Geometry Questions Pdf In Hindi
Vernie Montagna
240K MB shapefile. It's a large one - HEC-RAS inundation boundary generated from HEC-RAS. This is the US Army Corp's Hydraulic Modelling software (FYI). There is a geometry issue I found using (surprise, surprise) QGIS (and quickly). It's a self-intersection.
That's kind of big for a shapefile; I would export it to something a bit more performant like a geopackage, mobile geodatabase, or file geodatabase and the repair geometry tool should run faster. Then export it back to shapefile (if you absolutely have to have a shapefile); otherwise I'd stop using shapefiles altogether.
When shapefiles are loaded into a geodatabase, the features' geometries are copied as they are (with any existing geometry problems), so the same precaution that is needed when using shapefile data must be taken into account when using geodatabase feature classes. The exception is when the data is loaded into an enterprise geodatabase using an ArcGIS client application.
working with a file geodatabase would be preferable than the shapefile directly, but if you are working in an enterprise environment then the loading of the shapefile should address some/all of the issues.
If you would like us to take a look at your data to see if there is anything further we can do to improve the performance against your specific data you can either attach it here or call your support analyst and submit a request with them.
I am now doing an article survey on the field of information geometry started by S.Amari and Barndorff-Nielson. I want to know some research situation in this field. I have read (4) and parts of (3).
As the comment in (1)'s answer said"I am seeking more of an expert's perspective on the field." I want some comments from experts too since I am totally new to this field.(I would say that it is lying between math and statistics.)
It seems like a young branch which starts at 1960s and reached its peak at around 1990s(5). According to (2), I still feel there is much potentiality in this field at the first sight. However, the critiques (6) certainly make sense but I doubt that these small flaws (like the lack of independence assumption) will affect its future development since this can probably be remedied be adding slightly more restrictive priori assumptions.
My question is how the direction of researches is going on in the field of Information Geometry today? Is it a fancy field to be explored or it is just a dead end with some severe flaws I didn't catch? (If so, please point it out.)
(9)A most recent paper by Amari talking about the interplay between information geometry, statistics and machine learning.Information geometry in optimization, machine learning and statistical inference
I'm not sure I would say I'm an expert in information geometry. However, I worked for several years on the subject as a postdoc. As a disclaimer, this is entirely my own opinion and others may disagree.
Since you asked this question, the research situation in the field has improved. Firstly, two separate books ([1$ $], [2$ $]) have been published, both of which are good references for the material. In particular, the second gives a rigorous mathematical treatment for the basic theory. Secondly, a new journal, Information Geometry, has been released. Thus far several issues have been published and they contain some interesting papers.
However, information geometry is definitely a relatively niche mathematical field. As to the reason for this, in my opinion IG is really an interdisciplinary field and not simply a branch of mathematics. Many of the people working in the field are not mathematicians by background. As a result, information geometry embodies a wide range of research. Some papers are mathematical, but many others are really statistics, computer science, or some hybrid thereof. Many of the publishing conventions are differ from math, as well. For instance, it's common to publish short papers without proofs in conference proceedings and, generally speaking, the main theorems are not stated in the introduction.
While there is a lot of good work being done in the field, there is also too much research that is not really serious. Most of this is not done in bad faith, but due to a lack of experience and background in geometry. Furthermore, a lot of the work is published in a for-profit journal whose peer review process is minimal. Without giving examples, some papers boil down to slightly modifying known results and treating them as novel. Other papers try to use really big ideas without understanding the underlying theory or really proving anything. Furthermore, what is considered acceptable overlap between publications is far greater than in pure math. Needless to say, these issues create serious problems for the field, and makes it much less likely to be taken seriously.
Even with the good papers, they often seem to lack a good punchline. As was mentioned in the comments, the math in IG has built up a very general foundational theory, often without providing mathematical or statistical motivation for this theory. My impression is that quite a few of the researchers in the field were heavily influenced by the "structural point of view" pioneered by Nomizu and Kobayashi. I suppose the motivation for these structures might be self-evident to a statistician, but as a geometer oftentimes it's completely lost on me. In my experience, I only really started to understand what was going on when I worked through some important examples of statistical manifolds, instead of trying to learn the theory from the ground up.
Related to the point above, it's difficult to find explicit conjectures in the field. There isn't something similar to Yau's list of open problems in geometry to guide progress in the field. As such, when I was learning the field it was hard to tell what was considered an important problem and to understand the motivations for the research.
As a result of all of these factors, information geometry has remained a specialized sub-field. I think this will remain the case unless it is used to solve a big problem or it evolves to be more in line with standard mathematical conventions. All that being said, I've learned a lot from information geometry, and there is definitely a fair amount of low-hanging fruit to be picked. Furthermore, the field seems to be making progress in recent years, so hopefully my critiques will soon be obsolete.
To end on a positive note, let me give an example of a paper that I think does things well [3$ $]. This work studies necessary conditions for a Riemannian manifold to locally be written as the Hessian of a convex potential. I really like this paper and have found it helpful for my intuition.
There natural flaw with his reasoning is the fact that information geometry is not supposed to be used as relative entropy. Then he presents his 'examples' that are nothing more than unnecessarily complicated versions of problems in which usually relative entropy is used and the results from minimizing distances do not match. So what?
There are many surveys and books with open problems, but it would be nice to have a list of a dozen problems that are open and yet embarrasingly simple to state. A list that is "folklore" and that every graduate student in differential geometry should keep in his/her pocket.
2. If the volume of a Riemannian $3$-sphere is equal to 1, does it carry a closed geodesic whose length is less that $10^24$? Same question with $S^1 \times S^2$ if you like it better than the $3$-sphere.
There is also "Review of Geometry and Analysis" by S.-T. Yau (Asian Journal of Mathematics, vol. 4, no. 1, pp. 235-278, March 2000), where he discusses many big open problems in Riemannian geometry, symplectic geometry, algebraic geometry, and geometric analysis. This can keep you occupied for a long long time...
Again motivated by Mostow's strong rigidity theorem, Yau called for a notion of rank for general manifolds extending the one for locally symmetric spaces, and asked for rigidity properties for higher rank metrics. Advances in this direction have been made by Ballmann, Brin and Eberlein in their work on non-positive curved manifolds, Gromov's and Eberlein's metric rigidity theorems for higher rank locally symmetric spaces and the classification of closed higher rank manifolds of non-positive curvature by Ballmann and Burns-Spatzier. This leaves rank 1 manifolds of non-positive curvature as the focus of research. They behave more like manifolds of negative curvature, but remain poorly understood in many regards.
You can try one of these: All of them concern with nonnegatively curved Riemannian manifolds and Alexandrov geometry.In the same context I know a couple of surveys: and It has been conjectured (you can check in those papers) that any nonnegatively curved manifold is rationally elliptic. This is an important open problem in Riemannian geometry.
Notes: 1) I know next to nothing about algebraic geometry, although I am greatly interested in the field. 2) I realize that "constructive" might be a technical term, here I am using it only in an informal manner.
As an autodidact (I have an ongoing formal education in physics, but the amount of math we learn here is abysmal, so most of my mathematics knowledge is self-taught), I have noted that algebraic geometry seems to be really impenetrable for somebody who has no formal education in the field, unlike, say, differential geometry or functional analysis, which are areas I can effectively learn on my own.
Pretty much every time I encounter AG-related stuff on sites such as this one or math.se, I see layers and layers of abstractation on top of one another to the point where it makes me wonder, is this field of mathematics constructive, in the sense that can it be used to actually calculate anything or have any use outside mathematics?
The point I am trying to make is that, using differential geometry as an example, is is constructive. No matter how abstractly do I define manifolds, tensor fields, differential forms, connections, etc, they are always resoluble into component functions in some local trivializations, with which one can actually calculate stuff.
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