arnosimons/wikipedia-physics-corpus · Datasets at Hugging Face

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In 2014, the SAGE-experiment's GGNT-apparatus (gallium-germanium neutrino telescope) was upgraded to perform a very-short-baseline neutrino oscillation experiment BEST (Baksan Experiment on Sterile Transitions) with an intense artificial neutrino source based on Cr. In 2017, the BEST apparatus was completed, but the artificial neutrino source was missing. As of 2018, the BEST experiment was underway. As of 2018, a follow-up experiment BEST-2 where the source would be changed to Zn was under consideration. It uses two gallium chambers instead of one, to better determine whether the anomaly could be explained by the distance from the source of the neutrinos	663	Gallium_anomaly	https://en.wikipedia.org/wiki/Gallium_anomaly	102,300	5	2,024	8	10	54	36
In June 2022, the BEST experiment released two papers observing a 20-24% deficit in the production the isotope germanium expected from the reaction 71 Ga + ν ν e → → e − − + 71 Ge , confirming previous results from SAGE and GALLEX on the so called "gallium anomaly" pointing out that a sterile neutrino explanation can be consistent with the data. Further work have refined the precision for the cross section of the neutrino capture in 2023 as well as half-life of 71 G e in 2024 ruling them out as possible explanations for the anomaly.	538	Gallium_anomaly	https://en.wikipedia.org/wiki/Gallium_anomaly	102,301	6	2,024	8	10	54	36
SAGE is led by the following physicists :	41	Gallium_anomaly	https://en.wikipedia.org/wiki/Gallium_anomaly	102,302	7	2,024	8	10	54	36
43°16′32″N 42°41′25″E / 43.27556°N 42.69028°E / 43.27556; 42.69028	72	Gallium_anomaly	https://en.wikipedia.org/wiki/Gallium_anomaly	102,303	8	2,024	8	10	54	36
In mathematics, a normalized solution to an ordinary or partial differential equation is a solution with prescribed norm, that is, a solution which satisfies a condition like ∫ ∫ R N \| u ( x ) \| 2 d x = 1. In this article, the normalized solution is introduced by using the nonlinear Schrödinger equation. The nonlinear Schrödinger equation (NLSE) is a fundamental equation in quantum mechanics and other various fields of physics, describing the evolution of complex wave functions. In Quantum Physics, normalization means that the total probability of finding a quantum particle anywhere in the universe is unity.	615	Normalized_solution_(mathematics)	https://en.wikipedia.org/wiki/Normalized_solution_(mathematics)	102,304	0	2,024	8	10	54	37
In order to illustrate this concept, consider the following nonlinear Schrödinger equation with prescribed norm:	112	Normalized_solution_(mathematics)	https://en.wikipedia.org/wiki/Normalized_solution_(mathematics)	102,305	1	2,024	8	10	54	37
where Δ Δ is a Laplacian operator, N ≥ ≥ 1 , λ λ ∈ ∈ R is a Lagrange multiplier and f is a nonlinearity. If we want to find a normalized solution to the equation, we need to consider the following functional : Let I : H 0 1 ( R N ) → → R be defined by	251	Normalized_solution_(mathematics)	https://en.wikipedia.org/wiki/Normalized_solution_(mathematics)	102,306	2	2,024	8	10	54	37
with the constraint	19	Normalized_solution_(mathematics)	https://en.wikipedia.org/wiki/Normalized_solution_(mathematics)	102,307	3	2,024	8	10	54	37
where H 0 1 ( R N ) is the Hilbert space and F ( s ) is the primitive of f ( s ) .	82	Normalized_solution_(mathematics)	https://en.wikipedia.org/wiki/Normalized_solution_(mathematics)	102,308	4	2,024	8	10	54	37
A common method of finding normalized solutions is through variational methods, i.e., finding the maxima and minima of the corresponding functional with the prescribed norm. Thus, we can find the weak solution of the equation. Moreover, if it satisfies the constraint, it's a normalized solution.	296	Normalized_solution_(mathematics)	https://en.wikipedia.org/wiki/Normalized_solution_(mathematics)	102,309	5	2,024	8	10	54	37
On a Euclidean space R 3 , we define a function f : R 2 → → R :	63	Normalized_solution_(mathematics)	https://en.wikipedia.org/wiki/Normalized_solution_(mathematics)	102,310	6	2,024	8	10	54	37
f ( x , y ) = ( x + y ) 2 with the constraint x 2 + y 2 = 1 .	61	Normalized_solution_(mathematics)	https://en.wikipedia.org/wiki/Normalized_solution_(mathematics)	102,311	7	2,024	8	10	54	37
By direct calculation, it is not difficult to conclude that the constrained maximum is f = 2 , with solutions ( x , y ) = ( 2 2 , 2 2 ) and ( x , y ) = ( − − 2 2 , − − 2 2 ) , while the constrained minimum is f = 0 , with solutions ( x , y ) = ( − − 2 2 , 2 2 ) and ( x , y ) = ( 2 2 , − − 2 2 ) .	297	Normalized_solution_(mathematics)	https://en.wikipedia.org/wiki/Normalized_solution_(mathematics)	102,312	8	2,024	8	10	54	37
The exploration of normalized solutions for the nonlinear Schrödinger equation can be traced back to the study of standing wave solutions with prescribed L 2 -norm. Jürgen Moser firstly introduced the concept of normalized solutions in the study of regularity properties of solutions to elliptic partial differential equations (elliptic PDEs). Specifically, he used normalized sequences of functions to prove regularity results for solutions of elliptic equations, which was a significant contribution to the field. Inequalities developed by Emilio Gagliardo and Louis Nirenberg played a crucial role in the study of PDE solutions in L p spaces. These inequalities provided important tools and background for defining and understanding normalized solutions.	757	Normalized_solution_(mathematics)	https://en.wikipedia.org/wiki/Normalized_solution_(mathematics)	102,313	9	2,024	8	10	54	37
For the variational problem, early foundational work in this area includes the concentration-compactness principle introduced by Pierre-Louis Lions in 1984, which provided essential techniques for solving these problems.	220	Normalized_solution_(mathematics)	https://en.wikipedia.org/wiki/Normalized_solution_(mathematics)	102,314	10	2,024	8	10	54	37
For variational problems with prescribed mass, several methods commonly used to deal with unconstrained variational problems are no longer available. At the same time, a new critical exponent appeared, the L 2 -critical exponent. From the Gagliardo-Nirenberg inequality, we can find that the nonlinearity satisfying L 2 -subcritical or critical or supercritical leads to a different geometry for functional. In the case the functional is bounded below, i.e., L 2 subcritical case, the earliest result on this problem was obtained by Charles-Alexander Stuart using bifurcation methods to demonstrate the existence of solutions. Later, Thierry Cazenave and Pierre-Louis Lions obtained existence results using minimization methods. Then, Masataka Shibata considered Schrödinger equations with a general nonlinear term.	815	Normalized_solution_(mathematics)	https://en.wikipedia.org/wiki/Normalized_solution_(mathematics)	102,315	11	2,024	8	10	54	37
In the case the functional is not bounded below, i.e., L 2 supcritical case, some new difficulties arise. Firstly, since λ λ is unknown, it is impossible to construct the corresponding Nehari manifold. Secondly, it is not easy to obtain the boundedness of the Palais-Smale sequence. Furthermore, verifying the compactness of the Palais-Smale sequence is challenging because the embedding H 1 ( R N ) ↪ ↪ L 2 ( R N ) is not compact. In 1997, Louis Jeanjean using the following transform:	486	Normalized_solution_(mathematics)	https://en.wikipedia.org/wiki/Normalized_solution_(mathematics)	102,316	12	2,024	8	10	54	37
Thus, one has the following functional:	39	Normalized_solution_(mathematics)	https://en.wikipedia.org/wiki/Normalized_solution_(mathematics)	102,317	13	2,024	8	10	54	37
Then,	5	Normalized_solution_(mathematics)	https://en.wikipedia.org/wiki/Normalized_solution_(mathematics)	102,318	14	2,024	8	10	54	37
which corresponds exactly to the Pokhozhaev's identity of equation. Jeanjean used this additional condition to ensure the boundedness of the Palais-Smale sequence, thereby overcoming the difficulties mentioned earlier. As the first method to address the issue of normalized solutions in unbounded functional, Jeanjean's approach has become a common method for handling such problems and has been imitated and developed by subsequent researchers.	445	Normalized_solution_(mathematics)	https://en.wikipedia.org/wiki/Normalized_solution_(mathematics)	102,319	15	2,024	8	10	54	37
In the following decades, researchers expanded on these foundational results. Thomas Bartsch and Sébastien de Valeriola investigate the existence of multiple normalized solutions to nonlinear Schrödinger equations. The authors focus on finding solutions that satisfy a prescribed L 2 norm constraint. Recent advancements include the study of normalized ground states for NLS equations with combined nonlinearities by Nicola Soave in 2020, who examined both subcritical and critical cases. This research highlighted the intricate balance between different types of nonlinearities and their impact on the existence and multiplicity of solutions.	643	Normalized_solution_(mathematics)	https://en.wikipedia.org/wiki/Normalized_solution_(mathematics)	102,320	16	2,024	8	10	54	37
In bounded domain, the situation is very different. Let's define f ( s ) = \| s \| p − − 2 s where p ∈ ∈ ( 2 , 2 ∗ ∗ ) . Refer to Pokhozhaev's identity,	150	Normalized_solution_(mathematics)	https://en.wikipedia.org/wiki/Normalized_solution_(mathematics)	102,321	17	2,024	8	10	54	37
The boundary term will make it impossible to apply Jeanjean's method. This has led many scholars to explore the problem of normalized solutions on bounded domains in recent years. In addition, there have been a number of interesting results in recent years about normalized solutions in Schrödinger system, Choquard equation, or Dirac equation.	344	Normalized_solution_(mathematics)	https://en.wikipedia.org/wiki/Normalized_solution_(mathematics)	102,322	18	2,024	8	10	54	37
Let's consider the nonlinear term to be homogeneous, that is, let's define f ( s ) = \| s \| p − − 2 s where p ∈ ∈ ( 2 , 2 ∗ ∗ ) . Refer to Gagliardo-Nirenberg inequality: define	176	Normalized_solution_(mathematics)	https://en.wikipedia.org/wiki/Normalized_solution_(mathematics)	102,323	19	2,024	8	10	54	37
then there exists a constant C N , p such that for any u ∈ ∈ H 1 ( R N ) , the following inequality holds:	106	Normalized_solution_(mathematics)	https://en.wikipedia.org/wiki/Normalized_solution_(mathematics)	102,324	20	2,024	8	10	54	37
Thus, there's a concept of mass critical exponent,	50	Normalized_solution_(mathematics)	https://en.wikipedia.org/wiki/Normalized_solution_(mathematics)	102,325	21	2,024	8	10	54	37
From this, we can get different concepts about mass subcritical as well as mass supercritical. It is also useful to get whether the functional is bounded below or not.	167	Normalized_solution_(mathematics)	https://en.wikipedia.org/wiki/Normalized_solution_(mathematics)	102,326	22	2,024	8	10	54	37
Let X be a Banach space and I : X → → R be a functional. A sequence ( u n ) n ⊂ ⊂ X is called a Palais-Smale sequence for I at the level c ∈ ∈ R if it satisfies the following conditions:	186	Normalized_solution_(mathematics)	https://en.wikipedia.org/wiki/Normalized_solution_(mathematics)	102,327	23	2,024	8	10	54	37
1. Energy Bound: sup n I ( u n ) < ∞ ∞ .	40	Normalized_solution_(mathematics)	https://en.wikipedia.org/wiki/Normalized_solution_(mathematics)	102,328	24	2,024	8	10	54	37
2. Gradient Condition: ⟨ ⟨ I ′ ( u n ) , u n − − u ⟩ ⟩ → → 0 as n → → ∞ ∞ for some u ∈ ∈ X .	92	Normalized_solution_(mathematics)	https://en.wikipedia.org/wiki/Normalized_solution_(mathematics)	102,329	25	2,024	8	10	54	37
Here, I ′ denotes the Fréchet derivative of I , and ⟨ ⟨ ⋅ ⋅ , ⋅ ⋅ ⟩ ⟩ denotes the inner product in X . Palais-Smale sequence named after Richard Palais and Stephen Smale.	170	Normalized_solution_(mathematics)	https://en.wikipedia.org/wiki/Normalized_solution_(mathematics)	102,330	26	2,024	8	10	54	37
Radiation-enhanced diffusion is a phenomenon in materials science like physics and chemistry, wherein the presence of radiation accelerates the diffusion of atoms or ions within a material. The effect arises because of the creation of defects in the crystal lattice, such as vacancies or interstitials, by the radiation.	320	Radiation-enhanced_diffusion	https://en.wikipedia.org/wiki/Radiation-enhanced_diffusion	102,331	0	2,024	8	10	54	37
This physics -related article is a stub. You can help Wikipedia by expanding it.	80	Radiation-enhanced_diffusion	https://en.wikipedia.org/wiki/Radiation-enhanced_diffusion	102,332	1	2,024	8	10	54	37
The Fradkin tensor, or Jauch-Hill-Fradkin tensor, named after Josef-Maria Jauch and Edward Lee Hill and David M. Fradkin, is a conservation law used in the treatment of the isotropic multidimensional harmonic oscillator in classical mechanics. For the treatment of the quantum harmonic oscillator in quantum mechanics, it is replaced by the tensor-valued Fradkin operator.	372	Fradkin_tensor	https://en.wikipedia.org/wiki/Fradkin_tensor	102,333	0	2,024	8	10	54	37
The Fradkin tensor provides enough conserved quantities to make the oscillator's equations of motion maximally superintegrable. This implies that to determine the trajectory of the system, no differential equations need to be solved, only algebraic ones.	254	Fradkin_tensor	https://en.wikipedia.org/wiki/Fradkin_tensor	102,334	1	2,024	8	10	54	37
Similarly to the Laplace–Runge–Lenz vector in the Kepler problem, the Fradkin tensor arises from a hidden symmetry of the harmonic oscillator.	142	Fradkin_tensor	https://en.wikipedia.org/wiki/Fradkin_tensor	102,335	2	2,024	8	10	54	37
Suppose the Hamiltonian of a harmonic oscillator is given by	60	Fradkin_tensor	https://en.wikipedia.org/wiki/Fradkin_tensor	102,336	3	2,024	8	10	54	37
with	4	Fradkin_tensor	https://en.wikipedia.org/wiki/Fradkin_tensor	102,337	4	2,024	8	10	54	37
then the Fradkin tensor (up to an arbitrary normalisation) is defined as	72	Fradkin_tensor	https://en.wikipedia.org/wiki/Fradkin_tensor	102,338	5	2,024	8	10	54	37
In particular, H is given by the trace : H = Tr ⁡ ⁡ ( F ) . The Fradkin Tensor is a thus a symmetric matrix, and for an n -dimensional harmonic oscillator has n ( n + 1 ) 2 − − 1 independent entries, for example 5 in 3 dimensions.	230	Fradkin_tensor	https://en.wikipedia.org/wiki/Fradkin_tensor	102,339	6	2,024	8	10	54	37
In Hamiltonian mechanics, the time evolution of any function A defined on phase space is given by	97	Fradkin_tensor	https://en.wikipedia.org/wiki/Fradkin_tensor	102,340	7	2,024	8	10	54	37
so for the Fradkin tensor of the harmonic oscillator,	53	Fradkin_tensor	https://en.wikipedia.org/wiki/Fradkin_tensor	102,341	8	2,024	8	10	54	37
The Fradkin tensor is the conserved quantity associated to the transformation	77	Fradkin_tensor	https://en.wikipedia.org/wiki/Fradkin_tensor	102,342	9	2,024	8	10	54	37
by Noether's theorem.	21	Fradkin_tensor	https://en.wikipedia.org/wiki/Fradkin_tensor	102,343	10	2,024	8	10	54	37
In quantum mechanics, position and momentum are replaced by the position- and momentum operators and the Poisson brackets by the commutator. As such the Hamiltonian becomes the Hamiltonian operator, angular momentum the angular momentum operator, and the Fradkin tensor the Fradkin operator. All of the above properties continue to hold after making these replacements.	369	Fradkin_tensor	https://en.wikipedia.org/wiki/Fradkin_tensor	102,344	11	2,024	8	10	54	37
This page lists examples of magnetic moments produced by various sources, grouped by orders of magnitude. The magnetic moment of an object is an intrinsic property and does not change with distance, and thus can be used to measure "how strong" a magnet is. For example, Earth possesses an enormous magnetic moment, however we are very distant from its center and experience only a tiny magnetic flux density (measured in tesla) on its surface.	443	Orders_of_magnitude_(magnetic_moment)	https://en.wikipedia.org/wiki/Orders_of_magnitude_(magnetic_moment)	102,345	0	2,024	8	10	54	37
Knowing the magnetic moment of an object ( m ) and the distance from its centre ( r ) it is possible to calculate the magnetic flux density experienced on the surface ( B ) using the following approximation:	207	Orders_of_magnitude_(magnetic_moment)	https://en.wikipedia.org/wiki/Orders_of_magnitude_(magnetic_moment)	102,346	1	2,024	8	10	54	37
where μ μ o is the constant of vacuum permeability.	51	Orders_of_magnitude_(magnetic_moment)	https://en.wikipedia.org/wiki/Orders_of_magnitude_(magnetic_moment)	102,347	2	2,024	8	10	54	37
This electromagnetism -related article is a stub. You can help Wikipedia by expanding it.	89	Orders_of_magnitude_(magnetic_moment)	https://en.wikipedia.org/wiki/Orders_of_magnitude_(magnetic_moment)	102,348	3	2,024	8	10	54	37
In condensed matter physics and quantum information theory, the quantum double model, proposed by Alexei Kitaev, is a lattice model that exhibits topological excitations. This model can be regarded as a lattice gauge theory, and it has applications in many fields, like topological quantum computation, topological order, topological quantum memory, quantum error-correcting code, etc. The name "quantum double" come from the Drinfeld double of a finite groups and Hopf algebras. The most well-known example is the toric code model, which is a special case of quantum double model by setting input group as cyclic group Z 2 .	625	Quantum_double_model	https://en.wikipedia.org/wiki/Quantum_double_model	102,349	0	2,024	8	10	54	37
The input data for Kitaev quantum double is a finite group G . Consider a directed lattice Σ Σ , we put a Hilbert space C [ G ] spanned by group elements on each edge, there are four types of edge operators	206	Quantum_double_model	https://en.wikipedia.org/wiki/Quantum_double_model	102,350	1	2,024	8	10	54	37
L + g \| h ⟩ ⟩ = \| g h ⟩ ⟩ , L − − g \| h ⟩ ⟩ = \| h g − − 1 ⟩ ⟩ ,	63	Quantum_double_model	https://en.wikipedia.org/wiki/Quantum_double_model	102,351	2	2,024	8	10	54	37
T + g \| h ⟩ ⟩ = δ δ g , h \| h ⟩ ⟩ , L − − g \| h ⟩ ⟩ = T g − − 1 , h \| h ⟩ ⟩ .	77	Quantum_double_model	https://en.wikipedia.org/wiki/Quantum_double_model	102,352	3	2,024	8	10	54	37
For each vertex connecting to m edges e 1 , … … , e m , there is a vertex operator	82	Quantum_double_model	https://en.wikipedia.org/wiki/Quantum_double_model	102,353	4	2,024	8	10	54	37
A v = 1 \| G \| ∑ ∑ g ∈ ∈ G L g ( e 1 ) ⊗ ⊗ … … ⊗ ⊗ L g ( e m ) .	63	Quantum_double_model	https://en.wikipedia.org/wiki/Quantum_double_model	102,354	5	2,024	8	10	54	37
Notice each edge has an orientation, when v is the starting point of e k , the operator is set as L − − , otherwise, it is set as L + .	135	Quantum_double_model	https://en.wikipedia.org/wiki/Quantum_double_model	102,355	6	2,024	8	10	54	37
For each face surrounded by m edges e 1 , … … , e m , there is a face operator	78	Quantum_double_model	https://en.wikipedia.org/wiki/Quantum_double_model	102,356	7	2,024	8	10	54	37
B f = ∑ ∑ h 1 ⋯ ⋯ h m = 1 G ∏ ∏ k = 1 m T h k ( e k ) .	55	Quantum_double_model	https://en.wikipedia.org/wiki/Quantum_double_model	102,357	8	2,024	8	10	54	37
Similar to the vertex operator, due to the orientation of the edge, when face f is on the right-hand side when traversing the positive direction of e , we set T + ; otherwise, we set T − − in the above expression. Also, note that the order of edges surrounding the face is assumed to be counterclockwise.	304	Quantum_double_model	https://en.wikipedia.org/wiki/Quantum_double_model	102,358	9	2,024	8	10	54	37
The lattice Hamiltonian of quantum double model is given by	59	Quantum_double_model	https://en.wikipedia.org/wiki/Quantum_double_model	102,359	10	2,024	8	10	54	37
H = − − ∑ ∑ v A v − − ∑ ∑ f B f .	33	Quantum_double_model	https://en.wikipedia.org/wiki/Quantum_double_model	102,360	11	2,024	8	10	54	37
Both of A v and B f are Hermitian projectors, they are stabilizer when regard the model is a quantum error correcting code.	123	Quantum_double_model	https://en.wikipedia.org/wiki/Quantum_double_model	102,361	12	2,024	8	10	54	37
The topological excitations of the model is characterized by the representations of the quantum double of finite group G . The anyon types are given by irreducible representations. For the lattice model, the topological excitations are created by ribbon operators.	264	Quantum_double_model	https://en.wikipedia.org/wiki/Quantum_double_model	102,362	13	2,024	8	10	54	37
The gapped boundary theory of quantum double model can be constructed based on subgroups of G . There is a boundary-bulk duality for this model.	144	Quantum_double_model	https://en.wikipedia.org/wiki/Quantum_double_model	102,363	14	2,024	8	10	54	37
The topological excitation of the model is equivalent to that of the Levin-Wen string-net model with input given by the representation category of finite group G .	163	Quantum_double_model	https://en.wikipedia.org/wiki/Quantum_double_model	102,364	15	2,024	8	10	54	37
The quantum double model can be generalized to the case where the input data is given by a C* Hopf algebra. In this case, the face and vertex operators are constructed using the comultiplication of Hopf algebra. For each vertex, the Haar integral of the input Hopf algebra is used to construct the vertex operator. For each face, the Haar integral of the dual Hopf algebra of the input Hopf algebra is used to construct the face operator.	438	Quantum_double_model	https://en.wikipedia.org/wiki/Quantum_double_model	102,365	16	2,024	8	10	54	37
The topological excitation are created by ribbon operators.	59	Quantum_double_model	https://en.wikipedia.org/wiki/Quantum_double_model	102,366	17	2,024	8	10	54	37
A more general case arises when the input data is chosen as a weak Hopf algebra, resulting in the weak Hopf quantum double model.	129	Quantum_double_model	https://en.wikipedia.org/wiki/Quantum_double_model	102,367	18	2,024	8	10	54	37
An Introduction to Mechanics, commonly referred to as Kleppner and Kolenkow, is an undergraduate level textbook on classical mechanics coauthored by physicists Daniel Kleppner and Robert J. Kolenkow. It originated as the textbook for a one- semester mechanics course at the Massachusetts Institute of Technology, where both Kleppner and Kolenkow taught, intended to go deeper than an ordinary first year course. Since its introduction, it has expanded its reach to other universities to become one of the most popular mechanics textbooks.	538	An_Introduction_to_Mechanics	https://en.wikipedia.org/wiki/An_Introduction_to_Mechanics	102,368	0	2,024	8	10	54	37
The first edition was published in 1973 by McGraw Hill and republished in 2010 by Cambridge University. The second edition was published in 2013 by Cambridge.	158	An_Introduction_to_Mechanics	https://en.wikipedia.org/wiki/An_Introduction_to_Mechanics	102,369	1	2,024	8	10	54	37
The first edition of the book was criticized for sexism in the exercises, though this was improved in the second edition.	121	An_Introduction_to_Mechanics	https://en.wikipedia.org/wiki/An_Introduction_to_Mechanics	102,370	2	2,024	8	10	54	37
This article about a physics -related book is a stub. You can help Wikipedia by expanding it.	93	An_Introduction_to_Mechanics	https://en.wikipedia.org/wiki/An_Introduction_to_Mechanics	102,371	3	2,024	8	10	54	37
Beijing Institute of Mathematical Sciences and Applications (BIMSA), located in Huairou district of Beijing, is an international mathematics research institution, co-sponsored by Beijing Municipal Government and Tsinghua University.	232	Beijing_Institute_of_Mathematical_Sciences_and_Applications	https://en.wikipedia.org/wiki/Beijing_Institute_of_Mathematical_Sciences_and_Applications	102,372	0	2,024	8	10	54	38
BIMSA is dedicated to advancing research in mathematics, theoretical physics, computer science, and related fields. Its goal is to promote inter-disciplinary collaboration and to bridge the gap between fundamental research and real-world applications.	251	Beijing_Institute_of_Mathematical_Sciences_and_Applications	https://en.wikipedia.org/wiki/Beijing_Institute_of_Mathematical_Sciences_and_Applications	102,373	1	2,024	8	10	54	38
The International Congress of Basic Science (ICBS) is held at BIMSA.	68	Beijing_Institute_of_Mathematical_Sciences_and_Applications	https://en.wikipedia.org/wiki/Beijing_Institute_of_Mathematical_Sciences_and_Applications	102,374	2	2,024	8	10	54	38
On June 12, 2020, with the support of the Beijing Municipal Government and the efforts of the Beijing Municipal Science and Technology Commission and the Huairou District Government, the Beijing Yanqi Lake Institute of Applied Mathematics (referred to as "Beijing Institute of Applied Mathematics") was officially established. This new research and development institution leverages the mathematical resources of Tsinghua University and the Chinese Academy of Sciences and was initiated by world-renowned mathematician Professor Shing-Tung Yau, director of the Center of Mathematical Sciences at Tsinghua University. Tsinghua University signed a cooperation agreement with the newly established Beijing Institute of Applied Mathematics.	736	Beijing_Institute_of_Mathematical_Sciences_and_Applications	https://en.wikipedia.org/wiki/Beijing_Institute_of_Mathematical_Sciences_and_Applications	102,375	3	2,024	8	10	54	38
The first president is Shing-Tung Yau, a mathematician known for his work in differential geometry and mathematical physics. He is a Fields Medalist and a Wolf Prize in Mathematics laureate.	190	Beijing_Institute_of_Mathematical_Sciences_and_Applications	https://en.wikipedia.org/wiki/Beijing_Institute_of_Mathematical_Sciences_and_Applications	102,376	4	2,024	8	10	54	38
The BIMSA conducts research across a wide range of areas in mathematics, physics and information science, from theoretical to applied.	134	Beijing_Institute_of_Mathematical_Sciences_and_Applications	https://en.wikipedia.org/wiki/Beijing_Institute_of_Mathematical_Sciences_and_Applications	102,377	5	2,024	8	10	54	38
BIMSA has more than 10 research groups active across a range of topics. The research areas include:	99	Beijing_Institute_of_Mathematical_Sciences_and_Applications	https://en.wikipedia.org/wiki/Beijing_Institute_of_Mathematical_Sciences_and_Applications	102,378	6	2,024	8	10	54	38
BIMSA has established long-term cooperative relationships with various universities to jointly train doctoral students, leveraging the teaching and research strengths of both parties. Graduates receive doctoral degree certificates and diplomas from the respective universities. Currently, BIMSA is conducting joint doctoral training programs with the Academy of Mathematics and Systems Science of the Chinese Academy of Sciences, the University of the Chinese Academy of Sciences, and the Institute of Statistics and Big Data at Renmin University of China. The first cohort of joint doctoral students with the Academy of Mathematics and Systems Science of the Chinese Academy of Sciences, the University of the Chinese Academy of Sciences, and Beihang University officially started and enrolled in 2023.	803	Beijing_Institute_of_Mathematical_Sciences_and_Applications	https://en.wikipedia.org/wiki/Beijing_Institute_of_Mathematical_Sciences_and_Applications	102,379	7	2,024	8	10	54	38
Yip-Wah Chung (born 1950) is a materials scientist at Northwestern University. He is a professor of materials science & engineering, and, by courtesy, of mechanical engineering within the McCormick School of Engineering, and serves as co-director of the mechanical engineering–materials science & engineering Master of Science program.	335	Yip-Wah_Chung	https://en.wikipedia.org/wiki/Yip-Wah_Chung	102,380	0	2,024	8	10	54	38
Chung was raised in Hong Kong, and holds a B.S. and an M.S. in physics from the University of Hong Kong, as well as a Ph.D. in physics from the University of California, Berkeley. He joined Northwestern, after obtaining his doctorate; at Northwestern, he previously served as department chair of materials science & engineering (1992–1998).	340	Yip-Wah_Chung	https://en.wikipedia.org/wiki/Yip-Wah_Chung	102,381	1	2,024	8	10	54	38
His research includes work on energy efficiency, surface engineering, and tribology. In 2016, Chung, Jiaxing Huang, and other co-authors published an article in the Proceedings of the National Academy of Sciences describing how a lubricant containing crumpled graphene could provide higher lubrication performance than other lubricant oils. In 2017, Chung was featured in the Northwestern Engineering magazine for his research on improving energy efficiency. The article describes a development by Chung and others on reducing friction within automobiles. Their development, a lubricant additive, "can reduce friction by up to 70 percent and wear by up to 90 percent compared to conventional lubricant counterparts." In 2019, Chung was interviewed by Tribology & Lubrication Technology. In his interview, he expressed sentiment that communication skills are a vital part of career tribology, and are not emphasized enough in education.	935	Yip-Wah_Chung	https://en.wikipedia.org/wiki/Yip-Wah_Chung	102,382	2	2,024	8	10	54	38
In 2002, Chung, Leon M. Keer, and Kornel Ehmann won the Innovative Research Award, conferred by the tribology division of the American Society of Mechanical Engineers. For his contributions to surface engineering and coatings, Chung received the 2024 R.F. Bunshah Award from the Advanced Surface Engineering Division, American Vacuum Society. As of 2024, he is a fellow of ASM International, American Vacuum Society, and the Society of Tribologists and Lubrication Engineers.	475	Yip-Wah_Chung	https://en.wikipedia.org/wiki/Yip-Wah_Chung	102,383	3	2,024	8	10	54	38
This article about an engineer, inventor or industrial designer is a stub. You can help Wikipedia by expanding it.	114	Yip-Wah_Chung	https://en.wikipedia.org/wiki/Yip-Wah_Chung	102,384	4	2,024	8	10	54	38
This article about materials science is a stub. You can help Wikipedia by expanding it.	87	Yip-Wah_Chung	https://en.wikipedia.org/wiki/Yip-Wah_Chung	102,385	5	2,024	8	10	54	38
Vesna Sossi is a Canadian medical physicist at the University of British Columbia, where she is a professor in the Department of Physics and Astronomy and adjunct professor of medicine. Her research interests include theranostics and the use of positron emission tomography in the study of Parkinson's disease. She is president of the IEEE Nuclear & Plasma Sciences Society (NPSS), and a distinguished lecturer for NPSS.	420	Vesna_Sossi	https://en.wikipedia.org/wiki/Vesna_Sossi	102,386	0	2,024	8	10	54	38
Sossi studied high energy physics at the University of Trieste, earning a laurea (the Italian equivalent of a master's degree). She came to the University of British Columbia (UBC) for doctoral study in physics, completing her Ph.D. in 1991.	241	Vesna_Sossi	https://en.wikipedia.org/wiki/Vesna_Sossi	102,387	1	2,024	8	10	54	38
She has been an adjunct professor of medicine at UBC since 1997. After completing her doctorate, she continued at UBC as an assistant professor of physics from 2001 to 2004, associate professor from 2004 to 2009, and full professor since 2009.	243	Vesna_Sossi	https://en.wikipedia.org/wiki/Vesna_Sossi	102,388	2	2,024	8	10	54	38
Sossi was elected as an IEEE Fellow in 2023, "for contributions to quantitative and translational brain PET imaging".	117	Vesna_Sossi	https://en.wikipedia.org/wiki/Vesna_Sossi	102,389	3	2,024	8	10	54	38
The planar reentry equations are the equations of motion governing the unpowered reentry of a spacecraft, based on the assumptions of planar motion and constant mass, in an Earth-fixed reference frame.	201	Planar_reentry_equations	https://en.wikipedia.org/wiki/Planar_reentry_equations	102,390	0	2,024	8	10	54	38
{ d V d t = − − ρ ρ V 2 2 β β + g sin ⁡ ⁡ γ γ d γ γ d t = − − V cos ⁡ ⁡ γ γ r − − ρ ρ V 2 β β ( L D ) cos ⁡ ⁡ σ σ + g cos ⁡ ⁡ γ γ V d h d t = − − V sin ⁡ ⁡ γ γ	159	Planar_reentry_equations	https://en.wikipedia.org/wiki/Planar_reentry_equations	102,391	1	2,024	8	10	54	38
where the quantities in these equations are:	44	Planar_reentry_equations	https://en.wikipedia.org/wiki/Planar_reentry_equations	102,392	2	2,024	8	10	54	38
Harry Allen and Alfred Eggers, based on their studies of ICBM trajectories, were able to derive an analytical expression for the velocity as a function of altitude. They made several assumptions:	195	Planar_reentry_equations	https://en.wikipedia.org/wiki/Planar_reentry_equations	102,393	3	2,024	8	10	54	38
These assumptions are valid for hypersonic speeds, where the Mach number is greater than 5. Then the planar reentry equations for the spacecraft are:	149	Planar_reentry_equations	https://en.wikipedia.org/wiki/Planar_reentry_equations	102,394	4	2,024	8	10	54	38
Rearranging terms and integrating from the atmospheric interface conditions at the start of reentry ( V atm , h atm ) leads to the expression:	142	Planar_reentry_equations	https://en.wikipedia.org/wiki/Planar_reentry_equations	102,395	5	2,024	8	10	54	38
The term exp ⁡ ⁡ ( − − h atm / H ) is small and may be neglected, leading to the velocity:	90	Planar_reentry_equations	https://en.wikipedia.org/wiki/Planar_reentry_equations	102,396	6	2,024	8	10	54	38
Allen and Eggers were also able to calculate the deceleration along the trajectory, in terms of the number of g's experienced n = g 0 − − 1 ( d V / d t ) , where g 0 is the gravitational acceleration at the planet's surface. The altitude and velocity at maximum deceleration are:	279	Planar_reentry_equations	https://en.wikipedia.org/wiki/Planar_reentry_equations	102,397	7	2,024	8	10	54	38
Another commonly encountered simplification is a lifting entry with a shallow, slowly-varying, flight path angle. The velocity as a function of altitude can be derived from two assumptions:	189	Planar_reentry_equations	https://en.wikipedia.org/wiki/Planar_reentry_equations	102,398	8	2,024	8	10	54	38
From these two assumptions, we may infer from the second equation of motion that:	81	Planar_reentry_equations	https://en.wikipedia.org/wiki/Planar_reentry_equations	102,399	9	2,024	8	10	54	38

In 2014, the SAGE-experiment's GGNT-apparatus (gallium-germanium neutrino telescope) was upgraded to perform a very-short-baseline neutrino oscillation experiment BEST (Baksan Experiment on Sterile Transitions) with an intense artificial neutrino source based on Cr. In 2017, the BEST apparatus was completed, but the artificial neutrino source was missing. As of 2018, the BEST experiment was underway. As of 2018, a follow-up experiment BEST-2 where the source would be changed to Zn was under consideration. It uses two gallium chambers instead of one, to better determine whether the anomaly could be explained by the distance from the source of the neutrinos

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In June 2022, the BEST experiment released two papers observing a 20-24% deficit in the production the isotope germanium expected from the reaction 71 Ga + ν ν e → → e − − + 71 Ge , confirming previous results from SAGE and GALLEX on the so called "gallium anomaly" pointing out that a sterile neutrino explanation can be consistent with the data. Further work have refined the precision for the cross section of the neutrino capture in 2023 as well as half-life of 71 G e in 2024 ruling them out as possible explanations for the anomaly.

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SAGE is led by the following physicists :

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43°16′32″N 42°41′25″E / 43.27556°N 42.69028°E / 43.27556; 42.69028

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In mathematics, a normalized solution to an ordinary or partial differential equation is a solution with prescribed norm, that is, a solution which satisfies a condition like ∫ ∫ R N | u ( x ) | 2 d x = 1. In this article, the normalized solution is introduced by using the nonlinear Schrödinger equation. The nonlinear Schrödinger equation (NLSE) is a fundamental equation in quantum mechanics and other various fields of physics, describing the evolution of complex wave functions. In Quantum Physics, normalization means that the total probability of finding a quantum particle anywhere in the universe is unity.

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In order to illustrate this concept, consider the following nonlinear Schrödinger equation with prescribed norm:

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where Δ Δ is a Laplacian operator, N ≥ ≥ 1 , λ λ ∈ ∈ R is a Lagrange multiplier and f is a nonlinearity. If we want to find a normalized solution to the equation, we need to consider the following functional : Let I : H 0 1 ( R N ) → → R be defined by

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with the constraint

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where H 0 1 ( R N ) is the Hilbert space and F ( s ) is the primitive of f ( s ) .

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A common method of finding normalized solutions is through variational methods, i.e., finding the maxima and minima of the corresponding functional with the prescribed norm. Thus, we can find the weak solution of the equation. Moreover, if it satisfies the constraint, it's a normalized solution.

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On a Euclidean space R 3 , we define a function f : R 2 → → R :

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f ( x , y ) = ( x + y ) 2 with the constraint x 2 + y 2 = 1 .

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By direct calculation, it is not difficult to conclude that the constrained maximum is f = 2 , with solutions ( x , y ) = ( 2 2 , 2 2 ) and ( x , y ) = ( − − 2 2 , − − 2 2 ) , while the constrained minimum is f = 0 , with solutions ( x , y ) = ( − − 2 2 , 2 2 ) and ( x , y ) = ( 2 2 , − − 2 2 ) .

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The exploration of normalized solutions for the nonlinear Schrödinger equation can be traced back to the study of standing wave solutions with prescribed L 2 -norm. Jürgen Moser firstly introduced the concept of normalized solutions in the study of regularity properties of solutions to elliptic partial differential equations (elliptic PDEs). Specifically, he used normalized sequences of functions to prove regularity results for solutions of elliptic equations, which was a significant contribution to the field. Inequalities developed by Emilio Gagliardo and Louis Nirenberg played a crucial role in the study of PDE solutions in L p spaces. These inequalities provided important tools and background for defining and understanding normalized solutions.

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For the variational problem, early foundational work in this area includes the concentration-compactness principle introduced by Pierre-Louis Lions in 1984, which provided essential techniques for solving these problems.

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For variational problems with prescribed mass, several methods commonly used to deal with unconstrained variational problems are no longer available. At the same time, a new critical exponent appeared, the L 2 -critical exponent. From the Gagliardo-Nirenberg inequality, we can find that the nonlinearity satisfying L 2 -subcritical or critical or supercritical leads to a different geometry for functional. In the case the functional is bounded below, i.e., L 2 subcritical case, the earliest result on this problem was obtained by Charles-Alexander Stuart using bifurcation methods to demonstrate the existence of solutions. Later, Thierry Cazenave and Pierre-Louis Lions obtained existence results using minimization methods. Then, Masataka Shibata considered Schrödinger equations with a general nonlinear term.

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In the case the functional is not bounded below, i.e., L 2 supcritical case, some new difficulties arise. Firstly, since λ λ is unknown, it is impossible to construct the corresponding Nehari manifold. Secondly, it is not easy to obtain the boundedness of the Palais-Smale sequence. Furthermore, verifying the compactness of the Palais-Smale sequence is challenging because the embedding H 1 ( R N ) ↪ ↪ L 2 ( R N ) is not compact. In 1997, Louis Jeanjean using the following transform:

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Thus, one has the following functional:

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Then,

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which corresponds exactly to the Pokhozhaev's identity of equation. Jeanjean used this additional condition to ensure the boundedness of the Palais-Smale sequence, thereby overcoming the difficulties mentioned earlier. As the first method to address the issue of normalized solutions in unbounded functional, Jeanjean's approach has become a common method for handling such problems and has been imitated and developed by subsequent researchers.

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In the following decades, researchers expanded on these foundational results. Thomas Bartsch and Sébastien de Valeriola investigate the existence of multiple normalized solutions to nonlinear Schrödinger equations. The authors focus on finding solutions that satisfy a prescribed L 2 norm constraint. Recent advancements include the study of normalized ground states for NLS equations with combined nonlinearities by Nicola Soave in 2020, who examined both subcritical and critical cases. This research highlighted the intricate balance between different types of nonlinearities and their impact on the existence and multiplicity of solutions.

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In bounded domain, the situation is very different. Let's define f ( s ) = | s | p − − 2 s where p ∈ ∈ ( 2 , 2 ∗ ∗ ) . Refer to Pokhozhaev's identity,

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The boundary term will make it impossible to apply Jeanjean's method. This has led many scholars to explore the problem of normalized solutions on bounded domains in recent years. In addition, there have been a number of interesting results in recent years about normalized solutions in Schrödinger system, Choquard equation, or Dirac equation.

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Let's consider the nonlinear term to be homogeneous, that is, let's define f ( s ) = | s | p − − 2 s where p ∈ ∈ ( 2 , 2 ∗ ∗ ) . Refer to Gagliardo-Nirenberg inequality: define

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then there exists a constant C N , p such that for any u ∈ ∈ H 1 ( R N ) , the following inequality holds:

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Thus, there's a concept of mass critical exponent,

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From this, we can get different concepts about mass subcritical as well as mass supercritical. It is also useful to get whether the functional is bounded below or not.

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Let X be a Banach space and I : X → → R be a functional. A sequence ( u n ) n ⊂ ⊂ X is called a Palais-Smale sequence for I at the level c ∈ ∈ R if it satisfies the following conditions:

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1. Energy Bound: sup n I ( u n ) < ∞ ∞ .

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2. Gradient Condition: ⟨ ⟨ I ′ ( u n ) , u n − − u ⟩ ⟩ → → 0 as n → → ∞ ∞ for some u ∈ ∈ X .

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Here, I ′ denotes the Fréchet derivative of I , and ⟨ ⟨ ⋅ ⋅ , ⋅ ⋅ ⟩ ⟩ denotes the inner product in X . Palais-Smale sequence named after Richard Palais and Stephen Smale.

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Radiation-enhanced diffusion is a phenomenon in materials science like physics and chemistry, wherein the presence of radiation accelerates the diffusion of atoms or ions within a material. The effect arises because of the creation of defects in the crystal lattice, such as vacancies or interstitials, by the radiation.

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This physics -related article is a stub. You can help Wikipedia by expanding it.

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The Fradkin tensor, or Jauch-Hill-Fradkin tensor, named after Josef-Maria Jauch and Edward Lee Hill and David M. Fradkin, is a conservation law used in the treatment of the isotropic multidimensional harmonic oscillator in classical mechanics. For the treatment of the quantum harmonic oscillator in quantum mechanics, it is replaced by the tensor-valued Fradkin operator.

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The Fradkin tensor provides enough conserved quantities to make the oscillator's equations of motion maximally superintegrable. This implies that to determine the trajectory of the system, no differential equations need to be solved, only algebraic ones.

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Similarly to the Laplace–Runge–Lenz vector in the Kepler problem, the Fradkin tensor arises from a hidden symmetry of the harmonic oscillator.

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Suppose the Hamiltonian of a harmonic oscillator is given by

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with

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then the Fradkin tensor (up to an arbitrary normalisation) is defined as

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In particular, H is given by the trace : H = Tr ⁡ ⁡ ( F ) . The Fradkin Tensor is a thus a symmetric matrix, and for an n -dimensional harmonic oscillator has n ( n + 1 ) 2 − − 1 independent entries, for example 5 in 3 dimensions.

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In Hamiltonian mechanics, the time evolution of any function A defined on phase space is given by

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so for the Fradkin tensor of the harmonic oscillator,

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The Fradkin tensor is the conserved quantity associated to the transformation

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by Noether's theorem.

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In quantum mechanics, position and momentum are replaced by the position- and momentum operators and the Poisson brackets by the commutator. As such the Hamiltonian becomes the Hamiltonian operator, angular momentum the angular momentum operator, and the Fradkin tensor the Fradkin operator. All of the above properties continue to hold after making these replacements.

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This page lists examples of magnetic moments produced by various sources, grouped by orders of magnitude. The magnetic moment of an object is an intrinsic property and does not change with distance, and thus can be used to measure "how strong" a magnet is. For example, Earth possesses an enormous magnetic moment, however we are very distant from its center and experience only a tiny magnetic flux density (measured in tesla) on its surface.

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Knowing the magnetic moment of an object ( m ) and the distance from its centre ( r ) it is possible to calculate the magnetic flux density experienced on the surface ( B ) using the following approximation:

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where μ μ o is the constant of vacuum permeability.

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This electromagnetism -related article is a stub. You can help Wikipedia by expanding it.

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In condensed matter physics and quantum information theory, the quantum double model, proposed by Alexei Kitaev, is a lattice model that exhibits topological excitations. This model can be regarded as a lattice gauge theory, and it has applications in many fields, like topological quantum computation, topological order, topological quantum memory, quantum error-correcting code, etc. The name "quantum double" come from the Drinfeld double of a finite groups and Hopf algebras. The most well-known example is the toric code model, which is a special case of quantum double model by setting input group as cyclic group Z 2 .

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The input data for Kitaev quantum double is a finite group G . Consider a directed lattice Σ Σ , we put a Hilbert space C [ G ] spanned by group elements on each edge, there are four types of edge operators

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L + g | h ⟩ ⟩ = | g h ⟩ ⟩ , L − − g | h ⟩ ⟩ = | h g − − 1 ⟩ ⟩ ,

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https://en.wikipedia.org/wiki/Quantum_double_model

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T + g | h ⟩ ⟩ = δ δ g , h | h ⟩ ⟩ , L − − g | h ⟩ ⟩ = T g − − 1 , h | h ⟩ ⟩ .

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For each vertex connecting to m edges e 1 , … … , e m , there is a vertex operator

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Quantum_double_model

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A v = 1 | G | ∑ ∑ g ∈ ∈ G L g ( e 1 ) ⊗ ⊗ … … ⊗ ⊗ L g ( e m ) .

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Quantum_double_model

https://en.wikipedia.org/wiki/Quantum_double_model

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Notice each edge has an orientation, when v is the starting point of e k , the operator is set as L − − , otherwise, it is set as L + .

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Quantum_double_model

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For each face surrounded by m edges e 1 , … … , e m , there is a face operator

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Quantum_double_model

https://en.wikipedia.org/wiki/Quantum_double_model

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B f = ∑ ∑ h 1 ⋯ ⋯ h m = 1 G ∏ ∏ k = 1 m T h k ( e k ) .

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Quantum_double_model

https://en.wikipedia.org/wiki/Quantum_double_model

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Similar to the vertex operator, due to the orientation of the edge, when face f is on the right-hand side when traversing the positive direction of e , we set T + ; otherwise, we set T − − in the above expression. Also, note that the order of edges surrounding the face is assumed to be counterclockwise.

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Quantum_double_model

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The lattice Hamiltonian of quantum double model is given by

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Quantum_double_model

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H = − − ∑ ∑ v A v − − ∑ ∑ f B f .

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Quantum_double_model

https://en.wikipedia.org/wiki/Quantum_double_model

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Both of A v and B f are Hermitian projectors, they are stabilizer when regard the model is a quantum error correcting code.

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Quantum_double_model

https://en.wikipedia.org/wiki/Quantum_double_model

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The topological excitations of the model is characterized by the representations of the quantum double of finite group G . The anyon types are given by irreducible representations. For the lattice model, the topological excitations are created by ribbon operators.

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The gapped boundary theory of quantum double model can be constructed based on subgroups of G . There is a boundary-bulk duality for this model.

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The topological excitation of the model is equivalent to that of the Levin-Wen string-net model with input given by the representation category of finite group G .

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The quantum double model can be generalized to the case where the input data is given by a C* Hopf algebra. In this case, the face and vertex operators are constructed using the comultiplication of Hopf algebra. For each vertex, the Haar integral of the input Hopf algebra is used to construct the vertex operator. For each face, the Haar integral of the dual Hopf algebra of the input Hopf algebra is used to construct the face operator.

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Quantum_double_model

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The topological excitation are created by ribbon operators.

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Quantum_double_model

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A more general case arises when the input data is chosen as a weak Hopf algebra, resulting in the weak Hopf quantum double model.

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An Introduction to Mechanics, commonly referred to as Kleppner and Kolenkow, is an undergraduate level textbook on classical mechanics coauthored by physicists Daniel Kleppner and Robert J. Kolenkow. It originated as the textbook for a one- semester mechanics course at the Massachusetts Institute of Technology, where both Kleppner and Kolenkow taught, intended to go deeper than an ordinary first year course. Since its introduction, it has expanded its reach to other universities to become one of the most popular mechanics textbooks.

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An_Introduction_to_Mechanics

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The first edition was published in 1973 by McGraw Hill and republished in 2010 by Cambridge University. The second edition was published in 2013 by Cambridge.

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An_Introduction_to_Mechanics

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The first edition of the book was criticized for sexism in the exercises, though this was improved in the second edition.

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An_Introduction_to_Mechanics

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An_Introduction_to_Mechanics

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Beijing Institute of Mathematical Sciences and Applications (BIMSA), located in Huairou district of Beijing, is an international mathematics research institution, co-sponsored by Beijing Municipal Government and Tsinghua University.

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BIMSA is dedicated to advancing research in mathematics, theoretical physics, computer science, and related fields. Its goal is to promote inter-disciplinary collaboration and to bridge the gap between fundamental research and real-world applications.

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Beijing_Institute_of_Mathematical_Sciences_and_Applications

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The International Congress of Basic Science (ICBS) is held at BIMSA.

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Beijing_Institute_of_Mathematical_Sciences_and_Applications

https://en.wikipedia.org/wiki/Beijing_Institute_of_Mathematical_Sciences_and_Applications

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On June 12, 2020, with the support of the Beijing Municipal Government and the efforts of the Beijing Municipal Science and Technology Commission and the Huairou District Government, the Beijing Yanqi Lake Institute of Applied Mathematics (referred to as "Beijing Institute of Applied Mathematics") was officially established. This new research and development institution leverages the mathematical resources of Tsinghua University and the Chinese Academy of Sciences and was initiated by world-renowned mathematician Professor Shing-Tung Yau, director of the Center of Mathematical Sciences at Tsinghua University. Tsinghua University signed a cooperation agreement with the newly established Beijing Institute of Applied Mathematics.

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Beijing_Institute_of_Mathematical_Sciences_and_Applications

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The first president is Shing-Tung Yau, a mathematician known for his work in differential geometry and mathematical physics. He is a Fields Medalist and a Wolf Prize in Mathematics laureate.

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Beijing_Institute_of_Mathematical_Sciences_and_Applications

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The BIMSA conducts research across a wide range of areas in mathematics, physics and information science, from theoretical to applied.

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BIMSA has more than 10 research groups active across a range of topics. The research areas include:

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BIMSA has established long-term cooperative relationships with various universities to jointly train doctoral students, leveraging the teaching and research strengths of both parties. Graduates receive doctoral degree certificates and diplomas from the respective universities. Currently, BIMSA is conducting joint doctoral training programs with the Academy of Mathematics and Systems Science of the Chinese Academy of Sciences, the University of the Chinese Academy of Sciences, and the Institute of Statistics and Big Data at Renmin University of China. The first cohort of joint doctoral students with the Academy of Mathematics and Systems Science of the Chinese Academy of Sciences, the University of the Chinese Academy of Sciences, and Beihang University officially started and enrolled in 2023.

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Yip-Wah Chung (born 1950) is a materials scientist at Northwestern University. He is a professor of materials science & engineering, and, by courtesy, of mechanical engineering within the McCormick School of Engineering, and serves as co-director of the mechanical engineering–materials science & engineering Master of Science program.

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Chung was raised in Hong Kong, and holds a B.S. and an M.S. in physics from the University of Hong Kong, as well as a Ph.D. in physics from the University of California, Berkeley. He joined Northwestern, after obtaining his doctorate; at Northwestern, he previously served as department chair of materials science & engineering (1992–1998).

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Yip-Wah_Chung

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His research includes work on energy efficiency, surface engineering, and tribology. In 2016, Chung, Jiaxing Huang, and other co-authors published an article in the Proceedings of the National Academy of Sciences describing how a lubricant containing crumpled graphene could provide higher lubrication performance than other lubricant oils. In 2017, Chung was featured in the Northwestern Engineering magazine for his research on improving energy efficiency. The article describes a development by Chung and others on reducing friction within automobiles. Their development, a lubricant additive, "can reduce friction by up to 70 percent and wear by up to 90 percent compared to conventional lubricant counterparts." In 2019, Chung was interviewed by Tribology & Lubrication Technology. In his interview, he expressed sentiment that communication skills are a vital part of career tribology, and are not emphasized enough in education.

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Yip-Wah_Chung

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In 2002, Chung, Leon M. Keer, and Kornel Ehmann won the Innovative Research Award, conferred by the tribology division of the American Society of Mechanical Engineers. For his contributions to surface engineering and coatings, Chung received the 2024 R.F. Bunshah Award from the Advanced Surface Engineering Division, American Vacuum Society. As of 2024, he is a fellow of ASM International, American Vacuum Society, and the Society of Tribologists and Lubrication Engineers.

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Yip-Wah_Chung

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Vesna Sossi is a Canadian medical physicist at the University of British Columbia, where she is a professor in the Department of Physics and Astronomy and adjunct professor of medicine. Her research interests include theranostics and the use of positron emission tomography in the study of Parkinson's disease. She is president of the IEEE Nuclear & Plasma Sciences Society (NPSS), and a distinguished lecturer for NPSS.

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Vesna_Sossi

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Sossi studied high energy physics at the University of Trieste, earning a laurea (the Italian equivalent of a master's degree). She came to the University of British Columbia (UBC) for doctoral study in physics, completing her Ph.D. in 1991.

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Vesna_Sossi

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She has been an adjunct professor of medicine at UBC since 1997. After completing her doctorate, she continued at UBC as an assistant professor of physics from 2001 to 2004, associate professor from 2004 to 2009, and full professor since 2009.

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Vesna_Sossi

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Sossi was elected as an IEEE Fellow in 2023, "for contributions to quantitative and translational brain PET imaging".

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Vesna_Sossi

https://en.wikipedia.org/wiki/Vesna_Sossi

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The planar reentry equations are the equations of motion governing the unpowered reentry of a spacecraft, based on the assumptions of planar motion and constant mass, in an Earth-fixed reference frame.

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{ d V d t = − − ρ ρ V 2 2 β β + g sin ⁡ ⁡ γ γ d γ γ d t = − − V cos ⁡ ⁡ γ γ r − − ρ ρ V 2 β β ( L D ) cos ⁡ ⁡ σ σ + g cos ⁡ ⁡ γ γ V d h d t = − − V sin ⁡ ⁡ γ γ

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Planar_reentry_equations

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where the quantities in these equations are:

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Planar_reentry_equations

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Harry Allen and Alfred Eggers, based on their studies of ICBM trajectories, were able to derive an analytical expression for the velocity as a function of altitude. They made several assumptions:

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Planar_reentry_equations

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These assumptions are valid for hypersonic speeds, where the Mach number is greater than 5. Then the planar reentry equations for the spacecraft are:

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Planar_reentry_equations

https://en.wikipedia.org/wiki/Planar_reentry_equations

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Rearranging terms and integrating from the atmospheric interface conditions at the start of reentry ( V atm , h atm ) leads to the expression:

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The term exp ⁡ ⁡ ( − − h atm / H ) is small and may be neglected, leading to the velocity:

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Planar_reentry_equations

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Allen and Eggers were also able to calculate the deceleration along the trajectory, in terms of the number of g's experienced n = g 0 − − 1 ( d V / d t ) , where g 0 is the gravitational acceleration at the planet's surface. The altitude and velocity at maximum deceleration are:

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Another commonly encountered simplification is a lifting entry with a shallow, slowly-varying, flight path angle. The velocity as a function of altitude can be derived from two assumptions:

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From these two assumptions, we may infer from the second equation of motion that:

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