6 research outputs found
Groups with right-invariant multiorders
A Cayley object for a group G is a structure on which G acts regularly as a
group of automorphisms. The main theorem asserts that a necessary and
sufficient condition for the free abelian group G of rank m to have the generic
n-tuple of linear orders as a Cayley object is that m>n. The background to this
theorem is discussed. The proof uses Kronecker's Theorem on diophantine
approximation.Comment: 9 page
Multiorders in amenable group actions
The paper offers a thorough study of multiorders and their applications to
measure-preserving actions of countable amenable groups. By a~{\em multiorder}
on a~countable group we mean any probability measure on the collection
of linear orders of type on , invariant
under the natural action of on such orders. Every free measure-preserving
-action has a~multiorder as a
factor and has the same orbits as the -action , where
is the \emph{successor map} determined by the multiorder factor. Moreover, the
sub-sigma-algebra associated with the multiorder
factor is invariant under , which makes the corresponding -action
a factor of . We prove that the
entropy of any -process generated by a finite partition of , conditional
with respect to , is preserved by the orbit
equivalence with . Furthermore, this entropy can be computed in
terms of the so-called random past, by a formula analogous to known for -actions. The above fact is then applied to prove a variant of a result by
Rudolph and Weiss. The original theorem states that orbit equivalence between
free actions of countable amenable groups preserves conditional entropy with
respect to a~sub-sigma-algebra , as soon as the ``orbit change'' is
measurable with respect to . In our variant, we replace the
measurability assumption by a~simpler one: should be invariant under
both actions and the actions on the resulting factor should be free. In
conclusion we provide a characterization of the Pinsker sigma-algebra of any
-process in terms of an appropriately defined remote past arising from a
multiorder.Comment: 36 pages, 2 figures, Changes: slightly changed formulation and proof
of Theorem 7.4, some remarks adde
Oscillatory Integrals, Spectral Multiplier Operators, Semilinear Elliptic Equations, and Pseudodifferential Calculus on Carnot Manifolds
ํ์๋
ผ๋ฌธ (๋ฐ์ฌ)-- ์์ธ๋ํ๊ต ๋ํ์ : ์๋ฆฌ๊ณผํ๋ถ, 2015. 2. Raphael Ponge.๋
ผ๋ฌธ์ ๊ตฌ์ฑ์ ํฌ๊ฒ ๋ค์์ ์ธ ๋ถ๋ถ์ผ๋ก ๋๋์ด์ ธ ์๋ค์ ํ์์ฉ์์ ์ ๋ฐํ ๊ณ์ธก, ๋ฐ์ ํ ํ์ํ ๋ฐฉ์ ์, ๊ทธ๋ฆฌ๊ณ ์บ๋ ๋ค์์ฒด์์์์ ์๋ฏธ๋ถ ์ฐ์ฐ. ์ด ์ฃผ์ ๋ค์ ์ง์ ์ ์ด๊ฑฐ๋ ๊ฐ์ ์ ์ผ๋ก ์๋ก ์ฐ๊ด์ด ๋์ด์๋ค.
์ฒซ ๋ถ๋ถ์ ์ ์์ ๋
ผ๋ฌธ [CH1, CH2, CH3] ์ ๋ฐํ์ผ๋ก ํ๊ณ ์ง๋์์ฉ์์ ๋ถ๊ด ๊ณฑ ์ฐ์ฐ์์ ๊ดํ ์ ๋ฐ ๊ณ์ธก์ ์ป๋ ๊ฒ์ ๋ชฉํ๋ก ํ๋ค. ์ข ๋ ๊ตฌ์ฒด์ ์ผ๋ก, ์ฒซ๋ฒ์งธ ๋
ผ๋ฌธ [CH1]์์๋ ํ์ด์ ๋ฒ ๋ฅด๊ทธ ๊ตฐ์์ ์ ์๋ ๊ฐํ ํน์์ฑ์ ๊ฐ์ง ์์ฉ์์ ๊ณต๊ฐ๊ณผ ๊ณต๊ฐ์์์ ๋ฐ์ด๋๋ฅผ ๋ณด์ธ๋ค. ๊ณต๊ฐ ๋ฐ์ด๋๋ฅผ ์ํด ํดํ๋ ํํ์ ์ง๋์์ฉ์ ๊ณ์ธก์ ์ด์ฉํ๊ณ , ๊ณต๊ฐ ๋ฐ์ด๋๋ฅผ ์ํด์๋ ํ๋ ๊ณต๊ฐ์ ๋ถ์ ๋ถํด๋ฅผ ์ด์ฉํ๋ค. ๋๋ฒ์งธ ๋
ผ๋ฌธ [CH2] ์์๋ ์ธต์ํ๋ ๊ตฐ๋ค์์ ๊ณฑ ์์ฉ์๋ค์ ์ต๋ํจ์๋ค์ ๋ํ ์ ๋ฐํ๋ ๋ฐ์ด๋๋ฅผ ๊ตฌํ๋ค. ๋ํ ์ธต์ํ๋ ๊ตฐ๋ค์ ๊ณฑํํ์ ๊ตฐ์์๋ ๊ด๋ จ๋ ๋ฐ์ด๋๋ฅผ ์ป๊ณ , ํ๋์ ์์ฉ์ผ๋ก ํ์ด์ ๋ฒ ๋ฅด๊ทธ ๊ตฐ์์ ๊ฒฐํฉ ๋ถ๊ด ๊ณฑ ์์ฉ์๋ค์ ์ต๋ํจ์์ ๋ํด์๋ ์ ๋ฐํ๋ ๋ฐ์ด๋๋ฅผ ์ป๋๋ค. ์ธ๋ฒ์งธ ๋
ผ๋ฌธ [CH3]์์๋ ๋ฐ์ด๋๊ฐ ์๋ ์น๊ณจํ ๋ค์์ฒด ์์์ ์ ์๋ ์์ ์์ฒด ์๋ฐ ํ์ํ ๋ฏธ๋ถ ์์ฉ์ ๊ฐ ์์๋, ํค๋ฅด๋ง๋-๋ฏธํ๋ฆฐ ์กฐ๊ฑด์๋์์ ์ด ์์ฉ์์ ๊ด๋ จ๋ ๋ถ๊ด ๊ณฑ ์์ฉ์๋ค์ ์ต๋ํจ์์ ๋ํ ์ ๋ฐํ๋ ๋ฐ์ด๋๋ฅผ ๊ตฌํ๋ค.
๋๋ฒ์งธ ๋ถ๋ถ์ ๋ฐ์ ํ ํ์ํ ๋ฐฉ์ ์๋ค์ ๋ํ ๊ณต๋ถ์ด๊ณ , ๋
ผ๋ฌธ [CH4]์ ๊ณต๋ ๋
ผ๋ฌธ [CKL, CKL2, CS1]์ ๊ธฐ๋ฐ์ผ๋ก ๋์ด์๋ค.
๋
ผ๋ฌธ [CH4]์์ ์ฐ๋ฆฌ๋ ์ ํ ์์ญ๋ด์์ ๋ถ์ ๋ผํ๋ผ์์์ ํฌํจํ๋ฉฐ ๊ฐํ๊ฒ ์ฎ์ฌ์๋ ์์คํ
์ ๋ํด์ ์ฐ๊ตฌํ๋ค. ๊ตฌ์ฒด์ ์ผ๋ก, ์ฐ๋ฆฌ๋ ์กด์ฌ์ฑ๊ณผ ๋น์กด์ฌ์ฑ์ ๊ดํ ๊ฒฐ๊ณผ๋ค์ ๋ณด์ด๊ณ , ๊ธฐ๋ค์ค-์คํ๋ญ ํํ์ ์ ๊ณ์ธก, ๋์นญ ๊ตฌ์กฐ์ ๊ดํ ๊ฒฐ๊ณผ๋ฅผ ๋ณด์ธ๋ค. ์ฌ๊ธฐ์ ์ฐ๋ฆฌ๋ ๋
ผ๋ฌธ [CT, T]์์ ๋ณด์ฌ์ก๋ ๋น์ ํ ํ์ํ ๋ฐฉ์ ์๋ค์ ๋ํ ์ ๊ณ์ธก์ ๋ํด์ ์๋ก์ด ์ฆ๋ช
์ ์ป๋๋ค.
๊น์นํ ๋ฐ์ฌ๋, ์ด๊ธฐ์ ๊ต์๋๊ณผ์ ๊ณต๋ ๋
ผ๋ฌธ์ธ [CKL]์์๋ ๋ถ์ ๋ผํ๋ผ์์์ ํฌํจํ ๋น์ ํ ํ์ํ ๋ฐฉ์ ์๋ค์ ๋ํด์ ์๊ณ์ง์์ ๊ด๋ จ๋์ด ์ต์ ์๋์ง ํด๋ค์ ์ ๊ทผ ํ๋์ ๊ณต๋ถํ๊ณ , ๋ค์ค์ผ๋ก ๋ฒ๋ธ๋งํ๋ ํด๋ค์ ์กด์ฌ์ฑ์ ๊ณต๋ถํ๋ค. ์ด๊ฒ์ Han (1991) [H] ๊ณผ Rey (1990) [R] ๊ฒฐ๊ณผ์ ๋น๊ตญ๋ถ์ ๋ฒ์ ์ด๋ผ๊ณ ํ ์ ์๋ค.
์์ง๋ช
๊ต์๋๊ณผ ํจ๊ป ์ฐ๊ตฌํ ๋
ผ๋ฌธ [CS1]์์ ์ฐ๋ฆฌ๋ ์น๊ณจ์ฑ์ด ์๋ ๋น๊ตญ๋ถ์ ๋ฐ์ ํ ํ์ํ ๋ฐฉ์ ์์ ๋ํด์ ๊ณต๋ถํ๋ค. ๊ตฌ์ฒด์ ์ผ๋ก, ์ฐ๋ฆฌ๋ ์ ํ ์์ญ๋ด์์ ๋ถ์ ๊ณ์ ๋ฒ์ ์ ๋ธ๋ ์ง์ค-๋๋ ๋ฒ๋ฅด๊ทธ ๋ฌธ์ ๊ฐ ๋ฌดํํด๋ฅผ ๊ฐ๋๋ค๋ ๊ฒ์ ์ฆ๋ช
ํ๋ค.
์ด ํํธ์ ๋ง์ง๋ง ์ฑํฐ๋ ๊น์นํ ๋ฐ์ฌ๋, ์ด๊ธฐ์ ๊ต์๋๊ณผ์ ๊ณต๋ ์ฐ๊ตฌ ๋
ผ๋ฌธ [CKL2] ์ ๋ฐํ์ผ๋ก ํ๋ค. ์ด ๋
ผ๋ฌธ์ ๋ชฉ์ ์ 3์ฐจ์ ์ด์์์ ๋ ์ธ-์ ๋ด-ํ์ธ๋ฌ ๋ฐฉ์ ์์ ์๊ณ์ง์๊ทผ์ฒ์์ ๋ค์ค ๋ฒ๋ธ๋งํ๋ ํด๋ค์ ๋ํ ์ง์ ์ฑ์ง๋ค์ ์ป๋๋ฐ ์๋ค. ๊ฐ๊ฐ์ ๋ฒ๋ธ ํด๋ค์์ ์ ํํ๋ ๋ฌธ์ ๋ฅผ ๊ณต๋ถํ์ฌ, ์ฐ๋ฆฌ๋ ์ฒ์ ๊ฐ์ ๊ณ ์ ํจ์์ ๊ณ ์ ์น์ ๋ํด์ ์ ํํ ๊ณ์ธก๋ค์ ๋ณด์ธ๋ค. ํน๋ณํ, ์ฐ๋ฆฌ๋ 4์ฐจ์์ด์์์ ๋ค์ค ๋ฒ๋ธ ํด์ ๋ชจ์ค-์ธ๋ฑ์ค๊ฐ ๊ทธ๋ํจ์, ๋ก๋นํจ์๋ค์ ์ผ์ฐจ, ์ด์ฐจ ๋ฏธ๋ถ๋ค๋ก ์ด๋ฃจ์ด์ง ๋์นญ ํ๋ ฌ๋ค๋ก ๊ท๋ช
๋๋๋ค๋ Bahri-Li-Rey (1995) ์ ์ํ ๊ณ ์ ์ ์ธ ๊ฒฐ๊ณผ์ ๋ํ ์๋ก์ด ์ฆ๋ช
์ ์ ์ํ๋ค. ์ฐ๋ฆฌ์ ์ฆ๋ช
์ 3์ฐจ์์ผ ๊ฒฝ์ฐ์๋ ์ ์ฉ์ด ๋๋ค.
์ธ๋ฒ์งธ ํํธ๋ ๋ผํ์ ํฐ์ฆ๊ต์๋๊ณผ ํจ๊ปํ ๋
ผ๋ฌธ [CP1, CP2]๋ฅผ ๋ฐํ์ผ๋ก ์ฐ์ฌ์ก๋ค. ๋
ผ๋ฌธ [CP1] ์์๋ ์บ๋ ๋ค์์ฒด์ ๋ด๋ถ์ ์ผ๋ก ์ฃผ์ด์ง ์ ํ ๊ตฐ ๋ค๋ฐ๋ค์ ์ ์ํ๊ณ ์ฐ์ ์ ์ขํ์ ๋ํด์ ๊ณต๋ถ๋ฅผ ํ๋ค. ์ด๋ฅผ ํตํด์ ์บ๋ ๋ค์์ฒด์ ๋งค๋ํ ์ ์ด๊ตฐ์ ์ ์ํ๋ค. ์ด๋ฌํ ๊ณต๋ถ๋ค์ ๋ฐํ์ผ๋ก ๋
ผ๋ฌธ [CP2] ์์๋ ์บ๋ ๋ค์์ฒด์์์์ ์๋ฏธ๋ถ ์์ฉ์์ ๋ํ ๊ณต๋ถ๋ฅผ ํฉ๋๋ค. ์ ์ ํ ์๋ฏธ๋ถ ์์ฉ์๋ค์ ๋ชจ์์ ์ ์ํ๊ณ ์ด ์์ฉ์๋ค์ ๊ณ์ฐ๋ฒ์ ์ ํํ ๊ตฌํ๋ค. ๊ตฌ์ฒด์ ์ผ๋ก๋, ๊ฒฐํฉ, ์๋ฐ ์์ฉ์, ์ขํ ๋ณํ์ ๊ดํ ๊ตฌ์ฒด์ ์ธ ์ปค๋ ์ ๊ฐ๋ฅผ ๊ตฌํ๋ค. ์ด๊ฒ์ ํตํด ์ฐ๋ฆฌ๋ ์ฝํ ํ์์ฑ์ ๊ฐ์ง ๋ฏธ๋ถ ์์ฉ์๋ค์ ์ญ์ ๊ตฌ์ฒด์ ์ธ ์ปค๋ ์ ๊ฐ ํํ์ ์ป์ด๋ผ ์ ์๋ค. ๋ํ ๊ด๋ จ๋ ์ด ๋ฏธ๋ถ ์์ฉ์์ ๋ํ ์ด ์ปค๋ ์ ๊ฐ๋ ์ป์ ์ ์๋ค. ์ด๊ฒ์ ํ ์์ฉ์ผ๋ก ์ผ๋ ๋ค์์ฒด์์์์ ๋ถ๊ด ๋ฐด๋์ ์ฑ์ง์ ๊ณต๋ถํ ์ ์๋ค.1. Introduction
1.1 Oscillatory Integrals and Spectral Mutiplier Operators
1.1.1 L2 and Hp boundedness of strongly singular operators and oscillating operators on Heisenberg groups
1.1.2 Maximal multiplier on Stratified groups and compact manfiolds without boundary
1.2 Semilinear Elliptic Equations and Fractional Laplacians
1.2.1 On strongly indefinite systems involving the fractional Laplacian
1.2.2 behavior of solutions for nonlinear elliptic problems with the fractional Laplacian
1.2.3 Infinitely many solutions for semilinear nonlocal elliptic equations under noncompact settings
1.2.4 Qualitative properties of multi-bubble solutions for nonlinear elliptic equations involving critical exponents
1.3 Pseudodifferential Calculus on Carnot Manifolds
1.3.1 Privileged coordinates and Tangent groupoid for Carnot manifolds
1.3.2 Pseudodierential calculus on Carnot manifolds
Part 1. Oscillatory Integrals and Spectral Mutiplier Operators
2 L2 and Hp boundedness of strongly singular operators and oscillating operators on Heisenberg groups [Ch1]
2.1 Introduction
2.2 Dyadic decomposition and Localization
2.3 L2 estimates
2.4 Hardy spaces on the Heisenberg groups
2.5 Hp estimates
2.6 Necessary conditions
3 Maximal functions for multipliers on stratified groups [Ch2]
3.1 Introduction
3.2 Kernels of multipliers on Stratified groups
3.3 Martingales on homogeneous space and its application to maximal multipliers
3.4 Maximal multipliers on product spaces
3.5 Bound of maximal multiplier on product spaces
4 Maximal functions of multipliers on compact manifolds without boundary [Ch3]
4.1 Introduction
4.2 Preliminaries
4.3 The proof of Proposition 4.2.2
4.4 Localization of the operator A(mP)
4.5 Properties of the kernels and the Hardy-Littlewood maximal funtion
4.6 Martingale operators and the proof of Proposition 4.2.3
II Semilinear Elliptic Equations and Fractional Laplacians
5 On strongly indefinite systems involving the fractional Laplacian [Ch4]
5.1 Introduction
5.2 Preliminaries
5.2.1 Spectral definition of the fractional Sobolev spaces and fractional Laplacians
5.2.2 Extended problems of nonlinear systems
5.2.3 Definition of weak solutions
5.2.4 The sobolev embedding
5.2.5 Greens functions and the Robin function
5.3 The integral estimates
5.4 The proof of Theorem 5.1.1
5.5 On the nonlinear system (5.1)
6 Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional
Laplacian [CKL]
6.1 Introduction
6.2 Preliminaries
6.2.1 Sharp Sobolev and trace inequalities
6.2.2 Greens functions and the Robin function
6.2.3 Maximum principle
6.2.4 Properties of the Robin function
6.3 The asymptotic behavior
6.4 Uniform boundedness
6.5 Location of the blowup point
6.6 Construction of solutions for (6.1) concentrating at multiple points
6.6.1 Finite dimensional reduction
6.6.2 The reduced problem
6.6.3 Definition of stable critical sets and conclusion of the proofs of Theorems
6.7 The subcritical problem
Appendices
6.A Proof of Proposition 6.4.7
6.B Technical computations in the proof of Theorem 6.1.4
6.B.1 Estimation of the projected bubbles
6.B.2 Basic estimates
6.B.3 Proof of Proposition 6.6.4
7 Infinitely many solutions for semilinear nonlocal elliptic equations under noncompact settings [ChS]
7.1 Introduction
7.2 Mathematical frameworks and preliminaries
7.2.1 Fractional Sobolev spaces, fractional Laplacians and fractional harmonic extensions
7.2.2 Weighted Sobolev and Sobolev-trace inequalities
7.2.3 Useful lemmas
7.3 Settings and Ideas for the proof of Theorem 7.1.2
7.4 A refined norm estimate
7.5 Integral estimates
7.6 End of proofs of main theorems
Appendices
7.A Proof of Lemma 7.5.2
7.B A variant of Mosers iteration method
7.C Local Pohozaev identity
8 Qualitative properties of multi-bubble solutions for nonlinear elliptic equations involving critical exponents [CKL2]
8.1 Introduction
8.2 Preliminaries
8.3 Proof of Theorem 8.1.1
8.4 Upper bounds for the l-th eigenvalues and asymptotic behavior of the `-th eigenfunctions, m + 1 =< l =< (n + 1)m
8.5 A further analysis on asymptotic behavior of the l-th eigenfunctions, m + 1 =< l
=< (n + 1)m
8.6 Characterization of the `-th eigenvalues, m + 1 =< l =< (n + 1)m
8.7 Estimates for the `-th eigenvalues and eigenfunctions, (n + 1)(m + 1) =< l =< (n + 2)m
Appendices
8.A A moving sphere argument
III Pseudodifferential Calculus on Carnot Manifolds
9 Privileged Coordinates and Tangent Groupoid for Carnot Manifolds
9.1 Introduction
9.2 Carnot Manifolds: Definitions and Main Examples
9.3 The Tangent Group Bundle of a Carnot Manifold
9.3.1 The tangent Lie algebra bundle g M
9.3.2 The tangent Lie group bundle GM
9.3.3 Description of ga M in terms of left-invariant vector fields
9.4 Privileged Coordinates for Carnot Manifolds
9.5 Nilpotent Approximation of Vector Fields
9.6 Carnot Coordinates
9.7 The Tangent Groupoid of a Carnot Manifold
9.7.1 Differentiable groupoids
9.7.2 The tangent groupoid of a Carnot manifold
Appendices
9.A A matrix computation for degree
10 Pseudodifferential calculus
10.1 Classes of Symbol and Pseudodifferential operators
10.1.1 Definition of PsiHDOs
10.2 Convolutions on nilpotent Lie groups
10.3 Pseudodifferential calculus
10.3.1 Composition of Pseudodifferential operators on vector fields
10.3.2 Invariance theorem of peudodifferential operators
10.3.3 Adjoint of pseudodifferential operators
10.4 Mapping properties on Lp
spaces
10.5 Rockland condition and the construction of parametrix
10.6 Heat equation
10.7 Holomorphic families of PsiHDOs
10.7.1 Kernels of holomorphic PsiHDOs
10.8 Complex powers of PsiHDOs
10.9 Spectral asymptotics for Hypoelliptic operators
Appendices
10.A Review on the class of symbols and kernels given at a point
10.A.1 Micellaneous
10.B Technical computations
10.C Some properties of distributionsDocto
Personality Identification from Social Media Using Deep Learning: A Review
Social media helps in sharing of ideas and information among people scattered around the world and thus helps in creating communities, groups, and virtual networks. Identification of personality is significant in many types of applications such as in detecting the mental state or character of a person, predicting job satisfaction, professional and personal relationship success, in recommendation systems. Personality is also an important factor to determine individual variation in thoughts, feelings, and conduct systems. According to the survey of Global social media research in 2018, approximately 3.196 billion social media users are in worldwide. The numbers are estimated to grow rapidly further with the use of mobile smart devices and advancement in technology. Support vector machine (SVM), Naive Bayes (NB), Multilayer perceptron neural network, and convolutional neural network (CNN) are some of the machine learning techniques used for personality identification in the literature review. This paper presents various studies conducted in identifying the personality of social media users with the help of machine learning approaches and the recent studies that targeted to predict the personality of online social media (OSM) users are reviewed