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https://blog.csdn.net/qq_33471732/article/details/132440048
传递函数零极点对系统的影响_右半平面零点对系统的影响-CSDN博客 博客 下载 学习 社区 GitCode InsCodeAI 会议 搜索 AI 搜索 登录 登录后您可以： 复制代码和一键运行 与博主大V深度互动 解锁海量精选资源 获取前沿技术资讯 立即登录 会员·新人礼包 消息 历史 创作中心 创作 传递函数零极点对系统的影响 最新推荐文章于 2024-08-13 16:13:19 发布 原创 于 2023-08-23 01:02:38 发布·7.4k 阅读 · 1 · 17· CC 4.0 BY-SA版权 版权声明：本文为博主原创文章，遵循CC 4.0 BY-SA版权协议，转载请附上原文出处链接和本声明。 文章标签： #嵌入式硬件#arm开发#mcu 电源控制 专栏收录该内容 30 篇文章 订阅专栏 本文详细阐述了传递函数中零点和极点如何影响系统频率响应、稳定性及动态特性。零点和极点的位置决定了增益变化、相位特性和系统的稳定性，对控制系统设计至关重要。 传递函数的零点和极点分别对系统的影响的详细介绍： 零点（Zero）的影响：传递函数的零点是使得传递函数的分子为零的点。零点对系统的频率响应和稳定性产生影响。具体而言： 频率响应：零点的位置会影响系统在不同频率下的增益和相位特性。当传递函数的零点与频率轴上的某个频率相对应时，它会导致系统在该频率处的增益增加或相位提前。因此，通过调整零点的位置，可以调节系统在不同频率下的增益和相位特性。 稳定性：对于线性时不变（LTI）系统，如果所有的零点都位于左半平面，系统将是稳定的。如果存在零点位于右半平面，系统可能会变得不稳定。因此，在控制系统设计中，需要确保所有零点的实部位于左半平面，以保持系统的稳定性。 极点（Pole）的影响：传递函数的极点是使得传递函数的分母为零的点。极点对系统的频率响应、稳定性和动态特性产生影响。具体而言： 频率响应：极点的位置会影响系统在不同频率下的增益和相位特性。当传递函数的极点与频率轴上的某个频率相对应时，它会导致系统在该频率处的增益下降或相位延迟。因此，通过调整极点的位置，可以调节系统在不同频率下的增益和相位特性。 稳定性：对于线性时不变（LTI）系统，如果所有极点的实部都位于左半平面，系统将是稳定的。如果存在极点位于右半平面，系统可能会变得不稳定。因此，在控制系统设计中，需要确保所有极点的实部位于左半平面，以保持系统的稳定性。 动态特性：极点的位置会影响系统的动态特性，如响应时间、超调量和阻尼比等。通过调整极点的位置，可以实现更快的响应时间、更小的超调量或更好的阻尼特性。 综上所述，传递函数的零点和极点分别对系统的频率响应、稳定性和动态特性产生影响。在系统分析和控制设计中，对传递函数的零点和极点进行分析和调节是非常重要的。 最后一个bug 一个嵌入式技术进阶公众号，分享嵌入式技术 微信公众号 关注博主即可阅读全文 确定要放弃本次机会？ 福利倒计时 : : 立减 ¥ 普通VIP年卡可用 立即使用 最后一个bug 关注关注 1点赞 踩 17 收藏 觉得还不错? 一键收藏 0评论 分享复制链接 分享到 QQ 分享到新浪微博 扫一扫 打赏打赏 打赏举报 举报 专栏目录 武汉理工大学课程设计优秀报告_ 零极点 对系统 性能的 影响 分析 weixin_46325577的博客 03-19 2652 自动控制原理优秀开源报告---零极点 对系统 性能的 影响 分析 要求完成的主要任务: (包括课程设计工作量及其技术要求以及说明书撰写等具体要求) 当开环 传递函数 为G1(s)时，绘制系统的根轨迹和奈奎斯特曲线； 当开环 传递函数 为G1(s)时，a分别取0.01，0.1，1，10，100时，用Matlab计算系统阶跃响应的超调量和系统频率响应的谐振峰值，并分析两者的关系； 画出(2)中各a值的波特图； 当开环 传递函数 为G2(s)时，绘制系统的根轨迹和奈奎斯特曲线； 当开环 传递函数 为G2(s). 《自控原理》系统 传递函数 的 零极点 模型、分式模型、系统增益 最新发布 野生猿-群号1025127672 09-27 1044 （3-61）也一定可以化成（3-64）的形式，3-64其实更友好，可以看到，他把分母化成了一次因式和二次因式，一次因式对应的特征根都是实数（允许多重艮2），二次因式对应的特征根都是共轭复数。根据3-62的分母可以直接看出系统的特征根，不过要注意，3-62中的特征根si有可能是复数。(3-61)所示的是系统的 传递函数 的分式形式。他一定可以化成（3-62）形式。《自动控制原理》胡寿松，第六版。 参与评论 您还未登录，请先 登录 后发表或查看评论 右半 平面 零点 特性分析 9-20 影响:RHP 零点 引入额外的相位滞后,降低相位裕度,可能导致系统不稳定。 3. 总结: 幅频:先升后降,拐点在 零点 频率。 相频:相位先增后减,净减少180°,增加不稳定风险。 零点 与极点的定义特点解析_ 右半 平面 零点 资源 9-13 如果存在位于 右半 部分的极点,系统将会是不稳定的。此外,零点 和极点的位置还决定了系统的阻尼比和自然频率,进而 影响 系统的超调量和上升时间等性能指标。 为了详细解析 零点 与极点,我们还需要了解相关的数学公式和图形表示。传递函数 的 零点 和极点在s 平面(拉普拉斯变换域)上表示为复数,其中实部与系统的时间常数相关,虚部与... 零极点、相位裕度、系统稳定性 pickweb的博客 10-10 1万+ 在相位变化180°之前，系统增益降低为1，则系统稳定。 零极点 对系统 性能的 影响 分析 10-23 在自动控制系统中，对系统 各项性能如稳定性，动态性能和稳态性能等有一定的要求，稳定性是控制系统的本质，指的是控制系统偏离平衡状态后自动恢复到平衡状态的能力。系统动态性能是在零初始条件下通过阶跃响应来定义的，对于稳定的系统，动态性能一般指系统的超调量、超调时间、上升时间、调整时间，描述的是系统的最大偏差以及反应的快速性；稳态性能指的是系统的稳态误差，描述的是系统的控制精度。 在本文中，采用增加 零极点 并变化其值的思路，从时域和频域两个方面来研究高阶系统的各项性能指标，并借助工程软件matlab通过编程来绘制系统的根轨迹曲线、奈奎斯特曲线，阶跃响应曲线以及波特图曲线，研究系统的 零极点 对系统 性能的 影响。 天线技术与应用全解析 9-26 互调产物的存在会对通信系统产生干扰,特别是落在接收带内的互调产物将 对系统 的接收性能产生严重 影响,因此在GSM系统中对接头,电缆,天线等 无源部件的互调特性都有严格的要求。我们选用的厂家的接头的无源互调指 标可达到-150dBc,电缆的无源互调指标可达到-170dBc,天线的无源互调指 标可达到-150dBc。 【计算机视觉】复习笔记 (。・`ω´・)_如何获取波形特征内的锚... 9-27 Z的结果误差主要在分母(视差)那里。视差小的时候,视差的误差会对Z产生很大的 影响。此外T越小,误差越小 T越大,看到的范围越小(因为是取两眼图像的交叉部分) · 立体视觉的基本步骤 1. 恢复失真,消除畸变 2. 矫正相机,使图像在同一个 平面 上 3. 在两张图中找到对应的相同特征 ... 零极点 对系统 的 影响 qq_21512315的博客 07-17 5069 的（关于稳定性的问题与相位裕度也有关系），当极点（或 零点）位于坐标轴的 右半 平面 时，对应的信号一定是发散的，而当极点（或 零点）落于jw坐标轴上时，这实际上是对应的原函数的傅里叶变换，也就是该信号的成分中只包含sin wave，而没有exp wave项。因此我们必须把 零点 调节到频率的无穷远处或者移到左半 平面。根据 零极点 的公式以及根轨迹图便于我们理解 零极点 对系统 造成的 影响，这是我们分析系统稳定性的前提条件，而同时，基于 零极点 对系统 的 影响，我们可以针 对系统 波特图来判断系统内的节点存在 零极点，并且来控制它。 Z变换后的 零点、极点分布 对系统 函数的 影响 qq_42233059的博客 04-18 9015 3、如果 零点 分布在低频区域，则单位脉冲响应会有较长的持续时间，并且对低频信号的抑制能力较强。相反，如果 零点 分布在高频区域，则单位脉冲响应的持续时间会缩短，并且对高频信号的抑制能力较弱。变换里，零点 的位置表示系统的“谷”，极点的位置表示系统的“峰”，我们把有峰的地方看做信号可以通过的地方，而有谷的地方看做信号被截止的地方。2、我们选择单位圆为频域的一个周期，那么可以得出，如果无 零点 时，极点在虚轴左半边为高通，极点在虚轴右边为低通；4、如果同时有 零点 和极点，对于一阶系统，往往极点和 零点 靠的越近，其带宽越大。 我们为什么讨厌 右半 平面 零点?如何避免和转化? 9-26 ③ 零点 在左半 平面,且频率小于主极点,信号轻微向下震荡; ④ 零点 在左半 平面,且频率大于主极点,小于非主极点,信号完美收敛。 我有一篇博客中,写到了 零点 位于 右半 平面 的物理意义—— 但是没说它有什么 影响。如下图中,左侧则是 零点 在左半 平面,右侧是... 基于 零极点 分析法的谐振电路的频率响应研究(2006年) _右半 平面 零点... 9-14 - 零点:传递函数 的分子根,影响 系统的幅频响应形状。 ### 谐振电路的频率响应分析 1. 幅频特性: - 在谐振频率附近,幅频特性会出现峰值。 - 幅值随着频率偏离谐振频率而降低。 - 极点位置决定了峰值的宽度和高度,即品质因数(Q因子)。 - 对于具有复数极点的电路,峰值更加明显。 2. 相频特性... 对 传递函数 的 零极点、频率响应、稳定性的理解 qq_42702596的博客 05-06 2万+ 令 传递函数 分子为0求出 零点，令分母为0求出 零点。 传递函数 零极点 对系统 性能的 影响.doc 09-29 传递函数 零极点 对系统 性能的 影响.doc III型补偿误差放大器双 零点 双极点合集补充_三型补偿器资源-CSDN... 9-22 “双 零点”是指在误差放大器的频率响应中存在两个 零点。这些 零点 位于s 平面 的 右半 部分,有助于提升系统的相位裕度,从而提高其稳定性。零点 的位置可以通过设计电路参数来调整,以适应不同的系统需求。 “双极点”,则是指系统中的两个极点,它们决定了系统的时间常数和上升时间。极点位置的设置对于控制系统的响应速度至关... 二阶滤波器的标准 传递函数 零极点 分布以及幅频特性示意图PPT课件.pptx 10-11 二阶滤波器的标准 传递函数 零极点 分布以及幅频特性示意图PPT课件.pptx MATLAB实现控制系统模型（传递函数）的建立与转化，传递函数 模型与 零极点 增益模型的转化，连续系统与离散系统的转化，对比不同采样周期 对系统 性能的 影响 clear_lantern的博客 10-31 8951 MATLAB实现控制系统模型（传递函数）的建立与转化，传递函数 模型与 零极点 增益模型的转化，连续系统与离散系统的转化，对比不同采样周期 对系统 性能的 影响 二阶滤波器的标准 传递函数 零极点 分布以及幅频特性示意图PPT学习教案.pptx 10-02 "二阶滤波器的标准 传递函数 零极点 分布以及幅频特性示意图PPT学习... 本教案对二阶滤波器的标准 传递函数 零极点 分布以及幅频特性示意图进行了详细的介绍。 学习该教案可以帮助学生更好地理解滤波器的原理和设计方法。 零点 与极点对LTI系统的 影响 dream_201306的专栏 11-01 3518 1、 2、，也就是相频特性 3、极点其实也 影响 体统的相位，但我们一般不调整极点 4、极点越靠近单位圆，则在此极点附件的e^jw与此极点的向量模值越小，由于是位于分母相，就会倒置幅频响应就越大。 5、 6、极点与系统的因果性相关 7、极点与系统的稳定性相关：极点都必须包含在单位圆内。 ... 零极点 和系统稳定性关系 热门推荐 wanrenqi的博客 04-01 6万+ 1、零点 零点：使系统 传递函数 G(s)为0的s的值，其中s为复数。比如： 2、极点 极点：使系统 传递函数 G(s)分母为0的s的值，其中s为复数。比如： 由于系统有开环传函和闭环传函，因此有开环极点和闭环极点之分。注意开环 零极点 和闭环 零极点 在理论分析中都有用。 3、系统的稳定性 稳定性判断：在零初始条件下，当且仅当t→∞t\rightarrow \inftyt→∞,闭环系统的单位脉冲响应为零时，... 两级OTA 零点 位置 对系统 影响 分析 qq_46579389的博客 09-17 1283 分析了哈之前的一个问题，容易忽略掉的调零电阻虽然能够改两级OTA的 零点 位置，也会导致一个新的极点引入。 开环 零点 与闭环 零点 对系统 的 影响 weixin_44690490的博客 11-08 1万+ 在中，得到了小电机的 传递函数 以这个电机为例展开开环 零点 与闭环 零点 对电机响应的 影响。系统中极点是 影响 系统动态性能以及稳定性的主要因素，这里不做赘述。 零点 和极点到底 影响 了什么？什么是最小相位系统？ zhanghaijun2013的博客 01-28 2万+ 零点 和极点到底 影响 了什么？什么是最小相位系统？ 零点、极点、稳定、因果、最小相位是信号系统中经常听到名词，也许有的同学对这些概念有所了解，但对它们之间的关系却不甚了解，这篇文章我们就来看一下，它们之间到底有什么关系？零点 和极点是怎么 对系统 产生应 影响 的？ 下面我们先来看几个信号系统中的基本概念，知道了这几个概念才能继续深入下去。 1. 信号系统基本概念 1.1 静态系统和动态系统 如果一个离散系统在任意时刻n的输出至多依赖于同一时刻的输入样本，而与过去或者将来的输入样本无关，那么该系统就称为静态的 RC电路（三）：零点 和极点 既有随处可栖的江湖，也有追风逐梦的骁勇！ 08-13 1万+ 零点 和极点是在自动控制原理中用于描述系统特性的概念。‌零点（Zero）：‌在 传递函数 的分子多项式等于零的解。即当系统的输入信号等于零时，‌输出信号不为零的情况下，‌输入信号与输出信号相等的点。‌在数学和电学中，‌零点 也是系统程度上不能再超过的界限，‌具有特定的数学和物理意义。‌极点（Pole）：‌在 传递函数 的分母多项式等于零的解。即当系统的输入信号不为零时，‌输出信号为零的情况下，‌输入信号与输出信号相等的点。‌极点的性质决定系统的稳定性，‌对系统 的稳定性有直接 影响。‌。 《现代控制系统》第五章——反馈控制系统性能分析 5.4 二阶系统里面极点以及 零点 带来的 影响 zhelijun的博客 11-29 7289 图5.8里面描绘的曲线仅仅是针 对系统 方程为 的二阶系统来说的。但是这给我们提供了一个很好的例子：许多系统拥有主极点对，并且可以通过类似图5.8的关系来估计系统的阶跃响应。这个方法尽管只是一个估算，但却能在避免拉普拉斯转化的情况下提供一个对超调或者其他的性能参数的简单估算。举个例子，对于具有以下闭环转换方程的三阶系统： s域极点图如下： ... 开关电源的反馈回路有那么难吗？ helloworldsyf的博客 03-24 3019 有的 431反馈回路 芯片是isl6840 等效电路 Rf2可以忽略的原因是它只起到了直流偏置的作用，不会出现在交流分析当中 matlab 传递函数 零极点 对消 04-11 传递函数 的 零极点 对消可以通过使用MATLAB中的控制系统工具箱来实现。下面是一种常见的方法： 1. 首先，使用tf函数创建 传递函数 对象。例如，假设有一个 传递函数 为G(s) = (s+1)(s+2)/(s+3)(s+4)，可以使用以下代码... 关于我们 招贤纳士 商务合作 寻求报道 400-660-0108 kefu@csdn.net 在线客服 工作时间 8:30-22:00 公安备案号11010502030143 京ICP备19004658号 京网文〔2020〕1039-165号 经营性网站备案信息 北京互联网违法和不良信息举报中心 家长监护 网络110报警服务 中国互联网举报中心 Chrome商店下载 账号管理规范 版权与免责声明 版权申诉 出版物许可证 营业执照 ©1999-2025北京创新乐知网络技术有限公司 最后一个bug 博客等级 码龄10年 863 原创4262 点赞 6634 收藏 1万+粉丝 关注 私信 🆓注册即刻获得高达 $200 抵扣金☁️开始您的云计算之旅同时获得30余项永久免费服务广告 TA的精选 新 你用I2C总线时考虑过死锁问题吗？ 127 阅读 新 同事问 : VSCode中如何查找所有“数组[ ]“ ？ 388 阅读 热【MCU】可怕，别人把我MCU固件给反汇编了！(逆向) 18998 阅读 热 详解双闭环控制算法（理论篇） 15507 阅读 热【进阶】三种" 堆栈溢出检测 "方法，请拿去吹牛！ 14356 阅读 查看更多 2025 09月 13篇 08月 13篇 07月 47篇 06月 7篇 05月 15篇 04月 31篇 03月 60篇 02月 37篇 01月 38篇 2024年 137篇 2023年 300篇 2022年 40篇 2021年 59篇 2020年 66篇 大家在看 CSS定位流全解析：精准控制元素位置 免费AI聊天API无认证调用 Web暖场题解：路径包含漏洞利用 62 TCP/IP与HTTP协同工作原理 1030 Java实现大文件断点续传方案 分类专栏 高质量嵌入式Linux应用开发 付费36篇 yocto构建嵌入式linux系统 付费34篇 python高质量编程5篇 linux内核设计与实现35篇 linux应用程序开发102篇 嵌入式中的Lua脚本开发7篇 编程经验分享35篇 嵌入式C语言172篇 嵌入式网络技术22篇 Win32&MFC上位机开发11篇 C#与winform上位机开发2篇 FAT文件系统3篇 主流单片机开发(MCU)339篇 程序人生32篇 电源控制30篇 嵌入式linux/RTOS129篇 展开全部收起 上一篇： udp与can通信的选择与比较 下一篇： bode100测量频率响应的基本原理 🆓注册即刻获得高达 $200 抵扣金☁️开始您的云计算之旅同时获得30余项永久免费服务广告 上一篇： udp与can通信的选择与比较 下一篇： bode100测量频率响应的基本原理 分类专栏 高质量嵌入式Linux应用开发 付费36篇 yocto构建嵌入式linux系统 付费34篇 python高质量编程5篇 linux内核设计与实现35篇 linux应用程序开发102篇 嵌入式中的Lua脚本开发7篇 编程经验分享35篇 嵌入式C语言172篇 嵌入式网络技术22篇 Win32&MFC上位机开发11篇 C#与winform上位机开发2篇 FAT文件系统3篇 主流单片机开发(MCU)339篇 程序人生32篇 电源控制30篇 嵌入式linux/RTOS129篇 展开全部收起 登录后您可以享受以下权益： 免费复制代码 和博主大V互动 下载海量资源 发动态/写文章/加入社区 ×立即登录 评论 被折叠的 条评论 为什么被折叠?到【灌水乐园】发言 查看更多评论 添加红包 祝福语 请填写红包祝福语或标题 红包数量 个 红包个数最小为10个 红包总金额 元 红包金额最低5元 余额支付 当前余额 3.43 元 前往充值 > 需支付：10.00 元 取消 确定 成就一亿技术人! 领取后你会自动成为博主和红包主的粉丝 规则 hope_wisdom 发出的红包 打赏作者 最后一个bug 你的鼓励将是我创作的最大动力 ¥1¥2¥4¥6¥10¥20 扫码支付：¥1 获取中 扫码支付 您的余额不足，请更换扫码支付或充值 打赏作者 实付 元 使用余额支付 点击重新获取 扫码支付 钱包余额 0 抵扣说明： 1.余额是钱包充值的虚拟货币，按照1:1的比例进行支付金额的抵扣。 2.余额无法直接购买下载，可以购买VIP、付费专栏及课程。 余额充值 确定 取消 举报 选择你想要举报的内容（必选） 内容涉黄 政治相关 内容抄袭 涉嫌广告 内容侵权 侮辱谩骂 样式问题 其他 原文链接（必填） 请选择具体原因（必选） 包含不实信息 涉及个人隐私 请选择具体原因（必选） 侮辱谩骂 诽谤 请选择具体原因（必选） 搬家样式 博文样式 补充说明（选填） 取消 确定 点击体验 DeepSeekR1满血版 下载APP 程序员都在用的中文IT技术交流社区 公众号 专业的中文 IT 技术社区，与千万技术人共成长 视频号 关注【CSDN】视频号，行业资讯、技术分享精彩不断，直播好礼送不停！客服返回顶部 微信公众号 公众号名称：最后一个bug 微信扫码关注或搜索公众号名称 复制公众号名称
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https://www.reddit.com/r/ALevelChemistry/comments/1d33v4d/help_i_still_cant_get_a_brain_of_alcl3_and_al2cl6/
Help! I still can't get a brain of AlCl3 and Al2Cl6 : r/ALevelChemistry Skip to main contentHelp! I still can't get a brain of AlCl3 and Al2Cl6 : r/ALevelChemistry Open menu Open navigationGo to Reddit Home r/ALevelChemistry A chip A close button Log InLog in to Reddit Expand user menu Open settings menu Go to ALevelChemistry r/ALevelChemistry r/ALevelChemistry Help and resources for students taking A Level Chemistry. 13K Members Online •1 yr. ago hideki75 Help! I still can't get a brain of AlCl3 and Al2Cl6 Okay, all i know is AlCl3 has a structure of Giant Ionic Lattice. Al2Cl6 is a simple molecular structure and a dimer of Aluminium Chloride that is in solid state. what is the difference between AlCl3 and Al2Cl6? why does it has AlCl3 and Al2Cl6? does AlCl3 is made from Al2Cl6? does it has a different products from if each of it dissolve in water? Are the pH of solution produce are same? Sorry for a long question, I'm gladly and open hands if someone wanted to explain it even if it is out of syllabus context. If you could give me more information other than my question, I would appreciate it thanks! (⁠◠⁠‿⁠・⁠)⁠—⁠☆ Read more Share Related Answers Section Related Answers Structure and bonding of Al2Cl6 Ionic or covalent nature of Al2Cl6 Explanation of AlCl3 dimer formation Ionic or covalent bonding in AlCl3 Aluminium chloride bonding type New to Reddit? Create your account and connect with a world of communities. Continue with Google Continue with Google. Opens in new tab Continue with Email Continue With Phone Number By continuing, you agree to ourUser Agreementand acknowledge that you understand thePrivacy Policy. Public Anyone can view, post, and comment to this community 0 0 Top Posts Reddit reReddit: Top posts of May 29, 2024 Reddit reReddit: Top posts of May 2024 Reddit reReddit: Top posts of 2024 Reddit RulesPrivacy PolicyUser AgreementAccessibilityReddit, Inc. © 2025. All rights reserved. Expand Navigation Collapse Navigation
5202
https://bigpicture.one/blog/problem-solving-strategies/
Demo BigPicture Enterprise just got more strategic! Learn more July 28, 2023 Figure it out: 9 problem-solving strategies for managers Project Management Agnieszka Sienkiewicz “Okay Houston, we’ve had a problem here” – John “Jack” Swigert. In 1970, the Apollo 13 crew faced a number of serious obstacles — of which the loss of oxygen, which left them with under two hours of power generation remaining. NASA Mission Control had to quickly figure out how to keep their astronauts alive, and eventually, the results of these joint problem-solving efforts by engineers on the ground allowed three astronauts to safely return to Earth. In life — be it private or professional — we also face challenges, perhaps even on a daily basis. Knowing how to solve problems is extremely important, so if you’re interested in a complete, end-to-end process for developing solutions, here are nine effective problem-solving strategies we put together for you. What are problem-solving strategies? However obvious this may sound; to solve a problem means to produce solutions, then choose the best out of your alternatives. And to find the ultimate solution, you must first produce a plan of action, or strategy. Which may be different from the way you typically go about problem-solving; nonetheless it’s quite effective. Just like you can find many solutions to the same problem, you can also use different problem-solving strategies with different action plans associated with them. Importance of (using) problem-solving strategies Resolving business challenges is an important outcome of the successful application of problem-solving strategies; but there’s more to it. No matter the nature or magnitude of a problem, or decision you need to make, the stress can be palpable. Naturally, you want to do something about it — you want to resolve it; which is where problem-solving strategies come in. Knowing that there’s a suitable strategy to help remedy a situation can significantly reduce your anxiety and overall distress. Another important aspect to keep in mind when it comes to problem-solving strategies is personality. We all have different personalities and aptitudes, which means we think differently. For that reason, it’s better to have several strategies to choose from, as some may resonate better than others. Consequently, these will prove more effective. Likewise, some problem-solving strategies are more likely to help you achieve your goals when used specifically for certain situations or challenges. In other words, not every approach will work for every type of problem. Knowing at least a few different problem-solving strategies can increase your chances to find the right solution and achieve better outcomes. And finally, problem-solving strategies promote critical thinking, creativity, and oftentimes — collaboration within or across teams. All of these are essential skills in personal and professional contexts. Steps in the problem-solving process When facing a business problem such as handling complex dependencies or conflicting schedules — how do you solve it? Do you grind it for days or jump straight into thinking about potential solutions? The very process of problem-solving isn’t new; it includes a set of specific steps to guide you through the entire solution-discovery journey. Problem-solving strategies, on the other hand, reinforce this sequential approach so you can make the most out of each step. 1. Defining and analyzing the problem The first step in problem-solving is problem analysis and definition. At first, you know what the symptoms are, but you also want to know what triggers them. So, you gather the necessary data by using your observational skills and talking to colleagues. Be mindful, however, that some of the information you will gather might be more of an opinion than a fact; make sure to differentiate between the two. The goal of this step is to formulate the problem specifically and determine the underlying cause (or causes). For example, your organization is struggling with high employee turnover (problem formulation). As a hiring manager, you discover that new hires often leave after the trial period, due to a poor onboarding experience and lack of adequate support (problem causes). The process of collecting data from individual stakeholders is also important for breaking down a problem into smaller key elements. On one hand, this can get you to determine the causes of a problem; on the other hand, it will later enable you to generate more accurate solution alternatives. Corresponding strategies and tools: Fishbone diagram Flowcharts Kipling method Work backward SWOT analysis 2. Generating alternative solutions Now that you know why the problem has arisen, you want to generate possible solutions (on your own, or alongside involved individuals). For the time being, focus on the solutions alone — you can evaluate them later. The alternatives you will manage to produce should be consistent with your organizational goals, regardless of whether they’re short or long term. Corresponding strategies and tools: Algorithm Mind map Brainstorming What-if scenarios Divide and conquer Means-end analysis 3. Evaluating and selecting an alternative Once you have a list of potential solutions for your business or organizational problem, then you can pick the best alternative. Look at the solutions you and your colleagues have shortlisted, and evaluate them without bias. Which of them are relative to the goals you established, and which are most likely to produce the desired results? When selecting the alternative, consider all organizational constraints and the consequences it may bring forth. For example, a new solution could lead to a decrease in resource capacity due to implementing a buddy system that aims to help onboard new hires. Corresponding strategies and tools: Brainstorming Decision matrix analysis 4. Implementing and monitoring new solutions By now, you (and your colleagues) know what the real problem is, and how to remedy it. At this point, you’re ready to implement the solution and observe how the situation unfolds. Take the time to collect feedback from those affected by the solution, and seek their acceptance. What if the solution fails to produce the expected results? What if you encounter new challenges that will require you to change your original approach? In such a case, you’ll need to reassess the changes, then follow the problem-solving process steps (again) to create a new list of solutions, or to update the existing one. Corresponding strategies and tools: Trial and error Effective problem solving-techniques to try There are many different ways to approach problem-solving. Each suitable for different types of problems, or stages of the problem-solving process. The strategies you’ll find below are suggestions to help you get started. You may need to experiment with several strategies before you find a workable solution for your specific problem. Algorithm If you’re looking for a structured and procedural approach to problem-solving, an algorithm might be just what you need. An algorithm is a problem-solving strategy that provides you with step-by-step instructions that — if followed to the letter — can help you achieve your desired outcome. You can think of an algorithm as a cooking recipe (or function in a code) that describes instructions in high detail. An algorithm will return the same result every time you execute it. One benefit of this strategy is that it produces very accurate results. But it might not always be the best approach to problem-solving. That’s partly due to the fact that detecting certain patterns can be incredibly time-consuming. Still, if you need to deal with a group of similar problems, algorithms can help you figure out the common denominator, and find a workable solution for these problems. Heuristics Heuristics is a general problem-solving strategy; opposite to the algorithmic approach as it uses intuition and (judge)mental shortcuts (the “rule of thumb”) instead of a systematic approach. For that reason, heuristic approaches are less time and energy-consuming than algorithmic problem-solving. For example, hiring managers can apply heuristics when considering a pool of candidates. By following their intuition and experience, they can quickly choose whom to offer the job to. In certain cases, and despite these characteristics, heuristics may not be the best way to make rational decisions. Especially, in the cases where you need to process a lot of information, work under pressure, or lack the data needed to generate a solution. Taking a shortcut might be the most tempting thing to do under these conditions; it could also send you off course. Work backward Working backward is a useful heuristic problem-solving strategy. To work backward means to begin solving the problem by identifying the steps needed to achieve the end result. It’s like reverse engineering; where you’re focused on a solution to a problem, instead of a software or system. For example, if the product you manage has received several negative reviews lately, you can ask yourself “What has happened which has led to this situation?” Then, you can work backward, step by step. Kipling (5W1H) method This strategy is about asking a series of six questions in order to understand a problem better. The questions come from Rudyard Kipling’s poem, “I keep six honest serving men.” The six questions are: “What” – What is the problem? “Why” – Why is the problem important? “When” – When did the problem arise, and when do you need to solve it? “How” – How did the problem happen? “Where” – Where is the problem occurring? “Who” – Who does the problem affect? As a manager, you can use the Kipling method to identify all relevant factors and ensure you fully understand thr problem — before you start looking for solutions. 5W1H and 5 Whys You can use the Kipling and 5 Whys problem-solving methods interchangeably or together, because both of them help you identify existing problems. But keep in mind that there’s a significant difference between these two methods. The Kipling method asks about various key details regarding the problem; “Why” is just one of these questions. The 5 Whys method, on the other hand, repeats the “Why” five times. The goal is to get granular with the reasons behind the problem, asking one “Why” after another. If a problem is too complex and requires a more comprehensive analysis, you can combine both methods for a more solid outcome. This way, you get a higher chance to better understand the problem and find a solution. Trial and error When you have several possible solutions, and would like to test them in order to see which one works best, a trial-and-error strategy can be helpful. Using this approach, you can try different solutions until you find the right one. For example, you may need to figure out how to allocate your shared resources for the upcoming project phase. By testing various setups and comparing outcomes, you can identify the best solution. We recommend using what-if scenarios that will allow you to safely carry out your tests, without impacting your original plan. Brainstorming The problem you face is not necessarily a problem on its own. Your team or the other managers can provide valuable hints or help with solutions that you did not even consider. Consequently, the more people you gather to help solve a problem, the more potential solutions you get to produce together. The more, the merrier. The brainstorming strategy not only helps overcome critical challenges, but also stimulates creativity and encourages collaboration among your colleagues. Divide and conquer (and combine) This problem-solving strategy is about breaking down large complex problems (“divide”) into a set of smaller, more manageable subproblems that are similar to the original problem. You look into each subproblem individually and try to solve it one by one (“conquer”). Then, you merge those solutions back into one in order to create a solution to the original, larger problem (“combine”). Divide and conquer and Means-end analysis Means-end analysis is a problem-solving strategy where you consider the obstacles standing between the problem state and the end-result (solution) state. In other words, it helps you get from A to B by examining the obstacles along the way and finding solutions to them. Elimination of these obstacles produces the smaller subgoals you need to achieve. And when you achieve all these subgoals, i.e., when all of the obstacles are out of your way, you’ve reached your main goal (point B, or the solution state). The means-end analysis is a version of the divide-and-conquer strategy. The difference between the two is this: divide and conquer is recursive — when using this method, the subproblems you solve are always of the same type. Means-end analysis, on the other hand, is more flexible and less recursive; that’s because the subproblems you’re trying to solve don’t need to be similar type-wise. Walking the path always traveled: pitfalls to problem-solving Problem-solving strategies are helpful, nonetheless they cannot guarantee that you’ll find the solution you seek. Why is that? What could possibly stand in the way between you and your solution? Many things; for instance, your mindset. As Albert Einstein once said, “Insanity is doing the same thing over and over again and expecting a different result.” Imagine yourself identifying and dismantling a problem into the tiniest particles. Then, applying the same true and tried method over and over again, expecting to achieve your goal. Instead, you’re failing to resolve the problem every time and getting stuck, running in circles. What happens in such situations is that you persist to solve a problem in the same way you did before. The only difference is that this method doesn’t seem to be working this time. Such a mindset is called functional fixedness. It’s a state where you’re unable to see a method, object, or tool that could be used for something else than what it was designed for. In Apollo 13 mission, ground engineers had to overcome functional fixedness and figure out how to literally fit a square peg in a round hole. Their solution to this problem involved basic materials, such as spare plastic bags, tape, and air hoses. So, if you ever get stuck overcoming your challenge, it’s a good idea to step back and think about why your strategy isn’t working. Perhaps, there’s another, better approach you haven’t considered yet. Don’t be afraid to try different strategies or combine a few together; there’s no obviously-right or obviously-wrong way to solve problems. Problem-solving strategies: summary There are many different strategies you could use to solve a business problem. Typical problem-solving strategies include trial and error, applying algorithms, and using heuristics. When you need to solve a large, complex problem, it often helps to break the problem into smaller subproblems that you can then solve individually. The sum of these subsolutions can then get you to the general solution for your large problem. When trying to find a solution, be aware of major roadblocks including your very own mindset. Also, whenever you face a problem in the workplace, keep in mind that apart from strategies, you also have a variety of tools at your disposal. While neither of them will solve the problem for you, they will help you gain the necessary insights into the situation. BigPicture PPM software will provide you with the data you need to clearly identify the problem. Then, it will let you test different solutions so you can pick the best one. And finally, it will allow you to monitor your initiative all the way through the entire project lifecycle. Best part? BigPicture is used by NASA; but you don’t need to be a ground control engineer to benefit from it! 😉 Find out what BigPicture can do for you, get a 30-day free trial that you can start today. Or visit our demo page to play with the app straight in your browser — no registration or installation needed. Related posts #### 7 most common challenges in product management Product Managers (PMs), due to their responsibilities, commonly face various challenges. Knowing how to recognize those challenges and address them … #### 5 biggest challenges of a Product Owner A Product Owner (PO) role can be challenging. POs have many different responsibilities on their plate that highly depend on … #### 7 biggest UX designer challenges The “user experience” (UX) field is not entirely new. It has been around since the 1950s and has quite matured … #### Solving common challenges in resource allocation in Jira Allocation is one of the most popular resource management strategies for assigning work to people or non-human assets to a&… Agnieszka Sienkiewicz All articles by the author Content writer at BigPicture. Previously, Aggie worked for SaaS companies writing specifically about eCommerce and marketing. As a continuous learner and advocate for knowledge-sharing, she creates content for beginners as well as more advanced readers. She loves clean plant-based food and morning workouts. Project Management Share: What kind of demo would you like to get? Sign up for a live demo webinar Take a live guided tour, ask questions, and get answers. The session takes up to 60 minutes, including the Q&A part. Schedule an individual live session Are you representing an enterprise organization (1000+ users)? Discuss your needs with one of our solution advisors. Watch the demo! "" indicates required fields Go back Watch the demo! Watch in full screen Do you need more information? Sign up for a live demo webinar Take a live guided tour, ask questions, and get answers. The session takes up to 60 minutes, including the Q&A part. Get an individual live demo Go back Thank you! Your form was successfully submitted! We will review it and contact you shortly to find a suitable date and time. Close Register for a live demo webinar Go back Congratulations! We’ve just signed you up. Check your inbox for details. In the meantime, you can… Get a concise overview of BigPicture. Explore highlights in less than 10 minutes. Get started! "" indicates required fields Request offer "" indicates required fields Questionnaire "" indicates required fields Get started! Try for free See BigPicture Enterprise Contact us! Get Support Contact Customer Success Contact Partner Relations Contact us! 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https://aacrjournals.org/cancerres/article/43/10/4935/487145/Epidermal-Cell-Cycle-and-Transit-Times-during
Epidermal Cell Cycle and Transit Times during Hyperplastic Growth Induced by Abrasion or Treatment with 12-O-Tetradecanoylphorbol-13-acetate1 | Cancer Research | American Association for Cancer Research Skip to Main Content Advertisement Open Menu Close AACR Journals Open Menu Blood Cancer Discovery Cancer Discovery Cancer Epidemiology, Biomarkers & Prevention Cancer Immunology Research Cancer Prevention Research Cancer Research Cancer Research Communications Clinical Cancer Research Molecular Cancer Research Molecular Cancer Therapeutics For Authors Open Menu Information for Authors Author Services Journal Awards About Us Open Menu AACR Journals Advertising Permissions & Reprints Subscriptions Submit Article Collections Open Menu Cancer Hallmarks Collection Hot Topics Towards Degrader-Antibody Conjugates Advances in Tumor Immunology and Immunotherapy Editors' Picks Meeting Abstracts COVID-19 Alerts News Cancer Hallmarks Webinars Search Dropdown Menu header search search input Search input auto suggest Search Advanced Search User Tools Dropdown Register Sign In Open Menu Toggle Menu Menu About Open Menu The Journal Editorial Board AACR Journals Subscriptions Permissions and Reprints Articles Open Menu Online First Issues Meeting Abstracts Cancer Research Landmarks Collection: Editors' Picks COVID-19 For Authors Open Menu Information for Authors Author Services Early Career Award Submit Open External Link Alerts News Cancer Hallmarks Webinars Skip Nav Destination Close navigation menu Article navigation Volume 43, Issue 10 1 October 1983 Previous Article Next Article Article Contents Abstract Article Navigation Articles|October 01 1983 Epidermal Cell Cycle and Transit Times during Hyperplastic Growth Induced by Abrasion or Treatment with 12-O-Tetradecanoylphorbol-13-acetate1Free Rebecca Morris; Rebecca Morris Department of Biology, Syracuse University [R. M.] , and Department of Pathology, Upstate Medical Center [T. S. A.] , Syracuse, New York 13210 Search for other works by this author on: This Site PubMed Google Scholar Thomas S. Argyris Thomas S. Argyris Department of Biology, Syracuse University [R. M.] , and Department of Pathology, Upstate Medical Center [T. S. A.] , Syracuse, New York 13210 Search for other works by this author on: This Site PubMed Google Scholar Author & Article Information 2 Work performed in partial fulfillment of the requirements for the Ph.D. in biology in the Graduate School of Syracuse University, Syracuse, N. Y. Present address: The University of Texas System Cancer Center, Science Park Research Division, P. O. Box 389, Smithville, Texas 78957. 3 To whom requests for reprints should be addressed. Received:March 10 1983 Accepted:July 12 1983 Online ISSN: 1538-7445 Print ISSN: 0008-5472 ©1983 American Association for Cancer Research. 1983 Cancer Research, Inc. Cancer Res (1983) 43 (10): 4935–4942. Article history Received: March 10 1983 Accepted: July 12 1983 Split-Screen Views Icon Views Open Menu Article contents Open the PDF for in another window Share Icon Share Facebook X LinkedIn Bluesky Email Tools Icon Tools Open Menu Get Permissions Cite Icon Cite Search Site Article Versions Icon Versions Version of Record October 1 1983 Citation Rebecca Morris, Thomas S. Argyris; Epidermal Cell Cycle and Transit Times during Hyperplastic Growth Induced by Abrasion or Treatment with 12-O-Tetradecanoylphorbol-13-acetate1. _Cancer Res_ 1 October 1983; 43 (10): 4935–4942. Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest Search Advanced Search Abstract The purpose of this investigation was to determine the cellular kinetics of the epidermis during hyperplastic growth in CD-1 mice induced by treatments effective in skin tumor promotion: abrasion or the application of 12-O-tetradecanoylphorbol-13-acetate (TPA). Following removal by abrasion, the epidermis regenerated from intrafollicular epidermal cells. The maximal increase in the number of epidermal cells, over 2-fold from normal, was reached by 4 days after abrasion. Regression of the hyperplasia began about Day 7 and continued through Day 20, resulting in a nearly normal epidermis. The growth fraction measured at 3 days after abrasion did not change from its normal value of about 100%. However, the cell cycle time of basal keratinocytes 3 days after abrasion was drastically reduced to about 11 hr compared to 5 to 7 days in normal epidermis. The cell cycle time gradually increased to 14 hr at 5 days, 1 to 2 days at 7 days, and 4 to 5 days at 14 days after abrasion. The reduction in the length of the cell cycle time was primarily due to a decrease in the length of G 1. The rate of epidermal cell loss was measured by the epidermal transit time, the time required for [3 H]thymidine-labeled basal cells to reach the uppermost nucleated layer. Labeled nuclei were followed through the epidermal columns, when present. Transit time was dramatically reduced from 8 days in normal skin to 1 day by 3 days following abrasion, then rose to 2 days by 5 days postabrasion, 4 days by 7 days postabrasion, and by 6 days 14 days postabrasion. A single application of 17 nmol of TPA resulted, within 1 day, in over a 50% increase in the total number of epidermal cells. This thickness was maintained until about Day 4, and then regression began resulting in an essentially normal epidermis by 10 days. As after abrasion, the growth fraction 3 days after treatment with TPA was about 100%, that is, not changed from normal. The cell cycle time, however, was again dramatically reduced to 16 hr beginning at 1 hr, 25 hr at 3 days, 2 to 3 days at 5 days, and 3 to 5 days at 10 days after application of TPA. Also, as after abrasion-induced hyperplasia, the transit time was reduced to 2 days at 1 hr, 4 days at 3 days, and 5 days at 5 and 10 days following treatment with TPA. The decrease in the cell cycle time can be accounted for by the decrease in the length of G 1. The results of this investigation demonstrate that the production of an epidermal hyperplasia following abrasion or the application of 17 nmol of TPA is associated with both an increase in cell proliferation and cell loss. The increased cell proliferation can be accounted for by the decrease in the length of the cell cycle. It appears that the increase in epidermal cell loss may be also linked to the decrease in the length of the cell cycle, since all suprabasal cells arise from the basal cells. 1 This work supported by NIH Grants AM 18219 and AG 01324. This content is only available via PDF. Open the PDF for in another window ©1983 American Association for Cancer Research. 1983 Cancer Research, Inc. 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https://www.wizeprep.com/textbooks/high-school/mathematics/2812/sections/103192
Properties of Polygons - Wize High School Grade 9 Math Textbook | Wizeprep University MCAT LSAT DATHigh SchoolLog in Get Started for Free ###### Wize High School Grade 9 Math Textbook > 2D Geometry - Properties of 2D Figures Properties of Polygons Conjectures & Hypotheses Previous Section Line of Symmetry Next Section Review: QuadrilateralsExamplePracticePracticeReview: TrianglesExample Popular Courses Grade 9 Math Canada High School Find My Course 10 10 0:00 / 0:00 1X Quadrilaterals Quadrilaterals Show Solutions Example: Properties of Quadrilaterals Three conjectures about the diagonals of a quadrilateral are given: The diagonals are equal The diagonals bisect each other ("cuts each other in half") The diagonals meet at a 90°90\degree 90° angle (they are perpendicular) Come up with examples to support the conjecture or come up with a counterexample to disprove the conjecture. Then put a ✔ or ✖ in each of the following boxes to indicate whether the conjecture is likely true or not true for each quadrilateral. PAGE BREAK The diagonals are always equal For each type of quadrilateral, create a few different sizes and draw in the diagonals. If you measure these diagonals, you'll see if their lengths are the same. If they are, the diagonals are likely equal. If you are able to come up with even one example where the lengths are not the same, then we know for sure that the lengths are not always equal. Rectangles: ✔ Squares: ✔ Parallelograms: ✔ Isosceles Trapezoid: ✔ Rhombus: ✖ (they aren't always equal) Kite: ✖ (they aren't always equal) The diagonals always bisect each other For each type of quadrilateral, create a few different sizes and draw in the diagonals. Measure the lengths of each side of each diagonal, if they are the same, then the diagonals likely bisect each others. If you are able to come up with even one example where the lengths are not the same, then we know for sure that the diagonals do not always bisect each other. Rectangles: ✔ Squares: ✔ Parallelograms: ✔ Isosceles Trapezoid: ✖ Rhombus: ✖ (they don't always bisect each other) Kite: ✖ (they don't always bisect each other) The diagonals are always perpendicular (meet at 90°90\degree 90°) For each type of quadrilateral, create a few different sizes and draw in the diagonals. Measure the angle the diagonals make with one another, if they are 90°90\degree 90°, then the diagonals likely are always perpendicular. If you are able to come up with even one examlpe where they are not 90°90\degree 90°, then we know for sure that the diagonals are not always perpendicular Rectangles: ✖ (they aren't always perpendicular) Squares: ✔ Parallelograms: ✖ (they aren't always perpendicular) Isosceles Trapezoid: ✖ (they aren't always perpendicular) Rhombus: ✔ Kite: ✖ (they aren't always perpendicular) Practice: Properties of Quadrilaterals Adjacent angles are the angles that are "next to each other". Given the conjecture "the sum of the two adjacent angles is always 180°180\degree 180°", come up with examples to either support this conjecture or come up with a counterexample to disprove the conjecture for each of the following quadrilaterals. Put "yes" in the box if the conjecture for that quadrilateral is likely true, and put "no" in the box if the conjecture is not true for that quadrilateral. Rectangle Square Parallelogram Isosceles Trapezoid Rhombus Kite The sum of the two adjacent angles is always 180 for this type of quadrilateral [x] I don't know Check Submission Mark Yourself Question 1. Grab a piece of paper and try this problem yourself. 2. When you're done, check the "I have answered this question" box below. 3. View the solution and report whether you got it right or wrong. Practice: Properties of Quadrilaterlas. The midsegment in a polygon is a line segment that joins the midpoints of two adjacent sides in that polygon. The conjecture "the quadrilateral formed by connecting the four midsegments in any quadrilateral are parallelograms" is given. Either come up with multiple examples to support this conjecture or come up with one counterexample to disprove this conjecture. [x] I have answered this question Check Submission 10 10 0:00 / 0:00 1X Triangles Triangles PAGE BREAK Median: a line that connects a vertex to the midpoint of its opposite side Altitude: a line that connects a vertex to its opposite side, and is perpendicular to the opposite side (an altitude is often called the "height" of the triangle) Angle bisector: a line that connects a vertex to its opposite side, and it cuts the angle in half Perpendicular (right) bisector: a line that cuts a side in the triangle in half, and is perpendicular to that side Wize Tip In an equilateral triangle, the median, altitude, and right bisector from a single vertex are all the same! 10 10 0:00 / 0:00 1X Show Solutions Example: Properties of Triangles Based on the following angle measurements, come up with a conjecture for the relationship between exterior and interior angles of any triangle. Then try to disprove the conjecture by finding a counterexample, or actually prove the conjecture using angle properties of triangles. Right angle triangle PAGE BREAK Isosceles Triangle PAGE BREAK Scalene Triangle According to the 3 triangles given, it seems like the exterior angle is the sum of the two opposite interior angles. Although we have 3 examples here that support our conjecture, it's not enought to actually prove the conjecture because "what if there is an example out there that disproves this conjecture?" But we can really prove this conjecture by not looking at examples at all. Instead, we turn to properties and facts about angles that we already know to be true. Let's take a look at any triangle, we can label 4 angles as a,b,c\bct{a,\ b,\ c}a,b,c and d\bct d d: Recall: Supplemental angles that form a straight line must add up to 180°180\degree 180° The sum of the interior angles in any triangle is 180°180\degree 180° Using property 1: c+d=180°\colorbox{yellow}{$c+d=180\degree$}c+d=180°​ Using property 2: a+b+d=180°\colorbox{yellow}{$a+b+d=180\degree$}a+b+d=180°​ Since c+d c+d c+d and a+b+d a+b+d a+b+d both equal 180°180\degree 180°, we know that c+d c+d c+d and a+b+d a+b+d a+b+d must equal each other: c+d=a+b+d−d=−d c=a+b\begin{array}{rcl} c+d&=&a+b+d\ \scriptsize-d&=&~~~~~~~~~~~~\scriptsize-d\ c&=&a+b \end{array}c+d−d c​===​a+b+d−d a+b​ We see that for any triangle, c=a+b\boxed{c=a+b}c=a+b​, meaning that the exterior angle of any triangle is the sum of the two opposite interior angles! 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https://artofproblemsolving.com/wiki/index.php/Polynomial_Remainder_Theorem?srsltid=AfmBOooLi8Us2lVK_q0NIyhQpnBC5JVdlmCmLn18M-BCiujJMNjIoOCS
Art of Problem Solving Polynomial Remainder Theorem - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki Polynomial Remainder Theorem Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Polynomial Remainder Theorem In algebra, the Polynomial Remainder Theorem states that the remainder upon dividing any polynomial by a linear polynomial , both with complex coefficients, is equal to . Contents [hide] 1 Proof 2 Generalization 3 Problems 3.1 Example 1 3.2 More Problems 4 See Also Proof By polynomial division with dividend and divisor , that exist a quotient and remainder such that with . We wish to show that is equal to the constant . Because , . Therefore, , and so the is a constant. Let this constant be . We may substitute this into our original equation and rearrange to yield When , this equation becomes . Hence, the remainder upon diving by is equal to . Generalization The strategy used in the above proof can be generalized to divisors with degree greater than . A more general method, with any dividend and divisor , is to write , and then substitute the zeroes of to eliminate and find values of . Example 2 showcases this strategy. Problems Here are some problems with solutions that utilize the Polynomial Remainder Theorem and its generalization. Example 1 What is the remainder when is divided by ? Solution: Although one could use long or synthetic division, the Polynomial Remainder Theorem provides a significantly shorter solution. Note that , and . A common mistake is to forget to flip the negative sign and assume , but simplifying the linear equation yields . Thus, the answer is , or , which is equal to . . More Problems 1950 AHSME Problem 20 1961 AHSME Problem 22 1969 AHSME Problem 34 See Also Polynomial Factor theorem This article is a stub. Help us out by expanding it. Retrieved from " Categories: Algebra Polynomials Theorems Stubs Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
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https://ctlm.uni.edu/sites/default/files/inline-files/a_closer_lookmaking_sense_of_log_properties_1.pdf
2010 1 Making Sense of Log Properties A Closer Look at the Video After watching the Making Sense of Log Properties video, make sense of the mathematics by taking a closer look at the problem situations and solutions. Use the comments and questions in bold to help you investigate the key points of the video and develop a deeper understanding of properties of logarithms. Problem: You may use log properties, but to really understand why they work, you need to prove them. Use prior knowledge such as properties of exponents and the definition of logarithm to prove the following properties of logarithms: log (xy) = log (x) + log (y) log (xn) = n log (x). What do we need to do if we want to prove log (xy) = log (x) + log (y)? If we want to prove this, we need to show that the left-hand side of the equation is equivalent to the right-hand side. Figuring out where to begin is sometimes the hardest part. We can browse through our math manual, that is, think about what we already know, to help us get started. What information do we already know? We know the definition of logarithm, and since a logarithm is an exponent, it is also helpful to keep in mind the properties of exponents. Mathematics Manual Definition of a Logarithm: If ax = y then x = logay, and if x = logay then y = ax. Properties of Exponents: a0 = 1 anam = an+m = an-m (an)m = an⋅m Now, where might we begin? Since we want to prove log (xy) = log (x) + log (y), we need something with log (x) and log (y). We can try letting n = log x and m = log y. Then, using the definition of logarithms, if n = log x then x = 10n; this is also true for y = 10m. 2010 2 Making Sense of Log Properties A Closer Look at the Video Since we’re working with the log (xy), what would be a reasonable next step in our proof? We should multiply x and y. xy = (10n)(10m). Then, using the multiplication property of exponents, xy = 10(n + m). Now we have x times y, but we want an equation involving logarithms. What can we do to the equation, xy = 10(n + m) to change it into an equation with a logarithm? We should use the definition of logs again to write log (xy) = n + m. log (xy) = log (x) + log (y) Let n = log (x) and m = log (y) x = 10n and y = 10m log (xy) = log (x) + log (y) Let n = log x and m = log y x = 10n and y = 10m xy = (10n)(10m) xy = 10(n + m) log (xy) = log (x) + log (y) Let n = log x and m = log y x = 10n and y = 10m xy = (10n)(10m) xy = 10(n + m) log (xy) = n + m 2010 3 Making Sense of Log Properties A Closer Look at the Video Think back to how we started the proof. How did we define n and m? We let n = log x, and m = log y. How can we use that information to finish the proof? We can substitute those expressions and get log (xy) = log (x) + log (y). Now, we have proven our property. What connection is there between multiplying numbers with exponents and the logarithm of a product? When you multiply numbers with exponents, you add the exponents. Similarly, when you take the log of a product, you add the logs. Our math manual is continually growing by adding what we learn to what we already know. Let’s see if using our updated math manual can help us prove one more property: log (xn) = n log (x). What is another way log (xn) can be written that relates to the property of logarithms we have already proved? log (xy) = log (x) + log (y) Let n = log x and m = log y x = 10n and y = 10m xy = (10n)(10m) xy = 10(n + m) log (xy) = n + m log (xy) = log x + log y Property of Exponents: an⋅am = an+m Property of Logarithms: log (xy) = log (x) + log (y) 2010 4 Making Sense of Log Properties A Closer Look at the Video If you see log (xn), you might quickly recognize it is the same as the log of the quantity x⋅x⋅…⋅x, taken n times. Using the previous property, we know that the log of x⋅x⋅…⋅x, n times, is equal to log x + log x + … + log x, n times or n log (x). Unfortunately, we cannot use this as a proof because this only makes sense for whole number values of n. In proofs, we have to choose strategies that can work for any value. Let’s not get discouraged though; we’ll just consult our manual and try again. If we want to start the proof of this property in a similar way to the first proof, what should we do first? Let m = log x, and use the definition of logarithm, to write 10m = x. Since we are trying to prove something about a power, what might be a logical next step in our proof? We can raise both sides of the equation to the nth power. Then, we can use a property of exponents for powers to get 10m•n = xn. The property we are trying to prove has log (xn). How can we get from our current equation, 10(m⋅n) = xn to an equation with log (xn)? Using the definition of logarithms again, we can write m⋅n = log (xn). log (xn) = n log (x) Let m = log x 10m = x log (xn) = n log (x) Let m = log x 10m = x (10m)n = xn 10m⋅n = xn definition of logarithm raise both sides to nth power property of exponents 2010 5 Making Sense of Log Properties A Closer Look at the Video At the beginning of the proof we let m = log x. How can we use this to finish the proof? By substituting, we have log (x)•n = log xn or n•log (x) = log xn. Now, we can add this property to our math manual. log (xn) = n log (x) Let m = log x 10m = x (10m)n = xn 10m⋅n = xn m⋅n = log (xn) log (xn) = n log (x) Let m = log x 10m = x (10m)n = xn 10m⋅n = xn m⋅n = log (xn) n log x = log (xn) definition of logarithms
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https://www.cdc.gov/epiinfo/worddocs/userguide/11_nutritionalanthropometry.docx
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https://www.quora.com/How-do-you-solve-this-kind-of-equations-2-n-3n-+-1
Something went wrong. Wait a moment and try again. Solving Problems Linear Equations Elementary Math Problems Algebra Class Solving Quadratic Equatio... Solving Differential Equa... Basic Algebra Mathematical Problems 5 How do you solve this kind of equations: 2 n = 3 n + 1 ? Emad Noujeim Knowledge & science,reader,former teacher,multiple interests · Author has 2.2K answers and 10.7M answer views · 8y This problem can be solved with the help of Mathematica . The Mathematica buit-in function Solve[] can be used . The most general solution can be computed by using Reduce[]. Typing the code : Reduce[2^n == 3 n + 1, n] yields the result or answer : n=0 And n=−3Wc1(−ln(2)33√2)+ln(2)ln(8) , c1∈Z W(z) is the Lambert W function or the product log function . Wk(z) is the analytic continuation of the product log function . Typing : Reduce[2^n == 3 n + 1, n, Reals] yields three real solutions which are numerically : n=0 n≈ This problem can be solved with the help of Mathematica . The Mathematica buit-in function Solve[] can be used . The most general solution can be computed by using Reduce[]. Typing the code : Reduce[2^n == 3 n + 1, n] yields the result or answer : n=0 And n=−3Wc1(−ln(2)33√2)+ln(2)ln(8) , c1∈Z W(z) is the Lambert W function or the product log function . Wk(z) is the analytic continuation of the product log function . Typing : Reduce[2^n == 3 n + 1, n, Reals] yields three real solutions which are numerically : n=0 n≈5.3390441730248×10−17 n≈3.53767008075982791484447477 It is to be noticed that (after inspection with Mathematica) the first two real solutions are either equal or they have very close numerical values . Below is a plot of the two curves 2n and 3n+1 (made with Mathematica and a little Photoshop) : Using the Mathematica built-in function FindInstance[] and typing : SetPrecision[FindInstance[2^n == 3 n + 1, n, 10], 20] yields complex valued solutions to the given equation . Here are as an example four complex valued solutions : n≈12.159724920078226574−1525.1273701741173412i n≈13.293688538518636403+3347.142091212737379i n≈13.358624832557886297−3501.242567787459858i n≈14.422099088270069829−7317.492539846826285i Promoted by Coverage.com Johnny M Master's Degree from Harvard University (Graduated 2011) · Updated Sep 9 Does switching car insurance really save you money, or is that just marketing hype? This is one of those things that I didn’t expect to be worthwhile, but it was. You actually can save a solid chunk of money—if you use the right tool like this one. I ended up saving over $1,500/year, but I also insure four cars. I tested several comparison tools and while some of them ended up spamming me with junk, there were a couple like Coverage.com and these alternatives that I now recommend to my friend. Most insurance companies quietly raise your rate year after year. Nothing major, just enough that you don’t notice. They’re banking on you not shopping around—and to be honest, I didn’t. This is one of those things that I didn’t expect to be worthwhile, but it was. You actually can save a solid chunk of money—if you use the right tool like this one. I ended up saving over $1,500/year, but I also insure four cars. I tested several comparison tools and while some of them ended up spamming me with junk, there were a couple like Coverage.com and these alternatives that I now recommend to my friend. Most insurance companies quietly raise your rate year after year. Nothing major, just enough that you don’t notice. They’re banking on you not shopping around—and to be honest, I didn’t. It always sounded like a hassle. Dozens of tabs, endless forms, phone calls I didn’t want to take. But recently I decided to check so I used this quote tool, which compares everything in one place. It took maybe 2 minutes, tops. I just answered a few questions and it pulled up offers from multiple big-name providers, side by side. Prices, coverage details, even customer reviews—all laid out in a way that made the choice pretty obvious. They claimed I could save over $1,000 per year. I ended up exceeding that number and I cut my monthly premium by over $100. That’s over $1200 a year. For the exact same coverage. No phone tag. No junk emails. Just a better deal in less time than it takes to make coffee. Here’s the link to two comparison sites - the one I used and an alternative that I also tested. If it’s been a while since you’ve checked your rate, do it. You might be surprised at how much you’re overpaying. Related questions How do I solve ∫ 2 0 √ 1 + x 3 d x ? How can I solve this equation? lim x → ∞ x 3 ( √ x 2 + √ x 4 + 1 − x √ 2 ) How do i solve this equation: x 2 − x − 2 = √ 3 − x + √ x ? What is (N! = 1⋅2⋅3⋅…⋅n) mean in math? How can I solve this equation: 7 2 x + 1 − 2 ( 7 ) x + 1 = − 7 ? Awnon Bhowmik I can work with moderately tough equations. · Author has 3.7K answers and 11.2M answer views · 8y I just came across this question on my feed and saw that nobody wants to grind and find the actual result using the Lambert W function. I will write down a version of my solution for this. Suppose that 3x=3n+1 So that n=3x−13=x−13 2x−13=3x ⟹2x3√2=3x ⟹3x2x=13√2 ⟹3xe−xln2=13√2 ⟹xe−xln2=133√2 ⟹−xln2e−xln2=−ln233√2 ⟹−xln2=Wk(−ln233√2) \implies x=-\dfrac{W_k\left(-\dfrac{\ln 2} I just came across this question on my feed and saw that nobody wants to grind and find the actual result using the Lambert W function. I will write down a version of my solution for this. Suppose that 3x=3n+1 So that n=3x−13=x−13 2x−13=3x ⟹2x3√2=3x ⟹3x2x=13√2 ⟹3xe−xln2=13√2 ⟹xe−xln2=133√2 ⟹−xln2e−xln2=−ln233√2 ⟹−xln2=Wk(−ln233√2) ⟹x=−Wk(−ln233√2)ln2 ⟹3x=−3Wk(−ln233√2)ln2 ⟹3x−1=−3Wk(−ln233√2)ln2−1 ⟹3n=−3Wk(−ln233√2)+ln2ln2 ⟹n=−3Wk(−ln233√2)+ln23ln2 ⟹n=−3Wk(−ln233√2)+ln2ln8 where W(x) is the Lambert W or the ProductLog function and k∈Z And this is exactly what WolframAlpha gives. Assistant Bot · Sep 6 Solve 2^n = 3n + 1 for integer n (typical interpretation). Proceed with a mix of elementary reasoning, monotonicity/inequality checks, and modular constraints. Check small integers by direct substitution n = 0: 2^0 = 1, 3·0 + 1 = 1 → solution. n = 1: 2 = 4 → no. n = 2: 4 = 7 → no. n = 3: 8 = 10 → no. n = 4: 16 = 13 → no. n = 5: 32 = 16 → no. So far only n = 0 works. Monotonicity / growth argument for n ≥ 5 For n ≥ 5, compare growth rates: 2^n grows exponentially, 3n+1 linearly. More concretely, show 2^n > 3n + 1 for all n ≥ 5 by induction: Base n = 5: 2^5 = 32 > 16 = 3·5+1. Inducti Solve 2^n = 3n + 1 for integer n (typical interpretation). Proceed with a mix of elementary reasoning, monotonicity/inequality checks, and modular constraints. Check small integers by direct substitution n = 0: 2^0 = 1, 3·0 + 1 = 1 → solution. n = 1: 2 = 4 → no. n = 2: 4 = 7 → no. n = 3: 8 = 10 → no. n = 4: 16 = 13 → no. n = 5: 32 = 16 → no. So far only n = 0 works. Monotonicity / growth argument for n ≥ 5 For n ≥ 5, compare growth rates: 2^n grows exponentially, 3n+1 linearly. More concretely, show 2^n > 3n + 1 for all n ≥ 5 by induction: Base n = 5: 2^5 = 32 > 16 = 3·5+1. Inductive step: if 2^k > 3k+1 then 2^{k+1} = 2·2^k > 2(3k+1) = 6k+2. Since 6k+2 ≥ 3(k+1)+1 for k ≥ 1 (6k+2 − (3k+4) = 3k−2 ≥ 1 when k ≥ 1), we get 2^{k+1} > 3(k+1)+1. Thus inequality holds for all k ≥ 5. Therefore no solutions with n ≥ 5. Exclude negative integers For negative integer n = −m (m ≥ 1), 2^n = 2^{−m} is between 0 and 1, while 3n + 1 = 1 − 3m ≤ −2, so equality impossible. So no negative integer solutions. Conclusion The only integer solution is n = 0. Optional remark for real n As a real equation, f(n) = 2^n − 3n − 1 is continuous, f(0)=0, f'(n)=ln2·2^n − 3. f'(0)=ln2−3<0, derivative later becomes positive, so there may be a second real root >0. Numerical check: f(1)=−1, f(2)=−3, f(3)=−2, f(4)=3, so a real root exists between 3 and 4 (approx n ≈ 3.214). But for integer solutions only n = 0. Thomas Schürger M.C.S. · Author has 2.3K answers and 2.5M answer views · 8y One obvious solution is n=0. By sketching the graph of 2n and 3n+1 you can see that there must be another real solution between n=3 and n=4. For finding the other solution we can try solving this using the Lambert W function. We can do this if we manage to analytically transform the equation into the form XeX=Y, where X is an expression that depends on n and Y is an expression that does not depend on n (i.e. it is a constant). Then X=W(Y) and we can continue solving this for n. [math]\displaystyle 2^n = 3n+1\\iff e^{n\ln 2} = 3n+1\\displaystyle\iff 1 = (3n+1)e^{-n\ln 2} \\displaystyle\iff -[/math] One obvious solution is n=0. By sketching the graph of 2n and 3n+1 you can see that there must be another real solution between n=3 and n=4. For finding the other solution we can try solving this using the Lambert W function. We can do this if we manage to analytically transform the equation into the form XeX=Y, where X is an expression that depends on n and Y is an expression that does not depend on n (i.e. it is a constant). Then X=W(Y) and we can continue solving this for n. 2n=3n+1⟺enln2=3n+1⟺1=(3n+1)e−nln2⟺−13=(−n−13)e−nln2⟺−ln23=(−nln2−ln23)e−nln2⟺−ln23e−ln23=(−nln2−ln23)e−nln2−ln23 Now we have reached the form Y=XeX, so we can use X=W(Y). ⟺−nln2−ln23=W(−ln23e−ln23)⟺−nln2=W(−ln23e−ln23)+ln23⟺n=−W(−ln23e−ln23)+ln23ln2 Realizing that e−ln23=13√2: ⟺n=−W(−ln233√2)+ln23ln2 Using the laws of logarithms this is the same as ⟺n=−3W(−ln233√2)+ln2ln8≈3.5376700807598279 So the original equation has two real solutions: n=0 and n≈3.5376700807598279. Related questions How do I solve ∫ 2 0 √ 1 + x 3 d x ? How can I solve this equation? lim x → ∞ x 3 ( √ x 2 + √ x 4 + 1 − x √ 2 ) How do i solve this equation: x 2 − x − 2 = √ 3 − x + √ x ? What is (N! = 1⋅2⋅3⋅…⋅n) mean in math? How can I solve this equation: 7 2 x + 1 − 2 ( 7 ) x + 1 = − 7 ? How do I solve the following system of equations: ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ x 1 ( x 2 + x 3 + ⋯ + x n ) + 2 ⋅ 1 ( x 1 + x 2 + ⋯ + x n ) 2 = 9 a 2 , x 2 ( x 1 + x 3 + ⋯ + x n ) + 3 ⋅ 2 ( x 1 + x 2 + ⋯ + x n ) 2 = 25 a 2 , ⋮ x n ( x 1 + x 2 + ⋯ + x n − 1 ) + ( n + 1 ) ⋅ n ( x 1 + x 2 + ⋯ + x n ) 2 = ( 2 n + 1 ) 2 a 2 ? How do I solve this equation: cos 2 a + cos 2 √ π 2 − a 2 = 1 ? How can I solve this equation? 3 + 2 − x + 3 ⋅ 5 − x − 8 2 x ⋅ 5 x − 10 − x = 0 How can I solve the equation: [\math] 2^ {x^2+6} +3^ {x^2-x+1} =2^ {5x} +3^ {4x-6} [/math]? How do I solve the following equation: X 2 ( 2 − X ) 2 = 1 + 2 ( 2 − X ) 2 ? How do I solve the following equation for x : x 5 ( x − 1 ) 2 ( x + 1 ) = 25 ? How do solve the following system of equations: { ( x + √ x 2 + 1 ) ( y + √ y 2 + 1 ) = 1 , y + y √ x 2 − 1 + 35 12 = 0 ? What is ∑ ∞ n = 0 ( 1 3 n + 1 − 1 3 n + 2 ) ? How do you solve the following equations, x 2 − y z = a , y 2 − z x = b , z 2 − x y = c ? 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https://pmc.ncbi.nlm.nih.gov/articles/PMC11173655/
A 23-year-old woman with metabolic alkalosis and hypokalemia - PMC Skip to main content An official website of the United States government Here's how you know Here's how you know Official websites use .gov A .gov website belongs to an official government organization in the United States. Secure .gov websites use HTTPS A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites. PMC Search Update PMC Beta search will replace the current PMC search the week of September 7, 2025. Try out PMC Beta search now and give us your feedback. Learn more Search Log in Dashboard Publications Account settings Log out Search… Search NCBI Primary site navigation Search Logged in as: Dashboard Publications Account settings Log in Search PMC Full-Text Archive Search in PMC Journal List User Guide New Try this search in PMC Beta Search View on publisher site Download PDF Add to Collections Cite Permalink PERMALINK Copy As a library, NLM provides access to scientific literature. Inclusion in an NLM database does not imply endorsement of, or agreement with, the contents by NLM or the National Institutes of Health. Learn more: PMC Disclaimer | PMC Copyright Notice CMAJ . 2024 Jun 10;196(22):E760–E764. doi: 10.1503/cmaj.240163 Search in PMC Search in PubMed View in NLM Catalog Add to search A 23-year-old woman with metabolic alkalosis and hypokalemia Kamel S Kamel Kamel S Kamel, MD 1 Division of Nephrology, St. Michael’s Hospital, University of Toronto, Toronto, Ont. Find articles by Kamel S Kamel 1, Joshua Shapiro Joshua Shapiro, MD 1 Division of Nephrology, St. Michael’s Hospital, University of Toronto, Toronto, Ont. Find articles by Joshua Shapiro 1,#, Ziv Harel Ziv Harel, MD, MSc 1 Division of Nephrology, St. Michael’s Hospital, University of Toronto, Toronto, Ont. Find articles by Ziv Harel 1,✉,# Author information Article notes Copyright and License information 1 Division of Nephrology, St. Michael’s Hospital, University of Toronto, Toronto, Ont. ✉ Correspondence to: Ziv Harel, ziv.harel@unityhealth.to ✉ Corresponding author. Contributed equally. Issue date 2024 Jun 10. © 2024 CMA Impact Inc. or its licensors This is an Open Access article distributed in accordance with the terms of the Creative Commons Attribution (CC BY-NC-ND 4.0) licence, which permits use, distribution and reproduction in any medium, provided that the original publication is properly cited, the use is noncommercial (i.e., research or educational use), and no modifications or adaptations are made. See: PMC Copyright notice PMCID: PMC11173655 PMID: 38857937 This article has been corrected. See CMAJ. 2024 Sep 9;196(29):E1014. This article is also available in French. See "Alcalose métabolique et hypokaliémie chez une femme de 23 ans". KEY POINTS The most common causes of metabolic alkalosis are vomiting and diuretic use. Urine chloride levels are usually low among patients with vomiting and among those using diuretics, but may be high if the patient took diuretics recently. In patients with metabolic alkalosis and high urine chloride, assessment of effective arterial blood volume and blood pressure may distinguish those with metabolic alkalosis caused by a diuretic from those with primary excess mineralocorticoid effect. In patients with metabolic alkalosis, high urine chloride, and low effective arterial blood volume, measuring urine chloride repeatedly may distinguish Bartter syndrome or Gitelman syndrome from occult diuretic use. A 23-year-old woman with no notable medical history was seen in our nephrology clinic for electrolyte abnormalities detected on laboratory testing ordered by her primary care physician, who she had seen after noting mild muscle weakness during exercise. She reported no use of daily medications, including diuretics or laxatives, and had no history of vomiting or diarrhea. Her blood pressure was 95/70 mm Hg, her body mass index was 24, and her jugular venous pressure was 1 cm below the level of the sternal angle while supine. The results of laboratory testing are shown in Table 1. The patient had metabolic alkalosis, hypokalemia, and hypomagnesemia. The findings of low blood pressure, low extracellular fluid volume on physical examination, and elevated concentration of chloride ions in a random urine sample suggested that a diuretic effect was causing her electrolyte abnormalities. A diuretic effect can be caused by diuretic use (e.g., furosemide, thiazide) or by a genetic disorder that causes decreased reabsorption of sodium chloride in the loop of Henle (i.e., Bartter syndrome, which mimics the effects of loop diuretics) or in the distal convoluted tubule (i.e., Gitelman syndrome, which mimics the effects of thiazide diuretics). To distinguish between these 2 possibilities, we asked the patient to provide 5 random urine samples over 48 hours. Chloride was elevated In each of these samples, which was consistent with a genetic defect that impaired reabsorption of sodium chloride by the kidney. The hypomagnesemia and hypocalciuria suggested a diagnosis of Gitelman syndrome. Genetic testing revealed heterozygous, biallelic, pathogenic variants in SLCl2A3, the gene that encodes the sodium chloride cotransporter in the distal convoluted tubule, confirming the diagnosis of Gitelman syndrome. Table 1: Laboratory values in a 23-year-old woman with metabolic alkalosis | Laboratory test | Value | Reference range | :---: | Plasma | | Sodium, mmol/L | 136 | 135–145 | | Potassium, mmol/L | 3.2 | 3.5–5.0 | | Chloride, mmol/L | 90 | 96–106 | | Bicarbonate, mmol/L | 34 | 25–30 | | Calcium, mmol/L | 2.45 | 2.10–2.60 | | Urea, mmol/L | 6 | 3–7 | | Creatinine, μmol/L | 62 | 52–112 | | Magnesium, mmol/L | 0.52 | 0.63–0.94 | | Venous pH | 7.50 | 7.35–7.45 | | Venous PCO 2, mm Hg | 45 | 42–52 | | Urine | | pH | 6.0 | Varies | | Creatinine, mmol/L | 5.3 | Varies | | Potassium, mmol/L | 20 | Varies | | Sodium, mmol/L | 45 | Varies | | Chloride, mmol/L | 40 | Varies | | Calcium, mmol/L | 0.8 | Varies | | Calcium-to-creatinine ratio | 0.03 | Hypocalciuria: < 0.2 | | Chloride, mmol/L, from 5 random spot urine collections over 48 h | 52, 49, 61, 40, 43 | Varies | Open in a new tab We prescribed potassium chloride (40 mmol/d) and magnesium oxide (420 mg, twice daily). The patient’s plasma magnesium improved to 0.61 mmol/L, but she remained hypokalemic at 3.0 (normal 3.5–5.0) mmol/L. Accordingly, we prescribed amiloride (5 mg/d); 4 weeks later, her serum potassium had increased to 3.6 mmol/L, her bicarbonate was 27 (normal 25–30) mmol/L, and her magnesium was unchanged. However, because the patient developed postural lightheadedness, we stopped the amiloride and increased the potassium chloride to 80 mmol/d. One year later, she continues taking the same doses of potassium chloride and magnesium oxide. Her potassium and magnesium levels are in the low-to-normal range at 3.5 mmol/L and 0.65 mmol/L, respectively, and her bicarbonate is 29 mmol/L. Discussion Metabolic alkalosis is defined as a plasma bicarbonate concentration greater than 30 mmol/L and an arterial pH above 7.45. An elevated bicarbonate level can be a compensatory response to chronic respiratory acidosis, but a diagnosis of primary metabolic alkalosis can be made if the clinical history is not consistent with that condition. Measurement of the venous blood pH can confirm the diagnosis of metabolic alkalosis, noting that arterial blood pH is typically 0.03 higher than venous blood pH.1 Metabolic alkalosis represents an increase in the quantity of bicarbonate relative to the volume of water in the extracellular fluid. This may result from a loss of extracellular fluid (i.e., contraction alkalosis), the addition of bicarbonate, or both.2,3 Bicarbonate is added to the extracellular fluid from exogenous or endogenous sources. Exogenous sources include the ingestion or infusion of bicarbonate salts or of anions (e.g., lactate, citrate) that are converted to bicarbonate, but metabolic alkalosis will develop only if the glomerular filtration rate (GFR) is markedly low (i.e., < 15 mL/min). The 2 major endogenous sources of bicarbonate are the stomach and the kidneys. In the stomach, bicarbonate is generated when hydrochloric acid is secreted by the parietal cells into the gastric lumen. Under normal physiologic conditions, hydrochloric acid is subsequently titrated in the small intestine by an equal amount of bicarbonate secreted by the pancreas, thereby negating the previous bicarbonate gain. Metabolic alkalosis will occur if gastric fluid is lost from vomiting or suctioning of stomach contents. In the kidney, metabolism of glutamine in the proximal tubule cells produces ammonium and α-ketoglutarate anion, which is metabolized to bicarbonate (Figure 1A). To retain this newly generated bicarbonate, the kidney must excrete ammonium in the urine. If it cannot do so, ammonium is shunted to the liver and metabolized into urea in a process that consumes bicarbonate. Increased ammonium excretion in the urine, which results in more bicarbonate being added to the body, occurs when more ammonium is produced by the proximal tubule and when the secretion of hydrogen ions by the distal kidney tubule cells increases, which leads to more ammonium being trapped in the urine. Ammonium production is stimulated by an acidic milieu in the proximal tubule cell, which may occur because of hypokalemia or because of high partial pressure of carbon dioxide in the peritubular capillaries.4 Aldosterone increases ammonium excretion by causing hypokalemia, which, again, stimulates ammonium production in the proximal tubule, and by increasing the rate of hydrogen secretion from the distal tubules. Figure 1: Open in a new tab (A) Generation of bicarbonate in the proximal tubule cell. Metabolism of glutamine in the proximal tubule cells produces ammonium (NH 4+) and α-ketoglutarate, which is metabolized to bicarbonate (HCO 3−). This process is stimulated by a rise in concentration of hydrogen (H+) ions in proximal tubule cells. Ammonium (NH 4+) ions are secreted into the lumen by the sodium–hydrogen exchanger. The newly formed HCO 3− ions exit the cells with sodium ions (Na+) on a sodium–bicarbonate cotransporter. (B) Maintenance of metabolic alkalosis by bicarbonate reabsorption in the proximal tubule cell. Bicarbonate reabsorption in the proximal tubule is mediated by the sodium–hydrogen exchanger. This cation exchanger is stimulated by increased levels of angiotensin II and increased concentration of H+ ions in cells. Metabolic alkalosis is maintained only if the kidneys are unable to excrete the excess bicarbonate because of a decreased filtered load of bicarbonate, caused by a low GFR, an increased rate of bicarbonate reabsorption, or both. Reabsorption of bicarbonate occurs mainly in the proximal tubule and is mediated by a sodium–hydrogen exchanger, which reabsorbs sodium and secretes hydrogen into the lumen. The secreted hydrogen titrates the filtered bicarbonate to form carbon dioxide and water, which together enter cells to be converted back into bicarbonate, which is then returned to the blood. This exchanger is activated by increased angiotensin II levels and an acidic proximal tubule cell (Figure 1B). In the distal kidney tubule, pendrin — a chloride–bicarbonate exchanger — reabsorbs chloride and secretes bicarbonate into the lumen.5 Chloride depletion diminishes the delivery of chloride to the exchanger, which further maintains a state of metabolic alkalosis as less bicarbonate is secreted into the lumen. The evaluation of patients with metabolic alkalosis begins by taking a history to rule out the 2 most common causes of this condition, namely vomiting and diuretic use (Table 2).6 Some patients may deny vomiting or using diuretics, so assessment of chloride in a random urine sample helps determine the cause of metabolic alkalosis. A low urine chloride level (i.e., < 20 mmol/L) is expected among patients who have depleted effective arterial blood volume and chloride from vomiting, diuretics, chloride-wasting diarrhea, or loss of chloride in sweat. However, urine chloride will not be low if the urine sample was collected when a diuretic was still acting. If the urine chloride is not low, assessment of the effective arterial blood volume and the blood pressure may distinguish patients with metabolic alkalosis caused by a diuretic effect from those with a primary excess mineralocorticoid effect (Figure 2). Patients with the former condition do not usually have high blood pressure and their effective arterial blood volume may be contracted. An exception would be in patients with hypertension who take diuretics as they may develop metabolic alkalosis from hypokalemia and may not have contracted effective arterial blood volume. Those experiencing a mineralocorticoid effect may be hypertensive, and have either normal or expanded blood volume. Further characterization of the cause of the mineralocorticoid effect can be aided by measuring plasma renin and aldosterone levels. Table 2: Common causes of metabolic alkalosis | Cause | Mechanism | :---: | | Vomiting or nasogastric suctioning | Loss of gastric fluid causes a gain of bicarbonate because of the loss of hydrochloric acid. Angiotensin II — released in response to low EABV — and hypokalemia stimulate bicarbonate reabsorption in proximal tubule cells and maintain the increase in plasma bicarbonate. | | Diuretics and hereditary conditions with diuretic effects | Diuretics may lead to loss of extracellular fluid, hypokalemia, and increased angiotensin II levels. Diuretic effects can result from medications such as furosemide and thiazides, or from genetic disorders that involve the loop of Henle (Bartter syndrome) or the distal convoluted tubule (Gitelman syndrome).11 | | Chloride-wasting diarrhea | A loss of chloride in stool may be caused by reduced activity of the chloride-bicarbonate exchanger in the colon from a congenital or acquired defect (e.g., adenocarcinoma, inflammatory disorder involving the colon). | | Post-hypercapnia alkalosis | The correction of hypercapnia in patients with a chronic respiratory acidosis may lead to metabolic alkalosis if the excess bicarbonate caused by the kidney compensatory response to hypercapnia is retained because of low EABV. | | Calcium-alkali syndrome | This syndrome results from excessive use of calcium with absorbable alkali supplements, most often for the management of osteoporosis. Hypercalcemia causes vasoconstriction of kidney arterioles, which results in reduced GFR; activation of the calcium-sensing receptor in the thick ascending limb of the loop of Henle results in a furosemide-like diuretic effect.12 | | Cystic fibrosis | Patients with cystic fibrosis may develop metabolic alkalosis because of decreased EABV, hypokalemia, and impaired bicarbonate secretion by the distal kidney tubule as the defective cystic fibrosis transmembrane regulator protein fails to secret chloride into the lumen and provide pendrin with sufficient chloride to reabsorb in exchange for bicarbonate. | | Excess mineralocorticoid effect | An excess mineralocorticoid effect can result from renin production that stimulates the release of aldosterone (e.g., renal artery stenosis), primary hyperaldosteronism, conditions in which cortisol acts as a mineralocorticoid because of 11-β hydroxydehydrogenase impairment, and a constitutively active epithelial sodium channel in the distal kidney tubule (e.g., Liddle syndrome).13 This effect leads to the secretion of potassium, which stimulates ammonium production, and bicarbonate reabsorption. Mineralocorticoids also increase ammonium excretion by stimulating distal secretion of hydrogen ions. | Open in a new tab Note: EABV = effective arterial blood volume, GFR = glomerular filtration rate. Figure 2: Open in a new tab Approach to the patient with metabolic alkalosis. This approach has not been evaluated or validated in clinical studies. There are limitations of physical examination in the assessment of extracellular fluid (ECF) volume.7 †These conditions may be hereditary or acquired. Acquired Bartter syndrome may result from activation of the calcium-sensing receptor in the medullary thick ascending limb of the loop of Henle by calcium (among patients with hypercalcemia) or by other cationic ligands (e.g., amikacin, gentamicin, immunoglobulins). Acquired Gitelman syndrome has been associated with Sjögren syndrome and cisplatin use. ‡Cortisol acts as a mineralocorticoid if the enzyme 11-β hydroxydehydrogenase (which inactivates cortisol by metabolizing it to cortisone) is deficient (e.g., apparent mineralocorticoid excess syndrome), if it is inhibited (e.g., from ingestion of a compound containing glycyrrhizinic acid, such as licorice), or if its activity is overwhelmed by an excess production of cortisol (e.g., adrenocorticotropic hormone–producing tumour). Note: BP = blood pressure, EABV = effective arterial blood volume. Measurement of chloride in repeat random urine samples may distinguish patients with genetic tubulopathies from those who use diuretics, as the former will have persistently elevated urine chloride, whereas the latter will have intermittently elevated urine chloride. An assay for diuretics in the urine may be ordered if chloride levels are persistently high in random urine samples but surreptitious use of diuretics is still suspected. Genetic testing for Gitelman syndrome and Bartter syndrome, which is available through commercial laboratories, can help confirm those diagnoses. Treatment of patients with metabolic alkalosis involves addressing the processes causing the addition of bicarbonate and correcting the conditions preventing the kidney from excreting the excess bicarbonate (e.g., low effective arterial blood volume, hypokalemia). Gitelman syndrome is one of the most frequent inherited disorders of the kidney, with a prevalence of 1 per 40 000 people. It is caused by mutations in the SLC12A3 gene, which encodes the sodium chloride cotransporter in the distal convoluted tubule.8 Gitelman syndrome usually presents in adolescence and adulthood with symptoms linked to electrolyte disturbances, including salt craving, thirst, cramps, and muscle weakness. Biochemically, patients have hypokalemia and metabolic alkalosis. Hypomagnesemia (< 0.7 mmol/L) is also common and hypocalciuria (calcium-to-creatinine ratio < 0.2 from spot urine sample) is characteristic.8,9 Management of Gitelman syndrome involves lifelong supplementation with oral potassium chloride and magnesium at doses sufficient to reach levels of at least 3.0 mmol/L for potassium, and 0.6 mmol/L for magnesium.8 In patients in whom supplementation is insufficient or associated with intolerable adverse effects, amiloride (an epithelial sodium-channel blocker) or eplerenone (an aldosterone receptor antagonist) may be added, although these medications can cause postural symptoms. In an open-label, cross-over randomized trial of 6 weeks’ duration, these medications increased serum potassium levels by 0.3 mmol/L when added to potassium and magnesium supplementation.10 In the same study, indomethacin also raised the serum potassium level by a similar amount. However, its association with gastrointestinal intolerance and long-term kidney dysfunction limit its use. Metabolic alkalosis is a primary acid–base disorder characterized by elevated blood pH and plasma bicarbonate concentration. The most common causes of metabolic alkalosis are vomiting and the use of diuretics. Measurement of urine chloride and assessment of the effective arterial blood volume and blood pressure can help differentiate the different cause of alkalosis. Patients with high urine chloride levels may require special investigations, including genetic testing for Bartter and Gitelman syndromes and measurement of plasma renin and aldosterone levels. The management of metabolic alkalosis involves correcting the underlying disorder, leading to the addition of bicarbonate and correcting conditions that prevent the kidney from excreting the excess bicarbonate. The section Cases presents brief case reports that convey clear, practical lessons. Preference is given to common presentations of important rare conditions, and important unusual presentations of common problems. Articles start with a case presentation (500 words maximum), and a discussion of the underlying condition follows (1000 words maximum). Visual elements (e.g., tables of the differential diagnosis, clinical features or diagnostic approach) are encouraged. Consent from patients for publication of their story is a necessity. See information for authors at www.cmaj.ca. Acknowledgements The authors thank Joel G. Ray and Catherine Yu for their critique and helpful suggestions in the preparation of the manuscript. Footnotes Competing interests: None declared. This article has been peer reviewed. The authors have obtained patient consent. Contributors: All of the authors contributed to the conception and design of the work, drafted the manuscript, revised it critically for important intellectual content, gave final approval of the version to be published, and agreed to be accountable for all aspects of the work. Joshua Shapiro and Ziv Harel contributed equally to the work. References 1.Adrogué HJ, Madias NE. Secondary responses to altered acid-base status: the rules of engagement. J Am Soc Nephrol 2010;21:920–3. [DOI] [PubMed] [Google Scholar] 2.Emmett M. Metabolic alkalosis: a brief pathophysiologic review. Clin J Am Soc Nephrol 2020;15:1848–56. [DOI] [PMC free article] [PubMed] [Google Scholar] 3.Do C, Vasquez PC, Soleimani M. Metabolic alkalosis pathogenesis, diagnosis, and treatment: core curriculum 2022. Am J Kidney Dis 2022;80:536–51. [DOI] [PMC free article] [PubMed] [Google Scholar] 4.Kamel KS, Halperin ML. Fluid, electrolytes and acid-base physiology. a problembased approach. Philadelphia: Elsevier; 2017. [Google Scholar] 5.Soleimani M. The multiple roles of pendrin in the kidney. Nephrol Dial Transplant 2015;30:1257–66. [DOI] [PMC free article] [PubMed] [Google Scholar] 6.Kamel KS, Halperin ML. Interpertation of electrolytes and acid base parameters in blood and urine. In: Skorecki K, Chertow GM, Marsden PA, editors. Brenner and Rector’s the kidney 11th ed.Philadelphia: Elsevier; 2019:758–95. [Google Scholar] 7.McGee S, Abernethy WB, III, Simel DL. The rational clinical examination. Is this patient hypovolemic? JAMA 1999;281:1022–9. [DOI] [PubMed] [Google Scholar] 8.Blanchard A, Bockenhauer D, Bolignano D, et al. Gitelman syndrome: consensus and guidance from a Kidney Disease: Improving Global Outcomes (KDIGO) Controversies Conference. Kidney Int 2017;91:24–33. [DOI] [PubMed] [Google Scholar] 9.Riveira-Munoz E, Chang Q, Godefroid N, et al. Transcriptional and functional analyses of SLC12A3 mutations: new clues for the pathogenesis of Gitelman syndrome. J Am Soc Nephrol 2007;18:1271–83. [DOI] [PubMed] [Google Scholar] 10.Blanchard A, Vargas-Poussou R, Vallet M, et al. Indomethacin, amiloride, or eplerenone for treating hypokalemia in Gitelman syndrome. J Am Soc Nephrol 2015;26:468–75. [DOI] [PMC free article] [PubMed] [Google Scholar] 11.Downie ML, Lopez Garcia SC, Kleta R, et al. Inherited tubulopathies of the kidney: insights from genetics. Clin J Am Soc Nephrol 2021;16:620–30. [DOI] [PMC free article] [PubMed] [Google Scholar] 12.Arroyo M, Fenves AZ, Emmett M. The calcium-alkali syndrome. Proc Bayl Univ Med Cent 2013;26:179–81. [DOI] [PMC free article] [PubMed] [Google Scholar] 13.Kardalas E, Paschou SA, Anagnostis P, et al. Hypokalemia: a clinical update. Endocr Connect 2018;7:R135–46. 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https://www.convertunits.com/from/g-unit/to/m/s%5E2
Convert g-unit to metre/square second More information from the unit converter How many g-unit in 1 m/s^2? The answer is 0.10197162129779. We assume you are converting between g-unit and metre/square second. You can view more details on each measurement unit: g-unit or m/s^2 The SI derived unit for acceleration is the meter/square second. 1 g-unit is equal to 9.80665 meter/square second. Note that rounding errors may occur, so always check the results. Use this page to learn how to convert between g-units and meters/square second. Type in your own numbers in the form to convert the units! Quick conversion chart of g-unit to m/s^2 1 g-unit to m/s^2 = 9.80665 m/s^2 5 g-unit to m/s^2 = 49.03325 m/s^2 10 g-unit to m/s^2 = 98.0665 m/s^2 15 g-unit to m/s^2 = 147.09975 m/s^2 20 g-unit to m/s^2 = 196.133 m/s^2 25 g-unit to m/s^2 = 245.16625 m/s^2 30 g-unit to m/s^2 = 294.1995 m/s^2 40 g-unit to m/s^2 = 392.266 m/s^2 50 g-unit to m/s^2 = 490.3325 m/s^2 Want other units? You can do the reverse unit conversion from m/s^2 to g-unit, or enter any two units below: Common acceleration conversions g-unit to milligal g-unit to centigal g-unit to mile/square second g-unit to inch/square second g-unit to kilometer/square second g-unit to dekameter/square second g-unit to decimeter/square second g-unit to millimeter/square second g-unit to decigal g-unit to foot/square second Metric conversions and more ConvertUnits.com provides an online conversion calculator for all types of measurement units. You can find metric conversion tables for SI units, as well as English units, currency, and other data. Type in unit symbols, abbreviations, or full names for units of length, area, mass, pressure, and other types. Examples include mm, inch, 70 kg, 150 lbs, US fluid ounce, 6'3", 10 stone 4, cubic cm, metres squared, grams, moles, feet per second, and many more!
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https://www.mheducation.com/highered/mhp/product/schaum-s-outline-laplace-transforms.html
Schaum's Outline of Laplace Transforms Skip to main content Learning Levels PreK–12 PreK–12 ------- Back to Learning Levels PreK–12 Home Page Browse by Subject Browse by Subject Back to PreK–12 Language Arts Math Science Health Social Studies Intervention Supplemental World Languages Advanced Placement & Electives CTE Computing Purchase Purchase Back to PreK–12 Contact a Rep Request a Quote Create® EasyOrder Search by ISBN or Browse Programs for School and Home Quick Order Discover Resources for your Classroom & School Discover Resources for your Classroom & School Back to PreK–12 Product Trainings K–12 Mobile App Integration Services Free Educational Activities Science of Literacy McGraw Hill + Kahoot Small Schools Research Personalized Learning Discover Our Principles Discover Our Principles Back to PreK–12 What We Stand For Art of Teaching Equity in Action Connect Connect Back to PreK–12 Educator Communities Inspired Ideas (blog) Contact a RepGet Support Higher Ed Higher Ed --------- Back to Learning Levels Higher Ed Home Page Browse by Discipline Browse by Discipline Back to Higher Ed Showing 52 results Accounting Agriculture & Forestry American Government Anatomy & Physiology Anthropology Art Astronomy & Physical Science Biology - Majors Biology - Non-Majors Business Communication Business Law Business Mathematics Business Statistics & Analytics Career Development Cell/Molecular Biology & Genetics Chemistry Communication Composition Computer & Information Technology Criminal Justice Decision Sciences & Operations Management Developmental English Earth & Environmental Science Ecology Economics Education Engineering/Computer Science Engineering Technologies - Tech & Trade Film Finance Health & Human Performance Health Professions Humanities History Insurance & Real Estate Introduction to Business Keyboarding Management Management Information Systems Mathematics Marketing Microbiology Music Nutrition Plants & Animals Philosophy & Religion Physics Psychology Student Success Sociology Theater World Languages Digital Products Digital Products Back to Higher Ed Connect® McGraw Hill GO ALEKS® ALEKS® Placement, Preparation, and Learning SIMnet McGraw Hill eBook & ReadAnywhere App Sharpen: Study App Virtual Labs AI Reader MH Medical Services Services Back to Higher Ed Affordable Access Learning Management System Integration Content Collections powered by Create® Custom Courseware Solutions Education for All Business Program Professional Services Remote Proctoring Institutional Solutions Evergreen Campuswide Solutions Campuswide Solutions Back to Higher Ed Dual Enrollment Solutions Blog Events Events Back to Higher Ed Live Events On-Demand Events Support Support Back to Higher Ed General Help & Support Info Online Technical Support Center Support At Every Step Instructor Sample Requests Platform System Check Contact a RepGet Support Cart Sign In My Account My Account Details My Account My Information Security & Login Order History My Lists My Information Security & Login Order History My Digital Products Log In to My PreK-12 Platform Achieve3000 Actively Learn ALEKS AP/Honors & Electives ConnectED my.mheducation.com Open Learning Platform Log In to My Higher Ed Platform Connect ALEKS My Bookshelf (eBook Access) Sharpen Sign Out My cart Sign In My Account PreK–12 Support Higher Ed Support International Support Search ISBN10: 007060231X |ISBN13: 9780070602311 Schaum's Outline of Laplace Transforms, 1st Edition ISBN10: 007060231X ISBN13: 9780070602311 By Murray Spiegel ©1965 Published June 1, 1965 For Students For Instructors Print from $31.00 ISBN10:007060231X | ISBN13: 9780070602311 Features More information about this Print book can be found in the page section below. $31.00 View Price ISBN10:007060231X | ISBN13: 9780070602311 Features More information about this Print book can be found in the page section below. How to Access Your eBook Step 1 . DownloadAdobe Digital Editions to your PC or Mac desktop/laptop. Step 2. Register and authorize your Adobe ID (optional).To access your eBook on multiple devices, first create an Adobe ID at account.adobe.com. Then, open Adobe Digital Editions, go to the Help menu, and select "Authorize Computer" to link your Adobe ID. Step 3. Open Your eBook. Use Adobe Digital Editions to open the file. If the eBook doesn’t open, contact customer service for assistance. Overview Confusing Textbooks? Missed Lectures? Not Enough Time? Fortunately for you, there's Schaum's Outlines. More than 40 million students have trusted Schaum's to help them succeed in the classroom and on exams. Schaum's is the key to faster learning and higher grades in every subject. Each Outline presents all the essential course information in an easy-to-follow, topic-by-topic format. You also get hundreds of examples, solved problems, and practice exercises to test your skills. This Schaum's Outline gives you Practice problems with full explanations that reinforce knowledge Coverage of the most up-to-date developments in your course field In-depth review of practices and applications Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum's to shorten your study time-and get your best test scores! Schaum's Outlines-Problem Solved. Table of Contents The Laplace Transform. The Inverse Laplace Transform. Applications to Differential Equations. Applications to Integral and Difference Equations. Complex Variable Theory. Fourier Series and Integrals. The Complex Inversion Formula. Applications to Boundary-Value Problems. Appendix A: Table of General Properties of Laplace Transforms. Appendix B: Table of Special Laplace Transforms. Appendix C: Table of Special Functions. Need support?We're here to help -Get real-world support and resources every step of the way. 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https://physics.stackexchange.com/questions/111888/partition-function-total-internal-energy-vs-average-energy
Partition Function - TOTAL internal energy vs Average Energy - Physics Stack Exchange Join Physics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Loading… Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company, and our products current community Physics helpchat Physics Meta your communities Sign up or log in to customize your list. more stack exchange communities company blog Log in Sign up Home Questions Unanswered AI Assist Labs Tags Chat Users Teams Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Try Teams for freeExplore Teams 3. Teams 4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Hang on, you can't upvote just yet. You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Partition Function - TOTAL internal energy vs Average Energy Ask Question Asked 11 years, 4 months ago Modified11 years, 4 months ago Viewed 5k times This question shows research effort; it is useful and clear 1 Save this question. Show activity on this post. Given a partition function Z, the books sometimes uses the terms 'total internal energy' and 'average energy' exchangeably. It confuses me to no end. On one hand they say that Internal energy is average energy: U¯=−∂l n Z∂β U¯=−∂l n Z∂β Then they use this to calculate specific heat: C v=(∂U∂t)V C v=(∂U∂t)V. On the other hand they say internal energy is U=∑ϵ p n p=∫ϵ n p g(ϵ)d ϵ U=∑ϵ p n p=∫ϵ n p g(ϵ)d ϵ where n p n p is mean number of particles with energy p, g(E)g(E) is density of states. Then they use this to calculate heat capacity. Which is which?! partition-function energy Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Improve this question Follow Follow this question to receive notifications asked May 9, 2014 at 1:05 user44840user44840 941 14 14 silver badges 35 35 bronze badges 2 I might be remembering wrong, but aren't they both the same?DumpsterDoofus –DumpsterDoofus 2014-05-09 01:16:12 +00:00 Commented May 9, 2014 at 1:16 The second equation is correct only for free particles. I also suspect than in the second case, the Hamiltonian is defined as ∑p ϵ p N p∑p ϵ p N p (with N¯p=n p N¯p=n p), which shows what you want.Adam –Adam 2014-05-09 02:28:29 +00:00 Commented May 9, 2014 at 2:28 Add a comment| 2 Answers 2 Sorted by: Reset to default This answer is useful 4 Save this answer. Show activity on this post. The average energy is U¯¯¯¯=−∂∂β log(Z)=−1 Z∂Z∂β=∑p g p ϵ p exp(−ϵ p β)∑p g p exp(−ϵ p β)=∑p ϵ p P p U¯=−∂∂β log⁡(Z)=−1 Z∂Z∂β=∑p g p ϵ p exp⁡(−ϵ p β)∑p g p exp⁡(−ϵ p β)=∑p ϵ p P p where P p P p is the probability of being in the p th p th state. Multiplying this by N N (the total number of particles) and noting that n p=N P p n p=N P p gives the total energy N U¯¯¯¯=∑ϵ p n p=U.N U¯=∑ϵ p n p=U. This is where my thermo skills are rusty (or nonexistent); I'm not sure why there isn't a factor of N N in your original question (unless Z Z isn't a single-particle partition function). Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Improve this answer Follow Follow this answer to receive notifications edited May 9, 2014 at 2:31 answered May 9, 2014 at 1:24 DumpsterDoofusDumpsterDoofus 10.7k 1 1 gold badge 27 27 silver badges 32 32 bronze badges 4 how did you get from the second last step to the last step? n p n p is the occupation number.user44840 –user44840 2014-05-09 02:00:02 +00:00 Commented May 9, 2014 at 2:00 I'm bad at thermo so I'm probably leaving something out, but recall that for Boltzmann-distributed populations, you have P p=g p exp(−ϵ p)/Z P p=g p exp⁡(−ϵ p)/Z where P p P p is the probability of being in the p th p th state (Z Z is just a normalizing factor, like you learn about in any intro probability math course). I just replaced P p P p with n p=N P p n p=N P p, since it sounds like you're dealing with an entire system of N N particles, rather than a single particle. Not sure if that's valid, but I think the real answer probably isn't far off.DumpsterDoofus –DumpsterDoofus 2014-05-09 02:25:25 +00:00 Commented May 9, 2014 at 2:25 @user44840: In other words, my answer above shows that N U¯¯¯¯=U N U¯=U.DumpsterDoofus –DumpsterDoofus 2014-05-09 02:33:19 +00:00 Commented May 9, 2014 at 2:33 @DumpsterDoofus If I understand correctly, the index p p refers to single particle states, but Z Z should be a sum over all N N-particle states. I guess one should replace Z→Z N 1 Z→Z 1 N where Z 1 Z 1 is the single-particle partition function. This produces under the log the factor N N. But it is only valid for non-interacting systems. Not 100% sure though, feeling a little rusty myself.Nephente –Nephente 2014-05-09 07:30:07 +00:00 Commented May 9, 2014 at 7:30 Add a comment| This answer is useful 0 Save this answer. Show activity on this post. This is what I've learnt. According to conversation of energy... U=∑i P i ε i U=∑i P i ε i ...where P i=1 Z e x p(β ε i)P i=1 Z e x p(β ε i) and β=−1 k B T β=−1 k B T Put it all together and they give us... U=1 Z∑i ε i e x p(β ε i)U=1 Z∑i ε i e x p(β ε i) Then there's the partial derivative expression for U: U=∂l o g(Z)∂β U=∂l o g(Z)∂β ...where Z=∑i e x p(β ε i)Z=∑i e x p(β ε i) U=∂l o g(Z)∂β=U=∂l o g(Z)∂Z∂Z∂β=1 Z∑i ε i e x p(β ε i)U=∂l o g(Z)∂β=U=∂l o g(Z)∂Z∂Z∂β=1 Z∑i ε i e x p(β ε i) Anyway, that's just plain old derivation. I hope it helps clear some things up. To be honest, I'm still trying to understand the underlying physical implications. For me it's a case of failing to grasp the semantics of the math. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Improve this answer Follow Follow this answer to receive notifications answered May 9, 2014 at 1:36 FractualHallEffectFractualHallEffect 11 2 2 bronze badges 1 U=∑i P i ϵ i=U¯U=∑i P i ϵ i=U¯ is the average energy, not the total.user44840 –user44840 2014-05-09 01:59:30 +00:00 Commented May 9, 2014 at 1:59 Add a comment| Your Answer Thanks for contributing an answer to Physics Stack Exchange! Please be sure to answer the question. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. Making statements based on opinion; back them up with references or personal experience. Use MathJax to format equations. MathJax reference. To learn more, see our tips on writing great answers. 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5213
https://siepr.stanford.edu/publications/policy-brief/who-most-affected-inflation-consider-source
Skip to secondary navigation Stanford University (link is external) Institute for Economic Policy Research (SIEPR) Who is most affected by inflation? Consider the source Key Takeaways The impact of inflation depends on what’s causing it. Inflationary oil supply shocks tend to hurt the least affluent by more than the most affluent. Inflationary monetary shocks do the opposite: They hurt the most affluent more than the least affluent. This discrepancy is largely driven by the different response of asset prices: Monetary policy raises home and stock prices, which hurts those buying houses, while oil shocks do the opposite. For the first time in years, inflation has surged across the world. In the U.S. and Europe, consumer prices grew in 2022 by almost 9 percent after years of inflation rates around 2 percent or less. This surge has reignited interest in a longstanding question: Who is most hurt by increasing prices? There are two reasons why this seemingly simple question may be difficult to answer. First, inflation can arise from different sources. For instance, conventional wisdom says the inflation spike during the 1970s was caused by a rapid increase in the price of oil—an aggregate supply contraction. The subsequent disinflation during the 1980s is often attributed to hawkish monetary policy and specifically Paul Volcker’s decision to rapidly raise nominal interest rates—an aggregate demand shock. Inflationary episodes driven by aggregate supply and aggregate demand shocks need not produce the same winners and losers. A monetary expansion may be extremely different from an oil supply contraction, for instance. The second challenge is that the drivers of inflation affect more than just prices of goods and services. Both oil supply and monetary policy also affect unemployment, wages, and asset prices, which likely affect different households quite differently. It may be important, therefore, to consider movements in both prices and income when assessing whether inflation differentially affects the most or least affluent. Why might inflation affect different households differently? The classic view in macroeconomics is that inflation transfers resources from so-called net nominal savers to net nominal borrowers. To understand this, suppose that Alice lent Bob $100 at an interest rate of 5 percent. Bob then needs to pay Alice back $105 the next year. Bob is a nominal borrower and Alice is a nominal saver. If prices do not rise over the course of this year, then Bob’s repayment is worth 5 percent more goods and services than the amount he borrowed. However, if prices rise by 10 percent over this year, then Bob’s repayment buys fewer goods and services than his original $100 debt allowed him to purchase. In this sense, the inflation has made Bob richer and Alice poorer. Doepke and Schneider (2006) studied the scale of this redistribution and found that the main losers from inflation are old, rich households—the major bondholders in the economy. In contrast, young, middle-class households are the largest winners from inflation in the U.S., because the real value of their substantial fixed-rate mortgage debt is eroded by inflation. Focusing solely on this channel, inflation has often been considered to be a progressive force—it transfers resources from the wealthiest to borrowers. This logic also holds for governments, firms, and foreigners. Since the U.S. government issues nominal bonds (i.e., borrows) to finance deficit spending, inflation reduces the real value of what they owe. Much of U.S. government debt is held by foreigners and rich Americans, so this is a force for inflation to redistribute real resources from foreigners and rich Americans to the U.S. government. Of course, this is just one way in which inflation affects households. Most directly, inflation affects the price of goods and services that households purchase. If inflation tends to disproportionately affect the prices of goods that the least well-off households consume, then inflation could in principle be regressive. As a concrete example: If aggregate inflationary periods are accompanied by spikes in the prices of gasoline, households that spend a larger share of their income on fuel— which are largely less affluent households—will be more affected by inflation. Finally, household income may also respond to inflation. Wages are often annually adjusted to keep up with inflation, while Social Security benefits are usually indexed to inflation. These movements in income will naturally offset whatever pain you suffer from paying higher prices. As an extreme example, if prices rise by 2 percent and household income also rises by 2 percent, the inflation will have no real effect, as households will be able to afford exactly as many goods and services as they could without the inflation. Who is most affected by inflationary shocks? To properly assess the winners and losers from inflation, one needs to consider all of these effects —on prices, income, and wealth—on one scale. This is the goal of a recent paper I wrote with colleagues (del Canto et al., 2023). We write down a simple economic theory of a household choosing consumption of a variety of goods, supplying labor, and investing in a number of assets. We then consider how the household’s well-being changes when some shock affects the prices the household faces. This exercise shows that the impact of this shock on household well-being is summarized by a simple statistic: Did the price of the goods that the household consumes go up by more than its income in present value terms? Assessing the effect of inflationary shocks on household well-being therefore requires two inputs. First, one needs estimates of the effect of the shock on the prices of a variety of goods and assets, as well as its effects on income. Fortunately, there is a large existing toolbox of techniques precisely designed to estimate such effects. Second, one needs to measure consumption of a variety of goods by households, the assets they invest in, and the salaries they receive. Such measures can be produced from a variety of cross-sectional surveys, such as the Consumer Expenditure Survey, the Survey of Consumer Finances, and the Current Population Survey. Our paper studies the distributional effects of two major drivers of short-run inflation fluctuations—oil supply and monetary policy shocks. We estimate the impact of these shocks using a standard Structural Vector Autoregression (SVAR) approach, where we isolate surprises in oil supply (monetary policy) using high-frequency movements in oil prices (treasury yields) around OPEC supply announcements (FOMC meetings). Figure 1 shows the total losses to various households arising from inflations tied to a 10 percent increase in the price of oil (Panel A) and a 25 basis point cut in nominal interest rates (Panel B). The horizontal axis plots the age of the household head, while the three lines indicate three education groups—high school or less (dark blue), some college (light blue), or at least a bachelor’s degree (red). Figure 1 – Welfare Losses from Inflationary Shocks Figure 1 shows that inflationary monetary and oil supply shocks have vastly different distributional consequences. This is despite both shocks being scaled so that they generate the same aggregate inflation. The figure shows that when inflation is driven by the Fed unexpectedly cutting interest rates, young and middle-aged college-educated households lose the most, while older and less-educated households are largely unaffected or even benefit. This is in sharp contrast with what happens when inflation is driven by a jump in oil prices. Such inflations lead to welfare losses, which are largest for younger, less-educated households and for retirement-age college-educated households. Meanwhile, young and middle-aged college-educated households actually benefit from the inflationary oil shock. Quantitatively, a 10 percent jump in oil prices means less-educated households must be paid around 0.5 percent of their quarterly consumption to afford their no-shock choices, while college-educated households would be willing to pay up to 0.25 percent of quarterly consumption to have the shock. Figure 2 – Channels of Welfare Losses What drives these patterns? Figure 2 decomposes the welfare losses into components coming from consumption prices, movements in labor income, movements from asset prices and dividends, and shifts in government transfer income such as Social Security benefits. Panel A presents the effects arising from an inflationary monetary policy shock. The top left figure shows that all households are about equally hurt from rising prices of consumption after a monetary shock. This is ultimately because differences in consumption bundles across households are small: Even though gas prices do rise more than the price of, say, clothes after a monetary inflation, motor fuel occupies a similar share—around 4 to 6 percent—of consumption for all household groups. Paychecks, likewise, are not the primary driver of the distributional consequences of monetary inflation—all three education groups see similar increases in pay (and thus negative losses) from monetary expansions, though the gains are perhaps slightly larger for the least educated. Labor income increases nearly exactly offset the losses households see from increased expenditures. The exception is for older non-working households, which do not see changes in earnings. These households, however, see increases in their transfer income—specifically, Social Security benefits— which offset the price increases. Monetary inflations therefore principally have distributional effects because of differences in households’ asset portfolio decisions. This can be seen by the fact that the portfolio channel in the bottom left mirrors the total welfare effect from Figure 1. Monetary policy has a couple of key effects on asset prices. A cut in interest rates pushes up the stock market and house prices. These higher prices benefit households that are selling stocks and houses, but hurt those buying stocks and houses. Younger college-educated households are precisely those that buy houses and equities. Meanwhile, older households that may be selling assets from their retirement accounts or downsizing their home benefit from these high asset prices. These effects are partially offset by a decline in mortgage rates and the fact that dividends on the S&P500 fall, which hurts older households holding a lot of equities. However, these dividend and mortgage rate effects turn out to be smaller than the effect from asset accumulation. Compared to monetary shocks, inflationary oil shocks have an almost opposite impact. First, oil price spikes are slightly more regressive on the consumption price side, mostly because oil price spikes have big impacts on the price of motor fuel and utilities, both of which occupy a larger share of less-educated households’ consumption. Labor income falls after a contractionary oil shock, as unemployment rates rise, but transfer income—which is largely indexed to inflation—still rises for older households. But the major difference between monetary and oil shocks arises because of their different effects on asset prices. Oil shocks tend to hurt the stock market and have limited effects on mortgage rates or housing markets. This generates a portfolio effect on households that is the opposite to that of a monetary inflation. This is the major reason inflationary oil shocks are regressive while monetary inflations are progressive. Policy considerations Inflations driven by oil supply and monetary shocks have historically had opposite distributional consequences—oil supply shocks most hurt the least affluent while inflationary monetary policy most hurts the most affluent. An implication of this is that disinflationary monetary shocks—an increase in interest rates—likely have the same distributional consequences as inflationary oil shocks. This could present a challenge for policymakers: If the Fed responds to inflation driven by reductions in global oil supply by raising interest rates, that could exacerbate the regressivity of the initial oil shock. It should be noted that this conclusion requires some more research since we are only able to estimate the effects of a surprise monetary shock rather than an anticipated policy rule. Regardless, there is no simple answer to the question: “Is inflation regressive?” About the Author John Grigsby is a Visiting Fellow at SIEPR. He is an Assistant Professor in the Department of Economics and School of Public and International Affairs at Princeton University. He is an empirical macroeconomist studying a broad set of questions related to wage and unemployment dynamics, the drivers of historical innovation, the functioning of mortgage markets and the distributional consequences of inflation. Footnote Consumer Price Index for All Urban Consumers: All Items in U.S. City Average and Inflation, consumer prices for the Euro Area This conclusion would be different in countries where most mortgage debt is subject to an adjustable interest rate, as is the case in many European countries and the UK. Note that all monetary effects are estimated based on surprises to the policy rate and should not be interpreted as the effects of changing the policy rule. References del Canto, Felipe, John Grigsby, Eric Qian, and Conor Walsh. “Are Inflationary Shocks Regressive? A Feasible Set Approach.” NBER Working Paper #31124 (2023). Doepke, Matthias, and Martin Schneider. “Inflation and the Redistribution of Nominal Wealth.” Journal of Political Economy (2006). 114(6). Author(s) John Grigsby Publication Date March, 2024 View this Policy Brief Related Topics Policy Brief More Publications Publication ## The Appropriate Design of Collective Bargaining Systems: Learning from the Experience of Britain, Australia, and New Zealand Working Paper Publication ## Faculty Retirement Incentives by Colleges and Universities Working Paper Publication ## Heterogeneity in Financial Incentives for High and Low Income School Districts Working Paper See All Publications
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Synonyms of delay a : the act of postponing, hindering, or causing something to occur more slowly than normal : the state of being delayed get started without delay b : an instance of being delayed apologized for the delay a rain delay : the time during which something is delayed waited out a delay of 30 minutes delayed; delaying; delays transitive verb 1 : put off, postpone delay a departure They're delaying marriage or, increasingly, not getting married at all …—Irin Carmon 2 : to stop, detain, or hinder for a time The mails were delayed by heavy snows. … issued executive orders delaying the release of records from Ronald Reagan's administration …—Editor & Publisher 3 : to cause to be slower or to occur more slowly than normal delay a child's development … a drug that not only can extend life by delaying the onset of aging-related diseases …—Bill Gifford intransitive verb : to move or act slowly This offer ends soon, so don't delay. delayed in responding to my message also : to cause delay delayer noun Synonyms Noun wait Verb linger drag crawl poke lag creep See All Synonyms & Antonyms in Thesaurus Choose the Right Synonym for delay delay, retard, slow, slacken, detain mean to cause to be late or behind in movement or progress. delay implies a holding back, usually by interference, from completion or arrival. bad weather delayed our arrival retard suggests reduction of speed without actual stopping. treatment that retards tumor growth slow and slacken also imply a reduction of speed, slow often suggesting deliberate intention she closed her eyes and slowed her breathing , slacken an easing up or relaxing of power or effort. on hot days runners slacken their pace detain implies a holding back beyond a reasonable or appointed time. unexpected business had detained her delay, procrastinate, lag, loiter, dawdle, dally mean to move or act slowly so as to fall behind. delay usually implies a putting off of something (such as a beginning or departure). we cannot delay any longer procrastinate implies blameworthy delay especially through laziness or apathy. procrastinates about making decisions lag implies failure to maintain a speed set by others. lagging behind in technology loiter and dawdle imply delay while in progress, especially in walking, but dawdle more clearly suggests an aimless wasting of time. loitered at several store windows children dawdling on their way home from school dally suggests delay through trifling or vacillation when promptness is necessary. stop dallying and get to work Examples of delay in a Sentence Noun Do you know what's causing the delay? a number of flight delays After months of delay, construction on the new school began. Airline travelers are experiencing delays of up to three hours. Verb The doctor wants to delay surgery for a few weeks. She's planning to delay her retirement. He delayed too long, and now it's too late. “Don't delay! Sale ends Saturday.” Production problems delayed the introduction of the new model by several months. Recent Examples on the Web Examples are automatically compiled from online sources to show current usage. Read More Opinions expressed in the examples do not represent those of Merriam-Webster or its editors. Send us feedback. Noun The story of the day will largely be that weather delay and the seemingly avoidable circumstances around it. —Daniel Sperry, Kansas City Star, 17 Aug. 2025 The manager keeps coming up with new delays and excuses why this basic amenity has been ignored. —Ticked Off, The Orlando Sentinel, 17 Aug. 2025 Verb Financial Narrative Control: In one case that stood out, a retailer’s communications team, reporting to the CFO, delayed announcing a new technology partnership for a year to better align with the company's financial narrative. —Kiri Masters, Forbes.com, 18 Aug. 2025 Thunderstorms delayed Thursday's opening round, softening the greens and testing the pros. —Devlina Sarkar, MSNBC Newsweek, 18 Aug. 2025 See All Example Sentences for delay Word History Etymology Verb and Noun Middle English, from Anglo-French delaier, from de- + laier to leave, from lai-, present and future stem of lesser, laisser to leave, from Latin laxare to slacken, from laxus loose — more at slack First Known Use Noun 14th century, in the meaning defined at sense 1a Verb 14th century, in the meaning defined at transitive sense 1 Time Traveler The first known use of delay was in the 14th century See more words from the same century Phrases Containing delay delay tactic without delay Rhymes for delay airway allay archway array ashtray astray aue away ballet belay beltway beret See All Rhymes for delay Browse Nearby Words De La Warr delay DeLay See all Nearby Words Cite this Entry “Delay.” Merriam-Webster.com Dictionary, Merriam-Webster, Accessed 28 Aug. 2025. Copy Citation Share Kids Definition delay 1 of 2 noun de·​lay di-ˈlā 1 : the act of delaying : the state of being delayed start without delay 2 : the time during which something is delayed a delay of 30 minutes delay 2 of 2 verb 1 : postpone, put off delay a trip 2 : to stop, detain, or hinder for a time delayed by a storm 3 : to move or act slowly delayer noun Biographical Definition DeLay biographical name De·​Lay di-ˈlā Thomas (Dale) 1947–     American politician More from Merriam-Webster on delay Nglish: Translation of delay for Spanish Speakers Last Updated: - Updated example sentences Love words? Need even more definitions? Subscribe to America's largest dictionary and get thousands more definitions and advanced search—ad free! Merriam-Webster unabridged More from Merriam-Webster ### Can you solve 4 words at once? Can you solve 4 words at once? Word of the Day diminution See Definitions and Examples » Get Word of the Day daily email! Popular in Grammar & Usage See More ### 31 Useful Rhetorical Devices ### Merriam-Webster’s Great Big List of Words You Love to Hate ### How to Use Em Dashes (—), En Dashes (–) , and Hyphens (-) ### The Difference Between 'i.e.' and 'e.g.' ### Why is '-ed' sometimes pronounced at the end of a word? See More Popular in Wordplay See More ### Our Best Historical Slang Terms ### Even More Bird Names that Sound Like Insults (and Sometimes Are) ### Words That Turned 100 in 2025 ### 'Za' and 9 Other Words to Help You Win at SCRABBLE ### 12 Words Whose History Will Surprise You See More Popular See More ### 31 Useful Rhetorical Devices ### Our Best Historical Slang Terms ### Even More Bird Names that Sound Like Insults (and Sometimes Are) See More
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The Role Polysemy can Play in Evoking Action: the Case of "recovery" in Economic Discourse Citation Petrella, Joyce Popovich. 2011. The Role Polysemy can Play in Evoking Action: the Case of "recovery" in Economic Discourse. Master's thesis, Harvard University, Extension School. Link Terms of use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material (LAA), as set forth at Accessibility Share Your Story The Harvard community has made this article openly available. Please share how this access benefits you. Submit a story The Role Polysemy can Play in Evoking Action: the Case of “ recovery” in Economic Discourse Joyce P. Petrella A Thesis in the Field of Linguistics for the Degree of Master of Liberal Arts in Extension Studies Harvard University November 2011 Copyright © 2011 Joyce Petrella Abstract This thesis investigates a specific case of polysemy, the word “ recovery,” as it occurs in economic discourse and the role it can play in evoking responses in a speech community. The analysis explores the following set of questions: what are the linguistic dynamics underlying the marked emergence of this metaphor during the 2008 financial crisis? If the metaphor emerged during prior financial crises, what role did it play? Using both statistical and discourse analysis techniques, the study traces the attestations of “ recovery ” and its broader conceptual metaphor THE ECONOMY IS A PATIENT over four centuries. By applying the tenets of conceptual metaphor theory, the analysis reveals that THE ECONOMY IS A PATIENT metaphor carries significant cognitive freight via its conceptual blend attributes. Over time, these attributes are reactivated through economic and cultural narratives prompted by the stress of the financial environment. The cognitive freight conveys a powerful ideological and emotive force which can call on the speech community to take political, social, and moral action. The study concludes that, while often overlooked, diachronic analysis can be a powerful linguistic tool to study the cognitive as well as rhetorical and ideological dynamics underlying polysemy in a given culture. Biographical Sketch Joyce Petrella is a graduate of Trinity College in Hartford, Connecticut where she earned a Bachelor of Science degree in Mathematics and English. Joyce started her professional career in the field of Actuarial Science at the Travelers Company, in Hartford, Connecticut. She later joined Aetna Life & Casualty, achieving increasing roles in executive development and organizational renewal. It was during this time, as Aetna transformed itself into a leading health care company, that she began her interest in the interplay between the written and spoken word and the actions taken in financial settings. Most recently, during the 2008 global financial crisis, she has been working with the senior leadership at The Hartford, a financial services company, during their transformation. Her personal experience in analyzing narratives emerging during times of economic strain led to the motivation for this thesis. iv Dedication I wish to dedicate this work to my family: husband, Mario, and two sons, Michael and David, for their unrelenting support and patience as I researched and crafted versions of this thesis during evenings, weekends, and vacations. I also make a special dedication to my father, Michael Popovich, for his inspiration, enduring wisdom, and continual encouragement in every goal I attempt. v Acknowledgments I wish to acknowledge Steve Caton of Harvard University for his generous academic advice and direction in preparing this thesis; Lori Gensler, for her careful and meticulous proofreading; and Dierdre McCloskey, for seminal research in economic rhetoric and unwitting inspiration for this thesis. the chemist or economist must start with some interesting gunk in a test tube or some story about how a particular economy has developed - that is to say, with conceptions on which she has a tacit, experiential, diachronic grasp. The experience (in literary terms, the narrative, or in novelistic terms, perhaps the dialogue) is the phenomenon to be theorized about. You have to have a direct grasp of the diachronic subject to have something to be synchronic about. 1 1Deirdre N. McCloskey, The Rhetoric of Economics , 2 nd ed. (Madison: The University of Wisconsin Press, 1998) 31. vi Table of Contents Biographical Sketch ……………………………………………………………………iv Dedication …………………………………………………………………………..…...v Acknowledgments ………..………………………………………………………..…..vi List of Tables …………………………………………………………………………...ix List of Figures/Graphs ……………………………………………………………..…..x I. Introduction …………………………………………………………………..……...1 Background: Metaphor and Polysemy in Economic Discourse ………..….1 Purpose and Significance of the Study ………………………………..……..3 Research Questions and Hypothesis ……………..…………………...….....5 Research Methodology …..……………………………………..…….6 II. Research Review ……..……………………………………………………...……9 Economic Metaphors …………………………………………………….….…9 The Economists’ Views of Metaphors: Rhetoric in Economic Discourse ……………………………………………...……………….11 Linguistic Analyses of Metaphor ………..…………………………………...13 Socio-cultural Analyses of Metaphor …………………………………….....28 Friedrich’s Analytic Model for Revealing Ideologies in Language..38 Joining Disciplines to Analyze HEALTH-WEALTH Metaphor Domains ...40 III. Limitations of Synchronic Analysis …………………..………………………...48 IV. Research Findings ……………………………………………………..…….….52 Organization of Research Findings ……...………………………….……...52 vii Findings Summary ……………………………….……………….…..53 The Twenty-first and Twentieth Centuries – The 2008 Economic Crisis and the Market Crash of 1929 ………………………………………..……..55 Nineteenth – late Eighteenth-Century Contexts ………...……………......66 Eighteenth-and-Seventeenth-Century Contexts …...…………………..…78 Etymological Review ……………..………………………..........................101 V. Discussion and Implications of the Findings …..……..……………………...108 VI. Conclusion ………………………..……………………….………………...….118 Appendix A Terminology……………..………………………….………………….120 Appendix B Conceptual blending network for interaction of “ recovery ” in economic discourse .........................................……………………123 Appendix C Instances of Metaphor ……………………………………………….124 Bibliography ……....……………………………………………………..…………...125 viii List of Tables Table 1 Instances of Economic Metaphor ……………...………………………124 ix x List of Figures/Graphs Fig. 1 Conceptual Blending Basic Diagram ….……………………..……………..19 Fig. 2 COCA occurrences of “ recover ” over time ….……………………..……….58 Fig. 3 Google Trends data for “ economic recovery” …………….…………..…….59 Fig. 4 Google Trends data for “ financial recovery” ……………….………..……...60 Fig. 5 COHA data for collocated occurrences of “ recovery ” and “ economy ”…....62 Fig. 6 Conceptual blending network for interaction of “ recovery ” in economic discourse ..……………………………………..………………………………123 1 Chapter I Introduction As linguistic expressions, metaphors are regularly found in economic discourse. Economic theory is a highly conceptual construct so it is not surprising to find metaphors used to convey the concepts in this domain. “People tend to invoke metaphors most commonly when trying to understand or comprehend things which are remote or abstract,” states Keith Shimko. 2 In economic texts, anthropomorphic metaphors are particularly prominent. Background: Metaphor and Polysemy in Economic Discourse A variety of metaphors applying human attributes are often noted in economic discourse. In fact, during the financial crisis of 2008, a specific anthropomorphic metaphor, THE ECONOMY IS A PATIENT, emerged with particular prominence. Observe The Economist’ s depiction of the language used by Ben Bernanke to describe the US economy in the fall of 2008 (italics are mine): American congressmen are used to hyperbole, but they were left speechless by the dire scenario Ben Bernanke, the chairman of the Federal Reserve, painted for them on the night of September 18 th . He ‘told us that our American economy’s arteries , our financial system is clogged , and if we don’t act, the 2Keith L. Shimko, “Metaphors and Foreign Policy Decision Making,” Political Psychology 15.4 (1994): 668. 2 patient will surely suffer a heart attack , maybe next week, maybe in six months, but it will happen,’ according to Charles Schumer, a Democratic senator from New York. 3 Similarly, the metaphor’s related conventional or dead metaphor “ recovery” and its verbalized form, “ recover,” were also pronounced both in specialized financial publications and in general public media. In fact, one of the US government’s interventions to “ save ” the “ ailing ” economy was the “ The American Recovery and Reinvestment Act of 2009. ” 4 “Recovery ” is a specific and robust case of polysemy taking meanings from health as well as economic domains. The multiple senses for “ recovery ” in economic discourse provide a unique opportunity to consider what conceptual category is at play and what impact it may have on the speech community. Lakoff and Johnson’s studies highlight the impact metaphors can have in society writing, “Metaphor may create realities for us, especially social realities. A metaphor may thus be a guide for future action. Such actions will, of course, fit the metaphor.” 5 The case of THE ECONOMY IS A PATIENT is also at the nexus of Irvine and others’ explication of linguistic ideologies and the linguistic philosophies of Foucault, Derrida and their predecessors. Predicated on Woolard and 3“America’s Bail-out Plan: The Doctors’ Bill,” The Economist , 25 Sep 2008, 10 March 2010 < >. 4“Recovery.gov: Track the Money” 9 April 2011 . 5George Lakoff and Mark Johnson, Metaphors We Live By (Chicago: University of Chicago Press, 1980) 156. 3 Shieffelin’s belief that “cultural frames have social histories,” 6 the case study draws on the social contexts in which the metaphors emerge and is intended to deepen our “sometimes superficial understanding of linguistic form and its cultural variability in political economic studies of discourse.” 7 Stated more specifically, the linguistic forms within the metaphor THE ECONOMIC IS A PATIENT conceptual domain can illuminate our understanding of the embedded political and cultural ideologies of the community in which they are spoken. Purpose and Significance of the Study The study of polysemy is typically conducted through examination of lexical semantics in discourse at a specific point in time, that is, synchronically. However, as noted in the quotation by Dierdre McCloskey in the Acknowledgments, in order to understand the synchronic phenomenon of “recovery ” and the efficacy of its polysemy in contemporary economic discourse, the linguist would benefit from diachronic analysis, i.e., through time. If we are to create a theory of the phenomenon, “you have to have a direct grasp of the diachronic subject to have something synchronic about.” 8 This thesis shows that by neglecting a diachronic view, a rich amount of data is overlooked that could enhance our understanding of polysemy and its sociolinguistic imp act. This analysis exists at the intersection of pragmatics and conceptual 6Kathryn Woolard and Bambi Schieffelin, “Language Ideology,” Annual Review of Anthropology 23 (1994): 58. 7Woolard and Schieffelin 72. 8McCloskey (1998) 31 .4 metaphor theory (CMT). While neither addresses the mechanics of language’s cognitive and emotive power alone, together they elucidate the linguistic dynamics underlying each. The analysis described in this study fills a significant gap in current metaphor and polysemy research. Christopher Hart, in his Critical Discourse Analysis and Metaphor: Toward a Theoretical Framework , observes, “Metaphor… has been largely neglected in mainstream CDA (Critical Discourse Analysis).” 9 This study provides an opportunity to better understand the interplay between the two. In fact, Hart argues that “in attending to metaphor in CDA, cognitive linguistics is indeed the perfect tool.” 10 He continues, “Here we may make a Saussurian distinction between language (langue) and discourse (parole), where langue refers to a language system and parole refers to the use of that system for communicative purposes.” 11 Jonathan Charteris-Black, a linguist focusing on “the rhetorical motivation and influence of metaphor choice in discourse types that include political speeches, religious texts and other persuasive areas of language use,” 12 confirms that metaphor is “central to critical discourse analysis since it is concerned with forming a coherent view of reality.” 13 The tools of CMT and related themes such as embodiment, motiv ation 9Christopher Hart, “Critical Discourse Analysis and Metaphor: Toward a Theoretical Framework,” Critical Discourse Studies 5.2 (2008): 2. 10 Hart 3. 11 Hart 4. 12 Jonathan Charteris-Black, School of Humanities, Languages and Social Sciences , University of the West of England website 13 Jonathan Charteris-Black, Corpus Approaches to Critical Metaphor Analysis (Basingstoke & New York: Palgrave-MacMillan, 2004) 28. 5 and conceptual blending theory become paramount in this research. Using a variety of linguistic disciplines can reveal how secondary metaphors emerge what socio-cognitive influences the emergence of lexical items and how habitually used metaphors become polysemous. These conventional metapho which have become a part of our daily lexicon become embedded in the culture. Lakoff and Johnson contend that “primary metaphors are part of the cog unconscious.” ,rs nitive 14 It is the basic primary metaphors, based on our embodied experience, they argue is central to our speech and central to our interpretation. Just as importantly, when re-invoked in modern discourse, we will see what impact the primary metaphors turned polysemes can have on the speech community and what ideologies are revealed about the culture and the speakers. It is my hope that the insights gained from this analysis offer additional phenomenological data to the current body of work in metaphor study and further promote the diachronic approach. Research Questions and Hypothesis The research for this thesis is based on an in-depth analysis of the metaphor, THE ECONOMY IS A PATIENT, in economic texts. The case study explores the polyseme “ recover ” and its nominalized form “ recovery ”diachronically over its long history of use. The examination probes whether polysemeous words play an ideological role in discourse. The case explores questions such as: What are the contexts under which the metaphor is found? 14 George Lakoff and Mark Johnson, Philosophy in the Flesh (New York: Basic Books, 1999) 56. 6 What are the linguistic dynamics underlying the marked emergence of this metaphor during the 2008 financial crisis? What role, if any, has the polysemy of “recovery” in the health and economic domains played in the metaphor’s efficacy? What can the conceptual categories of the metaphor reveal about the linguistic and economic ideologies? What can this case reveal about the connection between metaphor and “thought and action” overall? My hypothesis is that the metaphor THE ECONOMY IS A PATIENT is found in historical texts during times of economic and political stress and that the metaphor is cognitively significant. Further, using CMT analysis we find that the metaphor reflects conceptual categories that reveal semantic “baggage” from its earlier use. These linguistic dynamics heighten the metaphor’s emotional impact which can have a powerful influence on the speech community during economic pressure. From this analysis, I believe we can discern that the metaphor has a strong illocutionary force. In fact, the metaphor’s use in these contexts reveals the set of ideologies at work within the cultural environment in which it is spoken. The ideologies embedded in the language exert influence on the speech community to act. This analysis identifies some of the underlying linguistic dynamics why this may be so. Research Methodology My research method consists of using a diachronic approach to linguistic analysis to analyze THE ECONOMY AS A PATIENT metaphor with a focused examination of the conventional metaphors, “ recover” and “ recovery.” The 7 analysis centers on occurrences of “ recover” and “ recovery” as they are found in contemporary and historical primary sources . The examination traces the etymology of these dead metaphors from early attestations to their contemporary use during the 2008 financial crisis. The study considers the social and linguistic contexts in which they emerged. The case study method enables us to reveal detailed components of a linguistic phenomenon in a variety of natural settings, events and conditions. This qualitative research method will refine prior theoretical frameworks deduced from quantitative research. Robert K. Yin, a researcher and author of Case Study Research: Design and Methods , defends the case study method as an “essential form of social science inquiry” 15 and defines the method as an effective approach to investigating “a contemporary phenomenon within its real-life context; when the boundaries between phenomenon and context are not clearly evident and, in which multiple sources of evidence are used.” 16 As such, the research outlined in this thesis is intended to elucidate linguistic phenomena in a complex matrix of social and cognitive domains while maintaining analytical rigor. The analysis draws from the theoretical bases of Conceptual Metaphor Theory, Critical Discourse Analysis and philosophy of language. By juxtaposing the contemporary metaphors against their use historically, the analysis from these disciplines reveals how THE ECONOMY AS A PATIENT metaphor 15 Robert K. Yin, Case Study Research: Design and Methods (Newbury Park: Sage, 1984) xi. 16 Yin 23. 8 became effective in economic discourse. More broadly, this method provides us with a deeper understanding of the way in which polysemy can spur action, thus illustrating Lakoff and Johnson’s “thought and action” theory at work. In addition, this research approach provides empirical examples of Friedrich’s notion of linking social culture, political ideology with linguistic forms. 9 Chapter II Research Review The primary body of research in metaphors in economic discourse has been conducted by economists, linguists, rhetoricians and philosophers of language. In most cases, the research has centered on the metaphor’s efficacy in conveying abstractions in economics theory and describing financial market phenomena. While the groups agree metaphors are linguistically powerful, their analysis derives from different perspectives. To begin this research review, let us consider what economic metaphors are observed in discourse and where “recovery ” fits within the corpora. Economic Metaphors Regardless of Adam Smith’s seminal metaphor of the “ invisible hand, ”many other economic metaphors are more prominent. Corpora analysis research conducted by Jonathan Charteris-Black, Andreas Musolff, Willie Henderson, Frank Boers and Michael White, all linguists concerned with metaphors in economic texts, identify a number of highly recurring metaphors. For example, in his study of the metaphor and polyseme, “ growth, ” 17 Michael White identified the general metaphor THE ECONOMY IS A LIVING ORGANISM as the primary sense for “ growth ” in the conceptual domain of living things. 17 Michael White, “Metaphor and Economics: the Case of growth, ”English for Specific Purposes 22 (2003): 131-151. 10 Subordinate extensions of the metaphor, THE ECONOMY IS A PLANT, THE ECONOMY IS AN ANIMAL and THE ECONOMY IS A HUMAN are highly attested. Other metaphors in these studies include: THE ECONOMY IS A MECHANICAL PROCESS, THE ECONOMY IS AN AUCTION, THE ECONOMY IS A LIQUID, THE ECONOMY IS A PATH, and THE ECONOMY IS A BUBBLE, THE ECONOMY IS A BALL, THE ECONOMY IS A FUN FAIR RIDE. Relevant to this thesis is Charteris-Black’s analysis of economic metaphors in his 2000 article, “Metaphor and Vocabulary Teaching in ESP Economics.” 18 Charteris-Black portrays the high frequency of THE ECONOMY IS AN ORGANISM found in “The Economist” and other U.K. magazines from January, 1995 to September 1997. The table is reproduced in Table 1 in Appendix C. The data in Table 1 reveal the high level of productivity of THE ECONOMY IS AN ORGANISM in generating discrete extensions in the health and illness domains. Charteris-Black’s analysis of these findings concludes that THE ECONOMY IS A PATIENT is an extended metaphor and relates to the “cyclical stages of health and sickness.” 19 He notes the polysemes in this domain, “ grow ” and “ recover, ” are the highest and second highest in frequency within the corpus. In a later study exploring THE ECONOMY IS A PATIENT, Charteris-Black and fellow linguist Andreas Musolff show the metaphor can also be found in German economic publications. In fact, the polyseme “ recover ” is 18 Jonathan Charteris-Black, “Metaphor and Vocabulary Teaching in ESP Economics,” English for Specific Purposes 19 (2000): 149-165. 19 Charteris-Black, Metaphor 156. 11 attested as “ erholen ” or “ Erholung. ” 20 Similarly, Kosei Fukuda finds the polyseme, “ kaifuku, ” 21 “recover, ” in Japanese economics texts. There are a myriad of additional polysemes in the economic domain. Expert and lay speakers utter them easily in everyday speech. Consider: “bank, ” “ check, ” “ market, ” “ trust, ” “ flow, ” as well as the word “ economy ” itself. Clearly, polysemes find economic discourse a congenial environment. The Economists’ Views of Metaphors: Rhetoric in Economic Discourse Economists tend to focus their research on the rhetorical and instructional use of metaphors as means to convey economic theory. Alfred Marshall’s influential work in 1898 raised the awareness of mechanical and biological analogies used to describe many economic constructs. 22 He noted these metaphors are useful and effective in the pedagogy of economic theory. More recently, economist Deirdre McCloskey’s writings explore the power of language in economic argumentation. Noting that “linguistics is an appropriate model for economic science,” 23 she surveyed the similarities between metaphor use in the sciences and those observed in economic treatises. She writes that economists are unselfconscious, that is, unaware of their use of metaphor and yet “no 20 Jonathan Charteris-Black and Andreas Musolff, “’Battered hero’ or ‘innocent victim’? A Comparative Study of Metaphors for Euro Trading in British and German Financial Reporting,” English for Specific Purposes 22 (2003): 153-176. 21 Kosei Fukuda, “A Comparative Study of Metaphors Represent the U.S. and Japanese Economies, ” Journal of Pragmatics 41 (2009): 1693-1702. 22 Alfred Marshall, “Mechanical and Biological Analogies in Economics (1898),” A.Pigou Memorials of Alfred Marshall (1925): 312-318. 23 McCloskey (1998) 31. 12 economist could speak without metaphor.” 24 Her 1995 article, “Metaphors Economists Live By,” cited many metaphorical allusions to war, sports (e.g., “competing,” “beating,” “battle”), organisms and mechanisms (e.g., calculators) used as rhetorical devices in economics. “Adam Smith,” she writes, “knew at the beginning of economics that an economy was illuminated by a metaphor of speaking,” noting that Smith began his career as a teacher of rhetoric. 25 As rhetorical devices, metaphors are not only pervasive in economic discourse but also highly persuasive in conveying ideologies. In fact, McCloskey argues, Smith connected this propensity for rhetoric in economic metaphors to persuasion. Quoting Mark Perlman in a 1978 article in the Journal of Economic Literature , McCloskey confirms “economists are not experts; they are basically persuaders.” 26 Research in the rhetorical power of economic metaphors bridges the gap between science and political will. Kenneth Burke, in his work on rhetoric and persuasion, argues that economists commit “rhetoric of casuistry” under the auspices of scientific truths and do so with the best of rhetoricians: historians, sociologists and the like. 27 If this is so, the metaphors in use have significant power to persuade and the power to enact. McCloskey agrees, “We believe and act on what persuades us.” 28 24 Deirdre N. McCloskey (1998) 79. 25 Deirdre N. McCloskey, "Metaphors Economists Live By," Social Research 62.2 (1995): 215-237. 26 Mark Perlman, “Review of Hutchison’s Knowledge and Ignorance in Economics,” Journal of Economic Literature 16 (June 1978): 582. 27 Kenneth Burke, Rhetoric of Motives (Berkeley: University of California Press, 1969) 73. 28 McCloskey (1998) 179. 13 McCloskey’s observations about the use of metaphor in economic discourse are substantiated by linguistic research. For example, in a quantitative analysis of economic metaphors in financial and business journals, linguists Hanna Skorczynska and Alice Deignan found that the choice of metaphors is influenced by the text’s intended readership and its purpose. 29 Their analysis reveals that while perhaps “unselfconscious,” there is much more to the use of metaphor than mere convention. Skorszynska and Deignan suggest that metaphor selection is driven by not only social contexts but also the purpose of a text. Their study corroborates the view that metaphor choices are neither merely enhancing nor are they arbitrary. They most likely are linked to the rhetorical function. Linguistic Analyses of Metaphor In the linguistic discipline of English for Specific Purposes (ESP), economists and linguists alike analyze the use of language as it pertains to instruction. Typically, the focus of their work is to determine which pedagogical means are most effective. They analyze what techniques facilitate teaching English to foreign speakers who are preparing for careers in English-speaking countries. In this view, metaphors are useful aids in conveying complex and abstract concepts. The utility of metaphors in teaching results in establishing a definitive economics lexicon. The metaphors themselves become a natural and habitual part of the economics discourse. As the metaphors become highly 29 Hanna Skorczynska and Alice Deignan, “Readership and Purpose in the Choice of Economics Metaphors,” Metaphor and Symbol 21.2 (2006): 97. 14 ingrained in discourse through praxis, hidden ideologies become entrenched. As a result, the ideology is so deeply embedded in the culture that the economic concepts are perceived as “givens” or truths. These truths are rarely questioned and are highly persuasive. Researchers in ESP such as Jonathan Charteris-Black warn instructors to be careful with the power they hold over their students and to be conscious of the ideology they convey as they teach. He warns the students to stay alert as well. He writes, “The pragmatic approach to metaphor highlights the rhetorical importance of metaphors because they influence opinions. It is important for ESP learners to be aware of the cognitive and pragmatics differences in the purposes to which metaphors can be put.” 30 This power of linguistic “performativity,” as Austin outlined, and its ability to enact states of reality is at play here. The metaphors uttered in economics classrooms and the idealized view of the economic world become fixed in economic principles. As the metaphors are habitually reinforced through continual learning, the related ideologies become even more persuasive and deeply embedded in the culture. We will return to this topic again as we analyze the history of THE ECONOMY IS A PATIENT metaphor and the ideologies carried over time. With the advent of cognitive linguistics and the seminal work of George Lakoff and Mark Johnson, metaphors are seen as a window into the nature of language and human cognition. In their book, Metaphors We Live By , Lakoff and Johnson introduce the conceptual metaphor theory (CMT) acknowledging that metaphors are “pervasive in everyday life, not just in language but in thought and 30 Jonathan Charteris-Black and Andreas Musolff (2003): 153. 15 action.” 31 Metaphors are ubiquitous in virtually all areas of human discourse and, in this view, play a fundamental role in how we envision and operate in the world. Lakoff and Johnson’s work has advanced research in the area of cognitive semantics and generated a set of theoretical frameworks and analytical tools to study language. Fundamental to CMT is the premise that “our bodily experience and the way we use imaginative mechanisms are central to how we construct categories to make sense of experience.” 32 These mechanisms include metaphors and their related linguistic counterparts such as metonymy and polysemy. They “structure thought and are used in forming categories and in reasoning.” 33 In particular, polysemy “arises from the fact that there are systematic relationships between different cognitive models and between elements of the same model. The same word is often used for elements that stand in such cognitive relations to one another.” 34 Research has revealed that these systematic relationships or categories are, more often than not, automatic and unconscious. They are learned early in language acquisition and span topics from generic terms to emotions. They tie the conceptual to the experiential, the intangible to the concrete. In his analysis of polysemous expressions, Lakoff argues the categories representing multiple senses may not be predictable based on the traditional notion of “shared attributes” but are motivated by 31 George Lakoff and Mark Johnson, Metaphors We Live By (Chicago: University of Chicago Press, 1980) 3. 32 George Lakoff, Women, Fire, and Dangerous Things (Chicago: University of Chicago Press, 1987) xii. 33 Lakoff 13. 34 Lakoff 13. 16 metaphorical models in the minds of the speakers. The categories in this regard are not arbitrary. For cognitive linguists, the classic approach for analyzing metaphoric phenomena is based on the Conceptual Metaphor Theory (also referred to as Cognitive Metaphor Theory or CMT). CMT states that metaphor is primarily conceptual. Researchers in this discipline analyze the interaction between language, the mind, and culture. For example, Lakoff, Johnson, Fauconnier, Turner and others consider how metaphors may affect the speech community, motivate societal behavior, and spur action. Much of contemporary linguistic analysis, therefore, is based on the foundation that cognition and language are interrelated. Metaphor becomes more than a form of speech that is “not literal”. “In the cognitive linguistic view, metaphor is defined as understanding one conceptual domain in terms of another conceptual domain.” 35 It is a phenomenon of cognition that bridges the tangible (Lakoff calls “embodied”) experience with the abstract. The metaphor itself plays a cognitive as well as a linguistic role in bridging the two domains. As we have seen, the choices speakers make in selecting the domains can reveal much about not only their pragmatic intentions, but also their cognitive processes and cultural influences. In order to study the underlying dynamics of metaphors and polysemes in economics discourse, it is important to review the progress cognitive linguists are making in understanding the language-mind connection. Current research in cognitive linguistics has revealed growing evidence 35 Zoltán Kövecses, Metaphor: A Practical Introduction , 2 nd ed. (Oxford: Oxford University Press, 2010) 4. 17 suggesting metaphor use reflects humans’ innate ability to understand abstract concepts and experiences as well as communicate them. Fundamental to the Conceptual Metaphor Theory is the notion that metaphors establish links or maps between a “source” domain and “target” domain. The mapping process semantically transfers the characteristics from the source to the target domain. As a result, the target acquires an enhanced meaning and access to secondary characteristics that are meaningful and expansive. When the receiver conceptualizes the new attributes in the target domain and translates its meaning in the new sense, the metaphor is considered effective. For example, a common metaphor, LIFE IS A JOURNEY, has been used to illustrate the conceptual metaphor concept. First posed by George Lakoff, the example easily demonstrates the power of metaphorical mapping between the characteristics of a “journey,” the “source” domain, and the abstract notion of one’s life, the “target” domain. Consider the following sample sentences from the LIFE IS A JOURNEY conceptual metaphor: “He’s got a head start in life.” “He’s without direction in his life.” “I’m at a cross-roads.” 36 The attributes of a journey have been mapped to a human’s life. The resulting metaphor conveys meanings beyond the literal translation to an abstraction of an ontological meaning and experience of being. It does so by assigning the specific characterizations of “life” to how one experiences an actual journey. Consider the attributes mapped below: 36 Lakoff (1993) 223. 18 Source: JOURNEY Target: LIFE Head start  Early assistance in achieving success in life Direction  Life is pursuing a destination Cross-roads  Life requires choosing between destinations The mapping of corresponding attributes confers a set of systematic references to the target domain. These sets of correspondences are often described as categorizations or prototypes, concepts borrowed from psychology (Eleanor Rosch 37 ). The prototypes include additional sets of relationships the metaphor may later use to build future maps . Note, however, that not all attributes logically map from the source to the target domain. To illustrate this critical point, consider the LIFE IS A JOURNEY metaphor more deeply. One characteristic of an actual physical journey is the ability to return to the original destination (a “round trip,” if you will). Yet in life, no one can physically return to childhood or infancy. As a result, this attribute is not considered appropriate so is not mapped to the target domain “LIFE.” The cognitive and linguistic selectivity of attributes is an extremely important dynamic in understanding how metaphor mappings achieve the desired outcomes between speaker and listener. In a landmark work combining Conceptual Metaphor Theory with a deeper view of cognition, Gilles Fauconnier 37 Eleanor Rosch and Barbara B. Lloyd, eds., “Principles of Characterization,” Cognition and Categorization (Hillsdale: Lawrence Erlbaum Associates Publishers, 1978): 27-48. 19 and Mark Turner introduced the Blending Theory (BT) 38 which accounts for selectivity. Under the BT framework, the blending process (also referred to as “conceptual integration”) yields a new emergent form, the conceptual blend. The construct of conceptual blend enables us to explain how only those connections that “work” between source and target domains are mapped in a given metaphor. In addition, as we will see later, BT also rationalizes how new mappings occur regardless of selectivity. Fauconnier and Turner describe how conceptual blends are created and the selectivity of attributes using a simple graphic represented in Figure 1. Generic Space Input I 1Input I 2 Blend Figure 1. Conceptual Blending Basic Diagram . This figure is based on the model introduced in Fauconnier & Turner’s The Way We Think , 2002. In Figure 1, the spheres represent a conceptual integration network of 38 Gilles Fauconnier and Mark Turner, The Way We Think (New York: Basic Books, 2002). 20 mental spaces from which metaphors arise. The input domains, I 1 and I 2 (i.e., source and target, generically) contain attributes represented by the black dots within the spheres. During the metaphoric mapping, a generic space structure captures the inputs shared by the source and target domains, which in turn, maps onto each of the inputs. 39 The lines in the diagram represent the mappings linking pairs in the two input spaces. The solid lines depict connections produced by the matching process. These are the elements that fit the metaphor mapping as intended by the sender. Note that not all attributes are matched. Other partial matches between input spaces are depicted by the dotted lines. The metaphor is created when an emergent structure, the blend, is created from the integration of the two input mental spaces. The solid square in the blended space represents this emergent structure. We have all experienced the development of an emergent structure when uttering or comprehending a metaphor. Fauconnier and Turner describe it this way, “What comes into consciousness is the flash of comprehension. And it seems magical precisely because the elaborate imaginative work is all unconscious.” 40 While it appears magical, the conceptual integration network and metaphors they produce do not arise out of nothing. It is in the process of language production that the conceptual integration network activates. 41 The context, the words and sentences of 39 The blending process has been described sequentially for simplicity. However, the process is recursive and Fauconnier and Turner warn that the sequence is “not meant to reflect actual stages of the process” (Fauconnier and Turner 46). 40 Fauconnier and Turner, 44. 41 While there has been significant work done on the importance of both the nonverbal as well as verbal entailments of language on culture (e.g., in Bronislov Malinowski’s “The Problem of Meaning in Primitive Languages”), the scope of this thesis focuses specifically on the verbal. 21 common discourse, guide the routes of connections within the network, firing ne combinations from the blended space. Cedric Boeckx writes, “Words and sentences can thus be thought of as procedures that impose certain mental traffic patterns among concepts; that is to say, they enforce perspectives on the way we think, … the reader should think of them as commands to activate certa mental concepts and combine them according to instructions implicit in structure of sentences.” win the ot only ts reveals al 42 This also explains why the metaphors remain consistent syntactically within a given language system. Blending theory explains how and why metaphors continue to be productive, that is, generate novel metaphors from the conceptual network. From the emergent structure and within the conceptual integrated network, the storage of attributes serves as a warehouse from which novel mappings can be created. Blending theory n reconciles how speakers can use metaphor effectively in a variety of contex through the selection of attributes, but it also explains how metaphors are highly productive. The process of selectivity will be explored in more detail as it the categories or conceptual frames of the speech community. This conceptu blend construct will be used to analyze THE ECONOMY IS A PATIENT in Chapter 4. The question now arises within CMT– How do we explain the emergence of polysemes? Fauconnier and Turner describe polysemy as “not just an accident of history or of synchrony, but rather an essential manifestation of the flexibility, adaptability, and richness in meaning potential that lie at the very heart 42 Cedric Boeckx, Language in Cognition: Uncovering Mental Structures and the Rules Behind Them (Oxford: Wiley-Blackwell, 2010) 117. 22 of what a language is and what it is for. It is also a symptom (rather than a primitive component) of the way in which various cognitive operations allow for creativity at many levels.” 43 Polysemes are the result of a non-arbitrary linguistic process originating from once novel metaphors. Bowdle and Gentner contend that some metaphors evolve to the point when they lose “all s metaphoricity” becoming dead metaphors. ense of 44 In their theoretical framework, “The Career of Metaphor,” 45 Bowdle and Gentner propose that metaphors follow a logical evolutionary path from initial target-source domain mapping to conventionalization (i.e., treated by speakers as a lexical item with minimal reliance on the original source domain’s attributes). Our lexicons are full of conventional metaphors. However, not all metaphors achieve conventionalization. Some stay novel and effective for only a period of time until its meaning becomes outmoded or irrelevant. By contrast, conventional metaphors are the result of habitual use over time. When this use has become so entrenched and customary, the base terms of the mapping refer to both aliteral concept and to an associated metaphoric category. It has become a dead or conventional metaphor. Bowdle and Gentner argue that this is the very process that enables a word to take on multiple meanings. The metaphor, as a single lexical item, becomes polysemous. Bowdle and Gentner and other cognitive linguists agree, “it is often claimed that metaphors are a primary source 43 Gilles Fauconnier and Mark Turner, “Polysemy and Conceptual Blending,” Polysemy: Flexible Patterns of Meaning in Mind and Language , Brigitte Nerlich et al, eds. (Berlin: De Gruyter, 2003) 80. 44 Bowdle and Gentner 209. 45 Bowdle and Gentner 193-216. 23 of polysemy.” 46 They cite the word, “ blueprint ,” as a good example of this phenomenon. “ Blueprint ” can mean both the literal source domain of a blue and white photographic print of an architectural plan as well as “anything that provides a plan.” 47 Through continual and habitual use over time, the second sense, “anything that provides a plan,” becomes as conventionalized as the literal meaning. Over time, the linguistic form becomes a full member of the lexicon, or “lexicalized” as the polyseme which can be applied in a variety of contexts. In fact, Fauconnier and Turner contend that the contents of the conceptual blend can also become entrenched in long-term memory. This makes the metaphors more accessible and available for future use. Regardless of their status as lexical items, polysemes are hardly dead but active and productive. As active members in the conceptual integration network of the blended space, they remain productive in generating new extensions by accessing the rich reservoir of attributes in the network. The polysemes can be viewed as category extensions that share “an inner coherence,” a “unified gestalt” 48 with other members in the network. The unified relationships and frequently emerging maps of related metaphors sustain this vibrant gestalt. As dynamic as the network appears, there is an internal order that enables the polysemes to be productive, but do so in a meaningful way. If the conceptual network has so many connections and attributes 46 Bowdle and Gentner 198. 47 Bowdle and Gentner 199. 48 D.A. Cruse, “Polysemy and Related Phenomena from a Cognitive Linguistic Viewpoint,” Computational Lexical Semantics , eds., Patrick Saint-Dizier and Evelyne Viegas (Cambridge: Cambridge University Press, 1995) 46. 24 available and ready, what happens cognitively when a polysemous word is encountered in discourse? How do listeners process multiple senses and determine the “correct” meaning? Stated differently, how do we resolve ambiguous meanings? Raymond Gibbs and Teenie Matlock cite a relevant study by psycholinguist, John Williams, 49 suggesting that “speakers might be activating a network of related senses when they hear polysemous words, part of which remains active even when the contextually appropriate sense of a word has been determined.” 50 Williams appears to confirm that in processing polysemes, multiple senses are accessible from the mental dictionary and operate simultaneously. The mind leaves its options open, if you will, to address any semantic trigger it may encounter. The entire database of meanings must be readily accessible to respond. Why is this important to understand? As we review metaphors in economics discourse, we will see that context alone is not sufficient to arrive at a singular meaning. Raymond Gibbs and Teenie Matlock’s own research in this area illustrates that a polyseme “gets its meanings both through interaction with specific lexical items in ordinary linguistic expressions and via speakers’ specific characterizations of discourse situations.” 51 While context is helpful, Gibbs and Matlock find that polysemes cooperate either 49 John N. Williams, “Processing Polysemous Words in Context: Evidence for Interrelated Meanings,” Journal of Psycholinguistic Research 21.3 (1992): 193-218. 50 Raymond Gibbs and Teenie Matlock, “Psycholinguistic Perspectives on Polysemy,” Polysemy in Cognitive Linguistics: Selected Papers from the International 5 th Cognitive Linguistics Conference , eds., H. Cuyckens, Britta Zawada (Amsterdam: John Benjamins, 1997) 215. 51 Gibbs and Matlock 218. 25 syntactically or semantically with other frequently co-occurring lexical items (called “collocation”). By keeping the conceptual integrated network active and interpreting the collocated items in discourse, there is a sifting and sorting process to identify and select the most likely meaning. Linguists critically analyzing a polyseme in natural discourse consider the “company it keeps” as well as the context to thoroughly catalogue the semantic connections. Salience is an important concept when analyzing polysemes. It has been found that in the process of polyseme creation, the new emergent form gains its salience from the source domain. 52 As a result, the polysemous word becomes marked in discourse by absorbing the low salience referents. It is this salience that enables a polyseme to be comprehensible time and time again by the speech community, thus becoming entrenched as a conventional, dead metaphor. In use, linguists have found that the most salient meaning is activated first by the receiver not always what would be considered the original meaning. 53 Over time, even the primary meaning may lose its prominence in the speech community. Recall the “ blueprint ” example. As most individuals do not handle physical blueprints on a daily basis, the metaphorical meaning, “a plan,” is more salient. Still, the original meaning is accessible in long-term memory. At any point in time, the multiple senses are ready for activation. Language users can take advantage of this phenomenon, creating double entendres, idioms and puns very 52 For fascinating case studies in salience and polysemy, refer to Cecil Brown and Stanley Witkowski’s research on “eye/seed” and “eye/fruit” polysemes in the Uto-Aztecan languages in the article, “ Polysemy, Lexical Change and Cultural Importance,” Royal Anthropological Institute of Great Britain and Ireland 18.1 (March 1983): 72-89. 53 Rachel Giora, “Literal vs. Figurative Language: Different or Equal,” Journal of Pragmatics 34 (2002): 487-506 .26 effectively in new contexts. Certainly readers of newspaper headlines find this to be true. As we have seen, salience and productivity enable supposedly dead metaphors to activate new semantic extensions as novel metaphors. Over time, the new extensions, if successful and relevant, can become so entrenched they result permanent semantic change. Cognitive linguists and sociologists believe that analyzing semantic shifts can reveal much about the collective thinking of a society and the evolution of its culture. Let’s examine now what influences semantic change. While there are a number of influences that drive change in a word’s meaning, cognitive research finds that the primary driver is the speaker’s own intuitions about the real world. These intuitions, of course, are derived from their personal experiences and those around them. As we have seen, the most basic influence is our interpretation of human physical attributes. Gibbs provides an excellent example of how our embodied experience informs polysemy in his analysis of the word, “ stand. ” 54 In this case study, Gibbs shows that our innate sense of VERTICALITY and BALANCE shapes how we interpret phrases such as “ to stand firm, ” “ we stand on 30 years of experience ” or “ to stand to profit. ” He proposes that people tacitly recognize connections between their bodily experience and different metaphorical / polysemic meanings. Prototypes and categories that involve the body are considered more primitive and, therefore, 54 Raymond Gibbs, “Embodied Standing and the Psychological Semantics of Stand ,” The Linguistics of Sitting, Standing, and Lying , ed. John Newman (Amsterdam: John Benjamins, 2002) 347-400. 27 more central than others. Speakers and receivers of such polysemies process them intuitively and unconsciously. When a novel embodied metaphor evolves to convention, the semantic change inherits a high level of saliency. As a result, primitive polysemes such as “ stand ” and others relating to the human body may be more significantly marked and effective in conveying semantic content. Recent empirical research 55 appears to substantiate philosophical and rhetorical theories that embodied metaphors are also predictive of emotional responses. Linguistic anthropologists and linguists from the interaction or constructivist views would argue that while we intuit certain meanings from our bodily experiences, these intuitions are not without influence from the society in which we function. Max Black, for example, writes of a “system of associated commonplaces” which is shared by a speech community. They are a collective set of opinions that become a part of the public gestalt and experience within the culture. So while we may universally possess and experience human bodies, our mental models of the body are constructed by that experience and its relation to what is already known or learned. In the act of metaphor production, the interaction between metaphor speaker and receiver that involves “shifts in meaning of words belonging to the same family or system as the metaphorical expression.” 56 That interaction can “enable us to see aspects of reality that the metaph production helps to constitute. But that is no longer surprising if one believes that or’s 55 Reference the work by Raymond Gibbs, Paula Lenz Costa Lima, and Edson Francozo, “Metaphor is Grounded in Embodied Experience,” Journal of Pragmatics 36 (2004): 1189-1210; and Raymond Gibbs, “Embodied Experience and Linguistic Meaning,” Brain and Language 84 (2003): 1-15. 56 Max Black, Models and Metaphors , (Ithaca, Cornell University Press, 1962) 45. 28 the world is necessarily a world under a certain description – or a world seen from a certain perspective. Some metaphors can create such a perspective.” 57 An example of this phenomenon in the health domain has been studied by Arthur Kleinman, MD., in his study of patients’ stories related to their illness. He writes, …commercialized symbolic meanings, which, like all cultural systems, orients the person to body and self experiences and to the priorities and expectations of the group. Indeed, through these embodied values social control is internalized and political ideology materializes as corporeal feeling and physiological needs. To understand symptoms and illnesses have meaning, therefore, we first must understand normative conceptions of the body in relation to the self and world. These integral aspects of local social systems inform how we feel, how we perceive mundane bodily processes, and how we interpret those feelings and processes. 58 Metaphors, in this view, are both linguistically and socially constructed. We will find these perspectives extremely relevant as we review polysemy in economic discourse. Socio-cultural Analyses of Metaphor Socio-linguists consider the cultural influences on polysemy creation and interpretation. Research in this arena considers the collective un consciousness of a speech community. In “Ambiguities We Live By: Towards a Pragmatics of Polysemy,” psychologists Brigitte Nerlich and David Clarke attest that historical 57 Max Black, “More on Metaphor,” Metaphor and Thought , 2 nd ed., ed. Andrew Ortony (Cambridge, Cambridge University Press, 1993) 38. 58 Arthur Kleinman, MD., The Illness Narratives: Suffering, Healing, And The Human Condition (New York, Basic Books, 1988) 13. 29 explorations of the semantic fields and frames 59 can contribute greatly to the understanding of culturally derived categories / prototypes underlying the polysemes. By analyzing the progression of a specific polyseme over its life cycle, we can gain significant new insight to the integrated conceptual network from which it emerged. Nerlich and Clarke write, “constructions and deconstructions of shared (multiple) meanings in conversation have not only the function of keeping meanings and conversations alive, in the long run they also contribute to the steady diachronic changes in the polysemy of words, some meaning come to the fore, some dropping away….” 60 Nerlich and Clarke believe in some cases these diachronic changes can be made transparent in linguistic study. They acknowledge “what has been overlooked so far in polysemy research is that (‘synchronic’ and ‘diachronic’) polysemy is very much ‘alive’, that it can be studied in its natural habitat, in the course of conversations and so on.” 61 Dirk Geeraerts has applied the tools of cognitive linguistics to reveal that “synchronic links that exist between the various senses of an item coincide with diachronic mechanisms of semantic extension such as metaphor and metonymy.” 62 While there are too many variables to conclusively identify nor predict cause-effect relationships between linguistic mechanisms and 59 Nerlich and Clarke’s research expands Charles Fillmore and B.T.S. Atkins’ theory of frame semantics as they define it, i.e., “a word’s meaning is understood ‘with reference to a structured background of experience, beliefs, or practices, constituting a kind of conceptual pre-requisite for understating the meaning” with a diachronic view (1992) 76. 60 Brigitte Nerlich and David Clarke, “Ambiguities We Live By: Towards a Pragmatics of Polysemy,” Journal of Pragmatics 333 (2001): 5. 61 Nerlich and Clarke 6. 62 Dirk Geeraerts, Diachronic Prototype Semantics: A Contribution to Historical Lexicology (Oxford: Clarendon Press, 1997) 6. 30 polysemy outcomes, Geeraerts acknowledges that by examining the conceptual domains or prototypes of metaphors, we can reveal the underlying social and historical dynamics of the semantic change. In a comprehensive article, Gábor Gy őri outlines what linguistic, cognitive and social factors are involved: Change in usage relies on general cognitive mechanisms like analogy, association, categorization, etc. The cognitive basis of this innovative language use is the exploitation of familiar knowledge. This provides the motivational support for both the production and the comprehension of occasion-bound meanings. The familiar knowledge that can be exploited resides in conventional expressions and in the connotations attached to them by the speech community, strongly suggesting that encyclopedic information is part of semantic knowledge. The individual speaker turns to these expressions in order to form occasion-bound meaning for efficient but economical reference and representation in an effort to adapt the language to new circumstances be they internal (linguistic) or external (natural, social, cultural, etc.) The choice and path of semantic modification of the available expressions is influenced by four cognitive factors: cue-validity, cognitive economy, perceived world structure, and conjunctivity. In this way the effect of the environment on semantic structure will be largely filtered through the speakers’ cognitive systems. These ad hoc modifications of meaning will provide the variations for selection by the speech community if the communicative circumstances that trigger a new usage persist and take on cultural dimensions. Then conventionalization of the ad hoc meaning will take place through the stages of pragmatic ambiguity, polysemy, and finally full semantic change. 63 Gy őri recognizes that semantic change cannot be generalized. Many semantic innovations are unpredictable and are often phonological rather than 63 Gábor Gy őri, “Semantic Change and Cognition,” Cognitive Linguistics 13.2 (2002): 159-160. 31 cognitive. He does posit, however, that metaphor and meaning extensions parallel the changes in cultural categories in response to changing external environments. “Etymologies,” he writes, “reveal categorization processes.” 64 Semantic shifts are hardly arbitrary. Language reflects the speech communities’ effort to describe new experiences and concepts. If no word exists to express a new idea, speakers will often refer to apt prior experiences or familiar things in analogy. Analogies and, by extension, metaphors are economical and one of the most efficient communication methods. This efficiency can create momentum to trigger semantic change. Once again, we refer to the principles of conceptual mapping processes and selectivity. Gy őri writes, “the choice of properties must also be governed by some kind of joint salience for both speaker and hearer based on a common ground.” 65 The characteristics with the highest salience in the culture most likely to trigger recall of the category from the blended space. The shared world view and how it works (“perceived world structure”) significantly contributes to the metaphorical mapping. Gy őri writes, “Conjunctive relationships [between features in the metaphor map] are psychologically more salient on logical grounds and this may have possible cognitive priority.” 66 In this view, polysemy is influenced by sociological factors and the shared cultural perspectives. Language is naturally economical. It takes the easiest path to understanding and makes familiar, salient connections. As the culture evolves over time, what is salient to the culture is salient in the language. Within the 64 Gy őri 135. 65 Gy őri 141. 66 Gy őri 145. 32 fossilized polyseme is a history of its diachronic evolution and the encyclopedic information locked inside it. A relevant example of semantic change is Sophia Mamaridou’s recent analysis of the case of the Greek polyseme, “ psyche ” (“ Ψνχή ”). 67 In Mararidou’s examination of “ psyche, ” she reveals the cognitive and cultural constructs underlying its multiple meanings. “ Psyche ”, she writes, “defines the ontology of man according to a particular cultural model of the self.” 68 It is this cultural model or ideology that is “closely associated with a network of image-schemas, metaphors and metonymies.” 69 Marmaridou illustrates the productivity of the conventionalized metaphor. Her research shows how new metaphors and extensions of meaning are motivated. She lays bare the cultural legacy of the word and the relevant semantic contents accompanying its polysemy in current use. Most intriguing, is her finding that the embedded ideologies carry powerful cognitive and emotional impact. Marmaridou claims that over time, the semantics of “ psyche ” were motivated by the “interaction of conceptual metaphors and cultural models.” 70 Her research bears this out. In numerous examples, Mamaridou illustrates how “ psyche ” draws semantic information from related conceptual domains and derives multiple metaphors such as PSYCHE IS BODY, PSYCHE IS CONTAINER, PSYCHE IS A MOVING BODY and PSYCHE 67 Sophia Mamaridou, “Cognitive, Cultural, and Constructional Motivations of Polysemy and Semantic Change,” Pragmatics & Cognition 18 (2010): 68-110. 68 Marmaridou 68. 69 Marmaridou 69. 70 Marmaridou 77. 33 IS A VALUED POSSESSION among many others. Marmaridou’s work reveals the role familiar cultural encyclopedic knowledge and salience play in semantic change. Of particular interest, this case study illustrates how related meanings are derived from evolving ideologies. Note how “ psyche ” is used in new contexts over time and collocated with other semantic forms such as innovative idioms in the sentences below: 71 (1) o an θtropos aotelite apo soma ke psixi The man consists-3 rd sg.pres.pass. of body and soul Man consists of body and soul. (2) xa θikan 500 psixis Lose-3 rd pl.past.pass. 500 psyches 500 souls were lost. (3) Ti psixi exi ena penindraiko What soul have-3 rd sg. pres a fifty-drachma coin have? What value does a fifty-drachma coin have? In (1), we see the familiar use of the word “ psyche ” revealing the ideology of the human self consisting of two parts, body and soul. In (2), the word has emerged in a metonymous form, representing not only the whole of the individual but representing people in general. Finally, we observe a metaphorical extension to the conceptual domain of commerce in (3). Marmaridou explains that this example reveals the cultural and ideological base on which PSYCHE IS A VALUED POSSESSION is based. In the integrated cognitive network, the attributes of “value” are mapped from the source domain, PSYCHE (soul) to the 71 All examples used here are taken from Marmaridou’s article cited earlier. 34 target domain, POSSESSION. In this case, the possession is “money.” Marmaridou writes, “the metonymic extension of “ psyche ” to refer to value is motivated by the cultural narrative of existence and the particular fairly productive construction. Given that this construction expresses both the speaker’s evaluation of an entity and her wish to prompt the address to do something, is highly expressive.” 72 Marmaridou emphasizes the emotive power of these constructions. The case of “ psyche ” catalyzes the interactional relationships between conceptual networks, cultural ideologies and polysemy. Her study highlights how polysemes carry their history of cultural meaning as cognitive freight and is extended and repurposed in new and innovative contexts. 73 Cognitive linguists recognize what rhetoricians and philosophers of language have theorized: that culture and language are engaged in a two-way synergistic relationship that impacts what is said, and in what semantic forms. We have seen that the integrated conceptual network facilitates the development of metaphors and polysemes. We have also seen that encyclopedic knowledge is accessible for metaphor creation and semantic change. It also influences interpretation of polysemes as Jörg Zinken writes, “Encyclopedic knowledge that is frequently relevant in the usage of a particular construction becomes more accessible. This leads to a situation where different kinds of knowledge can be immediately accessible through one particular lexical item in appropriate 72 Marmaridou 91. 73 Other relevant diachronic analyses of semantic change include Aijmer’s research of the English auxiliary “ will ” and Grygiel’s deconstruction of the word “ boy. ”35 contexts, i.e., to polysemy.” 74 If such encyclopedic knowledge is so easily accessible between speakers, what are the implications of the cognitive freight that accompany such words? Sally McConnell-Ginet explores the notion of conceptual freight or conceptual baggage in detail. She writes, “words serve as pointers to or place-holders for not only language users’ semantic representations and their (perhaps limited) knowledge of referents but also their understanding of what is widely presumed about those referents and their place in various kinds if scenarios, including how the words (and also, of course, their referents) figure in various kinds of social practices. Conceptual baggage, I propose, attaches to a word as it figures in various discourses and is deployed in social and cultural projects… Importantly, there is a general past discursive history as well as situated future development that matters to insertion of particular inferential fodder into the conceptual baggage associated with a particular word (or in some cases, perhaps, with a family of words – kinship terminology, for example).” 75 Historic conceptual baggage remains in the integrated conceptual network. This is particularly true when as, Zinken notes, it is habitually reactivated in discourse. This is a distinguishing feature of metaphorical forms in discourse. And, Zinken argues, we find that metaphors of this nature carry ideological content as well. They are entrenched and motivated by experiences 74 Jörg Zinken, “Discourse Metaphors: The Link Between Figurative Language and Habitual Analogies,” Cognitive Linguistics 18.4 (2007): 452. 75 Sally McConnell-Ginet, “Words in the World: How and Why Meanings Can Matter,” Language 84.3 (2008): 515. 36 within the culture. 76 Michael White and Honesto Herrara corroborates Zinken’s hypothesis in a review of metaphors found in press coverage of European telecommunications mergers and acquisitions. Their research shows how “the discourse, easily taken as a more or less objective or neutral account of events, in fact reveals a very marked though hidden ideology.” 77 Their analysis demonstrates how tools of cognitive linguistics can expose unconscious or, as they state, “covert” ideologies. White and Herrara’s analysis “shows us how such metaphors, while operative merely on a basis of their concrete logic and entailments, actually tie up with and reinforce a long tradition within economic thought (e.g., Social Darwinism) or entrenched cultural values (e.g., gender roles).” 78 Their deconstruction of economic metaphors such as BUSINESS IS A JUNGLE and MONOPOLIES ARE DINOSAURS used in M & A contexts revea ideological positions. For example, they cite how the references to metaphorical dinosaurs and allusions to Darwin’s “survival of the fittest” are unconsciously promoting a free market ideology. Further analyses in similar texts also find additional hidden ideologies such as historic colonialism and gender related norms. The result, they write, is an “emotional force and persuasiveness accruing to an argument calling for the defence of national territory against an l 76 See Jörg Zinken, “Ideological Imagination: Intertextual and Correlational Metaphors in Political Discourse,” Discourse Society 14.4 (2007): 507-523 and, Christopher Hart, “Critical Discourse Analysis and Conceptualization: Mental Spaces, Blended Spaces and Discourse Spaces,” Cognitive Linguistics in Critical Discourse Analysis: Application and Theory (Cambridge: Cambridge University Press, 2008). 77 Michael White and Honesto Herrera, “Metaphor and Ideology in the Press Coverage of Telecom Corporate Consolidations,” Cognitive Linguistics Research: Cognitive Models in Language and Thought ,eds. René Dirven, Roslyn Frank and Martin Pütz (2003): 277. 78 White and Herrara 277. 37 external threat. Moreover, the fact that BT’s 79 nationality associated it with the colonizing power historically operative in the U.S. intensifies the emotional charge. Thus framing the issue in this way makes latent ideology a very powerful element of persuasion.” 80 White and Herrara illustrate the power metaphoric language can have to activate emotive ideologies. White and Herrara’s deep analysis of the cognitive blend resulting from the metaphor process unpacks inherent cultural prototypes influencing the map. As Zinken proposes, metaphors can contain historical ideological content; White and Herrara have shown that by analyzing the conceptual maps, the accompanying cognitive freight and related ideologies can be revealed. It is not always possible or desirable to research the etymology of a word to gain new insights. In most cases of polysemy, the original metaphorical map has been bleached or totally forgotten. The risk of hypothesizing folk etymologies is too great. However, in some unique cases, where the metaphor is active, productive and the associated attributes are especially salient, diachronic analyses can reveal the cognitive associations that have been built up over time. Cognitive linguists attest that polysemes and similar lexical items undergo semantic changes, shifts and leaps that are traceable. 81 Unique insights can be gained by analyzing the semantic evolution of individual words 79 “BT” is the acronym used for “British Telecom” in this context. 80 White and Herrara 303. 81 Reference sources on the topic of semantic change, metaphor, polysemy by cognitive linguistics such as Vanhove’s From Polysemy to Semantic Change (Amsterdam: John Benjamins Publishing, 2008); Seana Coulson’s Semantic Leaps: Frame-Shifting and Conceptual Blending in Meaning Construction (Cambridge: Cambridge University, 2001); and related work of Jörg Zinken, Brigitte Nerlich et al. 38 and phrases over time. Friedrich’s Analytic Model for Revealing Ideologies in Language Paul Friedrich states there are two approaches to critical analysis of ideologies in discourse: the analytical-scientific and the emotional-ethical. He argues that “the two approaches are essential to each other, for both are ‘critical’ and concerned with ‘values,’ although meaning of these terms differ greatly in context: a scientific criticism is always implicitly ethical to a significant degree, and an ethical criticism is almost always scientific to some extent. In other words, the scientific approach is primarily rooted in the cognitive (e.g., the logic of experiment) and is concerned with diverse levels of knowledge and the ethical-emotive is rooted in the affective as well as being overtly focused in such phenomena, but it is also commonplace for a cognitive analysis to arrive from an ethical concern, and for an analysis of affect to arise from the libido cognosciendi (the drive to know).” 82 Freidrich’s model addresses the “interdependencies among language, ideology, and political economy.” 83 Friedrich believes that major theorists view political economy as determining, intertwining and, possibly, meditating language and ideology in the economic domain. Their models propose that not only is language and political economy so inextricably intertwined, but that “ideology emerges as a primary output – perhaps the 82 Paul Friedrich, “Language, Ideology, and Political Economy,” American Anthropologist 91 (1989) 296. 83 Friedrich 297. 39 primary output governing human acts and attitudes.” 84 Since the goal of this thesis is to reveal the underlying cognitive, linguistic and emotive power of THE ECONOMY IS A PATIENT and the related polyseme, “ recovery, ” Friedrich’s dual model is most appropriate for this analysis. Let us define Friedrich’s model in more detail. Friedrich outlines the three components of his framework as follows: Political economy – involves resource allocation in the sense, for example, of control over goods. Political economy involves the generic economic process of the production, distribution, and consumption of goods, including ‘non-material’ ones, and the patterns and culture of power that control or influence these processes. 85 Ideology – the basic notions or ideas that the members of a society hold about a fairly definite, if not bounded set or area such as honor, matrilineal affiliation, or the division of labor, and the interrelations and implications of such sets of notions. It is “a system, or at least an amalgam, of ideas, strategies, tactics, and practical symbols for promoting, perpetuating, or changing a social and cultural order; in brief, it is political ideas in action. 86 Language – a verbal process by which the individual relates ideas and emotions to sound and other material symbolism in terms of a code and in the context of a society and its culture, and their respective, interrelated histories. 87 Freidrich’s framework, thus defined, prepares us for analyzing the interplay between language and economic constructs within a given culture. He views ideologies as “political ideas in action” which “arise from the engagement of creative individuals with practical problems and necessarily reflect or express the 84 Friedrich 297. 85 Friedrich 297. 86 Friedrich 300. 87 Friedrich 302. 40 will and interests for control or change of some social group or class – notably, its economic interests.” 88 He cites symbols in language such as metaphors and the wider family of lexical and semantic forms as highly effective tools in sending key messages to the speech community. The result is a “linguaculture” whereby language and culture “constitute a single universe of its own kind.” 89 Friedrich’s “linguaculture” is consistent in substance with the cognitive linguistic not “social cognitions” of Fauconnier and Turner or Van Dijk ions of pedic 90 and the encyclo knowledge of Gy őri and Zinken. They agree that conceptual domains reside in the collective minds of the speech community. The concepts are shared through a historical memory and sustained via habitual use of linguistic forms in the culture. Joining Disciplines to Analyze HEALTH-WEALTH Metaphor Domains The frameworks presented here illustrate the cognitive, linguistic, socio-cultural and rhetorical views of metaphor research. Together they provide a comprehensive set of lenses to view metaphors in economic discourse. In the conceptual blend of THE ECONOMY IS A PATIENT, we have an opportunity to unpack the underlying dynamics that has lead to its emergence and reveal any ideologies that may have historically influenced its creation. The progress in cognitive linguistics analyses has provided us the tools to now confidently 88 Friedrich 301. 89 Friedrich 306. 90 Tuen A. Van Dijk, “Social Cognition and Discourse,” Handbook of Social Psychology and Language, eds. H. Giles and R. P. Robinson (Chichester: Wiley, 1989): 163-183. 41 consider the question posed in this thesis: can metaphorical polysemes activate conceptual networks to spur action within a community? The current research suggests that it can. In the seminal book, Philosophy in the Flesh , Lakoff and Johnson write compellingly that an overriding metaphor in Western economics is WELL-BEING IS WEALTH. They write, “we all conceptualize well-being as wealth.” 91 If this is true, then fundamental ideology of Western thought maps the idealized notion of well-being from the source conceptual domain of wealth. We realize that metaphors sustain the ideology by being habitually ingrained in common discourse. As in Freidrich’s model, the language, the ideology and the culture become intertwined. Consider Lakoff and Johnson’s observation and how they describe the WELL-BEING IS WEALTH metaphor as, “the basis for a massive metaphor system by which we understand our moral interactions, obligation, and responsibilities… Thus moral action is conceptualized in terms of financial transaction.” 92 The metaphor is deeply integrated in not only the language but the political economy and economic ideologies. They go on to write that the underlying ideology and morality is “grounded” in our “embodied experience of well-being: health, strength, wealth, purity, control, nurturance, empathy, and so forth.“ 93 It is no surprise, then that the conceptual maps of HEALTH and WEALTH share the blended conceptual domain of WELL-BEING. The mapping 91 George Lakoff and Mark Johnson, Philosophy in the Flesh (New York: Basic Books, 1999) 292. 92 Lakoff and Johnson, 292. 93 Lakoff and Johnson 331. 42 is logical and simple as it reflects the underlying sense of safety and welfare. As a result, the extended conceptual domain yields the related metaphor, WELL-BEING IS HEALTH. Lakoff and Johnson conclude that in the economics domain and from an ideological point of view, the pursuit of well-being translates to both a moral and political right to pursue the state of sustained well-being. When out of alignment, through economic crisis or imbalance, there is not only an economic crisis but a moral one. It is a macro-economic dilemma that threatens the very well-being of the entire society and its citizens as individuals. The influence of this metaphor on the speech community is predictable and inevitable: a call to act. Similarly, in his critical analysis, The Order of Things: An Archaeology of the Human Sciences , Foucault refers to the use of body metaphors in the early economic treatises of the seventeenth century illustrating the intersection of metaphoric language, mercantilist ideology and the political economy of the time. Consider this excerpt in his description of money exchange as a signifier of Hobbesian mercantilist ideology from the Leviathan (italics are mine): According to Hobbes , the venous circulation of money is that of duties and taxes, which levy a certain mass of bullion upon all merchandise transported, bought, or sold; the bullion levied is conveyed to the heart of Man – Leviathan – in other words, into the coffers of the state. It is there that the metal is ‘made vital’ : the state can, in effect, melt it down or send it back into circulation. But at all events it is the state’s authority alone that can give it currency: and redistributed among private persons (in the form of pensions, salaries or enumeration for provisions bought by the state), it will stimulate, in its second, arterial circuit, exchanges of wealth, manufactures, and agriculture. Thus circulation becomes one of the fundamental categories of analysis. 43 But the transference of this physiological metaphor was made possible only by the more profound opening up of a space common to both money and signs, to both wealth and representations. The metaphor of the city and the body , so assiduously put to work in our Western culture, derived its imaginary powers only from the much deeper foundation of archeological necessities. 94 Note how Foucault’s analysis exemplifies Friedrich’s approach. Like Lakoff and Johnson’s example, Foucault’s excerpt is an apt illustration of Friedrich’s framework for analysis. He illustrates the interconnections of language, political economy and culture by explaining how the metaphoric mapping reinforces the ideology in the Western economic “linguaculture.” His references to the embodied polysemes “ heart, ” “ arterial, ” “ venous, ” “ circulation ” in these contexts illustrate the conceptual metaphor mapping of “ city and body .” Again, the language, the political economy and ideologies can be parsed from the original text in the Leviathan to disclose the embedded dynamics at work (e.g., “state’s authority alone to give currency”). Consider also the following stunning statements made by Lakoff in 1991 regarding the Gulf War: Metaphors can kill. The discourse over whether to go to war in the gulf was a panorama of metaphor. Secretary of State Baker saw Saddam Hussein as ‘sitting on our economic lifeline.’ President Bush portrayed him as having a "stranglehold" on our economy. General Schwarzkopf characterized the occupation of Kuwait as a "rape" that was ongoing. The President said that the US was in the gulf to "protect freedom, protect our future, and protect the innocent", and that we had to "push Saddam Hussein back." Saddam Hussein was painted as a Hitler. It is 94 Michel Foucault, The Order of Things: An Archaeology of the Human Sciences (London: Routledge, 2002) 194. 44 vital, literally vital, to understand just what role metaphorical thought played in bringing us in this war. 95 For Lakoff, it is without question that metaphor is a linguistic force with strong embedded ideologies capable of influencing a nation and mobilizing an army. In this treatise, he makes the case powerfully that the Gulf War and the “well-being” of the U.S. economy are inextricably linked: Well-being is wealth. The general well-being of a state is understood in economic terms: its economic health. A serious threat to economic health can thus be seen as a death threat. To the extent that a nation's economy depends on foreign oil, that oil supply becomes a “lifeline” (reinforced by the image of an oil pipeline). 96 Again we see the ideology, language and political economy coming to life in rhetorical fashion. The interconnections of health and wealth conceptual domains tap into the power of embodied metaphors which we have seen have great emotive power. In a relevant study for this thesis, Boers and Demecheleer on English, Dutch, and French economic metaphors specifically analyzed the use of THE ECONOMY IS A PATIENT metaphor. They confirm the efficacy of the metaphor noting how the attributes associated with health are applied. They write, “healthy people tend to be active and energetic. Consequently, a high level of activity is usually valued positively. Illness or injury typically undermines the patient’s energy and mobility. Hence, immobility or low levels of activity are considered as 95 George Lakoff, “Metaphor and War: The Metaphor System Used to Justify War in the Gulf,” The Sixties Project: Viet Nam Generation Journal & Newsletter, Part I, 3.3, (1991), ed. Kalí Tal, 22 March 2010 < Scholarly/Lakoff _Gulf_Metaphor_1.html >. 96 Lakoff 1991. 45 negative symptoms… As long as one is in good health, there is no need for medical treatment. When one’s health breaks down, however, the illness has to be diagnosed and medical treatment prescribed…. If the treatment is successful, the patient will recover and be active again.” 97 Their analysis reveals the political economic view that only when the economy is inactive or in trouble, is there a need for government intervention. Once “well” again, the treatment is considered a success. While this metaphor appears across English, Dutch and French texts, they conclude that metaphors reveal cultural diversity and are “reflections of that community’s conventional patterns of thought and its prevailing ideologies (at that time).” 98 Like Lakoff, Boers and Demechleer suggest “the inherent bias involved in metaphor can be exploited for purposes of persuasion.” 99 They believe preserved in the metaphorical map is an inference that neither a patient nor an economy can be blamed for its illness: “this may serve as a very convenient excuse for the government of a country suffering from depression or for private enterprises asking for injections of public capital.” 100 The metaphor THE ECONOMY IS A PATIENT will be the focus of the analysis presented in Chapter 4. 97 Frank Boers and Murielle Demecheleer, “Travellers, Patients and Warriors in English, Dutch and French Economic Discourse,” Revue Belge de Philology et d’Histoire 73 (1995): 685-686. 98 Boers and Demecheleer 673. 99 Boers and Demecheleer 678. 100 Boers and Demecheleer 686. 46 One last question remains in this research review: within a complex and highly integrated conceptual network such as economics, can a single lexeme, such as a polysemous word, spur action within a community? Frank Boers, in a 1997 study of metaphors specifically within the HEALTH and WEALTH domains of economics observes “metaphors can guide participants’ decision-making processes about socioeconomic issues.” He continues, “this applies to conventional as well as creative figurative expressions.” 101 We can conclude from Boers’ analysis in this domain that not only are metaphors persuasive but their evolved form as polysemes can as well. Consider Susan Sontag’s monograph, Illness as Metaphor . In this work, Sontag identified a critical link between metaphors used and judgments made by society. Like Kleinman, she observes the impact language has in creating the patient’s view of “reality” in the world of medicine and disease. Specifically, she describes the influence the word (and polyseme) “ cancer ” can have on patients of the disease. She characterizes the word as “encumbered by the trappings of metaphor.” 102 “Cancer ” carries a heavy cognitive freight, loaded by historic and etymological meaning that drives conscious human behavior. She argues the use of the word itself changes the interaction between cancer patients and the health care system. It raises perceptions of guilt and punishment. In rhetorical scenarios, we will see that “ cancer ” has a long history of damning language, a 101 Frank Boers, “ ’No Pain, No Gain’ in a Free Market Rhetoric: A Test for Cognitive Semantics?” Metaphor and Symbol 12.4 (1997): 238. 102 Susan Sontag, Illness as Metaphor and Aids and Its Metaphors (1978; New York: Farrar, Straus and Giroux, 1988) 5. 47 reflection of past cognitive maps that continue to this day. The research and theoretical models presented in this section are relevant to this thesis in order to effectively deconstruct the cognitive and linguistic dynamics underlying the conceptual blend, THE ECONOMY IS A PATIENT metaphor. They also explain how polysemy carries cognitive freight from the culture in which it emerges that can be traced through diachronic analyses. In Chapter 4, we will consider how the polyseme “ recover ” emerged in the economic discourse over time and in what sociological and linguistic environments. As the above examples attest, the ECONOMY IS A PATIENT metaphor can have immense rhetorical implications. The cultural ideologies embedded in the metaphor and what semantic freight can be discerned from the polyseme, “ recover ” is linked to its ability persuade the speech community. 48 Chapter III Limitations of Synchronic Analysis Cognitive linguistics has a long tradition of synchronic analysis. 103 With few exceptions, the primary analyses of semantics and specific lexemes in discourse have sought to understand the mental processes involved in language and meaning, metaphor generation and interpretation. As a result, research focusing on metaphors has revealed valuable insight into the organizational structures of the mind and how language in discourse lays bare shared models of conceptualization. Lackoff, Turner, Fauconnier and Johnson, early pioneers in metaphor research, focused much of their attention on reifying the theories of categorization, embodiment, encyclopedic knowledge and semantic fields. Their work has expanded the field’s understanding of the cognitive and cultural underpinnings of language and provided evidence of universal processes governing human thought. However, many questions remain regarding meaning variations, the mechanics of semantic change, polysemy, metonymy and related linguistic forms. A purely synchronic view of these forms provides little insight into the genesis of semantic prototypes and how they evolve over time. While we have since learned from Bowdle and Gentner that polysemy can result from metaphoric processes, without diachronic analysis it is virtually impossible to 103 Jan Nuyts, “Cognitive Linguistics and Functional Linguistics,” The Oxford Handbook of Cognitive Linguistics (Oxford: Oxford University Press, 2007) 550. 49 either research the source of semantic innovation or derive directionality of meaning extensions. Synchronic analyses limit our ability to deconstruct the synergies between sociological influences and language shifts and theorize the influence both have on present linguistic forms. Joan Bybee writes, As language is viewed less as a structured, tight-knit system and more as a variable, negotiated set of social and cognitive behaviors, the importance of the study of language change increases. Language change provides evidence for the nature of linguistic representation and processing, and thus provides a window on the synchronic mental representation and the forces that create grammar. Moreover, since all synchronic states are the result of a long chain of diachronic developments, the construction of complete explanations for linguistic structures requires attention to the diachronic dimension. 104 There are a number of prominent examples where new ground-breaking insight is gained by taking a diachronic approach. In each case, diachronic analysis enables us to analyze the systemic structure of cognition. Eve Sweetser has been an early adopter of diachronic analysis. Her research illustrates patterns in polysemes revealing their metaphorical connections between semantic fields. Andreas Blank 105 and Peter Koch are re-opening and raising the respectability of an old linguistic discussion about semasiology (study of meaning change) and onomasiology (study of the means of expression) using diachronic linguistic techniques. Elizabeth Closs Traugott’s extensive research demonstrates the regularity of semantic change and the links between meaning 104 Joan Bybee, “ Diachronic Linguistics ,” The Oxford Handbook of Cognitive Linguistics, eds., Dirk Geeraerts and Herbert Cuyckens (Oxford: Oxford University Press, 2007) 945. 105 Andreas Blank, “ Words and Concepts in Time: Diachronic Cognitive Onomasiology ,” Words in Time: Diachronic Semantics from Different Points of View , eds., Regine Eckardt, et al. (Berlin: Walter de Gruyter, 2003) 37-65. 50 and the speaker’s subjective belief state/attitude toward the proposition (subjectification). In her studies of polysemy, Traugott demonstrates empirically the theory that polysemes are informed by experience and can be lexicalized into conventional forms through continued use in specific contexts. She shows that polysemes experience accretion of meaning from their historical senses and exposes their potential to carrying historic as well as contemporary cognitive content. Similarly, Geeraerts’ work in semantics and prototype theory reveals the causes of semantic change by analyzing case studies of individual linguistic forms diachronically. His specific focus is polysemy. He writes, “one of the major things cognitive semantics is interested in is polysemy – and polysemy is, roughly, the synchronic reflection of diachronic-semantic change. The interest of theoretical semanticists working within the framework of cognitive semantics in the study of meaning changes derives from their interest in polysemy, if only because the synchronic links that exists between the various senses of an item coincide with diachronic mechanisms of semantic extension such as metaphor and metonymy.” 106 The limitations of synchronic linguistic analysis are apparent when considering the power of polysemes in discourse. Recall the insights revealed by Marmiadou’s case of ‘ psyche’ cited earlier in this paper. Marmiadou’s analysis would hardly be as rich if she had confined her work to contemporary contexts. As Bybee notes, “Mechanisms of change that create grammar are built into the language ability: they occur synchronically, as language is used. Thus 106 Dirk Geeraerts, Diachronic Prototype Semantics: A Contribution to Historical Linguistics (Oxford: Clarendon Press, 1997) 6. 51 explanations for linguistic structure must make crucial reference to diachronic changes and the mechanisms that propel that change.” 107 Or, as McCloskey writes, “You have to have a direct grasp of the diachronic subject to have something to be synchronic about.” 108 107 Bybee 981. 108 McCloskey (1998) 31. 52 Chapter IV Research Findings The following research is a diachronic examination of THE ECONOMY IS A PATIENT metaphor as it has occurred in economic discourse since its adoption in the English lexicon in Anglo-Norman form (circa 11 th century). Organization of Research Findings The research findings trace the occurrences of the metaphor regressively in order to illustrate the historical evolution of the metaphor over time. The data highlights specific examples of the word “ recover ” in their original contexts. These samples are grouped by the centuries in which they occur to underscore the macro-economic and historical settings in which the language was used. Recall that THE ECONOMY IS A PATIENT is a conceptual blend emerging from the metaphor mapping between the source domain of HEALTH and the target domain, ECONOMY. The blend is a complex map deriving attributes from the larger anthropomorphic domain, THE ECONOMY IS AN ORGANISM. The polysemes “ recover ” and “ recovery ” reside in the conceptual blend and share selected anthropomorphic attributes very specific to illness, remediation and return to wellness. The excerpts are analyzed considering the metaphor’s network of conceptual domains, collocates and other linguistic rhetorical constructs. 53 The research includes brief analyses using Freidrich’s “linguaculture” framework (Language, Political Economy and Ideology) to deconstruct the metaphor at the given a period of time. The framework is an apt tool to reveal the underlying ideology of the speaker. Findings Summary A diachronic analysis of the metaphor indicates that THE ECONOMY IS A PATIENT metaphor emerged early in discourse and continues to be salient and relevant in spite of the evolving economic environments. Similar rhetorical and linguistic patterns can be observed in the use of the metaphor, THE ECONOMY IS A PATIENT, in a variety of contexts. Specifically, the metaphor arises in highly emotive narratives regardless of the author’s intent. For example, the metaphor is found in popular press, academic analyses and political speeches alike. The metaphor is also consistently collocated with other lexical items from the health domain which primes the metaphor for interpretation as well as expanding the integrated conceptual network. As a result, the metaphor is highly productive. In many cases the extended metaphors become conventionalized and polysemic within economic discourse (such as “ heal, ”“remedy, ” and “ cure ”). In some cases, the polyseme’s dual meanings are ambiguous in context. That is, both the health and economic senses are meaningful. In some instances, the speaker leverages the ambiguity (as a pun or idiom) to drive a 54 rhetorical effect. 109 The diachronic analysis reveals the polyseme, “ recover ” and its nominalized form, “ recovery” entered the English lexicon very early. Its etymology traces its roots to the Latin verb, “recuperare” meaning “to take back.” In comparison to its cognate “ recuperate ” the semantic evolution of “ recovery ”challenges our thinking about the unidirectional nature of source/target metaphorical mapping and semantic change. Using quantitative methods, the research also shows that the metaphor has retained its efficacy in economic discourse into modern contexts. The data shows that metaphor emerges during times of economic stress or financial failure. Linguistically, we find THE ECONOMY IS A PATIENT metaphor is a highly complex conceptual blend derived from health, economic, life, well-being conceptual domains. The blend derives attributes from extensions of the integrated conceptual network including the larger anthropomorphic domain, THE ECONOMY IS AN ORGANISM. As a result, the metaphor has nearly a limitless amount of cognitive content from which to draw. Jonathan Gil Harris likens the health-economy conceptual blend to a double helix 110 of mutually reinforcing semantic domains. Harris’ DNA metaphor is very appropriate; the research suggests that the metaphor contains significant 109 See also Billig and MacMillan’s “Metaphor, Idiom and Ideology: the Search for ‘No Smoking Guns’ Across Time”; Goatly’s Washing the Brain: Metaphor and Hidden Ideology ;Horn’s “Idioms, Metaphors and Syntactic Mobility”; and the Language Teaching Research article on idioms and etymology by Boers, et al. 110 This effective metaphor for the health-economy conceptual blend was coined by Jonathan Gil Harris in his book, Sick Economies , which will be discussed in this chapter. 55 generational content or semantic freight preserved in the cognitive and cultural encyclopedic memory of the discourse speakers. The research reveals a long history of the metaphor as a vehicle for invoking political calls of action to the dominant players in the society (either centralized government or the highly powerful financial institutions). The calls request an intervention to restore economic stability. Freidrich’s “linguaculture” framework (Language, Political Economy and Ideology) enables us to deconstruct the metaphor at a given a period of time. The emergence of THE ECONOMY IS A PATIENT in times of financial tension reveals a political economic ideology deeply embedded in the psyche of Western society. Freidrich’s framework reveals how the speakers make the most of the emotive power of the metaphor, including the polysemes, derived from the anthropomorphic health domain to evoke action of the speaking community. A diachronic analysis of the metaphor indicates that THE ECONOMY IS A PATIENT has retained its salience and relevancy over time in spite of the evolving economic environments. The polyseme “ recover ” and its nominalized form “ recovery” has similarly retained its efficacy in economic discourse into modern contexts. The Twenty-first and Twentieth Centuries – The 2008 Economic Crisis and the Market Crash of 1929 We saw earlier The Economist’ s depiction of Ben Bernanke’s description of US economy as a “ patient ” suffering a near “ heart attack” from “clogged” 56 “arteries.” 111 THE ECONOMY IS A PATIENT metaphor was so powerful, an extension of the metaphor found its way into the Joint Economic Committee Research Report record filed by Congressman Jim Saxton when he wrote, “During the week of September 13-20, 2008, the United States confronted the worst global financial crisis in almost a century. Credit markets, which are the circulatory system of the U.S. economy, seized up .” 112 The metaphor was significant in setting the tone and framing discussions about the crisis. In an analysis of President Barack Obama’s speeches during his first hundred days, Joshua Scacco, while a graduate student at Georgetown University, cited a number of health metaphors in Obama’s language. Scacco writes (italics are his), 113 Obama talks of the role of government (as a doctor) to “stem the spread of foreclosures and falling home values,” to “ clean up the credit crisis that has severely weakened our financial system,” and to “ quarantine ”institutions that pose “ a systemic risk that could bring down the financial system” (ARRA2-17-09; SOTN2-24-09; EconPress3-24-09). What emerges here is an image of an active government-doctor role that aggressively treats, quarantines sick financial institutions, and acts “boldly and wisely” to not only stop the spreading crisis, but to reverse its effects (SOTN2-24-09). The president must, as the doctor-in-chief, give a prognosis and the effect that his policy treatments will have on the economic patient. “Recovery” becomes a 111 “America’s Bail-out Plan: The Doctors’ Bill,” The Economist , 25 Sep 2008, 10 March 2010 < >. 112 United States, Congress Joint Economic Committee, “Financial Meltdown and Policy Response: Research Report #110-25,”by Congressman Jim Saxton, September 2008, 25 April 2010 113 Scacco’s speeches cited in this excerpt are as follows: Remarks on the Economy & Executive Pay (EEP2-4-09); Remarks Upon Signing the American Recovery & Reinvestment Act (ARRA2-17-09); The State of the Nation: Address to a Joint Session of Congress (SOTN2-24-09); More on the Economy: The Second Prime-Time Press Conference (EconPress3-24-09); Progress of the American Economy: Speech at Georgetown University (GU4-14-09). 57 recurring word, in his speeches and the labeling of his “stimulus” legislation, The American Recovery and Reinvestment Act. His stimulus is the equivalent of a shot-in-the-arm for the economy. Obama speaks of moving “this economy from recession to recovery , and ultimately to prosperity ,” “long-term fiscal health ,” the stimulus as “ not merely a prescription for short-term spending,” and his desire to “ heal our financial system” (EconPress3-24-09; SOTN2-24-09; EEP2-4-09; GU4-14-09). Under media criticism in the early days of his administration for being economically “pessimistic,” the language of recovery is positive, forward looking, and engenders public confidence in the economy and Obama’s policies. 114 Scacco’s observations illustrate the salience of the metaphor in Obama’s political rhetoric and its power to shape the ideology and collective thinking of a nation from the highest ranking official. Popular press released numerous books on the financial crisis. John R. Talbott’s Contagion: The Financial Epidemic That is Sweeping the Global Economy and How to Protect Yourself From It , leverages THE ECONOMY IS A PATIENT metaphor extensively. Talbott’s use of the metaphor is set in context with the notion that the economic crisis is predicated on a contagious illness of greed on a national and global scale. The contagion has disabled the economy and a remedy is clearly called for to mitigate its further spread. Note how Talbott weaves the metaphor and the polyseme “ recover ” in the short excerpt below: So it is important to understand the underlying health of the U.S. economy to see just how well positioned it is to take this housing hit. If the housing crisis causes the expected global recession, it might also 114 Joshua Scacco, “Shaping Economic Reality: A Critical Metaphor Analysis of President Barack Obama’s Economic Language During His First 100 Days,” Gnovis Journal , ed. Lydia Kelow-Bennett, Georgetown University 10.1 (2009) Web 14 January 2011. . 58 further damage the U.S.’s financial position so that it makes the recovery more distant into the future, The U.S. government is an important player in containing the housing crisis, preventing the contagion to other asset classes and other markets and, hopefully, ending this financial crisis. 115 Talbott’s call to action is directed to the U.S. government to “treat” the economy and stem the contagion. As we will see later in this chapter, his metaphor of a financial epidemic emerged centuries before. During the timeframe between Bernanke’s meeting with Congress and the end of 2009, a period less than 18 months, the polyseme, “ recovery, ” surged in use. Brigham Young University’s Corpus of Contemporary American English (COCA) 116 illustrates this phenomenon in Figure 2: Figure 2. COCA occurrences of “ recover ” over time . The graph illustrates the frequency of occurrences of the word “ recover ” and its cognates (such as “ recovery, ” “ recovering, ” etc.) in the COCA corpus. < americancorpus.org/ >. The tokens represented in the graph represented in Figure 2 are tallied only if collocated with strings related to the “ economy. ” The data reveal the word’s 115 John R. Talbott, Contagion (Hoboken, NJ, John Wiley & Sons, 2009) 72. 116 Corpus of Contemporary American English (COCA) Brigham Young University, ed. Mark Davies, 11 April 2011 < >. SEE ALL SUB-SECTIONS AT ONCE 8.63 3.21 0.00 0.00 0.00 0.65 2.13 0.68 0.02 2.29 PER MIL 219 328 00056 185 61 2206 EQ FR 2010-2011 2005-2009 2000-2004 1995-1999 1990-1994 ACADEMIC NEWSPAPER MAGAZINE FICTION SPOKEN SECTION SEE ALL SUB-SECTIONS AT ONCE 8.63 3.21 0.00 0.00 0.00 0.65 2.13 0.68 0.02 2.29 PER MIL 219 328 00056 185 61 2206 FR EQ 2010-2011 2005-2009 2000-2004 1995-1999 1990-1994 ACADEMIC NEWSPAPER MAGAZINE SECTION FICTION SPOKEN 59 frequency increased sharply in total of all forms i.e., spoken, fiction, magazine, newspaper as well as academic literature, between 2005-9 timeframe and 2010-2011, the years immediately following the 2008 market crisis. Similarly, the Google Trends 117 tool, which illustrates Internet search and News reference volume, corroborates these data. The graph in Figure 3 demonstrates the frequencies for the string, “ economic recovery ”: Figure 3. Google Trends data for “ economic recovery. ” This graphic represents Google Trends search results for text string, “ economic recovery. ” Note the spike in Google search volume for the term “ economic recovery ” just following the market crisis in the second half of 2008 and early first quarter of 2009. This was a time of significant market volatility and public anxiety over the state of the global financial environment. 117 Google Trends, Google, Inc., Mountain View, CA, 11 April 2011 . 60 The next graph represents Google Trends results for text string “ financial recovery ”: Figure 4. Google Trends data for “ financial recovery. ” This graphic represents Google Trends search results for text string, “ financial recovery. ” The frequency of “ recovery ” in the economic domain remains at relatively high levels at the time of this writing. Notably, the world economy has yet to be restored to pre-2008 levels. While not statistically correlated, the data does suggest an increase in the use of the term (collocated with terms such as “economic ” and “ financial ”) during the 2008 market crisis and the subsequent years. By contrast, the metaphor used in popular press immediately following the market crash of 1929 was of a “violent storm” rather than a “sick patient.” 61 Observe the storm metaphor in the following excerpts (italics are mine throughout the following section): With speculative nerves rubbed raw under the persistent hammering of bearish traders, a renewed wave of public liquidation swept over the stock market yesterday… The World , October 20, 1929. 118 After the stock market had come crashing down again in a veritable deluge of forced and hysterical liquidation… The World , October 29, 1929. 119 the second hurricane of liquidation… the storm has now blown itself out. New York Times , October 29, 1929. 120 The Wall Street tornado , although this country was very slightly affected directly, has had a serious indirect effect on the Bourse and now on industry in general. The Economist, November 9, 1929. 121 “Recover ” or “ recovery ” occurs infrequently in the writings of the time. It is not until later in the Depression that we find references to “ recovery ” in the press. The word “ recovery ” begins to emerge following years of sustained financial stress after the Market Crash of 1929. Again, the Brigham Young corpus data illustrate the rise in the frequency of the word “ recover ” and its cognates during the financial crisis. The analysis in Figure 5 draws historical data from the Corpus of Historical 118 “American Experience: The Crash of 1929: Primary Resources: Headlines,” Public Broadcasting Service (PBS), 24 Feb 2010 < features/primary-resources/crash-headlines/ >. 119 “American Experience: The Crash” < features/primary-resources/crash-headlines/ >. 120 Floyd Norris, “Looking Back at the Crash of ’29,” The New York Times on the Web 1999, 24 Feb 2010 121 “France, Politics, Budget, Wall Street Repercussions, Money,” The Economist 9November 1929 . 62 American English (COHA). 122 1.25 35 1990 3.32 84 1980 2.23 53 1970 0.92 22 1960 1.14 28 1950 1.36 33 1940 2.93 72 1930 0.58 15 1920 0.00 0 1910 SEE ALL YEARS AT ONCE 0.98 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 PER MIL 29 000000000FREQ 2000 1900 1890 1870 1860 1850 1840 1830 1820 1810 SECTION 1.25 35 1990 3.32 84 1980 2.23 53 1970 0.92 22 1960 1.14 28 1950 1.36 33 1940 2.93 72 1930 0.58 15 1920 0.00 0 1910 SEE ALL YEARS AT ONCE 0.98 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 PER MIL 29 000000000FREQ 2000 1900 1890 1870 1860 1850 1840 1830 1820 1810 SECTION SECTION 1930 TOKENS 72 SIZE 24,602,615 PER MILLION 2.93 Figure 5. COHA data for collocated occurrences of “ recovery ” and “economy. ” This graph reflects only occurrences of “ recover ” when collocated with forms of “ economy. ” < >. In Figure 5, note the significant increase in the frequency from 1556 tokens per million words during the 1920 decade to 2358 tokens in the 1930 decade, the era of the Depression. The word also rose in use during the 1980’s, a time when there were a series of financial crises including the savings and loan crisis and the Market Crash of 1987. Note that the frequency of the word fell during the decades that followed the crises, only to spike again in use when the economy is under stress. These data suggest there is a notable rise in use of the word “ recovery ” during the times of financial crisis. In Irving Fisher’s 1933 article in Econometrica , we see a new attestation of THE ECONOMY IS A PATIENT metaphor. Irving writes about the lax U.S. government policy during the Depression and accuses the leaders of allowing the economy to come close to a metaphorical death: 122 Corpus of Historical American English (COHA) Brigham Young University, ed. Mark Davies, 20 April 2011 < coha/>. 63 If even then our rulers should still have insisted on ‘leaving recovery to nature’ and should still have refused to inflate in any way, should vainly have tried to balance the budget and discharge more government employees, to raise taxes, to float, or try to float, more loans, they would have ceased to be our rulers. For we would have insolvency of our national government itself, and probably some form of political revolution without waiting for the next legal election. The mid-west farmers had already begun to defy the law. If all this is true, it would be as silly and immoral to ‘let nature take her course’ as for a physician to neglect a case of pneumonia. It would also be a libel on economic science, which has it therapeutics as truly as medical science. If reflation can now so easily and quickly reverse the deadly down-swing of deflation after nearly four years, when it was gathering increased momentum, it would have been easier, and at any time, to have stopped it earlier. In fact, under President Hoover, recovery was apparently well started by the Deferral Resource open-market purchase, which revived prices and business from May to September 1932. The efforts were not kept up and recovery was stopped by various circumstances, including the political ‘campaign of fear.’ It would have been still easier to have prevented the depression almost altogether. In fact, in my opinion, this would have been done had Governor Strong of the Federal Reserve Bank of New York loved, or had his policies been embraced by other banks and the Federal Reserve Board and pursue consistently after his death. In that case, there would have been nothing worse than the first crash. We would have had the debt disease , but not the dollar disease – the bad cold but not pneumonia . 123 Note the emergence of “ recovery ” in the first line and the references to the government as the neglectful “physician.” Irving’s language reveals his ideology of central government oversight as a means to prevent severe swings in the economy’s health and to avert market crashes. 123 Irving Fisher, “The Debt-Deflation Theory of Great Depressions,” Econometrica: Journal of the Econometric Society 1.4 (October, 1933): 347. 64 Similarly, John Maynard Keynes, in reflecting on the protraction of the Depression, writes a retrospective view of the expanding economy prior to the Crash. Like Irving, he advocates a more vigilant monitoring of the economy and sensible intervention. However, he injects THE ECONOMY IS A PATIENT metaphor in warning that the wrong intervention or metaphorical remedy to prevent economic disaster can have deleterious effects if too extreme (italics are mine): Correct foresight would have brought down the marginal efficiency of capital to an unprecedentedly low figure; so that the ‘boom’ could not have continued on a sound basis except with a very low long-term rate of interest, and an avoidance of misdirected investment in the particular directions which were in danger of being over-exploited. In fact, the rate of interest was high enough to deter new investment except in those particular directions which were under the influence of speculative excitement and, therefore, in special danger of being over-exploited; and a rate of interest, high enough to overcome the speculative excitement, would have checked, at the same time, every kind of reasonable new investment. Thus an increase in the rate of interest, as a remedy for the state of affairs arising out of a prolonged period of abnormally heavy new investment, belongs to the species of remedy which cures the disease by killing the patient . 124 Keynes uses “ recovery ” a total of eleven times in the treatise. In each instance “recovery ” is collocated with other polysemes from the health-economic blended conceptual domain (e.g., slump, weakening, collapse, chronic condition, sagging, aid, remedy, relief ). Note the example below: If a reduction in the rate of interest was capable of proving an effective remedy by itself; it might be possible 124 John Maynard Keynes, The General Theory of Employment Interest and Money 1936 (New York: Classic Books America, 2009) 277. 65 to achieve a recovery without the elapse of any considerable interval of time and by means more or less directly under the control of the monetary authority. But, in fact, this is not usually the case; and it is not so easy to revive the marginal efficiency of capital, determined, as it is, by the uncontrollable and disobedient psychology of the business world. It is the return of confidence, to speak in ordinary language, which is so insusceptible to control in an economy of individualistic capitalism. This is the aspect of the slump which bankers and business men have been right in emphasizing, and which the economists who have put their faith in a ‘purely monetary’ remedy have underestimated. 125 True to his ideology of free-market capitalism, Keynes warns of overtreating the patient through heavily centralized monetary involvement. As we have seen, the above analysis of the polyseme “ recovery ” suggests that THE ECONOMY IS A PATIENT metaphor appears to increase during times of significant economic pressure. This would suggest that the metaphor is highly effective in this environment. While this finding is notable, using Friedrich’s framework can provide further insight to its prominence in discourse during these financial crises. For example, why should the metaphor not appear at the onset of the 1929 Market Crash but emerge later as the Depression wore on? What cognitive domains underlie the metaphor of an economic storm? And, does the fact that Bernanke found the ECONOMY IS A PATIENT metaphor appropriate for his message to Congress eighty years later have any significance to this analysis? A close analysis of the texts provided above reveal two common political ideologies in U.S. economics in the twentieth and twenty-first centuries: 1) the free market concept which believes the economy operates and grows under 125 Keynes 271. 66 natural and self-regulating means has a boundary beyond which it will collapse, 2) there is a moral and social underpinning to ensure that, when the economy can not regulate its self-control and edges dangerously close to that boundary, an intervention by central government is required to correct the situation immediately. Unlike the “storm” metaphor, which brings the cognitive attributes of an “outside force,” “beyond one’s control” and “temporary danger that will pass,” the metaphor, THE ECONOMY IS A PATIENT, evokes a human sense of urgency to act from attributes in the health domain. Lakoff, Boers and Charteris-Black acknowledge that the Western economies we study are active in a culture where “fitness” and “self-discipline” are highly regarded. It is not surprising, then, that the political economy reflects the ideology of self-restraint, which, when not in force, threatens the survival of the society. There is an urgency to act. Bernanke espouses this ideology as evidenced by his use of the metaphor during the 2008 crisis. THE ECONOMY IS A PATIENT metaphor has maintained its relevancy throughout the recent economic crises and from prior financial calamities as the following data reveals. Nineteenth – late Eighteenth Century Contexts Texts from the nineteenth century contain rich examples of THE ECONOMY IS A PATIENT during the financial strains of the time. An entry in an 1891 “The Economic Journal” reads with a tone eerily reminiscent of the 2008 economic crisis and similar fall of the financial institution, Lehman Brothers: “The year 1890 will be remembered as the year in which the great house of Baring 67 Brothers and Co. collapsed.” 126 Note THE ECONOMY AS A PATIENT metaphor and the use of the dead metaphor “ recovery ” (italics and underline are mine): The Stock Exchange opened with a feeling of unusual uneasiness … This feeling of vague terror became much more precise on Wednesday… The crisis once over the money market rapidly became very easy, and a recovery 127 took place on the Stock Exchange, o the repurchase by operator for the fall of par of the stock sold by them in view of a highly probable catastrophe 128 wing to . The market collapse of 1890 in London was precipitated by the default of a loan issued to Argentina years earlier. Following a revolution in Argentina, the loan default created significant panic in the Bank. In the text, the anonymous writer critiques the Baring Brothers for not taking the necessary precautions and discipline in underwriting the loan. Other banks had followed Baring Brothers’ lead and similarly invested in the country assuming Baring had conducted a rigorous analysis of the risks. They are not alone in the blame, the writer comments, noting that the British economic policy enabled such risky behavior. He writes, The banks have got thoroughly into the habit of regarding the amount of the reserve of the Bank of England as of no importance, although it is the sole fund in existence to enable them to meet their liabilities if called upon. They fully believe that, if the worst comes to the worst, the Bank 126 “The Crisis of 1890,” The Economic Journal 1.1 (March 1891): 192. 127 Note the semantic ambiguity of ‘ recovery’ in this text; it is unclear whether the metaphor is being used in the financial/economic sense or has continued the health sense from the prior paragraph. We will see this phenomenon again in other excerpts using ‘ recover’ in economic discourse. 128 “The Crisis of 1890” 194-195. 68 will not be allowed to fail, 129 as an ordinary bank would fail if the whole of its cash were gone, and the experience of more than one commercial crisis has shown that this calculation is a safe one, and has consequently encouraged the banks to adhere to their present dangerous practice. 130 Following Baring Brothers’ collapse and its ripple effect on other banks in the British system, foreign funds were infused to stabilize the economy and avert subsequent panic. In describing the stabilization, the writer introduces the word “recovery ” to describe the resulting return to normalcy. The writer explicitly states that action of this kind is necessary due to the severity of the situation and the impact it would have on the entire banking system. Reeling from the Baring example, the writer exhorts in closing, “This is not a thoroughly healthy state of things.” 131 Like the twentieth century examples, the Baring excerpts provide another example of THE ECONOMY IS A PATIENT metaphor emerging following a harrowing crisis in the financial system. Again we see, the term, “ recovery, ”found within the context of an intervention designed to correct a dire situation. And, once again, “ recovery ” is collocated with words from the health domain, i.e., “not thoroughly a health y state of things.” As noted earlier, “ recovery ” is a 129 Note the similarity of this language to the “too big to fail” phrase from the 2008 financial crisis. While written in 2004, the seminal book, Too Big to Fail: The Hazards of Bank Bailouts by Gary Stern, Ron Feldman and Paul Volcker solidified the phrase and became entrenched in 2008 (so frequently used, it was often shortened to “TBTF”). This was epitomized by the award-winning book, Too Big to Fail: The Inside Story of How Wall Street and Washington Fought to Save the Financial System – and Themselves , by Andrew Ross Sorkin published in 2009. 130 “The Crisis of 1890” 193. 131 “The Crisis of 1890” 196. 69 conventional metaphor. The word can be interpreted in either the health domain or in the financial domain. In the excerpt above, we are unclear which sense is intended. However, this ambiguity can be an effective rhetorical device. In either sense, the effect is dramatic and emotive. The writer conveys an underlying ideology through this seemingly dead metaphor. Clearly, he declares the government has a role in maintaining the safety of securities. In a culture of free markets, good economic public policy should enable good behaviors that prevent similar crises and “threatened panic.” 132 By contrast, when bad policy is instituted as in the given example of the Bankruptcy Act of 1883 (where insolvent firms can continue to conduct business), the “state of things” becomes unhealthy. The economy, in effect, requires an intervention to stabilize and return to health. If left on its own, the author is warning, the outcome would be disastrous. The treatise using THE ECONOMY IS A PATIENT metaphor is a rhetorical call for help. Mathew Carey was a prolific American writer of economic treatises in the late-eighteenth and early-nineteenth century. Born in Ireland, Carey moved to Philadelphia to open a printing shop and newspaper. He had immediate access to virtually unlimited printing resources. Carey composed a series of economic analyses advocating an ideology that supported central banking for the new country’s fledgling economy. His use of THE ECONOMY IS A PATIENT metaphor was most vibrant in his Essays in Banking written in 1816. Carey’s book was written in response to the economic depression following the War of 132 “The Crisis of 1890” 195. 70 Critical of banks who misuse their power by controlling and manipulating funds at their discretion, Carey applied numerous health-related metaphors to his rhetoric. Note the use of “ recovery ” in the excerpts from his “Essays on Banking” (italics and underline are mine): 133 And the commercial world now appears to be in a convalescent state. But it will require a long period, to recover the vigour, and tone, and elasticity which were destroyed by the depleting system so long pursued. (vii) And I confess such an epidemic appears to me to prevail at present. There is one proof, cogent and irresistible. It is, that the boards of directors of the Banks, sober, grave, and many of them intelligent and estimable men, are tampering with, and trying a depletory system on, patients , already exhausted by phlebotomy , and who only require a liberal and generous treatment , to restore them to perfect sanity. (xii) It is obvious, as I have stated, that these subjects affect deeply and vitally the interest, mediately or immediately, not merely of the trading part of the community, but of the whole mass of our citizens, and even of a great portion of the State, of which Philadelphia is the heart. It is impossible that the heart can be materially disordered without the extremities partaking of the ailment . (14) Whether the Directors of the banks were the first to feel the terror, or were infected by with disorder from abroad, I cannot discover. Suffice it to say, that the panic was sensibly felt at the boards: and the necessity of applying an immediate and powerful remedy , was insisted on and appeared to be admitted as incontrovertible. (29) But unfortunately a determination was formed to apply a violent remedy to a disorder , whose existence to any considerable extent was not clearly established, and which a mild regimen would have radically removed in a short space of time. (31) Unfortunately the remedy to which recourse was had, 133 Mathew Carey, Essays on Banking (Philadelphia, 1816). 71 was curtailment. How much more reasonable, and sound, and worthy of enlightened men, deserving to be entrusted with banking establishments, would it have been, to have sold a large portion of this extravagant amount of stock, and thus absorbed the superabundant bank notes.” This would have averted the stroke which that fatal measure gave to the hopes and prospects of its citizens – and to the commerce and prosperity of the city. (75) That had the banks sold the whole or chief part of this stock, the awful depreciation of property – the paralysis of industry – the stagnation of trade, commerce and manufactures – which have arisen from the injudicious measures of curtailment, would have been avoided. (89) The banks may on this subject derive a useful lesson from the case recorded of a man, who, when in a high state of health , was desirous of improving his constitution, and put himself into the hands of a quack doctor , who soon closed his career. The dying man ordered for his tombstone the following admonitory epitaph: “I was well. I would be better. Here I am.” (132) Carey’s use of THE ECONOMY IS A PATIENT metaphor found its way into a number of publically delivered speeches. In his “Address Delivered Before the Philadelphia Society for Promoting Agriculture on the Twentieth of July, 1824,” we see Carey’s metaphor most clearly (italics and underlines are mine): Nothing is so pernicious to a patient , whether a nation or an individual, when labouring over serious disorders , than a belief of the existence of robust health – and the more morbid the state, the greater the danger of the error. This point, therefore, demands a severe scrutiny, which, although an ungracious office, I venture to undertake, because a serious conviction of disorder is a necessary preliminary to the application of any remedies . 134 THE ECONOMY IS A PATIENT metaphor is used so frequently in his 134 Mathew Carey, “Address Delivered Before the Philadelphia Society for Promoting Agriculture on the Twentieth of July, 1824”, Hume Tracts (1827): 23 JSTOR Accessed 13 June 2010. 72 treatises that it suggests he found it not only an apt portrayal of the economy, but also conveyed his ideological message well. The metaphor is found most regularly in his chastisement of his adopted country for the conditions of those living in poverty. His use of the metaphor illustrates the profound connection between employment, financial sustainability and health. He writes, “Every individual industriously employed, in a useful occupation, has an indisputable claim to healthful and comfortable support.” 135 He extends the metaphor to the economy at large, writing: “This state of things calls for a remedy whereby burdens, the benefits of which are enjoyed by all, should be more equally distributed.” 136 In each these instances, Carey uses the metaphor in the context of a financial crisis. The young nation faced numerous financial crises following the War of Independence and the War of 1812. Carey’s most productive writings came at times that roused his disgust and rage against the established banking system and wealthy elite. His use of the anthropomorphic metaphor for the financial system bridges the social and the moral conscience with economic domain. In the midst of the Depression of 1819, he writes of recommendations to reduce imports to restart local economies in emphatic metaphoric language: To depend on this salutary effect being produced by the restorations of the spirit of economy which is to result from general distress, or from the forbearance of our merchants to import, is to allow a violent fever to rage in the body politic ,and exhaust itself, or the national strength, without the 135 Mathew Carey, “Essays on the Public Charities of Philadelphia,” Hume Tracts (1829): viii. 136 Carey viii. 73 application of any remedy to arrest its destructive career. 137 Later, in his search for the root cause for the economic collapse he writes, Several causes, we found, had combined to produce this calamitous result. The prosperity of the country had engendered a spirit of extravagance - and the inordinate spirit of banking, carried in many cases to a most culpable excess, had done much mischief. But the great paramount evil, in comparison with which all the rest sink into insignificance, is the immoderate extent of our importations, whereby we are involved in debts, for which our produce, at the highest prices, would have been inadequate to pay; and their great recent reduction, of course, increased our disabilities. The evils arising from other sources would have gradually cured themselves –or involved in ruin only deluded parties. Whereas the loss of our industry, the drain of our specie, and the consequent impoverishment of our country, affect all classes of citizens, the economical and the extravagant – the labourer, the artisan, the cultivator of the soil, as well as the landholder, the manufacturer, the trader and the merchant. On the most mature consideration we have given the subject, we are persuaded that the only radical remedy for those evils is to limit the importation of such articles as we can manufacture ourselves, and thus foster our domestic industry. Other measures may be adopted to co-operate and aid in this great work. But without the grand restorative of ‘buying less than we sell,’ which a proper tariff alone can effect (sic), they will operate as mere palliatives of an evil whose immense extent and magnitude require prompt and decisive remedies . All our efforts have been directed to convince our fellow citizens of this truth, so important to their virtue, their happiness, their independence. “We warmly recommend associations throughout the county to carry its salutary objects into operation, and thus arrest the impoverishment of our citizens. Should they be general – should the plan proposed be faithfully adhered to, and the tariff be properly modified – the thick clouds that environ our horizon will disappear – the sun 137 Mathew Carey, et al, Addresses of the Philadelphia Society for the Promotion of National Industry (Philadelphia: James Maxwell Publisher, 1820): 79. 74 of prosperity will again shine on us – we shall recover from our disastrous situation – and only remember our sufferings to warn to avoid the fatal source, a false and mistaken policy, from whence they burst forth on us with destructive violence. Delaware claims the high honour of having first adopted the federal constitution. It will be another just cause of pride, that she has taken the lead on this occasion, more particularly should the sound views she has given of the causes of our distresses, and the excellent remedies she has prescribed, lead to their radical cure . 138 Carey’s use of THE ECONOMY IS A PATIENT metaphor reflects his moral judgment that excess spending and the taking on of debt has brought the country in need of a cure. His language echoes language from his first published work in the United States in 1793, a personal account of the Yellow Fever epidemic in Philadelphia. 139 He opens his narrative with a brief portrayal of the ebullient Philadelphia economy prior to the onset of the epidemic. Within a matter of weeks, he notes how dramatically the Philadelphia citizens are now suffering both physically and economically. Following a painfully detailed account of Yellow Fever symptoms, illness and deaths in the text, THE ECONOMY IS A PATIENT metaphor is used most fittingly in its blended conceptual form (italics and underlines are mine): Business, therefore, has languished in many parts of the union, and it is probable, that, considering the matter merely in a commercial point of light, the shock caused by the fever , has been felt far to the south and west of this State. 140 138 Mathew Carey, et al. 97-98. 139 Mathew Carey, Short Account of the Malignant Fever which Prevailed in Philadelphia, 1793 (Philadelphia, 1793). 140 Mathew Carey, 1793. 75 Carey uses the word “ recover ” twenty-five times in this short piece of only forty thousand words. All occurrences are used solely in a health context. The economy had not yet fully returned to its prior state at the time of his writing. A careful analysis of Carey’s opening paragraph to the “State of Philadelphia previous to the appearance of the Malignant Fever” is more than a literary device to illustrate a contrast between the environment before and after the Yellow Fever outbreak. Jennifer Baker believes Carey’s opening is actually a rhetorical opinion piece opportunistically positioned to suggest the epidemic was a form of God’s retribution for Philadelphia’s overindulgence and pride. She writes, Mathew Carey’s famous chronicle of the 1793 yellow fever opens with a telling indictment of the city’s financial overreaching… Carey wonders, hesitantly, if the plague was not, in fact divine punishment for the city’s bloated economy: ‘And although it were presumption to attempt to scan the decrees of heaven, yet few, I believe, will pretend to deny, that something was wanting to humble the pride of a city, which was running on in full canter, to the goals of prodigality and dissipation.’ Carey considers the prevailing medical theories about the plague’s origins, but here he cannot resist assigning providential significance to the fact that credit-based expansion was leveled by catastrophe. 141 Baker argues that Carey’s text reveals a prevailing ideology that ties a moral “cause and effect” connection between the yellow fever epidemic and the crisis in the Philadelphian economy. The result is an effective metaphor that carries an old yet prevalent view that excess invokes God’s punishment. In a culture of expanding prosperity and growing demand for foreign trade, the notion that the 141 Jennifer J. Baker, Securing the Commonwealth (Baltimore: Johns Hopkins University Press, 2005) 119. 76 epidemic was caused by the influx of goods through Philadelphia’s docks was too fitting for Carey to resist. His ideology was simple: restrain from purchasing foreign luxury goods while the poor at home suffer financially. 142 The economic crisis spurred by the Yellow Fever epidemic in Philadelphia was followed by additional monetary crises, thus giving Mathew Carey a lifetime of material for his essays. He easily found efficacy for THE ECONOMY IS A PATIENT metaphor from his early experience in 1793. While Carey’s work was predominantly economic in focus, Jennifer Baker is quick to point out that many other writers of the time took THE ECONOMY IS A PATIENT metaphor to a higher, more explicitly allegorical form in fiction and plays. Baker explores Charles Brockden Brown’s fictional work, “Arthur Mervyn,” which also uses the Yellow Fever epidemic as a literary device to “make a dramatic statement about the potentially toxic effects of burgeoning commerce on the city’s moral fiber.” 143 While some ideologies, like Brockden Brown’s, were rooted in moral post-Puritan era beliefs, merchants and businessmen wrote their exposés with a simple and genuine concern for the poor among them. Consider the following excerpt from Patrick Colquhoun’s 1806 “A Treatise on Indigence: Exhibiting a 142 For an in-depth analysis of the interconnection of health, economic and religious metaphors in Biblical texts, consider Seong-Hyuk Hong’s The Metaphor or Illness and Healing in Hosea and Its Significance in the Socio-Economic Context of Eighth-Century Israel and Judah .(New York: Peter Lang Publishing, 2006): 148. While out of the scope of this thesis, Hong’s analysis points to metaphors in the Bible’s book of Hosea that connect health, and socio-economic metaphors in the context of morality and religiousity. Hong writes, “the foreign nature of monarchy in Hosea’s eyes is characterized by commercialization and internationalization through foreign alliances, centralization of political and economic power to maximize the leaders’ profits at the cost of ordinary subjects….” These themes are reminiscent of seventeenth and eighteenth century the mercantilists. 143 Baker 120. 77 General view of the National Resources for Productive Labour Propositions for Ameliorating the Condition of the Poor by Regulations of Political Economy” and observe the polysemy of “ recover” and “ recovery” in the text (italics and underlines are mine): Here, indeed, the pauper cannot be removed until his recovery from sickness , and the expense incurred during this interval is to be refunded by the parish where he is legally settled; but in order to recover this expense the object of it must be actually removed, although sufficiently recovered 144 to result his labour, and when so removed he must never again return to the parish where he was in a situation to gain a subsistence, on pain for being treated as a rogue and vag abond. This is the individual punished and the country deprived of his labour, where it was most wanted and could be rendered the most productive: -- for what cause? merely because debility and distress came upon him for a time, requiring temporary relief, which, however, by the act could not be recovered from his own parish, until the additional expense was incurred of removing his as a pauper. Surely this never could have been the intention of the legislature. After the individual ceased to be chargeable, the money advanced as a temporary relief , could have been recovered without making it a necessary condition that he should be actually removed. This act, while it extends a privilege to the labouring people during health , imposed the greatest of hardships upon them in the event of sickness requiring temporary relief . 145 This text illustrates unique characteristics of THE ECONOMY IS A PATIENT metaphor. In this example, “ recover” is used in the same paragraph as “dead” metaphors in two conceptual domains, financial and health, with no apparent 144 Note the ambiguity of the word “recovered” in this sentence. 145 Patrick Colquhoun, “A Treatise on Indigence: Exhibiting a General View of the National Resources for Productive Labour Propositions for Ameliorating the Condition of the Poor by Regulations of Political Economy,” Bristol Selected Pamphlet (London, 1806) 214. 78 ambiguity for the reader. How does the reader disambiguate the multiple senses when the context is inadequate in providing clues? Recall Williams’ research which reveals that, in processing polysemes, the multiple senses are activated simultaneously in cognition. As in this case of the pauper recovering from illness and recovering his livelihood, Williams’ research suggests that in the encyclopedic knowledge, both health and economic domains are processing. As partners in this discourse, the domains remain active and effective. The metaphor is both cognitively salient and rhetorically robust. Eighteenth-and-Seventeenth-Century Contexts The eighteenth century saw a proliferation of THE ECONOMY IS A PATIENT metaphor as new economic treatises met the needs of the expanding world view. In the United States, leaders found their democratic ideologies challenged by the troubled economy following the War of Independence and the new governance structure of the new nation. The most notable composition is Adam Smith’s The Wealth of Nations written in 1776. Smith’s language is clear, concise and to the point. So, when he uses metaphors in his treatise, it is safe to assume it is not for literary value but to serve a pragmatic purpose. Smith does employ THE ECONOMY IS A PATIENT metaphor as an efficient tool to explain his ideology. Note how he leverages the polysemy of the “body politic” and “disease” as an effective trope (all italics and underlines in this section are mine): This frugality and good conduct, however, is, upon most occasions, it appears from experience, sufficient to compensate, not only the private prodigality and misconduct of individuals, but the public extravagance of 79 government. The uniform, constant, and uninterrupted effort of every man to better his condition, the principle from which public and national, as well as private opulence is originally derived, is frequently powerful enough to maintain the natural progress of things towards improvement, in spite both of the extravagance of government, and of the greatest errors of administration. Like the unknown principle of animal life, it frequently restores health and vigour to the constitution, in spite not only of the disease , but of the absurd prescriptions of the doctor . 146 His ideology is equally clear: “absurd prescriptions of the doctor” can disrupt the self-correcting nature of his free market economy. There is no call to action. No plea for a governing physician to intervene and assist the patient. No use of the word, “ recovery .” He states his case for laissez-faire trade policy by extending THE ECONOMY IS A PATIENT as well: The monopoly of the colony trade, besides, by forcing towards it a much greater proportion of the capital of Great Britain than what would naturally have gone to it, seems to have broken altogether that natural balance which would otherwise have taken place among all the different branches of British industry… But the whole system of her industry and commerce has thereby been rendered less secure; the whole state of her body politic less healthful than it otherwise would have been. In her present condition, Great Britain resembles one of those unwholesome bodies in which some of the vital parts are overgrown, and which, upon that account, are liable to many dangerous disorders , scarce incident to those in which all the parts are more properly proportioned. A small stop in that great blood-vessel , which has been artificially swelled beyond its natural dimensions, and through which an unnatural proportion of the industry and commerce of the country has been forced to circulate , is very likely to bring on the most dangerous disorders upon the whole body politic. The expectation of a rupture with the colonies, accordingly, has struck the people of Great 146 Adam Smith, The Wealth of Nations 1776, Web. Library of Economics and Liberty, 15 October 2010 Book II, 3.31. 80 Britain with more terror than they ever felt for a Spanish armada, or a French invasion. 147 Smith’s language exploits the anthropomorphic nature of the “body politic” and maps salient attributes of a bloated and unhealthful human body to Britain’s economic state (as an overweight monopoly). By extending the metaphor to convey the benefits of healthy circulation and blood-flow, Smith illustrates the virtue of a vigorous political economy that enables unhindered trade. Note in the following sample from Smith’s The Wealth of Nations how he characterizes the physician as constraining the patient with too strict a course of therapy. Smith’s exhortation is not a call for intervention. It is a call for in action. Let the economic body “remedy” itself: Some speculative physicians seem to have imagined that the health of the human body could be preserved only by a certain precise regimen of diet and exercise , of which every, the smallest violation, necessarily occasioned some degree of disease or disorder proportionate to the degree of the violation. Experience, however, would seem to shew, that the human body frequently preserves , to all appearance at least, the most perfect state of health under a vast variety of different regimens ; even under some which are generally believed to be very far from being perfectly wholesome. But the healthful state of the human body , it would seem, contains in itself some unknown principle of preservation, capable either of preventing or of correcting, in many respects, the bad effects even of a very faulty regimen. Mr Quesnai, who was himself a physician ,and a very speculative physician, seems to have entertained a notion of the same kind concerning the political body , and to have imagined that it would thrive and prosper only under a certain precise regimen, the exact regimen of perfect liberty and perfect justice. He seems not to have considered, that in the political body , the natural effort which every man is continually making to better his own condition, is a principle of preservation capable of 147 Smith, Book IV, 7.129. 81 preventing and correcting, in many respects, the bad effects of a political economy, in some degree both partial and oppressive. Such a political economy, though it no doubt retards more or less, is not always capable of stopping altogether, the natural progress of a nation towards wealth and prosperity, and still less of making it go backwards. If a nation could not prosper without the enjoyment of perfect liberty and perfect justice, there is not in the world a nation which could ever have prospered. In the political body , however, the wisdom of nature has fortunately made ample provision for remedying many of the bad effects of the folly and injustice of man; it the same manner as it has done in the natural body , for remedying those of his sloth and intemperance. 148 However, Smith’s The Wealth of Nations is not devoid of the words, “ recover ”and “ recovery. ” He uses them unambiguously in the financial sense as in “recovering a debt” or “earning back from an investment” as seen in the sample texts below: The uncertainty of recovering his money makes the lender exact the same usurious interest which is usually required from bankrupts.” (54) The high rate of interest among all Mahometan nations is accounted for by M. Montesquieu, not from their poverty, but partly from this, and partly from the difficulty of recovering the money. (54) Italy seems not to have gone backwards. The fall of Italy preceded the conquest of Peru. Since that time it seems rather to have recovered a little. (116). “When such farmers have a lease for a term of years, they may sometimes find it for their interest to lay out part of their capital in the further improvement of the farm; because they may sometimes expect to recover it, with a large profit, before the expiration of the lease. (217) Even in England, the country, perhaps of Europe, where the yeomanry has always been most respected, it was 148 Smith, Book IV, 9.28. 82 not till about the 14th of Henry VII, that the action of ejectment was invented, by which the tenant recovers ,not damages only, but possession, and in which his claim is not necessarily concluded by the uncertain decision of a single assize. (218) His tenants could agree to this upon one condition only, that they should be secured in their possession for such a term of years as might give them time to recover , with profit, whatever they should lay not in the further improvement of the land. (230) Smith’s use of “ recovery ” in its sole financial sense is a critical distinction from other economic treatises using THE ECONOMY IS A PATIENT metaphor. In spite of the fact that Smith uses THE ECONOMY IS A PATIENT metaphor extensively in The Wealth of Nations , why should he not use “ recovery ”polysemously as a rhetorical device? What can we infer from this distinction? Friedrich’s framework may give us some insight to answering this question. Friedrich indicates that an ideology is the output from the interaction between political economy, culture and language. The language used is a verbal process, a code in the context of a society. A close analysis of Smith’s language in the texts above reveals the new culture of his fledgling country. The United States of 1776 is stretching its new political beliefs of democratic self-governance into its fiscal and economic policies. Former embodied metaphors such as the “body politic” are still relevant. However, we see them in a new light. Adam Smith’s belief in the natural balance of supply and demand and, like the circulation of blood in the human body, the natural flow of trade and through the circulation of commerce find their way in his use of THE ECONOMY IS A PATIENT metaphor. The attributes within the integrated conceptual network are 83 now filtered through a new set of selection criteria. Smith has little use for a heavy-handed physician to help the patient recover. Nature is the ultimate healer. His ideology screens out the attributes pertaining to medical oversight and treatment. But why would Smith use “ remedy ” rather than “ recover? ” The answer may lie in the two words’ etymologies and semantic freight they carry. In contrast to the etymology of “ recover ” which has a long and history in the economic domain, (recall “ recover ” is derived from “recupare,” “to take back” as in debt), “ remedy ” is etymologically derived from the Latin, “ remedium, ” “to heal again.” The cognates of “ remedy ” include words such as “medical” and “medication” with strong physician /intervention associations. Note also how, in Smith’s text, “ remedy ” is collocated with negative words such as, “bad,” “injustice,” “sloth.” He draws these cognitive meanings into the conceptual blend with the “medical” attributes. Whether conscious or not, as Smith preserves “recover ” solely to its financial sense, and applies “ remedy ” polysemously with its dual senses in THE ECONOMY IS A PATIENT blend, he conveys his ideology. Smith projects a free market ideology which, even when financial strains are experienced, is capable of correcting itself to become healthy again. As he states metaphorically, “the wisdom of nature has fortunately made ample provision for remedying many of the bad effects of the folly and injustice of man.” 149 For Smith, “ recover ” resides in the single domain of “economic man” and his financial transactions. Smith’s text reflects a new linguaculture for a new country. Prior to Smith’s publication of The Wealth of Nations , a number of writers 149 Smith, Book IV, 9.28 84 plied THE ECONOMY IS A PATIENT metaphor and used “ recovery ” in creative and polysemous ways. In the same book, Securing the Commonwealth , Jennifer Baker finds examples of THE ECONOMY IS A PATIENT metaphor in the writings of Cotton Mather, Robinson Crusoe and others whose writings hold very different embedded economic ideologies. While the writers may differ in opinion, she writes, “they used the currency crises to resist laissez-faire economics and urge a return to an economy regulated for the common good.” 150 Unlike Adam Smith promoting free trade and the benefits of self-interest, Mather and Crusoe use THE ECONOMY IS A PATIENT to convey the goals of the commonwealth. This is an era of high rhetoric with strong emotive content. Economic discourse was a targeted beneficiary of the language. Innovative narratives in literary form were artfully crafted to convey strong political, moral and economic biases. “ Recovery ”in these texts is highly productive linguistically in health and economic senses, generating novel metaphors in both domains. In 1647, an anonymous writer produced a short political piece criticizing the policies of the time. The title itself carries a strong metaphorical message: “The Plague at Westminster or, An Order for the Visitation of a Sick Parliament, Grievously Troubled with a new Disease, called the Consumption of their Members.” The treatise is designed to chastise the government for excessive taxation. THE ECONOMY IS A PATIENT metaphor is extended to include the disease of consumption to convey how taxation has infected the local economy. The writer highlights how the taxation/infection is affecting the most vulnerable of 150 Baker 31. 85 society, the poor, into states of metaphorical illness (italics and underlines are mine): poore wretched and languishing wretches, mounting to the number of millions of millions, being sufficiently humbled by all these plagues , and punishment (cry to your honours for redress) besides the large prorion of our blouds which from the earth cries unto your honours, even as Abels did to heaven, so we to you Mighty Lords, we therefore humbly preay and beseech you, that your honours would be graciously pleased (in your omnipotent power) to raise to life again by halfe a dozen thousand poor widows their deer husbands and many fatherlesse children now in a languishing condition, will forever magnifie your Honours for the same, or else your Honours must expect the cry of the Widow to Heaven against you, the Curse of the Fatherlesse and the Cry of the Earth, which already begins to vomit up that bluud in your faces, which so rebel’ously and unchristianly you have stained hers withal; shee hath yet been a pace of pleasure unto you, yielding no contagious ayre to infect you with these consuming diseases , that now reign amongst your Honours, besides so many sorrows, distractions, disorders or passions, that visit your Honours consciences; all earthly creatures have been obedient until you mights Lords. And the call to action is explicit: We humbly beseech you the Knights and Burgesses chosen and put in trust by your severall Countreys, to redress our grievances; (not to make us new grievances, to cure our Maladies , not in a desperate madnesse; to kill us instead of curing us) to keep us from robbing not to rob us your selves. 151 The writer draws from the religious orthodoxy of the day and extends a new metaphor, that of the Lords of Parliament as God exacting punishment on a 151 “The Plague at Westminster or, An Order for the Visitation of a Sick Parliament, Grievously Troubled with a new Disease, called the Consumption of their Members” ( London, 1647). 86 sinful nation. The pun, “Lord,” in its dual meaning for the Lords of Parliament and Lord, God is evident and striking: Almighty and everlasting Lords, we acknowledge and confess from the bottome of our hearts, that you have most justly plagued us these full seven years for our manifold sins and iniquities. 152 Like Carey, centuries later, the writer of “The Plague of Westminster” views economic crisis is a form of punishment from God. Is the economic crisis and plague a sign of moral decay in the seventeenth century? What does THE ECONOMY IS A PATIENT conceptual network reveal by blending health and wealth with the religious? The economic treatises written in seventeenth century Britain illustrate an undercurrent of internal and ideological conflict between the morality of honest trade and the sinfulness of greed. At the intersection of the health, the economic and the moral metaphorical domains, the economic discourse explores the divine balance of commercial exchange. Observe the connection of the three metaphor domains in the following excerpt from Edward Misselden’s The Circle of Commerce, or The Balance of Trade in Defense of Free Trade : Commodities, moneyes, and exchange of monies may be aptly compared to the Bodie, Soule, and Spirit of traffique. The first, as the Body, upheld the world by Commutation and bartering, until money was devised to be coyned. The second, as the Soule in the Body, did infuse life to traffique by the meanes of equalities and equity, preventing advantage between Buyer and Sellers. 152 “The Plague at Westminster” 1. 87 The third, as the Spirit and faculty of the Soule, being seated everywhere, corroberateth the vitall Spirit of traffique, directing and controlling by iust proportions, the prises and values of Commodities and monies. 153 The metaphors sit on the edge of simile and allegory. In these contexts, the writings have a moral foundation rooted in the religious and political discourse of the era. At the time of Misselden’s writing, England had been suffering from a severe economic depression since 1620. Misselden had already been engaged in a year of fierce rhetorical exchanges between himself and fellow economist, Gerard de Malynes. In this latest round, Misselden responds to Malynes’ Maintenance of Free Trade 154 which, in itself, is a counter to Misselden’s earlier critique. This war of words and ideologies began in 1622. Misselden had published his treatise, Free Trade 155 which closed with a short commentary on an earlier work of Malynes. The ensuing exchange highlights the deep and entrenched metaphors in the health-economy-morality conceptual blend of the time. In these texts we see more than a tussle of political economies, but rather a fundamental sorting out of responses to a financial crisis in England’s trading status. We also see THE ECONOMY IS A PATIENT metaphor used as a shared vehicle for revealing their ideological differences. Misselden and Malynes both chose diseases to convey their ideological thinking, but used different 153 Edward Misselden, Circle of Commerce or The Ballance of Trade in Defense of Free Trade (London, 1623) 19. 154 Gerard de Malynes, Maintenance of Free Trade (London, 1622). 155 Edward Misselden, Free Trade or, The Meanes to Make Trade Florish: Wherein, The Causes of the Decay of Trade in the Kingdome, are Discovered: And the Remedies also to Remooue the Same, are Represented (London, 1622). 88 pathological attributes. For Misselden, we see an extensive use of blood disease as metaphor to represent the disruption of the natural flow and balance of the body’s natural fluids, 156 while Malynes shows a proclivity for the metaphor of an open sore, or canker, which drains the strength of the body politic. In the series of exchanges, we see a fundamental struggle with the society’s collective thinking about England’s position in an increasingly global economy. Following a collapse in the cloth trade, the two argued for a more level balance of imports and exports. Misselden chastises Malynes for accusing merchants of causing the imbalance by selling cloth too inexpensively in exchange for more expensive cloths from abroad. The two men could be no closer in their desire to rebalance trade and no further apart in their thoughts on how to achieve that goal. Misselden writes about Malynes’ recommendation for trade equity or “par of exchange” managed centrally (italics and underlines in this section are mine): So that if there should be a stop in the Course of the Exchange, then either the English Merchant will forbeare to take up mony by Exchange; or els hee will looke to “recouer ” the loss of the Exchange, upon his Cloth. If he forbeare to take up mony by Exchange, then he can neither buy so much cloth, nor give ready mony for the same as he was wont. Wherby will follow a stand in Blackwell-Hall, which is wont much to be refreshed by the ready use of the Exchange. And if the English wil not take, the Stranger cannot deliver: and if he cannot deliver, of necessity he must be thrust upon the Transportation of Mony, more than ever he was before: and then the remedy will be far worse then the disease .And if the English Merchant must needs recouer the loss of the Exchange upon the Cloth; it must either be done in 156 The understanding of human physiology during the seventeenth century was still highly influenced by the writings of Galen, particularly regarding the circulatory system and the balance of humors in the body. Galen’s explanations of disease provided a rich reservoir of entrenched encyclopedic knowledge from which to draw numerous health-economic metaphors. 89 the buying of it at home, or selling of it abroad. But it cannot be done in the sale of the Cloth abroad: for the Cloth-trade grones already under the present burthen that lye’s upon it, which presseth it downe so sore, that it cannot recover it selfe: whereof there are 2 principall witnesses, the Quantity, and the Price of Cloth, both diminished. 157 Note Misselden’s use of “ remedy ” and “ recover ” in the above paragraph. Misselden uses the word “ remedy ” metaphorically as Adam Smith does to convey the threat of overtreatment by an authority as in the “ remedy will be far worse than the disease.” “ Remedy,” again as seen in Smith’s treatise, is collocated with negative words such as “worse” and “disease.” Misselden, too, preserves the word “ recover ” to its financial sense, as in ” recouer the loss of the Exchange upon the Cloth.” However, at the close of the paragraph Misselden applies “ recover ” polysemously as a pun: “The Cloth-trade grones already under the present burthen that lyes upon it, which presseth it downe so sore, that it cannot recover it selfe.” Misselden is resolute that Malynes’ proposal to control the trade exchange centrally will complicate matters when a simpler solution is available. He writes: So then, the End of the Balance of Trade, may be said to either be Propior, or Remotior. There’s one End neerer hand; There’s Another End farther off. One End of it is, to finde out The cause of the Malady : The other, to present a Medicable Remedy , for the decay of trade. 158 What’s the other End of it? Surely to direct us to the Remedy : which in a word, is nothing els, but to make our Importations lesse, and our Exportations more. 159 157 Misselden, Circle of Commerce 109-110. 158 Misselden, Circle of Commerce 131. 159 Misselden, Circle of Commerce 134. 90 Misselden believes nature will resolve the imbalance. Too much intervention may do more harm than good. Note how the two cognates, “ remedy ” and “medical, ” are collocated with the negative term, “decay,” in the text. Again, the etymologies reveal semantic freight which Misselden leverages to link the negative conceptual domains of “decay” with physician intervention. Misselden’s commentary, “the remedy will be far worse then the disease” will be echoed centuries later in works such as Carey’s dark humored epitaph and, nearly three hundred year later, Keynes’ statement, “Thus an increase in the rate of interest, as a remedy for the state of affairs arising out of a prolonged period of abnormally heavy new investment, belongs to the species of remedy which cures the disease by killing the patient .” 160 Misselden uses the polyseme “ recover ” throughout the text in the financial sense “to gain back.” Note its use in the following excerpt: … For it [balance of trade] will bring to God, glory: to the King, honour: to the Kingdome, treasure: to the Subjects, trade: to the poore, employment: and prove by Gods blessing, a most excellent meanes, to encrease our Exportations, and to recover our Balance of Trade. 161 And this is also another meanes, not inferiour unto any, for the recovery of our Exportations, in the Balance of Trade. 162 The text gets a positive lift by the collocation of “ recover ” with words such as “God,” “glory,” “honour” and “blessing.” For Misselden, “ recovery ” restores the 160 Keynes 277. 161 Misselden, Circle of Commerce 138. 162 Misselden, Circle of Commerce 141. 91 natural balance bestowed by God. He uses the religious with the economic to ply his argument for Mercantilism and free trade. Jonathan Gil Harris, in his book, Sick Economies , 163 highlights how Misselden “embraces a much more decentralized understanding of the national economy’s lifeblood” and its “conception of bullion flow,” all key concepts to the Mercantilism ideology. While Misselden and Malynes continued their heated exchange of ideologies, Thomas Mun, another adopter of Mercantilist principles, wrote a seminal piece titled, A Discourse of Trade . Steeped in his devotion for God and Country, Mun’s goal was to educate the reader on the root of the Depression of 1620. He used the metaphor ECONOMY IS A PATIENT with a particular penchant for the metaphorical disease of consumption. In Mun’s texts we find the polysemeous word “ consumption ” applied as a metaphor and pun for the reckless purchasing of imported goods which wastes the economy’s resources (especially its currency) and reduces its strength to the point of death: “the malady is grown mortal here with us, and therefore cries out for remedy ,” and “the Commodities of this Kingdome, and also forraine wares, are the more consumed and wasted , a double meanes to abate the Common-wealth.” 164 Mun’s use of the consumption metaphor in his later writings illustrates the efficacy of the metaphor for representing the wasting of England’s wealth. Mun’s fondness for using consumption as the disease wracking the British economy was not new as illustrated in his Englands Treasure by Forraign Trade; or The 163 Jonathan Gil Harris, Sick Economies (Philadelphia: University of Pennsylvania Press, 2004) 145. 164 Thomas Mun, A Discourse of Trade, From England Unto the East-Indies: Answering to Diverse Objections Which are Usually Made Against the Same (London, 1621) H1. 92 Ballance of Our Forraign Trade is The Rule of our Treasure 165 written nearly forty years later. In Sick Economies , Harris outlines sixteenth century references to consumption as a metaphor for an ailing political economy reaching even further back in the word’s history. For example, Harris cites Thomas Starkey’s Dialogue Between Reginald Pole and Thomas Lupset 166 in which Starkey applies the simile explicitly to explain a weakened body politic. Harris states, “Starkey’s attribution of the origins of economic ‘consumption’ to factors within the English body politic resonates with the prevailing humoral understanding of the disease in the early sixteenth century.” 167 From a linguistic perspective, Harris suggests the consumption metaphor in the early seventeenth century is at the cusp of a significant semantic change. He references a proclamation made by King James in 1622, attributing the deepening depression on the “consumption of Coyne & Bullion.” 168 Harris suggests the Proclamation reveals King James’ sense that the country’s “immoral behavior that leads to the wasteful depletion of the nation’s treasure” reveals an early condemnation of “conspicuous consumption, whereby luxury commodities are purchased by a new kind of subject, the individual consumer, so that they may be both privately owned and publicly flaunted.” 169 Recall that in Mun’s treatise, the metaphor 165 Thomas Mun, Englands Treasure by Forraign Trade; Or, the Balance of our Forraign Trade is The Rule of our Treasure (London, 1669). 166 Thomas Starkey, Dialogue Between Reginald Pole and Thomas Lupset (London, 1535). 167 Harris 166. 168 King James I, A Proclamation for Restraint of the Exportation Waste and Consumption of Coine and Bullion (London, 1622) 2. 169 Harris 167. 93 reveals an evolving ideology that views cross-country exchange of goods and currency, “consumption,” not only as a waste of currency and wealth of the nation but an immoral act. Yet, from a linguistic point of view, Harris suggests we are seeing semantic change occurring before our eyes. He cites other writings (such as those by William Petyt) whereby “consumption” is seen as not a depletion of strength, but as a necessity for economic well-being. Even Mun, himself, occasionally appears conflicted in his treatises. Mun writes, “for the commodities which are brought in, & after carried out vnto forren parts again, cannot hurt but doe greatly help the commonwealth, by encrease of his Maiesties Customes and Trades,” 170 while stating later, “the Commodities of this Kingdome, and also forraine wares, are the more consumed and wasted, a double meanes to abate the Common-wealth.” 171 Harris states, “the ‘wasteful’ and ‘conspicuous’ senses of consumption were to overlap for some time.” 172 He argues this moral and economic debate creates a platform from which Adam Smith builds the ideology that consumption is a means of economic growth in his Wealth of Nations . By 1776, the semantic change is complete. The habitual uses of the conventionalized metaphor over the century and a half, the evolving ideologies and changed political environments enabled the word, “consumption,” to evolve as well. “Consumption” became polysemous with a new and quite different meaning in the English lexicon. The semantic shift illustrated in this example of the ECONOMY IS A PATIENT metaphor aptly exemplifies the conceptual 170 Mun, Discourse of Trade G4. 171 Mun, Discourse of Trade H1. 172 Harris 167. 94 metaphor theory in action. The underlying dynamics of semantic change include not only the salience of the attributes and the collocation of indexed lexical items in the discourse, but also the political and ideological influences as identified in Friedrich’s linguaculture model. The research that follows shows that the polyseme “ recover ” has undergone a similar semantic change. The debate between Misselden and Gerard de Malynes began in a comment Misselden made regarding a pamphlet produced by Mayles in 1601. The work, titled A Treatise of the Canker of Englands Commonwealth. Devided into Three Parts: Wherein the Author Imitating the Rule of Good Phisitions, First declareth the Disease. Secondarily, sheweth the Efficient Cause Thereof. Lastly, a Remedy for the Same , was an early mercantilist thesis addressing the imbalance of trade. Malynes’ agenda is stated simply and overtly, invoking the ECONOMY IS A PATIENT from the outset (italics and underlines are mine): 173 Plato the Philisopher perceiving that equality would be the cause that every man should have enough, was of opinion and willed all tings in a common wealth to be common, whom sir Thomas Moore in his Utopian common weale seemeth to imitate, to the end that an infinite number of lawes already made, might be abolished: whereas all of them are not sufficient, for every man to enjoy, defend and know from another mans that which he calleth his owene proper and private goods. But this equality cannot be established, neither was there an such ever used in any age, or commaunded by the word of God, but that possessing these worldly good, we should so use them whit charity toward others, as thought we did not possess then at all: Nevertheless (as a commonwealth is nothing else but a great household of family yet the Prince (being as it were the 173 Gerard de Malynes, A Treatise of the Canker of Englands Commonwealth (London: 1601). Web, IDEAS, Christian Zimmerman, Ed. Department of Economics, University of Connecticut, 23 April 2011 < 2-3 .95 father of the family) ought to keep a certaine equality in the trade or traficke betwixt his realme and other countries, not suffering an overbalancing of forreine commodities with his home commodities, or in buying more then he selleth. For thereby his treasure and the wealth of the realme doth decrease, as it were his expenses become greater, or do surmount his incomes or revenues. This is the unknowne disease of the politicke body of our weale publicke before mentioned: the efficient cause whereof must be found out, before any remedy can be applied or devised. Malynes’ Canker of England’s Commonwealth was written in a uniquely difficult economic time following the Reformation in England. There was a scarcity of bullion which Malynes believed was the result, in part, to the outflow of British cash in the form of coin to other countries. This was true primarily due to the undervalued pound. Malynes introduces THE ECONOMY IS A PATIENT metaphor as a means of conveying the damage the depletion of wealth has had on the financial stability of the nation. Here we will explore his selection of the metaphor of the canker, an open sore that deforms as it spreads its decay: the abuse of the exchange for money to be the very efficient cause of this disease : wherewith as with a Canker the politike body of our weale publike is overtaken: the cause thereof being predominant & overruling the course both of commodities & mony. 174 Jonathan Gil Harris contends that Malynes use of the metaphor belies the complexity of the language in his discourse. Harris believes Malynes’ selection of the canker metaphor implies a new set of conceptual attributes in the minds of the Elizabethans; that of a disease caused by both imbalance of the humors and 174 Malynes 18. 96 the contagion from foreign sources. He writes, “Certain subtle semantic shifts in the meanings of canker had helped lend it new metaphorical possibilities… Instead of designating an exclusively endogenous, humoral disorder, the now multivalent term more readily suggested a hostile, even foreign organism that invades and consumes the body.” 175 Embedded in the metaphor and Malynes’ discourse are the roots of mercantilism ideology and a call to action before it is too late: “for it is hard to heale a sore that a man would not have opened to his Phisition , though he be never so skilful, and of great experience.” 176 Yet nowhere in this discourse do we see the word, “ recovery .” Rather, we see the more medical term “remedy” collocated with “phisition.” This short excerpt belies the cognitive freight carried by the word “Phisition.” In the same year, Malynes wrote Saint George for England, Allegorically Described , a narrative about a dragon named Gangrem (playing on the word, “gangrene”) who is threatening the English commonwealth. Gangrem is the allegorical form of usury which is destroying the wealth, harmony and charity of the English commerce. In the allegory, Malynes extends the metaphor of the physician mapping to the governors of England who are called on to smite the dragon. The role of the physician, Harris writes, “is to heal ‘the biles botches, cankers and sores thereof’; chief among these is the ‘venimous sore’ of usury.” 177 This is not an ideology of free trade and allowing the financial system to cure itself. The call to action is clear: central government, intervene and save the commonwealth. 175 Harris 92. 176 Malynes 124. 177 Harris 59. 97 Malynes’ ideology, made explicit in his metaphorical language, drives his political agenda. He draws on the common and shared experiences of his society and the salience of the disease metaphors to build a compelling case for economic change. Jonathan Gil Harris illustrates key examples of “economic discourse” embedded in the scripts of Elizabethan plays. For example, he cites Shakespeare’s plays which “repeatedly blurred the boundaries between what we know regard as the separate domains of the medical and the mercantile.” 178 Shakespeare’s talent for maximizing the power of polysemy for punning is renowned. It is here where we find economic and health polysemy of the word, “recover .” Harris illustrates how in Shakespeare’s “The Comedy of Errors,” syphilis is a metaphorical disease contracted from foreign contact that wastes the body and the financial resources of the nation. Alopecia, the loss of hair, is a visible side-effect of syphilis. Shakespeare uses the dual senses of “ recover ” in the following excerpt as a clever pun to convey the physical loss of hair with the financial loss and depreciation of wealth from foreign exchange: Dromio: There’s no time for a man to recover his hair that grows bald by nature. Antipholus: May he not do it fine and recovery ? Dromio: Yes, to pay a fine for a periwig and recover the lost hair of another man. (2.2.71-88) Linguistically, “ recover ” carries a significant amount of semantic freight in this passage. For audiences of Shakespeare, the pun not only plays its 178 Harris 21. 98 immediate comedic role, but also triggers the encyclopedic knowledge in long-term memory that sustains the metaphor and its multiple meanings. Shakespeare alludes to the syphilis metaphor in “A Midsummer Night’s Dream” (1.2.100) and in “Titus Andronicus” as well. The result is a highly effective metaphor that, as Gy őri suggests, exploits the familiar knowledge sustained in the minds of the speaking community. It reveals the era’s xenophobic sentiment and growing political economy driven by a fear of foreign contagion and the imbalance of trade. In describing the play, “An Interlude of Wealth and Health,” Harris illustrates very old roots of the ECONOMY IS A PATIENT metaphor played out by the allegorical characters of Health, Wealth and Remedy. The drama is tightly linked with the political ideology and language of the day. In the play, the character Remedy expels another character Hans who personifies a foreign immigrant. Harris writes, “In order to restore health and wealth to a polity that is more nationally than universally coded, Remedy expels Hans from England… Thus is economic health reconfigured in nationalist terms as liberation from invasive foreign bodies.” 179 Harris’ analysis explores the roots of mercantilism and centrist ideologies reflected in health-wealth metaphors used at that time. The treatises and plays alike are written as social commentaries in innovative language that call for a change in the political economy structure. “I argue,” Harris writes, “that our modern notions of economy have a decidedly pathological provenance and that our modern notions of disease cannot be disentangled from 179 Harris 23. 99 the development of transnational capitalism.” 180 His historical research suggests that THE ECONOMY IS A PATIENT metaphor embodied by a variety of diseases in this era was coincident with the development of a new ideology of the English nation, centralized religiously, politically and economically in the monarchy. “This development was partly inspired by a financial crisis,” 181 Harris writes. The linguistic analyses of these works reveal the underlying cognitive dynamics enabling the metaphor to carry ideological freight. In a relevant example from 1581, Harris highlights a treatise by an unknown author (although attributed to Sir Thomas Smith), “A Discourse of the Commonweal of The Realm of England.” In the economic text, the author highlights the role of the Doctor as diagnostician and source of a cure. Implicit in the text, we find attestations for calls for “judicious fiscal ‘ remedies’ implemented and policed by the national sovereign.” 182 There are numerous calls to action using the ECONOMY IS A PATIENT metaphor, even from physicians themselves. In an ironic twist, Timothy Bright, a medical doctor, wrote the Treatise: Wherein is Declared the Sufficiencie of English Medicines, for Cure of All Diseases in 1580. He calls the commonwealth to change their ways: I hope this my enterprise shall be a meanes to prouoke others to deal with the same argument more plentifully, and kindle in vs a greater diligence to inquire after the medicines of our owne countrie yeelde, and more care to put them in practice…. Hath God so dispense his blessings, that a medicine to cure the iawndie, or the greene sicknes, or ye rheume, or such like, should cost 180 Harris 27. 181 Harris 35. 182 Harris 38. 100 more oftentimes then one quarter of the substance yt the patient is worth?... is Physicke only made for rich men? 183 Harris argues, “Bright’s call for medical protectionism, therefore, neatly dovetails with mercantilist conceptions of economic protectionism.” 184 As Harris notes, Bright’s extended metaphors in the domains of disease carries significant cognitive freight for the Elizabethan audience. We see in these texts again the dual directionality of metaphor mapping (i.e., the source health domain maps to the target economic domain as readily as the attributes of the economic domain map to a target in the health domain). In either direction, THE ECONOMY IS A PATIENT and the related polysemes evoke an emotional cry for action. The aptness of the metaphor for this purpose is clear. The conjunctive relationships between illness, medicine, patient, physicians, financial strain, remedies, the economy and centralized power in the conceptual blend deliver a strong rhetorical punch. The economic treatises in the Elizabethan era contain remnants of ideologies from the medieval periods. The seeds of mercantilism were planted in the language of the metaphor over centuries of discourse in Europe. In each case, THE ECONOMY IS A PATIENT emerged in times of financial turmoil. The polysemy of “ recover ” emerges, not as one would imagine, uni-directionally from the health domain to the economic in these metaphorical texts, but rather “recover ” adopted the salient attributes of the health domain through collocation with medical and disease metaphors while conveying its own attributes to the 183 Harris 122, 125. 184 Harris 125. 101 health domain. In addition, the integrated conceptual blend of the metaphor was reinforced by the encyclopedic knowledge from health and religion and became an equal part of the culture’s “system of commonplaces.” “ Recover ” carries the semantic freight it accumulated from centuries of collocations with broader disease metaphors in economic treatises and literary works. A review of “recover ” in history through an etymological analysis can uncover some of that history and the implications for the cognitive freight it carries. Etymological Review The Oxford English Dictionary (OED) indicates that the word, “ recover, ”was already polysemous by the early fourteenth centuries. The earliest cited meaning in the OED is “to obtain (c1100), to regain, recover (c1100), to restore, re-establish” 185 reflecting its Latin source, “ recuperare” – literally, “to get or take back.” The OED states that the word is derived from the Anglo-Norman word, “recouverer,” borrowed from the Middle French with the primary meaning “to regain (something lost); to take back into one’s control or possession.” The earliest narrative samples in the OED are from works written in the first half of the fourteenth century. Note how these samples illustrate the multiple meanings of “recover ”: (1) Þe oþer rekeuerd oȝain wiþ main. Arthour & Merlin 185 "recover, v.1". OED Online . March 2011. Oxford University Press. 8 May 2011 102 (2) He flei ȝe into te valaye And recouerd mi ȝt. Arthour & Merlin (3) There is heraude mysse bee-falle: Loste he hath his men alle, And recouere he shall sone this; For grete socour him cometh ywis. Guy of Warwick, Caius (4) Whan ouer-gon was his smerte And rekeured was of is hertte sir Beues set him vp. Bevis of Hampton In the texts from Arthour & Merlin, (1) and (2) , “recovered” is used in both health and “to regain as a possession” senses. Samples (3) and (4) illustrate “ recover ”already entrenched in the English language pertaining to health. The OED suggests the word derived its health sense from the Middle French as in “recouvrer la santé,” literally, “to get back health.” It is clear that “ recover ” was already polysemous and highly productive by this time. A related cognate, “recuperate,” was similarly in the lexicon at this time, however, by contrast, the OED suggests “recuperate” possessed a health sense, “get well again,” as early as the 5 th century. The OED cites sample texts for “recuperate” that illustrate its polysemy: 186 (5) For the recuperacion of the holy londe & holy Cyte of Iherusalem. Caxton, in tr. Siege & Conqueste Jerusalem (1481) (6) Your grace recuperatyng your helth. A. Borde, Compend. Regyment Helth (1542) “Recovery ,” as a noun, is listed in the OED as a member of the legal domain (as 186 "recuperate, v." OED Online , March 2011, Oxford University Press, 8 May 2011 103 in to pay fines) from 1422. See the earliest cited OED entry below: 187 (7) recovered in the seide Maires Court, vnto the seide Maire and to such persone ȝ as the seide recovrees belongeth to of right. in T. Smith & L. T. Smith, Eng. Gilds (1422) The semantic domain of law in context of legal judgments and transactions were highly productive both in the verb and nominalized forms. Defined as “the fact or process of gaining or regaining possession of or a right to property, compensation, etc., by a legal process or judgement,” “ recovery ” in this sense draws entailments that lead to financial attributes. Consider the following entries from the OED: (8) Sir Roger schal relese and for ȝefe to the forsaide John Bagger_al the damagez that be recouered be the same assis. in H. M. Flasdieck, Mittelengl. Originalurkunden (1405) (9) Ȝif tei kittide tus openly here purses, tei chulden reckevere it bi comyn lawe. Wycliff, Sel. Eng. Wks (1383) The semantic leap to the financial and economic domains is not difficult to conceive. Note the early examples again from the OED: (10) Los of catel may recouered be But los of tyme shendeth vs. Chaucer, Man of Law's Prol. c1405 (1390) (11) Thou calledist vpone me that by me thou myghtstid thy loosse recouere . Bk. Found. St. Bartholomew's (c 1425) (12) Yf she lacke on one syde she recouerith it on that other. Caxton, Myrrour of Worlde I. xiv. 43 (1481) 187 "recovery, n." OED Online , March 2011, Oxford University Press, 8 May 2011 104 (13) They determyned to go thyder_to assay if they coude recover any parte of their domage. LD. BERNERS, Cronycles I. 506 (1523) By the late thirteenth and early fourteenth century, the stage for the ECONOMY IS A PATIENT had been set. Already in the integrated conceptual domain for “ recover ” and “ recovery, ” the lexicon shared the polyseme for multiple senses. This etymological analysis suggests the Latin cognates “ recover ” and “recovery ” were highly productive virtually from the inclusion of the Anglo-Norman forms following the Norman Conquest. 188 It is impossible to determine how English speakers of the time interpreted these multiple meanings. However, the entry for “ recover ” and “ recovery ” from Nathan Bailey’s An Universal Etymological English Dictionary , compiled in 1763 may provide a clue. His entries for “ recover ” and “ recovery ” read as follows (highlights are original from the text): 189 To RECOVER, [ recouvrir , F recuperare L] to get again, to restore to Health, to be on the mending Hand. RECOVERY [ recouverment , F, recuperate L.] a regaining or getting again, &c. Remedy, Help. True RECOVERY {in Law} is an obtaining any Thing by Judgment or Trial at Law. Feign.d RECOVERY is a certain Form, Common RECOVERY or Courte in Law, for the better assuring one’s Title to Lands or Tenements. 188 The Bosworth-Toller Anglo-Saxon Dictionary indicates “ recover ” in the Anglo-Saxon lexicon had multiple lexical entries for the health senses (e.g., “ batian ” as in “to recover from ill health, to get better,” “ ge-wyrpan, ” “ wirpan ” or “ wyrp ” as in “Godwin fell sick and got better again” and “ halian ” as in “to heal from wounds or illness”), none of which have extended senses in other semantic domains. 189 Nathan Bailey, An Universal Etymological English Dictionary 1721 (Oxford: Claredon Press, 1763): 698, 07 May 2011 Web. OpenLibrary Internet Archive < stream/universaletymolo00bail#page/698/mode/2up >. 105 While we need to be careful not to place the OED or other compendia of historical lexicons as all encompassing, 190 they do provide rich data that can illuminate our analysis for what is not included in them as much as what is. For example, Bailey’s entries suggest that “ recover ” and “ recovery ” had two very important senses at the time: health and legal. We can infer from his entries that these are the two most salient meanings in the eighteenth century, at least from Bailey’s point of view. In fact, the legal contexts had such a high degree of salience, the productivity of the word generated deeper sub-categories to differentiate specific purposes in the Law as illustrated by the multiple entries. In these two conceptual blends, “ recover ” and “ recovery ” are found in domains that share highly emotive attributes. The definitions provided by Bailey include phrases and words such as “to get again,” “Help,” “obtaining a Thing by Judgement,” and “assuring one’s Title” as in getting something back that was owed to you which required legal intervention on your behalf. In the health domain, Bailey includes the definition for “ recover, ” “to be on the mending Hand,” a familiar idiom of the time meaning “to be convalescing.” Why use such an idiom to define a word? How was “on the mending hand” used to connote healing? We find the phrase as a repeating refrain in a ditty, “The Merry Cobblers,” written one hundred years earlier by a popular balladeer, Martin Parker. The idiom, “to be on the mending Hand,” had significant 190 Kathryn Allan writes in the introduction to her book, Metaphor and Metonymy: A Diachronic Approach (Chichester: Wiley-Blackwell, 2008), pages 15-16, “it is generally acknowledged that the evidence that survives for English in earlier periods is uneven…Where only a single attestation can be found for a lexeme or a particular sense of a lexeme, this may reflect an idiosyncratic use by a particular writer, i.e., a nonce-word, but in many cases it is more likely to reflect more frequent use, either in written sources that have not survived or in speech.” 106 political and economic freight as well as health connotations during the seventeenth century. It was the time of the English Depression of 1620. In the song, the cobblers’ lyrics are the words of working men, not learned economists or sophisticated merchants. Consider how the lyrics use “on the mending hand” metaphorically in the following opening and closing verses (italics and underlines are mine): Come, follow, follow me! To th' alehouse weele march all three; Leave aule, last, threed and lether, and let's goe altogether; Our trade excells most trades i'th' land, for we are still on the mending hand. Though shoomakers us disdaine, yet 'tis approved plaine Our trade cannot be mist, let them say what they list; Though all grow worse quite through the land, yet we are still on the mending hand. Though shoomakers us disdaine, yet 'tis approved plaine Our trade cannot be mist, let them say what they list; Though all grow worse quite through the land, yet we are still on the mending hand. … While other callings great, for fraud and foule deceit, Are lookt unto by law, we need not weigh't a straw; Our honesty spreads though the land, for we are still on the mending hand. Therefore let's be of good cheere, though lether be something deare; The law some course will take, amends for all to make; And by their care we understand, the world is now on the mending hand. We pray for durty weather, and money to pay for lether, Which if we have, and health , a fig for worldly wealth ; Till men upon their heads doe stand, we shall be still on the mending hand. 107 M.P. FINIS 191 The “Three Merry Cobblers” does not contain the words, “ recover ” nor “ remedy ”in any of its stanzas. Instead we see the idiom repeated in the refrains. Note in the closing lines we see the dual domains, health and wealth, clearly collocated with “on the mending hand.” The lines from this ballad confirm that the health and economic metaphor domains were clearly entrenched. Bailey’s dictionary similarly confirms the efficacy of “ recover ” as a powerful and self-reinforcing polyseme in dual senses. The close bond between the two semantic domains is so strong, that it enables an idiomatic pun to define its etymology! While “ recover ” continues to evolve and produce additional senses in a variety of contexts, this diachronic review has illustrated its high level of salience and cognitive freight. The research above attests that “ recover ” has a strong rhetorical force in its use, influenced by the cognitive freight from his historical past that continues to influence economic discourse. The health-wealth conceptual blend can be a powerful sociological force. Its interconnectivity with moral semantics continues to evoke emotional responses calling for action when the economic well-being of the community is threatened by financial stress. 191 Martin Parker, “Three Merry Cobblers,” Roxburghe Collection (I. 408- 409): c1625, English Broadside Ballad Archive. University of California Santa Barbara. Ed. Patricia Fumerton 28 May 2011 Web. 108 Chapter V Discussion and Implications of the Findings By applying quantitative and CMT analytics, we have been able to observe the underlying cognitive and pragmatic dynamics of THE ECONOMY IS A PATIENT metaphor in common English economic discourse. The quantitative tools indicate that the metaphor as represented by the polyseme, “ recovery, ”does, in fact, emerge during eras of significant economic stress and recedes as the crisis subsides. Similarly, context analyses show that “ recover ” and “recovery ” in historical contexts, as well as in present day economic narratives, convey the urgency of the financial crisis. These analyses suggest that THE ECONOMY IS A PATIENT metaphor and polyseme are salient and significant during these specific economic conditions. The CMT analysis presented here reveals the complexity of the integrated conceptual network resulting from the metaphor blend. The health and financial domains are clear. However, the research demonstrates how the blend as a member of the broader domains, THE ECONOMY IS AN ORGANISM and THE ECONOMY IS A PERSON, leverages the broad array of conventionalized metaphors and idioms. The conceptual network is as productive now as it was in past history. Idioms such as “road to recovery,” for example, are possible because of the metaphor’s ability to generate a new conceptual blend, RECOVERY IS A JOURNEY. The blend is the product resulting from the 109 melding of attributes from the common LIFE IS A JOURNEY metaphor and THE ECONOMIC IS A PATIENT metaphor. We have seen similar prolific generations as in the early cases of “on the mending hand” and Shakespeare’s punning of “recover. ” In addition, upon deeper analysis diachronically, we see that the metaphor blend also reflects historic attributes from other conceptual domains including the Western model of “well-being,” morality and man’s ontological relationship with God. It is this relationship that the examination of “ recovery ” becomes most profound. Health and illness has had a long term cultural and linguistic relationship with morality and Judeo-Christian religious precepts, i.e., illness is a punishment, therefore, healing is a gift from God. The conceptual and semantic extension of financial stress to this domain has been metaphorically mapped for millennia. Researchers have found similar metaphor patterns in Biblical texts “where religion, family life, political and economic stability, and general health and welfare were seen as thoroughly intertwined.” 192 Recall the prominent financial/ legal senses of “ recovery ” illustrated in Chapter 4. Diego Garcia, in a 1992 study of early health ideologies, finds attestations of the metaphor blend created by the illness and debt domains in the Biblical book of Deuteronomy. Garcia writes, “in the language of Deuteronomy, health appears as a gift from Yahweh and illness as a debt owed to Him on account of the transgression of the law imposed on his people. Corporal or somatic illness is the manifestation of a 192 Seong-Hyuk Hong, The Metaphor of Illness and Healing in Hosea and Its Significance in the Socio-Economic Context of Eighth-Century Israel and Judah (New York: Peter Lang Publishing, 2006) 88. 110 sin of the heart: such is the ‘etiological diagnosis’ (Deut. 28:15). Besides this there exists a ‘differential diagnosis’, which leads to a differentiation between the different types of illness (Deut. 28: 21-22).” 193 In Garcia’s studies we see that the patient is a sinner, owing a debt to God. In this historical view, treatment and restoration to health has a moral sense that transcends the economic and the financial. 194 While economic discourse of the twenty-first century is viewed as a dispassionate, mathematically-based science, the roots of THE ECONOMY IS A PATIENT metaphor tells a different story. As the research in this thesis ha shown, encyclopedic knowledge of a speech community is readily accessible during the processing of metaphors. The interaction between the multiple domains of health, economic and morality enables the metaphor to call into action the multiplicity of senses whether intended or merely consequential. The data presented in this thesis illustrates how the metaphor’s extensions can account for its enduring power and continued productivity. The metaphor’s cognitive dynamics have created similarities in the economic and health domains. The analysis reifies Max Black’s interactionist theory and illustrates it in real time. s The diachronic review of “ recovery ” has also been instrumental in understanding not only the metaphor’s use over time, but also the evolution of political economic thinking. This case is an apt example of what Philip Seargeant 193 Diego Garcia, “The Ethics of Diagnosis in Early Christianity and the Middle Ages,” The Ethics of Diagnosis eds. José Luis Peset, and Diego Garcia, (Dordrecht: Kluwer Academic Publishers Group, 1992) 19. 194 My interpretation of the integrated conceptual network using Turner and Fauconnier’s graphing technique can be found in Appendix B. 111 calls “the historical ontology of language.” Seargeant defines historical ontology as allowing “us to examine how and why language has been conceptualized in the way that it has at particular times and places in history. It allows us to interrogate the processes that give rise to such conceptualization, and evaluate their politics and the contingencies that contributed to them.” 195 We have seen in this thesis how ideologies are tested in times of financial crisis and how the metaphor activates. Regardless of the writer’s political perspective, the metaphorical physician is called upon to cure the ailing economy. The metaphor heightens the text’s rhetorical and pragmatic impact. As Friedrich’s framework suggests, a critical analysis of ideologies in discourse intertwines the analytical-scientific and the emotional-ethical. We have seen how the polyseme “ recovery ”carries significant cognitive freight containing both analytical and moral attributes, that is, “concerned with ‘values.’” 196 The diachronic analysis has proven to be an illuminating approach to understanding how a linguaculture evolves over time in a community of speakers. In addition, we have seen how metaphors are active agents in the rhetoric designed to influence the society’s “economic process and resource allocation.” 197 Similarly, this “archaeological” approach demonstrates the value of Foucault’s notions of epistemes underlying the formation of discourse. We have seen how the metaphorical map of a troubled economy to an ailing patient has 195 Philip Seargeant, “The Historical Ontology of Language,” Language Sciences 32 (2008): 11. 196 Friedrich 296. 197 Friedrich 297. 112 evolved in conjunction with the historical ideologies and understandings of the body and disease at a given time. These cognitive and social constructs are, in turn, leveraged in and informed by economic discourse as nature’s “truths.” While the anthropomorphic metaphor of the body is particularly salient and enduring, the detailed attributes, based on the community’s comprehension of the body, is manifested in the metaphor. In Jack Amariglio’s explication of Foucault in economic discourse, he writes, “the notion of the body—indeed, the order of all things and words—was discursively organized and ordered according to a different ‘episteme.’” 198 Compare again the manner in which “ recover “and “recovery ” was used by Smith in 1776 at the dawn of a new economic experiment, the United States, to Misselden’s view hundred-fifty years earlier. While both chastise physician over-interventions, the metaphor’s meaning was highly influenced by the “episteme” of the era. Both agree free trade is self-correcting; the ailing economy will “cure itself.” For Smith, it is a rational solution; for Misselden, it is God’s way. Still, underpinning the discourse, the metaphor of an ECONOMY IS A PATIENT and an administering physician as an intervening government prevails. It has remained a relevant metaphor in Western political economic discourse well into the Obama era. We observe a meme that has been passed down from one generation to the next, keeping both the metaphor and the polyseme “ recover ” salient and relevant over time. How this occurs has been illustrated by the cognitive dynamics underlying the metaphor in the narratives excerpts provided in this thesis. The interrelations 198 Jack L. Amariglio, “The Body, Economic Discourse, and Power: an Economist’s Introduction to Foucault,” History of Political Economy 20.4. (1988): 586. 113 between health and financial conceptual domains are proven to be rich in meaning and heavy with cognitive freight. They enable the “political ideas in action” which, as Freidrich writes, “arise from the engagement of creative individuals with practical problems and necessarily reflect or express the will and interests for control or change of some social group or class – notably, its economic interests.” 199 When the collective minds of the speech community apply the conventionalized word “ recovery ” the research suggests the encyclopedic knowledge of the multiple senses is activated. The concepts shared through a historical memory and sustained via habitual use of linguistic forms in the culture. 200 Therefore, we find that the selection of polysemes in economic discourse is not arbitrary. Far from it: they serve a semantic and cultural purpose. They are highly emotive and, as a result, have high persuasive power for action. James Fernandez states this fact succinctly, “metaphors are not only rhetorical devices of persuasion; they can also lead to performance.” 201 The linguistic analysis enables us to better understand why and how THE ECONOMY IS A PATIENT metaphor can be called into action. The data in this analysis reveals that the polyseme “ recovery ” is frequently collocated with other lexical items relevant to the society. This, in turn, raises the salience of the metaphor to what is important and critical. Research indicates that lexical priming of this sort has a strong linguistic 199 Friedrich 301. 200 Anna Wierzbicka refers to this phenomenon as “cultural scripts” in her account of Russian conversational phrases and their role in revealing cultural norms. 201 James W. Fernández, Persuasions and Performances: the Play of Tropes in Culture (Bloomington: Indiana University Press, 1986) 42. 114 force. 202 Coupled with the power of social memory, speakers leverage the ambiguity to positively enhance the metaphor. This central dynamic underlies the use of idioms, conventionalization and semantic change and reveals the cognitive mechanisms in the mind of the speakers. Thus far, we have discussed the integral role of polysemes as lexical units in economic discourse. The samples provided in this thesis also reveal that “recovery ” plays a critical function in the narratives in which it is used. Philip Eubanks states that metaphors in rhetoric find their efficacy through “licensing stories.” He writes, “licensing stories are not merely supporting narratives that happen to be confluent with a given metaphor. They are individual and cultural keys that people use to establish some disposition toward a metaphor – either a conceptual metaphor or a specific instance of it.” 203 Eubanks concurs that metaphors have a social origin that the “licensing stories” confirm and give them value and “cognitive force.” He writes that the narratives confirm the suitability of the metaphor for the situation at hand, For us to regard any mapping as apt, it must comport with our licensing stories – our repertoire of ideologically inflected narratives, short and long, individual, professional, and cultural, that organize our sense of how the world works and how the world should work . 204 (Eubanks’ italics) 202 Michael Hoey’s Lexical Priming: A New Theory of Words and Language explores the intricacies of lexical priming encompassing universality of collocations and their semantic role. Of particular interest is his discussion on disambiguation which we have seen numerous examples in historic economic narratives. 203 Philip Eubanks, The War of Words in the Discourse of Trade: The Rhetorical Constitution of Metaphor (Carbondale: Southern Illinois University Press, 2000) 110. 204 Eubanks 111. 115 While Eubanks’ research establishes a compelling case for the mutually reinforcing 205 nature of conceptual metaphors and licensing stories both conceptually and rhetorically, his research is limited to modern discourse in a traditional synchronic manner. The licensing stories we have seen in this thesis significantly expand his theory. By analyzing a metaphor regressively through narratives and discourse, we see broader implications for cognitive metaphor research. The analysis presented above reveals that licensing stories can re-activate very old metaphor maps resident in the community’s encyclopedic knowledge making them salient again. Not only are the metaphors restored, but the conventional metaphors and polysemes within the integrated conceptual network are activated as well. In the case of “ recover ” and “ recovery, ” we have seen they carry cognitive freight and referents from history and historical ideologies with such emotive power they can spur action. In today’s economic discourse, the cognitive blend embodied by the terms, “ recover ” and “ recovery, ”raise a sense of urgency; the licensing story validates a call for emergency treatment, a call for government intervention. How can we suggest with such confidence that polysemic words such as “recovery ” can evoke action, particularly in the domain of economics and finance? A variety of new studies reveal a correlation between the language used in economic narratives and the actions individuals take in response. From 205 Philip Eubanks, “ An Analysis of Corporate Rule in Global Discourse, ”Rhetoric Review 27.3 (2008): 255. 116 influencing political and social attitudes 206 to stock trading behavior, 207 these studies illustrate the power metaphors can have in rhetorical discourse. Deirdre McCloskey’s bold statements in The Rhetoric of Economics reverberate in a 2005 study by Fabrizio Ferraro, Heffrey Pfeffer and Robert Sutton. Their research illustrates how economic language can influence decisions even when logic may suggest other courses of action. They write, The metaphors and other linguistic tropes used in a discipline coalesce into a more or less coherent knowledge structure that shapes how its members and those they influence construe reality… Acting on the basis of that language in ways consistent with those norms and assumptions, we do things that, in turn, will produce behavior on the part of others consistent with our linguistic frame. Language produced a social reality that reinforces and validates the terminology we use… 208 In the economic domain, they contend, the economic theories conveyed in discourse become self-fulfilling: “the dominant assumptions, language, and ideas of economics can exercise a subtle but powerful influence on behavior, including the behavior in an organization, through the formation of beliefs and norms about behavior that affect what people do and how they design institutions and management practices.” 209 In the case of “ recovery, ” its 206 Mark Landau, Daniel Sullivan and Jeff Greenberg, “Evidence That Self-Relevant Motives and Metaphoric Framing Interact to Influence Political and Social Attitudes,” Psychological Science 20.11 (2009): 1421-1427. 207 Michael W. Morris and Oliver J. Sheldon, Daniel R. Ames, Maia J. Young, "Metaphors and the Market: Consequences and Preconditions of Agent and Object Metaphors in Stock Market Commentary," Organizational Behavior and Human Decision Processes 102 (2007): 174-192. 208 Fabrizio Ferraro, Jeffrey Pfeffer, and Robert I. Sutton. “Economics Language and Assumptions: How Theories can Become Self-fulfilling,” Academy of Management Review 30.1 (2005): 15-16. 209 Ferraro et al, 20. 117 emotive rhetorical power is elicited through its semantic freight from multiple domains. The integrated conceptual network is rich with meaning derived from illness, vulnerability and moral judgment. The metaphor’s ability to leverage these senses is ancient and deeply rooted in Western culture. As a linguistic form, the ability of “ recovery ” to evoke action appears to be similarly self-fulfilling. In this second decade of the twenty-first century, “ recovery ” has again become an iconic metaphor for the ailing Western economy. Not only is the metaphor’s continued efficacy revealed today but also its ability to shape a linguaculture over time. 118 Chapter VI Conclusion The research provided in this thesis uncovers the conceptual and linguistic dynamics underlying polysemes and their illocutionary power. Linguistically, these findings reveal a powerful new insight. Through detailed diachronic analysis we see that a single lexical unit, a polyseme, can carry historical cognitive freight. When the polyseme is used in cultural narratives, it can have significant rhetorical effect in a speech community. For cognitive linguists, the diachronic exploration of metaphors can reveal how they remain productive over many centuries. In fact, we see that the continual use of the metaphor over time embeds the deeply held beliefs of the community in the language. The cognitive freight can convey not only current and novel political views, but also carry historic political ideologies. By using the tools of modern Conceptual Metaphor Theory in a diachronic analysis, the linguist can study how former uses of a single polyseme are reactivated by different environmental triggers to reveal the community’s encyclopedic knowledge and cultural frames. This type of analysis can illuminate the cognitive power of “licensing” narratives. The findings from this research suggest that diachronic analysis has significant potential to advance metaphor and critical discourse research. By revealing a polyseme’s historic evolution, the cognitive linguist may generate new insights into the cognitive and social underpinnings of a metaphor’s role in a 119 speech community’s thought and action. 120 Appendix A Terminology Cognitive linguistics – the branch of linguistics study that researches the connection between human cognition and language: how language is not an autonomous cognitive faculty and that knowledge of language emerges from language use. 210 Conceptual metaphor theory (CMT ) – theory proposing that metaphors are the manifestation of a conceptual mapping between two conceptual domains; the metaphorical expression is about a situation in one domain (the target domain) using concepts from another domain (the source domain). 211 Conceptual domain – a conceptual representation or knowledge of any coherent segment of experience; involves both the knowledge of basic elements that constitute a domain and knowledge that is rich in detail. 212 For example, the conceptual domain of “mother” includes attributes pertaining to motherhood, such as “pregnancy,” “children,” and ”nurturing.” Conventional metaphor – a metaphor whose base terms refer to both a literal concept and to an associated metaphoric category. For example, blueprint (as in “A gene is a blueprint” ) has two closely related senses: “a blue and 210 William Croft and D. Alan Cruse, Cognitive Linguistics (Cambridge: Cambridge University Press, 2004) 1. 211 Croft and Cruse 198. 212 Zoltán Kövecses, Metaphor: A Practical Introduction (Oxford: Oxford University Press, 2010) 324. 121 white photographic print showing an architect’s plan” and “anything that provides a plan.” 213 Critical Discourse Analysis – a form of linguistic research and analysis that focuses on the way language is used in discourse as a social and/or political force. “It is primarily interested and motivated by pressing social issues, which it hopes to better understand through discourse analysis.” 214 Dead metaphor – a word or phrase that was once metaphorical, but has since lost any sense of connection with the original base concept. For example, culture refers to a particular heritage or society and its use seems literal, but this sense is a metaphoric extension of another known sense of the word, “preparation for growth” (as in the bacteria culture ). The two meanings no longer seem related. 215 English for Specific Purposes – related to specific teaching situations of English for use in specific disciplines or professional work. 216 Language ideology – “the cultural system of ideas about social and linguistics relationships, together with their loading of moral and political interests.” 217 Linguistic/ language philosophy – the study of language from the perspective of 213 Brian F. Bowdle and Dedre Gentner, “The Career of Metaphor,” Psychological Review 112.1 (2005): 199. 214 Teun A. van Dijk, “Principles of Critical Discourse Analysis,” Discourse & Society (London: Sage Publishing, 1993) 252. 215 Bowdle and Gentner 209. 216 Tom Hutchinson and Alan Waters, English for Specific Purposes: A Learner-centered Approach (Cambridge: Cambridge University Press, 1987) 19. 217 Judith Irvine, “When Talk Isn’t Cheap: Language and Political Economy,” American Ethnologist 16.2 (1989): 255. 122 how language is used and the meaning derived from its use; highly influenced by J.L. Austin. 218 Performativity – the notion that the “issuing of [an] utterance is the performing of an action” 219 and not just the saying of it. Polysemy - when a word has several meanings which are (metaphorically) related. 220 Pragmatics - the study of meaning as communicated by a speaker (or writer) and interpreted by a listener (or reader). It is the study of speaker meaning, contextual meaning and how meaning is communicated. 221 Rhetoric – the use and study of language for persuasion. 222 218 J.T. Austin, How to do Things with Words , ed. J.O. Urmson (Oxford: Clarendon, 1962). 219 Austin 6. 220 Joost C. Van de Weijer, Glossary of Linguistic Terminology (October 28 2004), 14 February 2009 < > 29. 221 George Yule, Pragmatics (Oxford: Oxford University Press: 1996) 3. 222 Aristotle, Rhetoric , 1355b. 123 Appendix B Sin – Debt Retribution – Repayment Forgiveness – Acquittal Sin Retribution Forgiveness Debt Repayment Topic Theme Process Serious Illness – Market Crisis Patient - Economy Emergency - Government Treatment Intervention Serious Illness Patient Emergency Treatment Market Crisis Economy Government Intervention “recover” “recovery” Sin – Debt Retribution – Repayment Forgiveness – Acquittal Sin Retribution Forgiveness Debt Repayment Topic Theme Process Serious Illness – Market Crisis Patient - Economy Emergency - Government Treatment Intervention Serious Illness Patient Emergency Treatment Market Crisis Economy Government Intervention “recover” “recovery” Figure 6. Conceptual blending network for interaction of “ recovery ” in economic discourse. This graphic represents a possible mapping of cognitive attributes across the economic, health and moral conceptual domains. 124 Appendix C Lexical Item Economist (Jan ’95-Sept ’97) n = approx. 9.7 million Frequency words 223 per million words U.K. magazines n = 5 million words Frequency words 223 per million words Economic Growth 1576 162.5 10 2 Fat 328 33.8 476 97 Diet 149 15.4 236 48.1 Healthy Economy 133 13.7 0 0 Ailing 133 13.7 16 3.3 Remedy 99 10.2 32 6.5 Economic disease 94 9.7 0 0 Economic cure 80 8.2 0 0 Economic depression 45 4.6 1 0.2 Infant industry 32 3.3 0 0 Economic decay 30 3.1 0 0 Bleeding 23 2.4. 62 12.6 Haemorrhage 7 0.7 9 1.8 Table 1. Instances of Economic Metaphor . Source:Jonathan Charteris-Black, “Metaphor and Vocabulary Teaching in ESP Economics,” English for Specific Purposes 19 (2000). This table illustrates the number of instances each metaphor occurs within Charteris-Black’s research corpora. 223 Charteris-Black is referring to the frequency of occurrences per million words. 125 Bibliography Works Cited Aijmer, Karin. “The Semantic Development of Will. ” Historical Semantics, Historical Word-formation. ed. Jack Fisiak. Berlin: De Gruyter (1985): 11-21. Allan, Kathryn. Metaphor and Metonymy: A Diachronic Approach . Chichester: Wiley-Blackwell, 2008. Amariglio, Jack L. “The Body, Economic Discourse, and Power: an Economist’s Introduction to Foucault.” History of Political Economy 20.4 (1988): 583-613. “America’s Bail-out Plan: The Doctors’ Bill.” The Economist . Sep 25, 2008. 10 March 2010 < displaystory.cfm?story_id=E1_TNPDVSQG >. “American Experience: The Crash of 1929: Primary Resources: Headlines.” Public Broadcasting Service (PBS). 24 Feb 2010 . Aristotle, Rhetoric . Trans. W. Rhys Roberts . Iowa State University of Science and Technology Public Web Server . Iowa State University. Ed. Lee Honeycutt, 21 June 2004. Web. 27 Feb 2011. < /Rhetoric/ >. Austin, J.T. How to do Things with Words . Ed. J.O. Urmson. Oxford: Clarendon, 1962. Bailey, Nathan. An Universal Etymological English Dictionary . London, 1675. Web. Google Books. 26 January 2011 < >. Baker, Jennifer. Securing the Commonwealth: Debt, Speculation & Writing in the Making of Early America . Baltimore: The Johns Hopkins University Press, 2005. Billig, Michael, and Katie MacMillan. “Metaphor, Idiom and Ideology: the Search for ‘No Smoking Guns’ Across Time.” Discourse and Society 16.4 (2005): 459-480. 126 Black, Max. Models and Metaphors . Ithaca: Cornell University Press, 1962. ---. “ More About Metaphor.“ Metaphor and Thought . Ed. Andrew Ortony, Cambridge: Cambridge University Press (1993): 19-41. Blank, Andreas. “ Words and Concepts in Time: Diachronic Cognitive Onomasiology ,” Words in Time: Diachronic Semantics from Different Points of View . Eds. Regine Eckardt, et al. Berlin: Walter de Gruyter (2003): 37-65. Boeckx, Cedric. Language in Cognition: Uncovering Mental Structures and the Rules Behind Them . Oxford: Wiley-Blackwell, 2010. Boers, Frank. “’No Pain, No Gain’ in a Free Market Rhetoric: A Test of Cognitive Semantics?” Metaphor and Symbolic Activity 12.4 (1997): 231-241. Boers, Frank, and Murielle Demecheleer. “Travellers, Patients and Warriors in English, Dutch and French Economic Discourse.” Revue Belge de Philology et d’Histoire 73 (1995): 673-692. Boers, Frank, June Eyckmans, and Hélène Stengers. “Presenting Figurative Idioms with a Touch of Etymology: More than Mere Mnemonics?” Language Teaching Research 11.1 (2007): 43-62. Web. Sage Publications 25 March 2011 < >. Bosworth, Joseph. “An Anglo-Saxon Dictionary Online”. Thomas Northcote Toller and others, eds. Sean Christ and Ond řej Tichý, Comps. Faculty of Arts, Charles University in Prague. 11 Nov. 2010. Web. 14 January 2011. . Bowdle, Brian F., and Dedre Gentner. “The Career of Metaphor.” Psychological Review 112.1 (2005): 193-216. Brinley, Thomas. “Alfred Marshall on Economic Biology.” Review of Political Economy 3.1 (1991):1-14. Brown, Cecil H., and Stanley R. Witkowski. “Polysemy, Lexical Change and Cultural Importance.” Royal Anthropological Institute of Great Britain and Ireland 18.1 (March 1983): 72-89. Burke, Kenneth. Rhetoric of Motives . Berkeley: University of California Press, 1969. 127 Bybee, Joan. “ Diachronic Linguistics .” The Oxford Handbook of Cognitive Linguistics . Eds. Dirk Geeraerts, and Herbert Cuyckens. Oxford: Oxford University Press, (2007): 945-981. Carey, Mathew. “Address Delivered Before the Philadelphia Society for Promoting Agriculture, at its Meeting on the Twentieth of July, 1824.” Hume Tracts (1827) JSTOR. Web. UCL Library Services. 23 January 2011 ---. Essays on Banking . Philadelphia. 1816. ---. “Essays on the Public Charities of Philadelphia: Intended to Vindicate Benevolent Societies from the Charge of Encouraging Idleness.” Hume Tracts (1829) JSTOR. Web. UCL Library Services. 23 January 2011 Carey, Mathew, et al. Addresses of the Philadelphia Society for Promotion of National Industry. Philadelphia. 1820. Web. Google Books. 17 April 2011 . Charteris-Black, Jonathan. Corpus Approaches to Critical Metaphor Analysis .Basingstoke & New York: Palgrave-MacMillan 2004. ---. “Metaphor and Vocabulary Teaching in ESP Economics.” English for Specific Purposes 19 (2000): 149-165. ---. School of Humanities, Languages and Social Sciences. University of the West of England, Web. 10 April 2010. < charteris-black_j.shtml >. Charteris-Black, Jonathan, and Andreas Musolff. “’Battered hero’ or ‘innocent vicitm’? A Comparative Study of Metaphor or Euro Trading in British and German Financial Reporting.” English for Specific Purposes 22 (2003): 153-176. Colquhoun, Patrick. “A Treatise on Indigence: Exhibiting a General View of the National Resources for Productive Labour Propositions for Ameliorating the Condition of the Poor by Regulations of Political Economy.” Bristol Selected Pamphlets (1806): 1-302. JSTOR. Web. University of Bristol Library. 2 April 2010 Corpus of Contemporary American English (COCA). Brigham Young University. Mark Davies, ed. 11 April 2011 < >. Corpus of Historical American English (COHA). Brigham Young University, Mark Davies, ed. 20 April 2011 < coha/>. 128 Coulson, Seana. Semantic Leaps: Frame-Shifting and Conceptual Blending in Meaning Construction . Cambridge: Cambridge University, 2001. “The Crisis of 1890.” The Economic Journal . 1.1 (March 1891):192-196. Croft, William, and D. Alan Cruse. Cognitive Linguistics . Cambridge: Cambridge University Press, 2004. Cruse, D.A. “Polysemy and Related Phenomena from a Cognitive Linguistic Viewpoint.” Computational Lexical Semantics . Eds. Patrick Saint-Dizier and Evelyne Viegas. Cambridge: Cambridge University, (1995): 46. Daramola, Adeyemi. “A Child of Necessity: An Analysis of Political Discourse in Nigeria.” Pragmatics 18.3 (2008): 355-380. Deignan, A. “Persuasive Uses of Metaphor in Discourse about Business and the Economy.” Words in Context: A Tribute to John Sinclair on his Retirement . Eds. Chris Heffer, and Helen Saunton. Birmingham: University of Birmingham, (2000):156-168. Derrida, Jacques, and F.C.T. Moore. "White Mythology: Metaphor in the Text of Philosophy.” New Literary History 6.1 (1974): 5-74. Eubanks, Philip. “An Analysis of Corporate Rule in Globalization Discourse: Why We Need Rhetoric to Explain Conceptual Figures.” Rhetoric Review 27.3 (2008): 236-258. ---. “The Story of Conceptual Metaphor: What Motivates Metaphoric Mappings?” Poetics Today 20.3 (1999): 419-442. ---. A War of Words in the Discourse of Trade . Carbondale: Southern Illinois University, (2000). Fauconnier, Giles, and Mark Turner. “Polysemy and Conceptual Blending.” Polysemy: Flexible Patterns of Meaning in Mind and Language . Eds. Brigitte Nerlich et al. Berlin: De Gruyter, (2003): 79-94. ---. The Way We Think . New York: Basic Books, 2002. Fernández, James W. Persuasions and Performances: the Play of Tropes in Culture . Bloomington: Indiana University Press, 1986. Ferraro, Fabrizio, Jeffrey Pfeffer, and Robert I. Sutton. “Economics Language and Assumptions: How Theories can Become Self-fulfilling.” Academy of Management Review 30.1 (2005): 8-24. 129 Fillmore, Charles J. “Frame Semantics.” Cognitive Linguistics: Basic Readings .Ed. Dirk Geeraerts. Berlin: De Gruyter, (2006): 373-400. Fillmore, Charles J. and B. T. S. Atkins. “Towards a Frame-based Lexicon: The Semantics of RISK and its Neighbors.” Frames, Fields, and Contrast: New Essays in Semantics and Lexical Organization . Eds. A. Lehrer and E. Kittay. Hillsdale: Lawrence Erlbaum Associates, (1992): 75-102. Fisher, Irving. “The Debt-Deflation Theory of Great Depressions.” Econometrica 1.4 (October 1933): 337-357. Foucault, Michel. The Archaeology of Knowledge . 1969. London: Routledge, 2002. ---. The Order of Things: An Archaeology of the Human Sciences . 1989. London: Routledge, 2002. ---. Power/Knowledge: Selected Interviews and Other Writings, 1972-1977 .Translated by Colin Gordon. New York: Pantheon, 1980. “France, Politics, Budget, Wall Street Repercussions, Money.” The Economist 9November 1929 . Freidrich, Paul. “Language, Ideology, and Political Economy.” American Anthropologist . 91 (1989): 295-312. Fukuda, Kosei. “A Comparative Study of Metaphors Representing the U.S. and Japanese Economies.” Journal of Pragmatics 41 (2009): 1693-1702 . Geeraerts, Dirk. Diachronic Prototype Semantics: A Contribution to Historical Lexicology . Oxford: Clarendon Press, 1997. Gibbs, Raymond W., ed. The Cambridge Handbook of Metaphor and Thought . Cambridge: Cambridge University Press, 2008. ---. “Embodied Experience and Linguistic Meaning.” Brain and Language 84 (2003): 1-15. ---. “Embodied Standing and the Psychological Semantics of Stand .” The Linguistics of Sitting, Standing, and Lying. Ed. John Newman. Amsterdam: John Benjamins Publishing. (2002): 347-400. ---. The Poetics of Mind: Figurative Thought, Language, and Understanding .Cambridge: Cambridge University, 1994. 130 Gibbs, Raymond, Paula Lenz Costa Lima, and Edson Francozo. “Metaphor is Grounded in Embodied Experience.“ Journal of Pragmatics 36 (2004): 1189-1210. Gibbs, Raymond W., and Teenie Matlock. “Psycholinguistic Perspectives on Polysemy.” Polysemy in Cognitive Linguistics: Selected Papers from the International 5 th Cognitive Linguistics Conference . Eds. H. Cuyckens and Britta Zawada. Amsterdam: John Benjamins (1997): 213-240. Giora, Rachel. "Literal vs. Figurative Language: Different or Equal?" Journal of Pragmatics 34 (2002): 487-506. Goatly, Andrew. Washing the Brain: Metaphor and Hidden Ideology .Philadelphia: John Benjamins, 2007. Google Trends . ed. Google, Inc. 11 April 2011 Gracia, Diego. “The Ethics of Diagnosis in Early Christianity and the Middle Ages.” The Ethics of Diagnosis . Eds. José Luis Peset and Diego Gracia, Dordrecht: Kluwer Academic Publishers,1992. Grygiel, Marcin “Semantic Change as a Process of Conceptual Blending” Annual Review of Cognitive Linguistics 2 (2004): 285-304. Gy őri, Gábor. “Semantic Change and Cognition.” Cognitive Linguistics 13.2 (2002):123-166 . Harris, Jonathan Gil. Sick Economies . Philadelphia: University of Pennsylvania Press, 2004. Hart, Christopher. “Critical Discourse Analysis and Metaphor: Toward a Theoretical Framework.” Critical Discourse Studies 5.2 (2008): 91-106. Hart, Christopher, and D. Lukes, eds. “Critical Discourse Analysis and Conceptualisation: Mental Spaces, Blended Spaces and Discourse Spaces,” Cognitive Linguistics in Critical Discourse Analysis: Application and Theory . Cambridge: Cambridge University Press, 2008. Henderson, Willie. “Metaphor and Economics.” New Directions in Economic Methodology . Ed. Roger E. Backhouse. Routledge: London, (1994): 343-367. ---. “Metaphor in Economics.” Economics 18 (1982):147-157. ---. “Metaphor, Economics and ESP: Some Comments.” English for Specific Purposes 19 (2000): 167-173. 131 Hoey, Michael. Lexical Priming: A New Theory of Words and Language . New York: Routledge, 2005. Hong, Seong-Hyuk. The Metaphor of Illness and Healing in Hosea and Its Significance in the Socio-Economic Context of Eighth-Century Israel and Judah . New York: Peter Lang Publishing, 2006. Horn, George M. “Idioms, Metaphors and Syntactic Mobility.” Journal of Linguistics 39.2 (July 2003): 245-273. Hutchinson, Tom, and Alan Waters. English for Specific Purposes: A Learner- centered Approach . Cambridge: Cambridge University Press, 1987. Irvine, Judith. “When Talk Isn’t Cheap.” American Ethnologist 16.2 (1989): 248-267. Irvine, Judith, and Susan Gal. “Language Ideology and Linguistic Differentation,” Regimes of Language: Ideologies, Polities, and Identities . Ed. P. V. Kroskrity. (2000): 35-84. Keynes, John Maynard. The General Theory of Employment Interest and Money .1936 New York: Classic Books America, 2009. Kimmel, Michael. “Why We Mix Metaphors (and Mix Them Well): Discourse Coherence, Conceptual Metaphor, and Beyond.” Journal of Pragmatics 42 (2010): 97-115. King James I. A Proclamation for Restraint of the Exportation Waste and Consumption of Coine and Bullion . London, 1622. Kittay, Eva F. Metaphor: Its Cognitive Force and Linguistic Structure . Oxford: Oxford University Press, 1987. Kittay, Eva F., and Adrienne Lehrer. “Semantic Fields and the Structure of Metaphor.” Studies in Language 5.1 (1987): 31-63. Kleinman, Arthur, MD. The Illness Narratives: Suffering, Healing, And The Human Condition . New York: Basic Books, 1988. Kövecses, Zoltán. Metaphor: A Practical Introduction . 2 nd ed. Oxford: Oxford University Press, 2010. ---. Metaphor in Culture: Universality and Variation . Cambridge: Cambridge University Press, 2005. 132 Lakoff, George. “The Death of Dead Metaphor.” Metaphor and Symbolic Activity 2.2 (1987): 143-147. ---. “Metaphor and War: The Metaphor System Used to Justify War in the Gulf.” The Sixties Project: Viet Nam Generation Journal & Newsletter , Part I 3.3 (1991): Ed. Kalí Tal. University of Virginia. 22 March 2010 iath.virginia.edu/sixties/HTML_docs/Texts/Scholarly/Lakoff_Gulf_Metapho r_1.html. ---. The Political Mind . New York: Penguin Books, 2008. ---. Women, Fire, and Dangerous Things . Chicago: University of Chicago Press, 1987. Lakoff, George, and Mark Johnson. Metaphors We Live By . Chicago: University of Chicago Press, 1980. ---. Philosophy in the Flesh . New York: Basic Books, 1999. Landau, Mark L., Daniel Sullivan, and Jeff Greenberg. “Evidence that Self-Relevant Motives and Metaphoric Framing Interact to Influence Political and Social Attitudes.” Psychological Science 20.11 (2009): 1421-1427. Malinowski, Bronislov. “The Problem of Meaning in Primitive Languages.” The Meaning of Meaning . Eds. C.K. Ogden and Ian A. Richards. New York: Harcourt & Brace (1923): 296-336. Malynes, Gerard de. Maintenance of Free Trade . London,1622. ---. Saint George for England Allegorically Described . London,1601. ---. A Treatise of the Canker of Englands Common Wealth. Divided in Three Parts: Wherein the Author imitating the Rule of Good Phisitions, First, declareth the Disease. Secondarily, Sheweth the Efficient Cause thereof. Lastly, a Remedy for the Same . London, 1601. Marmaridou, Sophia. “Cognitive, Cultural, and Constructional Motivations of Polysemy and Semantic Change: The Case of Greek Psyche .” Pragmatics & Cognition 18 (2010): 68-110. Marshall, Alfred. “Mechanical and Biological Analogies in Economics (1898).” Memorials of Alfred Marshall . Ed. A . Pigou (1925): 312-318. Marshall, Arthur. Principles of Economics . New York: The Macmillan Co., 1898. 133 McCloskey, Deirdre N. Knowledge and Persuasion in Economics . Cambridge: Cambridge University Press, 1994. ---. "Metaphors Economists Live By." Social Research 62.2 (1995): 215-237. ---. The Rhetoric of Economics . 2 nd ed. Madison: The University of Wisconsin Press, 1998. McCloskey, Donald N. “The Rhetoric of Economics.” Journal of Economic Literature 21 (1983): 481-517. McConnell-Ginet, Sally. “Words in the World: How and Why Meanings can Matter.” Language 84.3 (2008): 497-525. McConnell-Ginet, Sally, and Gennaro Chierchia. Meaning and Grammar: An Introduction to Semantics . 2 nd ed. 1990. Cambridge: MIT Publishing, 2000. Milken, Michael, and Jonathan Simons. “Illness as Economic Metaphor.” Wall Street Journal . June 20, 2009. 23 February 2010 com/article/SB1245457861132323049.html. Misselden, E. The Circle of Commerce or The Ballance of Trade in Defense of Free Trade . London, 1623. ---. Free Trade or, The Meanes to Make Trade Florish: Wherein, The Causes of the Decay of Trade in this Kingdome, are Discovered: And the Remedies also to Remooue the Same, are Represented . London,1622. Morris, Michael W., Oliver J. Sheldon, Daniel R. Ames, and Maia J. Young."Metaphors and the Market: Consequences and Preconditions of Agent and Object Metaphors in Stock Market Commentary." Organizational Behavior and Human Decision Processes 102 (2007): 174-192. Mun, Thomas. A Discourse of Trade, From England Vnto the East-Indies: Ansering to Diuerse Obiections Which Are Usually Made Against the Same . London, 1621. ---. Englands Treasure by Forraign Trade; Or, The Balance of our Forraign Trade is The Rule of our Treasure . London, 1669. Musolff, Andreas. “Ideological Functions of Metaphors: The Conceptual Metaphors of Health and Illness in Public Discourse.” Cognitive Linguistics Research: Cognitive models in Language and Thought . Eds. René Dirven, Roslyn Frank, and Martin Pütz. (2003): 327-352. 134 ---. “Metaphor Scenarios in Public Discourse.” Metaphor and Symbolic Activity 21.2 (2006): 23-38. Nerlich, Brigitte, and David D. Clarke. "Ambiguities We Live By: Towards a Pragmatics of Polysemy." Journal of Pragmatics 33 (2001): 1-20. ---. “Semantic Fields and Frames: Historical explorations of the Interface Between Language, Action, and Cognition.” Journal of Pragmatics 32 (2000): 125-150. Norris, Floyd. “Looking Back at the Crash of ’29.” The New York Times on the Web. 1999 24 Feb 2010 financial/index-1929-crash.html. Nuyts, Jan. “Cognitive Linguistics and Functional Linguistics.” The Oxford Handbook of Cognitive Linguistics . Eds. Dirk Geeraerts, and Herbert Cuyckens. Oxford: Oxford University Press, 2007: 543-565. OED Online . 2010. Oxford University Press. “recover, v. 1” OED Online . 2010. Oxford University Press. 28 Mar. 2010. entry/501996 . “recovery, n. 3” OED Online . 2010. Oxford University Press. 28 Mar. 2010. entry/501996 . “recovery, n. 17 ” OED Online . 2010. Oxford University Press. 28 Mar. 2010. entry/501996 . “recuperate, v. 1” OED Online . 2010. Oxford University Press. 3 Apr. 2010. entry/5019967 . Ortony, Andrew. Metaphor and Thought . 2 nd ed. 1979. Cambridge: Cambridge University Press, 1993. Parker, Martin. “Three Merry Cobblers.” Roxburghe Collection (I. 408- 409): c1625. English Broadside Ballad Archive . University of California Santa Barbara. ed. Patricia Fumerton 28 May 2011 Web. english.ucsb.edu/ballad/30279 . Partington, Alan Scott. "A Linguistic Account of Wordplay: The Lexical Grammar of Punning." Journal of Pragmatics 41 (2009): 1794-1809. Perlman, Mark. “Review of Hutchison’s Knowledge and Ignorance in Economics.” Journal of Economic Literature 6 (June 1978): 582-585. 135 Pierce, Russell S., and Dan L. Chiappe. “The Roles of Aptness, Conventionality, and Working Memory in the Production of Metaphors and Similes.” Metaphor and Symbol 24 (2009): 1-19. “The Plague at Westminster or, An Order for the Visitation of a Sick Parliament.” London. 1647. Bell and Howell 13 June 2010. “recoveren.” Def. v2. Middle English Dictionary . 2001. University of Michigan. 3 April 2010. < /mec/med-idx?type=id&id=MED36240&egs=all>. “Recovery.gov: Track the Money.” 9 April 2011 < >. Reimer, M. “The Problem of Dead Metaphors.” Philosophical Studies: An International Journal for Philosophy in the Analytic Tradition 82.1 (1996): 13-25. Rosch, Eleanor, and Barbara B. Lloyd, eds. “Principles of Characterization.” Cognition and Categorization . Hillsdale: Lawrence Erlbaum Associates Publishers (1978): 27-48. Scacco, Joshua. “Shaping Economic Reality: A Critical Metaphor Analysis of President Barack Obama’s Economic Language During His First 100 Days.” Gnovis Journal . Ed. Lydia Kelow-Bennett. Georgetown University. 10.1 (2009) Web 14 January 2011. volume-10/fall. Scheff, Thomas J. “Micro-Linguistics and Social Structure: A Theory of Social Action.” Sociological Theory 4.1 (1986): 71-83. Seargeant, Philip. “The Historical Ontology of Language.” Language Sciences 32 (2008): 1-13. Shimko, Keith L. “Metaphors and Foreign Policy Decision Making .” Political Psychology 15.4 (1994): 655-671. Silverstein, Michael. “Language Structure and Language Ideology.” The Elements, a Parasession on Linguistic Units and Levels . Eds . P.R. Clyne, William F. Hanks, and Carol L. Hofbauer. Chicago: Chicago Linguistic Society (1979): 193-247. ---. “The Uses and Utility of Ideology.” Language Ideologies: Practice and Theory . Eds. Bambi B. Schieffelin, Kathryn A. Woolard, and Paul V. Kroskrity. Oxford: Oxford University Press (1998): 123-145. 136 Skorczynska, Hanna, and Alice Deignan. “Readership and Purpose in the Choice of Economics Metaphors.” Metaphor and Symbol 21.2 (2006): 87-104. Smith, Adam. Wealth of Nations . 1776. Web. Library of Economics and Liberty. 15 October 2010 < http:www.econlib.org >. Sontag, Susan. Illness as Metaphor and AIDS and Its Metaphors . 1978. New York: Farrar, Straus and Giroux, 1988. Sorkin, Andrew Ross. Too Big to Fail: The Inside Story of How Wall Street and Washington Fought to Save the Financial System – and Themselves . New York: Penguin, 2009. Starkey, Thomas. Dialogue Between Reginald Pole and Thomas Lupset .London, 1535. Stern, Gary, Ron Feldman, and Paul Volcker. Too Big to Fail: The Hazards of Bank Bailouts . Washington D.C.: The Brookings Institution, 2004. “Stocks Collapse in 16,4100,030-Share Day, Buy Rally at Close Cheers Brokers; Bankers Optimistic, To Continue Aid.” The New York Times . October 29, 1929. Web. 23 February 2010 NYT Learning <http:www.nytimes.com/ learning/general/onthisdat/991029onthisday>. Sweetser, Eve E. “Blended Spaces and Performativity.” Cognitive Linguistics 11.34 (2000): 305-333. ---. From Etymology to Pragmatics: Metaphorical and Cultural Aspects of Semantic Structure . Cambridge: Cambridge University Press, 1990. Talbott, John R. Contagion: The Financial Epidemic that is Sweeping the Global Economy and How to Protect Yourself from It . Hoboken, NJ: John Wiley & Sons, 2009. Thibodeau, Paul, and Frank H. Durgin. “Productive Figurative Communication: Conventional Metaphors Facilitate the Comprehension of Related Novel Metaphors.” Journal of Memory and Language 58 (2008): 521-540 . Traugott, Ellizabeth Closs, and Richard B, Dasher. Regularty of Semantic Change . Cambridge: Cambridge University Press, 2002. United States. Congress Joint Economic Committee. “Financial Meltdown and Policy Response: Research Report #110-25.” By Congressman Jim Saxton. September 2008. 25 April 2010 < Research%20Reports/2008/rr110-25.pdf>. 137 Van de Weijer, Joost C. Glossary of Linguistic Terminology . October 28 2004. 14 February 2009. < glossary.pdf >. Van Dijk, Tuen A. “Principles of Critical Discourse Analysis.” Discourse & Society 4.2 (1993): 249-283. Vanhove, Martine, ed. From Polysemy to Semantic Change . Amsterdam: John Benjamins Publishing, 2008. White, Michael. "Metaphor and Economics: the Case of growth. " English for Specific Purposes 22 (2003): 131-151. White, Michael, and Honesto Herrera. “Metaphor and Ideology in Press Coverage.” Cognitive Linguistics Research: Cognitive Models in Language and Thought . Eds. René Dirven, Roslyn Frank, Martin Pütz. Berlin: De Gruyter (2003): 277-324. Wierzbicka, Anna. “Russian Cultural Scripts: The Theory of Cultural Scripts and Its Applications.” Ethos 30.4 (2002):401-432. Williams, John N., “Processing Polysemous Words in Context: Evidence for Interrelated Meanings.” Journal of Psycholinguistic Research 21.3 (1992): 193-218. Woolard, Kathryn, and Bambi Schieffelin. “Language Ideology.” Annual Review of Anthropology 23 (1994): 55-82. Yin, Robert K. Case Study Research: Design and Methods . Newbury Park: Sage, 1984. Yule, George. Pragmatics . Oxford: Oxford University Press, 1996. Zinken, Jörg. “Discourse Metaphors: The Link Between Figurative Language and Habitual Analogies.” Cognitive Linguistics 18.4 (2007): 445-466. ---. “Ideological Imagination: Intertextual and Correlational Metaphors in Political Discourse.” Discourse Society 14.4 (2003): 507-523.
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https://www.hopkinsmedicine.org/health/conditions-and-diseases/radiculopathy
Skip to Main Content Health Health Radiculopathy What You Need to Know Radiculopathy describes a range of symptoms produced by the pinching of a nerve root in the spinal column. The pinched nerve can occur at different areas along the spine (cervical, thoracic or lumbar). Symptoms of radiculopathy vary by location but frequently include pain, weakness, numbness and tingling. A common cause of radiculopathy is narrowing of the space where nerve roots exit the spine, which can be a result of stenosis, bone spurs, disc herniation or other conditions. Radiculopathy symptoms can often be managed with nonsurgical treatments, but minimally invasive surgery can also help some patients. What is radiculopathy? Your spine is made of many bones called vertebrae, and your spinal cord runs through a canal in the center of these bones. Nerve roots split from the cord and travel between the vertebrae into various areas of your body. When these nerve roots become pinched or damaged, the resulting symptoms are called radiculopathy. Types of Radiculopathy Radiculopathy can have different symptoms and different names depending on where in the spine it occurs. Lumbar Radiculopathy When radiculopathy occurs in the lower back, it is known as lumbar radiculopathy, also referred to as sciatica because nerve roots that make up the sciatic nerve are often involved. The lower back is the area most frequently affected by radiculopathy. Radiculopathy Prevention While radiculopathy can’t always be prevented, staying physically fit and maintaining a healthy weight may reduce your risk of radiculopathy. Using best practices for good posture while sitting, playing sports, exercising or lifting heavy objects is also important for preventing injuries. Cervical Radiculopathy Cervical radiculopathy describes a compressed nerve root in the neck (cervical spine). Because the nerve roots in this area of the spine primarily control sensations in your arms and hands, this is where the symptoms are most likely to occur. Thoracic Radiculopathy Thoracic radiculopathy refers to a compressed nerve root in the thoracic area of the spine, which is your upper back. This is the least common location for radiculopathy. The symptoms often follow a dermatomal distribution, and can cause pain and numbness that wraps around to the front of your body. Symptoms of Radiculopathy When a nerve root is compressed, it becomes inflamed. This results in several unpleasant symptoms that may include: Sharp pain in the back, arms, legs or shoulders that may worsen with certain activities, even something as simple as coughing or sneezing Weakness or loss of reflexes in the arms or legs Numbness of the skin, “pins and needles,” or other abnormal sensations (paresthesia) in the arms or legs Your specific symptoms will depend on where in the spine the nerve root is pinched. However, it’s also possible that you don’t experience any symptoms or you go through periodic flare-ups of symptoms. Causes of Radiculopathy Radiculopathy is typically caused by changes in the tissues surrounding the nerve roots. These tissues include bones of the spinal vertebrae, tendons and intervertebral discs. When these tissues shift or change in size, they may narrow the spaces where the nerve roots travel inside the spine or exit the spine; these openings are called foramina. The narrowing of foramina is known as foraminal stenosis, which is very similar to spinal stenosis that affects the spinal cord. In most cases, foraminal stenosis is caused by gradual degeneration of the spine that happens as you age. But it can also be a result of a spinal injury. Herniated Discs One common cause of foraminal stenosis and radiculopathy is a bulging or herniated disc. Spinal discs act as cushions between your vertebrae. On occasion, these discs slip out of place or become damaged and press on nerves. This problem is most likely to occur in your lower back, but it can also affect your neck. Bone Spurs Another possible cause of radiculopathy that may lead to narrowing of foramina is bone spurs — areas of extra bone growth. Bone spurs can form in the spine due to inflammation from osteoarthritis, trauma or other degenerative conditions. Other Causes Thickening (ossification) of the spinal ligaments may also lead to narrowing of the space around the nerve roots and subsequent nerve compression. Less common causes of radiculopathy include spinal infections and various cancerous and noncancerous growths in the spine that may press against the nerve roots. Radiculopathy and Myelopathy Sometimes, radiculopathy can be accompanied by myelopathy — compression of the spinal cord itself. Herniated or bulging discs can sometimes press on the spinal cord and on the nerve roots. When the spinal cord is involved, the symptoms can be more severe, including poor coordination, trouble walking and paralysis. Radiculopathy Versus Neuropathy Radiculopathy symptoms may overlap with those of peripheral neuropathy, making it difficult to pinpoint the source of the problem. Peripheral neuropathy is the damage of the peripheral nervous system, such as carpal tunnel syndrome that involves trapped nerves in the wrist. Radiculopathy is the pinching of the nerves at the root, which sometimes can also produce pain, weakness and numbness in the wrist and hand. Consult a spine specialist for an accurate diagnosis. Radiculopathy Diagnosis Your doctor may take several steps to diagnose radiculopathy: A physical exam and physical tests may be used to check your muscle strength and reflexes. If you have pain with certain movements, this may help your doctor identify the affected nerve root. Imaging tests, such as an X-ray, CT scan or MRI scan, are used to better see the structures in the problem area. Nerve conduction studies, along with electromyography, can also be used to help pinpoint whether the problem is neurological or muscular. Radiculopathy Treatment Radiculopathy treatment will depend on the location and the cause of the condition as well as many other factors. Nonsurgical treatment is typically recommended first and may include: Medications, like nonsteroidal anti-inflammatory drugs, opioid medicines or muscle relaxants, to manage the symptoms Weight loss strategies to reduce pressure on the problem area Physical therapy to strengthen the muscles and prevent further damage Steroid injections to reduce inflammation and relieve pain Some people may need more advanced treatments, such as surgery. Surgery is typically used to reduce the pressure on the nerve root by widening the space where the nerve roots exit the spine. This may involve removing all or parts of a disc and/or vertebrae. Cervical posterior foraminotomy is one of the minimally invasive spine surgery options available. Specializing In: Spine Surgery Lower Back Injury Back Pain Physical Medicine and Rehabilitation Neck Pain Degenerative Disc Disease Spinal Stenosis Minimally Invasive Lumbar Decompression Cervical Degenerative Disc Herniation Cervical Degenerative Disc Disease Cervical Radiculopathy Thoracic Disc Herniation Transverse Myelitis Traumatic and Non-traumatic Spinal Cord Injury Acute Flaccid Myelitis Spinal Instability Spinal Pain Spondylolisthesis Spondyloarthropathies Disc Pain Degenerative Spine Disease Musculoskeletal Center Find Additional Treatment Centers at: Howard County Medical Center Sibley Memorial Hospital Suburban Hospital Related Minimally Invasive Lumbar Discectomy Spinal Stenosis Lumbar Spinal Stenosis Spinal Cord Compression Request an Appointment Find a Doctor Find a Doctor Related Minimally Invasive Lumbar Discectomy Spinal Stenosis Lumbar Spinal Stenosis Related Topics Back and Neck Pain Orthopedics Back and Spine Surgery
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https://www.usna.edu/Users/physics/mungan/_files/documents/Scholarship/EquilibriumForces.pdf
Mechanical Equilibrium of a Rigid Body—C.E. Mungan, Spring 2023 If exactly one nonzero external force acts on an extended rigid body, it cannot be in equilibrium. If exactly two forces act on it, we get equilibrium if and only if the two forces are equal & opposite and are directed along the same line of action. If four or more forces act on the body, we can choose an origin and then combine all forces Fi with 3 i ≥ into a single new force that equals the sum of these forces and whose torque equals the sum of their torques . So we are left with the question of when will exactly three forces1 result in equilibrium? The answer is that the forces must satisfy two conditions: (1) they must vectorially sum to zero, and (2) the three forces must be concurrent which means that their three lines of action must intersect at a common point (possibly at infinity). Equilibrium here means that two facts are true about the body. First, its center of mass will have zero translational acceleration. That will be true if and only if 3 1 0 i i= = ∑F . (1) Second, its angular acceleration about its center of mass (defining an origin O) must be zero, which requires 3 1 0 i i i= × = ∑r F (2) where ri is the position vector from O to the point of application of force Fi. We can prove that if both Eqs. (1) and (2) hold, then it also must be true that 3 1 0 i i i= ′× = ∑r F (3) where i′ r is the position vector from any other origin O′ to the point of application of force Fi. In other words, we can shift the origin by R for the calculation of the torques from O to O′ such that i i′ = + r r R (4) for all 1 i = to 3. The proof is straightforward. Substitute Eq. (4) into (2) to get 3 3 1 1 0 i i i i i = = ′× + × = ∑ ∑ r F R F . (5) But the second sum is zero according to Eq. (1), and thus Eq. (3) follows. There is nothing special about having chosen the origin O to coincide with the center of mass of the body; we could have started with any origin O we like. The key result is that if Eqs. (1) and (2) hold for 1If the object rotates about an axle, the axle is assumed to exert only a single reaction force on the object. For example, there cannot be a frictional force distributed around the circumference of the axle, nor a normal force distributed along the length of the axle. any one choice of origin, then they must hold for all possible choices of origin. We will apply this conclusion to concurrency. First let’s show that equilibrium requires the three forces to be coplanar. If any two force vectors were not contained in the same plane, then we could choose the origin to be located at the point of application of the third force so that it produces zero torque. But the two noncoplanar forces would produce noncoplanar torques about that origin, and thus the sum of the three torques would not be zero. Therefore, any two of the forces must be coplanar. In addition, the third force must equal the negative of the vector sum of those two forces, and thus the third force must also be in the same plane as them, so that all three forces are necessarily coplanar. Now start the concurrency proof by supposing that at least two of the three equilibrium forces are not parallel to each other. In that case, they must intersect at one specific point C in their common plane. Choose that point to be the origin so that neither of those two forces produces a torque about it. Then the only way the three torques can sum to zero is if the third force also has a line of action passing through that point of application C, which is therefore concurrent to all three forces. Finally, a special case is that C could be located at infinity, which happens if all three forces are parallel to each other . Choose that parallel direction to define the y axis. Now note that, for any choice of origin, the torque produced by a force F is unchanged if we translate the force along its line of action. (Originally the torque is × r F . Replacing r by m + r F , where m is any numerical multiple we like with units of m/N, gives exactly the same torque since 0 × = F F .) So, translate all three parallel forces such that their points of application all lie along the x axis in their common xy plane. Equilibrium now requires that 3 1 0 i yi i x F = = ∑ . (6) This sum reduces to at most two terms if the origin is chosen to pass through the line of action of one of the forces. An application is to a one-dimensional rigid rod like a see-saw. K.L. Goh, “Concurrent and coplanar forces that are in equilibrium,” Phys. Teach. 56, 384 (2018). A. Alameh, “Static equilibrium in a uniform gravitational field,” Phys. Teach. 61, 298 (2023).
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https://artofproblemsolving.com/wiki/index.php/Arithmetico-geometric_series?srsltid=AfmBOorBhQ1JcypIBHm1Le6NLgIAdxS-szxSLs79W3ImyQF7kkJFhlzn
Art of Problem Solving Arithmetico-geometric series - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki Arithmetico-geometric series Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Arithmetico-geometric series An arithmetico-geometric series is the sum of consecutive terms in an arithmetico-geometric sequence defined as: , where and are the th terms of arithmetic and geometric sequences, respectively. Contents [hide] 1 Finite Sum 2 Infinite Sum 3 Example Problems 4 See Also Finite Sum The sum of the first terms of an is , where is the common difference of and is the common ratio of . Or, , where is the sum of the first terms of . Proof: Let represent the sum of the first terms. Infinite Sum The sum of an infinite arithmetico-geometric sequence is , where is the common difference of and is the common ratio of (). Or, , where is the infinite sum of the . Example Problems Mock AIME 2 2006-2007 Problem 5 1994 AIME Problem 4 See Also Sequence Arithmetic sequence Geometric sequence Retrieved from " Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
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https://www.youtube.com/watch?v=TgKwz5Ikpc8
Abstract vector spaces | Chapter 16, Essence of linear algebra 3Blue1Brown 7710000 subscribers 56579 likes Description 1600539 views Posted: 24 Sep 2016 This is really the reason linear algebra is so powerful. Help fund future projects: An equally valuable form of support is to simply share some of the videos. Home page: Full series: Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. Thanks to these viewers for their contributions to translations Russian: e-p-h 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: Various social media stuffs: Website: Twitter: Patreon: Facebook: Reddit: Transcript: I'd like to revisit a deceptively simple question that I asked in the very first video of this series. What are vectors? Is a two-dimensional vector, for example, fundamentally an arrow on a flat plane that we can describe with coordinates for convenience? Or is it fundamentally that pair of real numbers which is just nicely visualized as an arrow on a flat plane? Or are both of these just manifestations of something deeper? On the one hand, defining vectors as primarily being a list of numbers feels clear-cut and unambiguous. It makes things like four-dimensional vectors or 100-dimensional vectors sound like real, concrete ideas that you can actually work with. When otherwise, an idea like four dimensions is just a vague geometric notion that's difficult to describe without waving your hands a bit. But on the other hand, a common sensation for those who actually work with linear algebra, especially as you get more fluent with changing your basis, is that you're dealing with a space that exists independently from the coordinates that you give it, and that coordinates are actually somewhat arbitrary, depending on what you happen to choose as your basis vectors. Core topics in linear algebra, like determinants and eigenvectors, seem indifferent to your choice of coordinate systems. The determinant tells you how much a transformation scales areas, and eigenvectors are the ones that stay on their own span during a transformation. But both of these properties are inherently spatial, and you can freely change your coordinate system without changing the underlying values of either one. But if vectors are not fundamentally lists of real numbers, and if their underlying essence is something more spatial, that just begs the question of what mathematicians mean when they use a word like space or spatial. To build up to where this is going, I'd actually like to spend the bulk of this video talking about something which is neither an arrow nor a list of numbers, but also has vector-ish qualities – functions. You see, there's a sense in which functions are actually just another type of vector. In the same way that you can add two vectors together, there's also a sensible notion for adding two functions, f and g, to get a new function, f plus g. It's one of those things where you kind of already know what it's going to be, but actually phrasing it is a mouthful. The output of this new function at any given input, like negative four, is the sum of the outputs of f and g when you evaluate them each at that same input, negative four. Or more generally, the value of the sum function at any given input x is the sum of the values f of x plus g of x. This is pretty similar to adding vectors coordinate by coordinate, it's just that there are, in a sense, infinitely many coordinates to deal with. Similarly, there's a sensible notion for scaling a function by a real number, just scale all of the outputs by that number. And again, this is analogous to scaling a vector coordinate by coordinate, it just feels like there's infinitely many coordinates. Now, given that the only thing vectors can really do is get added together or scaled, it feels like we should be able to take the same useful constructs and problem solving techniques of linear algebra that were originally thought about in the context of arrows and space and apply them to functions as well. For example, there's a perfectly reasonable notion of a linear transformation for functions, something that takes in one function and turns it into another. One familiar example comes from calculus, the derivative. It's something which transforms one function into another function. Sometimes in this context you'll hear these called operators instead of transformations, but the meaning is the same. A natural question you might want to ask is what it means for a transformation of functions to be linear. The formal definition of linearity is relatively abstract and symbolically driven compared to the way that I first talked about it in chapter 3 of this series. But the reward of abstractness is that we'll get something general enough to apply to functions as well as arrows. A transformation is linear if it satisfies two properties, commonly called additivity and scaling. Additivity means that if you add two vectors, v and w, then apply a transformation to their sum, you get the same result as if you added the transformed versions of v and w. The scaling property is that when you scale a vector v by some number, then apply the transformation, you get the same ultimate vector as if you scaled the transformed version of v by that same amount. The way you'll often hear this described is that linear transformations preserve the operations of vector addition and scalar multiplication. The idea of gridlines remaining parallel and evenly spaced that I've talked about in past videos is really just an illustration of what these two properties mean in the specific case of points in 2D space. One of the most important consequences of these properties, which makes matrix vector multiplication possible, is that a linear transformation is completely described by where it takes the basis vectors. Since any vector can be expressed by scaling and adding the basis vectors in some way, finding the transformed version of a vector comes down to scaling and adding the transformed versions of the basis vectors in that same way. As you'll see in just a moment, this is as true for functions as it is for arrows. For example, calculus students are always using the fact that the derivative is additive and has the scaling property, even if they haven't heard it phrased that way. If you add two functions, then take the derivative, it's the same as first taking the derivative of each one separately, then adding the result. Similarly, if you scale a function, then take the derivative, it's the same as first taking the derivative, then scaling the result. To really drill in the parallel, let's see what it might look like to describe the derivative with a matrix. This will be a little tricky, since function spaces have a tendency to be infinite dimensional, but I think this exercise is actually quite satisfying. Let's limit ourselves to polynomials, things like x squared plus 3x plus 5, or 4x to the seventh minus 5x squared. Each of the polynomials in our space will only have finitely many terms, but the full space is going to include polynomials with arbitrarily large degree. The first thing we need to do is give coordinates to this space, which requires choosing a basis. Since polynomials are already written down as the sum of scaled powers of the variable x, it's pretty natural to just choose pure powers of x as the basis function. In other words, our first basis function will be the constant function, b0 of x equals 1. The second basis function will be b1 of x equals x, then b2 of x equals x squared, then b3 of x equals x cubed, and so on. The role that these basis functions serve will be similar to the roles of i-hat, j-hat, and k-hat in the world of vectors as arrows. Since our polynomials can have arbitrarily large degree, this set of basis functions is infinite. But that's okay, it just means that when we treat our polynomials as vectors, they're going to have infinitely many coordinates. A polynomial like x squared plus 3x plus 5, for example, would be described with the coordinates 5, 3, 1, then infinitely many zeros. You'd read this as saying that it's 5 times the first basis function, plus 3 times that second basis function, plus 1 times the third basis function, and then none of the other basis functions should be added from that point on. The polynomial 4x to the seventh minus 5x squared would have the coordinates 0, 0, negative 5, 0, 0, 0, 0, 4, then an infinite string of zeros. In general, since every individual polynomial has only finitely many terms, its coordinates will be some finite string of numbers with an infinite tail of zeros. In this coordinate system, the derivative is described with an infinite matrix that's mostly full of zeros, but which has the positive integers counting down on this offset diagonal. I'll talk about how you could find this matrix in just a moment, but the best way to get a feel for it is to just watch it in action. Take the coordinates representing the polynomial x cubed plus 5x squared plus 4x plus 5, then put those coordinates on the right of the matrix. The only term that contributes to the first coordinate of the result is 1 times 4, which means the constant term in the result will be 4. This corresponds to the fact that the derivative of 4x is the constant 4. The only term contributing to the second coordinate of the matrix vector product is 2 times 5, which means the coefficient in front of x in the derivative is 10. That one corresponds to the derivative of 5x squared. Similarly, the third coordinate in the matrix vector product comes down to taking 3 times 1. This one corresponds to the derivative of x cubed being 3x squared. And after that, it'll be nothing but zeros. What makes this possible is that the derivative is linear. And for those of you who like to pause and ponder, you could construct this matrix by taking the derivative of each basis function and putting the coordinates of the results in each column. So, surprisingly, matrix vector multiplication and taking a derivative, which at first seem like completely different animals, are both just really members of the same family. In fact, most of the concepts I've talked about in this series with respect to vectors as arrows in space, things like the dot product or eigenvectors, have direct analogs in the world of functions, though sometimes they go by different names, things like inner product or eigenfunction. So back to the question of what is a vector. The point I want to make here is that there are lots of vectorish things in math. As long as you're dealing with a set of objects where there's a reasonable notion of scaling and adding, whether that's a set of arrows in space, lists of numbers, functions, or whatever other crazy thing you choose to define, all of the tools developed in linear algebra regarding vectors, linear transformations and all that stuff, should be able to apply. Take a moment to imagine yourself right now as a mathematician developing the theory of linear algebra. You want all of the definitions and discoveries of your work to apply to all of the vectorish things in full generality, not just to one specific case. These sets of vectorish things, like arrows or lists of numbers or functions, are called vector spaces. And what you as the mathematician might want to do is say, hey everyone, I don't want to have to think about all the different types of crazy vector spaces that you all might come up with. So what you do is establish a list of rules that vector addition and scaling have to abide by. These rules are called axioms, and in the modern theory of linear algebra, there are eight axioms that any vector space must satisfy if all of the theory and constructs that we've discovered are going to apply. I'll leave them on the screen here for anyone who wants to pause and ponder, but basically it's just a checklist to make sure that the notions of vector addition and scalar multiplication do the things that you'd expect them to do. These axioms are not so much fundamental rules of nature as they are an interface between you, the mathematician, discovering results, and other people who might want to apply those results to new sorts of vector spaces. If, for example, someone defines some crazy type of vector space, like the set of all pi creatures with some definition of adding and scaling pi creatures, these axioms are like a checklist of things that they need to verify about their definitions before they can start applying the results of linear algebra. And you, as the mathematician, never have to think about all the possible crazy vector spaces people might define. You just have to prove your results in terms of these axioms so anyone whose definitions satisfy those axioms can happily apply your results, even if you never thought about their situation. As a consequence, you'd tend to phrase all of your results pretty abstractly, which is to say, only in terms of these axioms, rather than centering on a specific type of vector, like arrows in space or functions. For example, this is why just about every textbook you'll find will define linear transformations in terms of additivity and scaling, rather than talking about gridlines remaining parallel and evenly spaced. Even though the latter is more intuitive, and at least in my view, more helpful for first-time learners, even if it is specific to one situation. So the mathematician's answer to what are vectors is to just ignore the question. In the modern theory, the form that vectors take doesn't really matter. Arrows, lists of numbers, functions, pi creatures, really, it can be anything, so long as there's some notion of adding and scaling vectors that follows these rules. It's like asking what the number 3 really is. Whenever it comes up concretely, it's in the context of some triplet of things, but in math, it's treated as an abstraction for all possible triplets of things, and lets you reason about all possible triplets using a single idea. Same goes with vectors, which have many embodiments, but math abstracts them all into a single, intangible notion of a vector space. But, as anyone watching this series knows, I think it's better to begin reasoning about vectors in a concrete, visualizable setting, like 2D space, with arrows rooted at the origin. But as you learn more linear algebra, know that these tools apply much more generally, and that this is the underlying reason why textbooks and lectures tend to be phrased, well, abstractly. So with that, folks, I think I'll call it an in to this essence of linear algebra series. If you've watched and understood the videos, I really do believe that you have a solid foundation in the underlying intuitions of linear algebra. This is not the same thing as learning the full topic, of course, that's something that can only really come from working through problems, but the learning you do moving forward could be substantially more efficient if you have all the right intuitions in place. So, have fun applying those intuitions, and best of luck with your future learning.
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http://teacher.pas.rochester.edu/phy121/lecturenotes/Chapter15/Chapter15.html
Chapter 15 15. OSCILLATIONS 15.1. Simple Harmonic Motion 15.2. Damped Simple Harmonic Motion 15.3. Driven Harmonic Motion 15. OSCILLATIONS 15.1. Simple Harmonic Motion Any motion that repeats itself at regular intervals is called harmonic motion. A particle experiences a simple harmonics motion if its displacement from the origin as function of time is given by where xm, [omega] and [phi] are constants, independent of time. The quantity xm is called the amplitude of the motion and is the maximum displacement of the mass. The time-varying quantity ([omega]t + [phi]) is called the phase of the motion and [phi] is called the phase constant. The phase constant is determined by the initial conditions. The angular frequency [omega] is a characteristic of the system, and does not depend on the initial conditions. The unit of angular frequency is rad/s. The period T of the motion is defined as the time required to complete one oscillation. Therefore, the displacement x(t) must return to its initial value after one period x(t) = x(t + T) This is equivalent to Using the relation it is immediately clear that The number of oscillations carried out per second is called the frequency of the oscillation. The symbol for frequency is [nu] and its unit is the Hertz (Hz): 1 Hz = 1 oscillation per second = 1 s-1 The period T and the frequency [nu] are related as follows The velocity of an object carrying out simple harmonic motion can be calculated easily The positive quantity [omega] xm is called the velocity amplitude and is the maximum velocity of the object. Note that the phase of the velocity and displacement differ by 90deg.. This means that the velocity is greatest when the displacement is zero and vice versa. The acceleration of an object carrying out simple harmonic motion is given by The positive quantity [omega]2 xm is the acceleration amplitude am. Using the expression for x(t), the expression for a(t) can be rewritten as This shows that the acceleration is proportional to the displacement, but opposite in sign. The force acting on the mass can be calculated using Newton's second law This equation of force is similar to the force exerted by a spring (Hooke's law) F = - k x Comparing these last two equations we conclude that k = m [omega]2 and " Simple harmonic motion is the motion executed by a particle of mass m, subject to a force F that is proportional to the displacement of the particle, but opposite in sign. " The system shown in Figure 15.1 forms a simple harmonic oscillator. It will oscillate with an angular frequency [omega] given by The period T of the oscillation is given by The total mechanical energy of the simple harmonic oscillator consist of potential and kinetic energy. The potential energy of the system is given by Figure 15.1. A simple harmonic oscillator. The kinetic energy of the system is given by The total mechanical energy of the system can now be calculated The total mechanical energy of the simple harmonic oscillator is constant (independent of time). However, the kinetic and potential energies are functions of time. Example: The torsion pendulum The operation of a torsion pendulum is associated with twisting a suspension wire. The motion described by the torsion pendulum is called angular simple harmonic motion. The restoring torque is given by where [kappa] is a constant that depends on the properties of the suspension wire (its length, diameter and material). For a given torque we can calculate the angular acceleration a or Comparing this equation with the relation between the linear acceleration and the linear displacement of an object, we conclude that The period of the torsion pendulum is given by Example: Classical simple pendulum The classical simple pendulum is shown in Figure 15.2. It consists out of a mass m suspended from a massless string of length L. The forces acting on the mass are the gravitational force m g and the tension T in the string. The radial component of the gravitational force, m g cos([theta]), determines the tension in the wire, but will not alter the motion of the mass. The tangential component of the gravitational force, m g sin([theta]), is always directed towards the rest position of the pendulum. This component of the gravitational force is called the restoring force: For small angles, sin([theta]) ~ [theta]. This shows that where s is the displacement of the mass along the arc. Again we conclude that the restoring force is proportional to the displacement, and of opposite sign. The motion is therefore that of a harmonic oscillator. The acceleration of the mass is related to the displacement s Figure 15.2. Classical simple pendulum. This immediately indicates that the angular frequency [omega] is given by and therefore, the period of the motion is given by Figure 15.3. The physical pendulum. Example: The Physical Pendulum In the real world pendulums are far from simple. In general, the mass of the pendulum is not concentrated in one point, but will be distributed. Figure 15.3 shows a physical pendulum. The physical pendulum is suspended through point O. The effect of the force of gravity can be replaced by the effect of a single force, whose magnitude is m g, acting on the center of gravity of the pendulum (which is equal to the center of mass if the gravitational acceleration is constant). The resulting torque (with respect to O) is given by where h is the distance between the rotation axis and the center of gravity. In the limit of small angles, this torque can be rewritten as The angular acceleration a of the pendulum is related to the torque [tau] and the rotational inertia I We therefore conclude that This is again the equation for harmonic motion with an angular frequency given by and a period equal to Note that the simple pendulum is a special case of the physical pendulum: h = L and I = m L2. The period of the oscillation is then given by Note: The equations of motion that describe harmonic motion all have the following form: The general solution of this differential equation is This can be shown easily by differentiating x(t) twice with respect to time and The simple harmonic motion is a special case in which the amplitudes A and B are equal. In that case, x(t) can be rewritten as This equation describes a simple harmonic motion with an angular frequency equal to [omega]. Example: Problem 33P Two springs are attached to a block of mass m and to fixed supports as shown in Figure 15.4. Show that the frequency of oscillation on the frictionless surface is given by Figure 15.4. Problem 33P. When spring 1 is extended by x, spring 2 is compressed by the same distance. The total force acting on the mass is the sum of the forces exerted by these two springs. Note that both forces are always pointing in the same direction. This is similar to the equation of motion of a simple harmonic oscillator. This equation can be rewritten as or We conclude that the angular frequency is given by and the period T by Example: Problem 35P Two springs are joined and connected to a mass m as shown in Figure 15.5. The surfaces are frictionless. If the springs each have a force constant k, show that the frequency of oscillation of m is Figure 15.5. Problem 35P Assume that the spring constants are not the same. As the mass oscillates, spring 1 is stretched or compressed by a distance x1; the corresponding distance for the other spring is called x2. By Newton's third law, the forces exerted by the springs on each other are equal in magnitude but pointed in opposite directions. The force exerted by spring 1 on spring 2 is given by This equation implies that if spring 1 is stretched (x1 > 0) the force exerted by spring 1 on spring 2 is pointed in the negative direction. The force exerted by spring 2 on spring 1 is given by This equation implies that if spring 2 is stretched (x2 > 0) the force exerted by spring 2 on spring 1 is pointed in the positive direction. Applying Newton's third law we conclude that The displacement of the mass itself is given by and therefore F1 is the only force acting on the mass, and F1 is equal to k1 x1. The previous relation can now be used to express the force F1 in terms of the displacement x: We conclude that two springs, with spring constant k1 and k2 and joined in the way shown in Figure 15.5, act like a single spring with spring constant k, where k is given by 15.2. Damped Simple Harmonic Motion Up to now we have discussed systems in which the force is proportional to the displacement, but pointed in an opposite direction. In these cases, the motion of the system can be described by simple harmonic motion. However, if we include the friction force, the motion will not be simple harmonic anymore. The system will still oscillate, but its amplitude will slowly decrease over time. Suppose the total force acting on a mass is not only proportional to its displacement, but also to its velocity. The total force can be represented in the following way In this formula, b is called the damping constant. Substituting the expression for the force in terms of the acceleration we obtain the following differential equation The general solution of this differential equation will have the form Substituting this expression in the differential equation we obtain This equation can be rewritten as and the solutions for [omega] are Substituting this in the expression for x(t) we obtain We see that the amplitude of the motion gradually decreases over time. This is also true for the kinetic energy of the oscillator. At any point the mechanical energy of the oscillator can be calculated using the expression for x(t): Example: Problem 87P A damped harmonic oscillator involves a block (m = 2 kg), a spring (k = 10 N/m), and a damping force F = - b v. Initially it oscillates with an amplitude of 0.25 m; because of the damping, the amplitude falls to three-fourths of its initial value after four complete cycles. (a) What is the value of b ? (b). How much energy is lost during these four cycles ? The time dependence of the amplitude of the oscillation is given by The period of one oscillation is given by The amplitude after 4 oscillations is therefore given by The angular frequency [omega] is related to the spring constant k and mass m in the following manner Using this expression we obtain for b The mechanical energy lost during these 4 oscillation can also be easily calculated 15.3. Driven Harmonic Motion The case of a harmonic oscillator driven by a sinusoidal varying force is an extremely important one in many branches of physics. In the previous sections we have discussed several examples of harmonic oscillators, and for each system we have been able to calculate the natural frequency [omega]0, (for example, for the spring [omega]02 = k/m). The equation of motion for an oscillator on which no damping force is working, and no external force is applied is given by Suppose an external force F(t) is applied to this system. The external force has an amplitude m F0 and an angular frequency [omega]. The equation of motion describing the system is now given by The steady state (the state of the system after any transient effects have died down) response of the system will be precisely at the driving frequency. Otherwise the relative phase between force an response would change with time. Thus, the steady-state response of a harmonic oscillator is at the driving frequency [omega] and not at the natural frequency [omega]0. The general solution of the equation of motion is Substituting this expression into the equation of motion we obtain This equation can be rewritten by using some trigonometric relations This equation can only be satisfied if the coefficients of cos([omega]t) and sin([omega]t) are zero. This implies that and In general A != 0 and [omega] != [omega]0. The first condition than shows that The second condition can now be rewritten as The amplitude of the harmonic oscillator is given by The amplitude of the oscillation of the system gets very large if [omega] approaches [omega]0. The system is said to be in resonance when this happens. Send comments, questions and/or suggestions via email to wolfs@pas.rochester.edu and/or visit the home page of Frank Wolfs.
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https://www.youtube.com/watch?v=7kNiVrL6uV8
Finding volume of an irregularly-shaped object using water displacement AllThingsChemistry 623 subscribers Description 15021 views Posted: 10 Feb 2014 We use water displacement to determine the volume of an irregularly-shaped object. Transcript: when calculating or trying to find the volume of an irregularly shaped object such as a sphere like a marble a rock piece of aluminum you can use what's known as water displacement simply pour a set amount of water into a graduated cylinder and it doesn't matter what amount that you um aim for we're going to say 70 milliters but what when you read the graduated cylinder be sure to read I level from the meniscus so 70 mL of water is placed into a graduated cylinder next drop your object into the cylinder and see how far up the water goes and when we read this at eye level we notice that it goes up to 77 Mill so it actually went from 70 M to 77 so we put 77 as our final volume minus 70 millit and we get 7 milliliters as the volume of our irregularly shaped object again which was our marble
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https://pmc.ncbi.nlm.nih.gov/articles/PMC7178847/
Cervical Lymphadenopathy - PMC Skip to main content An official website of the United States government Here's how you know Here's how you know Official websites use .gov A .gov website belongs to an official government organization in the United States. Secure .gov websites use HTTPS A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites. Search Log in Dashboard Publications Account settings Log out Search… Search NCBI Primary site navigation Search Logged in as: Dashboard Publications Account settings Log in Search PMC Full-Text Archive Search in PMC Journal List User Guide View on publisher site Download PDF Add to Collections Cite Permalink PERMALINK Copy As a library, NLM provides access to scientific literature. Inclusion in an NLM database does not imply endorsement of, or agreement with, the contents by NLM or the National Institutes of Health. Learn more: PMC Disclaimer | PMC Copyright Notice Fundamentals of Pediatric Surgery . 2010 Jul 28:213–219. doi: 10.1007/978-1-4419-6643-8_28 Search in PMC Search in PubMed View in NLM Catalog Add to search Cervical Lymphadenopathy Rajeev Prasad Rajeev Prasad 2 Department of Pediatric General Surgery, Drexel University College of Medicine, St. Christopher’s Hospital for Children, Erie Avenue at Front Street, Philadelphia, PA 19134 USA Find articles by Rajeev Prasad 2,✉, L Grier Arthur L Grier Arthur Find articles by L Grier Arthur Editor: Peter Mattei 1 Author information Copyright and License information 1 The Children's Hospital of Philadelphia, Assistant Professor of Surgery,, University of Pennsylvania School of Med, Philadelphia, 19104-4345 Pennsylvania USA 2 Department of Pediatric General Surgery, Drexel University College of Medicine, St. Christopher’s Hospital for Children, Erie Avenue at Front Street, Philadelphia, PA 19134 USA ✉ Corresponding author. © Springer Science+Business Media, LLC 2011 This article is made available via the PMC Open Access Subset for unrestricted research re-use and secondary analysis in any form or by any means with acknowledgement of the original source. These permissions are granted for the duration of the World Health Organization (WHO) declaration of COVID-19 as a global pandemic. PMC Copyright notice PMCID: PMC7178847 Abstract Cervical lymphadenopathy is common in children. The condition frequently results in a child’s referral to a pediatric surgeon for further evaluation, and surgical intervention is often required. The majority of these masses represent benign disease, but the possibility of a malignancy exists. Parents often experience a significant amount of anxiety, and so it is important that pediatric surgeons are comfortable with the evaluation and management of these common lesions. Keywords: Kawasaki Disease, Tuberculin Skin Test, Chronic Granulomatous Disease, Peak Expiratory Flow Rate, Pediatric Surgeon Cervical lymphadenopathy is common in children. The condition frequently results in a child’s referral to a pediatric surgeon for further evaluation, and surgical intervention is often required. The majority of these masses represent benign disease, but the possibility of a malignancy exists. Parents often experience a significant amount of anxiety, and so it is important that pediatric surgeons are comfortable with the evaluation and management of these common lesions. In general, neck masses in children can be congenital, neoplastic or inflammatory. Not all of these lesions cause cervical lymphadenopathy. Congenital lesions, including thyroglossal duct cysts, branchial cleft cysts, dermoid cysts, hemangiomas and lymphangiomas, are a part of the differential diagnosis of a neck mass in a child. Neoplastic causes of neck masses include relatively uncommon primary tumors such as neuroblastoma and rhabdomyosarcoma. Neoplasia that results in cervical lymphadenopathy is much more common and includes lymphoma and metastatic disease (most commonly thyroid cancer). Inflammatory lesions are the most common etiology of cervical lymphadenopathy. Acute lymphadenitis, either viral or bacterial, is most often seen. Pediatric surgeons, however, will also encounter cases of subacute or chronic lymphadenitis, and the management of these can differ significantly. Some of the causes of these more indolent infections include atypical mycobacteria, tuberculosis, Bartonella henselae (cat scratch disease), and rarer fungal, parasitic or opportunistic infections. Finally, there are several miscellaneous conditions that can cause cervical lymphadenopathy in children as well. Acute Cervical Lymphadenitis Acute cervical lymphadenitis in children is most commonly associated with a viral respiratory tract infection. The lymph nodes generally undergo reactive hyperplasia due to the viral infection and are usually bilateral, multiple, and small. Erythema is uncommon, and suppuration rarely occurs. Viral agents frequently associated with a respiratory tract illness include adenovirus, coronavirus, influenza virus, parainfluenza virus, reovirus, respiratory syncytial virus, and rhinovirus. Other common viruses that cause cervical lymphadenitis include Ebstein-Barr virus and cytomegalovirus. Less common causes include measles, mumps, rubella, and varicella. When bilateral enlarged lymph nodes appear in conjunction with typical upper respiratory tract infection symptoms, further work-up is not immediately necessary. The lymph node enlargement generally subsides spontaneously within 2–3 weeks. Enlarged nodes that persist beyond this time or continue to enlarge will likely require further investigation. Acute bacterial lymphadenitis in children usually occurs due to an infection by Staphylococcus aureus or Streptococcus pyogenes. In infants less than 1 year of age, Group B Streptococcus, Haemophilus influenza type B, and anaerobes (Bacteroides, Peptococcus, Peptostreptococcus species) are possible causative agents. The lymph node groups that are affected, in decreasing order of frequency, include the submandibular, upper cervical, submental, occipital, and lower cervical nodes. The adenopathy is occasionally bilateral but more often unilateral. The involved node is usually solitary, large, and tender. Erythema and suppuration are common. Other associated findings include fever, pharyngitis, malaise, otitis, tonsillitis, dental caries, or periodontal disease. Initial treatment is with antibiotics unless there is obvious suppuration requiring incision and drainage. Antibiotic therapy should be directed at the most likely organism. In general, an antibiotic with broad spectrum coverage, particularly against beta-lactamase producing organisms, is instituted first. However, coverage against methicillin-resistant S. aureus might be necessary given the increasing prevalence of this organism in the community. In about 25% of cases, one finds that an initially firm and tender lymph node, initially associated with mild overlying erythema, will suppurate after the institution of antibiotics. Incision and drainage or aspiration may then be required. Subacute and Chronic Cervical Lymphadenitis Lymphadenitis that persists beyond approximately 2 weeks is considered to be subacute or chronic. These localized infections can be caused by a variety of organisms. Atypical mycobacteria, specifically Mycobacterium avium intracellulare and M. scrofulaceum, are the most common cause of subacute lymphadenitis. Other less common strains include M. kansasii, M. fortuitum, and M. hemophilum. Patients with atypical mycobacterial lymphadenitis commonly present with a rapid onset of unilateral lymph node enlargement near the angle of the mandible. Typically, the nodes are only mildly tender and gradually increase in size over the course of 2–3 weeks. Erythema, induration, and fluctuance are often present. The skin overlying the nodes often becomes dry and flaky, and can develop a pink or purple hue. Patients rarely have other symptomatology, and a tuberculin skin test is at most mildly reactive. Antibiotic therapy is generally ineffective, and excision of the involved nodes is indicated. Simple drainage can sometimes lead to formation of a chronic draining fistula or simply a recurrence and therefore should be avoided. Chronic lymphadenitis due to M. tuberculosis has a similar appearance to the other atypical infections. However patients may have constitutional symptoms. There is usually systemic disease as evidenced by an abnormal chest radiograph. A tuberculin skin test is positive, and there is a history of contact with an infected individual. Multi-agent antituberculous antibiotic therapy for 12–18 months is indicated. Cat scratch disease is a lymphocutaneous disorder in which regional lymphadenitis occurs after infection with the bacterium B. henselae. There is usually a skin lesion in the area of inoculation. Over the course of days to weeks after inoculation, regional adenopathy occurs. The neck is the second most commonly affected area after the axilla. Patients sometimes have mild constitutional symptoms. Typically, there is a single enlarged lymph node in the chain that drains the area of inoculation. The lymph node is usually tender and firm, and suppuration can occur. If cat scratch disease is suspected, the diagnosis can be confirmed by serologic testing. The infection usually is self-limited, but antibiotic therapy with a macrolide antibiotic is often helpful in facilitating resolution of the adenopathy. Once the diagnosis has been confirmed, surgical intervention is not necessary unless purulence develops, in which case incision and drainage may be necessary. Fungal infections is occasionally the cause of cervical lymphadenopathy in children. Histoplasmosis, blastomycosis, and coccidiomycosis are examples of these infections and are caused by Histoplasma capsulatum, Blastomyces dermatitidis and Coccidioides immitis, respectively. These organisms, which are endemic to certain regions of the country, usually cause a pulmonary infection that subsequently leads to involvement of cervical lymph nodes. Most cases are self-limited, but severe infections require systemic anti-fungal therapy. Toxoplasmosis is caused by the consumption of tainted meat or milk products. The intracellular protozoan Toxoplasma gondii is the causative organism. Lymphadenopathy, which can be only mildly symptomatic, is the presenting symptom in 10% of patients. The diagnosis is confirmed by serologic testing. Severe cases should be treated with 4–6 weeks of antibiotics. Opportunistic infections in immunocompromised children can also be a cause of chronic cervical lymphadenopathy. For instance, Nocardia species are ubiquitous pathogens found in the environment that only cause infections in immunosuppressed hosts. The infections are acquired through the skin or by way of the respiratory tract. Direct skin contact may result in a localized pustule, which can be cultured to establish a diagnosis. Nocardia infections can also cause significant adenopathy, in which case biopsy and culture of the node itself establishes the diagnosis. Sulfonamides are the treatment of choice. Actinomyces species are oral commensal organisms in humans. However, in hosts with compromised defense barriers, local invasion results in craniofacial actinomycosis and cervical nodal involvement. The diagnosis is difficult to make, but sulfur granules may be seen on histologic examination of an involved lymph node. Human immunodeficiency virus in children is usually acquired by vertical transmission from mother to child. Adenopathy is often a prominent manifestation and is sometimes the presenting sign. The diagnosis is made by serology, and the treatment is medical. Malignancy Cervical lymphadenopathy can also be the result of neoplasia in children, although statistically this is much less common than an inflammatory cause. By far the most common etiology of neoplastic lymphadenopathy in the neck is lymphoma. The cervical lymph node chains may be the prominent lymph node basin harboring the systemic disease, or they may be associated with a mediastinal mass. Lymphomas generally fall into two histologic subtypes: Hodgkin’s disease and non-Hodgkin’s lymphoma. Hodgkin’s disease (HD) accounts for approximately 40% of childhood lymphomas. In the pediatric age group, HD generally occurs in adolescents, and is rare in children less than 10 years of age. It is characterized histologically by the pathognomonic Reed–Sternberg cells. The four classic subtypes include nodular sclerosing, mixed cellularity, lymphocyte predominant, and lymphocyte depleted. HD arises in the lymph node itself. Patients will often have constitutional (“B”) symptoms such as fever, night sweats, or unintentional 10% or greater weight loss in the preceding 6 months. The nodes are generally nontender, firm, and rubbery. Solitary nodes are usually mobile, whereas aggregates of nodes may be bulky and fixed to the underlying tissue. Non-Hodgkin’s lymphoma (NHL) accounts for approximately 60% of childhood lymphomas. It most commonly occurs in children 7–11 years of age, and there is a 3:1 male to female predominance. NHL are divided into small-cell noncleaved (Burkitt’s and non-Burkitt’s), lymphoblastic, and large cell lymphomas (anaplastic and diffuse large B cell). Ten percent of patients with NHL have head and neck involvement. The neoplasm itself may or may not arise in nodal tissue. Often there is an aggressively enlarging mass causing local symptoms due to invasion of bone, nerves or soft tissue, and constitutional symptoms may be present. Cervical lymphadenopathy in children may also be caused by metastatic disease. Metastasis from a thyroid carcinoma sometimes present as unilateral lymph node enlargement. When this occurs, it is important not to disregard the mass as ectopic thyroid tissue, and a search for a thyroid mass should be undertaken. Patients with stage 4 neuroblastoma sometimes present with cervical lymphadenopathy, often bilateral. In these cases, the diagnosis is made on biopsy of the enlarged lymph node. Miscellaneous Causes of Cervical Lymphadenopathy There are numerous other causes of cervical lymphadenopathy in children that a pediatric surgeon should be familiar with. When the more common inflammatory and neoplastic causes have been ruled out, one must consider some of these esoteric conditions. In general, a biopsy of an involved node is required to make the diagnosis of one of these diseases. Uncommon infections can lead to lymph node enlargement. An infection due to Francisella tularensis causes tularemia (“rabbit fever”), a serious infectious disease that occurs in humans after contact with infected rodents. Yersinia pestis is the causative organism of the plague. The vector of infection is a flea, and bites in the head and neck region can cause regional adenopathy. Pasteurella multocida, an organism transmitted from animal bites, is another unusual cause of cervical adenopathy. Sarcoidosis is a chronic granulomatous disease that can affect children. Pulmonary involvement is common, but peripheral lymphadenopathy also readily occurs. The involved lymph nodes are usually bilateral, firm, and rubbery. Children can have cervical lymphadenopathy from sarcoidosis, and the diagnosis is made by biopsy of one of the affected nodes. The treatment is medical. Kawasaki disease, or mucocutaneous lymph node syndrome, is an acute vasculitis in which there is inflammation of small and medium-sized blood vessels throughout the body. The peak age is between 1 and 2 years and 80% of cases occur before the age of 4. The etiology of the condition is unknown. Inflammation may occur in cervical lymph nodes early in the course of the disease. The involved nodes are usually confined to the anterior triangle of the neck on one side. The nodes are sometimes large (>1.5 cm) and are tender and non-fluctuant. The disease and the adenopathy are self-limited. Kikuchi–Fujimoto disease, also known as histiocytic necrotizing lymphadenitis, is a rare condition of unknown etiology. Patients present with bilateral, painful, enlarged lymph nodes in the posterior triangle of the neck. Constitutional symptoms are present, and children can develop splenomegaly as well as a skin rash. The diagnosis is confirmed by excisional biopsy of an affected lymph node. The treatment is supportive as the disease rarely causes significant morbidity and is self-limited. Rosai–Dorfman disease, also known as sinus histiocytosis and massive lymphadenopathy, is a rare disease of unknown etiology that occurs in young children. Cervical lymphadenopathy commonly occurs as proliferating histiocytes accumulate in lymph nodes. The lymph nodes are initially mobile and discrete. However, as the condition progresses, there is massive enlargement of the cervical lymph nodes as well as other nodal regions. The disease may resolve spontaneously, however progression of the disease requires chemotherapy to control associated histiocytosis and plasmacytosis. Castleman’s disease, or giant lymph node hyperplasia, may cause a unicentric or multicentric adenopathy. The disease is caused by the hypersecretion of the cytokine IL-6. Excision of the involved node can be curative. Periodic fever, aphthous stomatitis, pharyngitis and cervical adenitis (PFAPA) syndrome is a disease of unknown etiology that occurs in young children. Patients have cyclic recurrences of the above symptoms every 3–5 weeks, and are healthy in between episodes. Corticosteroids have been used to alleviate symptoms during flare-ups but the episodes generally abate with time. Preoperative Preparation The most important aspect of the work-up for cervical lymphadenopathy is a properly obtained history and a thorough physical examination. The history should elicit whether the adenopathy has occurred acutely or has become a chronic condition. It should be determined whether there has been a recent upper respiratory infection or if there has been contact with an individual with typical URI symptoms. Any recent travel and any contact with animals, especially cats, should be noted. The presence of other symptoms in the patient is also important. For example, the acute onset of pain and swelling should raise the suspicion of acute lymphadenitis. Related constitutional symptoms such as fever, night sweats, and weight loss might indicate a disseminated process such as a lymphoma. Physical examination, especially serial exams by the same practitioner, is of particular importance. Pertinent findings to note on exam include the laterality, size, number, and mobility of the lymph nodes. In addition, the presence of tenderness, overlying skin changes, erythema, induration, or fluctuance should be determined. Laboratory studies are also an important aspect of the work-up for cervical lymphadenopathy. Basic studies like a complete blood count with differential and a peripheral smear should be obtained. Serologic studies can be obtained to confirm infections due to CMV, EBV, or HIV. Serologies can also confirm toxoplasmosis and cat scratch disease. If fluid can be obtained from an infected lymph node it should be sent for gram stain and culture (aerobic, anaerobic, fungal, acid-fast bacilli). Finally a PPD skin test should be applied. Imaging studies should include at least a chest radiograph to evaluate for mediastinal lymphadenopathy, as this has important implications if a biopsy under anesthesia is being considered. If further imaging of the neck or lymph node itself is necessary, an ultrasound should be obtained. The ultrasound can give information about the lymph nodes such as size, number, and whether there is normal or abnormal architecture. Additionally, the relationship of the node to adjacent structures can be determined. It is sometimes difficult to determine whether the palpable mass is in fact a lymph node as opposed to either the parotid or submandibular salivary glands. In these cases, ultrasound is useful. Occasionally, a CT or MRI can give further information regarding the relationship of a palpable node to adjacent structures, but most of the time these costly studies do not add much information. Fine needle aspiration is a procedure to consider in helping to make a diagnosis. It is most useful for obtaining fluid from an abscessed lymph node, and can be useful for making the diagnosis of a malignancy. An FNA should also be considered when a family is very reluctant for their child to have an anesthetic and undergo an operation. However, there is the distinct possibility that the FNA will be non-diagnostic. An open biopsy will then be required to obtain an adequate amount of tissue to make a diagnosis, for confirmation of a diagnosis that was suggested on FNA or for other studies that may be necessary such as flow cytometry. It is our preference, therefore, to forego an FNA and to proceed with a single operative intervention, whether it is a drainage procedure or a biopsy, when the decision to obtain tissue has been made. Another very important aspect of the evaluation and ­management of cervical lymphadenopathy is a proper discussion with the parents of the child, particularly if the child’s adenopathy has become chronic. The parents are usually aware of the possibility of an infection or a malignancy, and so their anxiety level is already high. A thorough discussion of significant history and physical findings and the possible diagnoses is essential. Also, a logical explanation of the rationale for either a period of observation or proceeding with immediate operative intervention is necessary. In general, it is our practice to observe small (subcentimeter), mobile, bilateral cervical lymph nodes that have been present for less than 2–3 weeks, particularly in the presence of recent URI symptoms or a documented exposure to cats. This allows time for a proper workup, including laboratory/serology studies, a chest radiograph, and PPD. When unilateral, large (>1 cm), firm, fixed or matted nodes are present, particularly in the posterior triangle of the neck or in the supraclavicular area, we favor biopsy. It is especially important to expedite a biopsy if constitutional symptoms such as fever, night sweats, and weight loss have been present. The workup, particularly the laboratory studies and chest radiograph, also should be done expeditiously. Once it has been decided to proceed with operative intervention, the procedure planned must be explained to the parents. Options include an FNA only, a simple incision and drainage procedure, an incisional biopsy for large, fixed lesions, or an excisional biopsy. Risks of the operation must be discussed. These include bleeding, infection, injury to adjacent structures (such as the facial nerve for lymph nodes near the angle of the jaw), and the possible need for further treatment, including another procedure for recurrent infection, to obtain more tissue, or excision of a persistent fistulous tract in the case of an atypical mycobacterial infection. Further medical management, such as antibiotics for an infection or chemotherapy for a malignancy, may be necessary and should be thoroughly discussed. Surgical Technique There are several operative options available, and the appropriate technique depends on the clinical scenario. For a suspected abscessed lymph node, a simple incision and drainage procedure, either with a local or general anesthetic, is all that is required. Fluid is sent for gram stain and culture. Gentle curettage followed by irrigation and packing of the cavity to prevent premature skin closure is useful. When the abscess has been caused by typical bacteria, the cavity fills and the wound generally heals without the need for any further surgical therapy. However, in cases where the abscess has been caused by an atypical mycobacterial infection, the wound may persist and subsequently mature into a chronic draining fistula. In this case, antibiotics are generally ineffective and it is therefore necessary to reoperate for excision of the entire fistulous tract and any residual nodal tissue. When biopsy is required, one must consider what is easiest and safest for making a diagnosis. For large or fixed lesions, an incisional biopsy may be the best choice. A nerve stimulator is useful especially when the node is located near important nerves such as the facial nerve. For smaller, easily accessible lesions that are not fixed to adjacent structures, an excisional biopsy is safe. In cases where a lymphoma is diagnosed, further excision is unnecessary, and systemic chemotherapy is instituted. This is also the case when cat-scratch disease is diagnosed, as the disease is usually self-limited. A very important situation to consider is when a cervical lymph node biopsy is required in a patient with a large anterior mediastinal mass. This scenario is most often encountered in cases of lymphoblastic lymphoma and is one reason that a chest radiograph is an important part of the preoperative workup. In such cases, there is a significant possibility of life-threatening airway collapse upon induction of anesthesia. Once this occurs, there is little that can be done to re-establish the airway as the collapse is distal to the tip of a typical endotracheal tube. Therefore, in this situation, a CT scan is recommended preoperatively to evaluate the degree of either tracheal or bronchial compression. Some also suggest obtaining pulmonary function testing to evaluate peak expiratory flow rate (PEFR). It has been suggested that a decrease in either the tracheal cross-sectional area by one half or the PEFR by 50% of predicted for age places a patient at high risk for respiratory compromise during general anesthesia. It is the responsibility of the surgeon and anesthesiologist to recognize this danger when a cervical lymph node biopsy is requested and to plan to perform the biopsy as an awake procedure under local anesthesia. Alternatively, if a pleural effusion is present, one can forego the lymph node biopsy and perform thoracentesis, whereupon analysis of the fluid will make the diagnosis. Postoperative Care The post-operative management is relatively straight forward. For incision and drainage procedures, the packing should be removed in 24–48 h. Local wound care with a topical antibiotic and gauze dressing along with frequent washing is all that is necessary. Otherwise, wounds that have been primarily closed generally heal without incident. Culture and biopsy results are shared with the parents and further therapy, if necessary, can be planned. Cervical lymphadenopathy is common in children, and pediatric surgeons must be familiar with the evaluation and management of this condition. Lymphadenitis is the most common cause of lymph node enlargement in children. However, neoplasia and other uncommon disorders should also be considered. The patient’s history, physical findings, laboratory tests and imaging studies are all important in helping to make the diagnosis and to formulate a plan of care. If acute viral cervical lymphadenitis is suspected, the enlarged lymph nodes should be closely observed for 2–3 weeks. If a severe bacterial infection, neoplasm or other unusual condition is suspected, or if the adenopathy has become chronic, then surgical intervention must be considered. The operative technique chosen is based on the characteristics of the lymph node enlargement, and one should avoid a general anesthetic when a large anterior mediastinal mass is associated with the adenopathy. Finally, it is essential to have a thorough discussion with the parents of the child regarding the rationale for the treatment plan that has been instituted. Summary Points In general, neck masses in children can be congenital, inflammatory or neoplastic. Specifically, cervical lymphadenopathy is most commonly due to inflammation or infection and can be acute, subacute or chronic. Cervical lymphadenopathy due to neoplasia is most likely a lymphoma. Unusual causes of cervical lymphadenopathy should be considered once the more common inflammatory and neoplastic causes have been ruled out. Editor’s Comment Few clinical issues create more anxiety for parents than an enlarged cervical lymph node. They need to know that it is not cancer and they need to know today. The experienced pediatric surgeon usually has a good feel for whether an enlarged lymph node is something to be concerned about or can be safely observed and the parents reassured. Unfortunately the only option for sampling a lymph node in a child is a surgical procedure under general anesthesia, which is generally safe and usually straightforward, but entails a certain amount of risk and an obligatory scar. This means that the surgeon should have a high index of suspicion before recommending a biopsy. Fortunately, a period of observation is almost always safe, even in the case of a malignant process, so, when in doubt, a brief delay can help one to make the best recommendation. In children, FNA is simply not a good option for the evaluation of cervical masses: pediatric pathologists have little experience with the technique, most children will not let you come near them with a needle, and, most importantly, the most common neoplastic processes seen in children (lymphoma and leukemia) cannot be reliably differentiated from an inflammatory process by FNA. Likewise, a neoplastic process cannot be excluded on the basis of blood tests, serologies, or medical imaging. What we are left with then is the history, the physical examination, and the growth pattern of the node. A lymph node that is larger than 1.5 cm and continues to grow over time, especially if it is located in an unusual location (supraclavicular), should be excised. Likewise, the patient with constitutional symptoms (the presence or absence of which should be specifically documented at the initial visit) should undergo biopsy. A typical busy pediatric surgeon will see at least one or two children with an enlarged lymph node every week. Most can be simply observed with no further studies, but nearly all should be encouraged to return for at least one follow-up visit in 2–3 weeks. At the other extreme is the rare patient with systemic symptoms and a worrisome node that clearly needs to be excised for biopsy. These patients should be scheduled for surgery and at minimum have a CBC with differential and a chest X-ray to rule out a mediastinal mass. The remainder will have clearly pathologic lymphadenopathy but no clear indication that a neoplastic process is necessarily involved. These patients should be scheduled for follow up in no more than 2–3 weeks and should undergo a work up: CBC w/diff.; serologies for cat scratch, toxoplasmosis, and mononucleosis, depending on what is endemic in the area; and a chest X-ray. If there are risk factors, a PPD might be prudent. In some cases in which a bacterial lymphadenitis is suspected, an empiric 7-day trial of antibiotics is reasonable, albeit controversial. A node involved with tumor almost never get smaller without treatment, so a node that shrinks can probably be observed. However, lymphoma can regress rapidly when the patient is given corticosteroids (for example for a coincidental asthma flare), in which case a biopsy becomes imperative. Cervical lymph node biopsy is a delicate procedure not to be taken lightly. There is always the risk of nerve injury and attention should be paid to scar placement for cosmesis and comfort. A small transverse incision placed in a skin crease is preferred. Once the platysma has been breached, the remainder of the dissection should be by careful blunt dissection only. A curved hemostat should be used to gently push adjacent tissues away from the capsule of the lymph node and nothing should be cut or cauterized. With proper technique, the node will gently rise up to meet the incision and the vessels at the hilum can be ligated or cauterized with precision right at the capsule. The goal should be complete excision of the node, but this can be done in piece-meal fashion. Lymph nodes that surprise the surgeon by being excessively vascular can be assumed to represent metastatic thyroid carcinoma (or another even less common neoplasm). The incision should be closed only at the level of the platysma and the skin as deeper sutures are not necessary and increase the risk of nerve injury. Finally, the child with an enlarged lymph node that is highly suspicious for malignancy should be evaluated by a pediatric oncologist before surgery so that a proper work up can be initiated, including a bone marrow biopsy to be performed while the patient is under general anesthesia. Differential Diagnosis Acute cervical lymphadenitis Viral Bacterial Subacute and chronic cervical lymphadenitis Atypical mycobacterial Typical or tuberculous mycobacterial Cat scratch disease Fungal Parasitic Opportunistic Neoplasia as a cause of cervical lymphadenopathy Hodgkin’s disease Non-Hodgkin’s lymphoma Metastatic disease Uncommon causes of cervical lymphadenopathy Unusual infections Sarcoidosis Kawasaki’s disease Kikuchi–Fujimoto disease Castleman’s disease Rosai–Dorfman disease PFAPA syndrome Diagnostic Studies Complete blood count with differential Peripheral blood smear Serology testing PPD Chest radiograph Ultrasound CT, MRI (rarely necessary) Consider FNA Parental Preparation Frank discussion regarding possible etiologies. Discussion of treatment options: Period of observation Operative intervention Discussion of possible complications of surgical intervention: Bleeding Wound infection Nerve injury Recurrence Discussion of the possible need for further therapy: Antibiotics for infection Re-excision for recurrence or fistula Chemotherapy for lymphoma Preoperative Preparation Review results of laboratory and imaging studies Informed consent Technical Points Consider using a nerve stimulator. Perform incision and drainage for an abscess. A large mass should be sampled by incisional biopsy. Perform excisional biopsy for smaller lymph nodes. Perform complete excision of nodes for suspected atypical mycobacterial infection. If a lymph node biopsy is being performed in a patient with a large mediastinal mass, strongly consider performing the biopsy awake under local anesthesia to avoid life-threatening airway compromise. Suggested Reading Dickson PV, Davidoff AM. Malignant neoplasms of the head and neck. Semin Pediatr Surg. 2006;15:92–8. doi: 10.1053/j.sempedsurg.2006.02.006. [DOI] [PubMed] [Google Scholar] Gosche JR, Vick L. Acute, subacute, and chronic cervical lymphadenitis in children. Semin Pediatr Surg. 2006;15:99–106. doi: 10.1053/j.sempedsurg.2006.02.007. [DOI] [PMC free article] [PubMed] [Google Scholar] Moss RL, Skarsgard ED, Kosloske AM, Smith BM. Case studies in pediatric surgery. Philadelphia, PA: McGraw Hill; 2000: pp. 258–65. Shamberger RC, Holzman RS, Griscom NT, Tarbell NJ, Weinstein HJ, Wohl ME. Prospective evaluation by computed tomography and pulmonary function tests of children with mediastinal masses. Surgery. 1995;118:468–71. doi: 10.1016/S0039-6060(05)80360-5. [DOI] [PubMed] [Google Scholar] Tracy TF, Muratore CS. Management of common head and neck masses. Semin Pediatr Surg. 2007;16:3–13. doi: 10.1053/j.sempedsurg.2006.10.002. [DOI] [PubMed] [Google Scholar] Articles from Fundamentals of Pediatric Surgery are provided here courtesy of Nature Publishing Group ACTIONS View on publisher site PDF (1.5 MB) Cite Collections Permalink PERMALINK Copy RESOURCES Similar articles Cited by other articles Links to NCBI Databases On this page Abstract Acute Cervical Lymphadenitis Subacute and Chronic Cervical Lymphadenitis Malignancy Miscellaneous Causes of Cervical Lymphadenopathy Preoperative Preparation Surgical Technique Postoperative Care Suggested Reading Cite Copy Download .nbib.nbib Format: Add to Collections Create a new collection Add to an existing collection Name your collection Choose a collection Unable to load your collection due to an error Please try again Add Cancel Follow NCBI NCBI on X (formerly known as Twitter)NCBI on FacebookNCBI on LinkedInNCBI on GitHubNCBI RSS feed Connect with NLM NLM on X (formerly known as Twitter)NLM on FacebookNLM on YouTube National Library of Medicine 8600 Rockville Pike Bethesda, MD 20894 Web Policies FOIA HHS Vulnerability Disclosure Help Accessibility Careers NLM NIH HHS USA.gov Back to Top
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https://www.noor-book.com/en/ebook-Fivefigure-logarithmic-and-other-tables-pdf
download book five figure logarithmic and other tables pdf - Noor Library العربية اكتشاف المزيد كتب رقمية مجانية المكتبة بيع مؤلفات ابن القيم والمكتبات رفوف كتب بيع كتب إلكترونية خطب الزعيم عبد الكريم قاسم شراء كتب جديدة Home Book Categories Authors of books Menu Search Sign upSign in My Account My Profile Upload Book My Library Settings Log out Five-figure Logarithmic And Other Tables Account upgrade The download is free, but we offer some paid services. Support us by subscribing Delete ads and speed up browsing the library. The download starts with the click of a button without waiting for the book to be ready. No limits for download times. You can upload unlimited books in the library. Enable readers to download your books without waiting. Delete ads on the books that you publish. No problems with download links for your uploaded books. Upgrade account now The book is being prepared Close Post a review on "Five-figure Logarithmic And Other Tables" Add Cancel Post a quote from "Five-figure Logarithmic And Other Tables" The Author: Frank Castle The quote is the literal transfer from the source and no more than ten lines Add Cancel Rate "Five-figure Logarithmic And Other Tables" Report the book Report Type Report Details Report Copy UrlShare on WhatsappShare on FacebookShare on TwitterShare on TelegramShare on LinkedinClose Search for a book SearchUpload Book Book CategoriesAuthors of booksBook QuotesBook ReviewsEducated communityUpload BookClose Download Book Five Figure Logarithmic And Other Tables Pdf Home Frank Castle Mathematics Literature Translated Five-figure Logarithmic And Other Tables This book is in public domain This book was published with a Creative Commons license with a mention the author and source Five-figure Logarithmic And Other Tables Author:Frank Castle Category:Mathematics Literature Translated[Edit] Language:English Publisher:London Mcmillan Release Date:01 Jan 1920 Pages:72 File Size:4.92 MB Extension:PDF Creation Date:18 Dec 2011 Rank:533,236 No 1 most popular Short link:Copy More books like this book Rate Review Quote Download Share Reviews ( 0 ) Quotes ( 0 ) Score ( 16 ) #### Frank Castle The Author Book Five-figure Logarithmic And Other Tables and the author of 3 another books. Castle, Frank, 1856- Quotes of Books Frank CastleReviews of Books Frank Castle اكتشاف المزيد كتب غير متوفرة قاعدة بيانات كتب كتب صوتية كتب عربية والمكتبات تأليف كتب والتعليم كتب حديثة خدمات نشر ذاتي كتب مجانية the five namesfive days in parislogarithmic timethe five birds and other storiesmathematics exponential and logarithmic functionsarrangement and tables of scientific textsasia and spring scheduleslogarithmic scalelist of integrals of logarithmic functionslogarithmic derivativelogarithmic matchinglogarithmic expressions of functionslimits of logarithmic functionslogarithmic potential functionlogarithmic probability functionthe science of tables and numberslogarithmic scales of measurementlogarithmic spirallogarithmic integration functionlogarithmic spiral beach اكتشاف المزيد اشتراكات مكتبات رقمية مكتبة أجهزة قراءة إلكترونية والمكتبات كتب نادرة برامج تعلم اللغة العربية في قفص تنظيم مكتبة شخصية كتب حديثة حقائب كتب Read 7 Download 9 Search Five Figure Logarithmic And Other Tables Five-place Logarithmic And Trigonometric Tables Logarithmic Tables Five-place Logarithmic And Trigonometric Tables Five-place Logarithmic And Trigonometric Tables Five-place Logarithmic And Trigonometric Tables Five Place Logarithmic And Trigonometric Tables Logarithmic And Trigonometric Tables Logarithmic And Other Mathematical Tables Logarithmic Tables Of Numbers And Trigonometrical Functions Five-place Logarithmic And Trigonometric Tables Five-place Logarithmic And Trigonometric Tables Four Figure Mathematical Tables: Comprising Logarithmic And Trigonometrical ... Logarithmic Tables Four Figure Mathematical Tables; Comprising Logarithmic And Trigonometrical Tables, And Tables Of Squares, Square Roots, And Reciprocals Five-place Logarithmic And Trigonometric Tables Five-place Logarithmic And Trigonometric Tables More with book covers Share the book : اكتشاف المزيد كتاب إلكتروني اشتراك مكتبة رقمية كتب للقراءة مجاناً شراء كتاب صحيح البخاري أثاث مكتبة العثور على الكتب الأكثر مبيعًا اشتراك مكتبة كتب عربية الكتب الإلكترونية شراء كتاب البداية والنهاية Microsoft Planner Book Review "Five-figure Logarithmic And Other Tables" Be the first one to Rate, Review and Quote from the book Rate Review Quote Book Quotes "Five-figure Logarithmic And Other Tables" Be the first one to Rate, Review and Quote from the book Rate Review Quote Other books like "Five-figure Logarithmic And Other Tables" ### Five Figure Logarithmic And Other TablesAlexander McAulay ### Five-place Logarithmic And Trigonometric TablesJames Morford Taylor ### Logarithmic TablesGeorge William Jones ### Five-place Logarithmic And Trigonometric TablesG. 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(Earle Raymond) Hedrick (1) ### Logarithmic And Other Mathematical TablesWilliam Joseph Hussey ### Logarithmic Tables Of Numbers And Trigonometrical FunctionsFreiherr Von Georg Vega ### Five-place Logarithmic And Trigonometric TablesJames Morford Taylor ### Five-place Logarithmic And Trigonometric TablesJames Morford Taylor Other books for "Frank Castle" ### Elementary Practical Mathematics ### A Manual Of Machine Design ### Elementary Course In Practical Physics Hide Intellectual property is reserved to the author of the aforementioned book If there is a problem with the book, please report through one of the following links: Report the book or by Contact us E-books are complementary and supportive of paper books and never cancel it. With the click of a button, the e-book reaches anyone, anywhere in the world. E-books may weaken your eyesight due to the glare of the screen. Support the book publisher by purchasing his original paper book. 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Publish your book now for free Book Categories Novels And Literary StoriesIslamic ReligionHuman Development And Self-developmentHistoryIslamic FiqhLiteratureInterpretation Of The KoranPhilosophy And LogicIslamic FaithThe CultureBiographies, Translations And Life Of The Famous PeopleBiography Of The Prophet Muhammad, May God Bless Him And Grant Him PeacePsychologyTeaching English EnglishIslamic PhilosophyArabic LiteratureArabicSciences Of The Noble Qur’an And The Sunnah Of The ProphetPolitical Science And StrategyHistoryThe Holy Qur'anAcclamationEngineeringPoetry And PoetsEducationMore of Book Categories اكتشاف المزيد دورة تعلم اللغة الإنجليزية المكتبة والتعليم كتب رقمية مجانية كتاب إلكتروني التربيه بيع كتب محمد بن عبد الوهاب والمكتبات شراء كتب السيرة النبوية مكتبة Authors of books Mustafa MahmoudThe Words Of God AlmightyIbrahim AlFiqiMuhammad Bin Saleh AlUthaymeenAhmed Khaled TawfiqIbn Qayyim Al-JawziyyaFahad Amer AlAhmadiMuhammad Metwally AlShaarawiAli Bin Jaber AlFifiDale CarnegieIslam GamalSayed QutbMuhammad AlGhazaliAgatha ChristieOmar AlHouraniAbbas AlAkkadMuhammad Bin Ismail AlBukhariNajib MahfouzAbu Hamid AlGhazaliJalal AlDin AlSuyutiFyodor DostoevskyIbrahim Al-SukranIsmail Bin Omar Bin Katheer AlQurashi AldamashqiuJihad AlTurbaniBin TaymiyyahMore of Authors of books اكتشاف المزيد إحياء علوم الدين ` وبذيله كتاب المغني عن حمل الأسفار في الأسفار في تخريج ما في الإحياء من الأخبار نظرية الفستق كتاب إلكتروني مكتبة التربيه اشتراك مكتبة كتب عربية شراء كتاب البداية والنهاية بيع كتب دينية قارئ كتب إلكترونية شراء كتب لسان العرب Loading... 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https://www.reddit.com/r/calculus/comments/1kocfaf/trig_sub_for_integrating_sqrta2_x2/
trig sub for integrating sqrt(a^2 - x^2)... : r/calculus Skip to main contenttrig sub for integrating sqrt(a^2 - x^2)... : r/calculus Open menu Open navigationGo to Reddit Home r/calculus A chip A close button Log InLog in to Reddit Expand user menu Open settings menu Go to calculus r/calculus r/calculus Welcome to r/calculus - a space for learning calculus and related disciplines. Remember to read the rules before posting and flair your posts appropriately. 168K Members Online •5 mo. ago gorram1mhumped trig sub for integrating sqrt(a^2 - x^2)... Integral Calculus text is using x=asintheta ... sqrt(a^2 - x^2 = a|cos theta| for a triangle formed with an ellipse/circle with radius a i guess x could arbitrarily be the opposite or the adjacent, but so often x is gonna be the adjacent, i was surprised that they didn't use the sub x=acostheta, which would lead to equaling a|sin theta|. the implication of integrating the sqrt of a^2 - x^2 is that its isolating y in the function a^2 = y^2+x^2. so subbing x=asintheta implies y is the adjacent? just seems odd to me. Read more Share Related Answers Section Related Answers Applications of calculus in real life Visualizing multivariable calculus concepts Best methods to learn integration techniques Common mistakes in differential equations Calculus problems involving optimization New to Reddit? Create your account and connect with a world of communities. Continue with Email Continue With Phone Number By continuing, you agree to ourUser Agreementand acknowledge that you understand thePrivacy Policy. Public Anyone can view, post, and comment to this community 0 0 Reddit RulesPrivacy PolicyUser AgreementAccessibilityReddit, Inc. © 2025. All rights reserved. Expand Navigation Collapse Navigation
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https://www.tutor2u.net/psychology/reference/episodic-procedural-and-semantic-memory?srsltid=AfmBOookHP3sRm6XxDlRCcQ21bTX50AVruQs3TO28PQEYWuDx-q7Poim
Join us at the cinema! Our exam workshops are back in Leeds, Manchester, Birmingham and London this November Learn more → Students Teachers Got a code for an online course? Redeem your code Students Teachers Got a code for an online course? Redeem your code Psychology Reference Library Study notes, videos, interactive activities and more! Blog Psychology news, insights and enrichment Collections Currated collections of free resources Topics Browse resources by topic Resource Selections Currated lists of resources Study Notes Episodic, Procedural and Semantic Memory Last updated 22 Mar 2021 Long-Term Memory (LTM) includes any memories that are held for durations upwards of 30 seconds. LTM can be split up into declarative memories (explicit memories that can be inspected and recalled consciously) and procedural memories (which are implicit in that we are typically unable to consciously recall them). Declarative memory can be sub-categorised further into episodic and semantic memories, as shown in the diagram below. Episodic memory Episodic memory refers to any events that can be reported from a person’s life. This covers information such as any times, places involved – for example, when you went to the zoo with a friend last week. It is a type of ‘declarative’ memory, i.e. it can be explicitly inspected and recalled consciously. Episodic memory can be split further into autobiographical episodic memory (memories of specific episodes of one’s life) and experimental episodic memory (where learning a fact [a semantic memory, below] has been associated with memory of the specific life episode when it was learned). Flashbulb memories are detailed autobiographical episodic memories that are stored permanently in LTM when they are first learned, often because they were of emotional or historical importance in that person’s life (e.g. a birth or a death). Semantic memory Like episodic memory, semantic memory is also a type of ‘declarative’ (explicit, consciously recalled) memory. However, the conscious recall here is of facts that have meaning, as opposed to the recall of past life events associated with episodic memory. For instance, recalling that you listen to music using your ears does not require knowing when or where you first learned this fact. Procedural memory Procedural memory describes our implicit knowledge of tasks that usually do not require conscious recall to perform them. One example would be riding a bike –you might struggle to consciously recall how to manage the task, but we can [unconsciously] perform it with relative ease. You might also like Capacity Study Notes Coding & Encoding Study Notes Multi-Store Model of Memory Study Notes Proactive and Retroactive Interference Study Notes Proactive Interference - Keppel and Underwood (1962) Study Notes IB Psychology (BLOA): Animal Research May Inform Our Understanding of Human Behaviour Study Notes Q&A from AQA: Multi-Store Model - Research for Coding, Capacity and Duration 24th February 2017 Memory: Types of Long-term Memory | AQA A-Level Psychology Quizzes & Activities Our subjects Explore Contact Boston House, 214 High Street, Boston Spa, West Yorkshire, LS23 6AD Tel: 01937 848885 © 2002-2025 Tutor2u Limited. Company Reg no: 04489574. VAT reg no 816865400.
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https://fiveable.me/key-terms/introduction-chemical-engineering/complete-combustion
Complete combustion - (Intro to Chemical Engineering) - Vocab, Definition, Explanations | Fiveable | Fiveable new!Printable guides for educators Printable guides for educators. Bring Fiveable to your classroom ap study content toolsprintablespricing my subjectsupgrade All Key Terms Intro to Chemical Engineering Complete combustion 🦫intro to chemical engineering review key term - Complete combustion Citation: MLA Definition Complete combustion is a chemical reaction where a hydrocarbon fuel reacts fully with oxygen to produce carbon dioxide and water. This process is highly efficient, releasing maximum energy from the fuel while minimizing the production of harmful byproducts like soot and carbon monoxide. It plays a critical role in energy generation, engine operation, and environmental management. 5 Must Know Facts For Your Next Test Complete combustion requires an optimal fuel-to-oxygen ratio to ensure that all hydrocarbons are converted to carbon dioxide and water. It is characterized by a blue flame, indicating efficient burning with less soot production. Engines designed for complete combustion can achieve higher thermal efficiencies, translating to better fuel economy. Complete combustion can significantly reduce air pollutants, helping to meet environmental regulations. The efficiency of complete combustion is crucial in power plants, where maximizing energy output while minimizing emissions is essential. Review Questions How does complete combustion differ from incomplete combustion in terms of products formed and efficiency? Complete combustion occurs when a fuel burns fully with enough oxygen, producing only carbon dioxide and water. In contrast, incomplete combustion happens when there isn't enough oxygen, resulting in byproducts like carbon monoxide and soot. Complete combustion is more efficient as it releases more energy per unit of fuel consumed and minimizes harmful emissions, while incomplete combustion leads to energy waste and increased pollution. Discuss the importance of achieving complete combustion in internal combustion engines and its impact on engine performance. Achieving complete combustion in internal combustion engines is vital because it maximizes the conversion of fuel into useful energy, leading to improved engine performance and efficiency. When complete combustion occurs, engines can produce more power while consuming less fuel, which enhances fuel economy. Additionally, complete combustion reduces harmful emissions such as carbon monoxide and unburned hydrocarbons, contributing to cleaner air and compliance with environmental regulations. Evaluate the role of stoichiometry in determining the conditions necessary for complete combustion and its implications for engineering applications. Stoichiometry plays a crucial role in determining the precise amounts of fuel and oxygen needed for complete combustion. By calculating the ideal ratios based on the chemical formula of the hydrocarbon fuel, engineers can design systems that optimize fuel usage and minimize emissions. This understanding impacts various engineering applications, from designing efficient engines to developing power plants that meet stringent environmental standards, ultimately ensuring energy production is both effective and sustainable. Related terms Incomplete combustion:A reaction where there is insufficient oxygen available, leading to the formation of carbon monoxide and other byproducts instead of just carbon dioxide and water. Stoichiometry: The calculation of reactants and products in chemical reactions, essential for understanding the ratios needed for complete combustion. Thermodynamics: The study of energy transformations, which includes analyzing the heat released during combustion processes. 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5227
https://flexbooks.ck12.org/cbook/ck-12-precalculus-concepts-2.0/section/13.5/primary/lesson/apr-and-apy-nominal-and-effective-rates-pcalc/
Skip to content Elementary Math Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Math 6 Math 7 Math 8 Algebra I Geometry Algebra II Conventional Math 6 Math 7 Math 8 Algebra I Geometry Algebra II Probability & Statistics Trigonometry Math Analysis Precalculus Calculus What's the difference? 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Learn. Interact. eXplore. CCSS Math Concepts and FlexBooks aligned to Common Core NGSS Concepts aligned to Next Generation Science Standards Certified Educator Stand out as an educator. Become CK-12 Certified. Webinars Live and archived sessions to learn about CK-12 Other Resources CK-12 Resources Concept Map Testimonials CK-12 Mission Meet the Team CK-12 Helpdesk FlexLets Know the essentials. Pick a Subject Donate Sign Up 13.5 APR and APY (Nominal and Effective Rates) Written by:CK-12 | Mark Spong Fact-checked by:The CK-12 Editorial Team Last Modified: Sep 01, 2025 Lesson In looking at an advertisement for a car you might see 2.5% APR financing on a $20,000 car. What does APR mean? What rate are they really charging you for the loan? Different banks may offer 8.1% annually, 8% compounded monthly or 7.9% compounded continuously. How much would you really be making if you put $100 in each bank? Which bank has the best deal? Nominal and Effective Rates of Interest Anominal interest rate is an interest rate in name only since a method of compounding needs to be associated with it in order to get a true effective interest rate. APR rates are nominal. APR stands for Annual Percentage Rate. The compounding periods are usually monthly, so typically @$\begin{align}k=12\end{align}@$. An annual effective interest rate is the true interest that is being charged or earned. APY rates are effective rates. APY stands for Annual Percentage Yield. It is a true rate that states exactly how much money will be earned as interest. Banks, car dealerships and all companies will often advertise the interest rate that is most appealing to consumers who don’t know the difference between APR and APY. In places like loans where the interest rate is working against you, they advertise a nominal rate that is lower than the effective rate. On the other hand, banks want to advertise the highest rates possible on their savings accounts so that people believe they are earning more interest. In order to calculate what you are truly being charged, or how much money an account is truly making, it is necessary to use what you have learned about compounding interest and continuous interest. Then, you can make an informed decision about what is best. Take a credit card that advertises 19.9% APR (annual rate compounded monthly). Say you left $1000 unpaid, how much would you owe in a year? First recognize that 19.9% APR is a nominal rate compounded monthly. @$\begin{align}FV =? \ PV=1000, \ i=.199, \ k=12, \ t=1\\end{align}@$ @$$\begin{align}FV = 1000 \left(1+\frac{0.199}{12}\right)^{12} \approx \$1,218.19\end{align}@$$ Notice that $1,218.19 is an increase of about 21.82% on the original $1,000. Many consumers expect to pay only $199 in interest because they misunderstood the term APR. The effective interest on this account is about 21.82%, which is more than advertised. Another interesting note is that just like there are rounding conventions in this math text (4 significant digits or dollars and cents), there are legal conventions for rounding interest rate decimals. Many companies include an additional 0.0049% because it rounds down for advertising purposes, but adds additional cost when it is time to pay up. For the purposes of this concept, ignore this addition. Examples Example 1 Earlier, you were asked about financing a car and the difference between APR and APY. A loan that offers 2.5% APR that compounds monthly is really charging lightly more than 2.5% of the initial loan per year. @$\begin{align}\left(1+\frac{0.025}{12}\right)^{12} \approx 1.025288\end{align}@$ They are really charging about 2.529%. The table below shows the APY calculations for three different banks offering 8.1% annually, 8% compounded monthly and 7.9% compounded continuously. | | | | --- | Bank A | Bank B | Bank C | | @$$\begin{align}FV &= PV(1+i)^t\ FV &= 100(1+0.081)\ FV &= \$ 108.1\end{align}@$$ @$\begin{align}APY=8.1 \ \%\end{align}@$ | @$$\begin{align}FV &= PV \left(1+\frac{i}{k}\right)^{kt}\ FV &= 100 \left(1+\frac{0.08}{12}\right)^{12}\ FV &\approx 108.299\end{align}@$$ @$\begin{align}APY \approx 8.299 \%\end{align}@$ | @$$\begin{align}FV &= PV \cdot e^{rt}\ FV &= 100e^{0.079}\ FV & \approx 108.22\end{align}@$$ @$\begin{align}APY \approx 8.22 \%\end{align}@$ | Even though Bank B does not seem to offer the best interest rate, or the most advantageous compounding strategy, it still offers the highest yield to the consumer. Example 2 Three banks offer three slightly different savings accounts. Calculate the Annual Percentage Yield for each bank and choose which bank would be best to invest in. Bank A offers 7.1% annual interest. Bank B offers 7.0% annual interest compounded monthly. Bank C offers 6.98% annual interest compounded continuously. Since no initial amount is given, choose a @$\begin{align}PV\end{align}@$ that is easy to work with like $1 or $100 and test just one year so @$\begin{align}t=1\end{align}@$. Once you have the future value for 1 year, you can look at the percentage increase from the present value to determine the APY. | | | | --- | Bank A | Bank B | Bank C | | @$$\begin{align}FV &= PV(1+i)^t\ FV &= 100(1+0.071)\ FV &= \$ 107.1\end{align}@$$ @$\begin{align}APY=7.1 \%\end{align}@$ | @$$\begin{align}FV &= PV \left(1+\frac{i}{k}\right)^{kt}\ FV &= 100 \left(1+\frac{0.07}{12}\right)^{12}\ FV &\approx 107.229\end{align}@$$ @$\begin{align}APY \approx 7.2290 \%\end{align}@$ | @$$\begin{align}FV &= PV \cdot e^{rt}\ FV &= 100e^{.0698}\ FV &\approx 107.2294\end{align}@$$ @$\begin{align}APY \approx 7.2294 \%\end{align}@$ | Bank A compounded only once per year so the APY was exactly the starting interest rate. However, for both Bank B and Bank C, the APY was higher than the original interest rates. While the APY’s are very close, Bank C offers a slightly more favorable interest rate to an investor. Example 3 The APY for two banks are the same. What nominal interest rate would a monthly compounding bank need to offer to match another bank offering 4% compounding continuously? Solve for APY for the bank where all information is given, the continuously compounding bank. @$\begin{align}FV=PV \cdot e^{rt}=100 \cdot e^{0.04} \approx 104.08\end{align}@$ The APY is about 4.08%. Now you will set up an equation where you use the 104.08 you just calculated, but with the other banks interest rate. @$$\begin{align}FV &= PV \left(1+\frac{i}{k}\right)^{kt}\ 104.08 &= 100 \left(1+\frac{i}{12}\right)^{12}\ i &= 12 \left[ \left( \frac{104.08}{100}\right)^{\frac{1}{12}}-1\right] \approx 0.0400667\end{align}@$$ The second bank will need to offer slightly more than 4% to match the first bank. Example 4 Which bank offers the best deal to someone wishing to deposit money? Bank A, offering 4.5% annually compounded Bank B, offering 4.4% compounded quarterly Bank C, offering 4.3% compounding continuously The following table shows the APY calculations for the three banks. | | | | --- | Bank A | Bank B | Bank C | | @$$\begin{align}FV &= PV(1+i)^t\ FV &= 100(1+0.045)\end{align}@$$ @$\begin{align}APY=4.5\%\end{align}@$ | @$$\begin{align}FV &= PV \left(1+\frac{i}{k}\right)^{kt}\ FV &= 100 \left(1+\frac{0.044}{4}\right)^4\end{align}@$$ @$\begin{align}APY \approx 4.473\%\end{align}@$ | @$$\begin{align}FV &= PV \cdot e^{rt}\ FV &= 100e^{0.043}\end{align}@$$ @$\begin{align}APY \approx 4.394\%\end{align}@$ | Bank B offers the best interest rate. Example 5 What is the effective rate of a credit card interest charge of 34.99% APR compounded monthly? @$\begin{align}\left(1+\frac{.3499}{12}\right)^{12} \approx 1.4118\end{align}@$ or a 41.18% effective interest rate. | | | Summary | | APR (Annual Percentage Rate) is a nominal interest rate that usually compounds monthly (k = 12) APY (Annual Percentage Yield) is an effective interest rate, which accurately states how much money will be earned as interest. Companies often advertise nominal rates (APR) for loans and effective rates (APY) for savings accounts to make their offers more appealing to consumers. To calculate the true interest rate, it is necessary to use compounding interest and continuous interest formulas. | Review For problems 1-4, find the APY for each of the following bank accounts. Bank A, offering 3.5% annually compounded. Bank B, offering 3.4% compounded quarterly. Bank C, offering 3.3% compounded monthly. Bank D, offering 3.3% compounding continuously. What is the effective rate of a credit card interest charge of 21.99% APR compounded monthly? What is the effective rate of a credit card interest charge of 16.89% APR compounded monthly? What is the effective rate of a credit card interest charge of 18.49% APR compounded monthly? The APY for two banks are the same. What nominal interest rate would a monthly compounding bank need to offer to match another bank offering 3% compounding continuously? The APY for two banks are the same. What nominal interest rate would a quarterly compounding bank need to offer to match another bank offering 1.5% compounding continuously? The APY for two banks are the same. What nominal interest rate would a daily compounding bank need to offer to match another bank offering 2% compounding monthly? Explain the difference between APR and APY. Give an example of a situation where the APY is higher than the APR. Explain why the APY is higher. Give an example of a situation where the APY is the same as the APR. Explain why the APY is the same. Give an example of a situation where you would be looking for the highest possible APY. Give an example of a situation where you would be looking for the lowest possible APY. Review (Answers) Click HERE to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option. Asked by Students Here are the top questions that students are asking Flexi for this concept: Student Sign Up Are you a teacher? Having issues? Click here By signing up, I confirm that I have read and agree to the Terms of use and Privacy Policy Already have an account? Adaptive Practice Save this section to your Library in order to add a Practice or Quiz to it. (Edit Title)41/ 100 This lesson has been added to your library. |Searching in: | | | Looks like this FlexBook 2.0 has changed since you visited it last time. 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5228
https://www.chemicalbook.com/ChemicalProductProperty_EN_CB41259584.htm
hypobromous acid | 14380-62-2 Search Products: ChemicalBook>>CAS DataBase List>>hypobromous acid hypobromous acid hypobromous acid structure CAS No.14380-62-2Chemical Name:hypobromous acid Synonyms Hypobromite;hypobromous acid CBNumber:CB41259584 Molecular Formula:BrHO Lewis structure Molecular Weight:0 MDL Number:MOL File:14380-62-2.mol Last updated:2024-08-28 13:53:23 Request For Quotation PropertiesUsesSuppliers 3 PropertiesUsesSuppliers 3 Search Products: hypobromous acid Properties | pka | pKa: 8.60(25°C) | | PH | pKa1= 8.59(25℃) | | EWG's Food Scores | 1 | | FDA UNII | GHT9BV419J | hypobromous acid Chemical Properties,Uses,Production Physical properties Hypobromous acid is a weak, unstable acid with the chemical formula HBrO, where the bromine atom is in the ＋1 oxidation state. It is also called “bromanol” or “hydroxidobromine”. It occurs only in solution and has chemical and physical properties that are very similar to those of hypochlorous acid, HClO. In aqueous solution, hypobromous acid only partially dissociates into the hypobromite anion BrO- and the cation, H+. The salts of hypobromous acid are called hypobromites. Like the acid, these salts are unstable and when evaporated or boiled to dryness, they undergo a disproportionation reaction, yielding the respective bromate and bromide salts. Uses Bactericide and wastewater disinfectant. Definition ChEBI: A monovalent inorganic anion obtained by deprotonation of hypobromous acid. hypobromous acid Preparation Products And Raw materials Raw materials Preparation Products hypobromous acid Suppliers Global( 3)Suppliers | Supplier | Tel | Email | Country | ProdList | Advantage | --- --- --- | | Hubei Jusheng Technology Co.,Ltd. | 18871490254 | linda@hubeijusheng.com | CHINA | 28172 | 58 | | DAYANG CHEM (HANGZHOU) CO.,LTD | +86-88938639 +86-17705817739 | info@dycnchem.com | China | 53900 | 58 | | Supplier | Advantage | --- | | Hubei Jusheng Technology Co.,Ltd. | 58 | | DAYANG CHEM (HANGZHOU) CO.,LTD | 58 | 14380-62-2(hypobromous acid)Related Search: Sodium bromite hypobromous acid Hypobromite 14380-62-2 HomePage | About Us | Contact us | Privacy | Terms All products displayed on this website are only for non-medical purposes such as industrial applications or scientific research, and cannot be used for clinical diagnosis or treatment of humans or animals. They are not medicinal or edible. According to relevant laws and regulations and the regulations of this website, units or individuals who purchase hazardous materials should obtain valid qualifications and qualification conditions. Copyright © 2023 ChemicalBook All rights reserved. This site uses cookies This website uses cookies and similar technologies to store and retrieve information about your use of this website. This information helps us to provide, analyse and improve our services, which may include personalised content or advertising. We may share this information with Google and other third parties. This cookies are necessary for our website to work properly . By clicking "Continue" or continuing to browse our site you are agreeing to our and our partners use of cookies. Accept Ask a Question× Question QuestionMaximum of 255 characters. Name E-mail [x] Anonymous submission Post Question Submitted successfully Please wait for a reply Published issue
5229
https://constructiondefect.com/construction-defect-and-real-estate-litigation-statutes-of-limitation-on-claims-for-broker-agent-non-disclosure-in-real-estate-transactions/
Statutes of Limitation on Claims for Broker-Agent Non-Disclosure in Real Estate Transactions | Norton & Associates Skip to content info@constructiondefect.com 310-706-4134 NORTON & ASSOCIATES Home About Practice Areas Real Estate Litigation Construction Defect Litigation Contractor Breach of Contract Architectural Negligence Engineering Negligence FAQs Statutes of Limitations (Construction Defects) Statutes of Limitations (Real Estate Contracts and Litigation) Clients Contact Us Home About Practice Areas Real Estate Litigation Construction Defect Litigation Contractor Breach of Contract Architectural Negligence Engineering Negligence FAQs Statutes of Limitations (Construction Defects) Statutes of Limitations (Real Estate Contracts and Litigation) Clients Contact Us info@constructiondefect.com 310-706-4134 NORTON & ASSOCIATES Home About Practice Areas Real Estate Litigation Construction Defect Litigation Contractor Breach of Contract Architectural Negligence Engineering Negligence FAQs Statutes of Limitations (Construction Defects) Statutes of Limitations (Real Estate Contracts and Litigation) Clients Contact Us Home About Practice Areas Real Estate Litigation Construction Defect Litigation Contractor Breach of Contract Architectural Negligence Engineering Negligence FAQs Statutes of Limitations (Construction Defects) Statutes of Limitations (Real Estate Contracts and Litigation) Clients Contact Us Statutes of Limitation on Claims for Broker-Agent Non-Disclosure in Real Estate Transactions admin-norton November 20, 2020 No Comments Much of the litigation arising from real property sales transactions involves some form or non-disclosure or even concealment of information related to the condition of the property by the brokers and agents in the sale. There are specific disclosures required of the broker-agents by law and violating these disclosure duties gives rise to a cause of action. However, different statutes of limitation apply to claims arising from broker-agent’s disclosures depending on the relationship between the non-disclosing broker and the buyer. Knowing and understanding the statutes of limitation that apply to broker non-disclosure in real estate sale transactions is vital to protecting and maintaining the buyer’s right to bring such claims. Click Here To Request a Free Case Evaluation Broker-Agent Duty of Disclosure: 2-Year Statute of Limitations Civil Code § 2079.4 A broker’s duty of disclosure in California real property sales transactions is set forth in Civil Code § 2079, which requires a broker or salesperson to conduct a reasonably competent and diligent visual inspection of the property offered for sale and to disclose to the prospective buyer all facts affecting the value or desirability of the property that an investigation would reveal. The statute of limitations for actions for breach of this duty is found in Civil Code § 2079.4, which is 2 years, starting from the date of possession of the property, the date of recordation, close of escrow or occupancy, whichever occurs first. By its clear terms, the delayed discovery rule does not apply to actions brought under § 2079.4. Broker-Agent as Fiduciary: Statutes of Limitation Certain variations to the broker-client relationship will shift the statute of limitations that apply based the nature of the relationship. For example, where the broker represents the purchaser in a real estate sale transaction, either exclusively or as a dual agent, that broker is in a fiduciary relationship with the purchaser client, thus owing the client a fiduciary duty. Field v. Century 21 Klowden-Forness Realty(1998) 68 Cal.App.4 th 18, 24-27. As indicated in Assilzadeh v. California Federal Bank(2000) 82 Cal.App.4 th 399, 414, a broker in a dual agency relationship, representing both the seller and buyer, also has a fiduciary duty to both buyer and seller. Thus, any broker representing a client in a real estate transaction also acts as a fiduciary to that client, requiring the highest good faith, undivided service and loyalty, including a duty to learn the material facts that may affect the client’s decision. The broker is hired for their professional knowledge and skill and is expected to perform the necessary research and investigation in order to know those important matters that will affect the client’s decision and has a duty to counsel and advise the client regarding the propriety and ramifications of the decision. The broker must place themselves in the position of the client and ask himself the type of information required for the client to make a well-informed decision, including the investigation of facts not known to the agent. These fiduciary duties are broader and more expansive than the simple visual inspection and disclosure requirements of Civil Code § 2079, and the statute of limitations for a breach of fiduciary by a broker-agent in a real estate transaction is not governed by Civil Code § 2079.4, but is governed by the statutes of limitation that apply to breach of a fiduciary duty, such as negligence or fraud, in the context of a real property transaction, and thus falls within the 3-year statute of limitations set forth in Code of Civil Procedure § 338, governing actions for injury to real property or fraud. Further, as noted above, the discovery rule does not apply to a claim under Civil Code § 2079.4, but the discovery rule would apply to a claim for breach of fiduciary duty. In actions involving fiduciary obligations, where the discovery rule would apply, the start of the statute of limitations is triggered on the date the plaintiff discovers, or should have discovered, the negligence or breach, rather than at the date of the transaction, as provided in § 2079.4, thus giving the buyer a reasonable period of time to “discover” the harm, thereby extending start time to the date of discovery. As noted above, the relationship between the broker and the buyer is a key factor in determining which statutes of limitation apply. For instance, where the seller’s broker makes a negligent misrepresentation to a prospective buyer, where that broker is representing the seller exclusively and not the buyer, in that case, a negligent misrepresentation, although technically a species of fraud, would not take the case out of the purview of the 2-year statute of limitations in Civil Code § 2079.4. Thus, in that circumstance, the 2-year statute of limitations would apply, and the discovery rule would not. So, the claim would start to run from the close of the transaction, and the statute of limitations would bar the claim after 2 years from that date. Broker-Agent as Fiduciary: Fraud & Constructive Fraud: Statutes of Limitation As discussed above, the breach of a real estate agent-broker’s fiduciary duty to his or her client may constitute negligence or fraud, depending on the circumstances of the case, with fraud typically involving intentional misrepresentations or concealment, and with negligence intent being absent. However, a real estate agent or broker, as agent or fiduciary may also be liable to the client/principal for constructive fraud even though his conduct was not actually fraudulent. This is “constructive fraud”, a unique species of fraud applicable only to a fiduciary or confidential relationship, comprising any act, omission or concealment involving a breach of legal, equitable duty, trust or confidence that results in damage to another, even though the conduct is not intentional or fraudulent. In fact, most acts by an agent in breach of their fiduciary duties is constructive fraud, even though there is not fraudulent intent. In terms of the statute of limitations, this would fall under the 3-year statute of limitations for fraud. Code of Civil Procedure § 338(d). Hire a construction defect lawyer. For a consultation call (310) 706-4134 or visit us on the web at constructiondefect.com to make an appointment. Norton & Associates. Los Angeles Construction Defect and Real Estate Litigation Attorney. Experience. Excellence. Results. Can't Talk Now? Click Here To Request a Callback Instead 3 Steps to new cases We will be glad to provide necessary legal assistance. 1 Schedule A Call Please submit your contact information in the form to schedule a call with Timothy to discuss your case. Timothy will then reach out to find a time that works best for you. 2 Book An Appointment After you’ve spoken with Timothy over the phone, it may be necessary to have an in-person appointment (in Los Angeles or Manhattan Beach) to discuss specifics or observe the job site. Timothy will schedule that appointment if necessary. 3 Work with Timothy Norton After thorough review, if we believe we can help you with your case, we will begin working together on finding and executing the solution. Use the form below to schedule a call. We are always ready to help you SUBMIT Related Posts Construction Defect and Real Estate Litigation: 3 Ways to Contest Arbitration January 15, 2021 No Comments Although arbitration is strongly favored by the courts and the policy of the state of California, there are certain circumstances that provide a basis to challenge and defeat such a petition. This article presents 3 ways a party to an arbitration provision can contest and possibly defeat a petition to compel arbitration in court. 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https://www.teacherspayteachers.com/browse?search=repeated%20addition%20and%20multiplication%20word%20problems%20add
Repeated Addition and Multiplication Word Problems Add Array Word Problems with Repeated Addition and Multiplication Times Tables Multiplication & Division Word Problems Arrays Repeated Addition Multiple Step Word Problems Addition Subtraction Multiplication and Division Repeated Addition Task Cards - Equal Groups and Arrays - Multiplication Practice Multiplication With Equal Groups, Arrays, and Repeated Addition - Word Problems Intro to Multiplication Facts Game (Repeated Addition & Arrays) Multiplication Using Arrays, Equal Groups, Number Lines, and Repeated Addition Multiplication Arrays Repeated Addition Skip Counting Worksheets 3rd Grade 2nd & 3rd Grade Multiplication Array Worksheets & Activities | Repeated Addition 3rd Grade Multiplication Facts Games: Arrays and Repeated Addition (3.OA.1) 1 and 2 Step Money Store Word Problems with Adding, Subtracting and Multiplying Third Grade Math for Halloween, Multiplication, Addition, Subtraction, Rounding Multiplication Puzzles Math Center Arrays, Repeated Addition, Word Problems Single & Multi-Step Word Problems- Add, Subtract, Multiply, Divide - SET #1 Beginning Multiplication Worksheets Arrays Repeated Addition Equal Groups Daily Math Word Problem Worksheet Bundle | Addition, Subtraction, Place Value Multiplication Facts Practice with Arrays,Repeated Addition,Basic Multiplication Repeated Addition Arrays Worksheets | Fall Word Problems | Multplication Equal Groups, Repeated Addition, and Arrays in Multiplication Bundle! Halloween Multiplication Number Line and Repeated Addition (3rd Grade) Multiplication Practice Worksheets: Arrays, Repeated Addition, Word Problems Digital Beginning Multiplication & Repeated Addition Activities/Printables Back to School : Multiplication Arrays Worksheets with Repeated Addition Arrays and Repeated Addition Worksheets | 2nd Grade Multiplication Foundations
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https://www.khanacademy.org/math/5th-grade-illustrative-mathematics/xe7a2395079b692f7:more-decimal-and-fraction-operations
Published Time: Thu, 11 Sep 2025 13:16:25 GMT More decimal and fraction operations | Khan Academy Skip to main content If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains .kastatic.org and .kasandbox.org are unblocked. Explore Browse By Standards Explore Khanmigo Math: Pre-K - 8th grade Math: Illustrative Math-aligned Math: Eureka Math-aligned Math: Get ready courses Math: high school & college Math: Multiple grades Test prep Science Economics Reading & language arts Computing Life skills Social studies Partner courses Khan for educators Select a category to view its courses Search AI for Teachers FreeDonateLog inSign up Search for courses, skills, and videos 5th grade math (Illustrative Math-aligned)7 units · 145 skillsUnit 1 Finding volumeUnit 2 Fractions as quotients and fraction multiplicationUnit 3 Multiplying and dividing fractionsUnit 4 Wrapping up multiplication and division with multi-digit numbersUnit 5 Place value patterns and decimal operationsUnit 6 More decimal and fraction operationsUnit 7 Shapes on the coordinate grid Course challenge Test your knowledge of the skills in this course.Start Course challenge Math 5th grade math (Illustrative Math-aligned) Unit 6: More decimal and fraction operations 2,700 possible mastery points Mastered Proficient Familiar Attempted Not started Quiz Unit test Powers of ten Multiply and divide by powers of 10 Convert units (metrics) Convert units word problems (metrics) Convert units multi-step word problems (metric) Convert units word problems (US customary) Convert units multi-step word problems (US customary) More decimal and fraction operations: Quiz 1 Visually add and subtract fractions Estimate to add and subtract fractions with different denominators Common denominators Equivalent expressions with common denominators More decimal and fraction operations: Quiz 2 Add fractions with unlike denominators Subtracting fractions with unlike denominators Add and subtract fractions More decimal and fraction operations: Quiz 3 Add and subtract fractions word problems Add and subtract mixed numbers with unlike denominators (no regrouping) Add and subtract mixed numbers with unlike denominators (regrouping) More decimal and fraction operations: Quiz 4 Graph data on line plots (through 1/8 of a unit) Interpret line plots with fraction addition and subtraction Interpret dot plots with fraction operations Fraction multiplication as scaling More decimal and fraction operations: Quiz 5 Compare products without multiplying (fractions) Multiply and divide whole numbers by 10, 100, and 1000 Multiply and divide decimals by 10 Multiply and divide decimals by 10, 100, and 1000 Convert units of time Convert units (US customary) More decimal and fraction operations: Quiz 6 More decimal and fraction operations: Unit test Our mission is to provide a free, world-class education to anyone, anywhere. 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5232
https://www.youtube.com/watch?v=XfYltSlcago
Quadratic Function App: Find Profit from Revenue and Cost (Vertex) Mathispower4u 330000 subscribers 186 likes Description 47489 views Posted: 6 Mar 2018 This video explains how to determine the profit function. Then the vertex is found and the meaning of the values is explained. 1 comments Transcript: the cost in dollars to produce x designer dog leashes is given by c of x and the revenue function in dollars is given by r of x we are first asked to determine the profit function p of x remember the profit function is equal to the revenue function minus the cost function because the revenue function gives us the money coming in the cost function gives us the money going out and the difference or what's left over is the profit and because we are given the cost and revenue functions we can determine the profit function the profit function big p of x equals r of x minus c of x this gives us the quantity negative two x squared plus 59x minus the quantity 7x plus 10. it is important that we have the cost function in parentheses so that we subtract the entire cost function now we clear the parentheses and combine like terms if it's helpful we can think of distributing a positive one here and because of the subtraction we can think of distributing a negative one distributing positive one doesn't change anything we have negative two x squared plus 59 x when distributing negative one we have negative one times seven x which equals negative seven x giving us minus seven x and then negative one times ten is equal to negative 10 giving us minus 10. or we can just think of subtracting both terms inside which gives us minus 7x minus 10. and now we combine like terms there are two x terms so the profit function big p of x equals negative two x squared and then 59x minus seven x is 52x giving us plus 52x minus 10. this is our profit function next we're asked to find the number of leashes needed to be sold to maximize the profit and also determine the maximum profit well we should recognize p of x is a quadratic function and therefore the graph is a parabola and because a the leading coefficient is negative two the parabola opens down let's take a look at the graph of the profit function again we have a parabola opening down along the horizontal axis we have the number of dog leashes along the vertical axis we have the profit in dollars so we should be able to recognize that if we can determine the ordered pair for the vertex this point here the highest point on the graph the first value of the ordered pair is going to give us x the number of dog leeches that must be sold to maximize the profit and the second value of the ordered pair is going to be the output or function value which will give us the maximum profit and because the profit function is in general form or the form ax squared plus bx plus c we can use this formula here to determine the ordered pair for the vertex so for the profit function p of x a is equal to negative two b is equal to 52 and c is equal to negative 10. the x coordinate of the vertex is equal to negative b divided by 2a this equation also gives us the equation of the axis of symmetry performing substitution b is equal to 52 giving us negative 52 divided by two times a and a is negative two giving us two times negative two so we have x equals negative 52 divided by negative four which is equal to positive 13. so for the vertex we know the first value of the ordered pair or the x coordinate is 13 which means 13 dog leashes must be sold in order to maximize the profit and now we need to evaluate the profit function at 13 to determine the maximum profit so p of 13 is equal to negative two times the square of 13 plus 52 times 13 minus 10. 13 squared is equal to hundred sixty-nine giving us negative two times one hundred sixty-nine plus fifty-two times thirteen minus ten and now we multiply giving us negative three hundred thirty-eight plus six hundred seventy-six minus 10 and now we add and subtract from left to right which gives us 328 so now we know the maximum profit is 328 dollars again the ordered pair for the vertex is 13 comma 328 the 13 indicates 13 dog leashes must be sold to maximize the profit and the 328 indicates the maximum profit is 328 going back to our first slide 13 dog leashes must be sold to maximize the profit the maximum profit is 328 dollars and then finally we're asked to find the price to charge per leash to maximize the profit let's go back to our notes for a moment little p of x equals the price demand function in dollars where little p of x represents the selling price and we know x represents the quantity sold which means the revenue function r of x can be expressed as little p of x times x or if we want x times little p of x and we need to find a little p of x in order to determine the selling price to maximize the profit so we now know that the revenue function r of x can be expressed as little p of x times x or x times little p of x let's say x times little p of x and the revenue function is equal to negative two x squared plus 59 x so if we factor out an x we can determine the price demand function little p of x we factor out x we're left with the quantity negative two x plus 59 where again this is x and therefore little p of x must be equal to negative two x plus 59. so now that we know that little p of x is equal to negative two x plus 59 we can determine the selling price to maximize the profit because remember x is equal to 13 when the profit is maximized so we need to find little p of 13 which is equal to negative two times 13 plus 59 which equals negative 26 plus 59 which is equal to 33 so now we know the price per leash must be 33 to maximize the profit so going back to our first slide one last time again the price per leash must be 13 to maximize the profit i hope you found this helpful
5233
https://www.scirp.org/reference/referencespapers?referenceid=2309181
Carslaw, H.S. and Jaeger, J.C. (1948) Conduction of Heat in Solids. Clarendon Press, Oxford. - References - Scientific Research Publishing Login Login切换导航 Home Articles Journals Books News About Services Submit Home References Article citations Journals A-Z Journals by Subject Biomedical & Life Sci. Business & Economics Chemistry & Materials Sci. Computer Sci. & Commun. Earth & Environmental Sci. Engineering Medicine & Healthcare Physics & Mathematics Social Sci. & Humanities Journals by Subject Biomedical & Life Sciences Business & Economics Chemistry & Materials Science Computer Science & Communications Earth & Environmental Sciences Engineering Medicine & Healthcare Physics & Mathematics Social Sciences & Humanities Publish with us Paper Submission Information for Authors Peer-Review Resources Open Special Issues Open Access Statement Frequently Asked Questions Publish with us Paper Submission Information for Authors Peer-Review Resources Open Special Issues Open Access Statement Frequently Asked Questions Follow SCIRP Contact us customer@scirp.org +86 18163351462(WhatsApp) 1655362766 Paper Publishing WeChat Article citationsMore>> Carslaw, H.S. and Jaeger, J.C. (1948) Conduction of Heat in Solids. Clarendon Press, Oxford. has been cited by the following article: TITLE: Numerical Treatment of Initial-Boundary Value Problems with Mixed Boundary Conditions AUTHORS: Nawal Abdullah Alzaid, Huda Omar Bakodah KEYWORDS: Decomposition Method, Modified Adomian Decomposition Method, Linear and Nonlinear Partial Differential Equations, Mixed Boundary Conditions, Initial-Boundary Value Problem JOURNAL NAME: American Journal of Computational Mathematics, Vol.8 No.2, June 28, 2018 ABSTRACT: In this paper, we extend the reliable modification of the Adomian Decom-position Method coupled to the Lesnic’s approach to solve boundary value problems and initial boundary value problems with mixed boundary conditions for linear and nonlinear partial differential equations. The method is applied to different forms of heat and wave equations as illustrative examples to exhibit the effectiveness of the method. The method provides the solution in a rapidly convergent series with components that can be computed iteratively. The numerical results for the illustrative examples obtained show remarkable agreement with the exact solutions. We also provide some graphical representations for clear-cut comparisons between the solutions using Maple software. Follow SCIRP Contact us customer@scirp.org +86 18163351462(WhatsApp) 1655362766 Paper Publishing WeChat Free SCIRP Newsletters Add your e-mail address to receive free newsletters from SCIRP. Home Journals A-Z Subject Books Sitemap Contact Us About SCIRP Publication Fees For Authors Peer-Review Issues Special Issues News Service Manuscript Tracking System Subscription Translation & Proofreading FAQ Volume & Issue Policies Open Access Publication Ethics Preservation Retraction Privacy Policy Copyright © 2006-2025 Scientific Research Publishing Inc. All Rights Reserved. Top
5234
https://math.stackexchange.com/questions/298616/what-is-inverse-of-ia
Skip to main content What is inverse of I+A? Ask Question Asked Modified 1 year, 3 months ago Viewed 47k times This question shows research effort; it is useful and clear 26 Save this question. Show activity on this post. Assume A is a square invertible matrix and we have A−1. If we know that I+A is also invertible, do we have a close form for (I+A)−1 in terms of A−1 and A? Does it make it any easier if we know that sum of all rows are equal? linear-algebra matrices inverse Share CC BY-SA 3.0 Follow this question to receive notifications edited May 16, 2024 at 4:02 Martin Sleziak 56.2k2020 gold badges210210 silver badges391391 bronze badges asked Feb 9, 2013 at 9:26 user54626user54626 78711 gold badge66 silver badges1818 bronze badges 6 3 By substituting A′=A−I, you are basically asking for the inverse of all matrices. – akkkk Commented Feb 9, 2013 at 9:29 1 And of course, it might well be that A is invertible, while I+A is not. – Andreas Caranti Commented Feb 9, 2013 at 9:33 equal row sum won't help either. Consider the 2x2 matrix obtained from I by interchanging its rows. – Ittay Weiss Commented Feb 9, 2013 at 9:37 What does "convertible" mean? – Chris Eagle Commented Feb 9, 2013 at 9:39 3 The identity (A−1+I)−1=A−A(I+A)−1A (consequence of Woodbury matrix identity) does not answer you question, but perhaps you find it useful. – leonbloy Commented Feb 9, 2013 at 14:12 | Show 1 more comment 3 Answers 3 Reset to default This answer is useful 28 Save this answer. Show activity on this post. I know I'am a bit late, but I wanted to add the following for future readers. If you know the Eigendecomposition of your matrix A=QΛQ−1 the result is the following: (A+I)−1=(QΛQ−1+I)−1=(Q(Λ+I)Q−1)−1=Q(Λ+I)−1Q−1 I know that this is a very trivial calculation and the result is maybe not helpful to many of you, but I wasted countless hours using the Shermann-Morrison-Theorem because I forgot about the Eigendecomposition. Cheers, Lukas Share CC BY-SA 3.0 Follow this answer to receive notifications answered May 20, 2016 at 14:48 Lukas KoestlerLukas Koestler 39111 gold badge33 silver badges55 bronze badges 1 3 (+1) This answer was exactly what I was looking for. I have an iterative method with Ai=aiA0. So now inverting the matrix at step i just involves inverting a diagonal matrix! – knrumsey Commented Sep 20, 2018 at 17:12 Add a comment | This answer is useful 14 Save this answer. Show activity on this post. If the series S=I−A+A2−A3+⋯ converges, expand S(I+A) to find that it is equal to the identity. See here for details on when S converges. S clearly converges if Ak=0 for some positive integer k (nilpotency). As for invertibility, if A is diagonalizable, i.e. A=PDP−1 for some diagonal matrix D=diag(e1,e2,…,en), then by Sylvester's Determinant Theorem, det(I+(PD)P−1)=det(P−1(PD)+I)=∏i=1n(ei+1) Hence, I+A is invertible if no eigenvalue ei has a value of −1. If A is nilpotent, then its only eigenvalue is 0, so I+A is invertible. Is there a closed-form solution? I believe not. Basically, a closed-form expression of (I+A)−1 using A and A−1 would amount to a closed-form expression of (1+x)−1 using x and x−1, where x is real (or complex). A semi-rigorous articulation of this argument follows: Proposition: There exists no family of matrices {Xij}m×n, where every Xij is either equal to A, A−1 or a constant dependent on the dimension of A, such that (I+A)−1=∑mi=1(∏nj=1Xij) for all values of A. Proof: Assume there exists such a family. Let A be the 1×1 matrix x. Note that ∑mi=1(∏nj=1Xij) is a polynomial P(x), which apparently equals (1+x)−1 for all values of x. Hence, P(x) must be the taylor series 1−x+x2−x3+⋯, which contradicts the fact that P(x) has a finite number of terms. Share CC BY-SA 3.0 Follow this answer to receive notifications edited Feb 9, 2013 at 14:45 answered Feb 9, 2013 at 9:37 Herng YiHerng Yi 3,2362323 silver badges3535 bronze badges 3 1 Herng, thanks for your comment. if we know that (I+A) is invertible and also we have A−1. Then can we write a close form solution for (I+A)−1? – user54626 Commented Feb 9, 2013 at 10:10 2 I doubt so. How can you express 11+x using only 1x and x..? – Berci Commented Feb 9, 2013 at 14:18 haha I had the same idea as @Berci and edited it into the answer. – Herng Yi Commented Feb 9, 2013 at 14:30 Add a comment | This answer is useful 1 Save this answer. Show activity on this post. Check this question. The first answer presents a recursive formula to retrieve the inverse of a generic sum of matrices. So yours should be a special case. Share CC BY-SA 3.0 Follow this answer to receive notifications edited Apr 13, 2017 at 12:20 CommunityBot 1 answered Feb 9, 2013 at 14:31 FerpectFerpect 18899 bronze badges 3 In particular to answer the question on how to express 11+x using only 1,x and 1/x we have 11+x=1x−11+trace(1/x)1x2 – Ferpect Commented Feb 9, 2013 at 23:58 Thanks for your answer. I have checked that question, but the first answer suggest recursive formula when rank of one of the matrices is 1. But rank of I is not 1. – user54626 Commented Feb 10, 2013 at 10:27 And after that Lemma, the autor states the theorem for rank>1 – Ferpect Commented Feb 10, 2013 at 15:08 Add a comment | You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions linear-algebra matrices inverse See similar questions with these tags. Featured on Meta Community help needed to clean up goo.gl links (by August 25) Linked 241 Inverse of the sum of matrices 1 What is inverse of I+A given that A2=2I? 3 If A2=O, then prove that I+A is invertible and find (I+A)−1 0 power of square matrices 0 Is there close form for the eigendecomposition of a the kronecker product of a matrix with itself A⊗A 0 Trace of integral of a function of two matrices Related 0 Inverible matrix and canonical form relation 8 Inverse of orthogonal projection 2 About inverse matrixes 3 Under what circumstances can we take this inverse matrix? 0 Why the addition of the noise will almost certainly make the square matrix invertible? 2 Prove a matrix AQ−1AT is invertible 1 Inverse of a sum of special matrices Hot Network Questions Why does Wittgenstein use long, step-by-step chains of reasoning in his works? 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https://vanleeuwenlab.com/inc/pdf/van_Leeuwen_2015_CSHP_protocol.pdf
Protocol Rapid and Efficient Plasmid Construction by Homologous Recombination in Yeast Jolanda van Leeuwen,1,3 Brenda Andrews,1,2 Charles Boone,1,2 and Guihong Tan1,3 1Donnelly Centre for Cellular and Biomolecular Research, University of Toronto, Toronto, Ontario M5S 3E1, Canada; 2Department of Molecular Genetics, University of Toronto, Toronto, Ontario M5S 3E1, Canada The cloning of DNA fragments is a fundamental aspect of molecular biology. Traditional DNA cloning techniques rely on the ligation of an insert and a linearized plasmid that have been digested with restriction enzymes and the subsequent introduction of the ligated DNA into Escherichia coli for propagation. However, this method is limited by the availability of restriction sites, which often becomes problematic when cloning multiple or large DNA fragments. Furthermore, using traditional methods to clone multiple DNA fragments requires experience and multiple laborious steps. In this protocol, we describe a simple and efficient cloning method that relies on homologous recombination in the yeast Saccharomyces cerevisiae to assemble multiple DNA fragments, with 30-bp homology regions between the fragments, into one sophisticated construct. This method can easily be extended to clone plasmids for other organisms, such as bacteria, plants, and mammalian cells. MATERIALS It is essential that you consult the appropriate Material Safety Data Sheets and your institution’s Environmental Health and Safety Office for proper handling of equipment and hazardous material used in this protocol. RECIPES: Please see the end of this protocol for recipes indicated by . Additional recipes can be found online at Reagents Agarose gel (1.2%) and electrophoresis reagents Bacterial strains Any standard Escherichia coli strain such as Top10 will suffice. Deionized water (sterilized) Dimethyl sulfoxide (DMSO) dNTPs (10 mM) Ethanol (70%) Glass beads (0.4–0.6 mm) Glycerol (10%) Isopropanol Lithium acetate (LiAc) 3Correspondence: guihong.tan@utoronto.ca; jolanda.vanleeuwen@utoronto.ca © 2015 Cold Spring Harbor Laboratory Press Cite this protocol as Cold Spring Harb Protoc; doi:10.1101/pdb.prot085100 853 Cold Spring Harbor Laboratory Press on October 6, 2015 - Published by Downloaded from To prepare a 1 M solution, dissolve 10.2 g of LiAc in 100 mL of deionized water, filter-sterilize, and store at room temperature. Luria–Bertani (LB) medium plus ampicillin Lysis buffer for yeast Milli-Q water Oligonucleotides (desalted) For a list of all the oligonucleotides used in this protocol, see Table 1. Phenol (saturated) Phusion high-fidelity DNA polymerase and buffer (5×) For example, consider using Phusion Hot Start II DNA Polymerase (Thermo Scientific); kit includes bespoke buffers. Plasmid miniprep kit Polyethylene glycol (PEG) 3350 Prepare 100 mL of a 50% solution by dissolving 50 g of PEG3350 in 50 mL deionized water, adjusting the volume to 100 mL with deionized water, followed by filter sterilization. Single-stranded DNA (ssDNA; e.g., Sigma-Aldrich D8899) SmaI restriction endonuclease Synthetic amino-acid-dropout medium (SD-all) Synthetic lysine-dropout medium (SD-lys) Terrific broth (TB) medium Tris–EDTA (TE) buffer (10×) Add 0.2 mL of EDTA (0.5 M, pH 8.0) and 1 mL of Tris–Cl (1 M, pH 8.0) to 99 mL of deionized water. Filter-sterilize. Store at room temperature. Yeast extract-peptone-dextrose (YEPD) Yeast strains This protocol uses BY4709 MATα ura3Δ0 (ATCC200872), BY4712 MATα leu2Δ0 (ATCC200875), and BY4742 MATα his3Δ1 leu20 lys2Δ0 met15Δ0 ura3Δ0 (ATCC201389), but many other standard laboratory yeast strains can also be used. Equipment Agarose gel electrophoresis materials Benchtop centrifuge/microcentrifuge TABLE 1. Sequence of the oligonucleotides used in this protocol Name Sequence (5′–3′) ori-F GATACTAACGCCGCCATCCAGTTTCCCGGGaaaggcggtaatacggtta ori-R CCCGGGttgataatctcatgaccaaaatcc ampR-F TGGTCATGAGATTATCAACCCGGGaaaggatcttcacctagatcct ampR-R GGGcacttttcggggaaatgtgcg CEN-F GTTCCGCGCACATTTCCCCGAAAAGTGCCCGGGtccttttcatcacgtgc CEN-R GGGcttaggacggatcgcttgc LEU2-F AGTTACAGGCAAGCGATCCGTCCTAAGCCCGGGaactgtgggaatactcaggt LEU2-R Cgtgtcgtttctattatgaatttc LYS2-F TTTATAAATGAAATTCATAATAGAAACGACCCGGGcttcaatagttttgccagcg LYS2-R GCTCCCGGGcatatcatacgtaatgctca URA3-F TTGAGCATTACGTATGATATGCCCGGGagcttttcaattcatcttttttttttttgttc URA3-R GGGtaataactgatataattaaattgaagc HIS3-F GCTTCAATTTAATTATATCAGTTATTACCCGGGcttcattcaacgtttcccatt HIS3-R GGGtgatgcattaccttgtcatc kanR-F TACTGAAGATGACAAGGTAATGCATCACCCGGGtagcccatacatccccatgt kanR-R CCCGGGTAAATCACGCTAACATTTGA All oligonucleotides are desalted and <60 bp in length. The sequences shown in lower case are homologous to sequences upstream (F) or downstream (R) of the template gene or origin; the sequences shown in upper case are homologous to one of the other fragments; the SmaI sites are underlined. 854 Cite this protocol as Cold Spring Harb Protoc; doi:10.1101/pdb.prot085100 J. van Leeuwen et al. Cold Spring Harbor Laboratory Press on October 6, 2015 - Published by Downloaded from Electroporation system (e.g., MicroPulser by Bio-Rad) Ice Incubators (30˚C and 37˚C) Microcentrifuge tubes (1.5-mL) Orbital shaker PCR tubes Pipettes Plates Thermal cycler (PCR machine) Vortex Water bath (42˚C) METHOD To illustrate the cloning of multiple DNA fragments into one construct by homologous recombination, we describe the assembly of five different yeast selection markers (HIS3, LEU2, LYS2, URA3, and kanMX6) together with origins of replication for both yeast and E. coli (CEN6/ARS4 and ori) and an E. coli selection marker (ampR) into one plasmid (Fig. 1A). SD-lys SD-all YEPD+G418 C D ori ampR ... + + + + ... LYS2 LEU2 HIS3 ori CEN6/ARS4 URA3 ampR kanMX6 PCR LYS2 HIS3 URA3 LEU2 CEN6/ARS4 ampR ori kanMX6 1000 bp 3000 bp 250 bp 6000 bp 1000 bp 3000 bp 500 bp 6000 bp 1000 bp 3000 bp 500 bp 6000 bp A B FIGURE 1. Rapid and efficient plasmid construction by homologous recombination in yeast. (A) A schematic repre-sentation of the described assembly of eight DNA fragments in yeast using 30-bp recombination sequences. (B) The PCR fragments used in the featured assembly. (C) The selection of yeast transformants on synthetic lysine-dropout medium (SD-lys) and subsequent confirmation of the presence of the six fragments containing yeast sequences by replica plating on YEPD + G418 and synthetic-dropout medium without amino acids (SD-all). Only two transformants fail to grow after replica plating (yellow circles). (D) The SmaI profiles of the final plasmids isolated from 24 yeast colonies (upper panel) or 24 E. coli colonies (lower panel). Cite this protocol as Cold Spring Harb Protoc; doi:10.1101/pdb.prot085100 855 Multiple-Fragment Cloning Cold Spring Harbor Laboratory Press on October 6, 2015 - Published by Downloaded from Preparation of DNA Fragments This procedure should take 3–4 h on day 1. 1. Prepare genomic DNA from a mixture of the yeast strains BY4709 and BY4712 to serve as a template in the polymerase chain reaction (PCR) (see Steps 23–35). 2. Prepare a 20-µL PCR. Sterile deionized water 13.4 µL Buffer (5×) 4 µL dNTP mixture (10 mM) 0.4 µL Forward primer (10 µM) 0.5 µL Reverse primer (10 µM) 0.5 µL Template DNA 1 µL Phusion DNA polymerase 0.2 µL 3. Use the following PCR cycling conditions. Initial denaturation 60 sec at 98˚C 5 Cycles 10 sec at 98˚C, 20 sec at 55˚C, and 90 sec at 72˚C 25 Cycles 10 sec at 98˚C, 20 sec at 62˚C, and 90 sec at 72˚C HIS3, LEU2, LYS2, and URA3 can be amplified from ≏100 ng of genomic DNA of a mixture of the yeast strains BY4709 and BY4712. CEN6/ARS4, ampR, and ori can be amplified from 1 pg of pRS416 (Sikorski and Hieter 1989), and kanMX6 can be amplified from 1 pg of pFA6-kanMX6 (Bahler et al. 1998). 4. Run 2 µL of each PCR product on a 1.2% agarose gel (Fig. 1B). If a PCR product contains multiple fragments, all fragments can potentially be assembled in the final construct. In this case, purification of the correct PCR product from the gel might be necessary. Alternatively, the unpurified PCR product with multiple fragments can be used, in which case the number of colonies that have to be screened for the expected construct has to be increased. The PCR products can be stored at 4˚C or –20˚C. Preparation of Competent Yeast Cells The following procedure is a modified version of Gietz’s method (Gietz and Woods 2002) and should take 5 min on day 1 and 10–15 min on day 2. 5. Patch the yeast strain BY4742 on a 2-cm2 area on YEPD agar and incubate overnight at 30˚C. Alternatively, the yeast strain can be used to inoculate a 5-mL culture of liquid YEPD medium. Incubate overnight at 30˚C while shaking on an orbital shaker at 200 rpm. Any standard laboratory yeast strain, such as BY4741 or W303, can be used instead of BY4742. 6. Scrape a 50-µL portion of yeast cells from the YEPD plate and resuspend the cells in 1 mL of sterile deionized water in a 1.5-mL microcentrifuge tube. Alternatively, harvest 1–1.5 mL of a liquid culture. 7. Pellet the cells by centrifugation at 3000 rpm (800 rcf) for 1 min at room temperature in a micro-centrifuge and discard the supernatant. 8. Wash the pellet once with 1 mL of 0.1 M LiAc. 9. Estimate the volume of the cell pellet and resuspend the cells in an equal volume of 0.1 M LiAc supplemented with 10% glycerol. By using larger volumes of liquid culture in Steps 5 and 6, a large amount of competent cells can be made at once. These competent cells can be stored at –80˚C in 0.1 M LiAc supplemented with 10% glycerol. Yeast Transformation This procedure is a modified version of Gietz’s method (Gietz and Woods 2002) and should take 3 h (not including the time needed to grow the yeast cells) on day 2. 856 Cite this protocol as Cold Spring Harb Protoc; doi:10.1101/pdb.prot085100 J. van Leeuwen et al. Cold Spring Harbor Laboratory Press on October 6, 2015 - Published by Downloaded from 10. Prepare the transformation buffer by mixing 800 µL of 50% PEG3350, 100 µL of 1 M LiAc, 100 µL of 10× TE, and 50 µL of DMSO. 11. Boil the ssDNA (10 mg/mL) for 5 min and place it on ice. 12. Add ≏100 ng of each DNA fragment, 2 µL of ssDNA, and 12 µL of competent yeast cells into a 1.5-mL microcentrifuge tube; mix gently by pipetting up and down. If the total volume of all the DNA fragments together is >20 µL, the overall transformation efficiency will decrease. The volume can be reduced by air-drying the DNA overnight at room temperature. 13. Add 100 µL of the transformation buffer from Step 10 and vortex for ≏10 sec. 14. Incubate for 30 min at room temperature. 15. Incubate for 15 min in a 42˚C water bath. 16. Incubate on ice for 5 min. 17. Pellet the cells by centrifugation at 3000 rpm (800 rcf) for 1 min and remove the supernatant. 18. Resuspend the cells in 1 mL of YEPD. 19. (Optional) Incubate the cells for 2 h at 30˚C. This step allows the cells to produce the antibiotic resistance and/or auxotrophic marker proteins before applying selection and thereby increases the overall transformation efficiency. 20. Plate 250 µL of the cell suspension on SD-lys plates. Other media that select for one of the other cloned genes can also be used. 21. Incubate the plates for 2–3 d at 30˚C. 22. Replica-plate the colonies on SD-all and YEPD + G418 plates (Fig. 1C). The SD-lys plates select only for one of the yeast markers (LYS2) and for presence of the CEN6/ARS4. This replica-plating step tests whether the other yeast selection markers are present in the construct and functional. We have obtained 470 colonies on the SD-lys plate, only two of which failed to grow on the YEPD + G418 (which selects for kanMX6) and SD-all (which selects for HIS3, LEU2, and URA3) plates. Preparation of Yeast Genomic DNA This procedure should take 1 h on day 4. 23. Wash all the colonies from the SD-lys plate from Step 21 using 5 mL of sterile water and transfer 1 mL of cells to a 1.5-mL microcentrifuge tube. 24. Pellet the cells by centrifugation at 3000 rpm (800 rcf) for 1 min and remove the supernatant. 25. Resuspend the pellet in 250 µL of lysis buffer for yeast. 26. Add 250 µL of saturated phenol and ≏200 µL (≏200 mg) of glass beads. 27. Close the cap tightly and vortex for >2 min at room temperature. 28. Centrifuge at 13,000 rpm (15,700 rcf) for 5 min. 29. Transfer 150 µL of the aqueous top layer to a new 1.5-mL microcentrifuge tube and centrifuge again at 13,000 rpm (15,700 rcf) for 5 min. 30. Transfer 100 µL of the top layer to a new 1.5 mL microcentrifuge tube, add 100 µL of 100% isopropanol and mix thoroughly by inversion. 31. Centrifuge at 13,000 rpm (15,700 rcf) for 10 min and remove the supernatant. 32. Wash the pellet once with 500 µL of 70% ethanol. 33. Briefly centrifuge at 13,000 rpm (15,700 rcf) for 10 sec and remove the remaining ethanol. 34. Dry the pellet at room temperature for ≏10 min. 35. Dissolve the pellet in 40 µL of sterile water. Cite this protocol as Cold Spring Harb Protoc; doi:10.1101/pdb.prot085100 857 Multiple-Fragment Cloning Cold Spring Harbor Laboratory Press on October 6, 2015 - Published by Downloaded from Preparation of Competent E. coli Cells This procedure should take 10 min on day 4. 36. Inoculate 2 mL of TB using one bacterial colony and incubate overnight at 37˚C while shaking at 250 rpm. In the example described here, we use Top10 cells. 37. Add 2 mL of TB and incubate for another 25 min at 37˚C while shaking at 250 rpm in an orbital shaker. 38. Divide the culture into aliquots of 1 mL. This is approximately the amount of culture needed for one transformation. 39. Collect the cells by centrifuging at 13,000 rpm (15,700 rcf) for 1 min. 40. Wash the cells four times with Milli-Q water. 41. Estimate the volume of the cell pellet and resuspend the cells in an equal volume of Milli-Q water. More water (up to five times the volume of the cell pellet) can be used to obtain “more” competent cells. However, the transformation efficiency will slightly decrease because the cells are more diluted. By using greater volumes of liquid culture in Steps 36–41, a large number of competent cells can be made at once. These competent cells can be stored at –80˚C in 10% glycerol. Other methods can be used to make competent E. coli cells, but these will require longer preparation times. Also, with this method, all the steps after growing the culture can be performed at room temperature, which makes it more straightforward and robust. Plasmid Recovery from Escherichia coli This procedure should take ≏5 h on day 5 and day 6. 42. Add 20 µL of competent E. coli cells to a 200-µL PCR tube and add 0.5 µL of yeast genomic DNA. 43. Mix the DNA with the competent cells by pipetting up and down. 44. Transfer the mixture to an electroporation cuvette (1 mm gap). 45. Electroporate the cells according to the manufacturer’s manual. 46. Add 250 µL of LB and plate all the cells on an LB plate containing ampicillin. Note that kanMX6 is functional in both S. cerevisiae and E. coli. Therefore, transformants can also be selected on LB media containing kanamycin. 47. Incubate the plates overnight at 37˚C. 48. Purify the plasmid DNA with a plasmid miniprep kit. 49. Confirm the correct assembly of the fragments by a SmaI digestion (Fig. 1D). In the example described here, SmaI restriction sites were added to each fragment by means of the primers to show the accuracy of homologous recombination in yeast. However, the addition of restriction sites is not necessary for this method. DISCUSSION Homologous recombination in yeast has been successfully used in the cloning of both natural and synthetic DNA fragments (Ma et al. 1987; Wang 2000; Chen et al. 2005; Gibson et al. 2008a,b; Gibson 2009; Shao and Zhao 2009; Tan and Tan 2010; Tsvetanova et al. 2011; Liang et al. 2012; Shao and Zhao 2013). However, many of these methods are based on yeast–bacterial shuttle plasmids, which rather restricted the technology to a small niche within molecular biology. For an introduction to these technologies, see Introduction: Construction of Multifragment Plasmids by Homologous Recombi-858 Cite this protocol as Cold Spring Harb Protoc; doi:10.1101/pdb.prot085100 J. van Leeuwen et al. Cold Spring Harbor Laboratory Press on October 6, 2015 - Published by Downloaded from nation in Yeast (van Leeuwen et al. 2015). Recently, however, Kuijpers et al. (2013) have presented an efficient strategy for assembling a plasmid from mutiple DNA fragments by using overlapping ho-mology regions of 60 bp. As this method does not make use of a vector backbone, it can easily be extended to construct plasmids for other organisms. In this protocol, we have described the use of yeast as a host to assemble eight DNA fragments, with short homology regions (30 bp) between the fragments, into one plasmid (Fig. 1A). Equal amounts of PCR products were cotransformed into yeast, and transformants were selected on synthetic media lacking lysine (SD-lys), which confirms the presence and functionality of LYS2 and the CEN6/ARS4 origin of replication. The presence and activity of HIS3, LEU2, URA3, and kanMX6 were confirmed by replica plating (Step 22). Only two out of 470 transformants selected on SD-lys failed to grow after replica plating. The genomic DNA was individually purified from 24 randomly picked yeast colonies from the SD-lys plate and from 24 randomly picked E. coli colonies from Steps 46–48, followed by recovery of the plasmids. SmaI sites were introduced between all the fragments when the primers were designed, which enabled confirmation of the correct assembly of the construct by a SmaI digestion. In one out of the 48 isolated plasmids, a SmaI site was missing, which could be caused by impurities in the primers used, whereas the other 47 plasmids all showed the expected DNA bands (Fig. 1D). Based on the growth phenotypes on the different media and the SmaI digestion profiles, we can conclude that the overall assembly effi-ciency is >95%. This protocol uses short (<60 bp), desalted oligonucleotides and unpurified PCR products. This makes the method highly cost- and time-efficient, especially when cloning multiple fragments. We modified the yeast and bacterial transformation methods, which facilitates high-throughput cloning. However, this method has the common disadvantage that inserts that are toxic to either yeast or bacteria cannot be cloned. Also, because homologous recombination is so efficient in yeast, the cloning of inverted or repetitive sequences can be problematic. Interestingly, we have found that the kanMX6 selector module is functional both in yeast and bacteria, which reduces the number of required selection markers from two (one for yeast and one for E. coli) to one. Although, in the example described here, separate fragments are used for the E. coli selection marker and origin of replication (ampR and ori), we have routinely used the shortest back-bone of pBlueScript II (Alting-Mees and Short 1989), which contains both ori and ampR, as one fragment in our assemblies. Finally, this method can be extended to clone constructs for other organisms by combining one or more fragment(s) with the genes or sequences of interest with a fragment containing a linearized nonyeast plasmid, and another fragment containing a CEN/ARS origin and yeast selection marker. RECIPES Luria–Bertani (LB) Medium Plus Ampicillin Reagent Quantity Agar 20 g NaCl 10 g Tryptone 10 g Yeast extract 5 g Prepare the above-listed ingredients in 1 L of deionized water. Adjust the pH to 7.0 with 5 N NaOH. Autoclave for 20 min at 15 psi (1.05 kg/cm2). Cool to ≏60˚C and add ampicillin (final concentration 120 µg/mL). Pour the medium into Petri dishes (≏25 mL per 100-mm plate). Store the LB plates at 4˚C; they will keep for at least 4 mo. Cite this protocol as Cold Spring Harb Protoc; doi:10.1101/pdb.prot085100 859 Multiple-Fragment Cloning Cold Spring Harbor Laboratory Press on October 6, 2015 - Published by Downloaded from Lysis Buffer for Yeast Reagent Quantity Final concentration Triton X-100 10 mL 2% (v/v) SDS (10%) 50 mL 1% (w/v) NaCl (5 M) 10 mL 100 mM Tris-Cl (1 M, pH 8.0) 5 mL 10 mM EDTA (0.5 M, pH 8.0) 1 mL 1 mM Prepare in 500 mL of deionized water. Autoclave for 20 min at 15 psi (1.05 kg/cm2). Store at room temperature; it will keep for at least 1 yr. Supplements for SD-Lys Reagent Quantity L-Leucine 0.84 g L-Histidine HCl 0.42 g L-Methionine 0.42 g Uracil 0.25 g In separate tubes, prepare 100× stocks of each of the four reagents by dissolving the quantities indicated in 100 mL of deionized water and then filter-sterilizing. Store all stocks at room temperature; they will keep for at least 6 mo. Synthetic Amino-Acid-Dropout Medium (SD-All) Reagent Quantity Final concentration Difco yeast nitrogen base without amino acids 6.7 g 6.7 g/L Agar 20 g 20 g/L Dextrose (40%) 50 mL 20 g/L Add 950 mL of deionized water to 6.7 g Difco yeast nitrogen base without amino acids and 20 g of agar. Autoclave for 20 min at 15 psi (1.05 kg/cm2). After autoclaving, add 50 mL of a 40% dextrose solution. Cool the medium to ≏60˚C and pour into Petri dishes (≏25 mL per 100-mm plate). Store the SD-all plates at 4˚C; they will keep for at least 6 mo. Synthetic Lysine-Dropout Medium (SD-Lys) Reagent Quantity Final concentration Difco yeast nitrogen base without amino acids 6.7 g 6.7 g/L Agar 20 g 20 g/L Supplements for SD-lys: L-Leucine (100×) 10 mL 8.4 mg/L L-Histidine HCl (100×) 10 mL 4.2 mg/L L-Methionine (100×) 10 mL 4.2 mg/L Uracil (100×) 10 mL 2.5 mg/L Dextrose (40%) 50 mL 20 g/L Combine 10 mL of each of the SD-lys supplements (100× L-leucine, 100× L-histidine HCl, 100× L-methionine, 100× uracil) with 6.7 g of Difco yeast nitrogen base without amino acids and 20 g agar, and add 950 mL of deionized water. Autoclave for 20 min at 15 psi (1.05 kg/cm2). After auto-claving, add 50 mL of a 40% dextrose solution. Cool the medium to ≏60˚C and pour into Petri dishes (≏25 mL per 100-mm plate). Store the SD-lys plates at 4˚C; they will keep for at least 6 mo. (Note that this recipe is optimized for BY4742 strains. It should be adjusted if another yeast strain is used as a recombination host.) 860 Cite this protocol as Cold Spring Harb Protoc; doi:10.1101/pdb.prot085100 J. van Leeuwen et al. Cold Spring Harbor Laboratory Press on October 6, 2015 - Published by Downloaded from Terrific Broth (TB) Medium Reagent Quantity Final concentration Yeast extract 24 g 24 g/L Tryptone 20 g 20 g/L Glycerol 4 mL 4 mL/L Phosphate buffer (0.17 M KH2PO4, 0.72 M K2HPO4) 100 mL 0.017 M KH2PO4, 0.072 M K2HPO4 Add 900 mL of deionized water to 24 g of yeast extract, 20 g of tryptone, and 4 mL of glycerol. Shake or stir until the solutes have dissolved and sterilize by autoclaving for 20 min at 15 psi (1.05 kg/cm2). Allow the solution to cool to ≏60˚C and add 100 mL of sterile phosphate buffer. Store TB at room temperature; it will keep for at least 1 yr. Yeast Extract-Peptone-Dextrose (YEPD) Reagent Quantity Final concentration Bacto peptone 20 g 2% (w/v) Yeast extract 10 g 1% (w/v) Dextrose 20 g 2% (w/v) Agar (optional) 20 g 2% (w/v) G418 (200 mg/mL; optional) 1 mL 200 mg/L Add 1 L of deionized water to 20 g bacto peptone, 10 g yeast extract, and 20 g dextrose (and 20 g of agar for YEPD plates). Sterilize by autoclaving for 20 min at 15 psi (1.05 kg/cm2). To prepare YEPD plus G418 plates, allow the solution to cool to ≏60˚C, add 1 mL of G418 stock solution, and pour into Petri dishes (≏25 mL per 100-mm plate). Store YEPD medium without G418 at room temperature, and store YEPD containing medium G418 at 4˚C. ACKNOWLEDGMENTS This work was supported by grants HHMI 55007643, CIHR MOP-130358, and Ministry of Research and Innovation GL-01-022 to B.A. and C.B. and a CIHR fellowship held by J.v.L. REFERENCES Alting-Mees MA, Short JM. 1989. pBluescript II: Gene mapping vectors. Nucleic Acids Res 17: 9494. Bahler J, Wu JQ, Longtine MS, Shah NG, McKenzie A 3rd, Steever AB, Wach A, Philippsen P, Pringle JR. 1998. Heterologous modules for efficient and versatile PCR-based gene targeting in Schizosaccharomyces pombe. Yeast 14: 943–951. Chen X, Yuan H, He W, Hu X, Lu H, Li Y. 2005. Construction of a novel kind of expression plasmid by homologous recombination in Saccha-romyces cerevisiae. Sci China C Life Sci 48: 330–336. Gibson DG. 2009. Synthesis of DNA fragments in yeast by one-step assembly of overlapping oligonucleotides. Nucleic Acids Res 37: 6984–6990. Gibson DG, Benders GA, Andrews-Pfannkoch C, Denisova EA, Baden-Tillson H, Zaveri J, Stockwell TB, Brownley A, Thomas DW, Algire MA, et al. 2008a. Complete chemical synthesis, assembly, and cloning of a Mycoplasma genitalium genome. Science 319: 1215–1220. Gibson DG, Benders GA, Axelrod KC, Zaveri J, Algire MA, Moodie M, Montague MG, Venter JC, Smith HO, Hutchison CA 3rd. 2008b. One-step assembly in yeast of 25 overlapping DNA fragments to form a complete synthetic Mycoplasma genitalium genome. Proc Natl Acad Sci 105: 20404–20409. Gietz RD, Woods RA. 2002. Transformation of yeast by lithium acetate/ single-stranded carrier DNA/polyethylene glycol method. Methods Enzymol 350: 87–96. KuijpersNG,Solis-EscalanteD,BosmanL,vandenBroekM,PronkJT,Daran JM,Daran-LapujadeP.2013.Aversatile,efficientstrategyforassemblyof multi-fragment expression vectors in Saccharomyces cerevisiae using 60 bp synthetic recombination sequences. Microb Cell Fact 12: 47. Liang X, Peng L, Tsvetanova B, Li K, Yang JP, Ho T, Shirley J, Xu L, Potter J, Kudlicki W, et al. 2012. Recombination-based DNA assembly and mu-tagenesis methods for metabolic engineering. Methods Mol Biol 834: 93–109. Ma H, Kunes S, Schatz PJ, Botstein D. 1987. Plasmid construction by ho-mologous recombination in yeast. Gene 58: 201–216. Shao Z, Zhao H. 2009. DNA assembler, an in vivo genetic method for rapid construction of biochemical pathways. Nucleic Acids Res 37: e16. Shao Z, Zhao H. 2013. Construction and engineering of large biochemical pathways via DNA assembler. Methods Mol Biol 1073: 85–106. Sikorski RS, Hieter P. 1989. A system of shuttle vectors and yeast host strains designed for efficient manipulation of DNA in Saccharomyces cerevisiae. Genetics 122: 19–27. Tan G, Tan C. 2010. SMC, a simple method to rapidly assemble multiple fragments into one construct. Front Biosci (Elite Ed) 2: 1105–1114. Tsvetanova B, Peng L, Liang X, Li K, Yang JP, Ho T, Shirley J, Xu L, Potter J, Kudlicki W, et al. 2011. Genetic assembly tools for synthetic biology. Methods Enzymol 498: 327–348. van Leeuwen JS, Andrews BJ, Boone C, Tan G. 2015. Construction of multi-fragment plasmids by homologous recombination in yeast. Cold Spring Harb Protoc doi: 10.1101/pdb.top084111. Wang PL. 2000. Creating hybrid genes by homologous recombination. Dis Markers 16: 3–13. Cite this protocol as Cold Spring Harb Protoc; doi:10.1101/pdb.prot085100 861 Multiple-Fragment Cloning Cold Spring Harbor Laboratory Press on October 6, 2015 - Published by Downloaded from
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Translation of telo – Italian–English dictionary telo (Translation of telo from the GLOBAL Italian–English Dictionary © 2018 K Dictionaries Ltd) Browse Word of the Day take something back to admit that something you said was wrong Blog Calm and collected (The language of staying calm in a crisis) New Words vibe coding © Cambridge University Press & Assessment 2025 © Cambridge University Press & Assessment 2025 Learn more with +Plus Learn more with +Plus To add telo to a word list please sign up or log in. Add telo to one of your lists below, or create a new one. {{message}} {{message}} Something went wrong. {{message}} {{message}} Something went wrong. {{message}} {{message}} There was a problem sending your report. {{message}} {{message}} There was a problem sending your report.
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Skip to Main content My account Sign in Canonical Form In subject area:Computer Science Canonical form in Computer Science refers to a Boolean function that is expressed either as a sum of minterms or as a product of maxterms. It represents the standard form of a Boolean function that can be manipulated to simplify its expression. AI generated definition based on: Digital Logic Design (Fourth Edition), 2002 How useful is this definition? Add to Mendeley Also in subject areas: Earth and Planetary Sciences Engineering Mathematics Discover other topics Chapters and Articles You might find these chapters and articles relevant to this topic. Karnaugh maps and function simplification 2002, Digital Logic Design (Fourth Edition)B. HOLDSWORTH BSc (Eng), MSc, FIEE, R.C. WOODS MA, DPhil 3.3Canonical forms Also mentioned in section 2.5 is the concept of the canonical form, a term used to describe a Boolean function that is written either as a sum of minterms, or as a product of maxterms. For example, using three variables A, B, and C, the equation is not written explicitly as a sum of minterms (or a product of maxterms) and so is not in canonical form. Simple Boolean algebraic manipulation produces the same function in canonical form written as the logical sum of three minterms:while the following equation is written as the product of three maxterms and so is also in canonical form: View chapterExplore book Read full chapter URL: Book2002, Digital Logic Design (Fourth Edition)B. HOLDSWORTH BSc (Eng), MSc, FIEE, R.C. WOODS MA, DPhil Chapter Ordering and Reordering 2009, Disappearing Cryptography (Third Edition)Peter Wayner 13.5Canonical Forms Another solution is to create a “canonical form” for each element. That is, choose one version of the element that is the same. Then, the data is removed by converting it into the canonical form and the result is used to compute f. Here are some basic ones: • : If the data is hidden in the least significant bit of pixels or audio file elements, then the canonical form can be found by setting the least significant bit to zero. • : If the information-hiding process modifies the elements by no more than ±ɛ then canonical points can be established as they were above. Let {0, 2ɛ, 4ɛ, 6ɛ,…} be the set of canonical points. Only one will lie in the range xi – ɛ < xi ≤ xi + ɛ. • : Sentences can be put into canonical form. Mikhail Atallah and Victor Raskin use natural language processing algorithms to parse sentences. [ARC+01] The sentence “The dog chased the cat” takes this form in their LISP-based system: The letter ‘S’ stands for the beginning of a sentence, the letters ‘NP’ stands for a noun phrase, and the letters ‘VP’ stands for a verb phrase. If the sentence can't be parsed, it is ignored. Their solution hides information by applying a number of transformations like switching between the active and passive voice. One bit could be encoded by switching this example to read “The cat was chased by the dog”. Others solutions they use include moving the adjunct of a sentence, clefting the sentence, and inserting extra unnecessary words and phrases like “It seems that…” A canonical form can be defined by choosing one version of the transformation. In this example, the active voice might be the canonical form for the sentence. View chapterExplore book Read full chapter URL: Book2009, Disappearing Cryptography (Third Edition)Peter Wayner Chapter The EXPAND and COLLAPSE Operators 2014, Time and Relational Theory (Second Edition)C.J. Date, ... Nikos A. Lorentzos Preliminary Remarks The result sets produced by EXPAND and COLLAPSE can each be regarded as a particular canonical form for the corresponding input set—and the canonical forms in question both have an important role to play in the solutions we’re at last beginning to approach to the problems we identified in Chapters 4 and 5. Note: It might help to state up front that each of those two canonical forms has the property that every point represented in some interval in the input set is represented exactly once—i.e., as part of exactly one interval—in the corresponding result set. Aside: The notion of canonical form is central to many branches of mathematics and related disciplines. In case you’re unfamiliar with it, we digress for a moment to explain it briefly. The basic idea is this: Given a set s1, together with a stated notion of equivalence among the elements of that set, subset s2 of s1 is a set of canonical forms for s1 if and only if every element a1 of s1 is equivalent, under that notion of equivalence, to just one element a2 of s2 (and that element a2 is a canonical form for the element a1). We can also say, a trifle loosely, that the set s2 taken as a whole is a canonical form for the set s1 as such. Various ‘interesting’ properties that apply to s1 also apply to s2, mutatis mutandis; thus, we can study just the ‘small’ set s2, not the ‘large’ set s1, in order to prove a variety of interesting theorems or results. Here’s a simple example: ■ : Let s1 be the set of nonnegative integers {0,1,2,…}. ■ : Define equivalence among elements of s1 as follows: Elements a and b are equivalent if and only if they leave the same remainder on division by five. Thus, e.g., the integers 8, 13, 18, 23, etc., are all equivalent to 3 (as is 3 itself, of course). ■ : Under this definition of equivalence, then, s2 is the set {0,1,2,3,4}—every element of s1 is equivalent to just one of these five integers, and each of these five is in fact the canonical form for an infinite number of elements of s1. Note in particular that s2 here is finite, whereas s1 is infinite. ■ : As for an ‘interesting’ property that applies in this example, let a1, b1, and c1 be any three elements of s1, and let their canonical forms in s2 be a2, b2, and c2, respectively. Then the product b1 c1 is equivalent to a1 if and only if the product b2 c2 is equivalent to a2. > End of aside. The applicability of the foregoing ideas to sets of intervals in particular is explained in the next two sections. View chapterExplore book Read full chapter URL: Book2014, Time and Relational Theory (Second Edition)C.J. Date, ... Nikos A. Lorentzos Review article Optimization by Canonical Analysis in a Radial Basis Function 2015, Expert Systems with ApplicationsRolando J. Praga-Alejo, ... David S. González-González 2.2.3A or B canonical forms? When should the A or B canonical form must be applied? In design units, the stationary point distance (represented by the system center) from the design center O should be given by: (25) That is: (26) This means: – : If D is greater than 1, the A canonical form must be used to explore the stationary point neighborhood. – : If D is closer to zero or one, the B canonical form must be used. – : The A canonical form should be used in presence of a saddle point (Myers et al., 2009). View article Read full article URL: Journal2015, Expert Systems with ApplicationsRolando J. Praga-Alejo, ... David S. González-González Chapter Analysis of Combinatorial Neural Codes: An Algebraic Approach 2019, Algebraic and Combinatorial Computational BiologyCarina Curto, ... Nora Youngs 7.3.1Computing the Canonical Form Computing the canonical form for the neural ideal code can be done iteratively by codeword. Writing down the canonical form for the ideal consisting of no codewords is simple; thereafter, we add the codewords one at a time, updating the canonical form at each step. Here, we describe an algorithm which takes the canonical form for a given code , and a codeword c ∈{0, 1}n, and outputs the canonical form for . It is generally assumed that since otherwise the canonical form will not change; however, the success of the algorithm does not depend on this assumption. This algorithm can then be iterated to build the canonical form for a code. A brief explanation of the algorithm: the polynomials in the original canonical form will either vanish on the new codeword c, or they will not. In Step 1, we sort the polynomials by this feature. If the polynomial f does vanish on c, then f will remain in the new canonical form (add f to L); if not, we add f to the list M to potentially adjust f so that it will vanish on c. In Step 2, we perform this adjustment to each f, multiplying by a linear term which will certainly vanish on c; namely, (xi −ci). If the result of this multiplication is not a pseudo-monomial, or if it is a multiple of some pseudo-monomial in L and therefore redundant, then we will not retain it; otherwise, we add it to the list of new pseudo-monomials to add to ) (list N). To ensure that we obtain the entire new canonical form and not only a subset, it is necessary to consider all such linear terms (xi −ci) and add all the valid possible adjustments to the new canonical form. Finally, we output the collection of the old polynomials which are still valid (list L) and the newly-created polynomials (list N) as the new canonical form. A proof that this process will result in the correct canonical form for the new code can be found in . Example 7.11 Let . Observe that in this code we have the RF relationships U3 = ∅ since neuron 3 never fires, and since neurons 1 and 2 never fire together. The canonical form of is . We will compute , where , inputting and c = 110 into Algorithm 7.1: Algorithm 7.1 Input:The canonical form for a code , and a word c ∈{0, 1}n. Output: The canonical form . Step 0: Initialize empty lists L, M, and N. Step 1: For , if f(c) = 0, add f to L; otherwise, add f to M. Repeat for all polynomials . Step 2: For each polynomial g ∈ M and each 1 ≤ i ≤ n, if (i) : neither xi nor (1 − xi) divide g, and (ii) : (xi −ci)g is not a multiple of some polynomial in L, then add (xi −ci)g to N. Repeat for all polynomials g ∈ M and 1 ≤ i ≤ n. Step 3: Output . : Step 0: Set L = M = N = ∅. : Step 1 for x3, x1x2: Since f = x3 satisfies f(110) = 0, add x3 to L. Since f = x1x2 does not satisfy f(110) = 0, add x1x2 to M. Then, M = {x1x2} and L = {x3}. : Step 2 for x1x2 and i = 1, 2, 3: It is not true that neither x1 nor 1 − x1 divide x1x2. So N does not change. It is not true that neither x2 nor 1 − x2 divide x1x2. So N does not change. It is true that neither x3 nor 1 − x3 divide x1x2. However, (x3 − 0)x1x2 is a multiple of a polynomial in L. So N does not change. : Step 3: The output is . This ends Algorithm 7.1. Thus, we find that . This result makes sense, since our new code is characterized by the RF relationship U3 = ∅. Exercise 7.34 Let as in Example 7.11. Define . Use Algorithm 7.1 to show that . Algorithm 7.1 shows how to adjust a given canonical form to include one new codeword; computing the canonical form for a given code is simply an iteration of this process. We note that the canonical form for the empty code is . Then, we order our code , and add in the codewords one at a time, using Algorithm 7.1 each time to compute the new canonical form. Once all codewords have been added, the result is the canonical form for the complete code . Algorithm 7.2 describes this process. Algorithm 7.2 Input: A code . Output: The canonical form . Step 0: Arbitrarily order the codewords of (so ). Step 1: Define the code . The canonical form of is . Set j = 1. Step 2: Define . Input and cj into Algorithm 7.1; the output will be . If j = d, output . Otherwise, set j = j + 1 and repeat Step 2. Exercise 7.35 Using the iterative process described in Algorithm 7.2 and as in Example 7.11, show that . Then, reorder the codewords in a new way, recompute , and show that the result is the same. View chapterExplore book Read full chapter URL: Book2019, Algebraic and Combinatorial Computational BiologyCarina Curto, ... Nora Youngs Chapter EXPRESSION-INVARIANT THREE-DIMENSIONAL FACE RECOGNITION 2006, Face Processing 5.3.2Canonical Forms of Facial Surfaces When embedding is performed into a space of dimension m = 3, the canonical form can be plotted as a surface. Figure 5.7 depicts canonical forms of a person's face with different facial expressions. It demonstrates that, although the facial surface changes are substantial, the changes between the corresponding canonical forms are insignificant. Embedding into ℝ2 is a special case – in this case, the codimension of the canonical form in the embedding space is zero. Such an embedding can be thought of as an intrinsic parametrization of the facial surface, which leads to a “warping” of the facial texture. This serves as a way of performing geometry-based registration of 2D facial images . Flat embedding into ℝ2 was previously used for cortical surface matching in brain analysis, and adopted to texture mapping in computer graphics. View chapterExplore book Read full chapter URL: Book2006, Face Processing Chapter XML Basics 2003, Modeling Business Objects with XML SchemaBERTHOLD DAUM 4.3.6Canonical Attributes The canonical form of an attribute consists of ▪ : a space character ▪ : the attribute's qualified name ▪ : an equality sign ▪ : a double quote ▪ : the value as a canonical string with normalized whitespace (see Section 4.3.2) ▪ : a double quote Here, we have normalized the whitespace in the attribute value, removed whitespace between element name and attribute and between equality sign and quote, and replaced the single quotes with double quotes: | Noncanonical | Canonical | --- | View chapterExplore book Read full chapter URL: Book2003, Modeling Business Objects with XML SchemaBERTHOLD DAUM Review article Optimization by Canonical Analysis in a Radial Basis Function 2015, Expert Systems with ApplicationsRolando J. Praga-Alejo, ... David S. González-González 2.2.2The B canonical form This is achieved by differentiating and equating to zero Eq. (11) with respect to , and derived in relation to X the Eq. (16). In this way, it is possible finding the stationary point S coordinates: (19) Or of equal than Eq. (18). At this point, the fitted response is equal to: (20) Or, equivalently: (21) The new vectors: and where: – can be rewritten in coded variable as: (22) ; referring to the coordinates that were measured from stationary point S. The B canonical form is achieved by changing the modeling system origin at the stationary point () given by: (23) B Canonical form, which might be expressed through an expanding form: (24) The canonical form of the second-order model given by Eq. (24) is usually called the B Canonical form; these Eqs. (19)–(24) can be calculated by the RBFNN. It is very useful for determining the nature of the fitted response surface, particularly in identifying saddle systems and ridges (Myers et al., 2009). View article Read full article URL: Journal2015, Expert Systems with ApplicationsRolando J. Praga-Alejo, ... David S. González-González Chapter XML Basics 2003, Modeling Business Objects with XML SchemaBERTHOLD DAUM 4.3XML CANONICAL FORM The XML Information Set as discussed above represents XML documents in an abstract form. The lexical form of an XML document allows many variations for the same content. For example, attributes may appear in arbitrary order, redundant namespace declarations are possible, character content may be expressed with or without CDATA, and so on. This makes it difficult for humans and machines to determine whether two given XML documents are equivalent— not identical by the letter but equivalent by content. And this is not the only problem. Cryptographic methods used by message digests and digital signatures rely on the textual representation of a document. With different text representations, equivalent documents would, for example, have different digital signatures. The proposed W3C recommendation “XML-Signature Syntax and Processing” [Eastlake2002] therefore relies on the existing methods for producing canonical XML. The W3C recommendation “Canonical XML” [Boyer2001] defines a canonical form for XML documents, a syntactical form that allows simple character string comparison of two XML documents. However, this recommendation does not cover the new features introduced with XML Schema, such as the canonical form for the various new data types. Therefore, XML Schema itself defines a canonical form for the lexical representation of all built-in data types defined in XML Schema. 4.3.1Canonical Text Acceptable forms of text in XML documents meet the following requirements: ▪ : Canonical XML documents are always encoded in UTF-8. ▪ : All line breaks are normalized to #xA. ▪ : Character entity references are resolved—that is, they are replaced by the referenced entities. Here, the character entity euro is replaced with its definition in the ENTITY clause: | Noncanonical | Canonical | --- | | | ▪ : Canonical text does not contain the characters &, <,>, nor the carriage return character (#xD). These characters are replaced by &, <, >, and . ▪ : CDATA sections are replaced with their character content. Here, the CDATA wrapping around the if instruction is removed. By doing so, the unparsed character data becomes parsed character data, so we have to replace < with<. | Noncanonical | Canonical | --- | 4.3.2Canonical Whitespace Acceptable forms for whitespace in canonical XML documents meet the following requirements: ▪ : Whitespace outside of the document element and within start and end tags is normalized. (A character string is normalized by reducing whitespace within the string to a single whitespace character and by removing any whitespace from the beginning and end of the string.) ▪ : Attribute values are normalized; that is, whitespace within the attribute value string is reduced to a single whitespace character, and whitespace at the beginning and end of the string is removed. ▪ : All whitespace in character content is retained (excluding characters removed during line feed normalization—see above). Here, all whitespace within tags, and within the value of attribute ref, is normalized: | Noncanonical | Canonical | --- | 4.3.3Resolved References The resolution of parsed entity references involves the following steps: ▪ : Parsed entity references are replaced by the referenced entities. ▪ : Default attributes and fixed attributes are added to each element if not already present. Here, we have resolved entity legalDoc with the content of the file to which it refers. We have also included the default value of attribute review in the document content. | Noncanonical | Canonical | --- | 4.3.4Removal of Redundant Nodes In canonical XML, the XML declaration and Document Type Definition (DTD) are removed. The XML declaration is no longer necessary since the canonical document is always a standalone document in UTF-8 code. The DTD is no longer necessary, as all default and fixed values and all referenced entities have been resolved. Redundant namespace nodes are also removed. An element's namespace node is redundant when the nearest parent element has a namespace node in the node-set with the same local name and value. Here, we have removed the namespace declaration in element title because the parent element book already declared the same namespace. We have also removed the XML declaration and the DOCTYPE declaration. | Noncanonical | Canonical | --- | 4.3.5Canonical Elements The child nodes of an element (elements, attributes, processing instructions, comments, character data, unparsed and unexpanded entities, and in-scope namespaces) are ordered in the following sequence: 1. : The element itself. 2. : Namespaces: Namespaces are ordered in lexical sequence. 3. : Attributes: The attribute nodes are sorted lexicographically, with the namespace URI as the primary sort criterion and the local name as the secondary sort criterion. 4. : Child elements in the given sequence. Empty elements are expanded to start-end tag pairs. Here, we have brought namespace declarations and attributes into the correct order. We have also expanded the empty element to a start tag and end tag. | Noncanonical | Canonical | --- | 4.3.6Canonical Attributes The canonical form of an attribute consists of ▪ : a space character ▪ : the attribute's qualified name ▪ : an equality sign ▪ : a double quote ▪ : the value as a canonical string with normalized whitespace (see Section 4.3.2) ▪ : a double quote Here, we have normalized the whitespace in the attribute value, removed whitespace between element name and attribute and between equality sign and quote, and replaced the single quotes with double quotes: | Noncanonical | Canonical | --- | 4.3.7Canonical Processing Instructions Processing instruction (PI) nodes consist of ▪ : the opening PI symbol () ▪ : the PI target name of the node ▪ : a leading space and the string value if the string value is not empty ▪ : the closing PI symbol (?>) ▪ : for PIs outside the document element, a separating #xA character between processing instruction and document element Here, we have removed unnecessary whitespace: | Noncanonical | Canonical | --- | 4.3.8Canonical Comments For canonical XML without comments, all comments are removed. For canonical XML with comments, comments are normalized in the following way: ▪ : the opening comment symbol ( ▪ : the string value of the node ▪ : the closing comment symbol (–>) ▪ : for comments outside the document element, a separating #xA character between comment and document element View chapterExplore book Read full chapter URL: Book2003, Modeling Business Objects with XML SchemaBERTHOLD DAUM Chapter The PACK and UNPACK Operators I: The Single-Attribute Case 2014, Time and Relational Theory (Second Edition)C.J. Date, ... Nikos A. Lorentzos This chapter and the next build on the notions introduced in the previous chapter—viz., expanded form and collapsed form, which apply to sets of intervals as such—to define two further canonical forms, viz., unpacked and packed form, which apply to relations with zero or more attributes of some interval type. The present chapter considers what’s probably the most important special case, viz., the case of unpacking or packing a relation on the basis of exactly one interval attribute. In particular, the usefulness of the two canonical forms, and the corresponding UNPACK and PACK operators, in dealing with the sample queries from Chapter 5 is clearly demonstrated. View chapterExplore book Read full chapter URL: Book2014, Time and Relational Theory (Second Edition)C.J. Date, ... Nikos A. Lorentzos Related terms: Approximation (Algorithm) Boolean Function Feasible Solution Truth Table Minterm Linear Programming Problem Eigenvalue Eigenvector Logical Operator Canonical Model View all Topics
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https://www.quora.com/Between-ship-and-submarine-which-one-is-faster-and-consumes-less-fuel-in-transporting-goods
Between ship and submarine, which one is faster and consumes less fuel in transporting goods.? - Quora Something went wrong. Wait a moment and try again. Try again Skip to content Skip to search Sign In Travel and Transportation Naval Architecture Submarine Merchant Ships Speed Fuel Consumption Marine Engineering Naval Vessels (Naval Ship... Ships and Submarines 5 Between ship and submarine, which one is faster and consumes less fuel in transporting goods.? All related (33) Sort Recommended Chris Rentsch Chemical engineer, oil refining, lithium ion batteries · Author has 322 answers and 2.3M answer views ·9y A ship will go faster and consume less fuel than a submarine, owing to the fact that the bulk of its cross-sectional area travels through air instead of water. Air has a density that is 0.1% of water's and a viscosity that is 1% of water's. This is intuitive if you consider whether one can swim faster with or without a body board that keeps the body floating on the surface (vs 95% submerged). Upvote · 99 13 9 2 Sponsored by Grammarly 92% of professionals who use Grammarly say it has saved them time Work faster with AI, while ensuring your writing always makes the right impression. Download 999 210 Related questions More answers below Why aren't submarines and ships used for transportation? Why don't we use submarines instead of ships for transportation? What is the average fuel consumption rate of a submarine per minute? How does a submarine's depth affect its fuel consumption? How much does it cost to fill a submarine ship with fuel? Randy Topechka Locomotive Mechanic at Canadian National Railway Company (2017–present) · Author has 587 answers and 12.2M answer views ·6y Related Is a submarine faster than a ship? Short answer: Yes, Some submarines can travel faster then some ships. Long answer: as with everything, it depends. What submarine are we talking about and what ship. They both come in a wide range of speed capabilities. For example: Submarines This is the North Korean Yono class submarine. This little green midget submarine displaces 130 tons, has a crew of 2 plus 6 or 7 special forces. It can travel submerged at a speed of 8 knots. This, as I’m sure you can recognize, is an American 688 flight 3, or improved Los Angeles class nuclear powered attack submarine. While the speed of this sub is classi Continue Reading Short answer: Yes, Some submarines can travel faster then some ships. Long answer: as with everything, it depends. What submarine are we talking about and what ship. They both come in a wide range of speed capabilities. For example: Submarines This is the North Korean Yono class submarine. This little green midget submarine displaces 130 tons, has a crew of 2 plus 6 or 7 special forces. It can travel submerged at a speed of 8 knots. This, as I’m sure you can recognize, is an American 688 flight 3, or improved Los Angeles class nuclear powered attack submarine. While the speed of this sub is classified, we can assume from the speculation that it can travel submerged at a Speed of 25–30 knots. And this is the currently hailed record holder of submarine speed. This is the Soviet Project 661 Anchar Nuclear Attack / Guided missile submarine, known to NATO as the Papa class. Only one submarine of this class was built, the K-162 or K-222 as she was later redesignated. She could travel at a submerged speed of 44.7 knots. Now in our other corner, we have our ships. First one I will bring up is your general, run of the mill supertanker. This particular vessel is the MT Hellespont Alhambra. Big and bulky, she can attain a speed of 16.5 knots. Next up I have perhaps the most recognizable warship in the world, the renowned Arleigh Burke class guided missile destroyer. Possibly the most capable destroyer on the seas, she can travel at a speed in excess of 30 knots. This here now is a runner for possibly the fastest ship in the world. Ladies and gentlemen, I give you the High speed passenger ferry HSC Francisco. Powered by large aircraft style jet engines, this ferry can carry its passengers at a speed of up to 58 knots! So in the end, it depends on what submarine and what ship. That’s my input, let’s see what the others have to say Upvote · 99 66 99 12 9 3 Austin Bugden B.Sc. in Physics&Mathematics, Memorial University of Newfoundland (Graduated 2018) · Author has 8.3K answers and 13.5M answer views ·Updated 8y Related Which one has less drag force at the same volume, a submarine or a ship? Image 1: Vessels of “Similar” Displacements It’s a bit of a trick question. The short answer is volume and displacement are different things when referring to ships. Shape and length are important. Air resistance compared to water resistance is almost trivial. Subs have roughly the density of water and surface ships have a density much less than that of water. Since ships are less dense, they displace less water, and thus need to move less water out of the way for each hull length they travel, therefore less drag. If you read on, it’s not even close. For vessels of equal volume, the sub has more Continue Reading Image 1: Vessels of “Similar” Displacements It’s a bit of a trick question. The short answer is volume and displacement are different things when referring to ships. Shape and length are important. Air resistance compared to water resistance is almost trivial. Subs have roughly the density of water and surface ships have a density much less than that of water. Since ships are less dense, they displace less water, and thus need to move less water out of the way for each hull length they travel, therefore less drag. If you read on, it’s not even close. For vessels of equal volume, the sub has more drag and in fact have more in common with airplanes than surface ships. First there really isn’t anyway to say “all things being equal”. Let us say it anyway. Let’s compare two ocean race cars. (there’s a reason why the US invites these frigates to it’s “parties”) Hard to get open stats on vessel volume. Displacement and profile is easier. To illustrate this I took the Virginia Class sub and placed it next to a similar displacement vessel. I chose a Halifax Class frigate or CPF. They both have classified top speeds, but Wikipedia puts them in the same ballpark. Really I needed TWO CPFs to match the displacement of one Virginia class. Two CPF occupy a much larger volume than one Virginia class sub. The scaling in the diagram is close, not perfect. At rated ship horse power, Two CPFs have roughly 2.8 times the shaft horsepower “available” than the Virginia Class. Although I don’t have firm numbers on total volume, I think it is a fair theory that there is far more than 2.8 times the volume in two CPF’s than one Virginia Class Submarine. Image 2: Similar Displacement I’ll ignore that submarines have a fair amount of induced drag as they “fly” and push away from the surface as lifting bodies in addition to using buoyancy. One can look up the USS Thresher and thermoclines for two independent reasons why. A ship floats so doesn't need to fly. (it does squat though) So Induced drag from lift, strike one against submarine. The total net density of a displacement hull of a surface ship is much less than the density of water. To give it enough buoyancy, it only has to immerse a small fraction of its surface area in water to displace sufficient water for it to float. For a submarine it’s density the same as water. So a submarine of the same volume as a displacement hull will present 100% of it’s hull as wetted surface area. From a volumetric point of view, it’s not really that close. It’s not that simple but wetted surface area is one metric of drag. Its not that simple but that is the gross detail. It’s a bit of an apples and oranges question. A great question all the same. Thank you for asking. Upvote · 9 8 Assistant Bot · 1y When comparing ships and submarines in terms of speed and fuel consumption for transporting goods, ships are generally faster and more fuel-efficient. Speed: Ships: Typically, cargo ships travel at speeds of around 15-25 knots (17-29 mph or 28-46 km/h), depending on the type of vessel. Submarines: While military submarines can reach speeds of over 30 knots (34.5 mph or 55.5 km/h) when submerged, they are not designed for transporting goods and are usually slower when operating at cruising speeds. Additionally, civilian submarines, which are rare, are slower and less efficient for cargo transport. Continue Reading When comparing ships and submarines in terms of speed and fuel consumption for transporting goods, ships are generally faster and more fuel-efficient. Speed: Ships: Typically, cargo ships travel at speeds of around 15-25 knots (17-29 mph or 28-46 km/h), depending on the type of vessel. Submarines: While military submarines can reach speeds of over 30 knots (34.5 mph or 55.5 km/h) when submerged, they are not designed for transporting goods and are usually slower when operating at cruising speeds. Additionally, civilian submarines, which are rare, are slower and less efficient for cargo transport. Fuel Consumption: Ships: Cargo ships are designed for efficiency in transporting large quantities of goods over long distances. They typically consume less fuel per ton of cargo compared to submarines, especially when operating at optimal speeds. Submarines: Submarines consume more fuel relative to the amount of cargo they could theoretically carry, as they are designed for stealth and military operations rather than bulk transport. Conclusion: Overall, for transporting goods, conventional ships are faster and more fuel-efficient than submarines. Submarines are specialized vessels that serve different purposes, primarily military rather than commercial. Upvote · Related questions Why aren't submarines and ships used for transportation? Why don't we use submarines instead of ships for transportation? What is the average fuel consumption rate of a submarine per minute? How does a submarine's depth affect its fuel consumption? How much does it cost to fill a submarine ship with fuel? Can a ship be made that carries a submarine under it? What is the difference between a ship and a submarine? At what depth does a submarine use the least fuel per distance travelled? What distinguishes a submarine from a ship? Which types of ships are capable of prolonged submersion in water? What happens if a submarine runs out of fuel? How would it be brought back to shore for repairs, etc.? What is the difference in fuel usage per day between a submarine and a plane? What if a submarine lost their fuel while being under the sea? What is the typical range that a submarine can cover before having to be refueled? How much fuel does a container ship use at idle? What is the cost of running a small submarine for one hour? How much fuel is consumed per mile travelled in average sea conditions? Related questions Why aren't submarines and ships used for transportation? Why don't we use submarines instead of ships for transportation? What is the average fuel consumption rate of a submarine per minute? How does a submarine's depth affect its fuel consumption? How much does it cost to fill a submarine ship with fuel? Can a ship be made that carries a submarine under it? 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https://pubmed.ncbi.nlm.nih.gov/19882076/
[The role of serotonin in haemostasis] - PubMed Clipboard, Search History, and several other advanced features are temporarily unavailable. Skip to main page content An official website of the United States government Here's how you know The .gov means it’s official. Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site. The site is secure. The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely. Log inShow account info Close Account Logged in as: username Dashboard Publications Account settings Log out Access keysNCBI HomepageMyNCBI HomepageMain ContentMain Navigation Search: Search AdvancedClipboard User Guide Save Email Send to Clipboard My Bibliography Collections Citation manager Display options Display options Format Save citation to file Format: Create file Cancel Email citation Email address has not been verified. Go to My NCBI account settings to confirm your email and then refresh this page. 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Report format: Send at most: [x] Send even when there aren't any new results Optional text in email: Save Cancel Create a file for external citation management software Create file Cancel Your RSS Feed Name of RSS Feed: Number of items displayed: Create RSS Cancel RSS Link Copy Actions Cite Collections Add to Collections Create a new collection Add to an existing collection Name your collection: Name must be less than 100 characters Choose a collection: Unable to load your collection due to an error Please try again Add Cancel Permalink Permalink Copy Display options Display options Format Page navigation Title & authors Abstract Similar articles Cited by Publication types MeSH terms Substances Related information Hamostaseologie Actions Search in PubMed Search in NLM Catalog Add to Search . 2009 Nov;29(4):356-9. [The role of serotonin in haemostasis] [Article in German] D Duerschmied1,C Bode Affiliations Expand Affiliation 1 Kardiologie und Angiologie, Universitätsklinikum Freiburg, Freiburg, Germany. daniel.duerschmied@uniklinik-freiburg.de PMID: 19882076 Item in Clipboard [The role of serotonin in haemostasis] [Article in German] D Duerschmied et al. Hamostaseologie.2009 Nov. Show details Display options Display options Format Hamostaseologie Actions Search in PubMed Search in NLM Catalog Add to Search . 2009 Nov;29(4):356-9. Authors D Duerschmied1,C Bode Affiliation 1 Kardiologie und Angiologie, Universitätsklinikum Freiburg, Freiburg, Germany. daniel.duerschmied@uniklinik-freiburg.de PMID: 19882076 Item in Clipboard Cite Display options Display options Format Abstract Serotonin is transported by platelets and released upon activation. This induces constriction of injured blood vessels and enhances platelet aggregation to minimize blood loss. Consequently, serotonin receptor antagonists have been tested for their anti-ischemic potency in atherothrombotic disease. Unfortunately, the results have been contradictory. Recent murine studies found that activation of the platelet serotonin receptor induces shedding of important adhesion molecules. As a consequence, platelets lose their ability to contribute to thrombus formation and may be cleared from the circulation. Serotonin effects on platelets are not only mediated by receptor binding but also by covalently binding effector proteins (serotonylation) in the platelet cytoplasm and on the platelet surface. In conclusion, the effects of serotonin on haemostasis are complex and new antithrombotic strategies have to account for this complexity. PubMed Disclaimer Similar articles Mechanisms of platelet activation: need for new strategies to protect against platelet-mediated atherothrombosis.Jennings LK.Jennings LK.Thromb Haemost. 2009 Aug;102(2):248-57. doi: 10.1160/TH09-03-0192.Thromb Haemost. 2009.PMID: 19652875 Review. Activated platelets and atherosclerosis.Aukrust P, Halvorsen B, Ueland T, Michelsen AE, Skjelland M, Gullestad L, Yndestad A, Otterdal K.Aukrust P, et al.Expert Rev Cardiovasc Ther. 2010 Sep;8(9):1297-307. doi: 10.1586/erc.10.92.Expert Rev Cardiovasc Ther. 2010.PMID: 20828352 Review. Platelet membrane glycoproteins in haemostasis.Thomas S.Thomas S.Clin Lab. 2002;48(5-6):247-62.Clin Lab. 2002.PMID: 12071575 Review. Primary haemostasis and its assessment by laboratory tests.Reininger AJ.Reininger AJ.Hamostaseologie. 2006 Jan;26(1):42-4, 46-7.Hamostaseologie. 2006.PMID: 16444321 Platelet activation by ADP: the role of ADP antagonists.Gachet C.Gachet C.Ann Med. 2000 Dec;32 Suppl 1:15-20.Ann Med. 2000.PMID: 11209975 Review. See all similar articles Cited by Modulation of autism-associated serotonin transporters by palmitoylation: Insights into the molecular pathogenesis and targeted therapies for autism spectrum disorder.Brown CR, Foster JD.Brown CR, et al.bioRxiv [Preprint]. 2025 Mar 13:2025.03.12.642908. doi: 10.1101/2025.03.12.642908.bioRxiv. 2025.PMID: 40161745 Free PMC article.Preprint. The Role of Serotonin during Skin Healing in Post-Thermal Injury.Sadiq A, Shah A, Jeschke MG, Belo C, Qasim Hayat M, Murad S, Amini-Nik S.Sadiq A, et al.Int J Mol Sci. 2018 Mar 29;19(4):1034. doi: 10.3390/ijms19041034.Int J Mol Sci. 2018.PMID: 29596386 Free PMC article. Serotonergic Control of Metabolic Homeostasis.Wyler SC, Lord CC, Lee S, Elmquist JK, Liu C.Wyler SC, et al.Front Cell Neurosci. 2017 Sep 20;11:277. doi: 10.3389/fncel.2017.00277. eCollection 2017.Front Cell Neurosci. 2017.PMID: 28979187 Free PMC article.Review. An Insight into Recent Advances on Platelet Function in Health and Disease.Chaudhary PK, Kim S, Kim S.Chaudhary PK, et al.Int J Mol Sci. 2022 May 27;23(11):6022. doi: 10.3390/ijms23116022.Int J Mol Sci. 2022.PMID: 35682700 Free PMC article.Review. 5-HT1A Receptor Function Makes Wound Healing a Happier Process.Sadiq A, Menchetti I, Shah A, Jeschke MG, Belo C, Carlos-Alcalde W, Hayat MQ, Amini-Nik S.Sadiq A, et al.Front Pharmacol. 2018 Dec 11;9:1406. doi: 10.3389/fphar.2018.01406. eCollection 2018.Front Pharmacol. 2018.PMID: 30618734 Free PMC article. See all "Cited by" articles Publication types English Abstract Actions Search in PubMed Search in MeSH Add to Search MeSH terms Atherosclerosis / drug therapy Actions Search in PubMed Search in MeSH Add to Search Blood Platelets / physiology Actions Search in PubMed Search in MeSH Add to Search Hemostasis / physiology Actions Search in PubMed Search in MeSH Add to Search Humans Actions Search in PubMed Search in MeSH Add to Search Platelet Activation Actions Search in PubMed Search in MeSH Add to Search Platelet Aggregation Inhibitors / therapeutic use Actions Search in PubMed Search in MeSH Add to Search Receptors, Serotonin / drug effects Actions Search in PubMed Search in MeSH Add to Search Serotonin / physiology Actions Search in PubMed Search in MeSH Add to Search Substances Platelet Aggregation Inhibitors Actions Search in PubMed Search in MeSH Add to Search Receptors, Serotonin Actions Search in PubMed Search in MeSH Add to Search Serotonin Actions Search in PubMed Search in MeSH Add to Search Related information PubChem Compound PubChem Compound (MeSH Keyword) PubChem Substance [x] Cite Copy Download .nbib.nbib Format: Send To Clipboard Email Save My Bibliography Collections Citation Manager [x] NCBI Literature Resources MeSHPMCBookshelfDisclaimer The PubMed wordmark and PubMed logo are registered trademarks of the U.S. Department of Health and Human Services (HHS). 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| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- | | Table of Contents | Introduction | Home P l a n e G e o m e t r y An Adventure in Language and Logic based on THE THEORY OF PARALLEL LINESBook I. PROPOSITIONS 29, 30, and POSTULATE 5 Proposition 29 Postulate 5 versus Playfair's Axiom Proposition 30 HERE AGAIN is Proposition 27. If a straight line that meets two straight lines makes the alternate angles equal, then the two straight lines are parallel. We are about to prove Proposition 29, which is its converse: If two straight lines are parallel, then a straight line that meets them makes the alternate angles equal. To prove that the alternate angles are equal, we must have a sufficient condition for their being equal. But this proposition is the condition. Therefore we will have to prove this proposition indirectly. Now, a sufficient condition for the alternate angles not being equal is the following partial inverse to Proposition 28: If a straight line that meets two straight lines makes the interior angles on the same side less than two right angles, then those two straight lines, if extended, will meet on that same side. That is, if the angles BGH, GHD are not equal to two right angles, then the straight lines AB, CD are not parallel. The question is: Is it possible to prove that from the existing first principles? The answer, the student may be surprised to learn, is no. Mathematicians tried for over two thousand years. But from the existing first principles, no one could. It is in fact Euclid's famous, and controversial, Postulate 5. Here then is Proposition 29. It is the converse of Proposition 27 and Proposition 28 combined. The indirect proof assumes that the alternate angles are not equal. It will then turn out the straight lines are not parallel -- which by hypothesis they are! The proof will require Postulate 5. PROPOSITION 29. THEOREM | | | If two straight lines are parallel, then a straight line that meets them makes the alternate angles equal, it makes the exterior angle equal to the opposite interior angle on the same side, and it makes the interior angles on the same side equal to two right angles. | | | | Let the straight lines AB, CD be parallel, and let the straight line EF meet them;then it makes the alternate angles AGH, GHD equal; it makes the exterior angle EGB equal to the opposite interior angle GHD; and it makes the interior angles on the same side, angles BGH, GHD, equal to two right angles. | | | | | | | (We will prove only up to "makes the alternate angles equal." The rest will be left to the student. Problem 4.) | | | | For, if angle AGH is not equal to angle GHD, then one of them is greater. | | | | Let angle AGH be greater than angle GHD; | | | | and to each of them join angle BGH; | | | | then angles BGH, AGH are greater than angles BGH, GHD. | | | | But angles BGH, AGH are together equal to two right angles; | (I. 13) | | | | therefore angles BGH, GHD are together less than two right angles. | | | | But straight lines if extended from angles less than two right angles, will meet. | (Postulate 5) | | | | Therefore AB, CD, if extended, will meet. | | | | But they do not meet because by hypothesis they are parallel. | | | | Therefore angle AGH is not unequal to angle GHD; | | | | it is equal to it. | | | | Therefore, if two straight lines etc. Q.E.D. | Postulate 5 versus Playfair's Axiom In his Notes on the Elements, Todhunter observes, The theory of parallel straight lines has always been considered the great difficulty of elementary geometry, and many attempts have been made to overcome the difficulty in a better way than Euclid had done. (Elibron Classics, page 262.) He is referring of course to the fifth Postulate. Nevertheless, he continues, . . . after the axiom is once admitted, the remaining process is simple and clear. We can gain some insight into Postulate 5 in this figure. The straight line EF is perpendicular to both AB and CD -- angles BEF, EFD are two right angles. And we can see that AB, CD are therefore parallel. But if GH is any straight line that meets AB, CD, and angles EGH, GHF are together equal to two right angles, then again AB, CD are parallel. (I-28) Finally, if, when EF meets JK and CD, it makes the interior angles on the same side less than two right angles, then JK, CD are not parallel, and they will meet on that same side. That is Postulate 5. Some writers however have replaced Postulate 5 with the following, which they think is intuitively simpler: Two intersecting straight lines cannot both be parallel to a third line. That is, through the point G it is impossible that both AB and KL are parallel to CD. The student should not be fooled by the word games of the non-Euclidean geometries, which also contain the term "straight line." They are non-Euclidean because the surface in question is not a plane. The proposition above has been called Playfair's Axiom, after the 18th century English mathematician John Playfair. If we grant it in place of Postulate 5, then, in Propostion 29 we would again assume that the alternate angles AGH, GHD are not equal. We would continue: Draw a straight line KL through G that makes angle KGH equal to angle GHD. Then, since those alternate angles are equal, KL is parallel to CD. (I. 27) Therefore the two straight lines KL, AB that intersect at G are both parallel to CD, which is impossible. (Playfair's Axiom) Therefore, angles AGH, GHD are not unequal; they are equal. Q.E.D. Now, upon granting Playfair's Axiom, it is possible to prove Postulate 5 as a theorem. And conversely, if we grant Postulate 5, then we can prove Playfair's Axiom. (Problem 5.) The choice of which propositions to take as the postulates and which to take as the theorems is simply that: a matter of choice. Ultimately, the choice is aesthetic. The postulates typically are those which appear most elementary, or which make the least strain on the intellect, or which in some way are immediately satisfying. (These are aesthetic choices, because a postulate must appear that way to someone.) Thus, as Heath points out (Dover, page 313), Playfair asks us to assume what is impossibile, namely that two intersecting lines cannot both be parallel to a third, while Euclid asserts what is possible, namely that two straight lines under certain conditions will meet. The two postulates nevertheless are equivalent to one another, in the sense that we can deduce each one from the other. We see therefore that there is no unique set of postulates from which to deduce the remainder of a theory. Hence there is never a unique reason why we say that a theorem is true. For, the theorems are true whether we prove them or not. They are not true because we prove them. We prove them because they are true. PROPOSITION 30. THEOREM | | | Straight lines that are parallel to the same straight line are parallel to one another. | We leave the rest to the student. (Problem 7) Please "turn" the page and do some Problems. Continue on to the next proposition. Previous proposition Table of Contents | Introduction | Home Please make a donation to keep TheMathPage online.Even $1 will help. Copyright © 2021 Lawrence Spector Questions or comments? E-mail: teacher@themathpage.com |
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Understanding Thermodynamics: Key Concepts and Energy Laws | Course Hero AI Chat with PDF AI Homework Help Expert Help Study Resources SchoolTextbook SolutionsLiterature TitleStudy GuidesGrammar CheckerParaphraserProofreaderSpell Checker Log in Join Chapter Notes for Thermodynamics .pdf - hapter 2 Energyand... Pages 5 University Of Connecticut ME ME 2233 CountBookTarsier38 7/28/2025 Chapter Notes for Thermodynamics .pdf View full document Students also studied ### ME260HW#1 Solutions Solutions Available University Of Connecticut ME 3264 ### Lab 4 Cross-Flow Heat Exchanger_rev2.pdf Solutions Available University Of Connecticut ME 3264 ### ME2234Hwk1Solutions.pdf Solutions Available University Of Connecticut ME 2234 ### Project#3_ME3276.pdf University Of Connecticut ME 3276 ### ME260HW#2 Solutions Solutions Available University Of Connecticut ME 3264 View More hapter 2 Energyand the first law of Thermodynamics vocab Tinetic energy the quanity 1mV Kinetic energy is a scalar quality the change in kinenergy Δ KE KE KE Im CV V ravitational potential energy the quanity mgz the change in gravitational potential nergy APE PE PE mg 22 2 hermodynamic def of work work is done by a system on its surroundings if ole effect on everything external to the system couldhave been the raising of a weig ower the rate of energy transfer W F v uasiequilibrium or quasistatic process is one in which the departure from thermodynamic quilibrium is at most infinitesimal Olytropic process when pV constant or pun constant where n is constant diabatic without heat transfer ourier's law the rate of heattransfer across any plane normal to the x direction Qx is roportional to the wall area A Q KA LI efan boltzmann law the rate at which energy is emitted Qe from a surface of are Qe EOAT ewton's law of cooling the rate of energy transfer from the surface to the air an be be Qc hA TbTf rst law of thermodynamics energy is conserved nergy balance Ez E Q w or AKE Δ PE AU Q w ime rate form of the energy balance IF Q w hermodynamic cycle is a sequence of processes that begins and ends at the same tate ower cycle Waycle Qin Q out hermal efficiency n 89 efrigeration and heat pump cycles Wcycle Q out Qin efrigeration cycles 13 8 1 heat pump cycle Y Efe notes 2.1 2.6 Im V V I Rdz f mydz ex mass 1kg increases 15ms to 30ms while its elevation decreases by 10m g 9.7 Δ KE Im Cv Vi Icing 303 15 5 life I m 0.344J Δ PE mg 22 21 kg 9.7 C 10 ftp t nm 0.10kt the term work doesnot refer to what is being transferred between systems r to what is stored within systems Energy is transferred and stored when work done W O work done by the system Want to read all 5 pages? Previewing 2 of 5 pages Upload your study docs or become a member. View full document Want to read all 5 pages? Previewing 2 of 5 pages Upload your study docs or become a member. View full document End of preview Want to read all 5 pages?Upload your study docs or become a member. View full document Recently submitted questions Nedfr en fjllbrant med lutningsvinkeln 28 grader glider en skidkare som drar en pulka. Skidkaren vger 85 kg och friktionskraften p honom r 12 % av normalkraften. Pulkan vger 58 kg och friktionskraften Death of a Salesman Excerpt quiz: Question 1 (4 points) In a conversation with Linda, Willy says: "I'm tired to the death" Question 1 options: Question 2 (6 points) In a conversation with Linda, Willy Company About Us Careers Q&A Archive Responsible AI Course Hero Español Get Course Hero iOS Android Chrome Extension Tutors Study Tools AI Chat with PDF Grammar Checker Paraphraser Proofreader Spell Checker Course Hero Quizzes Help Contact Us FAQ Feedback Legal Copyright Policy Academic Integrity Our Honor Code Privacy Policy Service Terms Attributions Do Not Sell or Share My Personal Info Community Guidelines Connect with Us College Life Facebook Twitter LinkedIn YouTube Instagram Course Hero, a Learneo, Inc. business © Learneo, Inc. 2025. Course Hero is not sponsored or endorsed by any college or university.
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https://ocw.mit.edu/courses/18-05-introduction-to-probability-and-statistics-spring-2022/mit18_05_s22_lec05a.pdf
Variance; Continuous Random Variables 18.05 Spring 2022 1/21 Announcements/Agenda Announcements • None Agenda • Studio 2 comments • Variance and standard deviation for discrete variables • Calculus warmup • Continuous random variables - probability densities • Exponential distribution 2/21 Studio 2 comments • Graded studio 2 code is posted in the usual place. • Excellent job overall! • Please suppress stray printouts. We will start penalizing these. • If you used loops, please look at the solutions to see how R lets you do array operations without loops. • Expected value of payoff IS NOT payoff of expected value. Example. 𝑋 ∼ Bernoulli(𝑝) and 𝑌 = 𝑋2 + 𝑋. 𝐸[𝑋] = 𝑝, 𝐸[𝑌 ] = 2𝑝, 𝐸[𝑋]2 + 𝐸[𝑋] = 𝑝2 + 𝑝 ≠ 𝐸[𝑌 ]. 3/21 Variance and standard deviation 𝑋 a discrete random variable with mean 𝐸[𝑋] = 𝜇. • Meaning: spread of probability mass about the mean. • Definition as expectation (weighted sum): Var(𝑋) = 𝐸[(𝑋 − 𝜇)2]. • Computation as sum: 𝑛 Var(𝑋) = ∑ 𝑝(𝑥𝑖)(𝑥𝑖 − 𝜇)2. 𝑖=1 • Standard deviation 𝜎 = √Var(𝑋). Units for standard deviation = units of 𝑋. 4/21 1. Concept question: Order the variance The graphs below give the pmf for 3 random variables. 𝑥 1 2 3 4 5 (A) 𝑥 1 2 3 4 5 (B) 𝑥 1 2 3 4 5 (C) Order them by size of standard deviation from biggest to smallest. (Assume 𝑥 has the same units in all three.) 1. ABC 2. ACB 3. BAC 4. BCA 5. CAB 6. CBA 5/21 2. Concept question: Zero variance Suppose 𝑋 is a discrete random variable, True or False: If Var(𝑋) = 0 then 𝑋 is constant. 1. True 2. False 6/21 Computation from tables Example. Compute the variance and standard deviation of 𝑋. values 𝑥 1 2 3 4 5 pmf 𝑝(𝑥) 1/10 2/10 4/10 2/10 1/10 7/21 Computation from tables Example. Compute the variance and standard deviation of 𝑋. values 𝑥 1 2 3 4 5 pmf 𝑝(𝑥) 1/10 2/10 4/10 2/10 1/10 A very useful formula The following formula is often easier to use than the definition. 𝑛 Var(𝑋) = 𝐸[𝑋2] −𝐸[𝑋]2 = (∑𝑝(𝑥𝑖)𝑥2 𝑖) −𝜇2. 𝑖=1 Redo the above computation using this formula. (Written solution with posted solutions ) 7/21 3. Concept question: Standard deviation Make an intuitive guess: Which pmf has the bigger standard deviation? (Assume 𝑤 and 𝑦 have the same units.) 𝑦 𝑝(𝑦) 0 3 -3 1/2 pmf for 𝑌 𝑤 𝑝(𝑊) 10 20 30 40 50 0.1 0.2 0.4 pmf for 𝑊 1. 𝑌 2. 𝑊 8/21 3. Concept question: Standard deviation Make an intuitive guess: Which pmf has the bigger standard deviation? (Assume 𝑤 and 𝑦 have the same units.) 𝑦 𝑝(𝑦) 0 3 -3 1/2 pmf for 𝑌 𝑤 𝑝(𝑊) 10 20 30 40 50 0.1 0.2 0.4 pmf for 𝑊 1. 𝑌 2. 𝑊 Table question: make probability tables for 𝑌 and 𝑊 and compute their standard deviations. 8/21 Algebraic properties of variance If 𝑎 and 𝑏 are constants then Var(𝑎𝑋 + 𝑏) = 𝑎2 Var(𝑋), 𝜎𝑎𝑋+𝑏 = |𝑎| 𝜎𝑋. If 𝑋 and 𝑌 are independent random variables then Var(𝑋 + 𝑌) = Var(𝑋) + Var(𝑌 ). 9/21 Board questions (a) Let 𝑋 ∼ Bernoulli(𝑝). Compute Var(𝑋). (b) Let 𝑌 ∼ Bin(𝑛, 𝑝). Show Var(𝑌) = 𝑛𝑝(1 −𝑝). (c) Suppose 𝑋1, 𝑋2, … , 𝑋𝑛 are independent and all have the same standard deviation 𝜎 = 2. Let 𝑋 be the average of 𝑋1, … , 𝑋𝑛. What is the standard deviation of 𝑋? 10/21 Continuous random variables • Like discrete, except take a continuous range of values • Replace probability mass function by probability density function • Replace sums by integrals 11/21 Calculus warmup for continuous random variables 1. ∫ 𝑏 𝑓(𝑥) 𝑑𝑥 = area under the curve 𝑦 = 𝑓(𝑥). 𝑎 2. ∫ 𝑏 𝑓(𝑥) 𝑑𝑥 = ‘sum of 𝑓(𝑥) 𝑑𝑥’. 𝑎 Connection between the two views: 𝑥 𝑦 𝑎 𝑏 𝑦= 𝑓(𝑥) 𝑥 𝑦 𝑥0 𝑥1 𝑥2 𝑥𝑛 Δ𝑥 ⋯ 𝑎 𝑏 𝑦= 𝑓(𝑥) Area = 𝑓(𝑥𝑖)Δ𝑥 Area is approximately the sum of rectangles: 𝑏 𝑛 ∫ 𝑓(𝑥) 𝑑𝑥 ≈ 𝑓(𝑥1)Δ𝑥+ 𝑓(𝑥2)Δ𝑥+ … + 𝑓(𝑥𝑛)Δ𝑥 = ∑𝑓(𝑥𝑖)Δ𝑥. 𝑎 1 12/21 Continuous random variables: pdf and cdf • Continuous range of values: [0, 1], [𝑎, 𝑏], [0, ∞), (−∞, ∞). • Probability density function (pdf) 𝑓(𝑥) ≥0; 𝑃(𝑐≤𝑋≤𝑑) = ∫ 𝑑 𝑓(𝑥) 𝑑𝑥 = ‘sum’ of 𝑓(𝑥)𝑑𝑥. 𝑐 prob. Units for the pdf are (This explains the term density.) unit of x • Cumulative distribution function (cdf) 𝐹(𝑥) = 𝑃(𝑋 ≤ 𝑥) = ∫ 𝑥 𝑓(𝑡) 𝑑𝑡. −∞ 13/21 Visualization 𝑥 𝑓(𝑥) 𝑐 𝑑 𝑃(𝑐≤𝑋≤𝑑) pdf and probability 𝑥 𝑓(𝑥) 𝑥 𝐹(𝑥) = 𝑃(𝑋≤𝑥) pdf and cdf 14/21 Properties of the cdf (Same as for discrete distributions) • (Definition) 𝐹(𝑥) = 𝑃(𝑋 ≤ 𝑥) = ∫ 𝑥 𝑓(𝑢) 𝑑𝑢. • 0 ≤𝐹(𝑥) ≤1. −∞ • non-decreasing. • 0 to the left: lim 𝐹(𝑥) = 0. 𝑥→−∞ • 1 to the right: lim 𝐹(𝑥) = 1. 𝑥→∞ • 𝑃(𝑐< 𝑋≤𝑑) = 𝐹(𝑑) −𝐹(𝑐). • 𝐹 ′(𝑥) = 𝑓(𝑥). 15/21 Board questions 1. Suppose 𝑋 has range [0, 2] and pdf 𝑓(𝑥) = 𝑐 𝑥2. (a) What is the value of 𝑐? (b) Compute the cdf 𝐹 (𝑥). (c) Compute 𝑃(1 ≤𝑋 ≤2). (d) Plot the pdf and use it to illustrate part (c). 2. Suppose 𝑌 has range [0, 𝑏] and cdf 𝐹 (𝑦) = 𝑦2/9. (a) What is 𝑏? (b) Find the pdf of 𝑌 . 16/21 4. Discussion questions Suppose 𝑋 is a continuous random variable. (a) If the pdf of 𝑋 is 𝑓(𝑥) can there be an 𝑥 where 𝑓(𝑥) = 10? 17/21 4. Discussion questions Suppose 𝑋 is a continuous random variable. (a) If the pdf of 𝑋 is 𝑓(𝑥) can there be an 𝑥 where 𝑓(𝑥) = 10? (b) What is 𝑃(𝑋 = 𝑎)? 17/21 4. Discussion questions Suppose 𝑋 is a continuous random variable. (a) If the pdf of 𝑋 is 𝑓(𝑥) can there be an 𝑥 where 𝑓(𝑥) = 10? (b) What is 𝑃(𝑋 = 𝑎)? (c) Does 𝑃(𝑋 = 𝑎) = 0 mean 𝑋 never equals 𝑎? 17/21 Discussion questions Which of the following are graphs of valid cumulative distribution functions? 𝑥 −4 −2 2 4 −0.5 1 A. 𝑥 −4 −2 2 4 −0.5 1 B. 𝑥 −4 −2 2 4 −0.5 1 C. 𝑥 −4 −2 2 4 −0.5 1 D. 18/21 Exponential Random Variables Parameter: 𝜆 (called the rate parameter). Range: [0, ∞). Notation: exponential(𝜆) or exp(𝜆). Density: 𝑓(𝑥) = 𝜆e−𝜆𝑥 for 0 ≤ 𝑥. Models: Waiting time 𝑥 𝑃(3 < 𝑋< 7) 2 4 6 8 10 12 14 16 0.1 𝑓(𝑥) = 𝜆e−𝜆𝑥 𝑥 𝐹(𝑥) = 1 −e−𝜆𝑥 2 4 6 8 10 12 14 16 1 Continuous analogue of geometric distribution! 19/21 Board question I’ve noticed that taxis drive past 77 Mass. Ave. on the average of once every 10 minutes. Suppose time spent waiting for a taxi is modeled by an exponential random variable 𝑋 ∼ Exponential(1/10); 𝑓(𝑥) = 10 1 e−𝑥/10 (a) Sketch the pdf of this distribution (b) Shade the region which represents the probability of waiting between 3 and 7 minutes (c) Compute the probability of waiting between between 3 and 7 minutes for a taxi (d) Compute and sketch the cdf. 20/21 MIT OpenCourseWare 18.05 Introduction to Probability and Statistics Spring 2022 For information about citing these materials or our Terms of Use, visit: 21/21
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Please get in touch with us Vedantu Store Maths Tangent Forms in Parabola: Concepts, Formulas & Examples Tangent Forms in Parabola: Concepts, Formulas & Examples Reviewed by: Rama Sharma Download PDF NCERT Solutions NCERT Solutions for Class 12 NCERT Solutions for Class 11 NCERT Solutions for Class 10 NCERT Solutions for class 9 NCERT Solutions for class 8 NCERT Solutions for class 7 NCERT Solutions for class 6 NCERT Solutions for class 5 NCERT Solutions for class 4 NCERT Solutions for Class 3 NCERT Solutions for Class 2 NCERT Solutions for Class 1 CBSE CBSE class 3 CBSE class 4 CBSE class 5 CBSE class 6 CBSE class 7 CBSE class 8 CBSE class 9 CBSE class 10 CBSE class 11 CBSE class 12 NCERT CBSE Study Material CBSE Sample Papers CBSE Syllabus CBSE Previous Year Question Paper CBSE Important Questions Marking Scheme Textbook Solutions RD Sharma Solutions Lakhmir Singh Solutions HC Verma Solutions TS Grewal Solutions DK Goel Solutions NCERT Exemplar Solutions CBSE Notes CBSE Notes for class 12 CBSE Notes for class 11 CBSE Notes for class 10 CBSE Notes for class 9 CBSE Notes for class 8 CBSE Notes for class 7 CBSE Notes for class 6 How to Identify and Write Tangent Equations for Any Parabola A tangent to a parabola is only possible when the condition of tangency is satisfied. Different types of parabolas have different conditions of tangency. To represent a tangent to a parabola, there are three forms: Point, Slope and Parametric Forms. One needs to solve the equation of tangent and parabola, try to eliminate the variable, and use the discriminant rule b 2=4 a c if the equation of a parabola is in a general form. In this article, we will understand the concept of a tangent to a parabola and the different forms of the parabola. Solve some questions regarding the different forms of a tangent to a parabola. Different Types of Parabolas and Their Graphical Representation There are several parabolas. The first type of parabola is when the origin of the parabola is at the origin (0,0), and the other type is the general parabola, where the centre of the parabola is anywhere. Upward parabola (Faces upward) Downward parabola (Faces downward) Right parabola (Faces right) Left parabola (Faces left) General parabola Starting with the common standard upward parabola, The equation of an upward parabola is x 2=4 a y, where a is the distance of focus from the origin. The graph of the upward parabola is as follows, Graphical representation of upward parabola. The equation of the right parabola is y 2=4 a x. The graph of the right parabola is as follows, Graphical representation of right parabola. The equation of the left parabola is y 2=−4 a x. The graph of the left parabola is as follows, Graphical representation of left parabola. The equation of the downward parabola is x 2=−4 a y. The graph of the left parabola is as follows, Graphical representation of downward parabola. The equation parabola at center (h,k) will be y=a(x−h)2+k. Condition of Tangency of a Parabola It is necessary to adhere to the tangency condition to draw a tangent to a parabola. Finding the tangent to a parabola requires adhering to three different sorts of conditions, especially for right, upward, and parametric forms. The condition of tangency are as follows: The line y=m x+c is a tangent to a parabola y 2=4 a x only if c=a m. The line y=m x+c is a tangent to a parabola x 2=4 a y only if c=−a m 2. The line x cos⁡θ+y sin⁡θ=p is a tangent to a parabola y 2=4 a x only if sin⁡2 θ+p cos⁡θ=0. Various Forms of Tangent to a Parabola There are various forms of the tangent to a parabola. If the equation of a parabola is given in the standard form. Then, the equation of a tangent to parabola can be easily found by following the given steps: First, you need to find the point of intersection of the parabola and tangent and the slope of the tangent. The slope of the tangent is given by differentiating the equation of a parabola and putting the point of intersection on the derivative. Then, use the Point-slope form of a line to write the equation of tangent. There are three forms of a tangent to a parabola, Slope form, Point form, and Parametric form. Slope form of Tangent to a Parabola: In slope forms, the equation of tangent must be written in the slope-intercept form of a line. The line y=m x+a m is a tangent to a parabola y 2=4 a x. Similarly, if the parabola is x 2=4 a y, then the equation of tangent will be y=m x−a m 2. Point Form of Tangent to Parabola: In point form, the equation of a tangent to a parabola y 2=4 a x at some point (x 1,y 1) is given by y y 1=2 a(x+x 1). The parametric form of Tangent to a Parabola: In parametric form, the equation of a tangent to parabola y 2=4 a x at (a t 2,2 a t) is given by t y=x+a t 2. Example: Write the equation tangent to a parabola with equation y 2=16 x at (1,4)in slope form. Solution: Comparing the equation of the parabola with the standard equation and differentiating the equation to get the slope of the line: By comparing the equation with the standard equation, we get y 2=4×4 x a=4 Differentiating the curve equation, d(y 2)d x=d(16 x)d x 2 y d y d x=16 d y d x=8 y d y d x(1,4)=8 4 m=2 Now, using the condition of tangency of the parabola, c=a m c=4 2 c=2 So, the equation of a tangent to parabola in slope form is y=2 x+2. Point of Contact of Tangent and Parabola The point of contact between the tangent and parabola is a point where the tangent and curve meet. The point of contact is easily given by following some basic steps: First, you have to solve the tangent equation and a parabola equation. The value of x and y is the point of contact. The point of contact of tangent y=m x+a m right parabola y 2=4 a x is (a m 2,2 a m). The point of contact of tangent y=m x−a m 2 right parabola x 2=4 a y is (2 a m,a m 2). Where a is the length of focus and m is the slope of the tangent. Example: Verify the condition of tangency and find the point of contact of tangent 3 x+4 y=5 to a parabola 80 y=−9 x 2. Solution: First, verify the condition of tangency and then find the point of contact: Finding the value of a, 9 x 2=−80 y x 2=−80 9 y x 2=−4×20 9 y Hence, the value of a is 20 9. Now, finding the intercept and slope of a given tangent by writing it in slope-intercept form, 3 x+4 y=5 4 y=−3 x+5 y=−3 4 x+5 4 Hence, the slope of the tangent is −3 4 and the intercept is 5 4. The condition of tangency is c=a m 2, putting the values. 5 4=20 9×(−3 4)2 5 4=20 9×9 16 5 4=5 4 Hence, the condition of tangency is verified. Now, the point of contact of a tangent to the parabola is given as (2 a m,a m 2). Putting the values, (2×−20 9×−3 4,−20 9(−3 4)2)=(10 3,−5 4) Hence, the point of contact of parabola and tangent is (10 3,−5 4). Solved Problems Q.1 Find the equation of tangent to parabola x=14 t;y=7 t 2 at t=3. Ans. Given: The Parametric equation of parabola is x=14 t;y=7 t 2. To find: The equation of tangent. By looking at the equation of the parabola, it is clear that the parabola is upward. We know that the condition of tangency of an upward parabola is c=−a m 2. By comparing the equation of parabola with standard equation x=2 a t;y=a t 2, we get a=7 The point of contact of tangent and parabola is (a t 2,2 a t). (a t 2,2 a t)=(7⋅3 2,2⋅7⋅3)=(63,42) Now, using the parametric form of tangent to a parabola t y=x+a t 2, t y=x+a t 2 3 y=x+7(3)2 3 y=x+63 y=1 3 x+21 Hence, the equation of tangent is y=1 3 x+21. Q.2 Find the point of contact of tangent y=5 x+4 5 to a parabola y 2=20 x. Ans. Given: The equation of parabola is y 2=20 x and the equation of tangent is y=5 x+4 5. To find: The point of contact of tangent and parabola. The point of contact of a tangent to a parabola is given by (a m 2,2 a m). By comparing the equations with standard forms, we get a=4 m=5 Putting the values in standard point of contact, (a m 2,2 a m)=(4 5 2,2(4)5)=(4 25,8 5) Hence, the point of contact of tangent and parabola is (4 25,8 5). Summary The tangent to a parabola can be written in point form, slope form and parametric form. In point slope-form, the line y=m x+a m is a tangent to a parabola y 2=4 a x. Similarly, if the parabola is x 2=4 a y then the equation of tangent will be y=m x−a m 2. In point form, the tangent equation to a parabola y 2=4 a x at some point (x 1,y 1) is given by y y 1=2 a(x+x 1) and in parametric form, the equation of a tangent to parabola y 2=4 a x at (a t 2,2 a t) is given by t y=x+a t 2. Each form of the equation of tangent has some advantages. If the equation is given in a parametric form and also some value is given as a parameter, then using the parametric form of a tangent, the equation of tangent can be written easily. If a point is given, then the point form of the tangent to the parabola easily gives the equation. Practice Questions Verify the condition of tangency for parabola y=36 x 2 and the tangent y+9=x. Ans. The condition can be verified by finding the value of slope and focal distance. Find the equation of a tangent to a parabola y=56 x 2 with a slope equal to 7,in slope form. Ans. y=7x-686 List of Related Articles What is Parabola? Equation of tangent to a parabola. 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No, a tangent to any curve can intersect the curve at only one point. Can tangent lines cross through the graphs or not? A tangent line only appears at one location on a graph, and that location's slope is the same as the graph's overall slope. Visually, the graph at x = a bounces off the tangent line to f at a: Some points on the graph may also have tangent lines that pass through them. What is the tangent to a parabola at its vertex in slope form? As there will be either a maximum or a minimum at the parabola's vertex, the tangent at that point will be a straight line. The equation will be y=y o in slope form. 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Art of Problem Solving: Counting Pairs Art of Problem Solving 103000 subscribers 121 likes Description 24688 views Posted: 28 Dec 2011 Art of Problem Solving's Richard Rusczyk explains how to count pairs of people. This video is part of our AoPS curriculum. Take your math skills to the next level with our advanced materials: 📚 AoPS Prealgebra Textbook: 🖥️ AoPS Prealgebra 1 Course (Textbook Chapters 1-7): 🖥️ AoPS Prealgebra 2 Course (Textbook Chapters 8-15): More interested in counting? Checkout our Introduction to Counting & Probability materials: 📚 AoPS Introduction to Counting & Probability Textbook: 🖥️ AoPS Introduction to Counting & Probability Course: 🔔 Subscribe to our channel for more engaging math videos and updates Transcript:
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79 ENGINEERING ECONOMICS Factor Name Converts Symbol Formula Single Payment Compound Amount to F given P (F/P, i%, n) (1 + i)n Single Payment Present Worth to P given F (P/F, i%, n) (1 + i) –n Uniform Series Sinking Fund to A given F (A/F, i%, n) ( ) 1 1 − + n i i Capital Recovery to A given P (A/P, i%, n) ( ) ( ) 1 1 1 − + + n n i i i Uniform Series Compound Amount to F given A (F/A, i%, n) ( ) i i n 1 1 − + Uniform Series Present Worth to P given A (P/A, i%, n) ( ) ( )n n i i i + − + 1 1 1 Uniform Gradient Present Worth to P given G (P/G, i%, n) ( ) ( ) ( )n n n i i n i i i + − + − + 1 1 1 1 2 Uniform Gradient † Future Worth to F given G (F/G, i%, n) ( ) i n i i n − − + 2 1 1 Uniform Gradient ‡ Uniform Series to A given G (A/G, i%, n) ( ) 1 1 1 − + − n i n i NOMENCLATURE AND DEFINITIONS A.......... Uniform amount per interest period B.......... Benefit BV ....... Book Value C.......... Cost d .......... Combined interest rate per interest period Dj......... Depreciation in year j F.......... Future worth, value, or amount f ........... General inflation rate per interest period G ......... Uniform gradient amount per interest period i ........... Interest rate per interest period ie.......... Annual effective interest rate m ......... Number of compounding periods per year n .......... Number of compounding periods; or the expected life of an asset P.......... Present worth, value, or amount r........... Nominal annual interest rate Sn......... Expected salvage value in year n Subscripts j ........... at time j n .......... at time n ........ P/G = (F/G)/(F/P) = (P/A) × (A/G) † .......... F/G = (F/A – n)/i = (F/A) × (A/G) ‡ .......... A/G = [1 – n(A/F)]/i NON-ANNUAL COMPOUNDING 1 1 −     + = m e m r i Discount Factors for Continuous Compounding (n is the number of years) (F/P, r%, n) = er n (P/F, r%, n) = e–r n (A/F, r%, n) = 1 1 − − n r r e e (F/A, r%, n) = 1 1 − − r n r e e (A/P, r%, n) = n r r e e − − − 1 1 (P/A, r%, n) = 1 1 − − − r n r e e BOOK VALUE BV = initial cost – Σ Dj ENGINEERING ECONOMICS (continued) 80 DEPRECIATION Straight Line n S C D n j − = Accelerated Cost Recovery System (ACRS) Dj = (factor) C A table of modified factors is provided below. CAPITALIZED COSTS Capitalized costs are present worth values using an assumed perpetual period of time. Capitalized Costs = P = i A BONDS Bond Value equals the present worth of the payments the purchaser (or holder of the bond) receives during the life of the bond at some interest rate i. Bond Yield equals the computed interest rate of the bond value when compared with the bond cost. RATE-OF-RETURN The minimum acceptable rate-of-return is that interest rate that one is willing to accept, or the rate one desires to earn on investments. The rate-of-return on an investment is the interest rate that makes the benefits and costs equal. BREAK-EVEN ANALYSIS By altering the value of any one of the variables in a situation, holding all of the other values constant, it is possible to find a value for that variable that makes the two alternatives equally economical. This value is the break-even point. Break-even analysis is used to describe the percentage of capacity of operation for a manufacturing plant at which income will just cover expenses. The payback period is the period of time required for the profit or other benefits of an investment to equal the cost of the investment. INFLATION To account for inflation, the dollars are deflated by the general inflation rate per interest period f, and then they are shifted over the time scale using the interest rate per interest period i. Use a combined interest rate per interest period d for computing present worth values P and Net P. The formula for d is d = i + f + (i × f) BENEFIT-COST ANALYSIS In a benefit-cost analysis, the benefits B of a project should exceed the estimated costs C. B – C ≥ 0, or B/C ≥ 1 MODIFIED ACRS FACTORS Recovery Period (Years) 3 5 7 10 Year Recovery Rate (Percent) 1 33.3 20.0 14.3 10.0 2 44.5 32.0 24.5 18.0 3 14.8 19.2 17.5 14.4 4 7.4 11.5 12.5 11.5 5 11.5 8.9 9.2 6 5.8 8.9 7.4 7 8.9 6.6 8 4.5 6.6 9 6.5 10 6.5 11 3.3 ENGINEERING ECONOMICS (continued) 81 Factor Table - i = 0.50% n P/F P/A P/G F/P F/A A/P A/F A/G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 30 40 50 60 100 0.9950 0.9901 0.9851 0.9802 0.9754 0.9705 0.9657 0.9609 0.9561 0.9513 0.9466 0.9419 0.9372 0.9326 0.9279 0.9233 0.9187 0.9141 0.9096 0.9051 0.9006 0.8961 0.8916 0.8872 0.8828 0.8610 0.8191 0.7793 0.7414 0.6073 0.9950 1.9851 2.9702 3.9505 4.9259 5.8964 6.8621 7.8230 8.7791 9.7304 10.6770 11.6189 12.5562 13.4887 14.4166 15.3399 16.2586 17.1728 18.0824 18.9874 19.8880 20.7841 21.6757 22.5629 23.4456 27.7941 36.1722 44.1428 51.7256 78.5426 0.0000 0.9901 2.9604 5.9011 9.8026 14.6552 20.4493 27.1755 34.8244 43.3865 52.8526 63.2136 74.4602 86.5835 99.5743 113.4238 128.1231 143.6634 160.0360 177.2322 195.2434 214.0611 233.6768 254.0820 275.2686 392.6324 681.3347 1,035.6966 1,448.6458 3,562.7934 1.0050 1.0100 1.0151 1.0202 1.0253 1.0304 1.0355 1.0407 1.0459 1.0511 1.0564 1.0617 1.0670 1.0723 1.0777 1.0831 1.0885 1.0939 1.0994 1.1049 1.1104 1.1160 1.1216 1.1272 1.1328 1.1614 1.2208 1.2832 1.3489 1.6467 1.0000 2.0050 3.0150 4.0301 5.0503 6.0755 7.1059 8.1414 9.1821 10.2280 11.2792 12.3356 13.3972 14.4642 15.5365 16.6142 17.6973 18.7858 19.8797 20.9791 22.0840 23.1944 24.3104 25.4320 26.5591 32.2800 44.1588 56.6452 69.7700 129.3337 1.0050 0.5038 0.3367 0.2531 0.2030 0.1696 0.1457 0.1278 0.1139 0.1028 0.0937 0.0861 0.0796 0.0741 0.0694 0.0652 0.0615 0.0582 0.0553 0.0527 0.0503 0.0481 0.0461 0.0443 0.0427 0.0360 0.0276 0.0227 0.0193 0.0127 1.0000 0.4988 0.3317 0.2481 0.1980 0.1646 0.1407 0.1228 0.1089 0.0978 0.0887 0.0811 0.0746 0.0691 0.0644 0.0602 0.0565 0.0532 0.0503 0.0477 0.0453 0.0431 0.0411 0.0393 0.0377 0.0310 0.0226 0.0177 0.0143 0.0077 0.0000 0.4988 0.9967 1.4938 1.9900 2.4855 2.9801 3.4738 3.9668 4.4589 4.9501 5.4406 5.9302 6.4190 6.9069 7.3940 7.8803 8.3658 8.8504 9.3342 9.8172 10.2993 10.7806 11.2611 11.7407 14.1265 18.8359 23.4624 28.0064 45.3613 Factor Table - i = 1.00% n P/F P/A P/G F/P F/A A/P A/F A/G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 30 40 50 60 100 0.9901 0.9803 0.9706 0.9610 0.9515 0.9420 0.9327 0.9235 0.9143 0.9053 0.8963 0.8874 0.8787 0.8700 0.8613 0.8528 0.8444 0.8360 0.8277 0.8195 0.8114 0.8034 0.7954 0.7876 0.7798 0.7419 0.6717 0.6080 0.5504 0.3697 0.9901 1.9704 2.9410 3.9020 4.8534 5.7955 6.7282 7.6517 8.5650 9.4713 10.3676 11.2551 12.1337 13.0037 13.8651 14.7179 15.5623 16.3983 17.2260 18.0456 18.8570 19.6604 20.4558 21.2434 22.0232 25.8077 32.8347 39.1961 44.9550 63.0289 0.0000 0.9803 2.9215 5.8044 9.6103 14.3205 19.9168 26.3812 33.6959 41.8435 50.8067 60.5687 71.1126 82.4221 94.4810 107.2734 120.7834 134.9957 149.8950 165.4664 181.6950 198.5663 216.0660 234.1800 252.8945 355.0021 596.8561 879.4176 1,192.8061 2,605.7758 1.0100 1.0201 1.0303 1.0406 1.0510 1.0615 1.0721 1.0829 1.0937 1.1046 1.1157 1.1268 1.1381 1.1495 1.1610 1.1726 1.1843 1.1961 1.2081 1.2202 1.2324 1.2447 1.2572 1.2697 1.2824 1.3478 1.4889 1.6446 1.8167 2.7048 1.0000 2.0100 3.0301 4.0604 5.1010 6.1520 7.2135 8.2857 9.3685 10.4622 11.5668 12.6825 13.8093 14.9474 16.0969 17.2579 18.4304 19.6147 20.8109 22.0190 23.2392 24.4716 25.7163 26.9735 28.2432 34.7849 48.8864 64.4632 81.6697 170.4814 1.0100 0.5075 0.3400 0.2563 0.2060 0.1725 0.1486 0.1307 0.1167 0.1056 0.0965 0.0888 0.0824 0.0769 0.0721 0.0679 0.0643 0.0610 0.0581 0.0554 0.0530 0.0509 0.0489 0.0471 0.0454 0.0387 0.0305 0.0255 0.0222 0.0159 1.0000 0.4975 0.3300 0.2463 0.1960 0.1625 0.1386 0.1207 0.1067 0.0956 0.0865 0.0788 0.0724 0.0669 0.0621 0.0579 0.0543 0.0510 0.0481 0.0454 0.0430 0.0409 0.0389 0.0371 0.0354 0.0277 0.0205 0.0155 0.0122 0.0059 0.0000 0.4975 0.9934 1.4876 1.9801 2.4710 2.9602 3.4478 3.9337 4.4179 4.9005 5.3815 5.8607 6.3384 6.8143 7.2886 7.7613 8.2323 8.7017 9.1694 9.6354 10.0998 10.5626 11.0237 11.4831 13.7557 18.1776 22.4363 26.5333 41.3426 ENGINEERING ECONOMICS (continued) 82 Factor Table - i = 1.50% n P/F P/A P/G F/P F/A A/P A/F A/G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 30 40 50 60 100 0.9852 0.9707 0.9563 0.9422 0.9283 0.9145 0.9010 0.8877 0.8746 0.8617 0.8489 0.8364 0.8240 0.8118 0.7999 0.7880 0.7764 0.7649 0.7536 0.7425 0.7315 0.7207 0.7100 0.6995 0.6892 0.6398 0.5513 0.4750 0.4093 0.2256 0.9852 1.9559 2.9122 3.8544 4.7826 5.6972 6.5982 7.4859 8.3605 9.2222 10.0711 10.9075 11.7315 12.5434 13.3432 14.1313 14.9076 15.6726 16.4262 17.1686 17.9001 18.6208 19.3309 20.0304 20.7196 24.0158 29.9158 34.9997 39.3803 51.6247 0.0000 0.9707 2.8833 5.7098 9.4229 13.9956 19.4018 26.6157 32.6125 40.3675 48.8568 58.0571 67.9454 78.4994 89.6974 101.5178 113.9400 126.9435 140.5084 154.6154 169.2453 184.3798 200.0006 216.0901 232.6310 321.5310 524.3568 749.9636 988.1674 1,937.4506 1.0150 1.0302 1.0457 1.0614 1.0773 1.0934 1.1098 1.1265 1.1434 1.1605 1.1779 1.1956 1.2136 1.2318 1.2502 1.2690 1.2880 1.3073 1.3270 1.3469 1.3671 1.3876 1.4084 1.4295 1.4509 1.5631 1.8140 2.1052 2.4432 4.4320 1.0000 2.0150 3.0452 4.0909 5.1523 6.2296 7.3230 8.4328 9.5593 10.7027 11.8633 13.0412 14.2368 15.4504 16.6821 17.9324 19.2014 20.4894 21.7967 23.1237 24.4705 25.8376 27.2251 28.6335 30.0630 37.5387 54.2679 73.6828 96.2147 228.8030 1.0150 0.5113 0.3434 0.2594 0.2091 0.1755 0.1516 0.1336 0.1196 0.1084 0.0993 0.0917 0.0852 0.0797 0.0749 0.0708 0.0671 0.0638 0.0609 0.0582 0.0559 0.0537 0.0517 0.0499 0.0483 0.0416 0.0334 0.0286 0.0254 0.0194 1.0000 0.4963 0.3284 0.2444 0.1941 0.1605 0.1366 0.1186 0.1046 0.0934 0.0843 0.0767 0.0702 0.0647 0.0599 0.0558 0.0521 0.0488 0.0459 0.0432 0.0409 0.0387 0.0367 0.0349 0.0333 0.0266 0.0184 0.0136 0.0104 0.0044 0.0000 0.4963 0.9901 1.4814 1.9702 2.4566 2.9405 3.4219 3.9008 4.3772 4.8512 5.3227 5.7917 6.2582 6.7223 7.1839 7.6431 8.0997 8.5539 9.0057 9.4550 9.9018 10.3462 10.7881 11.2276 13.3883 17.5277 21.4277 25.0930 37.5295 Factor Table - i = 2.00% n P/F P/A P/G F/P F/A A/P A/F A/G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 30 40 50 60 100 0.9804 0.9612 0.9423 0.9238 0.9057 0.8880 0.8706 0.8535 0.8368 0.8203 0.8043 0.7885 0.7730 0.7579 0.7430 0.7284 0.7142 0.7002 0.6864 0.6730 0.6598 0.6468 0.6342 0.6217 0.6095 0.5521 0.4529 0.3715 0.3048 0.1380 0.9804 1.9416 2.8839 3.8077 4.7135 5.6014 6.4720 7.3255 8.1622 8.9826 9.7868 10.5753 11.3484 12.1062 12.8493 13.5777 14.2919 14.9920 15.6785 16.3514 17.0112 17.6580 18.2922 18.9139 19.5235 22.3965 27.3555 31.4236 34.7609 43.0984 0.0000 0.9612 2.8458 5.6173 9.2403 13.6801 18.9035 24.8779 31.5720 38.9551 46.9977 55.6712 64.9475 74.7999 85.2021 96.1288 107.5554 119.4581 131.8139 144.6003 157.7959 171.3795 185.3309 199.6305 214.2592 291.7164 461.9931 642.3606 823.6975 1,464.7527 1.0200 1.0404 1.0612 1.0824 1.1041 1.1262 1.1487 1.1717 1.1951 1.2190 1.2434 1.2682 1.2936 1.3195 1.3459 1.3728 1.4002 1.4282 1.4568 1.4859 1.5157 1.5460 1.5769 1.6084 1.6406 1.8114 2.2080 2.6916 3.2810 7.2446 1.0000 2.0200 3.0604 4.1216 5.2040 6.3081 7.4343 8.5830 9.7546 10.9497 12.1687 13.4121 14.6803 15.9739 17.2934 18.6393 20.0121 21.4123 22.8406 24.2974 25.7833 27.2990 28.8450 30.4219 32.0303 40.5681 60.4020 84.5794 114.0515 312.2323 1.0200 0.5150 0.3468 0.2626 0.2122 0.1785 0.1545 0.1365 0.1225 0.1113 0.1022 0.0946 0.0881 0.0826 0.0778 0.0737 0.0700 0.0667 0.0638 0.0612 0.0588 0.0566 0.0547 0.0529 0.0512 0.0446 0.0366 0.0318 0.0288 0.0232 1.0000 0.4950 0.3268 0.2426 0.1922 0.1585 0.1345 0.1165 0.1025 0.0913 0.0822 0.0746 0.0681 0.0626 0.0578 0.0537 0.0500 0.0467 0.0438 0.0412 0.0388 0.0366 0.0347 0.0329 0.0312 0.0246 0.0166 0.0118 0.0088 0.0032 0.0000 0.4950 0.9868 1.4752 1.9604 2.4423 2.9208 3.3961 3.8681 4.3367 4.8021 5.2642 5.7231 6.1786 6.6309 7.0799 7.5256 7.9681 8.4073 8.8433 9.2760 9.7055 10.1317 10.5547 10.9745 13.0251 16.8885 20.4420 23.6961 33.9863 ENGINEERING ECONOMICS (continued) 83 Factor Table - i = 4.00% n P/F P/A P/G F/P F/A A/P A/F A/G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 30 40 50 60 100 0.9615 0.9246 0.8890 0.8548 0.8219 0.7903 0.7599 0.7307 0.7026 0.6756 0.6496 0.6246 0.6006 0.5775 0.5553 0.5339 0.5134 0.4936 0.4746 0.4564 0.4388 0.4220 0.4057 0.3901 0.3751 0.3083 0.2083 0.1407 0.0951 0.0198 0.9615 1.8861 2.7751 3.6299 4.4518 5.2421 6.0021 6.7327 7.4353 8.1109 8.7605 9.3851 9.9856 10.5631 11.1184 11.6523 12.1657 12.6593 13.1339 13.5903 14.0292 14.4511 14.8568 15.2470 15.6221 17.2920 19.7928 21.4822 22.6235 24.5050 0.0000 0.9246 2.7025 5.2670 8.5547 12.5062 17.0657 22.1806 27.8013 33.8814 40.3772 47.2477 54.4546 61.9618 69.7355 77.7441 85.9581 94.3498 102.8933 111.5647 120.3414 129.2024 138.1284 147.1012 156.1040 201.0618 286.5303 361.1638 422.9966 563.1249 1.0400 1.0816 1.1249 1.1699 1.2167 1.2653 1.3159 1.3686 1.4233 1.4802 1.5395 1.6010 1.6651 1.7317 1.8009 1.8730 1.9479 2.0258 2.1068 2.1911 2.2788 2.3699 2.4647 2.5633 2.6658 3.2434 4.8010 7.1067 10.5196 50.5049 1.0000 2.0400 3.1216 4.2465 5.4163 6.6330 7.8983 9.2142 10.5828 12.0061 13.4864 15.0258 16.6268 18.2919 20.0236 21.8245 23.6975 25.6454 27.6712 29.7781 31.9692 34.2480 36.6179 39.0826 41.6459 56.0849 95.0255 152.6671 237.9907 1,237.6237 1.0400 0.5302 0.3603 0.2755 0.2246 0.1908 0.1666 0.1485 0.1345 0.1233 0.1141 0.1066 0.1001 0.0947 0.0899 0.0858 0.0822 0.0790 0.0761 0.0736 0.0713 0.0692 0.0673 0.0656 0.0640 0.0578 0.0505 0.0466 0.0442 0.0408 1.0000 0.4902 0.3203 0.2355 0.1846 0.1508 0.1266 0.1085 0.0945 0.0833 0.0741 0.0666 0.0601 0.0547 0.0499 0.0458 0.0422 0.0390 0.0361 0.0336 0.0313 0.0292 0.0273 0.0256 0.0240 0.0178 0.0105 0.0066 0.0042 0.0008 0.0000 0.4902 0.9739 1.4510 1.9216 2.3857 2.8433 3.2944 3.7391 4.1773 4.6090 5.0343 5.4533 5.8659 6.2721 6.6720 7.0656 7.4530 7.8342 8.2091 8.5779 8.9407 9.2973 9.6479 9.9925 11.6274 14.4765 16.8122 18.6972 22.9800 Factor Table - i = 6.00% n P/F P/A P/G F/P F/A A/P A/F A/G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 30 40 50 60 100 0.9434 0.8900 0.8396 0.7921 0.7473 0.7050 0.6651 0.6274 0.5919 0.5584 0.5268 0.4970 0.4688 0.4423 0.4173 0.3936 0.3714 0.3505 0.3305 0.3118 0.2942 0.2775 0.2618 0.2470 0.2330 0.1741 0.0972 0.0543 0.0303 0.0029 0.9434 1.8334 2.6730 3.4651 4.2124 4.9173 5.5824 6.2098 6.8017 7.3601 7.8869 8.3838 8.8527 9.2950 9.7122 10.1059 10.4773 10.8276 11.1581 11.4699 11.7641 12.0416 12.3034 12.5504 12.7834 13.7648 15.0463 15.7619 16.1614 16.6175 0.0000 0.8900 2.5692 4.9455 7.9345 11.4594 15.4497 19.8416 24.5768 29.6023 34.8702 40.3369 45.9629 51.7128 57.5546 63.4592 69.4011 75.3569 81.3062 87.2304 93.1136 98.9412 104.7007 110.3812 115.9732 142.3588 185.9568 217.4574 239.0428 272.0471 1.0600 1.1236 1.1910 1.2625 1.3382 1.4185 1.5036 1.5938 1.6895 1.7908 1.8983 2.0122 2.1329 2.2609 2.3966 2.5404 2.6928 2.8543 3.0256 3.2071 3.3996 3.6035 3.8197 4.0489 4.2919 5.7435 10.2857 18.4202 32.9877 339.3021 1.0000 2.0600 3.1836 4.3746 5.6371 6.9753 8.3938 9.8975 11.4913 13.1808 14.9716 16.8699 18.8821 21.0151 23.2760 25.6725 28.2129 30.9057 33.7600 36.7856 39.9927 43.3923 46.9958 50.8156 54.8645 79.0582 154.7620 290.3359 533.1282 5,638.3681 1.0600 0.5454 0.3741 0.2886 0.2374 0.2034 0.1791 0.1610 0.1470 0.1359 0.1268 0.1193 0.1130 0.1076 0.1030 0.0990 0.0954 0.0924 0.0896 0.0872 0.0850 0.0830 0.0813 0.0797 0.0782 0.0726 0.0665 0.0634 0.0619 0.0602 1.0000 0.4854 0.3141 0.2286 0.1774 0.1434 0.1191 0.1010 0.0870 0.0759 0.0668 0.0593 0.0530 0.0476 0.0430 0.0390 0.0354 0.0324 0.0296 0.0272 0.0250 0.0230 0.0213 0.0197 0.0182 0.0126 0.0065 0.0034 0.0019 0.0002 0.0000 0.4854 0.9612 1.4272 1.8836 2.3304 2.7676 3.1952 3.6133 4.0220 4.4213 4.8113 5.1920 5.5635 5.9260 6.2794 6.6240 6.9597 7.2867 7.6051 7.9151 8.2166 8.5099 8.7951 9.0722 10.3422 12.3590 13.7964 14.7909 16.3711 ENGINEERING ECONOMICS (continued) 84 Factor Table - i = 8.00% n P/F P/A P/G F/P F/A A/P A/F A/G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 30 40 50 60 100 0.9259 0.8573 0.7938 0.7350 0.6806 0.6302 0.5835 0.5403 0.5002 0.4632 0.4289 0.3971 0.3677 0.3405 0.3152 0.2919 0.2703 0.2502 0.2317 0.2145 0.1987 0.1839 0.1703 0.1577 0.1460 0.0994 0.0460 0.0213 0.0099 0.0005 0.9259 1.7833 2.5771 3.3121 3.9927 4.6229 5.2064 5.7466 6.2469 6.7101 7.1390 7.5361 7.9038 8.2442 8.5595 8.8514 9.1216 9.3719 9.6036 9.8181 10.0168 10.2007 10.3711 10.5288 10.6748 11.2578 11.9246 12.2335 12.3766 12.4943 0.0000 0.8573 2.4450 4.6501 7.3724 10.5233 14.0242 17.8061 21.8081 25.9768 30.2657 34.6339 39.0463 43.4723 47.8857 52.2640 56.5883 60.8426 65.0134 69.0898 73.0629 76.9257 80.6726 84.2997 87.8041 103.4558 126.0422 139.5928 147.3000 155.6107 1.0800 1.1664 1.2597 1.3605 1.4693 1.5869 1.7138 1.8509 1.9990 2.1589 2.3316 2.5182 2.7196 2.9372 3.1722 3.4259 3.7000 3.9960 4.3157 4.6610 5.0338 5.4365 5.8715 6.3412 6.8485 10.0627 21.7245 46.9016 101.2571 2,199.7613 1.0000 2.0800 3.2464 4.5061 5.8666 7.3359 8.9228 10.6366 12.4876 14.4866 16.6455 18.9771 21.4953 24.2149 27.1521 30.3243 33.7502 37.4502 41.4463 45.7620 50.4229 55.4568 60.8933 66.7648 73.1059 113.2832 259.0565 573.7702 1,253.2133 27,484.5157 1.0800 0.5608 0.3880 0.3019 0.2505 0.2163 0.1921 0.1740 0.1601 0.1490 0.1401 0.1327 0.1265 0.1213 0.1168 0.1130 0.1096 0.1067 0.1041 0.1019 0.0998 0.0980 0.0964 0.0950 0.0937 0.0888 0.0839 0.0817 0.0808 0.0800 1.0000 0.4808 0.3080 0.2219 0.1705 0.1363 0.1121 0.0940 0.0801 0.0690 0.0601 0.0527 0.0465 0.0413 0.0368 0.0330 0.0296 0.0267 0.0241 0.0219 0.0198 0.0180 0.0164 0.0150 0.0137 0.0088 0.0039 0.0017 0.0008 0.0000 0.4808 0.9487 1.4040 1.8465 2.2763 2.6937 3.0985 3.4910 3.8713 4.2395 4.5957 4.9402 5.2731 5.5945 5.9046 6.2037 6.4920 6.7697 7.0369 7.2940 7.5412 7.7786 8.0066 8.2254 9.1897 10.5699 11.4107 11.9015 12.4545 Factor Table - i = 10.00% n P/F P/A P/G F/P F/A A/P A/F A/G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 30 40 50 60 100 0.9091 0.8264 0.7513 0.6830 0.6209 0.5645 0.5132 0.4665 0.4241 0.3855 0.3505 0.3186 0.2897 0.2633 0.2394 0.2176 0.1978 0.1799 0.1635 0.1486 0.1351 0.1228 0.1117 0.1015 0.0923 0.0573 0.0221 0.0085 0.0033 0.0001 0.9091 1.7355 2.4869 3.1699 3.7908 4.3553 4.8684 5.3349 5.7590 6.1446 6.4951 6.8137 7.1034 7.3667 7.6061 7.8237 8.0216 8.2014 8.3649 8.5136 8.6487 8.7715 8.8832 8.9847 9.0770 9.4269 9.7791 9.9148 9.9672 9.9993 0.0000 0.8264 2.3291 4.3781 6.8618 9.6842 12.7631 16.0287 19.4215 22.8913 26.3962 29.9012 33.3772 36.8005 40.1520 43.4164 46.5819 49.6395 52.5827 55.4069 58.1095 60.6893 63.1462 65.4813 67.6964 77.0766 88.9525 94.8889 97.7010 99.9202 1.1000 1.2100 1.3310 1.4641 1.6105 1.7716 1.9487 2.1436 2.3579 2.5937 2.8531 3.1384 3.4523 3.7975 4.1772 4.5950 5.5045 5.5599 6.1159 6.7275 7.4002 8.1403 8.9543 9.8497 10.8347 17.4494 45.2593 117.3909 304.4816 13,780.6123 1.0000 2.1000 3.3100 4.6410 6.1051 7.7156 9.4872 11.4359 13.5735 15.9374 18.5312 21.3843 24.5227 27.9750 31.7725 35.9497 40.5447 45.5992 51.1591 57.2750 64.0025 71.4027 79.5430 88.4973 98.3471 164.4940 442.5926 1,163.9085 3,034.8164 137,796.1234 1.1000 0.5762 0.4021 0.3155 0.2638 0.2296 0.2054 0.1874 0.1736 0.1627 0.1540 0.1468 0.1408 0.1357 0.1315 0.1278 0.1247 0.1219 0.1195 0.1175 0.1156 0.1140 0.1126 0.1113 0.1102 0.1061 0.1023 0.1009 0.1003 0.1000 1.0000 0.4762 0.3021 0.2155 0.1638 0.1296 0.1054 0.0874 0.0736 0.0627 0.0540 0.0468 0.0408 0.0357 0.0315 0.0278 0.0247 0.0219 0.0195 0.0175 0.0156 0.0140 0.0126 0.0113 0.0102 0.0061 0.0023 0.0009 0.0003 0.0000 0.4762 0.9366 1.3812 1.8101 2.2236 2.6216 3.0045 3.3724 3.7255 4.0641 4.3884 4.6988 4.9955 5.2789 5.5493 5.8071 6.0526 6.2861 6.5081 6.7189 6.9189 7.1085 7.2881 7.4580 8.1762 9.0962 9.5704 9.8023 9.9927 ENGINEERING ECONOMICS (continued) 85 Factor Table - i = 12.00% n P/F P/A P/G F/P F/A A/P A/F A/G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 30 40 50 60 100 0.8929 0.7972 0.7118 0.6355 0.5674 0.5066 0.4523 0.4039 0.3606 0.3220 0.2875 0.2567 0.2292 0.2046 0.1827 0.1631 0.1456 0.1300 0.1161 0.1037 0.0926 0.0826 0.0738 0.0659 0.0588 0.0334 0.0107 0.0035 0.0011 0.8929 1.6901 2.4018 3.0373 3.6048 4.1114 4.5638 4.9676 5.3282 5.6502 5.9377 6.1944 6.4235 6.6282 6.8109 6.9740 7.1196 7.2497 7.3658 7.4694 7.5620 7.6446 7.7184 7.7843 7.8431 8.0552 8.2438 8.3045 8.3240 8.3332 0.0000 0.7972 2.2208 4.1273 6.3970 8.9302 11.6443 14.4714 17.3563 20.2541 23.1288 25.9523 28.7024 31.3624 33.9202 36.3670 38.6973 40.9080 42.9979 44.9676 46.8188 48.5543 50.1776 51.6929 53.1046 58.7821 65.1159 67.7624 68.8100 69.4336 1.1200 1.2544 1.4049 1.5735 1.7623 1.9738 2.2107 2.4760 2.7731 3.1058 3.4785 3.8960 4.3635 4.8871 5.4736 6.1304 6.8660 7.6900 8.6128 9.6463 10.8038 12.1003 13.5523 15.1786 17.0001 29.9599 93.0510 289.0022 897.5969 83,522.2657 1.0000 2.1200 3.3744 4.7793 6.3528 8.1152 10.0890 12.2997 14.7757 17.5487 20.6546 24.1331 28.0291 32.3926 37.2797 42.7533 48.8837 55.7497 63.4397 72.0524 81.6987 92.5026 104.6029 118.1552 133.3339 241.3327 767.0914 2,400.0182 7,471.6411 696,010.5477 1.1200 0.5917 0.4163 0.3292 0.2774 0.2432 0.2191 0.2013 0.1877 0.1770 0.1684 0.1614 0.1557 0.1509 0.1468 0.1434 0.1405 0.1379 0.1358 0.1339 0.1322 0.1308 0.1296 0.1285 0.1275 0.1241 0.1213 0.1204 0.1201 0.1200 1.0000 0.4717 0.2963 0.2092 0.1574 0.1232 0.0991 0.0813 0.0677 0.0570 0.0484 0.0414 0.0357 0.0309 0.0268 0.0234 0.0205 0.0179 0.0158 0.0139 0.0122 0.0108 0.0096 0.0085 0.0075 0.0041 0.0013 0.0004 0.0001 0.0000 0.4717 0.9246 1.3589 1.7746 2.1720 2.5515 2.9131 3.2574 3.5847 3.8953 4.1897 4.4683 4.7317 4.9803 5.2147 5.4353 5.6427 5.8375 6.0202 6.1913 6.3514 6.5010 6.6406 6.7708 7.2974 7.8988 8.1597 8.2664 8.3321 Factor Table - i = 18.00% n P/F P/A P/G F/P F/A A/P A/F A/G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 30 40 50 60 100 0.8475 0.7182 0.6086 0.5158 0.4371 0.3704 0.3139 0.2660 0.2255 0.1911 0.1619 0.1372 0.1163 0.0985 0.0835 0.0708 0.0600 0.0508 0.0431 0.0365 0.0309 0.0262 0.0222 0.0188 0.0159 0.0070 0.0013 0.0003 0.0001 0.8475 1.5656 2.1743 2.6901 3.1272 3.4976 3.8115 4.0776 4.3030 4.4941 4.6560 4.7932 4.9095 5.0081 5.0916 5.1624 5.2223 5.2732 5.3162 5.3527 5.3837 5.4099 5.4321 5.4509 5.4669 5.5168 5.5482 5.5541 5.5553 5.5556 0.0000 0.7182 1.9354 3.4828 5.2312 7.0834 8.9670 10.8292 12.6329 14.3525 15.9716 17.4811 18.8765 20.1576 21.3269 22.3885 23.3482 24.2123 24.9877 25.6813 26.3000 26.8506 27.3394 27.7725 28.1555 29.4864 30.5269 30.7856 30.8465 30.8642 1.1800 1.3924 1.6430 1.9388 2.2878 2.6996 3.1855 3.7589 4.4355 5.2338 6.1759 7.2876 8.5994 10.1472 11.9737 14.1290 16.6722 19.6731 23.2144 27.3930 32.3238 38.1421 45.0076 53.1090 62.6686 143.3706 750.3783 3,927.3569 20,555.1400 15,424,131.91 1.0000 2.1800 3.5724 5.2154 7.1542 9.4423 12.1415 15.3270 19.0859 23.5213 28.7551 34.9311 42.2187 50.8180 60.9653 72.9390 87.0680 103.7403 123.4135 146.6280 174.0210 206.3448 244.4868 289.4944 342.6035 790.9480 4,163.2130 21,813.0937 114,189.6665 85,689,616.17 1.1800 0.6387 0.4599 0.3717 0.3198 0.2859 0.2624 0.2452 0.2324 0.2225 0.2148 0.2086 0.2037 0.1997 0.1964 0.1937 0.1915 0.1896 0.1881 0.1868 0.1857 0.1848 0.1841 0.1835 0.1829 0.1813 0.1802 0.1800 0.1800 0.1800 1.0000 0.4587 0.2799 0.1917 0.1398 0.1059 0.0824 0.0652 0.0524 0.0425 0.0348 0.0286 0.0237 0.0197 0.0164 0.0137 0.0115 0.0096 0.0081 0.0068 0.0057 0.0048 0.0041 0.0035 0.0029 0.0013 0.0002 0.0000 0.4587 0.8902 1.2947 1.6728 2.0252 2.3526 2.6558 2.9358 3.1936 3.4303 3.6470 3.8449 4.0250 4.1887 4.3369 4.4708 4.5916 4.7003 4.7978 4.8851 4.9632 5.0329 5.0950 5.1502 5.3448 5.5022 5.5428 5.5526 5.5555
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https://www.cuemath.com/numbers/adding-fractions-with-unlike-denominators/
Adding Fractions With Unlike Denominators Adding fractions with unlike denominators means we need to add fractions that have different denominators. In this case, we convert the given fractions to like fractions to get common denominators so that it becomes easier to add them. This is done by finding the Least Common Multiple (LCM) of the given denominators. We convert each fraction in such a way so that we have a common denominator, and then we add the numerators to get the sum. In this article, we will learn how to add fractions with different denominators stepwise. We will also discuss the addition of mixed fractions with unlike denominators along with a few solved examples for a better understanding of the concept. | | | --- | | 1. | What is Adding Fractions with Unlike Denominators? | | 2. | Steps For Adding Fractions with Unlike Denominators | | 3. | How to Add Fractions with Unlike Denominators? | | 4. | Adding Mixed Numbers with Unlike Denominators | | 5. | FAQs on Adding Fractions with Unlike Denominators | What is Adding Fractions with Unlike Denominators? When the denominators are not the same, the fractions are known as unlike fractions. For example, 3/5 and 6/7 are called unlike fractions because they have different denominators. To add two or more given fractions, whose denominators are unlike or different, we need to find the Least Common Multiple (LCM) of the denominators. After finding the LCM, we multiply the given fractions with such a number so that their denominators remain common. After making the denominators equal, we can simply add the numerators. Steps For Adding Fractions with Unlike Denominators The following steps show the procedure for adding fractions with unlike denominators. Step 1: First, we find out the Least Common Multiple (LCM) of the given denominators. Step 2: Then, we write down each fraction in a form such that the LCM becomes the common denominator. For this, we multiply the numerator and denominator with a common number with the help of the LCM. Step 3: After this step, we add the numerators of these like fractions (which have common denominators now). Step 4: Finally, we reduce the resultant fraction to its lowest terms, if needed. These steps can be understood with the help of the example given in the following section. How to Add Fractions with Unlike Denominators? Now, let us learn how to add fractions with unlike denominators by following the steps given above. Example: Add 5/6 + 7/3 Solution: Step 1: Since the fractions have different denominators, we find the LCM of 6 and 3. The LCM of 6 and 3 is 6. Step 2: Now, convert the given fractions to equivalent fractions such that the LCM becomes their common denominator. As we can see 5/6 already has the LCM as its denominator, so we will only change the fraction 7/3 and make it an equivalent fraction, which will be 14/6. Step 3: After this, we can add the numerators of both the fractions since the denominators are the same. Step 4: 5/6 + 14/6 = (5 + 14)/6 = 19/6. This can be converted to a mixed fraction and written as 316 Adding 3 Fractions with Different Denominators For adding three or more fractions with unlike denominators we apply the same steps as given above. Let us learn how to add 3 fractions with different denominators with the help of the following example. Example: Add 1/2 + 3/5 + 7/3 Step 1: First, we will find the LCM of 2, 5, and 3, which is 30. Step 2: Now, we will make each fraction an equivalent fraction in such a way that the LCM 30 becomes the denominator of each fraction. Step 3: The equivalent fractions with denominator 30 are 15/30, 18/30, and 70/30. Step 4: Add all the numerators (15 + 18 + 70)/30 = 103/30. This can be converted to a mixed fraction and written as 31330 Adding Mixed Numbers with Unlike Denominators If we need to add mixed numbers with unlike denominators, we convert the mixed fraction to an improper fraction. After this, we can add the fractions by following the steps given above. Let us understand this with the help of the following example. Example: Add 517 and 415 First, convert the mixed number into an improper fraction. 517 = 36/7 415 = 21/5 Now add 36/7 and 21/5 LCM of 7 and 5 is 35 Equivalent fractions 36/7 = 180/35 21/5 = 147/35 Now add the numerators, (180 + 147)/35 = 327/35 = 91235 Important Notes on Adding Fractions with Unlike Denominators For adding fractions with unlike denominators, we take the LCM of the different denominators and convert them to like fractions and then add the numerators. For adding mixed fractions with unlike denominators, we convert them into improper fractions and then add. ☛ Related Articles Adding Fractions with Unlike Denominators Worksheets Addition and Subtraction of Fractions Subtracting Fractions with Unlike Denominators Adding Mixed Fractions Read More Download FREE Study Materials Adding fractions with unlike denominators Adding fractions with unlike denominators Questions of adding unlike fractions Adding Fractions with Unlike Denominators Examples Example 1: Add 8/9 + 1/3 + 7/6 Solution: For adding fractions with unlike denominators, first, we will find out the LCM of numbers 9, 3 and 6 LCM = 18 Now, we will convert each fraction into equivalent fractions by taking the denominator as the LCM 8/9 = 16/18 1/3 = 6/18 7/6 = 21/18 Now, we can add the numerators. (16 + 6 + 21)/18 = 43/18. This can be converted to a mixed fraction 2718 2. Example 2: Add 5/8 + 1/5 Solution: To add fractions with unlike denominators, first, we will find out the LCM of numbers 8 and 5 LCM = 40 Then, we will convert each fraction into equivalent fractions by taking the denominator as the LCM 40 5/8 = 25/40 1/5 = 8/40 Now, we can add the numerators. (25 + 8)/40 = 33/40 So, the sum of 5/8 + 1/5 is 33/40 3. Example 3: Add mixed fractions 314 and 223 Solution: To add the given mixed fractions, we first convert them into improper fractions. 314 = 13/4 and 223 = 8/3 Now, since the denominators are unlike, we take the LCM of 3 and 4. LCM (3, 4) = 12. Now, convert the fractions into like fractions. 13/4 = 39/12 and 8/3 = 32/12. Now add the two fractions. 39/12 + 32/12 = 71/12 = 51112 Answer: 314 + 223 = 51112 View Answer > Breakdown tough concepts through simple visuals. Math will no longer be a tough subject, especially when you understand the concepts through visualizations. Book a Free Trial Class Adding Fractions With Unlike Denominators Questions Check Answer > FAQs on Adding Fractions with Unlike Denominators How to Add Fractions with Different Denominators? Adding fractions with unlike denominators means that two fractions that have different denominators need to be added. In this case, we convert the given fractions to like fractions to get common denominators so that it becomes easier to add them. This is done by finding the Least Common Multiple (LCM) of the given denominators. We need to convert each fraction in such a way so that we have a common denominator and after that, we add the numerators to get the sum. What are the Examples of Adding Fractions with Unlike Denominators? An example of adding fractions with unlike denominators is given as follows. Let us add 1/3 + 6/5. We can see that the denominators are not the same, hence, we need to make the denominators equal, after which we can add the fractions. In this example, 1/3 + 6/5, we will first find the LCM of the denominators 3 and 5 which is 15. Then, we will multiply both the fractions with such a number so that the denominators remain the same. This results in (5 + 18)/15 = 23/15. Now, let us convert the improper fraction to a mixed number: 23/15 = 1815 What is the Strategy for Adding Fractions with Unlike Denominators? The strategy for adding fractions with unlike denominators is to find the LCM of the given denominators and make each fraction as an equivalent fraction with the LCM as the denominator. What are the Steps For Adding Fractions with Unlike Denominators? The steps for adding fractions with unlike denominators are given below. Let us understand this with an example and add 1/5 + 1/10 Step 1: First, we will find the LCM of 5 and 10 which is 10. Step 2: Now, make each fraction an equivalent fraction in such a way so that the LCM (10) becomes the denominator of each fraction. Step 3: The equivalent fractions with denominator 10 will be 2/10 and 1/10 Step 4: Add the numerator part (2 + 1)/10 = 3/10 Can we Add Fractions with Unlike Denominators Without Using LCM? No, we cannot add fractions with unlike denominators without using the LCM. What is the Rule for Adding Fractions with Unlike Denominators? The basic rule for adding fractions with unlike denominators is to find the LCM of the different denominators and convert the given unlike fractions into like fractions. This can be done by changing their denominators equal to the LCM. Once the denominators become the same, the numerators can be added. How to Add 3 Fractions with Different Denominators? In order to add 3 fractions with different denominators, we use the same rules that are used for adding 2 fractions with different denominators. For example, let us add 1/6 + 1/3 + 1/2 We need to find the LCM of 6, 3, and 2 which is 6. Now, we will convert each fraction into an equivalent fraction using LCM as the denominator. 1/6 = 1/6 1/3 = 2/6 1/2 = 3/6 Now, add the numerators of these fractions which is: 1/6 + 2/6 + 3/6 = (1 + 2 + 3)/6 = 6/6. This fraction can be further reduced to 1 after simplifying. How to Add and Subtract Fractions with Different Denominators? In order to add and subtract fractions with different denominators, we use the same rules. We need to find the LCM of the different denominators and convert the given unlike fractions into like fractions. After this step, we add or subtract the numerators according to the question. How to Add Fractions with Whole Numbers and Different Denominators? In order to add fractions with whole numbers and different denominators, we should write the whole number in its fraction form, that is, write 1 as its denominator and then add it using the same rules for the addition of fractions. For example, let us add 6 + 3/4 + 1/2. In this case, 6 is the whole number and we can write it as 6/1. So, let us rewrite the fractions as 6/1 + 3/4 + 1/2. Then, we will find the LCM of the denominators so that they are converted to like fractions. The LCM of 1, 4, and 2 will be 4. Now, the fractions can be written as 6/1 + 3/4 + 1/2 = (24 + 3 + 2)/4 = 29/4. This can be converted to mixed fraction as 714 How to Add Improper Fractions with Different Denominators? In order to add improper fractions with different denominators, we use the same rules that are used for the addition of fractions. For example, let us add these improper fractions with different denominators: 7/2 + 8/3. We will first find the LCM of the denominators. The LCM of 2 and 3 is 6. Then, we will convert the given fractions into equivalent fractions. So, 7/2 will become 21/6, and 8/3 will become 16/6. Now, we can add them because their denominators are the same. 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SANS-MIRI: Review of linear algebra and applications to data science Jorge Garcia Vidal, Jose M. Barcelo Ordinas and Pau Ferrer Cid July 17, 2023 The course SANS (Statistical Analysis of Networks and Systems) belongs to the Master MIRI (Master of Innovation and Research in Computer Science) of the Faculty of Computer Science of Barcelona. The course is an introduction to some mathematical foundations used in data science. The course content includes an introduction to probability, linear algebra, and estimation. These lecture notes of the course are devoted to linear algebra concepts applied to data science, and is divided into the following topics: 1. Basics in Linear Algebra: vector spaces, matrices and applications (linear equations and least squares equations); 2. Eigendecomposition of square matrices: eigenvectors and eigenvalues, pos-itive definite matrices and the trace operator; 3. Quadratic forms. Multivariate Gaussian distribution; 4. Eigendecomposition of square matrices: singular value decomposition, pseudoinverses, matrix norms, Eckart-Young approximation (low-rank ap-proximation of matrices); 5. Principal component analysis (PCA), the eigenfaces problem; 6. Fourier Transform and its applications; 7. Graph signal processing (GSP) and its applications. Linear Algebra is a classical topic, and there are many very good books covering the material that we need for our course at different levels of deep. Two books that we especially like are: ”Introduction to Linear Algebra” by Gilbert Strang, and ”Linear Algebra and Learning from Data” by Strang, Gilbert. Other books related to this course is ”Data-Driven Science and Engineering: Machine Learn-ing, Dynamical Systems, and Control” by Steven L. Brunton and J. Nathan Kutz, or ” Introduction to Applied Linear Algebra – Vectors, Matrices, and Least Squares” by Stephen Boyd and Lieven Vandenberghe. 11 Some basic facts that you probably know about vectors and linear maps 1.1 Vector spaces and sub-spaces Let V be a set in which we have defined the addition operation and the multipli-cation by a scalar (in this course, the scalars will be usually real numbers). We say that V is a vector space if the addition and scalar multiplication operations satisfy the following properties: ∀u, v, w ∈V : • Associativity: u + ( v + w) = ( u + v) + w, • Commutativity: u + v = v + u, • Additive identity: ∃0∈V such that v + 0 = v, , • Existence of inverse: ∀v∈V, ∃ −v ∈V such that v + ( −v) = 0. ∀ a, b ∈ R and ∀u, v ∈V : • Associativity of scalar multiplication: a(bv) = ( ab )v, • Scalar multiplication identity: 1 v = v, • Distributivity of scalar sums: ( a + b)v = av + bv, • Distributivity of vector sums: a(u + v) = au + av.The elements of a vector space are called vectors . Some examples of vector spaces are: Example 1.1 (Coordinate space) The set of vectors x∈Rn with coordinates x=( x1, ..., x n) and t ∈R is a vector space. Example 1.2 (Set of matrices) The set of matrices A∈Rm×n is a vector space over R, (addition of matrices and multiplication of scalar over matrices). Example 1.3 (Set of polynomials) The set of polynomials Pn over R (co-efficients in R) and of order less or equal of n is a vector space. Example 1.4 (Set of continuous functions) The set of continuous functions f : Rn −→ R, where ( f +g)( x) = f (x)+ g(x) and (a f )(x)=a f (x) is a vector space. 2If S is a subset of V (S⊂V ) which is closed respect the operations of sum of vectors (if u, v∈S, then u + v∈S), multiplication by an scalar (if u∈S and a scalar then au∈S), and the zero vector is in S, we say that S is a vector subspace of V . Some examples/non-examples of vector subspaces V are: Example 1.5 A line or a plane containing the origin is a vector subspace. Example 1.6 A line or a plane non containing the origin is not a vector sub-space. Example 1.7 A quadrant is not a vector subspace (fails to be close under scalar multiplication). Example 1.8 A circle is not a vector subspace (fails to be close under scalar or vector multiplication and does not contain the zero). 1.2 Linear combinations and independence If we have a set of vectors {v1, ..., vk}, a linear combination of these vectors is an expression of the form v = P i aivi for some scalars ai.If we have a set of vectors belonging to a vector space, {v1, ..., vk}, the set of all linear combinations of these vectors is a vector subspace. This subspace is called: span {v1, ..., vk} = { k X j=1 βj vj ; with β j ∈R} (1.2.1) One vector w is linearly independent of a set of vectors {v1, ..., vk} when w cannot be expressed as linear combination of the vectors {v1, ..., vk}, or in other words, when w /∈ span {v1, ..., vk}. A set of vectors {v1, ..., vk} are linearly independent when the only linear combination that produces the vector 0 is the one with all coefficients equal to zero. Summarizing, let us assume that Pkj=1 aj vj = 0; if some aj̸ =0 then the vectors are linearly dependent (l.d.), while if all aj =0 then the vectors are linearly independent (l.i.). 1.3 Bases and dimension There are many examples of vector spaces: Rn, Cn, or Pn(R), the set of poly-nomials of n degree with real coefficients. These spaces are examples of finite dimensional vector spaces. This means that there is a set of linearly indepen-dent vectors of V , {ui}i=1 ,...,n (a base of V ) such that all other vectors of V can be expressed as v = Pni=1 viui. A vector space has in general an infinite 3number of possible bases, but the number of elements in each of those basis is always the same. We say that this number n the dimension of V , dim( V ) = n.Moreover, we can use these coefficients to represent v as a column vector: v = [ v1, v 2, ..., v n]⊤ =  v1 v2 ... vn  (1.3.1) The same definitions apply to vector subspaces. Some vector spaces have infinite dimensions, for instance, P∞(R), the set of polynomials of an arbitrary order, C[0 , 1], the set of continuous functions defined on the interval [0 , 1], or L2[0 , 1], the set of square integrable functions f for which R 10 |f |2dμ < ∞. In this course, we will deal with the finite dimensional case only. The infinite dimensional case is studied in functional analysis , and is important, for instance, when dealing with stochastic processes. 1.4 Scalar product, orthogonality, and vector norms The scalar product is an operation that takes two vectors u, v ∈ V and returns a scalar; i.e., ⟨· , ·⟩ : V ×V −→ R. For this operation to qualify as a scalar product, it must fulfill the following properties: • commutative: ⟨u, v⟩ = ⟨v, u⟩, • distributive: ⟨u, v + w⟩ = ⟨u, v⟩ + ⟨u, w⟩, • linearity in any argument: ⟨au, v⟩ = ⟨u, a v⟩ = a⟨u, v⟩, • positive definiteness: if u̸ = 0, then ⟨u, u⟩ > 0. The most commonly used definition of the scalar product for two column vectors u and v is: ⟨u, v⟩ = u⊤v. (1.4.1) Note, however, that we can define other scalar products. For instance, if S is a symmetric positive definite matrix (see section 2.6), we can define a scalar product as: ⟨u, v⟩ = u⊤Sv . (1.4.2) Another important example is the scalar product of two square matrices A and B defined as (see section 3.3 for the definition of trace ): ⟨A, B⟩ = tr (A⊤B). (1.4.3) 4A very important property of the scalar product is the Cauchy-Schwarz in-equality : |⟨ u, v⟩| 2 ≤ ⟨ u, u⟩ ⟨ v, v⟩. (1.4.4) We have equality only when u = av for some scalar a.A simple proof of this important fact is the following: consider an arbitrary scalar a and two non-zero vectors u, v. Assume first that there is no scalar a for which au = v. Then, the positive-definiteness property of the scalar product implies that for any a:0 < ⟨au − v, a u − v⟩ = ⟨u, u⟩a2 − 2⟨u, v⟩a + ⟨v, v⟩. (1.4.5) But this second-degree polynomial on a is always non-negative for all a only when: 4( ⟨u, v⟩)2 − 4⟨u, u⟩⟨ v, v⟩ < 0, (1.4.6) and from this, we arrive at the strict Cauchy-Schwarz inequality. Assume now that there is a scalar a∗ for which a∗u = v. In this case, we have 0 ≤ ⟨ au − v, a u − v⟩ = ⟨u, u⟩a2 − 2⟨u, v⟩a + ⟨v, v⟩. (1.4.7) with equality for a = a∗. This means that the second-degree polynomial in a has a double root at a∗, which is only possible if: 4( ⟨u, v⟩)2 − 4⟨u, u⟩⟨ v, v⟩ = 0 , (1.4.8) and from this, we arrive at the equality case of Cauchy-Schwarz inequality when v is colinear with u.We can define the angle α between two non-zero vectors u, v as: cos (α) = ⟨u, v⟩ p⟨u, u⟩ ⟨ v, v⟩ . (1.4.9) Two non-zero vectors are orthogonal when its scalar product is zero: ⟨u, v⟩ = 0 . (1.4.10) Thus, the angle between two orthogonal vectors is α = 90 ◦.A norm is a function ∥·∥ : Rn −→ R, such that for each vector v∈Rn, the following conditions are satisfied: • ∥ · ∥ is non-negative, ∥v∥ ≥ 0 • ∥ · ∥ is definite, ∥v∥ = 0 iif v = 0, 5• ∥ · ∥ is homogeneous ∥(av)∥ = |a|∥ v∥ with a scalar, • ∥ · ∥ satisfies the triangle inequality (subadditivity property) ∥u + v∥ ≤ ∥u∥ + ∥v∥.The Euclidean norm or also called L2-norm represents the length of a vector v, and is defined as the non-negative number: ∥v∥ = + p⟨v, v⟩ (1.4.11) Other well-known norms are represented by a subscript ∥ · ∥ p, where p∈R+ (R+ means any real number ≥0). Examples of useful norms are: • ∥v∥0 (zero-norm or L0-norm) defined as the number of non-zero coordi-nates of vector v (or also as the Hamming distance of the vector from zero), • ∥v∥1 (Taxicab norm or Manhattan norm or sum-absolute norm) defined as ∥v∥1 = Pni=1 |vi|, • ∥v∥2 (Euclidean norm or L2-norm) defined as ∥v∥2 = pP ni=1 |vi|2 =√v⊤v, • ∥v∥p (Lp-norm) defined as ∥v∥p = ( Pni=1 |vi|p)1/p , • ∥v∥∞ (maximum norm or infinity norm or Chebyshev norm) defined as ∥v∥∞ = max i=1 ...n {| vi|} . Example 1.9 (vector norms) Let us assume vector v=[1,0,0,4,2,-3]. Then, ∥v∥0 = 4 , ∥v∥1 = 10 , ∥v∥2=√30 = 5 .477 , ∥v∥3.2 = 4 .558 and ∥v∥∞ = 4 The Cauchy-Schwarz inequality for the Euclidean norm reads: |u⊤v| ≤ ∥ u∥ ∥ v∥ (1.4.12) Let us define the unit circle or unit ball as the set of all vectors v of norm 1 (∥v∥p = 1). Then plot, as an exercise , the unit circle for norms ∥v∥0, ∥v∥1, ∥v∥2, ∥v∥∞, and ∥v∥p for any p. 1.5 Linear maps and matrices A linear map L is an application L: V → W , where V (domain) and W (co-domain) are vector spaces, that satisfies ∀u, v∈V , and ∀a, b ∈R: L(u + v) = L(u) + L(v) and L (au) = aL (u), (1.5.1) 6or (identical definition) L(au + bv) = aL (u) + bL (v). (1.5.2) For example; L(x 1,x 2)=(x 1+x 2,x 1,x 2+1) is not a linear map, as it fails in the condition L(au) = aL (u). A linear map always transforms 0 to 0, i.e., L(0) = 0.Assume that {vi}i=1 ,...,n is a base of a n dimensional space V , while {wi}i=1 ,...,m is a base of a m dimensional space W . The image of the ui base vector by the linear map L, i.e. L(ui), is a vector of W , that we can express in the base {wi}i=1 ,...,m : L(ui) = a1iw1 + a2iw2 + ... + ami wm (1.5.3) An m×n matrix A is an arrangement of these numbers aij into an m×n array A = [ aij ]. For an arbitrary vector v, expressed as a column vector in the base {vi}i=1 ,...,n , the product Av gives a result the vector w, which is the image of the vector v by the linear map L expressed in the base {wi}i=1 ,...,m .This is a bit confusing at first. Let’s think on the following example: {1, x, x 2} is a possible base of P2(R). Using this base, we can represent the polynomial p(x) = a + bx + cx 2 by the R3 vector p = [ a, b, c ]⊤. Let us define the linear map ddx : P2(R) → P2(R) that assigns to a polynomial p(x) its derivative (another polynomial). If we represent the polynomials in the image and domain sets by the same base {1, x, x 2}, we can thus represent this linear map by the matrix: D = 0 1 00 0 20 0 0  . (1.5.4) Usually, we will use matrices to represent linear maps. Very often, we express a matrix A as an arrangement of its columns (or its rows) considered vectors. For instance if A = [ aij ] and let us define the n column vectors ai = [ ai1, ..., a im ]⊤ for i = 1 , ..., n . We can write the matrix as: A = [ a1, ..., an] (1.5.5) Similarly, we can write A using its row vectors ri⊤ = [ a1i, ..., a ni ]: A =  — r1⊤ —...— rm⊤ —  . (1.5.6) If we have a matrix A and a column vector v = [ v1, ..., v n]⊤, we can express its matrix times vector product as w = Av = [ a1, ..., an]  v1 ... vn  = X i=1 ,..,n viai, (1.5.7) 7which is saying that the image vector w is a linear combination of the column vectors ai. 1.6 Rank of a matrix The image f of all vectors of a subspace S of V by means of a matrix A is also a subspace T of W ; f :S⊂V → T ⊂W .In the special case when the subspace S is V itself, the generated subspace is span {a1, ..., an}. The dimension of this subspace is the rank of the matrix A,rank( A), and it is the number of linearly independent columns. Very surprisingly, this number is also the number of linearly independent rows of the matrix A, meaning that A and A⊤ have the same rank .In terms of rows and columns of a matrix A∈Rm×n and matrix A⊤∈Rn×m,we can state the actions of an m×n matrix remembering that matrix A can be expressed as column vectors {a1, ..., an} or matrix A can be expressed as row vectors {r1⊤, ..., rm⊤}. In the same way matrix A⊤ can be expressed as column vectors {r1, ..., rm} or matrix A⊤ can be expressed as row vectors {a1⊤, ..., an⊤}.For any matrix, we can associate several subspaces: the null space (NS), the column space (CS), the row space (RS), and the left null space (LNS). Let us define the Range( A) or Im( A)=Image( A) as the image of the linear transformation f :Rn⊂V → Rm⊂W : Im (A) = Range (A) = {y∈Rm|y = Ax f or some x∈Rn} (1.6.1) and we can observe, Range( A)= Im( A) is a subspace of Rm. The Im( A⊤) is defined in a similar way and is a subspace of Rn. Im (A⊤) = Range (A⊤) = {x∈Rn|x = A⊤y f or some y∈Rm} (1.6.2) and, Range( A⊤)= Im( A⊤) is a subspace of Rn.Define the column space of A, CS( A), as the linear combination of its columns: CS( A)= {w∈Rm, w=c1a1 + · · · + cnan}, where ai∈Rm. The CS( A) is then a subspace of Rm. The CS( A) is the Range( A) or Im( A) of the linear transfor-mation f :Rn⊂V → Rm⊂W .In a similar way, we define the row space of A, RS( A), as the linear combina-tion of its rows: RS( A)= {v∈Rn, w=d1r1 + · · · + dnrm}, where ri∈Rn. The RS( A) is then a subspace of Rn, and then it is the Range( A⊤) or Im( A⊤). We know that dim(RS( A))= rank( A) and that dim(CS( A))= rank( A), which makes dim(RS( A))=dim(CS( A))= rank( A). Since the columns of A are the rows of A⊤, finding a basis for CS( A) is equivalent to finding a basis for RS( A⊤). 8The null space NS( A) or Ker(f) is defined as: N S (A) = {x∈Rn|Ax = 0 } (1.6.3) and we can observe that NS( A) is a subspace of Rn. The NS( A⊤) (also called the Left Null Space of A, LNS( A) or the CoKer(f)) is defined in a similar way and is a subspace of Rm. LN S (A) = N S (A⊤) = {y∈Rm|A⊤y = 0 } = {y∈Rm|y⊤A = 0 } (1.6.4) If A=[ a1, ..., an] is a column partitioning, and rank( A)=span {a1, ..., an}. Now, since the rank( A) is the dimension of the image; rank( A)= dim(Range( A))= dim(Im( A))=dim(CS( A)), and we know that rank( A)=rank( A⊤). We say that the matrix A∈Rm×n is rank deficient if rank( A)<min {m,n }, and therefore: dim (RS (A)) + dim (N S (A)) = rank (A) + dim (N S (A)) = n (1.6.5) We can state that RS( A)= Im( A⊤) ⊂Rn ⊥ NS( A)⊂Rn. On the other hand, in terms of the CS and LNS: CS (A) = Im (A)⊂Rm ⊥ LN S (A) = N S (A⊤)⊂Rm (1.6.6) and dim (CS (A)) + dim (LN S (A)) = rank (A) + dim (LN S (A)) = m (1.6.7) You can see this in the following way: A⊤y=0 (or y⊤A=0), so rows of A⊤ multiplied by vectors y in the null space are equal to 0 (or vectors y of the left null space multiplied by columns of A are equal to 0), so they are orthogonal. Example 1.10 (Row space, column space, null space and left null space) Let’s see an example of a matrix A∈Rm×n, with m=3 and n=4: A =  2 1 1 13 0 0 21 3 6 0  The rank of this matrix is r=3. It has 3 row vectors; r⊤ 1 = [2 , 1, 2, 1] , r⊤ 2 =[3 , 0, 0, 2] and r⊤ 3 = [1 , 3, 6, 0] . They form a basis of the row space that is a sub-space of Rn = R4. The null space has 1 vector with basis rns = [ −2/3, 4/9, −1/9, 1] .Now, vectors r1, r1, r1 and rns form a basis of the space Rn = R4.Matrix A has 4 column vectors; a⊤ 1 = [2 , 3, 1] , a⊤ 2 = [1 , 0, 3] , a⊤ 3 = [1 , 0, 6] and a⊤ 4 = [1 , 2, 0] . vectors a2 and a3 are linear dependent, thus, vectors a1, a2 and a4 form a basis of the column space that is a subspace of Rm = R3, and the left null space only contains vector alns = [0 , 0, 0] . 1.7 Applications Let us see some applications where these concepts appear. 9Figure 1: Fundamental theorem of linear algebra. 1.7.1 Linear equations Let us consider the Figure 1, where we have plotted the orthogonality of the subspaces and we consider the system of equations: Ax = b (1.7.1) where A∈Rm×n, x∈Rn and b∈Rm.We can observe that the action of matrix A over a vector xr in the RS is to transform it in a vector b in the CS. On the other hand, we can observe that the action of matrix A over a vector xn in the NullSpace is to transform it in vector 0.An interesting property is that a vector xp = xr + xns that is the sum of a vector in the RS and a vector of the NullSpace goes to the CS, since Ax p= A(xr + xns ) = Ax r + Ax ns = b + 0 = b.The conclusion of these facts are that the particular solution of Ax = b is xr , the homogeneous solution of Ax = b is xns , and the general solution of Ax = b is xp = xr + xns .10 1.7.2 Least squares equations Let us consider the Figure 2, where we have plotted the orthogonality of the subspaces and we consider the system of equations: Ax = b where A∈Rm×n, x∈Rn and b∈Rm. The objective is to find a vector x that satisfies the equation Ax = b. Let us consider three cases: Figure 2: Least squares equations. • Undetermined case: This is the case in which there are more vari-ables than equations, meaning m < n . In this case, there exists infinity solutions, since ˆx = {x: Ax =b} = {xr + xns } with xr ∈RS (A) and xns ∈N ull (A). This is because rank( A)= r=m, so the Null( A⊤) only con-tains the 0 vector and Null( A) has n-m=n-r vectors (assuming that A is full row rank). The best that can be done is to find the vector x in the RS which when transformed by matrix A is closest to CS, it is to say: minimize ∥x∥22 subject to Ax = b variable x (1.7.2) This vector is given (we will prove it in TOML-MIRI when we will study non-linear optimization) by the right pseudo-inverse : A† = A⊤(AA ⊤)−1 (1.7.3) We call it right pseudo-inverse because AA † =I (A† is on the right). We will come back to a derivation of the left pseudo-inverse when we study the singular value decomposition (SVD). 11 Example 1.11 (Undertermined case) Let us consider the following lin-ear system: Ax = b → 2 1 01 3 1  x1 x2 x3  = 31  A null space vector is xns = [1 , −2, 5] . The solution xr is given by xr = A⊤(AA ⊤)−1b: xr = A⊤(AA ⊤)−1b =  2 11 30 1  ( 2 1 01 3 1   2 11 30 1 )−1 31  and: xr =  2 11 30 1 11 /30 −1/6 −1/6 1/6   31  = 1 /30  46 −2 −10  and a solution of this linear system of equations is in the form of x = xr + cxns , with c any real constant. • Unique solution: This is the case in which there are the same number of variables as equations, meaning m = n. In this case, there are several possibilities: – If matrix A is non-singular (invertible), then there exists a unique solution: ˆx = A−1b (1.7.4) – If matrix A is singular (non-invertible), then rank( A)= r < (m=n), and there exists infinity number of solutions given by: ˆx = A†b + xns (1.7.5) with xns ∈N S (A), and A† the pseudinverse of A. • Overdetermined case: This is the case in which there are fewer variables than equations, meaning m > n . In this case, there exists no solution. The best that can be done is to make the error e=Ax − b as small as possible. Since Ax can never leave the CS, we have to find a vector x such that Ax is closest to b, or in other words this point is the projection p= Aˆx of b in the CS. In this way e=b-p is the smaller length if: minimize ∥Ax − b∥22 variable x (1.7.6) 12 This vector is given (we will prove it in TOML-MIRI) by the left pseudo-inverse A† = ( A⊤A)−1A⊤ (1.7.7) We call it left pseudo-inverse because A†A= I (A† is on the left). How-ever, we can derive it from the interpretation of Figure 1. We have to note that e=b-p is perpendicular to the CS, so e is in the left null space. Then: A⊤e = A⊤(Ax − b) = 0 (1.7.8) From here, we obtain: A⊤Ax = A⊤b (1.7.9) And finally: ˆx = ( A⊤A)−1A⊤b = A†b (1.7.10) We will come back to a derivation of the left pseudo-inverse when we study the singular value decomposition (SVD). Example 1.12 (Overdetermined case) Let us consider the following linear system: Ax = b →  2 11 30 1 x1 x2  =  314  The solution xr is given by x= (A⊤A)−1A⊤b: x = ( A⊤A)−1A⊤b = ( 2 1 01 3 1   2 11 30 1 )−1 2 1 01 3 1   314  and: x = 11 /30 −1/6 −1/6 1/6   2 1 01 3 1   314  = 1 /30 27 15  and a solution of this linear system of equations is in the form of x =1/30[27 , 15] = [0 .9, 0.5] . 2 Eigenvectors and eigenvalues An eigenvector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The 13 corresponding eigenvalue λ is the factor by which the eigenvector is scaled. The eigenvector points in a direction in which the vector is scaled by the trans-formation and the eigenvalue is the factor by which the vector is scaled. Amain application is the decomposition of a matrix by using eigenvectors and eigenvalues (called de eigenvector decomposition, EVD). Other applications ap-pear when solving differential equations, dimensionality reduction (e.g. princi-pal component analysis, PCA), denoising (e.g. eigenfaces), data compression, spectral graph theory, signal reconstruction, etc. The EVD (applied to squared matrices) is connected to another important decomposition called singular value decomposition (SVD) when the matrices are not-squared. 2.0.1 Linear transformations If T is a linear transformation on a vector space over itself, i.e., T :V → V , an eigenvector is a vector that satisfies: T (v) = λv (2.0.1) Assuming that the linear transformation (as seen in previous lectures) can be related to a matrix. Let, then, A∈Rn×n be a square real matrix (although the matrix A has real components, in this section is better to think that vectors can have complex coefficients and that scalars are also in general complex numbers). A non zero vector v is an eigenvector , and the scalar λ is an eigenvalue of the matrix A when: Av = λv (2.0.2) Eigenvalues must fulfill the condition: Av − λv = ( A − λI)v = 0. As v is non-zero, this is only possible if rank( A) < n , which is equivalent to the condition det( A − λI) = 0. This determinant is in general a polynomial on λ of degree n, meaning that has n complex roots (if we count the multiplicity of roots). For very small matrices we can find the roots of this polynomial to compute eigenvalues. For larger matrices, there are more computationally efficient methods. Once we know the eigenvalues, we can find the associated eigenvectors by solving the undetermined system of linear equations ( A − λI)v = 0. We can set, for instance, the condition ∥v∥ = 1 to find unique solutions (up to the sign). In the case of eigenvalues of multiplicity larger than 1, we can have several linearly independent associated eigenvectors. The dimension of the generated subspace must be less or equal to the multiplicity of the root. When this di-mension is strictly lower than the multiplicity of λ, we say that the matrix is defective . You can find detailed discussions on this in any text on linear al-gebra. We will be interested mainly in symmetric matrices, which are never defective. 14 2.1 Diagonalization and eigendecomposition Let A be a square n×n matrix with n linearly independent eigenvectors vi (where i=1, . . . , n). Then A can be factorized as: A = VΛV −1 (2.1.1) where V is a squared n×n matrix with column vectors vi (eigenvectors), and Λ is a squared n×n diagonal matrix whose diagonal elements Λ ii =λi (are the eigenvalues). This is easy to see since if v i an eigenvector: Av = λvAV = VΛ A = VΛV −1 (2.1.2) See the following property. The linearly independent eigenvectors associated with the eigenvalues different of 0, form the range(A) or Image(A), and they are the basis of the column space CS(A). The linearly independent eigenvectors associated with eigenvalues equal to 0, form the basis of the null space of A. 2.2 Some important types of matrices • An square n×n matrix Λ with 0 off-diagonal elements is a diagonal ma-trix . The product Λ v produces a stretching (or directional scaling) of the different components of the vector v according to the corresponding values of the diagonal. If some components of the diagonal of Λ are zero, the product collapses to zero the corresponding components of v. The inverse of Λ is obtained by simply inverting the diagonal elements (if any of those elements is zero, then the matrix is not invertible). • A square matrix Q with columns that are orthonormal vectors (i.e. or-thogonal and with norm 1) is called an orthonormal matrix (or very often simply orthogonal matrix, as we will assume the norm 1 condi-tion). The product Qv produces a rotation, a reflection, or a combination of both operations (called roto-reflection), on the vector v. Orthonormal matrices are always invertible and Q−1 = Q⊤. • A square matrix A that fulfills A=A⊤ is a symmetric matrix . Sym-metric matrices satisfy the following properties: i) sum (or difference) of symmetric matrices is symmetric, ii) if A and B are symmetric, then AB is symmetric only if AB =BA ; iii) if A−1 exists, then it is symmetric if and only if A is symmetric; • If v is a column vector of V , the square n×n matrix vv ⊤ is a matrix (do not confuse with the scalar product v⊤v which is a number) of rank 1. 15 • We define a projection as a linear operator P :V → V such that P 2=P .If the vector space V is finite-dimensional, a square matrix P is called projection matrix if P 2=P . Moreover, if P is real, and P 2=P =P ⊤ then P is called a orthogonal projection matrix . For the general case of a non-unitary vector v, the projection matrix is defined as: P = v < v, v >−1 v⊤ (2.2.1) so the projection of a vector t on the vector v, would be given by: ˆt = P t = v < v, v >−1 v⊤t (2.2.2) Example 2.1 (Projection matrix for a vector) We want to project vector t= [1 , 2] over vector v= [2 , 1] . We first obtain the projection matrix P : P = 1 ∥v∥22 vv ⊤ = 15 21  2 1 = 15 4 22 1  Then, now, the projection of vector t on the vector v, would be given by: ˆt = P t = 15 4 22 1   12  = 15 64  = 1.20.8  and the projected vector will be ˆt = [1 .2, 0.8] . We can observe that the matrix P satisfies property P 2=P =P ⊤: P 2 = 15 4 22 1  15 4 22 1  = 125 20 10 10 5  = 15 4 22 1  Finally, observe that since we project a vector over a vector, matrix P has rank r = 1 . In the special case in which v is unitary, < v, v >=1, and P = vv ⊤. The projection of vector t on the unitary vector v will then be ˆt = P t = vv ⊤t.In the case that the we want to project a vector t on a subspace generated by the matrix A, then the projection matrix will be given by: P = A(A⊤A)−1A⊤ (2.2.3) and the projection of the vector t on the subspace generated by the matrix A will be given by: ˆt = A(A⊤A)−1A⊤t (2.2.4) Example 2.2 (Projection matrix for a matrix) We want to project vector t= [1 , 2] over the space generated by matrix: A =  2 11 30 1  16 The projection matrix P will be given P = A(A⊤A)−1A⊤: P =  2 11 30 1  ( 2 1 01 3 1   2 11 30 1 )−1 2 1 01 3 1  = 130  29 2 −52 26 10 −5 10 5  and we can observe that matrix P has rank r = 2 (the same as matrix A). 2.3 An n×m matrix of rank r maps a sphere of dimension n into an ellipsoid of dimension r A basic fact of linear algebra is the following: Assume that we have an n×m matrix A with rank( A)= r. Assume that we compute the products y = Ax ,where x is an n-dimensional vector that lies in a sphere of radius 1 in the space Rn. Then the locus (set of points) of all the generated vector y lies in an ellipsoid of dimension r embedded in the space Rm.As an example, assume a 2 ×2 matrix A of rank 2. If we compute y = Ax for x = ( cos (θ), sin (θ)) ⊤ with θ ∈ [0 , 2π), the vector y will lie in an ellipsoid centered in the origin in R2. If rank( A)=1, the ellipsoid will collapse one of its dimensions, resulting in a segment that crosses the origin. 2.4 Matrix factorization In this course, we deal with two important matrix factorizations: • S = QΛQ⊤, for symmetric matrices. • A = U ΣV ⊤, Singular Value Decomposition (SVD) for general matrices. We are interested in the first factorization as covariance matrices are symmetric, and we are interested in the second factorization as it allows us to approximate clouds of points in high-dimensional spaces by clouds of points in lower dimen-sional spaces. 2.5 Diagonalization of symmetric matrices Assume that S is an n×n symmetric matrix , i.e. S = S⊤, i.e., aij = aji ∀i, j .Symmetric matrices have the following properties: i) S1 +S2 (sum) is symmetric if S1, S2 are symmetric, ii) S1S2 (product) is not necessarily symmetric even if S1, S2 are symmetric, iii) if S−1 exists, is symmetric if and only if S1 is symmetric. The spectral theorem tells us when a linear operator or matrix can be diag-onalized. In the case of S (symmetric matrix), the finite-dimensional spectral 17 theorem says that any symmetric matrix S whose entries are real can be diag-onalized by an orthogonal matrix (so, the eigenvectors are orthonormal). • S has n real eigenvalues, λi, (counting possible multiplicities) • The n associated eigenvectors, qi are orthonormal. • These matrices are not defective (defective means that rank( A)=r <n, then rank( S)=n) if they are positive definite. Let us prove it for the case in which the matrix has non-repeated eigenvalues. This result can be extended for the repeated eigenvalues case by using continuity arguments. 2.5.1 Eigenvectors are orthogonal Assume that Sv = λv and Su = μu different eigenvalues λ and μ. We have: v⊤Su = μv⊤u (2.5.1) and u⊤Sv = λu⊤v (2.5.2) but u⊤Sv = v⊤S⊤u = v⊤Su as S is symmetric, and u⊤v = λv⊤u, which implies λu⊤v = μu⊤v, and from this we get u⊤v = 0 (eigenvectors are orthog-onal). 2.5.2 Eigenvalues are real Assume that λ is a complex eigenvalue. As S is real, λ∗ (i.e. its complex conjugate) must also be an eigenvalue: Sv = λv (2.5.3) and: Sv ∗ = λ∗v∗ (2.5.4) Then we have: v∗T Sv = λv∗T v = λ (2.5.5) since eigenvectors v are orthonormal ( v∗v = 1), and: vT Sv∗ = λ∗vT v∗ = λ∗ (2.5.6) 18 But λ∗ = vT Sv ∗ = ( vT Sv ∗)⊤ = v∗T Sv = λ, meaning λ = λ∗.As a consequence, we can write S as: S=QΛQ⊤, where Q is an n×n matrix with columns the eigenvectors of S, Q=[ q1,. . . , qn], and Λ is a diagonal matrix with diagonal elements Λ i,i = λi.An alternative way of expressing this is by the formula: S = P i λiqiqit. Recall that the terms qiqit are rank 1 matrices. If rank (S)= n, the eigenvalues λi must be different from 0 (if not, the corre-spondent eigenvector qi would belong to the null space and rank(S) <n). In this case, S has an inverse, which is also a symmetric matrix, and: S−1=QΛ−1Q⊤ =P i1 λi qiqi⊤. Example 2.3 (Symmetric matrices) Let us assume the following symmet-ric matrix A: A =  2 1 0 11 3 4 50 4 1 41 5 4 4  Then the eigendecomposition of matrix A will A = QΛQ⊤, with: Q =  −0.128 −0.978 −0.155 −0.051 −0.596 0.021 0.553 −0.582 −0.463 0.204 −0.812 −0.29 −0.643 0.028 0.103 0.758  and Λ =  11 .712 0.0 0.0 0.00.0 1.95 0.0 0.00.0 0.0 −2.228 0.00.0 0.0 0.0 −1.434  and as we can observe, all eigenvalues are real, and the eigenvectors are oth-ornormal; q⊤ i qi = 1 and q⊤ i qj = 0 with i̸ =j, e.g. q⊤ 0 q0= [−0.128 , −0.596 , −0.463 , −0.643] ⊤ [−0.128 , −0.596 , −0.463 , −0.643] = 1.0 and q⊤ 0 q2= [−0.128 , −0.596 , −0.463 , −0.643] ⊤ [−0.155 , 0.553 , −0.812 , 0.103] = 0.0.Observe also that the matrix Q is a rotation matrix; for example Qq i = [0 , . . . , 1, . . . , 0] , a unitary vector with a 1 at position i, 0’s in the remainder. 2.6 Positive definite matrices When all eigenvalues S are positive, we say that the symmetric matrix is pos-itive definite . In this case, S−1 is also a symmetric positive definite matrix. These matrices somehow play the role of positive numbers in the matrix world. 19 For example, the variance-covariance matrix Σ = E[( X1, .., X n)( X1, ..., X n)⊤]of multivariate Gaussian distributions is a positive definite matrix. The inverse of the covariance matrix, called precision matrix is also positive definite. • An n×n symmetric real matrix A is said to be positive definite if x⊤Ax >0 (2.6.1) for all non-zero vectors x∈Rn and it is said negative definite if x⊤Ax <0. (2.6.2) • An n×n symmetric real matrix A is said to be positive semi-definite if x⊤Ax ≥0 (2.6.3) x⊤Ax ≥0 for all non-zero vectors x∈Rn and it is said negative semi-definite if x⊤Ax ≤0 (2.6.4) • An n×n symmetric real matrix which is neither positive semi-definite nor negative semi-definite is called indefinite. • A matrix A is positive (negative) definite if and only if all of its eigenvalues are >0 ( <0). • A matrix A is positive (negative) semi-definite if and only if all of its eigenvalues are ≥0 ( ≤0). You can find several interesting properties such as: i) if A and B are positive definite matrices, then the sum A + B is a positive definite matrix; or ii) if A is a positive definite matrix, then the inverse A−1 is a positive definite matrix. Finally, an interesting result is the following: let be a m×n A matrix. The matrix S=A⊤A is positive definite (and then symmetric), and therefore S has orthonormal eigenvectors and positive eigenvalues. Example 2.4 (Positive definite matrices) Let us consider matrices A, B, C and D. Check positive definiteness. A = −1 1 −11 0 10 1 1  ; B = −3 0 −10 −5 −2 −1 −2 −3  C =  3 1 −10 4 −2 −1 −2 3  ; D = 3 1 −10 4 −23 4 −3  20 Let us obtain the eigenvalues. The eigenvalues of matrix A are λ=[−1.879 , 0.347 , 1.532] . The matrix is neither positive or negative definite. The eigenval-ues of matrix B are λ=[−1.319 , −3.358 , −6.323] are all negative, and then the matrix is negative definite. The eigenvalues of matrix C are λ=[1 .097 , 3.194 , 5.709] are all positive, and then the matrix is positive definite. The eigenval-ues of matrix D are λ=[0 .0, 0.586 , 3.414] are all positive or equal to zero, and then the matrix is positive semidefinite. Moreover, since there are one eigen-value equal to zero, the rank of this matrix is 2. On the other hand, the rank of matrices A, B, C is 3. 3 Useful properties of matrices 3.1 Geometric interpretation for symmetric matrices: Ro-tation/Reflection, Stretching, Rotation/Reflection −1 Figure 3: Symmetric matrix applied to a disc of radius 1. We know that matrix multiplication can be interpreted as linear map com-position, meaning that the geometric interpretation of Sv = QΛQ⊤v for an arbitrary vector v is: • Rotate the coordinate system to align it with the set of vectors qi which form the columns of the matrix Q. The vector v in this new coordinate system has the expression Q⊤v (note that the vectors qi expressed in the 21 new coordinate system have the expression Q⊤qi = [0 , ..., 1, ... 0] as we expect). • Stretch each component of the resulting vector Q⊤v according with the diagonal elements of the matrix Λ, obtaining the vector Λ Q⊤v. This stretching causes in general a change of direction of the vector, but if v is aligned with a vector qi, it does not change its direction. • Apply the inverse rotation to the coordinate system. If we express the resulting Λ Q⊤v in this new coordinate system we obtain QΛQ⊤v.For instance, a positive definite matrix S will map an n-dimensional sphere of radius 1 to an n-dimensional ellipsoid with axis given by its eigenvectors qi, and axis lengths given by its eigenvalues λi, (think on this, Figure 3). 3.2 Derivatives with vectors and matrices First, some matrix manipulations typically appear when working with differen-tiation. • (AB )−1 = B−1A−1 • (AB )⊤ = B⊤A⊤ • (a⊤Ab )⊤ = b⊤A⊤a • a⊤b = b⊤a • (A + B)C = AC + BC • AB ̸ = BA • a⊤b = b⊤a • (a + b)⊤C = a⊤C + b⊤C The Hadamard product (also known as the element-wise product, entry-wise product, or Schur product) returns a matrix of the multiplied corresponding elements. It is defined with the symbol ⊙ (sometimes also with symbol ◦): (A ◦ B)ij = ( A ⊙ B)ij = ( A)ij (B)ij . (3.2.1) Example 3.1 (Hadamard product of matrices) Consider matrices A and B. A =  2 1 −21 0 20 1 3  ; B =  −3 0 −12 −5 −2 −1 2 4  22 Its Hadamard product is given by: A ⊙ B =  2 1 −21 0 20 1 3  ⊙  −3 0 −12 −5 −2 −1 2 4  =  −6 0 22 0 −40 2 12  Let us assume vector x and we want to obtain vector derivatives over the func-tion f( x) of the form df (x)/d x. We use the denominator layout (meaning that f ⊤ and x) • f( x) = x⊤a −→ df (x)/d x = a • f( x) = x⊤A −→ df (x)/d x = A • f( x) = Ax −→ df (x)/d x = A⊤ • f( x) = x⊤x −→ df (x)/d x = 2 x • f( x) = x⊤Ax −→ df (x)/d x = 2 Ax if A is symmetric • f( x) = x⊤Ax −→ df (x)/d x = ( A + A⊤)x 3.3 The trace operator For a square matrix n×n A we define the trace as the sum of the elements of its diagonal: tr( A) = n X k=1 akk = a11 + a22 + · · · + ann (3.3.1) The trace of the n×n I identity matrix is the dimension of the space, namely n: tr ( In) = n. The following relationships are satisfied: tr( A + B) = tr( A) + tr( B)tr( cA) = c tr( A) (3.3.2) for all square matrices A and B, and all scalars c. Moreover: tr( A) = tr AT (3.3.3) 3.3.1 The trace of a matrix is the sum of its eigenvalues counting multiplicities As we know, the eigenvalues λk are the solutions of the equation: det (A − λI ) = ( −1) nλn + ( −1) n−1tr (A)λn−1 + ... = 0 (3.3.4) 23 The eigenvalues λk are the roots of the polynomial in λ, meaning that we have: (−1) n(λ − λ1)( λ − λ2)... = 0 (3.3.5) and from this we obtain: X k λk = tr (A) (3.3.6) Example 3.2 (Symmetric matrices) Let us assume the following symmet-ric matrix A: A =  2 1 0 11 3 4 50 4 1 41 5 4 4  The eigenvalues are λ=[11 .712 , 2.228 , 1.95 , 1.434] . We can observe that P i λi= 11 .712 + 1 .95 + ( −2.228) + ( −1.434) = 10 .0 and that tr (A) = 2 + 3 + 1 + 4 = 10 3.3.2 The trace operator is cyclic Let A and B be general non-square matrices of sizes n×m and m×n. The diagonal elements of P = AB can be found as: pkk = X i=1 ,...,m aki bik (3.3.7) while that for Q = BA we have: qii = X k=1 ,...,n aik bki (3.3.8) We have then tr (AB ) = X k=1 ,..,n X i=1 ,...,m aki bik = X i=1 ,..,m X k=1 ,...,n aki bik = tr (BA ) (3.3.9) If we have now three arbitrary matrices of the right sizes to produce a square matrix in its product BAC we have: tr (BAC ) = tr (( BA )C) = tr (C(BA )) = tr (CBA ) (3.3.10) Moreover for real column vectors a ∈ Rn and b ∈ Rn, the trace of the outer product is equivalent to the inner product: tr ba T = aTb (3.3.11) 24 3.3.3 Derivatives of a trace Let A∈Rn×m and X∈Rm×n matrices. Then: ddX tr (AX ) = ddX tr (XA ) = A⊤ (3.3.12) Moreover, if A, X∈Rn×m: ddX tr (AX ⊤) = ddX tr (X⊤A) = A (3.3.13) Then, assuming correct matrices sizes A, B, X: ddX tr (AXB ) = ddX tr (BAX ) = (BA )⊤ (3.3.14) On the other hand, for deriving ( X⊤AX ) with respect to X, we first fix one of the X and then the other (e.g. fix one X and substitute the other X by Y and derive with respect Y , and then repeat exchanging the order): ddX tr (X⊤AX ) = ddY tr (Y ⊤AX )+ ddY tr (XAY ) = (A + A⊤)X (3.3.15) Using these rules, we can obtain the derivative of more complex trace expres-sions. 3.4 Quadratic forms, sub-level sets, paraboloids and ellip-soids A quadratic function has the form of f (x) = 12 x⊤P x + b⊤x + c (3.4.1) where P is a n×n symmetric matrix, b and x are n-dim vectors, and c is a real number. Define now a α sub-level set C α of a function f :Rn → R, as Cα = {x∈dom {f }| f (x) ≤ α} (3.4.2) Let us now remember the equation of an n-dim ellipsoid as x21 a21 x22 a22 · · · + x2 n a2 n = 1 Thus, we can see that the α sub-level set of a quadratic form is an ellipsoid. In fact, ϵ = {x|(x − xc)⊤P −1(x − xc) ≤ 1} (3.4.3) 25 where P is symmetric and positive definite, and the vector xc is the center of the ellipsoid. The matrix P defines how far the ellipsoid extends in every direction from xc (directions given by eigenvectors of P ), and the length of the semi-axes of ϵ are given by √λi (with λi the eigenvalues of P .If we consider now a new dimension xn+1 , and make xn+1 = x21 a21 x22 a22 · · · + x2 n a2 n (3.4.4) we obtain the equation of a paraboloid. 3.5 Multivariate Gaussian distribution Recall that we say that X follows a multivariate gaussian distribution of pa-rameters μ and Σ, where μ∈Rn and Σ is a positive definite matrix , if the joint probability density function of X is of the form: fX (x) = 1(2 π)n/ 2|Σ|1/2 e− 12 (x−μ)⊤Σ−1(x−μ) (3.5.1) Many times, we will express the multivariate gaussian distribution as: p{x|μ, Σ} = N {x|μ, Σ}, or X ∼ N {μ, Σ}. 3.5.1 The quadratic form 12 (x − μ)⊤Σ−1(x − μ)If S is a definite positive matrix, then the graph of the quadratic form z = 12 (x − μ)⊤Σ−1(x − μ) (3.5.2) is a n + 1 dimensional paraboloid that takes always values of z which are non-negative and the only point at which z = 0 is μ. Example 3.3 (Precission matrix Σ−1) For instance, let Σ =  32 − 12 − 1232  .To diagonalize Σ, we find the eigenvalues and orthonormal eigenvectors: Σ = " √22 − √22√22 √22 1 00 2  " √22 √22 − √22 √22 Meaning that: Σ−1 = " √22 − √22√22 √22 1 00 12  " √22 √22 − √22 √22 =  34141434  . Figure 4 we plot the 3-d parabole 12 (x − μ)⊤Σ−1(x − μ) for μ = [5 , 5] ⊤:26 Figure 4: Paraboloid of a quadratic form given in the example. 3.5.2 Isocontour lines (or α-level sets) We define the isocontour lines (or α-level sets) as the surface that represents points of a constant value within a volume of space: (x − μ)⊤Σ−1(x − μ) = c2 (3.5.3) The equation of an ellipsoid centered at the origin and of semi-axis given by ai oriented according to the orthonormal vectors qi is x⊤Q  1 a21 0 . . . 00 1 a22 . . . 00 0 . . . 0 . . . . . . . . . 1 a2 n  Q⊤x = 1 (3.5.4) If we plot the geometrical locus of the points that fulfill the equation: (x − μ)⊤Σ−1(x − μ) = c2 we would obtain an n dimensional ellipsoid, centered at the point μ, with axis aligned with the (orthonormal) eigenvectors qi of the matrix Σ, and with semi-axis length in the axis pointed by qi equal to c√λi, where λi is the eigenvalue associated with qi.27 Figure 5: Ellipse equations in terms of a quadratic form. Figure 6: Ellipse equations in terms of eigenvalue-eigenvectors. 3.5.3 Isotropic Gaussian distribution The special case of having μ = 0 and Σ=I, we say that we have the multivariate standard normal distribution , and if Σ= σ2I, the distribution is called a isotropic Gaussian distribution , meaning that instead of ellipsoids, we will obtain hyper-spheres. 28 3.5.4 Datasets generated from independent sampling of a multivari-ate Gaussian distribution If we perform a number of independent sampling of a multivariate Gaussian distribution, we will obtain clouds of points following the previously described ellipsoids: 3.5.5 Expressing 12n x⊤Σ−1x as the trace of the product of two ma-trices Assume that x is a vector with 0 mean (otherwise, we would use x−m instead). Scalars are special cases of square matrices, meaning that 12n x⊤Σ−1x = tr ( 12n x⊤Σ−1x) = tr ( 12n xx ⊤Σ−1) = 12 tr (SnΣ−1) (3.5.5) where Sn = 1 n xx ⊤ is the sample covariance-variance matrix. 3.5.6 The term |Σ|1/2 The determinant of a matrix is the product of its eigenvalues, meaning that |Σ|1/2 = ( Q i λi)1/2. 4 The Singular Value Decomposition (SVD) The diagonalization of a symmetric matrix is an extremely important result, that tells us that a symmetric matrix has as a ”core” a diagonal matrix with real diagonal elements. A similar result can be generalized for some other square matrices, but it cannot be applied to general (possibly non-square) matrices. There is however another factorization that can be applied to general matrices, even for non-square matrices which is known as Singular Value Decomposition 29 (SVD), that has a lot of applications, including dimensionality reduction appli-cations (principal component analysis, PCA), obtaining the effective rank of a matrix (closest rank approximation of a matrix, e.g. Eckart-Young theorem), calculate the generalized inverse of a matrix (pseudoinverse), or in linear least squares problems. 4.1 The SVD As we have seen if S is a symmetric n×n matrix, we can find a set of orthonormal vectors uk, which are left and right eigenvectors of S associated with the real eigenvalues λk, meaning that they fulfill the equations: Su i = λiui, i ∈ { 1, .., n } (4.1.1) The SVD generalizes these equalities for a general m×n (i.e. m rows and n columns) matrix A∈Rm×n of rank r≤min (m, n ). The idea is to find orthonor-mal matrices U ∈Rm×m and V ∈Rn×n and diagonal matrix Σ∈Rm×n, such as A = U ΣV ⊤ (Figure 7). The elements of the diagonal matrix Σii = σi are real positive numbers called singular values and which we will order as a nonincreas-ing order σ1 ≥ σ2 ≥ ... ≥ σr ≥ 0. Σ =  Σr 00 0  (4.1.2) with Σr =diag( σ1, . . . , σr ). For example, let us see the structure of matrix Σ for rank 1 and 2 in a 5 ×5 matrix:  σ1 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0  ;  σ1 0 0 0 00 σ2 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0  (4.1.3) We can observe the similitude with the EVD (eigenvector decomposition) if put matrix V on the left and see that it is satisfied the following expression: Av i = σiui, i ∈ { 1, .., n } (4.1.4) Surprisingly, we will see that this decomposition is always possible . The values σk are called singular values of A, while the columns of U and V are called the left and right singular vectors of A.Remember that orthonormality of singular vectors mean that U U ⊤ = U ⊤U = Im (4.1.5) and V V ⊤ = V ⊤V = In (4.1.6) 30 Figure 7: SVD for a m×n A matrix. 4.2 Relation between EVD (eigenvalue decomposition) and SVD (singular value decomposition) Let us have a non-symmetric real matrix A∈Rm×n, and assume a SVD as A = U ΣV ⊤. We can see that the following relationships hold: • Case A⊤A: in this case we obtain a n×n square matrix and: A⊤A = ( U ΣV ⊤)⊤U ΣV ⊤ = V Σ⊤U ⊤U ΣV ⊤ = V Σ⊤ΣV ⊤ (4.2.1) which says that A⊤A has as eigenvectors the columns of V (right singular vectors). • Case AA ⊤: in this case we obtain a m×m square matrix and: AA ⊤ = U ΣV ⊤(U ΣV ⊤)⊤ = U ΣV ⊤V ⊤Σ⊤U ⊤ = U ΣΣ ⊤U ⊤ (4.2.2) which says that AA ⊤ has as eigenvectors the columns of U (left singular vectors). Finally, we can observe that the eigenvalues of A⊤A and AA ⊤ are the squares of the singular values of A: σ2 i =λi or σi=√λi. 4.3 Selecting the orthonormal basis in the SVD Let us see how do we select the orthonormal basis {u1, . . . , um} for the range and the orthonormal basis {v1, . . . , vn} for the domain (remember that we have a function f :Rn → Rm associated to matrix A∈Rm×n). Let us assume that rank( A)= r; then, we will have: 31 • The left eigenvectors u1, . . . , ur with ui∈Rm are the orthonormal basis of the Image( A)= CS( A); • The left eigenvectors ur+1 , . . . , um with ui∈Rm are the orthonormal basis of the Left Null Space LNS( A)= NS( A⊤); U = [ U r | ˜U m−r ] = [ u1, . . . , ur | {z } Ur , ur+1 , . . . , um | {z } ˜Ur ] (4.3.1) • The right eigenvectors v1, . . . , vr with vi∈Rn are the orthonormal basis of the row space RS( A); • The right eigenvectors vr+1 , . . . , vn with vi∈Rn are the orthonormal basis of the null space NS( A); V = [ V r | ˜V n−r ] = [ v1, . . . , vr | {z } Vr , vr+1 , . . . , vn | {z } ˜Vr ] (4.3.2) We first assume symmetric matrix A⊤A∈Rn×n, and its eigenvalue decomposi-tion (obtain its eigenvalues and eigenvectors since it is a squared n×n matrix): A⊤A = V DV ⊤, where D is a diagonal matrix with eigenvalues λi arranged in nonincreasing order ( λ1 ≥ λ2 ≥ · · · ≥ 0), and the columns of V (eigenvectors of A⊤A) are the orthonormal vectors {v1, . . . , vn}. We can observe that vectors {v1, . . . , vn} form a basis of Rn. Moreover, since our matrix A has rank( A)= r,the right singular vectors associated with non-zero singular values (there are r of them) form a basis of RS( A), while the n-r right singular vectors associated with the zero n-r singular values form a basis of the NS( A). Remember that the Im (A) = Range (A) = {y∈Rm|y = Ax f or some x∈Rn},and then: Av i·Av j = ( Av i)⊤·Av j = vi⊤(A⊤Av j ) = vi⊤(λj vj ) = λj vi⊤vj = 0 (4.3.3) since vi and vj are orthonormal. Then Av i and Av j also are orthogonal (not necessarily orthonormal). Then the eigenvectors of A⊤A and their images under A form a basis for the range or image of A (or the column space CS(A)). We must check now that the vectors uk defined as Avk = σkuk are also or-thonormal: u⊤ i uj = 1 σiσj v⊤ i A⊤Avj = σ2 j σiσj v⊤ i vj = 0 , i̸ = j (4.3.4) In order to complete the orthonormal bases of the range, we have to normalize, thus: ui = Av i |Av i| = Av i √λi = Av i σi ; 1 ≤ i ≤ r (4.3.5) 32 and defining σ2 i = λi, we obtain Av i = σiui, with 1 ≤ i ≤ r, which shows that AV = U Σ taking ui the columns of U and vi the columns of V and Σii = σi.Finally, we can express the matrix A as: A = [ U r , ˜U r ]  Σr 00 0   V r ˜V r  (4.3.6) The dimensions of the matrices are U r ∈Rm×r , ˜U r ∈Rm×(m−r), V r ∈Rn×r , and ˜V r ∈Rn×(n−r). Observe that the following conditions are satisfied: U ⊤ r U r = Ir ˜U ⊤ r ˜U r = Im−r U ⊤ r ˜U r = 0 U ⊤ r U r + ˜U ⊤ r ˜U r = Im V ⊤ r V r = Ir ˜V ⊤ r ˜V r = In−r V ⊤ r ˜V r = 0 V ⊤ r V r + ˜V ⊤ r ˜V r = In (4.3.7) Example 4.1 (SVD example) Let’s calculate the SVD for matrix A∈R3×2: A =  2 13 01 3  =  −0.531 −0.113 −0.839 −0.595 −0.654 0.466 −0.602 0.748 0.279  4.169 00 2.572 0 0 −0.560 0.828 0.828 0.56  where we have used the full matrix decomposition; A = U ΣV ⊤ . If we consider the SVD with rank r=2, we can use the economy SVD A = U r Σr V ⊤ r : A =  2 13 01 3  =  −0.531 −0.113 −0.595 −0.654 −0.602 0.748 4.169 00 2.572   −0.560 0.828 0.828 0.56  It can be easily seen that u⊤ i ui = 1 and u⊤ i uj = 0 with i̸ =j, e.g. u⊤ 0 u0= [−0.531 , −0.595 , −0.602] ⊤[−0.531 , −0.595 , −0.602] = 1 .0 and u⊤ 0 u1= [−0.531 , −0.595 , −0.602] ⊤ [−0.113 , −0.654 , 0.748] = 0.0. 4.4 Geometric interpretation Remember that we have a function f :Rn → Rm associated to matrix A∈Rm×n.What happens if we apply matrix A to a unit sphere Rn (we assume m≤n) ? Let us assume that x∈Rn are the vectors in the unit sphere. Remembering that when we apply an orthonormal matrix to a vector we rotate the vector, then: Ax = U ΣV ⊤x (4.4.1) which means that we first apply a rotation V ⊤ to vectors x (still in Rn), then we stretch or shrink in each direction (since we multiply vectors by singular values larger or than 0), producing an ellipsoid in Rm, and finally we rotate again using U in Rm.In other words, we produce a linear mapping in which a sphere in Rn is converted to an ellipsoid in Rm (make some figures showing the geometric interpretation). 33 4.5 SVD, pseudo-inverse and projection matrices Let us consider a linear system of equations: Ax = y (4.5.1) where A∈Rm×n, x∈Rn and y∈Rm.If m=n we have a two-sided inverse of matrix A, i.e., AA −1 = A−1A = I,that is what we call inverse of A if the nullspace( A) and nullspace( A⊤) only contain the zero vector. Let us consider the case in which the rank( A)= r=n, and then the nullspace( A)only contains the zero vector. In this case m>n (overdetermined case ) in which case we can not produce an inverse. However, A⊤A has inverse since it is a symmetric n×n matrix. From here and the SVD we can deduce what is called the left pseudo-inverse : A−1 lef t = A† = ( A⊤A)−1A⊤ (4.5.2) Deduction: Ax = y → A⊤Ax = A⊤y (4.5.3) We can observe that: A⊤A = V Σ⊤ΣV ⊤, and: V Σ⊤ΣV ⊤x = A⊤y → x = V (Σ⊤Σ)−1V ⊤A⊤y (4.5.4) since ( A⊤A)−1 = V (Σ⊤Σ)−1V ⊤, we conclude that: x = ( A⊤A)−1A⊤y = A−1 lef t y (4.5.5) Note that A−1 lef t A = In and that AA −1 lef t only is In if n = m. As a statement, a non-symmetric matrix can not have a two-sided since A or A⊤ has null-space different of the zero vector. In the same way, let us consider the case in which the rank( A)= r=m, and then the nullspace( A⊤) only contains the zero vector. In this case m<n (underde-termined case ) in which case we can not produce an inverse. However, AA ⊤ has inverse since it is a symmetric n×n matrix. From here and the SVD we can deduce what is called the right pseudo-inverse : A−1 right = A† = A⊤(AA ⊤)−1 (4.5.6) Note that AA −1 right = Im and that A−1 right A only is Im if n = m. As a statement, a non-symmetric matrix can not have a two-sided since A or A⊤ has null-space different of the zero vector. Finally, If A is full column rank ( r = n), and A−1 lef t = ( A⊤A)−1A⊤, then matrix P = AA −1 lef t = A(A⊤A)−1A⊤ (4.5.7) 34 projects Rm into the column space of A.In a similar way, If A is full row rank ( r = m), and A−1 right = A⊤(A⊤A)−1,then matrix P = A−1 right A = A⊤(A⊤A)−1A (4.5.8) projects Rn into the row space of A. What is the pseudo-inverse in terms of the SVD? The pseudo-inverse is a matrix that satisfies that x = AA †x. In other words: U ΣV ⊤A† = I, and remembering that matrices U and V are othornormal (and easy to invert), then A† = U ⊤Σ†V . The best approximation of Σ† is a matrix with diagonal values for i = 1 , . . . , r for the first r rows (or columns), with r the rank of A and the rest of diagonal values are zero. Example 4.2 (Pseudo-inverse using the SVD) Remembering example 1.12, we obtained the pseudoinverse of matrix AAx = 2 11 30 1  as A† = ( A⊤A)−1A⊤: A† = ( 2 1 01 3 1   2 11 30 1 )−1 2 1 01 3 1  =  0.567 −0.133 −0.167 −0.167 0.333 0.167  then, if we want to solve Ax = b, with b = [3 , 1, 4] , we obtain x=A†b= [0 .9, 0.5] .Using the SVD, A† = U ⊤Σ†V . Since (see example 4.1) U =  −0.499 0.847 0.183 −0.834 −0.413 −0.365 −0.234 −0.335 0.913  ; Σ =  3.719 00 1.473 0 0  ; V −0.493 −0.87 0.87 −0.493  and considering the inverse of the Σ matrix ( Σ†) as: Σ† =  0.268 00 0.679 0 0  Then, now: A† = U ⊤Σ†V =  0.567 −0.133 −0.167 −0.167 0.333 0.167  35 4.6 Economy, compact and truncated SVD Remember that matrix Σ is a diagonal matrix with r singular values in positions Σii , with i = 1 , . . . , r and zeros otherwise: Σ =  σ1 0 0 ... 0 00 σ2 0 ... 0 0 ... ... ... ... ... ... 0 0 ... σr 0 ... 0 0 ... ... 0 ... 0 0 ... ... ... 0  (4.6.1) while V = [ v1, ..., vn] and U = [ u1, ..., um]. Let us assume that k = min (m, n ). Then, A can be written as: A = U ΣV ⊤ = [ σ1u1, ..., σ kuk, 0, ..., 0][ v1, ..., vn]⊤ = X j=1 ,..k σj uj v⊤ j X j=k+1 ,..n 0 v⊤ j = X j=1 ,..k σj uj v⊤ j (4.6.2) Defining Σj = diag (σ1, ..., σ k), Uk = [ u1, ..., uk], and Vk = [ v1, ..., vk], we obtain: A = U kΣkV ⊤ k (4.6.3) which is known as the economy SVD . On the other hand, if the rank r of A is r<min (m, n ), then there only are r singular values different of zero, and then A = U r Σr V ⊤ r (4.6.4) which is known as the compact SVD . This fact is useful when obtaining matrices U and V since only ur and vr vectors have to be calculated. Finally, in low-rank aproximation, only t singular values (the t highest ones) are considered, and: A = U tΣtV ⊤ t (4.6.5) which is known as the truncated SVD . 4.7 The SVD as the sum of r matrices of rank 1 Let us take the economy SVD expression: A=U r Σr V ⊤ r . and express matrix A as the sum of r matrices of rank 1: A = U r Σr V ⊤ r = r X k=1 σkukv⊤ k (4.7.1) 36 This will be a key fact to find the best low-rank approximation (in the following sections) of a matrix by using the SVD. 4.8 Matrix norms Remember that norms assign a real number (a length) to an element of vector space. The four (non-negativity, positive definiteness, absolute homogeneity and subadditivity or triangle inequality) defining properties of any norm applied to matrices A and B are (assume correct dimensions of the matrices): • ∥A∥ ≥ 0 (positive-valued), • ∥A∥ = 0 only if A=0 mn (definite), • ∥λA∥ = |λ| ∥ A∥ (absolutely homogeneous), • ∥A + B∥ ≤ ∥ A∥ + ∥B∥ (triangle inequality). For matrix norms we introduce the additional condition: • ∥AB ∥ ≤ ∥ A∥ ∥ B∥ (sub-multiplicative). 4.8.1 Matrix norms induced by vectors p-norms One way of defining a matrix norm, is using vector norms. The matrix norm measures how much a vector (assuming a vector p-norm) can increase in size when it is multiplied by A. Observe that in the definition we use a vector x that is a unitary vector (i.e., with norm ∥ · ∥ = 1). From this we can define: • l−1 norm : maximum absolute column sum of A, i.e., sum all columns and take the highest one: ∥A∥1 = max ∥Ax∥1 ∥x∥1 = max ∥x∥1=1 ∥Ax∥1 = max 1≤j≤nm X i=1 |aij | (4.8.1) • l−2 norm : maximum singular value of A: ∥A∥2 = max ∥Ax∥2 ∥x∥2 = max ∥x∥2=1 ∥Ax∥2 = q λmax (A⊤A) = σmax (A)(4.8.2) This can be seen because A=U ΣV ⊤, and the vector with maximum length is v1. Then ∥Av 1∥2 = ∥σ1u1∥2 = σ1∥u1∥2, taking into account that ∥v1∥2=1 and ∥u1∥2=1, which yields the result; 37 • l−∞ norm : maximum absolute row sum of A, i.e., sum all rows and take the highest one: ∥A∥∞ = max ∥Ax∥∞ ∥x∥∞ = max ∥x∥∞=1 ∥Ax∥∞ = max 1≤i≤mn X j=1 |aij | (4.8.3) You can find geometrically the value of a matrix norm for a given matrix A geometrically by: • Plotting the unit sphere for the matrix • Finding the image under the transformation y = A = x • Finding the maximum of ∥y∥ That means that an induced matrix norm ∥A∥ is how much a matrix can stretch a vector to a maximum. If norm of a matrix is say number d; it means it can stretch a vector x by d maximum. Example 4.3 (Matrix norms) Let us take as example the matrix: A = 1 20 2  and plot figures for ∥A∥1, ∥A∥2 and ∥A∥inf ty (Figure 8). Observe that σ1 =2.9208 and σ2 = 0 .6847 . Thus ∥A∥1 = 4 , ∥A∥2 = 2 .9208 and ∥A∥∞ = 3 . Ob-serve, also, how vectors (1,0) transforms to (1,0) and (0,1) transforms in (2,2), that precisely is ymax for ∥A∥1 and ∥A∥2. However, ymax =(3,2) corresponds to point (1,1) for ∥A∥∞. Applying the norm definition to ymax , you should obtain the same norm than applying the matrix p-norm definitions. 4.8.2 Schatten norms: Matrix norms that can be expressed in terms of singular values Other possible matrix norms are the Schatten norms, which are defined in terms of the singular values, and which in some cases can be expressed in terms of the trace operator. The spectral norm is the operator norm induced by the vector 2-norm. Then, this norm coincides with the induced vector p = 2 norm: ∥A∥2 = max ∥Ax∥2 ∥x∥2 = σmax (A) (4.8.4) 38 Figure 8: Induced matrix norms representation. The Frobenius norm can be defined in several ways (in terms of sum of all absolute coefficient values of the matrix, in terms of the trace of ( AA ⊤) or trace of ( A⊤A), and in terms of the sum of singular values): ∥A∥F = sX k X i |aik |2 = q tr (A⊤A) = q tr (AA ⊤) = sX k σ2 k (4.8.5) Frobenius norm is often easier to compute than induced norms, and has the use-ful property of being invariant under rotations (and unitary operations in gen-eral), meaning that if U is a rotation (unitary matrix), then ∥A∥F = ∥AU ∥F = ∥U A ∥F . From here, we get the connection between the expression ”summatory of all matrix aij coefficients” and the summatory of square singular values, since A=U ΣV ⊤.This norm is used in many applications, such as regularization in machine learn-ing/optimization problems when using matrices. An example is in obtaining the Laplacian matrix coefficients in graph signal processing (GSP) from the data measurements (we will see this application in some days). For the semidefinite matrix A⊤A we can define an square root as a matrix B 39 Figure 9: Spectral and Frobenius norms representation (2-D). The nuclear norm is the sum of σ1 +σ2 and thus is the perimeter of the paralelogram (sum of orange and blue arrows). for which B2 = A⊤A. We define the nuclear (or Ky-Fan) norm or trace norm as: ∥A∥N = X k σk = tr ( q (A⊤A)) (4.8.6) these norms usually appear in infinite dimensional spaces, and also it is often used in mathematical optimization to search for low-rank matrices (measures the ”amount of rank-1 matrices” needed to construct A). It has applications in deep learning (chooses the best weights in gradient descent when there is more weights than samples), and also appears in other applications such as compressive sensing (express a vector in a ”compressed” way with many zero entries). Example 4.4 (Schatten norms) Let us assume matrix AA =  2 1 0 11 3 4 20 1 3 3  Their singular values are σ1 = 6 .909 , σ2 = 2 .238 and σ3 = 1 .501 . The spectral norm is ∥A∥2 = σmax (A) = σ1 = 6 .909 . The Frobenius norm is ∥A∥F =pP k σ2 k = 7 .416 , and the nuclear norm is ∥A∥N = P k σk = 10 .648 . 40 4.9 Condition number of a matrix The condition number of a matrix A∈Rn×n characterizes the sensitivity of the solution of a linear system Ax = b to small changes in A and b. Let us take derivatives at both sides of the linear system: Adx + ( dA)x = db ⇒ dx = A−1(db − (dA)x) (4.9.1) Now, taking the Euclidean norm at both sides: ∥dx∥ ≤ ∥ A−1∥ ∥ (db − (dA)x)∥ ≤ ∥ A−1∥ (∥db∥ + ∥dA∥ ∥ x∥) (4.9.2) Now, we can use the inequality ∥b∥ = ∥Ax ∥ ≤ ∥ A∥ ∥ x∥, we get: ∥d(x)∥∥x∥ ≤ κ(A) ( ∥dA∥∥A∥ + ∥db∥∥b∥ ) (4.9.3) We define the condition number of matrix A as: κ(A) = ∥A∥2∥A−1∥2 = σmax σmin (4.9.4) where σmax and σmax are the maximum and minimum singular values of matrix A. Large condition number κ(A) results in a highly sensitive system, that is, small changes in A or b may result in very large changes in the solution x. On the other hand, a large condition number κ(A) implies that σmax >> σ min ,and then the matrix A is almost singular (is not invertible). Example 4.5 (Condition number) Let us assume matrix A0 (it is a singu-lar matrix) and an approximated matrix A. A0 = 2 41 2  ; A = 2.0002 3.9999 0.9996 2.0002  The SVD of matrix A is: A = 2.0002 3.9999 0.9996 2.0002  = −0.894 −0.447 −0.447 0.894   4.999 00 0.00049   −0.447 −0.894 −0.894 0.447  The condition number is κ(A) = σmax /σ min = 4 .999 /0.00049 = 1000 .00 . Let us calculate A†: A† =  800 .08 −1599 .96 −399 .84 800 .08  = Let us solve the linear system Ax = b with several b’s: 2.0002 3.9999 0.9996 2.0002   x1 x2  = b1 b2  For example b = [2 , 1] results in x = [0 .2, 0.4] , while a small change in component b1 such as b = [2 .05 , 1] results in x = [40 .204 , −19 .592] , or a small change in component b2 such as b = [2 , 1.02] results in x = [ −31 .79 , 16 .402] . 41 4.10 Eckart-Young approximation (low-rank approxima-tion) Let A∈Rm×n be a matrix with n columns and m rows with m≥n (thus it is full-rank when rank r=n). Suppose that A = AΣV ⊤ is the SVD, with U and V are orthonormal matrices, and Σ is an m×n diagonal matrix with entries (σ1, σ 2, · · · , σ n) such that σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0. The Eckart-Young Th. says that the best rank k approximation (a matrix B with rank k≤n) to matrix A in the spectral norm ∥A∥2 is given by the truncated SVD: B = Ak = k X i=1 σiuivti (4.10.1) This result can be extended to the Frobenius and the Nuclear norms. We can prove it for the special cases n=m=2 and k=1. For the general case, the proof follows the same reasoning. Let A = σ1u1v⊤ 1 σ2u2v⊤ 2 , and A1 = σ1u1v⊤ 1 . We can easily see that ∥A − A1∥ = σ2.Let B an arbitrary rank-1 2 ×2 general matrix. We can thus express B as: B = ρ1x1y⊤ 1 . Let w be an element of Ker (B) of length 1, which in our case would be an orthonormal vector to y. We can express w in terms of v1 and v2 as w = γ1v1 + γ2v2, with γ21 + γ22 = 1, with Bw = 0. Using the definition of the spectral norm (which coincides with the induced norm with p = 2) we have: ∥A − B∥22 ≥ ∥ (A − B)w∥22 = ∥Aw ∥22 = σ21 γ21 + σ21 γ22 ≥ σ22 (4.10.2) meaning that: ∥A − B∥22 ≥ ∥ A − A1∥22 (4.10.3) Example 4.6 (Low rank approximation example) Let us consider matrix A. Find the best rank-1, rank-2 and rank-3 approximations. A =  2 1 0 11 3 4 20 1 3 3  First, we obtain the SVD of matrix A = U ΣV ⊤: U = −0.188 −0.881 0.434 −0.78 −0.134 −0.611 −0.596 0.454 0.662  42 V =  −0.167 −0.452 −0.711 −0.512 −0.847 −0.371 0.368 0.094 0.172 −0.49 −0.304 0.799 −0.474 0.646 −0.517 0.302  and the singular values are: Σ =  6.909 0 0 00 2.238 0 00 0 1.501 0  The best rank-1 approximation is given by A1 = σu1v⊤ 1 A1 = σ1u1v⊤ 1 = 6 .909  −0.188 −0.78 −0.596  −0.167 −0.452 −0.711 −0.512 =  0.217 0.588 0.923 0.665 0.902 2.439 3.832 2.761 0.69 1.864 2.929 2.11  In the same way, we can obtain rank-2 ( A2) and rank-3 ( A3) approximations: A2 = σ1u1v⊤ 1 σ2u2v⊤ 2 =  1.888 1.319 0.198 0.479 1.157 2.551 3.721 2.732 −0.171 1.487 3.302 2.206  A3 = σ1u1v⊤ 1 σ2u2v⊤ 2 σ3u3v⊤ 3 =  2 1 0 11 3 4 20 1 3 3  5 Principal component analysis (PCA) Large datasets are common in many data science applications. In order to interpret such datasets, it is useful to drastically reduce their dimensionality in an interpretable way, such that most of the information in the data is preserved. One of the oldest and most widely used technique is principal component analysis (PCA) , which reduces the dimensionality of a dataset by solving an eigenvalue/eigenvector problem, while preserving as much ”variability” - i.e., statistical information - as possible. Principal component analysis is basically used as an exploratory tool for data analysis, although there exist several adaptations to other applications such as functional PCA (continuous variables), robust PCA (to avoid sensitiveness to the presence of outliers), etc. Thus, PCA is a dimensionality reduction method that is typically used to reduce the dimensionality of large data sets. The reduced dimensional representation retains the information conveyed by the large dimensional representation. 43 PCA is an orthogonal linear transformation that transforms the data into a new coordinate system such that the largest variance by some scalar projection of the data is placed in the first coordinate (called the first principal component), the second largest variance in the second coordinate, etc. 5.1 Interpretation 1: maximizing directions with maxi-mum variability In other words: the principal components of a set of data X∈Rm×n (assum-ing m≤n) provide a sequence of best linear approximations to that data, of all ranks q≤n. In other words, given x1, . . . , xm measurements, we want to find a q-rank linear model for representing them. y1 = a11 x1 + a12 x2 + · · · + a1nxn y2 = a21 x1 + a22 x2 + · · · + a2nxn . . . yq = aq1x1 + aq2x2 + · · · + aqn xn (5.1.1) The new axes represent the directions with maximum variability and provide simpler more concise description of the covariance structure. Let us assume in general that we have matrices C and X. Let us assume that X is a ran-dom variable with mean μX and covariance matrix ΣX and let’s take linear combinations Y =CX . We know the following properties (seen in probability sections): μY = E[Y ] = E[CX ] = CE[X] = Cμ X (5.1.2) ΣY = Cov [Y ] = Cov [CX ] = CΣX C⊤ (5.1.3) Then, the PCA are those uncorrelated linear combinations y1, . . . , y q whose vari-ances are as large as possible, meaning to maximize Var( y1)= a⊤ 1 ΣX a1. We should take care with the length of vectors a’s, since multiplying a1 by any con-stant will increase the variance. Then, we have to restrict to vectors a’s whose lengths are unitary: a⊤ 1 a1=1. Then the algorithm has to solve the following: Fist component: y1 = a⊤ 1 X that maximizes Var( a⊤ 1 X) subject to a⊤ 1 a1=1. Second component: y2 = a⊤ 2 X that maximizes Var( a⊤ 2 X) subject to a⊤ 2 a2=1 and Cov( a⊤ 2 X,a⊤ 1 X)=0. q-th component: yq = a⊤ q X that maximizes Var( a⊤ q X) subject to a⊤ q aq =1 and Cov( a⊤ q X,a⊤ j X)=0 for j < q . Result: Let ΣX be the covariance matrix associated to X with random vari-able X, and have ΣX the eigenvalue-eigenvector pairs ( e1,λ1), . . . , ( en,λn) with λ1≥λ2≥ . . . ≥λn≥0. Then, the i − th principal component is given by: yi = e⊤ i X (5.1.4) with Var( yi)= e⊤ i ΣX ei=λi and Covar( yi,yj )= e⊤ i ΣX ej =0. That means that the first component is the one with the largest eigenvector. 44 5.2 Interpretation 2: projecting in a subspace Let us assume that we have m data measurements, each of dimension n. We arrange the data in a matrix X∈Rm×n. Let us assume that we want to project this data in 1-D subspace, whose direction is defined by vector u1. Without loss of generality, we assume that this vector has length equal to 1 ( u⊤ 1 u1=1) since we are interested in the direction and not in the length. We now project each data point xi, i=1 , . . . , m in the 1-D subspace defined by u1. This projection amounts to u⊤ 1 xi, the projected mean is given by u⊤ 1 ¯x,with ¯x the sample mean: ¯x = 1 m m X i=1 xi (5.2.1) and the projected variance σ2 u1 is given by: σ2 u1 = 1 m m X i=1 (u⊤ 1 xi − u⊤ 1 ¯x)2=u⊤ 1 Σu1 (5.2.2) with Σ = XX ⊤ the covariance matrix. If we want to maximize the projected variance u⊤ 1 Σu1 we have to consider the constraint u⊤ 1 u1=1, and add a Lagrange multiplier (we will see the meaning of Lagrange multipliers in TOML, non-linear optimization). Then we multiply the constraint by a scalar λ1 and add it to the variance objective function: u⊤ 1 Σu1 + λ1(1 − u⊤ 1 u1) (5.2.3) To maximize this expression, we obtain the derivatives with respect to u1 equal to 0, and obtain the following expression we obtain: Σu1 = λ1u1 (5.2.4) which says that u1 is an eigenvector with eigenvalue λ1 and they maximize the variance of the projected data on subspace defined by u1. We can extend easily this idea to higher dimensional projected spaces (more principal components). 5.3 Connection PCA-SVD PCA and SVD are closely related approaches and can be both applied to de-compose any rectangular matrices. Let us assume our data matrix X∈Rm×n,and consider the covariance matrix S=X⊤X/(n − 1). Then: S = X⊤X (n − 1) = V ΣU ⊤U ΣV ⊤ (n − 1) = V Σ2V ⊤ (n − 1) = V Σ2V −1 (n − 1) (5.3.1) 45 since V is a unitary matrix and V ⊤ = V −1, and we know that Λ = Σ2/(n − 1), meaning that we can perform PCA using SVD or viceversa. In general, when performing PCA is computationally easier to use SVD than EVD, due to the economic/truncated SVD representations. 5.4 Amount of total variance explained Once we have obtained the principal components, the proportion of total vari-ance explained by the i − th principal component is given by: λi Pni=1 λi (5.4.1) For example, if the first two-three principal components explain 80-90% of the variability, then it could be worth replacing the n features by these principal components. 5.5 Eigenfaces We are going to use the SVD/PCA in a denoising application using a set of images. The application is called eigenfaces . Let us assume that we have a set of images of people where one person images are taken from different angles, e.g. different bright/lighting conditions or different poses. For example 40 people with 50 images of each person, forming a set of K images. Each image has p×q pixels that are vectorized. Consider that we have L1, . . . , LM faces, where each Li is a vector representing a face in RN , with N =p×q (a vectorized representation of an image of p×q pixels). The average face is given by Ψ=1 /M PMi=1 Li, and now we obtain the difference of each face with respect the average face: xi=Li − Ψ with i=1 , . . . , M . We then organize our database of faces in a matrix X=[ x1, . . . , xM ]∈RN ×M . we obtain the SVD of X as: X = U ΣV ⊤ (5.5.1) Where U ∈RN ×N and V ∈RM ×M are the left and right singular vector matri-ces, and Σ∈RN ×M is the diagonal singular value matrix, with singular values σ1≥σ2≥ . . . ≥σN ≥0. Then, applying the Eckart–Young theorem, the best r-rank approximation of matrix X can be obtained taking the singular vectors related to the r-largest singular values, Xr =Ur Σr V⊤ r , with Ur ∈RN×r , Vr ∈RM×r and Σr ∈Rr×r ex-pressed in their truncated form. Afterwards, any new image taken can be pro-jected onto the subspace generated by the left-singular vectors Ur . The idea behind this operation lies in projecting the image into a subspace generated by the most important latent patterns of face images encoded in the database. 46 Figure 10: Eigenfaces database (taken from Brunton book ”Data driven Science & Engineering). (Left:) several faces from different people; (Right:) faces from the same person. Suppose now that we have a new face image. The aim is to denoise the image, encoded in the vector xc∈RN, by projecting it onto the subspace generated by Ur , and then perform a signal reconstruction. First, we find ˆxc=xc−Ψ, the difference between the daily in-situ calibrated LCS data with the new image and the average of faces in the database. The new estimated vector will be given by: ˜xc = Ψ + Ur U⊤ r ˆxc A key parameter is the best r-rank approximation of matrix X, or in other words what is the optimal hard threshold r to denoise and reconstruct the images. We assume that matrix X is the sum of a true value and some noise : X = Xtrue + Xnosie (5.5.2) where entries in Xnosie are identically and independently distributed with Gaus-sian random variables of zero mean and variance γ. If the magnitude of γ is known , then: • If X∈Rn×n (square), then r = 4 √3 √nγ (5.5.3) • If If X∈Rm×n (non-square) and m<<n , then the fraction 4/ √3 is sub-stituted by a function λ(β), with β=m/n (you can find the expression in 47 papers or in the book of S. Brunton ”Data driven Science & Engineering”). r = λ(β)√nγ (5.5.4) r = λ(β)=(2( β + 1) + 8β (β + 1) + ( β2 + 14 β + 1) 1/2 )1/2 (5.5.5) If n<<m , then β=n/m , and of β=1 the expression reduces to the previous one. If the magnitude of γ is unknown , and X∈Rm×n (non-square) then the optimal threshold is given by: ˜σ = w(β)σmed (5.5.6) where σmed is the median of the singular values, while w(β) is obtained as: w(β) ≈ 0.56 β3 − 0.95 β2 + 1 .82 β + 1 .43 (5.5.7) where β=m /n. Finally, r corresponds to the number of singular values that are greater than the threshold ˜ σ. Figure 11: Eigenfaces database (taken from Brunton book ”Data driven Science & Engineering”). The approximation improves for r≥400. 6 Fourier transform and its applications The idea of Fourier series and Fourier transforms is to decompose functions into their basic components. Fourier transforms have many applications such as noise filtering, spectral derivatives, transforming partial differential equations, 48 Figure 12: Eigenfaces database (taken from Brunton book ”Data driven Science & Engineering”). Since faces have mouth, eyes, cheeks, and a lot of features, and there are more than 1600 faces representing a lot of situations, the approx-imation works pretty well for a dog. Figure 13: Eigenfaces database (taken from Brunton book ”Data driven Science & Engineering”). Approximation for a cappuccino, works well because the 1600 faces also represent non-localized spatial features. image processing, and a way of express vector data in generic or universal bases, in contrast with vector data expressed in tailored bases, in which we were able to compress data using the SVD (reduction of dimensionality). Sparsity consists in expressing a signal with a vector in which many components 49 are zero, Although the fast Fourier transform is a technology that allows a signal to be reconstructed from its sparse coefficients, it is not the only way to do so, e.g. Fourier is the basis of JPEG or MPEG. The Fourier modes are generic or universal bases, in the sense that nearly all natural images or audio signals are sparse in these bases. It is also possible to compress signals using the SVD, resulting in a tailored basis. 6.1 Fourier series As in finite-dimensional vector spaces, the inner product may be used to project a function into an new coordinate system defined by a basis of orthogonal func-tions. A Fourier series representation of a function f is precisely a projection of this function onto the orthogonal set of sine and cosine functions with integer period on the domain [a, b]. An important result is that if f (x) is periodic and piecewise smooth, then it can be written as a Fourier series. For example, if f (x) is L-periodic in [0,L), then: f (x) = a0 2 + ∞ X k=1 (akcos ( 2πkx L ) + bksin ( 2πkx L )) (6.1.1) ak = 2 L Z L 0 f (x)cos ( 2πkx L )dx = < f (x), cos ( 2πkx L ) > ∥cos ( 2πkx L )∥2 (6.1.2) bk = 2 L Z L 0 f (x)sin ( 2πkx L )dx = < f (x), sin ( 2πkx L ) > ∥sin ( 2πkx L )∥2 (6.1.3) These coefficients may be viewed as the coordinates obtained by projecting the function onto the orthogonal cosine and sine basis {cos (kx ), sin (kx )}∞ k=0 .Since we can write the Fourier series in complex form using the facts that eikx = cos ( 2πkx L ) + i sin ( 2πkx L ) and ck = ak + i b k: f (x) = ∞ X k=−∞ ckeikx = ∞ X k=−∞ ckψk(x) (6.1.4) Thus, a Fourier series is just a change of coordinates of a function f (x) into an infinite-dimensional orthogonal function space spanned by sines and cosines. 6.2 Fourier transform The Fourier series is defined for periodic functions, so that outside the domain of definition, the function repeats itself forever. The Fourier transform integral is essentially the limit of a Fourier series as the length of the domain goes to infinity, which allows us to define a function defined on ( −∞ , ∞). 50 Thus, we represent the set of frequencies as wk=kπ/L , and taking the limit L → ∞ , such as k/L → f , w=2 πf , ∆ w=2 π/L , and ∆ w → 0, we will arrive to the classical Fourier transforms formulas (not necessary to proof here the pass from Fourier series to Fourier transform, see books for that proof if interested): F (w) = Z ∞−∞ f (x) e−iwx dx (6.2.1) f (x) = Z ∞−∞ F (w) eiwx dw (6.2.2) Example 6.1 (Fourier Transform example) For example, assume a box sig-nal, Figure 14, f(x)=A if x∈[−T / 2, T / 2] and 0 otherwise, has Fourier Transform F(f )=ATsinc(fT)= ATsin( πf T /( πf T )), where the sinc(f )=sin( πf )/( πf ). (a) Box signal. (b) Sinc signal Figure 14: The box signal f(x)=A if x∈[−T / 2, T / 2] and 0 otherwise, has Fourier Transform F(f)=ATsinc(fT)= ATsin( πf T )/( πf T )). 51 6.2.1 Some basic properties of the FT Let us assume that we have functions f(x), g(x), and their Fourier Transform F(w), G(w). The following basic properties hold: • Linearity: the Fourier transform of sum of two or more functions that are multiplied by a constant is the sum of the Fourier transforms of the functions (af (x) + bg (x)) = ⇒ (aF (w) + bG (w)) (6.2.3) • Scaling: if we stretch a function by the factor in the time domain then squeeze the Fourier transform by the same factor in the frequency domain: f (ax ) = ⇒ (1 /|a|)F (w/a ) (6.2.4) • Derivative: differentiating function with respect to time yields to the constant multiple of the initial function: df (x)/dx =⇒ (jw )F (w) (6.2.5) • Convolution: the Fourier transform of a convolution of two functions is the point-wise product of their respective Fourier transforms: f (x)∗g(x) = ⇒ F (w)G(w) (6.2.6) Note: the convolution of two functions is a mathematical operation that says how the shape of a function is changed by the other function, a convolution in continuous time is expressed as f (x)∗g(x) = Z ∞−∞ f (y)g(x − y) dy = Z ∞−∞ f (x − y)g(y) dy (6.2.7) • Time shift: a linear displacement in time corresponds to a linear phase factor in the frequency domain: f (x − x′) = ⇒ F (w)e−jwx ′ (6.2.8) • Frequency shift: frequency is shifted according to the co-ordinates: f (x)ejw ′x =⇒ F (w − w′) (6.2.9) 6.2.2 Applications There are many applications in engineering in which Fourier Transforms are used. Among them, we can list: i) solving of partial differential equations such as the heat equation d2f(x,t)/ dx 2= df(x,t)/dt or the wave equation d2f(x,t)/ dx 2= d2f(x,t)/ dt 2; ii) spectral analysis of time-series (e.g. to find the response of the LTI ( linear time invariant ) systems); iii) filtering (lowpass, bandpass or highpass); iv) etc, etc, etc. We will see some examples after seeing the discrete version of the FT. 52 6.3 Discrete Fourier Transform and Fast Fourier Trans-form We have seen how to obtain the Fourier transform when the signal f(x) is a continuous function. However, in general, we discretize the analog/continuous signals, e.g., using an ADC that samples the signal to a specific sampling rate or spacing ∆ x, having a vector of data [ f0, f 1, . . . , f N −1]. Thus, it is necessary, when dealing with vectors of data, to approximate the Fourier transform for dealing with discrete vectors. This is called the discrete Fourier transform, DFT . Remember that the domain of the function was called x and then the data is sampled at point xi and the evaluation of the function (samples) are then called fi. However, for simplicity, from now, we will express our vectors of sampled data [ f0, f 1, . . . , f N −1] as [ x0, x 1, . . . , x N −1], to be coherent with the variables expressed during the course (meaning that vector x is the sample data and not the domain of the function). Let us define the complex number wN = ej 2πN = cos ( 2πN ) + j sin ( 2πN ), i.e. is the first N -th root of −1. The Fourier matrix FN is defined as: FN = 1 √N  1 1 1 ... 11 w−1 N w−2 N ... w−(N −1) N 1 w−2 N w−4 N ... w−2( N −1) N 1 w−2 N w−6 N ... w−3( N −1) N ... ... ... ... ... 1 w−(N −1) N w−2( N −1) N ... w−(N −1)( N −1) N  (6.3.1) Sometimes the factor 1√N is omitted in the definition. This matrix is unitary , meaning that FN H FN =FN FN H =I. The matrix AH is called hermitian if it is a complex square matrix that is equal to its own conjugate transpose, i.e., aij = ¯ aji (remember complex conjugate; if aij =3+j4, then ¯ aji =3-j4) or A=AH (special case is when the component is a real number in which A=A⊤). In other words, hermitian matrices can be understood as the complex extension of real symmetric matrices. Remember that as we are dealing with complex vectors and matrices, the scalar product between two vectors x and y (e.g. two columns of FN ) is defined as xH y. Do not forget to use the conjugate transpose instead of simply transpose! .For instance, if N = 4 we have: F4 = 12  1 1 1 11 −j −1 j 1 −1 1 −11 j −1 −j  (6.3.2) The Fourier matrix is one of the most important matrices in applied mathemat- 53 ics and engineering . 6.3.1 The Discrete Fourier Transform (DFT) The discrete Fourier transform (DFT), ˆx, is the operation of multiplying a vector of data x by the matrix FN : ˆx = FN x = 1 √N  1 1 1 ... 11 w−1 N w−2 N ... w−(N −1) N 1 w−2 N w−4 N ... w−2( N −1) N 1 w−2 N w−6 N ... w−3( N −1) N ... ... ... ... ... 1 w−(N −1) N w−2( N −1) N ... w−(N −1)( N −1) N  x0 x1 x2 x3 . . . xN −1  (6.3.3) This operation is the same as: ˆxk = N−1 X i=0 xie−j2πik/N ∀k = 0 , . . . , N − 1 (6.3.4) meaning that: x=[ x0, x 1, . . . , x N −1] DF T =⇒ ˆx=[ˆ x0, ˆx1, . . . , ˆxN −1] (6.3.5) The result of this multiplication ˆx is called the transformed vector . For com-modity, we will use the term signal for the vector x and Fourier transform for the vector ˆx. Recall that both vectors have dimension N . We can observe that the DFT is a linear operator (a matrix) that maps data points x in the frequency domain ˆx. It is to say, we see the DFT operation corresponds to finding the components of vector x expressed in the unitary base created by the columns of FN (the so called Fourier base ). The Inverse DFT is defined as: x = F HN ˆx (6.3.6) and this operation is the same as: xi = 1 N N−1 X k=0 ˆxkej2πki/N ∀i = 0 , . . . , N − 1 (6.3.7) meaning that: ˆx=[ˆ x0, ˆx1, . . . , ˆxN −1] IDF T =⇒ x=[ x0, x 1, . . . , x N −1] (6.3.8) For a 2D signal X (e.g. a picture represented by a matrix with M rows and N columns) we define the DFT as a two-step process: First find the DFT of the 54 columns, and then find the DFT of the columns of the resulting matrix (the order can be changed): ˆX = F HM XF N (6.3.9) 6.3.2 The Fast Fourier Transform (FFT) In general, if A is a N ×N matrix, and x is a vector with N components, the operation Ax requires N 2 multiplications. In the special case of FN x, the FFT algorithm gives a method for computing this product in only 12 N log 2N multiplications. This is a huge improvement .Some people (e.g. G. Strang) considers that the FFT is the most important numerical algorithm of the XX th century. 6.3.3 The complex conjugate of the columns of the Fourier matrix are the eigenvectors of the circulant matrices Another extraordinary property of the columns of the Fourier matrix is that the complex conjugate of its columns are the eigenvectors of circulant matrices .A circulant matrix has the following form: C =  h0 hN −1 hN −2 ... h1 h1 h0 hN −1 ... h2 ... ... ... ... ... hN −1 hN −2 hN −3 ... h0  (6.3.10) Circulant matrices are especially important as they can be used to represent a very rich and important class of linear system, the linear systems that are invariant to time shifts. The eigenvector (1 , w i, w 2i, ..., w (N −1) i)⊤ has associated the eigenvalue λi = h0 + h1w−i + h2w−2i + ... + hN −1w(N −1) i. Defining Λ = diag (λ0, ..., λ N −1), we obtain the decomposition: C = F HN ΛFN (6.3.11) The output signal for an input x can be obtained very efficiently by applying the FFT algorithm. Example 6.2 (Filtering noise (denoising) in a signal) A first example es filtering noise (denoising) in a signal. For example consider figure 15 where we have a signal f(t)= sin(2 πf 1t) + sin(2 πf 2t) for f1=50 and f2=120 (black curve). We add some Gaussian noise distributed with zero mean and σ2 vari-ance, i.e., N(0, σ2), (red curve). If we obtain the FT, we can observe the two 55 Figure 15: Denoising a signal (Figure taken from S.L. Brunton book, ”Data driven science & engineering”). peaks centered at f1=50 and f2=120 . We can then filter the signal keeping frequencies lower than f2=120 , and then removing high frequency components. We can observe in the third figure the original (without noise) signal and the denoised filtered signal. An example of a filter is to pass the signal f(t) by a linear system h(t). The output is the convolution y(t)=f(t) ∗h(t) −→ Y(W)=F(W)H(w). If H(w) is a filter allowing to pass frequencies f1 and f2 and not allowing the rest of frequencies, we are denoising (filtering noise) our signal. Example 6.3 (Spectral derivative) A second example is the spectral deriva-tive. We know that the FT of the derivative of a function in continuous time is F(df(x)/dx)=jwF(f(x)). If we discretize, then we can substitute jw → jκ , with κ = 2 πk/n , assuming n components. For example, let us assume the function: f (x) = cos (x)e−x2/25 =⇒ df (x)/dx = −sin (x)e−x2/25 − 225 xf (x) (6.3.12) 56 (a) Derivative methods with n=128 points. (b) Error as a function of n Figure 16: Taking the derivative of a signal (Figure taken from S.L. Brunton book, ”Data driven science & engineering”). One way of obtaining the derivative is to use finite differences: df dx (xk) = f (xk+1 ) − f (xk)∆x (6.3.13) for some ∆x. So, if we want to obtain the derivative of our signal, we have two options: i) use finite differences, and ii) take our signal, discretize with n samples, obtain the FFT of the signal, multiple by jκ , and obtain the IFFT. We can observe in Figure 16.a) the result of the true derivative (black), the finite difference method (blue) and the FFT (red). Both, the finite difference method (blue) and the FFT (red) are very close to the true derivative, but we can observe in Figure 16.b) the error in both methods, in which is seen how the FFT performs much better than the finite Euler method. 6.4 Most natural signals are sparse in the Fourier base Let us assume a vector of data x∈Rn, a vector x is called sparse if it has a large number of components equal to zero (or a small number of nonzero components). A vector x is called dense if it has a large number of components non equal to 57 zero. A vector x is called k-sparse if it has n − k components equal to zero, and k components equal to zero. A fundamental property in science and engineering is that many natural signals (pictures, video, audio, music, physical magnitudes such as heat, pressure, etc) are sparse when expressed in the Fourier base . This is consequence of a fact of nature that many of these signals are smooth .On the contrary, if we generate a random vector, for instance with i.i.d compo-nents sampled from a Gaussian distribution, we would have that with very high probability the components of the vector expressed in Fourier base will be not sparse. We call this sequence white noise . 7 Graph signal processing and its applications 7.1 Introduction The framework of graph signal processing (GSP) was conceived in the last decade with the ambition of generalizing the tools from classical digital signal processing to the case in which the signal is defined over an irregular structure modeled by a graph. Let us take the example (taken from Ljubisa Stankovic et al. survey paper, ”Understanding the basis of graph signal processing via an intuitive example-driven approach”, arXiv, May 2019) where a set of temperature sensors are deployed over a large region. We are interested, Figure 17.a), of finding the local neighborhood of nodes, in suchc a way that we will find a graph that represents the network, Figure 17.b). We have been able to connect (we will see later how), for example, node 20 with nodes 19, 22, 23, and node 29 with nodes 27, 28, 51, 59, and so on. If x(n) is the temperature value of node n, then we can now draw the temper-ature using bars, Figure 18.a) or even with color in the vertices, Figure 18.b). The objective is that now we can consider that the signal in a given node n is related to the node itself and its neighborhood: y(n) = x(n) + X m∈N(n) x(n) (7.1.1) with N( n) the neighborhood of node n. For example, for node n=20: y(20) = x(20) + x(19) + x(22) + x(23) (7.1.2) For convenience, we will write this expression as: y = x + Ax (7.1.3) 58 (a) Local neighborhood of single nodes. (b) Local neighborhood for all sensors. Figure 17: Multisensor IoT example, where nodes measure temperature (Fig-ure taken from L. Stankovic et al ”Understanding the Basis of Graph Signal Processing via an Intuitive Example-Driven Approach” paper). Graph repre-sentation. (a) Bars indicate temperature values. (b) Temperature values as bars (left) and ver-tex color (right) Figure 18: Multisensor IoT example, where nodes measure temperature (Fig-ure taken from L. Stankovic et al ”Understanding the Basis of Graph Signal Processing via an Intuitive Example-Driven Approach” paper). Colour repre-sentation of the temperature field. where matrix A is the adjacency matrix. There are several ways of creating a graph. The key is in discovering which are the relationships among the nodes. Example 7.1 (Subgraph of node 29 ) The subgraph represented by node 29 59 with neighbors 27, 28, 51, 59 in Figure 18.b) would be represented by matrix: A =  0 1 1 1 11 0 1 1 01 1 0 1 11 1 1 0 11 0 1 1 0  (7.1.4) The objective of GSP is to give a framework able to operate over such a graph making use of these relationships. 7.2 Adjacency matrix, the weighted matrix and the Lapla-cian matrix The adjacency matrix is not the only way of representing the graph. Other ways is using a weighted matrix and the Laplacian matrix. The weighted matrix W has coefficients wij >0 if node i is connected to node j, and zero otherwise. The idea behind the weighted matrix is that the cost of the edges between connected nodes is not equal to one, thus considering that are nodes better connected than others. In this case: y(n) = x(n) + X m∈N(n) wmn x(n) −→ y = x + W x (7.2.1) We can observe that the operator A is a special case of the operator W , in which all weights are considered of the same value. Finally, we can use a third operator called the Laplacian matrix L, which is obtained as: L = D − W (7.2.2) where D is the degree matrix and it has coefficients in the diagonal dii =P j̸=i wij (sum of row except the value at the diagonal) and the rest are 0. Example 7.2 (Laplacian matrix) We want to obtain the laplacian matrix L, from the weight matrix W : W =  0 .6 .3 0 .6 0 .1 .4 .3 .1 0 .20 .4 .2 0  (7.2.3) We first obtain the diagonal matrix D as sum of rows dii = P j̸=i wij : D =  .9 0 0 00 1.1 0 00 0 .6 00 0 0 .6  (7.2.4) 60 Then, the Laplacian will be: L = D − W =  .6 −.6 −.3 0 −.6 1.1 −.1 −.4 −.3 −.1 .6 −.20 −.4 −.2 0.6  (7.2.5) 7.3 Creating the graph There are several ways of creating a graph. The key is in discovering which are the relationships among the nodes: • Physically knowledge of the weights: there is an intrinsic knowledge of what are the weights, for example, circuits in electronic systems, social networks, etc; • Geometry of the vertex: use Euclidean distances. In this case it is built a decreasing function of the distance: wij = ed2 ij /α or wij = edij /α (7.3.1) • Obtain the weighted matrix or the Laplacian matrix from the data measured matrix X. An example is to use a non-linear optimiza-tion model: minimize L,Y ∥X − Y∥2 F | {z } data f idelity α tr (Y⊤LY ) | {z } smoothness +β∥L∥2 F | {z } sparsity subject to tr (L) = n, Lij = Lji ≤ 0, i̸ = j, L · 1 = 0. (7.3.2) We will study convex non-linear optimization in TOML-MIRI course, but we can say that this optimization problem is not convex (it has not global minimum). But it can be solved using in an alternating procedure, in which we first fix Y = X, so we find L from our data measurements X.We can play the sparsity (more L ij =0, meaning more nodes disconnected) of our solution using the β parameter. Finally when L is found, we can solved again fixing the L found and finding Y that will be a filtered version of our data X.61 Figure 19: Eigenvalues and eigenvectors of an N=8 node network (Figure taken from L. Stankovic et al ”Understanding the Basis of Graph Signal Processing via an Intuitive Example-Driven Approach” paper). 62 7.4 Spectral graph theory Since adjacency matrix A is squared and symmetric ( A=A⊤), is i) diagonizable, ii) its eigenvalues are real ( Λ=diag (λ0, . . . , λ N −1) is a diagonal matrix), and iii) its eigenvectors are orthogonal ( ∥uk∥22 = 1): A = U ΛU −1 = U ΛU ⊤ (7.4.1) 7.5 The adjacency matrix and the graph signal shift Let us consider N samples of a signal expressed as a vector x=[ x0, x 1, . . . , x N −1]. A signal shift on a graph can be defined as the movement of the signal sample, xn, from its original vertex, n, along all walks of length one that start at vertex n. If we define x(1) as the signal shifted, then: x(1) = Ax (7.5.1) In case we go on shifting the signal: x(2) = Ax (1) = A2x (7.5.2) and thus, m shiftings result in: x(m) = Ax (m−1) = A2x(m−2) = · · · = Am−1x(1) = Amx (7.5.3) You can recall that instead of the adjacency matrix A we can use the Laplacian matrix L, and consider the adjacency matrix a special case of the Laplacian matrix, and thus: x(m) = Lmx (7.5.4) 7.6 The graph discrete Fourier transform The graph discrete Fourier transform (GDFT) of a signal, x, is defined as: X = U −1x (7.6.1) where X denotes a vector of the GDFT coefficients, and U is a matrix whose columns represent the eigenvectors of the adjacency matrix, A or the Laplacian matrix L. Let us assume that we take the adjacency matrix (same considerations but different behavior if we use the Laplacian matrix), then vector X has k=0,1, . . . , N-1 coefficients. Since matrix A is symmetric ( A⊤ =A), we have that U −1 =U ⊤, and: X = U ⊤x (7.6.2) 63 The inverse graph discrete Fourier transform (IGDFT) of a signal, x, is thus defined as: x = U X (7.6.3) In case of having a circular graph, the GDFT reduces to the classical DFT. Given that we have defined a GDFT/IGDFT, we can define filters, convolutions, spectral analysis, signal reconstruction, denoising, and different operations per-formed in classical signal processing. 7.7 Signal reconstruction in a IoT network We consider the problem of having a subset of nodes (vertices of the graph) with samples and we would like to estimate the signal of the graph in the other vertices so that the resulting signal is smooth. Let is call M the set of nodes in the graph with observed data, and U the set of nodes in the graph with observed data (missing data). Thus the objective is to find the unobserved data from the observed neighboring nodes. This can be seen as a signal reconstruction problem that can be solved using methods from various fields. Let us consider the following methods; Laplacian interpolation and GSP low-pass based graph signal reconstruction. Laplacian interpolation is a graph-based semi-supervised learning algorithm whose goal is regression with graph regularization assuming smoothness with respect to the Laplacian matrix. This method regresses a function f :V→ R over the graph G, assuming partial information, it is to say, information for M nodes. GSP low-pass based graph signal reconstruction is a graph signal processing reconstruction method that considers subsampling low-pass graph signals, thus assuming a sparse Fourier coefficient vector. 7.7.1 Laplacian interpolation Also known as graph interpolated regularization by Belkin et al. , this method minimizes the quadratic form of the Laplacian matrix with respect to the graph signal x, which is a measure of signal smoothness, given that the observed measurements {xm: ∀m∈M} remain unchanged. This reconstruction results in a linear combination of the observations weighted by the Laplacian matrix entries Lij . M inimize y y⊤Ly s.t. ym = xm, ∀m∈M (7.7.1) 7.7.2 Graph Signal Processing (GSP) low-pass reconstruction This technique recovers a set of unobserved nodes {xu: ∀u∈U} given that the graph discrete Fourier transform of the complete signal is sparse and of low-pass 64 nature, meaning that it has K nonzero components corresponding to the lowest frequencies (smallest eigenvalues λi of the Laplacian matrix). Given that the Laplacian matrix admits the eigendecomposition L=UΛU ⊤, the graph discrete Fourier transform (GDFT) of a graph signal x can be computed as: X = U−1x (7.7.2) Now, a K-sparse GDFT coefficient vector of the following form is to be recovered: X = ( X(0) , . . . , X (K − 1) , 0, . . . , 0) ⊤ (7.7.3) For this purpose a subset of measurements M are used to recover the sparse coefficient vector by solving the following system: xM = UMK XK (7.7.4) Since the system is overdetermined, the solution of the above system in the least squares sense is given by XK = U†MK xM, where U†MK = ( U⊤MK UMK )−1U⊤MK is the matrix pseudo-inverse of UMK ; the nonzero coefficients are obtained, and after appending the corresponding zero coefficients, the inverse graph discrete Fourier transform (IGDFT) x=UX is computed to obtain the complete set of measurements x at all vertices. Example 7.3 (GSP low pass reconstruction) Let us assume a network with N = 8 nodes, in which we only have M = 4 measurements, e.g. x( 2), x( 4), x( 5)and x( 7), and we want to reconstruct the signal with K = 2 coefficients. Then, we have to find X(0) and X(1) that satisfies:  x(4) x(5) x(7)  =  u0(4) u1(4) u0(5) u1(5) u0(7) u1(7) X(0) X(1)  (7.7.5) For finding X(O) and X(1) , we solve the overdetermined system, and when we have these GDFT coefficients, we can obtain the original ones applying the IGDFT x=UX , with X=[X(0) , X (1) , 0, 0, 0, 0, 0, 0] . 65
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R Sine Theta Plus Alpha MADE EASY! Adding Sine and Cosine Waves with Phase Shifts. Maths4Engineers - James Cleves 746 subscribers 32 likes Description 996 views Posted: 15 Apr 2021 00:00 Welcome! 00:13 Sine or Cosine? 01:23 Adding Sine and Cosine Components explained 04:33 R Sin Theta Plus Alpha STEPS 05:34 Compound Angle Formula Setting up the addition 10:01 FINDING R and ALPHA 11:29 Checking we got it right by graphing 12:31 Understand 'Omega t' 12:32 Adding Sine and Cos with Excel! Transcript: Welcome! so welcome to this video on adding phase shifted sine and cosine waves and expressing the resultant voltage in the form of v equals r sine theta plus alpha Sine or Cosine? so it's a sine this is a sine wave because it starts at the origin and goes up or is it because it could be a phase shifted cosine wave it could be lagging by 90 degrees or pi over 2 radians this looks like a cos wave because it starts at 1 and then goes down but it's possible that it is a leading sine wave that's leading by 90 degrees or pi over 2 radians this one here what could it be is it a sine or a cosine and we're not quite sure what do you think it starts at 0.5 and goes up and the answer is it's neither and it's both it's a sine and a cosine or it could be it could be y equals sine omega t plus phi or sine theta plus the phase difference that could be true or it could be y equals cos theta plus the phase difference or y equals cos omega t they're both true but for each case the phase difference would be different Adding Sine and Cosine Components explained so we're going to add some sine waves and cos waves together v1 is 3 sine omega t plus pi over 4 radians and because there's a phase shift think of it as having a sine component and a cosine component and a phase shift of phi which is pi over 4 radians on 45 degrees and to find out the amplitude we would square the sine component and square the cosine component add them together then square roots using pythagoras and they would add up to three so the amplitude a is the hypotenuse and that's three volts but we can see it's three volts from what v1 says it's three times the sine so that means 3 is the amplitude so when we add our sine waves and cos waves we add the sine components together and the cosine components together square them add them and square root them to find the resultant voltage v2 is 2 cos omega t minus pi over 8 radians which is 22 and a half degrees so it's lagging and the same goes because there's a phase shift it'll have a sine component to it and it'll have a cosine component to it as well and if we square them add them and then square root using pythagoras we'll find that the amplitude a is 2 volts and using 10 to the minus 1 the phase shift would be minus pi over eight and when we add v1 and v2 together to get v3 what we're really doing is adding the sine components of v1 and v2 together and the cosine components of v1 and v2 together then we'll square them add them and square root them and that will give us r which is the resultant voltage of v3 because we want to express the resultant in the form of r sine theta plus alpha or r sine omega t plus alpha which is the engineering way so let's look at v1 as a graph it's a sine wave it's leading because the phase difference is positive and it has an amplitude of a volts v2 in blue it's a cosine and it's lagging it's lagging behind the y-axis because phi is negative and it also has an amplitude of a volts whatever a is and then v3 is the green one it's the sum of the other two at any point it's a sine wave and it's leading because the phase difference what we call alpha is positive because it's to the left of the y-axis and the amplitude of v3 we call r so we're going to find the resultant amplitude aha and the resultant phase shift alpha for v3 which is the sum of the other two voltages we could say r sine theta plus alpha which is like a mathematical way of doing it but engineers tend to say r sine omega t plus alpha but theta and omega t are just the angle on the x axis R Sin Theta Plus Alpha STEPS so the steps to express in the form of r sine theta plus alpha step one resolve each phase shifted waveform into its sine and cosine components using a compound angle formula and there's a few of these formulas and it's worth saying that if your sine or cosine wave has no phase difference or no phase shift then it'll only be say a sine wave will only have a sine component and a cosine without a phase shift will only have a cosine component but if we have phase shifts in any wave it'll have a sine and a cosine component step two is to use pythagoras's theorem to find the resultant amplitude ah for the third voltage v3 and then we use simple trigonometry to find the resultant phase shift using tan to the minus one and this is called alpha and it's easy when you know how so let's have a go Compound Angle Formula Setting up the addition so know your formulae compound ankle formulas there's a few of them we have v1 here 3 sine omega t plus pi over 4 radians and i'm going to use sine a plus b is sine a cos b plus cos a sine b v two is equal to two cos omega t minus pi over eight so there's two compound angle formulas i could use i'm going to use cos a plus b but if i do i need to make sure that my angle b is written as minus pi over eight radians i only like to use the positive versions of the compound angle formula so i only need to remember two i could have used cos a minus b in which case the angle b would be pi over eight but the way i it's just my preference that i like to use the positive version so i'm using cos a plus b is cos a cos b minus sine a sine b but i need to make sure that the angle b is always written and calculated as minus pi over eight because a plus b is a plus minus pi over eight so i'll use a compound angle formula for v1 to resolve it v1 is 3 sine omega t plus pi over 4 radians i'll use sine a plus b is sine a cos b plus cos a sine b i'm gonna let a be omega t and angle b is pi over four so v one is three lots of sine a cos b plus cos a sine b and that's equal to three lots of sine omega t cos pi over 4 plus cos omega t sine pi over 4 which is equal to 3 lots of sine omega t times 0.707 plus cos omega t times 0.707 so don't forget to multiply out by the 3 in front of the brackets and this becomes v1 is 2.121 sine omega t plus 2.121 cos omega t so we've resolved it in the sine and cos with no phase shift and the same for v2 v2 is 2 cos omega t minus pi over 8 radians i'm going to use like i said earlier cos a plus b is cos a cos b minus sine a sine b i'll let a be omega t but i must let b equal minus pi over eight if i use this formula so v2 is two lots of cos a cos b minus sine a sine b which is equal to two lots of cos omega t times cos minus pi over eight minus sine omega t times sine minus pi over eight and watch out for the double negative on the right hand side so v2 equals two lots of cos omega t times 0.9239 from our calculator minus sine omega t times minus 0.3827 and don't forget to multiply out by the two and this gives us v2 equals 0.7654 sine omega t plus 1.8478 cos omega t so we've resolved v2 into its sine and cosine components with only one angle to manage now omega t on the x-axis we don't have to worry about the phase difference anymore v3 is equal to the sum of the other two voltages so we add the sine components and we add the cosine components and we combine the like terms so v1 is 2.121 sine omega t plus 2.121 cos omega t v 2 is 0.765 sine omega t plus 1.848 cos omega t to three decimal places and we add sine to sine and goes to cos and this gives us v3 which is the sum of the other two voltages is 2.886 sine omega t plus 3.969 cos omega t and v3 is a resultant over the other two FINDING R and ALPHA so there's v3 with its sine and cosine components so just think of v3 like the other two voltages it has a sine component and a cosine component and a resultant amplitude which is the hypotenuse and we can work it out with pythagoras so what we do with pythagoras we square the sine component and square the cosine component add them together and square root and that tells us that the amplitude of v3 which we call r is 4.9 volts and then we use 10 to the minus 1 of the opposite over the adjacent which is 10 to the minus 1 of the cosine component divided by the sine component and that gives us the angle alpha for the resultant voltage v3 and that's 0.942 radians and that's positive so that means it's leading the y-axis so to express v3 in the form of our sine theta plus alpha or our sine omega t plus or minus alpha depending on the phase shift we can do it now because it's equal to 4.9 times the sine of omega t plus 0.942 radians or you could say 4.9 sine theta plus 54 degrees so we've done it and it hasn't been too painful it's been pretty straightforward Checking we got it right by graphing so i've used desmos to graph the three voltages we have v1 in red v2 in blue and v3 the resultant in green so if i zoom in to look at the three waveforms we can see v3 in green and we can see the amplitude r is 4.9 which is what we calculated and the phase shift alpha is 0.942 radians leading the y-axis to the left so we've got it correct so the steps to express just as a recap resolve the phase shifted waveforms into sine and cos components using the compound angle formula and you only need to do this if they have a phase shift step two use pythagoras theorem to find the resultant amplitude are for v3 and use simple trig 10 to the minus 1 to find v3s phase shift alpha Adding Sine and Cos with Excel! so thanks for watching like and subscribe and look at the other videos at the top to understand more about omega and how to graph using excel all the best
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https://www.atlas.org/solution/25248916-1b47-4181-9f5c-c9d33b22398e/prove-that-pv-is-constant-in-an-adiabatic-transformation-where-the-symbols-have-their-usual-meaning
Solved: Prove that PV^γ is constant in an adiabatic transformation, where the symbols have their usual meaning. Question Prove that PV^γ is constant in an adiabatic transformation, where the symbols have their usual meaning. Answer 􀊀100% (1 rated) PV^γ is constant in an adiabatic transformation, as derived from the first law of thermodynamics for an ideal gas undergoing adiabatic changes. Steps 1. To prove that PVγ is constant in an adiabatic transformation, we start with the definition of an adiabatic process. An adiabatic process is one in which no heat is exchanged with the surroundings, meaning dQ=0. 2. From the first law of thermodynamics, we have the relation dU=dQ−dW. Since dQ=0 for an adiabatic process, this simplifies to dU=−dW. The work done by the gas during volume change is given by dW=PdV. 3. Substituting for work, we get dU=−PdV. For an ideal gas, the change in internal energy dU can be expressed as dU=nCV​dT, where n is the number of moles and CV​ is the specific heat capacity at constant volume. 4. Now we can relate the changes in pressure, volume, and temperature during an adiabatic process. Using the ideal gas law, we express the temperature in terms of pressure and volume: T=nRPV​. Differentiating this gives us dT=nRPdV+VdP​. 5. Substituting dT into the equation for internal energy yields nCV​nRPdV+VdP​=−PdV. We can cancel n in our calculation: CV​RPdV+VdP​=−PdV. This manipulation will assist in obtaining our desired relation. 6. Rearranging leads to: CV​PdV+CV​VdP=−RPdV. Bringing terms involving dV together gives us dV(CV​P+RP)=−CV​VdP. 7. Factoring out P, we find dV(P(CV​+R))=−CV​VdP. Noticing that γ=CV​Cp​​, we see that Cp​=CV​+R, leading to further simplification. The additivity of specific heats allows us to express dV in relationship to dP. 8. Next, we can integrate the resulting differential equation. This eventually leads to a relation between P and V stating that PVγ=constant, as required. The constant can vary depending on the specific process initiated but remains true across specific instances in an adiabatic transformation. Related Simplify the expression 3×3−473×7−3​. 􀊀 63%(5 rated)What is the result of the expression 3x3-3:3+3? 􀊀 83%(5 rated)Evaluate, ( 5/7 × 2/3) + (5/6-8/9) ÷ 7/15 of 5/6 (3 marks) 􀊀 100%(2 rated)There are 7 characteristics of living things? 􀊀 80%(4 rated)Calculate the number of positive integers not exceeding 1200 that are divisible by 2, 3, 5 or by 7 􀊀 100%(2 rated)Given that tan x degrees = 3/7, find cos(90-x) degrees, giving the answer to 4 significant figures 􀊀 100%(2 rated)Given p x( ) = −4(x − 15)2 + 2, what is the value of p(7)? 􀊀 100%(3 rated)What is the meaning of "ameliorated" as it is used in paragraph 7? 􀊀 100%(2 rated)
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https://www.metric-conversions.org/speed/miles-per-minute-to-miles-per-hour.htm
Miles per minute to Miles per hour conversion Miles per minute to Miles per hour Language Miles per minute to mph Miles per minute to Miles per hour conversion calculator → 1miles/min = 60.00005mph Accuracy Note: Fractional results are rounded to the nearest 1/64. For a more accurate answer please select "decimal" from the options above the result. Note: You can increase or decrease the accuracy of this answer by selecting the number of significant figures required from the options above the result. Note: For a pure decimal result please select "decimal" from the options above the result. mph to Miles per minute (Swap units) Miles per minute to Miles per hour conversion formula Miles per hour = Miles per minute 60.00005364 Miles per minute to Miles per hour calculation Miles per hour = Miles per minute 60.00005364 Miles per hour = 1 60.000053644759 Miles per hour = 60.00005 Miles per minute (mi/min) Miles per minute (mi/min) is a unit of speed and this unit is the number of miles travelled in one minute. Miles per hour (mph, mi/h) Miles per hour (mph or mi/h) is a unit of speed. Mph is a unit commonly used to express the speed of vehicles, such as cars and trains in countries that have traditionally used the imperial system such as the United States and United Kingdom. Temperature conversionLength conversionArea conversionVolume conversionWeight conversionSpeed conversionTime conversionAngle conversionPressure conversionEnergy and power conversioniPhone and Android appMetric Conversion TableMiles per hour to Miles per minuteMiles per hour to Kilometers per hourKilometers per hour to Miles per hour Miles per minute to Miles per hour table Starting value Increment Accuracy Miles per minute Miles per hour 0miles/min 0.00000mph 1miles/min 60.00005mph 2miles/min 120.00011mph 3miles/min 180.00016mph 4miles/min 240.00021mph 5miles/min 300.00027mph 6miles/min 360.00032mph 7miles/min 420.00038mph 8miles/min 480.00043mph 9miles/min 540.00048mph 10miles/min 600.00054mph 11miles/min 660.00059mph 12miles/min 720.00064mph 13miles/min 780.00070mph 14miles/min 840.00075mph 15miles/min 900.00080mph 16miles/min 960.00086mph 17miles/min 1,020.00091mph 18miles/min 1,080.00097mph 19miles/min 1,140.00102mph Miles per minute Miles per hour 20miles/min 1,200.00107mph 21miles/min 1,260.00113mph 22miles/min 1,320.00118mph 23miles/min 1,380.00123mph 24miles/min 1,440.00129mph 25miles/min 1,500.00134mph 26miles/min 1,560.00139mph 27miles/min 1,620.00145mph 28miles/min 1,680.00150mph 29miles/min 1,740.00156mph 30miles/min 1,800.00161mph 31miles/min 1,860.00166mph 32miles/min 1,920.00172mph 33miles/min 1,980.00177mph 34miles/min 2,040.00182mph 35miles/min 2,100.00188mph 36miles/min 2,160.00193mph 37miles/min 2,220.00198mph 38miles/min 2,280.00204mph 39miles/min 2,340.00209mph Miles per minute Miles per hour 40miles/min 2,400.00215mph 41miles/min 2,460.00220mph 42miles/min 2,520.00225mph 43miles/min 2,580.00231mph 44miles/min 2,640.00236mph 45miles/min 2,700.00241mph 46miles/min 2,760.00247mph 47miles/min 2,820.00252mph 48miles/min 2,880.00257mph 49miles/min 2,940.00263mph 50miles/min 3,000.00268mph 51miles/min 3,060.00274mph 52miles/min 3,120.00279mph 53miles/min 3,180.00284mph 54miles/min 3,240.00290mph 55miles/min 3,300.00295mph 56miles/min 3,360.00300mph 57miles/min 3,420.00306mph 58miles/min 3,480.00311mph 59miles/min 3,540.00317mph Miles per minute Miles per hour 60miles/min 3,600.00322mph 61miles/min 3,660.00327mph 62miles/min 3,720.00333mph 63miles/min 3,780.00338mph 64miles/min 3,840.00343mph 65miles/min 3,900.00349mph 66miles/min 3,960.00354mph 67miles/min 4,020.00359mph 68miles/min 4,080.00365mph 69miles/min 4,140.00370mph 70miles/min 4,200.00376mph 71miles/min 4,260.00381mph 72miles/min 4,320.00386mph 73miles/min 4,380.00392mph 74miles/min 4,440.00397mph 75miles/min 4,500.00402mph 76miles/min 4,560.00408mph 77miles/min 4,620.00413mph 78miles/min 4,680.00418mph 79miles/min 4,740.00424mph Metric Conversion Metric Converter Site map Contact us This site is owned and maintained by Wight Hat Ltd ©2003-2025. D Our full terms & conditions can be found by clicking here. Whilst every effort has been made to ensure the accuracy of the metric calculators and charts given on this site, we cannot make a guarantee or be held responsible for any errors that have been made. Our full terms & conditions can be found by clicking here (in English).
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https://brilliant.org/wiki/surface-area-sphere/
Surface Area of a Sphere Sign up with Facebook or Sign up manually Already have an account? Log in here. Worranat Pakornrat, Abhineet Goel, Thaddeus Abiy, and Arron Kau Mahindra Jain Jongheun Lee Geoff Pilling Yash Dev Lamba Gene Keun Chung Andres Gonzalez Jimin Khim contributed A sphere is a perfectly round geometrical 3-dimensional object. It can be characterized as the set of all points located distance r (radius) away from a given point (center). It is perfectly symmetrical, and has no edges or vertices. A sphere with radius r has a volume of 34​πr3 and a surface area of 4πr2. A sphere has several interesting properties, one of which is that, of all shapes with the same surface area, the sphere has the largest volume. Contents Proof Archimedes' Hat-Box Theorem Practice Problems Proof To prove that the surface area of a sphere of radius r is 4πr2, one straightforward method we can use is calculus. We first have to realize that for a curve parameterized by x(t) and y(t), the arc length is S=∫ab​(dtdy​)2+(dtdx​)2​dt. From this we can derive the formula for the surface area of the solid obtained by rotating this about the x-axis. This turns out to be A=2π∫ab​y(dtdy​)2+(dtdx​)2​dt. We can obtain a sphere by revolving half a circle about the x-axis. This circle can be parameterized as x(t)=rcos(t) and y(t)=rsin(t) for 0≤t≤π. From this, we get dtdx​=−rsin(t),dtdy​=rcos(t). Substituting in our equations for surface area gives A​=2π∫0π​rsin(t)(−rsin(t))2+(rcos(t))2​ dt=2π∫0π​rsin(t)r2(sin(t)2+cos(t)2)​ dt=2π∫0π​r2sin(t) dt=2πr2∫0π​sin(t) dt=4πr2. □​​ Archimedes' Hat-Box Theorem Archimedes' Hat-Box Theorem Archimedes' hat-box theorem states that for any sphere section, its lateral surface will equal that of the cylinder with the same height as the section and the same radius of the sphere. Let us recall our last proof section. After revolving the semicircle around the x-axis, we will obtain a sphere's surface area, and if we cut just a partial section with parallel bases, the new surface area will be demonstrated in the image below: From the image, the section's lateral surface area is colored light blue with 2 circular bases of different radii. In order to visualize the section's height better, this section will be rotated by 90 degrees, as shown below: Now inside the section, there are 2 variable angles, ∠a and ∠b, which appear as the integral borders of the cut section. From the proof's conclusion, the surface area of the section (A′) can be calculated as A′​=2πr2∫ab​sin(t) dt=2πr2[−cos(t)∣ab​]=(2πr)r[cos(a)−cos(b)].​ Considering the right triangles with radius r (thick red) in the image, it is obvious that r is the hypotenuse side for both. As a result, the vertical sides can be calculated as r×cos(a) and r×cos(b) for the left and right triangles, respectively. Hence, the height of the section is h=(r×cos(a))−(r×cos(b))=r[cos(a)−cos(b)]. Substituting this term to the previous equation gives A′=(2πr)r[cos(a)−cos(b)]=2πrh. Clearly, this is the formula for the cylinder's lateral surface with radius r and height h! That means the lateral surface area of the sphere section equals the lateral surface area of the cylinder with radius r and height h, as shown in the image, and this holds true for any level of the sphere involved. □​ Tomato slice Cucumber slice Each slice of both kinds has the same lateral surface area Uncertain, depending on the level of cut A spherical tomato and a cylindrical portion of a cucumber have the same height and radius. Then they are chopped into slices of equal thickness, as shown above. Comparing each slice of both kinds, which slice will have more lateral surface area of the peel? The correct answer is: Each slice of both kinds has the same lateral surface area The melon plus the plate Both options have the same surface area Not enough information The blue dome A small green circle is inscribed within the section of a bigger blue circle, touching the mid-chord, as shown above left. Then the graphs are revolved around the y-axis to generate three figures: a blue cover dome, a green spherical melon, and a red serving plate. Which of the following options will have more surface area? I. The blue dome II. The melon plus the plate The correct answer is: Both options have the same surface area A sweets shop sells candies in 2 different styles: a spherical ball and a dome. The dome-like shape is a spherical section of a larger sphere with height h and base radius R, as shown above, while the candy ball has radius r with 2r=R+h. If both shapes have the same total surface area, what is the ratio hR​? The correct answer is: 2 Practice Problems What is the surface area of a sphere of radius 3? The surface area is 4π×32=36π. □​ If the volume of a sphere is 36π, what is the surface area of the sphere? Observe that the volume of the sphere can be rewritten as 36π=34​π×33. Then, since the volume of a sphere with radius r is 34​πr3, it follows that the radius of the sphere in this problem is r=3. Hence, its the surface area is 4πr2=4π×32=36π. □​ The volume of a sphere has grown 8 times. Then how many times has the surface area grown in the meanwhile? Observe that the volume of the sphere is 34​πr3. This implies that it is proportional to r3, that is34​πr3∝r3. Then 8 times growth in the volume of the sphere implies 2 times growth in the radius of sphere. Then, since the surface area of sphere is 4πr2∝r2, the surface area of the sphere has grown 22=4 times. □​ You have a watermelon whose volume is 288 cm3. If you cut the watermelon into halves, what is the surface area of one half of the watermelon? (Assume that the watermelon is a perfect sphere.) From the formula V=34​πr3 for the volume of a sphere with radius r, you know that the radius of the watermelon is r=6 cm. Since you cut the watermelon into two exact halves, you may think that the surface area of a half watermelon is also exactly half the surface area of the whole watermelon. However, this thinking is wrong. As shown int the above diagram, the surface area of a half watermelon is bigger than half the surface area of a whole watermelon, by the area the cross section A. Thus, the surface area of a half watermelon is (Half the surface area of the watermelon)+(Area of A). Since A is a circle whose radius is the same as the radius of the watermelon, our answer is 21​×4π×62+π×62=108π. □​ Cite as: Surface Area of a Sphere. Brilliant.org. Retrieved 07:27, August 30, 2025, from
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https://artofproblemsolving.com/wiki/index.php/Complementary_counting?srsltid=AfmBOoqbevxYYQ-iX1gKCQjnX2LznMA-0BEanEXNuo7mcADlmcyGMQIE
Art of Problem Solving Complementary counting - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki Complementary counting Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Complementary counting In combinatorics, complementary counting is a counting method where one counts what they don't want, then subtracts that from the total number of possibilities. In problems that involve complex or tedious casework, complementary counting is often a far simpler approach. A large hint that complementary counting may lead to a quick solution is the phrase "not" or "at least" within a problem statement. More formally, if is a subset of , complementary counting exploits the property that , where is the complement of . In most instances, though, is obvious from context and is committed from mention. Contents 1 Complementary Probability 2 Examples 2.1 Example 1 2.2 Example 2 2.3 Example 3 2.4 Example 4 2.5 Example 5 3 Resources 4 See also Complementary Probability There is a probability equivalent of complementary counting. For any event, the probability it happens plus the probability it does not happen is one. Thus, we have the identity Like its counting analog, complementary probability often vastly simplifies tedious casework. Unlike complementary counting, though, it sees frequent use as an intermediate step, primarily because computing complements is much easier in probability than in counting. Examples Here are some examples that demonstrate complementary counting and probability in action. It is worth noting that complementary probability is not typically an intermediate step, but a framework upon which a solution is built. Example 1 How many positive integers less than are not a multiple of five? Solution: We use a complementary approach. The total number of positive integers, with no restrictions, is integers. What we don't want are the multiples of five. These are or ; it's easy to see that there are of them. Thus, our answer is is . Example 2 2006 AMC 10A Problem 21: How many four-digit positive integers have at least one digit that is a 2 or a 3? Solution: We use a complementary approach. With no restrictions, there are four-digit positive integers. What we don't want are the four-digit integers with no digit that is a two or three, Using a constructive approach, the first digit can be one of seven integers; and . Note that the first digit cannot be zero, or else it ceases to have four digits. However, the second, third, and fourth digits can be zero; as a result, they have eight options. So, our total number of two-and-three-free numbers is . Hence, our final answer is , as desired. Example 3 Sally is drawing seven houses. She has four crayons, but she can only color any house a single color. In how many ways can she color the seven houses if at least one pair of consecutive houses must have the same color? Solution: Use complementary counting. First, we find the total colorings without restriction, which we do by constructing them. She has four options for what color the first house can be, four options for the second, and so on. Hence, there are ways she can color the four houses. Next, we find the possibilities where every house's next-door neighbor is a different color. Using a constructive approach, she has four options for the color of the first house. We have to make sure the next house is a different color; as a result, there are only three options for the color of the second house, with the color of the first house unavailable. By similar logic, there are 3 options for the third house, and so on for every other house. Combining these yields possibilities if every house must be a different color. Putting these two together, there are ways she can color the seven houses four colors if at least one pair of consecutive houses must be the same color. Example 4 2002 AIME I Problem 1 Many states use a sequence of three letters followed by a sequence of three digits as their standard license-plate pattern. Given that each three-letter three-digit arrangement is equally likely, the probability that such a license plate will contain at least one palindrome (a three-letter arrangement or a three-digit arrangement that reads the same left-to-right as it does right-to-left) is , where and are relatively prime positive integers. Find Solution: We use complementary probability. So first, we find the probability that a license plate has no palindromes; in other words, the probability that the first and third numbers and letters are distinct. The probability the numbers are distinct is , as for every digit of the first number, one out of ten is the same digit; similarly, for the letters, it is . Multiplying these together gives that the probability that a license plate has no palindromes is Taking the complement of this, the probability a license plate has a palindrome is thus . Hence, our answer is . Example 5 2008 AMC 12B Problem 22: A parking lot has 16 spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers chose spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires 2 adjacent spaces. What is the probability that she is able to park? Solution: Use complementary probability, where we find the probability that no two open spots are consecutive. With no restrictions, the total number of ways the 12 cars could park is , where is a combination. Finding how many permutations of the cars and spaces leave no two spaces next to each other is a more challenging task, though. One might eventually think to treat this problem like a distribution, a stars-and-bars approach. Let the 12 cars be stars and the 4 spaces be bars. Here is one arrangement of our stars and bars; The question mandates that no two bars sit next to each other. Thus, we have 13 "slots" where the bars could go (eleven between the stars, two at the endpoints), where only one bar can fit in each slot. It follows that the number of ways to insert these bars is . Then the probability that Auntie Em cannot park is . Finally, our answer is . Resources AoPS Complementary Counting Part 1 AoPS Complementary Counting Part 2 AoPS Complementary Probability Part 1 AoPS Complementary Probability Part 2 AoPS Complementary Probability Part 3 AoPS Complementary Probability Part 4 See also Casework Constructive counting Overcounting Retrieved from " Categories: Combinatorics Definition Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
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https://unacademy.com/content/question-answer/chemistry/what-is-the-relation-between-mole-fraction-and-molality/
What is the relation between mole fraction and molality Access free live classes and tests on the app Download + IIT JEE AFCAT AP EAMCET Bank Exam BPSC CA Foundation CAPF CAT CBSE Class 11 CBSE Class 12 CDS CLAT CSIR UGC GATE IIT JAM JEE Karnataka CET Karnataka PSC Kerala PSC MHT CET MPPSC NDA NEET PG NEET UG NTA UGC Railway Exam SSC TS EAMCET UPSC WBPSC CFA Search for: LoginJoin for Free Profile Settings Refer your friends Sign out Terms & conditions • Privacy policy About • Careers • Blog © 2023 Sorting Hat Technologies Pvt Ltd Question & Answer » Chemistry Questions » What is the Relation Between Mole Fraction and Molality What is the Relation Between Mole Fraction and Molality Share Answer: The ratio of the moles of a component present in the solution to the total number of moles of all the components in the solution is called the mole fraction of that component. While molality is termed as the measure of the number of moles of solute in the solution that corresponds to 1000g or 1kg of solvent Mole fraction= number of moles of solute number of moles of solvent mole fraction= nB / nA + nB Molality= moles of solute / Moles of solvent Molality= nB/WA Relation between mole fraction and molality- Let us consider a solution with A= solvent and B= solute. So, Mole fraction of solvent = x A Mole fraction of solute = x B Number of moles of solvent = n A Number of moles of solute = n B Mass of solvent = W A Mass of solute = W B Molar mass of solvent = M A The molar mass of solute = M B Mole fraction of solute = x B=n B / n B + n A Mole fraction of solvent = x A=n A/n B + n A Now, let us divide equations (1) and (2) Unacademy is India’s largest online learning platform. Download our apps to start learning Starting your preparation? Call us and we will answer all your questions about learning on Unacademy ##### Call +91 8585858585 Company About usShikshodayaCareers we're hiringBlogsPrivacy PolicyTerms and Conditions Help & support User GuidelinesSite MapRefund PolicyTakedown PolicyGrievance Redressal Products Learner appEducator appParent app Popular goals IIT JEEUPSCSSCCSIR UGC NETNEET UG Trending exams GATECATCANTA UGC NETBank Exams Study material UPSC Study MaterialNEET UG Study MaterialCA Foundation Study MaterialJEE Study MaterialSSC Study Material © 2025 Sorting Hat Technologies Pvt Ltd Goals AFCAT AP EAMCET Bank Exam BPSC CA Foundation CAPF CAT CBSE Class 11 CBSE Class 12 CDS CLAT CSIR UGC GATE IIT JAM JEE Karnataka CET Karnataka PSC Kerala PSC MHT CET MPPSC NDA NEET PG NEET UG NTA UGC Railway Exam SSC TS EAMCET UPSC WBPSC CFA Share via COPY
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https://artofproblemsolving.com/wiki/index.php/2019_AMC_12B_Problems/Problem_25?srsltid=AfmBOornGUGUn6e60OeI3fQBtiFZYbDsvW6lCLuBLrPq-i288UQFxlJC
Art of Problem Solving 2019 AMC 12B Problems/Problem 25 - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki 2019 AMC 12B Problems/Problem 25 Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search 2019 AMC 12B Problems/Problem 25 Contents [hide] 1 Problem 2 Solution 1 (vectors) 3 Solution 2 4 Solution 3 (Complex Numbers) 5 Solution 4 (Homothety) 6 Video Solution by MOP 2024 6.1 Solution 5 7 See Also Problem Let be a convex quadrilateral with and Suppose that the centroids of and form the vertices of an equilateral triangle. What is the maximum possible value of the area of ? Solution 1 (vectors) Place an origin at , and assign position vectors of and . Since is not parallel to , vectors and are linearly independent, so we can write for some constants and . Now, recall that the centroid of a triangle has position vector . Thus the centroid of is ; the centroid of is ; and the centroid of is . Hence , , and . For to be equilateral, we need . Further, . Hence we have , so is equilateral. Now let the side length of be , and let . By the Law of Cosines in , we have . Since is equilateral, its area is , while the area of is . Thus the total area of is , where in the last step we used the subtraction formula for . Alternatively, we can use calculus to find the local maximum. Observe that has maximum value when e.g. , which is a valid configuration, so the maximum area is . Solution 2 Let , , be the centroids of , , and respectively, and let be the midpoint of . , , and are collinear due to well-known properties of the centroid. Likewise, , , and are collinear as well. Because (as is also well-known) and , we have . This implies that is parallel to , and in terms of lengths, . (SAS Similarity) We can apply the same argument to the pair of triangles and , concluding that is parallel to and . Because (due to the triangle being equilateral), , and the pair of parallel lines preserve the angle, meaning . Therefore is equilateral. At this point, we can finish as in Solution 1, or, to avoid using trigonometry, we can continue as follows: Let , where due to the Triangle Inequality in . By breaking the quadrilateral into and , we can create an expression for the area of . We use the formula for the area of an equilateral triangle given its side length to find the area of and Heron's formula to find the area of . After simplifying, Substituting , the expression becomes We can ignore the for now and focus on . By the Cauchy-Schwarz inequality, The RHS simplifies to , meaning the maximum value of is . Thus the maximum possible area of is . Solution 3 (Complex Numbers) Let , , , and correspond to the complex numbers , , , and , respectively. Then, the complex representations of the centroids are , , and . The pairwise distances between the centroids are , , and , all equal. Thus, , so . Hence, is equilateral. By the Law of Cosines, . . Thus, the maximum possible area of is . ~ Leo.Euler Solution 4 (Homothety) Let , and be the centroids of , and , respectively, and let and be the midpoints of and , respectively. Note that and are of the way from to and , respectively, by a well-known property of centroids. Then a homothety centered at with ratio maps and to and , respectively, implying that is equilateral too. But is the medial triangle of , so is also equilateral. We may finish with the methods in the solutions above. ~ numberwhiz While the solutions above have attempted the problem in general, knowing the fact that is equilateral greatly reduces the effort to find the final answer, hence I propose an alternative after this. Let and . By cosine rule on : Thus, the total area of the quadrilateral is supposedly: Where the inequality comes from a common trigonometric identity, ~ SouradipClash_03 Video Solution by MOP 2024 ~r00tsOfUnity Solution 5 Let be the centroids of respectively, then , since , , since by midsegment theorem, so Similarly, , So is an equilateral triangle Assume , then , the area The maximal value happens when , and the value is , and the answer is . ~szhangmath See Also 2019 AMC 12B (Problems • Answer Key • Resources) Preceded by Problem 24Followed by Last Problem 1•2•3•4•5•6•7•8•9•10•11•12•13•14•15•16•17•18•19•20•21•22•23•24•25 All AMC 12 Problems and Solutions These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Retrieved from " Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
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https://www.quora.com/A-4-digit-number-multiplied-by-4-gives-same-number-in-reverse-order-What-is-the-four-digit-number-and-how-do-I-prove-it
A 4-digit number multiplied by 4 gives same number in reverse order. What is the four digit number and how do I prove it? - Quora Something went wrong. Wait a moment and try again. Try again Skip to content Skip to search Sign In Mathematics Reverse Four-digit Numbers Logical Math Puzzles Proofs (mathematics) Arithmetic Number Theory Number Puzzles Algebraic Puzzles 5 A 4-digit number multiplied by 4 gives same number in reverse order. What is the four digit number and how do I prove it? All related (48) Sort Recommended Sunil Kumar Gopal Author has 4.5K answers and 29.1M answer views ·12y Originally Answered: A 4-digit number multiplied by 4 gives same number in reverse order. What is the four digit no. and how to prove it? · Lets say the number is written as ABCD (1000a+100b+10c+d). Multiplying it by 4, gives you DCBA (1000d+100c+10b+a) By divisibility of 4, BA is a two-digit number divisible by 4. Now, the largest possible value of ABCD is 2499, because anything more will result in a five digit number when multiplied by 4. Out of the two possibilites 1 and 2 for A, follows that A is even, so 2. From , D is either 8 or 9. However, since D times 4 would end in 2, D is 8. Using , 4000a+400b+40c+4d = 1000d+100c+10b+a, which would simplify to 2c-13b = 1 after substituting A = 2 and D = 8. The only possible value of Continue Reading Lets say the number is written as ABCD (1000a+100b+10c+d). Multiplying it by 4, gives you DCBA (1000d+100c+10b+a) By divisibility of 4, BA is a two-digit number divisible by 4. Now, the largest possible value of ABCD is 2499, because anything more will result in a five digit number when multiplied by 4. Out of the two possibilites 1 and 2 for A, follows that A is even, so 2. From , D is either 8 or 9. However, since D times 4 would end in 2, D is 8. Using , 4000a+400b+40c+4d = 1000d+100c+10b+a, which would simplify to 2c-13b = 1 after substituting A = 2 and D = 8. The only possible value of (b,c) where they would both be single digit integers is (1,7). Also to note from , B is odd. And from , B is either 1 or 3. The answer you're looking for is 2178, multiplied by 4 gives 8712. Upvote · 999 127 99 11 Promoted by Betterbuck Anthony Madden Writer for Betterbuck ·Updated Aug 15 What are the weirdest mistakes people make on the internet right now? Here are a couple of the worst mistakes I’ve seen people make: Not using an ad blocker If you aren’t using an ad blocker yet, you definitely should be. A good ad blocking app will eliminate virtually all of the ads you’d see on the internet before they load. No more YouTube ads, no more banner ads, no more pop-up ads, etc. Most people I know use Total Adblock (link here) - it’s about £2/month, but there are plenty of solid options. Ads also typically take a while to load, so using an ad blocker reduces loading times (typically by 50% or more). They also block ad tracking pixels to protect your pr Continue Reading Here are a couple of the worst mistakes I’ve seen people make: Not using an ad blocker If you aren’t using an ad blocker yet, you definitely should be. A good ad blocking app will eliminate virtually all of the ads you’d see on the internet before they load. No more YouTube ads, no more banner ads, no more pop-up ads, etc. Most people I know use Total Adblock (link here) - it’s about £2/month, but there are plenty of solid options. Ads also typically take a while to load, so using an ad blocker reduces loading times (typically by 50% or more). They also block ad tracking pixels to protect your privacy, which is nice. More often than not, it saves even more than 50% on load times - here’s a test I ran: Using an ad blocker saved a whopping 6.5+ seconds of load time. Here’s a link to Total Adblock, if you’re interested. Not getting paid for your screentime Apps like Freecash will pay you to test new games on your phone. Some testers get paid as much as £270/game. Here are a few examples right now (from Freecash's website): You don't need any kind of prior experience or degree or anything: all you need is a smartphone (Android or IOS). If you're scrolling on your phone anyway, why not get paid for it? I've used Freecash in the past - it’s solid. (They also gave me a £3 bonus instantly when I installed my first game, which was cool). Upvote · 999 557 99 60 9 4 Assistant Bot · 1y Let's denote the 4-digit number as N N. We can express N N in terms of its digits: N=1000 a+100 b+10 c+d N=1000 a+100 b+10 c+d where a,b,c,d a,b,c,d are the digits of the number, and a a (the first digit) cannot be zero (since it is a 4-digit number). According to the problem, multiplying N N by 4 gives the reverse of N N: 4 N=1000 d+100 c+10 b+a 4 N=1000 d+100 c+10 b+a Substituting the expression for N N into the equation, we get: 4(1000 a+100 b+10 c+d)=1000 d+100 c+10 b+a 4(1000 a+100 b+10 c+d)=1000 d+100 c+10 b+a Expanding the left side: 4000 a+400 b+40 c+4 d=1000 d+100 c+10 b+a 4000 a+400 b+40 c+4 d=1000 d+100 c+10 b+a Now, rearranging this equation results in: 4000 a+400 b+40 c+4 d−1000 d−100 c−10 b−a=0 4000 a+400 b+40 c+4 d−1000 d−100 c−10 b−a=0 Combinin Continue Reading Let's denote the 4-digit number as N N. We can express N N in terms of its digits: N=1000 a+100 b+10 c+d N=1000 a+100 b+10 c+d where a,b,c,d a,b,c,d are the digits of the number, and a a (the first digit) cannot be zero (since it is a 4-digit number). According to the problem, multiplying N N by 4 gives the reverse of N N: 4 N=1000 d+100 c+10 b+a 4 N=1000 d+100 c+10 b+a Substituting the expression for N N into the equation, we get: 4(1000 a+100 b+10 c+d)=1000 d+100 c+10 b+a 4(1000 a+100 b+10 c+d)=1000 d+100 c+10 b+a Expanding the left side: 4000 a+400 b+40 c+4 d=1000 d+100 c+10 b+a 4000 a+400 b+40 c+4 d=1000 d+100 c+10 b+a Now, rearranging this equation results in: 4000 a+400 b+40 c+4 d−1000 d−100 c−10 b−a=0 4000 a+400 b+40 c+4 d−1000 d−100 c−10 b−a=0 Combining like terms gives: 3999 a+390 b−60 c−996 d=0 3999 a+390 b−60 c−996 d=0 To simplify, we can divide the entire equation by 3: 1333 a+130 b−20 c−332 d=0 1333 a+130 b−20 c−332 d=0 Now, we need to find integer solutions for a,b,c,d a,b,c,d where a,b,c,d a,b,c,d are digits (0 to 9) and a a is from 1 to 9. Rearranging gives: 20 c=1333 a+130 b−332 d 20 c=1333 a+130 b−332 d This means 1333 a+130 b−332 d 1333 a+130 b−332 d must be divisible by 20. Finding Valid Combinations Finding a a: Since a a must be from 1 to 9, we can try different values for a a and calculate possible values for b,c,d b,c,d. Trying values for a a: For a=2 a=2: 20 c=2666+130 b−332 d 20 c=2666+130 b−332 d We can test values of b b and d d to find c c. Continuing this process, we find: When a=2 a=2, b=1 b=1, c=7 c=7, d=8 d=8: We check if N=2178 N=2178: 4×2178=8712 4×2178=8712 The reverse of 2178 is indeed 8712. Conclusion Thus, the 4-digit number that, when multiplied by 4, gives the same number in reverse order is: 2178 2178 This proves that 4×2178=8712 4×2178=8712, which is the reverse of 2178. Upvote · Sabir Akhtar Works at Amazon (company) (2019–present) ·9y 2178 is the desired number. The proof is given below. When a certain four-digit number is multiplied by 4 its digits appear in reverse order. It also has the following properties. It's first digit is a quarter of the last one. And its second digit is one less than the first. Let the four digit number = 1000a + 100b + 10c + d -------- eqn(i) "When a four-digit number is multiplied by 4 its digits appear in reverse order." 4(1000a+100b+10c+d) = 1000d + 100c + 10b + a or, 4000a + 400b + 40c + 4d = 1000d + 100c + 10b + a or, 4000a - a + 400b - 10b = 100c - 40c + 1000d - 4d or, 3999a + 390b = 60c + Continue Reading 2178 is the desired number. The proof is given below. When a certain four-digit number is multiplied by 4 its digits appear in reverse order. It also has the following properties. It's first digit is a quarter of the last one. And its second digit is one less than the first. Let the four digit number = 1000a + 100b + 10c + d -------- eqn(i) "When a four-digit number is multiplied by 4 its digits appear in reverse order." 4(1000a+100b+10c+d) = 1000d + 100c + 10b + a or, 4000a + 400b + 40c + 4d = 1000d + 100c + 10b + a or, 4000a - a + 400b - 10b = 100c - 40c + 1000d - 4d or, 3999a + 390b = 60c + 996d or, 1333a + 130b = 20c + 332d ------------------------------------eqn(ii) Now remember the two properies. Its first digit is a quarter of the last one a = .25d or, d = 4a The second digit is one less than the first. b = (a-1) Now, from eqn (ii) 1333a + 130b = 20c + 332d Replace b and d or, 1333a + 130(a-1) = 20c + 332(4a) or, 1333a + 130a - 130 = 20c + 1328a or, 1333a + 130a - 1328a = 20c + 130 or, 135a = 20c + 130 27a = 4c + 26 or, a =(4/27) c + (26/27) A close look at the equation shows that there is only one single digit integer solution c = 7; a = 2 Now, b = 2 - 1 or, b = 1 and d = 4(2) or, d = 8 2178 is the number 4 2178 = 8712 Upvote · 99 17 9 2 9 1 Related questions More answers below Which 4-digit number multiplied by 4 gives the same number in reverse? A five digit number, when multiplied by 4, gives a product which is in the reverse order of the number. What is the number? What number times 4 is its reverse? Can you find a four-digit number which is reversed when multiplied by 9? What six-digit number has the same digits when multiplied by 1, 2, 3, 4, 5, or 6? Bhawan Rai B. Sc. in Mathematics, Sssihl (Graduated 2017) ·8y Okay many have given answers with so much math now let me try to do the same thing with just arguments so lets say that the required 4 digit number is xyzw where each letter represents a digit therefore 4xyzw = wzyx possibilities of x are only 1 or 2 as 3 or more would result in 5-digit number when multiplied by 4 and now note that x can’t be 1 also as ther is no number which gives 1 in the right most digit => x = 2 Now consider w, possibilities are 3 or 8 to give 2 in the right most digit when multiplied by 4 as 34 = 12 or 48 =32 but it can’t be 3 as it should be at least 4x = 42 = 8 thus Continue Reading Okay many have given answers with so much math now let me try to do the same thing with just arguments so lets say that the required 4 digit number is xyzw where each letter represents a digit therefore 4xyzw = wzyx possibilities of x are only 1 or 2 as 3 or more would result in 5-digit number when multiplied by 4 and now note that x can’t be 1 also as ther is no number which gives 1 in the right most digit => x = 2 Now consider w, possibilities are 3 or 8 to give 2 in the right most digit when multiplied by 4 as 34 = 12 or 48 =32 but it can’t be 3 as it should be at least 4x = 42 = 8 thus w=8 Now take both yz is such that 4yz = zy - 3 (where 3 is to compensate carry from 4w=48=32 ) and y is such that it will not give carry over when multiplied by 4, so possibilities of y are 1 or 2, and y can’t be 2 because when multiply 4 we can never get an odd number so that when we add carry over 3 it would be 2 at the last digit thus y=1, Now z is such that it gives 1 in the last digit when it is multiplied by 4 and added carry over 3, So possibilities of z are 2 (as 42+3=11 ) or 7 (as 47+3=31) BUT z cant be 2 because we should at least more than 41=4 thus z=7 , Hence the required number is xyzw = 2178 and 42178=8712 as required Upvote · 99 21 Vivek Gangwar Studied at Indian Institute of Science, Bangalore (IISc) (Graduated 2020) ·Updated 6y Originally Answered: A 4-digit number multiplied by 4 gives same number in reverse order. What is the four digit no. and how to prove it? · 2178. Let the number be a b c d a b c d. a b c d=1000 a+100 b+10 c+d a b c d=1000 a+100 b+10 c+d No need to say that a,b,c a,b,c and d d range from 0 to 9. As per the problem: 4∗(a b c d)=d c b a 4∗(a b c d)=d c b a ...(1) i.e., 4000 a+400 b+40 c+4 d 4000 a+400 b+40 c+4 d=1000 d+100 c+10 b+a=1000 d+100 c+10 b+a or 3999 a+390 b=996 d+60 c 3999 a+390 b=996 d+60 c To satisfy unit places of numbers of both sides of equation (1), a=2 a=2 and d=8 d=8 (and not 3, as it won't give b and c from 0 to 9) So finally we have, 1+13 b=2 c 1+13 b=2 c Since b and c are from 0 to 9, the above equation is satisfied only for b = 1 and c = 7. Hence the answer is 2178 Upvote · 99 23 9 2 9 1 Sponsored by Innovation Vista Virtual CIOs: 100% of the Expertise, a Fraction of the Cost ®. Innovation Vista's Virtual CIOs make IT a driver of business results, for far less than a full-time CIO. Learn More 99 87 Anil Bapat Lives in Mumbai, Maharashtra, India · Author has 2.8K answers and 3.8M answer views ·7y The Number is 2178. 2178 4 = 8712. 8712 is the reverse of 2178. Since it’s 4 Digit Number and since after multiplication also it has to be a 4 Digit Number, it’s easy to see that the Number has to be less than 2500 since 2500 4 = 10000. The Number also can’t be starting with 1, as any Number when multiplied by 4, can’t result into an odd number. So the Number has to start with 2 and end with 8 as 84 is 32 thus 2 of 32 matches with 2 of the Number. The Number can’t have second Digit as 0, so the second Digit has to be 1 or above. This is because if the last Digit is 8 and when 8 is multulied by Continue Reading The Number is 2178. 2178 4 = 8712. 8712 is the reverse of 2178. Since it’s 4 Digit Number and since after multiplication also it has to be a 4 Digit Number, it’s easy to see that the Number has to be less than 2500 since 2500 4 = 10000. The Number also can’t be starting with 1, as any Number when multiplied by 4, can’t result into an odd number. So the Number has to start with 2 and end with 8 as 84 is 32 thus 2 of 32 matches with 2 of the Number. The Number can’t have second Digit as 0, so the second Digit has to be 1 or above. This is because if the last Digit is 8 and when 8 is multulied by 4 gives the carry of 3 and to make the next Digit (from right) as 0, some Digit when multiplied by 4 has to result into a number ending with 7, which is impossible. If we try with 2108, 2118, 2138 etc., we will see that 2178 satisfies the required condition. Thus 2178 is the required Number. At the stage of 28, this can be very easily solved by considering the Digits as x and y i. e. considering the Number as 2xy8. On solving for x and y, we get x as 1 and y as 7. Since there are 2 variables and only 1 equation, there would be many answers and we will have to eliminate non-integer answers to arrive at x = 1 and y =7 as the only possible values for x and y, thus leading us to the final answer of 2178. You can also solve this by considering the Number itself as XY i. e. 10X+Y. So necessarily X is greater than 100 (to make it a 4 Digit Number) and typically Y is anything. Applying the given conditions, and solving for X and Y, we get X as 132 and Y as 858. So the actual Number which is 10X + Y is nothing but 1320+ 858 = 2178! Here again there would be many answers but the answer that works in the given situation is when X is 132 and Y is 858 taking us to 2178. Upvote · 9 8 Related questions More answers below Can multiplying a four-digit number by 2 ever reverse its digits? If yes, then produce an example. If not, then prove it is not possible. Is an additional 3 digit number multiplied by 4 so the answer is reversed? Which two digit number when multiplied by 32 gives the same number when reverse is multiplied with 23? What is the number, between 1 and 1 million, that can be multiplied by 2, 3, 4, 5 or 6 and the result will include the same digits in the same order as in the original number? What is the three-digit number that when multiplied by a four-digit number gives us a five-digit number with the same first four digits as the second one? Anonymous 12y Originally Answered: A 4-digit number multiplied by 4 gives same number in reverse order. What is the four digit no. and how to prove it? · abcd is a 4-digit no. such that 4abcd = dcba Clearly a can only be 1 or 2 4000a+400b+40c+4d = 1000d+100b+10c+a Working mod 10, this gives: 4d=a (mod 10) If a=1, this equation has no solution. a must be 2. This means d can be 3 or 8. Equation is now: 8000+400b+40c+4d=1000d+100c+10b+2 LHS is atleast 8000. If d=3, RHS can attain a maximum value of 3992. Hence d=8. Equation is now: 8000+400b+40c+32=8000+100c+10b+2 390b-60c+30=0 2c-13b=1 One solution is (7,1) Any other solution must be of the form (7+13k,1-2k) where k is an integer. (This is a diophantine equation). So (7,1) is the only solution in single Continue Reading abcd is a 4-digit no. such that 4abcd = dcba Clearly a can only be 1 or 2 4000a+400b+40c+4d = 1000d+100b+10c+a Working mod 10, this gives: 4d=a (mod 10) If a=1, this equation has no solution. a must be 2. This means d can be 3 or 8. Equation is now: 8000+400b+40c+4d=1000d+100c+10b+2 LHS is atleast 8000. If d=3, RHS can attain a maximum value of 3992. Hence d=8. Equation is now: 8000+400b+40c+32=8000+100c+10b+2 390b-60c+30=0 2c-13b=1 One solution is (7,1) Any other solution must be of the form (7+13k,1-2k) where k is an integer. (This is a diophantine equation). So (7,1) is the only solution in single digit integers. Hence our answer is 2178. Upvote · 9 6 9 2 Promoted by JH Simon JH Simon Author of 'How To Kill A Narcissist' ·Updated Fri How do we become aware of and heal the broken parts of us that attract(ed) and allow(ed) narcissistic abuse? Isolation is where you should begin. Like being a fish in water, entering into a narcissistic relationship happens seamlessly. By spending extended periods alone without a narcissist to muddy your mental and emotional space, you can grow your awareness of the deep longing which drives you into the arms of a narcissist in the first place. Also, you can see how this longing is deeply rooted in pain from your past. Go to the movies alone, go for a long walk alone, travel alone for extended periods, sit alone in your room and meditate, go shopping alone, go to an event alone. During all of these ac Continue Reading Isolation is where you should begin. Like being a fish in water, entering into a narcissistic relationship happens seamlessly. By spending extended periods alone without a narcissist to muddy your mental and emotional space, you can grow your awareness of the deep longing which drives you into the arms of a narcissist in the first place. Also, you can see how this longing is deeply rooted in pain from your past. Go to the movies alone, go for a long walk alone, travel alone for extended periods, sit alone in your room and meditate, go shopping alone, go to an event alone. During all of these activities, be mindful of how you feel. Shame, fear and uncertainty will all show up. Accepting them and being with them in a mindful way is at first unsettling and deeply painful. But if you pay attention, you’ll see how these echoes from the past are actually the wounds which lead you into abusive relationships as a form of escape. When they come up, don’t panic. Breathe deeply and focus on them. Don’t just pay attention to your thoughts about them. Instead, really go into them. Where are they in your body? In your chest? In your belly? In your legs as tension? What qualities do they have? Is the pain dull or sharp? Does the pain originate in one place or is it all over your body? Breathe deep into your belly and stay with the feeling. Does the feeling lead to total confusion in the mind and make you think you are going crazy? Right, stay with that confusion. Go deeper into it for as long as you can. You won’t go mad. Does the impulse to run away from your inner state arise? Go deep into that impulse. Do 1001 judgements and reasons to distract yourself come up? Pay careful attention to each one of them. Don’t react. Don’t do anything. Just be mindful. For as long as you can tolerate. When you have had enough, back away and do something else. Then come back to it. The longer you stay with the feelings and confusion, the more you strengthen your capacity to tolerate them, and the closer you come to transcendence. Once you transcend these feelings and mind states, you can then see how they map out into unhealthy behaviour. You’ll also come to learn about your belief system and on occasion you will realise that your beliefs about yourself are not gospel. Secondly, pay attention to who you are as a person when you are alone. Beyond the shame, fear and uncertainty, what else do you notice? At first you may not notice much else. The longing for someone to save you might be all you have. But as you practice solitude, something else will enter that abundant spiritual space inside you. If you focus on it long enough, your relationship with it will transform, and you will notice new states of being arise. For example, if you travel alone and have a day full of fun and adventure, you’ll realise that you were the one who did that. By challenging yourself in this way, you will call on new resources to help you cope. The ‘you’ which emerges from this challenge is the you which doesn’t need a special someone to make things happen. You learn that it is ok to be ‘you,’ pain and all. Best of all, you learn that you can handle the pain, embrace it, and use it to transform your being. In short, go at it alone in as many challenging ways as you can think of, be mindful, be with your painful emotions, try to look past the longing, and trust that you will evolve to meet every challenge. With enough of these experiences, you will have developed a ‘you’ which has nothing to do with a narcissist. In this way, you will finally have a choice in the matter. You will have a deep knowledge that life is far richer without a narcissist in your life, and that richness is deep within you. If you have just started your narcissistic abuse recovery journey, check out How To Kill A Narcissist. Or if you wish to immunise yourself against narcissists and move on for good, take a look at How To Bury A Narcissist. Upvote · 999 550 99 52 99 26 Jyoti Charan not acquired, but learned. · Author has 393 answers and 1.1M answer views ·9y Can it be solved with less calculations ? possible numbers: 2..8 only because rest higher number would become greater than 4 digits when multiplied by 4, and the last digit cannot be 1 so that we can skip the smaller numbers as well. Reverse of 2..8 is 8..2 (2000+100 x+10 y+8)∗4=(8000+100 y+10 x+2)(2000+100 x+10 y+8)∗4=(8000+100 y+10 x+2) 8000+400 x+40 y+32=8000+100 y+10 x+2 8000+400 x+40 y+32=8000+100 y+10 x+2 390 x+30=60 y 390 x+30=60 y 195 x+15=30 y 195 x+15=30 y Possible solution:x=1,y=7 x=1,y=7 Number:2178 Upvote · 99 12 9 2 Hélio Waldman Retired Professor, Collaborating Professor at University of Campinas (UNICAMP) (1973–present) · Author has 482 answers and 469.4K answer views ·5y Originally Answered: Which 4-digit number multiplied by 4 gives the same number in reverse? · If the 4 digits are a, b, c and d in this order, we must have: 4000a+400b+10c+4d = 1000d+100c+10b+a, with a>0 and d>0 so that both sides are 4-digit numbers, and a<3 so that the left side is not a 5-digit number. Then, a must be either 1 or 2. If a=1, then (4d) mod 10 = a = 1, which has no solution for d because a multiple of 4 cannot have an odd last digit. Therefore, a=2. If a=2, then 8000+400b+40c+4d = 1000d+100c+10b+2, so that (4d)mod 10 = 2 Therefore, d must be either 3 or 8, since these are the only digits that will yield a last digit equal to 2 when they are multiplied by 4. Howev Continue Reading If the 4 digits are a, b, c and d in this order, we must have: 4000a+400b+10c+4d = 1000d+100c+10b+a, with a>0 and d>0 so that both sides are 4-digit numbers, and a<3 so that the left side is not a 5-digit number. Then, a must be either 1 or 2. If a=1, then (4d) mod 10 = a = 1, which has no solution for d because a multiple of 4 cannot have an odd last digit. Therefore, a=2. If a=2, then 8000+400b+40c+4d = 1000d+100c+10b+2, so that (4d)mod 10 = 2 Therefore, d must be either 3 or 8, since these are the only digits that will yield a last digit equal to 2 when they are multiplied by 4. However, if d=3, the right side would be at most 3992, whereas the left side is at least 8000. Therefore, d=8, so we may write: 8000+400b+40c+32 = 8000+100c+10b+2 400b+40c+30 = 100c+10b 60c = 390b+30 2c = 13b+1 c=(13b+1)/2 If b=1, then c=7, so we have a solution at number 2178, which yields 8712 when multiplied by 4. Even values of b would not yield an integer solution for c. Notice also that b must be less than 5 in order for the number to be less than 10000/4 = 2500, otherwise it would yield a 5-digit number when multiplied by 4. Therefore, the only other possible solution would be given by b=3, which yields c=20, which is not a single digit, so it cannot be accepted. We conclude that the only possible solution is 2178. Upvote · 9 3 9 1 Sponsored by Best Gadget Advice Here Are The 33 Coolest Gifts For This Year. We've put together a list of incredible gifts that are selling out fast. Get these before they're gone! Learn More 999 160 Shubh Kumar Studied Artificial Intelligence&Machine Learning (Graduated 2016) · Author has 77 answers and 143.5K answer views ·8y Let the number be 1000a+100b+10c+d We know that , the number multiplied by 4 reverses it. => 4000a+400b+40c+4d = 1000d+100c+10b+a =>3999a+390b-60c-3996d = 0 =>1333a+130b-20c-1332d = 0 => 1333a+130b = 20c+1332d Therefore, Our task now is to find all possible quadruples, {a,b,c,d}, where a,b,c,d <=9 It could be verified that this is possible in only one situation, By proving that (1333a+130b-20c)/1332 could be an integer, and that could be done by doing it per pair of a and b one through 9 and then through c, only when a = 2, b = 7, c = 1 and d = 8. Therefore, 2718 is such a number. Upvote · 9 1 Ed Schneeflock Software Engineer: (tinker in writing, art, graphic design) ·4y I will not attempt a proof, but if you would like a challenge, what constant(s) other than 4 will yield a solution? Do the same constant(s) work when the number of digits is greater than 4? Here is another challenge. what 40-digit numbers multiplied by 4 yield the reverse of that number. Upvote · Ronan Mandra Works at Retirement · Author has 53 answers and 33.1K answer views ·5y Related What 5-digit number multiplied by 4 is itself backwards? Using Solver in Excel, the numbers are Continue Reading Using Solver in Excel, the numbers are Upvote · 99 11 9 2 Subham Kumar Gupta Bachelor of Technology in Information Technology, Dr. B. C. Roy Engineering College, Durgapur (Graduated 2020) · Author has 402 answers and 573.6K answer views ·5y Originally Answered: Which 4-digit number multiplied by 4 gives the same number in reverse? · The Four Digit Number is 2178. 2178 4 = 8712. 8712 is the reverse of 2178. Since it's 4 Digit Number and since after multiplication also it has to be a 4 Digit Number, it's easy to see that the Number has to be less than 2500 since 2500 4 = 10000. Upvote · 9 5 9 1 Related questions Which 4-digit number multiplied by 4 gives the same number in reverse? A five digit number, when multiplied by 4, gives a product which is in the reverse order of the number. What is the number? What number times 4 is its reverse? Can you find a four-digit number which is reversed when multiplied by 9? What six-digit number has the same digits when multiplied by 1, 2, 3, 4, 5, or 6? Can multiplying a four-digit number by 2 ever reverse its digits? If yes, then produce an example. If not, then prove it is not possible. Is an additional 3 digit number multiplied by 4 so the answer is reversed? Which two digit number when multiplied by 32 gives the same number when reverse is multiplied with 23? What is the number, between 1 and 1 million, that can be multiplied by 2, 3, 4, 5 or 6 and the result will include the same digits in the same order as in the original number? What is the three-digit number that when multiplied by a four-digit number gives us a five-digit number with the same first four digits as the second one? What is the number which when multiplied by any two-digit number the product obtained is the two times repetition of the two digit number? What will be the summation of all the numbers (1 digit, 2 digit, 3 digit & 4 digit) by using 2, 3, 4, 5 once in a number? How can I find a 4-digit number with 4 different digits which when subtracted from the number obtained by reversing the digits gives the lowest possible number? The two digits of a two-digit number differ by 4. what is the difference between the number and the number formed by reversing its digits? A three-digit number is multiplied by a two-digit number whose tens digit is 9. The product is a four-digit number whose hundreds digit is 2. How many three-digit numbers satisfy this condition? Related questions Which 4-digit number multiplied by 4 gives the same number in reverse? A five digit number, when multiplied by 4, gives a product which is in the reverse order of the number. What is the number? What number times 4 is its reverse? Can you find a four-digit number which is reversed when multiplied by 9? What six-digit number has the same digits when multiplied by 1, 2, 3, 4, 5, or 6? Can multiplying a four-digit number by 2 ever reverse its digits? If yes, then produce an example. If not, then prove it is not possible. Advertisement About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025 Privacy Preference Center When you visit any website, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized web experience. 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https://pmc.ncbi.nlm.nih.gov/articles/PMC12363244/
Genome integration of human DNA oncoviruses - PMC Skip to main content An official website of the United States government Here's how you know Here's how you know Official websites use .gov A .gov website belongs to an official government organization in the United States. Secure .gov websites use HTTPS A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites. Search Log in Dashboard Publications Account settings Log out Search… Search NCBI Primary site navigation Search Logged in as: Dashboard Publications Account settings Log in Search PMC Full-Text Archive Search in PMC Journal List User Guide View on publisher site Download PDF Add to Collections Cite Permalink PERMALINK Copy As a library, NLM provides access to scientific literature. Inclusion in an NLM database does not imply endorsement of, or agreement with, the contents by NLM or the National Institutes of Health. Learn more: PMC Disclaimer | PMC Copyright Notice J Virol . 2025 Jul 23;99(8):e00562-25. doi: 10.1128/jvi.00562-25 Search in PMC Search in PubMed View in NLM Catalog Add to search Genome integration of human DNA oncoviruses Zuzana Vojtechova Zuzana Vojtechova 1 Department of Genetics and Microbiology, Faculty of Science BIOCEV, Charles University, Prague, Czech Republic Conceptualization, Writing – original draft, Writing – review and editing Find articles by Zuzana Vojtechova 1, Ruth Tachezy Ruth Tachezy 1 Department of Genetics and Microbiology, Faculty of Science BIOCEV, Charles University, Prague, Czech Republic Conceptualization, Funding acquisition, Supervision, Writing – review and editing Find articles by Ruth Tachezy 1,✉ Editor: Suchetana Mukhopadhyay 2 Author information Article notes Copyright and License information 1 Department of Genetics and Microbiology, Faculty of Science BIOCEV, Charles University, Prague, Czech Republic 2 Indiana University Bloomington, Bloomington, Indiana, USA ✉ Address correspondence to Ruth Tachezy, ruth.tachezy@natur.cuni.cz The authors declare no conflict of interest. Roles Zuzana Vojtechova: Conceptualization, Writing – original draft, Writing – review and editing Ruth Tachezy: Conceptualization, Funding acquisition, Supervision, Writing – review and editing Suchetana Mukhopadhyay: Editor Collection date 2025 Aug. Copyright © 2025 Vojtechova and Tachezy. This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International license. PMC Copyright notice PMCID: PMC12363244 PMID: 40699151 ABSTRACT Tumors of infectious origin globally represent 13%. Oncogenic DNA viruses such as human papillomavirus (HPV), hepatitis B virus (HBV), and Epstein-Barr virus (EBV) are responsible for approximately 60% of these tumors. These oncoviruses are extensively studied to understand their role in cancer development, particularly through viral genome integration into the host DNA. Retroviruses require integration mediated by viral integrase for persistence, whereas DNA oncoviruses do not need integration for replication; instead, integration occurs incidentally. This process often targets fragile sites in the human genome, causing structural rearrangements that disrupt genes, activate proto-oncogenes, and increase genomic instability, all contributing to tumorigenesis. Integration near promoter regions and active genes is closely linked to carcinogenesis, highlighting its importance in developing diagnostic and therapeutic strategies. This review summarizes viral integration’s role in oncogenesis, mechanisms of integration, and methods to study this process, focusing on DNA tumor viruses such as HBV, EBV, HPV, and Merkel cell polyomavirus. KEYWORDS: virus, integration, hepatitis B virus, Epstein-Barr virus, human papillomavirus INTRODUCTION More than 13% of global tumor malignancies are linked to pathogens (1), with Helicobacter pylori being the leading cause, responsible for over 35% of these cases. Nearly 60% of tumors of infectious origin are associated with DNA tumor viruses like human papillomavirus (HPV), hepatitis B virus (HBV), and Epstein-Barr virus (EBV) (1). Oncoviruses are extensively studied to understand their mechanisms in malignancy development. For some viruses, such as retroviruses, integration of the viral genome into the host genome is essential and facilitated by the enzyme integrase, enabling lifelong persistence. Retroviral integration has been well-studied since 1990 (2–5). In contrast, DNA oncoviruses do not require integration for replication; it occurs incidentally and can disrupt genes, rearrange chromosomes, and promote oncogenesis. Despite significant research, many questions about viral integration remain, necessitating further study. Viral integration into the human genome often targets fragile sites, causing structural rearrangements in viral DNA and host chromosomes. This process can alter gene expression, activate proto-oncogenes, and increase genomic instability, contributing to tumorigenesis. Integration near promoter regions and transcriptionally active genes is closely linked to carcinogenesis, highlighting its importance in developing diagnostic and therapeutic strategies for virus-associated cancers. A unique case is human herpesvirus 6 (HHV-6), which integrates into the telomeric regions of germline cells and can be inherited as chromosomally integrated HHV-6 (iciHHV-6). IciHHV-6 has been identified as a risk factor for angina pectoris and acute graft-versus-host disease in hematopoietic cell transplant donor-recipient pairs (6, 7), though no statistically significant link to oncogenesis has been found (8). This review will summarize the role of viral integration in oncogenesis, mechanisms of integration into the host genome, and methods for studying this process, focusing on DNA tumor viruses such as HBV, EBV, HPV, and Merkel cell polyomavirus (MCPyV). HEPATITIS B VIRUS Human HBV, a small DNA virus in the Hepadnaviridae family, infects the liver and causes viral hepatitis. Despite the introduction of effective vaccination in the 1980s, 1.2 million new hepatitis B cases occur annually (9). HBV is transmitted via infectious blood or body fluids, and vertically from mother to child. Perinatal or childhood infection significantly increases the risk of chronic infection to 90% and 16–30%, respectively (10). While most adult infections resolve spontaneously, 5–10% become chronic, leading to cirrhosis or hepatocellular carcinoma (HCC) (11). HCC is the sixth most common cancer globally with over 900,000 new cases and 830,000 deaths annually (12). HBV accounts for approximately 46% of HCC cases and 56% of liver cancer deaths worldwide (12, 13). The HBV genome consists of a partially double-stranded circular DNA (3.2 kbp) with an incomplete (+) strand. It encodes four overlapping open reading frames: preS1/preS2/S for surface proteins, preC/C for core protein and E antigen, P for polymerase, and X for HBx protein. The virus enters hepatocytes via the sodium taurocholate co-transporting polypeptide receptor, interacting with the preS1 domain of its large surface protein (14). After entry, the nucleocapsid is directed to the nucleus where relaxed circular DNA (rcDNA) is released. Host factors convert it into covalently closed circular DNA, serving as a template for viral RNA transcription (15–17). Reverse transcription of pre-genomic RNA produces new rcDNA or double-stranded linear DNA, which can integrate into the host genome but does not produce infectious particles (18, 19). Integration occurs at a frequency of ~1 per 10³–10⁴ hepatocytes in vitro (20). HBV integration was first identified in the 1980s (21, 22) and is now recognized as a non-essential but early event in infection. It is detected in up to 90% of cirrhosis or HCC cases, but rarely in acute hepatitis patients (23–27). Integration contributes to HCC development by altering host gene expression, promoting chromosomal instability, and deregulating immune responses (28–30). The HBx protein plays a key role by modulating signaling pathways such as mitogen-activated protein kinase, nuclear factor κB (NF-κB), and Janus kinase/signal transducers and activators of transcription pathway to promote carcinogenesis (31, 32). HBV integration frequently targets cancer-associated genes like human telomerase gene (TERT), the lysin methyltransferase 2B (KMT2B), cyclin E1 (CCNE1), and cyclin A2 (CCNA2), fragile genomic sites, and repetitive sequences, enhancing HCC risk (26, 27, 33–37). Chromosomal rearrangements associated with integration often involve gains at chr8q or chr5p or losses at chr17p affecting proto-oncogenes like MYC or tumor suppressors like TP53 (26). However, many integrations are passenger events without functional consequences. The distribution of integrated HBV on chromosomes is random in both tumor and non-malignant tissues. Integrated HBV sequences often encode C-terminally truncated HBx proteins that disrupt cellular processes by activating signaling pathways or binding tumor suppressors like TP53, promoting tumorigenesis through angiogenesis, immune evasion, inflammation, and altered metabolism (32, 38–40). Integration is clonal in >48% of tumor cells. In adjacent non-tumor tissues, the integration occurs less frequently (<31%) (26, 36). In contrast, Péneau et al. identified HBV integration breakpoints in 84% of non-tumor tissues of patients with HCC, and most integration breakpoints in normal tissues were unique events (<3% of cells shared the same integration breakpoint) (26). Altogether, non-malignant and tumor tissues are concordant in less than 5% of samples (37, 41, 42). Integration seems to impact clinicopathological features such as disease severity and prognosis (43). Tumors with high HBV integration levels are associated with poor survival outcomes independent of other factors like tumor size or differentiation (26, 36). Cell-free virus-host chimeric DNA detected postoperatively correlates with recurrence risk in HCC patients, highlighting its potential as a biomarker (44). HBV integration confers resistance to antiviral therapy but provides stable targets for therapeutic intervention or prognostic biomarkers. Gene therapies targeting integration-derived transcripts offer promising strategies for managing HBV-related diseases (45). EPSTEIN-BARR VIRUS EBV (human herpesvirus 4) is a human DNA oncovirus from the family Herpesviridae, genus Gammaherpesvirinae. Discovered in Burkitt’s lymphoma (BL) cells in 1964 (46), it is linked to 1.5% of all malignancies globally. EBV infects over 90% of adults early in life and typically enters a latent state after primary infection with potential for reactivation. While primary infection is often asymptomatic, it can manifest as infectious mononucleosis in young adults (47). EBV is associated with various epithelial and B cell-derived malignancies, including nasopharyngeal carcinoma (NPC) (48), BL (49), NK/T cell lymphoma (NKTCL) (50), Hodgkin’s lymphoma (HL) (51), and gastric cancer (52). According to Globocan, EBV ranks as the fifth most common cancer-related infection (1) causing over 160,000 deaths annually (53). Its prevalence in BL varies geographically from 20% in low-incidence areas to 95% in endemic regions (53). EBV is linked to 75–100% of NPC cases depending on region (54–56), 25–75% of NKTCL cases (54, 57), 31% of HL cases (57), and about 10% of gastric carcinomas (55). However, only a small fraction of infections result in malignancy, with factors like immunodeficiency, genetic predisposition, and environmental influences playing roles. EBV is also implicated in non-malignant diseases such as multiple sclerosis and oral hairy leukoplakia (58, 59). EBV is an enveloped virus with a linear double-stranded DNA genome of 170–180 kbp encoding over 85 proteins and more than 40 non-coding RNAs (60–63). Transmitted via saliva, it infects the pharyngeal epithelium and resting B cells in underlying tissues (64). Unlike most enveloped viruses that use one or two glycoproteins for adhesion, EBV employs multiple glycoproteins like gp350/220, gH, gp42, and BMRF2 for cell entry (64, 65). Entry mechanisms differ between cell types: endocytosis for B cells and direct membrane fusion for epithelial cells (66). Once inside the host cell, the nucleocapsid travels to the nucleus where the viral genome circularizes (64, 67, 68). During the lytic cycle, immediate early genes are expressed first, followed by early genes for DNA replication and late genes for capsid and envelope proteins, culminating in mature virion release. After primary infection, EBV can enter latency in B cells without producing viral particles but with altered gene expression. The viral genome persists as a circular episome during latency, with latent genes expressed alongside host DNA. Latency types (0, I, II, III) depend on the differentiation state of infected B cells (63). Latent genes include those encoding EBV nuclear antigens, latent membrane proteins, and non-coding RNAs like Epstein-Barr virus-encoded small RNAs (EBERs), microRNAs, or BamHI fragment A rightward transcripts. Reactivation from latency occurs under conditions such as T cell immunity attenuation, switching the virus back to the lytic cycle (69–71). The carcinogenic mechanisms in EBV-associated tumors are primarily driven by viral latent gene products and non-coding RNAs, which promote cell proliferation, DNA damage, genomic instability, and chronic inflammation, all of which support tumor progression (72–76). While the latent state of EBV infection increases tumor development risk, evidence also suggests that lytic replication contributes to genetic instability and tumorigenesis (77). During latency, the EBV genome persists as an episome but can also integrate into the host genome (78–81). Since the first reports in the 1980s (82–84), EBV integration has been confirmed in lymphomas and carcinomas, though its role in carcinogenesis remains incompletely understood. EBV integrates at a lower rate than other DNA viruses like HBV or HPV. Xu et al. found the highest integration rates in gastric carcinoma (26%) (85), followed by HL and NKTCL (18% and 16%, respectively). Late-stage NPC tumors and large gastric cancers showed increased integration rates, while Ohshima et al. identified integration in 11% of various tumor samples (79). Coexistence of episomal and integrated viral DNA has been observed in NPC, NKTCL, and BL cell lines (79, 81, 86). Hurley et al. reported higher integration frequency than the presence of an episomal form of the virus in persistently infected activated B cells (78). Challenges in studying EBV integration include methylated DNA, interference from episomes, and the large viral genome size (87–89). However, studies have revealed key mechanisms linking integration to carcinogenesis. In BL-derived cell lines, viral integration near oncogenes like REL and BCL11A or tumor suppressor genes like BACH2 alters gene expression or disrupts tumor suppressor functions, promoting proliferation and lymphomagenesis (88, 90). In NKTCL samples, integration sites were enriched in repetitive regions such as short interspersed nuclear elements (SINEs) and long interspersed nuclear elements (LINEs), forming chimeric transcripts that disrupt DNA repair or increase genome instability (81). Genome-wide profiling revealed integration near fragile sites or microsatellite repeats linked to DNA damage, affecting tumor suppressor genes like KANK1, RB1CC1, PTEN, FHIT, and DLEC1 while dysregulating inflammatory pathways such as NF-κB or tumor necrosis factor alpha (TNF-alpha) apoptosis (85). EBV integration near fragile sites was observed in approximately one-third of cases (91, 92), with non-random distribution favouring certain chromosomal bands (91, 93). Chakravorty et al.’s interactome map showed preferential integration near highly expressed genes with super-enhancer regions and ribosomal RNA genes, significantly influencing host gene expression (54). Conversely, Xiao et al.’s findings suggested random integration correlated with structural variations in the host genome that increase instability (80). EBV breakpoints are distributed throughout its genome with hotspots near oriP and terminal repeats. The identified microhomologies and insertions near integration sites suggest involvement of mediated repair pathways during integration (85). The integration of a full-length viral genome remains unclear due to read-length limitations in next-generation sequencing methods (85). Like other viruses, EBV genome integration promotes carcinogenesis through multiple mechanisms, though its impact on patient prognosis remains unexplored. Interestingly, patients with EBV-associated malignancies tend to have better outcomes than those with EBV-negative tumors (94–96). HUMAN PAPILLOMAVIRUS HPVs are small DNA tumor viruses of the Papillomaviridae family that infect epithelial cells. Over 450 HPV types have been identified, with 226 genomes cloned and stored in the International HPV Reference Center (97). Low-risk types cause benign growths like warts, while high-risk (HR) oncogenic types are crucial in the development of premalignant lesions, cervical cancer (CC), and other HPV-associated squamous cell carcinomas in the anogenital, head, and neck regions. HPV is the most common sexually transmitted viral infection globally, affecting both men and women, primarily through sexual or skin-to-skin contact. Vertical transmission from mother to child is rare (98). Studies have focused on women due to HPV’s high affinity for cervical cells. The WHO estimates a global HPV prevalence of 12% among women, peaking at 24% in sub-Saharan Africa (99, 100). A meta-analysis by Bruni et al. (101) revealed a global HR-HPV prevalence of 21% among men, highlighting the need for inclusive prevention strategies. Most HPV infections are asymptomatic and resolve spontaneously. About 25% of infected individuals develop precancerous lesions, with less than 1% progressing to invasive cancer (102). HPV accounts for approximately 5% of all cancers worldwide, with over 690,000 new cases annually (1). Cervical cancer is nearly entirely caused by HPV and ranks as the fourth most common cancer and cause of cancer-related death in women, with an estimated 604,000 new cases and 342,000 deaths in 2020 (12). It is also the second most common cancer among women aged 15–44 (103). HPVs contribute to other anogenital cancers, such as those of the vagina, vulva, anus, and penis, with a combined incidence of 150,000 cases annually worldwide—nearly half attributable to HPV (1, 12). HR-HPV also plays a major role in head and neck cancers (HNC), particularly oropharyngeal squamous cell carcinoma (OPSCC), which affects the tonsils and tongue base. HNC is the eighth most common malignancy globally, with over 850,000 new cases annually (12). Approximately 30% of OPSCC cases are linked to HPV infection (104, 105), though proportions vary by region. The HPV genome consists of ~8 kbp circular double-stranded DNA organized into three regions: the early region (E), encoding proteins essential for viral replication; the late region (L), encoding capsid proteins L1 and L2; and the non-coding long control region (LCR) containing regulatory sequences like replication origins and transcription factor binding sites (106). The viral life cycle depends on keratinocyte differentiation and host cell machinery. HPVs infect basal epithelial layers via microtraumas, with entry mediated by interactions between capsid protein L1 and heparan sulfate proteoglycans on basal cell membranes (107). In basal cells, the viral genome persists as an episome at low copy numbers but replicates extensively during epithelial differentiation. Capsid proteins are expressed during terminal differentiation, leading to genome encapsidation and virion release (108). Persistent HR-HPV infection is essential for the development of premalignant lesions and their progression to invasive carcinoma (109). Low-grade cervical lesions (cervical intraepithelial lesion [CIN1]) are often transient and resolve within months (110). HPV oncogenes drive the progression to high-grade lesions (CIN2, CIN3) (111, 112), with E6 and E7 being key oncoproteins involved in cancer hallmarks (113). E6 forms a complex with E6AP and p53, leading to p53 degradation and loss of its tumor-suppressor functions (114–116). It also disrupts p53 binding to CBP/p300, activates telomerase, degrades PDZ-domain proteins, and modulates immune responses via IRF3 (117–121). E7 targets pRb for degradation, releasing E2F to promote cell cycle progression and interacting with other proteins to influence immune responses and cell regulation (122–126). E5 complements E6/E7 by enhancing EGFR signaling, inhibiting apoptosis, and promoting transformation (127–132). Although most HPV infections are transient and asymptomatic, persistent infections can evade immune responses, enabling progression to carcinoma due to limited antigen production and reduced inflammation (133). Viral replication requires host DNA damage repair factors, but dysregulation of these pathways increases genomic instability and mutation rates through mechanisms such as reactive oxygen species (ROS) elevation by the E6 variant (134–138). HPV oncoproteins also induce centrosome accumulation, leading to aneuploidy (139–141). Viral DNA integration into the host genome is considered an important event in HPV-associated carcinogenesis. While HPV initially exists as an extrachromosomal episome, integration frequency rises with disease severity (142, 143). Integrated forms are common in CCs (~80%) and head-and-neck squamous cell carcinomas (50–70%) (144–148). Integration disrupts regulatory genes like E2/E1, upregulating E6/E7 expression, while episomes may increase oncogene expression via copy number amplification or structural rearrangements (149, 150). Epigenetic modifications like DNA methylation may also enhance oncogene expression, though findings are inconsistent (150–153). Non-integrative carcinogenesis involving increased E2/E4/E5 expression has been proposed for certain cancers, suggesting alternative mechanisms for tumorigenesis (154). Integration of HPV DNA into the host genome requires both viral and host DNA breakage, with the rate of integration influenced by factors such as inflammation, reactive oxygen/nitrogen species, environmental toxins, and apolipoprotein B mRNA editing enzyme (APOBEC) polypeptides (155–158). HPV integration occurs across almost all human chromosomes but is recurrently observed at sites like PDL1, MYC, MACROD2, and KLF5 (145, 148, 159–163). These sites are often located near common fragile sites (CFSs), which are prone to chromosome breakage (164–166). Studies report HPV integration in CFSs in 38% of HPV-associated tumors and cell lines (167), consistent with findings in cervical carcinomas and tonsillar tumors (145, 168). HPV integration can occur in intergenic regions or coding regions, often affecting cancer-associated genes such as tumor suppressors or oncogenes. Akagi et al. mapped breakpoints to intergenic loci (30%), exons (3%), introns (40%), and gene ends (28%) (169). Zhao et al. also observed a preference for intronic and intergenic regions (170). Integration can disrupt key genes like RAD51B, leading to loss of function and structural rearrangements that alter gene expression, such as TP63 amplification or MYC overexpression via the viral promoter (171). Fusion transcripts driven by the viral LCR are highly expressed at productive integration sites, as shown in CC studies (162). In OPSCC, integration affects tumor-related pathways in nearly half of cases analyzed (172). Tian et al. identified HPV host extrachromosomal DNA containing super-enhancers that dysregulate chromosomal gene expression, proposing novel oncogenic functions for extrachromosomal DNA reservoirs (173). Epigenetic changes also play a role in HPV integration’s oncogenic effects. Methylation analyses suggest that integrated HPV genomes share methylation patterns with flanking human sequences, impacting gene expression (174). Tandem integrations may involve silenced complete viral copies, while incomplete sequences at breakpoints affect host genes (175, 176). Zeng et al. found that HPV integration into enhancer regions, such as MIR205HG, reduces methylation and upregulates expression, contributing to carcinogenesis (163). The disruption of the viral E2 gene during integration is thought to weaken E2-mediated regulation of oncogene expression, but studies suggest that constitutive rather than high-level oncogene expression is sufficient for oncogenesis (177). Deletions in other viral genes like E1, E5, L1, or L2 have also been observed upon integration (148), with breakpoints often occurring in L1/L2 rather than E2 regions in CC (170). The prognostic significance of HPV integration remains unclear due to methodological differences and small cohorts studied across HNC and CCs. Several studies reported no significant difference in disease-specific survival between patients with integrated vs. episomal or mixed forms of HPV but noted trends favoring mixed forms for longer recurrence-free survival (145, 178, 179). Contrastingly, Koneva et al., using TCGA data, found poor survival associated with integration, particularly in older patients (180, 181). Conversely, Pinatti et al.’s recent findings suggest HPV integration could serve as a marker of good prognosis (182). In summary, while increased expression of viral oncogenes is a key mechanism in HPV-driven oncogenesis, genomic alterations, changes in the genome that result from the integration of the virus, contribute significantly to this multifaceted process through structural rearrangements, epigenetic changes, and dysregulated gene expression pathways. MERKEL CELL POLYOMAVIRUS MCPyV is a small, non-enveloped, double-stranded DNA virus first identified in 2008 (183). It belongs to the Polyomaviridae family and is the only polyomavirus linked to human tumorigenesis. MCPyV is found in 80% of Merkel cell carcinomas (MCC), a rare but aggressive skin cancer (184). The cellular origin of MCC remains controversial, with viral and non-viral MCC likely arising from different skin cell types (185). MCC incidence is approximately three per million globally, with geographical variations (186). In the U.S., MCC incidence rose 3.5-fold from 1991 to 2016, continuing to increase with age due to immune senescence (187). The highest rates are in Australia, at 3.9 per 100,000 men and 2.5 per 100,000 women annually (188). Non-viral MCC is linked to UV-induced mutations, while viral MCC involves MCPyV infection and virus-induced carcinogenesis. MCPyV has also been detected in non-MCC skin lesions such as melanoma (4%), squamous cell carcinoma (15%), basal cell carcinoma (14%), and actinic keratosis (6%), as well as in normal skin, with prevalence ranging from 11-68% across studies (184, 189–192). It has also been found in 10% of non-malignant tonsillar tissues (193). Seroprevalence data suggest MCPyV is widespread, with initial infections occurring in childhood and increasing with age (194–196). The MCPyV genome is a circular, double-stranded DNA of 5.4 kbp comprising two transcription units and a non-coding control region. The E encodes small and large T antigens (sT and LT), a splice variant (57kT), an alternative LT open reading frame, and a viral miRNA (MCV miR M1) (183, 197, 198). The L encodes structural proteins VP1 and VP2 but lacks VP3, unlike other polyomaviruses (199, 200). Viral attachment involves VP1 binding to sulfated glycosaminoglycans and sialic acid-containing co-receptors, followed by transport through endosomes to the nucleus via nucleopores (201–203). Viral gene expression is restricted to human dermal fibroblasts, with early genes initiating replication and late genes regulating capsid protein expression for virion assembly and release (204–206). MCPyV-associated MCC development requires two key mutagenic events: truncation of the LT antigen sequence and viral genome integration into the host genome (183, 207, 208). LT truncation impairs helicase activity but preserves pRb-binding ability, disrupting tumor suppressor function and promoting oncogenesis (209). sT antigen expression remains intact, contributing to oncogenesis by inhibiting proteasomal degradation of oncoproteins like cyclin E or c-Myc (210–212). Viral integration occurs randomly in the host genome, often near repeat elements like SINEs or LINEs, with chromosome 5 being a frequent site (213–215). Integration disrupts LT gene sequences downstream of the pRb-binding site while preserving its oncogenic functions. Integration often occurs as single copies or tandem concatemers by the mechanism of rolling circle amplification without cleavage of unit-length genomes (215, 216). Most integration sites are intronic or intergenic regions, occasionally affecting genes directly involved in carcinogenesis (213, 217). Integrated MCPyV genomes frequently undergo large deletions, with over one-third of tumors showing deletions of at least half the viral genome (214). Integration is an early event in MCC carcinogenesis, as primary tumors and metastases often share integration patterns. Patients with MCPyV-positive MCC generally have better prognoses than those with non-viral MCCs, though the impact of integration on survival remains underexplored due to the rarity of MCPyV-associated tumors with extensive genomic changes (218, 219). MECHANISMS OF DNA TUMOR VIRUS INTEGRATION Viral DNA integration into the host genome primarily occurs through the repair of DNA double-stranded breaks (DSBs) at sites of genomic instability or damage, mainly via non-homologous end joining (NHEJ) without sequence homology between virus and host DNA. A secondary mechanism involves microhomology-mediated end joining (MMEJ) (Fig. 1). Both processes are observed in HBV integration (20, 220). Integrated viral DNA fragments range from 28 bp to the full-length HBV sequence (221). Around 37–40% of HBV breakpoints map between short repetitive sequences DR1 and DR2 (41, 222). In EBV, sequencing and computational analysis revealed that integration is primarily mediated by canonical NHEJ, followed by synthesis-dependent end joining, and other alternative mechanisms like MMEJ or replication fork stalling and template switching during DSB repair (223). Fig 1. Open in a new tab Diagram illustrating the main mechanisms of viral integration and the host genome regions that can be affected by viral integration, resulting in changes in gene expression. Created with BioRender.com. For HPV, integration mechanisms remain less understood, but two models have been proposed. The “looping model” suggests HPV DNA forms a loop between two host DNA fragments with DSBs, leading to amplification, rearrangement into concatemers, or excision into viral/host fusion episomes (169, 224–226). The second model involves direct integration via MMEJ, where HPV oncoproteins interfere with DNA damage repair pathways, leading to integration at microhomologous regions enriched near breakpoints (159). MMEJ is an error-prone backup when homologous repair (HR) fails to repair DSBs (227, 228). While HPV uses HR for replication, its role in integration remains unproven, and the switch from high-fidelity HR to error-prone MMEJ during integration is not yet elucidated (225). Two integration patterns have been proposed for MCPyV: NHEJ, resulting in a linear integration pattern (Fig. 2A) with minor host DNA loss and MMEJ, creating a Z-pattern integration (Fig. 2B) with distant breakpoints and duplication of adjacent host sequences after rolling cycle amplification (215, 216). These events may also involve additional rearrangements or amplifications. Fig 2. Open in a new tab Patterns of MCPyV integration. (A) Linear integration pattern (L—left integration breakpoint and R—right integration breakpoint) and (B) Z-pattern integration. (Adapted from reference 215.) METHODS USED FOR VIRAL INTEGRATION ANALYSIS Many methodological approaches have been reported for viral integration analysis. Historically, Southern blot hybridization has been used (21, 229). While inexpensive, it is limited in detecting viral copies per cell and cannot determine the nucleotide sequence of the integration site. Southern blot remains a standard for determining HPV genome status, including concatemers, but it is time-consuming and requires large amounts of fresh DNA, restricting its use in large or retrospective studies (145, 230). For EBV integration, fractionation of DNA on a cesium cloride (CsCl) density gradient followed by a Southern blot has been employed (83). Electrophoresis-based methods, such as those described by Gardella et al., detect linear and episomal EBV DNA (231–233). Pulse-field gel electrophoresis further improved the separation of long DNA molecules for EBV analysis (87). Southern blot analysis has been complemented by fluorescence in situ hybridization (FISH) (79, 232), which is now widely used. Hybridization-based methods like in situ hybridization (ISH) or FISH detect HPV integration at the single-cell level and determine whether integrated viral sequences are transcribed or silent (234–236). These methods are sensitive and applicable to fixed tissues but require prior knowledge of the sequence of interest for probe design and are relatively expensive compared to PCR-based techniques developed later. ISH with radiolabeled probes has also been used for HBV integrants' chromosomal positioning but with low sensitivity (237). Fiber FISH, combining molecular combing and FISH, characterizes HPV genomic integration sites containing tandemly integrated genomes interspersed with host DNA (238). Direct cloning and Sanger sequencing were initially used to identify exact integration sequences but are technically demanding, low-throughput, and reliant on correct restriction enzymes (239). More sensitive PCR-based methods combined with sequencing now allow detailed resolution of host-cell genome integration (80, 81, 85). For HBV analysis, primers targeting Alu repetitive elements, which constitute over 10% of the human genome, were paired with HBV-specific primers (25, 240). However, this method is limited to detecting integrations near Alu sequences without quantifying junctions. Inverse PCR detects single-copy HBV cell junctions and quantifies their absolute number but depends on restriction enzyme sites, limiting its scope (20, 241). DIPS PCR (detection of integrated papillomavirus sequences), developed for HPV analysis, maps chimeric virus-host DNA sequences through single-side ligation-mediated PCR followed by sequencing, enabling determination of HPV genome status and integration sites but requiring high-quality DNA and optimization due to reliance on restriction enzymes (242, 243). This method has also been applied to MCPyV integration in MCC (213, 217, 244). The amplification of papilloma virus oncogene transcripts (APOT) assay is a highly sensitive RNA-based method using modified 3′ rapid amplification of cDNA ends (RACE) to analyze actively transcribed sequences but requires high-quality RNA (145, 172, 245, 246). It involves reverse transcription with adaptor-linked oligo dT primers followed by two PCR steps with adaptor primers and HPV-specific primers. Amplification products can be hybridized to HPV-specific probes, or the fusion transcripts can be sequenced to determine the exact integration breakpoint. Quantitative PCR, which measures the E2/E6 DNA ratio based on frequent E2 gene deletions during integration, is commonly used but cannot distinguish between tandem repeats and extrachromosomal forms of HPV (247–249). Next-generation sequencing (NGS) has revolutionized viral integration detection by revealing exact sites with high sensitivity without prior sequence knowledge. NGS methods include whole-genome sequencing, whole-exome sequencing, RNA sequencing, capture-enriched NGS, or RAISING for HBV analysis (26, 27, 40, 215, 224, 250–255). Despite their sensitivity and applicability to diverse biological materials like formalin-fixed paraffin embedded (FFPE) tissue or blood, NGS methods remain costly and require complex bioinformatics analyses. Nanopore sequencing provides long DNA reads with lower accuracy but reduces experimental artefacts due to unnecessary or less intense fragmentation of the DNA (256, 257). Nanochannel sequencing has recently been applied to MCPyV integration mapping single DNA molecules with rapid determination of copy numbers and adjacent host sequences' patterns (215). For EBV integration, DeepEBV employs deep learning to predict integration sites based solely on DNA sequences, offering a promising tool for precise EBV research when combined with known results (258). CONCLUDING REMARKS DNA oncoviruses are extensively studied to elucidate their oncogenic pathways and molecular effects. A key mechanism involves integration into the host genome, altering cellular pathways to promote cell cycle deregulation and tumorigenesis. This review highlights carcinogenesis linked to prevalent DNA tumor viruses and examines how genomic integration affects tumor development and prognosis (for summary see Fig. 3). Fig 3. Open in a new tab Comparison of the characteristics of particular DNA tumor viruses. Created with BioRender.com. Viral DNA integration is critical in HBV-associated cirrhosis and HCC, where it occurs in over 70% of cases. Alongside chronic infection and inflammation, this process drives HBV progression to HCC. In contrast, EBV integrates less frequently, suggesting its oncogenic role arises from alternative mechanisms, such as modulation of cellular signaling by EBV-encoded proteins. HPV integration is well-studied in cervical and HNC. HPV oncoproteins deregulate the cell cycle and apoptosis, leading to cellular transformation. Similarly, MCPyV integration is common in MCC, where it facilitates tumor initiation alongside LT mutations. This review also explores viral genome integration mechanisms and the methods used to analyze them, emphasizing the advantages and limitations of these approaches. Advances in NGS provide sensitive tools for studying viral integration, despite challenges such as complex bioinformatics analyses. These methods promise to uncover previously unexplored data. Despite progress, many questions remain. Future research on viral integration could identify new therapeutic targets to improve patient outcomes. However, focused studies on viral integration are needed before clinical application. ACKNOWLEDGMENTS This work was supported by the Project National Institute of Virology and Bacteriology (Programme EXCELES, ID Project No. LX22NPO5103) - Funded by the European Union - Next Generation EU. Z.V. prepared the initial draft of the manuscript and R.T. revised the manuscript. Both authors approved the final version of the manuscript. Biographies Zuzana Vojtechova graduated with a Master’s degree in Microbiology from Charles University in Prague in 2012. She then completed seven years of doctoral studies in Molecular Biology, Genetics, and Virology at the same university. Her doctoral research, conducted under the supervision of Prof. Ruth Tachezy, began at the Institute of Hematology and Blood Transfusion and later continued at the Faculty of Science, Charles University Biocev in Vestec, where she worked until the end of 2024. Since the beginning of her doctoral studies, Zuzana has focused on viral integration and virus-associated oncogenesis. Her doctoral thesis investigated HPV integration and miRNA in HPV-positive and HPV-negative head and neck tumors. She is currently employed as a bioanalyst in the Microbiology Department at the Hospital in Jihlava. Ruth Tachezy holds a Ph.D. in Molecular Virology and an Associate Professorship from Charles University (CU). She is the founding director of the National Reference Laboratory for Papillomaviruses and Polyomaviruses and serves as the Head of the Department of Genetics and Microbiology at the Faculty of Science, CU, where she also leads her own research group. Dr. Tachezy has held various visiting research positions, including at the University of Kuopio (Finland), Yeshiva University (New York), and the University of Leuven (Belgium). She has supervised numerous bachelor’s, master’s, and doctoral students in biomedical programs and teaches courses at both the Medical Faculties and the Faculty of Science at CU. She has received numerous research grants and, since 2023, has served as the principal investigator for CU on the EU4H-2021-PJ4 project, "Delivering a Unified Research Alliance of Biomedical and Public Health Laboratories against Epidemics." Her research initially focused on oncogenic viruses and has recently expanded to include studies of viromes. Contributor Information Ruth Tachezy, Email: ruth.tachezy@natur.cuni.cz. Suchetana Mukhopadhyay, Indiana University Bloomington, Bloomington, Indiana, USA. REFERENCES de Martel C, Georges D, Bray F, Ferlay J, Clifford GM. 2020. Global burden of cancer attributable to infections in 2018: a worldwide incidence analysis. Lancet Glob Health 8:e180–e190. doi: 10.1016/S2214-109X(19)30488-7 [DOI] [PubMed] [Google Scholar] Grandgenett DP, Mumm SR. 1990. Unraveling retrovirus integration. Cell 60:3–4. doi: 10.1016/0092-8674(90)90707-l [DOI] [PubMed] [Google Scholar] Serrao E, Engelman AN. 2016. Sites of retroviral DNA integration: from basic research to clinical applications. Crit Rev Biochem Mol Biol 51:26–42. doi: 10.3109/10409238.2015.1102859 [DOI] [PMC free article] [PubMed] [Google Scholar] Lesbats P, Engelman AN, Cherepanov P. 2016. Retroviral DNA integration. Chem Rev 116:12730–12757. doi: 10.1021/acs.chemrev.6b00125 [DOI] [PMC free article] [PubMed] [Google Scholar] Maertens GN, Engelman AN, Cherepanov P. 2022. Structure and function of retroviral integrase. Nat Rev Microbiol 20:20–34. doi: 10.1038/s41579-021-00586-9 [DOI] [PMC free article] [PubMed] [Google Scholar] Gravel A, Dubuc I, Morissette G, Sedlak RH, Jerome KR, Flamand L. 2015. Inherited chromosomally integrated human herpesvirus 6 as a predisposing risk factor for the development of angina pectoris. Proc Natl Acad Sci USA 112:8058–8063. doi: 10.1073/pnas.1502741112 [DOI] [PMC free article] [PubMed] [Google Scholar] Hill JA, Magaret AS, Hall-Sedlak R, Mikhaylova A, Huang M-L, Sandmaier BM, Hansen JA, Jerome KR, Zerr DM, Boeckh M. 2017. Outcomes of hematopoietic cell transplantation using donors or recipients with inherited chromosomally integrated HHV-6. Blood 130:1062–1069. doi: 10.1182/blood-2017-03-775759 [DOI] [PMC free article] [PubMed] [Google Scholar] Gravel A, Dubuc I, Brooks-Wilson A, Aronson KJ, Simard J, Velásquez-García HA, Spinelli JJ, Flamand L. 2017. Inherited chromosomally integrated human herpesvirus 6 and breast cancer. Cancer Epidemiol Biomarkers Prev 26:425–427. doi: 10.1158/1055-9965.EPI-16-0735 [DOI] [PubMed] [Google Scholar] Global hepatitis report. 2024. Available from: Retrieved 15 Jul 2025. Pollicino T, Caminiti G. 2021. HBV-integration studies in the clinic: role in the natural history of infection. Viruses 13:368. doi: 10.3390/v13030368 [DOI] [PMC free article] [PubMed] [Google Scholar] Shi Y-H, Shi C-H. 2009. Molecular characteristics and stages of chronic hepatitis B virus infection. World J Gastroenterol 15:3099–3105. doi: 10.3748/wjg.15.3099 [DOI] [PMC free article] [PubMed] [Google Scholar] Sung H, Ferlay J, Siegel RL, Laversanne M, Soerjomataram I, Jemal A, Bray F. 2021. Global Cancer Statistics 2020: GLOBOCAN estimates of incidence and mortality worldwide for 36 cancers in 185 countries. CA Cancer J Clin 71:209–249. doi: 10.3322/caac.21660 [DOI] [PubMed] [Google Scholar] Plummer M, de Martel C, Vignat J, Ferlay J, Bray F, Franceschi S. 2016. Global burden of cancers attributable to infections in 2012: a synthetic analysis. Lancet Glob Health 4:e609–e616. doi: 10.1016/S2214-109X(16)30143-7 [DOI] [PubMed] [Google Scholar] Yan H, Zhong G, Xu G, He W, Jing Z, Gao Z, Huang Y, Qi Y, Peng B, Wang H, Fu L, Song M, Chen P, Gao W, Ren B, Sun Y, Cai T, Feng X, Sui J, Li W. 2012. Sodium taurocholate cotransporting polypeptide is a functional receptor for human hepatitis B and D virus. Elife 3:e00049. doi: 10.7554/eLife.00049 [DOI] [PMC free article] [PubMed] [Google Scholar] Long Q, Yan R, Hu J, Cai D, Mitra B, Kim ES, Marchetti A, Zhang H, Wang S, Liu Y, Huang A, Guo H. 2017. The role of host DNA ligases in hepadnavirus covalently closed circular DNA formation. PLoS Pathog 13:e1006784. doi: 10.1371/journal.ppat.1006784 [DOI] [PMC free article] [PubMed] [Google Scholar] Kitamura K, Que L, Shimadu M, Koura M, Ishihara Y, Wakae K, Nakamura T, Watashi K, Wakita T, Muramatsu M. 2018. Flap endonuclease 1 is involved in cccDNA formation in the hepatitis B virus. PLoS Pathog 14:e1007124. doi: 10.1371/journal.ppat.1007124 [DOI] [PMC free article] [PubMed] [Google Scholar] Tang L, Sheraz M, McGrane M, Chang J, Guo J-T. 2019. DNA polymerase alpha is essential for intracellular amplification of hepatitis B virus covalently closed circular DNA. PLoS Pathog 15:e1007742. doi: 10.1371/journal.ppat.1007742 [DOI] [PMC free article] [PubMed] [Google Scholar] Haines KM, Loeb DD. 2007. The sequence of the RNA primer and the DNA template influence the initiation of plus-strand DNA synthesis in hepatitis B virus. J Mol Biol 370:471–480. doi: 10.1016/j.jmb.2007.04.057 [DOI] [PMC free article] [PubMed] [Google Scholar] Zhao X-L, Yang J-R, Lin S-Z, Ma H, Guo F, Yang R-F, Zhang H-H, Han J-C, Wei L, Pan X-B. 2016. Serum viral duplex-linear DNA proportion increases with the progression of liver disease in patients infected with HBV. Gut 65:502–511. doi: 10.1136/gutjnl-2014-308989 [DOI] [PubMed] [Google Scholar] Tu T, Budzinska MA, Vondran FWR, Shackel NA, Urban S. 2018. Hepatitis B virus DNA integration occurs early in the viral life cycle in an in vitro infection model via sodium taurocholate cotransporting polypeptide-dependent uptake of enveloped virus particles. J Virol 92:e02007-17. doi: 10.1128/JVI.02007-17 [DOI] [PMC free article] [PubMed] [Google Scholar] Brechot C, Pourcel C, Louise A, Rain B, Tiollais P. 1980. Presence of integrated hepatitis B virus DNA sequences in cellular DNA of human hepatocellular carcinoma. Nature 286:533–535. doi: 10.1038/286533a0 [DOI] [PubMed] [Google Scholar] Shafritz DA, Shouval D, Sherman HI, Hadziyannis SJ, Kew MC. 1981. Integration of hepatitis B virus DNA into the genome of liver cells in chronic liver disease and hepatocellular carcinoma. Studies in percutaneous liver biopsies and post-mortem tissue specimens. N Engl J Med 305:1067–1073. doi: 10.1056/NEJM198110293051807 [DOI] [PubMed] [Google Scholar] Kimbi GC, Kramvis A, Kew MC. 2005. Integration of hepatitis B virus DNA into chromosomal DNA during acute hepatitis B. World J Gastroenterol 11:6416–6421. doi: 10.3748/wjg.v11.i41.6416 [DOI] [PMC free article] [PubMed] [Google Scholar] Bréchot C. 2004. Pathogenesis of hepatitis B virus-related hepatocellular carcinoma: old and new paradigms. Gastroenterology 127:S56–S61. doi: 10.1053/j.gastro.2004.09.016 [DOI] [PubMed] [Google Scholar] Murakami Y, Saigo K, Takashima H, Minami M, Okanoue T, Bréchot C, Paterlini-Bréchot P. 2005. Large scaled analysis of hepatitis B virus (HBV) DNA integration in HBV related hepatocellular carcinomas. Gut 54:1162–1168. doi: 10.1136/gut.2004.054452 [DOI] [PMC free article] [PubMed] [Google Scholar] Péneau C, Imbeaud S, La Bella T, Hirsch TZ, Caruso S, Calderaro J, Paradis V, Blanc J-F, Letouzé E, Nault J-C, Amaddeo G, Zucman-Rossi J. 2022. Hepatitis B virus integrations promote local and distant oncogenic driver alterations in hepatocellular carcinoma. Gut 71:616–626. doi: 10.1136/gutjnl-2020-323153 [DOI] [PMC free article] [PubMed] [Google Scholar] Zhao L-H, Liu X, Yan H-X, Li W-Y, Zeng X, Yang Y, Zhao J, Liu S-P, Zhuang X-H, Lin C, et al. 2016. Genomic and oncogenic preference of HBV integration in hepatocellular carcinoma. Nat Commun 7:12992. doi: 10.1038/ncomms12992 [DOI] [PMC free article] [PubMed] [Google Scholar] Berasain C, Castillo J, Perugorria MJ, Latasa MU, Prieto J, Avila MA. 2009. Inflammation and liver cancer: new molecular links. Ann N Y Acad Sci 1155:206–221. doi: 10.1111/j.1749-6632.2009.03704.x [DOI] [PubMed] [Google Scholar] Chen Y, Tian Z. 2019. HBV-induced immune imbalance in the development of HCC. Front Immunol 10:2048. doi: 10.3389/fimmu.2019.02048 [DOI] [PMC free article] [PubMed] [Google Scholar] Zhang HH, Mei MH, Fei R, Liu F, Wang JH, Liao WJ, Qin LL, Wei L, Chen HS. 2010. Regulatory T cells in chronic hepatitis B patients affect the immunopathogenesis of hepatocellular carcinoma by suppressing the anti-tumour immune responses. J Viral Hepat 17:34–43. doi: 10.1111/j.1365-2893.2010.01269.x [DOI] [PubMed] [Google Scholar] Kim CM, Koike K, Saito I, Miyamura T, Jay G. 1991. HBx gene of hepatitis B virus induces liver cancer in transgenic mice. Nature 351:317–320. doi: 10.1038/351317a0 [DOI] [PubMed] [Google Scholar] Sivasudhan E, Blake N, Lu Z, Meng J, Rong R. 2022. Hepatitis B viral protein HBx and the molecular mechanisms modulating the hallmarks of hepatocellular carcinoma: a comprehensive review. Cells 11:741. doi: 10.3390/cells11040741 [DOI] [PMC free article] [PubMed] [Google Scholar] Horikawa I, Barrett JC. 2001. cis-Activation of the human telomerase gene (hTERT) by the hepatitis B virus genome. J Natl Cancer Inst 93:1171–1173. doi: 10.1093/jnci/93.15.1171 [DOI] [PubMed] [Google Scholar] Lin SY, Zhang A, Lian J, Wang J, Chang T-T, Lin Y-J, Song W, Su Y-H. 2021. Recurrent HBV integration targets as potential drivers in hepatocellular carcinoma. Cells 10:1294. doi: 10.3390/cells10061294 [DOI] [PMC free article] [PubMed] [Google Scholar] Feitelson MA, Lee J. 2007. Hepatitis B virus integration, fragile sites, and hepatocarcinogenesis. Cancer Lett 252:157–170. doi: 10.1016/j.canlet.2006.11.010 [DOI] [PubMed] [Google Scholar] Sung W-K, Zheng H, Li S, Chen R, Liu X, Li Y, Lee NP, Lee WH, Ariyaratne PN, Tennakoon C, et al. 2012. Genome-wide survey of recurrent HBV integration in hepatocellular carcinoma. Nat Genet 44:765–769. doi: 10.1038/ng.2295 [DOI] [PubMed] [Google Scholar] Jang J-W, Kim H-S, Kim J-S, Lee S-K, Han J-W, Sung P-S, Bae S-H, Choi J-Y, Yoon S-K, Han D-J, Kim T-M, Roberts LR. 2021. Distinct patterns of HBV integration and TERT alterations between in tumor and non-tumor tissue in patients with hepatocellular carcinoma. Int J Mol Sci 22:7056. doi: 10.3390/ijms22137056 [DOI] [PMC free article] [PubMed] [Google Scholar] Schlüter V, Meyer M, Hofschneider PH, Koshy R, Caselmann WH. 1994. Integrated hepatitis B virus X and 3’ truncated preS/S sequences derived from human hepatomas encode functionally active transactivators. Oncogene 9:3335–3344. [PubMed] [Google Scholar] Liu X-H, Lin J, Zhang S-H, Zhang S-M, Feitelson M-A, Gao H-J, Zhu M-H. 2008. COOH-terminal deletion of HBx gene is a frequent event in HBV-associated hepatocellular carcinoma. World J Gastroenterol 14:1346. doi: 10.3748/wjg.14.1346 [DOI] [PMC free article] [PubMed] [Google Scholar] Fujimoto A, Totoki Y, Abe T, Boroevich KA, Hosoda F, Nguyen HH, Aoki M, Hosono N, Kubo M, Miya F, et al. 2012. Whole-genome sequencing of liver cancers identifies etiological influences on mutation patterns and recurrent mutations in chromatin regulators. Nat Genet 44:760–764. doi: 10.1038/ng.2291 [DOI] [PubMed] [Google Scholar] Jiang S, Yang Z, Li W, Li X, Wang Y, Zhang J, Xu C, Chen P-J, Hou J, McCrae MA, Chen X, Zhuang H, Lu F. 2012. Re-evaluation of the carcinogenic significance of hepatitis B virus integration in hepatocarcinogenesis. PLoS One 7:e40363. doi: 10.1371/journal.pone.0040363 [DOI] [PMC free article] [PubMed] [Google Scholar] Yang X, Wu L, Lin J, Wang A, Wan X, Wu Y, Robson SC, Sang X, Zhao H. 2017. Distinct hepatitis B virus integration patterns in hepatocellular carcinoma and adjacent normal liver tissue. Int J Cancer 140:1324–1330. doi: 10.1002/ijc.30547 [DOI] [PubMed] [Google Scholar] Katoh H, Shibata T, Kokubu A, Ojima H, Loukopoulos P, Kanai Y, Kosuge T, Fukayama M, Kondo T, Sakamoto M, Hosoda F, Ohki M, Imoto I, Inazawa J, Hirohashi S. 2005. Genetic profile of hepatocellular carcinoma revealed by array-based comparative genomic hybridization: identification of genetic indicators to predict patient outcome. J Hepatol 43:863–874. doi: 10.1016/j.jhep.2005.05.033 [DOI] [PubMed] [Google Scholar] Li C-L, Ho M-C, Lin Y-Y, Tzeng S-T, Chen Y-J, Pai H-Y, Wang Y-C, Chen C-L, Lee Y-H, Chen D-S, Yeh S-H, Chen P-J. 2020. Cell-free virus-host chimera DNA from hepatitis B virus integration sites as a circulating biomarker of hepatocellular cancer. Hepatology 72:2063–2076. doi: 10.1002/hep.31230 [DOI] [PubMed] [Google Scholar] Salpini R, D’Anna S, Benedetti L, Piermatteo L, Gill U, Svicher V, Kennedy PTF. 2022. Hepatitis B virus DNA integration as a novel biomarker of hepatitis B virus-mediated pathogenetic properties and a barrier to the current strategies for hepatitis B virus cure. Front Microbiol 13:972687. doi: 10.3389/fmicb.2022.972687 [DOI] [PMC free article] [PubMed] [Google Scholar] Epstein MA, Achong BG, Barr YM. 1964. Virus particles in cultured lymphoblasts from Burkitt’s lymphoma. Lancet 1:702–703. doi: 10.1016/s0140-6736(64)91524-7 [DOI] [PubMed] [Google Scholar] Rostgaard K, Balfour HH, Jarrett R, Erikstrup C, Pedersen O, Ullum H, Nielsen LP, Voldstedlund M, Hjalgrim H. 2019. Primary Epstein-Barr virus infection with and without infectious mononucleosis. PLoS One 14:e0226436. doi: 10.1371/journal.pone.0226436 [DOI] [PMC free article] [PubMed] [Google Scholar] Su ZY, Siak PY, Leong C-O, Cheah S-C. 2023. The role of Epstein-Barr virus in nasopharyngeal carcinoma. Front Microbiol 14:1116143. doi: 10.3389/fmicb.2023.1116143 [DOI] [PMC free article] [PubMed] [Google Scholar] Campanero MR. 2008. Mechanisms involved in Burkitt’s tumor formation. Clin Transl Oncol 10:250–255. doi: 10.1007/s12094-008-0193-x [DOI] [PubMed] [Google Scholar] Allen PB, Lechowicz MJ. 2019. Management of NK/T-cell lymphoma, nasal type. J Oncol Pract 15:513–520. doi: 10.1200/JOP.18.00719 [DOI] [PMC free article] [PubMed] [Google Scholar] Farrell K, Jarrett RF. 2011. The molecular pathogenesis of Hodgkin lymphoma. Histopathology 58:15–25. doi: 10.1111/j.1365-2559.2010.03705.x [DOI] [PubMed] [Google Scholar] Sun K, Jia K, Lv H, Wang S-Q, Wu Y, Lei H, Chen X. 2020. EBV-positive gastric cancer: current knowledge and future perspectives. Front Oncol 10:583463. doi: 10.3389/fonc.2020.583463 [DOI] [PMC free article] [PubMed] [Google Scholar] Khan G, Fitzmaurice C, Naghavi M, Ahmed LA. 2020. Global and regional incidence, mortality and disability-adjusted life-years for Epstein-Barr virus-attributable malignancies, 1990-2017. BMJ Open 10:e037505. doi: 10.1136/bmjopen-2020-037505 [DOI] [PMC free article] [PubMed] [Google Scholar] Chakravorty S, Yan B, Wang C, Wang L, Quaid JT, Lin CF, Briggs SD, Majumder J, Canaria DA, Chauss D, Chopra G, Olson MR, Zhao B, Afzali B, Kazemian M. 2019. Integrated pan-cancer map of EBV-associated neoplasms reveals functional host-virus interactions. Cancer Res 79:6010–6023. doi: 10.1158/0008-5472.CAN-19-0615 [DOI] [PMC free article] [PubMed] [Google Scholar] Wong Y, Meehan MT, Burrows SR, Doolan DL, Miles JJ. 2022. Estimating the global burden of Epstein–Barr virus-related cancers. J Cancer Res Clin Oncol 148:31–46. doi: 10.1007/s00432-021-03824-y [DOI] [PMC free article] [PubMed] [Google Scholar] Al-Anazi AE, Alanazi BS, Alshanbari HM, Masuadi E, Hamed ME, Dandachi I, Alkathiri A, Hanif A, Nour I, Fatani H, Alsaran H, AlKhareeb F, Al Zahrani A, Alsharm AA, Eifan S, Alosaimi B. 2023. Increased prevalence of EBV infection in nasopharyngeal carcinoma patients: a six-year cross-sectional study. Cancers (Basel) 15:643. doi: 10.3390/cancers15030643 [DOI] [PMC free article] [PubMed] [Google Scholar] Donzel M, Bonjour M, Combes J-D, Broussais F, Sesques P, Traverse-Glehen A, de Martel C. 2022. Lymphomas associated with Epstein-Barr virus infection in 2020: results from a large, unselected case series in France. EClinicalMedicine 54:101674. doi: 10.1016/j.eclinm.2022.101674 [DOI] [PMC free article] [PubMed] [Google Scholar] Farisyi MA, Sufiawati I. 2020. Detection of Epstein–Barr virus DNA in saliva of HIV-1-infected individuals with oral hairy leukoplakia. Oral Dis 26:158–160. doi: 10.1111/odi.13400 [DOI] [PubMed] [Google Scholar] Soldan SS, Lieberman PM. 2023. Epstein–Barr virus and multiple sclerosis. Nat Rev Microbiol 21:51–64. doi: 10.1038/s41579-022-00770-5 [DOI] [PMC free article] [PubMed] [Google Scholar] Tarbouriech N, Buisson M, Géoui T, Daenke S, Cusack S, Burmeister WP. 2006. Structural genomics of the Epstein–Barr virus. Acta Crystallogr D Biol Crystallogr 62:1276–1285. doi: 10.1107/S0907444906030034 [DOI] [PubMed] [Google Scholar] Moss WN, Lee N, Pimienta G, Steitz JA. 2014. RNA families in Epstein-Barr virus. RNA Biol 11:10–17. doi: 10.4161/rna.27488 [DOI] [PMC free article] [PubMed] [Google Scholar] Skalsky RL, Cullen BR. 2015. EBV noncoding RNAs. Curr Top Microbiol Immunol 391:181–217. doi: 10.1007/978-3-319-22834-1_6 [DOI] [PMC free article] [PubMed] [Google Scholar] Farrell PJ. 2019. Epstein-Barr virus and cancer. Annu Rev Pathol 14:29–53. doi: 10.1146/annurev-pathmechdis-012418-013023 [DOI] [PubMed] [Google Scholar] Chesnokova LS, Jiang R, Hutt-Fletcher LM. 2015. Viral entry. Curr Top Microbiol Immunol 391:221–235. doi: 10.1007/978-3-319-22834-1_7 [DOI] [PubMed] [Google Scholar] Jean-Pierre V, Lupo J, Buisson M, Morand P, Germi R. 2021. Main targets of interest for the development of a prophylactic or therapeutic Epstein-Barr virus vaccine. Front Microbiol 12:701611. doi: 10.3389/fmicb.2021.701611 [DOI] [PMC free article] [PubMed] [Google Scholar] Miller N, Hutt-Fletcher LM. 1992. Epstein-Barr virus enters B cells and epithelial cells by different routes. J Virol 66:3409–3414. doi: 10.1128/JVI.66.6.3409-3414.1992 [DOI] [PMC free article] [PubMed] [Google Scholar] Valencia SM, Hutt-Fletcher LM. 2012. Important but differential roles for actin in trafficking of Epstein-Barr virus in B cells and epithelial cells. J Virol 86:2–10. doi: 10.1128/JVI.05883-11 [DOI] [PMC free article] [PubMed] [Google Scholar] Lee C-P, Chen M-R. 2021. Conquering the nuclear envelope barriers by EBV lytic replication. Viruses 13:702. doi: 10.3390/v13040702 [DOI] [PMC free article] [PubMed] [Google Scholar] Murata T. 2014. Regulation of Epstein–Barr virus reactivation from latency. Microbiol Immunol 58:307–317. doi: 10.1111/1348-0421.12155 [DOI] [PubMed] [Google Scholar] Hammerschmidt W. 2015. The epigenetic life cycle of Epstein-Barr virus. Curr Top Microbiol Immunol 390:103–117. doi: 10.1007/978-3-319-22822-8_6 [DOI] [PubMed] [Google Scholar] Sausen DG, Bhutta MS, Gallo ES, Dahari H, Borenstein R. 2021. Stress-induced Epstein-Barr virus reactivation. Biomolecules 11:1380. doi: 10.3390/biom11091380 [DOI] [PMC free article] [PubMed] [Google Scholar] Gruhne B, Sompallae R, Masucci MG. 2009. Three Epstein-Barr virus latency proteins independently promote genomic instability by inducing DNA damage, inhibiting DNA repair and inactivating cell cycle checkpoints. Oncogene 28:3997–4008. doi: 10.1038/onc.2009.258 [DOI] [PubMed] [Google Scholar] Deng W, Pang PS, Tsang CM, Hau PM, Yip YL, Cheung ALM, Tsao SW. 2012. Epstein-Barr virus-encoded latent membrane protein 1 impairs G2 checkpoint in human nasopharyngeal epithelial cells through defective Chk1 activation. PLoS One 7:e39095. doi: 10.1371/journal.pone.0039095 [DOI] [PMC free article] [PubMed] [Google Scholar] Dolcetti R, Dal Col J, Martorelli D, Carbone A, Klein E. 2013. Interplay among viral antigens, cellular pathways and tumor microenvironment in the pathogenesis of EBV-driven lymphomas. Semin Cancer Biol 23:441–456. doi: 10.1016/j.semcancer.2013.07.005 [DOI] [PubMed] [Google Scholar] Tsao S-W, Tsang CM, To K-F, Lo K-W. 2015. The role of Epstein–Barr virus in epithelial malignancies. J Pathol 235:323–333. doi: 10.1002/path.4448 [DOI] [PMC free article] [PubMed] [Google Scholar] Tsang CM, Lui VWY, Bruce JP, Pugh TJ, Lo KW. 2020. Translational genomics of nasopharyngeal cancer. Semin Cancer Biol 61:84–100. doi: 10.1016/j.semcancer.2019.09.006 [DOI] [PubMed] [Google Scholar] Shumilov A, Tsai M-H, Schlosser YT, Kratz A-S, Bernhardt K, Fink S, Mizani T, Lin X, Jauch A, Mautner J, Kopp-Schneider A, Feederle R, Hoffmann I, Delecluse H-J. 2017. Epstein-Barr virus particles induce centrosome amplification and chromosomal instability. Nat Commun 8:14257. doi: 10.1038/ncomms14257 [DOI] [PMC free article] [PubMed] [Google Scholar] Hurley EA, Agger S, McNeil JA, Lawrence JB, Calendar A, Lenoir G, Thorley-Lawson DA. 1991. When Epstein-Barr virus persistently infects B-cell lines, it frequently integrates. J Virol 65:1245–1254. doi: 10.1128/JVI.65.3.1245-1254.1991 [DOI] [PMC free article] [PubMed] [Google Scholar] Ohshima K, Suzumiya J, Kanda M, Kato A, Kikuchi M. 1998. Integrated and episomal forms of Epstein–Barr virus (EBV) in EBV associated disease. Cancer Lett 122:43–50. doi: 10.1016/S0304-3835(97)00368-6 [DOI] [PubMed] [Google Scholar] Xiao K, Yu Z, Li X, Li X, Tang K, Tu C, Qi P, Liao Q, Chen P, Zeng Z, Li G, Xiong W. 2016. Genome-wide analysis of Epstein-Barr Virus (EBV) integration and strain in C666-1 and Raji cells. J Cancer 7:214–224. doi: 10.7150/jca.13150 [DOI] [PMC free article] [PubMed] [Google Scholar] Peng R-J, Han B-W, Cai Q-Q, Zuo X-Y, Xia T, Chen J-R, Feng L-N, Lim JQ, Chen S-W, Zeng M-S, Guo Y-M, Li B, Xia X-J, Xia Y, Laurensia Y, Chia BKH, Huang H-Q, Young KH, Lim ST, Ong CK, Zeng Y-X, Bei J-X. 2019. Genomic and transcriptomic landscapes of Epstein-Barr virus in extranodal natural killer T-cell lymphoma. Leukemia 33:1451–1462. doi: 10.1038/s41375-018-0324-5 [DOI] [PMC free article] [PubMed] [Google Scholar] Koliais SI. 1979. Mode of integration of Epstein-Barr virus genome into host DNA in Burkitt lymphoma cells. J Gen Virol 44:573–576. doi: 10.1099/0022-1317-44-2-573 [DOI] [PubMed] [Google Scholar] Anvret M, Karlsson A, Bjursell G. 1984. Evidence for integrated EBV genomes in Raji cellular DNA. Nucleic Acids Res 12:1149–1161. doi: 10.1093/nar/12.2.1149 [DOI] [PMC free article] [PubMed] [Google Scholar] Kieff E, Hennessy K, Fennewald S, Matsuo T, Dambaugh T, Heller M, Hummel M. 1985. Biochemistry of latent Epstein-Barr virus infection and associated cell growth transformation, p 323–339. IARC scientific publications. [PubMed] [Google Scholar] Xu M, Zhang W-L, Zhu Q, Zhang S, Yao Y-Y, Xiang T, Feng Q-S, Zhang Z, Peng R-J, Jia W-H, He G-P, Feng L, Zeng Z-L, Luo B, Xu R-H, Zeng M-S, Zhao W-L, Chen S-J, Zeng Y-X, Jiao Y. 2019. Genome-wide profiling of Epstein-Barr virus integration by targeted sequencing in Epstein-Barr virus associated malignancies. Theranostics 9:1115–1124. doi: 10.7150/thno.29622 [DOI] [PMC free article] [PubMed] [Google Scholar] Delecluse HJ, Bartnizke S, Hammerschmidt W, Bullerdiek J, Bornkamm GW. 1993. Episomal and integrated copies of Epstein-Barr virus coexist in Burkitt lymphoma cell lines. J Virol 67:1292–1299. doi: 10.1128/JVI.67.3.1292-1299.1993 [DOI] [PMC free article] [PubMed] [Google Scholar] Kripalani-Joshi S, Law HY. 1994. Identification of integrated Epstein-Barr virus in nasopharyngeal carcinoma using pulse field gel electrophoresis. Int J Cancer 56:187–192. doi: 10.1002/ijc.2910560207 [DOI] [PubMed] [Google Scholar] Takakuwa T, Luo W-J, Ham MF, Sakane-Ishikawa F, Wada N, Aozasa K. 2004. Integration of Epstein-Barr virus into chromosome 6q15 of Burkitt lymphoma cell line (Raji) induces loss of BACH2 expression. Am J Pathol 164:967–974. doi: 10.1016/S0002-9440(10)63184-7 [DOI] [PMC free article] [PubMed] [Google Scholar] Zhang L, Wang R, Xie Z. 2022. The roles of DNA methylation on the promotor of the Epstein–Barr virus (EBV) gene and the genome in patients with EBV-associated diseases. Appl Microbiol Biotechnol 106:4413–4426. doi: 10.1007/s00253-022-12029-3 [DOI] [PMC free article] [PubMed] [Google Scholar] Luo W-J, Takakuwa T, Ham MF, Wada N, Liu A, Fujita S, Sakane-Ishikawa E, Aozasa K. 2004. Epstein-Barr virus is integrated between REL and BCL-11A in American Burkitt lymphoma cell line (NAB-2). Lab Invest 84:1193–1199. doi: 10.1038/labinvest.3700152 [DOI] [PubMed] [Google Scholar] Janjetovic S, Hinke J, Balachandran S, Akyüz N, Behrmann P, Bokemeyer C, Dierlamm J, Murga Penas EM. 2022. Non-random pattern of integration for epstein-barr virus with preference for gene-poor genomic chromosomal regions into the genome of burkitt lymphoma cell lines. Viruses 14:86. doi: 10.3390/v14010086 [DOI] [PMC free article] [PubMed] [Google Scholar] Tang D, Li B, Xu T, Hu R, Tan D, Song X, Jia P, Zhao Z. 2020. VISDB: a manually curated database of viral integration sites in the human genome. Nucleic Acids Res 48:D633–D641. doi: 10.1093/nar/gkz867 [DOI] [PMC free article] [PubMed] [Google Scholar] Gao J, Luo X, Tang K, Li X, Li G. 2006. Epstein-Barr virus integrates frequently into chromosome 4q, 2q, 1q and 7q of Burkitt’s lymphoma cell line (Raji). J Virol Methods 136:193–199. doi: 10.1016/j.jviromet.2006.05.013 [DOI] [PubMed] [Google Scholar] Gasenko E, Isajevs S, Camargo MC, Offerhaus GJA, Polaka I, Gulley ML, Skapars R, Sivins A, Kojalo I, Kirsners A, Santare D, Pavlova J, Sjomina O, Liepina E, Tzivian L, Rabkin CS, Leja M. 2019. Clinicopathological characteristics of Epstein-Barr virus-positive gastric cancer in Latvia. Eur J Gastroenterol Hepatol 31:1328–1333. doi: 10.1097/MEG.0000000000001521 [DOI] [PMC free article] [PubMed] [Google Scholar] He C-Y, Qiu M-Z, Yang X-H, Zhou D-L, Ma J-J, Long Y-K, Ye Z-L, Xu B-H, Zhao Q, Jin Y, Lu S-X, Wang Z-Q, Guan W-L, Zhao B-W, Zhou Z-W, Shao J-Y, Xu R-H. 2020. Classification of gastric cancer by EBV status combined with molecular profiling predicts patient prognosis. Clin Transl Med 10:353–362. doi: 10.1002/ctm2.32 [DOI] [PMC free article] [PubMed] [Google Scholar] Qiu M-Z, He C-Y, Lu S-X, Guan W-L, Wang F, Wang X-J, Jin Y, Wang F-H, Li Y-H, Shao J-Y, Zhou Z-W, Yun J-P, Xu R-H. 2020. Prospective observation: clinical utility of plasma Epstein-Barr virus DNA load in EBV-associated gastric carcinoma patients. Int J Cancer 146:272–280. doi: 10.1002/ijc.32490 [DOI] [PubMed] [Google Scholar] PaVE . 2025. Available from: Retrieved 06 Mar 2025. Burchell AN, Winer RL, de Sanjosé S, Franco EL. 2006. Chapter 6: epidemiology and transmission dynamics of genital HPV infection. Vaccine (Auckl) 24:S3 doi: 10.1016/j.vaccine.2006.05.031 [DOI] [PubMed] [Google Scholar] Bruni L, Diaz M, Castellsagué X, Ferrer E, Bosch FX, de Sanjosé S. 2010. Cervical human papillomavirus prevalence in 5 continents: meta-analysis of 1 million women with normal cytological findings. J Infect Dis 202:1789–1799. doi: 10.1086/657321 [DOI] [PubMed] [Google Scholar] Human papillomavirus and cancer. 2024. Available from: Retrieved Jul 2025. Bruni L, Albero G, Rowley J, Alemany L, Arbyn M, Giuliano AR, Markowitz LE, Broutet N, Taylor M. 2023. Global and regional estimates of genital human papillomavirus prevalence among men: a systematic review and meta-analysis. Lancet Glob Health 11:e1345–e1362. doi: 10.1016/S2214-109X(23)00305-4 [DOI] [PMC free article] [PubMed] [Google Scholar] Malik H, Khan FH, Ahsan H. 2014. Human papillomavirus: current status and issues of vaccination. Arch Virol 159:199–205. doi: 10.1007/s00705-013-1827-z [DOI] [PubMed] [Google Scholar] 103.Serrano B, Albero G, Bruni L2025. Fact Sheet World 262. Available from: Retrieved 6 Jun 2025. de Martel C, Plummer M, Vignat J, Franceschi S. 2017. Worldwide burden of cancer attributable to HPV by site, country and HPV type. Int J Cancer 141:664–670. doi: 10.1002/ijc.30716 [DOI] [PMC free article] [PubMed] [Google Scholar] Ndon S, Singh A, Ha PK, Aswani J, Chan J-K, Xu MJ. 2023. Human papillomavirus-associated oropharyngeal cancer: global epidemiology and public policy implications. Cancers (Basel) 15:4080. doi: 10.3390/cancers15164080 [DOI] [PMC free article] [PubMed] [Google Scholar] de Sanjosé S, Brotons M, Pavón MA. 2018. The natural history of human papillomavirus infection. Best Pract Res Clin Obstet Gynaecol 47:2–13. doi: 10.1016/j.bpobgyn.2017.08.015 [DOI] [PubMed] [Google Scholar] Mikuličić S, Strunk J, Florin L. 2021. HPV16 entry into epithelial cells: running a gauntlet. Viruses 13:2460. doi: 10.3390/v13122460 [DOI] [PMC free article] [PubMed] [Google Scholar] McBride AA. 2008. Replication and partitioning of papillomavirus genomes. Adv Virus Res 72:155–205. doi: 10.1016/S0065-3527(08)00404-1 [DOI] [PMC free article] [PubMed] [Google Scholar] Radley D, Saah A, Stanley M. 2016. Persistent infection with human papillomavirus 16 or 18 is strongly linked with high-grade cervical disease. Hum Vaccin Immunother 12:768–772. doi: 10.1080/21645515.2015.1088616 [DOI] [PMC free article] [PubMed] [Google Scholar] Moscicki A-B, Shiboski S, Hills NK, Powell KJ, Jay N, Hanson EN, Miller S, Canjura-Clayton KL, Farhat S, Broering JM, Darragh TM. 2004. Regression of low-grade squamous intra-epithelial lesions in young women. Lancet 364:1678–1683. doi: 10.1016/S0140-6736(04)17354-6 [DOI] [PubMed] [Google Scholar] Ostör AG. 1993. Natural history of cervical intraepithelial neoplasia: a critical review. Int J Gynecol Pathol 12:186–192. [PubMed] [Google Scholar] Basu P, Taghavi K, Hu S-Y, Mogri S, Joshi S. 2018. Management of cervical premalignant lesions. Curr Probl Cancer 42:129–136. doi: 10.1016/j.currproblcancer.2018.01.010 [DOI] [PubMed] [Google Scholar] Bhattacharjee R, Das SS, Biswal SS, Nath A, Das D, Basu A, Malik S, Kumar L, Kar S, Singh SK, Upadhye VJ, Iqbal D, Almojam S, Roychoudhury S, Ojha S, Ruokolainen J, Jha NK, Kesari KK. 2022. Mechanistic role of HPV-associated early proteins in cervical cancer: molecular pathways and targeted therapeutic strategies. Crit Rev Oncol Hematol 174:103675. doi: 10.1016/j.critrevonc.2022.103675 [DOI] [PubMed] [Google Scholar] Scheffner M, Huibregtse JM, Vierstra RD, Howley PM. 1993. The HPV-16 E6 and E6-AP complex functions as a ubiquitin-protein ligase in the ubiquitination of p53. Cell 75:495–505. doi: 10.1016/0092-8674(93)90384-3 [DOI] [PubMed] [Google Scholar] Beaudenon S, Huibregtse JM. 2008. HPV E6, E6AP and cervical cancer. BMC Biochem 9:S4. doi: 10.1186/1471-2091-9-S1-S4 [DOI] [PMC free article] [PubMed] [Google Scholar] Mortensen F, Schneider D, Barbic T, Sladewska-Marquardt A, Kühnle S, Marx A, Scheffner M. 2015. Role of ubiquitin and the HPV E6 oncoprotein in E6AP-mediated ubiquitination. Proc Natl Acad Sci USA 112:9872–9877. doi: 10.1073/pnas.1505923112 [DOI] [PMC free article] [PubMed] [Google Scholar] Zimmermann H, Degenkolbe R, Bernard HU, O’Connor MJ. 1999. The human papillomavirus type 16 E6 oncoprotein can down-regulate p53 activity by targeting the transcriptional coactivator CBP/p300. J Virol 73:6209–6219. doi: 10.1128/JVI.73.8.6209-6219.1999 [DOI] [PMC free article] [PubMed] [Google Scholar] Patel D, Huang SM, Baglia LA, McCance DJ. 1999. The E6 protein of human papillomavirus type 16 binds to and inhibits co-activation by CBP and p300. EMBO J 18:5061–5072. doi: 10.1093/emboj/18.18.5061 [DOI] [PMC free article] [PubMed] [Google Scholar] Yoshimatsu Y, Nakahara T, Tanaka K, Inagawa Y, Narisawa-Saito M, Yugawa T, Ohno S-I, Fujita M, Nakagama H, Kiyono T. 2017. Roles of the PDZ-binding motif of HPV 16 E6 protein in oncogenic transformation of human cervical keratinocytes. Cancer Sci 108:1303–1309. doi: 10.1111/cas.13264 [DOI] [PMC free article] [PubMed] [Google Scholar] Ronco LV, Karpova AY, Vidal M, Howley PM. 1998. Human papillomavirus 16 E6 oncoprotein binds to interferon regulatory factor-3 and inhibits its transcriptional activity. Genes Dev 12:2061–2072. doi: 10.1101/gad.12.13.2061 [DOI] [PMC free article] [PubMed] [Google Scholar] Liu X, Dakic A, Zhang Y, Dai Y, Chen R, Schlegel R. 2009. HPV E6 protein interacts physically and functionally with the cellular telomerase complex. Proc Natl Acad Sci USA 106:18780–18785. doi: 10.1073/pnas.0906357106 [DOI] [PMC free article] [PubMed] [Google Scholar] Münger K, Howley PM. 2002. Human papillomavirus immortalization and transformation functions. Virus Res 89:213–228. doi: 10.1016/s0168-1702(02)00190-9 [DOI] [PubMed] [Google Scholar] Longworth MS, Laimins LA. 2004. The binding of histone deacetylases and the integrity of zinc finger-like motifs of the E7 protein are essential for the life cycle of human papillomavirus type 31. J Virol 78:3533–3541. doi: 10.1128/JVI.78.7.3533-3541.2004 [DOI] [PMC free article] [PubMed] [Google Scholar] McLaughlin-Drubin ME, Münger K. 2009. The human papillomavirus E7 oncoprotein. Virology (Auckl) 384:335–344. doi: 10.1016/j.virol.2008.10.006 [DOI] [PMC free article] [PubMed] [Google Scholar] Park JS, Kim EJ, Kwon HJ, Hwang ES, Namkoong SE, Um SJ. 2000. Inactivation of interferon regulatory factor-1 tumor suppressor protein by HPV E7 oncoprotein. Implication for the E7-mediated immune evasion mechanism in cervical carcinogenesis. J Biol Chem 275:6764–6769. doi: 10.1074/jbc.275.10.6764 [DOI] [PubMed] [Google Scholar] Um S-J, Rhyu J-W, Kim E-J, Jeon K-C, Hwang E-S, Park J-S. 2002. Abrogation of IRF-1 response by high-risk HPV E7 protein in vivo. Cancer Lett 179:205–212. doi: 10.1016/s0304-3835(01)00871-0 [DOI] [PubMed] [Google Scholar] Tomakidi P, Cheng H, Kohl A, Komposch G, Alonso A. 2000. Modulation of the epidermal growth factor receptor by the human papillomavirus type 16 E5 protein in raft cultures of human keratinocytes. Eur J Cell Biol 79:407–412. doi: 10.1078/0171-9335-00060 [DOI] [PubMed] [Google Scholar] Wasson CW, Morgan EL, Müller M, Ross RL, Hartley M, Roberts S, Macdonald A. 2017. Human papillomavirus type 18 E5 oncogene supports cell cycle progression and impairs epithelial differentiation by modulating growth factor receptor signalling during the virus life cycle. Oncotarget 8:103581–103600. doi: 10.18632/oncotarget.21658 [DOI] [PMC free article] [PubMed] [Google Scholar] Oh J-M, Kim S-H, Cho E-A, Song Y-S, Kim W-H, Juhnn Y-S. 2010. Human papillomavirus type 16 E5 protein inhibits hydrogen peroxide-induced apoptosis by stimulating ubiquitin–proteasome-mediated degradation of Bax in human cervical cancer cells. Carcinogenesis 31:402–410. doi: 10.1093/carcin/bgp318 [DOI] [PubMed] [Google Scholar] Kim S-H, Juhnn Y-S, Kang S, Park S-W, Sung M-W, Bang Y-J, Song Y-S. 2006. Human papillomavirus 16 E5 up-regulates the expression of vascular endothelial growth factor through the activation of epidermal growth factor receptor, MEK/ ERK1,2 and PI3K/Akt. Cell Mol Life Sci 63:930–938. doi: 10.1007/s00018-005-5561-x [DOI] [PMC free article] [PubMed] [Google Scholar] Ilahi NE, Bhatti A. 2020. Impact of HPV E5 on viral life cycle via EGFR signaling. Microb Pathog 139:103923. doi: 10.1016/j.micpath.2019.103923 [DOI] [PubMed] [Google Scholar] Kim S-H, Oh J-M, No J-H, Bang Y-J, Juhnn Y-S, Song Y-S. 2009. Involvement of NF-κB and AP-1 in COX-2 upregulation by human papillomavirus 16 E5 oncoprotein. Carcinogenesis 30:753–757. doi: 10.1093/carcin/bgp066 [DOI] [PubMed] [Google Scholar] Hatano T, Sano D, Takahashi H, Oridate N. 2021. Pathogenic role of immune evasion and integration of human papillomavirus in oropharyngeal cancer. Microorganisms 9:891. doi: 10.3390/microorganisms9050891 [DOI] [PMC free article] [PubMed] [Google Scholar] Moody CA, Laimins LA. 2009. Human papillomaviruses activate the ATM DNA damage pathway for viral genome amplification upon differentiation. PLoS Pathog 5:e1000605. doi: 10.1371/journal.ppat.1000605 [DOI] [PMC free article] [PubMed] [Google Scholar] Hong S, Laimins LA. 2013. Regulation of the life cycle of HPVs by differentiation and the DNA damage response. Future Microbiol 8:1547–1557. doi: 10.2217/fmb.13.127 [DOI] [PMC free article] [PubMed] [Google Scholar] Spriggs CC, Laimins LA. 2017. FANCD2 binds human papillomavirus genomes and associates with a distinct set of DNA repair proteins to regulate viral replication. mBio 8:e02340-16. doi: 10.1128/mBio.02340-16 [DOI] [PMC free article] [PubMed] [Google Scholar] Kono T, Laimins L. 2021. Genomic instability and DNA damage repair pathways induced by human papillomaviruses. Viruses 13:1821. doi: 10.3390/v13091821 [DOI] [PMC free article] [PubMed] [Google Scholar] Williams VM, Filippova M, Filippov V, Payne KJ, Duerksen-Hughes P. 2014. Human papillomavirus type 16 E6 induces oxidative stress and DNA damage. J Virol 88:6751–6761. doi: 10.1128/JVI.03355-13 [DOI] [PMC free article] [PubMed] [Google Scholar] Duensing S, Lee LY, Duensing A, Basile J, Piboonniyom S, Gonzalez S, Crum CP, Munger K. 2000. The human papillomavirus type 16 E6 and E7 oncoproteins cooperate to induce mitotic defects and genomic instability by uncoupling centrosome duplication from the cell division cycle. Proc Natl Acad Sci USA 97:10002–10007. doi: 10.1073/pnas.170093297 [DOI] [PMC free article] [PubMed] [Google Scholar] Duensing S, Duensing A, Crum CP, Münger K. 2001. Human papillomavirus type 16 E7 oncoprotein-induced abnormal centrosome synthesis is an early event in the evolving malignant phenotype. Cancer Res 61:2356–2360. [PubMed] [Google Scholar] Korzeniewski N, Spardy N, Duensing A, Duensing S. 2011. Genomic instability and cancer: lessons learned from human papillomaviruses. Cancer Lett 305:113–122. doi: 10.1016/j.canlet.2010.10.013 [DOI] [PMC free article] [PubMed] [Google Scholar] Hudelist G, Manavi M, Pischinger KID, Watkins-Riedel T, Singer CF, Kubista E, Czerwenka KF. 2004. Physical state and expression of HPV DNA in benign and dysplastic cervical tissue: different levels of viral integration are correlated with lesion grade. Gynecol Oncol 92:873–880. doi: 10.1016/j.ygyno.2003.11.035 [DOI] [PubMed] [Google Scholar] Briolat J, Dalstein V, Saunier M, Joseph K, Caudroy S, Prétet J-L, Birembaut P, Clavel C. 2007. HPV prevalence, viral load and physical state of HPV-16 in cervical smears of patients with different grades of CIN. Int J Cancer 121:2198–2204. doi: 10.1002/ijc.22959 [DOI] [PubMed] [Google Scholar] Khoury JD, Tannir NM, Williams MD, Chen Y, Yao H, Zhang J, Thompson EJ, TCGA Network, Meric-Bernstam F, Medeiros LJ, Weinstein JN, Su X. 2013. Landscape of DNA virus associations across human malignant cancers: analysis of 3,775 cases using RNA-Seq. J Virol 87:8916–8926. doi: 10.1128/JVI.00340-13 [DOI] [PMC free article] [PubMed] [Google Scholar] Vojtechova Z, Sabol I, Salakova M, Turek L, Grega M, Smahelova J, Vencalek O, Lukesova E, Klozar J, Tachezy R. 2016. Analysis of the integration of human papillomaviruses in head and neck tumours in relation to patients’ prognosis. Int J Cancer 138:386–395. doi: 10.1002/ijc.29712 [DOI] [PubMed] [Google Scholar] Cancer Genome Atlas Research Network, Albert Einstein College of Medicine, Analytical Biological Services, Barretos Cancer Hospital, Baylor College of Medicine, Beckman Research Institute of City of Hope, Buck Institute for Research on Aging Canada’s Michael Smith Genome Sciences Centre Harvard Medical School, Helen F. Graham Cancer Center & Research Institute at Christiana Care Health Services, et al. 2017. Integrated genomic and molecular characterization of cervical cancer. Nature 543:378–384. doi: 10.1038/nature21386 [DOI] [PMC free article] [PubMed] [Google Scholar] Tang KD, Baeten K, Kenny L, Frazer IH, Scheper G, Punyadeera C. 2019. Unlocking the potential of saliva-based test to detect HPV-16-driven oropharyngeal cancer. Cancers (Basel) 11:473. doi: 10.3390/cancers11040473 [DOI] [PMC free article] [PubMed] [Google Scholar] Mainguené J, Vacher S, Kamal M, Hamza A, Masliah-Planchon J, Baulande S, Ibadioune S, Borcoman E, Cacheux W, Calugaru V, et al. 2022. Human papilloma virus integration sites and genomic signatures in head and neck squamous cell carcinoma. Mol Oncol 16:3001–3016. doi: 10.1002/1878-0261.13219 [DOI] [PMC free article] [PubMed] [Google Scholar] Gray E, Pett MR, Ward D, Winder DM, Stanley MA, Roberts I, Scarpini CG, Coleman N. 2010. In vitro progression of human papillomavirus 16 episome-associated cervical neoplasia displays fundamental similarities to integrant-associated carcinogenesis. Cancer Res 70:4081–4091. doi: 10.1158/0008-5472.CAN-09-3335 [DOI] [PMC free article] [PubMed] [Google Scholar] Rossi NM, Dai J, Xie Y, Wangsa D, Heselmeyer-Haddad K, Lou H, Boland JF, Yeager M, Orozco R, Freites EA, Mirabello L, Gharzouzi E, Dean M. 2023. Extrachromosomal amplification of human papillomavirus episomes is a mechanism of cervical carcinogenesis. Cancer Res 83:1768–1781. doi: 10.1158/0008-5472.CAN-22-3030 [DOI] [PMC free article] [PubMed] [Google Scholar] Chaiwongkot A, Vinokurova S, Pientong C, Ekalaksananan T, Kongyingyoes B, Kleebkaow P, Chumworathayi B, Patarapadungkit N, Reuschenbach M, von Knebel Doeberitz M. 2013. Differential methylation of E2 binding sites in episomal and integrated HPV 16 genomes in preinvasive and invasive cervical lesions. Int J Cancer 132:2087–2094. doi: 10.1002/ijc.27906 [DOI] [PubMed] [Google Scholar] Pokrývková B, Saláková M, Šmahelová J, Vojtěchová Z, Novosadová V, Tachezy R. 2019. Detailed characteristics of tonsillar tumors with extrachromosomal or integrated form of human papillomavirus. Viruses 12:42. doi: 10.3390/v12010042 [DOI] [PMC free article] [PubMed] [Google Scholar] Cheung JLK, Cheung T-H, Yu MY, Chan PKS. 2013. Virological characteristics of cervical cancers carrying pure episomal form of HPV16 genome. Gynecol Oncol 131:374–379. doi: 10.1016/j.ygyno.2013.08.026 [DOI] [PubMed] [Google Scholar] Ren S, Gaykalova DA, Guo T, Favorov AV, Fertig EJ, Tamayo P, Callejas-Valera JL, Allevato M, Gilardi M, Santos J, Fukusumi T, Sakai A, Ando M, Sadat S, Liu C, Xu G, Fisch KM, Wang Z, Molinolo AA, Gutkind JS, Ideker T, Koch WM, Califano JA. 2020. HPV E2, E4, E5 drive alternative carcinogenic pathways in HPV positive cancers. Oncogene 39:6327–6339. doi: 10.1038/s41388-020-01431-8 [DOI] [PMC free article] [PubMed] [Google Scholar] Williams VM, Filippova M, Soto U, Duerksen-Hughes PJ. 2011. HPV-DNA integration and carcinogenesis: putative roles for inflammation and oxidative stress. Future Virol 6:45–57. doi: 10.2217/fvl.10.73 [DOI] [PMC free article] [PubMed] [Google Scholar] Visalli G, Riso R, Facciolà A, Mondello P, Caruso C, Picerno I, Di Pietro A, Spataro P, Bertuccio MP. 2016. Higher levels of oxidative DNA damage in cervical cells are correlated with the grade of dysplasia and HPV infection. J Med Virol 88:336–344. doi: 10.1002/jmv.24327 [DOI] [PubMed] [Google Scholar] Kondo S, Wakae K, Wakisaka N, Nakanishi Y, Ishikawa K, Komori T, Moriyama-Kita M, Endo K, Murono S, Wang Z, Kitamura K, Nishiyama T, Yamaguchi K, Shigenobu S, Muramatsu M, Yoshizaki T. 2017. APOBEC3A associates with human papillomavirus genome integration in oropharyngeal cancers. Oncogene 36:1687–1697. doi: 10.1038/onc.2016.335 [DOI] [PubMed] [Google Scholar] Zapatka M, Borozan I, Brewer DS, Iskar M, Grundhoff A, Alawi M, Desai N, Sültmann H, Moch H, Cooper CS, Eils R, Ferretti V, Lichter P, PCAWG Pathogens, PCAWG Consortium . 2020. The landscape of viral associations in human cancers. Nat Genet 52:320–330. doi: 10.1038/s41588-019-0558-9 [DOI] [PMC free article] [PubMed] [Google Scholar] Hu Z, Zhu D, Wang W, Li W, Jia W, Zeng X, Ding W, Yu L, Wang X, Wang L, et al. 2015. Genome-wide profiling of HPV integration in cervical cancer identifies clustered genomic hot spots and a potential microhomology-mediated integration mechanism. Nat Genet 47:158–163. doi: 10.1038/ng.3178 [DOI] [PubMed] [Google Scholar] Zhang R, Shen C, Zhao L, Wang J, McCrae M, Chen X, Lu F. 2016. Dysregulation of host cellular genes targeted by human papillomavirus (HPV) integration contributes to HPV-related cervical carcinogenesis. Int J Cancer 138:1163–1174. doi: 10.1002/ijc.29872 [DOI] [PMC free article] [PubMed] [Google Scholar] Kamal M, Lameiras S, Deloger M, Morel A, Vacher S, Lecerf C, Dupain C, Jeannot E, Girard E, Baulande S, Dubot C, Kenter G, Jordanova ES, Berns EMJJ, Bataillon G, Popovic M, Rouzier R, Cacheux W, Le Tourneau C, Nicolas A, Servant N, Scholl SM, Bièche I, RAIDs Consortium . 2021. Human papilloma virus (HPV) integration signature in Cervical cancer: identification of MACROD2 gene as HPV hot spot integration site. Br J Cancer 124:777–785. doi: 10.1038/s41416-020-01153-4 [DOI] [PMC free article] [PubMed] [Google Scholar] Fan J, Fu Y, Peng W, Li X, Shen Y, Guo E, Lu F, Zhou S, Liu S, Yang B, et al. 2023. Multi-omics characterization of silent and productive HPV integration in cervical cancer. Cell Genom 3:100211. doi: 10.1016/j.xgen.2022.100211 [DOI] [PMC free article] [PubMed] [Google Scholar] Zeng X, Wang Y, Liu B, Rao X, Cao C, Peng F, Zhi W, Wu P, Peng T, Wei Y, Chu T, Xu M, Xu Y, Ding W, Li G, Lin S, Wu P. 2023. Multi-omics data reveals novel impacts of human papillomavirus integration on the epigenomic and transcriptomic signatures of cervical tumorigenesis. J Med Virol 95:e28789. doi: 10.1002/jmv.28789 [DOI] [PubMed] [Google Scholar] Thorland EC, Myers SL, Persing DH, Sarkar G, McGovern RM, Gostout BS, Smith DI. 2000. Human papillomavirus type 16 integrations in cervical tumors frequently occur in common fragile sites. Cancer Res 60:5916–5921. [PubMed] [Google Scholar] Bodelon C, Untereiner ME, Machiela MJ, Vinokurova S, Wentzensen N. 2016. Genomic characterization of viral integration sites in HPV-related cancers. Int J Cancer 139:2001–2011. doi: 10.1002/ijc.30243 [DOI] [PMC free article] [PubMed] [Google Scholar] Gao G, Johnson SH, Vasmatzis G, Pauley CE, Tombers NM, Kasperbauer JL, Smith DI. 2017. Common fragile sites (CFS) and extremely large CFS genes are targets for human papillomavirus integrations and chromosome rearrangements in oropharyngeal squamous cell carcinoma. Genes Chromosomes Cancer 56:59–74. doi: 10.1002/gcc.22415 [DOI] [PubMed] [Google Scholar] Wentzensen N, Vinokurova S, von Knebel Doeberitz M. 2004. Systematic review of genomic integration sites of human papillomavirus genomes in epithelial dysplasia and invasive cancer of the female lower genital tract. Cancer Res 64:3878–3884. doi: 10.1158/0008-5472.CAN-04-0009 [DOI] [PubMed] [Google Scholar] Ragin CCR, Reshmi SC, Gollin SM. 2004. Mapping and analysis of HPV16 integration sites in a head and neck cancer cell line. Intl J Cancer 110:701–709. doi: 10.1002/ijc.20193 [DOI] [PubMed] [Google Scholar] Akagi K, Li J, Broutian TR, Padilla-Nash H, Xiao W, Jiang B, Rocco JW, Teknos TN, Kumar B, Wangsa D, He D, Ried T, Symer DE, Gillison ML. 2014. Genome-wide analysis of HPV integration in human cancers reveals recurrent, focal genomic instability. Genome Res 24:185–199. doi: 10.1101/gr.164806.113 [DOI] [PMC free article] [PubMed] [Google Scholar] Zhao J, Zheng W, Wang L, Jiang H, Wang X, Hou J, Xu A, Cong J. 2023. Human papillomavirus (HPV) integration signature in cervical lesions: identification of MACROD2 gene as HPV hot spot integration site. Arch Gynecol Obstet 307:1115–1123. doi: 10.1007/s00404-022-06748-1 [DOI] [PubMed] [Google Scholar] Parfenov M, Pedamallu CS, Gehlenborg N, Freeman SS, Danilova L, Bristow CA, Lee S, Hadjipanayis AG, Ivanova EV, Wilkerson MD, et al. 2014. Characterization of HPV and host genome interactions in primary head and neck cancers. Proc Natl Acad Sci USA 111:15544–15549. doi: 10.1073/pnas.1416074111 [DOI] [PMC free article] [PubMed] [Google Scholar] Olthof NC, Speel E-J, Kolligs J, Haesevoets A, Henfling M, Ramaekers FCS, Preuss SF, Drebber U, Wieland U, Silling S, Lam WL, Vucic EA, Kremer B, Klussmann J-P, Huebbers CU. 2014. Comprehensive analysis of HPV16 integration in OSCC reveals no significant impact of physical status on viral oncogene and virally disrupted human gene expression. PLoS One 9:e88718. doi: 10.1371/journal.pone.0088718 [DOI] [PMC free article] [PubMed] [Google Scholar] Tian R, Huang Z, Li L, Yuan J, Zhang Q, Meng L, Lang B, Hong Y, Zhong C, Tian X, Cui Z, Jin Z, Liu J, Huang Z, Wang Y, Chen Y, Hu Z. 2023. HPV integration generates a cellular super-enhancer which functions as ecDNA to regulate genome-wide transcription. Nucleic Acids Res 51:4237–4251. doi: 10.1093/nar/gkad105 [DOI] [PMC free article] [PubMed] [Google Scholar] Hatano T, Sano D, Takahashi H, Hyakusoku H, Isono Y, Shimada S, Sawakuma K, Takada K, Oikawa R, Watanabe Y, Yamamoto H, Itoh F, Myers JN, Oridate N. 2017. Identification of human papillomavirus (HPV) 16 DNA integration and the ensuing patterns of methylation in HPV-associated head and neck squamous cell carcinoma cell lines. Int J Cancer 140:1571–1580. doi: 10.1002/ijc.30589 [DOI] [PMC free article] [PubMed] [Google Scholar] Badal V, Chuang LSH, Tan E-H, Badal S, Villa LL, Wheeler CM, Li BFL, Bernard H-U. 2003. CpG methylation of human papillomavirus type 16 DNA in cervical cancer cell lines and in clinical specimens: genomic hypomethylation correlates with carcinogenic progression. J Virol 77:6227–6234. doi: 10.1128/jvi.77.11.6227-6234.2003 [DOI] [PMC free article] [PubMed] [Google Scholar] Rosendo-Chalma P, Antonio-Véjar V, Ortiz Tejedor JG, Ortiz Segarra J, Vega Crespo B, Bigoni-Ordóñez GD. 2024. The hallmarks of cervical cancer: molecular mechanisms induced by human papillomavirus. Biology (Basel) 13:77. doi: 10.3390/biology13020077 [DOI] [PMC free article] [PubMed] [Google Scholar] Balaji H, Demers I, Wuerdemann N, Schrijnder J, Kremer B, Klussmann JP, Huebbers CU, Speel E-JM. 2021. Causes and consequences of HPV integration in head and neck squamous cell carcinomas: state of the art. Cancers (Basel) 13:4089. doi: 10.3390/cancers13164089 [DOI] [PMC free article] [PubMed] [Google Scholar] Veitía D, Liuzzi J, Ávila M, Rodriguez I, Toro F, Correnti M. 2020. Association of viral load and physical status of HPV-16 with survival of patients with head and neck cancer. Ecancermedicalscience 14:1082. doi: 10.3332/ecancer.2020.1082 [DOI] [PMC free article] [PubMed] [Google Scholar] Nambaru L, Meenakumari B, Swaminathan R, Rajkumar T. 2009. Prognostic significance of HPV physical status and integration sites in cervical cancer. Asian Pac J Cancer Prev 10:355–360. [PubMed] [Google Scholar] Koneva LA, Zhang Y, Virani S, Hall PB, McHugh JB, Chepeha DB, Wolf GT, Carey TE, Rozek LS, Sartor MA. 2018. HPV integration in HNSCC correlates with survival outcomes, immune response signatures, and candidate drivers. Mol Cancer Res 16:90–102. doi: 10.1158/1541-7786.MCR-17-0153 [DOI] [PMC free article] [PubMed] [Google Scholar] Nulton TJ, Kim N-K, DiNardo LJ, Morgan IM, Windle B. 2018. Patients with integrated HPV16 in head and neck cancer show poor survival. Oral Oncol 80:52–55. doi: 10.1016/j.oraloncology.2018.03.015 [DOI] [PMC free article] [PubMed] [Google Scholar] Pinatti LM, Sinha HN, Brummel CV, Goudsmit CM, Geddes TJ, Wilson GD, Akervall JA, Brenner CJ, Walline HM, Carey TE. 2021. Association of human papillomavirus integration with better patient outcomes in oropharyngeal squamous cell carcinoma. Head Neck 43:544–557. doi: 10.1002/hed.26501 [DOI] [PMC free article] [PubMed] [Google Scholar] Feng H, Shuda M, Chang Y, Moore PS. 2008. Clonal integration of a polyomavirus in human Merkel cell carcinoma. Science 319:1096–1100. doi: 10.1126/science.1152586 [DOI] [PMC free article] [PubMed] [Google Scholar] Wijaya WA, Liu Y, Qing Y, Li Z. 2022. Prevalence of Merkel cell polyomavirus in normal and lesional skin: a systematic review and meta-analysis. Front Oncol 12:868781. doi: 10.3389/fonc.2022.868781 [DOI] [PMC free article] [PubMed] [Google Scholar] Sunshine JC, Jahchan NS, Sage J, Choi J. 2018. Are there multiple cells of origin of Merkel cell carcinoma? Oncogene 37:1409–1416. doi: 10.1038/s41388-017-0073-3 [DOI] [PMC free article] [PubMed] [Google Scholar] Juan HY, Khachemoune A. 2023. A review of Merkel cell carcinoma. JAAPA 36:11–16. doi: 10.1097/01.JAA.0000979460.69305.b7 [DOI] [PubMed] [Google Scholar] Jacobs D, Huang H, Olino K, Weiss S, Kluger H, Judson BL, Zhang Y. 2021. Assessment of age, period, and birth cohort effects and trends in Merkel cell carcinoma incidence in the United States. JAMA Dermatol 157:59. doi: 10.1001/jamadermatol.2020.4102 [DOI] [PMC free article] [PubMed] [Google Scholar] Stang A, Becker JC, Nghiem P, Ferlay J. 2018. The association between geographic location and incidence of Merkel cell carcinoma in comparison to melanoma: an international assessment. Eur J Cancer 94:47–60. doi: 10.1016/j.ejca.2018.02.003 [DOI] [PMC free article] [PubMed] [Google Scholar] Wieland U, Mauch C, Kreuter A, Krieg T, Pfister H. 2009. Merkel cell polyomavirus DNA in persons without Merkel cell carcinoma. Emerg Infect Dis 15:1496–1498. doi: 10.3201/eid1509.081575 [DOI] [PMC free article] [PubMed] [Google Scholar] Foulongne V, Dereure O, Kluger N, Molès JP, Guillot B, Segondy M. 2010. Merkel cell polyomavirus DNA detection in lesional and nonlesional skin from patients with Merkel cell carcinoma or other skin diseases. Br J Dermatol 162:59–63. doi: 10.1111/j.1365-2133.2009.09381.x [DOI] [PubMed] [Google Scholar] Schowalter RM, Pastrana DV, Pumphrey KA, Moyer AL, Buck CB. 2010. Merkel cell polyomavirus and two previously unknown polyomaviruses are chronically shed from human skin. Cell Host Microbe 7:509–515. doi: 10.1016/j.chom.2010.05.006 [DOI] [PMC free article] [PubMed] [Google Scholar] Bopp L, Wieland U, Hellmich M, Kreuter A, Pfister H, Silling S. 2021. Natural history of cutaneous human polyomavirus infection in healthy individuals. Front Microbiol 12:740947. doi: 10.3389/fmicb.2021.740947 [DOI] [PMC free article] [PubMed] [Google Scholar] Saláková M, Košlabová E, Vojtěchová Z, Tachezy R, Šroller V. 2016. Detection of human polyomaviruses MCPyV, HPyV6, and HPyV7 in malignant and non-malignant tonsillar tissues. J Med Virol 88:695–702. doi: 10.1002/jmv.24385 [DOI] [PubMed] [Google Scholar] Nicol JTJ, Robinot R, Carpentier A, Carandina G, Mazzoni E, Tognon M, Touzé A, Coursaget P. 2013. Age-specific seroprevalences of merkel cell polyomavirus, human polyomaviruses 6, 7, and 9, and trichodysplasia spinulosa-associated polyomavirus. Clin Vaccine Immunol 20:363–368. doi: 10.1128/CVI.00438-12 [DOI] [PMC free article] [PubMed] [Google Scholar] Zhang C, Liu F, He Z, Deng Q, Pan Y, Liu Y, Zhang C, Ning T, Guo C, Liang Y, Xu R, Zhang L, Cai H, Ke Y. 2014. Seroprevalence of Merkel cell polyomavirus in the general rural population of Anyang, China. PLoS One 9:e106430. doi: 10.1371/journal.pone.0106430 [DOI] [PMC free article] [PubMed] [Google Scholar] Šroller V, Hamšíková E, Ludvíková V, Vochozková P, Kojzarová M, Fraiberk M, Saláková M, Morávková A, Forstová J, Němečková Š. 2014. Seroprevalence rates of BKV, JCV, and MCPyV polyomaviruses in the general Czech Republic population. J Med Virol 86:1560–1568. doi: 10.1002/jmv.23841 [DOI] [PubMed] [Google Scholar] Carter JJ, Daugherty MD, Qi X, Bheda-Malge A, Wipf GC, Robinson K, Roman A, Malik HS, Galloway DA. 2013. Identification of an overprinting gene in Merkel cell polyomavirus provides evolutionary insight into the birth of viral genes. Proc Natl Acad Sci USA 110:12744–12749. doi: 10.1073/pnas.1303526110 [DOI] [PMC free article] [PubMed] [Google Scholar] Seo GJ, Chen CJ, Sullivan CS. 2009. Merkel cell polyomavirus encodes a microRNA with the ability to autoregulate viral gene expression. Virology (Auckl) 383:183–187. doi: 10.1016/j.virol.2008.11.001 [DOI] [PubMed] [Google Scholar] Schowalter RM, Buck CB. 2013. The Merkel cell polyomavirus minor capsid protein. PLoS Pathog 9:e1003558. doi: 10.1371/journal.ppat.1003558 [DOI] [PMC free article] [PubMed] [Google Scholar] Bayer NJ, Januliene D, Zocher G, Stehle T, Moeller A, Blaum BS. 2020. Structure of Merkel cell polyomavirus capsid and interaction with its glycosaminoglycan attachment receptor. J Virol 94:e01664-19. doi: 10.1128/JVI.01664-19 [DOI] [PMC free article] [PubMed] [Google Scholar] Schowalter RM, Pastrana DV, Buck CB. 2011. Glycosaminoglycans and sialylated glycans sequentially facilitate Merkel cell polyomavirus infectious entry. PLoS Pathog 7:e1002161. doi: 10.1371/journal.ppat.1002161 [DOI] [PMC free article] [PubMed] [Google Scholar] Neu U, Hengel H, Blaum BS, Schowalter RM, Macejak D, Gilbert M, Wakarchuk WW, Imamura A, Ando H, Kiso M, Arnberg N, Garcea RL, Peters T, Buck CB, Stehle T. 2012. Structures of Merkel cell polyomavirus VP1 complexes define a sialic acid binding site required for infection. PLoS Pathog 8:e1002738. doi: 10.1371/journal.ppat.1002738 [DOI] [PMC free article] [PubMed] [Google Scholar] Becker M, Dominguez M, Greune L, Soria-Martinez L, Pfleiderer MM, Schowalter R, Buck CB, Blaum BS, Schmidt MA, Schelhaas M. 2019. Infectious entry of Merkel cell polyomavirus. J Virol 93:e02004-18. doi: 10.1128/JVI.02004-18 [DOI] [PMC free article] [PubMed] [Google Scholar] Liu W, Yang R, Payne AS, Schowalter RM, Spurgeon ME, Lambert PF, Xu X, Buck CB, You J. 2016. Identifying the target cells and mechanisms of Merkel cell polyomavirus infection. Cell Host Microbe 19:775–787. doi: 10.1016/j.chom.2016.04.024 [DOI] [PMC free article] [PubMed] [Google Scholar] Liu W, Krump NA, MacDonald M, You J. 2018. Merkel cell polyomavirus infection of animal dermal fibroblasts. J Virol 92:e01610-17. doi: 10.1128/JVI.01610-17 [DOI] [PMC free article] [PubMed] [Google Scholar] Liu W, Krump NA, Buck CB, You J. 2019. Merkel cell polyomavirus infection and detection. J Vis Exp. doi: 10.3791/58950 [DOI] [PMC free article] [PubMed] [Google Scholar] Shuda M, Feng H, Kwun HJ, Rosen ST, Gjoerup O, Moore PS, Chang Y. 2008. T antigen mutations are a human tumor-specific signature for Merkel cell polyomavirus. Proc Natl Acad Sci USA 105:16272–16277. doi: 10.1073/pnas.0806526105 [DOI] [PMC free article] [PubMed] [Google Scholar] Passerini S, Prezioso C, Babini G, Ferlosio A, Cosio T, Campione E, Moens U, Ciotti M, Pietropaolo V. 2023. Detection of Merkel Cell Polyomavirus (MCPyV) DNA and transcripts in Merkel Cell Carcinoma (MCC). Pathogens 12:894. doi: 10.3390/pathogens12070894 [DOI] [PMC free article] [PubMed] [Google Scholar] Sihto H, Kukko H, Koljonen V, Sankila R, Böhling T, Joensuu H. 2011. Merkel cell polyomavirus infection, large T antigen, retinoblastoma protein and outcome in Merkel cell carcinoma. Clin Cancer Res 17:4806–4813. doi: 10.1158/1078-0432.CCR-10-3363 [DOI] [PubMed] [Google Scholar] Verhaegen ME, Mangelberger D, Harms PW, Vozheiko TD, Weick JW, Wilbert DM, Saunders TL, Ermilov AN, Bichakjian CK, Johnson TM, Imperiale MJ, Dlugosz AA. 2015. Merkel cell polyomavirus small T antigen is oncogenic in transgenic mice. J Invest Dermatol 135:1415–1424. doi: 10.1038/jid.2014.446 [DOI] [PMC free article] [PubMed] [Google Scholar] Dye KN, Welcker M, Clurman BE, Roman A, Galloway DA. 2019. Merkel cell polyomavirus Tumor antigens expressed in Merkel cell carcinoma function independently of the ubiquitin ligases Fbw7 and β-TrCP. PLoS Pathog 15:e1007543. doi: 10.1371/journal.ppat.1007543 [DOI] [PMC free article] [PubMed] [Google Scholar] Sergi MC, Lauricella E, Porta C, Tucci M, Cives M. 2023. An update on Merkel cell carcinoma. Biochim Biophys Acta Rev Cancer 1878:188880. doi: 10.1016/j.bbcan.2023.188880 [DOI] [PubMed] [Google Scholar] Martel-Jantin C, Filippone C, Cassar O, Peter M, Tomasic G, Vielh P, Brière J, Petrella T, Aubriot-Lorton MH, Mortier L, Jouvion G, Sastre-Garau X, Robert C, Gessain A. 2012. Genetic variability and integration of Merkel cell polyomavirus in Merkel cell carcinoma. Virology (Auckl) 426:134–142. doi: 10.1016/j.virol.2012.01.018 [DOI] [PubMed] [Google Scholar] Starrett GJ, Thakuria M, Chen T, Marcelus C, Cheng J, Nomburg J, Thorner AR, Slevin MK, Powers W, Burns RT, Perry C, Piris A, Kuo FC, Rabinowits G, Giobbie-Hurder A, MacConaill LE, DeCaprio JA. 2020. Clinical and molecular characterization of virus-positive and virus-negative Merkel cell carcinoma. Genome Med 12:30. doi: 10.1186/s13073-020-00727-4 [DOI] [PMC free article] [PubMed] [Google Scholar] Czech-Sioli M, Günther T, Therre M, Spohn M, Indenbirken D, Theiss J, Riethdorf S, Qi M, Alawi M, Wülbeck C, Fernandez-Cuesta I, Esmek F, Becker JC, Grundhoff A, Fischer N. 2020. High-resolution analysis of Merkel cell polyomavirus in Merkel cell carcinoma reveals distinct integration patterns and suggests NHEJ and MMBIR as underlying mechanisms. PLoS Pathog 16:e1008562. doi: 10.1371/journal.ppat.1008562 [DOI] [PMC free article] [PubMed] [Google Scholar] Starrett GJ, Marcelus C, Cantalupo PG, Katz JP, Cheng J, Akagi K, Thakuria M, Rabinowits G, Wang LC, Symer DE, Pipas JM, Harris RS, DeCaprio JA. 2017. Merkel cell polyomavirus exhibits dominant control of the tumor genome and transcriptome in virus-associated merkel cell carcinoma. mBio 8:e02079-16. doi: 10.1128/mBio.02079-16 [DOI] [PMC free article] [PubMed] [Google Scholar] Laude HC, Jonchère B, Maubec E, Carlotti A, Marinho E, Couturaud B, Peter M, Sastre-Garau X, Avril M-F, Dupin N, Rozenberg F. 2010. Distinct merkel cell polyomavirus molecular features in tumour and non tumour specimens from patients with merkel cell carcinoma. PLoS Pathog 6:e1001076. doi: 10.1371/journal.ppat.1001076 [DOI] [PMC free article] [PubMed] [Google Scholar] Nirenberg A, Steinman H, Dixon J, Dixon A. 2020. Merkel cell carcinoma update: the case for two tumours. J Eur Acad Dermatol Venereol 34:1425–1431. doi: 10.1111/jdv.16158 [DOI] [PubMed] [Google Scholar] Sundqvist BZ, Kilpinen SK, Böhling TO, Koljonen VSK, Sihto HJ. 2023. Transcriptomic analyses reveal three distinct molecular subgroups of Merkel cell carcinoma with differing prognoses. Int J Cancer 152:2099–2108. doi: 10.1002/ijc.34425 [DOI] [PubMed] [Google Scholar] Bill CA, Summers J. 2004. Genomic DNA double-strand breaks are targets for hepadnaviral DNA integration. Proc Natl Acad Sci USA 101:11135–11140. doi: 10.1073/pnas.0403925101 [DOI] [PMC free article] [PubMed] [Google Scholar] Yang L, Ye S, Zhao X, Ji L, Zhang Y, Zhou P, Sun J, Guan Y, Han Y, Ni C, Hu X, Liu W, Wang H, Zhou B, Huang J. 2018. Molecular characterization of HBV DNA integration in patients with hepatitis and hepatocellular carcinoma. J Cancer 9:3225–3235. doi: 10.7150/jca.26052 [DOI] [PMC free article] [PubMed] [Google Scholar] Li X, Zhang J, Yang Z, Kang J, Jiang S, Zhang T, Chen T, Li M, Lv Q, Chen X, McCrae MA, Zhuang H, Lu F. 2014. The function of targeted host genes determines the oncogenicity of HBV integration in hepatocellular carcinoma. J Hepatol 60:975–984. doi: 10.1016/j.jhep.2013.12.014 [DOI] [PubMed] [Google Scholar] Tian R, Wang Y, Li W, Cui Z, Pan T, Jin Z, Huang Z, Li L, Lang B, Wu J, Xie H, Lu Y, Tian X, Hu Z. 2022. Genome-wide virus-integration analysis reveals a common insertional mechanism of HPV, HBV and EBV. Clin Transl Med 12:e971. doi: 10.1002/ctm2.971 [DOI] [PMC free article] [PubMed] [Google Scholar] Groves IJ, Coleman N. 2018. Human papillomavirus genome integration in squamous carcinogenesis: what have next-generation sequencing studies taught us? J Pathol 245:9–18. doi: 10.1002/path.5058 [DOI] [PubMed] [Google Scholar] Porter VL, Marra MA. 2022. The drivers, mechanisms, and consequences of genome instability in HPV-driven cancers. Cancers (Basel) 14:4623. doi: 10.3390/cancers14194623 [DOI] [PMC free article] [PubMed] [Google Scholar] Akagi K, Symer DE, Mahmoud M, Jiang B, Goodwin S, Wangsa D, Li Z, Xiao W, Dunn JD, Ried T, Coombes KR, Sedlazeck FJ, Gillison ML. 2023. Intratumoral heterogeneity and clonal evolution induced by HPV integration. Cancer Discov 13:910–927. doi: 10.1158/2159-8290.CD-22-0900 [DOI] [PMC free article] [PubMed] [Google Scholar] Ceccaldi R, Liu JC, Amunugama R, Hajdu I, Primack B, Petalcorin MIR, O’Connor KW, Konstantinopoulos PA, Elledge SJ, Boulton SJ, Yusufzai T, D’Andrea AD. 2015. Homologous-recombination-deficient tumours are dependent on Polθ-mediated repair. Nature 518:258–262. doi: 10.1038/nature14184 [DOI] [PMC free article] [PubMed] [Google Scholar] Leeman JE, Li Y, Bell A, Hussain SS, Majumdar R, Rong-Mullins X, Blecua P, Damerla R, Narang H, Ravindran PT, Lee NY, Riaz N, Powell SN, Higginson DS. 2019. Human papillomavirus 16 promotes microhomology-mediated end-joining. Proc Natl Acad Sci USA 116:21573–21579. doi: 10.1073/pnas.1906120116 [DOI] [PMC free article] [PubMed] [Google Scholar] Chakraborty PR, Ruiz-Opazo N, Shouval D, Shafritz DA. 1980. Identification of integrated hepatitis B virus DNA and expression of viral RNA in an HBsAg-producing human hepatocellular carcinoma cell line. Nature 286:531–533. doi: 10.1038/286531a0 [DOI] [PubMed] [Google Scholar] Lace MJ, Anson JR, Klussmann JP, Wang DH, Smith EM, Haugen TH, Turek LP. 2011. Human papillomavirus type 16 (HPV-16) genomes integrated in head and neck cancers and in HPV-16-immortalized human keratinocyte clones express chimeric virus-cell mRNAs similar to those found in cervical cancers. J Virol 85:1645–1654. doi: 10.1128/JVI.02093-10 [DOI] [PMC free article] [PubMed] [Google Scholar] Gardella T, Medveczky P, Sairenji T, Mulder C. 1984. Detection of circular and linear herpesvirus DNA molecules in mammalian cells by gel electrophoresis. J Virol 50:248–254. doi: 10.1128/jvi.50.1.248-254.1984 [DOI] [PMC free article] [PubMed] [Google Scholar] Chang Y, Cheng S-D, Tsai C-H. 2002. Chromosomal integration of Epstein - Barr virus genomes in nasopharyngeal carcinoma cells. Head Neck 24:143–150. doi: 10.1002/hed.10039 [DOI] [PubMed] [Google Scholar] Whitehouse A. 2011. Gardella gel analysis to detect Herpesvirus saimiri episomal DNA. Cold Spring Harb Protoc 2011:1524–1526. doi: 10.1101/pdb.prot066969 [DOI] [PubMed] [Google Scholar] Begum S, Cao D, Gillison M, Zahurak M, Westra WH. 2005. Tissue distribution of human papillomavirus 16 DNA integration in patients with tonsillar carcinoma. Clin Cancer Res 11:5694–5699. doi: 10.1158/1078-0432.CCR-05-0587 [DOI] [PubMed] [Google Scholar] Brooks EG, Evans MF, Adamson C-C, Peng Z, Rajendran V, Laucirica R, Cooper K. 2012. In situ hybridization signal patterns in recurrent laryngeal squamous papillomas indicate that HPV integration occurs at an early stage. Head Neck Pathol 6:32–37. doi: 10.1007/s12105-011-0308-5 [DOI] [PMC free article] [PubMed] [Google Scholar] Xiong J, Cheng J, Shen H, Ren C, Wang L, Gao C, Zhu T, Li X, Ding W, Zhu D, Wang H. 2021. Detection of HPV and human chromosome sites by dual-color fluorescence in situ hybridization reveals recurrent HPV integration sites and heterogeneity in cervical cancer. Front Oncol 11:734758. doi: 10.3389/fonc.2021.734758 [DOI] [PMC free article] [PubMed] [Google Scholar] Tokino T, Matsubara K. 1991. Chromosomal sites for hepatitis B virus integration in human hepatocellular carcinoma. J Virol 65:6761–6764. doi: 10.1128/JVI.65.12.6761-6764.1991 [DOI] [PMC free article] [PubMed] [Google Scholar] Redmond CJ, Fu H, Aladjem MI, McBride AA. 2018. Human papillomavirus integration: analysis by molecular combing and fiber-FISH. Curr Protoc Microbiol 51:e61. doi: 10.1002/cpmc.61 [DOI] [PMC free article] [PubMed] [Google Scholar] Takada S, Gotoh Y, Hayashi S, Yoshida M, Koike K. 1990. Structural rearrangement of integrated hepatitis B virus DNA as well as cellular flanking DNA is present in chronically infected hepatic tissues. J Virol 64:822–828. doi: 10.1128/JVI.64.2.822-828.1990 [DOI] [PMC free article] [PubMed] [Google Scholar] Minami M, Poussin K, Bréchot C, Paterlini P. 1995. A novel PCR technique UsingAlu-specific primers to identify unknown flanking sequences from the human genome. Genomics 29:403–408. doi: 10.1006/geno.1995.9004 [DOI] [PubMed] [Google Scholar] Tu T, Jilbert AR. 2017. Detection of hepatocyte clones containing integrated hepatitis B virus DNA using inverse nested PCR. Methods Mol Biol 1540:97–118. doi: 10.1007/978-1-4939-6700-1_9 [DOI] [PubMed] [Google Scholar] Luft F, Klaes R, Nees M, Dürst M, Heilmann V, Melsheimer P, von Knebel Doeberitz M. 2001. Detection of integrated papillomavirus sequences by ligation-mediated PCR (DIPS-PCR) and molecular characterization in cervical cancer cells. Int J Cancer 92:9–17. doi: [DOI] [PubMed] [Google Scholar] Matovina M, Sabol I, Grubišić G, Gašperov NM, Grce M. 2009. Identification of human papillomavirus type 16 integration sites in high-grade precancerous cervical lesions. Gynecol Oncol 113:120–127. doi: 10.1016/j.ygyno.2008.12.004 [DOI] [PubMed] [Google Scholar] Schrama D, Sarosi E-M, Adam C, Ritter C, Kaemmerer U, Klopocki E, König E-M, Utikal J, Becker JC, Houben R. 2019. Characterization of six Merkel cell polyomavirus-positive Merkel cell carcinoma cell lines: integration pattern suggest that large T antigen truncating events occur before or during integration. Int J Cancer 145:1020–1032. doi: 10.1002/ijc.32280 [DOI] [PubMed] [Google Scholar] Klaes R, Woerner SM, Ridder R, Wentzensen N, Duerst M, Schneider A, Lotz B, Melsheimer P, von Knebel Doeberitz M. 1999. Detection of high-risk cervical intraepithelial neoplasia and cervical cancer by amplification of transcripts derived from integrated papillomavirus oncogenes. Cancer Res 59:6132–6136. [PubMed] [Google Scholar] Ziegert C, Wentzensen N, Vinokurova S, Kisseljov F, Einenkel J, Hoeckel M, von Knebel Doeberitz M. 2003. A comprehensive analysis of HPV integration loci in anogenital lesions combining transcript and genome-based amplification techniques. Oncogene 22:3977–3984. doi: 10.1038/sj.onc.1206629 [DOI] [PubMed] [Google Scholar] Kim S-H, Koo B-S, Kang S, Park K, Kim H, Lee KR, Lee MJ, Kim JM, Choi EC, Cho NH. 2007. HPV integration begins in the tonsillar crypt and leads to the alteration of p16, EGFR and c-myc during tumor formation. Int J Cancer 120:1418–1425. doi: 10.1002/ijc.22464 [DOI] [PubMed] [Google Scholar] Cricca M, Morselli-Labate AM, Venturoli S, Ambretti S, Gentilomi GA, Gallinella G, Costa S, Musiani M, Zerbini M. 2007. Viral DNA load, physical status and E2/E6 ratio as markers to grade HPV16 positive women for high-grade cervical lesions. Gynecol Oncol 106:549–557. doi: 10.1016/j.ygyno.2007.05.004 [DOI] [PubMed] [Google Scholar] Deng Z, Hasegawa M, Kiyuna A, Matayoshi S, Uehara T, Agena S, Yamashita Y, Ogawa K, Maeda H, Suzuki M. 2013. Viral load, physical status, and E6/E7 mRNA expression of human papillomavirus in head and neck squamous cell carcinoma. Head Neck 35:800–808. doi: 10.1002/hed.23034 [DOI] [PubMed] [Google Scholar] Wang Q, Jia P, Zhao Z. 2013. VirusFinder: software for efficient and accurate detection of viruses and their integration sites in host genomes through next generation sequencing data. PLoS One 8:e64465. doi: 10.1371/journal.pone.0064465 [DOI] [PMC free article] [PubMed] [Google Scholar] Fujimoto A, Furuta M, Totoki Y, Tsunoda T, Kato M, Shiraishi Y, Tanaka H, Taniguchi H, Kawakami Y, Ueno M, et al. 2016. Whole-genome mutational landscape and characterization of noncoding and structural mutations in liver cancer. Nat Genet 48:500–509. doi: 10.1038/ng.3547 [DOI] [PubMed] [Google Scholar] Wang A, Wu L, Lin J, Han L, Bian J, Wu Y, Robson SC, Xue L, Ge Y, Sang X, Wang W, Zhao H. 2018. Whole-exome sequencing reveals the origin and evolution of hepato-cholangiocarcinoma. Nat Commun 9:894. doi: 10.1038/s41467-018-03276-y [DOI] [PMC free article] [PubMed] [Google Scholar] Svicher V, Salpini R, Piermatteo L, Carioti L, Battisti A, Colagrossi L, Scutari R, Surdo M, Cacciafesta V, Nuccitelli A, Hansi N, Ceccherini Silberstein F, Perno CF, Gill US, Kennedy PTF. 2021. Whole exome HBV DNA integration is independent of the intrahepatic HBV reservoir in HBeAg-negative chronic hepatitis B. Gut 70:2337–2348. doi: 10.1136/gutjnl-2020-323300 [DOI] [PMC free article] [PubMed] [Google Scholar] Tuna M, Amos CI. 2017. Next generation sequencing and its applications in HPV-associated cancers. Oncotarget 8:8877–8889. doi: 10.18632/oncotarget.12830 [DOI] [PMC free article] [PubMed] [Google Scholar] Fukano K, Wakae K, Nao N, Saito M, Tsubota A, Toyoshima T, Aizaki H, Iijima H, Matsudaira T, Kimura M, Watashi K, Sugiura W, Muramatsu M. 2023. A versatile method to profile hepatitis B virus DNA integration. Hepatol Commun 7:e0328. doi: 10.1097/HC9.0000000000000328 [DOI] [PMC free article] [PubMed] [Google Scholar] Yang W, Liu Y, Dong R, Liu J, Lang J, Yang J, Wang W, Li J, Meng B, Tian G. 2020. Accurate detection of HPV integration sites in cervical cancer samples using the nanopore MinION sequencer without error correction. Front Genet 11:660. doi: 10.3389/fgene.2020.00660 [DOI] [PMC free article] [PubMed] [Google Scholar] Andersen K, Holm K, Tranberg M, Pedersen CL, Bønløkke S, Steiniche T, Andersen B, Stougaard M. 2022. Targeted next generation sequencing for human papillomavirus genotyping in cervical liquid-based cytology samples. Cancers (Basel) 14:652. doi: 10.3390/cancers14030652 [DOI] [PMC free article] [PubMed] [Google Scholar] Liang J, Cui Z, Wu C, Yu Y, Tian R, Xie H, Jin Z, Fan W, Xie W, Huang Z, Xu W, Zhu J, You Z, Guo X, Qiu X, Ye J, Lang B, Li M, Tan S, Hu Z. 2021. DeepEBV: a deep learning model to predict Epstein-Barr virus (EBV) integration sites. Bioinformatics 37:3405–3411. doi: 10.1093/bioinformatics/btab388 [DOI] [PubMed] [Google Scholar] Valmary-Degano S, Jacquin E, Prétet J-L, Monnien F, Girardo B, Arbez-Gindre F, Joly M, Bosset J-F, Kantelip B, Mougin C. 2013. Signature patterns of human papillomavirus type 16 in invasive anal carcinoma. Hum Pathol 44:992–1002. doi: 10.1016/j.humpath.2012.08.019 [DOI] [PubMed] [Google Scholar] Lang B, Dong D, Zhao T, Zhong R, Qin H, Cao C, Wang Y, Liu T, Liang W, Tian X, Yan Y, Hu Z. 2023. A cross-sectional study of human papillomavirus genotype distribution and integration status in penile cancer among Chinese population. Virology (Auckl) 584:53–57. doi: 10.1016/j.virol.2023.04.013 [DOI] [PubMed] [Google Scholar] Rodig SJ, Cheng J, Wardzala J, DoRosario A, Scanlon JJ, Laga AC, Martinez-Fernandez A, Barletta JA, Bellizzi AM, Sadasivam S, Holloway DT, Cooper DJ, Kupper TS, Wang LC, DeCaprio JA. 2012. Improved detection suggests all Merkel cell carcinomas harbor Merkel polyomavirus. J Clin Invest 122:4645–4653. doi: 10.1172/JCI64116 [DOI] [PMC free article] [PubMed] [Google Scholar] Lim MY, Dahlstrom KR, Sturgis EM, Li G. 2016. Human papillomavirus integration pattern and demographic, clinical, and survival characteristics of patients with oropharyngeal squamous cell carcinoma. Head Neck 38:1139–1144. doi: 10.1002/hed.24429 [DOI] [PubMed] [Google Scholar] Articles from Journal of Virology are provided here courtesy of American Society for Microbiology (ASM) ACTIONS View on publisher site PDF (3.4 MB) Cite Collections Permalink PERMALINK Copy RESOURCES Similar articles Cited by other articles Links to NCBI Databases On this page ABSTRACT INTRODUCTION HEPATITIS B VIRUS EPSTEIN-BARR VIRUS HUMAN PAPILLOMAVIRUS MERKEL CELL POLYOMAVIRUS MECHANISMS OF DNA TUMOR VIRUS INTEGRATION METHODS USED FOR VIRAL INTEGRATION ANALYSIS CONCLUDING REMARKS ACKNOWLEDGMENTS Biographies Contributor Information REFERENCES Cite Copy Download .nbib.nbib Format: Add to Collections Create a new collection Add to an existing collection Name your collection Choose a collection Unable to load your collection due to an error Please try again Add Cancel Follow NCBI NCBI on X (formerly known as Twitter)NCBI on FacebookNCBI on LinkedInNCBI on GitHubNCBI RSS feed Connect with NLM NLM on X (formerly known as Twitter)NLM on FacebookNLM on YouTube National Library of Medicine 8600 Rockville Pike Bethesda, MD 20894 Web Policies FOIA HHS Vulnerability Disclosure Help Accessibility Careers NLM NIH HHS USA.gov Back to Top
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https://www.wyzant.com/resources/answers/878913/solve-the-equation-on-the-interval-0-2-no-calculator
WYZANT TUTORING Mike O. Solve the equation on the interval [0,2𝜋). NO CALCULATOR. Solve the equation on the interval [0,2𝜋). 2cos(^2)𝑥 + 3cos𝑥+1 = 0 3 Answers By Expert Tutors Jacob K. answered • 12/14/21 McGill Grad for Nighttime Math Tutoring and Emergency Help 2cos2(x) + 3cos(x) + 1 = 0. Let's think of another way to look at this. This looks very similar to a quadratic function, of the form ax2+bx+c=0. If we let a dummy variable, say u = cos(x), then we can rewrite this as 2u2+3u+1=0 This becomes a quadratic that we know how to solve, using the quadratic formula a=2, b=3, c=1, -3±√(3)^2-4(2)(1) / 2(2) = -3±√(9-8)/4 = -3±√(1)/4 = (-3±1)/4 u1=(-3+1)/4=-2/4=-1/2 u2=(-3-1)/4=(-4)/4=-1 u1=-1/2 u2=-1 So now that we have our solutions of this, we have to solve them in the context of [0, 2π). First, u1: Solve cos(x)=-1/2, 0≤x<2π Take the arccos to get x=arccos(-1/2)=2π/3 and 4π/3. We do not need to include additional periods here, as we within our domain here and any multiple of 2π will take us out. Now, u2: Solve cos(x)=-1, 0≤x<2π x=arccos(-1)=π. This is the only solution which stays within our bounds. So, finally, we just now list all of our solutions, which are x=2π/3, x=π, x=4π/3. I hope this is clear and able to help you out! Good luck! Osman A. answered • 12/19/21 Professor of Engineering Mathematics – Trigonometry and Geometry Solve the equation on the interval [0, 2𝜋). NO CALCULATOR. 2 cos2 𝑥 + 3 cos 𝑥 + 1 = 0 Detailed Solution: 2 cos2 𝑥 + 3 cos 𝑥 + 1 = 0 [0, 2𝜋) 2 cos2 𝑥 + 3 cos 𝑥 + 1 = 0 ==> (cos 𝑥 + 1) (2 cos 𝑥 + 1) = 0 ==> cos 𝑥 + 1 = 0 and 2 cos 𝑥 + 1 = 0 cos 𝑥 + 1 = 0 ==> cos 𝑥 = −1 ==> 𝑥 = cos-1 (−1) ==> 𝑥 = 𝜋 2 cos 𝑥 + 1 = 0 ==> 2 cos 𝑥 = −1 ==> cos 𝑥 = −1/2 (Q 2 & Q3) Quadrant 2: cos 𝑥 = −1/2 ==> 𝑥 = cos-1 (−1/2) ==> 𝑥 = 𝜋 − 𝜋/3 ==> 𝑥 = 2𝜋/3 Quadrant 3: cos 𝑥 = −1/2 ==> 𝑥 = cos-1 (−1/2) ==> 𝑥 = 𝜋 + 𝜋/3 ==> 𝑥 = 4𝜋/3 Therefore: x = {2𝜋/3, 𝜋, 4𝜋/3} Mark M. answered • 12/14/21 Mathematics Teacher - NCLB Highly Qualified Factor using Algebraic techniques. Use zero-product rule Still looking for help? Get the right answer, fast. Get a free answer to a quick problem. Most questions answered within 4 hours. OR Choose an expert and meet online. No packages or subscriptions, pay only for the time you need. RELATED TOPICS RELATED QUESTIONS tan^2x-sin^2x=tan^2xsin^2x Answers · 5 I need help trying to sole tan^2 x =1 where x is more than or equal to 0 but x is less than or equal to pi Answers · 5 find all solutions to the equation in (0, 2pi) sin(6x)+sin(2x)=0 Answers · 4 (2sin18)(cos18) Answers · 5 how do i go about solving a trig identity; that simplify s 1-sin^2 theta/1-cos theta Answers · 2 RECOMMENDED TUTORS Priti S. Kubrat D. Nicholas P. find an online tutor Download our free app A link to the app was sent to your phone. Get to know us Learn with us Work with us Download our free app Let’s keep in touch Need more help? Learn more about how it works Tutors by Subject Tutors by Location IXL Comprehensive K-12 personalized learning Rosetta Stone Immersive learning for 25 languages Education.com 35,000 worksheets, games, and lesson plans TPT Marketplace for millions of educator-created resources Vocabulary.com Adaptive learning for English vocabulary ABCya Fun educational games for kids SpanishDictionary.com Spanish-English dictionary, translator, and learning Inglés.com Diccionario inglés-español, traductor y sitio de aprendizaje Emmersion Fast and accurate language certification
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https://www.youtube.com/watch?v=tb9jwxr0HyU
Art of Problem Solving: Complementary Counting Part 1 Art of Problem Solving 103000 subscribers 142 likes Description 22350 views Posted: 21 Dec 2011 Art of Problem Solving's Richard Rusczyk explains how to count what you want by counting what you don't want. This video is part of our AoPS Counting & Probability curriculum. Take your math skills to the next level with our Counting & Probability materials: 📚 AoPS Introduction to Counting & Probability Textbook: 🖥️ AoPS Introduction to Counting & Probability Course: 🔔 Subscribe to our channel for more engaging math videos and updates Transcript:
5260
https://oeis.org/A098148
A098148 - OEIS login The OEIS is supported by the many generous donors to the OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!) A098148 Number of real (0,1) n X n matrices such that some eigenvalues are strictly complex. 2 0, 0, 52, 22196, 21005094 (list; graph; refs; listen; history; text; internal format) OFFSET 1,3 LINKS Table of n, a(n) for n=1..5. Index entries for sequences related to binary matrices EXAMPLE The 3 X 3 matrix ((0,1,0),(0,0,1),(1,1,1)) has real eigenvalue 1.83929 and the complex pair -0.41964+-0.60629i. There are 12 (0,1) 3 X 3 matrices with these eigenvalues. There are 6 groups of 6 matrices having eigenvalues (1.3472,-0.66236+-0.56228i), (1.46557,-0.23279+-0.79255i),..., (2.32472,0.33764+-0.56228i). Two matrices (e.g. ((0,0,1),(1,0,0),(0,1,0)) ) have eigenvalues (1,-0.5+-0.5sqrt(3)i). Two matrices (e.g. ((1,1,0),(0,1,1),(1,0,1)) ) have eigenvalues (2,0.5+-0.5sqrt(3)i). Total: 12+66+2+2=52=a(3). MATHEMATICA a[n_] := Module[{M, iter, cnt=0}, M = Table[a[i, j], {i, 1, n}, {j, 1, n}]; iter = Thread[{Flatten[M], 0, 1}]; Do[If[AnyTrue[Eigenvalues[M], Im[#] != 0&], cnt++], Evaluate[Sequence @@ iter]]; cnt]; Do[Print[n, " ", a[n]], {n, 1, 4}] ( Jean-François Alcover, Dec 09 2018 ) CROSSREFS Cf. other counts for (0, 1) matrices: A003024 (positive eigenvalues), A055165 (nonsingular), A085656 (positive definite), A086510 (nonnegative eigenvalues). Sequence in context: A346934A200806A208305 A068255A230532A157280 Adjacent sequences: A098145A098146A098147 A098149A098150A098151 KEYWORD more,nonn AUTHOR Hugo Pfoertner, Sep 07 2004 EXTENSIONS a(5) corrected by Hugo Pfoertner, Sep 26 2017 STATUS approved LookupWelcomeWikiRegisterMusicPlot 2DemosIndexWebCamContributeFormatStyle SheetTransformsSuperseekerRecents The OEIS Community Maintained by The OEIS Foundation Inc. Last modified September 29 04:39 EDT 2025. Contains 388827 sequences. License Agreements, Terms of Use, Privacy Policy
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https://www.slideshare.net/slideshow/molecular-cell-biology-lodish-6thppt-chapter-10-biomembrane-structure-29705817/29705817
Molecular Cell Biology Lodish 6th.ppt - Chapter 10 biomembrane structure | PPT Download free for 30 days Sign in UploadLanguage (EN)Support BusinessMobileSocial MediaMarketingTechnologyArt & PhotosCareerDesignEducationPresentations & Public SpeakingGovernment & NonprofitHealthcareInternetLawLeadership & ManagementAutomotiveEngineeringSoftwareRecruiting & HRRetailSalesServicesScienceSmall Business & EntrepreneurshipFoodEnvironmentEconomy & FinanceData & AnalyticsInvestor RelationsSportsSpiritualNews & PoliticsTravelSelf ImprovementReal EstateEntertainment & HumorHealth & MedicineDevices & HardwareLifestyle Change Language Language English Español Português Français Deutsche Cancel Save Submit search EN Uploaded byNattawut Huayyai PPT, PDF 4,602 views Molecular Cell Biology Lodish 6th.ppt - Chapter 10 biomembrane structure AI-enhanced description This chapter discusses the structure of biomembranes. Biomembranes are made of a lipid bilayer with embedded and associated proteins. The lipid bilayer forms a barrier that separates the interior of the cell from the outside environment. Proteins perform important functions like transporting molecules across the membrane and acting as receptors to receive signals. Education◦Technology◦ Related topics: Cell Biology Overview• Read more 9 Save Share Embed Download Downloaded 469 times 1 / 44 2 / 44 3 / 44 4 / 44 5 / 44 6 / 44 7 / 44 8 / 44 9 / 44 10 / 44 11 / 44 12 / 44 13 / 44 14 / 44 15 / 44 16 / 44 17 / 44 18 / 44 19 / 44 20 / 44 21 / 44 22 / 44 23 / 44 24 / 44 25 / 44 26 / 44 27 / 44 28 / 44 29 / 44 30 / 44 Most read 31 / 44 32 / 44 33 / 44 34 / 44 35 / 44 36 / 44 37 / 44 38 / 44 39 / 44 40 / 44 41 / 44 42 / 44 43 / 44 44 / 44 Image 46: Lodish • Berk • Kaiser • Krieger • Scott • Bretscher •Ploegh • Matsudaira MOLECULAR CELL BIOLOGY SIXTH EDITION CHAPTER 10 Biomembrane Structure © 2008 W. H. Freeman and Company Copyright 2008 © W. H. 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5262
https://www.cuemath.com/measurement/surface-area-of-sphere/
Surface Area of Sphere The surface area of a sphere is the area occupied by the curved surface of the sphere. Circular shapes take the shape of a sphere when observed as three-dimensional structures. For example, a globe or a soccer ball. Let us learn about the formula of surface area of a sphere and how to calculate the surface area of a sphere in this lesson. | | | --- | | 1. | What is the Surface Area of Sphere? | | 2. | Derivation of Surface Area of Sphere | | 3. | Formula of Surface Area of Sphere | | 4. | How to Calculate the Surface Area of Sphere? | | 5. | FAQs on Surface Area of Sphere | What is the Surface Area of a Sphere? The area covered by the outer surface of the sphere is known as the surface area of a sphere. A sphere is a three-dimensional form of a circle. The difference between a sphere and a circle is that a circle is a 2-dimensional shape (2D shape), whereas a sphere is a 3-dimensional shape. The surface area of a sphere is expressed in square units. Observe the sphere given below which shows the center, the radius, and the diameter of a sphere. Sphere Definition Sphere is a three-dimensional round-shaped object with no vertices or edges. The important aspects of this shape are radius, diameter, circumference, and volume. Derivation of Surface Area of Sphere A sphere is round in shape, therefore to find its surface area, we relate it to a curved shape, such as the cylinder. A cylinder is a shape that has a curved surface along with flat surfaces. Now, if the radius of a cylinder is the same as the radius of a sphere, it means that the sphere can fit into the cylinder perfectly. This means that the height of the cylinder is equal to the height of the sphere. So, this height can also be called as the diameter of the sphere. Therefore, this fact was proved by a great mathematician, Archimedes, that if the radius of a cylinder and sphere is 'r', the surface area of a sphere is equal to the lateral surface area of the cylinder. Hence, the relation between the surface area of a sphere and lateral surface area of a cylinder is given as: Surface Area of Sphere = Lateral Surface Area of Cylinder The lateral surface area of a cylinder = 2πrh, where 'r' is the radius and 'h' is the height of the cylinder. Now, the height of the cylinder can also be called the diameter of the sphere because we are assuming that this sphere is perfectly fit in the cylinder. Hence, it can be said that height of the cylinder = diameter of sphere = 2r. So, in the formula, surface area of Sphere = 2πrh; 'h' can be replaced by the diameter, that is, 2r. Hence, surface area of sphere is 2πrh = 2πr(2r) = 4πr2 Formula of Surface Area of Sphere The formula of the surface area of the sphere depends on the radius of the sphere. If the radius of the sphere is r and the surface area of the sphere is S. Then, the surface area of the sphere is expressed as: Surface Area of Sphere = 4πr2; where 'r' is the radius of the sphere. In terms of diameter, the surface area of a sphere is expressed as S = 4π(d/2)2 where d is the diameter of the sphere. How to Calculate the Surface Area of Sphere? The surface area of a sphere is the space occupied by its surface. The surface area of the sphere can be calculated using the formula of the surface area of the sphere. The steps to calculate the surface area of a sphere are given below. Let us take an example to learn how to calculate the surface area of a sphere using its formula. Example: Find the surface area of a spherical ball that has a radius of 9 inches. Curved Surface Area of Sphere The curved surface area of a sphere is the total surface area of the sphere because a sphere has just one surface which is curved. Since there is no flat surface in a sphere, the curved surface area of a sphere is equal to the total surface area of the sphere. Therefore, the formula for the curved surface area of a sphere is expressed as, Curved surface area of sphere = 4πr2; where 'r' is the radius of the sphere. ☛ Related Articles Surface Area of Sphere Examples Example 1: If the radius of a sphere is 20 feet, find its surface area. (Use π = 3.14). Solution: Given, the radius 'r' of the sphere = 20 feet. The surface area of the sphere = 4πr2 = 4 × π × 202 = 5024 feet2 ∴ The surface area of the sphere is 5024 feet2 Example 2: Find the surface area of a sphere if its radius is given as 6 units. Solution: Given, the radius 'r' = 6 units. So, let us substitute the value of r = 6 units ⇒ The surface area of the sphere = 4πr2 = 4 × π × 62 = 4 × 3.14 × 36 = 452.16 unit2 ∴ The surface area of the sphere is 452.16 unit2 Example 3: State true or false. a.) A sphere is a three-dimensional form of a circle. b.) The curved surface area of a sphere is the total surface area of the sphere because a sphere has just one surface which is curved. Solution: a.) True, a sphere is a three-dimensional form of a circle. b.) True, the curved surface area of a sphere is the total surface area of the sphere because a sphere has just one surface which is curved. go to slidego to slidego to slide Book a Free Trial Class Practice Questions on Surface Area of Sphere go to slidego to slide FAQs on Surface Area of Sphere What is the Surface Area of Sphere in Math? The surface area of a sphere is the total area that is covered by its outer surface. The surface area of a sphere is always expressed in square units. The formula for the surface area of a sphere depends on the radius and the diameter of the sphere. It is mathematically expressed as 4πr2; where 'r' is the radius of the sphere. Why is the Surface Area of a Sphere 4 Times the Area of a Circle? A string that completely covers the surface area of a sphere can completely cover the area surface of exactly four circles. This way you can check that the surface area of a sphere is four times the area of a circle. When we write the formula for the surface area of a sphere, we write the surface area of a sphere = 4πr2 = 4(πr2) = 4 × area of a circle. How Many Sides and Vertices Does a Sphere Have? A sphere is a three-dimensional shape that is round like a circle. Hence, it has no sides, vertices, or faces. Does a Sphere have Infinite Faces? No, a sphere has no face. A face is a flat surface and a sphere has no flat surface. This makes the sphere a faceless three-dimensional shape (3D shape). What is the Curved Surface Area and Total Surface Area of a Sphere? A sphere has just one surface and that is curved. Since there is no flat surface in a sphere, the curved surface area of a sphere is equal to its total surface area of the sphere which is 4πr2. ☛Also Check: What is the Surface Area of a Sphere Formula in Terms of Diameter? The surface area of a sphere formula in terms of diameter is given as, πD2 where 'D' is the diameter of the sphere. It gives the relationship between the surface area of a sphere and the diameter of the sphere. How to Calculate the Surface Area of a Sphere With the Volume? The surface area of a sphere can be easily calculated with the help of the volume of the sphere. In this case, we should know the value of the radius of the sphere. The radius of the sphere can be calculated from the formula of the volume of the sphere, that is, Volume of a sphere = 4/3 × πr3. From this, the radius can be calculated and then its value is substituted in the formula for the surface area. We know that the surface area of the sphere = 4πr2. Another way to follow this is as follows. From the volume formula, we can derive that, r3 = 3V/4π, or r = (3V/4π)1/3. After this, we can substitute the value of r in the surface area of the sphere formula. What is the Surface Area of Sphere Calculator? Surface area of sphere calculator is an online tool available for kids to ease their calculations. It is system generated tool where the surface area formula is pre-fixed all we have to do is enter the value of the given parameters, such as radius and we get the surface area of the sphere. Try now Cuemath's surface area of a sphere calculator and get your answers in a few seconds. How Does the Surface Area of Sphere Change When the Radius is Halved? The surface area of the sphere gets one-fourth when the radius is halved because r becomes r/2. As, the surface area of a sphere = 4πr2, so, if we replace 'r' with r/2, the formula becomes 4π(r/2)2 = πr2 which is one-fourth of the surface area. Thus, the surface area of the sphere gets one-fourth as soon as its radius gets halved. How does the Surface Area of a Sphere Change When the Radius is Tripled? The surface area of the sphere becomes 36πr2 when the radius is tripled because 'r' becomes 3r'. We know that the surface area of a sphere = 4πr2, so if we replace 'r' with 3r, we get the formula as, surface area = 4π(3r)2 = 4π × 9r2 = 36πr2
5263
https://prase.cz/kalva/putnam.html
Putnam problems The first 5 competitions had a total of 77 problems, the next 17 had 14 each, and the last 41 (thru 2002) had 12 each, for a total of 819. I have now got up all the problems, and solutions for all years except some of 1941, 1942, 1948 and 1949 (41/B7(2), 42/A6, 42/B3, 48/A6(1), 48/B2, 48/B4, 48/B6(1), 49/A1, 49/A2, 49/A4, 49/B6). Archive 1st Putnam 1938 2nd Putnam 1939 3rd Putnam 1940 4th Putnam 1941 5th Putnam 1942 No Putnam 1943! No Putnam 1944! No Putnam 1945! 6th Putnam 1946 7th Putnam 1947 8th Putnam 1948 9th Putnam 194910th Putnam 1950 11th Putnam 1951 12th Putnam 1952 13th Putnam 1953 14th Putnam 1954 15th Putnam 1955 16th Putnam 1956 17th Putnam 1957 18th Putnam 1958 19th Putnam 1958 20th Putnam 1959 21st Putnam 1960 22nd Putnam 1961 23rd Putnam 1962 24th Putnam 1963 25th Putnam 1964 26th Putnam 1965 27th Putnam 1966 28th Putnam 1967 29th Putnam 196830th Putnam 1969 31st Putnam 1970 32nd Putnam 1971 33rd Putnam 1972 34th Putnam 1973 35th Putnam 1974 36th Putnam 1975 37th Putnam 1976 38th Putnam 1977 39th Putnam 1978 40th Putnam 1979 41st Putnam 1980 42nd Putnam 1981 43rd Putnam 1982 44th Putnam 1983 45th Putnam 1984 46th Putnam 1985 47th Putnam 1986 48th Putnam 1987 49th Putnam 198850th Putnam 1989 51st Putnam 1990 52nd Putnam 1991 53rd Putnam 1992 54th Putnam 1993 55th Putnam 1994 56th Putnam 1995 57th Putnam 1996 58th Putnam 1997 59th Putnam 1998 60th Putnam 1999 61st Putnam 2000 62nd Putnam 2001 63rd Putnam 2002 64th Putnam 2003 The 1st - 18th Putnams were held in March (or occasionally April). The 19th onwards were held in December (or occasionally November). Home John Scholes jscholes@kalva.demon.co.uk 4 Mar 2002 Last corrected/updated 9 Dec 2003
5264
https://fiveable.me/key-terms/hs-physical-science/thermal-expansion
Thermal expansion - (Physical Science) - Vocab, Definition, Explanations | Fiveable | Fiveable new!Printable guides for educators Printable guides for educators. Bring Fiveable to your classroom ap study content toolsprintablespricing my subjectsupgrade All Key Terms Physical Science Thermal expansion 🫴physical science review key term - Thermal expansion Citation: MLA Definition Thermal expansion is the increase in volume or size of a substance as it is heated. When temperature rises, the kinetic energy of particles increases, causing them to move apart and occupy a larger space. This property affects solids, liquids, and gases, influencing many practical applications like engineering, construction, and everyday objects. 5 Must Know Facts For Your Next Test All materials expand when heated and contract when cooled, but the degree of expansion varies by material. The coefficient of thermal expansion helps predict how much a material will expand or contract with temperature changes. In engineering, thermal expansion must be considered in construction projects to avoid structural failure due to temperature fluctuations. Different materials can expand at different rates, which can cause problems if they are joined together (like metal and glass). Thermal expansion plays a critical role in phenomena such as the weathering of rocks and the functioning of thermometers. Review Questions How does thermal expansion impact the design of structures and materials? Thermal expansion significantly influences the design of structures and materials because engineers must account for the changes in size and shape that occur with temperature variations. For instance, bridges and railways incorporate expansion joints to accommodate the expansion and contraction of materials due to heat. Failure to consider thermal expansion can lead to structural damage or even catastrophic failure. Analyze how the coefficient of thermal expansion varies between different materials and its implications. The coefficient of thermal expansion varies widely among different materials, affecting how they respond to temperature changes. For example, metals typically have higher coefficients than ceramics or glass. This variance is crucial for applications where multiple materials are combined, as mismatched expansions can lead to stresses and potential failure at the joints. Understanding these differences allows for better material selection and design. Evaluate the role of thermal expansion in everyday life, providing examples of both beneficial and detrimental effects. Thermal expansion plays a vital role in everyday life with both positive and negative impacts. For instance, thermometers utilize thermal expansion for accurate temperature readings, while road surfaces may buckle due to extreme heat if not properly designed. Similarly, metal railroad tracks are engineered with expansion joints to prevent warping. On the other hand, unanticipated expansion in buildings can lead to cracks or misalignments. Recognizing these effects helps in making informed decisions regarding material use and environmental considerations. Related terms contraction:The decrease in volume or size of a substance as it cools down, causing particles to move closer together. thermal conductivity: A measure of a material's ability to conduct heat, which impacts how quickly thermal expansion occurs. coefficient of thermal expansion:A numerical value that quantifies how much a material expands per degree of temperature change. 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5265
https://www.teacherspayteachers.com/Product/Addition-and-Subtraction-Comparing-Word-Problems-472103
Addition and Subtraction Comparing Word Problems What others say Description Help children practice their reading comprehension of comparison math problems, using easy numbers so they can tell immediately if their answers make sense. How often have we seen children do a page of addition or subtraction problems without hesitation, but get stumped by a word problem? Their math computation skills are fine, but they are hindered by reading comprehension skills. Word problems are tricky for most students, and comparing problems are the hardest of all. As a second grade teacher, I know my students need lots of practice with all types of word problems (so I’ve written a lot of them over the years!) but especially with comparing problems. These practice pages will help your students focus on comparing. These ten pages (100 problems in all!) can be used as reading, math, or homework. The math computation is easy (all sums/differences are within 10) because each problem requires students to read and to think carefully before solving. To solve story problems, children need to be able to: • Determine/visualize the story/problem • Identify the question that needs to be answered • Figure out what to do with the numbers given • Do the math • Explain what the answer means ("4" doesn't answer the question: Students must determine if the answer is, "Jessa has 4 dogs," "Andrew has 4 dogs," "Jessa has 4 more dogs than Andrew," "Andrew has 4 more dogs than Jessa," or "Jessa and Andrew have 4 dogs altogether.") The “Addition and Subtraction Comparing Word Problems" pages address these skills! There are five kinds of problems on each practice page. Each type of problem appears twice (once in problems 1-5, again in problems 6-10). There are two formats for each page: • The single-sided pages have all ten problems on one side, with no space for showing work. They are great if you need to save paper, if most of your students can do the math mentally, or if you’ve already used several of these pages and your students know what to do. • The double-sided pages have the same ten problems, but with only five on each side and space to solve/show beside each problem. The answer keys provided include possible equations in case you choose to ask children to write an equation for each problem in addition to their sentence. The ten pages can be used in any order. You might also like these word problem packets: Reading for Math: Building Comprehension of Story Problems (word problems with sums and differences within 20) Math Story Problems: Two Digit Addition and Subtraction (word problems within 100) Choose Your Math: Two Step Word Problems (students can choose whether to use simple or complex numbers for each problem) Want to save money on your TpT purchases? Leave feedback, earn credits, and apply those credits toward your next checkout! Follow me on TpT so you can see when I add new products or have a sale! Have questions? Please contact me at SecondGradeSuccessbyAmandaK@gmail.com about this resource or anything in my store! Happy teaching! :) Amanda Addition and Subtraction Comparing Word Problems Save even more with bundles Reviews Questions & Answers Standards
5266
https://www.le.ac.uk/users/dsgp1/COURSES/LEISTATS/A06ans.pdf
EXERCISES IN STATISTICS Series A, No. 6 1. Let x1 and x2 have the joint p.d.f f(x1, x2) = x1 + x2 with 0 ≤x1, x2 ≤1. Find the conditional mean of x2 given x1. Answer. First let us show that Z x2 Z x1 f(x1, x2)dx2dx1 = 1 : We have Z x2 Z x1 (x1 + x2)dx1dx2 = Z x2 ·x2 1 2 + x2x1 ¸1 0 dx2 = Z x2 µ1 2 + x2 ¶ dx2 = ·x2 2 + x2 2 2 ¸1 0 = 1. From this we can see that f(x2) = [ 1 2 + x2]. Likewise, by an argument of symmetry, we have f(x1) = [ 1 2 + x1]. Next consider E(x2|x1) = Z x2 x2 f(x1, x2) f(x1) dx2 = 1 ( 1 2 + x1) Z x2 x2(x1 + x2)dx2 = 1 ( 1 2 + x1) ·x1x2 2 2 + x3 2 3 ¸1 x2=0 = 1 (x1 + 1 2) (3x1 + 2) 6 = 3x1 + 2 6x1 + 3. 2. Find the P(x < y|x < 2y) when f(x, y) = e−(x+y). Draw a diagram to represent the event. Answer. P(x < ay) = Z ∞ y=0 Z ay x=0 f(x, y)dxdy = Z ∞ y=0 e−y ½Z ay x=0 e−x ¾ dy = Z ∞ y=0 e−y £ −e−x¤ay 0 dy = Z ∞ y=0 e−y £ 1 −e−ay¤ dy = · −e−y + e−(1+a)y 1 + a ¸∞ 0 = 1 − 1 1 + a 1 SERIES A No.6, ANSWERS Hence we get P(x < y) = 1 −1 2 = 1 2 P(x < 2y) = 1 −1 3 = 2 3 P(x < y|x < 2y) = P(x < y) P(x < 2y) = 1 2.3 2 = 3 4. 3. The variance of x1 + x2 is V (x1) + V (x2) + 2C(x1, x2). Extend this result to find the variance of x1 + x2 + x3. Answer. V (x1 + x2 + x3) = V © (x1 + x2) + x3 ª = V (x1 + x2) + V (x3) + 2C(x1 + x2, x3) = V (x1) + V (x2) + V (x3) + 2C(x1 + x2, x3) + 2C(x1, x2). But C(x1 + x2, x3) = E h© (x1 + x2) −E(x1 + x2) ª© x3 −E(x3) ªi = E h© x1 −E(x1) ª + © x2 −E(x2) ªih x3 −E(x3) i = E h© x1 −E(x1) ª© x3 −E(x3) ªi + E h© x2 −E(x2) ª© x3 −E(x3) ªi = C(x1, x3) + C(x2, x3) Substituting this result into the expression for V (x1 + x2 + x3) gives V (x1+x2+x3) = V (x1)+V (x2)+V (x3)+2C(x1, x2)+2C(x1, x3)+2C(x2, x3). 4. Find V (x1 −x2). Prove that C(x1, x2) ≤ p V (x1)V (x2) ≤1 2 © V (x1) + V (x2) ª . Answer. For the first part, V (x1 −x2) = E © (x1 −x2) −E(x1 −x2) ª2 = E h© x1 −E(x1) ª − © x2 −E(x2) ªi2 = E © x1 −E(x1) ª2 −2E h© x1 −E(x1) ª© x2 −E(x2) ªi + E © x2 −E(x2) ª2 = V (x1) + V (x2) −2C(x1, x2) 2 SERIES A No.6, ANSWERS For the second part, we may assume that −1 ≤ρ = C(x1, x2) p V (x1)V (x2) ≤1. It follows that C(x1, x2) ≤ p V (x1) + V (x2) Also we have 0 ≤ ¡p V (x1) − p V (x2) ¢2 = V (x1) −2 p V (x1) p V (x2) + V (x2) whence 1 2 © V (x1) + V (x2) ª ≥ p V (x1) + V (x2) and the result follows from combining the inequalities. 3
5267
https://web.njit.edu/~sirenko/Phys-446/Lecture2-SSP-2007.pdf
1 Lecture 2 Andrei Sirenko, NJIT 1 Phys 446: Solid State Physics / Optical Properties Fall 2015 Lecture 2 Andrei Sirenko, NJIT 2 Solid State Physics Lecture 2 (Ch. 2.1-2.3, 2.6-2.7) Last week: • Crystals, • Crystal Lattice, • Reciprocal Lattice Today: • Types of bonds in crystals Diffraction from crystals • Importance of the reciprocal lattice concept 2 Lecture 2 Andrei Sirenko, NJIT 3 (3) The Hexagonal Closed-packed (HCP) structure • The HCP structure is made up of stacking spheres in a ABABAB… configuration • The HCP structure has the primitive cell of the hexagonal lattice, with a basis of two identical atoms • Atom positions: 000, 2/3 1/3 ½ (remember, the unit axes are not all perpendicular) • The number of nearest-neighbours is 12 • The ideal ratio of c/a for this packing is (8/3)1/2 = 1.633 . Be, Sc, Te, Co, Zn, Y, Zr, Tc, Ru, Gd,Tb, Py, Ho, Er, Tm, Lu, Hf, Re, Os, Tl Rotated three times Conventional HCP unit cell Lecture 2 Andrei Sirenko, NJIT 4 Crystal Lattice 3 Lecture 2 Andrei Sirenko, NJIT 5 Reciprocal Lattice Lecture 2 Andrei Sirenko, NJIT 6 Some examples of reciprocal lattices 1. Reciprocal lattice to simple cubic lattice a1 = ax, a2 = ay, a3 = az V = a1·(a2a3) = a3  b1 = (2/a)x, b2 = (2/a)y, b3 = (2/a)z  reciprocal lattice is also cubic with lattice constant 2/a 2. Reciprocal lattice to bcc lattice   z y x a     a 2 1 1   z y x a    a 2 1 2   z y x a    a 2 1 3 3 3 2 1 2 1 a V     a a a    z y b  a  2 1   z x b  a  2 2   y x b  a  2 3 4 Lecture 2 Andrei Sirenko, NJIT 7   z y b  a  2 1   z x b  a  2 2   y x b  a  2 3 got but these are primitive vectors of fcc lattice So, the reciprocal lattice to bcc is fcc. Analogously, show that the reciprocal lattice to fcc is bcc Lecture 2 Andrei Sirenko, NJIT 8 Brillouin zones Determine all the perpendicular bisecting planes in the reciprocal lattice First Brillouin zone - the Wigner-Seitz cell of the reciprocal lattice Higher Brillouin zones: Second Brillouin zone: 5 Lecture 2 Andrei Sirenko, NJIT 9 Brillouin zones of cubic lattices First BZ of a BCC lattice First BZ of an FCC lattice Lecture 2 Andrei Sirenko, NJIT 10 Summary Reciprocal lattice is defined by primitive vectors: A reciprocal lattice vector has the form G = hb1 + kb2 + lb3 It is normal to (hkl) planes of direct lattice First Brillouin zone is the Wigner-Seitz primitive cell of the reciprocal lattice Simple cubic cube; bcc Rhombic dodecahedron; fcc truncated octahedron (figures on the previous slide) 6 Lecture 2 Andrei Sirenko, NJIT 11 Indexing system for crystal planes • Since crystal structures are obtained from diffraction experiments (in which particles diffract from planes of atoms), it is useful to develop a system for indexing lattice planes. • We can use the lattice constants a1, a2, a3, but it turns out to be more useful to use what are called Miller Indices. Index Lecture 2 Andrei Sirenko, NJIT 12 Rules for determining Miller Indices • (1) Find the intercepts on the axes in terms of the lattice constants a1, a2, a3. • (2) Take the reciprocals of these numbers and then reduce to three integers having the same ratio, usually the smallest of the three integers. The result, listed as (hkl), is called the index of the plane. An example: Intercepts: a, ∞,∞ Reciprocals: a/a, a/∞, a/∞ = 1, 0, 0 Miller index for this plane : (1 0 0) (note: this is the normal vector for this plane) 7 Lecture 2 Andrei Sirenko, NJIT 13 Examples of Miller Indices Intercepts: a, a,∞ Reciprocals: a/a, a/a, a/∞ = 1, 1, 0 Miller index for this plane : (1 1 0) Intercepts: a,a,a Reciprocals: a/a, a/a, a/a = 1, 1, 1 Miller index for this plane : (1 1 1) Lecture 2 Andrei Sirenko, NJIT 14 Examples of Miller Indices Intercepts: 1/2a, a,∞ Reciprocals: 2a/a, a/a, a/∞ = 2, 1, 0 Miller index for this plane : (2 1 0) 8 Lecture 2 Andrei Sirenko, NJIT 15 Notes on notation • (hkl) might mean a single plane, or a set of planes • If a plane cuts a negative axis, we have minus signs in the (hkl) (ie. (hkl)) • Planes are denoted with curly brackets (hkl) • A set of faces are denoted {hkl} • The direction of a crystal (for example, along x for a cubic crystal) is denoted with [uvw] (ie. The direction) • In cubic crystals, the direction [hkl] is perpendicular to the plane (hkl) having the same indices, but this isn’t necessarily true for other crystal systems direction {001} face direction Lecture 2 Andrei Sirenko, NJIT 16 Inter-atomic forces and types of bonds in solids. 9 Lecture 2 Andrei Sirenko, NJIT 17 Interatomic forces What holds a crystal together? Attractive electrostatic interaction between electrons and nuclei – the force responsible for cohesion of solids equilibrium position R0 binding energy Interatomic distance R Interatomic potential V Force: R R V R F     ) ( ) ( F(R) < 0 for R > R0 : attraction F(R) > 0 for R < R0 : repulsion Lecture 2 Andrei Sirenko, NJIT 18 Types of bonding I. Ionic crystals Usually involve atoms of strongly different electro-negativities (Example: alkali halides). n R A N R e N R U    0 2 4 ) (   KCl: energy per molecule vs R (from Kittel) attractive (Coulomb) repulsive Ionic bond is strong (binding energy - few eV/pair) hardness, high melting T electrons are strongly localized insulators in solid form Typical crystal structures: NaCl, CsCl 10 Lecture 2 Andrei Sirenko, NJIT 19 II. Covalent crystals – Electron pair bond: usually one electron from each atom – Electrons tend to be localized in part between the two atoms – The spins of electrons in the bond are anti-parallel – Gap between fully occupied and unoccupied states dielectrics or semiconductors Directionality of covalent bonds. Example: carbon Hybridization. 2s22p2 2s2px2py2pz : sp3 tetrahedral configuration Also possible sp2: 2s2px2py – planar (graphite, fullerenes) remaining pz : interlayer -bonding Covalent polar bond (many compound semiconductors) – intermediate case between non-polar and ionic bond. Described by effective ionic charge or fractional ionic character (between 0 and 1: 0 for Si or diamond, 0.94 for NaCl). Covalent bond is also strong, binding energies of several eV per atom Lecture 2 Andrei Sirenko, NJIT 20 III. Metals – Most elements in periodic table – High electrical and thermal conductivity – High density – Considerable mechanical strength, but plasticity These properties can be explained considering the metallic type of bond Example: alkali metals – single electron weakly bound to atom – easily delocalized. In contrast to covalent bonding, electronic wave functions are very extended compared to interatomic distances. Why total energy is reduced ? Partially occupied electronic bands – electrons can move freely Group II metals – two s electrons – should be fully occupied... but overlapped with empty p-states Transition metals: d-electrons are more localized – form covalent-like bonds; s and p-electrons again form a common band Metals crystallize in closely packed structures (hcp, fcc, bcc) 11 Lecture 2 Andrei Sirenko, NJIT 21 IV. Van der Waals bonds Inert gases: outer electronic shells are full – no ionic or covalent forces Weak interatomic forces due to quantum fluctuations of charge arising dipole moments cause a weak attractive force Can be described in the quantum-mechanical model of two linear oscillators (given in Kittel) results in R-6 dependence of potential Binding energy in order of 0.1 eV Crystal structures are fcc (electronic distribution is spherical, atoms pack as closely as possible) Van der Waals forces are also responsible for bonding in organic molecular crystals. Molecules are weakly bound; often form low-symmetry crystals They also exist in covalent or ionic crystals, but negligible V. Hydrogen bonds Formed between two strongly electronegative atoms (F, O, N) via H Example: ice Binding energy is also ~0.1 eV Lecture 2 Andrei Sirenko, NJIT 22 Summary Repulsive interaction between atoms is primarily due to electrostatic repulsion of overlapping charge distributions and Pauli principle Several types of attractive forces: • Ionic crystals – electrostatic forces between "+" and "-" ions • Covalent bond: overlap of charge distributions with antiparallel spin • Metals: reduction of kinetic energy of electrons in free state compared to the localized state of a single atom • Secondary forces (Van der Waals, hydrogen) become significant when the other bonds are impossible, e.g. in inert gases Physical properties are closely related to the type of bonding 12 Lecture 2 Andrei Sirenko, NJIT 23 DIFFRACTION Lecture 2 Andrei Sirenko, NJIT 24 Diffraction of waves by crystal lattice • Most methods for determining the atomic structure of crystals are based on scattering of particles/radiation. • X-rays is one of the types of the radiation which can be used • Other types include electrons and neutrons • The wavelength of the radiation should be comparable to a typical interatomic distance of a few Å (1 Å =10-10 m) E hc hc h E        (Å) = 12398/E(eV)  few keV is suitable energy for = 1 Å • X-rays are scattered mostly by electronic shells of atoms in a solid. Nuclei are too heavy to respond. • Reflectivity of x-rays ~10-3-10-5 deep penetration into the solid x-rays serve as a bulk probe 13 Lecture 2 Andrei Sirenko, NJIT 25 The Bragg Law Conditions for a sharp peak in the intensity of the scattered radiation: 1) the x-rays should be specularly reflected by the atoms in one plane 2) the reflected rays from the successive planes interfere constructively The path difference between the two x-rays: 2d·sinθ  the Bragg formula: 2d·sinθ = mλ The model used to get the Bragg law are greatly oversimplified (but it works!). – It says nothing about intensity and width of x-ray diffraction peaks – neglects differences in scattering from different atoms – assumes single atom in every lattice point – neglects distribution of charge around atoms Lecture 2 Andrei Sirenko, NJIT 26 The Bragg Law and Diffraction grating Compare Bragg Law 2d·sinθ = mλ X-ray Diffraction 14 Lecture 2 Andrei Sirenko, NJIT 27 Meaning of d for 2D d 2d·sinθ = mλ Lecture 2 Andrei Sirenko, NJIT 28 Meaning of d for 3D Intercepts: a,a,a Reciprocals: a/a, a/a, a/a = 1, 1, 1 Miller index for this plane : (1 1 1) 2 2 2 2 2 2 hkl n d h k l a b c    d 111 111 3.13 A for Si with 5.431 A 3 n a d a     2d·sinθ = mλ 15 Lecture 2 Andrei Sirenko, NJIT 29 Lecture 2 Andrei Sirenko, NJIT 30 X-rays are EM waves 16 Lecture 2 Andrei Sirenko, NJIT 31 The most important information arises when the wavelength of the radiation is similar to, or smaller than, the size of the spacing between the objects being studied. Lecture 2 Andrei Sirenko, NJIT 32 X-rays and X-ray tube Bragg Law 2d·sinθ = mλ for m=1 2d > λ Electronic transitions X-ray tube 17 Lecture 2 Andrei Sirenko, NJIT 33 X-rays and Synchrotrons Bragg Law 2d·sinθ = mλ Synchrotron radiation Natural Synchrotron Radiation Accelerating electron emits light Stars and Galaxies Lecture 2 Andrei Sirenko, NJIT 34 •Synchrotron Radiation from a storage ring is the most bright manmade source of white light •Useful for materials studies from Far Infrared and UV to X-ray Synchrotron Radiation produced by relativistic electrons in accelerators (since 1947) 18 Lecture 2 Andrei Sirenko, NJIT 35 Diffraction condition and reciprocal lattice Von Laue approach: – crystal is composed of identical atoms placed at the lattice sites T – each atom can reradiate the incident radiation in all directions. – Sharp peaks are observed only in the directions for which the x-rays scattered from all lattice points interfere constructively. Consider two scatterers separated by a lattice vector T. Incident x-rays: wavelength λ, wavevector k; |k| = k = 2/; Assume elastic scattering: scattered x-rays have same energy (same λ)  wavevector k' has the same magnitude |k'| = k = 2/ Condition of constructive interference: Define k = k' - k - scattering wave vector Then k = G , where G is defined as such a vector for which G·T = 2m k k k   k' k' ' k     m  2    T k k' Lecture 2 Andrei Sirenko, NJIT 36 We obtained the diffraction (Laue) condition: k = G where G·T = 2m Vectors G which satisfy this relation form a reciprocal lattice A reciprocal lattice is defined with reference to a particular Bravais lattice, which is determined by a set of lattice vectors T. Constricting the reciprocal lattice from the direct lattice: Let a1, a2, a3 - primitive vectors of the direct lattice; T = n1a1 + n2a2 + n3a3 Then reciprocal lattice can be generated using the primitive vectors where V = a1·(a2a3) is the volume of the unit cell Then vector G = m1b1 + m2b2 + m3b3 We have bi·aj = δij Therefore, G·T = (m1b1 + m2b2 + m3b3)·(n1a1 + n2a2 + n3a3) = 2(m1n1+ m2n2+ m3n3) = 2m The set of reciprocal lattice vectors determines the possible scattering wave vectors for diffraction 19 Lecture 2 Andrei Sirenko, NJIT 37 We got k = k' – k = G |k'|2 = |k|2 + |G|2 +2k·G G2 +2k·G = 0 2k·G = G2 – another expression for diffraction condition Now, show that the reciprocal lattice vector G = hb1 + kb2 + lb3 is orthogonal to the plane represented by Miller indices (hkl) plane (hkl) intercepts axes at points x, y, and z given in units a1, a2 and a3 By the definition of the Miller indices: define plane by two non-collinear vectors u and v lying within this plane: prove that G is orthogonal to u and v: analogously show Lecture 2 Andrei Sirenko, NJIT 38 Now, prove that the distance between two adjacent parallel planes of the direct lattice is d = 2π/G. The interplanar distance is given by the projection of the one of the vectors xa1, ya2, za3, to the direction normal to the (hkl) plane, which is the direction of the unit vector G/G   k k' k The reciprocal vector G(hkl) is associated with the crystal planes (hkl) and is normal to these planes. The separation between these planes is 2π/G 2k·G = G2 2|k|Gsin= G2 2·2sin/= 2/d 2dsin=  2dsin= m- get Bragg law 20 Lecture 2 Andrei Sirenko, NJIT 39 Ewald Construction for Diffraction Condition and reciprocal space Lecture 2 Andrei Sirenko, NJIT 40 Reciprocal Space: Accessible Area for Diffraction 21 Lecture 2 Andrei Sirenko, NJIT 41 Summary Various statements of the Bragg condition: 2d·sinθ = mλ ; k = G ; 2k·G = G2 Reciprocal lattice is defined by primitive vectors: A reciprocal lattice vector has the form G = hb1 + kb2 + lb3 It is normal to (hkl) planes of direct lattice Only waves whose wave vector drawn from the origin terminates on a surface of the Brillouin zone can be diffracted by the crystal First BZ of fcc lattice First BZ of bcc lattice Lecture 2 Andrei Sirenko, NJIT 42 Summary 22 Lecture 2 Andrei Sirenko, NJIT 43 Rotating crystal method – for single crystals, epitaxial films -2, rocking curve, - scan Powder diffraction Laue method – white x-ray beam used most often used for mounting single crystals in a precisely known orientation Experimental XRD techniques Lecture 2 Andrei Sirenko, NJIT 44 Applications of X-ray Diffraction for crystal and thin-film analysis 23 Lecture 2 Andrei Sirenko, NJIT 45 Applications of X-ray Diffraction for hetero-structures (one or more crystalline films grown on a substrate) Lecture 2 Andrei Sirenko, NJIT 46 X-ray Diffraction Setup 24 Lecture 2 Andrei Sirenko, NJIT 47 High Angular Resolution X-ray Diffraction Setup B11 Tiernan Lecture 2 Andrei Sirenko, NJIT 48 Example of High Angular Resolution X-ray Diffraction analysis of a SiGe film on Si substrate
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https://artofproblemsolving.com/wiki/index.php/Divisibility_rules?srsltid=AfmBOor__PE35gpYSDHWiTJy-R7HmtCdo3EtKi8lKgv00GM0IRzWYgas
Art of Problem Solving Divisibility rules - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki Divisibility rules Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Divisibility rules These divisibility rules help determine when positive integers are divisible by particular other integers. All of these rules apply for base-10only -- other bases have their own, different versions of these rules. Contents [hide] 1 Divisibility Videos 2 Basics 2.1 Divisibility Rule for 2 and Powers of 2 2.2 Divisibility Rule for 3 and 9 2.3 Divisibility Rule for 5 and Powers of 5 2.4 Divisibility Rule for 7 2.5 Divisibility Rule for 10 and Powers of 10 2.6 Divisibility Rule for 11 2.7 General Rule for Composites 2.7.1 Example 3 Advanced 3.1 General Rule for Primes 3.2 Divisibility Rule for 13 3.3 Divisibility Rule for 17 3.4 Divisibility Rule for 19 3.5 Divisibility Rule for 29 3.6 Divisibility Rule for 49 4 Special 4.1 Mod-preserving tests 4.1.1 Mod-preserving for 7 4.1.2 Mod-preserving for 13 4.2 Block tests 4.2.1 Small blocks -- 101 and 1001 4.2.2 Bigger blocks -- 10001 and 10000001 4.2.3 Type 2 blocks -- 111 and 11111 5 Problems 6 Resources 6.1 Books 6.2 Classes 7 See also Divisibility Videos Basics Divisibility Rule for 2 and Powers of 2 A number is divisible by if and only if the last digits of the number are divisible by . Thus, in particular, a number is divisible by 2 if and only if its units digit is divisible by 2, i.e. if the number ends in 0, 2, 4, 6 or 8. Proof Divisibility Rule for 3 and 9 A number is divisible by 3 or 9 if and only if the sum of its digits is divisible by 3 or 9, respectively. Note that this does not work for higher powers of 3. For instance, the sum of the digits of 1899 is divisible by 27, but 1899 is not itself divisible by 27. Proof Divisibility Rule for 5 and Powers of 5 A number is divisible by if and only if the last digits are divisible by that power of 5. Proof Divisibility Rule for 7 Rule 1: Partition into 3 digit numbers from the right (). The alternating sum () is divisible by 7 if and only if is divisible by 7. Proof Rule 2: Truncate the last digit of , double that digit, and subtract it from the rest of the number (or vice-versa). is divisible by 7 if and only if the result is divisible by 7. Proof Rule 3: "Tail-End divisibility." Note. This only tells you if it is divisible and NOT the remainder. Take a number say 12345. Look at the last digit and add or subtract a multiple of 7 to make it zero. In this case we get 12380 or 12310 (both are acceptable; I am using the former). Lop off the ending 0's and repeat. 1238 - 28 ==> 1210 ==> 121 - 21 ==> 100 ==> 1 NOPE. Works in general with numbers that are relatively prime to the base (and works GREAT in binary). Here's one that works. 12348 - 28 ==> 12320 ==> 1232 +28 ==> 1260 ==> 126 + 14 ==> 14 YAY! Tiny extension to tell you the remainder: Count how many zeroes you lop off and mod 6. Multiply mod 7 with the corresponding number Zeroes (mod 6) Number to multiply by 0 1 1 3 2 2 3 6 4 4 5 5 And that's the remainder. Divisibility Rule for 10 and Powers of 10 If a number is power of 10, define it as a power of 10. The exponent is the number of zeros that should be at the end of a number for it to be divisible by that power of 10. Example: A number needs to have 6 zeroes at the end of it to be divisible by 1,000,000 because . Divisibility Rule for 11 A number is divisible by 11 if the alternating sum of the digits is divisible by 11. Proof General Rule for Composites A number is divisible by , where the prime factorization of is , if the number is divisible by each of . Example For the example, we will check if 55682168544 is divisible by 36. The prime factorization of 36 to be . Thus we must check for divisibility by 4 and 9 to see if it's divisible by 36. Since the last two digits, 44, of the number is divisible by 4, so is the entire number. To check for divisibility by 9, we look to see if the sum of the digits is divisible by 9. The sum of the digits is 54 which is divisible by 9. Thus, the number is divisible by both 4 and 9 and must be divisible by 36. Advanced General Rule for Primes For every prime number other than 2 and 5, there exists a rule similar to rule 2 for divisibility by 7. For a general prime , there exists some number such that an integer is divisible by if and only if truncating the last digit, multiplying it by and subtracting it from the remaining number gives us a result divisible by . Divisibility rule 2 for 7 says that for , . The divisibility rule for 11 is equivalent to choosing . The divisibility rule for 3 is equivalent to choosing . These rules can also be found under the appropriate conditions in number bases other than 10. Also note that these rules exist in two forms: if is replaced by then subtraction may be replaced with addition. We see one instance of this in the divisibility rule for 13: we could multiply by 9 and subtract rather than multiplying by 4 and adding. is any number so that Divisibility Rule for 13 Rule 1: Truncate the last digit, multiply it by 4 and add it to the rest of the number. The result is divisible by 13 if and only if the original number was divisble by 13. This process can be repeated for large numbers, as with the second divisibility rule for 7. Proof Rule 2: Partition into 3 digit numbers from the right (). The alternating sum () is divisible by 13 if and only if is divisible by 13. Proof Rule 3: Works for . Let . If is odd add 39 to . Round up to the nearest multiple of 80, call the result . Find . Check: Is . Proof Divisibility Rule for 17 Truncate the last digit, multiply it by 5 and subtract from the remaining leading number. The number is divisible if and only if the result is divisible. The process can be repeated for any number. Proof Divisibility Rule for 19 Truncate the last digit, multiply it by 2 and add to the remaining leading number. The number is divisible if and only if the result is divisible. This can also be repeated for large numbers. Proof Divisibility Rule for 29 Truncate the last digit, multiply it by 3 and add to the remaining leading number. The number is divisible if and only if the result is divisible. This can also be repeated for large numbers. Proof Divisibility Rule for 49 Why 49? For taking pesky out of a root. Useful below 4900. Round up to a multiple of 50, call it , and subtract the original number, call this . If , it is divisible by 49. Examples: Round up: . Difference: . ? Yes! Round up: . Difference: . ? No! Round up: . Difference: . ? Yes! Extension to work for all numbers. Floor divide by 4950, multiply by 50, and add to before calculating Proof Special Mod-preserving tests These tests allow you take the modulo operation easily. Mod-preserving for 7 Multiply the first digit by 3 and add it to the rest. Mod-preserving for 13 Multiply the first digit by 3 and subtract it from the rest Block tests As a bonus, these are also mod-preserving Small blocks -- 101 and 1001 The divisibility for 101 test is simple: Alternate adding and subtracting blocks of two digits starting from the end two, which are added. Ex. 1102314 by 101 01 + 10 - 23 + 14 ← last block is always two digits and positive =0 so 1102314 is divisible by 101 The divisibility for 1001 is the same, but with blocks of three. (Starting with the end three, this time) The 1001 test also works for all it's divisors. The most useful are 7, 11, and 13. Bigger blocks -- 10001 and 10000001 10001 has block size length 4, and factors nicely into 73137. 1000001 has block size 6, and factors into 175882353. 5882353 isn't much use, but 17 is, when we're testing a large number. Type 2 blocks -- 111 and 11111 A different type of test can be yielded from adding all the blocks, but again starting with the end. 111 has a block length of three, and factors into 37 and 3. 11111 has a length of five, and factors to 41 and 271. 1111111, with a length of seven, can provide a test for 239 and 4649, if you ever need it. Problems Practice Problems on Alcumus Divisibility (Prealgebra) 2000 AMC 8 Problems/Problem 11 2006 AMC 10B Problems/Problem 25 Resources Books The AoPS Introduction to Number Theory by Mathew Crawford. The Art of Problem Solving by Sandor Lehoczky and Richard Rusczyk. Classes AoPS Introduction to Number Theory Course See also Number theory Modular arithmetic Math books Mathematics competitions Retrieved from " Category: Divisibility Rules Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
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https://www.ck12.org/flexi/algebra/histograms/lesspgreaterthe-weekly-wages-(in-rs-)-of-30-workers-in-a-factory-are-lessmathgreater830835890810835836869845898890820less-by-mathgreater-lessmathgreater860832833855845804808812840885835less-by-mathgreater-835-836-878-840-8-44989c0/
Flexi answers - The weekly wages (in Rs.) of 30 workers in a factory are @$\begin{align}830,835,890,810,835,836,869,845,898,890,820\end{align}@$ , @$\begin{align}860,832,833,855,845,804,808,812,840,885,835\end{align}@$ , 835, 836, 878, 840, 868, 890, 806, 840. Using tally marks, make a frequency distribution table with class intervals @$\begin{align}800-810,810-820\end{align}@$ and so on. | CK-12 Foundation Subjects Explore Donate Sign InSign Up All Subjects Algebra I Histograms Question The weekly wages (in Rs.) of 30 workers in a factory are 830,835,890,810,835,836,869,845,898,890,820 , 860,832,833,855,845,804,808,812,840,885,835 , 835, 836, 878, 840, 868, 890, 806, 840. Using tally marks, make a frequency distribution table with class intervals 800−810,810−820 and so on. Flexi Says: To create a frequency distribution table using tally marks, we first need to organize the data into class intervals. The given class intervals are 800-810, 810-820, and so on. Let's go through the steps: List the Class Intervals: 800-810 810-820 820-830 830-840 840-850 850-860 860-870 870-880 880-890 890-900 Tally the Data: Count how many wages fall into each class interval and use tally marks to represent them. Create the Frequency Distribution Table: | Class Interval | Tally | Frequency | --- | 800-810 | | | | 810-820 | | | | 820-830 | | | | 830-840 | | | | 840-850 | | | | 850-860 | | | | 860-870 | | | | 870-880 | | | | 880-890 | | | | 890-900 | | | Each tally mark represents one worker's wage in the corresponding class interval. The frequency is the total number of tally marks in each interval. Analogy / Example Try Asking: What are the advantages of the graphical representation of data?Following is the frequency distribution of total marks obtained by the students of different sections of Class VIII. | Marks | Unexpected text node: '100-150' | Unexpected text node: '150-200' | Unexpected text node: '200-300' | Unexpected text node: '300-500' | Unexpected text node: '500-800' | :---: :---: :---: | | Number of students | 60 | 100 | 100 | 80 | 180 | | Draw a histogram for the distribution above. | | | | | |How to read histogram? How can Flexi help? By messaging Flexi, you agree to our Terms and Privacy Policy
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https://stackoverflow.com/questions/77940100/efficient-trigonometric-functions-in-c
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Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Efficient trigonometric functions in C Ask Question Asked 1 year, 7 months ago Modified4 months ago Viewed 246 times This question shows research effort; it is useful and clear 1 Save this question. Show activity on this post. I am implementing some algorithms and I need to use cos(), sin() and atan2(). Especially in loops these functions are very slow. For atan2() I am using atan() with some modifications: c double atan2_custom(double y, double x) { if (x > 0) { return atan(y / x); } else if (x < 0 && y >= 0) { return atan(y / x) + M_PI; } else if (x < 0 && y < 0) { return atan(y / x) - M_PI; } else if (x == 0 && y > 0) { return M_PI / 2; } else if (x == 0 && y < 0) { return -M_PI / 2; } else { return 0; // x and y are both zero } } Are there any alternatives that are more efficient to these math functions? c performance runtime trigonometry Share Share a link to this question Copy linkCC BY-SA 4.0 Improve this question Follow Follow this question to receive notifications edited May 4 at 4:48 chux 157k 17 17 gold badges 160 160 silver badges 307 307 bronze badges asked Feb 5, 2024 at 10:09 phwphw 111 5 5 bronze badges 15 2 Note that atan(y / x) has big drawbacks compared to atan2(y, x), even under the assumption x > 0.Stef –Stef 2024-02-05 10:24:35 +00:00 Commented Feb 5, 2024 at 10:24 1 OT: Some of the conditions are redundant (first y<0 and the two x==0).nielsen –nielsen 2024-02-05 10:29:29 +00:00 Commented Feb 5, 2024 at 10:29 3 You have to define over what range of arguments you want your trig functions to work. For my problems I can guarantee a priori that -pi <= x <= pi which means I can avoid the argument range reduction step entirely. Likewise for tan I can engineer things so that my argument will always be -pi/4 <= x <= pi/4 which allows many shortcuts and can also avoid doing a divide. Atan is about the most tricky and DIY code is unlikely to cut it. System implementations are close to as fast as you can get at full numerical precision. To go faster something has to give! What platform and what CPU ?Martin Brown –Martin Brown 2024-02-05 10:37:38 +00:00 Commented Feb 5, 2024 at 10:37 2 If you have both x and y, you should be using atan2. That will eliminate most of your checks (possibly all of them), since having both x and y makes the quadrant clear. You should always prefer atan2 over atan unless you really only have the ratio, which is less information than having the x and y values. So basically, the answer to this question is this: Use atan2.Tom Karzes –Tom Karzes 2024-02-05 11:09:00 +00:00 Commented Feb 5, 2024 at 11:09 3 Library implementors generally have already put work into making the trigonometric routines faster for general purposes. There is usually little opportunity to improve on that for general purposes. There may be opportunity to improve on it for special purposes, such as when your code does not need support for NaNs, does not need support for infinities, does not need support for arguments out of a reduced range, and so on. For assistance with that, you should specify what your special purposes are.Eric Postpischil –Eric Postpischil 2024-02-05 12:53:39 +00:00 Commented Feb 5, 2024 at 12:53 |Show 10 more comments 3 Answers 3 Sorted by: Reset to default This answer is useful 1 Save this answer. Show activity on this post. Are there any alternatives that are more efficient to these math functions? Thanks in advance. One of the method: Computes the trigonometric sine function using a combination of table lookup and linear interpolation. c Calculation of the nearest integer table index Compute the fractional portion (fract) of the table index. The final result equals (1.0f-fract)a + fractb; where b = Table[index]; c = Table[index+1]; Source: CMSIS-DSP documentation And some example implementation: Share Share a link to this answer Copy linkCC BY-SA 4.0 Improve this answer Follow Follow this answer to receive notifications answered Feb 5, 2024 at 10:15 gulprgulpr 4,619 3 3 silver badges 19 19 bronze badges Comments Add a comment This answer is useful 1 Save this answer. Show activity on this post. Are there any alternatives that are more efficient to these math functions? If one is willing to forego precision, then there are readily other time efficient alternates. If one is willing to forego correctness, then there are even more time efficient alternates. Get functionality right first, then precision, before trying for speed. So what ever alternates you choose, first consider testing it for 1) correctness & 2) preciseness and rate your solution. OP has selected one of the two most challenging functions (the other pow()) to code as it has many corner cases and precision issue in the extremes. Following is a basic test harness. It demos, among other short-comings, a common issue: atan2_custom(small_negative_y_even_minus_0, large_negative_x) should return -pi/2, not +pi/2. ```c include include include include include include ifndef M_PI define M_PI 3.1415926535897932384626433832795 endif double atan2_custom(double y, double x) { if (x > 0) { return atan(y / x); } else if (x < 0 && y >= 0) { return atan(y / x) + M_PI; } else if (x < 0 && y < 0) { return atan(y / x) - M_PI; } else if (x == 0 && y > 0) { return M_PI / 2; } else if (x == 0 && y < 0) { return -M_PI / 2; } else { return 0; // x and y are both zero } assert(0); return 0; } static_assert(sizeof(double) == sizeof(uint64_t),"Unexpected sizes"); union double_bits { double d; uint64_t u64; }; unsigned long long test_atan2(double (f)(double, double), double x, double y) { union double_bits a0 = {.d = atan2(y, x)}; union double_bits a1 = {.d = f(y, x)}; if (a0.u64 == a1.u64) { return 0; } if (a0.u64 > a1.u64) { return a0.u64 - a1.u64; } return a1.u64 - a0.u64; } int main() { double (f)(double, double) = atan2_custom; unsigned long long emax = 0; size_t eother = 0; const unsigned long long threshold = 1; static const double d[] = { // 0, DBL_TRUE_MIN, DBL_MIN, DBL_EPSILON, 1.0, 2.0, 3.0, 5.0, 7.0, DBL_MAX, INFINITY}; size_t n = sizeof d / sizeof d; for (unsigned signx = 0; signx < 2; signx++) { for (size_t xi = 0; xi < n; xi++) { double x = d[xi]; if (signx) x = -x; for (unsigned signy = 0; signy < 2; signy++) { for (size_t yi = 0; yi < n; yi++) { double y = d[yi]; if (signy) y = -y; unsigned long long e = test_atan2(f, x, y); if (e > emax || e > threshold) { if (e > emax) emax = e; printf( "x:% 24.17g y:% 24.17g, a0:% 24.17g a1: % 24.17g: error:%llu\n", x, y, atan2(y, x), f(y, x), e); } else if (e > 0) { eother++; } } } } } printf("Max error:%llu\n", emax); printf("Small other error count:%zu\n", eother); } ``` Output c x: 0 y: -0, a0: -0 a1: 0: error:9223372036854775808 x: inf y: inf, a0: 0.78539816339744828 a1: -nan: error:13839242816598561512 x: inf y: -inf, a0: -0.78539816339744828 a1: -nan: error:4615870779743785704 x: -0 y: 0, a0: 3.1415926535897931 a1: 0: error:4614256656552045848 x: -0 y: -0, a0: -3.1415926535897931 a1: 0: error:13837628693406821656 x:-4.9406564584124654e-324 y: -0, a0: -3.1415926535897931 a1: 3.1415926535897931: error:9223372036854775808 x:-2.2250738585072014e-308 y: -0, a0: -3.1415926535897931 a1: 3.1415926535897931: error:9223372036854775808 x: -2.2204460492503131e-16 y: -0, a0: -3.1415926535897931 a1: 3.1415926535897931: error:9223372036854775808 x: -1 y: -0, a0: -3.1415926535897931 a1: 3.1415926535897931: error:9223372036854775808 x: -2 y: -0, a0: -3.1415926535897931 a1: 3.1415926535897931: error:9223372036854775808 x: -3 y: -0, a0: -3.1415926535897931 a1: 3.1415926535897931: error:9223372036854775808 x: -5 y: -0, a0: -3.1415926535897931 a1: 3.1415926535897931: error:9223372036854775808 x: -7 y: -0, a0: -3.1415926535897931 a1: 3.1415926535897931: error:9223372036854775808 x:-1.7976931348623157e+308 y: -0, a0: -3.1415926535897931 a1: 3.1415926535897931: error:9223372036854775808 x: -inf y: inf, a0: 2.3561944901923448 a1: -nan: error:13832004176781827630 x: -inf y: -0, a0: -3.1415926535897931 a1: 3.1415926535897931: error:9223372036854775808 x: -inf y: -inf, a0: -2.3561944901923448 a1: -nan: error:4608632139927051822 Max error:13839242816598561512 Small other error count:68 Follows matches atan2() in various corner cases. ```c include double atan2_custom_alt1(double y, double x) { bool ysignbit = signbit(y); if (ysignbit) { y = -y; } bool xsignbit = signbit(x); if (xsignbit) { x = -x; } double arctan; if (x > y) { arctan = atan(y/x); } else if (y > x) { arctan = M_PI/2 - atan(x/y); } else if (x == 0) { // zero/zero arctan = 0.0; } else { // positive/zero arctan = M_PI/4; } if (xsignbit) { arctan = M_PI - arctan; } if (ysignbit) { arctan = -arctan; } return arctan; } ``` Perhaps later I'll try coding a better alternative, yet I would first like to know more about OP's functionality and precision requirements. Share Share a link to this answer Copy linkCC BY-SA 4.0 Improve this answer Follow Follow this answer to receive notifications edited May 3 at 2:07 answered Feb 22, 2024 at 5:15 chuxchux 157k 17 17 gold badges 160 160 silver badges 307 307 bronze badges Comments Add a comment This answer is useful 0 Save this answer. Show activity on this post. You can try a vectorized math libary like SLEEF to make use of your processor's SIMD capabilities. Share Share a link to this answer Copy linkCC BY-SA 4.0 Improve this answer Follow Follow this answer to receive notifications answered Feb 5, 2024 at 19:20 Mark AdlerMark Adler 115k 15 15 gold badges 136 136 silver badges 182 182 bronze badges Comments Add a comment Your Answer Thanks for contributing an answer to Stack Overflow! Please be sure to answer the question. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. 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5271
https://www.ijord.com/index.php/ijord/article/view/1513
A study of dermoscopic patterns of Wickham's striae in lichen planus | International Journal of Research in Dermatology Skip to main contentSkip to main navigation menuSkip to site footer Open Menu International Journal of Research in Dermatology Home About About the Journal Submissions Editorial Team Privacy Statement Contact Login Register Articles In Press Current Archives Author Guidelines Search Home/ Archives/ Vol. 8 No. 2 (2022): March-April 2022/ Original Research Articles A study of dermoscopic patterns of Wickham's striae in lichen planus Authors Nandini A. S. Department of Dermatology, Kempegowda Institute of Medical Sciences, Bangalore, Karnataka, India Mariet Zacharias Department of Dermatology, Kempegowda Institute of Medical Sciences, Bangalore, Karnataka, India DOI: Keywords: Lichen planus, Dermoscopy, Wickham’s striae Abstract Background: Wickham striae corresponds to fine white or gray lines or dots seen on the top of the papular rash and oral mucosal lesions of lichen planus. The documented types of Wickham’s striae dermoscopic patterns are linear, reticular, circular, globular, star-burst, radial and leaf-like. Aim was to study and describe the dermoscopic patterns of Wickham’s striae in lichen planus. Methods: A prospective study was conducted between December 2019 to March 2020, among outpatients in the department of dermatology, venereology and leprosy in a tertiary care hospital. The lichen planus lesions including Wickham striae pattern were visualised with the dermatoscope (DermLite DL 4) and documented. Results: The 20% had the circular pattern, 10% radial streaming, 4% leaf venation and 2% starry sky appearance. Other pattern observed were linear which was seen in 6% of the patients. They were seen associated with lesions over the lip. Vascular patterns were noted only in 20% of the patients with most of them having dotted patchy type of vessels. Conclusions: With dermoscopy, the Wickham’s striae are the diagnostic key to differentiate lichen planus from other papulosquamous disorders. In our study we were able to see various patterns of Wickham’s striae including a less reported form like leaf venation. Metrics PDF views 1,320 daily (first 30) | monthly | yearly References James WD, Berger TG, Elston DM. Andrews Diseases of the skin: clinical dermatology. 12th ed. Vol 1. Philadelphia, PA: Elsevier. 2016;201. Steffen C, Dupree ML. Louis-Frédéric Wickham and the Wickham's striae of lichen planus. Skinmed. 2004;3:287-9. Sachdeva S, Sachdeva S, Kapoor P. Wickham striae: etiopathogenensis and clinical significance. Indian J Dermatol. 2011;56(4):442-3. Vázquez-López F, Alvarez-Cuesta C, Hidalgo-García Y, Pérez-Oliva N. The handheld dermatoscope improves the recognition of Wickham striae and capillaries in Lichen planus lesions. Arch Dermatol. 2001;137(10):1376. Friedman P, Sabban EC, Marcucci C. Dermoscopic findings in different clinical variants of lichen planus. Is dermoscopy useful? Dermatol Pract Concept. 2015;5(4):51-5. Makhecha M, Singh T, Malladi N, Rambhia K. Dermoscopic features of various stages of lichen planus. Indian J Dermatol Venereol Leprol. 2020;86:191-4. Güngör Ş, Topal IO, Göncü EK. Dermoscopic patterns in active and regressive lichen planus and lichen planus variants: A morphological study. Dermatol Pract Concept. 2015;5:45-53. Lallas A, Argenziano G. Dermatoscope-the dermatologist’s stethoscope. Indian J Dermatol Venereol Leprol. 2014;80(6):493-94. Rivers JK, Jackson R, Orizaga M. Who was Wickham and what are his striae? Int J Dermatol. 1986;25:611-3. Summerly R, Wilson Jones E. The Microarchitecture of Wickham's Stirae. Trans St. Jhon's Hosp Dermatol Soc. 1964;50:157-61. Ryan TJ. The direction of the growth of the epithelium. Br J Dermatol. 1966;78:403-15. Tan C, Min ZS, Xue Y, Zhu WY. Spectrum of dermoscopic patterns in lichen planus: a case series from China. J Cutan Med Surg. 2014:18(1):28-32. Litaiem N, Mansour Y, Jones M. Dermoscopic signs of lichen planus. Case Rep. 2016;2016:bcr2015213923. Downloads PDF Published 2022-02-24 How to Cite S., N. A., & Zacharias, M. (2022). A study of dermoscopic patterns of Wickham’s striae in lichen planus. International Journal of Research in Dermatology, 8(2), 206–211. More Citation Formats ACM ACS APA ABNT Chicago Harvard IEEE MLA Turabian Vancouver Download Citation Endnote/Zotero/Mendeley (RIS) BibTeX Issue Vol. 8 No. 2 (2022): March-April 2022 Section Original Research Articles 1 1 Prathibha Kuchana, Vishal Gaurav, Rachita Misri (2025) Quiz questions from dermoscopy of inflammatory dermatoses (inflammoscopy). Journal of Skin and Sexually Transmitted Diseases, 7, 140. 10.25259/JSSTD_91_2024 Make a Submission Make a Submission Information For Readers For Authors For Librarians Keywords Current Issue International Journal of Research in Dermatology. Copyright © 2025. ISSN: 2455-4529 medipeditor@gmail.com,editor@ijord.com
5272
https://www.droracle.ai/articles/260470/whats-the-management-for-adrenal-crisis
What is the management for adrenal crisis? ​▼ What is the management for adrenal crisis? Medical Advisory BoardAll articles are reviewed for accuracy by our Medical Advisory Board Educational purpose only • Exercise caution as content is pending human review Article Review Status Submitted Under Review Approved Last updated: August 13, 2025 • View editorial policy Management of Adrenal Crisis Adrenal crisis requires immediate treatment with intravenous hydrocortisone 100 mg bolus followed by continuous infusion of 200 mg over 24 hours, along with rapid fluid resuscitation using isotonic saline.1 Immediate Management First-line Treatment Hydrocortisone administration: Immediate IV/IM hydrocortisone 100 mg bolus 1 Follow with continuous IV infusion of 200 mg over 24 hours 2, 1 If continuous infusion not available, administer 100 mg IV/IM every 6-8 hours until clinical improvement 1 Fluid Resuscitation Rapid administration of isotonic (0.9%) saline: 1000 ml within the first hour 1 Total of 3-4 L as needed based on hemodynamic status 1 Monitor for fluid overload in susceptible patients Monitoring Continuous monitoring of: Hemodynamic parameters (blood pressure, heart rate) Electrolytes (particularly sodium and potassium) Blood glucose Clinical response to treatment Underlying Cause Management Identify and treat precipitating factors: Infections (most common trigger) 3 Gastroenteritis 1 Surgery or trauma Medication omission Emotional stress Transition to Maintenance Therapy Post-Crisis Management Once stabilized and able to take oral medications: Double the regular oral replacement dose of hydrocortisone for 48 hours 2, 1 Continue increased dose for up to one week following major stress events 2 Example: If usual dose is 10-5-5 mg hydrocortisone, increase to 20-10-10 mg 2 Gradually taper to maintenance dose over a period of up to one week 1 Maintenance Therapy Resume regular replacement therapy once recovered: Hydrocortisone 15-25 mg daily in divided doses 1 For primary adrenal insufficiency: add fludrocortisone 50-200 μg daily 1 Prevention of Future Adrenal Crises Patient Education (Critical Component) Instruct patients on: Early recognition of adrenal crisis symptoms 1 Stress dosing protocols for illness or stress 1 Proper use of emergency hydrocortisone injection kit 1 Importance of wearing medical alert identification 1 Carrying a steroid alert card 1 Stress Dosing Protocol Minor illness/stress: Double or triple usual daily dose 1 Moderate stress: Hydrocortisone 50-75 mg/day in divided doses 1 Severe stress: Hydrocortisone 100 mg IV immediately followed by 200-300 mg/day 1 Special Considerations Perioperative Management Pre-operative: Hydrocortisone 100 mg IV just before anesthesia 1 Post-operative: Continue hydrocortisone 100 mg IV every 6 hours until able to eat and drink 1 Common Pitfalls to Avoid Delaying treatment while awaiting diagnostic confirmation - treatment should never be delayed 1 Inadequate fluid resuscitation - hypotension may not respond to steroids alone Failure to identify and treat the underlying trigger Premature reduction of steroid doses before full recovery Inadequate patient education on prevention strategies Adrenal crisis carries significant mortality risk (0.5/100 patient-years) 3, making prompt recognition and aggressive treatment essential for survival. The continuous IV infusion method has been shown to be superior to intermittent bolus administration in maintaining appropriate cortisol levels during major stress 4. References 1 Guideline Adrenal Insufficiency Management Praxis Medical Insights: Practical Summaries of Clinical Guidelines, 2025 2 Guideline Guideline Directed Topic Overview Dr.Oracle Medical Advisory Board & Editors, 2025 3 Research Extensive expertise in endocrinology. Adrenal crisis. European journal of endocrinology, 2015 4 Research Prevention of Adrenal Crisis: Cortisol Responses to Major Stress Compared to Stress Dose Hydrocortisone Delivery. The Journal of clinical endocrinology and metabolism, 2020 Related Questions What is the appropriate management for an adrenal crisis?What is the treatment for an adrenal crisis?Can we administer D25 (50% dextrose solution) infusion in an adrenal crisis?What is the management of acute adrenal crisis?What is the dosing regimen for stress dosing of Solucortef (hydrocortisone) in adrenal insufficiency?What is the protocol for rheumatoid arthritis treatment?How can I adjust Torsemide and Metolazone to improve impaired renal function?Is neurosyphilis treatable?What is the treatment for balanitis?What is the management approach for a retrograde P wave?What is the first line treatment for anemia? Professional Medical Disclaimer This information is intended for healthcare professionals. Any medical decision-making should rely on clinical judgment and independently verified information. The content provided herein does not replace professional discretion and should be considered supplementary to established clinical guidelines. Healthcare providers should verify all information against primary literature and current practice standards before application in patient care. Dr.Oracle assumes no liability for clinical decisions based on this content. Have a follow-up question? Our Medical A.I. is used by practicing medical doctors at top research institutions around the world. Ask any follow up question and get world-class guideline-backed answers instantly. Ask Question Original text Rate this translation Your feedback will be used to help improve Google Translate
5273
https://feaforall.com/calculate-safety-factor/
from basics to Advanced Safety factor: How do I calculate that? This article talks explains the concept of safety factor and how it is calculated. It’s a fundamental notion that every mechanical engineer have to understand well. How do I know if the design of my part is « safe » enough? That’s a difficult question that many designers asked themselves. One of the potential answers involves the measure of the safety factor. For certain people, this notion is still not well understood, so I decided to write about it! Important note: This article talks mainly about “stress-based” safety factors, but you should know that there are different definitions of “safety factor” margins, which are not all necessarily related to the state of stress. Additionally, stress definitions can vary… there is Von Mises Stress, but also Tresca, etc…. (Thanks to Boyd McKay for his contribution in improving that article by mentioning those) Getting safe designs First of all, I think it is not too much to remind that one of the purpose of simulation is to get safe designs… When a part fails, it involves of course a risk for the life of people, but also a huge financial loss for the company who created this part (just think about the explosive Samsung batteries and you will understand what I am talking about) FEA simulation helps to understands why a design fails, where it failed and how to improve it. That’s why FEA is so important for companies who design products. To assess the safety of a design, designers need a simple factor which will help in understanding if a design is safe enough. This factor is called the safety factor. How is the safety factor calculated The definition of the safety factor is simple. It is defined as the ratio between the strength of the material and the maximum stress in the part. When the stress in a specific position becomes superior to the strength of the material, the safety factor ratio becomes inferior to 1, this when there is danger What it tells us basically is that in a specific area of the model, the stress is higher than the strength the material can bear. When the stress in the model remains much inferior to the strength of the material, the safety factor stays superior to 1 and the model is « safe ». Keep in mind that if the safety factor is way superior to 1 everywhere in your model, this is also indicating that your part may be over-engineered. In this case, this is not desirable either, because you are just wasting material resources and increasing the cost. Now, let’s talk about the 2 important values that you need to calculate this safety factor: Stress and Strength What is stress ? If you still have some doubts about that, no shame, it’s not a concept easy to grasp for beginners, but it is an essential one. In short, stress is a value that mesure the inner pressure inside a solid which is cause by an external loading. If stress is too high inside a part, the part may fail. The notion of stress is not so different with what we experience everyday at work… When we receive a load of work, we become stressed. If we are too stressed, we may experience a nervous breakdown and many health problems. If you want to understand more about stress and how stress is actually calculated, I wrote a full article about that few months ago. Read the article: What is stress What is the strength of a material? Stress and Strength are different and that’s where many people don’t get it. Stress in a body is always a function of the applied loading and cross-section, whereas strength is an inherent property of the body’s material/ manufacturing process. Strength is obtained similarly to other material properties, by doing for example a standard tensile test which subjects a sample bar to uniaxial stress. Then we can draw the material stress-strain curve by extracting the deformation data and plotting it in function of the load data. Note that if you need some high accuracy, the test should be performed under conditions similar to the operating conditions of the part or the system (Temperature, strain rate, material grain, flow direction,…) There are several important points to understand on this curve: The point P is the proportional limit, it limits the portion of the curve which governed by Hooke’s law The point E is the elastic limit. The material will continue to behave elastically up to point E, but stress and strain won’t be proportional anymore. The point Y is the yield point which corresponds to the yield strength of the material The point U indicates the maximum stress that can be achieved by the material. It corresponds to its ultimate or tensile strength. The point F is the fracture point. Note that the points E and Y may coincide for some types of materials such as ferrous materials. The yield point is not necessarily very clear, and it is generally obtained by an offset method: Y is considered to be the intersection of an offset line, parallel to the linear portion of the stress-strain curve typically at 0.002 axial strain, and the plastic portion of the curve. As you read, there are several material strength values: the yield strength, the ultimate strength and the fracture strength. The safety factor is calculated with the yield strength so this is the parameter you need to know in priority. Is this ratio a perfect indicator of a model safety? I’d like to say that nothing is really perfect… As engineers, we have to learn to live with errors ;-) Errors are everywhere: In the testing process that will provide you with the stress-strain material curve and the yield strength used to calculate the safety factor In the FE model that you build, it is probable that the boundary conditions and/or the meshing will cause a certain amount of error In the FEA software itself and the algorithms it uses, error is included (and hopefully controlled) That’s why it’s always better to consider a safety factor which is not exactly 1, but maybe a little higher (2-3) depending on the hypothesis you take. Additional note:The safety factor only describe material failure. In some designs, it is sufficient, but if you are designing a slender element some form of stability failure (i.e. buckling) may occur. Such safety factor do not take that into account since buckling can happen when stress is much smaller than limit stress of the material. Comments worth mentionning from other FEA specialists David Backhouse (Backhouse Technical Service LTD): There is no set ‘safety factor’ as such. That is too simple a concept as there are many modes of failure. There are instances, for example, where stresses above yield are acceptable. You should really refer to the appropriate design standard for the structure, its use, and the classification of the stresses. Karl Van Aswegen (Fluid Codes FZ LLE): Safety Factors are not necessarily max stress/yield. Many times the industry you work in will dictate how you calculate design safety factors. More often than not fatigue is your biggest problem not yield. Eric Lee (Austal): In my opinion, safety factors are really only important in certain cases. In the industries that I’ve worked in (shipbuilding/offshore) safety factors are hardly the criteria we work to. We do have our allowable stresses, but safety factors never govern because there are always stress concentrations that are allowed to be waived because of the geometry, mesh size/aspect ratio, loading conditions, etc. The only real place where safety factors absolutely drive the design is in lifting applications where you need a SF of 3 to 5. That said, it’s always a good thing to check, especially if you’re doing approximate hand calculations. Jeff Finlayson (Boeing): It is best to understand what a safety factor really is and understand what the requirements actually state. In general, the ultimate safety factor is Ultimate load/applied load, and for yield SF is Yield load/applied load. Stresses may not be linearly related to the load due to local plasticity effects. Static preloads may receive no safety or a small one depending on requirements for uncertainties. Vlad Kerchman (Independent Consultant): In evaluating the possible ultimate loads/stresses people frequently look for quasi-static or steady-state conditions with extreme overload or bias. In real service/ life it typically happens under drastic change of loading – in dynamics, say, vehicle impacting a road obstacle or bump, or seismic loading on a structure, explosion, etc.. That’s where FEA can really help to replace difficult and expensive testing. Neil Grant (Allen-Vanguard): I think safety factor needs to be expressed with respect to something. For a one time use, it can be with respect to ultimate strength. Similarly, for many cycles, it can be expressed with respect to the fatigue limit strength. //////////////////////////////////////////////////////////////////////////////////// Do you like this article? Was it useful for you to understand the concept of safety factor? If it was useful and you think that others can benefit for it, can I ask you to share it with your network on Linkedin or facebook? Also, don’t hesitate to write me a comment, so I know it was useful for you and let me know what else you would like to learn about more in details! If you are a beginner in FEA simulation, read this article. Comments Abhinav T says Excellent explanation. Thanks. Reply Cyprien says You’re welcome Abhinav! :) Reply sunil says nicely explained sir ,thank you Reply Cyprien says You’re welcome Sunil! Reply 2. Andrei Dragoș says It very interesting and easy to understand. What your article do not touch enough is when we have overstressed area in the part. How big those can be and element can be cosiderred safe? How big the stress can be in those areas but the part can still considered safe? In the FEA analysis it is very rare the situation when the entire part stress is below yielding point. This is the hardest part in analysis of the model, to know house far you can go and to keep the model in the safety limits. Thank you for you article and a very nice Holidays. Reply Cyprien says Thanks for taking the time to posting those questions Dragos! It would indeed be interesting to go further and talk about that as well… maybe in a next article ;-) Reply Vinny says Superb Reply Vinny says Can you touch on how safety factors are determined and how reliable they are ? What work is done before we calculate stress expected / maximum stress ? In terms of determining material properties, are book values used ? These can vary a lot. What if failure is simply not an option, how to we be 100% accurate on material properties ? Instead of using book values ? How does all this affect how reliable safety factors are? How do safety factors include degrading of the material, possible overloading, miss use, there are many factors which make safety factors unreliable. Reply 3. Khashayar says The stress and strain must be of the same type and unit and must be referring to the same (critical) point. Reply Cyprien says You’re totally right. I just didn’t want to complexify the explanation, but there are more factors to take in account! Reply 4. Vali says Nice article in simple language! Choosing the right safety factor depends on application of the part which is being analysed and the impact that failure creates in the field. It would be nice to give more info about industry practices and standards we usually follow in choosing safety factor. Thank you in advance! Reply 5. Gaurang Dave says Nice and informative article and also comments of experts. Reply 6. aizacky says Thank you so much. God bless you Reply 7. Moataz Aly says Hey I am designing a Bicycle Trailer and I need to select a proper Factor of safety according to any Standard, I am in Germany So you know which standard I should follow and If you have the link for it I will be Glad. Thanks in advance Reply 8. john says Thanks Cyprien Reply 9. Abdul Waheed Mahar says Very nice way of explaining Reply 10. mechanic410 says Excellent Reply longpham says Excuse me, Mr. Cyprien. Is the Safety factor true with the compressive cases? For example, when I carry out an experiment about compressive of connecting rod in the engine, I want to evaluate its safety, so can I use the Safety factor with custom value in Stress limit type? Thank you Reply Cyprien says It depends on your system. Some materials do not behave the same in tension and compression (like concrete) so you have to be careful with that. Also make sure your system isn’t failing for another reason like buckling for example. Reply 12. Phd_2022 says When a part fails, it involves of course a risk for the life of people, but also a huge financial loss for the company who created this part (just think about the explosive Samsung batteries and you will understand what I am talking about) Keep in mind that if the safety factor is way superior to 1 everywhere in your model, this is also indicating that your part may be over-engineered. In this case, this is not desirable either, because you are just wasting material resources and increasing the cost. Reply 13. Majd Thabit says Excellent work, what is your recommendation for safety factors for calculating strength in facilities for a person such as a leg prosthesis I will share the artical to my students thank you Reply Trackbacks […] awesome is that? But the concept of uncertainty is definitely not limited to them – simply check out this post about mechanical engineering for an […] Reply Leave a Reply Cancel reply Read previous post: What is frequency response analysis in FEA In the last article, I described in depth Modal Analysis, Eigenvalues and all what that means and in this article,... Close
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https://www.merriam-webster.com/thesaurus/oppose
OPPOSE Synonyms: 57 Similar and Opposite Words | Merriam-Webster Thesaurus Chatbot Chatbot Games Word of the Day Grammar Word Finder Slang NewNewsletters Wordplay Rhymes Thesaurus Join MWU More Games Word of the Day Grammar Wordplay Slang Rhymes Word Finder Newsletters New Thesaurus Join MWU Shop Books Merch Log In Username My Words Recents Account Log Out Est. 1828 Thesaurus as in toresist as in tofight as in to resist as in to fight Synonym Chooser Example Sentences Entries Near Cite this Entry Citation Share More from M-W Show more Show more Citation Share More from M-W Save Word To save this word, you'll need to log in.Log In Synonyms of oppose oppose verb ə-ˈpōz Definition of oppose 1 as in to resist to refuse to give in to I will continue to oppose any attempts to infringe upon our civil liberties Synonyms & Similar Words Relevance resist fight withstand repel defy thwart contradict combat contend (with) hinder challenge buck counter contest obstruct dispute frustrate stem balk check battle foil baffle Antonyms & Near Antonyms submit (to) yield (to) surrender (to) succumb (to) bow (to) give in (to) stoop (to) capitulate (to) knuckle under (to) 2 as in to fight to strive to reduce or eliminate we must oppose ignorance and prejudice wherever and whenever they arise Synonyms & Similar Words fight combat resist confront counter withstand battle contend (with) thwart oppugn frustrate face foil defy baffle meet checkmate Antonyms & Near Antonyms encourage promote forward advance cultivate foster further suffer endure support abide bear nurture endorse nourish advocate uphold back indorse champion See More Synonym Chooser How does the verb oppose differ from other similar words? Some common synonyms of oppose are combat, resist, and withstand. While all these words mean "to set oneself against someone or something," oppose can apply to any conflict, from mere objection to bitter hostility or warfare. opposed the plan When can combat be used instead of oppose? Although the words combat and oppose have much in common, combat stresses the forceful or urgent countering of something. combat disease When is resist a more appropriate choice than oppose? While in some cases nearly identical to oppose, resist implies an overt recognition of a hostile or threatening force and a positive effort to counteract or repel it. resisting temptation In what contexts can withstand take the place of oppose? The words withstand and oppose can be used in similar contexts, but withstand suggests a more passive resistance. trying to withstand peer pressure Example Sentences Examples are automatically compiled from online sources to show current usage.Read More Opinions expressed in the examples do not represent those of Merriam-Webster or its editors. Send us feedback. Recent Examples of oppose White told The Star that the recall effort didn’t seriously concern him until April 2024, when White publicly opposed a ballot measure that failed but if passed would have helped pay for a new Royals ballpark and renovations for Chiefs.—Ilana Arougheti, Kansas City Star, 27 Sep. 2025 In addition to a frosty reception from heads of state opposed to Israel's ongoing military campaign in Gaza, Netanyahu is expected to face protests while in New York.—ABC News, 26 Sep. 2025 Kris Seals opposed the project, saying that other apartment developments in the city, such as the upscale The Ruby at Brookfield Square, should already be considered an affordable housing option.—Bridget Fogarty, jsonline.com, 26 Sep. 2025 The team has developed an atomically thin material that allows two opposing magnetic forces to coexist, reducing energy use in memory devices by a factor of ten.—Neetika Walter, Interesting Engineering, 26 Sep. 2025 See All Example Sentences for oppose Recent Examples of Synonyms for oppose 1. resist 2. fight 3. combat Verb Criminal court records reviewed by PEOPLE show the 35-year-old faces charges of public intoxication, disorderly conduct, resisting arrest and a false report. — Colson Thayer, PEOPLE, 27 Sep. 2025 But previous presidents have resisted the temptation to exploit them, and the law has served as a powerful norm notwithstanding its weaknesses. — Elizabeth Goitein, Time, 27 Sep. 2025 Definition of resist Verb Two consecutive completions to rookie tight end Oronde Gadsden, who fought through contests and contact on both catches. — Daniel Popper, New York Times, 22 Sep. 2025 The poll found opinions on the administration's expansion of its deployment of military forces to fight crime more divided, with 42% supporting the idea and 46% opposing it. — Kathryn Palmer, USA Today, 21 Sep. 2025 Definition of fight Verb To combat that challenge, many younger buyers are considering buying (or continuing to rent) with friends or family. — Sydney Lake, Fortune, 24 Sep. 2025 Eilish will also donate $1 from each ticket sold to Reverb, the organization dedicated to providing support to combat food insecurity and climate change causes. — Larisha Paul, Rolling Stone, 24 Sep. 2025 Definition of combat Browse Nearby Words opportunity oppose opposed See all Nearby Words Cite this Entry Style “Oppose.” Merriam-Webster.com Thesaurus, Merriam-Webster, Accessed 28 Sep. 2025. Copy Citation Share More from Merriam-Webster on oppose Nglish: Translation of oppose for Spanish Speakers Last Updated:28 Sep 2025 - Updated example sentences Love words? Need even more definitions? Subscribe to America's largest dictionary and get thousands more definitions and advanced search—ad free! Merriam-Webster unabridged More from Merriam-Webster ### Can you solve 4 words at once? Play Play ### Can you solve 4 words at once? Play Play Word of the Day kerfuffle See Definitions and Examples » Get Word of the Day daily email! 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https://www.medmastery.com/magazine/how-measure-pulsus-paradoxus?srsltid=AfmBOorOEkAOSTRP-8w02la6XfbKcqnu_RJndI-YEvHyokIU3Uq-cBq0
How to measure pulsus paradoxus Discover the key to confidently diagnosing cardiac tamponade with this quick guide on measuring pulsus paradoxus. When faced with a patient possibly suffering from cardiac tamponade, obtaining a pulsus paradoxus measurement can be a game-changer. It's a vital tool in confirming the diagnosis, and distinguishing tamponade from pericardial effusion or other conditions like asthma or COPD exacerbations. Preferred method Here’s a step-by-step guide to measuring pulsus paradoxus: Understanding the numbers The difference in pressures reflects the dynamics of inspiration and expiration. During expiration, higher pressure allows blood flow into the artery, producing the Korotkoff sound. Conversely, during inspiration, lower arterial pressure halts blood flow, resulting in silence. Diagnosing cardiac tamponade Mastering the art of measuring pulsus paradoxus empowers clinicians in swiftly diagnosing cardiac tamponade, saving critical time in emergencies. Stay tuned for more insights into tamponade diagnosis with electrocardiogram and echocardiogram findings. Unlock the secrets of cardiac emergencies with our comprehensive course. Start the first chapter of our Cardiac Tamponade course for free About the author Become an expert Cardiac Tamponade Related articles [UPDATE] ECG Mastery Yellow and Blue Belt courses Ultrasound of Aortic Endografts: Postoperative Protocol Unveiled Identifying Abdominal Aortic Dissections
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https://community.powerbi.com/t5/Desktop/calculate-sum-based-on-distinct-values-of-another-column/td-p/2255429
skip to main content calculate sum based on distinct values of another column powerbiss Helper I Mark as New Bookmark Subscribe Mute Subscribe to RSS Feed Permalink Print Report Inappropriate Content calculate sum based on distinct values of another column ‎12-24-2021 07:20 AM Hi, Is it possible to get some help in creating a DAX formula for the attached problem that I came across in power bi. I have two columns in my sample database (expense_id and amount). I have duplicate values in expense_id column and as well as duplicate values in amount column. I want to sum the amount column that has distinct values in expense_id column. I have attached a screenshot. Really appreciate all your help. Thanks Solved! Go to Solution. Labels: Labels: Need Help Message 1 of 8 18,591 Views 0 Reply 1 ACCEPTED SOLUTION smpa01 Super User In response to powerbiss Mark as New Bookmark Subscribe Mute Subscribe to RSS Feed Permalink Print Report Inappropriate Content ‎12-24-2021 07:46 AM @powerbiss not sure what you meant Did I answer your question? Mark my post as a solution! Proud to be a Super User! My custom visualization projects Plotting Live Sound: Viz1 Beautiful News:Viz1, Viz2, Viz3 Visual Capitalist: Working Hrs Others:Easing Graph, Animated Calendar View solution in original post Message 4 of 8 18,546 Views 0 Reply All forum topics Previous Topic Next Topic 7 REPLIES 7 parry2k Super User Mark as New Bookmark Subscribe Mute Subscribe to RSS Feed Permalink Print Report Inappropriate Content ‎12-24-2021 07:46 AM @powerbiss here is the measure: Unique Sum Amount = SUMX ( SUMMARIZE ('Table','Table'[expense_id], 'Table'[amount] ), 'Table'[amount] ) ✨ Follow us on LinkedIn Learn about conditional formatting at Microsoft Reactor My latest blog post The Power of Using Calculation Groups with Inactive Relationships (Part 1) (perytus.com) I would ❤ Kudos if my solution helped. 👉 If you can spend time posting the question, you can also make efforts to give Kudos to whoever helped to solve your problem. It is a token of appreciation! ⚡ Visit us at your one-stop-shop for Power BI-related projects/training/consultancy.⚡ Subscribe to the @PowerBIHowTo YT channel for an upcoming video on List and Record functions in Power Query!! Learn Power BI and Fabric - subscribe to our YT channel - Click here: @PowerBIHowTo If my solution proved useful, I'd be delighted to receive Kudos. When you put effort into asking a question, it's equally thoughtful to acknowledge and give Kudos to the individual who helped you solve the problem. It's a small gesture that shows appreciation and encouragement! ❤ Did I answer your question? Mark my post as a solution. Proud to be a Super User! Appreciate your Kudos 🙂 Feel free to email me with any of your BI needs. Message 7 of 8 18,546 Views 4 Reply Syndicate_Admin Administrator In response to parry2k Mark as New Bookmark Subscribe Mute Subscribe to RSS Feed Permalink Print Report Inappropriate Content ‎03-15-2023 07:15 AM Source Community: Power BI Spanish | Source Author Name: PVVBl00 Hello good afternoon, which is what I should do if I want to add the values of more than 1 single column, as shown in the example, what should I do if I have 2 columns called amount Message 8 of 8 18,136 Views 0 Reply smpa01 Super User Mark as New Bookmark Subscribe Mute Subscribe to RSS Feed Permalink Print Report Inappropriate Content ‎12-24-2021 07:26 AM @powerbiss try this measure Measure = SUMX(GROUPBY('Table','Table'[expense id],'Table'[amount]),[amount]) Did I answer your question? Mark my post as a solution! Proud to be a Super User! My custom visualization projects Plotting Live Sound: Viz1 Beautiful News:Viz1, Viz2, Viz3 Visual Capitalist: Working Hrs Others:Easing Graph, Animated Calendar Message 2 of 8 18,561 Views 0 Reply Syndicate_Admin Administrator In response to smpa01 Mark as New Bookmark Subscribe Mute Subscribe to RSS Feed Permalink Print Report Inappropriate Content ‎03-15-2023 07:19 AM Source Community: Power BI Spanish | Source Author Name: PVVBl00 Hello good afternoon, which is what I should do if I want to add the values of more than 1 single column, as shown in the example, what should I do if I have 2 columns called amount Message 6 of 8 18,134 Views 0 Reply powerbiss Helper I In response to smpa01 Mark as New Bookmark Subscribe Mute Subscribe to RSS Feed Permalink Print Report Inappropriate Content ‎12-24-2021 07:37 AM Hello @powerbiss For some reason it is showing 285.80 in the all rows. It is not showing 250 for 17 and 35.8 for 19. Message 3 of 8 18,557 Views 0 Reply smpa01 Super User In response to powerbiss Mark as New Bookmark Subscribe Mute Subscribe to RSS Feed Permalink Print Report Inappropriate Content ‎12-24-2021 07:46 AM @powerbiss not sure what you meant Did I answer your question? Mark my post as a solution! Proud to be a Super User! My custom visualization projects Plotting Live Sound: Viz1 Beautiful News:Viz1, Viz2, Viz3 Visual Capitalist: Working Hrs Others:Easing Graph, Animated Calendar Message 4 of 8 18,547 Views 0 Reply powerbiss Helper I In response to smpa01 Mark as New Bookmark Subscribe Mute Subscribe to RSS Feed Permalink Print Report Inappropriate Content ‎12-24-2021 09:22 AM Hey @smpa01 That worked. had some issues with creating the relationship with the tables. Message 5 of 8 18,524 Views 0 Reply Helpful resources Announcements Power BI Monthly Update - August 2025 Check out the August 2025 Power BI update to learn about new features. Learn more Fabric Community Update - August 2025 Find out what's new and trending in the Fabric community. Read more PlayStop PlayStop PreviousNext View All Recommendations | | Subject | Author | Posted | --- --- | | | SUM based on distinct values ​​from another column | | ‎01-23-2023 06:35 AM | | | calculate sum based on another column distinct va... | Anonymous | ‎06-17-2021 04:20 AM | | | Create New Table Based on DIstinct Values of One C... | | ‎03-23-2022 03:58 PM | | | calculate sum distinct values based on another co... | Anonymous | ‎07-14-2021 09:00 PM | | | Find Most Recent Date Relative to another Date Col... | | ‎08-07-2025 07:58 AM | Auto-suggest helps you quickly narrow down your search results by suggesting possible matches as you type.
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https://old.maa.org/press/periodicals/convergence/geometrical-representation-of-arithmetic-series-terminology-and-trapezia
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E. Shaw Group AMC 8 Awards & Certificates Maryam Mirzakhani AMC 10 A Awards & Certificates Two Sigma AMC 10 B Awards & Certificates Jane Street AMC 12 A Awards & Certificates Akamai AMC 12 B Awards & Certificates High School Teachers News Our Blog MAA Social Media RSS You are here Home » MAA Publications » Periodicals » Convergence » Geometrical Representation of Arithmetic Series – Terminology and Trapezia Geometrical Representation of Arithmetic Series – Terminology and Trapezia ‹ Geometrical Representation of Arithmetic Series – Introduction up Geometrical Representation of Arithmetic Series – The Twisted Trapezium › Author(s): Gautami Bhowmik (Université de Lille) The systematic study of series, śreḍhī-vyavahāra (literally "series-practices") in Sanskrit, forms part of texts of arithmetic and mensuration called pāṭīgaṇita. Here we will consider finite arithmetic progressions where the difference between two consecutive entries is always the same. To explain the historical descriptions, we will use modern notations when necessary. We note that the original texts contain no mathematical symbols and descriptions are given entirely in words. Thus the sum of a series is described as follows (Pāṭīgaṇita, v. 85.1): The common difference multiplied by one-half of the number of terms minus one, increased by the first term, then multiplied by the number of terms, gives the sum of the series. The Sanskrit terms used here are ādi for the first term, caya for the common difference, pada for the number of terms, and gaṇita for the sum. Rewritten in terms familiar to us: a generic arithmetic series containing (n) terms, beginning with a first term (a) and increasing by a common difference (d,) has the sum (S_n,) given by [S_n=\left(\left(\frac{n-1}{2}\right)\,d +a\right)n.] We now recognize the familiar result! Associated to a series (S,) we will describe a geometric figure whose total area equals (S_n) and whose partial areas equal the partial sums of the series (the units are neglected, since an area and a sum have necessarily different units). Such a figure is known as śreḍhī-kṣetra, which literally means “series-area.” Notice the following comparison, which brings to mind our teacup (Pāṭīgaṇita, v. 79): As for an earthen pot [śarāva in Sanskrit], the width of the base is smaller and the top larger, so is it for a series-area. Figure 5. Geometric figure representing the sum of an arithmetic progression, or “series-area” If we geometrically represent each term of the series by a rectangle with the same length as that term and with breadth 1, then the sum of the areas of the pile of rectangles would give the sum of the series. The representation in Figure 5 looks more like the front view of a staircase than a regular geometric figure. Figure 6. Staircase in Lille, France But what happens when we rearrange the rectangles a little? Imagine inverting the figure (so it looks more like an earthen pot or teacup) and smoothing it by drawing a line to remove the protruding parts (green) of the rectangles on the right and pasting these “extra” triangles on the other side (red), as shown in Figure 7. Now we get a trapezium with the same area as that of the staircase. Figure 7. “Staircase” shape inverted and smoothed into a trapezium Figure 8. Trapezium with area equal to the sum of the arithmetic progression Furthermore, each term of the series, beginning with the first, is represented as the area of a trapezium. The following text describes this correspondence for the simplest case, i.e. when there is one term (Pāṭīgaṇita, v. 81). The number of terms which is one is the altitude [lamba] of the series-figure, the first term of the series diminished by half the common difference of the series is the base [dharā], and [the base] increased by the common difference is the face [vaktra]. Let (b,) (f,) and (h) be the base (lower edge in Figure 8), face (top edge in Figure 8), and altitude (or height) of the figure. Then (b=a-\frac{d}{2}) and (f=b+d=a-\frac{d}{2}+d) for (h=n=1.) This is precisely what we have obtained and the trapezium obtained does resemble a two-dimensional teacup! We emphasize the fact that these descriptions were written in verse with strict metrical rules (lost in translation, alas!) meant to be learnt by heart. Thus they are cryptic and more “rules” than “proofs.” The reader must convince himself that these rules are indeed valid. For instance the rearrangement we described above (which is a reconstruction of how the ancients might have argued) can be symbolically written as | | | --- | | (S_n) | (= a+(a+d)+\cdots +(a+(n-1)d)) | | | (={\left(a-\frac{d}{2}\right)}+{\frac{d}{2}}+{\left(a+d-\frac{d}{2}\right)}+{\frac{d}{2}}+\cdots+{\left(a+(n-1)d-\frac{d}{2}\right)} +{\frac{d}{2}}.) | To each term of a given series is associated a rectangle (in blue) with height 1 and two triangles (in red) the sum of whose areas is the corresponding term of the series. To obtain the area of the series-figure we use the following simple fact (Pāṭīgaṇita, v. 85.2): The area [phala] of the [corresponding] series-figure is equal to the product of half of the sum of the base and face, and the altitude. This is the familiar formula for the area of a trapezium. Thus, when the base is (a-\frac{d}{2},) the face (a-\frac d{2}+nd,) and the altitude (n,) we get the area [\frac 1 {2}\left(a-\frac d{2}+a-\frac d{2}+nd\right)n=\left(a+(n-1)\frac{d}{2}\right) n=S_n.] The area of each rectangle with its two triangles gives the corresponding term of the series. Thus the lowest rectangle of length (a-\frac {d}{2}) and height (1) is (a-\frac{d}{2}.) The sum of the areas of the two triangles is (\frac d {2}) and these two areas add up to give (a, ) the first term of the series. The area of the trapezium above the lowest one is (\frac 1{2}(a+\frac d{2}+a+\frac {3d}{2})=a+d,) the second term of the series, and so on. Notice that if we used the first term of the series as the base and the (n)th term as the face of the trapezium, its area would still give the sum of the (n) terms but the previous terms would no longer correspond nicely to its subsections. Figure 9. Series-area, or series-figure, with base (of length (a)) not reduced Gautami Bhowmik (Université de Lille), "Geometrical Representation of Arithmetic Series – Terminology and Trapezia," Convergence (December 2015) Convergence Tags: History of Mathematics Non-Western Cultures Printer-friendly version Dummy View - NOT TO BE DELETED Get Ready: Our Brand New Website is Coming Soon! 2024 MAA Awards & Prize Winners Announced! Register for our OPEN Math Summer Workshops Register for MathFest 2024! 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5278
https://www.thieme-connect.de/products/ebooks/pdf/10.1055/b-0038-160826.pdf
Share / Bookmark Download PDF 3.2.3 Tension band principle Book Editors: Buckley, Richard E; Moran, Christopher G; Apivatthakakul, Theerachai Authors: Rüedi, Thomas P; Arraf, John; Babst, Reto; Balogh, Zsolt J; Barbosa, Paulo; Barla, Jorge Daniel; Baumgaertel, Friedrich; Bernstein, Brian; Blauth, Michael; Borens, Olivier; Campell, Douglas A; Capo, John T; Chong, Keenwai; Boer, Piet de; Dhillon, Mandeep; Dimai, Hans Peter; Forward, Daren; Giannoudis, Peter; Gosch, Markus; Grujic, Les; Gueorguiev-Rüegg, Boyko; Hahn, David M; Harder, Yves; Hessmann, Martin H; Höntzsch, Dankward; Hunter, James B; Jiang, Chunyan; Kastelec, Matej; Kates, Stephen L; Kellam, James F; Kfuri, Mauricio; Khaled, Sherif A; Kreder, Hans J; Kwek, Ernest; Lee, Mark A; Liu, Fan; Luger, Thomas J; Luo, Cong-Feng; Ma, Ching-Hou; McKee, Michael; Mosheiff, Rami; Nijs, Stefaan; Nousiainen, Markku; Odat, Mahmoud M; Oh, Chang-Wug; Oh, Jong-Keon; Pesántez, Rodrigo; Phornphutkul, Chanakarn; Porteous, Matthew; Richards, R. Geoff; Ring, David; Schütz, Michael; Simmermacher, Rogier KJ; Sirkin, Michael S; Slongo, Theddy; Smith, R Malcolm; Snape, Susan; Sommer, Christoph; Stannard, James; Stoddart, Martin; Taha, Wa'el; Williams, John R Title: AO Principles of Fracture Management Subtitle: Vol. 1: Principles, Vol. 2: Specific fractures Subtitle: Vol. 1: Principles, Vol. 2: Specific fractures Print ISBN: 9783132423091; Online ISBN: 9783132423107; Book DOI: 10.1055/b-006-149767 Third Edition © 2018 Georg Thieme Verlag KG Georg Thieme Verlag KG, Stuttgart Subjects: Orthopaedics and Trauma Surgery Thieme Clinical Collections (English Language) Preview pages
5279
https://prowritingaid.com/analogy
Cookie Policy The Grammar Guide Analogy: Definition & Meaning (with Examples) Analogy is one of the most common types of literary devices. It's also one of the hardest to understand because it's similar to other types of figurative language. Today, we're going to dive into the meaning of analogy with in-depth explanations and examples. Analogy Definition: What Is an Analogy? Let's start with the dictionary definition of an analogy. According to Merriam-Webster, an analogy is "a comparison of two otherwise unlike things based on a resemblance of a particular aspect." We use analogies all the time in speaking and fields like history and science. They help us illustrate a point that might be hard to comprehend. For example, we might make an analogy between the Trail of Tears of U.S. History and the Jewish Diaspora of World History. In biology, you might discuss the analogous relationship of bat wings and bird wings. As a literary device, however, analogy's meaning has more nuance that separates it from other types of rhetorical devices. Let's look at the literary meaning of analogy in more detail in the next section. Analogy Meaning As a rhetorical device, analogy compares two unlike things with the purpose of both illustrating a comparison and explaining it. You aren't just trying to show a similarity when you use an analogy. You are also trying to make a point about this similarity. Analogies can be useful to explain complex concepts by comparing them to a familiar idea. Analogies also help paint a picture in a reader's mind and add emphasis to important ideas in writing. Let's take a look at a popular example of an analogy from the movie Forrest Gump. In the movie, Forrest says that his mother always told him, "Life is like a box of chocolates—you never know what you're gonna get." If Forrest just said that life was like a box of chocolates, we would wonder what the similarity is. What point is he trying to get across? It might be that life is sweet or that life is a gift from someone who loves you. But then he explains that we never know where life will take us or what circumstances we will fall into. We don't know until we get there; we can't see the future. He's not just painting a picture about life. He's making a point about the uncertainty of life and the many twists and turns it takes. This analogy goes further and illustrates the entire premise of the film. Forrest goes from being a boy in leg braces to an international ping pong champion to a dad. No one could have predicted that! Analogies look similar to other types of figurative language. So, what's the difference? Analogy vs. Metaphor A metaphor is a figure of speech that shows a likeness between two otherwise different things. The point of a metaphor is comparison. For example, we can say, "the kids were a bunch of monkeys today." We are comparing kids to monkeys. An analogy not only compares but explains. "The kids were a bunch of monkeys today, climbing all over the furniture, running all over the house, and shrieking." As you can see, an analogy might feature a metaphor, but it goes further in making a point. This is also different from an extended metaphor. An extended metaphor continues to use a comparison to illustrate similarities of two objects. An analogy requires some explicit explanation to make its point. Analogy vs. Simile A simile is a type of metaphor, but it uses "like" or "as" to draw comparisons. Just like with a metaphor, an analogy might use a simile to compare two things, but then the analogy goes on to explain the idea behind it. The Forrest Gump quote is an example of this. The part of the quote, "Life is like a box of chocolates," is a simile. The next part of the quote that tells how life and a box of chocolates are related is what makes this quote an analogy. Many analogies use similes and metaphors to make comparisons, but it is not required. Analogy vs. Allusion Another figurative language element that is easy to confuse with analogy is allusion. Allusion is a mention of a person, place, or thing that is considered common knowledge. It's often a reference to a famous person or event or a well-known story, like fairy tales, myths, or religious parables. Allusion is a way to compare two things. Let's look at an example: "The books on the top shelf were forbidden fruit." Forbidden fruit refers to the fruit of the Tree of Knowledge in the Bible. This is an allusion. We can draw enough conclusions from this allusion to understand what the books represent. Allusions can be part of analogies, too. Remember, where allusions compare, analogies explain: "If the library was Eden, the books on the top shelf were forbidden fruit. They opened my eyes to a world beyond the life I had always known." All of these are useful types of figurative language. The difference lies in the purpose. If the goal is to explain an idea or get a specific point across, it's an analogy. A grammar guru, style editor, and writing mentor in one package. Try it for free! Analogies, Idioms, and Clichés Sometimes analogies become so well-known that they become part of our everyday lexicon. Idioms are phrases that don't make sense literally, but they do make sense figuratively. Overused idioms can become clichés. An example of an analogy that is a cliché is "she's as blind as a bat." It's a very overused comparison. Use ProWritingAid's Clichés Report to help identify the clichés and idioms in your writing. While some common analogies might help you get your point across, some can actually hinder your writing's clarity, especially to non-native speakers. Try the Cliché Report with a free ProWritingAid account. Analogy Examples You probably hear or read analogy examples all the time—they're a common rhetorical device. Today, we'll take a look at some analogy examples from everyday sentences and literature. Examples of Analogy in a Sentence Humans love figurative language, and we create analogies in our everyday speech. Here are some examples. Ordering clothes online is like playing the lottery. Some fit great, and some are a complete waste of money when they don't even go over your head! His voice was warm honey on toast, sweet and comforting and familiar. She thought the sound of babies crying was as annoying as fingernails on a chalkboard. Babies definitely weren't for her. Can you create any analogies? Examples of Analogy in Literature Analogy is a powerful rhetorical device. Here are some famous examples of analogy in literature: "All the world’s a stage / And all the men and women merely players / They have their exits and their entrances / And one man in his time plays many parts / His acts being seven ages."—As You Like It, William Shakespeare “I can admire the perfect murderer—I can also admire a tiger—that splendid tawny-striped beast. But I will admire him from outside his cage. I will not go inside . . . . That is to say, not unless it is my duty to do so. For you see, Mr. Shaitana, the tiger might spring . . . .”—Cards on the Table, Agatha Christie "Memory is to love what the saucer is to the cup."—The House in Paris, Elizabeth Bowen Tips on How to Write an Analogy When you're writing an analogy to express an idea, it's important to keep two things in mind. First, make sure that at least one of the two things you're comparing is familiar and easy to understand. An analogy should make your point clear to the reader, not leave them confused! Animal or nature imagery, allusions to well-known tales, and everyday objects are good things to use in your analogies. Secondly, make sure that your comparison is clear without much explanation. If you compare a shy, demure princess to a tiger, you need to explain what specific aspects of the princess and the tiger are similar. Is she ferocious when her loved ones are attacked? Does she prefer to spend time alone outdoors and seethe when caged? If it takes too many sentences to explain the analogy, try using different imagery that is simpler to understand. Different Types of Analogy There are two main types of analogy. These are based on how closely related the two things being compared are. Literal Analogy The first type of analogy is a literal analogy. When two things are very closely related, we compare them with literal analogies. These are the types of analogies commonly used in science. Literal analogies can help scientists draw comparisons or make a logical argument. For example, a virologist might compare the viral structure of two different viruses. If the virus has a similar structure and similar symptoms to another, they are analogous. This will help them theorize that the second virus can be treated similarly to the first. Literal analogies don't have to be just for science! If you're a baker, you might know that you can make a cheesecake out of either cream cheese or mascarpone. As an analogy, we can say that mascarpone is a lot like cream cheese. They both have high fat content, are very soft, and are not aged. Literal analogies are the type you might see on standardized tests. They used to feature on the SAT and looked like this: A:B::C:D. You can read literal analogies as "A is to B as C is to D." Here's a simple example: Night is to sleep as morning is to wake. Standardized tests would have one or more of the words blank, and you had to determine the connection in the analogy. How are they connected? Figurative Analogy A figurative analogy makes a comparison between two or more things that aren't necessarily that similar at first glance. The analogy focuses on making a comparison based on a specific aspect of the unrelated things. This is called shared abstraction. Take a look at the following analogy: "Giving candy and coffee as appreciation gifts is just rearranging deck chairs on the Titanic. It's not actually fixing the issues that are causing low morale, like low pay, long hours, and micromanaging." If the Titanic is sinking, it's pointless to rearrange deck chairs. Likewise, this quote suggests spending money on little gifts is pointless because it's not addressing the real issues at hand that are causing employee dissatisfaction. No one is suggesting that low morale at a company is the same thing as the hundreds of lives lost on the Titanic. The shared abstraction is doing something pointless in the face of a disaster. Should You Use Analogies in Your Writing? Analogies are powerful literary devices because they create an image in the reader's mind while making a point in a deeper way than a metaphor. Remember, an analogy compares two objects with the purpose of explaining a deeper idea. A literal analogy compares two very similar objects, while a figurative analogy relies on shared abstractions. 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5280
https://math-angel.io/lessons/thales-theorem/
Previous Lesson Next Lesson {results_count} Math videos for {phrase} Displaying {results_count} results of {results_count_total} Thales’ Theorem Courses Overview 🎬 Video: Thales' Theorem Definition and Application What is Thales' Theorem? (0:01) Thales’s theorem states that if a triangle is formed using the diameter of a circle and any point on the semicircle, it will always be a right-angled triangle. In the diagram, AB is the diameter of the circle, and C is a point on the semicircle. No matter where C is, the angle at C is always 90°. How to Apply Thales' Theorem? (0:37) Step 1: Use Thales’s Theorem Since $AB$ is the diameter, and vertex $C$ is a point on the semicircle. Thus, $$ \angle C = 90^\circ $$ Step 2: Find the Remaining Angle The sum of all angles in a triangle is: $$ \angle A + \angle B + \angle C = 180^\circ $$ Since we know $ \angle C = 90^\circ $ and $ \angle A = 35^\circ $, we can find $ \angle B $: $$ \angle B = 180^\circ – 90^\circ – 35^\circ = 55^\circ $$ 📂 Flashcards: Thales' Theorem Examples with Triangles 🍪 Quiz: Test Your Skills with Thales' Theorem 0% 🎩 Curious About Circle Geometry? Try AI Math Solver Need math help? Chat with our AI Math Solver at the bottom right — available 24/7 for instant answers. Previous Lesson Back to Course Next Lesson 5 1 vote Article Rating 0 Comments Newest Oldest Most Voted Inline Feedbacks View all comments
5281
https://cp-algorithms.com/game_theory/sprague-grundy-nim.html
Skip to content Last update: October 12, 2024  Translated From: e-maxx.ru Sprague-Grundy theorem. Nim¶ Introduction¶ This theorem describes the so-called impartial two-player game, i.e. those in which the available moves and winning/losing depends only on the state of the game. In other words, the only difference between the two players is that one of them moves first. Additionally, we assume that the game has perfect information, i.e. no information is hidden from the players (they know the rules and the possible moves). It is assumed that the game is finite, i.e. after a certain number of moves, one of the players will end up in a losing position — from which they can't move to another position. On the other side, the player who set up this position for the opponent wins. Understandably, there are no draws in this game. Such games can be completely described by a directed acyclic graph: the vertices are game states and the edges are transitions (moves). A vertex without outgoing edges is a losing vertex (a player who must make a move from this vertex loses). Since there are no draws, we can classify all game states as either winning or losing. Winning states are those from which there is a move that causes inevitable defeat of the other player, even with their best response. Losing states are those from which all moves lead to winning states for the other player. Summarizing, a state is winning if there is at least one transition to a losing state and is losing if there isn't at least one transition to a losing state. Our task is to classify the states of a given game. The theory of such games was independently developed by Roland Sprague in 1935 and Patrick Michael Grundy in 1939. Nim¶ This game obeys the restrictions described above. Moreover, any perfect-information impartial two-player game can be reduced to the game of Nim. Studying this game will allow us to solve all other similar games, but more on that later. Historically this game was popular in ancient times. Its origin is probably in China — or at least the game Jianshizi is very similar to it. In Europe the earliest references to it are from the 16th century. The name was given by Charles Bouton, who in 1901 published a full analysis of this game. Game description¶ There are several piles, each with several stones. In a move a player can take any positive number of stones from any one pile and throw them away. A player loses if they can't make a move, which happens when all the piles are empty. The game state is unambiguously described by a multiset of positive integers. A move consists of strictly decreasing a chosen integer (if it becomes zero, it is removed from the set). The solution¶ The solution by Charles L. Bouton looks like this: Theorem. The current player has a winning strategy if and only if the xor-sum of the pile sizes is non-zero. The xor-sum of a sequence $a$ is $a_1 \oplus a_2 \oplus \ldots \oplus a_n$, where $\oplus$ is the bitwise exclusive or. Proof. The key to the proof is the presence of a symmetric strategy for the opponent. We show that a once in a position with the xor-sum equal to zero, the player won't be able to make it non-zero in the long term — if they transition to a position with a non-zero xor-sum, the opponent will always have a move returning the xor-sum back to zero. We will prove the theorem by mathematical induction. For an empty Nim (where all the piles are empty i.e. the multiset is empty) the xor-sum is zero and the theorem is true. Now suppose we are in a non-empty state. Using the assumption of induction (and the acyclicity of the game) we assume that the theorem is proven for all states reachable from the current one. Then the proof splits into two parts: if for the current position the xor-sum $s = 0$, we have to prove that this state is losing, i.e. all reachable states have xor-sum $t \neq 0$. If $s \neq 0$, we have to prove that there is a move leading to a state with $t = 0$. Let $s = 0$ and let's consider any move. This move reduces the size of a pile $x$ to a size $y$. Using elementary properties of $\oplus$, we have $$ t = s \oplus x \oplus y = 0 \oplus x \oplus y = x \oplus y $$ Since $y < x$, $y \oplus x$ can't be zero, so $t \neq 0$. That means any reachable state is a winning one (by the assumption of induction), so we are in a losing position. Let $s \neq 0$. Consider the binary representation of the number $s$. Let $d$ be the index of its leading (biggest value) non-zero bit. Our move will be on a pile whose size's bit number $d$ is set (it must exist, otherwise the bit wouldn't be set in $s$). We will reduce its size $x$ to $y = x \oplus s$. All bits at positions greater than $d$ in $x$ and $y$ match and bit $d$ is set in $x$ but not set in $y$. Therefore, $y < x$, which is all we need for a move to be legal. Now we have: $$ t = s \oplus x \oplus y = s \oplus x \oplus (s \oplus x) = 0 $$ This means we found a reachable losing state (by the assumption of induction) and the current state is winning. Corollary. Any state of Nim can be replaced by an equivalent state as long as the xor-sum doesn't change. Moreover, when analyzing a Nim with several piles, we can replace it with a single pile of size $s$. Misère Game¶ In a misère game, the goal of the game is opposite, so the player who removes the last stick loses the game. It turns out that the misère nim game can be optimally played almost like a standard nim game. The idea is to first play the misère game like the standard game, but change the strategy at the end of the game. The new strategy will be introduced in a situation where each heap would contain at most one stick after the next move. In the standard game, we should choose a move after which there is an even number of heaps with one stick. However, in the misère game,we choose a move so that there is an odd number of heaps with one stick. This strategy works because a state where the strategy changes always appears in the game, and this state is a winning state, because it contains exactly one heap that has more than one stick so the nim sum is not 0. The equivalence of impartial games and Nim (Sprague-Grundy theorem)¶ Now we will learn how to find, for any game state of any impartial game, a corresponding state of Nim. Lemma about Nim with increases¶ We consider the following modification to Nim: we also allow adding stones to a chosen pile. The exact rules about how and when increasing is allowed do not interest us, however the rules should keep our game acyclic. In later sections, example games are considered. Lemma. The addition of increasing to Nim doesn't change how winning and losing states are determined. In other words, increases are useless, and we don't have to use them in a winning strategy. Proof. Suppose a player added stones to a pile. Then his opponent can simply undo his move — decrease the number back to the previous value. Since the game is acyclic, sooner or later the current player won't be able to use an increase move and will have to do the usual Nim move. Sprague-Grundy theorem¶ Let's consider a state $v$ of a two-player impartial game and let $v_i$ be the states reachable from it (where $i \in { 1, 2, \dots, k } , k \ge 0$). To this state, we can assign a fully equivalent game of Nim with one pile of size $x$. The number $x$ is called the Grundy value or nim-value of state $v$. Moreover, this number can be found in the following recursive way: $$ x = \text{mex}\ { x_1, \ldots, x_k }, $$ where $x_i$ is the Grundy value for state $v_i$ and the function $\text{mex}$ (minimum excludant) is the smallest non-negative integer not found in the given set. Viewing the game as a graph, we can gradually calculate the Grundy values starting from vertices without outgoing edges. Grundy value being equal to zero means a state is losing. Proof. We will use a proof by induction. For vertices without a move, the value $x$ is the $\text{mex}$ of an empty set, which is zero. That is correct, since an empty Nim is losing. Now consider any other vertex $v$. By induction, we assume the values $x_i$ corresponding to its reachable vertices are already calculated. Let $p = \text{mex}\ { x_1, \ldots, x_k }$. Then we know that for any integer $i \in [0, p)$ there exists a reachable vertex with Grundy value $i$. This means $v$ is equivalent to a state of the game of Nim with increases with one pile of size $p$. In such a game we have transitions to piles of every size smaller than $p$ and possibly transitions to piles with sizes greater than $p$. Therefore, $p$ is indeed the desired Grundy value for the currently considered state. Application of the theorem¶ Finally, we describe an algorithm to determine the win/loss outcome of a game, which is applicable to any impartial two-player game. To calculate the Grundy value of a given state you need to: Get all possible transitions from this state Each transition can lead to a sum of independent games (one game in the degenerate case). Calculate the Grundy value for each independent game and xor-sum them. Of course xor does nothing if there is just one game. After we calculated Grundy values for each transition we find the state's value as the $\text{mex}$ of these numbers. If the value is zero, then the current state is losing, otherwise it is winning. In comparison to the previous section, we take into account the fact that there can be transitions to combined games. We consider them a Nim with pile sizes equal to the independent games' Grundy values. We can xor-sum them just like usual Nim according to Bouton's theorem. Patterns in Grundy values¶ Very often when solving specific tasks using Grundy values, it may be beneficial to study the table of the values in search of patterns. In many games, which may seem rather difficult for theoretical analysis, the Grundy values turn out to be periodic or of an easily understandable form. In the overwhelming majority of cases the observed pattern turns out to be true and can be proved by induction if desired. However, Grundy values are far from always containing such regularities and even for some very simple games, the problem asking if those regularities exist is still open (e.g. "Grundy's game"). Example games¶ Crosses-crosses¶ The rules. Consider a checkered strip of size $1 \times n$. In one move, the player must put one cross, but it is forbidden to put two crosses next to each other (in adjacent cells). As usual, the player without a valid move loses. The solution. When a player puts a cross in any cell, we can think of the strip being split into two independent parts: to the left of the cross and to the right of it. In this case, the cell with a cross, as well as its left and right neighbours are destroyed — nothing more can be put in them. Therefore, if we number the cells from $1$ to $n$ then putting the cross in position $1 < i < n$ breaks the strip into two strips of length $i-2$ and $n-i-1$ i.e. we go to the sum of games $i-2$ and $n-i-1$. For the edge case of the cross being marked on position $1$ or $n$, we go to the game $n-2$. Thus, the Grundy value $g(n)$ has the form: $$g(n) = \text{mex} \Bigl( { g(n-2) } \cup {g(i-2) \oplus g(n-i-1) \mid 2 \leq i \leq n-1} \Bigr) .$$ So we've got a $O(n^2)$ solution. In fact, $g(n)$ has a period of length 34 starting with $n=52$. Practice Problems¶ KATTIS S-Nim CodeForces - Marbles (2018-2019 ACM-ICPC Brazil Subregional) KATTIS - Cuboid Slicing Game HackerRank - Tower Breakers, Revisited! HackerRank - Tower Breakers, Again! HackerRank - Chessboard Game, Again! Atcoder - ABC368F - Dividing Game Contributors: wikku (69.96%) AbhijeetKrishnan (13.9%) AniketR10 (4.93%) iagorrr (3.14%) adamant-pwn (2.69%) jakobkogler (2.69%) Kakalinn (1.35%) conlacda (0.45%) JustAnAverageGuy (0.45%) hoke-t (0.45%)
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https://www.physicsclassroom.com/mmedia/vectors/sat.cfm
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Satellite as a Projectile Satellite Motion A satellite is often thought of as being a projectile which is orbiting the Earth. But how can a projectile orbit the Earth? Doesn't a projectile accelerate towards the Earth under the influence of gravity? And as such, wouldn't any projectile ultimately fall towards the Earth and collide with the Earth, thus ceasing its orbit? These are all good questions and represent stumbling blocks for many students of physics. We will discuss each question here. First, an orbiting satellite is a projectile in the sense that the only force acting upon an orbiting satellite is the force of gravity. Most Earth-orbiting satellites are orbiting at a distance high above the Earth such that their motion is unaffected by forces of air resistance. Indeed, a satellite is a projectile. Second, a satellite is acted upon by the force of gravity and this force does accelerate it towards the Earth. In the absence of gravity a satellite would move in a straight line path tangent to the Earth. In the absence of any forces whatsoever, an object in motion (such as a satellite) would continue in motion with the same speed and in the same direction. This is the law of inertia. The force of gravity acts upon a high speed satellite to deviate its trajectory from a straight-line inertial path. Indeed, a satellite is accelerating towards the Earth due to the force of gravity. Finally, a satellite does fall towards the Earth; only it never falls into the Earth. To understand this concept, we have to remind ourselves of the fact that the Earth is round; that is the Earth curves. In fact, scientists know that on average, the Earth curves approximately 5 meters downward for every 8000 meters along its horizon. If you were to look out horizontally along the horizon of the Earth for 8000 meters, you would observe that the Earth curves downwards below this straight-line path a distance of 5 meters. In order for a satellite to successfully orbit the Earth, it must travel a horizontal distance of 8000 meters before falling a vertical distance of 5 meters. A horizontally launched projectile falls a vertical distance of 5 meters in its first second of motion. To avoid hitting the Earth, an orbiting projectile must be launched with a horizontal speed of 8000 m/s. When launched at this speed, the projectile will fall towards the Earth with a trajectory which matches the curvature of the Earth. As such, the projectile will fall around the Earth, always accelerating towards it under the influence of gravity, yet never colliding into it since the Earth is constantly curving at the same rate. Such a projectile is an orbiting satellite. To further understanding the concept of a projectile orbiting around the Earth, consider the following thought experiment. Suppose that a very powerful cannon was mounted on top of a very tall mountain. Suppose that the mountain was so tall that any object set in motion from the mountaintop would be unaffected by air drag. Suppose that several cannonballs were fired from the cannon at various speeds - say speeds of 8000 m/s, less than 8000 m/s, and more than 8000 m/s. A cannonball launched with speeds less than 8000 m/s would eventually fall to the Earth. A cannonball launched with a speed of 8000 m/s would orbit the Earth in a circular path. Finally, a cannonball launched with a speed greater than 8000 m/s would orbit the Earth in an elliptical path. The animations below depict these ideas. Launch Speed less than 8000 m/s Projectile falls to Earth Launch Speed less than 8000 m/s Projectile falls to Earth Launch Speed equal to 8000 m/s Projectile orbits Earth - Circular Path Launch Speed greater than 8000 m/s Projectile orbits Earth - Elliptical Path Two final notes should be made about satellite motion. First, the 8000 m/s figure used in the above discussion applies to satellites launched from heights just above Earth's surface. Since gravitational influences decrease with the height above the Earth, the orbital speed required for a circular orbit is less than 8000 m/s at significantly greater heights above Earth's surface. Second, there is an upper limit on the orbital speed of a satellite. If launched with too great of a speed, a projectile will escape Earth's gravitational influences and continue in motion without actually orbiting the Earth. Such a projectile will continue in motion until influenced by the gravitational influences of other celestial bodies. For more information on physical descriptions of motion, visit The Physics Classroom Tutorial. Detailed information is available there on the following topics: Newton's Law of Inertia Acceleration of Gravity Projectiles Characteristics of a Projectile's Trajectory Return to List of Animations Privacy Manager privacy contact home about terms © 1996-2025 The Physics Classroom, All rights reserved. By using this website, you agree to our use of cookies. We use cookies to provide you with a great experience and to help our website run effectively. Freestar.comReport This Ad
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https://pmc.ncbi.nlm.nih.gov/articles/PMC10150256/
Slope Identification and Decision Making: A Comparison of Linear and Ratio Graphs - PMC Skip to main content An official website of the United States government Here's how you know Here's how you know Official websites use .gov A .gov website belongs to an official government organization in the United States. Secure .gov websites use HTTPS A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites. Search Log in Dashboard Publications Account settings Log out Search… Search NCBI Primary site navigation Search Logged in as: Dashboard Publications Account settings Log in Search PMC Full-Text Archive Search in PMC Advanced Search Journal List User Guide New Try this search in PMC Beta Search PERMALINK Copy As a library, NLM provides access to scientific literature. Inclusion in an NLM database does not imply endorsement of, or agreement with, the contents by NLM or the National Institutes of Health. Learn more: PMC Disclaimer | PMC Copyright Notice Behav Modif . 2022 Nov 13;47(3):615–643. doi: 10.1177/01454455221130002 Search in PMC Search in PubMed View in NLM Catalog Add to search Slope Identification and Decision Making: A Comparison of Linear and Ratio Graphs Richard M Kubina Jr Richard M Kubina Jr 1 The Pennsylvania State University, University Park, USA Find articles by Richard M Kubina Jr 1,✉, Seth A King Seth A King 2 University of Iowa, USA Find articles by Seth A King 2, Madeline Halkowski Madeline Halkowski 1 The Pennsylvania State University, University Park, USA Find articles by Madeline Halkowski 1, Shawn Quigley Shawn Quigley 3 Melmark, Berwyn, PA, USA Find articles by Shawn Quigley 3, Tracy Kettering Tracy Kettering 4 Bancroft, Cherry Hill, NJ, USA Find articles by Tracy Kettering 4 Author information Article notes Copyright and License information 1 The Pennsylvania State University, University Park, USA 2 University of Iowa, USA 3 Melmark, Berwyn, PA, USA 4 Bancroft, Cherry Hill, NJ, USA ✉ Richard M. Kubina Jr., The Pennsylvania State University, 209 CEDAR Building, University Park, PA 16802, USA. Email: rmk11@psu.edu Issue date 2023 May. © The Author(s) 2022 This article is distributed under the terms of the Creative Commons Attribution 4.0 License ( which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access page ( PMC Copyright notice PMCID: PMC10150256 PMID: 36373492 Abstract Applied behavior analysts have traditionally relied on visual analysis of graphic data displays to determine the extent of functional relations between variables and guide treatment implementation. The present study assessed the influence of graph type on behavior analysts’ (n = 51) ratings of trend magnitude, treatment decisions based on changes in trend, and their confidence in decision making. Participants examined simulated data presented on linear graphs featuring equal-interval scales as well as graphs with ratio scales (i.e., multiply/divide or logarithmic vertical axis) and numeric indicators of celeration. Standard rules for interpreting trends using each display accompanied the assessment items. Results suggested participants maintained significantly higher levels of agreement on evaluations of trend magnitude and treatment decisions and reported higher levels of confidence in making decisions when using ratio graphs. Furthermore, decision making occurred most efficiently with ratio charts and a celeration value. The findings have implications for research and practice. Keywords: trend lines, ratio graphs, linear graphs, decision making, slope identification Single-case experimental design (SCED) encompasses a range of within-participant experimental methodologies (e.g., ABAB design, multiple-baseline design) that involve repeated assessment over time and the replication of intervention effects across conditions, individuals, or groups (Kazdin, 2021a). Strongly associated with behavior analysis, research involving SCED also appears in disciplines where the absence of sufficient sample sizes or the emphasis on performance at the individual level (e.g., special education) renders group experimental designs impossible or impractical (Hurtado-Parrado & López-López, 2015). The detection of experimental effects in SCED occurs through visual analysis of data displayed on a linear graph in which the vertical axis (i.e., y-axis) depicts changes in an outcome and the horizontal axis (i.e., x-axis) depicts a unit of time (Cooper et al., 2020; Kazdin, 2021a). Visual analysis requires an examination of level (e.g., relation of data to the vertical axis), trend, stability (e.g., consistency of data over time), overlap (i.e., extent to which values maintain across separate conditions), and immediacy of change (i.e., time until apparent effect of intervention; Cooper et al., 2020; Horner & Spaulding, 2010; Kazdin, 2021b). Assessment of intervention effects eschews formal statistical tests, instead relying upon visual comparison of data in one condition (e.g., non-treatment) to an adjacent condition (e.g., treatment) as a means of determining a functional relation (i.e., causal link between changes in the dependent variable and introduction of an intervention; Barton et al., 2018; Johnston et al., 2020). Visual analysis appears on the list of foundational skills applied behavior analysts must master before obtaining certification (Behavior Analyst Certification Board, 2017) and represents a critical competency among researchers and practitioners who use analogs of SCED. Despite the traditional emphasis on visual analysis in behavior analysis and related fields, researchers have increasingly questioned its technical adequacy and utility as a means of making treatment decisions in practice (e.g., Ninci et al., 2015; Riley-Tillman et al., 2020). Researchers also suggest the limitations of visual analysis may relate to graphing conventions (Kinney et al., 2022). Graphs used in practice typically have horizontal axes that show time linearly (i.e., equal amounts of space indicate equivalent amounts of change in time). However, researchers and practitioners in behavior analysis and related fields typically rely on linear graphs (i.e., equal interval graphs) with a vertical axis scaled linearly. In other words, equal distance or space on the vertical axis shows equal amounts of change (Schmid, 1986, 1992). Though relatively less prominent, practitioners in fields complementary to behavior analysis (e.g., precision teaching) have traditionally employed variations of ratio graphs (i.e., semilogarithmic graphs), in which equal distance or space between two values on the vertical axis represents an equal ratio of change. Comparisons between the two graphing formats has produced mixed results. Some of the results favor linear graphs while others demonstrate an advantage for ratio graphs (Bailey, 1984; Fuchs & Fuchs, 1986; Kinney et al., 2022; Lefebre et al., 2008; Mawhinney & Austin, 1999). However, researchers have yet to examine the effect ratio graphs, quantification of slope (i.e., trend or celeration line and celeration value), and associated interpretation rules have on decision making. The current article describes the limitations of visual analysis as applied to linear graphs in research and practice. We then the relative advantages of linear and ratio graphs. Finally, we describe the results of a survey study examining the influence of graphing format and associated decision-making rules on practitioners’ interpretation of graphic displays. Issues with Traditional Visual Analysis Although long considered the gold standard for analyzing SCED data (e.g., Manolov et al., 2016), recent developments suggest the dominance of visual analysis in fields associated with SCED (e.g., special education, applied behavior analysis; Hurtado-Parrado & López-López, 2015) has begun to waver. The What Works Clearinghouse (WWC, 2020), the leading research evaluation initiative of the US Department of Education, recently signaled a greater acceptance of SCED by removing the pilot designation from their design standards. In a departure from previous iterations of their SCED evaluation standards, however, the WWC explicitly limited the role of visual analysis in favor of statistically derived measures of effect comparable to those used in traditional group designs. Behavior analysts have a long tradition of rejecting statistical approaches to SCED (e.g., Baer, 1977; Graf, 1982; Johnston et al., 2020; Perone, 1999; Sidman, 1960); nonetheless, supplements to visual analysis have become far more acceptable within the field in recent years (e.g., Hantula, 2016; Killeen, 2019; Kyonka et al., 2019). Openness to substitutes for visual analysis likely stems from the body of evidence concerning its unreliability and adverse sensitivity (Ninci et al., 2015). Factors such as the expertise of the research team (Hojem & Ottenbacher, 1988), data characteristics (e.g., variability; Van Norman & Christ, 2016), and study context (e.g., social significance of the dependent variable; Ottenbacher, 1990) can lead to disparate interpretations of the same graph. Research increasingly suggests elements of the graphic display, such as the scale depicted on the vertical axis or the relative length of axes, may also distort visual analysis (Dart & Radley, 2017; Radley et al., 2018). Low agreement poses serious questions for the use of visual analysis as a means of supporting the effectiveness of interventions. The absence of transparent or consistent guidelines for the procedure in the research literature represents an additional explanation for the lack of consensus among analysts may occur due to (Barton et al., 2019; King et al., 2020). Many authors have advocated for more explicit descriptions of techniques associated with visual analysis to appear in published research (i.e., systematic visual analysis; Ninci et al., 2015; Nelson et al., 2017). Lane and Gast (2014) describe approaches for operationalizing aspects of visual design, such as the stability envelope, which determines the presence of variability within a data path by identifying the number of observations within ±25% of the median. Analysts further suggest supplementing qualitative assessments of trend with the split-middle method, which involves bisecting a data path, identifying the midpoint of each segment, and drawing a line through the median vertical axis value for each segment at each midpoint. Calculating the relative level change, which involves subtracting the median vertical axis value from the first data path segment from the median of the second data path segment, represents an additional quantitative approach to evaluating trend. Although transparent approaches to visual analysis may hold promise in terms of increasing agreement among researchers and their audiences, such procedures may have limited relevance for practitioners. For example, the emphasis on identifying a functional relation represents a major limitation of visual analysis (Browder et al., 1989; Ninci, 2019). Practitioners value the attainment of meaningful student achievement, relative to various instructional goals, more than verification of their instruction as the sole source of changes in behavior (Riley-Tillman et al., 2020). Systematic approaches to visual analysis (e.g., Lane & Gast, 2014) has the potential to reduce disagreement among observers regarding the quantifiable aspects of a line graph but do not aid in determining the magnitude or practical significance of changes in trend. Procedural descriptions of the split-middle method and similar approaches generally do not include guidelines related to the classification of trend magnitude. Likewise, indicators of treatment effect commonly employed as supplements to visual analysis (e.g., percentage of nonoverlapping data) usually assess the extent of nonoverlap between data in baseline and treatment conditions rather than the magnitude of effect (Wolery et al., 2010). Consequently, such metrics do not represent the best approach to guiding practice or assessing the instructional value of interventions. Tools for deriving suitable, data-based objectives and evaluating the effect of an intervention in the context of progress needed for the attainment of objectives have historically varied with the types of graphs employed in practice. Comparisons of Linear and Ratio Graphs Notwithstanding historical attempts to disseminate idiosyncratic data-based decision making rules for typical linear graphs (e.g., Browder et al., 1989), precision teaching (PT) has produced behavior analytic literature concerning practitioner-oriented data-based decision making (PT; see Binder, 1990; Johnson & Street, 2013; Kubina, 2019). PT emphasizes the importance of changes in performance and alters instruction and other behavioral interventions based on the frequent assessment of progress relative to quantified targets (Kubina, 2019). Yet PT uses the standard celeration chart, a formalized example of a ratio graph. The ratio graph depicts trends without the need for complex calculations. Additionally, practitioners have historically applied PT graphing and decision making procedures to a wide array of behaviors. The increasing demand for data-based decision making (e.g., Fuchs et al., 2021) warrants a closer inspection of how linear and ratio graphs may contribute to visual analysis. Linear and ratio graphs arguably facilitate different objectives based solely on their construction features, most notably the vertical axis. Linear graphs provide a clear view of absolute change and excel at visually presenting commensurate amounts of change to the graph reader. Yet absolute changes do not promote equal ratios of change. Figure 1 displays three comparable magnitudes of change. The space allocation for all three intervals appears the same because all three add or subtract 10 depending on the value and direction of movement (e.g., 25 + 10 = 35 or 35–10 = 25). The relative change expressed as a percent growth or decay and a multiplier or divider indicates unequal changes for the three different change distances though visually, they all appear identical. Linear graphs represent how the sheer number of counted units of behavior change have occurred but not the degree of growth or decay. Figure 1. Open in a new tab A comparison of a linear and ratio graph. Note. Bolded values represent how each graph creates quantitative equivalencies based on visual representation of the distance between two values. Conversely, a ratio graph has relative change as its primary feature. Equal distance of space between two values on the vertical axis represents an equal ratio of change and display data changing relative to one another. Figure 1 shows three equal distances between two values on the vertical axis (i.e., 1–2, 4–8, and 20–40). Additively the values of +1, + 4, and + 20, respectively appear visually equivalent because all have the same ratio or proportion of change (Schmid, 1986, 1992). Therefore, the ability to view proportional change relative to an initial rate of response represents one of the primary benefits of ratio graphs. The emphasis on proportional change in visual presentation prevents concealing the significance of nominally small performance changes and overstating the importance of nominally large changes. A series of data points graphed across time also changes according to the dictates of linearity. A linear graph has its basis in a cartesian coordinate system, and the trend line delineates change based on a slope-intercept equation, y = mx + b. However, the use of an equal-interval scale based on the range of participant responses also represents a primary drawback of linear scaling. The graph allows for easy determination of the direction of trend yet cannot conveniently facilitate the precise quantification of trend (Kinney et al., 2022). In contrast, ratio graphs feature vertical scales based on a multiply/divide scale and logarithms with a slope determined by log y = mx + log b (Giesecke et al., 2016; Prochaska & Theodore, 2018). The slope on a ratio graph demonstrates growth or decay. The steepness of the slope communicates how fast a quantity changes (Schmid, 1992). A horizontal slope has no growth changing at 0% or ×1.0. Behavior moving upward or downward at a uniform rate will show up as straight and have a percentage change and multiplier/divider associated with the values. The values of growth or decay will depend on the graphed quantities. There exist many differences between linear and ratio graphs which fall beyond the scope of the present discussion; interest readers may wish to examine historical and current sources showcasing the benefits of ratio graphs (Fisher, 1917; Giesecke et al, 2016; Griffin & Bowden, 1963; Harris, 1999; Schmid, 1986, 1992). As an engineering student, Lindsley created Precision Teaching (PT) and featured the standard celeration chart (SCC) in his system as the driver for data display and decision making. Lindsley based the SCC on Skinner’s use of a standard visual display (i.e., cumulative response recorder) and the benefits of a ratio graph (Lindsley, 1991; Potts et al., 1993). The paper SCC comprises a standard ratio graph that displays data up to 140 days and covers values ranging from 1 per day to 100 per minute or the full range of observable behavior (Kubina & Yurich, 2012). The SCC shows celeration or a measure of growth depicting the change in responding over a period of time (Johnston et al., 2020; Pennypacker et al., 2003). PT practitioners have established a framework for data-based decision making founded on the SCC and the extent of change over time. From the 1970s to the present, PT researchers and teachers established guidelines that indicated when to change or continue an intervention. One such rule involved using a celeration line, or graphic representation of the change in the rate of behavior over time, to determine if progress meets the change-across-time value (Liberty, 2019; White, 1984). A celeration aim value of ×1.5 means the line represents an increase of 50% per week (Johnston & Street, 2013). Therefore, a decision rule could state, “For a celeration of ×1.5 or greater, continue the intervention. For a celeration less than ×1.5, make a change.” Decision rules based on a quantified value has facilitated objective, clear, data-based actions for chart users (Johnson & Street, 2013; Kubina, 2019). Despite apparent differences in the properties and application, few studies have empirically evaluated the differences in linear and ratio graphs (Kinney et al., 2022). The bulk of the literature suggests the decision to use a linear or ratio graph should be left to practitioner preference due to minor differences in the effect of graph type on student achievement (Fuchs & Fuchs, 1986) or the interpretation of data features (e.g., trend, level; Knapp, 1983; Bailey, 1984). Kinney et al.’s (2022) comparison of practitioner performance on multiple tasks using linear and ratio graphs supports the notion that different displays may lend themselves to different tasks. Raters (n = 74) more accurately identified worsening or improving trends using ratio graphs, whereas linear graphs resulted in higher levels of performance on tasks such as identifying or plotting specific points in a data series. Evidence supporting the use of linear graphs in identifying trend provides some support for using ratio displays in instructional decision making. However, the extent to which either linear or ratio graphs facilitate practitioner decision making, particularly in the context of the forms of analysis typically associated with the two displays, remains unclear. Experimental Questions Much SCED research emphasizes the analysis of linear graphs and places a premium on level. The extent to which the conventional visual analysis of linear graphs facilitates more nuanced decision making based on trends that, though positive, may not indicate the effectiveness of treatment remains unclear. In contrast, ratio graphs and decision making rules may result in more consistent, confident assessments of data patterns and treatment decisions among practitioners due to the addition of quantification and objective guidelines. As they receive explicit training in visual analysis and routinely analyze data in practice, applied behavior analysts may represent an ideal population to evaluate the efficacy of procedural variations in visual analysis. The present study compared the decision making of applied behavior analysts based on the graphical displays and decision-making rules that typically accompany linear and ratio graphs. Specific questions included: To what extent does agreement concerning the magnitude of data trends (e.g., low, high) and appropriate treatment decisions (e.g., change, maintain) vary based on the types of graphs used and their accompanying decision rules? How does graph type influence efficiency of decision making (i.e., speed at which participants respond)? How does graph type influence behavior analysts’ confidence in their evaluations of trend magnitude and treatment decisions? Method Participants and Settings We recruited participants from multiple behavior analytic service providers across the northeastern United States. Cooperating administrators within the organization distributed an email with a survey link to professionals. The email indicated that (a) participation would require approximately 30 minutes, and (b) respondents would have the opportunity to participate in a drawing for a $20 gift card. Fifty-one participants agreed to participate and completed the survey in its entirety. Specific details concerning participants’ credentials, practical experience, and data interpretation methods appear in Table 1. An a priori power-analysis conducted in G-power indicated that a sample of 42 participants would be sufficient to detect a small effect size (e.g., r = .3) at the recommended level of statistical power (.80). Table 1. Participant Demographics. | Characteristics | n | % | :---: | Gender | | Female | 35 | 66.04 | | Male | 19 | 33.96 | | Certification level | | BCaBA | 66 | 81.48 | | BCBA | 10 | 12.35 | | BCBA-D | 4 | 4.94 | | Not certified | 1 | 1.23 | | Year of experience | | 0-5 | 29 | 54.72 | | 6-10 | 13 | 24.53 | | 11-15 | 5 | 9.43 | | 16+ | 6 | 11.32 | | Work settinga | | Administration | 18 | 27.27 | | Clinical | 21 | 31.82 | | Higher education | 2 | 3.03 | | Home | 9 | 13.64 | | Schools | 16 | 24.24 | | Employment status | | Full-time (40 + hour/week) | 50 | 94.34 | | Part-time (<40 hours/week) | 2 | 3.77 | | Unemployed | 1 | 1.89 | | Self-employed | 9 | 18 | | Retired | 0 | 0 | | Preferred method to graph/Interpret dataa | 17 | 34 | | Home-made (e.g., excel) | 39 | 39 | | Visual analysis | 39 | 39 | | Program-made (e.g., Chartlytics) | 15 | 15 | | Celeration lines | 7 | 7 | Open in a new tab Note. N = 53. a Reflects the number and percent of participants selecting “yes” to each option. Survey question allowed for multiple answer per participant creating subtotal greater than 53 (i.e., total number of participants). Instrument Graph generation We used Adobe Illustrator and Microsoft Excel to generate the graphs. The first author created blank graph templates which showed successive calendar days on the horizontal axis and either an equal-interval scale (linear graphs; that is, distance moving up or down on the vertical axis depict additive or subtractive change) or a ratio scale (ratio graphs; that is, distance moving up or down on the vertical axis depict multiplicative or divisional change) on the vertical axis (See Figure 2 below for examples of each type graph type). Each graph included nine data points displayed in a positive linear relationship. The pattern of the data points, though, remained appeared identical to one another in each of the three conditions: linear with no slope value, linear graph with a slope value, ratio graph with a celeration value. For the linear graphs with a slope value, the values ranged from 4/7 to 5 2/7 (i.e., slope expressed as rise over run and in the form of fractions and mixed numbers). For the ratio graphs, the celeration values ranged from ×1.1 to ×1.6. Figure 2. Open in a new tab Three different graph types presented to participants. To ensure that the physical characteristics of data paths presented in each condition did not influence responding, we assessed the gradient of the slope, in degrees, of lines in the ratio (M = 19.50; Range = 6–35; SD = 9.64) and linear/slope conditions (M = 19.25; R = 4–36; SD = 9.64). Results of an independent sample t-test identified no significant differences in line gradient across conditions (t = .04, p = .962, d = .02). Furthermore, we maintained all data meet a variability standard which included low variability as measured on the ratio graphs (i.e., ×2.5 to ×4 total variability) with a corresponding identical physical distance for variability ranges on linear graphs. And last, we controlled for level by have the overall level for each graph containing and equal distribution of low, medium, and high levels (i.e., range from 5 to 41). Survey We collected demographic questions, graphs, and related rating scales with a single electronic survey using Qualtrics. For demographic items (Table 1), participants completed several items related to their demographic characteristics, including their identified gender, year of experience, certification status, and general work responsibilities. Additional items concerned the approaches respondents used in analyzing graphed data (e.g., visual analysis, split-middle method, celeration lines; Ledford & Gast, 2018). We further assessed respondents’ level of familiarity with approaches to data interpretation using a 4-point Likert-type scale, with “1” indicating no familiarity and “4” indicating high familiarity with a specific method (Table 2). Table 2. Types of Graphs/Methods Participants Use to Interpret Data. | Graphs/Methods | Not at all familiar | Slightly familiar | Moderately familiar | Very familiar | :---: :---: | n | % | n | % | n | % | n | % | | Split-middle method | 32 | 60.38 | 13 | 24.53 | 7 | 13.21 | 1 | 1.89 | | Visual analysis | 1 | 1.89 | 4 | 7.55 | 10 | 18.87 | 38 | 71.70 | | Home-made | 2 | 3.77 | 1 | 1.89 | 8 | 15.09 | 42 | 79.25 | | Program-madea | 12 | 23.53 | 15 | 29.41 | 12 | 23.53 | 12 | 23.53 | | Celeration linesb | 7 | 13.46 | 27 | 51.92 | 15 | 28.85 | 3 | 5.77 | Open in a new tab Note. N = 53. a N = 51. b N = 52. After completing the previously described items, participants viewed six instructional videos (each approximately 60 seconds): (1) Visual Analysis and Decision Rules Instruction (0:37); (2) Within and Between Condition Analysis (0:40); (3) Estimating and Quantifying Slope (2:52); (4) Differences Between Ratio and Linear Graphs (0:50); (5) Decision Making for Linear Graphs (0:46); and (6) Decision Making for Ratio Graphs (0:30). The videos explained basic components of visual analysis, how to estimate slope, how to read slopes with either a slope value (linear graphs) or a celeration value (ratio graph), and accompanying decision-making rules (Links to the videos available from the first author). A sample multiple-choice quiz followed each video to assess comprehension and familiarize participants with survey items. Participants repeated incorrect sections until achieving 100% accuracy. We did not include sample items in the analysis. For each graph, respondents described the trend associated with the data path (e.g., a specific value for ratio graphs “low,” “medium,” or “high” for linear graphs with and without slope values). The difference in the trend identification tasks varied based on the graph type and their associated rules. For ratio values, practitioners only needed to determine the objective celeration value corresponding with change over time, which indicated a specific treatment action. The quantitative value (e.g., ×1.2, ×1.6) reflects a standard rate of change across any trend regardless of level or variability; a characteristic unique to ratio graphs and a means of fostering objectivity). To analyze linear and slope graphs, respondents had to make a qualitative determination regarding the change in performance of assessment sessions (i.e., standard practice in the field and as specified in behavior analytic textbooks). As with many published SCED, linear graphs did not have any form of quantitative information regarding the characteristic of the line. Slope graphs depicted the slope of the line as a fraction, whole number, or mixed number to control for the possibility of quantitative information improving decision making or improving confidence. Unlike the celeration values for ratio graphs, the ratios included with slope graphs did not correspond with historical rules for interpretation. Respondents then indicated whether they would continue or change a hypothetical treatment based on the performance depicted in the data path. To protect against order effects (i.e., possibility responses varied based on the order of completion), respondents received items in a random order as arranged through Qualtrics. Dependent Variables The present study had six dependent variables. Two dependent variables concerned the extent of each respondent’s agreement regarding trend and treatment decision. We calculated agreement for each participant by determining a respondent’s answer for a specific item, determining the number of respondents who selected the same response, and dividing the number by the total number of responses. We then multiplied the number by 100 to yield a percentage agreement score. The remaining dependent variables concerned the respondents’ confidence in their ratings for trend and treatment decisions. Respondents indicated their confidence in each decision using a 6-point Likert-type scale, with “1” representing the lowest level of confidence and “6” representing the highest level of confidence. Additionally, we determined the efficiency of participants’ responses using two hidden timing questions in the Qualtrics survey. The first timer measured initial response time as the number of seconds between the question section appearing on the screen and the participants’ first answer selection (i.e., latency). The second timer measured participants’ total response time as the time between the section loading and the participants’ selection of the “Next” button at the bottom of the screen. We averaged participants’ initial response time and total response measures for each graph type. Analysis We evaluated differences in the within-subject factor of graph type (Ratio, Linear, Slope) for all variables using the Friedman Test, a non-parametric alternative to an ANOVA of repeated measures data. In the event of a significant finding, we examined the significance of comparisons between individual groups using the non-parametric Wilcoxon signed-rank test. Effect sizes were determined using Kendall’s W (for Friedman Test) and Pearson’s r (for Wilcoxon signed-rank test), with scores exceeding .5 representing a large effect, scores between .49 and .3 representing a moderate effect, and scores between .29 and .1 representing a small effect. We observed a significance level of p = .05 for the Friedman Test. However, we adjusted the significance level for pairwise comparisons involving dependent variables using the Benjamini-Hochberg procedure, with a false-discovery rate of 5%. Survey logic did not permit respondents to skip questions, and we discarded surveys in which respondents discontinued the survey prior to completion; thus, the analysis did not need to account for missing data. All analyses were performed in SPSS. Response Rate and Reliability The survey was disseminated to potential participants described above (i.e., multiple behavior analytic companies). Of those, 81 began the survey. A total of 51 eligible professionals completed the survey in its entirety. We obtained internal consistency data related to confidence scales and decision making using Cronbach’s alpha. Results for scales related to trend assessment confidence (24 items; α = .975) and decision making confidence (24 items; α = .980) were acceptable. Results Agreement Descriptive statistics for agreement measures appear in Table 3. The column graph in Figure 3 graphically portrays participants’ percent agreement identifying trends in each condition and their subsequent treatment decision. Figure 3 shows respondents agreed more frequently on trend on ratio graphs than on either linear or linear plus slope value graphs. Participants agreed the least when viewing data on linear graphs with the slope value assigned to a trend. The participants then made treatments decisions based on their assessment of data and a decision-making rule. The results show both linear graphs with and without slope values evoked the same level of agreement. The ratio graph decision-making rule produced almost the same agreement as participants during the identifying trend condition, 99% versus 100%, respectively. Table 3. Descriptive Statistics for Measures of Confidence, Agreement, and Efficiency across Graph Type. | Graph | Mean confidence (Range/SD) | Mean agreement (Range/SD) | Mean efficiency (Range/SD) | :---: :---: | | Trend | Decision | Trend | Decision | Initial response | Total response | | Ratio | 5.17 (2.50–6.00/1.10) | 5.24 (2.50–6.00/1.05) | 100 (-/-) | 97.48 (68.38–99.00/5.24) | 4.88 (0.63–19.50/3.72) | 18.34 (10.55–43.69/7.67) | | Linear | 4.54 (2.13–6.00/.95) | 4.82 (2.25–6.00/1.01) | 65.85 (42.50–74.38/7.52) | 82.279 (63.25–99.00/6.26) | 7.92 (0.73–34.65/6.53) | 22.20 (8.31–68.11/11.92) | | Slope | 4.61 (2.00–6.00/.94) | 4.83 (2.00–6.00/1.01) | 66.26 (44.88–75.50/9.37) | 83.179 (61.00–87.00/6.31) | 12.22 (1.02–118.94/30.58) | 25.85 (9.89–131.03/21.29) | Open in a new tab Note. Confidence ratings based on a 6-point Likert-type scale, with 1 representing low confidence. Agreement represents a percentage of responses in which a participants identified the same choice in regard to the assessment of trend or instructional decisions. Response time for efficiency items presented in seconds. Figure 3. Open in a new tab A column graph displaying agreement identifying trend and decision. We analyzed data using a Friedman Test with a within-subject factor of graph type (Ratio, Linear, Slope). Results indicated a large, significant effect of graph type on agreement for trend, χ 2(2) = 78.157, p< .000, r = .766), and decision making, χ 2(2) = 71.892, p< .000, r = .705. In order to determine the difference between pairs, we conducted pairwise comparisons for each type of graph. Results revealed moderate, significant differences in respondent agreement for trend between the ratio and linear graphs at an adjusted significance level of .005 (Z = −6.217, p<.000, r = .439). We observed additional moderate, significant differences between agreement for trend in the ratio and slope value graphs at an adjusted significance level of .002 (Z = −6.219, p<.000, r = .440). We did not observe significant differences in agreement on trend for the linear and slope value graphs (Z = −.619, p = .536, r = .044) Pairwise comparisons also revealed similar differences in agreement on decision making. Results indicated moderate, significant differences in respondent agreement on treatment decisions for ratio and linear graphs at an adjusted significance level of .011 (Z = −5.787, p< .000, r = .409). We also observed moderate, significant differences between agreement for treatment decision in the ratio and slope value graphs at an adjusted significance level of .008 (Z = −5.908, p< .000, r = .418). Results revealed significant differences in treatment decision agreement between slope and linear graphs at an adjusted significance level of .036 (Z = −2.234, p = .026, r = .158). Confidence Descriptive statistics for confidence appear in Table 2, while a 100% stacked bar graph (Figure 4; Harris, 1999) provides a granular view of confidence ratings. In general, respondents reported higher confidence levels for trend and treatment decision ratings on ratio graphs relative to the other graph types. Comparing “completely confident” ratings across the three graph types, the ranges for percent-of-the-whole for ratio graphs plus celeration value, linear graph with trend, and linear graph with trend and slope value came to, respectively, 60%, 23%, and 28%. The total confidence rating averages for ratio graphs appeared 2.6 times higher than linear graphs with trend, and 2.1 times higher than linear graphs with trend and slope value. Figure 4. Open in a new tab A column graph showing efficiency analyzing data on different graph types. Results of the Friedman Test indicated moderate, significant effects of graph type on confidence in ratings for trend χ 2(2) = 37.780, p< .000, r = .370, and decision making, χ 2(2) = 27.445, p< .000, r = .269. Results of pairwise comparisons related to confidence in trend revealed moderate, significant difference between ratings on ratio and linear graphs (Z = −4.885, p< .000, r = .339) as well as ratio and slope graphs (Z = −4.328, p< .000, r = .312) at significance levels of .013 and .019, respectively. Analyses of decision confidence ratings indicated moderate differences between ratio and linear graphs at an adjusted significance level of .027 (Z = −3.994, p< .000, r = .282); likewise, results suggested a moderate difference in confidence on decisions regarding ratio and slope graphs at an adjusted significance level of .025 (Z = −4.051, p< .000, r = .286). We did not observe significance differences on confidence ratings related to trend or decision making between linear and slope value graphs (Z = −.949, p = .343, r = .067; Z = −.232, p = .817, r = .016). Efficiency Descriptive statistics for efficiency appear in Table 2. Figure 5 presents a column graph showing the participants’ average duration in seconds for the total time for each section and the first click after seeing a response item (i.e., a single instance of the three graph types). The results show participants clicked on their first answer most quickly when presented with a ratio graph and a celeration value. The second fastest click occurred with the linear graph with a trend, while the linear graph with a trend and slope value fostered the longest time to click. Total time on each section showed a similar order for efficiency. Participants spent the least amount of time with ratio graphs with a celeration value, more time for linear graphs with a trend, and the most time for linear graphs with a trend and slope value. Figure 5. Open in a new tab A column graph showing efficiency analyzing data on different graph types. Results of the Friedman Test indicated moderate, significant effects of graph type on efficiency for initial responses, χ 2(2) = 34.980, p< .000, r = .343, and a small effect on total response time, χ 2(2) = 12.118, p = .002, r = .119. Results of pairwise comparison related to initial response revealed significant differences between responses on ratio and linear graphs (Z = −4.227, p< .000, r = .591) as well as ratio and slope graphs (Z = −4.771, p< .000, r = .668), with adjusted significance levels of .022 and .017, respectively. For total response time, pairwise comparisons likewise revealed significant differences between ratio and linear graphs (Z = −2.925, p = .003, r = .409) at an adjusted significance level of .033 as well as ratio and slope graphs (Z = −3.946, p< .000, r = .552) at an adjusted significance level of 031. We did not observe significant differences between initial (Z = −1.509, p = .131, r = .211) or total response times (Z = −.591, p = .555, r = .082) for linear and slope graphs. Discussion The current study examined the effect of different graphic displays and their associated interpretation rules on assessments made by applied behavior analysts. A within condition analysis can focus on many aspects of the data, such as the number of data points, trend, variability/stability, level, outliers, and changes to trend within a condition (Barton et al., 2018; Cooper et al., 2020; Parsonson & Baer, 1978, 1992). We focused mainly on trend while controlling for variability/stability and level. Ratio graphs with an objective celeration value (i.e., Figure 2, graph type 3) resulted in a higher agreement among analysts regarding the magnitude of trend and the decision to change or maintain treatment compared to linear graphs requiring purely subjective appraisals of magnitude (i.e., graph type 1) or accompanied by a slope value (i.e., graph type 2). Results further suggest that professionals trained in visual analysis (i.e., behavior analysts) assessed graphs more efficiently and reported higher degrees of confidence in their assessment of trend and related treatment decisions when provided with ratio graphs. The literature has repeatedly demonstrated the difficulty of reliably describing presented data patterns using visual analysis (e.g., Danov & Symons, 2008; DeProspero & Cohen, 1979; Ninci et al., 2015; Ottenbacher, 1993; Wolfe et al., 2016). Although some exceptions suggest high reliability can occur under certain conditions (e.g., expert visual analysts; Kahng et al., 2010), the present study found behavior analysts with years of experience and high familiarity with visual analysis achieved low agreement (65.85%) when examining trend data under typical conditions. The data for linear graphs with trend represented how the behavior analytic field currently engages in visual analysis; the graph reader inspects the trend and subjectively determines if the slope fits a low, medium, or high category (Kennedy, 2005). As such, the mean agreement appeared very close to other studies that employed similar methods (e.g., 67%; Normand & Bailey, 2006). In contrast, the present results show quantifying trend with a celeration value, and providing objective guidelines for interpretation offers a consistent means for determining the magnitude of behavioral change. One hundred percent of the participants identified the celeration value of each trend line correctly. The objectivity of quantifying trend line with a celeration value compared to the subjectivity of qualitatively estimating a trend line explains the difference in results. The results also clarify why previous research did not come to similar conclusions. For example, Bailey (1984) indicated chart type (i.e., ratio, aka semilogarithmic, vs. linear) did not affect ratings of significance for lines of progress. However, Bailey, like many other researchers who compared the two graph types (Lefebre et al., 2008; Kinney et al., 2022; Knapp, 1983; Marston, 1988; Mawhinney & Austin, 1999), asked participants to apply their qualitative judgment of trend lines to both conditions. None of the past research explicitly provided celeration values, one of the main benefits of a ratio graph. Quantifying the trend with slope created a condition where participants performed most poorly (i.e., 59% agreement). Even though the brief instructional video explained the rise and run and how to interpret trend with slope value, the fractions and mixed numbers appeared to confuse participants when integrating the values with visual analysis. We could not find any published literature in textbooks or behavioral articles with the trend quantified with a slope value. Therefore, the likely combination of no experience and brief instruction did not improve participants' understanding of the trend. Furthermore, using slope values could have confounded participants due to the varied propensity to portray progress, growth, or decay without a constant ratio. A slope value across a series of data 1, 2, 3, 4, 5, 6, 7 and 51, 52, 53, 54, 55, 56, 57 would appear visually similar but represent two progressions seeing 600% and 12% increases across time respectively. Nonetheless, graphing data on a ratio display does not automatically produce an easily interpretable value. The celeration value found on a standard celeration chart offers a standard, well-defined, easily calculated quantification of change across time when understood. Though not a primary focus of this study, the inability of quantitative values to improve decision making in the absence of standardization represents a potential consideration when considering the merits of systematic approaches to visual analysis based on traditional linear graphs (e.g., Barton et al., 2019; Manolov, 2017; Odom et al., 2018; Shadish, 2014). We also examined the impact of graph-type on the efficiency of analysis and decision making (Figure 5). Even though most participants (n = 65%) reported “not at all familiar” or “slightly familiar” with celeration lines, the overall data suggest the celeration line and value produced the least amount of time determining its direction, value, and appropriate decision contingent upon the decision rule. Efficiency and high agreement suggest two favorable outcomes; (1) behavior analysts will spend less time parsing significant from insignificant trends and (2) past visual analysis problems (e.g., Danov & Symons, 2008; DeProspero & Cohen, 1979; Ninci et al., 2015; Ottenbacher, 1993; Wolfe et al., 2016) have a solution with an objective, quantified trend. Some researchers may argue we put forward an unfair comparison given that linear graphs, with the exception of the slope sample, lacked quantification and less obvious decision making rules. Yet our study hopes to make such an important point—the existing literature regarding traditional analysis does not provide precise rules regarding the assessment of trend, and the use of linear scales, and variable graphing conventions prevents the formulation of decision making rules based solely on data. Ratio graphs, on the hand, provide a standard quantification of trend such as percentage of growth or rates of change (Devesa et al., 1995; Giesecke et al., 2016). Certain types of ratio graphs like the standard celeration chart have an associated body of literature supporting the circumstances in which data support specific treatment decisions. The use of the latter method might self-evidently be associated with higher practitioner confidence and consensus calls into question the continued use of the former, less precise method. We concede that convention departures must have accompanying evidence supporting new approaches and have conducted the present study in that spirit. Another fairness argument surrounding the graph type comparison centers on the reduction of ratio data trend into an easily interpretable number. The slope condition suggests the presence of numeric value alone did not increase confidence or agreement among practitioners. Like the intricate methods of trend quantification that accompany visual analysis, and which the sample reported having little use for in practice, the slopes of the line likely served as ineffective tools for decision making because they would ultimately possess the limitations of linear graphs and absolute change. Additionally, a survey asked Behavior Analyst Certification Board verified course sequence instructors how they teach students to estimate trend (Wolfe & McCammon, 2022). Eighty-one percent reported eyeballing or visualizing the data, 51% taught how to fit data with the split-middle line, and 16% of the respondents reported using other methods like linear regression to produce a line of best fit. No data spoke to teaching slope or otherwise quantifying the trend line. Limitations The present study has notable limitations. First, the sample consisted of behavior professionals contacted through their association with clinics in the northeastern US and does not necessarily represent the broader profession. As the study aimed to provide a preliminary, experimental assessment of the impact of graphing conventions (i.e., different graph types and quantified trend), rather than claims regarding the facility of BCBAs more generally, the experiment, therefore, has value despite the use of a relatively small convenience sample. The inclusion of verified “experts” may have resulted in different findings (e.g., Ledford et al., 2019). However, all participants had some form of credential from the national certifying entity for applied behavior analysts. Given the centrality of visual analysis to services delivered by practitioners of ABA, results from certified professionals may have more meaningful implications for the state of practice relative to those of purported experts in the field. Future Directions The control exercised in the current study may expand to cover more varied data characteristics. For example, all variability visually ranged from very stable to stable (or a ×2 to ×4 distance on a standard celeration chart). Increasing the scope of variability and varying level more widely would demonstrate the robustness of experimental findings. Likewise, including more behavior analysts in number, location, and certification level would indicate whether the present study has generality. Extending the study to between condition analyses and different experimental designs could likewise yield interesting findings. The current findings provide support for the use of ratio graphs and related decision making rules; future scholarship has the potential to verify findings under more authentic conditions. Kuntz et al. (in press) recently found that the presence of lines connecting levels of student performance in baseline to a long-term goal, or airlines, which are frequently used in DBI and more easily constructed in ratio graphs, significantly improved the ability of preservice teachers to make instructional decisions. Similar studies using ratio graphs, which are increasingly acceptable to educators (e.g., Kinney et al., 2022), could provide further evidence of the advantages of alternatives to traditional visual analysis (i.e., subjective appraisal of data). Such studies would preferably compare the utility of displays across entire cases, with much of the responsibility transferred to professionals using standard tools rather than isolated items or tasks mediated by researchers. Studies asking behavior analysts to estimate trend typically led to low agreement due to several conditions in experience and graph production. First, no behavior analytic textbooks (e.g., Cooper et al., 2020; Mayer et al., 2019) provide standard, precise rules for specifying the degree of a trend’s slope beyond estimation. A representative example states, “Trend can further be characterized by magnitude, and is often described as steep or gradual and paired with direction (e.g., steep accelerating trend or gradual decelerating trend) bold in original, Barton et al., 2018, p. 185).” Second, almost every published article with a line graph varies from one article to the next in terms of graph construction features such as the length of axes, the proportion of one axis to the other, and scaling of axes (Kubina et al., 2017). The combination of no objective standards or guidance from authoritative sources such as textbooks, journals, or professional organizations and the accepted practice of idiosyncratically constructed graphs promotes a lack of consistent pattern recognition and lower rates of trend identification. Whether through the adoption of standard ratio displays, systematic visual analysis, or more dedicated training, improving the integrity of visual analysis represents a priority for behavior analysis and other disciplines that rely on graphic displays. Supplemental Material sj-pdf-1-bmo-10.1177_01454455221130002 – Supplemental material for Slope Identification and Decision Making: A Comparison of Linear and Ratio Graphs Click here for additional data file. (28.6KB, pdf) Supplemental material, sj-pdf-1-bmo-10.1177_01454455221130002 for Slope Identification and Decision Making: A Comparison of Linear and Ratio Graphs by Richard M. Kubina, Seth A. King, Madeline Halkowski, Shawn Quigley and Tracy Kettering in Behavior Modification Author Biographies Richard M. Kubina Jr. is a Professor of Special Education in the Department of Educational Psychology, Counseling, and Special Education at The Pennsylvania State University. His research explores the intersection between learning, science, and technology. He has studied explicit instruction, instructional design, Precision Teaching, video modeling, and robotics. Seth A. King is an Associate Professor of Special Education in the Department of Teaching and Learning at the University of Iowa, where he serves as the coordinator of the applied behavior analysis program. His research interests include academic and behavioral interventions for children with disabilities and training methods for education personnel. His most recent scholarship involves incorporates virtual reality and artificial intelligence into instructional simulations. Madeline Halkowski, MEd, BCBA, is a PhD candidate in the Department of Educational Psychology, Counseling, and Special Education at the Pennsylvania State University. She conducts research in the areas of measurement, technology, and literacy using mixed methods and single-case research designs. Shawn Quigley, PhD, BCBA-D, CDE is the Chief Operating Officer for Melmark, which operates in Massachusetts, Pennsylvania, and North Carolina. His research interests are training, organizational practices, and challenging behavior. Tracy Kettering, PhD, BCBA-D, is the Director of the Applied Behavior Analysis Center of Excellence at Bancroft in Cherry Hill, NJ, where she oversees clinical quality, professional development, and training for behavior analysts working with individuals with autism and developmental disabilities. She is also an adjunct professor and research advisor at Rider University. Footnotes The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding: The author(s) received no financial support for the research, authorship, and/or publication of this article. ORCID iD: Richard M. Kubina Supplemental Material: Supplemental material for this article is available online. References Baer D. M. (1977). Perhaps it would be better not to know everything. Journal of Applied Behavior Analysis, 10(1), 167–172. 10.1901/jaba.1977.10-167 [DOI] [PMC free article] [PubMed] [Google Scholar] Bailey D. B. (1984). Effects of lines of progress and semilogarithmic charts on ratings of charted data. Journal of Applied Behavior Analysis, 17(3), 359–365. 10.1901/jaba.1984.17-359 [DOI] [PMC free article] [PubMed] [Google Scholar] Barton E. E., Lloyd B. P., Spriggs A. D., Gast D. L. (2018). Visual analysis of graphic data. In Ledford J., Gast D. (Eds.), Single case research methodology: Applications in special education and behavioral sciences (pp. 179–214). Routledge. [Google Scholar] Barton E. E., Meadan H., Fettig A. (2019). Comparison of visual analysis, non-overlap methods, and effect sizes in the evaluation of parent implemented functional assessment based interventions. Research in Developmental Disabilities, 85, 31–41. 10.1016/j.ridd.2018.11.001 [DOI] [PubMed] [Google Scholar] Behavior Analyst Certification Board. (2017). BCBA/BCaBA task list (5th ed.). Author. [Google Scholar] Binder C. (1990). Precision teaching and curriculum based measurement. Journal of Precision Teaching, 7(2), 33–35. [Google Scholar] Browder D., Demchak M. A., Heller M., King D. (1989). An in vivo evaluation of the use of data-based rules to guide instructional decisions. Journal of the Association for Persons with Severe Handicaps, 14(3), 234–240. 10.1177/154079698901400309 [DOI] [Google Scholar] Cooper J. O., Heron T. E., Heward W. L. (2020). Applied behavior analysis (3rd ed.). Pearson Education. [Google Scholar] Danov S. E., Symons F. J. (2008). A survey evaluation of the reliability of visual inspection and functional analysis graphs. Behavior Modification, 32(6), 828–839. 10.1177/0145445508318606 [DOI] [PubMed] [Google Scholar] Dart E. H., Radley K. C. (2017). The impact of ordinate scaling on the visual analysis of single-case data. Journal of School Psychology, 63, 105–118. 10.1016/j.jsp.2017.03.008 [DOI] [PubMed] [Google Scholar] DeProspero A., Cohen S. (1979). Inconsistent visual analyses of intrasubject data. Journal of Applied Behavior Analysis, 12(4), 573–579. 10.1901/jaba.1979.12-573 [DOI] [PMC free article] [PubMed] [Google Scholar] Devesa S. S., Donaldson J., Fears T. (1995). Graphical presentation of trends in rates. American Journal of Epidemiology, 141(4), 300–304. 10.1093/aje/141.4.300 [DOI] [PubMed] [Google Scholar] Fisher I. (1917). The “ratio” chart for plotting statistics. Publications of the American Statistical Association, 15(118), 577–601. 10.2307/2965173 [DOI] [Google Scholar] Fuchs L. S., Fuchs D. (1986). The relation between methods of graphing student performance data and achievement: A meta-analysis. Journal of Special Education Technology, 8(3), 5–13. 10.1177/016264348700800302 [DOI] [Google Scholar] Fuchs L. S., Fuchs D., Hamlett C. L., Stecker P. M. (2021). Bringing data-based individualization to scale: A call for the next-generation technology of teacher supports. Journal of Learning Disabilities, 54(5), 319–333. 10.1177/0022219420950654 [DOI] [PubMed] [Google Scholar] Giesecke F. E., Mitchell A., Spencer H. C., Hill I. L., Dygdon J. T., Novak J. E., Loving R. O., Lockhart S. E., Johnson L., Goodman M. (2016). Technical drawing with engineering graphics (15th ed.). Peachpit Press. [Google Scholar] Graf S. A. (1982). Is this the right road? A review of Kratochwill’s single subject research: Strategies for evaluating change. Perspectives on Behavior Science, 5(1), 95–99. 10.1007/BF03393143 [DOI] [Google Scholar] Griffin D. W., Bowden L. W. (1963). Semi-logarithmic graphs in geography. The Professional Geographer, 15(5), 19–23. 10.1111/j.0033-0124.1963.019_e.x [DOI] [Google Scholar] Hantula D. A. (2016). Editorial: A very special issue. Perspectives on Behavior Science, 39(1), 1–5. 10.1007/s40614-018-0163-8 [DOI] [PMC free article] [PubMed] [Google Scholar] Harris R. L. (1999). Information graphics: A comprehensive illustrated reference. Oxford University Press. [Google Scholar] Hojem M. A., Ottenbacher K. J. (1988). Empirical investigation of visual-inspection versus trend-line analysis of single-subject data. Physical Therapy, 68(6), 983–988. 10.1093/ptj/68.6.983 [DOI] [PubMed] [Google Scholar] Horner R., Spaulding S. (2010). Single-case research designs. In Salkind N. J. (Ed.), Encyclopedia of research design (pp. 1386–1394). Sage Publications. [Google Scholar] Hurtado-Parrado C., López-López W. (2015). Single-case research methods: History and suitability for a psychological science in need of alternatives. Integrative Psychological and Behavioral Science, 49(3), 323–349. 10.1007/s12124-014-9290-2 [DOI] [PubMed] [Google Scholar] Johnson K., Street E. M. (2013). Response to intervention and precision teaching: Creating synergy in the classroom. Guilford Press. [Google Scholar] Johnston J. M., Pennypacker H. S., Green G. (2020). Strategies and tactics of behavioral research and practice (4th ed.). Routledge. [Google Scholar] Kahng S. W., Chung K. M., Gutshall K., Pitts S. C., Kao J., Girolami K. (2010). Consistent visual analysese of intrasubject data. Journal of Applied Behavior Analysis, 43(1), 35–45. 10.1901/jaba.2010. 43–35 [DOI] [PMC free article] [PubMed] [Google Scholar] Kazdin A. E. (2021. a). Single-case research designs: Methods for clinical and applied settings (3rd ed.). Oxford University Press. [Google Scholar] Kazdin A. E. (2021. b). Single-case experimental designs: Characteristics, changes, and challenges. Journal of the Experimental Analysis of Behavior, 115(1), 56–85. [DOI] [PubMed] [Google Scholar] Kennedy C. H. (2005). Single-case designs for educational research. Allyn and Bacon. [Google Scholar] Killeen P. R. (2019). Predict, control, and replicate to understand: How statistics can foster the fundamental goals of science. Perspectives on Behavior Science, 42(1), 109–132. 10.1007/s40614-018-0171-8 [DOI] [PMC free article] [PubMed] [Google Scholar] King S. A., Kostewicz D., Enders O., Burch T., Chitiyo A., Taylor J., DeMaria S., Reid M. (2020). Search and selection procedures of literature reviews in behavior analysis. Perspectives on Behavior Science, 43(4), 725–760. 10.1007/s40614-020-00265-9 [DOI] [PMC free article] [PubMed] [Google Scholar] Kinney C. E. L., Begeny J. C., Stage S. A., Patterson S., Johnson A. (2022). Three alternatives for graphing behavioral data: A comparison of usability and acceptability. Behavior Modification, 46(1), 3–35. 10.1177/0145445520946321 [DOI] [PubMed] [Google Scholar] Knapp T. (1983). Behavior analysts’ visual appraisal of behavior change in graphic display. Behavioral Assessment, 5(2), 155–164. [Google Scholar] Kubina R. M. (2019). The precision teaching implementation manual. Greatness Achieved Publishing Company. [Google Scholar] Kubina R. M., Yurich K. K. (2012). Precision teaching book. Greatness Achieved Publishing Company. [Google Scholar] Kubina R. M., Kostewicz D. E., Brennan K. M., King S. A. (2017). A critical review of line graphs in behavior analytic journals. Educational Psychology Review, 29(3), 583–598. 10.1007/s10648-015-9339-x [DOI] [Google Scholar] Kuntz E., Massy C., Peliter C. P., Barczak M., Crowson M. (2022). Graph manipulation and the impact on pre-service teachers’ accuracy in evaluating progress monitoring data. Teacher Education and Special Education. 10.1177/08884064221086991 [DOI] Kyonka E. G., Mitchell S. H., Bizo L. A. (2019). Beyond inference by eye: Statistical and graphing practices in JEAB, 1992–2017. Journal of the Experimental Analysis of Behavior, 111(2), 155–165. 10.1002/jeab.509 [DOI] [PubMed] [Google Scholar] Lane J. D., Gast D. L. (2014). Visual analysis in single case experimental design studies: Brief review and guidelines. Neuropsychological Rehabilitation, 24(3–4), 445–463. 10.1080/09602011.2013.815636 [DOI] [PubMed] [Google Scholar] Ledford J. R., Gast D. L. (2018). Single case research methodology: Application in special education and behavioral sciences (3rd ed.). Routledge. [Google Scholar] Ledford J. R., Barton E. E., Severini K. E., Zimmerman K. N., Pokorski E. A. (2019). Visual display of graphic data in single case design studies. Education and Training in Autism and Developmental Disabilities, 54(4), 315–327. [Google Scholar] Lefebre E., Fabrizio M., Merbitz C. (2008). Accuracy and efficiency of data interpretation: A comparison of data display methods. Journal of Precision Teaching and Celeration, 24, 2–20. [Google Scholar] Liberty K. A. (2019). Decision rules we learned through precision teaching. In Haring N., White M., Neely M. (Eds.), Precision teaching – A practical science of education (pp. 100–126). Sloan Publishing. [Google Scholar] Lindsley O. R. (1991). Precision teaching’s unique legacy from B. F. Skinner. Journal of Behavioral Education, 1(2), 253–266. 10.1007/BF00957007 [DOI] [Google Scholar] Manolov R. (2017). Reporting single-case design studies: Advice in relation to the designs’ methodological and analytical peculiarities. Anuario De Psicología, 47(1), 45–55. 10.1016/j.anpsic.2017.05.004 [DOI] [Google Scholar] Manolov R., Losada J. L., Chacón-Moscoso S., Sanduvete-Chaves S. (2016). Analyzing two-phase single-case data with non-overlap and mean difference indices: Illustration, software tools, and alternatives. Frontiers in Psychology, 7, Article ID 32. 10.3389/fpsyg.2016.00032 [DOI] [PMC free article] [PubMed] Marston D. (1988). Measuring progress on IEPs: A comparison of graphing approaches. Exceptional Children, 55(1), 38–44. 10.1177/001440298805500104 [DOI] [PubMed] [Google Scholar] Mawhinney T. C., Austin J. (1999). Speed and accuracy of data analysts’ behavior using methods of equal interval graphic data charts, standard celeration charts, and statistical process control charts. Journal of Organizational Behavior Management, 18, 5–45. 10.1300/J075v18n04_02 [DOI] [Google Scholar] Mayer G. R., Sulzer-Azaroff B., Wallace M. (2019). Behavior analysis for lasting change (4th ed.). Sloan Publishing. [Google Scholar] Nelson P. M., Van Normand E. R., Christ T. J. (2017). Visual analysis among novices: Training and trend lines as graphic aids. Contemporary School Psychology, 21(2), 93–102. 10.1007/s40688-016-0107-9 [DOI] [Google Scholar] Ninci J. (2019). Single-case data analysis: A practitioner guide for accurate and reliable decisions. Behavior Modification, Advance online publication. 10.1177/0145445519867054 [DOI] [PubMed] Ninci J., Vannest K. J., Wilson V., Zhang N. (2015). Interrater agreement between visual analysts of single-case data: A meta-analysis. Behavior Modification, 39(4), 510–541. 10.1177/0145445515581327 [DOI] [PubMed] [Google Scholar] Normand M. P., Bailey J. S. (2006). The effects of celeration lines on visual data analysis. Behavior Modification, 30(3), 295–314. 10.1177/0145445503262406 [DOI] [PubMed] [Google Scholar] Odom S. L., Barton E. E., Reichow B., Swaminathan H., Pustejovsky J. E. (2018). Between-case standardized effect size analysis of single case designs: Examination of the two methods. Research in Developmental Disabilities, 79, 88–96. 10.1016/j.ridd.2018.05.009 [DOI] [PubMed] [Google Scholar] Ottenbacher K. J. (1990). Visual inspection of single-subject data: An empirical analysis. Mental Retardation, 28(5), 283–290. [PubMed] [Google Scholar] Ottenbacher K. J. (1993). Interrater agreement of visual analysis in single-subject decisions: Quantitative review and analysis. American Journal on Mental Retardation, 98(1), 135–142. [PubMed] [Google Scholar] Parsonson B. S., Baer D. M. (1978). The analysis and presentation of graphic data. In Kratochwill T. R. (Ed.), Single-subject research: Strategies for evaluating change (pp. 101–165). Academic Press. [Google Scholar] Parsonson B. S., Baer D. M. (1992). The visual analysis of data, and current research into the stimuli controlling it. In Kratochwill T. R., Levin J. R. (Eds.), Single-case research design and analysis (pp. 15–40). Routledge. [Google Scholar] Pennypacker H. S., Gutierrez A., Lindsley O. R. (2003). Handbook of the Standard Celeration Chart. Cambridge Center for Behavioral Studies. [Google Scholar] Perone M. (1999). Statistical inference in behavior analysis: Experimental control is better. Perspectives on Behavior Science, 22(2), 109–116. 10.1007/BF03391988 [DOI] [PMC free article] [PubMed] [Google Scholar] Potts L., Eshleman J. W., Cooper J. O. (1993). Ogden R. Lindsley and the historical development of precision teaching. The Behavior Analyst, 16(2), 177–189. 10.1007/BF03392622 [DOI] [PMC free article] [PubMed] [Google Scholar] Prochaska C., Theodore L. (2018). Introduction to mathematical methods for environmental engineers and scientists. Scrivener Publishing. [Google Scholar] Radley K. C., Dart E. H., Wright S. J. (2018). The effect of data points per x- to y-axis ratio on visual analysts evaluation of single-case graphs. School Psychology Quarterly, 33(2), 314–322. 10.1037/spq0000243 [DOI] [PubMed] [Google Scholar] Riley-Tillman T. C., Burns M. K., Kilgus S. P. (2020). Evaluating educational interventions: Single-case design for measuring response to interventions (2nd ed.). The Guilford Press. [Google Scholar] Schmid C. F. (1986). Whatever has happened to the semilogarithmic chart? The American Statistician, 40(3), 238–244. 10.1080/00031305.1986.10475401 [DOI] [Google Scholar] Schmid C. F. (1992). Statistical graphics: Design principles and practices. Wiley. [Google Scholar] Shadish W. R. (2014). Statistical analyses of single-case designs: The shape of things to come. Current Directions in Psychological Science, 23(2), 139–146. 10.1177/0963721414524773 [DOI] [Google Scholar] Sidman M. (1960). Tactics of scientific research: Evaluating experimental data in psychology. Basic Books. [DOI] [PMC free article] [PubMed] [Google Scholar] Van Norman E. R., Christ T. J. (2016). How accurate are interpretations of curriculum-based measurement progress monitoring data? Visual analysis versus decision rules. Journal of School Psychology, 58, 41–55. 10.1016/j.jsp.2016.07.003 [DOI] [PubMed] [Google Scholar] What Works Clearinghouse (WWC). (2020). What works clearinghouse standards handbook, version 4.1. U.S. Department of Education, Institute of Education Sciences, National Center for Education Evaluation and Regional Assistance. [Google Scholar] White O. R. (1984). Performance based decisions: When and what to change. In West R., Young K. (Eds.), Precision teaching: Instructional decision-making, curriculum and management, and research. Department of Special Education, Utah State University. [Google Scholar] Wolery M., Busick M., Reichow B., Barton E. E. (2010). Comparison of overlap methods for quantitatively synthesizing single-subject data. The Journal of Special Education, 44(1), 18–28. 10.1177/0022466908328009 [DOI] [Google Scholar] Wolfe K., McCammon M.N. (2022). The analysis of single-case research data: Current instructional practices. Journal of Behavioral Education, 31(1), 28–42. 10.1007/s10864-020-09403-4 [DOI] [Google Scholar] Wolfe K., Seaman M.A., Drasgow E. (2016). Interrater agreement on the visual analysis of individual tiers and functional relations in multiple baseline designs. Behavior Modification, 40(6), 852–873. 10.1177/0145445516644699 [DOI] [PubMed] [Google Scholar] Associated Data This section collects any data citations, data availability statements, or supplementary materials included in this article. Supplementary Materials sj-pdf-1-bmo-10.1177_01454455221130002 – Supplemental material for Slope Identification and Decision Making: A Comparison of Linear and Ratio Graphs Click here for additional data file. (28.6KB, pdf) Supplemental material, sj-pdf-1-bmo-10.1177_01454455221130002 for Slope Identification and Decision Making: A Comparison of Linear and Ratio Graphs by Richard M. Kubina, Seth A. King, Madeline Halkowski, Shawn Quigley and Tracy Kettering in Behavior Modification Articles from Behavior Modification are provided here courtesy of SAGE Publications ACTIONS View on publisher site PDF (955.6 KB) Cite Collections Permalink PERMALINK Copy RESOURCES Similar articles Cited by other articles Links to NCBI Databases Cite Copy Download .nbib.nbib Format: Add to Collections Create a new collection Add to an existing collection Name your collection Choose a collection Unable to load your collection due to an error Please try again Add Cancel Follow NCBI NCBI on X (formerly known as Twitter)NCBI on FacebookNCBI on LinkedInNCBI on GitHubNCBI RSS feed Connect with NLM NLM on X (formerly known as Twitter)NLM on FacebookNLM on YouTube National Library of Medicine 8600 Rockville Pike Bethesda, MD 20894 Web Policies FOIA HHS Vulnerability Disclosure Help Accessibility Careers NLM NIH HHS USA.gov Back to Top
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[FREE] Use the Distance Formula to write an equation of the parabola with vertex (0,0) and focus (0, -10). An - brainly.com 4 Search Learning Mode Cancel Log in / Join for free Browser ExtensionTest PrepBrainly App Brainly TutorFor StudentsFor TeachersFor ParentsHonor CodeTextbook Solutions Log in Join for free Tutoring Session +57,4k Smart guidance, rooted in what you’re studying Get Guidance Test Prep +43,6k Ace exams faster, with practice that adapts to you Practice Worksheets +7,1k Guided help for every grade, topic or textbook Complete See more / Mathematics Textbook & Expert-Verified Textbook & Expert-Verified Use the Distance Formula to write an equation of the parabola with vertex (0,0) and focus (0,−10). An equation of the parabola is y= 2 See answers Explain with Learning Companion NEW Asked by gabriellaizhere • 10/26/2020 0:00 / -- Read More Community by Students Brainly by Experts ChatGPT by OpenAI Gemini Google AI Community Answer This answer helped 28323117 people 28M 5.0 1 Upload your school material for a more relevant answer Use the Distance Formula to write an equation of the parabola with vertex (0,0) and focus (0, −10). The equation of the parabola is y = -(1/40)x². What is a distance formula? It is defined as the formula for finding the distance between two points. It has given the shortest path distance between two points. The distance formula can be given as: d=(x 2​−x 1​)2+(y 2​−y 1​)2​ It is given that the vertex is (0,0) and the focus is (0, −10). (x - h)² = 4a(y - k) (h, k) is the vertex of the parabola: (x - 0)² = 4a(y - 0) x² =4ay a = √[(0-0)² + (0-(-10)²] (c, d) is the focus of the parabola: a =10 As a result, the above equation is y = -(1/40)x². Thus, use the Distance Formula to write an equation of the parabola with vertex (0,0) and focus (0, −10). The equation of the parabola is y = -(1/40)x². Learn more about the distance formula here: brainly.com/question/18296211 SPJ2 Answered by maheshpatelvVT •7.9K answers•28.3M people helped Thanks 1 5.0 (2 votes) Textbook &Expert-Verified⬈(opens in a new tab) This answer helped 28323117 people 28M 5.0 1 Graduate Classical Mechanics - Michael Fowler Celestial Mechanics - Jeremy Tatum Classical Mechanics - Jeremy Tatum Upload your school material for a more relevant answer The equation of the parabola with vertex (0,0) and focus (0, -10) is given by the formula y=−40 1​x 2. This represents a downward-opening parabola. The distance p from the vertex to the focus is 10, denoting the downward orientation of the parabola. Explanation To find the equation of a parabola with its vertex at the origin (0,0) and focus at (0, -10), we will first recall how a parabola is defined in relation to its focus and directrix. A parabola can be described as the set of all points (x, y) that are equidistant from both a point called the focus and a line called the directrix. In our case, the focus is at (0, -10) and the directrix will be the line y = 10, which is located 10 units above the vertex. Since the focus is located below the vertex, this parabola opens downwards. For a parabola with its vertex at (h, k) and focus at (h, k - p), the standard form of the equation is: (x−h)2=4 p(y−k) Here: The vertex (h, k) is (0, 0), so h = 0 and k = 0. The distance p is the distance from the vertex to the focus, which in this case is 10 (the distance from (0, 0) to (0, -10)). Since it's opening downward, we take p = -10. Substituting these values into the equation gives: (x−0)2=4(−10)(y−0) This simplifies to: x 2=−40 y Thus, the equation of the parabola is: y=−40 1​x 2 Examples & Evidence For example, if we plot points that satisfy the equation y=−40 1​x 2 such as (0,0), (20,-10), and (-20,-10), we can see that they lie on the curve of the parabola. This helps visualize how the parabola opens downward with the vertex at (0,0) and focus at (0,-10). The standard equation of a parabola derived from its focus and directrix supports the formulated equation, confirming that the distance from any point on the curve to the focus equals its distance to the directrix, which is consistent with the properties of parabolas. Thanks 1 5.0 (2 votes) Advertisement Community Answer This answer helped 760 people 760 5.0 4 Answer: -−40 1​x 2 Eplanation: The quation form is: y=4(p)1​x 2 The dielectrick is: y=−p=10 ⇒ p=−10 The equation is: y=4(−10)1​x 2=−40 1​x 2 Answered by laviniadm05 •2 answers•760 people helped Thanks 4 5.0 (4 votes) Advertisement ### Free Mathematics solutions and answers Community Answer 4.8 7 The vertex of a parabola is (0,0)and the focus is (1/8, 0) . What is the equation of the parabola? Community Answer Write an equation of a parabola with the given information. vertex (0,0) , focus (0, 1/2) Community Answer 4.8 49 The vertex of a parabola is (0,0) and the focus is (0,2) what is the equation of the parabola Community Answer What is an equation of a parabola with the given vertex and focus? vertex 0,0 focus 2.5 0? Community Answer 4.6 13 A parabola and its focus are shown on the graph. The vertex of the parabola is at (0,0). On a coordinate plane, a parabola opens to the right. It has a vertex at (0, 0) and a focus (3, 0). What is the equation of the directrix of the parabola? y = 3 y = –3 x = 3 x = –3 Community Answer a. What is an equation of the parabola with vertex (0,0) and focus (0,-1.5) ? Community Answer 4.6 12 Jonathan and his sister Jennifer have a combined age of 48. If Jonathan is twice as old as his sister, how old is Jennifer Community Answer 11 What is the present value of a cash inflow of 1250 four years from now if the required rate of return is 8% (Rounded to 2 decimal places)? Community Answer 13 Where can you find your state-specific Lottery information to sell Lottery tickets and redeem winning Lottery tickets? (Select all that apply.) 1. Barcode and Quick Reference Guide 2. Lottery Terminal Handbook 3. Lottery vending machine 4. OneWalmart using Handheld/BYOD New questions in Mathematics Simplify 3 8 x 6​ completely given x>0. Simplify 3 27 x 15​ completely given x>0. Simplify. (v 3)−2 Write your answer without using negative exponents. Suppose X is a binomial random variable with 37 trials and a probability of success of 0.43. Find P(X=14). Round your answer to four decimal places. Solve for x: 3(x−1)(x+3)−7 x=9+x Previous questionNext question Learn Practice Test Open in Learning Companion Company Copyright Policy Privacy Policy Cookie Preferences Insights: The Brainly Blog Advertise with us Careers Homework Questions & Answers Help Terms of Use Help Center Safety Center Responsible Disclosure Agreement Connect with us (opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab) Brainly.com Dismiss Materials from your teacher, like lecture notes or study guides, help Brainly adjust this answer to fit your needs. Dismiss
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https://pmc.ncbi.nlm.nih.gov/articles/PMC4611579/
Comparison of four proton pump inhibitors for the short-term treatment of esophagitis in elderly patients - PMC Skip to main content An official website of the United States government Here's how you know Here's how you know Official websites use .gov A .gov website belongs to an official government organization in the United States. Secure .gov websites use HTTPS A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites. Search Log in Dashboard Publications Account settings Log out Search… Search NCBI Primary site navigation Search Logged in as: Dashboard Publications Account settings Log in Search PMC Full-Text Archive Search in PMC Journal List User Guide PERMALINK Copy As a library, NLM provides access to scientific literature. Inclusion in an NLM database does not imply endorsement of, or agreement with, the contents by NLM or the National Institutes of Health. Learn more: PMC Disclaimer | PMC Copyright Notice World J Gastroenterol . 2007 Sep 7;13(33):4467–4472. doi: 10.3748/wjg.v13.i33.4467 Search in PMC Search in PubMed View in NLM Catalog Add to search Comparison of four proton pump inhibitors for the short-term treatment of esophagitis in elderly patients Alberto Pilotto Alberto Pilotto 1 Alberto Pilotto, Marilisa Franceschi, Carlo Scarcelli, Luigi Piero D’Ambrosio, Francesco Paris, Davide Seripa, Geriatric Unit, Department of Medical Sciences & Gerontology and Geriatrics Laboratories, Department of Research, IRCCS “Casa Sollievo della Sofferenza”, San Giovanni Rotondo (FG), Italy 2 Marilisa Franceschi, Francesco Di Mario, Gastroenterology Department, University of Parma, Parma, Italy 3 Gioacchino Leandro, Gastroenterology Unit, IRCCS De Bellis, Castellana Grotte, Bari, Italy 4 Angelo Andriulli, Vito Annese, Gastroenterology Unit, Department of Medical Sciences, IRCCS “Casa Sollievo della Sofferenza”, San Giovanni Rotondo (FG), Italy Find articles by Alberto Pilotto 1,2,3,4, Marilisa Franceschi Marilisa Franceschi 1 Alberto Pilotto, Marilisa Franceschi, Carlo Scarcelli, Luigi Piero D’Ambrosio, Francesco Paris, Davide Seripa, Geriatric Unit, Department of Medical Sciences & Gerontology and Geriatrics Laboratories, Department of Research, IRCCS “Casa Sollievo della Sofferenza”, San Giovanni Rotondo (FG), Italy 2 Marilisa Franceschi, Francesco Di Mario, Gastroenterology Department, University of Parma, Parma, Italy 3 Gioacchino Leandro, Gastroenterology Unit, IRCCS De Bellis, Castellana Grotte, Bari, Italy 4 Angelo Andriulli, Vito Annese, Gastroenterology Unit, Department of Medical Sciences, IRCCS “Casa Sollievo della Sofferenza”, San Giovanni Rotondo (FG), Italy Find articles by Marilisa Franceschi 1,2,3,4, Gioacchino Leandro Gioacchino Leandro 1 Alberto Pilotto, Marilisa Franceschi, Carlo Scarcelli, Luigi Piero D’Ambrosio, Francesco Paris, Davide Seripa, Geriatric Unit, Department of Medical Sciences & Gerontology and Geriatrics Laboratories, Department of Research, IRCCS “Casa Sollievo della Sofferenza”, San Giovanni Rotondo (FG), Italy 2 Marilisa Franceschi, Francesco Di Mario, Gastroenterology Department, University of Parma, Parma, Italy 3 Gioacchino Leandro, Gastroenterology Unit, IRCCS De Bellis, Castellana Grotte, Bari, Italy 4 Angelo Andriulli, Vito Annese, Gastroenterology Unit, Department of Medical Sciences, IRCCS “Casa Sollievo della Sofferenza”, San Giovanni Rotondo (FG), Italy Find articles by Gioacchino Leandro 1,2,3,4, Carlo Scarcelli Carlo Scarcelli 1 Alberto Pilotto, Marilisa Franceschi, Carlo Scarcelli, Luigi Piero D’Ambrosio, Francesco Paris, Davide Seripa, Geriatric Unit, Department of Medical Sciences & Gerontology and Geriatrics Laboratories, Department of Research, IRCCS “Casa Sollievo della Sofferenza”, San Giovanni Rotondo (FG), Italy 2 Marilisa Franceschi, Francesco Di Mario, Gastroenterology Department, University of Parma, Parma, Italy 3 Gioacchino Leandro, Gastroenterology Unit, IRCCS De Bellis, Castellana Grotte, Bari, Italy 4 Angelo Andriulli, Vito Annese, Gastroenterology Unit, Department of Medical Sciences, IRCCS “Casa Sollievo della Sofferenza”, San Giovanni Rotondo (FG), Italy Find articles by Carlo Scarcelli 1,2,3,4, Luigi Piero D’Ambrosio Luigi Piero D’Ambrosio 1 Alberto Pilotto, Marilisa Franceschi, Carlo Scarcelli, Luigi Piero D’Ambrosio, Francesco Paris, Davide Seripa, Geriatric Unit, Department of Medical Sciences & Gerontology and Geriatrics Laboratories, Department of Research, IRCCS “Casa Sollievo della Sofferenza”, San Giovanni Rotondo (FG), Italy 2 Marilisa Franceschi, Francesco Di Mario, Gastroenterology Department, University of Parma, Parma, Italy 3 Gioacchino Leandro, Gastroenterology Unit, IRCCS De Bellis, Castellana Grotte, Bari, Italy 4 Angelo Andriulli, Vito Annese, Gastroenterology Unit, Department of Medical Sciences, IRCCS “Casa Sollievo della Sofferenza”, San Giovanni Rotondo (FG), Italy Find articles by Luigi Piero D’Ambrosio 1,2,3,4, Francesco Paris Francesco Paris 1 Alberto Pilotto, Marilisa Franceschi, Carlo Scarcelli, Luigi Piero D’Ambrosio, Francesco Paris, Davide Seripa, Geriatric Unit, Department of Medical Sciences & Gerontology and Geriatrics Laboratories, Department of Research, IRCCS “Casa Sollievo della Sofferenza”, San Giovanni Rotondo (FG), Italy 2 Marilisa Franceschi, Francesco Di Mario, Gastroenterology Department, University of Parma, Parma, Italy 3 Gioacchino Leandro, Gastroenterology Unit, IRCCS De Bellis, Castellana Grotte, Bari, Italy 4 Angelo Andriulli, Vito Annese, Gastroenterology Unit, Department of Medical Sciences, IRCCS “Casa Sollievo della Sofferenza”, San Giovanni Rotondo (FG), Italy Find articles by Francesco Paris 1,2,3,4, Vito Annese Vito Annese 1 Alberto Pilotto, Marilisa Franceschi, Carlo Scarcelli, Luigi Piero D’Ambrosio, Francesco Paris, Davide Seripa, Geriatric Unit, Department of Medical Sciences & Gerontology and Geriatrics Laboratories, Department of Research, IRCCS “Casa Sollievo della Sofferenza”, San Giovanni Rotondo (FG), Italy 2 Marilisa Franceschi, Francesco Di Mario, Gastroenterology Department, University of Parma, Parma, Italy 3 Gioacchino Leandro, Gastroenterology Unit, IRCCS De Bellis, Castellana Grotte, Bari, Italy 4 Angelo Andriulli, Vito Annese, Gastroenterology Unit, Department of Medical Sciences, IRCCS “Casa Sollievo della Sofferenza”, San Giovanni Rotondo (FG), Italy Find articles by Vito Annese 1,2,3,4, Davide Seripa Davide Seripa 1 Alberto Pilotto, Marilisa Franceschi, Carlo Scarcelli, Luigi Piero D’Ambrosio, Francesco Paris, Davide Seripa, Geriatric Unit, Department of Medical Sciences & Gerontology and Geriatrics Laboratories, Department of Research, IRCCS “Casa Sollievo della Sofferenza”, San Giovanni Rotondo (FG), Italy 2 Marilisa Franceschi, Francesco Di Mario, Gastroenterology Department, University of Parma, Parma, Italy 3 Gioacchino Leandro, Gastroenterology Unit, IRCCS De Bellis, Castellana Grotte, Bari, Italy 4 Angelo Andriulli, Vito Annese, Gastroenterology Unit, Department of Medical Sciences, IRCCS “Casa Sollievo della Sofferenza”, San Giovanni Rotondo (FG), Italy Find articles by Davide Seripa 1,2,3,4, Angelo Andriulli Angelo Andriulli 1 Alberto Pilotto, Marilisa Franceschi, Carlo Scarcelli, Luigi Piero D’Ambrosio, Francesco Paris, Davide Seripa, Geriatric Unit, Department of Medical Sciences & Gerontology and Geriatrics Laboratories, Department of Research, IRCCS “Casa Sollievo della Sofferenza”, San Giovanni Rotondo (FG), Italy 2 Marilisa Franceschi, Francesco Di Mario, Gastroenterology Department, University of Parma, Parma, Italy 3 Gioacchino Leandro, Gastroenterology Unit, IRCCS De Bellis, Castellana Grotte, Bari, Italy 4 Angelo Andriulli, Vito Annese, Gastroenterology Unit, Department of Medical Sciences, IRCCS “Casa Sollievo della Sofferenza”, San Giovanni Rotondo (FG), Italy Find articles by Angelo Andriulli 1,2,3,4, Francesco Di Mario Francesco Di Mario 1 Alberto Pilotto, Marilisa Franceschi, Carlo Scarcelli, Luigi Piero D’Ambrosio, Francesco Paris, Davide Seripa, Geriatric Unit, Department of Medical Sciences & Gerontology and Geriatrics Laboratories, Department of Research, IRCCS “Casa Sollievo della Sofferenza”, San Giovanni Rotondo (FG), Italy 2 Marilisa Franceschi, Francesco Di Mario, Gastroenterology Department, University of Parma, Parma, Italy 3 Gioacchino Leandro, Gastroenterology Unit, IRCCS De Bellis, Castellana Grotte, Bari, Italy 4 Angelo Andriulli, Vito Annese, Gastroenterology Unit, Department of Medical Sciences, IRCCS “Casa Sollievo della Sofferenza”, San Giovanni Rotondo (FG), Italy Find articles by Francesco Di Mario 1,2,3,4 Author information Article notes Copyright and License information 1 Alberto Pilotto, Marilisa Franceschi, Carlo Scarcelli, Luigi Piero D’Ambrosio, Francesco Paris, Davide Seripa, Geriatric Unit, Department of Medical Sciences & Gerontology and Geriatrics Laboratories, Department of Research, IRCCS “Casa Sollievo della Sofferenza”, San Giovanni Rotondo (FG), Italy 2 Marilisa Franceschi, Francesco Di Mario, Gastroenterology Department, University of Parma, Parma, Italy 3 Gioacchino Leandro, Gastroenterology Unit, IRCCS De Bellis, Castellana Grotte, Bari, Italy 4 Angelo Andriulli, Vito Annese, Gastroenterology Unit, Department of Medical Sciences, IRCCS “Casa Sollievo della Sofferenza”, San Giovanni Rotondo (FG), Italy Author contributions: All authors contributed equally to the work. Correspondence to: Alberto Pilotto, Geriatric Unit, IRCCS "Casa Sollievo della Sofferenza", Viale Cappuccini, 71013 San Giovanni Rotondo (FG), Italy. alberto.pilotto@operapadrepio.it Telephone: +39-882-410271 Fax: +39-882-410271 Received 2007 May 16; Revised 2007 Jun 1; Accepted 2007 Jun 4; Issue date 2007 Sep 7. ©2007 Baishideng Publishing Group Co., Limited. All rights reserved. PMC Copyright notice PMCID: PMC4611579 PMID: 17724802 Abstract AIM: To compare efficacy and tolerability of four proton pump inhibitors (PPIs) commonly used in the short-term therapy of esophagitis in elderly patients. METHODS: A total of 320 patients over 65 years with endoscopically diagnosed esophagitis were randomly assigned to one of the following treatments for 8 wk: (1) omeprazole 20 mg/d; (2) lansoprazole 30 mg/d; (3) pantoprazole 40 mg/d, or (4) rabeprazole 20 mg/d. Major symptoms, compliance, and adverse events were recorded. After 8 wk, endoscopy and clinical evaluation were repeated. RESULTS: Per protocol and intention to treat healing rates of esophagitis were: omeprazole = 81.0% and 75.0%, lansoprazole = 90.7% (P = 0.143 vs omeprazole) and 85.0%, pantoprazole = 93.5% (P = 0.04 vs omeprazole) and 90.0% (P = 0.02 vs omeprazole), rabeprazole = 94.6% (P = 0.02 vs omeprazole) and 88.8% (P = 0.04 vs omeprazole). Dividing patients according to the grades of esophagitis, omeprazole was significantly less effective than the three other PPIs in healing grade 1 esophagitis (healing rates: 81.8% vs 100%, 100% and 100%, respectively, P = 0.012). Pantoprazole and rabeprazole (100%) were more effective vs omeprazole (89.6%, P = 0.0001) and lansoprazole (82.4%, P = 0.0001) in decreasing heartburn. Pantoprazole and rabeprazole (92.2% and 90.1%, respectively) were also more effective vs lansoprazole (75.0%, P< 0.05) in decreasing acid regurgitation. Finally, pantoprazole and rabeprazole (95.2% and 100%) were also more effective vs lansoprazole (82.6%, P< 0.05) in decreasing epigastric pain. CONCLUSION: In elderly patients, pantoprazole and rabeprazole were significantly more effective than omeprazole in healing esophagitis and than omeprazole or lansoprazole in improving symptoms. H pylori infection did not influence the healing rates of esophagitis after a short-term treatment with PPI. Keywords: Elderly, Esophagitis, Proton pump inhibitors INTRODUCTION Old age is known to be a significant risk factor for severe esophagitis[1,2], chronic relapses, as well as severe complications of the disease[1,4]. Clinical features of esophagitis in elderly patients are quite different from those of young or adult subjects. Indeed, elderly patients present less frequently the typical symptoms of heartburn, acid regurgitation and/or epigastric pain. Conversely, the prevalence of other non-specific symptoms, i.e. anorexia, weight loss, anaemia, and/or vomiting significantly increases with age. Thus, the diagnosis of reflux esophagitis may be missed in the elderly, and a substantial number of patients may suffer subclinical relapses of the disease. The treatment of esophagitis is based on gastric acid suppression with antisecretory drugs. Proton pump inhibitors (PPIs) are widely used and their effectiveness and safety have been demonstrated also in patients of old age. Currently, five PPIs are available on the market: omeprazole, lansoprazole, rabeprazole, pantoprazole, and esomeprazole. Some age-associated differences in pharmacokinetics and pharmacodynamics of the PPIs have been reported. However, it is unknown if these differences are associated with different clinical effects, i.e. healing rates and/or symptom relief, particularly in older patients. The aim of this study was to compare the clinical efficacy and tolerability of four PPIs used for the short-term therapy of esophagitis in elderly patients. MATERIALS AND METHODS Study design This was an open, single-centre, randomized study including elderly subjects that consecutively underwent an upper gastrointestinal endoscopy. It was conducted according to the Declaration of Helsinki and the guidelines for Good Clinical Practice. All patients gave their informed consent prior to participation in the study. The inclusion criteria were: (1) age 65 years or over and (2) endoscopic diagnosis of esophagitis grade I to IV according to the Savary-Miller classification. Major exclusion criteria were: history of Zollinger-Ellison syndrome, pyloric stenosis, previous surgery of the esophagus and/or gastrointestinal tract (except for appendectomy and cholecystectomy), and gastrointestinal malignancy. Patients were excluded if they had received antacids, sucralfate, prokinetics, H2-blockers, and/or PPIs for more than 7 d in the four weeks prior to the start of the study. Assessments At the initial visit, demographic data, medical history, clinical symptoms, non-steroidal anti-inflammatory drug (NSAID) use, and antisecretory therapy were recorded. At study entry, an endoscopy was performed to diagnose acute esophagitis (inclusion criteria). After 2 mo of treatment, endoscopy was repeated to evaluate healing of acute esophagitis. All patients were examined during therapy to record side effects and to count tablets. Compliance was defined as “good” when more than 90% of the tablets had been taken by the patients. Adverse events were rated by the investigator as not related, unlikely, possibly related, or likely related to the medication. Endoscopic diagnoses Reflux esophagitis was endoscopically defined by epithelial defects according to the Savary-Miller criteria and classified as grade I: non-confluent erosions; grade II: confluent erosions; grade III: lesions extending to the entire circumference of the lower esophagus; and grade IV: deep ulcer or esophagitis with complications, i.e. stenosis and/or hemorrhagic lesions. Patients with diffuse erythema and/or fragility of the lower esophagus were not included. Hiatus hernia was diagnosed when the Z-line and the gastric folds extended 2 cm or more above the diaphragmatic hiatus. Patients with Barrett’s esophagus were not included unless erosive esophagitis was also present. Histology and H pylori infection During endoscopy, six gastric biopsies were taken from both the antrum (three biopsies), and from the body (three biopsies). Two antral and two body biopsies were used for histological analysis, while one from each site was used for the rapid urease test (CLO test, Delta West Pty Ltd, Western Australia). For histological examination, biopsy specimens were immediately fixed in buffered neutral formalin and embedded in paraffin. Sections were stained with hematoxylin-eosin and modified Giemsa for the detection of H pylori and evaluated according to the Sydney classification. Patients were considered H pylori negative if both histology and the rapid urease test were negative; patients were considered H pylori positive if either their histology or rapid urease test, or both, were positive for Hp infection. Symptomatology Symptoms were assessed during a structured interview. The patient was questioned about the principal symptoms, i.e. acid regurgitation, heartburn, and other symptoms of reflux esophagitis, i.e. epigastric pain, dysphagia, vomiting, and anaemia (loss of ≥ 3 grams of haemoglobin during the last 3 mo) and expressed as absent/present. Treatments Patients included in the study were consecutively assigned to one of the following regimens for two months: omeprazole 20 mg once daily, lansoprazole 30 mg once daily; pantoprazole 40 mg once daily, or rabeprazole 20 mg once daily. Randomization was performed by a computer-generated list in blocks of four with a 1:1:1:1 ratio. All PPIs were taken in the morning fasting just before breakfast. Patients who resulted H pylori positive were treated with the PPI plus two antibiotics i.e., amoxicillin 1g twice daily and claritromycin 250 mg twice daily or metronidazole 250 mg four times daily for 7 d. Statistical analysis Statistical analysis was performed by means of the SPSS version 13. Results were evaluated using both "per protocol" (PP) and "intention-to-treat" (ITT) analyses; the 95% confidence intervals (95% CI) were also calculated. The ITT population was defined as all patients initially enrolled who had taken at least one dose of study medication. Statistical analysis was performed using the χ 2 test (comparison of outcomes with the treatments) and Fisher exact test (healing rates related to H pylori infection, symptoms). All p values were two-tailed with statistical significance indicated by a value of P< 0.05. RESULTS A total of 320 consecutive elderly (156 males and 164 females, mean age 77.4 ± 7.9 years, range from 65 to 93 years) with an endoscopic diagnosis of acute esophagitis, grades 1 to 4 according to the Savary-Miller classification, were included in the study. Demographic and clinical characteristics of patients are shown in Table 1. Table 1. Demographic and clinical characteristics of the study population All patientsOmeprazoleLansoprazolePantoprazoleRabeprazole Number of patients 320 80 80 80 80 Males/Females 156/164 44/36 36/44 39/41 37/43 Mean age (yr)77.4 ± 7.9 77.9 ± 6.4 77.8 ± 9.2 76.8 ± 6.1 77.0 ± 9.5 Age Range (yr)65-93 65-93 65-92 65-88 65-93 Esophagitis n (%) -Grade I°96 (30.0)34 (42.5)26 (32.5)20 (25.0)16 (20.0) -Grade II°152 (47.5)27 (33.8)33 (41.3)42 (52.5)50 (62.5) -Grade III°-IV°72 (22.5)19 (23.8)21 (26.2)18 (22.5)14 (27.6) Hiatus hernia n (%)194 (60.6)43 (53.8)48 (60.0)50 (62.5)53 (66.3) H pylori infection n/n (%)202/306 (66.0)52/76 (68.4)61/76 (80.3)51/77 (66.2)38/77 (49.3) NSAIDs/Aspirin use n (%)78 (24.4)18 (22.5)17 (21.3)26 (32.5)17 (21.3) Open in a new tab Nineteen patients (5.9% of the total population) dropped-out from the study due to: adverse events (2 patients), low compliance (11 patients), and refusal of endoscopy after two months of treatment (6 patients). Among the 301 patients who completed the study, 271 had healed esophagitis and 30 were unhealed. The overall PP and ITT healing rates of esophagitis were 90.0% (95% CI = 86.6-93.4) and 84.7% (95% CI = 80.7-88.6), respectively. Dividing patients according to treatments, the PP and ITT healing rates of esophagitis were: omeprazole = 81.0% and 75.0%, lansoprazole = 90.7% (P = 0.143 vs omeprazole) and 85% (P = 0.167 vs omeprazole), pantoprazole = 93.5% (P = 0.04 vs omeprazole) and 90.0% (P = 0.02 vs omeprazole), rabeprazole = 94.6% (P = 0.02 vs omeprazole) and 88.8% (P = 0.04 vs omeprazole) respectively (Table 2). Dividing patients according to the grades of esophagitis, a significantly lower healing rate was observed in patients with grade 1 esophagitis treated with omeprazole compared to patients treated with lansoprazole, pantoprazole, or rabeprazole (healing rates: 81.8% vs 100%, 100% and 100%, respectively, P = 0.012). Omeprazole was less effective than the three other PPIs also in patients with grade 2 esophagitis (healing rates: 81.8% vs 96.5% vs 90% vs 95.8%, respectively) and than pantoprazole and rabeprazole in grade 3-4 esophagitis (healing rates: 78.9% vs 94.1% vs 84.6%, respectively); probably due to the low number of patients, however, the differences were no statistically significant (Table 3). Table 2. Healing rates, drop-out patients, and side effects in elderly patients divided according to the different PPI regimens Per protocol analysis Intention to treat analysis RegimenNo. of patientsCure rates %(N° of patients)95% CICure rates %(N° of patients)95% CIDrop outsSide effects Omeprazole 80 81.0 72.0–89.9 75.0 65.0–84.0 6 (7.5)1 60/74 60/80 Lansoprazole 80 90.7 84.1–97.2 85.0 77.0-92.8 5 (6.3)1 68/75 68/80 Pantoprazole 80 93.5187.9–99.0 90.01 83.4-96.5 3 (3.8)1 72/77 72/80 Rabeprazole 80 94.6289.4–99.7 88.82 81.5-95.6 5 (6.3)1 71/75 71/80 Total 320 90.0 86.6-93.4 84.7 80.7-88.6 19 (5.9)4 271/301 271/320(1.4) Open in a new tab 1 Pantoprazole vs Omeprazole: PP analysis: P = 0.039, ITT analysis P = 0.022; 2 Rabeprazole vs Omeprazole: PP analysis: P = 0.022, ITT analysis P = 0.040. Table 3. Healing rates of esophagitis after eight weeks of PPI treatment in elderly patients with esophagitis divided according to the grades of severity of esophagitis according to the Savary-Miller classification Severity gradesOmeprazoleLansoprazolePantoprazoleRabeprazole n%n%n%n% I gradea27/33 81.8 25/25 100 20/20 100 14/14 100 II gradec18/22 81.8 28/29 96.5 36/40 90.0 46/48 95.8 III-IV gradese15/19 78.9 15/21 71.4 16/17 94.1 11/13 84.6 Open in a new tab a P = 0.012; c P = 0.215; e P = 0.458. At baseline 188 of 288 patients (65.3%) were identified as infected with H pylori in the gastric mucosa. No differences were observed in the healing rates of esophagitis between H pylori positive and H pylori negative patients (90.4% vs 89.0%, P = NS). After two months, 149 of 188 (79.3%) who were treated with triple therapies for one week were H pylori negative while 39 patients (20.7%) remained H pylori positive after treatment. No significant differences in the healing rates of esophagitis were observed between successfully and unsuccessfully treated H pylori patients (negative H pylori vs still-positive after treatment: 89.9% vs 92.3%, P = NS) (Table 4). Table 4. Healing rates of esophagitis in elderly patients divided according to H pylori infection Omeprazole Lansoprazole Pantoprazole Rabeprazole All (n = 71)(n = 71)(n = 74)(n = 72)(n = 288) H pylori positive 38/49 54/57 45/48 33/34 170/188 n = 188 77.6%94.7%93.8%97.1%90.4% H pylori negative 19/22 11/14 24/26 35/38 89/100 n = 100 86.4%78.6%92.3%92.1%89.0% Omeprazole Lansoprazole Pantoprazole Rabeprazole All (n = 49)(n = 57)(n = 48)(n = 34)(n = 188) H pylori cured 24/32 43/46 39/42 28/29 134/149 n = 149 75.0%93.5%92.6%96.6%89.9% H pylori still-positive 14/17 11/11 6/6 5/5 36/39 n = 39 82.4%100%100%100%92.3% Open in a new tab After two months of PPI treatment, a significant reduction of symptoms as compared to baseline was observed both in healed and in unhealed patients. While heartburn improved significantly more effectively in healed patients than unhealed patients (rates of heartburn disappearance = 96.7% vs 80%, P = 0.001), other symptoms improved significantly both in healed and unhealed patients (Table 5). The rates of symptom disappearance in the four treatment groups, i.e. omeprazole, lansoprazole, pantoprazole, and rabeprazole, were 86.9%, 82.4%, 100%, and 100% for heartburn, 100%, 75.0%, 92.9%, and 90.1% for acid regurgitation, and 95.0%, 82.6%, 95.2, and 100% for epigastric pain, respectively (Table 6). Comparisons between the four PPIs demonstrated that pantoprazole and rabeprazole were more effective than omeprazole (100% vs 86.9, and 100% vs 86.9%, respectively, P< 0.05) and than lansoprazole (100% vs 82.4%, P = 0.0001 and 100% vs 82.4%, P = 0.005, respectively) in decreasing heartburn. Lansoprazole was less effective in improving acid regurgitation and epigastric pain than omeprazole (P = 0.0001, P = 0.033, respectively), pantoprazole (P = 0.005, P = 0.028, respectively), and rabeprazole (P = 0.026, P = 0.0001, respectively) (Table 6). Table 5. Symptoms in elderly patients with esophagitis before and after two months of PPI therapy SymptomsBefore therapyAfter therapy All Healed Unhealed Healed vs unhealed n = 301 n = 271 n = 30 P value Heartburn (n, %)131 (43.5)9 (3.3)6 (20.0)0.0001 Acid regurgitation (n, %)39 (13.0)4 (1.5)0 (0.0)0.874 Epigastric pain (n , %)143 (47.5)10 (3.7)2 (6.6)0.781 Dysphagia (n, %)10 (3.3)0 (0.0)0 (0.0)-- Vomiting (n, %)60 (19.9)0 (0.0)0 (0.0)-- Anaemia (n, %)28 (9.3)0 (0.0)0 (0.0)-- Open in a new tab Table 6. Symptom disappearance after therapy in elderly patients divided according to PPI regimens % OmeprazoleLansoprazolePantoprazoleRabeprazole Heartburn 86.9a82.4bd100 100 Acid regurgitation 100 75.0cf92.2 90.1 Epigastric pain 95 82.6eh95.2 100 Dysphagia 100 100 100 100 Vomiting 100 100 100 100 Aenemia 100 100 100 100 Open in a new tab a P< 0.05 Omeprazole vs Pantoprazole and Omeprazole vs Rabeprazole; b P = 0.0001 Lansoprazole vs Pantoprazole; d P = 0.005 Lansoprazole vs Rabeprazole; f P = 0.0001 Lansoprazole vs Omeprazole; c P< 0.05 Lansoprazole vs Pantoprazole, Lansoprazole vs Rabeprazole; e P< 0.05 Lansoprazole vs Omeprazole and Lansoprazole vs Pantoprazole; h P = 0.0001 Lansoprazole vs Rabeprazole. All four PPIs were well tolerated. Adverse events were reported only by four patients (1.3%): orticaria, glossitis, nausea, and headache. Two patients discontinued therapy due to treatment-related side effects. No significant differences were found in the prevalence of adverse events among the four treatment groups. DISCUSSION This study demonstrates that in patients over 65 years PPI therapy for 2 mo is very effective in healing acute esophagitis. The pooled ITT and PP healing rates were 84.7% and 90.0%, respectively: These are comparable to previous data from double-blind studies carried out in non-elderly subjects treated for 8 wk with omeprazole 20 mg or lansoprazole 30 mg daily, pantoprazole 40 mg daily, or rabeprazole 20 mg daily. In this population of older patients, pantoprazole and rabeprazole were significantly more effective in healing esophagitis than omeprazole. Moreover, pantoprazole and rabeprazole were more effective than lansoprazole and omeprazole in improving heartburn, and than lansoprazole in improving acid regurgitation and epigastric pain. Previous studies were focused on potential discrepancies in efficacy among the different PPIs used for treatment of reflux esophagitis. While some previous reports suggest that acid-suppressive effect of the four PPIs is different on the basis of equivalent molecular dose, clinical studies that support such a different efficacy in healing esophagitis or improving symptoms of GERD on a PPI-equivalent molecular doses are lacking. A meta-analysis of 38 studies evaluating acute therapy of esophagitis reported that the PPIs were superior to ranitidine and placebo in healing erosive esophagitis, without significant differences in efficacy between omeprazole 20 mg daily and lansoprazole 30 mg daily, or pantoprazole 40 mg daily, or rabeprazole 20 mg daily. Similarly, in another meta-analysis, no differences in healing rates of esophagitis were reported between standard doses of lansoprazole, pantoprazole, rabeprazole, and omeprazole. More recently, a meta-analysis of eleven studies with 23 treatment arms reported no significant difference in the two-month healing rates of esophagitis between omeprazole 20 mg daily (n = 3.137 patients, pooled healing rate = 84.5%) and other PPIs, including lansoprazole, pantoprazole, rabeprazole, and esomeprazole at standard doses (n = 3.397 patients, pooled healing rate = 89.4%). However, none of the studies included in these meta-analyses were carried out specifically in elderly patients. Indeed, to our knowledge, this is the first study that compared the efficacy of different PPIs in curing esophagitis and improving symptoms in elderly patients. Why pantoprazole and rabeprazole were more effective than omeprazole in healing esophagitis and than omeprazole and lansoprazole in improving symptoms in elderly patients is not clear. Very recently it was suggested that omeprazole has considerable potential for drug interactions since it has high affinity for the cytochrome CYP2C19 and a lower affinity for the cytochrome CYP3A4, while pantoprazole, and maybe rabeprazole, appear to have lower potential for interactions with other drugs. Data from this study cannot confirm this hypothesis since no information was collected on concomitant treatments, with the exception of NSAID and aspirin. Interestingly, a previous multicentre study, carried out in 164 elderly patients with esophagitis reported that a 2-month therapy with pantoprazole 40 mg/d was highly effective in healing reflux esophagitis (81.1% and 93.7% by ITT and PP analyses, respectively), although the majority of patients received other drugs for concomitant illnesses (76.2% of patients), without that the presence of concomitant treatments adversely affected the efficacy or tolerability of pantoprazole. Very recently, a systematic review of randomized controlled trials in patients with reflux esophagitis reported that esomeprazole demonstrated higher short-term healing rates when compared with standard dose PPIs. While no data were reported comparing rabeprazole 20 mg to esomeprazole 40 mg, two studies included in the analysis compared pantoprazole 40 mg daily to esomperazole 40 mg daily. This comparison found no differences in healing rates between the two treatments both in patients with moderate-severe esophagitis, i.e. Los Angeles grades B and C (healing rates with pantoprazole = 83.2% vs esomeprazole = 80.7%, P = NS) and in the subgroup of 550 patients aged 65 years or over included in the large multicenter EXPO study (healing rates with pantoprazole = 87.4% vs esomeprazole = 90.4%, P = NS). Unfortunately, information on esomeprazole was not available for the present study. In this elderly population, H pylori infection did not influence the response to short-term treatment with PPIs. This finding confirms the data of previous studies performed in elderly populations showing that H pylori infection does not have a negative effect on healing of esophagitis, nor does it worsen reflux symptoms at two-month follow up. It is also evident from our study that H pylori eradication does not affect the cure rate of esophagitis during a two-month course of PPI, in agreement with a recent multicentre randomized study also performed in elderly patients. In conclusion, PPIs are highly effective and well tolerated in curing gastroesophageal reflux disease in elderly patients. Pantoprazole and rabeprazole were significantly more effective than omeprazole in healing esophagitis and than omeprazole or lansoprazole in improving symptoms. H pylori infection does not influence the healing rates of esophagitis after a short-term treatment with PPI. Footnotes Supported by “Ministero della Salute”, IRCCS Research Program, Ricerca Corrente 2006-2008, Linea n. 2 “Malattie di rilevanza sociale” S- Editor Liu Y L- Editor Mihm S E- Editor Li JL References 1.Richter JE. Gastroesophageal reflux disease in the older patient: presentation, treatment, and complications. Am J Gastroenterol. 2000;95:368–373. doi: 10.1111/j.1572-0241.2000.t01-1-01791.x. [DOI] [PubMed] [Google Scholar] 2.Pilotto A, Franceschi M, Leandro G, Scarcelli C, D'Ambrosio LP, Seripa D, Perri F, Niro V, Paris F, Andriulli A, et al. Clinical features of reflux esophagitis in older people: a study of 840 consecutive patients. J Am Geriatr Soc. 2006;54:1537–1542. doi: 10.1111/j.1532-5415.2006.00899.x. [DOI] [PubMed] [Google Scholar] 3.Pilotto A, Franceschi M, Leandro G, Novello R, Di Mario F, Valerio G. Long-term clinical outcome of elderly patients with reflux esophagitis: a six-month to three-year follow-up study. Am J Ther. 2002;9:295–300. doi: 10.1097/00045391-200207000-00006. [DOI] [PubMed] [Google Scholar] 4.Zimmerman J, Shohat V, Tsvang E, Arnon R, Safadi R, Wengrower D. Esophagitis is a major cause of upper gastrointestinal hemorrhage in the elderly. Scand J Gastroenterol. 1997;32:906–909. doi: 10.3109/00365529709011200. [DOI] [PubMed] [Google Scholar] 5.Maekawa T, Kinoshita Y, Okada A, Fukui H, Waki S, Hassan S, Matsushima Y, Kawanami C, Kishi K, Chiba T. Relationship between severity and symptoms of reflux oesophagitis in elderly patients in Japan. J Gastroenterol Hepatol. 1998;13:927–930. doi: 10.1111/j.1440-1746.1998.tb00763.x. [DOI] [PubMed] [Google Scholar] 6.Pilotto A, Franceschi M, Paris F. Recent advances in the treatment of GERD in the elderly: focus on proton pump inhibitors. Int J Clin Pract. 2005;59:1204–1209. doi: 10.1111/j.1368-5031.2005.00639.x. [DOI] [PubMed] [Google Scholar] 7.Klotz U. Effect of aging on the pharmacokinetics of gastrointestinal drugs. In: Aging and the Gastrointestinal Tract., editor. Pilotto A, Malfertheiner P, Holt P, editors. Switzerland: Karger Press Basel; 2003. pp. 28–39. [Google Scholar] 8.Ollyo JB, Lang F, Fontolliet CH, Brossard E. Savary-Miller's new endoscopic grading of reflux-oesophagitis: a simple, reproducible, logical, complete and useful classification. Gastroenterology. 1990;98 Supp l:A–100. [Google Scholar] 9.Kaul B, Petersen H, Myrvold HE, Grette K, Røysland P, Halvorsen T. Hiatus hernia in gastroesophageal reflux disease. Scand J Gastroenterol. 1986;21:31–34. doi: 10.3109/00365528609034617. [DOI] [PubMed] [Google Scholar] 10.Misiewicz JJ, Tytgat GNJ, Goodwin CS et al. The Sydney system: a new classification of gastritis. Working Party Reports of the 9th World Congress of Gastroenterology, Melbourne: Blackwell Scientific; 1990. pp. 1–10. [Google Scholar] 11.Pilotto A, Salles N. Helicobacter pylori infection in geriatrics. Helicobacter. 2002;7 Suppl 1:56–62. doi: 10.1046/j.1523-5378.7.s1.9.x. [DOI] [PubMed] [Google Scholar] 12.Pilotto A, Franceschi M, Perri F, Orsitto G, Di Mario F. Treatment options for H pylori infection in the elderly. Aging Health. 2006;2:661–668. [Google Scholar] 13.Sharma VK, Leontiadis GI, Howden CW. Meta-analysis of randomized controlled trials comparing standard clinical doses of omeprazole and lansoprazole in erosive oesophagitis. Aliment Pharmacol Ther. 2001;15:227–231. doi: 10.1046/j.1365-2036.2001.00904.x. [DOI] [PubMed] [Google Scholar] 14.Holtmann G, Cain C, Malfertheiner P. Gastric Helicobacter pylori infection accelerates healing of reflux esophagitis during treatment with the proton pump inhibitor pantoprazole. Gastroenterology. 1999;117:11–16. doi: 10.1016/s0016-5085(99)70544-5. [DOI] [PubMed] [Google Scholar] 15.Pace F, Annese V, Prada A, Zambelli A, Casalini S, Nardini P, Bianchi Porro G. Rabeprazole is equivalent to omeprazole in the treatment of erosive gastro-oesophageal reflux disease. A randomised, double-blind, comparative study of rabeprazole and omeprazole 20 mg in acute treatment of reflux oesophagitis, followed by a maintenance open-label, low-dose therapy with rabeprazole. Dig Liver Dis. 2005;37:741–750. doi: 10.1016/j.dld.2005.04.026. [DOI] [PubMed] [Google Scholar] 16.Caro JJ, Salas M, Ward A. Healing and relapse rates in gastroesophageal reflux disease treated with the newer proton-pump inhibitors lansoprazole, rabeprazole, and pantoprazole compared with omeprazole, ranitidine, and placebo: evidence from randomized clinical trials. Clin Ther. 2001;23:998–1017. doi: 10.1016/s0149-2918(01)80087-4. [DOI] [PubMed] [Google Scholar] 17.Edwards SJ, Lind T, Lundell L. Systematic review of proton pump inhibitors for the acute treatment of reflux oesophagitis. Aliment Pharmacol Ther. 2001;15:1729–1736. doi: 10.1046/j.1365-2036.2001.01128.x. [DOI] [PubMed] [Google Scholar] 18.Wang WH, Huang JQ, Zheng GF, Xia HH, Wong WM, Lam SK, Wong BC. Head-to-head comparison of H2-receptor antagonists and proton pump inhibitors in the treatment of erosive esophagitis: a meta-analysis. World J Gastroenterol. 2005;11:4067–4077. doi: 10.3748/wjg.v11.i26.4067. [DOI] [PMC free article] [PubMed] [Google Scholar] 19.Blume H, Donath F, Warnke A, Schug BS. Pharmacokinetic drug interaction profiles of proton pump inhibitors. Drug Saf. 2006;29:769–784. doi: 10.2165/00002018-200629090-00002. [DOI] [PubMed] [Google Scholar] 20.Pilotto A, Leandro G, Franceschi M. Short- and long-term therapy for reflux oesophagitis in the elderly: a multi-centre, placebo-controlled study with pantoprazole. Aliment Pharmacol Ther. 2003;17:1399–1406. doi: 10.1046/j.1365-2036.2003.01593.x. [DOI] [PubMed] [Google Scholar] 21.Edwards SJ, Lind T, Lundell L. Systematic review: proton pump inhibitors (PPIs) for the healing of reflux oesophagitis - a comparison of esomeprazole with other PPIs. Aliment Pharmacol Ther. 2006;24:743–750. doi: 10.1111/j.1365-2036.2006.03074.x. [DOI] [PubMed] [Google Scholar] 22.Gillessen A, Beil W, Modlin IM, Gatz G, Hole U. 40 mg pantoprazole and 40 mg esomeprazole are equivalent in the healing of esophageal lesions and relief from gastroesophageal reflux disease-related symptoms. J Clin Gastroenterol. 2004;38:332–340. doi: 10.1097/00004836-200404000-00007. [DOI] [PubMed] [Google Scholar] 23.Labenz J, Armstrong D, Lauritsen K, Katelaris P, Schmidt S, Schütze K, Wallner G, Juergens H, Preiksaitis H, Keeling N, et al. A randomized comparative study of esomeprazole 40 mg versus pantoprazole 40 mg for healing erosive oesophagitis: the EXPO study. Aliment Pharmacol Ther. 2005;21:739–746. doi: 10.1111/j.1365-2036.2005.02368.x. [DOI] [PubMed] [Google Scholar] 24.Pilotto A, Franceschi M, Leandro G, Rassu M, Bozzola L, Valerio G, Di Mario F. Influence of Helicobacter pylori infection on severity of oesophagitis and response to therapy in the elderly. Dig Liver Dis. 2002;34:328–331. doi: 10.1016/s1590-8658(02)80125-6. [DOI] [PubMed] [Google Scholar] 25.Pilotto A, Perri F, Leandro G, Franceschi M. Effect of Helicobacter pylori eradication on the outcome of reflux esophagitis and chronic gastritis in the elderly. A randomized, multicenter, eight-month study. Gerontology. 2006;52:99–106. doi: 10.1159/000090955. 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https://stats.stackexchange.com/questions/153726/proof-of-algebraic-formula-for-the-sum-of-two-dice-toss-as-a-convolution
probability - Proof of Algebraic Formula for the Sum of Two-Dice Toss as a Convolution - Cross Validated Join Cross Validated By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Loading… Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company, and our products current community Cross Validated helpchat Cross Validated Meta your communities Sign up or log in to customize your list. more stack exchange communities company blog Log in Sign up Home Questions Unanswered AI Assist Labs Tags Chat Users Teams Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Try Teams for freeExplore Teams 3. Teams 4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Hang on, you can't upvote just yet. You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Proof of Algebraic Formula for the Sum of Two-Dice Toss as a Convolution Ask Question Asked 10 years, 4 months ago Modified6 years, 11 months ago Viewed 4k times This question shows research effort; it is useful and clear 3 Save this question. Show activity on this post. To figure out exactly the expected frequency of a given sum in a dice toss (given a certain number of dice and sides/dice), the following formula is posted here by @Glen_b (adapted to dice of six sides, and two dice tossed) the multiplication of the probability of each outcome 1 6 2 1 6 2 times whichever is less of either the t o t a l(n)−1 t o t a l(n)−1, or the maximum possible attainable (12) minus the total under consideration ((m a x+1)−n)((m a x+1)−n): p(x)=P(X=n)=1 36 min(n−1,13−n)p(x)=P(X=n)=1 36 min(n−1,13−n) On the same post we are reminded that the problem is really a convolution. I have two questions about this formula: 1. Can it be generalized to any number of sides and tosses? And 2. It obviously works, but how can you prove it, perhaps making the connection to the mechanics of a convolution? If the question is still OK, I'd like to follow up with yet a third problem: How is this formula connected to the proof by @whuber in here based on probability generating functions? probability convolution dice probability-generating-fn Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Improve this question Follow Follow this question to receive notifications edited Oct 26, 2018 at 15:07 kjetil b halvorsen♦ 85.2k 32 32 gold badges 215 215 silver badges 694 694 bronze badges asked May 23, 2015 at 17:46 Antoni ParelladaAntoni Parellada 27.2k 18 18 gold badges 124 124 silver badges 238 238 bronze badges 2 3 Hint: Use your calculator to compute the square of the integer 111111 111111. Then try to check the answer via ordinary multiplication that is taught in elementary schools everywhere in the world except the US where it is slowly disappearing from the middle school curriculum.Dilip Sarwate –Dilip Sarwate 2015-05-23 19:49:47 +00:00 Commented May 23, 2015 at 19:49 1 For an explicit connection and extensive generalization, see my post at stats.stackexchange.com/a/116913/919. It includes all the R code needed to carry out these convolutions and much more.whuber –whuber♦ 2018-10-26 15:19:53 +00:00 Commented Oct 26, 2018 at 15:19 Add a comment| 2 Answers 2 Sorted by: Reset to default This answer is useful 3 Save this answer. Show activity on this post. That formula doesn't directly generalize for any number of dice, but yes, that formula for two dice generalizes directly for different numbers of faces as long as the two dice are the same "size". More generally for m m dice, you get piecewise polynomial functions, with m m polynomials of degree m−1 m−1. e.g. for 3 dice you'll get three quadratic pieces. Let's look at an example for m=3 m=3. Here's three (six-sided) dice: x x: 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 6 3 P(X=x)6 3 P(X=x): 1 3 6 10 15 21 25 27 27 25 21 15 10 6 3 1 Here are the corresponding 3d regions, shown in two different views; the 'red' points on the red-blue boundary should also be blue, and the 'blue' points on the blue-green boundary should also be green. On the right-side image, height "up" the screen roughly corresponds to x x in the previous plot of the pmf. Looking at it as a convolution of one die with the sum of two, we can describe what goes on in the three parts (left, right, center): Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Improve this answer Follow Follow this answer to receive notifications edited Oct 26, 2018 at 15:09 kjetil b halvorsen♦ 85.2k 32 32 gold badges 215 215 silver badges 694 694 bronze badges answered May 24, 2015 at 11:14 Glen_bGlen_b 296k 37 37 gold badges 674 674 silver badges 1.1k 1.1k bronze badges 0 Add a comment| This answer is useful 2 Save this answer. Show activity on this post. I've been looking for a visual explanation to tie up the formula in my OP with the idea of the total sum of dice tosses being an example of a convolution of discrete variables. There are some posts on-line that display some interesting graphs, but yet they do the actual calculations on the side. Likewise some code including either the combinatorics concept of composition, or tackling explicitly this idea through an outer product such in R, outer(1:6,1:6,"+"), also separate the actual computation of permutations adding up to the total, as opposed to performing one single convolution. This makes a lot of sense, but it doesn't quite help get a good visual. So I worked through this, and hope that this answer may help others also without strong mathematical background grasp the idea of convolutions. We have two 6-sided dice (d1 and d2) and the probability distribution of the addition of their values will be given by a convolution, such that: p(d 1+d 2=n)=∑k=0 k=n p(d 1=k)⋅p(d 2=n−k)p(d 1+d 2=n)=∑k=0 k=n p(d 1=k)⋅p(d 2=n−k) Since d1 and d2 are to independently distributed random variables, the multiplication is justified. On the other hand, the sigma notation is meant to include all the situations (compositions) where the total addition will be equal to n n, given that d 1=k d 1=k and d 2=n−k d 2=n−k, and hence their sum equals k+(n−k)=n k+(n−k)=n. For total sum values less or equal to the number of sides (6 6) the pattern can be extracted by running one of the convolutions, for example, for the sum, n=4 n=4. Implementing the summation in the equation above, starting for the probability values to compute: d1 d2 k = 0 n-k = 4 k = 1 n-k = 3 k = 2 n-k = 2 k = 3 n-k = 1 k = 4 n-k = 0 we see that since we are working with fair dice, the probability for all compositions is going to naturally be 1 6⋅1 6=1 36 1 6⋅1 6=1 36, except for the multiplications including k=0 k=0 or n−k=0 n−k=0, which indicate impossible situations (no zeros on the sides of the dice). Therefore only three of the five permutations on the table above will contribute to the convolution. We can get a visualization of the zipper-like process of the convolution on the animation below: This explains that for values of the sum including permutations of dice below 7 7 (those corresponding to the physical sides) the equation in the OP calls for n−1 n−1. We "lose" one of the contributions due to the presence of a zero in the convolution. The situation is slightly more complex when the total sum allows permutations that include dice results above those physically possible (i.e. greater than 6 6). In this part of the convolutions that generate the PMF of the total sum, the counting will have to reflect that the probability of any dice showing values of 7, 8, 9, 10, 11 or 12 is 0. That, plus, the fact that the definition of the convolution includes a zero, hence, we are not counting 12, but 13. 13−n 13−n will equal the remaining integers that will not be neutralized by a product in the sigma computation including a zero probability. Again, here is an animation with an example (n=11 n=11) (hover over or click on it): And for the middle value of 7, either n−1 n−1 or 13−n 13−n will produce identical results. I'm still working through the last part of my OP, but hopefully someone will address that part. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Improve this answer Follow Follow this answer to receive notifications edited Oct 26, 2018 at 15:15 kjetil b halvorsen♦ 85.2k 32 32 gold badges 215 215 silver badges 694 694 bronze badges answered May 24, 2015 at 7:03 Antoni ParelladaAntoni Parellada 27.2k 18 18 gold badges 124 124 silver badges 238 238 bronze badges Add a comment| Your Answer Thanks for contributing an answer to Cross Validated! Please be sure to answer the question. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. Making statements based on opinion; back them up with references or personal experience. Use MathJax to format equations. MathJax reference. To learn more, see our tips on writing great answers. Draft saved Draft discarded Sign up or log in Sign up using Google Sign up using Email and Password Submit Post as a guest Name Email Required, but never shown Post Your Answer Discard By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions probability convolution dice probability-generating-fn See similar questions with these tags. 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5288
https://spanish.kwiziq.com/questions/view/personal-a-with-animals
Personal a with animals | Spanish Q & A | Kwiziq Spanish How Kwiziq works Spanish learning library - Grammar lessons - Vocabulary themes - Listening practice - Reading practice - Speaking practice - Writing practice - Fill-in-the-blanks Try Our App! Spanish Q&A Forum Leaderboards For teachers Testimonials Spanish learning Blog FAQs Game Library For teachers Pricing Sign in How Kwiziq works Explore Spanish learning library - Grammar lessons - Vocabulary themes - Listening practice - Reading practice - Speaking practice - Writing practice - Fill-in-the-blanks Try Our App! Spanish Q&A Forum Leaderboards For teachers Testimonials Spanish learning Blog FAQs Game Library For teachers Pricing Sign in French Spanish More Get started for FREEJoin FREE Spanish» Spanish Q&A» personal a with animals personal a with animals « Back to Q&A Forum « Previous questionNext question » Gerald R.B2 Kwiziq Q&A regular contributor personal a with animals If the direct object of a verb is an animal, but not a pet or cherished animal, for example a tiger, is the personal a still indicated? Busco a un tigre. or, Busco un tigre. This question relates to:Spanish lesson "Spanish personal "a" verbs (ver, visitar, buscar, conocer)" Asked 3 years ago Like 0Answer 1 Share InmaKwiziq Head of Spanish, Native Spanish Teacher Hola Gerald The use of the personal a in Spanish with animals as direct objects is not a clear-cut rule. We generally use the personal a if we're talking about pets, as it is a more affectionate relationship (as if they were people) and normally don't use it with animals in general, especially if the animals are not "identified" and "not specific". In your sentences: Busco a un tigre Busco un tigre The use of the personal a in the first one makes us think you have a tiger in mind that you are looking for, however the second one is more general. Having said this, this rule is a bit flexible sometimes and we see some cases as optional. Saludos Inma Like 0 3 years ago Share Gerald R. asked:View original personal a with animals If the direct object of a verb is an animal, but not a pet or cherished animal, for example a tiger, is the personal a still indicated? Busco a un tigre. or, Busco un tigre. Sign in to submit your answer Don't have an account yet? Join today « Using cualquier/a to say any in SpanishUse of Vosotros » Ask a question Find your Spanish level for FREE Test your Spanish to the CEFR standard Find your Spanish level How it works FAQ Testimonials Pricing Spanish resource library Buy gift voucher Redeem gift voucher About us Blog Jobs at Kwiziq Press Contact Privacy policy © 2025 Kwiziq Ltd.Kwiziq Spanish is a product of and © Kwiziq Ltd 2025 Thinking...
5289
https://www.thoughtco.com/equation-for-the-reaction-of-baking-soda-and-vinegar-604043
Equation for Baking Soda and Vinegar Reaction Skip to content Menu Home Science, Tech, Math Science Math Social Sciences Computer Science Animals & Nature Humanities History & Culture Visual Arts Literature English Geography Philosophy Issues Languages English as a Second Language Spanish French German Italian Japanese Mandarin Russian Resources For Students & Parents For Educators For Adult Learners About Us Search Close Search the site GO Science, Tech, Math Science Math Social Sciences Computer Science Animals & Nature Humanities History & Culture Visual Arts Literature English Geography Philosophy Issues Languages English as a Second Language Spanish French German Italian Japanese Mandarin Russian Resources For Students & Parents For Educators For Adult Learners About Us Contact Us Editorial Guidelines Privacy Policy Science, Tech, Math› Science› Chemistry› Projects & Experiments› Equation for the Reaction Between Baking Soda and Vinegar Print Sidekick/Getty Images Science Chemistry Projects & Experiments Basics Chemical Laws Molecules Periodic Table Scientific Method Biochemistry Physical Chemistry Medical Chemistry Chemistry In Everyday Life Famous Chemists Activities for Kids Abbreviations & Acronyms Biology Physics Geology Astronomy Weather & Climate By Anne Marie Helmenstine, Ph.D. Anne Marie Helmenstine, Ph.D. Chemistry Expert Ph.D., Biomedical Sciences, University of Tennessee at Knoxville B.A., Physics and Mathematics, Hastings College Dr. Helmenstine holds a Ph.D. in biomedical sciences and is a science writer, educator, and consultant. She has taught science courses at the high school, college, and graduate levels. Learn about ourEditorial Process Updated on June 07, 2024 Close The reaction between baking soda (sodium bicarbonate) and vinegar (dilute acetic acid) generates carbon dioxide gas, which is used in chemical volcanoes and other projects. Here is a look at the reaction between vinegar and baking soda and the equation for the reaction. Key Takeaways: Reaction Between Baking Soda and Vinegar The overall chemical reaction between baking soda (sodium bicarbonate) and vinegar (weak acetic acid) is one mole of solid sodium bicarbonate reacts with one mole of liquid acetic acid to produce one mole each of carbon dioxide gas, liquid water, sodium ions, and acetate ions. The reaction proceeds in two steps. The first reaction is a double displacement reaction, while the second reaction is a decomposition reaction. The baking soda and vinegar reaction can be used to produce sodium acetate, by boiling off or evaporating all the liquid water. How the Reaction Works What happens when vinegar reacts with baking soda occurs in two steps, but the overall process can be summarized by the following word equation: baking soda (sodium bicarbonate) plus vinegar (acetic acid) yields carbon dioxide + water + sodium ion + acetate ion The chemical equation for the overall reaction is: NaHCO 3(s) + CH 3 COOH(l) → CO 2(g) + H 2 O(l) + Na+(aq) + CH 3 COO-(aq) with s = solid, l = liquid, g = gas, aq = aqueous or in water solution Another common way to write this reaction is: NaHCO 3 + HC 2 H 3 O 2 → NaC 2 H 3 O 2 + H 2 O + CO 2 The above reaction, while technically correct, does not account for the dissociation of the sodium acetate in water. The chemical reaction actually occurs in two steps. First, there is a double displacementreaction in which acetic acid in the vinegar reacts with sodium bicarbonate to form sodium acetate and carbonic acid: NaHCO 3 + HC 2 H 3 O 2 → NaC 2 H 3 O 2 + H 2 CO 3 Carbonic acid is unstable and undergoes a decomposition reactionto produce the carbon dioxide gas: H 2 CO 3 → H 2 O + CO 2 The carbon dioxide escapes the solution as bubbles. The bubbles are heavier than air, so the carbon dioxide collects at the surface of the container or overflows it. In a baking soda volcano, detergent is usually added to collect the gas and form bubbles that flow somewhat like lava down the side of the 'volcano.' A diluted sodium acetate solution remains after the reaction. If the water is boiled off of this solution, a supersaturated solution of sodium acetate forms. This "hot ice" will spontaneously crystallize, releasing heat and forming a solid that resembles water ice. The carbon dioxide released by the baking soda and vinegar reaction has other uses besides making a chemical volcano. It can be collected and used as a simple chemical fire extinguisher. Because carbon dioxide is heavier than air, it displaces it. This starves a fire of the oxygen needed for combustion. Cite this Article Format mlaapachicago Your Citation Helmenstine, Anne Marie, Ph.D. "Equation for the Reaction Between Baking Soda and Vinegar." ThoughtCo, Jun. 7, 2024, thoughtco.com/equation-for-the-reaction-of-baking-soda-and-vinegar-604043.Helmenstine, Anne Marie, Ph.D. (2024, June 7). Equation for the Reaction Between Baking Soda and Vinegar. Retrieved from Helmenstine, Anne Marie, Ph.D. "Equation for the Reaction Between Baking Soda and Vinegar." ThoughtCo. 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5290
https://www.cuemath.com/geometry/congruent-lines/
LearnPracticeDownload Congruent Lines Congruent lines in geometry refer to the segments which are identical in their measure. They are 1- dimensional superimposable segments having equal lengths. Congruent lines can have different alignments and orientations. We will be studying the properties and theorems of congruent lines in this article. | | | --- | | 1. | What are Congruent Lines Segments? | | 2. | Congruent Segments Properties | | 3. | FAQs on Congruent Lines | What are Congruent Line Segments? Congruent line segments are 1-dimensional geometrical figures having equal measures. The word "congruent" with respect to congruent lines in geometry is defined as the equality between the two line segments. Two lines are said to be congruent when they have the same length. Congruent segments are superimposable figures, which completely overlap when placed one over the other. On turning, flipping, or rotating the congruent segments, they still remain to be congruent. The symbol used to depict congruence between any two congruent line segments is ≅. To brief it, congruent segments is just another name given to congruent line segments or congruent lines in geometry. All three terms are mathematically the same. Now, let us look into some examples of congruent line segments we find in mathematics. Examples of congruent line segments: The radii of the same circle. Opposite sides of a parallelogram, rectangle, square, and rhombus. Sides of an equilateral triangle. Let's look into the diagram below showing congruent line segments. Here line segment PQ ≅ XY since the double vertical bars on each line segment, PQ and XY depict their equality. Congruent Segments Properties To understand more about the congruent line segments, we will be looking into their properties as listed below. The properties followed by congruent lines in geometry are: (i) Reflexive (ii) Symmetric (iii) Transitive Let us have a brief understanding of each of the properties individually. Reflexive Property Reflexive property is based on the principle of comparing an object to itself. It states that a line segment is always congruent to itself as the lengths remain the same. This can be understood by taking a real-time example. When the height of a pole is compared to itself, they are exactly equal and hence congruent. Therefore, the pole follows reflexive property. Consider a line segment XY = 5cm. By Reflexive Property, XY ≅ XY. Symmetric Property Symmetric property is based on the principle that, if A = B then, B = A. It states that if a line segment is congruent to another line segment, then the second line segment is also congruent to the first line segment. To understand this better let's think about a practical scenario. If Ron's height is equal to Kelly's height then, Kelly's height is also equal to Ron's height. Below is a pictorial representation that depicts the symmetric property of congruent lines. From the above figure, the length of MN and EF is 8 cm. Thus, by the symmetric property, MN ≅ EF EF ≅ MN Transitive Property Transitive property is based on the principle that, if A = B and B = C then, A = C. It states that, if a line segment is congruent to another line segment and the second line segment is congruent to the third line segment then, the first line segment is congruent to the third line segment. Let's understand this better by looking into the diagram shown below. From the above figure, we see that AB ≅ XY and XY ≅ PQ. Thus, using transitive property, we can conclude that AB ≅ PQ. Related Articles on Congruent Lines Check these articles related to the concept of congruent lines in geometry. Line Segment Congruent Angles Congruent Read More Download FREE Study Materials Worksheet on Congruent Lines Congruent Lines Worksheet Worksheet on Congruent Triangles Congruent Lines Examples Example 1: If AB ≅ CD and CD ≅ XY then AB will be congruent to which line segment? Solution: Given that, AB ≅ CD and CD ≅ XY Using the transitive property of congruent lines, we can say that AB ≅ XY because as per this property, when one quantity is equal to another and the second quantity is equal to the third quantity, the first quantity will be equal to the third quantity. Thus, the line segment AB is congruent to XY. 2. Example 2: Out of a parallelogram and rhombus, which geometrical figure will have all sides as congruent segments? Solution: A parallelogram is a quadrilateral with opposite sides to be equal whereas, in a rhombus all the sides are equal. Thus, we can say that out of a parallelogram and a rhombus, a rhombus will have all its sides as congruent line segments. View Answer > Breakdown tough concepts through simple visuals. Math will no longer be a tough subject, especially when you understand the concepts through visualizations. Book a Free Trial Class Practice Questions on Congruent Lines Check Answer > FAQs on Congruent Lines What are Congruent Lines in Geometry? Two line segments having lengths of equal measure are known as congruent lines. For example, if we have two line segments AB and PQ of 6 cm each, then we can say AB ≅ PQ and hence they are congruent. What is the Definition of Congruent Line Segment? A congruent line segment is defined as any line segment having equal measure. For example, the sides of an equilateral triangle are known as congruent line segments as all of them have the same measure. When is a Line Segment Congruent to itself? When a line segment is compared to itself, the line segment is congruent to itself since they exactly measure the same. This is the reflexive property followed by a congruent line segment. For example, consider a line segment, MN of length 7.5 cm. When this line segment is compared to itself they become congruent i.e., MN ≅ MN. Why are Segments Congruent? Segments are congruent due to their equality in length. The orientation or the inclination does not affect their congruency. Thus, two segments are congruent when they have equal lengths. What are Congruent Lines Angles? When two line segments exactly measure the same, they are known as congruent lines. For example, two line segments XY and AB have a length of 5 inches and are hence known as congruent lines. When two angles exactly measure the same, they are known as congruent angles. For example, the internal angles of a square are congruent as each angle measures 90º. How to Prove two Line Segments are Congruent? Given two line segments, the lengths can be measured using a ruler which helps us to compare their equality. If the lengths of two line segments are equal, they are known to be congruent. For example, sides of an equilateral triangle are congruent as all the three sides are of equal measure. The distance between two lines which are line segments and are congruent have a distance of zero units between them. What is Congruent Segment Midpoint? The congruent segment midpoint is defined as that point on a line segment that exactly divides the line segment into two parts of equal length and hence the two newly formed segments are congruent to each other. For example, for a line segment of length 10 cm, the midpoint will exactly be at 5 cm and the newly formed segments will be 5 cm each. 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5291
https://openstax.org/books/f%C3%ADsica-universitaria-volumen-1/pages/4-resumen
Published Time: Tue, 17 Jun 2025 15:28:12 GMT Cap. 4Resumen - Física universitaria volumen 1 | OpenStax Valoramos tu privacidad Usamos cookies para mejorar su experiencia de navegación, mostrarle anuncios o contenidos personalizados y analizar nuestro tráfico. Al hacer clic en “Aceptar todo” usted da su consentimiento a nuestro uso de las cookies. Personalizar Rechazar todo Aceptar todo Personalizar las preferencias de consentimiento Usamos cookies para ayudarle a navegar de manera eficiente y realizar ciertas funciones. Encontrará información detallada sobre cada una de las cookies bajo cada categoría de consentimiento a continuación. 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Rechazar todo Guardar mis preferencias Aceptar todo Omitir e ir al contenidoIr a la página de accesibilidadMenú de atajos de teclado Iniciar sesión Física universitaria volumen 1 Resumen Física universitaria volumen 1Resumen Índice Índice Textos resaltados Índice Prefacio Mecánica 1 Unidades y medidas 2 Vectores 3 Movimiento rectilíneo 4 Movimiento en dos y tres dimensiones Introducción 4.1 Vectores de desplazamiento y velocidad 4.2 Vector de aceleración 4.3 Movimiento de proyectil 4.4 Movimiento circular uniforme 4.5 Movimiento relativo en una y dos dimensiones Revisión Del Capítulo Términos clave Ecuaciones clave Resumen Preguntas Conceptuales Problemas Problemas Adicionales Problemas De Desafío 5 Leyes del movimiento de Newton 6 Aplicaciones de las leyes de Newton 7 Trabajo y energía cinética 8 Energía potencial y conservación de la energía 9 Momento lineal y colisiones 10 Rotación de un eje fijo 11 Momento angular 12 Equilibrio estático y elasticidad 13 Gravitación 14 Mecánica de fluidos Ondas y acústica A Unidades B Factores de conversión C Constantes fundamentales D Datos astronómicos E Fórmulas matemáticas F Química G El alfabeto griego Clave de respuestas Índice Buscar términos clave o texto. Cerrar Resumen 4.1 Vectores de desplazamiento y velocidad ------------------------------------------ La función de posición r⃗(t)r→(t)r→(t) da la posición en función del tiempo de una partícula que se mueve en dos o tres dimensiones. Gráficamente, es un vector desde el origen de un sistema de coordenadas elegido hasta el punto en el que se encuentra la partícula en un momento determinado. El vector de desplazamiento Δ r⃗Δ r→Δ r→ da la distancia más corta entre dos puntos cualquiera de la trayectoria de una partícula en dos o tres dimensiones. La velocidad instantánea da la rapidez y la dirección de una partícula en un momento determinado de su trayectoria en dos o tres dimensiones, y es un vector en dos y tres dimensiones. El vector velocidad es tangente a la trayectoria de la partícula. El desplazamiento r⃗(t)r→(t)r→(t) puede escribirse como una suma vectorial de los desplazamientos unidimensionales x⃗(t),y⃗(t),z⃗(t)x→(t),y→(t),z→(t)x→(t),y→(t),z→(t) a lo largo de las direcciones de la x, la y y la z. La velocidad v⃗(t)v→(t)v→(t) puede escribirse como una suma vectorial de las velocidades unidimensionales v x(t),v y(t),v z(t)v x(t),v y(t),v z(t)v x(t),v y(t),v z(t) a lo largo de las direcciones de la x, la y y la z. El movimiento en una dirección determinada es independiente del movimiento en una dirección perpendicular. 4.2 Vector de aceleración ------------------------- En dos y tres dimensiones, el vector de aceleración puede tener una dirección arbitraria y no apunta necesariamente a lo largo de un componente determinado de la velocidad. La aceleración instantánea se produce por un cambio de velocidad tomado en un tiempo muy corto (infinitesimal). La aceleración instantánea es un vector en dos o tres dimensiones. Se encuentra al tomar la derivada de la función de velocidad con respecto al tiempo. En tres dimensiones, la aceleración a⃗(t)a→(t)a→(t) puede escribirse como una suma vectorial de las aceleraciones unidimensionales a x(t),a y(t),y a z(t)a x(t),a y(t),y a z(t)a x(t),a y(t),y a z(t) a lo largo de los ejes de la x, la y y la z. Las ecuaciones cinemáticas para la aceleración constante pueden escribirse como la suma vectorial de las ecuaciones de aceleración constante en las direcciones de la x, la y y la z. 4.3 Movimiento de proyectil --------------------------- El movimiento de proyectil es el movimiento de un objeto sometido únicamente a la aceleración de la gravedad, cuando la aceleración es constante, como ocurre cerca de la superficie de la Tierra. Para resolver problemas de movimiento de proyectil, analizamos el movimiento del proyectil en las direcciones horizontal y vertical mediante el empleo de las ecuaciones cinemáticas unidimensionales para x y y. El tiempo de vuelo de un proyectil lanzado con velocidad vertical inicial v 0 y v 0 y v 0 y en una superficie llana está dado por T t o f=2(v 0 sen θ)g.T t o f=2(v 0 sen θ)g.T t o f=2(v 0 sen θ)g. Esta ecuación es válida únicamente cuando el proyectil cae a la misma altura desde la que se lanzó. La distancia horizontal máxima que recorre un proyectil se denomina alcance. Una vez más, la ecuación del alcance es válida únicamente cuando el proyectil cae a la misma altura desde la que se lanzó. 4.4 Movimiento circular uniforme -------------------------------- El movimiento circular uniforme es el movimiento en un círculo a rapidez constante. La aceleración centrípeta a⃗C a→C a→C es la aceleración que debe tener una partícula para seguir una trayectoria circular. La aceleración centrípeta siempre apunta hacia el centro de rotación y tiene una magnitud a C=v 2/r.a C=v 2/r.a C=v 2/r. El movimiento circular no uniforme se produce cuando hay aceleración tangencial de un objeto, que ejecuta un movimiento circular, de tal manera que la rapidez del objeto cambia. Esta aceleración recibe el nombre de aceleración tangencial a⃗T.a→T.a→T. La magnitud de la aceleración tangencial es la tasa de tiempo del cambio de la magnitud de la velocidad. El vector de aceleración tangencial es tangente al círculo, mientras que el vector de aceleración centrípeta apunta radialmente hacia el centro del círculo. La aceleración total es la suma vectorial de las aceleraciones tangencial y centrípeta. Un objeto que ejecuta un movimiento circular uniforme puede describirse con ecuaciones de movimiento. El vector de posición del objeto es r⃗(t)=A cos ω t i ˆ+A sen ω t j ˆ,r→(t)=A cos ω t i^+A sen ω t j^,r→(t)=A cos ω t i^+A sen ω t j^, donde A es la magnitud |r⃗(t)|,|r→(t)|,|r→(t)|, que es también el radio del círculo, y ω ω ω es la frecuencia angular. 4.5 Movimiento relativo en una y dos dimensiones ------------------------------------------------ Al analizar el movimiento de un objeto, es necesario especificar el marco de referencia en términos de posición, velocidad y aceleración. La velocidad relativa es la velocidad de un objeto observada desde un marco de referencia concreto, y varía con la elección del marco de referencia. Si S y S′S′S′ son dos marcos de referencia que se mueven uno respecto al otro a velocidad constante, entonces la velocidad de un objeto respecto a S es igual a su velocidad respecto a S′S′S′ más la velocidad de S′S′S′ en relación con S. Si dos marcos de referencia se mueven uno respecto al otro a velocidad constante, entonces la aceleración de un objeto que se observa en ambos marcos de referencia es igual. AnteriorSiguiente Solicitar una copia impresa Cita/Atribución Este libro no puede ser utilizado en la formación de grandes modelos de lenguaje ni incorporado de otra manera en grandes modelos de lenguaje u ofertas de IA generativa sin el permiso de OpenStax. ¿Desea citar, compartir o modificar este libro? Este libro utiliza la Creative Commons Attribution License y debe atribuir a OpenStax. Información de atribución Si redistribuye todo o parte de este libro en formato impreso, debe incluir en cada página física la siguiente atribución: Acceso gratis en Si redistribuye todo o parte de este libro en formato digital, debe incluir en cada vista de la página digital la siguiente atribución: Acceso gratuito en Información sobre citas Utilice la siguiente información para crear una cita. Recomendamos utilizar una herramienta de citas como this one. Autores: William Moebs, Samuel J. Ling, Jeff Sanny Editorial/sitio web: OpenStax Título del libro: Física universitaria volumen 1 Fecha de publicación: 28 sept 2021 Ubicación: Houston, Texas URL del libro: URL de la sección: © 13 abr 2022 OpenStax. El contenido de los libros de texto que produce OpenStax tiene una licencia de Creative Commons Attribution License . El nombre de OpenStax, el logotipo de OpenStax, las portadas de libros de OpenStax, el nombre de OpenStax CNX y el logotipo de OpenStax CNX no están sujetos a la licencia de Creative Commons y no se pueden reproducir sin el previo y expreso consentimiento por escrito de Rice University. Nuestra misión es mejorar el acceso a la educación y el aprendizaje para todos. OpenStax forma parte de Rice University, una organización sin fines de lucro 501 (c) (3). Done hoy y ayúdenos a llegar a más estudiantes. 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https://www.youtube.com/watch?v=Z44J_cudaCU
rank(a) = rank(transpose of a) | Matrix transformations | Linear Algebra | Khan Academy ailabhcmus 462 subscribers 1 likes Description 396 views Posted: 21 Nov 2016 Transcript: a couple of videos ago I made the statement that the rank the rank of a matrix a is equal to the rank of its transpose and I made a bit of a handwavy argument it was at the end of the video and I was tired it was actually the end of the day and I thought it was it'd be worthwhile to maybe flush this out a little bit because it's an important takeaway and'll help us in understand everything we've learned a little bit better so let's just understand what the actually I'm going to start with the rank of a transpose the rank of a transpose the rank the rank of a transpose is equal to the dimension of the column space of a transpose that's the definition of the rank and what is it the dimension of the column space of a transpose is the number of basis vectors basis vectors for the column space of a transpose that's what dimension is for any Subspace you figure out how many basis vectors you need in that Subspace and you count them and that's your dimension so it's the number of basis vectors for the column space of a transpose which is of course the same thing this thing we've seen multiple times is the same thing as the row space of a row space of a right The Columns of a transpose are the same things as the rows of a just because you switch the rows and the columns now how can we figure out the number of basis vectors we need for the column space of a transpose or the row space of a so let's just think about what the column space of a transpose is telling us so it's equivalent to so let's say let me draw a like this let me draw a have some Matrix a let's say it's an M by n Matrix and let me just write it as a bunch of row vectors I could also write as a bunch of column vectors but right now let's just stick to the row vectors so we'd have Row one they're the transpose of column vectors but we can just write that's Row one and we're going to have row two and I have row two and we're going to go all the way down to row M right it's an M byn Matrix each of these vectors are members of RN because they're going to have n entries in them because we have n columns so that's what a is going to look like a is going to look like that and then a transpose a transpose all of these rows are going to become columns a transpose is going to look like this R1 R2 all the way to RM and this is of course going to be an N bym Matrix you swap these out so all these rows are going to be columns right and obviously the column space or maybe not so obviously the column space of a transpose the column space of a transpose is equal to the span the span of R1 R2 all the way to RM right it's equal to the span of these things or you could equivalent call it it's it's equal to the span of the rows of a and that's why it's called so called the row space so this is equal to the span of the rows rows of a these two things are equivalent now these are the span that means this is some Subspace that's all of the linear combinations of these columns or all the linear combinations of these rows if we want the basis for it we want to find a minimum set of linearly independent vectors that we could use to construct any of these columns or that we could use to construct any of these rows right here now what happens when we put a into reduced row Echelon form so let's just we we do a bunch of row operations row operations to put it into reduced row Echelon form right you do a bunch of row operations and you eventually you'll get something like this you'll get the reduced row Echelon form of a the reduced row Echelon form of a is going to look something like this you're going to have some pivot rows some rows that have pivot entries let's say that's one of them and let's say the second let's say that's one of them this will all have zeros all the way down this one will have zeros your pivot entry has to be the only non-zero entry in its column and everything to the left of it also has to be zero let's say that this one isn't these are some nonzero values these are zero say we have another pivot entry over here everything else is zero and let's say everything else are non-pivot entries so you come here and you at a certain number of pivot rows or a certain number of pivot entries right and you got there by performing linear row operations on these guys so those linear row operations you know I take three times row two and I add it to row one and I you know and that's going to become my new row two and you keep doing that and you get these things here so these things here are linear combinations of those guys or another way to do it you could reverse those row operations I could start with these guys right here and I could just as easily perform the reverse row operations I you know you any linear operation you could perform the reverse of it we've seen that multiple times you could perform row operations you could perform row operations with these guys to get all of these guys or another way to view it is these vectors here these row vectors right here they span all of these all of these or all of these row vectors can be represented as linear combinations of your pivot rows right here obviously you're going to have your non-pivot rows are going to be all zeros are going to be all zeros and those are useless those are useless but your pivot rows if you take linear combinations of them you can clearly put you can clearly kind of do reverse row Echelon form and get back to your Matrix so all of these guys can be represented as linear combinations of them and all of these pivot entries are by definition are are by well almost by definition they're linearly independent right because I've got a one here no one else has a one there so no one else can be so this guy can definitely not be represented as a linear combination of the other guy so why am I going through this whole exercise well we started off saying we wanted a basis we wanted a basis for the row space we wanted some minimum set of linearly independent vectors that spans everything that these guys can span well if all of these guys can be represented as linear combinations of these row vectors in reduced row Echelon form or these pivot rows in reduced in reduced row Echelon form and these guys are all linearly independent then they are a reasonable basis so these pivot rows right here that's one of them this is the second one uh this is the third one maybe they're the only three this is just my particular example that would be a suitable basis for the row space so let me write this down the pivot rows pivot rows in reduced row Echelon form of a are a basis are a basis for the row space row space of a and the row space of a is the same thing you know or the column space of a transpose row space of a is the same thing as a column space of a transpose we've seen that multiple times now so if we want to know the dimension of your column space well you just count the number of pivot rows you have so you just count the number of pivot rows so the dimension of your row space which is the same thing as a column space of a transpose is going to be the number of pivot rows you have in reduced row Echelon form or even simpler the number of pivot entries you have because every pivot entry has a pivot row so we can say we can write that the rank the rank of a transpose is equal to the number of pivot entries pivot entries in reduced row Echelon form of a right because every pivot entry corresponds to a pivot row those pivot rows are a suitable basis for the entire row space because every row can be made with a linear combination of these guys and since all of these can be then anything that these guys can construct these guys can construct fair enough now what is the rank of a this is the rank of a transpose that we've been dealing with so far the rank of a the rank the rank of a is equal to the dimension the dimension of the column space of a or you could say it's the number of vectors in the basis for the column space of a so if we take that same that same Matrix a that we used above and we instead we write as a bunch of column vectors so C1 C2 all the way to CN we have n columns right there the column space is essentially the Subspace that's spanned by all of these characters right here right spanned by each of these column vectors so the column space of a is equal to the span of C1 C2 all the way to CN that's the definition of it but we want to know the number of basis vectors and we've seen before we've done this multiple times the basis vectors for a suitable basis vectors could be if you put this into reduced row Echelon form and you have some pivot entries and they're corresponding pivot columns so some pivot entries with their corresponding pivot columns just like that maybe that's like that and then maybe this one isn't one and then this one is so you have a certain number of pivot columns you have a certain number of pivot columns let me do them in another color right here when you put a into reduced row Echelon form we learned that the basis vectors or the basis columns that form a basis for your column space are the columns that Cor Corr respond to the pivot columns so this the First Column here is a pivot column so this guy could be a basis Vector the second column is so this guy could be a pivot vector and maybe the fourth one right here so this guy could be a pivot Vector so in general you just say hey how many if you want to if you want to count the number of basis vectors because we don't have to know what they are to figure out the rank we just have to know the number they are well you say well for every pivot column here we have a basis Vector over there so we could just count the number of pivot columns but the number of pivot columns is equivalent to just the number of pivot entries we have because every pivot entry gets its own column so we could say that the rank of a the rank of a is equal to the number number of pivot entries of pivot entries in the reduced row Echelon form of a and as you can see very clearly that's the exact same thing that we deduced was equivalent to the rank of a transpose or the dimension of the column space of a a transposer the dimension of the row space of a so we can now write our conclusion the rank of a is definitely the same thing as the rank of a transpose
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https://constantinides.net/2020/08/09/some-notes-on-metric-spaces/
Some Notes on Metric Spaces – Thinking Skip to content Thinking A personal blog from @gconstantinides Some Notes on Metric Spaces George ConstantinidesBook Review, EducationAugust 9, 2020 This post contains some summary informal notes of key ideas from my reading of Mícheál Ó Searcóid’s Metric Spaces (Springer, 2007). These notes are here as a reference for me, my students, and any others who may be interested. They are by no means exhaustive, but rather cover topics that seemed interesting to me on first reading. By way of a brief book review, it’s worth noting that Ó Searcóid’s approach is excellent for learning a subject. He has a few useful tricks up his sleeve, in particular: Chapters will often start with a theorem proving equivalence of various statements (e.g. Theorem 8.1.1, Criteria for Continuity at a Point). Only then will he choose one of these statements as a definition, and he explains this choice carefully, often via reference to other mathematics. The usual definition-theorem-proof style is supplemented with ‘question’ – these are relatively informally-stated questions and their answers. They have been carefully chosen to highlight some questions the reader might be wondering about at that point in the text and to demonstrate key (and sometimes surprising) answers before the formal theorem statement. The writing is pleasant, even playful at times though never lacking formality. This is a neat trick to pull off. There are plenty of exercises, and solutions are provided. These features combine to produce an excellent learning experience. 1. Some Basic Definitions A metric on a set X is a function such that: Positivity: with equality iff Symmetry: Triangle inequality: The combination of such a metric and a the corresponding set is a metric space. Given a metric space , the point function at is . A pointlike function is one where For metric spaces and , is a metric subspace of iff and is a restriction of . For metric spaces and , an isometry is a function such that . The metric subspace is an isometric copy of . Some standard constructions of metrics for product spaces: A conserving metric on a product space is one where . Ó Searcóid calls these conserving metrics because they conserve an isometric copy of the individual spaces, recoverable by projection (I don’t think this is a commonly used term). This can be seen because fixing elements of all-but-one of the constituent spaces makes the upper and lower bound coincide, resulting in recovery of the original metric. A norm on a linear space over or is a real function such that for and scalar: with equality iff The metric defined by the norm is . 2. Distances The diameter of a set of metric space is . The distance of a pointfrom a set is . An isolated point where is one for which . An accumulation point or limit point of is one for which . Note that doesn’t need to be in . A good example is , , . The distance from subset to subset of a metric space is defined as . A nearest point of to is one for which . Note that nearest points don’t need to exist, because is defined via the infimum. If a metric space is empty or admits a nearest point to each point in every metric superspace, it is said to have the nearest-point property. 3. Boundaries A point is a boundary point of in iff . The collection of these points is the boundary . Metric spaces with no proper non-trivial subset with empty boundary are connected. An example of a disconnected metric space is as a metric subspace of , while itself is certainly connected. Closed sets are those that contain their boundary. The closure of in is . The interior is . The exterior is . Interior, boundary, and exterior are mutually disjoint and their union is . 4. Sub- and super-spaces A subset is dense in iff , or equivalently if for every , . The archetypal example is that is dense in . A complete metric space is one that is closed in every metric superspace of . An example is . 5. Balls Let ![Image 89: ba;r) = { x \in X | d(a,x) < r } denote an open ball and similarly denote a closed ball. In the special case of normed linear spaces, ![Image 91: b[a;r) = a + rb0;1) and similarly for closed balls, so the important object is this unit ball – all others have the same shape. A norm on a space is actually defined by three properties such balls must have: Convexity Balanced (i.e. ) For each , the set , is nonempty must have real supremum 6. Convergence The th tail of a sequence is the set . Suppose is a metric space, and is a sequence in . Sequence converges to in , denoted iff every open subset of that contains includes a tail of . In this situation, is unique and is called the limit of the sequence, denoted . It follows that for a metric space, and a sequence in , the sequence converges to in iff the real sequence converges to in . For real sequences, we can define the: limit superior, and limit inferior, . It can be shown that iff . Clearly sequences in superspaces converge to the same limit – the same is true in subspaces if the limit point is in the subspace itself. Sequences in finite product spaces equipped with product metrics converge in the product space iff their projections onto the individual spaces converge. Every subsequence of a convergent sequence converges to the same limit as the parent sequence, but the picture for non-convergent parent sequences is more complicated, as we can still have convergent subsequences. There are various equivalent ways of characterising these limits of subsequences, e.g. centres of balls containing an infinite number of terms of the parent sequence. A sequence is Cauchy iff for every , there is a ball of radius that includes a tail of . Every convergent sequence is Cauchy. The converse is not true, but only if the what should be the limit point is missing from the space — adding this point and extending the metric appropriately yields a convergent sequence. It can be shown that a space is complete (see above for definition) iff every Cauchy sequence is also a convergent sequence in that space. 7. Bounds A subset of a metric space is a bounded subset iff or is included in some ball of . A metric space is bounded iff it is a bounded subset of itself. An alternative characterisation of a bounded subset is that it has finite diameter. The Hausdorff metric is defined on the set of all non-empty closed bounded subsets of a set equipped with metric . It is given by . Given a set and a metric space , is a bounded function iff is a bounded subset of . The set of bounded functions from to is denoted . There is a standard metric on bounded functions, where is the metric on . Let be a nonempty set and be a nonempty metric space. Let be a sequence of functions from to and . Then: converges pointwise to iff converges to for all converges uniformly to iff is real for each and the sequence converges to zero in . It’s interesting to look at these two different notions of convergence because the second is stronger. Every uniformly-convergent sequence of functions converges pointwise, but the converse is not true. An example is the sequence given by . This converges pointwise but not uniformly to the zero function. A stronger notion than boundedness is total boundedness. A subset of a metric space is totally bounded iff for each , there is a finite collection of balls of of radius that covers . An example of a bounded but not totally bounded subset is any infinite subset of a space with the discrete metric. Total boundedness carries over to subspaces and finite unions. Conserving metrics play an important role in bounds, allowing bounds on product spaces to be equivalent to bounds on the projections to the individual spaces. This goes for both boundedness and total boundedness. 8. Continuity Given metric spaces and , a point and a function , the function is said to be continuous at iff for each open subset with , there exists and open subset of with such that . Extending from points to the whole domain, the function is said to be continuous on iff for each open subset , is open in . Continuity is not determined by the codomain, in the sense that a continuous function is continuous on any metric superspace of its range. It is preserved by function composition and by restriction. Continuity plays well with product spaces, in the sense that if the product space is endowed with a product metric, a function mapping into the product space is continuous iff its compositions with the natural projections are all continuous. For and metric spaces, denotes the metric space of continuous bounded functions from to with the supremum metric . is closed in the space of bounded functions from to . Nicely, we can talk about convergence using the language of continuity. In particular, let be a metric space, and . Endow with the inverse metric for , and . Let . Then is continuous iff the sequence converges in to . In particular, the function extending each convergent sequence with its limit is an isometry from the space of convergent sequences in to the metric space of continuous bounded functions from to . 9. Uniform Continuity Here we explore increasing strengths of continuity: Lipschitz continuity > uniform continuity > continuity. Ó Searcóid also adds strong contractions into this hierarchy, as the strongest class studied. Uniform continuity requires the in the epsilon-delta definition of continuity to extend across a whole set. Consider metric spaces and , a function , and a metric subspace . The function is uniformly continuous on iff for every there exists a s.t. for every for which , it holds that . If is a metric space with the nearest-point property and is continuous, then is also uniformly continuous on every bounded subset of . A good example might be a polynomial on . Uniformly continuous functions map compact metric spaces into compact metric spaces. They preserve total boundedness and Cauchy sequences. This isn’t necessarily true for continuous functions, e.g. on ![Image 237: (0,1]]( does not preserve the Cauchy property of the sequence . There is a remarkable relationship between the Cantor Set and uniform continuity. Consider a nonempty metric space . Then is totally bounded iff there exists a bijective uniformly continuous function from a subset of the Cantor Set to . As Ó Searcóid notes, this means that totally bounded metric spaces are quite small, in the sense that none can have cardinality greater than that of the reals. Consider metric spaces and and function . The function is called Lipschitz with Lipschitz constant iff for all . Note here the difference to uniform continuity: Lipschitz continuity restricts uniform continuity by describing a relationship that must exist between the s and s – uniform leaves this open. A nice example from Ó Searcóid of a uniformly continuous non-Lipschitz function is on Image 251: 0,1). Lipschitz functions preserve boundedness, and the Lipschitz property is preserved by function composition. There is a relationship between Lipschitz functions on the reals and their differentials. Let be a non-degenerate intervals of and . Then is Lipschitz on iff is bounded on . A function with Lipschitz constant less than one is called a strong contraction. Unlike the case for continuity, not every product metric gives rise to uniformly continuous natural projections, but this does hold for conserving metrics. 10. Completeness Let be a metric space and . The function is called a virtual point iff: for all We saw earlier that a metric space is complete iff it is closed in every metric superspace of . There are a number of equivalent characterisations, including that every Cauchy sequence in converses in . Consider a metric space . A subset of is a complete subset of iff is a complete metric space. If is a complete metric space and , then is complete iff is closed in . Conserving metrics ensure that finite products of complete metric spaces are complete. A non-empty metric space is complete iff is complete, where denotes the collection of all non-empty closed bounded subsets of and denotes the Hausdorff metric. For a non-empty set and a metric space, the metric space of bounded functions from to with the supremum metric is a complete metric space iff is complete. An example is that the space of bounded sequences in is complete due to completeness of . We can extend uniformly continuous functions from dense subsets to complete spaces to unique uniformly continuous functions from the whole: Consider metric spaces and with the latter being complete. Let be a dense subset of and be a uniformly continuous function. Then there exists a uniformly continuous function such that . There are no other continuous extensions of to . (Banach’s Fixed-Point Theorem). Let be a non-empty complete metric space and be a strong contraction on with Lipschitz constant . Then has a unique fixed point in and, for each , the sequence converges to the fixed point. Beautiful examples of this abound, of course. Ó Searcóid discusses IFS fractals – computer scientists will be familiar with applications in the semantics of programming languages. A metric space is called a completion of metric space iff is complete and is isometric to a dense subspace of . We can complete any metric space. Let be a metric space. Define where denotes the set of all point functions in and denotes the set of all virtual points in . We can endow with the metric given by . Then is a completion of . Here the subspace of forms the subspace isometric to . 11. Connectedness A metric space is a connected metric space iff cannot be expressed as the union of two disjoint nonempty open subsets of itself. An example is with its usual metric. As usual, Ó Searcóid gives a number of equivalent criteria: Every proper nonempty subset of has nonempty boundary in No proper nonempty subset of is both open and closed in is not the union of two disjoint nonempty closed subsets of itself Either or the only continuous functions from to the discrete space are the two constant functions Connectedness is not a property that is relative to any metric superspace. In particular, if is a metric space, is a metric subspace of and , then the subspace of is a connected metric space iff the subspace of is a connected metric space. Moreover, for a connected subspace of with , the subspace is connected. In particular, itself is connected. Every continuous image of a connected metric space is connected. In particular, for nonempty , is connected iff is an interval. This is a generalisation of the Intermediate Value Theorem (to see this, consider the continuous functions . Finite products of connected subsets endowed with a product metric are connected. Unions of chained collections (i.e. sequences of subsets whose sequence neighbours are non-disjoint) of connected subsets are themselves connected. A connected component of a metric space is a subset that is connected and which has no proper superset that is also connected – a kind of maximal connected subset. It turns out that the connected components of a metric space are mutually disjoint, all closed in , and is the union of its connected components. A path in metric space is a continuous function . (These functions turn out to be uniformly continuous.) This definition allows us to consider a stronger notion of connectedness: a metric space is pathwise connected iff for each there is a path in with endpoints and . An example given by Ó Searcóid of a space that is connected but not pathwise connected is the closure in of . From one of the results above, is connected because is connected. But there is no path from, say, (which nevertheless is in ) to any point in . Every continuous image of a pathwise connected metric space is itself pathwise connected. For a linear space, an even stronger notion of connectedness is polygonal connectedness. For a linear space with subset and , a polygonal connection from to in is an -tuple of points s.t. , and for each , . We then say a space is polygonally connected iff there exists a polygonal connection between every two points in the space. Ó Searcóid gives the example of as a pathwise connected but not polygonally connected subset of . Although in general these three notions of connectedness are distinct, they coincide for open connected subsets of normed linear spaces. 12. Compactness Ó Searcóid gives a number of equivalent characterisations of compact non-empty metric spaces , some of the ones I found most interesting and useful for the following material include: Every open cover for has a finite subcover is complete and totally bounded is a continuous image of the Cantor set Every real continuous function defined on is bounded and attains its bounds The example is given of closed bounded intervals of as archetypal compact sets. An interesting observation is given that ‘most’ metric spaces cannot be extended to compact metric spaces, simply because there aren’t many compact metric spaces — as noted above in the section on bounds, there are certainly no more than , given they’re all images of the Cantor set. If is a compact metric space and then is compact iff is closed in . This follows because inherits total boundedness from , and completeness follows also if is closed. The Inverse Function Theorem states that for and metric spaces with compact, and for injective and continuous, is uniformly continuous. Compactness plays well with intersections, finite unions, and finite products endowed with a product metric. The latter is interesting, given that we noted above that for non conserving product metrics, total boundedness doesn’t necessarily carry forward. Things get trickier when dealing with infinite-dimension spaces. The following statement of the Arzelà-Ascoli Theorem is given, which allows us to characterise the compactness of a closed, bounded subset of for compact metric spaces and : For each , define by for each . Let . Then: and is compact iff from to is continuous 13. Equivalence Consider a set and the various metrics we can equip it with. We can define a partial order on these metrics in the following way. is topologically stronger than , iff every open subset of is open in . We then get an induced notion of topological equivalence of two metrics, when and . As well as obviously admitting the same open subsets, topologically equivalent metrics admit the same closed subsets, dense subsets, compact subsets, connected subsets, convergent sequences, limits, and continuous functions to/from that set. It turns out that two metrics are topologically equivalent iff the identity functions from to and vice versa are both continuous. Following the discussion above relating to continuity, this hints at potentially stronger notions of comparability – and hence of equivalence – of metrics, which indeed exist. In particular is uniformly stronger than iff the identify function from to is uniformly continuous. Also, is Lipschitz stronger than iff the identity function from to is Lipschitz. The stronger notion of a uniformly equivalent metric is important because these metrics additionally admit the same Cauchy sequences, totally bounded subsets and complete subsets. Lipschitz equivalence is even stronger, additionally providing the same bounded subsets and subsets with the nearest-point property. The various notions of equivalence discussed here collapse to a single one when dealing with norms. For a linear space , two norms on are topologically equivalent iff they are Lipschitz equivalent, so we can just refer to norms as being equivalent. All norms on finite-dimensional linear spaces are equivalent. Finally, some notes on the more general idea of equivalent metric spaces (rather than equivalent metrics.) Again, these are provided in three flavours: topologically equivalent metric spaces and are those for which there exists a continuous bijection with continuous inverse (a homeomorphism) from to . for uniformly equivalent metric spaces, we strengthen the requirement to uniform continuity for Lipschitz equivalent metric spaces, we strengthen the requirement to Lipschitz continuity strongest of all, isometries are discussed above Note that given the definitions above, the metric space is equivalent to the metric space if and are equivalent, but the converse is not necessarily true. For equivalent metric spaces, we require existence of a function — for equivalent metrics this is required to be the identity. Share this: Click to share on X (Opens in new window)X Click to share on Facebook (Opens in new window)Facebook Like Loading... 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https://wdv.com/Calculus/Lecture01.pdf
0.5 1.0 1.5 2.0 2.5 3.0 3.5 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0 -3.5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 -0.5 -1.0 x+h Q P x f(x) f(x+h) Lecture 1: Explicit, Implicit and Parametric Equations Chapter 1 – Functions and Equations Lecture 1 – Explicit, Implicit and Parametric Equations 2 LECTURE TOPIC 0 GEOMETRY EXPRESSIONS™ WARM-UP 1 EXPLICIT, IMPLICIT AND PARAMETRIC EQUATIONS 2 A SHORT ATLAS OF CURVES 3 SYSTEMS OF EQUATIONS 4 INVERTIBILITY, UNIQUENESS AND CLOSURE Chapter 1: Functions and Equations Learning Calculus with Geometry Expressions™ 3 Louis Eric Wasserman used a novel approach for attacking the Clay Math Prize of P vs. NP. He examined problem complexity. Louis calculated the least number of gates needed to compute explicit functions using only AND and OR, the basic atoms of computation. He produced a characterization of P, a class of problems that can be solved in by computer in polynomial time. He also likes ultimate Frisbee™. Calculus Inspiration Chapter 1 – Functions and Equations Lecture 1 – Explicit, Implicit and Parametric Equations 4 EXPLICIT FUNCTIONS Mathematicians like Louis use the term “explicit function” to express the idea that we have one dependent variable on the left-hand side of an equation, and all the independent variables and constants on the right-hand side of the equation. For example, the equation of a line is: Where m is the slope and b is the y-intercept. Explicit functions GENERATE y values from x values. LINEAR EQUATIONS In the linear equation above, m and b are considered constants. If we wanted we could declare m and b to be variables and x to be a constant. Switching the way we look at things is often useful, but for now we’ll start simply. It is important to declare definitions and assumptions from the outset to avoid mistakes. Learning Calculus with Geometry Expressions™ 5 FUNCTIONAL NOTATION Every book that talks about linear functions has to say the following: The values assigned to the independent variable x are the domain of the function. The values of the dependent variable y are the range of the function. Functions can have names other than y. If we want to give the function a name, like f, we write: allowing the explicit form of a linear equation to be written as: When f(x) has this definition, the graph is a line. Chapter 1 – Functions and Equations Lecture 1 – Explicit, Implicit and Parametric Equations 6 CONVENIENT FORM The equation for a line in convenient form is: Notice we have switched the order of terms and given them different names. Now a is the y-intercept and b is the slope. Writing an explicit equation in this way has an advantage. We could write: Which would mean that y is a constant, unchanging for any value of x. We can escalate complexity in a convenient way by first writing: And then writing the polynomial: and so on. The constants a, b, and c are called coefficients of the polynomial. Be prepared to work in both traditional forms and convenient forms interchangeably. Learning Calculus with Geometry Expressions™ 7 2 4 6 8 -2 -4 -6 -8 -10 2 4 6 8 -2 -4 B C A Y=a+X·b+X2·c (a,0) Y=a Y=a+X·b (b,0) (c,0) Lecture01-ConvenientForm.gx Convenient Form Chapter 1 – Functions and Equations Lecture 1 – Explicit, Implicit and Parametric Equations 8 2 4 6 8 -2 -4 -6 -8 -10 2 4 6 8 -2 -4 B C A Y=a+X·b+X2·c (a,0) Y=a Y=a+X·b (b,0) (c,0) EXERCISE 1) Open the example. FileOpenLecture01-ConvenientForm.gx Convenient Form Learning Calculus with Geometry Expressions™ 9 2 4 6 8 -2 -4 -6 -8 -10 2 4 6 8 -2 -4 B C A Y=a+X·b+X2·c (a,0) Y=a Y=a+X·b (b,0) (c,0) EXERCISE 1) Click on the variable named a. 2) Click Play to Animate a. 3) Repeat for variables b and c. Lecture01-ConvenientForm.gx Animation Chapter 1 – Functions and Equations Lecture 1 – Explicit, Implicit and Parametric Equations 10 2 4 6 8 10 -2 -4 -6 -8 -10 2 4 6 8 -2 -4 -6 Y=a+X·b+X2·c+X3·d Y=a+X·b+X2·c Y=a+X·b Y=a Y=a+X·b+X2·c+X3·d+X4·e EXERCISES 1) Open the example. 2) Extend the example by adding: 3) Animate the coefficients using the animation window. 4) Describe what you observe. FileOpenLecture01-ConvenientForm.gx Extending An Example Learning Calculus with Geometry Expressions™ 11 1) Click DrawText. The cursor will change to a this: 2) Select a region as shown. 3) Enter text in dialog box. 4) Click OK 5) Double-Click text to edit it. 6) Right-Click text to change its display properties. Labeling Chapter 1 – Functions and Equations Lecture 1 – Explicit, Implicit and Parametric Equations 12 Polynomials in Convenient Form 2 4 6 8 10 -2 -4 -6 -8 -10 2 4 6 8 -2 -4 -6 Polynomials in Convenient Form Y=a+X·b+X2·c+X3·d+X4·e Y=a+X·b+X2·c Y=a+X·b Y=a Y=a+X·b+X2·c+X3·d Lecture01-ConvenientFormB.gx Convenient Form Learning Calculus with Geometry Expressions™ 13 “Blank Looks Are Still Free” This dialog appears when a drawing is “overconstrained”. Chapter 1 – Functions and Equations Lecture 1 – Explicit, Implicit and Parametric Equations 14 A Popular Explicit Function The explicit function: occurs frequently in mathematics, physics, and electronics. It says that y is a function of A, , t, and . These one-letter symbols originated in the days where equations were written by hand, on chalkboards, and economy of communication was the priority. A stands for Amplitude, for frequency, t for time and for phase. We could just as well have written: Amplitude, frequency and phase are taken to be constants, but they need not. They can change with time to model many natural phenomena. When using the DrawFunction tool, we will use X to stand for time as the next page shows: Learning Calculus with Geometry Expressions™ 15 2 4 6 8 10 -2 -4 -6 -8 -10 2 4 6 8 -2 -4 -6 -8 Y=A·sin(φ+X·ω) Exercise 1) Create a new worksheet using FileNew. 2) Draw the sine function using DrawFunction. 3) Use names like “phi” at first then press Enter. 4) Double-click to select formula. 5) Use Symbols menu to enter Greek letters such as to replace names. 6) Vary the parameters: A, , and . How do they affect the sine wave? 7) Replace the A term with X. What happens? A Popular Explicit Function Chapter 1 – Functions and Equations Lecture 1 – Explicit, Implicit and Parametric Equations 16 sin(x)2 + cos(y)2 + sin(z) – 1 = 0 Implicit Equations Learning Calculus with Geometry Expressions™ 17 Implicit Equations Sometimes we encounter a complex equation that can’t be explicitly solved for the variable we want. When it is not possible to isolate the variable we want we use the term equation rather than function. More on that later. An implicit equation is an algebraic curve formed by points that satisfy an equation. Implicits TEST their input X and Y values and return true if the implicit relationship is satisfied. The general solution of implicit equations requires a search! When we have any equation of the form: We say that the equation is implicit. Setting any expression equal to zero makes it implicit, even if it was explicit before. A good example of an implicit equation is the equation of the circle: It is not possible to solve this equation explicitly for either x or y and obtain the entire circle. However, GX™ provides us tools to draw implicit equations. Chapter 1 – Functions and Equations Lecture 1 – Explicit, Implicit and Parametric Equations 18 2 4 6 8 10 -2 -4 -6 -8 -10 2 4 6 8 -2 -4 -6 -8 A B Steps in Drawing A Circle 1) FileNew 2) Click DrawCircle 3) Click to create center point. 4) Click to create a radius point. 5) Press ESC The circle appears. Lecture01-DrawCircle.gx Circle Drawing Learning Calculus with Geometry Expressions™ 19 2 4 6 8 10 -2 -4 -6 -8 -10 2 4 6 8 -2 -4 -6 -8 A B X2+Y2+X·d0+Y·e0+f0=0 Steps to Obtain Implicit Equation 1) Click on circle boundary. 2) Click ConstrainImplicit Equation Lecture01-DrawCircle.gx Circle Equation Chapter 1 – Functions and Equations Lecture 1 – Explicit, Implicit and Parametric Equations 20 2 4 6 8 10 12 14 16 -2 -4 2 4 6 -2 -4 -6 -8 2 4 6 8 10 12 14 16 -2 -4 2 4 6 -2 -4 -6 -8 A B Steps to Calculate Implicit Equation 1) Select Point A. 2) Constrain its Coordinate as (h, k). 3) Select Point A and Point B . 4) Constrain its Distance as r. 5) CalculateSymbolicImplicit. Lecture01-CircleDrawing.gx General Circle Equation Learning Calculus with Geometry Expressions™ 21 The General Form of a Circle … enables one to specify the radius and center (h, k). This general form is: But this doesn’t look anything like the form that GX™ gave us… until we expand the first equation like so: and compare similar terms. Now we can read off coefficients: In later chapters we will dive further into the fantastic properties of implicit functions, a more plentiful class than their explicit counterparts. Chapter 1 – Functions and Equations Lecture 1 – Explicit, Implicit and Parametric Equations 22 Exercises 1) Draw an infinite line using the tool, DrawInfinite Line: 2) Use the ConstrainImplicit Curve tool to produce the equation of the line: 3) The implicit formula for an explicit line translated by a distance (h,k) from the origin is: Using the technique described for the circle, how do the coefficients of the implicit line relate to the GX™ form. That is, what are the values of C0, A0 and B0 in terms of m, b, h and k? 4) Animate each of the variables in the Variables Dialog and record the effect they have on the implicit version of the infinite line. How do these compare with the rise and run animation done at the beginning of the book? Learning Calculus with Geometry Expressions™ 23 The Ladder There is a fascinating progression and yes, its musical. An explicit function can always be converted to an implicit function, but not the other way round about. The implicit form of : is: and this can be made a topographic surface: We find the contours of this topographic tale by setting z = 0. The “roots”, are “zero-crossings” of the x-y surface at the plane z=0. These roots are curves. We can do this for the four basic operations of mathematics: The constants b and c in front of x and y scale or distort these graphs. The constant a acts to move them up and down by a fixed amount. Chapter 1 – Functions and Equations Lecture 1 – Explicit, Implicit and Parametric Equations 24 Learning Calculus with Geometry Expressions™ 25 The Rungs Perhaps you perceive the progression. Here are a few rungs in the ladder: ax + by = 0 ax + by = z ax + by + cz =0 ax + by + cz = w ax + by + cz +dw =0 Each rung leads to a higher dimensional space. The implicit equation we are currently using is but a contour plot for an explicit equation in the next higher dimension. It goes on like a ladder forever! Implicit equations are contour plots, and require a search to discover. Explicit equations are deterministic and can simply be drawn. So we connect a search, with a deterministic guarantee of solution. So now you have seen the ladder, and its pretty cool, don’t you think? Chapter 1 – Functions and Equations Lecture 1 – Explicit, Implicit and Parametric Equations 26 Parametric Equations We have discussed explicit functions where a variable on the left is expressed in terms on the right hand side of the equation. We covered implicit functions, algebraic curves defined by setting a collection of terms equal to zero. We will now describe parametric functions, which could also be called generating functions – they generate coordinate pairs as output, given parameter values as input. In a parametric equation each geometric coordinates, x, y or z is written in terms of a parameter, often named s, t, or . A parametric equation is really a set of equations, one for each coordinate we are drawing, x, y, z, etc. Since we are doing two-dimensional geometry in the plane we will just use x and y. We could instantly loft our curves into space by specifying one more function for z. The parametric equations for a circle are: In these as we vary the parameter from 0 to , we GENERATE x and y coordinates for the top half of a circle. If we run from 0 to 2we get the whole circle. We can stop short or keep retracing the curve. Compare the three representations for a circle in the next slide: Learning Calculus with Geometry Expressions™ 27 Representations of A Circle: Explicit Implicit Parametric 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 -1 Representations of A Circle: Explicit Implicit Parametric C Þ X2+Y2-2·X·h+h2-2·Y·k+k2-r2=0 Þ r r (H,K) Y=1+ r2-(-1+X)2 (H+r·cos(θ),K+r·sin(θ)) (h,k) Lecture01-CircleReps.gx Three Forms of a Circle Chapter 1 – Functions and Equations Lecture 1 – Explicit, Implicit and Parametric Equations 28 0.5 1.0 1.5 -0.5 -1.0 -1.5 0.5 1.0 1.5 2.0 -0.5 A (r·cos(θ),r·sin(θ)) θ EXERCISE 1) Create a new worksheet. 2) Create Point A. Press ESC. 3) Select Point A 4) ConstrainCoordinate and type the function shown. 5) Select Point A 6) Choose ConstructLocus. 7) Compare with this example. Lecture01-ParametricArc.gx Parametric Curve Drawing Learning Calculus with Geometry Expressions™ 29 Parametric Equations In the case above, the parameter had an intuitive geometric meaning – the angle between the x-axis and a ray through a point on the circle. This is not always true. Great effort has been expended to make parameters intuitive, as in “arc-length parameterizations”. A common parameter name is s, for the arc length of the path being drawn. We might use the parameter t, to represent time. Whatever makes the point clear. Unlike their implicit cousins which provide a test of truth but no method, parametric equations are generators of paths through space. Though each equation is a function, the generated curve can loop over itself and thus may not be a function. They enable flexible shape generation with the guarantee of being able to compute an answer more rapidly than the an implicit search. We write the coordinate pair (x, y) as a pair of functions (r cos(), r sin()). Parametric functions are often called vector-valued functions, since a function is required for each coordinate we are specifying. All explicit functions can be represented as parametric functions. The variety of parametric functions one can generate with GX™ is truly amazing. For example. If we draw a line between a point on a parabola and a point on a line, what is the curve traced out by the midpoint? Is it linear? It is parabolic? If we average the two what shape do we obtain? Chapter 1 – Functions and Equations Lecture 1 – Explicit, Implicit and Parametric Equations 30 0.5 1.0 1.5 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0 0.5 1.0 1.5 2.0 -0.5 -1.0 -1.5 B A C Þ t, t+t2 2 t,t2 (t,t) Curve C is the average of a parabola and a line. Lecture01-AvgParametrics.gx The Average of Two Curves Learning Calculus with Geometry Expressions™ 31 Parametric Power Tools What we just did has a very rich set of possibilities, and we started with a single point! By specifying the behavior of one point, we can, using the locus tool, see the family of points generated by the parametric equations. You could stop now and enjoy an entire career of studying the behavior and utility of these equations. But wait… there’s more. What happens if we assign two points A, and B, their own independent functions? A and B don’t know about each other, and each has their own personal locus. But if we draw a line segment to connect A and B something happens. The line segment AB has to serve two masters. Line AB has to satisfy the A functions at the A end, AND it must also obey the B functions at the B end! Line AB is pulled in two directions at once. If you’ve ever felt jerked around, this is your equation! This is a powerful idea in life and in math. There is one point on the line segment that will evenly blend the effects of its two endpoints and that point is the midpoint. We can compound the expressive richness of this idea. We could have another line segment driven by its own points C, and D and ask what the relationship is between the midpoint of the line segment connecting the two line segments. The possibilities are endless. Chapter 1 – Functions and Equations Lecture 1 – Explicit, Implicit and Parametric Equations 32 1 2 3 4 5 6 -1 -2 -3 1 2 3 -1 -2 -3 M D C B I N A (4+c·sin(2·θ),d·cos(3·θ)) (a·sin(θ),b·cos(θ)) (4+a·sin(θ),b·cos(θ)) (c·sin(θ),d·cos(θ)) EXERCISE 1) Animate  Lecture01-PowerTools.gx Power Tools Learning Calculus with Geometry Expressions™ 33 End
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https://artofproblemsolving.com/wiki/index.php/Lifting_the_Exponent_Lemma?srsltid=AfmBOoqkS9vQAP6s2A-Aex_yMl5dWzdtLxstVZyQC4GDJjdVpPpQSd9C
Art of Problem Solving Lifting the Exponent Lemma - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki Lifting the Exponent Lemma Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Lifting the Exponent Lemma Lifting the exponent allows one to calculate the highest power of an integer that divides various numbers given certain information. It is extremely powerful and can sometimes "blow up" otherwise challenging problems. Let be a prime such that and . LTE comprises of the following identities (where represents the largest power of that divides ): When is odd: , if . , if and is odd. , if and is even. When : , if and is even. if and is odd. Corollaries: if . , if and is even. , if and is odd. External Links This article is a stub. Help us out by expanding it. Retrieved from " Category: Stubs Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
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https://www.cut-the-knot.org/Curriculum/Algebra/Figurate.shtml
Examples with series of figurate numbers Site What's new Content page Front page Index page About Privacy policy Help with math Subjects Arithmetic Algebra Geometry Probability Trigonometry Visual illusions Articles Cut the knot! What is what? Inventor's paradox Math as language Problem solving Collections Outline mathematics Book reviews Interactive activities Did you know? Eye opener Analogue gadgets Proofs in mathematics Things impossible Index/Glossary Simple math Fast Arithmetic Tips Stories for young Word problems Games and puzzles Our logo Make an identity Elementary geometry Examples with series of figurate numbers There are many patterns in the common multiplication table. Some are suggested by the applet below. If you are reading this, your browser is not set to run Java applets. Try IE11 or Safari and declare the site as trusted in the Java setup. What if applet does not run? Just move the mouse over. Discussion |Activities||Contact||Front page||Contents||Algebra| Copyright © 1996-2018 Alexander Bogomolny Identities in the Multiplication Table The applet points to four identities: (1)(1 + 2 + 3 + ... + n)2 = [n(n+1)/2]2 (2)2n(1 + 2 + 3 + ... + n) - n 2 = n 3 (3)1 2 + 2 2 + 3 2 + ... + n 2 = n(n + 1)(2n + 1)/6 (4)n·1 + (n - 1)·2 + (n - 2)·3 + ... + 2·(n - 1) + 1·n = n(n + 1)(n + 2)/6 The sum of all entries in an n×n multiplication table equals [n(n+1)/2]2. Indeed, the sum of the entries in the first row is 1 + 2 + ... + n = n(n + 1)/2. Then grouping the entries by rows be get | Row | | Sum | :--- | | 1 | | 1 + 2 + ... + n = 1·n(n + 1)/2 | | 2 | | 2 + 4 + ... + 2n = 2·n(n + 1)/2 | | 3 | | 3 + 6 + ... + 3n = 3·n(n + 1)/2 | | | | ... | | n | | n + 2n + ... + n·n = n·n(n + 1)/2 | | Total | | (1 + 2 + ... + n)·n(n + 1)/2 = [n(n+1)/2]2 | This settles (1). (2) arises as the sum of entries in the n th column and in the n th row, the diagonal term n 2 being counted only once. The entries in both the n th row and the n th column add up to n·(1 + 2 + ... + n) = n 2(n + 1)/2. Together, but with the exclusion of the diagonal entry, we get 2·n 2(n + 1)/2 - n 2 = (n 3 + n 2) - n 2 = n 3. As a bonus, a combination of (1) and (2) leads to the following curious identity, known as Nicomachus' theorem (1 + 2 + ... + n)2 = 1 3 + 2 3 + ... + n 3. (3) is the sum of the diagonal entries, i.e., the sum of the first n squares. The formula is well known and is easily proven by induction. Mathematical induction, however, though a powerful tool, is limited to the cases when the formula to be proven is known (or has been guessed) up front. But how does one get those formulas in the first place? Here is one possible approach based on the theory of generating functions. Denote q n = 1 2 + 2 2 + 3 2 + ... + n 2. Then obviously the sequence {q n} satisfies the recurrence relation q n = q n-1 + n 2, q 0 = 0. Multiply the relation by x n-1 and add up the first n: Q(x)/x = Q(x) + ∑n n 2 x n-1, where Q(x) is the generating function of the sequence {q n}. This would be determined from (5)Q(x)(1 - x)/x = ∑n n 2 x n-1 if we could express the sum on the right in terms of x. Let's try, ∑n n 2 x n = ∑n n(n + 1)x n-1 - ∑n nx n-1 The sum ∑n nx n-1has been found to equal 1/(1 - x)2 as the formal derivative of the sum of the geometric series 1/(1 - x) = 1 + x + x 2 + ... We shall apply the same idea to the series ∑n n(n + 1)x n-1. ∑n n(n + 1)x n-1= [1/(1 - x)]'' = [1/(1 - x)2]' = 2/(1 - x)3. From (5) then Q(x)= x/(1 - x)·(2/(1 - x)3 - 1/(1 - x)2) = x/(1 - x)·(1 + x)/(1 - x)3 = x(1 + x)/(1 - x)4. We know that ∑n C(n,k)x n = x k/(1-x)k+1, which implies [x n]Q(x)= [x n]x/(1 - x)4 + [x n]x 2/(1 - x)4 = C(n+2,3) + C(n+1,3) = (n + 2)(n + 1)n/6 + (n + 1)n(n - 1)/6 = n(n + 1)(2n + 1)/6. In (4), the coefficients are obtained by convolution of the terms of the plain sequence {n}. Convolutions arise when power series are multiplied term by term according to the definition. Let t 0 = 0 and t 1 = 0 and, for n = 2, 3, ...,, define t n = (n - 1)·1 + (n - 2)·2 + (n - 3)·3 + ... + 2·(n - 2) + 1·(n - 1). Introduce the generating function T(x) = ∑n t n x n. T(x)= (x + 2x 2 + 3x 3 + ...)·(x + 2x 2 + 3x 3 + ...) = x/(1 - x)2·x/(1 - x)2 = x 2/(1 - x)4. Using the same identity as above, [x n]T(x) = [x n]x 2/(1 - x)4 = C(n+1,3) = (n + 1)n(n - 1)/6. Thus t n = (n + 1)n(n - 1)/6, which corresponds to the sum of terms in the (n - 1)st diagonal. Generating Functions Generating Functions A Property of the Powers of 2 An USAMTS problem with light switches Examples with series of figurate numbers Euler's derivation of the binary representation Examples with finite sums with binomial coefficients Fast Power Indices and Coin Change Number of elements of various dimensions in a tesseract Straight Tromino on a Chessboard Ways To Count Probability Generating Functions Sicherman Dice Finite Sums of Terms 2^(n-i) i^2 Sylvester's Problem, a Second Look Generating Functions from Recurrences Binet's Formula via Generating Functions Number of Trials to First Success Another Binomial Identity with Proofs Matching Socks in Dark Room |Activities||Contact||Front page||Contents||Algebra| Copyright © 1996-2018 Alexander Bogomolny 73255325
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https://linux-blog.anracom.com/2023/09/09/properties-of-ellipses-by-matrix-coefficients-iii-coordinates-of-points-with-extremal-radii/
Properties of ellipses by matrix coefficients – III – coordinates of points with extremal radii | Linux-Blog – Dr. Mönchmeyer / anracon We value your privacy This site uses technically required cookies, only. We do not follow your activities or register personal data. Customize Reject All Accept All Customize Consent Preferences We use technically required cookies, only. The cookies that are categorized as "Necessary" are stored on your browser as they are essential for enabling the basic functionalities of the site. Necessary Always Active Necessary cookies are required to enable the basic features of this site, such as providing secure log-in or adjusting your consent preferences. These cookies do not store any personally identifiable data. No cookies to display. Functional Functional cookies help perform certain functionalities like sharing the content of the website on social media platforms, collecting feedback, and other third-party features. No cookies to display. 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Reject All Save My Preferences Accept All Skip to primary content Linux-Blog – Dr. Mönchmeyer / anracon Notes about Linux, ML and some simple math … Search Main menu Home About me Certificates Dr. Mönchmeyer Impressum Datenschutz Post navigation ← PreviousNext → Properties of ellipses by matrix coefficients – III – coordinates of points with extremal radii Posted on 9. September 2023 by eremo This post requires Javascript to display formulas! A centered, rotated ellipse can be defined by matrices which operate on position-vectors for points on the ellipse. The topic of this post series is the relation of the coefficients of such matrices to some basic geometrical properties of an ellipse. In the previous posts Properties of ellipses by matrix coefficients – I – Two defining matrices and eigenvalues Properties of ellipses by matrix coefficients – II – coordinates of points with extremal y-values we have found that we can use (at least) two matrix based approaches: One reflects a combination of two affine operations applied to a unit circle. This approach led us to a non-symmetric matrix, which we called AE. Its coefficients ((a, b), (c, d)) depend on the lengths of the ellipses’ principal axes and trigonometric functions of its rotation angle. The second approach is based on coefficients of a quadratic form which describes an ellipse as a special type of a conic section. We got a symmetric matrix, which we called Aq. We have shown how the coefficients α, β, γ of Aq can be expressed in terms of the coefficients of AE. Another major result was that the eigenvalues and eigenvectors of Aq completely control the ellipse’s properties. Furthermore, we have derived equations for the lengths σ 1, σ 2 of the ellipse’s principal axes and the rotation angle by which the major axis is rotated against the x-axis of the Cartesian coordinate system [CCS] we work with. We have also found equations for the components of the position vectors to those points of the ellipse with maximum y-values. In this post we determine the components of the vectors to the end-points of the ellipse’s principal axes in terms of the coefficients of Aq. Afterward we shall test our formulas by a Python program and plots for a specific example. Reduced matrix equation for an ellipse Our centered, but rotated ellipse is defined by a quadratic form, i.e. by a polynomial equation with quadratic terms in the components x e and y e of position vectors to points on the ellipse: α x 2 e+β x e y e+γ y 2 e=1.α x e 2+β x e y e+γ y e 2=1. The quadratic polynomial can be formulated as a matrix operation applied to position vectors vE = (x E, y E)T. With the the quadratic and symmetric matrix Aq A A q=(α β/2 β/2 γ)A A q=(α β/2 β/2 γ) we can rewrite the polynomial equation for the centered ellipse as v v T E∘A A q∘v v E=1,with v E v E=(x E y E).v v E T∘A A q∘v v E=1,with v E v E=(x E y E). Method 1 to determine the vectors to the principal axes’ end points My readers have certainly noticed that we have already gathered all required information to solve our task. In the first post of this series we have performed an eigendecomposition of our symmetric matrix Aq. We found that the two eigenvectors of Aq for respective eigenvalues λ 1 and λ 2 point along the principal axes of our rotated ellipse: λ 1:ξ 1 ξ 1=(1 β((α−γ)−[β 2+(γ−α)2]1/2),1)T,λ 2:ξ 2 ξ 2=(1 β((α−γ)+[β 2+(γ−α)2]1/2),1)T.λ 1:ξ 1 ξ 1=(1 β((α−γ)−[β 2+(γ−α)2]1/2),1)T,λ 2:ξ 2 ξ 2=(1 β((α−γ)+[β 2+(γ−α)2]1/2),1)T. The T symbolizes a transposition operation. The eigenvalues are related to the Aq-coefficients by the following equations: λ 1=1 2((α+γ)−[β 2+(γ−α)2]1/2),λ 2=1 2((α+γ)+[β 2+(γ−α)2]1/2).λ 1=1 2((α+γ)−[β 2+(γ−α)2]1/2),λ 2=1 2((α+γ)+[β 2+(γ−α)2]1/2). These eigenvalues correspond to the squares of the lengths of the ellipse’s axes. λ 1=1 σ 2 1,λ 2=1 σ 2 2.λ 1=1 σ 1 2,λ 2=1 σ 2 2. Therefore, we can simply take the components of the normalized vectors λ 1:ξ n 1 ξ n 1=1∥ξ 1 ξ 1∥ξ 1 ξ 1,λ 2:ξ n 2 ξ n 2=1∥ξ 2 ξ 2∥ξ 2 ξ 2 λ 1:ξ 1 n ξ 1 n=1‖ξ 1 ξ 1‖ξ 1 ξ 1,λ 2:ξ 2 n ξ 2 n=1‖ξ 2 ξ 2‖ξ 2 ξ 2 and multiply them with the square-root of the respective eigenvalues to get the vector components to the end-points of the ellipse’s axes: ξ 1 ξ 1 r m a x=(1/√λ 1)∗ξ n 1 ξ n 1,ξ 2 ξ 2 r m a x=(1/√λ 2)∗ξ n 1 ξ n 1.ξ 1 ξ 1 r m a x=(1/λ 1)∗ξ 1 n ξ 1 n,ξ 2 ξ 2 r m a x=(1/λ 2)∗ξ 1 n ξ 1 n. This is trivial regarding the algebraic operations, but results in lengthy (and boring) expressions in terms of the matrix coefficients. So, I skip to write down all the terms. (We do not need it for setting up ordered numerical programs.) Remember that you could in addition replace (α, β, γ) by coefficients (a, b, c, d) of matrix AE. See the first post of this series for the formulas. This would, however, produce even longer equation terms. Equation for points with maximum radius values We define again some convenience variables: a h=α γ,b h=1 2 β γ,d h=1 γ,g h=a h−b 2 h,f h=1+b 2 h−g h(a h=α γ,b h=1 2 β γ,d h=1 γ,g h=a h−b h 2,f h=1+b h 2−g h( and ξ h=[4 d h g h b 2 h+d h f 2 h]24 b 2 h g 2 h+g h f 2 hx 2 E]1/2=–b h x E±[d h−g h x 2 E]1/2(.y E=–b h x E±[d h−(a h−b h 2)x E 2]1/2=–b h x E±[d h−g h x E 2]1/2(. We pick the y E with the positive term in the following steps. (The way for the solution with the negative term in y E is analogous.) The square of y E is: y 2 E=–d h+(b 2−g)x 2 E−2 b h x E[d−g x 2 E]1/2.y E 2=–d h+(b 2−g)x E 2−2 b h x E[d−g x E 2]1/2. To find an extremal value of the radius we differentiate and set the derivative to zero: ∂(y 2 E+x 2 E)∂x E=0⇒∂(y E 2+x E 2)∂x E=0⇒ (1+b 2 h−g h)x E−b h[d h−g h x 2 E]1/2+b h g h x 2 E 1[d h−g h x 2 E]1/2=0,.(1+b h 2−g h)x E−b h[d h−g h x E 2]1/2+b h g h x E 2 1[d h−g h x E 2]1/2=0,. This results in f h x E[d h−g h x 2 E]1/2=b h d h−2 b h g h x 2 E.f h x E[d h−g h x E 2]1/2=b h d h−2 b h g h x E 2. Solution for x e-values of the end-points of the principal axes We take the square of both sides and reorder terms to get [4 b 2 h g 2 h+g h f 2 h]x 4 E−[4 d h g h b 2 h+d h f 2 h]x 2 E+b 2 h d 2 h=0,[4 b h 2 g h 2+g h f h 2]x E 4−[4 d h g h b h 2+d h f h 2]x E 2+b h 2 d h 2=0, x 4 E−[4 d h g h b 2 h+d h f 2 h][4 b 2 h g 2 h+g h f 2 h]x 2 E=−b 2 h d 2 h[4 b 2 h g 2 h+g h f 2 h]x E 4−[4 d h g h b h 2+d h f h 2][4 b h 2 g h 2+g h f h 2]x E 2=−b h 2 d h 2[4 b h 2 g h 2+g h f h 2] With ξ h=[4 d h g h b 2 h+d h f 2 h]2[4 b 2 h g 2 h+g h f 2 h]η h=b 2 h d 2 h[4 b 2 h g 2 h+g h f 2 h])A ξ h=[4 d h g h b h 2+d h f h 2]2[4 b h 2 g h 2+g h f h 2]η h=b h 2 d h 2[4 b h 2 g h 2+g h f h 2])A we have x 4 E−2 ξ h E 2 e=–η h.x E 4−2 ξ h E e 2=–η h. With the help of a quadratic supplement we get [x 2 E−ξ h]2=ξ 2 h−η h[x E 2−ξ h]2=ξ h 2−η h and find the solution x E=±√ξ h±√ξ 2 h−η h.x E=±ξ h±ξ h 2−η h. A detailed analysis also for the other y E-expression (see above) leads to further solutions for the coordinates (=vector component values) of points with extremal values for the radii. These are the end-points of the principal axes of the ellipse: x r m a x E 1=−√ξ h−√ξ 2 h−η,y r m a x E 1=+√d h−g h(x r m a x e 1)2−b h x r m a x e 1,x r m a x E 2=−x r m a x e 1),y r m a x E 2=−y r m a x e 1,x r m a x E 3=+√ξ h+√ξ 2 h−η),y r m a x E 3=+√d h−g h(x r m a x e 3)2−b h x r m a x e 3,x r m a x E 4=−x r m a x e 3)y r m a x E 4=−y r m a x e 3.x E 1 r m a x=−ξ h−ξ h 2−η,y E 1 r m a x=+d h−g h(x e 1 r m a x)2−b h x e 1 r m a x,x E 2 r m a x=−x e 1 r m a x),y E 2 r m a x=−y e 1 r m a x,x E 3 r m a x=+ξ h+ξ h 2−η),y E 3 r m a x=+d h−g h(x e 3 r m a x)2−b h x e 3 r m a x,x E 4 r m a x=−x e 3 r m a x)y E 4 r m a x=−y e 3 r m a x. I leave it to the reader to expand the convenience variables into terms containing the original coefficients α, β, γ. Plots It is easy to write a Python program, which calculates and plots the data of an ellipse and the special points with extremal values of the radii and extremal values of y e. The general steps which I followed were: Step 0: Create 100 points a unit circle. Save the coordinates in Python lists (or Numpy arrays). Use Matplotlib’s plot(x,y)-function to plot the vectors. Step 1: Create an axis-parallel ellipse with values for the axes ha = 2.0 and hb = 1.0 along the x- and the y-axis of the Cartesian coordinate system [CCS]. Do this by applying a diagonal scaling matrix Dσ1, σ2 (see the first post of this series). Step 2: Rotate the ellipse bei π/3 (60 °). Do this by applying a rotation matrix Rπ/3 to the position vectors of your ellipse (with the help of Numpy). Alternatively, you can first create the matrices, perform a matrix multiplication and then apply the resulting matrix to the position vectors of your unit circle. (The limiting lines have been calculated by the formulas given above.) Step 3: Determine the coefficients of combined matrix AE = Rπ/3 ○ Dσ1, σ2 I got for the coefficients ( (a, b), (c, d) ) of AE : A_ell = [[ 1. -0.8660254 ] [ 1.73205081 0.5 ]] Step 3: Determine the coefficients of the matrix Aq by the formulas given in the first post of this series. I got A_q = [[ 3.25 -1.29903811] [-1.29903811 1.75 ]] For δ I got: delta = 4.0 which is consistent with the length-values of the principal axes. Step 4: Determine values for the eigenvalues λ 1 and λ 2 from the Aq-coefficients by the formulas given in the first post. Also calculate them by using Numpy’s eigenvalues, eigenvectors = numpy.linalg.eig(A_q). Theory tells us that these values should be exactly λ 1 = 4 and λ 1 = 1. I got Eigenvalues from A_q: lambda_1 = 4. :: lambda_2 = 1. Step 5: Determine the components of the normalized eigenvectors with the help of numpy.linalg.eig(A_q). I got: Components of normalized eigenvectors by theoretical formulas from A_q coefficients: ev_1_n : -0.8660254037844386 : 0.5000000000000002 ev_2_n : 0.5000000000000001 : 0.8660254037844385 Eigenvectors from A_q via numpyy.linalg.eig(): ev_1_num : 0.8660254037844387 : -0.5000000000000001 ev_2_num : 0.5000000000000001 : 0.8660254037844387 The deviation between ev_1_n and ev_1_num is just due to a difference by -1. This is correct as the eigenvectors are unique only up to a minus-sign in all components. Step 6: Calculate the sinus of the rotation angle of our ellipse from Aq– and Aq-coefficients. The theoretical value is sin(2 π/3) = sin(2 pi/3) = 0.8660254037844387. I got: sin(2. rotation angle) of major axis of the ellipse against the CCS x-axis from A_E coefficients: sin_2phi-A_E = 0.8660254037844388 sin(2. rotation angle) of major axis of the ellipse against the CCS x-axis from from eigenvectors of A_q: sin_2phi-ev_A_q = 0.8660254037844387 sin(2. rotation angle) of major axis of the ellipse against the CCS x-axis from A_q-coefficients: sin_2phi-coeff-A_q = 0.8660254037844388 Perfect! Step 7: Plot the end-points of the normalized eigenvectors of Aq: Note that in our example case the end-point of the eigenvector along the minor axis must be located exactly on the elliptic curve as the ellipses minor axes has a length of b=1! Step 8: Calculate the components of the vectors to data-points of the ellipse with maximal absolute y e-values from the Aq-coefficients given in the previous post. Plot these data-points (here in green color). Step 9: Calculate the components of the vectors to data-points of the ellipse with maximal values of the radii with the help of the complex formulas presented in this post and plot these points in addition. Conclusion In this mini-series of posts we have performed some small mathematical exercises with respect to centered and rotated ellipses. We have calculated basic geometrical properties of such ellipses from the coefficients of matrices which define ellipses in algebraic form. Linear Algebra helped us to understand that the eigenvectors and eigenvalues of a symmetric matrix, whose coefficients stem from a quadratic equation (for a conic section), control both the orientation and the lengths of the ellipse’s axes completely. This knowledge is useful in some Machine Learning [ML] context where elliptic data appear as projections of multivariate normal distributions. Multivariate Gaussian probability functions control properties of a lot of natural objects. Experience shows that certain types of neural networks may transform such data into multivariate normal distributions in latent spaces. An evaluation of the numerical data coming from such ML-experiments often delivers the coefficients of defining matrices for ellipses. In my blog I now return to the study of with shearing operations applied to circles, spheres, ellipses and 3-dimensional ellipsoids. Later I will continue with the study of multivariate normal distributions in latent spaces of Autoencoders. For both of these topics the knowledge we have gathered regarding the matrices behind ellipses will help us a lot. This entry was posted in Machine Learning, Mathematics and tagged defining matrices of an ellipse, eigenvalues, eigenvectors, ellipses, end-points of principal axes of an ellipse, matrices, matrix coefficients, points with maximum radii, points with maximum y-value, position vectors, principal axes of an ellipse, quadratic forms by eremo. Bookmark the permalink. 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https://www.illumefertility.com/fertility-blog/fertility-testing-day-21
Get $100 OFF Your First Fertility Consultation BOOK NOW Open Search Schedule Your Consult Search for topics or resources Enter your search below and hit enter or click the search icon. Close Search Fertility Care Get Started Fertility Testing LGBTQ+ Family Building For International Patients PCOS & Fertility Oncofertility Fertility Treatment Egg Freezing In Vitro Fertilization (IVF) Intrauterine Insemination (IUI) Gestational Surrogacy Reciprocal IVF (RIVF) Timed Intercourse Ovulation Induction Donor Conception Fertility Surgeries All Treatment Options Wellness Acupuncture Nutrition Mental Health Support Groups & Events All Wellness Support Pricing & Insurance Treatment Costs Insurance & Mandates Resources Education Learning Center Free Guides & Tools Ask a Fertility Nurse Fertility FAQs Fertility Glossary Community Support Groups & Events Patient Stories Loss Support Adoption Resources About Contact Locations Norwalk, CT Stamford, CT Danbury, CT Trumbull, CT Harrison, NY Meet Our Team IVF Success Rates Press & Media Our Practice Careers Learning Center Events Gay Parents To Be Español Login Schedule Your Consult 203-750-7400 « View All Posts Fertility Basics / Fertility Testing / Video Day 21 Fertility Testing: Progesterone Levels, FAQs & What to Expect Fertility testing performed on day 21 of your cycle provides critical information your doctor needs to help you succeed. February 6th, 2025 | 11 min. read By Sarah Waters, RN, WHNP In this article: Why is fertility testing necessary? What is day 21 fertility testing? Day 21 Testing for Irregular Menstrual Cycles What cycle day do you ovulate? Why is day 21 progesterone testing important? What are normal progesterone levels on day 21? What does it mean if my progesterone is low on day 21? What happens after day 21 testing if my progesterone is low? What are the effects of low progesterone? What if I don't ovulate on my own? Can I test my progesterone at home on day 21? Does insurance cover day 21 fertility testing? How can I increase my progesterone levels naturally? One of the first steps on any family-building journey is a comprehensive fertility workup. This assessment includes blood work and other tests performed at various points in your menstrual cycle. Day 21 fertility testing is an essential part of that process - here's why. Why is fertility testing necessary? When you decide to seek help from a fertility specialist after trying to conceive on your own without success, your doctor's first priority will be getting the most accurate picture of what you might need to achieve fertility treatment success. As part of your fertility testing workup, your reproductive endocrinologist will order tests like a hysterosalpingogram (HSG) and saline sonogram (SHG) to evaluate the health of your uterus and fallopian tubes. Many of these tests need to be performed at particular points during your menstrual cycle. One of those important tests is referred to as the "day 21" test. What is day 21 fertility testing? "Day 21" of your menstrual cycle can be a useful day to evaluate several different processes, particularly for those having trouble conceiving on their own. This evaluation involves checking the levels of certain hormones (like progesterone and estradiol) via bloodwork and measuring the thickness of your endometrium (uterine lining). The Menstrual Cycle, Explained For someone with a 28-day cycle, it takes 14 days to develop a follicle and ovulate the oocyte (egg), and then 14 days of the luteal phase, ending with a menses on the 28th day of the cycle. So in this textbook 28-day cycle, day 21 falls in the middle of the luteal phase. For those with more frequent menses, the follicular phase is shorter. For those with longer cycles, the follicular phase is longer. The luteal phase is much more predictable than the follicular phase. But what happens if you don’t get a menstrual cycle every 28 days? Maybe you get your menses once every two to three months, or maybe it never seems to come at all. Perhaps your period is only regular when you are taking an oral contraceptive (birth control). Or maybe your menses occur more frequently. Let's talk about how "day 21" looks for those with irregular or absent cycles. Day 21 Testing for Irregular Menstrual Cycles We often refer to "day 21" in quotes because it doesn’t really make sense for the many people who don’t have typical 28-day cycles. Variability in cycle length is primarily due to the follicular phase. This means that the number of days it takes to grow and develop a dominant follicle that is ready to ovulate can be longer or shorter, depending on the person. While the follicular phase can vary, the luteal phase always takes about 14 days. Your peak progesterone day should still be seven days after ovulation and seven days before your period begins. How day 21 testing works if you have longer or shorter cycles: If you have 35-day cycles, then you ovulate around day 21, and your peak progesterone level would be checked around day 28. If your cycle typically lasts 25 days, your peak progesterone level would be checked around day 18. Your reproductive endocrinologist will work with you and your individual cycle to determine the optimal time to perform "day 21" testing or other fertility assessments. Remember: Many people don't have a typical 28-day cycle! There's nothing wrong with you if you don't fit into that textbook 28-day cycle category. 90-Day Preconception Guide Want to increase your chances of a healthy pregnancy? Get our free 3 month checklist for trying to conceive to learn how. What cycle day do you ovulate? People who have regular cycles presumably ovulate every month at a predictable time. In a typical 28-day cycle, it takes about 14 days to grow and develop a dominant follicle. If you are monitoring for ovulation, the leutenizing hormone (LH) surge comes roughly 24 to 44 hours before ovulation. In a 28-day cycle, ovulation of the oocyte (egg) occurs around day 14. The luteal phase starts once the follicle releases the oocyte, and generally lasts for 14 days. The area of ovulation on the ovary changes to form the "corpus luteum," which secretes progesterone. How to know if you're ovulating: Ovulation can typically be confirmed after it has occurred by testing the levels of the hormones estrogen and progesterone via a simple, in-office blood test. Why is day 21 progesterone testing important? Progesterone is a vital part of the conception process. It changes the uterine lining into its secretory phase, making the uterine lining receptive and hospitable to the implantation of an embryo. Without implantation of an embryo, the corpus luteum and its secretion of progesterone will recede within 14 days. Once ovulation has occurred, your menses (or period) should begin about 14 days later. As you can see, day 21 is meant to be a marker for when you're in the middle of the luteal phase of your cycle, and when progesterone production is at its peak. If your reproductive endocrinologist is concerned about whether your luteal phase is adequate, (i.e. whether the corpus luteum makes enough progesterone to support a healthy secretory endometrium and implantation of an embryo), they may want to check your progesterone level on "day 21," at the luteal phase peak. If your reproductive endocrinologist is concerned that you may not be ovulating at all, "day 21" is also a good day to check progesterone levels, as a level above 5ng/ml will confirm that ovulation has taken place. What are normal progesterone levels on day 21? If your progesterone level is high (above 5 ng/ml) this confirms that you have indeed ovulated and entered the luteal phase, the second half of the menstrual cycle. Progesterone rises after ovulation, reaching a peak around Day 21 of a 28-day cycle. Peak luteal phase progesterone levels can vary from cycle to cycle, and from person to person. Ideally, your "day 21" peak luteal progesterone levels should be 10ng/ml or higher. What does it mean if my progesterone is low on day 21? Always discuss test results with your doctor, as they will be able to provide the most accurate guidance for your personal situation. If your progesterone levels are low on day 21 of your menstrual cycle, it may indicate one of the following issues: You didn't ovulate: Also referred to as an anovulatory cycle, low progesterone on day 21 may suggest that ovulation simply did not occur. Progesterone is primarily produced after ovulation, so low levels on day 21 could signal that the egg was not released. A luteal phase defect: This refers to a situation where the corpus luteum (the structure formed after ovulation) does not produce enough progesterone. This can affect the uterine lining's ability to prepare for implantation, potentially leading to difficulties with fertility or early pregnancy loss (miscarriage). Other hormonal imbalances: Low progesterone could be tied to other underlying hormonal issues that need further investigation. Some common contributors to low progesterone levels are high body mass index (BMI), insulin resistance, high stress levels, poor diet, and lack of exercise. What happens after day 21 testing if my progesterone is low? If your fertility specialist is concerned that you are not ovulating or that your progesterone or estrogen levels might be too low to support a lining conducive to implantation, they will discuss options with you. If your day 21 testing results show that you are not ovulating, don't panic! There are many ways we can help to boost your hormone levels and create a more receptive endometrium (uterine lining). Most importantly, a low progesterone level doesn't necessarily mean you won't be able to get pregnant. Rest assured that your doctor will help you get those levels where they need to be so you can have the best chance at a healthy pregnancy. Progesterone levels can be supplemented with vaginal or injectable progesterone supplements. Estradiol levels can be supplemented with oral, vaginal, or transdermal estrogen. When is progesterone and estrogen supplementation recommended? Your fertility specialist may also check your peak luteal progesterone and estrogen levels during a treatment cycle. For certain treatments, such as in vitro fertilization (IVF), we recommend estrogen and progesterone supplementation in the luteal phase to most of our patients to boost hormone levels and ensure the best chances of embryo implantation. In ovulation induction and natural cycles, we typically only supplement if we check your estrogen and progesterone levels and have found them to be too low in the luteal phase of the cycle. What are the effects of low progesterone? Low progesterone levels can affect both the menstrual cycle and a person's overall fertility, since progesterone helps create a good environment for a pregnancy to develop. When progesterone levels are low, it’s harder for an embryo to implant and grow. Low progesterone can also contribute to: Pregnancy loss (miscarriage) Absence of menstruation Poor ovarian function Free Fertility Assessment Not sure where to begin? Take a short quiz to receive personalized insights into your reproductive health and best treatment options. What if I don't ovulate on my own? If you rarely or never get your menses (i.e. have a period), it means that you may be ovulating very rarely, or you may not ovulate on your own at all, which means that your natural "day 21" progesterone levels can’t be checked. If you are not ovulating or are rarely ovulating, your doctor will prescribe medication to induce ovulation, which may include the following: Clomiphene citrate (i.e. Clomid) Injectables (i.e. Ovidrel, hCG, or FSH) Letrozole Once ovulation has occurred in a treatment cycle, we can then assess your peak progesterone levels to confirm ovulation and whether your luteal progesterone and estrogen levels are adequate to support a healthy embryo implantation. Day 21 Fertility Testing FAQs Have other lingering questions? Let's answer them! Can I test my progesterone at home on day 21? Yes, you can test your progesterone at home on day 21 of your cycle using at-home test kits, as long as you have a 28-day period. However, if you have irregular or absent periods, your natural "day 21" progesterone levels may not be detectable. Note: While at-home hormone testing kits are convenient, having tests performed at your doctor's office or fertility clinic is the most reliable way to get accurate results. Does insurance cover day 21 fertility testing? It depends! Many insurance plans do cover diagnostics such as day 21 testing, even those that don't cover subsequent fertility treatment. Speak to an insurance representative or your HR department to determine what fertility tests your policy covers. How can I increase my progesterone levels naturally? There are a variety of ways to encourage progesterone production in the body, including eating a healthy diet, maintaining a health weight, incorporating exercise, getting acupuncture, and working to reduce your stress levels. Research has shown that processed foods, trans fats, refined sugars, and excess added sugars are linked with hormonal disruption and increased rates of infertility. In contrast, there is evidence to suggest that the Mediterranean diet and similar styles of eating (which include lean proteins, healthy fats, and lots of veggies) help promote healthy progesterone levels. Note: Always consult with your doctor to determine which approaches are best for you. Success Begins With Knowledge Fertility assessments like day 21 testing help to give you the very best chance at achieving fertility treatment success. No matter which treatment path you're on, day 21 fertility testing is a vital step in understanding your cycles, hormone fluctuations, and determining the best course of treatment for you. If you're concerned about low progesterone, it's important to consult with your healthcare provider. They can assess your individual situation, order appropriate tests, and discuss potential causes and treatment options if necessary. Just getting started on your fertility journey? Be sure to check out our comprehensive guide to your first consultation and download your free worksheet. We wish you all the best on your journey to baby! Sarah Waters, RN, WHNP Sarah Waters is a registered nurse and Women's Health Nurse Practitioner (WHNP) at Illume Fertility. Her main areas of focus are clinical research, quality assurance, and fertility patient education. Sarah has been an integral member of Illume's nursing team since 2005 and has over 20 years of experience working in the field of reproductive health. Struggling to conceive? Take our 3-minute quiz and get personalized recommendations on your best fertility treatment options. Ready to take the first step? Our team of board-certified reproductive endocrinologists are ready to help you build the family of your dreams. Schedule your consult today to learn how Illume Fertility can help you achieve your goals. Talk to An Expert More Fertility Resources ### Are At-Home Sperm Tests Accurate? 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A triangle OAB is determined by the vectors →aand→b as shown if fig. Show that the triangle has the area given by 12√|a|2|b|2−(a.b)2. More from this Exercise To find the area of triangle OAB determined by the vectors →a and →b, we can use the formula for the area of a triangle formed by two vectors. Here’s the step-by-step solution: Step 1: Area of Triangle Formula The area Δ of triangle OAB formed by vectors →a and →b can be expressed as: Δ=12|→a×→b| where |→a×→b| is the magnitude of the cross product of vectors →a and →b. Step 2: Magnitude of Cross Product The magnitude of the cross product |→a×→b| can be calculated using the formula: |→a×→b|=|→a||→b|sinC where C is the angle between vectors →a and →b. Step 3: Substitute into Area Formula Substituting the expression for the magnitude of the cross product into the area formula gives: Δ=12|→a||→b|sinC Step 4: Expressing sinC We can express sinC in terms of the dot product: sin2C=1−cos2C Using the relationship between the dot product and cosine: cosC=→a⋅→b|→a||→b| Thus, sin2C=1−(→a⋅→b|→a||→b|)2 Step 5: Substitute sinC into Area Formula Substituting sinC into the area formula: Δ=12|→a||→b| ⎷1−(→a⋅→b|→a||→b|)2 Step 6: Simplifying the Area Expression This simplifies to: Δ=12|→a||→b| ⎷|→a|2|→b|2−(→a⋅→b)2|→a|2|→b|2 =12√|→a|2|→b|2−(→a⋅→b)2 Conclusion Thus, we have shown that the area of triangle OAB is given by: Δ=12√|→a|2|→b|2−(→a⋅→b)2 To find the area of triangle OAB determined by the vectors →a and →b, we can use the formula for the area of a triangle formed by two vectors. Here’s the step-by-step solution: Step 1: Area of Triangle Formula The area Δ of triangle OAB formed by vectors →a and →b can be expressed as: Δ=12|→a×→b| where |→a×→b| is the magnitude of the cross product of vectors →a and →b. Step 2: Magnitude of Cross Product The magnitude of the cross product |→a×→b| can be calculated using the formula: |→a×→b|=|→a||→b|sinC where C is the angle between vectors →a and →b. Step 3: Substitute into Area Formula Substituting the expression for the magnitude of the cross product into the area formula gives: Δ=12|→a||→b|sinC Step 4: Expressing sinC We can express sinC in terms of the dot product: sin2C=1−cos2C Using the relationship between the dot product and cosine: cosC=→a⋅→b|→a||→b| Thus, sin2C=1−(→a⋅→b|→a||→b|)2 Step 5: Substitute sinC into Area Formula Substituting sinC into the area formula: Δ=12|→a||→b| ⎷1−(→a⋅→b|→a||→b|)2 Step 6: Simplifying the Area Expression This simplifies to: Δ=12|→a||→b| ⎷|→a|2|→b|2−(→a⋅→b)2|→a|2|→b|2 =12√|→a|2|→b|2−(→a⋅→b)2 Conclusion Thus, we have shown that the area of triangle OAB is given by: Δ=12√|→a|2|→b|2−(→a⋅→b)2 Topper's Solved these Questions Explore 305 Videos Similar Questions If the angel between unit vectors →aand→b600 , then find the value of ∣∣→a−→b∣∣. For any vector →a and →b prove that ∣∣→a+→b∣∣≤|→a|+∣∣→b∣∣. What is the angle between vectors →a and →b with magnitudes 2 and 3 respectively ? Given →a.→b=√3 . Find the angle between two vectors →aand→b with magnitudes √3 and 2 respectively and such that →a.→b=√6. The vertices A,B,C of triangle ABC have respectively position vectors →a,→b,→c with respect to a given origin O . Show that the point D where the bisector of ∠A meets BC has position vector →d=β→b+γ→cβ+γ, where β=|→c−→a| and, γ=∣∣→a−→b∣∣. Find the angle between two vectors →a and →b with magnitudes 1 and 2 respectively and when ∣∣→a×→b∣∣=√3. The vertices A,B,C of triangle ABC have respectively position vectors →a,→b,→c with respect to a given origin O . Show that the point D where the bisector of ∠A meets BC has position vector →d=β→b+γ→cβ+γ, where β=|→c−→a| and, γ=∣∣→a−→b∣∣. Hence, deduce that incentre I has position vector α→a+β→b+γ→cα+β+γ where α=∣∣→b−→c∣∣ For any two vectors →a and →b , show that : (→a+→b).→a−→b=0,when|→a|=∣∣→b∣∣. If →a and →b are unit vectors then write the value of ∣∣→a×→b∣∣2+(→a.→b)2. If →a and →b are unit vectors then write the value of ∣∣→a×→b∣∣2+(→a.→b)2. RD SHARMA ENGLISH-VECTOR OR CROSS PRODUCT -All Questions If vec a , vec b , vec c are three vectors such that | vec a+ vec b+ ... Let vec a=2 hat i+ hat k , vec b= hat i+ hat j+ hat ka n d vec c=4 ha... A triangle O A B is determined by the vectors vec aa n d vec b as sho... If A ,B ,C ,D be any four points in space, prove that | vec A Bxx vec ... If vec a , vec ba n d vec c are three non coplanar vectors, then pr... Let vec a , vec b , vec c be unit vectors such that vec adot vec b= ... Prove by vector method that sin(A-B)=sinAcosB-cosAsinB and sin(A+B)=si... In a triangle A B C , prove by vector method that a/(sinA)=b/(sinB)=c/... Show that ( vec axx vec b)^2=| vec a|^2| vec b|^2-( vec adot vec b)^2=... Given | vec a|=10 ,| vec b|=2 "and" vecadot vec b=12 ,"find"| vec axx ... If A(0,1,1),B(2,3,-2),C(22 ,19 ,-5)a n dD(1,-12 ,1) are the vertices o... Find the area of the parallelogram determined by the vectors hat i+2 ... Find a unit vector perpendicular to the plane A B C , where the coordi... Find a vector of magnitude 9, which is perpendicular to both vectors 4... Find a unit vector perpendicular to both the vectors hat i-2 hat j+3 ... For any vector vec a , prove that | vec axx hat i|^2+| vec axx hat j|... Find the magnitude of vec a give by vec a=( hat i+2 hat j-2 hat k)xx... Find vec axx vec b ,\ if\ vec a=2 hat i+ hat k\ a n d\ vec b= hat i... Find lambda and mu if (2 hat i+6 hat j+27 hat k)xx( hat i+lambda hat j... If vec r=x hat i+y hat j+z hat k ,\ find the value of ( vec rxx hat ... 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