Existence Of Such Maps Research Articles

Lie Ideals with Multiplicative Homomultipliers on Near-Rings

In this study, the new types of mappings on the rings and near-rings are defined, which are named multiplicative homoleft multipliers, multiplicative homoright multipliers and multiplicative homo multipliers. We also present examples that show the existence of such mappings. Moreover, we study how these mappings effected on the construction of near-rings.

Conformal pointwise slant Riemannian maps from or to Kähler manifolds

In this article, we study Conformal pointwise-slant Riemannian maps (CPSRM) from or to Kähler manifolds to or from Riemannian manifolds. To check the existence of such maps, we provide some non-trivial examples. We derive some important results for these maps. We discuss the integrability and totally geodesicness of the distributions. Further, we investigate the conditions for homotheticity and harmonicity of these maps. Finally, we study some inequalities for these maps.

On special generic maps of rational homology spheres into Euclidean spaces

Special generic maps are smooth maps between smooth manifolds with only definite fold points as their singularities. The problem of whether a closed $n$-manifold admits a special generic map into Euclidean $p$-space for $1 \leq p \leq n$ was studied by several authors including Burlet, de Rham, Porto, Furuya, Eliasberg, Saeki, and Sakuma. In this paper, we study rational homology $n$-spheres that admit special generic maps into $\mathbb{R}^{p}$ for $p<n$. We use the technique of Stein factorization to derive a necessary homological condition for the existence of such maps for odd $n$. We examine our condition for concrete rational homology spheres including lens spaces and total spaces of linear $S^{3}$-bundles over $S^{4}$, and obtain new results on the (non-)existence of special generic maps.

Constructions of $k$-regular maps using finite local schemes

A continuous map \mathbb R^m \to \mathbb R^N or \mathbb C^m \to \mathbb C^N is called k -regular if the images of any k points are linearly independent. Given integers m and k a problem going back to Chebyshev and Borsuk is to determine the minimal value of N for which such maps exist. The methods of algebraic topology provide lower bounds for N , but there are very few results on the existence of such maps for particular values m and k . Using methods of algebraic geometry we construct k -regular maps. We relate the upper bounds on N with the dimension of the locus of certain Gorenstein schemes in the punctual Hilbert scheme. The computations of the dimension of this family is explicit for k \leq 9 , and we provide explicit examples for k \leq 5 . We also provide upper bounds for arbitrary m and k .

Holomorphically projective mappings between generalized m-parabolic Kähler manifolds

Generalized m-parabolic K?hler manifolds are defined and holomorphically projective mappings between such manifolds have been considered. Two non-linear systems of PDE?s in covariant derivatives of the first and second kind for the existence of such mappings are given. Also, relations between five linearly independent curvature tensors of generalized m-parabolic K?hler manifolds with respect to these mappings are examined.

Homotopy moment maps

Associated to any manifold equipped with a closed form of degree >1 is an ‘L∞-algebra of observables’ which acts as a higher/homotopy analog of the Poisson algebra of functions on a symplectic manifold. In order to study Lie group actions on these manifolds, we introduce a theory of homotopy moment maps. Such a map is a L∞-morphism from the Lie algebra of the group into the observables which lifts the infinitesimal action. We establish the relationship between homotopy moment maps and equivariant de Rham cohomology, and analyze the obstruction theory for the existence of such maps. This allows us to easily and explicitly construct a large number of examples. These include results concerning group actions on loop spaces and moduli spaces of flat connections. Relationships are also established with previous work by others in classical field theory, algebroid theory, and dg geometry. Furthermore, we use our theory to geometrically construct various L∞-algebras as higher central extensions of Lie algebras, in analogy with Kostant's quantization theory. In particular, the so-called ‘string Lie 2-algebra’ arises this way.

On highly regular embeddings

A continuous map $\mathbb {R}^d\to \mathbb {R}^N$ is $k$-regular if it maps any $k$ pairwise distinct points to $k$ linearly independent vectors. Our main result on $k$-regular maps is the following lower bound for the existence of such maps between Euclidean spaces, in which $\alpha (k)$ denotes the number of ones in the dyadic expansion of $k$: For $d\geq 1$ and $k\geq 1$ there is no $k$-regular map $\mathbb {R}^d\to \mathbb {R}^N$ for $N<d(k-\alpha (k))+\alpha (k)$. This reproduces a result of Cohen & Handel from 1978 for $d=2$ and the extension by Chisholm from 1979 to the case when $d$ is a power of $2$; for the other values of $d$ our bounds are in general better than Karasev’s (2010), who had only recently gone beyond Chisholm’s special case. In particular, our lower bound turns out to be tight for $k\le 3$. A framework of Cohen & Handel (1979) relates the existence of a $k$-regular map to the existence of a low-dimensional inverse of a certain vector bundle. Thus the non-existence of regular maps into $\mathbb {R}^N$ for small $N$ follows from the non-vanishing of specific dual Stiefel–Whitney classes. This we prove using the general Borsuk–Ulam–Bourgin–Yang theorem combined with a key observation by Hung (1990) about the cohomology algebras of configuration spaces. Our study produces similar lower bound results also for the existence of $\ell$-skew embeddings $\mathbb {R}^d\to \mathbb {R}^N$, for which we require that the images of the tangent spaces of any $\ell$ distinct points are skew affine subspaces. This extends work by Ghomi & Tabachnikov (2008) for the case $\ell =2$.

On centrally-extended maps on rings

Let \(R\) be a ring with center \(Z\). A map \(D\) of \(R\) (resp. \(T\) of \(R\)) is called a centrally-extended derivation (resp. a centrally-extended endomorphism) if for each \(x,y\in R, D(x+y)-D(x)-D(y)\in Z\) and \(D(xy)-D(x)y-xD(y)\in Z\) (resp. \(T(x+y)-T(x)-T(y)\in Z\) and \(T(xy)-T(x)T(y)\in Z\)). We discuss existence of such maps which are not derivations or endomorphisms, we study their effect on \(Z\), and we give some commutativity results.

Non-continuous linear functionals on topological vector spaces

In this article we study the existence of non-continuous linear functionals on topological vector spaces. Both sufficient and necessary conditions for the existence of such maps are found.

On a connection between Lorentzian and Euclidean metrics

We investigate connections between pairs of pseudo-Riemannian metrics whose sum is a (tensor) product of a covector field with itself. A bijective mapping between the classes of Euclidean and Lorentzian metrics is constructed as a special result. The existence of such maps on a differentiable manifold is discussed. Similar relations for metrics of arbitrary signature on a manifold are considered.

𝐾-theory of Eilenberg-Mac Lane spaces and cell-like mapping problem

There exist cell-like dimension raising maps of 6 6 -dimensional manifolds. The existence of such maps is proved by using K K -theory of Eilenberg-Mac Lane complexes.

Arcs with Positive Measure and a Space-Filling Curve

In this note we construct a one-parameter family of arcs in R2 with positive two-dimensional Lebesgue measure, which converge uniformly to a space-filling curve. By an arc we mean a homeomorphism of I = [0, 1] into I x I. The existence of such maps is not news, but our procedure for constructing them is so transparently simple that it can be presented in undergraduate or graduate courses in advanced calculus, real analysis, or topology. Furthermore, each of these curves is the uniform limit of a recursively defined sequence of very simple functions. The examples of space filling curves of Peano or Hilbert, which are generally cited (e.g., [1], [21, or [3]), are also constructed recursively, but the recursions are more complicated. All of them pass from one stage to the next by splicing together reduced and rotated copies of an initial figure. In our construction, no rotation is necessary to get them to fit, so the recursion has a much simpler description. Our construction is similar to one found in Section 10.10 of [1]. This work was motivated by problem E 2975, posed by F. B. Jones in the December, 1982, issue of the American Mathematical Monthly. We wish to thank the referee for several helpful suggestions. Let us first describe a typical member of our family of arcs intuitively. Imagine a necklace composed of four congruent squares and three connecting line segments arranged as in FIGURE 1. In each square we put another necklace, but one in which the + shaped region is proportionately much thinner. By leaving the original

Mapping of trajectories of test bodies in the pseudo-Riemann spaces V4, and �V4

In general coordination and for identical initial conditions all time-like geodesies of the pseudo-Riemann space V4 are mapped with time-like world lines of the space ¯V4 due to a pseudo-Lorentzian force. The mapping of all time-like world lines of ¯V4 with the geodesies of the space V4, is also considered. Necessary and sufficient conditions for the existence of such mappings are found.

Historical Importance of the Ino's Special Small-Scale Map of Japan and Discovery of the Middle-Scale Maps with Extraordinary Detailed Information

1) The late Prof. S. Ayusawa of Yokohama Municipal University, an eminent historical geographer, died in 1964, kept a manuscript copy of Inô's map of special small-scale (1.864, 000) and presented it to his intimate colleague, Prof. Y. Kagose, as the memento immediately before his death. The author has been given an opportunity to examine carefully the map and revealed and proved that this was the original source of the maps of Japan which Von Siebold had, with conspicuous features such as simplified for a foreigner to get the general view of Japan as well as place-names were written in katakana (the square form of the Japanese syllabary) instead of kanji (Chinese characters), presented secretly by K. Takahashi Sakuzayemon, an excellent official of the astronomical observatory and the supervisor of the completion of Inô's maps of the entire country, in 1827. Von Siebold failed to smuggle them out and deprived of them by the Tokugawa Government. Today these maps are kept in the National Diet Library. The author's conclusion is based on cartographical comparison and analysis of the two kinds of maps mentioned above.2) The Library of Gakushuin University in Tokyo received recently a gift from Mr. Y. Hori, a former teacher of geography, of a series of Ino's manuscript maps of middle-scale (1: 216, 000), which contain nearly half of Ino's entire surveyed routes, i. e. his survey of 1800 to 1809. Maps provide abundant information along the Ino's routes with extraordinary small characters and compete actually with Ino's maps of large-scale (1: 36, 000) in their content. No documents are left which tell that such a kind of middle-scale maps were made and no one has ever reported about the existence of such maps. They were probably made trially by an expert of writing small characters in the Edo period, without presenting them to the Tokugawa Government.

Correction to “Measurable gambling houses”

I am indebted to Mr. William D. Sudderth for pointing out an error in my paper Measurable gambling houses, Trans. Amer. Math. Soc. 126 (1967), 64-72. The definition of a measurable gambling house (Definition 1, p. 65) is not sufficient to insure the existence of a selection map a: F -* P such that a(f) E F(f) for all f The existence of such maps was implicitly assumed in the proofs of Lemma 1, Theorem 3, and the second result in Theorem 1, and those results are therefore incorrect as stated. The existence of such selection maps is a necessary and sufficient condition for the existence of measurable strategies and of random strategies (Blackwell and Ryll-Nardezewski, Non-existence of everywhere proper conditional distributions, Ann. Math. Statist. 34 (1963), 223-225). A gambling house F is said to be leavable if 8(f) E F(f) for all f, where S(f) is the gamble which assigns probability one to the fortune f. If F is leavable, the required selection maps exist, and the above theorems are true. The second conclusion of Theorem 1 (p. 66) should therefore be modified to read