On dimensionally exotic maps

We call a value y = f(x) of a map f: X → Y dimensionally regular if dimX ≤ dim(Y × f ^?1(y)). It was shown in [6] that if a map f: X → Y between compact metric spaces does not have dimensionally regular values, then X is a Boltyanskii compactum, i.e., a compactum satisfying the equality dim(X × X) = 2dim X ? 1. In this paper we prove that every Boltyanskii compactum X of dimension dim X ≥ 6 admits a map f: X → Y without dimensionally regular values. We show that the converse does not hold by constructing a 4-dimensional Boltyanskii compactum for which every map has a dimensionally regular value.     

Closed maps on metric spaces

The following result is proved: Let Y be the image of a metric space X under a closed map f. Then every ?f^-1(y) is Lindelöf if and only if Y has a point-countable k-network.     

Birational composition of quadratic forms over a local field

Let f(X) and g(Y) be nondegenerate quadratic forms of dimensions m and n, respectively, over K, char K ≠ 2. The problem of birational composition of f(X) and g(Y) is considered: When is the product f(X) · g(Y) birationally equivalent over K to a quadratic form h(Z) over K of dimension m + n? The solution of the birational composition problem for anisotropic quadratic forms over K in the case of m = n = 2 is given. The main result of the paper is the complete solution of the birational composition problem for forms f(X) and g(Y) over a local field P, char P ≠ 2.     

A parametrization theorem

The following result, and a closely related one, is proved: If u:X → Y is an open, perfect surjection, with X metrizable and with dim X = 0 or dim Y = 0, then there exists a perfect surjection $\text{h: Y×}\text{S}\text{→X}$ such that u ° h = π_Y (where S in the Cantor set and $\text{π : Y×}\text{S}\text{→ Y}$ is the projection). If moreover, u^-1(y) is homeomorphic to S for all y?Y, then h can be chosen to be a homeomorphism.     

Lelek maps and n-dimensional maps from compacta to polyhedra

For a natural number m?0, a map from a compactum X to a metric space Y is an m-dimensional Lelek map if the union of all non-trivial continua contained in the fibers of f is of dimension ?m. In [M. Levin, Certain finite-dimensional maps and their application to hyperspaces, Israel J. Math. 105 (1998) 257-262], Levin proved that in the space C(X,I) of all maps of an n-dimensional compactum X to the unit interval I=[0,1], almost all maps are (n−1)-dimensional Lelek maps. Moreover, he showed that in the space C(X,Ik) of all maps of an n-dimensional compactum X to the k-dimensional cube Ik (k?1), almost all maps are (n−k)-dimensional Lelek maps. In this paper, we generalize Levin's result. For any (separable) metric space Y, we define the piecewise embedding dimension ped(Y) of Y and we prove that in the space C(X,Y) of all maps of an n-dimensional compactum X to a complete metric ANR Y, almost all maps are (n−k)-dimensional Lelek maps, where k=ped(Y). As a corollary, we prove that in the space C(X,Y) of all maps of an n-dimensional compactum X to a Peano curve Y, almost all maps are (n−1)-dimensional Lelek maps and in the space C(X,M) of all maps of an n-dimensional compactum X to a k-dimensional Menger manifold M, almost all maps are (n−k)-dimensional Lelek maps. It is known that k-dimensional Lelek maps are k-dimensional maps for k?0.     

A counterexample in ANR theory

There is an AR X and a non-AR Y and a continuous surjection f:X→Y so that each point-inverse f^-1(y) is an AR. This solves a problem of Borsuk.     

Continuous selections avoiding extreme points

It is shown that if X is a countably paracompact collectionwise normal space, Y is a Banach space and φ:X→Y² is a lower semicontinuous mapping such that φ(x) is Y or a compact convex subset with Cardφ(x)>1 for each x∈X, then φ admits a continuous selection f:X→Y such that f(x) is not an extreme point of φ(x) for each x∈X. This is an affirmative answer to the problem posed by V. Gutev, H. Ohta and K. Yamazaki [V. Gutev, H. Ohta and K. Yamazaki, Selections and sandwich-like properties via semi-continuous Banach-valued functions, J. Math. Soc. Japan 55 (2003) 499-521].     

Finite-to-one maps into Euclidean manifolds and spaces with disjoint disks properties

It is shown that every Euclidean manifold M has the following property for any m?1: If f:X→Y is a perfect surjection between finite-dimensional metric spaces, then the mapping space C(X,M) with the source limitation topology contains a dense Gδ-subset of maps g such that dimBm(g)?mdimf+dimY−(m−1)dimM. Here, Bm(g)={(y,z)∈Y×M||f−1(y)∩g−1(z)|?m}. The existence of residual sets of finite-to-one maps into product of manifolds and spaces having disjoint disks properties is also obtained.     

Parametric bing and Krasinkiewicz maps

It is shown that if is a perfect map between metrizable spaces and Y is a C-space, then the function space C(X,I) with the source limitation topology contains a dense Gδ-subset of maps g such that every restriction map gy=g|f⁻¹(y), y∈Y, satisfies the following condition: all fibers of gy are hereditarily indecomposable and any continuum in f⁻¹(y) either contains a component of a fiber of gy or is contained in a fiber of gy.     

On the root closedness of continuous function algebras

For a compact Hausdorff space X, C(X) denotes the algebra of all complex-valued continuous functions on X. For a positive integer n, we say that C(X) is n-th root closed if, for each f∈C(X), there exists g∈C(X) such that f=gn. It is shown that, for each integer m?2, there exists a compact Hausdorff space Xm such that C(Xm) is m-th root closed, but not n-th root closed for each integer n relatively prime to m. This answers a question posed by Countryman Jr. [R.S. Countryman Jr., On the characterization of compact Hausdorff X for which C(X) is algebraically closed, Pacific J. Math. 20 (1967) 433-438] et al.     

Extension theorems for functional equations with bisymmetric operations

Summary. In this paper we deal with the extension of the following functional equation¶¶ f (x) = M (f (m₁(x, y)), ..., f (m_k(x, y)))        (x, y ? K) f (x) = M \bigl(f (m_{1}(x, y)), \dots, f (m_{k}(x, y))\bigr) \qquad (x, y \in K) , (*)¶ where M is a k-variable operation on the image space Y, m₁,..., m_k are binary operations on X, K ì X K \subset X is closed under the operations m₁,..., m_k, and f : K ? Y f : K \rightarrow Y is considered as an unknown function.¶ The main result of this paper states that if the operations m₁,..., m_k, M satisfy certain commutativity relations and f satisfies (*) then there exists a unique extension of f to the (m₁,..., m_k)-affine hull K^* of K, such that (*) holds over K^*. (The set K^* is defined as the smallest subset of X that contains K and is (m₁,..., m_k)-affine, i.e., if x ? X x \in X , and there exists y ? K^* y \in K^* such that m₁(x, y), ?, m_k(x, y) ? K^* m_{1}(x, y), \ldots, m_{k}(x, y) \in K^* then x ? K^* x \in K^* ). As applications, extension theorems for functional equations on Abelian semigroups, convex sets, and symmetric convex sets are obtained.     

A duality relation for entrance and exit laws for Markov processes

Markov processes X_t on (X, F_X) and Y_t on (Y, F_Y) are said to be dual with respect to the function f(x, y) if E_xf(X_t, y) = E_yf(x, Y_t for all x ? X, y ? Y, t ? 0. It is shown that this duality reverses the role of entrance and exit laws for the processes, and that two previously published results of the authors are dual in precisely this sense. The duality relation for the function f(x, y) = 1_{x<y} is established for one-dimensional diffusions, and several new results on entrance and exit laws for diffusions, birth-death processes, and discrete time birth-death chains are obtained.     

Optimal Pricing for Optimal Transport

Suppose that c(x, y) is the cost of transporting a unit of mass from x ∈ X to y ∈ Y and suppose that a mass distribution μ on X is transported optimally (so that the total cost of transportation is minimal) to the mass distribution ν on Y. Then, roughly speaking, the Kantorovich duality theorem asserts that there is a price f(x) for a unit of mass sold (say by the producer to the distributor) at x and a price g(y) for a unit of mass sold (say by the distributor to the end consumer) at y such that for any x ∈ X and y ∈ Y, the price difference g(y) ? f(x) is not greater than the cost of transportation c(x, y) and such that there is equality g(y) ? f(x) = c(x, y) if indeed a nonzero mass was transported (via the optimal transportation plan) from x to y. We consider the following optimal pricing problem: suppose that a new pricing policy is to be determined while keeping a part of the optimal transportation plan fixed and, in addition, some prices at the sources of this part are also kept fixed. From the producers’ side, what would then be the highest compatible pricing policy possible? From the consumers’ side, what would then be the lowest compatible pricing policy possible? We have recently introduced and studied settings in c-convexity theory which gave rise to families of c-convex c-antiderivatives, and, in particular, we established the existence of optimal c-convex c-antiderivatives and explicit constructions of these optimizers were presented. In applications, it has turned out that this is a unifying language for phenomena in analysis which used to be considered quite apart. In the present paper we employ optimal c-convex c-antiderivatives and conclude that these are natural solutions to the optimal pricing problems mentioned above. This type of problems drew attention in the past and existence results were previously established in the case where X = Y = ? ⁿ under various specifications. We solve the above problem for general spaces X, Y and real-valued, lower semicontinuous cost functions c. Furthermore, an explicit construction of solutions to the general problem is presented.     

A non-separable Christensen's theorem and set tri-quotient maps

For every space X let K(X) be the set of all compact subsets of X. Christensen [J.P.R. Christensen, Necessary and sufficient conditions for measurability of certain sets of closed subsets, Math. Ann. 200 (1973) 189-193] proved that if X,Y are separable metrizable spaces and F:K(X)→K(Y) is a monotone map such that any L∈K(Y) is covered by F(K) for some K∈K(X), then Y is complete provided X is complete. It is well known [J. Baars, J. de Groot, J. Pelant, Function spaces of completely metrizable space, Trans. Amer. Math. Soc. 340 (1993) 871-879] that this result is not true for non-separable spaces. In this paper we discuss some additional properties of F which guarantee the validity of Christensen's result for more general spaces.     

A wage-type example with a pseudocompact factor

The following example is constructed without any set-theoretic assumptions beyond ZFC: There exist a hereditarily separable hereditarily Lindelöf space X and a first-countable locally compact separable pseudocompact space Y such that dim X = dimY = 0, while dim(X × Y)>0.     

Finite graphs have unique symmetric products

Let X be a metric continuum and let Fn(X) be the nth symmetric product of X (Fn(X) is the hyperspace of nonempty subsets of X with at most n points). In this paper we prove that if Fn(X) is homeomorphic to Fn(Y), where X is a finite graph and Y is a continuum, then X is homeomorphic to Y.     

On a conjecture of Kippenhahn about the characteristic polynomial of a pencil generated by two Hermitian matrices. I

Let A be an n × n complex matrix, and write A = H + iK, where i² = ?1 and H and K are Hermitian matrices. The characteristic polynomial of the pencil xH + yK is f(x, y, z) = det(zI ? xH ? yK). Suppose f(x, y, z) is factored into a product of irreducible polynomials. Kippenhahn [5, p. 212] conjectured that if there is a repeated factor, then there is a unitary matrix U such that U^?1AU is block diagonal. We prove that if f(x, y, z) has a linear factor of multiplicity greater than n?3, then H and K have a common eigenvector. This may be viewed as a special case of Kippenhahn’s conjecture.     

The disconnection number of a graph

The disconnection number d(X) is the least number of points in a connected topological graph X such that removal of d(X) points will disconnect X (Nadler, 1993 [6]). Let Dn denote the set of all homeomorphism classes of topological graphs with disconnection number n. The main result characterizes the members of Dn+1 in terms of four possible operations on members of Dn. In addition, if X and Y are topological graphs and X is a subspace of Y with no endpoints, then d(X)?d(Y) and Y obtains from X with exactly d(Y)−d(X) operations. Some upper and lower bounds on the size of Dn are discussed.The algorithm of the main result has been implemented to construct the classes Dn for n?8, to estimate the size of D₉, and to obtain information on certain subclasses such as non-planar graphs (n?9) and regular graphs (n?10).