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# what is category example

Simple Category Scale: Definition, Example. Immediately, the surrounding retail shops such as groceries and hardware will see a drop in sale. Category tactics will often include price changes, so it’s necessary to know how other products are priced, what other stores are doing, and what the market trends and consumer expectations are when it comes to price. if both $\text{ob}(\mathsf{C})$ (all the objects in $\mathsf{C}$) and $\text{hom}_{\mathsf{C}}(x,y)$ are honest-to-goodness sets. Another example of a category formed from existing ones is \mathsf {Cat}. Therefore there is an arrow $x\overset{\text{id}_x}{\longrightarrow} x$ satisfying $f\circ\text{id}_x=f$ for any morphism $f$. Matt is the Sr. (Or, again, is it merely a function?). La documentation de ce modèle est générée par le modèle {{Documentation}} . Examples of category in a Sentence. Furthermore, management can make changes at the category-level. Professionals need to know what is going on at the shelf-level. En tant qu'outil, il correspond bien aux objectifs de l'ECR. be sure to follow the link to find out! alexmarcelo 2259465 The two are not in the same category. 1. No matter what, effective category management requires actionable data to support and guide your business decisions. Here, the objects are categories*** $\mathsf{C},\mathsf{D},\mathsf{E}, \mathsf{F},\ldots,$ and the morphisms between them are functors. Given two categories $\mathsf{C}$ and $\mathsf{D}$, you might try defining their product $\mathsf{C}\times\mathsf{D}$. Paramètres du modèle La mise en forme sur une seule ligne est préférée pour ce modèle. You can also construct a new category $\mathsf{D}^\mathsf{C}$ from given categories $\mathsf{C}$ and $\mathsf{D}$ by declaring the objects to be functors $F:\mathsf{C}\to\mathsf{D}$. (What would the picture for this look like?) I also mentioned briefly that every poset (a set with a binary relation that is reflexive, transitive, and antisymmetric) is a category. Beyond that, this type of category role also includes in-depth analysis. The objects are the elements of $P$ and there is a morphism $x\to y$ whenever $x\leq y$. This is one of the harder retail concepts to define, as category management can mean different things to different people. Its purchasing power allows it to offer lower prices than the competition, and the size of its retail spaces allows it to compete with a wider variety of categories. It has the capacity to add … The Word "Category" in Example Sentences Page 1 . Category:Example. Below is a listing of all of the current examples, which will be added to over time. Is this composition associative? _undertoad 55111 These specimens are divided into several categories. *A category $\mathsf{C}$ where for all objects $x,y$, $\text{hom}_{\mathsf{C}}(x,y)$ is small enough to be an honest-to-goodness set is called a locally small category. Furthermore, management can make changes at the category-level. The category Setwith objects sets and morphisms the usualfunctions. Additionally, Lowes created different ways to shop (color, size, look), that make it easy for users to find the right product. 36. What do you think the objects in $\mathsf{C}\times\mathsf{D}$ should be? Also learn the facts to easily understand math glossary with fun math worksheet online at SplashLearn. Benjamin C. Pierce , p. 8 dans Basic Category Theory for Computer Scientists, considère un langage de programmation fonctionnelle où les types sont Bool (pour les booléens, i.e. CK 1 1006157 This doesn't fit into any category. A product category is a type of product or service. Product categories are typically created by a firm or industry organization to organize products. Definition and Examples for Retail Professionals, Understanding the market and consumer behavior, Planning changes and improvements to the category, Implementing the plan and measuring the results. Stakeholders will also want to monitor prices on the products in their categories. This can centralize procurement under one person or team. Category management is the process of bundling like products into a singular category, or business unit, and then addressing procurement, merchandising, sales, and other retail efforts on the category as a whole. Nouns, for instance, can be further subdivided into number, gender , case, and countability. We deployed tools such as Markov Chain Analysis, and clustering to help us to better understand the products in the category as well as how customers make decisions about products. … ** In particular one could ask, How can I make a new category from old ones? 2196900 How many categories are there? (This paves the way for monoidal categories.) Lastly, it’s a must to have high-quality insights into shopper behavior. You can use categories to sort and group your blog posts into different sections. For example, an online shoe store that offers a single product in many types of styles and brands should use a mega menu. Category Management is a strategic approach to procurement where organisations segment their spend into areas which contain similar or related products enabling focus opportunities for consolidation and efficiency. pour les valeurs de vérité), Int (pour les nombres entiers), Real (pour les nombres réels) et Unit (un type avec un seul élément). Elle est incluse depuis sa sous-page de documentation . In general, category management is the process of bundling like products into a singular category, or business unit, and then addressing procurement, merchandising, sales, and other retail efforts on the category as a whole. For example, respondents on a political issue may be asked whether they agree or … This category page design example works because it fits with Lowes’ brand and keeps the brand image consistent. In other words, a category is locally small if there aren't too many morphisms between the objects. But we should check: is the composition of two group homomorphisms another group homomorphism? Administrators: Please do not delete this category even if it is empty! Le category management repose par conséquent sur un partenariat réel et efficace avec les fournisseurs de l'enseigne, via un transfert d'informations bidirectionnel et continu. We could have deduced this from the next example as well. Every poset $P$ forms a category. If a mining company is planning to expand its operations in the next 3 years, for example, category management in heavy machinery and equipment gives a more streamlined strategy, as the company will be able to identify suppliers and capital in advance. Thankfully, there are plenty of tools you can leverage to take control of your category management program and set your business up for success. In this example from Sephora, they display main categories on top, followed by a row showcasing ‘just arrived’ items below. in fact, now Walmart has such an assortment of goods that you hardly need to step out of the store to get anything. The identity-morphism property holds because of reflexivity: $x\leq x$ is always true. Pages in category "Examples" The following 59 pages are in this category, out of 59 total. In grocery, for example, most stores are organized by category (dairy, produce, meat, and more) which makes it easy to navigate stores and find specific products. For starters, an organization can procure goods on a category level, not a product, brand, or store level. 32. Managers need to know the data behind their category—what is working, what isn’t, and what can be done to improve. Which products resonate with consumers? (The notation $\mathsf{D}^\mathsf{C}$ is quite suggestive.) Product categories can be organized as a hierarchy of categories that resemble a tree structure or can be a flat structure such as a list of product types. We'll chat about functors next week and natural transformations the following week. Examples.com includes examples (definitions, guidelines, how-to’s, real-life situations) of business-related topics such as human resources, marketing, policies, checklists, contracts, agreement templates, and budgets. Category:Example The SEBoK provides a variety of examples that demonstrate how systems engineering principles and practices have been applied in a real world context. And is the identity function from a group to itself a group homomorphism? A mega menu can offer a smoother and a simpler way for the users to browse shoes in different brands, types, and sizes directly from the menu. I'm jumping the gun a bit on this one since I haven't defined functors on the blog yet, but I think it's worth a mention. For example, to form the category of groups, we declare the objects to be groups and the morphisms to be group homomorphisms. Les délimitations des catégories de produits correspondent généralement à un univers de consommation donné comme par exemple les produits liés au petit déjeuner. Les éditeurs peuvent travailler dans le bac à sable (créer) et la page de test (créer) . Veuillez placer les catégories sur cette page-là. In the special case when $\mathsf{D}=\mathsf{Set}$, the objects of $\mathsf{Set}^{\mathsf{C}^{op}}$ are given the name presheaves. Associativty holds too. Therefore, they need accurate, real-time store insights on their categories. For some, it’s all about procurement: purchasing the right products and services for that category that will generate the most sales and profit for the business. For instance, a word can be both plural and feminine. They can launch new promotions, planograms, and much more across a single business unit. ***For size considerations, the categories in $\mathsf{Cat}$ are assumed to be small. It is not clear, for example, to which category it is necessary to refer the excretory organs of Arenicola, or Polynoe. Most of the categories you're familiar with--$\mathsf{Set},\mathsf{Top},\mathsf{Group}$, etc.--are locally small. But along with that, you could feature best selling items from one or two categories to further bolster sales. Another example of a category formed from existing ones is  $\mathsf{Cat}$. Categories: 1 Albums: 1,823 Uploaded Media: 13,293 Embedded Media: 9 Comments: 1,692 The … In particular, we've just shown that $\mathbb{Z},\mathbb{Q}$, and $\mathbb{R}$ are all categories! Again, I'm jumping the gun here, but I expect these examples to resurface later on the blog, so now is a good time to mention them. How a company approaches this strategic concept really depends on the organization in question. This can be data on planogram compliance, out-of-stocks, display compliance, facings, adjacencies, and much more. Your category scorecard is NOT the same as corporate monthly reports that measure overall business results. is_category( array( 9, 'blue-cheese', 'Stinky Cheeses' ) ); // Returns true when the category of posts being displayed is either term_ID 9, // or slug "blue-cheese", or name "Stinky Cheeses". For example, if you’re an apparel store, you’d still display categories by gender, type, season, etc. Il s'utilise principalement dans les descriptions de catégories ou sur d'autres page en dehors de l'espace encyclopédique. For example, a news website might have categories for their articles filed under News, Opinion, Weather, and Sports. Strategy In the last phase of the process we aimed to set the Category Definition and Segmentation. (Check that $\mathsf{C}^{op}$ really does satisfy the definition of a category.) An example category. Finally, the group identity $e\in G$ serves as the identity morphism for $\bullet$, and associativity holds because the group operation is associative. Almost every known example of a mathematical structure with theappropriate structure-preserving map yields a category. The morphisms? $$This turns out to be a useful example, so let's be explicit. As we all know, this isn’t as simple as it sounds. Utilisation. That means you can create your own strategic approach that fits your needs. In reality, there is much more to category management. Ce modèle permet de créer un ou plusieurs liens vers une ou plusieurs catégories connexes, c’est-à-dire proches du sujet de la catégorie actuelle. If you're just now tuning in and are wondering what is category theory, anyway? There are variants here: one can consider partial functionsinstead, or injective functions or again surjective functio… Example, in the luxury cars product category, cars such as the BMW, Mercedes and Audi compete with each other.$$. As promised, here is the first in our triad of posts on basic category theory definitions: categories, functors, and natural transformations. With that in mind, a category manager job could easily involve: Effective category management can then rely on a wide range of tools and resources in the retail industry. They can launch new promotions, planograms, and much more across a single business unit. Le category management est une procédure d'optimisation des ventes d'une catégorie de produits qui est généralement menée conjointement par une enseigne et un ou plusieurs fabricants industriels. The last two works, if not actually the works of eye-witnesses, are at any rate first-hand, and belong to the category of primary writers rather than to that of later compilations. Don't miss out on our latest blog posts, whitepapers and infographics. CM 939738 The category of a hurricane depends on its wind speed. In other words, homCop(x, y) = homC(y, x). Verbs can be subdivided by tense, aspect, or voice . (Or is it merely a function?) Thus Walmart in essence is a retail category killer. I encourage you to revisit Example #3 once you're familiar with these notions. Home / What is Category Management? Effective category managers need data on planogram compliance, out-of-stocks, display compliance, facings, adjacencies, and much more. Manager, Marketing & Demand at Wiser, the leading provider of actionable data for better decisions. Jump to navigation Jump to search. All of which lead to a surprisingly innovative looking attribute-driven Category Decision Tree which almost resembled a traditional decision tree – inverted. functors, between categories. Definition and Examples for Retail Professionals. At a broad level, category management is simple: to improve the customer experience, drive efficiencies for businesses, and deliver quality products and services that generate profits. \text{hom}_{\mathsf{C}^{op}}(x,y)=\text{hom}_{\mathsf{C}}(y,x). Category is one of the default taxonomies in WordPress. Any business benefits a lot from adopting category management as one of its best practices in procurement. All the photos are the same style, creating a cohesive product category design. The cars belong to the same category. All in all, category management is what you make of it. Below, we'll summarize the category review with an example Category Strategy & Scorecard for Jones Grocery's ice cream category. A single Walmart store may have groceries, electronics, home decor, and gardening supplies. datrukup 939740 Category 5 hurricanes are the worst. Depending on who you ask, this concept can be defined, approached, and completed in different ways. Each subcategory has a wide array of examples users can use as guidelines or even as an inspiration to write their own business documents. First of all, what is category management? Definition of Category explained with real life illustrated examples. In its most basic form, attitude scaling requires that an individual agrees or disagrees with a statement or responds to a single question. Brands and retailers need to survey shoppers regularly to find out or turn to data providers to gather information on consumer behavior and buying trends. Which promotions and displays are most effective? A category is essentially any group of similar items which the company wishes to buy under a single deal. Adding a post or page to a website should always include assigning it to a category, and you don't have to limit it to just one. In other words, A category $\mathsf{C}$ is called small if it doesn't contain too many objects and if there aren't too many morphisms between those objects, i.e. An Example – Imagine a Walmart opening in your neighborhood. She competed for the award in her age category. So naturally, we might take interest in relationships, i.e. That is, the objects in $\mathsf{C}^{op}$ are exactly the same as objects in $\mathsf{C}$, but there is a morphism $x\to y$ in $\mathsf{C}^{op}$ whenever there is morphism $y\to x$ in $\mathsf{C}$. Pour citer une catégorie de Catégorie:modèle de source, ou catégoriser un modèle.Utile pour insérer un Modèle de page sans que ce modèle soit lui même catégorisé (exemple d’utilisation sur Utilisateur:TomT0m/exemples) . Here, the objects are categories*** \mathsf {C},\mathsf {D},\mathsf {E}, \mathsf {F},\ldots, and the morphisms between them are functors. Category management can also be used to improve the customer experience. Here a morphism between two functors $F:\mathsf{C}\to\mathsf{D}$ and $G:\mathsf{C}\to\mathsf{D}$ is a natural transformation, often denoted by a double arrow $F\Longrightarrow G$. Additionally, every category $\mathsf{C}$ has an opposite category $\mathsf{C}^{op}$. is_category( 'blue-cheese' ); // When the archive page for the Category with Category Slug "blue-cheese" is being displayed. Recent Examples on the Web Not surprisingly tech companies were the biggest category—accounting for 14 of the 53 Blue Ribbon Companies. In future posts, let's always assume our categories have this property, unless otherwise stated. This category may be empty occasionally or even most of the time. That is, if $g,h\in G$ so that $\bullet\overset{g}{\longrightarrow}\bullet\overset{h}{\longrightarrow}\bullet$ are two composable morphisms, then there is a third morphism $\bullet\overset{hg}{\longrightarrow}\bullet$ that is their composition, namely the one corresponding to the element $hg\in G$. You could focus on purchasing, assortment, visual merchandising, pricing, inventory management, promotions, or something else! Category management can also be used to improve the customer experience. A very natural question that you might ask upon discovering new mathematical ideas is How can I construct new objects from given ones? 35. Last time I listed a few examples of categories but never provided the proofs. Taxpayers fall into one of several categories. What Category Scorecard should NOT DO? Composition holds because of transitivity: if $x\leq y$ and $y\leq z$ then $x\leq z$. // Note: the array ability was added in version 2.5. There are several ways to go about this. What does category management look like in real life? The gray arrow at each object indicates its identity morphism. A central idea behind this strategy is to emphasize the benefits of the category for the consumer and remove inefficiencies and unprofitable competition among brands and suppliers within a category. How the planograms look, which displays and promotions drive the most sales, and which products are most desirable for consumers. **For instance, the intersection of sets is another set, the direct product of groups is another group, the tensor product of vector spaces is another vector space, the product of compact topological spaces is another compact topological space, and so on. You can see it in action on the Nike website’s mega menu. A word can be classified into more than one grammatical category. Categories: 1 Albums: 1,824 Uploaded Media: 13,300 Embedded Media: 9 Comments: 1,692 To make sure you're comfortable with the definition, try proving a few of them. What do they consider good value for a product in this category? This concept gives category managers multiple options to solve retail challenges. Walmart is an example of a retail store that effectively uses large spaces to achieve category-killing market dominance. This can include a hierarchy of categories that resemble a tree structure. Every group $G$ can be viewed as a category---called $\mathsf{B}G$ (for cool reasons)---with a single object which we'll denote by $\bullet$. An example category. For instance, this approach can help retailers lift profits on similar products in multiple ways, including by organizing procurement efforts under a single category instead of by individual brand or supplier. Customer Examples Discover which leading brands choose BigCommerce; Help Center ... and category pages serve as an index of all the pages and posts that belong to that particular subject. 34. Perhaps you focus on all of these items. Alternatively, product categories can be a flat structure such as a list of product types. There is a morphism $\bullet\overset{g}{\longrightarrow}\bullet$ for each element $g\in G$, and composition holds since $G$ is closed under the group operation. A category $\mathsf{C}$ consists of some data that satisfy certain properties: Here's a picture of a category with four objects--depicted as bold dots instead of letters--and some morphisms between them. How you can use this category page design for inspiration. ©2017-2021 Wiser Solutions, Inc. All Rights Reserved, What is Category Management? For others, it’s also about the category on the shelf. As we discussed last time, it's helpful to think of a morphism/arrow as a relationship between its domain and codomain. That can simplify negotiations with suppliers and save the business time and money. This way, category managers can provide the most value to both their customers and their companies. See More. In grocery, for example, most stores are organized by category (dairy, produce, meat, and more) which makes it easy to navigate stores and find specific products. The only difference between a category and its opposite is the direction in which the arrows/morphisms point.


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Longwood Station

A pleasure to see this station so empty. Something here is just so captivating. IMG_3705
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chapter  12
Apraxia and the Inability to Learn: A Repiise of Previous Work
Pages 12

The possibility that maladaptation might be caused by psychological deficits, rather than by intrapsychic conflict alone, was considered as early as 75 years ago, when Ferenczi set out to trace the course of ego development (see Gedo, 1976). In the course of his pioneering studies of syndromes be yond the boundaries of the neuroses, he came to realize that psychoanalysis had no explana tion for the occurrence of catastrophic struc tural changes, such as the breakthrough of ahomosexual libido" in paranoia. In 191 1 Fe renczi concluded that the conflict about ho mosexuality to be found in these patients might not constitute the fundamental element of their psychopathology. Freud's famous dic tum (1905b) that a neurosis is like the unega tive" of a perversion (a photographic meta phor!) characterizes perversion as a structural deficiency. Subsequently, ego psychologists

(Hartmann, 1953) conceptualized the psychoses and alloplastic charac ter problems as conditions of aego weakness/' that is, a failure to develop adequate adaptive/defensive structures (see also Gitelson, 1958).


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# Collaborative Filtering

Since datasets on conventional recommender systems are commonly represented as a user-item matrix $R \in \mathbb{R}^{|\mathcal{U}| \times |\mathcal{I}|}$, this page specially focuses on the matrix completion techniques for recommendation. A key idea of matrix-completion-based recommendation is to predict missing values in $R$ and figure out the most promising items to a target user as follows:

To give an example, for a user $a$, item $2$ and $4$ are unobserved. In this case, what a recommender does is to predict the missing elements and decides which item is more likely to be preferred by the user.

In particular, collaborative filtering (CF) is one of the most popular matrix-completion-based recommenders which was originally introduced by Goldberg et al. in 1992. The goal of CF algorithm is to suggest new items for a particular user by modeling user-user and item-item similarities. From a users' perspective, CF assumes that users who behaved similarly on a service share common tastes for items. On the other hand, items which resemble each other are likely to be preferred by the same users.

## $k$-Nearest Neighbor

A $k$-nearest neighbor ($k$-NN) approach, one of the simplest CF algorithms, runs in two-fold. First, missing values in $R$ is predicted based on the past observations. Here, a $(u, i)$ element between a target user $u$ and item $i$ is estimated by explicitly computing the similarities of users (items). Second, a recommender chooses top-$k$ items from the results of the prediction step.

Importantly, $k$-NN can be classified into a user-based and item-based algorithm. In a user-based algorithm, user-user similarities are computed for every pairs of rows in $R$. By contrast, item-based CF stands on column-wise similarities between items. Following figure illustrates how $k$-NN works on a user-item matrix $R$:

Note that the figure is inspired by a paper written by Sarwar et al.

The elements are ratings in a $[1, 5]$ range for each user-item pair, so $1$ and $2$ mean relatively negative feedback and vice versa. In the figure, user $a$ and $c$ seem to have similar tastes because both of them gave nearly identical feedback to item $1$, $4$ and $6$. From an item-item perspective, item $4$ and $6$ are similarly rated by user $a$, $b$ and $c$.

In order to measure the similarities between rows (columns), the Pearson correlation and cosine similarity are widely used. For $d$-dimensional vectors $\mathbf{x}, \mathbf{y} \in \mathbb{R}^d$, the Pearson correlation $\mathrm{corr}(\mathbf{x}, \mathbf{y})$ and cosine similarity $\mathrm{cos}(\mathbf{x}, \mathbf{y})$ are respectively defined as:

$\mathrm{corr}(\mathbf{x}, \mathbf{y}) = \frac{\sum_i (x_{i} - \overline{x})(y_{i} - \overline{y})}{\sqrt{\sum_i (x_{i} - \overline{x})^2} \sqrt{\sum_i (y_{i} - \overline{y})^2}},$
$\mathrm{cos}(\mathbf{x}, \mathbf{y}) = \frac{\mathbf{x} \cdot \mathbf{y}}{\| \mathbf{x} \| \| \mathbf{y} \|} = \frac{\sum_i x_{i} y_{i}}{\sqrt{\sum_i x_{i}^2} \sqrt{\sum_i y_{i}^2}},$

where $\overline{x} = \frac{1}{d} \sum^d_{i=1} x_i$ and $\overline{y} = \frac{1}{d} \sum^d_{i=1} y_i$ denote mean values of the elements in a vector. Additionally, in a context of data mining, elements in $\mathbf{x}$ and $\mathbf{y}$ can be distributed on a different scale, so mean-centering of the vectors usually leads better results as shown by Sarwar et al. Notice that cosine similarity between the mean-centered vectors, $\hat{\mathbf{x}} = (x_1 - \overline{x}, x_2 - \overline{x}, \dots, x_n - \overline{x})$ and $\hat{\mathbf{y}} = (y_1 - \overline{y}, y_2 - \overline{y}, \dots, y_n - \overline{y})$, is mathematically equivalent to the Pearson correlation $\mathrm{corr}(\mathbf{x}, \mathbf{y})$ as follows:

$\mathrm{cos}(\hat{\mathbf{x}}, \hat{\mathbf{y}}) = \frac{\sum_i (x_{i} - \overline{x})(y_{i} - \overline{y})}{\sqrt{\sum_i (x_{i} - \overline{x})^2} \sqrt{\sum_i (y_{i} - \overline{y})^2}} = \mathrm{corr}(\mathbf{x}, \mathbf{y}).$
Recommendation.UserKNNType
UserKNN(
data::DataAccessor,
k::Int,
normalize::Bool=false
)

User-based CF using the Pearson correlation. k represents number of neighbors, and normalize specifies if weighted sum of neighbors' rating is normalized.

The technique gives a weight to a user-user pair by the following equation:

$s_{u, v} = \frac{ \sum_{i \in \mathcal{I}^+_{u \cap v}} (r_{u, i} - \overline{r}_u) (r_{v, i} - \overline{r}_v)} { \sqrt{\sum_{i \in \mathcal{I}^+_{u \cap v}} (r_{u,i} - \overline{r}_u)^2} \sqrt{\sum_{i \in \mathcal{I}^+_{u \cap v}} (r_{v, i} - \overline{r}_v)^2} },$

where $\mathcal{I}^+_{u \cap v}$ is a set of items which were observed by both user $u$ and $v$, and $r_{u, i}$ indicates a $(u, i)$ element in $R$. Plus, $\overline{r}_u$ and $\overline{r}_v$ are respectively mean values of $r_{u, i}$ and $r_{v, i}$ over $i \in \mathcal{I}^+_{u \cap v}$. By using the weights, user-based CF selects the top-$k$ highest-weighted users (i.e., nearest neighbors) of a target user $u$, and predicts missing $r_{u, i}$ for $i \in \mathcal{I}^-_u$ as:

$r_{u, i} = \overline{r}_u + \frac{\sum^k_{t=1} \left(r_{\sigma(t), i} - \overline{r}_{\sigma(t)}\right) \cdot s_{u,\sigma(t)} }{ \sum^k_{t=1} s_{u,\sigma(t)} },$

where $\sigma(t)$ denotes the $t$-th nearest-neighborhood user. Ultimately, sorting items $\mathcal{I}^-_u$ by the predicted values enables us to make recommendation to a user $u$.

It should be noted that user-based CF is highly inefficient because gradually increasing massive users and their dynamic tastes require us to frequently recompute the similarities for every pairs of users.

source
Recommendation.ItemKNNType
ItemKNN(
data::DataAccessor,
k::Int
)

Item-based CF that provides a way to model item-item concepts by utilizing the similarities of items in the CF paradigm. k represents number of neighbors.

Item properties are relatively stable compared to the users' tastes, and the number of items is generally smaller than the number of users. Hence, while user-based CF successfully captures the similarities of users' complex tastes, modeling item-item concepts could be much more promising in terms of both scalability and overall accuracy.

Item-based CF defines a similarity between an item $i$ and $j$ as:

$s_{i,j} = \frac{ \sum_{u \in \mathcal{U}_{i \cap j}} (r_{u, i} - \overline{r}_i) (r_{u, j} - \overline{r}_j)} { \sqrt{\sum_{u \in \mathcal{U}_{i \cap j}} (r_{u,i} - \overline{r}_i)^2} \sqrt{\sum_{u \in \mathcal{U}_{i \cap j}} (r_{u, j} - \overline{r}_j)^2} },$

where $\mathcal{U}_{i \cap j}$ is a set of users that both of $r_{u,i}$ and $r_{u, j}$ are not missing, and $\overline{r}_i, \overline{r}_j$ are mean values of $i$-th and $j$-th column in $R$. Similarly to the user-based algorithm, for the $t$-th nearest-neighborhood item $\tau(t)$, prediction can be done by top-$k$ weighted sum of target user's feedbacks:

$r_{u,i} = \frac{\sum^k_{t=1} s_{i,\tau(t)} \cdot r_{u,\tau(t)} }{ \sum^k_{t=1} s_{i,\tau(t)} }.$

In case that the number of items is smaller than users, item-based CF could be a more reasonable choice than the user-based approach.

source

## Singular Value Decomposition

Along with the development of the $k$-NN techniques, researchers noticed that handling the original huge user-item matrices is computationally expensive. Moreover, $k$-NN-based recommendation leads overfitting to individual taste due to the sparsity of $R$. Thus, dimensionality reduction techniques were applied to recommendation in order to capture more abstract preferences.

Singular value decomposition (SVD) is one of the most popular dimensionality reduction techniques which decomposes an $m$-by-$n$ matrix $A$ to $U \in \mathbb{R}^{m \times m}$, $\Sigma \in \mathbb{R}^{m \times n}$ and $V \in \mathbb{R}^{n \times n}$:

$\mathrm{SVD}(A) = U \Sigma V^{\mathrm{T}} = \left[\mathbf{u}_1, \mathbf{u}_2, \cdots, \mathbf{u}_m\right] \mathrm{diag}\left(\sigma_1, \sigma_2, \dots, \sigma_{\min(m, n)}\right) \left[\mathbf{v}_1, \mathbf{v}_2, \cdots, \mathbf{v}_n\right]^{\mathrm{T}},$

by letting $\sigma_1 \geq \sigma_2 \geq \cdots \geq \sigma_{\min(m, n)} \geq 0$. An orthogonal matrix $U$ ($V$) is called left (right) singular vectors which represents characteristics of columns (rows) in $R$, and a diagonal matrix $\Sigma$ holds singular values on the diagonal elements as weights of each singular vector.

The reason why SVD can be applied to dimensionality reduction is that the most lower singular values of real-world matrices are very close to zero. Hence, using only top-$k$ singular values $\Sigma_k \in \mathbb{R}^{k \times k}$ and corresponding singular vectors $U_k \in \mathbb{R}^{m \times k}$, $V_k \in \mathbb{R}^{n \times k}$ is sufficient to make reasonable rank-$k$ approximation of a matrix $A$. The following figure demonstrates how SVD decomposes a matrix into rank-$k$ matrices.

On top of that, $\mathrm{SVD}_k(A)$ is known as the best rank-$k$ approximation of the matrix $A$ in both the spectral and Frobenius norm. Note that the spectral norm of a matrix equals to its largest singular value.

Recommendation.SVDType
SVD(
data::DataAccessor,
k::Int
)

Recommendation based on SVD of a user-item matrix $R \in \mathbb{R}^{|\mathcal{U}| \times |\mathcal{I}|}$, which was originally studied by Sarwar et al. Rank $k$ is configured by k.

In a context of recommendation, $U_k \in \mathbb{R}^{|\mathcal{U}| \times k}$, $V \in \mathbb{R}^{|\mathcal{I}| \times k}$ and $\Sigma \in \mathbb{R}^{k \times k}$ are respectively seen as $k$ user/item feature vectors and corresponding weights. The idea of low-rank approximation that discards lower singular values intuitively works as compression or denoising of the original matrix; that is, each element in a rank-$k$ matrix $A_k$ holds the best compressed (or denoised) value of the original element in $A$. Thus, $R_k = \mathrm{SVD}_k(R)$, the best rank-$k$ approximation of $R$, captures as much as possible of underlying users' preferences. Once $R$ is decomposed into $U, \Sigma$ and $V$, a $(u, i)$ element of $R_k$ calculated by $\sum^k_{j=1} \sigma_j u_{u, j} v_{i, j}$ could be a prediction for the user-item pair.

source

SVD actually works well on user-item matrices, but the method poses some difficulties in practice. First, finding optimal $k$ is not straightforward. Even though $R_k$ is the best rank-$k$ approximation of $R$, too small $k$ obviously loses too much information of the original matrix, and it decreases the accuracy of recommendation. At the same time, too large $k$ prevents taking advantage of dimensionality reduction. Hence, carefully choosing $k$ by evaluation on a local environment is practically important.

Furthermore, the following points require us to take an extra effort to make SVD feasible for recommendation:

• We need to preliminarily fill missing values in $R$ to launch SVD, and there are several options such as mean imputation and zero filling.
• Although the original matrix $R$ is a huge matrix in both rows and columns, SVD is a computationally expensive operation.
• Explaining how the recommendation was generated is practically important, but it is hard for SVD-based recommendation unless mathematical background is explained.

## Matrix Factorization

Even though dimensionality reduction is a promising approach to make effective recommendation, SVD is not feasible enough in terms of running time and missing value imputation. As a result, a new technique generally called matrix factorization (MF) was introduced by Koren et al.

The initial MF technique was invented by Funk during the Netflix Prize, and the method is called regularized SVD because it can be seen as an extension of the conventional SVD-based recommendation. Regularized SVD and the other MF techniques achieved efficient approximation of the original SVD based on gradient descent optimization, and we do not need to care about missing values in $R$ any more. The basic idea of MF is to factorize a user-item matrix $R$ to a user factored matrix $P \in \mathbb{R}^{|\mathcal{U}| \times k}$ and item factored matrix $Q \in \mathbb{R}^{|\mathcal{I}| \times k}$ as shown below. Eventually, $R$ is approximated by $PQ^{\mathrm{T}}$. Unlike SVD, singular values in $\Sigma$ are embedded to the factored matrices, and orthogonality is lost. Thus, taking advantage of mathematically tractable properties of SVD is unfortunately impossible for MF.

Recommendation.MFType
MF(
data::DataAccessor,
k::Int
)

Recommendation based on matrix factorization (MF). Number of factors is configured by k.

MF solves the following minimization problem for a set of observed user-item interactions $\mathcal{S} = \{(u, i) \in \mathcal{U} \times \mathcal{I}\}$:

$\min_{P, Q} \sum_{(u, i) \in \mathcal{S}} \left( r_{u,i} - \mathbf{p}_u^{\mathrm{T}} \mathbf{q}_i \right)^2 + \lambda \ (\|\mathbf{p}_u\|^2 + \|\mathbf{q}_i\|^2),$

where $\mathbf{p}_u, \mathbf{q}_i \in \mathbb{R}^k$ are respectively a factorized user and item vector, and $\lambda$ is a regularization parameter to avoid overfitting. An optimal solution will be found by stochastic gradient descent (SGD). Ultimately, we can predict missing values in $R$ by just computing $PQ^{\mathrm{T}}$, and the prediction directly leads recommendation.

source

MF is attractive in terms of not only efficiency but extensibility. In the above formulation, prediction for each user-item pair can be written by a simple vector product as $r_{u,i} = \mathbf{p}_u^{\mathrm{T}} \mathbf{q}_i$, and extension of the formula is also possible. That is, we can incorporate different features (e.g., biases and temporal factors) into the model as linear combinations. For example, let $\mu$ be a global mean of all elements in $R$, and $b_u, b_i$ be respectively a user and item bias term. Here, we assume that each observation can be represented as $r_{u,i} = \mu + b_u + b_i + \mathbf{p}_u^{\mathrm{T}} \mathbf{q}_i$. This formulation is known as biased MF, and it is possible to capture more information than the original MF even on the same set of events $\mathcal{S}$.

Additionally, options for loss functions are also abundant. To give an example, Chen et al. showed various types of features and loss functions which can be incorporated into a MF scheme. An appropriate choice of their combinations is likely to lead surprisingly better accuracy compared to the classical MF.

It should be noted that the technique has many limitations behind great success of MF-based recommendation. Since extension of MF only allows us to append new features as linear combinations, representing more complex relationships between features is not straightforward. As a consequence, modern recommenders tend to use more complex models which are higher dimensional and hard to optimize such as tensor factorization.


text:
# Diophantine equation with quartic polynomial

What are all integral solutions to $$y^2=x^4+x^3+x^2+x+1$$

The RHS could become $$(x^2-x+1)(x^2+x+1)+x(x^2+1)$$ or $$\frac{x^5-1}{x-1}$$ I have no idea how to manipulate the equation into something useful or what the first step should be. Also, A quartic diophantine equation looks useful, but none of the answers completely solve the question? Thanks!

If $$x \gt 0$$, the only solution is $$x=3, y=11$$.

Note: Wolfy was used extensively getting this answer.

$$y^2 =x^4+x^3+x^2+x+1$$

$$y^2 =x^4+x^3+x^2+x+1 \gt x^4 \implies y > x^2$$.

$$(x+1)^4 =x^4+4x^4+,,,$$ so $$y^2 \lt (x+1)^4$$ or $$y < (x+1)^2$$.

$$(x^2+x/2+3/8)^2 =x^4 + x^3 + x^2 + (3 x)/8 + 9/64 \lt y^2$$ so $$y > x^2+x/2+3/8$$.

$$(x^2+x/2+1)^2 =x^4 + x^3 + (9 x^2)/4 + x + 1 \gt y^2$$ so $$y < x^2+x/2+1$$.

If $$x = 2n$$ then $$4n^2+n+3/8 \lt y \lt 4n^2+n+1$$, so there can be no such integer $$y$$.

If $$x = 2n+1$$ then $$x^2+x/2+3/8 =4n^2+4n+1+n+1/2+3/8 =4n^2+5n+15/8$$ and $$x^2+x/2+1 =4n^2+4n+1+n+1/2+1 =4n^2+5n+5/2$$ so $$y = 4n^2+5n+2$$.

But $$y^2 =16 n^4 + 40 n^3 + 41 n^2 + 20 n + 4$$ and $$x^4+x^3+x^2+x+1 =\dfrac{x^5-1}{x-1} =\dfrac{(2n+1)^5-1}{2n} =16 n^4 + 40 n^3 + 40 n^2 + 20 n + 5$$ and the difference is $$n^2-1$$ so they are never equal unless $$n = 1$$ so $$x = 3$$.

• A minor point, but I think your start of "If $x \gt 0$, ..." should be something like "If $x,y \gt 0$, ..." instead since otherwise, for example, $x = 3, y = -11$ is also a solution. Oct 22 '19 at 1:56
• Very good solution! I believe $n=-1$ yields another case, right? Oct 22 '19 at 16:52
• @Baker013273213 Change $x$ to $-z$ and assume $z>0$. $z$ cannot be even, otherwise $$(z^2-z/2)^2 < z^4-z^3+z^2-z+1=y^2 < (z^2-z/2+1)^2$$ With $z$ odd, we have $$(z^2 - (z+1)/2)^2 < z^4-z^3+z^2-z+1=y^2 < (z^2 - (z+1)/2+1)^2$$ if $z > 1$. So only possibility is $z=1$ giving $x=-1$. Oct 23 '19 at 2:59
• You should make this an answer. Oct 23 '19 at 14:35


text:
# Backing up a PostgreSQL database (Windows)¶

Note

Content contributed by DJ Platten

These instructions are based on PostgreSQL 9.1 and OpenREM 0.5.0 running on Windows Server 2008. The database restore has been tested on Ubuntu 12.04 LTS.

Create a PostgreSQL user called backup with a password of backup. This is easiest to do using the pgAdminIII tool: you’ll need to create a new login role. In the role privileges ensure that at least Can login, Superuser and Can initiate streaming replication and backups are checked.

The pgAdminIII tool is available by default on Windows, but needs to be explicitly installed if using Ubuntu with the following command:

sudo apt-get install pgadmin3

For the remainder of this article I’m going to assume that your OpenREM database is called openrempostgresql.

To backup the contents of openrempostgresql to a file called backup.sql run the following at the command line in a command prompt (Windows), or terminal window (Ubuntu):

pg_dump -U backup -F c -b -v -f backup.sql openrempostgresql

You will need to add your C:\path\to\postgres\bin folder to the path environment variable for this to work. Make sure to use the actual path to your PostgreSQL bin folder rather than the example text provided here. See http://www.computerhope.com/issues/ch000549.htm for instructions on editing the path environment variable.

The -U backup indicates that the backup user is to carry out the task. The -F c option archives in a suitable format for input into the pg_restore command. Further information on pg_dump and backing up a PostgreSQL database can be found here: http://www.postgresql.org/docs/9.3/static/app-pgdump.html and here: http://www.postgresql.org/docs/9.3/static/backup-dump.html

# Restoring a PostgreSQL database (Windows)¶

The pg_restore command can be used to restore the database using one of the backed-up SQL files that were produced using the pg_dump command.

Use the pgAdminIII tool to ensure that there is a PostgreSQL user called openremuser.

Use pgAdminIII to create a database called openrempostgresql; set the owner to openremuser and the encoding to UTF8.

Run the following command in a command prompt window (Windows) or terminal window (Ubuntu) to restore the contents of backupFile to the openrempostgresql database, where backupFile is the file created by the pg_dump command:

pg_restore -U postgres -d openrempostgresql backupFile

Ensure that openremuser has an entry in PostgreSQL’s pg_hpa.conf file for md5 authentication:

local all openremuser md5

The PostgreSQL server will need to be restarted if you have changed pg_hpa.conf.

See http://www.postgresql.org/docs/9.3/static/backup-dump.html#BACKUP-DUMP-RESTORE for further details.


text:
Question

# What is inert pair effect?

Hint: Inert pair takes into account the poor screening/shielding effect of ‘d’ and ‘f’ orbital electrons. This effect is easily observed in p- block elements (group 4 and group 5 elements) where the stability of lower oxidation state increases down the group.

Complete Step by Step Solution:
Inert pair effect is the reluctance of ‘s’ electrons to take part in bonding. This is due to the poor screening effect of ‘d’ and ‘f’ orbitals.
The inert pair theory was proposed by Sidgwick.
Sidwick along with Powell accounted for the shapes of several molecules and correlated the shapes with some of their physical properties.
Now, we will look at some examples of inert pair effects.
The inert pair effect can be easily observed among group 4 and group 5 elements.
In group 4 , the electronic configuration is ${ ns }^{ 2 }{ np }^{ 2 }$. Here, we can see that the group oxidation state is +3 and the other oxidation state is +1.
The oxidation state is two units less than the group oxidation state became more stable for heavier elements in each group and down the group the stability of lower oxidation state increases, this is due to inert pair effect.
Because of this in group 4, on moving down the group only ‘p’ electrons are involved in bonding.
In the case of Ga, Tl and In , they can show both +1 and +3 oxidation state. But the stability of +1 oxidation state increases down the group.
In Tl, +1 oxidation state is more predominant whereas +3 oxidation state is highly oxidising in nature.

Shielding effect is a different concept when compared with inert pair effect.
Shielding effect is basically , the outermost electrons will have an attractive pull from the nucleus and at the same time, they will be experiencing a repulsive force from the inner electrons. And thereby it results in a decrease in the effective nuclear charge.
We can have look at the equation of effective nuclear charge:
${ Z }^{ \ast }={ Z }-\sigma$
Where ${ Z }^{ \ast }$ is the effective nuclear charge
Z is the number of protons.
$\sigma$ is the screening constant or shielding constant.

The difference between shielding effect and inert pair effect is given below:

 Inert pair Effect Shielding effect The effect explains the reluctance of s-electrons in the valence shell to participate in the bonding. This effect decreases the effective nuclear charge on the outer shell electrons , this is due to the repulsion between the inner electrons and the outer electrons. The inert pair effect helps in understanding the stability of a particular oxidation state for a particular element. Shielding effect explains the ease of removal of valence electrons from an atom.

Note: Screening effect is additionally referred to as shielding effect. This effect occurs when there is an imbalance between charges. The outermost electrons experience an attractive pull from the nucleus but at the same time, they experience a repulsion from the inner electrons as well, and thus forms the effective nuclear charge.


text:
# What is the Opaque Component of Venus' Atmosphere

My understanding is that carbon dioxide makes up the majority of the Venusian atmosphere, and carbon dioxide is transparent. Even liquid CO2 is transparent!

Yet despite this Venus is enveloped in a thick all encompassing opaque atmosphere that conceals the surface, unlike Earth.

What is the primary cause for this? Why does Venus have an opaque atmosphere?

• This is a question from the category: "Why is the sky blue if the air is transparent?" – A. Rumlin Sep 21 '20 at 19:39
• @A.Rumlin that question has an answer, I too would like an answer, but a Venusian answer. Don't forget, you can see large sections of earths surface from orbit in the visible spectrum, you can't see the surface of Venus though. I am not interested in the colour of Venus, but it's opacity – Tom J Nowell Sep 21 '20 at 22:21
• about Raleigh scattering in Venus' atmosphere, there's @TomSpilker's authoritative answer – uhoh Sep 21 '20 at 23:30
• re Raleigh scattering in Venus' atmosphere, there's @TomSpilker's authoritative answer. This is a good question, but there might be more than one answer depending on if you mean opaque as viewed from above (e.g. orbiter looking down) or near the surface (e.g. lander looking into the distance, trying to see the "horizon". Or maybe it will turn out to be the same answer. I put horizon in quotes because at a certain altitude, light parallel to the surface already bends around the planet. Can't find that answer right now but it's somewhere in SE... – uhoh Sep 21 '20 at 23:35
• @TomJNowell Unfortunately, I have now not been able to find a Soviet publication on the properties of the lower layer of the atmosphere of Venus. As far as I remember, at these temperatures and pressures, carbon dioxide has amazing optical and physical properties. Something like a liquid. – A. Rumlin Sep 22 '20 at 6:30

• great! fun = len(questions.on_topic()) I still have to find that invisible-horizon-due-to-refraction post, right now it's hiding beyond my local horizon. The question asked at what altitude would the radius of curvature of a ray bent by refraction result in a circle around the planet, and I think it was around 20 km, but right now local SE opacity is too high due to Me scattering off of other questions. (all puns intended) – uhoh Sep 23 '20 at 6:03


text:
New Question

# Revision history [back]

Hello Muthu, did you install FreeRDP-WebConnect on the compute-node?(if yes, the host can be localhost, but it's not indicated). I suppose there is a problem with your openstack controller, as it should return you the proper compute-node ip. Could you enter inside Event Viewer inside the compute-node, under Application, search for source: wsgate and you should have a list of Information level entries. There are some specific ones to look for:

RDP Port:               '2179'
RDP Pcb:               '4F325E7A-7909-4537-8A5D-D85F150B6890'
RDP Host:              'some ip'

Those should coincide with the info related to the particular VM you're trying to view. Also, can you show the wsgate.ini file(remove the admin password) inside

C:\Program Files\Cloudbase Solutions\FreeRDP-WebConnect\etc\wsgate.ini

Hello Muthu, did you install FreeRDP-WebConnect on the compute-node?(if yes, the host can be localhost, but it's not indicated). I suppose there is a problem with your openstack controller, as it should return you the proper compute-node ip. Could you enter inside Event Viewer inside the compute-node, under Application, search for source: wsgate and you should have a list of Information level entries. There are some specific ones to look for:

RDP Port:               '2179'
RDP Pcb:               '4F325E7A-7909-4537-8A5D-D85F150B6890'
RDP Host:              'some ip'

Those should coincide with the info related to the particular VM you're trying to view. Also, can you show the wsgate.ini file(remove the admin password) inside

C:\Program Files\Cloudbase Solutions\FreeRDP-WebConnect\etc\wsgate.ini

Hello Muthu, did you install FreeRDP-WebConnect on the compute-node?(if yes, the host can be localhost, but it's not indicated). I suppose there is a problem with your openstack controller, as it should return you the proper compute-node ip. Could you enter inside Event Viewer inside the compute-node, under Application, search for source: wsgate and you should have a list of Information level entries. There are some specific ones to look for:

RDP Port:              '2179'


text:
# Graphs with unique zero forcing sets and Grundy dominating sets

The concept of zero forcing was introduced in the context of linear algebra, and was further studied by both graph theorists and linear algebraists. It is based on the process of activating vertices of a graph G starting from a set of vertices that are already active, and applying the rule that an active vertex with exactly one non-active neighbor forces that neighbor to become active. A set S⊂ V(G) is called a zero forcing set of G if initially only vertices of S are active and the described process enforces all vertices of G to become active. The size of a minimum zero forcing set in G is called the zero forcing number of G. While a minimum zero forcing set can only be unique in edgeless graphs, we consider the weaker uniqueness condition, notably that for every two minimum zero forcing sets in a graph G there is an automorphism that maps one to the other. We characterize the class of trees that enjoy this condition by using properties of minimum path covers of trees. In addition, we investigate both variations of uniqueness for several concepts of Grundy domination, which first appeared in the context of domination games, yet they are also closely related to zero forcing. For each of the four variations of Grundy domination we characterize the graphs that have only one Grundy dominating set of the given type, and characterize those forests that enjoy the weaker (isomorphism based) condition of uniqueness. The latter characterizations lead to efficient algorithms for recognizing the corresponding classes of forests.

## Authors

• 1 publication
• 1 publication
09/01/2020

### Reconfiguration graphs of zero forcing sets

This paper begins the study of reconfiguration of zero forcing sets, and...
05/07/2018

### Domination Cover Number of Graphs

A set D ⊆ V for the graph G=(V, E) is called a dominating set if any ver...
09/06/2019

### On the vertices belonging to all, some, none minimum dominating set

We characterize the vertices belonging to all minimum dominating sets, t...
09/25/2017

### On the error of a priori sampling: zero forcing sets and propagation time

Zero forcing is an iterative process on a graph used to bound the maximu...
11/17/2021

### On the coalition number of graphs

Let G be a graph with vertex set V. Two disjoint sets V_1, V_2 ⊆ V form ...
10/14/2017

### On complexity of mutlidistance graph recognition in R^1

Let A be a set of positive numbers. A graph G is called an A-embeddable ...
09/17/2019

### Rotational Uniqueness Conditions Under Oblique Factor Correlation Metric

In an addendum to his seminal 1969 article Jöreskog stated two sets of c...
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text:
# parasys.net

Home > Error Of > Error Of The Mean Wiki

# Error Of The Mean Wiki

## Contents

These all summarize performance in ways that disregard the direction of over- or under- prediction; a measure that does place emphasis on this is the mean signed difference. so that ( n − 1 ) S n − 1 2 σ 2 ∼ χ n − 1 2 {\displaystyle {\frac {(n-1)S_{n-1}^{2}}{\sigma ^{2}}}\sim \chi _{n-1}^{2}} . Gambrill, W., "False Positives on Newborns' Disease Tests Worry Parents", Health Day, (5 June 2006). 34471.html[dead link] Kaiser, H.F., "Directional Statistical Decisions", Psychological Review, Vol.67, No.3, (May 1960), pp.160–167. A positive correct outcome occurs when convicting a guilty person.

Consider a sample of n=16 runners selected at random from the 9,732. For an upcoming national election, 2000 voters are chosen at random and asked if they will vote for candidate A or candidate B. TypeII error False negative Freed! Introduction to the Theory of Statistics (3rd ed.). https://en.wikipedia.org/wiki/Standard_error

## Wiki Standard Error Of The Mean

When the sampling fraction is large (approximately at 5% or more) in an enumerative study, the estimate of the standard error must be corrected by multiplying by a "finite population correction"[9] Sampling always refers to a procedure of gathering data from a small aggregation of individuals that is purportedly representative of a larger grouping which must in principle be capable of being In other words, it is the standard deviation of the sampling distribution of the sample statistic. Unbiased estimators may not produce estimates with the smallest total variation (as measured by MSE): the MSE of S n − 1 2 {\displaystyle S_{n-1}^{2}} is larger than that of S

This edition is known as the 'Judas' Bible because in Matthew c26 v36 'Judas' appears instead of 'Jesus'. The following expressions can be used to calculate the upper and lower 95% confidence limits, where x ¯ {\displaystyle {\bar {x}}} is equal to the sample mean, S E {\displaystyle SE} Related measures The mean absolute error is one of a number of ways of comparing forecasts with their eventual outcomes. Equation For Standard Error Of The Mean Sampling from a distribution with a large standard deviation The first data set consists of the ages of 9,732 women who completed the 2012 Cherry Blossom run, a 10-mile race held

In this scenario, the 400 patients are a sample of all patients who may be treated with the drug. Standard Deviation Wiki Sampling from a distribution with a small standard deviation The second data set consists of the age at first marriage of 5,534 US women who responded to the National Survey of Notice that the population standard deviation of 4.72 years for age at first marriage is about half the standard deviation of 9.27 years for the runners. In science and engineering in general an error is defined as a difference between the desired and actual performance or behavior of a system or object.

## Mean Median Mode Wiki

References ^ a b Lehmann, E. False positive mammograms are costly, with over \$100million spent annually in the U.S. ISBN 0-521-81099-X ↑ Gurland, J; Tripathi RC (1971). "A simple approximation for unbiased estimation of the standard deviation". One can then also calculate the mean square of the model by dividing the sum of squares of the model minus the degrees of freedom, which is just the number of Sampling Error Wiki

The result of the test may be negative, relative to the null hypothesis (not healthy, guilty, broken) or positive (healthy, not guilty, not broken). Standard errors are easy to calculate and are used a lot because: If the standard error of several individual quantities is known then the standard error of some function of the Retrieved from "https://simple.wikipedia.org/w/index.php?title=Standard_error&oldid=5431478" Category: StatisticsHidden categories: All articles with unsourced statementsArticles with unsourced statements from October 2011 Navigation menu Personal tools Not logged inTalkContributionsCreate accountLog in Namespaces Page Talk Variants Views pp.401–424.

www.otexts.org. Central Limit Theorem Wiki The US rate of false positive mammograms is up to 15%, the highest in world. The true standard error of the mean, using σ = 9.27, is σ x ¯   = σ n = 9.27 16 = 2.32 {\displaystyle \sigma _{\bar {x}}\ ={\frac {\sigma }{\sqrt

## Applied Linear Regression (2nd ed.).

For the purpose of hypothesis testing or estimating confidence intervals, the standard error is primarily of use when the sampling distribution is normally distributed, or approximately normally distributed. Note that alternative formulations may include relative frequencies as weight factors. Although they display a high rate of false positives, the screening tests are considered valuable because they greatly increase the likelihood of detecting these disorders at a far earlier stage.[Note 1] Null Hypothesis Wiki ISBN0-387-96098-8.

A Type II error is committed when we fail to believe a truth.[7] In terms of folk tales, an investigator may fail to see the wolf ("failing to raise an alarm"). In regression analysis, the term "standard error" is also used in the phrase standard error of the regression to mean the ordinary least squares estimate of the standard deviation of the ISBN0840058012. ^ Cisco Secure IPS– Excluding False Positive Alarms http://www.cisco.com/en/US/products/hw/vpndevc/ps4077/products_tech_note09186a008009404e.shtml ^ a b Lindenmayer, David; Burgman, Mark A. (2005). "Monitoring, assessment and indicators". ISBN9780471879572.

For related, but non-synonymous terms in binary classification and testing generally, see false positives and false negatives. A typeII error may be compared with a so-called false negative (where an actual 'hit' was disregarded by the test and seen as a 'miss') in a test checking for a With n = 2 the underestimate is about 25%, but for n = 6 the underestimate is only 5%. It is asserting something that is absent, a false hit.

Such errors may or may not be scarce or rare. Contents 1 Introduction 2 In univariate distributions 2.1 Remark 3 Regressions 4 Other uses of the word "error" in statistics 5 See also 6 References Introduction Suppose there is a series Moreover, this formula works for positive and negative ρ alike.[10] See also unbiased estimation of standard deviation for more discussion. Note that the sum of the residuals within a random sample is necessarily zero, and thus the residuals are necessarily not independent.

n is the size (number of observations) of the sample.


text:
# Uniform continuity explained

f

of real numbers is said to be uniformly continuous if there is a positive real number

\delta

such that function values over any function domain interval of the size

\delta

are close to each other as we want, e.g., we want function value differences to be less than any positive real number

\epsilon

, then there is

\delta

such that

|f(x)-f(y)|<\epsilon

at any

x

and

y

over any interval of the size

\delta

.

The difference between uniform continuity and (ordinary) continuity is that, in uniform continuity there is a globally applicable

\delta

(the size of an interval over which function value differences are less than

\epsilon

) that depends on only

\varepsilon

while in (ordinary) continuity there is a locally applicable

\delta

that depends on the both

\varepsilon

and

x

. So uniform continuity is a stronger continuity condition than continuity; a function that is uniformly continuous is continuous, but a function that is continuous is not necessarily uniformly continuous. The concepts of uniform continuity and continuity can be expanded to functions defined between metric spaces.

Continuous functions can fail to be uniformly continuous if they are unbounded on a bounded domain, such as

f(x)=\tfrac1x

on

(0,1)

, or if their slopes become unbounded on an infinite domain, such as

f(x)=x2

on the real line. However, any Lipschitz map between metric spaces is uniformly continuous, in particular any isometry (distance-preserving map).

Although continuity can be defined for functions between general topological spaces, defining uniform continuity requires more structure. The concept relies on comparing the sizes of neighbourhoods of distinct points, so it requires a metric space, or more generally a uniform space.

## Definition for functions on metric spaces

For a function

f:X\toY

with metric spaces

(X,d1)

and

(Y,d2)

, the following definitions of uniform continuity and (ordinary) continuity hold.

### Definition of uniform continuity

f

is called uniformly continuous if for every real number

\varepsilon>0

there exists a real number

\delta>0

such that for every

x,y\inX

with

d1(x,y)<\delta

, we have

d2(f(x),f(y))<\varepsilon

. The set

\{y\inX:d1(x,y)<\delta\}

for each

x

is a neighbourhood of

x

and the set

\{x\inX:d1(x,y)<\delta\}

for each

y

is a neighbourhood of

y

by the definition of a neighbourhood in a metric space.
• If

X

and

Y

are subsets of the real line, then

d1

and

d2

can be the standard one-dimensional Euclidean distance, yielding the following definition: for every real number

\varepsilon>0

there exists a real number

\delta>0

such that for every

x,y\inX

,

|x-y|<\delta\implies|f(x)-f(y)|<\varepsilon

(where

A\impliesB

is a material conditional statement saying "if

A

, then

B

").
• Equivalently,

f

is said to be uniformly continuous if

. Here quantifications (

\forall\varepsilon>0

,

\exists\delta>0

,

\forallx\inX

, and

\forally\inX

) are used.
• Alternatively,

f

is said to be uniformly continuous if there is a function of all positive real numbers

\varepsilon

,

\delta(\varepsilon)

representing the maximum positive real number, such that for every

x,y\inX

if

d1(x,y)<\delta(\varepsilon)

then

d2(f(x),f(y))<\varepsilon

.

\delta(\varepsilon)

is a monotonically non-decreasing function.

### Definition of (ordinary) continuity

f

is called continuous at

x

if for every real number

\varepsilon>0

there exists a real number

\delta>0

such that for every

y\inX

with

d1(x,y)<\delta

, we have

d2(f(x),f(y))<\varepsilon

. The set

\{y\inX:d1(x,y)<\delta\}

is a neighbourhood of

x

.
• Equivalently, a function

f

is said to be continuous if

.
• Alternatively, a function

f

is said to be continuous if there is a function of all positive real numbers

\varepsilon

and

x\inX

,

\delta(\varepsilon,x)

representing the maximum positive real number, such that at each

x

if

y\inX

satisfies

d1(x,y)<\delta(\varepsilon,x)

then

d2(f(x),f(y))<\varepsilon

. At every

x

,

\delta(\varepsilon,x)

is a monotonically non-decreasing function.

## Local continuity versus global uniform continuity

In the definitions, the difference between uniform continuity and continuity is that, in uniform continuity there is a globally applicable

\delta

(the size of a neighbourhood in

X

over which values of the metric for function values in

Y

are less than

\varepsilon

) that depends on only

\varepsilon

while in continuity there is a locally applicable

\delta

that depends on the both

\varepsilon

and

x

. Continuity is a local property of a function — that is, a function

f

is continuous, or not, at a particular point

x

of the function domain

X

, and this can be determined by looking at only the values of the function in an arbitrarily small neighbourhood of that point. When we speak of a function being continuous on an interval, we mean that the function is continuous at every point of the interval. In contrast, uniform continuity is a global property of

f

, in the sense that the standard definition of uniform continuity refers to every point of

X

. On the other hand, it is possible to give a definition that is local in terms of the natural extension

f*

(the characteristics of which at nonstandard points are determined by the global properties of

f

), although it is not possible to give a local definition of uniform continuity for an arbitrary hyperreal-valued function, see below.

A mathematical definition that a function

f

is continuous on an interval

I

and a definition that

f

is uniformly continuous on

I

are structurally similar as shown in the following.

Continuity of a function

f:X\toY

for metric spaces

(X,d1)

and

(Y,d2)

at every point

x

of an interval

I\subseteqX

(i.e., continuity of

f

on the interval

I

) is expressed by a formula starting with quantifications

,(metrics

d1(x,y)

and

d2(f(x),f(y))

are

|x-y|

and

|f(x)-f(y)|

for

f:R\toR

for the set of real numbers

R

).

For uniform continuity, the order of the first, second, and third quantifications (

\forallx\inI

,

\forall\varepsilon>0

, and

\exists\delta>0

) are rotated:

.Thus for continuity on the interval, one takes an arbitrary point

x

of the interval, and then there must exist a distance

\delta

,

\forallx\exists\delta,

while for uniform continuity, a single

\delta

must work uniformly for all points

x

of the interval,

\exists\delta\forallx.

## Examples and counterexamples

(-\pi/2,\pi/2)

but is not uniformly continuous on that interval.
• The exponential function

x\toex

is continuous everywhere on the real line but is not uniformly continuous on the line.

## Properties

Every uniformly continuous function is continuous, but the converse does not hold. Consider for instance the continuous function

f\colonRR,x \mapstox2

where

R

is the set of real numbers. Given a positive real number

\varepsilon

, uniform continuity requires the existence of a positive real number

\delta

such that for all

x1,x2\inR

with

|x1-x2|<\delta

, we have

|f(x1)-f(x2)|<\varepsilon

. But

f\left(x+\delta\right)-f(x)=2x\delta+\delta2,

and as

x

goes to be a higher and higher value,

\delta

needs to be lower and lower to satisfy

|f(x+\beta)-f(x)|<\varepsilon

for positive real numbers

\beta<\delta

and the given

\varepsilon

. This means that there is no specifiable (no matter how small it is) positive real number

\delta

to satisfy the condition for

f

to be uniformly continuous so

f

is not uniformly continuous.

Any absolutely continuous function (over a compact interval) is uniformly continuous. On the other hand, the Cantor function is uniformly continuous but not absolutely continuous.

The image of a totally bounded subset under a uniformly continuous function is totally bounded. However, the image of a bounded subset of an arbitrary metric space under a uniformly continuous function need not be bounded: as a counterexample, consider the identity function from the integers endowed with the discrete metric to the integers endowed with the usual Euclidean metric.

The Heine–Cantor theorem asserts that every continuous function on a compact set is uniformly continuous. In particular, if a function is continuous on a closed bounded interval of the real line, it is uniformly continuous on that interval. The Darboux integrability of continuous functions follows almost immediately from this theorem.

If a real-valued function

f

is continuous on

[0,infty)

and

\limxf(x)

exists (and is finite), then

f

is uniformly continuous. In particular, every element of

C0(R)

, the space of continuous functions on

R

that vanish at infinity, is uniformly continuous. This is a generalization of the Heine-Cantor theorem mentioned above, since

Cc(R)\subsetC0(R)

.

## Visualization

For a uniformly continuous function, for every positive real number

\varepsilon>0

there is a positive real number

\delta>0

such that two function values

f(x)

and

f(y)

have the maximum distance

\varepsilon

whenever

x

and

y

are within the maximum distance

\delta

. Thus at each point

(x,f(x))

of the graph, if we draw a rectangle with a height slightly less than

2\varepsilon

and width a slightly less than

2\delta

around that point, then the graph lies completely within the height of the rectangle, i.e., the graph do not pass through the top or the bottom side of the rectangle. For functions that are not uniformly continuous, this isn't possible; for these functions, the graph might lie inside the height of the rectangle at some point on the graph but there is a point on the graph where the graph lies above or below the rectangle. (the graph penetrates the top or bottom side of the rectangle.)

## History

The first published definition of uniform continuity was by Heine in 1870, and in 1872 he published a proof that a continuous function on an open interval need not be uniformly continuous. The proofs are almost verbatim given by Dirichlet in his lectures on definite integrals in 1854. The definition of uniform continuity appears earlier in the work of Bolzano where he also proved that continuous functions on an open interval do not need to be uniformly continuous. In addition he also states that a continuous function on a closed interval is uniformly continuous, but he does not give a complete proof.

## Other characterizations

### Non-standard analysis

In non-standard analysis, a real-valued function

f

of a real variable is microcontinuous at a point

a

precisely if the difference

f*(a+\delta)-f*(a)

is infinitesimal whenever

\delta

is infinitesimal. Thus

f

is continuous on a set

A

in

R

precisely if

f*

is microcontinuous at every real point

a\inA

. Uniform continuity can be expressed as the condition that (the natural extension of)

f

is microcontinuous not only at real points in

A

, but at all points in its non-standard counterpart (natural extension)

*A

in

*R

. Note that there exist hyperreal-valued functions which meet this criterion but are not uniformly continuous, as well as uniformly continuous hyperreal-valued functions which do not meet this criterion, however, such functions cannot be expressed in the form

f*

for any real-valued function

f

. (see non-standard calculus for more details and examples).

### Cauchy continuity

For a function between metric spaces, uniform continuity implies Cauchy continuity . More specifically, let

A

be a subset of

Rn

. If a function

f:A\toRn

is uniformly continuous then for every pair of sequences

xn

and

yn

such that

\limn\toinfty|xn-yn|=0

we have

\limn\toinfty|f(xn)-f(yn)|=0.

## Relations with the extension problem

Let

X

be a metric space,

S

a subset of

X

,

R

a complete metric space, and

f:SR

a continuous function. A question to answer: When can

f

be extended to a continuous function on all of

X

?

If

S

is closed in

X

, the answer is given by the Tietze extension theorem. So it is necessary and sufficient to extend

f

to the closure of

S

in

X

: that is, we may assume without loss of generality that

S

is dense in

X

, and this has the further pleasant consequence that if the extension exists, it is unique. A sufficient condition for

f

to extend to a continuous function

f:XR

is that it is Cauchy-continuous, i.e., the image under

f

of a Cauchy sequence remains Cauchy. If

X

is complete (and thus the completion of

S

), then every continuous function from

X

to a metric space

Y

is Cauchy-continuous. Therefore when

X

is complete,

f

extends to a continuous function

f:XR

if and only if

f

is Cauchy-continuous.

It is easy to see that every uniformly continuous function is Cauchy-continuous and thus extends to

X

. The converse does not hold, since the function

f:RR,x\mapstox2

is, as seen above, not uniformly continuous, but it is continuous and thus Cauchy continuous. In general, for functions defined on unbounded spaces like

R

, uniform continuity is a rather strong condition. It is desirable to have a weaker condition from which to deduce extendability.

For example, suppose

a>1

is a real number. At the precalculus level, the function

f:x\mapstoax

can be given a precise definition only for rational values of

x

(assuming the existence of qth roots of positive real numbers, an application of the Intermediate Value Theorem). One would like to extend

f

to a function defined on all of

R

. The identity

f(x+\delta)-f(x)=ax\left(a\delta-1\right)

shows that

f

is not uniformly continuous on the set

Q

of all rational numbers; however for any bounded interval

I

the restriction of

f

to

Q\capI

is uniformly continuous, hence Cauchy-continuous, hence

f

extends to a continuous function on

I

. But since this holds for every

I

, there is then a unique extension of

f

to a continuous function on all of

R

.

More generally, a continuous function

f:SR

whose restriction to every bounded subset of

S

is uniformly continuous is extendable to

X

, and the converse holds if

X

is locally compact.

A typical application of the extendability of a uniformly continuous function is the proof of the inverse Fourier transformation formula. We first prove that the formula is true for test functions, there are densely many of them. We then extend the inverse map to the whole space using the fact that linear map is continuous; thus, uniformly continuous.

## Generalization to topological vector spaces

V

and

W

, the notion of uniform continuity of a map

f:V\toW

becomes: for any neighborhood

B

of zero in

W

, there exists a neighborhood

A

of zero in

V

such that

v1-v2\inA

implies

f(v1)-f(v2)\inB.

f:V\toW

, uniform continuity is equivalent to continuity. This fact is frequently used implicitly in functional analysis to extend a linear map off a dense subspace of a Banach space.

## Generalization to uniform spaces

Just as the most natural and general setting for continuity is topological spaces, the most natural and general setting for the study of uniform continuity are the uniform spaces. A function

f:X\toY

between uniform spaces is called uniformly continuous if for every entourage

V

in

Y

there exists an entourage

U

in

X

such that for every

(x1,x2)

in

U

we have

(f(x1),f(x2))

in

V

.

In this setting, it is also true that uniformly continuous maps transform Cauchy sequences into Cauchy sequences.

Each compact Hausdorff space possesses exactly one uniform structure compatible with the topology. A consequence is a generalization of the Heine-Cantor theorem: each continuous function from a compact Hausdorff space to a uniform space is uniformly continuous.


text:
# RD Sharma Solutions for Class 11 Chapter 29 - Limits Exercise 29.11

Here, in this exercise, we shall study the concept of evaluation of limits of the form $1^{\infty }$ with suitable examples to understand the concepts. Students facing any difficulties in understanding concepts of this exercise can refer to RD Sharma Class 11 Maths Solutions, which is developed by subject matter experts at BYJU’S. All the solutions are created with an aim to boost confidence among students for their exam preparations. The resource also improves problem-solving skills of the students, which are essential from the exam point of view. The PDF of RD Sharma Solutions for Class 11 Maths Chapter 29 Exercise 29.11 can be accessed, from the links given below.

## Download the pdf of RD Sharma Solutions for Class 11 Maths Exercise 29.11 Chapter 29 – Limits

### Access answers to RD Sharma Solutions for Class 11 Maths Exercise 29.11 Chapter 29 – Limits

#### EXERCISE 29.11 PAGE NO: 29.71

Evaluate the following limits:

So here,

f (x) = cos x + sin x – 1

g (x) = x

Then,


text:
# Quantum sophistry

A few days ago, we got to know the members of the St. Hypotheticus drinking and nerdery club. Today we listen in on their inaugural meeting. They have just finished the administrative part and are now about to enter on the drinking and nerdery.

Carl: (Knocking at a beer keg) All right, I already prepared the drinking …

[applause]

Carl: …now for the nerdery. Any ideas?

Bob: I have a question about quantum mechanics and philosophy.

Albert: (groan) You know, most I heard about this is bunk. It has far less philosophical consequences than you might think.

Bob: I know. The thing is, shortly I was accosted by some young folks at the college who where looking for an analytical philosopher…

Matthew: …Uhu…

Bob: An analytical Thomist does totally count, and besides if it was so important they could have asked, couldn’t they? Anyway, they asked if one could empirically prove two particles indistinguishable even in principle and I said no. Then they walked away laughing and talking about configurations and flows and distinguishable wave functions. Most of it went over my head, but I googled it and it turns out they were working from a script. I brought some copies, so I wondered what y’all have to say about that.

Matthew: Argh, it’s the…

Albert: … many-worlders …

Clara: … Bayesians …

Matthew: …Gnostics.

Bob: Um yes, there is a lot of strange stuff on that site, but I don’t even know how much of that the nice folks who followed the script actually believe. So can we look at the point? That script claims:

If you have a particle P1 and a particle P2, and it’s possible in the experiment for both P1 and P2 to end up in either of two possible locations L1 or L2, then the observed distribution of results will depend on whether “P1 at L1, P2 at L2” and “P1 at L2, P2 at L1” is the same configuration, or two distinct configurations. If they’re the same configuration, we add up the amplitudes flowing in, then take the squared modulus. If they’re different configurations, we keep the amplitudes separate, take the squared moduli separately, then add the resulting probabilities. As (1 + 1)2 != (12 + 12), it’s not hard to distinguish the experimental results after a few trials.

Clara: Well first of all someone should tell them Bayesian geniuses that neither $$\left(1+1\right)^2=4$$ nor $$\left(1^2+1^2\right)=2$$ are acceptable values for a probability. You know, those are supposed to be between 0 and 1.

Carl: Yeah, but the point seems to be that addition and modulus squaring don’t commute, and that much seems true.

Albert: Is there some spin $$\frac{1}{2}$$ stuff in there?

Bob: It explicitly says half-integer spins don’t matter.

Albert: That sure is a relief, otherwise our author might have wounded his fingers typing up all the math. Anyway, other than in the detail Clara just noted, that quote is quite true. But the claim it is supposed to support just doesn’t follow from it, except with an implicit wrong assumption that it couldn’t be just one configuration even if the particles are distinguishable in principle.

Bob: I’m not quite sure I get that. Can we look at this in more detail?

Albert: OK, the most famous mathematical objects we have in quantum mechanics are wave functions.  In the simplest case they describe  single spinless particles and then they are just  functions that gives you a complex number for every point in  space.

Jenny:  Why complex?

Albert: So that they can interfere. But don’t worry about that now, it isn’t that important for what we’re talking about today.

Jenny: OK.

Kate: Fine, so what do we do with these complex numbers?

Albert: Take their modulus (don’t worry about that step for now) and square it. And that gives us a probability density.

Matthew: Um, density?

Albert: Yeah, there are infinitely many points, so we can’t give probability to every one, or else we would have trouble with them adding up to more than one, and we can’t be more than sure the particle is somewhere. But the math trick isn’t that important, if there where only finitely many points this would simply be a probability. Now there are some neat things we can do with wave functions. One thing we can do with them is multiply them with (complex) numbers. So for example if $$\psi$$ is a valid wave function, then so is $$5\psi$$.

Clara: Whoa! That can’t work, you’re bringing in those probabilities larger than one again!

Albert: OK, we do some normalization. We said for a wave function $$\psi\left(\vec x\right)$$ the probability density for finding the particle at $$\vec x$$ is $$\left|\psi\left(\vec x\right)\right|^2$$.  Now if those probabilities would add up to more than one, whatever they add up to is called $$\left\langle\psi\middle|\psi\right\rangle$$. There’s a meaning to that notation, but today we’ll just treat it as a name. Then instead of $$\left|\psi\left(\vec x\right)\right|^2$$ our probability density is actually $$\frac{\left|\psi\left(\vec x\right)\right|^2}{\left\langle\psi\middle|\psi\right\rangle}$$.

Clara: OK, now you just canceled out the factor you had allowed in. What is this good for again?

Albert: Quite true, alone it would be useless. But we can also add wave functions. So if $$\psi_1$$ and $$\psi_2$$ are valid wave functions, then so is $$\psi$$ with $$\psi\left(\vec x\right):=\psi_1\left(\vec x\right)+\psi_2\left(\vec x\right)$$.

Clara: Aha! So you’re telling me they form a vector space.

Albert: Yes, though I didn’t want to confuse everyone else with that terminology. So now we can write some wave functions in terms of others. What we do next is select some set of wave functions as a base[1] and then write all the others by multiplying and adding those. To put the same in more pompous words, every wave function can be written as a linear combination of base functions.

Now, for many-particle systems it works in basically the same way.

Bob: Care to elaborate on that?

Albert: I think discussing it for two particles will do.  The principles generalize to an arbitrary number of particles, but then the math gets a bit more complicated.

Bob: Fine by me.

Albert: The way it works for two particles is is basically the same as for one. We just get a wave function depending on two locations and then the probability density for particle 1 being at $${\vec x}_1$$ and particle 2 being at $${\vec x}_2$$ is $$\left|\psi\left({\vec x}_1,{\vec x}_2\right)\right|^2$$.

Bob: Well mathematically it sounds very similar, but only because we haven’t talked about any particular wave functions and operators in either case. If we don’t have any connection between the particular objects for different numbers of particles, then the one-particle case just isn’t a simplification of the two-particle case.  And then by extension it would never make sense to talk of a part of a whole because the whole wouldn’t just be more than but actually unrelated to the sum of its parts. Since we can’t conceive of the whole universe at once that would make all physics impossible.

Albert: Right, but luckily[2] there is such a connection. It turns out, we can express the base functions of the two-particle system by those of two one-particle systems. Now how we do that depends on whether the particles are distinguishable and that is what those folks you met where getting at.

Bob: Now it gets interesting.

Albert: Yup. If the particles are distinguishable it’s really easy. In that case if we have a base function $$\psi_1\left({\vec x}_1\right)$$ for particle one and a base function $$\psi_2\left({\vec x}_2\right)$$ for particle two, then $$\psi\left({\vec x}_1,{\vec x}_2\right)=\psi_1\left({\vec x}_1\right)\psi_2\left({\vec x}_2\right)$$ is a base function for the two-particle system and all two-particle base functions can be expressed in that way. Since we don’t always want to mention the arguments I’ll introduce some notation here: Instead of $$\psi\left({\vec x}_1,{\vec x}_2\right)=\psi_1\left({\vec x}_1\right)\psi_2\left({\vec x}_2\right)$$ we’ll write $$\psi=\psi_1\otimes\psi_2$$.

Bob: I psi[3]. But of course this can’t work if they are indistinguishable, because if $$\psi_1\neq\psi_2$$ we would have the two particles distinguished right there.

Albert: Right. In the two-particle case it turns out that if $$\psi_1$$ and $$\psi_2$$ are one-particle base functions, then  $$\psi=\frac{1}{\sqrt{2}}\left(\psi_1\otimes\psi_2\pm\psi_2\otimes\psi_1\right)$$ is a two-particle base function. It’s only one base function, it’s either $$+$$ or $$-$$ depending on what kind of particle we’re talking about.

Bob: Slow now, I don’t quite understand this form. First, what is $$\frac{1}{\sqrt{2}}$$ doing in there?

Albert: You remember how a few minutes ago we talked about dividing out $$\langle\psi|\psi\rangle$$ to make sure all the probabilities added up to 1? Well, most of the time it is easier to multiply the whole function  by some constant that makes  $$\langle\psi|\psi\rangle=1$$, so that we don’t have to bother with that step. In this case $$\frac{1}{\sqrt{2}}$$ is that constant.

Bob: Fine. And the part in the parentheses?

Albert: That part takes care of the indistinguishability. See, if even in principle nobody knows which particle is where, then the probabilities really shouldn’t change if we swap them. So we want $$\left|\psi\left({\vec x}_1,{\vec x}_2\right)\right|=\left|\psi\left({\vec x}_2,{\vec x}_1\right)\right|$$ and this form guarantees precisely that.

Bob: Great.

Albert: There is one more feature I would like to draw your attention to. Please note $$\psi_1\otimes\psi_2$$ and $$\psi_2\otimes\psi_1$$ are base functions for the system of two distinguishable particles. So the base functions for the system of indistinguishable particles are linear combinations of those for the system of distinguishable particles.  That means all wave functions for indistinguishable particles are also valid wave functions for distinguishable particles.

Bob: But not the other way around?

Albert: Nope, not the other way around. In fact if $$\psi_1\neq\psi_2$$ then $$\psi_1\otimes\psi_2$$ and $$\psi_2\otimes\psi_1$$ themselves are wave functions for the system of distinguishable particles that don’t work for indistinguishable particles.

Bob: You know, the wave functions for indistinguishable particles being a subset of those for distinguishable ones makes sense.  After all we can always not distinguish distinguishable particles, but distinguishing indistinguishable ones is, by definition, impossible.

Albert: Exactly. But now look at this: Even if they would be distinguishable in principle, we can prepare two particles we can’t practically distinguish into an indistinguishable configuration. But if we could prepare them into any of the additional states available to distinguishable particles that would amount to distinguishing them. Since we have assumed we know of no method of distinguishing them, we also know of no method of preparing them to any of the additional states they might have available if they are distinguishable in principle. So whether our inability to distinguish them is an intrinsic feature or our ignorance, the set of configurations we can prepare is identical. Both cases correspond to only one alternative of the experiment they were proposing and therefore it can’t be used to tell the difference between those two cases.

Bob: So you can’t be sure the particles we now hold indistinguishable will never turn out to be distinguishable after all?

Albert: I can think that very unlikely chiefly because of Friar Ockham’s razor, but no, I can’t be absolutely sure of it.

Bob: It’s nice to be right.

Albert: Ahem. The argument you gave for your position was actually invalid. It’s only by luck that you turned out to be technically correct.

Bob: That is, of course, always the best kind of correct.

Footnotes    (↵ returns to text)

1. Well, complete orthonormal system actually, since some of the vectors might be infinite linear combinations of the base vectors. But if you know that difference you don’t need me to explain it and if you don’t you won’t care.
2. Actually the connection is not by luck or even by providence but by mathematical necessity. But we won’t go into that now.
3. ha, ha
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### 4 Responses to Quantum sophistry

1. Ronak says:

You’re right about the distinguishability argument; but as you yourself point out, the guy thinking in everyday logic was right only by coincidence; that doesn’t really contradict Yudkowsky’s point that everyday logic* need not work in all cases.

*I guess I should say everyday premises for logic.

• Gilbert says:

Well yes, human reasoning is highly fallible. I don’t know of anyone actually denying that and I certainly don’t.

Since I didn’t give relevancy reasons originally, let me point out why I actually thought this worth noting.

First, I appreciated the irony. Look at the two things Yudkowsky is scolding his hypothetical Bob for: (a) “Bob failed to imagine the evidence which falsified his basic and invisibly assumed ontology” and therefore couldn’t imagine any evidence possibly falsifying his theory. Then, in the same post, he says two identical-seeming particles being distinguishable after all “requires the observed universe to be a lie” and in one of the comments even notes he “can’t imagine any possible overthrow of QM that would resurrect the idea of two electrons having distinct individual identities”. All this while while the hypothetical experiments he claims to already know the results of easily could actually establish such a distinction without upsetting anything we already know about physics. And (b) Bob thinks being an expert on epistemology lets him put limits on what physicist can know. This in a sequence about how Yudkowsky’s superior knowledge of his prefered version of epistemology makes him better equipped to figure out the interpretation of quantum mechanics than the majority of physicists disagreeing with the MWI. And his misguided absolute certainty of configurations being more fundamental than particles actually plays a major role in that, because it is part of what overcomes the implausibility of there being many worlds. (B.t.w. I agree configurations are probably fundamental, but not with anything near the confidence that would allow positing infinitely many worlds.) So on both counts he’s doing exactly what he’s accusing his fictional whipping boy of.

The second reason was admittedly calculating: Philosophy debates are notoriously open-ended. This particular point is not essential to Yudkowskyanism, but it has the great advantage of allowing for settlement in a single blog post. I figured if some Yudkowskyan stumbled over my blog and saw I had caught Yudkowsky in an unimportant but clear-cut mistake, that might make them stay around for more interesting debates. To be honest that part doesn’t seem to work, perhaps because most Yudkowskyans don’t care about quantum mechanical details very much.

• Ronak says:

First point, fair enough; just would like to correct you that MWI does not actually postulate many worlds. The statement is about decoherence (or the cancellation of interference terms in the density matrix of the universe) – so that, different bits of the state can’t interact with each other*.

Though, I just remembered reading your comment reply that the field theoretic language just does not allow for even conceptualising distinguishable electrons; in NRQM indistinguishability was an ad hoc addition whereas it’s embedded into the idea of every particle being an excitation of the same field. I’m never comfortable equating language and ontology, however, and I’m not sure whether indistinguishability played a role in formulating the field idea or not, so I won’t say anything more than that.

(I care about quantum mechanical details a lot. It’s part of the job I’m training for)

*Disclaimer: I really haven’t looked into the math properly on this one, but skims give me this impression and decent popular presentations seem to corroborate it.

• Gilbert says:

To be honest I’m near the edges of my own expertize here, so take my words with a a few grains of salt.

My understanding is that the MWI is amendable to several meta-interpretations. It isn’t unique in this, for example even among Copenhagenians there are loads of incompatible understandings of what complementary means too. So I agree there are many-worlders who don’t actually believe in many worlds. Eliezer Yudkowsky, however, clearly does.

I suppose you must be mixing me up with someone else, because I never wrote a comment about field theoretic language. But you basically sound right there. As I said, I don’t believe in a second species of electrons either. I’m just denying the claim that we have near-certain proof of such a thing being totally impossible. More generally, I’m very skeptical of people being proud of their inability to imagine something.