Search

Conditional on a strong form of the Goldbach conjecture, we determine all finite connected components of the aliquot graph containing a number less than $10^9$, as well as those containing an amicable pair below $10^{14}$ or one of the known perfect or sociable cycles below $10^{17}$. Along the way we develop a fast algorithm for computing the inverse image of an even number under the sum-of-proper-divisors function., 10 pages, to appear in Mathematics of Computation

In this work we apply the polynomial inverse image method to derive the limit behavior of L p Christoffel functions on the basis of the results obtained by Levin, Lubinsky, and Totik. We consider two situations: measures supported on the real line and measures supported on systems of Jordan curves.

Let $${\mathfrak {F}}$$ be a category of subanalytic subsets of real analytic manifolds that is closed under basic set-theoretical operations (locally finite unions, difference and product) and basic topological operations (taking connected components and closures). Let M be a real analytic manifold and denote $${\mathfrak {F}}(M)$$ the family of the subsets of M that belong to the category $${\mathfrak {F}}$$. Let $$f:X\rightarrow \mathbb {R}$$ be a subanalytic function on a subset $$X\in {\mathfrak {F}}(M)$$ such that the inverse image under f of each interval of $$\mathbb {R}$$ belongs to $${\mathfrak {F}}(M)$$. Let $$\mathrm{Max}(f)$$ be the set of local maxima of f and consider its level sets $$\mathrm{Max}_\lambda (f):=\mathrm{Max}(f)\cap \{f=\lambda \}=\{f=\lambda \}{\setminus }{\text {Cl}}(\{f>\lambda \})$$ for each $$\lambda \in \mathbb {R}$$. In this work we show that if f is continuous, then $$\mathrm{Max}(f)=\bigsqcup _{\lambda \in \mathbb {R}}\mathrm{Max}_\lambda (f)\in {\mathfrak {F}}(M)$$ if and only if the family $$\{\mathrm{Max}_\lambda (f)\}_{\lambda \in \mathbb {R}}$$ is locally finite in M. If we erase continuity condition, there exist subanalytic functions $$f:X\rightarrow M$$ such that $$\mathrm{Max}(f)\in {\mathfrak {F}}(M)$$, but the family $$\{\mathrm{Max}_\lambda (f)\}_{\lambda \in \mathbb {R}}$$ is not locally finite in M or such that $$\mathrm{Max}(f)$$ is connected but it is not even subanalytic. We show in addition that if $${\mathfrak {F}}$$ is the category of subanalytic sets and $$f:X\rightarrow \mathbb {R}$$ is a (non-necessarily continuous) subanalytic map f that maps relatively compact subsets of M contained in X to bounded subsets of $$\mathbb {R}$$, then $$\mathrm{Max}(f)\in {\mathfrak {F}}(M)$$ and the family $$\{\mathrm{Max}_\lambda (f)\}_{\lambda \in \mathbb {R}}$$ is locally finite in M. An example of this type of functions are continuous subanalytic functions on closed subanalytic subsets of M. The previous results imply that if $${\mathfrak {F}}$$ is either the category of semianalytic sets or the category of C-semianalytic sets and f is the restriction to an $${\mathfrak {F}}$$-subset of M of an analytic function on M, then the family $$\{\mathrm{Max}_\lambda (f)\}_{\lambda \in \mathbb {R}}$$ is locally finite in M and $$\mathrm{Max}(f)=\bigsqcup _{\lambda \in \mathbb {R}}\mathrm{Max}_\lambda (f)\in {\mathfrak {F}}(M)$$. We also show that if the category $${\mathfrak {F}}$$ contains the intersections of algebraic sets with real analytic submanifolds and $$X\in {\mathfrak {F}}(M)$$ is not closed in M, then there exists a continuous subanalytic function $$f:X\rightarrow \mathbb {R}$$ with graph belonging to $${\mathfrak {F}}(M\times \mathbb {R})$$ such that inverse images under f of the intervals of $$\mathbb {R}$$ belong to $${\mathfrak {F}}(M)$$ but $$\mathrm{Max}(f)$$ does not belong to $${\mathfrak {F}}(M)$$. As subanalytic sets are locally connected, the set of non-openness points of a continuous subanalytic function $$f:X\rightarrow \mathbb {R}$$ coincides with the set of local extrema $$\mathrm{Extr}(f):=\mathrm{Max}(f)\cup \mathrm{Min}(f)$$. This means that if $$f:X\rightarrow \mathbb {R}$$ is a continuous subanalytic function defined on a closed set $$X\in {\mathfrak {F}}(M)$$ such that the inverse image under f of each interval of $$\mathbb {R}$$ belongs to $${\mathfrak {F}}(M)$$, then the set $$\mathrm{Op}(f)$$ of openness points of f belongs to $${\mathfrak {F}}(M)$$. Again the closedness of X in M is crucial to guarantee that $$\mathrm{Op}(f)$$ belongs to $${\mathfrak {F}}(M)$$. The type of results stated above are straightforward if $${\mathfrak {F}}$$ is an o-minimal structure of subanalytic sets. However, the proof of the previous results requires further work for a category $${\mathfrak {F}}$$ of subanalytic sets that does not constitute an o-minimal structure.

In this paper, the concept of cubic set to subalgebras, ideals and closed ideals of B -algebras are introduced. Relations among cubic subalgebras with cubic ideals and cubic closed ideals of B -algebras investigated. The homomorphic image and inverse image of cubic subalgebras, ideals are studied and some related properties are investigated. Also, the product of cubic B -algebras are investigated.

We study the normality of families of meromorphic mappings of a domain into the complex projective space having the same inverse image (ignoring counting multiplicities) of moving hypersurfaces.

Suppose that the inverse image of the zero vector by a continuous map $f:{\mathbb R}^n\to{\mathbb R}^q$ has an isolated point $P$. There is a local obstruction to removing this isolated zero by a small perturbation, generalizing the notion of index for vector fields, the $q=n$ case. The existence of a continuous map $g$ which approximates $f$ but is nonvanishing near $P$ is equivalent to a topological property we call "locally inessential," and for dimensions $n$, $q$ where $\pi_{n-1}(S^{q-1})$ is trivial, every isolated zero is locally inessential. We consider the problem of constructing such an approximation $g$, and show that there exists a continuous homotopy from $f$ to $g$ through locally nonvanishing maps. If $f$ is a semialgebraic map, then there exists such a homotopy which is also semialgebraic. For $q=2$ and $f$ real analytic with a locally inessential isolated zero, there exists a H\"older continuous homotopy $F(x,t)$ which, for $(x,t)\ne(P,0)$, is real analytic and nonvanishing. The existence of a smooth homotopy, given a smooth map $f$, is stated as an open question., Comment: to appear in Topological Methods in Nonlinear Analysis

If \(f\in L^{p}(\mathbb{R}^{d})\) is a bounded real valued continuous function which has a unique maximum or a unique minimum at a point \(x_{0}\in \mathbb{R}^{d}\) and if the inverse image of the neighborhoods of f(x0) shrinks regularly to x0, then \(\mathrm{ span }\{f^{m}(x-2^{-m}\varSigma_{i=1}^{d} j_{i} e_{i})\mid m\in\mathbb{N}, j_{i}\in\mathbb{Z}\}\) is a dense subset of \(L^{p}(\mathbb{R}^{d}), 1\le p

We propose the concept of fuzzy soft hypergroups and explore two particular classes of them, namely fuzzy soft closed hypergroups and fuzzy soft ultraclosed hypergroups. Moreover, we study the image and inverse image of a fuzzy soft closed, ultraclosed hypergroup under a fuzzy soft map.

The well-known Pogorelov theorem stating the cylindricity of any C 1-smooth, complete, developable surface of bounded exterior curvature in ℝ3 was generalized by Stocker to C 2-smooth surfaces with a more general notion of completeness. We extend Stocker’s result to C 1-smooth surfaces that are normal developable in the Burago-Shefel’ sense.

In this paper we examine the Bernstein–Markov inequality on special compact subsets of the complex plane, namely on lemniscates. Sharp constants are obtained which involve the Green function of the complement and the density of equilibrium measure of the compact set. Using lemniscates is a useful tool because of the possibility of taking inverse images. The proof also uses so-called peaking polynomials which will be constructed.

In this paper, a special kind of partial algebras called projective partial groupoids is defined. It is proved that the inverse image of all projections of a fundamental weak regular *-semigroup under the homomorphism induced by the maximum idempotent-separating congruence of a weak regular *-semigroup has a projective partial groupoid structure. Moreover, a weak regular *-product which connects a fundamental weak regular *-semigroup with corresponding projective partial groupoid is defined and characterized. It is finally proved that every weak regular *-product is in fact a weak regular *-semigroup and any weak regular *-semigroup is constructed in this way.

In this chapter, we will precisely define a random variable. A random variable is a real valued function with domain Ω which has an extra property called measurability that allows us to make probability statements about the random variables.

In this paper, we consider polynomials of degree $n$, for which the inverse image of $[-1,1]$ consists of two Jordan arcs. We prove that the four endpoints of these arcs form an ${\cal O}(1/n)$-net in the complex plane.

Given two distinct crossings of a knot or link projection, we consider the question: Under what conditions can we obtain the unlink by changing both crossings simultaneously? More generally, for which simultaneous twistings at the crossings is the genus reduced? Though several examples show that the answer must be complicated, they also suggest the correct necessary conditions on the twisting numbers. Let L be an oriented link in s3with a generic projection onto the plane R~. Let a be a short arc in R2 transverse to both strands of L at a crossing, so that the strands pass through a in opposite directions. Then the inverse image of a contains a disk punctured twice, with opposite orientation, by L. Define a crossing disk D for a link L in S3 to be a disk which intersects L in precisely two points, of opposite orientation. It is easy to see that any crossing disk arises in the manner described. Twisting the link q times as it passes through D is equivalent to doing llq surgery on dD in S3 and adds 29 crossings to this projection of L. We say that this new link L(q) is obtained by adding q twists at D. Call dD a crossing circle for L. A crossing disk D , and its boundary do,are essential if dD bounds no disk disjoint from L. This is equivalent to the requirement that L cannot be isotoped off D in S3-OD. Here we examine how a link can be turned into the unlink via simultaneous twists on disjoint crossing disks. In particular, we prove an analogue for pairs of crossing disks of the following theorem, a much more general version of which is proven in (ST, 1.41 (see also (Gal)

In this paper we show that there is no context-free language L such that L is an inverse image of a finite nonempty subset of a group with a noncontext-free word problem. Furthermore we can prove that any finitely generated group which is a syntactic monoid of a context-sensitive language L such that L is a union of finitely many syntactic classes always has a context-sensitive word problem.

In this paper we describe algorithms to solve different problems in commutative algebra. These algorithms are linked by the fact that they are useful in considering integral extensions (integral closure, weak integral closure, etc.). The first algorithm computes the conductor of an extension of rings. The algorithm is based on the computation of the inverse image of a submodule relative to a module homomorphism. This problem is solved lifting suitably to free modules over polynomial rings. To complete the computation of the conductor one has to represent an integral extension as a finitely presented module; we give an algorithm for this, that at the same time verifies that the extension is integral. Other algorithms are given, that test subring inclusion and birational equivalence. These algorithms are not strictly necessary to perform the conductor computation, but are in some sense connected to the former problem, since if the answer to these tests is negative then the conductor is trivial.

The distinguishing number $\Delta(X)$ of a graph $X$ is the least positive integer $n$ for which there exists a function $f:V(X)\to\{0,1,2,\cdots,n-1\}$ such that no nonidentity element of $\hbox{Aut}(X)$ fixes (setwise) every inverse image $f^{-1}(k)$, $k\in\{0,1,2,\cdots,n-1\}$. All infinite, locally finite trees without pendant vertices are shown to be 2-distinguishable. A proof is indicated that extends 2-distinguishability to locally countable trees without pendant vertices. It is shown that every infinite, locally finite tree $T$ with finite distinguishing number contains a finite subtree $J$ such that $\Delta(J)=\Delta(T)$. Analogous results are obtained for the distinguishing chromatic number, namely the least positive integer $n$ such that the function $f$ is also a proper vertex-coloring.

On the basis of the concept of the interval valued intuitionistic fuzzy sets introduced by K.Atanassov, the notion of interval valued intuitionistic fuzzy $H_v$-submodules of an $H_v$-module with respect to $t$-norm $T$ and $s$-norm $S$ is given and the characteristic properties are described. The homomorphic image and the inverse image are investigated.In particular, the connections between interval valued intuitionistic $(S,T)$-fuzzy $H_v$-submodules and interval valued intuitionistic $(S,T)$-fuzzy submodules are discussed.

This paper studies conditions under which classes of Borel mappings (i.e., mappings such that the inverse image of open sets are Borel sets) coincide with certain classes of Baire functions (i.e., functions which belong to the smallest family containing the continuous functions and closed with respect to pointwise limits). Generalizations of the classical Lebesgue-Hausdorff and Banach theorems are obtained for the class of mappings which we call “ σ \sigma -discrete". These results are then applied to the problem of extending Borel mappings over Borel sets, and generalizations of the theorems of Lavrentiev and Kuratowski are obtained.

In this paper we deduce some relations concerning the tangent sets to the inverse image of a set under a function and we use these relations to derive necessary conditions for extremality in the form of a multiplier rule.

In this paper, the notions of fuzzy dot subalgebras is introduced together with fuzzy normal dot subalgebras and fuzzy dot ideals of B G -algebras. The homomorphic image and inverse image are investigated in fuzzy dot subalgebras and fuzzy dot ideals of B G -algebras. Also, the notion of fuzzy relations on the family of fuzzy dot subalgebras and fuzzy dot ideals of B G -algebras are introduced with some related properties.

The inverse image of 11 under A is denoted by IA. In [7, p. 129] Steinhaus proved, that if A is a regular (i.e., limitpreserving) matrix, then there is a sequence of 0's and l's such that Ax is not convergent. It follows immediately that a regular matrix cannot sum every bounded sequence. (Cf. [6].) The principal result of this paper is the analogue of this theorem for 1-1 matrices. The author is indebted to H. I. Brown for several helpful comments and observations.

Given a branched cover $f:Y\to X$ between smooth projective curves over a non-archimedian mixed-characteristic local field and an open rigid disk $D\subset X$, we study the question under which conditions the inverse image $f^{-1}(D)$ is again an open disk. More generally, if the cover $f$ varies in an analytic family, is this true at least for some member of the family? Our main result gives a criterion for this to happen., Comment: Final version, to appear in Research in the Mathematical Sciences. 29 pages

A locally convex space E is barrelled if and only if L,(E, F) is quasicomplete for every Banach space F. Recently A. Wilansky [5] has shown that a locally convex space E is barrelled if for every compact Hausdorff space X and linear map T: E C(X) with closed graph, T is continuous. His proof runs as follows: For any barrel B in E, endow its absolute polar H : = BO with the topology induced by the weak topology a(E', E), form the Banach space F = C*(H) of bounded continuous functions on H (which can be identified with ( f(,H)), and show that the linear map T: E -* F, T(x)(h): = h(x) for x E E, h E H, has closed graph. By hypothesis the continuity of T follows, and so B, which is the inverse image of the unit disc in F, is a neighborhood of 0. Now let LS(E, F) be the space of continuous linear maps of E into F, s indicating the topology of pointwise convergence. We shall show that the continuity of T can be inferred as well, if quasicompleteness of LS(E, F) is supposed. To this end we shall define for every finite set A c E and e > 0 a linear map TA, E L(E, F) such that (1) sup{IITA,(x)II;A c Efinite,e > 0) < oo forx E E,and (2) IITAC(x) T(x)jj < e forx E A. Then from (1), (2) and our hypothesis, the continuity of T will be clear. Let d be the pseudo-metric defined by d (h, h') : = sup{ Ih (x) h'(x)I ; x E A on H. Then the uniformity of (H, d) is obviously coarser than the uniformity induced by u(E', E) on H, so (H, d) is precompact. Therefore, there exist hl . . . , hn E H such that H c u "-IK(hi;e) with K(hi; e) {h E H; d(hi, h) < e}. From [3, Corollary 5.35 and Problem 5.W] applied to (H, d) and (K(hi, e); i = 1, ... ., n), we get ri E e*(H), i = 1, . .. ., n, with n (3) 0 < mi

We prove that the Klein bottle embeds in the total space E E of an orientable S 1 {S^1} -bundle over an orientable 2 2 -manifold M M if and only if M = S 2 M = {S^2} and E = S 1 × S 2 E = {S^1} \times {S^2} or the lens space L ( 4 , 1 ) L(4,1) .

We exhibit a pull-back construction for Schunck class projectors in finite groups and some of its consequences. In [3] a pull-back construction for projectors of saturated formations in finite soluble groups has been given1, based on a result of Huppert that does not generalise to Schunck classes. Therefore (and with a specific application in mind-see [2]), in the present note we give an alternative approach to this construction and show that, in fact, pull-backs exist for Schunck class projectors in arbitrary finite groups. Here we rely on the theory of Schunck classes and their projectors in finite groups as developed in [1] and assume the reader to be familiar with terminology as well as main results of [1] mentioning only that, in [1, 4.2], we have shown that, for every Schunck class X' and any finite group G, the set fH (G) and N 1 G implies that H E Proj + (HN). THEOREM. Let H be a subdirect subgroup of G1 x G2 and put K = (H n G1) x (H n G2). Further, let X, e Proj,+(Gi) (i = 1, 2) and let Hi be the inverse image of X, in H with respect to the projection map vri: H -* Gi. Assume that H1K = H2K and put X = H1 rn H2. Then for each Y E Projj+(X) we have (i) Y E Proj +(H), and (ii) X and Y are subdirect subgroups of XI x X2. PROOF. By definition of H, H rn G3 -, < Hi (i = 1, 2). Therefore, applying Dedekind's modular law and using H1K = H2K, we get X(H n G3-,) = (H1 n H2)(H n G3-i) = H, n [H3-,(H n G3rJ] = H, n [H3-i(H n Gi)(H n G3 = Hi n H3_,K = H, n H,K = H,. Consequently, X/X n G3-i = X/X n (H n G3-i) X(H n G3Ji)/(H n G3-,) = HII( nt G3 -s I XI C VY, Received by the editors May 11, 1984. 1980 Mathematics Subject Classification. Primary 20D10. 'Perhaps, we may remark that this construction is superfluous for the applications given in [3], in constrast to what is done below. ?1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page

If X is a space, the deleted product space, X*, is Xx XD, where D is the diagonal. If Y is a space and f is a continuous map from X to Y, then Xf/ is the inverse image of Y* under the map f x f taking Xx X into Yx Y. In this paper, we investigate the following questions: "What maps f are such that Xf* is homotopically equivalent to X* ", and "What maps f are such that Xf* is homotopically equivalent to f(X)* ?" If X is the union of two nondisjoint simplexes and f is a simplicial map from Xx X such that f lf(X) is one-to-one, we obtain necessary and sufficient conditions for Xf* and f(X)* to be homotopically equivalent. If X is the union of nondisjoint simplexes A and B with dim B= 1+ dim (A r) B), we obtain necessary and sufficient conditions for X* and Xf to be homotopically equivalent if f is in the class of maps mentioned.