odd perfect numbers diophantine equations and upper bounds we obtain a new upper bound for odd multiperfect numbers if n is an odd perfect number with k distinct prime divisors and p is its largest prime divisor we find as a corollary that 1012p 2n 10 and n has at least 101 prime factors counting multiplicity if k is the number of distinct prime factors then as proved in 12 13 we have k 9 and n 10 starting in 2 readers should be familiar with basic facts about odd perfect numbers including knowledge of congruence restrictions related to the special prime as this paper is an extension of the methods used in 13 starting in 3 the reader should be familiar with the ideas in that paper 1 a better upper bound let n be a positive integer following the literature n is said in increasing order of generality to be perfect when n n 2 multiperfect when n n z and n d perfect when n n n d for simplicity we will always assume n d z 0 note that n d does not need to be in lowest terms writing n k i 1 p ei i where p1 pk are the prime divisors of n the equation n n n d 2010 mathematics subject classification primary 11n25 secondary 11y50

notes on bipolar fuzzy graphs in this note we show by examples that definitions 3 1 5 6 and 5 7 example 5 8 propositions 5 9 and 5 10 and partial proof of propositions 5 11 and 5 12 in a previous paper by akram muhammad akram bipolar fuzzy graphs inform sci 181 2011 5548 5564 contain some flaws and then provide the correct versions finally we introduce a generalized bipolar fuzzy graph

odd perfect numbers are greater than 10 1500 brent cohen and te riele proved in 1991 that an odd perfect number n is greater than 10 300 we modify their method to obtain n 10 1500 we also obtain that n has at least 101 not necessarily distinct prime factors and that its largest component i e divisor p a with p prime is greater than 10 62

odd perfect numbers have a prime factor exceeding 10 8 jenkins in 2003 showed that every odd perfect number is divisible by a prime exceeding 10 7 using the properties of cyclotomic polynomials we improve this result to show that every perfect number is divisible by a prime exceeding 10 8

improved techniques for lower bounds for odd perfect numbers if n is an odd perfect number and qk n q prime k even then it is almost immediate that n q2k we prove here that subject to certain conditions verifiable in polynomial time in fact n q5k 2 using this and related results we are able to extend the computations in an earlier paper 1 to show that n 10300 comments only the abstract is given here the full paper appeared as 2 the main part of the proof that there is no odd perfect number n less than 10300 is a very large tree each of whose 12655 leaves gives either a contradiction or a sufficiently large lower bound on n the proof tree is available by anonymous ftp in a separate file rpb116p dvi z the integer factorizations used in the proof are also available by anonymous ftp see 3 references 1 r p brent and g l cohen a new lower bound for odd perfect numbers mathematics of computation 53 1989 431 437 supplement ibid s7 s24 mr 89m 11008 also appeared as report tr cs 88 05 computer sciences laboratory anu february 1988 50 pp rpb100 2 r p brent g l cohen and h j j te riele improved techniques for lower bounds for odd perfect numbers mathematics of computation 57 1991 857 868 mr 92c 11004 also appeared as report nm r8921 centrum voor wiskunde en informatica amsterdam october 1989 13 pp a longer version was submitted to the mathematics of computation umt file and appeared as report cma r50 89 cma anu october 1989 198 pp rpb116 3 r p brent factor an integer factorization program for the ibm pc report tr cs 89 23 computer sciences laboratory anu october 1989 7 pp rpb117 brent computer sciences laboratory australian national university canberra e mail address rpb cslab anu edu au cohen school of mathematical sciences university of technology sydney po box 123 broadway nsw 2007 australia te riele centrum voor wiskunde en informatica kruislaan 413 1098 sj amsterdam the nether lands e mail address herman cwi nl 1991 mathematics subject classification primary 11a25 secondary 11 04 11y05 11y70

stochastic switching for partially observable dynamics and optimal asset allocation abstractin industrial applications optimal control problems frequently appear in the context of decision making under incomplete information in such framework decisions must be adapted dynamically to account for possible regime changes of the underlying dynamics using stochastic filtering theory markovian evolution can be modelled in terms of latent variables which naturally leads to high dimensional state space making practical solutions to these control problems notoriously challenging in our approach we utilise a specific structure of this problem class to present a solution in terms of simple reliable and fast algorithms the algorithms presented in this paper have already been implemented in an r package

more on the total number of prime factors of an odd perfect number let a n denote the sum of the positive divisors of n we say that n is perfect if a n 2n currently there are no known odd perfect numbers it is known that if an odd perfect number exists then it must be of the form n p k j 1 q 2 j j where p q 1 q k are distinct primes and p 1 mod 4 define the total number of prime factors of n as n a 2 k j 1 j sayers showed that n 29 this was later extended by iannucci and sorli to show that n 37 this paper extends these results to show that n 47

on odd perfect quasiperfect and odd almost perfect numbers 2 12 a n ni 10 14 from this it follows immediately that if m is an odd integer o m 2m l and if il mi 6 op quasiperfect qp numbers i e c n 2n 1 and odd almost perfect oap numbers i e u n 2n 1 are such examples also it can be proved from 2 that if m 1 p is op p6 2 1014 r 5 however if we consider only those n p p q in 2 for which l i p is op then exponents a are restricted and hence we have a better lower bound in 2 consequently we have a better upper bound for p6 in this paper we prove

fuzzy planar graphs fuzzy planar graph is a very important subclass of fuzzy graph in this paper two types of edges are mentioned for fuzzy graphs effective edges and considerable edges in addition a comparative study between kuratowski s graphs and fuzzy planar graph is made a new concept of a strong fuzzy planar graph is introduced some related results are established these results have certain applications in subway tunnels routes oil gas pipelines representation etc it is also shown that an image can be represented by a fuzzy planar graph and contraction of such an image can be made with the help of a fuzzy planar graph

optimal stopping under partial observation near value iteration we propose a new approximate value iteration method namely near value iteration nvi to solve continuous state optimal stopping problems under partial observation which in general cannot be solved analytically and also pose a great challenge to numerical solutions nvi is motivated by the expression of the value function as the supremum over an uncountable set of linear functions in the belief state after a smart manipulation of the operations in the updating equation for the value function we reduce the set to only two functions at every time step so as to achieve significant computational savings nvi yields a value function approximation bounded by the tightest lower and upper bounds that can be achieved by existing algorithms in the same class so the nvi approximation is closer to the true value function than at least one of these bounds we demonstrate the effectiveness of our approach on an example of pricing american options under stochastic volatility

on the largest prime divisor of an odd harmonic number a positive integer is called a ore s harmonic number if its positive divisors have integral harmonic mean ore conjectured that every harmonic number greater than 1 is even if ore s conjecture is true there exist no odd perfect numbers in this paper we prove that every odd harmonic number greater than 1 must be divisible by a prime greater than 10 5

on multi adjoint concept lattices based on heterogeneous conjunctors sets of attributes and objects in fuzzy formal concept analysis are usually different and hence it might not make sense to evaluate them on the same carrier in this context the operators used to obtain the concept lattice could be defined by associating different lattices to attributes and objects several reasons exist for which we need to evaluate the sets of attributes and objects in the same carrier following this direction we introduce a new definition of a concept lattice where objects and attributes are evaluated on the same lattice l although operators evaluating objects and attributes in different carriers are used moreover we study the relationship between this new concept lattice and the alternative one which can be obtained directly by using different carriers for the sets of attributes and objects

odd perfect numbers not divisible by 3 ii we prove that odd perfect numbers not divisible by 3 have at least eleven distinct prime factors 1 n is called a perfect number if u n 2n where o n is the sum of positive divisors of n twenty seven even perfect numbers are known however no odd perfect op numbers have been found suppose n is op and o n is the number of distinct prime factors of n gradstein 1925 kuihnel 1949 weber 1951 and the author 1978 5 proved that w n 6 pomerance 1972 7 and robbins 1972 proved that w n 7 hagis 1975 2 and chein 1978 1 proved that na 8 hagis and mcdaniel 3 proved that the largest prime factor of n 100129 and pomerance 8 proved that the second largest prime factor of n 7 139 if 31 n then kanold 1949 proved that o n 9 and the author 1977 4 proved that c n 10 in this paper we prove theorem if n is op and 31 n then w1n 1 1 2 in the remainder of this paper we assume that n is op and

a new lower bound for odd perfect numbers we describe an algorithm for proving that there is no odd perfect number less than a given bound k or finding such a number if one exists a program implementing the algorithm has been run successfully with k 10160 with an elliptic curve method used for the vast number of factorizations required comments only the abstract is given here the full paper appeared as 1 for a sequel which extended the result to k 10300 see 2 the integer factorizations used in the proofs are available by anonymous ftp 3 references 1 r p brent and g l cohen a new lower bound for odd perfect numbers mathematics of computation 53 1989 431 437 supplement ibid s7 s24 mr 89m 11008 also appeared as report tr cs 88 05 computer sciences laboratory anu february 1988 50 pp rpb100 2 r p brent g l cohen and h j j te riele improved techniques for lower bounds for odd perfect numbers mathematics of computation 57 1991 857 868 mr 92c 11004 rpb116 3 r p brent factor an integer factorization program for the ibm pc report tr cs 89 23 computer sciences laboratory anu october 1989 7 pp rpb117 brent computer sciences laboratory australian national university canberra australia e mail address rpb cslab anu edu au cohen school of mathematical sciences university of technology sydney po box 123 broadway nsw 2007 australia e mail address glcohen utscsd oz 1991 mathematics subject classification primary 11 04 secondary 11a25 11a51 11y05 11y70

the third largest prime divisor of an odd perfect number exceeds one hundred let n denote the sum of positive divisors of the natural number n such a number is said to be perfect if n 2n it is well known that a number is even and perfect if and only if it has the form 2 p 1 2 p 1 where 2 p 1 is prime it is unknown whether or not odd perfect numbers exist although many conditions necessary for their existence have been found for example cohen and hagis have shown that the largest prime divisor of an odd perfect number must exceed 10 6 and iannucci showed that the second largest must exceed 10 4 in this paper we prove that the third largest prime divisor of an odd perfect number must exceed 100

every odd perfect number has a prime factor which exceeds 10 6 it is proved here that every odd perfect number is divisible by a prime greater than 10 6

fuzzy graph representation of a fuzzy concept lattice in this paper we present a new relationship between fuzzy concept lattice and fuzzy graph given a fuzzy binary relation we define an underlying fuzzy graph and establish a new relation between the set of elements of the fuzzy concept lattice built from the fuzzy relation and the set of minimal disconnections of the associated fuzzy graph this link is important from the point of view of application and computation

the second largest prime divisor of an odd perfect number exceeds ten thousand let n denote the sum of positive divisors of the natural number n such a number is said to be perfect if n 2n it is well known that a number is even and perfect if and only if it has the form 2 p 1 2 p 1 where 2 p 1 is prime no odd perfect numbers are known nor has any proof of their nonexistence ever been given in the meantime much work has been done in establishing conditions necessary for their existence one class of necessary conditions would be lower bounds for the distinct prime divisors of an odd perfect number for example cohen and hagis have shown that the largest prime divisor of an odd perfect number must exceed 10 6 and hagis showed that the second largest must exceed 10 3 in this paper we improve the latter bound in particular we prove the statement in the title of this paper

bipolar fuzzy graph representation of concept lattice formal concept analysis fca is a mathematical framework for knowledge processing tasks fca has been successfully incorporated into fuzzy setting and its extension interval valued fuzzy set for handling vagueness and impreciseness in data however the analysis in such settings is restricted to unipolar space recently some applications of bipolar information are shown in bipolar fuzzy graph lattice theory as well as in fca the adequate analysis of bipolar information using fca requires incorporation of bipolar fuzzy set and an appropriate lattice structure for this purpose we propose an algorithm for generating the bipolar fuzzy formal concepts a method for cut of bipolar fuzzy formal context and its implications with illustrative examples

factorization of the tenth fermat number we describe the complete factorization of the tenth fermat number f10 by the elliptic curve method ecm f10 is a product of four prime factors with 8 10 40 and 252 decimal digits the 40 digit factor was found after about 140 mflop years of computation we also discuss the complete factorization of other fermat numbers by ecm and summarize the factorizations

on the total number of prime factors of an odd perfect number we say n n is perfect if n 2n where n denotes the sum of the positive divisors of n no odd perfect numbers are known but it is well known that if such a number exists it must have prime factorization of the form n p j 1k qj2 j where p q1 qk are distinct primes and p 1 mod 4 we prove that if j 1 mod 3 or j 2 mod 5 for all j 1 j k then 3 n we also prove as our main result that n 37 where n 2 j 1k j this improves a result of sayers n 29 given in 1986

optimal stopping of partially observable markov processes a filtering based duality approach in this note we develop a numerical approach to the problem of optimal stopping of discrete time continuous state partially observable markov processes pomps our motivation is to find approximate solutions that provide lower and upper bounds on the value function such that the gap between the bounds can provide a practical measure of the quality of the solutions to this end we develop a filtering based duality approach which relies on the martingale duality formulation of the optimal stopping problem and the particle filtering technique we show that this approach complements an asymptotic lower bound derived from a suboptimal stopping time with an asymptotic upper bound on the value function we carry out error analysis and illustrate the effectiveness of our method on an example of pricing american options under partial observation of stochastic volatility

reducing concept lattices by means of a weaker notion of congruence abstract attribute and size reductions are key issues in formal concept analysis in this paper we consider a special kind of equivalence relation to reduce concept lattices which will be called local congruence this equivalence relation is based on the notion of congruence on lattices with the goal of losing as less information as possible and being suitable for the reduction of concept lattices we analyze how the equivalence classes obtained from a local congruence can be ordered moreover different properties related to the algebraic structure of the whole set of local congruences are also presented finally a procedure to reduce concept lattices by the new weaker notion of congruence is introduced this procedure can be applied to the classical and fuzzy formal concept analysis frameworks

sieve methods for odd perfect numbers using a new factor chain argument we show that 5 does not divide an odd perfect number indivisible by a sixth power applying sieve techniques we also find an upper bound on the smallest prime divisor putting this together we prove that an odd perfect number must be divisible by the sixth power of a prime or its smallest prime factor lies in the range 10 8 p 10 1000 these results are generalized to much broader situations

odd perfect numbers have at least nine distinct prime factors an odd perfect number n is shown to have at least nine distinct prime factors if 3 l n then n must have at least twelve distinct prime divisors the proof ultimately avoids previous computational results for odd perfect numbers

on the number of prime factors of an odd perfect number let n and n denote respectively the total number of prime factors and the number of distinct prime factors of the integer n euler proved that an odd perfect number n is of the form n p m where p e 1 mod 4 p is prime and p m this implies that n 2 n 1 we we prove that n 18 n 31 7and n 2 n 51

new techniques for bounds on the total number of prime factors of an odd perfect number let n denote the sum of the positive divisors of n we say that n is perfect if n 2n currently there are no known odd perfect numbers it is known that if an odd perfect number exists then it must

construction of the l fuzzy concept lattice we propose two processes to obtain l fuzzy concepts based on finite l fuzzy contexts and the theory of cousot and cousot pacific j math 82 1979 43 the first algorithm calculates the l fuzzy concepts derived from an l fuzzy set and the second one constructs the whole l fuzzy concept lattice we also represent graphically the l fuzzy concept lattice

evolution in time of l fuzzy context sequences in this work we consider a complete lattice l and we study l fuzzy context sequences which represent the evolution in time of an l fuzzy context to carry out this study in the first part of the paper we consider n ary owa operators in complete lattices which enable us to make a general analysis and a temporal analysis at any moment in time of l fuzzy context sequences after that evolution in time of the relationship between the objects and the attributes is considered in particular we analyze the concepts of trend and persistent formal contexts finally we illustrate our results with an example where we consider the particular lattice l j 0 1

solutions of the congruence 1 1 mod to supplement existing data solutions of a p 1 1 mod p 2 are tabulated for primes a p with 100 2 32 are presented one of these p 188748146801 for a 5 also satisfies the reverse congruence p a 1 1 mod a 2 an effective procedure for searching for such double solutions is described and applied to the range a 2 and fixed prime p to see where the smallest prime solution a occurs

odd harmonic numbers exceed 10 24 a number n 1 is harmonic if n n n where n and n are the number of positive divisors of n and their sum respectively it is known that there are no odd harmonic numbers up to 10 15 we show here that for any odd number n 10 6 n p 3a 2 for any prime power divisor p a of n and we have used this in showing that n 10 18 we subsequently showed that for any odd number n 9 10 17 n p 8a 5 with p a as before and we use this improved result in showing that n 10 24

formal concept analysis based on fuzzy granularity base for different granulations this paper introduces granular computing grc into formal concept analysis fca it provides a unified model for concept lattice building and rule extraction on a fuzzy granularity base for different granulations one of the strengths of grc is that larger granulations help to hide some specific details whereas fca in a grc context can prevent losses due to concept lattice complexity however the number of superfluous rules increases exponentially with the scale of the decision context to overcome this we present some inference rules and maximal rules and prove that the set of all these maximal rules is complete and nonredundant thus users who want to obtain decision rules should generate maximal rules examples demonstrate that application of the method is valid and practicable in summary this approach utilizes fca in a grc context and provides a practical basis for data analysis and processing