on polynomial factorial diophantine equations we study equations of the form p x n and show that for some classes of polynomials p the equation has only finitely many solutions this is the case say if p is irreducible of degree greater than 1 or has an irreducible factor of relatively large degree this is also the case if the factorization of p contains some large power s of irreducible s for example we can show that the equation x r x 1 n has only finitely many solutions for r 4 but not that this is the case for 1 r 3 although it undoubtedly should be we also study the equation p x h n where h n is one of several other highly divisible sequences proving again that for various classes of polynomials these equations have only finitely many solutions

robust stability of a class of polynomials with coefficients depending multilinearly on perturbations necessary and sufficient conditions are given for robust stability of a family of polynomials each polynomial is obtained by a multilinearity perturbation structure restrictions on the multilinearity are involved but in contrast to existing literature these restrictions are derived from physical considerations stemming from analysis of a closed loop interval feedback system the main result indicates that all polynomials in the family of polynomials have their zeros in the strict left half plane if and only if two requirements are satisfied at each frequency the first requirement is the zero exclusion condition involving four kharitonov rectangles the second requirement is that a specially constructed theta 0 parameterized set of 16 intervals must cover the positive reals for each theta epsilon 0 2 pi

method of robust stability analysis with highly structured uncertainties this paper presents a method of analyzing the stability of linear time invariant interconnected systems with uncertainties each subsystem is single input single output and its vector locus lies inside a polygon at each frequency it is shown that the convex closure of the image of the cartesian product of these polygons under the mapping phi det i fh agrees with the convex closure of the image of the vertices of these polygons under the mapping from this result the image can be estimated by a finite dumber of calculations a sufficient stability condition is obtained by applying this result to the multivariable nyquist stability criterion

soft p ideals of soft bci algebras molodtsov d molodtsov soft set theory first results comput math appl 37 1999 19 31 introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches jun y b jun soft bck bci algebras comput math appl 56 2008 1408 1413 applied first the notion of soft sets by molodtsov to the theory of bck bci algebras in this paper we introduce the notion of soft p ideals and p idealistic soft bci algebras and then investigate their basic properties using soft sets we give characterizations of fuzzy p ideals in bci algebras we provide relations between fuzzy p ideals and p idealistic soft bci algebras

extremal robustness properties of multilinear interval systems this paper deals with robustness properties of a control system subject to parameter uncertainty when the parameters appear affine multilinearly in the coefficients of the characteristic polynomial the robust hurwitz stability of such a multilinear interval polynomial family reduces to that of the stability of a set of manifolds derived from the kharitonov polynomials several extremal properties of these stability manifolds are derived these properties are useful in determining the worst case parametric h and nonlinear sector bounded stability margins for control systems containing interval parametric uncertainty they generalize previous results obtained by the authors for the linear case

fuzzy equivalence relations abstract in this note we define and discuss properties of fuzzy equivalence relations on a set in particular we show that there exists a correspondence between fuzzy equivalence relations and certain classes of fuzzy subsets we also study the cuts of fuzzy equivalence relations finally certain lattice theoretic properties of fuzzy equivalence relations are discussed

fuzzy subinclines ideals of incline algebras the fuzzification of subinclines ideals in inclines are considered and some related properties are investigated we also state the product of fuzzy subinclines ideals and the projections of fuzzy subinclines ideals we discuss fuzzy relations fuzzy characteristic subinclines ideals

parameter partitioning via shaping conditions for the stability of families of polynomials it is shown that if parameter uncertainty enters a polynomial in a manner which allows an appropriate partitioning generated by certain shaping conditions then the q locus becomes at each frequency a convex parpolygon in the complex plane the results generalize some earlier work and expand the class of problems for which simple robust stability tests exist these results imply that when uncertainty affects polynomial coefficients in a linear manner the q locus is guaranteed to be a convex parpolygon at each frequency

on the robust stability of polynomials with no cross coupling between the perturbations in the coefficients of even odd powers abstract some criteria are presented for the stability in hurwitz sense of a family of polynomials with even and odd coefficients subject to uncoupled perturbations it is shown that the stability of an appropriate subset of extreme polynomials guarantees the stability of the entire family in particular a polytope of polynomials with the even odd uncoupling property is stable provided a certain subset of its vertices is for the case of an arbitrary set of polynomials this result often gives a less conservative sufficient condition than that provided by kharitonov s theorem

a class of multilinear uncertain polynomials to which the edge theorem is applicable this paper gives a class of multilinear uncertain polynomials to which the edge theorem is applicable in the paper a notion of symmetric uncertainty structure is introduced the main result is as follows if a multilinear uncertain polynomial has a symmetric uncertainty structure and the uncertain parameters are nonoverlapping then the corresponding value sets will be convex polygons this means that the use of the zero exclusion condition to test the robust stability of such uncertain polynomials is feasible since we need only to calculate the extreme points of the convex polygons furthermore a version of the edge theorem can be used to test the robust stability of such uncertain polynomials

the difference between consecutive primes ii with enough effort the value of x0 could be determined effectively the paper has much in common with 1 in particular we use the sieve method of harman 4 5 we no longer use zero density estimates however but rather mean value results on dirichlet polynomials similar to those that give rise to such estimates compare for example work of iwaniec and pintz 9 and baker harman and pintz 2 much of the improvement over 1 arises from the use of watt s theorem 11 on a particular kind of mean value more accurate estimates for six dimensional integrals are also used to good effect there is in addition a device which uses a two dimensional sieve to get an asymptotic formula for a one dimensionally sieved set see lemmas 16 17 unfortunately these lemmas which would be of great signi cance for v 0 53 are not very numerically signi cant when v drops to 0 525 the same applies to the roale reversals discussed below let us introduce enough notation to permit an outline of the proof when e is a nite sequence of positive integers counted with multiplicity we write jej for the number of terms of e and ed fm dm 2 eg let p z y

fuzzy ideals on a distributive lattice abstract we investigate the relationship between fuzzy ideals and fuzzy congruences on a distributive lattice and obtain that the lattice of fuzzy ideals is isomorphic to the lattice of fuzzy congruence on a generalized boolean algebra finally consider the products of fuzzy ideals

the distribution of integers with a divisor in a given interval we determine the order of magnitude of h x y z the number of integers n x having a divisor in y z for all x y and z we also study hr x y z the number of integers n x having exactly r divisors in y z whenr 1 we establish the order of magnitude ofh1 x y z for allx y z satisfying z x 1 2 for every r 2 c 1 and 0 we determine the order of magnitude of hr x y z uniformly for y large and y y logy log 4 1 z min y c x 1 2 as a consequence of these bounds we settle a 1960 conjecture of erd os and some conjectures of tenenbaum one key element of the proofs is a new result on the distribution of uniform order statistics

expansion of det a b and robustness analysis of uncertain state space systems in this note an expansion for the determinant of the sum of two matrices det a b is given a robustness analysis approach for state space systems with uncertain parameters is investigated by using kharitonov s theorem some useful results are obtained several illustrative examples are given

soft bck bci algebras molodtsov d molodtsov soft set theory first results comput math appl 37 1999 19 31 introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches in this paper we apply the notion of soft sets by molodtsov to the theory of bck bci algebras the notion of soft bck bci algebras and soft subalgebras are introduced and their basic properties are derived

a generalization of kharitonov s four polynomial concept for robust stability problems with linearly dependent coefficient perturbations kharitonov s four polynomial concept is generalized to the case of linearly dependent coefficient perturbations and more general zero location regions to this end a specially constructed scalar function of a scalar variable is instrumental to the robustness analysis the present work is motivated by two fundamental limitations of kharitonov s theorem namely 1 the theorem only applies to polynomials with independent coefficient perturbations and 2 it only applies to zeros in the left hand plane

robust analysis and design of control systems using interval arithmetic abstract several robustness problems such as stability and performance robustness analysis of feedback systems and robust design of control systems in the presence of mixed nonlinear parametric and nonparametric perturbations can be solved by means of algorithms based on interval arithmetic computation some of the main algorithms available in the literature are presented and their efficiency is tested on some examples of robustness analysis and design of control systems 1997 elsevier science ltd

when is the multiaffine image of a cube a convex polygon abstract we give two simple sufficient conditions under which the multiaffine image in the complex plane of an m dimensional cube is a convex polygon a third condition which in some generic sense is necessary and sufficient is then obtained our conditions involve checking the locations of the image of the vertices of the cube these results help determine whether a parameterized family of polynomials th stable

fuzzy soft set theory applied to bck bci algebras abstract in the paper p k maji r biswas a r roy fuzzy soft sets j fuzzy math 9 3 2001 589 602 maji et al introduced the concept of fuzzy soft sets as a generalization of the standard soft sets and presented an application of fuzzy soft sets in a decision making problem the aim of this manuscript is to apply fuzzy soft set for dealing with several kinds of theories in bck bci algebras the notions of fuzzy soft bck bci algebras closed fuzzy soft ideals and fuzzy soft p ideals are introduced and related properties are investigated

inclines of algebraic structures an incline is a generalization of a boolean algebra or fuzzy algebra consisting of a semiring satisfying additive idempotence and the incline axiom xy x x xy y y the ideals in a ring or semigroup form an incline as do the topologizing filters in a ring we generalize the latter result to semirings and obtain the structure of these inclines in specific cases

soft set theory applied to ideals in d algebras the notions of transitive soft d algebras soft edge d algebras soft d algebras soft d ideals soft d ideals soft d ideals and d idealistic d idealistic or d idealistic soft d algebras are introduced also their related properties are surveyed

robust stability under a class of nonlinear parametric perturbations this paper considers the robust stability verification of linear time invariant systems admitting a class of nonlinear parametric perturbations the general setting is one of determining the closed loop stability of systems whose open loop transfer functions consist of powers products and ratios of polytopes of polynomials apart from this general setting two special cases of independent interest are also considered the first special case concerns uncertainties in the open loop gain and real poles and zeros while the second special case treats uncertainties in the open loop gain and complex poles and zeros in light of the zero exclusion principle robust stability is equivalent to zero exclusion of the value sets of the system characteristic function a value set consists of the values of the characteristic functions at a fixed frequency the main results of this paper are as follows 1 the value set of the characteristic function at each fixed frequency is determined by the edges and some frequency dependent internal line segments 2 consequently hurwitz invariance verification simplifies to that of checking certain continuous scalar functions for avoidance of the negative real axis 3 for the case of real zero pole gain variations the critical lines are all frequency independent and therefore the determination of the robust stability is even simpler 4 for the case of complex zero pole gain variations the critical internal lines are shown to be either frequency independent or to be confined in certain two dimensional planes or three dimensional boxes

robust stability manifolds for multilinear interval systems the stability of a class of multilinearly perturbed families of systems is considered it is shown how the problem of checking the stability of the entire family can be reduced to that of checking certain subsets that are independent of the degrees of the polynomials involved the extremal property of these subsets is established the results point to the need for a complete study of the stability of manifolds of polynomials composed of products of simple surfaces

lattice of fuzzy subalgebras and closure systems in i x abstract let x be a non empty set and i be the unit interval of real numbers the elements of i x are called fuzzy subsets of x i x is a lattice with the pointwise ordering of i now if a x f is an universal algebra with a specified set f of finitary operations on x then each operation can be extended to a finitary operation on i x an element of i x is defined to be a fuzzy subalgebra of a using these extended operations we also define algebraic closure system in i x we then show that the lattice of fuzzy subalgebras of a is an algebraic closure system in i x

fuzzy congruence relations abstract if a x f is an universal algebra and if r x denotes the set of all fuzzy subsets of x x i e fuzzy relations on x then the algebraic operations fg f on x are extended to algebraic operations f on r x f is an algebra similar to a it is called relational algebra a fuzzy equivalence relation which is at the same time a fuzzy subalgebra of r x f is called a fuzzy congruence relation on x in this paper we study fuzzy congruence relations amongst other things we prove that the set of all fuzzy congruence relations on an algebra is a complete lattice and an algebraic closure system

an elementary proof of kharitonov s stability theorem with extensions gives an elementary proof of kharitonov s theorem using simple complex plane geometry without invoking the hermite bieler theorem kharitonov s theorem is a stability result for classes of polynomials defined by letting each coefficient vary independently in an arbitrary interval the result states that the whole class is hurwitz if and only if four special well defined polynomials are hurwitz the paper also gives elementary proofs of two previously known extensions for polynomials of degree less than six the requirement is reduced to fewer than four polynomials and the theorem is generalized to polynomials with complex coefficients

computation of algebraic combinations of uncertainty value sets a robust analysis approach based on the computation of boundaries of uncertainty value sets is presented the idea is to use the strongly connected subsystem separation and the tree structured decomposition tsd concept to extend the class of uncertainty structures to be considered to this end we compute the exact boundaries of regions produced by operations spl times spl divide among boundaries of simply connected value sets in polynomial time and without incrementing the total amount of input data parametric dynamic or general multiply connected value set descriptions may be accommodated