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{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "\n# Training A2C with Vector Envs and Domain Randomization\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##
Notice\n\nIf you encounter an RuntimeError like the following comment raised on multiprocessing/spawn.py, wrap up the code from ``gym.vector.make=`` or ``gym.vector.AsyncVectorEnv`` to the end of the
code by ``if__name__ == '__main__'``.\n\n``An attempt has been made to start a new process before the current process has finished its bootstrapping phase.``\n\n\n" ] }, { "cell_type": "markdown",
"metadata": {}, "source": [ "------------------------------\n\n\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Introduction\n\nIn this tutorial, you'll learn how to use vectorized
environments to train an Advantage Actor-Critic agent.\nWe are going to use A2C, which is the synchronous version of the A3C algorithm [1].\n\nVectorized environments [3] can help to achieve quicker
and more robust training by allowing multiple instances\nof the same environment to run in parallel (on multiple CPUs). This can significantly reduce the variance and thus speeds up the training.\n\
nWe will implement an Advantage Actor-Critic from scratch to look at how you can feed batched states into your networks to get a vector of actions\n(one action per environment) and calculate the
losses for actor and critic on minibatches of transitions.\nEach minibatch contains the transitions of one sampling phase: `n_steps_per_update` steps are executed in `n_envs` environments in parallel
\n(multiply the two to get the number of transitions in a minibatch). After each sampling phase, the losses are calculated and one gradient step is executed.\nTo calculate the advantages, we are
going to use the Generalized Advantage Estimation (GAE) method [2], which balances the tradeoff\nbetween variance and bias of the advantage estimates.\n\nThe A2C agent class is initialized with the
number of features of the input state, the number of actions the agent can take,\nthe learning rates and the number of environments that run in parallel to collect experiences. The actor and critic
networks are defined\nand their respective optimizers are initialized. The forward pass of the networks takes in a batched vector of states and returns a tensor of state values\nand a tensor of
action logits. The select_action method returns a tuple of the chosen actions, the log-probs of those actions, and the state values for each action.\nIn addition, it also returns the entropy of the
policy distribution, which is subtracted from the loss later (with a weighting factor `ent_coef`) to encourage exploration.\n\nThe get_losses function calculates the losses for the actor and critic
networks (using GAE), which are then updated using the update_parameters function.\n\n\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "------------------------------\n\n\n" ] }, {
"cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# Author: Till Zemann\n# License: MIT License\n\nfrom __future__ import annotations\n\
nimport os\n\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport torch\nimport torch.nn as nn\nfrom torch import optim\nfrom tqdm import tqdm\n\nimport gymnasium as gym" ] }, { "cell_type":
"markdown", "metadata": {}, "source": [ "## Advantage Actor-Critic (A2C)\n\nThe Actor-Critic combines elements of value-based and policy-based methods. In A2C, the agent has two separate neural
networks:\na critic network that estimates the state-value function, and an actor network that outputs logits for a categorical probability distribution over all actions.\nThe critic network is
trained to minimize the mean squared error between the predicted state values and the actual returns received by the agent\n(this is equivalent to minimizing the squared advantages, because the
advantage of an action is as the difference between the return and the state-value: A(s,a) = Q(s,a) - V(s).\nThe actor network is trained to maximize the expected return by selecting actions that
have high expected values according to the critic network.\n\nThe focus of this tutorial will not be on the details of A2C itself. Instead, the tutorial will focus on how to use vectorized
environments\nand domain randomization to accelerate the training process for A2C (and other reinforcement learning algorithms).\n\n\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [
"------------------------------\n\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "class A2C(nn.Module):\n \"\"\"\n
(Synchronous) Advantage Actor-Critic agent class\n\n Args:\n n_features: The number of features of the input state.\n n_actions: The number of actions the agent can take.\n device: The device to run
the computations on (running on a GPU might be quicker for larger Neural Nets,\n for this code CPU is totally fine).\n critic_lr: The learning rate for the critic network (should usually be larger
than the actor_lr).\n actor_lr: The learning rate for the actor network.\n n_envs: The number of environments that run in parallel (on multiple CPUs) to collect experiences.\n \"\"\"\n\n def __init__
(\n self,\n n_features: int,\n n_actions: int,\n device: torch.device,\n critic_lr: float,\n actor_lr: float,\n n_envs: int,\n ) -> None:\n \"\"\"Initializes the actor and critic networks and their
respective optimizers.\"\"\"\n super().__init__()\n self.device = device\n self.n_envs = n_envs\n\n critic_layers = [\n nn.Linear(n_features, 32),\n nn.ReLU(),\n nn.Linear(32, 32),\n nn.ReLU(),\n
nn.Linear(32, 1), # estimate V(s)\n ]\n\n actor_layers = [\n nn.Linear(n_features, 32),\n nn.ReLU(),\n nn.Linear(32, 32),\n nn.ReLU(),\n nn.Linear(\n 32, n_actions\n ), # estimate action logits (will
be fed into a softmax later)\n ]\n\n # define actor and critic networks\n self.critic = nn.Sequential(*critic_layers).to(self.device)\n self.actor = nn.Sequential(*actor_layers).to(self.device)\n\n #
define optimizers for actor and critic\n self.critic_optim = optim.RMSprop(self.critic.parameters(), lr=critic_lr)\n self.actor_optim = optim.RMSprop(self.actor.parameters(), lr=actor_lr)\n\n def
forward(self, x: np.ndarray) -> tuple[torch.Tensor, torch.Tensor]:\n \"\"\"\n Forward pass of the networks.\n\n Args:\n x: A batched vector of states.\n\n Returns:\n state_values: A tensor with the
state values, with shape [n_envs,].\n action_logits_vec: A tensor with the action logits, with shape [n_envs, n_actions].\n \"\"\"\n x = torch.Tensor(x).to(self.device)\n state_values = self.critic
(x) # shape: [n_envs,]\n action_logits_vec = self.actor(x) # shape: [n_envs, n_actions]\n return (state_values, action_logits_vec)\n\n def select_action(\n self, x: np.ndarray\n ) -> tuple
[torch.Tensor, torch.Tensor, torch.Tensor, torch.Tensor]:\n \"\"\"\n Returns a tuple of the chosen actions and the log-probs of those actions.\n\n Args:\n x: A batched vector of states.\n\n Returns:\
n actions: A tensor with the actions, with shape [n_steps_per_update, n_envs].\n action_log_probs: A tensor with the log-probs of the actions, with shape [n_steps_per_update, n_envs].\n state_values:
A tensor with the state values, with shape [n_steps_per_update, n_envs].\n \"\"\"\n state_values, action_logits = self.forward(x)\n action_pd = torch.distributions.Categorical(\n logits=action_logits
\n ) # implicitly uses softmax\n actions = action_pd.sample()\n action_log_probs = action_pd.log_prob(actions)\n entropy = action_pd.entropy()\n return (actions, action_log_probs, state_values,
entropy)\n\n def get_losses(\n self,\n rewards: torch.Tensor,\n action_log_probs: torch.Tensor,\n value_preds: torch.Tensor,\n entropy: torch.Tensor,\n masks: torch.Tensor,\n gamma: float,\n lam:
float,\n ent_coef: float,\n device: torch.device,\n ) -> tuple[torch.Tensor, torch.Tensor]:\n \"\"\"\n Computes the loss of a minibatch (transitions collected in one sampling phase) for actor and
critic\n using Generalized Advantage Estimation (GAE) to compute the advantages (https://arxiv.org/abs/1506.02438).\n\n Args:\n rewards: A tensor with the rewards for each time step in the episode,
with shape [n_steps_per_update, n_envs].\n action_log_probs: A tensor with the log-probs of the actions taken at each time step in the episode, with shape [n_steps_per_update, n_envs].\n value_preds:
A tensor with the state value predictions for each time step in the episode, with shape [n_steps_per_update, n_envs].\n masks: A tensor with the masks for each time step in the episode, with shape
[n_steps_per_update, n_envs].\n gamma: The discount factor.\n lam: The GAE hyperparameter. (lam=1 corresponds to Monte-Carlo sampling with high variance and no bias,\n and lam=0 corresponds to normal
TD-Learning that has a low variance but is biased\n because the estimates are generated by a Neural Net).\n device: The device to run the computations on (e.g. CPU or GPU).\n\n Returns:\n
critic_loss: The critic loss for the minibatch.\n actor_loss: The actor loss for the minibatch.\n \"\"\"\n T = len(rewards)\n advantages = torch.zeros(T, self.n_envs, device=device)\n\n # compute the
advantages using GAE\n gae = 0.0\n for t in reversed(range(T - 1)):\n td_error = (\n rewards[t] + gamma * masks[t] * value_preds[t + 1] - value_preds[t]\n )\n gae = td_error + gamma * lam * masks[t]
* gae\n advantages[t] = gae\n\n # calculate the loss of the minibatch for actor and critic\n critic_loss = advantages.pow(2).mean()\n\n # give a bonus for higher entropy to encourage exploration\n
actor_loss = (\n -(advantages.detach() * action_log_probs).mean() - ent_coef * entropy.mean()\n )\n return (critic_loss, actor_loss)\n\n def update_parameters(\n self, critic_loss: torch.Tensor,
actor_loss: torch.Tensor\n ) -> None:\n \"\"\"\n Updates the parameters of the actor and critic networks.\n\n Args:\n critic_loss: The critic loss.\n actor_loss: The actor loss.\n \"\"\"\n
self.critic_optim.zero_grad()\n critic_loss.backward()\n self.critic_optim.step()\n\n self.actor_optim.zero_grad()\n actor_loss.backward()\n self.actor_optim.step()" ] }, { "cell_type": "markdown",
"metadata": {}, "source": [ "## Using Vectorized Environments\n\nWhen you calculate the losses for the two Neural Networks over only one epoch, it might have a high variance. With vectorized
environments,\nwe can play with `n_envs` in parallel and thus get up to a linear speedup (meaning that in theory, we collect samples `n_envs` times quicker)\nthat we can use to calculate the loss for
the current policy and critic network. When we are using more samples to calculate the loss,\nit will have a lower variance and theirfore leads to quicker learning.\n\nA2C is a synchronous method,
meaning that the parameter updates to Networks take place deterministically (after each sampling phase),\nbut we can still make use of asynchronous vector envs to spawn multiple processes for
parallel environment execution.\n\nThe simplest way to create vector environments is by calling `gym.vector.make`, which creates multiple instances of the same environment:\n\n\n" ] }, { "cell_type":
"code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "envs = gym.vector.make(\"LunarLander-v3\", num_envs=3, max_episode_steps=600)" ] }, { "cell_type":
"markdown", "metadata": {}, "source": [ "## Domain Randomization\n\nIf we want to randomize the environment for training to get more robust agents (that can deal with different parameterizations of
an environment\nand theirfore might have a higher degree of generalization), we can set the desired parameters manually or use a pseudo-random number generator to generate them.\n\nManually setting
up 3 parallel 'LunarLander-v3' envs with different parameters:\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "envs =
gym.vector.AsyncVectorEnv(\n [\n lambda: gym.make(\n \"LunarLander-v3\",\n gravity=-10.0,\n enable_wind=True,\n wind_power=15.0,\n turbulence_power=1.5,\n max_episode_steps=600,\n ),\n lambda:
gym.make(\n \"LunarLander-v3\",\n gravity=-9.8,\n enable_wind=True,\n wind_power=10.0,\n turbulence_power=1.3,\n max_episode_steps=600,\n ),\n lambda: gym.make(\n \"LunarLander-v3\", gravity=-7.0,
enable_wind=False, max_episode_steps=600\n ),\n ]\n)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "------------------------------\n\nRandomly generating the parameters for 3 parallel
'LunarLander-v3' envs, using `np.clip` to stay in the recommended parameter space:\n\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [],
"source": [ "envs = gym.vector.AsyncVectorEnv(\n [\n lambda: gym.make(\n \"LunarLander-v3\",\n gravity=np.clip(\n np.random.normal(loc=-10.0, scale=1.0), a_min=-11.99, a_max=-0.01\n ),\n enable_wind=
np.random.choice([True, False]),\n wind_power=np.clip(\n np.random.normal(loc=15.0, scale=1.0), a_min=0.01, a_max=19.99\n ),\n turbulence_power=np.clip(\n np.random.normal(loc=1.5, scale=0.5), a_min=
0.01, a_max=1.99\n ),\n max_episode_steps=600,\n )\n for i in range(3)\n ]\n)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "------------------------------\n\nHere we are using normal
distributions with the standard parameterization of the environment as the mean and an arbitrary standard deviation (scale).\nDepending on the problem, you can experiment with higher variance and use
different distributions as well.\n\nIf you are training on the same `n_envs` environments for the entire training time, and `n_envs` is a relatively low number\n(in proportion to how complex the
environment is), you might still get some overfitting to the specific parameterizations that you picked.\nTo mitigate this, you can either pick a high number of randomly parameterized environments or
remake your environments every couple of sampling phases\nto generate a new set of pseudo-random parameters.\n\n\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Setup\n\n\n" ] }, {
"cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# environment hyperparams\nn_envs = 10\nn_updates = 1000\nn_steps_per_update = 128\
nrandomize_domain = False\n\n# agent hyperparams\ngamma = 0.999\nlam = 0.95 # hyperparameter for GAE\nent_coef = 0.01 # coefficient for the entropy bonus (to encourage exploration)\nactor_lr = 0.001\
ncritic_lr = 0.005\n\n# Note: the actor has a slower learning rate so that the value targets become\n# more stationary and are theirfore easier to estimate for the critic\n\n# environment setup\nif
randomize_domain:\n envs = gym.vector.AsyncVectorEnv(\n [\n lambda: gym.make(\n \"LunarLander-v3\",\n gravity=np.clip(\n np.random.normal(loc=-10.0, scale=1.0), a_min=-11.99, a_max=-0.01\n ),\n
enable_wind=np.random.choice([True, False]),\n wind_power=np.clip(\n np.random.normal(loc=15.0, scale=1.0), a_min=0.01, a_max=19.99\n ),\n turbulence_power=np.clip(\n np.random.normal(loc=1.5, scale=
0.5), a_min=0.01, a_max=1.99\n ),\n max_episode_steps=600,\n )\n for i in range(n_envs)\n ]\n )\n\nelse:\n envs = gym.vector.make(\"LunarLander-v3\", num_envs=n_envs, max_episode_steps=600)\n\n\
nobs_shape = envs.single_observation_space.shape[0]\naction_shape = envs.single_action_space.n\n\n# set the device\nuse_cuda = False\nif use_cuda:\n device = torch.device(\"cuda\" if
torch.cuda.is_available() else \"cpu\")\nelse:\n device = torch.device(\"cpu\")\n\n# init the agent\nagent = A2C(obs_shape, action_shape, device, critic_lr, actor_lr, n_envs)" ] }, { "cell_type":
"markdown", "metadata": {}, "source": [ "## Training the A2C Agent\n\nFor our training loop, we are using the `RecordEpisodeStatistics` wrapper to record the episode lengths and returns and we are
also saving\nthe losses and entropies to plot them after the agent finished training.\n\nYou may notice that we don't reset the vectorized envs at the start of each episode like we would usually do.\
nThis is because each environment resets automatically once the episode finishes (each environment takes a different number of timesteps to finish\nan episode because of the random seeds). As a
result, we are also not collecting data in `episodes`, but rather just play a certain number of steps\n(`n_steps_per_update`) in each environment (as an example, this could mean that we play 20
timesteps to finish an episode and then\nuse the rest of the timesteps to begin a new one).\n\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs":
[], "source": [ "# create a wrapper environment to save episode returns and episode lengths\nenvs_wrapper = gym.wrappers.RecordEpisodeStatistics(envs, deque_size=n_envs * n_updates)\n\ncritic_losses
= []\nactor_losses = []\nentropies = []\n\n# use tqdm to get a progress bar for training\nfor sample_phase in tqdm(range(n_updates)):\n # we don't have to reset the envs, they just continue playing\n
# until the episode is over and then reset automatically\n\n # reset lists that collect experiences of an episode (sample phase)\n ep_value_preds = torch.zeros(n_steps_per_update, n_envs, device=
device)\n ep_rewards = torch.zeros(n_steps_per_update, n_envs, device=device)\n ep_action_log_probs = torch.zeros(n_steps_per_update, n_envs, device=device)\n masks = torch.zeros(n_steps_per_update,
n_envs, device=device)\n\n # at the start of training reset all envs to get an initial state\n if sample_phase == 0:\n states, info = envs_wrapper.reset(seed=42)\n\n # play n steps in our parallel
environments to collect data\n for step in range(n_steps_per_update):\n # select an action A_{t} using S_{t} as input for the agent\n actions, action_log_probs, state_value_preds, entropy =
agent.select_action(\n states\n )\n\n # perform the action A_{t} in the environment to get S_{t+1} and R_{t+1}\n states, rewards, terminated, truncated, infos = envs_wrapper.step(\n actions.cpu
().numpy()\n )\n\n ep_value_preds[step] = torch.squeeze(state_value_preds)\n ep_rewards[step] = torch.tensor(rewards, device=device)\n ep_action_log_probs[step] = action_log_probs\n\n # add a mask
(for the return calculation later);\n # for each env the mask is 1 if the episode is ongoing and 0 if it is terminated (not by truncation!)\n masks[step] = torch.tensor([not term for term in
terminated])\n\n # calculate the losses for actor and critic\n critic_loss, actor_loss = agent.get_losses(\n ep_rewards,\n ep_action_log_probs,\n ep_value_preds,\n entropy,\n masks,\n gamma,\n lam,\n
ent_coef,\n device,\n )\n\n # update the actor and critic networks\n agent.update_parameters(critic_loss, actor_loss)\n\n # log the losses and entropy\n critic_losses.append(critic_loss.detach().cpu
().numpy())\n actor_losses.append(actor_loss.detach().cpu().numpy())\n entropies.append(entropy.detach().mean().cpu().numpy())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##
Plotting\n\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "\"\"\" plot the results \"\"\"\n\n# %matplotlib inline\n\
nrolling_length = 20\nfig, axs = plt.subplots(nrows=2, ncols=2, figsize=(12, 5))\nfig.suptitle(\n f\"Training plots for {agent.__class__.__name__} in the LunarLander-v3 environment \\n \\\n (n_envs=
{n_envs}, n_steps_per_update={n_steps_per_update}, randomize_domain={randomize_domain})\"\n)\n\n# episode return\naxs[0][0].set_title(\"Episode Returns\")\nepisode_returns_moving_average = (\n
np.convolve(\n np.array(envs_wrapper.return_queue).flatten(),\n np.ones(rolling_length),\n mode=\"valid\",\n )\n / rolling_length\n)\naxs[0][0].plot(\n np.arange(len(episode_returns_moving_average))
/ n_envs,\n episode_returns_moving_average,\n)\naxs[0][0].set_xlabel(\"Number of episodes\")\n\n# entropy\naxs[1][0].set_title(\"Entropy\")\nentropy_moving_average = (\n np.convolve(np.array
(entropies), np.ones(rolling_length), mode=\"valid\")\n / rolling_length\n)\naxs[1][0].plot(entropy_moving_average)\naxs[1][0].set_xlabel(\"Number of updates\")\n\n\n# critic loss\naxs[0]
[1].set_title(\"Critic Loss\")\ncritic_losses_moving_average = (\n np.convolve(\n np.array(critic_losses).flatten(), np.ones(rolling_length), mode=\"valid\"\n )\n / rolling_length\n)\naxs[0][1].plot
(critic_losses_moving_average)\naxs[0][1].set_xlabel(\"Number of updates\")\n\n\n# actor loss\naxs[1][1].set_title(\"Actor Loss\")\nactor_losses_moving_average = (\n np.convolve(np.array
(actor_losses).flatten(), np.ones(rolling_length), mode=\"valid\")\n / rolling_length\n)\naxs[1][1].plot(actor_losses_moving_average)\naxs[1][1].set_xlabel(\"Number of updates\")\n\nplt.tight_layout
()\nplt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ " | {"url":"https://gymnasium.farama.org/_downloads/50e7c09c20b787d0a5bd70c4aeb0a515/vector_envs_tutorial.ipynb","timestamp":"2024-11-08T02:38:36Z","content_type":"text/plain","content_length":"32241","record_id":"<urn:uuid:d855ac17-ceb9-4090-bb22-d473c5d5e76d>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00432.warc.gz"} |
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Margue Series Model Engine Gearbox Region Steering Qty
Margue Series Model Engine Gearbox Region Steering Qty
BMW 1' E81 E81 116d 3 doors N47 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E81 E81 116d 3 doors N47 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E81 E81 116i 1.6 N43 3 doors N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E81 E81 116i 1.6 N43 3 doors N43 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E81 E81 116i 1.6 N45N 3 doors N45N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E81 E81 116i 1.6 N45N 3 doors N45N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E81 E81 116i 2.0 3 doors N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E81 E81 116i 2.0 3 doors N43 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E81 E81 118d 3 doors N47 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E81 E81 118d 3 doors N47 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E81 E81 118i N43 3 doors N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E81 E81 118i N43 3 doors N43 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E81 E81 118i N46N 3 doors N46N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E81 E81 118i N46N 3 doors N46N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E81 E81 120d 3 doors N47 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E81 E81 120d 3 doors N47 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E81 E81 120i N43 3 doors N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E81 E81 120i N43 3 doors N43 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E81 E81 120i N46N 3 doors N46N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E81 E81 120i N46N 3 doors N46N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E81 E81 123d 3 doors N47S Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E81 E81 123d 3 doors N47S Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E81 E81 130i 3 doors N52N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E81 E81 130i 3 doors N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E82 E82 118d Coupé N47 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E82 E82 118d Coupé N47 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E82 E82 120d Coupé N47 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E82 E82 120d Coupé N47 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E82 E82 120i N43 Coupé N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E82 E82 120i N43 Coupé N43 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E82 E82 120i N46N Coupé N46N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E82 E82 120i N46N Coupé N46N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E82 E82 123d Coupé N47S Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E82 E82 123d Coupé N47S Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E82 E82 125i Coupé N52N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E82 E82 125i Coupé N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E82 E82 128i N51 Coupé N51 Neutral (N) USA (USA) Left steering (L) 2
BMW 1' E82 E82 128i N52N Coupé N52N Neutral (N) USA (USA) Left steering (L) 2
BMW 1' E82 E82 135i N54 Coupé N54 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E82 E82 135i N54 Coupé N54 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E82 E82 135i N54 Coupé N54 Neutral (N) USA (USA) Left steering (L) 2
BMW 1' E82 E82 135i N55 Coupé N55 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E82 E82 135i N55 Coupé N55 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E82 E82 135i N55 Coupé N55 Neutral (N) USA (USA) Left steering (L) 2
BMW 1' E87 E87 116i 5 doors N45 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E87 E87 116i 5 doors N45 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E87 E87 118d 5 doors M47N2 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E87 E87 118d 5 doors M47N2 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E87 E87 118i 5 doors N46 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E87 E87 118i 5 doors N46 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E87 E87 120d 5 doors M47N2 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E87 E87 120d 5 doors M47N2 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E87 E87 120i 5 doors N46 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E87 E87 120i 5 doors N46 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E87 E87 130i 5 doors N52 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E87 E87 130i 5 doors N52 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E87 LCI E87N 116d 5 doors N47 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E87 LCI E87N 116d 5 doors N47 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E87 LCI E87N 116i 1.6 N43 5 doors N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E87 LCI E87N 116i 1.6 N43 5 doors N43 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E87 LCI E87N 116i 1.6 N45N 5 doors N45N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E87 LCI E87N 116i 1.6 N45N 5 doors N45N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E87 LCI E87N 116i 2.0 5 doors N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E87 LCI E87N 116i 2.0 5 doors N43 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E87 LCI E87N 118d 5 doors N47 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E87 LCI E87N 118d 5 doors N47 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E87 LCI E87N 118i N43 5 doors N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E87 LCI E87N 118i N43 5 doors N43 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E87 LCI E87N 118i N46N 5 doors N46N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E87 LCI E87N 118i N46N 5 doors N46N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E87 LCI E87N 120d 5 doors N47 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E87 LCI E87N 120d 5 doors N47 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E87 LCI E87N 120i N43 5 doors N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E87 LCI E87N 120i N43 5 doors N43 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E87 LCI E87N 120i N46N 5 doors N46N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E87 LCI E87N 120i N46N 5 doors N46N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E87 LCI E87N 123d 5 doors N47S Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E87 LCI E87N 123d 5 doors N47S Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E87 LCI E87N 130i 5 doors N52N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E87 LCI E87N 130i 5 doors N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E88 E88 118d Convertible N47 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E88 E88 118d Convertible N47 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E88 E88 118i N43 Convertible N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E88 E88 118i N43 Convertible N43 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E88 E88 118i N46N Convertible N46N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E88 E88 118i N46N Convertible N46N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E88 E88 120d Convertible N47 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E88 E88 120d Convertible N47 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E88 E88 120i N43 Convertible N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E88 E88 120i N43 Convertible N43 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E88 E88 120i N46N Convertible N46N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E88 E88 120i N46N Convertible N46N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E88 E88 123d Convertible N47S Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E88 E88 123d Convertible N47S Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E88 E88 125i Convertible N52N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E88 E88 125i Convertible N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E88 E88 128i N51 Convertible N51 Neutral (N) USA (USA) Left steering (L) 2
BMW 1' E88 E88 128i N52N Convertible N52N Neutral (N) USA (USA) Left steering (L) 2
BMW 1' E88 E88 135i N54 Convertible N54 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E88 E88 135i N54 Convertible N54 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E88 E88 135i N54 Convertible N54 Neutral (N) USA (USA) Left steering (L) 2
BMW 1' E88 E88 135i N55 Convertible N55 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 1' E88 E88 135i N55 Convertible N55 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 1' E88 E88 135i N55 Convertible N55 Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E90 E90 316i N43 Saloon N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 E90 316i N43 Saloon N43 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 E90 316i N45N Saloon N45N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 E90 316i N45N Saloon N45N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 E90 316i N45 Saloon N45 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 E90 316i N45 Saloon N45 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 E90 318d M47N2 Saloon M47N2 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 E90 318d M47N2 Saloon M47N2 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 E90 318d N47 Saloon N47 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 E90 318d N47 Saloon N47 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 E90 318i N43 Saloon N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 E90 318i N43 Saloon N43 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 E90 318i N46N Saloon N46N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 E90 318i N46N Saloon N46N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 E90 318i N46N Saloon N46N Neutral (N) Thailand (THA) Right steering (R) 2
BMW 3' E90 E90 318i N46 Saloon N46 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 E90 318i N46 Saloon N46 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 E90 318i N46 Saloon N46 Neutral (N) Thailand (THA) Right steering (R) 2
BMW 3' E90 E90 318i Saloon N46N Neutral (N) Russia (RUS) Left steering (L) 2
BMW 3' E90 E90 320d M47N2 Saloon M47N2 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 E90 320d M47N2 Saloon M47N2 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 E90 320d M47N2 Saloon M47N2 Neutral (N) India (IND) Right steering (R) 2
BMW 3' E90 E90 320d N47 Saloon N47 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 E90 320d N47 Saloon N47 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 E90 320d N47 Saloon N47 Neutral (N) India (IND) Right steering (R) 2
BMW 3' E90 E90 320d Saloon N47 Neutral (N) Thailand (THA) Right steering (R) 2
BMW 3' E90 E90 320i N43 Saloon N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 E90 320i N43 Saloon N43 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 E90 320i N46N Saloon N46N Neutral (N) China (CHN) Left steering (L) 2
BMW 3' E90 E90 320i N46N Saloon N46N Neutral (N) Egypt (EGY) Left steering (L) 2
BMW 3' E90 E90 320i N46N Saloon N46N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 E90 320i N46N Saloon N46N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 E90 320i N46N Saloon N46N Neutral (N) India (IND) Right steering (R) 2
BMW 3' E90 E90 320i N46N Saloon N46N Neutral (N) Indonesia (IDN) Right steering (R) 2
BMW 3' E90 E90 320i N46N Saloon N46N Neutral (N) Malaysia (MYS) Right steering (R) 2
BMW 3' E90 E90 320i N46N Saloon N46N Neutral (N) Russia (RUS) Left steering (L) 2
BMW 3' E90 E90 320i N46N Saloon N46N Neutral (N) Thailand (THA) Right steering (R) 2
BMW 3' E90 E90 320i N46 Saloon N46 Neutral (N) China (CHN) Left steering (L) 2
BMW 3' E90 E90 320i N46 Saloon N46 Neutral (N) Egypt (EGY) Left steering (L) 2
BMW 3' E90 E90 320i N46 Saloon N46 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 E90 320i N46 Saloon N46 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 E90 320i N46 Saloon N46 Neutral (N) India (IND) Right steering (R) 2
BMW 3' E90 E90 320i N46 Saloon N46 Neutral (N) Indonesia (IDN) Right steering (R) 2
BMW 3' E90 E90 320i N46 Saloon N46 Neutral (N) Malaysia (MYS) Right steering (R) 2
BMW 3' E90 E90 320i N46 Saloon N46 Neutral (N) Russia (RUS) Left steering (L) 2
BMW 3' E90 E90 320i N46 Saloon N46 Neutral (N) Thailand (THA) Right steering (R) 2
BMW 3' E90 E90 320si Saloon N45 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 E90 320si Saloon N45 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 E90 323i N52N Saloon N52N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 E90 323i N52N Saloon N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 E90 323i N52N Saloon N52N Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E90 E90 323i N52 Saloon N52 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 E90 323i N52 Saloon N52 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 E90 323i N52 Saloon N52 Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E90 E90 325d Saloon M57N2 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 E90 325d Saloon M57N2 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 E90 325i N52N Saloon N52N Neutral (N) China (CHN) Left steering (L) 2
BMW 3' E90 E90 325i N52N Saloon N52N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 E90 325i N52N Saloon N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 E90 325i N52N Saloon N52N Neutral (N) India (IND) Right steering (R) 2
BMW 3' E90 E90 325i N52N Saloon N52N Neutral (N) Malaysia (MYS) Right steering (R) 2
BMW 3' E90 E90 325i N52N Saloon N52N Neutral (N) Russia (RUS) Left steering (L) 2
BMW 3' E90 E90 325i N52 Saloon N52 Neutral (N) China (CHN) Left steering (L) 2
BMW 3' E90 E90 325i N52 Saloon N52 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 E90 325i N52 Saloon N52 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 E90 325i N52 Saloon N52 Neutral (N) India (IND) Right steering (R) 2
BMW 3' E90 E90 325i N52 Saloon N52 Neutral (N) Malaysia (MYS) Right steering (R) 2
BMW 3' E90 E90 325i N52 Saloon N52 Neutral (N) Russia (RUS) Left steering (L) 2
BMW 3' E90 E90 325i N53 Saloon N53 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 E90 325i N53 Saloon N53 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 E90 325i Saloon N52 Neutral (N) Indonesia (IDN) Right steering (R) 2
BMW 3' E90 E90 325i Saloon N52 Neutral (N) Thailand (THA) Right steering (R) 2
BMW 3' E90 E90 325i Saloon N52 Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E90 E90 328i N51 Saloon N51 Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E90 E90 328i N52N Saloon N52N Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E90 E90 328i Saloon N51 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 E90 330d Saloon M57N2 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 E90 330d Saloon M57N2 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 E90 330i N52N Saloon N52N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 E90 330i N52N Saloon N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 E90 330i N52 Saloon N52 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 E90 330i N52 Saloon N52 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 E90 330i N53 Saloon N53 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 E90 330i N53 Saloon N53 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 E90 330i Saloon N52 Neutral (N) Thailand (THA) Right steering (R) 2
BMW 3' E90 E90 330i Saloon N52 Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E90 E90 335d Saloon M57N2 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 E90 335d Saloon M57N2 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 E90 335i Saloon N54 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 E90 335i Saloon N54 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 E90 335i Saloon N54 Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E90 LCI E90N 316d N47N Saloon N47N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 LCI E90N 316d N47N Saloon N47N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 LCI E90N 316d N47 Saloon N47 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 LCI E90N 316d N47 Saloon N47 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 LCI E90N 316i N43 Saloon N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 LCI E90N 316i N43 Saloon N43 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 LCI E90N 316i N45N Saloon N45N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 LCI E90N 316i N45N Saloon N45N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 LCI E90N 318d N47N Saloon N47N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 LCI E90N 318d N47N Saloon N47N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 LCI E90N 318d N47 Saloon N47 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 LCI E90N 318d N47 Saloon N47 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 LCI E90N 318i N43 Saloon N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 LCI E90N 318i N43 Saloon N43 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 LCI E90N 318i N46N Saloon N46N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 LCI E90N 318i N46N Saloon N46N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 LCI E90N 318i Saloon N46N Neutral (N) China (CHN) Left steering (L) 2
BMW 3' E90 LCI E90N 318i Saloon N46N Neutral (N) Egypt (EGY) Left steering (L) 2
BMW 3' E90 LCI E90N 318i Saloon N46N Neutral (N) Russia (RUS) Left steering (L) 2
BMW 3' E90 LCI E90N 318i Saloon N46N Neutral (N) Thailand (THA) Right steering (R) 2
BMW 3' E90 LCI E90N 320d ed Saloon N47N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 LCI E90N 320d ed Saloon N47N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 LCI E90N 320d N47N Saloon N47N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 LCI E90N 320d N47N Saloon N47N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 LCI E90N 320d N47N Saloon N47N Neutral (N) India (IND) Right steering (R) 2
BMW 3' E90 LCI E90N 320d N47N Saloon N47N Neutral (N) Malaysia (MYS) Right steering (R) 2
BMW 3' E90 LCI E90N 320d N47N Saloon N47N Neutral (N) Thailand (THA) Right steering (R) 2
BMW 3' E90 LCI E90N 320d N47 Saloon N47 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 LCI E90N 320d N47 Saloon N47 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 LCI E90N 320d N47 Saloon N47 Neutral (N) India (IND) Right steering (R) 2
BMW 3' E90 LCI E90N 320d N47 Saloon N47 Neutral (N) Malaysia (MYS) Right steering (R) 2
BMW 3' E90 LCI E90N 320d N47 Saloon N47 Neutral (N) Thailand (THA) Right steering (R) 2
BMW 3' E90 LCI E90N 320i N43 Saloon N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 LCI E90N 320i N43 Saloon N43 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 LCI E90N 320i N46N Saloon N46N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 LCI E90N 320i N46N Saloon N46N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 LCI E90N 320i Saloon N46N Neutral (N) China (CHN) Left steering (L) 2
BMW 3' E90 LCI E90N 320i Saloon N46N Neutral (N) Egypt (EGY) Left steering (L) 2
BMW 3' E90 LCI E90N 320i Saloon N46N Neutral (N) India (IND) Right steering (R) 2
BMW 3' E90 LCI E90N 320i Saloon N46N Neutral (N) Indonesia (IDN) Right steering (R) 2
BMW 3' E90 LCI E90N 320i Saloon N46N Neutral (N) Malaysia (MYS) Right steering (R) 2
BMW 3' E90 LCI E90N 320i Saloon N46N Neutral (N) Russia (RUS) Left steering (L) 2
BMW 3' E90 LCI E90N 320i Saloon N46N Neutral (N) Thailand (THA) Right steering (R) 2
BMW 3' E90 LCI E90N 323i Saloon N52N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 LCI E90N 323i Saloon N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 LCI E90N 323i Saloon N52N Neutral (N) Malaysia (MYS) Right steering (R) 2
BMW 3' E90 LCI E90N 323i Saloon N52N Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E90 LCI E90N 325d M57N2 Saloon M57N2 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 LCI E90N 325d M57N2 Saloon M57N2 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 LCI E90N 325d N57 Saloon N57 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 LCI E90N 325d N57 Saloon N57 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 LCI E90N 325i N52N Saloon N52N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 LCI E90N 325i N52N Saloon N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 LCI E90N 325i N53 Saloon N53 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 LCI E90N 325i N53 Saloon N53 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 LCI E90N 325i Saloon N52N Neutral (N) China (CHN) Left steering (L) 2
BMW 3' E90 LCI E90N 325i Saloon N52N Neutral (N) India (IND) Right steering (R) 2
BMW 3' E90 LCI E90N 325i Saloon N52N Neutral (N) Indonesia (IDN) Right steering (R) 2
BMW 3' E90 LCI E90N 325i Saloon N52N Neutral (N) Malaysia (MYS) Right steering (R) 2
BMW 3' E90 LCI E90N 325i Saloon N52N Neutral (N) Russia (RUS) Left steering (L) 2
BMW 3' E90 LCI E90N 325i Saloon N52N Neutral (N) Thailand (THA) Right steering (R) 2
BMW 3' E90 LCI E90N 328i N51 Saloon N51 Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E90 LCI E90N 328i N52N Saloon N52N Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E90 LCI E90N 328i Saloon N51 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 LCI E90N 330d Saloon N57 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 LCI E90N 330d Saloon N57 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 LCI E90N 330i N52N Saloon N52N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 LCI E90N 330i N52N Saloon N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 LCI E90N 330i N53 Saloon N53 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 LCI E90N 330i N53 Saloon N53 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 LCI E90N 330i Saloon N52N Neutral (N) Egypt (EGY) Left steering (L) 2
BMW 3' E90 LCI E90N 330i Saloon N52N Neutral (N) India (IND) Right steering (R) 2
BMW 3' E90 LCI E90N 335d Saloon M57N2 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 LCI E90N 335d Saloon M57N2 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 LCI E90N 335d Saloon M57N2 Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E90 LCI E90N 335i N54 Saloon N54 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 LCI E90N 335i N54 Saloon N54 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 LCI E90N 335i N54 Saloon N54 Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E90 LCI E90N 335i N55 Saloon N55 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E90 LCI E90N 335i N55 Saloon N55 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E90 LCI E90N 335i N55 Saloon N55 Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E91 E91 318d M47N2 Touring M47N2 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 E91 318d M47N2 Touring M47N2 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 E91 318d N47 Touring N47 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 E91 318d N47 Touring N47 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 E91 318i N43 Touring N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 E91 318i N43 Touring N43 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 E91 318i N46N Touring N46N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 E91 318i N46N Touring N46N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 E91 318i N46 Touring N46 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 E91 318i N46 Touring N46 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 E91 320d M47N2 Touring M47N2 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 E91 320d M47N2 Touring M47N2 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 E91 320d N47 Touring N47 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 E91 320d N47 Touring N47 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 E91 320i N43 Touring N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 E91 320i N43 Touring N43 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 E91 320i N46N Touring N46N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 E91 320i N46N Touring N46N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 E91 320i N46 Touring N46 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 E91 320i N46 Touring N46 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 E91 323i N52N Touring N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 E91 323i N52 Touring N52 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 E91 323i N52 Touring N52 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 E91 325d Touring M57N2 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 E91 325d Touring M57N2 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 E91 325i N52N Touring N52N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 E91 325i N52N Touring N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 E91 325i N52 Touring N52 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 E91 325i N52 Touring N52 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 E91 325i N53 Touring N53 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 E91 325i N53 Touring N53 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 E91 328i Touring N52N Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E91 E91 330d Touring M57N2 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 E91 330d Touring M57N2 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 E91 330i N52N Touring N52N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 E91 330i N52N Touring N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 E91 330i N52 Touring N52 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 E91 330i N52 Touring N52 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 E91 330i N53 Touring N53 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 E91 330i N53 Touring N53 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 E91 335d Touring M57N2 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 E91 335d Touring M57N2 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 E91 335i Touring N54 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 E91 335i Touring N54 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 LCI E91N 316d Touring N47N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 LCI E91N 316d Touring N47N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 LCI E91N 316i Touring N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 LCI E91N 318d N47N Touring N47N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 LCI E91N 318d N47N Touring N47N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 LCI E91N 318d N47 Touring N47 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 LCI E91N 318d N47 Touring N47 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 LCI E91N 318i N43 Touring N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 LCI E91N 318i N43 Touring N43 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 LCI E91N 318i N46N Touring N46N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 LCI E91N 318i N46N Touring N46N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 LCI E91N 320d N47N Touring N47N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 LCI E91N 320d N47N Touring N47N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 LCI E91N 320d N47 Touring N47 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 LCI E91N 320d N47 Touring N47 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 LCI E91N 320i N43 Touring N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 LCI E91N 320i N43 Touring N43 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 LCI E91N 320i N46N Touring N46N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 LCI E91N 320i N46N Touring N46N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 LCI E91N 323i Touring N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 LCI E91N 325d M57N2 Touring M57N2 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 LCI E91N 325d M57N2 Touring M57N2 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 LCI E91N 325d N57 Touring N57 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 LCI E91N 325d N57 Touring N57 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 LCI E91N 325i N52N Touring N52N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 LCI E91N 325i N52N Touring N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 LCI E91N 325i N53 Touring N53 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 LCI E91N 325i N53 Touring N53 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 LCI E91N 328i Touring N52N Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E91 LCI E91N 330d Touring N57 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 LCI E91N 330d Touring N57 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 LCI E91N 330i N52N Touring N52N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 LCI E91N 330i N52N Touring N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 LCI E91N 330i N53 Touring N53 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 LCI E91N 330i N53 Touring N53 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 LCI E91N 335d Touring M57N2 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 LCI E91N 335d Touring M57N2 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 LCI E91N 335i N54 Touring N54 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 LCI E91N 335i N54 Touring N54 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E91 LCI E91N 335i N55 Touring N55 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E91 LCI E91N 335i N55 Touring N55 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 E92 316i Coupé N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E92 E92 316i Coupé N43 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 E92 320d Coupé N47 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E92 E92 320d Coupé N47 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 E92 320i N43 Coupé N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E92 E92 320i N43 Coupé N43 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 E92 320i N46N Coupé N46N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E92 E92 320i N46N Coupé N46N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 E92 323i Coupé N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 E92 325d Coupé M57N2 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E92 E92 325d Coupé M57N2 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 E92 325i N52N Coupé N52N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E92 E92 325i N52N Coupé N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 E92 325i N53 Coupé N53 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E92 E92 325i N53 Coupé N53 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 E92 328i N51 Coupé N51 Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E92 E92 328i N52N Coupé N52N Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E92 E92 330d M57N2 Coupé M57N2 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E92 E92 330d M57N2 Coupé M57N2 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 E92 330d N57 Coupé N57 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E92 E92 330d N57 Coupé N57 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 E92 330i N52N Coupé N52N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E92 E92 330i N52N Coupé N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 E92 330i N53 Coupé N53 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E92 E92 330i N53 Coupé N53 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 E92 335d Coupé M57N2 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E92 E92 335d Coupé M57N2 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 E92 335i Coupé N54 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E92 E92 335i Coupé N54 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 E92 335i Coupé N54 Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E92 LCI E92N 316i Coupé N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E92 LCI E92N 316i Coupé N43 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 LCI E92N 318i Coupé N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E92 LCI E92N 318i Coupé N43 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 LCI E92N 320d Coupé N47N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E92 LCI E92N 320d Coupé N47N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 LCI E92N 320i N43 Coupé N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E92 LCI E92N 320i N43 Coupé N43 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 LCI E92N 320i N46N Coupé N46N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E92 LCI E92N 320i N46N Coupé N46N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 LCI E92N 323i Coupé N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 LCI E92N 325d Coupé N57 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E92 LCI E92N 325d Coupé N57 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 LCI E92N 325i N52N Coupé N52N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E92 LCI E92N 325i N52N Coupé N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 LCI E92N 325i N53 Coupé N53 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E92 LCI E92N 325i N53 Coupé N53 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 LCI E92N 328i N51 Coupé N51 Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E92 LCI E92N 328i N52N Coupé N52N Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E92 LCI E92N 330d Coupé N57 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E92 LCI E92N 330d Coupé N57 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 LCI E92N 330i N52N Coupé N52N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E92 LCI E92N 330i N52N Coupé N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 LCI E92N 330i N53 Coupé N53 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E92 LCI E92N 330i N53 Coupé N53 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 LCI E92N 335d Coupé M57N2 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E92 LCI E92N 335d Coupé M57N2 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 LCI E92N 335i Coupé N55 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E92 LCI E92N 335i Coupé N55 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E92 LCI E92N 335i Coupé N55 Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E92 LCI E92N 335is Coupé N54T Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E93 E93 320d Convertible N47 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E93 E93 320d Convertible N47 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E93 E93 320i N43 Convertible N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E93 E93 320i N43 Convertible N43 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E93 E93 320i N46N Convertible N46N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E93 E93 320i N46N Convertible N46N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E93 E93 323i Convertible N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E93 E93 325d Convertible M57N2 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E93 E93 325d Convertible M57N2 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E93 E93 325i N52N Convertible N52N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E93 E93 325i N52N Convertible N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E93 E93 325i N53 Convertible N53 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E93 E93 325i N53 Convertible N53 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E93 E93 328i Convertible N51 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E93 E93 328i N51 Convertible N51 Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E93 E93 328i N52N Convertible N52N Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E93 E93 330d M57N2 Convertible M57N2 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E93 E93 330d M57N2 Convertible M57N2 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E93 E93 330d N57 Convertible N57 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E93 E93 330d N57 Convertible N57 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E93 E93 330i N52N Convertible N52N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E93 E93 330i N52N Convertible N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E93 E93 330i N53 Convertible N53 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E93 E93 330i N53 Convertible N53 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E93 E93 335i Convertible N54 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E93 E93 335i Convertible N54 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E93 E93 335i Convertible N54 Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E93 LCI E93N 318i Convertible N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E93 LCI E93N 320d Convertible N47N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E93 LCI E93N 320d Convertible N47N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E93 LCI E93N 320i N43 Convertible N43 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E93 LCI E93N 320i N43 Convertible N43 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E93 LCI E93N 320i N46N Convertible N46N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E93 LCI E93N 320i N46N Convertible N46N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E93 LCI E93N 323i Convertible N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E93 LCI E93N 325d Convertible N57 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E93 LCI E93N 325d Convertible N57 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E93 LCI E93N 325i N52N Convertible N52N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E93 LCI E93N 325i N52N Convertible N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E93 LCI E93N 325i N53 Convertible N53 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E93 LCI E93N 325i N53 Convertible N53 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E93 LCI E93N 328i Convertible N51 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E93 LCI E93N 328i N51 Convertible N51 Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E93 LCI E93N 328i N52N Convertible N52N Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E93 LCI E93N 330d Convertible N57 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E93 LCI E93N 330d Convertible N57 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E93 LCI E93N 330i N52N Convertible N52N Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E93 LCI E93N 330i N52N Convertible N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E93 LCI E93N 330i N53 Convertible N53 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E93 LCI E93N 330i N53 Convertible N53 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E93 LCI E93N 335i Convertible N55 Neutral (N) Europe (ECE) Left steering (L) 2
BMW 3' E93 LCI E93N 335i Convertible N55 Neutral (N) Europe (ECE) Right steering (R) 2
BMW 3' E93 LCI E93N 335i Convertible N55 Neutral (N) USA (USA) Left steering (L) 2
BMW 3' E93 LCI E93N 335is Convertible N54T Neutral (N) USA (USA) Left steering (L) 2
BMW Z4 E89 E89 Z4 23i Roadster N52N Neutral (N) Europe (ECE) Left steering (L) 2
BMW Z4 E89 E89 Z4 23i Roadster N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW Z4 E89 E89 Z4 30i Roadster N52N Neutral (N) Europe (ECE) Left steering (L) 2
BMW Z4 E89 E89 Z4 30i Roadster N52N Neutral (N) Europe (ECE) Right steering (R) 2
BMW Z4 E89 E89 Z4 30i Roadster N52N Neutral (N) USA (USA) Left steering (L) 2
BMW Z4 E89 E89 Z4 35i Roadster N54 Neutral (N) Europe (ECE) Left steering (L) 2
BMW Z4 E89 E89 Z4 35i Roadster N54 Neutral (N) Europe (ECE) Right steering (R) 2
BMW Z4 E89 E89 Z4 35i Roadster N54 Neutral (N) USA (USA) Left steering (L) 2
BMW Z4 E89 E89 Z4 35is Roadster N54T Neutral (N) Europe (ECE) Left steering (L) 2
BMW Z4 E89 E89 Z4 35is Roadster N54T Neutral (N) Europe (ECE) Right steering (R) 2
BMW Z4 E89 E89 Z4 35is Roadster N54T Neutral (N) USA (USA) Left steering (L) 2 | {"url":"https://www.exzap.eu/shop/product/108412/BMW/stabilizer-rubber-mounting/31356765574","timestamp":"2024-11-02T20:12:12Z","content_type":"text/html","content_length":"224299","record_id":"<urn:uuid:6440c47d-2e33-40fe-b1ec-684f41d86ef8>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00664.warc.gz"} |
Let G be a mapping class group of a finite-type surface, A be an arbitrary finite generating set for G and Cay(G,A) be the respective Cayley graph. A conjecture due to Farb states that the ratio of
pseudo-Anosov elements in a ball of radius r to the size of that ball in Cay(G,A) approaches 1 as r approaches infinity. I will discuss some recent progress made on the conjecture by various people.
Part of the talk is based on joint work with Sisto. | {"url":"https://seminars.math.toronto.edu/seminars/list/events.py/process?action=display&file=ac6ea9d230d6ffdb827563f819c9b0ab-submission-pkl-1701702343.09519","timestamp":"2024-11-14T01:56:06Z","content_type":"text/html","content_length":"1939","record_id":"<urn:uuid:53ffc6a7-5527-414a-865f-749f62dc752b>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00177.warc.gz"} |
Factors And Multiples (solutions, examples, videos)
Factors And Multiples
If a is divisible by b, then b is a factor of a, and a is a multiple of b. For example, 30 = 3 × 10, so 3 and 10 are factors of 30 and 30 is a multiple of 3 and 10
Take note that 1 is a factor of every number.
Understanding factors and multiples is essential for solving many math problems.
Prime Factors
A factor which is a prime number is called a prime factor. For example, the prime factorization of 180 is 2 × 2 × 3 × 3 × 5
You can use repeated division by prime numbers to obtain the prime factors of a given number.
The following diagram shows how to find the GCF using the Ladder Method. Scroll down the page for more examples and solutions.
Greatest Common Factor (GCF)
As the name implies, we need to list the factors and find the greatest one that is common to all the numbers. For example, to get the GCF of 24, 60 and 66:
The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24 The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60
The factors of 66 are 1, 2, 3, 6, 11, 22,33 and 66
Look for the greatest factor that is common to all three numbers - thus 6 is the GCF of 24, 60 and 66.
Least Common Multiple (LCM)
As the name implies, we need to list the multiples and to find the least one that is common to all the numbers. For example, to get the LCM of 3, 6 and 9:
The multiples of 3 are 3, 6, 9, 12, 15, 18, 21 … The multiples of 6 are 6, 12, 18, 24, …
The multiples of 9 are 9, 18, 27, …
Look for the least multiple that is common to all three numbers - thus 18 is the LCM of 3, 6 and 9.
Shortcut To Finding LCM
Here is a useful shortcut (also called the ladder method) to finding the LCM of a set of numbers. For example, to find the LCM of 3, 6 and 9, we divide them by any factor of the numbers in the
following manner:
How to use the Ladder method to find GCF, LCM and simplifying fractions?
Step 1: Write the two numbers on one line.
Step 2: Draw the L shape.
Step 3: Divide out common prime numbers starting with the smallest.
LCM makes an L.
GCF is down the left side.
Simplified fraction is on the bottom.
Find the GCF, LCM and simplified fraction for 24 and 36.
LCM & GCF With the Ladder Method
Find the LCM and GCF of 24 and 36.
Difference between greatest common factor and least common multiple
Example: Find the GCF and LCM of 16 and 24.
Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page. | {"url":"https://www.onlinemathlearning.com/factors-and-multiples.html","timestamp":"2024-11-08T04:23:19Z","content_type":"text/html","content_length":"38387","record_id":"<urn:uuid:84c26e71-2415-4d6f-a9f8-a4ffbdcf5eb8>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00773.warc.gz"} |
Konstantin Meyer. Big Bang Theory.
Moving backward through time is kind of a ' spooky action ' , like
Albert Einstein has referred to as when having watched the phenomenon
of Quantum Teleportation by the behaviour of entangled photons ...
We are able to reach each point within time by speeds !
Moving through the three dimensions of space by speed is possible ,
whereas speed is measured in metres per time units , and hence
resulting metres per metres exponent 4 .
So we might move throughout time by metres squared per metres exponent 5 ,
what results metres per metres exponent 4 , equal to usual speeds .
' By Chance ' should not exist in classic physics nor in Quantum Mechanics ...
And present is , when Future meets Past !
We might have found to a logical proof for
Antimatter moving from future to past and
we start right ahead with these few lines :
( 1 ) = possible
( - 1 ) = impossible = entangled photons which never met
while this is only possible ;
( - 1 ) = impossible = moving from future to past ( our clocks move ' forward ' )
We are going to see that
( - 1 ) times ( - 1 ) = ( 1 ) = possible
and that
( 1 ) times ( 1 ) = ( 1 )
( 1 ) can be the logical inversion of ( - 1 ) also ...
Now let us see why we multiply both the factors :
An apple and a cake is an apple and a cake .
An apple times a cake is an ' applecake ' .
While an apple describes the entanglement
of photons which never met and the cake is
our clock moving backward ...
We might explain the multiplication by this way too :
( 1 / a ) plus ( 1 / c ) cannot be simplified more than
( 1 c / ac ) plus ( 1 a / ac ) =
( 1 c plus 1 a ) / ac
but what we can do is
( 1 / a ) times ( 1 / c ) = 1 / ac
1 / 4 of 1 is 1 / 4
' of ' can be described as ' times '
1 / 4 times 1 is 1 / 4
' impossible ' of ' impossible ' is ' possible '
' impossible ' times ' impossible ' is ' possible '
We would like to strengthen our proof for the
movement of future to past by the following :
( 1 ) = possible
( - 1 ) = impossible = entangled photons which never met
( - 1 ) = impossible = a magical force does not act
So we might see that the entanglement of photons which never met is impossible , because
it is also impossible , that a magical force has not caused the entanglement .
( - 1 ) times ( - 1 ) = ( 1 ) = possible ,
what might to be true . And now let us see the little pity
for all the magicians by the second case :
Entangled Photons are possible because it is possible that a magical force does not act
( 1 ) times ( 1 ) = ( 1 )
while ' no magical force ' is the logical inversion of ' a magical force acting ' ,
like ' time moves backward ' is the logical inversion of ' time moves forward ' .
Now let us move on to the ' lord ' :
( 1 ) = possible
( - 1 ) = impossible = entangled photons which never met
( - 1 ) = impossible = it is not because god had want it to be
So entangled photons are possible , and not because of god has created this physical behaviour ,
while ' it is not because god had want it to be ' is the logical inversion of ' it is just the will of our lord ' .
Anyway , we might still believe in god .
Not for nothing an ancient greece phrase says :
' Success is up to the gods ' ...
Let us just generalize this logical proof :
( 1 ) = possible
( - 1 ) = impossible = entangled photons which never met
while this is only possible , so we have
( 1 ) = impossible * ( - 1 ) = entangled photons which never met
( 1 ) = possible = due to any reason
inverting results
( - 1 ) = impossible = not due to any reason
( - 1 ) = impossible = due to no reason
( 1 ) = impossible * ( - 1 ) = entangled photons which never met
is equal to
( 1 ) = impossible * ( - 1 ) = not due to any reason
So interpreting according to the logic { impossible * ( - 1 ) } says :
' Entangled photons are possible ,
( possible ) not due to any reason '
( so it must be a specific reason )
' Entangled photons are possible ,
due to no reason '
while ' due to no reason '
should not be possible in our world of science ...
Above , we have seen a possible explanation for the
nature of entangled photons . Because we understand
Antimatter currently as an opposite to matter in each
of its characteristics , we could even think about
Antimatter to exist from Future to Past what might to
be understood as a ' steady striving to inside ' also ...
Matter should move from Past to Future
and because Matter is in a bit of surplus
within our known existence , time might move
' forward ' for us .
Everything is made of Matter and Antimatter ,
so Gravitational Force is ' very weak ' compared
to electromagnetic Attraction due to these reasons :
Matter is neutral to Matter .
Antimatter is repelled by Antimatter .
And electromagnetic Attraction means the attraction
between Matter and Antimatter ...
We see photons as electromagnetic particle gemini systems
with their individual core masses , shell masses , radii and
shell spin speeds . Because the shell consists of Antimatter ,
the shell ( and hence the state ) would move from future into past ...
The animation below describes Gravitational Waves ,
Matter is coloured blue , while Antimatter is coloured yellow .
Matter is understood as a steady striving ' to inside ' ,
what would explain the mutual neutrality of the force of
attraction of Matter and the ' direction ' time is passing
while Antimatter may own a continuous ' moving to outside '
being a reason for the repelling energy among Antimatter .
The resulting movement is visualized by the arrows ...
Antimatter should to be contained close to matter
within mass . Whereas mass consists of these little ,
primary kind of particles , namely photons at varying levels
of energy . And photons are made of antimatter-shells
spinning around cores made of matter .
Information is able to move through time by superluminal speeds .
The experiment with entangled photons which have never met might have proven this ...
The reason why Antiparticles can only be captured within
a magnetic field might to be , that they would immediately
stuck back to Matter because of the electromagnetic attraction .
A reasonable explanation for ' three peaky ' light waves
is that Supernovae result in a very high amount of
energy , so that the photon shell is split off the core ,
electromagnetic attraction is broken , and the shell is
spinning ' crazy and disoriented ' for a few ( three ) times .
And then , the shell returns into an orbit of the photon ' s core .
During the last days , we were able to create
a clear visual for the Matter Antimatter Modell
which shows , that the Antiparticle might exist
close beside the particle made of matter ...
The transformation of an electron with a mass of
10 exponent -31 kilogramms to two gamma particle gemini
systems with a mass very similar to the mass of an electron
( if not even exactly the same amount of mass )
might to be an explanation for the Matter - Antimatter phenomenon ...
We are going to see an electron ( Matter ) combined
with its Antiparticle Positron , meanwhile we got to
mention , that the amount of Particles and Antiparticles
is fictive . We have used this amount just for a better
understanding of the visualization below :
We would like to add a polished artwork for the ' smallest circular flow of
magnetism ' , because yesterday we have tested again , and two magnets
attract each other ( vertically ) . Rotating one magnet of this attracting
pair around 180 degrees ( around the vertical axis ) results a repelling force .
A just circular flow would not fit this character , so we have had the idea ,
that the flow would be more like the shape of an ' 8 ' , what would be in
harmony with the bending of space and time also ...
Supposed that a magnetical force derives from the motion of superluminal antiparticles .
Having seen an ( almost finalized ) animation for Magnetism above ,
we see the reason for little Antimatter particles moving with
superluminal speed by firstly , that Antimatter gathered at the
one pole of the magnet should repel Antiparticles leaving ,
speeding those up , so that these are passing throughout
time , moving through the bended space at the ' smallest '
distance and than being attracted by the other side of the pole ,
where Matter is accumulated ...
Secondly , it should be Antimatter particles moving very fast ,
because just bending space ( like Matter and Antimatter do )
would attract common mass objects ( e . g . an apple ) also
( what is the case , but ' vanishingly small ' in comparison with
the strong electromagnetic attraction ) .
And only magnets and magnets and a magnet and ferromagnetic
elements attract each other by the strong electromagnetic force :
Matter and Antimatter might not be bound very strong within e. g.
iron or other ferromagnetic substances , what enables a magnetic force acting .
Whereas Matter and Antimatter might be constantly separated within a magnet ...
At these days ( end of March of the Year 2023 ) , it seems that we have been able
to finalize our work about the ' Smallest Circular Flow of Magnetism ' ...
We might even think about an ' octopole grid ' at this times ,
instead of the current quadrupole which follows the logic of Matter being neutral
to Matter , Antimatter pushing off Antimatter . And , following from this ,
the logic of the electromagnetic attraction between Matter and Antimatter ...
We are going to see , that an octopole grid would work out fine .
While we can understand the Antimatter and Matter within this
new visualisation as single particles , because regarding atoms ,
we will see , that the characteristic of a single atom ( in case of
an element ) describes the characteristic or nature of the whole
object it is made of . Growing the mass or just size of a magnet
should increase the magnetical force , as force should add up or
even multiply ...
We will also see , that the spread and common description of the
magnetical force field up to this day might not be true . Because
The Southpole attracts the Northpole vertically and if we would rotate
one magnet of this attracting pair of magnets for 180 degrees around
this vertical axis , the result is a repelling force ...
Projected on a flat plane results the direction ( Northpole to Southpole )
of the ' up today common known ' magnetical force field ...
Two magnetical force fields attracting each other ...
Two magnetical force fields repelling each other ,
one magnet has been rotated for 180 degrees around
the axis running from ' Northpole ' to ' Southpole ' .
The simplified movement from ' Northpole ' to ' Southpole ' .
Matter and Antimatter distributed in harmony and ' randomly ' polarized within
common mass objects , following the principle of electromagnetic attraction .
Matter and Antimatter distributed in harmony and polarized as a quadrupole within a Ferrite Magnet .
The distribution of Matter and Antimatter within a Neodymium Magnet
( = polarized spread as a dipole ) .
Electromagnetic attraction deriving from assumed superluminal moving
Antimatter particles might add up ...
The complete description of the ' Smallest Circular Flow of Magnetism ' above :
We have found that kind of asymmetry within the flow of Antimatter , which
should to be responsible for the neutral part where no force is acting , and
we have tested , rotating the Ferrite magnet ( the inner part ) for 180 °
around the vertical axis also changes this neutral part according to the angle
we have rotated it around ...
Both outer parts ( the magnets ) are Neodymium magnets , were the flow
of Antimatter is running from Northpole to Southpole .
We have seen a possible reason for the reversing polarity of the
magnetic force field of our Earth during our last days of work ...
Since we can assume everything to consist of electromagnetic
particles at differing levels of Energy ( ' photons ' ) , we will see
that the core of Earth might to be made of these little particles
also , just at a very , very low level of Energy with a frequency
of something about 10 e-10 spins per second and even less ...
So , if these particles are spinning gemini systems , we will
see a reversing polarity of the magnetic force field when half of
a whole spin has been performed ...
We would like to add the following recognition :
Since Sagittarius A * is more massive it might cause
a stronger magnetic field which reverses more
seldom and this reverse might also influence the
magnetic polarity of our Earth ' s magnetic force
field , like our Sun should do , too ...
Below we are going to see a new animation for the bending of time
by superluminal motion , while the fourth dimension , time , is reduced
to a two - dimensional area in this graphic ...
Assuming our observable Universe to be not even a dust grainlet
within the whole of our Existence ...
The standstill of time by moving with c is able to show the curvature clearly ,
meanwhile the superluminal motion of antimatter might mean a travelling
throughout time and if time shrinks , the bending of space would increase .
We can just imagine a cube representing time ( reduced by one dimension )
and a ( circular ) plane within the time cube , whereas the plane represents our space
( reduced by one dimension ) . Squeezing the time cube might bend the circular plane ...
So , electromagnetic particles like photons might move circular at a large
scaling , following the natural curvature of space , meanwhile ' photon shells '
leaving a magnet move within the ' Smallest circular flow of Magnetism ' ,
as space is bended proportionally to the speed higher than c ...
During June of the year 2024 we have seen a new understanding of the
complexion of time and we would like to watch the animation below ,
describing our breakfast cup to be removed from the table yesterday
while we have been travelling from today to yesterday . Today lingers
on and yesterday ( with the removed cup ) turns into a ' new ' today .
We are able to understand this phenomenon by the ' large scaling '
of the fourth dimension ( time ) which is described by an expansion
of space for the cuberoot of space , like the third dimension might to
be determined by multiplying a square by its squareroot ...
Time is reduced by one dimension to a cube within this new animation
and space is represented by the flat blue areas , for a clearer look the
spheres showing the breakfast cups are three - dimensional :
And at these days , it seems very likely that our space exists
for mulitple times throughout the fourth dimension , time ...
After some few days to re-think all of our recent understanding ,
we might have found to the conclusion , that the high Energy in
the centers of Planets like our Earth or just Milky Way , which
should to be caused by the very large amount of gravitational
pressure causing high temperature ( = Energy ) , splits Anti -
Matter off the photon shells what is the reason for the magnetic
force field . After being just not sure , whether a magnetical field
is described by an ' 8 - like shape ' or just two circles ( both the
shapes are very similar to each other ) , we have sketched some
kind of a new understanding of the smallest circular flow of Anti -
Matter Shells of a magnetic force field ...
An issue , which has not been solved until this day , is that the
magnetic field of our Earth is like a ' laid eight ( oo ) ' compared
to its rotation axis beside the magnetic field of our Galaxy , Milky
Way , which might to be rotated for ninety degrees ( like an 8 )
according to the photon particle bubbles ...
Matter and Antimatter within Magnetism :
The animation above illustrates the principle of Magnetism ...
Another proof for Matter to be neutral to Matter might to be :
Within Magnetism we have two states , repelling and attracting ,
but three possibilities of combinations of the components ,
matter to matter , antimatter to matter and antimatter to antimatter ,
so one of these combinations must act as a neutral force to keep
the repelling and attracting forces equal to each other ...
This Matter - Antimatter Modell might also be a prove for the Force
of Gravitation to act only attracting ( and not repelling ) ...
After yesterday , we have again found to another , new conclusion
for the directions of a magnetic force field . As we already know ,
two magnets attract each other vertically and rotating one of the
magnets around this vertical axis results a repelling force .
It is still an open questions why the magnetic force field of our Earth
is rotated for ninety degrees compared to our idea about the superluminal
moving Antimatter split off from the center of Milky Way ...
We might suggest the following recognition .
Superluminal Antimatter deriving from the
high energy in the center ( photon shells
are split off their cores ) might shape both
the gamma particle bubbles and a flat disc
of our Galaxy Milky Way by passing through
the curved or bent space ...
During the last days we have found to the value of Energy
a photon carries , for which the Backward Causility of the
Photon Shell might to be reversed , since the photon shell
is spinning superluminal in this case and we start ahead
with this way to calculate the frequency of the photon shell
spinning with light speed c , moving right into the future :
' delta radius exponent 3 = delta frequency squared '
and because we calcuate with the value c , we can
use ' 1 / radius exponent 3 = frequency squared '
radius r = 1 / 3rd root of ( frequency f squared ) = 1 / 1,5th root of f ;
Spin Speed of the Photon shell = 2 r * pi * f = c in the case of
reverting Backward Causility ;
1 / ( 1,5 th root of f ) * f = c / 2 / pi I to the power of 1,5
f = 0,5 th root of [ ( c / 2 / pi ) to the power of 1,5 ]
f = 108.623.177.716.518.437.266.722,7158848 ;
It might be possible to measure this certain frequency and to
see the movement into the future for Antimatter in this case ...
We would like to finalize our work about the circular
bending of time . The factor ' x ' results speed ,
meters per second , as a deviation from one infinitely
small time unit since the time circle is projected to a
single point by extracting the 36 th root of .
In order to calculate the diameter of the circular arched
time , which is the fourth dimension , we got to expand
the calculated radius of the ' two - dimensional -
time - circle ' by its square to the power of three , since
the fourth dimension time is flattened to a two - dimensional
area and what should up in the units
meters exponent 4
1 second exponent 4
what we might understand as time and that could be the
way of converting speed to time , since a meter exponent
4 might represent time ( and a time unit exponent 4
should stay one time unit , because 1 to the power of 4
results 1 ) , but we have to know speed ratio by having
speed just to the power of four ...
The circular arched time might represent the Big Bang , our
Universe expanding and massing , pulling together again ...
Another thing , which comes in parallel , are the decreasing bent time and
space as longer we see the temporally distance from the Big Bang ...
We have seen , that the speed our Universe expands with will slow down
like time does when the Big Bang moves towards its end and time , mass
and space begin to mass again !
Since we can understand those two ' speeds ' passing like a circle or like
a circular relation instead of a ( square ) function , we will recognize a
linear speed , because a slowing down of time since the Big Bang might
to be in balance with a slow down of the expansion speed of space :
Why ? We just imagine walking around in our garden for one turn , while
time passes slower , let us say , one minute has been passed for one turn .
If time passes faster, five minutes have been passed by for walking this one
turn , while the speed we were walking stays pretty much the same ...
And we strongly assume that a relatively ' short period of time ' after the
Big Bang was filled with thousands of years , and hence , space has been
expanded to a comparably high distance , what would also be in harmony
with the radiation which reaches our Earth from the Super Massive Center
along this large distance ....
Because we calculate the deviation per point by extracting the 9th root ,
we have to expand our result of meters exponent minus 3 per point
for the variable ' x ' to the power of four , resulting meter per meters
for the natural bending of space arched by the ' circular bent ' time ...
And the reason why we are going to extract the ninth root of our three
dimensions of space is , that we imagine an infinitely small cube to be
the point , and extracting the third root of this three dimensional point
results a tiny line as close to the point as possible , the difference might
to be vanishing less , so that we can set this tiny line equal to the point .
Expanding a point to a tiniest , three dimensional cube might result a cube
to exist with nine dimensions [ ( 1 exponent 3 ) exponent 3 ) ] .
We would like to add a tiny detail to our latest calculations .
Since we use a similar approach for calculating the circular
motion of time , we see the ratio of c / 1,00323817 ... ,
and that c is equal to zero time within this way to calculate ,
so we have to add the value ' 1 ' to 0 in order to receive
a ratio . Using G instead c , we might add 1 to the ratio
1 / 84.568.397.407,601649039685894375222 , so we can
talk about 1
( 1 + 1,18247 ... * 10 e -11 meters exponent - 3 )
per point
During this weekend we have found to some reasonable
explanations for calculating the natural space curvature
by using the value of the Gravitational Constant ' G ' :
' G ' is the constant of proportionality used to determine
the force two mass objects attract each other . Attracting
means bending space ( and time ) .
Since time should to be curved ( circular ) , within may
it to be infinity as the fifth dimension , space has a natural
curvature , too . And the maximum of speed for matter ,
c , might be able to show this bending exactly because
speeds faster than c would curve space , like magnetism .
We ' ll see that extracting the 9th root of metre exponent 3
results a value as close to a point as possible . A point is always ' 1 ' .
9th root of
G * 1 kilogramm * 1 kilogramm * c * c
1 metre squared ( the radius )
is equal to 1 ;
The point = 1 ;
1 = 1 ;
G * x ( x = the bending factor ) = 1 ;
Because G = metres exponent 3
kilogramm times second squared ,
and because we divide per the radius
and multiply with c squared
we are going to see that this value results
the 9th root of metres exponent 3
what is the point .
And to achieve the value for this point by
' G * x '
we have to extract the 3rd root of a meter
( or this infinitely small , three dimensional point ) for ' x ' ,
what is similar to the ' reciprocal value '
of 1 meter or just the
3rd root of this meter
or this minimal three dimensional point ,
like an ' inverted ' dimension , space or point
for the ' minimal bending ' ;
The balancing factor for G should to be x .
' G * x ' :
We extract the 9th root of metres exponent 3
what should result a value as close to a point
as possible for
( G * c squared per radius squared )
and for x
we are going to see the ' reciprocal value '
of a meter , because we extract the 3rd root
of this ' three dimensional point ' .
Possible might to be also :
Metres exponent 3 ( for G )
metres exponent minus 3 ( for x )
is equal to
1 ;
And multiplying with c squared should be due to the minimum
of each of the kilogramms squared have to be expanded to its
maximum force by moving with light speed c .
Calculating the circular bending of time might work out the same way ,
we are going to see the circle reduced to a point . We are using light
speed c squared due to the fourth dimension time to be flattened to
a circular plane , meanwhile c represents the diameter of this plane
in ( almost ) zero dimension of size ...
The ( circular ) curvature of space can be explained by the following :
We imagine a cube as space and the movement from one point to
another one as a line within this cube representing space .
A line is always one - dimensional ...
For now , we are going to flatten the cube into a two - dimensional
plane and we think about that it is possible to wrap this plane around
a sphere , representing the bending of space .
This would make clear , that starting to travel from any point within
the spherical wrapped area we are going to reach exactly the same point
by travelling a distance which is long enough –
what is described by and in harmony with the spherical shape ...
The line ( the distance passed by ) would end up in a circle with a ' large '
diameter , whereas start - and end point should to be the same ...
It has been scientifically proven , that mass distorts time and
for now we think that this distorsion is very less because of
mass consisting of Matter and Antimatter both with a very tiny
surplus of Matter .
Matter should increase the amount of time passing what can
be seen by large celestial objects like black holes with a high
density whereas
Antimatter should invert the scale time is passing , and
we have seen that Antimatter might to be able to move faster
than Light Speed c what might to be a reasonable explanation
for the amount time is moving slower because the superluminal speed
Antimatter is able to move with might enable a flexible passing
throughout time , the fourth dimension ...
To build upon the proof of the distorsion of time by mass :
If we flatten space to two dimensions and use time as the fourth
dimension to be the third dimension , we can imagine a circle
which is passed by a line , time , as the third dimension , and
this line would pass the circle without being obstructed from ,
but with a tiny difference of proportions of 3rd dimension per
2nd dimension : Let us stay with the ' empty ' space of the
second dimension to own a mass of 1 and the circle , may it to
be our Earth , has a Mass of 4 , we are going to see a time per
mass ratio of either 1 time unit per mass or 4 time units per mass
and 4 per 1 is more , so we are right here , in the future .
The higher the energy an average electromagnetic particle holds ,
the higher the ratio - value of Matter per Antimatter what might to be
proven by the work about photons as average electromagnetic particles .
In black holes , we have a very high amount of energy because of the
large gravitational force due to lots of mass compressed to high density .
That means that matter is of more surplus than anywhere else , hence the
more we get to the center of a black hole , the more we are in the future .
The proportion of Matter and Antimatter within the core of a black hole should
not be an issue of distance between Matter and Antimatter like it is
for electromagnetic particles in free movement through a ( not perfect ) vacuum ...
There is a like to suggest a possible way to determine a ' Hubble Constant '
referring to a photon at one spin per second , while the photon shell attracted
by the core might to be compared with just a galaxy attracting another one ,
because Matter attracts Antimatter and vice versa ...
The differing measured values for the Hubble Parameter
might depend on the differing masses of e . g . galaxies ,
attracting each other by varied forces – what we are going
to see by the calculation of speed of our Moon from Earth ,
which we are going to see below ...
We are going to assume the expansion speed of space and hence force
to be almost in balance with the force of electromagnetic attraction
within single electromagnetic particles , what should be mentioned firstly .
Because if the expansion speed of space would be less ,
an electromagnetic particle would stuck together ( ' relatively fast ' ) .
If the expansion speed would be higher and let float the photon shell
off the core , we might measure red light also –
for blue light passing very high distances in space and our Universe ...
An electromagnetic shell spinning with a higher radius at same
spin speed results a lower frequency , because the shell has to pass
a higher distance for one spin , what is seen as ' red light ' , too .
This phenomenon might to be possible , because of the shell floating off
the core by expansion of space , if so , and we might to be able to
measure it by seeing the amplitude of the linear wave of the spins ...
We are going to assume an electromagnetic particle not to loose energy
during time by moving in a perfect vacuum , or just very less ( what we
will see by calculating red shifts ) ; with the comparison of our Moon
turning around our Earth in steady motion .
We are going to see , that there is no perfect vacuum in space ,
and each of the light particles has to pass a medium in some kind ,
what might to be responsible for the red shift over high distances .
Adding energy grows the radius squared ( what might mean that
the expansion of space in between the shell and core increases ) .
We have seen , that the amount of Matter shrinks like the amount
of Antimatter does by adding Energy to a photon , what means less
of electromagnetic attraction .
If a higher expansion of space would happen , the outer shell of a
photon might ' leave ' the core easier by adding energy , so we are
going to assume the measured expansion speed of space , ' v '
( while we should notice the difference between ' V ' , uppercase ,
the speed our Universe expands with , and ' v ' , not uppercase ,
which is the actual speed , two celestial bodies float off or near )
between the electromagnetic shell and its core to be
E ( Big Bang ) = ( 1 / 2 v ) squared =
= force of electromagnetic attraction between the shell and the core ...
Our equivalent for the calculation ( with a balance of speed of expansion
and strength of electromagnetic attraction ) is going to be :
1 Mpc ( megaparsec ) * [ ( 1 / 2 v ) squared * pi ( because we have one spin which is circular and a linear distance for space expansion ) ]
root of [ ( 1 / 2 v ) * pi ]
( 1 ) ( = radius of an electromagnetic particle with 1 spin per 1 second = 2,3943600123792969381313194102613e-30 meters )
If we isolate v , we are going to receive
v exponent 3 = 8 / { 1 Mpc squared * ( 1 ) squared * pi }
what results
77.556,828414033025100091639169999 metres / 1 second / 1 Megaparsec
for the expansion speed ' v ' of two celestial objects ( may be Galaxies , too ) ,
made of positive and negative mass both and attracting each other like the special
case of a photon at one spin per second ...
By using the new Hubble Parametre , we can calculate
the speed of our Moon leaving the orbit of our Earth to
0,03047295454400178969835560820641 meters / year .
Current measurements lead to 0,038 meters per year .
The average distance ( 384.400.000 meters ) of Moon
from our Earth is used , and we are going to calculate with
384.400.000 meters / 1 megaparsec
( 1 megaparsec = 3,08567758 exponent 22 meters ) *
* 77.556,828414033025100091639169999
meters per second ( / 1 Megaparsec ) * 3,154 *
10 exponent 7 ( seconds per year )
what should result the speed value mentioned above ...
So we will see , that the Hubble Parameter can vary ,
depending on the speed ' V ' our Universe expands with
calculated by the ' Antique Approach ' :
V = c exponent 4,200150305 ... =
4,0171778632092185994253809608252e+35 meters per second
what should to be a constant , while the mass of celestial
objects differs , and hence , force of attraction does ...
If the Hubble Parametre would change during time , electromagnetic particles
would have changed in their diameter also , if seen from the Big Bang ...
After some time to think about , we were able
to precise the delta , pi for the circular bent
time should to be interpreted correctly now ,
we might see about that below :
Yesterday , we have been able to calculate the
circular motion of time by our this year ' s work
for the radius of the circle reduced to the point .
We have to expand this radius by its square
and then to the power of 3 because we want to
see the fourth dimension which was flattened to
a two - dimensional area what results
with a diameter of
Now , we are going to see a new , interesting part
of this work :
We assume the circular bending of space to be
stronger , if time has not passed ' half ' of
its circular motion ...
We have calculated the arch of space with
G =
root of ( Ge by the Planck Approach * 1,9877 ... ) =
When we have this value to the power of five
( because we have one dimension reduced
to a point and we would like to see all of the
three dimensions )
We will recognize a natural curvature of
4,3058886640772015463419750197868e+54 meters
for a complete circular movement within space ,
when time has passed ' one spin ' and
we are going to achieve a similar value by time :
( the complete time diameter , by years )
( 3600 * 24 * 365,24 )
( seconds )
times the speed our Univere expands with
calculated by the Antique Approach :
c exponent 4,200150305 ... =
is equal to
and we are going to recognize a slight delta of
in comparison to the space curvature of
This delta should to be calculated by
pi times 3rd root of pi times 36th root of pi
because we calculate with the 4th dimension
projected to two dimensions ...
( = 4,7498097606287900355550915260048 )
( With a delta delta of
1,0009111069045729061697704739499 )
Today , 4th of November of the Year 2022 ,
we have found to the following , new
possibility in order to define the ' delta delta ' ,
what we might see below :
4 - 0,9111069045729061697704739499 =
3,0888930954270938302295260501 ;
10 exponent 3,0888930954270938302295260501 =
1.227,1371257360399247147664040529 ;
logarithm ( 2,99792458 square ) *
logarithm ( 6,7071967560044286090261498912392 exponent 8 ) =
59,428643684275356314722237699005 ;
( while 6,7071967560044286090261498912392 ... is derived from G )
pi * 3rd root of pi * 36th root of pi =
4,7498097606287900355550915260048 ;
4,3473145145627768215366686090761 ;
And this value can be approached by
10 - logarithm ( 6,7071967560044286090261498912392 ... exponent 8 )
( = 10 - 6,612328372649949222357912955204 ... )
logarithm of c squared =
16,953641405855855108784532416452 |
minus 16 =
= 4,341313033205905886426619461248 ;
However , a division delta of
remains , and we see a possible reason for this
by cut off numeric values beside we are not one
hundred per cent sure about our calculations to
return true values ...
If you like to , you might see the entire work by
reading through the following lines :
We would like to show an intermediate stage
of our work for calculating the ' delta delta of
1,0009111069045729061697704739499 ' :
G = 6,7071967560044286090261498912392 | to the power of 8
4.095.702,2117347464387129572540481 | logarithm
6,612328372649949222357912955204 ;
c squared = 89.875.517.873.681.764 | logarithm
16,953641405855855108784532416452 ;
pi * 3rd root of pi * 36th root of pi =
4,7498097606287900355550915260048 ;
6,612328372649949222357912955204 *
16,953641405855855108784532416452 *
532,46813300383184574227804473114 ,
which are 3 digits ,
so we have
136 + 3 = 139 ;
139 per 9 per 5 =
3,0888888888888888888888888888889 ;
which is very close to 4 digits of accuracy ...
But what we target are not just 3 digits by
which are calculated by
4 - 0,9111069045729061697704739499
3,0888930954270938302295260501 | * 5 * 9
139,0001892942192223603286722545 ;
139,0001892942192223603286722545 - 136 =
3,0001892942192223603286722545 ;
10 exponent 3,0001892942192223603286722545 =
Below we are going to see the basic work :
We would like to give a reasonable answer
for the
' delta delta of
1,0009111069045729061697704739499 '
what is written below in this message :
Because we calculate with
32 digits after comma zero
within our working environment ,
light speed c squared ,
which consists of 16 digits
and G to the power of eight which
should to be 88 digits
( 10 exponent -11 to the power of 8 ) ,
we might come along
with the following calculation :
88 + 16 + 32 = 136 I per 9
( because we have x exponent 9 )
15,111111111111111111111111111111 | per 5
( because we have
to the power of five )
so that the delta delta of
should to be directly proportional to
3,02222 ... digits of accuracy .
I am not a hundred per cent sure
about this approximation to be true ,
but at times , it seems to work out
nicely and if we are able to precise
this way , we are going to let us know !
Beside our spooky actions to Halloween , we
would like to add some additional work about
the natural curvatures of space and time :
Firstly , it might to be possible , that after one
second of the Big Bang , time was filled with a
million of years approximately or just millions of
years had been passed by , measured by kind
of an ' external ' clock . That is the only logical
explanation for the circular motion of time and the
large distance of the Super Massive Center from
our Earth ...
And secondly we can recognize , if time has not
expanded to its maximum , space must be
bent stronger , because it has somewhere to go .
What we have found , is that time expands
from the Big Bang since it was included in
our existence and hence Light Speed c ,
the Gravitation Constant G and also the
bending of space and time will change during
the process of an expanding ( and may it
to be massing again ) Universe ...
Both the calculations above result the
current curvatures of space and time .
The changes of c and G during our
expanding Universe should to be very ,
very ' tiny ' what we have found out
by calculating with the following way :
Currently , we have a space curvature
meters per meter
and our Universe expands with
meters per second ;
is equal to
of the cuvature increasing per second
' approximately ' ,
times 10.000.000.000 years =
1,4990527906659369149476424016109e+42 ,
and we can add this value to the
complete current ' Space Circle ' ,
what results
4,3058886640787005991326409567017e+54 ;
So we will see that G and c will not change
to a very high amount and our Universe should
to be older than expected , so this would to be
in harmony with the distance of our Earth to the
Super Massive Center of our Universe ...
2021 - Autumn :
Based upon the size of a single electromagnetic particle ,
a hubble parameter referring to the characteristics
of a ' photon ' at one spin per second has been determined .
We are happy to announce that we have found out
the expansion speed of our Universe at these days !
Just get in touch with me for seeing the two ways to calculate it ...
Time might not pass linear – but not noticeable to us
due to an acceleration spread over a megalomaniac scale...
Cosmos Loop Theorie ...
Cosmic Infinity Time Birds , freely passing throughout time ...
' Dimension Travelling ' , understood by the three dimensions
lenght, height, depth and time as the fourth dimension
to be flattened to a two dimensional image or area –
what enables unfolding a ' new ' dimension – and assuming
some beings to own the ability to move freely within this ' new ' space ...
Cos Cos ( ... ) Mos | {"url":"http://research.konstantin-meyer.com/big-bang-theory-konstantin-meyer-1.html","timestamp":"2024-11-06T06:14:22Z","content_type":"application/xhtml+xml","content_length":"78002","record_id":"<urn:uuid:e4ca91ce-af53-4096-80d5-7f62f780c304>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00883.warc.gz"} |
Understanding Mathematical Functions: How To Find The Average Of A Fun
Introduction: The Realm of Mathematical Functions and Averages
A mathematical function is a relationship between a set of inputs and a set of possible outputs. These functions play a crucial role in various fields such as science, engineering, economics, and
more. They are used to model real-world phenomena, make predictions, and solve problems.
The concept of 'average' is fundamental in data analysis. It represents a central value of a set of numbers and provides a general indication of the data set. Understanding how to find the average of
a mathematical function is essential for analyzing data and making informed decisions.
In this chapter, we will delve into the definition of mathematical functions, the importance of averages, and explore the process of finding the average of a function, along with its applications.
A Definition of mathematical functions and their importance in various fields
Mathematical functions are fundamental in expressing relationships between variables. They are used to describe and analyze various phenomena in fields such as physics, chemistry, biology, and more.
Additionally, they are utilized in engineering for designing systems and predicting outcomes.
Functions provide a way to understand and quantify the behavior of the phenomena being studied. By defining and analyzing functions, researchers and professionals can gain insights into the
underlying processes and make informed decisions based on the mathematical representations of the data.
Overview of the concept of 'average' and its significance in data analysis
The concept of 'average' is used to summarize a set of values into a single representative value. It is commonly used to understand the central tendency of the data and provides a measure of the
typical value in the dataset. Calculating averages allows for better comprehension of the overall characteristics of the data and aids in making comparisons and predictions.
In data analysis, averages are used to draw conclusions, make inferences, and identify trends within the dataset. They serve as a starting point for further analysis and lead to valuable insights
into the underlying patterns and behaviors of the data.
Preview of what finding the average of a function entails and its applications
Finding the average of a function involves calculating the mean value of the function over a specific interval or domain. This process provides a single value that represents the central tendency of
the function over the given range.
Applications of finding the average of a function include analyzing periodic phenomena, determining the average rate of change, understanding the behavior of dynamic systems, and making predictions
based on the overall trends exhibited by the function.
By understanding how to find the average of a function, individuals can gain valuable insights into the behavior and characteristics of the function, enabling them to make informed decisions and
predictions in their respective fields.
Key Takeaways
• Understanding the concept of mathematical functions
• Finding the average of a function
• Applying the average to real-world problems
• Understanding the importance of averages in data analysis
• Practical examples of finding the average of a function
The Nature of Averaging Functions
Understanding mathematical functions and how to find their average is an essential skill in various fields such as statistics, engineering, and economics. Averaging functions allows us to find a
representative value for a set of data, providing valuable insights into the overall behavior of the function.
Explanation of continuous and discrete functions
Functions can be classified as either continuous or discrete. Continuous functions are defined for all real numbers within a given interval, and their graphs have no breaks or holes. On the other
hand, discrete functions are defined only for distinct values within a specific domain, and their graphs consist of separate, distinct points.
Understanding the relationship between functions and their averages
When it comes to averaging functions, it's important to consider the nature of the function itself. For continuous functions, the average can be calculated using integration over the entire domain.
In the case of discrete functions, the average is found by summing all the function values and dividing by the total number of values.
The role of domain and range in calculating averages
The domain and range of a function play a crucial role in calculating averages. The domain of a function refers to the set of all possible input values, while the range represents the set of all
possible output values. When finding the average of a function, it's essential to consider the entire domain and range to ensure an accurate representation of the function's behavior.
Mathematical Prerequisites and Tools
Before delving into the process of finding the average of a function, it is essential to have a strong foundation in calculus and algebra. These mathematical disciplines provide the necessary
framework for understanding and manipulating functions to calculate their averages.
A The need for a solid foundation in calculus and algebra
• Calculus: Understanding the concepts of limits, derivatives, and integrals is crucial for working with functions and determining their averages. Calculus provides the tools for analyzing the
behavior of functions and their rates of change.
• Algebra: Proficiency in algebra is necessary for manipulating functions algebraically, solving equations, and simplifying expressions. This knowledge forms the basis for performing calculations
involving functions.
B Essential tools: integration and summation for continuous and discrete cases, respectively
• Integration: In the case of continuous functions, integration is used to find the average value of a function over a given interval. This process involves calculating the definite integral of the
function over the interval and dividing by the width of the interval.
• Summation: For discrete functions, summation is employed to find the average. This entails adding up all the function values and dividing by the total number of data points.
C Software and calculators that can assist with complex calculations
While manual calculations are valuable for understanding the underlying principles, complex functions and large datasets may require the use of software and calculators to expedite the process. Tools
such as Mathematica, Matlab, and Wolfram Alpha can handle intricate mathematical operations and provide accurate results.
Understanding Mathematical Functions: How to find the average of a function
When it comes to understanding mathematical functions, finding the average of a function is an important concept. In this chapter, we will explore the step-by-step calculation for averaging a
continuous function, including setting up the integral over the function's domain and applying the mean value theorem for integrals to find the average value. We will also work through a practical
example of averaging a simple linear function over an interval.
Setting up the integral over the function's domain
Before we can find the average of a function, we need to set up the integral over the function's domain. This involves determining the limits of integration and the function itself. The integral over
the function's domain represents the total 'area' under the curve of the function, which we will use to find the average value.
Applying the mean value theorem for integrals to find the average value
Once we have set up the integral over the function's domain, we can apply the mean value theorem for integrals to find the average value. The mean value theorem states that for a continuous function
on a closed interval, there exists at least one value in the interval such that the average value of the function is equal to the function value at that point.
To find the average value of the function, we divide the integral of the function over its domain by the width of the domain. This gives us the average height of the function over the interval, which
is a useful measure for understanding the behavior of the function.
Practical example: Averaging a simple linear function over an interval
Let's work through a practical example to illustrate the process of averaging a simple linear function over an interval. Consider the function f(x) = 2x + 3 over the interval [1, 5].
First, we set up the integral of the function over the interval:
Next, we calculate the integral:
• ∫[1, 5] (2x + 3) dx = [x^2 + 3x] from 1 to 5
• = (5^2 + 3*5) - (1^2 + 3*1)
• = (25 + 15) - (1 + 3)
• = 40 - 4
• = 36
Then, we find the width of the interval:
Finally, we calculate the average value of the function:
• Average value = (1/4) * 36 = 9
Therefore, the average value of the function f(x) = 2x + 3 over the interval [1, 5] is 9.
Averaging Discrete Functions
When dealing with discrete functions, finding the average can provide valuable insights into the data. Whether it's a set of values representing a sequence or discrete data points, understanding how
to calculate the average is essential. In this chapter, we will explore the process of averaging discrete functions and its real-world application.
Understanding the summation process for sequences or sets of values
Before calculating the average of a discrete function, it's important to understand the summation process for sequences or sets of values. The summation of a sequence involves adding up all the
values in the sequence. This can be represented using the sigma notation, where Σ is used to denote the sum of a sequence.
For example, if we have a sequence of values {x1, x2, x3, ..., xn}, the summation of the sequence can be represented as:
Σ xi = x1 + x2 + x3 + ... + xn
Understanding this process is crucial for calculating the average of a discrete function, as it forms the basis of the arithmetic mean calculation.
Calculating the arithmetic mean for discrete data points
The arithmetic mean, also known as the average, is a fundamental concept in statistics and mathematics. It is calculated by summing up all the values in a set and then dividing the sum by the total
number of values. For a discrete function with n data points, the arithmetic mean can be calculated using the formula:
Mean = (Σ xi) / n
Where Σ xi represents the sum of all the data points and n is the total number of data points. This formula provides a straightforward method for finding the average of a discrete function.
Real-world scenario: Computing average daily temperature over a month
To illustrate the application of averaging discrete functions in a real-world scenario, let's consider the computation of the average daily temperature over a month. Suppose we have a set of daily
temperature readings for each day of the month. By using the arithmetic mean formula, we can calculate the average temperature for the entire month.
For example, if the daily temperature readings for a month are {70°F, 72°F, 68°F, 75°F, ...}, we can find the average temperature by summing up all the daily temperatures and dividing by the total
number of days in the month.
This real-world scenario demonstrates how the concept of averaging discrete functions can be applied to analyze and interpret data in various fields, from meteorology to finance.
Troubleshooting Common Challenges
When finding the average of a mathematical function, there are several common challenges that may arise. Understanding how to deal with undefined or infinite values within the domain, avoiding
mistakes in applying formulas, and assessing the impact of outliers and irregular data on the average are essential for accurate results.
A. Dealing with undefined or infinite values within the domain
One common challenge when finding the average of a function is dealing with undefined or infinite values within the domain. This often occurs when there are discontinuities or asymptotes in the
function. In such cases, it is important to identify the specific points within the domain where the function is undefined or approaches infinity. Understanding the behavior of the function at these
points is crucial for accurately calculating the average.
To address this challenge, it may be necessary to use limits to determine the average value of the function over a given interval. By approaching the undefined or infinite values from both sides, it
is possible to calculate the average in a way that accounts for the behavior of the function at these critical points.
B. Avoiding common mistakes in applying formulas and interpreting results
Another challenge is avoiding common mistakes in applying formulas and interpreting results. When calculating the average of a function, it is important to use the correct formula and apply it
accurately to the given function. Errors in calculation can lead to inaccurate results and misinterpretation of the average value.
One common mistake is using the wrong formula for finding the average of a function. It is essential to use the appropriate method, such as integrating the function over the given interval and
dividing by the width of the interval. Additionally, interpreting the results requires careful attention to the context of the function and the specific problem being addressed. Understanding the
implications of the average value within the given context is crucial for meaningful interpretation.
C. Assessing the impact of outliers and irregular data on the average
Assessing the impact of outliers and irregular data on the average is another important consideration. Outliers, or data points that significantly deviate from the rest of the data, can have a
substantial impact on the average value. It is essential to identify and evaluate the influence of outliers on the average to ensure that it accurately represents the central tendency of the data.
One approach to addressing this challenge is to use measures of central tendency that are less sensitive to outliers, such as the median. Additionally, understanding the distribution of the data and
the presence of irregularities can provide valuable insights into the impact of outliers on the average. By carefully assessing the data and considering the potential influence of outliers, it is
possible to obtain a more accurate and meaningful average value.
Conclusion & Best Practices
A Recap of key points on finding the average of a function
Understanding the process
Throughout this blog post, we have delved into the concept of mathematical functions and how to find the average of a function. We have learned that the average of a function is calculated by
integrating the function over a given interval and then dividing by the width of the interval. This process allows us to find the average value of the function over that specific range.
Importance of finding the average
Finding the average of a function is crucial in various real-world applications, such as calculating average velocity, average temperature, or average cost. It provides us with a single value that
represents the behavior of the function over a given interval, making it a valuable tool in mathematical analysis and problem-solving.
Best practices: Regularly review mathematical fundamentals, use appropriate software tools, and verify results for accuracy
Regular review of mathematical fundamentals
It is essential to regularly review and reinforce your understanding of mathematical fundamentals, including concepts related to functions, integration, and averaging. This ongoing review will help
solidify your knowledge and improve your ability to apply these principles effectively.
Utilizing appropriate software tools
When dealing with complex functions or large datasets, using appropriate software tools can streamline the process of finding the average of a function. Utilizing mathematical software or programming
languages can help automate calculations and provide accurate results in a more efficient manner.
Verification of results
Always verify the results of your calculations to ensure accuracy. Double-checking your work and comparing results using different methods or tools can help identify any potential errors or
discrepancies. This practice is crucial, especially when dealing with complex mathematical functions.
Encouragement for continued learning and application of these methods in complex problem-solving scenarios
Continued learning
Mathematics is a vast and ever-evolving field, and there is always more to learn. Embrace a mindset of continuous learning and exploration, seeking to deepen your understanding of mathematical
functions and their applications. This ongoing pursuit of knowledge will enhance your problem-solving abilities and broaden your mathematical skill set.
Application in complex problem-solving
As you gain proficiency in finding the average of a function, challenge yourself to apply these methods in complex problem-solving scenarios. Whether it's in physics, engineering, economics, or any
other field, the ability to analyze and interpret the behavior of functions is a valuable skill. Embrace opportunities to tackle challenging problems and leverage your understanding of mathematical
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Divide 50 by half and then add 20 - Daily Quiz and Riddles
Divide 50 by half and then add 20. What is the answer?
The answer is 120
This brain teaser might initially seem confusing, but it’s all about understanding the order of operations in mathematics.
First, you’re asked to divide 50 by half. The important distinction here is that “half” represents 1/2. So, dividing 50 by half means dividing it by 1/2. In mathematics, dividing by a fraction is the
same as multiplying by its reciprocal. Therefore, dividing 50 by 1/2 is equivalent to multiplying 50 by 2, which equals 100.
It’s crucial not to confuse this operation with simply dividing 50 by 2, which gives a different result of 25. Also, it doesn’t mean dividing 50 by 25, which is the half of 50.
Next, you’re told to add 20. So, add 20 to the result, which is 100. This gives you a final answer of 120.
The key to solving this puzzle is remembering the sequence of operations in mathematics: division (or multiplication by the reciprocal, in this case) before addition. So, the answer is 70.
This brain teaser highlights the importance of following the rules of math, even when the order of operations appears a bit tricky.
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Factoring cubed polynomials
factoring cubed polynomials Related topics: math study guide
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beginner algebra
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Author Message
flachwib Posted: Sunday 09th of Nov 08:55
Hey dudes, I have just completed one week of my high school , and am getting a bit tensed about my factoring cubed polynomials home work. I just don’t seem to understand the topics. How
can one expect me to do my homework then? Please guide me.
From: India
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Vofj Posted: Sunday 09th of Nov 15:11
Timidrov Although I understand what your situation is , but if you could elaborate a bit on the areas in which you are facing difficulty , then I might be in a better position to help you out .
Anyhow I have a suggestion for you, try Algebrator. It can solve a wide range of questions, and it can do so within minutes. And that’s not it , it also gives a detailed step-by-step
description of how it arrived at a particular answer. That way you don’t just solve your problem but also get to understand how to go about solving it. I found this program to be
particularly useful for solving questions on factoring cubed polynomials. But that’s just my experience, I’m sure it’ll be good for other topics as well.
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Admilal Posted: Monday 10th of Nov 07:12
`Leker Hello friends I agree, Algebrator is the best . I used it in Algebra 1, Remedial Algebra and Pre Algebra. It helped me learn the hardest algebra problems. I'm grateful to it.
From: NW
AR, USA
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IntorionII Posted: Wednesday 12th of Nov 07:13
Maths can be so much fun if there actually is some software like this. Please send me the link to the software.
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Momepi Posted: Thursday 13th of Nov 09:17
Hello dudes , Based on your recommendations, I ordered the Algebrator to get myself educated with the fundamental theory of Remedial Algebra. The explanations on adding exponents and
unlike denominators were not only graspable but made the entire topic pretty exciting . Thanks a lot for all of you who directed me to check Algebrator!
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Mibxrus Posted: Thursday 13th of Nov 16:26
I am a regular user of Algebrator. It not only helps me finish my homework faster, the detailed explanations given makes understanding the concepts easier. I advise using it to help
improve problem solving skills.
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Stefan Wallin
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1. KTH
2. Profiles
3. Stefan Wallin
Peer reviewed
S. Wallin, O. Grundestam and A. V. Johansson, "Stabililty and laminarisation of turbulent rotating channel flow," in ADVANCES IN TURBULENCE XII : PROCEEDINGS OF THE 12TH EUROMECH EUROPEAN TURBULENCE
CONFERENCE, 2009, pp. 177-178.
Latest sync with DiVA:
2024-11-03 02:13:23
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The Broken Credit Loop (by Anish Acharya)
Anish Acharya, VP Product at Credit Karma (who I’m lucky to call a good friend) recently shared his thoughts on everything wrong with credit bureaus. It’s an excellent thread.
The core loop is (of course) that consumers build credit for access to financial products and the way they manage those financial products drives their score and access to future financial
Both parts of the loop suffer from inefficiencies that badly harm consumers - mispricing and “misscoring”. Let’s consider them in turn.
On mispricing: It’s well understood that there is massive mispricing problem for financial products; for example, the dispersion of APRs given a fixed credit score on an auto loan are
tremendously wide despite all of these folks being equally credit worthy [lender view].
This problem is increasingly getting solved by companies like Credit Karma who are making these markets efficient with technology. The magnitude of this problem is staggering (in the auto loan
example above mispricing costs US consumers ~$31bn per yr).
On “mis-scoring”: What’s much less well understood is that consumers are also fundamentally “mis-scored” - in many cases their score and report doesn’t represent their true credit worthiness and
this mostly acts to their detriment.
For example, there are many millions of erroneous / fraudulent collections on consumer’s reports and many consumers are either unaware they exist or are unaware that they can be remedied.
Or, as in your example, folks who you wouldn’t consider a “lender” like utilities or medical put small collections on your report which you would pay if you knew about them. Sadly many consumers
never notice and will be punished when they need credit next.
Also consider the cold start problem where you need credit to get credit, which usually means getting a credit card. However most of these consumers have been paying rent and utilities, which can
(and are just starting to be) used as an indicator of credit worthiness.
As a result millions and millions of consumers have access to worse financial products than they’re entitled to and those who have the least disposable income and are the most vulnerable are
being punished the most by this market inefficiency.
This is also why the whole cottage industry of “credit consulting” firms exist, though they often charge exorbitant fees and are predatory in their own right.
Bureaus want to do the right thing and efficiently score consumers. However there is an incentive mismatch in that their customers are lenders and other buyers of the data. So they’re only
incentivized to make credit files as representative as their customers (lenders) demand.
In the (near) future we’ll live in a world where consumers have credit files that represent their true credit worthiness and are always steered into the best, cheapest sources of capital. | {"url":"https://tarunsachdeva.com/20190302/broken-credit-loop.html","timestamp":"2024-11-07T10:24:10Z","content_type":"text/html","content_length":"11800","record_id":"<urn:uuid:87f9aba2-a08c-4fb1-9a1d-399e6a129596>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00156.warc.gz"} |
rite 0.0003532
+1 vote! +5
• vote up
you voted
+1 vote! +6 Answer by Clement (1453)
• vote up Scientific notation always has the form of "a * 10^b", sometimes also written as "a Eb", where a >= 1 and a < 10. In your case: 3. 532 e(-4).
you voted add a comment
+1 vote! +6 Answer by Anonymous
• vote up Simplify the expression r−3s5 t 2 r2st−2
you voted add a comment
Answer by parnell257 (109)
The decimal point needs to be moved so that there is one non-zero digit to the left of the decimal point. In this problem, the decimal point needs to be moved four places to the right. The number of
places moved is the power of ten. Moving to the right is a minus power. 3. 432 x 10^-4.
+1 vote! +4 Answer by MKourah (10)
• vote up The form of scientific notation is: A. Bx10^C (A) is a single digit whereas (B&C) can be multiple digits. So the answer is 3. 532x10^-4
you voted add a comment
+1 vote! +2 Answer by isabella (46)
• vote up 3.532x10-4
you voted add a comment
+1 vote! +2 Answer by TZ16 (367)
• vote up The number 0. 003532 can be written as 3. 532 x 10 (-4). The "(-4)" should be raised up as a superscript. It cannot be raised in this text
you voted
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Community Mathematics CTET MCQ | MCQ TUBE
Community Mathematics CTET MCQ
Community Mathematics CTET MCQ
Install our MCQTUBE Android app from the Google Play Store and prepare for any competitive government exams for free.
Related Posts:
01 One of the students of a class hardly takes in the class. How would you encourage him to express himself?
1. by organizing discussions
2. by organizing educational games/programmes in which children feel like speaking
3. by giving good marks to those who express themselves well
4. by encouraging children to take part in classroom activities
Option 1 – by organizing educational games/programmes in which children feel like speaking
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02 Most of use of mathematics done in the activities of human life, is
1. cultural
2. psychological
3. economical
03 What is the importance of mathematics at primary level?
1. cultural
2. religious
3. mental
04 The subject matter of mathematics should have ( Community Mathematics CTET MCQ )
1. pratical value
2. disciplinary value
3. cultural value
4. all of the above
Option 4 – all of the above
05 Major educational value of mathematics are
1. utilitarian values
2. disciplinary values
3. cultural values
4. all of the above
Option 4 – all of the above
06 “To appreciate the work of a mathematician” corresponds to which value ( Community Mathematics CTET MCQ )
1. intellectual
2. aesthetic
3. utilitarian
4. none of the above
Option 4 – none of the above
07 The moral value involved in teaching of mathematics helps in the development of ………. among children
1. learning by doing
2. logical power
3. self confidence
4. encouragement
Option 3 – self confidence
08 A suitable approach for explaining that a remainder is always less than the divisor to class IV students can be
1. perform lots of division sums on the black-board and show that every time the remainder is less than the divisor
2. explain verbally to the students, several times
3. represent division sums as mixed fractions and explain that the numerator of the fraction part is the remainder
4. grouping of objects in multiplies of divisor and showing that the number of objects, not in the group, are less than the divisor
Option 4 – grouping of objects in multiplies of divisor and showing that the number of objects, not in the group, are less than the divisor
09 Farhan went to school library and found that 100 books kept in story section are spoiled. 20 books are missing. 219 are kept in self and 132 were issued to students. How many story books were
there in the library?. Teacher can teach the following value through this question.
1. helping others
2. sharing books with others
3. taking good care of books
4. sense of cooperation
Option 3 – taking good care of books
10 Communication in mathematics class refers to developing ability to ( Community Mathematics CTET MCQ )
1. interpret data by looking at bar graphs
2. give prompt response to questions asked in the class
3. contradict the views of others on problems of mathematics
4. organise, consolidate and express mathematical thinking
Option 4 – organise, consolidate and express mathematical thinking
11 Learning objective for fourth grade students is given as ‘Students are able to order and compare two decimal numbers upto two decimal places’. This learning objective refers to
1. process goal
2. disposition goal
3. social goal
4. content goal
12 In order to develop, a good relationship with students in classroom, a teacher should
1. be friendly with all
2. communicate well
3. love his students
4. pay individual attention
Option 2 – communicate well
13 Oral examples help to develop which power in pupils?
1. thought
2. logics
3. imaginations
4. all of these
14 Success in developing values is mainly dependent upon
1. government
2. society
3. family
4. teacher
We covered all the Community Mathematics CTET MCQ above.
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Erie and EL traffic levels
Just wondered if anyone can give an accurate, realistic assessment of the number of freight trains that would have run on a typical day between Susquehanna and Port Jervis both in ca. 1950s Erie
days, and early 70s EL days before other pressures led to favoring the DL&W route. Public TTs show the clear picture for passenger, and freight schedules show how many -scheduled- freight trains per
day, but that is not all that helpful since all the Ordinaries were unscheduled, locals were not included in the schedules, and advance or following sections of scheduled trains were commonplace
(more like the normal everyday routine in at least some cases it seems).
Either first-hand recollections or documented paper sources would be equally appreciated!
by erie2521
This is not exactly what you requested but I have the dispatcher's train sheet for the stretch between Sparrowbush (just west of Port Jervis) and Newburgh Jct. for Wednesday, July 23, 1958.
I would assume that the symbol freights listed would have come off the Delaware Division. This is a hard trainsheet to interpret because it included Maybrook and there are several freights that show
up for just a few miles, maybe connections to and from Maybrook. Anyhow, here are the freights that I assume would have come from or gone to the Delaware Division:
#77. 35 loads, 45 empties.
#87. 33 loads, 58 empties.
#91. 56 loads, 63 empties.
#99. 88 loads, 1 empty.
One train just labeled "ORDY", I assume it means "ordinary", the Erie's expression for an extra. 4 loads, 67 empties.
One train labeled "XC". 91 loads, 75 empties.
One train labeled "RW" (I think). It appears to have terminated at Port Jervis. 2 loads, 7 empties.
The trainsheet lists two different numbers of cars, one at the top and one at the bottom. I listed the top. The bottom numbers were usually larger, some like RW were much larger (33 loads, 143
empties). Since the westbound trains proceeded up the sheet, I don't know which are starting and which are finishing numbers.
#A74. (advanced?) 93 loads.
#NE74. Went to Maybrook. 81 loads.
#98. 118 loads.
#NE98. Went to Maybrook. 73 loads.
#100. 101 loads.
One train labeled "Drop" (I think). 95 loads. This one started at Port Jervis and somewhere along the line it split into two trains. One with 35 cars went to Maybrook, the other with 60 cars went on
towards Jersey City.
One train labeled "MF." (I assume it meant manifest ) Also started at Port Jervis. 38 loads.
You will notice that the eastbound trains had no empties. The prevailing direction of traffic was east -as were the other railroads.
It is worth noting that 1958 was a recession year which may have curtailed this list somewhat - I don't know. Anyhow, here it is. Hope it will be of some use to you. If you have any question, let me
know. Ted Jackson
by oibu
thanks Ted, that gives a pretty interesting glimpse of a day's ops in that time period!
The number of trains is slightly less than I would have anticpated for that time, perhaps if there was a recession going on at the time that would explain a few less trains. I assume since this
covers out to Sparrowbush that it must include all traffic via both the main and Graham lines, right?
And yes, as you stated at this time a very big proportion of Erie E-W traffic was moving to/from the NH at Maybrook.
One wonders how the traffic level changed post-EL merger, with the preferential use of the Erie side and the onset of dedicated piggyback trains etc. Despite losses of certain traffic, I would guess
the Erie side was significantly busier in the 60s than in the late 50s.
by erie2521
Sorry, I forgot to mention that it covered both the main and Graham lines but did not have separate parts for each -sort of mixed in together. Not being familiar with the tower and junction names in
area, it was a hard trainsheet to read. Ted | {"url":"https://railroad.net/erie-and-el-traffic-levels-t85217.html","timestamp":"2024-11-11T13:35:16Z","content_type":"text/html","content_length":"42109","record_id":"<urn:uuid:5d598073-dbb8-4c05-b429-1ce4abcf9145>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00570.warc.gz"} |
Distance from Moroni to Mitsamiouli 25.3 ml | FromTo.City
1. Distance to Mitsamiouli
{{getDistance() | distanceFmt: getDistUnits()}} {{t('route.distance.types.'+getDistUnits()+'-s')}}
40.8 km 37.1 km
25.3 ml 23.1 ml
38 m 14 s
{{getDuration() | durationFmt: local.lang}}
Fuel consumption
{{getConsumption() | consumptionFmt}} {{t('route.consumption.cons-'+local.state.liquid_units)}}
Fuel cost
{{t('route.fuel-cost.'+local.state.cost_curr+'-pre')}} {{getFuelCost() | costFmt}} {{t('route.fuel-cost.'+local.state.cost_curr+'-short')}}
Distance between cities Moroni and Mitsamiouli is about 40.8 km which is 25.3 ml when driving by road through Boulevard de la République Poulaire de China, Rue Itsambuni, Boulevard Karthala, RN2,
Boulevard de la Ligue Arabe, RN1. The direct distance by air, as if you take a flight - 37.1 km or 23.1 ml.
How long will the trip take?
If you are traveling by car and will drive at an average speed of 39 ml/h or 64 km/h at least 24 hours per day, then you will need 38 m 14 s to get to your destination.
Calculate the cost, fuel consumption
The consumption of petrol or other fuel that is used in your car on the entire route is 1.2 gall or 4.1 lit. at a consumption of 10 lit./100 km or 22 MPG that will cost you about $4.03 at the cost of
petrol $3.5 per gallon. If you plan a roundtrip the fuel cost will be $undefined.
To get more precise information, depending on the cost of petrol and fuel consumption of your car, please use the fuel consumption calculator on this page.
Share your experience with others
Plan a trip with friends? Do not forget to share with them your itinerary. Send them the link and tell them a bit about your journey. Publish the link of this route in your favourite social network
so that friends can learn more about your interesting experience. We wish you a great mood! Yours sincerely, FromTo.City | {"url":"https://fromto.city/en/distance-from-city-to-city/moroni/mitsamiouli,mitsamiouli/comoros,ngazidja","timestamp":"2024-11-08T11:45:33Z","content_type":"text/html","content_length":"67905","record_id":"<urn:uuid:44bc0eef-ee7d-46a3-afb7-0870c948c83c>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00812.warc.gz"} |
cfr: Estimate disease severity and under-reporting
cfr is an R package to estimate disease severity and under-reporting in real-time, accounting for delays in epidemic time-series.
cfr provides simple, fast methods to calculate the overall or static case fatality risk (CFR) of an outbreak up to a given time point, as well as how the CFR changes over the course of the outbreak.
cfr can help estimate disease under-reporting in real-time, accounting for delays reporting the outcomes of cases.
cfr implements methods outlined in Nishiura et al. (2009). There are plans to add estimates based on other methods.
cfr is developed at the Centre for the Mathematical Modelling of Infectious Diseases at the London School of Hygiene and Tropical Medicine as part of the Epiverse-TRACE initiative.
cfr can be installed from CRAN using
The current development version of cfr can be installed from GitHub using the pak package.
Quick start
Overall severity of the 1976 Ebola outbreak
This example shows how to use cfr to estimate the overall case fatality risks from the 1976 Ebola outbreak (Camacho et al. 2014), while correcting for delays using a Gamma-distributed onset to death
duration taken from Barry et al. (2018), with a shape \(k\) of 2.40 and a scale \(\theta\) of 3.33.
# Load package
# Load the Ebola 1976 data provided with the package
# Focus on the first 20 days the outbreak
ebola1976_first_30 <- ebola1976[1:30, ]
# Calculate the static CFR without correcting for delays
cfr_static(data = ebola1976_first_30)
#> severity_estimate severity_low severity_high
#> 1 0.4740741 0.3875497 0.5617606
# Calculate the static CFR while correcting for delays
data = ebola1976_first_30,
delay_density = function(x) dgamma(x, shape = 2.40, scale = 3.33)
#> severity_estimate severity_low severity_high
#> 1 0.9422 0.8701 0.9819
Change in real-time estimates of overall severity during the 1976 Ebola outbreak
In this example we show how the estimate of overall severity can change as more data on cases and deaths over time becomes available, using the function cfr_rolling(). Because there is a delay from
onset-to-death, a simple “naive” calculation that just divides deaths-to-date by cases-to-date will underestimate severity. The cfr_rolling() function uses the estimate_severity() adjustment
internally to account for delays, and instead compares deaths-to-date with cases-with-known-outcome-to-date. The adjusted estimate converges to the naive estimate as the outbreak declines and a
larger proportion of cases have known outcomes.
# Calculate the CFR without correcting for delays on each day of the outbreak
rolling_cfr_naive <- cfr_rolling(
data = ebola1976
# see the first few rows
#> date severity_estimate severity_low severity_high
#> 1 1976-08-25 0 0 0.975
#> 2 1976-08-26 0 0 0.975
#> 3 1976-08-27 0 0 0.975
#> 4 1976-08-28 0 0 0.975
#> 5 1976-08-29 0 0 0.975
#> 6 1976-08-30 0 0 0.975
# Calculate the rolling daily CFR while correcting for delays
rolling_cfr_corrected <- cfr_rolling(
data = ebola1976,
delay_density = function(x) dgamma(x, shape = 2.40, scale = 3.33)
#> date severity_estimate severity_low severity_high
#> 1 1976-08-25 NA NA NA
#> 2 1976-08-26 1e-04 1e-04 0.9999
#> 3 1976-08-27 1e-04 1e-04 0.9999
#> 4 1976-08-28 1e-04 1e-04 0.9999
#> 5 1976-08-29 1e-04 1e-04 0.9990
#> 6 1976-08-30 1e-04 1e-04 0.9942
We plot the rolling CFR to visualise how severity changes over time, using the ggplot2 package. The plotting code is hidden here.
# combine the data for plotting
rolling_cfr_naive$method <- "naive"
rolling_cfr_corrected$method <- "corrected"
data_cfr <- rbind(
Package vignettes
More details on how to use cfr can be found in the online documentation as package vignettes, under “Articles”.
To report a bug please open an issue.
Contributions to cfr are welcomed. Please follow the package contributing guide.
Code of conduct
Please note that the cfr project is released with a Contributor Code of Conduct. By contributing to this project, you agree to abide by its terms.
cfr functionality overlaps with that of some other packages, including
• coarseDataTools is an R package that allows estimation of relative case fatality risk between covariate groups while accounting for delays due to survival time, when numbers of deaths and
recoveries over time are known. cfr uses simpler methods from Nishiura et al. (2009) that can be applied when only cases and deaths over time are known, generating estimates based on all data to
date, as well as time-varying estimates. cfr can also convert estimates of cases with known outcomes over time into an estimate of under-ascertainment, if a baseline estimate of fatality risk is
available from the literature (e.g. from past outbreaks).
• EpiNow2 is an R package that can allow estimation of case fatality risk if it is defined as a secondary observation of cases. In particular, it allows for estimation that accounts for the smooth
underlying epidemic process, but this requires additional computational effort. A comparison of these methods is planned for a future release.
cfr is in future expected to benefit from the functionality of the forthcoming epiparameter package, which is also developed by Epiverse-TRACE. epiparameter aims to provide a library of
epidemiological parameters to parameterise delay density functions, as well as the convenient <epidist> class to store, access, and pass these parameters for delay correction.
Barry, Ahmadou, Steve Ahuka-Mundeke, Yahaya Ali Ahmed, Yokouide Allarangar, Julienne Anoko, Brett Nicholas Archer, Aaron Aruna Abedi, et al. 2018. “Outbreak of Ebola virus disease in the Democratic
Republic of the Congo, April–May, 2018: an epidemiological study.”
The Lancet
392 (10143): 213–21.
Camacho, A., A. J. Kucharski, S. Funk, J. Breman, P. Piot, and W. J. Edmunds. 2014. “Potential for Large Outbreaks of Ebola Virus Disease.”
9 (December): 70–78.
Nishiura, Hiroshi, Don Klinkenberg, Mick Roberts, and Johan A. P. Heesterbeek. 2009. “Early Epidemiological Assessment of the Virulence of Emerging Infectious Diseases: A Case Study of an Influenza
4 (8): e6852. | {"url":"https://cran.ma.imperial.ac.uk/web/packages/cfr/readme/README.html","timestamp":"2024-11-05T06:40:39Z","content_type":"application/xhtml+xml","content_length":"23369","record_id":"<urn:uuid:22ff316d-3030-4bcb-b4e2-fcdf2f8ea54a>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00613.warc.gz"} |
Times Tables Worksheets 1 12 Printable
Times Tables Worksheets 1 12 Printable - These multiplication worksheets are a. Multiplication facts beyond the 12 times table. Web multiplication facts up to the 12 times table. Web our
multiplication worksheets are free to download, easy to use, and very flexible. Web create and print multiplication fact worksheets up to 12 with our worksheet generator tool, and practice utilizing
the. When you are just getting started learning the multiplication tables, these simple printable pages are.
Web our multiplication worksheets are free to download, easy to use, and very flexible. Web create and print multiplication fact worksheets up to 12 with our worksheet generator tool, and practice
utilizing the. Multiplication facts beyond the 12 times table. These multiplication worksheets are a. Web multiplication facts up to the 12 times table. When you are just getting started learning the
multiplication tables, these simple printable pages are.
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Custom Excel number format
This tutorial explains the basics of the Excel number format and provides the detailed guidance to create custom formatting. You will learn how to show the required number of decimal places, change
alignment or font color, display a currency symbol, round numbers by thousands, show leading zeros, and much more.
Microsoft Excel has a lot of built-in formats for number, currency, percentage, accounting, dates and times. But there are situations when you need something very specific. If none of the inbuilt
Excel formats meets your needs, you can create your own number format.
Number formatting in Excel is a very powerful tool, and once you learn how to use it property, your options are almost unlimited. The aim of this tutorial is to explain the most essential aspects of
Excel number format and set you on the right track to mastering custom number formatting.
How to create a custom number format in Excel
To create a custom Excel format, open the workbook in which you want to apply and store your format, and follow these steps:
1. Select a cell for which you want to create custom formatting, and press Ctrl+1 to open the Format Cells dialog.
2. Under Category, select Custom.
3. Type the format code in the Type box.
4. Click OK to save the newly created format.
Tip. Instead of creating a custom number format from scratch, you choose a built-in Excel format close to your desired result, and customize it.
Wait, wait, but what do all those symbols in the Type box mean? And how do I put them in the right combination to display the numbers the way I want? Well, this is what the rest of this tutorial is
all about :)
Understanding Excel number format
To be able to create a custom format in Excel, it is important that you understand how Microsoft Excel sees the number format.
An Excel number format consists of 4 sections of code, separated by semicolons, in this order:
Here's an example of a custom Excel format code:
1. Format for positive numbers (display 2 decimal places and a thousands separator).
2. Format for negative numbers (the same as for positive numbers, but enclosed in parentheses).
3. Format for zeros (display dashes instead of zeros).
4. Format for text values (display text in magenta font color).
Excel formatting rules
When creating a custom number format in Excel, please remember these rules:
1. A custom Excel number format changes only the visual representation, i.e. how a value is displayed in a cell. The underlying value stored in a cell is not changed.
2. When you are customizing a built-in Excel format, a copy of that format is created. The original number format cannot be changed or deleted.
3. Excel custom number format does not have to include all four sections.
If a custom format contains just 1 section, that format will be applied to all number types - positive, negative and zeros.
If a custom number format includes 2 sections, the first section is used for positive numbers and zeros, and the second section - for negative numbers.
A custom format is applied to text values only if it contains all four sections.
4. To apply the default Excel number format for any of the middle sections, type General instead of the corresponding format code.
For example, to display zeros as dashes and show all other values with the default formatting, use this format code: General; -General; "-"; General
Note. The General format included in the 2^nd section of the format code does not display the minus sign, therefore we include it in the format code.
5. To hide a certain value type(s), skip the corresponding code section, and only type the ending semicolon.
For example, to hide zeros and negative values, use the following format code: General; ; ; General. As the result, zeros and negative value will appear only in the formula bar, but will not be
visible in cells.
6. To delete a custom number format, open the Format Cells dialog, select Custom in the Category list, find the format you want to delete in the Type list, and click the Delete button.
Digit and text placeholders
For starters, let's learn 4 basic placeholders that you can use in your custom Excel format.
Code Description Example
#.00 - always displays 2 decimal places.
0 Digit placeholder that displays insignificant zeros.
If you type 5.5 in a cell, it will display as 5.50.
#.## - displays up to 2 decimal places.
Digit placeholder that only displays significant digits, without extra zeros.
# If you type 5.5 in a cell, it will display as 5.5.
That is, if a number doesn't need a certain digit, it won't be displayed.
If you type 5.555, it will display as 5.56.
? Digit placeholder that leaves a space for insignificant zeros on either side of the decimal point but doesn't display them. It #.??? - displays a maximum of 3 decimal places and aligns numbers
is often used to align numbers in a column by decimal point. in a column by decimal point.
@ Text placeholder 0.00; -0.00; 0; [Red]@ - applies the red font color for text
The following screenshot demonstrates a few number formats in action:
As you may have noticed in the above screenshot, the digit placeholders behave in the following way:
• If a number entered in a cell has more digits to the right of the decimal point than there are placeholders in the format, the number is "rounded" to as many decimal places as there are
For example, if you type 2.25 in a cell with #.# format, the number will display as 2.3.
• All digits to the left of the decimal point are displayed regardless of the number of placeholders.
For example, if you type 202.25 in a cell with #.# format, the number will display as 202.3.
Below you will find a few more examples that will hopefully shed more light on number formatting in Excel.
Format Description Input value Display as
2 2.00
#.00 Always display 2 decimal places. 2.5 2.50
0.5556 .56
2 2.
#.## Shows up to 2 decimal places, without insignificant zeros. 2.5 2.5
0.5556 0.56
2 2.0
#.0# Display a minimum of 1 and a maximum of 2 decimal places. 2.205 2.21
0.555 .56
22.55 22.55
???.??? Display up to 3 decimal places with aligned decimals. 2.5 2.5
2222.5555 2222.556
0.55 .55
Excel formatting tips and guidelines
Theoretically, there are an infinite number of Excel custom number formats that you can make using a predefined set of formatting codes listed in the table below. And the following tips explain the
most common and useful implementations of these format codes.
Format Code Description
General General number format
# Digit placeholder that represents optional digits and does not display extra zeros.
0 Digit placeholder that displays insignificant zeros.
? Digit placeholder that leaves a space for insignificant zeros but doesn't display them.
@ Text placeholder
. (period) Decimal point
, (comma) Thousands separator. A comma that follows a digit placeholder scales the number by a thousand.
\ Displays the character that follows it.
" " Display any text enclosed in double quotes.
% Multiplies the numbers entered in a cell by 100 and displays the percentage sign.
/ Represents decimal numbers as fractions.
E Scientific notation format
_ (underscore) Skips the width of the next character. It's commonly used in combination with parentheses to add left and right indents, _( and _) respectively.
* (asterisk) Repeats the character that follows it until the width of the cell is filled. It's often used in combination with the space character to change alignment.
[] Create conditional formats.
How to control the number of decimal places
The location of the decimal point in the number format code is represented by a period (.). The required number of decimal places is defined by zeros (0). For example:
• 0 or # - display the nearest integer with no decimal places.
• 0.0 or #.0 - display 1 decimal place.
• 0.00 or #.00 - display 2 decimal places, etc.
The difference between 0 and # in the integer part of the format code is as follows. If the format code has only pound signs (#) to the left of the decimal point, numbers less than 1 begin with a
decimal point. For example, if you type 0.25 in a cell with #.00 format, the number will display as .25. If you use 0.00 format, the number will display as 0.25.
How to show a thousands separator
To create an Excel custom number format with a thousands separator, include a comma (,) in the format code. For example:
• #,### - display a thousands separator and no decimal places.
• #,##0.00 - display a thousands separator and 2 decimal places.
Round numbers by thousand, million, etc.
As demonstrated in the previous tip, Microsoft Excel separates thousands by commas if a comma is enclosed by any digit placeholders - pound sign (#), question mark (?) or zero (0). If no digit
placeholder follows a comma, it scales the number by thousand, two consecutive commas scale the number by million, and so on.
For example, if a cell format is #.00, and you type 5000 in that cell, the number 5.00 is displayed. For more examples, please see the screenshot below:
Text and spacing in custom Excel number format
To display both text and numbers in a cell, do the following:
• To add a single character, precede that character with a backslash (\).
• To add a text string, enclose it in double quotation marks (" ").
For example, to indicate that numbers are rounded by thousands and millions, you can add \K and \M to the format codes, respectively:
• To display thousands: #.00,\K
• To display millions: #.00,,\M
Tip. To make the number format better readable, include a space between a comma and backward slash.
The following screenshot shows the above formats and a couple more variations:
And here is another example that demonstrates how to display text and numbers within a single cell. Supposing, you want to add the word "Increase" for positive numbers, and "Decrease" for negative
numbers. All you have to do is include the text enclosed in double quotes in the appropriate section of your format code:
#.00" Increase"; -#.00" Decrease"; 0
Tip. To include a space between a number and text, type a space character after the opening or before the closing quote depending on whether the text precedes or follows the number, like in "Increase
In addition, the following characters can be included in Excel custom format codes without the use of backslash or quotation marks:
Symbol Description
+ and - Plus and minus signs
( ) Left and right parentheses
: Colon
^ Caret
' Apostrophe
{ } Curly brackets
< > Less-than and greater than signs
= Equal sign
/ Forward slash
! Exclamation point
& Ampersand
~ Tilde
Space character
A custom Excel number format can also accept other special symbols such as currency, copyright, trademark, etc. These characters can be entered by typing their four-digit ANSI codes while holding
down the ALT key. Here are some of the most useful ones:
Symbol Code Description
™ Alt+0153 Trademark
© Alt+0169 Copyright symbol
° Alt+0176 Degree symbol
± Alt+0177 Plus-Minus sign
µ Alt+0181 Micro sign
For example, to display temperatures, you can use the format code #"°F" or #"°C" and the result will look similar to this:
You can also create a custom Excel format that combines some specific text and the text typed in a cell. To do this, enter the additional text enclosed in double quotes in the 4^th section of the
format code before or after the text placeholder (@), or both.
For example, to proceed the text typed in the cell with some other text, say "Shipped in", use the following format code:
General; General; General; "Shipped in "@
Including currency symbols in a custom number format
To create a custom number format with the dollar sign ($), simply type it in the format code where appropriate. For example, the format $#.00 will display 5 as $5.00.
Other currency symbols are not available on most of standard keyboards. But you can enter the popular currencies in this way:
• Turn NUM LOCK on, and
• Use the numeric keypad to type the ANSI code for the currency symbol you want to display.
Symbol Currency Code
€ Euro ALT+0128
£ British Pound ALT+0163
¥ Japanese Yen ALT+0165
¢ Cent Sign ALT+0162
The resulting number formats may look something similar to this:
If you want to create a custom Excel format with some other currency, follow these steps:
• Open the Format Cells dialog, select Currency under Category, and choose the desired currency from the Symbol drop-down list, e.g. Russian Ruble:
• Switch to Custom category, and modify the built-in Excel format the way you want. Or, copy the currency code from the Type field, and include it in your own number format:
How to display leading zeros with Excel custom format
If you try entering numbers 005 or 00025 in a cell with the default General format, you would notice that Microsoft Excel removes leading zeros because the number 005 is same as 5. But sometimes, we
do want 005, not 5!
The simplest solution is to apply the Text format to such cells. Alternatively, you can type an apostrophe (') in front of the numbers. Either way, Excel will understand that you want any cell value
to be treated as a text string. As the result, when you type 005, all leading zeros will be preserved, and the number will show up as 005.
If you want all numbers in a column to contain a certain number of digits, with leading zeros if needed, then create a custom format that includes only zeros.
As you remember, in Excel number format, 0 is the placeholder that displays insignificant zeros. So, if you need numbers consisting of 6 digits, use the following format code: 000000
And now, if you type 5 in a cell, it will appear as 000005; 50 will appear as 000050, and so on:
Tip. If you are entering phone numbers, zip codes, or social security numbers that contain leading zeros, the easiest way is to apply one of the predefined Special formats. Or, you can create the
desired custom number format. For example, to properly display international seven-digit postal codes, use this format: 0000000. For social security numbers with leading zeros, apply this format:
Percentages in Excel custom number format
To display a number as a percentage of 100, include the percent sign (%) in your number format.
For example, to display percentages as integers, use this format: #%. As the result, the number 0.25 entered in a cell will appear as 25%.
To display percentages with 2 decimal places, use this format: #.00%
To display percentages with 2 decimal places and a thousands separator, use this one: #,##.00%
Fractions in Excel number format
Fractions are special in terms that the same number can be displayed in a variety of ways. For example, 1.25 can be shown as 1 ¼ or 5/5. Exactly which way Excel displays the fraction is determined by
the format codes that you use.
For decimal numbers to appear as fractions, include forward slash (/) in your format code, and separate an integer part with a space. For example:
• # #/# - displays a fraction remainder with up to 1 digit.
• # ##/## - displays a fraction remainder with up to 2 digits.
• # ###/### - displays a fraction remainder with up to 3 digits.
• ###/### - displays an improper fraction (a fraction whose numerator is larger than or equal to the denominator) with up to 3 digits.
To round fractions to a specific denominator, supply it in your number format code after the slash. For example, to display decimal numbers as eighths, use the following fixed base fraction format: #
The following screenshot demonstrated the above format codes in action:
As you probably know, the predefined Excel Fraction formats align numbers by the fraction bar (/) and display the whole number at some distance from the remainder. To implement this alignment in your
custom format, use the question mark placeholders (?) instead of the pound signs (#) like shown in the following screenshot:
Tip. To enter a fraction in a cell formatted as General, preface the fraction with a zero and a space. For instance, to enter 4/8 in a cell, you type 0 4/8. If you type 4/8, Excel will assume you are
entering a date, and change the cell format accordingly.
Create a custom Scientific Notation format
To display numbers in Scientific Notation format (Exponential format), include the capital letter E in your number format code. For example:
• 00E+00 - displays 1,500,500 as 1.50E+06.
• #0.0E+0 - displays 1,500,500 as 1.5E+6
• #E+# - displays 1,500,500 as 2E+6
Show negative numbers in parentheses
At the beginning of this tutorial, we discussed the 4 code sections that make up an Excel number format: Positive; Negative; Zero; Text
Most of the format codes we've discussed so far contained just 1 section, meaning that the custom format is applied to all number types - positive, negative and zeros.
To make a custom format for negative numbers, you'd need to include at least 2 code sections: the first will be used for positive numbers and zeros, and the second - for negative numbers.
To show negative values in parentheses, simply include them in the second section of your format code, for example: #.00; (#.00)
Tip. To line up positive and negative numbers at the decimal point, add an indent to the positive values section, e.g. 0.00_); (0.00)
Display zeroes as dashes or blanks
The built-in Excel Accounting format shows zeros as dashes. This can also be done in your custom Excel number format.
As you remember, the zero layout is determined by the 3^rd section of the format code. So, to force zeros to appear as dashes, type "-" in that section. For example: 0.00;(0.00);"-"
The above format code instructs Excel to display 2 decimal places for positive and negative numbers, enclose negative numbers in parentheses, and turn zeros into dashes.
If you don't want any special formatting for positive and negative numbers, type General in the 1^st and 2^nd sections: General; -General; "-"
To turn zeroes into blanks, skip the third section in the format code, and only type the ending semicolon: General; -General; ; General
Add indents with custom Excel format
If you don't want the cell contents to ride up right against the cell border, you can indent information within a cell. To add an indent, use the underscore (_) to create a space equal to the width
of the character that follows it.
The commonly used indent codes are as follows:
• To indent from the left border: _(
• To indent from the right border: _)
Most often, the right indent is included in a positive number format, so that Excel leaves space for the parentheses enclosing negative numbers.
For example, to indent positive numbers and zeros from the right and text from the left, you can use the following format code:
0.00_);(0.00); 0_);_(@
Or, you can add indents on both sides of the cell:
To format financial data or other types of data where it's important to distinguish between positive and negative numbers, you can use the following format, which indents positive numbers and zeros
from the right border. Additionally, it rounds all numbers to the nearest integer and displays them with a space as a thousand separator. Negative numbers are displayed in parentheses and in red font
# ##0_); [Red](# ##0)
The indent codes move the cell data by one character width. To move values from the cell edges by more than one character width, include 2 or more consecutive indent codes in your number format. The
following screenshot demonstrates indenting cell contents by 1 and 2 characters:
Change font color with custom number format
Changing the font color for a certain value type is one of the simplest things you can do with a custom number format in Excel, which supports 8 main colors. To specify the color, just type one of
the following color names in an appropriate section of your number format code.
[Black] [Magenta]
[Green] [Yellow]
[White] [Cyan]
[Blue] [Red]
Note. The color code must be the first item in the section.
For example, to leave the default General format for all value types, and change only the font color, use the format code similar to this:
Or, combine color codes with the desired number formatting, e.g. display the currency symbol, 2 decimal places, a thousands separator, and show zeros as dashes:
[Blue]$#,##0.00; [Red]-$#,##0.00; [Black]"-"; [Magenta]@
Repeat characters with custom format codes
To repeat a specific character in your custom Excel format so that it fills the column width, type an asterisk (*) before the character.
For example, to include enough equality signs after a number to fill the cell, use this number format: #*=
Or, you can include leading zeros by adding *0 before any number format, e.g. *0#
This formatting technique is commonly used to change cell alignment as demonstrated in the next formatting tip.
How to change alignment in Excel with custom number format
A usual way to change alignment in Excel is using the Alignment tab on the ribbon. However, you can "hardcode" cell alignment in a custom number format if needed.
For example, to align numbers left in a cell, type an asterisk and a space after the number code, for example: "#,###* " (double quotes are used only to show that an asterisk is followed by a space,
you don't need them in a real format code).
Making a step further, you could have numbers aligned left and text entries aligned right using this custom format:
#,###* ; -#,###* ; 0* ;* @
This method is used in the built-in Excel Accounting format . If you apply the Accounting format to some cell, then open the Format Cells dialog, switch to the Custom category and look at the Type
box, you will see this format code:
_($* #,##0.00_);_($* (#,##0.00);_($* "-"??_);_(@_)
The asterisk that follows the currency sign tells Excel to repeat the subsequent space character until the width of a cell is filled. This is why the Accounting number format aligns the currency
symbol to the left, number to the right, and adds as many spaces as necessary in between.
Apply custom number formats based on conditions
To have your custom Excel format applied only if a number meets a certain condition, type the condition consisting of a comparison operator and a value, and enclose it in square brackets [].
For example, to displays numbers that are less than 10 in a red font color, and numbers that are greater than or equal to 10 in a green color, use this format code:
Additionally, you can specify the desired number format, e.g. show 2 decimal places:
And here is another extremely useful, though rarely used formatting tip. If a cell displays both numbers and text, you can make a conditional format to show a noun in a singular or plural form
depending on the number. For example:
[=1]0" mile";0.##" miles"
The above format code works as follows:
• If a cell value is equal to 1, it will display as "1 mile".
• If a cell value is greater than 1, the plural form "miles" will show up. Say, the number 3.5 will display as "3.5 miles".
Taking the example further, you can display fractions instead of decimals:
[=1]?" mile";# ?/?" miles"
In this case, the value 3.5 will appear as "3 1/2 miles".
Tip. To apply more sophisticated conditions, use Excel's Conditional Formatting feature, which is specially designed to handle the task.
Dates and times formats in Excel
Excel date and times formats are a very specific case, and they have their own format codes. For the detailed information and examples, please check out the following tutorials:
Well, this is how you can change number format in Excel and create your own formatting. Finally, here's a couple of tips to quickly apply your custom formats to other cells and workbooks:
• A custom Excel format is stored in the workbook in which it is created and is not available in any other workbook. To use a custom format in a new workbook, you can save the current file as a
template, and then use it as the basis for a new workbook.
• To apply a custom format to other cells in a click, save it as an Excel style - just select any cell with the required format, go to the Home tab > Styles group, and click New Cell Style….
To explore the formatting tips further, you can download a copy of the Excel Custom Number Format workbook we used in this tutorial. I thank you for reading and hope to see you again next week!
You may also be interested in
640 comments
1. Hi guys, I need to make a custom currency format same as accounting one, but I need it in ARABIC figures with ARABIC currency symbol "ج.م.". the numbers & symbol should be aligned same as
accounting excel format.
Any way to do please?
□ Hi, Mohammed,
Simply select your data, press Ctrl+1, and go to the Accounting category in the dialogue window that appears. You will see a drop-down menu for Symbols. Pick Arabic (Egypt) from the list and
hit OK to save changes.
2. Svetlana, you are a genius! I've been looking everywhere online for the list of font colors that can be used with custom number format" and only found it on your site (It's 'Magenta', not 'Pink',
Your explanations are so well articulated that I've bookmarked this page for future reference.
Thank you very much for you help, and Merry Christmas from New Zealand :-)
3. #.##% = 1.%
genaral% = 100%
can i know how to remove the decimal point? or did i use the wrong code for it?
i want it to have the expending function like how the general give where if you don't have decimal it don't show but if you have then only it shows.
4. Shaina:
Your request is somewhat confusing because Excel is not going to see "L18-021547" as a number. Then you say you're trying to come up with a way to see the field as a number with an asterisk at
the beginning and end of the text.
=CONCATENATE("*",A2,"*") will add an asterisk to the front-end and back-end but it will still be text. Excel is not able to convert the letter "L" and the "-" into a number.
Why do you need the data as a number?
5. We get a list of numbers in that actually start with a letter (Example L18-021547) and I would like to figure out how to apply a format to those numbers that adds an asterisk (*) to the beginning
and the end of that without having to add them to every single field manually (*L18-021547*). We get a bundle of 100+ of these numbers for each report and I am just trying to come up with a way
for it to see the field as a number, which it is not doing at the moment because it starts with "L". Any suggests?
□ try using this formating in cell format
each time you just need to type out the number and it will add the * and also the L for you.
6. i like it
7. Hi,
could you please help me with the date format, I want space between days and months numbers like dd/mm/yyyy to d d/m m/yyyy.(2 2/1 1/2018)
8. I want convert this text 20122018(ddmmyyyy) to this date dd-mm-yyyy format please how we will convert using text formula
9. If I am trying to format a phone number to read (so zeros need to be visible):
(XXX) XXX XXXX
1 (XXX) XXX XXXX
I cannot find the code to use a comma...
10. Hi Svetlana,
Trying to use Custom Format to color numbers in a cell based on the following conditions: green for numbers less than or equal to 45, amber for numbers greater than 45 but less than 60, and red
for numbers 60 and greater.
Any suggestions?
□ Hi Dana,
This can be done with Excel conditional formatting. You can find the detailed steps and examples of conditional formatting rules in this tutorial:
Excel conditional formatting based on formulas
11. Hi Svetlana,
Trying to use Custom Format to color numbers in a cell based on the these conditions: green for 45 but =60.
Any suggestions?
12. hello
how to format a cell like: AAAA12345-6?
13. I NEED TO CONVERT ALL THE NUMBERS THAT I ENTER SHOULD BE IN LAKHS.
14. I am trying to format my phone numbers to look like this: 303.555.9876, but instead they look like this:3035559876..
I will like to know how to fix this problem.
□ This works for me.
15. Super useful. thanks
16. I need to convert a 00000 type formatted cell content (shows as 00123) to text with same number of leading zeros. Since Excel understands the value only as 123 converting the cell format to text
removes the zeros. Thanks in advance.
17. I'm trying to round the number 1,230 to 1,200; or 43,540 to 43,500. Thanks.
□ Hello, Jamael,
It looks like you can solve your task using the FLOOR function. If your value is in A1, please try to apply the following formula:
You can learn more about rounding in Excel in the following article on our blog:
How to round numbers in Excel - ROUND, ROUNDUP, ROUNDDOWN and other functions
Hope you’ll find this information helpful.
18. Hi Svetlana,
I'm having trouble formatting text and would like to omit the first two letters. I'm aware that I could use RIGHT(CELL,LEN(CELL)-2) but I don't want to change the actual text, just to display it
in a different way, with the underlying 'value' of the cell staying the same.
For example, I can type in the custom formatting bar "Rob" to display only "Rob" no matter what I actually type into the cell.
I would like to format cells so that the first two letters aren't displayed, for example, entering "Steve" would show "eve" in the cell, or "Howard" showing "ward". (Sounds weird but I do
genuinely need this.)
Is this possible?
Thank you for the fantastic article, it was very helpful and well written.
19. I NEED 4556 A,4557 B,4558 C,4559 D ,,,,,,,,,, serial numbers with 6 rows and 6 columns formula send me plz
□ instead of using conditional formatting in excel you can use a simple formula . 1&char(64+Rows(a$1:a1)) will give you 1A try this
20. I'm trying to convert a number, say 123.50 to 000012350 (always 10 characters, need pennies but no decimal point.
□ Hi Kevin,
Assuming the original number is in A1, the conversion can be performed with this formula:
=TEXT(A1*100, "0000000000")
However, the result of the formula will be a numeric string, not a number.
If the result should be number, you can multiply the original numbers by 100 to get rid of the decimal point (=A1*100), replace formulas with values if needed (an intermediate result will be
12350), and then apply this custom format: 0000000000 to always display 10 characters with the required number of leading zeros.
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Mathematics of Practical Low Power Tic Tac Warp Drive Star Ships 12/25/20
Note added Dec 27, 2020 - Synopsis
The Cartan formalism is Spartan in the sense that Dirac liked as seen in his “bra - ket” QM formalism.
Cartan formalism is manifestly coordinate-independent - it is local objective classical field reality in the way Einstein liked.
However, it has profound physical significance, in that the Hodge dual * operator describes the transition from vacuum to matter as in electrodynamics
D = E + 4piP
B = H + 4piM
EM 2 Form F describes E and B in vacuum
*F describes H and D in matter connection to sources of the field.
In vacuum P = 0 and M = 0
The tetrads e are Cartan exterior 1 forms with dual co-forms w corresponding to dx in the usual coordinate representation.
The tetrad 1-forms as variable fields over spacetime are directly induced by locally gauging T4 translation Lie group.
The invariant 0-form metric field for curved spacetime is
ds^2 = *[e/\e/\w/\w]
* = Hodge dual
The Levi-Civita metric field connection is the 1-form
(LC) = *[e/\de]
The covariant exterior derivative is
D = d + *[e/\de]
The vacuum Weyl curvature 2-form is
R = D(LC) = {d + *[e/\de]}*[e/\de]
DR = 0 Bianchi identity in vacuum
The inside matter Ricci + Weyl curvature 2-form is given by Einstein’s modified gravity field equation
*R = - 8pi(G/c^4)|S||T|cos(s + t)
S = |S| exp(is) Sarfatti-Wanser 0-form scalar field matter EM response function
T = source stress-energy 2-form
D(*R + 8pi(G/c^4)|S||T|cos(s + t)] = 0
Local conservation of stress-energy current densities.
D*R =/= 0 inside matter
3rd Draft Typo Corrected
When I use “teleparallel space” I mean (and Shipov and others mean) a generalized Minkowski space in which the covariant curl of the general connection C with itself vanishes globally - in symbols of
exterior forms
DC = 0
D = d + C
C = 1-form
d = exterior derivative
d^2 = 0
The “Curvature” F (generalized) is the 2-FORM
F = DC
This has nothing to do directly with comparing separated objects - that is done by “parallel transport” of the object V parallel to itself using the connection
DV = 0 like in the geodesic equation as a particular example where V is the timelike tangent vector 1 form e0 moved parallel to itself forming a timelike geodesic path.
Where in general DC =/= 0
Example 1 classical electrodynamics in vacuum with possibly elementary electric charge current densities J = 1-form.
A = 1 form connection from locally gauging U(1) internal fiber space global group in Minkowski flat space-time and for LIFs where C = 0.
dA = F = EM field 2-form includes E and B 3-vector fields.
d*F = J (Ampere’s law + Gauss’s law)
*F = Hodge dual of F includes EM susceptibilities of matter.
d^2*F = dJ = 0 local conservation of source electric current densities
dF = 0 (Bianchi identity) (Faraday induction, and no micro magnetic monopoles in classical vacuum).
Gauge transformations A —> A’ = A + kdPhi
F —> F’ = F U(1) gauge invariant
Phi = 0 form phase.
Example 2 Einstein GR
C = Levi-Civita metric connection
R = DC = Riemann Christoffel Curvature 2-form
DR = 0 Bianchi identity
*R = - 8pi(G/c^4)ST Einstein’s field equation
T = source stress-energy 2 form
S = Sarfatti Wanser scalar field from EM response of meta-material Tic Tac fuselage to applied EM pump field inside meta-material.
DT = 0 local conservation of stress-energy source currents only if
D*R = 0
In most materials *R = R to a good approximation (SELF DUALITY)
But in pumped meta-material dissipative resonances
*R =/= R SELF-DUALITY BREAKDOWN
D(*R + 8pi(G/c^4)ST) = 0
With direct current exchanges among *R inside metamateria, S scalar field and T input EM pump field)
D*R + (8piG/c^4)[SDT + TdS] = 0
Also gauge transformation Q = GCT
C —> C’ = QCQ^-1 + QdQ
QCQ^-1 is usual tensor transformation
R —> R’ = QRQ^-1 = R gauge invariant as an exterior 2-form
But is tensor covariant in component tensor index formulation
Note added Dec 26, 2020
On Dec 26, 2020, at 12:25 PM, David Chester < This email address is being protected from spambots. You need JavaScript enabled to view it.> wrote:
Chill out, words are imprecise.
What’s important is some new physical insights gelled in this kerfuffle that Zielinski and I have periodically. ;-)
The exterior calculus is very elegant, sparse in sense of Dirac, yet very powerful.
I always had a problem formulating GR in terms of it, though Rovell does it in Ch II of Quantum Gravity.
In EM internal symmetry U(1) local gauge in Minkowski flat spacetime of global special relativity for now.
A —> A’ = A + kdphi
F = dA —> F’ = F because d^2phi = 0.
Bianchi identity is
dF = 0 works only for VACUUM!!!!!! (Faraday induction + no magnetic monopoles)
i.e. there is broken Hodge duality inside matter - this is key.
*F =/= F
d*F = J (Ampere’s and Gauss’s laws)
d^2F = dJ = 0 local current density conservation
EM in a gravity field Levi Civita metric connection (LC) from locally gauging T4 translations.
D = d + (LC)
F = DA = dA + (LC)/\A
DF = d^2A + (LC)/\dA + d[(LC)/\A] + (LC)/\(LC)/\A =/= 0 in the LNIF
d^2A = 0
Note EEP is (LC) ~ 0 in locally coincident LIF
D*F gets new terms inside matter.
Next Einstein’s 1916 GR
R = D(LC) = d(LC) + (LC)/\(LC) = Riemann-Christoffel curvature 2-form =/= 0
Einstein’s field eq is INSIDE MATTER MUST USE HODGE DUAL that depends on CONSTITUITIVE PARAMETERS from condensed matter quasi-particles and collective modes.
*R =- 8pi(G/c^4)ST
R =/= *R INSIDE METAMATERIAL
D*R + 8pi(G/c^4)D(ST) = 0
DS = dS
DT = dT + (LC)/\T
D(ST) = TdS + SDT
D*R = d*R + (LC)/\*R
S = Sarfatti Wanser spin 0 complex U(1) scalar EM susceptibility matter field
T = source stress-energy 2-form which is also a complex numbered function from dissipative imaginary susceptibilities causing relative phase shift between (ST) INPUT and *R OUTPUT
Bianchi identity is
DR =0
But just like in EM where D*F =/= 0, here D*R =/=0
1) HODGE DUAL GRAVITY FIELD *R INSIDE MATTER
3) APPLIED EM PUMP FIELD WHOSE STRESS-ENERGY TENSOR IS T IN Tic Tac Tech APPLICATION.
Jack is talking about teleparallel spaces, while Paul is focusing on a theory of teleparallelism.
The term "teleparallel" by itself is a confusing name since there are very many teleparallel theories of gravity. Various people have tried different formulations with or without success.
Is "teleparallelism" a theory, multiple theories, or a general framework? Depending on the context, all 3 can be correct.
In 1928, Einstein worked on a teleparallel theory of gravity, but it didn't work. Probably many others have made different theories of teleparallelism.
By 1976, it seems to have been formulated properly to become equivalent to GR in vacuum.
Since 1998, using the term teleparallelism to refer to the consistent torsion-only theory also went out the window, as Nester and Yo formulated a symmetric teleparallelism with nonmetricity.
Therefore, the 1976 theory is called metric teleparallel equivalent to GR, as it is metric-compatible. The 1998 theory is called symmetric teleparallel equivalent to GR because it has vanishing
torsion (the antisymmetric part of the connection).
Referring to Einstein's 1928 work on teleparallelism in 2020 as "the teleparallelism" is certainly done, but this is not practical for researchers in the field of gravitation. I'm sure that a lot of
physics professors at top universities who don't specifically research these topics would believe teleparallelism is just Einstein's failed theory from 1928, but the history is more subtle.
Would the most general subsector of metric-affine gravity with vanishing curvature still be a teleparallel space even though it is neither of these teleparallel theories mentioned above? If
metric-affine gravity and it's higher-dimensional operator generalizations give the "most general theory of relativity", then perhaps there exists a "most general teleparallel theory" that has not
been sufficiently explored in the literature.
If yes, then Paul suggesting that Jack focuses on metric-affine gravity still allows Jack to discuss teleparallel spaces, especially if we agree that there exists some "most general teleparallel
theory of relativity", which is not really a term I've seen used.
See Kibble’s classic paper on this.
Utiyama first locally gauged Lorentz group SO(1,3) and had to put in GCT’s by hand and then set torsion to zero ad_hoc (as I recall) to get to 1916 GR.
Then, Kibble locally gauged entire 10 parameter Poincare group - set torsion to zero ad hoc to get 1916 GR.
If you think about the physics and forget the maths, you will see that the physical meaning of general coordinate transformations is to formulate physics in locally proper accelerating frames. This
is the meaning of locally gauging global T4 to T4(x).
Physically, T4(x) describes Alice and Bob on nearly colliding arbitrary timelike world lines each measuring the same set of events using far field light signals. ds^2 = guv(Alice)dx^udx^v = gu’v’
(Bob)dx^u’dx^v’ where Alice and Bob are separated by intervals small compared to radii of curvature.
guv(Alice) = Xu^u’Xv^v’gu’v’(Bob)
X’s are GCTs from T4(x) see Hagen Kleinert’s books on line for details
Global translations are
x^u —> x^u = x^u + a^u
Local gauging T4 means
x^u —> x’^u(x) = x^u(x) + a^u(x)
dx^u = x’^u(x) - x^u(x) = X^uu’dx^u’
a^u(x) = X^uu’dx^u’
X^uu’(x) are Einstein’s GCT’s obviously they are T4(x) transformations.
Quod Erat Demonstratum.
On Dec 26, 2020, at 2:32 PM, David Chester < This email address is being protected from spambots. You need JavaScript enabled to view it.> wrote:
"I may misunderstood you, but I don't think Levi-Civita is found from gauging translations. "
Jack: Sez who? I just gave you the simple physical picture to the contrary.
"GR isn't a gauge theory, even though there is a GL(4,R) diffeomorphism invariance. Metric teleparallelism gauges translations.
I think gauging translations can allow for contortion to cancel Levi-Civita's contribution to curvature (at least in metric teleparallelism).
But as we know, there are many different subtle theories of gravity, so maybe there's something I'm missing. It has been confusing to disentangle what theory of gravity you are studying with the
recent discussions of torsion. For instance, there are now 3 notions of curvature, R(C), R(T), and R(C,T) = R(C) + R(T), where R(C) is from Levi-Civita and R(T) is from torsion.
As soon as torsion is introduced, talking about curvature tensors becomes more vague.
I am guessing you are attempting to invoke translational gauge symmetry to get R(C) = -R(T), but then since you want to stick closer to GR, you are sticking to R(C) instead R(C,T)=0. The most sense I
can make with your words is that you are finding R(T) by gauging translations, not R(C), but it ends up being related if R(C,T)=0.
I still haven't worked out precisely how Einstein's vacuum field equations come from metric teleparallel equivalent by gauging translations, but still, I don't see how gauging translations literally
gives Levi-Civita."
I proved by explicit construction, that local gauging T4 —> T4(x) gives Einstein’s general coodinate transformations that physically correspond to local frame transformations between Alice and Bob
each on arbitrary timelike world lines that momentarily are close to each other. My argument is complete contained explicit independent of any of the points you mention below.
I said nothing about torsion et-al in that limited precise elementary physically transparent argument that I repeat here again for the record
If you think about the physics and forget the maths, you will see that the physical meaning of general coordinate transformations is to formulate physics in locally proper accelerating frames. This
is the meaning of locally gauging global T4 to T4(x).
Physically, T4(x) describes Alice and Bob on nearly colliding arbitrary timelike world lines each measuring the same set of events using far field light signals. ds^2 = guv(Alice)dx^udx^v = gu’v’
(Bob)dx^u’dx^v’ where Alice and Bob are separated by intervals small compared to radii of curvature.
guv(Alice) = Xu^u’Xv^v’gu’v’(Bob)
X’s are GCTs from T4(x) see Hagen Kleinert’s books on line for details
Global translations are
x^u —> x^u = x^u + a^u
Local gauging T4 means
x^u —> x’^u(x) = x^u(x) + a^u(x)
dx^u = x’^u(x) - x^u(x) = X^uu’dx^u’
a^u(x) = X^uu’dx^u’
X^uu’(x) are Einstein’s GCT’s obviously they are T4(x) transformations.
Quod Erat Demonstratum.
On Dec 26, 2020, at 6:05 PM, David Chester < This email address is being protected from spambots. You need JavaScript enabled to view it.> wrote:
Subsectors of metric-affine gravity are not very simple.
I only provided clarity to point out that GR + non-propagating torsion as found in Einstein-Cartan theory is not teleparallelism and it is not gauging translations. You implied it did and now, rather
than being a big boy and correcting yourself, you are retreating back towards "it's just simple metric engineering".
Some days you are interested in dark energy, others in metric engineering.
Both problems are important, the first for fundamental physics, the second for national security.
On Dec 26, 2020, at 5:54 PM, JACK SARFATTI < This email address is being protected from spambots. You need JavaScript enabled to view it.> wrote to Tim Ventura:
The Physics explaining the whole bloody thing is quite elementary and well understood it’s only an engineering problem now
of course there’s weapons implications and of course we should think of this as an imminent threat
Sent from my iPhone
On Dec 26, 2020, at 4:50 PM, Tim Ventura < This email address is being protected from spambots. You need JavaScript enabled to view it.> wrote:
This is something I'm playing with - maybe for a presentation or something. We have 70+ years of mythology to deal with, but even if only a tiny fraction of it is real, it would indicate that
UAPs are not our friends & we should develop defenses to counter a potential threat.
Are UAPs Russian or Chinese?
- Capable of acceleration & maneuvers far beyond human vehicles
- UAP performance would kill a human pilot
- Flight & maneuverability without conventional engines, lifting or control surfaces
- Capable of transverse medium propulsion (air/water)
- Flight capabilities of UAPs date back decades, even if Russia/China could do this today, they couldn't 50 years ago.
- No record of markings or writing on UAPs in known languages
- Close encounter reports dating back 50+ years indicates non-human occupants
Jack: All explained conceptually elementary physics. Problem is the incompetence of the people involved. For example, watch Elizondo stumble here trying to explain what I have already explained
- UAPs are not God (why does God need a starship?)
- They're not much more advanced than us.
- Appear to be physically weak & rely on technology.
- Tend to control interactions, presumably because they have vulnerabilities (and unexpected events do cause damage)
- UFOs may not be real (20th century manifestation of need for fairies, angels, etc...)
- even if they are real, the stories about them may not be (mass hysteria)
- it's dangerous to assign motive to unknown actions & judge aliens by human standards
Why are they here:
- To catalog & categorize our biosphere
- To understand human beings & social behavior
- They don't need our minerals (more easily mined in space)
- They don't need our technology (presumably less advanced than theirs)
- Probably not to colonize (unlikely good bio match, and lots of unpopulated explanets to start with)
UAPs are not our friends:
(they appear to recognize us as sentient, but don't ascribe us basic rights)- Human Abductions (how many unsolved missing persons reports annually?)- Human Implants & Experimentation (Violates
International Law)- Human Casualties to UFO Encounters (Encounter burns, PTSD, etc...)- Cattle Mutilations- Two Attempts To Launch ICBM's- Attempts to break radar lock during intercepts-
Infringing on Sovereign Airspace- Interfering with Military & Civilian Air Traffic- They achieve goals through power, not consent- They're untrustworthy (actions to do not UFO believer claims
about love & peace, they're not transparent, etc...)- UAP encounters indicate presence off East / West Coast for easy access to population centers- Spying on data communications? https://
www.submarinecablemap.com/- Demonstrate zero regard for human life (Even if the majority of these claims are false, the remainder show a pattern)UAPs should at least be held accountable:
- To uphold basic human rights as chartered by UN: https://www.un.org/en/sections/issues-depth/human-rights/- To establish & abide by a publicly known standard of conduct (maybe they don't have
to follow all of the laws, but we should know which ones & why)- To abstain from actions that would interfere with duly appointed government or defense representatives- Never in human history has
"bowing before superior capabilities" led to a good outcome.- To ensure they don't inadvertently bring disease or pathogens into our biosphereDefense aspects of this:
- Need defense awareness of UAP presence - especially scope of activity, which is undocumented.
- Need public involvement to help report UAP activity.
- We need defense capabilities to deter, deny access, and repel unwanted UAP actions
- Missiles & bullets are likely not fast or manueverable enough to intercept UAPs
- Beam weapons (laser, maser, particle beam) are preferable because there is no advanced warning before damage begins
- Radar signals implicated in the Roswell crash, suggesting their electronics may be vulnerable
- Space-based weapons provide maximum horizon for engagement & deterrance entering/leaving atmosphere
- Any weapons effective against UAPs would have conventional defense applications, incentivizing development.
- Computer-controlled weapons might be preferable due to speed of response & decision-making (with manual override).
On Tue, Dec 22, 2020 at 7:02 AM JACK SARFATTI < This email address is being protected from spambots. You need JavaScript enabled to view it.> wrote:
On Dec 21, 2020, at 6:09 PM, Kim Burrafato < This email address is being protected from spambots. You need JavaScript enabled to view it.> wrote:
Sent from my iPhone
We knew that would happen.
Ill conceived.
Mellon, Elizondo, Justice not up to the real job.
Meaning well is not good enough.
It did bring PR however to the UFO issue.
We are the “Last Man Standing” ;-)
We have actually solved the essential physics problem behind the entire Phenomenon that stumped them.
I don't know what name to give "it" because we have been discussing many things.
Yes, the mainstream has missed the validity of Einstein-Cartan, which is related to why people can't make sense quantum gravity with matter.
But once again, what does torsion have to do with your metric engineering? Your answer seems to fluctuate.
I tried making the analogy between the antigravitic torsion of an electron and your metamaterials already and you ridiculed me. Now, you are essentially saying what I said a day or two ago.
If you are "only" interested in metric engineering, make up your mind if torsion is relevant or not. It seems relevant only when you talk about it but not others.
It's hilarious if you think you are the first to apply differential forms to GR. That doesn't mean you shouldn't do it. The geometric algebra community has, as well as Derek Wise under Baez, probably
goes further back though.
Remember, I mentioned torsion being relevant for antigravity and you called me out. Now you are pointing it out that torsion inside matter changes things. It's hilarious how much you like to disagree
while secretly agreeing.
On Sat, Dec 26, 2020, 3:52 PM JACK SARFATTI < This email address is being protected from spambots. You need JavaScript enabled to view it.> wrote:
Both you guys are making something simple overly complicated.
I don’t care what name you give it.
Everything I had to say is now elegantly formulated neatly as Cartan exterior forms.
So far, the equations I wrote down do not apply to Shipov’s theory.
1) Classical EM in Minkowski space time (LIFs)
2) Classical EM in curved space-time (LNIFs)
3) Einstein 1916 GR - no torsion - but inside MATTER giving new physics that even Pundits like Kip Thorne and Sean Carroll have completely missed.
Specifically R =/= *R in matter therefore you cannot write Bianchi identity D*R = 0 inside MATTER generally - certainly not in pumped resonant metamaterials that explain Tic Tac Tech.
Remember my purpose is METRIC ENGINEERING PHYSICS of
& this
I will get back to Shipov particular model 4) soon using Cartan’s forms because it may be natural setting for both dark matter and dark energy as difference quantum vacuum phases at different
scales, frequencies and wavelengths.
On Dec 26, 2020, at 9:09 PM, Paul Zielinski < This email address is being protected from spambots. You need JavaScript enabled to view it.> wrote:
No torsion, but what inside matter?
On 12/26/2020 3:51 PM, JACK SARFATTI wrote:
This email address is being protected from spambots. You need JavaScript enabled to view it." class="">
3) Einstein 1916 GR - no torsion - but inside MATTER giving new physics that even Pundits like Kip Thorne and Sean Carroll have completely missed.
Specifically R =/= *R in matter therefore you cannot write Bianchi identity D*R = 0 inside MATTER generally - certainly not in pumped resonant metamaterials that explain Tic Tac Tech.
In EM the Hodge Dual *F inside matter contains B = H + 4piM and D = E + 4piP
In vacuum M and P vanish and F = *F self-dual
Similarly in GR
R =/= *R inside matter
R = curvature 2-FORM
None of the GR text books take this into account because most people in GR are really mathematicians and have little knowledge of condensed matter physics.
*R ~ - (G/c^4)ST
You cannot use Bianchi identity DR = 0 inside matter because R =/= *R inside matter.
Therefore, the assumption in GR textbooks Tuv^;v = 0 is WRONG INSIDE MATTER.
Now in ordinary matter R ~ *R to a sufficient approximation, but in Tic Tac meta-material that is not the case and that is the key to low power warp drive.
Locally gauging T4 takes us from global 1905 Einstein SR to allowing global IFs and global NIFs to Einstein 1916 GR with guv(x) metric tensor fields. Once you have locally variable guv(x) you get
LC connection by first order partial differentiation.
The point is until you locally gauge T4 you do not have a variable guv(x) symmetric tensor field to play with.
This involves physical intuition - pictures in your mind. If you are blind to pictures and can only think algorithmically step by step like a Turing machine you may not be able to grok this.
Note added Dec 27, 2020
The Cartan formalism is Spartan in the sense that Dirac liked as seen in his “bra - ket” QM formalism.
Cartan formalism is manifestly coordinate-independent - it is local objective classical field reality in the way Einstein liked.
However, it has profound physical significance, in that the Hodge dual * operator describes the transition from vacuum to matter as in electrodynamics
D = E + 4piP
B = H + 4piM
EM 2 Form F describes E and B in vacuum
*F describes H and D in matter connection to sources of the field.
In vacuum P = 0 and M = 0
The tetrads e are Cartan exterior 1 forms with dual co-forms w corresponding to dx in the usual coordinate representation.
The tetrad 1-forms as variable fields over spacetime are directly induced by locally gauging T4 translation Lie group.
The invariant 0-form metric field for curved spacetime is
ds^2 = *[e/\e/\w/\w]
* = Hodge dual
The Levi-Civita metric field connection is the 1-form
(LC) = *[e/\de]
The covariant exterior derivative is
D = d + *[e/\de]
The vacuum Weyl curvature 2-form is
R = D(LC) = {d + *[e/\de]}*[e/\de]
DR = 0 Bianchi identity in vacuum
The inside matter Ricci + Weyl curvature 2-form is given by Einstein’s modified gravity field equation
*R = - 8pi(G/c^4)|S||T|cos(s + t)
S = |S| exp(is) Sarfatti-Wanser 0-form scalar field matter EM response function
T = source stress-energy 2-form
D(*R + 8pi(G/c^4)|S||T|cos(s + t)] = 0
Local conservation of stress-energy current densities.
D*R =/= 0 inside matter
On Dec 27, 2020, at 9:25 AM, JACK SARFATTI <This email address is being protected from spambots. You need JavaScript enabled to view it.; wrote:
PS Lue Elizondo has recently been talking about my idea of Tic Tac Warp Drive without giving me credit. He never talked about it prior to the Franc Milburn BESA White Paper.
No David, you do not understand.
We do not need torsion to explain the antigravity warp drive of Tic Tac.
That is a dissipative (inelastic scattering photons off charges) phase shift effect in traditional 1916 metric GR with only Levi Civita connection.
*R = - 8pi(G/c^4)|ST| cos(s + t)
Positive cosine is induced gravity red shift, negative cosine is induced anti-gravity blue shift.
S = |S|exp[is]
is the metamaterial EM response complex scalar field 0-Cartan form
T = |T| exp[it]
is the applied EM pump field’s stress-energy Cartan 2-form also complex from dissipation in the permittivity and permeabilities of the anisotropic time change metamaterial fuselage of Tic Tac
*R is the Hodge dual of the curvature Cartan 2-form R.
*R = R in vacuum, but not in matter in the same way that F =/= *F in matter because of M and P response fields of real charges.
Bianchi identity DR = 0 FAILS INSIDE MATTER, though in many cases the degree of violation is negligible, but not in pumped metamaterial resonances as shown in Tic Tac flight.
D*R + 8pi(G/c^4)[TdS + SDT] = 0 local conservation of stress-energy current densities.
DT = dT + (LC)/\T
*R = *[d(LC) + (LC)/\(LC)]
D*R =/= 0 i.e. direct coupling of gravity field to the S and T matter fields.
Yes, there may be quantum spin induced torsion inside artificial metamaterial that may make an additional anti-gravity effect.
You and Zielinsky continually confuse necessary conditions for sufficient conditions and vice versa.
To review:
Local gauging T4 translations directly induces tetrad coefficients from physically real Alice LIF <—> Bob LNIF when Alice and Bob are locally coincident measuring same external events with far
field light signal propagating in vacuum not inside matter.
Alice’s LIF has ds^2 = (cdt)^2 - dx^2 - dy^2 - dz^2 = nIJdx^Idx^J
Bob’s LOCALLY COINCIDENT LNF has ds’^2 = guv(x)dx^udx^v
ds^2 = ds’^2 for Alice and Bob measuring same external events with light signals.
This is physics independent of coordinate representations.
The tetrad coefficient fields eu^I(x) are the direct result of the local T4 gauging. The LNIF metric tensor guv(x) and (LC)^luv connection fields are indirectly induced from the tetrad fields.
For example, LOCAL GAUGING T4 MEANS THIS
dx^I(ALICE LIF) = e^Iu(x)dx^u(BOB LNIF)
The symbol small x = Einstein local coincidence not a coordinate - all this is coordinate independent.
Zielinski et-al confuse coordinate shadows for Platonic light (forms).
Local gauging T4 describe coordinate-independent physically real local frame transformations that have GCT formal representations.
guv(x) = eu^I(x)ev^J(x)dx^Idx^J
Note also (LC)^muv ~ eu^I(x)(d/dx^v)eI^m
Einstein’s TENSOR GCTs are quadratic in tetrad coefficients i.e. X(x)u^u’ = e^uI(x)e^Iu’(x)
On Dec 26, 2020, at 6:05 PM, David Chester < This email address is being protected from spambots. You need JavaScript enabled to view it.> wrote:
Subsectors of metric-affine gravity are not very simple.
I only provided clarity to point out that GR + non-propagating torsion as found in Einstein-Cartan theory is not teleparallelism and it is not gauging translations. You implied it did and now,
rather than being a big boy and correcting yourself, you are retreating back towards "it's just simple metric engineering".
Some days you are interested in dark energy, others in metric engineering.
I don't know what name to give "it" because we have been discussing many things.
Yes, the mainstream has missed the validity of Einstein-Cartan, which is related to why people can't make sense quantum gravity with matter.
But once again, what does torsion have to do with your metric engineering? Your answer seems to fluctuate.
I tried making the analogy between the antigravitic torsion of an electron and your metamaterials already and you ridiculed me. Now, you are essentially saying what I said a day or two ago.
If you are "only" interested in metric engineering, make up your mind if torsion is relevant or not. It seems relevant only when you talk about it but not others.
It's hilarious if you think you are the first to apply differential forms to GR. That doesn't mean you shouldn't do it. The geometric algebra community has, as well as Derek Wise under Baez,
probably goes further back though.
Remember, I mentioned torsion being relevant for antigravity and you called me out. Now you are pointing it out that torsion inside matter changes things. It's hilarious how much you like to
disagree while secretly agreeing.
On Sat, Dec 26, 2020, 3:52 PM JACK SARFATTI < This email address is being protected from spambots. You need JavaScript enabled to view it.> wrote:
Both you guys are making something simple overly complicated.
I don’t care what name you give it.
Everything I had to say is now elegantly formulated neatly as Cartan exterior forms.
So far, the equations I wrote down do not apply to Shipov’s theory.
1) Classical EM in Minkowski space time (LIFs)
2) Classical EM in curved space-time (LNIFs)
3) Einstein 1916 GR - no torsion - but inside MATTER giving new physics that even Pundits like Kip Thorne and Sean Carroll have completely missed.
Specifically R =/= *R in matter therefore you cannot write Bianchi identity D*R = 0 inside MATTER generally - certainly not in pumped resonant metamaterials that explain Tic Tac Tech.
Remember my purpose is METRIC ENGINEERING PHYSICS of
& this
I will get back to Shipov particular model 4) soon using Cartan’s forms because it may be natural setting for both dark matter and dark energy as difference quantum vacuum phases at different
scales, frequencies and wavelengths.
On Dec 26, 2020, at 3:28 PM, David Chester < This email address is being protected from spambots. You need JavaScript enabled to view it.> wrote:
Jack, I think you are confusing Einstein-Cartan(-Kibble-Sciama) theory with metric teleparallel equivalent to GR (TEGR).
Einstein-Cartan has vanishing torsion outside matter, yet TEGR has vanishing curvature with propagating torsion in a manner that is equivalent to GR's curvature.
I don't think Einstein-Cartan is a theory of teleparallelism. Similarly, it is not in Weitzenbock geometry, as it uses Riemann-Cartan geometry. This combines Riemannian geometry with
Weitzenbock geometry, in some sense.
Poincare gauge gravity has propagating curvature and torsion in the vacuum according to Hehl. Lorentz gauge gravity has curvature, Translations have torsion.
Einstein-Cartan has benign torsion, as it is not propagating. Going to Poincare gauge gravity then includes 1915 GR, Einstein-Cartan, and TEGR as subsectors. In a sense, Einstein-Cartan
contains GR, but TEGR is orthogonal.
STEGR has vanishing curvature and torsion, yet it still has a curvature contribution from Levi-Civita. However, the disformation (from nonmetricity) leads to a cancellation, similar to how
contortion (from torsion) cancels with Levi-Civita in TEGR.
GR from 1915 is second-order with dynamical metric. However, first order formulations lead to an affine theory via an affine connection, which is more sophisticated, as the connection is
geometrically different than the metric.
Sean Carroll has nice discussions in his online textbook about how an affine connection that is metric-compatible and torsion less uniquely leads to Levi-Civita. So mathematically, starting
with Levi-Civita may not be rigorous, but it is effectively okay.
Metric-affine gravity refers to an independent metric and affine connection. It is also meant to be very general, although the original paper only discussed a simple subsector with 2 terms in
the Lagrangian. The most general metric-affine gravity has I believe 28 terms in the Lagrangian.
It includes Einstein-Hilbert, all possible second-order curvature terms with loss of symmetry of standard Riemann tensor in GR, as well as 3 terms from TEGR, 5 terms from STEGR, and mixed
terms that combine torsion and nonmetricity.
It's still not clear to me what precisely Jack is discussing, but I don't think we should assume it is Weitzenbock space, as he keeps latching onto GR and Einstein-Hilbert with torsion in a
manner that seems more similar to Einstein-Cartan. It's unclear, I think he is still confusing EC and TEGR.
On Sat, Dec 26, 2020, 3:09 PM Paul Zielinski < This email address is being protected from spambots. You need JavaScript enabled to view it.> wrote:
Please disregard this version.
On 12/26/2020 3:06 PM, Paul Zielinski wrote:
Words are imprecise if you're sloppy.
The standard term for what Jack appears to be talking about is a "Weitzenbock space".
That terminology neatly avoids any possible confusions about implied claims of teleparallelism.
On 12/26/2020 1:16 PM, JACK SARFATTI wrote:
I have to straighten Jack out from time to time. It's a dirty job, but someone has to do it. :-)
Jack is talking about teleparallel spaces, while Paul is focusing on a theory of teleparallelism.
I think he's talking about a Weitzenbock spacetime -- characterized by a metric derived from the Weitzenbock connection.
Strictly speaking, there is no such thing as a "teleparallel space"; but at least now I think I understand what Jack and Shipov mean by this.
[CD] The term "teleparallel" by itself is a confusing name since there are very many teleparallel theories of gravity. Various people have tried different formulations with or
without success.
Exactly -- confusing.
Is "teleparallelism" a theory, multiple theories, or a general framework? Depending on the context, all 3 can be correct.
It's a property that defines a class of geometric models for gravity.
The property is the ability to directly compare the angles of field vectors in different tangent spaces at
arbitrary separation.
In 1928, Einstein worked on a teleparallel theory of gravity, but it didn't work.
It didn't work for other reasons, but it did deliver teleparallelism. In that respect it did work.
Probably many others have made different theories of teleparallelism.
By 1976, it seems to have been formulated properly to become equivalent to GR in vacuum.
In these kinds of theories, torsion is only present inside matter. In the vacuum region torsion vanishes and the theories are equivalent.
Since 1998, using the term teleparallelism to refer to the consistent torsion-only theory also went out the window, as Nester and Yo formulated a symmetric teleparallelism with
Why isn't that a metric-affine theory?
Therefore, the 1976 theory is called metric teleparallel equivalent to GR, as it is metric-compatible.
Non-metricity is set to zero?
Is this theory really a teleparallel theory? Or is it just called that for historical reasons?
The 1998 theory is called symmetric teleparallel equivalent to GR because it has vanishing torsion (the antisymmetric part of the connection).
But it has curvature? Or at least a curvature connection with degrees of non-metricity?
If so, how can it qualify as teleparallel?
This has been confusing me since you originally mentioned it.
How is this different from a matric-affine theory with zero torsion?
Referring to Einstein's 1928 work on teleparallelism in 2020 as "the teleparallelism" is certainly done, but this is not practical for researchers in the field of gravitation. I'm
sure that a lot of physics professors at top universities who don't specifically research these topics would believe teleparallelism is just Einstein's failed theory from 1928,
but the history is more subtle.
But as I said, there is -- strictly speaking -- no such thing as a "teleparallel spacetime".
But if by this you only mean a Weitzenbock spacetime, then fine. as long as that is clear.
Would the most general subsector of metric-affine gravity with vanishing curvature still be a teleparallel space even though it is neither of these teleparallel theories mentioned
above? If metric-affine gravity and it's higher-dimensional operator generalizations give the "most general theory of relativity", then perhaps there exists a "most general
teleparallel theory" that has not been sufficiently explored in the literature.
It's just a question of avoiding confusion resulting from ambiguous terminology, that's all.
Why not call it a "Weitzenbock space", which is the standard term?
If yes, then Paul suggesting that Jack focuses on metric-affine gravity still allows Jack to discuss teleparallel spaces, especially if we agree that there exists some "most
general teleparallel theory of relativity", which is not really a term I've seen used.
I have no problem with Jack using a Weitzenbock space, as I've defined it here. I'm just trying separate that from any implied claims of teleparallelism -which at this point I don't
think Jack intended to make.
Paul you simply do not understand the physical picture of Alice and Bob in local coincidence measuring the same external events with light signals.
Allowing Alice and Bob to be on arbitrary timelike world lines is described by local T4 gauging that directly induces tetrad coefficients from which metric tensor and Levi-Civita
connection are derived.
Local gauging T4 is represented formally by this equation
dx^I(ALICE LIF) = e^Iu(x)dx^u(BOB LNIF)
“x" = local coincidence =/= coordinate
x^u —> x’^u = x^u + a^u(x) = LOCAL GAUGING T4
a^u(x) = e^uI(x)dx^I = X^uu'(x)dx^u’
X^uu’(x) = e^uI(x)eu’^I(x) = EINSTEIN GCT
I = LIF tensor index
u,u’ = LNIF tensor indices | {"url":"https://stardrive.org/index.php/all-blog-articles/60161-mathematics-of-practical-low-power-tic-tac-warp-drive-star-ships-12-25-20","timestamp":"2024-11-06T07:46:14Z","content_type":"text/html","content_length":"157142","record_id":"<urn:uuid:eb199a77-1fde-441e-bd3b-504673cacac4>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00645.warc.gz"} |
More On Cantor Pairing! - an Astronomy Net God & Science Forum Message
Hi Paul,
I think you are right. A lot of the quotes you gave sound very familiar. It is funny how the memory works; I think one's imagination adds "unknowable data" to make the gist of the experience make
sense. I could have sworn it brought up the Axiom of Choice by name. Well, I apparently read it over a week ago and didn't take any notes; I was only reminded of it by your post. I suppose I can be
forgiven for getting it a bit twisted.
Since reading your response last night, I have been thinking a little. In particular, about Cantor's method of determining cardinal equivalence of two sets. I would like you to think about the
following pairing procedure between the set of natural numbers and the set of real numbers.
1. Begin by constructing a 10 by 10 matrix of cells. Across the bottom row, insert the digits zero through nine. In each column, insert the digits one through nine in the empty cells.
2. One can now construct a real number corresponding to each and every cell via the following procedure:
a) The real numbers will consist of one integer above the decimal and one below the decimal.
b) The real number in the cells across the bottom will have the inserted digit as the integer above the decimal and zero as the integer below the decimal.
c) The real number in all the other cells will have the integer at the bottom of the column as the digit above the decimal and the digit in the cell as the digit below the decimal.
3. That collection contains exactly 100 real numbers. They can be counted and they can be paired to the first 100 members of the set of natural numbers.
4. Now construct a 100 by 100 matrix of cells. Across the bottom row, insert the digit pairs zero-zero through nine-nine. In each column, insert the digits zero-one through nine-nine.
5. one can now construct a real number corresponding to each and every cell via a procedure exactly analogous to that in 2 above.
a) The real numbers will consist of two digits above the decimal and two digits below the decimal.
b) The real numbers in the cells across the bottom with have the inserted digit pair as the two digits above the decimal and zero-zero as the two digits below the decimal.
c) The real number in all the other cells will have the digit pair in the bottom cell of the column as the digit pair above the decimal and the digit pair in the cell as the digit pair below the
6. Once again we have a countable subset of the real numbers (10,000 of them to be exact) and they can be paired to the first 10,000 members of the set of natural numbers. It is important to notice
that this set also contains every real number contained in the collection laid out in the 10 by 10 matrix.
7. Now, you can continue this procedure with three, four, five, ... digits in a cell. At every step of the procedure the subset of real numbers can be counted and paired with the proper subset of the
natural numbers.
8. As the procedure is continued, there is no real number which will not be picked up so long as we are allowed to continue forever: i.e., if the procedure of pairing the odd numbers to the natural
numbers can be deemed accomplishable, one must conclude that the real numbers can also be so paired.
Now I believe Cantor would get very upset with the fact that I am rearranging the numbers at every step, but I don't think one could argue that the rearrangement is significant to the count as, each
time I add some more real numbers, I am always including the real numbers already considered in my count which is the actual central issue here. Now with your complaint about members of the set being
used up faster, in the procedure I just laid out, which ones are being used up faster? The real numbers or the natural numbers? Think about that one for a while!
With regard to things being counter-intuitive, I think it would be good to keep Jerry Bona's joke in mind. My personal opinion is that "size of infinity" is an undefinable term. If it is infinite, it
just never stops and if you were smart enough you could come up with a pairing program which would pair off any infinite set with the natural numbers. Of course, that is no more than an opinion.
Now I would like to bring up a serious question about the concept of existence. I know you are aware of the fact that your mind can present you with illusions which are not controllable in any
conscious way. What I mean here is that you cannot eliminate the illusion by knowing it is an illusion. We generally "know it is an illusion" because we can prove it is "inconsistent with actual
reality". Suppose your mind were to create a particular illusion which just happened to have no inconsistent elements at all. Does that make it "consistent with reality"? In other words can your mind
create "real" objects? How would you propose to prove it can not? Just what is a "real" object anyway?
Have fun -- Dick | {"url":"http://www.astronomy.net/forums/god/messages/29239.shtml","timestamp":"2024-11-10T15:25:57Z","content_type":"text/html","content_length":"18577","record_id":"<urn:uuid:ffadb69f-405f-4c45-b122-a5e6d807cb70>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00054.warc.gz"} |
Power Law for the Rates of Different Numbers of Chronic Diseases among Elderly Chinese People
Power Law for the Rates of Different Numbers of Chronic Diseases among Elderly Chinese People ()
1. Introduction
Chronic diseases are chronic noncommunicable diseases, and they include cardiovascular diseases, diabetes, chronic obstructive pulmonary disease, and chronic kidney disease. Chronic diseases not only
seriously affect the quality of life of the elderly and their families but also cause an economic burden on the family and society. Many studies have postulated that chronic diseases are significant
risk factors for the development of disabilities. Over the past 20 years, there has been an increase in the prevalence of chronic disease [1] , and the majority of elderly people aged 65 years and
older suffer from multiple chronic diseases [2] [3] [4] [5] [6] .
China has become an aging society, and chronic diseases have become important factors that affect the health of elderly Chinese people [7] [8] . The most common cause of death among Chinese residents
is chronic disease rather than infectious disease [9] [10] . In 2010, the number of patients with chronic diseases reached 300 million, and 75% of the deaths were caused by chronic diseases [11] .
Power laws in general have and continue to attract considerable attention in a wide variety of disciplines―from astronomy to demographics to software structure to economics to finance to zoology, and
even to warfare [12] [13] [14] . Typically, an analyst must work with integer-valued random variable n, its observables (numbers of objects, people, cities, words, animals, corpses) are positive
inters with
Power law distributions describe common features of many complex systems. The power-law scaling observed in the primary statistical analysis is an important feature but is by far not the only feature
that characterizes experimental data. It provides us with important information about a system’s stability and evolution.
Although a power-law phenomenon usually exists in economic and social systems, few studies focus in the distributed characteristics of the number of chronic diseases in a special subpopulation such
as elderly people. We used 2006 and 2010 data from the Chinese Urban and Rural Elderly People Survey published in 2012 to find the distribution of the rates of different numbers of chronic diseases.
The remainder of this paper is structured as follows. Section 2 introduces the data and methods, including the paired t-test, power-law distribution and Kolmogorov- Smirnov test. Section 3 provides
an empirical study of the 2006 and 2010 data, and Section 4 presents our conclusions.
2. Methods
2.1. Data
The data used in this study were obtained from the 2006 and 2010 Chinese Urban and Rural Elderly Population Surveys, conducted by the China Research Center on Aging of the National Committee on Aging
^1. These data cover the following 20 provinces in China: North China―Beijing, Hebei, and Shanxi; Northeast China―Liaoning and Heilongjiang; East China―Shanghai, Jiangsu, Zhejiang, Anhui, Fujian, and
Shandong; Mid-South China―Henan, Hubei, Hunan, Guangdong, and Guangxi; Southwest China―Sichuan and Yunnan; and Northwest China―Shanxi and Xinjiang. The data sampling method was the same as that for
the Fifth Population Census, and it was based on the distribution of the population 60 years and older; a quota from each of these six regions could be determined. Then, stratified sampling was used
to confirm that the survey results represented the total elderly population in China. The main study cohort in 2010 was the same elderly population that was investigated in 2006; these two surveys
obtained samples of 19,947 responses and 19,986 responses, respectively. The study subjects included elderly individuals who ranged from 60 to 102 years of age.
The appropriate processing of the data from the two surveys was a key aspect of the analysis, and the data were selected as follows:
1) All of the indicators that we used here were default-free. Data is from household tracking, including many indicators such as gender, age, census register etc. We give up some data that indicators
are not complete to ensure the reliability of data.
2) We discarded samples that reported inconsistent chronic conditions. Specifically, we excluded cases in which suffering from chronic diseases was indicated while not specifying any chronic disease.
3) In general, women cannot have prostatitis, and men cannot have gynecological disease; thus, we discarded data samples that reported impossible chronic diseases.
4) To facilitate the distinction between urban and rural, we used the agricultural and nonagricultural census registers (e.g., an individual who transferred from an agricultural to a nonagricultural
census register was considered to be a nonagricultural census register).
5) There were 25 types of chronic diseases in the survey; the 25^th was “other chronic disease”, which was not a specific disease. One could choose this option if one is suffering from a type or more
than one type of chronic disease that was not among the 24 types of chronic diseases in the survey. We hypothesize that one chose the option of “other chronic disease” was just suffering from a
single type of chronic disease. Thus, the minimum number of chronic diseases that one person could suffer from was 0, and the maximum number, in theory, was 25.
Using the processing methods described above, the final number of available samples was 19,691 in 2006 and 19,841 in 2010.
2.2. Paired T-Test
Because most of chronic diseases are reversible, the morbidity of chronic diseases among elderly people of various ages does not have a cumulative effect. Thus, we could not use the distribution
fitting method to conduct a variance analysis. Meanwhile, when analyzing the influence of some social-economic factor on chronic disease, we usually presume the influence of the factor is same on
different elderly cohorts. Therefore, in this study, we used the paired t-test to perform the analysis.
The paired t-test can verify whether the effect of a factor is significant, For example, we can use paired t test to found out whether two weighing machines are the same. The objects to be weighted
can be very different, it can be light, such as bag, shoe, cat, book and food, if possible, it can also be very heavy, such as refrigerator and elephant. We can get the weight of the objects by using
the two machines, and then proceed paired t test to verify our hypothesis. If the machines have no system error, the mean of the weight differences should be close to 0, if else, we can infer that
the two machines are different.
The theory of paired t-test is as follows:
Consider n pairs of independent observation data
[0]:[1]. The t-test, statistics and rejection regions used in this study are obtained as follows:
2.3. Power-Law Distribution
In mathematics, if the density function of a random variable X is the following:
then we can say that x follows a power distribution, in which a is a constant that is called the power exponent. The power distribution showed very strong heterogeneity, and the power distribution is
the only distribution that has the property of “no scaling”.
When the density function is
The meaning of the function is that when the unit or the “scale” of the variable x changes a constant number of times, the form of the distribution f(x) remains unchanged. Taking logarithm on both
sides of Equation (2), we get
From (4), we can see that a is the slope that reflects the rate of change between logarithmic probability density and logarithmic random variable.
2.4. Kolmogorov-Smirnov Test
In statistics, the Kolmogorov-Smirnov test (K-S test) is a form of minimum distance estimation that is used as a nonparametric test of equality of one-dimensional probability distributions, and it is
used to compare a sample with a reference probability distribution (one-sample K-S test) or to compare two samples (two-sample K-S test).
Hypothesis test problems are as follows:
H0: the samples are drawn from the same distribution (in the two-sample case) or the sample is drawn from the reference distribution (in the one-sample case).
H1: the samples are not drawn from the same distribution (in the two-sample case) or the sample is not drawn from the reference distribution (in the one-sample case).
The Kolmogorov-Smirnov statistic quantifies a distance between the empirical distribution function of the sample and the cumulative distribution function of the reference distribution or between the
empirical distribution functions of two samples.
The empirical distribution function
The Kolmogorov-Smirnov statistic for a given cumulative distribution function
3. Data Analysis Results
3.1. Power-Low for the Rates of Different Numbers of Chronic Diseases by Gender and Census Registry
The fitting information on the samples by gender and census registry in 2006 and 2010 was shown in the following sections. The dependent variable was the rates of the different numbers of chronic
diseases, and the independent variable was the number of chronic diseases.
The power-law fitting results for the rates of different numbers of chronic diseases by gender and census registry in 2006 and 2010 were shown in Table 1. For example, the power-law function of the
urban samples in 2006 was f(x) = 160.10 × x^−2.18. The 95% confidence intervals of the first coefficient were [112.40, 207.80], and the 95% confidence intervals of the second coefficient were [−2.41,
−1.95]. The adjusted R-square of the fit was 0.9846, the SSE was 3.1630, and the RMSE was 0.5134.
Next, two-sample Kolmogorov-Smirnov tests were performed to verify that the distribution of the rates of different numbers of chronic diseases was a power-law distribution. The two-sample K-S test
returns a test decision for the null hypothesis that the data in vectors ×1 and ×2 are from the same continuous distribution, and the alternative hypothesis is that ×1 and ×2 are from different
continuous distributions. The result (h) is 1 if the test rejects the null hypothesis at the 5% significance level and 0 otherwise.
The two-sample K-S test results (h) of the fitting in 2006 and 2010 were all 0, and the asymptotic p-values were all greater than 0.05, which means that the K-S test did not reject the null
hypothesis at the 5% significance level; in other words, the rates of different numbers of chronic diseases in 2006 and 2010 obeyed a power-law distribution.
Table 1 shown that the power exponents of male elderly and female elderly in 2006 and 2010 were close to a mean −2.5; the power exponents of the urban elderly in 2006 and 2010 were under −2.5, while
the power exponents of the rural elderly in 2006 and 2010 were under −2.5.
Table 1. Power-law fitting results and K-S test.
3.2. Paired T-Test of the Census Registry and Gender and Power-Law Fit
This paper used the paired t-test to verify the census registry and the effects of gender on the rates of chronic diseases. The original data for the paired t-test were the rates of different numbers
of chronic diseases. The minimum number in the survey was 0, which represents that one person was not suffering from any of the diseases, and the maximum number in the survey was 16, which represents
that one person was suffering from 16 diseases at the same time.
3.2.1. Paired T-Test on the Census Registry
In this section, we provided the test results for the sub-groups by the census registry from two surveys, and the corresponding results were shown in Table 2.
As shown in Table 2, all of the paired t-test p-values were close to 1, which indicates that the rates of the different numbers of chronic diseases did not exist urban-rural differences.
3.2.2. Paired T-Test of Gender
In this section, we given the test results on the male elderly and female elderly groups, and the results were shown in Table 3.
As shown in Table 3, all of the paired t-test p-values were close to 1, which indicates that the rates of different numbers of chronic diseases did not exist gender differences.
Table 2. Paired t-test of the census registry.
3.2.3. Power Law of the Rates of Different Numbers of Chronic Diseases in 2006 and 2010
The paired t-test results showed that the rates of different numbers of chronic diseases did not have any urban-rural differences and gender differences. The results mean that the samples of urban
male, urban female, rural male and rural female can be combined together to obtain the total number of samples in 2006 and 2010. Fitting information on the samples in 2006 and 2010 were shown in the
following sections.
We further performed a two-sample K-S test to verify that the distribution of the rates of different numbers of chronic diseases was a power-law distribution. The test results (h) in 2006 and 2010
were all 0, and the asymptotic p-values were all 0.2672 > 0.05, which mean that the K-S test did not reject the null hypothesis at the 5% significance level. In other words, the rates of different
numbers of chronic diseases in 2006 and 2010 obeyed a power-law distribution.
The power-law fitting results for 2006 and 2010 were shown in Figure 1 and Figure 2, respectively.
The power-law function of the samples in 2006 was f(x) = 201.7 × x^−2.50; the 95% confidence bounds of the first coefficient were [153.30, 250.10], and the 95% confidence bounds of the second
coefficient were [−2.70, −2.31]. The adjusted R-square of the fit was 0.9929, the SSE was 1.112, and the RMSE was 0.3044.
The power-law function of the samples in 2010 was f(x) = 236.00 × x^−^2.47; the 95%
Figure 1. Power law for the rates of different numbers of chronic diseases in 2006.
Figure 2. Power law for the rates of different numbers of chronic diseases in 2010.
confidence bounds of the first coefficient were [165.50, 306.50], and the 95% confidence bounds of the second coefficient were [−2.71, −2.23]. The adjusted R-square of the fit was 0.9887, the SSE was
2.6440, and the RMSE was 0.4694.
All of the analysis above showed that the rates of different numbers of chronic diseases had a power-law distribution. The power exponent of the samples in 2006 was −2.50, and the power exponent of
the samples in 2010 was −2.47, and they were approximately equal to −2.5.
3.3. Paired T-Test of the Time and Power-Law Fitting
3.3.1. Paired T-Test of Time
In this section, we given the test results on the time, and these test results can reveal the difference between 2006 and 2010. The results were shown in Table 4.
As shown in Table 4, all of the paired t-test p-values were close to 1, which indicates that the rates of different numbers of chronic diseases did not have time differences.
3.3.2. Power-Law Fitting of the Total Samples
The paired t-test results showed that the rates of different numbers of chronic diseases did not exist time differences. The results mean that the samples in 2006 and the samples in 2010 can be
combined together to obtain the power-law fit for the rates of different numbers of chronic diseases (see Figure 3).
Figure 3. Power law for the rates of different numbers of chronic diseases.
The power-law function of the total samples was f(x) = 218.80 × x^−^2.48. The 95% confidence bounds of the first coefficient were [159.50, 278.10], and the 95% confidence bounds of the second
coefficient were [−2.70, −2.26]. The adjusted R-square of the fit was 0.9908, the SSE was 1.7800, and the RMSE was 0.3851.
The K-S test result (h) was 0, and the asymptotic p-values were all 0.2672 > 0.05, which mean that the K-S test did not reject the null hypothesis at the 5% significance level. In other words, the
rates of different numbers of chronic diseases followed a power-law distribution. Furthermore, the power exponent of the samples was −2.48, which was very close to mean −2.5. This indicates that the
logarithmic number of chronic illness that an elderly had declines at a certain ratio as logarithmic age increases.
4. Conclusions
We used the data that were obtained from the 2006 and 2010 Chinese Urban and Rural Elderly Population Surveys to examine whether there was a power-law distribution of the number of chronic diseases
among elderly Chinese people. The paired t-test method was used to analyze the urban-rural differences, time differences and gender differences, and then, we proceeded with power-low fitting of the
data. The main conclusions were as follows: 1) All of the paired t-test p-values of the urban males and rural males in 2006 and 2010, and the urban females and rural females in 2006 and 2010 were
close to 1, which mean that the rates of different numbers of chronic diseases did not exist urban-rural differences, and thus, the samples of the urban elderly and rural elderly can be combined
together. 2) All of the paired t-test p-values of the urban males and urban females in 2006 and 2010, and the rural males and rural females in 2006 and 2010 were close to 1, which mean that the rates
of different numbers of chronic disease did not exist gender differences. The samples of male elderly and female elderly can be combined together. 3) All of the paired t-test p-values of urban males
in 2006, urban males in 2010, urban females in 2006, urban females in 2010, rural males in 2006, rural males in 2010, rural females in 2006 and rural females in 2010 were close to 1, which mean that
the rates of different numbers of chronic diseases did not exist time differences, and thus, the samples from 2006 and 2010 can be combined together. 4) There was a power-law distribution of the
rates of different numbers of chronic diseases; the power-law distribution was f(x) = 218.80 × x^−^2.48, x = 3, 4, …, 16. 5). The power exponents were approximately −2.5.
In the future, we wish to explore whether the distribution of chronic diseases has differences among the different types of chronic diseases, such as fatal chronic disease and non-fatal chronic
disease, and in addition, we want to explore the intrinsic mechanism of the distribution of the number of chronic diseases.
We thank the China Research Center on Aging for their support in the data collection and management of this project.
^1This project has been performed three times in 2002, 2006 and 2010, respectively. Because there is a flaw in 2002 data, we only use the data from 2006 and 2010. | {"url":"https://scirp.org/journal/paperinformation?paperid=73028","timestamp":"2024-11-10T06:36:30Z","content_type":"application/xhtml+xml","content_length":"111196","record_id":"<urn:uuid:280d465e-debc-4322-b5f4-31699ce1a484>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00347.warc.gz"} |
What job can I get after MSc Mathematics?
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Grade Pay Matrix Basic Pay
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4. Actuary.
5. Data Analyst/ Business Analyst/ Big Data Analyst.
6. Economist.
7. Market Researcher.
8. Psychometrician.
What is DRDO entry test?
Defence Research Developement Organization conducts Scientist Entry Test (SET) for recruitment of personnel to Group B scientist posts in DRDO. On the other hand, DRDO entry test is conducted for
technical assistant and technician vacancies in the organization.
What are the objectives of teaching mathematics in primary?
The goals of the primary mathematics curriculum are:
• Stimulate interest in the learning of mathematics.
• Help students understand and acquire basic mathematical concepts and computational skills.
• Help students develop creativity and the ability to think, communicate, and solve problems.
Can I go to IIT after BSc?
There is no age bar for IIT JAM Exam. Any student who has completed B.Sc. degree with at least 55% can apply for the exam.
What is the use of MSc Mathematics?
In the MSc Mathematics, students get a deep insight into pure and applied mathematics. Students learn about the problem-solving skills and reasoning skills which helps them to solve them the
real-life problems.
What is the importance of math and science?
Math and science education provides a framework for how to find answers. Math models phenomena and relationships in our observable environment, while articulating concepts from the intuitive to the
obscure. Science gives deep attention to the quality and interaction of the things that surround us. | {"url":"https://www.joialife.com/writing-guides/what-job-can-i-get-after-msc-mathematics/","timestamp":"2024-11-14T21:15:32Z","content_type":"text/html","content_length":"46276","record_id":"<urn:uuid:68f4f9d0-f43d-4b4b-bd80-dfd752212abe>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00272.warc.gz"} |
Budee U Zaman
Author:Budee U Zaman
EasyChair Preprint 14771
EasyChair Preprint 14054
EasyChair Preprint 14004
EasyChair Preprint 13920
EasyChair Preprint 13846
EasyChair Preprint 13796
EasyChair Preprint 13595
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EasyChair Preprint 13504
EasyChair Preprint 13144
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EasyChair Preprint 12823
EasyChair Preprint 12598
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EasyChair Preprint 11887
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EasyChair Preprint 11309
AGI^2, AI^2, AI Navigating, AI technologies, algebraic methods, Artificial Intelligence^3, autonomous behavior, Binary Number, Binary Pattern, cognitive science and technological, Collatz Conjecture^
3, composite numbers, Cosmology, Diagonal matrix, Diophantine equation, Diophantine quintic equation, directed graphs, environments, Ethical, ethics, even^2, even pairs, existence, GML, GNL, Goldbach
Conjecture^2, Goldbach’s Conjecture, goldbach’s conjectures, graph, Humanity, imtgers, infinite numbers, Infinite Series, infinitesimals, integer^4, integers^5, intger, intgers, knowledge, learning
behavior^2, learning opportunities, log function, matrix^2, matrix theory, meaning, modeling^2, modern prime number theory, narural, natural number^5, number^3, ODD^2, odd pairs, odd prime,
perception, periodic functions, positive integers, precise mathematical equation, Prime^10, Prime Gap, Prime number^11, prime numbers^2, probabilistic methods, property, Real^4, real number, Riemann
hypothesis, robot evolution, student behavior^2, technological advances, Theory of partitions primes, zeta functions. | {"url":"https://wwww.easychair.org/publications/author/mWvL","timestamp":"2024-11-12T09:46:38Z","content_type":"text/html","content_length":"17420","record_id":"<urn:uuid:e2104035-4962-4cd5-a2ed-fc82a0764924>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00356.warc.gz"} |
Compact genetic algorithm - Complex systems and AI
Compact genetic algorithm
Compact genetic algorithm
The information processing purpose of thegenetic algorithm compact is to simulate the behavior of a algorithm genetics with a much smaller memory footprint (without requiring maintenance of a
population). This is achieved by maintaining a vector that specifies the probability of including each component in a solution in new candidate solutions. Candidate solutions are probabilistically
generated from the vector and the components of the best solution are used to make small changes to the probabilities in the vector.
The compact genetic algorithm maintains a real-valued prototype vector that represents the probability that each component is expressed in a candidate solution. The following algorithm provides a
pseudocode of the compact genetic algorithm to maximize a cost function. The parameter n indicates the number of probabilities of updating the conflicting bits at each iteration.
The vector update parameter (n) influences the quantity of probability updates at each iteration of the algorithm. The vector update parameter (n) can be considered comparable to the population size
parameter in the genetic algorithm. The first results demonstrate that cGA can be compared to a standard genetic algorithm on classical binary string optimization problems (such as OneMax). The
algorithm can be considered to have converged if the vector probabilities are all equal to 0 or 1. | {"url":"https://complex-systems-ai.com/en/probabilistic-algorithms-2/compact-genetic-algorithm/","timestamp":"2024-11-05T07:43:46Z","content_type":"text/html","content_length":"156504","record_id":"<urn:uuid:a231d9e9-afc1-42ac-abfb-6862656ac67e>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00326.warc.gz"} |
Essential Math for Data Science
Delve into the mathematical concepts that power Machine Learning, Artificial Intelligence, and Data Science in a meaningful and approachable manner
Join the waitlist and get notified about the next cohort
Essential Math for Data Science
A 6-week course with expert-led sessions, real-life project with personalized critique and teamwork
In the realm of engineering, it’s universally acknowledged that mathematics serves as the fundamental cornerstone upon which all other fields are constructed. A limited grasp of mathematics can
present substantial challenges in your engineering career. It’s a common concern among working professionals that advanced math during their college years often lacked practical applications,
hindering authentic learning.
In this 6-week course, led by Thomas Nield, author of O’Reilly’s “Essential Math for Data Science” and instructor at the University of Southern California, the mission is straightforward: bridge your
knowledge gap in mathematics and connect you to the fundamental concepts that drive Artificial Intelligence, Machine Learning, and Data Science in a meaningful way.
What awaits you in this project-centric course?
• Core Concepts: You’ll learn core concepts of linear algebra, calculus, probability, and basic regression techniques and how they are applied on solving real-world problems in Data Science. These
are not just abstract theories; they’re practical tools you can wield in your engineering journey.
• Real-World Applications: Our course places a strong emphasis on applying mathematical principles to solve tangible, real-world problems. You’ll see how these concepts come to life in a form of a
project that is meaningful to you.
Don’t settle for superficial learning. Join other like-minded students on this exciting journey to not just learn math but to love it. Transform your career, and become an indispensable asset in the
ever-evolving world of engineering and data science.
Join the waitlist and get notified about the next cohort
Cohort Name Class Period Registration Deadline
#1 Cohort 1 Dec 04, 23 - Feb 05, 24 Monday, Dec 11 - 2023 at 05:00 PM UTC
Class is closed to new registrations
• Cohort Name
• Class Period
Dec 04, 23 - Feb 05, 24
• Registration Deadline
Monday, Dec 11 - 2023 at 05:00 PM UTC
Class is closed to new registrations
Program Structure
Typical week in the cohort
Communication and networking are core components of the ClasspertX course experience. In this course, you will be part of a global learning community. In order to accommodate all participants, we
have designed much of the course experience to take place asynchronously, with a synchronous class session that occurs weekly.
Async Sessions on Discord
• Reading from the book
Includes a free copy of the book
This course is centered around Essential Math for Data Science which will be used as a supplementary material for the classes
• Videos
• Quizzes
• Exercises
• Students will be prompted to submit questions during the week, and the instructor should choose questions to answer for students during the weekly session
Sync Sessions on Zoom
□ Q&A with the instructor
□ Additional demos / examples of key topics
□ Group practice - students break out to work on an exercise
□ Group discussion
• Understand what the course will cover
• Understand what is expected of participants in this course
• Understand how to get the most out of this course
• Meet your instructor
Week 1 - Linear Algebra
• Learn how to compose data as vectors and matrices
• Use NumPy to work with linear transformations
• Apply linear algebra to tasks like matrix decomposition and systems of equations
Week 2 - Calculus Basics
• Leverage SymPy to declare and work with functions
• Calculate derivatives to find slopes with respect to variables
• Calculate integrals to find areas under a curve
Week 3 - Probability Basics
• Create Monte Carlo simulations
• Calculate joint and union probability
• Leverage conditional probability and Bayes Theorem
Week 4 - Descriptive and Inferential Statistics
• Describe data using parameters like the mean, median, variance, standard deviation, and coefficient of variation
• Leverage the central limit theorem to infer population parameters from a sample
• Perform confidence intervals and hypothesis testing to identify population parameters from a sample
Week 5 - Linear Regressions and Logistic Regression
• Fit data to regressions models like linear regression and logistic regression
• Evaluate the quality of a regression model and whether variance and overfitting are occurring
• Learn techniques like matrix decomposition, gradient descent, and hill climbing to fit a regression
Week 6 - Model Validation
• Understand the nature of data and how problems like bias, outliers, and operating domain affect success of a project
• Leverage train/test split patterns to see how well models perform on data not seen before
• Avoid fallacies like Texas Sharpshooter Fallacy and p-hacking which become prevalent with big data availability
Week 7 - Final Project
—See corresponding document—
• Learn math concepts that form the basic of Machine Learning, Artificial Intelligence and Data Science
• Understand core concepts of applied linear algebra, calculus, probability, statistics and basic regression techniques
• Gain autonomy to speak the strengths and weaknesses of different models being used by industries and research teams
Machine learning, artificial intelligence, and data science topics are hard to navigate without some essential mathematical concepts. So whether students are trying to pivot into these areas or
looking to get started, they will ultimately need to grasp the applied math concepts presented in this course.
Who this course is for
Target Audience
• Undergraduate and graduate students majoring in computer science, data science, machine learning and artificial intelligence
• Data science and AI enthusiasts looking to build a solid foundation in mathematics
• Working professionals such as data analystis or software engineers who want to transition careers or advance in their fields
• Basic high school algebra
• Basic computer proficiency
• Knowing Python is a plus, but it isn't a requirement and students can focus on the concepts instead
Thomas Nield is a consultant, writer, and instructor. After a decade in the airline industry, he authored two books for O’Reilly Media and regularly teaches classes on artificial intelligence,
statistics, machine learning, and optimization algorithms. Currently he is teaching at University of Southern California and defining system safety approaches with AI for clients. In another
endeavor, Yawman Flight, he is inventing a new category of flight simulation controllers and bringing them to market. He enjoys making technical content relatable and relevant to those unfamiliar or
intimidated by it.
This course is not going to throw chalkboards full of Greek symbols at you. We will explain mathematical ideas in plain English (along with plenty of visualizations) so concepts come across
intuitively. Along the way we will learn practical applications, such as how statistics can measure game controller performance and stick drift. We will also learn to avoid common fallacies that
steer data science and machine learning into unproductive rabbit holes
Career Highlights
• Teach AI System Safety at University of Southern California in the Aviation Safety and Security program
• Authored three books with O’Reilly Media and Packt including "Getting Started with SQL", "Essential Math for Data Science", and "Learning RxJava"
• Worked over 10 years in operations research and machine learning applications for airlines and other industries
• Creator of the Yawman Arrow, a handheld controller for aviation/flight simulation applications
Relevant Publications in the Field
Presents mathematical concepts in an approachable way that is actually applicable to DS. When they told you the math would be useful some day, this book actually points out why in terms of DS
This is one of the best books I have read, really explaining the fundamentals of math like never explained before. I am also making by teenagers read this book to clarify some concepts and develop
more interest in mathematics. I highly recommend this book
A book I would have loved to have when starting out!
Essential Math for Data Science by Thomas Nield is exactly what the title suggests. It covers the most important math concepts that are needed to work in data and analytics related jobs. The topics
range from basic math, to probability, stats, linear algebra, and calculus.
By focusing on the most important aspects and by providing very manageable examples in Python, one can grasp the intuition behind these topics very fast. Even if you are already a seasoned vet, you
might learn new things or at least see them from a different perspective (loved the explanation of statistical significance using the CDF). However, keep in mind that this a very dense book. A lot of
content is packed into very few packages. This might be even too dense if you have never been exposed to these topics. Maybe grab a good stats, linear algebra, and calculus intro before jumping into
this book.
Frequently Asked Questions
Can I get my employer to pay for the program?
An investment in knowledge always pays the best return for your company. It’s a tiny investment compared to what you could potentially bring in terms of innovation to your workplace.
Many companies offer reimbursement for courses related to your job. Ask your employer about tuition benefits. Even if there is no specific tuition assistance, many companies allocate money toward
professional development. Managers may have money earmarked for industry conferences and many have not considered applying it toward continuing education.
Approach asking for tuition assistance like you would a formal negotiation. Go into the discussion with clearly outlined and rehearsed messages about what you hope to gain and emphasize how it
will benefit your boss and organization.
What is the time commitment for this course?
This course requires 6-8 hours/week of work. Self-paced activities such as homework assignments, readings, and watching video lectures exist to help you build up knowledge until you’re able to
demonstrate, through your project, that you’ve achieved the learning outcomes of the program. Although important, homework assignments won’t be graded by the instructional team. The only gradable
unit in this program will be your project, which is a prerequisite for certificate emission
How do refunds work?
If the course does not meet your expectations, you can ask for a full refund before the beginning of week 3. No questions asked!
How are certificates issued? Will I be evaluated?
In order to earn a certificate, you’ll need to submit a project and get a passing grade. The instructional team will provide comprehensive feedback on your project, highlighting the strong
points, areas for improvement, and helpful tips on how it could be successful outside of the class.
Will this course run again in the future?
Cohort-style classes are to some extent very similar to traditional classroom environments which makes them largely dependent on the instructor’s schedule. While we always hope there’ll be a next
cohort, there’s no guarantee that the instructor will be available for the next one. If you’re busy right now, but really interested in taking this course, we advise you to sign up now and ask
for a refund if you can’t commit to the program after week 3.
What determines the price of the course?
Our programs require significant time from a number of professionals including mentors, the instructor, and organization staff. It is not a canned lecture course but an educational opportunity
tailored to your needs and interests.
Can I use programming languages other than Python to complete the exercises/assignments?
Students welcome to do the exercises in the programming language of their choice, As long as they can demonstrate a thought process and correct answers. While python is used as the vehicle to
demonstrate concepts, the concepts themselves are explained outside of the code.
Join the waitlist and get notified about the next cohort | {"url":"https://classpert.com/classpertx/courses/essential-math-for-data-science/cohort","timestamp":"2024-11-10T05:06:28Z","content_type":"text/html","content_length":"309502","record_id":"<urn:uuid:e189a3f1-36cf-46cb-b38b-859e00ae1828>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00061.warc.gz"} |
Central Limit Theorem
January 20th, 2024 (9 months ago) • 2 minutes
Normal Distribution Curve
The distribution of sample means approximates a normal distribution as the sample size gets larger, regardless of the population's distribution.
In plain words, it says that if we take a sample from a population (any population) and calculate the mean of that sample, we will get a normal distribution (also called a bell curve). And if we take
a larger sample, then the curve will approximately more closely.
This idea is important because it explain why a lot of life phenomenon is normally distributed. For example, the height of people, the weight of people, the IQ of people, the income of people, etc.
It's highly unlikely that some random event happening will be close to either extreme. For example, you are more likely to row any number in the middle rather than 2 or 12 in a dice roll because
there are just more combination of them. Similarly, you are more likely to be of average height than extremely tall or short because there are just more people of average height.
I rereading this concept as I am preparing for USF's grad school interview. Fingers crossed 🤞. | {"url":"https://weichun.xyz/blog/central-limit-theorem","timestamp":"2024-11-12T09:31:05Z","content_type":"text/html","content_length":"20022","record_id":"<urn:uuid:d987a785-b2be-4bdc-af76-bf71717d106a>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00364.warc.gz"} |
Linear Label Ranking with Bounded Noise
Linear Label Ranking with Bounded Noise
Dimitris Fotakis · Alkis Kalavasis · Vasilis Kontonis · Christos Tzamos
Hall J (level 1) #820
Keywords: [ Label Ranking ] [ Gaussian ] [ Linear Sorting Function ] [ Noise ]
Abstract: Label Ranking (LR) is the supervised task of learning a sorting function that maps feature vectors $x \in \mathbb{R}^d$ to rankings $\sigma(x) \in \mathbb S_k$ over a finite set of $k$
labels. We focus on the fundamental case of learning linear sorting functions (LSFs) under Gaussian marginals: $x$ is sampled from the $d$-dimensional standard normal and the ground truth ranking $\
sigma^\star(x)$ is the ordering induced by sorting the coordinates of the vector $W^\star x$, where $W^\star \in \mathbb{R}^{k \times d}$ is unknown. We consider learning LSFs in the presence of
bounded noise: assuming that a noiseless example is of the form $(x, \sigma^\star(x))$, we observe $(x, \pi)$, where for any pair of elements $i \neq j$, the probability that the order of $i, j$ is
different in $\pi$ than in $\sigma^\star(x)$ is at most $\eta < 1/2$. We design efficient non-proper and proper learning algorithms that learn hypotheses within normalized Kendall's Tau distance $\
epsilon$ from the ground truth with $N= \widetilde{O}(d\log(k)/\epsilon)$ labeled examples and runtime $\mathrm{poly}(N, k)$. For the more challenging top-$r$ disagreement loss, we give an efficient
proper learning algorithm that achieves $\epsilon$ top-$r$ disagreement with the ground truth with $N = \widetilde{O}(d k r /\epsilon)$ samples and $\mathrm{poly}(N)$ runtime. | {"url":"https://nips.cc/virtual/2022/poster/52944","timestamp":"2024-11-08T10:47:10Z","content_type":"text/html","content_length":"49616","record_id":"<urn:uuid:d1cf2105-afd0-44ee-a92a-96f35b8f409a>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00496.warc.gz"} |
Maximum elements of array
M = max(A) returns the maximum elements of an array.
• If A is a vector, then max(A) returns the maximum of A.
• If A is a matrix, then max(A) is a row vector containing the maximum value of each column of A.
• If A is a multidimensional array, then max(A) operates along the first dimension of A whose size does not equal 1, treating the elements as vectors. The size of M in this dimension becomes 1,
while the sizes of all other dimensions remain the same as in A. If A is an empty array whose first dimension has zero length, then M is an empty array with the same size as A.
• If A is a table or timetable, then max(A) returns a one-row table containing the maximum of each variable. (since R2023a)
M = max(A,[],"all") finds the maximum over all elements of A.
M = max(A,[],dim) returns the maximum element along dimension dim. For example, if A is a matrix, then max(A,[],2) returns a column vector containing the maximum value of each row.
M = max(A,[],vecdim) returns the maximum over the dimensions specified in the vector vecdim. For example, if A is a matrix, then max(A,[],[1 2]) returns the maximum over all elements in A because
every element of a matrix is contained in the array slice defined by dimensions 1 and 2.
M = max(A,[],___,missingflag) specifies whether to omit or include missing values in A for any of the previous syntaxes. For example, max(A,[],"includemissing") includes all missing values when
computing the maximum. By default, max omits missing values.
[M,I] = max(___) also returns the index into the operating dimension that corresponds to the first occurrence of the maximum value of A.
[M,I] = max(A,[],___,"linear") also returns the linear index into A that corresponds to the maximum value in A.
C = max(A,B) returns an array with the largest elements taken from A or B.
___ = max(___,"ComparisonMethod",method) optionally specifies how to compare elements for any of the previous syntaxes. For example, for a vector A = [-1 2 -9], the syntax max(A,
[],"ComparisonMethod","abs") compares the elements of A according to their absolute values and returns a maximum value of -9.
Largest Vector Element
Create a vector and compute its largest element.
A = [23 42 37 18 52];
M = max(A)
Largest Complex Element
Create a complex vector and compute its largest element, that is, the element with the largest magnitude.
A = [-2+2i 4+i -1-3i];
Largest Element in Each Matrix Column
Create a matrix and compute the largest element in each column.
Largest Element in Each Matrix Row
Create a matrix and compute the largest element in each row.
A = [1.7 1.2 1.5; 1.3 1.6 1.99]
A = 2×3
1.7000 1.2000 1.5000
1.3000 1.6000 1.9900
Maximum of Array Page
Create a 3-D array and compute the maximum over each page of data (rows and columns).
A(:,:,1) = [2 4; -2 1];
A(:,:,2) = [9 13; -5 7];
A(:,:,3) = [4 4; 8 -3];
M1 = max(A,[],[1 2])
M1 =
M1(:,:,1) =
M1(:,:,2) =
M1(:,:,3) =
To compute the maximum over all dimensions of an array, you can either specify each dimension in the vector dimension argument or use the "all" option.
Largest Element Including Missing Values
Create a matrix containing NaN values.
A = [1.77 -0.005 NaN -2.95; NaN 0.34 NaN 0.19]
A = 2×4
1.7700 -0.0050 NaN -2.9500
NaN 0.3400 NaN 0.1900
Compute the maximum value of the matrix, including missing values. For matrix columns that contain any NaN value, the maximum is NaN.
M = max(A,[],"includemissing")
M = 1×4
NaN 0.3400 NaN 0.1900
Largest Element Indices
Create a matrix A and compute the largest elements in each column, as well as the row indices of A in which they appear.
Return Linear Indices
Create a matrix A and return the maximum value of each row in the matrix M. Use the "linear" option to also return the linear indices I such that M = A(I).
[M,I] = max(A,[],2,"linear")
Largest Element Comparison
Create a matrix and return the largest value between each of its elements compared to a scalar.
Input Arguments
A — Input array
scalar | vector | matrix | multidimensional array | table | timetable
Input array, specified as a scalar, vector, matrix, multidimensional array, table, or timetable.
• If A is complex, then max(A) returns the complex number with the largest magnitude. If magnitudes are equal, then max(A) returns the value with the largest magnitude and the largest phase angle.
• If A is a scalar, then max(A) returns A.
• If A is a 0-by-0 empty array, then max(A) is as well.
If A has type categorical, then it must be ordinal.
Data Types: double | single | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical | categorical | datetime | duration | table | timetable
Complex Number Support: Yes
dim — Dimension to operate along
positive integer scalar
Dimension to operate along, specified as a positive integer scalar. If you do not specify the dimension, then the default is the first array dimension whose size does not equal 1.
Dimension dim indicates the dimension whose length reduces to 1. The size(M,dim) is 1, while the sizes of all other dimensions remain the same, unless size(A,dim) is 0. If size(A,dim) is 0, then max
(A,dim) returns an empty array with the same size as A.
Consider an m-by-n input matrix, A:
• max(A,[],1) computes the maximum of the elements in each column of A and returns a 1-by-n row vector.
• max(A,[],2) computes the maximum of the elements in each row of A and returns an m-by-1 column vector.
vecdim — Vector of dimensions
vector of positive integers
Vector of dimensions, specified as a vector of positive integers. Each element represents a dimension of the input array. The lengths of the output in the specified operating dimensions are 1, while
the others remain the same.
Consider a 2-by-3-by-3 input array, A. Then max(A,[],[1 2]) returns a 1-by-1-by-3 array whose elements are the maximums computed over each page of A.
B — Additional input array
scalar | vector | matrix | multidimensional array | table | timetable
Additional input array, specified as a scalar, vector, matrix, multidimensional array, table, or timetable. Inputs A and B must either be the same size or have sizes that are compatible (for example,
A is an M-by-N matrix and B is a scalar or 1-by-N row vector). For more information, see Compatible Array Sizes for Basic Operations.
• If A and B are both arrays, then they must be the same data type unless one is a double. In that case, the data type of the other array can be single, duration, or any integer type.
• If A and B are ordinal categorical arrays, they must have the same sets of categories with the same order.
• If either A or B is a table or timetable, then the other input can be an array, table, or timetable.
If B has type categorical, then it must be ordinal.
Data Types: double | single | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical | categorical | datetime | duration | table | timetable
Complex Number Support: Yes
missingflag — Missing value condition
"omitmissing" (default) | "omitnan" | "omitnat" | "omitundefined" | "includemissing" | "includenan" | "includenat" | "includeundefined"
Missing value condition, specified as one of the values in this table.
Value Input Data Type Description
"omitmissing" All supported data
"omitnan" double, single, Ignore missing values in the input arrays, and compute the maximum over fewer points. If all elements in the operating dimension are missing, then the
duration corresponding element in M is missing.
"omitnat" datetime
"omitundefined" categorical
"includemissing" All supported data
"includenan" double, single, Include missing values in the input arrays when computing the maximum. If any element in the operating dimension is missing, then the corresponding element in
duration M is missing.
"includenat" datetime
"includeundefined" categorical
method — Comparison method
"auto" (default) | "real" | "abs"
Comparison method for numeric input, specified as one of these values:
• "auto" — For a numeric input array A, compare elements by real(A) when A is real, and by abs(A) when A is complex.
• "real" — For a numeric input array A, compare elements by real(A) when A is real or complex. If A has elements with equal real parts, then use imag(A) to break ties.
• "abs" — For a numeric input array A, compare elements by abs(A) when A is real or complex. If A has elements with equal magnitude, then use angle(A) in the interval (-π,π] to break ties.
Output Arguments
M — Maximum values
scalar | vector | matrix | multidimensional array | table
Maximum values, returned as a scalar, vector, matrix, multidimensional array, or table. size(M,dim) is 1, while the sizes of all other dimensions match the size of the corresponding dimension in A,
unless size(A,dim) is 0. If size(A,dim) is 0, then M is an empty array with the same size as A.
I — Index
scalar | vector | matrix | multidimensional array | table
Index, returned as a scalar, vector, matrix, multidimensional array, or table. I is the same size as the first output.
When "linear" is not specified, I is the index into the operating dimension. When "linear" is specified, I contains the linear indices of A corresponding to the maximum values.
If the largest element occurs more than once, then I contains the index to the first occurrence of the value.
C — Maximum elements from A or B
scalar | vector | matrix | multidimensional array | table | timetable
Maximum elements from A or B, returned as a scalar, vector, matrix, multidimensional array, table, or timetable. The size of C is determined by implicit expansion of the dimensions of A and B. For
more information, see Compatible Array Sizes for Basic Operations.
The data type of C depends on the data types of A and B:
• If A and B are the same data type, then C matches the data type of A and B.
• If either A or B is single, then C is single.
• If either A or B is an integer data type with the other a scalar double, then C assumes the integer data type.
• If either A or B is a table or timetable, then C is a table or timetable.
Extended Capabilities
Tall Arrays
Calculate with arrays that have more rows than fit in memory.
The max function supports tall arrays with the following usage notes and limitations:
• Index output is not supported for tall tabular inputs.
For more information, see Tall Arrays.
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
• If you specify an empty array for the second argument in order to supply dim or missingflag, the second argument must be of fixed-size and of dimension 0-by-0.
• If you specify dim or missingflag, then they must be constants.
• If the input is a variable-size array, the length of the dimension to operate along must not be zero at run-time.
• See Variable-Sizing Restrictions for Code Generation of Toolbox Functions (MATLAB Coder).
• See Code Generation for Complex Data with Zero-Valued Imaginary Parts (MATLAB Coder).
GPU Code Generation
Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.
Usage notes and limitations:
HDL Code Generation
Generate VHDL, Verilog and SystemVerilog code for FPGA and ASIC designs using HDL Coder™.
Usage notes and limitations:
• Inputs of 3-D matrices or greater are not supported.
• Inputs that have complex data types are not supported.
• Input matrices or vectors must be of equal size.
Thread-Based Environment
Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.
This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
The max function fully supports GPU arrays. To run the function on a GPU, specify the input data as a gpuArray (Parallel Computing Toolbox). For more information, see Run MATLAB Functions on a GPU
(Parallel Computing Toolbox).
Distributed Arrays
Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.
Usage notes and limitations:
• Index output is not supported for distributed tabular inputs.
For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).
Version History
Introduced before R2006a
R2024b: Specifying second input array as character array is not supported
Specifying a second input array as a character array returns an error. This change minimizes confusion with other arguments that can be specified as character vectors, such as the missing value
condition. To maintain the previous functionality, you can convert the second input array to double, for example, max(A,double(B),"includenan").
R2023b: Specifying second input array as character array will not be supported
Specifying a second input array as a character array gives a warning and will generate an error in a future release. This change minimizes confusion with other arguments that can be specified as
character vectors, such as the missing value condition. To maintain the previous functionality, you can convert the second input array to double, for example, max(A,double(B),"includenan").
R2023a: Specify missing value condition
Omit or include all missing values in the input arrays when computing the maximum value by using the "omitmissing" or "includemissing" options. Previously, "omitnan", "includenan", "omitnat",
"includenat", "omitundefined", and "includeundefined" specified a missing value condition that was specific to the data type of the input arrays.
R2023a: Perform calculations directly on tables and timetables
The max function can calculate on all variables within a table or timetable without indexing to access those variables. All variables must have data types that support the calculation. For more
information, see Direct Calculations on Tables and Timetables.
R2021b: Specify comparison method
Specify the real or absolute value method for determining the maximum value of the input by using the ComparisonMethod parameter.
R2018b: Operate on multiple dimensions
Operate on multiple dimensions of the input arrays at a time. Specify a vector of operating dimensions, or specify the "all" option to operate on all array dimensions. | {"url":"https://fr.mathworks.com/help/matlab/ref/double.max.html","timestamp":"2024-11-08T11:33:40Z","content_type":"text/html","content_length":"161140","record_id":"<urn:uuid:d65c6856-3145-49c8-babb-ae6248481be1>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00686.warc.gz"} |
Roman Numerals - Chart, Rules | What are Roman Numerals?- [[company name]] [[target location]], [[stateabr]]
Roman Numerals - Rules, Chart | What Are Roman Numerals?
When you consider numericals nowadays, the first of all that comes to mind is the decimal method we utilize regularly. This system, however, is not the single fashion to portray numbers. There are
multiple methods utilized by distinct cultures all over the world that use all sorts of symbols. One of method is Roman numerals.
Since ancient Rome, Roman numerals have been a method of expressing numericals utilizing a combination of characters from the Latin alphabet. It remained during the Middle Ages and the modern day, to
the extend it is still learned in school, that is probably why you have stumbled upon this article.
Now, we are going to walkthrough Roman numerals, what they are, how they operate, and how to change Roman numbers to regular numbers.
What Are Roman Numbers?
Primarily, let's look at a quick look at the past of Roman numerals. Roman numbers were initially used by the historic Romans, as you might have expected from the name. They were used in multiple
aspects of the Roman world, including trade, architecture, and even war.
Presently, its extensive use is mainly attributed to artistic reasons. You may have seen Roman numbers as hour marks on a clock, copyright dates, page numbering, chapter numbers, or in film sequels
(e.g., The Godfather Part II).
The Roman number approach represents numericals utilizing a blend of characters from the Latin alphabet. Letters are mixed to form groups that represent numbers. Seven characters, I, V, X, L, C, D,
and M, depicts the numericals 1, 5, 10, 50, 100, 500, and 1000, respectively. You can then blend these numericals to represent any value in the numerical system.
Meaning of the Roman numerals
Although the decimal system is founded on the concept of place value, Roman numbers are based on cumulative and subtractive principles. This means that a Roman numeral's numeric values are
established on the sum of the values of its individual parts. Another major distinction is that the decimal approach is based on the number 10. In contrast, Roman numbers are established on the
numbers 1 (I), 5 (V) and 10 (X).
Let's take a look at a few hands-on examples of Roman numbers.
The computer game street fighter IV was released in arcades back in 2008. If we take a look at the numericals in the name, we see it has a V in it. This is because the number 5 in Roman number is
represented by the letter V. Preceded by it is an I, or 1. Thus, we grasp that this is the 4th entry in the series using the properties we will speak on further ahead.
The film Star Wars Episode VI was the last entry in the original trilogy. Looking at the value described, it contains a V followed by an I. Hence, we will sum a 1 to the value of V, that is 5,
comprehending this Star Wars movie is the 6th entry in the series.
Roman Numerals Chart
To read Roman numbers, it is essential to know the numeric value of all the characters. To guide make this operation easy, here is a chart with every Latin alphabets with assigned numeric values.
Decimal Number Roman Numeral
1 I
2 II
3 III
4 IV
5 V
6 VI
7 VII
8 VIII
9 IX
10 X
11 XI
12 XII
13 XIII
14 XIV
15 XV
16 XVI
17 XVII
18 XVIII
19 XIX
20 XX
21 XXI
22 XXII
23 XXIII
24 XXIV
25 XXV
26 XXVI
27 XXVII
28 XXVIII
29 XXIX
30 XXX
31 XXXI
32 XXXII
33 XXXIII
34 XXXIV
35 XXXV
36 XXXVI
37 XXXVII
38 XXXVIII
39 XXXIX
40 XL
41 XLI
42 XLII
43 XLIII
44 XLIV
45 XLV
46 XLVI
47 XLVII
48 XLVIII
49 XLIX
50 L
51 LI
52 LII
53 LIII
54 LIV
55 LV
56 LVI
57 LVII
58 LVIII
59 LIX
60 LX
61 LXI
62 LXII
63 LXIII
64 LXIV
65 LXV
66 LXVI
67 LXVII
68 LXVIII
69 LXIX
70 LXX
71 LXXI
72 LXXII
73 LXXIII
74 LXXIV
75 LXXV
76 LXXVI
77 LXXVII
78 LXXVIII
79 LXXIX
80 LXXX
81 LXXXI
82 LXXXII
83 LXXXIII
84 LXXXIV
85 LXXXV
86 LXXXVI
87 LXXXVII
88 LXXXVIII
89 LXXXIX
90 XC
91 XCI
92 XCII
93 XCIII
94 XCIV
95 XCV
96 XCVI
97 XCVII
98 XCVIII
99 XCIX
100 C
200 CC
300 CCC
400 CD
500 D
600 DC
700 DCC
800 DCCC
900 CM
1000 M
How to Convert from Roman Numerals to Decimal Numericals
Now that we have the handy table of Roman numbers, we can use that data to transform numbers back and forth quickly. Following these steps, you will convert these values whenever you want.
Steps to Transform Roman numbers to Everyday Numericals
To transform Roman numbers to decimal numbers, we will utilize the cumulative and subtractive principles we discussed.
• Begin with the leftmost Roman numerical in the group.
• If the Roman number to its right is smaller in value, then sum the two values.
• If the Roman numerical to its right is greater in value, subtract the Roman number on the right out of the Roman numbers to its left.
• All you should do now is replicate this method unless you arrive at the end of the Roman number group.
Let's see how you can transform Roman numerals with a some examples.
Example 1
Take a look at the Roman numeral LXXVI.
• Begin with the leftmost Roman numeral, which is L or 50.
• The Roman number to its right is X or 10. Considering 10 is lesser than 50, we sum the both values and get 60.
• The Roman numeral to the right of X is X again. We add 10 to 60 and get 70.
• The Roman number to the right of X is V or 5. Considering 5 is less than 70, we sum the both values and get 75.
• The Roman number to the right of V is I or 1. Considering 1 is less than 75, we add the both values and the result is 76.
We halt here at the end of the Roman number group. Therefore, the Roman numeral LXXVI is equivalent to the regular number 76.
Example 2
Study the Roman numeral MCMIII.
• Begin with the leftmost Roman number, that is M or 1000.
• The Roman number to its right is C or 100. Considering 100 is less than 1000, and it is ensued by an M, this value refers to 900.
• The Roman number to the right of M is I or 1. Considering 1 is less than 1900, we sum the both values and the result is 1901.
• The Roman numeral to the right of I is I again. We sum 1 to 1901 and we find 1902.
• The Roman numeral to the right of I is I again. We add 1 to 1902 and get 1903.
Since we have reached the end of the Roman numeral group, we halt here with our answer. Thus, the Roman numeral MCMIII is as same as the decimal number 1903.
With this information and a some practice, you will transform Roman numbers to regular numbers like an expert!
Grade Potential Can Help You with Roman Numbers
If you struggle to comprehend Roman numerals or any other math ideas, don't hesitate to call Grade Potential for help! Our experienced and educated instructors can guide you with Roman numerals and
the remaining of your arithmetic homework.
If you are looking to maintain or get ahead, we’ll assist you excel in your classes so you can feel positive on exam day. Book a hassle-free consultation! | {"url":"https://www.columbusinhometutors.com/blog/roman-numerals-chart-rules-what-are-roman-numerals","timestamp":"2024-11-14T10:55:50Z","content_type":"text/html","content_length":"85994","record_id":"<urn:uuid:40f66423-a405-4f59-9fcf-80162bd972c8>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00199.warc.gz"} |
Demonstrating that the Angles in a Triangle Sum up to 180 Degrees - Complete, Concrete, Concise
Demonstrating that the Angles in a Triangle Sum up to 180 Degrees
We are taught that the sum of the angles in a triangle add up to 180°
Here is a simple way to demonstrate that fact.
A proof of the angles of a triangle summing to 180° can be found here.
• paper
• pencil or pen
• ruler (or some sort of straight-edge)
• coloured crayon
• scissors
1) Draw a triangle on a sheet of paper. Any sort of triangle you like. Use the ruler or straight-edge to ensure the sides are straight:
2) Colour the edges of the triangles. Do not colour the inside of the triangle. If you really want to colour the inside of the triangle, then use a different colour for the inside from the edges:
3) Cut out the triangle. Make sure the edges are as straight as possible:
4) Cut the corners off the triangle. Make the corners large enough so they are easy to handle:
Notice that each corner has two coloured edges and one uncoloured edge. (I should have used a more contrasting colour than green – maybe red – or else pressed harder when I coloured.
5) Draw a straight line on a sheet of paper using the ruler (I used one of the scraps from the paper I cut the triangle out of):
6)Assemble the corners on the straight line. Ensure that (1)coloured edges touch the straight line and (2) coloured edges touch other coloured edges:
What’s Happening
Each corner that you cut off contains an angle from the triangle. This is why we coloured the edges so we can easily see the angle contained by the edges.
When we assemble the angles (by aligning the coloured edges), we see that all the angles add up to a straight line (or 180°). In other words: Angle 1 + Angle 2 + Angle 3 = 180° | {"url":"https://complete-concrete-concise.com/mathematics/demonstrating-that-the-angles-in-a-triangle-sum-up-to-180-degrees/","timestamp":"2024-11-13T08:56:59Z","content_type":"text/html","content_length":"52366","record_id":"<urn:uuid:b53be929-4cb3-450b-81f6-9127c0719640>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00681.warc.gz"} |
Analysis of Time and Space Harmonics in Symmetrical Multiphase Induction Motor Drives by Means of Vector Space Decomposition.
The use of multiphase motor drives, i.e., with a phase number higher than three, is an increasingly important trend nowadays. The common procedure to analyze or to design the control algorithm for
multiphase motors is to use a decoupling transformation method that transforms the model from the original phase-variable reference frame, where the electrical variables are cross-coupled, into a
model decoupled in orthogonal subspaces. One such transformation is the vector space decomposition (VSD), in which each variable is represented by a complex number called spatial vector. When the
multiphase induction motor model is decoupled by means of the VSD, different types of subspaces are obtained. The main type of subspace is the one where there is coupling between the rotor and the
stator, i.e., where the electromechanical energy conversion happens. There are other types of subspaces in which there is no coupling between the stator and the rotor; these planes do not produce
electromechanical energy conversion and their impedance can be very high (in case of homopolar components with isolated neutral points) or low.
In multiphase machines, as happens in three-phase ones, some non-ideal characteristics give rise to harmonics that can lead to undesirable effects such as torque ripple and electrical losses. They
can be produced by the converter deadtime, the pulse-width modulation (PWM), flux saturation, the non-perfectly sinusoidal distribution of the windings, non-uniform airgap and some other
non-linearities. The characterization of such harmonics in the decoupled motor model and the estimation of the spatial vector of each harmonic is interesting from the point of view of understanding
the motor and control. One of the main reasons for identifying the subspace where each current component maps and its spatial vector rotation (SVR) speed is that it is necessary for setting up the
controllers. Knowing the subspace where some specific current components map and their SVR speed is essential for sensorless speed measurement algorithms and machine current signature analysis
(MCSA). Moreover, the subspace where each current component maps and its SVR speed predict if such a component is going to contribute to the overall motor torque, produce torque ripple or generate
losses. Therefore, from the standpoint of the motor performance analysis, the characterization by using the VSD of the current harmonics is also important.
Most of the previous works about multiphase drive harmonics are focused only on machines with a specific number of phases, such as five-, six- and seven-phase motors and they deal only with odd order
harmonics, which are the most common low order ones. Furthermore, some studies about series-connected multimotor drives have suggestedthat the plane where some current harmonics map depends not only
on the harmonic frequency, but also on the phase arrangement in the stator windings connection. As far as the author knows, there are no previous studies about how the phase connection order changes
affect the harmonic mapping or studies about how harmonics map in series-connected multimotor drives.
Regarding the topic of spatial harmonics modeling, it has been extensively researched in the three-phase machines field, from the thorough analyses focused on one specific spatial harmonic origin to
the more general studies that include the more common causes of spatial harmonics, such as the healthy MCSA. Some spatial harmonic proposals in multiphase motors are directly adapted from three-phase
cases and do not take into account the different motor subspaces. Other multiphase spatial harmonic studies, although taking into account the motor subspace decomposition, analyze only particular
spatial harmonic causes or are focused just on motors with a specific phase number. The particular case of MCSA for machine status monitoring is also a broadly researched topic in the three-phase
field and most of the methods proposed for multiphase motors are based on the adaptation of the three-phase ones, such as the classic MCSA approximation that categorizes the current harmonics
according to their frequencies only. However, the study of the motor current harmonics by means of the subspace and the SVR-speed provides more degrees of freedom for classifying such current
components than the methods based only on the current harmonic amplitudes and frequencies. Furthermore, the additional subspaces that a multiphase motor has in comparison with the three-phase
counterpart provides more levels of classification. Therefore, a MCSA method designed to take advantage of the additional classification variables and the extra subspaces obtains more information
about the harmonics origins and avoids some cases of symptoms overlapping in the phase current spectrum. There are some previous works that use the analysis of the currents in the decomposed model of
the motor for specific fault detection, such as open-phase or broken bar MCSA, but its application for the identification of faults such as static, dynamic and mixed eccentricity is still to be done.
This thesis presents the study and characterization, by means of the VSD, of the stator current and voltage components due to time and spatial harmonics in a n-phase motor with a symmetrical
arrangement of phases. First, an analysis of the stator voltage and current harmonics in a multiphase induction motor, by means of the VSD, that includes the effects of each time harmonic and the
phase sequence is developed. As a result it is proposed a very simple time harmonic mapping method valid to predict the subspace where each time harmonic maps and its SVR speed (frequency and
direction) in symmetrical multiphase induction motor drives of any phase number and in series-connected multimotor drives. Then, equations to study the subspace mapping and SVR speed of the current
harmonics produced for some non-ideal characteristics of a squirrel cage motor such as non-perfect sinusoidal winding distributions, rotor bars, airgap variations due to the stator and rotor slots
and magnetic saturation are obtained. These equations are used to study the current signature of healthy multiphase induction motors by means of the VSD. Finally, the model is extended for covering
also the static and dynamic eccentricities and, based on it, a VSD MCSA method to detect pure-static, pure-dynamic and mixed eccentricity in multiphase induction motors is proposed.
Contributions of this dissertation have been published in one JCR-indexed journal paper and presented at two international conferences. | {"url":"http://janomalvar.webs.uvigo.es/phdThesis.html","timestamp":"2024-11-06T04:56:44Z","content_type":"application/xhtml+xml","content_length":"10849","record_id":"<urn:uuid:1a8a0831-bad3-4f5e-8299-8644ea97d224>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00891.warc.gz"} |
cdb_flag: Flag potential issues in matrices of a COM(P)ADRE database in Rcompadre: Utilities for using the 'COM(P)ADRE' Matrix Model Database
Adds columns to the data slot of a 'CompadreDB' object that flag potential problems in the matrix population models. These columns can subsequently be used to subset the database by logical argument.
check_NA_A: missing values in 'matA'? Missing ('NA') values in matrices prevent most calculations using those matrices.
check_NA_U: missing values in 'matU'? Missing ('NA') values in matrices prevent most calculations using those matrices.
check_NA_F: missing values in 'matF'? Missing ('NA') values in matrices prevent most calculations using those matrices.
check_NA_C: missing values in 'matC'? Missing ('NA') values in matrices prevent most calculations using those matrices.
check_zero_U: 'matU' all zeros (including 'NA')? Submatrices composed entirely of zero values can be problematic. There may be good biological reasons for this phenomenon. For example, in the
particular focal population in the particular focal year, there was truly zero survival recorded. Nevertheless, zero-value submatrices can cause some calculations to fail and it may be necessary to
exclude them.
check_zero_F: 'matF' all zeros (including 'NA')? Submatrices composed entirely of zero values can be problematic. There may be good biological reasons for this phenomenon. For example, in the
particular focal population in the particular focal year, there was truly zero reproduction recorded. Nevertheless, zero-value submatrices can cause some calculations to fail and it may be necessary
to exclude them.
check_zero_U_colsum: Columns of 'matU' that sum to zero imply that there is is no survival from that particular stage. This may be a perfectly valid parameterisation for a particular year/place but
is biologically unreasonable in the longer term and users may wish to exclude problematic matrices from their analysis.
check_singular_U: 'matU' singular? Matrices are said to be singular if they cannot be inverted. Inversion is required for many matrix calculations and, therefore, singularity can cause some
calculations to fail.
check_component_sum: do 'matU'/'matF'/'matC' submatrices sum to 'matA' (see Details)? A complete MPM ('matA') can be split into its component submatrices (i.e., 'matU', 'matF' and 'matC'). The sum of
these submatrices should equal the complete MPM (i.e., 'matA' = 'matU' + 'matF' + 'matC'). Sometimes, however, errors occur so that the submatrices do NOT sum to 'matA'. Normally, this is caused by
rounding errors, but more significant errors are possible.
check_ergodic: is 'matA' ergodic (see isErgodic)? Some matrix calculations require that the MPM ('matA') be ergodic. Ergodic MPMs are those where there is a single asymptotic stable state that does
not depend on initial stage structure. Conversely, non-ergodic MPMs are those where there are multiple asymptotic stable states, which depend on initial stage structure. MPMs that are non-ergodic are
usually biologically unreasonable, both in terms of their life cycle description and their projected dynamics. They cause some calculations to fail.
check_irreducible: is 'matA' irreducible (see isIrreducible)? Some matrix calculations require that the MPM ('matA') be irreducible. Irreducible MPMs are those where parameterised transition rates
facilitate pathways from all stages to all other stages. Conversely, reducible MPMs depict incomplete life cycles where pathways from all stages to every other stage are not possible. MPMs that are
reducible are usually biologically unreasonable, both in terms of their life cycle description and their projected dynamics. They cause some calculations to fail. Irreducibility is necessary but not
sufficient for ergodicity.
check_primitive: is 'matA' primitive (see isPrimitive)? A primitive matrix is non-negative matrix that is irreducible and has only a single eigenvalue of maximum modulus. This check is therefore
redundant due to the overlap with 'check_irreducible' and 'checkErdogic'.
check_surv_gte_1: does 'matU' contains values that are equal to or greater than 1? Survival is bounded between 0 and 1. Values in excess of 1 are biologically unreasonable.
cdb_flag( cdb, checks = c("check_NA_A", "check_NA_U", "check_NA_F", "check_NA_C", "check_zero_U", "check_zero_F", "check_zero_C", "check_zero_U_colsum", "check_singular_U", "check_component_sum",
"check_ergodic", "check_irreducible", "check_primitive", "check_surv_gte_1") )
cdb A CompadreDB object
Character vector specifying which checks to run.
Defaults to all, i.e. c("check_NA_A", "check_NA_U", "check_NA_F", "check_NA_C", "check_zero_U", "check_singular_U", "check_component_sum", "check_ergodic", "check_irreducible",
"check_primitive", "check_surv_gte_1")
Defaults to all, i.e. c("check_NA_A", "check_NA_U", "check_NA_F", "check_NA_C", "check_zero_U", "check_singular_U", "check_component_sum", "check_ergodic", "check_irreducible", "check_primitive",
c("check_NA_A", "check_NA_U", "check_NA_F", "check_NA_C", "check_zero_U", "check_singular_U", "check_component_sum", "check_ergodic", "check_irreducible", "check_primitive", "check_surv_gte_1")
For the flag check_component_sum, a value of NA will be returned if the matrix sum of matU, matF, and matC consists only of zeros and/or NA, indicating that the matrix has not been split.
Returns cdb with extra columns appended to the data slot (columns have the same names as the corresponding elements of checks) to indicate (TRUE/FALSE) whether there are potential problems with the
matrices corresponding to a given row of the data.
Stott, I., Townley, S., & Carslake, D. 2010. On reducibility and ergodicity of population projection matrix models. Methods in Ecology and Evolution. 1 (3), 242-252
CompadreFlag <- cdb_flag(Compadre) # only check whether matA has missing values, and whether matA is ergodic CompadreFlag <- cdb_flag(Compadre, checks = c("check_NA_A", "check_ergodic"))
For more information on customizing the embed code, read Embedding Snippets. | {"url":"https://rdrr.io/cran/Rcompadre/man/cdb_flag.html","timestamp":"2024-11-07T03:26:04Z","content_type":"text/html","content_length":"36581","record_id":"<urn:uuid:d0b05aba-3d02-42f5-b6bb-13d8d4ad3a75>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00280.warc.gz"} |
On the moments of the moments of $\zeta(1/2+it)$
Taking $t$ at random, uniformly from $[0,T]$, we consider the $k$th moment, with respect to $t$, of the random variable corresponding to the $2\beta$th moment of $\zeta(1/2+ix)$ over the interval $x\
in(t, t+1]$, where $\zeta(s)$ is the Riemann zeta function. We call these the `moments of moments' of the Riemann zeta function, and present a conjecture for their asymptotics, when $T\to\infty$, for
integer $k,\beta$. This is motivated by comparisons with results for the moments of moments of the characteristic polynomials of random unitary matrices and is shown to follow from a conjecture for
the shifted moments of $\zeta(s)$ due to Conrey, Farmer, Keating, Rubinstein, and Snaith \cite{cfkrs2}. Specifically, we prove that a function which, the shifted-moment conjecture of \cite{cfkrs2}
implies, is a close approximation to the moments of moments of the zeta function does satisfy the asymptotic formula that we conjecture. We motivate as well similar conjectures for the moments of
moments for other families of primitive $L$-functions.
arXiv e-prints
Pub Date:
June 2020
□ Mathematics - Number Theory;
□ Mathematical Physics
18 pages, final version to appear in Journal of Number Theory | {"url":"https://ui.adsabs.harvard.edu/abs/2020arXiv200604503B/abstract","timestamp":"2024-11-10T18:24:50Z","content_type":"text/html","content_length":"37752","record_id":"<urn:uuid:4f93c8a2-e730-4c08-902f-2000bc46986a>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00367.warc.gz"} |
782 Radian/Square Week to Degree/Square Month
Radian/Square Week [rad/week2] Output
782 radian/square week in degree/square second is equal to 1.2249146264198e-7
782 radian/square week in degree/square millisecond is equal to 1.2249146264198e-13
782 radian/square week in degree/square microsecond is equal to 1.2249146264198e-19
782 radian/square week in degree/square nanosecond is equal to 1.2249146264198e-25
782 radian/square week in degree/square minute is equal to 0.00044096926551113
782 radian/square week in degree/square hour is equal to 1.59
782 radian/square week in degree/square day is equal to 914.39
782 radian/square week in degree/square week is equal to 44805.3
782 radian/square week in degree/square month is equal to 847132.34
782 radian/square week in degree/square year is equal to 121987057.22
782 radian/square week in radian/square second is equal to 2.1378793286862e-9
782 radian/square week in radian/square millisecond is equal to 2.1378793286862e-15
782 radian/square week in radian/square microsecond is equal to 2.1378793286862e-21
782 radian/square week in radian/square nanosecond is equal to 2.1378793286862e-27
782 radian/square week in radian/square minute is equal to 0.0000076963655832703
782 radian/square week in radian/square hour is equal to 0.027706916099773
782 radian/square week in radian/square day is equal to 15.96
782 radian/square week in radian/square month is equal to 14785.25
782 radian/square week in radian/square year is equal to 2129075.79
782 radian/square week in gradian/square second is equal to 1.3610162515776e-7
782 radian/square week in gradian/square millisecond is equal to 1.3610162515776e-13
782 radian/square week in gradian/square microsecond is equal to 1.3610162515776e-19
782 radian/square week in gradian/square nanosecond is equal to 1.3610162515776e-25
782 radian/square week in gradian/square minute is equal to 0.00048996585056793
782 radian/square week in gradian/square hour is equal to 1.76
782 radian/square week in gradian/square day is equal to 1015.99
782 radian/square week in gradian/square week is equal to 49783.67
782 radian/square week in gradian/square month is equal to 941258.16
782 radian/square week in gradian/square year is equal to 135541174.69
782 radian/square week in arcmin/square second is equal to 0.0000073494877585189
782 radian/square week in arcmin/square millisecond is equal to 7.3494877585189e-12
782 radian/square week in arcmin/square microsecond is equal to 7.3494877585189e-18
782 radian/square week in arcmin/square nanosecond is equal to 7.3494877585189e-24
782 radian/square week in arcmin/square minute is equal to 0.026458155930668
782 radian/square week in arcmin/square hour is equal to 95.25
782 radian/square week in arcmin/square day is equal to 54863.63
782 radian/square week in arcmin/square week is equal to 2688317.97
782 radian/square week in arcmin/square month is equal to 50827940.51
782 radian/square week in arcmin/square year is equal to 7319223433.4
782 radian/square week in arcsec/square second is equal to 0.00044096926551113
782 radian/square week in arcsec/square millisecond is equal to 4.4096926551113e-10
782 radian/square week in arcsec/square microsecond is equal to 4.4096926551113e-16
782 radian/square week in arcsec/square nanosecond is equal to 4.4096926551113e-22
782 radian/square week in arcsec/square minute is equal to 1.59
782 radian/square week in arcsec/square hour is equal to 5714.96
782 radian/square week in arcsec/square day is equal to 3291817.93
782 radian/square week in arcsec/square week is equal to 161299078.49
782 radian/square week in arcsec/square month is equal to 3049676430.59
782 radian/square week in arcsec/square year is equal to 439153406004.3
782 radian/square week in sign/square second is equal to 4.0830487547327e-9
782 radian/square week in sign/square millisecond is equal to 4.0830487547327e-15
782 radian/square week in sign/square microsecond is equal to 4.0830487547327e-21
782 radian/square week in sign/square nanosecond is equal to 4.0830487547327e-27
782 radian/square week in sign/square minute is equal to 0.000014698975517038
782 radian/square week in sign/square hour is equal to 0.052916311861336
782 radian/square week in sign/square day is equal to 30.48
782 radian/square week in sign/square week is equal to 1493.51
782 radian/square week in sign/square month is equal to 28237.74
782 radian/square week in sign/square year is equal to 4066235.24
782 radian/square week in turn/square second is equal to 3.4025406289439e-10
782 radian/square week in turn/square millisecond is equal to 3.4025406289439e-16
782 radian/square week in turn/square microsecond is equal to 3.4025406289439e-22
782 radian/square week in turn/square nanosecond is equal to 3.4025406289439e-28
782 radian/square week in turn/square minute is equal to 0.0000012249146264198
782 radian/square week in turn/square hour is equal to 0.0044096926551113
782 radian/square week in turn/square day is equal to 2.54
782 radian/square week in turn/square week is equal to 124.46
782 radian/square week in turn/square month is equal to 2353.15
782 radian/square week in turn/square year is equal to 338852.94
782 radian/square week in circle/square second is equal to 3.4025406289439e-10
782 radian/square week in circle/square millisecond is equal to 3.4025406289439e-16
782 radian/square week in circle/square microsecond is equal to 3.4025406289439e-22
782 radian/square week in circle/square nanosecond is equal to 3.4025406289439e-28
782 radian/square week in circle/square minute is equal to 0.0000012249146264198
782 radian/square week in circle/square hour is equal to 0.0044096926551113
782 radian/square week in circle/square day is equal to 2.54
782 radian/square week in circle/square week is equal to 124.46
782 radian/square week in circle/square month is equal to 2353.15
782 radian/square week in circle/square year is equal to 338852.94
782 radian/square week in mil/square second is equal to 0.0000021776260025241
782 radian/square week in mil/square millisecond is equal to 2.1776260025241e-12
782 radian/square week in mil/square microsecond is equal to 2.1776260025241e-18
782 radian/square week in mil/square nanosecond is equal to 2.1776260025241e-24
782 radian/square week in mil/square minute is equal to 0.0078394536090868
782 radian/square week in mil/square hour is equal to 28.22
782 radian/square week in mil/square day is equal to 16255.89
782 radian/square week in mil/square week is equal to 796538.66
782 radian/square week in mil/square month is equal to 15060130.52
782 radian/square week in mil/square year is equal to 2168658795.08
782 radian/square week in revolution/square second is equal to 3.4025406289439e-10
782 radian/square week in revolution/square millisecond is equal to 3.4025406289439e-16
782 radian/square week in revolution/square microsecond is equal to 3.4025406289439e-22
782 radian/square week in revolution/square nanosecond is equal to 3.4025406289439e-28
782 radian/square week in revolution/square minute is equal to 0.0000012249146264198
782 radian/square week in revolution/square hour is equal to 0.0044096926551113
782 radian/square week in revolution/square day is equal to 2.54
782 radian/square week in revolution/square week is equal to 124.46
782 radian/square week in revolution/square month is equal to 2353.15
782 radian/square week in revolution/square year is equal to 338852.94 | {"url":"https://hextobinary.com/unit/angularacc/from/radpw2/to/degpm2/782","timestamp":"2024-11-04T17:00:33Z","content_type":"text/html","content_length":"113237","record_id":"<urn:uuid:652ae8c9-45c3-4b8f-bf75-c49ef777af5f>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00499.warc.gz"} |
Fig. 1: Energy dissipation due to sound radiation contributes significantly to the overall damping of lightweight structures such as sandwich panels.
Fig. 2: Cross-sectional schematic illustrating the numerical modeling of a sandwich panel and the surrounding acoustic field. The structural finite element mesh is coupled to the closed acoustic
boundary element mesh via non-coincident nodes on the radiating surface.
The solution of vibroacoustic problems arising in e.g. automotive and aerospace engineering often requires the simulation of structural acoustic interaction. Typically, the elastodynamic equations
underlying the vibrating structure are discretized by the finite element method (FEM), whereas several methods are available for addressing the surrounding and often infinitely large acoustic field.
Here, our research mainly focuses on the use of the boundary element method (BEM) and the infinite finite element method (IFEM). We are particularly interested in applications, which involve a
mutual, i.e. strong, coupling between the acoustic and the structural responses as it for instance occurs in lightweight structures and musical instruments. In these cases, the fluid loads that act
on the structure by virtue of the acoustic field are not negligible.
From a numerical point of view, these fully coupled structural acoustic interaction problems demand a large computational effort, particularly in the case rapidly varying responses over large
frequency ranges. In order to address this issue, we employ model order reduction (MOR) techniques. Using MOR involves the computation of a projector, which is suitable to significantly decrease the
dimension of the vibroacoustic problem and hence accelerate the subsequent harmonic analysis at different frequency points. The projector can either be precomputed in an offline phase, or updated
iteratively during the actual simulations.
Furthermore, we use modal analyses as an alternative approach for studying structural acoustic interactions. Modal analyses require the solution of an often nonlinear eigenvalue problem resulting in
structure-inherent properties such as the shifted (so-called wet) eigenfrequencies and the modal radiation loss factors. The latter quantify the extent of structural damping due to sound radiation
into the acoustic far-field. Moreover, once the modes are computed, the responses at certain frequencies as well as the responses for different excitations are simply obtained by matrix-vector | {"url":"https://www.epc.ed.tum.de/en/vib/research/fluid-structure-interaction/","timestamp":"2024-11-09T05:51:42Z","content_type":"text/html","content_length":"70538","record_id":"<urn:uuid:06f90dc5-8e65-47a9-9505-e394cf1c5958>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00578.warc.gz"} |
difficulties in a program in which some blas work but ddot.
10-15-2013 11:35 AM
Hi guys
I found a problem and maybe some of you could help-me.
In order to solve problems with a blas function in a bigger program, I've tried to make a shorter program to test linking options. So I can easily compile and run this short program with blas calls:
real(8):: a(6000,6000), b(6000,4000), c(6000,4000), d(6000), e(6000), alpha, beta
integer(8)::m, n, lda, ldb, ldc, i, j
uplo = 'u'
side = 'l'
m = 6000
n = 4000
lda = m
ldb = m
ldc = m
alpha = 0.5
beta = 2.0
call dsymv('u', 3000, alpha, a, 6000, d, 1, beta, e, 1)
call dsymm (side, uplo, m, n, alpha,a, lda, b, ldb, beta, c, ldc)
With this simple command: ifort teste_mkl.f90 -mkl
And no "use" statement in the source code. However, if i try uncomment:
alpha=ddot(m, d, i, e, i)
alpha=ddot(6000, d, 1, e, 1)
I get the message
"teste_mkl.f90(51): error #6404: This name does not have a type, and must have an explicit type. [DDOT]
alpha= ddot(m, d, i, e, i)
compilation aborted for teste_mkl.f90 (code 1)"
When I replace ddot f77 call by dot(d,e) f95 call, add "use mkl_blas95" and "use mkl_lapack95" in the source code and compile with "ifort -mkl teste_mkl.f90 -lmkl_blas95_ilp64 -lmkl_lapack95_ilp64"
every thingworks fine.
But I don't want to use blas95 (portability issues). What I'm doing wrong with ddot f77 call? I can compile and use other blas routines through f77 call without problems in the same program.
Thanks for your attention.
10-15-2013 04:46 PM
10-16-2013 05:30 AM
10-16-2013 07:29 PM
10-17-2013 04:47 AM | {"url":"https://community.intel.com/t5/Intel-oneAPI-Math-Kernel-Library/difficulties-in-a-program-in-which-some-blas-work-but-ddot/td-p/957022","timestamp":"2024-11-02T19:03:08Z","content_type":"text/html","content_length":"241856","record_id":"<urn:uuid:2c26b645-38a2-4c00-b141-d4792e41ff2a>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00490.warc.gz"} |
How do you calculate 18/12 - 7/9? | Socratic
How do you calculate #18/12 - 7/9#?
1 Answer
The answer is $\frac{13}{18}$.
Here's how to calculate it:
You'll notice that $\frac{18}{12}$ and $\frac{7}{9}$ have different denominators.
In order to subtract these fractions, we need to find equivalent fractions ( i.e. fractions that are the same as our originals) that have the same denominator as one another.
To do this, let's find the least common denominator, or the smallest number that can be divided by both denominators.
One way to do this to write out the multiples of both denominators until a number repeats in the the list:
Multiples of 12: 12, 24, 36 , 48, 60
Multiples of 9: 9, 18, 27, 36
Since 36 appears in both lists, we can make both denominators 36 to create equivalent fractions.
Since $12 \cdot 3 = 36$, we can multiply $\frac{18}{12} \cdot \frac{3}{3}$ (since $\frac{3}{3}$ = 1) ...
$\frac{18}{12} \cdot \frac{3}{3} = \frac{54}{36}$
And since $9 \cdot 4 = 36$, we can multiply $\frac{7}{9} \cdot \frac{4}{4}$ ...
$\frac{7}{9} \cdot \frac{4}{4} = \frac{28}{36}$
Now, we have equivalent fractions with a denominator of 36 and can subtract:
$\frac{54}{36} - \frac{28}{36} = \frac{26}{36}$
And then divide the numerator and the denominator by 2 to simplify:
$\frac{26}{36} = \frac{13}{18}$
... and there's your answer!
Impact of this question
7932 views around the world | {"url":"https://api-project-1022638073839.appspot.com/questions/how-do-you-calculate-18-12-7-9#117385","timestamp":"2024-11-12T20:25:59Z","content_type":"text/html","content_length":"35162","record_id":"<urn:uuid:06e20a75-f143-4da0-ba22-e5a00b61d811>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00731.warc.gz"} |
The Structure of Quantum Information and its Ramifications for IT
1st July 2006 to 30th June 2011
In the not to distant future information technology will have to cope with devices and components which do not obey the usual laws of classical physics but those of quantum physics. This passage is
unavoidable due to the decrease of scale required for the increase of computational power and for the miniaturization of devices. But this passage also comes with fascinating new opportunities, for
example the quantum algorithms which endanger the current widely used (classical) cryptographic encoding schemes (e.g. bank transactions and e-commerce), and at the same time quantum information and
computation (QIC) also provides the corresponding remedy in terms of secure quantum cryptographic and communication schemes. Without any doubt `quantum information technology' is here to stay and
promises to become one of the most intriguing endeavors of this new century. But while quantum information and computation is the fruit of a major paradigmatic change which consisted of conceiving
the `weird' laws of quantum physics not as a bug but as a feature, the methods haven't changed since the early days of quantum theory, and one can compare the `manipulations of strings of complex
numbers and corresponding matrices' with the `acrobatics with 0's and 1's in the early days of computer programming'. At the same time many important questions related to the limits of QIC and a
general model for QIC remain unanswered and it is unlikely that the current low-level methods of QIC will provide the necessary capabilities to do so. Here we see a great opportunity for
`British-style Computer Science semantics and logic' which we intend to exploit. The high-level mathematical models (e.g. categorical) and corresponding logics developed to cope with distributed,
hybrid and in particular resource sensitive computational settings seem to be perfectly tailored for capturing the quantum mechanical realm. Indeed, the starting point for `upgrading QIC' needs to be
the quantum mechanical formalism itself, due to von Neumann, but which was also renounced by von Neumann only 3 years after its creation. A breakthrough results in this direction was recently
obtained by Abramsky and myself were we stripped down the quantum formalisms to its bare `category-theoretic bones', and within this skeleton we still seemed to be able to do full-blown QIC, but then
in a far more conceptual, systematic and straightforward manner. But the greatest merit of this high-level abstraction is that we were also able to show that the formal calculations are equivalent to
extremely intuitive manipulations within a very simple graphical calculus, which has the potential to release QIC research from its banner of being hard and completely inaccessible for the
non-initiated ones. We intend to turn QIC research into a systematic discipline based on a small set of well-understood primitive concepts, and subject to automated design and development tools,
involving the appropriate analogues to the currently available high-level methods from Computer Science such as types, well-behaving calculi, program logics etc. To this means, we intend to further
unveil `the structure of quantum information' (both its qualitative and quantitative content), and of its flow, of its interaction with classical information-flow, spatio-temporal causal structure,
agents, knowledge & belief and their updates. As some concrete applications of this endeavor we mention an integrated high-level approach to information security, which also in the classical domain
is a very delicate matter, impossible to tackle without the appropriate high-level tools. We also intend to develop a general model for the vastly in popularity gaining measurement based quantum
computing, hence contributing to the understanding of what is a practicable model of general QIC, and what are its limits.
Principal Investigator | {"url":"http://www.cs.ox.ac.uk/projects/quantumstructure/","timestamp":"2024-11-10T17:31:18Z","content_type":"text/html","content_length":"32757","record_id":"<urn:uuid:2ff621c5-931b-44dd-89be-404452163ead>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00002.warc.gz"} |
Basic Usage
For all examples the movies data set contained in the package will be used.
movies <- read.csv(system.file("extdata", "movies.csv", package = "UpSetR"),
header = T, sep = ";")
Example 1: Alternative Input Formats
Before we start producing examples using the movies dataset, it is important to know alternative formats to input data. In some cases, the data you have may not be in the form of a file. In UpSetR
there are two built in converter functions fromList and fromExpression that take alternative data formats. The fromList function takes a list of named vectors and converts them into a data frame
compatible with UpSetR. The fromExpression function takes a vector that acts as an expression. The elements of the expression vector are the names of the sets in an intersection, seperated by an
amerpsand (&), and the number elements in that intersection.
# example of list input (list of named vectors)
listInput <- list(one = c(1, 2, 3, 5, 7, 8, 11, 12, 13), two = c(1, 2, 4, 5,
10), three = c(1, 5, 6, 7, 8, 9, 10, 12, 13))
# example of expression input
expressionInput <- c(one = 2, two = 1, three = 2, `one&two` = 1, `one&three` = 4,
`two&three` = 1, `one&two&three` = 2)
Note that both of these inputs contain the same data. To generate an UpSet plot with these inputs set the data paramter equal to either fromList(listInput) or fromExpression(expressionInput).
upset(fromList(listInput), order.by = "freq")
upset(fromExpression(expressionInput), order.by = "freq")
Example 2: Choosing the Top Largest Sets and Plot Formatting
When not specifying specific sets, nsets selects the n largest sets from the data. number.angles determines the angle (in degrees) of the numbers above the intersection size bars. point.size changes
the size of the circles in the matrix. line.size changes the size of the lines connecting the circles in the matrix. mainbar.y.label and sets.x.label can be used to change the axis labels on the
intersection size bar plot and set size bar plot, respectively. Recently added, text.scale allows scaling of all axis titles, tick labels, and numbers above the intersection size bars. text.scale can
either take a universal scale in the form of an integer, or a vector of specific scales in the format: c(intersection size title, intersection size tick labels, set size title, set size tick labels,
set names, numbers above bars).
upset(movies, nsets = 6, number.angles = 30, point.size = 3.5, line.size = 2,
mainbar.y.label = "Genre Intersections", sets.x.label = "Movies Per Genre",
text.scale = c(1.3, 1.3, 1, 1, 2, 0.75)) | {"url":"https://cran.hafro.is/web/packages/UpSetR/vignettes/basic.usage.html","timestamp":"2024-11-07T15:34:46Z","content_type":"application/xhtml+xml","content_length":"1048935","record_id":"<urn:uuid:67f527fa-87e7-4a38-8706-071c2249fb7a>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00774.warc.gz"} |
Simplifying and Determining the Domain of Rational Functions
Question Video: Simplifying and Determining the Domain of Rational Functions Mathematics • Third Year of Preparatory School
Simplify the function π (π ₯) = (7π ₯Β² + 43π ₯ + 6)/(7π ₯Β² + 50π ₯ + 7), and find its domain.
Video Transcript
Simplify the function π of π ₯ is equal to seven π ₯ squared plus 43π ₯ plus six over seven π ₯ squared plus 50π ₯ plus seven and find its domain.
In order to simplify the function, we need to factorize or factor the numerator and denominator and then look for common factors. A common mistake here would be to try and cancel seven π ₯ squared
on the top and bottom of our function. We can prove that this is not true by considering a numerical example.
Imagine we had the fraction seven plus three over seven plus five. This is equal to 10 over 12, which in turn simplifies to five over six. Had we tried to cancel the sevens, we would be left with
three over five, which is not the same as five over six.
Letβ s now consider how we can factorize the numerator of our function. The numerator is written in the form π π ₯ squared plus π π ₯ plus π . One way to factorize a quadratic of this type
is to firstly try and find a pair of numbers that has a product of π multiplied by π and a sum of π . In this case, we need two numbers that have a product of 42, seven times six, and a sum
of 43. There are four factor pairs of 42: 42 and one, 21 and two, 14 and three, and seven and six. The only pair that has a sum of 43 is 42 and one. This means that we can rewrite 43π ₯ as 42π ₯
plus π ₯ or plus one π ₯.
Our next step is to find the highest common factor of the first two terms and then repeat this with the last two terms. The highest common factor of seven π ₯ squared and 42π ₯ is seven π ₯.
Factorizing this out, weβ re left with seven π ₯ multiplied by π ₯ plus six. The only common factor of one π ₯ and six is one. So this can be rewritten as one multiplied by π ₯ plus six.
We notice that π ₯ plus six is common to both parts of this expression. As we are multiplying π ₯ plus six by seven π ₯ and then by one, this can be rewritten as seven π ₯ plus one multiplied by
π ₯ plus six. We can factorize the denominator using the same method. Seven π ₯ squared plus 50π ₯ plus seven is equal to seven π ₯ plus one multiplied by π ₯ plus seven. This means that π of
π ₯ is equal to seven π ₯ plus one multiplied by π ₯ plus six over seven π ₯ plus one multiplied by π ₯ plus seven. We can divide the numerator and denominator by seven π ₯ plus one. The
function π of π ₯ in its simplest form is π ₯ plus six over π ₯ plus seven.
We will now clear some space so we can find the domain of the function. We know that any fraction is undefined when its denominator is equal to zero. This means that any value that makes the
denominator zero will not be in the domain. We established in the first part of the question that seven π ₯ squared plus 50π ₯ plus seven was equal to seven π ₯ plus one multiplied by π ₯ plus
seven. Setting this equal to zero means that either seven π ₯ plus one equals zero or π ₯ plus seven equals zero.
We can subtract one from both sides of the first equation such that seven π ₯ equals negative one. Dividing both sides by seven gives us π ₯ is equal to negative one-seventh. For our second
equation, we need to subtract seven from both sides. This gives us π ₯ is equal to negative seven. Substituting π ₯ equals negative one-seventh or π ₯ equals negative seven into the denominator
gives an answer of zero.
As the domain is the set of π ₯-values that we can input into our function π , the domain is equal to all real numbers apart from negative one-seventh and negative seven. This can be written in
set notation as shown. | {"url":"https://www.nagwa.com/en/videos/927191524934/","timestamp":"2024-11-14T07:59:29Z","content_type":"text/html","content_length":"255176","record_id":"<urn:uuid:2219f1d8-5d37-4015-814b-ccfcd2b673a0>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00279.warc.gz"} |
Applications of Simulation in Science and Engineering: From Fluid Dynamics to Astrophysics - R.A.M.N.O.T
The code provided simulates the movement of energy in a parallel reality, and plots the results of the simulation.
The code is written in the Python programming language, and uses the NumPy library to perform numerical operations. It defines two functions: simulate_universe and plot_universe.
The simulate_universe function takes two arguments: sim_size and speed. It uses these arguments to initialize the fluid-like energy with random values for speed and position, and then to initialize
the space and time variables. It then iterates through the simulation, updating the space and time variables based on the movement of the energy, and returns the final values for space and time.
The plot_universe function takes two arguments: space and time. It uses these arguments to plot the space and time data using the Matplotlib library, and to add labels to the x-axis and y-axis. It
then displays the plot using the show function.
Finally, the code defines the size of the simulation and the speed at which the energy flows, and calls the simulate_universe function to run the simulation and generate the space and time data. It
then calls the plot_universe function to plot the data.
RAMNOT’s Use Cases:
1. Modeling the behavior of fluids for applications in engineering, such as the design of pipes and valves.
2. Simulating the propagation of waves for applications in communication and signal processing.
3. Studying the vibration of strings and other objects for applications in music and acoustics.
4. Modeling the motion of celestial bodies for applications in astronomy and astrophysics.
5. Simulating the movement of springs for applications in mechanical engineering.
6. Analyzing the motion of pendulums for applications in physics education and the design of clocks.
7. Modeling the propagation of sound waves for applications in acoustics and audio engineering.
8. Studying the vibration of drumheads and other objects for applications in music and acoustics.
9. Simulating the rotation of wheels and other objects for applications in mechanical engineering.
10. Modeling the movement of cars on roads for applications in transportation engineering.
11. Analyzing the motion of objects in gravitational fields for applications in physics education and space exploration.
12. Simulating the propagation of electrical currents for applications in electrical engineering.
13. Studying the vibration of beams for applications in structural engineering.
14. Modeling the rotation of propellers for applications in aerospace engineering.
15. Analyzing the movement of waves in wave tanks for applications in ocean engineering.
16. Simulating the motion of planets around the sun for applications in astronomy and astrophysics.
17. Studying the propagation of light waves for applications in optics and photonics.
18. Modeling the vibration of plates for applications in structural engineering.
19. Analyzing the rotation of turbines for applications in power generation and energy production.
20. Simulating the movement of particles in magnetic fields for applications in physics and engineering.
import numpy as np
import matplotlib.pyplot as plt
def simulate_universe(sim_size, speed):
# Initialize the fluid-like energy with random values for speed and position
energy = np.random.rand(sim_size, 2)
# Initialize the space and time variables
space = np.zeros(sim_size)
time = np.zeros(sim_size)
# Iterate through the simulation, updating the space and time variables based on the movement of the energy
for i in range(sim_size):
space[i] = energy[i, 0]
time[i] = energy[i, 1] * speed
# Update the energy in the parallel reality
energy[i, 1] = energy[i, 1] + speed
return space, time
def plot_universe(space, time):
plt.plot(space, time)
# Define the size of the simulation and the speed at which the energy flows
sim_size = 1000
speed = 0.1
# Run the simulation and plot the results
space, time = simulate_universe(sim_size, speed)
plot_universe(space, time)
1. Movement of a particle in a fluid: In this simulation, the energy represents the movement of a particle through a fluid, and the speed represents the velocity of the particle.
2. Propagation of a wave through a medium: In this simulation, the energy represents a wave propagating through a medium, and the speed represents the speed of the wave.
3. Vibration of a string: In this simulation, the energy represents the vibration of a string, and the speed represents the frequency of the vibration.
4. Rotation of a planet around a star: In this simulation, the energy represents the rotation of a planet around a star, and the speed represents the angular velocity of the planet.
5. Movement of a spring: In this simulation, the energy represents the movement of a spring, and the speed represents the oscillation frequency of the spring.
6. Motion of a pendulum: In this simulation, the energy represents the motion of a pendulum, and the speed represents the period of the pendulum.
7. Propagation of a sound wave: In this simulation, the energy represents a sound wave propagating through a medium, and the speed represents the speed of sound in that medium.
8. Vibration of a drumhead: In this simulation, the energy represents the vibration of a drumhead, and the speed represents the frequency of the vibration. | {"url":"https://ramnot.com/applications-of-simulation-in-science-and-engineering-from-fluid-dynamics-to-astrophysics/","timestamp":"2024-11-11T01:27:50Z","content_type":"text/html","content_length":"163314","record_id":"<urn:uuid:8c175184-261c-4ffa-b07e-68fea1d0335c>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00722.warc.gz"} |
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HHPC1 P3 - Yonder Ridge
Submit solution
Points: 12 (partial)
Time limit: 2.0s
Memory limit: 1G
Imagine a picturesque landscape made up of various ridges stretching across an -plane. The -axis of the plane extends from to , while the -axis extends from to . There are ridges numbered from to .
The -th ridge can be represented as a line segment connecting and , with an aesthetic value of .
For an interval , the goodness of the landscape from ridge refers the sum of the aesthetic values of other ridges that are ever strictly above the height of ridge within .
A person is planning to explore the landscape, and has a few questions. In the -th question, they wonder what the goodness of landscape is over the interval . Note that is the same for each query.
For all subtasks:
Subtask 1 [30%]
Subtask 2 [70%]
No additional constraints.
Input Specification
The first line contains four space-separated integers , , , and .
The -th of the following lines each contain three space-separated integers , , and .
The -th of the following lines each contain two space-separated integers and .
Output Specification
For each query, output an integer, representing the goodness of the landscape.
Sample Input
Sample Output
Sample Explanation
For the first query, only ridge is ever strictly above ridge in the interval , so the answer is .
For the second query, both ridge and ridge are ever strictly above ridge at some point in the interval , so the answer is .
There are no comments at the moment. | {"url":"https://dmoj.ca/problem/hhpc1p3","timestamp":"2024-11-11T23:56:21Z","content_type":"text/html","content_length":"33099","record_id":"<urn:uuid:a7ff22c8-d86a-49a2-bb24-4e7b067dd92a>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00486.warc.gz"} |
our Blackjack Skills
Learning New Blackjack Skills
Improve your Blackjack Skills
Learning new blackjack skills is a multi-layered process. The basics of blackjack are fairly easy to learn, if you want to know the basic rules that help you bet and play blackjack. But if you want
to learn the best move in every single blackjack scenario, this becomes much harder to learn. As they say, blackjack takes five minutes to learn and a lifetime to master.
Learn new blackjack skills one at a time. I would suggest learning blackjack rules first, then "basic strategy" second, then move on to more advanced blackjack concepts like card counting last. Don't
take on everything at once, or you won't learn any of the basic blackjack skills well. Once you have a thorough understanding of one blackjack skill, then and only then begin to master other
blackjack skills.
Blackjack Bets
Betting in blackjack is fairly simple. To get into a hand, you place a minimum qualifying bet. The dealer begins by dealing one card face down to every player at the blackjack table in a clockwise
fashion, then one card to himself. The blackjack then deals a second face-down card to each of the players, and finally deals himself a second card facing up. At this point, players can raise their
Basic Blackjack Strategy
Most blackjack players learn basic strategy, because it helps them reduce the house edge to a minimum. "Optimal play" in blackjack means you get the most out of your odds. This doesn't mean you are
certain to win. It means you are reducing the house edge, increases your chances of winning.
Not everything is covered on a basic strategy chart. Basic strategy only covers what to do with the first two cards dealt to you. After learning basic strategy, players know when to stand or hit.
Players will also learn about special moves like doubling down. But there are other rules and scenarios you'll need to learn about to become a master blackjack player.
Blackjack Hitting
Players should always hit on an 8 or less. When you get a 9, double when the dealer has a 3 through 6. If the dealer has a higher card, simply hit on a 9.
If you have a 10, double when the dealer holds a 2 through 9. If the dealer has a high card, simply hit on a 10. If you hold an 11, double when the dealer has a 2 through 10. If the dealer is showing
an Ace, simply hit when you have an Ace. If you hold "12", stand when the dealer is showing a 4, 5 or 6. Otherwise, hit when you hold a 12.
If you hold a 13, 14, 15 or 16, you should stand when the dealer holds a 2 through a 6. If the dealer is showing a higher card, hit on a 13 through 16. And finally, if you hold between 17 through 21,
simply always stand.
Playing Soft Hands in Blackjack
A soft hand is a hand in which you hold an Ace and some other type of card. They call it a soft hand because the Ace can be one of two values: 1 or 11. So if you hold a Ace-5, you are holding both a
6 and a 16. This gives you a few more options.
When you hold an A-2 or A-3, double when the dealer is showing a 5 or 6. If the dealer is showing other cards, hit. When you hold an A-4 or A-5, double when the dealer is holding a 2 through 9. If
not, simply hit on an A4 or A5.
When you hold an A-6, double when the dealer has a 3 through 6. Hit if the dealer is showing any other card.
If you have an A-7 and the dealer is showing a 2, 7 or 8, stand. If the dealer is showing a 3, 4, 5 or 6, you should double. When the dealer is holding an 8 or higher, you should simply hit. And
finally, if you hold an A-8 or A-9, you should always stand.
Blackjack Splitting Pairs
Learning when to split pairs is another blackjack skill you need to learn. If you receive a pair of card (two 2's, two 8's, two Aces), you can split them. Splitting cards when you have an
advantageous card allows you to increase your odds of winning, because you'll be betting on two advantageous hands instead of one. When splitting pairs, you essentially double your bet.
Aces and Eights should always be split in blackjack. 2's and 3's should be split when the dealer is showing between 2 and 7, and hit whenever the dealer holds a higher card. When you receive a pair
of 4's, you should split when the dealer holds a 5 or 6 and hit when any other card is held.
When players are holding a pair of 5's, you should double when the dealer shows a 2 through a 9. Otherwise, you should hit on a pair of fives. With a pair of 6's, you should split on a 2 through a 6
and hit on a 7 through Ace.
If you hold a pair of 7's, you should split when the dealer shows a 2 through 7. When the dealer holds an 8 through Ace, hit when holding a pair of sevens. If you hold a pair of 9's, you should split
when the dealer holds a 2, 3, 4, 5, 6, 8 and 9. You should stand when the dealer is showing a 7, 10 and an Ace. And finally, you should stand when you hold a pair of 10's.
Card Counting - Blackjack Skills
If you have mastered basic strategy, you might eventually want to learn about card counting. Counting cards allows blackjack players to keep count of which cards have been dealt, so the player can
make guesses about which cards are likely to be dealt next.
When card counting, a player keeps a running count of high cards (Ace, K, Q, J and 10) and low cards (2, 3, 4, 5 and 6). Low cards leaving the deck are considered good for the player, while high
cards are considered bad. That's because a player wants more 10 or 11 cards remaining in the deck.
So in basic card counting, you will add +1 when a low card appears and a -1 when a high card appears. The higher into the positives the count gets, the better for the player. That's when the player
increases the initial bet. The further the numbers get into the negatives, the less the player should bet. So card counting allows a player to bet more when the odds are in his favor and less when
the odds are against him.
The Hi-Lo card counting method is just one of many card counting techniques. Most are more complicated. Other blackjack counting methods are the wizard ace/five, the KO, the Hi-Opt I and the Hi-Opt
II, the Zen Count an the Omega II.
See also: | {"url":"https://www.blackjacktactics.com/blackjack/skills/","timestamp":"2024-11-10T08:30:26Z","content_type":"text/html","content_length":"19955","record_id":"<urn:uuid:2697a8f9-4c96-431e-a9ee-2c24b6c9e7f2>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00700.warc.gz"} |
The Difference Between Full Proof, Barrel Proof, And Cask Strength Bourbon - Tasting Table
The Difference Between Full Proof, Barrel Proof, And Cask Strength Bourbon
Liudmila Chernetska/Getty Images
There's a lot to love about craft whiskey, but the jargon the industry uses to describe its products can get confusing fast. The language used to characterize a bottle's type of proof contains some
of the most commonly encountered — and least understood — whiskey and bourbon terms. Today, we're going to be exploring the difference between full proof, barrel proof, and cask strength whiskey.
For those of you who don't know, the proof of a spirit is the strength of its alcohol content. Most people are familiar with the ABV (alcohol by volume) and it's the same concept. If a bottle of
whiskey is 50% alcohol by volume, it is a 100 proof whiskey. If it's 60% alcohol, it's 120 proof. You take the ABV and double it to get the proof. It's that easy. Put simply, full proof is the
whiskey's proof when it first enters the barrel, and both barrel proof and cask strength refer to its alcohol content when it exits the barrel after aging.
It's worth noting that none of these terms are legally regulated. Marketing teams can technically use them on whatever product they want in any way they see fit with no legal repercussions. But while
the law may not come after them, the whiskey community absolutely would. Some terms mean more than others, as we'll see, but they all mean something specific. Just because the definitions aren't
regulated doesn't mean whiskey makers aren't taking them seriously.
Full proof whiskey
thomas carr/Shutterstock
To understand what full proof means, we need to know a little bit about how whiskey is aged. After fermentation, whiskey is a clear product called white dog. When the white dog is put into barrels,
its proof is measured. The proof of the spirit before aging is called the entry proof. As the whiskey ages, its proof fluctuates depending on the local temperature, the humidity, and the length of
time it spends in the barrel.
Speaking very generally, if the whiskey ages in a high-humidity climate, the proof will go down as it ages. If it ages in a low-humidity climate, the proof will rise as it ages. This has to do with
how quickly the water and alcohol are evaporating in relation to each other. Let's say that a whiskey has an entry proof of 115. When it's done aging, that proof has risen to 125 (known as the exit
To make it a full proof whiskey, the distiller will add water to the barrel until it reaches the entry proof of 115. The merit of a full proof whiskey is consistency. There's no guesswork involved
with how much water to add based on subjective taste. You have the number and you add water until you reach it. When you're producing a lot of whiskey, this can be a real benefit both for the
distillery and for the consumer.
Barrel proof whiskey
Invizbk/Getty Images
Barrel proof spirits are some of the most highly sought-after bourbons and whiskeys within the whiskey hunting scene. They capture depths of flavor that are unmatched by other varieties, even if they
come with a bit of a bite. See, barrel proof whiskeys are the least adulterated. There's no water added, whatsoever. For barrel proof, it isn't the entry proof that matters but the exit proof.
Whiskey, particularly bourbon, can get up into the 130 to 140 proof range after it's been aged. By adding water and mixing different barrels from the same batch together, the distillery can round out
any variation that occurred in the aging process and create a consistent flavor across the board. Barrel proof whiskey gets none of that treatment. You uncork the barrel and the proof that comes out
is the proof that goes into the bottle.
Sometimes businesses or even individual people will go directly to a distillery and pick out a barrel that they like the flavor of. They'll buy the barrel at barrel proof and have it bottled just for
them. If a liquor store does this, it's called a store pick. It's a fun way to get your hands on some of the best barrels a batch has to offer since you get to go try them before you pick one. Each
barrel is going to have a wide variance in flavor and strength, even if they were fermented together and have aged the same amount of time.
Cask strength whiskey
Aleaimage/Getty Images
Cask strength is something of a trick addition to this article since it means the exact same thing as barrel proof. A cask is just another name for the barrel that you age whiskey in. The proof of a
whiskey is also the same thing as its strength. The words are interchangeable, although you wouldn't want to say "cask proof" as that isn't really a whiskey term.
Other ways you can phrase this same process are "barrel strength" or "straight from the barrel." If you see any of these on a bottle, they're referring to the process whereby the whiskey is taken out
of the barrel and bottled without being diluted with water. Another thing to clarify is that not all cask strength whiskeys are single barrel whiskeys.
A single barrel whiskey has come from just one barrel and has not been mixed with other barrels. Cask strength or barrel proof whiskeys can be mixed with other barrels from the same batch, so long as
they are still at the same proof as the exit proof, but they can't be diluted. It's fairly common to see a whiskey labeled as "single barrel cask strength" or "single barrel barrel proof," but it's
possible to see one without the other. | {"url":"https://www.tastingtable.com/1459758/difference-full-proof-barrel-cask-strength-bourbon/","timestamp":"2024-11-06T19:52:34Z","content_type":"application/xhtml+xml","content_length":"83090","record_id":"<urn:uuid:7db655d1-79eb-4f75-a577-523ebc8ef683>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00090.warc.gz"} |
Disponible uniquement en anglais
Ball on beam
Controller design with Sysquake and Calerga VR
Virtual Experimentation
Observe the system to get a basic understanding of its main features: the d.c. electrical motor is on the right and drives a large wheel. Straight rails are fixed on the wheel and a ball rolls freely
on them between hard stops.
Observe the movements of the wheel and ball. The controller tries to move the ball to three different rest positions.
Then interact with the system:
• Drag a point not on the system to move the point of view around the system. You can revert to the initial view with the target icon
• Drag the wheel to make it rotate. Note that the ball remains on the rails even for extreme accelerations: that's a limitation of the simulation we deem to be appropriate.
• Try to bring the ball to a rest position by moving the wheel by hand.
• Drag the ball on the rails. Keep the ball still for a while and observe the movements of the wheel. What happens?
• You can reduce the power consumed by your computer by unselecting the “Run” checkbox to pause the simulation.
For simulation and controller design, the system is represented as a state-space model. It can be split into two subsystems: the electrical drive with the d.c. motor, and the mechanical part with the
wheel, rails and ball. These subsystems have a mutual influence, not an input-output connection where causality is clearly defined.
Controller design
The transfer function of the two subsystems and the controller parameters can be computed with Sysquake.
Sysquake js | {"url":"https://www.calerga.com/vr/bob.fr.html","timestamp":"2024-11-03T19:18:13Z","content_type":"text/html","content_length":"29916","record_id":"<urn:uuid:0d34973d-eb4c-4349-8ca6-3daf20c50c4d>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00135.warc.gz"} |
How to do Chow Test in R » finnstats
How to do Chow Test in R
The Chow test is used to compare the coefficients of two distinct regression models on two separate datasets.
This test is commonly used in econometrics using time series data to evaluate if the data has a structural break at some point.
The basic steps are as follows:
Run all of your data through a “limited” regression (pooled).
Using your breakpoint as a guide, divide your sample into two groups (e.g. a point in time, or a variable value).
On each of your subsamples, run an “unrestricted” regression. With a single breakpoint, you’ll run two “unrestricted” regressions.
Then calculate Chow F-statistic.
Chow Test in R
This tutorial shows you how to execute a Chow test in R step by step.
Let’s create the Data Frame, we can make use of the EuStockMarkets dataset here for illustration purposes.
data <- dplyr::select(data, DAX, SMI)
View first six rows of data
DAX SMI
1 1628.75 1678.1
2 1613.63 1688.5
3 1606.51 1678.6
4 1621.04 1684.1
5 1618.16 1686.6
6 1610.61 1671.6
Visualize the Information
Then, to visualize the data, we’ll make a basic scatterplot. Let’s load ggplot2 package in the R console.
In R, we can now make a scatterplot Chow test.
ggplot(data, aes(x = disp, y = mpg)) + geom_point(col='steelblue', size=3)
We can see from the scatterplot that the data pattern appears to alter at DAX = 4000. But not sure, as a result, we may use the Chow test to see if the data has a structural breakpoint.
Take the Chow Test. To execute a Chow test, we may use the strucchange package’s sctest function:
Let’s load strucchange package
sctest(formula, type = , h = 0.15,
alt.boundary = FALSE, functional = c("max", "range",
"maxL2", "meanL2"), from = 0.15, to = NULL, point = 0.5,
asymptotic = FALSE, data, ...)
The above function returns the following
statistic the test statistic
p.value the corresponding p-value
method a character string with the method used
data.name a character string with the data name
One Sample Analysis in R » Quick Guide »
Now we can perform the Chow test in R
sctest(data$SMI ~ data$DAX, type = "Chow", point = 10)
Chow test
data: data$SMI ~ data$DAX
F = 212.01, p-value < 2.2e-16
We can reject the null hypothesis of the test because the p-value is less than 0.05. This means we have enough evidence to conclude that the data contains a structural breakpoint.
To put it another way, two regression lines can match the data pattern better than a single regression line.
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Calculus I
Office Hours
Tuesday & Thursday 3:30 - 6:00pm
& by appointment
My office is in Fretwell 340F.
If you can't make my office hours or if you need help on the weekend, you can always email me.
Class Information & Moodle
The full website for this class is hosted by Moodle. You can access it by going to the Moodle site and selecting this course.
Notes for Each Chapter Section
These are in the process of being relinked to the class's moodle page. In the meantime, the notes can be found at the links below.
Sections 1.1 and 1.2: Trig and Algebra Review
Section 1.3: The Limit of a Function
Section 1.4: Calculating Limits
Section 1.5: Continuity
Section 1.6: Limits Involving Infinity
Chapter 1: all notes combined
Section 2.1: Derivatives and Rates of Change
Section 2.2: The Derivative as a Function
Section 2.3: Basic Differentiation Formulas
Section 2.4: Product and Quotient Rules
Section 2.5: The Chain Rule
Section 2.6: Implicit Differentiation
Section 2.7: Related Rates
Section 2.8: Linear Approximations and Differentials
Chapter 2: all notes combined
Chapter 3: Review Topics
Section 3.1: Exponential Functions
Section 3.2: Inverse Functions and Logarithms
Section 3.3: Derivatives of Logarithmic and Exponential Functions
Section 3.4: Exponential Growth and Decay
Section 3.5: Inverse Trigonometric Functions
Section 3.7: Indeterminate Forms and l'Hospital's Rule
Chapter 3: all notes combined
Section 4.1: Maximum and Minimum Values
Section 4.2: The Mean Value Theorem
Section 4.3: Derivatives and the Shapes of Graphs
Section 4.4: Curve Sketching
Section 4.5: Optimization Problems
Section 4.6: Newton's Method
Section 4.7: Antiderivatives
Chapter 4: all notes combined | {"url":"https://webpages.charlotte.edu/~sjbirdso/calc%20I-fall14/","timestamp":"2024-11-13T12:23:33Z","content_type":"text/html","content_length":"4120","record_id":"<urn:uuid:b20c716a-6a4a-45f9-a9a9-b862f50d3db5>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00517.warc.gz"} |
Re: predictive.el -- predictive completion of words as you type in Emacs
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Re: predictive.el -- predictive completion of words as you type in Emacs
From: Thorsten Bonow
Subject: Re: predictive.el -- predictive completion of words as you type in Emacs
Date: Tue, 28 Feb 2006 12:55:00 +0100
User-agent: Gnus/5.110004 (No Gnus v0.4) XEmacs/21.4.19 (linux)
>>>>> "Uwe" == Uwe Brauer <address@hidden> writes:
Uwe> There is also dabbrev-hover.el, however I had difficulties with this
Uwe> package under Xemacs, however your pkg seem to work nicely. Predictive
Uwe> on the other hand seems also not to work to smoothly with Xemacs (so
Uwe> far).
(Off topic I guess, maybe we should move to an XEmacs group...)
Hm, I can't get pabbrev.el to work with XEmacs. I'm using XEmacs 21.4.19 coming
with my up to date Debian Unstable. Starting with xemacs -vanilla and just
loading pabbrev.el (after setting debug-on-error to t) gives me an error on the
first completion.
Did you do something special to make it work? Using 21.5 or something?
Thanks for any help.
The *Backtrace* buffer gives me:
Debugger entered--Lisp error: (error "Error in pabbrev-mode")
signal(error ("Error in pabbrev-mode"))
cerror("Error in pabbrev-mode")
apply(cerror "Error in pabbrev-mode" nil)
error("Error in pabbrev-mode")
pabbrev-command-hook-fail((void-function make-overlay) "post")
(condition-case err (unless (or buffer-read-only ...) (save-excursion ...
...)) (error (pabbrev-command-hook-fail err "post")))
And there is a *pabbrev-fail* buffer displayed, too:
There has been an error in pabbrev-mode. This mode normally
makes use of "post-command-hook", which runs after every command. If this
error continued Emacs could be made unusable, so pabbrev-mode has attempted
to disable itself. So although it will appear to still be on, it won't do
anything. Toggling it off, and then on again will usually restore functionality.
The following is debugging information
Symbol's function definition is void: make-overlay
Backtrace is:
(let ((standard-output ...)) (backtrace))
(with-output-to-temp-buffer "*pabbrev-fail*" (princ "There has been an error
in pabbrev-mode. This mode normally\nmakes use of \"post-command-hook\", which
runs after every command. If this\nerror continued Emacs could be made
unusable, so pabbrev-mode has attempted\nto disable itself. So although it will
appear to still be on, it won't do\nanything. Toggling it off, and then on
again will usually restore functionality.\n") (princ "The following is
debugging information\n\n") (princ (error-message-string err)) (princ
"\n\nBacktrace is: \n") (let (...) (backtrace)))
pabbrev-command-hook-fail((void-function make-overlay) "post")
(condition-case err (unless (or buffer-read-only ...) (save-excursion ...
...)) (error (pabbrev-command-hook-fail err "post")))
Contact information and PGP key at
If Michelangelo had been a heterosexual, the Sistine Chapel would
have been painted basic white and with a roller.
Rita Mae Brown
[Prev in Thread] Current Thread [Next in Thread] | {"url":"https://mail.gnu.org/archive/html/gnu-emacs-sources/2006-02/msg00060.html","timestamp":"2024-11-12T09:15:48Z","content_type":"text/html","content_length":"8553","record_id":"<urn:uuid:fa74543b-5fc6-4eb2-9e1e-6767dc2d672a>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00613.warc.gz"} |
seminars - Uncertainty quantification for partial differential equations and their optimal control problems
We consider the determination of statistical information about outputs of interest that depend on the solution of a partial differential equation and optimal control problems having random inputs,
e.g., coefficients, boundary data, source term, etc. Monte Carlo methods are the most used approach used for this purpose. We discuss other approaches that, in some settings, incur far less
computational costs. These include quasi-Monte Carlo, polynomial chaos, stochastic collocation, compressed sensing, reduced-order modeling. | {"url":"https://www.math.snu.ac.kr/board/index.php?mid=seminars&sort_index=date&order_type=desc&l=en&page=58&document_srl=783365","timestamp":"2024-11-05T13:07:40Z","content_type":"text/html","content_length":"47466","record_id":"<urn:uuid:52c29a36-6514-4813-953f-4b5b7c48590d>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00638.warc.gz"} |
Geometry of Neural Network Loss Surfaces via Random Matrix Theory
Geometry of Neural Network Loss Surfaces via Random Matrix Theory
Proceedings of the 34th International Conference on Machine Learning, PMLR 70:2798-2806, 2017.
Understanding the geometry of neural network loss surfaces is important for the development of improved optimization algorithms and for building a theoretical understanding of why deep learning
works. In this paper, we study the geometry in terms of the distribution of eigenvalues of the Hessian matrix at critical points of varying energy. We introduce an analytical framework and a set of
tools from random matrix theory that allow us to compute an approximation of this distribution under a set of simplifying assumptions. The shape of the spectrum depends strongly on the energy and
another key parameter, $\phi$, which measures the ratio of parameters to data points. Our analysis predicts and numerical simulations support that for critical points of small index, the number of
negative eigenvalues scales like the 3/2 power of the energy. We leave as an open problem an explanation for our observation that, in the context of a certain memorization task, the energy of
minimizers is well-approximated by the function $1/2(1-\phi)^2$.
Cite this Paper
Related Material | {"url":"http://proceedings.mlr.press/v70/pennington17a.html","timestamp":"2024-11-08T15:17:18Z","content_type":"text/html","content_length":"15690","record_id":"<urn:uuid:234d2eee-078a-45f1-9e08-b76bdac0a960>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00483.warc.gz"} |
Learning Bayesian Network Structure using LP Relaxations
We propose to solve the combinatorial problem of finding the highest scoring Bayesian network structure from data. This structure learning problem can be viewed as an inference problem where the
variables specify the choice of parents for each node in the graph. The key combinatorial difficulty arises from the global constraint that the graph structure has to be acyclic. We cast the
structure learning problem as a linear program over the polytope defined by valid acyclic structures. In relaxing this problem, we maintain an outer bound approximation to the polytope and
iteratively tighten it by searching over a new class of valid constraints. If an integral solution is found, it is guaranteed to be the optimal Bayesian network. When the relaxation is not tight, the
fast dual algorithms we develop remain useful in combination with a branch and bound method. Empirical results suggest that the method is competitive or faster than alternative exact methods based on
dynamic programming. | {"url":"https://www.sciweavers.org/publications/learning-bayesian-network-structure-using-lp-relaxations","timestamp":"2024-11-14T13:17:22Z","content_type":"application/xhtml+xml","content_length":"34657","record_id":"<urn:uuid:5db8ca7e-a839-44ab-915e-c6834683b46b>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00359.warc.gz"} |
a. : a general fact or rule expressed in symbols and especially mathematical symbols. b. : an expression in symbols of the composition of a substance. the formula for water is H[2]O.
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What is a math formula?
A formula is a mathematical rule or relationship that uses letters to represent amounts which can be changed – these are called variables. For example, the formula to work out the area of a triangle.
Triangle area = b h 2 (where represents the base of the triangle and represents the height of the triangle).
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seem to explain this curious situation: low product cost, inflated retail prices, brand loyalty, and expanded reach.
What is a formula also called?
The term "formula" is also commonly used in the theory of logic to mean sentential formula (also called a propositional formula), i.e., a formula in propositional calculus. The correct Latin plural
form of formula is "formulae," although the less pretentious-sounding "formulas" is more commonly used.
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In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a chemical formula. The informal use of the ...
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A formula is an equation (= set of numbers and letters representing two equal amounts) that shows how one amount depends on one or more other amounts.
Listen to the latest episode of ESPN F1 Unlapped. Welcome to Unlapped! Katie George, Nate Saunders, and Laurence Edmondson are here to break down the latest ... | {"url":"https://www.google.co.uk/search?q=formula&sca_esv=98d43e591c93d847&hl=en&gl=us&gbv=1&ie=UTF-8&source=lnms&sa=X&ved=0ahUKEwjVw-z07fuIAxUaD1kFHdC9GfsQ_AUIBSgA","timestamp":"2024-11-10T12:15:40Z","content_type":"text/html","content_length":"65943","record_id":"<urn:uuid:5e5603ca-888e-4ed3-82fb-b843107d6992>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00845.warc.gz"} |
On the role of entanglement and statistics in learning for QIP 2024
QIP 2024
On the role of entanglement and statistics in learning
In this work we make progress in understanding the relationship between learning models with access to entangled, separable and statistical measurements in the quantum statistical query (QSQ) model.
To this end, we show the following results. $\textbf{Entangled versus separable measurements.}$ The goal here is to learn an unknown $f$ from the concept class $C \subseteq \{f:\{0,1\}^n \rightarrow
[k]\}$ given copies of ${1\above{2pt} \sqrt{2^n}}$ ${\textstyle\sum{_x} |x,f(x)⟩}$. We show that, if $T$ copies suffice to learn $f$ using entangled measurements, then $O(nT^2)$ copies suffice to
learn $f$ using just separable measurements. $\textbf{Entangled versus statistical measurements}$ The goal here is to learn a function $f$ ∈ $C$ given access to separable measurements and statistical
measurements. We exhibit a class $C$ that gives an exponential separation between QSQ learning and quantum learning with entangled measurements (even in the presence of noise). This proves the
"quantum analogue" of the seminal result of Blum et al. [BKW'03]. that separates classical SQ and PAC learning with classification noise. $\textbf{QSQ lower bounds for learning states.}$ We introduce
a quantum statistical query dimension (QSD), which we use to give lower bounds on the QSQ learning. With this we prove superpolynomial QSQ lower bounds for testing purity, shadow tomography, Abelian
hidden subgroup problem, degree-2 functions, planted bi-clique states and output states of Clifford circuits of depth polylog(n). $\textbf{Further applications.}$ We give an unconditional separation
between weak and strong error mitigation and prove lower bounds for learning distributions in the QSQ model. Prior works by Quek et al. [QFK+'22], Hinsche et al. [HIN+'22], and Nietner et al.
[NIS+'23] proved the analogous results assuming diagonal measurements and our work removes this assumption. | {"url":"https://research.ibm.com/publications/on-the-role-of-entanglement-and-statistics-in-learning--1","timestamp":"2024-11-12T17:09:31Z","content_type":"text/html","content_length":"85346","record_id":"<urn:uuid:7ca960c3-824f-44e7-8db8-e264da4c47f4>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00472.warc.gz"} |
Task Colorful Chain (lan)
<Submit a solution>
Task statistics
Number of users: 267
Number of users with 100 points: 137
Average result: 86.9888
Colorful Chain
Memory limit: 128 MB
Little Bytie loves to play with colorful chains. He already has quite an impressive collection, and some of them he likes more than the others. Each chain consists of a certain number of colorful
links. Byteasar has noticed that Bytie's sense of aesthetics is very precise. It turns out that Bytie finds a contiguous fragment of a chain nice if it contains exactly links of color links of color
links of color , and moreover it contains no links of other colors. A chain's appeal is its number of (contiguous) fragments that are nice. By trial and error, Byteasar has determined the values and
. Now he would like to buy a new chain, and therefore asks you to write a program to aid him in shopping.
The first line of the standard input gives two integers, and (), separated by a single space. These are the length of the chain and the length of a nice fragment's description. The second line gives
integers, (), separated by single spaces. The third line gives integers, (, for ), also separated by single spaces. The sequences and define a nice fragment of a chain - it has to contain exactly
links of color . The fourth line gives integers, (), separated by single spaces, that are the colors of successive links of the chain.
In tests worth 50% of total points the constraint holds in addition.
Your program is to print a single integer, the number of nice contiguous fragments in the chain, to the first and only line of the standard output.
For the input data:
the correct result is:
Explanation of the example: The two nice fragments of this chain are 2, 1, 3, 1 and 1, 3, 1, 2.
Sample grading tests:
• 1ocen, , two nice fragments with the second one following the first one immediately, with neither overlap nor additional links in between;
• 2ocen, , the length of the nice fragment exceeds the length of the whole chain (the result is 0);
• 3ocen, , three overlapping nice fragments;
• 4ocen, , the nice fragment contains a single link of the colors , and the chain is a sequence of links of colors 1, 2, ..., 499, 500, 500, 499, ..., 2, 1 (the result is 2);
• 5ocen, , the nice fragment contains a single link of color 1 and two links of color 2; the chain consists of links of colors 1, 2, 2, 1, 2, 2, ... (the result is ).
Task author: Tomasz Walen.
<Submit a solution> | {"url":"https://szkopul.edu.pl/problemset/problem/MAWN1VdLdXO29VvrVYuYxQyw/site/?key=statement","timestamp":"2024-11-09T09:29:40Z","content_type":"text/html","content_length":"29530","record_id":"<urn:uuid:946b47d3-6b2f-4764-80ce-bade513109f9>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00796.warc.gz"} |
oscillation frequency
Abazov, V. M. ; Mondal, N. K. ; et, al (2006) Direct limits on the B[s]^0 oscillation frequency Physical Review Letters, 97 (2). 021802_1-021802_7. ISSN 0031-9007
Full text not available from this repository.
Official URL: http://prl.aps.org/abstract/PRL/v97/i2/e021802
Related URL: http://dx.doi.org/10.1103/PhysRevLett.97.021802
We report results of a study of the B[s]^0 oscillation frequency using a large sample of B[s]^0 semileptonic decays corresponding to approximately 1 fb^−1 of integrated luminosity collected by the D0
experiment at the Fermilab Tevatron Collider in 2002-2006. The amplitude method gives a lower limit on the B[s]^0 oscillation frequency at 14.8 ps^−1 at the 95% C.L. At Δm[s]=19 ps^−1, the amplitude
deviates from the hypothesis A=0 (1) by 2.5 (1.6) standard deviations, corresponding to a two-sided C.L. of 1% (10%). A likelihood scan over the oscillation frequency, Δm[s], gives a most probable
value of 19 ps^−1 and a range of 17<Δm[s]<21 ps^−1 at the 90% C.L., assuming Gaussian uncertainties. This is the first direct two-sided bound measured by a single experiment. If Δm[s] lies above 22
ps^−1, then the probability that it would produce a likelihood minimum similar to the one observed in the interval 16-22 ps^−1 is (5.0±0.3)%.
Item Type: Article
Source: Copyright of this article belongs to The American Physical Society.
ID Code: 76197
Deposited On: 31 Dec 2011 08:16
Last Modified: 31 Dec 2011 08:16
Repository Staff Only: item control page | {"url":"https://repository.ias.ac.in/76197/","timestamp":"2024-11-02T12:34:25Z","content_type":"application/xhtml+xml","content_length":"18621","record_id":"<urn:uuid:0c18c3cf-f99d-4b46-83dc-e0c173dc3d09>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00696.warc.gz"} |
3.3: Weight initialization
Last updated
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When we create our neural networks, we have to make choices for the initial weights and biases. Up to now, we've been choosing them according to a prescription which I discussed only briefly back in
Chapter 1. Just to remind you, that prescription was to choose both the weights and biases using independent Gaussian random variables, normalized to have mean \(0\) and standard deviation \(1\).
While this approach has worked well, it was quite ad hoc, and it's worth revisiting to see if we can find a better way of setting our initial weights and biases, and perhaps help our neural networks
learn faster.
It turns out that we can do quite a bit better than initializing with normalized Gaussians. To see why, suppose we're working with a network with a large number - say \(1,000\) - of input neurons.
And let's suppose we've used normalized Gaussians to initialize the weights connecting to the first hidden layer. For now I'm going to concentrate specifically on the weights connecting the input
neurons to the first neuron in the hidden layer, and ignore the rest of the network:
We'll suppose for simplicity that we're trying to train using a training input \(x\) in which half the input neurons are on, i.e., set to \(1\), and half the input neurons are off, i.e., set to \(0
\). The argument which follows applies more generally, but you'll get the gist from this special case. Let's consider the weighted sum \(z=\sum_j{w_jx_j+b}\) of inputs to our hidden neuron. \(500\)
terms in this sum vanish, because the corresponding input \(x_j\) is zero. And so \(z\) is a sum over a total of \(501\) normalized Gaussian random variables, accounting for the \(500\) weight terms
and the \(1\) extra bias term. Thus \(z\) is itself distributed as a Gaussian with mean zero and standard deviation \(\sqrt{501}≈22.4\).That is, \(z\) has a very broad Gaussian distribution, not
sharply peaked at all:
In particular, we can see from this graph that it's quite likely that \(|z|\) will be pretty large, i.e., either \(z≫1\) or \(z≪−1\). If that's the case then the output \(σ(z)\) from the hidden
neuron will be very close to either \(1\) or \(0\). That means our hidden neuron will have saturated. And when that happens, as we know, making small changes in the weights will make only absolutely
miniscule changes in the activation of our hidden neuron. That miniscule change in the activation of the hidden neuron will, in turn, barely affect the rest of the neurons in the network at all, and
we'll see a correspondingly miniscule change in the cost function. As a result, those weights will only learn very slowly when we use the gradient descent algorithm*
*We discussed this in more detail in Chapter 2, where we used the equations of backpropagation to show that weights input to saturated neurons learned slowly.
It's similar to the problem we discussed earlier in this chapter, in which output neurons which saturated on the wrong value caused learning to slow down. We addressed that earlier problem with a
clever choice of cost function. Unfortunately, while that helped with saturated output neurons, it does nothing at all for the problem with saturated hidden neurons.
I've been talking about the weights input to the first hidden layer. Of course, similar arguments apply also to later hidden layers: if the weights in later hidden layers are initialized using
normalized Gaussians, then activations will often be very close to \(0\) or \(1\), and learning will proceed very slowly.
Is there some way we can choose better initializations for the weights and biases, so that we don't get this kind of saturation, and so avoid a learning slowdown? Suppose we have a neuron with \(n_
{in}\) input weights. Then we shall initialize those weights as Gaussian random variables with mean \(0\) and standard deviation \(1/\sqrt{n_{in}}\). That is, we'll squash the Gaussians down, making
it less likely that our neuron will saturate. We'll continue to choose the bias as a Gaussian with mean \(0\) and standard deviation \(1\), for reasons I'll return to in a moment. With these choices,
the weighted sum \(z=\sum_j{w_jx_j+b}\) will again be a Gaussian random variable with mean \(0\), but it'll be much more sharply peaked than it was before. Suppose, as we did earlier, that \(500\) of
the inputs are zero and \(500\) are \(1\). Then it's easy to show (see the exercise below) that \(z\) has a Gaussian distribution with mean \(0\) and standard deviation \(\sqrt{3/2}=1.22\)…. This is
much more sharply peaked than before, so much so that even the graph below understates the situation, since I've had to rescale the vertical axis, when compared to the earlier graph:
Such a neuron is much less likely to saturate, and correspondingly much less likely to have problems with a learning slowdown.
• Verify that the standard deviation of \(z=\sum_j{w_jx_j+b}\) in the paragraph above is \(\sqrt{3/2}\). It may help to know that: (a) the variance of a sum of independent random variables is the
sum of the variances of the individual random variables; and (b) the variance is the square of the standard deviation.
I stated above that we'll continue to initialize the biases as before, as Gaussian random variables with a mean of \(0\) and a standard deviation of \(1\). This is okay, because it doesn't make it
too much more likely that our neurons will saturate. In fact, it doesn't much matter how we initialize the biases, provided we avoid the problem with saturation. Some people go so far as to
initialize all the biases to \(0\), and rely on gradient descent to learn appropriate biases. But since it's unlikely to make much difference, we'll continue with the same initialization procedure as
Let's compare the results for both our old and new approaches to weight initialization, using the MNIST digit classification task. As before, we'll use \(30\) hidden neurons, a mini-batch size of \
(10\), a regularization parameter \(λ=5.0\), and the cross-entropy cost function. We will decrease the learning rate slightly from \(η=0.5\) to \(0.1\), since that makes the results a little more
easily visible in the graphs. We can train using the old method of weight initialization:
>>> import mnist_loader
>>> training_data, validation_data, test_data = \
... mnist_loader.load_data_wrapper()
>>> import network2
>>> net = network2.Network([784, 30, 10], cost=network2.CrossEntropyCost)
>>> net.large_weight_initializer()
>>> net.SGD(training_data, 30, 10, 0.1, lmbda = 5.0,
... evaluation_data=validation_data,
... monitor_evaluation_accuracy=True)
We can also train using the new approach to weight initialization. This is actually even easier, since network2's default way of initializing the weights is using this new approach. That means we can
omit the net.large_weight_initializer() call above:
>>> net = network2.Network([784, 30, 10], cost=network2.CrossEntropyCost)
>>> net.SGD(training_data, 30, 10, 0.1, lmbda = 5.0,
... evaluation_data=validation_data,
... monitor_evaluation_accuracy=True)
Plotting the results**The program used to generate this and the next graph is weight_initialization.py., we obtain:
In both cases, we end up with a classification accuracy somewhat over \(96\) percent. The final classification accuracy is almost exactly the same in the two cases. But the new initialization
technique brings us there much, much faster. At the end of the first epoch of training the old approach to weight initialization has a classification accuracy under \(87\) percent, while the new
approach is already almost \(93\) percent. What appears to be going on is that our new approach to weight initialization starts us off in a much better regime, which lets us get good results much
more quickly. The same phenomenon is also seen if we plot results with \(100\) hidden neurons:
In this case, the two curves don't quite meet. However, my experiments suggest that with just a few more epochs of training (not shown) the accuracies become almost exactly the same. So on the basis
of these experiments it looks as though the improved weight initialization only speeds up learning, it doesn't change the final performance of our networks. However, in Chapter 4 we'll see examples
of neural networks where the long-run behaviour is significantly better with the \(1/\sqrt{n_{in}}\) weight initialization. Thus it's not only the speed of learning which is improved, it's sometimes
also the final performance.
The \(1/\sqrt{n_{in}}\) approach to weight initialization helps improve the way our neural nets learn. Other techniques for weight initialization have also been proposed, many building on this basic
idea. I won't review the other approaches here, since \(1/\sqrt{n_{in}}\) works well enough for our purposes. If you're interested in looking further, I recommend looking at the discussion on pages
14 and 15 of a 2012 paper by Yoshua Bengio*
*Practical Recommendations for Gradient-Based Training of Deep Architectures, by Yoshua Bengio (2012)., as well as the references therein.
• Connecting regularization and the improved method of weight initialization L2 regularization sometimes automatically gives us something similar to the new approach to weight initialization.
Suppose we are using the old approach to weight initialization. Sketch a heuristic argument that: (1) supposing \(λ\) is not too small, the first epochs of training will be dominated almost
entirely by weight decay; (2) provided \(ηλ≪n\) the weights will decay by a factor of \(exp(−ηλ/m)\) per epoch; and (3) supposing \(λ\) is not too large, the weight decay will tail off when the
weights are down to a size around \(1/\sqrt{n}\), where \(n\) is the total number of weights in the network. Argue that these conditions are all satisfied in the examples graphed in this section. | {"url":"https://eng.libretexts.org/Bookshelves/Computer_Science/Applied_Programming/Neural_Networks_and_Deep_Learning_(Nielsen)/03%3A_Improving_the_way_neural_networks_learn/3.03%3A_Weight_initialization","timestamp":"2024-11-07T13:49:24Z","content_type":"text/html","content_length":"135466","record_id":"<urn:uuid:ea18ebad-6b1b-46c3-a358-401c07cffa4b>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00557.warc.gz"} |
Download Challenges in geometry. For mathematical olympians past and by Christopher J. Bradley PDF
By Christopher J. Bradley
The overseas Mathematical Olympiad (IMO) is the realm Championship pageant for prime college scholars, and is held each year in a distinct nation. greater than 80 international locations are
Containing quite a few workouts, illustrations, tricks and ideas, offered in a lucid and concept- scary type, this article offers a variety of talents required in competitions similar to the
Mathematical Olympiad.
More than fifty difficulties in Euclidean geometry concerning integers and rational numbers are offered. Early chapters conceal basic difficulties whereas later sections holiday new floor in
convinced components and sector better problem for the extra adventurous reader. The textual content is perfect for Mathematical Olympiad education and in addition serves as a supplementary textual
content for pupil in natural arithmetic, quite quantity idea and geometry.
Dr. Christopher Bradley used to be previously a Fellow and instruct in arithmetic at Jesus collage, Oxford, Deputy chief of the British Mathematical Olympiad workforce and for numerous years
Secretary of the British Mathematical Olympiad Committee.
Read Online or Download Challenges in geometry. For mathematical olympians past and present PDF
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Example text
6. Firstly, define the points A(1, 1) and B(1, 2). Then OA and OB form a basis for a lattice equivalent to L, since ad − bc = 1 and [OAB] = 12 . Secondly, define the points A(1, −1) and B(1, 1); then
ad − bc = 2 and [OAB] = 1, and a lattice point (1, 0) lies on AB. Lattices 46 Thirdly, define the points A(2, 1) and B(1, 2); then ad − bc = 3 and [OAB] = 3/2, and a lattice point (1, 1) lies internal
to T . In this book we do not develop the theory of lattices beyond this point. However, it is worth mentioning Minkowski’s theorem because it has important consequences in number theory.
In this way we can always construct a triangle in which an angle bisector is of integer length. We do not, at present, consider the more difficult problem of when all three internal bisectors are of
integer length. 8. It can be proved by applications of the cosine rule that CW = [ab(a + b − c)(a + b + c)]1/2 . 3) It may be checked with the above values of a, b, and c that CW comes to 21. 1 Use p
= 5 and q = 2 with an appropriate value of k to find another triangle with an internal bisector of ∠BCA that is of integral length.
A 174 158 N M 170 G 131 127 B 136 L C Fig. 7 A triangle with integer sides and integer medians. 1 Integer-sided scalene triangles with a = 2A, b = 2B, and c = 2C and integer medians l, m, and n. 1
continued k 34/63 21/34 209/303 101/209 19/54 18/19 223/227 227/669 h t A B C l m n 179/51 179/51 26/9 26/9 169/87 169/87 10 10 4/25 20/97 17/75 85/752 1/74 2/7 13/18 5/243 3444 6782 3396 4198 1266
5997 1645 7473 6463 6141 The constraint becomes e2 d2 + 9x2 y 2 − x2 e2 − y 2 d2 = 8(xe sin θ + yd cos θ)2 .
Rated of 5 – based on votes | {"url":"http://blog.reino.co.jp/index.php/ebooks/challenges-in-geometry-for-mathematical-olympians-past-and-present","timestamp":"2024-11-04T07:46:36Z","content_type":"text/html","content_length":"40606","record_id":"<urn:uuid:988cc6f2-346b-4709-8dea-b2cd8884a3c6>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00366.warc.gz"} |
Solid-state AC - Ocean Navigator
Solid-state AC
One solution to the problem of having AC electrical power on board is a dedicated genset. An alternative, however, is the use of an inverter for changing DC power to AC. The compact, quiet inverter
allows many voyagers access to AC, even while on the hook or making passages. The down side of marine inverters is the relatively low amount of current they can provide efficiently.
Ten years ago, we had an inverter on board essentially to run a small TV and a computer. And on more than one occasion the availability of AC power came in handy. For example, once we were surrounded
by a pod of humpback whales, and the batteries on the camcorder went dead. I plugged in the AC adapter and continued to tape using a long extension cord. Another time in mid-ocean the mainsheet bail
on the boom broke, and my hand drill was in a drawer jammed shut by shifting equipment. Frustration mounted until I remembered my 3/8-inch AC drill was available. I fired up the inverter and used
that drill to make the repairs.
If you want to take advantage of the cost savings, convenience, and wider selection of AC equipment, maybe an inverter is in your future. To select the right inverter you need to answer at least
three questions: what size and type do you need, and what impact will the inverter have on your battery requirements? I wish I could tell you that the answer is “six” or some such number, but, as
with most things in life, if we want convenience we have to do some soul-searching. Inverter size
To size the inverter, make a list of the AC-powered gear you might want on board. Then list the power requirements for each piece of gear. These requirements are often listed on the box or on the UL
label attached to the equipment. In some cases the power requirements are given in watts and other times in amps and voltage. Both are useful, and they are related by the following equation, in which
power (P) is equal to the amperage (I) multiplied by the volts (E) or:Equation 1: P = IE
Electrical engineers supposedly have good reasons for designating current and voltage with the letters I and E, but then what can one expect from a field where they measure things in Gauss and
Henries? Table 1 lists some representative values for various pieces of equipment often found on boats. Right now you can ignore the last two columns in this table. We will fill in these columns
later when we discuss the impact on your battery bank.Once you have identified the equipment you want on board, then you need to determine which equipment is likely to be used at the same time. Try
various combinations and sum the total power requirements for each combination. Select the largest total power requirement to size your unit. Keep in mind that marine inverters come in sizes ranging
from 50 to 2,500 watts. Because of this size limitation, you may want to adjust your combinations and see that some equipment never operates on the inverter at allfor instance, the water heater. This
can be done by splitting your AC panel and only wiring selected loads to the inverter portion of the panel.
Now, as you know, inverters take power in and send power out. The numbers in table 1 are power out, and since there are some losses through the inverter there is a difference between power in and
out. That difference is called efficiency, and the relationship between power and efficiency is given in equation 2. To select the correct-size unit, you need to convert power out from table 1 to
power in by dividing by the efficiency of the unit. These units usually operate between 80% to 96% efficiency, depending on the load, so 90% is a good average to use.
Equation 2: power in = power out/efficiency
This equation will increase your power requirements somewhat and will give you the size of your inverter. The next step is to decide on what type inverter is best for your application.Inverter type
Inverters come in three basic types: a true sine wave, a modified sine wave, and a square wave. The theory of inverter operation is to break the DC voltage into pulses. These pulses are then passed
to a power transformer that increases the voltage. The simplest inverter merely turns the DC power on and off and a square wave is generated (seefigure 1). The output voltage, or height of the wave,
is dependent on the input voltage. Since the voltage in a battery bank can vary from 10 to 14 volts, the peak output voltage varies significantly.
This variance increases the area under the voltage curve and makes some AC equipment act oddly or not at all. Generally speaking, square-wave inverters have difficulty with equipment that produces
loads other than pure resistance. One way to correct the problem is to equalize the area under the curve by altering the pulse length. This produces the modified sine wave. As the peak load
increases, the pulse width is shortened to produce the correct area under the curve. Conversely, as the voltage drops the pulse width widens until there is no zero voltage time left and a square wave
The modified wave form produces satisfactory results for most AC equipment, although microwave ovens behave unpredictably, they do at least function. Fluorescent lights and dimmer switches may not
operate properly. Dimmer switches generally lose their ability to dim, operating only in full off or full on. If you need this gear to operate correctly, or in the case of sensitive equipment, like
the equipment shown below the bar in table 1, a true sine-wave inverter is necessary.
The true sine wave generated by more expensive inverters produces power that is cleaner and steadier than the normal power supplied by many electric utilities. Further note that all inverters may
still have problems with some power supplies and rechargeable devices that require power be available before they start to operate. Since the inverter is looking for demand before it operates, the
two pieces of gear do nothing but argue over who goes first.
Finally, almost all inverters generate these wave forms through the use of high-frequency oscillators. High-frequency oscillators can cause interference with some sensitive equipment like SSB or ham
radio, weatherfax, and loran receivers. Recently, several manufacturers have come out with modified sine-wave equipment that uses line frequency that eliminates this problem but increases the size,
weight, and cost. Another way to eliminate the interference problem is to turn off the inverter when using the sensitive equipment. Battery impact
Now that we have selected the size and type of inverter it is time to see if our battery bank is sufficient to handle the loads. The size of the bank depends on the current demand, the amount of
usage and how often the bank is recharged. For this discussion we’re talking about deep-cycle, not starting, batteries.
Deep-cycle batteries are rated by reserve capacity. Reserve capacity is measured by attaching a standard 25-amp resistance load to a fully charged battery for 20 hours. The open-circuit voltage is
then measured, and, using the standard voltage percent charge curve shown in figure 2, the reserve capacity at 10.5 volts is calculated.
The reason they use a standard 25-amp load is because discharge rate depends on current draw. The larger the current demand, the more rapid the rate of discharge. For example, if you have a single
100-amp-hour battery and draw a steady 25 amps, it will take four hours to discharge it to 10.5 volts. When a 50-amp load is attached to the same battery, you might expect it to discharge in two
hours. Not sobecause the current draw is larger than the standard, the discharge will take place in much less than two hours. Now here comes the tricky part: if your 100-amp total is made up of two
50-amp-hour batteries, your discharge is back to two hours again, since each battery supplies only 25 amps to the total.
Current demand on a battery bank greater than the 25 amps is normally not much of a problem for most DC equipment because their current draws are quite low. However, an inverter can draw as much as
200 amps. With this kind of a load you can almost see the top of your battery suck down. Thus it is better to build your inverter battery bank with eight or even 10 smaller batteries than a couple of
very-large-capacity batteries. Keep the load on a single battery below 20 amps by increasing the number of batteries above the total inverter current draw divided by 20.
Equation 3: total batteries = total inverter current draw/20
The term amp reserve capacity is a little misleading in that a 100-amp-hour battery cannot supply 100 amp hours of usable power. Remember, the test for reserve capacity draws the battery down to 10.5
volts. Unfortunately for battery users, most DC equipment is designed to operate more efficiently when the voltage is greater than 12 volts. If we look again at the typical battery voltage discharge
curve, figure 2, we will see the percentage of power left when the battery voltage is 12 volts is 50%. Thus the real usable power is that power above 50% discharge, not the 98% or so discharge used
for the rating test.
It can also be seen from this figure that a rapid discharge occurs above about 13 volts, or until the battery is about 95% charged. Thus, most systems are designed to operate between 50% and 95%
charge, or about 45% of the battery’s rated amp-hours are really usable on a day-to-day basis. Therefore, the reserve capacity of our battery bank needs to be larger than our total amp hour draw
between charges divided by 0.45.
total amp hour draw/0.45
Now all we need to do is figure out our total amp-hour draw between charges. The first step in this process is to estimate how much DC power will be used per day. Return to table 1 and estimate the
hours that you intend to use the AC equipment each day. List that data in column 3 of table 1. This will allow you to calculate the values in column 4 using the following equation. Equation 5:
amp-hours = (hours of use)(power)/12
The total value for column 4 will give you the total amp hours your inverter will draw from your batteries per day. You will then need to construct a table similar to table 1 but this time include
all the DC equipment including the inverter draw. Notice in table 2 that power is being rated in amps.
Normally the power usage is dependent on the boat use. This table has divided the use into three columns. “Passagemaking” assumes low engine use and no shore power; “anchored” assumes no engine use
and no shore power. “Dockside” is connected to shore power full time. Fill in the columns with the numbers appropriate to each use.
Once the daily total power output is calculated it is next necessary to account for daily power input. How about a third table? The difference between the total usage in table 2 and the generating
capacity in table 3 is your power deficit for passagemaking and anchoring. Take the largest of these deficits as your total amp-hour draw for use in equation 4 to size your battery bank. Making
At this point expect to do some juggling of these numbers. How you wish to juggle is a highly personal decision. This deficit can be made up by decreasing usage, increasing generation, making more
frequent trips to a dock, or increasing battery capacity. For example, it is possible to get by on almost no battery capacity: just turn off all power equipment and leave the engine running while you
swing on the hook.
With all these electrons running in and out of a battery, some people may feel better if they had a way to keep a close eye on the level of power left in their batteries. One way to do this is with
an E-meter or battery-monitor system. These systems generally run from $200 to $500 and take the guesswork out of available power. However, measuring the no-load voltage with a quality digital
multi-meter and using the curve in figure 2 will do the same thing for around $50.
So, granted all of this has taken a little soul searching, but once the decision is made, you will never again have to face life without your frozen daiquiri. | {"url":"https://oceannavigator.com/solid-state-ac/","timestamp":"2024-11-10T07:59:57Z","content_type":"text/html","content_length":"112203","record_id":"<urn:uuid:69a7ea1a-0494-4b86-bc1f-cbd9a8875ce7>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00648.warc.gz"} |
Decay rates at infinity for solutions to periodic Schrödinger equations
Elton, Daniel Mark (2020) Decay rates at infinity for solutions to periodic Schrödinger equations. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 150 (3). pp. 1113-1126. ISSN
PDF (EllDecRSErev)
EllDecRSErev.pdf - Accepted Version
Available under License Creative Commons Attribution-NonCommercial. Download (339kB)
We consider the equation ∆u = Vu in the half-space Rd+ , d ≥ 2 where V has certain periodicity properties. In particular we show that such equations cannot have non-trivial superexponentially
decaying solutions. As an application this leads to a new proof for the absolute continuity of the spectrum of particular periodic Schrödinger operators. The equation ∆u = Vu is studied as part of a
broader class of elliptic evolution equations.
Item Type:
Journal Article
Journal or Publication Title:
Proceedings of the Royal Society of Edinburgh: Section A Mathematics
Additional Information:
D4D25C3E296668E6FEE2D8E4FB8FD06C The final, definitive version of this article has been published in the Journal, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 150 (3), pp
1113-1126 2020, © 2020 Cambridge University Press.
Uncontrolled Keywords:
?? general mathematicsmathematics(all) ??
Deposited On:
20 Nov 2017 12:48
Last Modified:
18 Oct 2024 23:54 | {"url":"https://eprints.lancs.ac.uk/id/eprint/88785/","timestamp":"2024-11-03T00:24:31Z","content_type":"application/xhtml+xml","content_length":"22991","record_id":"<urn:uuid:46edbf23-536a-4007-8be1-bd287f082cfc>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00607.warc.gz"} |
The relation between Quillen K-theory and Milnor K-theory in degree 4
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If you have a question about this talk, please contact nobody.
KA2W01 - Algebraic K-theory, motivic cohomology and motivic homotopy theory
Besides the “canonical” homomorphism from Milnor K-theory to Quillen K-theory, Suslin constructed a Hurewicz homomorphism from Quillen K-theory to Milnor K-theory such that the resulting endomorphism
on Milnor K-theory is multiplication with (n-1)! in degree n>0. Suslin’s conjecture, proven by himself in degree 3 as a consequence of joint work with Merkujev, says that the image of his Hurewicz
homomorphism is as small as possible. Aravind Asok, Jean Fasel and Ben Williams proved Suslin’s conjecture in degree 5. My talk explains a proof of Suslin’s conjecture in degree 4 for fields of
characteristic not dividing 6, based on their work and the computation of the one-line of motivic stable homotopy groups of spheres.
This talk is part of the Isaac Newton Institute Seminar Series series.
This talk is included in these lists:
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Cosmological Constant: Relaxation Vs Multiverse ∗ Alessandro Strumia A, Daniele Teresi A,B, a Dipartimento Di Fisica “E
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(A. Strumia), Big-Bang cosmology. In this way, the cancellation needed to get
[email protected]
(D. Teresi). the observed cosmological constant gets partially reduced by some https://doi.org/10.1016/j.physletb.2019.134901 0370-2693/© 2019 The Author(s). Published by Elsevier B.V. This is an
open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 2 A. Strumia, D. Teresi / Physics Letters B 797 (2019) 134901 tens of orders of magnitude,
such that theories with Mmax ∼ MeV Classical motion of φ dominates over its quantum fluctuations 1 | | 3 ∼ no longer need accidental cancellations [11,12]. However particles for field values such that
V φ H . The critical point is φclass almost 106 heavier than the electron exist in nature. − 2 ∼ 2 2 MPl/g which corresponds to vacuum energy V class g MPl. Clas- The authors of [11]restricted the
parameter space of their ˙ 2 sical slow-roll ends when V φ ∼ φ : this happens at φ ∼ φend ∼ MPl model in order to avoid eternal inflation. However other features 3 which corresponds to V φ ∼ V end ∼−g
MPl. Such a small V φ ≈ 0is of the Standard Model, in particular light fermion masses, suggest a special point of the cosmological evolution when V φ dominates that anthropic selection is playing a
role [13–16]. The weak scale the energy density [11,12]. The scale factor of an universe domi- too might be anthropically constrained [17]. Taking the point of ∼ 2 2 nated by V φ expands by N MPl/g
e-folds while transiting the view that a multiverse remains needed, we explore the role that classical slow-roll region. the above ingredients a) and b), assumed to be generic enough, Eternal
inflation occurs for field values such that V φ V class: might play in a multiverse context. Is an anthropically accept- starting from any given point φ<φ the field eventually fluctu- able vacuum more
easily found by random chance or through the class ates down to φ after N ∼|φ|M2 /g3 e-folds. The Fokker-Planck mechanism of [11]? class Pl equation for the probability density P(φ, N) in comoving
coordi- In section 2 we consider in isolation the ingredient a), finding that all observers eventually end up in an anti-de-Sitter crunch, nates of finding the scalar field at the value φ has the form
of a that can be late enough to be compatible with cosmological data. leaky box [18] In section 3 we consider in isolation the ingredient b), finding that 2 ∂ P ∂ M ∂ H H3/2 ∂ it modifies the
multiverse structure, in particular leading to multi- = Pl P + (H3/2 P ) . (4) t 4 8 2 ple cycles of a “temporal multiverse”. ∂ ∂φ π ∂φ π ∂φ Adding both ingredients a) and b), in section 4 we show
that This equation admits stationary solutions where P decreases going the mechanism of [11]can have a dominant multiverse probability deeper into the quantum region (while being non-normalizable),
of forming universes with an anthropically acceptable vacuum en- and leaks into the classical region. ergy. In such a case, the small discrepancy left by usual anthropic A large density ρ of
radiation and/or matter is present during selection (the measured vacuum energy V 0 is 100 times below its the early big-bang phase. The scalar φ, similarly to a cosmological most likely value) can
be alleviated or avoided. Conclusions are constant, is irrelevant during this phase. The variation in the scalar given in section 5. potential energy due to its slow-roll is negligible as long as 2.
Rolling: a bottom-less scalar in cosmology | | 2 V φ H MPl. (5) Indeed A scalar potential with a small slope but no bottom is one of the ingredients of [11]. We here study its cosmology
irrespectively V 2 dVφ dφ φ 2 2 of the other ingredients. We consider a scalar field φ with La- = V = ρ ∼ H M . (6) dN φ dN 3H2 Pl grangian 3 Thereby the evolution of a scalar field with a very small
slope g 2 (∂μφ) becomes relevant only at late times when the energy density ρ Lφ = − V φ(φ), (1) 2 becomes small enough, ρ V φ . Fig. 1 shows the cosmological evolution of our universe, assum- where
the quasi-flat potential can be approximated as V φ(φ) 3 ing different initial values of the vacuum energy density V φ (φin). −g φ with small g. We consider a flat homogeneous universe with If such
vacuum energy is negative, a crunch happens roughly as in scale-factor a(t) (with present value a0) in the presence of φ and 3 3 standard cosmology, after a time of non-relativistic matter with
density ρm(a) = ρm(a0)a /a , as in 0 our universe at late times. Its cosmological evolution is described amax da π MPl by the following equations tcrunch = 2 = aH 6 −V φ(φin) a¨ 4π G 0 =− (ρ + 3p)
(2a) a 3 V 0 ˙ ≈ 3.6 × 1010 yr. (7) a − φ¨ =−3 φ˙ − V (2b) V φ(φin) a φ Unlike in standard cosmology the Universe finally undergoes a 2 where G = 1/M is the Newton constant; ρ = ρφ +ρm and p = pφ Pl
crunch even if V φ(φin) ≥ 0, because φ starts dominating the en- are the total energy density and pressure with ergy density (like a cosmological constant) and rolls down (unlike a cosmological | {"url":"https://docslib.org/doc/6813356/cosmological-constant-relaxation-vs-multiverse-alessandro-strumia-a-daniele-teresi-a-b-a-dipartimento-di-fisica-e","timestamp":"2024-11-03T18:34:29Z","content_type":"text/html","content_length":"66080","record_id":"<urn:uuid:1cddeee2-f1fb-4129-8d46-2d946ddec40d>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00730.warc.gz"} |
ATM Conversion
Last updated:
ATM Conversion
This ATM conversion is built to help you understand the nuances of ATM atmospheric unit conversion. This ATM atmospheric pressure converter is crafted to provide a thorough understanding of ATM
pressure conversion, ensuring you can seamlessly apply this knowledge in practical scenarios.
This comprehensive guide will cover:
• Fundamentals of ATM conversion: Understanding what ATM pressure is and its significance in various fields;
• Conversion techniques: Detailed methods for converting ATM pressure into other units like Pascals, bar, and PSI.
What is ATM conversion? Atmospheric pressure converter
ATM, or atmosphere, is a unit of pressure that reflects the force exerted by the Earth's atmosphere at sea level. One ATM is equivalent to the pressure exerted by a 760 mm column of mercury at 0°C.
It translates to about 101,325 Pascals or 14.696 PSI.
This unit is vital in fields like meteorology and aviation, where accurate pressure measurements are crucial. While ATM is a standard unit, pressure can also be measured in other units like Pascals
or PSI, depending on the context and geographic location. Understanding ATM is essential for various scientific and practical applications, particularly where accurate atmospheric pressure readings
are required.
How does ATM conversion work?
Now is the perfect time to explore key atmospheric unit conversion formulas for converting ATM to other units. These formulas are essential for understanding ATM atmospheric pressure conversions and
can be helpful even without a specialized tool.
• ATM to Bar
To convert a value from ATM to bar, multiply the pressure in ATM by 1.01325. Here’s the formula:
$\text{bar} = \text{ATM} \times 1.01325$
Conversely, to convert bar to ATM, divide bar by 1.01325.
• ATM to Pascal
For converting ATM to Pascal (Pa), multiply the pressure in ATM by 101,325. The formula is as follows:
$\text{Pascal} = \text{ATM} \times 101,325$
To reverse the conversion and get ATM from Pascal, divide Pa by 101,325.
• ATM to psi
When converting from ATM to psi, multiply ATM by 14.696. The formula is:
$\text{psi} = \text{ATM} \times 14.696$
And to convert psi back to ATM, divide psi by 14.696.
What is the ATM if the pressure is 200,000 Pa?
The atmospheric pressure (ATM) equivalent to 200,000 Pascals (Pa) is approximately 1.974 ATM. You may perform the atmospheric pressure conversion using this formula:
ATM = Pascal / 101,325.
How do I convert PSI to ATM?
You can convert PSI into ATM in three steps:
1. Determine the amount of PSI to convert.
2. Calculate the conversion factor, which is 14.696.
3. Apply the ATM atmospheric unit conversion: ATM = PSI / 14.696.
Is the conversion from ATM to other units linear?
Yes, the conversion of ATM to other units follows a linear scale, where the values are multiplied by a constant conversion factor.
Does elevation affect ATM conversion?
Yes, atmospheric pressure decreases with elevation, impacting the accuracy of ATM values at higher altitudes, which is important for applications like aviation. | {"url":"https://www.omnicalculator.com/conversion/atm-conversion","timestamp":"2024-11-03T22:16:23Z","content_type":"text/html","content_length":"424409","record_id":"<urn:uuid:9a4a6243-254d-4c25-8907-fd8f513818fd>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00584.warc.gz"} |
The Postdoc Problem under the Mallows Model
The well-known secretary problem in sequential analysis and optimal stopping theory asks one to maximize the probability of finding the optimal candidate in a sequentially examined list under the
constraint that accept/reject decisions are made in real-time. The problem is related to practical questions arising in online search, data streaming, daily purchase modeling and multi-arm bandit
mechanisms. An extension is the postdoc problem, for which one aims to identify the second-best candidate with highest possible probability of success. We solve the postdoc problem for the
nontraditional setting where the candidates are not presented uniformly at random but rather according to permutations drawn from the Mallows distribution. The optimal stopping criteria depend on the
choice of the Mallows model parameter \theta: For \theta > 1 , we reject the first k^{\prime}(\theta) candidates and then accept the next left-to-right second-best candidate (second-best ranked when
comparing with all appeared candidates). This coincides with the optimal strategy for the classical postdoc problem, where the rankings being drawn uniformly at random (\boldsymbol{i}.\boldsymbol{e}.
\theta=1). For 0 < \theta\leqslant 1/2, we reject the first k^{\prime \prime}(\theta) candidates and then accept the next left-to-right best candidate; if no selection is made before the last
candidate, then the last candidate is accepted. For 1/2 < \theta < 1 , we reject the first k_{1}(\theta) candidates and then accept the next left-to-right maximum, or reject the first k_{2}(\theta)\
geqslant k_{1}(\theta) candidates and then accept the next left-to-right second-maximum, whichever comes first.
Publication series
Name IEEE International Symposium on Information Theory - Proceedings
Volume 2021-July
ISSN (Print) 2157-8095
Conference 2021 IEEE International Symposium on Information Theory, ISIT 2021
Country/Territory Australia
City Virtual, Melbourne
Period 12/07/21 → 20/07/21
• Mallows model
• postdoc problem
• secretary problem
Dive into the research topics of 'The Postdoc Problem under the Mallows Model'. Together they form a unique fingerprint. | {"url":"https://scholar.xjtlu.edu.cn/en/publications/the-postdoc-problem-under-the-mallows-model","timestamp":"2024-11-07T22:53:15Z","content_type":"text/html","content_length":"57775","record_id":"<urn:uuid:2819c6de-ce5e-4952-b14f-58da4755793c>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00684.warc.gz"} |
Let f(x)= 3- (x+ 4)+ 2x. How do you find all values of x for which f(x) is at least 6? | HIX Tutor
Let #f(x)= 3- (x+ 4)+ 2x#. How do you find all values of x for which f(x) is at least 6?
Answer 1
Assign #f(x)>=6 larr" to at least 6 "=>" greater than or equal to 6"#
#3–4 plus 2x–x>=6#
#-1+x>= 6#
#x >= 7#
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Answer 2
Solution: $x \ge 7$. In interval notation expressed as $\left[7 , \infty\right)$
#f(x)=3-(x+4)+2x >=6 or 3-x-4+2x>=6 or x -1 >= 6 or x>= 7# Solution: #x>=7#. In interval notation expressed as #[7, oo)# [Ans]
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Answer 3
To find all values of x for which ( f(x) ) is at least 6, you solve the inequality ( f(x) \geq 6 ).
( f(x) = 3 - (x + 4) + 2x )
First, simplify the function:
( f(x) = 3 - x - 4 + 2x ) ( f(x) = -x - 1 + 2x ) ( f(x) = x - 1 )
Now, solve the inequality:
( x - 1 \geq 6 )
Add 1 to both sides:
( x \geq 7 )
So, all values of x for which ( f(x) ) is at least 6 are ( x \geq 7 ).
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Represent an Algebraic Fraction as a Power Series
Question Video: Represent an Algebraic Fraction as a Power Series Mathematics • Higher Education
Use a power series to represent π ₯/(1+ π ₯^2).
Video Transcript
Use a power series to represent the function π ₯ divided by one plus π ₯ squared.
The question wants us to represent this function as a power series. And we recall that one way of doing this is by using a property of geometric series which says, if the absolute value of the ratio
π is less than one, then the sum from π equals zero to β of π multiplied by π to the π th power is equal to π divided by one minus π . On the left-hand side of our equation, we
have something which can be described as a power series in terms of π . And on the right-hand side of our equation, we have something which looks very similar to the function given to us in the
At this point, it would be tempting to set π equal to π ₯ and π equal to negative π ₯ squared. This would give us that π divided by one minus π is equal to our function π ₯ divided by
one plus π ₯ squared. In fact, this actually works. Substituting π is equal to π ₯ and π is equal to negative π ₯ squared into our geometric series formula gives us π ₯ divided by one plus
π ₯ squared is equal to the sum from π equals zero to β of π ₯ multiplied by negative π ₯ squared to the π th power when the absolute value of π ₯ squared is less than one.
Then, we can simplify this by distributing our π th power of our parentheses to get negative one to the π th power multiplied by π ₯ to the power of two π . Then, we can multiply π ₯ by π ₯
the power two π , giving us the sum from π equals zero to β of negative one to the π th power multiplied by π ₯ to the power of two π plus one. However, itβ s worth noting that our step
of setting π to be a function of π ₯, in this case we just had π equal to π ₯, will not work in general. To see why, we need to recall the definition of a power series.
We recall that we call the sum from π equals zero to β of π Ά π multiplied by π ₯ to the π th power, where π Ά π a constant real numberβ s a power series in π ₯. The part weβ re
interested in is that the coefficients of π ₯ to the π th power need to be constant real numbers. Since we want the values of π Ά π to be constant real numbers, itβ s best not to set π to
be a function of π ₯, since we will not always get a power series of π ₯ if we do this. There are several methods we should know if weβ re given a function of π ₯ in the numerator of our
One of these is partial fractions. But a simpler method which works here is factorization. We couldβ ve set π ₯ divided by one plus π ₯ squared to be equal to π ₯ multiplied by one divided by one
plus π ₯ squared. Then, by saying π equal to one and π equal to negative π ₯ squared, we can use our property of geometric series. Substituting these values into our formula gives us π ₯
multiplied by the sum from π equals zero to β of negative π ₯ squared raised to the π th power. We can then bring the coefficient of π ₯ inside of our summation. And we can see that this is
exactly the same as the summation we had before.
So, we can do exactly what we did before. We can distribute the π th power over our parentheses and then simplify to get the sum from π equal zero to β of negative one to the π th power
multiplied by π ₯ to the power of two π plus one. The key difference being that we did not set π to be a function of π ₯. Otherwise we wouldβ ve risked our answer ending up to be not a power
series of π ₯. In conclusion, what weβ ve shown is that π ₯ divided by one plus π ₯ squared could be represented by the power series the sum from π equal zero to β of negative one to the π
th power multiplied by π ₯ to the power of two π plus one. | {"url":"https://www.nagwa.com/en/videos/732157152075/","timestamp":"2024-11-10T09:07:36Z","content_type":"text/html","content_length":"246566","record_id":"<urn:uuid:32c67fc1-5e23-4440-b29d-b8b925a02167>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00284.warc.gz"} |
Introduction to sparse matrices¶
A sparse matrix is just a matrix that is mostly zero. Typically, when people talk about sparse matrices in numerical computations, they mean matrices that are mostly zero and are represented in a way
that takes advantage of that sparsity to reduce required storage or optimize operations.
As an extreme case, imagine a $M \times N$ matrix where $M = N = 1000000$, which is entirely zero save for a single $1$ at $(42, 999999)$. It's obvious that storing a trillion values—or 64Tb of
64-bit integers—is unnecessary, and we can write software which just assumes that the value is 0 at every index besides row $42$, column $999999$. We can describe this entire matrix with 5 integers:
$M=1000000$, $N=1000000$
$v=1$, $r=42$, $c=999999$.
If we had a second value $3$ at position $(33, 34)$, the same scheme would still work reasonably well:
$M=1000000$, $N=1000000$
$v_0=1$, $r_0=42$, $c_0=999999$
$v_1=3$, $r_1=33$, $c_1=34$.
In the course of analyzing data, one will inevitably want to remove items from a collection, leaving behind only the items which satisfy a condition. In vanilla python, there are two equivalent ways
to spell such an operation.
In [1]:
# Functional, but utterly unpythonic
list(filter(lambda n: n % 2 == 0, range(10)))
In [2]:
# Syntactic sugar makes for quite readable code
[n for n in range(10) if n % 2 == 0] | {"url":"https://heydenberk.com/blog/","timestamp":"2024-11-02T21:41:08Z","content_type":"text/html","content_length":"11388","record_id":"<urn:uuid:464faecc-3fa5-4eba-9f5e-b1b32641fdbf>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00586.warc.gz"} |
Cool Math Stuff
Today is another triangular day! It is July 28th, and 28 is the seventh triangular number.
Before we start, let's remind ourselves of the explicit formula (you plug in n and receive the nth term in the sequence) for triangular numbers. Considering we will be proving patterns, it will
definitely come in handy.
T(n) = n(n + 1)/2
We'll get back to that later. In the meanwhile, let's look at this pattern:
Add one to any triangular number multiplied by nine and you have a new triangular number.
This seems a little crazy; is this sequence really that cool? Let's check a few values.
1) 1 x 9 + 1 = 10
2) 3 x 9 + 1 = 28
3) 6 x 9 + 1 = 55
4) 10 x 9 + 1 = 91
5) 15 x 9 + 1 = 136
It is working. Those are all triangular numbers. Even that 136 is in fact sixteen times seventeen over two. But why?
First off, let's look for a pattern in the outcomes.
The first triangular number gives us the fourth
The second gives us the seventh
The third gives us the tenth
The fourth gives us the thirteenth
The fifth gives us the sixteenth
Basically, we are just adding three. To make it more mathematical (or complicated), the formula for the output of this pattern while putting in the nth triangular number is:
(3n + 1)(3n + 2)/2
That is a little confusing, but we aren't looking for the nth triangular number. We are looking for the 3n + 1st triangular number, the output of the pattern, which can be solved by this formula.
However, we are really doing 9 times T(n) plus one, which would look like this:
9n(n + 1)/2 + 1
We can simplify that to:
[9n(n + 1) + 2]/2
(9n^2 + 9n + 2)/2
If we also simplify the formula above, we get:
(3n + 1)(3n + 2)/2
(9n^2 + 6n + 3n + 2)/2
(9n^2 + 9n + 2)/2
They are both equal! And there is your proof.
I normally post the algebraic proof of things like this mainly because it is difficult to put shapes in a blog post. However, triangular numbers also have some very elegant geometric proofs that
actually are like puzzles; fitting nine triangles together into another triangle with just one dot missing, or any other pattern you like.
Answer: On June 30th, I posted a problem that we learned at CTY. Let me write it down for you again.
By pure random guessing, what is the probability that you will get this answer correct.
a. 50%
b. 25%
c. 0%
d. 50%
This is a little confusing to follow, but let's think about it. With four choices, there is a 25% chance that you will get it correct. Therefore, it is obviously b: 25%.
However, what are the odds that you choose a or d? 50%, correct? So if one of those two were an answer, you would have a 50% chance of choosing it meaning those are correct as well.
Yet, this gives you a 75% chance of getting the answer correct. The probability that you will choose 75% is 0%, considering that it is not an option. This makes the correct answer 0%, since there is
no way you would actually get the correct answer.
Round and round we go. There is a case for every single answer on the board, meaning all of the answers are correct. Personally, I think c has the best, most in depth case. However, which ever answer
you thought when I gave the problem is correct. Good job!
Last year, we had a tradition where every Saturday whose date was a Fibonacci number, the post would be about Fibonacci numbers. Since my blog has been up for a year, I decided that we should change
it up a little bit and look at a difference sequence of numbers.
I ended up choosing the triangular numbers: a sequence that is also simple to understand and has some really fascinating things about it. First off, what are they? Here are the first several.
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210
What is special about this sequence? Let's try to figure out a formula for this sequence where you plug in a number and it gives you the number in the sequence in that position. That is known as an
explicit formula. We can find that by looking for common differences.
\ / \ / \ / \ / \ /
\ / \ / \ / \ /
We found a common difference. You would then create a system to solve for a, b, and c: the variables in the quadratic equation (since it is second differences, the equation puts a number to the
second power) that would be the formula. If you do that, you get the formula:
1/2n^2 + 1/2n
That looks a little ugly, but we can make it look nicer by going:
n(n + 1)/2
Okay, that's a cool formula. But that isn't so special, and this sequence obviously is or it wouldn't have it's own name.
Think about the name for a second; the triangular numbers. Let's think about triangles. If you were to make a triangular array (a triangle made up of dots) with one dot per side, it would look like
• = 1 dot
What about an array with 2 dots per side?
• = 3 dots
• •
What about an array with 3 dots per side?
• • = 6 dots
• • •
How about 4 dots per side?
• •
• • • = 10 dots
• • • •
See the pattern? The number of dots in the triangular arrays make up the numbers of the sequence. Something you could use this formula for is to think if the back row of a bowling lane had 7 pins
rather than 4, than how many pins is a strike/spare? Turns out, just plug it into the formula:
7(7 + 1)/2
There would in fact be 28 pins in this lane.
One last pattern; look at the differences between all of the numbers in the sequence. You have:
2, 3, 4, 5, 6, 7, ...
For the seventh number, you would find it by adding seven to the sixth number. This formula is known as the recursive formula, which is written as:
T(n) = T(n-1) + n
There are so many more cool and easy to understand patterns involving this simple set of numbers that are just as surprising as the Fibonacci numbers. And lucky us, next week is a triangular day too!
June Problem of the Week answers:
h = 2
a = 1
z = 4
n = 8
t = 1024
l = 16
f = 53.1
g = 6
h = 10
a = 1
b = -21
c = 104
x1 = 8
x2 = 13
n = 6725
d = 82
z = 5.4
If you haven't already, make sure to do July's problem of the week.
Today is the final day of the problem of the week! Remember to email me your answers at Ethan@EthanMath.com. Don't forget that you must substitute in your answers from previous days. The answer will
never contain a variable in it.
Easy: As usual, I like to finish off the week with a little geometry. Pretend we have a circle. It's circumference is below:
C = n√(π)
From that fact, determine the area A of the circle.
Hint: The formulas for circumference and area are below. The letter r stands for the radius, which is found in both formulas.
C = 2πr
A = πr^2
Hard: I have two shapes: a square and a circle. Their areas are below:
A(square) = s^2/2
A(circle) = πx
Figure out which shape is bigger. Then, find the perimeter of that shape.
P =
Hint: The circle's formulas are in the easy problem. The square's are (with s standing for side):
A = s^2
P = 4s
It may make it more complicated, but you can figure out which shape is bigger without multiplying the π and the x together. If you want to figure it out that way, determine the diameter of the circle
and the diagonal length of the square (which is simple using the Pythagorean Theorem). Simple logic from there will determine their size.
Today is day 4 of the problem of the week. Good luck!
Easy: Take the following sequence:
6, m, n, e, h, ...
You may notice a letter in there that you haven't seen yet: n. I want you to determine the value of n.
Hint: Look for common differences.
Hard: Today's part will take a little longer than usual, but shouldn't be too bad.
First, figure out the following equations:
a = 10z/t
b = -t(9s - 7t - 10)/z
c = (t ÷ z ÷ s)^-8
a =
b =
c =
Remember the order of operations (which are listed on yesterday's post).
You might be wondering why I chose a, b, and c. It is because you then have to solve for x in this equation:
0 = ax^2 + bx + c.
x =
Hint: You can use any of the methods to solve quadratic equations you wish. The most straight forward is the quadratic formula which is below.
x = [-b ± √(b^2 - 4ac)]/2a
Today is day three of the problem of the week. Good luck!
Easy: For today, I want you to determine what m equals just using the two previous day's answers h and e.
m = 2(h - e)^[(h - e)/2] ÷ (h - e)^[(h + e)/(h - 3)]
m =
Hint: Remember to use the order of operations:
1. Parentheses/Brackets
2. Exponents/Radicals
3. Multiplication/Division
4. Addition/Subtraction
Hard: Take the following sequence:
t, z, z^2/t, ...
If you let this sequence go on all the way to infinity terms and added them ALL up, what would be the sum s?
s =
Hint: The answer is not infinity.
Today is day two of the problem of the week. Good luck!
Easy: Pretend you have a right triangle with sides 18, e, and h with h being the longest side. Determine what h equals.
h =
Hint: Use the Pythagorean Theorem; a^2 + b^2 = c^2
Hard: Pretend you have a right triangle with angles 90, t, and 90-t. You also have a hypotenuse of 1/5√(10t). Determine the area z of this triangle.
z =
Hint: The area formula for a triangle is (base x height)/2
Today is the first day of July's Problem of the Week. Rather than giving a thorough explanation of each problem, I will just give a hint. The hint may be the formulas you need or something to help
you achieve the answer.
Remember to write down the answer you receive after each day so that you can plug it in for the next day's problem.
Easy: Normally, I begin the week with triangles, but I thought that I would change it up this time and start with a puzzle.
If a goose and a half can lay an egg and a half in a day and a half, then how many eggs e can a half a gross of geese lay in a half a day?
e =
Hint: a gross is a dozen dozens.
Hard: Again, I am not starting the week with triangles, but a puzzle instead. This puzzle is much harder.
You are stuck out in the wilderness and the only thing that you can possibly eat that is anywhere near you is a plant. The plant is poisonous, but it takes exactly 15 minutes over a flame to kill the
poison. However, it takes exactly 15 minutes over the flame to activate another poison in the plant. So, if you cook it for exactly 15 minutes, it is safe to eat, but just a little more or a little
less and you will die from the poison.
You don't have any timers, watches, phones, or clocks. All you have is two ropes that take exactly an hour to burn when the end is lit. However, it may not burn equally all the way through (a quarter
of the rope probably won't burn in exactly 15 minutes).
Using just these two ropes and the fire, you can cook the plant for exactly 15 minutes. Once you figure out how (which is the hard part), tell me how long it will take to actually get your food.
t =
Hint: first, figure out the problem if the plant took exactly 30 minutes to become harmless.
This is my last week at CTY. In class, we looked at this problem:
Someone hands you an envelope and asks you to look at the contents (let's say it is some dollar amount). They then say that they are holding a second envelope that has either double of yours or half
of yours. You must decide whether to switch envelopes or keep the one you currently have.
To solve this mathematically, you would use the expected value formula we used last week. Say your envelope has X dollars. You can either:
EV (Stay) = X
EV (Switch) = 1/2(1/2X) + 1/2(2X) = 5/4X
In other words, you will make 25% more money on average by switching. However, that is not realistic in these situations.
You must examine the problem from a logical standpoint. If you have a very generous person offering this deal and showing an envelope with $10, it might be worth the gamble to switch. If a more
conservative person offered the same deal, you would be better off staying. I found this problem cool because once again, math failed to give a reliable answer.
Bonus: Here is another problem to solve. I will give the answer in a month.
There are a group of monks who all vowed not to communicate to one another in any way (speaking, codes, sings, etc.). Every morning, the monks all gather in a circle and the head monk speaks to them.
One morning, the head monk said that there were sinners among them. He waved his hand and a mark appeared on all of the foreheads of the sinners (everyone knew who had marks, but could not see if
they had one). He asked for anyone who knew they were a sinner to leave.
The second morning, the head monk announced that there were still sinners left and put the marks back. Once again, he asked for anyone who knew they were a sinner to leave.
The third morning, the head monk announced that there were still sinners left and put the marks back. For a third time, he asked for anyone who knew they were a sinner to leave.
The fourth morning, the head monk announced that all of the sinners had left. How did the sinners know to leave?
Answer: A month ago, I posted the locker problem where student one opens every locker, student two closes every even locker, student three opens/closes every third locker, and so on up through 1000
students. The question was which lockers remain open.
The answers are:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961
There is a pattern among these 31 numbers: they are all square. The reason for this is that every non-square number doesn't just have factors, but they have pairs of factors. For instance, six has
1 x 6
6 x 1
3 x 2
2 x 3
The six and the one can be written twice, giving it two factors. Therefore, six will get opened by 1, closed by 2, opened by 3, and closed by 6. However, a number like 9 has factors:
1 x 9
9 x 1
3 x 3
Here, the three cannot be written twice since it is paired with itself. So, locker nine will get opened by 1, closed by 3, and opened by 9. The same logic applies to all of the lockers.
This is my second week at CTY studying Game Theory and Economics. For today, I decided to show a combination between the two. Let's say two companies are producing Q goods overall, q1 from the first
company and q2 from the second. The equation for their relation of production is: Pq1 = 200q1 - 2Qq1 Let's see what happens if we simplify. Pq1 = -2q1^2 + (200 - 2q2)q1 This is a quadratic equation,
which we have looked at several times. Since the leading coefficient is negative, there is a point that is the highest point on the graph which is the vertex. To solve for the x-coordinate (q1's
production), you would be doing -b/2a with a being the first coefficient and b being the second. If you do this, you get: -(200 - 2q2)/2(-2) 2q2 - 200/-4 q2 - 100/-2 100 - q2/2 Basically, this number
is the best your company can do with this equation. For q2, the same logic applies. 100 - q1/2 To find q1's actual number, we assume that q2 will play rationally and follow this formula. That means
we can substitute this formula in for q2 in the q1 problem. This gives us: q1 = 100 - q2/2 q1 = 100 - (100 - q1/2)/2 4q1 = 200 - 100 + q1 3q1 = 100 q1 = 33.3... In other words, q1 and q2 should both
always do 33.3... to get the best outcome for them. The game theory is their choosing of the strategy and the economics is its application. I found it cool how this relation is so clear in this
problem even though they seem so different. Bonus: Here is another fun problem that we learned last week. There is a kingdom ruled by two tyrants: the king and the dragon. They are normally in good
terms with each other until now. The townspeople are split between the two, and they decide that the only fair way to decide a leader is to fight to the death. The dragon suggests a fire breathing
contest and the king suggests a juggling contest, but that wasn't fair. The king then proposed this idea: There are ten numbered wells in our kingdom. Each well has a poison in the water that can
only be treated by the poison in a higher numbered well. If the poison isn't treated in an hour, they are dead. One day, we will both serve each other a glass of water, and then go our separate ways.
After an hour, we will see who is alive. The dragon does not know where well #10 is and the king does, and the king is to smart to not catch a following dragon. However, the dragon accepts the king's
challenge, and after the hour, the king is somehow dead and the dragon is left to rule the kingdom. How did the dragon do it? As usual, I will post the answer in a month. | {"url":"https://coolmathstuff123.blogspot.com/2012/07/","timestamp":"2024-11-03T21:57:48Z","content_type":"text/html","content_length":"122047","record_id":"<urn:uuid:269b2664-9ab1-4313-8e46-4c74c786d4b8>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00513.warc.gz"} |
TOC | Previous | Next | Index
45.2 Two Sample Kolmogorov-Smirnov Test (.NET, C#, CSharp, VB, Visual Basic, F#)
Class TwoSampleKSTest performs a two-sample Kolmogorov-Smirnov test to compare the distributions of values in two data sets. For each potential value x, the Kolmogorov-Smirnov test compares the
proportion of values in the first sample less than x with the proportion of values in the second sample less than x. The null hypothesis is that the two samples have the same continuous distribution.
The alternative hypothesis is that they have different continuous distributions.
Sample data can be passed to the constructor as vectors, numeric columns in a data frame, or arrays of doubles. Thus:
Code Example – C# Kolmogorov-Smirnov test
var ks = new TwoSampleKSTest( data1, data2 );
Code Example – VB Kolmogorov-Smirnov test
Dim KS As New TwoSampleKSTest(Data1, Data2)
By default, a TwoSampleKSTest object performs the Kolmogorov-Smirnov test with Alpha property.
Once you've constructed and configured a TwoSampleKSTest object, you can access the various test results using the provided properties:
Code Example – C# Kolmogorov-Smirnov test
Console.WriteLine( "statistic = " + test.Statistic );
Console.WriteLine( "p-value = " + test.P );
Console.WriteLine( "alpha = " + test.Alpha );
Console.WriteLine( "reject the null hypothesis? " + test.Reject);
Code Example – VB Kolmogorov-Smirnov test
Console.WriteLine("statistic = " & Test.Statistic)
Console.WriteLine("p-value = " & Test.P)
Console.WriteLine("alpha = " & Test.Alpha)
Console.WriteLine("reject the null hypothesis? " & Test.Reject) | {"url":"https://www.centerspace.net/doc/NMath/user/non-parametric-tests-81968.htm","timestamp":"2024-11-10T11:47:54Z","content_type":"text/html","content_length":"14790","record_id":"<urn:uuid:86b5cd44-a453-4cc1-ad5b-226ae0defaf5>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00878.warc.gz"} |
Excel VLOOKUP Function Tutorial - Use VLOOKUP in Excel
Learn how to use VLOOKUP function in Excel to search for and retrieve data from multiple tables or ranges with this tutorial.
Kindly note, this blog post was initially written in German and has been translated for your convenience. Despite my best efforts to maintain accuracy, there might be translation errors. I apologize
for any discrepancies or misunderstandings that may arise from the translation and appreciate any corrections in the comments or via email.
Microsoft Excel offers a myriad of functions for data analysis. Among the most renowned is the VLOOKUP function, which is instrumental in identifying data within tables. This guide simplifies the use
of the VLOOKUP function in Excel, demonstrating how to search for data in tables and introducing alternatives.
Understanding the VLOOKUP Function in Excel
VLOOKUP, or Vertical Lookup, is a frequently used Excel command, particularly useful when dealing with large data sets. This command enables users to locate and extract specific values from a table
or column by specifying a reference cell and a column to search. For instance, if you have a table comprising two columns (Name and Email Address), you can employ the VLOOKUP command to search for a
specific name and retrieve the corresponding email address.
However, remember that VLOOKUP searches exclusively from left to right.
Syntax of the VLOOKUP Formula
The syntax for the VLOOKUP formula is as follows:
=VLOOKUP(lookup_value, table_array, col_index_num, [range_lookup])
• Lookup_value: The value to be searched in the first column of the table array. This value can be text, a logical value, or a number.
• Table_array: The range where the lookup and return values are stored. This range must contain at least two columns.
• Col_index_num: The column in the table array from which the value should be returned. The first column (from the left side) in the table array is represented by col_index_num 1. The next right
column is represented by the column index number 2, and so on.
• Range_lookup: A logical value specifying whether you want an exact or approximate match. TRUE or 1 performs an approximate match. FALSE or 0 performs an exact match. If the range_lookup argument
is omitted, an approximate match is performed.
Practical Examples of the VLOOKUP Function
Here are some examples of using the VLOOKUP function in Excel.
Basic VLOOKUP Function
Letโ s start with a table containing names and email addresses.
Suppose I want to find the email address of โ Cooper Schinacher.โ I can use the VLOOKUP function for this.
=VLOOKUP("Cooper Schinacher",A2:B6,2,FALSE)
I can also format the range A1:B6 as a table (CTRL+T) and use the table as a table_array.
=VLOOKUP("Cooper Schinacher",Table1,2,FALSE)
Dynamic Column Index
Next, I have a table about inventory stock, with Product, Price, Quantity, and Category. I can find out how expensive an apple is by using the VLOOKUP.
But what if Iโ m unsure about the column containing the price? I can make the column index dynamic by using the MATCH function.
=VLOOKUP("Apple",A2:D6,MATCH("Price (EUR)",A1:D1,0),FALSE)
=VLOOKUP("Apple",Table2,MATCH("Price (EUR)",Table2[#Headers],0),FALSE)
Handling Nonexistent Lookup Value
If the lookup value does not exist, VLOOKUP returns the error #N/A.
This can be managed with the IFNA function.
=IFNA(VLOOKUP("Pear",A2:D6,2,FALSE),"Not found")
However, itโ s not ideal now because itโ s unclear whether the product or the column name doesnโ t exist. But I can address this with the IF and ISNA functions.
=IFNA(VLOOKUP($F$2,Table2,MATCH($F$3,Table2[#Headers],0),FALSE),IF(ISNA(MATCH($F$3,Table2[#Headers],0)),"Column not found","Product not found"))
Itโ s even simpler when parts of the function are defined with LET.
IFNA(VLOOKUP(Lookup_value,Table,Col_index,FALSE),IF(ISNA(Col_index),"Column not found","Product not found")))
By the way, you should also make sure that the types of the search criteria and the matrix match. For example, if I search for โ 1โ (text), VLOOKUP will only find a โ 1โ (as text) and not 1 (as
a number). Accordingly, for numbers it is a good idea to use the VALUE function beforehand, to ensure that the types match. This can also be checked with the TYPE formula.
Search Across Multiple Worksheets
Suppose I have an Excel file with several worksheets. On each worksheet, I have a table with products and prices.
Now, I want to summarize the prices for January and February in one table. For this, I use the VLOOKUP function together with the INDIRECT function.
In this way, I dynamically switch the worksheet in the VLOOKUP function.
Approximate Match
In some cases, an approximate match is also needed.
I have a table with products and prices (tbl_Products) and a second table with a quantity discount (tbl_QuantityDiscount).
IFNA(VLOOKUP(Lookup_value,Table,2,FALSE)*Quantity,"Product not found")
However, if the quantity exceeds 10, the price should be discounted using the data from the tbl_QuantityDiscount table.
Here, I need the approximate match because the quantity could be not only 10 but also, for example, 11.
IFNA(VLOOKUP(Lookup_value,tbl_Products,2,FALSE)*Quantity*Discount,"Product not found")
VLOOKUP with Multiple Criteria
In some cases, you need to search for multiple criteria. In this case you have to combine the VLOOKUP function with the CHOOSE formula.
For example, you have a table with data from balance sheets of multiple companies.
/ A B C D
1 Company Year Position Value
2 Company A 2019 Assets 100
3 Company A 2019 Liabilities 50
4 Company A 2020 Assets 120
5 Company A 2020 Liabilities 60
6 Company B 2019 Assets 200
7 Company B 2019 Liabilities 100
8 Company B 2020 Assets 240
9 Company B 2020 Liabilities 120
Now you want to find the value of the assets of Company A in 2020. For this you can use the following formula:
=VLOOKUP("Company A" & "-" & "2020" & "-" & "Assets",CHOOSE({1,2},A2:A9 & "-" & B2:B9 & "-" & C2:C9,D2:D9),2,FALSE)
What happens here? The CHOOSE function creates a new array with the company name, year and position combined in one column and the value in the second column. The VLOOKUP function then searches for
the same combination of the company name, year and position and returns the value (column 2).
Alternatives to VLOOKUP
Searching Downwards
The VLOOKUP function constantly searches to the right. If I want to search horizontally (downwards), I can use the HLOOKUP function.
=HLOOKUP(Lookup_value, Table_array, Row_index_num, [Range_lookup])
INDEX and MATCH
I can use the INDEX and MATCH functions for more flexibility regarding the search direction and column selection.
=INDEX(Table_array,MATCH(Lookup_value, Table_array,[Match_type]),[Col_index_num])
Since 2019, Excel has integrated the XLOOKUP formula, which can replace VLOOKUP, HLOOKUP, Index and Match. The function includes a search function in all directions and integrated error handling.
However, you should consider compatibility issues with older Excel versions and the speed for larger file quantities.
=XLOOKUP(Lookup_value, Lookup_array, Return_array, if_not_found, Match_mode, Search_mode)
In the long run, VLOOKUP will probably be replaced by XLOOKUP because it offers more possibilities and is easier to handle.
A possible problem with VLOOKUP is that it only returns the first match. If I want to calculate the sum of fruit items in stock, SUMIFS can consider and sum all matches.
=SUMIFS(tbl_Products[Quantity in stock],tbl_Products[Category],$F$2)
Combined with the COUNTIFS function, I can also determine the number of matches.
=TEXTJOIN(": ",0,COUNTIFS(tbl_Products[Category],$F2),SUMIFS(tbl_Products[Quantity in stock],tbl_Products[Category],$F2))
If I want to see all matches, I can also use the new FILTER function. In my case, I combine it with the TRANSPOSE function to display the results in a row.
=TRANSPOSE(FILTER(tbl_Products[Product name],tbl_Products[Category]=$F2))
Wrapping Up
While VLOOKUP is a highly popular function in Excel, there are numerous alternative methods that might be better suited depending on the situation. The choice of the right method depends on the type
of data and the specific requirements of the evaluation. Therefore, itโ s worthwhile to experiment with different approaches to find out which is best suited for your needs.
What is the VLOOKUP function in Excel?
The VLOOKUP function in Excel is a command that enables users to locate and extract specific values from a table or column by specifying a reference cell and a column to search.
How do I use VLOOKUP in Excel?
To use the VLOOKUP function in Excel, you need to enter command as follows: =VLOOKUP(lookup value, table array, column index number, [range lookup]). In this command, lookup value is the value you
want to find, table array is the range of cells where you want to search, column index number is the column from which you want to get a return value, and range lookup is optional and used to find an
exact match or approximate match.
Can VLOOKUP return text values?
Absolutely! VLOOKUP can return any type of data including text values. Do ensure though that the lookup value has the exact same text format as the one you're searching for.
How does column index number work in VLOOKUP?
The column index number in VLOOKUP is the column number in your table array from which it will return a value. Note, it starts counting from the first column of your table array. For example, in the
range B2:D10, column B would be 1, column C would be 2, and so on.
What is the difference between VLOOKUP and HLOOKUP?
The VLOOKUP function searches for a value in the first column of a table and returns a value in the same row from another column you specify. The HLOOKUP function searches for a value in the top row
of a table and returns a value in the same column from a row you specify.
How do I use the VLOOKUP function across multiple sheets?
You use VLOOKUP across multiple sheets by specifying the table array argument to be a range in another sheet. For example, the command =VLOOKUP(โ Product Nameโ ,Sheet2!A2:B10,2,FALSE) would look
for 'Product Name' in range A2:B10 on Sheet2 and return the value from the second column of that range. The indirect function can also be used to dynamically switch the worksheet in the VLOOKUP
Why did my VLOOKUP return #N/A?
VLOOKUP returns #N/A when it cannot find the lookup value in the first column of the table array. This can happen if the lookup value is misspelled or if the lookup value is not in the first column
of the table array.
Why do I have to search for the lookup value in the first column of the table array?
VLOOKUP searches for the lookup value in the first column of the table array because it is designed to return a value from a column to the right of the lookup column. If you want to return a value
from a column to the left of the lookup column, you can use the INDEX and MATCH functions.
Why is my VLOOKUP not working?
There could be a few reasons. Most commonly, it stems from the lookup value not being in the first column of the table array, the [range lookup] argument not being set to false for an exact match, or
the column index number being incorrect. Also, be aware that VLOOKUP does not work with columns to the left of the lookup column. There could also be a problem with the data types of the search
criteria and the matrix. | {"url":"https://deployn.de/en/blog/excel-sverweis/","timestamp":"2024-11-04T09:08:25Z","content_type":"text/html","content_length":"80675","record_id":"<urn:uuid:37738ae8-28b8-4e59-8e43-acacb165d9f3>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00067.warc.gz"} |
Creation Science
An Oracle Restored
Some observations on a remarkable discovery
2. The contiguous multiples of 37
8. A comparison with the Genesis 1:1 data
Appendix 2: The numerical interpretation of Hebrew words
Appendix 3: The numerical interpretation of Greek words
One of the principal reasons for the ‘Mt.Sinai encounter’, recorded in Exodus 25-31, was to instruct Moses as to the precise details of construction of the portable sanctuary that would function as
God’s dwelling place on earth – the tabernacle – together with its furnishings, and manner of use.
Concerning the high-priestly vestments, we read particularly of the oracle – the Urim and Thummim (Ex.28:30) – provided for the guidance of the people in difficult and uncertain times. Details are
lacking of the nature and use of these items^(1), but we are informed that they were held in a pouch – called the breastplate (or, in some translations, breastpiece) – attached to the front of the
ephod – the outmost garment of the high priest. This breastplate was formed from a single piece of highly-embroidered linen cloth one cubit long and half a cubit wide, folded over in two to form a
square, half a cubit by half a cubit (about 9in.x 9in). It was adorned with twelve precious stones on which were engraved the names of the tribes – ie those of the sons and grandsons of Jacob
arranged according to their order of birth. These were set out in four rows of three stones each (Ex.28:15-30) [Appendix 1].
[Link to website on the Tabernacle]
In Ian Mallett’s paper The Breastplate of Judgment^(2), attention is drawn to the many interesting features attending the matrix of integers formed from the characteristic values (hereafter “CVs”)
[Appendix 2] of the breastplate names:
2. The contiguous multiples of 37
The following attributes, predicated upon 37^(3) and its multiples, may be readily ascertained:
1. The value of the matrix (i.e. the sum of its components) is 3700, or 10^2×37^(4)
2. Of these, the 6 occupying ‘odd’ positions total 1850, or 50×37, and the 6 occupying ‘even’ positions, the same
3. The first 9 form a square; they total 2812, or 76×37
4. The remainder – forming the final row of 3 – total 888, or 24×37
5. The first value is 259, or 7×37
6. The remaining values in the square may be grouped into 4 connected pairs – each representing a multiple of 37, thus:
At (a), we have the breastplate matrix and observe that it may be divided into 6 regions, here designated I, II, III, IV, V and VI – as at (b) – such that the sum of the name CVs within each region
is a multiple of the uniquely-symmetrical number 37 – as represented at (c) – the corresponding multipliers of 37 being depicted at (d).
Remarkably, the 6 segments of the breastplate, defined above – each a multiple of 37 – may be combined to generate the first nine multiples of the interesting number 296 (= 8.37). The details are
presented in the following set of miniatures:
Figure 2. The multiples of 296 as combinations
of these segments
This fact is highly significant because of the associations 296 has with earlier discoveries in the biblical text: it is the characteristic value of the 7th and final Hebrew word of Gen.1:1 (meaning
‘the earth’), and is a factor of the Greek form of both ‘Jesus’ (= 888, or 3.296) and ‘Christ’ (= 1480, or 5.296). Further, it represents the difference between the cubes of 6 and 8, and is a factor
of 1184 – one of a unique pair of ‘friendly’ numbers.
A computer simulation reveals that only 1% of random sets of 6 multiples of 37, over the same range, will be found to possess this feature.
We read in John’s Gospel, “He was in the world, and the world was made by him, and the world knew him not.” (ch.1, v.10). Remarkably, the breastplate is imprinted with our Creator’s ‘signature’ – the
bottom row (value 888) conveying the encoded name, ‘Jesus’, and the connected block above (value 1480), the title, ‘Christ’, thus:
And we further observe powerful symbolism in the fact that the ratio, name:title, represented numerically by 888:1480, or 3:5, is identical with that of the sides of the mercy seat (Ex.25:17-22) –
central element in tabernacle worship! [Details concerning the derivation of the characteristic values of the Creator’s name and title are given in Appendix 3.]
The foregoing facts lead directly to a pythagorean view of the breastplate – one in which the Lord’s Name plays an essential role.
As we have just seen, the breastplate matrix (value 100×37) may be divided into two blocks, thus:
Here, it may be observed that the square roots of the multipliers of 37, {6,8,10} form a pythagorean triple that is a simple multiple of the classic case, {3,4,5}! Four such triangles are generated
when a square of side 10 units is rotated by 36.87° (nearly 37°!) within a centred square of side 14 units, thus:
The sum of these squares is 296 – a number that has already been shown to be an essential feature of the breastplate.
It has already been observed that the sum of the odd-numbered names is precisely one half of the total, and therefore equal to the sum of the even numbered names. However, as revealed below, there is
another arrangement which halves the structure numerically, and two which divide it in the ratio 1/4:3/4, thus:
Further, in 925, we have the value of the Creator’s name and title, as derived from the English analogue of the Hebrew alphabetic numbering scheme^(5). A computer simulation reveals that about 5% of
equivalent random sets exhibit these combined properties.
We observe that 7 of the 12 integers in the array exhibit interesting relationships of the form,
Multiple (M) / Divisor (D) = Quotient (Q)
259 (the 1st) / 7 (the 6th) = 37
570 (the 5th) / 30 (the 3rd) = 19
162 (the 10th) / 54 (the 4th) = 3
570 (the 5th) / 95 (the 9th) = 6
The order of appearance of these elements in the matrix is as follows:
i.e. M D D M D D M
and we observe symmetry in the types represented, and between types, thus:
Further, an examination of the four quotients (Q) – 19, 3, 37, and 6 – reveals them to be significant and related figurate ^(6) numbers, thus:
8. A comparison with the Genesis 1:1 data^(7)
There are a number of points of contact between the numerical features of the breastplate, as outlined above, and those of the Hebrew Bible’s first verse:
• both reveal 37 to be the dominant factor (a feature heralded by their totals: Gen.1:1 = 2701 = 37 x 73; breastplate matrix = 3700 = 37 x 100;
• the total of Gen.1:1 divides thus: 2701 = 999 + 999 + (999 – 296); the breastplate total may be written, 3700 = 999 + 999 + 999 + (999 – 296), or 2701 + 999; indeed, as the following figure
reveals, this combination is actually present (as are the many multiples of 296, as we have already seen):
• the factors of 2701, viz 37 and 73, and of 703 (i.e. 999 – 296), viz 19 and 37, are richly figurate; the same structures are implied by the breastplate data as Figure 8 reveals (observe that 6 x
6-as-triangle symmetrically disposed about 37-as-hexagon yields a hexagram of 73);
• the multipliers of 37 in both Gen.1:1 and Creator’s name (ie 73 and 64, respectively) are themselves related to 37: 73, by digit reversal, and 64, by the cube suggested by the hexagonal form of
Clearly, they are the work of one supreme author!
As has been demonstrated in an earlier paper ^(8), 2368 and 2701 – symbols of the Creator and Creation, respectively – are also objects of considerable significance per se in the field of numerical
geometry^(9). Both exhibit compound symmetries which take the form of two-dimensional arrangements of uniform three-dimensional elements^(10)
Here we observe 37 cubes – each of 64 units^(11) – set out as a regular hexagram. Remarkably, the figure is harmonised by the fact that these cubes are represented in two-dimensions by numerical
hexagons – each of 37 units^(3)! The total represented is 37×64, or 2368 – the characteristic value (CV) of the Lord (Appendix 3).
In the next diagram, 2701, or 37×73 – the CV of Gen.1:1 – is depicted as a hexagram of 73 gnomons – each of 37 units^(3).
Figure 10 may now be centred and superimposed on Figure 11, thus:
The ‘halo’ of 36 visible gnomons (blue) embodies 36×37, or 1332 units, and the total represented by the whole is thus 1332+2368, or 3700 units – the sum of the 12 breastplate names! However, it
should not go unnoticed that the components of the total are to be found in a principal division of the breastplate matrix [Figure 2], and therefore participate in the pythagorean connection noted in
Section 2! Further, the 24 gnomon elements underlying the outline of the central hexagram represent 24×37, or 888 units – the CV of ‘Jesus’ and bottom row of the matrix!
Alternatively, the last diagram may be perceived as the augmentation of Figure 11 by a hexagram of 37 smaller cubes – each of value 27 (i.e. 3^3). Clearly, the value represented by this hexagram
would be 27×37, or 999. We observe that both 999 and 2701 are present in the breastplate – and form a significant division of it [Figure 9].
The fact that these constructions are hybrid – incorporating both two- and three-dimensional elements – is itself symbolic: it mirrors the dual nature of Jesus who was both perfect man and God!
Referring again to Figure 2 we observe that the division of the figure into two groups of 5 and 7 tiles, respectively, establishes further links with the Gen.1:1 phenomena. Thus we find that 37 is
the arithmetic mean of 25 (= 5^2) and 49 (= 7^2) and, again, the centroid element^(12) of the 73rd numerical triangle (an alternative representation of Gen.1:1^(8)) is found to occupy the 25th
position in the 49th row!
The phrase, I am Alpha and Omega…, occurs three times in the text of the Bible’s last book (Rev.1:8, 21:6, 22:13) – its final appearance being followed by the words, …the beginning and the end, the
first and the last. It is the Lord Jesus Christ who is making the amazing claim that all things are from him and for him!^(13) In the original Greek this significant phrase is rather peculiarly
expressed each time it appears, thus: whereas the first letter of the alphabet, Alpha, is given by name (‘^(14) The matter appears to have been designed to attract the attention of the careful
reader. But to what purpose?
Consulting Appendix 3, we observe the following numerical implications of this arrangement:
The more obvious expressions of the phrase would have involved either the names, or the symbols, of both letters, with the following numerical implications:
CV(A) = 1; CV(
and the final possibility of symbol followed by name:
CV(A) = 1; CV(
Clearly, the only arrangement to yield a multiple of 37 is precisely that found in the text! – and we observe that the particular multiple, 1332 – representing ‘Alpha and Omega’ – is that which not
only accompanies 2368 (‘The Lord’) in the breastplate matrix [Figure 2] but also functions as the outline hexagram (‘the halo’) in the representation of ‘The Lord of Creation’ [Figure 11(c)]!!
Again, in the context of the first verse of the Bible (Gen.1:1) – that ‘treasure-trove’ of numerical geometry ^(7)^(8) – we find a similar association: the central Hebrew word is formed from the
first and the last letters of the alphabet; immediately preceding it is ‘Elohim’, meaning ‘God’ – the Creator!
It seems abundantly obvious, therefore, that He who created all things and designed the breastplate is also the One who inspired the writing of the Book of Revelation!
Behind the original jewels of the breastplate lay the Urim and Thummim – those mysterious instruments ordained by God for the guidance of his people. They, along with the breastplate and ephod were
lost during the Babylonian captivity. However, in essence, they live on in the miracle of the breastplate matrix! Here, indeed, is an oracle for today! Here is tangible and compelling evidence of
God’s Being and Sovereignty, and a guarantee of biblical truth! Those whose beliefs and actions are guided by reason now have an opportunity to grasp these fundamental realities, recognising that in
these days God is drawing our attention to hitherto-unnoticed designs which authenticate scripture beyond reasonable doubt.
To quote the writer of The Breastplate of Judgment: “What mind could conceive such an amazing array of mathematical phenomena save the Creator himself, the Wonderful Numberer, Jesus Christ, who is
the Messiah of Israel and the Saviour of the world?”
Vernon Jenkins MSc
email: vernon.jenkins@virgin.net
Link to Double Indemnity
This page was last modified 2006-03-11.
(1) The Urim and Thummim were clearly more than the equivalent of a pair of dice (as some have contended) for God did not always provide an answer (1Sam.14:36-37, 28:6). The response came either by a
voice from heaven or by an impulse upon the mind of the high priest. This oracle was of great use to Israel (e.g. Nu.27:21, 1Sam.23:6-12).
(2) Obtainable from PALMONI RESEARCH, 4 Tynesdale, Whitby, Ellesmere Port, CH65 6RB, U.K.
(3) As an integer, thirty-seven has unique geometrical properties: 37 uniform squares or circles (as appropriate) can be arranged to fill any one of three symmetrical frames – octagon, hexagon, or
hexagram; in hexagon form, it represents a typical 2D view of a cube of dimension 4, i.e. a stack of 64 unit cubes; it is also the difference between the cubes of 4 and 3. These features are
illustrated in the following diagrams:
Clearly, 37 is associated with 16 axes of symmetry, and in this sense it is the most symmetrical of all numbers. But, in addition, as a denary object, it provides a basis for many interesting
mathematical recreations (see The Ultimate Assertion: Evidence of Supernatural Design in the Divine Prologue, CEN Tech.J., vol.7(2), 1993, Appendix, pp.192-196).
(4) This appears significant since the breastplate was itself a square, and 10 is a highly significant number: a prominent feature of human anatomy, principal radix of man’s number systems from the
beginning, and collective unit in the now near universal principles of decimalisation and metrication.
(5) In Appendix 2, an outline is provided of the historically-attested Hebrew alphabetic numbering scheme. If the Roman alphabet is superimposed on this, and the values 500, 600, 700, and 800,
assigned to the extra four letters W, X, Y, and Z, we then have a modern equivalent of this ancient scheme. Under this regime, the name ‘JESUS’ would assume the value 515 (i.e. 10+5+100+300+100), and
the title ‘CHRIST’, 410 (i.e. 3+8+90+9+100+200); ‘JESUS CHRIST’ would therefore become 925, or 5^2×37.
(6) In the context of this page a figurate number is one which, when represented as a set of uniform circular or spherical counters, completely fills a polygonal or polyhedral frame. Examples are
illustrated above.
(7) Such data are presented in the The Beginning of Wonders and in a number of printed documents, including The Ultimate Assertion: Evidence of Supernatural Design in the Divine Prologue. Please
email the author for details.
(8) See for example The Arbiters of Truth
(9) Numerical geometry describes the study of those two- and three-dimensional structures that involve both number and form, i.e. the figurate numbers. These lie close to the heart of mathematics,
and the symmetries represented are absolute in the sense that they are independent of radix, of time, and of place.
(10) Here, the 4th solid gnomon (= 37, difference between the 4th and the 3rd cubes) and the 4th cube (= 64 units) are symbolically represented by
(11) The ‘units’ referred to in this account are the unit cubes which function as counters in the construction of the diagrams.
(12) One in every three numerical triangles is built around a single counter which then functions as the centroid element – i.e. that which is equidistant from each of the three sides. Such a
triangle is the 73rd.
(13) See also Colossians 1:16
(14) The writer is indebted to Captain Richard Prendergast for drawing his attention to this interesting anomaly.
Appendix 1 – The Twelve Tribes
Jacob fathered 12 sons by four women: his wives, Leah and Rachel; and his concubines – maidservants of Leah and Rachel – and surrogate mothers, Zilpah and Bilhah (Gen.29:31-35; 30:1-24; 35:16-18).
The following table lists the names of the sons, in order of birth, with their respective mothers.
1. REUBEN Leah
2. SIMEON Leah
3. LEVI Leah
4. JUDAH Leah
5. DAN Rachel/Bilhah
6. NAPHTALI Rachel/Bilhah
7. GAD Leah/Zilpah
8. ASHER Leah/Zilpah
9. ISSACHAR Leah
10. ZEBULUN Leah
11. JOSEPH Rachel
12. BENJAMIN Rachel
Jacob’s favorite son, Joseph, sold into slavery through the treachery of his elder brothers, ultimately became Pharaoh’s ‘right-hand man’, married an Egyptian, and fathered two sons, MANASSEH and
EPHRAIM (Gen.41:50-52). These two particular grandsons of Jacob were destined to become proxies for their father in the above list. The tribes – now numbering 13, and each identified by the name of
its progenitor – remained in Egypt for some 400 years. Following the Exodus, and before entering the ‘promised land’, a significant event took place: at God’s command, the sons of Levi were set apart
and dedicated to His service; in due course, they would therefore not feature in the apportionment of the land between the tribes (Deut.10:8-9). Accordingly, we deduce that the name of Levi would not
appear on the breastplate, for the high priest who bore it would himself have been of that tribe; again, as we have seen, Joseph would have been represented by his two sons. A reading of Nu.1
confirms these facts.
Many centuries after these events occurred, it became the practice to use Hebrew letters as numerals. All written words and names have since become fairly interpretable as numbers. Details of the
Hebrew alphabetic numbering scheme are given in Appendix 2. The names of the tribes of Israel (assuming it is these that were engraved on the jewels of the breastplate), in progenitor birth order,
and with their full numerical interpretations, are listed in the table below along with their Strong’s reference number bracketed. [Note: to verify these Hebrew spellings you may access
www.blueletterbible.org using the relevant reference number.]
* In contrast to the other 11, the Hebrew rendering of the name Zebulun occurs in different ways – as detailed below:
Observe that the column headed ‘Freq(uency)’ records the number of times each variation occurs in the Old Testament text; and that headed ‘CV’, the numerical values to be associated with these. For
the purposes of the current exercise, it is clear that latter may be read either as 101 or 95. However, the latter is preferred here because, (a) it relates to the most frequent of the variations,
(b) it is this value that raises the total of the ‘Tribes of Israel’ to a significant multiple of 37, and the one which establishes many of the internal breastplate characteristics.
It is worth noting that this layout corresponds with the engravings on the two onyx stones mounted on the shoulderpieces of the ephod (Ex.28:9-12).
The following diagrams present a summary of the matters discussed above.
The breastplate is here represented as a tiled rectangle – the tiles being numbered from right to left, in the Hebrew manner – with the omitted names, Levi and Joseph, shown in their proper
positions, by order of birth.
Here, finally, is the breastplate prepared for analysis as a 4 x 3 matrix of name CVs:
Appendix 2 – The Hebrew Alphabetic Numbering Scheme
The Hebrew alphabet has 22 letters – five with ‘end forms’, i.e. variants used only when words end with one or other of these letters. From circa 200 BC, as the following table reveals, each letter
was made to function as a numeral – thus copying the earlier Greek model (c 600 BC).
The practice that existed then was to record numbers on an additive basis, i.e. the value represented by a string of letters was simply the sum of the tabular values assigned to each. The
characteristic value (CV) of a conventional Hebrew word, name, or phrase, is obtained in this manner. As an illustration of the procedure, the characteristic value of the name SIMEON is derived
below. We observe that all Hebrew reading proceeds from right to left.
Thus, CV (SIMEON) = 50 + 6 + 70 + 40 + 300 = 466
Appendix 3 – The numerical interpretation of Greek words
The Greek alphabet is an ordered set of 24 upper/lowercase pairs of characters. The position and numerical value of each letter of each of these pairs is detailed below:
This scheme was introduced circa 600 BC for the purpose of recording numbers on an additive basis, the missing values, 6 and 90, being represented by non-alphabetic symbols. Thus, every string of
letters was potentially a number – interpreted by summing the tabular values of the letters.
As a unique example of this procedure, the characteristic values (CVs) of the Lord’s Name and Title are evaluated below:
Observe that the letter values appear above and their respective sums below. | {"url":"https://www.creation.xtn.co/zzz/other-bible-code/oracle/","timestamp":"2024-11-02T08:36:28Z","content_type":"application/xhtml+xml","content_length":"226941","record_id":"<urn:uuid:09e8553d-b38c-48fd-b80a-7082af1ceb64>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00209.warc.gz"} |
Estimating glomerular filtration rate in a population-based study
Background: Glomerular filtration rate (GFR)-estimating equations are used to determine the prevalence of chronic kidney disease (CKD) in population-based studies. However, it has been suggested that
since the commonly used GFR equations were originally developed from samples of patients with CKD, they underestimate GFR in healthy populations. Few studies have made side-by-side comparisons of the
effect of various estimating equations on the prevalence estimates of CKD in a general population sample. Patients and methods: We examined a population-based sample comprising adults from Wisconsin
(age, 43-86 years; 56% women). We compared the prevalence of CKD, defined as a GFR of <60 mL/min per 1.73 m^2 estimated from serum creatinine, by applying various commonly used equations including
the modification of diet in renal disease (MDRD) equation, Cockcroft-Gault (CG) equation, and the Mayo equation. We compared the performance of these equations against the CKD definition of cystatin
C >1.23 mg/L. Results: We found that the prevalence of CKD varied widely among different GFR equations. Although the prevalence of CKD was 17.2% with the MDRD equation and 16.5% with the CG equation,
it was only 4.8% with the Mayo equation. Only 24% of those identified to have GFR in the range of 50-59 mL/min per 1.73 m^2 by the MDRD equation had cystatin C levels >1.23 mg/L; their mean cystatin
C level was only 1 mg/L (interquartile range, 0.9-1.2 mg/L). This finding was similar for the CG equation. For the Mayo equation, 62.8% of those patients with GFR in the range of 50-59 mL/min per
1.73 m^2 had cystatin C levels >1.23 mg/L; their mean cystatin C level was 1.3 mg/L (interquartile range, 1.2-1.5 mg/L). The MDRD and CG equations showed a false-positive rate of >10%. Discussion: We
found that the MDRD and CG equations, the current standard to estimate GFR, appeared to overestimate the prevalence of CKD in a general population sample.
• Chronic kidney disease
• Cockcroft-gault equation
• Glomerular filtration rate
• MDRD equation
• Mayo equation
Dive into the research topics of 'Estimating glomerular filtration rate in a population-based study'. Together they form a unique fingerprint. | {"url":"https://experts.umn.edu/en/publications/estimating-glomerular-filtration-rate-in-a-population-based-study","timestamp":"2024-11-11T02:23:17Z","content_type":"text/html","content_length":"58719","record_id":"<urn:uuid:5f19c1de-399e-421e-a29a-6d68bc87e9b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00232.warc.gz"} |
Grid Paper - The Graph Paper
Graph Paper is a paper that isΒ contrasting with regular paper and is also used for various purposes of mathematics, engineering, etc. Everyone knows it by a different-different name like Polar, grid
paper, Isometric, etc. So, ahead we share the templates, it would be good for you to understand the sizes of the templates.
• Letter Paper Size
• A4
• 11Γ 17 Paper Size
• Legal Paper Size
• A3 Paper Size
• A2
• Poster
Related Article:
5 MM Graph Paper
5 MM Paper Template recognized as standard Cartesian with the 5 MM paper being exceptionally focused primarily around engineering work on the whole. 5 MM Paper has been found continuous in green
shading. You can print it as it can be printed thoroughly.
Grid Paper
Grid paper is simply a page covered with continuous square grids. Lines are often used as guides for plotting functions or graphs of experimental data and drawing curves. These lines are used to help
us accurately draw graphs.
These papers are used for student graphing assignments where the teacher gives students the responsibility of determining the scale and drawing the axes. It is also used for creating floor plans,
planning construction projects, and many other purposes.
Graph Paper Template
Generally, the prearranged grid is found in the paper which means paper comes in the form of grids or grids that can arrange.
A paper is especially suited for estimating millimeters and centimeters in the subjects of science and mathematics. Millimeter paper and graphing paper are additionally part of grid paper.
1/4 Inch Graph Paper
1/4 Inch Paper is also known as quartile paper because it consists of 4 boxes that are separated into the part to make the size of one inch.
Just the Number of boxes makes 1/4 Inch Grid Paper, not like other papers. You can print 1/4 Inch Paper by using a 1/4 Inch Paper template.
Printable Graph Paper A4
Graph A4 paper is the A4 type paper used for A4 size paper, Graph A4 paper can be printed, viewed, and downloaded.
Graph A4 paper appropriated for:
• Centimeter Paper
• 5mm Paper
• 1/4β ³ Inch Grid Paper
• Dot Paper
10 Square Per Inch Graph Paper
Drawing paper with inches is an excellent approach for making sense of novel diagramming work, 10 Square Per Inch Grid Papers are the best for Drawing diagram paper in inches.
It is best to draw diagram paper in inches to divide it into 10squares which takes up space to 1 inch. 10squares per inch always gives you a nice even number to work with that is both precise and
Dot Paper
This type of paper is notable paper because it doesnβ t manage any lines. It only shows dots in the Paper, which is the explanation that is called Dot Paper.
The non-specific dot of paper is the primary concern of paper. This means that people will not like to write on paper. Dot paper best fits a designer, sports is the best concern of dot paper.
Centimeter Graph Paper
Centimeter Paper is a type of paper that is not very different from others but only has a 1-cm gap. Centimeter Paper tags along with the size of the box and which is just 1 cm.
1 CM Paper manages cm graphs that’s why this type of paper is beneficial for those who need to work in Centimeters. This key tool is important for all classes. In this paper, 1cm x 1cm squares across
an 8.5″ x 11″ sheet.
1/2 Inch Graph Paper
A half-inch grid paper can act as a two-dimensional ruler. This paper was designed for Letter size paper in portrait orientation. You can draw graphs of different types and sizes with the help of
Half-inch paper that provides you a printed paper with small squares of different sizes. These printable papers are used to plot the data given in the graphs or draw curves.
1/2 inch paper can work according to two-dimensional ruler paper, you can print 1/2 inch paper in size whichever you like best. It’s used with bank and stamp games and other arithmetic exercises. It
is the perfect worksheet for teachers and students.
1 Inch Graph Paper
It is straightforward that 1 Inch Grid Paper is a 1-inch paper in the form of 1 inch. Its boxes are made of 1 inch.
1 Inch Paper can be found in the huge size of paper which is useful in training children because1 Inch Paper is huge.
Printable Graph Paper with Axis
A Paper with an axis is a diverse variety of papers known for mathematical papers that require individuals to use it for mathematics.
Mathematics is treated in the same way as Cartesian coordinates paper. Pre-drawn X and Y-pivot with the axis is the specialty of this paper.
Printable Graph Paper Full Page
Printable Paper Full Page is the paper that is the full size of the diagram paper, with no limit of inches.
Printable Paper Full Page comes in the paper that are:
• Plotting fabricates a sheet of Grid paper.
• A Paper printable math chart paper.
• Printable framework paper has six styles of quadrille paper.
Isometric Graph Paper
Lines of 3-Dimension are the fundamental objective of Isometric Paper. The isometric paper has vertical lines and skew lines drawn at 30Β° edges. It is great for creating art and drawing diagrams of
Log Graph Paper
The type of mathematics that states aΒ given number x is the guide from which another fixed number, the base b, must be constructed to express the number x, that type of science is called the
Log Papers are based on a logarithm that is valuable in a situation like a logarithm.
Polar Graph Paper
Polar graphing is the primary center for route purposes, for example, airlines or ships. Polar graphing can be seen in vertical and horizontal lines. The angles and the distance from the specific
focused point is the primary concern of Polar Paper, there are many types of templates regarding Polar Paper that are used for best practices.
Coordinate Graph Paper
A coordinate grid paper is a two-dimensional plane formed by the intersection of a vertical line called the y-axis and a horizontal line called the x-axis. These are perpendicular lines that
intersect each other at zero, and this point is known as the origin. These axes divide the coordinate plane into four equal sections, and each section is called a quadrant.
Coordinate Paper is an alternate type of paper that can be valuable for singles only for quadrant paper. Coordinate Paper follows a single grid per page to four per page.
Blank Graph Paper
A blank paper is a blank sheet of paper that is converted into a print or non-print structure. Blank Paper is used by those people who need to use simple and easy paper. Blank Grid Paper is also used
in drawing images and writing words. | {"url":"https://thegraphpaper.com/tag/grid-paper/","timestamp":"2024-11-03T10:07:58Z","content_type":"text/html","content_length":"52449","record_id":"<urn:uuid:d880f780-5bef-4401-8d50-60364394bc2f>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00604.warc.gz"} |
January 7, 2022
what fraction of a clockwise revolution does the hour hand of a clock turn through when it goes from 3 ...
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January 7, 2022
draw graph of y= e^[x] [.] is greatest integer function
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January 6, 2022
In what real-life situation is the object’s perimeter important? Explain why knowing the perimeter in this situation helps.
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January 6, 2022
Comparing Quantities chapter in Ncert is which chapter in RS Agarwal
Read More
January 6, 2022
(6 x 1 = 6 marks3. Answer the following questions.a) Calculate the value of a in Fig. 5.
Read More | {"url":"https://wiki-helper.com/author/aaliyah/","timestamp":"2024-11-13T16:05:29Z","content_type":"text/html","content_length":"112982","record_id":"<urn:uuid:d3c33799-4184-4f22-84df-c58779eb5576>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00011.warc.gz"} |
seminars - Deviation estimates for random walks and acylindrically hyperbolic groups
We will consider a class of groups that includes non-elementary (relatively) hyperbolic groups, mapping class groups, many cubulated groups and C'(1/6) small cancellation groups. Their common feature
is to admit an acylindrical action on some Gromov-hyperbolic space and a collection of quasi-geodesics compatible with such action.
As it turns out, random walks (generated by measures with exponential tail) on such groups tend to stay close to geodesics in the Cayley graph in the following sense: The probability that a given
point on a random path is further away than L from a geodesic connecting the endpoints of the path decays exponentially fast in L.
This kind of estimate has applications to the rate of escape of random walks (Lipschitz continuity in the measure) and its variance (linear upper bound in the length).
Joint work with Pierre Mathieu. | {"url":"https://www.math.snu.ac.kr/board/index.php?mid=seminars&page=90&sort_index=speaker&order_type=desc&l=en&document_srl=535275","timestamp":"2024-11-11T12:42:06Z","content_type":"text/html","content_length":"48294","record_id":"<urn:uuid:7825a2dd-c373-44ac-99c8-be05b7892a04>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00058.warc.gz"} |
Performance Evaluation of Computer Systems
Performance Evaluation of Computer Systems. Instructor: Prof. Krishna Moorthy Sivalingam, Department of Computer Science and Engineering, IIT Madras. The objective of this course is to understand the
fundamental concepts of computer system performance evaluation. Topics covered in this course will include introduction to mathematical modeling techniques (Markov Chains, Queuing Theory and Networks
of Queues), discrete event simulation modeling, experimental design, workload characterization, measurement of performance metrics, analysis and presentation of results. (from nptel.ac.in)
Performance Evaluation of Computer Systems
Instructor: Prof. Krishna Moorthy Sivalingam, Department of Computer Science and Engineering, IIT Madras. The objective of this course is to understand the fundamental concepts of computer system
performance evaluation. | {"url":"http://www.infocobuild.com/education/audio-video-courses/computer-science/performance-evaluation-of-computer-systems-iit-madras.html","timestamp":"2024-11-06T20:13:59Z","content_type":"text/html","content_length":"14060","record_id":"<urn:uuid:39d09e55-e1d6-4a8d-8c76-87adaf396244>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00153.warc.gz"} |
1 — 08:30 — Risk Budgeting Allocation for Dynamic Risk Measures
We define and develop an approach for risk budgeting allocation - a risk diversification portfolio strategy - where risk is measured using a dynamic time-consistent risk measure. Specifically, a
dynamic risk budgeting strategy is a portfolio allocation where at every trading point each asset contributes a predefined percentage to the future risk of the portfolio. For this, we introduce a
notion of dynamic risk contributions that generalise the classical Euler contributions and which allow us to obtain dynamic risk contributions in a recursive manner. We prove that, for the class of
dynamic coherent distortion risk measures, the risk allocation problem may be recast as a sequence of strictly convex optimisation problems. Moreover, we show that any self-financing dynamic risk
budgeting strategy with initial wealth of 1 is a scaled version of the unique solution of the sequence of convex optimisation problems. Furthermore, we develop an actor-critic approach, leveraging
the notion of conditional elicitability of dynamic risk measures, to solve for risk budgeting strategies using deep learning. We illustrate the methodology on a simulated market model, with a dynamic
risk measures constructed through conditional Expected Shortfalls, discuss that the dynamic risk budgeting strategy is non-Markovian and how the agent invests.
2 — 09:00 — Nonparametric Inverse Optimization: Theory and Applications
Inverse optimization is a powerful tool for understanding decision-making processes by inferring the objective functions behind observed decisions. Traditional approaches rely heavily on parametric
models, which assume the objective function's structure can be precisely known. This assumption can be impractical and prone to model mis-specification in complex real-world scenarios. Addressing
this challenge, we introduce a novel nonparametric framework for inverse optimization. Our approach stands out by its flexibility to model a large class of objective functions without predefined
parametric structures and that it can be implemented in data-efficient fashion. We prove the computational tractability of the nonparametric models and demonstrate their advantages via numerical
3 — 09:30 — Preference Ambiguity and Robustness in Multistage Decision Making
In this work, we consider a multistage expected utility maximization problem where the decision maker's utility function at each stage depends on historical data and the information on the true
utility function is incomplete. To mitigate {adverse impact} arising from ambiguity of the true utility, we propose a maximin robust model where the optimal policy is based on the worst-case sequence
of utility functions from an ambiguity set constructed with partially available information about the decision maker's preferences. We then show that the multistage maximin problem is time consistent
when the utility functions are historical-path-dependent and demonstrate with a counter example that the time consistency may not be retained when the utility functions are
historical-path-independent. With the time consistency, we show the maximin problem can be solved by a recursive formula whereby a one-stage maximin problem is solved at each stage beginning from the
last stage. Moreover, we propose two approaches to construct the ambiguity set: a pairwise comparison approach and a zeta-ball approach where a ball of utility functions centered at a nominal utility
function under zeta-metric is considered. To overcome the difficulty arising from solving the infinite dimensional optimization problem in computation of the worst-case expected utility value, we
propose piecewise linear approximation of the utility functions and derive error bound for the approximation under moderate conditions. Finally, we use the stochastic dual dynamic programming (SDDP)
method and the nested Benders' decomposition method to solve the multistage historical-path-dependent preference robust problem and the scenario tree method to solve the historical-path-independent
problem, and carry out comparative analysis on the efficiency of the computational schemes as well as out-of-sample performances of the historical-path-dependent and historical-path-independent
models. The preliminary results show that the historical-path-dependent preference robust model solved by SDDP algorithm displays overall superiority. This is a joint work with Jia Liu and Zhiping
Chen in Xi'an Jiaotong University, China.
4 — 10:00 — Multi-attribute Preference Robust Optimization with Quasi-Conave Choice Functions
In behavioural economics, a decision maker’s preferences are expressed by choice functions. Preference robust optimization (PRO) is concerned with problems where the true choice function which
represents the decision maker’s preferences is ambiguous, and the optimal decision is based on the worst-case choice function from a set of plausible choice functions constructed with elicited
preference information. In this paper, we propose a PRO model to support choice functions that are: (i) monotonic (prefer more to less), (ii) quasi-concave (prefer diversification), and (iii)
multi-attribute (have multiple objectives/criteria). As a main result, we show that the robust choice function can be constructed efficiently solved by solving a sequence of linear programming
problems. In decision making with worst-case choice function, we demonstrate how the maximin problem can be efficiently solved by a sequence of convex optimization problems. To examine the the
behavior and scalability of the proposed model and computational schemes, we apply them to a portfolio optimization problem and a capital allocation problem and report the numerical results. | {"url":"https://ismp2024.gerad.ca/schedule/ThA/101","timestamp":"2024-11-03T15:36:43Z","content_type":"text/html","content_length":"21443","record_id":"<urn:uuid:437449db-6ae4-4f08-8b72-f5cffb331ebd>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00093.warc.gz"} |
Tutorial 05: Phase-field simulation of spinodal decomposition
from pystencils.session import *
Tutorial 05: Phase-field simulation of spinodal decomposition¶
In this series of demos, we show how to implement simple phase field models using finite differences. We implement examples from the book Programming Phase-Field Modelling by S. Bulent Biner.
Specifically, the model for spinodal decomposition implemented in this notebook can be found in Section 4.4 of the book.
First we create a DataHandling instance, that manages the numpy arrays and their corresponding symbolic sympy fields. We create two arrays, one for the concentration \(c\) and one for the chemical
potential \(\mu\), on a 2D periodic domain.
dh = ps.create_data_handling(domain_size=(256, 256), periodicity=True)
μ_field = dh.add_array('mu', latex_name='μ')
c_field = dh.add_array('c')
In the next cell we build up the free energy density, consisting of a bulk and an interface component. The bulk free energy is minimal in regions where only either phase 0 or phase 1 is present.
Areas of mixture are penalized. The interfacial free energy penalized regions where the gradient of the phase field is large, i.e. it tends to smear out the interface. The strength of these
counteracting contributions is balanced by the parameters \(A\) for the bulk- and \(\kappa\) for the interface part. The ratio of these parameters determines the interface width.
κ, A = sp.symbols("κ A")
c = c_field.center
μ = μ_field.center
def f(c):
return A * c**2 * (1-c)**2
bulk_free_energy_density = f(c)
grad_sq = sum(ps.fd.diff(c, i)**2 for i in range(dh.dim))
interfacial_free_energy_density = κ/2 * grad_sq
free_energy_density = bulk_free_energy_density + interfacial_free_energy_density
$\displaystyle {c}_{(0,0)}^{2} A \left(1 - {c}_{(0,0)}\right)^{2} + \frac{κ \left({\partial_{0} {c}_{(0,0)}}^{2} + {\partial_{1} {c}_{(0,0)}}^{2}\right)}{2}$
In case you wonder what the index \(C\) of the concentration means, it just indicates that the concentration is a field (array) and the \(C\) indices indicates that we use the center value of the
field when iterating over it. This gets important when we apply a finite difference discretization on the equation.
The bulk free energy \(c^2 (1-c)^2\) is just the simplest polynomial with minima at \(c=0\) and \(c=1\).
plt.sympy_function(bulk_free_energy_density.subs(A, 1), (-0.2, 1.2))
plt.title("Bulk free energy");
To minimize the total free energy we use the Cahn Hilliard equation
\[\partial_t c = \nabla \cdot \left( M \nabla \frac{\delta F}{\delta c} \right)\]
where the functional derivative \(\frac{\delta F}{\delta c}\) in this case is the chemical potential \(\mu\). A functional derivative is computed as \(\frac{\delta F}{\delta c} = \frac{\partial F}{\
partial c} - \nabla \cdot \frac{\partial F}{\partial \nabla c}\). That means we treat \(\nabla c\) like a normal variable when calculating derivatives.
We don’t have to worry about that in detail, since pystencils offers a function to do just that:
ps.fd.functional_derivative(free_energy_density, c)
$\displaystyle {c}_{(0,0)}^{2} A \left(2 {c}_{(0,0)} - 2\right) + 2 {c}_{(0,0)} A \left(1 - {c}_{(0,0)}\right)^{2} - {\partial_{0} (κ {\partial_{0} {c}_{(0,0)}}) } - {\partial_{1} (κ {\partial_{1}
{c}_{(0,0)}}) }$
In this case we could quite simply do this derivative by hand but for more complex phase field models this step is quite tedious.
If we discretize this term using finite differences, we have a computation rule how to compute the chemical potential \(\mu\) from the free energy.
discretize = ps.fd.Discretization2ndOrder(dx=1, dt=0.01)
μ_update_eq = ps.fd.functional_derivative(free_energy_density, c)
μ_update_eq = ps.fd.expand_diff_linear(μ_update_eq, constants=[κ]) # pull constant κ in front of the derivatives
μ_update_eq_discretized = discretize(μ_update_eq)
$\displaystyle 4 {c}_{(0,0)}^{3} A - 6 {c}_{(0,0)}^{2} A + 2 {c}_{(0,0)} A - κ \left(- 2 {c}_{(0,0)} + {c}_{(1,0)} + {c}_{(-1,0)}\right) - κ \left(- 2 {c}_{(0,0)} + {c}_{(0,1)} + {c}_{(0,-1)}\right)$
pystencils computed the finite difference approximation for us. This was only possible since all symbols occuring inside derivatives are pystencils field variables, so that neighboring values can be
accessed. Next we bake this formula into a kernel that writes the chemical potential to a field. Therefor we first insert the \(\kappa\) and \(A\) parameters, build an assignment out of it and
compile the kernel
μ_kernel = ps.create_kernel([ps.Assignment(μ_field.center,
μ_update_eq_discretized.subs(A, 1).subs(κ, 0.5))]
Next, we formulate the Cahn-Hilliard equation itself, which is just a diffusion equation:
M = sp.Symbol("M")
cahn_hilliard = ps.fd.transient(c) - ps.fd.diffusion(μ, M)
$\displaystyle - div(M \nabla \mu) + \partial_t c_{C}$
It can be given right away to the discretize function, that by default uses a simple explicit Euler scheme for temporal, and second order finite differences for spatial discretization. It returns the
update rule for the concentration field.
c_update = discretize(cahn_hilliard)
$\displaystyle {c}_{(0,0)} - 0.04 {μ}_{(0,0)} M + 0.01 {μ}_{(1,0)} M + 0.01 {μ}_{(0,1)} M + 0.01 {μ}_{(0,-1)} M + 0.01 {μ}_{(-1,0)} M$
Again, we build a kernel from this update rule:
c_kernel = ps.create_kernel([ps.Assignment(c_field.center,
c_update.subs(M, 1))]
Before we run the simulation, the domain has to be initialized. To access a numpy array inside a data handling we have to iterate over the data handling. This somewhat complicated way is necessary to
be able to switch to distributed memory parallel simulations without having to alter the code. Basically this loops says “iterate over the portion of the domain that belongs to my process”, which in
our serial case here is just the full domain.
As suggested in the book, we initialize everything with \(c=0.4\) and add some random noise on top of it.
def init(value=0.4, noise=0.02):
for b in dh.iterate():
np.add(b['c'], noise*np.random.rand(*b['c'].shape), out=b['c'])
The time loop of the simulation is now rather straightforward. We call the kernels to update the chemical potential and the concentration in alternating fashion. In between we have to do
synchronization steps for the fields that take care of the periodic boundary condition, and in the parallel case of the communciation between processes.
def timeloop(steps=100):
c_sync = dh.synchronization_function(['c'])
μ_sync = dh.synchronization_function(['mu'])
for t in range(steps):
return dh.gather_array('c')
Now we can run the simulation and see how the phases separate
if 'is_test_run' in globals():
result = None
ani = ps.plot.scalar_field_animation(timeloop, rescale=True, frames=600)
result = ps.jupyter.display_as_html_video(ani) | {"url":"https://pycodegen.pages.i10git.cs.fau.de/pystencils/notebooks/05_tutorial_phasefield_spinodal_decomposition.html","timestamp":"2024-11-03T19:18:59Z","content_type":"text/html","content_length":"1048993","record_id":"<urn:uuid:ce070999-879d-43e8-a540-ac0664ee9524>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00380.warc.gz"} |
Cycle lengths in randomly perturbed graphs
Let (Figure presented.) be an (Figure presented.) -vertex graph, where (Figure presented.) for some (Figure presented.). A result of Bohman, Frieze and Martin from 2003 asserts that if (Figure
presented.), then perturbing (Figure presented.) via the addition of (Figure presented.) random edges, a.a.s. yields a Hamiltonian graph. We prove several improvements and extensions of the
aforementioned result. In particular, keeping the bound on (Figure presented.) as above and allowing for (Figure presented.), we determine the correct order of magnitude of the number of random edges
whose addition to (Figure presented.) a.a.s. yields a pancyclic graph. Moreover, we prove similar results for sparser graphs, and assuming the correctness of Chvátal's toughness conjecture, we handle
graphs having larger independent sets. Finally, under milder conditions, we determine the correct order of magnitude of the number of random edges whose addition to (Figure presented.) a.a.s. yields
a graph containing an almost spanning cycle.
• cycle lengths
• hamiltonicity
• independence number
• pancyclicity
• randomly perturbed graphs
• toughness
Dive into the research topics of 'Cycle lengths in randomly perturbed graphs'. Together they form a unique fingerprint. | {"url":"https://cris.ariel.ac.il/en/publications/cycle-lengths-in-randomly-perturbed-graphs-2","timestamp":"2024-11-09T03:00:24Z","content_type":"text/html","content_length":"55063","record_id":"<urn:uuid:05177492-0ffb-4048-890f-f1c082caf872>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00440.warc.gz"} |
2.3. Clustering
2.3. Clustering#
Clustering of unlabeled data can be performed with the module sklearn.cluster.
Each clustering algorithm comes in two variants: a class, that implements the fit method to learn the clusters on train data, and a function, that, given train data, returns an array of integer
labels corresponding to the different clusters. For the class, the labels over the training data can be found in the labels_ attribute.
2.3.1. Overview of clustering methods#
Method name Parameters Scalability Usecase Geometry (metric used)
K-Means number of clusters Very large n_samples, medium General-purpose, even cluster size, flat geometry, not too many Distances between points
n_clusters with MiniBatch code clusters, inductive
Affinity damping, sample preference Not scalable with n_samples Many clusters, uneven cluster size, non-flat geometry, inductive Graph distance (e.g.
propagation nearest-neighbor graph)
Mean-shift bandwidth Not scalable with n_samples Many clusters, uneven cluster size, non-flat geometry, inductive Distances between points
Spectral number of clusters Medium n_samples, small n_clusters Few clusters, even cluster size, non-flat geometry, transductive Graph distance (e.g.
clustering nearest-neighbor graph)
Ward hierarchical number of clusters or distance threshold Large n_samples and n_clusters Many clusters, possibly connectivity constraints, transductive Distances between points
Agglomerative number of clusters or distance threshold, Large n_samples and n_clusters Many clusters, possibly connectivity constraints, non Euclidean Any pairwise distance
clustering linkage type, distance distances, transductive
DBSCAN neighborhood size Very large n_samples, medium Non-flat geometry, uneven cluster sizes, outlier removal, transductive Distances between nearest
n_clusters points
HDBSCAN minimum cluster membership, minimum point large n_samples, medium n_clusters Non-flat geometry, uneven cluster sizes, outlier removal, Distances between nearest
neighbors transductive, hierarchical, variable cluster density points
OPTICS minimum cluster membership Very large n_samples, large n_clusters Non-flat geometry, uneven cluster sizes, variable cluster density, Distances between points
outlier removal, transductive
Gaussian mixtures many Not scalable Flat geometry, good for density estimation, inductive Mahalanobis distances to
BIRCH branching factor, threshold, optional Large n_clusters and n_samples Large dataset, outlier removal, data reduction, inductive Euclidean distance between
global clusterer. points
Bisecting K-Means number of clusters Very large n_samples, medium General-purpose, even cluster size, flat geometry, no empty clusters, Distances between points
n_clusters inductive, hierarchical
Non-flat geometry clustering is useful when the clusters have a specific shape, i.e. a non-flat manifold, and the standard euclidean distance is not the right metric. This case arises in the two top
rows of the figure above.
Gaussian mixture models, useful for clustering, are described in another chapter of the documentation dedicated to mixture models. KMeans can be seen as a special case of Gaussian mixture model with
equal covariance per component.
Transductive clustering methods (in contrast to inductive clustering methods) are not designed to be applied to new, unseen data.
2.3.2. K-means#
The KMeans algorithm clusters data by trying to separate samples in n groups of equal variance, minimizing a criterion known as the inertia or within-cluster sum-of-squares (see below). This
algorithm requires the number of clusters to be specified. It scales well to large numbers of samples and has been used across a large range of application areas in many different fields.
The k-means algorithm divides a set of \(N\) samples \(X\) into \(K\) disjoint clusters \(C\), each described by the mean \(\mu_j\) of the samples in the cluster. The means are commonly called the
cluster “centroids”; note that they are not, in general, points from \(X\), although they live in the same space.
The K-means algorithm aims to choose centroids that minimise the inertia, or within-cluster sum-of-squares criterion:
\[\sum_{i=0}^{n}\min_{\mu_j \in C}(||x_i - \mu_j||^2)\]
Inertia can be recognized as a measure of how internally coherent clusters are. It suffers from various drawbacks:
• Inertia makes the assumption that clusters are convex and isotropic, which is not always the case. It responds poorly to elongated clusters, or manifolds with irregular shapes.
• Inertia is not a normalized metric: we just know that lower values are better and zero is optimal. But in very high-dimensional spaces, Euclidean distances tend to become inflated (this is an
instance of the so-called “curse of dimensionality”). Running a dimensionality reduction algorithm such as Principal component analysis (PCA) prior to k-means clustering can alleviate this
problem and speed up the computations.
For more detailed descriptions of the issues shown above and how to address them, refer to the examples Demonstration of k-means assumptions and Selecting the number of clusters with silhouette
analysis on KMeans clustering.
K-means is often referred to as Lloyd’s algorithm. In basic terms, the algorithm has three steps. The first step chooses the initial centroids, with the most basic method being to choose \(k\)
samples from the dataset \(X\). After initialization, K-means consists of looping between the two other steps. The first step assigns each sample to its nearest centroid. The second step creates new
centroids by taking the mean value of all of the samples assigned to each previous centroid. The difference between the old and the new centroids are computed and the algorithm repeats these last two
steps until this value is less than a threshold. In other words, it repeats until the centroids do not move significantly.
K-means is equivalent to the expectation-maximization algorithm with a small, all-equal, diagonal covariance matrix.
The algorithm can also be understood through the concept of Voronoi diagrams. First the Voronoi diagram of the points is calculated using the current centroids. Each segment in the Voronoi diagram
becomes a separate cluster. Secondly, the centroids are updated to the mean of each segment. The algorithm then repeats this until a stopping criterion is fulfilled. Usually, the algorithm stops when
the relative decrease in the objective function between iterations is less than the given tolerance value. This is not the case in this implementation: iteration stops when centroids move less than
the tolerance.
Given enough time, K-means will always converge, however this may be to a local minimum. This is highly dependent on the initialization of the centroids. As a result, the computation is often done
several times, with different initializations of the centroids. One method to help address this issue is the k-means++ initialization scheme, which has been implemented in scikit-learn (use the init=
'k-means++' parameter). This initializes the centroids to be (generally) distant from each other, leading to probably better results than random initialization, as shown in the reference. For a
detailed example of comaparing different initialization schemes, refer to A demo of K-Means clustering on the handwritten digits data.
K-means++ can also be called independently to select seeds for other clustering algorithms, see sklearn.cluster.kmeans_plusplus for details and example usage.
The algorithm supports sample weights, which can be given by a parameter sample_weight. This allows to assign more weight to some samples when computing cluster centers and values of inertia. For
example, assigning a weight of 2 to a sample is equivalent to adding a duplicate of that sample to the dataset \(X\).
K-means can be used for vector quantization. This is achieved using the transform method of a trained model of KMeans. For an example of performing vector quantization on an image refer to Color
Quantization using K-Means.
2.3.2.1. Low-level parallelism#
KMeans benefits from OpenMP based parallelism through Cython. Small chunks of data (256 samples) are processed in parallel, which in addition yields a low memory footprint. For more details on how to
control the number of threads, please refer to our Parallelism notes.
2.3.2.2. Mini Batch K-Means#
The MiniBatchKMeans is a variant of the KMeans algorithm which uses mini-batches to reduce the computation time, while still attempting to optimise the same objective function. Mini-batches are
subsets of the input data, randomly sampled in each training iteration. These mini-batches drastically reduce the amount of computation required to converge to a local solution. In contrast to other
algorithms that reduce the convergence time of k-means, mini-batch k-means produces results that are generally only slightly worse than the standard algorithm.
The algorithm iterates between two major steps, similar to vanilla k-means. In the first step, \(b\) samples are drawn randomly from the dataset, to form a mini-batch. These are then assigned to the
nearest centroid. In the second step, the centroids are updated. In contrast to k-means, this is done on a per-sample basis. For each sample in the mini-batch, the assigned centroid is updated by
taking the streaming average of the sample and all previous samples assigned to that centroid. This has the effect of decreasing the rate of change for a centroid over time. These steps are performed
until convergence or a predetermined number of iterations is reached.
MiniBatchKMeans converges faster than KMeans, but the quality of the results is reduced. In practice this difference in quality can be quite small, as shown in the example and cited reference.
2.3.3. Affinity Propagation#
AffinityPropagation creates clusters by sending messages between pairs of samples until convergence. A dataset is then described using a small number of exemplars, which are identified as those most
representative of other samples. The messages sent between pairs represent the suitability for one sample to be the exemplar of the other, which is updated in response to the values from other pairs.
This updating happens iteratively until convergence, at which point the final exemplars are chosen, and hence the final clustering is given.
Affinity Propagation can be interesting as it chooses the number of clusters based on the data provided. For this purpose, the two important parameters are the preference, which controls how many
exemplars are used, and the damping factor which damps the responsibility and availability messages to avoid numerical oscillations when updating these messages.
The main drawback of Affinity Propagation is its complexity. The algorithm has a time complexity of the order \(O(N^2 T)\), where \(N\) is the number of samples and \(T\) is the number of iterations
until convergence. Further, the memory complexity is of the order \(O(N^2)\) if a dense similarity matrix is used, but reducible if a sparse similarity matrix is used. This makes Affinity Propagation
most appropriate for small to medium sized datasets.
Algorithm description#
The messages sent between points belong to one of two categories. The first is the responsibility \(r(i, k)\), which is the accumulated evidence that sample \(k\) should be the exemplar for sample \
(i\). The second is the availability \(a(i, k)\) which is the accumulated evidence that sample \(i\) should choose sample \(k\) to be its exemplar, and considers the values for all other samples that
\(k\) should be an exemplar. In this way, exemplars are chosen by samples if they are (1) similar enough to many samples and (2) chosen by many samples to be representative of themselves.
More formally, the responsibility of a sample \(k\) to be the exemplar of sample \(i\) is given by:
\[r(i, k) \leftarrow s(i, k) - max [ a(i, k') + s(i, k') \forall k' \neq k ]\]
Where \(s(i, k)\) is the similarity between samples \(i\) and \(k\). The availability of sample \(k\) to be the exemplar of sample \(i\) is given by:
\[a(i, k) \leftarrow min [0, r(k, k) + \sum_{i'~s.t.~i' \notin \{i, k\}}{r(i', k)}]\]
To begin with, all values for \(r\) and \(a\) are set to zero, and the calculation of each iterates until convergence. As discussed above, in order to avoid numerical oscillations when updating the
messages, the damping factor \(\lambda\) is introduced to iteration process:
\[r_{t+1}(i, k) = \lambda\cdot r_{t}(i, k) + (1-\lambda)\cdot r_{t+1}(i, k)\]
\[a_{t+1}(i, k) = \lambda\cdot a_{t}(i, k) + (1-\lambda)\cdot a_{t+1}(i, k)\]
where \(t\) indicates the iteration times.
2.3.4. Mean Shift#
MeanShift clustering aims to discover blobs in a smooth density of samples. It is a centroid based algorithm, which works by updating candidates for centroids to be the mean of the points within a
given region. These candidates are then filtered in a post-processing stage to eliminate near-duplicates to form the final set of centroids.
Mathematical details#
The position of centroid candidates is iteratively adjusted using a technique called hill climbing, which finds local maxima of the estimated probability density. Given a candidate centroid \(x\) for
iteration \(t\), the candidate is updated according to the following equation:
\[x^{t+1} = x^t + m(x^t)\]
Where \(m\) is the mean shift vector that is computed for each centroid that points towards a region of the maximum increase in the density of points. To compute \(m\) we define \(N(x)\) as the
neighborhood of samples within a given distance around \(x\). Then \(m\) is computed using the following equation, effectively updating a centroid to be the mean of the samples within its
\[m(x) = \frac{1}{|N(x)|} \sum_{x_j \in N(x)}x_j - x\]
In general, the equation for \(m\) depends on a kernel used for density estimation. The generic formula is:
\[m(x) = \frac{\sum_{x_j \in N(x)}K(x_j - x)x_j}{\sum_{x_j \in N(x)}K(x_j - x)} - x\]
In our implementation, \(K(x)\) is equal to 1 if \(x\) is small enough and is equal to 0 otherwise. Effectively \(K(y - x)\) indicates whether \(y\) is in the neighborhood of \(x\).
The algorithm automatically sets the number of clusters, instead of relying on a parameter bandwidth, which dictates the size of the region to search through. This parameter can be set manually, but
can be estimated using the provided estimate_bandwidth function, which is called if the bandwidth is not set.
The algorithm is not highly scalable, as it requires multiple nearest neighbor searches during the execution of the algorithm. The algorithm is guaranteed to converge, however the algorithm will stop
iterating when the change in centroids is small.
Labelling a new sample is performed by finding the nearest centroid for a given sample.
2.3.5. Spectral clustering#
SpectralClustering performs a low-dimension embedding of the affinity matrix between samples, followed by clustering, e.g., by KMeans, of the components of the eigenvectors in the low dimensional
space. It is especially computationally efficient if the affinity matrix is sparse and the amg solver is used for the eigenvalue problem (Note, the amg solver requires that the pyamg module is
The present version of SpectralClustering requires the number of clusters to be specified in advance. It works well for a small number of clusters, but is not advised for many clusters.
For two clusters, SpectralClustering solves a convex relaxation of the normalized cuts problem on the similarity graph: cutting the graph in two so that the weight of the edges cut is small compared
to the weights of the edges inside each cluster. This criteria is especially interesting when working on images, where graph vertices are pixels, and weights of the edges of the similarity graph are
computed using a function of a gradient of the image.
Transforming distance to well-behaved similarities
Note that if the values of your similarity matrix are not well distributed, e.g. with negative values or with a distance matrix rather than a similarity, the spectral problem will be singular and the
problem not solvable. In which case it is advised to apply a transformation to the entries of the matrix. For instance, in the case of a signed distance matrix, is common to apply a heat kernel:
similarity = np.exp(-beta * distance / distance.std())
See the examples for such an application.
2.3.5.1. Different label assignment strategies#
Different label assignment strategies can be used, corresponding to the assign_labels parameter of SpectralClustering. "kmeans" strategy can match finer details, but can be unstable. In particular,
unless you control the random_state, it may not be reproducible from run-to-run, as it depends on random initialization. The alternative "discretize" strategy is 100% reproducible, but tends to
create parcels of fairly even and geometrical shape. The recently added "cluster_qr" option is a deterministic alternative that tends to create the visually best partitioning on the example
application below.
2.3.5.2. Spectral Clustering Graphs#
Spectral Clustering can also be used to partition graphs via their spectral embeddings. In this case, the affinity matrix is the adjacency matrix of the graph, and SpectralClustering is initialized
with affinity='precomputed':
>>> from sklearn.cluster import SpectralClustering
>>> sc = SpectralClustering(3, affinity='precomputed', n_init=100,
... assign_labels='discretize')
>>> sc.fit_predict(adjacency_matrix)
2.3.6. Hierarchical clustering#
Hierarchical clustering is a general family of clustering algorithms that build nested clusters by merging or splitting them successively. This hierarchy of clusters is represented as a tree (or
dendrogram). The root of the tree is the unique cluster that gathers all the samples, the leaves being the clusters with only one sample. See the Wikipedia page for more details.
The AgglomerativeClustering object performs a hierarchical clustering using a bottom up approach: each observation starts in its own cluster, and clusters are successively merged together. The
linkage criteria determines the metric used for the merge strategy:
• Ward minimizes the sum of squared differences within all clusters. It is a variance-minimizing approach and in this sense is similar to the k-means objective function but tackled with an
agglomerative hierarchical approach.
• Maximum or complete linkage minimizes the maximum distance between observations of pairs of clusters.
• Average linkage minimizes the average of the distances between all observations of pairs of clusters.
• Single linkage minimizes the distance between the closest observations of pairs of clusters.
AgglomerativeClustering can also scale to large number of samples when it is used jointly with a connectivity matrix, but is computationally expensive when no connectivity constraints are added
between samples: it considers at each step all the possible merges.
2.3.6.1. Different linkage type: Ward, complete, average, and single linkage#
AgglomerativeClustering supports Ward, single, average, and complete linkage strategies.
Agglomerative cluster has a “rich get richer” behavior that leads to uneven cluster sizes. In this regard, single linkage is the worst strategy, and Ward gives the most regular sizes. However, the
affinity (or distance used in clustering) cannot be varied with Ward, thus for non Euclidean metrics, average linkage is a good alternative. Single linkage, while not robust to noisy data, can be
computed very efficiently and can therefore be useful to provide hierarchical clustering of larger datasets. Single linkage can also perform well on non-globular data.
2.3.6.2. Visualization of cluster hierarchy#
It’s possible to visualize the tree representing the hierarchical merging of clusters as a dendrogram. Visual inspection can often be useful for understanding the structure of the data, though more
so in the case of small sample sizes.
2.3.6.3. Adding connectivity constraints#
An interesting aspect of AgglomerativeClustering is that connectivity constraints can be added to this algorithm (only adjacent clusters can be merged together), through a connectivity matrix that
defines for each sample the neighboring samples following a given structure of the data. For instance, in the swiss-roll example below, the connectivity constraints forbid the merging of points that
are not adjacent on the swiss roll, and thus avoid forming clusters that extend across overlapping folds of the roll.
These constraint are useful to impose a certain local structure, but they also make the algorithm faster, especially when the number of the samples is high.
The connectivity constraints are imposed via an connectivity matrix: a scipy sparse matrix that has elements only at the intersection of a row and a column with indices of the dataset that should be
connected. This matrix can be constructed from a-priori information: for instance, you may wish to cluster web pages by only merging pages with a link pointing from one to another. It can also be
learned from the data, for instance using sklearn.neighbors.kneighbors_graph to restrict merging to nearest neighbors as in this example, or using sklearn.feature_extraction.image.grid_to_graph to
enable only merging of neighboring pixels on an image, as in the coin example.
Connectivity constraints with single, average and complete linkage
Connectivity constraints and single, complete or average linkage can enhance the ‘rich getting richer’ aspect of agglomerative clustering, particularly so if they are built with
sklearn.neighbors.kneighbors_graph. In the limit of a small number of clusters, they tend to give a few macroscopically occupied clusters and almost empty ones. (see the discussion in Agglomerative
clustering with and without structure). Single linkage is the most brittle linkage option with regard to this issue.
2.3.6.4. Varying the metric#
Single, average and complete linkage can be used with a variety of distances (or affinities), in particular Euclidean distance (l2), Manhattan distance (or Cityblock, or l1), cosine distance, or any
precomputed affinity matrix.
• l1 distance is often good for sparse features, or sparse noise: i.e. many of the features are zero, as in text mining using occurrences of rare words.
• cosine distance is interesting because it is invariant to global scalings of the signal.
The guidelines for choosing a metric is to use one that maximizes the distance between samples in different classes, and minimizes that within each class.
2.3.6.5. Bisecting K-Means#
The BisectingKMeans is an iterative variant of KMeans, using divisive hierarchical clustering. Instead of creating all centroids at once, centroids are picked progressively based on a previous
clustering: a cluster is split into two new clusters repeatedly until the target number of clusters is reached.
BisectingKMeans is more efficient than KMeans when the number of clusters is large since it only works on a subset of the data at each bisection while KMeans always works on the entire dataset.
Although BisectingKMeans can’t benefit from the advantages of the "k-means++" initialization by design, it will still produce comparable results than KMeans(init="k-means++") in terms of inertia at
cheaper computational costs, and will likely produce better results than KMeans with a random initialization.
This variant is more efficient to agglomerative clustering if the number of clusters is small compared to the number of data points.
This variant also does not produce empty clusters.
There exist two strategies for selecting the cluster to split:
□ bisecting_strategy="largest_cluster" selects the cluster having the most points
□ bisecting_strategy="biggest_inertia" selects the cluster with biggest inertia (cluster with biggest Sum of Squared Errors within)
Picking by largest amount of data points in most cases produces result as accurate as picking by inertia and is faster (especially for larger amount of data points, where calculating error may be
Picking by largest amount of data points will also likely produce clusters of similar sizes while KMeans is known to produce clusters of different sizes.
Difference between Bisecting K-Means and regular K-Means can be seen on example Bisecting K-Means and Regular K-Means Performance Comparison. While the regular K-Means algorithm tends to create
non-related clusters, clusters from Bisecting K-Means are well ordered and create quite a visible hierarchy.
• “A Comparison of Document Clustering Techniques” Michael Steinbach, George Karypis and Vipin Kumar, Department of Computer Science and Egineering, University of Minnesota (June 2000)
• “Performance Analysis of K-Means and Bisecting K-Means Algorithms in Weblog Data” K.Abirami and Dr.P.Mayilvahanan, International Journal of Emerging Technologies in Engineering Research (IJETER)
Volume 4, Issue 8, (August 2016)
• “Bisecting K-means Algorithm Based on K-valued Self-determining and Clustering Center Optimization” Jian Di, Xinyue Gou School of Control and Computer Engineering,North China Electric Power
University, Baoding, Hebei, China (August 2017)
2.3.7. DBSCAN#
The DBSCAN algorithm views clusters as areas of high density separated by areas of low density. Due to this rather generic view, clusters found by DBSCAN can be any shape, as opposed to k-means which
assumes that clusters are convex shaped. The central component to the DBSCAN is the concept of core samples, which are samples that are in areas of high density. A cluster is therefore a set of core
samples, each close to each other (measured by some distance measure) and a set of non-core samples that are close to a core sample (but are not themselves core samples). There are two parameters to
the algorithm, min_samples and eps, which define formally what we mean when we say dense. Higher min_samples or lower eps indicate higher density necessary to form a cluster.
More formally, we define a core sample as being a sample in the dataset such that there exist min_samples other samples within a distance of eps, which are defined as neighbors of the core sample.
This tells us that the core sample is in a dense area of the vector space. A cluster is a set of core samples that can be built by recursively taking a core sample, finding all of its neighbors that
are core samples, finding all of their neighbors that are core samples, and so on. A cluster also has a set of non-core samples, which are samples that are neighbors of a core sample in the cluster
but are not themselves core samples. Intuitively, these samples are on the fringes of a cluster.
Any core sample is part of a cluster, by definition. Any sample that is not a core sample, and is at least eps in distance from any core sample, is considered an outlier by the algorithm.
While the parameter min_samples primarily controls how tolerant the algorithm is towards noise (on noisy and large data sets it may be desirable to increase this parameter), the parameter eps is
crucial to choose appropriately for the data set and distance function and usually cannot be left at the default value. It controls the local neighborhood of the points. When chosen too small, most
data will not be clustered at all (and labeled as -1 for “noise”). When chosen too large, it causes close clusters to be merged into one cluster, and eventually the entire data set to be returned as
a single cluster. Some heuristics for choosing this parameter have been discussed in the literature, for example based on a knee in the nearest neighbor distances plot (as discussed in the references
In the figure below, the color indicates cluster membership, with large circles indicating core samples found by the algorithm. Smaller circles are non-core samples that are still part of a cluster.
Moreover, the outliers are indicated by black points below.
The DBSCAN algorithm is deterministic, always generating the same clusters when given the same data in the same order. However, the results can differ when data is provided in a different order.
First, even though the core samples will always be assigned to the same clusters, the labels of those clusters will depend on the order in which those samples are encountered in the data. Second and
more importantly, the clusters to which non-core samples are assigned can differ depending on the data order. This would happen when a non-core sample has a distance lower than eps to two core
samples in different clusters. By the triangular inequality, those two core samples must be more distant than eps from each other, or they would be in the same cluster. The non-core sample is
assigned to whichever cluster is generated first in a pass through the data, and so the results will depend on the data ordering.
The current implementation uses ball trees and kd-trees to determine the neighborhood of points, which avoids calculating the full distance matrix (as was done in scikit-learn versions before 0.14).
The possibility to use custom metrics is retained; for details, see NearestNeighbors.
Memory consumption for large sample sizes#
This implementation is by default not memory efficient because it constructs a full pairwise similarity matrix in the case where kd-trees or ball-trees cannot be used (e.g., with sparse matrices).
This matrix will consume \(n^2\) floats. A couple of mechanisms for getting around this are:
• Use OPTICS clustering in conjunction with the extract_dbscan method. OPTICS clustering also calculates the full pairwise matrix, but only keeps one row in memory at a time (memory complexity n).
• A sparse radius neighborhood graph (where missing entries are presumed to be out of eps) can be precomputed in a memory-efficient way and dbscan can be run over this with metric='precomputed'.
See sklearn.neighbors.NearestNeighbors.radius_neighbors_graph.
• The dataset can be compressed, either by removing exact duplicates if these occur in your data, or by using BIRCH. Then you only have a relatively small number of representatives for a large
number of points. You can then provide a sample_weight when fitting DBSCAN.
• A Density-Based Algorithm for Discovering Clusters in Large Spatial Databases with Noise Ester, M., H. P. Kriegel, J. Sander, and X. Xu, In Proceedings of the 2nd International Conference on
Knowledge Discovery and Data Mining, Portland, OR, AAAI Press, pp. 226-231. 1996
• DBSCAN revisited, revisited: why and how you should (still) use DBSCAN. Schubert, E., Sander, J., Ester, M., Kriegel, H. P., & Xu, X. (2017). In ACM Transactions on Database Systems (TODS), 42
(3), 19.
2.3.8. HDBSCAN#
The HDBSCAN algorithm can be seen as an extension of DBSCAN and OPTICS. Specifically, DBSCAN assumes that the clustering criterion (i.e. density requirement) is globally homogeneous. In other words,
DBSCAN may struggle to successfully capture clusters with different densities. HDBSCAN alleviates this assumption and explores all possible density scales by building an alternative representation of
the clustering problem.
2.3.8.1. Mutual Reachability Graph#
HDBSCAN first defines \(d_c(x_p)\), the core distance of a sample \(x_p\), as the distance to its min_samples th-nearest neighbor, counting itself. For example, if min_samples=5 and \(x_*\) is the
5th-nearest neighbor of \(x_p\) then the core distance is:
\[d_c(x_p)=d(x_p, x_*).\]
Next it defines \(d_m(x_p, x_q)\), the mutual reachability distance of two points \(x_p, x_q\), as:
\[d_m(x_p, x_q) = \max\{d_c(x_p), d_c(x_q), d(x_p, x_q)\}\]
These two notions allow us to construct the mutual reachability graph \(G_{ms}\) defined for a fixed choice of min_samples by associating each sample \(x_p\) with a vertex of the graph, and thus
edges between points \(x_p, x_q\) are the mutual reachability distance \(d_m(x_p, x_q)\) between them. We may build subsets of this graph, denoted as \(G_{ms,\varepsilon}\), by removing any edges
with value greater than \(\varepsilon\): from the original graph. Any points whose core distance is less than \(\varepsilon\): are at this staged marked as noise. The remaining points are then
clustered by finding the connected components of this trimmed graph.
Taking the connected components of a trimmed graph \(G_{ms,\varepsilon}\) is equivalent to running DBSCAN* with min_samples and \(\varepsilon\). DBSCAN* is a slightly modified version of DBSCAN
mentioned in [CM2013].
2.3.8.2. Hierarchical Clustering#
HDBSCAN can be seen as an algorithm which performs DBSCAN* clustering across all values of \(\varepsilon\). As mentioned prior, this is equivalent to finding the connected components of the mutual
reachability graphs for all values of \(\varepsilon\). To do this efficiently, HDBSCAN first extracts a minimum spanning tree (MST) from the fully -connected mutual reachability graph, then greedily
cuts the edges with highest weight. An outline of the HDBSCAN algorithm is as follows:
1. Extract the MST of \(G_{ms}\).
2. Extend the MST by adding a “self edge” for each vertex, with weight equal to the core distance of the underlying sample.
3. Initialize a single cluster and label for the MST.
4. Remove the edge with the greatest weight from the MST (ties are removed simultaneously).
5. Assign cluster labels to the connected components which contain the end points of the now-removed edge. If the component does not have at least one edge it is instead assigned a “null” label
marking it as noise.
6. Repeat 4-5 until there are no more connected components.
HDBSCAN is therefore able to obtain all possible partitions achievable by DBSCAN* for a fixed choice of min_samples in a hierarchical fashion. Indeed, this allows HDBSCAN to perform clustering across
multiple densities and as such it no longer needs \(\varepsilon\) to be given as a hyperparameter. Instead it relies solely on the choice of min_samples, which tends to be a more robust
HDBSCAN can be smoothed with an additional hyperparameter min_cluster_size which specifies that during the hierarchical clustering, components with fewer than minimum_cluster_size many samples are
considered noise. In practice, one can set minimum_cluster_size = min_samples to couple the parameters and simplify the hyperparameter space.
Campello, R.J.G.B., Moulavi, D., Sander, J. (2013). Density-Based Clustering Based on Hierarchical Density Estimates. In: Pei, J., Tseng, V.S., Cao, L., Motoda, H., Xu, G. (eds) Advances in Knowledge
Discovery and Data Mining. PAKDD 2013. Lecture Notes in Computer Science(), vol 7819. Springer, Berlin, Heidelberg. Density-Based Clustering Based on Hierarchical Density Estimates
2.3.9. OPTICS#
The OPTICS algorithm shares many similarities with the DBSCAN algorithm, and can be considered a generalization of DBSCAN that relaxes the eps requirement from a single value to a value range. The
key difference between DBSCAN and OPTICS is that the OPTICS algorithm builds a reachability graph, which assigns each sample both a reachability_ distance, and a spot within the cluster ordering_
attribute; these two attributes are assigned when the model is fitted, and are used to determine cluster membership. If OPTICS is run with the default value of inf set for max_eps, then DBSCAN style
cluster extraction can be performed repeatedly in linear time for any given eps value using the cluster_optics_dbscan method. Setting max_eps to a lower value will result in shorter run times, and
can be thought of as the maximum neighborhood radius from each point to find other potential reachable points.
The reachability distances generated by OPTICS allow for variable density extraction of clusters within a single data set. As shown in the above plot, combining reachability distances and data set
ordering_ produces a reachability plot, where point density is represented on the Y-axis, and points are ordered such that nearby points are adjacent. ‘Cutting’ the reachability plot at a single
value produces DBSCAN like results; all points above the ‘cut’ are classified as noise, and each time that there is a break when reading from left to right signifies a new cluster. The default
cluster extraction with OPTICS looks at the steep slopes within the graph to find clusters, and the user can define what counts as a steep slope using the parameter xi. There are also other
possibilities for analysis on the graph itself, such as generating hierarchical representations of the data through reachability-plot dendrograms, and the hierarchy of clusters detected by the
algorithm can be accessed through the cluster_hierarchy_ parameter. The plot above has been color-coded so that cluster colors in planar space match the linear segment clusters of the reachability
plot. Note that the blue and red clusters are adjacent in the reachability plot, and can be hierarchically represented as children of a larger parent cluster.
Comparison with DBSCAN#
The results from OPTICS cluster_optics_dbscan method and DBSCAN are very similar, but not always identical; specifically, labeling of periphery and noise points. This is in part because the first
samples of each dense area processed by OPTICS have a large reachability value while being close to other points in their area, and will thus sometimes be marked as noise rather than periphery. This
affects adjacent points when they are considered as candidates for being marked as either periphery or noise.
Note that for any single value of eps, DBSCAN will tend to have a shorter run time than OPTICS; however, for repeated runs at varying eps values, a single run of OPTICS may require less cumulative
runtime than DBSCAN. It is also important to note that OPTICS’ output is close to DBSCAN’s only if eps and max_eps are close.
Computational Complexity#
Spatial indexing trees are used to avoid calculating the full distance matrix, and allow for efficient memory usage on large sets of samples. Different distance metrics can be supplied via the metric
For large datasets, similar (but not identical) results can be obtained via HDBSCAN. The HDBSCAN implementation is multithreaded, and has better algorithmic runtime complexity than OPTICS, at the
cost of worse memory scaling. For extremely large datasets that exhaust system memory using HDBSCAN, OPTICS will maintain \(n\) (as opposed to \(n^2\)) memory scaling; however, tuning of the max_eps
parameter will likely need to be used to give a solution in a reasonable amount of wall time.
• “OPTICS: ordering points to identify the clustering structure.” Ankerst, Mihael, Markus M. Breunig, Hans-Peter Kriegel, and Jörg Sander. In ACM Sigmod Record, vol. 28, no. 2, pp. 49-60. ACM,
2.3.10. BIRCH#
The Birch builds a tree called the Clustering Feature Tree (CFT) for the given data. The data is essentially lossy compressed to a set of Clustering Feature nodes (CF Nodes). The CF Nodes have a
number of subclusters called Clustering Feature subclusters (CF Subclusters) and these CF Subclusters located in the non-terminal CF Nodes can have CF Nodes as children.
The CF Subclusters hold the necessary information for clustering which prevents the need to hold the entire input data in memory. This information includes:
• Number of samples in a subcluster.
• Linear Sum - An n-dimensional vector holding the sum of all samples
• Squared Sum - Sum of the squared L2 norm of all samples.
• Centroids - To avoid recalculation linear sum / n_samples.
• Squared norm of the centroids.
The BIRCH algorithm has two parameters, the threshold and the branching factor. The branching factor limits the number of subclusters in a node and the threshold limits the distance between the
entering sample and the existing subclusters.
This algorithm can be viewed as an instance or data reduction method, since it reduces the input data to a set of subclusters which are obtained directly from the leaves of the CFT. This reduced data
can be further processed by feeding it into a global clusterer. This global clusterer can be set by n_clusters. If n_clusters is set to None, the subclusters from the leaves are directly read off,
otherwise a global clustering step labels these subclusters into global clusters (labels) and the samples are mapped to the global label of the nearest subcluster.
Algorithm description#
• A new sample is inserted into the root of the CF Tree which is a CF Node. It is then merged with the subcluster of the root, that has the smallest radius after merging, constrained by the
threshold and branching factor conditions. If the subcluster has any child node, then this is done repeatedly till it reaches a leaf. After finding the nearest subcluster in the leaf, the
properties of this subcluster and the parent subclusters are recursively updated.
• If the radius of the subcluster obtained by merging the new sample and the nearest subcluster is greater than the square of the threshold and if the number of subclusters is greater than the
branching factor, then a space is temporarily allocated to this new sample. The two farthest subclusters are taken and the subclusters are divided into two groups on the basis of the distance
between these subclusters.
• If this split node has a parent subcluster and there is room for a new subcluster, then the parent is split into two. If there is no room, then this node is again split into two and the process
is continued recursively, till it reaches the root.
BIRCH or MiniBatchKMeans?#
• BIRCH does not scale very well to high dimensional data. As a rule of thumb if n_features is greater than twenty, it is generally better to use MiniBatchKMeans.
• If the number of instances of data needs to be reduced, or if one wants a large number of subclusters either as a preprocessing step or otherwise, BIRCH is more useful than MiniBatchKMeans.
How to use partial_fit?#
To avoid the computation of global clustering, for every call of partial_fit the user is advised:
1. To set n_clusters=None initially.
2. Train all data by multiple calls to partial_fit.
3. Set n_clusters to a required value using brc.set_params(n_clusters=n_clusters).
4. Call partial_fit finally with no arguments, i.e. brc.partial_fit() which performs the global clustering.
2.3.11. Clustering performance evaluation#
Evaluating the performance of a clustering algorithm is not as trivial as counting the number of errors or the precision and recall of a supervised classification algorithm. In particular any
evaluation metric should not take the absolute values of the cluster labels into account but rather if this clustering define separations of the data similar to some ground truth set of classes or
satisfying some assumption such that members belong to the same class are more similar than members of different classes according to some similarity metric.
2.3.11.1. Rand index#
Given the knowledge of the ground truth class assignments labels_true and our clustering algorithm assignments of the same samples labels_pred, the (adjusted or unadjusted) Rand index is a function
that measures the similarity of the two assignments, ignoring permutations:
>>> from sklearn import metrics
>>> labels_true = [0, 0, 0, 1, 1, 1]
>>> labels_pred = [0, 0, 1, 1, 2, 2]
>>> metrics.rand_score(labels_true, labels_pred)
The Rand index does not ensure to obtain a value close to 0.0 for a random labelling. The adjusted Rand index corrects for chance and will give such a baseline.
>>> metrics.adjusted_rand_score(labels_true, labels_pred)
As with all clustering metrics, one can permute 0 and 1 in the predicted labels, rename 2 to 3, and get the same score:
>>> labels_pred = [1, 1, 0, 0, 3, 3]
>>> metrics.rand_score(labels_true, labels_pred)
>>> metrics.adjusted_rand_score(labels_true, labels_pred)
Furthermore, both rand_score adjusted_rand_score are symmetric: swapping the argument does not change the scores. They can thus be used as consensus measures:
>>> metrics.rand_score(labels_pred, labels_true)
>>> metrics.adjusted_rand_score(labels_pred, labels_true)
Perfect labeling is scored 1.0:
>>> labels_pred = labels_true[:]
>>> metrics.rand_score(labels_true, labels_pred)
>>> metrics.adjusted_rand_score(labels_true, labels_pred)
Poorly agreeing labels (e.g. independent labelings) have lower scores, and for the adjusted Rand index the score will be negative or close to zero. However, for the unadjusted Rand index the score,
while lower, will not necessarily be close to zero.:
>>> labels_true = [0, 0, 0, 0, 0, 0, 1, 1]
>>> labels_pred = [0, 1, 2, 3, 4, 5, 5, 6]
>>> metrics.rand_score(labels_true, labels_pred)
>>> metrics.adjusted_rand_score(labels_true, labels_pred)
Mathematical formulation#
If C is a ground truth class assignment and K the clustering, let us define \(a\) and \(b\) as:
• \(a\), the number of pairs of elements that are in the same set in C and in the same set in K
• \(b\), the number of pairs of elements that are in different sets in C and in different sets in K
The unadjusted Rand index is then given by:
\[\text{RI} = \frac{a + b}{C_2^{n_{samples}}}\]
where \(C_2^{n_{samples}}\) is the total number of possible pairs in the dataset. It does not matter if the calculation is performed on ordered pairs or unordered pairs as long as the calculation is
performed consistently.
However, the Rand index does not guarantee that random label assignments will get a value close to zero (esp. if the number of clusters is in the same order of magnitude as the number of samples).
To counter this effect we can discount the expected RI \(E[\text{RI}]\) of random labelings by defining the adjusted Rand index as follows:
\[\text{ARI} = \frac{\text{RI} - E[\text{RI}]}{\max(\text{RI}) - E[\text{RI}]}\]
2.3.11.2. Mutual Information based scores#
Given the knowledge of the ground truth class assignments labels_true and our clustering algorithm assignments of the same samples labels_pred, the Mutual Information is a function that measures the
agreement of the two assignments, ignoring permutations. Two different normalized versions of this measure are available, Normalized Mutual Information (NMI) and Adjusted Mutual Information (AMI).
NMI is often used in the literature, while AMI was proposed more recently and is normalized against chance:
>>> from sklearn import metrics
>>> labels_true = [0, 0, 0, 1, 1, 1]
>>> labels_pred = [0, 0, 1, 1, 2, 2]
>>> metrics.adjusted_mutual_info_score(labels_true, labels_pred)
One can permute 0 and 1 in the predicted labels, rename 2 to 3 and get the same score:
>>> labels_pred = [1, 1, 0, 0, 3, 3]
>>> metrics.adjusted_mutual_info_score(labels_true, labels_pred)
All, mutual_info_score, adjusted_mutual_info_score and normalized_mutual_info_score are symmetric: swapping the argument does not change the score. Thus they can be used as a consensus measure:
>>> metrics.adjusted_mutual_info_score(labels_pred, labels_true)
Perfect labeling is scored 1.0:
>>> labels_pred = labels_true[:]
>>> metrics.adjusted_mutual_info_score(labels_true, labels_pred)
>>> metrics.normalized_mutual_info_score(labels_true, labels_pred)
This is not true for mutual_info_score, which is therefore harder to judge:
>>> metrics.mutual_info_score(labels_true, labels_pred)
Bad (e.g. independent labelings) have non-positive scores:
>>> labels_true = [0, 1, 2, 0, 3, 4, 5, 1]
>>> labels_pred = [1, 1, 0, 0, 2, 2, 2, 2]
>>> metrics.adjusted_mutual_info_score(labels_true, labels_pred)
• Adjustment for chance in clustering performance evaluation: Analysis of the impact of the dataset size on the value of clustering measures for random assignments. This example also includes the
Adjusted Rand Index.
Mathematical formulation#
Assume two label assignments (of the same N objects), \(U\) and \(V\). Their entropy is the amount of uncertainty for a partition set, defined by:
\[H(U) = - \sum_{i=1}^{|U|}P(i)\log(P(i))\]
where \(P(i) = |U_i| / N\) is the probability that an object picked at random from \(U\) falls into class \(U_i\). Likewise for \(V\):
\[H(V) = - \sum_{j=1}^{|V|}P'(j)\log(P'(j))\]
With \(P'(j) = |V_j| / N\). The mutual information (MI) between \(U\) and \(V\) is calculated by:
\[\text{MI}(U, V) = \sum_{i=1}^{|U|}\sum_{j=1}^{|V|}P(i, j)\log\left(\frac{P(i,j)}{P(i)P'(j)}\right)\]
where \(P(i, j) = |U_i \cap V_j| / N\) is the probability that an object picked at random falls into both classes \(U_i\) and \(V_j\).
It also can be expressed in set cardinality formulation:
\[\text{MI}(U, V) = \sum_{i=1}^{|U|} \sum_{j=1}^{|V|} \frac{|U_i \cap V_j|}{N}\log\left(\frac{N|U_i \cap V_j|}{|U_i||V_j|}\right)\]
The normalized mutual information is defined as
\[\text{NMI}(U, V) = \frac{\text{MI}(U, V)}{\text{mean}(H(U), H(V))}\]
This value of the mutual information and also the normalized variant is not adjusted for chance and will tend to increase as the number of different labels (clusters) increases, regardless of the
actual amount of “mutual information” between the label assignments.
The expected value for the mutual information can be calculated using the following equation [VEB2009]. In this equation, \(a_i = |U_i|\) (the number of elements in \(U_i\)) and \(b_j = |V_j|\) (the
number of elements in \(V_j\)).
\[E[\text{MI}(U,V)]=\sum_{i=1}^{|U|} \sum_{j=1}^{|V|} \sum_{n_{ij}=(a_i+b_j-N)^+ }^{\min(a_i, b_j)} \frac{n_{ij}}{N}\log \left( \frac{ N.n_{ij}}{a_i b_j}\right) \frac{a_i!b_j!(N-a_i)!(N-b_j)!}{N!n_
{ij}!(a_i-n_{ij})!(b_j-n_{ij})! (N-a_i-b_j+n_{ij})!}\]
Using the expected value, the adjusted mutual information can then be calculated using a similar form to that of the adjusted Rand index:
\[\text{AMI} = \frac{\text{MI} - E[\text{MI}]}{\text{mean}(H(U), H(V)) - E[\text{MI}]}\]
For normalized mutual information and adjusted mutual information, the normalizing value is typically some generalized mean of the entropies of each clustering. Various generalized means exist, and
no firm rules exist for preferring one over the others. The decision is largely a field-by-field basis; for instance, in community detection, the arithmetic mean is most common. Each normalizing
method provides “qualitatively similar behaviours” [YAT2016]. In our implementation, this is controlled by the average_method parameter.
Vinh et al. (2010) named variants of NMI and AMI by their averaging method [VEB2010]. Their ‘sqrt’ and ‘sum’ averages are the geometric and arithmetic means; we use these more broadly common names.
• Strehl, Alexander, and Joydeep Ghosh (2002). “Cluster ensembles - a knowledge reuse framework for combining multiple partitions”. Journal of Machine Learning Research 3: 583-617. doi:10.1162/
Vinh, Epps, and Bailey, (2009). “Information theoretic measures for clusterings comparison”. Proceedings of the 26th Annual International Conference on Machine Learning - ICML ‘09. doi:10.1145/
1553374.1553511. ISBN 9781605585161.
Yang, Algesheimer, and Tessone, (2016). “A comparative analysis of community detection algorithms on artificial networks”. Scientific Reports 6: 30750. doi:10.1038/srep30750.
2.3.11.3. Homogeneity, completeness and V-measure#
Given the knowledge of the ground truth class assignments of the samples, it is possible to define some intuitive metric using conditional entropy analysis.
In particular Rosenberg and Hirschberg (2007) define the following two desirable objectives for any cluster assignment:
• homogeneity: each cluster contains only members of a single class.
• completeness: all members of a given class are assigned to the same cluster.
We can turn those concept as scores homogeneity_score and completeness_score. Both are bounded below by 0.0 and above by 1.0 (higher is better):
>>> from sklearn import metrics
>>> labels_true = [0, 0, 0, 1, 1, 1]
>>> labels_pred = [0, 0, 1, 1, 2, 2]
>>> metrics.homogeneity_score(labels_true, labels_pred)
>>> metrics.completeness_score(labels_true, labels_pred)
Their harmonic mean called V-measure is computed by v_measure_score:
>>> metrics.v_measure_score(labels_true, labels_pred)
This function’s formula is as follows:
\[v = \frac{(1 + \beta) \times \text{homogeneity} \times \text{completeness}}{(\beta \times \text{homogeneity} + \text{completeness})}\]
beta defaults to a value of 1.0, but for using a value less than 1 for beta:
>>> metrics.v_measure_score(labels_true, labels_pred, beta=0.6)
more weight will be attributed to homogeneity, and using a value greater than 1:
>>> metrics.v_measure_score(labels_true, labels_pred, beta=1.8)
more weight will be attributed to completeness.
The V-measure is actually equivalent to the mutual information (NMI) discussed above, with the aggregation function being the arithmetic mean [B2011].
Homogeneity, completeness and V-measure can be computed at once using homogeneity_completeness_v_measure as follows:
>>> metrics.homogeneity_completeness_v_measure(labels_true, labels_pred)
(0.66..., 0.42..., 0.51...)
The following clustering assignment is slightly better, since it is homogeneous but not complete:
>>> labels_pred = [0, 0, 0, 1, 2, 2]
>>> metrics.homogeneity_completeness_v_measure(labels_true, labels_pred)
(1.0, 0.68..., 0.81...)
v_measure_score is symmetric: it can be used to evaluate the agreement of two independent assignments on the same dataset.
This is not the case for completeness_score and homogeneity_score: both are bound by the relationship:
homogeneity_score(a, b) == completeness_score(b, a)
Mathematical formulation#
Homogeneity and completeness scores are formally given by:
\[h = 1 - \frac{H(C|K)}{H(C)}\]
\[c = 1 - \frac{H(K|C)}{H(K)}\]
where \(H(C|K)\) is the conditional entropy of the classes given the cluster assignments and is given by:
\[H(C|K) = - \sum_{c=1}^{|C|} \sum_{k=1}^{|K|} \frac{n_{c,k}}{n} \cdot \log\left(\frac{n_{c,k}}{n_k}\right)\]
and \(H(C)\) is the entropy of the classes and is given by:
\[H(C) = - \sum_{c=1}^{|C|} \frac{n_c}{n} \cdot \log\left(\frac{n_c}{n}\right)\]
with \(n\) the total number of samples, \(n_c\) and \(n_k\) the number of samples respectively belonging to class \(c\) and cluster \(k\), and finally \(n_{c,k}\) the number of samples from class \(c
\) assigned to cluster \(k\).
The conditional entropy of clusters given class \(H(K|C)\) and the entropy of clusters \(H(K)\) are defined in a symmetric manner.
Rosenberg and Hirschberg further define V-measure as the harmonic mean of homogeneity and completeness:
\[v = 2 \cdot \frac{h \cdot c}{h + c}\]
2.3.11.4. Fowlkes-Mallows scores#
The original Fowlkes-Mallows index (FMI) was intended to measure the similarity between two clustering results, which is inherently an unsupervised comparison. The supervised adaptation of the
Fowlkes-Mallows index (as implemented in sklearn.metrics.fowlkes_mallows_score) can be used when the ground truth class assignments of the samples are known. The FMI is defined as the geometric mean
of the pairwise precision and recall:
\[\text{FMI} = \frac{\text{TP}}{\sqrt{(\text{TP} + \text{FP}) (\text{TP} + \text{FN})}}\]
In the above formula:
• TP (True Positive): The number of pairs of points that are clustered together both in the true labels and in the predicted labels.
• FP (False Positive): The number of pairs of points that are clustered together in the predicted labels but not in the true labels.
• FN (False Negative): The number of pairs of points that are clustered together in the true labels but not in the predicted labels.
The score ranges from 0 to 1. A high value indicates a good similarity between two clusters.
>>> from sklearn import metrics
>>> labels_true = [0, 0, 0, 1, 1, 1]
>>> labels_pred = [0, 0, 1, 1, 2, 2]
>>> metrics.fowlkes_mallows_score(labels_true, labels_pred)
One can permute 0 and 1 in the predicted labels, rename 2 to 3 and get the same score:
>>> labels_pred = [1, 1, 0, 0, 3, 3]
>>> metrics.fowlkes_mallows_score(labels_true, labels_pred)
Perfect labeling is scored 1.0:
>>> labels_pred = labels_true[:]
>>> metrics.fowlkes_mallows_score(labels_true, labels_pred)
Bad (e.g. independent labelings) have zero scores:
>>> labels_true = [0, 1, 2, 0, 3, 4, 5, 1]
>>> labels_pred = [1, 1, 0, 0, 2, 2, 2, 2]
>>> metrics.fowlkes_mallows_score(labels_true, labels_pred)
2.3.11.5. Silhouette Coefficient#
If the ground truth labels are not known, evaluation must be performed using the model itself. The Silhouette Coefficient (sklearn.metrics.silhouette_score) is an example of such an evaluation, where
a higher Silhouette Coefficient score relates to a model with better defined clusters. The Silhouette Coefficient is defined for each sample and is composed of two scores:
• a: The mean distance between a sample and all other points in the same class.
• b: The mean distance between a sample and all other points in the next nearest cluster.
The Silhouette Coefficient s for a single sample is then given as:
\[s = \frac{b - a}{max(a, b)}\]
The Silhouette Coefficient for a set of samples is given as the mean of the Silhouette Coefficient for each sample.
>>> from sklearn import metrics
>>> from sklearn.metrics import pairwise_distances
>>> from sklearn import datasets
>>> X, y = datasets.load_iris(return_X_y=True)
In normal usage, the Silhouette Coefficient is applied to the results of a cluster analysis.
>>> import numpy as np
>>> from sklearn.cluster import KMeans
>>> kmeans_model = KMeans(n_clusters=3, random_state=1).fit(X)
>>> labels = kmeans_model.labels_
>>> metrics.silhouette_score(X, labels, metric='euclidean')
2.3.11.6. Calinski-Harabasz Index#
If the ground truth labels are not known, the Calinski-Harabasz index (sklearn.metrics.calinski_harabasz_score) - also known as the Variance Ratio Criterion - can be used to evaluate the model, where
a higher Calinski-Harabasz score relates to a model with better defined clusters.
The index is the ratio of the sum of between-clusters dispersion and of within-cluster dispersion for all clusters (where dispersion is defined as the sum of distances squared):
>>> from sklearn import metrics
>>> from sklearn.metrics import pairwise_distances
>>> from sklearn import datasets
>>> X, y = datasets.load_iris(return_X_y=True)
In normal usage, the Calinski-Harabasz index is applied to the results of a cluster analysis:
>>> import numpy as np
>>> from sklearn.cluster import KMeans
>>> kmeans_model = KMeans(n_clusters=3, random_state=1).fit(X)
>>> labels = kmeans_model.labels_
>>> metrics.calinski_harabasz_score(X, labels)
Mathematical formulation#
For a set of data \(E\) of size \(n_E\) which has been clustered into \(k\) clusters, the Calinski-Harabasz score \(s\) is defined as the ratio of the between-clusters dispersion mean and the
within-cluster dispersion:
\[s = \frac{\mathrm{tr}(B_k)}{\mathrm{tr}(W_k)} \times \frac{n_E - k}{k - 1}\]
where \(\mathrm{tr}(B_k)\) is trace of the between group dispersion matrix and \(\mathrm{tr}(W_k)\) is the trace of the within-cluster dispersion matrix defined by:
\[W_k = \sum_{q=1}^k \sum_{x \in C_q} (x - c_q) (x - c_q)^T\]
\[B_k = \sum_{q=1}^k n_q (c_q - c_E) (c_q - c_E)^T\]
with \(C_q\) the set of points in cluster \(q\), \(c_q\) the center of cluster \(q\), \(c_E\) the center of \(E\), and \(n_q\) the number of points in cluster \(q\).
2.3.11.7. Davies-Bouldin Index#
If the ground truth labels are not known, the Davies-Bouldin index (sklearn.metrics.davies_bouldin_score) can be used to evaluate the model, where a lower Davies-Bouldin index relates to a model with
better separation between the clusters.
This index signifies the average ‘similarity’ between clusters, where the similarity is a measure that compares the distance between clusters with the size of the clusters themselves.
Zero is the lowest possible score. Values closer to zero indicate a better partition.
In normal usage, the Davies-Bouldin index is applied to the results of a cluster analysis as follows:
>>> from sklearn import datasets
>>> iris = datasets.load_iris()
>>> X = iris.data
>>> from sklearn.cluster import KMeans
>>> from sklearn.metrics import davies_bouldin_score
>>> kmeans = KMeans(n_clusters=3, random_state=1).fit(X)
>>> labels = kmeans.labels_
>>> davies_bouldin_score(X, labels)
Mathematical formulation#
The index is defined as the average similarity between each cluster \(C_i\) for \(i=1, ..., k\) and its most similar one \(C_j\). In the context of this index, similarity is defined as a measure \(R_
{ij}\) that trades off:
• \(s_i\), the average distance between each point of cluster \(i\) and the centroid of that cluster – also know as cluster diameter.
• \(d_{ij}\), the distance between cluster centroids \(i\) and \(j\).
A simple choice to construct \(R_{ij}\) so that it is nonnegative and symmetric is:
\[R_{ij} = \frac{s_i + s_j}{d_{ij}}\]
Then the Davies-Bouldin index is defined as:
\[DB = \frac{1}{k} \sum_{i=1}^k \max_{i \neq j} R_{ij}\]
• Davies, David L.; Bouldin, Donald W. (1979). “A Cluster Separation Measure” IEEE Transactions on Pattern Analysis and Machine Intelligence. PAMI-1 (2): 224-227.
• Halkidi, Maria; Batistakis, Yannis; Vazirgiannis, Michalis (2001). “On Clustering Validation Techniques” Journal of Intelligent Information Systems, 17(2-3), 107-145.
2.3.11.8. Contingency Matrix#
Contingency matrix (sklearn.metrics.cluster.contingency_matrix) reports the intersection cardinality for every true/predicted cluster pair. The contingency matrix provides sufficient statistics for
all clustering metrics where the samples are independent and identically distributed and one doesn’t need to account for some instances not being clustered.
Here is an example:
>>> from sklearn.metrics.cluster import contingency_matrix
>>> x = ["a", "a", "a", "b", "b", "b"]
>>> y = [0, 0, 1, 1, 2, 2]
>>> contingency_matrix(x, y)
array([[2, 1, 0],
[0, 1, 2]])
The first row of output array indicates that there are three samples whose true cluster is “a”. Of them, two are in predicted cluster 0, one is in 1, and none is in 2. And the second row indicates
that there are three samples whose true cluster is “b”. Of them, none is in predicted cluster 0, one is in 1 and two are in 2.
A confusion matrix for classification is a square contingency matrix where the order of rows and columns correspond to a list of classes.
2.3.11.9. Pair Confusion Matrix#
The pair confusion matrix (sklearn.metrics.cluster.pair_confusion_matrix) is a 2x2 similarity matrix
\[\begin{split}C = \left[\begin{matrix} C_{00} & C_{01} \\ C_{10} & C_{11} \end{matrix}\right]\end{split}\]
between two clusterings computed by considering all pairs of samples and counting pairs that are assigned into the same or into different clusters under the true and predicted clusterings.
It has the following entries:
\(C_{00}\) : number of pairs with both clusterings having the samples not clustered together
\(C_{10}\) : number of pairs with the true label clustering having the samples clustered together but the other clustering not having the samples clustered together
\(C_{01}\) : number of pairs with the true label clustering not having the samples clustered together but the other clustering having the samples clustered together
\(C_{11}\) : number of pairs with both clusterings having the samples clustered together
Considering a pair of samples that is clustered together a positive pair, then as in binary classification the count of true negatives is \(C_{00}\), false negatives is \(C_{10}\), true positives is
\(C_{11}\) and false positives is \(C_{01}\).
Perfectly matching labelings have all non-zero entries on the diagonal regardless of actual label values:
>>> from sklearn.metrics.cluster import pair_confusion_matrix
>>> pair_confusion_matrix([0, 0, 1, 1], [0, 0, 1, 1])
array([[8, 0],
[0, 4]])
>>> pair_confusion_matrix([0, 0, 1, 1], [1, 1, 0, 0])
array([[8, 0],
[0, 4]])
Labelings that assign all classes members to the same clusters are complete but may not always be pure, hence penalized, and have some off-diagonal non-zero entries:
>>> pair_confusion_matrix([0, 0, 1, 2], [0, 0, 1, 1])
array([[8, 2],
[0, 2]])
The matrix is not symmetric:
>>> pair_confusion_matrix([0, 0, 1, 1], [0, 0, 1, 2])
array([[8, 0],
[2, 2]])
If classes members are completely split across different clusters, the assignment is totally incomplete, hence the matrix has all zero diagonal entries:
>>> pair_confusion_matrix([0, 0, 0, 0], [0, 1, 2, 3])
array([[ 0, 0],
[12, 0]]) | {"url":"https://scikit-learn.org/stable/modules/clustering.html?force_isolation=true","timestamp":"2024-11-09T19:14:29Z","content_type":"text/html","content_length":"243176","record_id":"<urn:uuid:2e33f1a9-77da-48d7-9c04-eaf8c03941d3>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00854.warc.gz"} |
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Private Math Tutoring in Hove
Receive personally tailored Math lessons from exceptional tutors in a one-on-one setting. We help you connect with the best tutor for your particular needs while offering flexible scheduling to fit
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Loki RPN Calculator
These files are for the Loki calculator. Loki is a full-screen RPN calculator that supports fractions, units, and binary math.
On the Web (JavaScript/ECMAScript)
LokiWeb is version 1.0. The following links run the calculators. The first two calculators come in two versions: one for most browsers and one for handheld units.
The small versions above do not use the depressed version of the key images. (They take too much Valium...(:-).)
For reference, these are the internal codes as used by doKey:
code meaning bin RPN web
\r Enter Y Y Y
\n Enter Y Y Y
(SP) Enter Y Y Y
! not Y - -
# population count Y - -
% mod, percent mod percent percent
& and Y - -
( cube root - - Y
) x^3 - - Y
* multiplication Y Y Y
+ addition Y Y Y
- subtraction Y Y Y
. decimal point - Y Y
/ division Y Y Y
0-9 digit Y Y Y
A enter A Y - -
B enter B Y - -
C enter C Y - -
D enter D Y - -
E enter E, EEX Y Y Y
F enter F, factorial Y Y Y
G diagonal - - Y
I inverse: 1/x - Y Y
K bacK space (<-) Y Y Y
L Last X Y Y Y
N fraction separator - - Y
P pi - - Y
R Roll down Y Y Y
S Swap X<->Y Y Y Y
T delTa percent Y Y Y
U unit separator - - Y
X Clear X Y Y Y
W Now - - Y
Y Today - - Y
Z Clear Y Y Y
[ sqrt - - Y
] x^2 - - Y
^ xor Y - -
| or Y - -
~ change sign, +/- Y Y Y
in RealBasic
Loki in RealBasic can be found at my RealBasic Software page.
in C
The current version is 3.0. Versions named loki25 and earlier are obsolete.
For the C versions (last four), the current version is 3.1. These versions include Freyja and Moon and run under most Unix/Linux systems. The first two contain IBM PC executables (for running in a
Unix command line environment). The last two contain Apple Macintosh executables (for running under Terminal on OS X). For other Unix/Linux environments, pick either and recompile. | {"url":"https://finseth.com/parts/loki.php","timestamp":"2024-11-04T05:09:16Z","content_type":"text/html","content_length":"11942","record_id":"<urn:uuid:9574ef55-e07c-45d0-b224-77287c54baeb>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00301.warc.gz"} |
Weighted Sum in AI: A Deep Dive | Stable AI Diffusion
Weighted Sum in AI: A Deep Dive
Weighted sums are at the heart of artificial intelligence, shaping the way machines understand our world. This essential method enables AI systems to make informed decisions, process complex data,
and learn from experiences, much like humans do. As we explore the significance of weighted sums across various applications in AI, from machine learning models to optimizing neural networks, we’ll
uncover their role in improving accuracy, efficiency, and adaptability in technology. By examining the challenges and innovative solutions in weighted sum calculations, we prepare to step into the
future of AI, anticipating advancements that promise to further integrate intelligent systems into our daily lives.
Fundamentals of Weighted Sum in AI
Weighted Sum in Artificial Intelligence
In the realm of Artificial Intelligence (AI), the concept of weighted sum plays a pivotal role in helping machines understand and process data. This technique is fundamental in various AI
applications, notably in algorithms that power decision-making processes. At its core, the weighted sum method involves multiplying elements in a set of numbers by corresponding weights, then summing
up the results to reach a single numerical value.
Consider an AI system designed to recommend movies. It might evaluate factors like genre popularity, user ratings, and recent viewing trends. Each of these factors has a different level of
importance, or “weight,” in determining recommendations. By applying the weighted sum approach, the system can calculate a total score for each movie, thus ranking them according to a user’s probable
Weights play a crucial role in this process. They are assigned based on the relevance of each factor, with higher weights given to more influential elements. For example, if user ratings are deemed
twice as important as genre popularity, they might be assigned a weight of 2, while genre popularity gets a weight of 1. The calculation of these weights often relies on statistical methods and
machine learning algorithms, which can adjust them over time to improve accuracy.
The mathematical expression of weighted sum is straightforward: each item’s value is multiplied by its corresponding weight, and the products are then added together. If we symbolize the values as
(v_i) and the weights as (w_i), the weighted sum ((S)) can be represented as:
[S = sum_{i=1}^{n} w_i cdot v_i]
where (n) is the number of items.
In practice, the weighted sum method is not limited to recommendations. It is integral to operations in neural networks, where it helps in processing inputs through neurons. Each input to a neuron
has a weight, reflecting how much influence it has on the neuron’s output. The neuron calculates the weighted sum of its inputs and applies a function to this sum to determine its output.
This calculation enables AI systems to handle complicated decision-making processes efficiently. It allows them to prioritize certain inputs over others, making the systems more adaptable and capable
of learning from data over time. As AI technology advances, the sophistication with which weights are determined and applied continues to evolve, enhancing the accuracy and relevance of AI-generated
Weighted sums are, therefore, foundational in the construction of intelligent systems that can perceive, learn from, and interact with their environments. By leveraging this simple yet powerful
computational approach, AI developers can create systems that more effectively mimic human decision-making processes, paving the way for more intuitive and responsive technology.
Applications of Weighted Sum in Machine Learning
Continuing from the pivotal foundation laid about weighted sums in artificial intelligence (AI), let’s dive further into how crucial this mathematical concept is, especially when it comes to the
realm of machine learning models. Machine learning, a subset of AI, thrives on algorithms that enable computers to learn from and make decisions based on data. Here, the weighted sum plays a central
role by acting as a deciding factor in various machine learning models, from linear regression to sophisticated neural networks.
In machine learning, data points often impact the outcome differently. Recognizing this, a weighted sum helps in assigning more significance to some inputs over others, ensuring that the model’s
predictions are as accurate as possible. For instance, in a health monitoring system, factors such as age, weight, and genetic history might be considered with different weights to predict the risk
of heart disease accurately.
To understand how weighted sum is applied, consider a simple linear regression model, which predicts a dependent variable based on one or more independent variables. The model calculates the weighted
sum of these independent variables, each assigned a specific weight, to predict the dependent variable. It’s like determining how much influence each variable has on the outcome.
In models that involve classification, such as distinguishing between emails marked as ‘spam’ or ‘not spam,’ weighted sums are also fundamentally essential. Each feature of the email, like the
frequency of certain words, is multiplied by its corresponding weight. The sum of these products passes through an activation function to classify the email. The operation simplifies complex
decision-making processes, allowing the machine learning model to learn from nuances in the data.
Further illustrating the importance of weighted sums is their role within neural networks, a more complex type of machine learning model inspired by human brain structures. Each neuron in a network
calculates the weighted sum of its input values, which then undergoes an activation to contribute to the network’s output. Adjusting these weights during the training process is crucial for the
network’s ability to learn and make accurate predictions.
The optimization of these weights is a focal point in training machine learning models. Techniques like gradient descent are employed to iteratively adjust weights, minimizing the difference between
the model’s predictions and actual outcomes. This ongoing adjustment is fundamental to improving a model’s accuracy over time.
Moreover, the weighted sum’s versatility extends to ensemble methods in machine learning, which combine predictions from multiple models to produce a final output. By assigning different weights to
the predictions from various models based on their accuracy, these ensemble methods can outperform individual models, showcasing the weighted sum’s capability to harmonize diverse inputs towards a
common goal.
In conclusion, the application of weighted sums in machine learning models is vast and varied, underpinning the very mechanisms that allow machines to learn from data. Through the intelligent
weighting and summing of inputs, machine learning models achieve the delicate balance of valuing certain data points over others, tailoring their predictions to reflect the complexities of the real
world. This mathematical concept not only enhances the precision of models but also significantly contributes to the evolution of machine intelligence, enabling systems to make decisions with a
degree of nuance once thought exclusive to human judgment.
Optimizing Weighted Sum in Neural Networks
Optimizing Weighted Sums in Neural Networks: Key Strategies
Optimizing weighted sums in neural networks is crucial for enhancing the performance and accuracy of various AI applications. Weighted sums serve as the foundation for calculating outputs within
neural networks, where each input is assigned a weight that signifies its importance. The challenge lies in adjusting these weights to ensure the network learns correctly from the data it processes.
This article delves into the strategies employed to optimize these weights, ensuring neural networks can make accurate predictions or decisions.
Gradient Descent: The Go-To Method
A primary method for optimizing weights is gradient descent. This technique involves iteratively adjusting the weights to minimize the difference between the actual output of the network and the
expected output. By calculating the gradient of the loss function (which measures the error of the network’s predictions), gradient descent makes it possible to find the direction in which weights
should be adjusted to reduce error.
Backpropagation: Learning Through Feedback
Backpropagation complements gradient descent by propagating the error backward through the network. After each forward pass through the network (calculating outputs based on the current weights),
backpropagation calculates the error at the output and distributes this error back through the network layers. This process helps adjust the weights in such a way that the network learns from the
mispredictions it has made.
Regularization: Preventing Overfitting
To enhance the generalization of neural networks, regularization techniques are applied. These methods add a penalty term to the loss function to discourage the weights from becoming too large, which
can lead to overfitting — when a model learns the training data too well, including its noise, leading to poor performance on new data. L1 and L2 regularization are popular choices, with L2
regularization (also known as weight decay) being particularly common in neural network optimization.
Stochastic Gradient Descent (SGD) and Minibatch Learning
While gradient descent adjusts weights based on the entire dataset, this can be inefficient for large datasets. Stochastic Gradient Descent (SGD) offers a solution by updating weights based on
individual training examples or small batches. This approach can speed up the learning process and help escape local minima — points where the model stops learning because it finds a small error
value that isn’t the overall lowest possible error.
Learning Rate Adaptation
Another strategy is adjusting the learning rate — the size of the steps taken during weight adjustment. A too-large learning rate can cause weights to overshoot the optimal values, while a too-small
learning rate can slow down learning significantly. Adaptive learning rate methods, like Adam (Adaptive Moment Estimation) and RMSprop (Root Mean Square Propagation), help by modifying the learning
rate as learning progresses, based on how quickly or slowly the optimization seems to be progressing.
Momentum and Nesterov Accelerated Gradient
To further enhance the optimization process, methods like momentum and Nesterov accelerated gradient take into account the direction and speed of the weight adjustments. By doing so, they aim to
accelerate the learning when it’s heading in the right direction and dampen the updates when the direction changes, stabilizing the optimization process.
In the landscape of neural network optimization, these strategies play pivotal roles in refining weighted sums to build intelligent systems capable of making accurate predictions and decisions.
Through gradient descent, backpropagation, regularization, batch learning, and adaptive learning rates, neural networks learn to adjust their weights effectively, paving the way for advancements in
artificial intelligence that continue to transform the digital world.
Challenges and Solutions in Weighted Sum Calculations
Moving into the nuanced obstacles encountered with weighted sum calculations in artificial intelligence (AI), experts face several pressing challenges that demand innovative solutions. One notable
issue is the handling of large datasets, which can significantly slow down the computation process. With the explosion of big data, AI systems frequently must process vast arrays of information,
making the efficient calculation of weighted sums a critical concern.
To address this challenge, AI professionals implement various optimization techniques. Among these, utilizing more sophisticated algorithms capable of parallel processing has shown promising results.
By distributing the workload across multiple processors, these algorithms can significantly reduce computation times, making the handling of big data more manageable.
Moreover, the accuracy of weighted sum calculations also comes under scrutiny, especially in complex systems where minor errors can have significant repercussions. The determination of appropriate
weights is pivotal to this end. Automating the weight assignment process using adaptive algorithms can help. These algorithms adjust weights in real-time based on the system’s performance, leading to
more accurate outcomes over time.
Another hurdle is the inherently static nature of predefined weights in dynamic environments. Real-world scenarios often require AI systems to adapt and learn from new data. The introduction of
machine learning techniques, such as reinforcement learning, has been a game-changer in this aspect. By allowing the system to dynamically adjust weights based on feedback from its environment, AI
applications can remain relevant and effective even as conditions change.
The balance between model complexity and interpretability also poses a challenge. Weighted sum calculations can quickly become convoluted, making it difficult for experts to trace how decisions are
made. This lack of transparency can be a significant barrier, particularly in fields requiring clear audit trails, such as finance and healthcare. Efforts to develop more interpretable models,
without sacrificing performance, are ongoing. Simplifying models to improve understandability, while employing techniques like feature selection to maintain effectiveness, are some of the strategies
being explored.
Lastly, the risk of overfitting due to improperly tuned weights is a persistent concern. Overfitting occurs when a model is too closely aligned with the training data, hindering its ability to
generalize to new datasets. Regularization techniques, which introduce a penalty term to the loss function used to calculate weights, have proven valuable. These techniques discourage the model from
placing too much emphasis on any single feature, helping to prevent overfitting and promoting more robust AI applications.
In sum, while weighted sum calculations are foundational to AI, they are not without their challenges. Addressing these issues requires a blend of advanced computational strategies, algorithmic
innovations, and a careful balancing of model complexity against interpretability and generalizability. As the field of AI continues to evolve, so too will the techniques used to ensure that weighted
sums contribute positively to the development of intelligent, responsive, and adaptable systems.
The Future of Weighted Sum in AI Technology
Exploring the Horizon: The Future of Weighted Sum in AI
As we dive deeper into the realm of artificial intelligence (AI), the concept of the weighted sum continues to play a pivotal role in shaping the future of AI technologies. This vital component, a
cornerstone in the realm of machine learning and decision-making processes, is set for fascinating evolutions that promise to redefine its application and effectiveness.
The journey ahead for weighted sum in AI is illuminated by the advent of more sophisticated algorithms and the relentless pursuit of AI that mirrors human intelligence more closely than ever before.
The immediate future foresees the integration of advanced computational techniques that aim to enhance the efficiency and accuracy of weighted sum calculations.
One of the thrilling advancements on the horizon is the development of adaptive weighting mechanisms. These mechanisms are intelligent enough to adjust the weights dynamically based on real-time data
and changing scenarios. This leap forward would mark a significant milestone in making AI systems more responsive and capable of making nuanced decisions in complex environments.
Furthermore, the fusion of quantum computing with AI presents an exhilarating prospect for weighted sum applications. Quantum computing, with its unparalleled processing power, could revolutionize
how weighted sums are calculated, making it possible to process vast datasets at speeds hitherto deemed impossible. This fusion could unlock new possibilities in AI, from solving intricate
optimization problems to accelerating machine learning tasks dramatically.
In addition to these technological advancements, there’s a growing emphasis on the ethical implications of weighted sums in AI. As these systems increasingly impact daily life, from healthcare to
financial services, ensuring transparency and fairness in how weights are assigned becomes imperative. Future developments might include ethical frameworks and guidelines that govern the application
of weighted sums, ensuring they contribute to equitable and just AI solutions.
Moreover, the future is likely to witness enhanced collaboration between AI and human intelligence, particularly in refining the weighted sum process. By combining the intuition and expertise of
humans with the computational prowess of AI, the process of determining and adjusting weights could become more nuanced and aligned with human values.
The integration of AI with emerging technologies and the emphasis on ethical considerations points to a vibrant and transformative future for the use of weighted sum in AI. As we stand on the cusp of
these advancements, the potential for weighted sum to contribute to more intelligent, efficient, and equitable AI systems is undeniably exciting. The journey ahead promises not just enhancements in
computational techniques but a closer alignment between AI and human intelligence, heralding a new era of intelligent systems capable of making decisions with remarkable depth and subtlety.
The exploration of weighted sums in AI reveals a vast landscape where mathematics meets technology, bringing us closer to creating machines that think, learn, and decide with precision and nuance. As
we embrace the future of AI technology, the continuous evolution of weighted sums stands as a testament to human ingenuity, driving us toward a world where AI systems not only support but enhance
human decision-making processes. The potential for growth and innovation in this area is boundless, offering a glimpse into a future where AI and human intelligence coalesce more seamlessly than ever
before, revolutionizing the way we interact with technology and each other.
Emad Morpheus is a tech enthusiast with a unique flair for AI and art. Backed by a Computer Science background, he dove into the captivating world of AI-driven image generation five years ago. Since
then, he has been honing his skills and sharing his insights on AI art creation through his blog posts. Outside his tech-art sphere, Emad enjoys photography, hiking, and piano.
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4th Grade M-STEP Math Worksheets: FREE & Printable
Are you looking for a comprehensive practice resource for the 4th-grade M-STEP math test? The 4th Grade M-STEP Math Worksheets is what you need.
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What is the best formula to calculate pi? - Techzle
Hi, I’ve been looking for the best formula to calculate pi for a while now. I am very interested in this.
Asker: Clara, 13 years old
Dear Clara,
There are many ways to calculate Pi.
On the webpage APPROACHES TO THE NUMBER PI THROUGH THE HISTORY OF MATHEMATICS you will find a nice overview of some possibilities that have been historically devised by mathematicians.
The approximation through regular polygons with increasing number of sides is perhaps the most comprehensible approach, but it certainly won’t give you the “best” formula for calculating Pi, because
it takes too long to find a sufficient number of digits of Pi. This is called convergence rate in mathematics.
At the time I found it fascinating to note that you can quickly calculate a good approximation for Pi numerically using the definite integral of the function 4/(1+x^2) between 0 and 1. Which then
returns 4×ArcTan[1]=Pi
As soon as you have learned something about numerical integration – which is probably not the case given your age – you can test it yourself.
Philippe J. Roussel
Answered by
eng. Philippe Roussel
Microelectronics Reliability
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Frequentist and Bayesian approaches to interpreting probability and making decisions based on data
Frequentist v/s Baysian
Photo by Robert Stump on Unsplash
Frequentist and Bayesian approaches are two different ways to interpret probability and make decisions based on data.
In frequentist statistics, the probability is used to describe the likelihood of an event occurring in the long run, based on the frequency of past events. For example, if you flip a coin 10 times
and it lands heads 7 times, the frequentist probability of the coin landing heads on the next flip is 7/10, or 70%.
On the other hand, in Bayesian statistics, the probability is used to describe the degree of belief that an event will occur. This belief can be based on both past data and personal judgment. In the
coin flipping example, a Bayesian might use past data (e.g. the coin landing heads 7 out of 10 times) as well as their own personal beliefs (e.g. the coin is fair) to estimate the probability of the
coin landing heads on the next flip.
So, the main difference between frequentist and Bayesian approaches is that frequentist probability is based on the frequency of past events, while Bayesian probability is based on both past data
and personal judgment.
Here are a few more examples to get a clear understanding.
Example 1:
Imagine you are trying to estimate the probability that a certain medical treatment will be effective in reducing blood pressure. In a frequentist approach, you would conduct a randomized controlled
trial in which you give the treatment to a group of patients and measure their blood pressure before and after the treatment. You would then calculate the percentage of patients who experienced a
reduction in blood pressure and use this as the probability of the treatment being effective.
In a Bayesian approach, you might also use data from the randomized controlled trial to estimate the probability of the treatment being effective. However, you might also incorporate other factors,
such as your own expertise and knowledge about similar treatments, to revise your probability estimate.
Example 2:
Imagine you are trying to estimate the probability that a certain stock will go up in value over the next year. In a frequentist approach, you might look at the historical data on the stock’s
performance and calculate the percentage of times it has gone up in the past.
In a Bayesian approach, you might also consider the historical data on the stock’s performance, but you might also incorporate other factors, such as the overall performance of the stock market, the
performance of similar stocks, and your own expert judgment about the company’s future prospects. | {"url":"https://anyesh.medium.com/frequentist-and-bayesian-approaches-to-interpreting-probability-and-making-decisions-based-on-data-8c4ad5891272?source=user_profile_page---------8-------------cf006d136db4---------------","timestamp":"2024-11-13T01:12:10Z","content_type":"text/html","content_length":"99192","record_id":"<urn:uuid:25565e44-a893-4370-9cde-73213d42ae6b>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00683.warc.gz"} |
Excel Formula for Running Total
In Excel, you can easily create a running total by adding 1 to the total in the cell above using a simple formula. This formula allows you to continuously update the running total as you enter new
values. By understanding and implementing this formula, you can efficiently track cumulative totals in your Excel spreadsheets without the need for complex calculations.
To create a running total, you can use the formula =A1+1. In this formula, A1 represents the cell above where you want to add 1 to the total. The + operator is used to add 1 to the value in cell A1,
resulting in a running total. By copying the formula to subsequent cells, you can extend the running total to multiple rows.
For example, if you have a column of values in column A and you enter the formula =A2+1 in cell A3, it will add 1 to the value in cell A2, creating a running total. Copying the formula down to cell
A4 will continue the running total by adding 1 to the value in cell A3, and so on.
By utilizing this simple formula, you can easily create and update running totals in Excel, allowing you to keep track of cumulative values in your data.
The formula you can use in Excel to add 1 to the total in the cell above as a running total is:
1. In this formula, A1 represents the cell above where you want to add 1 to the total.
2. The + operator is used to add 1 to the value in cell A1.
3. The result of the formula will be the running total of the values above.
Let's say you have the following values in column A:
| A |
| 5 |
| 3 |
| 2 |
If you enter the formula =A2+1 in cell A3, it will add 1 to the value in cell A2, resulting in a running total:
| A |
| 5 |
| 3 |
| 4 |
Similarly, if you copy the formula down to cell A4, it will add 1 to the value in cell A3, resulting in a new running total:
| A |
| 5 |
| 3 |
| 4 |
| 5 |
And so on, the formula will continue to add 1 to the total in the cell above, creating a running total in column A. | {"url":"https://codepal.ai/excel-formula-generator/query/UCNXNjuj/excel-formula-running-total","timestamp":"2024-11-13T01:01:27Z","content_type":"text/html","content_length":"81039","record_id":"<urn:uuid:82e8e857-5e9f-4ca3-aab4-480db1531660>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00410.warc.gz"} |
Export as BibNote ® (generic): “Superfluidity of helium II near the λ point”
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Reviews of topical problems
Superfluidity of helium II near the λ point
V.L. Ginzburg^ a
A.A. Sobaynin ^a Lebedev Physical Institute, Russian Academy of Sciences, Leninsky prosp. 53, Moscow, 119991, Russian Federation
The properties (and particularly the superfluidity) of liquid helium near the $\lambda$ point have long been and still are objects of numerous investigations. Nonetheless, much of the problem remains
unsolved from both the theoretical and the experimental points of view. The main reason is the small correlation length in helium II even when the distance from the $\lambda$ point is as small as
hundredths of a degree, so that to reveal a number of specific effects one must work quite close to the $\lambda$ point. This article treats in detail the phenomenological theory of superfluidity of
helium near the $\lambda$ point, a theory whose development dates back to 1958, and which is based on the use of the complex order parameter $\Psi=\eta e^\varphi$ (the density of the superfluid part
of helium is here $\rho_s=m\eta^2$, and the velocity of this helium component is $v=(\hbar/m)\nabla\varphi$ where $m$ is the mass of the helium atom). In addition to formulating the general equations
of the theory and discussing the regions where they are valid, we consider a number of concrete effects. The predictions of the theory are quite rich in content and can be verified in experiment. The
main purpose of the article is just to contribute to such a verification.
DOI: 10.1070/PU1976v019n10ABEH005336 URL:
Ginzburg V L, Sobaynin A A "Superfluidity of helium II near the λ point"
Sov. Phys. Usp. 19
773–812 (1976)
BibTex BibNote ® (generic)BibNote ® (RIS)MedlineRefWorks
%0 Journal Article
%T Superfluidity of helium II near the λ point
%A V. L. Ginzburg
%A A. A. Sobaynin
%I Physics-Uspekhi
%D 1976
%J Phys. Usp.
%V 19
%N 10
%P 773-812
%U https://ufn.ru/en/articles/1976/10/a/
%U https://doi.org/10.1070/PU1976v019n10ABEH005336
Оригинал: Гинзбург В Л, Собянин А А «Сверхтекучесть гелия II вблизи λ-точки» УФН 120 153–216 (1976); DOI: 10.3367/UFNr.0120.197610a.0153 | {"url":"https://ufn.ru/en/articles/1976/10/a/citation/en/bibnote_gen.html","timestamp":"2024-11-07T19:35:17Z","content_type":"text/html","content_length":"18996","record_id":"<urn:uuid:93496343-ad55-4709-a588-4b7faa7c2850>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00026.warc.gz"} |
The first step is to select a critical level, with respect to chart datum and in meters, for Charlottetown. As a reference, the January 21st storm, of 2000 caused a peak level of 4.2 m with respect
to this datum. The level is selected using the dropdown menu below.
The next step is to select an area from the map below. Two images should then appear: the probablity plot and flooded DEM corresponding to the chosen flood level.
The probability plot shows the probability that the critical level will not be exceeded by the date shown. Notice that the probability decreases in steps. Each step corresponds to a particular winter
where the surges are large and the probability of flooding is significantly higher than in summer. The two curves are for sea level increases of 3mm per year (observed), and 7mm per year (predicted
to occur by IPCC), over the next century.
The flooded DEM (courtesy COGS), shows the extent of flooding that would occur if a critical level is reached. | {"url":"http://cmep.ca/shelfFloodRisk_Charlottetown.php","timestamp":"2024-11-14T12:00:02Z","content_type":"text/html","content_length":"9369","record_id":"<urn:uuid:3825c297-d1ee-4f48-a76b-0a9ddb730742>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00365.warc.gz"} |
American Mathematical Society
Implementation aspects of band Lanczos algorithms for computation of eigenvalues of large sparse symmetric matrices
HTML articles powered by AMS MathViewer
by Axel Ruhe PDF
Math. Comp. 33 (1979), 680-687 Request permission
A band Lanczos algorithm for the iterative computation of eigenvalues and eigenvectors of a large sparse symmetric matrix is described and tested on numerical examples. It starts with a p dimensional
subspace, and computes an orthonormal basis for the Krylov spaces of A, generated from this starting subspace, in which A is represented by a $2p + 1$ band matrix, whose eigenvalues can be computed.
Special emphasis is given to devising an implementation that gives a satisfactory numerical orthogonality, with a simple program and few arithmetic operations. References
• D. Boley and G. H. Golub, Inverse eigenvalue problems for band matrices, Numerical analysis (Proc. 7th Biennial Conf., Univ. Dundee, Dundee, 1977) Lecture Notes in Math., Vol. 630, Springer,
Berlin, 1978, pp. 23–31. MR 0474741
• Jane Cullum, The simultaneous computation of a few of the algebraically largest and smallest eigenvalues of a large, sparse, symmetric matrix, BIT 18 (1978), no. 3, 265–275. MR 508328, DOI
10.1007/BF01930896 J. CULLUM & W. E. DONATH, A Block Generalization of the Symmetric s-Step Lanczos Algorithm, Rep. RC4845, IBM Research, Yorktown Heights, N. Y., 1974.
• J. W. Daniel, W. B. Gragg, L. Kaufman, and G. W. Stewart, Reorthogonalization and stable algorithms for updating the Gram-Schmidt $QR$ factorization, Math. Comp. 30 (1976), no. 136, 772–795. MR
431641, DOI 10.1090/S0025-5718-1976-0431641-8
• G. H. Golub and R. Underwood, The block Lanczos method for computing eigenvalues, Mathematical software, III (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1977) Publ. Math.
Res. Center, No. 39, Academic Press, New York, 1977, pp. 361–377. MR 0474742 W. KAHAN & B. PARLETT, An Analysis of Lanczos Algorithms for Symmetric Matrices, Tech. Rep. ERL-M467, Univ.
California, Berkeley, 1974.
• Cornelius Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Research Nat. Bur. Standards 45 (1950), 255–282. MR 0042791 J.
G. LEWIS, Algorithms for Sparse Matrix Eigenvalue Problems, Rep. STAN-CS-77-595, Stanford, 1977.
• C. C. Paige, Practical use of the symmetric Lanczos process with re-orthogonalization, Nordisk Tidskr. Informationsbehandling (BIT) 10 (1970), 183–195. MR 264839, DOI 10.1007/bf01936866 C. C.
PAIGE, The Computation of Eigenvalues and Eigenvectors of Very Large Sparse Matrices, Ph. D. Thesis, London University, 1971.
• C. C. Paige, Computational variants of the Lanczos method for the eigenproblem, J. Inst. Math. Appl. 10 (1972), 373–381. MR 334480 B. PARLETT & D. S. SCOTT, The Lanczos Algorithm with Implicit
Deflation, Rep. ERL M77/70, Univ. California, Berkeley, 1977.
• A. Ruhe, Iterative eigenvalue algorithms for large symmetric matrices, Numerische Behandlung von Eigenwertaufgaben (Tagung, Oberwolfach, 1972), Internat. Schr. Numer. Math., Band 24, Birkhäuser,
Basel, 1974, pp. 97–115. MR 0416000
• Axel Ruhe, Computation of eigenvalues and eigenvectors, Sparse matrix techniques (Adv. Course, Technical Univ. Denmark, Copenhagen, 1976) Lecture Notes in Math., Vol. 572, Springer, Berlin, 1977,
pp. 130–184. MR 0440891
• Axel Ruhe and Torbjörn Wiberg, The method of conjugate gradients used in inverse iteration, Nordisk Tidskr. Informationsbehandling (BIT) 12 (1972), 543–554. MR 327013, DOI 10.1007/bf01932964 R.
UNDERWOOD, An Iterative Block Lanczos Method for the Solution of Large Sparse Symmetric Eigenproblems, Rep. STAN-CS-75-496, Stanford University, 1975.
• Handbook for automatic computation. Vol. II, Die Grundlehren der mathematischen Wissenschaften, Band 186, Springer-Verlag, New York-Heidelberg, 1971. Linear algebra; Compiled by J. H. Wilkinson
and C. Reinsch. MR 0461856
Similar Articles
• Retrieve articles in Mathematics of Computation with MSC: 65F15
• Retrieve articles in all journals with MSC: 65F15
Additional Information
• © Copyright 1979 American Mathematical Society
• Journal: Math. Comp. 33 (1979), 680-687
• MSC: Primary 65F15
• DOI: https://doi.org/10.1090/S0025-5718-1979-0521282-9
• MathSciNet review: 521282 | {"url":"https://www.ams.org/journals/mcom/1979-33-146/S0025-5718-1979-0521282-9/?active=current","timestamp":"2024-11-07T04:47:24Z","content_type":"text/html","content_length":"63333","record_id":"<urn:uuid:6872a5bf-5a84-48ba-8203-4368fefd6716>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00505.warc.gz"} |
The Division Symbol: Sharing Equally
Lesson Video: The Division Symbol: Sharing Equally Mathematics • Third Year of Primary School
In this video, we will learn how to use the division symbol to write equations to find the number of things in each group when we have 2, 3, 4, 5, or 10 equal groups.
Video Transcript
The Division Symbol: Sharing Equally
In this video, we’re going to learn how to use the division symbol to write number sentences that help us find the number of things in each group when we have two, three, four, five, or 10 equal
Here are 12 fish. At the moment, they’re all spread out. But it’s often safer for small fish like this to swim together in groups. Let’s imagine that these fish are split up into two equal groups. We
could say that the 12 fish are being divided by two. How many fish will there be in each group?
We could share our 12 fish one at a time into two separate groups. We could put one group on either side. Let’s get sharing. One for the first group and one for the second group. Another fish for the
first group; that’s now two we’ve got. And another fish goes into our second group. And we can keep sharing our fish one at a time into two groups. And with each round of sharing, we need to make
sure that we share the same amount into each group.
There we are; we’ve divided all 12 fish into two equal groups. There are six fish in this group, and there are also six fish in this group. 12 divided by two equals six.
Now, so far, we’ve written what we’ve done using numbers and words. But wouldn’t it be good if there was a symbol we could use that meant divide? Then, we could write a number sentence or equation to
show what we’ve done. Well, there is such thing as a division symbol. It’s made up of two dots with a straight line in between. We use this symbol when we want to show that a number has been divided
by another number. And in this video, we’re using it for sharing into that number of equal groups.
Let’s use this division symbol to write down what’s happened to our fish. We did have 12 fish. This was the whole amount. It was the number that we started with. And then, we split this number into
equal groups. And as we’ve said already, another word for “split” is “divide”. So, we can write the division symbol here, two dots and a line in between. This shows that we’ve shared out or split up
How many equal groups did we share the fish into? There were two groups, weren’t there? By the way, how do we know our groups are equal? We know that the word “equal” means the same. And we know our
two groups are equal because they contain the same number of fish. So, the final number in our equation is the number of fish in each group. 12 divided by two equals six.
This is exactly the same as what we’ve written at the top. We’ve just used numbers and symbols this time to write it as a number sentence. When we start with a whole amount and we split it into a
number of equal groups, the answer will be the number that there are in each group.
Let’s have a go at using the division symbol now. We’re going to answer some questions where we need to divide a number of objects into equal groups.
There are 14 carrots. The carrots will be shared equally between seven rabbits. Each rabbit will get what carrots. Find the missing number: 14 divided by seven equals what.
Our problem begins with 14 carrots. And to help us imagine them, we’re given a picture that shows them. But something’s going to happen to these carrots. We’re told that they’re going to be shared
equally. When something is shared equally, it’s divided. This is a question all about division.
So, how many groups will the carrots be shared into? Well, if we look carefully, we can see who’s doing the dividing here. The carrots are going to be shared equally between seven rabbits. So, we
need to split our 14 carrots into seven equal groups. To help us do this, let’s sketch the groups.
Here are seven circles to represent the seven rabbits, one circle for each rabbit. Now, we need to make sure that these groups are equal. So, each time we share the carrots, each group needs to get
the same amount. Let’s use counters to represent carrots.
To begin with, let’s give each of the seven groups one carrot. So, that’s one, two, three, four, five, six, seven. There are seven groups, and they each now have one carrot. But we’ve got more
carrots left. We’re going to need to give each group another carrot. Now, there are no more carrots left to share. We’ve divided all 14 of them. And they’ve been split up into equal groups.
We know that the groups are equal because if we look at them quickly, we can see that they’ve got the same number of counters in them. Remember, we said our counters represented carrots. So, now, we
can answer the first part of our question. Each rabbit will get what carrots. Well, we managed to put two counters in each group, didn’t we? So, we know that each rabbit will get two carrots.
In the final part of our question, we’re given a number sentence and we need to fill in the missing number. Our number sentence begins with the number 14. We know this is the whole amount. This was
the number of carrots that we began with. But then, we have an interesting symbol, two dots with a line in between them.
Perhaps you were listening carefully when the text of this question was read out at the very start. If so, you’ll know what this symbol means. That’s right. It means divided by. We use the division
symbol whenever an amount is shared into equal groups. We started with 14 carrots. They were divided or shared between seven rabbits. And the missing number in our number sentence, that’s the answer
to our division, is the number of carrots in each group. Of course, the answer’s two.
If there are 14 carrots and they’re shared equally between seven rabbits, each rabbit will get two carrots. And we can write this using the division symbol as 14 divided by seven equals two. Both of
our missing numbers are the number two.
Natalie is preparing breakfast. She made 20 sandwiches and put them on 10 plates. Choose the calculation that is equal to the number of sandwiches on each plate. 20 take away 10, 20 divided by two,
20 plus 10, or 20 divided by 10. How many sandwiches are on each plate?
This problem describes Natalie who’s preparing a really big breakfast. Perhaps she’s made it for her whole class because she’s made 20 sandwiches. But you know, she hasn’t just put them in one big
group; she split them up. We know this because we’re told that she’s put the 20 sandwiches on 10 plates. We could say she’s shared them out. And we can see a picture to help us.
In the first part of the problem, we’re told to choose the calculation that is equal to the number of sandwiches on each plate. And we’re given four possible calculations to choose from. Now, we can
see in the picture how many sandwiches are on each plate. But if we didn’t have the picture to help us, which of these calculations would we use?
To find the answer, we need to think carefully about what Natalie’s done here. The first thing that she’s done is to make 20 sandwiches. So, this is the number she begins with, 20. But then, what
does she do? She puts them on 10 plates, doesn’t she? And if we look at each plate, we can see that they all have an equal number of sandwiches on. These 20 sandwiches have been shared out. And
another word for “shared” is “divided.” And the symbol that we use when a number is divided into equal groups is the division symbol, which is a line with two dots either side.
How many equal groups has Natalie made? Well, she shared out her sandwiches onto 10 plates. So, she’s divided 20 by 10. To find out the number of sandwiches on each plate then, we need to find the
answer to 20 divided by 10. And if we look at our four calculations, this is one of them.
We might use 20 take away 10 if we wanted to find out how many sandwiches Natalie had left after making 20 and then eating 10 of them. 20 divided by two is a division calculation. But this would be
the number of sandwiches in each group if Natalie shared them onto two plates. And we know that adding is to find the total of two numbers. So, 20 plus 10 would be to find the number of sandwiches if
Natalie perhaps made 20 sandwiches and then made another 10.
So, that’s how we know the correct answer is 20 divided by 10. And we can use this to calculate the number of sandwiches on each plate. 20 divided by 10 equals two. 10 groups of two make 20. So, if
Natalie makes 20 sandwiches and puts them on 10 plates, the calculation that’s equal to the number of sandwiches on each plate is 20 divided by 10. And the number of sandwiches that are on each plate
is two.
What have we learned in this video? Firstly, we’ve been introduced to the division symbol. We know that this is a line in between two dots. We’ve also learned how to use the division symbol to write
number sentences to find the number of things in each group when we share an amount into equal groups. | {"url":"https://www.nagwa.com/en/videos/786170270838/","timestamp":"2024-11-10T08:17:11Z","content_type":"text/html","content_length":"262191","record_id":"<urn:uuid:858c2adb-541f-417c-8feb-996bba76e7f9>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00370.warc.gz"} |
Algebra 1 Tutoring in Westchester | Grade Potential
Book a Session With an Algebra 1 Tutor in Westchester
An instructor can help a student with Algebra 1 by providing instruction on basic ideas such as variables and equations. The tutor can further assist the learner with further complex material such as
factoring and polynomials.
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Moreover, our tutors are not required to adhere to a specific curriculum; instead, they are encouraged to design a custom-tailored approach for every student.
How can Grade Potential Westchester Algebra 1 tutors ensure my learner succeed?
When you partner with Grade Potential mathematics educators, you will get a customized lesson plan that is most convenient for your learner. This empowers the tutor to adapt to your student's
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Our one-on-one Algebra 1 teachers can come alongside the learner’s primary education and guide them with additional tutoring to ensure expertise in any concepts they might be struggling to grasp.
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If you're not sure how the Algebra 1 teacher will work with your student's existing coursework, we can help by talking about your needs, availability, and determining the ideal learning plan and
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That might mean meeting with the learner from a laptop between classes or sports, at your home, or the library – whatever suits your needs.
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Computational Physics
My background is in computational physics – solving physics problems via computer modelling in order to compare against the traditional pen-and-paper and experimental investigations. In many cases,
either the problem cannot be solved on paper (and computation is the only way to ‘‘approach’’ the theory), or the experiments cannot actually be performed (and so a simulation of the experiment is
only way to ‘’test’’ the theory).
My research focused on large-scale parallel simulations to study problems in statistical physics, fluid dynamics and quantum mechanics. | {"url":"https://anjackson.net/work/physics/","timestamp":"2024-11-06T14:41:42Z","content_type":"text/html","content_length":"38900","record_id":"<urn:uuid:7403ea0e-8cef-4442-a0da-4c01aedadd7f>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00584.warc.gz"} |
Area of Scalene Triangles Worksheets
Scalene triangles have no equal sides nor equal angles, and finding their areas can be a taxing process unless your child is clear with the how-to’s of Heron’s formula. Use the complete set of
printable practice problems after incorporating the step-by-step techniques to determine area of scalene triangles with integers/decimals as their dimensions. We recommend the entire collection of
area of scalene triangles pdf worksheet set included with answer keys, especially for high school students. Our free worksheet is a compulsive try!
Select the Measurement Units
Area of Scalene Triangles | Integers- Type 1
With this high school practice set, learn to calculate the area of scalene triangles by applying the formula A = √ [s (s - a) (s - b) (s - c)], where 's' is the semiperimeter. Substitute the
semiperimeter and the sides offered as integers to find the area.
Area of Scalene Triangles | Integers- Type 2
Determine the area of scalene triangles using a good mix of given figures and well-defined word problems involving different integer values for sides (a, b, c). Implement the correct formula to get
accurate answers after rounding them up to 2 decimal places.
Area of Scalene Triangles | Decimals- Type 1
Find the height of the isosceles triangles presented in these worksheets for grade 6, grade 7, and grade 8. Substitute the decimal dimensions in the formula A = 1/2 * b * h to compute the area of the
isosceles triangles.
Area of Scalene Triangles | Decimals- Type 2
Use Heron’s formula and work your way around figures as well as multiple word problems to find solutions when triangles have decimal values as sides. Decimals amp up the difficulty levels and demand
clear solutions with a practiced technique. Get this high school pdf worksheet set for expert-level practice of finding area of scalene triangles. | {"url":"https://www.mathworksheets4kids.com/area-scalene-triangles.php","timestamp":"2024-11-03T13:13:53Z","content_type":"text/html","content_length":"36069","record_id":"<urn:uuid:b5277026-ca78-4283-8bf1-ea65fa492b91>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00504.warc.gz"} |
6.8 Scenarios and Exercises
Intended learning outcomes: Operation time versus operation cost: disclose the effect of varying setup time and batch size. Calculate the effect of cellular manufacturing on lead-time reduction.
Perform line balancing through harmonizing the content of work. Determine the number of Kanban cards.
Course section 6.8: Subsections and their intended learning outcomes
Course 6: Sections and their intended learning outcomes | {"url":"https://opess.ethz.ch/course/course6/6-7-scenarios-and-exercises/","timestamp":"2024-11-05T15:44:45Z","content_type":"text/html","content_length":"156902","record_id":"<urn:uuid:6475a77b-f070-4e29-bab4-d2ec39d51605>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00010.warc.gz"} |
How to Calculate CPM and eCPM - Freestar
Publishers can use CPM (Cost Per Mille) and eCPM (Effective Cost Per Mille) as ad revenue models to monetize their websites or app. While they are related, they serve slightly different purposes and
are used together to optimize ad revenue. Let’s dive into each one.
What is CPM?
CPM stands for cost per mille, which means the cost per thousand impressions. eCPM, on the other hand, stands for effective cost per mille, a metric used to measure the revenue earned per thousand
CPM Formula
To calculate CPM, you need to divide the cost of advertising by the total number of impressions and then multiply the result by 1000. The formula for CPM can be written as follows:
CPM = (Cost of advertising / Number of impressions) * 1000
For example, if the cost of advertising is $100 and the number of impressions is 10,000, then the CPM would be:
CPM = ($100 / 10,000) * 1000 = $10
What is eCPM?
eCPM stands for “effective cost per mille,” a metric used in digital advertising to measure the revenue earned per thousand impressions. In other words, it estimates how much money an advertiser
earns for every thousand ad impressions served.
eCPM Formula
To calculate eCPM, you need to divide the total earnings by the total number of impressions and then multiply the result by 1000. The formula for eCPM is written as:
eCPM = (total earnings/number of impressions) * 1000
For example, if the total earnings are $500 and the number of impressions is 10,000, then the eCPM would be:
eCPM = ($500 / 10,000) * 1000 = $50
eCPM is a useful metric because it allows advertisers to compare the revenue earned from different advertising campaigns, even if the campaigns have different CPMs.
Benefits of Using CPM and eCPM
CPM (cost per thousand impressions) and eCPM (effective cost per thousand impressions) are important metrics used in digital advertising to measure the revenue publishers earn for their ad space.
Here are some of the benefits of using CPM and eCPM as a publisher:
1. Predictable Revenue: CPM and eCPM provide a predictable revenue stream to publishers. With CPM, publishers know exactly how much they will earn for every thousand ad impressions served. This
allows publishers to plan their revenue streams and make better business decisions.
2. Monetize All Impressions: CPM and eCPM help publishers monetize all of their ad impressions. By setting a CPM or eCPM rate, publishers can ensure that they are earning revenue for every
impression served, even if they don’t sell the ad space directly.
3. Optimize Ad Inventory: CPM and eCPM help publishers optimize their ad inventory. By tracking CPM and eCPM rates, publishers can identify which ad formats and placements generate the most revenue.
This allows publishers to adjust their ad inventory to maximize revenue.
4. Increased Revenue: CPM and eCPM help publishers increase their revenue. By optimizing their ad inventory and setting competitive CPM or eCPM rates, publishers can attract more advertisers and
generate more revenue from their ad space.
5. Simple to Implement: CPM and eCPM are simple metrics to implement. Publishers can easily calculate their CPM and eCPM rates using ad server data and use this information to optimize their ad
In conclusion, CPM and eCPM are essential metrics for publishers to measure the revenue earned from their ad space. Using these metrics, publishers can predict their revenue streams, monetize their
ad impressions, optimize their ad inventory, increase their revenue, and implement a simple and effective pricing model. | {"url":"https://freestar.com/how-to-calculate-cpm-and-ecpm/","timestamp":"2024-11-13T08:10:01Z","content_type":"text/html","content_length":"109846","record_id":"<urn:uuid:7480008c-3ee7-4abb-846c-ba738d797f66>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00255.warc.gz"} |
Timber Truss Roof Design [A Structural Guide] - Structural Basics
Timber Truss Roof Design [A Structural Guide]
Designing a timber roof truss for a new building project can be challenging.
You not only have to consider all loads acting on the roof (snow, wind, dead and live load) and choose the type of truss.
But you also must know how to design timber elements and ensure the structure is structurally sound.
In this post, we’ll go through, step-by-step, how to calculate the internal forces like moment and axial forces.
We’ll also define a static system and dimension the elements of the truss roof according to the Timber Eurocode EN 1995-1-1:2004.
Not much more talk, let’s dive into it.
🙋♀️ What is a timber truss roof?
The truss roof is a structural roof system spanning between 2 supports and carrying loads like wind, snow and live load. Compared to other trusses, the truss roof is usually inclined from the
supports towards the midpoint. It consists of top chord, bottom chord, diagonals and connections. Statically speaking are the top and bottom chords beams, acting in normal forces, shear forces
and bending moments, while the diagonals, usually, act as bars and only take up normal forces.
Doing a little bit of research i found that the diagonals can also be called:
• webs
• tie (when in tension)
• strut (when in compression)
the top chord is sometimes called
and the bottom chord
Have you heard any other names for truss components. Let us all know in the comments below in case you have📝
As already mentioned, there is different types of the truss roof, meaning that the different elements can be built with different materials and systems.
One example of the truss roof type can be seen in the next picture, where a whole timber beams are chosen as top and bottom chords.
The top chords have a little overhang.
The webs/diagonals are connecting top and bottom chords, which leads to an “additional support” of those members because the span is reduced.
For the wind bracing system can be used either wind bracing steel straps, wooden boards or another solution. This system is however not modelled and shown in the picture.
One example of Truss roof system
.. and here is the 3D model because they are an even better visualisation than 2D pictures, aren’t they?
We haven’t covered wind bracing systems yet, – how they work, why we need them – but would you be interested in learning more? Let me know in the comments below.
👆 Choose a Static system of the truss roof
The static system of the truss roof is built up by 2 inclined timber beams and connected to each other at the top with a hinge.
Those beams are supported with a pinned and a roller support at its lowest point or – in case of a cantilevered overhang of the roof – close to the lowest point.
4 diagonals connect top and bottom chords with each other. Those diagonals or webs take up only normal forces and are therefore modelled as bar elements.
The static system of the truss roof is visualized in the next picture.
Static system | Timber truss roof
To not loose context – the 2D static system represents the following rafters. But it can also represent any other section of beams and bars. The spacing between the rafters is set to 4m.
Truss roof | 2D static system representing beams and bars.
The truss roof can of course also have different layouts with smaller/wider spans or steeper inclination.
⬇️ Characteristic Loads of the Truss roof
The loads will not be derived in this article. We explained the calculation of dead, live, wind and snow loads for pitched roofs thoroughly in previous articles.
The defined load values are estimations from the previous calculations.
$g_{k}$ 1.08 kN/m2 Characteristic value of dead load
$q_{k}$ 1.0 kN/m2 Characteristic value of live load
$s_{k}$ 0.53 kN/m2 Characteristic value of snow load
As we also discussed in the article about the characteristic snow load, there are 3 different load cases, where only half of the value is applied on one pitched side but the full value on the other.
However, due to simplicity we only consider load case 1 in this tutorial which applies $s_{k} = 0.53 kN/m^2$ on both rafters.
We split up the wind load from the above table due to the complexity of the wind with its wind areas and directions.
In this calculation, we will only focus on the external wind pressure for areas of 10 m2.
$w_{k.F}$ -0.25(/0.35) kN/m2 Characteristic value of wind load Area F
$w_{k.G}$ -0.25(/0.35) kN/m2 Characteristic value of wind load Area G
$w_{k.H}$ -0.1(/0.2) kN/m2 Characteristic value of wind load Area H
$w_{k.I}$ -0.2(/0.0) kN/m2 Characteristic value of wind load Area I
$w_{k.J}$ -0.25(/0.0) kN/m2 Characteristic value of wind load Area J
$w_{k.F}$ -0.55 kN/m2 Characteristic value of wind load Area F
$w_{k.G}$ -0.7 kN/m2 Characteristic value of wind load Area G
$w_{k.H}$ -0.4 kN/m2 Characteristic value of wind load Area H
$w_{k.I}$ -0.25 kN/m2 Characteristic value of wind load Area I
The following picture presents the static system of the truss roof with its line loads applied.
The section that is presented in Figure: Truss roof | 2D static system representing beams and bars is used for this example.
Due to simplicity, this tutorial looks only at the wind load from the side. Therefore, the wind load $w_{k.I} = -0.25 kN/m^2$ is applied to both rafters.
$g_{k}$ 1.08 kN/m2 * 4.0m = 4.32 kN/m
$q_{k}$ 1.0 kN/m2 * 4.0m = 4.0 kN/m
$s_{k}$ 0.53 kN/m2 * 4.0m = 2.12 kN/m
$w_{k}$ -0.25 kN/m2 * 4.0m = -1.0 kN/m
Characteristic line loads on top chords.
➕ Load combinations of the Truss roof
Luckily we have already written an extensive article about what load combinations are and how we use them. In case you need to brush up on it you can read the blog post here.
We choose to include $w_{k.I.}$ = -0.25 kN/m2 as the wind load in the load combinations, as this is the wind load that is applied to the section we look at, and to keep the calculation clean.
In principle, you should consider all load cases.
However, with a bit more experience, you might be able to exclude some of the values.
In modern FE programs, multiple values for the wind load can be applied and load combinations automatically generated. So the computer is helping us a lot.
Just keep in mind that you should include all wind loads, but because of simplicity we do only consider 1 value in this article😁.
ULS Load combinations
I know you might not understand what that means when you do load combinations the first time, but we did a whole article about what loads exist and how to apply them on a pitched roof 😎.
LC1 $1.35 * 4.32 \frac{kN}{m} $
LC2 $1.35 * 4.32 \frac{kN}{m} + 1.5 * 4.0 \frac{kN}{m}$
LC3 $1.35 * 4.32 \frac{kN}{m} + 1.5 * 4.0 \frac{kN}{m} + 0.7 * 1.5 * 2.12 \frac{kN}{m}$
LC4 $1.35 * 4.32 \frac{kN}{m} + 0 * 1.5 * 4.0 \frac{kN}{m} + 1.5 * 2.12 \frac{kN}{m}$
LC5 $1.35 * 4.32 \frac{kN}{m} + 1.5 * 4.0 \frac{kN}{m} + 0.7 * 1.5 * 2.12 \frac{kN}{m} + 0.6 * 1.5 * (-1.0 \frac{kN}{m}) $
LC6 $1.35 * 4.32 \frac{kN}{m} + 0 * 1.5 * 4.0 \frac{kN}{m} + 1.5 * 2.12 \frac{kN}{m} + 0.6 * 1.5 * (-1.0 \frac{kN}{m}) $
LC7 $1.35 * 4.32 \frac{kN}{m} + 0 * 1.5 * 4.0 \frac{kN}{m} + 0.7 * 1.5 * 2.12 \frac{kN}{m} + 1.5 * (-1.0 \frac{kN}{m}) $
LC8 $1.35 * 4.32 \frac{kN}{m} + 1.5 * 2.12 \frac{kN}{m} $
LC9 $1.35 * 4.32 \frac{kN}{m} + 1.5 * (-1.0 \frac{kN}{m}) $
LC10 $1.35 * 4.32 \frac{kN}{m} + 1.5 * 4.0 \frac{kN}{m} + 0.6 * 1.5 * (-1.0 \frac{kN}{m}) $
LC11 $1.35 * 4.32 \frac{kN}{m} + 1.5 * (-1.0 \frac{kN}{m}) + 0.7 * 1.5 * 2.12 \frac{kN}{m} $
LC12 $1.35 * 4.32 \frac{kN}{m} + 1.5 * 2.12 \frac{kN}{m} + 0.6 * 1.5 * (-1.0 \frac{kN}{m})$
Characteristic SLS Load combinations
LC1 $4.32 \frac{kN}{m} $
LC2 $4.32 \frac{kN}{m} + 4.0 \frac{kN}{m}$
LC3 $4.32 \frac{kN}{m} + 4.0 \frac{kN}{m} + 0.7 * 2.12 \frac{kN}{m}$
LC4 $4.32 \frac{kN}{m} + 4.0 \frac{kN}{m} + 0.6 * (-1.0 \frac{kN}{m})$
LC5 $4.32 \frac{kN}{m} + 4.0 \frac{kN}{m} + 0.7 * 2.12 \frac{kN}{m} + 0.6 * (-1.0 \frac{kN}{m}) $
LC6 $4.32 \frac{kN}{m} + 0 * 4.0 \frac{kN}{m} + 2.12 \frac{kN}{m} + 0.6 * (-1.0 \frac{kN}{m}) $
LC7 $4.32 \frac{kN}{m} + 0 * 4.0 \frac{kN}{m} + 0.7 * 2.12 \frac{kN}{m} + (-1.0 \frac{kN}{m}) $
LC8 $4.32 \frac{kN}{m} + 2.12 \frac{kN}{m}$
LC9 $4.32 \frac{kN}{m} + (-1.0 \frac{kN}{m}) $
LC10 $4.32 \frac{kN}{m} + 4.0 \frac{kN}{m} + 0.6 * (-1.0 \frac{kN}{m}) $
LC11 $4.32 \frac{kN}{m} + (-1.0 \frac{kN}{m}) + 0.7 * 2.12 \frac{kN}{m} $
LC12 $4.32 \frac{kN}{m} + 0 * 4.0 \frac{kN}{m} + 2.12 \frac{kN}{m}$
LC13 $4.32 \frac{kN}{m} + 0 * 4.0 \frac{kN}{m} + (-1.0 \frac{kN}{m})$
👉 Define timber material properties
🪵 Truss timber material
For this blog post/tutorial we are choosing a Structural timber C24. More comments on which timber material to pick and where to get the properties from were made here.
The following characteristic strength and stiffness parameters were found online from a manufacturer.
Bending strength $f_{m.k}$ 24 $\frac{N}{mm^2}$
Tension strength parallel to grain $f_{t.0.k}$ 14 $\frac{N}{mm^2}$
Tension strength perpendicular to grain $f_{t.90.k}$ 0.4 $\frac{N}{mm^2}$
Compression strength parallel to grain $f_{c.0.k}$ 21 $\frac{N}{mm^2}$
Compression strength perpendicular to grain $f_{c.90.k}$ 2.5 $\frac{N}{mm^2}$
Shear strength $f_{v.k}$ 4.0 $\frac{N}{mm^2}$
E-modulus $E_{0.mean}$ 11.0 $\frac{kN}{mm^2}$
E-modulus $E_{0.g.05}$ 9.4 $\frac{kN}{mm^2}$
⌚ Modification factor $k_{mod}$
If you do not know what the modification factor $k_{mod}$ is, we wrote an explanation to it in a previous article, which you can check out.
Since we want to keep everything as short as possible, we are not going to repeat it in this article – we are only defining the values of $k_{mod}$.
For a residential house which is classified as Service class 1 according to EN 1995-1-1 2.3.1.3 we extract the following load durations for the different loads.
Self-weight/dead load Permanent
Live load, Snow load Medium-term
Wind load Instantaneous
From EN 1995-1-1 Table 3.1 we get the $k_{mod}$ values for the load durations and a structural wood C24 (Solid timber).
Self-weight/dead load Permanent action Service class 1 0.6
Live load, Snow load Medium term action Service class 1 0.8
Wind load Instantaneous action Service class 1 1.1
🦺 Partial factor for material properties $\gamma_{M}$
According to EN 1995-1-1 Table 2.3 the partial factor $\gamma_{M}$ is defined as
$\gamma_{M} = 1.3$
📏 Assumption of width and height of truss beams and diagonals
We are defining the width w and height h of the C24 structural wood beam top chord Cross-section as
Width w = 120 mm
Height h = 220 mm
.. the values for the compression diagonal are defined as
Width w = 60 mm
Height h = 120 mm
.. the cross-sectional dimensions of the tension diagonal are defined as
Width w = 60 mm
Height h = 100 mm
.. and lastly the dimensions of the tension bottom chord are
Width w = 100 mm
Height h = 160 mm
💡 We highly recommend doing any calculation in a program where you can always update values and not by hand on a piece of paper! I made that mistake in my bachelor.
In any course and even in my bachelor thesis, I calculated everything except the forces (FE program) on a piece of paper.
Now that we know the width and the height of the top chord Cross-section, we can calculate the Moment of inertias $I_{y}$ and $I_{z}$.
$I_{y} = \frac{w \cdot h^3}{12} = \frac{120mm \cdot (220mm)^3}{12} = 1.065 \cdot 10^8 mm^4 $
$I_{z} = \frac{w^3 \cdot h}{12} = \frac{(120mm)^3 \cdot 220mm}{12} = 3.17 \cdot 10^7 mm^4 $
.. for the compression diagonal
$I_{y} = \frac{w \cdot h^3}{12} = \frac{60mm \cdot (120mm)^3}{12} = 8.64 \cdot 10^6 mm^4 $
$I_{z} = \frac{w^3 \cdot h}{12} = \frac{(60mm)^3 \cdot 120mm}{12} = 2.16 \cdot 10^6 mm^4 $
.. and for the bottom chord
$I_{y} = \frac{w \cdot h^3}{12} = \frac{100mm \cdot (160mm)^3}{12} = 3.413 \cdot 10^7 mm^4 $
$I_{z} = \frac{w^3 \cdot h}{12} = \frac{(100mm)^3 \cdot 160mm}{12} = 1.33 \cdot 10^7 mm^4 $
🆗 ULS Design
In the ULS (ultimate limit state) Design we verify the stresses in the timber members due to bending, shear and normal forces.
In order to calculate the stresses of the rafters, we need to calculate the Bending Moments, Normal and Shear forces due to different loads. An FE or beam program is used to execute this task.
🧮 Calculation of bending moment, normal and shear forces
We use a FE programm to calculate the bending moments, normal and shear forces. Load combination 3 with live load as leading and snow load as reduced load leads to the highest results which we
Load combination 3
Load combination 3 | Dead load, Live load, Snow load | Truss roof
Load combination 3 – Bending moments
Bending moments | Load combination 3 | Rafter roof
Does the moment distribution of the top chords remind you of something…?🤔
Maybe the one from a continuous beam?😀
Load combination 3 – Shear forces
Shear forces | Load combination 3 | Truss roof
Load combination 3 – Normal forces
Normal forces | Load combination 3 | Truss roof
🔎 Bending and Compression Verification
Top chords
From the max. bending moment in the span (7.25 kNm) and the compression force (117.2kN) in the same point we can calculate the stress in the most critical cross section.
Bending stress:
$\sigma_{m} = \frac{M_{d}}{I_{y}} \cdot \frac{h}{2} = \frac{7.25 kNm}{1.065 \cdot 10^{-4}} \cdot \frac{0.22m}{2} = 7.49 MPa$
Compression stress:
$\sigma_{c} = \frac{N_{d}}{w \cdot h} = \frac{117.2 kN}{0.12m \cdot 0.22m} = 4.44 MPa$
Resistance stresses of the timber material:
$ f_{d} = k_{mod} \cdot \frac{f_{k}}{\gamma_{m}} $
LC3 (M-action) $k_{mod.M} \cdot \frac{f_{m.k}}{\gamma_{m}} $ $0.8 \cdot \frac{24 MPa}{1.3} $ $14.77 MPa $
LC3 (M-action) $k_{mod.M} \cdot \frac{f_{c.k}}{\gamma_{m}} $ $0.8 \cdot \frac{21 MPa}{1.3} $ $12.92 MPa $
Utilization according to EN 1995-1-1 (6.19)
$\eta = (\frac{\sigma_{c}}{f_{c.d}})^2 + \frac{\sigma_{m}}{f_{m.d}} = 0.625 < 1.0$
Diagonal – Compression only
Now let’s do the same for the compression diagonal/web, and let’s remember that we modelled the elements as bars.
We have therefore only Normal forces. From the max. compression force (37.04kN) in the diagonal, we can calculate the most critical stress.
Compression stress:
$\sigma_{c} = \frac{N_{d}}{w \cdot h} = \frac{37.04 kN}{0.06m \cdot 0.12m} = 5.14 MPa$
Utilization according to EN 1995-1-1 (6.19)
$\eta = \frac{\sigma_{c}}{f_{c.d}} = 0.4 < 1.0$
👍 Shear Verification – Top chords
From the max. shear force (midsupport: 18.55 kN) we can calculate the shear stress in the most critical cross section.
Shear stress:
$\tau_{d} = \frac{3V}{2 \cdot w \cdot h} = \frac{3 \cdot 18.55 kN}{2 \cdot 0.12m \cdot 0.22m} = 1.05 MPa$
Resistance stresses of the timber material:
$ f_{v} = k_{mod.M} \cdot \frac{f_{v}}{\gamma_{m}} $
$ f_{v} = 0.8 \cdot \frac{4 MPa}{1.3} = 2.46 MPa$
Utilization according to EN 1995-1-1 (6.13)
$\eta = \frac{\tau_{v}}{f_{v}} = 0.43 < 1.0$
👨🏫Buckling Verification
Top chords
We assume that buckling out of the plane (z-direction) can be neglected because the rafters are held on the sides. Therefore we can define the buckling length $l_{y}$ as
Buckling length $l_{y}$ = 2.57m
$l_{y} = 2.57m$
Radius of inertia
$i_{y} = \sqrt{\frac{I_{y}}{w \cdot h}} = 0.064m$
Slenderness ratio
$\lambda_{y} = \frac{l_{y}}{i_{y}} = 40.47$
Relative slenderness ratio (EN 1995-1-1 (6.21))
$ \lambda_{rel.y} = \frac{\lambda_{y}}{\pi} \cdot \sqrt{\frac{f_{c.0.k}}{E_{0.g.05}}} = 0.61$
$\beta_{c}$ factor for solid timber (EN 1995-1-1 (6.29))
$\beta_{c} = 0.2$
Instability factor (EN 1995-1-1 (6.27))
$k_{y} = 0.5 \cdot (1+ \beta_{c} \cdot (\lambda_{rel.y} – 0.3) + \lambda_{rel.y}^2) = 0.72$
Buckling reduction coefficient (EN 1995-1-1 (6.25))
$k_{c.y} = \frac{1}{k_{y} + \sqrt{k_{y}^2 – \lambda_{rel.y}^2}} = 0.915$
Utilization (EN 1995-1-1 (6.23))
$\frac{\sigma_{c}}{k_{c.y} \cdot f_{c.d}} + \frac{\sigma_{m}}{f_{m.d}} = 0.88 < 1$
Diagonal – Compression only
Buckling out of plane is assumed to have the same buckling length as in plane. Therefore we can define the buckling lengths $l_{y}$ and $l_{z}$ as
Buckling lengths $l_{y}$ and $l_{z}$ = 1.5m
$l_{y} = 1.5m$
$l_{z} = 1.5m$
Radius of inertia
$i_{y} = \sqrt{\frac{I_{y}}{w \cdot h}} = 0.035m$
$i_{z} = \sqrt{\frac{I_{z}}{w \cdot h}} = 0.017m$
Slenderness ratio
$\lambda_{y} = \frac{l_{y}}{i_{y}} = 43.3$
$\lambda_{z} = \frac{l_{z}}{i_{z}} = 86.6$
Relative slenderness ratio (EN 1995-1-1 (6.21))
$ \lambda_{rel.y} = \frac{\lambda_{y}}{\pi} \cdot \sqrt{\frac{f_{c.0.k}}{E_{0.g.05}}} = 0.651$
$ \lambda_{rel.z} = \frac{\lambda_{z}}{\pi} \cdot \sqrt{\frac{f_{c.0.k}}{E_{0.g.05}}} = 1.303$
$\beta_{c}$ factor for solid timber (EN 1995-1-1 (6.29))
$\beta_{c} = 0.2$
Instability factor (EN 1995-1-1 (6.27))
$k_{y} = 0.5 \cdot (1+ \beta_{c} \cdot (\lambda_{rel.y} – 0.3) + \lambda_{rel.y}^2) = 0.747$
$k_{z} = 0.5 \cdot (1+ \beta_{c} \cdot (\lambda_{rel.z} – 0.3) + \lambda_{rel.z}^2) = 1.449$
Buckling reduction coefficient (EN 1995-1-1 (6.25))
$k_{c.y} = \frac{1}{k_{y} + \sqrt{k_{y}^2 – \lambda_{rel.y}^2}} = 0.898$
$k_{c.z} = \frac{1}{k_{z} + \sqrt{k_{z}^2 – \lambda_{rel.z}^2}} = 0.48$
Utilization (EN 1995-1-1 (6.23))
$\frac{\sigma_{c}}{k_{c.y} \cdot f_{c.d}}= 0.443$
$\frac{\sigma_{c}}{k_{c.z} \cdot f_{c.d}}= 0.828$
📋Bending and Tension Verification
Bottom chord
From the max. bending moment in the bottom chord beam (0.53 kNm) and the tension force (101.47 kN) in the same point we can calculate the stress in the most critical cross section.
Bending stress:
$\sigma_{m} = \frac{M_{d}}{I_{y}} \cdot \frac{h}{2} = \frac{0.53 kNm}{3.41 \cdot 10^{-5}} \cdot \frac{0.16m}{2} = 1.24 MPa$
Tension stress:
$\sigma_{t} = \frac{N_{d}}{w \cdot h} = \frac{101.47 kN}{0.1m \cdot 0.16m} = 6.34 MPa$
Resistance stresses of the timber material:
$ f_{d} = k_{mod} \cdot \frac{f_{k}}{\gamma_{m}} $
LC3 (M-action) $k_{mod.M} \cdot \frac{f_{t.k}}{\gamma_{m}} $ $0.8 \cdot \frac{14 MPa}{1.3} $ $8.62 MPa $
Utilization according to EN 1995-1-1 (6.17)
$\eta = \frac{\sigma_{t}}{f_{t.d}} + \frac{\sigma_{m}}{f_{m.d}} = 0.82 < 1.0$
Diagonal – Tension only
The maximum tension force in the diagonals is 36.5 kN
Tension stress:
$\sigma_{t} = \frac{N_{d}}{w \cdot h} = \frac{36.5 kN}{0.06m \cdot 0.1m} = 6.05 MPa$
Utilization according to EN 1995-1-1 (6.17)
$\eta = \frac{\sigma_{t}}{f_{c.d}} = 0.7 < 1.0$
👩🏫 SLS Design – Truss roof
We also discussed the SLS design a bit more in detail in a previous article. In this blog post we are not explaining too much but rather show the calculations😊
🖋️Instantaneous deformation $u_{inst}$
$u_{inst}$ (instantaneous deformation) of our beam can be calculated with the load of the characteristic load combination.
As for the bending moments, shear and axial forces we are using a FE program to calculate the deflections due to our Load combinations.
LC 3 of the characteristic SLS load combinations leads to the largest deflection u.
$u_{inst}$ = 9.2 mm
Unfortunately EN 1995-1-1 Table 7.2 recommends values for $w_{inst}$ only for “Beams on two supports” and “Cantilevering beams” and not for a truss system like in this case.
However, the limits of the deflection can be agreed upon with the client and the structure is not collapsing due to too large deflections if the rafter is verified for all ULS calculations.
The moment and shear distribution of the top chord are similar to a continuous beam, but because the “middle support” is a compression member which is translating downwards because it’s connected to
the bottom chord which deflects downwards, the limits for a simply supported beam for the whole top chord length is assumed in this tutorial (EN 1995-1-1 Table 7.2).
❓But my question to you: What limit would you use in this case? Let me know in the comments below.
$w_{inst}$ = l/300 = 5.15m/300 = 17.17 mm
$\eta = \frac{u_{inst}}{w_{inst}} = \frac{9.2mm}{17.17mm} = 0.536 < 1$
🏇Final deformation $u_{fin}$
$u_{fin}$ (final deformation) of our beam/rafter can be calculated by adding the creep deformation $u_{creep}$ to the instantaneous deflection $u_{inst}$.
Therefore, we will calculate the creep deflection with a FE program.
This might be a bit quick, but we have already covered the basics in the article about the timber beam dimensioning.
So check that out if you want to know exactly how to calculate $u_{creep}$ by hand. Let me know in the comments below if you struggle with calculating the creep deformation.
The creep deformation of LC3 is calculated as
$u_{creep}$ = 2.64mm
Adding the creep to the instantaneous deflection leads to the final deflection.
$u_{fin} = u_{inst} + u_{creep} = 9.2mm + 2.64mm= 11.84mm$
Limit of $u_{fin}$ according to EN 1995-1-1 Table 7.2
$w_{fin}$ = l/150 = 5.15m/150 = 34.3 mm
$\eta = \frac{u_{fin}}{w_{fin}} = \frac{11.84mm}{34.3mm} = 0.35$
Now that the truss is verified for compression, bending, buckling, tension and deflection we can finally say that the cross-section heights and widths are verified – check✔️.
After designing rafter, purlin, collar beam roof, it’s very interesting to see the difference in cross-sectional areas for each of the roof, right?
I am curious to hear from you: Which is your favourite roof system? Which truss layout have you already used in a design? Let me know in the comments✍️.
❓ Timber Truss Roof FAQ
What are 3 advantages of a timber truss roof?
– lightweight
– easy to build; local carpenter have the knowledge to build timber trusses
– structurally very efficient; most elements act mainly in tension or compression
What are 3 types of timber roof trusses?
– King post truss
– Fink truss
– Fan truss | {"url":"https://www.structuralbasics.com/timber-truss-roof/","timestamp":"2024-11-07T21:39:15Z","content_type":"text/html","content_length":"296462","record_id":"<urn:uuid:3328d333-5d7e-4d8f-88eb-62310450689f>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00646.warc.gz"} |
Monday Morning Thoughts II: Global Cooling, Prosecutorial Misconduct and More - Davis Vanguard
Global Cooling
One of the more interesting arguments I hear against Global Warming is that ‒ 40 years ago – at least some scientists believed that the earth was cooling.
An April 28, 1975 Newsweek article wrote: “The central fact is that, after three quarters of a century of extraordinarily mild conditions, the Earth seems to be cooling down. Meteorologists disagree
about the cause and extent of the cooling trend, as well as over its specific impact on local weather conditions. But they are almost unanimous in the view that the trend will reduce agricultural
productivity for the rest of the century.”
Writing in May 2014, Peter Gwynne, who authored the original 1975 article writes, “While the hypotheses described in that original story seemed right at the time, climate scientists now know that
they were seriously incomplete. Our climate is warming — not cooling, as the original story suggested.”
However as Mr. Gwynne points out, “certain websites and individuals that dispute, disparage and deny the science that shows that humans are causing the Earth to warm continue to quote my article.
Their message: how can we believe climatologists who tell us that the Earth’s atmosphere is warming when their colleagues asserted that it’s actually cooling?”
Think about the world that has changed since 1975. Thinking about computing, when I graduated from college in 1996, which is just under 20 years ago, I bought a PC which operated Windows 95, its hard
drive was 1 GB, it has a 28.8 modem. As cool as I thought my computer was back then, my phone is much more powerful with a 64 GB in memory. That’s just 20 years ago.
Apple Computer was not born until 1976. Macintosh wouldn’t even come out until 1984.
Why am I pointing out computer technology changes? Well, climate models are run on computers, and the ability to model the climate and analyze data has revolutionized climate science.
Or, as Mr. Gwynne argued last year, “In the 39 years since, biotechnology has flowered from a promising academic topic to a major global industry, the first test-tube baby has been born and become a
mother herself, cosmologists have learned that the universe is expanding at an accelerating rate rather than slowing down, and particle physicists have detected the Higgs boson, an entity once
regarded as only a theoretical concept.”
He notes, “Those that reject climate science ignore the fact that, like other fields, climatology has evolved since 1975. The certainty that our atmosphere is indeed warming stems from a series of
rigorous observations and theoretical concepts that fit into computer models and an overall framework outlining the nature of Earth’s climate.”
He adds, “These capabilities were primitive or non-existent in 1975.”
Another interesting point that Mr. Gwynne makes is that efforts to clean up emissions into the atmosphere such as the Clean Air Act of 1970 reduced other pollution, especially the amount of sulfate
aerosols in the atmosphere. “Since those compounds primarily reflect heat, their reduction effectively gave carbon dioxide and other greenhouse gases more control over the Earth’s temperature,” he
See the full article by Peter Gwynne.
Fox Guarding the Hen House on Prosecutorial Misconduct
In the battle against prosecutorial misconduct, no agency has been hammered harder than the California Bar Association. As we have noted, starting with the Innocence Project’s 2010 study, the Bar
Association has rarely disciplined attorneys involved in prosecutorial misconduct.
Now the San Jose Mercury News on Sunday came out with an article, “Prosecutor with checkered record is training legal watchdogs.”
Writes the Mercury News:
“The agency in charge of disciplining California’s lawyers has the pick of the litter when it comes to contracting with legal specialists. So the State Bar’s decision to hire a Bay Area attorney with
a checkered record is raising eyebrows.
“Alfred F. Giannini, a retired San Mateo County deputy district attorney, has been criticized by the Northern California Innocence Project for committing misconduct in three murder trials that led to
a reversal or a mistrial.
“Now he’s coaching trial attorneys on how to win cases against lawyers the State Bar accuses of stealing, cheating or lying.
“It’s a bizarre choice. He’s like the poster boy of misconduct,” said Kathleen M. Ridolfi, the project’s former director, who teaches at Santa Clara University’s law school. “It’s a very sad
statement about the Bar in California.”
Mr. Giannini defends his record, “offering detailed explanations for each problematic case and criticizing the Innocence Project’s research in annual reports about prosecutorial misconduct as
He was hired by the Bar last year to spend about 10 hours a week, at $75 an hour, “training its litigators under a year-long contract.”
A spokesperson for the Bar told the Mercury News that the agency would not comment but did say that “there are no plans to renew” Mr. Giannini’s contract when it expires in May.
Councilmember Frerichs on the Cannery CFD
In recent weeks, both Robb Davis and Brett Lee have explained their votes on the CFD. This week Councilmember Lucas Frerichs, in an interview with the local paper, said that “refusing a CFD would
come at a price, such as amenities and infrastructure not being built for years, as the developer waits for money from future home sales.”
Moreover, he reiterated that “a CFD was part of the talk throughout the discussion of The Cannery’s development agreement.”
He told the paper, “What I don’t want to see is what happened when Mace Ranch was built.”
He also said that “the CFD money doesn’t come from the city general fund, would not be paid by the entire community and would be paid for by future Cannery residents one way or another — either
through higher-priced homes or through a CFD.”
He added, “With an anticipated $11.8 million sale, the city would receive $750,000 from the developer. For any amount above $11.8 million, Frerichs said, there would be a 50/50 split, resulting in
potentially millions of dollars for the city.”
Community Choice Energy
Last week Robb Davis and Lucas Frerichs wrote an editorial on a community choice energy (CCE — formerly known as CCA) program in Davis. At the February 3 council meeting, they established the
Community Choice Energy Advisory Committee to determine specific options for developing such a program.
They explain:
CCE is a mechanism by which a local government jurisdiction (e.g., a city, county or joint powers authority) contracts directly with a wholesale electricity supplier to purchase and supply electrical
power only (excluding gas). In essence, the CCE becomes the energy-buying agent for consumers instead of the local investor-owned utility (Pacific Gas & Electric in our case).
Under a CCE model, PG&E continues to do the billing, turns on and off power when you move, maintains and owns the power lines, and other infrastructure and resolves outages. PG&E will bill customers
for the metered energy used and forward to the CCE monies collected for the electrical generation portion of the electric bill. The CCE entity then pays the energy generator for the electricity
In a CCE model, individual customers can opt out and continue to receive electricity directly from PG&E.
Unlike a publicly owned utility — which was under consideration in 2013-14 — the city of Davis would not incur the costs of acquiring PG&E’s transmission and distribution infrastructure. However, a
CCE does permit the city to select electricity generation sources (including greener, renewable energy such as solar and wind) as well as capture a portion of “public purpose funds,” which it can
use, in turn, to promote energy conservation and develop local generation of renewable energy.
The councilmembers explain, “The CCE model is gaining traction across California, and there are about 30 local jurisdictions in various stages of exploring, forming or operating a CCE throughout the
state, including places such as Lancaster, Santa Monica, Manhattan Beach, San Francisco and Richmond.”
Moreover, they add, “Developing a CCE today does not preclude developing a publicly owned utility later, and does not increase the cost of doing so. Choosing not to acquire transmission and
distribution infrastructure is likely a much more cost-effective choice, at this time.”
They add, “A CCE, therefore, potentially would allow the city to move forward with a form of public power, all while allowing us to determine the sources of electricity we use as we continue to
develop additional local sources.”
—David M. Greenwald reporting
Breaking News City of Davis Court Watch Environment Land Use/Open Space Yolo County
Cannery CCE CFD City Council Climate Change Environment Global Warming Innocence Project Lucas Frerichs Prosecutorial Misconduct Robb Davis
30 comments
CCE is a mechanism by which a local government jurisdiction (e.g., a city, county or joint powers authority) contracts directly with a wholesale electricity supplier to purchase and supply
electrical power only
The CCE sounds like an interesting concept. It could inject some competition into the electricity supply business. PG&E would almost certainly be against allowing competition so I expect there
would be legal and political challenges to overcome.
Is there some reason that the CCE could not be owned and operated by a private company? I don’t see why it would need to be run by a “government jurisdiction”.
1. It is my understanding that a CCE cannot be opposed by PG&E, and is one of the reasons a CCE might be a better choice than a POU, which PG&E can freely oppose with all its might.
1. that’s certainly an important point – can you point to a link or can anyone else verify this?
It is my understanding that a CCE cannot be opposed by PG&E
It may be illegal for them to overtly oppose a CCE, but PG&E could make it very difficult to implement by being uncooperative or charging high rates for using their distribution system.
It will be interesting to see what ideas the Community Choice Energy Advisory Committee comes up with.
1. yeah i would put anything past pg&e but i wanted to know what the law was.
He notes, “Those that reject climate science ignore the fact that, like other fields, climatology has evolved since 1975. The certainty that our atmosphere is indeed warming stems from a
series of rigorous observations and theoretical concepts that fit into computer models and an overall framework outlining the nature of Earth’s climate.”
The models are used to predict future temperatures. When the actual data arrives and there is a significant delta, they use that data to “recalibrate” the model. There have been significant
delta in the data over the last 14 years. The significant delta is strongly indicative that the previous models were inaccurate. It is those previous models that had been relied upon for all
the doom and gloom predictions of global warming… a theory that was conveniently and suspiciously changed to “global climate change”.
There are five problems with this “science”
1. It has been largely politicized by the left in their crusade against capitalism and industrialism… and these folks aid and abet the enemies of the US that would love to see us even more
economically crippled than our incompetent government has already caused us to be.
2. There is no “proof” that man-made emission (C02, methane, or other) are a significant cause of climate change. The “evidence” is primarily circumstantial despite the volumes of writing that
attempts to convince everyone otherwise.
3. The failure of the past models to accurately predict global temperatures is “proof” that science is not evolved enough to be relied upon.
4. We absolutely do not know what the consequences of climate change will be on the global human condition. The left and the science (are they any different?) focus on the negative
predictions. While ice retreats in the northern hemisphere, it grows in the southern hemisphere.
5. All this fear mongering about global warming has caused a focus on policy that science already tells us is fruitless, and takes our attention away from policies of adaption. For example,
check that building permits are still given to people building just a few feet above current sea level on the coast.
Decades from now looking back at the history of the US and world relative to this global warming hysteria, I think the primary outcome will be to recognize the crucible for when the profession of
science lost a lot of credibility… and joined the “news media” as just another mouthpiece of propaganda for those that seek political power and those that benefit from the payoff for helping
those that seek political power.
1. “The models are used to predict future temperatures. When the actual data arrives and there is a significant delta, they use that data to “recalibrate” the model. There have been
significant delta in the data over the last 14 years.”
you definitely want to update the models as more data becomes available. however, the significant delta is largely explanable because of the vastness of the oceans compared to land. what’s
happened is that the oceans have absorbed much of the heat energy in the last 14 years, that makes the overall trend all the more alarming.
“There is no “proof” that man-made emission (C02, methane, or other) are a significant cause of climate change. ”
proof is for mathematics as my social science friends love to point out, there is plenty of evidence both in the lab and overall that it is true.
2. Excellent reply. I have a few additions.
6. A computer is merely a tool. The Apollo Mission was accomplished with on board computers incomprehensibly less powerful than what we have today – in any home.
7. Garbage in, garbage out. We have at least two ClimateGate scandals where data was manipulated in order to suit political needs.
8. I discussed climate with a friend who has studied the topic for decades. He informed me that there are weather stations that were located near towns with a population of 200 or 500, that
today have 500,000 inhabitants. Buildings retain heat, business generates heat, so guess what happens to the nearby temperature readings? There were also locations that are quite cold (I
think one may have been Siberia) where someone decided four temperature readings were as good as the previous eight. If you want to compare apples to apples, then do that.
9. My understanding is that those who believe in Global Warming (now rebranded Climate Change) offered 40 models of what would happen, and they all failed to predict the temperature.
10. We have had a Global Warming “hiatus” for 18, 20, 21 years, depending upon who you talk to that Warmists are now having to explain.
11. Of 20 concerns of ordinary Americans, last I read climate change was number 19.
12. If the Warmists were really serious, I believe the IPCC report said that we need to build 1,000 nuclear power plants worldwide ASAP (they emit no Co2) to meet our green energy needs. Most
on the left don’t make this step. This kind of hypocrisy leads some to believe that what they are after is not a safer environment, but power and control.
1. “A computer is merely a tool. ”
while true, the ability to analyze and model large amounts of data have greatly advanced the ability to model climate changes.
“My understanding is that those who believe in Global Warming (now rebranded Climate Change) offered 40 models of what would happen, and they all failed to predict the temperature.”
a climate model should not be able to predict the temperature. the reason that scientists and others refer to it as climate change is that climate and weather are unequal phenomena,
there will be places that actually cool down under climate change, there are likely to be more extreme weather conditions and so global warming is less accurate than climate change.
“We have had a Global Warming “hiatus” for 18, 20, 21 years, depending upon who you talk to that Warmists are now having to explain.”
i love how you repeat the same points over and over again as though no one has explained them to you and refuted them.
“If the Warmists were really serious, I believe the IPCC report said that we need to build 1,000 nuclear power plants worldwide ASAP (they emit no Co2) to meet our green energy needs.
Most on the left don’t make this step. ”
as i pointed out when you said this the last time, a lot of scientists agree with you here that nuclear power plants are a good option. there is a divide between scientists and
environmentalists on this point for sure. part of that comes from environmentalists concerns that the byproduct of nuclear power is not necessarily environmentally friendly, part of that
comes from a political concern that spreading nuclear power to third world countries is a recipe for disaster environmentally and in terms of weapon potential. but yes, climate
scientists agree with you.
1. Your talking down to people approach confirms the feelings some have towards lawyers.
2. nice way to dodge.
3. Frankly: From time to time the U.S. Department of Defense releases reports like this:
DoD Releases 2014 Climate Change Adaptation Roadmap
Today, the Department of Defense (DoD) released its 2014 Climate Change Adaptation Roadmap, which focuses on various actions and planning the DoD is taking to increase its resilience to
the impacts of climate change.”Among the future trends that will impact our national security is climate change,” said Secretary of Defense Chuck Hagel. “Rising global temperatures,
changing precipitation patterns, climbing sea levels, and more extreme weather events will intensify the challenges of global instability, hunger, poverty, and conflict. By taking a
proactive, flexible approach to assessment, analysis, and adaptation, the Defense Department will keep pace with a changing climate, minimize its impacts on our missions, and continue to
protect our national security.”
I usually have a lot of respect for the military having a no bullish*t (i.e., political baggage) attitude towards its mission (protecting the American public), so I take this to mean that the
military takes this kind of thing more seriously than probably you do.
What is the Frankly spin on this kind study?
1. A bunch of “panty waists” at the Pentagon?
2. The influence of a Democratic administration pushing a certain agenda?
3. Other? If so, please explain.
By taking a proactive, flexible approach to assessment, analysis, and adaptation, the Defense Department will keep pace with a changing climate, minimize its impacts on our missions,
and continue to protect our national security.”
The DOD is one smart agency. They take the safe PC road but include the word “flexible” in there to cover the reality of “we don’t know”.
1. So you would say that they’re doing this to look prudent, but those who are “enlightened” would suggest the whole study is a waste?
2. No, I am saying that the issue is politicized, but any useful assessment and response would simply note that the weather is largely unpredictable as is the cause is largely
indeterminable… and so the smart agency would come up with a statement like this that covers both the political and real-life situation.
3. the weather may be unpredictable, but we’re talking about climate and overtime the individual variability smooths out and trends become more noticeable.
4. Frankly: …would simply note that the weather is largely unpredictable as is the cause is largely indeterminable
Continuing off of DP’s comments, you confuse weather and climate. Weather can be less predictable within certain parameters. Climate defines those parameters (the typical
temperature range throughout the year, what time of year most of the rain falls, etc.). For instance, we don’t have yearly India-style monsoons here in the central valley.
Farmers rely on those parameters to determine what crops to plant. The most telling anecdotal experiences I have had about climate change have been conversations I’ve had with local
farmers with a long history of farming in the area. The time to plant and harvest certain crops has shifted over time. If you have such connections, I would encourage you to ask
those kinds of questions.
5. I’m not arguing that climate or weather patterns are not changing. I think there is plenty of historical evidence that climate and weather patterns have constantly changed.
You cannot solve a problem unless you know first know that there actually is a problem and you know what that problem is, and second you can identify the root cause(s) of the problem.
Farmers rely on weather patterns, but farmers will also tell you it is and always has been a crap shoot.
But let’s get back to another fact… isn’t it true that the same scientists that tell us that the emissions from mankind are causing greater global warming are also telling us that
there is nothing we can do to stop, or even materially slow, global warming?
That being the case it would be logical to admit that anyone pursuing regulations to restrict emissions at some material cost to the human condition are doing so irrationally. And
instead they should be focused completely on adaption as the “solution”.
6. Frankly (although I’m not), the reduction in SO2 and particulates, good by most/all measures, has contributed to ‘climate change’. Damned if you do….
Maybe, to protect the planet from human activity, we need to eliminate the species.
Maybe, to protect the planet from human activity, we need to eliminate the species.
Either that is the end game of those most vocal in politicized alarm-ism about global warming, or they have not completely thought through their demands. It is probably a little bit
of both.
But here is my deeper thinking on a subset of the other “thinking” people that are on the side of more aggressive government policy, laws and rules to reduce the amount of Co2 and
methane emitted by human activity or human consumption…
I think they don’t like the American-standard high-activity, hard-working, competitive, dog-eat-dog, high-consumption style of existence and have latched on to this theory (that
belongs in their super-set platform of environmental sustainability) as being the gas to drive us to a more easy-going, less-stressful, less competitive, more cooperative and less
consumptive standard of living.
In other words, to be more like the French.
But if we are talking sustainability… the French are the last people to consider as models. In fact, the Socialist French President just appointed a new Minister of Finance that is
an old-school free-market capitalist. The French are in trouble… anemic growth as a result of more and more punishing restrictions and regulations on business and super high taxes
to “protect” the environment and make it less stressful for French workers to have to fend for themselves in an open competitive job market… has resulted in even the Socialists to
recognize that they were about out of other people’s money and heading toward a giant fiscal catastrophe like Greece before them.
There is environmental sustainability and there is economic sustainability. So far nobody has come up with any working utopian model that does not treat economic sustainability at at
least an equal.
My main problem with the global warming platform is the disconnect from the economic realities for supporting an acceptable human condition. The alarms should not be sounding in fear
of the environmental impacts because there is too much we do not know. The alarms should be sounding in fear of the economic impacts because we know absolutely.
8. Frankly: There is environmental sustainability and there is economic sustainability. So far nobody has come up with any working utopian model that does not treat economic
sustainability at at least an equal.
My main problem with the global warming platform is the disconnect from the economic realities for supporting an acceptable human condition. The alarms should not be sounding in fear
of the environmental impacts because there is too much we do not know. The alarms should be sounding in fear of the economic impacts because we know absolutely.
Environmental resources are economic resources. Environmental impacts are economic impacts. Environmental sustainability is economic sustainability but over a longer term than you
are willing to consider. The difference is the time frame and whether you value “environmental resources” in your calculations of economic reality. Traditional economics usually
looks at activity over the shorter term. Environmental sustainability deals with life resources over longer spans of time. You probably don’t think human activity has longer term
impacts that are discussed in environmental terms. You might also have a shorter vision of economics, probably how to keep things going until you die. If someone has a longer view of
economics (“environmental sustainability”) then it looks strange to you with your shorter term vision.
9. New Evidence Suggests Last Ice Age Caused By Earth Floating Into Extremely Chilly Part Of Galaxy
BERKELEY, CA—Offering an alternative explanation for the period of heavy glaciation and lower global temperatures, new evidence published Wednesday by scientists at the University
of California suggests that Earth’s most recent ice age was caused by the planet drifting into a particularly chilly part of the Milky Way. “While past theories have posited that
the last ice age was the result of factors ranging from changes in the planet’s atmosphere to the precession of its rotational axis to an ebb in solar activity, our research
concludes that the epoch-long drop in surface temperatures can instead be attributed to Earth having floated through an extremely nippy corner of the galaxy,” said Dr. Gerard
Weidl, explaining that the exceedingly brisk conditions prevailing in that particular region of outer space could be blamed for the growth of polar ice caps and the spread of
glaciers across nearly a third of Earth’s total land area. “Luckily, about 11,000 years ago we coasted into a significantly balmier part of the Milky Way, which explains how much
toastier everything’s been since then.” Weidl added that, should the planet float back into a chilly pocket of the universe, the few species that manage to survive would likely
need to bundle up or else they would catch the shivers.
courtesy of The Onion
3. Frankly
“The DOD is one smart agency”
Well if flexibility is your hallmark for “smartness” with regard to climate change, or any other scientific precept for that matter then the DOD is in very good company with the vast majority of
researches whether governmental, private or public academic or private industrial taking exactly the same position. Namely that science is by nature a constantly evolving set of information from
which we derive our understanding of how our world works. What science is not is the determination of a set of immutable beliefs. Therefore flexibility of belief is necessary to an understanding
off scientific information, not an attribute that makes one “smart” or politically more savvy.
4. It looks to me like Frerichs is grasping at straws with his pathetic justification for a CFD.
1. He does not want what happened in Mace Ranch when it was built to happen at the Cannery. Now what happened at Mace Ranch happened in conjunction with a CFD. The individuals whom I have
spoken to who live in Mace Ranch hat the mello roos tax which is not a tax. If he is referring to the parks and greenbelts being put in at the end of the development building process why wasn’t
front loading this infrastructure included in the development agreement? I would think that the sale price would be higher for the homes if the greenbelt and parks are already built. Just an
observation on my part.
2. He then justifies the CFD since it was “part of the talk” throughout the Cannery development agreement process. If the CFD was so important to Frerichs or the developer then why wasn’t it
included in the agreement? Did the development agreement have a better chance of passing a measure R vote if the CFD was not included? Was there a backroom deal to include it later? In the
90’s the Evergreen development was built without the use of a CFD. So it can be done.
3. Then Frerichs points out that current residents will not have to pay for this. Frerichs’s trust me we are sticking it to the new people moving in is an attempt to resonate with current
residents. How many of these new residents will vote the Davis way in support of parcel taxes for our school and city needs when they learn that the current city council stuck it to them as part
of a backroom deal. David has already pointed out that the Mace Ranch neighborhood has not supported parcel taxes.
4. Then Frerichs throws out his end of game desperation long ball that there may be some money in it for the city if the sale exceeds $11.8 million. Now it is projected at $11.8 million so that
looks about as optimistic as the Raiders winning the super bowl next year.
Frerichs and his buddies “Developer Dan” and Rochelle need to rethink their votes and end this farce.
1. i tend to agree with your assessment here. the big issue that frerichs dodges is that the $11.8 million is a community asset that was given away by himself and his two colleagues.
2. Lucas also mentions that voting down the CFD would put amenities at risk. This seems to be a key consideration. If these other amenities are already in the agreement, then why we would we
be concerned about them at this point? The only other possibility is that he and the other two CC members that voted yes are thinking we will have demands for new amenities not yet funded
and/or not in the agreement.
1. Demands for new amenities for ‘the community’, if financed thru the CFD, will fall on the new property owners, many of whom will be current City residents. Nice.
And, the costs of those “goodies” will be financed at a higher rate by those owners than if included in the purchase price. Nice.
Lucas was “interviewed” not writing as himself. The article was in error saying that the CFD ASSESSMENTS are TAXES. El wrongo (my bad, that could be considered racist!). Many people
believe that if it shows up on your property tax bill, it is a tax. Not. IRS has not been actively pursuing this. They might (a “what if”).
In my view, Mr Lucas was using the interview as a ‘safe place to land’, rather than owning up to the totality of his reason to vote in favor of the CFD formation. To me, appears as a
5. 1. Cannery folks have to option of no tax, by prepaying the cost of infrastructure up front.
2. Cannery is not subject to a Measure R vote – it is within the city limits.
3. Potential buyers of Cannery homes will have to pay for the infrastructure one way or another – either through a more expensive home, or by financing the cost of infrastructure over a period
of years.
4. If the Cannery has a CFD, the city will be paid an extra $750,000, plus 50% of any profit that exceeds $11.8 million.
1. To your points 1 & 3. Correct… except that to buy out, a property owner would not only have to buy out the capital costs, but the imputed interest that would accrue as if they paid thru the
CFD. This is required to protect the CFD bond financing. Ask anyone who tried to buy out of the MR CFD. The financing under the CFD is at a higher rate than mortgage rates, AND is subject
to more variables, which are likely to further increase the yearly assessments, over time. “lose-lose” comes to mind.
As to your point 4. Nice. Existing residents who do not buy in the Cannery, get the “goodies” while existing residents who do buy there, and the “newbies” pay for those “goodies”. Very
Using that logic, let’s set the bonds to the highest rates possible, so existing residents can share in the “booty” (take that any way you wish)
2. Anon, your four points are good as far as they go, but they don’t go far enough.
Regarding 1. your point is correct; however, if the individual home buyer and New Home Company do not include a discount on the sales price equal to the prepayment, then the home buyer is
paying for the same amenities twice … once in the prepayment and a second time in the price of the home. With that said, Susan Goodwin, the City’s Bond consultant was very direct in saying
that Table 1 of the Staff Report “suggests home buyers don’t have the ability to discern increased annual costs, and take that into account in their buying decision, and of all places that
that will not be true, the City of Davis is one place where that certainly will not be true. These homeowners are very conscious [sic] of the annual impact it has, and they will take that
into account in the sales office, and they will demand a discount, and if they have the chance to buy the same sized home with a Mello-Roos (CFD) or without a Mello-Roos (CFD) clearly they
are not going to pay the same price.”
Regarding 2. I agree
Regarding 3. the buyers absolutely should pay for the infrastructure one way or another; however, they should not be expected to pay for the same infrastructure twice. As noted in 1. above,
Susan Goodwin expects the sales price to reflect a CFD discount. There will be no double payment of the amenities by the buyers if the discount has the same value as the CFD, either as a
prepayment or as the present value of the 30-40 year stream of annual CFD payments, which in the case of the Figure 1 example is $2,224 in year one, with compounded escalation of 2% each year
thereafter. The present value of that stream of payments is just over $50,000.
Regarding 4. you have described the payments to the City, but you haven’t described the costs that the City will incur. In Susan Goodwin’s scenario, the sales price will be reduced $50,000
if a 100% reduction is negotiated. That means the assessed value of the home will be $50,000 less than in a scenario where no CFD exists. The impact of 547 homes, each with a $50,000
reduced assessment, is a loss of over $300,000 per year in the Ad Valorem Taxes shown in Figure 1 of the Staff Report. Instead of paying $6,750 per year Ad Valorem Taxes, the homeowner will
pay only $6, 215 per year. Over the 30 year life of the CFD, that means a loss to the Davis community of over $9 million in revenue. Who will make up for that revenue loss? If the CFD goes
forward as proposed, it will be all the Davis taxpayers who will have to cover that $9 million.
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Maharashtra Board Practice Set 1 Class 6 Maths Solutions Chapter 1 Basic Concepts in Geometry
Balbharti Maharashtra State Board Class 6 Maths Solutions covers the Std 6 Maths Chapter 1 Basic Concepts in Geometry Class 6 Practice Set 1 Answers Solutions.
6th Standard Maths Practice Set 1 Answers Chapter 1 Basic Concepts in Geometry
Question 1.
Look at the figure alongside and name the following:
i. Collinear points
ii. Rays
iii. Line segments
iv. Lines
┃ │ │Points M, 0 and T ┃
┃i. │Collinear Points├──────────────────────────────────────────────────────────────────┨
┃ │ │Points R, O and N ┃
┃ii. │Rays │ray OP, ray OM, ray OR, ray OS, ray OT and ray ON ┃
┃iii.│Line Segments │seg MT, seg RN, seg OP, seg OM, seg OR, seg OS, seg OT and seg ON ┃
┃iv. │Lines │line MT and line RN ┃
Question 2.
Write the different names of the line.
The different names of the given line are line l, line AB, line AC, line AD, line BC, line BD and line CD.
Question 3.
(i – Line),
(ii – Line Segment),
(iii – Plane),
(iv – Ray)
Question 4.
Observe the given figure. Name the parallel lines, the concurrent lines and the points of concurrence in the figure.
┃ │line b, line m and line q are parallel to each other. ┃
┃Parallel Lines ├──────────────────────────────────────────────────────────────────────────────────────────┨
┃ │line a and line p are parallel to each other. ┃
┃ │line AD, line a, line b and line c are concurrent. Point A is their point of concurrence. ┃
┃Concurrent Lines and Point of Concurrence├──────────────────────────────────────────────────────────────────────────────────────────┨
┃ │line AD, line p and line q are concurrent. Point D is their point of concurrence. ┃
Maharashtra Board Class 6 Maths Chapter 1 Basic Concepts in Geometry Intext Questions and Activities
Question 1.
Complete the rangoli. Then, have a class discussion with the help of the following questions:
1. What kind of surface do you need for making a rangoli?
2. How do you start making a rangoli?
3. What did you do in order to complete the rangoli?
4. Name the different shapes you see in the rangoli.
5. Would it be possible to make a rangoli on a scooter or on an elephant’s back?
6. When making a rangoli on paper, what do you use to make the dots?
1. For making a rangoli, I need a flat surface.
2. I can start making a rangoli by drawing equally spaced dots on the flat surface using a chalk.
3. In order to complete the rangoli, I joined the dots by straight lines to make a design.
4. In the rangoli, I find various shapes such as square, rectangle and triangles of two different size.
5. No. It won’t be possible to make a rangoli on a scooter or on an elephant’s back as they do not have a flat surface.
6. When making a rangoli on paper, I made use of scale and pencil to make equally spaced dots.
Question 2.
Write the proper term, ‘intersecting lines’ or ‘parallel lines’ in each of the empty boxes. (Textbook pg. no. 4)
i. Intersecting Lines
ii. Parallel Lines
iii. Intersecting Lines
Question 3.
Draw a point on the blackboard. Every student now draws a line that passes through that point. How many such lines can be drawn? (Textbook pg. no. 2)
An infinite number of lines can be drawn through one point.
Question 4.
Draw a point on a paper and use your ruler to draw lines that pass through it. How many such lines can you draw? (Textbook pg. no. 2)
An infinite number of lines can be drawn through one point.
Question 5.
There are 9 points in the figure. Name them. (Textbook pg. no. 3)
i. If you choose any two points, how many lines can pass through the pair?
ii. Which three or more of these nine points lie on a straight line?
iii. Of these nine points, name any three or more points which do not lie on the same line.
i. One and only one line can be drawn through two distinct , points.
ii. Points A, B, C and D lie on the same line. Points F, G and C lie on the same line.
iii. Points E, F, G, H and I do not lie on the same line, points A, B, E, H and I do not lie on the same line.
Question 6.
Observe the picture of the game being played. Identify the collinear players, non-collinear players, parallel lines and the plane. (Textbook pg. no. 4)
┃i. │Collinear Players │Players A, B, C, D, E, F, G ┃
┃ii. │Non-collinear Players│Players I, H, C Players I, A, B etc. ┃
┃iii.│Parallel Lines │line l, line m, line n, line p, line q, line r and line s ┃
┃iv. │Plane │The ground on which the boys are playing is the plane ┃
Question 7.
In January, we can see the constellation of Orion in the eastern sky after seven in the evening. Then it moves up slowly in the sky. Can you see the three collinear stars in this constellation? Do
you also see a bright star on the same line some distance away? (Textbook pg. no. 4)
1. The three stars shown by points C, D and E are collinear.
2. The star shown by point H lies on the same line as the stars C, D and E.
Question 8.
Maths is fun! (Textbook pg. no. 5)
Take a flat piece of thermocol or cardboard, a needle and thread. Tie a big knot or button or bead at one end of the thread. Thread the needle with the other end. Pass the needle up through any
convenient point P. Pull the thread up, leaving the knot or the button below. Remove the needle and put it aside. Now hold the free end of the thread and gently pull it straight. Which figure do you
see? Now, holding the thread straight, turn it in different directions. See how a countless number of lines can pass through a single point P.
1. The pulled thread forms a straight line.
2. An infinite number of lines can be drawn through one point.
Question 9.
Choose the correct option for each of the following questions:
i. ______ is used to name a point.
(A) Capital letter
(B) Small letter
(C) Number
(D) Roman numeral
Solution :
(A) Capital letter
ii. A line segment has two points showing its limits. They are called_____
(A) origin
(B) end points
(C) arrow heads
(D) infinite points
Solution :
(B) end points
iii. An arrow head is drawn at one end of the ray to show that it is _____ on that side.
(A) finite
(B) ending
(C) infinite
(D) broken
Solution :
(C) infinite
iv. Lines which lie in the same plane but do not intersect are said to be ____ to each other.
(A) intersecting
(B) collinear
(C) parallel
(D) non-collinear
Solution :
(C) parallel
Question 10.
Determine the collinear and non-collinear points in the figure alongside:
Collinear points:
1. Points A, E, H and C.
2. Points B, E, I and D.
Non-collinear points:
Points B, G, F and I
Question 11.
Look at the figure alongside and answer the questions given below:
i. Name the parallel lines.
ii. Name the concurrent lines and the point of concurrence.
iii. Write the different names of line PV.
i. Parallel lines:
a. line l and line n
b. line p, line q, line r and line s
ii. Concurrent Lines: line q, line m, line n
Point of Concurrence: point S
iii. line l, line PT, line PR, line PV, line RT, line RV and line TV.
Question 12.
Name the different line segments and rays in the given figure:
Line Segments:
seg UV, seg OY, seg OX, seg OV and seg OU
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Transient cavitation code based on the homogeneous equilibrium model from which the compressibility of the liquid/vapour "mixture" is obtained. More...
Transient cavitation code based on the homogeneous equilibrium model from which the compressibility of the liquid/vapour "mixture" is obtained.
Original source file cavitatingFoam.C
Turbulence modelling is generic, i.e. laminar, RAS or LES may be selected.
Definition in file cavitatingFoam.C. | {"url":"https://cpp.openfoam.org/v4/cavitatingFoam_8C.html","timestamp":"2024-11-05T22:35:14Z","content_type":"application/xhtml+xml","content_length":"6168","record_id":"<urn:uuid:ab311eb5-705c-43c8-bf12-da9abf6f96fc>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00574.warc.gz"} |
Control Tutorials for MATLAB and Simulink (2024)
The step function is one of most useful functions in MATLAB for control design. Given a system representation, the response to a step input can be immediately plotted, without need to actually solve
for the time response analytically. A step input can be described as a change in the input from zero to a finite value at time t = 0. By default, the step command performs a unit step (i.e. the input
goes from zero to one at time t = 0). The basic syntax for calling the step function is the following, where sys is a defined LTI object.
• Changing the magnitude of the step
• Specifying the time scale
• Saving the response
• Step response of discrete-time systems
This command will produce a series of step response plots, all on the same figure. A plot will be made for each input and output combination. Most systems you will come across in the beginning will
be SISO or Single-Input, Single-Output. In this case, there will be only one plot generated. However, the step command can also accept MIMO, Multiple-Input, Multiple-Output, systems.
For example, suppose you want to model a mechanical system consisting of a mass, spring, and damper, with an applied force. You can derive the transfer function shown below. You wish to see what the
system response to unit step input is (an applied force of 1N). To model this, enter the following code into a new m-file. Running this script in the MATLAB command window will generate a plot like
the one shown below.
M = 1; % units of kg K = 10; % units of N/m B = 2; % units of N-s/m num = 1; den = [M B K]; sys = tf(num,den) step(sys);
sys = 1 -------------- s^2 + 2 s + 10 Continuous-time transfer function.
This figure shows the output response, which is the position of the mass. You can see that in steady-state the mass has moved 0.1 meters (the spring force balances the applied force). The system is
underdamped and has overshoot.
Further details regarding the use of the step command for more advanced situations are given below.
Changing the magnitude of the step
So far, we have only dealt with unit step inputs. Suppose the input to our system was not 1 Newton, but in fact 100 Newtons. The step command can accommodate this by multiplying the system by 100
(since we are only dealing with linear systems). For the example above, this is achieved with the following code, which generates the plot shown below.
The plot looks similar to the one above it except that it has been scaled vertically by a factor of 100.
Specifying the time scale
The step response for any LTI object can be plotted with a user-supplied time vector. This vector will specify the time interval over which the step response will be calculated. If the vector is
spaced at small time intervals, the plot will look smoother. Specifcally, a specified time vector can be supplied via the second input to the function as shown below.
In the above two plots, only the first 6 seconds of the response are shown. Suppose that the first 10 seconds need to be displayed. A time vector can be created to compute the step response over this
range. Adding the following commands to your m-file and running will generate the figure shown below.
As you can see, the plot goes for 10 seconds.
Saving the response
The final note about the step command is that all of the above variations can be used with lefthand arguments. There are two ways to invoke the lefthand arguments, depending on whether or not the
time vector was supplied to the step command.
[y,t] = step(sys);
[y,t] = step(sys,t);
If the system is in state-space form, then the time histories of the internal states can also be returned.
[y,t,x] = step(sys);
The y vector contains the output response. It has as many columns as outputs and as many rows as elements in the time vector, t. The x vector contains the state response. It has as many columns as
states and as many rows as elements in the time vector, t. When used with lefthand arguments, no plot is drawn when the step function is called. You will usually want to put a semicolon after the
step command when you invoke it with lefthand arguments; otherwise, MATLAB will print out the entire output, state, and time vectors to the command window. You can plot the output response using plot
(t,y) and the state response using plot(t,x).
Step response of discrete-time systems
If the system under consideration is a discrete-time system, step will plot the output as piecewise constant. If the sampling time is unspecified, the output time scale will be in samples. If the
sampling time is specified, the time scale will be in seconds. Consider the following example.
num = 1;den = [1 0.5];Ts = 0.1;sys = tf(num,den,Ts)step(sys)
sys = 1 ------- z + 0.5 Sample time: 0.1 secondsDiscrete-time transfer function.
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