共查询到20条相似文献,搜索用时 46 毫秒
1.
O. P. Filatov 《Mathematical Notes》1996,59(5):547-553
It is proved that the limit $$\mathop {\lim }\limits_{\Delta \to \infty } \mathop {\sup }\limits_\gamma \tfrac{1}{\Delta }\int_0^\Delta {f(\gamma (t))dt} $$ , wheref: ? → ? is a locally integrable (in the sense of Lebesgue) function with zero mean and the supremum is taken over all solutions of the generalized differential equation γ ∈ [ω1, ω2], coincides with the limit $$\mathop {\lim }\limits_{T \to \infty } \mathop {\sup }\limits_{c \geqslant 0} \varphi _f (k,{\mathbf{ }}T,{\mathbf{ }}c)$$ , where $$\varphi _f = \frac{{(k - 1)\bar I_f (T,c)}}{{1 + (k - 1)\bar \lambda _f (T,c)}},k = \frac{{\omega _2 }}{{\omega _1 }}$$ . Here ¯λf = λf /T, ¯ If =If/T, and λf is the Lebesgue measure of the set $$\{ \gamma \in [\gamma _0 ,\gamma _0 + T]:f(\gamma ) \geqslant c\} = A_f ,I_f = \int_{A_f } {f(\gamma )d\gamma } $$ . It is established that this limit always exists for almost-periodic functionsf. 相似文献
2.
We introduce the set of bicomplex numbers
which is a commutative ring with zero divisors defined by
where
We present the conjugates and the moduli associated with the bicomplex numbers. Then we study the bicomplex Schr?dinger equation
and found the continuity equations. The discrete symmetries of the system of equations describing the bicomplex Schr?dinger
equation are obtained. Finally, we study the bicomplex Born formulas under the discrete symmetries. We obtain the standard
Born’s formula for the class of bicomplex wave functions having a null hyperbolic angle. 相似文献
3.
J. W. Sander 《Monatshefte für Mathematik》1987,104(2):125-132
LetQ(x) denote a quadratic form over the rational integers in four variables (x=(x1,...,x4)). ThenQ is representable as a symmetric matrix. Assume this matrix to be non-singular modp(p≠2 prime); then the “inverse” quadratic formQ ?1 modp can be defined. Letf:?4→? be defined such that the Fourier transformf exists and the sum $$\sum\limits_{x \in \mathbb{Z}^4 } {f(c x), c \in \mathbb{R}, c \ne 0} $$ is convergent. Furthermore, letm=p 1...p k be the product ofk distinct primes withm>1, 2×m; let $$\varepsilon = \prod\limits_{i = 1}^k {\left( {\frac{{\det Q}}{{p_i }}} \right)} \ne 0$$ for the Legendre symbol $$\left( {\frac{ \cdot }{p}} \right)$$ ; define $$B_i (Q,x) = \left\{ {\begin{array}{*{20}c} {1 for Q(x) \equiv 0\bmod p_i } \\ , \\ {0 for Q(x)\not \equiv 0\bmod p_i } \\ \end{array} } \right.$$ and forr∈?,r>0, $$F(Q,f,r) = \sum\limits_{x \in \mathbb{Z}^4 } {\left( {\prod\limits_{i = 1}^k {\left( {B_i (Q,x) - \frac{1}{{p_i }}} \right)} } \right)f(r^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} x)} $$ Then we have $$F(Q,f,m) = \varepsilon F(Q^{ - 1} ,\hat f,m)$$ 相似文献
4.
пУстьE — ИжМЕРИМОЕ пО лЕБЕгУ ОгРАНИЧЕННОЕ МНОжЕстВО пОлОжИтЕльНОИ плОЩА ДИ mes2 E кОМплЕксНОИ плОск ОстИ с. кАк ОБыЧНО, пРИp≧1 ОБОжНАЧИМ ЧЕРЕжL p (E) БА НАхОВО пРОстРАНстВО ИжМЕРИ Мых пО лЕБЕгУ НАE кОМплЕксНОжНАЧНых Ф УНкцИИf с сУММИРУЕМО Иp—стЕпЕНьУ Их МОДУль И ОБыЧНОИ НОРМОИ \(\left\| \cdot \right\|_p = \left\| \cdot \right\|_{L_p (E)}\) . ЧЕР ЕжL p R n (f,E) ОБОжНАЧИМ НАИМЕН ьшЕЕ УклОНЕНИЕf?L p (E) От РАц ИОНАльНых ФУНкцИИ ст ЕпЕНИ ≦n кОМплЕксНОгО пЕРЕМЕ ННОгОz пО НОРМЕ ∥ · ∥. пОлОжИМf(z)=0 Дльz?¯CE,E δ —δ-ОкРЕстНОсть МНО жЕстВАE (δ>0), И $$\omega _p (\delta ,f) = \mathop {\sup {\mathbf{ }}}\limits_{\left| h \right|< \delta } \{ \int\limits_{E_\sigma } {\int {{\mathbf{ }}|f(z + h) - f(z)|^p } d\sigma } \} ^{1/p} .$$ тЕОРЕМА.пУсть 1≦p<2,f?L p (E),n≧4.тОгДА $$\begin{array}{*{20}c} {L^p R_n (f,E) \leqq 12\omega _p \left( {\frac{{\delta + \ln n}}{{\sqrt n }},f} \right){\mathbf{ }}npu{\mathbf{ }}p = 1,} \\ {L^p R_n (f,E) \leqq \frac{{24}}{{(p - 1)(2 - p)}}\omega _p (n^{(p - 2)/2p} ,f){\mathbf{ }}npu{\mathbf{ }}1< p< 2,} \\ {L^1 R_n (\bar z,[0,1] \times [0,1]) \geqq \frac{1}{{32\sqrt n }}.} \\ \end{array} $$ . 相似文献
5.
The integral $$\int_0^{{\pi \mathord{\left/ {\vphantom {\pi 4}} \right. \kern-\nulldelimiterspace} 4}} {\ln \left( {\cos ^{{m \mathord{\left/ {\vphantom {m n}} \right. \kern-\nulldelimiterspace} n}} \theta \pm \sin ^{{m \mathord{\left/ {\vphantom {m n}} \right. \kern-\nulldelimiterspace} n}} \theta } \right)d\theta } $$ where m and n are relatively prime positive integers, is evaluated exactly in terms of elementary functions and the Catalan constant G. 相似文献
6.
Piotr Migus 《Archiv der Mathematik》2019,112(4):395-405
Let
$$f,g:({\mathbb {R}}^n,0)\rightarrow ({\mathbb {R}}^m,0)$$
be
$$C^{r+1}$$
mappings and let
$$Z=\{x\in \mathbf {\mathbb {R}}^n:\nu (df (x))=0\}$$
,
$$0\in Z$$
,
$$m\le n$$
. We will show that if there exist a neighbourhood U of
$$0\in {\mathbb {R}}^n$$
and constants
$$C,C'>0$$
and
$$k>1$$
such that for
$$x\in U$$
$$\begin{aligned}&\nu (df(x))\ge C{\text {dist}}(x,Z)^{k-1}, \\&\left| \partial ^{s} (f_i-g_i)(x) \right| \le C'\nu (df(x))^{r+k-|s|}, \end{aligned}$$
for any
$$i\in \{1,\dots , m\}$$
and for any
$$s \in \mathbf {\mathbb {N}}^n_0$$
such that
$$|s|\le r$$
, then there exists a
$$C^r$$
diffeomorphism
$$\varphi :({\mathbb {R}}^n,0)\rightarrow ({\mathbb {R}}^n,0)$$
such that
$$f=g\circ \varphi $$
in a neighbourhood of
$$0\in {\mathbb {R}}^n$$
. By
$$\nu (df)$$
we denote the Rabier function. 相似文献
7.
Yu. V. Nesterenko 《Mathematical Notes》1974,15(3):234-240
For almost all pointsξ? Rm (m≥2) the inequality $$\sup {\text{ }}ln \frac{1}{{|P (\xi )|}} \ll (ln u)^{m + 2} ,$$ is valid, where the upper bound is taken over all nonzero polynomials P for which $$\exp {\text{ }}(\deg {\text{ }}P){\text{ }}L{\text{ }}(P) \leqslant u,$$ where L(P) is the sum of the moduli of the coefficients of P. When m=1 the exponent of the right side is equal to 2. 相似文献
8.
This paper is a continuation of [3]. Suppose f∈Hp(T), 0σ r σ f,σ=1/p?1. When p=1, it is just the partial Fourier sums Skf. In this paper we establish the sharp estimations on the degree of approximation: $$\left\{ { - \frac{1}{{logR}}\int\limits_1^R {\left\| {\sigma _r^\delta f - f} \right\|_{H^p (T)}^p \frac{{dr}}{r}} } \right\}^{1/p} \leqq C{\mathbf{ }}{}_p\omega \left( {f,{\mathbf{ }}( - \frac{1}{{logR}})^{1/p} } \right)_{H^p (T)} ,0< p< 1,$$ and \(\frac{1}{{\log L}}\sum\limits_{k - 1}^L {\frac{{\left\| {S_k f - f} \right\|_H 1_{(T)} }}{k} \leqq Cp\omega (f; - \frac{1}{{\log L}})_H 1_{(T)} } \) Where $$\omega (f,{\mathbf{ }}h)_{H^p (T)} \begin{array}{*{20}c} { = Sup} \\ {0 \leqq \left| u \right| \leqq h} \\ \end{array} \left\| {f( \cdot + u) - f( \cdot )} \right\|_{H^p (T).} $$ . 相似文献
9.
A. Khatamov 《Mathematical Notes》1977,21(3):198-207
The following inequalities are shown to hold for the least uniform rational deviations Rn(f) of a function f(x), continuous and convex in the interval [a, b]: $$R_n (f) \leqslant C(v)\Omega (f)n^{ - 1} \overbrace {\ln \ldots \ln }^{vtimes}n$$ (ν is an integer, C(ν) depends only on ν, and Ω(f) is the total oscillation of f); $$R_n (f) \leqslant C_1 n^{ - 1} \overbrace {\ln \ldots \ln }^{vtimes}n\mathop {\inf }\limits_{(b - a)\chi _n \leqslant \lambda< b - a} \left\{ {\omega (\lambda ,f) + M(f)n^{ - 1} \ln \frac{{b - a}}{\lambda }} \right\}$$ (ν is an integer, C1(ν) depends only on ν, xn = exp (-n/(500 In2n)), ω (δ,f) is the modulus of continuity of f, and M(f) = max¦f(x) ¦. 相似文献
10.
В. И. ДАНЧЕНкО 《Analysis Mathematica》1990,16(4):241-255
LetG be an arbitrary domain in \(\bar C\) ,f a function meromorphic inG, $$M_f \mathop = \limits^{def} \mathop {\lim \sup }\limits_{G \mathrel\backepsilon z \to \partial G} \left| {f(z)} \right|< \infty ,$$ andR the sum of the principal parts in the Laurent expansions off with respect to all its poles inG. We set $$f_G (z) = R(z) - \alpha ,{\mathbf{ }}where{\mathbf{ }}\alpha = \mathop {\lim }\limits_{z \to \infty } (f(z) - R(z))$$ in case ∞?G, andα=0 in case ∞?G. It is proved that $$\left\| {f_G } \right\|_{C(\partial G)} \leqq 50(\deg f_G )M_f ,{\mathbf{ }}\left\| {f'_G } \right\|_{L_1 (\partial G)} \leqq 50(\deg f_G )V(\partial G)M_f ,$$ where $$V(\partial G) = \sup \left\{ {\left\| {r'} \right\|_{L_1 (\partial G)} :r(z) = a/(z - b),{\mathbf{ }}\left\| r \right\|_{G(\partial G)} \leqq 1} \right\}.$$ 相似文献
11.
Yu. N. Subbotin 《Mathematical Notes》1996,59(1):83-96
The smallest numberA<∞ is found such that for any sequenceY={y k ,k ∈ ?} with ¦Δ n y k ¦≤1 there exists au(t), ¦u(t)¦ ≤ A, for which the equationy n (t)=u(t) (?∞<t<∞) has a solution satisfying the conditions $$y_k = \frac{1}{h}\int_{ - h/2}^{h/2} {y(k + 1){\mathbf{ }}dt} ,{\mathbf{ }}where{\mathbf{ }}k{\mathbf{ }} \in {\mathbf{ }}\mathbb{Z},{\mathbf{ }}1{\mathbf{ }}< {\mathbf{ }}h{\mathbf{ }}< {\mathbf{ }}2.$$ , wherek ∈ ?, 1<h<2. A similar problem is treated inL p (?∞, ∞). It is shown that forh=2m (m a natural number) no such finiteA exists. 相似文献
12.
赵晓强 《应用数学学报(英文版)》1990,6(1):40-43
In this paper, we consider nonlinear and nonautonomous systems with the trivial solution:dx/dt=f(t, x), x∈R~nwhere f(t+ω,x)+f(t, x), f(t, 0)=0. By using the theory of Brouwer topological degree, we obtainthe existence theorem of nontrivial ω-periodic solution. Finally the applied example of the theoremis cited. 相似文献
13.
D. Suryanarayana 《Periodica Mathematica Hungarica》1983,14(1):69-75
LetL(x) denote the number of square-full integers not exceedingx. It is well-known that $$L\left( x \right) \sim \frac{{\zeta \left( {{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \right)}}{{\zeta \left( 3 \right)}}x^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + \frac{{\zeta \left( {{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}} \right)}}{{\zeta \left( 2 \right)}}x^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} ,$$ whereζ(s) denotes the Riemann Zeta function, LetΔ(x) denote the error function in the asymptotic formula forL(x). On the assumption of the Riemann hypothesis (R.H.), it is known that $$\Delta x = O\left( {x^{13/81 + 8} } \right)$$ for everyε > 0. In this paper, we prove on the assumption of R.H. that $$\frac{1}{x}\int\limits_x^1 {\left| {\Delta \left( t \right)} \right|dt = O\left( {x^{1/10 + ^8 } } \right)} .$$ In fact, we prove a more general result. We conjecture that $$\Delta x = O\left( {x^{1/10 + ^8 } } \right)$$ under the assumption of the R.H. 相似文献
14.
We consider the question of evaluating the normalizing multiplier $$\gamma _{n,k} = \frac{1}{\pi }\int_{ - \pi }^\pi {\left( {\frac{{sin\tfrac{{nt}}{2}}}{{sin\tfrac{t}{2}}}} \right)^{2k} dt} $$ for the generalized Jackson kernel J n,k (t). We obtain the explicit formula $$\gamma _{n,k} = 2\sum\limits_{p = 0}^{\left[ {k - \tfrac{k}{n}} \right]} {( - 1)\left( {\begin{array}{*{20}c} {2k} \\ p \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {k(n + 1) - np - 1} \\ {k(n - 1) - np} \\ \end{array} } \right)} $$ and the representation $$\gamma _{n,k} = \sqrt {\frac{{24}}{\pi }} \cdot \frac{{(n - 1)^{2k - 1} }}{{\sqrt {2k - 1} }}\left[ {1\frac{1}{8} \cdot \frac{1}{{2k - 1}} + \omega (n,k)} \right],$$ , where $$\left| {\omega (n,k)} \right| < \frac{4}{{(2k - 1)\sqrt {ln(2k - 1)} }} + \sqrt {12\pi } \cdot \frac{{k^{\tfrac{3}{2}} }}{{n - 1}}\left( {1 + \frac{1}{{n - 1}}} \right)^{2k - 2} .$$ . 相似文献
15.
D. Suryanarayana 《Periodica Mathematica Hungarica》1979,10(4):261-271
LetL(x) denote the number of square full integers ≤x. By a square-full integer, we mean a positive integer all of whose prime factors have multiplicity at least two. It is well known that $$\left. {L(x)} \right| \sim \frac{{\zeta ({3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2})}}{{\zeta (3)}}x^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + \frac{{\zeta ({2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3})}}{{\zeta (2)}}x^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} ,$$ where ζ(s) denotes the Riemann Zeta function. Let Δ(x) denote the error function in the asymptotic formula forL(x). On the basis of the Riemann hypothesis (R.H.), it is known that \(\Delta (x) = O(x^{\tfrac{{13}}{{81}} + \varepsilon } )\) for every ε>0. In this paper, we prove the following results on the assumption of R.H.: (1) $$\frac{1}{x}\int\limits_1^x {\Delta (t)dt} = O(x^{\tfrac{1}{{12}} + \varepsilon } ),$$ (2) $$\int\limits_1^x {\frac{{\Delta (t)}}{t}\log } ^{v - 1} \left( {\frac{x}{t}} \right) = O(x^{\tfrac{1}{{12}} + \varepsilon } )$$ for any integer ν≥1. In fact, we prove some general results and deduce the above from them. On the basis of (1) and (2) above, we conjecture that \(\Delta (x) = O(x^{{1 \mathord{\left/ {\vphantom {1 {12}}} \right. \kern-0em} {12}} + \varepsilon } )\) under the assumption of R.H. 相似文献
16.
Mei Yue JIANG 《数学学报(英文版)》2005,21(5):1219-1228
In this paper, we give a Landesman-Lazer type theorem for periodic solutions of the asymmetric 1-dimensional p-Laplacian equation -(|x'|^p-2x')'=λ|x|^p-2x++μ|x|^p-2x-+f(t,x)with periodic boundary value. 相似文献
17.
We extend classical volume formulas for ellipsoids and zonoids to p-sums of segments $${vol}\left( {\sum\limits_{i=1}^m { \oplus_p } [ -x_i ,x_i ]} \right)^{1/n} \sim_{c_p} n^{ - \frac{1}{{p'}}} \left( {\sum\limits_{card(I) = n} {|\det (x_i)_i |^p}} \right)^{\frac{1}{{pn}}}$$ where x1,...,xm are m vectors in $\mathbb{R}^n ,\frac{1}{p} + \frac{1}{{p\prime }} = 1$ . According to the definition of Firey, the Minkowski p-sum of segments is given by $$\sum\limits_{i = 1}^m { \oplus _p [ - x_{i,} x_i ]} = \left\{ {\sum\limits_{i = 1}^m {\alpha _i } x_i \left| {\left( {\sum\limits_{i = 1}^m {|\alpha _i |^{p^\prime } } } \right)} \right.^{\frac{1}{{p^\prime }}} \leqslant 1} \right\}.$$ We describe related geometric properties of the Lewis maps associated to classical operator norms. 相似文献
18.
Alessio Porretta 《Annali di Matematica Pura ed Applicata》1999,177(1):143-172
Summary We prove existence results for the initial-boundary value problem for parabolic equations of the type
where ω is a bounded open subset ofR
N and T>0, A is a pseudomonotone operator of Leray-Lions type defined in L2(), T; H
0
1
(ω), u0 is in L1 (ω) and g(x, t, s) is only assumed to be a Carathéodory function satisfying a sign condition. The result is achieved by proving
the strong convergence in L2 (0, T; H
0
1
(ω)) of trucations of solutions of approximating problems with L1 converging data. To underline the importance of this tool, we show how it can be used for getting other existence theorems,
dealing in particular with the following class of Cauchy-Dirichlet problems:
where ΦεC0 (R, R
N) and the data f and u0 are still taken in L1(Q) and L1(ω) respectively.
Entrata in Redazione il 2 aprile 1998. 相似文献
19.
A. V. Bogomol'naya 《Journal of Mathematical Sciences》1995,73(6):633-637
We consider
,mE > 0,G(E) is a certain subspace of L
1
(E) consisting of functions concentrated on E and integrable, and {dk}, (k ∈ ℤ) in a summable sequence of positive numbers. It is proved that if G(E)=Lp(E), p≥2, then there exists f∈G(E) such that |f(n)|≥dn,
(one of the questions involved in the majorization problem). Sufficient conditions are obtained for certain other function
classes G(E). We study the question of partial majorization. Bibliography: 2 titles.
Translated fromProblemy Matematicheskogo Analiza, No. 13, 1992, pp. 42–48. 相似文献
20.
Our main results are:
- Let α ≠ 0 be a real number. The function (Γ ? exp) α is convex on ${\mathbf{R}}$ if and only if $$\alpha \geq \max_{0<{t}<{x_0}}\Big(-\frac{1}{t\psi(t)} - \frac{\psi'(t)}{\psi(t)^2}\Big) = 0.0258... .$$ Here, x 0 = 1.4616... denotes the only positive zero of ${\psi = \Gamma'/\Gamma}$ .
- Assume that a function f: (0, ∞) → (0, ∞) is bounded from above on a set of positive Lebesgue measure (or on a set of the second category with the Baire property) and satisfies $$f(x+1) = x f(x) \quad{\rm for}\quad{x > 0}\quad{\rm and}\quad{f(1) = 1}.$$