共查询到20条相似文献,搜索用时 31 毫秒
1.
A. M. Savchuk 《Proceedings of the Steklov Institute of Mathematics》2008,261(1):237-242
In the space L 2[0, π], the Sturm-Liouville operator L D(y) = ?y″ + q(x)y with the Dirichlet boundary conditions y(0) = y(π) = 0 is analyzed. The potential q is assumed to be singular; namely, q = σ′, where σ ∈ L 2[0, π], i.e., q ∈ W 2 ?1 [0, π]. The inverse problem of reconstructing the function σ from the spectrum of the operator L D is solved in the subspace of odd real functions σ(π/2 ? x) = ?σ(π/2 + x). The existence and uniqueness of a solution to this inverse problem is proved. A method is proposed that allows one to solve this problem numerically. 相似文献
2.
Maxim Vsemirnov 《中国科学 数学(英文版)》2018,61(11):2101-2110
Consider the sequence of algebraic integers un given by the starting values u0 = 0, u1 = 1 and the recurrence \(u_{n+1}=(4\rm{cos}^2(2\pi/7)-1)\it{u}_{n}-u_{n-\rm{1}}\). We prove that for any n ? {1, 2, 3, 5, 8, 12, 18, 28, 30} the n-th term of the sequence has a primitive divisor in \(\mathbb{Z}[2\rm{cos}(2\pi/7)]\). As a consequence we deduce that for any suffciently large n there exists a prime power q such that the group PSL2(q) can be generated by a pair x, y with \(x^2=y^3=(xy)^7=1\) and the order of the commutator [x, y] is exactly n. The latter result answers in affrmative a question of Holt and Plesken. 相似文献
3.
Let J be the Lévy density of a symmetric Lévy process in \(\mathbb {R}^{d}\) with its Lévy exponent satisfying a weak lower scaling condition at infinity. Consider the non-symmetric and non-local operatorwhere κ(x, z) is a Borel function on \(\mathbb {R}^{d}\times \mathbb {R}^{d}\) satisfying 0 < κ 0 ≤ κ(x, z) ≤ κ 1, κ(x, z) = κ(x,?z) and |κ(x, z) ? κ(y, z)|≤ κ 2|x ? y| β for some β ∈ (0, 1]. We construct the heat kernel p κ (t, x, y) of \(\mathcal {L}^{\kappa }\), establish its upper bound as well as its fractional derivative and gradient estimates. Under an additional weak upper scaling condition at infinity, we also establish a lower bound for the heat kernel p κ .
相似文献
$$\mathcal{L}^{\kappa}f(x):= \lim_{{\varepsilon} \downarrow 0} {\int}_{\{z \in \mathbb{R}^{d}: |z|>{\varepsilon}\}} (f(x+z)-f(x))\kappa(x,z)J(z)\, dz\, , $$
4.
Quanwu Mu 《Frontiers of Mathematics in China》2017,12(6):1457-1468
Let d ? 3 be an integer, and set r = 2d?1 + 1 for 3 ? d ? 4, \(\tfrac{{17}}{{32}} \cdot 2^d + 1\) for 5 ? d ? 6, r = d2+d+1 for 7 ? d ? 8, and r = d2+d+2 for d ? 9, respectively. Suppose that Φ i (x, y) ∈ ?[x, y] (1 ? i ? r) are homogeneous and nondegenerate binary forms of degree d. Suppose further that λ1, λ2,..., λ r are nonzero real numbers with λ1/λ2 irrational, and λ1Φ1(x1, y1) + λ2Φ2(x2, y2) + · · · + λ r Φ r (x r , y r ) is indefinite. Then for any given real η and σ with 0 < σ < 22?d, it is proved that the inequality has infinitely many solutions in integers x1, x2,..., x r , y1, y2,..., y r . This result constitutes an improvement upon that of B. Q. Xue.
相似文献
$$\left| {\sum\limits_{i = 1}^r {{\lambda _i}\Phi {}_i\left( {{x_i},{y_i}} \right) + \eta } } \right| < {\left( {\mathop {\max \left\{ {\left| {{x_i}} \right|,\left| {{y_i}} \right|} \right\}}\limits_{1 \leqslant i \leqslant r} } \right)^{ - \sigma }}$$
5.
Let g be a linear combination with quasipolynomial coefficients of shifts of the Jacobi theta function and its derivatives in the argument. All entire functions f: ? → ? satisfying f(x+y)g(x?y) = α1(x)β1(y)+· · ·+αr(x)βr(y) for some r ∈ ? and αj, βj: ? → ? are described. 相似文献
6.
A. V. Setukha 《Differential Equations》2018,54(9):1236-1255
We consider a bulk charge potential of the form where Ω is a layer of small thickness h > 0 located around the midsurface Σ, which can be either closed or open, and F(x ? y) is a function with a singularity of the form 1/|x ? y|. We prove that, under certain assumptions on the shape of the surface Σ, the kernel F, and the function g at each point x lying on the midsurface Σ (but not on its boundary), the second derivatives of the function u can be represented as where the function γij(x) does not exceed in absolute value a certain quantity of the order of h2, the surface integral is understood in the sense of Hadamard finite value, and the ni(x), i = 1, 2, 3, are the coordinates of the normal vector on the surface Σ at a point x.
相似文献
$$u(x) = \int\limits_\Omega {g(y)F(x - y)dy,x = ({x_1},{x_2},{x_3}) \in {\mathbb{R}^3},} $$
$$\frac{{{\partial ^2}u(x)}}{{\partial {x_i}\partial {x_j}}} = h\int\limits_\Sigma {g(y)\frac{{{\partial ^2}F(x - y)}}{{\partial {x_i}\partial {x_j}}}} dy - {n_i}(x){n_j}(x)g(x) + {\gamma _{ij}}(x),i,j = 1,2,3,$$
7.
We derive a Karhunen–Loève expansion of the Gauss process \( {B}_t-g(t){\int}_0^1{g}^{\hbox{'}}(u)\mathrm{d}{B}_u,t\in \left[0,1\right] \), where (Bt)t?∈?[0,?1] is a standardWiener process, and g?:?[0,?1]?→?? is a twice continuously differentiable function with g(0) = 0 and \( {\int}_0^1{\left(g\hbox{'}(u)\right)}^2\mathrm{d}u=1 \). This process is an important limit process in the theory of goodness-of-fit tests. We formulate two particular cases with the functions \( g(t)=\left(\sqrt{2}/\pi \right)\sin \left(\pi t\right),t\in \left[0,1\right] \), and g(t)?=?t, t?∈?[0,?1]. The latter corresponds to the Wiener bridge over [0, 1] from 0 to 0. 相似文献
8.
Yohei Sato 《Calculus of Variations and Partial Differential Equations》2007,29(3):365-395
We study the nonlinear Schrödinger equations: \(-\epsilon^{2}\Delta u + V(x)u=u^p,\quad u > 0\quad \mbox{in } {\bf R}^{N},\quad u\in H^{1} ({\bf R}^{N}).\) where p > 1 is a subcritical exponent and V(x) is nonnegative potential function which has “critical frequency” \(\inf_{x\in{\bf R}^{N}} V(x)=0\). We also assume that V(x) satisfies \(0 < \liminf_{|x|\to\infty}V(x)\le \sup_{x\in{\bf R}^{N}}V(x) < \infty\) and V(x) has k local or global minima. In critical frequency cases, Byeon-Wang [5,6] showed the existence of single-peak solutions which concentrating around global minimum of V(x). Their limiting profiles—which depend on the local behavior of the potential V(x)—are quite different features from non-critical frequency case. We show the existence of multi-peak positive solutions joining single-peak solutions which concentrate around prescribed local or global minima of V(x). Moreover, under additional conditions on the behavior of V(x), we state the limiting profiles of peaks of solutions u ε(x) as follows: rescaled function \(w_\epsilon(y)=\left(\frac{g(\epsilon)}{\epsilon}\right)^{\frac{2}{p-1}} u_\epsilon(g(\epsilon)y+x_\epsilon)\) converges to a least energy solution of ?Δw + V 0(y) w = w p , w > 0 in Ω0, \(w\in H^{1}_0(\Omega_0)\). Here g(ε), V 0(x) and Ω0 depend on the local behaviors of V(x). 相似文献
9.
Mordechay B. Levin 《Israel Journal of Mathematics》2010,178(1):61-106
In this paper we describe a third class of low discrepancy sequences. Using a lattice Γ ? ? s , we construct Kronecker-like and van der Corput-like ergodic transformations T 1,Γ and T 2,Γ of [0, 1) s . We prove that for admissible lattices Γ, (T ν,Γ n (x))n≥0 is a low discrepancy sequence for all x ∈ [0, 1) s and ν ∈ {1, 2}. We also prove that for an arbitrary polyhedron P ? [0, 1) s , for almost all lattices Γ ∈ L s = SL(s,?)/SL(s, ?) (in the sense of the invariant measure on L s ), the following asymptotic formula holds with arbitrary small ? > 0, for all x ∈ [0, 1) s , and ν ∈ {1, 2}.
相似文献
$\# \{ 0 \le n < N:T_{v,\Gamma }^n(x) \in P\} = NvolP + O({(\ln N)^{s + \varepsilon }}),N \to \infty$
10.
A. A. Illarionov 《Proceedings of the Steklov Institute of Mathematics》2017,299(1):96-108
Functional equations of the form f(x + y)g(x ? y) = Σ j=1 n α j (x)β j (y) as well as of the form f1(x + z)f2(y + z)f3(x + y ? z) = Σ j=1 m φ j (x, y)ψ j (z) are solved for unknown entire functions f, g,α j , β j : ? → ? and f1, f2, f3, ψ j : ? → ?, φ j : ?2 → ? in the cases of n = 3 and m = 4. 相似文献
11.
This paper describes the interaction of two solitons with nearly equal speeds for the quartic (gKdV) equation We call soliton a solution of (0.1) of the form u(t,x)=Q c (x?ct?y 0), where c>0, y 0∈? and \(Q_{c}''+Q_{c}^{4}=cQ_{c}\). Since (0.1) is not an integrable model, the general question of the collision of two given solitons \(Q_{c_{1}}(x-c_{1}t)\), \(Q_{c_{2}}(x-c_{2}t)\) with c 1≠c 2 is an open problem.
$\partial_tu+\partial_x(\partial_x^2u+u^4)=0,\quad t,x\in \mathbb{R}.$
(0.1)
We focus on the special case where the two solitons have nearly equal speeds: let U(t) be the solution of (0.1) satisfying for \(\mu_{0}=(c_{2}^{-}-c_{1}^{-})/(c_{1}^{-}+c_{2}^{-})>0\) small. By constructing an approximate solution of (0.1), we prove that, for all time t∈?, with y 1(t)?y 2(t)>2|ln?μ 0|+C, for some C∈?. These estimates mean that the two solitons are preserved by the interaction and that for all time they are separated by a large distance, as in the case of the integrable KdV equation in this regime.
$\lim_{t\to-\infty}\|{U}(t)-Q_{c_1^-}(.-c_1^-t)-Q_{c_2^-}(.-c_2^-t)\|_{H^1}=0,$
$\begin{array}{l}\displaystyle{U}(t)={Q}_{c_1(t)}(x-y_1(t))+{Q}_{c_2(t)}(x-y_2(t))+{w}(t)\\[6pt]\displaystyle\quad\mbox{where }\|w(t)\|_{H^1}\leq|\ln\mu_0|\mu_0^2,\end{array}$
However, unlike in the integrable case, we prove that the collision is not perfectly elastic, in the following sense, for some C>0, 相似文献
$\lim_{t\to+\infty}c_1(t)>c_2^-\biggl(1+\frac{\mu_0^5}{C}\biggr),\quad \lim_{t\to+\infty}c_2(t)
and \({w}(t)\not\to0\) in H 1 as t→+∞.12.
A. M. Savchuk 《Mathematical Notes》2016,99(5-6):715-728
It is well known that the potential q of the Sturm–Liouville operator Ly = ?y? + q(x)y on the finite interval [0, π] can be uniquely reconstructed from the spectrum \(\left\{ {{\lambda _k}} \right\}_1^\infty \) and the normalizing numbers \(\left\{ {{\alpha _k}} \right\}_1^\infty \) of the operator LD with the Dirichlet conditions. For an arbitrary real-valued potential q lying in the Sobolev space \(W_2^\theta \left[ {0,\pi } \right],\theta > - 1\), we construct a function qN providing a 2N-approximation to the potential on the basis of the finite spectral data set \(\left\{ {{\lambda _k}} \right\}_1^N \cup \left\{ {{\alpha _k}} \right\}_1^N\). The main result is that, for arbitrary τ in the interval ?1 ≤ τ < θ, the estimate \({\left\| {q - \left. {{q_N}} \right\|} \right._\tau } \leqslant C{N^{\tau - \theta }}\) is true, where \({\left\| {\left. \cdot \right\|} \right._\tau }\) is the norm on the Sobolev space \(W_2^\tau \). The constant C depends solely on \({\left\| {\left. q \right\|} \right._\theta }\). 相似文献
13.
We consider quadratic functions f that satisfy the additional equation y2 f(x) = x2 f(y) for the pairs \({ (x,y) \in \mathbb{R}^2}\) that fulfill the condition P(x, y) = 0 for some fixed polynomial P of two variables. If P(x, y) = ax + by + c with \({ a , b , c \in \mathbb{R}}\) and \({(a^2 + b^2)c \neq 0}\) or P(x,y) = x n ? y with a natural number \({n \geq 2}\), we prove that f(x) = f(1) x2 for all \({x \in \mathbb{R}}\). Some related problems, admitting quadratic functions generated by derivations, are considered as well. 相似文献
14.
Viviane Ribeiro Tomaz da Silva 《Israel Journal of Mathematics》2012,188(1):441-462
Let F be a field of characteristic zero and E be the unitary Grassmann algebra generated over an infinite-dimensional F-vector space L. Denote by \(\mathcal{E} = \mathcal{E}^{(0)} \oplus \mathcal{E}^{(1)}\) an arbitrary ?2-grading of E such that the subspace L is homogeneous. Given a superalgebra A = A (0) ⊕ A (1), define the superalgebra \(A\hat \otimes \mathcal{E}\) by \(A\hat \otimes \mathcal{E} = (A^{(0)} \otimes \mathcal{E}^{(0)} ) \oplus (A^{(1)} \otimes \mathcal{E}^{(1)} )\). Note that when E is the canonical grading of E then \(A\hat \otimes \mathcal{E}\) is the Grassmann envelope of A. In this work we find bases of ?2-graded identities and we describe the ?2-graded codimension and cocharacter sequences for the superalgebras \(UT_2 (F)\hat \otimes \mathcal{E}\), when the algebra UT 2(F) of 2 ×2 upper triangular matrices over F is endowed with its canonical grading. 相似文献
15.
Let R be a prime ring of char R ≠ 2, let d be a nonzero derivation of R, and let ρ be a nonzero right ideal of R such that [[d(x)x n , d(y)] m , [y, x] s ] t = 0 for all x, y ? ρ, where n ≥ 1, m ≥ 0, s ≥ 0, and t ≥ 1 are fixed integers. If [ρ, ρ]ρ ≠ 0 then d(ρ)ρ = 0. 相似文献
16.
F. M. Korkmasov 《Vestnik St. Petersburg University: Mathematics》2007,40(2):138-151
It is shown that if P m α,β (x) (α, β > ?1, m = 0, 1, 2, …) are the classical Jaboci polynomials, then the system of polynomials of two variables {Ψ mn α,β (x, y)} m,n=0 r = {P m α,β (x)P n α,β (y)} m, n=0 r (r = m + n ≤ N ? 1) is an orthogonal system on the set Ω N×N = ?ub;(x i , y i ) i,j=0 N , where x i and y i are the zeros of the Jacobi polynomial P n α,β (x). Given an arbitrary continuous function f(x, y) on the square [?1, 1]2, we construct the discrete partial Fourier-Jacobi sums of the rectangular type S m, n, N α,β (f; x, y) by the orthogonal system introduced above. We prove that the order of the Lebesgue constants ∥S m, n, N α,β ∥ of the discrete sums S m, n, N α,β (f; x, y) for ?1/2 < α, β < 1/2, m + n ≤ N ? 1 is O((mn) q + 1/2), where q = max?ub;α,β?ub;. As a consequence of this result, several approximate properties of the discrete sums S m, n, N α,β (f; x, y) are considered. 相似文献
17.
E. Yu. Bunkova 《Proceedings of the Steklov Institute of Mathematics》2016,294(1):201-214
The article is devoted to the theory of elliptic functions of level n. An elliptic function of level n determines a Hirzebruch genus called an elliptic genus of level n. Elliptic functions of level n are also of interest because they are solutions of the Hirzebruch functional equations. The elliptic function of level 2 is the Jacobi elliptic sine function, which determines the famous Ochanine–Witten genus. It is the exponential of the universal formal group of the form F(u, v) = (u2 ? v2)/(uB(v) ? vB(u)), B(0) = 1. The elliptic function of level 3 is the exponential of the universal formal group of the form F(u, v) = (u2A(v) ? v2A(u))/(uA(v)2 ? vA(u)2), A(0) = 1, A″(0) = 0. In the present study we show that the elliptic function of level 4 is the exponential of the universal formal group of the form F(u, v) = (u2A(v) ? v2A(u))/(uB(v) ? vB(u)), where A(0) = B(0) = 1 and for B′(0) = A″(0) = 0, A′(0) = A1, and B″(0) = 2B2 the following relation holds: (2B(u) + 3A1u)2 = 4A(u)3 ? (3A12 ? 8B2)u2A(u)2. To prove this result, we express the elliptic function of level 4 in terms of the Weierstrass elliptic functions. 相似文献
18.
Let D be an arbitrary skew field and K a central subfield of D. We prove that D can be embedded in a skew field Δ such that w(Δ)=Δ for every nonempty Lie word w on a set of variables y1,y2,.?.?. with coefficients in K; moreover, we have for the multiplicative group Δ* that v(Δ*)=Δ* for every nonempty word \(v=x_{1}^{\varepsilon_{1}}x_{2}^{\varepsilon_{2}}\ldots x_{n}^{\varepsilon_{n}}\) (?i=±1; i=1,2,.?.?.,n). 相似文献
19.
In the complete Perron effect of change of values of characteristic exponents, where all nontrivial solutions y(t, y0) of the perturbed two-dimensional differential system are infinitely extendible and have finite positive exponents (the exponents of the linear approximation system being negative), we prove that the Lyapunov exponent λ[y(·, y0)] of these solutions is a function of the second Baire class of their initial vectors y0 ∈ ?n {0}. 相似文献
20.
We consider the Schrödinger operatorwhere V is bounded from below and prove a lower bound on the first eigenvalue λ 1 in terms of sublevel estimates: if w V (y) = |{x ∈ [a, b] : V (x) ≤ y}|, thenThe result is sharp up to a universal constant if {x ∈ [a, b] : V(x) ≤ y} is an interval for the value of y solving the minimization problem. An immediate application is as follows: let \({\Omega } \subset \mathbb {R}^{2}\) be a convex domain and let \(u:{\Omega } \rightarrow \mathbb {R}\) be the first eigenfunction of the Laplacian ? Δ on Ω with Dirichlet boundary conditions on ?Ω. We provewhich answers a question of van den Berg in the special case of two dimensions.
相似文献
$$ \text{-} \frac{d^{2}}{d x^{2}} + V {\text{on an interval}}~~[a,b]~{\text{with Dirichlet boundary conditions}},$$
$$\lambda_{1} \geq \frac{1}{250} \min\limits_{y > \min V}{\left( \frac{1}{w_{V}(y)^{2}} + y\right)}.$$
$$\| u \|_{L^{\infty}({\Omega})} \lesssim \frac{1}{\text{inrad}({\Omega})} \left( \frac{\text{inrad}({\Omega})}{\text{diam}({\Omega})} \right)^{1/6} \|u\|_{L^{2}({\Omega})},$$