Can an Unbounded Function Be Lipschitz Continuous
Can all continuous functions on a compact set be approximated by 1-lipschitz functions?
- Thread starter Marko Karbevski
- Start date
- #1
Marko Karbevski Asks: Can all continuous functions on a compact set be approximated by 1-lipschitz functions?
As an exercise in my book, there are a few questions that should guide us through the fact that if $f$ is continuous and $g$ is Riemann integrable on a compact interval, then $f(g)$ is also Riemann integrable.
- Show that if $ f$ is 1-Lipschitz then $f(g)$ is Riemann integrable.
- Show that every continuous function can be written as a uniform limit of 1-Lipschitz (!) functions.
- Deduce that if $f$ is continuous then $f(g)$ is Riemann integrable.
Did the author maybe mean "uniform limit of Lipschitz functions"? Because if I am not mistaken, $\sqrt x$ cannot be represented as uniform limit of 1-Lipschitz functions on any compact neighborhood of $0$.
- Damila Whi Shadow
- Computer Science
- Replies: 0
Damila Whi Shadow Asks: getting very small loss and val_loss values, what does it mean for the model
I have built an lstm model and i get the following results during training.
Code:
Epoch 1/100 2848/2848 - 23s - loss: 6.2023e-04 - val_loss: 8.9059e-05 - 23s/epoch - 8ms/step Epoch 2/100 2848/2848 - 14s - loss: 2.0135e-04 - val_loss: 8.7615e-05 - 14s/epoch - 5ms/step Epoch 3/100 2848/2848 - 15s - loss: 1.6023e-04 - val_loss: 5.6420e-05 - 15s/epoch - 5ms/step Epoch 4/100 2848/2848 - 14s - loss: 1.3783e-04 - val_loss: 9.0530e-06 - 14s/epoch - 5ms/step Epoch 5/100 2848/2848 - 15s - loss: 1.1438e-04 - val_loss: 9.2465e-06 - 15s/epoch - 5ms/step Epoch 6/100 2848/2848 - 15s - loss: 9.9488e-05 - val_loss: 5.4534e-05 - 15s/epoch - 5ms/step Epoch 7/100 2848/2848 - 15s - loss: 8.3334e-05 - val_loss: 9.9122e-06 - 15s/epoch - 5ms/step Epoch 8/100 2848/2848 - 14s - loss: 8.2492e-05 - val_loss: 2.9109e-05 - 14s/epoch - 5ms/step Epoch 9/100 2848/2848 - 15s - loss: 8.0917e-05 - val_loss: 1.8096e-05 - 15s/epoch - 5ms/step Epoch 10/100 2848/2848 - 15s - loss: 7.0808e-05 - val_loss: 5.9948e-06 - 15s/epoch - 5ms/step Epoch 11/100 2848/2848 - 14s - loss: 6.9479e-05 - val_loss: 1.0694e-05 - 14s/epoch - 5ms/step Epoch 12/100 2848/2848 - 15s - loss: 6.5420e-05 - val_loss: 1.4020e-05 - 15s/epoch - 5ms/step Epoch 13/100 2848/2848 - 14s - loss: 6.3023e-05 - val_loss: 8.5589e-06 - 14s/epoch - 5ms/step Epoch 14/100 2848/2848 - 15s - loss: 5.9587e-05 - val_loss: 9.4285e-06 - 15s/epoch - 5ms/step Epoch 15/100 2848/2848 - 14s - loss: 6.0492e-05 - val_loss: 2.9430e-05 - 14s/epoch - 5ms/step Epoch 16/100 2848/2848 - 14s - loss: 6.3208e-05 - val_loss: 3.3270e-05 - 14s/epoch - 5ms/step Epoch 17/100 2848/2848 - 15s - loss: 5.2385e-05 - val_loss: 1.2137e-04 - 15s/epoch - 5ms/step Epoch 18/100 does this mean that my model is overfitting??
- saurabh
- Computer Science
- Replies: 0
saurabh Asks: Intuition behind the fact that SVM uses only measure of similarity between examples for classification
I have read about SVM and although I did not understand the math behind it completly, I know that it produces decision plane with maximum margin between examples of different classes and role of support vectors in the process. I also know that SVM is a kind of dual learing algorithm(algorithms that operate only using the dot product between examples). It uses kernel functions to calculate dot product(measure of similarity) between training examples.
What I want to understand in simple terms is that: Suppose I have a similarity matrix of all training examples specifying amount of similary between any(all) two examples in training sample. How Can I make a classifier or cluster based only on this information?
- Jon Oliver
- Chemistry
- Replies: 0
Jon Oliver Asks: Why is 4-ethyl-3,3-dimethylhexane written the way it is?
I'm attempting to study for a chemistry midterm and I was reviewing questions I may have gotten wrong on some assignments. 
I was provided with the image above and asked to name it using IUPAC nomenclature. The compound is an alkane, so I knew I just needed to find the parent chain, which should be the longest continious chain of carbons, and then add the suffix -ane. The ethyl and dimethyl branches are equidistant length apart from one another, that is, the lowest set of numbers will be 3,4,4 or 3,3,4, no matter what. I didn't know which to pick so I asked a teacher who then told me that, if this were to occur, I would put the branches in alphabetical order, and then assign the lowest number to whichever branch comes first alphabetically. Because of this I assigned the ethyl branch '3' and the methyl '4,4'
I ended up getting marked wrong, and was told the correct name was 4-ethyl-3,3-dimethylhexane, not 3-ethyl-4,4-dimethylhexane. My question is; why didn't the alphabetical rule apply here?
- Akash
- Chemistry
- Replies: 0
Akash Asks: Is there a molecular orbital equivalent of rehybridization?
I generally have seen the pyramidal inversion of NH3 explained in terms of rehybridization. The sp3 hybridized NH3 changes to sp2, with the lone pair in the p orbital, and then reverts to sp3 in the opposite configuration. On the other hand, something like PH3 has a higher barrier to inversion, because the poor hybridization leads to the lone pair being placed in an orbital with higher s character, and moving this lone pair into a p orbital would require more energy than in NH3.
However, hybridization is an approximation for MO theory, so I am curious as to whether there is a molecular orbital justification for why we can assume molecules rehybridize. Is there an analogous idea in molecular orbital theory that can explain the pyramidal inversion of NH3, as well as why PH3 has a higher barrier to inversion?
Thank you.
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