scholarly journals A tour through δ-invariants: From Nash’s embedding theorem to ideal immersions, best ways of living and beyond

2013 ◽  
Vol 94 (108) ◽  
pp. 67-80 ◽  
Author(s):  
Bang-Yen Chen

First I will explain my motivation to introduce the ?-invariants for Riemannian manifolds. I will also recall the notions of ideal immersions and best ways of living. Then I will present a few of the many applications of ?-invariants in several areas in mathematics. Finally, I will present two optimal inequalities involving ?-invariants for Lagrangian submanifolds obtained very recently in joint papers with F. Dillen, J. Van der Veken and L. Vrancken.

2018 ◽  
Vol 12 (3) ◽  
pp. 336-352
Author(s):  
John Paetsch

Why would Deleuze condemn the dialectic of the One and the Many? It is not simply to replace one set of categories with another. Rather, it is to make differential topology safe for the philosophy of time. If Deleuze affirms pure multiplicity, it is to overcome Henri Bergson's prohibition upon using mathematics to inquire into time. How else could Deleuze justify his monstrous identification of ‘continuous multiplicities’ with Riemannian manifolds?


2018 ◽  
Vol 2020 (11) ◽  
pp. 3229-3259 ◽  
Author(s):  
Siyuan Lu

Abstract We consider a priori estimates of Weyl’s embedding problem of $(\mathbb{S}^2, g)$ in general three-dimensional Riemannian manifold $(N^3, \bar g)$. We establish interior $C^2$ estimate under natural geometric assumption. Together with a recent work by Li and Wang [18], we obtain an isometric embedding of $(\mathbb{S}^2,g)$ in Riemannian manifold. In addition, we reprove Weyl’s embedding theorem in space form under the condition that $g\in C^2$ with $D^2g$ Dini continuous.


Author(s):  
Duan Hai-bao

SynopsisLetf: Mm→Nnbe a map from a Riemannianm-manifold(Mm, d)into ann-manifold Nn. The major purpose of this paper is to give a lower bound for the numberby examining the behaviour of the cohomology homomorphisms induced byf. This idea will be used to generalise the classical Newman theorem and present a geometric background for a well-known non-embedding theorem in topology.


2021 ◽  
pp. 353-382
Author(s):  
Min Hoon Kim ◽  
Patrick Orson ◽  
JungHwan Park ◽  
Arunima Ray

Open problems in the study of topological 4-manifolds are explained in detail. An important open problem is to determine whether the disc embedding theorem and its antecedents hold for all groups; in other words, whether all groups are good. The disc embedding conjecture and the surgery conjecture are stated. The relationships between these conjectures and their various reformulations are explained. Of particular interest are the reformulations in terms of freely slicing certain infinite families of links. In particular, the surgery conjecture is true if and only if all good boundary links are freely slice. Good boundary links are the many-component analogues of Alexander polynomial one knots.


2018 ◽  
Vol 41 ◽  
Author(s):  
Wei Ji Ma

AbstractGiven the many types of suboptimality in perception, I ask how one should test for multiple forms of suboptimality at the same time – or, more generally, how one should compare process models that can differ in any or all of the multiple components. In analogy to factorial experimental design, I advocate for factorial model comparison.


2020 ◽  
Vol 43 ◽  
Author(s):  
David Spurrett

Abstract Comprehensive accounts of resource-rational attempts to maximise utility shouldn't ignore the demands of constructing utility representations. This can be onerous when, as in humans, there are many rewarding modalities. Another thing best not ignored is the processing demands of making functional activity out of the many degrees of freedom of a body. The target article is almost silent on both.


2020 ◽  
Vol 43 ◽  
Author(s):  
Michael Tomasello

Abstract My response to the commentaries focuses on four issues: (1) the diversity both within and between cultures of the many different faces of obligation; (2) the possible evolutionary roots of the sense of obligation, including possible sources that I did not consider; (3) the possible ontogenetic roots of the sense of obligation, including especially children's understanding of groups from a third-party perspective (rather than through participation, as in my account); and (4) the relation between philosophical accounts of normative phenomena in general – which are pitched as not totally empirical – and empirical accounts such as my own. I have tried to distinguish comments that argue for extensions of the theory from those that represent genuine disagreement.


1997 ◽  
Vol 161 ◽  
pp. 179-187
Author(s):  
Clifford N. Matthews ◽  
Rose A. Pesce-Rodriguez ◽  
Shirley A. Liebman

AbstractHydrogen cyanide polymers – heterogeneous solids ranging in color from yellow to orange to brown to black – may be among the organic macromolecules most readily formed within the Solar System. The non-volatile black crust of comet Halley, for example, as well as the extensive orangebrown streaks in the atmosphere of Jupiter, might consist largely of such polymers synthesized from HCN formed by photolysis of methane and ammonia, the color observed depending on the concentration of HCN involved. Laboratory studies of these ubiquitous compounds point to the presence of polyamidine structures synthesized directly from hydrogen cyanide. These would be converted by water to polypeptides which can be further hydrolyzed to α-amino acids. Black polymers and multimers with conjugated ladder structures derived from HCN could also be formed and might well be the source of the many nitrogen heterocycles, adenine included, observed after pyrolysis. The dark brown color arising from the impacts of comet P/Shoemaker-Levy 9 on Jupiter might therefore be mainly caused by the presence of HCN polymers, whether originally present, deposited by the impactor or synthesized directly from HCN. Spectroscopic detection of these predicted macromolecules and their hydrolytic and pyrolytic by-products would strengthen significantly the hypothesis that cyanide polymerization is a preferred pathway for prebiotic and extraterrestrial chemistry.


Author(s):  
Benjamin F. Trump ◽  
Irene K. Berezesky ◽  
Raymond T. Jones

The role of electron microscopy and associated techniques is assured in diagnostic pathology. At the present time, most of the progress has been made on tissues examined by transmission electron microscopy (TEM) and correlated with light microscopy (LM) and by cytochemistry using both plastic and paraffin-embedded materials. As mentioned elsewhere in this symposium, this has revolutionized many fields of pathology including diagnostic, anatomic and clinical pathology. It began with the kidney; however, it has now been extended to most other organ systems and to tumor diagnosis in general. The results of the past few years tend to indicate the future directions and needs of this expanding field. Now, in addition to routine EM, pathologists have access to the many newly developed methods and instruments mentioned below which should aid considerably not only in diagnostic pathology but in investigative pathology as well.


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