text:
Science topic

# Combinatorics - Science topic

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size (enumerative combinatorics), deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), finding "largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial optimization), and studying combinatorial structures arising in an algebraic context, or applying algebraic techniques to combinatorial problems (algebraic combinatorics).
Questions related to Combinatorics
• asked a question related to Combinatorics
Question
Dear Researchers,
Do researchers/universities value students/researchers having published sequences to the OEIS?
Dear Marco Ripà ,
I have done both. I cited my work in the sequences and the sequences in my work.
• asked a question related to Combinatorics
Question
Let G be cubic plane graph such that every face boundary of G is of length divisible by four. By an indirect proof, it follows that the number of vertices in G must be divisible by eight.
Can you prove this by an elementary method? (which should be possible I think, though I don't have such a proof).
Disclaimer: Cross-posted in stackexchange (https://cstheory.stackexchange.com/q/50959/47855).
This is NOT an answer to the question; rather a comment on the answer by Peter Breuer .
Yes, a cubic graph is graph such that every vertex has degree three. And, by "face boundary" I mean the edges that form the boundary of a face, and "length" means the number of edges in that boundary.
Thank you for your effort. But, the argument "he number of edges must be some multiple of 4 times the number of faces" seems wrong to me. For instance, a graph can have exactly 3 faces with boundary lengths 4,8 and 8 respectively, yet 4+8+8 is not a multiple of F=3.
Remark: The above one is not an actual example though; a graph with 10 faces and face boundaries 8,8,4,4,...,4 is a proper counterexample to the argument.
• asked a question related to Combinatorics
Question
Is the reciprocal of the inverse tangent $\frac{1}{\arctan x}$ a (logarithmically) completely monotonic function on the right-half line?
If $\frac{1}{\arctan x}$ is a (logarithmically) completely monotonic function on $(0,\infty)$, can one give an explicit expression of the measure $\mu(t)$ in the integral representation in the Bernstein--Widder theorem for $f(x)=\frac{1}{\arctan x}$?
These questions have been stated in details at the website https://math.stackexchange.com/questions/4247090
It seems that a correct proof for this question has been announced at arxiv.org/abs/2112.09960v1.
Qi’s conjecture on logarithmically complete monotonicity of the reciprocal of the inverse tangent function
• asked a question related to Combinatorics
Question
Dear Colleagues
I need an inequality for the ratio of two Bernoulli numbers, see attached picture. Could you please help me to find it? Thank you very much.
Best regards
Feng Qi (F. Qi)
• asked a question related to Combinatorics
Question
Dear Collegues,
Could you please recommend conferences in 2019-2021 having Finite Group theory and Combinatorics in their topics?
I will be very thankful for any references!
Best wishes,
Natalia
In addition to the excellent suggestions of Mohamed Amine Bahayou I would recommend this conference list: faculty.math.illinois.edu/~west/meetlist.html
• asked a question related to Combinatorics
Question
Do you know the web sites of the journals Ars Combinatoria and Utilitas Mathematica? Are these two journals ceased? One of my manuscripts was accepted a long time ago for publishing in the first journal, but now I cannot contact any editor, I cannot find its website, and I cannot get any message of these two journals. So I wonder if these two journals have been ceased.
Dear Dr.,
The information above is provided for possible search. That is, the corresponding transitions to other sites needed, that are possible from the links provided by me.
Unfortunately, I don't have anything more yet.
Sincerely,
Sergey Klykov
• asked a question related to Combinatorics
Question
I need some suggestions what are the growing topics in algebraic combinatorics and graph theory for research? Thank you in advance to everyone who will answer.
（enhanced） power graphs of groups
• asked a question related to Combinatorics
Question
I have written an analysis article but cannot seem to find an appropriate journal.
Could you recommend average level journals as well as low level journals in analysis?
Thank you for participating in this discussion.
The journal “American Journal of Mathematical Analysis” http://www.sciepub.com/journal/ajmais published by Science and Education Publishing (SciEP) a publisher included in the Beall’s list of potentially predatory publishers: This is a red flag. There are more, for example: https://beallslist.net/
Better avoid.
Roudy El Haddad indeed the journal “Moroccan Journal of Pure and Applied Mathematics” 2605-6364 https://sciendo.com/journal/MJPAA is having a pretty solid publisher behind it and received a Scopus indexing https://www.scopus.com/sourceid/21101026926, so most likely a safe choice.
The journal “International Journal of Nonlinear Analysis and Applications” https://ijnaa.semnan.ac.ir/ seems a pretty safe choice as well. Scopus indexed https://www.scopus.com/sourceid/21100873480 and indexed in ESCI.
Best regards.
• asked a question related to Combinatorics
Question
I have confirmed that the Hessenberg determinant whose elements are the Bernoulli numbers $B_{2r}$ is negative. See the picture uploaded here. My question is: What is the accurate value of the Hessenberg determinant in the equation (10) in the picture? Can one find a simple formula for the Hessenberg determinant in the equation (10) in the picture? Perhaps it is easy for you, but right now it is difficult for me.
• asked a question related to Combinatorics
Question
In the 19 hundreds, many though that an explicit formula for the partition function was never going to be found. In 2011, finally, an explicit formula for the partition function was discovered.
For that reason, I am fascinated by how close do mathematicians think we are currently to discovering an explicit formula for prime numbers.
• asked a question related to Combinatorics
Question
Dear one of the way is using "Using the Stirling approximation"
etc
• asked a question related to Combinatorics
Question
How many more years do you predict it will take before the Riemann Hypothesis is solved?
Do you think we are close or does it seem that we are still very far?
I agree with you. Although year after year we are getting closer. A few year ago, it was proven that at least 40% of zeros have to be on the critical line. So a lot of progress has been done. Many conjecture have also been presented, which if proven, they would imply the Riemann Hypothesis. So have gotten some important result, however, we have not quite solved it yet.
• asked a question related to Combinatorics
Question
I find that multiple sums are a very interesting concept. However, an even more interesting concept for me is the sum of such sums (the sum of multiple sums). The sum of multiple sums can be turned into a simple product by the formula in the first attached image.
What mathematical applications could this formula have? Where could it be useful?
I have found 2 interesting applications:
1- sum of multiple zeta values (see image 2).
2- sum of multiple power sums (see image 3).
Could you suggest any applications for these 2 particular cases? Could you suggest additional particular cases that would be of interest to mathematicians or physicists?
• asked a question related to Combinatorics
Question
I have written two articles about a generalization of Multiple zeta values and Multiple zeta star values. I also presented applications for this generalization including partition identities, polynomial identities, a generalization of the Faulhaber formula, as well as MZV identities. If you are intrested check them out on my profile and give me your opinion.
Also if anyone knows someone that would be interested in my articles please do recommend it to them.
• asked a question related to Combinatorics
Question
I am considering to distribute N-kinds of different parts among M-different countries and I wan to know the "most probable" pattern of distribution. My question is in fact ambiguous, because I am not very sure how I can distinguish types or patterns.
Let me give an example. If I were to distribute 3 kinds of parts to 3 countries, the set of all distribution is given by a set
{aaa, aab, aac, aba, abb, abc aca, acb, acc, baa, bab, bac, bba, bbb, bbc, bca, bcb, bcc, caa, cab, cac, cba, cbb, cbc, cca, ccb, ccc}.
The number of elements is of course 33 = 27. I may distinguish three types of patterns:
(1) One country receives all parts:
aaa, bbb, ccc 3 cases
(2) One country receives 2 parts and another country receives 1 part:
aab, aac, aba, abb, aca, acc, baa, bab, bba, bbc, bcb, caa, cac, cbb, cbc, cca, ccb 17 cases
(3) Each county rceives one part respectively:
abc, acb, bac, bca, cab, cba 6 cases
These types may correspond to a partition of integer 3 with the condition that (a) number of summands must not exceed 3 (in general M). In fact, 3 have three partitions:
3, 2+1, 1+1+1
In the above case of 3×3, the number of types was the number of partitions of 3 (which is often noted p(n)). But I have to consider the case when M is smaller than N.
If I am right, the number of "different types" of distributions is the number of partitions of N with the number of summands less than M+1. Let us denote it as
p*(N, M) = p( N | the number of summands must not exceed M. )
N.B. * is added in order to avoid confusion with p(N, M), wwhich is the number of partitions with summands smaller than M+1.
Now, my question is the following:
Which type (a partition among p*(N, M)) has the greatest number of distributions?
Are there any results already known? If so, would you kindly teach me a paper or a book that explains the results and how to approach to the question?
A typical case that I want to know is N = 100, M = 10. In this simple case, is it most probable that each country receives 10 parts? But, I am also interested to cases when M and N are small, for example when M and N is less than 10.
Thank, Luis Daniel Torres Gonzalez , you for your contribution. My question does not ask the probability distribution. It asks what is the most probable "pattern" when we distribute N-items among M-boxes. I have illustrated the meaning of "pattern" by examples, but it seems it was not sufficient. Please read Romeo Meštrović 's comments above posted in March, 2019.
• asked a question related to Combinatorics
Question
I am a researcher in combinatorics and Semigroup in Ahmadu Bello University, Zaria seeking to know the binding line between both.
Thank you.
• asked a question related to Combinatorics
Question
The following formally published papers are related to this question:
[1] Feng Qi, Diagonal recurrence relations for the Stirling numbers of the first kind, Contributions to Discrete Mathematics 11 (2016), no. 1, 22--30; available online at https://doi.org/10.11575/cdm.v11i1.62389
[2] Feng Qi and Jiao-Lian Zhao, Some properties of the Bernoulli numbers of the second kind and their generating function, Bulletin of the Korean Mathematical Society 55 (2018), no. 6, 1909--1920; available online at https://doi.org/10.4134/bkms.b180039
[3] Feng Qi and Bai-Ni Guo, A diagonal recurrence relation for the Stirling numbers of the first kind, Applicable Analysis and Discrete Mathematics 12 (2018), no. 1, 153--165; available online at https://doi.org/10.2298/AADM170405004Q
[4] Feng Qi and Bai-Ni Guo, Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials, Mediterranean Journal of Mathematics 14 (2017), no. 3, Article 140, 14 pages; available online at https://doi.org/10.1007/s00009-017-0939-1
[5] Feng Qi, Da-Wei Niu, Dongkyu Lim, and Yong-Hong Yao, Special values of the Bell polynomials of the second kind for some sequences and functions, Journal of Mathematical Analysis and Applications 491 (2020), no. 2, Paper No. 124382, 31 pages; available online at https://doi.org/10.1016/j.jmaa.2020.124382
• asked a question related to Combinatorics
Question
The following formally published papers are related to this question:
[1] Feng Qi, Diagonal recurrence relations for the Stirling numbers of the first kind, Contributions to Discrete Mathematics 11 (2016), no. 1, 22--30; available online at https://doi.org/10.11575/cdm.v11i1.62389
[2] Feng Qi and Jiao-Lian Zhao, Some properties of the Bernoulli numbers of the second kind and their generating function, Bulletin of the Korean Mathematical Society 55 (2018), no. 6, 1909--1920; available online at https://doi.org/10.4134/bkms.b180039
[3] Feng Qi and Bai-Ni Guo, A diagonal recurrence relation for the Stirling numbers of the first kind, Applicable Analysis and Discrete Mathematics 12 (2018), no. 1, 153--165; available online at https://doi.org/10.2298/AADM170405004Q
[4] Feng Qi and Bai-Ni Guo, Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials, Mediterranean Journal of Mathematics 14 (2017), no. 3, Article 140, 14 pages; available online at https://doi.org/10.1007/s00009-017-0939-1
• asked a question related to Combinatorics
Question
Bell polynomials of the second kind Bn,k(x1,x2,...,xn-k+1) are also called the partial Bell polynomials, where n and k are positive integers. It is known that Bn,k(1,1,...,1) equals Stirling numbers of the second kind S(n,k).
What are the values of the special Bell polynomials of the second kind Bn,k(0,1,0,1,0,1,0,...) and Bn,k(1,0,1,0,1,0,...)? Where can I find answers to Bn,k(0,1,0,1,0,1,0,...) and Bn,k(1,0,1,0,1,0,...)? Do they exist somewhere?
The following formally published papers are related to this question:
[1] Feng Qi and Bai-Ni Guo, Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials, Mediterranean Journal of Mathematics 14 (2017), no. 3, Article 140, 14 pages; available online at https://doi.org/10.1007/s00009-017-0939-1
[2] Feng Qi, Da-Wei Niu, Dongkyu Lim, and Yong-Hong Yao, Special values of the Bell polynomials of the second kind for some sequences and functions, Journal of Mathematical Analysis and Applications 491 (2020), no. 2, Paper No. 124382, 31 pages; available online at https://doi.org/10.1016/j.jmaa.2020.124382
[3] Feng Qi, Da-Wei Niu, Dongkyu Lim, and Bai-Ni Guo, Closed formulas and identities for the Bell polynomials and falling factorials, Contributions to Discrete Mathematics 15 (2020), no. 1, 163--174; available online at https://doi.org/10.11575/cdm.v15i1.68111
• asked a question related to Combinatorics
Question
Dear Researchers,
Kindly let me know about 5 top most research problems in number theory which are concerning to
Double series and / or combinatorics
• asked a question related to Combinatorics
Question
How to prove or where to find a proof of the lower Hessenberg determinant showed by two pictures uploaded here?
Dear All Colleagues
The final version has been accepted for publication in the Mathematica Slovaca. For details, please click at the website:
• asked a question related to Combinatorics
Question
It is well known that for every nonnegative n, there exist Steiner triple systems STS(6n+1) and STS(6n+3). I see similarities between this statement and the one that I included in my question, but perhaps you can shed light on this issue. Thanks.
According to a theorem of Doyen and Wilson, the answer to this question is "yes".
• asked a question related to Combinatorics
Question
From a combinatorics perspective, what if what you are requiring of the optimization algorithm is too scattered?
In other words, from a combinatorics perspective, what if what you are requiring in terms of the optimization problem specification, also referring to the constraints, including equality constraints, implies too scattered and sparse coefficient (variable cell) ranges.
Is this possible? The algorithm may then struggle to locate the feasible regions or ranges.
I am particularly referring to a nonconvex nonlinear optimization problem.
Do you have bounded, e.g. binary, variables or unbounded ones. I the case of unbounded variables, the problem may not have an algorithm.
• asked a question related to Combinatorics
Question
I have three tasks for 3 participants - task 1 (EO), task 2 (BCST), task 3 (RMET). I use six measures - M1 (CD), M2 (LLE), M3 (HFD), M4 (MSE), M5 (MFDFA), M6 (Kc). I have 20 EEG channels. I use Wilcoxon signed rank to compare each channel pair between two states. I get p-values for 3 H: tests -1H0: EO=BCST, 2HO: BCST=RMET, 3H0: EO=BCST. The other three are reciprocal adding to 1. So comparing p-values from the tests I get 12 possible outcomes - 1: EO=BCST>RMET, 2: EO=BCST<RMET, 3: BCST=RMET>EO, 4: BCST=RMET<EO, 5: EO=BCST>RMET, 6: EO=BCST<RMET, 7: EO>BCST>RMET, 8: BCST>RMET>EO, 9: RMET>EO>BCST, 10: EO>RMET>BCST, 11: RMET>BCST>EO 12: BCST>EO>RMET.
Question: Given a confidence level of 0.05 is the probability of getting all 3 participants to have the same order based on p-values below the 0.05 CL threshold thus excluding any two tasks being equal based on the following: 0.05 x 0.05 x 0.05 x 6 x 6 = 0.0045?
• asked a question related to Combinatorics
Question
I have N lattice points which are arranged linearly and equally spaced. I want to make connections(say with some wire or thread) with each lattice site with another. The first one has N-1 possibilities and the second one has N-3 as each one cannot make connections itself and the first one has already formed one. So the total possibility is (N-1)(N-3)(N-5)(N-7)..... and so on. Now I want to impose two conditions.
Case(i): If I impose a condition that each site cannot be connected with the nearest neighbor, how many ways I can make the connections. How do complete this counting problem with this condition?
Case(ii): Apart from the above condition(immediate neighbors should not be connected), If I impose a further condition that each site can be connected with other or it can also be left unconnected. How do I count the number of ways doing this?
I know both case(i) and case(ii) will have different answers. I really don't know where to start this problem at all
Is your lattice a line with N vertices?
And you want to connect them by forwarding paths? To make your question is clear. Assume that your lattice includes 4 vertices.
Sketch some graph, then attach it as a pdf file. Then we may help.
Best regards
• asked a question related to Combinatorics
Question
I am working on the  construction of Barnette graphs for given diameter. I would like to know the reason why many cubic 3 connected planar , (not  a bipartite)  are  both non-Hamiltonian and Hamiltonian graphs. I found a unique property of those Hamiltonian graphs. I need the latest results related to my question.
If any one property of Barnatte graph is dropped it is non hamiltonion
• asked a question related to Combinatorics
Question
for k=1, 2, 3 we get 7, 13, 19 as primes.
for k=5,  6, 7 we get  31, 37,  43,  as primes.
Such arithmetic progression does not exist. Namely, suppose that k_n=a+(n-1)d is a desired arithmetic progression, where a is a fixed integer, d is a fixed positive integer, such that 6k_n+1 is a prime number for all n=1,2,3,… . This means that 6k_n+1=6(a+(n-1)d)+1=(6a+1)+(n-1)(6d) would be an arithmetic progression containing only prime numbers. However, this is obviously impossible by the well known Prime Number Theorem for Arithmetic Progressions.
• asked a question related to Combinatorics
Question
Do you think that the iThenticate/CrossCheck/Similarity Index would cause heavy and serious confusion in mathematics? Even destroy, ruin, damage Mathematics? Our mathematics and mathematicians should follow and inherite symbols, phrases, terminology, notions, notations in previous papers, but now we have to change these to avoid, to escape, to hide, to decrease the iThenticate/CrossCheck/Similarity Index! It’s very ridiculous for mathematics and mathematicians! Mathematics is disappearing! being damaged!
Yes! Even standard mathematical symbols and notations are captured in similarity index. The habit of using unconventional symbols and notations just to reduce similarity index is destroying the beauty and taste of mathematics.
• asked a question related to Combinatorics
Question
Hi.
Please change the picture of header site. That is "2th International Conference on Combinatorics, Cryptography and Computation".
Thank you.
:) They should change to 3rd (and it's 2nd not 2th).
Cheers.
• asked a question related to Combinatorics
Question
Many text books claim that particles that obey Boltzmann statistics have to be indistinguishable in order to ensure an extensive expression for entropy. However a first principle derivation using combinatorics gives the Boltzmann only for distinguishable and the Bose Einstein distribution for indistinguishable particles (see Beiser, Atkins or my own text on Research Gate). Is there any direct evidence that indistinguishable particles can obey Boltzmann statistics?
• asked a question related to Combinatorics
Question
Is it a variant of Vandermonde convolution formula for falling factorials? What is the answer? See the picture.
• asked a question related to Combinatorics
Question
akin to more to cauchy additivity versus local  kolmorgov additivity/normalization of subjective credence/utility,  in a simplex representation of subjective probability or utility ranked by objective probability  distinction? Ie in the 2 or more unit simplex (at least three atomic outcomes on each unit probability vector, finitely additive space) where every events is ranked globally within vectors and between distinct vectors by < > and especially '='
i [resume that one is mere representability and the other unique-ness
the distinction between the trival
(1)x+y+z x,y,z mutually exclusive and exhaustive F(x)+F(y)+F(z)=1
(2)or F(x u y) = F(x)+F(y) xu y ie F(A V B)=F(A)+F(B)=F(A)+ F(B) A, B disjoint
disjoint on samevector)
(3)F(A)+F(AC)=1 disjoint and mutually excluisve on the same unit vector
and more like this or the properties below something more these
to these (3a, 3b, 3C) which are uniqueness properties
forall x,y events in the simplex
(3.A) F(x+y)=F(x)+F(y) cauchy addivity(same vector or probability state, or not)
This needs no explaining
aritrarily in the simplex of interest (ie whether they are on the same vector or not)
or(B)  x+y =z+m=F(z)+F(m) (any arbitary two or more events with teh same objective sum must have the same credence sum, same vector or not) disjoint or not (almost jensens equality)
or (C)F(1-x-y)+F(x)+F(y)=1 *(any arbitrary three events in the simplex, same vector or not, must to one in credence if they sum to one in objective chance)
(D) F(1-x)+F(x)=1 any arbitary two events whose sum is one,in chance must sum to 1 in credence same probability space,/state/vector or not
global symmetry (distinct from complement additivity) it applies to non disjoint events on disitnct vectors to the equalities in the rank. 'rank equalities, plus complement addivitity' gives rise to this in a two outcome system, a
. It seems to be entailed by global modal cross world rank, so long as there at least three outcome, without use of mixtures, unions or tradeoffs. Iff ones domain is the entire simplex
that is adding up function values of sums of evenst on distinct vectors to the value of some other event on some non commutting (arguably) probability vector
F(x+y)=F(x)+F(y)
In the context of certain probabilistic and/or utility unique-ness theorems,where one takes one objective probability function  and tries to show that any other probability function, given ones' constraints, must be the same function.
and F(x+y)=F(x)+F(y)
In the context of certain probabilistic and/or utility unique-ness theorems,where one takes one objective probability function  and tries to show that any other probability function, given ones' constraints, must be the same function.
what is meant by state dependent additivity does that mean that instead of F(x U y)=F(x)+F(y) x,y disjoint (lie on the same vector) that (same finite probability triple) ; or instead of F(x)+F(y)+F(z)=1 iff x,y,z are elements of the very same vector (same triple)
in the simplex one literally instead has that F(x+y)=F(x)+F(y) over the entire domain of the function. ie adding up (arguably non commuting) elements of distinct vectors,   or F(x)+F(y)+F(1-x-y)=1 arbitarily over the simplex; or domain of interest, where the only restriction is that one can only add up elements as many times as they are present in the domain. Whilst with cauchy additivity one if one domain is merely a single vector. <1,3, 1,6, 1,2, unit event=1 > one so long 1/6 is in the domain (supposing that entire domain, probability vector space, just is that vector dom(F)={1,3 1,6, 1,2 1),  where if F(1)=1 one can arbitrarily add up 1=F(1)=F(1/6+1/6+1/6 ) six times= 6F(1/6), so F(1/6)=1/6.
I presume however, that if ones domain is the entire simplex there would not be any relevant difference, between outright Cauchy additivity and state independent additivity; and thus to presume would be outright presumptuous.  Or this a name for cross world global rank which entails its long as there at least three atomic events on each simplex (so long as the simplex is well constructed, and the rank is global and modal), even if finite local standard additivity is presumed). As one can transfer values of equiprobable events onto other vectors where they are disjoint?
if by state independent addivity  this means one can arbitrary add up the function values of F(1/6) for objective probability six times, to F(1)=1 the function value lets say at chance=1 to attain that F(1/6)=6 so long as those events are ranked equal and are present at least six times somewhere or other, even if in distinct state, or vectors (in the same system).
or does this apply to local additivity, where one has a global modally transitive rank over the simplex where n>=2 (ie the number of elements in each triple is at least three) because a cross world rank, with equalities, will entail this in any case, if justified. So if one can derive that cross world additivity must hold given finite additivity and global modal rank including equalities cross world/vector, on pain of either either local addivitity failing (probabilism)  or ones justified global and local  total rank  is violated, justified for whatever reason is violated)  including equalities must hold (for whatever reason) does this count as presumptious.
• asked a question related to Combinatorics
Question
Points are two-combinations of n phase names. Lines are three-combinations of phase names. Point and line are incident upon one another if the two phase names comprising the point are part of the name of the line (e.g., point AB and line ABE are incident upon one another). In attached figures, lines of perspective are red, perspective triangles are green, and points of perspective of pairs of green triangles are connected with orange lines. Brute force counting of adjacencies is doable for Desargues configuration (n=5) , but impractical as system number of phases increases.
It's been a while, but I've finally achieved an answer to my question. In the Euclidean plane, given points as two-combinations of n phase names and lines are three-combinations of the phase names there must be 1 point adjacent to 2n-4, 2 points adjacent to 2n-5, 3 points adjacent to 2n-6, ..., n-1 points adjacent to n-2.
• asked a question related to Combinatorics
Question
John-Tagore Tevet
Let us try to open the essence of graphs, from that's so far tried to circumvent.
1. What is a graph
Graph is an association of elements with relationships between these that has a certain structure.
Graphs are represented for different purposes. On the early rock paintings have been found the constellations show schemes. Graphs was used also for explain the theological tenets.
Example 1. Graph (structural formula) of isobutane C4H10:
Graphs began to investigate after then when Leonhard Euler in 1736 has solved the problem of routing on the seven bridges between four banks of Königsberg [1].
Example 2. Königsberg’s bridges and corresponding graph:
Also in present time used the graphs mainly for solving the problems of routing and flowing. Already in 1976 considered that such one-sided approach is a hindering factor for studying of graphs [2]. To the essence of graph, to its structure and symmetry properties has the interest practically non-existent. The last explorer was evidently Boris Weisfeiler in 1976 [9].
Definition a graph as an object consisting in node set V and edge set E, G=(V, E), is a half-truth that beget confusions. Essential is to explain the properties of inner organizing (inner building) or structure, i.e. identification of graphs.
Graph is presentable: 1) as a list L of adjacencies; 2) in the form of adjacency matrix E; 3) graphically G, where the elements to “nodes” and relations to “edges” called.
Example 3. List of adjacencies L, corresponding adjacency matrix E and for both corresponded graphs GA and GB:
Explanations:
The outward look and location of the enumerated elements in graph not have something meaning. But on the emotional level it rather engenders some confusion.
One graph can be differs from the other on its looking or its inner organizing (inner-building) or structure S what in ordinarily visually not be opened. Maybe just due to this is to the present days the existence of structure ignored.
We can here make sure that graphs GA and GB have the same structure and these are isomorphic GA @ GB. Ordinarily differentiate in the objects just the “outward” differences and refuse to see some common structure.

Propositions 1. Structure axioms:
P1.1.    Structure S is presentable as a graph G and each graph G has its certain structure S.
P1.2.    Isomorphic graphs have the same structure – structure is the complete invariant of isomorphic graphs.

Identification of graph is based on identification the binary relations between elements [3 - 8]. Binary relation can a “distance relation”, “circle relation”, “clique relation” etc. and is measurable. Binary relation characterized by corresponding binary sign.
2. Identification of the graph
For identification of the graphs uses two each others complementary ways:
Multiplicative identification (products of adjacency matrixes);
Heuristic identification.

Propositions 2. Multiplicative identification: multiplication the adjacency matrixes:
P2.1.       To multiplying the adjacency matrix with itself E´E´E´…=En and fixing in case of each degree n the number p of different multiplicative binary signs enij that as rule enlarges. Forming the sequence vectors ui of different multiplicative binary signs.
P2.2.       In each case if p enlarges (change) must transpose the rows and columns of En correspondingly to the obtained frequency vectors ui.
P2.3.       Stop the multiply if p more no enlarges and to present the current En and the following En+1.
Explanation: Multiplicative signs differentiate the binary signs but no characterize these.
Example 4. Adjacency matrix E and its transposed products E2, E3 of graphs on example 3:
1  2  3  4  5  6| i
0  1  0  1  0  1| 1
1  0  1  0  1  1| 2
E    0  1  0  1  0  1| 3
1  0  1  0  1  0| 4
0  1  0  1  0  1| 5
1  1  1  0  1  0| 6
ui
2  6| 1  3  5| 4     |    i    0 1 3 4   k
4  3| 1  1  1| 3     |    2    0 3 2 1   1
3  4| 1  1  1| 3     |    6    0 3 2 1   1
E2   1  1| 3  3  3| 0     |    1    1 2 3 0   2
1  1| 3  3  3| 0     |    3    1 2 3 0   2
1  1| 3  3  3| 0     |    5    1 2 3 0   2
3  3| 0  0  0| 3     |    4    3 0 3 0   3
ui
2  6| 1  3  5| 4|   i    0 2 3 6 7 9 10  k
6  7|10 10 10| 3|   2    0 0 1 1 1 0 3   1
7  6|10 10 10| 3|   6    0 0 1 1 1 0 3   1
E3   10 10| 2  2  2| 9|   1    0 3 0 0 0 1 2   2
10 10| 2  2  2| 9|   3    0 3 0 0 0 1 2   2
10 10| 2  2  2| 9|   5    0 3 0 0 0 1 2   2
3  3| 9  9  9| 0|   4    1 0 2 0 0 3 0   3
Explanations:
a)      The set of similar relations (and elements) recognize their position W in the structure. Position W is in group theory known as transitivity domain of automorphisnsms, equivalence class or orbit.
Multiplicative binary signs enij recognize here five positions of binary relations WR and on they base three positions of elements WV.

Propositions 3. Position axioms:
P3.1.       If structural elements (graph nodes) vi , vj , … have in graph G the same position WVk then corresponding sub-graphs (Gi=G\vi) @ (Gj=G\vj) @....   are isomorphic.
P3.2.       If relations (edges) eij, ei*j*, … have in graph G the same binary(+)position WR+n then corresponding greatest subgraphs (Gij=G\eij) @ (Gi*j*=G\ei*j*) @....  are  isomorphic.
P3.3.       If relations (“non-edges”) eij, ei*j*, … have in graph G the same binary(–)position WRn– then corresponding smallest supergraphs (Gij=GÈeij) @ (Gi*j*=GÈei*j*) @....  are isomorphic.

Before elaboration of the multiplicative identification way was elaborated a heuristic way.
Propositions 4. Heuristic identification:
P4.1.       Fix an element i and form its neighborhood Ni, where the elements, connected with i divide according to distance d to entries Cd.
P4.2.       Fix an element j and fix its neighborhood Nj by condition P4.1.
P4.3.       Fix the intersection Ni ÇNj as a binary graph gij, and fix the distance –d between i and j (in case of adjacency collateral distance +d), the number n of elements (nodes) in gij, number q of adjacencies (edges). Fixing the heuristic binary sign ±d.n.q.ij of obtained graph gij.
P4.4.       Realize P4.1 to P4.3 for each pair i,jÎ[1, |V|]. Obtained preliminary heuristic structure model SMH.
P4.5.       Fixing for each row i its frequency vector ui. Transpose the preliminary model SM by frequency vectors ui lexicographically to partial models SMk.
P4.6.       In the framework of SMk transpose the rows and columns lexicographically by position vectors si to complementary partial models. Repeat P4.6 up to complementary transposing no arises.
Explanation: Heuristic binary signs differentiate the binary signs and characterize these.
Example 5. On the Example 3 presented differently enumerated graphs GA and GB, their heuristic binary signs and structure models SMA and SMB with their common product E3:
ui
3  4| 1  4  5| 2|   iA
1  2| 1  2  5| 6|        iB    0 2 3 6 7 9 10  k
6  7|10 10 10| 3|   3    3    0 0 1 1 1 0 3   1
6|10 10 10| 3|   4    6    0 0 1 1 1 0 3   1
E3         | 2  2  2| 9|   1    1    0 3 0 0 0 1 2   2
…..  |    2  2| 9|   2    4    0 3 0 0 0 1 2   2
|       2| 9|   5    5    0 3 0 0 0 1 2   2
|        | 0|   6    2    1 0 2 0 0 3 0   3
Explanations:
”Diverse” graphs GA and GB have equivalent heuristic structure models SMA » SMB and the same multiplicative model E3. This means that structures are equivalent and all on the examples 2 and 4 presented graphs GA and GB are isomorphic GA @ GB.
The binaries are divided to five binary positions WRn, where the “adjacent pairs” or “edges” divided to three binary(+)positions (full line, a dotted, dashed-line) that coincide with heuristic binary signs C, D, E and corresponding multiplicy binary signs 10, 7, 2, and with two binary(–)positions with signs –A and –B and multiplicative signs 9 and 3. In base of these divide the structural elements to three positions WVk.
The column ui constitutes frequency vectors, where each element i characterize its relationships with other elements. On the base of frequency vectors ui obtained the positions of elements WVk.
The column si constitutes position vectors that represent the connecting of i with elements on the position k.
A principal theoretical algorithm of isomorphism recognition exists really – it consists in rearranging (transposing) the rows and columns of adjacency matrices EA of graph GA as yet these coincides with the EB of GB. But this has an essential lacking – it is too complicated, the number of steps can be up to factorial n!
Propositions 5. On the relationships between isomorphism and structural equivalence:
P5.1.    Isomorphism GA@GB is a such one-to-one correspondence, a bijection j: VA®VB, between elements what retains the structure GS of graphs GA and GB.
P5.2.    Isomorphism recognition does not recognize the structure GS and its properties (positions etc.), but the structure models SM and En recognize the structure and its properties with exactness up to isomorphism.
P5.3.    Structural equivalence SMA»SMB and EnA»EnB is a coincidence or bijection j: WA®WB on the level of binary positions WRn and positions of nodes (elements) WVk.
P5.4.    In the case of large symmetric graphs recognizes the products En the binary positions more exact than heuristic models SM, where need to use the binary signs of higher degree. That why it is necessary to treat both in together, bearing in mind also that the heuristic binary signs characterize the essence of relationship itself.
P5.5.    Recognition of the positions by the structure model is more effective than detecting the orbits on the base of the group AutG.
Example 6. To the recognition on the Example 1 represented structure of isobutane suffice use the heuristic model SM:
Explanation: Decomposing the elements C and H to four positions corresponds to actuality. The positions are visually appreciable also on the Example 1.
3. List of tasks that solving based on the identified graphs (structure)
To conclusion it should be emphasized that the recognition of graph’s structure (organizing) is based on the identification (distinction) of binary relations between elements. Binary relation can be measured as a “relation of the distance”, “circle relation”, “clique relation”, etc. Binary relation is recognizable by the corresponding binary sign.
The complex of tasks that are based to recognizing structures is broad, various and novel (differ from up to now set up) [3 - 8]. We list here some.
1.      The relations between structural positions, automorphismsm and group-theoretical orbits.
2.      Structural classification the symmetry properties of graphs.
3.      Measurement the symmetry of graphs.
4.      Analyzing different situations of structural equivalency and graphs isomorphism.
5.      Positional structures that open the “hidden sides” of graphs.
6.      Unknown sides of well-known graphs.
7.      Adjacent structures and reconstruction problem. It is connected with general solving the notorious Ulam’s Conjecture.
8.      Sequences of adjacent structures and their associations – the systems of graph structures.
9.      Probabilistic characteristics of graph’s systems.
10.    The relations of graph systems with classical attributes.
References
1.       Euler. L. Solutio problematis ad geometriam situs pertinentis. – Comment. Academiae Sci. I. Petropolitanae 8 (1736), 128-140.
2.       Mayer, J. Developments recents de la theorie des graphes. – Historia Mathematica 3 (1976) 55-62.
3.       Tevet, J.-T. Semiotic testing of the graphs: a constructive approach and development. S.E.R.R., Tallinn, 2001.
4.                     Hidden sides of the graphs. S.E.R.R. Talinn, 2010.
5.                     Semiotic modeling of the structure. ISBN 9781503367456, Amazon Books. 2014.
6.                     Süsteem. ISBN 9789949388844. S.E.R.R., Tallinn, 2016.
7.                     Systematizing of graphs with n nodes. ISBN 9789949812592. S.E.R.R., Tallinn, 2016.
8.                     What is a graph and how it to study. ISBN 9789949817559. S.E.R.R., Tallinn, 2017.
9.       Weisfeiler, B. On Construction and Identification of Graphs. Springer Lect. Notes Math., 558, 1976 (last issue 2006).
It is easy to prove that (quote wikipedia):
If A is an adjacency matrix of the directed or undirected graph G, then the matrix An (i.e., the matrix product of n copies of A) has an interesting interpretation: the element (i, j) gives the number of (directed or undirected) walks of length n from vertex i to vertex j. If n is the smallest nonnegative integer, such that for some i, j, the element (i, j) of An is positive, then n is the distance between vertex i and vertex j.
• asked a question related to Combinatorics
Question
I want to build a kind of guess game. I do not know the right name but the concept of the game is: person-1(P-1) thinks a name(of anything) and person-2 will have to predict that name by asking as less questions as possible. For example:
p1: thinks something(Steve jobs)
p2: Are you talking about man?
p1: yes.
p2: Is he/she media guy?
p1: No
P2: is he tech personality?
p1: yes
p2: steve jobs.
p1: yes.
So p2 has to ask 4 questions. It could be even more as number of predictors are infinite. Now I want to model this scenario. My Goal is to reduce the number of question. Note that the number of predictors are limited. So situation is not that broad.
I can think of decision tree. But question is, how can I decide where to split so that length of the brunch will be small.
Any suggestion/reference will be appreciated.
Maximize the entropy, the information gain
for every question.
Regards,
Joachim
• asked a question related to Combinatorics
Question
Assume we have a class of graphs. Now what does this sentence mean?
"each of the graphs in the class, monotonically should make no difference".
@Nazanin "A graph peoperty is monoton if every subgraph of a graph with property P, also has the property P"  is called "hereditary property".
As for the original (part of a) sentence: if we join it to another domain of mathematics, it makes sense such as it is, despite imprecise wording: e.g. in a class of exponential functions with a basis a>1...
• asked a question related to Combinatorics
Question
If there are k sets of vertices in a graph, with the condition that each vertex in a set should be connected to at least one vertex from each of the other sets, then what is the least number of complete sub-graphs $K_{k}$ in this graph?
Here you have another counterexample, is this what you mean?
I don't understand your question about being $\chi$-colorable. Usually, $\chi(G)$ stands for the chromatic number of G... In the case of the graph I sent, it's 2-colorable.
• asked a question related to Combinatorics
Question
A NSWE-path is  a path consisting of North, South, East and West steps of length 1 in the plane. Define a weight w for the paths by  w(N)=w(E)=1 and w(S)=w(W)=t. Define the height of a path as the y-coordinate of the endpoint. For example the path NEENWSSSEENN has length 12, height 1 and its weight is t^4.
Let B(n,k) be the weight of all non-negative NSEW-paths of length n (i.e. those which never cross the x-axis) with endpoint on height k.
With generating functions it can be shown that for each n the identity
(*)            B(n,0)+(1+t)B(n,1)+…+(1+t+…+t^n)B(n,n)=(2+2t)^n
holds. The right-hand side is the weight of all paths of length n.
Is there a combinatorial proof of this identity?
For example B(2,0)=1+3t+t^2 because the non-negative paths of length 2 with height 0 are EE with weight 1, EW+WE+NS with weight 3t, and WW with weight t^2.
B(2,1)=2+2t because the non-negative paths are NE+EN with weight 2 and NW+WN with weight 2t. And  B(2,2)=1 because w(NN)=1.
In this case we get the identity
B(2,0)+(1+t)B(2,1)+(1+t+t^2)B(2,2)=(2+2t)^2.
In the meantime I have found a combinatorial proof in the literature: Naiomi T. Cameron and Asamoah Nkwanta, On some (pseudo) involutions in the Riordan group,  J. Integer Sequences   8 (2005), Article 05.3.7, proof of identity 1. https://www.researchgate.net/profile/Asamoah_Nkwanta/publications?sorting=newest
Instead of NSWE-paths they use bicolored Motzkin paths, but their proof can easily be translated to the situation of NSWE-paths. So my question has been answered.
• asked a question related to Combinatorics
Question
Let $n\ge 4$ and $W=\{w_1,w_2,\cdots,w_{n-1}\}$ be a given sequence of positive numbers. Suppose that $w_i$ corresponds the weight of the edge $e_i$ of the weighted path $P_n$ for $i=1,2,\ldots, n-1$. Let also $M_i$ be an $i$-matching of $P_n$.
Denote by $W(M_i)$ the weight of such an matching, {\it i.e.,} the product of all weights of such an $i$-matching, and by $WM(i)$ the sum of all weights of $i$-matchings of $P_n$.
{\bf Question:} Who know the applications, including combinatorics and physics, of the
parameter $WM(i)$?
Read 《Integer Flows and Cycle Covers of Graphs》 and 《Circuit Double Covers of Graphs》, the author is Prof. Cunquan Zhang. Another field is circular flow theory, you may read Prof. Genghua Fan's papers.  Hope it is useful.
• asked a question related to Combinatorics
Question
Can someone suggest me a good software for drawing graphs? The suggested software should be one which can be used to draw the attached graph (together with mathematical symbols) .
Thanks everyone for your kind suggestions. I tried most of the suggested softwares and found ipe the best one. Special thanks to Seyyed Aliasghar Hosseini,  Imran Khaliq and Darren Strash.
• asked a question related to Combinatorics
Question
Or generally:
\sum _{i=1}^n \sum _{j=1}^i \left\lfloor \frac{-j+n+1}{2 i}\right\rfloor
I'm assuming from your question that you're looking for a closed form solution.
Let q = Floor[n/2i] and r = n - q*2i (quotient and remainder when n is divided by 2i). Therefore, 0 <= r < 2i.  Since i <= n,
sum_{j=1}^i Floor[(n-j+1)/2i] = iq - d,
where d=0 if i-1 <= r < 2i, and d=i-1-r if 0 <= r < i-1.
This gives a closed form for the inner summation.  Obtaining a closed form over both summations looks like a harder problem, but maybe possible.
• asked a question related to Combinatorics
Question
what are the classes of posets closed under taking ordinal sum of posets?
Let X and Y be two posets on the disjoint sets P and Q. The  disjoint union P+Q is defined as:
(1)if (x 'less than, or equal to' y) is in P, for x and y in P, then x <=  y in P+Q and
(2)if  (x 'less than, or equal to' y) is in Q, for x and y in Q, then x <= y in P+Q.
The ordinal sum P*Q is defined as the partial order on P U Q, that satisfy (1) and (2), and the additional condition: (3) (x 'less than, or equal to' y) in P*Q for x in P, and y in Q.
The disjoint union + is commutative, but the ordinal sum '*' is not.
• asked a question related to Combinatorics
Question
Since a uniform permutation has a probability of 1/n!, only distances of O(1/n^n) can be regarded as negligible. Section 2.2 of the attached paper gives a network with distance O(1/n^n) but the depth is O(n). Is there a permutation network with the same distance but an asymptotically smaller depth?
Please attach the file because the link is not redirecting to appropriate web page of MIT.
• asked a question related to Combinatorics
Question
Let the set of all repunits (repeated units) i.e., 1,11,111,... is X.
=> X = { Rn : n belongs to postive integer }
(Repunit of n defined by Rn. 1 is R!. 11 is R2, etc....)
Generally is not satisfy closed property.
1. Under what condition or when X satisfy closed property?
2. For which n, R2n+1 is prime?
Yes dear Srinivasan, put the second "word" just next the first one. In French we call it juxtaposition, I think the terminology is the same in English .
• asked a question related to Combinatorics
Question
Let q be an odd positive integer, and let Nq denote the number of integers a such that 0 < a < q/4 and gcd(a, q) = 1. How do I see that Nq is odd if and only if q is of the form pk with k a positive integer and p a prime congruent to 5 or 7 modulo 8?
I see that you, too, follow the Putnam Exam.
• asked a question related to Combinatorics
Question
Assuming we have a planar Graph G=(V,E). There is a function f(G) which returns a positive real number and we wish to assign either 0 or 1 to each node in order to minimize the function f(G). Trying all different combinations of assigning 0 or 1 to each node to find the combination which returns the minimum value of f(G) would be computational intractable. Hence, the question is whether there exists a algorithm to find (or approximate) the minimum for such a function.
Note that the function f has the properties that a solution exists and is unique. Further, it is known that the Graph that minimizes the function has 0 in the vast majority of nodes and the set where the nodes are 1 is connected.
Thilo,
One heuristic you might try, given that you have the "sum of squares of distances is small" phenomenon, is to start by looking at all vertices at distance <= k from a fixed vertex.  These sets of vertices, essentially closed balls, will often have small sums of squares of distances, I'd guess (I'm sure that there's an isoperimetric result out there of which I'm unaware...)   Then looking at local changes to these sets might enable you to find better --- and perhaps even optimal --- solutions.  At the very least they'd give you a good choice of upper bounds.
Neil
• asked a question related to Combinatorics
Question
Is there any provision to apply combinatorial analysis on image fusion?
• asked a question related to Combinatorics
Question
The exercise I'm dealing with asks me to show that by adding S = K to the usual reduction rules for the SKI-calculus, one obtains an inconsistent equivalence. This must be done without using Böhm's theorem.
Now, I've found two terms (not combinators) M and N with the following properties:
M = x
N = y
M can be obtained from N by replacing one or more occurrences of S with K
From the rule S = K we thereby get that M = N, and hence that x = y. Which means that any term can be proved equal to any other.
Do you think such an answer could work? Actually, Böhm's theorem (as shown in the book I am studying) establishes that, for distinct combinators G and H, there is a combinator D such that DxyG = x and DxyH=y. So, I feel it would have been more appropriate to find a combinator D such that Dx1x2S = x_i and Dx1x2K = x_j with i , j = 1 , 2 and i ≠ j. But I have HUGE problems in finding combinators, so I have only found terms respectively reducing to one of the variables, and obtainable one from another by replacing S with K or vice versa.
Yeah I see now there's a much simpler solution. But I wondered if I had "well done" the exercise. Actually, it seems to me it works. I found two terms, M and N, the first one reducing to an arbitrary x and the second one to an arbitrary y. Moreover, M can be obtained from N by replacing an occurrence of S with K and therefore, as soon as one admits S = K as a rule, it follows that M = N. Since M = x and N = y, by transitivity one should get x = y for every x and y.
Actually, M and N, as I built them, contain other terms which are in the end not relevant to respective reducibility to x and y. But that's should not be a problem. It should be like to have two terms with variables (x, y, z, w), possibly with z = w, respectively reducing to x and y.
That's how I did the exercise...
PS: inconsistent in the sense that one can trivialize the equality relation.
• asked a question related to Combinatorics
Question
recently, some researchers work on infinite structure of matroids.
independent axiom of matroids  have a first role in definition of codes on GF(q) that we can see this point in representable matroids. now , can we have any logical definition  of this point for related codes of infinite matroids?
Hi Hossein.
Section 2.6 seems to touch on the issue of infinite matroids being representable over a field.
If that field is GF(q), one is close to your question, I assume.
• asked a question related to Combinatorics
Question
When a bipartite complete graph Km,n is given, two subgraphs of Km,n are in the same class when the degree of each right vertex coincides. I want to know the number of all spanning trees in a given class.
Any spanning tree in Km,n has M+N-1 edges. A class whose right degrees do not sum up to M+N-1 does not contain any spanning tree. The number of classes with total degree M+N-1  is the repeated combination of country labels taken N -1 times. Thus the number take the form
(M + N - 2)! / (M - 1)! (N -1)! .
From Scoins' formula the number of all spanning trees in Km,n is
MN-1NM-1 .
As a consequence, there are in general many spanning trees in a class in which the right degrees sum up to M+N-1.  I want to know an explicit formula that gives the number of all spanning trees for a given class with degree sum M + N -1.
This question is derived in the course of Ricardian trade theory study.
Dear Robert A. Beeler,
thank you for your comment. I am expecting to hear from you good news.
Shiozawa
• asked a question related to Combinatorics
Question
We usually use differential equations, ordinary and partial, difference and delayed. But, could dynamics be captured using discrete mathematics structures, or combinatorics?
If you look applications of tools provided by discrete mathematics in dynamical systems then this book can be good place to start:
Lind, Douglas; Marcus, Brian. An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge, 1995. xvi+495 pp. ISBN: 0-521-55124-2; 0-521-55900-6
It provides a very smooth introduction into symbolic dynamics from point of view of discrete structures (graphs, formal languages, Perron-Frobenius theory, etc.) and then reveal connections with topology, mixing, topological entropy, etc.
• asked a question related to Combinatorics
Question
I found this problem without comments in a french exercise book of 1982, now out of print : {\em 1932 exercices de mathématiques} by Luc Moisotte, ISBN 2-04-015483-3.
• asked a question related to Combinatorics
Question
In 1969 , David Barnette conjectured that 3 regular , 3-connected , bipartite , planar graph is Hamiltonian. . I am interested  to generate Barnette  graph for given  even number of vertices. There are countably finite number of Barnette graphs available in the literature.
Consider a grid graph with 4 vertices (cycle C 4) which is Hamiltonian. Increasing dimension in one direction , we see that the resulting graph is always Hamiltonian but not Barnette. Can one generate  countably infinite number of  Barnette graphs from one small Barnette graph?
It has been shown that fullerenes are hamiltonian [1] and they form an infinite family of cubic, 3-connected, planar graphs with faces of size at most 6. There is at least one fullerene on 2k vertices for each k greater than 11. Of course these are not bipartite since they contain pentagons, so I guess you are mainly interested in the other Barnette's conjecture. But, the thesis by Jan Goedgebeur also talks about this second conjecture and not the one about bipartite graphs.
• asked a question related to Combinatorics
Question
I am trying to calculate the most compact way of grouping a set of pixels together. Does anyone have a readable guide on how to do this?
My initial results are given below for clusters of up to 10 pixels. Results are expressed in terms of the sum of unique interpixel distances for a given cluster (e.g. for a 3 pixel cluster it is the sum of the the distances ab, ac, bc).
1 = 0
2 = 1
3 = 3.4
4 = 6.8
5 = 13.5
6 = 21.1
7 = 31.4
8 = 44.1
9 = 58.9
10 = 78.5
I looked at the paper. So at least for L1 distances, we do not have uniqueness, the incremental approach partially fails (some optimal solutions do not extend to +1 optimal solutions, while some optimal solutions are not extension of -1 optimal solutions), and 4x4 square would yield L1 sum of 320, while the optimal result of 318 is displayed in the paper. Good luck.
• asked a question related to Combinatorics
Question
In \cite{Roman} page 25 we read that,  a sequence $s_n(x)$ is Sheffer for $(g(t), f(t))$, for some invertible $g(t)$, if and only if
$$s_n(x+y)=\sum\limits_{k=0}^{\infty}\binom nk p_k(y) s_{n-k}(x),$$
for all $y$ in complex numbers,  where $p_n(x)$ is associated to $f(t)$.
Noting to the fact that $e_q(x+y) \neq e_q(x)e_q(y)$, leads to conclude that $s_{n,q}(x+y) \neq e_q(yt)s_{n,q}(x)$, and ,therefore, we do not have the $q$-analogue of the identity above directly. Is it possible to express the $q$-analogue of the above mentioned identity in any other way, or should we neglect such an identity for $q$-Sheffer sequences at all?
Any contribution is appreciated in advance.
\bibitem{Roman}Roman S., Rota G. The umbral calculus. Advances Math. 1978;27:95–188.
Dear Dr. Waldemar W Koczkodaj
Thanks for your response. I faced with this question, for the first time, while I was studying the sequence of $q$-Appell polynomials. Since that time, this question has been in my mind for a long time and so far I have not been able to make myself convinced by a perfect answer to it. The reason to ask this question here, actually, is to consult with the experts of this field and read about their different ideas which are originated, clearly, from different points of view. Although I am enthusiastic for learning more and more and go forward through the science, in case that I found a good answer which helps me to publish my studies, I make you sure that I will definitely obey the publication rules and humanity.
• asked a question related to Combinatorics
Question
Some authorities like Davenport have already explained that traditional (Small?) data relates to corporate operations while Big data relates to corporate products and services.
In these papers, I have argued that small data analysis based on Gaussian statistics, while BIG data based on power law statistics, and small data analysis based on Euclidean geometry, while BIG data based on fractal geometry.
Jiang B. (2015b), Geospatial analysis requires a different way of thinking: The problem of spatial heterogeneity, GeoJournal, 80(1), 1-13, Preprint: http://arxiv.org/ftp/arxiv/papers/1401/1401.5889.pdf
Jiang B. (2015a), Head/tail breaks for visualization of city structure and dynamics, Cities, 43, 69-77, Preprint: http://arxiv.org/ftp/arxiv/papers/1501/1501.03046.pdf
Jiang B. and Miao Y. (2014), The evolution of natural cities from the perspective of location-based social media, The Professional Geographer, xx(xx), xx-xx, DOI: 10.1080/00330124.2014.968886, Preprint: http://arxiv.org/abs/1401.6756
• asked a question related to Combinatorics
Question
Last year, Dr. Yitang Zhang has published a paper for the upper bound of twin primes, which is 7*10^7. Does anyone has idea to achieve lower bounded gap?
Terence Tao and his collaborators did tons of work on this in Polymath 8 (mentioned ambiguously in the above comment). You can read part of what was done here:
Tao's complete corpus of work on that project is:
• asked a question related to Combinatorics
Question
In Maple 16,  how can we with the software combstruct,  to give the sentence about the recurrence formula,
A(x)=1+x[A(x)3+3A(x)A(x2)+2A(x3)]/6
Very strange, first the gf then the object to count!
• asked a question related to Combinatorics
Question
It is well known that the majority rule may not be transitive for some configurations of individual preferences. Domain restrictions are possible ways out. But what is known about maximal such domains (i) with respect to the cardinality? (ii) via set inclusion?
Thank you very much. I also like the job done on mutiple issues.
• asked a question related to Combinatorics
Question
I and another three colleagues have an ongoing paper about the application of sociometry to multiple human resource allocation to multiple projects. The problem is mathematically quite challenging but we are on a dead end concerning one of the outcomes we are dealing with.
Particularly, we have been applying meta-heuristics and evolutionary algorithms to solve the problem of allocating groups of people so as to maximize cohesion among them. The research is quite interesting and we think it is going to open multiple and very interesting research and industry application in the near future.
However, we are stuck in one part. We are trying to calculate the number of viable combinations of people who can work either full-time, part-time or not work at all in several simultaneous projects and we need a person with advanced knowledge in combinatorics to give us a hand. We are willing to pay or to put his/her name as co-author on the paper.
Getting straight to the point, the problem statement is as follows:
There are N people (i=1...N) who can be selected to work in P simultaneous projects (j=1...P). Each person can have a dedication of work full-time (1), half-time (0.5) or not work (0), that is, three possible allocations (0, 0.5, 1).
Now, we know that each project j requires Rj people. How many different and viable combinations are there?
Numerical Example:
People available  Project j=1     Project j=2      Project j=3=P
i=1                       0 or 0.5 or 1   0 or 0.5 or 1    0 or 0.5 or 1     Row sum =<1
i=2                       0 or 0.5 or 1   0 or 0.5 or 1    0 or 0.5 or 1     Row sum =<1
i=3                       0 or 0.5 or 1   0 or 0.5 or 1    0 or 0.5 or 1     Row sum =<1
i=4                       0 or 0.5 or 1   0 or 0.5 or 1    0 or 0.5 or 1     Row sum =<1
i=5                       0 or 0.5 or 1   0 or 0.5 or 1    0 or 0.5 or 1     Row sum =<1
Requirements      R_1=2            R_2=1             R_3=1              Row sum=4
,that is, we need 2+1+1 people working in these 3 projects and 1 out of the five available people is not used. We can use each person totally (1), half-time (0.5) or not use him/her, but the combinations must be feasible, i.e., each person cannot be assigned over 1 (full-time) and the requirements have to be fulfilled.
This problem is easier when the people can only work full-time or not work, but with half-times is far more complicated.
If anyone think that he/she is able to solve it, or just want further details, please  contact me.
Thanks
Thanks Peter. We really appreciate the time you have spend solving this problem. Your contribution will be duly ackowledged as we agreed.
Thanks too for all you laborious explanations trying to make that someone like me understood the mathematical and combinatorics vision of this problem.
• asked a question related to Combinatorics
Question
Skew room squares exist for all odd values greater than 5. If n is prime it is a simple matter to generate a skew starter. But 667 is not prime. 667=23*29 which means a computer search has to be done in order to generate one. I would be satisfied with the skew room square of side 667, even though we can show it exists we can't seem to construct it. Any suggestions on this particular problem are appreciated.
The reference provided 10 minutes ago may not be appropriate.
But, one may try some direct product construction as follows.
Start from skew starter S_1 in Z_{23} and skew starter S_2 in Z_{29}.
Find a permutation P=(p_0,p_1,...,p_{28}) such that
P-I = (p_0-0, p_1-1, ..., p_{28}-28)
And
P+I = (p_0+0, p_1+1, ..., p_{28}+28)
Are both permutations in Z_{29}. For example, take p_s=2s.
For each pair {x, y} in S_1, construct 29 pairs {(x,p_s), (y,s)}, s=0,1,…,28.
This generates 11x29 pairs.
Add 14 more pairs {(0,u),(0v)} such that {u,v} is from S_2.
It may be easy to verify if the 11x29+14 pairs form a skew starter in Z_{23}xZ_{29}.
If it still does not work, please let me know.
• asked a question related to Combinatorics
Question
Given a permutation A on the set [n], is there a way to determine the maximum number of disjoint cycles of AC where C ranges over all n-cycles on [n]? For which class of permutations A, this problem has been studied before?  Thanks
Hamming distance between two permutations a and b is defined to be the minimum number of transpositions needed to bring a to b. If you express a^{-1} b as a one-line sequence, then Hamming distance is n - the number of fixed points.
• asked a question related to Combinatorics
Question
What is the best software package available for carrying out computations related to algebraic coding theory, especially codes over rings.
• asked a question related to Combinatorics
Question
I need a good reason why improper uniform prior could be use as a prior in Serial Numbered Population (SNP) problem. Or maybe someone can tell me about improper uniform prior itself. An pdf link might be very helpful for me.
I attached the journal.
I'm not sure if Jochen's answer is entirely correct; "An improper prior can be used when the resulting posterior is proper."  I'd modify it to say that an improper prior can be used when the resulting posterior can be achieved with a *proper* prior and the limits of the distribution, after the posterior is calculated, is taken to infinity and the result is shown to be the same.  For example, you can use a proper, bounded uniform prior and then, after you calculate the posterior, take the limits as the bounds grow to infinity to show that if you used the improper prior in the first place it would give the same answer.
Seen this way, the improper prior is a short-cut to make the analysis cleaner.  For example, I personally find E T Jaynes' analyses very clear because he uses improper priors (when he can!), and warns that one has to be careful to do it properly if there is even a hint of trouble.  I find a paper like Bretthorst's "Difference of Means" paper to be a much harder read because he uses proper priors throughout, even in cases where I think the improper prior would work.
• asked a question related to Combinatorics
Question
Could anyone suggest me some references about the uniqueness of the addresses of an Iterated Function System? E.g. points of Cantor set can be coded by a unique address, how general is this property?
More precisely, I am interested in the sufficient conditions to have a one-to-one relation between the shift space of the Iterated Function Systems and  its attractor.
Dear Anna,
There are three kinds of attractors of iterated function systems, totally disconnected, just touching and overlapping. Totally disconnected attractors have metrically equivalent structure to the Cantor set. This can be found in Barnley's book Fractals Everywhere. The remaining cases may be treated with the help of lifted IFS, which is also explained in the book.
Best regards
Miroslav
• asked a question related to Combinatorics
Question
It is well known that the secant $\sec z$ may be expanded at $z=0$ into the power series
\label{secant-Series}
\sec z=\sum_{n=0}^\infty(-1)^nE_{2n}\frac{z^{2n}}{(2n)!}
for $|z|<\frac\pi2$, where $E_n$ for $n\ge0$ stand for the Euler numbers which are integers and may be defined by
What is the power series expansion at $0$ of the secant to the power of $3$? In other words, what are coefficients in the following power series?
It is clear that the secant to the third power $\sec^3z$ is even on the interval $\bigl(-\frac\pi2,\frac\pi2\bigr)$.
Dear Dr. Feng,
Take a look at:
Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1
in particular section: 7.1 Multinomial Euler numbers
The results in section 7.1 are for powers of sech, so you'll need to substitute t -> it.
The above paper is freely available from the Journal's website, and also from my ResearchGate pages.
Best wishes,
Ghislain Franssens
• asked a question related to Combinatorics
Question


text:
##### Fracture Characterization, Mineral Vein Evolution and the Tectonic Pattern of Igarra Syn-Tectonic Granite, Southwestern Nigeria

Oden, Michael I., Udinmwen Efosa

Current Journal of Applied Science and Technology, Page 2417-2429
DOI: 10.9734/BJAST/2014/8540

Extensive outcrops of the pan-African granite were studied in detail with particular interest in the joint pattern and mineral vein characteristics. The attitude of 57 tectonic joints and 60 mineral veins were measured, the length and width of the veins documented. Rose diagrams and graphs representing fractures and veins were plotted and analysed based on their behavior and preferred orientations. Two (2) basic trends of mode I fractures(joints) and veins were observed in the granites, these are the E-W trending joints and veins parallel to the maximum principal stress (σ1) direction and the N-S trending joints and veins parallel to the minimum principal stress (σ3) direction. These granites contain mostly Mode I fractures which require lower stress to initiate and propagate relative to Mode II and Mode III fractures, although other trends (NW-SE and NE-SW) of joints and veins were mapped, they are highly insignificant compared to the major E-W and N-S trending joints and veins within the plutons. Mineral veins in these syn-tectonic granites developed apertures that were sensitive to the orientations of the stresses; the widest veins being those oriented approximately in the E-W direction, parallel to the maximum principal stress (σ1). Mineral veins in N-S orientation are generally thin and non-persistent; their formation indicates fluid pressure-build up in the course of the orogeny. The occurrence of sheet/exfoliation joints lends credence to the occurrence of some sort of near-surface deformation related to cooling and exhumation or unroofing within the granite. The simple pattern of surface joints and veins indicates a fairly constant orientation of the principal stresses σ1 and σ3 in the E-W and N-S directions respectively, through most part of the orogeny in this area. From this study, it would appear that the pan-African stress configuration did not change throughout the period tracked by the evolution of these granites. This work serves to elucidate the joint pattern and mineral vein evolution of the syn-tectonic granite to further discuss the history of Pan-African tectonics of the Igarra region in southwestern Nigeria and it was carried out in Igarra and environs with further analysis done at the Department of Geology, University of Calabar, Nigeria for a period of 18 months.

##### The Potential of Indigenous People’s Forest: Ecological Benefits and Plants Diversity in Bru People’s Spiritual Forest, Northeastern Thailand

Tunwa Chaitieng, Tharest Srisatit

Current Journal of Applied Science and Technology, Page 2430-2439
DOI: 10.9734/BJAST/2014/9170

Aims: To study the ecological benefits and biodiversity of plants in the Bru’s spiritual forest and the relationship between the Bru’s wisdoms and their spiritual forest.
Study Design: This research applied the use of forest ecology and a qualitative research.
Place and Duration of Study: Sakon Nakhon Province, the upper Northeastern Thailand, between June 2011and June 2012.
Methodology: We surveyed plants diversity in 3 spiritual forests of Bru people’s villages and collected primary data about relationship between Bru people and spiritual forest from the site by in-depth interviewing, group discussions with key informants.
Result: The Bru People are a group of indigenous people near the Lower Mekong River and have a culture that is closely related and tied to the spiritual dimension. Their philosophy and fundamental world views are holistic. There are 11 Spiritual Forest in Bru communities of Sakhon Nakhon Province. These forests once serves as: an important source of biodiversity and native plants, a source of food and herb to support the Bru people’s economy and ecology, a spiritual centre of communities, and a foundation to create strength to maintain their idealism and ecological heritage in the Lower Mekhong basin.
Conclusion: Spiritual forests of Bru people have served as important reservoirs of biodiversity, preserving native species of plants, protection air pollution and supporting the ground economy and culture of Bru people.

##### An Empirical Investigation into Factors Influencing the Intention to Use E-learning System: An Extended Technology Acceptance Model

Haitham Alshibly

Current Journal of Applied Science and Technology, Page 2440-2457
DOI: 10.9734/BJAST/2014/10033

Aims: The aim of this research is to investigate the factors affecting the intention towards using e-learning in a Jordanian university, through an extension of the technology acceptance model (TAM).
Place and Duration of Study: Amman University College between May and July 2013.
Study Design: Adapted theoretical framework. This study incorporates perceived system quality and perceived self-efficacy into the TAM model in order to investigate the role of perceived system quality and perceived self-efficacy in educators’ intention towards using e-learning.
Methodology: The adapted model was applied to Amman University College; the study used primary data that was collected through self-administered questionnaires; Partial least squares analysis was used to observe the associations of proposed constructs.
Results: The results of the study confirm the original technology acceptance model findings. The research results show that all the hypotheses are supported, which indicates that the extended variables can effectively predict whether users will adopt an e-learning system.
Conclusion: This research advances theory and contributes to the foundation for future research aimed at improving our understanding of students' adoption behavior of the e-learning system. This study will help to determine what are the factors that are significant in explaining the intention towards e-learning.

##### Hydrogeochemical Characteristics and Evaluation of the Drinking and Irrigation Water Quality in Marand Plain, East Azerbaijan, NW Iran

Current Journal of Applied Science and Technology, Page 2458-2467
DOI: 10.9734/BJAST/2014/9426

In order to study hydrogeochemistry of groundwater, samples of water were collected from Marand located in Eastern Azerbaijan province, NW of Iran at July, 2008. This region has 800 km2 space and 87 locations. Samples which were collected from 74 water wells, 10 Qantas and 3 springs analyzed for major cations and anions processed by statistical methods. Parameters were compared with water quality standards and public health standards for domestic usage. Some of the locations are defined by high concentration of EC, TDS, Cl, Na and K. Half of the groundwater samples were contained Ca-Mg-Cl type of hydrochemical facies, followed by Ca-HCO3, Ca-Cl and Na-Cl types. Based on US salinity diagram, 42% of the samples fall in the field of C3-S1. Due to low sodium and medium-high salinity groundwater C3-S1 class can be used for irrigation on almost all soils with little danger of sodium problem. Majority of the samples are not suitable for domestic purposes meanwhile they were far from standards of drinking water. In the recent years, comparing average electrical conductivity and total dissolved solids of the studied area revealed that declines in water levels as a result of extensive agricultural activities and urbanization resulting in deterioration of groundwater quality in the major parts of the plain.

##### Rejuvenating the Economies of Ogbese Rural Region, Nigeria

J. O. Daramola, J. O. Basorun

Current Journal of Applied Science and Technology, Page 2468-2476
DOI: 10.9734/BJAST/2014/9087

Aims: To examine the level of socio-economic and physical development of Ogbese town vis-à-vis the regional development potentials.
Study Design: Reconnaissance survey and administration of questionnaires were carried out in the study.
Place and Duration of Study: Ogbese region in Akure Local Government Area of Ondo State, Nigeria, between July and August, 2013.
Methodology: We sampled 96 respondents in the study region (Ogbese 60, Elemo 8, Benin 7, Owode 9, Araromi 7 and Lisa 5). Data analysis was carried out using descriptive statistics such as frequency counts and percentages.
Results: Research findings show that agriculture as the spine of the rural regional economy is declining due to the financial constraints and failure of the farmers and traders to harness labor and facilities for effective operations.
Conclusion: Policy measures were recommended based on research findings to improve the economies of the rural environment.

##### No Frequency Reuse: Wearable Steerable MIMO Microstrip Antenna Array for Mobile Ad Hoc Applications

Taha A. Elwi, Sa’ad Al-Frieh, Mohammed Al-Bawi, Mohammed Noori

Current Journal of Applied Science and Technology, Page 2477-2488
DOI: 10.9734/BJAST/2014/8739

Aim: The principle of eliminating the frequency reuse in the mobile Ad Hoc system among the sectors of the unit cell using Multi Inputs Multi Outputs (MIMO) antenna array is investigated in this paper.
Antenna Design: The size of the proposed antenna array is 10×10 cm2 to obtain a bandwidth around 1 GHz. The single antenna element is constructed from sub-patches that are connected with feeding network through pin diodes, as switches, that are mounted on an FR-4 substrate.
Antenna Performance: The antenna elements are characterized from 0.8 GHz to 2 GHz in terms of S-parameters and radiation patterns with different switching OFF/ON categories.
Methodology: A numerical investigation based on Finite Integral Techniques (FIT) of Time Domain (TD) formulations is conducted using CST MWS to evaluate the antenna performance. A Frequency Domain (FD) solver based on CST formulations is conducted for validation.
Results: It is found the antenna shows insignificant coupling around 1.2 GHz and 1.6 GHz. Furthermore, the radiation patters of the antenna are found to be in the end fir direction with phase change of about 120o among antenna sectors. The antenna provides absolute gain of 2 dBi at 1.2 GHz and 2.9 dBi at 1.6 GHz. It is found that the proposed antenna behaves like highly directive end fire antenna at the main lobe direction around 1.6 GHz. Moreover, it is found that the antenna array exhibits insignificant coupling among each other around this frequency.
Conclusion: The performance and structure of the proposed design allows the use with wearable Ad Hoc mobile systems without the need for frequency reuse by steering the radiation patterns of the antenna sectors through switching antenna branches. Finally, an excellent agreement has been achieved between the regarded results from TD and FD solvers.

##### Improving Design of a Cryogenic Process by Exergy and Process Economic Analyses

A. N. Anozie, M. N. Braimah, O. J. Odejobi, O. P. Ayoola

Current Journal of Applied Science and Technology, Page 2489-2500
DOI: 10.9734/BJAST/2014/8646

Aim: This work compared four different designs of a cryogenic process (Linde methane liquefaction) using exergy and process economic analyses.
Study Design: Simulation, energy, exergy and process economic analyses were used.
Place and Duration of Study: Department of Chemical Engineering, Obafemi Awolowo University Ile-Ife and the duration of study was between March and May 2011.
Methodology: Hyprotech System Simulator (HYSYS) software was used in simulating the process plants and Microsoft Excel was used for exergy, energy and process economic analyses. Annual venture profit was used as economic index in the selection of best design among the available alternatives.
Result: Linde Methane Liquefaction for cryogenic process was investigated for improvement using exergy and process economic analyses. Four design cases were considered. The exergy efficiency for the base case (case 1) was 5.19%, 5.30% for case 2 when a recycle was added to the base case, 28.7% for case 3 when a recycle and a turbine were added to the base case and 39.5% for case 4 when a recycle was added and the valve was replaced with a turbine in the base case. Case 3 has the highest equipment and installation cost of $594649.14. The capital investment, operating and maintenance expenses were$88574.00/yr, $89687.30/yr,$94280.17/yr and $90672.23/yr for case 1, case 2, case 3 and case 4, respectively. Process economic analysis based on 1 kg/h throughput revealed case 2 with the highest annual venture profit of$4564265.09/yr as the best design.
Conclusion: Since it is known that throughput has effect on venture profit, it was concluded that case 4 with the highest exergetic efficiency could be the best design at higher throughput. This will be investigated in a follow up study.

##### Influence of Cropping Pattern Systems on Incidence of Scab and Septoria Leaf Spot Disease of Cowpea (Vigna unguiculata)

G. A. Mbong, C. N. Fokunang, O. Alabi, E. A. Tembe-Fokunang, M. Emechebe, M. D. Alegbejo, C. N. Akem

Current Journal of Applied Science and Technology, Page 2501-2512
DOI: 10.9734/BJAST/2014/6229

Aims: Scab caused by Sphaceloma sp and Septoria leaf spot caused by Septoria vignae were evaluated on one variety (Ife Brown) of cowpea: in this study. Ife Brown is highly susceptible to scab and moderately susceptible to Septoria leaf spot. Cowpea (Vignia unguiculata L. Walp) was planted simultaneously with maize (intercrop) and sorghum (intercrop). The scientific importance of intercropping pattern in this study was to develop an information based system on how crop pattern can influence disease incidence and severity and thus improve crop yield within an agro ecological zone
Study Design: The field layout was factorial concept in Randomized Complete Block Design (RBCD) with four replications
Place and Duration of Study: The two cereals used for this study were, maize variety TZEEW, and sorghum variety ICSV111. The investigations were carried out during three cropping seasons of 2004, 2005 and 2006 at Samaru and Shika in Zaria, Nigeria.
Methodology: Five cropping patterns: interrow, intrarow, doublerow, strip and sole cropping on the incidence and severity of scab and Septoria leaf spot fungi isolated from infected field plants, were studied. Disease incidence and severity for scab and Septoria leaf spot were recorded at weekly intervals starting from the first appearance of the symptoms. Data for the different plant parts (leaf, stem, peduncle, flower cushion and pod) infected by scab and leaves infected by Septoria leaf spot were recorded from four middle ridges for each plot. The data obtained was subjected to analyses of variance (ANOVA) and mean separation was by Student Newman Keuls Test (SNK).
Results: Infection of the cowpea and severity of scab and Septoria leaf spot on the crop significantly increased with increase in age of the variety tested. Symptoms of infection resulting from scab were seen on all the aerial parts of the crop from seedling stage to maturity, depending on the time of infection. Cropping patterns significantly affected the incidence and severity of the two fungi on cowpea in the order of sole<double-row<strip<inter-row<intra-row for scab and sole<intra-row<double-row<strip<inter-row for Septoria leaf spot. However, the reaction of the crop to the individual pathogens was different across the various cropping patterns with the rate of infection of scab higher compared with that of Septoria leaf spot. Cowpea sole cropping proves better for both pathogens compared to the other cropping patterns.
Conclusion: This study showed that sole cropping system was more ideal for disease managements and incidence of Scab and Septoria leaf spot diseases of cowpea with respect to cropping systems studied.

##### Geochemistry and Environmetric Approaches to Estimate Pollution Impacts on Soils around Okaba Coal Mines

E. G. Ameh, I. O. Imasuen, E. G. Imeokparia

Current Journal of Applied Science and Technology, Page 2513-2527
DOI: 10.9734/BJAST/2014/7051

Soil samples were collected during dry season from Okaba coal mine areas to establish the level of impact. Impact evaluations were carried out using correlation analysis, factor and cluster analyses, determination of enrichment factor (EF), contamination factor (CF), pollution load index (PLI), anthropogenic factor (AF) and index of geo-accumulation (Igeo). The AF gave this order: Pb>Ni>Cu>Zn>Fe>Cd. The Igeo index, EF and CF also revealed the same rank as AF. The pollution load index indicated that all sampled sites have been polluted with locations Ok06, Ok16, Ok17, Ok03, Ok10 and Ok08 as the most impacted. At P<0.01 level, very strong correlations exist between Na-K, K-Mg, Ca-Zn while at P =.05 level, strong correlations were recorded between Na-Mg, Ca-Fe, Na-Pb, Pb-Cd and Ni-Pb. The R-mode factor suggested that factors one and three were both natural and anthropogenic while factor two was due to natural processes. The R-mode cluster analysis suggested that cluster one was anthropogenic and clusters two and three were pseudo anthropogenic. Based on the indices, Pb, Ni and Cu were the most impacted heavy metals on soils. While detailed study is apt, the author recommend immediate and total reclamation of the ponds/mines and treatment of all affected areas.

##### The Importance of Targeted Drug Delivery Systems on Pancreatic Cancer

S. Kaleli, G. Güney

Current Journal of Applied Science and Technology, Page 2400-2416
DOI: 10.9734/BJAST/2014/9671

Pancreatic cancer (PC) is the fourth type of cancer-related death in the world. There is estimated 45,220 new cases of PC will be diagnosed and 38,460 people will be passed away from the disease in the near future. The absence of diagnostic and prognostic markers and also unresponsiveness to current therapies including radiation and chemotherapies are major barriers against successful therapy even in the initial stage. The current therapies is not sufficient to provide effective treatment for PC patients due torapiddrug solubility and metabolism, insufficient drug concentration at the specific site and adverse side effects. To eliminate this disadvantages, nanoparticle based drug delivery systems have been investigated.In this review, we try to reveal the genetic and molecular mechanism which cause PC and emphasize the importance of targeted drug delivery systems for the disease.

format: plain


text:
# How to randomly pick an object from a collection?

I am working on a modal that paints objects & then drops them using rigid body. I'd like to include support for collections, where the user can store a group of objects as a collection & then randomly choose one of those objects to paint.

how can I access objects in a collection & duplicate the chosen object?

Thanks

Use this code:

import bpy
from random import random

#select your collection by name (in this case the collection named "Collection")
col = bpy.data.collections["Collection"]

objNum = len(col.objects)

#Generate random number
randomIndex = round(random()*(objNum-1))
#use the random number to call a random object in the collection
obj = col.objects[randomIndex]

#creat a copy of the object and link it to the collection
newObj = obj.copy()

• You may also want to .copy() the object's .data. Sep 17 '19 at 7:51
• You can also use random.randint(0,objNum - 1). Sep 17 '19 at 7:51


text:
# a strange array

You wake up into an empty room, all you can see is a locked door. On the wall, a numeric pad, on its screen you see

remaining tries: 1. enter passcode _ _ _ _ _

next to it a strange inscription: 6e2d3e666f724e74684c6574746572496e576f7264

On the floor, lays a laptop, on its screen this strange picture:

After looking closely it seems the laptop is connected to the internet.

Which code do you type on the pad and why?

1 day without other guesses, here's a clue

LeftColor/MiddleColor/RightColor

clue 2

remember, you see this on a computer, this is important

clue 3

the 3 colors will tell you more about the computer

it's been a long time ... so, here's a new clue:

19-4052/11-4800/17-1463 is a flag, together with clue 3 and the "strange inscription" it should help you decypher ify yky non*i^

hop hop, another hint buddies

Nikhil Eshvar found the country linked to that flag, maybe you should have in mind the keyboard pattern used in that country, this will allow you to decypher ify yky non*i^

## 6 Answers

Going off of all the other answers
The answer should be

27325

I deduced this based on the findings of Reinier, and that the password is being put in via a number pad.

The word we got was BREAK, punched into a number pad with letters, like a phone, will result in 2(B) 7(R) 3(E) 2(A) 5(K)

• yep that was the expected answer, though I don't really know how to give the "answer" mark, because everyone did some work on it, but since you gave the final answer, I guess it's yours, I will think about it this night :p Apr 30 '18 at 22:40

I think the passcode might be

BREAK

Building on the answer of zekei:

If we use the bucket-tool of some image editing software on some white part of the picture, we get the following:

We can now put the characters in the array in the order denoted by the Roman numerals that appeared. The resulting string is:
ENJOYTHESILENCE-V1W4
There is a song called "Enjoy the silence" by Depeche Mode. If we look at verse 1, word 4, we see that this word is "Break", which has five characters, just like the passcode. So I guess this is the solution.

• yep, this is almost the answer, BREAK is the word to find, now you have to enter it on a numeric pad ;) Apr 30 '18 at 11:40
• Could that mean the old phones keyboard? May 3 '18 at 6:03
• @PaulKaram yep, Bobalobdob gave the answer ;) May 3 '18 at 15:27

Welp, the easiest part is changing the hex string to ASCII text, which results in:

n->forNthLetterInWord

• this is the first step indeed ;) Feb 19 '18 at 19:27

Partial Answer: Each of the numbers is a

pantone color. 19-4052 is classic blue 11-4800 is blanc de blanc 17-1463 is tangerine tango

Edit: thanks to Reinier finding the correct word:

The passcode is

27325
BREAK converts to 27325 when inputted on any normal phone keypad.
B = 2, R = 7, E = 3, A = 2, K = 5

• Welcome to Puzzling SE! Please remember to place your answers in spoiler tags. For more info, take the Tour! Feb 21 '18 at 5:03
• Yep, The last one could be a little confusing since I found this color by typing "red Pantone" on google, it is described as a reddish orange, but you should see it more red than orange ;) Feb 21 '18 at 7:39

A few observations not explicitly mentioned that I decided I'd throw down since I took a look at this. I'm guessing the AZERTY key layout used in France has to do with this, since the fact that it's a computer is relevant. Perhaps the n->forNthLetterInWord clue means shifting along the keyboard? And perhaps the spacing in the ciphertext separates the cipher into words, meaning that the n references the position in each word of the ciphertext (as opposed to the position in the entire string).

Edit:

Using azertyuiop^\$qsdfghjklm%*<wxcvbn,;:! as a representation of the AZERTY key layout, searching for each character in the string ify yky non*i^, and shifting each character to the left by its position in each word (starting from 1) gives use the bucket. I'm confident this is the correct string, but unsure how to combine it with the given matrix (array?). My question would be: should I interpret this as a hint, or as a string, or both?

• The array is to be used once the string is decyphered. The first part of your comment is straight to the point, but you can forget the part where you talk about the array ;) Apr 28 '18 at 10:18
• Awesome. Now I just need to figure out how to use the deciphered text with the array! Apr 28 '18 at 21:07
• You should do exactly what the string tells you to do :) Apr 29 '18 at 0:04
• nah - never mind Apr 30 '18 at 7:24

From Retro's answer:

We get that it is a Pan-slavic flag which could correspond to Atbash cipher since the language of Slavic is Latin. Which results in the solution: rub bpb mlm*r^. It's obvious that the * and ^ should be found...

New edit:

Just now realized about the left-middle-right flag. So it's France and perhaps Vignere cipher now...

• it is indeed a french flag, if you connect this with HollyLeaves' answer, you should be able to decypher ;) Mar 20 '18 at 11:59


text:
# Number Partitions

Is this series complete, using '0' is not allowed:

$6 = 6$ or $5+1$ or $1+5$ or $4+2$ or $2+4$ or $3+3$ or $4+1+1$ or $1+4+1$ or $1+1+4$ or $1+2+3$ or $1+3+2$ or $2+3+1$ or $2+1+3$ or $3+1+2$ or $3+2+1$ or $1+1+1+3$ or $1+1+3+1$ or $1+3+1+1$ or $3+1+1+1$ or $1+1+1+1+2$ or $1+1+1+2+1$ or $1+1+2+1+1$ or $1+2+1+1+1$ or $2+1+1+1+1$ or $1+1+1+1+1+1$

5 = 5 or 4+1 or 1+4 or 3+2 or 2+3 or 1+1+3 or 3+1+1 or 1+3+1 or 1+2+2 or 2+1+2 or 2+2+1 or 2+1+1+1 or 1+2+1+1 or 1+1+2+1 or 1+1+1+2 or 1+1+1+1+1

4 = 4 or 3+1 or 2+2 or 1+3 or 1+1+2 or 1+2+1 or 2+1+1 or 1+1+1+1

3 = 3 or 1+1+1 or 1+2 or 2+1

2 = 2 or 1+1

1 = 1

The number of partitions in this series is 1,2,4,8,16,25.

Have I missed any, as 25 seems odd to me.

• 2+2+2 has been missed. There are 6 others. In total, there are 32. – S.C.B. May 9 '16 at 15:50

You have missed seven ways. Mainly, $$2+2+2=6$$$$1+1+2+2=6$$$$1+2+1+2=6$$$$1+2+2+1=6$$$$2+1+2+1=6$$$$2+2+1+1=6$$$$2+1+1+2=6$$So there are $32$ ways. However, one thing worth mentioning is that your definition strays from the normal definition of partitions, where two sums that differ only in the order of their summands are considered the same partition. In fact, if the order matters, as in your case, the correct term is a composition.


text:
## bookymb143 Group Title well hero people i dont own this page but i need help on my work on summer school can u help me 3 years ago 3 years ago

1. TheMan12345 Group Title

yes?

2. Dalvoron Group Title

We can probably provide some help. When you get stuck on a question, post it here and we'll see what we can do to help you understand the maths.

3. bookymb143 Group Title

ok here my question what the 34%

4. yoshita Group Title

yeah sure! whats up?

5. bookymb143 Group Title

i dont think that dont make sence

6. bookymb143 Group Title

but it own my paper

7. bookymb143 Group Title

help me

8. yoshita Group Title

9. Dalvoron Group Title

Post the question word for word as it appears in your paper.

10. bookymb143 Group Title

im and the 5th

11. yoshita Group Title

12. bookymb143 Group Title

13. bookymb143 Group Title

35%

14. bookymb143 Group Title

i dont understand

15. Dalvoron Group Title

35% means 35 per 100. It can also be written as $$\frac{35}{100}$$

16. yoshita Group Title

ok thats a great grade to be in. i dont get what you re asking

17. yoshita Group Title

ok thanks dalvoron. i get what they mean now

18. yoshita Group Title

see ya