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By
Duyunova, Anna; Lychagin, Valentin; Tychkov, Sergey
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Algebras of symmetries and the corresponding algebras of differential invariants for plane flows of inviscid fluids are given. Their dependence on thermodynamical states of media are studied and a classification of thermodynamical states is given.
By
Chen, Cheng; Jiang, YaoLin
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In this paper Lie symmetry analysis method is applied to study nonlinear generalized Zakharov system which is the coupled nonlinear system of Schrödinger equations. With the aid of Lie point symmetry, nonlinear generalized Zakharov system is reduced into the ODEs and some group invariant solutions are obtained where some solutions are new, which are not reported in literatures. Then the bifurcation theory and qualitative theory are employed to investigate nonlinear generalized Zakharov system. Through the analysis of phase portraits, some Jacobielliptic function solutions are found, such as the periodicwave solutions, kinkshaped and bellshaped solitarywave solutions.
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By
Yurko, V.
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1 Citations
We study inverse spectral problems for ordinary differential equations on compact startype graphs when differential equations have different orders on different edges. As the main spectral characteristics we introduce and study the socalled Weyltype matrices which are generalizations of the Weyl function (mfunction) for the classical Sturm–Liouville operator. We provide a procedure for constructing the solution of the inverse problem and prove its uniqueness.
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By
Arefijamaal, Ali Akbar; Ghaani Farashahi, Arash
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7 Citations
Let
$$H$$
be a locally compact group and
$$K$$
be an LCA group also let
$$\tau :H\rightarrow Aut(K)$$
be a continuous homomorphism and
$$G_\tau =H\ltimes _\tau K$$
be the semidirect product of
$$H$$
and
$$K$$
with respect to
$$\tau $$
. In this article we define the Zak transform
$$\mathcal{Z }_L$$
on
$$L^2(G_\tau )$$
with respect to a
$$\tau $$
invariant uniform lattice
$$L$$
of
$$K$$
and we also show that the Zak transform satisfies the Plancherel formula. As an application we analyze how these technique apply for the semidirect product group
$$\mathrm SL (2,\mathbb{Z })\ltimes _\tau \mathbb{R }^2$$
and also the WeylHeisenberg groups.
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By
Jiang, Guowei; Liu, Yu
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We consider the Schrödinger operator
$$L = \Delta _{G}+V$$
on the stratified Lie group G, where
$$\Delta _{G}$$
is the subLaplacian and the nonnegative potential V belongs to the reverse Hölder class
$$ B_{q_{1}}$$
for
$$q_1\ge \frac{Q}{2}$$
, where Q is the homogeneous dimension of G. Let
$$q_2 = 1$$
when
$$q_1\ge Q$$
and
$$\frac{1}{q_2}=1\frac{1}{q_1}+\frac{1}{Q}$$
when
$$\frac{Q}{2}<q_1<Q$$
. The commutator
$$[b,\mathcal {R}]$$
is generated by a function
$$b\in \varLambda ^{\theta }_{\nu }(G)$$
for
$$\theta >0,0<\nu <1$$
, where
$$\varLambda ^{\theta }_{\nu }(G)$$
is a new function space on the stratified Lie group which is larger than the classical Companato space, and the Riesz transform
$$\mathcal {R}=\nabla _{G}(\Delta _{G}+V)^{\frac{1}{2}}$$
. We prove that the commutator
$$[b,\mathcal {R}]$$
is bounded from
$$L^{p}(G)$$
into
$$L^{q}(G)$$
for
$$1<p<q^{'}_{2}$$
, where
$$\frac{1}{q}=\frac{1}{p}\frac{\nu }{Q}$$
.
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By
Fenton, P. C.; Rossi, John
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Suppose that
$$u$$
is subharmonic in the plane and such that, for some
$$c>1$$
and sufficiently large
$$K_0=K_0(c)$$
,
$$u$$
is harmonic in the disc
$$\Delta (z,\tau (z)^{c})$$
whenever
$$u(z)>B(z,u)K_0\log \tau (z)$$
, where
$$\tau (z)=\max \{z,B(z,u)\}$$
and
$$B(r,u)=\max _{z=r}u(z)$$
. It is shown that if in addition
$$u$$
satisfies a certain lower growth condition, then there are ‘Wiman–Valiron discs’ in each of which
$$u$$
is the logarithm of the modulus of an analytic function, and that the derivatives of the analytic functions have regular asymptotic growth.
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By
Gorbunov, Oleg; Yurko, Viacheslav
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2 Citations
We study the nonselfadjoint Dirac system on a finite interval having nonintegrable regular singularities in interior points with additional matching conditions at these points. Properties of spectral characteristics are established, and the inverse spectral problem is investigated. We provide a constructive procedure for the solution of the inverse problem, and prove its uniqueness. Moreover, necessary and sufficient conditions for the global solvability of this nonlinear inverse problem are obtained.
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By
Wone, Oumar
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We study the geometry of real analytic second order ODEs under the local real analytic diffeomorphism of
$$\mathbb {R}^2$$
which are area preserving, through the method of Cartan. We obtain a subdivision into three “parts”. The first one is the most symmetric case. It is characterized by the vanishing of an areapreserving relative invariant namely
$$f_y+\dfrac{2}{9}f_{y^{\prime }}^{2}\dfrac{1}{3}\mathfrak {D}(f_{y^{\prime }})$$
. In this situation we associate a local affine normal Cartan connection on the first jet
$$J^{1}(\mathbb {R},\mathbb {R})$$
space whose curvature contains all the areapreserving relative differential invariants, to any second order ODE under study. The second case which includes all the Painlevé transcendents is given by the ODEs for which
$$f_y+\dfrac{2}{9}f_{y^{\prime }}^{2}\dfrac{1}{3}\mathfrak {D}(f_{y^{\prime }})\not \equiv 0$$
. In the latter case we give all necessary steps in order to obtain an
$$e$$
structure on
$$J^{1}(\mathbb {R},\mathbb {R})$$
for a generic second order ODE equation of that type. Finally we give the method to reduce to an
$$e$$
structure on
$$J^{1}$$
when
$$f_{y^\prime y^\prime y^\prime y^\prime }\not \equiv 0$$
.
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By
Feki, Ines; Jeribi, Aref; Sfaxi, Ridha
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1 Citations
In the present paper, we deal with the perturbed operator
$$\begin{aligned} T(\varepsilon ):=T_0+\varepsilon T _1+\varepsilon ^2T_2+\cdots +\varepsilon ^kT_k+\cdots , \end{aligned}$$
where
$$\varepsilon \in \mathbb C ,\,T_0$$
is a closed densely defined linear operator on a separable Banach space
$$X$$
with domain
$$\mathcal D (T_0),$$
while
$$T_1, T_2, \ldots $$
are linear operators on
$$X$$
with the same domain
$$\mathcal D \supset \mathcal D (T_0)$$
and satisfying a specific growing inequality. The basic idea here is to investigate under sufficient conditions assuring the invariance of the closure of the perturbed operator
$$T(\varepsilon )$$
which enables us to study the changed spectrum. Moreover, we prove that the system formed by some eigenvectors of
$$T(\varepsilon )$$
which are analytic on
$$\varepsilon ,$$
forms a Schauder basis in
$$X.$$
After that, we apply the obtained results to a nonselfadjoint problem describing the radiation of a vibrating structure in a light fluid and to a nonselfadjoint Gribov operator in Bargmann space.
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