density of d-block elements, or transition metals, varies widely based on their atomic masses, atomic radii, and crystal structures. Generally, these metals tend to be denser than typical metals due to their relatively high atomic masses and tightly packed crystal lattices. Transition metals often have densities ranging from around 2 to 22 grams per cubic centimeter (g/cm³). For instance: Sc < Ti < V < Cr < Mn < Fe < Co < Ni = Cu > Zn (Zn has a lower density because of its large atomic volume.) Minimum density in 3d series → Sc Maximum density in 3d series → Ni and Cu In Group : 3d < 4d < 5d Elements with the highest densities are- Osmium (Os) = 22.51 g/cm3, Iridium (Ir) = 22.61 g/cm3 8.0Oxidation States of d block Elements The d-block, or transition metals, exhibit various oxidation states due to their unique electronic configurations that allow them to lose or gain electrons from their d-orbitals. Here's an overview of their oxidation states: 1. Variable Oxidation States: Transition metals can adopt multiple oxidation states, often due to the availability of different numbers of electrons in their d-orbitals. These oxidation states typically range from +1 to +8, though some metals can go beyond these limits. 2. Common Oxidation States: Lower Oxidation States: Many transition metals exhibit lower oxidation states, such as +1, +2, or +3. For example, iron (Fe) commonly forms ions with +2 and +3 oxidation states. Higher Oxidation States: Some transition metals can achieve higher oxidation states, like +4, +5, +6, or even higher. Manganese (Mn), for instance, can form compounds with oxidation states ranging from +2 to +7. 3. Factors Influencing Oxidation States: Electronic Configurations: The number of electrons in the outermost d-orbitals determines the possible oxidation states for a transition metal. Chemical Environment: The presence of ligands or the nature of the chemical surroundings can influence the stability of different oxidation states. Example:- In Ni(CO)4 and Fe(CO)5, the oxidation state of nickel and iron is zero. They show variable oxidation states. Note- Underlined states are the most stable ones. 9.0Formation of Colored Compound Transition metals in the d-block exhibit vibrant colors in their ions due to the presence of partially filled d-orbitals, which undergo electronic transitions when absorbing light. Here's a concise overview: Formation of Colored Ions: d-orbital Configurations: Transition metal ions often have partially filled d-orbitals due to their variable oxidation states. Electrons in these orbitals can absorb specific wavelengths of light, leading to the display of colors. Electronic Transitions: When visible light interacts with these ions, electrons move between different energy levels within the d-orbitals. Absorption of certain wavelengths causes electrons to jump from lower to higher energy levels, leading to the observed colors. Color Characteristics: The specific color observed depends on the energy difference between the d-orbital electronic levels. Different transition metals or oxidation states display distinct colors due to their unique electronic structures. Examples of Colored Ions: Copper (Cu²⁺): Forms blue-colored ions due to the d-orbital transitions involving its partially filled orbitals. Chromium (Cr³⁺): Yields green-colored ions attributed to its d-orbital transitions. 10.0Important Compound of d-block Potassium Dichromate (K2Cr2O7) Preparation- Potassium dichromate (K2Cr2O7) is commonly prepared through the reaction between sodium dichromate (Na2Cr2O7) and potassium chloride (KCl) in an aqueous solution. The balanced chemical equation for this reaction is: Na2Cr2O7+2KCl→K2Cr2O7+2NaCl This method involves the exchange of ions between sodium dichromate and potassium chloride in a solution, resulting in the formation of potassium dichromate and sodium chloride as a byproduct. The potassium dichromate can then be isolated by crystallization. Potassium dichromate (K2Cr2O7) possesses several important properties that make it valuable in various applications: Oxidizing Agent: It is a powerful oxidizing agent, used in numerous oxidation reactions due to its ability to provide oxygen or accept electrons. Color: Potassium dichromate appears as bright orange-red crystals or powder, making it easily identifiable. Solubility: It is soluble in water, forming a vibrant orange solution. This solubility aids its use in various solutions and applications. Chemical Stability: It exhibits good stability under normal conditions, but it decomposes when exposed to heat or when in contact with reducing agents. Other Applications: Potassium dichromate finds use in various industries: Laboratory and Analytical Chemistry: Used in qualitative analysis and titrations as an oxidizing agent. Photography: Historically used in old photographic processes. Manufacturing: Employed in the production of pigments, dyes, and inorganic chemicals. Wood Preservation: Utilized to treat wood, acting as a biocide to protect against decay. Toxicity: It is highly toxic and carcinogenic. Inhalation or ingestion can cause severe health issues. Due to its toxicity, its use is restricted or regulated in many countries. Potassium Permanganate (KMnO4) Preparation- Potassium permanganate (KMnO4) is typically prepared through the reaction between manganese dioxide (MnO2) and potassium hydroxide (KOH) in the presence of an oxidizing agent like potassium chlorate (KClO3) or potassium nitrate (KNO3). The preparation involves several steps: Step 1. Conversion of Manganese Dioxide (MnO2): Manganese dioxide is reacted with a hot concentrated solution of potassium hydroxide (KOH) in the presence of an oxidizing agent (such as potassium chlorate or potassium nitrate). This reaction produces potassium manganate (K2MnO4). 2MnO2 + 4KOH + O2 → 2K2MnO4 + 2H2O Step 2. Conversion to Potassium Permanganate: The potassium manganate (K2MnO4) formed in the first step is then converted to potassium permanganate (KMnO4) by adding an acid, usually sulfuric acid (H2SO4), which oxidizes the manganate ion to permanganate ion. 3K2MnO4 + 4H2SO4 → 2KMnO4 + 2MnSO4 + K2SO4 + 4H2O After this process, the resulting solution contains potassium permanganate, which can be isolated by crystallization or other purification methods. Applications of Potassium Permanganate (KMnO4)- KMnO4 is a powerful oxidizing agent and chemical compound. It appears as dark purple crystals or a powder and is commonly used for various purposes: Water Treatment: It's used to remove impurities and odors from water due to its oxidizing properties. It can help in treating water for drinking and industrial purposes. Antiseptic: In a diluted form, it can be used as an antiseptic to clean wounds, disinfect surfaces, or treat certain skin conditions like dermatitis and fungal infections. Chemical Reactions: Potassium permanganate is a strong oxidizing agent and is involved in various chemical reactions in laboratories and industries. It's used in the synthesis of organic compounds and as an oxidizer in chemical reactions. Analytical Chemistry: It's employed as a reagent in qualitative analysis to identify compounds, especially those containing reducing agents. Table Of Contents: 1.0Electronic Configuration of d Block Elements 1.13d Series: The Start of Transition Metals 1.24d Series: Moving Beyond the Fourth Row 1.35d Series: Advancing Deeper into the Table 1.46d Series: Beyond the Main Section 2.0Atomic Radii of d Block Elements 3.0Magnetic Properties of d Block Elements 4.0Formation of Interstitial Compounds by d Block Elements 4.1Characteristics and Examples: 5.0Formation of Alloys 6.0Compounds Formed by d-block Elements 7.0Catalytic Properties of d block Elements 8.0Oxidation States of d block Elements 9.0Formation of Colored Compound 9.1Formation of Colored Ions: 10.0Important Compound of d-block 10.1Potassium Dichromate (K 10.2Potassium Permanganate (KMnO Frequently Asked Questions D-block elements have widespread applications in various fields, such as in industrial catalysts, alloys, electronics, and biochemical processes. They usually exhibit multiple oxidation states, form colorful compounds, act as good catalysts, and possess high melting and boiling points. They are called transition metals because they are positioned between the s-block and p-block elements in the periodic table, transitioning in properties, including ionic and atomic sizes and electronegativity. Join ALLEN! (Session 2025 - 26) Choose class Choose your goal Preferred Mode Choose State Related Articles:- ## The s-Block Elements The "s-block" in the periodic table refers to the two groups of elements located in the leftmost part of the periodic table: Group 1 and Group 2.## The p-Block Elements Group 13-14 The p-block elements definition involves groups 13 to 18 and include nonmetals, metalloids, and metals.## The p-Block Elements Group 15-18 Learn the p block elements in groups 15 to 18 of the periodic table.## f block elements The f-block elements, commonly known as the inner transition metals. They consist of two series: the lanthanides and the actinides. These...## The d and f-Block Elements The d-block of the periodic table includes elements from groups 3 to 12 and the f-bock contains elements where the 4f and 5f orbitals are filled up.## Periodic Table The periodic table systematically organizes all 118 known elements by atomic number, indicating the number of protons in the nucleus.## Alkaline Earth Metals Alkaline earth metals are a group of chemical elements found in Group 2 of the periodic table.## Halogen Halogen elements are a group of highly reactive nonmetals found in Group 17 (Group VIIA) of the periodic table.
12712	https://firmsconsulting.com/quarterly/consulting-brain-teasers-with-answers/	Download Chapter You will receive additional complimentary videos/updates by email. To complete the process please click the link in the email we will send you. Download Preview You will receive additional complimentary videos/updates by email. To complete the process please click the link in the email we will send you. Welcome back! Sign in with Google Sign in with Linkedin Or, sign in with your email Don’t have an account? Subscribe now Popular searches : Strategy Implementation Slides Case interviews The Strategy Journal Amazon best-seller. Sample chapter? Most watched strategy videos Implementing the Masterplan ex-McK. partner teaches problem solving Preview videos? One of the best episodes ever! So precise description of implementation! Pavel Charny Multiple platforms iOS + Android Apps, Apple TV, ROKU, iTunes, Google Podcasts, Amazon Kindle Huge library +6,200 strategy episodes all by ex-McKinsey, BCG et al partners. +480 podcasts. +6 books. + editable slides Unparalleled Advice from partners who have collectively advised +100 Fortune 100 CEOs Download The Strategy Journal Consulting Brain Teasers with Answers HomeCase InterviewsConsulting Brain Teasers with Answers 67 Consulting Brain Teasers With Answers Consulting case interviews can include full cases as well as brainstorming, estimation, brain teasers etc. So, what is a brain teaser? The definition of brainteaser is “a puzzle or problem whose solution requires great ingenuity.” In other words, a brain teaser is a short riddle designed to challenge your ability to think logically and make not immediately obvious connections. And your ability to solve a brain teaser gives an indication to the interviewer about how flexible your mind is and if you can think logically. Additionally, it can indicate if you give up easily when you encounter a difficult problem, quality firms certainly don’t want in future consultants. Before we dive into various types of brain teasers, and then 67 consulting brain teasers and answers, we would like to share with you an interactive download we prepared for you as a gift, based on FIRMSconsulting book on brain teasers for consulting, banking, and tech interviews. This download includes 20 brain teasers not covered in this article, including explanations on how to approach solving each of those brain teasers. You can get a link to download your copy below. It is completely free. Enjoy! FREE GIFT #1 BONUS TUTORIAL DOWNLOAD 20 Brain Teasers With Answers And Explanations Brain teasers as part of your consulting case interview preparation So, as is clear from the above, consulting brain teasers should be a part of your case interview preparation. It helps to keep your mind sharp and helps you develop a habit of solving problems calmly and confidently, but with a sense of urgency. And the most important part of solving a brain teaser is how you approach a problem. Just blurting out a correct answer because you already heard a particular brain teaser before will not win you points. Similarly, giving an incorrect or illogical answer or arguing with the interviewer saying your answer is correct will also usually hurt you. For example, imagine the brain teaser interviewer gives you is as follows. “It’s dark in your room. You have a lot of white socks and black socks in your drawer. You want to get a matching pair, but socks are not organized. All socks are identical except for the color. What is the minimum number of socks you need to take to get a matching pair to wear?” Now imagine you say, “I would say two. The task is to find a minimum number of socks. A minimum number of socks could be as followed. The first sock you get is black, the second sock you get is black. Similarly, the minimum number of socks could be the first sock you get is white, the second sock you get is white.” This is the actual answer we got from one of the members of FC community. This answer is incorrect. If you try to push for it you will fail. The question is, “What is the minimum number of socks you need to take to get a matching pair to wear?” You need to read between the lines. In other words, “What is the minimum number of socks you need to take to guarantee you get a matching pair to wear?” The correct answer to this question can be found below in brain teasers with answers section. Types of consulting brain teasers What are some types of consulting brain teasers I can expect during a case interview? The Estimation teaser or “How Many” Brain Teaser: Another type of brain teaser, known as an estimation teaser or the “how many” brain teaser, is especially common in the world of management consulting interviews, and it’s a skill consultants will use in their careers. Many of the techniques we use to solve and teach estimation cases were developed by FIRMSconsulting, based on our years in management consulting. Examples of these types of questions are practically infinite: “How many marbles can you fit in a standard Subaru?” “How many boxes of tissues were sold in the average American pharmacy in 2019?” “How many driver’s licenses does Wisconsin allocate in a typical summer?” One of the most common questions from this category is “How many gas stations are there in the U.S.?” You get the idea. These teasers can leave you entirely stumped. After all, how are you meant to determine answers when you’ve been given no solid data to work with? They can feel like non-starters, but that isn’t the case at all. Estimation teasers are one of the most important types of all because they’re the best at showcasing how you go about solving problems. The answer is not as important as the approach you use, the assumptions you made and how you test the final answer. There is a lot to teach here. Their lack of definite answers is exactly what makes them so valuable; there’s no way to cheat your way around them or to land a correct solution out of sheer good luck. As a matter of fact, the people who are asking you questions like this almost certainly don’t know the answers themselves. Because of this, you won’t find any estimation teasers in this post. Estimation brain teasers require an exchange of ideas by nature, between the interviewer and interviewee, something which a written page cannot accurately replicate. Any attempt to condense the solution to an estimation teaser into one small paragraph would defeat its purpose entirely. Refer to estimation cases in The Consulting Offer (for example The Consulting Offer, Season 1, with Felix) to learn how to answer these questions. You can enroll to receive access to all programs on StrategyTraining.com by becoming an Insider (Insider status is granted when you become an annual Premium member). Your ability to answer this type of question during consulting case interviews is crucial. This type of question may be asked separately or as part of a full business case. Though we won’t explore this type of brain teaser any further in this post, try to practice them regularly on your own. Even outside of the consulting world, they’re some of the most useful mental exercises that you can perform. If you need more details on how to solve estimation cases, we would like to share with you a comprehensive estimation cases guide download we prepared for you as a gift, based on FIRMSconsulting book on how to prepare for consulting case interviews. This download includes a step by step guide on how to solve estimation cases, including a detailed example with an answer. You can get a link to download your copy below. It is completely free. Enjoy! FREE GIFT #2 BONUS TUTORIAL DOWNLOAD A Comprehensive Estimation Cases Guide The Math Brain Teaser, Math Puzzles and Math Riddles: Examples of mathematical brain teasers are probability questions. This barely qualifies as a brain teaser since it tests your math skills vs. your critical thinking and problem-solving ability. Mathematical brain teasers can be intimidating, but they’re usually relatively straightforward in nature, even if they don’t initially seem to be. If you do a real deep dive, you’ll find some that require an extensive array of equations, formulas, and calculations in order to reach their solution. On the other hand, the simpler, less specialized ones tend to focus on probability. Interestingly, mathematical teasers can be a little difficult to identify since they often don’t involve numbers outright, whereas other types of brain teasers occasionally do. This can be a good thing since it encourages you to approach the problem with an open mind rather than an assumption that you’re going to have to adhere to tried and true math-solving techniques. These days, most people associate math with subjects like computer science, but it’s important to remember that it didn’t start that way. At its core, the system of mathematics is designed to help us understand functions of the natural world, not to overly complicate them. When it comes to brain teasers, math is your friend! The Riddle Brain Teaser: In such a brain teaser the interviewer gives you particular problematic situation and asks you to find a solution. Riddles are the type of brain teaser that most requires you to think in an unconventional manner. They typically feature language that tries to misdirect you and often involve words and phrases that don’t mean what they initially appear to. These tend to be the most difficult type of teaser since they’re the ones that most aggressively push you out of your comfort zone. They can also be the most rewarding; there are fewer things more satisfactory than coming up with a single simple solution to a riddle that first seemed impossible. One example that will appear later below is the following: “If a plane crashes directly on the border between the U.S. and Mexico, where will the survivors be buried?” Another example is a question about a farmer, fox, carrot and rabbit below. When solving a riddle, the most important thing that you can do is identify it as just that—a riddle. Once you recognize that a question is most likely trying to trick you, you can be much more wary, and thereby avoid falling into its traps. Each word in a riddle tends to be carefully considered so be sure not to overlook anything—not even the tiniest scrap of punctuation. To the best of your ability, try to consider the information presented to you in a vacuum. Assumptions and snap judgments are a riddle master’s worst enemy. Above all else, remind yourself that you can solve anything if you put your mind to it. Relax, take your time, and—of course—enjoy! “Why is” Brain Teaser: “Why”-type brain teasers are used to test your ability to analyze the world around you. They can be some of the most difficult teasers to approach since many people will often have trouble knowing where to even begin. They often take a widely accepted quirk of the world around you and challenge you to propose a reason for why that thing works the way it does. One example that will appear later in this book is the following: “Why is a tennis ball fuzzy?” This is a commonly used “why”-type teaser that highlights something which the majority of people have never questioned. Working through this type of teaser is a great way to begin thinking more critically about our surroundings, especially those which have been created or engineered by other people. We accept most modern conveniences for what they are, never thinking to question what made them that way—but nothing ever works a certain way “just because.” Everything that we use that is menmade is crafted with intention, and the more that we can train ourselves to recognize that fact, the more effectively we’ll be able to solve problems of our own—even when we don’t have a clue, for instance, about the history of tennis. Job Interview Questions That Are Not Brain Teasers Some job interview questions may be difficult but may not be a brain teaser. Such questions often ask something about you vs. about some external problem you need to solve. Examples include: “Are you more of a hunter or a gatherer?” “If you were a box of chocolates, what kind of chocolates would you be and why?” “How honest are you?” “How would you explain the internet to a 3-year-old?” These job interview questions can mostly be divided into two opposite realms: subjective and knowledge-based. A subjective job interview question tends to have no correct answer, or at least an answer that depends on the person answering it. In a job interview, this might look like the typical prompt of “What is your greatest weakness?” or “Do you see yourself as more of a leader or a follower?” There are certainly right and wrong ways to go about responding to a question like this, but you can succeed for the most part simply by being honest about your experiences and abilities. In other words, a subjective question will never be trying to trick you in the same way that a brain teaser often will. A knowledge-based job interview question measures your competence in a particular area; often a highly specialized one. When a job interviewer presents you with one of these, they’re trying to gauge your familiarity with a topic that will likely be central to the position for which you’re applying. You can think of these questions as the type that you might find on a high school history quiz. As long as you have your facts straight, you should be able to answer knowledge-based questions easily without taking too long to contemplate and problem-solve. Compared to these other types of questions, brain teasers can be both easier and harder, depending on how you tend to problem-solve. You’ll never have to make an argument for yourself as a candidate when you’re dealing with a brain teaser, and you also won’t have to study a pile of facts beforehand. You can, however, increase your general aptitude to solve brain teasers via practice. Think of brain teasers like exercises for your mind. Your brain is a muscle like any other, and the more you use it, the stronger it will get. That’s why this post exists: to provide you with an additional opportunity to practice and get better at solving brain teasers. With that in mind, that’s another element of brain teasers that differentiates them from many other types of job interview questions: When you relax and take your time, they can be a whole lot of fun! FREE GIFT #3 BONUS DOWNLOAD Proven Strategies for Effective Leadership and Results Practice consulting brain teasers with other FC members The good news is you can become better at solving brain teasers by practicing brain teasers on a daily basis. Below you will find some consulting brain teasers with answers. You can also join our Consulting Case Interviews facebook group where we post new brain teasers with answers on a daily basis and you can engage in conversation with other members of FC community about how to solve a particular brain teaser or about anything related to case interview prep. Everyone is welcome to join. Consulting brain teasers with answers Now, let’s look at some actual brain teasers. When you read each brain teaser try to calmly, confidently and with a sense of urgency solve it. Think through the problem and think through it as slow as your brain needs to get to the answer. I remember when I was studying difficult subjects in university if I would try to go through my reading fast I would get lost. Instead, what I needed is to slow down and carefully think through all the connections the author was making to explain a particular point. Taking the time to think through a problem in steps often leads to much faster and better results versus rushing through it. We see it all the time with members using The Consulting Offer training and even members using our advanced consulting skills training. Members who rush through the material, jump from episode to episode, from program to program, trying to find shortcuts and get through the material fast don’t do as well as members who are diligently working through the programs in the order in which we recommend it, carefully absorbing and mastering various concepts and approaches. This same logic is applicable in solving brain teasers. If you rush you will probably get to the wrong answer. For example, if we look again at a brain teaser we looked at above, the one about socks. Some candidates may immediately start panicking, “Oh, this is probably statistics. I need to understand probabilities. Oh my God, I don’t know how to solve it!” Yet, all you need to do is to imagine you are in a dark room, there are white and black socks in the drawer. You need to get a matching pair. What are the possible scenarios? Well, you can get 2 socks but then there is no guarantee they will match. Let’s say you get 3 socks, the next lowest number. What are the possible scenarios? Well, you could get 3 black socks, 3 white socks, 2 white socks, and 1 black sock or 2 black socks and 1 white sock. In each possible scenario, you end up with a matching pair. You see, when you calmly think through the problem, when you let your brain to process the information, the answer is easy and obvious. The key is to trust that you are smart and you will figure it out, and let your brain work through it at a speed it is comfortable at to figure it out. And have fun with it. As you solve any cases during your case interview preparation or during an actual case interview have fun with it. Try to enjoy it. Having a solid problem-solving skill is powerful. You can do a lot of good in the world if you are good at problem-solving. So enjoy the process and trust in your ability to figure things out. Practice consulting brain teasers It’s important to note that difficulty is subjective, especially when it comes to problems that are structured with the goal of getting you to think outside of the box. If you find yourself stuck on a brain teaser, try working on some different ones before returning to it, and rest assured that you’ll get faster and faster as you familiarize yourself with reliable methods of problem solving. If you want to check your solutions or if you just can’t seem to figure out a particular brain teaser no matter how much you scrutinize it, you can always flip to the bottom of the section, which contains solutions for every brain teaser. Each numbered solution will contain an “Answer” which is a straightforward statement providing the correct response to the teaser, and, where needed an “Explanation,” which goes into greater depth in examining the strategies or thinking pattern that one might employ in solving that particular teaser. You’re strongly advised to read the “Explanation” parts of the answer key, even if you were able to come to the correct conclusion by yourself. Oftentimes, these bits will highlight ways to optimize your thinking, making future brain teasers of a similar nature that much easier. With that being said, get ready: It’s time to hit the mental gym. Simple consulting brain teasers The following 41 brain teasers are grouped under the “Easy” label, but that doesn’t mean that they won’t be challenging. In fact, some people find brain teasers like this to be the most difficult of them all because of their simplicity. If you’re used to relying on calculations and complex lines of logic, you’re best advised to set aside any preconceptions about how you should go about addressing these problems. The most important tip to keep in mind for this section is to look very carefully at each question. If it seems too easy, it’s probably misleading you—and the same is true if it seems too hard. Keep an eye out for tricks of phrasing, double meanings, and convergence of the literal with the metaphorical. You’re looking for an “Aha!” moment that will frame the original teaser in an entirely new light. Let’s start with a simple riddle. Ready? Set. Go! (1) A farmer is trying to cross a river. He is taking with him a rabbit, carrots and a fox. He has a very small raft so he can only bring one item at a time across the river. How does he cross the river? Assume that the fox does not eat the rabbit if the farmer is there. Assume both rabbit and fox are not trying to run away. This riddle can be solved easily if you put yourself in place of Jason. (2) Jason’s father has 4 children; Suzan, Jeffrey, Jennifer. Who is the fourth? The answer to this one will likely immediately be intuitive and then you can check the answer by thinking through the steps logically. (3) The day before yesterday John was 17. Next year he will be 20. What day is his birthday? (4) Two fathers and two sons sat down to eat eggs for breakfast. They ate exactly three eggs, each person had an egg. Can you explain how? For this one, again, if you carefully think through connections the answer is obvious. (5) Your mother’s brother’s only brother-in-law is asleep on your couch. Who is asleep on your couch? This was the most popular riddle we posted so far in our consulting case interviews facebook group. We broke down this question above. (6) It’s dark in your room. You have a lot of white socks and black socks in your drawer. You want to get a matching pair, but socks are not organized. All socks are identical except for the color. What is the minimum number of socks you need to take to get a matching pair to wear? Easy one if you carefully think through connections. Assume a speaker is not looking in the mirror. (7) Brothers and sisters I have none but this man’s father is my father’s son. Who is the man? (8) What has a head and a tail, but does not have a body? (9) John’s father has three sons: Jeffrey, Jack and _____ ? This one is similar to the earlier brain teaser. (10) 2 mothers and 2 daughters were fishing. They managed to catch one small fish, one big fish, one medium fish. Since only 3 fish were caught, how is it that they each took home a fish? This one is a piece of cake, if you pay attention. (11) How many times can you subtract the number 5 from 25? Another fun, easy one. (12) What has many keys but cannot open any doors? This one is very intuitive. (13) What is something you can keep after giving it to someone else? (14) How many apples can you put into an empty containter? Here is another easy one to help you warm up. (15) When you add 2 letters, a 5 letter word becomes shorter. What is it? Another simple one if you visualize what is happening. (16) How far can a dog run into the woods? And another fun one. (17) A cowboy rides into town on Monday. He stays from 2 days and leaves on Monday. How is this possible? (18) A man went for a walk and got caught in the rain. The man did not have an umbrella and he was not wearing a hat. His clothes got soaked but not a single hair on his head got wet. How is it possible? And here is an obvious one. (19) How many months have 28 days? This one is intuitive. (20) What is an odd number where if you take away one letter from it it becomes even? And I love this one. (21) What invention lets you look right through a wall? (22) What never asks a single question but is often answered? (23) The more you take the more you leave behind. (24) What is full of holes but still holds water? (25) It goes in dry and it comes out wet. The longer it is in the stronger it gets. What is it? (26) A family lives in a large apartment building, on the 17th floor. They have a son. Every morning he takes the elevator from the 17th floor to the ground floor and goes to school. In the afternoon he uses the elevator to get to the 6th floor and then uses the stairs for the remaining floors. Why? (27) You hold it without using your hands or arms. What is it? (28) One brick is one kilogram and half a brick heavy. What is the weight of one brick? (29) If Rebecca’s daughter is my daughter’s mother. What am I to Rebecca? (30) Say speaker backward. (31) When you have me you immediately feel like sharing me. But once you share me you no longer have me. (32) If electric train is traveling south, which way is smoke going? (33) How many times can you subtract 10 from 100? (34) Amanda’s mother had 3 children. First one was named April, second child was named May. What is the name of the 3rd child? (35) The more there is, the less you see. What is it? (36) What always ends everything? (37) I have seas without water, coasts without sand, towns without people, and mountains without land. What am I? (38) You can find it in Mercury, Earth, Mars, Jupiter and Saturn, but not in Venus or Neptune. What is it? (39) What coat is best put on wet? (40) It’s at the center of gravity. You can also find it in Venus, but not in Mars. What is it? (41) A clerk in a butcher shop stands six feet 5 inches tall and wears size 11 shoes. What does he weigh? Answers to Simple consulting brain teasers Takes the rabbit, sails back. Takes the fox, and sails back with the rabbit. Drops off the rabbit and takes carrots; sails back and takes the rabbit. However, there is an alternative answer. Take rabbit, sail back to take carrots, sail back with the rabbit to take fox, come back again to take the rabbit. The carrots and fox in 2nd step are interchangeable. Jason. Jason, Suzan, Jeffrey, and Jennifer are siblings. December 31. The day before yesterday is 30th December when he was 17, yesterday was the 31st December when he turned 18, today is the 1st of January and he will be 19 this year, and next year he will be 20. Three generations of one family. One of the ‘fathers’ is both a son and a father. The answer is your dad, assuming your mother and father are married. You need at least 3 socks. If the first sock is black and the next sock is black you got a pair. If the first sock is black, the second sock is white, you will get a pair regardless of if the third sock is white or black. The man is the speaker’s son. A coin. John. 3 generations of people went fishing. Grandma, mom and daughter. Only 1 time. Piano or any keyboard. Your word. Only one. After that it will not be empty. ANSWER: Short. EXPLANATION: Speaking this brain teaser out loud makes it a lot more difficult since you don’t have the visual aid of the word “shorter.” When you write it down, however, the solution is right in front of you. A word doesn’t lessen in length if you add two letters to it; rather, it becomes “shorter”—it literally turns into the word “shorter.” And if you subtract those two letters, you end up with your answer: “short.” Only until a do gets to the middle of the woods. After that a dog will be running out of the woods. ANSWER: His horse is named Monday. EXPLANATION: There are multiple ways to read this question. Most people’s default will be to assume that it’s referencing a day of the week in which case it has no possible solution. Two days after Monday will always be Wednesday, regardless of the month, year, et cetera. With that in mind, how can we find our solution? Something like “time travel” may seem to be the only remaining possibility if “Monday” is in fact referring to the day of the week. But there’s no actual confirmation of this within the brain teaser itself. The information we’re given is that the a man rides into town on Monday. This must mean that he physically rides upon something called “Monday”—most likely a horse. ANSWER: The man is bald. EXPLANATION: This type of brain teaser makes us challenge our assumptions about the world. Of course everyone knows that some people are bald. However, due to bald men being the minority compared to those with hair, our default image of a “man” typically is not a bald one. Working with this default image, this brain teaser can be quite puzzling indeed—but if we allow ourselves to think outside the box, the answer becomes obvious. 12 months. Each month has at least 28 days. Seven. A window. A doorbell. Footsteps. A sponge. A tea-bag. Because he cannot reach buttons higher than 6. Your breath. 1 brick is 2 kg heavy. I am Rebecca’s daughter. “Speaker backward.” ANSWER: A secret. EXPLANATION: This is actually a very easy brain teaser to unravel, so long as you follow each step to its logical conclusion. First, recognize that you’re looking for an “it”—an object, entity, or concept. Next, consider the single fact about this “it” that’s been provided to you: By nature, it cannot be shared. What thing is defined by its exclusivity? A secret. If you tell a personal secret to someone, it is no longer a secret by definition. There is no smoke since it is an electric train. Only 1 time because the next time you will be subtracting 10 from 90. Amanda. Fog. The letter G. A map. The letter R. A coat of paint. The letter V. ANSWER: Meat. EXPLANATION: Oftentimes, brain teasers will try to mislead you with extraneous information, while simultaneously masking any facts that are of true importance. Here, the key to solving the riddle lies within the first three words: “John the butcher.” Part of a butcher’s professional duties is the weighing of meat to sell to customers. The brain teaser distracts you with information about John’s size, but that isn’t enough on its own to determine his weight; therefore, “weigh” must not be operating as a reflexive verb. John regularly calculates the weight of something else, and since he’s a butcher, that “something” is meat. Medium difficulty consulting brain teasers And here is a more difficult one for you. (1) What do you notice about the following sequence of numbers? – 8, 11, 4, 9, 1, 6, 3, 0 And here is another somewhat difficult one. (2) You have a 3-gallon jug and 5-gallon jug. You need to fill the 5-gallon jug with exactly 4 gallons of water. What would you do? Another relatively easy one if you keep your mind open. (3) How can you throw a ball, make it stop and travel in the opposite direction all while touching the ball only once. Similarly, this one is relatively easy if you keep your mind open. (4) Imagine you had to put a coin into an empty bottle, then close the bottle with a cork. How could you take the coin out without removing the cork or breaking the bottle? This one if one of my favorites. (5) You have 3 boxes. One is with apples, one is with oranges and one has a combination of apples and oranges. All boxes are incorrectly labeled. In other words the label identifies incorrect content of the box. You can only open 1 box and without looking at the box you can take out 1 fruit and look at it. How can you label all the boxes correctly? Here is another one of my favorites. (6) You’re in a room with 3 light switches. Each light switch controls one of 3 light bulbs in the next room. How will you determine which switch controls which bulb if you can inspect the other room only once and cannot see into the other room from the other? Here is another fun one. (7) How can a pant’s pocket be empty yet still have something in it? (8) Which weights more, a pound of feathers or a pound of bricks? An interesting one. (9) A little girl fell off 29-meter ladder but did not get hurt. How is it possible? And here you will need to do some math. (10) Using only addition how can you get to number 1,000 by adding up eight 8’s? (11) Heavy it is but reverse it’s not. What is it? (12) What comes once in a minute, twice in a moment, but never in a thousand years? (13) It’s as light as a feather. However, most people cannot hold it for longer than 2 minutes. What is it? (14) What gets more wet while it dries? (15) A man shaves several times a day but still has a beard. Who is he? (16) You’re driving a city bus. At the first stop, 3 women get on. Then at the second stop, two children get on. At the 3rd stop, one man gets off and a man gets on. The bus is yellow and it’s snowing outside in December. What is the hair color of the bus driver? (17) Two boxers are in a match scheduled for 12 rounds. One of the boxers gets knocked out after only 5 rounds, yet no man throws a punch. How is this possible? (18) If a plane crashes directly on the border between the U.S. and Mexico, where will the survivors be buried. Answers to Medium difficulty consulting brain teasers The digits are in alphabetical order. (Eight, eleven, four, nine, one, six, three, zero) Fill the 3-gallon jug. Pour it into the 5-gallon jug. Fill the 3-gallon jug again and pour slowly into the 5-gallon jug until it is full. Because the 5-gallon jug already had 3 gallons in it only 2 more gallons will fit. And now you have 1 gallon in the 3-gallon jug. Remove water from the 5-gallon jug, pour 1 gallon from the 3-gallon jug. Refill the 3-gallon jug and pour into the 5-gallon jug. You now have exactly 4 gallons of water in the 5-gallon jug. OR, there is an alternative way. Fill the 5-gallon jug, pass it over to the 3-gallon jug. Now you have 2-gallons remaining in the big jug. Empty the small jug and pass the 2-gallons from the big to the small jug. Now you have space for 1-gallon in the small jug. Fill the big jug completely then seek to fill the 3-gallon jug with the 1-gallon that was missing there. Now you have 4-gallons in the big jug. Throw it up. Push the cork in. Here is an actual long explanation from one of FC members: “Let’s call 3 label Os (all oranges), As (all apples), OA (a combination of orange and apple. Since all boxes are incorrectly labeled, possible fruits with each box are: OA (all oranges, all apples), As (all oranges, combination of both), Os (all apples, combination of both). Next, take out 1 fruit from one having label OA. Scenario 1: if it’s orange, then the box is with all oranges, correct label must be Os. Then the one with As label must have a combination of both fruits, correct label OA. The last box, correct label As. Scenario 2: if it’s an apple, similar logic.” This is a correct answer, but it will help to make it more crisp during an actual interview. An actual correct answer from one of FC members: “Let’s call 3 switches 1, 2, 3. Turn switch 1 for the longest time, for example, 2 minutes; switch 2 for 20s. Go to the other room. The hottest light bulb is controlled by switch 1, the slightly warm bulb switch 2, the remaining bulb switch 3.” There is an alternative faster way to do this. Turn switch one for a short time, long enough for the bulb to warm up. Put it off. Then turn on switch 2. Go into the room. Warm bulb which is not on will be switch 1, bulb that is on will be switch 2 and the other one will be switch 3. There is a hole in the empty pant’s pocket. Neither. They both weigth 1 pound. She fell off from the lowest step. 888+88+8+8+8. TON. The letter M. Breath. A towel. A barber. Your color since you are driving the bus. The boxers were women. ANSWER: They won’t be buried. At least, not any time soon.EXPLANATION: The issue of geographical boundaries is an irrelevant one because the survivors won’t be buried at all. The victims, of course, are another matter, but not one that matters to this particular question. Always be on the lookout for questions that, despite being confidently stated, don’t actually make any sense under closer examination! Hard consulting brain teasers Next, let’s take a look at this much more difficult riddle. (1) Who makes it, has no need for it. Yet, who buys it, has no use for it. And who uses it can neither see nor feel it. What is it? A sad and difficult one. (2) A woman was born in 1996 and died in 1957. How is this possible? This one is quite difficult. (3) I have a dog that has 3 puppies: Barney, Cookie, and Sun. What is the mother’s name. Here is a fun but slightly difficult one. (4) What was the world’s tallest mountain before Mount Everest was discovered? Here is a poetic but somewhat difficult brain teaser. (5) What is at the beginning of eternity, the beginning of every end, the end of every place, and the end of time? (6) What word looks the same backward and upside? Hint: There is more than 1 answer. One of the answers has something to do with water. (7) Why tennis ball is fuzzy? Answers to hard consulting brain teasers The answer is very dreary. A coffin. 1996 and 1957 were the room numbers in the hospital. What. Mount Everest. This is a lateral thinking problem. The answer here has nothing to do with the meaning of the words but instead with the words themselves. The answer is the letter e. SWIMS, NON. ANSWER: The fuzz on a tennis ball ensures that it moves more slowly through the air.EXPLANATION: This is a very common brain teaser, and one that initially leaves most people drawing a complete blank. Don’t panic—that’s the worst thing you can do in a situation like this. Your interviewer certainly won’t expect you to have an advanced knowledge of tennis ball construction techniques, so it’s completely okay if you don’t know where to start. Take a step back and consider your objective. It’s true that tennis balls are fuzzy, and that this characteristic distinguishes them from balls used in other types of sports. With this established, your next task is to figure out what unique features of tennis would require a modification to its particular type of ball. After puzzling this out for a while, you’ll come to the conclusion that it’s the only sport in which the ball’s time in the air is the most important factor. This means that the fuzz must be somehow related to that particular aspect. From here, you can conclude that it helps to slow the ball, thereby making the game actually playable. You can figure all of that out without even the slightest knowledge of physics! It is difficult to predict which brain teaser questions you will likely to get during case interviews. But it certainly does not hurt to work out your problem-solving and critical thinking “muscles” with daily practice. The “hardest logic puzzle ever” (originally published in 1996 by the Harvard Business Review) If worked through those brain teasers and riddles, and if you’re still concerned about interviews, or if you just feel up for an even bigger challenge, try looking up other pre-written brain teasers in your spare time. Though this brain teasers article contained brain teasers with various levels of difficulty, there are far more advanced brain teasers, riddles and logic puzzles all over the internet. Just one example is the so-called “hardest logic puzzle ever,” originally published in 1996 by the Harvard Business Review. Mathematician Richard Smullyan, with a nickname “the undisputed master of logical puzzles” developed a very challenging brainteaser. Smullyan’s colleague, an MIT logic professor named George Boolos, called it “the Hardest Logic Puzzle Ever” and it will become very clear why. If you want to put your new skills to the test, go ahead and give it a shot: You’re given the opportunity to communicate with three divine entities, which will be referred to in this brain teaser as A, B, and C. One of them always speaks the truth, one of them always lies, and one of them responds randomly with either the truth or a lie. They have a unique language that has never been translated by mortals, and though they can understand any language spoken to them, they will only respond with their own. Furthermore, the entities will only speak one of two words to you: “ja” and “da.” One of these words means yes and one of them means no, but you have no idea which is which. You may ask these entities a total of three questions, which they will answer in their language. Only one entity can be addressed with each question. Using this, find a way to determine the identity of all three of them. Before you start to brainstorm, take a moment to reflect on the different strategies that you’ve devised throughout your time spent working through previous brain teasers. The “hardest logic puzzle ever”: Solution If you find yourself scratching your head, know that you’re not alone; this brain teaser is cited as the world’s most difficult for a good reason! At the same time, though, it isn’t impossible. Everything that you need to know in order to find the correct solution is laid out in the riddle itself. There are no tricks or shortcuts this time; the steps that you need to take are purely logical. If you’re completely stumped, the internet can give you a hand—but try to avoid skipping to a solution right away. Instead, pull out a pen and paper and give it your best shot! Here are the three questions you should ask, according to Nautilus: To god A: “Does ‘da’ mean ‘yes’ if and only if you are True and if and only if B is Random?” (We supposed A said, “ja,” making B True or False). To god B: “Does “da” mean ‘yes’ if and only if Pluto is a dwarf planet?” (We supposed B said, “da,” making B True.) And to god B (True) again: “Does ‘da’ mean ‘yes’ if and only if A is Random?” Since B’s True, he must say “da,” which means A is Random, leaving C to be False. Don’t worry if you’re still confused. You can start thinking through the solution with this 2008 paper, which states they came up with the easiest answer to the brain teaser. Or maybe you’d rather not dive right into the biggest challenge possible, which is completely understandable. In any case, whether or not you feel confident in your brain teaser solving abilities, don’t stop practicing. Bodybuilders don’t quit the gym just because they achieved their ideal weight, and the same principle applies here. Keep it up, and don’t let yourself get out of shape. There are numerous books, articles, and websites dedicated to the compilation of brain teasers, and now that you’ve familiarized yourself with some of the essential techniques, you’ll be amazed at how quickly and easily you can solve many of them. Conclusion Whether you’re an aspiring consultant, a seasoned professional, or just a curious person with a bit of spare time, you can always benefit from a good cerebral workout. At the end of the day, you’re going to be the one to set your own challenges—and remember that even the toughest problems have their solutions. As long as you believe in yourself, anything is possible. Hope you enjoyed these consulting brain teasers with answers we selected for you to practice with. If you have good consulting brain teasers to add to this article please let us know in the comments below. 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12713	https://www.cuemath.com/numbers/greater-than-or-equal-to/	LearnPracticeDownload Greater Than or Equal to The greater than or equal to symbol is used to represent inequality in math. It tells us that the given variable is either greater than or equal to a particular value. For example, if x ≥ 3 is given, it means that x is either greater than or equal to 3. It defines a range of values that x can take which starts from 3 and goes up till infinity. \| \| \| --- \| \| 1. \| What does Greater Than or Equal to Mean? \| \| 2. \| Greater Than or Equal to Symbol \| \| 3. \| Greater Than or Equal to Application \| \| 4. \| FAQs on Greater Than or Equal to \| What Does Greater Than or Equal to Mean? 'Greater than or equal to', as the name suggests, means the variable is either greater than or equal to a particular value. The term 'greater than' is used to express that one quantity is greater than the other quantity. The term 'is equal to' is used to express that two quantities are equal. When these terms are combined with each other they make a new term that is Greater Than Or Equal To and this term is used to show that, the quantity or amount value limit could be equal to or greater than the given limit. For example, for a person to be elected as president he or she should be a minimum of 35 years old. This means that a person should be either greater than or equal to 35 years old. Greater Than or Equal to Symbol The 'Greater than or equal to' symbol is used in linear inequalities when we do not know whether the value of a variable is greater than or equal to a particular value. It is expressed by the symbol '≥'. This symbol is the 'greater than' symbol ( >) with a sleeping line under it. The sleeping line below the greater-than sign means 'equal to'. Here are a few examples for 'Greater than or equal to'. x ≥ 100 means the value of x should be greater than or equal to 100. a ≥ - 2 means the value of 'a' should be greater than or equal to -2. Greater Than or Equal to Application The 'Greater than or equal to' symbol is used in mathematics to express the relationship between 2 expressions. The following table shows where and how the 'greater than or equal to' symbol is used along with examples and meanings. \| Symbol \| Example \| Meaning \| --- \| Greater than or equal to, ≥ \| x ≥ 2 2 ≥ x ≥ −1 \| The value of x is greater than or equal to 2. The value of x is between -1 and 2 inclusive of both values. \| ☛Related Articles Comparing and Ordering Greater Than Calculator How To Make a Greater Than or Equal to Sign Less Than or Equal To Read More Download FREE Study Materials Greater Than Less Than Worksheet Greater Than Less Than Worksheet Greater Than Less Than Worksheet Greater Than or Equal to Examples Example 1: If the value of x is greater than or equal to 9, how will this be shown in an expression? Solution: The statement says that the value of x is greater than or equal to 9. So this can be expressed as, x ≥ 9 2. Example 2: State true or false: a.) If the value of 'a' is greater than or equal to 10, this can be expressed as, a ≥ 10 b.) 6 ≥ x ≥ -1 means that the value of x is between -1 and 6 inclusive of both values. Solution: a.) True, if the value of 'a' is greater than or equal to 10, this can be expressed as, a ≥ 10 b.) True, 6 ≥ x ≥ -1 means that the value of x is between -1 and 6 inclusive of both values. 3. Example 3: What does the following expression mean? x ≥ 3 where x ∈ N? Solution: We know that N is the symbol for the set of natural numbers like 1, 2, 3, 4, and so on It is given that x ≥ 3 and x ∈ N This means that x is a natural number that is greater than or equal to 3. Show Answer > Have questions on basic mathematical concepts? Become a problem-solving champ using logic, not rules. Learn the why behind math with our certified experts Book a Free Trial Class Practice Questions on Greater Than or Equal To Check Answer > FAQs on Greater Than or Equal to What is Greater Than or Equal to in Math? Greater than or equal to, as the name suggests, means something is either greater than or equal to some quantity. Greater than or equal to is represented by the symbol "≥". For example, x ≥ −2 means the value of x is greater than or equal to -2. What is the Symbol for Greater Than or Equal to? The symbol of 'greater than or equal to' looks like '≥ '. The open side of the symbol should be in front of the bigger value. The line below the symbol shows that the value could be equal to or more than the limit. For example, x ≥ 5. Here, the value of x should either equal to or more than 5. How do you Explain Greater Than or Equal to? Greater than or equal to is something either greater than or equal to a given quantity. For example, if a florist earns $5 or more than $5 in a day, then it can be either $5, or more than $5. Now, if we assume the earning to be x, then this can be expressed as x ≥ $5. What is the Difference Between Greater Than and Greater Than or Equal to? The greater than symbol is written as > whereas the greater than or equal to is represented as ≥. 'Greater than' means that some variable or number can have any value that is greater than the given limit. Whereas the 'greater than or equal to' symbol states that the number or variable can be equal to or more than the given limit. What is the Difference Between Greater Than or Equal to and Less Than or Equal to? The 'greater than or equal to' sign tells that the amount is either more than or equal to the minimum limit whereas the 'less than or equal to' sign is just the opposite of this sign. Less than or equal to means, the amount is equal to or less than the maximum limit. What is the Difference Between Greater Than or Equal to and Equal to? The 'Greater than or equal to' ( ≥) symbol signifies that the value is either more than or equal to the given limit; whereas the equal to (=) symbol means the quantity is fixed. It is neither less than nor greater than the given value, it is exactly equal to the value. Is 4 Greater Than or Equal to 3? No, we cannot say that 4 is greater than or equal to 3. Because 4 is bigger than 3 and not equal to 3. Therefore, the correct sentence will be 4 is greater than 3. What is the use of Greater Than or Equal to? Greater than or equal to is used to show that one variable is greater than or equal to a given quantity. For example, if a company has a policy to launch the product either at the same price or more than the old price. It can be said that the new product price is greater than or equal to the old price. What is the Solution to 3x + 2.4 Greater than or Equal to 3.0? In order to solve this inequality, we need to write it in the form of a mathematical expression, that is, 3x + 2.4 ≥ 3.0. After this, we can solve for the value of x and we get the result as follows. 3x + 2.4 ≥ 3.0 3x ≥ 3.0 - 2.4 3x ≥ 0.6 x ≥ 0.2 This means the value of x is greater than or equal to 0.2 Math worksheets and visual curriculum FOLLOW CUEMATH Facebook Youtube Instagram Twitter LinkedIn Tiktok MATH PROGRAM Online math classes Online Math Courses online math tutoring Online Math Program After School Tutoring Private math tutor Summer Math Programs Math Tutors Near Me Math Tuition Homeschool Math Online Solve Math Online Curriculum NEW OFFERINGS Coding SAT Science English MATH ONLINE CLASSES 1st Grade Math 2nd Grade Math 3rd Grade Math 4th Grade Math 5th Grade Math 6th Grade Math 7th Grade Math 8th Grade Math 9th Grade Math 10th Grade Math ABOUT US Our Mission Our Journey Our Team MATH TOPICS Algebra 1 Algebra 2 Geometry Calculus math Pre-calculus math Math olympiad Numbers QUICK LINKS Maths Games Maths Puzzles Our Pricing Math Questions Events MATH WORKSHEETS Kindergarten Worksheets 1st Grade Worksheets 2nd Grade Worksheets 3rd Grade Worksheets 4th Grade Worksheets 5th Grade Worksheets 6th Grade Worksheets 7th Grade Worksheets 8th Grade Worksheets 9th Grade Worksheets 10th Grade Worksheets Terms and ConditionsPrivacy Policy
12714	https://www.quora.com/How-do-you-evaluate-sums-in-mathematics-with-no-limits-infinite	Something went wrong. Wait a moment and try again. Visualizing Infinite Seri... Evaluating Hard Integrals... Convergent Series Math Infinites Calculus (Mathematics) Mathematical Concepts Infinite Summation 5 How do you evaluate sums in mathematics with no limits (infinite)? Dries Van Heeswijk Studies Math & Computer Science · 1y Before I answer this question, I want to tell you that re-arranging the terms in an infinite sum can sometimes lead to conflicting results, so you can't do that like we would do normally (for example 1 + 2 + 3 = 1 + 3 + 2) To evaluate an infinite sum, it's handy to see the numbers we're adding as part of a ‘sequence’. we want to look at how the sum progresses as we include more and more terms from our sequence. We can see all these different ‘partial sums’ as part of a new sequence. Now if THIS sequence converges to a finite number, we have our answer. Let's look at a concrete example instead of Before I answer this question, I want to tell you that re-arranging the terms in an infinite sum can sometimes lead to conflicting results, so you can't do that like we would do normally (for example 1 + 2 + 3 = 1 + 3 + 2) To evaluate an infinite sum, it's handy to see the numbers we're adding as part of a ‘sequence’. we want to look at how the sum progresses as we include more and more terms from our sequence. We can see all these different ‘partial sums’ as part of a new sequence. Now if THIS sequence converges to a finite number, we have our answer. Let's look at a concrete example instead of this word mess: let's add 1 + 0.5 + 0.25 + 0.125 + … We make a new sequence out of the partial sums: 1 = 1 1 + 0.5 = 1.5 1 + 0.5 + 0.25 = 1.75 1 + 0.5 + 0.25 + 0.125 = 1.875 Now we can see that this ‘sequence of partial sums’ approaches the number 2, which means that our infinite sum is equal to 2. Again, the result MUST be a finite number, not infinity. If you don't do this, there are some types of infinite sums that can equal different numbers depending on the order in which you add the (infinitely many) numbers. And by the way, ‘converges’ means a sequence (eventually) gets very close to a number, and stays very close to it. So, the sequence 0 1 0 1 0 1 … doesn't converge. This means that the sum 1 - 1 + 1 - 1 + 1 … has no result. I hope this explanation wasn't too confusing, this is a surprisingly weird result to such a good and natural question. Related questions How do I evaluate: n ∑ p = 1 ( n ∑ m = p C m ⋅ m C p ) ? What is the difference in mathematics between saying there is no limit and the limit is infinity? Doesn't saying the limit is infinity imply there is no limit? How do you evaluate ∑ n ≥ 1 1 2 n 2 ( 2 n + 1 ) ? How do you evaluate the sum ∞ ∑ n = 1 n − n ? How do you evaluate limits symbolically? Jes Klinke M.S. in Computer Science & Mathematics, University of Copenhagen (Graduated 2005) · Author has 442 answers and 63.5K answer views · 1y Originally Answered: Can an infinite sum be calculated without using limits? · You can sometimes do informal manipulations such as multiplying by two and subtracting one infinite sum from another. But since the value of an infinite sum is DEFINED by means of limits, you never truly get around them if you want to be formal. Brian Sittinger PhD in Mathematics, University of California, Santa Barbara (Graduated 2006) · Upvoted by Shahnaz , M.Sc Mathematics, Indian Institute of Technology Jodhpur (2024) and Erik Bergland , PhD Mathematics, Brown University (2024) · Author has 8.4K answers and 20.7M answer views · 4y Related How can I evaluate this infinite sum: ∑ ∞ n = 1 ( − 1 ) n 5 n − 1 ? We want to find the value of the (conditionally convergent) infinite series S=∞∑n=1(−1)n5n−1=∞∑n=0(−1)n+15n+4. One approach is to consider the related power series Then, observe that differentiating yields a geometric series: In order to retrieve We want to find the value of the (conditionally convergent) infinite series One approach is to consider the related power series Then, observe that differentiating yields a geometric series: In order to retrieve , noting that and , we simply integrate both sides from 0 to 1: To evaluate this integral, applying the substitution yields This allows for a slightly nicer partial fractions expansion (most painlessly obtained from using roots of unity): Completing the square in the denominators yields Now, we can perform the integration: Remark: A exact answer can be readily found from the work above, but it has already become a sprawling affair. Accordingly, I’ll stop here. Martin Brilliant Former MTS at Bell Labs (1966–1989) · Upvoted by Daniel Roebuck , Integrated Masters Mathematics, University of St Andrews (2024) and Justin Rising , PhD in statistics · Author has 13.6K answers and 12.4M answer views · 1y Related Why, mathematically, are we allowed to define an infinite sum as the limit of its partial sums? I get why the limit of an infinite sum is be equal to the limit of its partial sums, but why do we say the total infinite sum is equal to its limit? We say it because it works. tl;dr: Mathematically, you are allowed to define anything as long as it doesn’t create a contradiction with what you’ve already defined. For example, in real arithmetic there’s no such thing as the square root of minus one. So we define it, call it i, and work out the consequences. We end up with a self-consistent system with no internal contradictions. That system also turns out to be useful. So now we start out with an infinite series. To begin with, we can’t find the sum because doing so would require an infinite number of operations. The sum of the series doesn’t ex We say it because it works. tl;dr: Mathematically, you are allowed to define anything as long as it doesn’t create a contradiction with what you’ve already defined. For example, in real arithmetic there’s no such thing as the square root of minus one. So we define it, call it i, and work out the consequences. We end up with a self-consistent system with no internal contradictions. That system also turns out to be useful. So now we start out with an infinite series. To begin with, we can’t find the sum because doing so would require an infinite number of operations. The sum of the series doesn’t exist. It turns out that if we define the sum of an infinite series as the limit of its partial sums—provided that such a limit exists—we get a self-consistent system with no internal contradictions. That system also turns out to be useful. By the way, we are not talking about infinite sums. We are talking about sums of infinite series. Here is an example of infinite sums, one stanza from a 1775 song, “The King’s Own Regulars”: To Monongehela with fifes and with drums We march’d in fine order, with cannon and bombs: That great expedition cost infinite sums; But a few irregulars cut us all into crumbs. O the old Soldiers of the King, and the King’s own Regulars. Answer to: Why, mathematically, are we allowed to define an infinite sum as the limit of its partial sums? I get why the limit of an infinite sum is be equal to the limit of its partial sums, but why do we say the total infinite sum is equal to its limit? Sponsored by Mutual of Omaha Retiring soon and need Medicare advice? Be prepared for retirement with a recommendation from our Medicare Advice Center. Related questions How do I evaluate this limit as n approaches to infinity? What is an infinite sum that isn’t an infinite sum? How do you evaluate this infinitely nested square root expression? Why do some one-sided limits get evaluated to a finite value while others infinity? How do you evaluate a limit approaching infinity? Jered M. mathematics educator · Upvoted by Justin Rising , PhD in statistics · Author has 3K answers and 5.4M answer views · 1y Related Why, mathematically, are we allowed to define an infinite sum as the limit of its partial sums? I get why the limit of an infinite sum is be equal to the limit of its partial sums, but why do we say the total infinite sum is equal to its limit? While the other answers are correct in saying “because we can,” there is actually something more fundamental and profound going on: until we give it one, “infinite sum” doesn’t have any meaning at all! When we are being careful, we define, e.g., [math]3+5[/math] using some very basic properties and principles: maybe a successor function and finite recursion, for instance. We then define [math]3+5+7[/math] as either math+7[/math] or [math]3+(5+7),[/math] whichever we prefer. From here we can again use finite induction to define any finite sum. But what, in a formal sense, does [math]“3+5+7+9+\dots”[/math] mean? We have to explain carefully what it is su While the other answers are correct in saying “because we can,” there is actually something more fundamental and profound going on: until we give it one, “infinite sum” doesn’t have any meaning at all! When we are being careful, we define, e.g., [math]3+5[/math] using some very basic properties and principles: maybe a successor function and finite recursion, for instance. We then define [math]3+5+7[/math] as either math+7[/math] or [math]3+(5+7),[/math] whichever we prefer. From here we can again use finite induction to define any finite sum. But what, in a formal sense, does [math]“3+5+7+9+\dots”[/math] mean? We have to explain carefully what it is supposed to mean before we can evaluate it. To put this another way, it is perfectly obvious to most of us, once we learn what the notation means, what is meant by [math]\displaystyle \qquad \sum_{i=1}^{37} (2i+1).[/math] But that only gives us limited insight into what is meant by [math]\displaystyle \qquad \sum_{i=1}^{\infty} (2i+1).[/math] And if you think that it’s “obvious”, then what about [math]\displaystyle \qquad \sum_{i=1}^{\infty}\frac{1}{2i+1} \;?[/math] In mathematics, we define things because there is no alternative to defining things. Nathan Hannon Ph. D. in Mathematics, University of California, Davis (Graduated 2021) · Upvoted by Justin Rising , PhD in statistics · Author has 2K answers and 3.5M answer views · 1y Related Why, mathematically, are we allowed to define an infinite sum as the limit of its partial sums? I get why the limit of an infinite sum is be equal to the limit of its partial sums, but why do we say the total infinite sum is equal to its limit? Why, mathematically, are we allowed to define an infinite sum as the limit of its partial sums? I think this question reflects a misunderstanding of where definitions in mathematics come from. There are no math gods handing down definitions that we are required to work with. Definitions are written down by mathematicians. We are “allowed” to define an infinite sum however we want. We could, for example, decide that we want to define the sum of every infinite series to be [math]42[/math]. The problem is that this definition is neither interesting nor useful. That brings us to the second question: I get why th Why, mathematically, are we allowed to define an infinite sum as the limit of its partial sums? I think this question reflects a misunderstanding of where definitions in mathematics come from. There are no math gods handing down definitions that we are required to work with. Definitions are written down by mathematicians. We are “allowed” to define an infinite sum however we want. We could, for example, decide that we want to define the sum of every infinite series to be [math]42[/math]. The problem is that this definition is neither interesting nor useful. That brings us to the second question: I get why the limit of an infinite sum is be equal to the limit of its partial sums, but why do we say the total infinite sum is equal to its limit? Because that is a very useful way to define it. It has a lot of nice properties and can be used to do linear algebra on infinite-dimensional spaces in a way that makes sense. Note that it is not the only useful way to define the sum of an infinite series - there are others, such as Cesàro summation but, in most cases, we stick with the usual definition because for a lot of purposes, it works really well. Unfortunately, a lot of our mathematics education makes it seem like definitions are handed down from the math gods, and does not pay enough attention to why things are defined the way they are. ”We” here refers to mathematicians as a whole. You can define an operation that takes any infinite series and returns the number [math]42[/math], but you should not call it the sum - that would be the equivalent of you deciding to use the word “apple” in English to describe what everyone else has agreed to call an elephant. Footnotes Cesàro summation - Wikipedia Promoted by The Hartford The Hartford Updated 1h What is small business insurance? Small business insurance is a comprehensive type of coverage designed to help protect small businesses from various risks and liabilities. It encompasses a range of policies based on the different aspects of a business’s operations, allowing owners to focus on growth and success. The primary purpose of small business insurance is to help safeguard a business’s financial health. It acts as a safety net, helping to mitigate financial losses that could arise from the unexpected, such as property damage, lawsuits, or employee injuries. For small business owners, it’s important for recovering quickl Small business insurance is a comprehensive type of coverage designed to help protect small businesses from various risks and liabilities. It encompasses a range of policies based on the different aspects of a business’s operations, allowing owners to focus on growth and success. The primary purpose of small business insurance is to help safeguard a business’s financial health. It acts as a safety net, helping to mitigate financial losses that could arise from the unexpected, such as property damage, lawsuits, or employee injuries. For small business owners, it’s important for recovering quickly and maintaining operations. Choosing the right insurance for your small business involves assessing your unique needs and consulting with an advisor to pick from comprehensive policy options. With over 200 years of experience and more than 1 million small business owners served, The Hartford is dedicated to providing personalized solutions that help you focus on growth and success. Get a quote today! Hallie B.Sc in Mechanical Engineering, University of Pittsburgh (Graduated 2019) · Upvoted by Hoosain Ebrahim , BSc Mathematics & Mathematical Statistics (2021) · Author has 523 answers and 1.5M answer views · 4y Related How can I evaluate this infinite sum: [math]\sum_{n=1}^{\infty} \frac{(-1)^n}{5n-1}[/math] ? A convenient way to evaluate the infinite sum [math]\displaystyle I = \sum_{n=1}^{\infty}\frac{(-1)^{n}}{5n-1} \tag{}[/math] is by relating the sum in question to the digamma function and then applying Gauss’s digamma theorem. First we rewrite the sum into the form: [math]\displaystyle I = -\sum_{n=0}^{\infty}\frac{(-1)^n}{5n+4} = -\frac{1}{5}\sum_{n=0}^{\infty}\frac{(-1)^n}{n+4/5}. \tag{}[/math] where we demonstrate here that for positive integers [math]p,q[/math] and [math]p<q[/math], [math]\begin{align} &\sum_{n=0}^{\infty}\frac{(-1)^n}{n+p/q} \ &= \frac{1}{2}\left[\psi\left(\frac{p+q}{2q}\right)-\psi\left(\frac{p}{2q}\right)\right] \ &= \frac{\pi}[/math] A convenient way to evaluate the infinite sum [math]\displaystyle I = \sum_{n=1}^{\infty}\frac{(-1)^{n}}{5n-1} \tag{}[/math] is by relating the sum in question to the digamma function and then applying Gauss’s digamma theorem. First we rewrite the sum into the form: [math]\displaystyle I = -\sum_{n=0}^{\infty}\frac{(-1)^n}{5n+4} = -\frac{1}{5}\sum_{n=0}^{\infty}\frac{(-1)^n}{n+4/5}. \tag{}[/math] where we demonstrate here that for positive integers [math]p,q[/math] and [math]p<q[/math], [math]\begin{align} &\sum_{n=0}^{\infty}\frac{(-1)^n}{n+p/q} \ &= \frac{1}{2}\left[\psi\left(\frac{p+q}{2q}\right)-\psi\left(\frac{p}{2q}\right)\right] \ &= \frac{\pi}{4}\cot\left(\frac{\pi p}{2q}\right)-\frac{\pi}{4}\cot\left(\frac{\pi(p+q)}{2q}\right)+\sum_{k=1}^{q-1}\cos\left(\frac{\pi k(p+q)}{q}\right)\ln\sin\left(\frac{\pi k}{2q}\right)\& -\sum_{k=1}^{q-1}\cos\left(\frac{\pi kp}{q}\right)\ln\sin\left(\frac{\pi k}{2q}\right). \end{align}. \tag{}[/math] We use the above, substituting [math]p=4[/math] and [math]q=5[/math], to obtain [math]I[/math]: [math]\begin{align} I &= -\frac{1}{5}\Bigg[\frac{\pi}{4}\cot\left(\frac{4\pi}{10}\right)-\frac{\pi}{4}\cot\left(\frac{9\pi}{10}\right)+\sum_{k=1}^{4}\cos\left(\frac{9\pi k}{5}\right)\ln\sin\left(\frac{\pi k}{10}\right)\& -\sum_{k=1}^{4}\cos\left(\frac{4\pi k}{5}\right)\ln\sin\left(\frac{\pi k}{10}\right)\Bigg]. \end{align}. \tag{}[/math] This is a fine answer; there’s no reason to expand the finite summations into a sprawling mess. A quick script will tell you that the decimal approximation is [math]\displaystyle I \approx -0.180645759463806. \tag{}[/math] Nick Nicholas Studied at University of Hard Head Impacts · Author has 687 answers and 607.3K answer views · Updated 1y Related How do you evaluate an infinite sum/integral (math)? Usually by analysis, however some sums (and definite integrals) do yield to some simple numerical evaluations The first sum below math[/math] is infinite, the second math[/math] is [math]\dfrac {\pi^2} 6[/math] : tic;1:1E6;cumsum(1./ans); ans(1:1E5:end), toc1 12.0902 12.7833 13.1888 13.4764 13.6996 13.8819 14.0361 14.1696 14.2874Elapsed time is 0.0270009 seconds. tic;1:1E6;cumsum(1./ans.^2); ans(1:1E5:end), toc1 1.6449 1.6449 1.6449 1.6449 1.6449 1.6449 1.6449 1.6449 1.6449Elapsed time is 0.0360022 seconds. pi^2/6 % = 1.6449 And now a Usually by analysis, however some sums (and definite integrals) do yield to some simple numerical evaluations The first sum below math[/math] is infinite, the second math[/math] is [math]\dfrac {\pi^2} 6[/math] : tic;1:1E6;cumsum(1./ans); ans(1:1E5:end), toc1 12.0902 12.7833 13.1888 13.4764 13.6996 13.8819 14.0361 14.1696 14.2874Elapsed time is 0.0270009 seconds. tic;1:1E6;cumsum(1./ans.^2); ans(1:1E5:end), toc1 1.6449 1.6449 1.6449 1.6449 1.6449 1.6449 1.6449 1.6449 1.6449Elapsed time is 0.0360022 seconds. pi^2/6 % = 1.6449 And now a much more interesting example % zeta function partial sums to millionth term, evaluated at first non trivial Riemann-Zeta zerotic;1:1E6;cumsum( (0.5 +i 14.1347251417346937904572519835625).^(-ans)); ans([ [1:9] [10:end/10:0.999end end]]), toc first 9 values 0.002499-0.070659i -0.002487-0.071013i -0.002524-0.070661i -0.002500-0.070658i -0.002499-0.070659i -0.002499-0.070659i -0.002499-0.070659i -0.002499-0.070659i -0.002499-0.070659i values from 10 every 100'000 to a million -0.002499-0.070659i -0.002499-0.070659i -0.002499-0.070659i -0.002499-0.070659i -0.002499-0.070659i -0.002499-0.070659i -0.002499-0.070659i -0.002499-0.070659i -0.002499-0.070659i -0.002499-0.070659i -0.002499-0.070659i Elapsed time is 0.0400021 seconds. Sponsored by OrderlyMeds Is Your GLP-1 Personalized? Find GLP-1 plans tailored to your unique body needs. Brian Sittinger PhD in Mathematics, University of California, Santa Barbara (Graduated 2006) · Upvoted by Nathan Hannon , Ph. D. Mathematics, University of California, Davis (2021) · Author has 8.4K answers and 20.7M answer views · 1y Related How do I evaluate the sum of n(5/6) ^n from 0 to infinity? Let [math]S[/math] denote the infinite series in question. Since the [math]n = 0[/math] term contributes zero to the sum, we can express this as [math]S = \displaystyle \sum_{n=1}^{\infty} n\Big(\frac{5}{6}\Big)^n = \boxed{30}. \tag{}[/math] Multiply both sides by [math]\frac{5}{6}[/math] and reindexing the resulting sum yields [math]\displaystyle \frac{5}{6} S = \sum_{n=1}^{\infty} n\Big(\frac{5}{6}\Big)^{n+1} = \sum_{n=2}^{\infty} (n-1)\Big(\frac{5}{6}\Big)^n. \tag{}[/math] Then, subtracting the two series above yields [math]\displaystyle S - \frac{5}{6} S = \sum_{n=1}^{\infty} n\Big(\frac{5}{6}\Big)^n - \sum_{n=2}^{\infty} (n-1)\Big(\frac{5}{6}\Big)^n, \tag{}[/math] and Let [math]S[/math] denote the infinite series in question. Since the [math]n = 0[/math] term contributes zero to the sum, we can express this as [math]S = \displaystyle \sum_{n=1}^{\infty} n\Big(\frac{5}{6}\Big)^n = \boxed{30}. \tag{}[/math] Multiply both sides by [math]\frac{5}{6}[/math] and reindexing the resulting sum yields [math]\displaystyle \frac{5}{6} S = \sum_{n=1}^{\infty} n\Big(\frac{5}{6}\Big)^{n+1} = \sum_{n=2}^{\infty} (n-1)\Big(\frac{5}{6}\Big)^n. \tag{}[/math] Then, subtracting the two series above yields [math]\displaystyle S - \frac{5}{6} S = \sum_{n=1}^{\infty} n\Big(\frac{5}{6}\Big)^n - \sum_{n=2}^{\infty} (n-1)\Big(\frac{5}{6}\Big)^n, \tag{}[/math] and thus [math]\begin{align} \displaystyle \frac{1}{6} S &= 1 \cdot \frac{5}{6} + \sum_{n=2}^{\infty} \Big(\frac{5}{6}\Big)^n\ &= \frac{5}{6} + \frac{(\frac{5}{6})^2}{1 - \frac{5}{6}}\ &= 5. \end{align} \tag{}[/math] Solving for [math]S[/math], we conclude that [math]S = \displaystyle \sum_{n=0}^{\infty} n\Big(\frac{5}{6}\Big)^n = \boxed{30}. \tag{}[/math] Alexander Farrugia Loves numbers. · Upvoted by Justin Rising , PhD in statistics and Jeremy Collins , M.A. Mathematics, Trinity College, Cambridge · Author has 3.2K answers and 27.4M answer views · 6y Related Are infinite sums of any use? Have you ever used trigonometric functions like [math]\sin x[/math]? Or the exponential function [math]e^x[/math]? Then you know that infinite sums are quite useful. Because both [math]\sin x[/math] and [math]e^x[/math] are actually infinite sums: [math]\displaystyle\sin x=\sum_{k=0}^\infty \frac{(-1)^kx^{2k+1}}{(2k+1)!}.[/math] [math]\displaystyle e^x=\sum_{k=0}^\infty \frac{x^k}{k!}.[/math] And if you ever wondered how your calculator can evaluate [math]\sin 53^\circ[/math] or [math]e^{-1.6}[/math] to a nice number of decimal places, well now you know: it uses the above infinite sums until the required level of accuracy is reached. Mohammad Afzaal Butt B.Sc in Mathematics & Physics, Islamia College Gujranwala (Graduated 1977) · Author has 24.6K answers and 22.8M answer views · 6y Related How do you evaluate the sum of (2n) /(2^n) from n=1 to infinity? [math]\text{Let}\,\,S = \displaystyle\sum_{n = 0}^{\infty} \dfrac{2n}{2^n}[/math] [math]= \dfrac{2}{2} + \dfrac{4}{2^2} + \dfrac{6}{2^3} + \dfrac{8}{2^4} + \dfrac{10}{2^5} + \cdots[/math] [math]= 1 +\dfrac{4}{2^2} + \dfrac{6}{2^3} + \dfrac{8}{2^4} + \dfrac{10}{2^5} + \cdots\tag 1[/math] [math]\implies 2 S =2 +\dfrac{4}{2} + \dfrac{6}{2^2} + \dfrac{8}{2^3} + \dfrac{10}{2^4} + \cdots\tag 2[/math] [math]\text{by (2) - (1)}[/math] [math]S = 2 + 2 - 1 + \left(\dfrac{6}{2^2} - \dfrac{4}{2^2} \right) + \left(\dfrac{8}{2^3} - \dfrac{6}{2^3} \right) + \left(\dfrac{10}{2^4} - \dfrac{8}{2^4} \right) + \cdots[/math] [math]=3 + \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + \cdots[/math] [math]= 3 + \dfrac{[/math] [math]\text{Let}\,\,S = \displaystyle\sum_{n = 0}^{\infty} \dfrac{2n}{2^n}[/math] [math]= \dfrac{2}{2} + \dfrac{4}{2^2} + \dfrac{6}{2^3} + \dfrac{8}{2^4} + \dfrac{10}{2^5} + \cdots[/math] [math]= 1 +\dfrac{4}{2^2} + \dfrac{6}{2^3} + \dfrac{8}{2^4} + \dfrac{10}{2^5} + \cdots\tag 1[/math] [math]\implies 2 S =2 +\dfrac{4}{2} + \dfrac{6}{2^2} + \dfrac{8}{2^3} + \dfrac{10}{2^4} + \cdots\tag 2[/math] [math]\text{by (2) - (1)}[/math] [math]S = 2 + 2 - 1 + \left(\dfrac{6}{2^2} - \dfrac{4}{2^2} \right) + \left(\dfrac{8}{2^3} - \dfrac{6}{2^3} \right) + \left(\dfrac{10}{2^4} - \dfrac{8}{2^4} \right) + \cdots[/math] [math]=3 + \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + \cdots[/math] [math]= 3 + \dfrac{\dfrac{1}{2}}{1 - \dfrac{1}{2}}\quad\because \,\, S_{\infty} = \dfrac{a}{1 - r}[/math] [math]=3 + 1[/math] [math]= 4[/math] Lorenz Müller · 8y Related Why are Infinite Sums so hard? Though I agree that most infinite sums are hard and unintuitive to evaluate, I think some infinite sums are easy! The most famous one is probably the one from Zeno’s paradox of the arrow: An arrow is traveling between a bow (from which it is shot) to a target. Now you could imagine marking some locations along its track: As it is shot mark the half-way point. Then when the arrow has reached the half-way point (let’s call it 0.5), you mark the half-way point between where the arrow is now (0.5) and the target (a point we could call 0.75). Imagine repeating this procedure until the arrow arrives Though I agree that most infinite sums are hard and unintuitive to evaluate, I think some infinite sums are easy! The most famous one is probably the one from Zeno’s paradox of the arrow: An arrow is traveling between a bow (from which it is shot) to a target. Now you could imagine marking some locations along its track: As it is shot mark the half-way point. Then when the arrow has reached the half-way point (let’s call it 0.5), you mark the half-way point between where the arrow is now (0.5) and the target (a point we could call 0.75). Imagine repeating this procedure until the arrow arrives at its target. … Of course it’s a little tricky to mark all the points in this sequence, because there are infinitely many of them. But on the upside we have found an infinite series that clearly converges to 1.0, namely the sum of half plus half of a half plus half of a half of a half aso. or written more compactly [math]\lim_{n \rightarrow \infty} \sum_i^n 1/2^i =1[/math] Saad Abdoulaziz PG in Quantum Computing & Mathematics, Ain Shams University (Graduated 1998) · Apr 19 Related Is it possible to evaluate an infinite sum at infinity? What would happen if it were evaluated at that point? I think the real meaning of infinite summation of a certain function is not that the function has a descent to be finite when it tends to infinity, but the practical meaning is that the function tends to the periodic form when it tends to infinity, when we say cos(x), this function extends to infinity x, we cannot say that it ends at zero.. At the same time we cannot say that going to infinity gives infinity.. In nature we find the increase of the number tends to stability in a periodic form. Which is the periodic description. This does not mean that there is no beginning and no end because no I think the real meaning of infinite summation of a certain function is not that the function has a descent to be finite when it tends to infinity, but the practical meaning is that the function tends to the periodic form when it tends to infinity, when we say cos(x), this function extends to infinity x, we cannot say that it ends at zero.. At the same time we cannot say that going to infinity gives infinity.. In nature we find the increase of the number tends to stability in a periodic form. Which is the periodic description. This does not mean that there is no beginning and no end because nothing starts from nothing, just as there is no moving body in the physical world that has endless energy that continues to infinity. Related questions How do I evaluate: ? What is the difference in mathematics between saying there is no limit and the limit is infinity? Doesn't saying the limit is infinity imply there is no limit? How do you evaluate ? How do you evaluate the sum ? How do you evaluate limits symbolically? How do I evaluate this limit as n approaches to infinity? What is an infinite sum that isn’t an infinite sum? How do you evaluate this infinitely nested square root expression? Why do some one-sided limits get evaluated to a finite value while others infinity? How do you evaluate a limit approaching infinity? How do you evaluate an integral when the lower limit is negative? What is the concept of the limit of an infinite series and what occurs when we take its limit? What are some examples of numbers that have no limit but aren't infinite? What is the process for evaluating an infinite limit in calculus? Is infinite equal to infinite? About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
12715	https://www.youtube.com/watch?v=uNweU6I4Icw	Polynomial special products: difference of squares \| Algebra 2 \| Khan Academy Khan Academy 9090000 subscribers 98 likes Description 10160 views Posted: 26 Apr 2019 Keep going! Check out the next lesson and practice what you’re learning: The difference of squares pattern tells us that (a+b)(a-b)=a²-b². This can be used to expand (x+2)(x-2) as x²-4, but also to expand (3+5x⁴)(3-5x⁴) as 9-25x⁸, or (3y²+2y⁵)(3y²-2y⁵) as 9y⁴-4y¹⁰. View more lessons or practice this subject at Khan Academy is a nonprofit organization with the mission of providing a free, world-class education for anyone, anywhere. We offer quizzes, questions, instructional videos, and articles on a range of academic subjects, including math, biology, chemistry, physics, history, economics, finance, grammar, preschool learning, and more. We provide teachers with tools and data so they can help their students develop the skills, habits, and mindsets for success in school and beyond. Khan Academy has been translated into dozens of languages, and 15 million people around the globe learn on Khan Academy every month. As a 501(c)(3) nonprofit organization, we would love your help! Donate or volunteer today! Donate here: Volunteer here: 5 comments Transcript: earlier in our mathematical adventures we had expanded things like x plus y times x minus y just as a bit of review this is going to be equal to x times x which is x squared plus x times negative y which is negative x y plus y times x which is plus x y and then minus y times y or you could say y times a negative y so it's going to be minus y squared negative xy positive xy so this is just going to simplify to x squared minus y squared and this is all review we covered it and when we thought about factoring things that are differences of squares we thought about this when we were first learning to multiply binomials and what we're going to do now is essentially just do the same thing but do it with slightly more complicated expressions and so another way of expressing what we just did is we could also write something like a plus b times a minus b is going to be equal to what well it's going to be equal to a squared minus b squared the only difference between what i did up here and what i did over here is instead of an x i wrote an a instead of a y i wrote a b so given that let's see if we can expand and then combine like terms for if i'm multiplying these two expressions say i'm multiplying three plus five x to the fourth times three minus five x to the fourth pause this video and see if you can work this out all right well there's two ways to approach it you could just approach it exactly the way that i approached it up here but we already know that when we have this pattern where we have something plus something times that same original something minus the other something well that's going to be of the form of this thing squared minus this thing squared and remember the only reason why i'm applying that is i have a 3 right over here and here so the 3 is playing the role of the a so let me write that down that is our a and then the role of the b is being played by 5x to the fourth so that is our b right over there so this is going to be equal to a squared minus b squared but our a is 3 so it's going to be equal to 3 squared minus and then our b is 5x to the 4th minus 5x to the 4th squared now what does all of this simplify to well this is going to be equal to 3 squared is 9 and then minus 5x to the fourth squared let's see 5 squared is 25 and then x to the 4th squared well that is just going to be x to the 4 times x to the fourth which is just x to the 8th another way to think about it our exponent properties we are this is the same thing as 5 squared times x to the 4th squared if i raise them to an exponent and then raise that to another exponent i multiply the exponents and there you have it let's do another example let's say that i were to ask you what is three y squared plus two y to the fifth times three y squared minus 2y to the fifth pause this video and see if you can work that out well we're going to do it the same way you can of course always just try to expand it out the way we did originally but we could recognize here that hey i have an a plus a b times the a minus a b so that's going to be equal to our a squared so what's 3y squared well that's going to be 9 y to the 4th minus our b squared well what's 2y to the fifth squared well 2 squared is 4 and y to the 5th squared is y to the 5 times 2 y to the 10th power and there's no further simplification that i could do here i can't combine any like terms and so we are done here as well
12716	https://www.bartleby.com/solution-answer/chapter-3-problem-64pe-college-physics-1st-edition/9781938168000/a-use-the-distance-and-velocity-data-in-figure-364-to-find-the-rate-of-expansion-as-a-function-of/7f1546b6-7ded-11e9-8385-02ee952b546e	(a) Use the distance and velocity data in Figure 3.64 to find the rate of expansion as a function of distance. (b) If you extrapolate back in time, how long ago would all of the galaxies have been at approximately the same position? The two parts of this problem give you some idea of how the Hubble constant for universal expansion and the time back to the Big Bang are determined, respectively. Figure 3.64 Five galaxies on a straight line, showing their distances and velocities relative to the Milky Way (MW) Galaxy. The distances are in millions of light years (Mly), where a light year is the distance light travels in one year. The velocities are nearly proportional to the distances. The sizes of the galaxies are greatly exaggerated; an average galaxy is about 0.1 MlY across. \| bartleby Skip to main content close Homework Help is Here – Start Your Trial Now!arrow_forward Literature guidesConcept explainersWriting guidePopular textbooksPopular high school textbooksPopular Q&ABusinessAccountingBusiness LawEconomicsFinanceLeadershipManagementMarketingOperations ManagementEngineeringAI and Machine LearningBioengineeringChemical EngineeringCivil EngineeringComputer EngineeringComputer ScienceCybersecurityData Structures and AlgorithmsElectrical EngineeringMechanical EngineeringLanguageSpanishMathAdvanced MathAlgebraCalculusGeometryProbabilityStatisticsTrigonometryScienceAdvanced PhysicsAnatomy and PhysiologyBiochemistryBiologyChemistryEarth ScienceHealth & NutritionHealth ScienceNursingPhysicsSocial ScienceAnthropologyGeographyHistoryPolitical SciencePsychologySociology learn writing tools expand_more plus study resources expand_more Log In Sign Up expand_more menu SEARCH Homework help starts here! ASK AN EXPERT ASK SciencePhysicsCollege Physics (a) Use the distance and velocity data in Figure 3.64 to find the rate of expansion as a function of distance. (b) If you extrapolate back in time, how long ago would all of the galaxies have been at approximately the same position? The two parts of this problem give you some idea of how the Hubble constant for universal expansion and the time back to the Big Bang are determined, respectively. Figure 3.64 Five galaxies on a straight line, showing their distances and velocities relative to the Milky Way (MW) Galaxy. The distances are in millions of light years (Mly), where a light year is the distance light travels in one year. The velocities are nearly proportional to the distances. The sizes of the galaxies are greatly exaggerated; an average galaxy is about 0.1 MlY across. (a) Use the distance and velocity data in Figure 3.64 to find the rate of expansion as a function of distance. (b) If you extrapolate back in time, how long ago would all of the galaxies have been at approximately the same position? The two parts of this problem give you some idea of how the Hubble constant for universal expansion and the time back to the Big Bang are determined, respectively. Figure 3.64 Five galaxies on a straight line, showing their distances and velocities relative to the Milky Way (MW) Galaxy. The distances are in millions of light years (Mly), where a light year is the distance light travels in one year. The velocities are nearly proportional to the distances. The sizes of the galaxies are greatly exaggerated; an average galaxy is about 0.1 MlY across. BUY College Physics 1st Edition ISBN: 9781938168000 Author: Paul Peter Urone, Roger Hinrichs Publisher: OpenStax College expand_less expand_more Chapter 3 : Two-dimensional Kinematics expand_more Section: Chapter Questions format_list_bulleted Problem 64PE: (a) Use the distance and velocity data in Figure 3.64 to find the rate of expansion as a function of... See similar textbooks Concept explainers Displacement, Velocity and Acceleration In classical mechanics, kinematics deals with the motion of a particle. It deals only with the position, velocity, acceleration, and displacement of a particle. It has no concern about the source of motion. Linear Displacement The term "displacement" refers to when something shifts away from its original "location," and "linear" refers to a straight line. As a result, “Linear Displacement” can be described as the movement of an object in a straight line along a single axis, fo… Videos Textbook Question Chapter 3, Problem 64PE (a) Use the distance and velocity data in Figure 3.64 to find the rate of expansion as a function of distance. (b) If you extrapolate back in time, how long ago would all of the galaxies have been at approximately the same position? The two parts of this problem give you some idea of how the Hubble constant for universal expansion and the time back to the Big Bang are determined, respectively. Figure 3.64 Five galaxies on a straight line, showing their distances and velocities relative to the Milky Way (MW) Galaxy. The distances are in millions of light years (Mly), where a light year is the distance light travels in one year. The velocities are nearly proportional to the distances. The sizes of the galaxies are greatly exaggerated; an average galaxy is about 0.1 MlY across. Expert Solution & Answer Trending now This is a popular solution! See solution Check out sample textbook solution chevron_left Previous chevron_left Chapter 3, Problem 63PEchevron_right Next chevron_right Chapter 3, Problem 65PE Students have asked these similar questions Wailua falls has a height of 25.0 m . After a heay rainfall in March, the water above the falls is travelling at 1.30 m/s . add Starting from rest, an alpine coaster ascends a gently inclined track, tilted 20° from the horizontal, thanks to a rope that pulls with a force of 550 N at an angle of 35° relative to the track. Together the coaster and rider have mass 85 kg. During this motion, there is a friction force between the track and the coaster with magnitude 70 N.a. Draw a free body diagram for the coaster. For each force, be sure to include the type of force, which object the force is acting on,and which object is exerting the force.b. 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Chemistry: Structure and Properties (2nd Edition) Knowledge Booster Learn more about Displacement, velocity and acceleration Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below. Similar questions arrow_back_ios arrow_forward_ios 6.32 •• To stretch an ideal spring 3.00 cm from its unstretched length, 12.0 J of work must be done. (a) What is the force constant of this spring? (b) What magnitude force is needed to stretch the spring 3.00 cm from its unstretched length? (c) How much work must be done to compress this spring 4.00 cm from its unstretched length, and what force is needed to compress it this distance? arrow_forward6.31 ⚫ Stopping Distance. A car is traveling on a level road with speed U at the instant when the brakes lock, so that the tires slide rather than roll. (a) Use the work-energy theorem to calculate the minimum stop- ping distance of the car in terms of up, g, and the coefficient of kinetic friction μ between the tires and the road. (b) By what factor would the minimum stopping distance change if (i) the coefficient of kinetic fric- tion were doubled, or (ii) the initial speed were doubled, or (iii) both the coefficient of kinetic friction and the initial speed were doubled? arrow_forward6.25 ⚫ A sled with mass 12.00 kg moves in a straight line on a friction- less, horizontal surface. At one point in its path, its speed is 4.00 m/s; after it has traveled 2.50 m beyond this point, its speed is 6.00 m/s. Use the work-energy theorem to find the net force acting on the sled, as- suming that this force is constant and that it acts in the direction of the sled's motion. arrow_forward 6.22 ⚫. Use the work-energy theorem to solve each of these prob- lems. You can use Newton's laws to check your answers. (a) A skier moving at 5.00 m/s encounters a long, rough horizontal patch of snow having a coefficient of kinetic friction of 0.220 with her skis. How far does she travel on this patch before stopping? (b) Suppose the rough patch in part (a) was only 2.90 m long. How fast would the skier be moving when she reached the end of the patch? (c) At the base of a fric- tionless icy hill that rises at 25.0° above the horizontal, a toboggan has a speed of 12.0 m/s toward the hill. How high vertically above the base will it go before stopping? arrow_forward[1. Proton-beam therapy is a treatment in which a large number of high-energy protons are fired at a tumor. Upon colliding with the tumor, the protons' kinetic energy breaks apart the tumor's DNA and kill its cells. The protons acquire their kinetic energy by being accelerated from rest through the electric field of a capacitor, as shown below (assume that the gap the protons exit through is small enough to be irrelevant to any calculations). Suppose a particular tumor requires 1 PeV (peta- electronvolt; 1015 eV) worth of proton kinetic energy to be sufficiently destroyed, and that you have 1012 protons to throw at it. If the capacitor plates are squares with a side length of 10 cm that are separated by a distance of 0.1 mm, what is the minimum amount of charge you need to put on the capacitor plates in order to destroy the tumor by firing all of the protons you have at it. An order-of-magnitude estimate is sufficient. electric potential, potential energy, voltage across a capacitor,… arrow_forwardThe electric potential, which is a scalar field, may be represented graphically by equipotential curves. The lines of electric field may be obtained from those equipotentials.Several points have been indicated with black dots on the diagram below. Consider the direction of the electric field at each of those points, and place the best-choice arrow label, or the vectorE =0 label, if appropriate, to the corresponding bucket. arrow_forward 6.15 •• On a farm, you are pushing on a stubborn pig with a constant horizontal force with magnitude 30.0 N and direction 37.0° counter- clockwise from the +x-axis. How much work does this force do during a displacement of the pig that is (a) = (5.00 m); (b) = (c) = (2.00 m) + (4.00 m)ĵ? (6.00 m)ĵ; arrow_forward6.16 •• A 1.50 kg book is sliding along a rough horizontal surface. At point A it is moving at 3.21 m/s, and at point B it has slowed to 1.25 m/s. (a) How much total work was done on the book between A and B? (b) If -0.750 J of total work is done on the book from B to C, how fast is it moving at point C? (c) How fast would it be moving at C if +0.750 J of total work was done on it from B to C? arrow_forward6.1 •You push your physics book 1.50 m along a horizontal tabletop with a horizontal push of 2.40 N while the opposing force of friction is 0.600 N. How much work does each of the following forces do on the book: (a) your 2.40 N push, (b) the friction force, (c) the normal force from the tabletop, and (d) gravity? (e) What is the net work done on the book? arrow_forward 1. Two bodies with the masses m₁ and m2 are moving with the initial velocity V1 and V2, respectively, on a horizontal surface, as shown in the figure below. Friction is described by μk. The first body stops at a distance D1. and the second one stops at a distance D2. (a) (2p) Use kinematic equations to determine the value of the stopping distance for the first body, D₁. Show how you calculate the acceleration and all the steps to determine the stopping distance. SHOW YOUR WORK (starting with Newton's Second Law). (no numerical value is required if you solve in symbols). (b) (2p) Use kinematic equations to determine the value of the stopping distance for the second body, D₂. Show your work or explain your answer. (no numerical value is required if you solve in symbols). (c) (6p) Determine the ratio between D1 and D2. D1/D2. Show your work or explain your answer. (Note: Ratio is a number, the result of the division of two values.) Use g = 9.80 m/s² (only if you need it). All of the… arrow_forward1. Describe 6 misconceptions about nuclear power. Use the book Physics and Technology for Future Presidents by Richard A. Muller in chapter 5, explain why these are misconceptions. Remember, explain why the statements are misconceptions. 2. Explain 6 different advantages of nuclear power. Remember, explain why these are advantages. 3. Name the locations of nuclear power plants in Florida, United States. 4. Include references from the Book Physics and Technology for Future Presidents by Richard A. Muller and other references and provide the links respectively. arrow_forwardA planar insulating sheet with an infinitesimal thickness has charge per unit area σ and is parallel to the x−z plane, as shown. (The arrows indicate that the object extends for a long distance in the horizontal plane.) By the symmetry of the problem, the electric field may be expressed as vectorE=Ej^ . This problem relates the change in the electric field from the lower side to the upper side of the insulating sheet to the surface charge density. A) Provide a vector expression, in Cartesian unit-vector notation, for the electric field at points just above the planar insulating sheet. B) Provide a vector expression, in Cartesian unit-vector notation, for the electric field at points just below the planar insulating sheet. C) Provide an expression for the change in the y component of the electric field,ΔE, moving from infinitesimally below the plane to infinitesimally above the plane. arrow_forward arrow_back_ios SEE MORE QUESTIONS arrow_forward_ios Recommended textbooks for you arrow_back_ios arrow_forward_ios Principles of Physics: A Calculus-Based Text Physics ISBN:9781133104261 Author:Raymond A. Serway, John W. Jewett Publisher:Cengage Learning College Physics Physics ISBN:9781938168000 Author:Paul Peter Urone, Roger Hinrichs Publisher:OpenStax College University Physics Volume 3 Physics ISBN:9781938168185 Author:William Moebs, Jeff Sanny Publisher:OpenStax Modern Physics Physics ISBN:9781111794378 Author:Raymond A. Serway, Clement J. Moses, Curt A. Moyer Publisher:Cengage Learning Glencoe Physics: Principles and Problems, Student... Physics ISBN:9780078807213 Author:Paul W. Zitzewitz Publisher:Glencoe/McGraw-Hill College Physics Physics ISBN:9781305952300 Author:Raymond A. 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12717	https://en.wikipedia.org/wiki/Vector_projection	Jump to content Search Contents (Top) 1 Notation 2 Definitions based on angle alpha 2.1 Scalar projection 2.2 Vector projection 2.3 Vector rejection 3 Definitions in terms of a and b 3.1 Scalar projection 3.2 Vector projection 3.3 Scalar rejection 3.4 Vector rejection 4 Properties 4.1 Scalar projection 4.2 Vector projection 4.3 Vector rejection 5 Matrix representation 6 Uses 7 Generalizations 7.1 Vector projection on a plane 8 See also 9 References 10 External links Vector projection Български Ελληνικά Español Esperanto فارسی Nederlands 日本語 Português Türkçe Українська Tiếng Việt Edit links Article Talk Read Edit View history Tools Actions Read Edit View history General What links here Related changes Upload file Permanent link Page information Cite this page Get shortened URL Download QR code Print/export Download as PDF Printable version In other projects Wikidata item Appearance From Wikipedia, the free encyclopedia Concept in linear algebra For more general concepts, see Projection (linear algebra) and Projection (mathematics). \| \| \| This article's lead section may be too long. Please read the length guidelines and help move details into the article's body. (September 2024) (Learn how and when to remove this message) \| The vector projection (also known as the vector component or vector resolution) of a vector a on (or onto) a nonzero vector b is the orthogonal projection of a onto a straight line parallel to b. The projection of a onto b is often written as or a∥b. The vector component or vector resolute of a perpendicular to b, sometimes also called the vector rejection of a from b (denoted or a⊥b), is the orthogonal projection of a onto the plane (or, in general, hyperplane) that is orthogonal to b. Since both and are vectors, and their sum is equal to a, the rejection of a from b is given by: To simplify notation, this article defines and Thus, the vector is parallel to the vector is orthogonal to and The projection of a onto b can be decomposed into a direction and a scalar magnitude by writing it as where is a scalar, called the scalar projection of a onto b, and b̂ is the unit vector in the direction of b. The scalar projection is defined as where the operator ⋅ denotes a dot product, ‖a‖ is the length of a, and θ is the angle between a and b. The scalar projection is equal in absolute value to the length of the vector projection, with a minus sign if the direction of the projection is opposite to the direction of b, that is, if the angle between the vectors is more than 90 degrees. The vector projection can be calculated using the dot product of and as: Notation [edit] This article uses the convention that vectors are denoted in a bold font (e.g. a1), and scalars are written in normal font (e.g. a1). The dot product of vectors a and b is written as , the norm of a is written ‖a‖, the angle between a and b is denoted θ. Definitions based on angle alpha [edit] Scalar projection [edit] Main article: Scalar projection The scalar projection of a on b is a scalar equal to where θ is the angle between a and b. A scalar projection can be used as a scale factor to compute the corresponding vector projection. Vector projection [edit] The vector projection of a on b is a vector whose magnitude is the scalar projection of a on b with the same direction as b. Namely, it is defined as where is the corresponding scalar projection, as defined above, and is the unit vector with the same direction as b: Vector rejection [edit] By definition, the vector rejection of a on b is: Hence, Definitions in terms of a and b [edit] When θ is not known, the cosine of θ can be computed in terms of a and b, by the following property of the dot product a ⋅ b Scalar projection [edit] By the above-mentioned property of the dot product, the definition of the scalar projection becomes: In two dimensions, this becomes Vector projection [edit] Similarly, the definition of the vector projection of a onto b becomes: which is equivalent to either or Scalar rejection [edit] In two dimensions, the scalar rejection is equivalent to the projection of a onto , which is rotated 90° to the left. Hence, Such a dot product is called the "perp dot product." Vector rejection [edit] By definition, Hence, By using the Scalar rejection using the perp dot product this gives Properties [edit] Scalar projection [edit] Main article: Scalar projection The scalar projection a on b is a scalar which has a negative sign if 90 degrees < θ ≤ 180 degrees. It coincides with the length ‖c‖ of the vector projection if the angle is smaller than 90°. More exactly: a1 = ‖a1‖ if 0° ≤ θ ≤ 90°, a1 = −‖a1‖ if 90° < θ ≤ 180°. Vector projection [edit] The vector projection of a on b is a vector a1 which is either null or parallel to b. More exactly: a1 = 0 if θ = 90°, a1 and b have the same direction if 0° ≤ θ < 90°, a1 and b have opposite directions if 90° < θ ≤ 180°. Vector rejection [edit] The vector rejection of a on b is a vector a2 which is either null or orthogonal to b. More exactly: a2 = 0 if θ = 0° or θ = 180°, a2 is orthogonal to b if 0 < θ < 180°, Matrix representation [edit] The orthogonal projection can be represented by a projection matrix. To project a vector onto the unit vector a = (ax, ay, az), it would need to be multiplied with this projection matrix: Uses [edit] The vector projection is an important operation in the Gram–Schmidt orthonormalization of vector space bases. It is also used in the separating axis theorem to detect whether two convex shapes intersect. Generalizations [edit] Since the notions of vector length and angle between vectors can be generalized to any n-dimensional inner product space, this is also true for the notions of orthogonal projection of a vector, projection of a vector onto another, and rejection of a vector from another. Vector projection on a plane [edit] In some cases, the inner product coincides with the dot product. Whenever they don't coincide, the inner product is used instead of the dot product in the formal definitions of projection and rejection. For a three-dimensional inner product space, the notions of projection of a vector onto another and rejection of a vector from another can be generalized to the notions of projection of a vector onto a plane, and rejection of a vector from a plane. The projection of a vector on a plane is its orthogonal projection on that plane. The rejection of a vector from a plane is its orthogonal projection on a straight line which is orthogonal to that plane. Both are vectors. The first is parallel to the plane, the second is orthogonal. For a given vector and plane, the sum of projection and rejection is equal to the original vector. Similarly, for inner product spaces with more than three dimensions, the notions of projection onto a vector and rejection from a vector can be generalized to the notions of projection onto a hyperplane, and rejection from a hyperplane. In geometric algebra, they can be further generalized to the notions of projection and rejection of a general multivector onto/from any invertible k-blade. See also [edit] Scalar projection Vector notation References [edit] ^ Perwass, G. (2009). Geometric Algebra With Applications in Engineering. Springer. p. 83. ISBN 9783540890676. ^ a b c "Scalar and Vector Projections". www.ck12.org. Retrieved 2020-09-07. ^ "Dot Products and Projections". Archived from the original on 2016-05-31. Retrieved 2010-09-05. ^ M.J. Baker, 2012. Projection of a vector onto a plane. Published on www.euclideanspace.com. External links [edit] Projection of a vector onto a plane \| v t e Linear algebra \| \| Outline Glossary Template:Matrix classes \| \| Linear equations \| Linear equation System of linear equations Determinant Minor Cauchy–Binet formula Cramer's rule Gaussian elimination Gauss–Jordan elimination Overcompleteness Strassen algorithm \| \| Matrices \| Matrix Matrix addition Matrix multiplication Basis transformation matrix Characteristic polynomial Spectrum Trace Eigenvalue, eigenvector and eigenspace Cayley–Hamilton theorem Jordan normal form Weyr canonical form Rank Inverse, Pseudoinverse Adjugate, Transpose Dot product Symmetric matrix, Skew-symmetric matrix Orthogonal matrix, Unitary matrix Hermitian matrix, Antihermitian matrix Positive-(semi)definite Pfaffian Projection Spectral theorem Perron–Frobenius theorem Diagonal matrix, Triangular matrix, Tridiagonal matrix Block matrix Sparse matrix Hessenberg matrix, Hessian matrix Vandermonde matrix Stochastic matrix, Toeplitz matrix, Circulant matrix, Hankel matrix (0,1)-matrix List of matrices \| \| Matrix decompositions \| Cholesky decomposition LU decomposition QR decomposition Polar decomposition Spectral theorem Singular value decomposition Higher-order singular value decomposition Schur decomposition Schur complement Haynsworth inertia additivity formula Reducing subspace \| \| Relations and computations \| Matrix equivalence Matrix congruence Matrix similarity Matrix consimilarity Row equivalence Elementary row operations Householder transformation Least squares Linear least squares Gram–Schmidt process Woodbury matrix identity \| \| Vector spaces \| Vector space Linear combination Linear span Linear independence Basis, Hamel basis Change of basis Dimension theorem for vector spaces Hamel dimension Examples of vector spaces Linear map Shear mapping Squeeze mapping Linear subspace Row and column spaces, Null space Rank–nullity theorem Nullity theorem Cyclic subspace Dual space, Linear functional Category of vector spaces \| \| Structures \| Topological vector space Normed vector space Inner product space Euclidean space Orthogonality Orthogonal complement Orthogonal projection Orthogonal group Pseudo-Euclidean space Null vector Indefinite orthogonal group Orientation Improper rotation Symplectic structure \| \| Multilinear algebra \| Multilinear algebra Tensor Tensors (classical) Component-free treatment of tensors Outer product Tensor algebra Exterior algebra Symmetric algebra Clifford algebra Geometric algebra Bivector Multivector Gamas's theorem \| \| Affine and projective \| Affine space Affine transformation, Affine group, Affine geometry Affine coordinate system, Flat (geometry) Cartesian coordinate system Euclidean group Poincaré group Galilean group Projective space Projective transformation Projective geometry Projective linear group Quadric \| \| Numerical linear algebra \| Numerical linear algebra Floating-point arithmetic Numerical stability Basic Linear Algebra Subprograms Sparse matrix Comparison of linear algebra libraries \| \| \| Retrieved from " Categories: Operations on vectors Transformation (function) Functions and mappings Hidden categories: Articles with short description Short description matches Wikidata Wikipedia introduction cleanup from September 2024 All pages needing cleanup Articles covered by WikiProject Wikify from September 2024 All articles covered by WikiProject Wikify Vector projection Add topic
12718	https://pubmed.ncbi.nlm.nih.gov/20955095/	Comparison of first and second trimester ultrasound screening for fetal anomalies in the southeast region of Sweden - PubMed Clipboard, Search History, and several other advanced features are temporarily unavailable. Skip to main page content An official website of the United States government Here's how you know The .gov means it’s official. Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site. The site is secure. The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely. 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Comparison of first and second trimester ultrasound screening for fetal anomalies in the southeast region of Sweden Eric Hildebrand1,Anders Selbing,Marie Blomberg Affiliations Expand Affiliation 1 Department of Clinical and Experimental Medicine, University of Linköping, Sweden. eric.hildebrand@lio.se PMID: 20955095 DOI: 10.3109/00016349.2010.517307 Free article Item in Clipboard Comparative Study Comparison of first and second trimester ultrasound screening for fetal anomalies in the southeast region of Sweden Eric Hildebrand et al. Acta Obstet Gynecol Scand.2010 Nov. Free article Show details Display options Display options Format Acta Obstet Gynecol Scand Actions Search in PubMed Search in NLM Catalog Add to Search . 2010 Nov;89(11):1412-9. doi: 10.3109/00016349.2010.517307. Authors Eric Hildebrand1,Anders Selbing,Marie Blomberg Affiliation 1 Department of Clinical and Experimental Medicine, University of Linköping, Sweden. eric.hildebrand@lio.se PMID: 20955095 DOI: 10.3109/00016349.2010.517307 Item in Clipboard Full text links Cite Display options Display options Format Erratum in Acta Obstet Gynecol Scand. 2014 Apr;93(4):439 Abstract Objective: To assess and compare the sensitivity for detecting fetal anomalies and chromosomal aberrations by routine ultrasound examination performed in the second trimester with results from an examination performed at 11-14 weeks gestation. Design: Observational study. Setting: Five centers in the southeast region of Sweden. Population: A total of 21,189 unselected pregnant women. Methods: The scan was performed at one center in the first trimester and at the remaining four centers in the second trimester. Outcome measures resulting from first trimester scanning were compared with those from the second trimester scanning. Main outcome measures: Detection rates of fetal structural anomalies and chromosomal aberrations. Results. At the first trimester scan 13% of all anomalies were detected, and at the second trimester scan 29% were detected. Lethal anomalies were detected at a high level at both times: 88% in the first, 92% in the second. The percentage of chromosomal aberrations discovered at the early scan was 71%, in the later 42%. The percentage of heart malformations detected was surprisingly low. Conclusion: The results showed the advantages of the later scan in discovering anomalies of the heart, urinary tract and CNS, and of the early scan in discovering chromosomal aberrations. Lethal malformations were detected at a high level in both groups, but detection of heart malformations needs improvement. PubMed Disclaimer Similar articles 13-14-week fetal anatomy scan: a 5-year prospective study.Ebrashy A, El Kateb A, Momtaz M, El Sheikhah A, Aboulghar MM, Ibrahim M, Saad M.Ebrashy A, et al.Ultrasound Obstet Gynecol. 2010 Mar;35(3):292-6. doi: 10.1002/uog.7444.Ultrasound Obstet Gynecol. 2010.PMID: 20205205 Economic modelling of antenatal screening and ultrasound scanning programmes for identification of fetal abnormalities.Ritchie K, Bradbury I, Slattery J, Wright D, Iqbal K, Penney G.Ritchie K, et al.BJOG. 2005 Jul;112(7):866-74. doi: 10.1111/j.1471-0528.2005.00560.x.BJOG. 2005.PMID: 15957985 [First trimester ultrasound screening for fetal anomalies--classics and novelties].Markov D.Markov D.Akush Ginekol (Sofiia). 2009;48 Suppl 2:3-12.Akush Ginekol (Sofiia). 2009.PMID: 20380089 Review.Bulgarian. Comparisons of first and second trimester screening for fetal anomalies.D'Ottavio G, Mandruzzato G, Meir YJ, Rustico MA, Fischer-Tamaro L, Conoscenti G, Natale R.D'Ottavio G, et al.Ann N Y Acad Sci. 1998 Jun 18;847:200-9. doi: 10.1111/j.1749-6632.1998.tb08941.x.Ann N Y Acad Sci. 1998.PMID: 9668713 Should second trimester ultrasound be routine for all pregnancies?Makhlouf M, Saade G.Makhlouf M, et al.Semin Perinatol. 2013 Oct;37(5):323-6. doi: 10.1053/j.semperi.2013.06.008.Semin Perinatol. 2013.PMID: 24176154 Review. See all similar articles Cited by Early Detection of Structural Anomalies in a Primary Care Setting in the Netherlands.Bardi F, Smith E, Kuilman M, Snijders RJM, Bilardo CM.Bardi F, et al.Fetal Diagn Ther. 2019;46(1):12-19. doi: 10.1159/000490723. Epub 2018 Jul 25.Fetal Diagn Ther. 2019.PMID: 30045038 Free PMC article. Diagnostic accuracy of ultrasound screening for fetal structural abnormalities during the first and second trimester of pregnancy in low-risk and unselected populations.Buijtendijk MF, Bet BB, Leeflang MM, Shah H, Reuvekamp T, Goring T, Docter D, Timmerman MG, Dawood Y, Lugthart MA, Berends B, Limpens J, Pajkrt E, van den Hoff MJ, de Bakker BS.Buijtendijk MF, et al.Cochrane Database Syst Rev. 2024 May 9;5(5):CD014715. doi: 10.1002/14651858.CD014715.pub2.Cochrane Database Syst Rev. 2024.PMID: 38721874 Free PMC article.Review. Ultrasound and genetic findings in a case series of fetuses presenting vertebral defects.Xie W, Zhou L, Hu A, Chen J, Pang J, Xi H, Luo Y, Hu J, Yang S, Gao X, Kuang H, Tang W, Liu R, Wang S, Peng Y.Xie W, et al.BMC Pregnancy Childbirth. 2025 Aug 15;25(1):858. doi: 10.1186/s12884-025-07920-6.BMC Pregnancy Childbirth. 2025.PMID: 40817209 Free PMC article. Fetal structural anomalies diagnosed during the first, second and third trimesters of pregnancy using ultrasonography: a retrospective cohort study.Dulgheroff FF, Peixoto AB, Petrini CG, Caldas TMRDC, Ramos DR, Magalhães FO, Araujo Júnior E.Dulgheroff FF, et al.Sao Paulo Med J. 2019 Sep-Oct;137(5):391-400. doi: 10.1590/1516-3180.2019.026906082019.Sao Paulo Med J. 2019.PMID: 31939566 Free PMC article. Association Between Isolated Single Umbilical Artery and Perinatal Outcomes: A Meta-Analysis.Xu Y, Ren L, Zhai S, Luo X, Hong T, Liu R, Ran L, Zhang Y.Xu Y, et al.Med Sci Monit. 2016 Apr 30;22:1451-9. doi: 10.12659/msm.897324.Med Sci Monit. 2016.PMID: 27130891 Free PMC article. 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12719	https://www.mathsisfun.com/algebra/special-binomial-products.html	Algebra Index Show Ads Hide Ads \| About Ads We may use Cookies Special Binomial Products See what happens when we multiply some binomials ... Binomial A binomial is a polynomial with two terms example of a binomial Product Product means the result we get after multiplying. In Algebra xy means x multiplied by y And (a+b)(a−b) means (a+b) multiplied by (a−b). We use that a lot here! Special Binomial Products So when we multiply binomials we get ... Binomial Products! And we will look at three special cases of multiplying binomials ... so they are Special Binomial Products. 1. Multiplying a Binomial by Itself What happens when we square a binomial (in other words, multiply it by itself) .. ? (a+b)2 = (a+b)(a+b) = ... ? The result: (a+b)2 = a2 + 2ab + b2 This illustration shows why it works: 2. Subtract Times Subtract And what happens when we square a binomial with a minus inside? (a−b)2 = (a−b)(a−b) = ... ? The result: (a−b)2 = a2 − 2ab + b2 If you want to see why, then look at how the (a−b)2 square is equal to the big a2 square minus the other rectangles: (a−b)2 = a2 − 2b(a−b) − b2 = a2 − 2ab + 2b2 − b2 = a2 − 2ab + b2 3. Add Times Subtract And then there is one more special case ... what about (a+b) times (a−b) ? (a+b)(a−b) = ... ? The result: (a+b)(a−b) = a2 − b2 That was interesting! It ended up very simple. And it is called the "difference of two squares" (the two squares are a2 and b2). This illustration shows why it works: a2 − b2 can become (a+b)(a−b) Note: (a−b) can swap places with (a+b): (a−b)(a+b) = a2 − b2 The Three Cases Here are the three results we just got: (a+b)2= a2 + 2ab + b2 (a−b)2= a2 − 2ab + b2 (a+b)(a−b)= a2 − b2 The first two are the "perfect square trinomials" and the last is the "difference of squares" Remember those patterns, they will save you time and help you solve many algebra puzzles. Using Them So far we have just used "a" and "b", but they could be anything. Example: (y+1)2 We can use the (a+b)2 case where "a" is y, and "b" is 1: (y+1)2 = (y)2 + 2(y)(1) + (1)2 = y2 + 2y + 1 Example: (3x−4)2 We can use the (a-b)2 case where "a" is 3x, and "b" is 4: (3x−4)2 = (3x)2 − 2(3x)(4) + (4)2 = 9x2 − 24x + 16 Example: (4y+2)(4y−2) We know the result is the difference of two squares, because: (a+b)(a−b) = a2 − b2 so: (4y+2)(4y−2) = (4y)2 − (2)2 = 16y2 − 4 Sometimes we can see the pattern of the answer: Example: which binomials multiply to get 4x2 − 9 Hmmm... is that the difference of two squares? 4x2 is (2x)2, and 9 is (3)2, so we have: 4x2 − 9 = (2x)2 − (3)2 And that can be produced by the difference of squares formula: (a+b)(a−b) = a2 − b2 Like this ("a" is 2x, and "b" is 3): (2x+3)(2x−3) = (2x)2 − (3)2 = 4x2 − 9 So the answer is that we can multiply (2x+3) and (2x−3) to get 4x2 − 9 Algebra Index Copyright © 2024 Rod Pierce
12720	https://chem.libretexts.org/Courses/University_of_Arkansas_Little_Rock/Chem_1300%3A_Preparatory_Chemistry/Learning_Modules/08%3A_Solution_Chemistry/8.01%3A_Measuring_Concentrations_of_Solutions	8.1: Measuring Concentrations of Solutions - Chemistry LibreTexts Skip to main content Table of Contents menu search Search build_circle Toolbar fact_check Homework cancel Exit Reader Mode school Campus Bookshelves menu_book Bookshelves perm_media Learning Objects login Login how_to_reg Request Instructor Account hub Instructor Commons Search Search this book Submit Search x Text Color Reset Bright Blues Gray Inverted Text Size Reset +- Margin Size Reset +- Font Type Enable Dyslexic Font - [x] Downloads expand_more Download Page (PDF) Download Full Book (PDF) Resources expand_more Periodic Table Physics Constants Scientific Calculator Reference expand_more Reference & Cite Tools expand_more Help expand_more Get Help Feedback Readability x selected template will load here Error This action is not available. chrome_reader_mode Enter Reader Mode 8: Solution Chemistry Learning Modules { } { "8.01:_Measuring_Concentrations_of_Solutions" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "8.02:_Acid_and_Base_Concentrations-pH_scale" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "8.03:_Stoichiometry_of_Aqueous_Reactions" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" } { "00:_Front_Matter" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "00:_General_Information" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "01:_Introduction" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "02:_Mathematical_Fundamentals" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "03:_Atoms_and_Elements" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "04:_Compounds_and_Molecules" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "05:_Chemical_Reactions" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "06:_Counting_Molecules_through_Measurements" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "07:_Stoichiometry" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "08:_Solution_Chemistry" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "09:_Gases" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "10:_Thermodynamics" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "zz:_Back_Matter" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" } Sat, 20 Nov 2021 15:29:51 GMT 8.1: Measuring Concentrations of Solutions 283929 283929 Robert E. Belford { } Anonymous Anonymous User 2 false false [ "article:topic" ] [ "article:topic" ] Search site Search Search Go back to previous article Sign in Username Password Sign in Sign in Sign in Forgot password Contents 1. Home 2. Campus Bookshelves 3. University of Arkansas Little Rock 4. Chem 1300: Preparatory Chemistry 5. Learning Modules 6. 8: Solution Chemistry 7. 8.1: Measuring Concentrations of Solutions Expand/collapse global location Learning Modules Front Matter 0: General Information 1: Introduction 2: Mathematical Fundamentals 3: Atoms and Elements 4: Compounds and Molecules 5: Chemical Reactions 6: Counting Molecules through Measurements 7: Stoichiometry 8: Solution Chemistry 9: Gases 10: Thermodynamics Back Matter 8.1: Measuring Concentrations of Solutions Last updated Nov 20, 2021 Save as PDF 8: Solution Chemistry 8.2: Acid and Base Concentrations-pH scale Page ID 283929 ( \newcommand{\kernel}{\mathrm{null}\,}) Table of contents 1. Introduction 1. Concentration of a Solute 1. Making a Solution with a Solid Solute. Ion Concentrations in Solution Preparing Solutions with Stock Concentrated Solutions (Dilution). Contributors and Attributions Learning Objectives Define Molarity Differentiate between solute, solvent, and solution Calculate the molar concentration for solutes Outline the steps to make a solution of a desired concentration from a solid or aqueous solute Calculate the concentration of ions in a soluble ionic compound Perform stoichiometric calculations involving aqueous solutes Calculate the concentration of unknown solutes Introduction Up to this point we have used stoichiometry to "count" atoms, molecules and ions by measuring mass of pure substances and using molar masses to calculate the number of chemical entities (moles). Many of the reactions we have studied involve solutions, where we are interested in a solute that is dissolved in a solvent, and we can not measure the mass of the solute independent of the solvent. We also need to realize that many chemical reactions require the reactants to be mobile, where they can bump into each other, and this can occur when they are dissolved in a solvent. So it is very important that we can count chemical entities when the entity of interest is a solute dissolved in a solvent, and in which case we measure the mass of volume of the solution as a whole (solvent plus all solutes) of which volume is typically the easiest to measure. Concentration of a Solute There are two basic ways of reporting the concentration of a solute in a solvent, by reporting the mass of solute in a given volume, or the number of moles of solute in a given volume. These are effectively conversion factors that define the equivalent mass or moles of a solute to the volume of the solution. Mass Concentration: has typical units of g/L (8.1.1)m⁡a⁢s⁢s s⁢o⁢l⁢u⁢t⁢e v⁢o⁢l s⁢o⁢l⁢u⁢t⁢i⁢o⁢n=m⁡(g))V⁡(L)) Mole Concentration (Molarity): has units of mol/L or M . You can consider M as a shorthand notation for mol/L.(8.1.2)m⁢o⁢l⁢e⁢s s⁢o⁢l⁢u⁢t⁢e v⁢o⁢l s⁢o⁢l⁢u⁢t⁢i⁢o⁢n=n⁡(m⁢o⁢l⁢e⁢s))V⁡(L))=M o⁢r⁢(m⁢o⁢l/L) So a 3.0M solution of sucrose has 3.0 mole of sucrose per liter Figure 8.1.1: In order to avoid confusion we will use a small "m" to denote mass, and a large ""m" to denote Molarity. How do we convert between mass and mole based concentration? Simply by multiplying or dividing by the molar mass (g/mol). If you know the g/L, you simply divide by the molar mass, and if you know the moles/L and want the mass/L, you multiply by the molar mass (just use the molar mass as an equivalence statement). How do we "count" solute molecules and ions? By measuring the volume and knowing the molarity. This is analagous to measuring the mass and knowing the molar mass for a pure substance. (8.1.3)n⁡(m⁢o⁢l⁢e⁢s)=M⁡(m⁢o⁢l L)⁢(V⁡(L))⁢n=M⁡V This can be contrasted to how we "count" chemical entities that are pure solids. (8.1.4)n⁡(m⁡o⁢l⁢e⁢s)=m⁡(g)m⁡o⁢l⁢a⁢r⁢m⁡a⁢s⁡s⁡(g m⁡o⁢l)n=m f⁡w Making a Solution with a Solid Solute. Step 1: Calculate Mass of solute needed for desired volume Step 2: Quantitatively Transfer Mass to Volumetric Flask Step 3: Dilute to volume with solvent, ensuring that all of the solid has dissolved. Making 500.0 mL of 0.500 M Copper(II)Sulfate Step 1: Calculate Mass CuSO 4(s) needed. (8.1.5)0.5000⁢L⁡(0.500⁢m⁢o⁢l⁢C⁢u⁢S⁢O 4 L)⁢(159.6⁢g⁡C⁢u⁢S⁢O 4 m⁢o⁢l)=39.9⁢g⁡C⁢u⁢S⁢O 4 Step 2: Weight 39.9g CuSO 4(s) and quantitatively transfer to 500 mL graduated cylinder (which is calibrated to 500.0 mL), being sure all the salt is transferred. Step 3: Fill half way and mix, then dilute to volume. Make sure all the solute is dissolved, and recheck that solution level is at bottom of meniscus. Exercise 8.1.1 What is the molarity of a solution made when 66.2 g of C 6 H 12 O 6 are dissolved to make 235 mL of solution? Answer 1.57 M C 6 H 12 O 6 Why can't you just add 500.0mL of water to 39.9 g CuSO 4 to make 500.0 mL of 0.500M solution? First, this is not taking into account the volume of the solute, and second, there is usually a change in volume when you dissolve a solute into a solvent, and in the case of ionic compounds this is a contraction. Exercise 8.1.2 Using concentration as a conversion factor, perform the following calculation. How many liters of 0.0444 M CH 2 O are needed to obtain 0.0773 mol of CH 2 O? What mass of solute is present in 1.08 L of 0.0578 M H 2 SO 4? What volume of 1.50 M HCl solution contains 10.0 g of hydrogen chloride? Answer a 1.74 L (8.1.6)0.0773⁢m⁢o⁢l C⁢H 2⁡O⁡(L s⁢o⁢l⁢u⁢t⁢i⁢o⁢n 0.0444⁢m⁢o⁢l C⁢H 2⁡O)=1.7409⁢L Answer b 6.12 g (8.1.7)1.08⁢L s⁢o⁢l⁢u⁢t⁢i⁢o⁢n⁡(0.0578⁢m⁢o⁢l⁢H 2⁡S⁢O 4 L s⁢o⁢l⁢u⁢t⁢i⁢o⁢n)⁢(98.08⁢g⁡H 2⁡S⁢O 4 m⁢o⁢l)=6.122546⁢g⁡H 2⁡S⁢O 4 Answer c 183 mL or 0.183L (8.1.8)10.0⁢g⁡H⁡C⁢l⁡(m⁢o⁢l⁡H⁡C⁢l 36.46⁢g)⁢(L s⁢o⁢l⁡u⁢t⁢i⁢o⁢n 1.5⁢m⁢o⁢l⁡H⁡C⁢l)=0.182849⁢L Ion Concentrations in Solution When an ionic compound dissolves it breaks up into its ions. (NH 4)2 Cr 2 O 7(aq) --> 2NH 4+(aq) + Cr 2 O 7(aq) Dissolving 252.07 grams of ammonium dichromate (fw = 252.07 g/mol) results in a solution that is 1M ammonium dichromate, but when an ionic salt dissolves, it breaks up into ions, and so what you really have is a solution that is 2 M ammonium ion (NH 4+) and 1 M in dichromate ion (Cr 2 O 7+2) Dissolution of 1 mol of an Ionic Compound. In this case, dissolving 1 mol of (NH 4)2 Cr 2 O 7 produces a solution that contains 1 mol of Cr 2 O 7 2− ions and 2 mol of NH 4+ ions. (Water molecules are omitted from a molecular view of the solution for clarity.) Example 8.1.1 Solution Concentration of Iron(II)chloride Water is added to 2.16 g of the ionic compound ferrous chloride to make a solution with a total volume of 100.0 mL. Express the concentration of the salt solution, and that of its ions. What is the salt concentration? What are the ion concentrations? Solution Video 8.1.1 goes over this calculation. Ans: 0.170M FeCl 2(aq) 0.170M Fe+2(aq) 0.340M Cl-(aq) Yes, there are three answers! (that of the salt, and of the ions) Video 8.1.1: 3'36" YouTube calculating the concentration of a Ferrous Chloride in example 8.1.1. Exercise 8.1.3 You have a 1.50 M solution of Na 2 CO 3. What is the concentration of: A) sodium ions? B) carbonate ions? C) total ions? Answer A) [Na+] = 3.00 M B) [CO 3-2]= 1.50 M C) [Na+] + [CO 3-2] = 4.50 M ; or you can choose think for every 1 mole of Na 2 CO 3, there are 3 moles of ions (identify of the ions is irrelevant). Preparing Solutions with Stock Concentrated Solutions (Dilution). Many stockroom reagents come as concentrated solutions that can easily be diluted to a desired concentration by adding solvent. Adding a solvent does not change the moles of solute (n), so n initial = n final n i = n f M i V i = M f V f Step 1: Calculate initial volume of stock Step 2: Transfer to volumetric flask with a volumetric pipette Step 3: Dilute to Volume Example 8.1.2 How do you make 500.0 ml of 0.70M HCl solution from stock 11.6 M HCl? Solution Step 1: Calculate initial volume of 11.6 M hydrochloric acid needed. (8.1.9)M i⁢V i=M F⁢V F (8.1.10)V i=V f⁡(M f M i)=500.0⁢m⁢L⁡(0.70⁢M 11.6⁢M)=30.2⁢m⁢L Step 2: Quantitatively transfer this volume to a 500 mL volumetric flask (this has 4 significant Figures, that is, it is calibrated to 500.0 mL) Step 3: Dilute with water to mark Video 8.1.2: Dilution problem on creating 500.0 ml of a 0.700M HCl solutioin Exercise 8.1.3 What is the concentration of an HCl solution if 598 mL of 0.778 M HCl is diluted to 1.00 L? Answer M i⁢V i=M F⁢V F M f=M i⁡(V i V f)=0.778⁢M⁡(0.0598⁢L 1.00⁢L)=0.465⁢M Contributors and Attributions Robert E. Belford (University of Arkansas Little Rock; Department of Chemistry). The breadth, depth and veracity of this work is the responsibility of Robert E. Belford, rebelford@ualr.edu. You should contact him if you have any concerns. 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12721	https://www.creative-proteomics.com/blog/long-chain-fatty-acids-structure.htm	Long-Chain Fatty Acids Structure Explained: Impacts on Function - Creative Proteomics Blog Get Your Instant Quote Order Online Home Services Proteomics Service Protein Post-translational Modification Analysis Disulfide Bond Glycosylation Analysis of Protein Phosphoproteomics Service Ubiquitinated-proteomics S-Nitrosylation Methyl-proteomics Acetyl-proteomics Characterization of Protein SUMOylation Redox Proteomics Protein-Protein Interaction Analysis Service Co-immunoprecipitation/mass spectrometry (co-IP/MS) Crosslinking Protein Interaction Analysis Far-Western Blot Analysis Service Pull-Down Assay Label Transfer Protein Interaction Analysis BioID-MS Service Chemical Cross-linking Mass Spectrometry (CX-MS) Service Tandem Affinity Purification (TAP)-MS Service TurboID Service IP-MS Protein Interactomics Solution Biacore Service Protein Gel and Imaging Isoelectric Point Analysis Service Protein 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Support Chromatography Technology Spectroscopy Technology Mass Spectrometry Platform Resource Knowledge Bases Q&A Support Documents Videos Sample Guideline Company About Us Contact Us Distributor Career Order Online Get Your Instant Quote To all articles Metabolomics Long-Chain Fatty Acids Structure Explained: Impacts on Function Discover how long-chain fatty acids' unique structure affects their biological function and impact on health. Learn more in this detailed explanation. Creative Proteomics scientist 2025-08-05 Contents 1. Fundamental Concepts and Structural Attributes of Long-Chain Fatty Acids 2. Hierarchical Classification of Long-Chain Fatty Acids 3. Structure-Function Interrelationships in Fatty Acids 4. Physiological Networks of Long-Chain Fatty Acid Functions 5. Structure-Function Summary Long-chain fatty acids(LCFAs), defined as aliphatic carboxylic acids containing 12-22 carbon atoms, represent the predominant category of fatty acids in human physiology. These compounds fulfill critical roles across nutritional, biochemical, and metabolic domains, serving as the primary constituents of dietary lipids. 1. Fundamental Concepts and Structural Attributes of Long-Chain Fatty Acids 1.1 Definition and Classification Criteria Long-chain fatty acids (LCFAs) are defined as aliphatic carboxylic acids comprising 12-22 carbon atoms, representing the predominant fatty acid category in biological systems. Classification follows International Union of Biochemistry (IUPAC) standards: Short-chain: C2-C5 (e.g., acetic, propionic acids) Medium-chain: C6-C11 (e.g., caprylic, capric acids) Long-chain: C12-C22 (e.g., lauric to docosahexaenoic acids) Very-long-chain: ≥C23 (e.g., nervonic acid) 1.2 Core Structural Organization While sharing a common carboxyl-terminal framework, LCFAs exhibit distinctive features through chain elongation: Polar headgroup: Carboxyl moiety (-COOH) Ionizes to -COO⁻ at physiological pH Confers amphiphilic properties Forms ester bonds in glyceride synthesis Hydrocarbon chain: Composed of methylene (-CH₂-) units Chain length governs hydrophobicity C12-C22 range enables substantial hydrophobic interactions ω-Terminus (methyl end): Serves as carbon numbering origin Basis for ω-3/6/9 classification Site for specific oxidative modifications 1.3 Dimensions of Structural Diversity LCFA heterogeneity manifests through four critical parameters: Chain length: C12-C22 carbon permutations Degree of unsaturation: Saturated to polyunsaturated states Double bond position: Determines metabolic fate and functionality Stereoconfiguration: cis vs. trans isomeric forms 1.4 Topology of carbon chain determines physical properties Saturated linear chains (e.g., stearic acid, C18:0) display a high melting point (69°C) and marked hydrophobicity. Van der Waals interactions produce tightly packed crystalline arrangements, creating the rigid structural foundation essential for biofilms. In contrast, unsaturated chains with bends (e.g., docosahexaenoic acid/DHA, C22:6) contain ≥3 cis double bonds that induce 30–60° kinks. This configuration diminishes molecular packing order, substantially increasing membrane fluidity by approximately 300%. Meanwhile, extra-long saturated chains (e.g., lignoceric acid, C24:0) exhibit heightened hydrophobic character due to chain elongation. This promotes specialized "lipid raft" domain formation within sphingolipids, modulating membrane protein functionality. 1.5 The polar head group serves dual roles: As a hydrogen bond donor, the carboxyl group (-COOH) facilitates targeted binding to receptor proteins like GPR120. Double bond positioning critically influences biological activity: Omega-3 fatty acids (e.g., eicosapentaenoic acid/EPA) feature their first unsaturation at the third carbon from the methyl terminus, enabling anti-inflammatory effects; conversely, omega-6 compounds (e.g., arachidonic acid) possess their initial double bond at the sixth position, acting as pro-inflammatory precursors. For more on the difference between long-chain fatty acids and short-chain fatty acid, see"Long-Chain vs. Medium- and Short-Chain Fatty Acids: What's the Difference?". For more information on long-chain fatty acids, please refer to"What Are Long-Chain Fatty Acids? A Beginner's Guide". 2. Hierarchical Classification of Long-Chain Fatty Acids 2.1 Saturation-Based Categorization 2.1.1 Saturated Fatty Acids (SFAs) Structural hallmark:Absence of carbon-carbon double bonds Key representatives: Lauric acid (12:0): Primary constituent in coconut oil Myristic acid (14:0): Abundant in dairy lipids Palmitic acid (16:0): Predominant natural SFA Stearic acid (18:0): Major component of animal fats Physical behavior: Melting point elevation with chain elongation Typically solid at ambient temperature Significant van der Waals interactions 2.1.2 Monounsaturated Fatty Acids (MUFAs) Structural signature:Single double bond Notable examples: Palmitoleic acid (16:1n-7): Fish oils and select plants Oleic acid (18:1n-9): >75% composition in olive oil Erucic acid (22:1n-9): Historically high in rapeseed cultivars Bond attributes: Predominantly cis configuration Endogenous synthesis via desaturase enzymes 2.1.3 Polyunsaturated Fatty Acids (PUFAs) Structural definition:≥2 double bonds Subclass representatives: Dienoic: Linoleic (18:2n-6) Trienoic: α-Linolenic (18:3n-3) Tetraenoic: Arachidonic (20:4n-6) Hexaenoic: DHA (22:6n-3) Biological roles: Essential fatty acid carriers Eicosanoid biosynthesis precursors Critical membrane phospholipid constituents Long chain and very long-chain fatty acid biosynthesis in vertebrates(Naudí A et al., 2013) 2.2 Double Bond Position Classification 2.2.1 ω-3 Series Metabolic precursor:α-Linolenic acid (ALA, 18:3n-3) Elongation metabolites: EPA (20:5n-3): Potent anti-inflammatory agent DPA (22:5n-3): Neural protection mediator DHA (22:6n-3): Brain/retinal structural component Metabolic regulation: Δ6-desaturase as rate-limiting enzyme Competitive inhibition with ω-6 pathway 2.2.2 ω-6 Series Primary substrate:Linoleic acid (LA, 18:2n-6) Biosynthetic derivatives: γ-Linolenic (GLA, 18:3n-6) Dihomo-γ-linolenic (DGLA, 20:3n-6) Arachidonic (AA, 20:4n-6) Functional attributes: Pro-inflammatory eicosanoid precursors Homeostatic balance with ω-3 compounds 2.2.3 ω-7/ω-9 Series Characteristic members: Palmitoleic (16:1n-7) Oleic (18:1n-9) Mead (20:3n-9) Metabolic features: Endogenously synthesized Upregulated during essential FA deficiency 2.3 Geometric Isomer Differentiation 2.3.1 cis-Unsaturated FAs Natural configuration: Syn-orientation of hydrogen atoms ~30° chain bending at double bond Reduced molecular packing efficiency Nutritional relevance: Cardioprotective lipids in olive oil/fish oil Demonstrated cardiovascular benefits 2.3.2 trans-Unsaturated FAs Formation pathways: Industrial partial hydrogenation Rumen microbial biohydrogenation Structural consequences: Anti-orientation of hydrogen atoms Linear chain conformation retention SFA-like physicochemical behavior Health impacts: Elevate LDL-cholesterol concentrations Suppress HDL-cholesterol levels Promote inflammatory responses 3. Structure-Function Interrelationships in Fatty Acids 3.1 Carbon Chain Length Determinants 3.1.1 Absorption and Transport Mechanisms MCFAs: Portal vein direct absorption Bile-independent uptake Rapid energy substrate utilization LCFAs: Micellar solubilization prerequisite Intracellular re-esterification to triglycerides Chylomicron-mediated transport Lymphatic systemic entry 3.1.2 Metabolic Pathway Specificity β-Oxidation efficiency: C6-C10: Mitochondrial direct oxidation C12-C20: Carnitine shuttle-dependent ≥C22: Peroxisomal primary oxidation Tissue distribution patterns: C16-C18: Ubiquitous tissue incorporation ≥C20: Selective tissue partitioning (e.g., neural accretion) 3.2 Double Bond Configuration Effects 3.2.1 Membrane Biophysical Modulation Fluidity regulation: Saturated FAs elevate membrane order Each unsaturation decreases phase transition by 5-10°C DHA reduces bilayer thickness ≈15% Microdomain organization: Saturated FAs stabilize lipid rafts ω-3 PUFAs disrupt raft integrity Signal transduction modulation 3.2.2 Oxidative Vulnerability Structural susceptibility: Per-oxidation rate increases exponentially with unsaturation count Conjugated dienes exhibit enhanced reactivity ω-3 > ω-6 oxidation propensity Stabilization requirements: Vitamin E co-supplementation for PUFA-rich diets Aldehydic toxin generation during thermal processing 3.3 Stereochemical Functional Implications 3.3.1 cis-Configuration Advantages Biological functionality: Optimal membrane fluidity maintenance Membrane protein conformational support Lipid mediator bioavailability Metabolic features: Lipase recognition efficiency High oxidative metabolic yield Reduced vascular deposition 3.3.2 trans-Fat Pathogenicity Molecular deception mechanisms: Saturated FA-like metabolism Retained unsaturation reactivity Disease pathogenesis: Essential FA metabolic interference Atherogenic plaque potentiation Insulin sensitivity impairment 4. Physiological Networks of Long-Chain Fatty Acid Functions 4.1 Energy Metabolism Regulation 4.1.1 Optimized Energy Storage Caloric density comparison (per gram): Lipids: 9 kcal Proteins/carbohydrates: 4 kcal Ethanol: 7 kcal Storage efficiency: Anhydrous deposition 1 kg adipose tissue ≈ 7,700 kcal Sustains adult basal metabolism for 72-96 hours 4.1.2 Metabolic Homeostasis Lipolytic control: Hormone-sensitive lipase modulated by insulin/glucagon Stress-induced activation Liberates circulating free fatty acids Oxidative regulation: Carnitine palmitoyltransferase-mediated transport AMPK pathway modulation Reciprocal inhibition with glucose metabolism 4.2 Cellular Membrane Architecture 4.2.1 Phospholipid Asymmetry sn-1 position: Saturated FA dominance (e.g., palmitic/stearic acids) Membrane structural stabilization sn-2 position: Unsaturated FA enrichment (e.g., arachidonic/DHA) Fluid-phase maintenance 4.2.2 Membrane-Mediated Signaling Receptor modulation: GPCR conformational dynamics Ion channel gating probability Receptor dimerization states Signal transduction: Lipid raft compartmentalization Secondary messenger precursor generation Kinase spatial organization 4.3 Bioactive Lipid Precursors 4.3.1 Eicosanoid Biosynthesis Production cascade: Phospholipase A2-mediated AA/EPA release Cyclooxygenase → Prostaglandins Lipoxygenase → Leukotrienes Functional antagonism: ω-6 derivatives: Pro-inflammatory mediators ω-3 derivatives: Anti-inflammatory resolvers Homeostatic inflammation control 4.3.2 Endocannabinoid System Precursor conversion: AA → Anandamide (AEA)/2-AG DHA → Docosahexaenoyl ethanolamide (DHEA) Neuromodulatory effects: ω-6 metabolites: Anxiogenic properties ω-3 metabolites: Neuroprotective functions Cognitive-emotional regulation 4.4 Genomic Regulation Mechanisms 4.4.1 Nuclear Receptor Activation PPAR isoforms: PPARα: FA catabolism induction PPARγ: Adipogenesis promotion PPARδ: Metabolic equilibrium LXR/RXR heterodimers: Cholesterol homeostasis Bile acid biosynthesis Glucose metabolism involvement 4.4.2 Epigenetic Modulation Histone modifications: Acetyl-CoA substrate provision HDAC inhibition by SCFAs Non-coding RNA regulation: ω-3 suppression of oncogenic miR-21 Tumor suppression pathway modulation Fibrotic progression control Services You May Be Interested In: Phospholipid-derived fatty acids (PLFAs) Medium- and Long-Chain Fatty Acid Profiling Short Chain Fatty Acids Saturated Fatty Acids Analysis Total Fatty Acids Profiling Very Long Chain Fatty Acids Analysis Free Fatty Acids (FFAs) Analysis Service 5. Structure-Function Regulation of virulence:Pathogens like Salmonella and Clostridium difficile directly modulate virulence factor secretion (e.g., toxins, adhesins). Enteric pathogens encounter a dynamic nutritional landscape shaped by host secretions, microbial metabolism, and dietary components. Critically, metabolites—collectively termed long-chain fatty acids (LCFAs)—exhibit biogeographical concentration gradients along intestinal segments (e.g., jejunum versus colon), creating an environmental signaling network detected by pathogens. LCFAs suppress virulence via dual pathways: Allosteric Inhibition: Direct binding to virulence gene transcriptional activators (e.g., FadR proteins) diminishes their DNA affinity, thereby repressing virulence expression. Signaling Cascade Disruption: By altering histidine kinase receptor activity (e.g., EnvZ/OmpR system), LCFAs modify downstream phosphorylation cascades to inhibit virulence programs. Exemplifying this: Palmitic acid (C16:0) reduces Salmonella intestinal invasion by 40-60% via Hi1A/Hi1D two-component system inhibition Oleic acid (C18:1) downregulates the ToxT virulence regulator in Vibrio cholerae, suppressing toxin production(Mitchell MK et al., 2022). Mechanisms of virulence regulation by LCFA cytoplasmic sensors(Mitchell MK et al., 2022) Long-chain fatty acids (LCFAs) exhibit bidirectional immunomodulatory effects in inflammatory bowel disease, critically regulating immune-inflammatory equilibrium through two primary mechanisms: 1. Signaling Pathway Regulation Anti-inflammatory actions:Suppression of proinflammatory pathways via TLR4/MyD88/NF-κB axis downregulation, reducing TNF-α and IL-6 release; NLR inflammasome inhibition, preventing IL-1β maturation. Pro-resolution actions:PPAR-γ pathway upregulation enhances anti-inflammatory mediators (e.g., IL-10); Elevated resolvin and protectin biosynthesis. 2. Immune Cell Functional Reprogramming Macrophage repolarization toward M2 anti-inflammatory phenotypes; T cell lineage modulation: Th17 suppression and Treg functional enhancement; Neutrophil recruitment attenuation through reduced CXCL1/CXCL8 chemokine expression. 3. Dynamic Intestinal Barrier Modulation Protective mechanisms:Tight junction reinforcement via claudin-1/occludin upregulation; Goblet cell stimulation with consequent MUC2-mediated mucus hypersecretion; Epithelial regeneration potentiation through Wnt/β-catenin activation. Pathological risks:Saturated LCFAs (e.g., palmitic acid) can:Trigger endoplasmic reticulum stress; Elevate epithelial cell apoptotic rates; Compromise mucus layer structural integrity(Ma C etal., 2019). VLCFAs leveragetheir pronounced hydrophobicity and structural heterogeneity to execute critical plant functionsacross four domains: Physical Barrier Formation Cuticular and cork-associated waxes comprise intricate blends of VLCFA derivatives (aldehydes, alkanes, ketones) integrated with non-acyl cyclic compounds including terpenoids and flavonoids. Membrane Functional Modulation VLCFAs fine-tune signal perception and protein anchoring. Wattelet-Boyer et al. demonstrated via transmission electron microscopy that sphingolipid-VLCFA chain length governs trans-Golgi network (TGN) secretory vesicle morphology and connectivity in Arabidopsis roots. Critically, sphingolipids enable de novo polar sorting of auxin transporter PIN2 to apical membranes. This establishes a molecular linkage whereby TGN-localized VLCFA derivatives facilitate gravitropic responses through PIN2 trafficking in polarized cells. Developmental and Energetic Programming Anther endothelial expression of KCS6 and CER2 homologs is essential for generating ≥C26 VLCFAs. These lipid derivatives accumulate on pollen surfaces, functioning as hydration signals that activate σ-cell-mediated water transport during germination. Environmental Adaptation Evolution VLCFAs underpin disease, drought, and heavy metal resistance. Arabidopsis KCS1 mutants exhibit impaired seedling survival under low humidity , whereas BnKCS1-1 or BnKCS1-2 overexpression enhances cuticular wax deposition in rapeseed (Brassica napus), reducing transpirational water loss and improving drought tolerance(Batsale M et al., 2021). Biosynthesis and selective involvement of VLC acyl-CoAs in the different lipid biosynthesis pathways in Arabidopsis(Batsale M et al., 2021) Summary Structural FeaturePhysicochemical EffectCore Biological FunctionApplication Example Long-Chain Saturated StructureHigh melting point, strong hydrophobicity Membrane stability, energy storage Palm oil structural lipids Polyunsaturated BendingLow melting point, high fluidity Membrane dynamics, anti-inflammatory signaling Fish oil for cardiovascular disease prevention Very Long-Chain (>C22)Extreme hydrophobicity, high rigidity Cellular barrier, developmental signaling Stress-resistant crop breeding Carboxyl Polar HeadAmphiphilicity, self-assembly into membranes Basic structure of lipid bilayers Artificial membrane simulation systems The biological significance of long-chain fatty acids is rooted in their precise carbon chain configurations. Future research will further elucidate the dynamic interactions between structure and function, providing targets for metabolic disease treatment, functional lipid design, and synthetic biology. For more on the role of long-chain fatty acids in human disease, read"The Role of Long-Chain Fatty Acids in Human Health and Disease". References Naudí A, Jové M, Ayala V, Portero-Otín M, Barja G, Pamplona R. "Membrane lipid unsaturation as physiological adaptation to animal longevity."Front Physiol.2013 Dec 17;4:372. doi: 10.3389/fphys.2013.00372 Agbaga MP, Ahmad M. "Emerging insights into the function of very long chain fatty acids at cerebellar synapses."Neural Regen Res.2025 Jun 1;20(6):1709-1710. doi: 10.4103/NRR.NRR-D-24-00436 Lauritzen L, Brambilla P, Mazzocchi A, Harsløf LB, Ciappolino V, Agostoni C. "DHA Effects in Brain Development and Function."Nutrients.2016 Jan 4;8(1):6. doi: 10.3390/nu8010006 Mitchell MK, Ellermann M. "Long Chain Fatty Acids and Virulence Repression in Intestinal Bacterial Pathogens."Front Cell Infect Microbiol.2022 Jun 17;12:928503. doi: 10.3389/fcimb.2022.928503 Ma C, Vasu R, Zhang H. "The Role of Long-Chain Fatty Acids in Inflammatory Bowel Disease."Mediators Inflamm.2019 Nov 3;2019:8495913. doi: 10.1155/2019/8495913 Batsale M, Bahammou D, Fouillen L, Mongrand S, Joubès J, Domergue F. "Biosynthesis and Functions of Very-Long-Chain Fatty Acids in the Responses of Plants to Abiotic and Biotic Stresses."Cells.2021 May 21;10(6):1284. doi: 10.3390/cells10061284 Interested in our proteomics services? Get in touch with us today! Medium- and Long-Chain Fatty Acid Profiling Service More Articles Co-immunoprecipitation Guide: Key Steps & Tips for Success Learn about the key considerations and best practices for performing co-immunoprecipitation experiments, including sample preparation, antibody selection, and troubleshooting tips. 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12722	https://mbahouse.com/gre/gre-critical-reasoning-tips/	Skip to content Contact Us GRE Critical Reasoning Explained: Sample Questions & Top 10 Tips for Success MBA House \| GMAT Prep Course Table of Contents The Critical Reasoning section of the GRE is your chance to show off how well you can think on your feet. Imagine this: you’re given a statement or argument, and with a few key strategies, you can confidently break it down, analyze it, and draw meaningful conclusions. Sound challenging? It’s not as tough as it seems when you know what to expect and how to approach it. So, let’s keep it simple—What is Critical Reasoning in the GRE? At its core, it’s about logical thinking. Whether you’re asked to strengthen an argument, identify an assumption, or draw an inference, this section tests how well you can evaluate information and form an informed judgment. And here’s the good news: you’ve likely already done this kind of reasoning in your daily life. Have you ever watched a movie and predicted the ending based on a few subtle clues? That’s critical reasoning in action! But how do you pass a Critical Reasoning test with confidence? The key is to familiarize yourself with the question types—like strengthening or weakening an argument—and then practice applying logical thinking to each one. With the right strategies and enough practice, you’ll find these questions become much more approachable. If you’re gearing up for the exam and looking for some last-minute tips, check out 20+ Practical Last-Minute GRE Study ideas to give your preparation that final boost. In this blog, we’ll dive deep into the best ways to approach Critical Reasoning, offering practical tips and examples to help you ace this section. Let’s get started and make this part of the GRE your strength! What Are Critical Thinking Reasoning Questions on the GRE? When tackling critical thinking reasoning questions on the GRE, you’ll face different question types that test your ability to analyze, evaluate, and interpret arguments logically. While these questions may seem challenging at first, they follow consistent patterns. Let’s break down the key types of Critical Reasoning questions you’ll likely encounter. 1. Identify the Assumption In these questions, you’re asked to find the hidden assumption that the argument relies on. An assumption is something that is not explicitly stated but is essential for the argument to hold. Think of it as the invisible bridge between the evidence provided and the conclusion drawn. 2. Weaken the Argument Here, your job is to find a piece of information that would weaken or challenge the argument. It could be a fact or an observation that casts doubt on the logic of the argument, potentially unraveling its entire conclusion. 3. Strengthen the Argument In contrast to weakening the argument, these questions ask you to identify a statement that would bolster the argument’s conclusion. Your goal is to find evidence or reasoning that adds weight to the argument, making it more convincing. 4. Resolve a Paradox These questions present you with a situation that seems contradictory. Your task is to find a statement or explanation that resolves the apparent inconsistency, making sense of the paradox. 5. Inference Inference questions test your ability to draw logical conclusions based on the information provided. You’re not looking for what might be true, but for what must be true given the facts at hand. 6. Complete the Passage In these questions, you’re asked to choose the most logical statement to finish a sentence or thought at the end of a passage. This requires understanding the flow of the argument and predicting how it should conclude. 7. Method of Reasoning This type of question focuses on how an argument is structured. You’ll need to identify the method or strategy the author uses to develop their argument, rather than focusing solely on the content. 8. Boldface Boldface questions present two bolded portions of text, and you’re asked to determine the role each one plays in the argument. Does the bold text serve as evidence, a conclusion, or something else entirely? Understanding how these parts work together is key to answering these questions. Practice Makes Perfect The key to mastering these types of questions is practice, practice, practice!!! Working through GRE logic questions sample exercises will give you the familiarity and confidence to recognize patterns and apply the right strategy. The more you expose yourself to these different question types, the better you’ll become at spotting the right answers quickly and efficiently. Sample Critical Thinking Reasoning Questions for the GRE Question #1: Sample Critical Reasoning Question: Over the past year, brown bears that inhabit the nearby mountains have been coming into the Town of Silverton with greater regularity than ever before. About a year ago, three farms in the town began producing honey, which brown bears find irresistible. Therefore, it must be that the reason why brown bears are coming into the town with greater regularity is that they are coming to consume the honey produced by these farms. Which of the following, if true, most seriously weakens the argument? One year ago, several farms in the town began growing blueberries, another favorite food of brown bears. The town’s residents have grown increasingly anxious about the increased presence of the brown bears. Over the past year, the number of mountain lions sighted in the town has remained the same as in previous years. The owners of the honeybee farms have reported seeing brown bears raiding their beehives for honey. Brown bears are drawn to honey because it’s rich in sugar and calories, which they need to maintain their weight. Question #2: Sample Critical Reasoning Question: Last week, the Silver Bullet Tea Company, which recently opened a chain of high-end tea shops in the affluent City of Silverton, launched a marketing campaign involving television, newspaper, and internet ads, aimed at letting residents know its tea shops are open. Therefore, it’s expected that Silverton’s tea shops will soon be filled with residents purchasing tea. Which of the following, if true, most weakens the argument above? In Silverton, local laws only allow tea shops to operate under strict ordinances. The Silver Bullet Tea Company spent less than half of its original marketing budget for the City of Silverton. Most residents of Silverton are willing to purchase high-end tea. Residents of Silverton, who are outdoorsy, rarely watch TV, read newspapers, or use the internet. In a recent survey, many residents expressed a desire for more high-end tea and coffee shops. 10 Tips and Strategies to Score Well in GRE Critical Reasoning How do you pass a Critical Reasoning test? Here are some key strategies to keep in mind while tackling the Critical Reasoning section on the GRE. 1. Understand the Facts in the Argument Begin by identifying the premise and the conclusion of the argument. What is stated versus what is implied? Often, arguments include statistical figures and surveys that may not represent the general population. By clarifying these elements, you’ll have a solid foundation for analyzing the argument. 2. Preview Questions Before Reading the Passage Instead of speed reading, actively engage with the passage. Skim the text first to get a general idea and then read more carefully with the questions in mind. This approach saves time and helps you focus on key details that are crucial for answering the questions. 3. Use Your Own Words Simplify the passage’s main ideas by paraphrasing in your own words. If you can explain the argument as if to a younger audience, you’ll better grasp its essence and choose the most logical answer. 4. Answer the Question Asked Pay close attention to what the question is specifically asking. Avoid letting personal opinions or assumptions sway your choice. Ensuring you fully understand the question will help you select the most accurate answer. 5. For Inference Types – Mimic the Reasoning Try to follow the same line of reasoning as the passage. The correct answer often mirrors the structure and length of the argument. Look for options that restate or support the conclusion in a similar format. 6. Eliminate Word Complexity Simplify the problem by focusing on straightforward answers. Avoid choices that are overly complex or extreme. The simplest solution is often the correct one. 7. Know the Difference Between Assumptions and Conclusions Assumptions are unstated premises, while conclusions are the author’s hypotheses. For assumption questions, check if the conclusion holds true when you negate the assumption. For conclusion questions, find options that test the hypothesis with “IF A then not B” scenarios. 8. For Strengthening – Identify Supporting Information Look for answers that add supporting evidence to the argument. Strengthening questions often require you to connect new information to the existing evidence and conclusion. 9. For Weakening – Identify Premises and Conclusions In weakening questions, assume the answer choice is true and see if it exposes flaws in the argument. Look for gaps or errors between the scenarios presented and the premises. 10. Beware of Test Makers’ Traps Test creators may try to mislead you with incorrect conclusions or tricky wording. Stay alert for such traps to avoid making errors based on misleading information. Bringing It All Together “Success is the sum of small efforts, repeated day in and day out.” — Robert J. Collier As you gear up for the GRE, remember: don’t stress out, and take care of your health. Consistent practice and a focused mindset are your best allies in mastering the Critical Reasoning section. By understanding the types of questions and applying the strategies we’ve discussed, you’ll be well-prepared to tackle this challenging part of the test. If you’re searching for trusted GRE study courses, MBA House is here to support you. Our team of Ivy League-trained tutors and consultants is dedicated to guiding you through every step of your GRE preparation. Your Questions Answered: GRE Critical Reasoning Section Does the GRE have logical reasoning questions? The GRE doesn’t feature traditional logical reasoning questions, but it does include GRE logic questions within its reading comprehension sections. These questions require you to make inferences based on the passage’s information. To succeed, focus on understanding and analyzing the text carefully, as logical reasoning is crucial for drawing accurate conclusions. How many questions can you get wrong on the GRE for 170? To achieve a perfect score of 170 on the GRE, the number of questions you can get wrong varies between sections. For the Quantitative section, you can typically only afford to miss 1-2 questions to score a 170. In contrast, for the Verbal section, you can get about 3-6 questions wrong and still achieve a perfect score. It’s important to note that exact thresholds can vary based on the difficulty of the test and overall test-taker performance. How many critical reasoning questions are in the GRE? The GRE Critical Reasoning section features seven types of questions: Identify the Assumption Weaken the Argument Strengthen the Argument Inference Resolve the Paradox Complete the Passage Boldface MBA House \| GMAT Prep Course 1447 Comments Facebook Twitter LinkedIn Pinterest ##### MBA House At MBA House, we deliver effective strategies that allow our students to achieve winning results and gain admission to the school of their dreams! ##### MBA House At MBA House, we deliver effective strategies that allow our students to achieve winning results and gain admission to the school of their dreams! Categories GMAT GRE Executive Assessment MBA Admissions Business Schools Social Media Instagram Facebook Twitter Linkedin Pinterest Related Posts GRE Vocabulary: Mastering the Words That Matter Most MBA House \| GMAT Prep Course November 29, 2024 How Long Does It Take to Study for The GMAT? 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12723	https://www.quora.com/What-is-the-formula-for-a-parabola-area	Something went wrong. Wait a moment and try again. Parabolas (geometry) Formulas (functions) Quadratic Function Calculus (Mathematics) PLANE GEOMETRY Mathematical Equations Mathematical Sciences 5 What is the formula for a parabola area? · The area of a parabola itself is not typically defined as parabolas are curves rather than closed shapes. However, if you are referring to the area under a parabolic curve defined by a quadratic function, you can calculate that using integration. For a parabola represented by the equation , the area under the curve between two points and is given by the definite integral: To compute the area, you would evaluate the integral: Find the antiderivative of the function: The area of a parabola itself is not typically defined as parabolas are curves rather than closed shapes. However, if you are referring to the area under a parabolic curve defined by a quadratic function, you can calculate that using integration. For a parabola represented by the equation , the area under the curve between two points and is given by the definite integral: To compute the area, you would evaluate the integral: Find the antiderivative of the function: Evaluate the definite integral from to : This will give you the area under the parabola between the specified limits. If you meant something different regarding the area related to a parabola, please provide more details! Himanshu Kashyap Promoted by The Hartford The Hartford Updated Fri What is small business insurance? Small business insurance is a comprehensive type of coverage designed to help protect small businesses from various risks and liabilities. It encompasses a range of policies based on the different aspects of a business’s operations, allowing owners to focus on growth and success. The primary purpose of small business insurance is to help safeguard a business’s financial health. It acts as a safety net, helping to mitigate financial losses that could arise from the unexpected, such as property damage, lawsuits, or employee injuries. For small business owners, it’s important for recovering quickl Small business insurance is a comprehensive type of coverage designed to help protect small businesses from various risks and liabilities. It encompasses a range of policies based on the different aspects of a business’s operations, allowing owners to focus on growth and success. The primary purpose of small business insurance is to help safeguard a business’s financial health. It acts as a safety net, helping to mitigate financial losses that could arise from the unexpected, such as property damage, lawsuits, or employee injuries. For small business owners, it’s important for recovering quickly and maintaining operations. Choosing the right insurance for your small business involves assessing your unique needs and consulting with an advisor to pick from comprehensive policy options. With over 200 years of experience and more than 1 million small business owners served, The Hartford is dedicated to providing personalized solutions that help you focus on growth and success. Get a quote today! Kenneth Simmons Former HS Teacher (Math, Computer Math Computer Science) (1994–2006) · Author has 155 answers and 158.6K answer views · 5y Select any horizontal segment to be the base. Take the distance between the base and the vertex to be the height of the parabolic area. The area is ( 2 / 3 ) b h . If the sides are linear, the area is ( 1 / 2 ) b h . If the sides are cubic, the area is ( 3 / 4 ) b h . If the sides are quartic, the area is ( 4 / 5 ) b h . Do you see the pattern for polynomial curves? Related questions How do you define area of a parabola? How do you find the area under a parabola? What is the formula for the perimeter of a parabola? What is the formula of parabola? How do you calculate a parabola? ZAM B.E in Civil Engneering & Construction, Quaid-e-Awam University of Engineering, Science and Technology (Graduated 2020) · 6y I have also Some Confusion about this, however I know formula for the area of parabola is (A=1/3baseAltitude) but in some problems I have also seem this forumla (A=2/3baseAltitude) ???????????? Martin Liu Author has 900 answers and 812K answer views · 5y take a look at this diagram. The parabola is y=x^2; the area is multiple squares that increase linearly. The pyramid is also made by multiple squares that increase linearly. y=x^2 is from 0 to a w... take a look at this diagram. The parabola is y=x^2; the area is multiple squares that increase linearly. The pyramid is also made by multiple squares that increase linearly. y=x^2 is from 0 to a w... Related questions What is the formula for the area of parabola and hyperbola (for CET)? How do you calculate the surface area of a parabola or circle? What is an example of finding area using a formula? Why are formulas used to find areas? What are some applications of parabola in real life? What is the formula for area and perimeter? Graham Dolby Author has 3.2K answers and 1.7M answer views · 2y Originally Answered: Is there an equation for the area under a parabola? · Yes, thank you for asking. The blue equation is the area of the bounding rectangle and the red equation is the area of the bounded parabola. All of this is subject to the parabola producing an area with the x axis. Yes, thank you for asking. The blue equation is the area of the bounding rectangle and the red equation is the area of the bounded parabola. All of this is subject to the parabola producing an area with the x axis. Promoted by Grammarly Grammarly Great Writing, Simplified · Aug 18 Which are the best AI tools for students? There are a lot of AI tools out there right now—so how do you know which ones are actually worth your time? Which tools are built for students and school—not just for clicks or content generation? And more importantly, which ones help you sharpen what you already know instead of just doing the work for you? That’s where Grammarly comes in. It’s an all-in-one writing surface designed specifically for students, with tools that help you brainstorm, write, revise, and grow your skills—without cutting corners. Here are five AI tools inside Grammarly’s document editor that are worth checking out: Do There are a lot of AI tools out there right now—so how do you know which ones are actually worth your time? Which tools are built for students and school—not just for clicks or content generation? And more importantly, which ones help you sharpen what you already know instead of just doing the work for you? That’s where Grammarly comes in. It’s an all-in-one writing surface designed specifically for students, with tools that help you brainstorm, write, revise, and grow your skills—without cutting corners. Here are five AI tools inside Grammarly’s document editor that are worth checking out: Docs – Your all-in-one writing surface Think of docs as your smart notebook meets your favorite editor. It’s a writing surface where you can brainstorm, draft, organize your thoughts, and edit—all in one place. It comes with a panel of smart tools to help you refine your work at every step of the writing process and even includes AI Chat to help you get started or unstuck. Expert Review – Your built-in subject expert Need to make sure your ideas land with credibility? Expert Review gives you tailored, discipline-aware feedback grounded in your field—whether you're writing about a specific topic, looking for historical context, or looking for some extra back-up on a point. 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Let us assume the parabola goes through origo, otherwise translate it there. The most simple parabola is y=x^2. The primitive function (=integral) for that is x^3/3. So the area UNDER that curve from origo to some x=a, is a^3/3. To find the parabola area, you subtract that from the corresponding surrounding rectangle. Next we can do some stretching. If you extend it by a factor b, i You can look it up in any mathematical table of some class. But I think it is often more valuable to see how it can be calculated. Area is calculated by the integral, and first do it in the most simple way possible. Let us assume the parabola goes through origo, otherwise translate it there. The most simple parabola is y=x^2. The primitive function (=integral) for that is x^3/3. So the area UNDER that curve from origo to some x=a, is a^3/3. To find the parabola area, you subtract that from the corresponding surrounding rectangle. Next we can do some stretching. If you extend it by a factor b, in the X-direction the result is scaled accordingly, when the equation becomes y=(x/b)^2. Stretching in the y-direction by a factor c, scales in the same way, when the equation becomes y/c=(x/b)^2. Then you can identify your parameters b and c to whatever you may have. Mathematics Topology Tₙ © Answered by Octavious Morst · Author has 924 answers and 79.5K answer views · Mar 28, 2023 ∫ 𝑥² dx = 𝑓⁻¹ + c ∫ 𝑥² dx = 𝑓⁻¹ + c Promoted by Cash Canvas Ethan Anderson Senior Writer at CashCanvas (2024–present) · Updated Apr 21 What are the biggest missed opportunities for building wealth that most people don’t know about? Overpaying on Auto Insurance Believe it or not, the average American family still overspends by $461/year¹ on car insurance. Sometimes it’s even worse: I switched carriers last year and saved literally $1,300/year. 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For simplicity you can eliminate “bx…all that does is locates the apex on a graph. “c” locates the apex on the y-axis and it’s useful in this discussion. “a” scales the parabola so we’ll leave it be, also. All I wish to do is place the parabola symmetrically on a graph so it will be simple to work with. Let’s use the equation: y=-x^2 +25. I chose it because it opens downward and also I won’t wind up with messy I’m going to assume you had at least a primer in integral calculus. First, you need have a boundary for the domain, otherwise the area is indefinite. The general equation is ax^2 + bx + c. For simplicity you can eliminate “bx…all that does is locates the apex on a graph. “c” locates the apex on the y-axis and it’s useful in this discussion. “a” scales the parabola so we’ll leave it be, also. All I wish to do is place the parabola symmetrically on a graph so it will be simple to work with. Let’s use the equation: y=-x^2 +25. I chose it because it opens downward and also I won’t wind up with messy irrational numbers for my boundaries. So you have 3 principal points: The apex at (0,25) and your domain boundaries (-5,0) and (5,0). Our goal is to find the area under the curve between the two boundary points. Integrate the function: y= -x^2 +25 Int (from x=-5 to x=5) (-x^2 +25)dx =[-(1/3)x^3 +25x] from x=-5 to 5 =-(125/3) +125 -(-1/3)(-5)^3 -(-125) = 250- 250/3 =500/3 square units. Marty Green used to be the math guy on Community Access TV in Winnipeg back in the 90's.. · Author has 652 answers and 593.7K answer views · 7y Two-thirds the exscribed rectangle. It’s a very usefule forumla, it’s exact, and it’s a very good approximation for any kind of shallow arc. Sponsored by Zoho One Top business software to run your entire business - Zoho One. From sales to marketing, HR to finance, manage your complete business with one solution. Get free trial! Gary Ward MaEd in Education & Mathematics, Austin Peay State University (Graduated 1997) · Author has 4.9K answers and 7.5M answer views · Updated 9mo Related How was the formula for finding roots of a parabola developed? Why is it used in this process? How was the formula for finding roots of a parabola developed? Why is it used in this process? My guess is it came from completing the square with u-substitution to take care of making the coefficient of x-squared equal to the number one. U-sub not needed. Divide every term by ‘a’. [math]\displaystyle x^2+\frac{b}{a}x+\frac{c}{a}=0[/math] Subtract c/a from both sides, then make the right-hand side a perfect square by dividing the x- coefficient by two and adding to both sides to preserve equality. As an aside: [math]\displaystyle \left(x + \left(\frac{b}{2a}\right) \right)^2 = x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\r[/math] How was the formula for finding roots of a parabola developed? Why is it used in this process? My guess is it came from completing the square with u-substitution to take care of making the coefficient of x-squared equal to the number one. U-sub not needed. Divide every term by ‘a’. [math]\displaystyle x^2+\frac{b}{a}x+\frac{c}{a}=0[/math] Subtract c/a from both sides, then make the right-hand side a perfect square by dividing the x- coefficient by two and adding to both sides to preserve equality. As an aside: [math]\displaystyle \left(x + \left(\frac{b}{2a}\right) \right)^2 = x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2[/math] That is where the third term making it a perfect square comes from. [math]\displaystyle x^2+\frac{b}{a}x+\left(\frac{b}{2a}\right)^2=-\frac{c}{a}+\frac{b^2}{4a^2}[/math] Multiply the -c/a term top and bottom by 4a so that the fractions can share the denominator. Change the left side to its before squared form. [math]\displaystyle \left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}[/math] Take the square root of each side. [math]\displaystyle x +\frac{b}{2a}=\pm\frac{\sqrt{b^2-4ac}}{2a}[/math] Since the fractions on both sides have the same denominator, we can just add -b/2a to both sides. [math]\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/math] The [math]\frac{-b}{2a}[/math] part finds the axis of symmetry and the [math]\frac{\pm \sqrt{b^2–4ac}}{2a}[/math] finds how much left and right of the axis of symmetry the roots are at. The inside of the square root is the determinate. If b² - 4ac > 0, there are two real roots; if b² - 4ac = 0, then there is one root (or duality, two roots in one); and if b² - 4ac < 0, the roots are imaginary meaning the parabola does not cross the x-axis or y-axis in the case of x = ay² + by + c. Gerald Grenier Uses both American and Queen's English · Author has 1.4K answers and 346.4K answer views · 2y Related How do you calculate the surface area of a parabola or circle? Surface area is a property of 3d shape. parabola and circles are 2d shapes. 2d shapes just have areas. circle has the area equal to pi times its radius squared aka [math]A=\pi r^2[/math] a parabola being an open ended shape it technically is infinite area, you need to define it closing off. typically you can find the area via indefinate intergration of the equation of the parabola William Heller Teacher · Author has 1.2K answers and 328.4K answer views · 2y Related How do you calculate the surface area of a parabola or circle? A circle or a parabola are generally graphed on the two-dimensional plane. We often use “surface area” in relation to a three-dimensional object like a sphere or paraboloid. For a two-dimensional object, “area” will usually be sufficient. The area of a circle is πr^2, where r is the radius of the circle. The area under a parabola bounded by the x-axis can be determined using calculus. You would need the formula for the parabola to integrate the area. Related questions How do you define area of a parabola? How do you find the area under a parabola? What is the formula for the perimeter of a parabola? What is the formula of parabola? How do you calculate a parabola? What is the formula for the area of parabola and hyperbola (for CET)? How do you calculate the surface area of a parabola or circle? What is an example of finding area using a formula? Why are formulas used to find areas? What are some applications of parabola in real life? What is the formula for area and perimeter? What is the area of cylinder formula? What is the formula for finding out how many points lie on a given line, circle or parabola (without using coordinates)? How do you calculate the areas of a polygon? How do you calculate the area of a triangle? What is the formula to calculate the area of a land? About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
12724	https://emcrit.org/emcrit/stellate-ganglion-block/	EMCrit RACC Online Medical Education on Emergency Department (ED) Critical Care, Trauma, and Resuscitation You are here: Home / EMCrit / EMCrit 395 – Stellate Ganglion Block – Not Whether, but When? EMCrit 395 – Stellate Ganglion Block – Not Whether, but When? by Scott Weingart, MD FCCM 8 Comments Today, we cover an ultrasound nerve block that has been on my radar for a while: The Stellate Ganglion Nerve Block for electrical storm/refractory ventricular fibrillation. Of course, I need an expert to discuss an ultrasound nerve block topic, so I got… Robert Stenberg, MD Ultrasound Director and Ultrasound Fellowship Director Cleveland Clinic Akron General Emergency Medicine Robert Stenberg, MD FPD-AEMEUS is the Emergency Ultrasound Director at Cleveland Clinic Akron General. He completed medical school at the University of Wisconsin, residency at University of North Carolina, and ultrasound fellowship at Virginia Commonwealth University. Bob has RDMS, RDCS and FPD-AEMEUS certifications, and has passed critical care echo boards. He is passionate about teaching, resuscitation and procedures, including nerve blocks. His primary research focus is on multimodal pain control through regional anesthesia in the ED. Background Electrical storm (ES), sometimes called ventricular storm, is defined as the occurrence of three or more episodes of sustained ventricular tachycardia (VT) or ventricular fibrillation (VF) in a short period of time (often 24 hour window).1 While multiple underlying etiologies may be responsible for causing ES, it is additionally perpetuated by sympathetic activation which decreases the threshold for ventricular arrhythmias and refractoriness. Traditional management includes Advanced Cardiac Life Support (ACLS)-guideline driven resuscitation if the patient is pulseless, with amiodarone as first line antiarrhythmic therapy, followed by beta-blockade with either propranolol, esmolol, or metoprolol. One caveat to following ACLS guidelines is that attempts should be made to minimize use of epinephrine as much as possible considering treatment of ED revolves around sympathetic blockade, and beta-blockade in these patients has shown to have a higher survival rate compared to ACLS antiarrhythmics (67% vs. 5%).2 Other considerations may include use of dual sequential defibrillation, changing pad location (especially switching from anterolateral to anteroposterior), aggressive electrolyte repletion (i.e. magnesium, potassium, and calcium), veno-arterial extracorporeal membrane oxygenation (VA ECMO), and use of phenylephrine as a first line vasopressor as it is a pure alpha-1 adrenergic agonist and will not stimulate cardiac beta-receptors.3,4 Sometimes, these efforts are unsuccessful at terminating the arrhythmia. The Stellate Ganglion Block (SGB) provides an adjunct in treatment when ES is refractory to traditional management. It is considered one type of cervical sympathetic block; and has not only been proven to be a safe and effective treatment modality in ES, but is also used for a multitude of indications, including PTSD, phantom limb pain, complex regional pain syndrome, zoster of head/neck/upper extremity, and other autonomic-related conditions.5-9 There is also growing evidence for use in the neurocritical care patient population to reduce vasospasm for conditions such as aneurysmal subarachnoid hemorrhage.10 For Emergency Medicine, the primary indication for SGB is to abort electrical/ventricular storm. The goal of the SGB in ES is to cause cardiac sympathetic denervation through blockade of the sympathetic chain at the inferior ganglion (also known as the stellate ganglion due to its star-shaped appearance). Blockade at this level targets sympathetic outflow to the myocardium, subsequently increasing the threshold for arrhythmia and allowing for further resuscitative efforts to be successful. Key Recent Publications The STAR Study11 Multi-Center (19 centers), International Retrospective & Prospective Longitudinal Study 131 patients, 184 total SGB’s performed 82% patients had bolus only 42% of procedures were ultrasound-guided Anesthetic used (% of cases) Lidocaine alone, 28% Bupivacaine alone, 3% Lidocaine + bupivacaine, 9% Lidocaine + ropivacaine, 2% Primary Outcomes = Efficacy & Safety 92% of patients showed a reduction of antitachycardia pacing or defibrillation shocks >50% in the 12 hours following SBG Only 1 major complication occurred (0.5%) One case of Local Anesthetic Systemic Toxicity (LAST), that had respiratory suppression and was successfully corrected with lipid emulsion therapy Secondary Outcomes Anisocoria Not all patients with a successful block will have anisocoria: 56% did not have anisocoria with no difference in reduction of dysrhythmia between groups. Anatomic versus ultrasound-guided, bolus vs infusion Significant reduction of treated arrhythmias in both anatomical and ultrasound-guided groups No significant difference between bolus and continuous infusion groups High volume versus Low Volume Similar efficacy in high volume and low volume centers Dual antiplatelet therapy and anticoagulation No complications with patients on antiplatelet therapy or anticoagulation (67% blocks) Mild complications Minor Bradycardia 1 case (0.5%) Hypotension 1 case (0.5%) Expected Side Effects of Block Temporary brachial plexus 3 cases (1.6%) Hoarseness 2 cases (1.1%) Dysphonia/Neck Pain/Vomiting 1 case each (0.5%) Savastano S, Baldi E, Compagnoni S, et Electrical storm treatment by percutaneous stellate ganglion block: the STAR study. Eur Heart J. 2024;45(10):823-833. doi:10.1093/eurheartj/ehae021 Meta-analysis published this fall12 15 studies, 542 patients, 553 ES events Demonstrated a significant reduction in ventricular arrhythmia with SGB: 70% complete resolution of ventricular arrhythmia (VA) 19% had partial resolution of VA 7% had no resolution of VA Motazedian, P., Quinn, N., Wells, G.A. et al. Efficacy of stellate ganglion block in treatment of electrical storm: a systematic review and meta-analysis. Sci Rep 14, 24719 (2024). New case series in AJEM, July 202413 Two scenarios with resolution of ES by left SGB using 10mL 2% Lidocaine One after amiodarone bolus x2, lidocaine bolus, and esmolol Resolution of ventricular tachycardia within five minutes. One after amiodarone bolus and infusion with “pharmacologic cardioversion”; resolution of dysrhythmia after block Procedure: Left-Sided Lateral to Medial Stellate Ganglion Block Generally, the block should be performed on the left side, as there is more evidence for left-sided block; and, in some studies, there has been a decreased rate of hypotension with a left-sided block. In a review of 16,404 right SGB and 13,766 left SGB, the right SGB decreased blood pressure by 25-49mmHg in 10.9% of cases and more than 50mmHg in 0.67% with only 4% cases with increase in pressure; on the other than, the left SGB decreased blood pressure by 25-49mmHg in 8.4% and more than 50mmHg in 0.49% with an increase in blood pressure in 6.2% of cases. Further, the right SGB had a 30 beats per minute drop in heart rate relative to left SGB .14, 15 To determine whether the block was successful, the primarily means of evaluating such should be reduction in VA. Prior thoughts assessing for Horner’s syndrome (ptosis, miosis, anhidrosis) for block success were demonstrated to not be reliable in the STAR trial. 11 Figure 2, SGB Sonoanatomy and block target, by Dr. Brian Makowski Indications: Electrical Storm Contraindications: Anaphylaxis to the specific local anesthetic Anticoagulation is NOT a contraindication (refer to STAR Study above); recommend providers use risk benefit and clinical decision-making Medication Selection Although variable in literature, typically around 10 mL of a short-acting local anesthetic is used, such as 1% or 2% lidocaine withOUT epinephrine A longer acting anesthetic has the significant benefit of extending the duration of blockade; on the other hand, any complications will have a longer duration. Ropi 0.2% as a long-acting option makes the most sense if you go that route Some studies also used dexamethasone to further extend the action of blockade, typically 8mg; this should be preservative-free 16,17 Set-up/Positioning Get medications and nerve block equipment Nerve block needle, micropuncture needle or blunt-tipped needle strongly preferred to reduce trauma to surrounding structures Consider placing the ultrasound machine to the right side of the patient near the head/neck while standing on patient’s left for optimal ergonomics Turn patient's head turned contralaterally (to the right). This exposes the neck for easier needle entry and to have a clearer needle trajectory as it can displace internal jugular vein (IJ) and other vessels out of the way Procedure Place transducer, in a transverse orientation, along the left side of the neck around the level of the cricoid cartilage (C6), near where you would for an IJ central line (Figure 2) Some experts recommend tracking the brachial plexus nerve roots from the supraclavicular region cranially to track where the C6 nerve root originates as a means to ensure ultrasounding the appropriate level 18 Identify the carotid artery, and make small sweeps cranially and caudally until you can see a defined muscle belly deep to the carotid (longus colli muscle) without any vital structures lateral to the area for safe needle approach 19 Use color/power Doppler to ensure a safe needle trajectory, particularly watching for vessels such as inferior thyroid and vertebral artery Insert needle in-plane, lateral to medial, advancing needle towards the prevertebral fascia, along superficial aspect of the longus colli muscle just deep to carotid. Try to avoid nerve fascicles, vessels and the esophagus along the needle trajectory It can be hard at times to get around the anterior C6 tubercle (also known as Chassaignac’s tubercle), and some resources recommend trying to sweep just caudal to this area to access. Anchoring hands for transducer stability and needle control will be essential if performing during chest compressions Vertebral artery can run adjacent to or what looks like through the stellate ganglion, so be alert. Injection site is just below stellate ganglion or just above the long. colli muscle. Aspirate, then inject 10mL of anesthetic (Figure 3). If no improvement, one can consider blocking the right stellate ganglion in similar fashion (a 10 minute window is often considered a fair time to wait in between).20 Consider the total dose of local anesthetic the patient has received in attempts to avoid LAST Syndrome. If using lidocaine, may need to repeat bolus or switch to long-acting on second procedure Complications Most common complications are transient21 Most common are hoarseness (up to 28%) and lightheadedness (up to 6%) Others include hypertension, cough, horner's, brachial plexus block, and Please review STAR trial complications above, highlighting a modern practice: There were minimal major complications and a few transient expected complications.11 There are rare case reports of more serious outcomes, many of which were done without ultrasound guidance, and perhaps with outdated equipment (sharp needles, etcetera). There are rare case reports of seizure, cardiac arrest, pneumothorax, infection, dural puncture, hemomediastinum Consider having resuscitative equipment nearby (which is very likely the case, already) Figure 3, SGB HALO Procedure Pocket Card, by Dr. Brian Makowski The Challenge: not whether, but when? There is now a solid body of evidence showing SGB has good reduction in ventricular arrhythmia in the setting of ES. The challenging question is when in your algorithm does the SGB fit in? To the best of the author’s knowledge there is no consistent evidence or guidelines regarding where this piece fits in the algorithm. The only information found was from 2017 AHA/ACC/HRS Guidelines on VT/VT, highlighting the following: “”In patients with VT/VF storm in whom a beta blocker, other antiarrhythmic medications, and catheter ablation are ineffective, not tolerated, or not possible, cardiac sympathetic denervation may be reasonable.” 22 Many have considered it when reaching for or administering esmolol.23 One other thought, with the case of LAST syndrome in the STAR trial, there may be some consideration of performing the block prior to multiple doses of intravenous lidocaine; based on this, one could consider performing block after one intravenous bolus of lidocaine, perhaps while administering beta blocker intravenous bolus. Lastly, there is some concern whether this block is being done at a point in the resuscitation at which a chance of recovery is minimal, and it may be worth performing at a point of the resuscitation while the patient has a more meaningful chance of recovery. Summary There is now substantial evidence supporting SGB in managing ES, demonstrating its efficacy and safety profiles; it should be considered in refractory ES, and may be more impactful at a center that does not readily have access to VA ECMO. To the best of the authors’ knowledge, there is no evidence to demonstrate a mortality benefit. Hopefully, the randomized controlled study enrolling patients until December 2024, the GANGlion Stellate Block for Treatment of Electric storRm Trial (GANGSTER Trial) can further add to the evidence.24 While it is fair to perform a landmark-based approach, emergency physicians easily have the skillset to perform this under ultrasound guidance and there is evidence of decreased complications with such. A reasonable plan is a single injection of 10mL of 1% lidocaine along the left stellate ganglion; anisocoria is not a reliable marker for success and reduction in ventricular arrhythmia should be the primary focus for block success. A large challenge is navigating the timing upon which this is performed in the emergency department with other more standard measures. One can apply their clinical judgement based on the multitude of variables, or consider performing after having given two to three antiarrhythmic medications and making other adjustments. References Eifling M, Razavi M, Massumi The evaluation and management of electrical storm. Tex Heart Inst J. 2011;38(2):111-21. PMID: 21494516; PMCID: PMC3066819. Nademanee K, Taylor R, Bailey WE, Rieders DE, Kosar EM. Treating electrical storm : sympathetic blockade versus advanced cardiac life support-guided Circulation. 2000;102(7):742-747. doi:10.1161/01.cir.102.7.742 Lupton JR, Newgard CD, Dennis D, et Initial Defibrillator Pad Position and Outcomes for Shockable Out-of-Hospital Cardiac Arrest. JAMA Netw Open. 2024;7(9):e2431673. doi:10.1001/jamanetworkopen.2024.31673 Cheskes S, Verbeek PR, Drennan IR, McLeod SL, Turner L, Pinto R, Feldman M, Davis M, Vaillancourt C, Morrison LJ, Dorian P, Scales DC. Defibrillation Strategies for Refractory Ventricular Fibrillation. N Engl J Med. 2022 Nov 24;387(21):1947-1956. doi: 10.1056/NEJMoa2207304. Epub 2022 Nov 6. PMID: 36342151. Narouze, “Ultrasound-guided stellate ganglion block: safety and efficacy.” Current pain and headache reports 18 (2014): 1-5. CARRON, HAROLD, and ROGER “Stellate ganglion block.” Anesthesia & Analgesia 54.5 (1975): 567-570. Olmsted, Kristine L. Rae, et al. “Effect of stellate ganglion block treatment on posttraumatic stress disorder symptoms: a randomized clinical ” JAMA psychiatry77.2 (2020): 130-138. Elias, Maizin, “Cervical Sympathetic and Stellate Ganglion Blocks” Pain Physician, Volume 3, Number 3, pp 294-304, 2000, Association of Pain Management Anesthesiologists® Guy Feigin, Sofia Velasco Figueroa, Marina F Englesakis, Rohan D’Souza, Yasmine Hoydonckx, Anuj Bhatia, Stellate ganglion block for non-pain indications: a scoping review, Pain Medicine, Volume 24, Issue 7, July 2023, Pages 775–781, Bombardieri AM, Albers GW, Rodriguez S, et al Percutaneous cervical sympathetic block to treat cerebral vasospasm and delayed cerebral ischemia: a review of the evidence Journal of NeuroInterventional Surgery 2023;15:1212-1217 Savastano S, Baldi E, Compagnoni S, et Electrical storm treatment by percutaneous stellate ganglion block: the STAR study. Eur Heart J. 2024;45(10):823-833. doi:10.1093/eurheartj/ehae021 Motazedian, P., Quinn, N., Wells, G.A. et al. Efficacy of stellate ganglion block in treatment of electrical storm: a systematic review and meta-analysis. Sci Rep 14, 24719 (2024). Archana Nair, Sanjeev Bhoi, Yatharth Choudhary, “Cease the storm – Successful stellate ganglion block in terminating refractory electrical storm”, The American Journal of Emergency Medicine, Volume 81, 2024, Pages 160.e3-160.e7,ISSN 0735-6757, Yokota, C. Taneyama and H. Goto, “Different Effects of Right and Left Stellate Ganglion Block on Systolic Blood Pressure and Heart Rate,” Open Journal of Anesthesiology, Vol. 3 No. 3, 2013, pp. 143-147. doi: 10.4236/ojanes.2013.33033. Kirkpatrick K, Khan MH, Deng Y, Shah A Review of Stellate Ganglion Block as an Adjunctive Treatment Modality. Cureus. 2023 Feb 19;15(2):e35174. doi: 10.7759/cureus.35174. PMID: 36949968; PMCID: PMC10029323. Reinertsen, , Sabayon, M., Riso, M., Lloyd, M. & Spektor, B. Stellate ganglion blockade for treating refractory electrical storm: a historical cohort study. Can. J. Anaesth. 68, 1683–1689. (2021). Cardona, R., Butnaru, D. S. & Ribero, O. F. G. Bloqueo De Ganglio Estrellado como una estrategia de manejo en cuidado paliativo en paciente con arritmias cardíacas terminales: Serie De Casos. Rev. Chil. Anest. 53, 66–72 (2024). Tian Y, Wittwer ED, Kapa S, McLeod CJ, Xiao P, Noseworthy PA, Mulpuru SK, Deshmukh AJ, Lee HC, Ackerman MJ, Asirvatham SJ, Munger TM, Liu XP, Friedman PA, Cha Effective Use of Percutaneous Stellate Ganglion Blockade in Patients With Electrical Storm. Circ Arrhythm Electrophysiol. 2019 Sep;12(9):e007118. doi: 10.1161/CIRCEP.118.007118. Epub 2019 Sep 13. PMID: 31514529. Goel V, Patwardhan AM, Ibrahim M, Howe CL, Schultz DM, Shankar H. Complications associated with stellate ganglion nerve block: a systematic review. Reg Anesth Pain Med. 2019 Apr 16:rapm-2018-100127. doi: 1136/rapm-2018-100127. Epub ahead of print. PMID: 30992414; PMCID: PMC9034660. Al-Khatib SM, Stevenson WG, Ackerman MJ, Bryant WJ, Callans DJ, Curtis AB, Deal BJ, Dickfeld T, Field ME, Fonarow GC, Gillis AM, Granger CB, Hammill SC, Hlatky MA, Joglar JA, Kay GN, Matlock DD, Myerburg RJ, Page 2017 AHA/ACC/HRS Guideline for Management of Patients With Ventricular Arrhythmias and the Prevention of Sudden Cardiac Death: A Report of the American College of Cardiology/American Heart Association Task Force on Clinical Practice Guidelines and the Heart Rhythm Society. J Am Coll Cardiol. 2018 Oct 2;72(14):e91-e220. doi: 10.1016/j.jacc.2017.10.054. Epub 2018 Aug 16. Erratum in: J Am Coll Cardiol. 2018 Oct 2;72(14):1760. doi:10.1016/j.jacc.2018.08.2132. PMID: 29097296. Video Tutorials Regional Anesthesiology and Acute Pain Medicine YouTube Ultrasound-Guided Stellate Ganglion Block CoreUltrasound Mastering Ultrasound-Guided Stellate Ganglion Blocks: Step-by-Step Guide Additional Literature Experience in 37 Patients Secondary Analysis of the STAR Trial Stellate Ganglion Block Review Paper Shownotes Excerpted (with permissions) from: Riders on the Storm: Stellate Ganglion Block for Electrical Storm⚡Brian Makowski, DO; Joshua Jacquet, MD, Trent She, MD; Kevin Watkins, MD; Robert Stenberg, MDACEP News More on EMCrit EMCrit 392 – All Things Defibrillation with Sheldon Cheskes You Need an EMCrit Membership to see this content. Login here if you already have one. Forgot Password Author Recent Posts Scott Weingart, MD FCCM Editor-in-Chief, at EMCrit.org An ED Intensivist from NY. ProfessorNassau University Medical CenterNo conflicts of interest (coi). Latest posts by Scott Weingart, MD FCCM (see all) EMCrit 1:1 Nursing 005 – Medical Cardiac Arrest - September 18, 2025 EMCrit 408 – Behind the Scenes of an Infuriating Medical Malpractice Trial - September 13, 2025 EMCrit 407 – Massive Hemorrhage Protocol 2.0 with Petro - August 30, 2025 Share this: Click to share on Facebook (Opens in new window) Facebook Click to share on X (Opens in new window) X Click to share on Reddit (Opens in new window) Reddit Click to share on Pinterest (Opens in new window) Pinterest Click to email a link to a friend (Opens in new window) Email Click to print (Opens in new window) Print Cite this post as: Scott Weingart, MD FCCM. EMCrit 395 – Stellate Ganglion Block – Not Whether, but When?. EMCrit Blog. Published on February 23, 2025. Accessed on September 28th 2025. Available at [ ]. Financial Disclosures: The course director, Dr. Scott D. Weingart MD FCCM, reports no relevant financial relationships with ineligible companies. This episode’s speaker(s) report no relevant financial relationships with ineligible companies unless listed above. CME Review Original Release: February 23, 2025Date of Most Recent Review: Aug 1, 2025 Termination Date: Aug 1, 2028 You finished the 'cast,Now Join EMCrit! As a member, you can... Get CME hours Get the On Deeper Reflection Podcast Support the show Write it off on your taxes or get reimbursed by your department Join Now! . Get the EMCrit Newsletter If you enjoyed this post, you will almost certainly enjoy our others. Subscribe to our email list to keep informed on all of the Resuscitation and Critical Care goodness. This Post was by the EMCrit Crew, published 7 months ago. We never spam; we hate spammers! Spammers probably work for the Joint Commission. Login 8 Comments oldest newest most voted Inline Feedbacks View all comments Edwin Saunders 7 months ago The stellate ganglion block seems like a promising approach for treating conditions like PTSD and refractory arrhythmias. It’s great to see more discussion on alternative treatments that could offer real relief. snow rider What's Your Job? manager 0 Reply Remi Schweizer 6 months ago Thank you for this post. I had several comments: -To date, no comparative study exists concerning SGB to treat ES. Litterature is only made of case reports and case series. It seems to be the smallest level of scientific evidence that we can imagine, because we cannot differentiate treatment effect of the natural course of electrical storm -Association of intravenous Lidocaine infusion and a locoregional anesthesia is potentially toxic (ventricular arrythmias for example!). Current anesthesia guidelines suggest waiting 4 hours between the end of Lidocaine infusion and the locoregional anesthesia ( -Bilateral SGB in a non intubated patient in prohibited… Read more » What's Your Job? Intensivist -2 Reply Scott Weingart, MD FCCM Author Reply to Remi Schweizer 6 months ago did you listen to the podcast or are you just responding to the shownotes? What's Your Job? emcrit 0 Reply Patrick Ward 6 months ago I have done this multiple times with good results. Most of the time it will originate from the left side of the heart which a left SGB will work great, but there will be a rare chance that a PVC from the right will send them into VT and you will need to block the right SGB. From experience, block it before or during your central line placement. You WILL hate your life if you’re doing it after. I will use 0.5% Bupivicaine just to get it to last longer too and buy time to either cannulate or activate the… Read more » What's Your Job? Cardiac and Critical care Anesthesiology 0 Reply Suhrith 6 months ago I’m really glad that you are addressing this here Scott! I’ve done it a few times with really good results. I’ve had to do it in the ED while managing storms and its surprisingly a lot quicker than expected and very satisfying! What's Your Job? ED Doctor 0 Reply Suhrith Bhattaram Reply to Suhrith 6 months ago Star in the storm: percutaneous stellate ganglion blockade for drug-refractory electrical storm in the emergency department – PMC What's Your Job? ED Doctor 0 Reply Andrea Lorenzo Poggi 4 months ago Thank you for the great post!I would like to ask your opinion about the use of propofol in electrical storm.I used it twice in patients with refractory ventricular fibrillation during cardiac arrest, with excellent results (both patients survived).I would be very interested to know what you think about this approach. What's Your Job? Emergency medicine's physician 0 Reply Scott Weingart, MD FCCM Author Reply to Andrea Lorenzo Poggi 4 months ago Yes, think we mentioned, before you would do stellate ganglion block, pt should be intubated with heavy sedation as this often stops the storm! What's Your Job? emcrit 0 Reply Who We Are We are the EMCrit Project, a team of independent medical bloggers and podcasters joined together by our common love of cutting-edge care, iconoclastic ramblings, and FOAM. Subscribe by Email wpDiscuz 8 0 Would love your thoughts, please comment.x x \| Reply You are going to send email to Move Comment
12725	https://bookdown.org/blazej_kochanski/statistics2/chisq-tests.html	18 Chi-square tests \| Statistics 2. Lecture notes Type to search Statistics 2. Lecture notes 1 Introduction 2 Three worlds 2.1 Questions 3 Population and samples 3.1 Population and sample: examples 3.2 Sampling with and without replacement 3.3 Questions Probability calculus 4 Probability 4.1 Probability axioms 4.2 Probability interpretations: 4.3 Events – basic concepts 4.3.1 Elementary event 4.3.2 Compound event 4.3.3 Sample space (set of elementary events) 4.3.4 Union of events 4.3.5 Intersection of events 4.3.6 Complementary event 4.3.7 Conditional probability 4.3.8 Mutually exclusive (disjoint) events 4.3.9 Independent events 4.4 Probability – basic rules 4.4.1 Complement rule 4.4.2 Addition Rule 4.4.3 Multiplication rule 4.5 Exercises 5 Bayes’ formula 5.1 Law of total probability 5.2 Bayes’ theorem 5.3 Examples 5.4 Templates 5.5 Links 5.6 Exercises 6 Combinatorics 6.1 Variations 6.2 Combinations 6.3 Questions 6.4 Exercises 7 Random variables 7.1 Probability distribution 7.2 Expected value 7.3 Variance and standard deviation 7.4 Cumulative distribution function (CDF) 7.5 Transformations of random variables 7.5.1 Adding a constant to a random variable 7.5.2 Multiplying a random variable by a constant 7.5.3 Adding random variables 7.6 Templates 7.7 Exercises 8 Discrete distributions 8.1 Bernoulli distribution 8.2 Binomial distribution 8.3 Poisson distribution 8.4 Templates 8.5 Questions 8.6 Exercises 9 Continuous distributions 9.1 Density function 9.2 Uniform distribution 9.3 Gaussian distribution 9.3.1 Standardized normal distribution 9.3.2 Sum and difference of variables with normal distribution 9.3.3 Approximation of binomial distribution by normal distribution 9.4 Templates 9.5 Exercises Inferencial statistics 10 Sampling distributions 10.1 Statistics and parameters 10.2 Sampling distribution of the sample mean 10.3 Central limit theorem (CLT) 10.4 Point estimators – biased and unbiased 10.5 Statistical inference 10.6 Links 10.7 Questions 10.8 Exercises 11 Confidence intervals for a mean 11.1 Confidence interval and confidence level 11.2 Interval estimation of the population mean 11.3 Additional notes 11.4 Links 11.5 Templates 11.6 Questions 11.7 Exercises 12 Confidence intervals for a proportion 12.1 Formulas 12.2 Links 12.3 Templates 12.4 Exercises 13 Sample size 13.1 Determination of sample size when estimating the mean 13.2 Determination of sample size when estimating a proportion 13.3 Rounding 13.4 Templates 13.5 Exercises 14 Hypothesis testing — one mean 14.1 Hypotheses and statistical tests 14.2 Test statistic and rejection region 14.3 Errors 14.4 Do we accept the null hypothesis? 14.5 Elements of a typical statistical test 14.6 Types of alternatives 14.7 Notation of null hypotheses with non-strict inequalities 14.8 One-sample mean test based on the z z statistic 14.9 One-sample t t test 14.10 Practical use of t t and z z tests 14.11 Links 14.12 Templates 14.13 Exercises 15 Hypothesis testing — one proportion 15.1 Single proportion test based on the z z statistic 15.2p-value 15.3 Power of a test 15.4 Links 15.5 Templates 15.6 Exercises 16 Hypothesis testing — two means 16.1 Testing two means – sampling distribution 16.2 Two-sample z z-test for two means 16.3 Two-sample t t-test 16.4 Paired samples 16.5 Formulas 16.5.1 Confidence intervals for differences in parameters (formulas) 16.5.2 Tests for differences in means (formulas) 16.5.3 Difference in means – paired samples 16.5.4 Confidence intervals – paired samples 16.6 Confidence intervals versus hypothesis tests 16.7 Effect size 16.8 Links 16.9 Templates 16.10 Questions 16.11 Exercises 17 Hypothesis testing — two proportions 17.1 Two proportions test for large samples 17.2 Formulas 17.3 Effect size 17.4 Templates 17.5 Exercises 18 Chi-square tests 18.1 Applications of the chi-square tests 18.2 Formula 18.3 Hipotheses 18.4 Expected frequencies 18.5 Conditions for test application 18.6 Degrees of freedom 18.7 Rejection region 18.8 Chi-square tests versus proportion tests 18.9 Effect size in independence tests 18.10 Links 18.11 Templates 18.12 Exercises 19 Analysis of variance (ANOVA) 19.1 One-way analysis of variance – test 19.2 F statistic – formula 19.2.1 Between-group variability 19.2.2 Within-group variability 19.2.3 F Statistic 19.3 ANOVA results table 19.4 Effect size 19.5Post hoc procedure 19.6 Levene's test and Brown-Forsythe test 19.7 Two-way analysis of variance 19.8 Links 19.9 Templates 19.10 Exercises 20 Nonparametric tests 20.1 Runs test 20.2 Mann-Whitney test 20.3 Wilcoxon signed-rank test 20.4 Kruskal–Wallis test 20.5 Templates 20.6 Exercises 21 Linear regression 21.1 Linear regression model – assumptions 21.2 Linear regression model – estimation 21.2.1 Confidence intervals for model coefficients 21.2.2 Confidence intervals for the expected value of the dependent variable 21.3 Prediction intervals 21.4 Linear regression model – t-test and F-test 21.4.1 Significance tests for coefficients 21.4.2 F-Test 21.5 Diagnostic tests for the regression model 21.5.1 Testing linearity 21.5.2 Testing homoskedasticity 21.5.3 Testing normality of the error term 21.6 Templates 22 Other tests 22.1 Checking conformity with the Gaussian distribution 22.1.1 Kolmogrov-Smirnov test 22.1.2 Shapiro-Wilk-Royston test 22.1.3 Jarque-Bera test 22.1.4 Anderson-Darling test 22.2 Test and confidence interval for Pearson's correlation coefficient 22.3 Templates 22.4 Exercises 23 Simulations, bootstrapping, permutations tests 23.1 Simulations in proportion estimation 23.2 Bootstrapping 23.3 Permutation tests 23.4 Exercises Appendices A Formulas A.1 Basic probability formulas Union (“A or B”) — addition rule Intersection (“A and B”) — product (or chain) rule Complementary event Conditional probabilities (A \| B – “A given B”) Law of total probability Bayes’ formula A.2 Counting A.2.1 Variations of n n taken r r at a time (order matters) A.2.2 Combinations of n n taken r r at a time (order does not matter) A.3 Random variables A.3.1 General formulas for a discrete random variable A.3.2 General formulas for a continuous random variable A.3.3 Binomial distribution A.3.4 Poisson distribution A.3.5 Hypergeometric distribution A.3.6 Uniform (continuous) distribution A.3.7 Exponential distribution A.3.8 Gaussian (normal) distribution A.3.9 Standard normal distribution A.3.10 Sampling normal distribution of the sample mean (approximation for large n n) A.4 Transformations of random variables A.4.1 Adding a constant to a random variable A.4.2 Multiplying a random variable by a constant A.4.3 Addition of random variables A.4.4 Addition of n n independent and identically distributed (i.i.d.) random variables A.5 Statistical inference A.5.1 Confidence interval A.5.2 Sample size determination A.5.3 Hypothesis testing for a population parameter A.5.4 Hypothesis testing for a difference of population parameters A.5.5 Confidence intervals for a difference of population parameters A.5.6 Hypothesis tests for the difference of parameters – paired samples A.5.7 Confidence intervals for the difference of parameters – paired samples A.5.8 Chi-square test A.5.9 ANOVA (one-way) A.6 Linear regression A.6.1 The model A.6.2 Point estimates A.6.3 Confidence intervals for the coefficients A.6.4 Confidence intervals for the expected value of the dependent variable A.6.5 Prediction intervals B Using normal distribution C Templates C.1 Bayes' formula C.2 Distributions C.2.1 Discrete random variable C.2.2 Parametric discrete distributions C.2.3 Continuous distributions C.3 Confidence intervals C.3.1 Sample size C.4 Tests for means and proportions C.4.1 Tests for 1 mean and 1 proportion C.4.2 Tests and confidence intervals for 2 means C.4.3 Tests and confidence intervals for 2 proportions C.5 Chi-square test C.6 ANOVA and Levene test C.7 Nonparametric tests C.8 Other tests C.8.1 Regression C.8.2 Normality check C.8.3 Correlation D Tables D.1 Standard normal distribution (z) table D.2 Student-t distribution table D.3 Chi-square distribution table D.4 F distibution tables D.5 Other tables D.5.1 Standard normal distribution table – alternative version E Using R E.1 R – typical problems E.2 How to do it in R E.2.1 Reading data E.2.2 Probability distributions E.2.3 Simulating using R F Using spreadsheets F.1 Spreadsheets – typical problems F.1.1 Regional settings F.1.2 Array formulas F.1.3 Excel vs Google sheets F.2 How to do it in spreadsheets F.2.1 Probability distributions G Schedule of laboratory meetings G.1 Suggested schedule for economic analytics statistics class G.2 Bonus: Statistics songs G.2.1 Probability song G.2.2 Discrete distributions song G.2.3 Continuous distributions song G.2.4 Confidence intervals song G.2.5 Hypothesis testing song Literature Published with bookdown Facebook Twitter LinkedIn Weibo Instapaper A A Serif Sans White Sepia Night Spacing -Spacing + PDF EPUB Statistics 2. Lecture notes 18 Chi-square tests 18.1 Applications of the chi-square tests The chi-square test follows a similar structure each time, but serves many purposes. We will use the following applications of the chi-square test: goodness-of-fit tests: we check whether the distribution in a certain population is consistent with a theoretical distribution), homogeneity tests: we check whether the structure of qualitative (categorical) variables is the same, i.e., homogeneous, in two or more populations, independence tests we check whether two qualitative variables are independent of each other in the population. 18.2 Formula Although the test has various applications, we will see that the formula used to calculate the test statistic is similar in all cases. We can express it as follows: χ 2=∑i(O i−E i)2 E i,(18.1)(18.1)χ 2=∑i(O i−E i)2 E i, where: i i – is the index indicating the individual categories or the "cells" in a contingency table, O i O i - observed (actual) frequencies E i E i - expected frequencies determined based on the assumed distribution or on the assumption of independence, also known as theoretical frequencies. 18.3 Hipotheses In the goodness-of-fit test, H 0 H 0 states that the distribution of a qualitative variable in the population matches the assumed distribution, while H A H A states that it is different. In the homogeneity test, H 0 H 0 states that the distribution of the qualitative variable in two or more populations is the same ("homogeneous"), while H A H A states that it is different. In the independence test, H 0 H 0 states that two qualitative variables are independent in the population, while H A H A states that they are dependent. 18.4 Expected frequencies In the goodness-of-fit test, the expected frequencies arise from the assumed distribution. In the homogeneity and independence tests, the expected frequencies must be calculated. In both tests, the frequency tables take the form of a contingency table. The expected frequency must be calculated for each cell in this table. Expected frequency=Row Total×Column Total Total Expected frequency=Row Total×Column Total Total 18.5 Conditions for test application The chi-square test is an approximate test (in this regard, it is similar to the z test for means or proportions) and, as such, has specific requirements. In addition to the standard assumptions of randomness and sample representativeness, it also requires an adequate sample size. A common guideline is that the minimum expected frequency in each cell should be at least 5 to ensure the validity of the test. 18.6 Degrees of freedom The test statistic in the chi-square test follows a chi-square distribution with a specified "degree of freedom" (this is the only parameter of the chi-square distribution). In goodness-of-fit tests with a specific probability distribution, the degrees of freedom are k−1 k−1, where k k is the number of classes (categories). In goodness-of-fit tests with a distribution with specific parameters, when these parameters are estimated from the same data, the degrees of freedom are k−m−1 k−m−1, where k k is the number of categories, and m m is the number of estimated parameters. In homogeneity tests, the degrees of freedom are k−1 k−1 if one is testing two populations. For more populations, the degrees of freedom are (k−1)×(c−1)(k−1)×(c−1), where c c is the number of populations. In independence tests, the degrees of freedom are (r−1)×(c−1)(r−1)×(c−1), where r r is the number of rows in the contingency table (the number of categories of one variable), and c c is the number of columns (the number of categories of the second variable). 18.7 Rejection region In all chi-square tests mentioned in this chapter, the rejection region is right-tailed, i.e., we reject the null hypothesis if the test statistic exceeds the critical value. Note! In the case of the chi-square test, we do not refer to the alternative hypothesis as "left-tailed", "right-tailed", or "two-tailed" – the alternative hypothesis is rather "multi-tailed". 18.8 Chi-square tests versus proportion tests It can be noticed that chi-square goodness-of-fit and homogeneity tests are extensions of proportion tests. They can include more than two categories or (in the case of homogeneity tests) more than two populations. In goodness-of-fit tests, the extension lies in the fact that the test for one proportion limits us to a binary distribution (a distribution with only two categories), while the goodness-of-fit test can be applied for more than two categories. The two-tailed test for one proportion is practically equivalent to the chi-square goodness-of-fit test for a dichotomous variable.10 In homogeneity tests, the extension may involve examining more than two populations (a test for 3, 4, 5, 6 proportions) or increasing the number of categories, or both. The two-tailed test for two proportions is practically equivalent to the chi-square homogeneity test for two populations and a binary variable.11 18.9 Effect size in independence tests For contingency tables showing frequencies in an independence test, several measures of effect size have been proposed. The most commonly used measure is Cramér's V: V=√χ 2 n×min(c−1,r−1)(18.2)(18.2)V=χ 2 n×min(c−1,r−1) For 2×2 tables, the phi coefficient is used, which in absolute terms is numerically equivalent to Cramér's V but can also take negative values. 18.10 Links Independence tests – visualization: Chi-square test – web application: 18.11 Templates Spreadsheets Chi-square test — Google spreadsheet Chi-square test — Excel template Chi-square distribution calculator — Google spreadsheet R code Chi-square test ``` Indepenence/homogeneity test Data matrix Input vector m <- c( 21, 14, 3, 10 ) Number of rows in the data matrix nrow <- 2 Significance level alpha <- 0.05 Transformation of the vector to a matrix m <- matrix(data=m, nrow=nrow, byrow=TRUE) Chi-square test without Yates' correction test_chi <- chisq.test(m, correct=FALSE) Chi-square test with Yates' correction for 2x2 tables test_chi_corrected <- chisq.test(m) G test test_g <- AMR::g.test(m) Exact Fisher test exact_fisher<-fisher.test(m) print(c('Degrees of freedom' = test_chi$parameter, 'Critical value' = qchisq(1-alpha, test_chi$parameter), 'Chi^2 statistic' = unname(test_chi$statistic), 'Chi^2 test p-value' = test_chi$p.value, "Cramér's V" = unname(sqrt(test_chi$statistic/sum(m)/min(dim(m)-1))), 'Phi coefficient (2x2 tables)' = if(all(dim(m)==2)) {psych::phi(m, digits=10)}, "Chi^2 with Yates' correction test statistic" = unname(test_chi_corrected$statistic), "Chi^2 with Yates' correction p-value" = test_chi_corrected$p.value, 'G statistic' = unname(test_g$statistic), 'G test p-value' = test_g$p.value, 'Exact Fisher test p-value' = exact_fisher$p.value )) ``` ``` Degrees of freedom.df Critical value Chi^2 statistic 1.00000000 3.84145882 5.16923077 Chi^2 test p-value Cramér's V Phi coefficient (2x2 tables) 0.02299039 0.32816506 0.32816506 Chi^2 with Yates' correction test statistic Chi^2 with Yates' correction p-value G statistic 3.79780220 0.05131990 5.38600494 G test p-value Exact Fisher test p-value 0.02029889 0.04899141 ``` ``` Goodness-of-fit chi-square test Observed frequencies: observed <- c(70, 10, 20) Expected frequencies: expected <- c(80, 10, 10) Just in case: Adjustment of the expected frequencies so that their sum is equal to the sum of the observed frequencies: expected <- expected / sum(expected) sum(observed) Significance level alpha <- 0.05 test_chi <- chisq.test(x = observed, p = expected, rescale.p = TRUE) test_g <- AMR::g.test(x = observed, p = expected, rescale.p = TRUE) print(c('Degrees of freedom' = test_chi$parameter, 'Critical value' = qchisq(1-alpha, test_chi$parameter), 'Chi^2 statistic' = unname(test_chi$statistic), 'p-value (chi-square test)' = test_chi$p.value, 'G statistic' = unname(test_g$statistic), 'p-value (G test)' = test_g$p.value )) ``` ``` Degrees of freedom.df Critical value Chi^2 statistic p-value (chi-square test) G statistic p-value (G test) 2.000000000 5.991464547 11.250000000 0.003606563 9.031492255 0.010935443 ``` Python code Chi-square test ``` Indepenence/homogeneity test import numpy as np import scipy.stats as stats from scipy.stats import chi2 from statsmodels.stats.contingency_tables import Table2x2 Data matrix m = np.array([ [21, 14], [3, 10] ]) alpha = 0.05 Chi-square test without Yates' correction test_chi = stats.chi2_contingency(m, correction=False) Chi-square test with Yates' correction for 2x2 tables test_chi_corrected = stats.chi2_contingency(m) G test g, p, dof, expected = stats.chi2_contingency(m, lambda_="log-likelihood") Exact Fisher test exact_fisher = stats.fisher_exact(m) Cramér's V cramers_v = np.sqrt(test_chi / m.sum() / min(m.shape-1, m.shape-1)) Phi coefficient (2x2 tables) phi_coefficient = None if m.shape == (2, 2): phi_coefficient = cramers_vnp.sign(np.diagonal(m).prod()-np.diagonal(np.fliplr(m)).prod()) Results results = { 'Number of degrees of freedom': test_chi, 'Critical value': chi2.ppf(1-alpha, test_chi), 'Chi-square statistic': test_chi, 'p-value (chi-square test)': test_chi, "Cramer\'s V": cramers_v, 'Phi coefficient (for 2x2 table)': phi_coefficient, "Chi-square statistic with Yates' correction": test_chi_corrected, "p-value (chi-square test with Yates' correction)": test_chi_corrected, 'G statistic': g, 'p-value (G test)': p, 'p-value (Fisher’s exact test)': exact_fisher } for key, value in results.items(): print(f"{key}: {value}") ``` ``` Number of degrees of freedom: 1 Critical value: 3.841458820694124 Chi-square statistic: 5.169230769230769 p-value (chi-square test): 0.022990394092464842 Cramer's V: 0.3281650616569468 Phi coefficient (for 2x2 table): 0.3281650616569468 Chi-square statistic with Yates' correction: 3.7978021978021976 p-value (chi-square test with Yates' correction): 0.05131990358807137 G statistic: 3.9106978537750194 p-value (G test): 0.04797967015430134 p-value (Fisher’s exact test): 0.048991413058947844 ``` ``` Goodness-of-fit chi-square test from scipy.stats import chisquare, chi2 import numpy as np Observed frequencies: observed = np.array([70, 10, 20]) Expected frequencies: expected = np.array([80, 10, 10]) Correction of the expected frequencies so that their sum is definitely equal to the sum of the actual frequencies: expected = expected / expected.sum() observed.sum() Chi^2 test: chi_stat, chi_p = chisquare(f_obs=observed, f_exp=expected) Degrees of freedom df = len(observed) - 1 Significance level: alpha = 0.05 Critical value: critical_value = chi2.ppf(1 - alpha, df) G test: from scipy.stats import power_divergence g_stat, g_p = power_divergence(f_obs=observed, f_exp=expected, lambda_="log-likelihood") Results results = { 'Degrees of freedom': df, 'Critical value': critical_value, 'Chi^2 statistic': chi_stat, 'P-value (chi^2 test)': chi_p, 'G statistic': g_stat, 'P-value (G test)': g_p } for key, value in results.items(): print(f"{key}: {value}") ``` ``` Degrees of freedom: 2 Critical value: 5.991464547107979 Chi^2 statistic: 11.25 P-value (chi^2 test): 0.0036065631360157305 G statistic: 9.031492254964643 P-value (G test): 0.010935442847719828 ``` 18.12 Exercises Exercise 18.1A sample of bank customers was drawn, and they were asked about their income and the banking product they were most interested in. The following results were obtained: \| Income \| Loans \| Deposits \| Investments \| Total \| :---: :---: \| Low \| 34 \| 18 \| 10 \| 62 \| \| Medium \| 19 \| 30 \| 21 \| 70 \| \| High \| 20 \| 18 \| 31 \| 69 \| \| Total \| 73 \| 66 \| 62 \| 201 \| Is the type of interest in the bank's product offering independent of income level? Exercise 18.2(Aczel and Sounderpandian 2018) A certain study describes the analysis of 35 key product categories. At the time of this study, 72.9% of the products sold were national brand products, 23% were private-label products, and 4.1% were no-name products. Suppose we want to test whether this distribution is still valid in today’s market. We collect a random sample of 1000 products from the 35 analyzed categories and find that 610 items are national brand products, 290 are private-label products, and 100 are no-name products. Conduct a test and formulate conclusions. Exercise 18.3A total of 200 rolls of a six-sided die were performed. The results were: 38 ones, 35 twos, 35 threes, 27 fours, 31 fives, and 34 sixes. Using the chi-square test, verify the hypothesis that the die is "fair," i.e., properly balanced. (If you have a die at hand, you can test it instead.) Exercise 18.4Benford's Law describes the probability distribution of the first digit appearing in many empirical datasets. According to Benford's Law, the digit one appears as the first digit in log 10(1+1/1)≈30.1 log 10⁡(1+1/1)≈30.1 of cases, two appears in log 10(1+1/2)≈17.6 log 10⁡(1+1/2)≈17.6 of cases, three in log 10(1+1/3)≈12.5 log 10⁡(1+1/3)≈12.5, and so on. The goal is to verify whether the distribution of the first digits in the annual salary data (in British pounds) of football players in different leagues (Bundesliga, La Liga, Ligue 1, Premier League, and Serie A, separately for each league) conforms to Benford's distribution. Data: Data collected by Edd Webster Exercise 18.5(Aczel and Sounderpandian 2018) It is believed that the returns from a certain investment follow a normal distribution with a mean of 11% (annualized rate) and a standard deviation of 2%. A brokerage firm wants to test the null hypothesis that this statement is true and has collected the following return data (assumed to be a random sample): 8.0; 9.0; 9.5; 9.5; 8.6; 13.0; 14.5; 12.0; 12.4; 19.0; 9.0; 10.0; 10.0; 11.7; 15.0; 10.1; 12.7; 17.0; 8.0; 9.9; 11.0; 12.5; 12.8; 10.6; 8.8; 9.4; 10.0; 12.3; 12.9; 7.0. Conduct an analysis using the chi-square test with six intervals and formulate a conclusion. Exercise 18.6(Aczel and Sounderpandian 2018) Using the data from the previous exercise, test the null hypothesis that investment returns follow a normal distribution but with an unknown mean and an unknown standard deviation. Only test the validity of the normality assumption. How does this test differ from the one in the previous exercise? Exercise 18.7(Aczel and Sounderpandian 2018) As markets become increasingly international, many companies invest in research to determine the maximum potential sales range in foreign markets. An American coffee machine manufacturer wants to verify whether its market share and the shares of its two competitors are roughly the same in three European countries where all three companies export their products. The results of a market study are presented in the table below. The data come from random samples of 150 consumers in each country. \| \| France \| England \| Spain \| Total \| :---: :---: \| Company \| 55 \| 38 \| 24 \| 117 \| \| Competitor 1 \| 28 \| 30 \| 21 \| 79 \| \| Competitor 2 \| 20 \| 18 \| 31 \| 69 \| \| Others \| 47 \| 64 \| 74 \| 185 \| \| Total \| 150 \| 150 \| 150 \| 450 \| Exercise 18.8Using the chi-square test and data from exercise 15.1, test the null hypothesis that the distribution of head tilts during kissing follows the parameters p R=0.5 p R=0.5 and p L=0.5 p L=0.5 (p R p R is the probability of tilting the head to the right, p L p L is the probability of tilting the head to the left). What one-proportion test corresponds to the conducted chi-square test? Compare the p-value results. Exercise 18.9In 1972, 48 bank managers were given identical personnel files. Each of them was asked to assess whether the individual should be promoted to branch manager or whether other candidates should be interviewed. The files were identical, except that in half of the cases, the person was identified as a woman, and in the other half, as a man. Among the 24 "male" cases, promotion was recommended in 21 instances; among the 24 "female" cases, promotion was recommended in 14 (Rosen and Jerdee 1974). Solve the problem using the two-sided difference in proportions test and compare the results (test statistics, p-values) with the chi-square test. Exercise 18.10Return to the data from exercise 15.5. Test whether the hypothesis that the distribution of lost keys in the population (i.e., the data-generating process) is uniform (discrete uniform). Exercise 18.11The table below presents data on bomb hits in the southern part of London during World War II. The entire area was divided into 576 square sections, each covering 1/4 km². The bombing data is presented in the table below (Clarke 1946). The symbol n k represents the number of squares that were hit by exactly k bombs. \| \| \| \| \| \| \| \| \| \| --- --- --- --- \| k \| 0 \| 1 \| 2 \| 3 \| 4 \| 5 \| 6 \| 7 \| \| nₖ \| 229 \| 211 \| 93 \| 35 \| 7 \| 0 \| 0 \| 1 \| Based on the given data, can the hypothesis that the London bombings were random and did not concentrate on any specific area be rejected? Literature Aczel, A. D., and J. Sounderpandian. 2018. Statystyka w Zarz ą dzaniu. PWN. Clarke, R. D. 1946. “An Application of the Poisson Distribution.”Journal of the Institute of Actuaries 72 (3): 481–81. Rosen, Benson, and Thomas H. Jerdee. 1974. “Influence of Sex Role Stereotypes on Personnel Decisions.”Journal of Applied Psychology 59: 9–14. If the distribution is binary (follows a Bernoulli distribution, eg. yes/no, disease/no disease etc.) and we compare it with some theoretical distribution, the test for 1 proportion can be used (15), as well as the goodness-of-fit test (compare exercises 15.1 i 18.8).↩︎ If we compare two proportions, the test for 2 propotions can be used (17) or the chi-square homogeneity test – cf. exercises 17.2 and 18.9.↩︎
12726	https://proofwiki.org/wiki/Definition:Circle/Arc/Subtend	Definition:Circle/Arc/Subtend - ProofWiki Definition:Circle/Arc/Subtend From ProofWiki <Definition:Circle \| Arc Jump to navigationJump to search Definition The angle subtended by an arc of a circle is the angle formed by the two radii from the center of the circle to the two endpoints of the arc. In the above diagram: the arcB C B Csubtends the angle∠B A C∠B A C the arcB D B Dsubtends the angle∠B A D∠B A D and so on. Retrieved from " Category: Definitions/Circles Navigation menu Personal tools Log in Request account Namespaces Definition Discussion [x] English Views Read View source View history [x] More Search Navigation Main Page Community discussion Community portal Recent changes Random proof Help FAQ P r∞f W i k i P r∞f W i k i L A T E X L A T E X commands ProofWiki.org Proof Index Definition Index Symbol Index Axiom Index Mathematicians Books Sandbox All Categories Glossary Jokes To Do Proofread Articles Wanted Proofs More Wanted Proofs Help Needed Research Required Stub Articles Tidy Articles Improvements Invited Refactoring Missing Links Maintenance Tools What links here Related changes Special pages Printable version Permanent link Page information This page was last modified on 26 November 2015, at 21:20 and is 0 bytes Content is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Privacy policy About ProofWiki Disclaimers
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His aim is to present calculus as the first real encounter with mathematics: it is the place to learn how logical reasoning combined with fundamental concepts can be developed into a rigorous mathematical theory rather than a bunch of tools and techniques learned by rote. Since analysis is a subject students traditionally find difficult to grasp, Spivak provides leisurely explanations, a profusion of examples, a wide range of exercises and plenty of illustrations in an easy-going approach that enlightens difficult concepts and rewards effort. Calculus will continue to be regarded as a modern classic, ideal for honours students and mathematics majors, who seek an alternative to doorstop textbooks on calculus, and the more formidable introductions to real analysis.", "item_img_path" : " "price_data" : { "retail_price" : "84.00", "online_price" : "84.00", "our_price" : "84.00", "club_price" : "84.00", "savings_pct" : "0", "savings_amt" : "0.00", "club_savings_pct" : "0", "club_savings_amt" : "0.00", "discount_pct" : "10", "store_price" : "" } } Save Calculus by Michael Spivak 0.0 No RatingsWrite the First Review local_shippingShip to Me On Order. Usually ships in 2-4 weeks FREE Shipping for Club Membershelp storeIn-Store Pickup Hardcover - Revised Ed. $84.00 Add to Cart  "Add Calculus to your Cart")+ Add to Wishlist 4 interest-free payments of $21 with Affirm. Learn more Overview Spivak's celebrated textbook is widely held as one of the finest introductions to mathematical analysis. His aim is to present calculus as the first real encounter with mathematics: it is the place to learn how logical reasoning combined with fundamental concepts can be developed into a rigorous mathematical theory rather than a bunch of tools and techniques learned by rote. Since analysis is a subject students traditionally find difficult to grasp, Spivak provides leisurely explanations, a profusion of examples, a wide range of exercises and plenty of illustrations in an easy-going approach that enlightens difficult concepts and rewards effort. Calculus will continue to be regarded as a modern classic, ideal for honours students and mathematics majors, who seek an alternative to doorstop textbooks on calculus, and the more formidable introductions to real analysis. Customers Also Bought )x The Algebraic Eigenvalue Problem (Nmsc)J. Harvie Wilkinson$190.00 Paperback - Revised Ed. )x The Traveling Salesman Problem David L. 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Applegate$110.00 Hardcover )x Semantics Frank Robert Palmer$84.00 Paperback - Revised Ed. )x Introduction to Experimental Particle Physics Richard Clinton Fernow$59.00 Paperback - Revised Ed. )x Homology Theory Anne Hilton$102.00 Paperback )x Chaotic Dynamics Gregory L. Baker$550.00 )x First Course Mathematical Analysis David Alexander Brannan$84.00 Paperback - Revised Ed. )x Foundations of Algebraic Topology Samuel Eilenberg$68.00 Paperback 1 )x Mathematical Analysis K. G. Binmore$111.00 Paperback - Revised Ed. )x The Ideas of Particle Physics James E. Dodd$53.00 Paperback - Revised Ed. )x Topology from the Differentiable Viewpoint John Milnor$51.00 Paperback - Revised Ed. )x Introduction To Commutative Algebra Michael F. Atiyah$190.00 Hardcover )x The Algebraic Eigenvalue Problem (Nmsc)J. Harvie Wilkinson$190.00 Paperback - Revised Ed. )x The Traveling Salesman Problem David L. Applegate$110.00 Hardcover )x Semantics Frank Robert Palmer$84.00 Paperback - Revised Ed. )x Introduction to Experimental Particle Physics Richard Clinton Fernow$59.00 Paperback - Revised Ed. Details ISBN-13: 9780521867443 ISBN-10: 0521867444 Publisher: Cambridge University Press Publish Date: June 2006 Dimensions: 10.26 x 9.02 x 1.66 inches Shipping Weight: 4.02 pounds Page Count: 681 Related Categories Books Mathematics Calculus Books Mathematics Geometry - General Books Mathematics Mathematical Analysis BAM Customer Reviews Write the First Review Questions? Reach us via Live Chat Get the latest on trends, deals, and promotions Enter your email address Stay Connected with us when you sign up for text alerts. 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12728	https://artofproblemsolving.com/wiki/index.php/Complementary_counting?srsltid=AfmBOopRBGk9Ur_JsBvyS0ls8LqD1rYLvABPdHqv1XZAjNWj4XMpZc_x	Art of Problem Solving Complementary counting - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki Complementary counting Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Complementary counting In combinatorics, complementary counting is a counting method where one counts what they don't want, then subtracts that from the total number of possibilities. In problems that involve complex or tedious casework, complementary counting is often a far simpler approach. A large hint that complementary counting may lead to a quick solution is the phrase "not" or "at least" within a problem statement. More formally, if is a subset of , complementary counting exploits the property that , where is the complement of . In most instances, though, is obvious from context and is committed from mention. Contents [hide] 1 Complementary Probability 2 Examples 2.1 Example 1 2.2 Example 2 2.3 Example 3 2.4 Example 4 2.5 Example 5 3 Resources 4 See also Complementary Probability There is a probability equivalent of complementary counting. For any event, the probability it happens plus the probability it does not happen is one. Thus, we have the identity Like its counting analog, complementary probability often vastly simplifies tedious casework. Unlike complementary counting, though, it sees frequent use as an intermediate step, primarily because computing complements is much easier in probability than in counting. Examples Here are some examples that demonstrate complementary counting and probability in action. It is worth noting that complementary probability is not typically an intermediate step, but a framework upon which a solution is built. Example 1 How many positive integers less than are not a multiple of five? Solution: We use a complementary approach. The total number of positive integers, with no restrictions, is integers. What we don't want are the multiples of five. These are or ; it's easy to see that there are of them. Thus, our answer is is . Example 2 2006 AMC 10A Problem 21: How many four-digit positive integers have at least one digit that is a 2 or a 3? Solution: We use a complementary approach. With no restrictions, there are four-digit positive integers. What we don't want are the four-digit integers with no digit that is a two or three, Using a constructive approach, the first digit can be one of seven integers; and . Note that the first digit cannot be zero, or else it ceases to have four digits. However, the second, third, and fourth digits can be zero; as a result, they have eight options. So, our total number of two-and-three-free numbers is . Hence, our final answer is , as desired. Example 3 Sally is drawing seven houses. She has four crayons, but she can only color any house a single color. In how many ways can she color the seven houses if at least one pair of consecutive houses must have the same color? Solution: Use complementary counting. First, we find the total colorings without restriction, which we do by constructing them. She has four options for what color the first house can be, four options for the second, and so on. Hence, there are ways she can color the four houses. Next, we find the possibilities where every house's next-door neighbor is a different color. Using a constructive approach, she has four options for the color of the first house. We have to make sure the next house is a different color; as a result, there are only three options for the color of the second house, with the color of the first house unavailable. By similar logic, there are 3 options for the third house, and so on for every other house. Combining these yields possibilities if every house must be a different color. Putting these two together, there are ways she can color the seven houses four colors if at least one pair of consecutive houses must be the same color. Example 4 2002 AIME I Problem 1 Many states use a sequence of three letters followed by a sequence of three digits as their standard license-plate pattern. Given that each three-letter three-digit arrangement is equally likely, the probability that such a license plate will contain at least one palindrome (a three-letter arrangement or a three-digit arrangement that reads the same left-to-right as it does right-to-left) is , where and are relatively prime positive integers. Find Solution: We use complementary probability. So first, we find the probability that a license plate has no palindromes; in other words, the probability that the first and third numbers and letters are distinct. The probability the numbers are distinct is , as for every digit of the first number, one out of ten is the same digit; similarly, for the letters, it is . Multiplying these together gives that the probability that a license plate has no palindromes is Taking the complement of this, the probability a license plate has a palindrome is thus . Hence, our answer is . Example 5 2008 AMC 12B Problem 22: A parking lot has 16 spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers chose spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires 2 adjacent spaces. What is the probability that she is able to park? Solution: Use complementary probability, where we find the probability that no two open spots are consecutive. With no restrictions, the total number of ways the 12 cars could park is , where is a combination. Finding how many permutations of the cars and spaces leave no two spaces next to each other is a more challenging task, though. One might eventually think to treat this problem like a distribution, a stars-and-bars approach. Let the 12 cars be stars and the 4 spaces be bars. Here is one arrangement of our stars and bars; The question mandates that no two bars sit next to each other. Thus, we have 13 "slots" where the bars could go (eleven between the stars, two at the endpoints), where only one bar can fit in each slot. It follows that the number of ways to insert these bars is . Then the probability that Auntie Em cannot park is . Finally, our answer is . Resources AoPS Complementary Counting Part 1 AoPS Complementary Counting Part 2 AoPS Complementary Probability Part 1 AoPS Complementary Probability Part 2 AoPS Complementary Probability Part 3 AoPS Complementary Probability Part 4 See also Casework Constructive counting Overcounting Retrieved from " Categories: Combinatorics Definition Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
12729	http://jiffyclub.github.io/scipy/tutorial/stats.html	Statistics (scipy.stats) — SciPy v0.19.0 Reference Guide SciPy v0.19.0 Reference Guide SciPy Tutorial index modules next previous Table Of Contents Statistics (scipy.stats) Introduction Random Variables Getting Help Common Methods Shifting and Scaling Shape Parameters Freezing a Distribution Broadcasting Specific Points for Discrete Distributions Fitting Distributions Performance Issues and Cautionary Remarks Remaining Issues Building Specific Distributions Making a Continuous Distribution, i.e., Subclassing rv_continuous Subclassing rv_discrete Analysing One Sample Descriptive Statistics T-test and KS-test Tails of the distribution Special tests for normal distributions Comparing two samples Comparing means Kolmogorov-Smirnov test for two samples ks_2samp Kernel Density Estimation Univariate estimation Multivariate estimation Previous topic Spatial data structures and algorithms (scipy.spatial) Next topic Multidimensional image processing (scipy.ndimage) Statistics (scipy.stats)¶ Introduction¶ In this tutorial we discuss many, but certainly not all, features of scipy.stats. The intention here is to provide a user with a working knowledge of this package. We refer to the reference manual for further details. Note: This documentation is work in progress. Random Variables¶ There are two general distribution classes that have been implemented for encapsulating continuous random variables and discrete random variables . Over 80 continuous random variables (RVs) and 10 discrete random variables have been implemented using these classes. Besides this, new routines and distributions can easily added by the end user. (If you create one, please contribute it). All of the statistics functions are located in the sub-package scipy.stats and a fairly complete listing of these functions can be obtained using info(stats). The list of the random variables available can also be obtained from the docstring for the stats sub-package. In the discussion below we mostly focus on continuous RVs. Nearly all applies to discrete variables also, but we point out some differences here: Specific Points for Discrete Distributions. In the code samples below we assume that the scipy.stats package is imported as >>> from scipy import stats and in some cases we assume that individual objects are imported as >>> from scipy.stats import norm Getting Help¶ First of all, all distributions are accompanied with help functions. To obtain just some basic information we print the relevant docstring: print(stats.norm.__doc__). To find the support, i.e., upper and lower bound of the distribution, call: >>> print 'bounds of distribution lower: %s, upper: %s' % (norm.a, norm.b) bounds of distribution lower: -inf, upper: inf We can list all methods and properties of the distribution with dir(norm). As it turns out, some of the methods are private methods although they are not named as such (their name does not start with a leading underscore), for example veccdf, are only available for internal calculation (those methods will give warnings when one tries to use them, and will be removed at some point). To obtain the real main methods, we list the methods of the frozen distribution. (We explain the meaning of a frozen distribution below). >>> rv \= norm() >>> dir(rv) # reformatted ['__class__', '__delattr__', '__dict__', '__doc__', '__getattribute__', '__hash__', '__init__', '__module__', '__new__', '__reduce__', '__reduce_ex__', '__repr__', '__setattr__', '__str__', '__weakref__', 'args', 'cdf', 'dist', 'entropy', 'isf', 'kwds', 'moment', 'pdf', 'pmf', 'ppf', 'rvs', 'sf', 'stats'] Finally, we can obtain the list of available distribution through introspection: >>> import warnings >>> warnings.simplefilter('ignore', DeprecationWarning) >>> dist_continu \= [d for d in dir(stats) if ... isinstance(getattr(stats,d), stats.rv_continuous)] >>> dist_discrete \= [d for d in dir(stats) if ... isinstance(getattr(stats,d), stats.rv_discrete)] >>> print 'number of continuous distributions:', len(dist_continu) number of continuous distributions: 95 >>> print 'number of discrete distributions: ', len(dist_discrete) number of discrete distributions: 13 Common Methods¶ The main public methods for continuous RVs are: rvs: Random Variates pdf: Probability Density Function cdf: Cumulative Distribution Function sf: Survival Function (1-CDF) ppf: Percent Point Function (Inverse of CDF) isf: Inverse Survival Function (Inverse of SF) stats: Return mean, variance, (Fisher’s) skew, or (Fisher’s) kurtosis moment: non-central moments of the distribution Let’s take a normal RV as an example. >>> norm.cdf(0) 0.5 To compute the cdf at a number of points, we can pass a list or a numpy array. >>> norm.cdf([-1., 0, 1]) array([ 0.15865525, 0.5, 0.84134475]) >>> import numpy as np >>> norm.cdf(np.array([-1., 0, 1])) array([ 0.15865525, 0.5, 0.84134475]) Thus, the basic methods such as pdf, cdf, and so on are vectorized with np.vectorize. Other generally useful methods are supported too: >>> norm.mean(), norm.std(), norm.var() (0.0, 1.0, 1.0) >>> norm.stats(moments \= "mv") (array(0.0), array(1.0)) To find the median of a distribution we can use the percent point function ppf, which is the inverse of the cdf: >>> norm.ppf(0.5) 0.0 To generate a sequence of random variates, use the size keyword argument: >>> norm.rvs(size\=3) array([-0.35687759, 1.34347647, -0.11710531]) # random Note that drawing random numbers relies on generators from numpy.random package. In the example above, the specific stream of random numbers is not reproducible across runs. To achieve reproducibility, you can explicitly seed a global variable >>> np.random.seed(1234) Relying on a global state is not recommended though. A better way is to use the random_state parameter which accepts an instance of numpy.random.RandomState class, or an integer which is then used to seed an internal RandomState object: >>> norm.rvs(size\=5, random_state\=1234) array([ 0.47143516, -1.19097569, 1.43270697, -0.3126519 , -0.72058873]) Don’t think that norm.rvs(5) generates 5 variates: >>> norm.rvs(5) 5.471435163732493 Here, 5 with no keyword is being interpreted as the first possible keyword argument, loc, which is the first of a pair of keyword arguments taken by all continuous distributions. This brings us to the topic of the next subsection. Shifting and Scaling¶ All continuous distributions take loc and scale as keyword parameters to adjust the location and scale of the distribution, e.g. for the standard normal distribution the location is the mean and the scale is the standard deviation. >>> norm.stats(loc \= 3, scale \= 4, moments \= "mv") (array(3.0), array(16.0)) In many cases the standardized distribution for a random variable X is obtained through the transformation (X - loc) / scale. The default values are loc = 0 and scale = 1. Smart use of loc and scale can help modify the standard distributions in many ways. To illustrate the scaling further, the cdf of an exponentially distributed RV with mean 1/λ1/λ is given by F(x)\=1−exp(−λx)F(x)\=1−exp⁡(−λx) By applying the scaling rule above, it can be seen that by taking scale = 1./lambda we get the proper scale. >>> from scipy.stats import expon >>> expon.mean(scale\=3.) 3.0 Note Distributions that take shape parameters may require more than simple application of loc and/or scale to achieve the desired form. For example, the distribution of 2-D vector lengths given a constant vector of length RR perturbed by independent N(0, σ2σ2) deviations in each component is rice(R/σR/σ, scale= σσ). The first argument is a shape parameter that needs to be scaled along with xx. The uniform distribution is also interesting: >>> from scipy.stats import uniform >>> uniform.cdf([0, 1, 2, 3, 4, 5], loc \= 1, scale \= 4) array([ 0. , 0. , 0.25, 0.5 , 0.75, 1. ]) Finally, recall from the previous paragraph that we are left with the problem of the meaning of norm.rvs(5). As it turns out, calling a distribution like this, the first argument, i.e., the 5, gets passed to set the loc parameter. Let’s see: >>> np.mean(norm.rvs(5, size\=500)) 4.983550784784704 Thus, to explain the output of the example of the last section: norm.rvs(5) generates a single normally distributed random variate with mean loc=5, because of the default size=1. We recommend that you set loc and scale parameters explicitly, by passing the values as keywords rather than as arguments. Repetition can be minimized when calling more than one method of a given RV by using the technique of Freezing a Distribution, as explained below. Shape Parameters¶ While a general continuous random variable can be shifted and scaled with the loc and scale parameters, some distributions require additional shape parameters. For instance, the gamma distribution, with density γ(x,a)\=λ(λx)a−1Γ(a)e−λx,γ(x,a)\=λ(λx)a−1Γ(a)e−λx, requires the shape parameter aa. Observe that setting λλ can be obtained by setting the scale keyword to 1/λ1/λ. Let’s check the number and name of the shape parameters of the gamma distribution. (We know from the above that this should be 1.) >>> from scipy.stats import gamma >>> gamma.numargs 1 >>> gamma.shapes 'a' Now we set the value of the shape variable to 1 to obtain the exponential distribution, so that we compare easily whether we get the results we expect. >>> gamma(1, scale\=2.).stats(moments\="mv") (array(2.0), array(4.0)) Notice that we can also specify shape parameters as keywords: >>> gamma(a\=1, scale\=2.).stats(moments\="mv") (array(2.0), array(4.0)) Freezing a Distribution¶ Passing the loc and scale keywords time and again can become quite bothersome. The concept of freezing a RV is used to solve such problems. >>> rv \= gamma(1, scale\=2.) By using rv we no longer have to include the scale or the shape parameters anymore. Thus, distributions can be used in one of two ways, either by passing all distribution parameters to each method call (such as we did earlier) or by freezing the parameters for the instance of the distribution. Let us check this: >>> rv.mean(), rv.std() (2.0, 2.0) This is indeed what we should get. Broadcasting¶ The basic methods pdf and so on satisfy the usual numpy broadcasting rules. For example, we can calculate the critical values for the upper tail of the t distribution for different probabilities and degrees of freedom. >>> stats.t.isf([0.1, 0.05, 0.01], ) array([[ 1.37218364, 1.81246112, 2.76376946], [ 1.36343032, 1.79588482, 2.71807918]]) Here, the first row are the critical values for 10 degrees of freedom and the second row for 11 degrees of freedom (d.o.f.). Thus, the broadcasting rules give the same result of calling isf twice: >>> stats.t.isf([0.1, 0.05, 0.01], 10) array([ 1.37218364, 1.81246112, 2.76376946]) >>> stats.t.isf([0.1, 0.05, 0.01], 11) array([ 1.36343032, 1.79588482, 2.71807918]) If the array with probabilities, i.e., [0.1, 0.05, 0.01] and the array of degrees of freedom i.e., [10, 11, 12], have the same array shape, then element wise matching is used. As an example, we can obtain the 10% tail for 10 d.o.f., the 5% tail for 11 d.o.f. and the 1% tail for 12 d.o.f. by calling >>> stats.t.isf([0.1, 0.05, 0.01], [10, 11, 12]) array([ 1.37218364, 1.79588482, 2.68099799]) Specific Points for Discrete Distributions¶ Discrete distribution have mostly the same basic methods as the continuous distributions. However pdf is replaced the probability mass function pmf, no estimation methods, such as fit, are available, and scale is not a valid keyword parameter. The location parameter, keyword loc can still be used to shift the distribution. The computation of the cdf requires some extra attention. In the case of continuous distribution the cumulative distribution function is in most standard cases strictly monotonic increasing in the bounds (a,b) and has therefore a unique inverse. The cdf of a discrete distribution, however, is a step function, hence the inverse cdf, i.e., the percent point function, requires a different definition: ppf(q) \= min{x : cdf(x) >\= q, x integer} For further info, see the docs here. We can look at the hypergeometric distribution as an example >>> from scipy.stats import hypergeom >>> [M, n, N] \= [20, 7, 12] If we use the cdf at some integer points and then evaluate the ppf at those cdf values, we get the initial integers back, for example >>> x \= np.arange(4)2 >>> x array([0, 2, 4, 6]) >>> prb \= hypergeom.cdf(x, M, n, N) >>> prb array([ 0.0001031991744066, 0.0521155830753351, 0.6083591331269301, 0.9897832817337386]) >>> hypergeom.ppf(prb, M, n, N) array([ 0., 2., 4., 6.]) If we use values that are not at the kinks of the cdf step function, we get the next higher integer back: >>> hypergeom.ppf(prb + 1e-8, M, n, N) array([ 1., 3., 5., 7.]) >>> hypergeom.ppf(prb - 1e-8, M, n, N) array([ 0., 2., 4., 6.]) Fitting Distributions¶ The main additional methods of the not frozen distribution are related to the estimation of distribution parameters: fit: maximum likelihood estimation of distribution parameters, including location and scale fit_loc_scale: estimation of location and scale when shape parameters are given nnlf: negative log likelihood function expect: Calculate the expectation of a function against the pdf or pmf Performance Issues and Cautionary Remarks¶ The performance of the individual methods, in terms of speed, varies widely by distribution and method. The results of a method are obtained in one of two ways: either by explicit calculation, or by a generic algorithm that is independent of the specific distribution. Explicit calculation, on the one hand, requires that the method is directly specified for the given distribution, either through analytic formulas or through special functions in scipy.special or numpy.random for rvs. These are usually relatively fast calculations. The generic methods, on the other hand, are used if the distribution does not specify any explicit calculation. To define a distribution, only one of pdf or cdf is necessary; all other methods can be derived using numeric integration and root finding. However, these indirect methods can be very slow. As an example, rgh = stats.gausshyper.rvs(0.5, 2, 2, 2, size=100) creates random variables in a very indirect way and takes about 19 seconds for 100 random variables on my computer, while one million random variables from the standard normal or from the t distribution take just above one second. Remaining Issues¶ The distributions in scipy.stats have recently been corrected and improved and gained a considerable test suite, however a few issues remain: the distributions have been tested over some range of parameters, however in some corner ranges, a few incorrect results may remain. the maximum likelihood estimation in fit does not work with default starting parameters for all distributions and the user needs to supply good starting parameters. Also, for some distribution using a maximum likelihood estimator might inherently not be the best choice. Building Specific Distributions¶ The next examples shows how to build your own distributions. Further examples show the usage of the distributions and some statistical tests. Making a Continuous Distribution, i.e., Subclassing rv_continuous¶ Making continuous distributions is fairly simple. >>> from scipy import stats >>> class deterministic_gen(stats.rv_continuous): ... def _cdf(self, x): ... return np.where(x < 0, 0., 1.) ... def _stats(self): ... return 0., 0., 0., 0. >>> deterministic \= deterministic_gen(name\="deterministic") >>> deterministic.cdf(np.arange(-3, 3, 0.5)) array([ 0., 0., 0., 0., 0., 0., 1., 1., 1., 1., 1., 1.]) Interestingly, the pdf is now computed automatically: >>> deterministic.pdf(np.arange(-3, 3, 0.5)) array([ 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 5.83333333e+04, 4.16333634e-12, 4.16333634e-12, 4.16333634e-12, 4.16333634e-12, 4.16333634e-12]) Be aware of the performance issues mentions in Performance Issues and Cautionary Remarks. The computation of unspecified common methods can become very slow, since only general methods are called which, by their very nature, cannot use any specific information about the distribution. Thus, as a cautionary example: >>> from scipy.integrate import quad >>> quad(deterministic.pdf, -1e-1, 1e-1) (4.163336342344337e-13, 0.0) But this is not correct: the integral over this pdf should be 1. Let’s make the integration interval smaller: >>> quad(deterministic.pdf, -1e-3, 1e-3) # warning removed (1.000076872229173, 0.0010625571718182458) This looks better. However, the problem originated from the fact that the pdf is not specified in the class definition of the deterministic distribution. Subclassing rv_discrete¶ In the following we use stats.rv_discrete to generate a discrete distribution that has the probabilities of the truncated normal for the intervals centered around the integers. General Info From the docstring of rv_discrete, help(stats.rv_discrete), “You can construct an arbitrary discrete rv where P{X=xk} = pk by passing to the rv_discrete initialization method (through the values= keyword) a tuple of sequences (xk, pk) which describes only those values of X (xk) that occur with nonzero probability (pk).” Next to this, there are some further requirements for this approach to work: The keyword name is required. The support points of the distribution xk have to be integers. The number of significant digits (decimals) needs to be specified. In fact, if the last two requirements are not satisfied an exception may be raised or the resulting numbers may be incorrect. An Example Let’s do the work. First >>> npoints \= 20 # number of integer support points of the distribution minus 1 >>> npointsh \= npoints / 2 >>> npointsf \= float(npoints) >>> nbound \= 4 # bounds for the truncated normal >>> normbound \= (1+1/npointsf) nbound # actual bounds of truncated normal >>> grid \= np.arange(-npointsh, npointsh+2, 1) # integer grid >>> gridlimitsnorm \= (grid-0.5) / npointsh nbound # bin limits for the truncnorm >>> gridlimits \= grid - 0.5 # used later in the analysis >>> grid \= grid[:-1] >>> probs \= np.diff(stats.truncnorm.cdf(gridlimitsnorm, -normbound, normbound)) >>> gridint \= grid And finally we can subclass rv_discrete: >>> normdiscrete \= stats.rv_discrete(values\=(gridint, ... np.round(probs, decimals\=7)), name\='normdiscrete') Now that we have defined the distribution, we have access to all common methods of discrete distributions. >>> print 'mean = %6.4f, variance = %6.4f, skew = %6.4f, kurtosis = %6.4f'% \ ... normdiscrete.stats(moments \= 'mvsk') mean = -0.0000, variance = 6.3302, skew = 0.0000, kurtosis = -0.0076 >>> nd_std \= np.sqrt(normdiscrete.stats(moments\='v')) Testing the Implementation Let’s generate a random sample and compare observed frequencies with the probabilities. >>> n_sample \= 500 >>> np.random.seed(87655678) # fix the seed for replicability >>> rvs \= normdiscrete.rvs(size\=n_sample) >>> rvsnd \= rvs >>> f, l \= np.histogram(rvs, bins\=gridlimits) >>> sfreq \= np.vstack([gridint, f, probsn_sample]).T >>> print sfreq [[ -1.00000000e+01 0.00000000e+00 2.95019349e-02] [ -9.00000000e+00 0.00000000e+00 1.32294142e-01] [ -8.00000000e+00 0.00000000e+00 5.06497902e-01] [ -7.00000000e+00 2.00000000e+00 1.65568919e+00] [ -6.00000000e+00 1.00000000e+00 4.62125309e+00] [ -5.00000000e+00 9.00000000e+00 1.10137298e+01] [ -4.00000000e+00 2.60000000e+01 2.24137683e+01] [ -3.00000000e+00 3.70000000e+01 3.89503370e+01] [ -2.00000000e+00 5.10000000e+01 5.78004747e+01] [ -1.00000000e+00 7.10000000e+01 7.32455414e+01] [ 0.00000000e+00 7.40000000e+01 7.92618251e+01] [ 1.00000000e+00 8.90000000e+01 7.32455414e+01] [ 2.00000000e+00 5.50000000e+01 5.78004747e+01] [ 3.00000000e+00 5.00000000e+01 3.89503370e+01] [ 4.00000000e+00 1.70000000e+01 2.24137683e+01] [ 5.00000000e+00 1.10000000e+01 1.10137298e+01] [ 6.00000000e+00 4.00000000e+00 4.62125309e+00] [ 7.00000000e+00 3.00000000e+00 1.65568919e+00] [ 8.00000000e+00 0.00000000e+00 5.06497902e-01] [ 9.00000000e+00 0.00000000e+00 1.32294142e-01] [ 1.00000000e+01 0.00000000e+00 2.95019349e-02]] (Source code) (Source code) Next, we can test, whether our sample was generated by our normdiscrete distribution. This also verifies whether the random numbers are generated correctly. The chisquare test requires that there are a minimum number of observations in each bin. We combine the tail bins into larger bins so that they contain enough observations. >>> f2 \= np.hstack([f[:5].sum(), f[5:-5], f[-5:].sum()]) >>> p2 \= np.hstack([probs[:5].sum(), probs[5:-5], probs[-5:].sum()]) >>> ch2, pval \= stats.chisquare(f2, p2n_sample) >>> print 'chisquare for normdiscrete: chi2 = %6.3f pvalue = %6.4f' % (ch2, pval) chisquare for normdiscrete: chi2 = 12.466 pvalue = 0.4090 The pvalue in this case is high, so we can be quite confident that our random sample was actually generated by the distribution. Analysing One Sample¶ First, we create some random variables. We set a seed so that in each run we get identical results to look at. As an example we take a sample from the Student t distribution: >>> np.random.seed(282629734) >>> x \= stats.t.rvs(10, size\=1000) Here, we set the required shape parameter of the t distribution, which in statistics corresponds to the degrees of freedom, to 10. Using size=1000 means that our sample consists of 1000 independently drawn (pseudo) random numbers. Since we did not specify the keyword arguments loc and scale, those are set to their default values zero and one. Descriptive Statistics¶ x is a numpy array, and we have direct access to all array methods, e.g. >>> print x.max(), x.min() # equivalent to np.max(x), np.min(x) 5.26327732981 -3.78975572422 >>> print x.mean(), x.var() # equivalent to np.mean(x), np.var(x) 0.0140610663985 1.28899386208 How do the some sample properties compare to their theoretical counterparts? >>> m, v, s, k \= stats.t.stats(10, moments\='mvsk') >>> n, (smin, smax), sm, sv, ss, sk \= stats.describe(x) >>> print 'distribution:', distribution: >>> sstr \= 'mean = %6.4f, variance = %6.4f, skew = %6.4f, kurtosis = %6.4f' >>> print sstr %(m, v, s ,k) mean = 0.0000, variance = 1.2500, skew = 0.0000, kurtosis = 1.0000 >>> print 'sample: ', sample: >>> print sstr %(sm, sv, ss, sk) mean = 0.0141, variance = 1.2903, skew = 0.2165, kurtosis = 1.0556 Note: stats.describe uses the unbiased estimator for the variance, while np.var is the biased estimator. For our sample the sample statistics differ a by a small amount from their theoretical counterparts. T-test and KS-test¶ We can use the t-test to test whether the mean of our sample differs in a statistically significant way from the theoretical expectation. >>> print 't-statistic = %6.3f pvalue = %6.4f' % stats.ttest_1samp(x, m) t-statistic = 0.391 pvalue = 0.6955 The pvalue is 0.7, this means that with an alpha error of, for example, 10%, we cannot reject the hypothesis that the sample mean is equal to zero, the expectation of the standard t-distribution. As an exercise, we can calculate our ttest also directly without using the provided function, which should give us the same answer, and so it does: >>> tt \= (sm-m)/np.sqrt(sv/float(n)) # t-statistic for mean >>> pval \= stats.t.sf(np.abs(tt), n-1)2 # two-sided pvalue = Prob(abs(t)>tt) >>> print 't-statistic = %6.3f pvalue = %6.4f' % (tt, pval) t-statistic = 0.391 pvalue = 0.6955 The Kolmogorov-Smirnov test can be used to test the hypothesis that the sample comes from the standard t-distribution >>> print 'KS-statistic D = %6.3f pvalue = %6.4f' % stats.kstest(x, 't', (10,)) KS-statistic D = 0.016 pvalue = 0.9606 Again the p-value is high enough that we cannot reject the hypothesis that the random sample really is distributed according to the t-distribution. In real applications, we don’t know what the underlying distribution is. If we perform the Kolmogorov-Smirnov test of our sample against the standard normal distribution, then we also cannot reject the hypothesis that our sample was generated by the normal distribution given that in this example the p-value is almost 40%. >>> print 'KS-statistic D = %6.3f pvalue = %6.4f' % stats.kstest(x,'norm') KS-statistic D = 0.028 pvalue = 0.3949 However, the standard normal distribution has a variance of 1, while our sample has a variance of 1.29. If we standardize our sample and test it against the normal distribution, then the p-value is again large enough that we cannot reject the hypothesis that the sample came form the normal distribution. >>> d, pval \= stats.kstest((x-x.mean())/x.std(), 'norm') >>> print 'KS-statistic D = %6.3f pvalue = %6.4f' % (d, pval) KS-statistic D = 0.032 pvalue = 0.2402 Note: The Kolmogorov-Smirnov test assumes that we test against a distribution with given parameters, since in the last case we estimated mean and variance, this assumption is violated, and the distribution of the test statistic on which the p-value is based, is not correct. Tails of the distribution¶ Finally, we can check the upper tail of the distribution. We can use the percent point function ppf, which is the inverse of the cdf function, to obtain the critical values, or, more directly, we can use the inverse of the survival function >>> crit01, crit05, crit10 \= stats.t.ppf([1-0.01, 1-0.05, 1-0.10], 10) >>> print 'critical values from ppf at 1%%, 5%% and 10%% %8.4f %8.4f %8.4f'% (crit01, crit05, crit10) critical values from ppf at 1%, 5% and 10% 2.7638 1.8125 1.3722 >>> print 'critical values from isf at 1%%, 5%% and 10%% %8.4f %8.4f %8.4f'% tuple(stats.t.isf([0.01,0.05,0.10],10)) critical values from isf at 1%, 5% and 10% 2.7638 1.8125 1.3722 >>> freq01 \= np.sum(x>crit01) / float(n) 100 >>> freq05 \= np.sum(x>crit05) / float(n) 100 >>> freq10 \= np.sum(x>crit10) / float(n) 100 >>> print 'sample %%-frequency at 1%%, 5%% and 10%% tail %8.4f %8.4f %8.4f'% (freq01, freq05, freq10) sample %-frequency at 1%, 5% and 10% tail 1.4000 5.8000 10.5000 In all three cases, our sample has more weight in the top tail than the underlying distribution. We can briefly check a larger sample to see if we get a closer match. In this case the empirical frequency is quite close to the theoretical probability, but if we repeat this several times the fluctuations are still pretty large. >>> freq05l \= np.sum(stats.t.rvs(10, size\=10000) > crit05) / 10000.0 100 >>> print 'larger sample %%-frequency at 5%% tail %8.4f'% freq05l larger sample %-frequency at 5% tail 4.8000 We can also compare it with the tail of the normal distribution, which has less weight in the tails: >>> print 'tail prob. of normal at 1%%, 5%% and 10%% %8.4f %8.4f %8.4f'% \ ... tuple(stats.norm.sf([crit01, crit05, crit10])100) tail prob. of normal at 1%, 5% and 10% 0.2857 3.4957 8.5003 The chisquare test can be used to test, whether for a finite number of bins, the observed frequencies differ significantly from the probabilities of the hypothesized distribution. >>> quantiles \= [0.0, 0.01, 0.05, 0.1, 1-0.10, 1-0.05, 1-0.01, 1.0] >>> crit \= stats.t.ppf(quantiles, 10) >>> crit array([-Inf, -2.76376946, -1.81246112, -1.37218364, 1.37218364, 1.81246112, 2.76376946, Inf]) >>> n_sample \= x.size >>> freqcount \= np.histogram(x, bins\=crit) >>> tprob \= np.diff(quantiles) >>> nprob \= np.diff(stats.norm.cdf(crit)) >>> tch, tpval \= stats.chisquare(freqcount, tprobn_sample) >>> nch, npval \= stats.chisquare(freqcount, nprobn_sample) >>> print 'chisquare for t: chi2 = %6.2f pvalue = %6.4f' % (tch, tpval) chisquare for t: chi2 = 2.30 pvalue = 0.8901 >>> print 'chisquare for normal: chi2 = %6.2f pvalue = %6.4f' % (nch, npval) chisquare for normal: chi2 = 64.60 pvalue = 0.0000 We see that the standard normal distribution is clearly rejected while the standard t-distribution cannot be rejected. Since the variance of our sample differs from both standard distribution, we can again redo the test taking the estimate for scale and location into account. The fit method of the distributions can be used to estimate the parameters of the distribution, and the test is repeated using probabilities of the estimated distribution. >>> tdof, tloc, tscale \= stats.t.fit(x) >>> nloc, nscale \= stats.norm.fit(x) >>> tprob \= np.diff(stats.t.cdf(crit, tdof, loc\=tloc, scale\=tscale)) >>> nprob \= np.diff(stats.norm.cdf(crit, loc\=nloc, scale\=nscale)) >>> tch, tpval \= stats.chisquare(freqcount, tprobn_sample) >>> nch, npval \= stats.chisquare(freqcount, nprobn_sample) >>> print 'chisquare for t: chi2 = %6.2f pvalue = %6.4f' % (tch, tpval) chisquare for t: chi2 = 1.58 pvalue = 0.9542 >>> print 'chisquare for normal: chi2 = %6.2f pvalue = %6.4f' % (nch, npval) chisquare for normal: chi2 = 11.08 pvalue = 0.0858 Taking account of the estimated parameters, we can still reject the hypothesis that our sample came from a normal distribution (at the 5% level), but again, with a p-value of 0.95, we cannot reject the t distribution. Special tests for normal distributions¶ Since the normal distribution is the most common distribution in statistics, there are several additional functions available to test whether a sample could have been drawn from a normal distribution First we can test if skew and kurtosis of our sample differ significantly from those of a normal distribution: >>> print 'normal skewtest teststat = %6.3f pvalue = %6.4f' % stats.skewtest(x) normal skewtest teststat = 2.785 pvalue = 0.0054 >>> print 'normal kurtosistest teststat = %6.3f pvalue = %6.4f' % stats.kurtosistest(x) normal kurtosistest teststat = 4.757 pvalue = 0.0000 These two tests are combined in the normality test >>> print 'normaltest teststat = %6.3f pvalue = %6.4f' % stats.normaltest(x) normaltest teststat = 30.379 pvalue = 0.0000 In all three tests the p-values are very low and we can reject the hypothesis that the our sample has skew and kurtosis of the normal distribution. Since skew and kurtosis of our sample are based on central moments, we get exactly the same results if we test the standardized sample: >>> print 'normaltest teststat = %6.3f pvalue = %6.4f' % \ ... stats.normaltest((x-x.mean())/x.std()) normaltest teststat = 30.379 pvalue = 0.0000 Because normality is rejected so strongly, we can check whether the normaltest gives reasonable results for other cases: >>> print('normaltest teststat = %6.3f pvalue = %6.4f' % ... stats.normaltest(stats.t.rvs(10, size\=100))) normaltest teststat = 4.698 pvalue = 0.0955 >>> print('normaltest teststat = %6.3f pvalue = %6.4f' % ... stats.normaltest(stats.norm.rvs(size\=1000))) normaltest teststat = 0.613 pvalue = 0.7361 When testing for normality of a small sample of t-distributed observations and a large sample of normal distributed observation, then in neither case can we reject the null hypothesis that the sample comes from a normal distribution. In the first case this is because the test is not powerful enough to distinguish a t and a normally distributed random variable in a small sample. Comparing two samples¶ In the following, we are given two samples, which can come either from the same or from different distribution, and we want to test whether these samples have the same statistical properties. Comparing means¶ Test with sample with identical means: >>> rvs1 \= stats.norm.rvs(loc\=5, scale\=10, size\=500) >>> rvs2 \= stats.norm.rvs(loc\=5, scale\=10, size\=500) >>> stats.ttest_ind(rvs1, rvs2) (-0.54890361750888583, 0.5831943748663857) Test with sample with different means: >>> rvs3 \= stats.norm.rvs(loc\=8, scale\=10, size\=500) >>> stats.ttest_ind(rvs1, rvs3) (-4.5334142901750321, 6.507128186505895e-006) Kolmogorov-Smirnov test for two samples ks_2samp¶ For the example where both samples are drawn from the same distribution, we cannot reject the null hypothesis since the pvalue is high >>> stats.ks_2samp(rvs1, rvs2) (0.025999999999999995, 0.99541195173064878) In the second example, with different location, i.e. means, we can reject the null hypothesis since the pvalue is below 1% >>> stats.ks_2samp(rvs1, rvs3) (0.11399999999999999, 0.0027132103661283141) Kernel Density Estimation¶ A common task in statistics is to estimate the probability density function (PDF) of a random variable from a set of data samples. This task is called density estimation. The most well-known tool to do this is the histogram. A histogram is a useful tool for visualization (mainly because everyone understands it), but doesn’t use the available data very efficiently. Kernel density estimation (KDE) is a more efficient tool for the same task. The gaussian_kde estimator can be used to estimate the PDF of univariate as well as multivariate data. It works best if the data is unimodal. Univariate estimation¶ We start with a minimal amount of data in order to see how gaussian_kde works, and what the different options for bandwidth selection do. The data sampled from the PDF is show as blue dashes at the bottom of the figure (this is called a rug plot): >>> from scipy import stats >>> import matplotlib.pyplot as plt >>> x1 \= np.array([-7, -5, 1, 4, 5], dtype\=np.float) >>> kde1 \= stats.gaussian_kde(x1) >>> kde2 \= stats.gaussian_kde(x1, bw_method\='silverman') >>> fig \= plt.figure() >>> ax \= fig.add_subplot(111) >>> ax.plot(x1, np.zeros(x1.shape), 'b+', ms\=20) # rug plot >>> x_eval \= np.linspace(-10, 10, num\=200) >>> ax.plot(x_eval, kde1(x_eval), 'k-', label\="Scott's Rule") >>> ax.plot(x_eval, kde2(x_eval), 'r-', label\="Silverman's Rule") >>> plt.show() (Source code) We see that there is very little difference between Scott’s Rule and Silverman’s Rule, and that the bandwidth selection with a limited amount of data is probably a bit too wide. We can define our own bandwidth function to get a less smoothed out result. >>> def my_kde_bandwidth(obj, fac\=1./5): ... """We use Scott's Rule, multiplied by a constant factor.""" ... return np.power(obj.n, -1./(obj.d+4)) fac >>> fig \= plt.figure() >>> ax \= fig.add_subplot(111) >>> ax.plot(x1, np.zeros(x1.shape), 'b+', ms\=20) # rug plot >>> kde3 \= stats.gaussian_kde(x1, bw_method\=my_kde_bandwidth) >>> ax.plot(x_eval, kde3(x_eval), 'g-', label\="With smaller BW") >>> plt.show() (Source code) We see that if we set bandwidth to be very narrow, the obtained estimate for the probability density function (PDF) is simply the sum of Gaussians around each data point. We now take a more realistic example, and look at the difference between the two available bandwidth selection rules. Those rules are known to work well for (close to) normal distributions, but even for unimodal distributions that are quite strongly non-normal they work reasonably well. As a non-normal distribution we take a Student’s T distribution with 5 degrees of freedom. import numpy as np import matplotlib.pyplot as plt from scipy import stats np.random.seed(12456) x1 \= np.random.normal(size\=200) # random data, normal distribution xs \= np.linspace(x1.min()-1, x1.max()+1, 200) kde1 \= stats.gaussian_kde(x1) kde2 \= stats.gaussian_kde(x1, bw_method\='silverman') fig \= plt.figure(figsize\=(8, 6)) ax1 \= fig.add_subplot(211) ax1.plot(x1, np.zeros(x1.shape), 'b+', ms\=12) # rug plot ax1.plot(xs, kde1(xs), 'k-', label\="Scott's Rule") ax1.plot(xs, kde2(xs), 'b-', label\="Silverman's Rule") ax1.plot(xs, stats.norm.pdf(xs), 'r--', label\="True PDF") ax1.set_xlabel('x') ax1.set_ylabel('Density') ax1.set_title("Normal (top) and Student's T$_{df=5}$ (bottom) distributions") ax1.legend(loc\=1) x2 \= stats.t.rvs(5, size\=200) # random data, T distribution xs \= np.linspace(x2.min() - 1, x2.max() + 1, 200) kde3 \= stats.gaussian_kde(x2) kde4 \= stats.gaussian_kde(x2, bw_method\='silverman') ax2 \= fig.add_subplot(212) ax2.plot(x2, np.zeros(x2.shape), 'b+', ms\=12) # rug plot ax2.plot(xs, kde3(xs), 'k-', label\="Scott's Rule") ax2.plot(xs, kde4(xs), 'b-', label\="Silverman's Rule") ax2.plot(xs, stats.t.pdf(xs, 5), 'r--', label\="True PDF") ax2.set_xlabel('x') ax2.set_ylabel('Density') plt.show() (Source code) We now take a look at a bimodal distribution with one wider and one narrower Gaussian feature. We expect that this will be a more difficult density to approximate, due to the different bandwidths required to accurately resolve each feature. >>> from functools import partial >>> loc1, scale1, size1 \= (-2, 1, 175) >>> loc2, scale2, size2 \= (2, 0.2, 50) >>> x2 \= np.concatenate([np.random.normal(loc\=loc1, scale\=scale1, size\=size1), ... np.random.normal(loc\=loc2, scale\=scale2, size\=size2)]) >>> x_eval \= np.linspace(x2.min() - 1, x2.max() + 1, 500) >>> kde \= stats.gaussian_kde(x2) >>> kde2 \= stats.gaussian_kde(x2, bw_method\='silverman') >>> kde3 \= stats.gaussian_kde(x2, bw_method\=partial(my_kde_bandwidth, fac\=0.2)) >>> kde4 \= stats.gaussian_kde(x2, bw_method\=partial(my_kde_bandwidth, fac\=0.5)) >>> pdf \= stats.norm.pdf >>> bimodal_pdf \= pdf(x_eval, loc\=loc1, scale\=scale1) float(size1) / x2.size + \ ... pdf(x_eval, loc\=loc2, scale\=scale2) float(size2) / x2.size >>> fig \= plt.figure(figsize\=(8, 6)) >>> ax \= fig.add_subplot(111) >>> ax.plot(x2, np.zeros(x2.shape), 'b+', ms\=12) >>> ax.plot(x_eval, kde(x_eval), 'k-', label\="Scott's Rule") >>> ax.plot(x_eval, kde2(x_eval), 'b-', label\="Silverman's Rule") >>> ax.plot(x_eval, kde3(x_eval), 'g-', label\="Scott 0.2") >>> ax.plot(x_eval, kde4(x_eval), 'c-', label\="Scott 0.5") >>> ax.plot(x_eval, bimodal_pdf, 'r--', label\="Actual PDF") >>> ax.set_xlim([x_eval.min(), x_eval.max()]) >>> ax.legend(loc\=2) >>> ax.set_xlabel('x') >>> ax.set_ylabel('Density') >>> plt.show() (Source code) As expected, the KDE is not as close to the true PDF as we would like due to the different characteristic size of the two features of the bimodal distribution. By halving the default bandwidth (Scott 0.5) we can do somewhat better, while using a factor 5 smaller bandwidth than the default doesn’t smooth enough. What we really need though in this case is a non-uniform (adaptive) bandwidth. Multivariate estimation¶ With gaussian_kde we can perform multivariate as well as univariate estimation. We demonstrate the bivariate case. First we generate some random data with a model in which the two variates are correlated. >>> def measure(n): ... """Measurement model, return two coupled measurements.""" ... m1 \= np.random.normal(size\=n) ... m2 \= np.random.normal(scale\=0.5, size\=n) ... return m1+m2, m1-m2 >>> m1, m2 \= measure(2000) >>> xmin \= m1.min() >>> xmax \= m1.max() >>> ymin \= m2.min() >>> ymax \= m2.max() Then we apply the KDE to the data: >>> X, Y \= np.mgrid[xmin:xmax:100j, ymin:ymax:100j] >>> positions \= np.vstack([X.ravel(), Y.ravel()]) >>> values \= np.vstack([m1, m2]) >>> kernel \= stats.gaussian_kde(values) >>> Z \= np.reshape(kernel.evaluate(positions).T, X.shape) Finally we plot the estimated bivariate distribution as a colormap, and plot the individual data points on top. >>> fig \= plt.figure(figsize\=(8, 6)) >>> ax \= fig.add_subplot(111) >>> ax.imshow(np.rot90(Z), cmap\=plt.cm.gist_earth_r, ... extent\=[xmin, xmax, ymin, ymax]) >>> ax.plot(m1, m2, 'k.', markersize\=2) >>> ax.set_xlim([xmin, xmax]) >>> ax.set_ylim([ymin, ymax]) >>> plt.show() (Source code) © Copyright 2008-2016, The Scipy community. Last updated on Mar 15, 2017. Created using Sphinx 1.5.2.
12730	https://www.scribd.com/doc/2280493/Expansion-of-Liquid	Opens in a new window Opens an external website Opens an external website in a new window This website utilizes technologies such as cookies to enable essential site functionality, as well as for analytics, personalization, and targeted advertising. To learn more, view the following link: Privacy Policy Open navigation menu Upload 89%(9)89% found this document useful (9 votes) 36K views46 pages Expansion of Liquid This chapter in 'Heat and Themodynamics' has been written as a basic course for 10+2 std students. Some figures could not be provided(will be added in the next edn). Examples, exercises and… Uploaded by Abhijit Kar Gupta You are on page 1/ 46 Fundamental Physics -I Abhijit Kar Gupta (email: Thermal Expansion of Liquid We shall consider only the volume expansion of liquid in this chapter. The concept of thermal expansion of length or surface of a liquid is not meaningful as a liquid has no definite shape like that of a solid. Some Common Facts: The expansion in liquid is usually much more than in a solid for a same rise in temperature; on an average 10 times more. • The rate of expansion of a same liquid sometimes differs greatly in different temperature ranges. The amount of volume expansion of water in the range of C C 00 1110 is quite different from that in the range of C C 00 9493 − • Anomalous expansion of water in the temperature range of C C 00 40 − . Apparent and Real Expansion of Liquids: A liquid is heated while keeping it in a container. There occurs an expansion of the solid container along with the liquid. But the amount of expansion of the container is small compared to that of the liquid. Thus the expansion of a container is not usually noticeable. • When the expansion of liquid is considered ignoring the expansion of the container, it is called apparent expansion . • real expansion of liquid is calculated by adding the expansion of the part of the container containing liquid (before expansion) with the apparent expansion of liquid. The relation among apparent and real expansion of liquid and the expansion of the container can be demonstrated with the help of following experiment: Include Figure 3.1 fr A flask made of thick glass is filled with a coloured liquid (as shown in the figure). A cork is fitted through which a narrow glass tube is inserted. There is a scale marked on the tube. Some liquid will rise up inside the tube up to a point which is marked by O on the scale. Now the flask is placed inside a big vessel of boiling water which heats up the flask and as well as the liquid inside. It is observed that the liquid inside the tube first comes down to a position marked by A and later it rises again, goes past O and reaches at a point marked by B. The reason is that there is an expansion of the glass container at the beginning. As the glass is a bad conductor of heat, it takes some time for heat to reach the liquid inside the flask and because of this there is no expansion of liquid at the initial stage. Thus the liquid surface comes down from O to A inside the tube. After a while the heat starts flowing into the liquid for which the liquid expands and it begins to rise inside the tube. The expansion of liquid is more than the expansion of glass. Thus we see the final rise of liquid is at B above the point O. Fig. to be included Download to read ad-free Fundamental Physics -I by Dr. Abhijit Kar Gupta (email: kg.abhi@gmail.com ) 2As the level of water comes down from O to A for a short time, it appears that the water level directly expands from O to B. The expansion of water level OB, from its initial position O to the final position B is the apparent expansion . AB is the amount of real expansion . According to figure, OAOB AB += . The real expansion of the liquid = the apparent expansion of the liquid + the expansion of the container. Coefficients of Apparent and Real expansions of Liquid We have two different coefficients of expansion for a liquid corresponding to apparent and real expansions. Coefficient of apparent expansion for liquid: The amount of apparent expansion of unit volume of a liquid for a unit change of temperature is called coefficient of apparent expansion of liquid. If the initial volume of the liquid is and the final apparent volume which is observed to be V ′ due to an increase in temperature 0 T T T −=∆ , the apparent expansion of volume of the liquid is V ∆ = 0 V V −′ . ∴ The coefficient of apparent expansion of liquid can be written as T V V ∆∆=′ 0 γ …………………..(1) Thus we can write, [ ] [ ] )(11 000 T T V T V V −′+=∆′+=′ γ γ . It may be easily understood that the coefficient of apparent expansion of a liquid can not be a characteristic property of a liquid as the apparent expansions will be different when containers of different materials are taken. Coefficient of real expansion for liquid: The amount of real expansion of unit volume of a liquid for a unit change of temperature is called coefficient of real expansion of liquid. If the initial volume of the liquid is 0 V and the final real volume which is calculated to be V due to an increase in temperature 0 T T T −=∆ , the real expansion of the volume of liquid is V ∆ = 0 V V − . ∴ The coefficient of real expansion of liquid is T V V ∆∆= 0 γ ……………………..(2) Thus we can write, [ ] [ ] )(11 000 T T V T V V −+=∆+= γ γ . The coefficient of real expansion is a property of a liquid as the real expansion of volume of a liquid is not dependent on the expansion of the container. Download to read ad-free Fundamental Physics -I by Dr. Abhijit Kar Gupta (email: kg.abhi@gmail.com ) 3 Note: • The dimension of the coefficient of real or apparent expansion of liquid is dependent only on the temperature difference as can be seen from expressions (1) and (2). For example, the value of the coefficient of real expansion for mercury in Celsius and Fahrenheit scales are C 05 /1018.18 − × and F 05 /101.10 − × . • The coefficient of real expansion is a characteristic property of a liquid. Thus it is more important to know the coefficient of real expansion of a liquid than that for apparent expansion. • The value of the coefficient of linear expansion for a liquid may be slightly different at different temperatures. The value which is usually mentioned is assumed to be an average value within a temperature range. Table # 1: Coefficient of real expansion of different liquids Liquid γ 10 )( − C Mercury 18.18 5 10 − × Benzene 121 5 10 − × Alcohol 110 5 10 − × Sulphuric acid 57 5 10 − × Chloroform 126 5 10 − × Glycerine 53 5 10 − × Tarpin Oil 94 5 10 − × Olive Oil 70 5 10 − × Water ( C C 00 105 − ) 5.3 5 10 − × Water ( C C 00 2010 − ) 15 5 10 − × Water ( C C 00 4020 − ) 30.2 5 10 − × Water ( C C 00 6040 − ) 45.8 5 10 − × Relation between the coefficients of apparent and real expansions of liquid Let us assume the initial volume of a definite amount of liquid in a container is 0 V . Due to an increase in temperature T ∆ , the apparent volume of the liquid is measured to be V ′ and the real volume is found to be V . ∴ The apparent expansion of the liquid = 0 V V −′ and the real expansion of the liquid = 0 V V − . The expansion of the part of the container containing liquid is = V V ′− . We know, the real expansion of liquid = the apparent expansion of liquid + the expansion of the container. ∴ 0 V V − = ( 0 V V −′ ) + ( V V ′− ) Download to read ad-free Fundamental Physics -I by Dr. Abhijit Kar Gupta (email: kg.abhi@gmail.com ) 4Dividing both sides by T V ∆ 0 we find, T V V V ∆− 00 = T V V V ∆−′ 00 + T V V V ∆′− 0 Or, g γ γ γ +′= , where g γ is the coefficient of volume expansion of the material of the container. Therefore, we have the coefficient of real expansion of liquid = the coefficient of apparent expansion of liquid + the coefficient of volume expansion of the material of the container. Relation between the density and the coefficient of real expansion of liquid As there is an expansion of volume of a liquid due to increase in temperature, the density of it must decrease. The density depends on the coefficient of real expansion at different temperatures. Let us have a liquid of a certain mass m . The initial volume of this liquid is 0 V and the density is 0 ρ . The volume and the density are changed to V and ρ as the temperature is increased by T ∆ . We can write, ρ ρ V V m == 00 Or, 00 V V = ρ ρ …………………(1) If γ be the coefficient of real expansion of the liquid, [ ] T V V ∆+= .1 0 γ Or, T V V ∆+= .1 0 γ …………………(2) From (1) and (2) we get, T ∆+= .1 0 γ ρ ρ Or, [ ] T ∆+= .1 0 γ ρ ρ …………………….(3) We can also write (for small T ∆ ), T ∆+= .1 0 γ ρ ρ = ( ) 10 .1 − ∆+ T γ ρ = ( ) T ∆− .1 0 γ ρ [ Q γ is very small] Thus we see the density of a liquid is decreased due to increase in temperature. The relation (3) can be rewritten as T ∆=− .1 0 γ ρ ρ . ∴ T T ∆∆−=∆−= .. 0 ρ ρ ρ ρ ρ γ . The negative sign indicates that the density decreases with increasing temperature. The above can be used to calculate the coefficient of real expansion of a liquid when the density of it is known at two different temperatures. Numerical Problems with Solutions Example 1 Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Download to read ad-free Share this document Share on Facebook, opens a new window Share on LinkedIn, opens a new window Share with Email, opens mail client Millions of documents at your fingertips, ad-free Subscribe with a free trial You might also like 1st Year Physics According To COVID-19 Syllabus PDF No ratings yet 1st Year Physics According To COVID-19 Syllabus PDF 334 pages Thermal Properties of Matter-1 100% (1) Thermal Properties of Matter-1 25 pages Non Destructive Testing No ratings yet Non Destructive Testing 559 pages Thermal Expansion Powerpoint No ratings yet Thermal Expansion Powerpoint 19 pages Lecture - 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12731	https://www.quora.com/How-do-I-find-the-minimum-value-of-sin-x-cos-x-tan-x-cot-x-sec-x-csc-x-for-real-numbers-x	How to find the minimum value of [math]\|\sin{x} + \cos{x} + \tan{x} + \cot{x} + \sec{x} + \csc{x}\|[/math] for real numbers [math]x[/math] - Quora Something went wrong. Wait a moment and try again. Try again Skip to content Skip to search Sign In Mathematics Lowest Value Algebraic Expressions Optimization Theory Functions (mathematics) Real Numbers Maximum and Minimum Value... Minimum Element Algebra 5 How do I find the minimum value of \|sin x+cos x+tan x+cot x+sec x+csc x\|\|sin⁡x+cos⁡x+tan⁡x+cot⁡x+sec⁡x+csc⁡x\| for real numbers x x? All related (42) Sort Recommended Max Gretinski Studied Mathematics · Author has 6.6K answers and 2.5M answer views ·1y Let f(x) = sin(x) + cos(x) + tan(x) + cot(x) + sec(x) + csc(x), so that g(x) = \| f(x) \| is the function in which we are interested. The functions are undefined at all integer multiples of π 2 π 2, and they are continuous on intervals not containing the above values. The functions are periodic, with period 2 π π. Therefore, we only need to consider the behavior of the function on the intervals (0, π 2)U(π 2,π)U(π,3 π 2)U(3 π 2,2 π).π 2)U(π 2,π)U(π,3 π 2)U(3 π 2,2 π). These separate intervals we will label Quadrant I, Quadrant II, Quadrant III, and Quadrant IV, as in trigonometry Continue Reading Let f(x) = sin(x) + cos(x) + tan(x) + cot(x) + sec(x) + csc(x), so that g(x) = \| f(x) \| is the function in which we are interested. The functions are undefined at all integer multiples of π 2 π 2, and they are continuous on intervals not containing the above values. The functions are periodic, with period 2 π π. Therefore, we only need to consider the behavior of the function on the intervals (0, π 2)U(π 2,π)U(π,3 π 2)U(3 π 2,2 π).π 2)U(π 2,π)U(π,3 π 2)U(3 π 2,2 π). These separate intervals we will label Quadrant I, Quadrant II, Quadrant III, and Quadrant IV, as in trigonometry. Observation: In Quadrant I, all six trig functions have positive values. Therefore, the minimum value of f in QI is likely to be greater than in QII, QIII, or QIV. Now, by testing values, it is easy to determine that f(x) > 0 in QI and f(x) < 0 in the other three quadrants. This means that g(x) = f(x) in QI, and g(x) = -f(x) in the other three quadrants. Look at g(x) in the other three quadrants. Except at the points of discontinuity (above), g(x) is differentiable, and g(x) = -1(sin(x) + cos(x) + tan(x) + cot(x) + sec(x) + csc(x)), giving us g’(x) = -cosx + sinx -s e c 2(x)+c s c 2(x)s e c 2(x)+c s c 2(x)- secxtanx + cscxcotx To learn where g’(x) = 0, we set the above equal to zero. We may multiply by s i n 2(x)c o s 2(x)s i n 2(x)c o s 2(x) to convert as much as possible to sines and cosines. There is a WHOLE LOT OF ALGEBRAIC MANIPULATION after this point. −s i n 2 x c o s 3 x+s i n 3 x c o s 2 x−s i n 2(x)+c o s 2(x)−s i n 3 x+c o s 3 x−s i n 2 x c o s 3 x+s i n 3 x c o s 2 x−s i n 2(x)+c o s 2(x)−s i n 3 x+c o s 3 x = 0 −s i n 2 x c o s 2 x(c o s x−s i n x)+c o s 2(x)−s i n 2(x)+c o s 3 x−s i n 3 x−s i n 2 x c o s 2 x(c o s x−s i n x)+c o s 2(x)−s i n 2(x)+c o s 3 x−s i n 3 x = 0 −s i n 2 x c o s 2 x−s i n 2 x c o s 2 x(cosx - sinx) + (cosx + sinx)(cosx – sinx) + (cosx – sinx)(1 + sinxcosx) = 0 Then sinx = cosx OR -sin2xcos2x + cosx + sinx + 1 + sinxcosx = 0. FIRST…If sinx = cosx in QII through QIV, we have x = 3 π 4 3 π 4. This will turn out to be a relative maximum, which does not concern us. Look at… −s i n 2 x c o s 2 x−s i n 2 x c o s 2 x+ sinx + 1 + sinxcosx + cosx = 0 -s i n 2 x s i n 2 x(1 + sinx)(1 – sinx) + sinx + 1 + sinxcosx + cosx = 0 -s i n 2 x s i n 2 x(sinx + 1)(1 – sinx) + (sinx + 1) + cosx(sinx + 1) = 0 So sinx = -1 OR -s i n 2 x s i n 2 x(1 – sinx) + 1 + cosx = 0 At this point, we see that sinx = -1 at a point where g is undefined. Ignore it, and continue by converting s i n 2 x s i n 2 x to 1 - c o s 2 x c o s 2 x, and then factoring it. -(1 + cosx)(1 – cosx)(1 – sinx) + 1 + cosx = 0 (1 + cosx)(cosx - 1)(1 – sinx) + 1 + cosx = 0 (1 + cosx)[(cosx - 1)(1 – sinx) + 1]= 0 So cosx = -1 OR (cosx - 1)(1 – sinx) + 1 = 0 But again, cosx = -1 at a point where g is undefined. Ignore it, and continue with… cosx – sinxcosx – 1 + sinx + 1 = 0 cosx – sinxcosx + sinx = 0 That is sinx + cosx – sinxcosx = 0 sinx + cosx = sinxcosx Square both sides 1 + 2sinxcosx = s i n 2 x c o s 2 x s i n 2 x c o s 2 x Let y = sinxcosx, and we have a polynomial that is quadratic in form: 1 + 2y = y 2 y 2 y 2 y 2 - 2y - 1 = 0 y =2±√4+4 2 2±4+4 2 y =2±2√2 2 2±2 2 2 y = 1±√2 1±2 sinxcosx = 1±√2 1±2 Multiply by 2. 2sin(x)cos(x) = 2±2√2 2±2 2 sin(2x) = 2−2√2 2−2 2 2x = Arcsin(2−2√2 2−2 2) x = A r c s i n(2−2√2)2 A r c s i n(2−2 2)2 This value is negative in Quadrant IV. Add 2 π π to it to obtain the value between 0 and 2 π π. The corresponding value of x in QII is π π + A r c s i n(2−2√2)2 A r c s i n(2−2 2)2 These values of x (+2 π πn for any integer n) are the absolute minima for g(x). Now, finding the minimum VALUE involves using half-angle formulas, because you must evaluate all six trig functions at the above value of x. I will skip that part, because this post is quite long already. However, you can obtain an exact solution this way. The approximate minimum value is g(x) = 1.828. Upvote · 9 2 Sponsored by Best Gadget Advice Here Are The 33 Coolest Gifts For This Year. We've put together a list of incredible gifts that are selling out fast. Get these before they're gone! Learn More 999 160 Related questions More answers below What is the minimum value of \|sin x+cos x+tan x+csc x+sec x+cot x\|\|sin⁡x+cos⁡x+tan⁡x+csc⁡x+sec⁡x+cot⁡x\|? How do I evaluate the minimum value of \|sin x+cos x+tan x+cot x+sec x+csc x\|\|sin⁡x+cos⁡x+tan⁡x+cot⁡x+sec⁡x+csc⁡x\| for real number x x not multiple of π 2 π 2? If x,y x,y are real numbers, then how do I find the minimum value of the expression max{1,\|x+y\|,\|x y\|}max{1,\|x\|}max{1,\|y\|}max{1,\|x+y\|,\|x y\|}max{1,\|x\|}max{1,\|y\|}? How do I find the minimum value of ∣∣sin x+cos x+tan x+cot x+sec x+cosec x∣∣\|sin⁡x+cos⁡x+tan⁡x+cot⁡x+sec⁡x+cosec⁡x\| for real numbers x x not multiple of π 2 π 2? Given that 5 r+4 s+3 t+6 u=100,5 r+4 s+3 t+6 u=100, where r≥s≥t≥u≥0 r≥s≥t≥u≥0 are real numbers, how do I find, with proof, the maximum and minimum possible values of r+s+t+u r+s+t+u? Enrico Gregorio Associate professor in Algebra · Author has 18.4K answers and 16M answer views ·1y Let’s first consider the function without the absolute value, which we can rewrite as (sin x+cos x)+2 2 sin x cos x+2(sin x+cos x)2 sin x cos x(sin⁡x+cos⁡x)+2 2 sin⁡x cos⁡x+2(sin⁡x+cos⁡x)2 sin⁡x cos⁡x It’s now expedient to replace x=y+π/4,x=y+π/4, whereby sin x+cos x=√2 cos y sin⁡x+cos⁡x=2 cos⁡y 2 sin x cos x=sin 2 x=cos 2 y 2 sin⁡x cos⁡x=sin⁡2 x=cos⁡2 y and we’re led to find the minimum of the absolute value of √2 cos y+2 2 cos 2 y−1+2√2 cos y 2 cos 2 y−1 2 cos⁡y+2 2 cos 2⁡y−1+2 2 cos⁡y 2 cos 2⁡y−1 Now set z=√2 cos y,z=2 cos⁡y, so −√2≤z≤√2−2≤z≤2 and we want to find the minimum of the absolute value of f(z)=z+2 z 2−1+2 z z 2−1=z+2 z−1 f(z)=z+2 z 2−1+2 z z 2−1=z+2 z−1 over the domain [-\sqrt{2},\sqrt[-\sqrt{2},\sqrt Continue Reading Let’s first consider the function without the absolute value, which we can rewrite as (sin x+cos x)+2 2 sin x cos x+2(sin x+cos x)2 sin x cos x(sin⁡x+cos⁡x)+2 2 sin⁡x cos⁡x+2(sin⁡x+cos⁡x)2 sin⁡x cos⁡x It’s now expedient to replace x=y+π/4,x=y+π/4, whereby sin x+cos x=√2 cos y sin⁡x+cos⁡x=2 cos⁡y 2 sin x cos x=sin 2 x=cos 2 y 2 sin⁡x cos⁡x=sin⁡2 x=cos⁡2 y and we’re led to find the minimum of the absolute value of √2 cos y+2 2 cos 2 y−1+2√2 cos y 2 cos 2 y−1 2 cos⁡y+2 2 cos 2⁡y−1+2 2 cos⁡y 2 cos 2⁡y−1 Now set z=√2 cos y,z=2 cos⁡y, so −√2≤z≤√2−2≤z≤2 and we want to find the minimum of the absolute value of f(z)=z+2 z 2−1+2 z z 2−1=z+2 z−1 f(z)=z+2 z 2−1+2 z z 2−1=z+2 z−1 over the domain [−√2,√2]∖{−1,1}[−2,2]∖{−1,1} The numerator is everywhere positive, so the study of the absolute value is easier. We see that f′(z)=1−2(z−1)2=z 2−2 z−1(z−1)2 f′(z)=1−2(z−1)2=z 2−2 z−1(z−1)2 so the only critical point in the specified domain is 1−√2.1−2. For g(z)=\|f(z)\|g(z)=\|f(z)\| we have g(−√2)=3√2−2 g(−2)=3 2−2 g(1−√2)=2√2−1 g(1−2)=2 2−1 g(√2)=3√2+2 g(2)=3 2+2 The sought minimum is 2√2−1 2 2−1 Upvote · 99 11 9 1 Ragu Rajagopalan Passionate Maths solver ;Reviving knowledge after 3 decades · Author has 10.1K answers and 7.6M answer views ·3y Originally Answered: How do I evaluate the minimum value of \left\|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x\right\| for real number x not multiple of \dfrac\pi2? · y=\|sin x+cos x+tan x+cot x+sec x+csc x\|y=\|sin⁡x+cos⁡x+tan⁡x+cot⁡x+sec⁡x+csc⁡x\| Let t=sin x+cos x⟹sin x⋅cos x=t 2−1 2 Let t=sin⁡x+cos⁡x⟹sin⁡x⋅cos⁡x=t 2−1 2 ∴y=∣∣∣t+sin x cos x+cos x sin x+1 cos x+1 sin x∣∣∣∴y=\|t+sin⁡x cos⁡x+cos⁡x sin⁡x+1 cos⁡x+1 sin⁡x\| =∣∣∣t+sin 2 x+cos 2 x sin x⋅cos x+sin x+cos x sin x⋅cos x∣∣∣=\|t+sin 2⁡x+cos 2⁡x sin⁡x⋅cos⁡x+sin⁡x+cos⁡x sin⁡x⋅cos⁡x\| =∣∣∣t+2 t 2−1+2 t t 2−1∣∣∣=∣∣∣t+2 t−1∣∣∣=\|t+2 t 2−1+2 t t 2−1\|=\|t+2 t−1\| ⟹y′=1−2(t−1)2=0⟹y′=1−2(t−1)2=0 [math]\implies (t-1)^2 = 2 \implies t^2 - 2t - 1 = 0 \implies t = \dfrac{2 \pm 2[/math] Continue Reading y=\|sin x+cos x+tan x+cot x+sec x+csc x\|y=\|sin⁡x+cos⁡x+tan⁡x+cot⁡x+sec⁡x+csc⁡x\| Let t=sin x+cos x⟹sin x⋅cos x=t 2−1 2 Let t=sin⁡x+cos⁡x⟹sin⁡x⋅cos⁡x=t 2−1 2 ∴y=∣∣∣t+sin x cos x+cos x sin x+1 cos x+1 sin x∣∣∣∴y=\|t+sin⁡x cos⁡x+cos⁡x sin⁡x+1 cos⁡x+1 sin⁡x\| =∣∣∣t+sin 2 x+cos 2 x sin x⋅cos x+sin x+cos x sin x⋅cos x∣∣∣=\|t+sin 2⁡x+cos 2⁡x sin⁡x⋅cos⁡x+sin⁡x+cos⁡x sin⁡x⋅cos⁡x\| =∣∣∣t+2 t 2−1+2 t t 2−1∣∣∣=∣∣∣t+2 t−1∣∣∣=\|t+2 t 2−1+2 t t 2−1\|=\|t+2 t−1\| ⟹y′=1−2(t−1)2=0⟹y′=1−2(t−1)2=0 ⟹(t−1)2=2⟹t 2−2 t−1=0⟹t=2±2√2 2=1±√2⟹(t−1)2=2⟹t 2−2 t−1=0⟹t=2±2 2 2=1±2 t=sin x+cos x⟹t∈[−√2,√2]t=sin⁡x+cos⁡x⟹t∈[−2,2] ∴t=sin x+cos x=1−√2∴t=sin⁡x+cos⁡x=1−2 y m a x=∞when x =n π 2 y m a x=∞when x =n π 2 ∴y m i n=∣∣∣(1−√2)+2 1−√2−1∣∣∣∴y m i n=\|(1−2)+2 1−2−1\| =∣∣1−2√2∣∣=2√2−1≈1.828427=\|1−2 2\|=2 2−1≈1.828427 Upvote · 99 12 Brian Sittinger PhD in Mathematics, University of California, Santa Barbara (Graduated 2006) · Upvoted by David Joyce , Ph.D. Mathematics, University of Pennsylvania (1979) · Author has 8.5K answers and 21.1M answer views ·Updated May 21 Originally Answered: What is the minimum value of \|\sin{x}+\cos{x}+\tan{x}+\csc{x}+\sec{x}+\cot{x}\|? · Writing the sum of the six trigonometric functions in terms of sine and cosine yields ∣∣sin x+cos x+sin x cos x+1 sin x+1 cos x+cos x sin x∣∣=∣∣sin x+cos x+1+sin x+cos x sin x cos x∣∣.\|sin⁡x+cos⁡x+sin⁡x cos⁡x+1 sin⁡x+1 cos⁡x+cos⁡x sin⁡x\|=\|sin⁡x+cos⁡x+1+sin⁡x+cos⁡x sin⁡x cos⁡x\|. We can simplify this further by noting that sin x+cos x=√2 cos(π 4−x)sin⁡x+cos⁡x=2 cos⁡(π 4−x). Letting y=π 4−x y=π 4−x, we see that sin x cos x=1 2 sin(2 x)=1 2 cos(2 y).sin⁡x cos⁡x=1 2 sin⁡(2 x)=1 2 cos⁡(2 y). With this change of variable, Continue Reading Writing the sum of the six trigonometric functions in terms of sine and cosine yields ∣∣sin x+cos x+sin x cos x+1 sin x+1 cos x+cos x sin x∣∣=∣∣sin x+cos x+1+sin x+cos x sin x cos x∣∣.\|sin⁡x+cos⁡x+sin⁡x cos⁡x+1 sin⁡x+1 cos⁡x+cos⁡x sin⁡x\|=\|sin⁡x+cos⁡x+1+sin⁡x+cos⁡x sin⁡x cos⁡x\|. We can simplify this further by noting that sin x+cos x=√2 cos(π 4−x)sin⁡x+cos⁡x=2 cos⁡(π 4−x). Letting y=π 4−x y=π 4−x, we see that sin x cos x=1 2 sin(2 x)=1 2 cos(2 y).sin⁡x cos⁡x=1 2 sin⁡(2 x)=1 2 cos⁡(2 y). With this change of variable, it suffices to minimize ∣∣∣√2 cos y+2+2√2 cos y cos(2 y)∣∣∣.\|2 cos⁡y+2+2 2 cos⁡y cos⁡(2 y)\|. Next, noting that cos(2 y)=2 cos 2 y−1 cos⁡(2 y)=2 cos 2⁡y−1, by letting t=√2 cos y t=2 cos⁡y, it suffices to minimize the absolute value of f(t)=t+2+2 t t 2−1=t+2 t−1.f(t)=t+2+2 t t 2−1=t+2 t−1. Given t=√2 cos(π 4−x)t=2 cos⁡(π 4−x), we need to minimize \|f\|\|f\| on [−√2,√2].[−2,2]. Differentiating yields f′(t)=1−2(t−1)2 f′(t)=1−2(t−1)2, which vanishes when t=1±√2 t=1±2; we ignore t=1+√2 t=1+2, because it is out of the domain. By substitution, \|f(±√2)\|=3√2±2\|f(±2)\|=3 2±2 and \|f(1−√2)\|=2√2−1;\|f(1−2)\|=2 2−1; of these values, 2√2−1 2 2−1 is the smallest. Since lim t→1\|f(t)\|=∞lim t→1\|f(t)\|=∞, we conclude that the minimum value in question is indeed 2√2−1 2 2−1. Upvote · 999 112 9 3 99 12 Sponsored by MRPeasy Powerful yet simple MRP software for growing manufacturers. MRPeasy makes it easier for growing manufacturing businesses to keep on expanding successfully. Free Trial 99 65 Related questions More answers below What is the value of ∫e sec x.sec 3 x(sin 2 x+cos x+sin x+sin x.cos x)d x∫e sec⁡x.sec 3⁡x(sin 2⁡x+cos⁡x+sin⁡x+sin⁡x.cos⁡x)d x? Let (g∘f)(x)−f(x)=x 2+2 x−2,f(0)=1(g∘f)(x)−f(x)=x 2+2 x−2,f(0)=1 and g′(x)>1 g′(x)>1 for all real number x.x. How do I find the value of (f∘g)′(1)(f∘g)′(1)? Real numbers x,y,z x,y,z are such that x+y+z z+x=2 and x+y+z y+z=4.x+y+z z+x=2 and x+y+z y+z=4. How do I find the value of the expression x+y+z x+y x+y+z x+y? For n≥2 n≥2 and real numbers 0≤x 1≤x 2≤...≤x n≤n 0≤x 1≤x 2≤...≤x n≤n, what is the least possible value of f(x 1,x 2,...,x n)=√(x 1)2+0 2+√(x 2−x 1)2+1 2+√(x 3−x 2)2+2 2+...+√(x n−x n−1)2+(n−1)2+√(n−x n)2+n 2 f(x 1,x 2,...,x n)=(x 1)2+0 2+(x 2−x 1)2+1 2+(x 3−x 2)2+2 2+...+(x n−x n−1)2+(n−1)2+(n−x n)2+n 2? If x∈(0,π/2)x∈(0,π/2) then, what is the minimum value of tan(x)/cot(x)+cot(x)/csc(x)+csc(x)/sec(x)+sec(x)/tan(x)tan⁡(x)/cot⁡(x)+cot⁡(x)/csc⁡(x)+csc⁡(x)/sec⁡(x)+sec⁡(x)/tan⁡(x)? Eleftherios Argyropoulos B.S. in Mathematics&Physics, Northeastern University (Graduated 2002) · Author has 2K answers and 2.5M answer views ·Updated 6y Originally Answered: What is the minimum value of \|\sin{x}+\cos{x}+\tan{x}+\csc{x}+\sec{x}+\cot{x}\|? · We have: s i n x+c o s x+t a n x+c o t x+s e c x+c s c x=s i n x+c o s x+t a n x+c o t x+s e c x+c s c x= [s i n x+c o s x]+[(s i n x)/(c o s x)+(c o s x)/(s i n x)]+[1/(c o s x)+1/(s i n x)]=[s i n x+c o s x]+[(s i n x)/(c o s x)+(c o s x)/(s i n x)]+[1/(c o s x)+1/(s i n x)]= [s i n x+c o s x+((s i n x)2+(c o s x)2)/((c o s x)(s i n x))+(s i n x+c o s x)/((c o s x)(s i n x))]=[s i n x+c o s x+((s i n x)2+(c o s x)2)/((c o s x)(s i n x))+(s i n x+c o s x)/((c o s x)(s i n x))]= [s i n x+c o s x+1/((c o s x)(s i n x))+(s i n x+c o s x)/((c o s x)(s i n x))]=[s i n x+c o s x+1/((c o s x)(s i n x))+(s i n x+c o s x)/((c o s x)(s i n x))]= [s i n x+c o s x+(1+s i n x+c o s x)/((c o s x)(s i n x))]=[s i n x+c o s x+(1+s i n x+c o s x)/((c o s x)(s i n x))]= [s i n x+c o s x+2(1+s i n x+c o s x)/(s i n(2 x))]…(1)[s i n x+c o s x+2(1+s i n x+c o s x)/(s i n(2 x))]…(1) Now, we observe that: s i n(x+π/4)=(s i n x)[c o s(π/4)]+(c o s x)[s i n(π/4)]=(√(2)/2)(s i n x)+(√(2)/2)(c o s x)=s i n(x+π/4)=(s i n x)[c o s(π/4)]+(c o s x)[s i n(π/4)]=(√(2)/2)(s i n x)+(√(2)/2)(c o s x)= (√(2)/2)[s i n x+c o s x]…(2)(√(2)/2)[s i n x+c o s x]…(2) Now, we multiply both sides of (2) by √(2)√(2) and we take: √(2)[s i n(x+π/4)]=s i n x+c o s x…(3)√(2)[s i n(x+π/4)]=s i n x+c o s x…(3) We se Continue Reading We have: s i n x+c o s x+t a n x+c o t x+s e c x+c s c x=s i n x+c o s x+t a n x+c o t x+s e c x+c s c x= [s i n x+c o s x]+[(s i n x)/(c o s x)+(c o s x)/(s i n x)]+[1/(c o s x)+1/(s i n x)]=[s i n x+c o s x]+[(s i n x)/(c o s x)+(c o s x)/(s i n x)]+[1/(c o s x)+1/(s i n x)]= [s i n x+c o s x+((s i n x)2+(c o s x)2)/((c o s x)(s i n x))+(s i n x+c o s x)/((c o s x)(s i n x))]=[s i n x+c o s x+((s i n x)2+(c o s x)2)/((c o s x)(s i n x))+(s i n x+c o s x)/((c o s x)(s i n x))]= [s i n x+c o s x+1/((c o s x)(s i n x))+(s i n x+c o s x)/((c o s x)(s i n x))]=[s i n x+c o s x+1/((c o s x)(s i n x))+(s i n x+c o s x)/((c o s x)(s i n x))]= [s i n x+c o s x+(1+s i n x+c o s x)/((c o s x)(s i n x))]=[s i n x+c o s x+(1+s i n x+c o s x)/((c o s x)(s i n x))]= [s i n x+c o s x+2(1+s i n x+c o s x)/(s i n(2 x))]…(1)[s i n x+c o s x+2(1+s i n x+c o s x)/(s i n(2 x))]…(1) Now, we observe that: s i n(x+π/4)=(s i n x)[c o s(π/4)]+(c o s x)[s i n(π/4)]=(√(2)/2)(s i n x)+(√(2)/2)(c o s x)=s i n(x+π/4)=(s i n x)[c o s(π/4)]+(c o s x)[s i n(π/4)]=(√(2)/2)(s i n x)+(√(2)/2)(c o s x)= (√(2)/2)[s i n x+c o s x]…(2)(√(2)/2)[s i n x+c o s x]…(2) Now, we multiply both sides of (2) by √(2)√(2) and we take: √(2)[s i n(x+π/4)]=s i n x+c o s x…(3)√(2)[s i n(x+π/4)]=s i n x+c o s x…(3) We set √(2)s i n(x+π/4)=y√(2)s i n(x+π/4)=y and then (1) takes the form: s i n x+c o s x+t a n x+c o t x+s e c x+c s c x=y+[2(1+y)]/[s i n(2 x)]…(4)s i n x+c o s x+t a n x+c o t x+s e c x+c s c x=y+[2(1+y)]/[s i n(2 x)]…(4) Now, we have to express sin(2x) in terms of y. We have: √(2)s i n(x+π/4)=y=>s i n(x+π/4)=(√(2)/2)y…(5)√(2)s i n(x+π/4)=y=>s i n(x+π/4)=(√(2)/2)y…(5) Moreover, we obtain: s i n(2 x)=s i n[(x+π/4)+(x−π/4)]=s i n(2 x)=s i n[(x+π/4)+(x−π/4)]= [s i n(x+π/4)][c o s(x−π/4)]+[s i n(x−π/4)][c o s(x+π/4)]=[s i n(x+π/4)][c o s(x−π/4)]+[s i n(x−π/4)][c o s(x+π/4)]= [s i n(x−π/4)]2−[c o s(x+π/4)]2=(y 2)/2−[1−(y 2)/2]=y 2−1…(6)[s i n(x−π/4)]2−[c o s(x+π/4)]2=(y 2)/2−[1−(y 2)/2]=y 2−1…(6) Now, by (4) and (6), we obtain: s i n x+c o s x+t a n x+c o t x+s e c x+c s c x=y+[2(1+y)]/(y 2−1)=y+2/(y−1)…(7)s i n x+c o s x+t a n x+c o t x+s e c x+c s c x=y+[2(1+y)]/(y 2−1)=y+2/(y−1)…(7) The first derivative of the function f(y)=y+2/(y−1)f(y)=y+2/(y−1), is: f′(y)=(y 2−2 y−1)/(y−1)2 f′(y)=(y 2−2 y−1)/(y−1)2 We set f′(y)=0 f′(y)=0 and we take: y 2−2 y−1=0=>y=1−√(2)y 2−2 y−1=0=>y=1−√(2) and y=1+√(2)…(8)y=1+√(2)…(8) Now, since s i n(x+π/4)≤1,s i n(x+π/4)≤1,from√(2)s i n(x+π/4)=y√(2)s i n(x+π/4)=y, it follows that: √(2)s i n(x+π/4)≤√(2)=>y≤√(2)=>y=1−√(2)…(9)√(2)s i n(x+π/4)≤√(2)=>y≤√(2)=>y=1−√(2)…(9) We now set this value of y y in f(y)f(y) and we take: f(1−√(2))=1−2√(2)=>\|f(1−√(2))\|=2√(2)−1…(10)f(1−√(2))=1−2√(2)=>\|f(1−√(2))\|=2√(2)−1…(10) Therefore, by (8), (9) and (10), we conclude that the minimum value we seek for \|s i n x+c o s x+t a n x+c o t x+s e c x+c s c x\|\|s i n x+c o s x+t a n x+c o t x+s e c x+c s c x\|is 2√(2)−1.2√(2)−1. Upvote · 9 4 Pritam Ghoshal GATE 2020 AIR 400 Mechanical Engineering · Author has 188 answers and 633.9K answer views ·9y Originally Answered: What is the minimum value of \|\sin x+\cos x+\tan x+\csc x+\sec x+\cot x\|? · Let, sinx+cosx=p, then \|p\|≤2.5 2.5, \|p\|≠1 p 2 p 2=sin2x+2sinxcosx+cos2x=1+2sinxcosx, So, sinxcosx=(p 2−1)/2(p 2−1)/2 tanx+cotx=sin2x+cos2xsinxcosx=2/(p 2−1)2/(p 2−1) secx+cosecx=cosx+sinxsinxcosx=2 p/(p 2−1)2 p/(p 2−1) We write, y=sinx+cosx+tanx+cotx+secx+cosecx=p+2/(p 2−1)+2 p/(p 2−1)p+2/(p 2−1)+2 p/(p 2−1) =p−1+2/(p−1)+1 p−1+2/(p−1)+1 If, 1=2(2).5−1 p−1+2/(p−1)+1>=2(2).5−1 If −(2.5)−(2.5)≤p<−1 or −1<p<1 then −y=1−p+2/(1−p)−1>=2(2).5−1−y=1−p+2/(1−p)−1>=2(2).5−1 \|sinx+cosx+tanx+cotx+secx+cosecx\|≥2(2).5−1 2(2).5−1 The equality will hold true when p=1−2.5 1−2.5 So the minimum value is 2∗(2.5)−1 2∗(2.5)−1 Upvote · 9 6 9 2 Sponsored by Sync.com Protect your files, photos and data with Sync.com. Treat your data carefully. Use Sync.com to protect it. Try 5 GB for free. Learn More 2.7K 2.7K David Vanderschel PhD in Mathematics&Physics, Rice (Houston neighborhood) (Graduated 1970) · Author has 37.6K answers and 50.1M answer views ·1y A2A: I don’t know about you. But I submit a question like that to Wolfram Alpha. I did so, and I did get an answer. Something close to 1.8 looks reasonable. If I did not have Wolfram Alpha to fall back on, I’d have used some iterative minimization method. (E.g., gradient descent.) (The trigonometry, calculus, and algebra for an analytic approach looks like it would be too tedious to try to mess with.) Upvote · Sohel Zibara Studied at Doctor of Philosophy Degrees (Graduated 2000) · Author has 5.1K answers and 2.6M answer views ·May 8 Related How do I find the minimum value of ∣∣sin x+cos x+tan x+cot x+sec x+cosec x∣∣\|sin⁡x+cos⁡x+tan⁡x+cot⁡x+sec⁡x+cosec⁡x\| for real numbers x x not multiple of π 2 π 2? Set U(x)=∣∣sin x+cos x+tan x+cot x+sec x+csc x∣∣.We have:Set U(x)=\|sin x+cos x+tan x+cot x+sec x+csc x\|.We have: [Math Processing Error]\displaystyle{\color{darkgreen}{\color{darkred}{U\left(x\right)\,=\,\Bigg\|\sin\,x\,+\,\cos\,x\,+\,\frac{\sin\,x}{\cos\,x}\,+\,\frac{\cos\,x}{\sin\,x}\,+\,\frac{1}{\cos\,x}\,+\,\frac{1}{\sin\,x}\Bigg\|\,=\=\,\Bigg\|\sin\,x\,+\,\cos\,x\,+\,\frac{\sin^2x\,+\,\cos^2x}{\sin\,x\,\cos\,x}\,+\,\frac{\sin\,x\,+\,\cos\,x}{\sin\,x\,\cos\,x}\Bigg\|\,\underbrace{=}_{\color{darkgreen}{\sin^2x\,+\,\cos^2x\,=\,1}}\\underbrace Continue Reading Set U(x)=∣∣sin x+cos x+tan x+cot x+sec x+csc x∣∣.We have:Set U(x)=\|sin x+cos x+tan x+cot x+sec x+csc x\|.We have: U(x)=∣∣∣sin x+cos x+sin x cos x+cos x sin x+1 cos x+1 sin x∣∣∣==∣∣∣sin x+cos x+sin 2 x+cos 2 x sin x cos x+sin x+cos x sin x cos x∣∣∣=sin 2 x+cos 2 x=1=sin 2 x+cos 2 x=1∣∣∣sin x+cos x+1+sin x+cos x sin x cos x∣∣∣==∣∣∣sin x+cos x+2(1+sin x+cos x)2 sin x cos x∣∣∣=2 sin x cos x=(sin x+cos x)2–1 U(x)=\|sin x+cos x+sin x cos x+cos x sin x+1 cos x+1 sin x\|==\|sin x+cos x+sin 2⁡x+cos 2⁡x sin x cos x+sin x+cos x sin x cos x\|=⏟sin 2⁡x+cos 2⁡x=1=⏟sin 2⁡x+cos 2⁡x=1\|sin x+cos x+1+sin x+cos x sin x cos x\|==\|sin x+cos x+2(1+sin x+cos x)2 sin x cos x\|=⏟2 sin x cos x=(sin x+cos x)2–1 =2 sin x cos x=(sin x+cos x)2–1∣∣∣sin x+cos x+2(1+sin x+cos x)(sin x+cos x)2–1∣∣∣==∣∣∣sin x+cos x+2(1+sin x+cos x)(sin x+cos x–1)(sin x+cos x+1)∣∣∣==∣∣∣sin x+cos x+2 sin x+cos x–1∣∣∣==∣∣∣sin x+cos x–1+2 sin x+cos x–1+1∣∣∣=sin x+cos x=√2 sin(π 4+x)=⏟2 sin x cos x=(sin x+cos x)2–1\|sin x+cos x+2(1+sin x+cos x)(sin x+cos x)2–1\|==\|sin x+cos x+2(1+sin x+cos x)(sin x+cos x–1)(sin x+cos x+1)\|==\|sin x+cos x+2 sin x+cos x–1\|==\|sin x+cos x–1+2 sin x+cos x–1+1\|=⏟sin x+cos x=2 sin⁡(π 4+x) =sin x+cos x=√2 sin(π 4+x)=∣∣∣√2 sin(π 4+x)–1+2√2 sin(π 4+x)–1+1∣∣∣.=⏟sin x+cos x=2 sin⁡(π 4+x)=\|2 sin⁡(π 4+x)–1+2 2 sin⁡(π 4+x)–1+1\|. We have thus proved that :U(x)=∣∣∣√2 sin(π 4+x)–1+2√2 sin(π 4+x)–1+1∣∣∣We have thus proved that :U(x)=\|2 sin⁡(π 4+x)–1+2 2 sin⁡(π 4+x)–1+1\| Setting a=√2 sin(π 4+x)yields :U(x)=∣∣∣a–1+2 a–1+1∣∣∣Setting a=2 sin⁡(π 4+x)yields :U(x)=\|a–1+2 a–1+1\| Since–√2≤a≤√2,we will be considering two cases.Since–2≤a≤2,we will be considering two cases. Case I–√2≤a<1 This means that 1–a>0.Hence :U(x)=∣∣∣(1–a+2 1–a)–1∣∣∣≥AM–GM∣∣2√2–1∣∣=2√2–1,with equality iff 1–a=2 1–a iff a=1–√2 iff sin(π 4+x)=1 2(√2–2).Case I–2≤a<1 This means that 1–a>0.Hence :U(x)=\|(1–a+2 1–a)–1\|≥⏟AM–GM\|2 2–1\|=2 2–1,with equality iff 1–a=2 1–a iff a=1–2 iff sin⁡(π 4+x)=1 2(2–2). Case II 10.Hence :U(x)=∣∣∣(a–1+2 a–1)+1∣∣∣≥AM–GM 2√2+1.Case II 10.Hence :U(x)=\|(a–1+2 a–1)+1\|≥⏟AM–GM 2 2+1. It is clear now that the minimum of U(x)is 2√2–1.It is clear now that the minimum of U(x)is 2 2–1. Upvote · 9 8 Promoted by Bata India Dhruti Shah Visualiser \| Graphic Designer (2018–present) ·Sep 12 What are the best professional affordable and comfortable shoes for women? I usually look at three things when I’m buying work shoes: comfort, cushioning and arch support; how sturdy the sole is; and whether I can actually afford to get more than one pair if I want them in different colours. Ballerinas by Bata though, are what I wear the most. I didn’t know about them until recently, when a coworker recommended them to me, also spotted my favorite creator Siddhi Karwa styling them across Europe and I have been absolutely loving them. They’re professional enough for work wear but don’t feel heavy and keep me comfortable throughout the day, even when I’m commuting to the Continue Reading I usually look at three things when I’m buying work shoes: comfort, cushioning and arch support; how sturdy the sole is; and whether I can actually afford to get more than one pair if I want them in different colours. Ballerinas by Bata though, are what I wear the most. I didn’t know about them until recently, when a coworker recommended them to me, also spotted my favorite creator Siddhi Karwa styling them across Europe and I have been absolutely loving them. They’re professional enough for work wear but don’t feel heavy and keep me comfortable throughout the day, even when I’m commuting to the office. I got mine for around ₹999 from Bata, which felt like a steal compared to some other brands I looked at. They’ve held up really well, and I can easily pair them with trousers, skirts for my work outfits. If you’re on a budget but still want something that is comfortable and follows fashion trends, Ballerinas by Bata are the perfect choice. I picked up mine from a Bata store near me, you can grab yours too. Upvote · 1.1K 1.1K 99 86 99 13 Alon Amit PhD in Mathematics; Mathcircler. · Upvoted by Richard Farrer , MSc Mathematics, University of Bristol (1990) and Nathan Hannon , Ph. D. Mathematics, University of California, Davis (2021) · Author has 8.8K answers and 173.8M answer views ·2y Related If x,y x,y are real numbers, then how do I find the minimum value of the expression max{1,\|x+y\|,\|x y\|}max{1,\|x\|}max{1,\|y\|}max{1,\|x+y\|,\|x y\|}max{1,\|x\|}max{1,\|y\|}? This is a case study in case analysis and the use of symmetry. Here’s my attempt; there may well be more efficient approaches. Let f=f(x,y)f=f(x,y) be the expression in the question. The max(1,\|x\|)max(1,\|x\|) part compels us to know if x x is <−1<−1, >1>1 or in between. Let’s name those regions N N for negative, P P for positive and O O for neutral or “close to 0 0". The placement of x x and y y can then be described as N O N O, or P N P N, and so on: the first letter indicates where x x is, the second indicates y y. We can reduce the number of cases as follows: a simultaneous sign flip (x,y)↦(−x,−y)(x,y)↦(−x,−y) doesn’t change the value of f f. It mov Continue Reading This is a case study in case analysis and the use of symmetry. Here’s my attempt; there may well be more efficient approaches. Let f=f(x,y)f=f(x,y) be the expression in the question. The max(1,\|x\|)max(1,\|x\|) part compels us to know if x x is <−1<−1, >1>1 or in between. Let’s name those regions N N for negative, P P for positive and O O for neutral or “close to 0 0". The placement of x x and y y can then be described as N O N O, or P N P N, and so on: the first letter indicates where x x is, the second indicates y y. We can reduce the number of cases as follows: a simultaneous sign flip (x,y)↦(−x,−y)(x,y)↦(−x,−y) doesn’t change the value of f f. It moves N N N N to P P P P, so we might as well ignore N N N N. Similarly we may ignore O N O N and N O N O. A further reduction comes from the exchange (x,y)↦(y,x)(x,y)↦(y,x), which also leaves f f intact. We can therefore ignore N P N P (replacing it with P N P N) as well as O P O P. We are left with just four regions: O O O O, P N P N, P O P O and P P P P. The regions P P P P, P N P N and O O O O are trivial to analyze: all of them have f≥1 f≥1. The only remaining region is P O P O, and it’s the interesting one. Here x>1 x>1 and −1≤y≤1−1≤y≤1. If y≥0 y≥0 we have x+y≥x>1 x+y≥x>1 and x y<x x y<x so the max in the numerator is x+y x+y, and the denominator is x x, so the ratio is 1+y/x 1+y/x and again 1 1 is a lower bound. We are left to analyze the region −1≤y<0−1≤y<0. Set z=−y z=−y, so z z is non-negative and f f is now expressed as f(x,z)=max(1,x−z,x z)x f(x,z)=max(1,x−z,x z)x As x x and z z vary, the expressions 1 1, x−z x−z and x z x z are ordered one way or another, and for that order to change we must have two of them become equal: the critical points are 1=x−z 1=x−z, 1=x z 1=x z and x−z=x z x−z=x z. Away from these coincidences, there are no changes in ordering. Treating z z as a parameter, the critical values for x x are therefore x=1+z x=1+z, x=1 z x=1 z and x=z 1−z x=z 1−z. Curiously, these critical values all become equal when z=λ=√5−1 2 z=λ=5−1 2. This a strong hint that this may be the point we care about (and that turns out to be true), but right now it just means that we expect a different behavior when z<λ z<λ and when z>λ z>λ. Suppose first that z≤λ z≤λ. Then 1+z≤1 z 1+z≤1 z, so if x≤1+z x≤1+z then 1 1 prevails in the numerator (it beats both x−z x−z and x z x z), and f=1/x f=1/x. If x≥1+z x≥1+z, on the other hand, then x−z x−z beats the other two, and f=1−z/x f=1−z/x. To minimize f f you need to maximize x x in the first case and minimize it in the second, so in both cases x=1+z x=1+z is your best bet, and f=1/(1+z)f=1/(1+z). This is now minimized when z z is maximized, meaning at z=λ z=λ, yielding - surprise, surprise – f(1+λ,λ)=λ f(1+λ,λ)=λ. When z≥λ z≥λ, a similar analysis works but the structure is quite different: here, 1 1 wins when x≤1/z x≤1/z, x z x z wins when 1/z≤x≤z 1−z 1/z≤x≤z 1−z, and x−z x−z wins when x≥z 1−z x≥z 1−z. In all cases, the minimum is no less than z z, and z z is minimized once again when z=λ z=λ. All told, the minimum of f f is obtained when z=λ z=λ and x=1+z=1 z=z 1−z x=1+z=1 z=z 1−z, and the minimal value is just λ=√5−1 2 λ=5−1 2, and that’s your final answer. Upvote · 99 65 9 5 9 2 Dean Rubine Former Faculty at Carnegie Mellon School Of Computer Science (1991–1994) · Author has 10.6K answers and 23.7M answer views ·Sep 21 Related What is the minimum possible value of x 2+y 2+x y−x+y x 2+y 2+x y−x+y, where x x and y y are real numbers? Let’s solve this one first: minimize U(x,y)=x 2+y 2−x y+x+y U(x,y)=x 2+y 2−x y+x+y This one is symmetrical in x x and y y so likely has an extrema when x=y x=y x 2+x 2−x 2+x+x=x 2+2 x=(x+1)2−1 x 2+x 2−x 2+x+x=x 2+2 x=(x+1)2−1 so x=y=−1 x=y=−1 is a the minimum , which gives U min=−1 U min=−1 The one we want to solve is to minimize U(−x,y)U(−x,y) so we get −x=y=−1−x=y=−1 Answer: min −1−1 at (1,−1)(1,−1) Continue Reading Let’s solve this one first: minimize U(x,y)=x 2+y 2−x y+x+y U(x,y)=x 2+y 2−x y+x+y This one is symmetrical in x x and y y so likely has an extrema when x=y x=y x 2+x 2−x 2+x+x=x 2+2 x=(x+1)2−1 x 2+x 2−x 2+x+x=x 2+2 x=(x+1)2−1 so x=y=−1 x=y=−1 is a the minimum , which gives U min=−1 U min=−1 The one we want to solve is to minimize U(−x,y)U(−x,y) so we get −x=y=−1−x=y=−1 Answer: min −1−1 at (1,−1)(1,−1) Upvote · 99 11 Awnon Bhowmik Studied at University of Dhaka · Author has 3.7K answers and 11.2M answer views ·8y Related If we let y=(sin x+csc x)2+(cos x+sec x)2 y=(sin⁡x+csc⁡x)2+(cos⁡x+sec⁡x)2, then what is the minimum value of y y? Since I’m lazy, I’ll take the easy way out y=(sin x+csc x)2+(cos x+sec x)2=sin 2 x+csc 2 x+2+cos 2 x+sec 2 x+2=5+csc 2 x+sec 2 x=5+1 sin 2 x+1 cos 2 x=5+sin 2 x+cos 2 x sin 2 x cos 2 x=5+1 sin 2 x cos 2 x=5+1(1 2 sin 2 x)2=5+4 sin 2 2 x y=(sin⁡x+csc⁡x)2+(cos⁡x+sec⁡x)2=sin 2⁡x+csc 2⁡x+2+cos 2⁡x+sec 2⁡x+2=5+csc 2⁡x+sec 2⁡x=5+1 sin 2⁡x+1 cos 2⁡x=5+sin 2⁡x+cos 2⁡x sin 2⁡x cos 2⁡x=5+1 sin 2⁡x cos 2⁡x=5+1(1 2 sin⁡2 x)2=5+4 sin 2⁡2 x Taking a detour We know −1≤sin 2 x≤1−1≤sin⁡2 x≤1 So, 0≤sin 2 2 x≤1 0≤sin 2⁡2 x≤1 0≤sin 2 2 x≤1 1≤1 sin 2 2 x≤∞4≤4 sin 2 2 x≤∞9≤5+4 sin 2 2 x≤∞0≤sin 2⁡2 x≤1 1≤1 sin 2⁡2 x≤∞4≤4 sin 2⁡2 x≤∞9≤5+4 sin 2⁡2 x≤∞ Go ahead and pick u Continue Reading Since I’m lazy, I’ll take the easy way out y=(sin x+csc x)2+(cos x+sec x)2=sin 2 x+csc 2 x+2+cos 2 x+sec 2 x+2=5+csc 2 x+sec 2 x=5+1 sin 2 x+1 cos 2 x=5+sin 2 x+cos 2 x sin 2 x cos 2 x=5+1 sin 2 x cos 2 x=5+1(1 2 sin 2 x)2=5+4 sin 2 2 x y=(sin⁡x+csc⁡x)2+(cos⁡x+sec⁡x)2=sin 2⁡x+csc 2⁡x+2+cos 2⁡x+sec 2⁡x+2=5+csc 2⁡x+sec 2⁡x=5+1 sin 2⁡x+1 cos 2⁡x=5+sin 2⁡x+cos 2⁡x sin 2⁡x cos 2⁡x=5+1 sin 2⁡x cos 2⁡x=5+1(1 2 sin⁡2 x)2=5+4 sin 2⁡2 x Taking a detour We know −1≤sin 2 x≤1−1≤sin⁡2 x≤1 So, 0≤sin 2 2 x≤1 0≤sin 2⁡2 x≤1 0≤sin 2 2 x≤1 1≤1 sin 2 2 x≤∞4≤4 sin 2 2 x≤∞9≤5+4 sin 2 2 x≤∞0≤sin 2⁡2 x≤1 1≤1 sin 2⁡2 x≤∞4≤4 sin 2⁡2 x≤∞9≤5+4 sin 2⁡2 x≤∞ Go ahead and pick up the scraps. And here comes a Visual confirmation via Desmos Upvote · 99 27 9 1 Shi Ming 8y Originally Answered: What is the minimum value of \sin(x)+\cos(x)+\tan(x)+\cot(x)+\sec(x)+\csc(x)? · We mark the function F(x)=sin(x)+cos(x)+tan(x)+cot(x)+sec(x)+csc(x)F(x)=sin⁡(x)+cos⁡(x)+tan⁡(x)+cot⁡(x)+sec⁡(x)+csc⁡(x) We discuss the value of F(x) when x belongs to (-1,1) If x→0−x→0−, the following holds s i n(x)→0,c o s(x)→1,t a n(x)→0,c o t(x)→−∞,s e c(x)→1,c s c(x)→−∞s i n(x)→0,c o s(x)→1,t a n(x)→0,c o t(x)→−∞,s e c(x)→1,c s c(x)→−∞ So x→0−,x→0−, we have F(x)→−∞.F(x)→−∞. If x→0+x→0+, the following holds s i n(x)→0,c o s(x)→1,t a n(x)→0,c o t(x)→+∞,s e c(x)→1,c s c(x)→+∞s i n(x)→0,c o s(x)→1,t a n(x)→0,c o t(x)→+∞,s e c(x)→1,c s c(x)→+∞ So x→0+,w e h a v e F(x)→+∞.x→0+,w e h a v e F(x)→+∞. The minimum value of F(x)F(x) is−∞;−∞; the maximum value of F(x)F(x) is +∞.+∞. Upvote · 9 5 9 5 Doug Dillon Ph.D. Mathematics · Upvoted by Justin Rising , PhD in statistics · Author has 12.4K answers and 11.4M answer views ·6y Related What's the minimum value of s i n 2 x+c o s 2 x+t a n 2 x+s e c 2 x+c o s e c 2 x+c o t 2 x s i n 2 x+c o s 2 x+t a n 2 x+s e c 2 x+c o s e c 2 x+c o t 2 x ? Why can't we use AM-GM inequality? Upvote · 99 55 9 7 Related questions What is the minimum value of \|sin x+cos x+tan x+csc x+sec x+cot x\|\|sin⁡x+cos⁡x+tan⁡x+csc⁡x+sec⁡x+cot⁡x\|? How do I evaluate the minimum value of \|sin x+cos x+tan x+cot x+sec x+csc x\|\|sin⁡x+cos⁡x+tan⁡x+cot⁡x+sec⁡x+csc⁡x\| for real number x x not multiple of π 2 π 2? If x,y x,y are real numbers, then how do I find the minimum value of the expression max{1,\|x+y\|,\|x y\|}max{1,\|x\|}max{1,\|y\|}max{1,\|x+y\|,\|x y\|}max{1,\|x\|}max{1,\|y\|}? How do I find the minimum value of ∣∣sin x+cos x+tan x+cot x+sec x+cosec x∣∣\|sin⁡x+cos⁡x+tan⁡x+cot⁡x+sec⁡x+cosec⁡x\| for real numbers x x not multiple of π 2 π 2? Given that 5 r+4 s+3 t+6 u=100,5 r+4 s+3 t+6 u=100, where r≥s≥t≥u≥0 r≥s≥t≥u≥0 are real numbers, how do I find, with proof, the maximum and minimum possible values of r+s+t+u r+s+t+u? What is the value of ∫e sec x.sec 3 x(sin 2 x+cos x+sin x+sin x.cos x)d x∫e sec⁡x.sec 3⁡x(sin 2⁡x+cos⁡x+sin⁡x+sin⁡x.cos⁡x)d x? Let (g∘f)(x)−f(x)=x 2+2 x−2,f(0)=1(g∘f)(x)−f(x)=x 2+2 x−2,f(0)=1 and g′(x)>1 g′(x)>1 for all real number x.x. How do I find the value of (f∘g)′(1)(f∘g)′(1)? Real numbers x,y,z x,y,z are such that x+y+z z+x=2 and x+y+z y+z=4.x+y+z z+x=2 and x+y+z y+z=4. How do I find the value of the expression x+y+z x+y x+y+z x+y? For n≥2 n≥2 and real numbers 0≤x 1≤x 2≤...≤x n≤n 0≤x 1≤x 2≤...≤x n≤n, what is the least possible value of f(x 1,x 2,...,x n)=√(x 1)2+0 2+√(x 2−x 1)2+1 2+√(x 3−x 2)2+2 2+...+√(x n−x n−1)2+(n−1)2+√(n−x n)2+n 2 f(x 1,x 2,...,x n)=(x 1)2+0 2+(x 2−x 1)2+1 2+(x 3−x 2)2+2 2+...+(x n−x n−1)2+(n−1)2+(n−x n)2+n 2? If x∈(0,π/2)x∈(0,π/2) then, what is the minimum value of tan(x)/cot(x)+cot(x)/csc(x)+csc(x)/sec(x)+sec(x)/tan(x)tan⁡(x)/cot⁡(x)+cot⁡(x)/csc⁡(x)+csc⁡(x)/sec⁡(x)+sec⁡(x)/tan⁡(x)? If x,y x,y are two real numbers such that 4 x 3 y+x y 3=a 4 x 3 y+x y 3=a where a≥0 a≥0, then what is the maximum value of x⋅y x⋅y? What is the minimum value of sin(x)+cos(x)+tan(x)+cot(x)+sec(x)+csc(x)sin⁡(x)+cos⁡(x)+tan⁡(x)+cot⁡(x)+sec⁡(x)+csc⁡(x)? If positive real numbers x,y x,y satisfy 2(x+y)=1+x y,2(x+y)=1+x y, then how do I find the minimum value of expression A=x+1 x+y+1 y A=x+1 x+y+1 y? For every real numbers a,b such that a 2+b 2−4 a−6 b=2 a 2+b 2−4 a−6 b=2, what is the minimum and the maximum value of the expression √a 2+b 2−8 a−10 b+41 a 2+b 2−8 a−10 b+41 ? What symbol is bigger than " {}" in Math? Eg. {X+ [X+(X+X) +X] +X}. Related questions What is the minimum value of \|sin x+cos x+tan x+csc x+sec x+cot x\|\|sin⁡x+cos⁡x+tan⁡x+csc⁡x+sec⁡x+cot⁡x\|? How do I evaluate the minimum value of \|sin x+cos x+tan x+cot x+sec x+csc x\|\|sin⁡x+cos⁡x+tan⁡x+cot⁡x+sec⁡x+csc⁡x\| for real number x x not multiple of π 2 π 2? If x,y x,y are real numbers, then how do I find the minimum value of the expression max{1,\|x+y\|,\|x y\|}max{1,\|x\|}max{1,\|y\|}max{1,\|x+y\|,\|x y\|}max{1,\|x\|}max{1,\|y\|}? How do I find the minimum value of ∣∣sin x+cos x+tan x+cot x+sec x+cosec x∣∣\|sin⁡x+cos⁡x+tan⁡x+cot⁡x+sec⁡x+cosec⁡x\| for real numbers x x not multiple of π 2 π 2? Given that 5 r+4 s+3 t+6 u=100,5 r+4 s+3 t+6 u=100, where r≥s≥t≥u≥0 r≥s≥t≥u≥0 are real numbers, how do I find, with proof, the maximum and minimum possible values of r+s+t+u r+s+t+u? What is the value of ∫e sec x.sec 3 x(sin 2 x+cos x+sin x+sin x.cos x)d x∫e sec⁡x.sec 3⁡x(sin 2⁡x+cos⁡x+sin⁡x+sin⁡x.cos⁡x)d x? 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12732	https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_Introductory_Physics_-_Building_Models_to_Describe_Our_World_(Martin_Neary_Rinaldo_and_Woodman)/10%3A_Linear_Momentum_and_the_Center_of_Mass/10.02%3A_Collisions	10.2.1 10.2.2 10.2.3 10.2.1 10.2.4 10.2.5 Skip to main content 10.2: Collisions Last updated : Mar 28, 2024 Save as PDF 10.1: Momentum 10.3: The center of mass Page ID : 19430 ( \newcommand{\kernel}{\mathrm{null}\,}) In this section we go through a few examples of applying conservation of momentum to model collisions. Collisions can loosely be defined as events where the momenta of individual particles in a system are different before and after the event. We distinguish between two types of collisions: elastic and inelastic collisions. Elastic collisions are those for which the total mechanical energy of the system is conserved during the collision (i.e. it is the same before and after the collision). Inelastic collisions are those for which the total mechanical energy of the system is not conserved. In either case, to model the system, one chooses to define the system such that there are no external forces on the system so that total momentum is conserved. Inelastic collisions In this section, we give a few examples of modelling inelastic collisions. Inelastic collisions are usually easier to handle mathematically, because one only needs to consider conservation of momentum and does not use conservation of energy (which usually involves equations that are quadratic in the speeds because of the kinetic energy term). Example 10.2.110.2.1 You (mass msms) and your friend (mass mfmf) face each other on ice skates on an ice surface that is slippery enough that friction can be considered negligible, as shown in Figure 10.2.110.2.1. You shove your friend away from you so that he moves with velocity →vfv⃗ f away from you (the velocity is measured relative to the ice). Is the collision elastic? What is your speed relative to the ice after you shoved your friend? Solution We can consider the system as being comprised of you and your friend. There are no net external forces on the system (gravity and normal forces cancel each other), so the momentum of the system will be conserved. The mechanical energy will not be conserved. You had to use chemical potential energy stored in your muscles to shove your friend. Thus, external energy (i.e. not mechanical energy from you or your friend) was injected into the system, and we should expect the total mechanical energy to be larger after the collision. Before the collision, both you and your friend have zero speed, and thus zero kinetic energy and zero momentum. After the collision, your friend has a velocity →vfv⃗ f. We can use conservation of total momentum, →PP⃗ , to determine your velocity, →vsv⃗ s, after the collision. →P=→P′0=ms→vs+mf→vf∴→vs=−mfms→vf where primes (′) denote a quantity after the collision. We find that your velocity is in the opposite direction from that of your friend. Before the collision, the mechanical energy, E, of the system is zero (we can ignore gravitational potential energy, since everything is in the horizontal plane). After the collision, the mechanical energy, E′, is: E′=12msv2s+12mfv2f which is clearly bigger than the mechanical energy before the collision (i.e. 0), as we suspected it would be. Discussion We find that you recoil in the opposite direction, which makes sense. If you push your friend in one direction, Newton’s Third Law says that your friend pushes you in the opposite direction. Your speed furthermore depends on the ratio of your friend’s mass to yours. This also makes sense, because if you both feel the same force, the person with the smallest mass will have the highest speed; if your mass is higher than your friend’s, then your speed after the collision will be smaller than your friend’s. We also saw that mechanical energy was not conserved. In terms of energy, we can explain this by saying that you burned up chemical potential energy stored in your muscles in order to shove your friend. Because we included both you and your friend in the system, the shove was an internal force and momentum is conserved. Of course, if we had considered only you as the system, then your momentum would not have been conserved during the collision. The type of collision that we described here is also sometimes called an “explosion”. You can imagine all of the parts that make up a bomb as small particles. When the bomb explodes, chemical potential energy is converted into the kinetic energy of the bomb fragments. If you consider all of the particles/fragments of the bomb as a system, then the total momentum of all of the bomb fragments is conserved (and equal to zero if the bomb was initially at rest). Again, mechanical energy would not be conserved (and would increase) as the chemical potential energy is converted into mechanical energy. Example 10.2.2 A proton of mass mp and initial velocity →vp collides inelastically with a nucleus of mass mN at rest, as shown in Figure 10.2.2. A coordinate system is set up as shown, such that the initial velocity of the proton is in the x direction. After the collision, the proton’s speed is measured to be v′p and its velocity vector is found to make an angle θ with the x axis as shown. What is the velocity vector of the nucleus after the collision? Assume that the collision takes place in vacuum. Solution As a system, we consider the proton and the nucleus together, so that the total momentum of the system is conserved during the collision, as no other external forces are exerted on the two particles (since they are in vacuum). Because momentum is a vector, each component of the total momentum, →P, is conserved during the collision: →P=→P′∴Px=P′x∴Py=P′y where, as usual, primes (′) denote quantities after the collision. After the collision, both particles will have velocity vectors that have x and y components. Let the velocity vector of the nucleus after the collision be →v′N and let ϕ be the angle that it makes with the x axis, as shown in Figure 10.2.2. We can start by considering the conservation of the x component of the total momentum. The initial and final momenta in the x direction are given by: Px=mpvpP′x=mpv′pcosθ+mNv′Ncosϕ∴mpvp=mpv′pcosθ+mNv′Ncosϕ which gives us a first equation to determine the final velocity of the nucleus. The y component of the total momentum before the collision is zero since we chose the coordinate system such that the initial velocity of the proton is in the x direction. The initial and final momenta in the y direction are given by: Py=0P′y=mpv′psinθ−mNv′Nsinϕ∴mpv′psinθ=mNv′Nsinϕ which gives us a second equation to solve for the velocity of the nucleus. With the two equations from momentum conservation, we can solve for the magnitude and direction of the velocity of the nucleus. From the y component of momentum conservation, we can find an expression for the speed of the nucleus: mpv′psinθ=mNv′Nsinϕ∴v′N=mpmNv′psinθ1sinϕ which we can substitute into the x equation for momentum conservation to solve for the angle ϕ: mpvp=mpv′pcosθ+mNv′Ncosϕmpvp=mpv′pcosθ+mNmpmNv′psinθcosϕsinϕvp=v′pcosθ+v′psinθ1tanϕ∴tanϕ=v′psinθvp−v′pcosθ If we were given numbers for the initial and final speed of the proton, as well as the angle θ, we would be able to find a value for the angle ϕ, which we could then use to determine the final speed of the nucleus: v′N=mpmNv′psinθ1sinϕ Discussion: By using the conservation of momentum equation and writing out the x and y components, we were able to find two equations to determine the magnitude and direction of the nucleus’ velocity after the collision. In the limit where mN>>mp, the final speed of the nucleus would be very small (close to zero). Elastic collisions In this section, we give a few examples of modeling elastic collisions. Even though it is mechanical energy that is conserved in an elastic collision, one can almost always simplify this to only kinetic energy being conserved. If a collision takes place in a well localized position in space (i.e. before and after the collision are the same point in space), then the potential energies of the objects involved will not change, thus any change in their mechanical energy is due to a change in kinetic energy. Example 10.2.3 A block of mass M moves with velocity →vM in the x direction, as shown in Figure 10.2.3. A block of mass m is moving with velocity →vm also in the x direction and collides elastically with block M. Both blocks slide with no friction on the horizontal surface. What are the velocities of the two blocks after the collision? Solution Because this is an elastic collision, both the total momentum and total mechanical energy are conserved. Equating the total momentum before and after the collision, and considering only the x component gives the following equation: →P=→P′MvM+mvm=Mv′M+mv′m where the primes (′) correspond to the quantities after the collision. Note that, in principle, the x components of the velocities (vM, v′M, vm, v′m) could be negative numbers if the corresponding block is moving in the negative x direction. For the mechanical energy of the two blocks, we only need to consider their kinetic energy since their gravitational potential energies are the same before and after the collision on the horizontal surface. The total mechanical energy of the system, before and after the collision is given by: E=E′12Mv2M+12mv2m=12Mv′2M+12mv′2m∴Mv2M+mv2m=Mv′2M+mv′2m where we canceled the factor of one half in the last line. This gives two equations (conservation of energy and momentum) and two unknowns (the two speeds after the collision). This is not a linear system of equations, because the equation from conservation of energy is quadratic in the speeds. The following method allows many models for elastic collisions between two particles to be solved easily by converting the quadratic equation from energy conservation into an equation that is linear in the speeds. First, write both equations so that the quantities related to each particle are on opposite sides of the equation. For momentum, this gives: MvM+mvm=Mv′M+mv′m ∴M(vM−v′M)=m(v′m−vm) For conservation of energy, this gives: Mv2M+mv2m=Mv′2M+mv′2m ∴M(v2M−v′2M=M(v′2m−v2m) which we can re-write as: M(v2M−v′2M)=M(v′2m−v2m)M(vM−v′M)(vM+v′M)=M(v′m−vm)(v′m+vm) We can then divide Equation 10.2.3 and 10.2.4 by Equation 10.2.1 and 10.2.2: M(vM−v′M)(vM+v′M)M(vM−v′M)=M(v′m−vm)(v′m+vm)m(v′m−vm)∴vM+v′M=v′m+vm which gives us an equation that is much easier to work with, since it is linear in the speeds. If we re-arrange this last equation back so that quantities before and after the collision are on different sides of the equality: vM−vm=−(v′M−v′m) we can see that the relative speed between M and m is the same before and after the collision. That is, if block M “saw” block m approaching with a speed of 3m/s before the collision, it would “see” block m moving away with speed 3m/s after the collision, regardless of the actual directions and velocities of the block, if the collision was elastic. By using this equation with the original conservation of momentum equation, we now have two equations and two unknowns that are easy to solve: vM−vm=−(v′M−v′m)MvM+mvm=Mv′M+mv′m Solving for v′m in both equations gives: vM−vm=−(v′M−v′m)∴v′m=vM+v′M−vmMvM+mvm=Mv′M+mv′m∴v′m=1m(MvM+mvm−Mv′M) Equating the two expressions for v′m allows us to solve for v′M: 1m(MvM+mvm−Mv′M)=vM+v′M−vmMvM+mvm−Mv′M=mvM+mv′M−mvm(M−m)vM+2mvm=(M+m)v′M∴v′M=M−mM+mvM+2mM+mvm One can easily solve for the other speed, v′m: ∴v′m=m−MM+mvm+2MM+mvM And writing these together: v′M=M−mM+mvM+2mM+mvmv′m=m−MM+mvm+2MM+mvM Discussion The formulas that we obtained above are valid for any one dimensional elastic collision. Exercise 10.2.1 Two trains of equal masses collide elastically on a track. If train A had a speed v and train B was at rest, what are the speeds of the trains after the collision? Both trains A and B travel away from each other with speeds 12v. Train A will be at rest and train B will move away with a speed v. Both trains A and B will stick together and move at a speed of v. Train B will be at rest and train A will move away at a speed of v. Answer Example 10.2.4 A proton of mass m and initial velocity →v1 collides elastically with a second proton that is at rest. After the collision, the two protons have velocities →v′1 and →v′2, as shown in Figure 10.2.4. Show that the velocity vectors of the two protons are perpendicular after the collision. Solution This example highlights a particular feature of elastic collisions when the two objects have the same mass and one of the objects is initially at rest. The conservation of momentum for the system comprised of the two protons can be written as: m→v1=m→v′1+m→v′2→v1=→v′1+→v′2 where the left hand side corresponds to the initial total momentum and the right hand side to the total momentum after the collision. In the second line, we canceled out the mass, and obtained a vector relation between the velocity vectors. We can graphically illustrate the vector relation as in Figure 10.2.5 which shows the triangle that is formed by adding the two outgoing velocity vectors to obtain the initial velocity vector. Conservation of kinetic energy for the collision can be written as: 12mv21=12mv′21+12mv′22v21=v′21+v′22 where the left hand side corresponds to the initial kinetic energy and the right hand side to the final kinetic energy. We canceled the mass and factor of one half in the second line. This last equation gives a relation between the magnitudes of the velocity vectors. By comparing the equation above to Pythagoras’ theorem, and by inspecting the triangle in Figure 10.2.5, it is clear that the triangle must be a right angle triangle, and thus that →v′1 and →v′2 must be perpendicular. Frames of reference review topics Before proceeding, you may wish to review Sections 3.4 and 4.1 on expressing velocities in different frames of reference. Because the momentum of a particle is defined using the velocity of the particle, its value depends on the reference frame in which we chose to measure that velocity. In some cases, it is useful to apply momentum conservation in a frame of reference where the total momentum of the system is zero. For example, consider two particles of mass m1 and m2, moving towards each other with velocities →v1 and →v2, respectively, as measured in a frame of reference S, as illustrated in Figure 10.2.6. In the frame of reference S, the total momentum, →P, of the two particles can be written: →P=m1→v1+m2→v2 Consider a frame of reference, S′, that is moving with velocity, →vCM, relative to the frame of reference S. In that frame of reference, the velocities of the two particles are different and given by: →v′1=→v1−→vCM→v′2=→v2−→vCM The total momentum, →P′, in the frame of reference S′ is then given by1: →P′=m1→v′1+m2→v′2=m1(→v1−→vCM)+m2(→v2−→vCM)=m1→v1+m2→v2−(m1+m2)→vCM We can choose the velocity of the frame S′, →vCM, such that the total momentum in that frame of reference is zero: →P′=0m1→v1+m2→v2−(m1+m2)→vCM=0∴→vCM=m1→v1+m2→v2m1+m2 This “special” frame of reference, in which the total momentum of the system is zero, is called the “center of mass frame of reference”. The velocity of center of mass frame of reference can easily be obtained if there are N particles involved instead of two: ∴→vCM=m1→v1+m2→v2+m3→v3+...m1+m2+m3+...=∑mi→vi∑mi Again, you should note that because the above equation is a vector equation, it represents one equation per component of the vectors. For example, the x component of the velocity of the center of mass frame of reference is given by: ∴vCMx=m1v1x+m2v2x+m3v3x+…m1+m2+m3+…=∑mivix∑mi Example 10.2.5 In the frame of reference of a lab, a block of mass m has a velocity →v1 directed along the positive x axis and is approaching a second block of mass m that is at rest (→v2=0), as shown in Figure 10.2.7. What is the velocity of the center of mass frame? What is the velocity of each block in the center of mass frame? Verify that the total momentum is zero in the center of mass frame. Solution Since this is a one dimensional situation, we only need to evaluate the x component of the velocity of the center of mass: →vCM=m1→v1+m2→v2m1+m2∴vCMx=m1v1x+m2v2xm1+m2=mv1+m(0)m+m=12v1 The center of mass frame of reference is thus also moving along the positive direction of the x axis, but with a speed that is half of that of the moving block. In the center of mass frame of reference, it appears that the block on the left is slower than in the lab frame and that the block on the right is moving in the negative x direction. The velocities of the two blocks in the center of mass frame of reference are given by: v′1=v1−vCMx=12v1v′2=(0)−vCMx=−12v1 Thus, in the reference frame of the center of mass, the two block are approaching each other with the same speed (v1/2), which is only the case because the two blocks have the same mass. The blocks, as viewed in the center of mass frame of reference, are shown in Figure 10.2.8. Clearly, the total momentum is zero in the center of mass frame of reference: →P′=m→v′1+m→v′2=m(12→v1−12→v1)=0 Discussion As we have seen, in the center of mass frame of reference the total momentum is zero. If there are only two particles, and they have the same mass, then, in the center of mass frame of reference, they both have the same speed and move either towards or away from each other. Footnotes 1.Note that we are using primes (′) to denote quantities in a different reference frame, not after a collision. 10.1: Momentum 10.3: The center of mass
12733	https://www.youtube.com/watch?v=m2egqvfpz_Y	【解題】連乘的簡便計算應用乘法交換律與結合律）均一教育平台 Junyi Academy 184000 subscribers 10 likes Description 1022 views Posted: 2 Jan 2025 Transcript:
12734	https://www.youtube.com/watch?v=1WEd2yXEZWI	How To Represent Numbers in Standard, Expanded, and Word Form Miacademy & MiaPrep Learning Channel 328000 subscribers 2 likes Description 226 views Posted: 28 Jul 2025 Let's find a new way to represent numbers! In this math lesson for 2nd graders, students will review place value and represent numbers in standard, expanded, and word form. This lesson is from Miacademy’s Math: Level C course. Check out our playlists for more 2nd-grade math lessons! We hope you are enjoying our large selection of engaging core & elective K-12 learning videos. New videos are added all the time - make sure you come back often to learn more! If you'd like us to cover any additional topics, please let us know. For practice, assessment, and many interactive activities that go along with each video, as well as a teacher/parent dashboard, go to Miacademy.co for Grades K-8 or Miaprep.com for Grades 9-12! 🔗 Discount Link: Transcript: Math Quest: Write 114 in standard, expanded, and word form. Sydni: This is so exciting—our first math quest! We need to solve this math quest to get one step closer to the next level. I bet we already have some knowledge that can help us solve it. Let's get started! [Music] Our quest today is to write 114 in standard, expanded, and word form. One of these sounds familiar, but the others don’t. Let's think about what we already know that can help us solve this problem. We know there are many ways we can represent numbers. One is with numerals, like this. Anytime we write numbers with digits like this, we are representing it in standard form. When numbers are in standard form, we can use what we know about place value to understand how many ones, tens, or hundreds the number has. For example, the number 108 in standard form looks like this. It has three digits, each with a different value. Do you remember the value of this digit here? This is the ones place. And this one over here? Hundreds. And in the middle? Tens. Just by looking at this number in standard form, we know it has one hundred, zero tens, and eight ones. Okay, we've covered standard form. But how else can we represent numbers? We can also represent numbers using pictorial or visual models. We can represent numbers with pictures like this, or even in base-10 models like this. Let's represent 106 with a base-10 model. Because we understand place value, we can use 106 in its standard form to help us. By studying the digits in its standard form, we can tell that this number has six ones, zero tens, and one hundred. Awesome! Now we have a pictorial model to represent 106. Can you think of another way to represent numbers? So far, we have standard form, a pictorial model... Aha! The math quest also mentioned expanded form. Let's give that a shot. When we represent a number in expanded form, we decompose—or break—a number apart according to place value. So in a two-digit number, we would break it into two parts: tens and ones. The important part here is to think about the number that each place is representing. For example, to represent the number 47 in expanded form, you might take the 4 from the tens place and the 7 from the ones place. But wait—does 4 plus 7 equal 47? No, it equals 11. The number 47 has four tens. So think—what number are those four tens representing? 40. 47 in expanded form is 40 plus 7. Okay, we represented a two-digit number in expanded form. But what about a three-digit number? 120 has three digits, so we'll decompose it into three parts: hundreds, tens, and ones. Starting with the greatest place first—how many hundreds does 120 have? One, and that represents 100. And how many tens? Two, which represents the number 20. Finally, how many ones? Zero. Since there aren't any ones in 120, we can say that 120 in expanded form is simply 100 plus 20. Great job! We have one final way we need to practice representing numbers: word form. Let's take a moment to think—what could it mean to represent a number in word form? Hm... Are you thinking what I'm thinking? Could it be that simple? When we communicate with each other, we often use words just like these. When we write about numbers, we can also use words. Let's try to represent the number 78 with words. Here it is in standard form, and here is seventy-eight in word form. Wow! Let's try another one. Here is 109. What do you think this would be in word form? One hundred nine. Great job! Now that we've practiced word form, I think we're ready to solve our math quest. Can you represent 114 in standard, expanded, and word form? Pause this video to record your answer on your math quest. To represent 114 in standard form, we need to write it with digits—just like it's written in the math quest. For expanded form, we need to break it into parts based on place value: 100 plus 10 plus 4 equals 114. And finally, for word form, we need to write the words for each number: one hundred fourteen. Do your answers match? Awesome! Now it's time to input and check our answer. Sweet! We solved our first math quest and we learned how to represent numbers in word form. Remember—every problem has a solution, and we've got the skills to solve it. See you next time! [Music]
12735	https://www.teach-me-mommy.com/counting-with-robots/	Counting with Robots - Teach Me Mommy Search Teach Me Mommy Playful and easy activities with the aim to teach Kids Activities Teach my Baby Teach my Toddler Teach my Preschooler Teach me Fine Motor Skills Teach me Literacy/Numeracy Teach me Rhymes Teach me Sensory Teach me Science Holidays Valentine’s Day Easter Christmas Birthdays Gifts DIY Busy Bags Keepsakes Toys Recipes Food Recipes South African Recipes Play Recipes SHOP Counting with Robots February 16, 2017 by Nadia van ZylLeave a Comment We are a participant in the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for us to earn fees by linking to Amazon.com and affiliated sites. We did some fun counting, but not just counting, counting with robots! Learning while playing is the best way to learn, and that is exactly what this activity is about. My 4 year old boy took out the robot toys and used one robot to count a number line. We used ourupcycled spool chalk tableto write a number line. The robot then counted while moving forwards and backwards. Then, I drew six circles(because we have six robots) and played that the robots were put into jail. The number one ‘jail’ has only place for one robot, the number two ‘jail’, two robots and so forth. The jail idea was purely my 4 year olds idea, lol! My boy had to recognize the number and add the corresponding number of robots in the circle jail. Language development took place incidentally while using mathematical vocabulary like: “more”, “less”, “take away”, “altogether” etc. A simple number activity but done in a playful manner. The best way to learn, while playing! More Robot activities you will want to see: FROM LEFT TO RIGHT: Name Puzzle Robot // Still Playing School Robot Coloring Matching Clip Cards// Modern Preschool Robot Sensory Writing Tray // Fun-A-Day! Beginning Blends Robot Puzzles // Mom Inspired Life Robot Preposition Posters // Liz’s Early Learning Spot Free Robot Bingo// Powerful Mothering Robot Sums of Ten Puzzles // The Kindergarten Connection Robot Subtraction Cards// The STEM Laboratory Robot Coverall Game // Recipe for Teaching Robot CVC Word Puzzles // A Dab of Glue Will Do Robot Reading Buddy// Play and Learn Every Day Robot Beginning Digraph Puzzles // Letters of Literacy Robot Digraph Scratch// Adventures and Play Robot Rhyming Mats // Fairy Poppins Robot Addition Cards// Playdough to Plato Robot Pattern Task Box Activities // My Creative Inclusion Robot Beginning Sound Match // Sara J. Creations Related Posts Learning Activities using Craft Sticks Craft sticks (or you can recycle real popsicle sticks) are… Gingerbread Cookies Counting Make a gingerbread cookies counting activity tray from cardboard and… Simple Counting Fun Activity This simple counting activity is easy to setup, can be… Feel free to share! 74 shares Share Tweet Pin Filed Under: default, Teach me Literacy/NumeracyTagged With: counting, robots « Whiteboard Hack for Maths Activities Missing Letters Clip Sticks » Leave a Reply You must be logged in to post a comment. Welcome! Hi! I'm Nadia! A teacher and a mommy of two, who loves to be creative and share easy activities you can do at home with your kids too! Read More… Visit our TpT Store Let’s Connect! Categories Categories More About us/Contact Contact Freebies SHOP Sign-up to our Newsletter Thanks Freebies Sign-Up FREEBIES Access Sign up with your email address to receive access to our Freebies Page. (Remember to check your Junk Folder too.) First F name F c irst name Fi B rst name Fir v st name Last na v me r L ast name La n st name La m st name p E mail address Emai I l address Email O address Z E mail address Subscribe Thank you for subscribing! Copyright © 2025 · Delightful theme by Restored 316 Copyright ©2025 · Delightful Pro Theme on Genesis Framework · WordPress · Log in
12736	https://ti.inf.ethz.ch/ew/courses/Geo23/lecture/gca23-2-1.pdf	Chapter 2 Plane Embeddings Graphs can be represented in various ways, for instance, as an adjacency matrix or using adjacency lists. In this chapter we explore another class of representations that are quite diﬀerent in nature, namely geometric representations. In a geometric representation, vertices and edges are represented by geometric objects, for example points and curves. This approach is appealing because it succinctly visualizes a graph along with its many properties. We have many degrees of freedom in selecting the geometric objects and the details of their geometry. This freedom allows us to tailor the representation to meet speciﬁc goals, such as emphasizing certain structural aspects of the graph at hand or reducing the complexity of the obtained representation. The most common geometric graph representation is a drawing, where vertices are mapped to points and edges to curves in R2. It is desirable to make such a map injective by avoiding edge crossings, both from a mathematically aesthetic viewpoint and for the sake of the practical readability. Those graphs that allow such an embedding into the Euclidean plane are known as planar. Our goal is to study the interplay between abstract planar graphs and their plane embeddings. Speciﬁcally, we want to answer the following questions: What is the combinatorial complexity (that is, the number of edges and faces) of planar graphs? Under which conditions are plane embeddings unique (up to a certain sense of equivalence)? How can we represent plane embeddings in a data structure? What is the geometric complexity (that is, the encoding size of the geometric objects used to represent vertices and edges) of plane embeddings? Most deﬁnitions we use directly extend to multigraphs. But for simplicity, we use the term “graph” throughout. 13 Chapter 2. Plane Embeddings Geometry: C&A 2022 2.1 Drawings, Embeddings and Planarity A curve is a set C ⇢R2 of the form {γ(t) : 0 6 t 6 1}, where γ : [0, 1] ! R2 is a continuous function. The function γ is called a parameterization of C. The points γ(0) and γ(1) are the endpoints of the curve. A curve is closed if γ(0) = γ(1). A curve is simple if it admits a parameterization γ that is injective on [0, 1]; for a closed simple curve we allow as an exception that γ(0) = γ(1). The following famous theorem describes an important property of the plane. A proof can, for instance, be found in the book of Mohar and Thomassen . Theorem 2.1 (Jordan). Any simple closed curve C partitions the plane into exactly two regions (connected open sets), each bounded by C. Figure 2.1: Left: a simple closed curve in the plane and two points in one of its faces. Right: a simple closed curve that does not disconnect the torus. Observe that, for instance, on the torus there are simple closed curves that do not disconnect the surface, and thus the theorem does not hold there. Drawings. As a ﬁrst criterion for a reasonable geometric representation of a graph, we would like to have a clear separation between diﬀerent vertices and also between a vertex and nonincident edges. Formally, a drawing of a graph G = (V, E) in the plane is a function f that assigns a point f(v) 2 R2 to every vertex v 2 V and a simple curve f(uv) with endpoints f(u) and f(v) to every edge uv 2 E, such that (1) f is injective on V and (2) f(uv) \ f(V) = {f(u), f(v)}, for every edge uv 2 E. A common point f(e) \ f(e0) between two curves that represent distinct edges e, e0 2 E is called a crossing if it is not a common endpoint of e and e0. Commonly, when discussing a drawing of a graph G = (V, E), we do not diﬀerentiate a vertex/an edge from its geometric realization. That is, a vertex v 2 V is identiﬁed with the point f(v), and an edge e 2 E is identiﬁed with the curve f(e). For instance, the last 14 Geometry: C&A 2022 2.1. Drawings, Embeddings and Planarity sentence in the previous paragraph may be phrased as “A common point of two edges is called a crossing if it is not their common endpoint.” Often it is convenient to make additional assumptions about edge intersections in a drawing. For example, we may demand nondegeneracy in the sense that no three edges can meet at a single crossing, or that any two edges can intersect at only ﬁnitely many points. Planar vs. plane. A graph is planar if it admits a drawing in the plane without crossings. Such a drawing is also called a crossing-free drawing or a (plane) embedding of the graph. A planar graph together with a particular plane embedding is called a plane graph. Note the distinction between “planar” and “plane”: the former refers to an ab-stract graph and indicates the possibility of an embedding, whereas the latter refers to a concrete embedding (Figure 2.2). Figure 2.2: A planar graph (left) and a plane embedding of it (right). A geometric graph is a graph together with a drawing in which all edges are straight-line segments. Note that such a drawing is fully determined by the vertex positions. A plane graph which is also geometric is called a plane straight-line graph (PSLG). On the other hand, a plane graph whose edges are arbitrary simple curves is emphasized as topological plane graph. The faces of a plane graph G are the maximally connected regions of R2 \ G, that is, the plane without the points occupied by the embedding (as the image of a vertex or an edge). Each embedding of a ﬁnite graph has exactly one unbounded face, also called outer or inﬁnite face. Using stereographic projection, we could show that any face can be swapped out to serve as the unbounded face: Theorem 2.2. If a graph G has a plane embedding in which some face is bounded by a cycle (v1, . . . , vk), then G also has a plane embedding in which the unbounded face is bounded by the cycle (v1, . . . , vk). Proof Sketch. Take a plane embedding Γ of G and map it to the sphere using stereo-graphic projection: Imagine R2 being the x/y-plane in R3 and place a unit sphere S whose south pole touches the origin. We establish a bijection between R2 and S \ {n}, where n := (0, 0, 2) is the north pole position: A point p 2 R2 is mapped to the intersec-tion p0 of the segment pn and S, see Figure 2.3. The map is continuous, so it preserves incidence between vertices, edges and faces. 15 Chapter 2. Plane Embeddings Geometry: C&A 2022 n p p0 (a) Three-dimensional view. n p p0 q q0 0 (b) Cross-section view. Figure 2.3: Stereographic projection. Consider the resulting embedding Γ 0 of G on S: The inﬁnite face of Γ corresponds to the face of Γ 0 that contains the north pole n of S. Now rotate the embedding Γ 0 on S such that the desired face contains n. Mapping back to the plane using stereographic projection results in an embedding in which the desired face is the outer face. Exercise 2.3. Consider the plane graphs depicted in Figure 2.4. For both graphs give a plane embedding in which the cycle (1, 2, 3) bounds the outer face. 2 3 5 4 1 (a) 2 3 5 4 1 6 7 8 (b) Figure 2.4: Make (1, 2, 3) bound the outer face. Duality. Every plane graph G has a dual G⇤whose vertices are the faces of G. For every edge in G, we connect its two incident faces by an edge in the dual G⇤. Note that in general, G⇤is a multigraph (with loops and multiple edges) and may depend on the embedding. So an abstract planar graph G may have several nonisomorphic duals; see Figure 2.5 for an example. If G is a connected plane graph, then (G⇤)⇤= G. We will see later in Section 2.3 that the dual of a 3-connected planar graph is unique (up to isomorphism). The Euler Formula and its ramiﬁcations. One of the most important tools for planar graphs (and more generally, graphs embedded on a surface) is the Euler–Poincaré Formula. 16 Geometry: C&A 2022 2.1. Drawings, Embeddings and Planarity G1 G1 ⇤ G2 G2 ⇤ Figure 2.5: Two plane drawings G1 and G2 of the same abstract planar graph and their duals G1 ⇤and G2 ⇤with G1 ⇤6' G2 ⇤. (To see this, for instance, count the number of vertices of degree greater than three.) Theorem 2.4 (Euler’s Formula). For every connected plane graph with n vertices, e edges, and f faces, we have n - e + f = 2. Proof. Let G be a connected plane graph with n vertices, e edges, and f faces. Note that e > n - 1 as G is connected. We prove the statement by induction on e-n. In the base case e-n = -1, the graph G is a (plane) tree and contains exactly one (unbounded) face, and so n-e+f = 1+1 = 2 as claimed. In the general case, ﬁx a spanning tree T of G, pick an arbitrary edge e of G \ T, and consider the graph G- = G \ e. By construction it has n vertices and e - 1 edges. We claim that it has f-1 faces. To see this observe that G- ⊃T is connected. In particular, the endpoints of e are connected by a path in G-, which together with e forms a cycle in G. So in G, any two points suﬃciently close to but on opposite sides of e are in diﬀerent faces, whereas they are in the same face of G-. In other words, the two incident faces of e are distinct in G but merged into one in G-. All other faces remain untouched. It follows that G- has f - 1 faces, as claimed. Then by the inductive assumption on G-, we have n - e + f = n - (e - 1) + (f - 1) = 2, which concludes the induction. In particular, this shows that every plane embedding of a planar graph has the same number of faces. In other words, the number of faces is an invariant of an abstract planar graph. It also follows (as the corollary below) that planar graphs are sparse, that is, they have a linear number of edges and faces only. So the asymptotic complexity of a planar graph is already determined by its number of vertices. Corollary 2.5. A simple planar graph on n > 3 vertices has at most 3n - 6 edges and at most 2n - 4 faces. Proof. Without loss of generality we may assume that G is connected. (If not, add edges between components of G until the graph is connected. The number of edges increases and the number of faces remains unchanged.) The statement is easily checked for n = 3, where G is either a triangle or a path and therefore has no more than 3 6 3 · 3 - 6 edges and no more than 2 6 2 · 3 - 4 faces. Next consider a simple connected planar graph G 17 Chapter 2. Plane Embeddings Geometry: C&A 2022 on n > 4 vertices, and ﬁx any plane embedding of it. Denote by E its set of edges and by F its set of faces. Let X = {(e, f) 2 E ⇥F : e bounds f} denote the set of incident edge-face pairs. We count X in two diﬀerent ways. First note that each edge bounds at most two faces and so \|X\| 6 2 · \|E\|. Second note that every face is bounded by at least three edges: If G contains a cycle, then the boundary of every face shall contain a cycle and hence at least three edges. If G is acyclic, then it must be a tree since we assumed it to be connected. Its only face (the outer face) is bounded by all edges; and there are at least three since G contains at least four vertices. In both cases we have \|X\| > 3 · \|F\|. Therefore 3\|F\| 6 2\|E\|. Using Euler’s Formula we conclude that 4 = 2(n - \|E\| + \|F\|) 6 2n - 3\|F\| + 2\|F\| = 2n - \|F\| and 6 = 3(n - \|E\| + \|F\|) 6 3n - 3\|E\| + 2\|E\| = 3n - \|E\| , which yield the claimed bounds. Corollary 2.5 implies that the degree of a “typical” vertex in a planar graph is a small constant. Corollary 2.6. The average vertex degree in a simple planar graph is less than six. Exercise 2.7. Prove Corollary 2.6. There exist several variations of this statement, a few more of which we will encounter during this course. Exercise 2.8. Show that neither K5 (the complete graph on ﬁve vertices) nor K3,3 (the complete bipartite graph where both classes have three vertices) is planar. Exercise 2.9. Let P be a set of n > 3 points in the plane such that the distance between every pair of points is at least one. Show that there are at most 3n - 6 pairs of points in P at distance exactly one. Characterizing planarity. The classical theorems of Kuratowski and Wagner provide a char-acterization of planar graphs in terms of forbidden substructures. A subdivision of a graph G = (V, E) is obtained from G by replacing each edge with a path. Theorem 2.10 (Kuratowski [22, 31]). A graph is planar if and only if it does not contain a subdivision of K3,3 or K5. A minor of a graph G = (V, E) is obtained from G using zero or more edge contrac-tions, edge deletions, and/or vertex deletions. Theorem 2.11 (Wagner ). A graph is planar if and only if it does not contain K3,3 or K5 as a minor. 18 Geometry: C&A 2022 2.1. Drawings, Embeddings and Planarity In some sense, Wagner’s Theorem is a special instance1 of a much more general theorem. Theorem 2.12 (Graph Minor Theorem, Robertson/Seymour ). Every minor-closed family of graphs can be described in terms of a ﬁnite set of forbidden minors. Being minor-closed means that any minor of any graph from the family also belongs to the family. For instance, the family of planar graphs is minor-closed because planarity is preserved under removal of edges and vertices and under edge contractions. Exercise 2.13. A graph is 1-planar if it admits a drawing in the plane in which every edge has at most one crossing. Prove or disprove: The family of 1-planar graphs is minor-closed. The Graph Minor Theorem is a celebrated result established by Robertson and Sey-mour in a series of twenty papers, see also the survey by Lovász . They also describe an O(n3) algorithm (with horrendous constants, though) to decide whether a graph on n vertices contains a ﬁxed (constant-size) minor. As a consequence, every minor-closed property can be tested in polynomial time. Later, Kawarabayashi et al. showed that this problem can be solved in O(n2) time. Unfortunately, the Graph Minor Theorem is nonconstructive in the sense that in general we do not know how to obtain the set of forbidden minors for a given family. For instance, for the family of toroidal graphs (graphs that can be embedded without crossings on the torus) more than 160000 forbidden minors are known, and the theorem tells us that the number is ﬁnite, but we still do not know the concrete number. So while we know that there exists a quadratic time algorithm to test membership for minor-closed families, we have no idea what such an algorithm looks like in general. Graph families other than planar graphs for which the forbidden minors are known include forests (free of K3 minors) and outerplanar graphs (free of K2,3 and K4 minors). A graph is outerplanar if it admits a plane embedding in which all vertices appear on the outer face (Figure 2.6). Figure 2.6: An outerplanar graph (left) and a plane embedding of it in which all vertices are incident to the outer face (right). Exercise 2.14. (a) Give an example of a 6-connected planar graph or argue that no such graph exists. 1It is more than just a special instance because it also speciﬁes the forbidden minors explicitly. 19 Chapter 2. Plane Embeddings Geometry: C&A 2022 (b) Give an example of a 5-connected planar graph or argue that no such graph exists. (c) Give an example of a 3-connected outerplanar graph or argue that no such graph exists. Planarity testing. To test a given graph for planarity we do not have to contend ourselves with a quadratic-time algorithm. In fact, there exist a number of diﬀerent linear time algorithms that decide if a given abstract graph is planar; all of them—from a very high-level point of view—can be regarded as an annotated depth-ﬁrst-search. The ﬁrst such algorithm was described by Hopcroft and Tarjan , while the current state-of-the-art is probably among the “path searching” method by Boyer and Myrwold and the “LR-partition” method by de Fraysseix et al. . Although the overall idea in all these approaches is easy to convey, many technical details make an in-depth discussion rather painful to go through. 2.2 Graph Representations There are two standard representations for an abstract graph G = (V, E) on n = \|V\| vertices. For the adjacency matrix representation we consider the vertices to be ordered as V = {v1, . . . , vn}. The adjacency matrix of an undirected graph is a symmetric n ⇥n-matrix A = (aij)16i,j6n where aij = aji = 1, if {vi, vj} 2 E, and aij = aji = 0 otherwise. Storing such a matrix explicitly requires ⌦(n2) space, but it allows testing in constant time whether or not two given vertices are adjacent. In an adjacency list representation, we store for each vertex a list of its neighbors in G. This requires only O(n+\|E\|) storage, which is better than for the adjacency matrix in case that \|E\| = o(n2). On the other hand, the adjacency test for two given vertices is not a constant-time operation, because it requires a search in one of the lists. Depending on the implementation of the lists, the search time ranges from O(d) (for an unsorted list) to O(log d) (for a sorted dynamic data structure such as a balanced search tree), where d is the minimum degree of the two vertices. Both representations have their merits. The choice typically depends on what one wants to do with the graph. When dealing with embedded graphs, however, additional information about the embedding is needed beyond the pure incidence structure of the graph. The next section discusses a standard data structure to represent embedded graphs. 2.2.1 The Doubly-Connected Edge List The doubly-connected edge list (DCEL) is a data structure to represent a plane graph in such a way that it is easy to traverse and to manipulate. To avoid complications, let us discuss only connected graphs that contain at least two vertices. It is not hard to extend the data structure to be able to represent all plane graphs. We also assume 20 Geometry: C&A 2022 2.2. Graph Representations that we deal with a straight-line embedding and thus the geometry of edges is deﬁned by the positions of their endpoints already. For more general embeddings, the geometric description of edges has to be stored in addition. The main building block of a DCEL is a list of halfedges. Every actual edge is split into two halfedges going in opposite direction, and these are called twins, see Figure 2.7. Along the boundary of each face, halfedges are oriented counterclockwise, that is, the face always stays to the left. h next(h) prev(h) twin(h) target(h) face(h) Figure 2.7: A halfedge in a DCEL. A DCEL also stores a list of vertices and a list of faces. These three lists are unordered but interconnected by various pointers. A vertex v stores a pointer halfedge(v) to an arbitrary halfedge originating from v. Every vertex also records its coordinates point(v), that is, the point it is mapped to in the embedding. A face f stores a pointer halfedge(f) to an arbitrary halfedge within the face. A halfedge h stores ﬁve pointers: a pointer target(h) to its target vertex, a pointer face(h) to its incident face, a pointer twin(h) to its twin halfedge, a pointer next(h) to the halfedge following h along the boundary of face(h), and a pointer prev(h) to the halfedge preceding h along the boundary of face(h). A constant amount of information is stored for every vertex, (half-)edge, and face of the graph. Therefore the whole DCEL needs storage proportional to \|V\| + \|E\| + \|F\|, which is O(n) for a plane graph with n vertices by Corollary 2.5. This information is suﬃcient for most tasks. For example, traversing all edges around a face f can be done as follows: s halfedge(f) h s do 21 Chapter 2. Plane Embeddings Geometry: C&A 2022 something with h h next(h) while h 6= s Exercise 2.15. Give pseudocode to traverse all edges incident to a given vertex v of a DCEL. Exercise 2.16. Why is the previous halfedge prev(·) stored explicitly whereas the source vertex of a halfedge is not? 2.2.2 Manipulating a DCEL In many applications, plane graphs do not just appear as static objects but rather evolve over the course of an algorithm. Therefore the data structure must allow for eﬃcient updates. These include, but are not limited to, appending new vertices, edges and faces to the corresponding list within the DCEL and—symmetrically—the ability to delete an existing entity. First, it should be easy to add a new vertex v to the graph within a given face f and (as we maintain a connected graph) connect v to an existing vertex u. For such a connection to be valid, we require that the open line segment uv lies completely in f. Given that we need access to both f and u, it would be convenient to pass a halfedge h as an argument, which is supposed to satisfy face(h) = f and target(h) = u. Then our operation becomes add-vertex-at(v, h) Precondition: the open line segment point(v)point(u), where u := target(h), lies completely in f := face(h). Postcondition: the new vertex v has been inserted into f, connected by an edge to u. u v h f . . . . . . (a) before u v h f h1 h2 . . . . . . (b) after Figure 2.8: Add a new vertex connected to an existing vertex u. See also Figure 2.8. It can be realized by manipulating a constant number of pointers as follows. 22 Geometry: C&A 2022 2.2. Graph Representations add-vertex-at(v, h) { h1 a new halfedge h2 a new halfedge halfedge(v) h2 twin(h1) h2 twin(h2) h1 target(h1) v target(h2) u face(h1) f face(h2) f next(h1) h2 next(h2) next(h) prev(h1) h prev(h2) h1 next(h) h1 prev(next(h2)) h2 } Similarly, it should be possible to add an edge between two existing vertices u and v, provided the open line segment uv lies completely within a face f of the graph, see Figure 2.9. Since such an edge insertion splits f into two faces, the operation is called split-face. Again we pass as an argument the halfedge h satisfying face(h) = f and target(h) = u. split-face(h, v) Precondition: v is incident to f := face(h) but not adjacent to u := target(h). The open line segment point(v)point(u) lies completely in f. Postcondition: f has been split by a new edge uv. u v f h (a) before u v f1 h f2 h1 h2 (b) after Figure 2.9: Split a face by an edge uv. The implementation is slightly more complicated compared to add-vertex-at above, be-cause the face f is destroyed and so we have to update the face information of all incident 23 Chapter 2. Plane Embeddings Geometry: C&A 2022 halfedges. In particular, this is not a constant time operation and has complexity pro-portional to the size of f. split-face(h, v) { f1 a new face f2 a new face h1 a new halfedge h2 a new halfedge halfedge(f1) h1 halfedge(f2) h2 twin(h1) h2 twin(h2) h1 target(h1) v target(h2) u next(h2) next(h) prev(next(h2)) h2 prev(h1) h next(h) h1 i h2 loop face(i) f2 if target(i) = v break the loop i next(i) endloop next(h1) next(i) prev(next(h1)) h1 next(i) h2 prev(h2) i i h1 do face(i) f1 i next(i) until target(i) = u delete the face f } In a similar fashion one can realize the inverse operation join-face(h) that removes the edge represented by h, thereby joining the faces face(h) and face(twin(h)). It is easy to see that every connected plane graph on at least two vertices can be constructed using the operations add-vertex-at and split-face, starting from an embedding of K2 (two vertices connected by an edge). Exercise 2.17. Give pseudocode for the operation join-face(h). Specify preconditions if needed. 24 Geometry: C&A 2022 2.2. Graph Representations Exercise 2.18. Give pseudocode for the operation split-edge(h), that splits the edge represented by h into two by a new vertex w, see Figure 2.10. u v h f2 f1 (a) before u v w h2 h1 k1 k2 f2 f1 (b) after Figure 2.10: Split an edge by a new vertex. 2.2.3 Graphs with Unbounded Edges In some cases it is convenient to consider plane graphs in which some edges are not mapped to a line segment but to an unbounded curve, such as a ray. This setting is not really much diﬀerent from the one we studied before, except that one special vertex is placed “at inﬁnity”. One way to think of it is in terms of stereographic projection (see the proof of Theorem 2.2). The further away a point in R2 is from the origin, the closer its image on the sphere S gets to the north pole n of S. But there is no way to reach n except in the limit. Therefore, we can imagine drawing the graph on S instead of in R2 and putting the “inﬁnite vertex” at n. All this is just for the sake of a proper geometric interpretation. As far as a DCEL of such a graph is concerned, there is no need to consider spheres or anything beyond what we have discussed. The only diﬀerence to the case with all ﬁnite edges is that there is this special inﬁnite vertex, which does not have any point/coordinates associated to it. Other than that, the inﬁnite vertex is treated in exactly the same way as the ﬁnite vertices: it has in- and out-going halfedges along which the unbounded faces can be traversed (Figure 2.11). Remarks. It is actually not so easy to point exactly to where the DCEL data struc-ture originates from. Often Muller and Preparata are credited, but while they use the term DCEL, the data structure they describe is diﬀerent from what we discussed above and from what people usually consider a DCEL nowadays. Overall, there are a large number of variants of this data structure, which appear under the names winged edge data structure , halfedge data structure , or quad-edge data structure . Kettner provides a comparison of all these with some additional references. 25 Chapter 2. Plane Embeddings Geometry: C&A 2022 1 Figure 2.11: A DCEL with unbounded edges. Usually, we will not show the inﬁ-nite vertex and draw all edges as straight-line segments. This yields a geometric drawing, like the one within the gray box. 2.2.4 Combinatorial Embeddings The basic DCEL omits geometric aspects (that is, positions and shapes of a vertex/edge/face) and only stores incidences and adjacencies between vertices, edges, and faces of an em-bedding. We call such information the combinatorial embedding of the actual plane graph. Conventionally, we write it as a set of face boundaries, where each boundary is encoded as a circular sequence of vertices in counterclockwise order. For instance, the combinatorial embeddings of the plane graphs in Figure 2.12a are (a) : {(1, 2, 3), (1, 3, 6, 4, 5, 4), (1, 4, 6, 3, 2)} , (b) : {(1, 2, 3, 6, 4, 5, 4), (1, 3, 2), (1, 4, 6, 3)} , and (c) : {(1, 4, 5, 4, 6, 3), (1, 3, 2), (1, 2, 3, 6, 4)} . Note that a vertex can appear several times along the boundary of a face (if it is a cut-vertex). This view allows us to compare embeddings easily. Two embeddings (plane graphs) are combinatorially equivalent if their combinatorial embeddings are equal up to a global change of orientation (reversing the order of all sequences simultaneously). For example, (b) is not equivalent to (a) nor (c), because it is the only one with a face bounded by seven vertices. However, (a) and (c) turn out to be equivalent: after reverting orientations f1 takes the role of h2, f2 takes the role of h1, and f3 takes the role of h3. Exercise 2.19. Let G be a planar graph with vertex set {1, . . . , 9}. Try to ﬁnd an embedding corresponding to the following list of circular sequences of faces: 26 Geometry: C&A 2022 2.3. Unique Embeddings 1 2 3 4 5 6 f1 f2 f3 (a) 1 2 3 4 5 6 g2 g1 g3 (b) 1 2 3 4 6 5 h2 h1 h3 (c) Figure 2.12: Equivalent embeddings? (a) {(1, 4, 5, 6, 3), (1, 3, 6, 2), (1, 2, 6, 7, 8, 9, 7, 6, 5), (7, 9, 8), (1, 5, 4)} (b) {(1, 4, 5, 6, 3), (1, 3, 6, 2), (1, 2, 6, 7, 8, 9, 7, 6, 5), (7, 9, 8), (1, 4, 5)} Combinatorial embeddings are not only used to categorize plane graphs. They also play a role in algorithm design. Quite often, algorithms dealing with planar graphs do not need a full-ﬂedged embedding to proceed. It is suﬃcient to operate on a combinatorial embedding, which is more eﬃcient to handle. Many people prefer a dual representation which, instead of listing face boundaries, enumerates the neighbors of v in cyclic order for each vertex v. It can avoid the issue of a vertex appearing multiple times in the sequence. However, the following lemma shows that such an issue does not arise when dealing with biconnected graphs. Lemma 2.20. In a biconnected plane graph every face is bounded by a cycle. We leave the proof as an exercise. Intuitively the statement is clear, but we believe it is instructive to think about a formal argument. An easy consequence is stated below, whose proof is also an exercise. Corollary 2.21. For any vertex v in a 3-connected plane graph, there is a cycle that contains all neighbours of v. Exercise 2.22. Prove Lemma 2.20 and Corollary 2.21. Given Lemma 2.20, one might wonder the converse question: Which cycles in a planar graph G bound a face (in some plane embedding of G)? Such cycles are said to be facial; see Figure 2.13. Exercise 2.23. Describe a linear time algorithm that, given an abstract planar graph G and a cycle C in G, tests whether C is a facial cycle. (You may assume that planarity can be tested in linear time.) 2.3 Unique Embeddings As we have seen, an abstract planar graph may admit many diﬀerent embeddings, even in the combinatorial sense. Under what condition does it admit a unique combinatorial embedding? 27
12737	https://brainly.com/question/41908830	[FREE] A polygon with n sides has D diagonals, where D is given by the function: D(n) = \frac{n(n-3)}{2} - brainly.com 3 Search Learning Mode Cancel Log in / Join for free Browser ExtensionTest PrepBrainly App Brainly TutorFor StudentsFor TeachersFor ParentsHonor CodeTextbook Solutions Log in Join for free Tutoring Session +77,8k Smart guidance, rooted in what you’re studying Get Guidance Test Prep +34,1k Ace exams faster, with practice that adapts to you Practice Worksheets +8k Guided help for every grade, topic or textbook Complete See more / Mathematics Textbook & Expert-Verified Textbook & Expert-Verified A polygon with n sides has D diagonals, where D is given by the function: D(n)=2 n(n−3) Find the number of sides n if D=135. 1 See answer Explain with Learning Companion NEW Asked by summerBreeze55 • 11/07/2023 0:02 / 0:15 Read More Community by Students Brainly by Experts ChatGPT by OpenAI Gemini Google AI Community Answer This answer helped 9335 people 9K 0.0 0 Upload your school material for a more relevant answer To find the number of sides, n, of a polygon given the number of diagonals, D, we can use the formula D(n) = n(n-3)/2. Given that 135 < D < 594, the number of sides of the polygon is n > 18. Explanation To find the number of sides, n, of a polygon given the number of diagonals, D, we can use the formula D(n) = n(n-3)/2. Given that 135 < D < 594, we can substitute the formula into the inequality to solve for n: n(n-3)/2 > 135 n(n-3) > 270 n^2 - 3n - 270 > 0 Factoring the quadratic equation, we get (n - 18)(n + 15) > 0. The solutions to this inequality are n > 18 or n < -15. Since we are dealing with polygon sides, n cannot be negative. Therefore, the number of sides of the polygon is n > 18. Learn more about Polygon sides and diagonals here: brainly.com/question/23366307 Answered by autumnDust89 •159 answers•9.3K people helped Thanks 0 0.0 (0 votes) Textbook &Expert-Verified⬈(opens in a new tab) This answer helped 9335 people 9K 0.0 0 Essential Graduate Physics - Classical Electrodynamics - Konstantin K. Likharev The Foundations of Geometry - David Hilbert Conceptual Physics - Benjamin Crowell Upload your school material for a more relevant answer Using the diagonals formula D(n)=2 n(n−3) and setting it equal to 135 leads us to find that the number of sides of the polygon is n=18. Explanation To find the number of sides n of a polygon given the number of diagonals D, we will use the formula for diagonals, which is given by: D(n)=2 n(n−3) We know from the problem that D=135. Therefore, we can set up the equation: 2 n(n−3)=135 To eliminate the fraction, we can multiply both sides by 2: n(n−3)=270 This can be rearranged into a standard quadratic equation: n 2−3 n−270=0 Next, we can factor this quadratic equation. We are looking for two numbers that multiply to −270 and add to −3. These numbers are −18 and 15. Thus, we can factor the equation as: (n−18)(n+15)=0 Setting each factor equal to zero gives us the possible values for n: n−18=0 This results in n=18 n+15=0 This results in n=−15 Since the number of sides of a polygon cannot be negative, we discard n=−15. Therefore, the solution is: n=18 This means that the polygon has 18 sides. Examples & Evidence For example, a polygon with 18 sides is called an 18-gon. Each vertex of this polygon contributes to the calculation of diagonals as given in the formula, confirming that it aligns with the properties of such a shape. The formula for the number of diagonals in a polygon is well-established in geometry. The quadratic equation solved correctly affirms that only a positive integer is valid for the context of polygon sides. Thanks 0 0.0 (0 votes) Advertisement summerBreeze55 has a question! Can you help? Add your answer See Expert-Verified Answer ### Free Mathematics solutions and answers Community Answer n(n-3) A polygon with n sides has D diagonals, where D is given by the function D(n) = 2 Find the number of sides n if 44 ≤D≤230. Community Answer Number of diagonals. a polygon with n sides hasd diagonals, where d is given by the functiond1n2 =n1n - the number of sides n if27 … d … 230. Community Answer 4 the equation d=1/2n (n-3) gives the number of diagonals D for polygon with n sides. use this equation to find the number of sides n for a polygon that has 65 diagonals Community Answer the equation D=(1)/(2)n(n-3) gives the number 9f diagonals D for polygon with n sides. Find the number of sides fr polygon that has 54 diagonals. Community Answer Use Excel to find the following: The equation D=12n(n−3) gives the number of diagonals D for a polygon with n sides. Find the number of sides for polygon that has 119 diagonals. Community Answer The equation D= 1/2 n(n−3) gives the number of diagonals D for a polygon with n sides. Find the number of diagonals for polygon that has 14 sides. Community Answer The equation D=(1)/(2)n(n-3) gives the number of diagonals D for a polygon with n sides. Use this equation to find the number of sides n for a polygon that has 14 diagonals. Community Answer The equation D=(1)/(2)n(n-3) gives the number of diagonals D for a polygon with sides. Use this equation to find the number of diagonals for a polygon that has 15 sides. Community Answer 4.6 12 Jonathan and his sister Jennifer have a combined age of 48. If Jonathan is twice as old as his sister, how old is Jennifer Community Answer 11 What is the present value of a cash inflow of 1250 four years from now if the required rate of return is 8% (Rounded to 2 decimal places)? New questions in Mathematics Solve the rational equation. Express numbers as integers or simplified fractions. 7 y 1+9 1=21 y 19−7 1 Information is given about a polynomial f whose coefficients are real numbers. Find the remaining zeros of f. Degree 4; zeros: 8 i,6,−6 Find the indicated difference. Express your answer in mixed number form, and reduce if possible. 49−(−20 9 8) Solve the rational equation. Express numbers as integers or simplified fractions. 4 3 t−13=3 12 t Based on the type of equations in the system, what is the greatest possible number of solutions? {x 2+y 2=9 9 x+2 y=16 Previous questionNext question Learn Practice Test Open in Learning Companion Company Copyright Policy Privacy Policy Cookie Preferences Insights: The Brainly Blog Advertise with us Careers Homework Questions & Answers Help Terms of Use Help Center Safety Center Responsible Disclosure Agreement Connect with us (opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab) Brainly.com Dismiss Materials from your teacher, like lecture notes or study guides, help Brainly adjust this answer to fit your needs. Dismiss
12738	https://math.stackexchange.com/questions/2512339/solve-for-a-b-c-d-in-bbb-r-given-that-a2b2c2d2-ab-bc-cd-d-frac-25	Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Solve for $a,b,c,d \in \Bbb R$, given that $a^2+b^2+c^2+d^2-ab-bc-cd-d+\frac 25 =0$ Ask Question Asked Modified 5 years, 11 months ago Viewed 7k times 5 $\begingroup$ Today, I came across an equation in practice mock-test of my coaching institute, aiming for engineering entrance examination (The course for the test wasn't topic-specific, it was a test of complete high school mathematics). It was having four variables and only one equation. While analyzing my test paper, this is the only problem I (and my friends too) couldn't figure out even after giving this problem several hours. So I came here for some help. Question : Solve for $a,b,c,d \in \Bbb R$, given that $$a^2+b^2+c^2+d^2-ab-bc-cd-d+\frac 25 =0$$ Since only one equation is given, there must be involvement of making of perfect squares, such that they all add up to $0$. Thus, resulting in few more equations. But how to? I tried a lot of things, such as making $(a-b)^2 $ by adding the missing terms and subtracting again, but got no success. Thanks! algebra-precalculus polynomials systems-of-equations square-numbers Share edited Apr 11, 2018 at 12:21 Jaideep KhareJaideep Khare asked Nov 9, 2017 at 14:21 Jaideep KhareJaideep Khare 19.6k44 gold badges4444 silver badges8080 bronze badges $\endgroup$ 4 3 $\begingroup$ Is it really "$... -d + \frac{2}{5}$" instead of "$...-da +\frac{2}{5}$"? $\endgroup$ achille hui – achille hui 2017-11-09 14:23:53 +00:00 Commented Nov 9, 2017 at 14:23 2 $\begingroup$ @achillehui Yes. The equation is exactly what it's written. $\endgroup$ Jaideep Khare – Jaideep Khare 2017-11-09 14:28:45 +00:00 Commented Nov 9, 2017 at 14:28 $\begingroup$ The equation is equivalent to $(a-b)^2 + (b-c)^2 + (c-d)^2 + (d-a)^2 + 2ad - 2d + \frac{4}{5} = 0$, which at least put some restrictions on $a$ and $d$ (e.g., if $d$ is positive then $a$ is less than 1). $\endgroup$ Connor Harris – Connor Harris 2017-11-09 14:38:57 +00:00 Commented Nov 9, 2017 at 14:38 $\begingroup$ $(a,b,c,d) = (1/5,2/5,3/5,4/5)$ $\endgroup$ achille hui – achille hui 2017-11-09 14:45:11 +00:00 Commented Nov 9, 2017 at 14:45 Add a comment \| 6 Answers 6 Reset to default 7 $\begingroup$ Let $F(a,b,c,d) = a^2+b^2+c^2+d^2-ab-bc-cd-d+\frac25$. With help of a CAS, one can verify $$\begin{align} F\left(\frac15+p,\frac25+q,\frac35+r,\frac45+s\right) &= p^2 - pq + q^2 - qr + r^2 -rs + s^2\ &= \frac12\left(p^2 + (p-q)^2 + (q-r)^2 + (r-s)^2 + s^2\right)\end{align}$$ If one set $(a,b,c,d)$ to $\left(\frac15+p,\frac25+q,\frac35+r,\frac45+s\right)$, one find $$\begin{align} F(a,b,c,d) = 0 &\iff p = p-q = q-r = r-s = s = 0\ &\iff p = q = r = s = 0 \end{align} $$ This implies the equation at hand has a unique solution: $$(a,b,c,d) = \left(\frac15,\frac25,\frac35,\frac45\right)$$ Update About the question how I come up with this. I first write $F(a,b,c,d)$ as $$\begin{align} F(a,b,c,d) &= a^2 + b^2 + c^2 + d^2 - ab - bc - cd - da + d(a-1) + \frac25\ &= \frac12((a-b)^2+(b-c)^2+(c-d)^2+(d-a)^2) + d(a-1) + \frac25 \end{align}\tag{1} $$ To simplify the term $d(a-1)$, I introduce $\lambda, \mu$ such that $$\begin{cases} d &= \frac12 + \lambda + \mu\ a &= \frac12 + \lambda - \mu \end{cases} \quad\implies\quad d(a-1) = \lambda^2 - \left(\frac12+\mu\right)^2 $$ Now $d-a = 2\mu$ and $(a-b)^2 + (b-c)^2 + (c-d)^2 \ge 3\left(\frac{d-a}{3}\right)^2 = \frac43 \mu^2$. If one substitute this back into $(1)$, one find $$F(a,b,c,d) \ge \frac83\mu^2 + \lambda^2 - (\frac12 + \mu)^2 + \frac25 = \lambda^2 + \frac53\left(\mu - \frac{3}{10}\right)^2$$ In order for $F(a,b,c,d) = 0$, we need $$\lambda = 0,\quad\mu = \frac{3}{10} \quad\text{ and }\quad(a-b)^2 + (b-c)^2 + (c-d)^2 = \frac13(d-a)^2$$ The last condition forces $d-c = c-b = b-a = \frac13(d-a)$ and leads to the solution $(a,b,c,d) = \left(\frac15,\frac25,\frac35,\frac45\right)$. This is a little bit sloppy to describe, so I look at expansion of $F(a,b,c,d)$ near the solution and obtain a simpler description of $F$ in terms of $p,q,r,s$. Share edited Nov 9, 2017 at 21:00 answered Nov 9, 2017 at 14:55 achille huiachille hui 126k77 gold badges189189 silver badges365365 bronze badges $\endgroup$ 2 $\begingroup$ Amazing! (+1)BTW, How did you come up with this solution, I mean how did you think about doing so? $\endgroup$ Jaideep Khare – Jaideep Khare 2017-11-09 15:17:11 +00:00 Commented Nov 9, 2017 at 15:17 1 $\begingroup$ @JaideepKhare see update in answer. $\endgroup$ achille hui – achille hui 2017-11-09 15:42:37 +00:00 Commented Nov 9, 2017 at 15:42 Add a comment \| 5 $\begingroup$ Your idea of turning the right side into a sum of perfect squares is a good one. Observing that $$ (a-b)^2+(b-c)^2+(c-d)^2+(d-a)^2=2(a^2+b^2+c^2+d^2-ab-bc-cd-da) $$ we multiply $$ a^2+b^2+c^2+d^2-ab-bc-cd-d+\frac{2}{5}=0 $$ by $2$ and rewrite the result as $$ (a-b)^2+(b-c)^2+(c-d)^2+(d-a)^2+2ad-2d+\frac{4}{5}=0. $$ We define $x=a-b$, $y=b-c$, and $z=c-d$, which gives $a=x+y+z+d$. Substituting gives $$ x^2+y^2+z^2+(x+y+z)^2+2d(x+y+z+d-1)+\frac{4}{5}=0, $$ which can be rewritten as $$ x^2+y^2+z^2+(x+y+z+d)^2+d^2-2d+\frac{4}{5}=0 $$ or $$ x^2+y^2+z^2+(x+y+z+d)^2+(d-1)^2=\frac{1}{5}. $$ So a sum of five perfect squares equals $\frac{1}{5}$. We might hope for a solution in which each of the perfect squares is $\left(\pm\frac{1}{5}\right)^2$, and indeed we find that $x=y=z=-\frac{1}{5}$, $d=\frac{4}{5}$ provides such a solution. Now we ask whether the solution can be perturbed. Letting $x=-\frac{1}{5}+e$, $y=-\frac{1}{5}+f$, $z=-\frac{1}{5}+g$, $d=\frac{4}{5}+h$ and substituting, we get $$ (-1/5+e)^2+(-1/5+f)^2+(-1/5+g)^2+(1/5+e+f+g+h)^2+(-1/5+h)^2=\frac{1}{5}, $$ which simplifies to $$ e^2+f^2+g^2+h^2+(e+f+g+h)^2=0. $$ This forces $e=f=g=h=0$, and therefore the solution is unique. Share answered Apr 7, 2018 at 12:24 Will OrrickWill Orrick 19.2k22 gold badges5757 silver badges9090 bronze badges $\endgroup$ 1 $\begingroup$ Amazing! This is the one solution that is practically possible to think during examination. Thanks for this!ð $\endgroup$ Jaideep Khare – Jaideep Khare 2018-04-07 13:30:22 +00:00 Commented Apr 7, 2018 at 13:30 Add a comment \| 5 $\begingroup$ Multiply by $2$ and rearrange to \begin{align}(a-b)^2 + (b-c)^2 + (c-d)^2 + (d-a)^2 + 2ad - 2d + \frac{4}{5} = 0. \tag{$\star$}\end{align} For fixed $a$ and $d$, the minimum value of $(a-b)^2 + (b-c)^2 + (c-d)^2$ is $\frac{(d - a)^2}{3}$, with equality if and only if $a, b, c, d$ is an arithmetic progression by the lemma below, so the LHS of $(\star)$ is at least $$\frac{4}{3} (d-a)^2 + 2ad - 2d + \frac{4}{5} = \frac{4}{3} \left( a - \frac{d}{4} \right)^2 + \frac{5}{4} \left( d - \frac{4}{5}\right)^2 \tag{$\dagger$}.$$ But $(\dagger)$ is clearly non-negative, and it is zero if and only if $d = 4/5$ and $a = d/4 = 1/5$, but the LHS of $(\star)$ must be zero. From this, $b = 2/5$ and $c = 3/5$ follow, and there can be no other solution. Lemma. For fixed $x_0$ and $x_n$, the sum $\sum_{i=1}^n (x_i - x_{i-1})^2$ is minimized when the $x_i$ form an arithmetic progression $x_i = \frac{n-i}{n} x_0 + \frac{i}{n} x_n$. Proof. For $n = 2$, $(x_0 - x_1)^2 + (x_1 - x_2)^2$ can be rearranged as $$ \left( x_1 - \frac{x_0 + x_2}{2} \right)^2 +x_0^2 + x_2^2 - \frac{(x_0 + x_2)^2}{4}. $$ For $n > 2$, if some $x_k$ is not the midpoint of $x_{k-1}$ and $x_{k+1}$, then $(x_k - x_{k-1})^2 + (x_{k+1} - x_k)^2$ can be reduced by moving $x_k$ to the midpoint, leaving the other terms of $\sum_{i=1}^n (x_i - x_{i-1})^2$ alone. So if a minimum exists, it must have evenly spaced $x_i$. And proving that a minimum exists is simple: the possible values of $x_1, \ldots, x_{n-1}$ that can minimize $f(x_1, \ldots, x_{n-1}) = \sum_{i=1}^n (x_i - x_{i-1})^2$ can be bounded in some closed interval $[-R, R]$, and the image of a connected compact set $[-R, R]^n$ under a continuous function $f$ must be compact and connected (that is, a closed bounded interval). This lemma can be interpreted physically as stating that the potential energy of a chain of $n$ identical springs with unstretched length zero, with the endpoints of the whole chain anchored, is minimized (and thus the forces at each spring endpoint are in equilibrium) when each spring is stretched equally. Here, $x_0$ and $x_n$ are the fixed endpoints, and $x_{i-1}$ and $x_i$ are the endpoints of the $i$th spring. Share edited Oct 11, 2019 at 18:57 answered Nov 9, 2017 at 15:21 Connor HarrisConnor Harris 7,4391515 silver badges3131 bronze badges $\endgroup$ Add a comment \| 3 $\begingroup$ It's $$a^2-ab+\frac{b^2}{4}+\frac{3}{4}b^2-bc+\frac{1}{3}c^2+\frac{2}{3}c^2-cd+\frac{3}{8}d^2+\frac{5}{8}d^2-d+\frac{2}{5}=0$$ or $$\left(a-\frac{b}{2}\right)^2+\frac{3}{4}\left(b-\frac{2c}{3}\right)^2+\frac{2}{3}\left(c-\frac{3d}{4}\right)^2+\frac{5}{8}d^2-d+\frac{2}{5}=0$$ or $$\left(a-\frac{b}{2}\right)^2+\frac{3}{4}\left(b-\frac{2c}{3}\right)^2+\frac{2}{3}\left(c-\frac{3d}{4}\right)^2+\frac{5}{8}\left(d-\frac{4}{5}\right)^2=0,$$ which gives the answer. Share edited Nov 10, 2017 at 10:24 answered Nov 9, 2017 at 15:25 Michael RozenbergMichael Rozenberg 208k3131 gold badges171171 silver badges294294 bronze badges $\endgroup$ 2 $\begingroup$ Can you please explain what made you bring the equation in the perfect square form $\endgroup$ user471651 – user471651 2017-11-10 09:13:37 +00:00 Commented Nov 10, 2017 at 9:13 2 $\begingroup$ @user471651 I added something. See now. $\endgroup$ Michael Rozenberg – Michael Rozenberg 2017-11-10 10:25:01 +00:00 Commented Nov 10, 2017 at 10:25 Add a comment \| 2 $\begingroup$ The expression is a quadratic form. Experimenting a bit you can put $v^{T}=(a, b, c, d, 1)$ and let A be the 5 by 5 matrix with 1, 1, 1, 1, $\frac{2}{5}$ down the lead diagonal, with -1, -1, -1, -1 above that and 0 everywhere else. Then the expression is $v^{T}Av = 0$. The expression is also $v^{T}A^{T}v = 0$ and better still it is $v^{T}(A+A^{T})v = 0$ where $A+A^{T}$ is a symmetric matrix. In this form $(A+A^{T})v = 0$ has the unique solution v already given. It would be nice if it were obvious that this is the unique solution of $v^{T}(A+A^{T})v = 0$ but I can't see it. Share answered Nov 9, 2017 at 20:16 PaulPaul 1,5021111 silver badges99 bronze badges $\endgroup$ Add a comment \| 1 $\begingroup$ I think that the following is most natural way to solve this equation. $$a^2+b^2+c^2+d^2-ab-bc-cd-d+\frac 25 =0$$ Since it is quadratic on $a$ we have nonegative discriminant: $$ b^2- 4(b^2+c^2+d^2-bc-cd-d+\frac 25) \geq 0$$ so $$4c^2-4c(b+d)+4d^2+3b^2-4d +\frac85 \leq 0$$ so $$4c^2-4c(b+d)+(b+d)^2-(b+d)^2+4d^2+3b^2-4d +\frac85 \leq 0$$ so $$(2c-b-d)^2+3d^2+2b^2-2bd-4d+\frac85 \leq 0$$ so$$3d^2+2b^2-2bd-4d+\frac85 \leq 0\;\;\;/\cdot 2$$ so $$4b^2-4bd+d^2+5d^2-8d+\frac{16}5 \leq 0 \;\;\;/\cdot 5$$ so $$5(2b-d)^2+25d^2-40d+16 \leq 0 $$ so $$5(2b-d)^2+(5d-4)^2 \leq 0 $$ which means that $d=4/5$ and $b=2/5$ and $c=(b+d)/2=...$ and $a=...$ Share answered May 3, 2018 at 10:01 nonusernonuser 92.1k2020 gold badges110110 silver badges218218 bronze badges $\endgroup$ Add a comment \| You must log in to answer this question. 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12739	https://www.youtube.com/watch?v=JJklTox3iic	Use the diagram to determine if the statements are true or false. Vertical, supplementary angles Ms Shaws Math Class 51100 subscribers 5 likes Description 433 views Posted: 15 Oct 2019 Transcript: everyone we're going to use the diagram to determine whether the statement is true or false so it says that the measure of angle one is 47 degrees then the measure of angle 2 is 43 now that would be false because these two angles must add up to 180 and 47 plus 43 is less than a hundred so it's or actually it is 90 so this is false now since of measure of angle one is 47 then the measure of angle 3 is 47 degrees that's true based off your vertical angle theorem these two angles are congruent and at angle measure of angle two and the measure of angle 4 also congruent so now it's saying the measure of angle 1 and 3 equals the measure of angle 2 and 4 that's not true because let's see 2 and 4th the measure of angle 2 is 180 minus 47 so the measure of this angle is 133 so then would be the measure of angle 4 so if you add 133 plus 133 and I'll move it over here that does not equal 47 wait wait wait let me do it right all right so basically it's saying measure of angle 1 is 47 measure angle 3 is 47 and that does not equal 1 33 plus 133 just by looking don't even need to use a calculator so this is false and the last one is true because 47 plus 133 which is 180 equals 47 plus 133 degrees and so that would be true and that's it thank you have a nice day bye bye [Music] you
12740	https://www.quora.com/What-is-the-relationship-between-boiling-point-and-atmospheric-pressure	What is the relationship between boiling point and atmospheric pressure? - Quora Something went wrong. Wait a moment and try again. Try again Skip to content Skip to search Sign In What is the relationship between boiling point and atmospheric pressure? All related (45) Sort Recommended Assistant Bot · 11mo The boiling point of a liquid is the temperature at which its vapor pressure equals the surrounding atmospheric pressure. Here’s how atmospheric pressure affects boiling point: Increased Atmospheric Pressure: When atmospheric pressure increases, the boiling point of a liquid also increases. This is because a higher pressure requires the liquid to reach a higher temperature for its vapor pressure to match the surrounding pressure. For example, water boils at 100°C (212°F) at sea level (1 atm), but at higher pressures, such as in a pressure cooker, water can boil at temperatures above 100°C. Decre Continue Reading The boiling point of a liquid is the temperature at which its vapor pressure equals the surrounding atmospheric pressure. Here’s how atmospheric pressure affects boiling point: Increased Atmospheric Pressure: When atmospheric pressure increases, the boiling point of a liquid also increases. This is because a higher pressure requires the liquid to reach a higher temperature for its vapor pressure to match the surrounding pressure. For example, water boils at 100°C (212°F) at sea level (1 atm), but at higher pressures, such as in a pressure cooker, water can boil at temperatures above 100°C. Decreased Atmospheric Pressure: Conversely, when atmospheric pressure decreases, the boiling point of a liquid decreases. At high altitudes, where atmospheric pressure is lower, water boils at temperatures lower than 100°C. For instance, at an elevation of about 2,000 meters (approximately 6,600 feet), water boils at around 93.4°C (200.1°F). Summary Boiling Point Increases: With increased atmospheric pressure. Boiling Point Decreases: With decreased atmospheric pressure. This relationship is crucial in various applications, including cooking, distillation, and understanding weather patterns. Upvote · Dr Ashutosh Bijalwan ExxonMobil \| CIMNE-UPC \| SLB \| Eaton \| IIT D · Author has 58 answers and 303.8K answer views ·2y Imagine you have a vessel filled with water and its free surface is exposed to the atmospheric air. Now if we start heating the vessel from the bottom we will observe the following things: The bottom surface of the vessel starts forming bubbles Bubbles start rising and collapse at the top surface. Now let's explain what we observed. The surface of the vessel has micro-voids that serve as the nucleation site of bubbles. With the heat flux these bubbles start growing in size and exert vapor pressure on surrounding fluid (One can use the Laplace equation to estimate the internal bubble pressure from Continue Reading Imagine you have a vessel filled with water and its free surface is exposed to the atmospheric air. Now if we start heating the vessel from the bottom we will observe the following things: The bottom surface of the vessel starts forming bubbles Bubbles start rising and collapse at the top surface. Now let's explain what we observed. The surface of the vessel has micro-voids that serve as the nucleation site of bubbles. With the heat flux these bubbles start growing in size and exert vapor pressure on surrounding fluid (One can use the Laplace equation to estimate the internal bubble pressure from its size). Now mechanics come into the picture when the force exerted by atmospheric pressure just equilibrates the bubble vapor pressure+density x g x h, which is the point when we reach boiling. Any small addition of heat flux will overcome the atmospheric pressure head and the bubble start rising. Essentially, if we are on the hill then atmospheric pressure would be low and bubble internal vapor pressure will overcome ambient pressure easily and the boiling point would be lower than at sea level. On contrary, at high chamber pressure, we would expect a higher boiling point. Another interesting fact is that at phase transition we have one degree of freedom for pure substance (Gibbs phase rule). If ambient pressure (equilibrium pressure) is fixed, the transition temperature will automatically get fixed and so boiling point. Cheers. Upvote · 9 3 Sponsored by Complex Law Attention UK Drivers: Check PCP refund eligibility now. If you took out car finance between 2007–2021, use Complex Law to check your refund eligibility. Learn More 999 147 Christian Droßmann Freelance Trainer for German and English (2001–present) · Author has 1.2K answers and 953.4K answer views ·4y Originally Answered: How do atmospheric pressure and boiling points work? · Every element or substance has what we call a “vapor pressure”. In a nutshell this describes how badly said element or substance “wants” to turn into a gas. The atmospheric pressure works against that, by preventing the molecules to break free from the substance. The point at which a liquid starts to turn into gas spontaneously is called the “boiling point”. When this happens depends on two factors: atmospheric pressure and temperature. If you are at a constant pressure (let’s say the pressure at sea level, zero altitude) you need to apply heat to raise the temperature if the substance is not Continue Reading Every element or substance has what we call a “vapor pressure”. In a nutshell this describes how badly said element or substance “wants” to turn into a gas. The atmospheric pressure works against that, by preventing the molecules to break free from the substance. The point at which a liquid starts to turn into gas spontaneously is called the “boiling point”. When this happens depends on two factors: atmospheric pressure and temperature. If you are at a constant pressure (let’s say the pressure at sea level, zero altitude) you need to apply heat to raise the temperature if the substance is not already a gas under those conditions. Heat is essentially the movement of atoms and molecules and the faster they move the more liquid and subsequently gaseous the substance becomes. The other way to reach boiling point is to keep the temperature constant and lower the atmospheric pressure. At high altitude, such as in the Himalayan mountains, you will notice that water boils more quickly than at sea level when heat is applied. The atmospheric pressure is considerably lower at 8,000 metres of altitude. If you increased the altitude further, you would reach the point at which the substance’s vapor pressure is greater than the atmospheric pressure and the substance will start to boil although the temperature is relatively low. If you opened a can of beer in the vacuum of space the beer would evaporate instantaneously. Upvote · 9 1 Related questions More answers below What is the relationship between the atmospheric pressure and the boiling point of liq? What is the relationship between atmospheric pressure and boiling point? How much does atmospheric pressure affect boiling point? Is there an equation to calculate the boiling point in relation to atmospheric pressure? What is the relationship between boiling temperature and pressure? How does a change in pressure affect the boiling point of a compound? Robert Lockwood Ecologist at Federal Government Research Laboratory (1978–present) · Author has 1.1K answers and 2M answer views ·8y What is the relationship between boiling point and atmospheric pressure? This has been answered, more or less, by others on Quora so do a search. Chiro Polo provided a phase diagram for water. Note the line connecting D and E which shows how the boiling point increases with increasing pressure and, of course the opposite, how the boiling point decreases with decreasing pressure. To a first approximation the relationship is almost linear. At standard conditions (pressure of one atmosphere) water boils at approximately 100 degrees C, at a higher elevation the atmospheric pressure is less and the wa Continue Reading What is the relationship between boiling point and atmospheric pressure? This has been answered, more or less, by others on Quora so do a search. Chiro Polo provided a phase diagram for water. Note the line connecting D and E which shows how the boiling point increases with increasing pressure and, of course the opposite, how the boiling point decreases with decreasing pressure. To a first approximation the relationship is almost linear. At standard conditions (pressure of one atmosphere) water boils at approximately 100 degrees C, at a higher elevation the atmospheric pressure is less and the water boils at a slightly lower temperature. Consult specific phase diagrams for substances other than water. Upvote · 9 1 Mike Jones MAEd in Chemistry&Physics, Western Carolina University (Graduated 1974) · Author has 6.2K answers and 8.3M answer views ·5y Boiling occurs when the ambient pressure is equal to the vapor pressure of the liquid (or solution). For instance, the boiling point of water at sea level (760 torr) is 100C. At 100C the vapor pressure of water is 760 torr. If the water is located an an elevation where the ambient pressure is lower, the boiling point will be lower. Suppose the ambient pressure is 634 torr. Then the boiling point of water would be 95C. The vapor pressure of water at 95C is 634 torr. You can even get water to boil at room temperature by placing water at 20C in a vacuum pump bell jar and reducing the pressure to 17 Continue Reading Boiling occurs when the ambient pressure is equal to the vapor pressure of the liquid (or solution). For instance, the boiling point of water at sea level (760 torr) is 100C. At 100C the vapor pressure of water is 760 torr. If the water is located an an elevation where the ambient pressure is lower, the boiling point will be lower. Suppose the ambient pressure is 634 torr. Then the boiling point of water would be 95C. The vapor pressure of water at 95C is 634 torr. You can even get water to boil at room temperature by placing water at 20C in a vacuum pump bell jar and reducing the pressure to 17.5 torr or below. Upvote · 9 1 Sponsored by Fisher Investments 99 Tips for Your Retirement Once Your Portfolio Reaches £250k. For individuals with £250k+, get this guide and ongoing insights. Learn More Edward Willhoft PhD (1966) FIFST (1987) from King’s College London (Graduated 1966) · Author has 3.3K answers and 10M answer views ·Updated 5y There is a simple concept for defining the boiling point (bp) of a liquid determined by the constancy of temperature at the bp. When that happens, the saturation vapour pressure of the liquid equates to the total pressure on the liquid. If you lower the pressure then you lower the boiling point of a liquid and vice versa. As to a mathematical (empirical) relationship, that is generally available in the literature for many liquids and implicitly relating to a nominal atmospheric pressure of 1 bar or 1 atm. If you require improved accuracy then you need to: Ensure that the liquid is pure (solutes Continue Reading There is a simple concept for defining the boiling point (bp) of a liquid determined by the constancy of temperature at the bp. When that happens, the saturation vapour pressure of the liquid equates to the total pressure on the liquid. If you lower the pressure then you lower the boiling point of a liquid and vice versa. As to a mathematical (empirical) relationship, that is generally available in the literature for many liquids and implicitly relating to a nominal atmospheric pressure of 1 bar or 1 atm. If you require improved accuracy then you need to: Ensure that the liquid is pure (solutes raise the bp) and, determine the atmospheric pressure at the time of boiling the liquid; the value of the saturation vapour pressure (svp) of the liquid then equates to the atmospheric pressure. You then simply calculate the temperature of the pure liquid that coincides with the measured atmospheric pressure to give the boiling point at that pressure. An example for water, using the Antoine empirical equation, in the relationship between the svp and bp, is given by: log P = 8.07 - 1730.63/(233.43 + Tb) where Tb (C) and P (torr) are the boiling point and pressure, respectively, of the water. You would simply select the value of the atmospheric pressure (760 torr = 1 atm) to calculate the bp in the transposed form of the Antoine equation. Thus, in terms of temperature: Tb = [1730.63/(8.07 - log P)] - 233.43 P would be the measured atmospheric pressure. Boiling the water and measuring the temperature of the constant-boiling liquid would provide a way of determining the barometric pressure using the Antoine equation. Upvote · 9 2 9 2 9 1 Related questions More answers below What is the relationship between boiling point and vapour pressure? How is external pressure related with boiling point? How does air pressure affect the boiling point of water? What is the relationship between boiling point, atmospheric pressure and temperature? Why is boiling point changed due to atmospheric pressure? Ronald L Klaus Worked in the US space program and taught chemical engineering. · Author has 129 answers and 604.9K answer views ·7y The higher the pressure the higher is the boiling point of a liquid. To explain this simply: as you increase the temperature of a liquid, its molecules start moving faster and faster. They therefore have a greater tendency to escape. Thus you need a higher pressure to keep them in the liquid. This is why when you’re at a high altitude (lower pressure) water boils at a lower temperature, which means that if you’re cooking something in water, you likely need to cook it longer because the boiling water is at a lower temperature. Upvote · 99 12 9 1 Promoted by HP HP Tech Takes Tech Enthusiast \| Insights, Tips & Guidance ·Updated Jul 21 What is the best laser printer for Mac? When it comes to finding the best laser printer for a Mac, it’s all about compatibility, print quality, and ease of use. macOS users often value seamless wireless connectivity, native AirPrint support for printing directly from Apple devices, and reliable drivers that don’t leave you wrestling with installations. HP has a solid reputation for building printers that tick these boxes, and they offer a range of models from simple mono printers for everyday documents to robust multifunction color machines perfect for busy home offices or creative professionals. Here are a few of my top picks: HP Continue Reading When it comes to finding the best laser printer for a Mac, it’s all about compatibility, print quality, and ease of use. macOS users often value seamless wireless connectivity, native AirPrint support for printing directly from Apple devices, and reliable drivers that don’t leave you wrestling with installations. HP has a solid reputation for building printers that tick these boxes, and they offer a range of models from simple mono printers for everyday documents to robust multifunction color machines perfect for busy home offices or creative professionals. Here are a few of my top picks: HP LaserJet Pro MFP 4102dw Printer If you need crisp black-and-white prints, this mono laser printer is fully macOS compatible. Its wireless and mobile printing options make it perfect for any Mac setup. Plus, the automatic duplex printing helps save paper. Key Features: Quick printing, energy-saving, and simple to use, with AirPrint and Mac OS integration. HP LaserJet Pro 4002dn Printer The HP LaserJet Pro 4002dn is another solid monochrome printer that plays nicely with Macs, offering AirPrint and handling large print jobs easily. It's built tough for daily use, providing fast printing and sharp text. Key Features: Two paper trays, automatic double-sided printing, and a stylish design that fits right into any office. HP Color LaserJet Pro MFP 4301fdw If you are after a colour multifunction printer, I would recommend this one. You can print, scan, copy, and fax using this. It’s also fully Mac-compatible, ensuring smooth integration with your Apple devices. Key Features: Wireless and mobile colour printing, easy-to-use interface, and duplex printing. LaserJet Printers - Black & White or Color Document Printers Consider the HP LaserJet Pro MFP 4102dw and HP LaserJet Pro 4002dn for speedy, high-quality monochrome prints with easy Mac integration. And if colour printing is a necessity for you, the HP Color LaserJet Pro MFP 4301fdw is the way to go. Whether you're printing crisp text documents, scanning receipts, or wirelessly queuing up family photos from your iPhone, the right HP laser printer can make your workflow smoother and your tech life a little less frustrating with your Mac-based workspace. Check out this blog post if you’d like some more Colour LaserJet Printer recommendations: Top Affordable And Reliable Colour Laser Printers for Your Home Office Needs \| HP® Tech Takes By Emma - HP Tech Expert Upvote · 99 29 9 3 Martin Carr PhD. in Materials Chemistry, Cranfield University (Graduated 1996) · Author has 2.3K answers and 3.7M answer views ·5y A liquid will boil when its vapour pressure and atmospheric pressure are equal. This is why you can’t get a decent, hot cup of tea at the top of mount Everest, because water boils at about 70degC (Chris Bonnington). Upvote · 9 1 Fred Starr Former Retired Metallurgist and Energy Expert (1961–2010) · Author has 171 answers and 33K answer views ·1y Originally Answered: What is the relationship between external atmospheric pressure and the boiling point of a liquid? How does this relate to vapour pressure? · The vapour pressure of a liquid which is at boiling point is always equal to surrounding pressure. For water at boiling point, in a vessel exposed the atmosphere., the vapour pressure is equal to atmospheric pressure. At sea level to the boiling point is 100 deg C and the pressure is one atmosphere. Roughly 1 bar or 14.7 lb per sq inch. Upvote · Sponsored by BetterTROM.com Self-studied, self-applied & free! Learn the techniques previously known only to the few. Text, audio, & video supported. Start Now 99 25 Guy Clentsmith Chemistry tutor... at Self-Employment (2018–present) · Author has 26.5K answers and 19.4M answer views ·5y At the normal boiling point normal boiling point the vapour pressure exerted by the LIQUID is EQUAL to the atmospheric pressure, and bubbles of vapour form directly in the liquid. This is a very important principle to grasp, and you should consult your text. Upvote · Sameer Ranjan B.tech. from Guru Gobind Singh Indraprastha University (Graduated 2021) ·4y Related How does atmospheric pressure affect the boiling point? Short Answer : More the atmospheric pressure higher the boiling point, so directly proportional relation. Explanation : Water boils at 100°C, but this is true only if you are boiling it at sea level. To understand the atmospheric pressure effect on boiling, it's important to know what boiling is. Like all material water too has an internal force that holds it together, in ice it's highest so the molecules are closer but when you heat, the thermal energy leads to vibration of molecules and they tend to break from each other and move apart and become liquid and on further heating they move farther ap Continue Reading Short Answer : More the atmospheric pressure higher the boiling point, so directly proportional relation. Explanation : Water boils at 100°C, but this is true only if you are boiling it at sea level. To understand the atmospheric pressure effect on boiling, it's important to know what boiling is. Like all material water too has an internal force that holds it together, in ice it's highest so the molecules are closer but when you heat, the thermal energy leads to vibration of molecules and they tend to break from each other and move apart and become liquid and on further heating they move farther apart and start boiling and evaporating, now think of atmosphere as a weight, at sea level its weight equivalent to 1000kg ! [ Over your head ]. Now each molecule (vapour) of the water has to rise and evaporate against this huge force ( weight ) at sea level, so takes more temperature to do so, at higher altitudes the atmosphere gets thinner so lesser weight above the boiling water, vapours easily counter that in lesser heat and escape that is water boils at lesser temperature on higher altitudes. Upvote · 9 2 Barry Mcmullan BS in Chemical Engineering, Auburn University (Graduated 1969) · Author has 636 answers and 137.8K answer views ·2y Originally Answered: What is the relationship between boiling point, atmospheric pressure and temperature? · When the vapor pressure of the liquid equals the atmospheric pressure, 14.7 Pisa, boiling begins. The temperature of the liquid when vapor pressure is 14.7 will be the temperature Upvote · Raphael B.S. in Atmospheric Sciences, University of Washington (Graduated 2023) · Author has 2.5K answers and 1.9M answer views ·1y Related How atmospheric pressure can effect boiling point? Water boils when its saturation vapour pressure exceeds atmospheric pressure; this allows its molecules to escape from the liquid state and turn into a gas. The greater the temperature, the higher the saturation vapour pressure — and the curve is exponential. (This is also why hot summer days are often more humid than cold winter days — there is more water vapour in the air). So, by changing the pressure, one changes the boiling point. The two are directly correlated. The greater the atmospheric pressure, the higher the boiling point. Ever wonder why it takes so long to cook at high altitude? The Continue Reading Water boils when its saturation vapour pressure exceeds atmospheric pressure; this allows its molecules to escape from the liquid state and turn into a gas. The greater the temperature, the higher the saturation vapour pressure — and the curve is exponential. (This is also why hot summer days are often more humid than cold winter days — there is more water vapour in the air). So, by changing the pressure, one changes the boiling point. The two are directly correlated. The greater the atmospheric pressure, the higher the boiling point. Ever wonder why it takes so long to cook at high altitude? The lower pressure causes water to boil at a lower temperature; raising the heat will simply make the water boil away faster. It will not make the food get hotter until all water has boiled out of the container. Similarly, a pressure cooker is a device which (as the name implies) applies additional pressure to the container. Most pressure cookers are able to double the pressure inside, relative to atmospheric pressure. This raises the boiling point of water from about 100 to about 120 degrees Centigrade, allowing the food to cook faster. Upvote · Related questions What is the relationship between the atmospheric pressure and the boiling point of liq? What is the relationship between atmospheric pressure and boiling point? How much does atmospheric pressure affect boiling point? Is there an equation to calculate the boiling point in relation to atmospheric pressure? What is the relationship between boiling temperature and pressure? How does a change in pressure affect the boiling point of a compound? What is the relationship between boiling point and vapour pressure? How is external pressure related with boiling point? How does air pressure affect the boiling point of water? What is the relationship between boiling point, atmospheric pressure and temperature? Why is boiling point changed due to atmospheric pressure? How do atmospheric pressure and boiling points work? What is the relationship between atmospheric pressure and boiling point? How can this relationship be explained theoretically and experimentally? How can I indicate the relation between a boiling point and atmospheric pressure in an equation? At atmospheric pressure, what is the characteristic boiling point of water in degrees Celsius? Is the boiling point of distilled water always at 100 degrees Celsius at atmospheric pressure? Related questions What is the relationship between the atmospheric pressure and the boiling point of liq? What is the relationship between atmospheric pressure and boiling point? How much does atmospheric pressure affect boiling point? Is there an equation to calculate the boiling point in relation to atmospheric pressure? What is the relationship between boiling temperature and pressure? How does a change in pressure affect the boiling point of a compound? What is the relationship between boiling point and vapour pressure? Advertisement About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025 Privacy Preference Center When you visit any website, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized web experience. 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12741	https://www.vocabulary.com/dictionary/right%20wing	SKIP TO CONTENT Random Word right wing Add to list /ˌˈraɪt ˌˈwɪŋ/ /raɪt wɪŋ/ IPA guide Other forms: right wings Someone or something that's right wing is very politically conservative. A right wing priority might be reducing the size of government, for example, or opposing gun control measures. A right wing candidate is conservative, often (but not always) a member of the Republican Party in the U.S. At the opposite end of the spectrum are people who consider themselves liberal or progressive. An individual's freedom and autonomy tends to be the most important thing to a right wing voter, while those in the left wing focus on social equality. Before the political meaning, right wing described an army's formation. Definitions of right wing noun those who support political or social or economic conservatism; those who believe that things are better left unchanged synonyms: right see moresee less types: religious right United States political faction that advocates social and political conservatism, school prayer, and federal aid for religious groups and schools hard right the extreme right wing type of: faction, sect a dissenting clique Cite this entry Style: MLA MLA APA Chicago "Right wing." Vocabulary.com Dictionary, Vocabulary.com, wing. Accessed 12 Aug. 2025. Copy citation Expert-designed, NYT-trusted: Elevate your vocabulary with VocabTrainer! Start now Examples from books and articles All sources All sources Fiction Arts / culture News Business Sports Science / Med Technology He stripped his man clean in the open court, raced down the right wing and did this nasty reverse dunk. We Were Here by Matt De La Peña The right wing was all cavalry, some four thousand men, heavy with the weight of their armor. A Game of Thrones by George R.R. Martin I started on the right wing, and the whole thing looked messed up again. Okay for Now by Gary D. Schmidt We near the top of the tunnel, which means we are nearing the front tip of the mantis’s right wing. The Last Cuentista by Donna Barba Higuera < prev \| next > DISCLAIMER: These example sentences appear in various news sources and books to reflect the usage of the word ‘right wing'. Views expressed in the examples do not represent the opinion of Vocabulary.com or its editors. Send us feedback Word Family right wingright wings the "right wing" family 2 million people are mastering new words. Master a word Sign up now (it’s free!) Whether you’re a teacher or a learner, Vocabulary.com can put you or your class on the path to systematic vocabulary improvement. Get started
12742	https://thecontentauthority.com/blog/liaison-vs-liason	Liaison vs Liason: Which One Is The Correct One? Skip to Content Home Grammar Capitalization Definitions Idioms Parts of Speech Word Lists Word Usage Blog About Write For Us Login Liaison vs Liason: Which One Is The Correct One? Home » Grammar » Word Usage Have you ever been confused about whether to use liaison or liason? You’re not alone. These two words are often misspelled or used interchangeably, but they actually have different meanings and spellings. The proper word is liaison, which means a person who acts as a link to assist communication or cooperation between groups of people or organizations. On the other hand, liason is a misspelling of liaison. It’s important to use the correct spelling to avoid confusion and maintain professionalism in your writing. next stay CC Settings Off Arabic Chinese English French German Hindi Portuguese Spanish Font Color white Font Opacity 100%Font Size 100%Font Family Arial Text Shadow none Background Color black Background Opacity 50%Window Color black Window Opacity 0% White Black Red Green Blue Yellow Magenta Cyan 100%75%50%25% 200%175%150%125%100%75%50% Arial Georgia Garamond Courier New Tahoma Times New Roman Trebuchet MS Verdana None Raised Depressed Uniform Drop Shadow White Black Red Green Blue Yellow Magenta Cyan 100%75%50%25%0% White Black Red Green Blue Yellow Magenta Cyan 100%75%50%25%0% In this article, we’ll explore the differences between liaison and liason, as well as provide examples of how to use them correctly in your writing. Define Liaison Liaison is a noun that refers to a person who acts as a link or intermediary between two parties. It can also refer to the communication or coordination between two or more groups or individuals. A liaison can be an individual who facilitates communication between different departments within an organization or between an organization and external parties. They can also be individuals who work in a diplomatic capacity to foster relationships between nations or groups. For example, a liaison officer may be appointed by a government agency to work with other agencies or organizations to coordinate efforts on a specific project or initiative. In the business world, a liaison may be responsible for managing relationships with suppliers or other business partners. Define Liason Liason is a misspelling of the word liaison. It is a common mistake that is often made due to the phonetic pronunciation of the word. It is important to note that while liason is not a correct spelling of the word, it is sometimes used informally in certain contexts. However, in formal writing and communication, it is important to use the correct spelling of the word, which is liaison. Using the incorrect spelling of the word can be seen as unprofessional and can detract from the credibility of the writer or speaker. Therefore, it is important to double-check the spelling of the word before using it in any formal communication. How To Properly Use The Words In A Sentence Using words correctly in a sentence is crucial to effective communication. The words “liaison” and “liason” are often confused with each other due to their similar spelling and pronunciation. However, they have different meanings and should be used appropriately in a sentence. How To Use “Liaison” In A Sentence “Liaison” is a noun that refers to a person who acts as a link or intermediary between two groups or individuals. It can also refer to the communication or cooperation between two groups or individuals. Here are some examples of how to use “liaison” in a sentence: The marketing team serves as the liaison between the company and its customers. The teacher acts as a liaison between the students and their parents. The ambassador is responsible for maintaining a liaison between the two countries. As you can see from these examples, “liaison” is used to describe a person or a connection between two groups or individuals. How To Use “Liason” In A Sentence “Liason” is a misspelling of the word “liaison” and is not a correct spelling. Therefore, it should not be used in a sentence. If you want to use the word with the correct spelling, use “liaison” instead. It’s important to use words correctly in order to avoid confusion and effectively convey your message. By understanding the difference between “liaison” and “liason”, you can use them appropriately in your writing and communication. More Examples Of Liaison & Liason Used In Sentences In order to better understand the proper usage of the words liaison and liason, it is helpful to see them used in context. Below are examples of how each word can be used in a sentence. Examples Of Using liaison In A Sentence The marketing team will act as a liaison between the company and the media. She was appointed as the liaison between the government and the local community. The school counselor serves as a liaison between students and teachers. The CEO met with the liaison from the potential business partner. The ambassador acted as a liaison between the two countries during the negotiations. The project manager will be the liaison between the client and the development team. The customer service representative served as a liaison between the customer and the technical support team. The union representative acted as a liaison between the workers and management. The liaison officer was responsible for coordinating the joint military exercise. The public relations team will be the liaison between the company and the press. Examples Of Using liason In A Sentence The detective had a secret liason with the witness to gather more information. The politician’s scandalous liason was exposed by the media. The novel’s plot revolves around the romantic liason between the two main characters. The spy had a liason with the enemy agent to exchange information. The wealthy businessman had a liason with his mistress at his private villa. The play’s climax is the liason between the two star-crossed lovers. The film’s plot centers around the liason between the detective and the femme fatale. The prince had a secret liason with the commoner, causing a scandal in the royal court. The novel’s protagonist had a brief liason with the mysterious stranger she met at the bar. The artist’s muse was his liason with the world of beauty and inspiration. Common Mistakes To Avoid When it comes to using the words ‘liaison’ and ‘liason’, people often make the mistake of using them interchangeably. However, these two words have different meanings and spellings. Here are some common mistakes to avoid: Using ‘Liason’ Instead Of ‘Liaison’ One of the most common mistakes people make is misspelling ‘liaison’ as ‘liason’. While ‘liason’ may seem like the correct spelling, it is actually incorrect. ‘Liaison’ is the correct spelling and refers to a person who acts as a connection between two groups or organizations. Using ‘Liaison’ As A Verb Another mistake people make is using ‘liaison’ as a verb. However, ‘liaison’ is a noun and cannot be used as a verb. For example, it would be incorrect to say, “I will liaison with the marketing team.” Instead, you should say, “I will act as a liaison between the sales and marketing teams.” Using ‘Liaison’ To Refer To A Romantic Relationship Some people mistakenly use ‘liaison’ to refer to a romantic relationship. However, ‘liaison’ does not have this connotation and should not be used in this way. Instead, you can use words like ‘affair’ or ‘relationship’ to refer to romantic connections. Tips To Avoid These Mistakes To avoid making these common mistakes, it’s important to understand the correct spelling and usage of ‘liaison’. Here are some tips: Remember that ‘liaison’ is spelled with an ‘s’ and not a ‘c’. Use ‘liaison’ as a noun to refer to a person who acts as a connection between two groups or organizations. Avoid using ‘liaison’ as a verb. Do not use ‘liaison’ to refer to a romantic relationship. Context Matters Choosing between liaison and liason can be influenced by the context in which they are used. The two words have similar meanings, but their usage can vary depending on the situation. Examples Of Different Contexts Business: In a business setting, a liaison is often used to describe a person or group that acts as a link between different departments or organizations. For example, a company might have a liaison between its sales and marketing departments to ensure effective communication and collaboration. On the other hand, liason is not commonly used in a business context. Military: In military jargon, a liaison officer is someone who works with allied forces to coordinate efforts and exchange information. The term is also used in other contexts, such as disaster relief efforts, where different agencies need to work together. Liason is not typically used in these contexts. Romantic Relationships: In the context of romantic relationships, liaison is often used to describe a secret or illicit affair. For example, a married person might have a liaison with someone who is not their spouse. Liason is not commonly used in this context. As seen in the above examples, the choice between liaison and liason can depend on the context in which they are used. It is important to understand the nuances of these words to use them correctly in different situations. Exceptions To The Rules While the rules for using liaison and liason are generally straightforward, there are some exceptions where they may not apply. Here are a few examples: 1. Proper Nouns When using proper nouns, the rules for liaison and liason may not always apply. For example, if a company or organization has a name that includes the word “liaison” or “liason,” it may be spelled differently than the standard usage. Additionally, if a person’s name is spelled with one of these words, it may be spelled differently than the standard usage as well. Example: The company “Liaison Technologies” is spelled with one “s” in “liaison.” The name “Liaison Officer” may be spelled with one “s” in “liaison” depending on the individual’s preference. 2. Regional Variations The rules for using liaison and liason may also vary depending on the region or country. In some cases, one spelling may be more commonly used than the other. For example, in British English, “liaison” is the standard spelling, while in American English, “liason” is sometimes used as an alternative. Example: In the UK, it is more common to use “liaison” in formal writing. In the US, “liason” may be used in informal writing or speech. 3. Contextual Usage Finally, the rules for using liaison and liason may also depend on the specific context in which they are being used. In some cases, one spelling may be more appropriate than the other based on the intended meaning of the word. Example: In a military context, “liaison” may refer to a person who acts as a link between different units or organizations, while “liason” may refer to a sexual relationship between two people. In a business context, “liaison” may refer to a person who facilitates communication between different departments, while “liason” may be used more generally to refer to any type of connection or relationship. Overall, while the rules for using liaison and liason are generally straightforward, there are some exceptions where they may not apply. By understanding these exceptions and the contexts in which they occur, you can use these words more effectively in your writing and communication. Practice Exercises Improving one’s understanding and use of liaison and liason in sentences can be achieved through consistent practice. Here are some practice exercises to help readers enhance their knowledge: Exercise 1: Fill In The Blank \| Sentence \| Correct Spelling \| --- \| \| The _ officer was responsible for coordinating between the two departments. \| liaison \| \| She acted as a _ between the company and its clients. \| liaison \| \| The company hired a _ to help them with their marketing strategy. \| liason \| \| The _ between the two companies resulted in a successful merger. \| liaison \| \| He was appointed as the ____ between the government and the private sector. \| liaison \| Exercise 2: Identify The Correct Spelling Choose the correct spelling for the following sentences: The correct spelling is: liaison liason The correct spelling is: liaison liason The correct spelling is: liaison liason The correct spelling is: liaison liason The correct spelling is: liaison liason Answer Key: \| Sentence \| Correct Spelling \| --- \| \| The _ officer was responsible for coordinating between the two departments. \| liaison \| \| She acted as a _ between the company and its clients. \| liaison \| \| The company hired a _ to help them with their marketing strategy. \| liason \| \| The _ between the two companies resulted in a successful merger. \| liaison \| \| He was appointed as the ____ between the government and the private sector. \| liaison \| For exercise 2, the correct spelling for each sentence is: liaison liaison liason liaison liaison Conclusion After exploring the nuances of the words “liaison” and “liason,” it is clear that they are not interchangeable. “Liaison” refers to a connection or communication between two or more parties, while “liason” is a common misspelling of the former. It is important to pay attention to the spelling and usage of words, as proper grammar and language use can greatly enhance one’s credibility and professionalism. By continuing to learn and improve our language skills, we can communicate more effectively and efficiently. Remember to always double-check your spelling and usage, and don’t be afraid to consult a dictionary or style guide if you’re unsure. With practice and attention to detail, we can all become better communicators. Shawn Manaher Shawn Manaher is the founder and CEO of The Content Authority. He’s one part content manager, one part writing ninja organizer, and two parts leader of top content creators. You don’t even want to know what he calls pancakes. 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12743	https://mathworld.wolfram.com/AurifeuilleanFactorization.html	Aurifeuillean Factorization -- from Wolfram MathWorld TOPICS AlgebraApplied MathematicsCalculus and AnalysisDiscrete MathematicsFoundations of MathematicsGeometryHistory and TerminologyNumber TheoryProbability and StatisticsRecreational MathematicsTopologyAlphabetical IndexNew in MathWorld Number Theory Prime Numbers Prime Factorization Aurifeuillean Factorization A factorization of the form (1) The factorization for was discovered by Aurifeuille, and the general form was subsequently discovered by Lucas. The large factors are sometimes written as and as follows: (2) (3) which can be written (4) (5) (6) where and (7) (8) (9) See also Gauss's Cyclotomic Formula Explore with Wolfram\|Alpha More things to try: prime factorization 30-sided polyhedron Gamma(11/2) References Brillhart, J.; Lehmer, D.H.; Selfridge, J.; Wagstaff, S.S.Jr.; and Tuckerman, B. Factorizations of b-n+/-1, b=2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers, rev. ed. Providence, RI: Amer. Math. Soc., pp.lxviii-lxxii, 1988.Riesel, H. "Aurifeullian Factorization" in Appendix 6. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, pp.309-315, 1994.Wagstaff, S.S. Jr. "Aurifeullian Factorizations and the Period of the Bell Numbers Modulo a Prime." Math. Comput.65, 383-391, 1996. Referenced on Wolfram\|Alpha Aurifeuillean Factorization Cite this as: Weisstein, Eric W. "Aurifeuillean Factorization." From MathWorld--A Wolfram Resource. Subject classifications Number Theory Prime Numbers Prime Factorization About MathWorld MathWorld Classroom Contribute MathWorld Book wolfram.com 13,278 Entries Last Updated: Sun Sep 28 2025 ©1999–2025 Wolfram Research, Inc. Terms of Use wolfram.com Wolfram for Education Created, developed and nurtured by Eric Weisstein at Wolfram Research Created, developed and nurtured by Eric Weisstein at Wolfram Research
12744	https://www.wordhippo.com/what-is/the-opposite-of/craven.html	What is the opposite of craven? What's another word for What's the opposite of Meaning of the word Words that rhyme with Sentences with the word Translate to Find Words Use for blank tiles (max 2)Use for blank spacesAdvanced Word Finder Find the of Pronounce the word in Find Names Synonyms Antonyms Definitions Rhymes Sentences Translations Find Words Word Forms Pronunciations ☀ \| \| Appearance \| --- \| \| ✓ \| Use device theme \| \| ✓ \| Dark theme \| \| ✓ \| Light theme \| What is the opposite of craven? =================================== Need antonyms for craven? Here's a list of opposite words from our thesaurus that you can use instead. ================================================================================================================== Contexts ▼▲ Adjective Opposite of cowardly or lacking in courage Opposite of cowardly or weak in character Opposite of servile or submissive in nature or manner Opposite of obsequiously fawning in nature or character Opposite of humble or lower in rank or status Opposite of not honorable in character or purpose Noun Opposite of someone who displays a lack of courage in the face of danger Opposite of a person who lacks courage … more ▼▲ Adjective ▲ Opposite of cowardly or lacking in courage brave courageous daring dauntless doughty fearless gallant greathearted gutsy hardy heroic heroical intrepid lionhearted stalwart stout stouthearted valiant valorous bold strong mettlesome plucky unflinching manful ballsy unabashed daredevil spirited aweless unblenching unshrinking audacious nervy undaunted venturous adventuresome chivalrous spartan venturesome adventurous feisty spunky dashing determined confident dynamic indomitable unalarmed undismayed knightly lion-hearted stout-hearted go-ahead have-a-go rock-ribbed firm backboned tough hard resolute strong-willed virile unfearful unafraid assertive forceful resilient sure game fierce self-confident self-assertive gritty unfearing unfazed collected unruffled undauntable frightless temerarious assured forward uncompromising unwavering steadfast self-possessed aggressive shameless staunch dedicated unbendable brazen loud calm presumptuous able self-assured proud forthcoming extroverted egotistical reckless conceited bellicose tenacious driven ardent true defiant gutty sure of oneself resolved earnest unyielding unshakable principled passionate iron-willed unflagging unshaky unfaltering unbending fanatical firm in spirit strong-minded tough-minded hard-nosed Adjective ▲ Opposite of cowardly or weak in character manlike manly strong Adjective ▲ Opposite of servile or submissive in nature or manner august dignified distinguished elevated eminent exalted grand great high lofty noble patrician proud worthy Adjective ▲ Opposite of obsequiously fawning in nature or character assertive independent domineering masterful rebellious self-willed wilful willful Adjective ▲ Opposite of humble or lower in rank or status domineering demanding authoritarian authoritive dominant forceful harsh officious oppressive overbearing strict subjugating commanding “She became a shadow of herself during her relationship with a domineering partner.” Adjective ▲ Opposite of not honorable in character or purpose noble high high-minded honorableUS honourableUK lofty straight upright venerable virtuous dignified reputable respectable worthy Noun ▲ Opposite of someone who displays a lack of courage in the face of danger hero stalwart valiant supporter advocate adherent proponent follower Noun ▲ Opposite of a person who lacks courage braveheart hero stalwart brave lionheart lion champion valiant battler fighter man of courage trooper warrior soldier aggressor brave man brave person courageous person “Many lack the courage to attempt such a course, but for a braveheart like you, it should be a cinch.” Find more words! Use for blank tiles (max 2)Advanced SearchAdvanced Search Use for blank spacesAdvanced Search Advanced Word Finder Related Words and Phrases ------------------------- cravenness cravennesses cravenly cravens See Also -------- What is another word for craven? Sentences with the word craven Words that rhyme with craven What is the past tense of craven? What is the plural of craven? What is the adverb for craven? What is the adjective for craven? What is the noun for craven? Use our Antonym Finder Nearby Words cravenly cravenness cravennesses cravens craves craving crave for craved crave craters cratering cratered 6-letter Words Starting With c cr cra crav craveFind Antonyms go Word Tools Finders & Helpers Apps More Synonyms Synonyms Antonyms Rhymes Sentences Nouns VerbsAdjectives Adverbs Plural Singular Past Tense Present TenseWord Unscrambler Words With Friends Scrabble Crossword / CodewordiOS / Apple AndroidHome About Us Terms of Use Privacy Statement Don't Sell Personal DataABCDEFGHIJKLM NOPQRSTUVWXYZ Antonyms ABCDEFGHIJKLM NOPQRSTUVWXYZ Copyright WordHippo © 2025
12745	https://www.benfrederickson.com/calculating-the-intersection-of-3-or-more-circles/	Ben Frederickson Blog GitHub Twitter RSS Calculating the intersection area of 3+ circles While attempting to learn Javascript and D3.js a couple months ago, I wrote a little library for displaying area proportional venn diagrams. One thing this library didn’t do though is consider the intersection areas of 3 or more circles when placing each set in the venn diagram. Its a trickier problem than I first thought, mainly because of all the special cases that can arise when the number of circles gets large. While the 2 circle case is a simple calculus problem, I failed to extend this solution to calculate the intersection area of an arbitrary number of circles. The research papers I read on this both avoided calculating the circle intersection by using approximation techniques. One paper approximated the circles using polygons and used polygon intersection techniques to get the area, and the other approximated by plotting each circle and using binary indexing to compute the area. I tried out the latter approach, but found it to be too slow for realtime use. Since then, I’ve had some ideas on different approaches to this problem that I wanted to try out. To keep up with learning D3, I also thought I’d try visualizing each approach here. Monte Carlo Estimation One of these ideas was to run a Monte Carlo simulation to try to figure this out. Its an exceedingly simple approach: just randomly sample a bunch of points, and compute the ratio of points that are inside all the circles. The area of the intersection is approximately this ratio multiplied by the size of the bounding rectangle. I thought of using this method after seeing someone use a Monte Carlo simulation to estimate Pi - and it seemed like a pretty easy tweak to extend to handle multiple circles. I’ve included the 1 circle case to include that estimate here: Note: you need Javascript enabled and a SVG compatible browser to view the diagrams here! Its looking like you're missing javascript, or are running an old web browser that doesn't support SVG. If no diagrams show up then you might want to consider loading this up in a different browser. I've tested this with Chrome, Safari and Firefox only. 1 Circle 2 Circles 3 Circles 4 Circles 5 Circles 6 Circles 7 Circles 8 Circles \| \| \| --- \| \| Area: \| 0 \| \| Ratio: \| 0 / 0 \| \| Error: \| 0 % \| When I remove all the visualization overhead, my laptop can sample about 10 million points a second. Taking a sample of 10k points happens in around a millisecond, and is accurate to within an percentage point or so. Which isn’t really an awesome result, unless you’re the kind of person comfortable with your estimate of Pi being 3.15. Also even a millisecond evaluation time leads to noticeable lag in my use case, since I have to compute this hundreds of times. On the plus side, its a very easy method to implement. Quadtree Approximation Another idea I had was to decompose the intersection area recursively using a Quadtree style approach. Since the intersection areas here are convex, a rectangle is fully contained inside the intersection area if all four of its corners are inside. So the idea here is just to divide the region recursively into 4 rectangles, and check if the corners of these rectangles are contained in each circle. If all the corners are inside all the circles, we don’t need to recurse. Likewise we don’t need to recurse if all the corners are outside all the circles (assuming an initial bounding rectangle that is tight around the intersection area): 1 Circle 2 Circles 3 Circles 4 Circles 5 Circles 6 Circles 7 Circles 8 Circles \| \| \| --- \| \| Area: \| 0 +/- 1 \| \| Rectangle Count: \| 0 \| \| Tree Depth: \| 0 \| \| Error: \| 0 % \| I didn’t bother benchmarking this one, because as I was looking at it I finally figured out a way to calculate the intersection area without resorting to using an approximation. An Exact Solution The key here is that each intersection area is just a polygon, with an extra circle arc bulging outwards from every line segment: 1 Circles 2 Circles 3 Circles 4 Circles 5 Circles 6 Circles 7 Circles 8 Circles \| \| \| --- \| \| Area: \| 0 \| \| Polygon Area: \| 0 \| \| Arc Area: \| 0 \| 0 The polygon can be found by examining all the possible intersection points for all pairs of circles. The intersection points that are inside all the circles define the perimeter of the polygon. After sorting these points by their angle from the centre of the polygon, its relatively straightforward to calculate the area of the polygon. Calculating the area of each circle arc is a little trickier. For each line segment on the inner polygon, there can be many circles that link both points - and for each circle there are two different arcs between the two points. We need to pick the arc that lies in the right direction, which can be done in a bunch of ways: I’m doing this by picking the arc that has an angle between the angle of the two points of the line segment. The circle is the one where the arc has the smallest distance to the line segment. Finally the area is computed by integrating the circle up to the width of the arc. Final Thoughts I’ve put the code up on github, and changed my venn.js library to use the exact method when calculating the loss function for 3+ sets. While not fast enough or accurate enough for my use case, both of the approximation methods came in useful in testing out the exact solution. I generated a million random layouts and tested that the exact solution was within the error bound given by the quadtree approximation. If the answers didn’t match up, I investigated manually using the monte carlo estimate to figure out if it was the quadtree method or the exact method that had the issue. This caught a bunch of bugs that I don’t think I would have caught otherwise, especially with the quadtree estimate. In retrospect the exact solution seems really obvious. I think the only reason I didn’t come up with it originally is that I ended up googling for papers on laying out venn diagrams when I got stuck. Since all the papers I read used approximation techniques, I went that route instead. Published on 19 November 2013 Get new posts by email! 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12746	https://www.math.brown.edu/tbanchof/midpoint/tri.html	Midpoint Polygons The Problem Solution Original Discussion Pedagogy Demonstrations Bibliography### Triangles: the base case The first example, familiar to most students, concerns a triangle. The midpoint polygon divides the triangular region up into four congruent triangles so the area ratio is 1/4. Each edge of the midpoint triangle is half the length of its corresponding edge on the original, so the perimeter ratio is 1/2. Demonstration 1. Area ratio for midpoint triangles (Click on the picture or caption to go to the demonstration.) Next: convex quadrilaterals
12747	https://wayground.com/en-us/sequences-and-series-worksheets-grade-10	50+ Sequences and Series worksheets for 10th Grade on Quizizz \| Free & Printable Flashcards For Teachers Schools and Districts en-us Enter Code... Login Signup Enter Code Explore our full library of resources Including flashcards, videos, quizzes, and reading comprehension materials to enhance your students’ learning. Browse all resources Browse all resources Free Printable Sequences and Series Worksheets for 10th Grade Math Sequences and Series worksheets for Grade 10 students: Discover a comprehensive collection of free printable resources to help teachers effectively teach math concepts and enhance students' learning experience. grade 10Sequences and Series grade grade 10 grade 11 grade 12 Subjects ### MathFundamentals and Building Blocks Equations and Inequalities System of Equations and Quadratic Complex Numbers Matrices Polynomial Operations Functions Operations Logarithms Radical Expressions Rational Expressions Conic Sections Trigonometric Functions Statistics and Probabilities Sequences and Series algebra arithmetic and number theory calculus geometry probability and statistics trigonometry ### Science ### social studies ### ELA grade 10Sequences and Series Sequences and Series 12 Q 9th - 10th Sequences and Series Review 15 Q 10th - 11th Pre-Calculus Sequences and Series 17 Q 10th - 12th Sequences and Series 15 Q 10th - 12th sequences and series 21 Q 10th - 12th Geometric Sequences and Series 24 Q 10th - 12th Arithmetic Sequences and Series 10 Q 10th HPC Sequences and Series Test 13 Q 10th - 12th Sequences and Series 15 Q 10th - 12th Sequences and Series: Arithmetic and Geometric 20 Q 10th - 11th AI Sequences and Series 17 Q 8th - 10th Sequences and Series Bingo 24 Q 10th Sequences and Series 10 Q 10th Into to Sequences and Series 20 Q 10th - 12th Sequences and Series 10 Q 10th arithmetic and geometric sequences and series 20 Q 10th Algebra 2 - Test - Sequences and Series 14 Q 10th - 12th Algebra 2 Sequences and Series 14 Q 10th - 11th arithmetic sequences and series 15 Q 10th HPC Practice Test for Sequences and Series 23 Q 10th - 12th Unit 8: Sequences and Series Review 11 Q 10th - 12th Final Review - Sequences and Series 15 Q 10th - Uni IM3H Sequences and Series 38 Q 10th - 12th Sequences and Series 20 Q 10th - 12th PreviousNext Explore Sequences and Series Worksheets by Grades grade 10 grade 11 grade 12 Explore Other Subject Worksheets for grade 10 Math Science social studies ELA Browse resources by Grade Kindergarten 1st Grade 2nd Grade 3rd Grade 4th Grade 5th Grade 6th Grade 7th Grade 8th Grade 9th Grade 10th Grade 11th Grade 12th Grade Browse resources by Subject Explore printable Sequences and Series worksheets for 10th Grade Sequences and Series worksheets for Grade 10 are an excellent resource for teachers looking to challenge their students and enhance their understanding of mathematical concepts. These worksheets cover a wide range of topics, including arithmetic and geometric sequences, finite and infinite series, as well as the application of these concepts in real-world situations. By incorporating these worksheets into their lesson plans, teachers can provide their students with ample opportunities to practice and master the skills required for success in Grade 10 Math. Furthermore, these worksheets are designed to cater to different learning styles, ensuring that all students can benefit from the engaging and interactive activities provided. In addition to Sequences and Series worksheets for Grade 10, teachers can also utilize Quizizz, a popular online platform that offers a variety of educational resources, including quizzes, games, and interactive lessons. Quizizz allows teachers to create customized quizzes and assessments that align with their curriculum, making it an ideal tool for reinforcing the concepts taught in the classroom. Moreover, Quizizz provides teachers with real-time feedback on student performance, enabling them to identify areas where students may need additional support or practice. By incorporating Quizizz into their teaching strategies, educators can create a dynamic and engaging learning environment that not only supports the development of essential Grade 10 Math skills but also fosters a love for learning among their students. Features Quizizz super School & District NEW Quizizz for Work NEW Create a quiz Create a lesson Subjects Math Social Studies Science Physics Chemistry Biology About Our Story Quizizz Blog Media Kit Careers Support FAQ Contact us Help & Support Privacy Policy Terms of Service Teacher Resources Accessibility and Inclusion Download our app © 2025 Quizizz Inc. Sitemap
12748	https://convert-units.info/speed/meter-second/343	Convert speed: 343 m/s (meter / second) to ... Online unit conversion - speed Select measure: Temperature, Length, Weight, Speed, Angle, Area, Time, Volume, Pressure, Power, Energy&work, Acceleration, Force, Density, Flow rate, Fuel consumption, EV consumption, Torque(moment of force), Data storage, Metric prefixes Convert speed: 343 m/s (meter / second) to other units Select input unit of speed:343 m/s (meter / second) equals to: cm/s (centimeter / second) m/s (meter / second) m/min (meter / minute) km/s (kilometer / second) km/h (kilometer / hour) ft/s (foot / second) mph (mile / hour) yd/s (yard / second) kn (knot) M (mach - speed of sound) speed of light34300 cm/s (centimeter / second) 343 m/s (meter / second) 20580 m/min (meter / minute) 0.343 km/s (kilometer / second) 1234.8 km/h (kilometer / hour) 1125.3280839895 ft/s (foot / second) 767.2691481747 mph (mile / hour) 375.1093613298 yd/s (yard / second) 666.7386609072 kn (knot) 1 M (mach - speed of sound) 1.1441E-6 speed of light The speed value 343 m/s (meter / second) in words is "three hundred and fourty-three m/s (meter / second)". This is simple to use online converter of weights and measures. Simply select the input unit, enter the value and click "Convert" button. The value will be converted to all other units of the actual measure. You can simply convert for example between metric, UK imperial and US customary units system. danyk.czAbout MeContact MePrivacy Policy
12749	https://winter.group.shef.ac.uk/webelements/periodicity/elec_binding_energy/	WebElements Periodic Table » Periodicity » Electron binding energies » Periodic table gallery Full property list Abundances Atomic numbers Atomic radii Atomic spectra Atomic weights Block in periodic table Boiling point Bond enthalpies Bond length in element Bulk modulus CAS Registry ID Classification Colour Covalent radii Critical temperature Crystal structure Density of solid Description Discovery Effective nucl. charges Electron bind. energies Electrical resistivity Electron affinity Electron shell structure Electronegativity Electronic configuration Enthalpy: atomization Enthalpy of fusion Expansion coefficient Enthalpy: vaporization Geology Group Name Group numbers Hardness Health hazards History of the element Hydration enthalpies Hydrolysis constants Ionization energies Isolation Isotopes Lattice energies Liquid Range Mass in human Meaning of name Melting points Molar volume Names and symbols NMR Orbital_properties Oxidation numbers Period numbers Poisson's ratio Ionic radii Radii - ionic, Pauling Radii - metallic (12) Radii - valence orbital Radii - Van der Waals Reactions of elements Reduction potentials Reduction potentials Reflectivity Refractive index Rigidity modulus Standard state Superconductivity Term symbol Thermal conductivity Thermodynamics Uses Velocity of sound X-ray mass abs. coeff. Young's modulus Electron binding energies Electron binding energies The binding energies are quoted relative to the vacuum level for rare gases and H 2, N 2, O 2, F 2, and Cl 2 molecules; relative to the Fermi level for metals; and relative to the top of the valence band for semiconductors. Units eV Notes I am grateful to Gwyn Williams (then Brookhaven National Laboratory, USA) who provided the electron binding energy data. The data are adapted from references 1-3. They are tabulated elsewhere on the WWW (reference 4) and in paper form (reference 5). For access to other electron binding energies, select from: Electron binding energies Electron binding energies (K) Electron binding energies (L-I) Electron binding energies (L-II) Electron binding energies (L-III) Electron binding energies (M-I) Electron binding energies (M-II) Electron binding energies (M-III) Electron binding energies (M-IV) Electron binding energies (M-V) Electron binding energies (N-I) Electron binding energies (N-II) Electron binding energies (N-III) Electron binding energies (N-IV) Electron binding energies (N-V) Electron binding energies (N-VI) Electron binding energies (N-VII) Electron binding energies (O-I) Electron binding energies (O-II) Electron binding energies (O-III) Electron binding energies (O-IV) Electron binding energies (O-V) Electron binding energies (P-I) Electron binding energies (P-II) Electron binding energies (P-III) Literature sources J. A. Bearden and A. F. Burr, "Reevaluation of X-Ray Atomic Energy Levels," Rev. Mod. Phys., 1967, 39, 125. M. Cardona and L. Ley, Eds., Photoemission in Solids I: General Principles (Springer-Verlag, Berlin) with additional corrections, 1978. J. C. Fuggle and N. Mårtensson, "Core-Level Binding Energies in Metals," J. Electron Spectrosc. Relat. Phenom., 1980, 21, 275. Gwyn Williams (then Brookhaven National Laboratory, USA) D.R. Lide, (ed.) in Chemical Rubber Company handbook of chemistry and physics, CRC Press, Boca Raton, Florida, USA, 79th edition, 1998. Explore the element of your choice through this periodic table.\| 1 \| 2 \| \| 3 \| 4 \| 5 \| 6 \| 7 \| 8 \| 9 \| 10 \| 11 \| 12 \| 13 \| 14 \| 15 \| 16 \| 17 \| 18 \| --- --- --- --- --- --- --- --- --- \| 1 H \| \| \| 2 He \| \| 3 Li \| 4 Be \| \| 5 B \| 6 C \| 7 N \| 8 O \| 9 F \| 10 Ne \| \| 11 Na \| 12 Mg \| \| 13 Al \| 14 Si \| 15 P \| 16 S \| 17 Cl \| 18 Ar \| \| 19 K \| 20 Ca \| \| 21 Sc \| 22 Ti \| 23 V \| 24 Cr \| 25 Mn \| 26 Fe \| 27 Co \| 28 Ni \| 29 Cu \| 30 Zn \| 31 Ga \| 32 Ge \| 33 As \| 34 Se \| 35 Br \| 36 Kr \| \| 37 Rb \| 38 Sr \| \| 39 Y \| 40 Zr \| 41 Nb \| 42 Mo \| 43 Tc \| 44 Ru \| 45 Rh \| 46 Pd \| 47 Ag \| 48 Cd \| 49 In \| 50 Sn \| 51 Sb \| 52 Te \| 53 I \| 54 Xe \| \| 55 Cs \| 56 Ba \| \| 71 Lu \| 72 Hf \| 73 Ta \| 74 W \| 75 Re \| 76 Os \| 77 Ir \| 78 Pt \| 79 Au \| 80 Hg \| 81 Tl \| 82 Pb \| 83 Bi \| 84 Po \| 85 At \| 86 Rn \| \| 87 Fr \| 88 Ra \| \| 103 Lr \| 104 Rf \| 105 Db \| 106 Sg \| 107 Bh \| 108 Hs \| 109 Mt \| 110 Ds \| 111 Rg \| 112 Cn \| 113 Nh \| 114 Fl \| 115 Mc \| 116 Lv \| 117 Ts \| 118 Og \| \| \| \| Lanthanoids \| \| 57 La \| 58 Ce \| 59 Pr \| 60 Nd \| 61 Pm \| 62 Sm \| 63 Eu \| 64 Gd \| 65 Tb \| 66 Dy \| 67 Ho \| 68 Er \| 69 Tm \| 70 Yb \| \| \| Actinoids \| \| 89 Ac \| 90 Th \| 91 Pa \| 92 U \| 93 Np \| 94 Pu \| 95 Am \| 96 Cm \| 97 Bk \| 98 Cf \| 99 Es \| 100 Fm \| 101 Md \| 102 No \| \| ▸▸ Printable periodic table Orbitron Chemdex Chemputer VSEPR Facebook Twitter Pinterest Privacy About Copyright Thanks... WebElements: THE periodic table on the WWW [winter.group.shef.ac.uk/webelements/] Copyright 1993-2025 Prof Mark J. Winter (Department of Chemistry, The University of Sheffield). All rights reserved. You can reference the WebElements periodic table as follows: "WebElements, (last accessed September 2025)"
12750	https://www.thermaxxjackets.com/news/newton-laplace-equation-sound-velocity/	The Newton-Laplace Equation & Speed of Sound Skip to Content The Newton-Laplace Equation & Speed of Sound Posted on October 28, 2021 In order to determine the speed of sound in a particular medium, we need to know the medium’s elastic properties and its density. A medium’s elastic properties determine whether or not the medium will deform or lose its shape due to external forces. Sound travels faster in mediums with high elasticity and minimal deformity – like steel. Sound travels more slowly in less rigid mediums that deform easily – like rubber. Sound travels more slowly in mediums with greater density. Sir Isaac Newton proposed a simple formula for calculating the speed of sound. Years later Pierre-Simon Laplace would revise Newton’s formula and the new formula would be called the Newton-Laplace Equation. Developing the Equation In the latter half of the 17th century, Sir Isaac Newton published his famous work Principia Mathematica. Newton thought that he correctly predicted the speed of sound though a medium: solid, liquid, or gas. If he knew the density of the medium and pressure acting on the sound wave, he believed he could ascertain the speed of sound by calculating the square root of the pressure divided by the medium’s density: c = speed of sound P = pressure acting on the sound p = density of the medium On page 301 in The Principia Mathematica Newton states “in mediums of equal density and elastic force (the medium’s pressure in Pascals, or Newtons over area measured in meters squared), all pulses are equally swift”. In the earth’s atmosphere at sea level all sound will consistently travel at the same velocity. The same goes for other mediums like homogenius liquids and solids. Pierre-Simon Laplace Newton further states that “if density or the elastic force of the medium were increased, then, because the motive force is increased in the ratio of the elastic force, and the matter to be moved is increased in the ratio of density, the time which is necessary for producing the same motion as before will be increased in the subduplicate ratio of the density, and will be diminished in the in the subduplicate ratio of the elastic force.” Ultimately, Newton is saying that if the medium’s density increases, the sound velocity slows and inversely if the medium’s pressure increases the sound velocity accelerates. Later in the same century, French mathematician Pierre-Simon Laplace saw the flaw in Newton’s thinking and ultimately corrected Newton’s formula. He expanded Newton’s equation to include the idea that the process is not isothermic as Newton had thought, but it is adiabatic. Laplace slightly revised Newton’s formula by adding gamma to Newton’s pressure component. Laplace correction: Newton’s formula neglected the influence of heat on the speed of sound. Laplace made this correction. Laplace multiplied the gamma (heat component) x the pressure. Experimentation proved that Newton’s results were wrong. Results from Newton’s equations fell short of what really took place. Laplace’s fix hit the mark. A new equation was born: The Newton-Laplace Equations. Laplace shorted the equation by having K = gamma × pressure. K refers to the elastic bulk modulus. The formula is called the Newton-Laplace equation: c = speed of sound K = elastic bulk modulus p = density of the medium Putting the Equation to Work Determining Bulk Modulus The speed of sound in sea level atmosphere at 20° Celsius is 343.21 m/s. Since we know the density of air at sea level is 1.2041 kg/m³. We can solve for K. Squaring both sides leaves (rounding): or and finally: Determining Density The speed of sound in fresh water 1,428 m/s. Water’s elastic bulk modulus is 2.2 × 109 Pa. Knowing these two values, we can confirm water’s density. Squaring both sides leaves: or Determining the Speed of Sound The density of cold-rolled steel is 7861 kg/m3. Its elastic bulk modulus is 159 GPa. With these two values, we can calculate the speed of sound through cold-rolled steel. Conclusion If we know a particular medium’s elastic bulk modulus and its density, we can calculate the speed of sound traveling through it. Sound travels faster in mediums with higher elasticity like steel and iron as shown in the steel equation above. In mediums like rubber and fiberglass sound travels slower. These mediums easily deform when forces are applied. We can conclude that the sound wave is being attenuated and or absorbed when passing through solids that are easily deformed when a force is applied. The stiffer and less rigid the medium the faster sound will travel through it. Get more information about Thermaxx’s Sound Insulation or contact us! This entry was posted in Noise, Sound Insulation. 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12751	https://www2.stat.duke.edu/courses/Fall21/sta601.001/slides/05-hypothesis-testing-handout.pdf	Lecture 5: Basics of Bayesian Hypothesis Testing Merlise Clyde September 9, 2021 1 / 17 Hypothesis Testing Suppose we have univariate data goal is to test Frequentist testing - likelihood ratio, Wald, score, UMP, confidence regions, etc Need a test statistic (and its sampling distribution) p-value: Calculate the probability of seeing a dataset/test statistics as extreme or more extreme than the oberved data with repeated sampling under the null hypothesis yi iid ∼N (θ, 1) H0 : θ = 0; vs H1 : θ ≠0 T(y(n)) 2 / 17 Errors if p-value is less than a pre-specified then reject in favor of Type I error: falsely concluding in favor of when is true To maintain a Type I error rate of , then we reject in favor of when For this to be a valid frequents test the p-value must have a uniform distribution under Type II error: failing to conclude in favor of when is true 1 - P(Type II error) is the power of the test Note: we never conclude in favor of . We are looking for enough evidence to reject . But if we fail to reject we do not conclude that it is true! α H0 H1 H1 H0 α H0 H1 p < α H0 H1 H1 H0 H0 3 / 17 Bayesian Approach 1. Put a prior on , . 2. Compute posterior for updated parameters and . θ π(θ) = N (θ0, 1/τ 2 0 ) θ ∣y(n) ∼N (θn, 1/τ 2 n) θn τ 2 n 4 / 17 Informal Credible Intervals 1. Compute a 95% CI based on the posterior. 2. Reject if interval does not contain zero. Tail Areas: 1. Compute and 2. Report minimum of these probabilities as a "Bayesian p-value" Note: Tail probability is not the same as H0 Pr(θ > 0 ∣y(n)) Pr(θ < 0 ∣y(n)) Pr(H0 ∣y(n)) 5 / 17 Formal Bayesian Hypothesis Testing Unknowns are and Put a prior on the actual hypotheses/models, that is, on and . For example, set and , if a priori, we believe the two hypotheses are equally likely. Likelihood of the hypotheses H0 H1 π(H0) = Pr(H0 = True) π(H1) = Pr(H1 = True) π(H0) = 0.5 π(H1) = 0.5 L(Hi) ∝p(y(n) ∣Hi) p(y(n) ∣H0) = n ∏ i=1 (2π)−1/2 exp − (yi −0)2 1 2 p(y(n) ∣H1) = ∫ Θ p(y(n) ∣H1, θ)p(θ ∣H1) dθ 6 / 17 Bayesian Approach Priors on parameters under each hypothesis In our simple normal model, the only unknown parameter is under , with probability 1 under , Could take . Compute marginal likelihoods for each hypothesis, that is, and . Obtain posterior probabilities of and via Bayes Theorem. θ H0 θ = 0 H0 θ ∈R π(θ) = N (θ0, 1/τ 2 0 ) L(H0) L(H1) H0 H1 7 / 17 Bayesian Approach - Decisions Loss function for hypothesis testing is the chosen hypothesis is the true hypothesis, for short Two types of errors: Type I error: and Type II error: and Loss function: weights how bad making a Type I error weights how bad making a Type II error ^ H Htrue H ^ H = 1 H = 0 ^ H = 0 H = 1 L( ^ H, H) = w1 1( ^ H = 1, H = 0) + w2 1( ^ H = 0, H = 1) w1 w2 8 / 17 Loss Function Functions and Decisions Relative weights Special case known as 0-1 loss (most common) Bayes Risk (Posterior Expected Loss) Minimize loss by picking hypothesis with the highest posterior probability L( ^ H, H) = 1( ^ H = 1, H = 0) + w 1( ^ H = 0, H = 1) w = 1 L( ^ H, H) = 1( ^ H ≠H) EH∣y(n)[L( ^ H, H)] = 1( ^ H = 1)π(H0 ∣y(n)) + 1( ^ H = 0)π(H1 ∣y(n)) 9 / 17 Bayesian hypothesis testing Using Bayes theorem, where and are the marginal likelihoods hypotheses. If for example we set and a priori, then The ratio is known as the Bayes factor in favor of , and often written as . Similarly, we can compute . π(H1 ∣Y ) = , p(y(n) ∣H1)π(H1) p(y(n) ∣H0)π(H0) + p(y(n) ∣H1)π(H1) p(y(n) ∣H0) p(y(n) ∣H1) π(H0) = 0.5 π(H1) = 0.5 π(H1 ∣Y ) = = = . 0.5p(y(n) ∣H1) 0.5p(y(n) ∣H0) + 0.5p(y(n) ∣H1) p(y(n) ∣H1) p(y(n) ∣H0) + p(y(n) ∣H1) 1 + 1 p(y(n)∣H0) p(y(n)∣H1) p(y(n)∣H0) p(y(n)∣H1) H0 BF 01 BF 10 10 / 17 Bayes factors Bayes factor: is a ratio of marginal likelihoods and it provides a weight of evidence in the data in favor of one model over another. It is often used as an alternative to the frequentist p-value. Rule of thumb: is strong evidence for ; is decisive evidence for . Notice that for our example, the higher the value of , that is, the weight of evidence in the data in favor of , the lower the marginal posterior probability that is true. That is, here, as , . BF 01 > 10 H0 BF 01 > 100 H0 π(H1 ∣Y ) = = 1 + 1 p(y(n)∣H0) p(y(n)∣H1) 1 BF 01 + 1 BF 01 H0 H1 BF 01 ↑π(H1 ∣Y ) ↓ 11 / 17 Bayes factors Let's look at another way to think of Bayes factors. First, recall that so that Therefore, the Bayes factor can be thought of as the factor by which our prior odds change (towards the posterior odds) in the light of the data. π(H1 ∣Y ) = , p(y(n) ∣H1)π(H1) p(y(n) ∣H0)π(H0) + p(y(n) ∣H1)π(H1) = ÷ = × ∴  posterior odds =  prior odds ×  Bayes factor BF 01 π(H0\|Y ) π(H1\|Y ) p(y(n)\|H0)π(H0) p(y(n)\|H0)π(H0) + p(y(n)\|H1)π(H1) p(y(n)\|H1)π(H1) p(y(n)H0)π(H0) + p(y(n)\|H1)π(H1) p(y(n)\|H0)π(H0) p(y(n)\|H0)π(H0) + p(y(n)\|H1)π(H1) p(y(n)\|H0)π(H0) + p(y(n)\|H1)π(H1) p(y(n)\|H1)π(H1) π(H0 ∣Y ) π(H1 ∣Y ) π(H0) π(H1) p(y(n) ∣H0) p(y(n) ∣H1) 12 / 17 Likelihoods & Evidence Maximized Likelihood p-value = 0.05 13 / 17 Marginal Likelihoods & Evidence Maximized Likelihood = 1.73 BF 10 14 / 17 Candidate's Formula (Besag 1989) Alternative expression for Bayes Factor ratio of the prior to posterior densities for evaluated at zero Savage-Dickey Ratio = p(y(n) ∣H1) p(y(n) ∣H0) πθ(0 ∣H1) πθ(0 ∣y(n), H1) θ 15 / 17 Prior Plots were based on a centered at value for under (goes back to Jeffreys) "unit information prior" equivalent to a prior sample size is 1 What happens if ? What happens of ? θ ∣H1 ∼N(0, 1) θ H0 n →∞ τ0 →0 16 / 17 Precision Posterior Probability of = 0.9361 As the posterior probability of goes to 0! Bartlett's Paradox - the paradox is that a seemingly non-informative prior for is very informative about ! τ0 = 1/1000 H0 τ0 →0 H1 θ H 17 / 17
12752	https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_1e_(OpenStax)/07%3A_Trigonometric_Identities_and_Equations	Skip to main content 7: Trigonometric Identities and Equations Last updated : May 1, 2022 Save as PDF 6.R: Periodic Functions (Review) 7.0: Prelude to Trigonometric Identities and Equations Page ID : 1272 OpenStax OpenStax ( \newcommand{\kernel}{\mathrm{null}\,}) In this chapter, we discuss how to manipulate trigonometric equations algebraically by applying various formulas and trigonometric identities. We will also investigate some of the ways that trigonometric equations are used to model real-life phenomena. 7.0: Prelude to Trigonometric Identities and Equations : Math is everywhere, even in places we might not immediately recognize. For example, mathematical relationships describe the transmission of images, light, and sound. Such phenomena are described using trigonometric equations and functions. In this chapter, we discuss how to manipulate trigonometric equations algebraically by applying various formulas and trigonometric identities. 7.1: Solving Trigonometric Equations with Identities : In this section, we will begin an examination of the fundamental trigonometric identities, including how we can verify them and how we can use them to simplify trigonometric expressions. 7.2: Sum and Difference Identities : In this section, we will learn techniques that will enable us to solve useful problems. The formulas that follow will simplify many trigonometric expressions and equations. Keep in mind that, throughout this section, the termformula is used synonymously with the word identity. 7.3: Double-Angle, Half-Angle, and Reduction Formulas : In this section, we will investigate three additional categories of identities. Double-angle identities are derived from the sum formulas of the fundamental trigonometric functions: sine, cosine, and tangent. Reduction formulas are especially useful in calculus, as they allow us to reduce the power of the trigonometric term. Half-angle formulas allow us to find the value of trigonometric functions involving half-angles, whether the original angle is known or not. 7.4: Sum-to-Product and Product-to-Sum Formulas : From the sum and difference identities, we can derive the product-to-sum formulas and the sum-to-product formulas for sine and cosine. The product-to-sum formulas can rewrite products of sines, products of cosines, and products of sine and cosine as sums or differences of sines and cosines. We can also derive the sum-to-product identities from the product-to-sum identities using substitution. The sum-to-product formulas are used to rewrite sum or difference as products of sines and cosines. 7.5: Solving Trigonometric Equations : In earlier sections of this chapter, we looked at trigonometric identities. Identities are true for all values in the domain of the variable. In this section, we begin our study of trigonometric equations to study real-world scenarios such as the finding the dimensions of the pyramids. 7.6: Modeling with Trigonometric Equations : Many natural phenomena are also periodic. For example, the phases of the moon have a period of approximately 28 days, and birds know to fly south at about the same time each year. So how can we model an equation to reflect periodic behavior? First, we must collect and record data. We then find a function that resembles an observed pattern and alter the function to get adependable model. Here. we will take a deeper look at specific types of periodic behavior and model equations to fit data. 7.E: Trigonometric Identities and Equations (Exercises) 7.R: Trigonometric Identities and Equations (Review) 6.R: Periodic Functions (Review) 7.0: Prelude to Trigonometric Identities and Equations
12753	https://www.cia.gov/the-world-factbook/countries/italy/	Italy - The World Factbook Skip to main content Go to CIA.gov Search CIA.gov Search WFB The World Factbook Countries Maps References About Explore All Countries Italy Europe Page last updated: September 10, 2025 Photos of Italy view 53 photos Country Flag View Details Country Map View Details Special Country Products Country FactsheetTravel Facts Locator Map View Details Contents Introduction Geography People and Society Environment Government Economy Energy Communications Transportation Military and Security Space Terrorism Transnational Issues Introduction Background Italy became a nation-state in 1861 when the regional states of the peninsula, along with Sardinia and Sicily, were united under King Victor EMMANUEL II. An era of parliamentary government came to a close in the early 1920s when Benito MUSSOLINI established a Fascist dictatorship. His alliance with Nazi Germany led to Italy's defeat in World War II. A democratic republic replaced the monarchy in 1946, and economic revival followed. Italy is a charter member of NATO, as well as the European Economic Community (EEC) and its successors, the EC and the EU. It has been at the forefront of European economic and political unification, joining the Economic and Monetary Union in 1999. Persistent problems include sluggish economic growth, high youth and female unemployment, organized crime, corruption, and economic disparities between southern Italy and the more prosperous north. Tip Visit the Definitions and Notes page to view a description of each topic. Definitions and Notes Geography Location Southern Europe, a peninsula extending into the central Mediterranean Sea, northeast of Tunisia Geographic coordinates 42 50 N, 12 50 E Map references Europe Area total : 301,340 sq km land: 294,140 sq km water: 7,200 sq km note: includes Sardinia and Sicily comparison ranking: total 73 Area - comparative almost twice the size of Georgia; slightly larger than Arizona Area comparison map: Land boundaries total: 1,836.4 km border countries (6): Austria 404 km; France 476 km; Holy See (Vatican City) 3.4 km; San Marino 37 km; Slovenia 218 km; Switzerland 698 km Coastline 7,600 km Maritime claims territorial sea: 12 nm continental shelf: 200-m depth or to the depth of exploitation Climate predominantly Mediterranean; alpine in far north; hot, dry in south Terrain mostly rugged and mountainous; some plains, coastal lowlands Elevation highest point: Mont Blanc (Monte Bianco) de Courmayeur (a secondary peak of Mont Blanc) 4,748 m lowest point: Mediterranean Sea 0 m mean elevation: 538 m Natural resources coal, antimony, mercury, zinc, potash, marble, barite, asbestos, pumice, fluorspar, feldspar, pyrite (sulfur), natural gas and crude oil reserves, fish, arable land Land use agricultural land: 44% (2022 est.) arable land: 24% (2022 est.) permanent crops: 8.1% (2022 est.) permanent pasture: 11.9% (2022 est.) forest: 32.7% (2022 est.) other: 23.3% (2022 est.) Irrigated land 24,460 sq km (2021) Major watersheds (area sq km) Atlantic Ocean drainage: Rhine-Maas (198,735 sq km), (Black Sea) Danube (795,656 sq km), (Adriatic Sea) Po (76,997 sq km), (Mediterranean Sea) Rhone (100,543 sq km) Population distribution a fairly even population distribution exists throughout most of the country, with coastal areas, the Po River Valley, and urban centers (particularly Milan, Rome, and Naples) attracting larger and denser populations Natural hazards regional risks include landslides, mudflows, avalanches, earthquakes, volcanic eruptions, flooding; land subsidence in Venice volcanism: significant volcanic activity; Etna (3,330 m) is Europe's most active volcano, and its flank eruptions pose a threat to nearby Sicilian villages; Etna, along with the famous Vesuvius, have both been deemed Decade Volcanoes by the International Association of Volcanology and Chemistry of the Earth's Interior, worthy of study due to their explosive history and close proximity to human populations; Stromboli, on its namesake island, has also been continuously active with moderate volcanic activity; other historically active volcanoes include Campi Flegrei, Ischia, Larderello, Pantelleria, Vulcano, and Vulsini Geography - note strategic location dominating central Mediterranean, as well as southern sea and air approaches to Western Europe People and Society Population total: 60,964,931 (2024 est.) male: 29,414,065 female: 31,550,866 comparison rankings: total 24; male 25; female 24 Nationality noun: Italian(s) adjective: Italian Ethnic groups Italian (includes small clusters of German-, French-, and Slovene-Italians in the north, Albanian-Italians, Croat-Italians, and Greek-Italians in the south) Languages Italian (official), German (parts of Trentino-Alto Adige region are predominantly German-speaking), French (small French-speaking minority in Valle d'Aosta region), Slovene (Slovene-speaking minority in the Trieste-Gorizia area), Croatian (in Molise) major-language sample(s): L'Almanacco dei fatti del mondo, l'indispensabile fonte per le informazioni di base. (Italian) The World Factbook, the indispensable source for basic information. Italian audio sample: Religions Christian 80.8% (overwhelmingly Roman Catholic with very small groups of Jehovah's Witnesses and Protestants), Muslim 4.9%, unaffiliated 13.4%, other 0.9% (2020 est.) Age structure 0-14 years: 11.9% (male 3,699,167/female 3,531,734) 15-64 years: 64.5% (male 19,378,160/female 19,958,137) 65 years and over: 23.6% (2024 est.) (male 6,336,738/female 8,060,995) 2024 population pyramid: Dependency ratios total dependency ratio: 55 (2024 est.) youth dependency ratio: 18.4 (2024 est.) elderly dependency ratio: 36.6 (2024 est.) potential support ratio: 2.7 (2024 est.) Median age total: 48.4 years (2024 est.) male: 47.4 years female: 49.4 years comparison ranking: total 5 Population growth rate -0.08% (2024 est.) comparison ranking: 201 Birth rate 7.1 births/1,000 population (2024 est.) comparison ranking: 222 Death rate 11.2 deaths/1,000 population (2024 est.) comparison ranking: 22 Net migration rate 3.4 migrant(s)/1,000 population (2024 est.) comparison ranking: 33 Population distribution a fairly even population distribution exists throughout most of the country, with coastal areas, the Po River Valley, and urban centers (particularly Milan, Rome, and Naples) attracting larger and denser populations Urbanization urban population: 72% of total population (2023) rate of urbanization: 0.27% annual rate of change (2020-25 est.) Major urban areas - population 4.316 million ROME (capital), 3.155 million Milan, 2.179 million Naples, 1.802 million Turin, 913,000 Bergamo, 850,000 Palermo (2023) Sex ratio at birth: 1.06 male(s)/female 0-14 years: 1.05 male(s)/female 15-64 years: 0.97 male(s)/female 65 years and over: 0.79 male(s)/female total population: 0.93 male(s)/female (2024 est.) Mother's mean age at first birth 31.4 years (2020 est.) Maternal mortality ratio 6 deaths/100,000 live births (2023 est.) comparison ranking: 163 Infant mortality rate total: 3.1 deaths/1,000 live births (2024 est.) male: 3.2 deaths/1,000 live births female: 2.9 deaths/1,000 live births comparison ranking: total 207 Life expectancy at birth total population: 83 years (2024 est.) male: 80.7 years female: 85.5 years comparison ranking: total population 19 Total fertility rate 1.26 children born/woman (2024 est.) comparison ranking: 219 Gross reproduction rate 0.61 (2024 est.) Drinking water source improved: total: 99.9% of population (2022 est.) unimproved: total: 0.1% of population (2022 est.) Health expenditure 9% of GDP (2022) 11.8% of national budget (2022 est.) Physician density 4.19 physicians/1,000 population (2022) Hospital bed density 3.2 beds/1,000 population (2020 est.) Sanitation facility access improved: urban: 100% of population (2022 est.) rural: 100% of population (2022 est.) total: 100% of population (2022 est.) unimproved: urban: 0% of population (2022 est.) rural: 0% of population (2022 est.) total: 0% of population (2022 est.) Obesity - adult prevalence rate 19.9% (2016) comparison ranking: 108 Alcohol consumption per capita total: 7.65 liters of pure alcohol (2019 est.) beer: 1.99 liters of pure alcohol (2019 est.) wine: 4.83 liters of pure alcohol (2019 est.) spirits: 0.83 liters of pure alcohol (2019 est.) other alcohols: 0 liters of pure alcohol (2019 est.) comparison ranking: total 51 Tobacco use total: 19.8% (2025 est.) male: 23.2% (2025 est.) female: 16.6% (2025 est.) comparison ranking: total 71 Currently married women (ages 15-49) 52.5% (2023 est.) Education expenditure 4% of GDP (2022 est.) 7.2% national budget (2022 est.) comparison ranking: Education expenditure (% GDP) 105 Literacy total population: 99% (2019 est.) male: 99% (2019 est.) female: 99% (2019 est.) School life expectancy (primary to tertiary education) total: 17 years (2023 est.) male: 16 years (2023 est.) female: 17 years (2023 est.) Environment Environmental issues air pollution from industrial emissions; water pollution from industrial and agricultural effluents, as well as acid rain; inadequate industrial waste treatment and disposal facilities International environmental agreements party to: Air Pollution, Air Pollution-Nitrogen Oxides, Air Pollution-Persistent Organic Pollutants, Air Pollution-Sulphur 85, Air Pollution-Sulphur 94, Air Pollution-Volatile Organic Compounds, Antarctic-Environmental Protection, Antarctic-Marine Living Resources, Antarctic Seals, Antarctic Treaty, Biodiversity, Climate Change, Climate Change-Kyoto Protocol, Climate Change-Paris Agreement, Comprehensive Nuclear Test Ban, Desertification, Endangered Species, Environmental Modification, Hazardous Wastes, Law of the Sea, Marine Dumping-London Convention, Marine Dumping-London Protocol, Nuclear Test Ban, Ozone Layer Protection, Ship Pollution, Tropical Timber 2006, Wetlands, Whaling signed, but not ratified: Air Pollution-Heavy Metals, Air Pollution-Multi-effect Protocol Climate predominantly Mediterranean; alpine in far north; hot, dry in south Land use agricultural land: 44% (2022 est.) arable land: 24% (2022 est.) permanent crops: 8.1% (2022 est.) permanent pasture: 11.9% (2022 est.) forest: 32.7% (2022 est.) other: 23.3% (2022 est.) Urbanization urban population: 72% of total population (2023) rate of urbanization: 0.27% annual rate of change (2020-25 est.) Carbon dioxide emissions 307.442 million metric tonnes of CO2 (2023 est.) from coal and metallurgical coke: 26.15 million metric tonnes of CO2 (2023 est.) from petroleum and other liquids: 162.688 million metric tonnes of CO2 (2023 est.) from consumed natural gas: 118.604 million metric tonnes of CO2 (2023 est.) comparison ranking: total emissions 19 Particulate matter emissions 12.3 micrograms per cubic meter (2019 est.) Methane emissions energy: 276.4 kt (2022-2024 est.) agriculture: 764.9 kt (2019-2021 est.) waste: 523.4 kt (2019-2021 est.) other: 35.3 kt (2019-2021 est.) Waste and recycling municipal solid waste generated annually: 30.088 million tons (2024 est.) percent of municipal solid waste recycled: 39.9% (2022 est.) Total water withdrawal municipal: 9.19 billion cubic meters (2020 est.) industrial: 7.7 billion cubic meters (2020 est.) agricultural: 17 billion cubic meters (2020 est.) Total renewable water resources 191.3 billion cubic meters (2020 est.) Geoparks total global geoparks and regional networks: 12 (2025) global geoparks and regional networks: Adamello-Brenta; Alpi Apuane; Aspromonte; Beigua; Cilento, Vallo di Diano e Alburni; Madonie; Maiella; MurGEopark; Pollino; Rocca di Cerere; Sesia Val Grande; Tuscan Mining Park (2025) Government Country name conventional long form: Italian Republic conventional short form: Italy local long form: Repubblica Italiana local short form: Italia former: Kingdom of Italy etymology: derivation is unclear; traditionally said to come from the Vitali, a tribe that settled in what is now Calabria, and whose name is believed to be linked to the Latin word vitulus, or "calf;" alternatively, the name may derive from a local ruler known to the Romans as Italus Government type parliamentary republic Capital name: Rome geographic coordinates: 41 54 N, 12 29 E time difference: UTC+1 (6 hours ahead of Washington, DC, during Standard Time) daylight saving time: +1hr, begins last Sunday in March; ends last Sunday in October etymology: by tradition, named after Romulus, one of the legendary founders of the city, but the name Romulus may instead derive from the city's name; the name Rome may come from an Etruscan name for the Tiber River, which was Roma or Ruma Administrative divisions 15 regions (regioni, singular - regione) and 5 autonomous regions (regioni autonome, singular - regione autonoma) regions: Abruzzo, Basilicata, Calabria, Campania, Emilia-Romagna, Lazio (Latium), Liguria, Lombardia, Marche, Molise, Piemonte (Piedmont), Puglia (Apulia), Toscana (Tuscany), Umbria, Veneto autonomous regions: Friuli Venezia Giulia, Sardegna (Sardinia), Sicilia (Sicily), Trentino-Alto Adige (Trentino-South Tyrol) or Trentino-Suedtirol (German), Valle d'Aosta (Aosta Valley) or Vallée d'Aoste (French) Legal system civil law system; Constitutional Court reviews legislation under certain conditions Constitution history: previous 1848 (originally for the Kingdom of Sardinia and adopted by the Kingdom of Italy in 1861); latest enacted 22 December 1947, adopted 27 December 1947, entered into force 1 January 1948 amendment process: proposed by both houses of Parliament; passage requires two successive debates and approval by absolute majority of each house on the second vote; a referendum is only required when requested by one fifth of the members of either house, by voter petition, or by 5 Regional Councils (elected legislative assemblies of the 15 first-level administrative regions and 5 autonomous regions of Italy); referendum not required if an amendment has been approved by a two-thirds majority in each house in the second vote International law organization participation accepts compulsory ICJ jurisdiction with reservations; accepts ICCt jurisdiction Citizenship citizenship by birth: no citizenship by descent only: at least one parent must be a citizen of Italy dual citizenship recognized: yes residency requirement for naturalization: 4 years for EU nationals, 5 years for refugees and specified exceptions, 10 years for all others Suffrage 18 years of age; universal except in senatorial elections, where minimum age is 25 Executive branch chief of state: President Sergio MATTARELLA (since 3 February 2015) head of government: Prime Minister Giorgia MELONI (since 22 October 2022); the prime minister's official title is President of the Council of Ministers cabinet: Council of Ministers proposed by the prime minister, who is known officially as the President of the Council of Ministers and locally as the premier; nominated by the president election/appointment process: president indirectly elected by an electoral college consisting of both houses of Parliament and 58 regional representatives for a 7-year term (no term limits); prime minister appointed by the president, confirmed by parliament most recent election date: 24-29 January 2022 (eight rounds) election results: 2022: Sergio MATTARELLA (independent) reelected president; electoral college vote count in eighth round - 759 out of 1,009 (505 vote threshold) 2015: Sergio MATTARELLA (independent) elected president; electoral college vote count in fourth round - 665 out of 995 (505 vote threshold) expected date of next election: 2029 Legislative branch legislature name: Parliament (Il Parlamento) legislative structure: bicameral Legislative branch - lower chamber chamber name: Chamber of Deputies (Camera dei Deputati) number of seats: 400 (all directly elected) electoral system: mixed system scope of elections: full renewal term in office: 5 years most recent election date: 9/25/2022 parties elected and seats per party: Coalition Brothers of Italy (FdI) - Lega - Forza Italia - Us Moderates (Noi moderati, NM) (237); Democratic Party - Democratic and Progressive Italy (PD-IDP) - Greens and Left Alliance (AVS) - +EUROPA" - Civic Commitment (IC) (84); Five Star Movement (M5s) (52); Action - Italia Viva (21); Other (6) percentage of women in chamber: 32.8% expected date of next election: September 2027 Legislative branch - upper chamber chamber name: Senate (Senato della Repubblica) number of seats: 205 (200 directly elected; 5 appointed) electoral system: mixed system scope of elections: full renewal term in office: 5 years most recent election date: 9/25/2022 parties elected and seats per party: Coalition Brothers of Italy (FdI) - Lega - Forza Italia - Us Moderates (Noi moderati, NM) (115); Democratic Party - Democratic and Progressive Italy (PD-IDP) - Greens and Left Alliance (AVS) - +EUROPA" - Civic Commitment (IC) (44); Five Star Movement (M5s) (28); Other (13) percentage of women in chamber: 36.3% expected date of next election: September 2027 Judicial branch highest court(s): Supreme Court of Cassation or Corte Suprema di Cassazione (consists of the first president, deputy president, 54 justices presiding over 6 civil and 7 criminal divisions, and 288 judges; an additional 30 judges of lower courts serve as supporting judges; cases normally heard by 5-judge panels; more complex cases heard by 9-judge panels); Constitutional Court or Corte Costituzionale (consists of the court president and 14 judges) judge selection and term of office: Supreme Court judges appointed by the High Council of the Judiciary, headed by the president of the republic; judges may serve for life; Constitutional Court judges - 5 appointed by the president, 5 elected by Parliament, 5 elected by select higher courts; judges serve up to 9 years subordinate courts: various lower civil and criminal courts (primary and secondary tribunals and courts of appeal) Political parties Action-Italia Viva Associative Movement of Italians Abroad or MAIE Brothers of Italy or FdI Democratic Party or PD Five Star Movement or M5S Forza Italia or FI Free and Equal (Liberi e Uguali) or LeU Greens and Left Alliance or AVS Italexit League or Lega More Europe or +EU Popular Union or PU South calls North or ScN South Tyrolean Peoples Party or SVP other minor parties Diplomatic representation in the US chief of mission: Ambassador-designate Marco PERONACI; Chargé d’Affaires Alessandro GONZALES (since 4 July 2025) chancery: 3000 Whitehaven Street NW, Washington, DC 20008 telephone: (202) 612-4400 FAX: (202) 518-2154 email address and website: washington.ambasciata@esteri.it consulate(s) general: Boston, Chicago, Houston, Miami, New York, Los Angeles, Philadelphia, San Francisco consulate(s): Detroit Diplomatic representation from the US chief of mission: Ambassador Tilman J. FERTITTA (since 6 May 2025); note - also accredited to San Marino embassy: via Vittorio Veneto 121, 00187 Roma mailing address: 9500 Rome Place, Washington DC 20521-9500 telephone: 06-46741 FAX: 06-4674-2244 email address and website: uscitizenrome@state.gov consulate(s) general: Florence, Milan, Naples International organization participation ADB (nonregional member), AfDB (nonregional member), Arctic Council (observer), Australia Group, BIS, BSEC (observer), CBSS (observer), CD, CDB, CE, CEI, CERN, EAPC, EBRD, ECB, EIB, EITI (implementing country), EMU, ESA, EU, FAO, FATF, G-7, G-8, G-10, G-20, IADB, IAEA, IBRD, ICAO, ICC (national committees), ICCt, ICRM, IDA, IEA, IFAD, IFC, IFRCS, IGAD (partners), IHO, ILO, IMF, IMO, IMSO, Interpol, IOC, IOM, IPU, ISO, ITSO, ITU, ITUC (NGOs), LAIA (observer), MIGA, MINURSO, NATO, NEA, NSG, OAS (observer), OECD, OPCW, OSCE, Pacific Alliance (observer), Paris Club, PCA, PIF (partner), Schengen Convention, SELEC (observer), SICA (observer), UN, UNCTAD, UNESCO, UNHCR, UNIDO, UNIFIL, Union Latina, UNMOGIP, UNOOSA, UNRWA, UNTSO, UNWTO, UPU, Wassenaar Arrangement, WCO, WHO, WIPO, WMO, WTO, ZC Independence 17 March 1861 note: the Kingdom of Italy proclaimed on 17 March 1861, but Italy was not fully unified until 1871 National holiday Republic Day, 2 June (1946) Flag description: three equal vertical bands of green (left side), white, and red meaning: colors are those of Milan (red and white) combined with the green uniform color of the Milanese civic guard history: design inspired by the French flag that Napoleon brought to Italy in 1797 note: similar to the flags of Mexico (longer, darker shades of green and red, and has its coat of arms centered on the white band), Ireland (longer and with orange instead of red), and Cote d'Ivoire (colors reversed) National symbol(s) five-pointed white star (Stella d'Italia) National color(s) red, white, green National coat of arms this coat of arms has been a symbol of the Italian Republic since May 5, 1948, when Paolo Paschetto’s design won a two-year public competition; the olive branch symbolizes national and global peace; the oak branch stands for the strength and the dignity of the Italian people, and the steel cog-wheel for their hard work; the single star represents Italy’s solidarity National anthem(s) title: "Il Canto degli Italiani" (The Song of the Italians) lyrics/music: Goffredo MAMELI/Michele NOVARO history: adopted 2005; the anthem, originally written in 1847, is also known as "L'Inno di Mameli" (Mameli's Hymn), and "Fratelli d'Italia" (Brothers of Italy) National heritage total World Heritage Sites: 60 (54 cultural, 6 natural) selected World Heritage Site locales:Historic Center of Rome (c); Archaeological Areas of Pompeii, Herculaneum, and Torre Annunziata (c); Venice and its Lagoon (c); Historic Center of Florence (c); Piazza del Duomo, Pisa (c); Historic Centre of Naples (c); Portovenere, Cinque Terre, and the Islands (Palmaria, Tino and Tinetto)(c); Mount Etna (n); Cultural landscape of the Benedictine settlements in medieval Italy (c); Church and Dominican Convent of Santa Maria delle Grazie with “The Last Supper” by Leonardo da Vinci (c); City of Vicenza and the Palladian Villas of the Veneto (c); Crespi d'Adda (c); Early Christian Monuments of Ravenna (c); Historic Centre of the City of Pienza (c); Cathedral, Torre Civica and Piazza Grande, Modena (c); Costiera Amalfitana (c); Villa Romana del Casale (c); Archaeological Area and the Patriarchal Basilica of Aquileia (c); Cilento and Vallo di Diano National Park with the Archeological Sites of Paestum and Velia, and the Certosa di Padula (c); Historic Centre of Urbino (c); Villa Adriana (Tivoli) (c); Assisi, the Basilica of San Francesco and Other Franciscan Sites (c); City of Verona (c); Isole Eolie (Aeolian Islands) (n); Etruscan Necropolises of Cerveteri and Tarquinia (c); Val d'Orcia (c); Mantua and Sabbioneta (c); The Dolomites (n); Prehistoric Pile Dwellings around the Alps (c); Medici Villas and Gardens in Tuscany (c); Venetian Works of Defence between the 16th and 17th Centuries: Stato da Terra – Western Stato da Mar (c); Padua’s fourteenth-century fresco cycles (c); The Porticoes of Bologna (c); Evaporitic Karst and Caves of Northern Apennines (n); Via Appia: Regina Viarum (c) Economy Economic overview high-income, core EU economy; strong services, manufacturing, and tourism sectors; modest growth supported by net exports, low inflation, and public investments via EU funds; tight labor market with aging workforce and shortages in specialized skills; high public debt levels Real GDP (purchasing power parity) $3.133 trillion (2024 est.) $3.11 trillion (2023 est.) $3.088 trillion (2022 est.) note: data in 2021 dollars comparison ranking: 11 Real GDP growth rate 0.7% (2024 est.) 0.7% (2023 est.) 4.8% (2022 est.) note: annual GDP % growth based on constant local currency comparison ranking: 185 Real GDP per capita $53,100 (2024 est.) $52,700 (2023 est.) $52,300 (2022 est.) note: data in 2021 dollars comparison ranking: 37 GDP (official exchange rate) $2.373 trillion (2024 est.) note: data in current dollars at official exchange rate Inflation rate (consumer prices) 1% (2024 est.) 5.6% (2023 est.) 8.2% (2022 est.) note: annual % change based on consumer prices comparison ranking: 23 GDP - composition, by sector of origin agriculture: 2% (2024 est.) industry: 21.7% (2024 est.) services: 65.6% (2024 est.) note: figures may not total 100% due to non-allocated consumption not captured in sector-reported data comparison rankings: agriculture 151; industry 122; services 52 GDP - composition, by end use household consumption: 58.3% (2023 est.) government consumption: 17.8% (2023 est.) investment in fixed capital: 22.5% (2023 est.) investment in inventories: 0.4% (2023 est.) exports of goods and services: 33.5% (2023 est.) imports of goods and services: -32.1% (2023 est.) note: figures may not total 100% due to rounding or gaps in data collection Agricultural products milk, wheat, grapes, tomatoes, maize, olives, apples, oranges, sugar beets, rice (2023) note: top ten agricultural products based on tonnage Industries tourism, machinery, iron and steel, chemicals, food processing, textiles, motor vehicles, clothing, footwear, ceramics Industrial production growth rate 0.2% (2024 est.) note: annual % change in industrial value added based on constant local currency comparison ranking: 130 Labor force 25.828 million (2024 est.) note: number of people ages 15 or older who are employed or seeking work comparison ranking: 27 Unemployment rate 6.8% (2024 est.) 7.7% (2023 est.) 8.1% (2022 est.) note: % of labor force seeking employment comparison ranking: 123 Youth unemployment rate (ages 15-24) total: 21.8% (2024 est.) male: 19.9% (2024 est.) female: 24.8% (2024 est.) note: % of labor force ages 15-24 seeking employment comparison ranking: total 49 Population below poverty line 20.1% (2021 est.) note: % of population with income below national poverty line Gini Index coefficient - distribution of family income 33.7 (2022 est.) note: index (0-100) of income distribution; higher values represent greater inequality comparison ranking: 91 Average household expenditures on food: 14.7% of household expenditures (2023 est.) on alcohol and tobacco: 3.8% of household expenditures (2023 est.) Household income or consumption by percentage share lowest 10%: 2.5% (2022 est.) highest 10%: 25.3% (2022 est.) note: % share of income accruing to lowest and highest 10% of population Remittances 0.5% of GDP (2024 est.) 0.5% of GDP (2023 est.) 0.5% of GDP (2022 est.) note: personal transfers and compensation between resident and non-resident individuals/households/entities Budget revenues: $935.038 billion (2023 est.) expenditures: $1.104 trillion (2023 est.) note: central government revenues (excluding grants) and expenditures converted to US dollars at average official exchange rate for year indicated Public debt 131.8% of GDP (2017 est.) note: Italy reports its data on public debt according to guidelines set out in the Maastricht Treaty; general government gross debt is defined in the Maastricht Treaty as consolidated general government gross debt at nominal value, outstanding at the end of the year, in the following categories of government liabilities (as defined in ESA95): currency and deposits (AF.2), securities other than shares excluding financial derivatives (AF.3, excluding AF.34), and loans (AF.4); the general government sector comprises central, state, and local government and social security funds comparison ranking: 8 Taxes and other revenues 24.8% (of GDP) (2023 est.) note: central government tax revenue as a % of GDP comparison ranking: 19 Current account balance $26.76 billion (2024 est.) $3.261 billion (2023 est.) -$36.325 billion (2022 est.) note: balance of payments - net trade and primary/secondary income in current dollars comparison ranking: 20 Exports $778.898 billion (2024 est.) $774.311 billion (2023 est.) $737.083 billion (2022 est.) note: balance of payments - exports of goods and services in current dollars comparison ranking: 11 Exports - partners Germany 11%, USA 11%, France 10%, Spain 5%, UK 5% (2023) note: top five export partners based on percentage share of exports Exports - commodities packaged medicine, garments, cars, refined petroleum, vehicle parts/accessories (2023) note: top five export commodities based on value in dollars Imports $717.278 billion (2024 est.) $739.646 billion (2023 est.) $775.518 billion (2022 est.) note: balance of payments - imports of goods and services in current dollars comparison ranking: 13 Imports - partners Germany 15%, France 9%, China 8%, Netherlands 6%, Spain 5% (2023) note: top five import partners based on percentage share of imports Imports - commodities natural gas, crude petroleum, cars, packaged medicine, garments (2023) note: top five import commodities based on value in dollars Reserves of foreign exchange and gold $290.547 billion (2024 est.) $247.396 billion (2023 est.) $224.581 billion (2022 est.) note: holdings of gold (year-end prices)/foreign exchange/special drawing rights in current dollars comparison ranking: 13 Exchange rates euros (EUR) per US dollar - Exchange rates: 0.924 (2024 est.) 0.925 (2023 est.) 0.95 (2022 est.) 0.845 (2021 est.) 0.876 (2020 est.) Energy Electricity access electrification - total population: 100% (2022 est.) Electricity installed generating capacity: 128.692 million kW (2023 est.) consumption: 290.664 billion kWh (2023 est.) exports: 3.32 billion kWh (2023 est.) imports: 54.572 billion kWh (2023 est.) transmission/distribution losses: 17.62 billion kWh (2023 est.) comparison rankings: installed generating capacity 12; consumption 15; exports 45; imports 2; transmission/distribution losses 187 Electricity generation sources fossil fuels: 56% of total installed capacity (2023 est.) solar: 12% of total installed capacity (2023 est.) wind: 9.1% of total installed capacity (2023 est.) hydroelectricity: 14.7% of total installed capacity (2023 est.) geothermal: 2.1% of total installed capacity (2023 est.) biomass and waste: 6.2% of total installed capacity (2023 est.) Nuclear energy Number of nuclear reactors permanently shut down: 4 (2025) Coal production: 1.572 million metric tons (2023 est.) consumption: 12.424 million metric tons (2023 est.) exports: 304,000 metric tons (2023 est.) imports: 12.069 million metric tons (2023 est.) proven reserves: 609.999 million metric tons (2023 est.) Petroleum total petroleum production: 111,000 bbl/day (2023 est.) refined petroleum consumption: 1.245 million bbl/day (2024 est.) crude oil estimated reserves: 497.934 million barrels (2021 est.) Natural gas production: 2.778 billion cubic meters (2023 est.) consumption: 61.906 billion cubic meters (2023 est.) exports: 2.609 billion cubic meters (2023 est.) imports: 61.851 billion cubic meters (2023 est.) proven reserves: 45.76 billion cubic meters (2021 est.) Energy consumption per capita 96.797 million Btu/person (2023 est.) comparison ranking: 53 Communications Telephones - fixed lines total subscriptions: 20.107 million (2023 est.) subscriptions per 100 inhabitants: 34 (2023 est.) comparison ranking: total subscriptions 13 Telephones - mobile cellular total subscriptions: 78.5 million (2023 est.) subscriptions per 100 inhabitants: 133 (2022 est.) comparison ranking: total subscriptions 23 Broadcast media two Italian media giants dominate, with 3 national terrestrial stations; privately owned companies have 3 national terrestrial stations; a large number of private stations, a satellite TV network; 3 AM/FM nationwide radio stations; about 1,300 commercial radio stations Internet country code .it Internet users percent of population: 87% (2023 est.) Broadband - fixed subscriptions total: 20.1 million (2023 est.) subscriptions per 100 inhabitants: 34 (2023 est.) comparison ranking: total 13 Transportation Civil aircraft registration country code prefix I Airports 655 (2025) comparison ranking: 12 Heliports 163 (2025) comparison ranking: 16 Railways total: 18,475 km (2020) 12,936 km electrified 1289.3 0.950-mm gauge (151.3 km electrified) Merchant marine total: 1,276 (2023) by type: bulk carrier 17, container ship 6, general cargo 109, oil tanker 95, other 1,049 comparison ranking: total 18 Ports total ports: 123 (2024) large: 12 medium: 11 small: 71 very small: 28 size unknown: 1 ports with oil terminals: 33 key ports: Brindisi, Civitavecchia, Genova, Gioia Tauro, La Spezia, Livorno, Messina, Napoli, Porto di Lido-Venezia, Siracusa, Taranto, Trieste Military and Security Military and security forces Italian Armed Forces (Forze Armate Italiane): Army (Esercito Italiano, EI), Navy (Marina Militare Italiana, MMI; includes aviation, marines), Italian Air Force (Aeronautica Militare Italiana, AMI); Carabinieri Corps (Arma dei Carabinieri, CC) (2025) note 1: the National (or State) Police and Carabinieri (gendarmerie or military police) maintain internal security; the National Police reports to the Ministry of Interior while the Carabinieri reports to the Ministry of Defense but is also under the coordination of the Ministry of Interior; the Carabinieri is primarily a domestic police force organized along military lines, with some overseas responsibilities note 2: the Financial Guard (Guardia di Finanza) under the Ministry of Economy and Finance is a force with military status and nationwide remit for financial crime investigations, including narcotics trafficking, smuggling, and illegal immigration Military expenditures 2% of GDP (2025 est.) 1.5% of GDP (2024 est.) 1.5% of GDP (2023 est.) 1.5% of GDP (2022 est.) 1.5% of GDP (2021 est.) Military and security service personnel strengths approximately 170,000 active-duty military personnel; approximately 105,000 Carabinieri (2025) Military equipment inventories and acquisitions the military's inventory includes a mix of domestically manufactured, imported, and jointly produced weapons systems, mostly from Europe and the US; in recent years, the US has been the lead supplier of military hardware to Italy; the Italian defense industry is capable of producing equipment across all the military domains with particular strengths in aircraft, armored vehicles, and naval vessels; it also participates in joint development and production of advanced weapons systems with other European countries and the US (2024) Military service age and obligation 17-25 years of age for voluntary military service for men and women (some variations on age depending on the military branch); voluntary service is a minimum of 12 months with the option to extend in the Armed Forces or compete for positions in the Military Corps of the Italian Red Cross, the State Police, the Carabinieri, the Guardia di Finanza, the Penitentiary Police, or the National Fire Brigade; 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12754	https://smartaau.files.wordpress.com/2014/03/lecture-note-ch-4-uniform-flow.pdf	Lecture Note for Open Channel Hydraulics By Belete B. AAiT Department of Civil Engineering October 2010 Page 1 of 26 CAPTER Four UNIFORM FLOW Concept of Uniform Flow  Uniform flow is referring the steady uniform flow  Steady flow is characterized by no changes in time.  Uniform flow is characterized by the water cross section and depth remaining constant over a certain reach of the channel.  For any channel of given roughness, cross section and slope, there exists one and only one water depth, called the normal depth , at which the flow will be uniform.  Uniform equilibrium flow can occur only in a straight channel with a constant channel slope and cross-sectional shape, and a constant discharge.  The energy grade line Sf, water surface slope Sw and channel bed slope S0 are all parallel, i.e. Sf=Sw=So  For steady uniform channel flow, channel slope, depth and velocity all remain constant along the channel Establishment of Uniform Flow  When flow occurs in an open channel, resistance is encountered by the water as it flows downstream  A uniform flow will be developed if the resistance is balanced by the gravity forces.  If the water enters into a channel slowly, the velocity and the resistance are small, and the resistance is outbalanced by the gravity forces, resulting in an accelerating flow in the upstream reach.  The velocity and the resistance will gradually increase until a balance between resistance and gravity forces is reached. At this moment and afterward the flow becomes uniform. Lecture Note for Open Channel Hydraulics By Belete B. AAiT Department of Civil Engineering October 2010 Page 2 of 26  The upstream reach that is required for the establishment of uniform flow is known as the transitory zone. The Chezy Formula Consider the following stretch of channel P2  W Wsin V P1 L Y0 Y0 Y0 1 2 A V2/2g Y0 Lecture Note for Open Channel Hydraulics By Belete B. AAiT Department of Civil Engineering October 2010 Page 3 of 26  By definition there is no acceleration in uniform flow  By applying the momentum equation to control volume encompassing sections 1 and 2, distance L apart as shown in the figure. 1 2 2 1 sin M M P F W P f       ------------------- (4.1) Where: P1 and P2 are the pressure forces and M1 and M2 are the momentum fluxes at section 1 and 2 respectively W= weight to fluid in the control volume and Ff = shear force at the boundary Since the flow is uniform P1 =P2 and M1=M2 also W=AL and Ff= 0PL Where 0 = average shear stress on the wetted perimeter of length P = unit weight of water Replacing sin by S0 (bottom slope) equation (4.1) become 0 0 0 0 0 RS S P A PL ALS          ------------------ (4.2) Where R= A/P = hydraulic radius , which is a length parameter accounting for the shape of the channel. And it plays a very important role in developing flow equations which are common to all shapes of channels. Expressing the average shear stress 0 = kV2, where k=a coefficient which depends on the nature of the surface and flow parameters. Equation (4.2) can be written as 0 0 2 RS C V RS V k     ------------------------------ (4.3) Where k C 1    = a coefficient which depends on the nature of the surface Equation (4.3) is known as Chezy formula and the coefficient C is known as the chezy coefficient. We remember that for pipe flow, the Darcy –Weisbach equation is g V D L f h f 2 2  Where hf = head loss due to friction in a pipe of diameter D and length L f = Darcy-Weisbach friction factor Lecture Note for Open Channel Hydraulics By Belete B. AAiT Department of Civil Engineering October 2010 Page 4 of 26 For smooth pipes, f is found to be a function of the Reynolds number ( v VD  Re ) only. For rough turbulent flows, f is a function of the relative roughness (/D) and types of roughness, which independent of the Reynolds number. So for the case of Open Channel, we can be considered to be a conduit cut into two. The hydraulics radius would then be appropriate length parameter and prediction of friction factor f. So equation of head loss due to friction is written as L h R f g V g V R L f h f f / . . 8 2 4 2    -------------------- (4.4) Note that for uniform flow in an open channel hf/L = slope of the energy line = Sf = S0, it may be seen that equation (4.4) is the same as Chezy formula, equation (4.3). with f g C / 8  --------------------------------------------------------------(4.5) Equation (4.5) be use to develop different charts and empirical formulas that relate C with Re.                s R and v RV  4 4 Re 5146 . 1 Re log 80 . 1 1   f ----------------------------------------------- (4.6) And          9 . 0 Re 25 . 21 4 log 0 . 2 14 . 1 1 R f s  -------------------------------------------- (4.7) Equation (4.7) is valid for 5000 ≤ Re ≤ 108 and 10-6 < s/4R < 10-2 Generally, the open channels that are encountered in the field are very large in size and also in the magnitude of roughness elements. Due to scarcity of reliable experimental or field data on channels covering a wide range of parameters, values of s are not available to the same degree of confidence as for pipe materials. However, the following table will give the estimated values of s for some common open channel surfaces. Lecture Note for Open Channel Hydraulics By Belete B. AAiT Department of Civil Engineering October 2010 Page 5 of 26 Table 4. values of s for some common open channel surfaces S.No Surface Equivalent Roughness (s) in mm 1 Glass 3x10-4 2 Very smooth concrete surface 0.15 – 0.30 3 Rough concrete 3.0 - 4.5 4 Earth Channels (straight, uniform) 3.0 5 Rubble masonry 6.0 6 Untreated channel 3.0 -10.0 Example 4.1: A 2.0m wide rectangular channel carries water at 20oc at a depth of 0.5m. The channel is laid on a slope of 0.0004. Find the hydrodynamic nature of the surface if the channel is made of A. Very smooth concrete B. Rough concrete C. Estimate the discharges in the channel in both case using chezy formula with Dancy-Weisbach f. Solution: B=2.0m Y=0.5m S0 = 0.0004 T= 20oc R= A/P = (0.52)/(2+20.5)=0.33m o= RSo = (9.81x103)x0.333x0.0004 = 1.308N/m2 v = shear velocity = o/ = (1.308/103) = 0.03617m/sec Kinematic viscosity () @ 20oc = 10-6m2/s a). For a smooth concrete surface from the Equivalent roughness table, s = 0.25mm = 0.00025m s v/ = (0.000250.03617)/(10-6) = 9.04 > 4 but less than 60  Transition b) for a rough concrete From the Equivalent roughness table, s = 3.5mm = 0.0035m s v/ = (0.0035x0.03617)/(10-6) =126.6 > 60  Rough c). case (i) = smooth concrete channel s= 0.25mm and s/4R = (0.25)/(4x0.33x103) = 1.894x 10-4          9 . 0 Re 25 . 21 4 log 0 . 2 14 . 1 1 R f s  Since Re is known we use the moody chart or iterative trial and error methods, we get Lecture Note for Open Channel Hydraulics By Belete B. AAiT Department of Civil Engineering October 2010 Page 6 of 26 f= 0.0145 C= (8g/f) = (8x9.81/0.0145) = 73.6 V = C(RS0) = 73.6x(0.333x0.0004) = 0.850m/s Q= AV= (2x0.5)x0.850 = 0.85m3/se Case (ii) Rough concrete Channel s= 3.5mm and s/4R = (3.5)/(4x0.33x103) = 2.625x 10-3 f = 0.025 C = (8g/f) = (8x9.81/0.025) = 56.0 V = C(RS0) = 56x(0.333x0.0004) = 0.647m/s Q= AV= (2x0.5)x0.647 = 0.647m3/se The MANNING’S Formula The simplest resistance formula and the most widely used equation for the mean velocity calculation is the Manning equation which has been derived by Robert Manning (1890) by analyzing the experimental data obtained from his own experiments and from those of others. His equation is, 2 / 1 3 / 2 1 o S R n V  --------------------------------------------------------------- (4.8) Where V = mean velocity R= Hydraulic Radius So = channel slope n = Manning’s roughness coefficient if we equating Equations. (4.3) and (4.8), we get 2 / 1 3 / 2 2 / 1 2 / 1 1 o o S R n S CR  n R C 6 / 1  ----------------------------------------------------------------------- (4.9) Similarly we can be equating equations (4.5) and (4.9) we get   g R n f n R f g 8 8 3 / 1 2 6 / 1            ----------------------------- (4.10) Lecture Note for Open Channel Hydraulics By Belete B. AAiT Department of Civil Engineering October 2010 Page 7 of 26 From equation (4.10) we see         3 / 1 2 R n f  , it follows that 6 / 1 s n  . If so the both Manning’s formula and Darcy – Weisbach formula represent rough turbulent flow at 60   v OTHERS RESISTANCE FORMULAE Several forms for the chezy coefficient C have been proposed by different investigators in the past. Some of them are in use for problems in open channel flow. 1. Pavlovsik formula: n R C x  in which   10 . 0 75 . 0 13 . 0 5 . 2     n R n x and n= manning’s coefficient. This formula appears to be in use in Russia 2. Bazin’s formula R M C / 1 0 . 87   , in which M = a coefficient dependent on the surface roughness 3. Ganguillet and Kutter Formula R n S S n C o o            00155 . 0 23 1 00155 . 0 1 23 , in which n= manning’s coefficient MANNING’S Roughness Coefficient (n) In applying the Manning equation, the greatest difficulty lies in the determination of the roughness coefficient, n; there is no exact method of selecting the n value. Selecting a value of n actually means to estimate the resistance to flow in a given channel, which is really a matter of intangibles.(Chow, 1959) .To experienced engineers, this means the exercise of engineering judgment and experience; for a new engineer, it can be no more than a guess and different individuals will obtain different results. Factors Affecting Manning’s Roughness Coefficient It is not uncommon for engineers to think of a channel as having a single value of n for all occasions. Actually, the value of n is highly variable and depends on a number of factors. The factors that exert the greatest influence upon the roughness coefficient in both artificial and natural channels are described below. a) Surface Roughness: The surface roughness is represented by the size and shape of the grains of the material forming the wetted perimeter. This usually considered the only factor Lecture Note for Open Channel Hydraulics By Belete B. AAiT Department of Civil Engineering October 2010 Page 8 of 26 in selecting the roughness coefficient, but it is usually just one of the several factors. Generally, fine grains result in a relatively low value of n and coarse grains in a high value of n. b) Vegetation: Vegetation may be regarded as a kind of surface roughness, but it also reduces the capacity of the channel. This effect depends mainly on height, density, and type of vegetation. c) Channel Irregularity: Channel irregularity comprises irregularities in wetted perimeter and variations in cross-section, size, and shape along the channel length. d) Channel Alignment: Smooth curvature with large radius will give a relatively low value of n, whereas sharp curvature with severe meandering will increase n. e) Silting and Scouring: Generally speaking, silting may change a very irregular channel into a comparatively uniform one and decrease n, whereas scouring may do the reverse and increase n. f) Obstruction: The presence of logjams, bridge piers, and the like tends to increase n. g) Size and Shape of the Channel: There is no definite evidence about the size and shape of the channel as an important factor affecting the value of n. h) Stage and Discharge: The n value in most streams decreases with increase in stage and discharge. i) Seasonal Change: Owing to the seasonal growth of aquatic plants, the value of n may change from one season to another season. Determination of Manning’s Roughness Coefficient a. Cowan Method Taking into account primary factors affecting the roughness coefficient, Cowan (1956) developed a method for estimating the value of n. The value of n may be computed by, n =(n0 + n1 + n2 + n3 + n4)×m ---------------------------------------------------------(4.11) Where: n0 is a basic value for straight, uniform, smooth channel in the natural materials involved, n1 is a value added to n0 to correct for the effect of surface irregularities, n2 is a value for variations in shape and size of the channel cross-section, n3 is a value of obstructions, n4 is a value for vegetation and flow conditions, and m is a correction factor for meandering of channel. Lecture Note for Open Channel Hydraulics By Belete B. AAiT Department of Civil Engineering October 2010 Page 9 of 26 These coefficients are given in Table (4.2) depending on the channel characteristics. (French,1994). Table 4.2: Values of the manning’s coefficients of Cowan Method b. Empirical Formulae for n Many empirical formulae have been presented for estimating manning’s coefficient n in natural streams. These relate n to the bed-particle size. (Subramanya, 1997). The most popular one under this type is the Strickler formula, 1 . 21 6 / 1 50 d n  ----------------------------------------------------------------------------- (4.12) Where d50 is in meters and represents the particle size in which 50 per cent of the bed material is finer. For mixtures of bed materials with considerable coarse-grained sizes, 26 6 / 1 90 d n  ------------------------------------------------------------------------------- (4.13) Where d90 = size in meters in which 90 per cent of the particles are finer than d90. This equation is reported to be useful in predicting n in mountain streams paved with coarse gravel and cobbles. Lecture Note for Open Channel Hydraulics By Belete B. AAiT Department of Civil Engineering October 2010 Page 10 of 26 c. Equivalent Roughness In some channels different parts of the channel perimeter may have different roughnesses. Canals in which only the sides are lined, laboratory flumes with glass walls and rough beds, rivers with sand bed in deepwater portion and flood plains covered with vegetation, are some typical examples. For such channels it is necessary to determine an equivalent roughness coefficient that can be applied to the entire cross-sectional perimeter in using the Manning’s formula. This equivalent roughness, also called the composite roughness, represents a weighted average value for the roughness coefficient, n. A large number of formulae, proposed by various investigators for calculating equivalent roughness of multi-roughness channel are available in literature. All of them are based on some assumptions and approximately effective to the same degree. We see the derivation of one of the common formula called Horton’s Method and present others as table below. Horton’s method of Equivalent Roughness Estimation: Consider a channel having its perimeter composed of N types rough nesses. P1, P2,…., PN are the lengths of these N parts and n1, n2,…….., nN are the respective roughness coefficients as presented in the above figure. Let each part Pi be associated with a partial area Ai such that, Lecture Note for Open Channel Hydraulics By Belete B. AAiT Department of Civil Engineering October 2010 Page 11 of 26 It is assumed that the mean velocity in each partial area is the mean velocity V for the entire area of flow, By the Manning’s equation, , ---------------------(4.14) Where n = Equivalent roughness. From Equ. (4.14), ------------------------------------------------------- (4.15) This equation gives a means of estimating the equivalent roughness of a channel having multiple roughness types in its perimeters Table 4.3. Equations for equivalent Roughness Coefficient (adopt from K. Subramanya 2010) No Investigators ne Concept 1 Horton (1933); Einstein (1934)   3 / 2 2 / 3 1         i i P n P Mean Velocity is constant in all subareas 2 Pavloskii (1931), Muhlhofer(1933) Einstein and Banks (1950)   2 / 1 2 1         i i P n P Total resistance force F is sum of subarea resistance force, Fi 3 Lotter (1932)   i i i n R P PR 3 / 5 3 / 5 Total discharge is sum of subarea discharge 4 Yen (1991)   P P n i i   Total shear velocity is weighted sum of subarea shear velocity Lecture Note for Open Channel Hydraulics By Belete B. AAiT Department of Civil Engineering October 2010 Page 12 of 26 Table 4.4: values of Manning’s Roughness coefficient (adopt from ERA design manuals 2001) Type of Channel and Description EXCAVATED OR DREDGED a. Earth, straight and uniform 1. Clean, recently completed 2. Clean, after weathering 3. Gravel, uniform section, clean 4. With short grass, few weeds b. Earth, winding and sluggish 1. No vegetation 2. Grass, some weeds 3. Dense Weeds or aquatic plants in deep channels 4. Earth bottom and rubble sides 5. Stony bottom and weedy sides 6. Cobble bottom and clean sides c. Backhoe-excavated or dredged 1. No vegetation 2. Light brush on banks d. Rock cuts 1. Smooth and uniform 2. Jagged and irregular e. Channels not maintained, weeds and brush uncut 1. Dense weeds, high as flow depth 2 Clean bottom, brush on sides 3. Same, highest stage of flow 4. Dense brush, high stage NATURAL STREAMS 1 Minor streams (top width at flood stage < 30 m) a. Streams on Plain 1. Clean, straight, full stage, no rims or deep pools 2. Same as above, but more stones and weeds 3. Clean, winding, some pools and shoals 4. Same as above, but some weeds and stones 5. Same as above, lower stages, more ineffective slopes and sections 6. Same as 4, but more stones 7. Sluggish reaches, weedy, deep pools 8 Very weedy reaches, deep pools, or floodways with heavy stand of timber and underbrush b. Mountain streams, no vegetation in channel, banks usually steep, trees and brush along banks submerged at high stages 1. Bottom: gravel, cobbles, and few boulders 2. Bottom: cobbles with large boulders 2 Flood Plains a. Pasture, no brush 1. Short grass 2. High grass Minimum 0.016 0.018 0.022 0.022 0.023 0.025 0.030 0.025 0.025 0.030 0.025 0.035 0.025 0.035 0.050 0.040 0.045 0.080 0.025 0.030 0.033 0.035 0.040 0.045 0.050 0.075 0.030 0.040 0.025 0.030 Normal 0.018 0.022 0.025 0.027 0.025 0.030 0.035 0.030 0.035 0.040 0.028 0.050 0.035 0.040 0.080 0.050 0.070 0.100 0.030 0.035 0.040 0.045 0.048 0.050 0.070 0.100 0.040 0.050 0.030 0.035 Maximum 0.020 0.025 0.030 0.033 0.030 0.033 0.040 0.035 0.045 0.050 0.033 0.060 0.040 0.050 0.120 0.080 0.110 0.140 0.033 0.040 0.045 0.050 0.055 0.060 0.080 0.150 0.050 0.070 0.035 0.050 Lecture Note for Open Channel Hydraulics By Belete B. AAiT Department of Civil Engineering October 2010 Page 13 of 26 Type of Channel and Description b. Cultivated area 1. No crop 2. Mature row crops 3. Mature field crops c. Brush 1. Scattered brush, heavy weeds 2. Light brush and trees in winter 3. Light brush and trees, in summer 4. Medium to dense brush, in winter 5. Medium to dense brush, in summer d. Trees 1. Dense willows, summer, straight 2. Cleared land with tree stumps, no sprouts 3. Same as above, but with heavy growth of spouts 4. Heavy stand of timber, a few down trees, little undergrowth, flood stage below branches 5. Same as above, but with flood stage reaching branches 3 Major Streams (top width at flood stage > 30 m). The n value is less than that for minor streams of similar description, because banks offer less effective resistance. a. Regular section with no boulders or brush b. Irregular and rough section 4 Various Open Channel Surfaces a. Concrete b. Gravel bottom with: Concrete Mortared stone Riprap c. Natural Stream Channels Clean, straight stream Clean, winding stream Winding with weeds and pools With heavy brush and timber d. Flood Plains Pasture Field Crops Light Brush and Weeds Dense Brush Dense Trees Minimum 0.020 0.025 0.030 0.035 0.035 0.040 0.045 0.070 0.110 0.030 0.050 0.080 0.100 0.025 0.035 0.012- 0.020 0.023 0.033 0.030 0.040 0.050 0.100 0.035 0.040 0.050 0.070 0.100 Normal 0.030 0.035 0.040 0.050 0.050 0.060 0.070 0.100 0.150 0.040 0.060 0.100 0.120 -- -- 0.020 Maximum 0.040 0.045 0.050 0.070 0.060 0.080 0.110 0.160 0.200 0.050 0.080 0.120 0.160 0.060 0.100 Example 4.2: An earthen trapezoidal channel (n = 0.025) has a bottom width of 5.0 m, side slopes of 1.5 horizontal: 1 vertical and a uniform flow depth of 1.10 m. In an economic study to remedy excessive seepage from the canal two proposals, a) to line the sides only and, b) to line the bed only are considered. If the lining is of smooth concrete (n = 0.012), calculate the equivalent roughness in the above two cases. Lecture Note for Open Channel Hydraulics By Belete B. AAiT Department of Civil Engineering October 2010 Page 14 of 26 Solution : Earthen (n) = 0.025 B = 5.0m m= 1.5 y = 1.10m a). line the sides only b). line the bed only lining (n) = 0.012 Case a): Lining on the sides only, For the bed → n1 = 0.025 and P1 = 5.0 m. For the sides → n2 = 0.012 and Case b): Lining on the bottom only Lecture Note for Open Channel Hydraulics By Belete B. AAiT Department of Civil Engineering October 2010 Page 15 of 26 UNIFORM FLOW COMPUTATION The basic equations involved in the computation of uniform flow are the manning’s and the continuity equation. From continuity equation Q = AV, if we substitute the velocity from the manning’s formula we get the equation for discharge as 2 / 1 3 / 2 1 o S AR n Q  ------------------------------------------------ (4.16) o S K Q  --------------------------------------------------------- (4.17) Where 3 / 2 1 AR n K  is called the conveyance of the channel and express the discharge capacity of the channel per unit longitudinal slope. The term nK = AR2/3 is also called the section factor for uniform flow computations. The basic variables in uniform flow problems can be the discharge Q, velocity of flow V, normal depth y0, roughness coefficient n, channel slope S0 and the geometric elements (e.g. B and side slope m for a trapezoidal channel). There can be many other derived variables accompanied by corresponding relationships. From among the above, the following five types of basic problems are recognized. Table 4.5 Problem Types and the given and required variables in uniform flow Hydraulic Radius Hydraulic radius plays a prominent role in the equations of open-channel flow and therefore, the variation of hydraulic radius with depth and width of the channel becomes an important consideration. This is mainly a problem of section geometry. Lecture Note for Open Channel Hydraulics By Belete B. AAiT Department of Civil Engineering October 2010 Page 16 of 26 Figure on the relation of Hydraulic Radius with depth and width Consider first the variation of hydraulic radius with depth in a rectangular channel of width B Therefore the variation of R with y is as shown in Fig (a) above. From this comes a useful engineering approximation: for narrow deep cross-sections R≈ B/2. Since any (nonrectangular) section when deep and narrow approaches a rectangle, when a channel is deep and narrow, the hydraulic radius may be taken to be half of mean width for practical applications. Consider the variation of hydraulic radius with width in a rectangular channel of with a constant water depth y From this it may be concluded that for wide shallow rectangular cross-sections R≈y; for rectangular sections the approximation is also valid if the section is wide and shallow, here the hydraulic radius approaches the mean depth. Lecture Note for Open Channel Hydraulics By Belete B. AAiT Department of Civil Engineering October 2010 Page 17 of 26 Normal Depth Rectangular Channel Area A = Byo Wetted Perimeter P = B + 2yo Hydraulic Radius R= A/P B y y R o o 2 1   ---------- (4.18) a). wide Rectangular Channel As yo/B, the aspect ratio of the channel decrease, R yo. Such channels with large bed-widths as compared to their respective depths are known as wide rectangular channels. In these channels, the hydraulics radius approximates to the depth of flow. Considering a unit width of a wide rectangular channel, A= yo, R = yo and B = 1.0 Discharge per unit width Q/B = 2 / 1 3 / 5 1 o o S y n q   5 / 3          o o S qn y --------------------------- (4.19) This approximation of a wide rectangular channel is found applicable to rectangular channels yo/B < 0.02. b). Rectangular Channels with yo/B ≥ 0.02 For these channels 3 / 2 AR S Qn o  , and         3 / 8 3 / 2 3 / 5 3 / 2 3 / 5 3 / 2 / 2 1 / 2 B B y B y y B By AR o o o o           o o o o B AR B S Qn         3 / 2 3 / 5 3 / 8 3 / 2 3 / 8 2 1 ------------------------------------------------------ (4.20) Where B y0 0   , Tables of (o) Vs o will provide a non-dimensional graphical aid for general application. Since 3 / 8 B S Qn o   , one can easily find yo/B from this table for any combination of Q, n, So, and B in a rectangular channel. Y0 B Lecture Note for Open Channel Hydraulics By Belete B. AAiT Department of Civil Engineering October 2010 Page 18 of 26 Trapezoidal Channel Following a procedure similar to the above, for a trapezoidal section of side slope m:1. Area = A = (B + myo)yo Wetted Perimeter = P= (B +2yom2+1) Hydraulic Radius     o o o y m B y my B P A R 1 2 2          3 / 2 2 3 / 5 3 / 5 3 / 2 1 2 o o o o y m B y my B AR S Qn      , Non-dimensionalising the variables,     ) , ( 1 2 1 1 3 / 2 2 3 / 5 3 / 5 3 / 8 3 / 8 3 / 2 m y m m B S Qn B AR o o o o o           ----------------------------------------- (4.21) Where o = yo/B. Equation (4.21) represent as curves or tables of  vs o with m as the third parameter to provide a general normal depth solution aid. It may be noted m=0 is the case of rectangular. Lined canal Section Most lined channels and built-up channels can withstand erosion and reduce seepage. Exposed hard surface lining using materials, such as cement concrete, brick tiles, asphaltic concrete and stone masonry are the members of lined canal category and also carry large canal. Indian Standards (IS: 4745 -1968) consists of two standardized lined canal section (i.e., Trapezoidal and triangular with corners rounded off for discharge > 55m3/sec and < 55m3/sec). Actually the triangular section is the limiting case of the standard lined trapezoidal section with bottom width (B) = 0. 1 m yo B B m 1     r r =yo r =yo yo r m=cot Lecture Note for Open Channel Hydraulics By Belete B. AAiT Department of Civil Engineering October 2010 Page 19 of 26 Referring the above figure the full supply depth = normal depth at design discharge = yo at normal depth For standard trapezoidal case Area = A = Byo +myo 2 + yo 2 = (B +yo)yo , where            m m m 1 tan 1   Wetted perimeter =P = B +2myo +2yo = B +2yo Hydraulics radius =R= A/P =   o o y B y y B 2 ) ( 0   By manning’s formula 2 / 1 3 / 2 3 / 5 3 / 5 ) 2 ( ) ( 1 o o o o S y B y y B n Q            ---------------------------------------- (4.22) Non-dimensionalising the variables, 3 / 2 3 / 5 3 / 5 3 / 8 2 / 1 3 / 5 ) 2 1 ( ) 1 ( ) ( o o o o o B S Qn           ------------------------------------------------------ (4.23) For standard triangular case similarly we can drive 2 / 1 3 / 2 2 ) 2 / )( ( 1 o o o S y y n Q   ------------------------------------------------------------------------- (4.24) Example 4.3 A standard lined trapezoidal cannel section is to be designed to convey 100m3/sec of flow. The side slope is to be 1.5H:1V and the manning’s coefficient n= 0.016. The longitudinal slope of the bed is 1 in 5000m. If a bed width of 10m is preferred what would be the normal depth? Solution Q= 100m3/sec m= 1.5 n= 0.016 So = 0.0002 B= 10m yo =?  = m+ tan-1(1/m)= 1.5 +tan-1(1/1.5) = 2.088 8314 . 0 ) 0 . 10 ( ) 0002 . 0 ( ) 088 . 2 ( 016 . 0 100 3 / 8 2 / 1 3 / 5 3 / 8 2 / 1 3 / 5 1    X X B S Qn o   8314 . 0 ) 2 1 ( ) 1 ( 3 / 2 3 / 5 3 / 5 1     o o o     By solving trial and error o = 0.74 Then B yo  0 , which imply that 554 . 3 088 . 2 0 . 10 74 . 0 0 0    y X B y o   Lecture Note for Open Channel Hydraulics By Belete B. AAiT Department of Civil Engineering October 2010 Page 20 of 26 The Hydraulic Efficient Channel Section The best hydraulic (the most efficient) cross-section for a given Q, n, and S0 is the one with a minimum excavation and minimum lining cross-section. A = Amin and P = Pmin. The minimum cross-sectional area and the minimum lining area will reduce construction expenses and therefore that cross-section is economically the most efficient one. In other case the best hydraulic cross-section for a given A, n, and S0 is the cross-section that conveys maximum discharge. Thus the cross-section with the minimum wetted perimeter is the best hydraulic cross-section within the cross-sections with the same area since lining and maintenance expenses will reduce substantially. Rectangular channel section Since P=Pmin for the best hydraulic section, taking the derivative of P with respect to y The best rectangular hydraulic cross-section for a constant area is the one with B = 2y. The hydraulic radius of this cross- section is, For all best hydraulic cross-sections, the hydraulic radius should always be R = y/2 regardless of their shapes. Lecture Note for Open Channel Hydraulics By Belete B. AAiT Department of Civil Engineering October 2010 Page 21 of 26 Trapezoidal Channel Section a). For a given side slope m, what will be the water depth y for best hydraulic trapezoidal cross-section? For a given A , P = Pmin The hydraulic radius R, channel bottom width B, and free surface width L may be found as, my my Lecture Note for Open Channel Hydraulics By Belete B. AAiT Department of Civil Engineering October 2010 Page 22 of 26 b). For a given water depth y, what will be the side slope m for best hydraulic trapezoidal cross-section? The channel bottom width is equal one third of the wetted perimeter and therefore sides and channel width B are equal to each other at the best trapezoidal hydraulic cross-section. Since α = 600, the cross-section is half of the hexagon. Table 4.6 values of parameters in Efficient (best) hydraulic section Lecture Note for Open Channel Hydraulics By Belete B. AAiT Department of Civil Engineering October 2010 Page 23 of 26 Compound Sections Some channel sections may be formed as a combination of elementary sections. Typically natural channels, such as rivers, have flood plains which are wide and shallow compared to the main channel. The figure below represents a simplified section of a stream with flood banks. Consider the compound section to be divided into subsections by arbitrary lines. These can be extensions of the deep channel boundaries as in figure. Assuming the longitudinal slope to be same for all subsections, it is easy to see that the subsections will have different mean velocities depending upon the depth and roughness of the boundaries. Generally, overbanks have larger size roughness than the deeper main channel. If the mean velocities Vi in the various subsections are known then the total discharge is ΣViAi. If the depth of flow is confined to the deep channel only (y < h), calculation of discharge by using Manning’s equation is very simple. However, when the flow spills over the flood plain (y > h), the problem of discharge calculation is complicated as the calculation may give a smaller hydraulic radius for the whole stream section and hence the discharge may be underestimated. The following method of discharge estimation can be used. In this method, while calculating the wetted perimeter for the sub-areas, the imaginary divisions (FJ and CK in the Figure) are considered as boundaries for the deeper portion only and neglected completely in the calculation relating to the shallower portion. 1. The discharge is calculated as the sum of the partial discharges in the sub-areas; for e.g. units 1, 2 and 3 in Figuer 2. The discharge is also calculated by considering the whole section as one unit, (ABCDEFGH area in Figure), say Qw. 3. The larger of the above discharges, Qp and Qw, is adopted as the discharge at the depth y. Lecture Note for Open Channel Hydraulics By Belete B. AAiT Department of Civil Engineering October 2010 Page 24 of 26 Design of Irrigation Channels For a uniform flow in a canal, Where A and R are in general, functions of the geometric elements of the canal. If the canal is of trapezoidal cross-section, -------------------------------------- (4.25) Equ. (4.25) has six variables out of which one is a dependent variable and the rest five are independent ones. Similarly, for other channel shapes, the number of variables depends upon the channel geometry. In a channel design problem, the independent variables are known either explicitly or implicitly, or as inequalities, mostly in terms of empirical relationships. The canal-design practice given below is meant only for rigid-boundary channels, i.e. for lined and unlined non-erodible channels. Canal Section Normally a trapezoidal section is adopted. Rectangular cross-sections are also used in special situations, such as in rock cuts; steep chutes and in cross-drainage works. The side slope, expressed as m horizontal: 1 vertical, depends on the type of canal, (i.e. lined or unlined, nature and type of soil through which the canal is laid). The slopes are designed to withstand seepage forces under critical conditions, such as; 1. A canal running full with banks saturated due to rainfall, 2. The sudden drawdown of canal supply. Usually the slopes are steeper in cutting than in filling. For lined canals, the slopes roughly correspond to the angle of repose of the natural soil and the values of m range from 1.0 to 1.5 and rarely up to 2.0. The slopes recommended for unlined canals in cutting are given in Table (4.7). Table 4.7: Side slopes for unlined canals in cutting Lecture Note for Open Channel Hydraulics By Belete B. AAiT Department of Civil Engineering October 2010 Page 25 of 26 Longitudinal Slope The longitudinal slope is fixed on the basis of topography to command as much area as possible with the limiting velocities acting as constraints. Usually the slopes are of the order of 0.0001. For lined canals a velocity of about 2 m/sec is usually recommended. Roughness coefficient n Procedures for selecting n are discussed and values of n can be taken from Table (4.4). Permissible Velocities Since the cost for a given length of canal depends upon its size, if the available slope permits, it is economical to use highest safe velocities. High velocities may cause scour and erosion of the boundaries. As such, in unlined channels the maximum permissible velocities refer to the velocities that can be safely allowed in the channel without causing scour or erosion of the channel material. In lined canals, where the material of lining can withstand very high velocities, the maximum permissible velocity is determined by the stability and durability of the lining and also on the erosive action of any abrasive material that may be carried in the stream. The permissible maximum velocities normally adopted for a few soil types and lining materials are given in Table (4.8). Table 4.8 Permissible Maximum velocities In addition to the maximum velocities, a minimum velocity in the channel is also an important constraint in the canal design. Too low velocity would cause deposition of suspended material, like silt, which cannot only impair the carrying capacity but also increase the maintenance costs. Also, in unlined canals, too low a velocity may encourage weed growth. The minimum velocity in irrigation channels is of the order of 0.30 m/sec. Lecture Note for Open Channel Hydraulics By Belete B. AAiT Department of Civil Engineering October 2010 Page 26 of 26 Free Board Free board for lined canals is the vertical distance between the full supply levels to the top of lining (Fig. 4.17). For unlined canals, it is the vertical distance from the full supply level to the top of the bank. This distance should be sufficient to prevent overtopping of the canal lining or banks due to waves. The amount of free board provided depends on the canal size, location, velocity and depth of flow. Width to Depth Ratio The relationship between width and depth varies widely depending upon the design practice. If the hydraulically most-efficient channel cross-section is adopted, If any other value of m is use, the corresponding value of B/y0 for the efficient section would be In large channels it is necessary to limit the depth to avoid dangers of bank failure. Usually depths higher than about 4.0 m are applied only when it is absolutely necessary. For selection of width and depth, the usual procedure is to adopt a recommended value. Example 4.4.
12755	https://www.youtube.com/watch?v=-jmiW7X2c2Q	0-1 BFS and Multisource BFS \| Interview Preparation for Graph and Tree \| GeeksforGeeks Practice GeeksforGeeks Practice 84200 subscribers 20 likes Description 918 views Posted: 12 Dec 2022 In this session we will be covering 0-1 BFS and Multisource BFS, two algorithms used to find the shortest path between two nodes in a graph. We will discuss the differences between the two algorithms and how they can be used to solve problems in graph theory. We'll also look at some examples of how to implement these algorithms in code. So tag along with our mentor Gourav Kumar Shaw and clear your graph traversal concepts. Happy Learning! Slides for today's session: Data Structures and Algorithms - Self Paced: Solve the Problem of the day and test yourself: Follow us and stay updated on everything happening in the world of geeks: 📱 Twitter- 📝 LinkedIn- 🌐 Facebook- 📷 Instagram- GFGPractice #GeeksforGeeks #datastructures #graphs #trees Transcript: Introduction foreign [Music] [Music] foreign [Music] hi hi are you sure this course is just for 2499 yes and you know this is the only course with this expertise across the world no way yes way [Music] don't delay enroll now Jake's learning together yeah hello uh good evening everyone so am I properly Audible and visible can you please mention the same in the chat box so that we can begin so how many people are with me today so can you please write your name in the chat box to mark your presence and let us wait for a few minutes to the students to join and then we can start yeah so hello can you please mention in the chat box that my screen is visible and I am properly audible as well so that we can start foreign hello asset official blog Vlogs what is your name can you please write your real name in the chat box foreign [Music] okay your name is shilandra Kumar hello and hello Sumit as well so uh do you guys have any idea about uh zero one BFS Topic Introduction and multisource BFS we are going to study today so are you ready to study do you know about uh BFS breadth first search and uh hello Sumit and hello uh shilandra I would like to request you that kindly stay with me uh till the very end of the session this will be around 50 minutes to one hour long and I will be covering uh important Concepts like multisource BFS and 0 1 VFS I will also be discussing uh two important problems which are very frequently acts in interviews point of view that's very great Sumit uh so uh you are in your which year in college can you please write in the chat box are you there okay you are in your first year person that's very very good and uh that's very uh actually great because since you are in your first year and you already know BFS that's kind of uh very great so I would recommend you to continue studying data structures and algorithms and by the time you come to your third year you will be very very um by the time you come to your internship time it will be very much prepared okay so yeah so uh basically let me give you a small introduction of the session and then we can start in just a few minutes okay so uh my name is gaurav Kumar shop and I am presently in my third year at IIT kharagpur I'm doing my B Tech here and I'm also a mentor at Geeks for geeks and I am also an incoming software About the Series engineering intern at uh cure fit okay so uh basically let me tell you about the series this is the fourth session today and uh we will and in this session we we are basically covering all the important concepts related to graphs and trees which are important from interview point of view okay so we already had three sessions in which we discussed about representations of graph DFS VFS algorithm we discuss few problems as well and we also gave our homework as well so you will find the link of the slides in the description and you can check out the previous sessions as well and we will be continuing a total of 15 sessions in the entire month of December okay so this is a fourth session and we are meeting every alternate a day at 10 pm but today I came a bit early because uh because of some personal commitments so uh today I'm I'm live at 9 30. but usually I will be live at 10 pm only every alternate day so you can see me next one the coming Tuesday so today we will be studying the 0 1 BFS and multi Source BSS so first we will be studying multisource and then zero one so can anyone tell me the difference Normal vs Multisource BFS between a normal BFS and a multi Source BFS hello Amit hello Aryan so uh we are just about to start with uh 0 1 BFS and multisource VFS so I would request all of you to kindly stay with me till the end of the session this will be around 50 minutes to one hour long and I promise you that we will be discussing two to three very important questions as well as Concepts which are very frequently accident okay so kindly Stay With Me so and at the same time I would request you all to kindly uh keep a bit of interaction uh in the chat box so that uh things get pretty much easier while we are doing it okay so let us start then yeah so the very first problem that you are going to discuss today is about multisource BSS so let me give you a bit Multisource BFS of introduction about multisource BFS and what are those okay okay so uh basically you all know about BFS BFS basically means breakfast Source okay so what happens in breadth first search is that we are given a graph and uh we are given a q data structures so we start reversing From Any Given node and what it does it traverses the given graph level by level so basically what we'll do it will first of all it will Traverse all the nodes near adjacent to the given starting node okay so let us say uh this is our graph okay and uh yeah let me see this is a This Is My Graph okay so this is the starting point of the BFS so uh during the first iteration what it will do it will try to cover it will try to Traverse the first nodes that are present uh adjacent to it so we said basically these nodes okay so these three nodes come at a distance of one unit from the starting node so basically the BFS will try to cover these nodes first after it covers this one then it will try to cover the next nodes which are at a distance one BFS traversal unit from this node this node and this node so basically there is a level by level traversal level by level traversal okay so initially what happens it tries to cover the okay Amit uh that's very uh great and uh I would like to congratulate you for your placements and can you please mention the company in which you got placed yeah so uh anyway let us continue so uh basically BFS troubles what happens in VFS traversal is that we go by level by level okay so first the level one gets traversed and the level two gets troubles and so on and this entire concept of level by level traversal uh is basically implemented with the help of a queue data structure so what happens we have a Q data structure and you know what happens in Cube It's a first in first out kind of data structure so whatever will go uh so it's a first in first out okay so um this is not a queue Yeah so basically what happens whatever Elements which will go inside the queue first will come out first only so initially we put this element in the queue and after that what we do we take out this element and explore all its neighboring elements so this and this and this and then we put these three elements in the queue and remove that after that we will take one of this element again and again Traverse all its neighboring elements then then we will put those elements in the removing this element again this element and this element so we are observing uh certain important things here is that whatever element we are putting in first is becoming the source of the BFS okay so what you know here is that uh the first element that you are putting in the queue is becoming the source of the VFS and from that point onward the entire traversal is happening now since we are only putting one node in the brq but uh what can what will happen if we try to put multiple nodes in the queue okay whatever BFS you know is basically single Source BFS but we are discussing multi multi Source BFS so what will happen we if we try to put multiple Multiple sources sources in the VFS can anyone tell me hello Aryan hello uh shilendra hello Sumit are you all there with me yeah so okay so let us start then so let us continue yeah so let's okay thank you sherendra yeah so let's continue uh with our studies then okay so basically what did I just say what will happen if I'm trying to put Multisource VFS multiple sources in the uh in the queue okay so this will become multisource VFS now one thing you can observe from here is that [Music] yeah so uh kindly ignore the background noise uh I cannot do anything about that so let's see one thing so basically uh Sumit is asking a doubt he's saying that is this like leveler traversal of a tree yes exactly it is a level of traversal of a tree but uh BFS can be applied to any graft tree is just a subset of a graph culture so uh the idea is that BFS traversal is an iterative traversal and it traverses level by level and depth versus goes into depth so basically since BFS is traversing level wise level we can basically exploit this functionality and try to solve a variety of problems okay so uh let's say uh What uh one more one more observation that I would like to allow you to have is let's say uh in the starting node if let's say this is starting node we are putting initially in the queue and after this nodes gets processed the next few nodes will be definitely more than one let's say this node is connected to three nodes so after this node gets processed the queue will look something like this one two three it will have three nodes okay now we will process these three nodes so basically you see that from the second iteration onwards it is basically becoming a multi Source BFS only because there are multiple nodes in the queue okay so multi the concept of multisource VFS lies in this that initially we will be putting multiple sources in the VFS and we will be continuing BFS for each of the sources so let's say initially we put three nodes in the BFS and now for each of the nodes we will be traversing uh according to the rules of BFS that for every node we will look at for the its next loads which are present at the next level so on okay and since we are doing level wise level we will be trying to implement some kind of functionality to solve our problem so this will be better understood by solving a problem so the first problem that I will be solving is the problem name number of enclaves number of enclaves this is a very interesting Number of Enclaves problem and you will find that in the Geeks for geeks hotel now I am sharing my screen with that problem yeah so uh this is a problem I have shared on the screen so this is a problem that we will try to solve okay so let's see what exactly the problem says that now if you are given an N into M binary Matrix grid where 0 represents a C cell and one represents the land cell a move consists of walking from one land cell to another adjacent four directionally land cell or walking of the boundary of the grid okay find the number of lancels in grid for which we cannot walk of the boundary of the Grid in any number of moves okay so what what is this saying find the number of land cell in grid for which we cannot walk off the boundary of any of the grid let us digest the statement so it is saying that we have to find the number of land cells in grid for which we cannot walk off the boundary okay so for example let's say uh let me just uh show you something yeah in the first example you see that uh there are zeros and there are ones so you see that these three ones in the as marked highlighted in the explanation part when you are standing on any of these three ones then it means that you will always stay inside this island and you cannot walk off the boundary of this grid okay that is you cannot come out of this grid but let's say you are standing on this island and if you try to move to the next try to move one step to the left then you will come out of this grip grid okay but if you try to move one step left you will fall in the water okay so that is the reason uh these three are the answers the highlighter cells are representing the land cells which are which so if you're standing on those lenses you basically cannot come out of this grid instead you will fall in into the water of the sea so the output will be three okay so this is the question now let's see another question example so as you can see in this question also uh in this example also there are uh this is a land cell this is the land cell but uh you cannot put this in your answer because if you're Land Cell standing on this you will basically fall off uh you can basically come out of the grid by one step moving down so basically you cannot include this so it's kind of becoming obvious that you cannot include those cells which are basically present in the boundary of your given grid okay but what about those Solutions not present on the boundary but are connected to the boundary what about those so can anyone just uh give a write a one statement uh about how exactly are we going to apply the concept of VFS to solve this problem anyone uh Sumit you said you knew about BFS can you just write a concept how are we going to apply this concept of BFS to solve this Matrix related type problem people usually yeah so uh let me tell you one thing so let's come to the uh White Jambo part yeah so the thing was that let's say this is our grid and uh if there is a one present in the boundary of the grid and uh one more thing is the question it is mentioned that you can only move in these four directions okay so basically uh let's say if you are trying to move uh a one is present in the boundary another one is present here another one is present here another one is present here and another one is present here so uh basically and rest are zeros so do you know what will happen so basically the if you are present on this uh this land you can basically come out of this boundary easily by moving one step right but if you're present on this slide also then then also you can come out of the boundary because these lands are connected to each other okay so uh it doesn't mean that if you're standing on the boundary then only you can connect but basically if you're standing on any piece of land which basically gets connected to the boundary in either horizontal or vertically fashion you have to eliminate those options because you do not want to fall okay so it means that you can only stay in those piece of land which are basically surrounded by water in all the four directions basically let's say if uh this is a this is a let's say uh this is one land and it is surrounded by zero okay and here is one another one here is another one and here is another one so and here is another one okay so let's say uh this is zero this is zero this is zero zero zero zero zero zero zero and zero uh zero zero okay so what is happening here is that if you're standing on any one of these lands any one of these lands you there is no way you can basically come out of this grid okay so this is uh so your answer would be one two three four five so that is the uh you have to basically output that the number of land cells uh which basically if you're standing on any of those land cells you there is no way you can come out of the grid which means that it is not connected to the any of the land cells which is getting connected to the boundary okay so how are we going to solve this problem okay so let's see any ideas anyone how are we going to solve this problem yeah so let me tell you one thing what can we do here is that uh let's say uh this is our Matrix so let me just uh copy paste a few stuffs uh yeah so we are in a number of enclaves okay uh give me a minute foreign Multi Source BFS let us look into something like this let us say that we have our Matrix and our Matrix contains multiple ones okay there is another one here one here one here and another one here one here one here okay so what we will do we will apply the concept of BFS but this time we will apply the concept of multi-source BFS so if we are applying the concept of multi Source BFS it means that we have to put multiple sources in our queue so initially what we will do initially initially we will put all those ones which are at the boundary once which are at the boundary edit so you will be I will be putting all those ones which are at the boundary into my Q data structure so okay so this is my Q data structure I will put all the ones inside this boundary let it sorry okay why am I doing that because I need to find all the other ones which are basically not in the boundary but are connected to these boundaries and how can I do that I have to Traverse from the boundary okay so if I if I'm if I am putting this one into my queue this what will be FS group BFS will Traverse in all possible Direction and look for the next one that is present which is connected to this one and it will try and it will make it visited so that we can eliminate that one from our answer so basically what we will do initially we will put up all the ones which are at the boundary inside my queue and I will Mark all those visitors then I have to run a BFS batch for search from all of those uh points coordinates uh of the boundary ones and from that it will keep on marking visited all the ones which are directly connected to the given ones in any other one of the four directions so after this BFS gets over my uh my Matrix will contain will Mark all those ones visited which I want to eliminate and whatever once remains unvisited are basically my answer I hope everyone got it this is the beauty of breadth first so search so Sumit is saying that we need to find the number of ones that are not in boundary yes we need to find the number of funds that are not in boundary that is good so but we also need to find the number of ones which are in more way connected to boundary okay because there may be a one which is not in the boundary but there is a connection between that what the one that is in the boundary there is a direct path so we want to avoid that also so that is how the breast per search will help so initially we will put all those uh ones which are at the boundary inside the queue so uh and we will run a BFS so BFS will what will BFS do BFS will try to mark all those ones visited which are directly connected to the ones at the boundary therefore eliminating our uh we will eliminate all those ones which are not needed okay and after that after the BFS gets over whatever ones are unvisited we will have those ones uh as our answer we will count those one and we will print that one as our answer okay so this was the concept of multi Source BFS that we are going to apply okay so let's uh continue Declaring Queue yeah so I am sharing uh the this tab uh the problem tab so how exactly are we going to do it yeah so uh the first and foremost thing is that uh we need to declare a queue so can anyone tell me what will be uh like uh uh is this Q of integers yeah so let us look into this so at first I have declared a cube which contains up here so pair basically I need to store two values the coordinates because we have a our graph is given in the form of a matrix so Matrix basically has two coordinates I comma J so I will store in the queue now I have uh int n is equal to Green Dot size so this is basically the number of rows and in time is equal to grid of 0 dot size so basically this is the number of columns okay so I hope Declaring visited array everyone understood it n and then M okay yeah so uh now I will I will I also need to put up a visited array as well so this is also visited array a matrix you can say because it is two dimensional of n rows and M columns okay so uh basically yeah so we have declared our queue we have declared our visited array now what is next now next we need to Traverse our given row why do we need to Traverse our given roof uh because yeah because we need to find all those Elements which are present in the boundary of our given uh boundary of our given Matrix and so that we can put it inside our queue so basically what we are doing we are running two nested Loops to Traverse our given grid so we are checking whether it is present in the boundary or not and what does it mean if it is present in the boundary it means that the first it is either in the first row or in the First Column or in the last row or in the last column so if either I is equal to equal to 0 or J is equal to equal to 0 or I is equal to equal to n minus 1 or J is equal to equal to M minus 1 in these cases what will happen it is basically uh the boundary boundary grid boundary cells and we have to check whether it is equal to equal to one or not if it is equal to equal to one we will just push the coordinate into our Q so this becomes the initial sources of the multi Source VFS and at the same time you will Mark those cells as visited as well because we will we are not including those in our answer okay so it makes sense that we should Mark them visit it okay this is done now what we will do now we will do Rana BFS so how exactly are we going to run a BFS let's see so uh first of all we need to run a while loop okay so let me explain you to the BFS part BFS okay so uh yeah so let us look into the BFS part so what will happen we have initially put up the uh all the multiple sources of our uh of the boundary cells in our into our queue now we will run multiple VFS from each of one of those uh cells so we run a while loop and what does the while loop do it has to we declare two variables the first is the row variable which will basically take the first element of our uh of the of the coordinate which is basically the ith and the second element so I know the row this is these two values are giving me the coordinate of the of the node under consideration or the cylinder consideration and as usual we will remove that element now the thing is that we will be traversing in all four directions of this given uh cell that is given rho comma column we will traversing in all four Direction so this two with the help of these two arrays basically I can easily Traverse in all the four Direction so what will I do I will just add rho plus Del rho of I so initially it is minus 1 in Del rho and Del column it is 0. so what will happen initially initially it will be rho minus 1 and call plus 0. so basically if you want to Traverse in all the four direction from row comma column so basically it will be plus one minus 1 and in both horizontal and vertical Direction so running a for loop from 0 to 4 and doing this operation will help us in every iteration we are moving to a new path so initially basically we are moving to the coordinate rho plus one and column in the second iteration we will be moving to rho comma column plus one in the third iteration we will be moving to row plus one and uh column the fourth iteration will be moving to row and column minus one so in this way we are moving to all possible four Direction horizontally and vertically now the question that we need to check is whether it is a valid move or not so if n row is greater than equal to 0 that is a new row and rho basically means a new row is greater than equal to 0 and N Ro is less than equal to n it means that it is within the boundary of our grid and if n column is also greater than equal to 0 and less than M it means that it is in the boundary of our grid if it is in within the grid and the second thing if it is not visited okay if it is not visited this is also important condition and if the and if the number present in that grid is 1 okay so these three condition must be satisfied the first condition is that that must be a valid cell that in which we are moving the second condition is that it should not be visited okay that uh it should not be visited already so basically uh the visiting visit of Andro and end call should be zero so because it is not visited then we will only try to visit it by doing a BFS third condition is that it should be one because we are trying to eliminate all those ones which are connected directly to uh the boundary ones okay so if these three conditions are met then only we will push that into our queue so we'll push the neuro and new column into the queue and at the same time we will Mark those visited as well for the next BFS traversal so after this entire while loop gets over we will get all those cells marked as while or all those once marked as visited which we do not want to include in our answer because there somehow or the other connected to the boundary cell okay so after that whatever once remains will be our answer so now the answer is a simple we will maintain account variable and uh what we will do we will just uh look look for all those ones which are basically not yet visited okay so let's uh let's try the code snippet for that so this is the code snippet for that okay so in count equal to zero okay so uh what will I do uh I will be writing from I equal to 0 to n so basically I am traversing the entire Matrix and we're checking for the unvisited array so if it is a one if the cell is uh cell under consideration is one and it is not visited so what we will do we will do account plus plus and finally we'll return count okay so because that will be our answer so let's submit this code and check okay so uh this code is basically correct okay so uh I hope everyone understood it can you please do a plus one in the chat those all understood this concept so that we can move to the next part thank you Sumit uh for understanding hello uh cilantra are you there thank you so let us proceed to the next part so next I will be discussing about 0 1 VFS okay now uh I will be discussing zero one VFS in a quite concise manner uh because it is a kind of an advanced topic and we will be discussing more questions later on okay so I will just give you a guest of zero one BFS and later will uh if time permits will move to another question okay so what does 0 1 BFS means so uh we learned about normal BFS we learned about multisource BFS now what exactly is 0 1bfs so 0 1 BF is by the word it means that we will have two options okay so let us see something whenever we are let's say this is our node and this is the this is a particular node that is under consideration so what are the a normal BFS 2 it keeps on traversing it looks for all the adjacent nodes to this given nodes or all the nodes which are at a level one that is which are directly connected to my given node okay it tries to find that node right hello Sumit uh do you have any doubt okay so uh we can continue then yeah so uh yeah so BFS what does vfl does let's say this is our node under consideration okay this is our node under consideration and what the what will the BFS do BFS because this is inside of a queue so it takes this node out of our queue and it explores all the nodes which are at a distance of one unit from this given node basically all the all its adjacent mode so it will uh try to visit it will try to take this node and it will put in the queue it will type this node and put in the queue and this node it it will put in the queue and in this node but uh is there any order in which uh these four nodes are getting visited no basically whatever input we will get it will visit in that fashion only let's say if you have got a list and the list in the list that it has been said that uh this node is uh coming first so basically it will go in that fashion so there is no a priority okay it's not like that uh this node uh I want to keep this node early and then this node at the end it's not like that okay so whatever order it will go inside the queue it will be processed in that fashion only so let's say if this node is going first then only this node will come out first okay but what will happens in BFS I want to put up a preference order I want to put up a preference order okay so let's say if there are two nodes okay connected to the the node under consideration and I want that this node is uh is put in the queue at the beginning and this node is put in the queue at the end okay because what will happen I want this node to be a processed first before this node okay so uh in that case let's say this is my queue and uh so if if my BFS is basically traversing this node it will try it will try to put it at the end of the queue okay because Q is nothing but first in first out so basically this will be processed after all the nodes in the front are processed but not I don't want that I want that this node to be processed in the very first only okay so in that case what will I do I will put up a like kind of a virtual weights okay so I will say that the node connected these two nodes the H has a weight of Z1 and this has a weight of 0 okay so whatever uh if the weight is equal to equal to one what we will do we will put that in the front of the and if the weight is equal to equal to zero we will put in the back of the queue so basically we are kind of giving up a partiality to this high note okay there so that it can be processed early okay so how can we Implement uh this uh BFS because we cannot use a queue because there is no functionality in queue which may permits us to put up an element in at the beginning we must put it at the end but now I want that queue to be processed in the beginning so how can I Implement that can anyone tell a data structure which will help us do that data structure which can help us do that foreign Priority Queue got it now I think I am Audible right hello yeah so uh okay so can you submit can you explain how uh you are saying a priority queue like uh what happens in a priority queue so basically uh whatever element has a value uh highest it comes on top of the priority even it will be uh accessed first okay but we do not have any value of the node which you want to put we basically have only two options either we want this or we want this okay so how can we do that okay we will be doing that in the uh by using DQ okay so by using DQ as Manju said okay hello Manju are you there I would request you to kindly stay with me till the end so that we can learn this concept very well okay so basically we will be using a DQ okay now we will DQ just do a normal BFS but at the same time we will do one more thing what we will do we basically will check the weight of the connected edges so we will see if the weight is equal to equal to 1 it means that if the weight is 1 what will it do what will I do I want to give it a special treatment so basically what will I do I will push that node in front of the DQ okay because I want that node to be processed earlier then all the nodes which are which are basically inside my Cube and whatever nodes will have a value 0 The Edge will have a 0 I will put it in the back of the D Cube so DQ is a data structure which allows this functionality and we will be using this okay so this is the concept of 0 1 BFS but we have to do a problem to understand this concept so there is an article on geek squares which I'm going to explain it to you which will basically show you the functionality of this okay yeah so uh let's uh come have a look at this so uh basically yeah so uh the question says that you have to find the shortest path in a binary weight graph okay so I would Question request everyone to kindly uh read this article I am posting it in the chat okay so uh this is a excellent article and to understand how exactly the zero one VFS works so here it is saying that every graph has a edge has a weight of either zero or One A source that is vertex is also given the graph now we have to find the shortest part from The Source vertex to every other vertex so let's have a source vertex is equal to zero and our shortest path will look something like this okay so uh let's say uh our input is something like this shortest path with some we have to basically output the array distance I which will basically put up a shortest path that is shortest path from 0 to 0 will be zero the shortest way from zero to one will also going to be 0 because the weight is zero the shortest path from 0 to 2 will be one and so on according to the input okay so how exactly are we going to solve this problem okay so we are basically going to solve this problem by Solution using uh DF uh a zero one BFS how exactly let us see so uh first of all uh we basically will have a structure to represent the edges okay so this is a structure that is representing the edges it is denoted by two variables the first variable is basically the node it is denoting and the second is basically denoting a binary value it is either 0 or 1 okay and according to that this particular weight we will either put it in the front of the DQ or at the end of the DQ okay so the first thing is that we basically need to have a distance array okay we don't need to have visited array because that won't help us much so we will be having a visited array okay so and this is a vector which is basically storing our graph adjacency list okay uh what happens in a normal value normally uh what happens you have an integer value representing the nodes but we have put up a structure we have declared a structure which contains two value the node number and the weight associated with that node okay so this is how things will proceed okay so it contains two values okay so now we will be declaring a distance uh array and initially we are basically uh putting all the values as end Max okay why exactly because we want to minimize the distance right and it will help us to uh condition to check the condition of minimization so now we are declaring our DQ and we will do exactly those things which are basically we do in a normal BFS so we will initially put up the distance of the source as equal to zero okay just like we make a uh we can you can basically imagine this distance that is a visited array okay so now what we will do so now what we will do you will we will push back the source in the rdq okay since uh uh DQ is empty right now we will just push back one element into the DQ and we'll run a normal BFS so what we will do we will take the front element of the DQ because we want to process that early and we will just remove that element from the front okay now what we will do we will process all those Elements which are adjacent to the given element and how we will do that by running a for Loop so this for Loop will run from I equal to 0 to I is less than edges of V so edges of V Dot size For Loop basically means that it is running to through all the edges so if let's say three edges are connected to the starting node let's say we are starting from node number 0 and 3 nodes are connected node number one two and three are connected so it will run from I is equal to 0 to I equal to 3 okay so basically this Vector will this array will give us the three nodes okay now we will check the optimal distance now we have to basically what do we have to do we have to minimize the distance so we what we will do we will check whether the distance of the node we are trying to reach that is Yeah so basically what we will do uh we are trying to check the optimal distance so let's say uh the distance of this Edge so edges of V of I dot 2 dot 2 means the node we are trying to reach if that distance is greater than the distance of my presence node plus the weight of the edges if that is greater it means that we can update the distance because we're going to minimize and if it an already Update Distance greater distance is present so my work is now to minimize the distance so what will I do I will just update with this value okay so because let's say if you're standing on vertex number V and you want to reach to the vertex number uh V plus one so what will you do you will distance uh you will basically add the two distances the distance your till vertex number V plus the H from V to V plus 1 you will do that so basically if you see that the distance is in the distance that if it is greater but now let's say uh I want to just uh give you an input here that we have already declared our distance array with into max into max is a very high value so if let's say if you have never visited that place so basically this will contain a very high value into max so this will always be true and basically we are at every step we are trying to minimize the distance so we will update that distance now we are done okay we have updated a distance but our work is not done so basically what we want here is that we will see whether the weight is 0 or not if it is 0 we will push back in the front of the queue DQ and if it is equal to equal to one we'll push back in the back of the queue so what will happen the weight which are basically 0 will be processed first now so this Edge which has a basically weight 0 value process first and now again we will try to find the distance from here okay so this is exactly how we are finding the shortest distance and after that what we will do we will just print the short distance array and we are able to get the shortest distance from the starting point only okay so I hope everyone understood this so any doubts here yeah so this was the concept of uh multisource BFS and uh zero one BFS okay so I hope everyone got it so let us look into one more problem we have some amount of time left yeah so the next problem uh will involve the concept of multisource BF a second okay so this is the problem so the problem says that we have to find the nearest cell having one okay BFS Question so given a binary grid of n into M find the distance of the nearest one in the grid of the each cell so uh I will not be showing you the entire code of this question this is also a BFS question and this is this question will be added as a homework okay I am adding the link to this question in the in the uh slides okay so okay so I have added this question in the slide so I would request everyone to kindly uh solve this question after the session ends okay because we will not be having a Time to implement this entirely but I will show you uh I will tell you explain all the problem that I needed to solve this problem and it's also very easy problem similar to the first we did okay so let us see we are saying that we have we have a binary grid of size n into M okay now we have to find the distance of the nearest one in a grid of excel the distance is calculated as something like this that we will basically subtract the row numbers and the column numbers and we'll add them okay and we'll also take the absolute value of the differences of the rows and the columns and do an addition so where i1 and J1 are the row number and the column number of the current cell and I2 and J2 are the row number and the column number of the nearest cell having so let's say this is our input grid okay this is my input grid now uh this is our distance array basically so uh what do we the question is saying that we have to find the uh minimum nearest one negative for each cell find the distance of the nearest one so uh since uh this is a zero the the first cell is a zero so what is the distance of the nearest one as you can see uh this 0 is basically surrounded by zeros in its two directions so this is a distance at a distance one because if you just do a distance one this will take a zero one and this will also take a one distance okay so uh let me just uh paste this stuff here and explain it to you [Music] okay so this is my distance area I hope everyone can see this okay let me just do a bit of magnification yes Okay so the question is asking we have to basically find the for every cell we have to find the nearest distance of the nearest one okay we have to find the distance of the nearest one so let us consider this cell okay let us consider this cell so what is the distance of the nearest one and distance is calculated something like this that let's say if you're standing at two values i j i 1 J1 and I2 J2 so distance is something like this modulus of I 1 minus I 2 plus modulus of uh J1 minus J2 okay so this is our distance right so uh let us see what is the distance of the nearest one from this zero so as you can see it is at a distance one because one minus 0 will be 1 and 0 minus 0 will be one so the distance is 1 and this one also the distance one so basically the nearest one has a distance one so we will put our answer equal to one in the first cell okay now the second cell is already one so what is the minimum distance uh for uh nearest one so it will be 0 because it is itself a one so this will be zero this will also be zero okay now this is the zero and it sees that the nearest one is one again because it is present on the left side okay now this is the one so answer will be zero this is a one so answer will be zero now this is a zero but one is present on its top and left side so answer will be one because it is at a one unit distance now this 0 also has a unit one okay basically so it is present on the bottom so I will just put a one here as well okay now this is the 0 and this is the zero it will also have a distance of one unit okay now this will have a distance of 0 0 right so uh basically this will be a be our answer get it so as in the question also something like this has been given so this will this will be our answer but you know that's uh there may be possibility that something like this is present let's say there is a zero here it is a zero here it is a one and here it is a zero so if this Matrix is given and I want to write the answer what will you write in the first case it is one so basically the answer will be one in the second case it is connected just by a distance one unit so answer will be again one uh but sorry here uh since of in the zero comma zero itself the input is one so the answer will be zero because that is itself the nearest one that is present so in the first case it will be zero in the uh in this one it will be one and this one it will be one but do you know that in this case in this number uh 2 comma 2 uh the answer will be 2 because uh in order to uh this is the nearest one and you are not supposed to move diagonally you are supposed to move horizontally or vertically so let's say the coordinate of this number is zero comma 0 and the coordinate of this number is one comma one so the distance between two according to this formula will be 1 minus 0 mod plus one minus 0 mod so it will be so this will be our answer and you have to implement this concept using um so you have to implement this concept using uh how will you implement this concept okay so uh let me tell you that you can apply the concept of VFS to solve this problem okay so this question is basically your homework for you all and I would like you all to solve this question so I will also give you a small hint that uh what you can do here is that uh you can maintain a BFS and you can have a Q data structure and inside that queue data structure what will you do you will put up uh like um so basically what you will do here is that uh you will start a BFS with all the grid which contains one okay because uh initially we'll put up a distance array okay which will be your answer and fill the distance array wherever there is one you can fill the distance array with value zero because that's the answer will be zero now you do or you run a loop and you Traverse across all the cells and you see if it is equal to equal to one another if it is equal to one you just push that into the queue and Mark that visited and if it is not equal to one you just mark it zero okay now what you do you run up you run a multi Source BFS and if you run a multi Source BFS you you take any particular value of one and you go across all the four directions okay okay and when you go across all the four Direction just check whether that particular value is a one or a zero okay so if it is a valid uh if it is equal to equal to zero uh then what you do uh basically uh you mark that visited and you increase the steps also okay because uh if you are moving in four Direction so basically it means that uh you are uh moving in one step because the beauty of BFS lies in that the BFS algorithm versus in step by step fashion okay so if it is if the iteration in the iteration it is moving in one step it means that uh it has uh basically moved one step forward so you the so from the initial point the distance to that number is one so we will update in that fashion okay so uh you can just simply apply your multi Source BFS and similar to the problem we did in the first the very first problem that we did the final number of enclaves exactly the same problem is there but in that we have to just count but here we have to maintain a distance array and keep on increasing the distances as we move ahead in our iteration Okay so this brings to the end of our session uh do you have any questions to ask Okay so hello Sumit are you there did you like the session hello Manju are you there so uh Sumit is in first year so so uh Manju can you please mention in which college you are in an investment year you are in okay so much is asking you are in your first year I have solved around 380 plus question in DS Alcon week so this is still not feeling that fluency problem solvent can't solve all the problems in coding contest okay uh so uh basically I will Sumit recommend you to kindly follow a guided approach so what you can do is that you can start if you have stored already 380 problems uh I think if you've solved across all particular values so uh it seems like you have to keep on trying basically right see the quantity doesn't matter the quality matters okay so uh what you can do here is that uh what like uh if you already have gotten all the topics covered then I would Solutions recommend you to solve a question without looking at the solution okay it may happen that you are looking at a solution too early in that case the problem solving skills are not getting developed okay so that is not a good thing to have right you are in your final year and from Tamil Nadu okay Manju that's very great so did you like the session so I would uh request you to kindly uh like this session on the YouTube I also subscribe to our channels so basically we can bring up uh similar contents in the future as well yeah so uh Sumit uh you got it like you can basically try a problem but do not look at a solution okay see you if you solve three attic problems and if those creative problems covers all the concept it means that whenever you are trying to solve a new problem those uh the new problem the concept in that new problem will basically be somehow related to the old problems you have solved so either you are not able to relate the concept of the new problem to the old problem but you do not know that problem already right so if you do not know that problem Outro then learn if you already know that problem then try to remember how was the problem related to the previous problems you have solved like this only you will be able to improve your problem solving skills okay so uh so uh I guess we have moved to the end of the session and um thank you for being with me I will again meet up on Tuesday that is day after tomorrow and in the next session we will be learning about uh maybe we will do a bit of problem solving sessions solve a three or four more problems and then we will move to the uh next topics of graph and then finally we move to trees welcome thank you so much thank you Manju thank you shailendra for uh being with me today and a very good night to everyone and all the best for your career and preparations okay bye
12756	https://my.clevelandclinic.org/health/diseases/21741-thyrotoxicosis	Abu Dhabi\|Canada\|Florida\|London\|Nevada\|Ohio\| Home/ Health Library/ Diseases & Conditions/ Thyrotoxicosis AdvertisementAdvertisement Thyrotoxicosis Thyrotoxicosis is a treatable condition that happens when you have too much thyroid hormone in your body. Common symptoms include unexplained weight loss, a rapid heart rate and shakiness. The treatment for thyrotoxicosis depends on what’s causing it. Advertisement Cleveland Clinic is a non-profit academic medical center. Advertising on our site helps support our mission. We do not endorse non-Cleveland Clinic products or services.Policy Care at Cleveland Clinic Thyroid Disorders Treatment Find a Doctor and Specialists Make an Appointment ContentsOverviewSymptoms and CausesDiagnosis and TestsManagement and TreatmentOutlook / PrognosisPreventionLiving With Overview What is thyrotoxicosis? Thyrotoxicosis happens when you have too much thyroid hormone in your body. Advertisement Cleveland Clinic is a non-profit academic medical center. Advertising on our site helps support our mission. We do not endorse non-Cleveland Clinic products or services.Policy Thyroid hormone is the hormone that controls your body’s metabolism — the process in which your body transforms the food you eat into energy. With thyrotoxicosis, your metabolism becomes too fast, causing symptoms that can affect your entire body. You may feel like your body’s in overdrive or that you don’t have control over it. An overactive thyroid gland (hyperthyroidism) is the most common cause of thyrotoxicosis. But there are other ways you can have too much thyroid hormone in your body. Is thyrotoxicosis life-threatening? Despite having the word “toxic” in its name, only severe cases of thyrotoxicosis are life-threatening. This is called thyroid storm (thyroid crisis), and it’s rare. A sudden event or illness (like surgery or an infection) is the most common trigger of thyroid storm. It can also happen if you’re on anti-thyroid medication and suddenly stop taking it. It’s still important to get treatment for mild to moderate thyrotoxicosis, though. Untreated or undertreated thyrotoxicosis can lead to certain health issues. Symptoms and Causes What are the symptoms of thyrotoxicosis? Symptoms of thyrotoxicosis are generally the same in mild and moderate cases. But they’re more intense the more severe the thyrotoxicosis is. Signs and symptoms of mild and moderate thyrotoxicosis include: Advertisement Unexplained weight loss. Irregular heartbeat (arrhythmia). Rapid heart rate (tachycardia) — usually more than 100 beats per minute. Shakiness. Feeling nervous, anxious and/or irritable. Increased sensitivity to heat. Menstrual period changes, like lighter or missed periods. It’s important to see your healthcare provider if you have these symptoms. A severe case of thyrotoxicosis is a thyroid storm, or thyroid crisis. It’s rare. But it’s life-threatening and requires immediate medical attention. Symptoms of thyroid storm are more intense. For example, your heart rate may be over 140 beats per minute, and you may feel very agitated and confused. You may also have a high fever and even lose consciousness. What causes thyrotoxicosis? Many conditions and situations can cause thyrotoxicosis, including: Hyperthyroidism: Hyperthyroidism happens when your thyroid makes and releases too much thyroid hormone. It’s the most common cause of thyrotoxicosis. Causes of hyperthyroidism include Graves’ disease (an autoimmune condition) and overactive thyroid nodules (toxic multinodular goiter). Thyroid inflammation (thyroiditis): Certain bacterial and fungal infections, immune system issues and medications (like lithium and interferons) can cause thyroid inflammation (swelling). The inflammation causes your thyroid to leak stored thyroid hormone, resulting in higher levels of hormones than your body needs. Thyroiditis can also happen after giving birth (postpartum thyroiditis). Excess thyroid medication: Consuming excess thyroid medication (levothyroxine) can cause thyrotoxicosis. This can happen if you have hypothyroidism (underactive thyroid) and take too much of your thyroid medication, whether accidentally or intentionally. Consuming thyroid hormone from food: You can also have too much thyroid hormone in your body by consuming beef that’s contaminated with thyroid tissue from the cow’s neck. This is often called “hamburger thyrotoxicosis,” and it’s a very rare cause of thyrotoxicosis. What are the risk factors for thyrotoxicosis? Each cause of thyrotoxicosis has certain risk factors. In general, risk factors for thyrotoxicosis include: Having a biological family history of thyroid disease, especially Graves’ disease. Being female. Being over the age of 60. Having certain autoimmune diseases, like Type 1 diabetes, pernicious anemia and Addison’s disease. Having just delivered a baby. Having access to thyroid medication (levothyroxine) can also be a risk factor for thyrotoxicosis. Someone in your household may accidentally consume it or someone with factitious disorder may take it intentionally. Diagnosis and Tests How is thyrotoxicosis diagnosed? Healthcare providers diagnose thyrotoxicosis if blood tests show that you have elevated thyroid hormone levels and low or undetectable levels of thyroid-stimulating hormone (TSH). If you have thyrotoxicosis, your provider will also need to figure out what’s causing it, which could lead to another diagnosis. The following tests and assessments can lead to a diagnosis: A physical exam: To start, your provider will do a physical exam to check for signs of thyrotoxicosis, like a rapid heart rate and shakiness (tremor). Thyroid blood tests: Blood tests can check your thyroid hormone levels. When you have thyrotoxicosis, levels of the thyroid hormones triiodothyronine (T3) and thyroxine (T4) are above normal and TSH is lower than normal. Thyroid antibody blood tests can also check if Graves’ disease is the cause. Imaging tests: Various imaging tests of your thyroid can help diagnose the cause of thyrotoxicosis. They include a thyroid ultrasound and a radioactive iodine uptake (RAIU) test and scan. Your provider will go over the options and processes with you and recommend the test they think is best. Advertisement Management and Treatment What is the treatment for thyrotoxicosis? The treatment for thyrotoxicosis depends on the underlying cause. Your healthcare provider will need to find the cause to recommend the best treatment for you. Treatment may include: Anti-thyroid medications: Methimazole and propylthiouracil (PTU) block your thyroid from making hormones. These medications can treat many cases of hyperthyroidism. Radioactive iodine (RAI) therapy: RAI therapy destroys overactive thyroid cells. This usually leads to permanent destruction of your thyroid, which then causes hypothyroidism. Because of this, you’ll likely have to take replacement thyroid hormone medication for the rest of your life. Surgery: In some cases, your provider may recommend removing your thyroid gland through surgery (thyroidectomy). This will correct thyrotoxicosis that’s caused by an overactive thyroid gland, but it’ll usually cause long-term hypothyroidism. Beta-blockers: These medications can help manage thyrotoxicosis symptoms — like rapid heartbeat, nervousness and shakiness — in the short term. But they don’t change the level of hormones in your blood. Glucocorticoids (corticosteroids): These medications can help with inflammation and pain if you have thyroiditis. If outside sources of thyroid hormone are causing thyrotoxicosis — like medications or contaminated beef — thyrotoxicosis should go away once the excess hormones have cleared your system. Your provider may have you do follow-up blood tests to make sure your thyroid hormone levels have returned to a healthy range. Advertisement You need treatment for thyroid storm (severe thyrotoxicosis) in a hospital. Care at Cleveland Clinic Thyroid Disorders Treatment Find a Doctor and Specialists Make an Appointment Outlook / Prognosis What is the prognosis for people with thyrotoxicosis? The prognosis (outlook) for people with thyrotoxicosis is generally good if they receive treatment for it. There are several effective forms of treatment for thyrotoxicosis. Just like all treatments, they each have advantages and disadvantages. Together, you and your healthcare provider will determine the best plan for you. What are the complications of thyrotoxicosis? Complications from untreated or undertreated thyrotoxicosis include: Atrial fibrillation (Afib). Congestive heart failure. Ischemic stroke. Osteoporosis. Muscle weakness. These complications most commonly affect people who have untreated hyperthyroidism, especially Graves’ disease. Thyroid storm can lead to serious complications if you receive delayed or no treatment, including seizures, cardiovascular collapse and death. Prevention Can I prevent thyrotoxicosis? Most cases of thyrotoxicosis aren’t preventable. If you’re taking thyroid medication (levothyroxine), you can prevent thyrotoxicosis by never taking more than your healthcare provider prescribes. Taking too much thyroid medication can lead to thyrotoxicosis. It’s also important to store your medication safely away from children and pets. Living With When should I see my healthcare provider for thyrotoxicosis? If you’re experiencing symptoms of thyrotoxicosis, it’s important to see your healthcare provider so they can figure out the cause and recommend proper treatment. Advertisement If you have chronic thyrotoxicosis (usually a form of hyperthyroidism), you should see your provider regularly to make sure your treatment is working well. What questions should I ask my provider? If you’ve received a thyrotoxicosis diagnosis, it may be helpful to ask your provider the following questions: What’s causing my case of thyrotoxicosis? What treatment plan do you recommend? What are the pros and cons of my treatment plan for thyrotoxicosis? Can I get thyrotoxicosis again? Am I at risk for thyroid storm? Is this kind of thyrotoxicosis hereditary? A note from Cleveland Clinic Although it may sound scary, thyrotoxicosis is a manageable and treatable condition. If you’re experiencing symptoms of thyrotoxicosis — like your bodily functions are going too fast — see your healthcare provider. They can have you undergo some simple tests and recommend treatment to get your body back into balance. Care at Cleveland Clinic Cleveland Clinic’s experienced healthcare providers treat all kinds of thyroid disorders, including issues that cause hypothyroidism and hyperthyroidism. Thyroid Disorders Treatment Find a Doctor and Specialists Make an Appointment Medically Reviewed Last reviewed on 06/07/2024. Learn more about the Health Library and our editorial process. AdvertisementAdvertisement Ad Appointments216.444.6568 Appointments & Locations Request an Appointment Virtual Visits for Diabetes Rendered: Sun Sep 28 2025 03:17:30 GMT+0000 (Coordinated Universal Time)
12757	https://www.quora.com/How-do-you-convert-Kelvin-to-Celsius-or-Fahrenheit-without-a-chart	Something went wrong. Wait a moment and try again. Conversion Method Fahrenheit Temperature Units of Measure Degree Conversion Kelvin (unit of temperatu... Temperature Measurements 5 How do you convert Kelvin to Celsius or Fahrenheit without a chart? Susanna Viljanen Knows Finnish · Author has 18.2K answers and 314.9M answer views · Updated 5y Is Fahrenheit or Celsius better? Celsius. It is reasonable, rational and logical, it is decimal and it is based on the properties of water - the life giver. It works perfectly for weather as well. 0 is freezing. 5 is cold 10 is cool 15 is lukewarm 20 is comfortable 25 is warm 30 is definitely a beach weather 35 is hot. Get some ice-cream and quick! 40 is definitely hot. Get into shadow or turn on the A/C. 45 happens only in Australian outback, Iraq or Deep South 50 and you must be out in some desert, like Sahara 60 is sauna for children and foreigners 70 is sauna for young people 80 is sauna for women 90 is sauna for men 100 is sauna for swi Celsius. It is reasonable, rational and logical, it is decimal and it is based on the properties of water - the life giver. It works perfectly for weather as well. 0 is freezing. 5 is cold 10 is cool 15 is lukewarm 20 is comfortable 25 is warm 30 is definitely a beach weather 35 is hot. Get some ice-cream and quick! 40 is definitely hot. Get into shadow or turn on the A/C. 45 happens only in Australian outback, Iraq or Deep South 50 and you must be out in some desert, like Sahara 60 is sauna for children and foreigners 70 is sauna for young people 80 is sauna for women 90 is sauna for men 100 is sauna for swimmers and other afficionados Promoted by Coverage.com Johnny M Master's Degree from Harvard University (Graduated 2011) · Updated Sep 9 Does switching car insurance really save you money, or is that just marketing hype? This is one of those things that I didn’t expect to be worthwhile, but it was. You actually can save a solid chunk of money—if you use the right tool like this one. I ended up saving over $1,500/year, but I also insure four cars. I tested several comparison tools and while some of them ended up spamming me with junk, there were a couple like Coverage.com and these alternatives that I now recommend to my friend. Most insurance companies quietly raise your rate year after year. 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How do you convert a degree in Kelvin into a degree in Celsius or Fahrenheit? Joe Rovito Helped Son w/ science project on Thermal Conductivity. · Author has 2.9K answers and 7.8M answer views · 1y Originally Answered: How can one convert Celsius to Kelvin without a conversion table or formula? Is there a direct rule that can be used in this situation? · To convert Celsius to Kelvin, add 273.15 ex: Convert 0C to Kelvin 1) add 273.15: 0 + 273.15 2) evaluate the result: 0 + 273.15 = 273.15 I’ve got news for you - that’s a formula. Do you think you can handle that? I really can’t make it any easier. Joseph Donnelly 5 Yr apprenticeship and 50 years in refrigeration industry · 1y Originally Answered: How can one convert Celsius to Kelvin without a conversion table or formula? Is there a direct rule that can be used in this situation? · degrees Kelvin is a temperature scale starting at absolute zero which is the point where there is no thermal energy present. 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Related questions How do you convert Celsius to Kelvin? How do you convert Fahrenheit to Kelvin? How do you convert from Fahrenheit to Celsius without using a calculator or conversion chart (measurement)? How do you convert Fahrenheit to Celsius and Kelvin? How do I convert 195 degrees Celsius to Fahrenheit? How many degrees Celsius are in one Fahrenheit degree? How do you convert between Celsius and Fahrenheit without a calculator or a chart? Michael Oxley Former Quality Engineering (1970–2020) · Author has 1.5K answers and 969K answer views · 1y Originally Answered: What is the mathematical method for converting between Celsius, Fahrenheit, and Kelvin temperatures without using a calculator or conversion table? · To convert Celsius to Kelvin, just add 273.15. To convert Kelvin to Celsius, just subtract 273.15. Both the above can be done with mental arithmetic. Why would you want to mess about with the obsolescent Fahrenheit. Just donate your Fahrenheit thermometers to a museum and get some Celsius or Kelvin ones. Tiago Dioltas Doctoral Degree from Harvard University (Graduated 2015) · Author has 575 answers and 1.1M answer views · Updated 3y Related How do you convert Celsius to Fahrenheit without doing any math? Just memorize a few equivalence points between the two scales. I prepared a chart with the temperatures that I would recommend learning (below). It will be easier to memorize these Fahrenheit-Celsius equivalence temperatures if you associate them with something you’re familiar with in the real world. For that reason, I included temperatures that are noteworthy for some reason (freezing point of water, human body temperature, etc.). Once you learn these temperatures, you can extrapolate from them to figure out other temperatures as needed. The points I selected in the chart are somewhat evenly s Just memorize a few equivalence points between the two scales. I prepared a chart with the temperatures that I would recommend learning (below). It will be easier to memorize these Fahrenheit-Celsius equivalence temperatures if you associate them with something you’re familiar with in the real world. For that reason, I included temperatures that are noteworthy for some reason (freezing point of water, human body temperature, etc.). Once you learn these temperatures, you can extrapolate from them to figure out other temperatures as needed. The points I selected in the chart are somewhat evenly spread out from each other over the temperature range that you’re most likely to encounter, which makes extrapolation a little easier. For example, if you don’t know what 4o°C is, you might remember that 37°C = 98.6°F (body temperature), so 40°C is a few degrees above normal body temperature. For the Record: 40°C = 104°F = a very high fever After the chart, I have included some explanatory notes on the following topics: The “Double Meaning” of the Term Degrees A Disquisition Regarding Human Body Temperature Note Regarding Rounding and Precision As They Relate to Measurement The Double Meaning of the Term “Degrees” In the context of temperature, the term “degrees” can refer to a specific temperature reading on a given scale, or it can refer to a difference in temperature between two readings. At first glance, this might seem like a trivial distinction, but it’s not. To illustrate this fact, let me present you with the following scenario: At work, your boss asks you, “What is 10°C in degrees Fahrenheit?” You heard from your friend in HR that you’re currently being considered for a promotion, so you want to be extra sure of your answer in order make a good impression. You decide to ask Google, Siri, and Alexa, and they all tell you the same thing: They all say that 10°C equals 50°F. Therefore, you feel pretty confident about that answer. You tell your boss that 10°C equals 50°F, but she gives you a confused look and says, “No, I’m pretty sure that’s not correct. That wouldn’t make sense.” A moment later, your rival at work walks over and says, “I couldn’t help but overhear your conversation. I believe the answer you’re looking for is that 10°C is equal to 18°F.” Your boss replies, “Ah, yes, that makes much more sense. Thank you for being so helpful!” After this incident, you notice that your boss starts asking your rival for help more often. A couple of weeks later, you get passed by for the promotion, and your rival gets promoted instead. Where did you go wrong? Did Google, Siri, and Alexa all lie to you? If you can’t figure out what your mistake was, then you don’t understand the distinction between the two meanings of the term “degrees,” and so I suggest that you keep reading. As noted above, the term “degrees” can refer to a specific temperature reading on a given scale, or it can refer to a difference in temperature between two readings. The mathematical conversions between °F and °C for these two meanings are different. For example, if the high temperature (for the weather) yesterday was 50°F (10°C) and the high temperature today was 68°F (20°C), then today’s high was 18°F warmer than yesterday’s high (because 68°F - 50°F = 18°F difference). Using the Celsius scale, yesterday’s high was 10°C (50°F) and today’s high was 20°C (68°F), so today’s high was 10°C warmer than yesterday’s high (because 20°C - 10°C = 10°C). Thus, when we want to say how much warmer today’s high was compared to yesterday’s high, it’s a difference of 10°C, which equals a difference of 18°F. But if we want to say what yesterday’s high temperature was, it was a temperature reading of 10°C, which equals a temperature reading of 50°F. Thus, if we want to convert “10°C” to degrees Fahrenheit, the the correct answer is either 50°F or 18°F, depending on whether we are referring to a temperature reading or a difference between two temperature readings. For those are are interested in why this dichotomy exists: Ultimately it’s because neither the Fahrenheit nor Celsius scale has a “true zero.” When measurement scales have a true-zero, then this “split meaning” doesn’t exist. For example, there is a true zero when measuring length (or height), regardless of whether we are using meters or feet as the measurement units. A length of 6.0 feet is equivalent to a length of 1.8 meters. Similarly, if we are comparing two lengths, such as 20.0 feet (6.10 meters) and 14.0 feet (4.3 meters), the difference between the two lengths is (in feet) 20.0 feet - 14.0 feet = 6.0 feet or (in meters) 6.10 meters - 4.3 meters = 1.8 meters. Thus, 6.0 feet = 1.8 meters, regardless of whether we are talking about a specific measurement of length or a difference between two measured lengths. This is not true for measurements made using degrees Fahrenheit or Celsius. This is part of the reason why the base unit for temperature in the International System of Units (also known as “SI” or “the metric system”) is kelvins (a scale which has a true zero) instead of degrees Celsius (a scale which lacks a true zero). A Disquisition Regarding Human Body Temperature: Measuring body temperature and determining what should be considered a “normal” temperature are much more complicated of issues than most people realize. Numerous factors affect our measurement of body temperature, and these factors must all be taken into account if we are to attach any real meaning to a number that we record as someone’s “temperature.” As such, there is not just one body temperature that can be considered normal in all cases. In fact, there isn’t even a range of temperatures that could be considered to be normal for all people and in all circumstances. A temperature that’s normal in one set of circumstances may be abnormal in a different set of circumstances. I think it may help to consider a question unrelated to human biology before delving into the scientific reasons why the topic of “body temperature” is so complicated. Let’s think about the weather. You will often hear people say things like, “It’s really warm out this week, but this is not normal for us” or “It’s much colder than normal for this time of year.” People have a certain concept of what a “normal” temperature is when it comes to the weather. Imagine that there is a person from the (fictional) country of Thermoland who has never been to the United States, and this person asked you the following question: In the context of the weather, what’s a normal temperature in the United States? How would you answer that question? 65°F? 50°F? Perhaps a range of temperatures: 40–75°F? Oh, and is this daytime temperature we’re talking about? Nighttime? The high temperature? The low? The average? Do we measure it in the shade? In direct sunlight? Next to a body of water? Far away from water? Should we include the wind chill or heat index? Which season are we talking about? Or is this the average temperature for the entire year? And which part of the United States are we considering? Kansas (geographically in the middle)? Washington, DC (the nation’s capital city)? New York City (the most populous city)? California (the most populous state)? Alaska (the largest state by area)? Texas (the loudest state)? The questions could go on and on, yet each additional question would have a significant effect on what we pick as a “normal” temperature. You can probably see by this point that there is no easy way to give a “normal temperature” for the United States because there are so many different factors that must be considered. The more we try to clarify the issue, the more complicated it becomes. Any answer that’s considered “normal” for Miami or Honolulu is going to be off the mark for places like Chicago or Minneapolis, yet they’re all part of the United States. Stating an answer such as “60°F” is meaningless without a lot of additional clarifying information: What location? What time of day? What time of year? Weather conditions? Wind chill? Heat index? Is this an average or a single reading? The same general concept applies to the topic of determining what a “normal” human body temperature is. The question “What is a normal body temperature?” is too simplistic to accurately address what is, in fact, a complex issue. Many factors play a part in how we measure body temperature and how we decide what we should consider to be “normal.” Among the major factors that need to be taken into account are the following: Different people’s bodies have different set points for temperature. All else being equal, what’s normal for one person might be a little higher or lower than what’s normal for another person. Due to daily biological cycles (circadian rhythms), a given person’s temperature will go up or down depending on the time of day. It’s typical for a person’s body temperature to vary by about 0.9°F (about 0.5°C) throughout the day, with the lowest temperature in the early morning and the highest temperature in the late afternoon or evening. A person’s activities will also affect body temperature. For example, even though people’s temperature tends to be lowest in the early morning, the temperature measured at 6:00am of someone who just ran 5 miles on a treadmill is going to be higher than the temperature of an otherwise similar person who is still sleeping in bed. Consuming any type of food or beverage will temporarily change body temperature readings. There are two major reasons for this: Reason #1: Pretty much anything that we consume as food or drink will be hotter or colder than our current body temperature. This should be obvious in the case of things like ice cream (around o°F or -20°C), ice water (around 32°F or 0°C), or heated/cooked foods (typically at least 135°F or 60°C), but keep in mind that even something you eat that’s at room temperature (e.g., a banana that’s sitting out on your kitchen counter) is colder than your body temperature by around 30°F or 17° C. As a general food safety rule, it is recommended to avoid serving perishable foods at anything close to human body temperature, since our body temperature is also an ideal temperature for pathogenic microorganisms to grow. In order to prevent the growth of microbes, perishable foods are typically kept at temperatures that are either much colder or much warmer than our body temperature. Reason #2: The process of digestion tends to cause measurable increases in body temperature about 20–30 minutes after food is consumed, due to increases in the body’s metabolic rate. Being in a cold or warm environment (such as inside in the air conditioning versus outside on a hot summer day) changes our temperature somewhat. The method of measuring body temperature, including the specific location of the body measured, also affects temperature readings. Compared to a temperature taken orally (by mouth): a temperature taken in the ear (tympanic) or rectally will tend to be 0.5–1.0°F (0.3–0.6°C) higher. a temperature taken in the armpit (axillary) or on the forehead will tend to be 0.5–1.0°F (0.3–0.6°C) lower. Normal body temperature differs between men and women, and also varies based on age. For women who are menstruating, body temperature varies by about 0.5–1.1°F (0.3–0.7°C) based on where they are in the menstrual cycle. We have been discussing “normal” body temperature, but factors like illness, injury, emotional state, and medications/drugs can also change a person’s body temperature. Note Regarding Rounding and Precision As They Relate to Measurement: Even though people often think of 98.6°F (37°C) as “normal,” this fails to capture the range of normal variation among people. It’s not even particularly accurate as an average across all people. It is rather unlikely that our measurements of human body temperature would happen to have an average of exactly 37°C (the figure cited most often in a medical context), as opposed to something a little higher or lower like 36.8°C or 37.1°C. The fact of the matter is that the figure of “37°C” is typically the result of rounding to the nearest degree Celsius. Further evidence of the effect of rounding is that a fever is traditionally defined as being 38°C or higher. Note that it’s not 37.9°C and not 38.1°C, but “38°C.” While it’s true that all measurements of continuous quantities must be rounded in one way or another, there’s something misleading about the figure “37°C” being presented as “normal” given that body temperature is typically reported to the nearest tenth of a degree. This leads people to assume that normal is “37.0°C” only, and therefore that 36.9°C or 37.1°C are abnormal, even though that’s not an accurate or useful definition of “normal” for any context. If it’s not clear to you why those figures are rounded, consider this: If you go shopping at the grocery store and buy a bunch of food, it’s unlikely that your total bill would be exactly $100.00, down to the penny. Even if someone said, “I usually spend $100 on groceries per week,” the reality is that “$100” is almost always a rounded figure. Their actual bill might be $99.14 one week or $102.51 another week, but it’s unlikely that that the bill on any given week was exactly $100.00, or even that the average across many weeks was exactly $100.00. Instead, we can only assume that it was around $100. If we (somehow, and for some reason) obtained access to this person’s actual grocery receipts, we could compare that “data” to the “$100” figure that they told us. If we found that the weekly totals for the five most recent weeks of groceries were $62.30, $58.13, $52.97, $65.75, and $59.91, then it would be reasonable to conclude that the person was not being honest with us when they said they spend “$100.″ In such a case, it might have been more accurate for them to say “$60.” (The average of those five numbers is actually $59.81.) However, if the totals were actually $105.65, $93.98, $97.20, $99.84, and $106.49, then I would think that their estimate of $100 per week was reasonable. Even though the average of those five numbers is technically $100.63, it’s understood that that is included in the range when someone casually says “$100.” Similarly, even though people say that “37°C” is normal, that does not mean that your reading of “37.6°” is not normal. The number “37” implies “around 37,” and is actually reflective of a wider range of temperatures that cannot be expressed using a single numerical value. If we wanted to imply that we had more precision in our measurement, then we might say that normal is “37.00000°C,” but that level of precision is unjustified, both based on the limitations of our measurement tools and also the normal variation among different people. So we just say “37°C,” just like the person telling us about their grocery expenses says they spend “$100.” By the way: The temperatures often cited as “normal” and “fever” on the Fahrenheit scale are 98.6°F and 100.4°F, and although these may seem to be more precise figures, they really are not: 98.6°F and 100.4°F are simply the exact conversions of 37°C and 38°C to the Fahrenheit scale using the formula F = 1.8C + 32. This is an example of what’s known as false precision. What is false precision? Here’s another example that may help to explain this concept: If someone asks me how much I weigh, I might say, “170 lbs.” My bathroom scale is not extremely accurate or precise. Sometimes the reading looks a little above the “170” mark, and sometimes, it’s a little under the “170” mark, but the reading is generally around 170 lbs. Besides the issues regarding the bathroom scale itself, my body weight fluctuates small amounts throughout the day and from one day to another. The “170″ number implies that my weight is probably more than 160 lbs. and less than 180 lbs, but it’s possible that it’s actually 169 lbs. or 172 lbs. I might get a slightly different number each time I weigh myself, but those numbers are usually pretty close to 170, so that’s what I give as my weight. Using very precise standards, the conversion from pounds to kilograms is: 1 pound = 0.453592 kilograms. Since I said I weigh 170 lbs., does that mean that my weight in kilograms is 1700.453592 = 77.11064 kg? No. That would imply a false degree of precision down to the nearest 1/100th of a gram, or around the weight of a single sesame seed. In other words, if I reported my body weight with that much precision, I would be implying that if I was standing on the scale while I ate a single sesame seed, that my scale would reliably register the increase in weight from that one seed — but my scale has nothing even close to that level of precision. The scale’s reading before and after I ate the seed would look exactly the same. The level of precision needed to register a sesame seed would seem excessive when discussing someone’s body weight, yet it would be insufficient if we were measuring how much of a drug is in each pill of a medication: For example, being off by 1/100th of a gram (which is 10 mg) could be fatal for a medication that is supposed to contain 1 mg per pill. Thus, the amount of precision that we use depends on the situation (and on what the consequences will be if we are wrong). If I said that I weighed 77.11064 kg, that might suggest that I weigh more than 77.11059 kg but less than 77.11068 kg, but the fact of the matter is that I can’t be sure on such a detailed level because I only know that I weigh around 170 lbs. With a more precise weighing scale, my true weight could actually turn out to be: 169 lbs. (76.7 kg) or 171.3 lbs. (77.70 kg) or 170.004 lbs. (77.1125 kg) All of those weights would fall outside of the narrow range implied by “77.11059 kg.” And since I can’t guarantee such precision with the scale that I have, I don’t pretend to know more than I really do by giving numbers with 7 significant digits. Plus, there’s almost never any situation where it is necessary to know someone’s weight to such a precise degree. By saying “170 lbs.” (which technically has only 2 significant digits), I imply that my answer is accurate to the nearest 10 pounds. If I say “170 lbs.,” then I’m telling you that it’s somewhere in the range of 165–174 lbs, and that’s enough precision for most situations. If someone wanted to know (for some reason) if my weight was over or under 171 lbs., I could not say for sure. It’s PROBABLY under 171 lbs., but if we measured it more precisely and it was 171.2 lbs., I would not be shocked. But if it was 168.8 lbs., I also wouldn’t be shocked. Both are equally likely given the limitations of precision implied by the figure “170 pounds.” Therefore, if I weigh “170 lbs.,” then the most honest way to state that in kilograms would be with an equivalent degree of precision: “77 kg.” Applying that logic to the issue of temperature conversions: If a scientist recorded a measurement of “37°C” and wanted to convert that reading to degrees Fahrenheit, then the formula F = 1.8C + 32 would yield the mathematical result of “98.6°F.” However, the scientist should report this result with the same degree of precision as the measured data. The reason for this is that plugging a measured value into a mathematical formula cannot possibly add any real information about that measurement that wasn’t there to begin with. Any false precision that is added to a measurement in this way (whether intentionally or not) is artificial and not based on reality. It’s effectively an error. Since the original data (“37°C”) was measured to two significant figures, then the conversion to degrees Fahrenheit should also be reported using only two significant figures: “99°F.” Your response is private Was this worth your time? This helps us sort answers on the page. Absolutely not Definitely yes Sponsored by Grammarly Is your writing working as hard as your ideas? Grammarly’s AI brings research, clarity, and structure—so your writing gets sharper with every step. Loot Offers 24/7 Bachelor of technology in Electronics & JavaScript (programming language), Jawaharlal Nehru Technological University, Hyderabad (Graduated 2022) · 2y Originally Answered: How do you convert Fahrenheit to kelvin or Celsius without a calculator? · To convert Fahrenheit to Celsius, subtract 32 from the Fahrenheit temperature and then multiply the result by 5/9. To convert Fahrenheit to Kelvin, first convert to Celsius using the method above and then add 273.15. Ronald Richards Former Physics Professor at Universidad Ana G. Méndez (2015–2020) · Author has 2K answers and 699.7K answer views · 1y Originally Answered: What is the mathematical method for converting between Celsius, Fahrenheit, and Kelvin temperatures without using a calculator or conversion table? · F = 1.8C +32 or C = (F - 32)/1.8 K = C + 273.15 You should be able to use these equations without using a calculator. Promoted by Almedia Charlee Anthony Go-to Resource for Realistic, Side Hustle Ideas · Sep 22 How can I make an extra $200 a week online? This one good method to make extra income brought me over $3,000, and I still make $150–$200 every single week from Freecash. As a gig hunter, this app came natural to me. I hacked this Freecash platform so you don't have to. These GPT platforms can be hell to learn, so learn my top methods, rent free.👇 How I consistently make $200 a week online Here’s exactly how I made $200 a week online with zero upfront investment. 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How to start right now sign up and grab your $5 welcome bonus instantly complete your first high-paying offer tonight and cash out within hours. reinvest small amounts into in-app purchases to unlock bigger rewards. withdraw your first payout to PayPal before the week’s over. Freecash has become my go-to extra income stream. If you want to start seeing $200 weeks, then stop your doom scroll session and try Freecash now. Dug McDowell A parvenu of the dilettante sort. · Author has 1.1K answers and 2.1M answer views · 6y Related Is Fahrenheit or Celsius better? Is Fahrenheit or Celsius better? People look at the weather app or ask Google, Seri, or Alexa, “What’s the weather?”, so they know what to wear. Celsius is best for getting dressed. Give up Fahrenheit today, change all your apps to Celsius, I guarantee that after a week, you will never go back to Fahrenheit. I promise you will never have to do math. There is no need to convert Celsius to Fahrenheit or Fahrenheit to Celsius because Celsius converts to simple words. CELSIUS WORD CONVERSION: First memorize four numbers and the words they represent. That isn’t hard, is it? No doubling and adding 32, Is Fahrenheit or Celsius better? People look at the weather app or ask Google, Seri, or Alexa, “What’s the weather?”, so they know what to wear. Celsius is best for getting dressed. Give up Fahrenheit today, change all your apps to Celsius, I guarantee that after a week, you will never go back to Fahrenheit. I promise you will never have to do math. There is no need to convert Celsius to Fahrenheit or Fahrenheit to Celsius because Celsius converts to simple words. CELSIUS WORD CONVERSION: First memorize four numbers and the words they represent. That isn’t hard, is it? No doubling and adding 32, or finding the square root. Nope, just four numbers that increase by ten and now you know can plan what to wear. When you got these four numbers and the words they convert into down. When it takes no thinking to rattle out, “zero is freezing, ten is cold, twenty is warm, and thirty is hot”, you are now ready for the second phase of numbers and their conversion words. These will be easy and intuitive, these will be the numbers that let you get more detail. Here they are; Do you see what we did there? With the exception of the word cool, which was a new word, but all the others are just a matter of adding the word very to our other words. Very easy and very smart. Let’s go over our new numbers and words. Easy right? Eight numbers that convert to six words; freezing, cold, cool, warm, hot, and very. I have two bonus words for those who want to be ready for anywhere and any weather. That’s it. Try Celsius for a week and you will NEVER go back to Fahrenheit. In fact, within a month you won’t even understand Fahrenheit temperatures. They will not aid you in dressing or planning your day. Leave that all up to Celsius. And the bonus is that when you go to Canada, Brazil, Europe, Japan, actually anywhere outside the USA, you will understand what the weather report means. As a helpful guide, I will list all of the temperatures below, that you can print out, and hang on the fridge for easy reference. Have a nice day. Tony Christian Ratcliffe Former Electromechanical tech and trainer · Author has 4.5K answers and 16.4M answer views · 11mo Related Can Celsius be converted to Kelvin without using Fahrenheit as an intermediary? My understanding is that Celcius is actual temperatures where 0 degrees represents water at sea level whereas Kelvin temperatures are from absolute zero (-273C) but it is also used to indicate a change in temperature rather than an actual temperature. No conversion is needed because it’s a one to one ratio between the two but you need to know the starting temperature in Celcius. For example 20 degrees Celcius to 200 degrees Celcius. (20C to 200C) represents a rise of 180 Kelvin. This could also be given as a 180 Kelvin increase from a start point of 20 degrees Celcius. I hope that makes sense. My understanding is that Celcius is actual temperatures where 0 degrees represents water at sea level whereas Kelvin temperatures are from absolute zero (-273C) but it is also used to indicate a change in temperature rather than an actual temperature. No conversion is needed because it’s a one to one ratio between the two but you need to know the starting temperature in Celcius. For example 20 degrees Celcius to 200 degrees Celcius. (20C to 200C) represents a rise of 180 Kelvin. This could also be given as a 180 Kelvin increase from a start point of 20 degrees Celcius. I hope that makes sense. Roger Bliss Engineer / Programmer (1987–present) · Author has 21.4K answers and 8.4M answer views · 10mo kelvins to celsius - just add 273.15 celsius to fahrenheit is easy to remember double it subtract 10% add 32 e.g. 37 C 2x37 =74 , 74 -10% = 67.6, 67.6 + 32 = 98.6 M Adnan Ch BSCS from Virtual University of Pakistan (Graduated 2010) · 2y Originally Answered: How do you convert Fahrenheit to kelvin or Celsius without a calculator? · There are different formulas to convert temperatures between the three most common units: Fahrenheit, Celsius, and Kelvin. To convert Fahrenheit to Celsius, you can use the following formula: C = (F - 32) (5/9) To convert Fahrenheit to Kelvin, you can use the following formula: K = (F + 459.67) (5/9) for complete answer Click Here Mark Hughes Consulting Defective · Author has 1.8K answers and 1.1M answer views · 2y Zero degrees K is - 273 C (give or take). n degrees C is (nine fifths of n) plus 32 F. A lot of people say 2n + 30 and this is pretty close for everyday purposes Related questions How do I convert Kelvin to Fahrenheit or Celsius? How do we convert from Celsius to Kelvin using the Fahrenheit formula? How do you convert 100 degrees Fahrenheit to Kelvin? How do I quickly convert Celsius to Fahrenheit? How do you convert a degree in Kelvin into a degree in Celsius or Fahrenheit? How do you convert Celsius to Kelvin? How do you convert Fahrenheit to Kelvin? How do you convert from Fahrenheit to Celsius without using a calculator or conversion chart (measurement)? How do you convert Fahrenheit to Celsius and Kelvin? How do I convert 195 degrees Celsius to Fahrenheit? How many degrees Celsius are in one Fahrenheit degree? How do you convert between Celsius and Fahrenheit without a calculator or a chart? Can Celsius be converted to Kelvin without using Fahrenheit as an intermediary? How do I convert Fahrenheit to Kelvin? Is there any easy way to convert Fahrenheit to Celsius on the go? How does one convert from Fahrenheit to Kelvin? How do I convert 400 Celsius to Fahrenheit? About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
12758	https://www.youtube.com/watch?v=GftJjpBmhk8	Find the Minimum of a Quadratic Function Using Desmos Math and Stats Help 28700 subscribers 10 likes Description 4077 views Posted: 24 Jan 2018 Find the minimum of a quadratic function using the free online graphing calculator Desmos.com 2 comments Transcript: hi for this video what we're going to do is we're gonna find the minimum of a quadratic function using a free online graphing calculator called decimals to access this you just go to decimals calm they also have an app for your phone so you can download the graphing technology on your cell phone or tablet so what we're going to do is we're going to enter this equation 2x squared plus 8x minus 3 into the website so let me go ahead and pull up my browser and on here like I said that you can google decimals the graphing calculator or you can go directly to the site www.microsoft.com/downloads [Music]
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The orbit is a regular, repeating path and refers to the time it takes for a celestial body to orbit around its star, such as the Moon's orbit around the Earth or the time it takes for the Earth to orbit the Sun. Enter the Radius and Mass of the Central Object Radius of Orbit This is a required field. Mass of Central Object This is a required field. Calculate Reset Result Orbital Period s Orbital Period Minute Orbital Period Hours Click here to view image The orbital period for a circular orbit is calculated from Kepler’s third law (law of periods) which states that the square of the time period of any satellite is proportional to the cube of the semi-major axis of its elliptical orbit. Here for orbital period calculation, we assume a circular orbit with radius ‘r’. The orbital radius can be calculated for different orbits such as Low-Earth Orbit – Altitude from Earth's surface 200-2000 Km Medium-Earth Orbit – Altitude from Earth's surface 2000 Km to 35,786 Km GNSS Constellations - A list of all GNSS satellites by constellations BeiDou Galileo GLONASS GPS IRNSS beidou \| Satellite Name \| Orbit \| Date \| --- \| BeiDou-3 G4 \| Geostationary Orbit (GEO) \| 17 May, 2023 \| \| BeiDou-3 G2 \| Geostationary Orbit (GEO) \| 09 Mar, 2020 \| \| Compass-IGSO7 \| Inclined Geosynchronous Orbit (IGSO) \| 09 Feb, 2020 \| \| BeiDou-3 M19 \| Medium Earth Orbit (MEO) \| 16 Dec, 2019 \| \| BeiDou-3 M20 \| Medium Earth Orbit (MEO) \| 16 Dec, 2019 \| \| BeiDou-3 M21 \| Medium Earth Orbit (MEO) \| 23 Nov, 2019 \| \| BeiDou-3 M22 \| Medium Earth Orbit (MEO) \| 23 Nov, 2019 \| \| BeiDou-3 I3 \| Inclined Geosynchronous Orbit (IGSO) \| 04 Nov, 2019 \| \| BeiDou-3 M23 \| Medium Earth Orbit (MEO) \| 22 Sep, 2019 \| \| BeiDou-3 M24 \| Medium Earth Orbit (MEO) \| 22 Sep, 2019 \| View all BeiDou satellites galileo \| Satellite Name \| Orbit \| Date \| --- \| GSAT0223 \| MEO - Near-Circular \| 05 Dec, 2021 \| \| GSAT0224 \| MEO - Near-Circular \| 05 Dec, 2021 \| \| GSAT0219 \| MEO - Near-Circular \| 25 Jul, 2018 \| \| GSAT0220 \| MEO - Near-Circular \| 25 Jul, 2018 \| \| GSAT0221 \| MEO - Near-Circular \| 25 Jul, 2018 \| \| GSAT0222 \| MEO - Near-Circular \| 25 Jul, 2018 \| \| GSAT0215 \| MEO - Near-Circular \| 12 Dec, 2017 \| \| GSAT0216 \| MEO - Near-Circular \| 12 Dec, 2017 \| \| GSAT0217 \| MEO - Near-Circular \| 12 Dec, 2017 \| \| GSAT0218 \| MEO - Near-Circular \| 12 Dec, 2017 \| View all Galileo satellites glonass \| Satellite Name \| Orbit \| Date \| --- \| Kosmos 2569 07 Aug, 2023 \| \| Kosmos 2564 28 Nov, 2022 \| \| Kosmos 2559 10 Oct, 2022 \| \| Kosmos 2557 07 Jul, 2022 \| \| Kosmos 2547 25 Oct, 2020 \| \| Kosmos 2545 16 Mar, 2020 \| \| Kosmos 2544 11 Dec, 2019 \| \| Kosmos 2534 27 May, 2019 \| \| Kosmos 2529 03 Nov, 2018 \| \| Kosmos 2527 16 Jun, 2018 \| View all GLONASS satellites gps \| Satellite Name \| Orbit \| Date \| --- \| Navstar 82 \| Medium Earth Orbit \| 19 Jan, 2023 \| \| Navstar 81 \| Medium Earth Orbit \| 17 Jun, 2021 \| \| Navstar 78 \| Medium Earth Orbit \| 22 Aug, 2019 \| \| Navstar 77 \| Medium Earth Orbit \| 23 Dec, 2018 \| \| Navstar 76 \| Medium Earth Orbit \| 05 Feb, 2016 \| \| Navstar 75 \| Medium Earth Orbit \| 31 Oct, 2015 \| \| Navstar 74 \| Medium Earth Orbit \| 15 Jul, 2015 \| \| Navstar 73 \| Medium Earth Orbit \| 25 Mar, 2015 \| \| Navstar 72 \| Medium Earth Orbit \| 29 Oct, 2014 \| \| Navstar 71 \| Medium Earth Orbit \| 02 Aug, 2014 \| View all GPS satellites irnss \| Satellite Name \| Orbit \| Date \| --- \| NVS-01 \| Geostationary Orbit (GEO) \| 29 May, 2023 \| \| IRNSS-1I \| Inclined Geosynchronous Orbit (IGSO) \| 12 Apr, 2018 \| \| IRNSS-1H \| Sub Geosynchronous Transfer Orbit (Sub-GTO) \| 31 Aug, 2017 \| \| IRNSS-1G \| Geostationary Orbit (GEO) \| 28 Apr, 2016 \| \| IRNSS-1F \| Geostationary Orbit (GEO) \| 10 Mar, 2016 \| \| IRNSS-1E \| Geosynchronous Orbit (IGSO) \| 20 Jan, 2016 \| \| IRNSS-1D \| Inclined Geosynchronous Orbit (IGSO) \| 28 Mar, 2015 \| \| IRNSS-1C \| Geostationary Orbit (GEO) \| 16 Oct, 2014 \| \| IRNSS-1B \| Inclined Geosynchronous Orbit (IGSO) \| 04 Apr, 2014 \| \| IRNSS-1A \| Inclined Geosynchronous Orbit (IGSO) \| 01 Jul, 2013 \| View all IRNSS satellites Other Calculators Orbital Acceleration Calculator Rocket Acceleration Calculator Lumens Distance Calculator Star Distance Calculator Light Speed Distance Calculator Popular Calculators Antenna G/T Ratio Calculator Apogee and Perigee Distance Calculator Apogee Distance Calculator Circular Orbit Time Period Calculator Distance between Earth's Geocenter and Satellite Calculator Effective Antenna Aperture Calculator EIRP Calculator Elliptical Orbit Time Period Calculator Space Missions ESA ISRO JAXA NASA Altius 01 May, 2025 #### Hera 01 Oct, 2024 #### Arctic Weather Satellite 01 Jun, 2024 #### EarthCARE 29 May, 2024 #### Arctic Weather Satellite (AWS) 01 Mar, 2024 View all ESA missions INSAT-3DS 17 Feb, 2024 #### XPoSat 01 Jan, 2024 #### Aditya-L1 02 Sep, 2023 #### DS-SAR 30 Jul, 2023 #### Chandrayaan-3 14 Jul, 2023 View all ISRO missions VEP-4 17 Feb, 2024 #### TIRSAT 17 Feb, 2024 #### CE-SAT 1E 17 Feb, 2024 #### XRISM 07 Sep, 2023 #### SLIM 07 Sep, 2023 View all JAXA missions NEO Surveyor 01 Jun, 2028 #### Libera 01 Dec, 2027 #### Artemis III 30 Sep, 2026 #### Artemis II 30 Sep, 2025 #### Europa Clipper 10 Oct, 2024 View all NASA missions Quick Links Home Latest Products Add Your Company Technical Articles Contact us Resources Advertise with us Popular Categories ADCS Systems Star Trackers Sun Sensors Spacecraft Valves Earth/Horizon Sensor Launch Vehicle Antenna Thrusters Satellite Calculators Acute Value Calculator Antenna Efficiency Calculator Azimuth Angle Calculator Earth Station Altitude Calculator Free Space Path Loss Calculator Orbital Period Calculator Propeller Thrust Calculator Our Network everything RF CalcTown PCB Directory GoPhotonics EMC Directory 3D Directory everything PE Copyright 2025 © SatNow All Rights Reserved \|Privacy Need Help Finding a Product? 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12760	https://www.youtube.com/watch?v=RZY19vw_O4g	Solving Linear Equations Using Inverse Operations AlRichards314 20800 subscribers 13 likes Description 842 views Posted: 25 Feb 2024 This tutorial shows how to solve linear equations using inverse operations. The tutorial includes two one step equation and two 2 step equations examples. 2 comments Transcript: in this tutorial I'm going to take a look at uh explaining how you use inverse operations to solve linear equations and actually that's useful not just for linear ones we're going to take a look at linear equations here so the first equation is 2N = 28 now what an equation is it's a statement that two different things are the same or equal that's why we have equal sign here so we're saying that 2N is the same as 28 or equals 28 now when I say 2 N that means 2 multiplied by whatever n is so until you solve it we don't know what n is the whole point is to try to find what n is so N is a placeholder it has some numerical value and we're trying to find what that numerical value is so what this equation I'm going to use flowcharts here to show what the equation means and then how we use the inverse operations so n is our variable so we start with n and as I already said 2 N means mult the N is multiplied by two so we multiply by two and then we get 28 that's the result because that's what the equation says so we have some number n that when we multiply by two that we get 28 so to use Inver inverse operations we're going to start with the 28 and we're going to do the opposite of what we did to get the 28 so the opposite of multiplying is dividing so the opposite of multiplying by two would be to divide by two and so what that looks like algebraically is we divide both sides by two you have to do the same thing to both sides I can't just divide the left by two and not the right because that changes it to be a different equation that would have a different answer okay so you have to do the same thing to both sides now what happens on the left here is these twos are said to divide out you'll hear some people use the word cancel um that's not as mathematically uh correct term uh we cancel checks but see the twos divide out so after you divide that two by the two that we divided by two ID two is one so there's actually just a one n left on the left side and so and then of course 28 divided 2 is 14 so so the way we write this as n or I mean if you really want to write there's nothing algebra wise wrong with writing One n but n and one n are the same thing so we get Nal 14 so we think that the answer is 14 so I'm going to bring my calculator over now the original equation was 2 N = 28 so if I take two and multiply it by what I think n is okay 2 14 notice that does give me 28 so that demonstrates that n the value for n is 14 now let's say that you made a mistake and dividing 28 by two maybe thought the answer was 15 you missed it by a little bit so you see if I plug 15 in place of n 2 15 isn't 28 it's 30 so 15 does not uh make the equation at the same value on both sides or we use the words often satisfy satisfy means satisfying equation means it makes both sides have the same value so the 15 doesn't so 15 is not a solution but 14 does so 14 is a solution there are some equations have multiple Solutions linear equations always just have one but there are some kinds of equations that have more than one solution we'll be dealing with just one ones with ones with one here so second equation is x + 7 = 20 so we start with some number X we want to know what x is and you add seven and you get 20 okay so the now these first two are actually they're often called one step equations because there's one step to like we multiply by two to get 28 or in this case we're adding seven to whatever X is to get 20 so that's why they're called one step in this example here and the one on the next page there there's a second step they're often called two-step equations so this one here we started with X added seven we got 20 so we're going to start with 20 and we're going to do the inverse operation so the inverse or opposite operation of adding seven would be to subtract seven so algebra wise what that looks like is we take the equation and we subtract seven from both sides and see what that does on the left here see adding seven and subtracting seven uh that's the same as adding nothing or zero it's like if you gave someone $7 and then took the $7 right away in the end you really haven't given them anything because they have the same that they started with so this this plus seven and ne7 adds to zero and we just left with X on the left side and 20 - 7 is 13 so the result is 13 so we think that X is 13 13 and if we go to our back to our calculator again and plug 13 in place of x 13 + 7 you see we do get 20 so that is the correct answer you know if if I if I did the math correctly let's say I I did 20 minus 7 let's say I thought the answer was I don't know 15 for example if I do 15 + 7 I do not get 20 I get 22 so 15 doesn't satisfy the equation but 13 does the 13 is the solution okay second equation here or sorry third one 3 w- 5 = 19 so uh W is our unknown number so 3 w means to multiply W by three so we multiply by three and then we subtract five and you get 19 now the reason they're done in that order is bed Mass actually so uh see this is multiplying by three this is subtracting five so the multiplying is is done first and then the subtracting five is done after okay so so it's not the same as if we put subtract five and then multiply by three that would be different that would not be the same so uh inverse operations so we're going to start with the 19 and now we're doing the inverse operations in the inverse order okay see we we first multiply by three and then we subtracted by five so when we do the inverse we're going to have to undo that first and then this so the opp of subtracting five would be to add five so what we do in the equation is we add five to both sides and so subtracting five and adding five is really the same as doing nothing doing zero so that's why on the left we're just left with the 3 W and 19 + 5 is 24 and then we do the opposite of multiplying by three which would be to divide by three so we divide by three both sides and these threes divide out leaving us just with one W on the left left and we get 8 24 ID 3 is 8 so that would be the solution and again I can demonstrate that it is the solution see 3 w means 3times w so three times I think W is 8 minus the five okay 3 8 - 5 does give me 19 and if you know if I did the division wrong maybe maybe I thought that was nine okay if I made a mistake if I thought it was nine see when I do 3 9 - 5 I do no not any longer get 19 I get 22 so N9 doesn't satisfy the equation but eight does now um before I go to the last example so when there's more than one step here if there's something added or subtracted for from your term that has your unknown then you're always undoing that first so notice when I started doing the opposite operations here I did the adding five like the the opposite of subtracting five first and then I did the opposite of multiplying by W so you want to do something and something in this case is the add five first so that you get rid of the five so that on the left we just have the three W term and then you undo the three times okay at last okay so in the in this equation that's why I do the adding five first to get my w term by itself and then after that I'll do the dividing by three okay one more example on the second page here so this one says we've got some unknown number X we're going to divide it by five and add six and we get 18 so uh so we're going to start with 18 and then um you see we we did the add six second so we'll do that first and the way back okay the Ops of adding six to subtract six from both sides so that's what this looks like so we'll take six away from both sides and the adding six and subtracting six is nothing okay in the end that's really nothing so we're just left with X over 5 or x/ 5 and 18 minus 6 is 12 so what that does that subtracting six is it gets rid of this so we have just X over 5 that expression with x in all alone on itself on the left side that's the idea first now uh next we have to undo dividing by five so the opposite of dividing by five would be to multiply by five so this is what we're doing both sides and so these fives divide out here so let me show you what this looks like so when we multiply five by X you could write that as 5x and of course it's over the five in the denominator and how we simplify 5x / five is this five divides into this five going in one time so there actually a one coefficient left here by our X so that just gives us plain old X or if you want to write the one you can but X and 1 x are the same thing so that's why that simplifies the X on the left and on the right 12 5 works out to be 60 so the result is 60 so we think that X is 60 and one more time I can bring my calculator over so I think that X is 60 so if I go 60 / 5 and then + 6 okay see I do get 18 and you know if for some reason let's say you multiply that wrong and you thought the answer was 70 okay if I put 70 here in place of X 70 divided by five and then add six see it does not any longer work out to what's on the right side so 70 would not be a correct solution but 60 is 60 satisfies the equation because it makes both sides have a value of 18 and so that's how you use inverse operations to solve linear equations and I hope that you found that tutorial helpful and that's the end
12761	https://pubmed.ncbi.nlm.nih.gov/9237967/	Assessment of the nutritional effects of olestra, a nonabsorbed fat replacement: summary - PubMed Clipboard, Search History, and several other advanced features are temporarily unavailable. Skip to main page content An official website of the United States government Here's how you know The .gov means it’s official. Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site. The site is secure. The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely. 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Assessment of the nutritional effects of olestra, a nonabsorbed fat replacement: summary J C Peters1,K D Lawson,S J Middleton,K C Triebwasser Affiliations Expand Affiliation 1 The Procter & Gamble Company, Winton Hill Technical Center, Cincinnati, OH 45224, USA. PMID: 9237967 DOI: 10.1093/jn/127.8.1719S Item in Clipboard Assessment of the nutritional effects of olestra, a nonabsorbed fat replacement: summary J C Peters et al. J Nutr.1997 Aug. Show details Display options Display options Format J Nutr Actions Search in PubMed Search in NLM Catalog Add to Search . 1997 Aug;127(8 Suppl):1719S-1728S. doi: 10.1093/jn/127.8.1719S. Authors J C Peters1,K D Lawson,S J Middleton,K C Triebwasser Affiliation 1 The Procter & Gamble Company, Winton Hill Technical Center, Cincinnati, OH 45224, USA. PMID: 9237967 DOI: 10.1093/jn/127.8.1719S Item in Clipboard Cite Display options Display options Format Abstract Olestra is a zero-calorie fat replacement intended to replace 100% of the fat used in the preparation of savory snacks. Olestra can affect the absorption of other dietary components, especially highly lipophilic ones, when ingested at the same time. The potential effects of olestra on the absorption of essential fat-soluble and water-soluble dietary components have been investigated in pigs and in humans. In these studies, subjects were fed daily amounts of olestra up to 10 times the estimated mean intake from savory snacks and the olestra was eaten each day of the studies. In real life, snacks are eaten on average five times in a 14-d period. Olestra did not affect the availability of water-soluble micronutrients or the absorption and utilization of macronutrients. Olestra reduced the absorption of fat-soluble vitamins A, D, E and K; however, the effects can be offset by adding specified amounts of the vitamins to olestra foods. Olestra also reduced the absorption of carotenoids; analysis of dietary patterns showed that in real life the reduction will likely be <10%. Any effect on vitamin A stores caused by a reduction in carotenoid uptake is offset by the addition of vitamin A to olestra foods. Because of the olestra-to-nutrient ratios fed and the nutritional requirements of the test subjects, the effects of olestra on nutritional status of subgroups of the population are unlikely to be different than those measured in the studies. An analysis of lipophilicity showed that olestra is unlikely to significantly affect the uptake of potentially beneficial phytochemicals from fruits and vegetables. Some people eating large amounts of olestra snacks may experience common GI symptoms such as stomach discomfort or changes in stool consistency, similar to symptoms accompanying other dietary changes. These symptoms present no health risks. PubMed Disclaimer Similar articles Assessment of the nutritional effects of olestra, a nonabsorbed fat replacement: introduction and overview.Peters JC, Lawson KD, Middleton SJ, Triebwasser KC.Peters JC, et al.J Nutr. 1997 Aug;127(8 Suppl):1539S-1546S. doi: 10.1093/jn/127.8.1539S.J Nutr. 1997.PMID: 9237952 Olestra dose response on fat-soluble and water-soluble nutrients in humans.Schlagheck TG, Riccardi KA, Zorich NL, Torri SA, Dugan LD, Peters JC.Schlagheck TG, et al.J Nutr. 1997 Aug;127(8 Suppl):1646S-1665S. doi: 10.1093/jn/127.8.1646S.J Nutr. 1997.PMID: 9237961 Clinical Trial. Olestra, a nonabsorbed, noncaloric replacement for dietary fat: a review.Lawson KD, Middleton SJ, Hassall CD.Lawson KD, et al.Drug Metab Rev. 1997 Aug;29(3):651-703. doi: 10.3109/03602539709037594.Drug Metab Rev. 1997.PMID: 9262944 Review. Nutritional status of pigs fed olestra with and without increased dietary levels of vitamins A and E in long-term studies.Cooper DA, Berry DA, Spendel VA, Jones MB, Kiorpes AL, Peters JC.Cooper DA, et al.J Nutr. 1997 Aug;127(8 Suppl):1609S-1635S. doi: 10.1093/jn/127.8.1609S.J Nutr. 1997.PMID: 9237959 Review article: olestra and its gastrointestinal safety.Thomson AB, Hunt RH, Zorich NL.Thomson AB, et al.Aliment Pharmacol Ther. 1998 Dec;12(12):1185-200. doi: 10.1046/j.1365-2036.1998.00415.x.Aliment Pharmacol Ther. 1998.PMID: 9882026 Review. See all similar articles Cited by Bioavailability of vitamin D2 from enriched mushrooms in prediabetic adults: a randomized controlled trial.Mehrotra A, Calvo MS, Beelman RB, Levy E, Siuty J, Kalaras MD, Uribarri J.Mehrotra A, et al.Eur J Clin Nutr. 2014 Oct;68(10):1154-60. doi: 10.1038/ejcn.2014.157. Epub 2014 Aug 13.Eur J Clin Nutr. 2014.PMID: 25117997 Clinical Trial. Authors' financial relationships with the food and beverage industry and their published positions on the fat substitute olestra.Levine J, Gussow JD, Hastings D, Eccher A.Levine J, et al.Am J Public Health. 2003 Apr;93(4):664-9. doi: 10.2105/ajph.93.4.664.Am J Public Health. 2003.PMID: 12660215 Free PMC article. An assessment of the intestinal lumen as a site for intervention in reducing body burdens of organochlorine compounds.Jandacek RJ, Genuis SJ.Jandacek RJ, et al.ScientificWorldJournal. 2013;2013:205621. doi: 10.1155/2013/205621. Epub 2013 Feb 7.ScientificWorldJournal. 2013.PMID: 23476122 Free PMC article.Review. Gender-specific differences in the kinetics of nonfasting TRL, IDL, and LDL apolipoprotein B-100 in men and premenopausal women.Matthan NR, Jalbert SM, Barrett PH, Dolnikowski GG, Schaefer EJ, Lichtenstein AH.Matthan NR, et al.Arterioscler Thromb Vasc Biol. 2008 Oct;28(10):1838-43. doi: 10.1161/ATVBAHA.108.163931. Epub 2008 Jul 24.Arterioscler Thromb Vasc Biol. 2008.PMID: 18658047 Free PMC article. 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Yemen YER ﷼ Zambia USD $ Zimbabwe USD $ X (Twitter) Pinterest Instagram Sacred Geometry Shop Wall Art Sculptures Artisan Originals Accessories Digital Collection Blog Sacred Geometry Technology Art & Design About About Us Artist CV Contact Search Cart Item added to your cart View cart Check out Continue shopping Vesica Piscis March 28, 2023 Share Share Link Close share Copy link Vesica Piscis Geometric and Mathematical Exploration Formation of the Vesica Piscis: The vesica piscis emerges from the intersection of two equal circles such that each circle's center lies on the circumference of the other. This configuration creates a symmetrical lens-shaped almond in the middle (the vesica piscis itself) where the two disks overlap. Geometrically, the overlap is rich in structure – for instance, the centers and intersection points of the circles form two back-to-back equilateral triangles inside the vesica piscis, revealing inherent harmonic proportions. The figure's name literally means "fish's bladder" in Latin, a nod to its fish-like shape. Mathematically, this simple shape encodes fundamental ratios: it contains the square roots of 2, 3, and 5 hidden in its dimensions, numbers which fascinated Pythagoreans for their role in creating form and proportion in nature. Such properties underscore why the vesica piscis is often deemed the "womb" of sacred geometry – a genesis point from which further geometric patterns (like the Seed of Life and Flower of Life) can be constructed. Relationships to Other Symbols: The vesica piscis is a foundational template that appears in or inspires many other symbolic shapes: Ichthys (Jesus Fish): When outlined as two curved lines, the vesica piscis closely mirrors the Ichthys – the "Jesus fish." In early Christianity, the Ichthys (Greek for "fish") consisted of two intersecting arcs resembling the profile of a fish, used by secret congregations as a sign of faith and community. The almond-shaped vesica region forms the body of the fish, with the tail created by the crossing of the lines. This overlap of circles – literally a shared space – resonated with early Christians as a symbol of unity in belief. Thus the vesica piscis, by evoking the Ichthys, symbolizes not only Christ himself but also the coming together of believers (two circles joining as one). It carried connotations of protection and identity for an underground community and has remained an enduring emblem of faith. Triquetra: By adding a third circle of the same radius to the two of a vesica piscis, one obtains the triquetra or "Trinity knot." The triquetra design – three interlaced loops – can be understood as three overlapping vesica piscis lens shapes interlocked in symmetry. The central area where all three overlaps meet is a Reuleaux triangle (a curved equilateral triangle). In Celtic and pagan symbolism, the triquetra represents unity in triplicity (such as mind, body, and spirit or the Triple Goddess) and an interwoven existence where three elements are distinct yet continuously flowing into one another. The geometry of three overlapping circles visually reinforces this concept: each pair of circles creates a vesica piscis, and all three together create a balanced knot of three vesicae. In essence, the triquetra extends the dual unity of the vesica piscis into a trinity – symbolizing that beyond the union of two there can emerge a third force or harmonious wholeness embracing all three. Reuleaux Triangle: The Reuleaux triangle is the curved triangular region found at the heart of three overlapping circles (as in the triquetra). Uniquely, a Reuleaux triangle is a curve of constant width – meaning it's not a "true" triangle but a shape formed by circular arcs, with the same distance across any two parallel supporting lines. In the context of the vesica piscis, if one rotates the vesica shape 120° twice (like three vesicae in a ring), the central intersection is a Reuleaux triangle. This shape demonstrates remarkable mathematical properties: it has the smallest area for a given constant width and can rotate smoothly within a square, which has even led to its use in engineering (e.g. drill bits that cut square holes). The Reuleaux triangle's presence in sacred geometry (via the triquetra) adds a layer of meaning – it embodies strength through curved unity, and shows how circles (symbols of eternity) can combine to produce a stable, equitable shape. Its arcs are literally pieces of the original circles, emphasizing the idea that the "third force" or offspring of a union (here, of three overlapping pairs) inherits qualities of its parents. Mandorla: In art and sacred iconography, the vesica piscis shape is often called a mandorla (Italian for "almond"). A mandorla is the almond-shaped aureole created by two intersecting circles, frequently seen as the halo or frame enclosing holy figures in Christian art. Symbolically, the mandorla represents the intersection of two realms – for example, heaven and earth or divine and human. Placing Christ or the Virgin Mary within a vesica-shaped mandorla in medieval paintings was a visual theology: it showed the holy figure as the "common ground" between God and the world. This usage highlights the vesica piscis as a portal or vesicle between dualities. In a broader sense, the mandorla conveys the union of opposites (spirit and matter, masculine and feminine, etc.) and the birth of a unified third space that transcends both – a powerful concept in mysticism and Jungian thought alike. The very shape of the vesica (two circles overlapping) illustrates that when two entities overlap in mutuality, a new sacred space is born at their intersection. Sacred Geometry, Proportion, and Symmetry: Each of the above shapes – from the Ichthys to the triquetra – can trace its form back to the simple act of two circles joining. This speaks to the vesica piscis' pivotal role in sacred geometry. It is often one of the first constructions in classical geometric art, because its proportions generate foundational structures: an equilateral triangle (by connecting the circle centers and one intersection point), the square roots and harmonic ratios noted earlier, and the basis for constructing the Flower of Life pattern. The vesica's width-to-height ratio is linked to √3, lending it a pleasing balance that architects and artists have used in designs of everything from church windows to temple layouts. Symmetrically, the shape has a vertical axis of mirror symmetry, reinforcing its connotation of balance. It visually balances two equal circles – a representation of duality resolved into oneness. In summary, the vesica piscis and its related shapes illustrate how simple symmetric principles underlie much of sacred geometry. They teach us that by combining two equal wholes in perfect proportion, one yields a third form that is mathematically elegant and symbolically profound. This principle of union creating a transcendent third echoes through art, nature, and spirituality, deepening our understanding of harmony and proportion. Historical and Cultural Context Ancient Roots and Sacred Art: The vesica piscis is truly a global and timeless symbol, appearing across different civilizations in religious art, architecture, and spiritual diagrams. It has been found in ancient Mesopotamian and Indus Valley artifacts as a motif associated with the Mother Goddess and fertility (owing to its yonic shape). Later, in classical and medieval art, it became prominent in Christian iconography as the mandorla, framing Christ in Majesty and the Virgin Mary to signify their bridging of heaven and earth. Throughout Gothic architecture, the vesica piscis inspired the pointed arch and window designs of cathedrals, literally building the geometry of the divine into stone and glass. For example, the almond-shaped window and portal designs in many European cathedrals use vesica outlines to draw the eye upward in spiritual yearning. In the East, overlapping circle patterns (comparable to vesica shapes) appear in Islamic art and Buddhist mandalas, suggesting the shape's universal appeal as a representation of unity. By the Renaissance, scholars like Leonardo da Vinci studied the vesica piscis when exploring geometric ratios and the human form, further cementing its status in the canon of sacred geometry. The 4-Circle Configuration Here we see four circles arranged to create a cross-like or square symmetry. This generates six vesica piscis formations and a complex central region where all four circles overlap. This pattern relates to the four elements, cardinal directions, and seasons. It represents stability, materiality, and earthly completeness. The central overlapping region forms a particularly intricate sacred geometric pattern. Symbolism in Religious Traditions: In Christian tradition, the vesica piscis took on layered meanings. Because of its similarity to the Ichthys fish, it became linked to Jesus Christ – hence, it's sometimes colloquially called the "Jesus fish" symbol itself. Early Christians under persecution used the sign of the fish (essentially a stylized vesica piscis drawn with a single stroke) as a secret code to identify one another and gather safely. Thus, the vesica came to represent not only Christ as the "Fisher of Men" but also the unity and fellowship of believers. Meanwhile, the shape's obvious resemblance to female genitalia connected it to the divine feminine. In many ancient and pagan beliefs, an oval vesica form was a womb symbol – the vesica piscis was identified with the vulva of the Goddess, representing fertility, birth, and the generative power of the feminine. This dual identity – as both the Jesus fish and the Goddess womb – is not seen as a contradiction but rather a complementary balance. It speaks to a balance of opposites: masculine and feminine, heaven and earth, spiritual and material. Christian mystics noted that the two circles could represent Christ's dual nature (divine and human) overlapping in one being. In a more metaphysical interpretation, some saw the circles as God the Father and God the Mother, whose intersection (the vesica) is Christ the Son – an esoteric Trinitarian view aligning with the idea of creation emanating from the union of duality. Gnostic and early Christian sects often embraced this symbolism of the vesica piscis as the doorway through which the divine manifests in the world (hence its frequent appearance in baptismal fonts and church portals). Across traditions, whenever opposites needed reconciliation – whether the yin and yang in Eastern thought or the sun and moon in alchemical symbolism – the vesica piscis offered a visual language for their union, the mandorla between. The 5-Circle Configuration This pentagonal arrangement creates ten vesica piscis shapes and an even more elaborate central mandala. The five-fold symmetry connects to the golden ratio, the pentagon, and the pentagram. This configuration appears in nature (like five-petaled flowers) and has significance in sacred geometry as representing the human form, the five elements in various traditions, and dynamic balance between stability and change. Notable Uses and Interpretations: Many influential figures and cultures have employed the vesica piscis in their symbolic vocabulary. The Pythagoreans (6th century BCE) studied geometry as a sacred science and would have recognized the vesica's role in constructing polygons and harmonic ratios. Early geometric texts describe the construction of the vesica piscis as fundamental to inscribing triangles and hexagrams, suggesting its use in ancient Egypt and Greece for design and measurement. In the medieval period, the architects of Gothic Europe (often monks or geometers themselves) intentionally incorporated vesica piscis geometry into church layouts – for instance, the famous rose windows often feature a petal pattern derived from repeated vesica overlaps. The symbol was also adopted by esoteric groups. The overlap of the circles can be found in the emblem of certain secret societies and alchemical illustrations, denoting the conjunction of dual principles (Sol and Luna, sulfur and mercury, etc.). On a more humanistic note, artists like Leonardo da Vinci and later Johannes Kepler delved into the vesica's proportions when exploring the harmony of the cosmos; Kepler even noted the vesica piscis in his Mysterium Cosmographicum when relating planetary orbits. In the East, the concept of a sacred almond-shaped aura appears in Tibetan Buddhism (though the shapes there are often more circular mandalas, the idea of an interpenetrating space of enlightenment is analogous). Fast forward to the modern era, and we see corporations tapping into sacred geometry: the Coco Chanel logo, for example, with its two interlocking C's, echoes a vesica piscis shape (two circles overlapping) – perhaps unconsciously invoking notions of unity and luxury meeting in the middle. Whether in the context of divine architecture, spiritual iconography, or cultural artistry, the vesica piscis has been a quiet yet powerful teacher of the idea that two forces in harmony can create something new, balanced, and transcendent. Applications and Meditative Practice Modern Design and Architecture: The enduring legacy of the vesica piscis in design speaks to its aesthetic and symbolic power. Contemporary architects and designers continue to apply its proportions to create a sense of balance and "sacredness" in spaces. The vesica's silhouette often appears in logos, jewelry, and décor as a subtle nod to unity – for instance, it's a popular motif in pendants and ceramics aiming to convey harmony. In architecture, some modern sacred spaces (churches, temples, meditation halls) incorporate vesica piscis windows, doorways, or floor plans to imbue the structure with a naturally pleasing symmetry. Because the shape automatically brings to mind a vesica or mandorla, its presence can subconsciously evoke feelings of entering a holy or balanced environment. Even urban design has flirted with it: garden layouts or fountains sometimes use overlapping circular arcs to delineate a vesica-shaped pool or plaza, inviting people to literally inhabit that "common ground" between two circles. These applications show how the vesica piscis has moved beyond pure symbolism into practical use – it's functional sacred geometry. Designers find that the shape's proportions (width to height ~1.732, related to √3) are naturally elegant, and thus a useful template for creating visually satisfying compositions. Personal Growth and Meditation: In New Age and holistic practices, the vesica piscis is often used as a tool for meditation and spiritual growth. Its form is seen as a portal between worlds – the physical and the spiritual – and meditating upon it can help one access deeper states of consciousness. One simple exercise is to visualize oneself sitting at the center of a vesica piscis, the overlap, with one circle representing everyday reality and the other representing the divine realm. Breathing slowly, one imagines the two spheres overlapping within one's heart, fostering a feeling of union between one's human self and higher self. This meditation on the "sacred intersection" can bring about insights into balancing opposites in one's life (for example, work and personal life, or rational mind and intuitive mind). Practitioners also place the vesica piscis symbol on altars or in healing spaces to invite feminine energy and creativity. Because the almond shape echoes the female yoni, it is used as a focus to reconnect with goddess energy and creativity – one might gaze at or draw a vesica piscis and reflect on qualities of receptivity, birth, and creation. Another practical use is in crystal grids and mandalas: the vesica outline can be made with stones or drawn on paper, and crystals are laid out along its form to amplify intentions of unity or new beginnings. For example, placing motivating stones on the two circles and a grounding stone in the center could symbolically "bridge" those energies, manifesting a desired integration. Some guided meditations prompt individuals to contemplate the vesica piscis as the "womb of the universe" – envisioning it as the initial spark where the One became two and then birthed creation. This can be a profound contemplation on creation and one's place in the cosmos. By focusing on this symbol, many report a sense of centering – as if aligning with the universal pattern of creation and balance that the vesica encapsulates. Self-Reflection and Integration: Beyond formal meditation, the vesica piscis can inspire everyday reflective practices. For instance, journalers might draw two overlapping circles and label one "what I show the world" and the other "my inner self," then use the overlapping region to write qualities or desires that reconcile the two. This is a way to visually explore one's identity and find the "sweet spot" that is authentic. In conflict resolution or relationship counseling, the vesica diagram is sometimes used to identify common ground: two parties list their needs in two circles and then fill the intersection with shared values or goals – a technique directly inspired by the vesica piscis concept of unity in difference. Even in yoga or movement practice, some instructors use the vesica shape as a template for positioning the body or moving energy between two points (imagine tracing a figure-eight or infinity symbol, which is essentially a dynamic vesica). All these applications reinforce the vesica piscis' role as more than a static image – it is living symbolism that individuals can engage with to foster balance, unity, and creative genesis in their own lives. 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12764	https://www.quora.com/What-is-residual-velocity-loss-in-a-steam-turbine	What is residual velocity loss in a steam turbine? - Quora Something went wrong. Wait a moment and try again. Try again Skip to content Skip to search Sign In Mechanical Engineering Frictional Losses Power Conversion Steam Turbines Thermodynamics Thermal Efficiency Energy Generation Mechanical Efficiency Steam Power 5 What is residual velocity loss in a steam turbine? All related (31) Sort Recommended Mick Cane BSc in Physics&Psychology, London (Graduated 1975) · Author has 1.8K answers and 826.5K answer views ·3y Disclosure. I am not an engineer so I may not have got this right, but as a physicist I’m reasonably sure this is what is meant by residual velocity loss. A steam turbine extracts kinetic energy from the steam that passes through it. For a given mass (m) of steam entering the turbine at a velocity (V) the total amount of Kinetic Energy is E(k) = ½m.V². However, the steam has to exhaust from the turbine so not all of that V² can be used up. Let’s suppose the same gas is coming out at a velocity, v. Then the energy absorbed by the turbine is ½m.V²-½m.v². In other words ½m.v² of the total energy is Continue Reading Disclosure. I am not an engineer so I may not have got this right, but as a physicist I’m reasonably sure this is what is meant by residual velocity loss. A steam turbine extracts kinetic energy from the steam that passes through it. For a given mass (m) of steam entering the turbine at a velocity (V) the total amount of Kinetic Energy is E(k) = ½m.V². However, the steam has to exhaust from the turbine so not all of that V² can be used up. Let’s suppose the same gas is coming out at a velocity, v. Then the energy absorbed by the turbine is ½m.V²-½m.v². In other words ½m.v² of the total energy is lost to the exhaust. This is residual velocity loss. If we express this as a fraction (½m.v²)/(½m.V²), we can cancel the ½m and get the fractional loss as v²/V ². Multiply by 100 to get a percentage residual velocity loss. Upvote · Sponsored by Grammarly 92% of professionals who use Grammarly say it has saved them time Work faster with AI, while ensuring your writing always makes the right impression. Download 999 207 Related questions More answers below How much steam pressure is required to rotate a turbine at 1500 RPM? What are the interlocks for steam turbine? What is the temperature of steam at the exit of a turbine? How do you align a steam turbine? How tripped steam turbine on over speed? Vicky R B.E in Mechanical Engineering, Srinivasan Engineering College (SEC), Perambalur · Author has 265 answers and 2.9M answer views ·Updated 8y Related What is a back-pressure steam turbine? Back Pressure Steam Turbine is also called as Non Condensing Steam Turbine is mostly used in process steam industries or co-generation plants. The non-condensing steam turbine uses high-pressure steam for the rotation of blades. This steam then leaves the turbine at the atmospheric pressure or Higher Pressure. Advantages: The configuration of this steam turbine is very simple It is rel Continue Reading Back Pressure Steam Turbine is also called as Non Condensing Steam Turbine is mostly used in process steam industries or co-generation plants. The non-condensing steam turbine uses high-pressure steam for the rotation of blades. This steam then leaves the turbine at the atmospheric pressure or Higher Pressure. Advantages: The configuration of this steam turbine is very simple It is relatively inexpensive as compared to extraction steam turbine It requires very less or no cooling water Its efficiency is higher as it does not reject heat in the condensation process Disadvantages: The biggest disadvantage of this type of steam turbine is that it is highly inflexible. The output of this tu... Upvote · 99 55 9 4 9 2 Anurag Rao Manager (Turbine Maintenance Department) at NSPCL (A Joint Venture of NTPC & SAIL) (2010–present) ·5y Related What is Axial shift of steam turbine? Steam turbines work under high temperature and pressure The inlet pressure of steam turbine is very high pressure when compared to outlet pressure of steam turbine This difference in pressure creates a net resultant axial force on the steam turbine rotor called axial thrust and tries to move rotor in a certain direction or rather the direction of flow of steam The movement of turbine rotor in running condition is known as axial shift. It is generally measured in +ve or -ve depending on the movement towards flow of steam (+ve) or in opposite direction of flow of steam (-ve) There is generally a bal Continue Reading Steam turbines work under high temperature and pressure The inlet pressure of steam turbine is very high pressure when compared to outlet pressure of steam turbine This difference in pressure creates a net resultant axial force on the steam turbine rotor called axial thrust and tries to move rotor in a certain direction or rather the direction of flow of steam The movement of turbine rotor in running condition is known as axial shift. It is generally measured in +ve or -ve depending on the movement towards flow of steam (+ve) or in opposite direction of flow of steam (-ve) There is generally a balancing drum with labyrinth seals and combined thrust and journal bearing to take care of axial thrust in a steam turbine If the axial shift goes beyond a permissible limit, the turbine trips Upvote · 99 23 9 2 Robert Goodrick Former Steam Turbine Specialist (1981-2015) (1981–2015) · Author has 1.7K answers and 2.5M answer views ·4y Related What are steam turbine protections? There are numerous conditions under which steam turbines are normally protected by various instruments. Here follows a list of the most common threats in response to which a turbine is normally protected, by either alarms or trips. Overspeed; Low bearing oil pressure; High bearing oil inlet temperature; High thrust or journal bearing drain oil or metal temperature; High vibration, either of the rotor shaft or the bearing support structures; High exhaust pressure (condensing exhaust); High exhaust temperature (condensing exhaust); Excessive axial movement of the thrust collar within the confines of the Continue Reading There are numerous conditions under which steam turbines are normally protected by various instruments. Here follows a list of the most common threats in response to which a turbine is normally protected, by either alarms or trips. Overspeed; Low bearing oil pressure; High bearing oil inlet temperature; High thrust or journal bearing drain oil or metal temperature; High vibration, either of the rotor shaft or the bearing support structures; High exhaust pressure (condensing exhaust); High exhaust temperature (condensing exhaust); Excessive axial movement of the thrust collar within the confines of the thrust bearing (beyond that which would normally be associated with the axial clearance of the collar between the babbit coated thrust faces); Excessive differential expansion (between shell and rotor); Rapidly decaying inlet steam pressure; High turbine inner/outer casing skin temperature differential; There are a variety of conditions against which generators driven by turbines are also protected, though in many cases these act to trip the generator breaker only, and not the turbine. Upvote · 9 7 9 7 Sponsored by All Out Kill Dengue, Malaria and Chikungunya with New 30% Faster All Out. Chance Mat Lo, Naya All Out Lo - Recommended by Indian Medical Association. Shop Now 999 621 Related questions More answers below What is a balance drum in a steam turbine? Why does temperature decrease in a steam turbine? What the application of steam turbines? What is grid Synchronisation in steam turbine? Why does vibration occur in steam turbine? Ladislav Tabery Former Retired Process Control Commissioning Engineer. (1991–2020) · Author has 2.4K answers and 1.5M answer views ·5y Related What are some effects of steam turbine over speeding? Steam turbines are typically designed to survive quite large (30%) overspeed. Beyond that parts might start flying out. It doesn’t happen often but it does happen. I personally was involved as an expert in one event where a guy was killed by self-destructing steam turbine. I also know about another event when four guys were killed during overspeed testing. Soo to answer your question: The extreme effect is blades flying out. At normal circumstances the turbine speed governor trips the “trip and throttle valve” (TT valve) and shuts off steam supply. The TT valve is mounted very close to the turb Continue Reading Steam turbines are typically designed to survive quite large (30%) overspeed. Beyond that parts might start flying out. It doesn’t happen often but it does happen. I personally was involved as an expert in one event where a guy was killed by self-destructing steam turbine. I also know about another event when four guys were killed during overspeed testing. Soo to answer your question: The extreme effect is blades flying out. At normal circumstances the turbine speed governor trips the “trip and throttle valve” (TT valve) and shuts off steam supply. The TT valve is mounted very close to the turbine so the volume of steam downstream is minimized. There are circumstances when the turbine overspeeds even when the TT valve is tripped. In example the turbine drives a compressor or pump. The machine is tripped but the discharge check valve fails to close. The compressor or the pump became turbines due to back flow and the whole machine can accelerate within few seconds to a speed that will damage or destroy the machine. It is usually the compressor or the pump that fails first though. Upvote · 9 1 9 1 Thomas Holderread Former Mech Engr and Programmer (1976–2018) · Author has 1.4K answers and 1.3M answer views ·5y Related What will happen if a steam turbine is not running in constant speed? Turbines are designed with some kind of governor to maintain their RPM (revolutions per minute) at a “standard” value. In the USA that is 60 cycles per second (60 hertz). I think it is 50 cycle in Europe. If a turbine is not running at its calibrated target speed then it may be adjusting to changes in electrical load. I’ve learned that the plant operator will allow some tolerance to turbine speed to meet load. For example, in the morning when people are awakening and set on the coffee, run the electric razor, turn up the heat on the furnace, etc. more load than the night time demand will be reg Continue Reading Turbines are designed with some kind of governor to maintain their RPM (revolutions per minute) at a “standard” value. In the USA that is 60 cycles per second (60 hertz). I think it is 50 cycle in Europe. If a turbine is not running at its calibrated target speed then it may be adjusting to changes in electrical load. I’ve learned that the plant operator will allow some tolerance to turbine speed to meet load. For example, in the morning when people are awakening and set on the coffee, run the electric razor, turn up the heat on the furnace, etc. more load than the night time demand will be registered at the power plant and the turbine speed will increase slightly above 60 cycles, for example to match demand with power supplied through torque. This means also that during periods (ramps) of higher demand, the governing equipment will also demand more steam to the turbine and open the throttle more, allowing more steam volume to enter the turbine per unit time. Eventually, as throttle pressure and steam flow rate are enough to balance the load, the turbine will again “seek” its target RPM of 60 hertz. As long as the “new”, higher load or electrical demand stays steady, the “new” throttle setting and steam flow to the turbine will also be steady state. The reverse happens when demand drops and initially, before steady state is regained, the turbine will run slightly slower than 60 hertz. Apparently “all” motors, appliances, fans, etc. can tolerate a small deviation from 60 hertz. Even electric clocks have internal mechanisms that maintain the speed of the clock hands or digits. The principle is somewhat analogous to the gates of a dam’s spillway. You supply a power “potential” to the clock that is a little more than needed. The clock’s governing device lets some of the power (just what is needed to measure off proper time increments) “spill over” or “bleed” from the supplied power “reservoir”. Finally, if the turbine speed changes significantly due to some error or failure of some component it could be supplying more or less power than demand. The turbine could race (over-speed) until it destroyed itself or nearby components. If it ran too slow for demand, it could cause power brownouts at the source. Your response is private Was this worth your time? This helps us sort answers on the page. Absolutely not Definitely yes Upvote · 9 2 9 2 Sponsored by State Bank of India This Festive Season, Make Your Dream Home a Reality! Apply for an SBI Home Loan—easy, quick & hassle-free. Celebrate in a home you can call your own. Learn More 99 66 Nishu Yadav Mechanical engineer · Author has 217 answers and 2.8M answer views ·9y Related How do I calculate steam turbine efficiency? Here :- Alpha (a) = Nozzle angle or guide vane angle. V 1=absolute velocity at inlet V 2= absolute velocity at exit. Vf= flow velocity Vw= whirl velocity m(mass flow rate) = q.a.Vr1 Here - a=area q=row, density Vr1= relative velocity. In x-direction Fx=m.V1.cos(a)-(-m.V2.cos(b)) =m(V1cos(a)+V2cos(b)) Here. V1cosa=vw1 V2cosb=vw2 Now Fx=m(Vw1+Vw2) Work done /sec =m(Vw1+Vw2)u1 Efficiency (n)=work done per sec /w.p =m(Vw1+Vw2)u1100 /.5m(Nozzle). Vi^2 In y-direction Fy=m(vf1-Vf2) Thanks Continue Reading Here :- Alpha (a) = Nozzle angle or guide vane angle. V 1=absolute velocity at inlet V 2= absolute velocity at exit. Vf= flow velocity Vw= whirl velocity m(mass flow rate) = q.a.Vr1 Here - a=area q=row, density Vr1= relative velocity. In x-direction Fx=m.V1.cos(a)-(-m.V2.cos(b)) =m(V1cos(a)+V2cos(b)) Here. V1cosa=vw1 V2cosb=vw2 Now Fx=m(Vw1+Vw2) Work done /sec =m(Vw1+Vw2)u1 Efficiency (n)=work done per sec /w.p =m(Vw1+Vw2)u1100 /.5m(Nozzle). Vi^2 In y-direction Fy=m(vf1-Vf2) Thanks Upvote · 99 18 S.K. Mani B.E. in Mechanical Engineering, PSG College of Technology, Tamil Nadu, India (Graduated 1964) · Author has 3.2K answers and 5.2M answer views ·7y Related What is Axial shift of steam turbine? Steam turbine utilises Steam at high temperature to convert Thermal Energy to kinetic Energy for say Power generation or as a prime mover for other machines. The steam has an effect of expanding all parts of any machine by altering all dimensions. The rotor is also subject to axial thrust due to the flow of steam while in operation. Due to manufacturing clearances and Thermal expansion all parts (Both Stator and Rotors) are subject alteration of dimensions. These are different for different parts as materials of construction are different Mountings thrust and reactions also vary. To allow for this Continue Reading Steam turbine utilises Steam at high temperature to convert Thermal Energy to kinetic Energy for say Power generation or as a prime mover for other machines. The steam has an effect of expanding all parts of any machine by altering all dimensions. The rotor is also subject to axial thrust due to the flow of steam while in operation. Due to manufacturing clearances and Thermal expansion all parts (Both Stator and Rotors) are subject alteration of dimensions. These are different for different parts as materials of construction are different Mountings thrust and reactions also vary. To allow for this particularly most sensitive, Critical and costly part of Turbine is mounted carefully with one end fixed and the other end floating. Usually the floating has Complex Labyrinth seal to seal steam leakage to the minimum and a journal bearing to allow the expansion of the rotor and pressure oil sealing cum cooling system to take awy heat, reduce steam leakage and prevent disaster. This permitted axial shifts also prevent even minor the bending of the rotor which will cause Severe vibrations which are extremely deleterious for the turbine installation and safety of the people working in the near and far vicinity. This phenomenon is called the axial shift of turbine and is closely monitored and the machine is tripped if any of the critical points is breached. I thought I would give you link that may be useful for you to learn about Turbines. Upvote · 99 24 9 2 9 1 Sponsored by RedHat Customize AI for your needs, with simpler model alignment tools. Your AI needs context, not common knowledge. Learn More 9 7 Robert Goodrick Former Steam Turbine Specialist (1981-2015) (1981–2015) · Author has 1.7K answers and 2.5M answer views ·3y Related What is the rated speed in a steam turbine? “Rated speed” for a steam turbine is typically that which is specified by the manufacturer for the intended application. If a turbine is driving a 2-pole generator connected to the North American utility grid then its rated speed is 3600 RPM (1800 RPM for a 4-pole generator). There is a wide range of potential rated speeds for steam turbines in industrial mechanical drive applications and many units are capable of operating at speeds well in excess of the manufacturer’s “Rated Speed”. However, rated speed is an important distinction, as it serves as a benchmark to establish allowable maximum s Continue Reading “Rated speed” for a steam turbine is typically that which is specified by the manufacturer for the intended application. If a turbine is driving a 2-pole generator connected to the North American utility grid then its rated speed is 3600 RPM (1800 RPM for a 4-pole generator). There is a wide range of potential rated speeds for steam turbines in industrial mechanical drive applications and many units are capable of operating at speeds well in excess of the manufacturer’s “Rated Speed”. However, rated speed is an important distinction, as it serves as a benchmark to establish allowable maximum speed (“Overspeed Trip Limit”) which is typically specified at 110% to 112% of rated speed. This is a crucially important consideration, as the radial forces acting upon steam turbine rotor components vary as the square of the speed, so at 110% or rated speed the blade roots - by which blades are attached to wheels - and wheel shrink fits on the rotor shaft (stacked rotors) will sustain 121% of the centrifugal stresses of rated speed. Allowing a turbine rotor to operate at excessive speed can - as has often occurred - destroy these machines utterly, with significant potential for human injury and loss of life and property damage soaring into the millions of dollars. Upvote · 9 1 9 1 Inquisitive Indian Chemical Engineer · Author has 326 answers and 790.5K answer views ·7y Related How load is increased in steam turbine when speed is constant? The catch lies in the methodology used to maintain the constant speed. It is often done using a governing mechanism which can be either mechanical or electronic. The objective of this system would be to increase the inlet steam valve opening in order to allow a higher inflow of steam into the turbine that provides the required driving force for the drive to run. However, this only compensates for the increasing load requirement of the process equipment on the driven end say compressor/pump/blower etc. As the load requirement from the process increases (due to increased flow through the system) Continue Reading The catch lies in the methodology used to maintain the constant speed. It is often done using a governing mechanism which can be either mechanical or electronic. The objective of this system would be to increase the inlet steam valve opening in order to allow a higher inflow of steam into the turbine that provides the required driving force for the drive to run. However, this only compensates for the increasing load requirement of the process equipment on the driven end say compressor/pump/blower etc. As the load requirement from the process increases (due to increased flow through the system) the torque component on the driven end increases and hence the speed tends to fall to match the power being generated. Power = Torque Speed To cope up for the fall in speed the governing system lets in excess steam through the steam inlet valve (governing valve) and thus increasing the power. The increased load requirement is thus compensated. TM. Upvote · 9 5 Forbes Marshall Author has 307 answers and 265.1K answer views ·5y Related What will happen if a steam turbine is not running in constant speed? Steam turbines in normal operating conditions run at constant speed. Turbines with synchronous generators have a governing system which maintains the speed of the turbine and hence the connected generator. In abnormal conditions, like malfunction of the governing system, the turbine may not operate at constant speed which means that the generator won't operate at constant speed. Generators need to operate at constant speed for synchronization with the grid. In such a scenario, synchronization would break and the turbine system would trip / stop. Turbines with asynchronous generators, speed is Continue Reading Steam turbines in normal operating conditions run at constant speed. Turbines with synchronous generators have a governing system which maintains the speed of the turbine and hence the connected generator. In abnormal conditions, like malfunction of the governing system, the turbine may not operate at constant speed which means that the generator won't operate at constant speed. Generators need to operate at constant speed for synchronization with the grid. In such a scenario, synchronization would break and the turbine system would trip / stop. Turbines with asynchronous generators, speed is controlled by the generator & the grid. Turbine would always operate at constant speed till it's generator is connected to grid. If the grid fails turbine speed may overshoot and this moment turbine control system should close the steam inlet valve to prevent turbine runaway. Upvote · 9 1 Robert Cook Nuclear Service Engineer, Power Plant Ops, Repair, (1974–present) · Author has 12.2K answers and 7.8M answer views ·5y Related What is the efficiency of steam turbine and gas turbine? No. I disagree with the values below. A conventional medium to large steam turbine running from a superheated boiler burning fossil fuels (today, almost all are coal-fired, though some wood-burning and wood-residue plants are available) has an overall plant efficiency of 42 - 45%. No steam turbine has a thermal efficiency over 48%. A once-through gas turbine for electrical production has a thermal efficiency of 28–35%. The newer the turbine blade design, the hotter the GT temperature on exiting the burners, the closer-to-failure the GT exhaust blade coatings and geometry are, the higher the effi Continue Reading No. I disagree with the values below. A conventional medium to large steam turbine running from a superheated boiler burning fossil fuels (today, almost all are coal-fired, though some wood-burning and wood-residue plants are available) has an overall plant efficiency of 42 - 45%. No steam turbine has a thermal efficiency over 48%. A once-through gas turbine for electrical production has a thermal efficiency of 28–35%. The newer the turbine blade design, the hotter the GT temperature on exiting the burners, the closer-to-failure the GT exhaust blade coatings and geometry are, the higher the efficiency. The ONLY power plant exceeding 50% thermal efficiency are the dual-cycle units. There, a 250–300 Meg Gas turbine exhausts its high-temperature, low pressure, lower velocity exhaust through a Heat Recovery Steam Generator on the outlet of the first power plant. In the HRSG, the exhaust gases (previously wasted directly to the air) are used to boil then superheat the feedwater for an entirely separate second steam turbine. That second, independent steam turbine then generates power through a second generator using the ”free” energy from the first. Thermal efficiency for the entire paired unit continues to improve: The latest pairs are now 63–66% efficient in converting theoretical energy into usable, reliable electricity 24x7x365. Upvote · 9 4 S.K. Mani B.E. in Mechanical Engineering, PSG College of Technology, Tamil Nadu, India (Graduated 1964) · Author has 3.2K answers and 5.2M answer views ·8y Related What is critical rpm of steam turbine? In any rotating machinery. a major component is rotating. The effect is like vibration of a taut wire whic when excited vibrates as in musical instruments such as violin, Piano, Harp Saranghi. Sarod, Veena etc and the strings have a particular pitch depending the gauge of wire and its tautness. But all vibrations are accompanied with harmonic frequencies of 3rd, 5th, 7th and so on progressively reducing in amplitude and this gives the distinct sound of every voice and sound. These are called the natural frequencies and its harmonics. When a rotating component is rotating at its or near its natur Continue Reading In any rotating machinery. a major component is rotating. The effect is like vibration of a taut wire whic when excited vibrates as in musical instruments such as violin, Piano, Harp Saranghi. Sarod, Veena etc and the strings have a particular pitch depending the gauge of wire and its tautness. But all vibrations are accompanied with harmonic frequencies of 3rd, 5th, 7th and so on progressively reducing in amplitude and this gives the distinct sound of every voice and sound. These are called the natural frequencies and its harmonics. When a rotating component is rotating at its or near its natural frequency, the amplitude of the vibration is at its highest. This could lead to failure. It has been said roman soldiers marching unison are known to have caused collapse of bridges. In modern times suspension bridges have collapsed due to wind effect and causing major disaster. The rotors of turbines are massive and spinning at high speed and it could fail at critical speed. It is difficult to calculate these without high speed computing and as rotors consists of several components any one component’s failure can cascade into a major failure. Failure is actually a fatigue failure and so the strategy is to keep as low as possible from the operating speed. Upvote · 9 4 9 1 Related questions How much steam pressure is required to rotate a turbine at 1500 RPM? What are the interlocks for steam turbine? What is the temperature of steam at the exit of a turbine? How do you align a steam turbine? How tripped steam turbine on over speed? What is a balance drum in a steam turbine? Why does temperature decrease in a steam turbine? What the application of steam turbines? What is grid Synchronisation in steam turbine? Why does vibration occur in steam turbine? How many interlocking do we normally use in a steam turbine? Why? Is the Curtis turbine a pressure velocity compounded steam turbine or a velocity compounded steam turbine? What is gland steam used in steam turbines? What is motoring operation of steam turbine? How can I build a steam turbine? Related questions How much steam pressure is required to rotate a turbine at 1500 RPM? What are the interlocks for steam turbine? What is the temperature of steam at the exit of a turbine? How do you align a steam turbine? How tripped steam turbine on over speed? What is a balance drum in a steam turbine? Advertisement About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
12765	https://www.math.northwestern.edu/~scanez/courses/320/notes/riemann-lebesgue.pdf	The Riemann-Lebesgue Theorem (or, a brief introduction to Measure Theory) Our study of integration naturally leads us to ask: which functions are integrable? This is silly, since the answer is “those satisfying the deﬁnition of integrability”. A better question to ask is: is there a (quick) way to tell, just by looking at a function and without too much work, whether or not it is integrable? As long as we can determine where the function is continuous, which is often simpler to do than trying to establish integrability from scratch, the Riemann-Lebesgue Theorem will give us a way to do this. The key notion is that of a zero set, which ﬁts into the broader framework of measure theory. The ﬁnal section on Lebesgue integration is not material which may appear on the ﬁnal. 1 Zero Sets Intuitively, zero sets are the subsets of R which have zero “length”. To make this precise, we have to deﬁne what we mean by the “length” of an arbitrary subset of R. This is easy to do for intervals or unions of intervals, but trickier to do for general subsets; for example, what is the “length” of the set of rational numbers, or of the set of irrational numbers? For now, let us deﬁne what mean by “zero length”; we will come back to arbitrary “length” a bit later when we talk about measure theory. Deﬁnition 1. A subset Z ⊆R is a zero set if for any ϵ > 0 there exists a countable collection {(ai, bi)} of intervals which cover Z such that ∞ X i=1 (bi −ai) ≤ϵ. This sum is called the total length of the collection {(ai, bi)}. We will also say that a zero set has measure zero. Let us wrap our heads around this deﬁnition. Given a countable collection of open intervals, its total length is exactly what is sounds like: we are just adding up the lengths of all intervals in the collection. (Of course we should only consider collections where this sum actually exists, i.e. such that P(bi −ai) converges.) If a set Z is covered by such a collection, clearly the “length” of Z should be smaller than or equal to the total length of the collection. The above deﬁnition says that a zero set is a set whose “length” is smaller than or equal to any ϵ > 0, so that the “length” of a zero set should actually be zero. To summarize: a zero set is one which can be covered by collections of open intervals of arbi-trarily small total length. Example 1. Any ﬁnite subset of R is a zero set. Indeed, suppose that Z = {x1, . . . , xn} is a ﬁnite subset of R and ﬁx ϵ > 0. For each i, let Ii = (xi −ϵ 2n, xi + ϵ 2n) be the interval of radius ϵ 2n around xi. Then the the collection {Ii} covers Z and its total length is X length(Ii) = n X i=1 ϵ n = ϵ. Hence Z is a zero set. Note that this makes sense intuitively: the “length” of a single point is zero, and the “length” of Z is obtained by adding together the lengths of {xi} for each i. Example 2. More interestingly, any countable set is a zero set. We have already shown this for countable sets which are ﬁnite, so suppose that Z is countably inﬁnite. Since Z is countable, we can list its elements as x1, x2, x3, . . . . Let ϵ > 0 and for each i let Ii be an interval of length ϵ 2i around xi; so, I1 is an interval of length ϵ 2 around x1, I2 is an interval of length ϵ 4 around x2, I3 has length ϵ 8 around x3, and so on. Then the collection {Ii} covers Z and its total length is ∞ X n=1 length(Ii) = ∞ X n=1 ϵ 2i = ϵ ∞ X n=1 1 2 i = ϵ, where we use properties of geometric series to compute the ﬁnal sum. Thus Z is a zero set. Thus, for instance, the set of rational numbers is a zero set. The set of irrational numbers, however, is not a zero set, since if it were its union with Q would be a zero set as a consequence of the following proposition; this union is all of R, and R is not a zero set since it has “inﬁnite length”. Proposition 1. The countable union of zero sets is a zero set, as is any subset of a zero set. Proof. Suppose ﬁrst that Z1 and Z2 are both zero sets, and let ϵ > 0. Then we can countable covers {Ii} ad {Ji} of Z1 and Z2 respectively each of total length less than or equal to ϵ 2. The collection we get by taking the union of {Ii} and {Ji} then covers Z1 ∪Z2 and has total length less than or equal to ϵ, so Z1 ∪Z2 is a zero set. The same idea works for a ﬁnite union of zero sets, and for the union of countably many zero sets we use the same “ ϵ 2i ”-trick we used to show that a countably inﬁnite set was a zero set. If A ⊂Z and Z is a zero set, then for any ϵ > 0 we can ﬁnd a collection {Ii} of open intervals covering Z with total length ≤ϵ. This same collection also covers A, so it follows that A is also a zero set. Note that the above facts are perfectly intuitive: the ﬁnite (or countable) union of sets of “length” zero should still have length zero, as should any subset of a set of “length” zero. It may seem surprising that there are uncountable zero sets, but there are—see the optional problem on the ﬁnal homework assignment for an example. 2 Riemann-Lebesgue Theorem Now we can give a complete characterization of integrable functions. For a function f : [a, b] →R, let D(f) denote its discontinuity set: D(f) = {x ∈[a, b] \| f is not continuous at x}. So, for example, a continuous function has an empty discontinuity set. Theorem 1. A function f : [a, b] →R is (Riemann) integrable if and only if it is bounded and its set of discontinuity points D(f) is a zero set. So, whether or not a function is integrable is completely determined by whether or not it is discontinuous at “too many” points, or by whether or not the set of points where it is discontinuous has “length” zero. In particular, if a function is integrable on [a, b], then it is in fact continuous 2 at an uncountable number of points in [a, b]; more precisely, it is continuous “outside of a set of measure zero”, so we say that it is continuous almost everywhere. The proof of this theorem is not easy but is still pretty manageable, it just requires a bit of setup. To save time, we will just assume this to be true and leave proofs to Wikipedia or other texts. As the following examples now show, this theorem in general gives us a quicker way of determining integrability. Example 3. Since the discontinuity set of a continuous function is empty and the empty set has measure zero, the Riemann-Lebesgue theorem immediately implies that continuous functions on closed intervals are always integrable. Example 4. A piecewise continuous function has a ﬁnite set of discontinuity points. Since ﬁnite sets are always zero sets, Riemann-Lebesgue again implies that a piecewise continuous function on [a, b] is integrable. Example 5. Consider the function f : [0, 1] →R deﬁned by f(x) = ( 1 if x ∈Q 0 if x ∈R\Q. Then f is discontinuous everywhere, so D(f) = [0, 1] is not a zero set. Thus f is not integrable. Example 6. My favorite function on [0, 1] is discontinuous at each rational. Thus its set of discontinuity points is contained in a zero set (the set of all rationals) and so is a zero set itself. Hence my favorite function on [0, 1] is integrable by the Riemann-Lebesgue Theorem. The moral is that an integrable function is one whose discontinuity set is not “too large” in the sense that it has length zero. 3 Lebesgue Integration Here is another way to think about the Riemann-Lebesgue Theorem. Suppose that f : [a, b] →R is bounded. If f were integrable, we could “split” its integral up into one over the subset of points of [a, b] where f is continuous and the subset D(f) where it is not: Z b a f “=” Z [a,b]\D(f) f + Z D(f) f. Of course, so far we have only deﬁned what it means for a function to be integrable on an interval, so integrating over arbitrary subsets of R (as in the two expressions on the right) does not (yet) make sense. However, let’s ignore this for now. Since f is continuous on [a, b]\D(f), the ﬁrst integral above exists. Thus, whether or not f is integrable over [a, b] is completely determined by whether or not it is “integrable” over D(f). If D(f) is a zero set, any integral over it is zero since an integral is supposed to give the area under the graph of the function, and such an area is always zero if the “base” of the region has “length” zero. In this case then, R D(f) f = 0, so the points in D(f) contribute nothing to the integral of f. Thus, the Riemann-Lebesgue theorem says that an integrable function is one for which the points where it is not continuous contribute nothing to the value of integral. To make this precise would require us to develop a theory of integration over more general subsets of R. In fact, Example 5 also shows that we need such a theory: the function f given there is not integrable and yet I claim there is a well-deﬁned area under its graph. 3 Example 7. Consider the function f of Example 5. The region under its graph consists of vertical line segments lying over each rational in [0, 1]. Intuitively, I claim that the “area” of this region is zero. Indeed, as we know, the set of rationals in [0, 1] is a zero set, so for any ϵ > 0 we can ﬁnd a countable collection of intervals {Ii} covering [0, 1] ∩Q such that X length(Ii) ≤ϵ. Consider the collection {Ri} of rectangles of height one where Ri has Ii as its base. Then the area of Ri is height · length = 1 · length(Ii) = length(Ii). Thus the total area enclosed by these rectangles is X area(Ri) = X length(Ii) ≤ϵ. But the region A under the graph of f is contained in the region enclosed by all the rectangles Ri, so the area of A is ≤ϵ. Since this is true for all ϵ > 0, we must have that the area of A is zero as claimed. The point again is that the region under the graph of this function has a well-deﬁned area, but the theory of Riemann integration is not strong enough to detect this. We need a theory of integration which can “integrate” the same functions which Riemann integration does, but can also handle other functions which “should” be integrable. This consideration leads to what is called the Lebesgue integral and gives a glimpse into what is more generally known as measure theory. This is essentially the most general theory of integration available, and allows one to deﬁne integration over a vast variety of diﬀerent types of spaces all at once. We will outline how this works in the case of R via the Lebesgue integral. The starting point is deﬁning a general notion of “length”. We mimick the deﬁnition we gave for zero sets, only modiﬁed to allow for positive measure: Deﬁnition 2. Given A ⊆R, consider all possible collections of intervals covering A. The Lebesgue (outer) measure of A is the inﬁmum µ(A) of the total lengths of all such collections: µ(A) := inf nX (bi −ai) the collection {(ai, bi)} covers A o . If this inﬁmum does not exist—i.e. if all possible total lengths P(bi −ai) are inﬁnite—then we say that A has inﬁnite measure. The idea is similar to the one given after the deﬁnition of a zero set: the measure (or “length”) of A should be ≤the total length of any countable covering of A by open intervals, and the actual measure of A is the smallest possible such total length. A zero set is then one which has measure zero. We use “outer” to describe this measure since we are measuring the length of A from the “outside” by looking at collections of intervals which contain A in their union; there is a similar notion of inner measure where we measure the length of A from the “inside” by looking at collections of intervals whose union is contained in A. This distinction between outer and inner measure is important in the full theory of Lebesgue integration and is related to the notion of what it means for a set to be measurable, but we will skip this distinction here and focus on outer measure. Example 8. An interval [a, b], (a, b), (a, b], or [a, b) has measure b −a, as one would expect of a notion of “length”. R has inﬁnite measure, and since Q has measure zero, R\Q also has inﬁnite measure. However, since [0, 1] has measure 1, the set of irrationals in [0, 1] also has measure 1. 4 Now that we have a general notion of “length”, we can outline how this leads to a more general theory of integration. To start, let A ⊆R and consider the function χA deﬁned by χA(x) = ( 1 if x ∈A 0 if x / ∈A. We call this the indicator function of A since it “picks out” which numbers belong to A. The region under the graph of χA consists of vertical line segments lying above each element of A, and as a result the “area” of this region should intuitively be the height 1 times the length of the “base” A, so we deﬁne Z R χA dµ := µ(A) and call this the Lebesgue integral of the function χA. (The dµ is just notation referring to the fact that we are integrating with respect to Lebesgue measure.) Example 9. The function of Example 5 is precisely the indicator function of Q ∩[0, 1], and hence its Lebesgue integral is µ(Q∩[0, 1]) = 0. So, although this function is not Riemann integrable, it is Lebesgue integrable (a notion which we admittedly have not and will not deﬁne) and its Lebesgue integral is precisely what the area of the region under its graph should be, namely 0. Example 10. As a consequence of the fact that µ(Q ∩[0, 1]) = 0 and µ([0, 1]) = 1, we see that µ((R\Q) ∩[0, 1]) = 1. Thus the function g on [0, 1] which is zero at all rationals and 1 at all irrationals has Lebesgue integral 1. To deﬁne the integrals of more general functions, we proceed as follows. First, for any linear combination nχA + mχB of indicator functions (called a simple function), we deﬁne its Lebesgue integral as Z (nχA + mχB) dµ := n Z χA dµ + m Z χBdµ = nµ(A) + mµ(B). Of course, this “linearity” property is one we would expect an integral to have, so we are deﬁning the Lebesgue integral precisely so that this property is forced to hold. Note that now we already know how to integrate step functions (since we can express such a function as a linear combination of indicator functions), and the result is going to be equal to the total area enclosed by rectangles whose bases are the intervals where the “steps” occur. Finally, given a nonnegative function f, we deﬁne its integral to be the supremum of the integrals of all simple functions ≤f: Z f dµ := sup Z s dµ s is a simple function such that s ≤f . In other words, to deﬁne the Lebesgue integral of a nonnegative function, we approximate it from below by simple functions whose integrals we know, and take the supremum of the values of these integrals. One can then go on to deﬁne integrals for functions which take on both positive and negative values. The key result is the following: Theorem 2. A function which is Riemann integrable is Lebesgue integrable and its Lebesgue integral agrees with its Riemann integral. So, the upshot is that if a function is Riemann integrable to begin with, it remains integrable in the Lebesgue sense, but now Lebesgue integration allows one to integrate functions which are not Riemann integrable. This is just the tip of the iceberg, and measure theory in general has vast applications. In particular, if you ever take a more advanced probability or statistics course, expect to see some of these ideas again. 5
12766	https://www.etymonline.com/word/save	Save - Etymology, Origin & Meaning Search Log in Columns Forum Apps Premium Log in Origin and history of save save(v.) c. 1200, saven, "to deliver from some danger; rescue from peril, bring to safety," also "prevent the death of;" also "to deliver from sin or its consequences; admit to eternal life; gain salvation," from Old French sauver "keep (safe), protect, redeem," from Late Latin salvare "make safe, secure," from Latin salvus "safe" (from PIE root sol- "whole, well-kept"). From c. 1300 as "reserve for future use, hold back, store up instead of spending;" hence "keep possession of" (late 14c.). As a quasi-preposition from c. 1300, "without prejudice or harm to," on model of French and Latin cognates. To save face (1898) first was used among the British community in China and is said to be from Chinese; it has not been found in Chinese, but tiu lien "to lose face" does occur. To save appearances "do something to prevent exposure, embarrassment, etc." is by 1711; earlier save (the) appearances, a term in philosophy that goes back to ancient Greek in reference to a theory which explains the observed facts. To not (do something) to save one's life is recorded from 1848. To save (one's) breath "cease talking or arguing in a lost cause" is from 1926. also from c. 1200 save(n.) in the sports sense of "act of preventing opponent from scoring," 1890, from save (v.). The verb save in a sporting sense of "prevent the opposing side from gaining (a run, goal, etc.)" is by 1816. also from 1890 save(prep., conj.) c. 1300, sauf, "except for" (with noun as object), "with the exception of, not including," from safe (adj.), which had save (adj.) as a variant form. The evolution parallels that of Old French sauf "safe," prepositional use of the adjective, in phrases such as saulve l'honneur "save (our) honor;" also a use in Latin (salva lege, etc.). also from c. 1300 Entries linking to save safe(adj.) c. 1300, sauf, "unscathed, unhurt, uninjured; free from danger or molestation, in safety, secure; saved spiritually, redeemed, not damned;" from Old French sauf "protected, watched-over; assured of salvation," from Latin salvus "uninjured, in good health, safe," which is related to salus "good health," saluber "healthful" (all from PIE solwos from root sol- "whole, well-kept"). For the phonological development of safe from sauf, OED compares gage from Old North French gauge. From late 14c. as "rescued, delivered; protected; left alive, unkilled." The meaning "not exposed to danger" (of places, later of valuables) is attested from late 14c.; in reference to actions, etc., the meaning "free from risk," is recorded by 1580s. The sense of "sure, reliable, not a danger" is from c. 1600. The sense of "conservative, cautious" is from 1823. It has been paired alliteratively with sound (adj.) from c. 1300. In Middle English it also meant "in good health," also "delivered from sin or damnation." Related: Safeness. appearance(n.) late 14c., "visible state or form, figure; mere show," from Anglo-French apparaunce, Old French aparance "appearance, display, pomp" (13c.), from Latin apparentia, abstract noun from aparentem, past participle of apparere "come in sight, make an appearance," especially "be evident, be seen in public, show oneself" (see appear). The meaning "semblance" is recorded from early 15c.; that of "action of coming into view" is by mid-15c.; that of "a coming before the public or an audience" is from 1670s. The phrase keeping up appearances is attested from 1751 (save appearances in a similar sense is by 1711; see save (v.)). face life-saving salvage salvation salver savable saved saver saving savior sol- See All Related Words (13) Advertisement Close Trends of save adapted from books.google.com/ngrams/ with a 7-year moving average; ngrams are probably unreliable. More to explore face c. 1300, "the human face, a face; facial appearance or expression; likeness, image," from Old French face "face, countenance, look, appearance" (12c.), from Vulgar Latin facia (source also of Italian faccia), from Latin facies "appearance, form, figure," and secondarily "visage, salvage 1640s, "payment for saving a ship from wreck or capture," from French salvage (15c.), from Old French salver "to save" (see save (v.)). The general sense of "the saving of property from danger" is attested from 1878. Meaning "recycling of waste material" is from 1918, from the Br salvation c. 1200, savacioun, saluatiun, sauvacioun, etc., originally in the Christian sense, "the saving of the soul, deliverance from the power of sin and admission to eternal bliss," from Old French salvaciun and directly from Late Latin salvationem (nominative salvatio, a Church Latin reservoir figurative, from French réservoir "storehouse," from Old French reserver "set aside, withhold," from Latin reservare "keep back, save...up; retain, preserve," from re- "back" (see re-) + servare "to keep, save, preserve, protect" (from PIE root ser- (1) "... deliver c. 1200, deliveren, "save, rescue, set free, liberate," from Old French delivrer "to set free; remove; save, preserve; hand... reservation Late Latin reservationem (nominative reservatio), noun of action from past-participle stem of Latin reservare "keep back, save...up; retain, preserve," from re- "back" (see re-) + servare "to keep, save, preserve, protect" (from PIE root ser- (1) "... redeem The general sense of "rescue, deliver, save" is from late 15c. The meaning "make amends for" is from 1520s....The sense of "save (time) from being lost" (Tyndale, Shakespeare, Young, Cowper, Eliot) is after Ephesians v.16, Colossians... salve "medicinal ointment or adhesive preparation for external use on wounds and sores," Old English sealf "healing ointment," from West Germanic salbo- "oily substance" (source also of Old Saxon salba, Middle Dutch salve, Dutch zalf, Old High German salba, German salbe "ointment"), f, f") refrain mid-14c., refreinen, transitive, "exercise control over, restrain; hold (someone or something) back from action," senses now obsolete, also "exercise control over" (thoughts, desires, feelings, vices, etc.); from Old French refraigner, refrener, refreiner "restrain, repress, keep prod 1530s, "to poke with a stick," of uncertain origin; possibly [Barnhart, Century Dictionary] a variant of brod, from Middle English brodden "to goad," from Old Norse broddr "shaft, spike" (see brad), or perhaps imitative [OED]. Compare dialectal prog "pointed instrument for poking Share save ‘cite’ Page URL: Copy HTML Link: Etymology of save by etymonlineCopy APA Style: Harper, D. (n.d.). Etymology of save. Online Etymology Dictionary. Retrieved September 29, 2025, from Copy Chicago Style: Harper Douglas, "Etymology of save," Online Etymology Dictionary, accessed September 29, 2025, MLA Style: Harper, Douglas. "Etymology of save." Online Etymology Dictionary, Accessed 29 September, 2025.Copy IEEE Style: D. Harper. "Etymology of save." Online Etymology Dictionary. (accessed September 29, 2025).Copy Advertisement Remove Ads Trending 1.nice 2.Fritz 3.leniency 4.flapjack 5.madam 6.politics 7.money 8.ravishing 9.knock up 10.Christian Dictionary entries near save savageness savagery savannah savant savate save saved saveloy saver Saville Row saving Advertisement Close Want to remove ads? Log in to see fewer ads, and become a Premium Member to remove all ads. ABCDEFGHIJKLMNOPQRSTUVWXYZ Quick and reliable accounts of the origin and history of English words. Scholarly, yet simple. About Who Did This Sources Introduction Links Support Premium Patreon Merch Apps Get Chrome Extension Get iOS App Get Android App Dark Auto Light Terms of ServicesPrivacy Policy English (English) © 2001 - 2025 Douglas Harper
12767	https://math.stackexchange.com/questions/441571/lights-out-variant-flipping-the-whole-row-and-column	combinatorics - Lights Out Variant: Flipping the whole row and column. - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Lights Out Variant: Flipping the whole row and column. Ask Question Asked 12 years, 2 months ago Modified7 years, 2 months ago Viewed 8k times This question shows research effort; it is useful and clear 7 Save this question. Show activity on this post. So I found this puzzle similar to Lights Out, if any of you have ever played that. Basically the puzzle works in a grid of lights like so: 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 When you selected a light (the X), it toggled itself and all the lights in its row and column: 1 0 1 0 1 1 X 1 0 1 1 0 0 0 0 0 This got me wondering how one could tell whether there was a solution for a given setup and grid size, and if so, what was it? I can't seem to get anywhere. Could anyone push me in the right direction? combinatorics algorithms puzzle Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications edited Dec 13, 2014 at 6:30 rhymes_with_dorange 107 3 3 bronze badges asked Jul 11, 2013 at 20:25 NumeriNumeri 194 1 1 silver badge 9 9 bronze badges 4 For starters, if the grid is m×n m×n where exactly one of m m and n n are even, then a move can never change the parity of the number of lights that are on. Other than that, I'm intrigued myself.Arthur –Arthur 2013-07-11 20:37:35 +00:00 Commented Jul 11, 2013 at 20:37 Where did you find this problem?Aryabhata –Aryabhata 2013-07-12 00:30:08 +00:00 Commented Jul 12, 2013 at 0:30 I found this in a game on the iPod touch, actually. It is called Doors & Rooms. It's a puzzle game.Numeri –Numeri 2013-07-12 02:46:58 +00:00 Commented Jul 12, 2013 at 2:46 You did not mention the end of the operation. To make the whole board 1?Royi –Royi 2024-11-03 08:18:05 +00:00 Commented Nov 3, 2024 at 8:18 Add a comment\| 2 Answers 2 Sorted by: Reset to default This answer is useful 10 Save this answer. Show activity on this post. First, consider the n×n n×n case. I claim the following: Claim: If n n is even, there is always a solution given any starting configuration. If n n is odd, there is a solution iff the 'on' lights parities for each row and column are the same. i.e. if the lights were 1 1 for on and 0 0 for off, then modulo 2 2, the sum of each individual row, and sum of each individual column must be the same. Proof: Case I:n n is even. You can toggle just a particular light by toggling all the lights in its row and column. So you can switch off all the lights by going after individual lights. Case II:n n is odd. Notice that if n n is odd, on any single operation, all the row and column parities change simultaneously. Thus if they are not all the same, we can never achieve all lights off. This shows the necessity of the parities being the same. To prove sufficiency, consider an arrangement of (2 k+1)×(2 k+1)(2 k+1)×(2 k+1) which has all row and column parities the same. Consider the 2 k×2 k 2 k×2 k subgrid which does not include the bottom row and the right column. Use the above even n n case algorithm to switch off all the lights in that 2 k×2 k 2 k×2 k grid. Since the row and column parities all flip together and were initially all the same, we must have that the remaining lights, in the bottom row and right column, are all the same (including the bottom right corner). Now we toggle the bottom right corner, if needed. ∘∘ Now, the above can be generalized to the m×n m×n case, when m m and n n have the same parity. If m m and n n have different parities, there is more work to be done. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications edited Jul 12, 2013 at 0:48 answered Jul 12, 2013 at 0:29 AryabhataAryabhata 84.1k 11 11 gold badges 192 192 silver badges 281 281 bronze badges 1 Thanks for your help! I was looking for hints, not answers, otherwise I would have marked this as the answer. :D Numeri –Numeri 2013-07-12 17:42:31 +00:00 Commented Jul 12, 2013 at 17:42 Add a comment\| This answer is useful 7 Save this answer. Show activity on this post. Think of each light as a variable taking values in 0,1 0,1. Flipping a switch has the effect of adding 1 1 to each corresponding light, modulo 2 2. To the switch at position i,j i,j make correspond a variable s i,j s i,j representing the number of times you flip that switch. Now the final value of each light is equal to its initial value plus each s i,j s i,j that shares a column or row with it - this is a linear equation in the variables s i,j s i,j. For an m×n m×n grid, we thus obtain m n m n linear equations (one for each light) in m n m n variables (one for each switch). The integers modulo 2 2 are a field, so all of the usual results of linear algebra should apply. Example. Let's works out the 2×2 2×2 case. Which grids are solvable? If we represent "on" as 1 1 and "off" as 0 0, then note that we need to add each light to itself. That is, if a light starts at value 0 0, we need to flip the switches so as to add 0 0 to that light (because we don't want it to change), but if it starts at value 1 1, we need to flip the switches so as to add 1 1 to that light. So the equation for each light is going to look like sum of associated switches = initial state. Writing the initial state of the i,j i,j light as l i,j l i,j, we therefore need to solve: l 1,1=s 1,1+s 1,2+s 2,1 l 1,1=s 1,1+s 1,2+s 2,1 l 1,2=s 1,2+s 1,1+s 2,2 l 1,2=s 1,2+s 1,1+s 2,2 l 2,1=s 2,1+s 1,1+s 2,2 l 2,1=s 2,1+s 1,1+s 2,2 l 2,2=s 2,2+s 1,2+s 2,1 l 2,2=s 2,2+s 1,2+s 2,1 The matrix for this system looks like this (omitting zeros for readability): 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Gaussian Elimination works easily on this matrix (tip: remember that mod 2 2, a+a=0 a+a=0 for any a a) and we obtain an explicit solution for the general 2×2 2×2 grid: s 1,1=l 1,1+l 1,2+l 2,1 s 1,1=l 1,1+l 1,2+l 2,1 s 1,2=l 1,2+l 1,1+l 2,2 s 1,2=l 1,2+l 1,1+l 2,2 s 2,1=l 2,1+l 1,1+l 2,2 s 2,1=l 2,1+l 1,1+l 2,2 s 2,2=l 2,2+l 1,2+l 2,1 s 2,2=l 2,2+l 1,2+l 2,1 So the answer is that all 2×2 2×2 grids are solvable, and there's the solution (concurring with Aryabhata's proposition). As the number of equations is equal to the number of cells on the grid, this method quickly becomes too tedious and complicated to perform by hand. I would try to investigate the determinant of these matrices in the abstract case and see if you come up with anything. That would at least tell you which grid sizes have a guaranteed solution. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications edited Jul 12, 2013 at 21:04 answered Jul 11, 2013 at 20:58 Jack MJack M 28.6k 7 7 gold badges 71 71 silver badges 136 136 bronze badges 1 @Numeri No problem. Epilogue: Although the calculations are indeed too heavy to perform by hand, I did get the above method set up in Sage, and tested it out against a Flash game with complete success.Jack M –Jack M 2013-07-12 20:43:42 +00:00 Commented Jul 12, 2013 at 20:43 Add a comment\| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions combinatorics algorithms puzzle See similar questions with these tags. Featured on Meta Introducing a new proactive anti-spam measure Spevacus has joined us as a Community Manager stackoverflow.ai - rebuilt for attribution Community Asks Sprint Announcement - September 2025 Report this ad Linked 9Minimal number of moves needed to solve a "Lights Out" variant 2Characterize the nullspace of a given matrix in F n×n 2 F 2 n×n 3Looking for solution to "lights out" puzzle variant with multiple states 2Minimize the sum of solution of linear equation 0Strategy to zerofy a matrix Related 9Minimal number of moves needed to solve a "Lights Out" variant 6Number of valid NxN Takuzu Boards a.k.a 0h h1 (details inside)? 7A gameshow logic puzzle 1Is there a mathematical way to check if a KenKen (Mathdoku / Calcdoku) puzzle has at most one solution? 2N lights, N levers puzzle design - finding interesting connections for a given solution 04x4 matrix with colours and symbols - no repeat in row or column 2Are there instances of the "Lights Out" puzzle that can be solved faster with the square-root-of-NOT? 3"Humane" algorithm for 5×5 5×5 'Lights Out' game Hot Network Questions What can be said? 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12768	https://journalfeed.org/article-a-day/2017/why-ng-aspiration-for-upper-gi-bleed-is-unnecessary/	Why NG Aspiration for Upper GI Bleed Is Unnecessary - JournalFeed Skip to content Want to stay smart? Sign up for a FREE 7-day trial today! Learn Read Listen Watch Weekly Quiz Ask Alex Resus Videos Landmark eBook LLSA Pricing EBM Tools Ask Alex PE Workup Calculator PubMetric Speed Read EBM Calculators About Our Approach FAQ Groups CME Login Why NG Aspiration for Upper GI Bleed Is Unnecessary Emergency MedicineGILandmark March 18, 2017 October 15, 2024 On the Shoulders of Giants NG tubes are just mean if they don’t help patients. Don’t do this noxious procedure to patients if it doesn’t help them. This systematic review identified three retrospective studies that were useful to determine whether NG tube aspiration, +/- lavage, helped differentiate an upper or lower GI source of bleeding in patients with melena or hematochezia but without hematemesis. They found the prevalence of an upper GI bleed (UGIB) ranged from 32-74%. “[Sensitivity of NG aspiration] ranged from 42% to 84%, the specificity from 54% to 91%, and negative likelihood ratio from 0.62 to 0.20.” Let’s do an example of a mid-range value from each of these numbers Pre-test probability (or prevalence), 53% Negative LR, 0.41 Post-test probability for UGIB,31.6% (calculation here). In this scenario, there remains a roughly 1 in 3 chance of an UGIB source with negative NG aspiration – and they still need an EGD. Spoon Feed NG tube aspiration or lavage for UGI bleeding is unhelpful as a diagnostic tool. It’s a noxious procedure that is best avoided for this indication. See this for an in-depth review of this article. Abstract Acad Emerg Med. 2010 Feb;17(2):126-32. doi: 10.1111/j.1553-2712.2009.00609.x. Nasogastric aspiration and lavage in emergency department patients with hematochezia or melena without hematemesis. Palamidessi N1, Sinert R, Falzon L, Zehtabchi S. Author information: 1Department of Emergency Medicine, State University of New York, Downstate Medical Center, Brooklyn, New York, USA. npdessi@hotmail.com Abstract OBJECTIVES: The utility of nasogastric aspiration and lavage in the emergency management of patients with melena or hematochezia without hematemesis is controversial. This evidence-based emergency medicine review evaluates the following question: does nasogastric aspiration and lavage in patients with melena or hematochezia and no hematemesis differentiate an upper from lower source of gastrointestinal (GI) bleeding? METHODS: MEDLINE, EMBASE, the Cochrane Library, and other databases were searched. Studies were selected for inclusion in the review if the authors had performed nasogastric aspiration (with or without lavage) in all patients with hematochezia or melena and performed esophagogastroduodenal endoscopy (EGD) in all patients. Studies were excluded if they enrolled patients with history of esophageal varices or included patients with hematemesis or coffee ground emesis (unless the data for patients without hematemesis or coffee ground emesis could be separated out). The outcome was identifying upper GI hemorrhage (active bleeding or high-risk lesions potentially responsible for hemorrhage) and the rate of complications associated with the nasogastric tube insertion. Quality of the included studies was assessed using standard criteria for diagnostic accuracy studies. RESULTS: Three retrospective studies met our inclusion and exclusion criteria. The prevalence of an upper GI source for patients with melena or hematochezia without hematemesis was 32% to 74%. According to the included studies, the diagnostic performance of the nasogastric aspiration and lavage for predicting upper GI bleeding is poor. The sensitivity of this test ranged from 42% to 84%, the specificity from 54% to 91%, and negative likelihood ratios from 0.62 to 0.20. Only one study reported the rate complications associated with nasogastric aspiration and lavage (1.6%). CONCLUSIONS: Nasogastric aspiration, with or without lavage, has a low sensitivity and poor negative likelihood ratio, which limits its utility in ruling out an upper GI source of bleeding in patients with melena or hematochezia without hematemesis. (c) 2010 by the Society for Academic Emergency Medicine. PMID: 20370741 [PubMed – indexed for MEDLINE] Related What are your thoughts?Cancel reply CME Disclosures & Relevant Financial Relationships 2023-2024 Activity2025 EM Activity2025 IM/Primary Care Activity2025 Gen Peds Activity2025 POCUS Activity Core Specialties Emergency Medicine Critical CarePediatric Emergency POCUS All Categories Back to all posts Categories Past Posts Archives Login Helping you stay smart FacebookInstagramLinkedinYoutubeEnvelope ©2025 JournalFeed All Rights Reserved Terms of Use Privacy Terms of Use Privacy Keep up the easy way... Join 21,000. EM, CC, PEM, POCUS. You're current in 5 min/day Anti-bot: 9 + 3 = ? Select your JournalFeed topics: [x] Emergency Medicine - [x] Critical Care - [x] Pediatric Emergency - [x] POCUS [x] I agree to the Terms of Service & Privacy Policy. Subscribe We respect your privacy. Unsubscribe at any time. 💬 I'm Alex, Your AI Research Assistant Ask medical questions • Get evidence-based answers I'm Alex - How can I help? 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12769	https://stats.stackexchange.com/questions/41896/varx-is-known-how-to-calculate-var1-x	Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Var(X) is known, how to calculate Var(1/X)? Ask Question Asked Modified 6 months ago Viewed 44k times 20 $\begingroup$ If I have only $\mathrm{Var}(X)$, how can I calculate $\mathrm{Var}(\frac{1}{X})$? I do not have any information about the distribution of $X$, so I cannot use transformation, or any other methods which use the probability distribution of $X$. distributions variance data-transformation Share Improve this question edited Sep 5, 2013 at 5:12 Glen_b 296k3737 gold badges674674 silver badges1.1k1.1k bronze badges asked Nov 5, 2012 at 9:15 ARATARAT 71011 gold badge99 silver badges2525 bronze badges $\endgroup$ 1 1 $\begingroup$ I think that this might help you. $\endgroup$ Christoph_J – Christoph_J 2012-11-05 09:27:35 +00:00 Commented Nov 5, 2012 at 9:27 Add a comment \| 2 Answers 2 Reset to default 18 $\begingroup$ It is impossible. Consider a sequence $X_n$ of random variables, where $$P(X_n=n-1)=P(X_n=n+1)=0.5$$ Then: $$\newcommand{\Var}{\mathrm{Var}}\Var(X_n)=1 \quad \text{for all $n$}$$ But $\Var\left(\frac{1}{X_n}\right)$ approaches zero as $n$ goes to infinity: $$\Var\left(\frac{1}{X_n}\right)=\left(0.5\left(\frac{1}{n+1}-\frac{1}{n-1}\right)\right)^2$$ This example uses the fact that $\Var(X)$ is invariant under translations of $X$, but $\Var\left(\frac{1}{X}\right)$ is not. But even if we assume $\mathrm{E}(X)=0$, we can't compute $\Var\left(\frac{1}{X}\right)$: Let $$P(X_n=-1)=P(X_n=1)=0.5\left(1-\frac{1}{n}\right)$$ and $$P(X_n=0)=\frac{1}{n} \quad \text{for $n>0$} $$ Then $\Var(X_n)$ approaches 1 as $n$ goes to infinity, but $\Var\left(\frac{1}{X_n}\right)=\infty$ for all $n$. Share Improve this answer edited Aug 31, 2013 at 20:15 Comp_Warrior 2,19322 gold badges2424 silver badges3737 bronze badges answered Nov 5, 2012 at 9:57 mogronmogron 89888 silver badges1313 bronze badges $\endgroup$ 0 Add a comment \| 28 $\begingroup$ You can use Taylor series to get an approximation of the low order moments of a transformed random variable, if the distribution is fairly 'tight' around the mean (in a particular sense). In such cases the approximation can be pretty good. So for example $$g(X) = g(\mu) + (X-\mu) g'(\mu) + \frac{(X-\mu)^2}{2} g''(\mu) + \ldots$$ so \begin{eqnarray} \text{Var}[g(X)] &=& \text{Var}[g(\mu) + (X-\mu) g'(\mu) + \frac{(X-\mu)^2}{2} g''(\mu) + \ldots]\ &=& \text{Var}[(X-\mu) g'(\mu) + \frac{(X-\mu)^2}{2} g''(\mu) + \ldots]\ &=& g'(\mu)^2 \text{Var}[(X-\mu)] + 2g'(\mu)\text{Cov}[(X-\mu),\frac{(X-\mu)^2}{2} g''(\mu) + \ldots] \& &\quad+ \text{Var}[\frac{(X-\mu)^2}{2} g''(\mu) + \ldots]\ \end{eqnarray} often only the first term is taken $$\text{Var}[g(X)] \approx g'(\mu)^2 \text{Var}(X)$$ In this case (assuming I didn't make a mistake), with $g(X)=\frac{1}{X}$, $\text{Var}[\frac{1}{X}] \approx \frac{1}{\mu^4} \text{Var}(X)$. Wikipedia: Taylor expansions for the moments of functions of random variables Some examples to illustrate this. I'll generate two (gamma-distributed) samples in R, one with a 'not-so-tight' distribution about the mean and one a bit tighter. a <- rgamma(1000,10,1) # mean and variance 10; the mean is not many sds from 0 var(a) 10.20819 # reasonably close to the population variance The approximation suggests the variance of $1/a$ should be close to $(1/10)^4 \times 10 = 0.001$ var(1/a) 0.00147171 Algebraic calculation has that the actual population variance is $1/648 \approx 0.00154$ Now for the tighter one: a <- rgamma(1000,100,10) # should have mean 10 and variance 1 var(a) 1.069147 The approximation suggests the variance of $1/a$ should be close to $(1/10)^4 \times 1 = 0.0001$ var(1/a) 0.0001122586 Algebraic calculation shows that the population variance of the reciprocal is $\frac{10^2}{99^2\times 98} \approx 0.000104$. Share Improve this answer edited Mar 20 at 19:42 answered Nov 5, 2012 at 10:47 Glen_bGlen_b 296k3737 gold badges674674 silver badges1.1k1.1k bronze badges $\endgroup$ 4 1 $\begingroup$ Note that in this case, a quite weak hypothesis leads to the conclusion that no mean (whence variance) for $1/X$ will exist, i.e., that the approximation in the answer will be rather misleading. :-) An example hypothesis is that $X$ has a density $f$ that is continuous in an interval around zero and such that $f(0) \neq 0$. The result then follows because the density will be bounded away from zero on some interval $[-\epsilon,\epsilon]$. The hypothesis just given is not the weakest possible, of course. $\endgroup$ cardinal – cardinal 2013-09-15 13:53:12 +00:00 Commented Sep 15, 2013 at 13:53 2 $\begingroup$ The reason the Taylor series argument then fails is because $\approx$ hides the remainder (error) term, which in this case is $$R(x,\mu) = \frac{(x+\mu)(x-\mu)^2}{x\mu} >,$$ and this behaves badly around $x = 0$. $\endgroup$ cardinal – cardinal 2013-09-15 13:57:46 +00:00 Commented Sep 15, 2013 at 13:57 $\begingroup$ One must indeed be careful about the behavior of the density near 0. Note that in the above gamma examples, the distribution of the inverse is inverse gamma, for which having a finite mean requires $\alpha>1$ ($\alpha$ being the shape parameter of the gamma we're inverting). The two examples had $\alpha = 10$ and $\alpha = 100$. Even so (with "nice" distributions for inverting) neglect of higher terms can introduce a noticeable bias. $\endgroup$ Glen_b – Glen_b 2019-11-06 22:41:47 +00:00 Commented Nov 6, 2019 at 22:41 $\begingroup$ this seems in the right direction, of a reciprocal shifted normal distribution instead of a reciprocal standard normal distribution: en.wikipedia.org/wiki/… $\endgroup$ Felipe G. Nievinski – Felipe G. Nievinski 2019-11-25 15:59:22 +00:00 Commented Nov 25, 2019 at 15:59 Add a comment \| Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions distributions variance data-transformation See similar questions with these tags. 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12770	https://www.letuelezioni.it/blog/fisica-moderna-maxwell-einstein-velocita-luce	Maxwell, Einstein e la velocità della luce: un viaggio nella fisica moderna Politica dei cookie Letuelezioni e i nostri partner digitali utilizzano i cookie sul sito. Alcuni sono strettamente necessari per il funzionamento del sito, e altri cookie opzionali sono utilizzati per capire come vengono utilizzati i nostri servizi e per adattare la pubblicità ai tuoi gusti e preferenze. Cliccando su "Accetta", accetti di abilitare questi cookie. Se preferisci impostarli tu stesso o rifiutarli, clicca su 'Pannello cookies'. Per ulteriori informazioni potete visitare il nostro Politica dei cookie e Politica della privacy. Pannello cookies Accettare Accettare la selezione Accettare tutti Cerca Migliaia di annunci Insegnanti privati Lezioni private online Lezioni per aziende Alunni per le tue lezioni Accedi Cambiare vedere profilo AccediSei nuovo?registrarsi I tuoi messaggi Le tue notifiche La tua zona personale Disconnettersi Registrati Accedi Dare lezioni private Blog Fisica Maxwell, Einstein e la velocità della l... Maxwell, Einstein e la velocità della luce: un viaggio nella fisica moderna Riccardo Bertoldo Contattare Mi piace Attorno al 1870, Maxwell, oltre al prevedere l'esistenza delle onde elettromagnetiche (un fenomeno fino ad allora sconosciuto), dimostrò che esse si propagano nel vuoto con una certa velocità v, che dipende dalla costante dielettrica del vuoto e la permeabilità nel vuoto. Nello specifico, mediante una serie di calcoli e sostituzioni, si ottiene che la velocità che tali onde assumono nel vuoto è numericamente uguale a quella della luce c. Nel dettaglio, indicando con Cd la costante dielettrica del vuoto e con Pm la permeabilità magnetica del vuoto, si ottiene la seguente formula: v = c = 1/[radice quadrata di (Cd x Pm)] Tale risultato suggerisce una conclusione di fondamentale importanza:la luce è costituita da onde elettromagnetiche, ossia da onde trasversali formate da campi elettrici e campi magnetici oscillanti perpendicolari tra di loro. Tali campi risultano perpendicolari alla direzione di propagazione dell'onda. Dunque, secondo la teoria di Maxwell, la luce è un onda elettromagnetica che si propaga nel vuoto alla velocità c = 3,00x10^8 m/s. Inoltre,la velocità della luce c è indipendente dalla velocità della sorgente che emette la luce e dal moto relativo tra sorgente e osservatore. Ciò, tuttavia, è inconciliabile con la legge di composizione delle velocità per la quale, quando un corpo è soggetto a due movimenti contemporanei con velocità rispettivamente v1 e v2, la velocità totale sarà v = v1 + v2. Per superare la difficoltà nel render compatibili meccanica e elettromagnetismo, Einstein nel 1905 formulò, come parte della famosa teoria della relatività ristretta (o speciale), il cosiddetto principio di invarianza della velocità della luce,secondo cui la velocità della luce nel vuoto, misurata in qualsiasi sistema inerziale, ha sempre lo stesso valore c, indipendentemente dalla velocità relativa tra la sorgente di luce e l'osservatore. La luce, in altre parole, viaggia alla stessa velocità c per qualsiasi osservatore, in accordo appunto con tale principio di invarianza della velocità della luce. Uno delle più importanti conseguenze della teoria della relatività ristretta è che gli oggetti dotati di massa non possono raggiungere la velocità della luce nel vuoto essendo che è impossibile compiere un lavoro infinito (che, secondo il teorema dell'energia cinetica, sarebbero necessario al fine di fornire al corpo una energia cinetica infinita); c è dunque la più alta velocità possibile! Per quanto concerne il calcolo della velocità della luce, le prime misure sufficientemente accurate furono fatte da Foucault, col metodo dello specchio rotante, perfezionato poi da Michelson. Michelson utilizzò uno specchio rotante con otto lati ed uno specchio fisso: la minima velocità angolare deve esser tale che un lato dello specchio ruoti per un ottavo di giro nel tempo che la luce impiega a percorrere il cammino fra gli specchi. In un suo esperimento del 1926, mise tale specchio fisso a una distanza di 35km da quello rotante, ottenendo il valore c = (2,99796 +- 0,00004) x 10^8 m/s Oggi la velocità della luce è nota con un grado di accuratezza tale che viene utilizzata per definire il metro; essa è definita come: Velocità della luce nel vuoto c=299792458 m/s (anche se il valore approssimato a 3,00x10^8 m/s è più che sufficiente nella maggior parte dei casi) fisica Riportare l' errore Ti è piaciuto? Condividilo Mi piace Riccardo Bertoldo Insegnante di Fisica a Cassano Magnago, Fagnano Olona, Castronno, Busto Arsizio, Gallarate, Samarate, Somma Lombardo, Tradate, Vanzaghello, Buguggiate. Specializzato/a nell'offerta di lezioni di lezioni presenziali e lezioni a domicilio, adattate alle esigenze individuali di ogni studente. 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12771	https://flexbooks.ck12.org/cbook/ck-12-interactive-middle-school-math-7-for-ccss/section/1.6/primary/lesson/using-equations-to-represent-proportional-relationships-math-7-ccss-msm7-ccss/	Skip to content Math Elementary Math Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Interactive Math 6 Math 7 Math 8 Algebra I Geometry Algebra II Conventional Math 6 Math 7 Math 8 Algebra I Geometry Algebra II Probability & Statistics Trigonometry Math Analysis Precalculus Calculus What's the difference? Science Grade K to 5 Earth Science Life Science Physical Science Biology Chemistry Physics Advanced Biology FlexLets Math FlexLets Science FlexLets English Writing Spelling Social Studies Economics Geography Government History World History Philosophy Sociology More Astronomy Engineering Health Photography Technology College College Algebra College Precalculus Linear Algebra College Human Biology The Universe Adult Education Basic Education High School Diploma High School Equivalency Career Technical Ed English as 2nd Language Country Bhutan Brasil Chile Georgia India Translations Spanish Korean Deutsch Chinese Greek Polski EXPLORE Flexi A FREE Digital Tutor for Every Student FlexBooks 2.0 Customizable, digital textbooks in a new, interactive platform FlexBooks Customizable, digital textbooks Schools FlexBooks from schools and districts near you Study Guides Quick review with key information for each concept Adaptive Practice Building knowledge at each student’s skill level Simulations Interactive Physics & Chemistry Simulations PLIX Play. Learn. Interact. eXplore. CCSS Math Concepts and FlexBooks aligned to Common Core NGSS Concepts aligned to Next Generation Science Standards Certified Educator Stand out as an educator. Become CK-12 Certified. Webinars Live and archived sessions to learn about CK-12 Other Resources CK-12 Resources Concept Map Testimonials CK-12 Mission Meet the Team CK-12 Helpdesk FlexLets Know the essentials. Pick a Subject Donate Sign Up 1.6 Using Equations to Represent Proportional Relationships Written by:Larame Spence \| Lori Jordan Fact-checked by:The CK-12 Editorial Team Last Modified: Sep 01, 2025 Lesson Proportional Relationships in Equations You will be introduced to the equation of a line. As shown in lesson Unit Rate and the Constant of Proportionality, lines that pass through the origin have a constant of proportionality, @$\begin{align}k.\end{align}@$ For any proportional relationship, @$\begin{align}k=\tfrac{y}{x}\end{align}@$ (such as a line that passes through the origin). Find the equation of the line by solving for @$\begin{align}y \end{align}@$ in the constant of proportionality equation. @$$\begin{align}k &=\frac{y}{x} \ x\cdot k &=\frac{y}{\cancel x} \cdot \cancel x \ kx&=y\end{align}@$$ This equation @$\begin{align}y=kx\end{align}@$ is another representation of a proportional relationship. Digital Downloads A website offers music downloads for $0.79 per song. Use the slider to see how the cost changes as you increase the number of songs you buy. Use the record button to mark different price points on the table in the interactive. Then use the data you record to answer the questions below. Likes Per Minute On a social media site, the number of likes you receive for a recent post are in the table below. \| \| \| \| \| \| --- --- \| minutes \| 1 \| 5 \| 7 \| 10 \| \| number of likes \| 3 \| 15 \| 21 \| 30 \| Take the values in the table and plot them in the graph. Then, use the graph to determine the constant of proportionality and the equation of the line. Discussion Question A proportional relationship can be represented by a table of values, a graph, and an equation. Discuss how you can find one representation if you are given one of the other two. Try asking Flexi for some examples! How are They the Same? You can represent a proportional relationship in different ways. Below is a Venn diagram with three circles, one for each representation. Drag each of the words, graph, table, and equation to a circle. Then drag the appropriate descriptors to each word to each circle. If something is a property of two or three of the representations, put that descriptor in the one of the overlapping areas. Discussion Questions Why do you think there are different representations of a proportional relationship? How does your Venn Diagram compare to the Venn diagrams of other students? Summary The equation that represents a proportional relationship, or a line, is @$\begin{align}y = kx, \end{align}@$ where @$\begin{align}k \end{align}@$ is the constant of proportionality. Use @$\begin{align}k=\tfrac{y}{x}\end{align}@$ from either a table or a graph to find k and create the equation. Proportional relationships can be represented by tables, graphs and equations. Asked by Students Here are the top questions that students are asking Flexi for this concept: Overview The equation that represents a proportional relationship, or a line, is @$y = kx, @$ where @$k @$ is the constant of proportionality. Use @$k=\tfrac{y}{x}@$ from either a table or a graph to find k and create the equation. Proportional relationships can be represented by tables, graphs and equations. Vocabulary Equation Constant of Proportionality Property Asked by Students Here are the top questions that students are asking Flexi for this concept: Related Content Direct Variation Writing Direct Variation Equations - Overview What Goes Up Must Come Down! Student Sign Up Are you a teacher? Having issues? Click here By signing up, I confirm that I have read and agree to the Terms of use and Privacy Policy Already have an account? Adaptive Practice Save this section to your Library in order to add a Practice or Quiz to it. (Edit Title)55/ 100 This lesson has been added to your library. \|Searching in: \| \| \| Looks like this FlexBook 2.0 has changed since you visited it last time. We found the following sections in the book that match the one you are looking for: Go to the Table of Contents No Results Found Your search did not match anything in .
12772	https://www.quora.com/How-do-you-convert-135-degrees-to-radians-1	How to convert 135 degrees to radians - Quora Something went wrong. Wait a moment and try again. Try again Skip to content Skip to search Sign In Mathematics Angle Value Conversion Method Radians Degrees (mathematics) Angles Angle Measurement Measurement Conversion Angle In Radians 5 How do you convert 135 degrees to radians? All related (33) Sort Recommended Manendra Singh It is good and give the information about all the things ·2y To convert degrees to radians, you can use the formula: radians = degrees × (π / 180) For example, to convert 135 degrees to radians: radians = 135 × (π / 180) ≈ 2.35619 radians So, 135 degrees is approximately equal to 2.35619 radians. Upvote · Sponsored by RedHat Know what your AI knows, with open source models. Your AI should keep records, not secrets. Learn More 99 36 Related questions More answers below What is 90 degrees in radians? How do I convert 1/4 radians to degree? How do I convert (-27°30'31") into radians? How do I convert radians to degrees in terms of pi? What are 60 degrees equal in radians? Sapna Kumari M.Sc from Jamshedpur Co-operative College (Graduated 2003) · Author has 381 answers and 90.9K answer views ·2y Since, 180 degrees= π radian So, 1 degree = π / 180 radian So, 135 degrees =( π×135 )/180radians = ( π×3)/4 radians = (3π/4) radians Upvote · Srinivasan Narasimhan Retired from Govt.,service after serving for 34 years in varc · Author has 4K answers and 2.3M answer views ·2y 180 degrees = Pi radians. Therefore 135 degrees = (135 / 180)× Pi. radians. = 3/4 Pi radians. Upvote · Philip Lloyd Former Specialist Calculus Teacher and Mentor.. · Author has 6.8K answers and 52.8M answer views ·1y Related How do you convert radians to degrees? To change from radians to degrees, I like to use the simple idea of simple proportion. If 3 cakes cost $15 then 1 cake costs $5 Continue Reading To change from radians to degrees, I like to use the simple idea of simple proportion. If 3 cakes cost $15 then 1 cake costs $5 Upvote · 9 2 Related questions What is 90 degrees in radians? How do I convert 1/4 radians to degree? How do I convert (-27°30'31") into radians? How do I convert radians to degrees in terms of pi? What are 60 degrees equal in radians? How many radians is -135Â°? How do you convert degrees to radians on a calculator? How do you convert an angle from radians to degrees or from degrees to radians? 1 radian is equal to how many degrees? How do I convert 14 degrees 24 minutes into radian? How do I convert 3.4 radians into degrees (round to the nearest 10 minutes)? How is the formula for converting radians to degrees determined? How many radians are in one degree? How do I find radians from degrees? How do you convert radians to degrees, and vice versa? Related questions What is 90 degrees in radians? How do I convert 1/4 radians to degree? How do I convert (-27°30'31") into radians? How do I convert radians to degrees in terms of pi? What are 60 degrees equal in radians? How many radians is -135Â°? Advertisement About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
12773	https://www.cuemath.com/ncert-solutions/prove-that-the-logarithmic-function-is-strictly-increasing-on-0-/	Prove that the logarithmic function is strictly increasing on (0, ∞) Solution: Increasing functions are those functions that increase monotonically within a particular domain, and decreasing functions are those which decrease monotonically within a particular domain. Logs (or) logarithms are nothing but another way of expressing exponents. Understanding logs is not so difficult. To understand logs, it is sufficient to know that a logarithmic equation is just another way of writing an exponential equation The given function is f (x) = log x Therefore, f'(x) = 1/x For, x > 0, f' (x) = 1/x > 0 Thus, the logarithmic function is strictly increasing in the interval (0, ∞) NCERT Solutions Class 12 Maths - Chapter 6 Exercise 6.2 Question 10 Prove that the logarithmic function is strictly increasing on (0, ∞). Summary: Hence we have concluded that the logarithmic function is strictly increasing on (0, ∞)
12774	https://www.quora.com/What-is-the-limit-n-n-when-n-infinity	What is the limit (-n) ^n when n ->infinity? - Quora Something went wrong. Wait a moment and try again. Try again Skip to content Skip to search Sign In Mathematics Towards Infinity Limits of Functions Calculus (Mathematics) Mathematical Concepts Mathematical Functions Limits (mathematics) Derivatives and Limits Limits at Infinity 5 What is the limit (-n) ^n when n ->infinity? All related (38) Sort Recommended Konstantinos Michailidis BSc Mathematics ; MSc · Author has 1K answers and 2.4M answer views ·6y Originally Answered: What is the limit (-1) ^n when n ->infinity? · It does not exist If a sequence converges all its subsequences converge to the same limit. But if you take the subsequences below and their respective limits [math]lim(-1)^{2n}\rightarrow 1[/math] and [math]lim(-1)^{2n+1}\rightarrow -1[/math] You see the contradiction hence the limit does not exist Upvote · 9 2 Related questions More answers below How do I calculate this limit: limit n^(n!) /(n!) ^n , n -> +infinity? What is the limit (-1) ^n when n ->infinity? What is the limit [n/(1-n)] ^n when n ->infinity? What is the limit of n tends to infinity (1/2) ^n? What is the limit of [math]\frac{(n!)^{1/n}}{n}[/math] as n tends to infinity? Frank Levinson Veterinarian (1985–present) ·6y Originally Answered: What is the limit (-1/n!) ^n when n ->infinity? · The values oscillate from negative to positive but approach 0 from both sides. For n = 10 the value is less than 10^65 and for n = 11 the negative value is within 10^83 of 0. It converges quickly. Upvote · 9 3 Eric Platt Ph.D in Mathematics · Author has 2.7K answers and 13.3M answer views ·6y [math]\lim_{n \to \infty} (-n)^n[/math] does not exist. There are many reasons why. The sequences magnitude is growing without bound. For any value we choose eventually it will be larger than it in magnitude. It is an oscillating sequence that isn’t approaching zero. Oscillatory sequences that do not approach zero have no limit. Upvote · 9 5 Lukas Schmidinger I have graduate CS and my studies included math courses. · Author has 27.7K answers and 14.9M answer views ·6y Originally Answered: What is the limit (-1/n!) ^n when n ->infinity? · Zero. [math]\left (\frac{-1}{n!} \right )^n=\frac{(-1)^n}{n!^n}[/math] And eventhough math^n[/math] doesn’t converge it is surely [math]-1[/math] or [math]1[/math] in either case we have [math]\frac{1}{n'!^{n'}}[/math] or [math]\frac{-1}{n''!^{n''}}[/math] which both converge to zero. Upvote · 9 2 Related questions How do I calculate this limit: limit n^(n!) /(n!) ^n , n -> +infinity? What is the limit (-1) ^n when n ->infinity? What is the limit [n/(1-n)] ^n when n ->infinity? What is the limit of n tends to infinity (1/2) ^n? What is the limit of [math]\frac{(n!)^{1/n}}{n}[/math] as n tends to infinity? What is the limit n tends to infinity (n! /n)? What will be the value of limit n--> infinity (n^2 -1) / (n-1)? What is the limit of (a+b) ^n as n approaches infinity? What is the limit of (1+1/n)^n as n approaches negative infinity? What is the limit of the expression (1+2n) ^(-n) as n approaches infinity? How can we prove that [math]\frac{\zeta(k)}{\zeta(k+1)}=\sum\limits_{n=1}^{\infty}\frac{\|\mu(n)\|\cdot\varphi(n)}{n\cdot J_{k}(n)}[/math]? What is the limit of [math]\sqrt{n-1}-\sqrt{n}[/math] as [math]n[/math] approaches infinity? What is the limit as n approaches infinity in the formula [math]\dfrac{2^{n}}{n!}[/math]? What is the limit of (n!) ^2/((2n)!)? What is the limit of (Fibnonacci(n) ^(1/n)) when n-> infinity? Related questions How do I calculate this limit: limit n^(n!) /(n!) ^n , n -> +infinity? What is the limit (-1) ^n when n ->infinity? What is the limit [n/(1-n)] ^n when n ->infinity? What is the limit of n tends to infinity (1/2) ^n? What is the limit of [math]\frac{(n!)^{1/n}}{n}[/math] as n tends to infinity? What is the limit n tends to infinity (n! /n)? Advertisement About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
12775	https://stackoverflow.com/questions/66906806/find-the-median-in-my-array-while-also-sorting-the-numbers-from-smallest-to-larg	java - Find the median in my array while also sorting the numbers from smallest to largest - Stack Overflow Join Stack Overflow By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google Sign up with GitHub OR Email Password Sign up Already have an account? Log in Skip to main content Stack Overflow 1. About 2. Products 3. 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Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Find the median in my array while also sorting the numbers from smallest to largest [duplicate] Ask Question Asked 4 years, 6 months ago Modified4 years, 6 months ago Viewed 2k times This question shows research effort; it is useful and clear 2 Save this question. Show activity on this post. This question already has answers here: How to calculate the median of an array? (16 answers) Closed 4 years ago. This is what I've done so far; I was able to calculate the average, but I'm not sure how to find the median. I also know that I need to sort the array to make it easier. java public class SAA { public static void main(String[] args) { int[] num = {60, 70, 82, 1216, 57, 82, 34, 560, 91, 86}; int total = 0; for (int i = 0; i < num.length; i++) { if ((num[i] > 0) && (num[i] < 100)) { total += num[i]; } } System.out.println(total / 10); } } This is my attempt at using bubble sort: java public class Bubblesort { public static void main(String[] args) { int[] num = {60, 70, 82, 1216, 57, 82, 34, 560, 91, 86}; int temp = 0; int[] add = new int[num.length + 1]; for (int i = 0; i < num.length; i++) { for (int j = i + 1; j < num.length; j++) { if (num[i] > num[j]) { temp = num[i]; num[i] = num[j]; num[j] = temp; } } } for (int i = 0; i < num.length; i++) { System.out.print(num[i] + " "); } } } java bubble-sort median Share Share a link to this question Copy linkCC BY-SA 4.0 Improve this question Follow Follow this question to receive notifications edited Apr 2, 2021 at 14:10 Mark Rotteveel 110k 239 239 gold badges 158 158 silver badges 229 229 bronze badges asked Apr 1, 2021 at 14:53 LucyLucy 29 8 8 bronze badges 8 I'm a beginner so I just need to know how to do it using bubble sort and a basic formula. I can't use the median function in Java. I've formulated a plan but I'm not fully sure how to do it; Sort the numbers from smallest to largest, write the total number of indices and make a for loop that goes through the code and looks for the index thats in the center Lucy –Lucy 2021-04-01 15:00:32 +00:00 Commented Apr 1, 2021 at 15:00 Did you not search the Internet for java bubble sort ?Abra –Abra 2021-04-01 15:02:29 +00:00 Commented Apr 1, 2021 at 15:02 Why would you need a loop for the index at the center? You know the array's length so you can calculate that (and if there is no exact center you'd need to either choose or calculate the average of the 2 center indices).Thomas –Thomas 2021-04-01 15:03:13 +00:00 Commented Apr 1, 2021 at 15:03 "I just need to know how to do it using bubble sort" - I'd say if you "just" need to know (i.e. use) something then Arrays.sort() would be the better option. I assume you need to use bubble sort, do you?Thomas –Thomas 2021-04-01 15:04:43 +00:00 Commented Apr 1, 2021 at 15:04 1 This is clearly a homework assignment and the OP hasn't even attempted to write the bubble sort yet. Start by writing your sort. Once the array is sorted, it's easy to find the middle value.Phaelax –Phaelax 2021-04-01 15:07:08 +00:00 Commented Apr 1, 2021 at 15:07 \|Show 3 more comments 2 Answers 2 Sorted by: Reset to default This answer is useful 2 Save this answer. Show activity on this post. Median is the middle element of the sorted array if the array size is odd, else it is the average of the middle two elements. The following code snippet finds the median of the array. ```java public static void main(String[] args) { int[]num = {60, 70, 82, 1216, 57, 82, 34, 560, 91, 86}; int median; int len = num.length; Arrays.sort(num); // sorts the array //check if the length is odd if(len%2 != 0) median = num[len/2]; else // length is even median = (num[(len - 1) / 2] + num[len / 2])/2; System.out.println(median); } ``` Share Share a link to this answer Copy linkCC BY-SA 4.0 Improve this answer Follow Follow this answer to receive notifications answered Apr 1, 2021 at 15:20 Sr_33Sr_33 56 1 1 silver badge 7 7 bronze badges Comments Add a comment This answer is useful 2 Save this answer. Show activity on this post. ```java int min = 0; int max = 0; for (int i = 0; i < numbers.length; i++) { for (int j = 0; j < numbers.length-1; j++) { min = Math.min(numbers[j], numbers[j+1]); //using min and max methods max = Math.max(numbers[j], numbers[j+1]); numbers[j] = min; //placing the smaller value on the left numbers[j+1] = max; //placing the larger value on the right } } ``` This is the program to bubble sort an array in a short and simpler way, you should use this along with @Sr_33's answer as well! Share Share a link to this answer Copy linkCC BY-SA 4.0 Improve this answer Follow Follow this answer to receive notifications answered Apr 1, 2021 at 15:34 TheSjTheSj 386 1 1 silver badge 12 12 bronze badges Comments Add a comment Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions java bubble-sort median See similar questions with these tags. 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12776	https://webbook.nist.gov/cgi/inchi?ID=C14581921&Units=CAL	Magnesium ion (1+) Jump to content National Institute of Standards and Technology NIST Chemistry WebBook, SRD 69 Home Search Name Formula IUPAC identifier CAS number More options NIST Data SRD Program Science Data Portal Office of Data and Informatics About FAQ Credits More documentation Magnesium ion (1+) Formula: Mg+ Molecular weight: 24.3045 IUPAC Standard InChI:InChI=1S/Mg/q+1 Copy IUPAC Standard InChIKey:XYAGEBWLYIDCRX-UHFFFAOYSA-N Copy CAS Registry Number: 14581-92-1 Chemical structure: This structure is also available as a 2d Mol file Other names: Magnesium cation Permanent link for this species. Use this link for bookmarking this species for future reference. Information on this page: Notes Other data available: Gas phase thermochemistry data Reaction thermochemistry data Ion clustering data Options: Switch to SI units Notes Go To:Top Data from NIST Standard Reference Database 69: NIST Chemistry WebBook The National Institute of Standards and Technology (NIST) uses its best efforts to deliver a high quality copy of the Database and to verify that the data contained therein have been selected on the basis of sound scientific judgment. However, NIST makes no warranties to that effect, and NIST shall not be liable for any damage that may result from errors or omissions in the Database. Customer support for NIST Standard Reference Data products. © 2025 by the U.S. Secretary of Commerce on behalf of the United States of America. All rights reserved. Copyright for NIST Standard Reference Data is governed by the Standard Reference Data Act. Privacy Statement Privacy Policy Security Notice Disclaimer (Note: This site is covered by copyright.) Accessibility Statement FOIA Contact Us
12777	https://www.schwabmoneywise.com/teaching-kids/money-math-for-middle-school-students	Skip to content Money Math for Middle School Students Curriculum for middle school math classes (grades 7-9). Money Math: Lessons for Life is a four-lesson curriculum supplement for middle school math classes, teaching grade 7-9 math concepts using real-life examples from personal finance. Free to teachers, the curriculum was developed by the Center for Entrepreneurship and Economic Education at the University of Missouri-St. Louis in accordance with national school mathematics standards. It consists of an 86-page teacher's guide with lesson plans, reproducible activity pages, and teaching tips. You only need one copy to teach several classes of students. And you don't need to be an expert in personal finance—questions and answers are clearly provided in the book The curriculum is distributed by the U.S. Treasury Department and endorsed by the President's Advisory Council on Financial Capability. » Go to the Money Math: Lessons for Life website1 to get the complete curriculum. What's inside? You can also get each of the four lessons individually. Here are the descriptions to help you select the right one: Introduction and Correlations See how the curriculum measures up to the National K-12 Personal Finance Standards and NCTM Principles and Standards of Mathematics. The Secret to Becoming a Millionaire—Lesson 1 Students learn how saving helps people become wealthy. They develop "rules to become a millionaire" as they work through a series of exercises, learning that it is important to: (1) save early and often, (2) save as much as possible, (3) earn compound interest, (4) try to earn a high interest rate, (5) leave deposits and interest earned in the account as long as possible, and (6) choose accounts for which interest is compounded often. This lesson assumes that students have worked with percents and decimal equivalents. Wallpaper Woes—Lesson 2 Students hear a story about Tom, a middle-school student who wants to redecorate his bedroom. They measure the classroom wall dimensions, draw a scale model, and incorporate measurements for windows and doors to determine the area that could be covered by wallpaper. Students then hear more about Tom’s redecorating adventure, learning about expenses, budget constraints, and trade-offs. For assessment, students measure their rooms at home. This lesson requires that students know how to measure, or a review may be necessary before teaching. Math and Taxes: A Pair to Count On—Lesson 3 Students examine careers and reflect on how workers use math in their occupations. They study selected occupations, learning about the work skills (human capital) that different workers possess and salaries that those workers earn. Next, students learn about how taxes are paid on income that people earn and how income tax is calculated. They learn how the progressive federal income tax is based on the ability-to-pay principle. Spreading the Budget—Lesson 4 Students develop a budget for a college student, using a spreadsheet. They examine the student’s fixed, variable, and periodic expenses and revise to adjust for cash flow problems that appear on the first spreadsheet. This lesson is designed to increase student awareness and appreciation of the efficiency of using computer technology in math applications. Schwab MoneyWise® does not endorse any third-party website, its sponsors or any of the products or services offered on the website. (0709-9815) Investment and Insurance Products Are: Not FDIC Insured • Not Insured by Any Federal Government Agency • Not a Deposit or Other Obligation of, or Guaranteed by, the Bank or any of its Affiliates • Subject to Investment Risks, Including Possible Loss of Principal Amount Invested
12778	https://en.wikipedia.org/wiki/Signed_number_representations	Jump to content Search Contents (Top) 1 History 2 Sign–magnitude 3 Ones' complement 4 Two's complement 5 Offset binary 6 Base −2 7 Comparison table 8 Other systems 9 See also 10 References Signed number representations العربية Català Čeština Español Esperanto Français 한국어 Bahasa Indonesia Italiano 日本語 Português Simple English Slovenščina ไทย Українська Tiếng Việt 粵語中文 Edit links Article Talk Read Edit View history Tools Actions Read Edit View history General What links here Related changes Upload file Permanent link Page information Cite this page Get shortened URL Download QR code Print/export Download as PDF Printable version In other projects Wikidata item Appearance From Wikipedia, the free encyclopedia Encoding of negative numbers in binary number systems \| \| \| This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: "Signed number representations" – news · newspapers · books · scholar · JSTOR (April 2013) (Learn how and when to remove this message) \| Not to be confused with Signed-digit representation. In computing, signed number representations are required to encode negative numbers in binary number systems. In mathematics, negative numbers in any base are represented by prefixing them with a minus sign ("−"). However, in RAM or CPU registers, numbers are represented only as sequences of bits, without extra symbols. The four best-known methods of extending the binary numeral system to represent signed numbers are: sign–magnitude, ones' complement, whether positive, negative, fractional, or other elaborations on such themes. There is no definitive criterion by which any of the representations is universally superior. For integers, the representation used in most current computing devices is two's complement, although the Unisys ClearPath Dorado series mainframes use ones' complement. History [edit] The early days of digital computing were marked by competing ideas about both hardware technology and mathematics technology (numbering systems). One of the great debates was the format of negative numbers, with some of the era's top experts expressing very strong and differing opinions.[citation needed] One camp supported two's complement, the system that is dominant today. Another camp supported ones' complement, where a negative value is formed by inverting all of the bits in its positive equivalent. A third group supported sign–magnitude, where a value is changed from positive to negative simply by toggling the word's highest-order bit. There were arguments for and against each of the systems. Sign–magnitude allowed for easier tracing of memory dumps (a common process in the 1960s) as small numeric values use fewer 1 bits. These systems did ones' complement math internally, so numbers would have to be converted to ones' complement values when they were transmitted from a register to the math unit and then converted back to sign–magnitude when the result was transmitted back to the register. The electronics required more gates than the other systems – a key concern when the cost and packaging of discrete transistors were critical. IBM was one of the early supporters of sign–magnitude, with their 704, 709 and 709x series computers being perhaps the best-known systems to use it. Ones' complement allowed for somewhat simpler hardware designs, as there was no need to convert values when passed to and from the math unit. But it also shared an undesirable characteristic with sign–magnitude: the ability to represent negative zero (−0). Negative zero behaves exactly like positive zero: when used as an operand in any calculation, the result will be the same whether an operand is positive or negative zero. The disadvantage is that the existence of two forms of the same value necessitates two comparisons when checking for equality with zero. Ones' complement subtraction can also result in an end-around borrow (described below). It can be argued that this makes the addition and subtraction logic more complicated or that it makes it simpler, as a subtraction requires simply inverting the bits of the second operand as it is passed to the adder. The PDP-1, CDC 160 series, CDC 3000 series, CDC 6000 series, UNIVAC 1100 series, and LINC computer use ones' complement representation. Two's complement is the easiest to implement in hardware, which may be the ultimate reason for its widespread popularity. Processors on the early mainframes often consisted of thousands of transistors, so eliminating a significant number of transistors was a significant cost savings. Mainframes such as the IBM System/360, the GE-600 series, and the PDP-6 and PDP-10 use two's complement, as did minicomputers such as the PDP-5 and PDP-8 and the PDP-11 and VAX machines. The architects of the early integrated-circuit-based CPUs (Intel 8080, etc.) also chose to use two's complement math. As IC technology advanced, two's complement technology was adopted in virtually all processors, including x86, m68k, Power ISA, MIPS, SPARC, ARM, Itanium, PA-RISC, and DEC Alpha. Sign–magnitude [edit] Eight-bit sign–magnitude \| Binary value \| Sign–magnitude interpretation \| Unsigned interpretation \| \| 00000000 \| 0 \| 0 \| \| 00000001 \| 1 \| 1 \| \| ⋮ \| ⋮ \| ⋮ \| \| 01111101 \| 125 \| 125 \| \| 01111110 \| 126 \| 126 \| \| 01111111 \| 127 \| 127 \| \| 10000000 \| −0 \| 128 \| \| 10000001 \| −1 \| 129 \| \| 10000010 \| −2 \| 130 \| \| ⋮ \| ⋮ \| ⋮ \| \| 11111101 \| −125 \| 253 \| \| 11111110 \| −126 \| 254 \| \| 11111111 \| −127 \| 255 \| In the sign–magnitude representation, also called sign-and-magnitude or signed magnitude, a signed number is represented by the bit pattern corresponding to the sign of the number for the sign bit (often the most significant bit, set to 0 for a positive number and to 1 for a negative number), and the magnitude of the number (or absolute value) for the remaining bits. For example, in an eight-bit byte, only seven bits represent the magnitude, which can range from 0000000 (0) to 1111111 (127). Thus numbers ranging from −12710 to +12710 can be represented once the sign bit (the eighth bit) is added. For example, −4310 encoded in an eight-bit byte is 10101011 while 4310 is 00101011. Using sign–magnitude representation has multiple consequences which makes them more intricate to implement: There are two ways to represent zero, 00000000 (0) and 10000000 (−0). Addition and subtraction require different behavior depending on the sign bit, whereas ones' complement can ignore the sign bit and just do an end-around carry, and two's complement can ignore the sign bit and depend on the overflow behavior. Comparison also requires inspecting the sign bit, whereas in two's complement, one can simply subtract the two numbers, and check if the outcome is positive or negative. The minimum negative number is −127, instead of −128 as in the case of two's complement. This approach is directly comparable to the common way of showing a sign (placing a "+" or "−" next to the number's magnitude). Some early binary computers (e.g., IBM 7090) use this representation, perhaps because of its natural relation to common usage. Sign–magnitude is the most common way of representing the significand in floating-point values. Ones' complement [edit] Main article: Ones' complement Eight-bit ones' complement \| Binary value \| Ones' complement interpretation \| Unsigned interpretation \| \| 00000000 \| 0 \| 0 \| \| 00000001 \| 1 \| 1 \| \| ⋮ \| ⋮ \| ⋮ \| \| 01111101 \| 125 \| 125 \| \| 01111110 \| 126 \| 126 \| \| 01111111 \| 127 \| 127 \| \| 10000000 \| −127 \| 128 \| \| 10000001 \| −126 \| 129 \| \| 10000010 \| −125 \| 130 \| \| ⋮ \| ⋮ \| ⋮ \| \| 11111101 \| −2 \| 253 \| \| 11111110 \| −1 \| 254 \| \| 11111111 \| −0 \| 255 \| In the ones' complement representation, a negative number is represented by the bit pattern corresponding to the bitwise NOT (i.e. the "complement") of the positive number. Like sign–magnitude representation, ones' complement has two representations of 0: 00000000 (+0) and 11111111 (−0). As an example, the ones' complement form of 00101011 (4310) becomes 11010100 (−4310). The range of signed numbers using ones' complement is represented by −(2N−1 − 1) to (2N−1 − 1) and ±0. A conventional eight-bit byte is −12710 to +12710 with zero being either 00000000 (+0) or 11111111 (−0). To add two numbers represented in this system, one does a conventional binary addition, but it is then necessary to do an end-around carry: that is, add any resulting carry back into the resulting sum. To see why this is necessary, consider the following example showing the case of the addition of −1 (11111110) to +2 (00000010): binary decimal 11111110 −1 + 00000010 +2 ─────────── ── 1 00000000 0 ← Incorrect answer 1 +1 ← Add carry ─────────── ── 00000001 1 ← Correct answer In the previous example, the first binary addition gives 00000000, which is incorrect. The correct result (00000001) only appears when the carry is added back in. A remark on terminology: The system is referred to as "ones' complement" because the negation of a positive value x (represented as the bitwise NOT of x) can also be formed by subtracting x from the ones' complement representation of zero that is a long sequence of ones (−0). Two's complement arithmetic, on the other hand, forms the negation of x by subtracting x from a single large power of two that is congruent to +0. Therefore, ones' complement and two's complement representations of the same negative value will differ by one. Note that the ones' complement representation of a negative number can be obtained from the sign–magnitude representation merely by bitwise complementing the magnitude (inverting all the bits after the first). For example, the decimal number −125 with its sign–magnitude representation 11111101 can be represented in ones' complement form as 10000010. Two's complement [edit] Main article: Two's complement Eight-bit two's complement \| Binary value \| Two's complement interpretation \| Unsigned interpretation \| \| 00000000 \| 0 \| 0 \| \| 00000001 \| 1 \| 1 \| \| ⋮ \| ⋮ \| ⋮ \| \| 01111110 \| 126 \| 126 \| \| 01111111 \| 127 \| 127 \| \| 10000000 \| −128 \| 128 \| \| 10000001 \| −127 \| 129 \| \| 10000010 \| −126 \| 130 \| \| ⋮ \| ⋮ \| ⋮ \| \| 11111110 \| −2 \| 254 \| \| 11111111 \| −1 \| 255 \| In the two's complement representation, a negative number is represented by the bit pattern corresponding to the bitwise NOT (i.e. the "complement") of the positive number plus one, i.e. to the ones' complement plus one. It circumvents the problems of multiple representations of 0 and the need for the end-around carry of the ones' complement representation. This can also be thought of as the most significant bit representing the inverse of its value in an unsigned integer; in an 8-bit unsigned byte, the most significant bit represents the 128ths place, where in two's complement that bit would represent −128. In two's-complement, there is only one zero, represented as 00000000. Negating a number (whether negative or positive) is done by inverting all the bits and then adding one to that result. This actually reflects the ring structure on all integers modulo 2N: . Addition of a pair of two's-complement integers is the same as addition of a pair of unsigned numbers (except for detection of overflow, if that is done); the same is true for subtraction and even for N lowest significant bits of a product (value of multiplication). For instance, a two's-complement addition of 127 and −128 gives the same binary bit pattern as an unsigned addition of 127 and 128, as can be seen from the 8-bit two's complement table. An easier method to get the negation of a number in two's complement is as follows: \| \| Example 1 \| Example 2 \| --- \| 1. Starting from the right, find the first "1" \| 00101001 \| 00101100 \| \| 2. Invert all of the bits to the left of that "1" \| 11010111 \| 11010100 \| Method two: Invert all the bits through the number. This computes the same result as subtracting from negative one. Add one Example: for +2, which is 00000010 in binary (the ~ character is the C bitwise NOT operator, so ~X means "invert all the bits in X"): ~00000010 → 11111101 11111101 + 1 → 11111110 (−2 in two's complement) Offset binary [edit] Main article: Offset binary Eight-bit excess-128 \| Binary value \| Excess-128 interpretation \| Unsigned interpretation \| \| 00000000 \| −128 \| 0 \| \| 00000001 \| −127 \| 1 \| \| ⋮ \| ⋮ \| ⋮ \| \| 01111111 \| −1 \| 127 \| \| 10000000 \| 0 \| 128 \| \| 10000001 \| 1 \| 129 \| \| ⋮ \| ⋮ \| ⋮ \| \| 11111111 \| 127 \| 255 \| In the offset binary representation, also called excess-K or biased, a signed number is represented by the bit pattern corresponding to the unsigned number plus K, with K being the biasing value or offset. Thus 0 is represented by K, and −K is represented by an all-zero bit pattern. This can be seen as a slight modification and generalization of the aforementioned two's-complement, which is virtually the excess-(2N−1) representation with negated most significant bit. Biased representations are now primarily used for the exponent of floating-point numbers. The IEEE 754 floating-point standard defines the exponent field of a single-precision (32-bit) number as an 8-bit excess-127 field. The double-precision (64-bit) exponent field is an 11-bit excess-1023 field; see exponent bias. It also had use for binary-coded decimal numbers as excess-3. Base −2 [edit] See also: Negative base Eight-bit base −2 \| Binary value \| Base −2 interpretation \| Unsigned interpretation \| \| 00000000 \| 0 \| 0 \| \| 00000001 \| 1 \| 1 \| \| ⋮ \| ⋮ \| ⋮ \| \| 01111111 \| 43 \| 127 \| \| 10000000 \| −128 \| 128 \| \| 10000001 \| −127 \| 129 \| \| ⋮ \| ⋮ \| ⋮ \| \| 11111111 \| −85 \| 255 \| In the base −2 representation, a signed number is represented using a number system with base −2. In conventional binary number systems, the base, or radix, is 2; thus the rightmost bit represents 20, the next bit represents 21, the next bit 22, and so on. However, a binary number system with base −2 is also possible. The rightmost bit represents (−2)0 = +1, the next bit represents (−2)1 = −2, the next bit (−2)2 = +4 and so on, with alternating sign. The numbers that can be represented with four bits are shown in the comparison table below. The range of numbers that can be represented is asymmetric. If the word has an even number of bits, the magnitude of the largest negative number that can be represented is twice as large as the largest positive number that can be represented, and vice versa if the word has an odd number of bits. Comparison table [edit] The following table shows the positive and negative integers that can be represented using four bits. Four-bit integer representations \| Decimal \| Unsigned \| Sign–magnitude \| Ones' complement \| Two's complement \| Excess-8 (biased) \| Base −2 \| \| 16 \| — \| — \| — \| — \| — \| — \| \| 15 \| 1111 \| — \| — \| — \| — \| — \| \| 14 \| 1110 \| — \| — \| — \| — \| — \| \| 13 \| 1101 \| — \| — \| — \| — \| — \| \| 12 \| 1100 \| — \| — \| — \| — \| — \| \| 11 \| 1011 \| — \| — \| — \| — \| — \| \| 10 \| 1010 \| — \| — \| — \| — \| — \| \| 9 \| 1001 \| — \| — \| — \| — \| — \| \| 8 \| 1000 \| — \| — \| — \| — \| — \| \| 7 \| 0111 \| 0111 \| 0111 \| 0111 \| 1111 \| — \| \| 6 \| 0110 \| 0110 \| 0110 \| 0110 \| 1110 \| — \| \| 5 \| 0101 \| 0101 \| 0101 \| 0101 \| 1101 \| 0101 \| \| 4 \| 0100 \| 0100 \| 0100 \| 0100 \| 1100 \| 0100 \| \| 3 \| 0011 \| 0011 \| 0011 \| 0011 \| 1011 \| 0111 \| \| 2 \| 0010 \| 0010 \| 0010 \| 0010 \| 1010 \| 0110 \| \| 1 \| 0001 \| 0001 \| 0001 \| 0001 \| 1001 \| 0001 \| \| 0 \| 0000 \| 0000 \| 0000 \| 0000 \| 1000 \| 0000 \| \| −0 \| 1000 \| 1111 \| \| −1 \| — \| 1001 \| 1110 \| 1111 \| 0111 \| 0011 \| \| −2 \| — \| 1010 \| 1101 \| 1110 \| 0110 \| 0010 \| \| −3 \| — \| 1011 \| 1100 \| 1101 \| 0101 \| 1101 \| \| −4 \| — \| 1100 \| 1011 \| 1100 \| 0100 \| 1100 \| \| −5 \| — \| 1101 \| 1010 \| 1011 \| 0011 \| 1111 \| \| −6 \| — \| 1110 \| 1001 \| 1010 \| 0010 \| 1110 \| \| −7 \| — \| 1111 \| 1000 \| 1001 \| 0001 \| 1001 \| \| −8 \| — \| — \| — \| 1000 \| 0000 \| 1000 \| \| −9 \| — \| — \| — \| — \| — \| 1011 \| \| −10 \| — \| — \| — \| — \| — \| 1010 \| \| −11 \| — \| — \| — \| — \| — \| — \| Same table, as viewed from "given these binary bits, what is the number as interpreted by the representation system": \| Binary \| Unsigned \| Sign–magnitude \| Ones' complement \| Two's complement \| Excess-8 \| Base −2 \| --- --- --- \| 0000 \| 0 \| 0 \| 0 \| 0 \| −8 \| 0 \| \| 0001 \| 1 \| 1 \| 1 \| 1 \| −7 \| 1 \| \| 0010 \| 2 \| 2 \| 2 \| 2 \| −6 \| −2 \| \| 0011 \| 3 \| 3 \| 3 \| 3 \| −5 \| −1 \| \| 0100 \| 4 \| 4 \| 4 \| 4 \| −4 \| 4 \| \| 0101 \| 5 \| 5 \| 5 \| 5 \| −3 \| 5 \| \| 0110 \| 6 \| 6 \| 6 \| 6 \| −2 \| 2 \| \| 0111 \| 7 \| 7 \| 7 \| 7 \| −1 \| 3 \| \| 1000 \| 8 \| −0 \| −7 \| −8 \| 0 \| −8 \| \| 1001 \| 9 \| −1 \| −6 \| −7 \| 1 \| −7 \| \| 1010 \| 10 \| −2 \| −5 \| −6 \| 2 \| −10 \| \| 1011 \| 11 \| −3 \| −4 \| −5 \| 3 \| −9 \| \| 1100 \| 12 \| −4 \| −3 \| −4 \| 4 \| −4 \| \| 1101 \| 13 \| −5 \| −2 \| −3 \| 5 \| −3 \| \| 1110 \| 14 \| −6 \| −1 \| −2 \| 6 \| −6 \| \| 1111 \| 15 \| −7 \| −0 \| −1 \| 7 \| −5 \| Other systems [edit] Google's Protocol Buffers "zig-zag encoding" is a system similar to sign–magnitude, but uses the least significant bit to represent the sign and has a single representation of zero. This allows a variable-length quantity encoding intended for nonnegative (unsigned) integers to be used efficiently for signed integers. A similar method is used in the Advanced Video Coding/H.264 and High Efficiency Video Coding/H.265 video compression standards to extend exponential-Golomb coding to negative numbers. In that extension, the least significant bit is almost a sign bit; zero has the same least significant bit (0) as all the negative numbers. This choice results in the largest magnitude representable positive number being one higher than the largest magnitude negative number, unlike in two's complement or the Protocol Buffers zig-zag encoding. Another approach is to give each digit a sign, yielding the signed-digit representation. For instance, in 1726, John Colson advocated reducing expressions to "small numbers", numerals 1, 2, 3, 4, and 5. In 1840, Augustin Cauchy also expressed preference for such modified decimal numbers to reduce errors in computation. See also [edit] Balanced ternary Binary-coded decimal Computer number format Method of complements Signedness References [edit] ^ Choo, Hunsoo; Muhammad, K.; Roy, K. (February 2003). "Two's complement computation sharing multiplier and its applications to high performance DFE". IEEE Transactions on Signal Processing. 51 (2): 458–469. Bibcode:2003ITSP...51..458C. doi:10.1109/TSP.2002.806984. ^ GE-625 / 635 Programming Reference Manual. General Electric. January 1966. Retrieved August 15, 2013. ^ Intel 64 and IA-32 Architectures Software Developer's Manual (PDF). Intel. Section 4.2.1. Retrieved August 6, 2013. ^ Power ISA Version 2.07 (PDF). Power.org. Section 1.4. Retrieved November 2, 2023., ^ Bacon, Jason W. (2010–2011). "Computer Science 315 Lecture Notes". Archived from the original on 14 February 2020. Retrieved 21 February 2020. ^ US 4484301, "Array multiplier operating in one's complement format", issued 1981-03-10 ^ US 6760440, "One's complement cryptographic combiner", issued 1999-12-11 ^ Shedletsky, John J. (1977). "Comment on the Sequential and Indeterminate Behavior of an End-Around-Carry Adder". IEEE Transactions on Computers. 26 (3): 271–272. doi:10.1109/TC.1977.1674817. S2CID 14661474. ^ Donald Knuth: The Art of Computer Programming, Volume 2: Seminumerical Algorithms, chapter 4.1 ^ Thomas Finley (April 2000). "Two's Complement". Cornell University. Retrieved 15 September 2015. ^ Protocol Buffers: Signed Integers Ivan Flores, The Logic of Computer Arithmetic, Prentice-Hall (1963) Israel Koren, Computer Arithmetic Algorithms, A.K. Peters (2002), ISBN 1-56881-160-8 Retrieved from " Category: Computer arithmetic Hidden categories: Articles with short description Short description matches Wikidata Articles needing additional references from April 2013 All articles needing additional references All articles with unsourced statements Articles with unsourced statements from May 2012 Signed number representations Add topic
12779	https://pubmed.ncbi.nlm.nih.gov/33830682/	Using the swimbladder as a respiratory organ and/or a buoyancy structure-Benefits and consequences - PubMed Clipboard, Search History, and several other advanced features are temporarily unavailable. Skip to main page content An official website of the United States government Here's how you know The .gov means it’s official. Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site. The site is secure. The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely. Log inShow account info Close Account Logged in as: username Dashboard Publications Account settings Log out Access keysNCBI HomepageMyNCBI HomepageMain ContentMain Navigation Search: Search AdvancedClipboard User Guide Save Email Send to Clipboard My Bibliography Collections Citation manager Display options Display options Format Save citation to file Format: Create file Cancel Email citation Email address has not been verified. Go to My NCBI account settings to confirm your email and then refresh this page. 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Using the swimbladder as a respiratory organ and/or a buoyancy structure-Benefits and consequences Bernd Pelster12 Affiliations Expand Affiliations 1 Institute of Zoology, University of Innsbruck, Innsbruck, Austria. 2 Center for Molecular Biosciences, University Innsbruck, Innsbruck, Austria. PMID: 33830682 DOI: 10.1002/jez.2460 Item in Clipboard Review Using the swimbladder as a respiratory organ and/or a buoyancy structure-Benefits and consequences Bernd Pelster. J Exp Zool A Ecol Integr Physiol.2021 Nov. Show details Display options Display options Format J Exp Zool A Ecol Integr Physiol Actions Search in PubMed Search in NLM Catalog Add to Search . 2021 Nov;335(9-10):831-842. doi: 10.1002/jez.2460. Epub 2021 Apr 8. Author Bernd Pelster12 Affiliations 1 Institute of Zoology, University of Innsbruck, Innsbruck, Austria. 2 Center for Molecular Biosciences, University Innsbruck, Innsbruck, Austria. PMID: 33830682 DOI: 10.1002/jez.2460 Item in Clipboard Cite Display options Display options Format Abstract A swimbladder is a special organ present in several orders of Actinopterygians. As a gas-filled cavity it contributes to a reduction in overall density, but on descend from the water surface its contribution as a buoyancy device is very limited because the swimbladder is compressed by increasing hydrostatic pressure. It serves, however, as a very efficient organ for aerial gas exchange. To avoid the loss of oxygen to hypoxic water at the gills many air-breathing fish show a reduced gill surface area. This, in turn, also reduces surface area available for other functions, so that breathing air is connected to a number of physiological adjustments with respect to ion homeostasis, acid-base regulation and nitrogen excretion. Using the swimbladder as a buoyancy structure resulted in the loss of its function as an air-breathing organ and required the development of a gas secreting mechanism. This was achieved via the Root effect and a countercurrent arrangement of the blood supply to the swimbladder. In addition, a detachable air space with separated blood supply was necessary to allow the resorption of gas from the swimbladder. Gas secretion as well as gas resorption are slow phenomena, so that rapid changes in depth cannot instantaneously be compensated by appropriate volume changes. As gas-filled cavities the respiratory swimbladder and the buoyancy device require surfactant. Due to high oxygen partial pressures inside the bladder air-exposed tissues need an effective reactive oxygen species defense system, which is particularly important for a swimbladder at depth. Keywords: aerial respiration; air-breathing fish; buoyancy; gills; ion regulation; respiration; swimbladder. © 2021 The Authors. Journal of Experimental Zoology Part A: Ecological Genetics and Physiology published by Wiley Periodicals LLC. PubMed Disclaimer Similar articles Hyperbaric Physics.Jones MW, Brett K, Han N, Cooper JS, Wyatt HA.Jones MW, et al.2024 Jan 31. In: StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2025 Jan–.2024 Jan 31. In: StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2025 Jan–.PMID: 28846268 Free Books & Documents. Improved ROS defense in the swimbladder of a facultative air-breathing erythrinid fish, jeju, compared to a non-air-breathing close relative, traira.Pelster B, Giacomin M, Wood CM, Val AL.Pelster B, et al.J Comp Physiol B. 2016 Jul;186(5):615-24. doi: 10.1007/s00360-016-0981-5. Epub 2016 Apr 5.J Comp Physiol B. 2016.PMID: 27048554 Free PMC article. Ionoregulatory and oxidative stress issues associated with the evolution of air-breathing.Pelster B, Wood CM.Pelster B, et al.Acta Histochem. 2018 Oct;120(7):667-679. doi: 10.1016/j.acthis.2018.08.012. Epub 2018 Aug 31.Acta Histochem. 2018.PMID: 30177382 Gas exchange in the fish swimbladder.Scheid P, Pelster B, Kobayashi H.Scheid P, et al.Adv Exp Med Biol. 1990;277:735-42. doi: 10.1007/978-1-4684-8181-5_84.Adv Exp Med Biol. 1990.PMID: 1965765 Review. Fundamental structural aspects and features in the bioengineering of the gas exchangers: comparative perspectives.Maina JN.Maina JN.Adv Anat Embryol Cell Biol. 2002;163:III-XII, 1-108. doi: 10.1007/978-3-642-55917-4.Adv Anat Embryol Cell Biol. 2002.PMID: 11892241 Review. See all similar articles Cited by In situ observation of a macrourid fish at 7259 m in the Japan Trench: swimbladder buoyancy at extreme depth.Priede IG, Jamieson AJ, Bond T, Kitazato H.Priede IG, et al.J Exp Biol. 2024 Feb 1;227(3):jeb246522. doi: 10.1242/jeb.246522. Epub 2024 Feb 1.J Exp Biol. 2024.PMID: 38230425 Free PMC article. Morphogenic, molecular and cellular adaptations for unidirectional airflow in the chicken lung.Gerner-Mauro KN, Vila Ellis L, Wang G, Nayak R, Lwigale PY, Poché RA, Chen J.Gerner-Mauro KN, et al.Development. 2025 Apr 15;152(8):dev204346. doi: 10.1242/dev.204346. Epub 2025 Apr 28.Development. 2025.PMID: 40177910 Aquaporin expression and cholesterol content in eel swimbladder tissue.Drechsel V, Schneebauer G, Fiechtner B, Cutler CP, Pelster B.Drechsel V, et al.J Fish Biol. 2022 Mar;100(3):609-618. doi: 10.1111/jfb.14973. Epub 2021 Dec 21.J Fish Biol. 2022.PMID: 34882794 Free PMC article. Christian Bohr. Discoverer of Homotropic and Heterotopic Allostery: Periods Leading up to Bohr's Life, Work, and Beyond. Was Bohr a Vitalist and Right About a "Specific Activity"?Bindslev N.Bindslev N.Acta Physiol (Oxf). 2025 Jul;241 Suppl 734(Suppl 734):e70016. doi: 10.1111/apha.70016.Acta Physiol (Oxf). 2025.PMID: 40665777 Free PMC article.Review. Ecotoxicity of commonly used oilfield-based emulsifiers on Guinean Tilapia (Tilapia guineensis) using histopathology and behavioral alterations as protocol.Chris DI, Wokeh OK, Téllez-Isaías G, Kari ZA, Azra MN.Chris DI, et al.Sci Prog. 2024 Jan-Mar;107(1):368504241231663. doi: 10.1177/00368504241231663.Sci Prog. 2024.PMID: 38490166 Free PMC article. See all "Cited by" articles References REFERENCES Aarestrup, K., Okland, F., Hansen, M. M., Righton, D., Gargan, P., Castonguay, M., Bernatchez, L., Howey, P., Sparholt, H., Pedersen, M. I., & McKinley, R. S. (2009). Oceanic spawning migration of the European Eel (Anguilla anguilla). Science, 325(5948), 1660. Alexander, R. M. (1966). Physical aspects of swimbladder function. Biological Reviews, 41, 141-176. Alexander, R. M. (1971). Swimbladder gas secretion and energy expenditure in vertically migrating fishes. In G. B. Farquhar (Ed.), Proceedings of the International Symposium on Biological Sound Scattering in the Ocean: Vol. MC Report (pp. 75-86). US Government Printing Office. Alexander, R. M. (1972). The energetics of vertical migration by fishes. In M. A. Sleigh & A. G. MacDonald (Eds.), The effect of pressure (pp. 273-294). Alexander, R. M. (1990). Size, speed and buoyancy adaptations in aquatic animals. American Zoologist, 30, 189-196. Show all 106 references Publication types Research Support, Non-U.S. Gov't Actions Search in PubMed Search in MeSH Add to Search Review Actions Search in PubMed Search in MeSH Add to Search MeSH terms Air Sacs Actions Search in PubMed Search in MeSH Add to Search Animals Actions Search in PubMed Search in MeSH Add to Search Fishes Actions Search in PubMed Search in MeSH Add to Search Gills Actions Search in PubMed Search in MeSH Add to Search Oxygen Actions Search in PubMed Search in MeSH Add to Search Reactive Oxygen Species Actions Search in PubMed Search in MeSH Add to Search Substances Reactive Oxygen Species Actions Search in PubMed Search in MeSH Add to Search Oxygen Actions Search in PubMed Search in MeSH Add to Search Related information PubChem Compound (MeSH Keyword) Grants and funding I 2984/FWF_/Austrian Science Fund FWF/Austria I12984-B25 P26363-25/FWF_/Austrian Science Fund FWF/Austria [x] Cite Copy Download .nbib.nbib Format: Send To Clipboard Email Save My Bibliography Collections Citation Manager [x] NCBI Literature Resources MeSHPMCBookshelfDisclaimer The PubMed wordmark and PubMed logo are registered trademarks of the U.S. Department of Health and Human Services (HHS). 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12780	https://www.dictionary.com/browse/bounce	Daily Crossword Word Puzzle Word Finder All games Word of the Day Word of the Year New words Language stories All featured Slang Emoji Memes Acronyms Gender and sexuality All culture Writing tips Writing hub Grammar essentials Commonly confused All writing tips Games Featured Culture Writing tips Advertisement View synonyms for bounce bounce [bouns] verb (used without object) bounced, bouncing to spring back from a surface in a lively manner. The ball bounced off the wall. 2. to strike the ground or other surface, and rebound. The ball bounced once before he caught it. 3. to move or walk in a lively, exuberant, or energetic manner. She bounced into the room. 4. to move along in a lively manner, repeatedly striking the surface below and rebounding. The box bounced down the stairs. 5. to move about or enter or leave noisily or angrily (followed by around, about, out, out of, into, etc.). He bounced out of the room in a huff. 6. (of a check or the like) to fail to be honored by the bank against which it was drawn, due to lack of sufficient funds. 1 / 46 Lucid: Word of the Day 3K 12 Video Player is loading. Duration 0:00 / Current Time 0:00 Loaded: 0% Remaining Time -0:00 This is a modal window. No compatible source was found for this media. Beginning of dialog window. Escape will cancel and close the window. End of dialog window. Share Settings Playback Speed Normal Closed Captions Off TOP ARTICLES Powered by AnyClip Privacy Policy Keyboard Shortcuts Lucid: Word of the Day NOW PLAYING UP NEXT Mukbang: Word of the Day NOW PLAYING UP NEXT Tycoon: Word of the Day NOW PLAYING UP NEXT Banshee: Word of the Day NOW PLAYING UP NEXT Gleam: Word of the Day NOW PLAYING UP NEXT Leviathan: Word of the Day NOW PLAYING UP NEXT Myriad: Word of the Day NOW PLAYING UP NEXT Caravan: Word of the Day NOW PLAYING UP NEXT Veld: Word of the Day NOW PLAYING UP NEXT Pidgin: Word of the Day NOW PLAYING UP NEXT Talisman: Word of the Day NOW PLAYING UP NEXT Verboten: Word of the Day NOW PLAYING UP NEXT Vigilante: Word of the Day NOW PLAYING UP NEXT Panache: Word of the Day NOW PLAYING UP NEXT Gusto: Word of the Day NOW PLAYING UP NEXT Pigeonhole: Word of the Day NOW PLAYING UP NEXT Amain: Word of the Day NOW PLAYING UP NEXT Sidereal: Word of the Day NOW PLAYING UP NEXT Carrack: Word of the Day NOW PLAYING UP NEXT Stymie: Word of the Day NOW PLAYING UP NEXT Betimes: Word of the Day NOW PLAYING UP NEXT Gadzookery: Word of the Day NOW PLAYING UP NEXT Mistryst: Word of the Day NOW PLAYING UP NEXT Demitasse: Word of the Day NOW PLAYING UP NEXT Flexuous: Word of the Day NOW PLAYING UP NEXT Riposte: Word of the Day NOW PLAYING UP NEXT Etiolate: Word of the Day NOW PLAYING UP NEXT Blithesome: Word of the Day NOW PLAYING UP NEXT Remora: Word of the Day NOW PLAYING UP NEXT Primogeniture: Word of the Day NOW PLAYING UP NEXT Arenaceous: Word of the Day NOW PLAYING UP NEXT Soliloquy: Word of the Day NOW PLAYING UP NEXT Haply: Word of the Day NOW PLAYING UP NEXT Tip-of-the-Tongue Poetry Challenge NOW PLAYING UP NEXT "Empathy" vs. "Sympathy": Here's The Key Difference NOW PLAYING UP NEXT Excelsior! How Do You Use This Lofty Word? NOW PLAYING UP NEXT Surprise! These Phrases Are Repetitive NOW PLAYING UP NEXT What's DoggoLingo? Speak Like A Dog (Or Cat)! NOW PLAYING UP NEXT Who Created The Hijab Emoji? NOW PLAYING UP NEXT What Exactly Is An Adverb? NOW PLAYING UP NEXT Ew. Do You Use These Terms For Bodily Functions? NOW PLAYING UP NEXT Can You Answer These Capitalization Questions Correctly? NOW PLAYING UP NEXT Hey, Y'all: How Do YOU Pluralize "You"? NOW PLAYING UP NEXT How Well Do You Know Your Baby Animals? NOW PLAYING UP NEXT Ready For These Fun Synonyms? Here's How To Liven Up Your Vocab NOW PLAYING UP NEXT This Or That: Jealous vs. Envious NOW PLAYING UP NEXT Lucid: Word of the Day verb (used with object) bounced, bouncing to cause to bound and rebound. to bounce a ball; to bounce a child on one's knee; to bounce a signal off a satellite. 2. to refuse payment on (a check) because of insufficient funds. The bank bounced my rent check. 3. to give (a bad check) as payment. That's the first time anyone bounced a check on me. 4. Slang., to eject, expel, or dismiss summarily or forcibly. noun a bound or rebound. to catch a ball on the first bounce. 2. a sudden spring or leap. In one bounce he was at the door. 3. ability to rebound; resilience. This tennis ball has no more bounce. 4. vitality; energy; liveliness. There is bounce in his step. This soda water has more bounce to it. Synonyms: zip, vigor, pep, spirit, life, vivacity, animation 5. the fluctuation in magnitude of target echoes on a radarscope. 6. Slang., a dismissal, rejection, or expulsion. He's gotten the bounce from three different jobs. adverb with a bounce; suddenly. verb phrase bounce back, to recover quickly. After losing the first game of the double-header, the team bounced back to win the second. bounce / baʊns / verb (intr) (of an elastic object, such as a ball) to rebound from an impact (tr) to cause (such an object) to hit a solid surface and spring back to rebound or cause to rebound repeatedly to move or cause to move suddenly, excitedly, or violently; spring she bounced up from her chair 5. slang, (of a bank) to send (a cheque) back or (of a cheque) to be sent back unredeemed because of lack of funds in the drawer's account 6. (of an internet service provider) to send (an email message) back or (of an email message) to be sent back to the sender, for example because the recipient's email account is full 7. slang, (tr) to force (a person) to leave (a place or job); throw out; eject 8. (tr) to hustle (a person) into believing or doing something noun the action of rebounding from an impact a leap; jump; bound the quality of being able to rebound; springiness informal, vitality; vigour; resilience swagger or impudence informal, a temporary increase or rise Australian rules football the start of play at the beginning of each quarter or after a goal informal, to dismiss or be dismissed from a job informal, in succession; one after the other they have lost nine games on the bounce Discover More Other Word Forms bounceable adjective bounceably adverb Discover More Word History and Origins Origin of bounce1 1175–1225; Middle English buncin, bounsen, variant of bunkin, apparently cognate with Dutch bonken to thump, belabor, bonzen to knock, bump Discover More Word History and Origins Origin of bounce1 C13: probably of imitative origin; compare Low German bunsen to beat, Dutch bonken to thump Discover More Idioms and Phrases get the ax (bounce) more bounce for the ounce that's how the ball bounces Discover More Example Sentences Examples have not been reviewed. However, McDonald’s has managed to bounce back with sales doing better in recent months. From Salon Hyeseong Kim, whom the Dodgers have bounced uneasily between the infield and the outfield, between the starting lineup and the bench, between the major leagues and minor leagues. From Los Angeles Times Adams noted that Nacua “bounced up” after taking hits against the Texans, came back after receiving stitches and continued to excel. From Los Angeles Times I didn't think that would happen either, but Thomas Frank's teams are usually so well-balanced and I am expecting them to bounce back. From BBC The Dolphins looked awful last week but should bounce back offensively. From Los Angeles Times Advertisement Discover More Related Words bound bump carom hop jump leap rebound ricochet Word of the Day Global Connections Series [loo-sid] Meaning and examples Start each day with the Word of the Day in your inbox! 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12781	https://www.teacherspayteachers.com/browse/free?search=multiply%20by%203%20maths%20puzzles	Multiply by 3 Maths Puzzles \| TPT Log InSign Up Cart is empty Total: $0.00 View Wish ListView Cart Grade Elementary Preschool Kindergarten 1st grade 2nd grade 3rd grade 4th grade 5th grade Middle school 6th grade 7th grade 8th grade High school 9th grade 10th grade 11th grade 12th grade Adult education Resource type Student practice Independent work packet Worksheets Assessment Graphic organizers Task cards Flash cards Teacher tools Classroom management Teacher manuals Outlines Rubrics Syllabi Unit plans Lessons Activities Games Centers Projects Laboratory Songs Clip art Classroom decor Bulletin board ideas Posters Word walls Printables Seasonal Holiday Black History Month Christmas-Chanukah-Kwanzaa Earth Day Easter Halloween Hispanic Heritage Month Martin Luther King Day Presidents' Day St. Patrick's Day Thanksgiving New Year Valentine's Day Women's History Month Seasonal Autumn Winter Spring Summer Back to school End of year ELA ELA by grade PreK ELA Kindergarten ELA 1st grade ELA 2nd grade ELA 3rd grade ELA 4th grade ELA 5th grade ELA 6th grade ELA 7th grade ELA 8th grade ELA High school ELA Elementary ELA Reading Writing Phonics Vocabulary Grammar Spelling Poetry ELA test prep Middle school ELA Literature Informational text Writing Creative writing Writing-essays ELA test prep High school ELA Literature Informational text Writing Creative writing Writing-essays ELA test prep Math Math by grade PreK math Kindergarten math 1st grade math 2nd grade math 3rd grade math 4th grade math 5th grade math 6th grade math 7th grade math 8th grade math High school math Elementary math Basic operations Numbers Geometry Measurement Mental math Place value Arithmetic Fractions Decimals Math test prep Middle school math Algebra Basic operations Decimals Fractions Geometry Math test prep High school math Algebra Algebra 2 Geometry Math test prep Statistics Precalculus Calculus Science Science by grade PreK science Kindergarten science 1st grade science 2nd grade science 3rd grade science 4th grade science 5th grade science 6th grade science 7th grade science 8th grade science High school science By topic Astronomy Biology Chemistry Earth sciences Physics Physical science Social studies Social studies by grade PreK social studies Kindergarten social studies 1st grade social studies 2nd grade social studies 3rd grade social studies 4th grade social studies 5th grade social studies 6th grade social studies 7th grade social studies 8th grade social studies High school social studies Social studies by topic Ancient history Economics European history Government Geography Native Americans Middle ages Psychology U.S. History World history Languages Languages American sign language Arabic Chinese French German Italian Japanese Latin Portuguese Spanish Arts Arts Art history Graphic arts Visual arts Other (arts) Performing arts Dance Drama Instrumental music Music Music composition Vocal music Special education Speech therapy Social emotional Social emotional Character education Classroom community School counseling School psychology Social emotional learning Specialty Specialty Career and technical education Child care Coaching Cooking Health Life skills Occupational therapy Physical education Physical therapy Professional development Service learning Vocational education Other (specialty) Multiply by 3 Maths Puzzles 104+results Sort by: Relevance Relevance Rating Rating Count Price (Ascending) Price (Descending) Most Recent Sort by: Relevance Relevance Rating Rating Count Price (Ascending) Price (Descending) Most Recent Search Grade Subject Supports Free Format All filters (1) Filters Free Clear all Grade Elementary Preschool Kindergarten 1st grade 2nd grade 3rd grade 4th grade 5th grade Middle school 6th grade 7th grade 8th grade High school 9th grade 10th grade 11th grade 12th grade Not grade specific More Subject Art Graphic arts Visual arts Other (arts) English language arts Reading Reading strategies Writing Math Algebra Algebra 2 Applied math Arithmetic Basic operations Calculus Decimals Geometry Math test prep Mental math Numbers Order of operations Place value Precalculus Other (math) Performing arts Music Science Biology Computer science - 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Games and Puzzle Sheet Created by Games 4 Learning This Freebie from Games 4 Learning includes 4 St. Patrick's Day math activities. These are NO PREP activities that are fun, engaging and worthwhile activities that review important math skills. Here's What's Included: Make a Ten is a a math board game to practice basic addition to 10. Shamrock Race Triples is a St. Patrick's Day math board game that reviews multiplying 1-6 by 3. The second Shamrock Race Triples is a St. Patrick's Day math board game that reviews multiplying 2-12 by 3. St. Pat 1 st - 3 rd Math FREE Rated 4.71 out of 5, based on 223 reviews 4.7(223) Log in to Download Wish List FREE Math and Logic Mystery Puzzle - Multiplying by Multiples of 10 Created by Mrs Thompson's Treasures Multiplying multiples of 10, logic, mystery, and fun, all in one FREE resource! With this engaging activity, students start with a mystery story. They will then work their way through 5 pages of math problems to discover clues. They use the clues and the story to logically figure out the solution to the mystery! Clue 1 - Multiply by multiples of 10Clue 2 - Multiply by multiples of 100Clue 3 - Find the missing number (multiples of 10)Clue 4 - Find the missing number (multiples of 100)Clue 5 - M 3 rd - 4 th Math CCSS 3.NBT.A.3 Also included in:3rd Grade Standards Mystery Math Puzzles – Logic & Critical Thinking Bundle FREE Rated 4.9 out of 5, based on 20 reviews 4.9(20) Log in to Download Wish List Get more with resources under $5 See all Multiplication 2,3,4 by 1-digit MATH MYSTERY DRAW PUZZLE BUNDLE! FUN! Orange Square Feet $3.50 Price $3.50$4.50 Original Price $4.50 Rated 5 out of 5, based on 1 reviews 5.0(1) Multiplying Multi Digit Numbers 3 Digit by 1 Digit Digital Math Jigsaw Puzzle Caffeinated in Fourth $3.00 Original Price $3.00 3 Digit by 1 Digit Multiplication Game: Math Tarsia Puzzle Science Spot $2.50 Original Price $2.50 Rated 5 out of 5, based on 3 reviews 5.0(3) 3-Digit By 2-Digit Multiplication \| Math Escape Room & Kakurasu Puzzles Gamified EduTech $3.00 Original Price $3.00 Multiplying Pumpkins FREE Fall Activities and Worksheets for Multiplying By 1 Created by Primarily Blue Multiplying Pumpkins-Free Activities and Worksheets for Multiplying By 1 is a resource that is easy to prep and reinforces multiplication facts for X 1. This resource covers equal groups, arrays, repeated addition, groups of, and writing multiplication number stories. What a teacher like you says: “This resource was very engaging for my students and a helpful supplement for our unit lesson! My kids really enjoyed coloring in and actively working on the problems!” In this resource, you w 2 nd - 3 rd Basic Operations, Math CCSS 2.OA.C.4 , 3.OA.A.1 , 3.OA.A.3 Also included in:A BUNDLE of Halloween Multiplication Resources Facts for X 0 to X 12 FREE Rated 5 out of 5, based on 5 reviews 5.0(5) Log in to Download Wish List Math Picture Puzzle Games: Powers of 10 Created by Practice Made Purposeful A Fun and DIFFERENTIATED Way to Practice the Skill of MULTIPLYING AND DIVIDING BY POWERS OF 10! The 5 puzzles included in this product gradually increase in difficulty, so they are appropriate for varied learning levels in your classroom! A CHALLENGE puzzle is even included for more advanced learners! - Puzzle 1: Multiplying by Powers of 10 - Puzzle 2: Dividing by Powers of 10 - Puzzle 3: Multiplying and Dividing by Powers of 10 - Puzzle 4: Multiplying and Dividing by Powers of 10 - Puzzle 5: 4 th - 6 th Decimals, Math Test Prep, Numbers CCSS 5.NBT.A.2 FREE Rated 4.75 out of 5, based on 16 reviews 4.8(16) Log in to Download Wish List Multi-Digit Multiplication Puzzle Freebie \| 2-Digit by 1-Digit Created by Ms Turai's Learning Corner Are your students ready for multi-digit multiplication? Are you looking for a fun activity to work on 2-digit by 1-digit multiplication? Do you need a differentiated resource for your math class? Either way, this 3-puzzle resource is the perfect way to make math class fun and engaging. ⭐ You can find more puzzles here. ⭐ What you can expect:You will find 3 puzzles in this resource. The puzzle sizes are 3x3, 3x4, and 4x4 grids. Each puzzle has a different theme. Themes: campingclassroombaseb 3 rd - 4 th Basic Operations CCSS 4.NBT.B.5 FREE Rated 5 out of 5, based on 5 reviews 5.0(5) Log in to Download Wish List 3& 4 digits by 1 digit - Multiplication Tiling Puzzle - FREE Created by Raki's Rad Resources Are you looking for a creative new way to have your students practice multiplication? Try Tiling! In Tiling activities, students will use the provided "digit tiles" to place the numbers 0 - 9 to fill in the holes left in provided multiplication equations. While trying to figure out where the missing numbers go, students will work on basic facts, properties of multiplication and problem solving skills. Simple and easy to use as a homework assignment or a math center. This is just one puzzle, 4 th - 6 th Basic Operations, Math CCSS 4.NBT.B.5 , 5.NBT.B.5 FREE Rated 4.9 out of 5, based on 21 reviews 4.9(21) Log in to Download Wish List 2023 Math Multiplication/Division Review Puzzles Created by Tales of a 3rd Grade ELA and SS Teacher Puzzles include review for: Multiplying by Powers of TenRoundingMultiplying by 100/1,000/10,000Dividing by 0.1/0.01/0.001Multiplying by 2 digit numbersDividing by 2 digit numbersMultiplying by 3 digit numbersDividing by 3 digit numbersPrint these and cut them apart to differentiate or give students multiple puzzles to work through. You can print these for students to work on in stations or as partners for a collaborative activity to help students get back to learning after winter break. 4 th - 6 th Math CCSS 5.NBT.A.4 , 5.NBT.B.5 , 5.NBT.B.6 FREE Log in to Download Wish List Multiplication Practice 3rd Grade \| Creative Math Puzzles Multiply By 5 Shark Created by LittleYellowStar Print & Go creative math puzzles to practice Multiply by 5! Color, Draw, Go through a Maze, and Solve the riddle, all designed to make multiplication practice fun! This is the Free Sample of the Creative Math Puzzles Packet In this download, you will only get Color By Code & Solve The Riddle. Learn more in the download on how to get the full packet for free. Math Skill Focus: Multiply by 5 (5 Times Table) Grade Level Recommendation: 3rd Grade and 4th Grade This Free Sample Resource Incl 2 nd - 4 th Basic Operations, Math, Other (Math) CCSS 4.NBT.B.5 , 3.OA.C.7 Also included in:Creative Math Puzzles Bundle: Multiplication Practice 3rd Grade 12 Times Tables FREE Rated 4.75 out of 5, based on 12 reviews 4.8(12) Log in to Download Wish List Winter Math Puzzles Bundle Created by Jessica Senatore This amazing Winter/Christmas Math Puzzles Bundle is all you need to have some math fun this December! The activities are both challenging and engaging for students! Included in this bundle: 5 Hidden Picture Multiplication and Division Math Puzzles (Christmas tree - fours family, Santa - sevens family, snowman - sixes family, reindeer - eights family, elf - nines family) 3 Skip Count Mazes (3's, 4's, 5's) 3 Solve the Joke Scoot! Activities (1 add/subtract within 1,000, 1 multiply and divi 3 rd - 9 th Basic Operations FREE Rated 4.72 out of 5, based on 18 reviews 4.7(18) Log in to Download Wish List Winter Holiday Fun Puzzles, Cross Math Puzzle Worksheets, Free Printables Created by Alexander Frans Winter Holiday Fun Addition, Subtraction, Multiplication, Division and Mixed Cross Math Puzzles. Six pages with answers included. See other free puzzles, by author, for more Addition, Subtraction, Multiplication and Division. Your votes and/or comments much appreciated. 2 nd - 7 th Arithmetic, Basic Operations, Mental Math CCSS 2.NBT.A.2 , 3.NBT.A.2 , 4.NBT.A.2 +1 FREE Rated 4.89 out of 5, based on 28 reviews 4.9(28) Log in to Download Wish List Summer Holiday Fun Puzzles - Math Puzzle Worksheets - Free Printables Created by Alexander Frans Free Summer Holiday Fun Addition, Subtraction, Multiplication, Division and Mixed Cross Math Puzzles. Fifteen (15) pages with answers included. See link to other free puzzles, by author, for more Addition, Subtraction, Multiplication and Division of various levels of difficulty. Your votes and/or comments much appreciated. 3 rd - 7 th Arithmetic, Basic Operations, Mental Math CCSS 3.NBT.A.2 , 4.NBT.A.2 , 5.NBT.A.2 FREE Rated 4.63 out of 5, based on 41 reviews 4.6(41) Log in to Download Wish List Order of Operations Middle School Math Puzzle Created by Michael Royal This puzzle requires students to solve a variety of problems involving the order of operations. I purposely included problems such as 6(3+2) and 4(2) to introduce my students to the concept that a number outside a set of parentheses should be multiplied by the value inside the parentheses. If I simply gave my students the 32 problems in this activity on a worksheet, they would GROAN! But every time I use this puzzle I'm impressed at how motivated students are to work through the problems in or 5 th - 7 th Basic Operations, Math, Order of Operations FREE Rated 4.7 out of 5, based on 81 reviews 4.7(81) Log in to Download Wish List Math Computation Pumpkin Puzzle Created by Go On and Go Fourth Students will use their basic math understanding to navigate out of the pumpkin patch. Focusing on addition and subtraction of multi-digit whole numbers, multi-digit by single-digit multiplication and division with remainders, and place value- this fun and engaging math puzzle is perfect for fall, Thanksgiving, or Halloween! Use as an early finisher activity, or activity to celebrate your final days before fall break. Math problems fit with 4th grade Module 1 and 3 of Eureka Math curriculum. 3 rd - 5 th Math CCSS 4.NBT.A.1 , 4.NBT.B.4 , 4.NBT.B.5 +1 FREE Rated 4.76 out of 5, based on 5 reviews 4.8(5) Log in to Download Wish List Multiplication Fact Color by Code Snowman Free Math Puzzle Created by Teacher Tech Studio - Danna Rodebush You know your students need practice with multiplication fact fluency. Why not give them an engaging activity to help them practice! The snowman math color by code puzzle includes 50 random multiplication facts for students to solve. They will then use the color key to create the hidden snowman picture. This activity is great for a quick math center or a holiday party puzzle for upper elementary students in the days leading up to Christmas break or when they come back in January! Included insi 3 rd - 5 th Math CCSS 3.NBT.A.3 FREE Rated 4 out of 5, based on 1 reviews 4.0(1) Log in to Download Wish List FREE!! Space Theme Math Noggle \| Multiplication& Division Game for Grades 3-5. Created by My happy brain FREE!! Space Theme Math Noggle game! Perfect for Grades 3–5, this 5x5 math puzzle combines multiplication and division practice with a fun, space-inspired twist. Students solve problems by connecting numbers in the grid to reach target equations, just like a math version of Boggle. This low-prep resource is ideal for math centers, early finishers, small groups, or whole-class challenges. Simply print and play for an engaging way to reinforce multiplication and division skills while building crit 3 rd - 5 th Basic Operations, Mental Math, Other (Math) FREE Log in to Download Wish List Multiplication Puzzle FREEBIE Created by Miss American Pi Your students will love practicing multiplication with this fun, no-prep puzzle. Students multiply two- and three-digit numbers by one-digit numbers and match them to answers. It's ideal for student centers, small group work, or independent practice. You'll receive two different versions for easy differentiation. Teaching multiplication? Check out these great resources to help your sudents! Multiplication Maze (3-digits by 1-digit) Multiplication Poster: Box MethodMultiplication Maze (2-dig 3 rd - 6 th Math FREE Log in to Download Wish List Multiplication Puzzle Race: Multiplying by 2 - FREEBIE! Created by Teaching to the Child Group your students into teams and send them on a hunt for hidden puzzle pieces. The first team to find all of their pieces and match them correctly is the winner! This game is a great way to review math facts with your students. The following pages include: • 3 Sets of puzzle pieces multiplying by 2 (8 Pairs of Puzzle Pieces per set) • Team Cards • Directions If you enjoy this produ 2 nd - 4 th Basic Operations, Math, Mental Math CCSS 2.OA.C.3 , 3.OA.C.7 , 4.OA.A.1 FREE Rated 4.88 out of 5, based on 16 reviews 4.9(16) Log in to Download Wish List Multiplying by Facts 7 Created by Djurdjica Stojkovic Here’s a polished, ready-to-copy English description for your Multiplying by Facts – 7 resource—perfect for Teachers Pay Teachers: MULTIPLYING BY 7 – FACT FLUENCY PRACTICEBoost students' multiplication skills with focused practice on the 7s times table! This engaging resource offers a variety of strategies and practice formats to help learners master multiplying by 7. HOW TO USE IT: Print & Prepare – Choose color or B&W and select appropriate practice pages. Drill & Practice – Students com 1 st - 2 nd Math FREE Rated 5 out of 5, based on 4 reviews 5.0(4) Log in to Download Wish List FREE Math Multiplication Facts Mystery Picture Puzzle Activities Created by Aussie Waves These 12 digital task cards are perfect Math centre activities, revising Times Tables, or multiplication facts. Tasks involve matching answers with facts to reveal a little more of a hidden, mystery picture. Great fun for students Math centres and distance learning. These 12 cards ask students to use their knowledge of simple multiplication facts, and is a sample of my similar, full decks.the first is an instruction/ explanationthe next 9 are click the correct answer (questions involve mul 2 nd - 4 th Math CCSS 3.OA.A.3 FREE Log in to Download Wish List 3& 4 digits by 1 digit - Multiplication Tiling Puzzle -FREE - Distance Learning Created by Raki's Rad Resources Are you looking for a creative new way to have your students practice multiplication? Try Tiling! In Tiling activities, students will use the provided "digit tiles" to place the numbers 0 - 9 to fill in the holes left in provided multiplication equations. While trying to figure out where the missing numbers go, students will work on basic facts, properties of multiplication and problem solving skills. Simple and easy to use as a homework assignment or a math center. Using this digital version, s 4 th - 6 th Applied Math, Basic Operations, Math CCSS 4.NBT.B.5 , 4.NBT.B.6 , 5.NBT.B.5 +5 FREE Add to Google Drive Wish List Multiplication Coloring Sheets Math Freebie Created by A Spot of Curriculum The multiplication coloring pages are puzzles that give your students extra practice. These no-prep worksheets help you incorporate fun into your curriculum. They are perfect printables for centers, independent work, early finishers, or when you plan for a sub. You can use them during them any time of the year. There are four puzzles and each page practices a specific set of multiplication facts. By the time students complete all the puzzles, they have practiced all facts 2's through 12's. T 3 rd - 5 th Basic Operations, Math CCSS 3.OA.C.7 Also included in:Multiplication Coloring Sheets Coloring Pages Year Long Bundle - Color by Number FREE Rated 4.79 out of 5, based on 203 reviews 4.8(203) Log in to Download Wish List Multiplication Color by Number Freebie Created by A Spot of Curriculum These color by code multiplication puzzles give your students extra practice. These no-prep coloring pages help you incorporate fun into your curriculum. They are perfect printables for centers, independent work, or when you plan for a sub. This set is a sample from six different color by code sets, however this line of products actually has 13 different sets. There are six puzzles and the first five practice a specific set of multiplication facts. The last puzzle is a mixed review of all 3 rd - 5 th Math CCSS 3.OA.C.7 FREE Rated 4.71 out of 5, based on 139 reviews 4.7(139) Log in to Download Wish List Multiplication Facts Worksheets - Skip Counting by 3s Dot to Dots - Owl FREEBIE Created by Making Math Class Fun Two cute Owl dot-to-dot puzzles are included in this sample set and are from my larger multiplication practice resource. The puzzles are set at two levels and offer differentiation. The full resource includes 18 puzzles counting by 2s, through to 10s. Great for math centers, early finishers or morning work! To complete the FIRST owl puzzle: Students start at 0 in the HEART They find the CLOSEST first number (e.g. 3), then look for the CLOSEST (in distance) next multiple/product of 3 (e.g. 6. On 2 nd - 6 th Basic Operations, Math, Mental Math FREE Rated 4.75 out of 5, based on 16 reviews 4.8(16) Log in to Download Wish List FREE Polar Express Day Math Puzzle \| Christmas Math Activity \| Polar Express Day Created by Crafting Curiosity Looking for a FREE Polar Express Day Math Activity to include in your 3rd, 4th, or 5th Grade Polar Express Day Plans? Add this Polar Express math puzzle to your Polar Express Day! With options to solve addition, subtraction, multiplication, or division, this puzzle will cover Polar Express Day needs for 3rd, 4th, and 5th grade classrooms! Make it a FULL Polar Express Day by including these Polar Express Math Escape Rooms that your students will love! 3rd Grade Escape Room4th Grade Escape Room5t 3 rd - 5 th Math CCSS 3.NBT.A.2 , 4.NBT.B.5 , 5.NBT.B.6 Also included in:4th Grade Christmas Math Activity Bundle \| Crafts, Puzzles, and Color by Numbers FREE Rated 5 out of 5, based on 7 reviews 5.0(7) Log in to Download Wish List 1 2 3 4 5 Showing 1-24 of 104+results TPT is the largest marketplace for PreK-12 resources, powered by a community of educators. Facebook Instagram Pinterest Twitter About Who we are We're hiring Press Blog Gift Cards Support Help & FAQ Security Privacy policy Student privacy Terms of service Tell us what you think Updates Get our weekly newsletter with free resources, updates, and special offers. 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12782	https://www.youtube.com/watch?v=nisuQBJZ3fc	Geometry 5-2: Bisectors in Triangles MissJohnsonOLG 4019 subscribers 125 likes Description 9131 views Posted: 24 Nov 2018 14 comments Transcript: Geometry 5-2: Bisectors in Triangles hello everyone welcome to geometry lesson 5 - which is all about bisectors and triangles by the end of this lesson you should be able to use triangle bisectors to solve problems alright so we have a lot of vocabulary today I highly recommend that you write this down maybe you make flashcards if you don't know what these words mean you're not gonna know what the questions are asking you this vocabulary is super important ok so let's start with the word concurrent concurrent is when you have three or more lines that intersect at a point so if I had these three lines those three lines would be concurrent because they meet at one point they meet at this single point and that point is called the point of concurrency so concurrent when three or more lines intersect at one point that point is called point of concurrency circumcenter is when you have a triangle and it's the point where three perpendicular bisectors of the triangle meet okay so if we look at this picture over on the right with the perpendicular bisectors you can see that perpendicular bisectors intersect at the circumcenter so remember perpendicular means that it meets at our right angle or 90 degree and bisector means that it cuts in half so if we look this perpendicular bisector cut that side and half this other perpendicular bisector cut this side half and the third perpendicular bisector cut that third side in half so that point where they all met is called the circumcenter and to go along with circum centers remember all of your sides end up getting cut in half that's important information to know because that's the information you will use in order to solve problems okay so if you have circumcenter that means that you're you have perpendicular bisectors of the the triangle and the last important bit of information with the circumcenter is that the circumcenter is the same distance from the circumcenter to each vertice so pc would be the same distance as PA which is the same distance as PB circumcircle scribed the circle contains all three vertices of a given triangle so if we look there's a picture of a triangle that is drawn inside of a circle and that just kind of illustrates what I was just saying about how if this was P it's a C a and B all those lines end up being congruent or equal equidistant to of each other from the circumcenter to each of the vertices so THEOREM 5-5 Concurrency of Perpendicular Bisectors let's talk about that theorem the theorem 5/5 is about the concurrency of perpendicular bisectors remember concurrency simply means that we have three or more lines intersecting since we're talking about a triangle we're only gonna have three lines great so concurrency of perpendicular bisectors so where are those perpendicular bisectors meet so kind of what I just said here's a perpendicular bisector here's a perpendicular bisector and here's another perpendicular bisector and if you have that then each of the blue lines going from circumcenter to the vertices are going to be congruent and you're also going to have those sides that have been cut in half okay all super important information I know I basically just repeated myself on this slide but you need to know that and the more I say it the more likely you are to get annoyed to become annoyed and therefore remember this information or just get angry at me we'll see you let me know then we have more important vocabulary okay so two more words inscribed is when the circle inside the triangle so inscribed is when we have that circle inside our figure so the circle that intersects each side of a triangle at exactly one point and has no points outside of the triangle so inscribed this circle touches the triangle at exactly one point on each side and if we draw a line from the center of that circle to that point where the circle touches the triangle we would get three 90-degree angles and we would get three equal lines because if you think about it those three lines are all the same because they're each a radius in a circle okay so each of those lines are going to be equal to each other and that point where they all meet is called the in-center okay so that's called the in-center the other bit of information that's important about the in-center so this would be my in Center is that each of those lines connecting from the in center to the vertice cuts that angle in half so angle a this large angle was just cut in half angle B was cut in half and go C was cut in half okay that's information that you will use then if you're looking for how that connects to what I said on the last example these purple lines would be the information that we were just talking about where those lines would end up being equal to each other okay so we have equal angles because they were bisected in center has bisected angles and then the lines that are perpendicular end up being the going from the side of the triangle to the in center are equal and this is another way THEOREM 5-6 Concurrency of Angle Bisectors to say that this is all theorem five six concurrency of angle bisectors meaning you're in center so again my in center going from the in center to the outside of the triangle meets at 90 degrees and each of those are equal to each other and then going from the in center to the vertices of each going to man Center to each vertice means that we have angles that are cut in half or bisected so we have equal angles and we have some equal sides all right so if we look at example one circum centers Point P is the circumcenter of the triangle below lists any segments or segments congruent to those listed below so if P is my in center the drawings a little bad that's point P and I want to know what segments are congruent to segment VR so remember circumcenter means that we have a perpendicular bisector our sides are cut in half so BR would be equal to AR if I look at the next one TC here's my perpendicular bisector that would be equal to TB in my last one AP so going from a vertice to the circumcenter remember is the same as going from any vertice to the circumcenter so ap would be the same as BP and CP so those are really similar to the questions I'll ask you both on your worksheet tonight or today and your test okay now let's look at what some of the questions within centers would look like so within centers if we know the measure of angle B a F equals 15 and the measure of angle C B F equals 52 what is the measure of angle a C F so if Example 2: Incenters we look remember in centers has to do with angle bisectors so B a F is half of my large angle a so if I draw a b c i would have 15 and another 15 right that would be 30 and then measure of angle c BF would be 52 so if i double that to get this other part here at 104 excuse me and I want to know what AC is AC f is my halfway or half of my angle C so if I find angle C I can simply cut it in half so I know that there are 180 degrees in a triangle so I can do 180 minus 104 minus 30 and I'm left with 46 degrees for angle C and if I cut that in half I'm left with 23 degrees for angle ade see that then the other part of in centers the other information we know is if I have EF so notice that's going from a side to the in center and DF is going from a side to the end Center they both meet at 90 degrees what is the distance from F to a B so going from the side a B to the insider we always go on the line that's the perpendicular because that's the shortest possible distance okay also within centers those are all equal to each other so I can say EF equals D F equals that third side I want so EF would be 3y minus five I can set that equal to dia 2y plus four and if I solve I can subtract 2y from both sides and I'm left with Y minus five equals four add five to both sides I get y equals nine include all of these sides are equal I can plug it in anywhere I want so if I did three times nine minus five and have 27 minus five or 22 units if you have any questions at all on these problems please be sure to reach out for some help like I said the vocab in this section is of the utmost importance you really need to make sure you know it because I will not answer those questions if you come and ask me during your test I'm unable to tell you because it gives away the information that you would need to solve okay so make sure you study if you have any questions at all I know I already said that but I want to make sure you know please be sure to reach out for some help and have a great day
12783	https://www.reddit.com/r/askmath/comments/1heednf/probability_of_two_strong_players_facing_each/	Probability of two strong players facing each other in a tournament without seeding? : r/askmath Skip to main contentProbability of two strong players facing each other in a tournament without seeding? : r/askmath Open menu Open navigationGo to Reddit Home r/askmath A chip A close button Log InLog in to Reddit Expand user menu Open settings menu Go to askmath r/askmath r/askmath This subreddit is for questions of a mathematical nature. Please read the subreddit rules below before posting. 209K Members Online •10 mo. ago stevensterkddd Probability of two strong players facing each other in a tournament without seeding? Probability Let's say a tournament has 64 players, 6 are strong, 58 are weak. Given no seeding, what are the odds at least one pairing involves 2 strong players facing each other in round 1? Would simply 5/64+5/64+5/64+5/64+5/64+5/64 be a close approximation of the answer? Which would mean the chance would be close to 47%. It feels intuitively right. I saw in a birthday calculation thread that you actually have to do it the other way around. Find the probabilities of not facing each other and multiple them like this 58/64 6 which is 55%, in that case the probability of them facing each other is 45% Am i missing something here? Read more Archived post. New comments cannot be posted and votes cannot be cast. Share Related Answers Section Related Answers Understanding limits using intuitive examples How to approach word problems in algebra Exploring the Fibonacci sequence in nature Strategies for mastering calculus derivatives Using matrices to solve systems of equations New to Reddit? Create your account and connect with a world of communities. Continue with Email Continue With Phone Number By continuing, you agree to ourUser Agreementand acknowledge that you understand thePrivacy Policy. Public Anyone can view, post, and comment to this community 0 0 Top Posts Reddit reReddit: Top posts of December 14, 2024 Reddit reReddit: Top posts of December 2024 Reddit reReddit: Top posts of 2024 Reddit RulesPrivacy PolicyUser AgreementAccessibilityReddit, Inc. © 2025. All rights reserved. Expand Navigation Collapse Navigation
12784	https://wuu.wikipedia.org/wiki/%E7%BB%84%E5%90%88%E6%95%B0%E5%AD%A6	组合数学 - 维基百科跳转到内容 [x] 主菜单主菜单移至侧栏囥起來导航封面社区门堂近段辰光个事体近段辰光个改动随机页面帮忙特别页面搜寻搜寻 [x] 外观捐款建账号登录 [x] 私人家伙捐款建账号登录目录移至侧栏囥起來首段 1 组合数学中个著名问题 2 排列 3 组合 4 总结 5 参见 6 參攷文獻 7 外部連結 [x] 开关目录组合数学 [x] 79種閒話 አማርኛ العربية Asturianu Azərbaycanca Žemaitėška Беларуская Беларуская (тарашкевіца) Български বাংলা Bosanski Català Čeština Чӑвашла Cymraeg Dansk Deutsch Ελληνικά English Esperanto Español Eesti Euskara فارسی Suomi Français 贛語 Kriyòl gwiyannen Galego עברית हिन्दी Hrvatski Magyar Հայերեն Jaku Iban Bahasa Indonesia Ido Íslenska Italiano 日本語 Patois ქართული Қазақша 한국어 Кыргызча Latina Lingua Franca Nova Lietuvių Latviešu Македонски Монгол Bahasa Melayu မြန်မာဘာသာ Nederlands Norsk nynorsk Norsk bokmål Polski Português Română Русский Саха тыла Srpskohrvatski / српскохрватски Simple English Slovenčina Slovenščina Shqip Српски / srpski Svenska தமிழ் ไทย Tagalog Türkçe Татарча / tatarça Українська اردو Oʻzbekcha / ўзбекча Tiếng Việt ייִדיש 中文粵語编辑链接文章讨论 [x] 原文原文简体正體阅读编辑望历史 [x] 家生家生移至侧栏囥起來操作阅读编辑望历史常规链进来点啥搭界个改动上传文件老世链接页面信息引用箇篇文章获取短链接下载二维码打印/导出创建书本作为PDF下载打印版别个项目里向维基共享资源维基数据项外观移至侧栏囥起來出自维基百科，自由个百科全书广义个组合数学（英语：Combinatorics）就是离散数学，狭义个组合数学是组合计数、图论、代数结构、数理逻辑等个总称。但昰些衹畢過弗同学者在叫法上个区别。总之，组合数学是一门研究可數或离散对象个科学。随著计算机科学个日益发展，组合数学个重要性也日渐凸显，因为计算机科学个核心内容是使用算法处理离散数据。狭义个组合数学主要研究满足一定条件个组态（也称组合模型）个存在、计数以及构造等方面个问题。组合数学个主要内容有得组合计数、组合设计、组合矩阵、组合优化（最佳組合）等。组合数学中个著名问题 [编辑] 計算一眼物品在特定條件下分組个方法數目。昰些是關於排列、組合搭整數分拆个。地图着色问题：对世界地图着色，每一個国家使用一种颜色。如果要求相邻国家个颜色相异，是否总共衹需四种颜色？昰個是圖論个問題。船夫过河问题：船夫要擔一匹狼、一只羊搭一棵白菜运过河。衹要船夫弗在场，羊就会得吃白菜、狼就会得吃羊。船夫个船每趟衹能运送一种物事。如然擔所有物事儕运过河？昰個是線性規劃个問題。中国邮差问题：由中国组合数学家管梅谷教授提出來。邮递员要穿过城市个每一条路至少一趟，如然樣子走嚜走过个路程頂短？昰個弗是一個NP完全问题，存在多项式复杂度算法：先求出度为奇数个点，用匹配算法算出昰些点间个连接方式，然后再用欧拉路径算法求解。昰個也是圖論个問題。任务分配问题（也称分配问题）：有一些员工要完成一些任务。各個员工完成弗同任务所花费个时间儕弗同。每個员工衹分配一项任务。每项任务衹畀分配给一個员工。如然分配员工与任务以使所花费个时间頂頂少？昰個是線性規劃个問題。如何構造幻方。幻方爲一方陣，填入弗重複个自然數，並使其中每一縱列、橫列、對角線內數字之總和儕相同。排列 [编辑] 从 n{\displaystyle n}隻元素中取出 k{\displaystyle k}隻元素，k{\displaystyle k}隻元素个排列數量爲： P k n=n!(n−k)!{\displaystyle P_{k}^{n}={\frac {n!}{(n-k)!}}} 以賽馬爲例，有8匹马参加比赛，玩家需要在彩票上填入前三胜出个马匹个号码，從8匹馬中取出3匹馬來排前3名，排列數量爲： P 3 8=8!(8−3)!=8×7×6=336{\displaystyle P_{3}^{8}={\frac {8!}{(8-3)!}}=8\times 7\times 6=336} 因为一共存在336种可能性，因此玩家在一次填入中中奖个概率应该是： P=1 336=0.00298{\displaystyle P={\frac {1}{336}}=0.00298} 上面个例子是建立在取出元素弗重複出現狀況。從 n{\displaystyle n}個元素中取出 k{\displaystyle k}個元素，k{\displaystyle k}個元素可以重复出现，昰排列數量爲： U k n=n k{\displaystyle U_{k}^{n}=n^{k}} 以四星彩爲例，10個數字取4個數字，因可能重複所以排列數量爲：昰歇个一次性添入中奖个概率就应该是： P=1 10000=0.0001{\displaystyle P={\frac {1}{10000}}=0.0001} 组合 [编辑] 搭排列弗同个是，组合取出元素个顺序弗考虑。从 n{\displaystyle n}隻元素中取出 k{\displaystyle k}隻元素，k{\displaystyle k}隻元素个组合數量为： C k n=(n k)=P k n k!=n!k!(n−k)!{\displaystyle C_{k}^{n}={n \choose k}={\frac {P_{k}^{n}}{k!}}={\frac {n!}{k!(n-k)!}}} 以六合彩爲例。在六合彩中从49顆球裏向取出6顆球个组合數量为： C 6 49=(49 6)=49!6!43!=13983816{\displaystyle C_{6}^{49}={49 \choose 6}={\frac {49!}{6!43!}}=13983816} 如同排列，上面个例子是建立在取出元素弗重複出現狀況。从 n{\displaystyle n}隻元素中取出 k{\displaystyle k}隻元素，k{\displaystyle k}隻元素可以重複出現，隻组合數量为：以取色球爲例，每種顏色个球有無限多顆，從8種色球中取出5顆球，昰組合數量爲： H 5 8=C 5 8+5−1=C 5 12=12!5!7!=792{\displaystyle H_{5}^{8}=C_{5}^{8+5-1}=C_{5}^{12}={\frac {12!}{5!7!}}=792} 因爲組合數量公式特性，重複組合轉換成組合有另一種公式爲： H k n=C k n+k−1=(n+k−1)!k!(n−1)!=C n−1 n+k−1{\displaystyle H_{k}^{n}=C_{k}^{n+k-1}={\frac {(n+k-1)!}{k!(n-1)!}}=C_{n-1}^{n+k-1}} 另外 H k n{\displaystyle H_{k}^{n}}也可以記爲 F k n{\displaystyle F_{k}^{n}} F k n=H k n{\displaystyle F_{k}^{n}=H_{k}^{n}} 总结 [编辑] n{\displaystyle n}中取 k{\displaystyle k}直線排列（考慮順序）环状排列组合（弗考慮順序）弗重复出现（弗擺回去）P k n=n!(n−k)!{\displaystyle P_{k}^{n}={\frac {n!}{(n-k)!}}}n!k⋅(n−k)!{\displaystyle {\frac {n!}{k\cdot (n-k)!}}}C k n=n!k!⋅(n−k)!{\displaystyle C_{k}^{n}={\frac {n!}{k!\cdot (n-k)!}}} 可重复出现（再擺回去）U k n=n k{\displaystyle U_{k}^{n}=n^{k}}∑r\|k(r⋅φ(r)⋅n k r)k{\displaystyle {\frac {\sum {r\|k}(r\cdot \varphi (r)\cdot n^{\frac {k}{r}})}{k}}}H k n=(n+k−1)!k!⋅(n−1)!{\displaystyle H{k}^{n}={\frac {(n+k-1)!}{k!\cdot (n-1)!}}} 参见 [编辑] 排列组合符号阶乘阶乘符号排列參攷文獻 [编辑] ↑組合數學 ─算法與分析─ ．九章出版社，29． OCLC:44527392 ↑組合數學 ─算法與分析─ ．九章出版社，33． OCLC:44527392 外部連結 [编辑] The Combinatorics Net档案，存勒互联网档案馆当中。（2021年5月3号） Electronic Journal of Combinatorics 点算个奥秘取自“ 分类：数学箇只页面阿末趟编辑来拉2024年3月7号 (四) 02:08。文字内容采用知识共享“署名-相同方式共享”许可协议授权；作兴会应用附加条款。详情见使用条款。隐私政策有关维基百科免责声明行为准则开发者统计 Cookie声明手机版视图编辑预览设置搜寻搜寻 [x] 开关目录组合数学 79種閒話加话题
12785	https://demmelearning.com/support-center/manipulatives-strategies-for-gamma/	Demme Learning Building Lifelong Learners Search Sort by Category Customer Service: M-Th 8:30am - 6pm ET Live Chat • 888-854-6284 • Email Support Center NEW! Get started with the updated Digital Toolbox here Find Errata and Print Corrections here Need additional help? Click here Math-U-See General Questions Levels Primer Alpha Beta Gamma Delta Epsilon Zeta Pre-Algebra Algebra 1: Principles of Secondary Mathematics Algebra 1: Legacy Geometry Algebra 2 PreCalculus Calculus Manipulatives Digital Access & Tools Spelling You See General Questions Levels Listen and Write Jack and Jill Wild Tales Americana American Spirit Modern Milestones Analytical Grammar General Questions Levels Level 1: Grammar Basics Level 2: Mechanics Level 3: Parts of Speech Level 4: Phrases and Clauses Level 5: Punctuation and Usage Beyond the Book Report The Eternal Argument Previous Editions Analytical Grammar Junior Analytical Grammar Junior Analytical Grammar Mechanics Accelerated Individualized Mastery General Questions Levels AIM for Addition and Subtraction AIM for Multiplication with a Bridge to Division WriteShop General Questions Downloadable Products WriteShop Primary WriteShop Junior WriteShop I & II Customer Service Schools Schools Digital Tools General Questions Administrator Account Teacher Accounts Student Accounts Digital Tools General Questions Levels Algebra 2 Stewardship Building Faith and Family Stewardship Math Group Learning Strategies for Gamma: Multi-Digit Multiplication with Manipulatives Lesson 21 introduces one of the more challenging concepts in multiplication, especially for students who are just beginning to grasp multi-digit multiplication. Instructors often encounter difficulty when students reach this point, requiring a deeper understanding of multiplication concepts and fluency with basic facts. To effectively guide students through this lesson, instructors must approach it with patience, a solid understanding of the teaching tools, and a readiness to revisit fundamental skills if necessary. Here are three essential strategies to keep in mind: 1. Leverage the Manipulatives The manipulatives are central to the Math-U-See approach; this level is no exception. These physical tools help students visualize the process of multiplication in a way that abstract numbers alone cannot. Further, using manipulatives reinforces the understanding of place value in multi-digit multiplication, ensuring that students can accurately break down and organize the numbers. This hands-on approach helps them internalize the concept, making it easier to retrieve and apply when solving complex problems later on. Using the Hundred Block As you move into equations with factors larger than 10, it is important to know how to use the hundred block. For example, when we build 12 x 13, we know we have an over factor of 12 and an up factor of 13. When you ask your student what they should start with, they may intuitively know to start with the hundred block, but don’t panic if you have to guide them through this scenario. You can say something like, “We know our 10 X 10 block, also known as our hundred block, is 10 over and 10 up” so your student understands that they need to use the 10 over and 10 up block to start our building process. Then, since that block is 10 over and 10 up and we need to get 12 over and 13 up, we can add 2 10’s vertically next to the hundred block, and we can place 3 10’s horizontally on top of the hundred block. We will still see a gap in the upper right-hand corner, so we must fill that in for a complete rectangle. We can do this with two 3 blocks.” Good news! These multiple-digit multiplication problems are simply bigger rectangles and involve multiple steps of solving smaller multiplication problems that should be familiar to the student once broken down step-by-step. These problems also become multi-step because they often include addition to solve and arrive at the complete solution. This is a familiar (but also new) skill set, so give your student plenty of modeling and practice to gain confidence. Build, Write, Say As an instructor, it’s vital to continue using the Build, Write, Say method, which reinforces concepts through multiple learning modes. Speaking the process out loud (the SAY of Build, Write, Say) allows the brain to properly catalog the information for later recall and helps expose errors in the multi-step process. Demonstrating multi-digit multiplication with the manipulative blocks helps students see how the numbers interact, making the process clearer and more intuitive. Skipping this step can lead to confusion, so it’s essential to integrate manipulatives into every part of the lesson. 2. Address Gaps in Multiplication Facts Mastery If students haven’t fully mastered their basic multiplication facts and still rely on skip-counting, they will likely struggle with the multi-digit multiplication introduced in Lesson 23. This is a critical point where you may need to pause the progression and invest time in strengthening their multiplication fact fluency. You can use our practice sheet generator in the Digital Toolbox for a refresh. If your student needs to develop a more complete mastery of the facts, our Accelerated Individualized Mastery (AIM) for Multiplication program may be right for your student. It’s an excellent intervention tool to help students catch up and become more confident in their skills. Taking time to ensure students have this foundation will set them up for success in this lesson and in future ones. 3. Stick with the Math-U-See Methodology It can be tempting to revert to the way you learned multiplication when your students begin to struggle. However, Math-U-See’s methodologies are designed with long-term understanding in mind, and stepping away from them can hinder progress. Take the time to revisit and understand the concepts yourself if needed, ensuring that you’re demonstrating and explaining the process in a way that aligns with Math-U-See’s structured approach. This helps students grasp the material and avoids confusion that can come from switching between teaching methods. Prior to this lesson, our focus was on the basic multiplication facts using the method of “over factor times up factor equals area.” We do it this way because it is the perfect way to visualize this concept as we build our rectangles. If you haven’t modeled building the multiplication tables with your student, we highly recommend this be done before moving forward. This process of Build, Write, Say takes them through the full process of connecting the abstract equation to concrete and tangible tools. The more senses we can engage in this learning process, the more successful the outcome and retention. Building is critical to their success. By remembering these three strategies, instructors can more easily navigate the challenges of more difficult Gamma lessons, helping students build a solid foundation in multiplication and prepare for more advanced math concepts. 4.9 out Of 5 Stars 4 ratings \| \| \| \| --- \| 5 Stars \| \| 75% \| \| 4 Stars \| \| 25% \| \| 3 Stars \| \| 0% \| \| 2 Stars \| \| 0% \| \| 1 Stars \| \| 0% \| How can we improve this article? Please submit the reason for your vote so that we can improve the article. Recent Articles Still need support? 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12786	https://www.freemathhelp.com/forum/threads/dna-nucleotides-permutations-and-combinations.55102/	New posts Search forums Menu Log in Register Install the app Forums Free Math Help Probability / Statistics You are using an out of date browser. It may not display this or other websites correctly.You should upgrade or use an alternative browser. DNA Nucleotides - Permutations and Combinations Thread starter karrate7785 Start date K karrate7785 New member Joined : Feb 9, 2008 Messages : 3 #1 I have been working on this problem for 2 days and keep coming back to square 1: I don't know what to do!The problem reads as follows:DNA is made of nucleotides and each nucleotide can contain any one of these nitrogenous bases: A,G,C,T. One of those four bases must be selected three times to form a linear triplet. How many different triplets are possible? Note that all four bases can be selected for each of the three components of the triplet.I just figured out this morning that this means we have the letters ACGT which can be arranged in either 35 or 256 ways (I'm not sure if order matters in this case). For instance, AAAT, AGTA, AAGG, GGGT, GGGA and so on. However, I got this far by plugging it into a formula online and I want to know how to do it by hand. Additionally, I'm not sure how to go about figuring how many of those possible arrangements are linear triplets?Any help would be GREATLY appreciated as this assignment is due today and this is the last problem I have left - the only one I can't figure out!Thank you. R royhaas Full Member Joined : Dec 14, 2005 Messages : 832 #2 For a triplet, you have 4 choices for each of the 3 positions. pka Elite Member Joined : Jan 29, 2005 Messages : 11,990 #3 Using the letters A, G, C, T there are four positions in any string and each position can be filled in four ways thus there are (\displaystyle {4}^{4}={256}) ways to have a string of four.I am not quite sure about what a linear triple actually is.Having said that, if AAAG & GAAA are linear triples and AAGA is not then there are (4)(3)(2) ways to pick them. Four ways to pick the letter to be tripled, three ways to pick the single letter and two ways to arrange the.But I still have two questions. Is AAAA a possible triple? What about AAGA? K karrate7785 New member Joined : Feb 9, 2008 Messages : 3 #4 Thanks for your replies! The answer was 4^3, though I'm still not sure why. Thanks again! pka Elite Member Joined : Jan 29, 2005 Messages : 11,990 #5 karrate7785 said: The answer was 4^3, though I'm still not sure why. Click to expand... No that is not the answer to the question as you framed it. What is the correct wording of the question? It must be different from what you posted. Otherwise that is a wrong answer! K kasie-tutor Banned Joined : Jan 20, 2008 Messages : 36 #6 Isn't (\displaystyle 4^3) equal to the number of possible codons?All these excerpts from Wikipedia at "...the code defines a mapping between tri-nucleotide sequences called codons and amino acids; every triplet of nucleotides in a nucleic acid sequence specifies a single amino acid." "There are 4³ = 64 different codon combinations possible with a triplet codon of three nucleotides. In reality, all 64 codons of the standard genetic code are assigned for either amino acids or stop signals during translation." The article gives examples like this as codons: AAA, AGA, ACT, or any arrangement of any of the three nucleotide bases. I still don't know what a "linear triplet" is either. Is it a codon made of all one letter, like AAA? In that case, wouldn't the answer be 4? Though, it seems like a linear triplet is basically a codon. Then, you would have 4 ways to fill-in each blank of the triplet. You would just use the "counting rule". These problems that have to do with topics that not everyone is familiar with can really throw our minds through a mobius strip. Don't ask me a math question related to baseball or any other sport. ~Kasie K karrate7785 New member Joined : Feb 9, 2008 Messages : 3 #7 I worded it exactly as it is in my text and that is the answer that was given by the instructor after the assignment was graded.Like I said before, I don't understand the problem which is why I came searching for help. Honestly, I still don't understand the question, let alone how to arrive at the answer. You must log in or register to reply here. Share: Facebook Twitter Reddit Pinterest Tumblr WhatsApp Email Share Link Forums Free Math Help Probability / Statistics This site uses cookies to help personalise content, tailor your experience and to keep you logged in if you register.By continuing to use this site, you are consenting to our use of cookies. Accept Learn more…
12787	https://www.stylemanual.gov.au/grammar-punctuation-and-conventions/numbers-and-measurements/dates-and-time	Dates and time \| Style Manual Skip to main content Australian Government Style Manual Home Search Search Main Menu Close Home About Style Manual resources Accessibility Writing content Grammar, punctuation and conventions Content types Structuring content Referencing and attribution Style Manual Home Grammar, punctuation and conventions Numbers and measurements Dates and time Secondary navigation In this section About the Style Manual Style Manual resources Accessible and inclusive content Writing and designing content Grammar, punctuation and conventions Types of words Parts of sentences Punctuation Spelling Shortened words and phrases Numbers and measurements Choosing numerals or words Currency Dates and time Fractions and decimals Mathematical relationships Measurement and units Ordinal numbers Percentages Telephone numbers Italics Names and terms Titles, honours, forms of address Content types Structuring content Referencing and attribution Dates and time Dates and expressions of time need to be readable and clear, particularly in content that contains detailed timelines. Write, abbreviate and punctuate dates and times consistently so people can understand your content. Follow international and Australian standards to write dates and times for data systems and international communication. Guidance Follow Australian conventions for dates Combine numerals and words for dates in body text Use shortened forms for dates when space is limited Don’t write dates as numerals unless space is limited Don’t use an apostrophe for decades Use words for spans of years in body text Use en dashes for particular types of year spans Use words for spans of days and months in body text Use en dashes for spans of days and months when space is limited Refer to specific days, events and periods with capitals Use numerals for times of day Use en dashes for spans of time when space is limited Follow the manual’s number rules for duration Use shortened forms for units of time when space is limited Avoid using ‘bi’to mean either 2 or twice How to combine dates and times Meet standards for data systems and information interchange Insert a non-breaking space correctly Release notes About this page Help us improve the Style Manual Follow Australian conventions for dates There are Australian conventions for writing dates in words and numerals, and in numeric formats. These conventions include how to sequence elements of the date. Use numerals and words for dates in most content. Use numeric dates when space is limited and in content types like tables. Combine numerals and words for dates in body text In Australia, the conventional sequence for dates is ‘day month year’. Use this sequence when expressing dates in numerals and words. For dates in body text, use numerals for the day and year and spell out the name of the month. Don’t include a comma or any other punctuation. Spell out the name of the day if it is being used, but don’t include a comma after the day. The names of months and days start with an initial capital because they are proper nouns. Write this 31 December 2020 Thursday 31 December 2020 Not this December 31, 2020 Thursday, 31 December 2020 Insert a non-breaking space between the day and the month so they stay together on one line. A non-breaking space means that a line break will split the date before the year. Keeping the day and month together allows people to identify the information appearing before the line break as a date. Write this Please find attached the new agenda for the extraordinary general meeting, 2 pm 8 November 2022. [With a non-breaking space between ‘8’ and ‘November’] Not this Please find attached the draft minutes of the extraordinary general meeting held in Sydney, 8 November 2022. [With no non-breaking space between ‘8’ and ‘November’] Don’t use an ordinal number (12th, 21st etc.) when writing dates in body text. Write this 1 May 1997 Schedule 3 commences on the first 1 July after the bill receives Royal Assent. [Extract from an explanatory memorandum] Not this 1 st May 1997 Schedule 3 commences on the first 1 st July after the bill receives Royal Assent. Incomplete dates Follow the general rules above when writing incomplete dates. Spell out the month in words if you need to leave out either the day or the year. Example The winning yacht usually reaches Hobart on 27 December. More than 1,700 jobs have been created since January 2018. There is an exception to the general rule for writing dates in body text. If you refer to the day but not the month, use an ordinal number. Don’t put the ordinal suffix (‘st’, ‘nd’, ‘rd’ or ‘th’) in superscript. Superscript can cause problems for people who use screen readers. Write this She will leave by the 20 th. Not this She will leave by the 20 th. If you refer to the year only, use the full numerical year. Don’t abbreviate it. Write this 1945 Not this ‘45 45 Use shortened forms for dates when space is limited Only use abbreviations if space is limited – for example, in tables, illustrations, charts and notes. Ensure that it is obvious to users which days of the week or months you are referring to. The standard abbreviations for the days of the week are: Monday – ‘Mon’ or ‘M’ Tuesday – ‘Tues’ (‘Tue’) or ‘Tu’ Wednesday – ‘Wed’ or ‘W’ Thursday – 'Thurs’ (‘Thur’,‘Thu’) or ‘Th’ Friday – ‘Fri’ or ‘F’ Saturday – ‘Sat’ or ‘Sa’ Sunday – ‘Sun’ or ‘Su’. The abbreviations in parentheses are alternatives for the standard abbreviation they follow. Only use the alternatives when the context ensures their meaning is clear. Note: Style Manual lists Monday as the first day of the week. This is consistent with the order of calendar days in a calendar week as defined in the international standard adopted by Australia. The standard abbreviations for the months are: January – ‘Jan’ February – ‘Feb’ March – ‘Mar’ April – ‘Apr’ May – retain as ‘May’ June – retain as ‘June’ or shorten to ‘Jun’ July – retain as ‘July’ or shorten to ‘Jul’ August – ‘Aug’ September – ‘Sept’ or shorten to ‘Sep’ October – ‘Oct’ November – ‘Nov’ December – ‘Dec’. Only use the shortest form of days and months – ‘F’, ‘M’, ‘N’ and so on –in limited applications. An example is a time-series chart where the context and order allow users to understand the difference between each capital letter. Don’t use a full stop after shortened days and months. No full stop is the correct Australian Government style for all abbreviations, acronyms and initialisms and contractions. Don’t abbreviate dates in body text Avoid abbreviated words when writing dates in body text. Words written in full are usually easier to read and understand. Write this The Labor Party called an urgent conference on Saturday 22 December. Not this The Labor Party called an urgent conference on Sat 22 Dec. Don’t write dates as numerals unless space is limited Avoid writing dates entirely in numerals for general content. Use numeric dates only when space is limited (such as in tables). Numeric dates can be confusing because their order and format differs between countries. Use Australia’s conventional order of ‘day month year’ unless you are writing for users in a country that uses a different style. Use a forward slash in numeric dates Separate the numbers in a numeric date with an unspaced forward slash, using the format ‘day/month/year’. This format uses single digits for single-digit days and months. Write this 4/6/2021 [Australia: d/m/yyyy] 7/12/2020 Not this 12/7/2020 [USA: m/d/yyyy] 2021-06-04 [Sweden: yyyy-mm-dd] Numeric dates can have 2-digit elements You can also use 2 digits for each element. Only use this style for the year: in financial data if it is clear which century you are referring to if users understand the order of the elements (‘day month year’ for Australian users). Example 07/12/20 30/06/22 Whichever style you use for date formats, use it consistently. Full stops in computer applications Many computer systems and applications use a full stop in numeric dates. Use 2 digits for the day and month and 4 digits for the year: ‘dd.mm.yyyy’. Example 07.12.2020 10.09.2021 Don’t use an apostrophe for decades Write decades with an ‘s’ on the end. Don’t use an apostrophe to show the plural. Correct 2010s 1980s Incorrect 2010’s 1980’s In more casual writing, you can use expressions such as‘the eighties’. You can also use an apostrophe to show the missing numerals in a decade – for example, ‘In the ’80s, all my jackets had shoulder pads.’ Use words for spans of years in body text As a general rule, write spans of years in words, using ‘to’, ‘from … to’ or ‘between … and’. Write the years out in full, not as abbreviations. Don’t replace the word between the years with an en dash. Write this the years 2015 to 2019 from 2015 to 2019 between 2015 and 2019 Not this the years 2015–2019 from 2015–2019 between 2015–2019 Use en dashes for particular types of year spans Government content often includes spans of years. Some year spans are easier to read and understand if they contain an unspaced en dash rather than words. This is particularly true in content that contains multiple spans of years. In this case, using en dashes makes the content easier to scan. Use an unspaced en dash for a: financial year calendar year span of years in the titles of publications and programs span of years written in parentheses, such as for a term of office and the years of birth and death. Always include the phrases ‘financial year’, ‘financial years’, ‘calendar year’ or ‘calendar years’ unless the context makes the meaning clear. You can also introduce the relevant phrase at first mention and just write the year span, without the phrase, in later mentions. Finally, exercise your judgement. Consider using en dashes for year spans when using words makes the content harder to read. Example This document includes expenditure and revenue estimates for the 2021–22 financial year. For comparison, the attachment contains estimates for 2020–21 and 2019–20. The agency measures injury hospitalisations and deaths over 2 calendar years. Data showed a small increase in injury hospitalisations for 2017–2018 and 2019–2020. Injury deaths declined over the earlier period, but showed a marked increase for 2019–2020. The library holds a reference copy of the Inclusion and diversity strategy 2022–24. Alfred Deakin was Prime Minister for much of Australia’s 2nd Parliament (1903–1906). Sidney Nolan (1917–1992) had 3 younger siblings. Always use unspaced en dashes in spans of years. Don’t use forward slashes. Write this National Road Safety Action Plan 2018–20 Australia’s energy consumption rose by 0.6% in 2018–19 and fell by 2.9% in 2019–20. Not this National Road Safety Action Plan 2018/20 Australia’s energy consumption rose by 0.6% in 2018/19 and fell by 2.9% in 2019/20. Use words for spans of days and months in body text As a general rule, use ‘from … to’ and ‘between … and’ in spans of days and months. If it’s appropriate for your content, keep elements of the span together by inserting non-breaking spaces between them. When you include a year, insert a non-breaking space between the day and month. A non-breaking space keeps the day and month together while the line break splits the date before the year. Example Parliament is scheduled to sit from 3 to 21 December. [Non-breaking spaces between ‘3’, ‘to’, ‘21’ and ‘December’] We will do snap inspections between 6 and 8 September. [Non-breaking spaces between ‘6’, ‘and’, ‘8’ and ‘September’] The exhibition will run from 30 November to 23 February 2022. [Non-breaking space between ‘30’ and ‘November’, and between ‘23’ and ‘February’] Use en dashes for spans of days and months when space is limited Only use en dashes for spans of days and months if you have limited space. This could be in display text, tables, lists or in social media posts. But also exercise your judgement – consider using en dashes if using words makes the content harder to read. The en dash is spaced when the day and month appear on both sides of the span. Example The en dash is unspaced when the month only appears at the end of the span. Example Symposium Series 2025 Plain language: 3–5 March Accessible tables: 9–11 June Content design: 8–10 September Refer to specific days, events and periods with capitals Treat specific days, public events and periods in history as proper nouns and use initial capitals. Use lower case for ‘the’ and any prepositions, unless they are capitalised as part of a proprietary name. In body text, use lower case for generic terms like the names of seasons –‘autumn’ – and astronomical events such as ‘equinox’ and ‘solstice’. Write ‘century’ and ‘centuries’ in lower case. Holidays and events Use initial capitals for all institutional holidays, religious days and public events. Follow the capitalisation of proprietary names. Example New Year’s Day Good Friday Ramadan Yom Kippur the City2Surf the AFL Grand Final Labour Day the Adelaide Festival Party In The Paddock 2024 Anzac Day [‘Anzac’ stands for Australian and New Zealand Army Corps, but only takes an initial capital. The acronym appears in legislation with an initial capital and this has become the commonly-used form.] Periods and events of historical importance Use initial capitals for specific periods and events of historical importance, but not when you abbreviate them to a generic term. Example the Renaissance the Bronze Age the French Revolution, the revolution the Battle of Long Tan, the battle World wars There are 2 acceptable styles for referring to the world wars. Use either style, but be consistent in your content. Example First World War, Second World War World War I, World War II [With roman numerals] Use 2 non-breaking spaces to keep each name together on one line of text. Write ‘World War I’ and ‘World War II’ with roman numerals. The accepted way to write most roman numerals is by typing letters of the alphabet. Use a capital ‘i’ – I – for the world wars. The Style Manual acknowledges that the house styles of some government agencies require the use of the arabic numerals ‘1’ and ‘2’. User research will always guide an agency’s style choices. Our recommendation to use roman numerals comes from corpus evidence. Shortened forms for world wars In content with many mentions of the world wars, you might decide to use a shortened form. Write the name of the war out in full the first time you use it. Include the shortened form in parentheses immediately afterwards and use the shortened form from then on. Use roman numerals in the shortened form. Example The Second World War (WWII) started in 1939. Many Australians died in WWII. World War I (WWI) started in 1914. Many Australians enlisted for service in WWI. Eras and periods Use initial capitals for the actual name of a geological era or period but not for broad historical and cultural times. Example the Lower Jurassic period the Mesozoic era the colonial era baroque ornamentation Centuries Use numerals, not words, for centuries. This is an exception to the rule for ordinal numbers. Write ‘century’ and ‘centuries’ in lower case. Don’t use shortened forms, such as 18C or C18, unless you have limited space. Don’t use superscript for the ordinal suffix. Write this the 18th century in the 2nd and 3rd centuries a 19th-century writer an 8th-century monastery Not this the eighteenth century in 2C and 3C a nineteenth-century writer an eighth-century monastery an 8 th-century monastery Use CE and BCE to represent the common era (CE) and the time before the common era (BCE). There is no ‘year 0’ in this system. The years progress from 1 BCE to 1 CE. Write ‘CE’ and ‘BCE’ without full stops and with a non-breaking space separating them from the year or century. Example 44 BCE 1452 CE the 3rd century BCE the 3rd century CE Seasons and seasonal events Use lower case for the seasons and recurrent seasonal events. Example winter summer solstice Use numerals for times of day In most documents, numerals give a clearer expression of time. Write times of day using numerals, especially when you need to convey precise times. Use a colon between the hours and minutes. The use of a colon as the separator, rather than a full stop, reflects a shift in contemporary Australian usage. A colon ensures that the time isn’t confused with a decimal number. For example, ‘10.50’ can be read as ‘10 and a half’ as well as ‘50 minutes past 10’. Screen reader users will probably hear ‘10.50’ as ‘10 point 5’. Example The bus leaves at 8:22 am. The broadcast will run from 9:45 am to 11:45 am. Times using ‘am’ and ‘pm’ The initialisms ‘am’ and ‘pm’ come from the Latin phrases ante meridiem (before noon) and post meridiem (after noon). Write ‘am’ and ‘pm’ in lower case. Separate the numbers and the initialism with a non-breaking space. Don’t use ‘am’ and ‘pm’ with words that duplicate their meaning, for example ‘morning’ and ‘afternoon’. Correct Please respond by 10 am tomorrow. Let’s meet at 6:30 pm. Incorrect Please respond by 10 am tomorrow morning. Let’s meet at 6:30 pm in the evening. You can use 2 zeros to show the full hour, but they aren’t essential. Use 2 zeros if that is consistent with other expressions of time in your content, such as in running sheets. Example 9 pm 9:00 pm Noon, midday and midnight Use ‘noon’, ‘midday’ or ‘midnight’ instead of ‘12 am’ or ‘12 pm’. This makes it easier for people to understand the time. Write this We have extended the closing date to midnight Friday 7 October 2022. Not this We have extended the closing date to 12 am Friday 7 October 2022. Write ‘o’clock’ only when you quote someone directly or transcribe a recording. In these situations, use numerals and the word ‘o’clock’. Example ‘The minister is speaking at about 10 o’clock,’ they said. The 24-hour clock Use the 24-hour clock if it helps people understand your content. This is important when referring to times in these contexts: travel certain scientific fields the armed services content written for countries using the 24-hour clock. The 24-hour clock is also useful in content where space is limited. This is because it uses fewer characters than times with ‘am’ and ‘pm’. For this reason, it is often used in timetables and schedules. This system numbers the hours from 00:00 hours (midnight) to 23:59. It always uses at least 4 digits. It can have 6 digits if seconds are included: The first 2 digits are the hours. The next 2 digits are the minutes. The last 2 digits are the seconds, if you include them. Always use a ‘leading zero’ for hours under 10 – for example, write ‘05:30’ not ‘5:30’. Use a colon to separate the hours, minutes and seconds in the 24-hour clock. Example 00:45 [12:45 am] 07:38 [7:38 am] 23:18 [11:18 pm] 23:59:17 [11:59:17 pm–includes hours, minutes and seconds] Don’t add ‘am’ or ‘pm’ to times written in 24-hour clock format. Write this 06:45 23:18 Not this 06:45 am 23:18 pm Some government agencies that produce technical and scientific content don’t use a colon for the 24-hour clock – for example, 2300 and 0430. This is the ‘basic format’ used for international communication. If space is limited and you use the 24-hour clock in general content, we recommend inserting a colon for clarity. This is the ‘extended format’ used for international communication. Time zones You might need to define which time zone you are referring to. Time zones are usually written with the 24-hour clock. The main Australian zones are: ACST (Australian Central Standard Time) ACDT (Australian Central Daylight saving Time) AEST (Australian Eastern Standard Time) AEDT (Australian Eastern Daylight saving Time) AWST (Australian Western Standard Time) LHST (Lord Howe Standard Time) LHDT (Lord Howe Daylight Time). Daylight saving time is not observed in the Northern Territory, Queensland and Western Australia. There are also time zones for some Australian external territories. Example The meeting will commence at 15:30 AEDT on 17 November 2022. Coordinated Universal Time Coordinated Universal Time (UTC) is a time standard used as the basis for regulating world timekeeping. UTC expresses the unadjusted local time at 0° longitude. It is not adjusted for daylight saving. Local standard time at longitudes around the world is represented by an offset to UTC. UTC is based on International Atomic Time (TAI), which is a weighted average of atomic clocks located around the world, including in Australia. TAI does not take into account changes in the earth’s rotation, so leap seconds are occasionally added to UTC. UTC is the standard and legal reference for times of day in Australia. The UTC(AUS) standard is maintained by the Australian Government’s National Measurement Institute. Time zones are written as positive or negative offsets to UTC. Write the initialism ‘UTC’, followed by ‘+’ or ‘−’, followed by the time offset to UTC in 24-hour system format. Example During winter, the time in Sydney is UTC+10:00. São Paulo is in the Brasília Time Zone which is UTC−03:00. Use en dashes for spans of time when space is limited Only use en dashes for time spans if you have limited space. This could be in display text, tables, lists or in social media posts. But also exercise your judgement – consider using en dashes if using words makes the content harder to read. The spacing of the en dash depends on the elements of the span. Use an unspaced en dash: if the ‘am’ or ‘pm’ appears only at the end of the span for spans of time in the 24-hour clock format. Use a spaced en dash: when ‘am’ or ‘pm’ appears on both sides of the span if ‘noon’, ‘midday’ or ‘midnight’ appears in the span. Example Soccer training this Sat: 8–9 am Available appointment times are: 08:00–08:15 13:30–13:45 16:45–17:00 Help desk opening hours: Monday to Thursday: 7 am–4 pm Friday: 9 am–midday Don’t combine words and the en dash. Write this Closed 11 am–2 pm. Not this Closed between 11 am–2 pm. Follow the manual’s number rules for duration When expressing duration (lengths of time) in body text, follow Style Manual rules about choosing words or numerals. This means writing the words ‘zero’ and ‘one’ and using numerals for ‘2’ and above. The rules also say to use the numerals ‘0’ and ‘1’ in specific situations. For duration, the specific situations are likely to be when: you compare numbers in a sentence a sentence contains a series of numbers. Write the words ‘zero’ and ‘one’ in sentences that don’t contain other numerals. Write the numerals ‘0’ and ‘1’ in sentences that contain numerals from 2 and above, or where all numbers show duration. Spell out the units of time: ‘hours’, ‘minutes’ and ‘seconds’. Example There are 7 minutes and 30 seconds remaining. [Numerals for 2 and above] The hearing adjourned for 2 hours. [Numerals for 2 and above] The committee will break for lunch for one hour. [Words for zero and one] Although we allowed candidates 1 minute to answer each question, most took over 2 minutes. [Numerals when comparing duration] I noted faults at these time stamps: 0 minutes 45 seconds, 1 minute 4 seconds and 3 minutes 8 seconds. [Numerals for a series of numbers showing duration] Use fractions written in words if users only need a general idea of values. Only use decimal numbers if they are the best way to explain what people need to know. But be aware that some people might not understand the decimal’s time value. For example, 1.25 hours is 1 hour 15 minutes, not 1 hour 25 minutes. It is usually better to avoid decimals and include the number and unit of time. Example The session finished about a quarter of an hour early. [Words for a fraction – gives a general idea of the duration] I clocked her at 15 minutes and 12 seconds. [Easier to understand than 15.2 minutes] They broke the record by 0.04 seconds. [Numeral – a decimal gives people the information that is appropriate in this context. But the words ‘… by 4 hundredths of a second’ might be clearer to some users.] Use shortened forms for units of time when space is limited It is usually better to spell out the units that measure time. This is particularly so in general content. Only use short forms if space is limited and the short forms are easy to identify correctly. If you need to abbreviate time, use the following units as shortened forms: second – s minute – min hour – h day – d week – wk month – mo year – y or yr (Choose one and use it consistently in your document.) The International System of Units – SI – is the international standard for measurement. The unit for second – symbol ‘s’ – is the SI base unit for time. Other time measures are not SI units – but ‘min’, ‘h’ and ‘d’ are used with ‘s’ and recognised as legal units of measurement in Australia. They are listed in Schedules 1 and 2 of the National Measurement Regulations. There are also commonly-used shortened forms for time measures – for example, ‘wk’ (week), ‘mo’ (month) and ‘yr’ or ‘y’ (year). These are not legal units of time, but are likely to be understood when used alongside other time units. If your expression contains one time measure only, insert a non-breaking space between the number and unit. Never add an ‘s’ to show a plural. Write this 35 s 15 min 6 h 5 d 3 wk 8 mo 2 y or 2 yr [There are non-breaking spaces between numbers and units. There is no ‘s’ to show plurals.] Not this 35s 6h s 15min s 5d 3wk s 8mo 2y or 2yr [There are no non-breaking spaces between numbers and units. There is an ‘s’ after ‘min’, ‘hr’ and ‘wk’.] But don’t space a number and its unit when your expression contains more than one time measure. Use a non-breaking space between each time value instead. Never add an ‘s’ to show a plural. Write this 11min 12s 7h 8min 30s [There is no space between a number and its unit. There is a non-breaking space between each time measure in the expression.] Not this 11min12s [There is no (non-breaking) space between ‘11min’ and ‘12s’.] 11 min 12 s [There is a non-breaking space between ‘11’ and ‘min’, and between ‘12’ and ‘s’.] 7h s 8min s 30s [There is an ‘s’ after ‘7h’ and ‘8min’.] Data systems support specific (and usually several) shortened forms for units of time. For example, hours might be: h, hh, hr, hours, hrs. They might also be case-sensitive or case-insensitive. You will need to check system specifications. Avoid using ‘bi’to mean either 2 or twice The prefix ‘bi’ can be confusing when used with expressions of time: ‘Bimonthly’ can mean either every 2 months or twice a month. ‘Biannual’ means twice a year. ‘Biennial’ means every 2 years. Instead of using these words, be clear about the frequency and period of time you mean. Write this We meet once every 2 years. Not this We meet biennially. How to combine dates and times There is no fixed rule about the order of dates and times when combining them in body text. You can choose whether the date or the time should come first. The order doesn’t matter as long as the information is clear and the sentence flows logically. But make sure that: you follow the style rules for each element the time doesn’t come between the day and the date you use the same style consistently throughout your document. Write this They will appear before the committee at 3 pm on Wednesday 7 August 2024. They will appear before the committee on Wednesday 7 August 2024 at 3 pm. [There is a non-breaking space between ‘3’ and ‘pm’, no comma after ‘Wednesday’ and a non-breaking space between ‘7’ and ‘August’.] Not this They will appear before the committee at Wednesday 3 pm on 7 August. [The time appears between day and date.] They will appear before the committee on Wednesday, 7 August at 3 pm. [There is a comma between day and date.] Combining dates and times when space is limited Use the same approach to combine the date and time when you have limited space. This might be in a table, social media post, or in a display or presentation context (display text). We always recommend using minimal punctuation, but exercise your judgement. A comma between the date and time can make information easier to scan if you haven't used a preposition like ‘at’ or ‘on’. You can also use shortened forms for the date. Only do this if you are sure users will understand what you mean. Example Lunch and Learn: Financial security in the 1980s Friday 1 March at midday Level 3 seminar room Lunch and Learn: Financial security in the 1980s Midday, Friday 1 March Level 3 seminar room Content Meetup Tue 23 Apr at 4 pm All welcome! Meet standards for data systems and information interchange Follow international standards when writing dates and time: for international communication to transfer data between systems. The International Organization for Standardization (ISO) develops and publishes the international standard for dates and time format – the ISO 8601 series. Australia and New Zealand have adopted the international standard. The Australia – New Zealand standard is Date and time: representations for information interchange. It is published as the AS/NZS ISO 8601 series in 2 parts: Part 1: basic rules Part 2: extensions You can purchase copies of ISO 8601 and AS/NZS ISO 8601 from Standards Australia. Standards are voluntary International and Australian – New Zealand standards on dates and time are voluntary. The standards ensure that data systems and humans can exchange date and time information internationally and across time zones in a recognised format. A description of all the dates and times standards is beyond the scope of the Style Manual. We cover these standards from Part 1: basic rules: calendar date ordinal date local time of day combined calendar date and local time of day. We don’t cover Part 2: extensions. Basic and extended formats There are 2 format options for each standard: ‘basic format’ and ‘extended format’. Basic format does not have separators between units of dates and time. Avoid using basic format in body text. It is easy for computers to read but harder for humans. Extended format is also for computers, but it includes separators between units to make it easier for humans to read. We show both formats in our examples. Calendar date The standard sets out a descending order of year (4 digits), month (2 digits) and day (2 digits) for a complete representation of calendar date. The numbers can be unspaced (basic format) or separated by a hyphen (extended format). Example 20201207 [Calendar date, basic format] 2020-12-07 [Calendar date, extended format] ISO format is becoming more common, especially in software. Ordinal date Ordinal dates are often used when transferring data between data systems. This is because they are easy for simple systems to read. Ordinal dates are not the same as ordinal numbers. Ordinal dates have 7 digits: The first 4 digits represent the year. The next 3 digits represent the day. This is the ISO standard. Some older systems might use 2 digits for the year, not 4. Days are numbered from 1 to 365 (366 for a leap year). For example, 20 September is the 263rd day of the year (264th in a leap year) and its 3 digits are 263. The ordinal date can be unspaced (basic format) or separated by a hyphen (extended format). If you use dates in electronic data transfer, use basic format and don’t separate the numbers. If people, rather than a computer, will read the information, use extended format and insert a hyphen between the year and day. Example Write 7 January 2023 as: 2023007 [The 4-digit form for the year, basic format] 23007 [The 2-digit form for the year, basic format]. Write 31 July 2020 as: 2020-213 [The 4-digit form for the (leap) year, extended format] 20-213 [The 2-digit form for the (leap) year, extended format]. Local time of day The standard for a complete representation of local time of day starts with the letter ‘T’ followed by 2-digit numbers for the hour, minute and second. The numbers can be unspaced (basic format) or separated by a colon (extended format). You can omit the ‘T’ if there’s no possibility of confusion. A ‘reduced precision’ time of 15 minutes after 7 pm in basic format – ‘1915’ – can easily be confused with the calendar year 1915. In these cases, it is better to write time of day as T1915 (basic format) or 19:15 (extended format). Note: The format for local time of day doesn’t allow for daylight saving. The format showing daylight saving includes the time shift between local time and Coordinated Universal Time. This is beyond the scope of the Style Manual. Example 153020 [Local time of day, basic format – 15h 30m 20s, that is 20 seconds after 3:30 pm] 15:30:20 [Local time of day, extended format] Combined date and local time of day Combine date and local time of day using the style standards set out above. Always use the letter ‘T’ between the date and time. The basic format for international communication is unspaced. The extended format uses hyphens for the date and colons for the time of day. Example 20230811T121505 [Basic format – 11 August 2023 at 15 minutes and 5 seconds after midday.] 2023-08-11T12:15:05 [Extended format] 2023223T121505 [Ordinal date, basic format – same date and time as above] 2023-223T12:15:05 [Ordinal date, extended format] Insert a non-breaking space correctly You can insert a non-breaking space using the Unicode character U+00A0. In HTML, use the entity to insert a non-breaking space. The keyboard shortcut in Word is Ctrl+Shift+Spacebar. Release notes The digital edition adds new content including rules and examples for: date spans time spans and lengths shortened forms Coordinated Universal Time combined dates and times. There is expanded guidance on standards for data systems and information interchange. The digital edition changes the punctuation used with expressions of dates and time. There are no full stops for shortened forms of months and days of the week. This is consistent with the new general rule for abbreviations. Unlike the sixth edition, but consistent with the Content Guide, the digital edition recommends using a colon rather than a full stop when expressing times. The Latin shortened forms, ‘am’ and ‘pm’ do not have punctuation. This is consistent with the sixth edition and the Content Guide. Consistent with the Content Guide, the digital edition recommends specifying noon or midnight for 12 o’clock (instead of using ‘am’ or ‘pm’). The recommendation to express a date span using a phrase, rather than an en dash, aligns with the Content Guide. Like the sixth edition, the digital edition recommends the use of non-breaking spaces between the day and month in dates. It differs from the sixth edition by recommending the use of a numeral with centuries. The digital edition recommends 2 acceptable styles for references to the world wars, while the sixth edition does not have an explicit rule. The Content Guide illustrated guidance, but did not have explicit advice, on the use of spacing for dates and time. The use of a space between time and ‘am’ or ‘pm’ is consistent with the sixth edition, but a departure from examples given in the Content Guide. About this page Evidence Australian Broadcasting Corporation (2024) ‘Wars’, The ABC style guide, ABC website, accessed 14 March 2024. Biotext Pty Ltd (2024) ‘Date and time systems’, Australian manual of style, AMOS website, accessed 4 January 2024. Bureau International des Poids et Mesures (BIPM) (n.d.) SI base unit: second (s), BIPM website, accessed 4 October 2023. Butterfield J (ed) (2015) Fowler’s dictionary of modern English usage, 4th edn, Oxford University Press, Oxford. Joint Standards Australia/Standards New Zealand Committee IT-019 (2023) Date and time: representations for information exchange. Part 1: Basic rules, AS/NZS ISO 8601.1:2021 (at February 2023), Standards Australia Limited/Standards New Zealand. National Measurement Regulations 1999 (Cth), Schedule 1. Oxford University Press (2016) ‘5.7: Events’, New Oxford style manual, Oxford. Oxford University Press (2016) ‘11.3: Times of day’, New Oxford style manual, Oxford Peters P (2007) ‘World war’, The Cambridge Australian English style guide, Cambridge University Press, Cambridge. Treasury Board of Canada Secretariat (2024) ‘Times’, Canada.ca content style guide, Canada.ca, accessed 14 March 2024. University of Chicago (2017) ‘8.113: Wars and revolutions’, Chicago manual of style, 17th edn, University of Chicago Press, Chicago. University of Chicago (2017) ‘9.37: Numerals versus words for time of day’, Chicago manual of style, 17th edn, University of Chicago Press, Chicago. University of Chicago (2017) ’10.39: Abbreviations for months’,Chicago manual of style, 17th edn, University of Chicago Press, Chicago. University of Chicago (2017) ’10.40: Abbreviations for days of the week’, Chicago manual of style, 17th edn, University of Chicago Press, Chicago. US Government Publishing Office (2017) ‘12.9.b. Clock time’, Government Publishing Office style manual, US Government Publishing Office website, accessed 30 March 2022. References Australian Broadcasting Corporation (2024) ‘Dates and times’, The ABC style guide, ABC website, accessed 7 March 2024. Australian Institute of Health and Welfare (2023) Injury in Australia, AIHW website, accessed 15 August 2023. Beard R (2013) ‘The past and future of Coordinated Universal Time [PDF 3.38 MB]’, ITU News, 7:9–12, accessed 10 October 2023. Bikos K and Buckle A (2024) What Is International Atomic Time (TAI)?, Time and Date AS website, accessed 14 March 2024. British Broadcasting Corporation (2024) ‘World war’, News style guide, BBC website, accessed 14 March 2024. Btb Translation Bureau (2022) ‘Numerical expressions’, The Canadian style, Btb Translation Bureau website, accessed 24 February 2023. Bureau International des Poids et Mesures (BIPM) (n.d.) Annual reports, BIPM website, accessed 13 September 2023. Bureau of Meteorology (2021)Daylight Saving Time and weather observations, Bureau of Meteorology website, accessed 13 September 2023. Bureau of Meteorology (2014) Time conventions, Bureau of Meteorology website, accessed 7 April 2022. Content Design London (2020) ‘Grammar points: numbers’, Content Design London readability guidelines, Content Design London website, accessed 24 February 2022. Department of Climate Change, Energy, the Environment and Water (DCCEEW) (2022) ‘Australian energy update 2022’, Australian Energy Statistics, DCCEEW, Australian Government, accessed 2 August 2023. GOV.UK (2024) ‘A to Z: dates’, Style guide, GOV.UK, accessed 14 March 2024. GOV.UK (2024) ‘A to Z: times’, Style guide, GOV.UK, accessed 14 March 2024. GOV.UK (2024) ‘A to Z: World War 1, World War 2’, Style guide, GOV.UK, accessed 14 March 2024. International Organization for Standardization (ISO) (n.d.) ISO 8601: date and time format, ISO website, accessed 9 August 2023. International Telecommunication Union (2024) ITU radiocommunication sector, ITU website, accessed 14 March 2024. Lyons G (2019) Which day do you consider the start of the week?, ABC News website, accessed 3 January 2024. National Institute of Standards and Technology (2023) NIST time frequently asked questions (FAQ), NIST website, accessed 14 September 2023. National Measurement Institute (n.d.) Time and frequency services, Department of Industry, Science, Energy and Resources website, accessed 30 March 2022. New Zealand Government (2020) ‘Numbers’, Content design guidance, Digital.govt.nz, accessed 30 March 2022. Oxford University Press (2016) ‘10.2.6: Upper- and lower-case abbreviations’, New Oxford style manual, Oxford University Press, Oxford. Oxford University Press (2016) ‘11.5: Date forms’, New Oxford style manual, Oxford University Press, Oxford. Parliamentary Library (2017) ‘Anzac Day traditions and rituals: a quick guide’, Research Papers 2016–2017, Parliament of Australia website, accessed 20 July 2022. Tavella P (31 July 2023) ‘Coordinated Universal Time: an overview’, ITU News Magazine, ITU website, accessed 10 October 2023. University of Chicago (2017) ‘Dates’, Chicago manual of style, 17th edn, University of Chicago Press, Chicago. University of Chicago (2017) ‘Time of day’, Chicago manual of style, 17th edn, University of Chicago Press, Chicago. Last updated This page was updated Monday 18 August 2025. Help us improve the Style Manual Did you find this page useful? Yes No Partially Do you have any other feedback? 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12788	https://www.shmoop.com/math-shack/pre-algebra/calculating-percentages-from-bar-graphs/	Calculating Percentages from Bar Graphs - Math Shack Skip to navigation Skip to content © 2025 Shmoop University, Inc. All rights reserved. Register\|Login Test Prep Learning Guides Video For Teachers Courses All of Shmoop Test Prep Learning Guides Video For Teachers Courses All of Shmoop Home/ Math Shack Math Shack < Pick a Different Problem I'm Feeling Lucky View Your Progress 0 Calculating Percentages from Bar Graphs Pre-algebra Answer questions correctly to move the progress bar forward. Once the progress bar is complete, you've mastered the topic. Your browser does not support HTML5 Canvas. To get the most out of Math Shack you will need to upgrade to a more modern browser. The graph below illustrates the number of cookies sold by 4 sellers: Pi, Lincoln, Tom, and Horatio. What percentage to the nearest whole number of the total cookies sold does Tom sell? Your browser does not support HTML5 Canvas. To practice this topic you will need to update your browser. % answer format Submit Give Me a Hint Calculator show buttons C inverse functions !%√^ log 10 ln EE e sin cos tan sin-1 cos-1 tan-1 π abs()÷ 7 8 9× 4 5 6- 1 2 3+ 0.= You Will Earn 16 Shmoints When You Master This Topic! You've already mastered this topic. You can't earn more Shmoints for the same topic, but feel free to keep practicing. × Having Trouble? Looks like you're stuck. Do you want to review the topic and come back to practice later? REVIEWCANCEL Email Got Math Shack feedback? Share it with us! × You came, you saw, you conquered. You completed {{MODULE_TITLE}} and have earned Login OR Register for free so you can strut your stuff. {{BADGE_TITLE}} × Good effort! You've already earned points for these correct answers. Try getting them all correct, or take another quiz. × You're so close! You're so close to scoring some shmoints! 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12789	https://books.google.com/books/about/Sampling_Techniques.html?id=62O4AAAAIAAJ	Sign in Try the new Google Books Check out the new look and enjoy easier access to your favorite features Try it now No thanks Try the new Google Books Try the new Google Books My library Help Advanced Book Search Get print book No eBook available Amazon.com Barnes&Noble.com Books-A-Million IndieBound Find in a library All sellers » ### Get Textbooks on Google Play Rent and save from the world's largest eBookstore. Read, highlight, and take notes, across web, tablet, and phone. Go to Google Play Now » \| \| \| Sampling Techniques William Gemmell Cochran Wiley, 1963 - Mathematics - 413 pages Clearly demonstrates a wide range of sampling methods now in use by governments, in business, market and operations research, social science, medicine, public health, agriculture, and accounting. Gives proofs of all the theoretical results used in modern sampling practice. New topics in this edition include the approximate methods developed for the problem of attaching standard errors or confidence limits to nonlinear estimates made from the results of surveys with complex plans. \| From inside the book Contents \| \| \| --- \| \| CHAPTER PAGE \| 1 \| \| \| \| SIMPLE RANDOM SAMPLING \| 18 \| \| \| \| \| \| --- \| \| SAMPLING FOR PROPORTIONS AND PERCENTAGES \| 49 \| \| \| \| Copyright \| \| \| 12 other sections not shown Other editions - View all \| \| \| Sampling TechniquesWilliam Gemmell CochranSnippet view - 1953 \| \| \| \| Sampling TechniquesWilliam Gemmell CochranSnippet view - 1953 \| \| \| \| Sampling TechniquesWilliam Gemmell CochranSnippet view - 1963 \| View all » Common terms and phrases a analysis of variance apply assumed average bias biased binomial cluster cluster sampling compute confidence limits correlation double sampling drawn equation estimated variance example farm finite population follows formula gain in precision given gives Hence households interviewer ith unit Jour linear M mean per element mean square mean square error method mopt n n negligible nonresponse number of units obtained optimum allocation p population mean population total primary units probability proportional proportional allocation ratio estimate regression estimate relative precision replacement S sample estimate sample mean sample survey sampling fraction sampling units selected self-weighting simple random sample sizes standard error Stat strata stratified random sampling stratified sampling stratum stratum h subsample systematic sample Table term theorem total number unbiased estimate V(yst variate W W x y z Î£Î References to this book \| \| \| Multiple Imputation for Nonresponse in SurveysDonald B. RubinLimited preview - 2004 \| \| \| \| Meta-Analysis in Social ResearchGene V. Glass,Gene V Glass,Barry McGaw,Mary Lee SmithSnippet view - 1981 \| All Book Search results » Bibliographic information \| \| \| --- \| \| Title \| Sampling TechniquesVolume 134 of WILEY SERIES in PROBABILITY and STATISTICS: APPLIED PROBABILITY and STATIST ICS SECTION SeriesWiley publication in applied statisticsWiley series in probability and mathematical statisticsWiley series in probability and mathematical statistics. Applied probability and statistics \| \| Author \| William Gemmell Cochran \| \| Edition \| 2 \| \| Publisher \| Wiley, 1963 \| \| Original from \| the University of California \| \| Digitized \| Aug 26, 2008 \| \| ISBN \| 0471162388, 9780471162384 \| \| Length \| 413 pages \| \| \| \| \| Export Citation \| BiBTeX EndNote RefMan \| About Google Books - Privacy Policy - Terms of Service - Information for Publishers - Report an issue - Help - Google Home
12790	https://www.caliper.com/glossary/what-is-a-map-projection.htm?srsltid=AfmBOooXzSXVYv2mrAOJI_1uMXXTMeDVUiRIkHxdBSzo6XrN_LcJNctm	What is a Map Projection - Map Projection Definition This website uses cookies to ensure you get the best experience on our website. Learn more Got it! Toggle navigation Desktop Maptitude Desktop Overview Features Examples API Map Your Data Data Included Latest Version Free Add-in Tools Brochure (PDF) Redistricting (Electoral, USA) Online Pricing Pricing Purchase/Store Learn Options Training Video Tutorials Free Webinars Tech Tips Learning Portal Glossary Blog Resources Testimonials Blog Case Studies Featured Maps Newsletters Articles News About Us Contact Us Solutions Banking Business Census Data Mapping Geocoding GIS Mapping Government Healthcare Marketing & Sales Microsoft Integration Online Mapping Political Mapping Real Estate Retail Resources & Utilities Territory Mapping Three Dimensional & GPS Mapping Transportation USA Data Australia Data Brazil Data Canada Data UK Data Europe Data Other Country Packages Deutsch English Español Français Italiano Portugêse العربية 中文 עברית Indonesia اردو Call Us 617-527-4700ReviewsFree TrialRequest Demo Mapping Software and GIS Glossary DEFINITION What is a Map Projection? A map projection is a method for taking the curved surface of the earth and displaying it on something flat, like a computer screen or a piece of paper. Map makers have devised methods for taking points on the curved surface of the earth and "projecting" them onto a flat surface. These methods enable map makers to control the distortion that results from creating a flat map of the round earth. Every map projection has some distortion. Equal area projections attempt to show regions that are the same size on the Earth the same size on the map but may distort the shape. Conformal projections favor the shape of features on the map but may distort the size. What is a map projection? The Equal Area Projection on the left correctly shows the relative sizes of regions on the map. For example, Alaska is about twice the size of Texas. The Conformal Projection on the right correctly shows the shapes of areas such as Alaska and Texas. However, the sizes of areas on the map are distorted. Alaska now looks much larger than it really is. GIS Software Maptitude Mapping Software gives you all of the tools, maps, and data you need to analyze and understand how geography affects you and your business. Maptitude supports dozens of projections to suit your needs. Learn MoreFree TrialFree for Students/Teachers Related Articles Using Map Projections with Maptitude Using Other Geographic File Formats in Maptitude Home Maptitude Mapping Software Learning Glossary Map Projection 617-527-4700 1172 Beacon St.,Suite 300 Newton MA 02461 USA sales@caliper.com Caliper Home Maptitude Desktop Online Tutorial Videos Learning Pricing & Requirements Solutions TestimonialsTransCAD About Pricing Learning Caliper Online Store Buy Maptitude Buy TransModeler SE Buy Maptitude DataMaptitude for Redistricting About Online TransModeler About Pricing TransModeler SE AboutCompany About Contact Career Opportunities Free Mapping Resources Legal Media Relations Press Releases Publications Products Home\|Products\|Contact\|Secure Store Copyright © Caliper Corporation
12791	https://courses.lumenlearning.com/waymakercollegealgebra/chapter/graphing-parabolas-with-vertices-not-at-the-origin/	Parabolas with Vertices Not at the Origin Learning Outcomes Identify and label the vertex, axis of symmetry, focus, directrix, and endpoints of the focal diameter given the equation of a parabola in standard form. Find the equation of a parabolic shaped object given dimensions. Like other graphs we’ve worked with, the graph of a parabola can be translated. If a parabola is translated h units horizontally and k units vertically, the vertex will be (h,k). This translation results in the standard form of the equation we saw previously with x replaced by (x−h) and y replaced by (y−k). To graph parabolas with a vertex (h,k) other than the origin, we use the standard form (y−k)2=4p(x−h) for parabolas that have an axis of symmetry parallel to the x-axis, and (x−h)2=4p(y−k) for parabolas that have an axis of symmetry parallel to the y-axis. These standard forms are given below, along with their general graphs and key features. A General Note: Standard Forms of Parabolas with Vertex (h, k) The table summarizes the standard features of parabolas with a vertex at a point (h,k). \| \| \| \| \| \| --- --- \| Axis of Symmetry \| Equation \| Focus \| Directrix \| Endpoints of focal diameter \| \| y=k \| (y−k)2=4p(x−h) \| (h+p, k) \| x=h−p \| (h+p, k±2p) \| \| x=h \| (x−h)2=4p(y−k) \| (h, k+p) \| y=k−p \| (h±2p, k+p) \| (a) When p>0, the parabola opens right. (b) When p<0, the parabola opens left. (c) When p>0, the parabola opens up. (d) When p<0, the parabola opens down. How To: Given a standard form equation for a parabola centered at (h, k), sketch the graph. Determine which of the standard forms applies to the given equation: (y−k)2=4p(x−h) or (x−h)2=4p(y−k). Use the standard form identified in Step 1 to determine the vertex, axis of symmetry, focus, equation of the directrix, and endpoints of the focal diameter. If the equation is in the form (y−k)2=4p(x−h), then: use the given equation to identify h and k for the vertex, (h,k) use the value of k to determine the axis of symmetry, y=k set 4p equal to the coefficient of (x−h) in the given equation to solve for p. If p>0, the parabola opens right. If p<0, the parabola opens left. use h,k, and p to find the coordinates of the focus, (h+p, k) use h and p to find the equation of the directrix, x=h−p use h,k, and p to find the endpoints of the focal diameter, (h+p,k±2p) If the equation is in the form (x−h)2=4p(y−k), then: use the given equation to identify h and k for the vertex, (h,k) use the value of h to determine the axis of symmetry, x=h set 4p equal to the coefficient of (y−k) in the given equation to solve for p. If p>0, the parabola opens up. If p<0, the parabola opens down. use h,k, and p to find the coordinates of the focus, (h, k+p) use k and p to find the equation of the directrix, y=k−p use h,k, and p to find the endpoints of the focal diameter, (h±2p, k+p) Plot the vertex, axis of symmetry, focus, directrix, and focal diameter, and draw a smooth curve to form the parabola. Example: Graphing a Parabola with Vertex (h, k) and Axis of Symmetry Parallel to the x-axis Graph (y−1)2=−16(x+3). Identify and label the vertex, axis of symmetry, focus, directrix, and endpoints of the focal diameter. Show Solution The standard form that applies to the given equation is (y−k)2=4p(x−h). Thus, the axis of symmetry is parallel to the x-axis. It follows that: the vertex is (h,k)=(−3,1) the axis of symmetry is y=k=1 −16=4p, so p=−4. Since p<0, the parabola opens left. the coordinates of the focus are (h+p,k)=(−3+(−4),1)=(−7,1) the equation of the directrix is x=h−p=−3−(−4)=1 the endpoints of the focal diameter are (h+p,k±2p)=(−3+(−4),1±2(−4)), or (−7,−7) and (−7,9) Next we plot the vertex, axis of symmetry, focus, directrix, and focal diameter, and draw a smooth curve to form the parabola. Try It Graph (y+1)2=4(x−8). Identify and label the vertex, axis of symmetry, focus, directrix, and endpoints of the focal diameter. Show Solution Vertex: (8,−1); Axis of symmetry: y=−1; Focus: (9,−1); Directrix: x=7; Endpoints of the latus rectum: (9,−3) and (9,1). Example: Graphing a Parabola from an Equation Given in General Form Graph x2−8x−28y−208=0. Identify and label the vertex, axis of symmetry, focus, directrix, and endpoints of the focal diameter. Show Solution Start by writing the equation of the parabola in standard form. The standard form that applies to the given equation is (x−h)2=4p(y−k). Thus, the axis of symmetry is parallel to the y-axis. To express the equation of the parabola in this form, we begin by isolating the terms that contain the variable x in order to complete the square. x2−8x−28y−208=0x2−8x=28y+208x2−8x+16=28y+208+16(x−4)2=28y+224(x−4)2=28(y+8)(x−4)2=4⋅7⋅(y+8) It follows that: the vertex is (h,k)=(4,−8) the axis of symmetry is x=h=4 since p=7,p>0 and so the parabola opens up the coordinates of the focus are (h,k+p)=(4,−8+7)=(4,−1) the equation of the directrix is y=k−p=−8−7=−15 the endpoints of the focal diameter are (h±2p,k+p)=(4±2(7),−8+7), or (−10,−1) and (18,−1) Next we plot the vertex, axis of symmetry, focus, directrix, and focal diameter, and draw a smooth curve to form the parabola. Try It Graph (x+2)2=−20(y−3). Identify and label the vertex, axis of symmetry, focus, directrix, and endpoints of the focal diameter. Show Solution Vertex: (−2,3); Axis of symmetry: x=−2; Focus: (−2,−2); Directrix: y=8; Endpoints of the latus rectum: (−12,−2) and (8,−2). Solving Applied Problems Involving Parabolas As we mentioned at the beginning of the section, parabolas are used to design many objects we use every day, such as telescopes, suspension bridges, microphones, and radar equipment. Parabolic mirrors, such as the one used to light the Olympic torch, have a very unique reflecting property. When rays of light parallel to the parabola’s axis of symmetry are directed toward any surface of the mirror, the light is reflected directly to the focus. This is why the Olympic torch is ignited when it is held at the focus of the parabolic mirror. Reflecting property of parabolas Parabolic mirrors have the ability to focus the sun’s energy to a single point, raising the temperature hundreds of degrees in a matter of seconds. Thus parabolic mirrors are featured in many low-cost, energy efficient solar products, such as solar cookers, solar heaters, and even travel-sized fire starters. Example: Solving Applied Problems Involving Parabolas A cross-section of a design for a travel-sized solar fire starter. The sun’s rays reflect off the parabolic mirror toward an object attached to the igniter. Because the igniter is located at the focus of the parabola, the reflected rays cause the object to burn in just seconds. Find the equation of the parabola that models the fire starter. Assume that the vertex of the parabolic mirror is the origin of the coordinate plane. Use the equation found in part (a) to find the depth of the fire starter. Cross-section of a travel-sized solar fire starter Show Solution The vertex of the dish is the origin of the coordinate plane, so the parabola will take the standard form x2=4py, where p>0. The igniter, which is the focus, is 1.7 inches above the vertex of the dish. Thus we have p=1.7. x2=4pyStandard form of upward-facing parabola with vertex (0,0)x2=4(1.7)ySubstitute 1.7 for p.x2=6.8yMultiply. The dish extends 4.52=2.25 inches on either side of the origin. We can substitute 2.25 for x in the equation from part (a) to find the depth of the dish. x2=6.8yEquation found in part (a).(2.25)2=6.8ySubstitute 2.25 for x.y≈0.74Solve for y. The dish is about 0.74 inches deep. Try It Balcony-sized solar cookers have been designed for families living in India. The top of a dish has a diameter of 1600 mm. The sun’s rays reflect off the parabolic mirror toward the “cooker,” which is placed 320 mm from the base. Find an equation that models a cross-section of the solar cooker. Assume that the vertex of the parabolic mirror is the origin of the coordinate plane, and that the parabola opens to the right (i.e., has the x-axis as its axis of symmetry). Use the equation found in part (a) to find the depth of the cooker. Show Solution y2=1280x The depth of the cooker is 500 mm Contribute! Did you have an idea for improving this content? We’d love your input. Improve this pageLearn More Candela Citations CC licensed content, Original Revision and Adaptation. Provided by: Lumen Learning. License: CC BY: Attribution CC licensed content, Shared previously College Algebra. Authored by: Abramson, Jay et al.. Provided by: OpenStax. Located at: License: CC BY: Attribution. License Terms: Download for free at Question ID 86148, 86124, 87082. Authored by: Shahbazian, Roy. License: CC BY: Attribution. License Terms: IMathAS Community License CC-BY + GPL CC licensed content, Specific attribution Precalculus. Authored by: OpenStax College. Provided by: OpenStax. Located at: License: CC BY: Attribution Licenses and Attributions CC licensed content, Original Revision and Adaptation. Provided by: Lumen Learning. License: CC BY: Attribution CC licensed content, Shared previously College Algebra. Authored by: Abramson, Jay et al.. Provided by: OpenStax. Located at: License: CC BY: Attribution. License Terms: Download for free at Question ID 86148, 86124, 87082. Authored by: Shahbazian, Roy. License: CC BY: Attribution. License Terms: IMathAS Community License CC-BY + GPL CC licensed content, Specific attribution Precalculus. Authored by: OpenStax College. Provided by: OpenStax. Located at: License: CC BY: Attribution
12792	https://math.libretexts.org/Bookshelves/Algebra/Algebra_and_Trigonometry_1e_(OpenStax)/06%3A_Exponential_and_Logarithmic_Functions/6.05%3A_Logarithmic_Properties	6.5: Logarithmic Properties - Mathematics LibreTexts Skip to main content Table of Contents menu search Search build_circle Toolbar fact_check Homework cancel Exit Reader Mode school Campus Bookshelves menu_book Bookshelves perm_media Learning Objects login Login how_to_reg Request Instructor Account hub Instructor Commons Search Search this book Submit Search x Text Color Reset Bright Blues Gray Inverted Text Size Reset +- Margin Size Reset +- Font Type Enable Dyslexic Font - [x] Downloads expand_more Download Page (PDF) Download Full Book (PDF) Resources expand_more Periodic Table Physics Constants Scientific Calculator Reference expand_more Reference & Cite Tools expand_more Help expand_more Get Help Feedback Readability x selected template will load here Error This action is not available. chrome_reader_mode Enter 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Home 2. Bookshelves 3. Algebra 4. Algebra and Trigonometry 1e (OpenStax) 5. 6: Exponential and Logarithmic Functions 6. 6.5: Logarithmic Properties Expand/collapse global location 6.5: Logarithmic Properties Last updated Dec 26, 2024 Save as PDF 6.4E: Graphs of Logarithmic Functions (Exercises) 6.5E: Logarithmic Properties (Exercises) Page ID 1509 OpenStax OpenStax ( \newcommand{\kernel}{\mathrm{null}\,}) Table of contents 1. Learning Objectives 2. Using the Product Rule for Logarithms 1. The Product Rule for Logarithms 2. How to: Given the logarithm of a product, use the product rule of logarithms to write an equivalent sum of logarithms 3. Example 6.5.1: Using the Product Rule for Logarithms 1. Solution 4. Exercise 6.5.1/06:_Exponential_and_Logarithmic_Functions/6.05:_Logarithmic_Properties#Exercise_.5C(.5CPageIndex.7B1.7D.5C)) Using the Quotient Rule for Logarithms The Quotient Rule for Logarithms Proof How to: Given the logarithm of a quotient, use the quotient rule of logarithms to write an equivalent difference of logarithms Example 6.5.2: Using the Quotient Rule for Logarithms Solution Exercise 6.5.2 Using the Power Rule for Logarithms The Power Rule for Logarithms How to: Given the logarithm of a power, use the power rule of logarithms to write an equivalent product of a factor and a logarithm Example 6.5.3: Expanding a Logarithm with Powers Solution Exercise 6.5.3 Example 6.5.4: Rewriting an Expression as a Power before Using the Power Rule Solution Exercise 6.5.4 Example 6.5.5: Using the Power Rule in Reverse Solution Exercise 6.5.5 Expanding Logarithmic Expressions Example 6.5.6: Expanding Logarithms Using Product, Quotient, and Power Rules Solution Exercise 6.5.6 Example 6.5.7: Using the Power Rule for Logarithms to Simplify the Logarithm of a Radical Expression Solution Exercise 6.5.7 Q&A: Can we expand ln⁡(x 2+y 2)? Example 6.5.8: Expanding Complex Logarithmic Expressions Solution Exercise 6.5.8 Condensing Logarithmic Expressions How to: Given a sum, difference, or product of logarithms with the same base, write an equivalent expression as a single logarithm Example 6.5.9: Using the Product and Quotient Rules to Combine Logarithms Solution Exercise 6.5.9 Example 6.5.10: Condensing Complex Logarithmic Expressions Solution Example 6.5.11: Rewriting as a Single Logarithm Solution Exercise 6.5.10 Exercise 6.5.11 Example 6.5.12: Applying of the Laws of Logs Solution Exercise 6.5.12 Using the Change-of-Base Formula for Logarithms THE CHANGE-OF-BASE FORMULA How to: Given a logarithm with the form log b⁡M, use the change-of-base formula to rewrite it as a quotient of logs with any positive base n, where n≠1 Example 6.5.13: Changing Logarithmic Expressions to Expressions Involving Only Natural Logs Solution Exercise 6.5.13 Q&A: Can we change common logarithms to natural logarithms? Example 6.5.14: Using the Change-of-Base Formula with a Calculator Solution Exercise 6.5.14 Media Key Equations Key Concepts Learning Objectives Use the product rule for logarithms. Use the quotient rule for logarithms. Use the power rule for logarithms. Expand logarithmic expressions. Condense logarithmic expressions. Use the change-of-base formula for logarithms. In chemistry, the pH scale is used as a measure of the acidity or alkalinity of a substance. Substances with a pH less than 7 are considered acidic, and substances with a pH greater than 7 are said to be alkaline. Our bodies, for instance, must maintain a pH close to 7.35 in order for enzymes to work properly. To get a feel for what is acidic and what is alkaline, consider the following pH levels of some common substances: Battery acid: 0.8 Stomach acid: 2.7 Orange juice: 3.3 Pure water: 7 at 25∘⁢C Human blood: 7.35 Fresh coconut: 7.8 Sodium hydroxide (lye): 14 To determine whether a solution is acidic or alkaline, we find its pH, which is a measure of the number of active positive hydrogen ions in the solution. The pH is defined by the following formula, where [H⁡A+] is the concentration of hydrogen ions in the solution (6.5.1)p⁢H=−log⁡([H⁡A+])(6.5.2)=log⁡(1[H⁡A+]) The equivalence of Equations 6.5.1 and 6.5.2 is one of the logarithm properties we will examine in this section. Figure 6.5.1: The pH of hydrochloric acid is tested with litmus paper. (credit: David Berardan). Using the Product Rule for Logarithms Recall that the logarithmic and exponential functions “undo” each other. This means that logarithms have similar properties to exponents. Some important properties of logarithms are given here. First, the following properties are easy to prove. log b⁡1=0 log b⁡b=1 For example, log 5⁡1=0 since 5 0=1. And log 5⁡5=1 since 5 1=5. Next, we have the inverse property. log b⁡(b x)=x b log b⁡x=x,x>0 For example, to evaluate log⁡(100), we can rewrite the logarithm as log 10⁡(10 2), and then apply the inverse property log b⁡(b x)=x to get log 10⁡(10 2)=2. To evaluate e ln⁡(7), we can rewrite the logarithm as e log e⁡7, and then apply the inverse property b log b⁡x=x to get e log e⁡7=7. Finally, we have the one-to-one property. (6.5.3)log b⁡M=log b⁡N if and only if M=N We can use the one-to-one property to solve the equation log 3⁡(3⁢x)=log 3⁡(2⁢x+5) for x. Since the bases are the same, we can apply the one-to-one property by setting the arguments equal and solving for x: 3⁢x=2⁢x+5 Set the arguments equal. x=5 Subtract 2⁢x. But what about the equation log 3⁡(3⁢x)+log 3⁡(2⁢x+5)=2? The one-to-one property does not help us in this instance. Before we can solve an equation like this, we need a method for combining terms on the left side of the equation. Recall that we use the product rule of exponents to combine the product of exponents by adding: x a⁢x b=x a+b. We have a similar property for logarithms, called the product rule for logarithms, which says that the logarithm of a product is equal to a sum of logarithms. Because logs are exponents, and we multiply like bases, we can add the exponents. We will use the inverse property to derive the product rule below. Given any real number x and positive real numbers M, N, and b, where b≠1, we will show log b⁡(M⁢N)=log b⁡(M)+log b⁡(N). Let m=log b⁡M and n=log b⁡N. In exponential form, these equations are b m=M and b n=N. It follows that log b⁡(M⁢N)=log b⁡(b m⁢b n)Substitute for M and N=log b⁡(b m+n)Apply the product rule for exponents=m+n Apply the inverse property of logs=log b⁡(M)+log b⁡(N)Substitute for m and n Note that repeated applications of the product rule for logarithms allow us to simplify the logarithm of the product of any number of factors. For example, consider log b⁡(w⁢x⁢y⁢z). Using the product rule for logarithms, we can rewrite this logarithm of a product as the sum of logarithms of its factors: log b⁡(w⁢x⁢y⁢z)=log b⁡w+log b⁡x+log b⁡y+log b⁡z The Product Rule for Logarithms The product rule for logarithms can be used to simplify a logarithm of a product by rewriting it as a sum of individual logarithms. (6.5.4)log b⁡(M⁢N)=log b⁡(M)+log b⁡(N)for b>0 How to: Given the logarithm of a product, use the product rule of logarithms to write an equivalent sum of logarithms Factor the argument completely, expressing each whole number factor as a product of primes. Write the equivalent expression by summing the logarithms of each factor. Example 6.5.1: Using the Product Rule for Logarithms Expand log 3⁡(30⁢x⁢(3⁢x+4)). Solution We begin by factoring the argument completely, expressing 30 as a product of primes. log 3⁡(30⁢x⁢(3⁢x+4))=log 3⁡(2⋅3⋅5⋅x⋅(3⁢x+4)) Next we write the equivalent equation by summing the logarithms of each factor. log 3⁡(30⁢x⁢(3⁢x+4))=log 3⁡(2)+log 3⁡(3)+log 3⁡(5)+log 3⁡(x)+log 3⁡(3⁢x+4) Exercise 6.5.1 Expand log b⁡(8⁢k). Answer log b⁡2+log b⁡2+log b⁡2+log b⁡k=3⁢log b⁡2+log b⁡k Using the Quotient Rule for Logarithms For quotients, we have a similar rule for logarithms. Recall that we use the quotient rule of exponents to combine the quotient of exponents by subtracting: x a b=x a−b. The quotient rule for logarithms says that the logarithm of a quotient is equal to a difference of logarithms. The Quotient Rule for Logarithms The quotient rule for logarithms can be used to simplify a logarithm or a quotient by rewriting it as the difference of individual logarithms. (6.5.5)log b⁡(M N)=log b⁡M−log b⁡N Just as with the product rule, we can use the inverse property to derive the quotient rule. Proof Given any real number x and positive real numbers M, N, and b, b, where b≠1, we will show log b⁡(M N)=log b⁡(M)−log b⁡(N). Let m=log b⁡M and n=log b⁡N. In exponential form, these equations are b m=M and b n=N. It follows that log b⁡(M N)=log b⁡(b m b n)Substitute for M and N=log b⁡(b m−n)Apply the quotient rule for exponents=m−n Apply the inverse property of logs=log b⁡(M)−log b⁡(N)Substitute for m and n For example, to expand log⁡(2⁢x 2+6⁢x 3⁢x+9), we must first express the quotient in lowest terms. Factoring and canceling we get, log⁡(2⁢x 2+6⁢x 3⁢x+9)=log⁡(2⁢x⁢(x+3)3⁢(x+3))Factor the numerator and denominator=log⁡(2⁢x 3)Cancel the common factors Next we apply the quotient rule by subtracting the logarithm of the denominator from the logarithm of the numerator. Then we apply the product rule. log⁡(2⁢x 3)=log⁡(2⁢x)−log⁡(3)=log⁡(2)+log⁡(x)−log⁡(3) How to: Given the logarithm of a quotient, use the quotient rule of logarithms to write an equivalent difference of logarithms Express the argument in lowest terms by factoring the numerator and denominator and canceling common terms. Write the equivalent expression by subtracting the logarithm of the denominator from the logarithm of the numerator. Check to see that each term is fully expanded. If not, apply the product rule for logarithms to expand completely. Example 6.5.2: Using the Quotient Rule for Logarithms Expand l o g 2⁡(15⁢x⁢(x−1)(3⁢x+4)⁢(2−x)). Solution First we note that the quotient is factored and in lowest terms, so we apply the quotient rule. log 2⁡(15⁢x⁢(x−1)(3⁢x+4)⁢(2−x))=log 2⁡(15⁢x⁢(x−1))−log 2⁡((3⁢x+4)⁢(2−x)) Notice that the resulting terms are logarithms of products. To expand completely, we apply the product rule, noting that the prime factors of the factor 15 are 3 and 5. log 2⁡(15⁢x⁢(x−1))−log 2⁡((3⁢x+4)⁢(2−x))=[log 2⁡(3)+log 2⁡(5)+log 2⁡(x)+log 2⁡(x−1)]−[log 2⁡(3⁢x+4)+log 2⁡(2−x)]=log 2⁡(3)+log 2⁡(5)+log 2⁡(x)+log 2⁡(x−1)−log 2⁡(3⁢x+4)−log 2⁡(2−x) Analysis There are exceptions to consider in this and later examples. First, because denominators must never be zero, this expression is not defined for x=−4 3 and x=2. Also, since the argument of a logarithm must be positive, we note as we observe the expanded logarithm, that x>0, x>1, x>−4 3, and x<2. Combining these conditions is beyond the scope of this section, and we will not consider them here or in subsequent exercises. Exercise 6.5.2 Expand log 3⁡(7⁢x 2+21⁢x 7⁢x⁢(x−1)⁢(x−2)). Answer log 3⁡(x+3)−log 3⁡(x−1)−log 3⁡(x−2) Using the Power Rule for Logarithms We’ve explored the product rule and the quotient rule, but how can we take the logarithm of a power, such as x 2? One method is as follows: log b⁡(x 2)=log b⁡(x⋅x)=log b⁡x+log b⁡x=2⁢log b⁡x Notice that we used the product rule for logarithms to find a solution for the example above. By doing so, we have derived the power rule for logarithms, which says that the log of a power is equal to the exponent times the log of the base. Keep in mind that, although the input to a logarithm may not be written as a power, we may be able to change it to a power. For example, 100=10 2 3=3 1 2 1 e=e−1 The Power Rule for Logarithms The power rule for logarithms can be used to simplify the logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base. (6.5.6)log b⁡(M n)=n⁢log b⁡M How to: Given the logarithm of a power, use the power rule of logarithms to write an equivalent product of a factor and a logarithm Express the argument as a power, if needed. Write the equivalent expression by multiplying the exponent times the logarithm of the base. Example 6.5.3: Expanding a Logarithm with Powers Expand log 2⁡x 5. Solution The argument is already written as a power, so we identify the exponent, 5, and the base, x, and rewrite the equivalent expression by multiplying the exponent times the logarithm of the base. log 2⁡(x 5)=5⁢log 2⁡x Exercise 6.5.3 Expand ln⁡x 2. Answer 2⁢ln⁡x Example 6.5.4: Rewriting an Expression as a Power before Using the Power Rule Expand log 3⁡(25) using the power rule for logs. Solution Expressing the argument as a power, we get log 3⁡(25)=log 3⁡(5 2). Next we identify the exponent, 2, and the base, 5, and rewrite the equivalent expression by multiplying the exponent times the logarithm of the base. log 3⁡(52)=2⁢log 3⁡(5) Exercise 6.5.4 Expand ln⁡(1 x 2). Answer −2⁢ln⁡(x) Example 6.5.5: Using the Power Rule in Reverse Rewrite 4⁢ln⁡(x) using the power rule for logs to a single logarithm with a leading coefficient of 1. Solution Because the logarithm of a power is the product of the exponent times the logarithm of the base, it follows that the product of a number and a logarithm can be written as a power. For the expression 4⁢ln⁡(x), we identify the factor, 4, as the exponent and the argument, x, as the base, and rewrite the product as a logarithm of a power: 4⁢ln⁡(x)=ln⁡(x 4). Exercise 6.5.5 Rewrite 2⁢log 3⁡4 using the power rule for logs to a single logarithm with a leading coefficient of 1. Answer log 3⁡16 Expanding Logarithmic Expressions Taken together, the product rule, quotient rule, and power rule are often called “laws of logs.” Sometimes we apply more than one rule in order to simplify an expression. For example: log b⁡(6⁢x y)=log b⁡(6⁢x)−log b⁡y=log b⁡6+log b⁡x−log b⁡y We can use the power rule to expand logarithmic expressions involving negative and fractional exponents. Here is an alternate proof of the quotient rule for logarithms using the fact that a reciprocal is a negative power: log b⁡(A C)=log b⁡(A⁢C−1)=log b⁡(A)+log b⁡(C−1)=log b⁡A+(−1)⁢log b⁡C=log b⁡A−log b⁡C We can also apply the product rule to express a sum or difference of logarithms as the logarithm of a product. With practice, we can look at a logarithmic expression and expand it mentally, writing the final answer. Remember, however, that we can only do this with products, quotients, powers, and roots—never with addition or subtraction inside the argument of the logarithm. Example 6.5.6: Expanding Logarithms Using Product, Quotient, and Power Rules Rewrite l⁢n⁡(x 4⁢y 7) as a sum or difference of logs. Solution First, because we have a quotient of two expressions, we can use the quotient rule: ln⁡(x 4⁢y 7)=ln⁡(x 4⁢y)−ln⁡(7) Then seeing the product in the first term, we use the product rule: ln⁡(x 4⁢y)−ln⁡(7)=ln⁡(x 4)+ln⁡(y)−ln⁡(7) Finally, we use the power rule on the first term: ln⁡(x 4)+ln⁡(y)−ln⁡(7)=4⁢ln⁡(x)+ln⁡(y)−ln⁡(7) Exercise 6.5.6 Expand log⁡(x 2⁢y 3 z 4). Answer 2⁢log⁡x+3⁢log⁡y−4⁢log⁡z Example 6.5.7: Using the Power Rule for Logarithms to Simplify the Logarithm of a Radical Expression Expand log⁡(x). Solution log⁡(x)=log⁡x(1 2)=1 2⁢log⁡x Exercise 6.5.7 Expand ln⁡(x 2 3). Answer 2 3⁢ln⁡x Q&A: Can we expand ln⁡(x 2+y 2)? No. There is no way to expand the logarithm of a sum or difference inside the argument of the logarithm. Example 6.5.8: Expanding Complex Logarithmic Expressions Expand log 6⁡(64⁢x 3⁢(4⁢x+1)(2⁢x−1)). Solution We can expand by applying the Product and Quotient Rules. log 6⁡(64⁢x 3⁢(4⁢x+1)(2⁢x−1))=log 6⁡64+log 6⁡x 3+log 6⁡(4⁢x+1)−log 6⁡(2⁢x−1)Apply the Quotient Rule=log 6⁡26+log 6⁡x 3+log 6⁡(4⁢x+1)−log 6⁡(2⁢x−1)Simplify by writing 64 as 2 6=6⁢log 6⁡2+3⁢log 6⁡x+log 6⁡(4⁢x+1)−log 6⁡(2⁢x−1)Apply the Power Rule Exercise 6.5.8 Expand ln⁡((x−1)⁢(2⁢x+1)2(x 2−9)). Answer 1 2⁢ln⁡(x−1)+ln⁡(2⁢x+1)−ln⁡(x+3)−ln⁡(x−3) Condensing Logarithmic Expressions We can use the rules of logarithms we just learned to condense sums, differences, and products with the same base as a single logarithm. It is important to remember that the logarithms must have the same base to be combined. We will learn later how to change the base of any logarithm before condensing. How to: Given a sum, difference, or product of logarithms with the same base, write an equivalent expression as a single logarithm Apply the power property first. Identify terms that are products of factors and a logarithm, and rewrite each as the logarithm of a power. Next apply the product property. Rewrite sums of logarithms as the logarithm of a product. Apply the quotient property last. Rewrite differences of logarithms as the logarithm of a quotient. Example 6.5.9: Using the Product and Quotient Rules to Combine Logarithms Write log 3⁡(5)+log 3⁡(8)−log 3⁡(2) as a single logarithm. Solution Using the product and quotient rules log 3⁡(5)+log 3⁡(8)=log 3⁡(5⋅8)=log 3⁡(40) This reduces our original expression to log 3⁡(40)−log 3⁡(2) Then, using the quotient rule log 3⁡(40)−log 3⁡(2)=log 3⁡(40 2)=log 3⁡(20) Exercise 6.5.9 Condense log 3−log 4+log 5−log 6. Answer log⁡(3⋅5 4⋅6); can also be written log⁡(5 8) by reducing the fraction to lowest terms. Example 6.5.10: Condensing Complex Logarithmic Expressions Condense log 2⁡(x 2)+1 2⁢log 2⁡(x−1)−3⁢log 2⁡((x+3)2). Solution We apply the power rule first: log 2⁡(x 2)+1 2⁢log 2⁡(x−1)−3⁢log 2⁡((x+3)2)=log 2⁡(x 2)+log 2⁡(x−1)−log 2⁡((x+3)6) Next we apply the product rule to the sum: log 2⁡(x 2)+log 2⁡(x−1)−log 2⁡((x+3)6)=log 2⁡(x 2⁢x−1)−log 2⁡((x+3)6) Finally, we apply the quotient rule to the difference: log 2⁡(x 2⁢x−1)−log 2⁡((x+3)6)=log 2⁡x 2⁢x−1(x+3)6 Example 6.5.11: Rewriting as a Single Logarithm Rewrite 2⁢log⁡x−4⁢log⁡(x+5)+1 x⁢log⁡(3⁢x+5) as a single logarithm. Solution We apply the power rule first: 2⁢log⁡x−4⁢log⁡(x+5)+1 x⁢log⁡(3⁢x+5)=log⁡(x 2)−log⁡(x+5)4+log⁡((3⁢x+5)x−1) Next we rearrange and apply the product rule to the sum: log⁡(x 2)−log⁡(x+5)4+log⁡((3⁢x+5)x−1)=log⁡(x 2)+log⁡((3⁢x+5)x−1−log⁡(x+5)4=log⁡(x 2⁢(3⁢x+5)x−1)−log⁡(x+5)4=log⁡x 2⁢(3⁢x+5)x−1(x+5)4 Apply the quotient rule to the difference Exercise 6.5.10 Rewrite log⁡(5)+0.5⁢log⁡(x)−log⁡(7⁢x−1)+3⁢log⁡(x−1) as a single logarithm. Answer log⁡5⁢(x−1)3⁢x(7⁢x−1) Exercise 6.5.11 Condense 4⁢(3⁢log⁡(x)+log⁡(x+5)−log⁡(2⁢x+3)). Answer log⁡x 12⁢(x+5)4(2⁢x+3)4; this answer could also be written log⁡(x 3⁢(x+5)(2⁢x+3))4 Example 6.5.12: Applying of the Laws of Logs Recall that, in chemistry, p⁢H=−log⁡[H+]. If the concentration of hydrogen ions in a liquid is doubled, what is the effect on pH? Solution Suppose C is the original concentration of hydrogen ions, and P is the original pH of the liquid. Then P=–log⁡(C). If the concentration is doubled, the new concentration is 2⁢C. Then the pH of the new liquid is p⁢H=−log⁡(2⁢C) Using the product rule of logs p⁢H=−log⁡(2⁢C)=−(log⁡(2)+log⁡(C))=−log⁡(2)−log⁡(C) Since P=–log⁡(C),the new pH is p⁢H=P−log⁡(2)≈P−0.301 Exercise 6.5.12 When the concentration of hydrogen ions is doubled, the pH decreases by about 0.301. How does the pH change when the concentration of positive hydrogen ions is decreased by half? Answer The pH increases by about 0.301. Using the Change-of-Base Formula for Logarithms Most calculators can evaluate only common and natural logs. In order to evaluate logarithms with a base other than 10 ore, e,we use the change-of-baseformula to rewrite the logarithm as the quotient of logarithms of any other base; when using a calculator, we would change them to common or natural logs. To derive the change-of-base formula, we use the one-to-one property and power rule for logarithms. Given any positive real numbers M, b, and n, where n≠1 and b≠1,we show log b⁡M=log n⁡M log n⁡b Let y=log b⁡M. By taking the log base n of both sides of the equation, we arrive at an exponential form, namely b y=M. It follows that log n⁡(b y)=log n⁡M Apply the one-to-one property y⁢log n⁡b=log n⁡M Apply the power rule for logarithms y=log n⁡M log n⁡b Isolate y log b⁡M=log n⁡M log n⁡b Substitute for y For example, to evaluate log 5⁡36 using a calculator, we must first rewrite the expression as a quotient of common or natural logs. We will use the common log. log 5⁡36=log⁡(36)log⁡(5)Apply the change of base formula using base 10≈2.2266 Use a calculator to evaluate to 4 decimal places THE CHANGE-OF-BASE FORMULA The change-of-base formula can be used to evaluate a logarithm with any base. For any positive real numbers M, b, and n, where n≠1 and b≠1, (6.5.7)log b⁡M=log n⁡M log n⁡b It follows that the change-of-base formula can be used to rewrite a logarithm with any base as the quotient of common or natural logs. (6.5.8)log b⁡M=ln⁡M ln⁡b and (6.5.9)log b⁡M=log⁡M log⁡b How to: Given a logarithm with the form log b⁡M, use the change-of-base formula to rewrite it as a quotient of logs with any positive base n, where n≠1 Determine the new base n, remembering that the common log, log⁡(x), has base 10, and the natural log, ln⁡(x),has base e. Rewrite the log as a quotient using the change-of-base formula The numerator of the quotient will be a logarithm with base n and argument M. The denominator of the quotient will be a logarithm with base n and argument b. Example 6.5.13: Changing Logarithmic Expressions to Expressions Involving Only Natural Logs Change log 5⁡3 to a quotient of natural logarithms. Solution Because we will be expressing log 5⁡3 as a quotient of natural logarithms, the new base, n=e. We rewrite the log as a quotient using the change-of-base formula. The numerator of the quotient will be the natural log with argument 3. The denominator of the quotient will be the natural log with argument 5. log b⁡M=ln⁡M ln⁡b log 5⁡3=ln⁡3 ln⁡5 Exercise 6.5.13 Change log⁡0.58 to a quotient of natural logarithms. Answer ln⁡8 ln⁡0.5 Q&A: Can we change common logarithms to natural logarithms? Yes. Remember that log⁡9 meansl log 10⁡9. So, log⁡9=ln⁡9 ln⁡10. Example 6.5.14: Using the Change-of-Base Formula with a Calculator Evaluate log 2⁡(10) using the change-of-base formula with a calculator. Solution According to the change-of-base formula, we can rewrite the log base 2 as a logarithm of any other base. Since our calculators can evaluate the natural log, we might choose to use the natural logarithm, which is the log base e. log 2⁡10=ln⁡10 ln⁡2 Apply the change of base formula using base e≈3.3219 Use a calculator to evaluate to 4 decimal places Exercise 6.5.14 Evaluate log 5⁡(100) using the change-of-base formula. Answer ln⁡100 ln⁡5≈4.6051 1.6094=2.861 Media Access these online resources for additional instruction and practice with laws of logarithms. The Properties of Logarithms Expand Logarithmic Expressions Evaluate a Natural Logarithmic Expression Key Equations The Product Rule for Logarithms log b⁡(M⁢N)=log b⁡(M)+log b⁡(N) The Quotient Rule for Logarithms log b⁡(M N)=log b⁡M−log b⁡N The Power Rule for Logarithms log b⁡(M n)=n⁢log b⁡M The Change-of-Base Formula log b⁡M=log n⁡M log n⁡b n>0, n≠1, b≠1 Key Concepts We can use the product rule of logarithms to rewrite the log of a product as a sum of logarithms. See Example 6.5.1. We can use the quotient rule of logarithms to rewrite the log of a quotient as a difference of logarithms. See Example 6.5.2. We can use the power rule for logarithms to rewrite the log of a power as the product of the exponent and the log of its base. See Example 6.5.3, Example 6.5.4, and Example 6.5.5. We can use the product rule, the quotient rule, and the power rule together to combine or expand a logarithm with a complex input. See Example 6.5.6, Example 6.5.7,and Example 6.5.8. The rules of logarithms can also be used to condense sums, differences, and products with the same base as a single logarithm. See Example 6.5.9, Example 6.5.10, Example 6.5.11, and Example 6.5.12. We can convert a logarithm with any base to a quotient of logarithms with any other base using the change-of-base formula. See Example 6.5.13. The change-of-base formula is often used to rewrite a logarithm with a base other than 10 and e as the quotient of natural or common logs. That way a calculator can be used to evaluate. See Example 6.5.14. This page titled 6.5: Logarithmic Properties is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform. Back to top 6.4E: Graphs of Logarithmic Functions (Exercises) 6.5E: Logarithmic Properties (Exercises) Was this article helpful? Yes No Recommended articles 6.6: Logarithmic PropertiesRecall that the logarithmic and exponential functions “undo” each other. This means that logarithms have similar properties to exponents. Some importa... 5.6: Logarithmic PropertiesRecall that the logarithmic and exponential functions “undo” each other. This means that logarithms have similar properties to exponents. Some importa... 6.6: Logarithmic PropertiesRecall that the logarithmic and exponential functions “undo” each other. This means that logarithms have similar properties to exponents. Some importa... 6.6: Logarithmic PropertiesRecall that the logarithmic and exponential functions “undo” each other. This means that logarithms have similar properties to exponents. Some importa... 6.6: Logarithmic PropertiesRecall that the logarithmic and exponential functions “undo” each other. This means that logarithms have similar properties to exponents. Some importa... 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12793	https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Electron_Paramagnetic_Resonance_(Jenschke)/04%3A_Hyperfine_Interaction/4.01%3A_Physical_origin_of_the_hyperfine_interaction	Skip to main content 4.1: Physical origin of the hyperfine interaction Last updated : Mar 24, 2022 Save as PDF 4: Hyperfine Interaction 4.2: Hyperfine Hamiltonian Page ID : 370931 Gunnar Jeschke ETH Zürich ( \newcommand{\kernel}{\mathrm{null}\,}) The magnetic moments of an electron and a nuclear spin couple by the magnetic dipole-dipole interaction; similar to the dipole-dipole interaction between nuclear spins discussed in the NMR part of the lecture course. The main difference to the NMR case is that, in many cases, a point-dipole description is not a good approximation for the electron spin, as the electron is distributed over the SOMO. The nucleus under consideration can be considered as well localized in space. We now picture the SOMO as a linear combination of atomic orbitals. Contributions from spin density in an atomic orbital of another nucleus (population of the unpaired electron in such an atomic orbital) can be approximated by assuming that the unpaired electron is a point-dipole localized at this other nucleus. For spin density in atomic orbitals on the same nucleus, we have to distinguish between types of atomic orbitals. In s orbitals, the unpaired electron has finite probability density for residing at the nucleus, at zero distance rSI to the nuclear spin. This leads to a singularity of the dipole-dipole interaction, since this interaction scales with r−3SI. The singularity has been treated by Fermi. The contribution to the hyperfine coupling from spin density in s orbitals on the nucleus under consideration is therefor called Fermi contact interaction. Because of the spherical symmetry of s orbitals, the Fermi contact interaction is purely isotropic. For spin density in other orbitals ( p,d,f orbitals) on the nucleus under consideration, the dipole-dipole interaction must be averaged over the spatial distribution of the electron spin in these orbitals. This average has no isotropic contribution. Therefore, spin density in p,d,f orbitals does not influence spectra of fast tumbling radicals or metal complexes in liquid solution and neither does spin density in s orbitals of other nuclei. The isotropic couplings detected in solution result only from the Fermi contact interaction. Since the isotropic and purely anisotropic contributions to the hyperfine coupling have different physical origin, we separate these contributions in the hyperfine tensor Aki that describes the interaction between electron spin Sk and nuclear spin Ii : Aki=Aiso,ki⎛⎝⎜100010001⎞⎠⎟+Tki(4.1.1) where Aiso, ki is the isotropic hyperfine coupling and Tki the purely anisotropic coupling. In the following, we drop the electron and nuclear spin indices k and i. Dipole-dipole hyperfine interaction The anisotropic hyperfine coupling tensor T of a given nucleus can be computed from the ground state wavefunction ψ0 by applying the correspondence principle to the classical interaction between two point dipoles Tij=μ04πℏgeμBgnμn⟨ψ0∣∣∣3rirj−δijr2r5∣∣∣ψ0⟩(4.1.2) Such computations are implemented in quantum chemistry programs such as ORCA, ADF, or Gaussian. If the SOMO is considered as a linear combination of atomic orbitals, the contributions from an individual orbital can be expressed as the product of spin density in this orbital with a spatial factor that can be computed once for all. The spatial factors have been tabulated [KM85]. In general, nuclei of elements with larger electronegativity have larger spatial factors. At the same spatial factor, such as for isotopes of the same element, the hyperfine coupling is proportional to the nuclear g value gn and thus proportional to the gyromagnetic ratio of the nucleus. Hence, a deuterium coupling can be computed from a known proton coupling or vice versa. A special situation applies to protons, alkali metals and earth alkaline metals, which have no significant spin densities in p−,d−, or f-orbitals. In this case, the anisotropic contribution can only arise from through-space dipole-dipole coupling to centers of spin density at other nuclei. In a point-dipole approximation the hyperfine tensor is then given by T=μ04πℏgeμBgnμn∑j≠iρj3n⃗ ijn⃗ Tij−1→R3ij(4.1.3) where the sum runs over all nuclei j with significant spin density ρj (summed over all orbitals at this nucleus) other than nucleus i under consideration. The Rij are distances between the nucleus under consideration and the centers of spin density, and the n⃗ ij are unit vectors along the direction from the considered nucleus to the center of spin density. For protons in transition metal complexes it is often a good approximation to consider spin density only at the central metal ion. The distance R from the proton to the central ion can then be directly inferred from the anisotropic part of the hyperfine coupling. Hyperfine tensor contributions T computed by any of these ways must be corrected for the influence of SOC if the g tensor is strongly anisotropic. If the dominant contribution to SOC arises at a single nucleus, the hyperfine tensor at this nucleus 1 can be corrected by T(g)=gTge(4.1.4) The product g T may have an isotropic part, although T is purely anisotropic. This isotropic pseudocontact contribution depends on the relative orientation of the g tensor and the spin-only dipole-dipole hyperfine tensor T. The correction is negligible for most organic radicals, but not for paramagnetic metal ions. If contributions to SOC arise from several centers, the necessary correction cannot be written as a function of the g tensor. Fermi contact interaction The Fermi contact contribution takes the form Aiso =ρs⋅23μ0ℏgeμBgnμn\|ψ0(0)\|2(4.1.5) 1 Most literature holds that the correction should be done for all nuclei. As pointed out by Frank Neese, this is not true. An earlier discussion of this point is found in [Lef67] where ρs is the spin density in the s orbital under consideration, gn the nuclear g value and μn=βn=5.05078317(20)⋅10−27 J T−1 the nuclear magneton (gnμn=γnℏ). The factor \|ψ0(0)\|2 denotes the probability to find the electron at this nucleus in the ground state with wave function ψ0 and has been tabulated [KM85]. Spin polarization The contributions to the hyperfine coupling discussed up to this point can be understood and computed in a single-electron picture. Further contributions arise from correlation of electrons in a molecule. Assume that the pz orbital on a carbon atom contributes to the SOMO, so that the α spin state of the electron is preferred in that orbital (Fig. 4.1). Electrons in other orbitals on the same atom will then also have a slight preference for the α state (left panel), as electrons with the same spin tend to avoid each other and thus have less electrostatic repulsion. 2 In particular, this means that the spin configuration in the left panel of Fig. 4.1 is slightly more preferable than the one in the right panel. According to the Pauli principle, the two electrons that share the s bond orbital of the C−H bond must have antiparallel spin. Thus, the electron in the s orbital of the hydrogen atom that is bound to the spin-carrying carbon atom has a slight preference for the β state. This corresponds to a negative isotropic hyperfine coupling of the directly bound α proton, which is induced by the positive hyperfine coupling of the adjacent carbon atom. The effect is termed "spin polarization", although it has no physical relation to the polarization of electron spin transitions in an external magnetic field. Spin polarization is important, as it transfers spin density from p orbitals, where it is invisible in liquid solution and from carbon atoms with low natural abundance of the magnetic isotope 13C to s orbitals on protons, where it can be easily observed in liquid solution. This transfer occurs, both, in σ radicals, where the unpaired electron is localized on a single atom, and in π radicals, where it is distributed over the π system. The latter case is of larger interest, as the distribution of the π orbital over the nuclei can be mapped by measuring and assigning the isotropic proton hyperfine couplings. This coupling can be predicted by the McConnell equation Aiso,H=QHρπ(4.1.6) where ρπ is the spin density at the adjacent carbon atom and QH is a parameter of the order of −2.5mT, which slightly depends on structure of the π system. 2 This preference for electrons on the same atom to have parallel spin is also the basis of Hund’s rule. The McConnell equation is mainly applied for mapping the LUMO and HOMO of aromatic molecules (Figure 4.2). An unpaired electron can be put into these orbitals by one-electron reduction or oxidation, respectively, without perturbing the orbitals too strongly. The isotropic hyperfine couplings of the hydrogen atom directly bound to a carbon atom report on the contribution of the pz orbital of this carbon atom to the π orbital. The challenges in this mapping are twofold. First, it is hard to assign the observed couplings to the hydrogen atoms unless a model for the distribution of the π orbital is already available. Second, the method is blind to carbon atoms without a directly bonded hydrogen atom. 4: Hyperfine Interaction 4.2: Hyperfine Hamiltonian
12794	https://math.stackexchange.com/questions/1044837/sum-to-infinity-of-an-and-sk	sequences and series - Sum to infinity of An and Sk - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Sum to infinity of An and Sk Ask Question Asked 10 years, 10 months ago Modified10 years, 10 months ago Viewed 958 times This question shows research effort; it is useful and clear 1 Save this question. Show activity on this post. Consider a series ∑∞n=1 A n∑n=1∞A n for which S k=k+1 k S k=k+1 k∀k∈N∀k∈N. Find A n A n and the value of ∑∞n=1 A n∑n=1∞A n. I think i have worked out that ∑∞n=1 A n∑n=1∞A n is 1 1. But i don't know how to work out A n A n or show fully how i got my answer for the sum to infinity. S k=k+1 k S k=k+1 k hence lim k→∞S k=1 lim k→∞S k=1. which implies that ∑∞n=1 A n∑n=1∞A n is 1 1. sequences-and-series Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications edited Nov 30, 2014 at 11:38 user99914 asked Nov 30, 2014 at 11:14 DumbanddumberDumbanddumber 13 5 5 bronze badges 2 Hi, I edited your question. Please see if I accidentally chenged your meaning. BTW, what is the relationship between S k S k and A n A n?user99914 –user99914 2014-11-30 11:39:38 +00:00 Commented Nov 30, 2014 at 11:39 Thank you for editing it. The meaning hasn't changed. Sk is a sequence of partial sums of An.Dumbanddumber –Dumbanddumber 2014-11-30 11:41:35 +00:00 Commented Nov 30, 2014 at 11:41 Add a comment\| 3 Answers 3 Sorted by: Reset to default This answer is useful 1 Save this answer. Show activity on this post. By definition, ∑n=1∞a n=lim n→+∞∑k=1 n a k.∑n=1∞a n=lim n→+∞∑k=1 n a k. If you have really proved that S n=∑k=1 n a k=n+1 n S n=∑k=1 n a k=n+1 n for every k k, then your conclusion is perfectly legitimate, since lim n→+∞n+1 n=1 lim n→+∞n+1 n=1. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Nov 30, 2014 at 11:41 SiminoreSiminore 35.8k 3 3 gold badges 55 55 silver badges 85 85 bronze badges 3 Thank you. How do i find An though?Dumbanddumber –Dumbanddumber 2014-11-30 11:45:24 +00:00 Commented Nov 30, 2014 at 11:45 Well, a n=s n−s n−1 a n=s n−s n−1, just by canceling out all the terms except the last one.Siminore –Siminore 2014-11-30 11:54:21 +00:00 Commented Nov 30, 2014 at 11:54 Come on, it's elementary algebra!Siminore –Siminore 2014-12-01 09:39:35 +00:00 Commented Dec 1, 2014 at 9:39 Add a comment\| This answer is useful 0 Save this answer. Show activity on this post. You are on the right trace. You should know that the sum is the limit of S k S k. And A n=S n−S n−1 A n=S n−S n−1. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Nov 30, 2014 at 12:01 PaulPaul 21.2k 6 6 gold badges 38 38 silver badges 77 77 bronze badges 0 Add a comment\| This answer is useful -1 Save this answer. Show activity on this post. a k=S k−S k−1 for k>1 a k=S k−S k−1 for k>1 Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications edited Nov 30, 2014 at 11:57 Aditya Hase 8,931 2 2 gold badges 43 43 silver badges 53 53 bronze badges answered Nov 30, 2014 at 11:48 Monika GargMonika Garg 51 1 1 silver badge 2 2 bronze badges Add a comment\| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions sequences-and-series See similar questions with these tags. Featured on Meta Introducing a new proactive anti-spam measure Spevacus has joined us as a Community Manager stackoverflow.ai - rebuilt for attribution Community Asks Sprint Announcement - September 2025 Report this ad Related 10Is ∑n=1∞n+1√(n+1)!e n+5∑n=1∞n+1(n+1)!e n+5 convergent or divergent 1Find a n a n using the partial sum of series 2partial sum of convergent series 4Prob. 11, Chap. 3, in Baby Rudin: If a n>0 a n>0 and ∑a n∑a n diverges, then how do we show that ∑a n 1+a n∑a n 1+a n too diverges? 1partial sum of series which is convergent but not absolutely convergent 1given Sn, find a_n and sum of a_n 1Find sequence (a n)(a n) from partial sum of a series. 2(a)∑∞n=2 3 n 2+2 n(a)∑n=2∞3 n 2+2 n Determine whether it converges and find sum 0Finding a formula for the kth partial sum Hot Network Questions What's the expectation around asking to be invited to invitation-only workshops? Are there any world leaders who are/were good at chess? 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AO1 AO3 Blog Contact Resources LONG TERM MEMORY \| \| \| --- \| \| \| Tulving's ideas are identified in the Specification, so you need to know about the different types of declarative and non-declarative memory, examples of them in use as well as how to apply them and evaluate the theory as a whole. Clive Wearing does NOT feature in the Specification so the Exam will never ask you a question about him. However, he is a good example of the differences between types of long term memory. \| \| \| \| --- \| \| \| TULVING (1967)TYPES OF LONG TERM MEMORY This theory was proposed by Endel Tulving, one of the leading figures in memory research. It is based on the Multi-Store Model idea of LTM, but it suggests there is a difference between episodic memory (eg remembering a family holiday in Disneyland) and more general memory (eg knowing that Disneyland is in Florida).This theory is significant for students in other ways: It shows how scientific research proceeds, because Tulving’s distinction is an advance on the Multi Store Model. It also ties in with Baddeley’s research into semantic encoding in LTM and the case studies of H.M. It illustrates features of the Cognitive Approach, since it expresses the processes of memory as a diagram or flowchart, which resembles the sort of information processing used by a computer It ties in to your Key Question in Cognitive Psychology, since it helps explain Alzheimer’s It shows the importance of neuroscience which combines the Cognitive and Biological approaches, because functions of Semantic LTM have been located in parts of the brain (eg the Contemporary Study by Schmolck et al.) DECLARATIVE MEMORY Tulving makes a distinction between different types of LTM: procedural memory and declarative memory. Procedural memory is the memory of how to do things. It includes tying shoelaces, writing, tapping in your banking PIN and using a knife and fork. You may retain procedural memories even after you have forgotten being taught to do these things in the first place. Declarative memory is the memory of meaningful events. You might remember being taught to play the guitar, even if you’ve forgotten how to do it. Tulving splits declarative memory into two sub-types: Episodic memory is the memory of particular events and specific information: events, names and dates. It includes memories of things that have happened to you and information like a person’s address. Episodic memories seem to be perceptually encoded – they are linked to the 5 senses which is why they can be triggered (“cued”) by a sight or a sound or a smell. Tulving gives examples like remembering he has an appointment with a student the next day or recalling words from a list studied earlier as well as autobiographical memories (remembering details from your own past). Semantic memory is the memory of relationships and how things fit together. It includes the memory that you have brothers or sisters, where things are located and what they do. Semantic memory is needed for language because words have meaning – learning words in the first place involves episodic memory but once they are learned they go into the semantic store. Tulving gives examples like knowing that summers are hot in Kathmandu and knowing that July is the month after June. \| \| \| \| --- \| \| \| RESEARCH INTO LONG TERM MEMORYTHE CASE OF CLIVE WEARING Clive Wearing is a musician who suffered brain damage from a viral infection (herpes simplex encephalitis) in 1985. He suffered almost complete amnesia. He also lost the ability to encode new long term memories. Clive Wearing forgets everything within 30 seconds and is always “coming into consciousness”, feeling he is waking up for the first time.However, although Clive Wearing has lost his episodic memory, he still has semantic memory. When his wife Deborah enters the room he greets her joyously, believing he hasn’t seen her for years or even that they are meeting for the first time (even if she has only been gone for a minute). Although he has no episodic memories of Deborah, he has semantic knowledge of her: he remembers that he loves her. Similarly, although he cannot remember their names or ages, Clive Wearing knows that he is a father and that he has children.Clive Wearing also has intact procedural memory. He can still play piano and conduct a choir – although he cannot remember his musical education and as soon as the music stops he forgets he was performing and suffers a shaking fit.Sir Colin Blakemore (1988) carried out a case study on Clive Wearing. Blakemore discovered that damage to Clive Wearing’s brain had been to the hippocampus, which seems to be the part of the brain where the Short Term Memory (STM) rehearses information to encode it into LTM. The Contemporary Study by Schmolck et al. (2002) looks at other patients with amnesia because of damage to the hippocampus; it includes patients like Clive Wearing who suffered herpes encephalitis infection. These patients also struggled with semantic memory because of damage to the wider temporal cortex. Clive Wearing's case is fascinating and heartbreaking. However, Clive Wearing does NOT feature in the Edexcel Specification. The patient who does is H.M. (Henry Molaison) and you must not get the two men confused.Clive Wearing is the pianist who loves his wife and was struck down by a brain virusH.M. (Henry Molaison) lost his memory after brain surgery and was the subject of many case studies \| \| \| \| --- \| \| \| APPLYING LONG TERM MEMORY (AO2)MEMORY IN THE REAL WORLD Jogging Your MemoryTulving argues that episodic memory is encoded based on how it was experienced (the encoding specificity principle). This means that when a memory is stored details of time and space (when and where) are stored with it. This is why people can normally answer the question “When did that happen?” or “Where were you with that happened?” Even if they cannot give exact dates or places, they can reply, “Before the summer holidays,” or “At my old school.”This means that episodic memory can be “jogged” by context cues – things that remind you of when/where the original memory was encoded. Godden & Baddeley (1974) tested this and found that divers who learned words underwater recalled them better underwater than back on dry land.Semantic memory doesn’t seem to be organised this way. Instead, it seems to work by using rules. For example, you might remember how to spell “receipt” by applying the rule “’i’ before ‘e’ except after ‘c’.”Episodic memory seems to be changed by being used. For example, when people recall and event, it gets re-encoded into LTM and may get altered as a result. This is how false memories occur. Semantic memory doesn’t seem to work like this. Your memory of relationships and meanings is not changed by being used and it can be quite separate from episodes. Dementia & Alzheimer'sThe most common symptom of dementia is difficulty to make new memories. STM (which rehearses information) is the first type of memory to go. Episodic memory is the next to go, as sufferers begin to forget autobiographical events. Usually, recent episodes are lost first, but sufferers still remember episodes from their young adulthood and youth. Semantic memory is lost later, when sufferers struggle with language and no longer recognise family members. As the disease advances, parts of memory which were previously intact also become impaired. Eventually all reasoning and language abilities are disrupted.Patients tend to display a loss of knowledge of semantic categories. Initially, they lose the ability to distinguish fine categories, such as species of animals or types of objects, but, over time, this lack of discrimination becomes more general. At first, a patient with advanced dementia may see a spaniel and say, “That is a dog.” Later, they may just say, “That is an animal”.You can use all of this in your Cognitive Key Question. \| \| \| \| --- \| \| \| EVALUATING LONG TERM MEMORY (AO3)CODA CredibilityThere’s a lot of research in support of Tulving’s distinctions. Some of this is case studies of amnesia patients like Clive Wearing who have lost episodic memory but still have semantic memory. The deterioration of dementia patients also suggests that episodic and semantic memory are separate because episodic memory is lost first and semantic memory last.The Classic Cognitive Study by Baddeley (1966b) also supports the existence of semantic memory. Baddeley found that participants struggled with word lists linked by a common theme, which suggests the semantic similarity confused LTM. Unrelated word lists were not confusing. This suggests at least part of LTM works semantically.The Contemporary Study by Schmolck et al. (2002) also supports the idea of long term memory being located in a specific part of the brain – the temporal cortex.Tulving carried out a case study of Kent Cochrane (K.C.) who suffered brain damage in a motor accident in 1981. Like Clive Wearing. K.C.'s hippocampus was destroyed in the injury and he lost all episodic memory. However, K.C. could still remember things he had learned in books, like dates or definitions (such as the difference between a stalagmite and a stalactite) - in other words, his semantic memory was still intact. This is evidence for a difference between episodic and semantic memory. Short video of an interview with K.C., who remembers facts but not personal experiences Great article on K.C.'s memory An example of this is the case study of K.F. who suffered brain damage in a motorbike accident. Like Clive Wearing and K.C., K.F. suffered damage to the temporal lobe which made it almost impossible for him to rehearse new memories. However, Shallice & Warrington (1970) report that K.F. could still remember episodes. This is a problem for the multi store model (on which Tulving's ideas are based) but it does suggest that episodic memory is a special type of LTM. Wait - K.C. and K.F.? Are they the same? No - they're different, but they both had names beginning with K and suffered brain damage from a motorbike accident. ObjectionsIt seems as if semantic and episodic memory both rely on each other and might not be all that separate. For example, if you learn that you husband or wife is unfaithful (episodic memory) you will probably trust them less (semantic memory) – which suggests that the two are linked.Damage to the temporal cortex of the brain seems to cause problems with both types of memory, as does dementia. This suggests declarative and non-declarative memory are located in the same place and may turn out to be the same thing working in different ways.Squire & Zola (1998) put this to the test. They examined children with amnesia (who never got a chance to acquire a semantic store in the first place) and adults with amnesia (who had semantic and episodic memories from before suffering brain damage). The participants' episodic and semantic memories seem to be equally impaired which supports the idea that the two memory functions are linked or even the same thing. episodic and semantic memory depend similarly on the medial temporal lobe - Squire & Zola What about K.C.? Squire & Zola propose that K.C.'s problems were due to damage to his frontal lobe - in other words, not a problem with his memories as such, more a problem with his ability to understand and make sense of his own memories.This leads ton the final criticism of Tulving's ideas - that it's really hard to define episodic and semantic memory in a measurable way. This means that Tulving's concepts are not operationalisable.. DifferencesTulving’s ideas tie in closely with Atkinson & Shiffrin's Multi Store Model of Memory, which proposes that LTM is a separate memory store from STM and that LTM is created through rehearsal. Tulving would agree, but argues there are different types of encoding, episodic and semantic. Shiffrin seems to have come round to this view and added Elaborative Rehearsal to his model in 2003.These ideas also link to the theory of Reconstructive Memory and Bartlett’s ideas about schemas. Schemas are meaningful patterns of information: they can be stereotypes, but they are also categories (“farm animals”, “kitchen appliances”) which might differ from person to person and culture to culture. In other words, they are separate semantic stores. If Tulving’s ideas are true, this makes Reconstructive Memory more plausible. If Reconstructive Memory is true, then semantic memory might have much more influence over episodic memory than Tulving imagined, because schemas influence how we reconstruct our memories. ApplicationsThe distinction between semantic memory and episodic memory helps us understand patients with memory loss like Clive Wearing, K.C. or people in the early stages of dementia. Though they may be confused by their amnesia, they might still remember relationships and meanings and this could be used to calm and focus them. Showing these patients meaningful things and getting them to talk about the meaning can be a type of Cognitive Stimulation Therapy – such as getting them to talk about how familiar songs or activities make them feel.The distinction should help you with your revision. No matter how charming or colourful your teacher’s explanations are, those are episodic memories that are specific to the time and place you encoded them – your Psychology lesson, not the exam hall. Semantic knowledge can be recalled anywhere, without needing “cues”, but to encode things semantically you have to understand them. This means revising by creating your own mind maps, category lists and charts. \| \| \| \| --- \| \| Don't start with description: start with evaluation. Evaluation point + evidence = "logical chain of reasoning" 4 of these "logical chains" are enough for a 8-mark question Conclusions are good if they are balanced - so long as you don't repeat yourself! \| EXEMPLAR ESSAYHow to write a 8-mark answer Evaluate Tulving's theory of Long Term Memory. (8 marks) A 8-mark “evaluate” question awards 4 marks for AO1 (Describe) and 4 marks for AO3 (Evaluate). You must include a conclusion to get the top band (7-8 marks) Tulving’s ideas are credible because they are supported by lab experiments like Baddeley (1966b). Baddeley showed that LTM is confused by word lists with similar meanings. LTM must be encoded semantically because similar sounding word lists had no such effect.However, an objection is that the supporting studies lack ecological validity because they are unrealistic. Learning lists of similar sounding words is not an ordinary activity. This means the theory is based on research that lacks validity.Tulving's theory is good because it has real-world applications. You could help dementia patients by giving activities like singing songs that are semantically meaningful for them then asking them about their feelings. This is Cognitive Stimulation Therapy.Semantic Memory is an improvement on the Multi Store Model. It suggests there may be different types of LTM just the way Working Memory suggests there may be different processes going on in STM. It also has similarities with Reconstructive Memory because semantic stores seem to be similar to schemas. In conclusion, semantic memory is a very important idea because it explains how we link our memories together and how we learn things like language. However, sometimes episodic and semantic memory seem very similar. For example, if you know the end of a joke (episodic memory) you stop finding it funny (semantic memory), which suggests the two may not be so different. Notice that for a 8-mark answer you don’t have to include everything in the theory. I haven’t mentioned procedural memory or the different parts of the brain. I haven’t described Schmolck’s research into semantic LTM and brain damage. But I have tried to make the two halves – Description and Evaluation – evenly balanced. Apply the the concepts of long term memory. A 4-mark “apply” question awards 4 marks for AO2 (Application) and gives you a piece of stimulus material. Greta has colour coded her Psychology revision, using blue ink for studies, red for theories, green for applications and pink for evaluations. Nigel figures he’ll remember his Psychology work in the exam because his teacher is always coming out with funny anecdotes. When they get their results, Greta has a much higher grade than Nigel.Using your knowledge of psychology, explain why Greta remembered her Psychology work better than Nigel. (4 marks)Semantic Memory would explain Greta’s memory. By colour coding her revision, she is putting it into semantic categories in LTM. Semantic categories are meaningful groupings.Whereas Nigel is depending on episodic LTM which contains memories of particular events. In the exam, he will have to try to recall a particular occasion when his teacher explained something.Tulving argues that episodic memory is perceptually and specifically encoded. This means it is hard to access episodic memories when you are in a different place and when there isn’t a context cue from the 5 senses. This would be a problem for Nigel in the exam hall.Semantic memory isn’t linked to any context. This makes it easier for Greta to recall information when she is not in her classroom and is not hearing her teacher’s voice. Notice that the question doesn't specify which memory theories you have to use. To get 4 marks for AO2, I’m making 4 clear and different applications of episodic and/or semantic memory I’m writing 4 paragraphs, hoping to get a point for each. Because this isn’t a 8-mark or 12-mark essay, I don’t need a conclusion. 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12796	https://my.clevelandclinic.org/health/body/23204-keratin	Abu Dhabi\|Canada\|Florida\|London\|Nevada\|Ohio\| Home/ Health Library/ Body Systems & Organs/ Keratin AdvertisementAdvertisement Keratin Your body naturally produces keratin, and keratin helps form your hair, nails and skin. Keratin products and treatments can help strengthen your hair and make it look brighter and feel softer. You can help your body produce keratin by eating keratin-rich foods. Advertisement Cleveland Clinic is a non-profit academic medical center. Advertising on our site helps support our mission. We do not endorse non-Cleveland Clinic products or services.Policy Care at Cleveland Clinic Find a Primary Care Provider Schedule an Appointment ContentsOverviewFunctionAnatomyConditions and DisordersCare Overview What is keratin? Keratin is a protein that helps form hair, nails and your skin’s outer layer (epidermis). It helps support your skin, heal wounds and keep your nails and hair healthy. Advertisement Cleveland Clinic is a non-profit academic medical center. Advertising on our site helps support our mission. We do not endorse non-Cleveland Clinic products or services.Policy There are 54 kinds of keratin in your body. There are two types: Type I: Of the 54 kinds of keratins in your body, 28 of them are type I. Of those, 17 are skin cell (epithelial) keratins, and 11 are hair keratins. Most type I keratins (cytokeratins) consist of acidic, low-weight proteins. They have many functions, including helping protect cells from internal forces in your body (mechanical stress). Type II: The other 26 kinds of keratins in your body are type II. Of those, 20 are skin cell keratins, and six are hair keratins. They consist of basic-neutral, high-weight proteins. Their basic-neutral pH helps balance type I keratins and govern cell activity. There are two forms of keratin: Alpha-keratin: Alpha-keratin is in the hair, epidermis, horns and nails of mammals. Type I and type II keratins are alpha-keratins. Beta-keratin: Beta-keratin is in the feathers, claws, beaks and scales of birds and reptiles. Care at Cleveland Clinic Find a Primary Care Provider Schedule an Appointment Function What does keratin do to the body? Keratin provides support and protection in your body. Your hair, nails and skin rely on the amount of keratin in your body for their overall health. Your glands and organs also contain keratin. Keratin is strong, so it won’t dissolve in diluted acids, alkalines, solvents or waters. Your body has many chemicals in it, and none of them affect keratin. Therefore, many believe that keratin treatments are beneficial for their hair, nails and skin. Advertisement Do I need keratin for my hair? Your body produces keratin naturally. Animal fur, feathers, hooves and horns also consist of keratin. The keratin in keratin hair treatments usually comes from ground-up animal parts, so if you’re a vegetarian, you may not want to use these products. Keratin is the primary component of hair, so many people believe that taking keratin supplements makes their hair stronger. However, there are no studies that conclude that keratin supplements make your hair stronger. Talk to your healthcare provider if you’re thinking of taking keratin supplements to discuss any risks and benefits. Many people also believe that shampoos and conditioners infused with keratin oil make their hair healthier. Studies have shown that shampoos and conditioners that contain keratin hydrolysates can make your hair stronger, brighter and softer. Getting a keratin hair treatment is a personal decision. It makes your hair shiny and silky while also reducing frizz. However, the treatment is usually expensive, and it may have negative effects on your body. Keratin hair treatments are chemical protein treatments that make your hair shiny and silky while also reducing frizz. They’re sometimes called Brazilian blowouts. During a keratin hair treatment, your hairstylist: Washes your hair with a special shampoo and then dries your hair with a towel. Applies a liquid keratin solution to your hair in small sections. Blow dries your hair. Runs a flat iron set to a high temperature through your hair. This seals the keratin solution to your hair. After your hair treatment, you shouldn’t get your hair wet for several days or pull it back with hair clips, scrunchies (elastics), hats or sunglasses. You may have to use special shampoos and other hair products to maintain the treatment. When performed by a hair care professional, your hair will be smooth and voluminous for up to six months. Anatomy Where is keratin located? Keratin is in your hair, nails and your skin’s outer layer, and it’s also in your glands and organs. What does keratin look like? Keratin can exist as alpha-keratins and beta-keratins according to the configuration of its polypeptide chains (the series of amino acids attached by peptide bonds). Alpha-keratins are mostly fibrous, and their structure looks like the thread of a screw (helical). Beta-keratins are sheets of polypeptide chains that extend in the same directions and never overlap (parallel). This construction gives beta-keratins their tough, rigid structure. What color is keratin? Hair, and the keratin within it, contains a pigment called melanin. Melanocytes are special cells that make melanin. Once made, melanin travels to other cells throughout your body. Melanin provides the pigment in your skin, hair and eyes. Advertisement Melanocytes produce two types of melanin that help determine the overall pigmentation that you have: Eumelanin: This type of melanin primarily makes dark colors in your hair, skin and eyes. Pheomelanin: This type of melanin primarily makes pink or red colors in your body, including lips, nipples, the vagina and the bulbous structure at the end of the penis (glans), as well as hair. As you get older, the amount of pigment in your hair’s keratin reduces. As a result, your hair turns gray and eventually white. What is keratin made of? Keratin isn’t a single substance. It consists of many different proteins, including various types of keratins, keratin-associated proteins (KFAPs) and enzymes drawn from animal tissues. Conditions and Disorders Can keratin hair treatments cause hair loss? Though uncommon, keratin hair treatments can sometimes cause hair damage and loss. Many keratin hair treatments contain formaldehyde, which is a chemical used as a germicide, fungicide and disinfectant. Funeral homes and medical laboratories also use formaldehyde in dead bodies so they don’t break down as quickly. Long-term exposure to formaldehyde may cause cancer. In addition to hair damage or hair loss, keratin hair treatments may cause: Chest pain. Coughing. Eye irritation. Nausea and vomiting. Rashes. Respiratory system problems. Advertisement What are conditions and disorders that affect keratin? Some conditions and disorders that affect keratin include: Epidermolysis bullosa simplex (EBS). EBS is a disease group in which your skin is delicate and develops blisters easily. Keratin gene mutations are most often the cause of EBS. Keratin cysts. Keratin cysts (epidermal inclusion cysts) are a common dome-shaped lump filled with keratin. Keratosis pilaris (KP). Keratosis pilaris is a common skin condition where small bumps develop on your arms, legs or butt. An excess of keratin clogs your pores, which causes the bumps. Monilethrix. Monilethrix is a rare disorder in which your hair breaks easily. It appears within the first few months of life, and it may also affect your eyebrows, eyelashes and body hair. A gene mutation in type II keratins causes monilethrix. Palmoplantar keratoderma (PPK). PPK is a disorder in which the top layer of your epidermis (stratum corneum) on the palms of your hands and soles of your feet becomes very thick. Keratin gene mutations cause PPK. What are common treatments for keratin conditions? Certain conditions, like KP, can be treated with over-the-counter moisturizing lotions, medicated creams, gently scrubbing (exfoliating) your skin or laser treatments. More severe conditions may require: A life-long application of urea ointments or other skin-smoothing creams. Avoiding environments that worsen side effects. Experimental treatments, like gene transfer. Advertisement Care Simple lifestyle tips to help maintain keratin Many foods help your body produce keratin. You can help your body boost its production of keratin by eating: Broccoli. Carrots. Eggs. Garlic. Kale. Salmon. Sweet potatoes. A note from Cleveland Clinic Keratin is a protein that your body produces naturally, and it helps keep your hair, skin and nails healthy and strong. Your body produces keratin naturally, but keratin shampoos and conditioners that contain keratin hydrolysates may strengthen your hair and improve its appearance. Keratin hair treatments can also improve your hair’s appearance and reduce frizz. Talk to your healthcare provider if you experience hair loss, coughing, eye irritation, rashes or other side effects after a keratin hair treatment. Care at Cleveland Clinic Cleveland Clinic’s primary care providers offer lifelong medical care. From sinus infections and high blood pressure to preventive screening, we’re here for you. Find a Primary Care Provider Schedule an Appointment Medically Reviewed Last reviewed on 06/09/2022. Learn more about the Health Library and our editorial process. 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12797	https://web.physics.ucsb.edu/~lecturedemonstrations/Composer/Pages/68.39.html	68.39 -- Force between current-carrying wires A video of this demonstration is available at this link. Two wires running parallel to each other are connected to a car battery through a double-pole, double-throw (DPDT) switch and a tap switch. When the DPDT switch is in one position, the currents in the wires flow in the same direction, so that when you close the tap switch, the wires attract each other. When you flip the DPDT switch, the currents in the wires flow in opposite directions, and when you press the tap switch, the wires repel each other. In 1820, Hans Christian Oersted – while performing a classroom demonstration! – discovered that a current in a wire produces a magnetic field around the wire. The relationship between this magnetic field and the current that produces it is given by Ampère’s law, which states ∮B ·dl = μ 0 i, where dl is an element of a circular path around the wire, μ 0 is the permeability constant, which equals 4π × 10-7 tesla·meter/ampere, and i is the current in the wire. By convention, current refers to the flow of positive charge, which is in the opposite direction to the flow of electrons. Evaluating the integral gives the magnitude of the magnetic field. ∮B ·dl = (B)(2π r) = μ 0 i, or B = (μ 0 i/2π r). The right-hand rule (see demonstration 68.13 – Right-hand rule model) gives its direction. Place your right hand near the current-carrying wire, with the thumb pointing in the direction of the current. Now curl the fingers around the wire. Your fingertips point in the direction of the magnetic field (north). In a way, this explains half of this demonstration. The other half is that a charge moving in a direction perpendicular to a magnetic field (or in a direction that has a component that is perpendicular to the magnetic field), experiences a sideways force. So, then, does a current flowing in a wire. This demonstration shows the effects of forces that act on a pair of parallel wires as a result of the current in each wire flowing through the magnetic field produced by the current in the other wire. In this demonstration, when you press the tap switch, current flows simultaneously in two parallel wires mounted next to each other. Each wire thus sits in the magnetic field produced by the current in the other wire. With the wires sitting as shown in the photograph above, in the space between the wires, the magnetic field produced by the current in each wire points either toward the back of the table, or out from the front of the table, depending on the direction of the current. With the red lead and tap switch connected to the positive terminal of the car battery, the current in the right-hand wire always flows downward. (Switching the DPDT switch changes the direction of the current in the left-hand wire.) So the magnetic field produced by the current in the right-hand wire always points toward the back of the table. We first note that a charge q moving in the left-hand wire, experiences a force Fq = qv × B, whose magnitude is F q = qvB sin θ. Since the current in the wire is perpendicular to the magnetic field, θ = 90°. If we consider electron flow, q = e, and v = v d, the drift speed of the electron in the wire, and F e = ev d B.v d = j/ne, where j is the current density (i/A, where A is the cross-sectional area of the wire), and n is the number of conduction electrons per unit volume in the wire. Thus, F e = jB/n. The length of wire in the field, l, contains nAl conduction electrons, so the total force on the wire is F = (nAl)F e = nAl(jB/n). Since jA = i, this gives F = ilB. When the current-carrying wire is not perpendicular to B, the general equation is F = il × B, which we must use if we wish to find the direction of the force. (If we are interested only in the direction of the force, considering a single charge and using F = qv × B gives, of course, the same result. Also, as noted above, by convention, current refers to the flow of positive charge. Electrons flow in the opposite direction, but they are also negative. The double change in sign means that whether we use conventional current or flow of electrons, we obtain the same direction for the force on the wire.) We now consider the operation of the demonstration. In all cases, the direction of B refers to its tangent between the two wires, where the field lines intersect the line between the two wires. Current flowing in opposite directions in the two wires: Current in the left-hand wire flows upward. B from the current in the right-hand wire points toward the back of the table, so for the left-hand wire, il × B points to the left. B from the current in the left-hand wire also points toward the back of the table. Current in the right-hand wire flows downward, so for the right-hand wire, il × B points to the right. With a leftward force on the left-hand wire, and a rightward force on the right-hand wire, the wires are pushed apart. Current flowing in the same direction in both wires: Current in the left-hand wire flows downward. B from the current in the right-hand wire points toward the back of the table, so for the left-hand wire, il × B points to the right. B from the current in the left-hand wire points out from the front of the table. Current in the right-hand wire flows downward, so for the right-hand wire, il × B points to the left. With a rightward force on the left-hand wire, and a leftward force on the right-hand wire, the wires are pulled together. The definition of the Ampere: The ampere (A) is the SI unit of current. One ampere equals one coulomb per second (C/s). From 1948 until 2019, the official definition of the ampere was based on an ideal version of an experiment performed in the 1820s by André-Marie Ampère, and was stated: The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2 × 10-7 newton per meter of length. As noted above, B = (μ 0 i/2π r), and F = ilB. So if we consider a wire carrying a current i, sitting in the magnetic field of a wire carrying a current i′, the force per unit length is F/l = (μ 0 ii′/2π r). From these equations and this definition of the ampere, it follows that μ 0 = 4π × 10-7 weber/(ampere·meter). The earlier definition of the ampere, adopted in the U.S. in 1893, was the current that would cause silver to deposit on a cathode at a rate of 1.118 mg/s. When it was adopted internationally in 1908, it was restated as the current equal to (1.118 mg/M) elementary charges per second, where M is the mass of a silver atom (in mg/mol). The new definition sets the value of the ampere by fixing the value of the elementary charge at 1.602176634 × 10-19 C, so that 1 s·A = 1 C. A fun fact: Power transformers comprise tightly-wound coils that have many turns of wire, with no space between the turns. All the turns in each coil are, of course, wound in the same direction, so current flows through all of them in the same direction. A DC current would cause all the turns to be pulled together. The current flowing through a power transformer, however, is AC at the line frequency of 60 Hz. Every half cycle, the current reaches a maximum. Between maxima, the current decreases, passes through zero and increases again. This means that every half cycle, the coil turns are pulled together, and between half cycles, they relax. This is why, no matter how tightly wound they are, power transformers hum. References: 1) Halliday, David and Resnick, Robert. Physics, Part Two, Third Edition (New York: John Wiley and Sons, 1977), pp. 677, 719, 721-2, 746-9. 2) Sears, Francis Weston and Zemansky, Mark W. College Physics, Third Edition (Reading, Massachusetts: 1960), pp. 645-7. 3) 4) 5) (Keller, Mark W., Elementary again).
12798	https://ned.ipac.caltech.edu/level5/March14/Mortonson/Mortonson4.html	4. CURRENT CONSTRAINTS ON EXPANSION, GROWTH, AND DARK ENERGY The last decade has seen dramatic progress in measurements of the cosmic expansion history and structure growth, leading to much tighter constraints on the parameters of dark energy models. CMB data from the WMAP and Planck satellites and from higher resolution ground-based experiments have provided an exquisitely detailed picture of structure at the recombination epoch and the first CMB-based measures of low redshift structure through lensing and SZ cluster counts. Cosmological supernova samples have increased in size from tens to many hundreds, with continuous coverage from z = 0 to z ≈ 1.4, alongside major improvements in data quality, analysis methods, and detailed understanding of local populations. BAO measurements have advanced from the first detections to 2% precision at multiple redshifts, with increasingly sophisticated methods for testing systematics, fitting models, and evaluating statistical errors. Constraints on low redshift structure from galaxy clusters have become more robust, with improved X-ray and SZ data and weak lensing mass calibrations, and they have been joined by the first precise structure constraints from cosmic shear weak lensing, galaxy-galaxy lensing, and redshift-space distortions. The precision of direct H0 measurements has sharpened from the ~ 10% error of the HST Key Project to 3-4% in some recent analyses. As an illustration of current measurements of the cosmic expansion history, Figure 1 compares distance-redshift measurements from SN and BAO data to the predictions for a flat universe with a cosmological constant. SN cosmology relies on compilation analyses that try to bring data from different surveys probing distinct redshift ranges to a common scale. The most influential current compilations are SNLS3 , which combines data from the 3-year Supernova Legacy Survey sample and the 1st-year SDSS-II Supernova Survey sample with local calibrators and high-redshift SNe from HST surveys, and Union2.1 , which has a broader selection of data, including some but not all of the sources in SNLS3. Here we have used binned distance measurements from Union2.1, but we caution that the different sample selections and analysis methodologies lead to systematic differences comparable to the statistical uncertainties, and it is not obvious which compilation, if either, should be preferred. Because the peak luminosity of a fiducial SN Ia is an unknown free parameter, the SN distance measurements could all be shifted up and down by a constant multiplicative factor; cosmological information resides in the relative distances as a function of redshift. The four BAO data points are taken from analyses of the 6dFGS survey , SDSS-II , BOSS , and WiggleZ . For the BAO measurements we have adopted the sound horizon scale rs = 147.49 Mpc from Planck CMB data, whose 0.4% uncertainty is small compared to the current BAO measurement errors . We have converted both SN luminosity distances and BAO Dv distances to an equivalent comoving angular diameter distance. \| \| \| \| Figure 1. The distance-redshift relation measured from Type Ia SNe and BAO compared to the predictions (gray curve) of a flat ΛCDM model with the best-fit parameters inferred from Planck+WP CMB data. Circles show binned luminosity distances from the Union2.1 SN sample, multiplied by (1 + z)-1 to convert to comoving angular diameter distance. Squares show BAO distance measurements, converted to DA,c(z) for the Planck+WP cosmology and sound horizon, from the references given in the text. The lower panel plots residuals from the Planck+WP ΛCDM prediction, with dashed curves that show the effect of changing w by ± 0.1 while all other parameters are held fixed. Note that the SN data points can be shifted up or down by a constant factor to account for freedom in the peak luminosity, while the BAO points are calibrated to 0.4% precision by the sound horizon scale computed from Planck+WP data. \| The plotted cosmological model has Ωm = 0.315 and h = 0.673, the best-fit values from Planck+WP CMB data assuming w = -1 and Ωtot = 1. Specifically, here and below we use parameter values and MCMC chains from the "Planck + WP" analysis of , which combines the Planck temperature power spectrum with low multipole polarization measurements from WMAP . In contrast to the Cosmological Parameters article of this Review, we do not use the CMB data set that includes higher resolution ground-based results because the corresponding chains are not available for all of the cases we wish to examine, but differences in cases where they are available are small. The SN, BAO, and CMB data sets, probing a wide range of redshifts with radically different techniques, are mutually consistent with the predictions of a flat ΛCDM cosmology. We have not included the z=2.5 BAO measurement from the BOSS Lyman-α forest on this plot, but it is also consistent with this fiducial model. Other curves in the lower panel of Figure 1 show the effect of changing w by ± 0.1 with all other parameters held fixed. However, such a single-parameter comparison does not capture the impact of parameter degeneracies or the ability of complementary data sets to break them, and if one instead forces a match to CMB data by changing h and Ωm when changing w then the predicted BAO distances diverge at z = 0 rather than converging there. Figure 2a plots joint constraints on Ωm and ΩΛ in a ΛCDM cosmological model, assuming w = -1 but not requiring spatial flatness. The SN constraints are computed from the Union2 sample, and the CMB, CMB+BAO, and CMB+BAO+SN constraints are taken from MCMC chains provided by the Planck Collaboration . We do not examine BAO constraints separately from CMB, because the constraining power of BAO relies heavily on the CMB calibration of rs. The SN data or CMB data on their own are sufficient to reject an ΩΛ = 0 universe, but individually they allow a wide range of Ωm and significant non-zero curvature. The CMB+BAO combination zeroes in on a tightly constrained region with Ωm = 0.309 ± 0.011 and Ωtot = 1.000 ± 0.0033. Combining SN with CMB would lead to a consistent constraint with around 3-4× larger errors. Adding the SN data to the CMB+BAO combination makes only a small difference to the constraints in this restricted model space. \| \| \| \| Figure 2. Constraints on the present matter fraction Ωm and dark energy model parameters. Dark and light shaded regions indicate 68.3% and 95.4% confidence levels, respectively. "CMB" is Planck+WP, "BAO" is the combination of SDSS-II, BOSS, and 6dFGS, and "SN" is Union2. (a) The present dark energy fraction ΩΛ vs. Ωm, assuming a ΛCDM model. CMB data, especially when combined with BAO constraints, strongly favor a flat universe (diagonal dashed line). (b) The dark energy equation of state w vs. Ωm, assuming a constant value of w. The dashed contours show the 68.3% and 95.4% CL regions for the combination of WMAP9 and BAO data. Curves on the left vertical axis show the probability distributions for w (normalized arbitrarily), after marginalizing over Ωm, for the CMB+BAO and CMB+BAO+SN combinations (yellow and black, respectively), using Planck+WP CMB data, and for the WMAP9+BAO combination (dashed black). (c) Constraints on the two parameters of the dark energy model with a time-dependent equation of state given by Equation 4: w(z = 0.5) and wa = -dw / da. \| Figure 2b plots constraints in the Ωm - w space, where we now consider models with constant w(z) and (in contrast to panel a) assume spatial flatness. CMB data alone allow a wide range of w, but combination with BAO narrows the allowed range sharply. The preferred region is consistent with the orthogonal SN constraint, and the combination of the three data sets yields smaller uncertainties. The black curve on the left axis shows the posterior p.d.f. for w after marginalizing (with a flat prior) over Ωm; we find w = -1.10 ± 0.08 at 68.3% CL and -1.10 ± 0.15 at 95.4% CL. The dashed contours and dashed marginal curve show the impact of substituting WMAP9 data for Planck+WP in the CMB+BAO combination. The two constraints are compatible, but the shift from WMAP to Planck+WP has reduced the uncertainty in w and pulled the best-fit value lower. Figure 2c considers a model space with time varying w, evolving according to the linear parameterization w(a) = w0 + wa(1 - a), again assuming flat space. Instead of w0 we show constraints on w(z = 0.5), approximately the pivot redshift where w is best determined and covariance with wa is minimized. This plot shows that even the combination of current CMB, BAO, and SN data places only weak constraints on time evolution of the equation of state, still allowing order unity changes in w between z = 1 and z = 0 (Δa = 0.5). The value of w(z = 0.5), on the other hand, is reasonably well constrained, with errors only slightly larger than those for the constant-w model of panel b. Errors on w0 = w(z = 0.5) - 0.333wa are much larger and are strongly correlated with the wa errors. While the CMB, BAO, and SN data sets considered here are mutually consistent with a flat ΛCDM model, tensions arise when other cosmological measurements enter the mix. Blue and yellow contours in Figure 3a show CMB and CMB+BAO constraints in the Ωm - H0 plane, assuming w = -1 and Ωtot = 1. Red horizontal bars represent the direct estimate H0 = 73.8 ± 2.4 km s-1 Mpc-1 from Ref. , who use SN Ia distances to galaxies in the Hubble flow with the Ia luminosity scale calibrated by HST observations of Cepheids in nearby SN host galaxies. Another recent estimate by Ref. , which employs 3.6 μm Cepheid observations to recalibrate the HST Key Project distance ladder and reduce its uncertainties, yields a similar central value and estimated error, H0 = 74.3 ± 2.1 km s-1 Mpc-1. Figure 3a indicates a roughly 2σ tension between these direct measurements and the CMB+BAO predictions. The tension was already present with WMAP CMB data, as shown in Figure 3b, but it has become stiffer with Planck+WP, because of smaller CMB+BAO errors and a shift of central values to slightly higher Ωm and lower H0. In models with free, constant w (still assuming Ωtot = 1), the tension can be lifted by going to w < -1 and lower Ωm, as illustrated in Figure 3c. CMB data determine Ωm h2 with high precision from the heights of the acoustic peaks, essentially independent of w. Within the flat ΛCDM framework, the well determined distance to the last scattering surface pins down a specific combination of (Ωm, h), but with free w one can obtain the same distance from other combinations along the Ωm h2 degeneracy axis. \| \| \| \| Figure 3. Constraints on the present matter fraction Ωm and the Hubble constant H0 from various combinations of data, assuming flat ΛCDM (left and middle panels) or a constant dark energy equation of state w (right panel). Dark and light shaded regions indicate 68.3% and 95.4% confidence levels, respectively. The right panel also shows 100 Monte Carlo samples from the CMB+BAO constraints with the value of w indicated by the colors of the dots. "CMB" is Planck+WP in the outer panels and WMAP9 in the middle panel, "BAO" is the combination of SDSS-II, BOSS, and 6dFGS, and "H0 (HST)" is the HST constraint from . \| One should not immediately conclude from Figure 3 that w ≠ -1, but this comparison highlights the importance of fully understanding (and reducing) systematic uncertainties in direct H0 measurements. If errors were reduced and the central value remained close to that plotted in Figure 3, then the implications would be striking. Other recent H0 determinations exhibit less tension with CMB+BAO, because of lower central values and/or larger errors [42, 43], including the values of H0 = 69 ± 7 km s-1 Mpc-1 and 68 ± 9 km s-1 Mpc-1 from Refs. [44, 45], who circumvent the traditional distance ladder by using maser distances to galaxies in the Hubble flow. Gravitational lens time delays offer another alternative to the traditional distance ladder, and their precision could become competitive over the next few years, with increasing sample sizes and better constrained lens models. The amplitude of CMB anisotropies is proportional to the amplitude of density fluctuations present at recombination, and by assuming GR and a specified dark energy model one can extrapolate the growth of structure forward to the present day to predict σ8. As discussed in Section 3 probes of low redshift structure typically constrain the combination σ8Ωmα with α ≈ 0.3-0.5. Figure 4 displays constraints in the σ8-Ωm plane from CMB+BAO data and from weak lensing and cluster surveys . Planck data themselves reveal a CMB lensing signature that constrains low redshift matter clustering and suggests a fluctuation amplitude somewhat lower than the extrapolated value for flat ΛCDM. However, including the CMB lensing signal only slightly alters the Planck+WP confidence interval for ΛCDM (purple vs. yellow contours in Fig. 4a). Allowing free w (gray contours) expands this interval, primarily in the direction of lower Ωm and higher σ8 (with w < -1). \| \| \| \| Figure 4. Constraints on the present matter fraction Ωm and the present matter fluctuation amplitude σ8. Dark and light shaded regions indicate 68.3% and 95.4% confidence levels, respectively. The upper left panel compares CMB+BAO constraints (using the same data sets as in Fig. 2) for ΛCDM with and without CMB lensing, and for a constant w model (including CMB lensing). The other three panels compare flat ΛCDM constraints between various dark energy probes, including weak lensing (upper right panel) and clusters (lower panels). \| The red contours in Figure 4b plot the constraint σ8(Ωm / 0.27)0.46 = 0.774-0.041+0.032 inferred from tomographic cosmic shear measurements in the CFHTLens survey . An independent analysis of galaxy-galaxy lensing and galaxy clustering in the SDSS yields a similar result , σ8(Ωm / 0.27)0.57 = 0.77 ± 0.05. Note that σ8 and Ωm refer to z = 0 values; the weak lensing samples and the cluster samples discussed below are not at zero redshift, but the values of σ8 are effectively extrapolated to z=0 for a fiducial cosmology. (Within current parameter bounds, the uncertainty in extrapolating growth from z = 0.5 to z = 0 is 1-2%, small compared to the observational uncertainties.) There is approximately 2σ tension between the σ8 - Ωm combination predicted by Planck+WP CMB+BAO for ΛCDM and the lower value implied by the weak lensing measurements. This tension was weaker for WMAP+BAO data (dotted contour) because of the larger error and slightly lower best-fit parameter values. Additional contours in Figures 4c and d show σ8 - Ωm constraints inferred from three representative cluster analyses : σ8(Ωm / 0.27)0.47 = 0.784 ± 0.027 (CPPP), σ8(Ωm / 0.27)0.41 = 0.806 ± 0.032 (MaxBCG), and σ8(Ωm / 0.27)0.32 = 0.782± 0.010 (PlanckSZ). The basic mass calibration comes from X-ray data in CPPP, from weak lensing data in MaxBCG, and from SZ data in PlanckSZ. Because the PlanckSZ constraint itself incorporates BAO data, we have replaced the CMB+BAO contour with a CMB-only contour in panel d. The σ8Ωmα constraints from recent cluster analyses are not in perfect agreement, and the examples shown here are far from exhaustive. Nonetheless, on balance the cluster analyses, like the weak lensing analyses, favor lower σ8Ωmα than the value extrapolated forward from Planck+WP assuming flat ΛCDM. Redshift-space distortion analyses also tend to favor lower σ8Ωmα, though statistical errors are still fairly large. For example, find f(z)σ8(z) = 0.415 ± 0.034 from SDSS-III BOSS galaxies at z = 0.57, while the best-fit Planck+WP+BAO flat ΛCDM model predicts f(z)σ8(z) = 0.478 ± 0.008 at this redshift. With somewhat more aggressive modeling assumptions, infer f(z) σ8(z) from the WiggleZ survey at z = 0.22, 0.41, 0.60, and 0.78, with ≈ 10% errors in the three highest redshift bins (and 17% at z = 0.22), finding excellent agreement with a flat ΛCDM model that has Ωm = 0.27 and σ8 = 0.8, and thus with the structure measurements plotted in Figure 4. Going from ΛCDM to wCDM does not readily resolve this tension, because the CMB degeneracy direction with free w is roughly parallel to the σ8 Ωmα tracks from low redshift structure (though the tracks themselves could shift or widen for w ≠ -1). Each of the low redshift probes has significant systematic uncertainties that may not be fully represented in the quoted observational errors, and the tensions are only about 2σ in the first place, so they may be resolved by larger samples, better data, and better modeling. However, it is notable that all of the discrepancies are in the same direction. On the CMB side, the tensions would be reduced if the value of Ωm or the optical depth τ (and thus the predicted σ8) has been systematically overestimated. The most exciting but speculative possibility is that these tensions reflect a deviation from GR-predicted structure growth, pointing towards a gravitational explanation of cosmic acceleration. Other possible physical resolutions could come from dark energy models with significant time evolution, from a massive neutrino component that suppresses low redshift structure growth, or from decaying dark matter that reduces Ωm at low z. Table 1 summarizes key results from Figures 2 - 4, with marginalized constraints on Ωm, Ωtot, w, h, and σ8(Ωm / 0.27)0.4 for the Planck+WP+BAO, Planck+WP+BAO+SN, and WMAP9+BAO combinations. We list 68.3% errors, and also 95.4% errors for WMAP9+BAO constraints on wCDM; in all other cases, the 95.4% errors are very close to double the 68.3% errors. For ΛCDM the Planck+WP combinations give Ωtot = 1.000 with an error of 0.3% and they predict, approximately, h = 0.68 ± 0.01 and σ8(Ωm / 0.27)0.4 = 0.87 ± 0.02. Note that the Ωm and h constraints are not identical to those in Table 21.1 of the Cosmological Parameters article of this Review because those values assume spatial flatness. For wCDM, where flatness is assumed, the Planck+WP+BAO+SN combination yields w = -1.10-0.07+0.08, consistent with a cosmological constant at 1.2σ. With free w the best-fit h increases and its error roughly doubles, but the error in σ8(Ωm / 0.27)0.4 grows only slightly, and its best-fit value moves a bit further away from the lower amplitudes suggested by measurements of low redshift structure. Table 1. Constraints on selected parameters from various combinations of CMB, BAO, and SN data, given as mean values ± 68.3% CL limits (and ± 95.4% CL limits for WMAP9-wCDM). "Planck+WP" combines the Planck temperature power spectrum with WMAP large scale polarization. "BAO" combines the measurements of SDSS-II, BOSS, and 6dFGS. "SN" refers to the Union2.1 compilation. The upper (lower) half of the table assumes a ΛCDM (flat wCDM) cosmological model. \| \| Data combination \| \| Parameter \| Planck+WP+BAO \| Planck+WP+BAO+SN \| WMAP9+BAO \| \| \| ΛCDM \| \| \| Ωm \| 0.309-0.011+0.010 \| 0.307-0.010+0.011 \| 0.295-0.012+0.012 \| \| Ωtot \| 1.000-0.0033+0.0033 \| 1.000-0.0033+0.0032 \| 1.003-0.004+0.004 \| \| h \| 0.678-0.010+0.011 \| 0.679-0.011+0.010 \| 0.681-0.011+0.011 \| \| σ8 (Ωm/0.27)0.4 \| 0.871-0.021+0.020 \| 0.869-0.021+0.020 \| 0.836-0.033+0.033 \| \| \| wCDM (flat) \| \| Ωm \| 0.287-0.021+0.021 \| 0.294-0.014+0.014 \| 0.299-0.019+0.022(-0.042+0.045) \| \| w \| -1.13-0.11+0.13 \| -1.10-0.07+0.08 \| -0.98-0.12+0.16(-0.29+0.33) \| \| h \| 0.708-0.030+0.026 \| 0.699-0.018+0.017 \| 0.681-0.032+0.025(-0.066+0.060) \| \| σ8 (Ωm/0.27)0.4 \| 0.888-0.025+0.025 \| 0.885-0.023+0.023 \| 0.84-0.05+0.05(-0.09+0.09) \| \|
12799	https://www.scribd.com/presentation/620158991/Lecture-2-Series-telescoping-and-type-of-series-stu	Lecture 2 - Series - Telescoping and Type of Series Stu Lecture 2 - Series - Telescoping and Type of Series Stu Uploaded by AI-enhanced title and description Lecture 2 - Series - Telescoping and Type of Series Stu This document provides an overview of Lecture 2 which discusses sequences, series, and telescoping series. It includes examples of determining if a series converges or diverges and examples of using telescoping series to find the sum of infinite series. It also lists different types of series including geometric, harmonic, and p-series. The examples demonstrate setting up partial fractions, comparing coefficients to solve simultaneous equations, and evaluating limits to determine if the series converges. Lecture 2 - Series - Telescoping and Type of Series Stu Lecture 2 - Series - Telescoping and Type of Series Stu Uploaded by AI-enhanced title and description Share this document Footer menu About Support Legal Social Get our free apps About Legal Support Social Get our free apps

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