Is there a way to determine the roughness of an n-dimensional optimization surface (n > 3)? Preferably a method that uses measures from fractal geometry/chaos theory...
I am currently working on a model which takes two parameters and produces a measurement statistic. Think of it as Z = f(X,Y).   Z is a matrix of my statistics and I am creating a surface plot of it in matlab. Basically, I am looking for a mathematical/analytical way of determining if the surface is smooth, or if it is jagged. Do large values tend to be clustered together or are they dispersed throughout the matrix? - that is my question. Basically, how mixed up are the values of my matrix?  I need to run the model over different parameter sets and I want to be able to analytically determine which one of my surfaces is the smoothest, has the greatest clustering of large values, and ideally, has no negative values.   Any help will be greatly appreciated and please let me know if you need any further information.   Cheers
Suppose I have some theorem in the paper  Section 1 Theorem 1.1. Let ...  And then later in the paper I want to recall the theorem by reprinting it  Section 4 We recall Theorem 1.1: Theorem 1.1. Let ...  What's the proper way to do this? I have been looking at a few scripts other people wrote (specifically Red Hat), and a lot of their variables are assigned using the following notation VARIABLE1="${VARIABLE1:-some_val}" or some expand other variables VARIABLE2="${VARIABLE2:-`echo $VARIABLE1`}"  What is the point of using this notation instead of just declaring the values directly (e.g., VARIABLE1=some_val)?  Are there benefits to this notation or possible errors that would be prevented?  Does the :- have specific meaning in this context? Determine which circuits are parallel and which are in series I need to find the Equivalent Resistance across A and B. The problem is, I don't know which one's are in series and which ones are in parallel. How do I identify which ones are parallel or series? Is there a formal proof for the superposition theorem? Is there any point of using "return !0" in javascript? I had an application for returning closest matches to certain values in a large  cluster of values( as in my earlier )   and I chose a VBA solution. While in the course of using the said application, I observed that the results for the value of 0.5 were not correct. I had been using the VBA Round function which I found to be returning 0 for 0.5 rounded to integer whereas the worksheet round function returns 1. Strangely, the VBA round function returns 2 for 1.5. I had to substitute the worksheet function in place of the VBA one.  Am I missing something? How to add ground to DC circuit? I'm just a dabbler in electronic circuits and I'm really still kind of a noob. I am working on a project called the "Atari Punk Console" (Originally known as the "Stepped Tone Generator") but there are some parts of the wiring schematic that I don't understand, specifically the parts with the ground symbol.  Here is the schematic:   See those two ground symbols? I don't understand how to properly connect the circuit to ground at these points. How can I do this and be sure that I did it properly. I don't understand grounding at all so any pointers or clarification is also welcome. Thanks in advance. Why Cooper or Dr. Brand can't send message? Is past simple ever justified with 'by'? Can anyone shed some light on the following aspect?  By 5 PM I gave him the book.  By 5 PM I had given him the book.  In both the sentences the book was given before 5 PM and no later than 5 PM. But I can't understand then what the difference between them is. a Also is it right to use simple past with 'by' at all? If yes, then why and if not, then why is it incorrect to use simple past with 'by'? A Particular Two-Variable System in a Group Suppose $a$ and $b$ are elements of a group $G$. If $a^{-1}b^{2}a=b^{3}$ and $b^{-1}a^{2}b=a^{3}$, prove $a=e=b$.   I've been trying to prove but still inconclusive. Please prove to me.  Thanks very much for proof. Let $X,Y,Z$ be subspaces of $V$ so that $X$ is a subspace of $Y$. Prove that $Y\cap (X+Z)=X+(Y\cap Z)$ Where do electrons in electricity come from? Where do the electrons come from when an electric generator is making electricity?  Is from the air?  Would a generator work in a vacuum?  Electrons have mass so where would they be pulled from if there is nothing? Inspired by a StackOverflow question ().  Is there a DBMS that actually supports the = NULL syntax? How to calculate time difference in java? I found this math "problem" on the internet, and I'm wondering if it has an answer:     Question: If you choose an answer to this question at random, what is the probability that you will be correct?      a. $25\%$      b. $50\%$      c. $0\%$      d. $25\%$   Does this question have a correct answer? As I understand it, electromagnetic waves have two components which are the result of each other, i.e., when a moving electric charge creates a changing magnetic field at point X then a changing electric field is created at point Y and this repeating process is what creates EM waves, so therefore, it requires no medium. Is my understanding correct?  One thing that I'm surprised to know is that light is also called an electromagnetic wave.  Does this include light of any kind, for example: light from a bulb, a tube, and also from the Sun? How do they contain electric and magnetic fields? Non-empty intersection. Let $A_1,A_2,...,A_{2019}$ be sets s.t. $|A_i|$=45 for each $i=\overline{1,2019}$ and $|A_i\cap A_j|=1$ for every $i\neq j$. Is it true that $$\Big|\bigcap _{i=1}^{2019} A_i\Big|=1?$$ What I thought of doing is proving by induction by i, the base is obvious. However I'm stuck on the next step. ActiveResource timeout not functioning How does the Comma Operator work Identify which OrgWideEmailAddress records are valid for current user I'd like to create a popup list on one of my Visualforce pages, for a user to select from a list of OrgWideEmailAddress records.  My goal is to limit the list to only those records that they're allowed to use.  I'm going to save the selection for use in later automatic emails.  Based on what I've read, I won't be able to send emails to certain OrgWideEmailAddresses:   The address hasn't been verified yet.  I don't see any fields in the schema that appear to show verification state. The address is only available to profiles that the user isn't a member of.  I don't see where the profile list is stored for each record.   Some of my problem could just be due to not understanding the standard objects so well, and perhaps there's a way to get at this through UserInfo, or by making my query on the User object which may have a child relationship to OrgWideEmailAddress since it's listed in the schema viewer of the Force.com IDE.  Here's the test code I'm working with:  for (OrgWideEmailAddress addr : [SELECT DisplayName, Address          FROM OrgWideEmailAddress ORDER BY DisplayName]) {     System.debug( addr.DisplayName + ' &lt;' + addr.Address + '&gt;'); }   I tried creating a new Messaging.SingleEmailMessage and then calling setOrgWideEmailAddressId() on it with each OrgWideEmailAddress Id, in hopes that invalid Ids would throw an exception, but to no avail.  My goal is to modify the test code to provide just the subset of records that the current user will be able to use.