A new characterization for some linear groups

Let G be a group and π _e (G) be the set of element orders of G. Let k ? p_e(G){k\in\pi_e(G)} and m _k be the number of elements of order k in G. Let nse(G) = {m_k|k ? p_e(G)}{{\rm nse}(G) = \{m_k|k\in\pi_e(G)\}} . In Shen et al. (Monatsh Math, 2009), the authors proved that A₄ @ PSL(2, 3), A₅ @ PSL(2, 4) @ PSL(2,5){A_4\cong {\rm PSL}(2, 3), A_5\cong \rm{PSL}(2, 4)\cong \rm{PSL}(2,5)} and A₆ @ PSL(2,9){A_6\cong \rm{PSL}(2,9)} are uniquely determined by nse(G). In this paper, we prove that if G is a group such that nse(G) = nse(PSL(2, q)), where q ? {7,8,11,13}{q\in\{7,8,11,13\}} , then G @ PSL(2,q){G\cong {PSL}(2,q)} .     

Some Poincaré-type inequalities for functions of bounded deformation involving the deviatoric part of the symmetric gradient

It is proved that if Ω ⊂ Rⁿ {R^n}  is a bounded Lipschitz domain, then the inequality || u ||₁ \leqslant c(n)\textdiam( W)ò_W | e^D(u) | {\left\| u \right\|_1} \leqslant c(n){\text{diam}}\left( \Omega \right)\int\limits_\Omega {\left| {{\varepsilon^D}(u)} \right|} is valid for functions of bounded deformation vanishing on ∂Ω. Here e^D(u) {\varepsilon^D}(u) denotes the deviatoric part of the symmetric gradient and ò_W | e^D(u) | \int\limits_\Omega {\left| {{\varepsilon^D}(u)} \right|} stands for the total variation of the tensor-valued measure e^D(u) {\varepsilon^D}(u) . Further results concern possible extensions of this Poincaré-type inequality. Bibliography: 27 titles.     

Connected Domination Number of a Graph and its Complement

A set S of vertices in a graph G is a connected dominating set if every vertex not in S is adjacent to some vertex in S and the subgraph induced by S is connected. The connected domination number γ _c (G) is the minimum size of such a set. Let d^*(G)=min{d(G),d([`(G)])}{\delta^*(G)={\rm min}\{\delta(G),\delta({\overline{G}})\}} , where [`(G)]{{\overline{G}}} is the complement of G and δ(G) is the minimum vertex degree. We prove that when G and [`(G)]{{\overline{G}}} are both connected, g<sub>c</sub>(G)+g<sub>c</sub>([`(G)]) ￡ d^*(G)+4-(g_c(G)-3)(g_c([`(G)])-3){{\gamma_c}(G)+{\gamma_c}({\overline{G}})\le \delta^*(G)+4-({\gamma_c}(G)-3)({\gamma_c}({\overline{G}})-3)} . As a corollary, g<sub>c</sub>(G)+g<sub>c</sub>([`(G)]) ￡ \frac3n4{{\gamma_c}(G)+{\gamma_c}({\overline{G}})\le \frac{3n}{4}} when δ*(G) ≥ 3 and n ≥ 14, where G has n vertices. We also prove that g_c(G)+g_c([`(G)]) ￡ d<sup>*</sup>(G)+2{{\gamma_c}(G)+{\gamma_c}({\overline{G}})\le \delta^*(G)+2} when g<sub>c</sub>(G),g<sub>c</sub>([`(G)]) 3 4{{\gamma_c}(G),{\gamma_c}({\overline{G}})\ge 4} . This bound is sharp when δ*(G) = 6, and equality can only hold when δ*(G) = 6. Finally, we prove that g_c(G)g_c([`(G)]) ￡ 2n-4{{\gamma_c}(G){\gamma_c}({\overline{G}})\le 2n-4} when n ≥ 7, with equality only for paths and cycles.     

Vertex-Distinguishing Edge Colorings of Graphs with Degree Sum Conditions

An edge coloring is called vertex-distinguishing if every two distinct vertices are incident to different sets of colored edges. The minimum number of colors required for a vertex-distinguishing proper edge coloring of a simple graph G is denoted by c￠_vd(G){\chi'_{vd}(G)}. It is proved that c￠_vd(G) ￡ D(G)+5{\chi'_{vd}(G)\leq\Delta(G)+5} if G is a connected graph of order n ≥ 3 and s₂(G) 3 \frac2n3{\sigma_{2}(G)\geq\frac{2n}{3}}, where σ ₂(G) denotes the minimum degree sum of two nonadjacent vertices in G.     

A New Regularity Criterion in Terms of the Direction of the Velocity for the MHD Equations

In this paper we obtain a new regularity criterion for weak solutions to the 3D MHD equations. It is proved that if div( \fracu|u|) \mathrm{div}( \frac{u}{|u|}) belongs to L^\frac21-r( 0,T;[(X)\dot]_r( \mathbbR³) ) L^{\frac{2}{1-r}}( 0,T;\dot{X}_{r}( \mathbb{R}^{3}) ) with 0≤r≤1, then the weak solution actually is regular and unique.     

Simple exceptional groups of Lie type are determined by their character degrees

Let G be a finite group. Denote by Irr(G) the set of all irreducible complex characters of G. Let cd(G)={c(1)  |  c ? Irr(G)}{{\rm cd}(G)=\{\chi(1)\;|\;\chi\in {\rm Irr}(G)\}} be the set of all irreducible complex character degrees of G forgetting multiplicities, and let X₁(G) be the set of all irreducible complex character degrees of G counting multiplicities. Let H be any non-abelian simple exceptional group of Lie type. In this paper, we will show that if S is a non-abelian simple group and cd(S) í cd(H){{\rm cd}(S)\subseteq {\rm cd}(H)} then S must be isomorphic to H. As a consequence, we show that if G is a finite group with X₁(G) í X₁(H){{\rm X}_1(G)\subseteq {\rm X}_1(H)} then G is isomorphic to H. In particular, this implies that the simple exceptional groups of Lie type are uniquely determined by the structure of their complex group algebras.     

Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential

We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ${u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1}We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation u_t + Lu + a(x) |u|^q-1u=0, 0 < q < 1{u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1} with a(x) ≥ 0 bounded in the bounded domain W ì \mathbb R^N{\Omega \subset \mathbb R^N}. We prove that if N 1 2m{N \ne 2m} and ò₀¹ s^-1 (meas\nolimits {x ? W: |a(x)| ￡ s })^q ds < ￥, q = min(\frac2mN,1){\int_0^1 s^{-1} (\mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \})^\theta {\rm d}s < \infty,\ \theta=\min\left(\frac{2m}N,1\right)}, then the solution u vanishes in a finite time. When N = 2m, the same property holds if ${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}.     

The Cycle Discrepancy of Three-Regular Graphs

Let G = (V, E) be an undirected graph and C(G){{\mathcal C}(G)} denote the set of all cycles in G. We introduce a graph invariant cycle discrepancy, which we define as

Solution of Initial Value Problems with Monogenic Initial Functions in Banach Spaces with L p -Norm

This paper deals with the initial value problem of the type

On the Uniform Convergence of the Generalized Bieberbach Polynomials in Regions with K-Quasiconformal Boundary

Let G be a finite domain in the complex plane with K-quasicon formal boundary, z ₀ be an arbitrary fixed point in G and p>0. Let j_p ( z ): = ò_x0 ^x [ f( z) ]^2/8 dz\varphi _p \left( z \right): = \int_{x_0 }^x {\left[ {\phi \left( \zeta \right)} \right]^{2/8} } d\zeta , and let \iint_c | j_p ( z ) - P_x¹ (z) |^p d0_x \iint\limits_c {\left| {\varphi _p \left( z \right) - P_x^1 (z)} \right|^p d0_x } in the class \mathop ?_n \mathop \prod \limits_n of all polynomials of degree [`(G)]\bar G in case of $p > 2 - \frac{{K^2 + 1}}{{2K^4 }}$p > 2 - \frac{{K^2 + 1}}{{2K^4 }} .     

Gaussian upper bounds for fundamental solutions of a family of parabolic equations

We study the family of divergence-type second-order parabolic equations w_e(x)\frac?u?t=div(a(x)w_e(x) ?u), x ? \mathbbRⁿ{\omega_\varepsilon(x)\frac{\partial u}{\partial t}={\rm div}(a(x)\omega_\varepsilon(x) \nabla u), x \in \mathbb{R}^n} , with parameter ${\varepsilon >0 }${\varepsilon >0 } , where a(x) is uniformly elliptic matrix and w_e=1{\omega_\varepsilon=1} for x _n  < 0 and w_e=e{\omega_\varepsilon=\varepsilon} for x _n  > 0. We show that the fundamental solution obeys the Gaussian upper bound uniformly with respect to e{\varepsilon} .     

Multiplicity of solutions for a fourth order equation with power-type nonlinearity

Let B be the unit ball in ${\mathbb{R}^N}Let B be the unit ball in \mathbbR^N{\mathbb{R}^N}, N ≥ 3 and n be the exterior unit normal vector on the boundary. We consider radial solutions to

D2 u = l(1+ sign(p)u)p     in  B,     u = 0,     \frac?u?n = 0     on  ?B\Delta^2 u = \lambda(1+ {\rm sign}(p)u)^{p} \quad {\rm in} \, B, \quad u = 0, \quad \frac{\partial{u}}{\partial{n}} = 0 \quad {\rm on} \, \partial B      13. Arithmetic mean of differences of Dedekind sums      Emre Alkan  Maosheng Xiong  Alexandru Zaharescu 《Monatshefte für Mathematik》2007,342(1):175-187 Recently, Girstmair and Schoissengeier studied the asymptotic behavior of the arithmetic mean of Dedekind sums \frac1j(N) ? 0 ￡ m < Ngcd(m,N)=1 |S(m,N)|\frac{1}{\varphi(N)} \sum_{\mathop{\mathop{ 0 \le m< N}}\limits_{\gcd(m,N)=1}} \vert S(m,N)\vert , as N → ∞. In this paper we consider the arithmetic mean of weighted differences of Dedekind sums in the form Ah(Q)=\frac1?\fracaq ? FQh(\fracaq) ×?\fracaq ? FQh(\fracaq) |s(a￠,q￠)-s(a,q)|A_{h}(Q)=\frac{1}{\sum_{\frac{a}{q} \in {\cal F}_{Q}}h\left(\frac{a}{q}\right)} \times \sum_{\frac{a}{q} \in {\cal F}_{\!Q}}h\left(\frac{a}{q}\right) \vert s(a^{\prime},q^{\prime})-s(a,q)\vert , where h:[0,1] ? \Bbb Ch:[0,1] \rightarrow {\Bbb C} is a continuous function with ò01 h(t)  d t 1 0\int_0^1 h(t) \, {\rm d} t \ne 0 , \fracaq{\frac{a}{q}} runs over FQ{\cal F}_{\!Q} , the set of Farey fractions of order Q in the unit interval [0,1] and \fracaq < \fraca￠q￠{\frac{a}{q}}<\frac{a^{\prime}}{q^{\prime}} are consecutive elements of FQ{\cal F}_{\!Q} . We show that the limit lim Q→∞ A h (Q) exists and is independent of h.      14. Cleaning Random d-Regular Graphs with Brooms      Pawe? Pra?at 《Graphs and Combinatorics》2011,27(4):567-584 A model for cleaning a graph with brushes was recently introduced. Let α = (v 1, v 2, . . . , v n ) be a permutation of the vertices of G; for each vertex v i let ${N^+(v_i)=\{j: v_j v_i \in E {\rm and} j>\,i\}}${N^+(v_i)=\{j: v_j v_i \in E {\rm and} j>\,i\}} and N-(vi)={j: vj vi ? E and j <  i}{N^-(v_i)=\{j: v_j v_i \in E {\rm and} j<\,i\}} ; finally let ba(G)=?i=1n max{|N+(vi)|-|N-(vi)|,0}{b_{\alpha}(G)=\sum_{i=1}^n {\rm max}\{|N^+(v_i)|-|N^-(v_i)|,0\}}. The Broom number is given by B(G) =  max α b α (G). We consider the Broom number of d-regular graphs, focusing on the asymptotic number for random d-regular graphs. Various lower and upper bounds are proposed. To get an asymptotically almost sure lower bound we use a degree-greedy algorithm to clean a random d-regular graph on n vertices (with dn even) and analyze it using the differential equations method (for fixed d). We further show that for any d-regular graph on n vertices there is a cleaning sequence such at least n(d + 1)/4 brushes are needed to clean a graph using this sequence. For an asymptotically almost sure upper bound, the pairing model is used to show that at most n(d+2?{d ln2})/4{n(d+2\sqrt{d \ln 2})/4} brushes can be used when a random d-regular graph is cleaned. This implies that for fixed large d, the Broom number of a random d-regular graph on n vertices is asymptotically almost surely \fracn4(d+Q(?d)){\frac{n}{4}(d+\Theta(\sqrt{d}))}.      15. Recurrent approach to Blaschke’s problem      E. Gečiauskas 《Geometriae Dedicata》2006,121(1):9-18 We have obtained a recurrence formula $I_{n+3} = \frac{4(n+3)}{\pi(n+4)}VI_{n+1}We have obtained a recurrence formula In+3 = \frac4(n+3)p(n+4)VIn+1I_{n+3} = \frac{4(n+3)}{\pi(n+4)}VI_{n+1} for integro-geometric moments in the case of a circle with the area V, where In = ò\nolimitsK ?Gsnd GI_n = \int \nolimits_{K \cap G}\sigma^{n}{\rm d} G, while in the case of a convex domain K with the perimeter S and area V the recurrence formula In+3 = \frac8(n+3)V2(n+1)(n+4)p[\fracd In+1d V - \fracIn+1S \fracd Sd V ] I_{n+3} = \frac{8(n+3)V^2}{(n+1)(n+4)\pi}\Big[\frac{{\rm d} I_{n+1}}{{\rm d} V} - \frac{I_{n+1}}{S} \frac{{\rm d} S}{{\rm d} V} \Big]      16. Asymptotic behaviour for a nonlocal diffusion equation on a lattice      Liviu I. Ignat  Julio D. Rossi 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2008,170(1):918-925 In this paper we study the asymptotic behaviour as t → ∞ of solutions to a nonlocal diffusion problem on a lattice, namely, u￠n(t) = ?j ? \mathbbZd Jn-juj(t)-un(t)u^{\prime}_{n}(t) = \sum_{{j\in}{{{\mathbb{Z}}}^{d}}} J_{n-j}u_{j}(t)-u_{n}(t) with t ≥ 0 and n ? \mathbbZdn \in {\mathbb{Z}}^{d}. We assume that J is nonnegative and verifies ?n ? \mathbbZdJn = 1\sum_{{n \in {\mathbb{Z}}}^{d}}J_{n}= 1. We find that solutions decay to zero as t → ∞ and prove an optimal decay rate using, as our main tool, the discrete Fourier transform.      17. On the Euler–Lagrange equation for a variational problem: the general case II      Stefano Bianchini  Matteo Gloyer 《Mathematische Zeitschrift》2010,265(4):889-923 In this paper we study the existence of a solution in ${L^\infty_{\rm loc}(\Omega)}In this paper we study the existence of a solution in L￥loc(W){L^\infty_{\rm loc}(\Omega)} to the Euler–Lagrange equation for the variational problem inf[`(u)] + W1,￥0(W) òW (ID(?u) + g(u)) dx,                   (0.1)\inf_{\bar u + W^{1,\infty}_0(\Omega)} \int\limits_{\Omega} ({\bf I}_D(\nabla u) + g(u)) dx,\quad \quad \quad \quad \quad(0.1)      18. Pointwise multipliers of Orlicz spaces      Lech Maligranda  Eiichi Nakai 《Archiv der Mathematik》2010,95(3):251-256 We show that the result on multipliers of Orlicz spaces holds in general. Namely, under the assumption that three Young functions Φ1, Φ2 and Φ, generating corresponding Orlicz spaces, satisfy the estimate ${\Phi^{-1}(u) \leq C \Phi_1^{-1}(u)\, \Phi_2^{-1}(u)}We show that the result on multipliers of Orlicz spaces holds in general. Namely, under the assumption that three Young functions Φ1, Φ2 and Φ, generating corresponding Orlicz spaces, satisfy the estimate F-1(u) ￡ C F1-1(u) F2-1(u){\Phi^{-1}(u) \leq C \Phi_1^{-1}(u)\, \Phi_2^{-1}(u)} for all u > 0, we prove that if the pointwise product xy belongs to L Φ(μ) for all y ? LF1(m){y \in L^{\Phi_1}(\mu)}, then x ? LF2(m){x \in L^{\Phi_2}(\mu)}. The result with some restrictions either on Young functions or on the measure μ was proved by Maligranda and Persson (Indag. Math. 51 (1989), 323–338). Our result holds for any collection of three Young functions satisfying the above estimate and for an arbitrary complete σ-finite measure μ.      19. Probability measures on [SIN] groups and some related ideals in group algebras      Wojciech Jaworski 《Monatshefte für Mathematik》2008,336(4):135-144 Given a locally compact group G, let J(G){\cal J}(G) denote the set of closed left ideals in L 1(G), of the form J μ = [L1(G) * (δ e − μ)]−, where μ is a probability measure on G. Let Jd(G)={\cal J}_d(G)= {Jm;m is discrete}\{J_{\mu};\mu\ {\rm is discrete}\} , Ja(G)={Jm;m is absolutely continuous}{\cal J}_a(G)=\{J_{\mu};\mu\ {\rm is absolutely continuous}\} . When G is a second countable [SIN] group, we prove that J(G)=Jd(G){\cal J}(G)={\cal J}_d(G) and that Ja(G){\cal J}_a(G) , being a proper subset of J(G){\cal J}(G) when G is nondiscrete, contains every maximal element of J(G){\cal J}(G) . Some results concerning the ideals J μ in general locally compact second countable groups are also obtained.      20. Properties of Riesz fractional derivatives for Korteweg–de Vries solitons      Vladimir Varlamov 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,138(6):1017-1031 Riesz fractional derivatives are defined as fractional powers of the Laplacian, D α  = (?Δ) α/2 for ${\alpha \in \mathbb{R}}Riesz fractional derivatives are defined as fractional powers of the Laplacian, D α  = (−Δ) α/2 for a ? \mathbbR{\alpha \in \mathbb{R}}. For the soliton solution of the Korteweg–de Vries equation, u 0(X) with X = x − 4t, these derivatives, u α (X) = D α u 0(X), and their Hilbert transforms, v α (X) = −HD α u 0(X), can be expressed in terms of the full range Hurwitz Zeta functions ζ+(s, a) and ζ−(s, a), respectively. New properties are established for u α (X) and v α (X). It is proved that the functions w α (X) = u α (X) + iv α (X) with α > −1 are solutions of the differential equation -\fracddX(Pa(X)\fracdwdX)+Qa(X)w = lra(X)w,       X ? \mathbbR,-\frac{\rm d}{{\rm d}X}\left(P_{\alpha}(X)\frac{{\rm d}w}{{\rm d}X}\right)+Q_{\alpha}(X)w = \lambda\rho_{\alpha}(X)w,\qquad X \in \mathbb{R},      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Arithmetic mean of differences of Dedekind sums

Recently, Girstmair and Schoissengeier studied the asymptotic behavior of the arithmetic mean of Dedekind sums \frac1j(N) ? _{0 ￡ m < Ngcd(m,N)=1} |S(m,N)|\frac{1}{\varphi(N)} \sum_{\mathop{\mathop{ 0 \le m< N}}\limits_{\gcd(m,N)=1}} \vert S(m,N)\vert , as N → ∞. In this paper we consider the arithmetic mean of weighted differences of Dedekind sums in the form A_h(Q)=\frac1?_{\fracaq ? FQ}h(\fracaq) ×?_{\fracaq ? FQ}h(\fracaq) |s(a^￠,q^￠)-s(a,q)|A_{h}(Q)=\frac{1}{\sum_{\frac{a}{q} \in {\cal F}_{Q}}h\left(\frac{a}{q}\right)} \times \sum_{\frac{a}{q} \in {\cal F}_{\!Q}}h\left(\frac{a}{q}\right) \vert s(a^{\prime},q^{\prime})-s(a,q)\vert , where h:[0,1] ? \Bbb Ch:[0,1] \rightarrow {\Bbb C} is a continuous function with ò₀¹ h(t)  d t 1 0\int_0^1 h(t) \, {\rm d} t \ne 0 , \fracaq{\frac{a}{q}} runs over F_Q{\cal F}_{\!Q} , the set of Farey fractions of order Q in the unit interval [0,1] and \fracaq < \fraca^￠q^￠{\frac{a}{q}}<\frac{a^{\prime}}{q^{\prime}} are consecutive elements of F_Q{\cal F}_{\!Q} . We show that the limit lim _Q→∞ A _h (Q) exists and is independent of h.     

Cleaning Random d-Regular Graphs with Brooms

A model for cleaning a graph with brushes was recently introduced. Let α = (v ₁, v ₂, . . . , v _n ) be a permutation of the vertices of G; for each vertex v _i let ${N^+(v_i)=\{j: v_j v_i \in E {\rm and} j>\,i\}}${N^+(v_i)=\{j: v_j v_i \in E {\rm and} j>\,i\}} and N^-(v_i)={j: v_j v_i ? E and j <  i}{N^-(v_i)=\{j: v_j v_i \in E {\rm and} j<\,i\}} ; finally let b_a(G)=?_i=1ⁿ max{|N⁺(v_i)|-|N^-(v_i)|,0}{b_{\alpha}(G)=\sum_{i=1}^n {\rm max}\{|N^+(v_i)|-|N^-(v_i)|,0\}}. The Broom number is given by B(G) =  max _α b _α (G). We consider the Broom number of d-regular graphs, focusing on the asymptotic number for random d-regular graphs. Various lower and upper bounds are proposed. To get an asymptotically almost sure lower bound we use a degree-greedy algorithm to clean a random d-regular graph on n vertices (with dn even) and analyze it using the differential equations method (for fixed d). We further show that for any d-regular graph on n vertices there is a cleaning sequence such at least n(d + 1)/4 brushes are needed to clean a graph using this sequence. For an asymptotically almost sure upper bound, the pairing model is used to show that at most n(d+2?{d ln2})/4{n(d+2\sqrt{d \ln 2})/4} brushes can be used when a random d-regular graph is cleaned. This implies that for fixed large d, the Broom number of a random d-regular graph on n vertices is asymptotically almost surely \fracn4(d+Q(?d)){\frac{n}{4}(d+\Theta(\sqrt{d}))}.     

Recurrent approach to Blaschke’s problem

We have obtained a recurrence formula $I_{n+3} = \frac{4(n+3)}{\pi(n+4)}VI_{n+1}We have obtained a recurrence formula I_n+3 = \frac4(n+3)p(n+4)VI_n+1I_{n+3} = \frac{4(n+3)}{\pi(n+4)}VI_{n+1} for integro-geometric moments in the case of a circle with the area V, where I_n = ò\nolimits_{K ?G}sⁿd GI_n = \int \nolimits_{K \cap G}\sigma^{n}{\rm d} G, while in the case of a convex domain K with the perimeter S and area V the recurrence formula

In+3 = \frac8(n+3)V2(n+1)(n+4)p[\fracd In+1d V - \fracIn+1S \fracd Sd V ] I_{n+3} = \frac{8(n+3)V^2}{(n+1)(n+4)\pi}\Big[\frac{{\rm d} I_{n+1}}{{\rm d} V} - \frac{I_{n+1}}{S} \frac{{\rm d} S}{{\rm d} V} \Big]      16. Asymptotic behaviour for a nonlocal diffusion equation on a lattice      Liviu I. Ignat  Julio D. Rossi 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2008,170(1):918-925 In this paper we study the asymptotic behaviour as t → ∞ of solutions to a nonlocal diffusion problem on a lattice, namely, u￠n(t) = ?j ? \mathbbZd Jn-juj(t)-un(t)u^{\prime}_{n}(t) = \sum_{{j\in}{{{\mathbb{Z}}}^{d}}} J_{n-j}u_{j}(t)-u_{n}(t) with t ≥ 0 and n ? \mathbbZdn \in {\mathbb{Z}}^{d}. We assume that J is nonnegative and verifies ?n ? \mathbbZdJn = 1\sum_{{n \in {\mathbb{Z}}}^{d}}J_{n}= 1. We find that solutions decay to zero as t → ∞ and prove an optimal decay rate using, as our main tool, the discrete Fourier transform.      17. On the Euler–Lagrange equation for a variational problem: the general case II      Stefano Bianchini  Matteo Gloyer 《Mathematische Zeitschrift》2010,265(4):889-923 In this paper we study the existence of a solution in ${L^\infty_{\rm loc}(\Omega)}In this paper we study the existence of a solution in L￥loc(W){L^\infty_{\rm loc}(\Omega)} to the Euler–Lagrange equation for the variational problem inf[`(u)] + W1,￥0(W) òW (ID(?u) + g(u)) dx,                   (0.1)\inf_{\bar u + W^{1,\infty}_0(\Omega)} \int\limits_{\Omega} ({\bf I}_D(\nabla u) + g(u)) dx,\quad \quad \quad \quad \quad(0.1)      18. Pointwise multipliers of Orlicz spaces      Lech Maligranda  Eiichi Nakai 《Archiv der Mathematik》2010,95(3):251-256 We show that the result on multipliers of Orlicz spaces holds in general. Namely, under the assumption that three Young functions Φ1, Φ2 and Φ, generating corresponding Orlicz spaces, satisfy the estimate ${\Phi^{-1}(u) \leq C \Phi_1^{-1}(u)\, \Phi_2^{-1}(u)}We show that the result on multipliers of Orlicz spaces holds in general. Namely, under the assumption that three Young functions Φ1, Φ2 and Φ, generating corresponding Orlicz spaces, satisfy the estimate F-1(u) ￡ C F1-1(u) F2-1(u){\Phi^{-1}(u) \leq C \Phi_1^{-1}(u)\, \Phi_2^{-1}(u)} for all u > 0, we prove that if the pointwise product xy belongs to L Φ(μ) for all y ? LF1(m){y \in L^{\Phi_1}(\mu)}, then x ? LF2(m){x \in L^{\Phi_2}(\mu)}. The result with some restrictions either on Young functions or on the measure μ was proved by Maligranda and Persson (Indag. Math. 51 (1989), 323–338). Our result holds for any collection of three Young functions satisfying the above estimate and for an arbitrary complete σ-finite measure μ.      19. Probability measures on [SIN] groups and some related ideals in group algebras      Wojciech Jaworski 《Monatshefte für Mathematik》2008,336(4):135-144 Given a locally compact group G, let J(G){\cal J}(G) denote the set of closed left ideals in L 1(G), of the form J μ = [L1(G) * (δ e − μ)]−, where μ is a probability measure on G. Let Jd(G)={\cal J}_d(G)= {Jm;m is discrete}\{J_{\mu};\mu\ {\rm is discrete}\} , Ja(G)={Jm;m is absolutely continuous}{\cal J}_a(G)=\{J_{\mu};\mu\ {\rm is absolutely continuous}\} . When G is a second countable [SIN] group, we prove that J(G)=Jd(G){\cal J}(G)={\cal J}_d(G) and that Ja(G){\cal J}_a(G) , being a proper subset of J(G){\cal J}(G) when G is nondiscrete, contains every maximal element of J(G){\cal J}(G) . Some results concerning the ideals J μ in general locally compact second countable groups are also obtained.      20. Properties of Riesz fractional derivatives for Korteweg–de Vries solitons      Vladimir Varlamov 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,138(6):1017-1031 Riesz fractional derivatives are defined as fractional powers of the Laplacian, D α  = (?Δ) α/2 for ${\alpha \in \mathbb{R}}Riesz fractional derivatives are defined as fractional powers of the Laplacian, D α  = (−Δ) α/2 for a ? \mathbbR{\alpha \in \mathbb{R}}. For the soliton solution of the Korteweg–de Vries equation, u 0(X) with X = x − 4t, these derivatives, u α (X) = D α u 0(X), and their Hilbert transforms, v α (X) = −HD α u 0(X), can be expressed in terms of the full range Hurwitz Zeta functions ζ+(s, a) and ζ−(s, a), respectively. New properties are established for u α (X) and v α (X). It is proved that the functions w α (X) = u α (X) + iv α (X) with α > −1 are solutions of the differential equation -\fracddX(Pa(X)\fracdwdX)+Qa(X)w = lra(X)w,       X ? \mathbbR,-\frac{\rm d}{{\rm d}X}\left(P_{\alpha}(X)\frac{{\rm d}w}{{\rm d}X}\right)+Q_{\alpha}(X)w = \lambda\rho_{\alpha}(X)w,\qquad X \in \mathbb{R},      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Asymptotic behaviour for a nonlocal diffusion equation on a lattice

In this paper we study the asymptotic behaviour as t → ∞ of solutions to a nonlocal diffusion problem on a lattice, namely, u^￠_n(t) = ?_{j ? \mathbbZ^d} J_n-ju_j(t)-u_n(t)u^{\prime}_{n}(t) = \sum_{{j\in}{{{\mathbb{Z}}}^{d}}} J_{n-j}u_{j}(t)-u_{n}(t) with t ≥ 0 and n ? \mathbbZ^dn \in {\mathbb{Z}}^{d}. We assume that J is nonnegative and verifies ?_{n ? \mathbbZ^d}J_n = 1\sum_{{n \in {\mathbb{Z}}}^{d}}J_{n}= 1. We find that solutions decay to zero as t → ∞ and prove an optimal decay rate using, as our main tool, the discrete Fourier transform.     

On the Euler–Lagrange equation for a variational problem: the general case II

In this paper we study the existence of a solution in ${L^\infty_{\rm loc}(\Omega)}In this paper we study the existence of a solution in L^￥_loc(W){L^\infty_{\rm loc}(\Omega)} to the Euler–Lagrange equation for the variational problem

inf[`(u)] + W1,￥0(W) òW (ID(?u) + g(u)) dx,                   (0.1)\inf_{\bar u + W^{1,\infty}_0(\Omega)} \int\limits_{\Omega} ({\bf I}_D(\nabla u) + g(u)) dx,\quad \quad \quad \quad \quad(0.1)      18. Pointwise multipliers of Orlicz spaces      Lech Maligranda  Eiichi Nakai 《Archiv der Mathematik》2010,95(3):251-256 We show that the result on multipliers of Orlicz spaces holds in general. Namely, under the assumption that three Young functions Φ1, Φ2 and Φ, generating corresponding Orlicz spaces, satisfy the estimate ${\Phi^{-1}(u) \leq C \Phi_1^{-1}(u)\, \Phi_2^{-1}(u)}We show that the result on multipliers of Orlicz spaces holds in general. Namely, under the assumption that three Young functions Φ1, Φ2 and Φ, generating corresponding Orlicz spaces, satisfy the estimate F-1(u) ￡ C F1-1(u) F2-1(u){\Phi^{-1}(u) \leq C \Phi_1^{-1}(u)\, \Phi_2^{-1}(u)} for all u > 0, we prove that if the pointwise product xy belongs to L Φ(μ) for all y ? LF1(m){y \in L^{\Phi_1}(\mu)}, then x ? LF2(m){x \in L^{\Phi_2}(\mu)}. The result with some restrictions either on Young functions or on the measure μ was proved by Maligranda and Persson (Indag. Math. 51 (1989), 323–338). Our result holds for any collection of three Young functions satisfying the above estimate and for an arbitrary complete σ-finite measure μ.      19. Probability measures on [SIN] groups and some related ideals in group algebras      Wojciech Jaworski 《Monatshefte für Mathematik》2008,336(4):135-144 Given a locally compact group G, let J(G){\cal J}(G) denote the set of closed left ideals in L 1(G), of the form J μ = [L1(G) * (δ e − μ)]−, where μ is a probability measure on G. Let Jd(G)={\cal J}_d(G)= {Jm;m is discrete}\{J_{\mu};\mu\ {\rm is discrete}\} , Ja(G)={Jm;m is absolutely continuous}{\cal J}_a(G)=\{J_{\mu};\mu\ {\rm is absolutely continuous}\} . When G is a second countable [SIN] group, we prove that J(G)=Jd(G){\cal J}(G)={\cal J}_d(G) and that Ja(G){\cal J}_a(G) , being a proper subset of J(G){\cal J}(G) when G is nondiscrete, contains every maximal element of J(G){\cal J}(G) . Some results concerning the ideals J μ in general locally compact second countable groups are also obtained.      20. Properties of Riesz fractional derivatives for Korteweg–de Vries solitons      Vladimir Varlamov 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,138(6):1017-1031 Riesz fractional derivatives are defined as fractional powers of the Laplacian, D α  = (?Δ) α/2 for ${\alpha \in \mathbb{R}}Riesz fractional derivatives are defined as fractional powers of the Laplacian, D α  = (−Δ) α/2 for a ? \mathbbR{\alpha \in \mathbb{R}}. For the soliton solution of the Korteweg–de Vries equation, u 0(X) with X = x − 4t, these derivatives, u α (X) = D α u 0(X), and their Hilbert transforms, v α (X) = −HD α u 0(X), can be expressed in terms of the full range Hurwitz Zeta functions ζ+(s, a) and ζ−(s, a), respectively. New properties are established for u α (X) and v α (X). It is proved that the functions w α (X) = u α (X) + iv α (X) with α > −1 are solutions of the differential equation -\fracddX(Pa(X)\fracdwdX)+Qa(X)w = lra(X)w,       X ? \mathbbR,-\frac{\rm d}{{\rm d}X}\left(P_{\alpha}(X)\frac{{\rm d}w}{{\rm d}X}\right)+Q_{\alpha}(X)w = \lambda\rho_{\alpha}(X)w,\qquad X \in \mathbb{R},

设为首页 | 免责声明 | 关于勤云 | 加入收藏

Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Pointwise multipliers of Orlicz spaces

We show that the result on multipliers of Orlicz spaces holds in general. Namely, under the assumption that three Young functions Φ₁, Φ₂ and Φ, generating corresponding Orlicz spaces, satisfy the estimate ${\Phi^{-1}(u) \leq C \Phi_1^{-1}(u)\, \Phi_2^{-1}(u)}We show that the result on multipliers of Orlicz spaces holds in general. Namely, under the assumption that three Young functions Φ₁, Φ₂ and Φ, generating corresponding Orlicz spaces, satisfy the estimate F^-1(u) ￡ C F₁^-1(u) F₂^-1(u){\Phi^{-1}(u) \leq C \Phi_1^{-1}(u)\, \Phi_2^{-1}(u)} for all u > 0, we prove that if the pointwise product xy belongs to L ^Φ(μ) for all y ? L^F₁(m){y \in L^{\Phi_1}(\mu)}, then x ? L^F₂(m){x \in L^{\Phi_2}(\mu)}. The result with some restrictions either on Young functions or on the measure μ was proved by Maligranda and Persson (Indag. Math. 51 (1989), 323–338). Our result holds for any collection of three Young functions satisfying the above estimate and for an arbitrary complete σ-finite measure μ.     

Probability measures on [SIN] groups and some related ideals in group algebras

Given a locally compact group G, let J(G){\cal J}(G) denote the set of closed left ideals in L ¹(G), of the form J _μ = [L¹(G) * (δ _e − μ)]⁻, where μ is a probability measure on G. Let J_d(G)={\cal J}_d(G)= {J_m;m is discrete}\{J_{\mu};\mu\ {\rm is discrete}\} , J_a(G)={J_m;m is absolutely continuous}{\cal J}_a(G)=\{J_{\mu};\mu\ {\rm is absolutely continuous}\} . When G is a second countable [SIN] group, we prove that J(G)=J_d(G){\cal J}(G)={\cal J}_d(G) and that J_a(G){\cal J}_a(G) , being a proper subset of J(G){\cal J}(G) when G is nondiscrete, contains every maximal element of J(G){\cal J}(G) . Some results concerning the ideals J _μ in general locally compact second countable groups are also obtained.     

Properties of Riesz fractional derivatives for Korteweg–de Vries solitons

Riesz fractional derivatives are defined as fractional powers of the Laplacian, D ^α  = (?Δ) ^α/2 for ${\alpha \in \mathbb{R}}Riesz fractional derivatives are defined as fractional powers of the Laplacian, D ^α  = (−Δ) ^α/2 for a ? \mathbbR{\alpha \in \mathbb{R}}. For the soliton solution of the Korteweg–de Vries equation, u ₀(X) with X = x − 4t, these derivatives, u _α (X) = D ^α u ₀(X), and their Hilbert transforms, v _α (X) = −HD ^α u ₀(X), can be expressed in terms of the full range Hurwitz Zeta functions ζ₊(s, a) and ζ₋(s, a), respectively. New properties are established for u _α (X) and v _α (X). It is proved that the functions w _α (X) = u _α (X) + iv _α (X) with α > −1 are solutions of the differential equation

-\fracddX(Pa(X)\fracdwdX)+Qa(X)w = lra(X)w,       X ? \mathbbR,-\frac{\rm d}{{\rm d}X}\left(P_{\alpha}(X)\frac{{\rm d}w}{{\rm d}X}\right)+Q_{\alpha}(X)w = \lambda\rho_{\alpha}(X)w,\qquad X \in \mathbb{R},