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Recent questions in Types Of Research Studies
sondestiny120g sondestiny120g 2022-08-16

New kind of identities?
I found a new kind of identities which are half logic and half algebraic while working on a proof of NP-completeness. They are like this:
a + m b n + m < a n b < a n
It's the first time, I encounter such an identity. It has the amusing and counter-intuitive property that the right member is not modified, but the two left members are not algebraically equivalents.
Can you tell me if you ever encountereed such identities before? These ones in particular? (There are 9 in total: equality, strict inequality or not, changing way if n + m and m have same sign or not.) If not would you have suggestions for naming this kind of identities?
Moreover it would be interesting to know if there is a finite number of identities of this kind, an infinite number but recursively enumerable, etc.
I do not ask for a proof [!] but only whether somebody knows of a systematic treatment of equivalences of this kind in the literature.
Feel free to search one but let me a few months to search by myself. If I don't have any idea to prove it, I'll update my question to let you know anyone is welcome to publish on the subject.
Update 2013/07/26:
egreg definitely found the good idea behind these identities. Let me update my question as follows : Does there exist an identity of the type
a ? b a ? f ( a , b )
where f(a,b) is not a weighted mean of a and b (? may be =, <, >, <=, >=)?
Update 2013/07/30:
Thanks to zyx we now have a very simple way to construct an infinity of such identities. There remains a "few" questions:
Since it doesn't appear anyone know an article or a book that explicitely remarked these kind of identities as particular, would you have suggestions for naming this kind of identities? (Question A)
Are all such identities as zyx described (in the meaning ( b a ) n × )? (Question B) If not, are these identities recursively enumerable? (Question C)
What are the examples of such identities that are important, useful for demonstrating mathematical results? (Question D) We already know that these identities from weigthed means are useful. Do you have other examples from classical proofs? (Question D')
Note that the answer to question B is no if you change (or extend) the rules ;). For example on rings instead of fields and if the inequality may be strict or not in the same identity. As an example, if we take the ring 2Z of even integers, we have the identity
b < a ( b a ) 3 + 8 + a a .
So far, I only had the following idea for naming this kind of identities: "semi-invariant identities". It shouldn't be hard to find a better name.
Update 2013/07/31 Clarification :
For now, I would like to answer the previous questions on fields only BUT the final word on these identities is out of reach. Why?
The broader framework I see for these identities is the framework of universal algebras. Given such an algebra A of domain A and two binary relations ? 1 and ? 2 on A such that ? 1 and ? 2 are orders or equivalence relations (yes, you can mix an equivalence relation and an order relation if you want but I have no idea if it could yield something interesting on some algebra), the idea is to study the identities that are as follows:
Let n N , an n- ? 1 - ? 2 -semi-invariant identity on A is defined by three terms f 1 , f 2 , f 3 on A such that:
( a 1 , , a n ) A n , f 1 ( a 1 , , a n ) ? 1 f 2 ( a 1 , , a n ) f 1 ( a 1 , , a n ) ? 2 f 3 ( a 1 , , a n )
where f 2 and f 3 are not algebraically equivalents (as I noted in the original question).
Clearly, this is too broad to ask a question in this framework right now. These identities may be interesting on algebras used to construct graphs, etc. But not many people would see an interest in it or see what I'm talking about. I would like to see what can be useful with these identities on the most standard algebras (fields is a good starting point) and it could benefit to many people not only a few dozens of specialists of some particular research domain. Following part of zyx idea, one could go even further by considering arbitrary relations.

empalhaviyt empalhaviyt 2022-08-14

Linear multivariate recurrences with constant coefficients
In the theory of univariate linear recurrences with constant coefficients, there is a general method of solving initial value problems based on characteristic polynomials. I would like to ask, if any similar method is known for multivariate linear recurrences with constant coefficients. E.g., if there is a general method for solving recurrences like this:
f ( n + 1 , m + 1 ) = 2 f ( n + 1 , m ) + 3 f ( n , m ) f ( n 1 , m ) , f ( n , 0 ) = 1 , f ( 0 , m ) = m + 2.
Moreover, is their any method for solving recurrences in several variables, when the recurrence goes only by one of the variables? E.g., recurrences like this:
f ( n + 1 , m ) = f ( n , 2 m ) + f ( n 1 , 0 ) , f ( 0 , m ) = m .
This second question is equivalent to the question, if there is a method of solving infinite systems of linear univariate recurrences with constant coefficients. That is, using these optics, the second recurrence becomes f m ( n + 1 ) = f 2 m ( n ) + f 0 ( n 1 ) , f m ( 0 ) = m , m = 0 , 1 , .
I am not interested in a solution of any specific recurrence, but in solving such recurrences in general, or at least in finding out some of the properties of possible solutions. For instance, for univariate linear recurrences, each solution has a form c 1 p 1 ( n ) z 1 n + + c k p k ( n ) z k n ,, where c i 's are constants, p i 's are polynomials and z i 's are complex numbers. Does any similar property hold for some class of recurrences similar to what I have written?
I have been googling a lot, but have found only methods for some very special cases (in monographs on partial difference equations, etc.), but nothing general enough. I am not asking for a detailed explanation of any method, but references to the literature would be helpful. I don't know much about transforms (like discrete Fourier transform or z-transform), but I found certain hints that there could be a method based on these techniques. Is it possible to develop something general enough using transform, i.e., is the study of transforms worth an effort (in the context of solving these types of recurrences)? However, it seems to me that the generalization of the characteristic polynomial method (perhaps, some operator-theoretic approach) could lead to more general results. Has there been any research on this topic?

Research MethodologyAnswered question
Rose Graves Rose Graves 2022-08-11

Is anybody researching "ternary" groups?
As someone who has an undergraduate education in mathematics, but didn't take it any further, I have often wondered something.
Of course mathematicians like to generalize ideas. i.e. it is often better to define and write proofs for a wider scope of objects than for a specific type of object. A kind of "paradox" if you will - the more general your ideas, often the deeper the proofs (quoting a professor).
Anyway I used to often wonder about group theory especially the idea of it being a set with a list of axioms and a binary function a b = c. But has anybody done research on tertiary (or is that trinary/ternary) groups? As in, the same definition of a group, but with a ( a , b , c ) = d function.
Is there such a discipline? Perhaps it reduces to standard group theory or triviality and is provably of no interest. But since many results in Finite Groups are very difficult, notably the classification of simple groups, has anybody studied a way to generalize a group in such a way that a classification theorem becomes simpler? As a trite example: Algebra was pretty tricky before the study of imaginary numbers. Or to be even more trite: the Riemann ζ function wasn't doing much before it was extended to the whole complex plane.
EDIT: Just to expand what I mean. In standard groups there is an operation : G × G G I am asking about the case with an operation : G × G × G G

Research MethodologyAnswered question
Tirimwb Tirimwb 2022-07-14

Probability of 1 billion monkeys typing a sentence if they type for 10 billion years
Suppose a billion monkeys type on word processors at a rate of 10 symbols per second. Assume that the word processors produce 27 symbols, namely, 26 letters of the English alphabet and a space. These monkeys type for 10 billion years. What is the probability that they can type the first sentence of Lincoln’s “Gettysburg Address”?
Four score and seven years ago our fathers brought forth on this continent a new nation conceived in liberty and dedicated to the proposition that all men are created equal.
Hint: Look up Boole’s inequality to provide an upper bound for the probability!
This is a homework question. I just want some pointers how to move forward from what I have done so far. Below I will explain my research so far.
First I calculated the probability of the monkey 1 typing the sentence (this question helped me do that); let's say that probability is p:
P ( Monkey 1 types our sentence ) = P ( M 1 ) = p
Now let's say that the monkeys are labeled M 1 to M 10 9 , so given the hint in the question I calculated the upper bound for the probabilities of union of all P ( M i ) (the probability that i-th monkey types the sentence) using Boole's inequality.
Since P ( M i ) = P ( M 1 ) = p,
P ( i M i ) i = 1 10 9 P ( M i ) = 10 9 p = 10 9 p
Am I correct till this point? If yes, what can I do more in this question? I tried to study Bonferroni inequality for lower bounds but was unsuccessful to obtain a logical step. If not, how to approach the problem?

Research studies are an essential part of understanding mathematics. Through the use of questions, equations, and answers, research studies help to build a better understanding of the subject. There are a variety of types of research studies to choose from, each designed to help students gain a better understanding of mathematics. Experimental research studies are designed to test mathematical theories and discover new ones. Descriptive research studies focus on analyzing and describing existing data. Analytical research studies use existing data to draw conclusions about mathematical theories. Theoretical research studies attempt to prove or disprove existing mathematical theories. All types of research studies are valuable tools in furthering the understanding of mathematics.