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The Lieb-Thirring inequalities: Recent results and open problems [article]

Rupert L. Frank
2020 arXiv   pre-print
We attempt to survey recent results and open problems connected to Lieb-Thirring inequalities.  ...  For further results and open problems, see [10, 152, 11, 12] . For applications of these bounds to the dynamics of quantum many-body systems, see, for instance, [121, 122] .  ...  The Lieb-Thirring theorem leads naturally to the following challenge, which is a famous open problem in the field. Open Problem 2. Find the optimal constant K d in (1).  ... 
arXiv:2007.09326v1 fatcat:qrlo54yfhjbefa7v3ox2dsxowa

Homogeneous manifolds whose geodesics are orbits. Recent results and some open problems [article]

Andreas Arvanitoyeorgos
2017 arXiv   pre-print
We give an exposition on the subject, by presenting techniques that have been used so far and a wide selection of previous and recent results. We also present some open problems.  ...  Introduction The aim of the present article is to give an exposition on developments about homogeneous geodesics in Riemannian homogeneous spaces, to present various recent results and discuss some open  ...  Low dimensional examples The problem of a complete classification of g.o. manifolds is open.  ... 
arXiv:1706.09618v1 fatcat:rjobu7zjubh5fce46zkwcthc2q

Some recent results and open problems on sets of lengths of Krull monoids with finite class group [article]

Wolfgang Schmid
2015 arXiv   pre-print
These include the elasticity and related notions, the set of distances, and the structure theorem for sets of lengths. Several open problems are mentioned.  ...  Some of the fundamental notions related to sets of lengths of Krull monoids with finite class group are discussed, and a survey of recent results is given.  ...  Acknowledgment The author is very grateful to the referee for many useful remarks and corrections.  ... 
arXiv:1511.08080v1 fatcat:2wxm4rk3zjchply5v6bnzhryhu

Homogeneous manifolds whose geodesics are orbits. Recent results and some open problems

A. Arvanitoyeorgos
2017 Irish Mathematical Society Bulletin  
We also present some open problems.  ...  We give an exposition on the subject, by presenting techniques that have been used so far and a wide selection of previous and recent results.  ...  Finally, in Section 10 we present some open problems.  ... 
doi:10.33232/bims.0079.5.29 fatcat:cbdovwicrbh2bkd4w6phyae43m

From Individual to Collective Behavior of Unicellular Organisms: Recent Results and Open Problems

Chuan Xue, Hans G. Othmer, Radek Erban, Robert M. Sinclair, Klaus M. Stiefel
2009 AIP Conference Proceedings  
Here we summarize recent progress on this question in the context of bacterial and amoeboid chemotaxis, and formulate several open problems.  ...  However, these models are often based on phenomenological descriptions of cell fluxes with no direct correspondence to individual cell processes such signal transduction and cell movement.  ...  Summary We have summarized recent results and open problems in modeling the behavior of unicellular organisms, with a focus on the derivation of macroscopic equations from individual-based models.  ... 
doi:10.1063/1.3246413 fatcat:vhsteqswifbhzaktnxotdgw6ty

Information-based complexity

Edward W. Packel, J. F. Traub
1987 Nature  
Recently, Blum, Shub and Smale [89] obtained the first NP-completeness result over the reals for a problem with complete information.  ...  The complexity of approximating the largest eigenvalue in the randomized setting is open.  ... 
doi:10.1038/328029a0 fatcat:p55xu6essbgxjerpf2w37lyck4

Information-based complexity

1989 Mathematics and Computers in Simulation  
Recently, Blum, Shub and Smale [89] obtained the first NP-completeness result over the reals for a problem with complete information.  ...  The complexity of approximating the largest eigenvalue in the randomized setting is open.  ... 
doi:10.1016/0378-4754(89)90072-4 fatcat:unyblw2m7zdithdyelkwxlsoim

Information-Based Complexity

1989 Science  
Recently, Blum, Shub and Smale [89] obtained the first NP-completeness result over the reals for a problem with complete information.  ...  The complexity of approximating the largest eigenvalue in the randomized setting is open.  ... 
doi:10.1126/science.243.4895.1142-a pmid:17799891 fatcat:yt5qwb6sqjg4lhyqoicfre5oyi

Information-Based Complexity

H Wozniakowski
1986 Annual Review of Computer Science  
Recently, Blum, Shub and Smale [89] obtained the first NP-completeness result over the reals for a problem with complete information.  ...  The complexity of approximating the largest eigenvalue in the randomized setting is open.  ... 
doi:10.1146/annurev.cs.01.060186.001535 fatcat:a7ogeqrrdbaunnqdjmcnjvx3uq

Recent developments in information-based complexity

Edward W. Packel, Henryk Woźniakowski
1987 Bulletin of the American Mathematical Society  
Interspersed throughout the paper and in the concluding section we also Hst ten open problems from information-based complexity that we think will be of interest to mathematicians.  ...  We believe, and our short Hst of open problems is one testimony to this, that information-based complexity offers much to interest and challenge mathematicians working in a variety of fields. e(*,AO =  ... 
doi:10.1090/s0273-0979-1987-15511-x fatcat:tphezvnz2bdpria5i7kadzfftm

Perspectives on Information-Based Complexity

J. F. Traub, H. Wo\'zniakowski
1992 Bulletin of the American Mathematical Society  
Recently, Blum, Shub and Smale [89] obtained the first NP-completeness result over the reals for a problem with complete information.  ...  The complexity of approximating the largest eigenvalue in the randomized setting is open.  ... 
doi:10.1090/s0273-0979-1992-00240-9 fatcat:cg6dj5jfmfadjdu4kxzguasd24

A survey of information-based complexity

H Woźniakowski
1985 Journal of Complexity  
We survey some recent results in information-based complexity. We focus on the worst case setting and also indicate some average case results. 0 1985 Academic PWSS, Inc.  ...  Wasilkowski, and A. Werschulz for valuable comments and suggestions concerning this paper.  ...  An interesting open problem is to find the e-complexity of these nonlinear optimization problems. AVERAGE CASE SEITNG So far we discussed information-based complexity in the worst case setting.  ... 
doi:10.1016/0885-064x(85)90020-2 fatcat:o733w5xs2rgu5fca6bjwbiw5j4

Perspectives on information-based complexity [article]

J. F. Traub, Henryk Woźniakowski
1992 arXiv   pre-print
The authors discuss information-based complexity theory, which is a model of finite-precision computations with real numbers, and its applications to numerical analysis.  ...  For simplicity, we will not consider contaminated information in this paper. PERSPECTIVES ON INFORMATION-BASED COMPLEXITY J. F. TRAUB AND H.  ...  The complexity of approximating the largest eigenvalue in the randomized setting is open.  ... 
arXiv:math/9201269v1 fatcat:bzi6xz4szvfrjdx3w7ej3x64h4

s-Numbers in information-based complexity

P Mathé
1990 Journal of Complexity  
Using the ideas of s-scales and function analytic methods we reprove some known results and obtain some new ones, thus answering questions posed in several papers on this subject. o 1990 Academic press  ...  We focus our interest on two natural questions: (a) For what problems (the dependence on the underlying Banach spaces turns out to be crucial) do different classes of information or algorithms, respectively  ...  , 1988) and the survey papers "Informationbased complexity" (Woiniakowski, 1986) and "Recent developments in information-based complexity" (Packel and Woiniakowski, 1987) .  ... 
doi:10.1016/0885-064x(90)90011-2 fatcat:cv2gd2mq2fgf7j7u2vh7rh45n4

Probabilistic setting of information-based complexity

H Woźniakowski
1986 Journal of Complexity  
In particular, we study the approximation and integration problems.  ...  We study the probabilistic (E, b)-complexity for linear problems equipped with Gaussian measures.  ...  Wasilkowski, and A. G. Werschulz for valuable comments on this paper.  ... 
doi:10.1016/0885-064x(86)90005-1 fatcat:2zjaem7afjbvzk6k725xanhweu
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