Around splitting and reaping for partitions of ω

We investigate splitting number and reaping number for the structure (ω) ^ω of infinite partitions of ω. We prove that \mathfrakr_d ￡ non(M),non(N),\mathfrakd{\mathfrak{r}_{d}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N}),\mathfrak{d}} and \mathfraks_d 3 \mathfrakb{\mathfrak{s}_{d}\geq\mathfrak{b}} . We also show the consistency results ${\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})}${\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})} and ${\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})}${\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})} . To prove the consistency \mathfrakr_d < add(M){\mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})} and \mathfraks_d < cof(M){\mathfrak{s}_{d} < \mathsf{cof}(\mathcal{M})} we introduce new cardinal invariants \mathfrakr_pair{\mathfrak{r}_{pair}} and \mathfraks_pair{\mathfrak{s}_{pair}} . We also study the relation between \mathfrakr_pair, \mathfraks_pair{\mathfrak{r}_{pair}, \mathfrak{s}_{pair}} and other cardinal invariants. We show that cov(M),cov(N) ￡ \mathfrakr_pair ￡ \mathfraks_d,\mathfrakr{\mathsf{cov}(\mathcal{M}),\mathsf{cov}(\mathcal{N})\leq\mathfrak{r}_{pair}\leq\mathfrak{s}_{d},\mathfrak{r}} and \mathfraks ￡ \mathfraks_pair ￡ non(M),non(N){\mathfrak{s}\leq\mathfrak{s}_{pair}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N})} .     

Lyapunov exponents for products of matrices and multifractal analysis. Part I: Positive matrices

Let (Σ,σ) be a full shift space on an alphabet consisting ofm symbols and letM: Σ→L ⁺(ℝ ^d , ℝ ^d ) be a continuous function taking values in the set ofd×d positive matrices. Denote by λ _M (x) the upper Lyapunov exponent ofM atx. The set of possible Lyapunov exponents is just an interval. For any possible Lyapunov exponentα, we prove the following variational formula, , where dim is the Hausdorff dimension or the packing dimension,P _M(q) is the pressure function ofM, μ is aσ-invariant Borel probability measure on Σ,h(μ) is the entropy ofμ, and . The author was partially supported by a HK RGC grant in Hong Kong and the Special Funds for Major State Basic Research Projects in China.     

Two Linear Transformations each Tridiagonal with Respect to an Eigenbasis of the other; Comments on the Parameter Array

Let ${\mathbb K} $ denote a field. Let it d denote a nonnegative integer and consider a sequence p=( $\theta_i, \theta^*_i,i=0...d; \varphi_j, \phi_j,j=1...{\it d})$ consisting of scalars taken from ${\mathbb K} $ . We call p a parameter array whenever: (PA1) $\theta_i \not=\theta_j, \; \theta^*_i\not=\theta^*_j$ if $$i\not=j$, $(0 \leq i, j\leq d)$; (PA2) $ \varphi_i\not=0$, $\phi_i\not=0$ $(1 \leq i \leq d)$; (PA3) $\varphi_i = \phi_1 \sum_{h=0}^{i-1} ({\theta_h-\theta_{d-h}})/({\theta_0-\theta_d}) + (\theta^*_i-\theta^*_0)(\theta_{i-1}-\theta_d)$ $(1 \leq i \leq d)$; (PA4) $\phi_i = \varphi_1 \sum_{h=0}^{i-1} ({\theta_h-\theta_{d-h}})/({\theta_0-\theta_d}) + (\theta^*_i-\theta^*_0)(\theta_{d-i+1}-\theta_0)$ $(1 \leq i \leq d)$; (PA5) $(\theta_{i-2}-\theta_{i+1})(\theta_{i-1}-\theta_i)^{-1}$, $(\theta^*_{i-2}-\theta^*_{i+1})(\theta^*_{i-1}-\theta^*_i)^{-1}$$ are equal and independent of i for $2 \leq i \leq d-1$ . In Terwilliger, J. Terwilliger, Linear Algebra Appl., Vol. 330(2001) p. 155 we showed the parameter arrays are in bijection with the isomorphism classes of Leonard systems. Using this bijection we obtain the following two characterizations of parameter arrays. Assume p satisfies PA1 and PA2. Let A, B,A^*, B^* denote the matrices in ${Mat}_{{\it d}+1}$ ( ${\mathbb K} $ ) which have entries A _ii =θ _i , B _ii =θ _d-i , A ^* _ii =θ^* _i , B ^* _ii =θ^* _i (0 ≤ i ≤ d), A _i,i-1=1, B _i,i-1=1, A ^* _i-1,i =φ _i , B ^* _i-1,i =? _i (1 ≤ i ≤ d), and all other entries 0. We show the following are equivalent: (i) p satisfies PA3–PA5; (ii) there exists an invertible G ∈Mat _d+1( ${\mathbb K} $ ) such that G ^?1 AG=B and G ^?1 A ^* G=B ^*; (iii) for 0 ≤ i ≤ d the polynomial $$ \sum_{n=0}^i \frac{ (\lambda-\theta_0) (\lambda-\theta_1) \cdots (\lambda-\theta_{n-1}) (\theta^*_i-\theta^*_0) (\theta^*_i-\theta^*_1) \cdots (\theta^*_i-\theta^*_{n-1}) } {\varphi_1\varphi_2\cdots \varphi_n}$$ is a scalar multiple of the polynomial $$\sum_{n=0}^i \frac{ (\lambda-\theta_d) (\lambda-\theta_{d-1}) \cdots (\lambda-\theta_{d-n+1}) (\theta^*_i-\theta^*_0) (\theta^*_i-\theta^*_1) \cdots (\theta^*_i-\theta^*_{n-1}) } {\phi_1\phi_2\cdots \phi_n}.$$ We display all the parameter arrays in parametric form. For each array we compute the above polynomials. The resulting polynomials form a class consisting of the q-Racah, q-Hahn, dual q-Hahn, q-Krawtchouk, dual q-Krawtchouk, quantum q-Krawtchouk, affine q-Krawtchouk, Racah, Hahn, dual-Hahn, Krawtchouk, Bannai/Ito, and Orphan polynomials. The Bannai/Ito polynomials can be obtained from the q-Racah polynomials by letting q tend to ?1. The Orphan polynomials have maximal degree 3 and exist for ( ${\mathbb K} $ )=2 only. For each of the polynomials listed above we give the orthogonality, 3-term recurrence, and difference equation in terms of the parameter array.     

On the Distance to the Closest Matrix with Triple Zero Eigenvalue

The 2-norm distance from a matrix A to the set ${\mathcal{M}}$ of n × n matrices with a zero eigenvalue of multiplicity ≥3 is estimated. If $$Q(\gamma _1 ,\gamma _2 ,\gamma _3 ) = \left( {\begin{array}{*{20}c} A &amp; {\gamma _1 I_n } &amp; {\gamma _3 I_n } \\ 0 &amp; A &amp; {\gamma _2 I_n } \\ 0 &amp; 0 &amp; A \\ \end{array} } \right), n \geqslant 3,$$ then $$\rho _2 (A,{\mathcal{M}}) \geqslant {\mathop {max}\limits_{\gamma _1 ,\gamma _2 \geqslant 0,\gamma _3 \in {\mathbb{C}}}} \sigma _{3n - 2} (Q(\gamma _1 ,\gamma _2 ,\gamma _3 )),$$ where σⁱ(·)is the ith singular value of the corresponding matrix in the decreasing order of singular values. Moreover, if the maximum on the right-hand side is attained at the point $\gamma ^ * = (\gamma _1^ * ,\gamma _2^ * ,\gamma _3^ * )$ , where $\gamma _1^ * \gamma _2^ * \ne 0$ , then, in fact, one has the exact equality $$\rho _2 (A,{\mathcal{M}}) = \sigma _{3n - 2} (Q(\gamma _1^ * ,\gamma _2^ * ,\gamma _3^ * )).$$ This result can be regarded as an extension of Malyshev's formula, which gives the 2-norm distance from A to the set of matrices with a multiple zero eigenvalue.     

On the regular growth of Dirichlet series absolutely convergent in a half-plane

For the Dirichlet series F(s) = ?_{n = 1}^￥ a_nexp{ sl_n } F(s) = \sum\nolimits_{n = 1}^\infty {{a_n}\exp \left\{ {s{\lambda_n}} \right\}} with abscissa of absolute convergence σ _a =0, we establish conditions for (λ _n ) and (a _n ) under which lnM( s, F ) = T_R( 1 + o(1) )exp{ r_R

The Cubic Complex Moment Problem

Let \({s = \{s_{jk}\}_{0 \leq j+k \leq 3}}\) be a given complex-valued sequence. The cubic complex moment problem involves determining necessary and sufficient conditions for the existence of a positive Borel measure \({\sigma}\) on \({\mathbb{C}}\) (called a representing measure for s) such that \({s_{jk} = \int_{\mathbb{C}}\bar{z}^j z^k d\sigma(z)}\) for \({0 \leq j + k \leq 3}\) . Put $$\Phi = \left(\begin{array}{lll} s_{00} & s_{01} & s_{10} \\s_{10} & s_{11} & s_{20} \\s_{01} & s_{02} & s_{11}\end{array}\right), \quad \Phi_z = \left(\begin{array}{lll}s_{01} & s_{02} & s_{11} \\s_{10} & s_{12} & s_{21} \\s_{02} & s_{03} & s_{12}\end{array} \right)\quad {\rm and}\quad\Phi_{\bar{z}} = (\Phi_z)^*.$$ If \({\Phi \succ 0}\) , then the commutativity of \({\Phi^{-1} \Phi_z}\) and \({\Phi^{-1} \Phi_{\bar{z}}}\) is necessary and sufficient for the existence a 3-atomic representing measure for s. If \({\Phi^{-1} \Phi_z}\) and \({\Phi^{-1} \Phi_{\bar{z}}}\) do not commute, then we show that s has a 4-atomic representing measure. The proof is constructive in nature and yields a concrete parametrization of all 4-atomic representing measures of s. Consequently, given a set \({K \subseteq \mathbb{C}}\) necessary and sufficient conditions are obtained for s to have a 4-atomic representing measure \({\sigma}\) which satisfies \({{\rm supp} \sigma \cap K \neq \emptyset}\) or \({{\rm supp} \sigma \subseteq K}\) . The cases when \({K = \overline{\mathbb{D}}}\) and \({K = \mathbb{T}}\) are considered in detail.     

Asymptotic Behavior of Variable-Coefficient Toeplitz Determinants

For a continuous function s\sigma defined on [0,1]×\mathbbT[0,1]\times\mathbb{T}, let \ops\op\sigma stand for the (n+1)×(n+1)(n+1)\times(n+1) matrix whose (j,k)(j,k)-entries are equal to \frac1 2pò₀^2p s( \fracjn,e^iq) e^-i(j-k)q  dq,        j,k = 0,1,...,n . \displaystyle \frac{1} {2\pi}\int_0^{2\pi} \sigma \left( \frac{j}{n},e^{i\theta}\right) e^{-i(j-k)\theta} \,d\theta, \qquad j,k =0,1,\dots,n~. These matrices can be thought of as variable-coefficient Toeplitz matrices or as the discrete analogue of pseudodifferential operators. Under the assumption that the function s\sigma possesses a logarithm which is sufficiently smooth on [0,1]×\mathbbT[0,1]\times\mathbb{T}, we prove that the asymptotics of the determinants of \ops\op\sigma are given by det[\ops] ~ G[s]⁽ⁿ⁺¹⁾E[s]     \text as   n?￥ , \det \left[\op\sigma\right] \sim G[\sigma]^{(n+1)}E[\sigma] \quad \text{ as \ } n\to\infty~, where G[s]G[\sigma] and E[s]E[\sigma] are explicitly determined constants. This formula is a generalization of the Szegö Limit Theorem. In comparison with the classical theory of Toeplitz determinants some new features appear.     

Lp theory for the multidimensional aggregation equation

We consider well‐posedness of the aggregation equation ∂_tu + div(uv) = 0, v = −▿K * u with initial data in \input amssym ${\cal P}_2 {\rm (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$ in dimensions 2 and higher. We consider radially symmetric kernels where the singularity at the origin is of order |x|^α, α > 2 − d, and prove local well‐posedness in \input amssym ${\cal P}_2 { (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$ for sufficiently large p < p_s. In the special case of K(x) = |x|, the exponent p_s = d/(d = 1) is sharp for local well‐posedness in that solutions can instantaneously concentrate mass for initial data in \input amssym ${\cal P}_2 { (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$ with p < p_s. We also give an Osgood condition on the potential K(x) that guarantees global existence and uniqueness in \input amssym ${\cal P}_2 { (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$ . © 2010 Wiley Periodicals, Inc.     

Linear and Algebraic Independence of q-Zeta Values

In the paper, results on linear and algebraic independence of q-series of the form $\varsigma _q (s) = \sum\nolimits_{n = 1}^\infty {\sigma _{s - 1} (n)q^n }$ over the field ?(q) are obtained, where $\sigma _{s - 1} (n) = \sum\nolimits_{d|n} {d^{s - 1} }$ , s = 1, 2,... .     

Diffeomorphism classification of smooth weighted complete intersections

Xn(d1, . . . , dr-1, dr; w) and Xn(e1, . . . , er-1, dr; w) are two complex odd-dimensional smooth weighted complete intersections defined in a smooth weighted hypersurface Xn+r-1(dr; w). We prove that they are diffeomorphic if and only if they have the same total degree d, the Pontrjagin classes and the Euler characteristic, under the following assumptions: the weights w = (ω0, . . . , ωn+r) are pairwise relatively prime and odd, νp(d/dr) ≥ 2n+1/ 2(p-1) + 1 for all primes p with p(p-1) ≤ n + 1, where νp(d/dr) satisfies d/dr =Ⅱp prime pνp (d/dr).     

(<Emphasis Type="Italic">o</Emphasis>)-Topology in *-algebras of locally measurable operators

We consider the topology t( M ) t\left( \mathcal{M} \right) of convergence locally in measure in the *-algebra LS( M ) LS\left( \mathcal{M} \right) of all locally measurable operators affiliated to the von Neumann algebra M \mathcal{M} . We prove that t( M ) t\left( \mathcal{M} \right) coincides with the (o)-topology in LS_h( M ) = { T ? LS( M ):T* = T } L{S_h}\left( \mathcal{M} \right) = \left\{ {T \in LS\left( \mathcal{M} \right):T* = T} \right\} if and only if the algebra M \mathcal{M} is σ-finite and is of finite type. We also establish relations between t( M ) t\left( \mathcal{M} \right) and various topologies generated by a faithful normal semifinite trace on M \mathcal{M} .     

Generalized Knopp identities for homogeneous Hardy sums and Cochrane-Hardy sums

Let q, h, a, b be integers with q > 0. The classical and the homogeneous Dedekind sums are defined by $$s(h,q) = \sum\limits_{j = 1}^q {\left( {\left( {{j \over q}} \right)} \right)\left( {\left( {{{hj} \over q}} \right)} \right),{\rm{ }}s(a,b,q) = \sum\limits_{j = 1}^q {\left( {\left( {{{aj} \over q}} \right)} \right)\left( {\left( {{{bj} \over q}} \right)} \right),} } $$ respectively, where $((x)) = \left\{ \begin{gathered} x - [x] - \tfrac{1} {2},if x is not an integer; \hfill \\ 0,if x is an integer. \hfill \\ \end{gathered} \right. $ The Knopp identities for the classical and the homogeneous Dedekind sum were the following: $$\sum\limits_{d|n} {\sum\limits_{r = 1}^d {s\left( {{n \over d}a + rq,dq} \right) = \sigma (n)s(a,q),} } $$ $$\sum\limits_{d|n} {\sum\limits_{{r_1} = 1}^d {\sum\limits_{{r_2} = 1}^d s \left( {{n \over d}a + {r_1}q,{n \over d}b + {r_2}q,dq} \right) = n\sigma (n)s(a,b,q),} } $$ where σ(n) =Σ _d|n d. In this paper generalized homogeneous Hardy sums and Cochrane-Hardy sums are defined, and their arithmetic properties are studied. Generalized Knopp identities for homogeneous Hardy sums and Cochrane-Hardy sums are given.     

Density theorems and the mean value of arithmetical functions in short intervals

Let Γ=SL₂(?) and let Z_Γ(s) be the Selberg zeta function. Set π_Γ(P)=∑_N(P)≤P1, where P is a primitive hyperbolic class of conjugate elements in Γ and N(P) is the norm of P. It is shown that for P^1/2+θ=Q, 1≤θ≤1/2, we have $\pi _\Gamma \left( {P + Q} \right) - \pi _\Gamma \left( P \right) = \int\limits_P^{P + Q} {\frac{{du}}{{\log u}} + O_ \in \left( {QP^{ - \sigma \left( \theta \right) + \in } } \right),} $ where σ(θ)=θ²/2+O(¸₃), θ→0. Thus, a conjecture of Iwaniec (1984) is proved. Similar asymptotic formulas are obtained for the sums ∑_{Ph(-d)/ $\sqrt d $ and ∑Pr3(n)/ $\sqrt n $ , where h(?d) is the class number of the imaginary quadratic field of discriminant ?d<0 and r3(n) is the number of representations of n by the sum of three squares.}     

On the Generation of Certain Bundles of the Projective Space

Extending a result of Manivel, we prove the following: THEOREM. Suppose $\sum\limits_i {b_i \geqslant } \sum\limits_i {a_i } + n$ and $$\sum\limits_i {b_i } [n + d_i d_i - 1] \geqslant \sum\limits_i {a_i } [n + l_i l_i - 1] + n.$$ Then the kernel E(d) of the general morphism: $$\mathop \oplus \limits_{i = 1}^v (Bi \otimes O_{P^n } (d_i )) \to \mathop \oplus \limits_{j = 1}^v (A_j \otimes O_{P^n } (l_i ))$$ (l ₁>...>l _s>d ₁>...>d _v) is a globally generated vector bundle, except for at most finitely many sets $\left\{ {b_i ,a_i } \right\}$ .     

Relative average widths of Sobolev spaces in L 2(ℝ d )

In this paper, in order to consider the problems of relative width on ? ^d , we proposed definitions of relative average width which combine the ideas of the relative width and the average width. We established the smallest number M which make the following equality $$ \overline K _\sigma (U(W_2^\alpha ),M(W_2^\alpha ),L_2 ({\mathbb{R}}^d )) = \overline d _\sigma (U(W_2^\alpha ),L_2 ({\mathbb{R}}^d )) $$ hold, where U(W ₂ ^α ) is the Riesz potential or Bessel potential of the unit ball in L ²(? ^k ) and the notations $ \overline K _\sigma $ (·, ·,L ₂(? ^d )) and $ \overline d _\sigma $ (·, L ₂(? ^d )) denote respectively the relative average width in the sense of Kolmogorov and the average width in the sense of Kolmogorov in their given order. In 2001, Subbotin and Telyakovskii got similar results on the relative width of Kolmogorov type. We also proved that $$ \overline K _\sigma (U(W_2^\alpha ) \cap B(L_2 (\mathbb{R}^d )),U(W_2^\beta ) \cap B(L_2 (\mathbb{R}^d ))L_2 (\mathbb{R}^d )) = \overline d _\sigma (U(W_2^\alpha ),L_2 (\mathbb{R}^d )), $$ where 0 × β × α.     

Permanent Versus Determinant over a Finite Field

Let $ \mathbb{F} $ be a finite field of characteristic different from 2. We study the cardinality of sets of matrices with a given determinant or a given permanent for the set of Hermitian matrices $ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $ and for the whole matrix space M _n ( $ \mathbb{F} $ ). It is known that for n = 2, there are bijective linear maps Φ on $ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $ and M _n ( $ \mathbb{F} $ ) satisfying the condition per A = det Φ(A). As an application of the obtained results, we show that if n ≥ 3, then the situation is completely different and already for n = 3, there is no pair of maps (Φ, ?), where Φ is an arbitrary bijective map on matrices and ? : $ \mathbb{F} $ → $ \mathbb{F} $ is an arbitrary map such that per A = ?(det Φ(A)) for all matrices A from the spaces $ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $ and M _n ( $ \mathbb{F} $ ), respectively. Moreover, for the space M _n ( $ \mathbb{F} $ ), we show that such a pair of transformations does not exist also for an arbitrary n > 3 if the field $ \mathbb{F} $ contains sufficiently many elements (depending on n). Our results are illustrated by a number of examples.     

Quantum Diffusion and Delocalization for Band Matrices with General Distribution

We consider Hermitian and symmetric random band matrices H in d \geqslant 1{d \geqslant 1} dimensions. The matrix elements H _xy , indexed by x,y ? L ì \mathbbZ^d{x,y \in \Lambda \subset \mathbb{Z}^d}, are independent and their variances satisfy s_xy²:=\mathbbE |H_xy|² = W^-d f((x - y)/W){\sigma_{xy}^2:=\mathbb{E} |{H_{xy}}|^2 = W^{-d} f((x - y)/W)} for some probability density f. We assume that the law of each matrix element H _xy is symmetric and exhibits subexponential decay. We prove that the time evolution of a quantum particle subject to the Hamiltonian H is diffusive on time scales t << W^d/3{t\ll W^{d/3}} . We also show that the localization length of the eigenvectors of H is larger than a factor W^d/6{W^{d/6}} times the band width W. All results are uniform in the size |Λ| of the matrix. This extends our recent result (Erdős and Knowles in Commun. Math. Phys., 2011) to general band matrices. As another consequence of our proof we show that, for a larger class of random matrices satisfying ?_xs_xy²=1{\sum_x\sigma_{xy}^2=1} for all y, the largest eigenvalue of H is bounded with high probability by 2 + M^{-2/3 + e}{2 + M^{-2/3 + \varepsilon}} for any ${\varepsilon > 0}${\varepsilon > 0}, where M : = 1 / (max_x,ys_xy²){M := 1 / (\max_{x,y}\sigma_{xy}^2)} .     

Mixing via the extended family

In this paper,the relationship between the extended family and several mixing properties in measuretheoretical dynamical systems is investigated.The extended family eF related to a given family F can be regarded as the collection of all sets obtained as"piecewise shifted"members of F.For a measure preserving transformation T on a Lebesgue space(X,B,μ),the sets of"accurate intersections of order k"defined below are studied,Nε(A0,A1,...,Ak)=n∈Z+:μk i=0T inAiμ(A0)μ(A1)μ(Ak)ε,for k∈N,A0,A1,...,Ak∈B and ε0.It is shown that if T is weakly mixing(mildly mixing)then for any k∈N,all the sets Nε(A0,A1,...,Ak)have Banach density 1(are in(eFip),i.e.,the dual of the extended family related to IP-sets).     

Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group

We study Sobolev-type metrics of fractional order s ≥ 0 on the group Diff _c (M) of compactly supported diffeomorphisms of a manifold M. We show that for the important special case M = S ¹, the geodesic distance on Diff _c (S ¹) vanishes if and only if ${s\leq\frac12}$ . For other manifolds, we obtain a partial characterization: the geodesic distance on Diff _c (M) vanishes for ${M=\mathbb{R}\times N, s < \frac12}$ and for ${M=S^1\times N, s\leq\frac12}$ , with N being a compact Riemannian manifold. On the other hand, the geodesic distance on Diff _c (M) is positive for ${{\rm dim}(M)=1, s > \frac12}$ and dim(M) ≥ 2, s ≥ 1. For ${M=\mathbb{R}^n}$ , we discuss the geodesic equations for these metrics. For n = 1, we obtain some well-known PDEs of hydrodynamics: Burgers’ equation for s = 0, the modified Constantin–Lax–Majda equation for ${s=\frac12}$ , and the Camassa–Holm equation for s = 1.