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Case Report

# Digital Transformation for Efficient Communication in the Workplace: Analyzing the Flow Coworking Tool

Hyun Young Cho1,*, Hak-June Lee2
1Dongduk Women’s University, Seoul, Korea
*Corresponding author: Hyun Young Cho, Department of Business Administration, Dongduk Women’s University, 60 Hwarang-ro 13-gil, Seongbuk-gu, Seoul 02748, Korea, Tel: +82-2-940-4423, E-mail:hycho12@dongduk.ac.kr

Received: Dec 02, 2021 ; Revised: Dec 13, 2021 ; Accepted: Dec 28, 2021

Published Online: Jan 31, 2022

## Abstract

Objectives:

The coronavirus disease (COVID-19) pandemic has necessitated new ways of conducting business, including hybrid work, which requires the active digital transformation of workplaces. This study examined cases in which the coworking tool Flow was adopted and explained Flow’s characteristics and user experiences.

Methods:

Flow is a coworking tool developed by Madras Check to improve communication efficiency between workers. This case report was based on interviews and surveys of Flow users.

Results:

Flow was designed to increase productivity related to tasks and efficient management. According to client interviews, surveys, and our case analysis, Flow appears to achieve its purpose. In a survey of 1,429 respondents, Flow was found to reduce the amount of time spent checking messages via email and messenger applications, and respondents felt that Flow was a user-friendly tool and were willing to recommend Flow to others.

Conclusions:

The present case report shows how the Flow coworking tool contributed to fostering a digital work environment. Our analysis of the benefits and user experiences of Flow indicates that such coworking tools are necessary for digital transformation so that organizations can create a hybrid work environment, which is the primary work format following the COVID-19 pandemic.

Keywords: Flow; Coworking Tool; Digital Transformation; Hybrid Work; Virtual Office

## Introduction

According to an analysis by Trend Korea 2022 (Kim et al., 2021), four distinctive social trends are expected to arise in 2022, and these trends are likely to be strengthened due to the coronavirus disease 2019 (COVID-19) pandemic. The trends are 1) nano society, 2) rustic lifestyles, 3) the extended presence of technology, and 4) the reemergence of Generation X. The nano society trend refers to the phenomenon of widespread individualism due to an increase in smartphone, social media, and algorithm usage; this trend is becoming further intensified by the COVID-19 pandemic. Second, the rustic lifestyle trend refers to individuals’ ability to pursue a leisurely lifestyle as if they lived in a rural area even when residing in a city. Third, the extended presence of technology trend refers to the increased use of live commerce and metaverse technologies. Finally, Generation X, the members of which experienced both the analog and digital era, has emerged as a core group with a high degree of purchasing power. Each of these four distinctive trends can inspire changes in the way business is conducted. Due to deepening individualism, employee evaluations will increasingly be based on individual achievement rather than cooperative ability. The increasing preference for a rustic lifestyle will lead to a wider variety of workplaces in addition to a typical office setting and flexible working time. In addition, the extended presence of technology enables new types of workplaces through the use of metaverse-based coworking tools and virtual offices. Finally, members of Generation X are now old enough to be middle managers, connecting Millennials and members of Generation Z to older generations in organizations and enhancing communication between them.

In response to such changes in the business environment, many firms are trying to change the work environment. For example, global companies such as Google and Apple have extended telecommuting or work-from-home policies in response to the COVID-19 pandemic, and Korean firms such as Naver Line and Hyundai have begun to implement hybrid work models. In an analysis of 61 major listed corporations in the US, MarketWatch reported that 69% of them planned to adopt a hybrid work model. The proportion of companies in South Korea that used a hybrid work model increased from 4% in 2015 to 15% in 2020 (Wiltermuth, 2021).

Along with these shifts in work trends in response to the COVID-19 pandemic, companies now understand the need to reestablish their overall business strategies beyond temporary measures in response to the pandemic, as long-term changes are taking place in the business environment. Hybrid work is one of the work trends in 2022 that is guiding new business strategies (Offin, 2022). Hybrid work combines traditional office-based working with remote work that has been widely used after COVID-19, and 83% of people wanted to work in hybrid format after pandemic according to one study in May 2021 (Christian, 2022). Hybrid work enables the flexible use of digital technology and company resources and allows employees to experience work from any place at any time. Hybrid work enables an increase in the flexibility of the work environment as well as employee autonomy concerning decision-making. Hybrid work also allows companies to retain teams and human resources for productive cooperation, enabling new opportunities for growth.

As a result, hybrid work has been suggested as a new form of work. For example, Gartner Top Strategic Technology Trends for 2022 predicted that 75% of the companies that implement hybrid work will have a 25% faster growth rate than their competitors by 2023. In addition, a predicted 9 out of 10 companies will combine remote work and fieldwork in the future (Lund, Madgavkar, Manvika, & Smit, 2021), and Global Workplace Analytics (2022) predicted that 70% of all human resource workers will work from home by 2025.

As such, the COVID-19 pandemic has increased the need for new ways of conducting business, such as hybrid work. Hybrid work calls for active digital transformation in the workplace. There are several ways to increase productivity under hybrid work environment, and digital transformation is one of them (Pinola, 2021). Specifically, it was suggested that workers should go paperless as much as possible, without getting around the hassle of hauling of paper files, create schedule managers for efficient time management, and stay connected with co-workers for efficient hybrid work (Pinola, 2021). These suggestions are closely related to digital transformation in the organization, which can be enabled by coworking tools where workers can simultaneously communicate on the given project without constrained by time and place. Thus, it appears that COVID-19 pandemic facilitates the need for coworking tools. In fact, it has been reported that coworking tool market is expected to grow from $47.2 billion in 2021 to$85.8 billion in 2026, recording a compound annual growth rate of 12.7% (Kim, 2021).

The present case report analyzes the coworking tool Flow, developed by Madras Check, to help prepare companies for digital transformation. In the next section, we describe the background concerning companies’ shift to digital transformation in greater detail and introduce the key features and benefits of Flow. Next, we describe the experiences of companies that implemented Flow and discuss the implications of digital transformation for business strategies.

## Case Description

Coworking Solutions Proposed by Flow

Flow is a coworking tool developed by Madras Check that helps companies digitally transform the work environment. Flow refers to the experience of immersion in a task, enabling the user to proceed toward a goal. Madras Check created Flow to help workers concentrate on tasks without being overwhelmed by so many communication routes. Approximately 3,000 companies, ranging from large to small in size, have adopted Flow, and approximately 200 companies newly adopt Flow each month. As such, Flow ranks first among similar coworking tools, and the demand for Flow has increased by three times since the beginning of the COVID-19 pandemic in January 2020.

Flow has several distinctive features that differentiate it from its competitors. First, employees can create project rooms based on individual project topics and invite related personnel for timeline-based collaboration like Facebook. Flow provides not only messenger services but also posting-based cooperation tools. Though communication through messenger is convenient, it can be hard to continuously keep track of messages. In contrast, posting-based communication makes it easier to keep track of messages and feedback by enabling comments under posts that are relevant to ongoing projects.

Since Flow is available on both PC and mobile, files can be opened using a mobile application, and large files can be shared without problems. In addition, Flow enables jobs to be tracked across five stages: “request,” “in-process,” “feedback,” “completion,” and “on hold.” A powerful filtering function allows important tasks to be handled without error. Finally, Flow includes useful supplementary functions such as a smart calendar that workers can use to manage their schedules and the schedules of others, to-do lists, and strong security functions. Flow is an all-in-one coworking tool that can be operated intuitively in a manner that is familiar to Koreans; thus, anyone can easily use it without a long tutorial. Figure 1 and Figure 2 summarize Flow’s distinctive features compared to other coworking tools and examples of how to use Flow for each job, respectively.

Figure 1. Features of Flow compared to other coworking tools.
Figure 2. Application of Flow across various departments.
Rules of Successful Coworking Fulfilled by Flow

The use of Flow in a workplace can help fulfill the seven principles of successful coworking, which include place, connection, real-time tracking, objectives, security, partnership, and growth (Flow, 2021b).

Place

Flow can turn any location into a workplace in seconds . For successful collaboration to take place, there must be no restrictions on where an individual works. Flow provides a flexible workplace. LIG Nex1 was the first aerospace manufacturer and defense industry company to implement a work-from-home policy, and flexible non-face-to-face work has progressed smoothly since the implementation of Flow.

Connection

It is crucial in a collaboration that all employees can communicate smoothly. Flow allows all employees to connect with each other while working in different locations and at different times. Flow unifies scattered communication channels via systematic linkage with existing in-house systems such as groupware, enterprise resource planning, mail, organization charts, and chatbots. In doing this, Flow strengthens communication by integrating data services provided by different and dispersed platforms into one platform (chatbot). The CEO of Busan-based GENTZ Bakery has stated that the use of Flow allowed GENTZ Bakery to enter Market Kurly, one of the largest online grocery services, stating that, while many tasks must be performed simultaneously and promptly to run a bakery, the company was able to manage various projects systematically because of Flow.

Real-Time Tracking

Prompt decision-making is required in collaborative work. Flow provides workers with real-time tracking of the process of each task. Specifically, Flow has a function to assign the priority level of tasks, the person in charge, and the deadline for each task. This allows for rapid decision-making and business processing.

Objectives

Flow allows each employee to determine the overall objective of the organization, which in turn enables them to create detailed task goals based on the overall objective.

Security

Due to the development of information technology, the speed and convenience of conducting business have improved; however, document sharing and data leakage have become easier, and even a single USB can cause great damage. Flow has a strong security system with various functions including a screen capture prevention setting, 2-factor authentication, a duplicate login limit, and data encryption that minimize the possibility of such issues. Hyundai Mobis, one of Flow’s clients, secures information by requiring employees to agree to an information protection pledge when creating a new project and penalizes those who violate this pledge. In addition, Flow’s text security authentication procedure severely restricts login by non-members.

Partnership

With Flow, organizations can invite an unlimited number of partners for free, allowing for close and systematic collaboration. For example, POSCO and E-LAND invited marketing agencies using Flow to systematically collaborate on marketing/advertisement-related projects.

Growth

Flow can store all project-related data indefinitely, so workers can easily find the data they want at any time. Thus, even members who join a project late can refer to the project history. In addition, data from past projects can help users determine what worked and what did not, providing the opportunity to grow. COOCON, a company that provides a fintech API platform and business information, strongly recommended Flow based on its experience of growth after COOCON was able to enhance the quality of its core customer-oriented products using the work history stored by Flow.

Benefits of Flow Based on a Survey

In this section, we examine the results of a survey conducted by Flow of 1,429 workers (Flow, 2021d) and discuss the benefits of Flow. First, after using Flow, the average scores for the quality of collaborative tasks and the management of each worker’s performance increased from 58 to 80 and 58 to 89, respectively (Figure 3). Thus, Flow enhanced two of the basic principles of coworking—productivity and management efficiency—by enabling business to be conducted in innovative ways. Second, survey respondents found that Flow helped reduce wasted time spent on tasks that were unrelated to workers’ main tasks.

Figure 3. Improvement of quality in collarative tasks and performance management.

Specifically, before using Flow, 40.6% of the respondents felt that about 30 minutes to 1 hour per day was wasted due to inefficient collaboration methods used before the implementation of Flow. In particular, the respondents said that they wasted a lot of time replying to messages over messenger applications and emails. After the implementation of Flow, 38.2% of the respondents said that approximately 30 minutes to 1 hour per day was saved (Figure 4). Finally, the average overall satisfaction score exceeded 4 out of 5 points. Specifically, the average ratings for “ease of operation,” “operation speed,” “service function,” and “user-friendliness” were 4 points, 4 points, 4.1 points, and 4.1 points, respectively, and 83% of the respondents said that they were willing to recommend Flow to others (Figure 5). Overall, wasted time was greatly reduced through the use of Flow, and the efficiency of collaboration increased.

Figure 4. Perceived efficiency with Flow tool.
Figure 5. Overall satisfaction with Flow.
Instances of Digital Transformation after the Adoption of Flow
Case A

Company A adopted Flow after searching for a coworking tool that could be connected to both PC and mobile devices. Initially this company has used Jira and Conflience but they are suitable only for those who work in R & D department. What Company A has looked for is the toolthat workers across the whole organization could use regardless of whether they worked in the R&D department. Since personnel information is automatically connected to Flow, the handling of employees’ on- and off-boarding information was automated. In addition, the company’s performance appraisal system connected to Flow so that workers could register their work output and track their work history. Thus, workers could seamlessly communicate with their colleagues using Flow and simultaneously post their work outputs to receive and share feedback.

Case B

Comapny B adopted Flow for two reasons. First, the company found communication through messenger and email to be inefficient, which led them to adopt an Objectives and Key Results (OKR) system. The company then searched for a coworking tool that could be connected to the OKR system and subsequently adopted Flow. Second, as the format of work shifted toward a hybrid work model, the company realized that performance appraisals, as well as communication channels, should be revised in an innovative way. Workers now communicate through Flow, and their work outputs are reported using the OKR system. Flow was also connected to Company B’s video conference system, Nos.

Case C

Company C adopted Flow to enhance work efficiency and transparently evaluate employees’ work performance. In a face-to-face work model, it was easy to evaluate individual performance, but it became difficult to evaluate an individual’s work performance in a non-face-to-face work setting. Therefore, this company chose Flow as a coworking tool to help track workers’ performance in a non-face-to-face context. The telecommuting daily work record system used by the company was connected with Flow so that each individual’s work history was automatically recorded and synced to Flow, enabling each employee to easily share what they do with other employees. In addition, employees could open small rooms in Flow to communicate with each other and form groups to interact with Flow as a small community.

Case D

Company D adopted Flow to efficiently develop and operate information technology projects with its R&D department and partners. In addition to collaboration, Flow was used to enhance security related to mobile device management. In particular, company’s own Mobile Device Management (MDM) solution is connected to Flow so that the internal security system is advanced. Also, this comapny customized the security system in Flow so that a warning notification appeared when sensitive information such as an individual’s social security number or account number was entered while workers chatted or shared files.

Case E

Company E adopted Flow to enable digital transformation throughout its organization. In particular, the company chose Flow as a result of its search for a tool that enabled systematic communication across various brands and the efficient management of each brand’s business. All employees were highly satisfied with Flow since it was introduced after its usability was verified after a 3-month proof of concept (POC) that included employees and managers. In addition, Flow was connected with Zoom, a video conferencing tool that was previously used by the company, thus increasing Flow’s usability.

Case F

Company F made plans for digital transformation across the entire organization to manage the work of various branches scattered across the country. In addition, the messenger application previously used by the company only worked on PC, and the company needed a coworking tool that enabled communication via mobile devices. As a result, Flow was introduced, and it is used not only for all employees in the head office, but also for collaboration with partners. Flow was connected with the groupware, schedule management, and email applications used by this company.

Case G

Company G introduced Flow to reorganize a smart communication platform that can be used with the next-generation version of ERP. The second reason for introducing Flow is to efficiently communicate with many stores branches across the country and to manage the performance of those branches. This company linked the chat function of Flow with SAP ERP, which was used previously, to enable inventory inquiry and sales inquiry through Flow’s chatbot.

In addition, Flow is installed and operated in AWS private cloud.

## Discussion

This case report showed how the Flow coworking tool contributed to companies’ attempts to establish digital work environments by describing the characteristics of Flow and companies’ experiences using Flow. The need for coworking tools has increased due to growing uncertainty about how to best conduct business in the modern era and was accelerated as a result of the COVID-19 pandemic (Schubert & Williams, 2022). Approximately 87% of companies have not determined a business plan for the next year due to business uncertainty (KDI Economic Policy Uncertainty, 2020), and 56.2% of 80 major companies and 236 small and medium-sized companies have not yet established investment plans (Korea Chamber of Commerce and Industry, 2021).

As uncertainty in business environments grows, businesses must stick to the basics—productivity and management efficiency. In addition, innovation is required to promptly respond to uncertainties in the business environment. Introducing a coworking tool is a way to prioritize these basic rules in innovative ways. Hybrid work is an example of an innovation related to how work is conducted. In the COVID-19 era, during which non-face-to-face work environments are dominant (de Lucas Ancillo, del Val Núñez, & Gavrila, 2021; Newman & Ford, 2021), hybrid work presents a suitable work format for this context (Yang, Kim, & Hong, 2021). Implementing a hybrid work model requires a digital transformation in organizations (Gabryelczyk, 2020; Soto-Acosta, 2020), and the need for digital transformation often facilitates the adoption of coworking tools.

An increase in demand for coworking tools is inevitable since communication tools have become more varied since the late 2010s. From the 1980s to early 2010s, the major communication tools used in workplaces were emails and PC messenger applications, while groupware, video (Zoom conferencing and remote meetings), and metaverse applications have become more prevalent beginning in the late 2010s. As a result, employees often answer phone calls while replying to messages and emails, which increases the complexity of their work and reduces work efficiency (Flow, 2021a). In order to enhance work efficiency, coworking tools have been developed. The usefulness of coworking tools for enhancing smooth communication and work efficiency is supported by the existing literature.

Since the COVID-19 pandemic, more people have begun to work from home, reducing employee engagement, which is one of the most prominent issues for human resource managers in the COVID-19 era (Chanana & Sangeeta, 2020). Specifically, working from home makes it difficult to focus compared to working in an office setting. In addition to interruptions by family members and anxiety about the inability to separate work from life, some employees do not have access to the equipment needed to complete their work at home. Because of these factors, telecommuting can reduce employee engagement (Chanana & Sangeeta, 2020).

In response to this issue, various engagement practices have been introduced in organizations such as virtual learning and development, online team-building activities, webinars by industry experts, online weekly alignment sessions, and team meet-ups over video conferences during lunch. Many of these activities have been successful, helping employees learn new skills and grow and increasing employees’ work productivity and sense of commitment (Chanana & Sangeeta, 2020). Coworking tools can also be understood as tools for engagement since they increase users’ motivation to collaborate and increase work efficiency.

In addition, in response to the acceleration of digital transformation caused by the COVID-19 pandemic, many believe that working digitally will be the dominant work format in the future (Nagel, 2020). Our case analysis of companies across various industries revealed that working in a virtual office setting using Flow helped improve business efficiency, supporting Nagel’s (2020) prediction. Finally, the notion of the workplace has changed from a building or place where people perform their job to a wider concept encompassing more dimensions such as social relationships, gathering, training, motivation, and engagement (Fisher, 2010; Gill, 2019). Therefore, a virtual office setting that addresses these various dimensions is important. Coworking tools that enable work in a virtual office setting will increase in relevance in the coming years.

Coworking tools like Flow enable companies to fulfill the three requirements for successful implementation of a hybrid work model: making digital tools available to foster a flexible work environment, improving the efficiency of work processes, and cultivating an innovative organizational culture (Flow, 2021c). Based on the survey and case analysis, Flow helped to change the overall work environments of organizations in an innovative way. Specifically, Flow enabled seamless communication between employees regardless of location. Instead of going to an office to check and respond to emails, workers became capable of communicating using mobile devices and in real-time with Flow.

In addition, employees who worked on the same project could share their objectives and work in accordance with those objectives, and external partners could easily be invited to join the project. Flow’s strong security functions enabled workers to keep important information safe, and by tracking past work history, workers could learn from past successes and failures and use data on Flow as a tool for growth. Therefore, Flow contributes to innovation in workplaces undergoing a digital transformation. Based on this case analysis, companies across various industries can benefit from using Flow. By using Flow, the companies in the present study agreed that communication between colleagues was smoother, work efficiency increased, and transparency with regard to employee evaluations improved since one’s work could be monitored in real-time. In the future case report, it will be interesting to examine in detail how Flow can be differentiated when compared with competitor’s coworking tools such as Notion, Base Camp, and Jira.

## Conclusion

Coworking tools are expected to lead to changes in work processes that are markedly different from previous ones. Specifically, in the past, one-way communication via phone or email in physical spaces was the dominant work format, and collaboration was conducted as teams. Using coworking tools, however, anyone can freely communicate in both directions in an online virtual space, and organizations can conduct business as project units rather than team units. In order to build such work processes, both employees and managers need to cultivate an organizational culture with a digital mindset. Based on this case analysis of Flow, it can be concluded that coworking tools are necessary for the digital transformation of organizations. Companies should actively adopt coworking tools in response to the current need to cultivate a hybrid work environment. Such digital transformation efforts in which people can perform their jobs without time or place constraints will ultimately increase the autonomy and productivity of workers.

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Chanana, N., & Sangeeta (2020). Employee engagement practices during COVID‐19 lockdown. Journal of Public Affairs, 21(4), e2508. .

2.

Christian, A. (2022, January 21). A part-remote, part-office schedule has been hailed as the future of work. Yet in this hybrid set-up, some employees have never been so tired. Retrieved from https://www.bbc.com/worklife/article/20220120-why-hybrid-work-is-emotionally-exhausting#:~:text=In%20theory%2C%20hybrid%20offers%20the,for%20focused%20work%20at%20home.

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de Lucas Ancillo, A., del Val Núñez, M. T., & Gavrila, S. G. (2021). Workplace change within the COVID-19 context: A grounded theory approach. Economic Research-Ekonomska Istraživanja, 34(1), 2297-2316. .

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Fisher, C. D. (2010). Happiness at work. International Journal of Management Reviews, 12(4), 384-412. .

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Flow (2021a, May 20). Coworking tools, essential conditions for companies to achieve their goals. Retrieved from https://post.flow.team/insight/4505.

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Flow (2021b, June 7). 10 Construction strategies for 2021 digital work trends. Retrieved from https://post.flow.team/insight/5005.

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Flow (2021c, June 21). Defining coworking tool and strategy for constructing 3 components of digital transformation. Retrieved from https://post.flow.team/insight/5262.

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# Proving Inverse Matrices?

1. Mar 16, 2010

### swuster

1. The problem statement, all variables and given/known data
I'm trying to show that a given matrix is the inverse of the other, by showing that multiplying them together generates the identity matrix. I can't see a way to simplify the last step and I feel like I'm missing something..? Any input on this would be helpful. Thanks!

2. Relevant equations
see below
j = sqrt(-1)

3. The attempt at a solution

How do I prove that the rest of the terms are 0?

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2. Mar 16, 2010

### LCKurtz

It looks to me like the last sentence, which you cut off at the comma, might give you a hint. What does it say?

3. Mar 17, 2010

### swuster

It's not cut off. It's as far as I have gotten with the proof, haha. There is nothing obvious that jumps out at me when I consider the case where the exponential argument isn't 0.

4. Mar 17, 2010

### jbunniii

Try using

$$\sum_{m=0}^{M-1} z^m = \frac{1 - z^{M}}{1 - z}$$

which holds for any complex number $z \neq 1$.

5. Mar 17, 2010

### swuster

Thanks so much! That did the trick!

6. Mar 17, 2010

### jbunniii

Looks good, nice job.

It's also insightful to consider what is going on geometrically.

This sum:

$$\sum_{i=0}^{M-1}\frac{1}{M}e^{j\left(\frac{2\pi(\lambda-\kappa)i}{M}\right)$$

is calculating the average of $M$ complex numbers. Furthermore, since $\lambda - \kappa$ is a nonzero integer, these complex numbers are evenly spaced samples around the unit circle, so their average is zero.

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# Continuity & differentiability of an improper integral of discontinuous function.

My question regards the continuity & differentiability of an improper integral of a discontinuous function. Let the function be defined as $$g(t) = \int_{0}^{t}f(x)\,\mathrm{d}x$$

Assume that $f(x)$ is bounded and continuous $\forall x \in \mathbb{R},x\neq 0$. Since the number of discontinuties of $f(x)$ are countable the function is Riemann integrable on $\mathbb{R}$. Since the function is Reimann integrable is that enough to state that the function $g(x)$ is continuous on $\mathbb{R}$?

As for differentiability, is knowing that the function $f(x)$ is not continuous on $\mathbb{R}$ enough to conclude that the function g(x) is not differentiable on $\mathbb{R}$ as we wouldn't be able to apply the Fundamental Theorem of Calculus?

• Why would anyone down vote this? – zhw. Sep 3 '17 at 15:47
• Yes, $g$ is continuous. This is basic and easy to prove. Not sure why you removed the $\cos(1/x)$ example. Indeed $g$ will be differentiable everywhere in this case. – zhw. Sep 3 '17 at 15:56
• Do you mean for $x$ to be both the variable of integration and one of the limits of integration? Or do you mean something like $g(x) := \int_{0}^{x} f(t)\mathrm{d}t$? – Xander Henderson Sep 4 '17 at 2:15
• @XanderHenderson yes I did mean something like that. Edited. – werdho Sep 4 '17 at 2:48
• By the way your integral is not improper so removing the tag. – Paramanand Singh Sep 7 '17 at 7:41

Let us consider the function $x\mapsto g(x)=\int_{x_0}^x f(s) \; \mathrm d s$. Checking the proof of the fundamental theorem of calculus one can see that the main idea is to calculate $(g(x+h)-g(x))/h$ and then to calculate the limit $h\to 0$ with the help of the mean value theorem for definite integrals. Since the mean value theorem also holds for so called regulated functions (which are in general not continuous) $f$ does not have to be continuous.

• No, in this special case, $g$ is indeed differentiable at each $x\in\mathbb R$ with $g'(x)=f(x)$, even for $x=0$, while $f$ is not continuous at $x=0$. However, if you change $f$ at $x=0$ by setting $f(0)=17$, say, then $g$ is still differentable everywhere, but now $g'(0)=0\neq 17=f(0)$. So in this case, $f$ would have the "wrong" value at $x=0$. – sranthrop Sep 5 '17 at 3:04

Depending on what kind of integral you use, the following statements are true:

• If $f$ is continuous (and hence Riemann-integrable), then $g$ is differentiable everywhere with $g'=f$ everywhere.
• If $f$ is Lebegsue-integrable, then $g$ is absolutely continuous and almost everywhere differentiable with $g'=f$ almost everywhere.
• If $f$ is Henstock-integrable, then $g$ is continuous and almost everywhere differentiable with $g'=f$ almost everywhere.

In particular, in either of the above cases, $g$ is continuous. For the differentiability part, the following is true.

• $g$ is differentiable everywhere with $g'=f$, if and only if $f$ is Henstock-integrable and satisfies \begin{align*} g(x)=\lim_{h\to0}\frac{1}{h}\int_x^{x+h}f(t)dt \end{align*} for each $x$.

However, there are examples of very discontinuous $f$, while $g$ is still everywhere differentiable. A nice discussion can be found here: How discontinuous can a derivative be? In particular, there are derivatives which are neither continuous, nor Riemann- or Lebesgue-integrable, nor bounded.

Moreover, from Darboux-Theorem we know that each derivative has the Intermediate Value Property. This implies that if $f$ is a derivative of $g$ which is discontinuous at some point $x_0$, then $x_0$ is an essential discontinuity, which means that at least one of the one-sided limits does not exist. In particular, a regulated function $f$ is a derivative of $g$, if and only if $f$ is continuous.

Let's assume that the integral involved is Riemann integral. If $f$ is Riemann integrable then $g$ is continuous and of bounded variation. Further $g$ is differentiable at points of continuity of $f$ and at those points $g'=f$. Since a Riemann integrable is continuous almost everywhere, you will have $g'=f$ almost everywhere.

It is possible that $g$ is differentiable even at points where $f$ is discontinuous but then it is not guaranteed that $g'=f$. Further if $f$ has a jump discontinuity at a point, then it is guaranteed that $g$ is not differentiable at that point.

A proper discussion of Fundamental Theorem of Calculus should cover the proof of all the statements mentioned above (perhaps leaving the part about almost everywhere) and not just deal with the simpler and special case when $f$ is continuous everywhere.

Also the two parts of Fundamental Theorem of Calculus are equivalent if the function $f$ is continuous. These parts are different if the function $f$ being integrated is discontinuous. You may have a look at this answer for more details.


text:
[Ilugc] re: how to display c, c++, php, html in latex

Kumar Appaiah a.kumar at alumni.iitm.ac.in
Sat Dec 13 10:42:42 IST 2008

On Fri, Dec 12, 2008 at 11:40:53PM -0500, Ravi Jaya wrote:
> Recently i started  with latex, I got struck when i need to display the
> source code as a part of the article.  For example, if i need to display the
> following set of html tags in the latex article.
> <html>
> <body>
> foo...
> </body>
> </html>

\begin{verbatim} ... \end{verbatim}


text:
Continued Fraction Proofs of $m$-versions of Some Identities of Rogers-Ramanujan-Slater Type. (arXiv:1901.05888v1 [math.NT])

We derive two general transformations for certain basic hypergeometric series from the recurrence formulae for the partial numerators and denominators of two $q$-continued fractions previously investigated by the authors. By then specializing certain free parameters in these transformations, and employing various identities of Rogers-Ramanujan type, we derive \emph{$m$-versions} of these identities. Some of the identities thus found are new, and some have been derived previously by other authors, using different methods. By applying certain transformations due to Watson, Heine and Ramanujan, we derive still more examples of such $m$-versions of Rogers-Ramanujan-type identities.查看全文

Solidot 文章翻译

 你的名字 留空匿名提交 你的Email或网站 用户可以联系你 标题 简单描述 内容 We derive two general transformations for certain basic hypergeometric series from the recurrence formulae for the partial numerators and denominators of two $q$-continued fractions previously investigated by the authors. By then specializing certain free parameters in these transformations, and employing various identities of Rogers-Ramanujan type, we derive \emph{$m$-versions} of these identities. Some of the identities thus found are new, and some have been derived previously by other authors, using different methods. By applying certain transformations due to Watson, Heine and Ramanujan, we derive still more examples of such $m$-versions of Rogers-Ramanujan-type identities.


text:
Definition 59.64.1. Let $X$ be a scheme. Let $\mathcal{F}$ be a sheaf of sets on $X_{\acute{e}tale}$.

1. Let $E$ be a set. We say $\mathcal{F}$ is the constant sheaf with value $E$ if $\mathcal{F}$ is the sheafification of the presheaf $U \mapsto E$. Notation: $\underline{E}_ X$ or $\underline{E}$.

2. We say $\mathcal{F}$ is a constant sheaf if it is isomorphic to a sheaf as in (1).

3. We say $\mathcal{F}$ is locally constant if there exists a covering $\{ U_ i \to X\}$ such that $\mathcal{F}|_{U_ i}$ is a constant sheaf.

4. We say that $\mathcal{F}$ is finite locally constant if it is locally constant and the values are finite sets.

Let $\mathcal{F}$ be a sheaf of abelian groups on $X_{\acute{e}tale}$.

1. Let $A$ be an abelian group. We say $\mathcal{F}$ is the constant sheaf with value $A$ if $\mathcal{F}$ is the sheafification of the presheaf $U \mapsto A$. Notation: $\underline{A}_ X$ or $\underline{A}$.

2. We say $\mathcal{F}$ is a constant sheaf if it is isomorphic as an abelian sheaf to a sheaf as in (1).

3. We say $\mathcal{F}$ is locally constant if there exists a covering $\{ U_ i \to X\}$ such that $\mathcal{F}|_{U_ i}$ is a constant sheaf.

4. We say that $\mathcal{F}$ is finite locally constant if it is locally constant and the values are finite abelian groups.

Let $\Lambda$ be a ring. Let $\mathcal{F}$ be a sheaf of $\Lambda$-modules on $X_{\acute{e}tale}$.

1. Let $M$ be a $\Lambda$-module. We say $\mathcal{F}$ is the constant sheaf with value $M$ if $\mathcal{F}$ is the sheafification of the presheaf $U \mapsto M$. Notation: $\underline{M}_ X$ or $\underline{M}$.

2. We say $\mathcal{F}$ is a constant sheaf if it is isomorphic as a sheaf of $\Lambda$-modules to a sheaf as in (1).

3. We say $\mathcal{F}$ is locally constant if there exists a covering $\{ U_ i \to X\}$ such that $\mathcal{F}|_{U_ i}$ is a constant sheaf.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).


text: When multiplying a given matrix with an identity matrix, the values remain the same. For the multiplication of numbers, the identity is 1. All the elements of the matrix apart from the diagonal are zero. By definition, when you multiply two matrices that are inverses of each other, then you will get the identity matrix. Learn about the properties of matrix multiplication (like the distributive property) and how they relate to real number multiplication. Hat Matrix: Properties and Interpretation Week 5, Lecture 1 1 Hat Matrix 1.1 From Observed to Fitted Values The OLS estimator was found to be given by the (p 1) vector, b= (XT X) 1XT y: The predicted values ybcan then be written as, by= X b= X(XT X) 1XT y =: Hy; where H := X(XT X) 1XT is an n nmatrix, which \puts the hat on y" and is therefore referred to as the hat matrix. One concept studied heavily in mathematics is the concept of invertible matrices, which are those matrices that have an inverse. Identity matrix is always in the form of a square matrix. (The matrices that have inverses are called invertible.) If any matrix is multiplied with the identity matrix, the result will be given matrix. No, the given matrix is not an identity matrix since it is not a square matrix. The elements of the given matrix remain unchanged. Open Live Script. And it will remain unchanged. If you multiply two matrices that are inverses of each other you would get an identity matrix. Not every square matrix has an inverse! Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share â¦ 2. Triangular Matrices. Vedantu academic counsellor will be calling you shortly for your Online Counselling session. Properties of an Identity Matrix. Pro Lite, Vedantu For any whole number $$n$$, there is a corresponding $$n \times n$$ identity matrix. In this presentation we shall review the Identity properties and use these properties to solve some simple example problems. An identity matrix is a square matrix in which all the elements of principal diagonals are one, and all other elements are zeros. Now, for some of the binary operations, we have what is called an identity element. These matrices are said to be square since there is always the same number of rows and columns. Finally, an identity matrix is said to be idempotent. An identity matrix, also known as a unit matrix, is a square matrix in which all of the elements of the principle diagonal are ones, and the rest are zeros. 1. The identity matrix is a fundamental idea when working with matrices – whether you are working with just multiplication, inverses, or even solving matrix equations. For example, consider the following matrix. Solution: THE IDENTITY MATRIX In this case, the rref of A is the identity matrix, denoted In characterized by the diagonal row of 1's surrounded by zeros in a square matrix. In particular, elementary row operations involve nonsingular matrices and, hence, do not change the rank of the matrix being transformed. This MATLAB function returns the scalar 1 with the same fixed-point properties and complexity (real or complex) as the prototype argument, p. In linear algebra, the identity matrix (sometimes ambiguously called a unit matrix) of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere. If A is a non-singular square matrix, there is an existence of n x n matrix A-1, which is called the inverse of a matrix A such that it satisfies the property:. It is also referred to as a unit matrix or an elementary matrix. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. So in the figure above, the $$2 \times 2$$ identity could be referred to as $$I_2$$ and the $$3 \times 3$$ identity could be referred to as $$I_3$$. 2. The identity matrix is known as the matrix that is in the form of the n Ã n square matrix in which the diagonal contains the ones and all the other elements are zeros. In this article, we will learn about what is an identity matrix, the determinant of identity matrix, identity matrix properties, the identity matrix in c, and learn about the identity matrix example. It acts just like the multiplication of the real numbers by 1. When a vector is multiplied by an identity matrix of the same dimension, the product is the vector itself, Inv = v. rref( )A = 1 0 0 0 1 0 0 0 1 The order of the matrix comes from its dimensions, and the main diagonal is the array of the elements that are inside the matrix that form the inclined line starting from the top left corner and extending to the bottom right corner. It is the matrix that leaves another matrix alone when it is multiplied by it. This is one of the midterm 1 problems of Linear Algebra at the Ohio State University in Spring 2018. This is the associative law for matrix addition. Search. The given matrix is of the order 2 x 3. Consider the example below where $$B$$ is a $$2 \times 2$$ matrix. Sign up to get occasional emails (once every couple or three weeks) letting you know what's new! This indicates that the square of an identity matrix is equal to itself. Given the characteristics of the identity matrix, you can also conclude that these type of matrices are also called as diagonal matrices. This is a $$2 \times 4$$ matrix since there are 2 rows and 4 columns. Linear algebra is a branch of mathematics that deals with vectors and operations on vectors. Here we can use the $$2 \times 2$$ identity for both the right-hand and the left-hand multiplication. Courses. For any whole number n, there is a corresponding n×nidentity matrix. Zero matrix: we denote by 0 the matrix of all zeroes (of relevant size). Three properties of matrix rank are of general interest to matrix algebra: 1. The matrix multiplication also contains an identity element. This tutorial covers the basics of vectors and matrices, as well as the concepts that are required for data science and machine â¦ They will come in handy when you want to simplify an expression before di erentiating. or just I, wherein n is the size of the square matrix. C = $\begin{bmatrix} 1Â & 0Â & 0 \\ 0Â & 0 & 1 \end {bmatrix}$. Same with the zero matrix, which is the zero element such that Z+A=A+Z=A. The main diagonal divides a square matrix in to two triangles. We are given an expression using three matrices and their inverse matrices. Sorry!, This page is not available for now to bookmark. The identity matrix is for square matrices what 1 is to numbers, that is, they are both identity elements such that IA=AI=A. It is denoted as. In addition, the identity is 0. In this article, we will learn about what is an identity matrix, the determinant of identity matrix, identity matrix properties, the identity matrix in c, and learn about the identity matrix example. An identity matrix is always an square matrix:As seen in equations 1 and 2, the order of an identity matrix is always n, which refers to the dimensions nxn (meaning there is always the same amount of rows and columns in the matrix). Inverse: if A is a square matrix, then its inverse A 1 is a matrix of the same size. You cannot find the determinant of a rectangular matrix because determinant is only defined for the square matrices. [MUSIC] (In) to the power of k = In where k = 1, 2 and so forth. Hence, I = C = g(t) = e(A+B)te Bte At for all t. After multiplying by eAteBt on both sides we have eAteBt = e(A+B)t. Exercises: 1. While we say “the identity matrix”, we are often talking about “an” identity matrix. Symmetric Matrix and Skew Symmetric Matrix, Solutions â Definition, Examples, Properties and Types, Classification of Elements and Periodicity in Properties, Physical Properties of Alkanes and Their Variations, NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties, NCERT Solutions for Class 11 Chemistry Chapter 3, NCERT Solutions for Class 11 Physics Chapter 11, NCERT Solutions for Class 11 Physics Chapter 10, NCERT Solutions for Class 11 Physics Chapter 9, NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties Exercise 6.3, NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and Its Properties In Hindi, NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties (EX 6.2) Exercise 6.2, NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties (EX 6.5) Exercise 6.5, NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties (EX 6.4) Exercise 6.4, CBSE Class 7 Maths Chapter 6 - Triangle and Its Properties Formulas, Class 11 Chemistry Revision Notes for Chapter 3 - Classification of Elements and Periodicity in Properties, CBSE Class 11 Physics Thermal Properties of Matter Formulas, CBSE Class 11 Physics Mechanical Properties of Solids Formulas, CBSE Class 11 Physics Mechanical Properties of Fluids Formulas, CBSE Class 7 Maths Revision Notes Chapter 6 - The Triangle and Its Properties, Class 11 Physics Revision Notes for Chapter 10 - Mechanical Properties of Fluids, Class 11 Physics Revision Notes for Chapter 9 - Mechanical Properties of Solids, Class 11 Physics Revision Notes for Chapter 11 - Thermal Properties of Matter, CBSE Class 7 Maths Chapter 2 - Fractions and Decimals Formulas, Vedantu So in the figure above, the 2×2 identity could be referred to as I2 and the 3×3 identity could be referred to as I3. Properties of matrix addition We restrict attention to the set of all m nmatrices. Large datasets are often comprised of hundreds to millions of individual data items. In this lesson, we will look at this property and some other important idea associated with identity matrices. In other words, if all the main diagonal of a square matrix are 1âs and rest â¦ The number of rows is not equal to the number of columns. For any given whole number n, the identity matrix is given by n x n. In general we say that. It is easier to work with this data and operate on it when it is represented in the form of vectors and matrices. And this property refers to the fact the main identity matrix can be multiplied by itself any number of time. The zero matrix O, the same size as A, is the additive identity for matrices the same size as A. (MA2): A+ O= A= O+ A. The above matrix is a 2 x 4 matrix since it contains 2 rows and 4 columns. Also, in the case of the tall rectangular matrix such as 3 x 2 you have two 3-dimensional vectors that cannot form the sides of a parallelepiped. When multiplying two inverse matrices, you would get an identity matrix. 2. A square matrix having zeros at all positions below the main diagonal. If I is a left identity matrix for a given matrix A, then the matrix product I.A = A. It is denoted by the notation âI nâ or simply âIâ. You can study this idea more here: inverse matrices. Identity Matrix Definition. It is also referred to as a unit matrix or an elementary matrix. The rank of a matrix is not changed by its premultiplication (or postmultiplication) by a nonsingular matrix. An identity matrix is a matrix whose product with another matrix A equals the same matrix A. If I is a right identity matrix for A, then the matrix product A.I = A. This is a special element that leaves things alone under that particular operation. The identity matrix is called a square matrix because it has the same number of the rows and the columns. Matrix derivatives cheat sheet Kirsty McNaught October 2017 1 Matrix/vector manipulation You should be comfortable with these rules. The identity matrix is known as the matrix that is in the form of the n Ã n square matrix in which the diagonal contains the ones and all the other elements are zeros. The transposed matrix is denoted by AT. Properties of matrices This is a version of part of Section 8.2. For a matrix B that is m x n, the identity element for matrix multiplication e = I should satisfy IB = BI = B. AA-1 = A-1 A = I, where I is the Identity matrix. All bold capitals are matrices, bold lowercase are vectors. Transpose Matrix. The identity matrix is special in that when it is applied to vertices, they are unchanged. How to find the determinant of a rectangular matrix such as a 2 x 3 matrix? Thus, in the case of a wide rectangular matrix such as 2 x 3, it does not make any sense to define the determinant, since there are three 2-dimensional vectors that do not form a parallelogram when it is drawn from the origin in the cartesian coordinate system. A square matrix A is called orthogonal if AAT = I, where I is the identity matrix. The order of the matrix comes from its dimensions, and the main diagonal is the array of the elements that are inside the matrix that form the inclined line starting from the top left corner and extending to the bottom right corner. Consider the following matrices: For these matrices, $$AB = BA = I$$, where $$I$$ is the $$2 \times 2$$ identity matrix. Pro Lite, CBSE Previous Year Question Paper for Class 10, CBSE Previous Year Question Paper for Class 12. An identity matrix refers to a type of the square matrix in which its diagonal entries are equal to 1 and the off-diagonal entries are equal to 0. Some examples of identity matrices are as follows: I = $\begin{bmatrix} 1Â & 0 \\ 0 & 1 \end {bmatrix}$. The identity matrix for the 2 x 2 matrix is given by $$I=\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}$$ n is the n n identity matrix; its diagonal elements are equal to 1 and its o diagonal elements are equal to 0. Since the multiplication is not always defined, the size of the matrix matters when you work on the matrix multiplication. Using properties of inverse matrices, simplify the expression. The columns of A T are rows of A. The identity matrix of order 3 is represented in the following manner: I = $\begin{bmatrix} 1Â & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1\end {bmatrix}$, The identity matrix of the order 4 x 4 is given as, I = $\begin{bmatrix} 1Â & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0Â & 1 & 0\\ 0 & 0 & 0 & 1 \end {bmatrix}$. Any matrix typically has two different identity matrices: a left identity matrix and a right identity matrix. We are always posting new free lessons and adding more study guides, calculator guides, and problem packs. Because an identity matrix is a square matrix, its number of rows matches its number of columns. Example 1: If , then find M × I, where I is an identity matrix. Any number multiplied by 1 results in the same number. In particular, their role in the matrix multiplication is similar to the role that is played by the number 1 when it comes to the multiplication of the real numbers: The real number remains unchanged if it is multiplied by 1, The matrix remains unchanged if it is multiplied by an identity matrix. Transpose of a matrix If the rows and columns in a matrix A are interchanged, the new matrix is called the transpose of the original matrix A. Let us discuss the properties of the identity matrix. While we say âthe identity matrixâ, we are often talking about âanâ identity matrix. Therefore $$A$$ and $$B$$ are inverse matrices. For an m × n matrix A: Im A = A In = A. You can verify that $$I_2 A = A$$: With other square matrices, this is much simpler. A = I. In short, the identity matrix functions in the same â¦ These matrices are said to be square since there is always the same number of rows and columns. When working with matrix multiplication, the size of a matrix is important as the multiplication is not always defined. Use the deï¬nition (1) of the matrix exponential to prove the basic properties listed in Proposition 2. In linear algebra and functional analysis, a projection is a linear transformation from a vector space to itself such that =.That is, whenever is applied twice to any value, it gives the same result as if it were applied once ().It leaves its image unchanged. (MA1):(A+ B) + C= A+ (B+ C). Pro Lite, Vedantu A square matrix having zeros at all positions above the main diagonal. The identity matrix is called a square matrix because it has the same number of the rows and the columns. Because the identity matrix you need for any particular matrix multiplication will depend upon the size of the matrix against which the identity is being multiplied, and perhaps also the side against which you're doing the multiplication (because, for a non-square matrix, right-multiplication and left-multiplication will require a different-size identity matrix). (1 â¦ Transpose of matrix A is denoted by A T. Two rows of A T are the columns of A. Determine if the given matrix is an identity matrix or not. Matrix multiplication is a type of a binary operation. Less frequently, some mathematics books use U or E to represent the identity matrix, meaning "unit matrix" and the German word Einheitsmatrix respectively. The determinant is said to represent the size of the n-dimensional hyperspace that is occupied by the n-dimensional rectangular parallelepiped which is having the column vectors of the square matrix in the form of its sides.Â. Linear algebra is thus an important prerequisite for machine learning and data processing algorithms. 1. To explain the identity matrix definition part by part, let us start by reminding you that the square matrix refers is the matrix that contains the same amount of the rows and the columns. If A2 = 0, the zero matrix, prove that eA = I + A. C Program to check Matrix is an Identity Matrix Example. Take two elements from a given set and then follow some rules and combine them together to some other element of the given set. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context. Given the characteristics of the identity matrix, you can also conclude that these type of matrices are also called as diagonal matrices. In some fields, such as quantum mechanics, the identity matrix is denoted by a boldface one, 1; otherwise it is identical to I. As you study these types of topics, be sure that you have a fundamental understanding of this matrix. What do you mean by an identity matrix? To prevent confusion, a subscript is often used. An identity matrix has a value of 1 in coefficients [1,1],[2,2],[3,3] and a value of 0 in the rest of the coefficients. Copyright 2010- 2017 MathBootCamps | Privacy Policy, Click to share on Twitter (Opens in new window), Click to share on Facebook (Opens in new window), Click to share on Google+ (Opens in new window). If you're seeing this message, it means we're having trouble loading external resources on our website. It is denoted as In or just I, wherein n is the size of the square matrix. Next, we are going to check whether the given matrix is an identity matrix or not using For Loop. Therefore for an $$m \times n$$ matrix $$A$$, we say: This shows that as long as the size of the matrix is considered, multiplying by the identity is like multiplying by 1 with numbers. Create Identity Matrix with Fixed-Point Properties. For any given whole number n, the identity matrix is given by n x n. Multiplying a given matrix with the identity matrix would result in the matrix itself. To prevent confusion, a subscript is often used. Though abstract, this definition of "projection" formalizes and generalizes the idea of graphical projection. Any number when added to 0 results in the same number. Identity matrices play a vital role in the linear algebra. For example, for the given m x n matrix C, you get, C = $\begin{bmatrix} 1Â & 2 & 3 & 4\\ 5 & 6 & 7Â & 8 \end {bmatrix}$. Identity matrix is always in the form of a square matrix. In an identity matrix, all coefficients are 0 except the [1,1][2,2][3,3][4,4] coefficients, which are set to 1. the identity matrix. The rank of an n × n identity matrix I n × n, is equal to n. 2. To explain the identity matrix definition part by part, let us start by reminding you that the square matrix refers is the matrix that contains the same amount of the rows and the columns. This program allows the user to enter the number of rows and columns of a Matrix. Other examples of the binary operations include the addition of the real numbers and the multiplication of the real numbers. The identity matrix is a square matrix that has 1’s along the main diagonal and 0’s for all other entries. This correlates to setting M11 and M22 to 1 and M12, M21, OffsetX, and OffsetY to 0. Properties & Relations (3) Use DiagonalMatrix for general diagonal matrices: The KroneckerProduct of a matrix with the identity matrix is a block diagonal matrix: The WorkingPrecision option is equivalent to creating the matrix, then applying N: Possible Issues (1) IdentityMatrix gives a matrix with dense storage. This matrix is often written simply as $$I$$, and is special in that it acts like 1 in matrix multiplication. For a set S, there is an identity element e with respect to operation * such that for an element a in S: a*e = e*a = a. An identity matrix is capable of multiplying any matrix with any order (dimensions) as long as it follows the next rules: 1. Let us discuss the properties of the identity matrix.


text:
Fraction as an operator 'of':

Look at these similar squares

Each shaded portion represents 1/2 of 1.

So, the three shaded portions represent 1/2 of 3.

Combine the 3 shaded parts.

It represents 1 1/2 "i.e.," 3/2.

So, 1/2  "of"  3  "is"  3/2.

Also, 1/2 xx 3 = 3/2.

Thus, 1/2 "of"  3 = 1/2 × 3 = 3/2.

So we see that 'of' represents multiplication.

Example

What is 1/2 of 10?

1/2 of 10

= 1/2 xx 10

= (1 xx 10)/2

= 10/2

= 5.

Example

What is 2/5 of 20?

2/5 of 20

= 2/5 xx 20

= (2 xx 20)/5

= 40/5

= 8.

Example

What is 2/3 of 60.

2/3 of 60

= 2/3 xx 60

= (2 xx 60)/3

= (120)/3

= 40.

Example

In a class of 40 students 1/5 of the total number of students like to study English, 25 of the total number like to study Mathematics and the remaining students like to study Science.
(i) How many students like to study English?
(ii) How many students like to study Mathematics?
(iii) What fraction of the total number of students like to study Science?

Total number of students in the class = 40.

(i) Of these 1/5 of the total number of students like to study English.

Thus, the number of students who like to study English = 1/5 of 40

= 1/5 xx 40 = 8 students.

(ii) 2/5 "of"  40 = 2/5 xx 40 = (2 xx 40)/5 = 80/5 = 16 Students.

(iii) The number of students who like English and Mathematics = 8 + 16 = 24.
Thus, the number of students who like Science = 40 – 24 = 16.
Thus, the required fraction is 16/40.

If you would like to contribute notes or other learning material, please submit them using the button below.

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text:
• # question_answer A couple of ${{10}^{5}}N-m$is applied to a flywheel of mass 10 kg and radius of gyration 50m. Then the angular acceleration of the wheel is (in$rad/{{s}^{2}}$) A)  200               B)  50C)  10                D)  4

Angular acceleration is $\alpha =\frac{\tau }{I}=\frac{\tau }{M{{K}^{2}}}=\frac{{{10}^{5}}}{10\times {{50}^{2}}}=4\frac{rad}{{{s}^{2}}}$


text:
# Simple and Compound Surds

We will discuss about the simple and compound surds.

Definition of Simple Surd:

A surd having a single term only is called a monomial or simple surd.

For example, each of the surds √2, ∛7, ∜6, 7√3, 2√a, 5∛3, m∛n, 5 ∙ 7^$$^{3/5}$$ etc. is a simple surd.

Definition of Compound Surd:

The algebraic sum of two or more simple surds or the algebraic sum of a rational number and simple surds is called a compound scud.

For example, each of the surds (√5 + √7), (√5 - √7), (5√8 - ∛7), (∜6 + 9), (∛7 + ∜6), (x∛y - b)  is a compound surd.

Note: The compound surd is also known as binomial surd. That is, the algebraic sum of two surds or a surd and a rational number is called a binomial surd.

For example, each of the surds (√5 + 2), (5 - ∜6), (√2 + ∛7) etc. is a binomial surd.

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text:
## PREP 2014 Question Authoring - Archived

### Multivariable functions

by Adam Graham-Squire -
Number of replies: 1
How do you define a multivariable function?  And how do you take its partial derivatives?  And where else could I find this information?  I have been trying to figure out breakout exercise 2 from Day 2, and I can't seem to be able to figure it out or figure out where I can find code from someone else who has done something similar.

### Re: Multivariable functions

by Gavin LaRose -

Context("Numeric")->variables->add( y => 'Real' );
$f = Compute( "sin(x y)" );$fx = $f->D('x');$fy = $f->D('y'); Context()->texStrings; BEGIN_TEXT If $$f(x,y) = f$$, then$BR
$$f_x(x,y) =$$ \{ $fx->ans_rule() \}$BR
$$f_y(x,y) =$$ \{ $fy->ans_rule() \} END_TEXT Context()->normalStrings; ANS($fx->cmp() );


text:
# Ubuntu – PDF font falls back to DejaVu Sans, need a monospace replacement

16.04evincefontspdf

I have a PDF that features code fragments and it references Courier Std (which is not available on my system and I am not planning to buy it either). The problem is that in the Evince Document Viewer (3.18.2) this font is replaced by DejaVu Sans, which is not a monospace font and code that is note set in a monospace font is just hard to read. (the idiosyncratic kerning adjustment just makes it even harder).

I have installed the msttcorefonts in the hope that it might help, it does not. How can replace DejaVu Sans with a monospace font?

UPDATE: If I load the PDF in Firefox, the font gets correctly replaced by a monospace font.

• This issue should be patched in the next version of Ubuntu (21.04). There is also an upstream fontconfig merge request #128 pending.

In the meantime, the following steps should help address some PDF font substitution problems:

• Install fonts-urw-base35.

• Create the file ~/.config/fontconfig/conf.d/10-pdf-aliases.conf with the following contents:

  <?xml version="1.0"?>
<!DOCTYPE fontconfig SYSTEM "/etc/fonts/conf.d/fonts.dtd">
<fontconfig>

<alias binding="same">
<family>CourierStd</family>
<accept>
<family>Courier</family>
</accept>
</alias>

<alias>
<family>CourierStd</family>
<default><family>monospace</family></default>
</alias>

</fontconfig>

Then run fc-cache (may not be necessary).

You can add other font aliases to the file as needed.


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# Volatility Processes¶

A volatility process is added to a mean model to capture time-varying volatility.

 ConstantVariance() Constant volatility process GARCH([p, o, q, power]) GARCH and related model estimation FIGARCH([p, q, power, truncation]) FIGARCH model EGARCH([p, o, q]) EGARCH model estimation HARCH([lags]) Heterogeneous ARCH process MIDASHyperbolic([m, asym]) MIDAS Hyperbolic ARCH process ARCH([p]) ARCH process

## Parameterless Variance Processes¶

Some volatility processes use fixed parameters and so have no parameters that are estimable.

 EWMAVariance([lam]) Exponentially Weighted Moving-Average (RiskMetrics) Variance process RiskMetrics2006([tau0, tau1, kmax, rho]) RiskMetrics 2006 Variance process

## FixedVariance¶

The FixedVariance class is a special-purpose volatility process that allows the so-called zig-zag algorithm to be used. See the example for usage.

 FixedVariance(variance[, unit_scale]) Fixed volatility process

## Writing New Volatility Processes¶

All volatility processes must inherit from :class:VolatilityProcess and provide all public methods.

 VolatilityProcess() Abstract base class for ARCH models.


text:
# Where i can find complete data for material elastic constants?

1. ### yzou_ua

11
Where i can find complete data for material elastic constants??

I need elastic constants, piezoelectric coefficients & permittivity for different semiconductor alloy?
Someone could kindly help...

2. ### ZapperZ

30,149
Staff Emeritus
Re: Where i can find complete data for material elastic constants??

Have you tried the CRC Handbook?

Zz.

3. ### Andy Resnick

5,891
Re: Where i can find complete data for material elastic constants??

NIST has some online information, and you could contact the relevant person directly for additional information which is not online or available for a fee.

http://www.nist.gov/srd/index.htm
http://www.nist.gov/srd/materials.htm


text:
# What formula or rule has been used here?

by mettled mike   Last Updated July 17, 2017 10:20 AM

I was in between proving a trigonometric identity but couldn't succeed. I went through the solution and saw this in between

$\frac{\cos A \cos B}{\sin A \sin B}= \frac{3}{1}$

$\frac{\cos A \cos B +\sin A \sin B}{\cos A \cos B - \sin A \sin B}= \frac{3+1}{3-1}$

What happened there in the second step?

Tags :

If we have $$\frac xy = \frac31$$ then this means that $x = 3y$. This gives $$\frac{x+y}{x-y} =\frac{3y+y}{3y-y} = \frac{3+1}{3-1}$$ In your case, $x = \cos A\cos B$ and $y = \sin A\sin B$.


text:
8:43 AM
I have edited this recent question that had the deprecated (geometry) tag: Do Minkowski sums have anything like calculus? Maybe somebody has some suggestions how to improve the choice of tags.
3

Is there anything resembling differential calculus over the space of (nicely behaved) regions in $\mathbb{R}^d$, where addition is interpreted in terms of Minkowski sums? For example, it is known that Minkowski sums act linearly on the perimeter of two-dimensional convex bodies. Is there any se...

Here is another recent question with a deprecated tag: Two convex sets' intersection in a discrete domain. The tag (set-theory) does not seem like a suitable tag for this question, either.
0

I'm considering a set which is described by the intersection of two convex sets in a discrete domain. We assume that the sum of elements of each vector is the same. For instance, $S = \{(2, 2, 2, 2), (3, 2, 2, 1), (2, 3, 2, 1), (2, 2, 3, 1), (3, 2, 3, 0)\}$ is not convex set in a discrete sense b...

I will mention that the tag (discrete-mathematics) is deprecated on MathOverflow, see the tag-info. Also, it is recommended to include at least one top-level tag. I will leave the choice of tags to more experienced users (and to the OP). — Martin Sleziak 2 mins ago
0

I'm considering a set which is described by the intersection of two convex sets in a discrete domain. We assume that the sum of elements of each vector is the same. For instance, $S = \{(2, 2, 2, 2), (3, 2, 2, 1), (2, 3, 2, 1), (2, 2, 3, 1), (3, 2, 3, 0)\}$ is not convex set in a discrete sense b...

All right, thank you. — Sato 48 secs ago

1 hour later…
10:06 AM
in MathOverflow, Sep 6 at 22:25, by Federico Poloni
I disagree on your idea that someone can decide the capitalization of their own name. See for instance the take of this of this Washington Post copy editor: http://www.theslot.com/webnames.html
The above was in connection with MathOverflow vs. Mathoverflow. It seems that the same link came up in connection with colors in names.
I had cause to do exactly the abuse in my previous comment, here: mathoverflow.net/questions/313527/… as there is a proof assistant whose correctly formatted name involves half of the letters coloured red. — David Roberts Oct 24 '18 at 21:44
@DavidRoberts I don't like it either. I more or less agree with this article on this kind of stunts in logos and names. — Federico Poloni Oct 25 '18 at 7:56


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# Move object at angle like a sine wave

I currently have this code, which moves an object in a straight line at an angle, where "dir" is the angle.

velX = (float)(Math.cos(Math.toRadians(dir))) * speed;

velY = (float)(Math.sin(Math.toRadians(dir))) * speed;

x += velX;

y += velY;

What I need now is to have the object move like a sine wave, do I have to change the way I calculate velX and velY or is there some way of changing dir that would make it work?

I tried this but it didn't work, my objects would just got bunched up near 0, 0.
Move a sprite in a sine wave along different angles

• velY = (float)(Math.sin(Math.toRadians(dir) + dT)) * speed;, where dT is the elapsed time - probably milliseconds - since the last update. This will oscillate on the Y axis. – 3Dave Aug 14 '18 at 21:22

Let's take a unit vector $\vec u$ in the direction we want to travel, and a vector perpendicular to it, $\vec v$:

$$\vec u = \begin{bmatrix}cos \Theta\\sin \Theta\end{bmatrix}, \vec v = \begin{bmatrix}-sin\Theta\\cos \Theta\end{bmatrix}$$

Then we can define a point on a sinusoidal curve through starting point $\vec p_0$, travelling in the direction $\vec u$, with a given speed, frequency, and amplitude, after time $t$ as:

$$\vec p(t) = \vec p_0 + \vec u \cdot speed \cdot t + \vec v \cdot amplitude \cdot sin(frequency \cdot t)$$

Taking the derivative, we get:

$$\frac{\delta \vec p(t)}{\delta t} = \vec u \cdot speed + \vec v \cdot amplitude \cdot cos(frequency \cdot t) \cdot frequency$$

So, we can rewrite your velocity expressions as...

c = (float)(Math.cos(Math.toRadians(dir)));

Note that steering the object purely by velocity, you can accumulate rounding and integration errors. While the path will have the overall shape, it may stray from the parametric equation $\vec p(t)$ above. To fix this, you could instead compute your next position directly from the parametric equation, then compute a velocity to reach it.


text:
# Homework Help: Laplace Unit Step function

1. Dec 4, 2012

### gbuitkus

1. The problem statement, all variables and given/known data
y''-4y'-32y={1 when 0<=t<1 and 0 when 1<=t
y(0)=y'(0)=0

2. Relevant equations

3. The attempt at a solution
s2L(y) -4sL(y)-32 L(y)=u1(t)

I am just struggling to figure out if my unit step function is correct.

Solving for L(y) I get:
(e-s) / (s(s2 -4s-32))
but it says I am wrong so im pretty sure I am doing something wrong with the unit step function

2. Dec 4, 2012

### LCKurtz

You didn't explain what your notation $u_1(t)$ means. Is it $u(t-1)$? What you want on the right side is $u(t)-u(t-1)$.