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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
Selmi, Bilel
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We prove a decomposition theorem of Besicovitch’s type for the relative multifractal Hausdorff measure and packing measure in a probability space. By obtaining a new necessary condition for the strong regularity with the multifractal measures in a more general framework, we extend in this paper the density theorem of Dai and Li (A multifractal formalism in a probability space. Chaos Solitons Fractals 27:57–73, 2006). In particular, this result is more refined than those found in Dai and Taylor (Defining fractal in a probability space. Ill J Math 38:480–500, 1994).
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By
Gumenyuk, Pavel; Prause, István
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Becker (J Reine Angew Math 255:23–43, 1972) discovered a sufficient condition for quasiconformal extendibility of Loewner chains. Many known conditions for quasiconformal extendibility of holomorphic functions in the unit disk can be deduced from his result. We give a new proof of (a generalization of) Becker’s result based on Slodkowski’s Extended
$$\lambda $$
Lemma. Moreover, we characterize all quasiconformal extensions produced by Becker’s (classical) construction and use that to obtain examples in which Becker’s extension is extremal (i.e. optimal in the sense of maximal dilatation) or, on the contrary, fails to be extremal.
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