2,241 results
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2. Remarks on a paper about functional inequalities for polynomials and Bernoulli numbers
- Author
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Jens Schwaiger
- Subjects
Combinatorics ,Polynomial ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Discrete Mathematics and Combinatorics ,Arithmetic function ,Context (language use) ,Limit (mathematics) ,Function (mathematics) ,Bernoulli number ,Mathematics - Abstract
The authors of [KMM] consider a system of two functional inequalities for a function $$f : {\mathbb{R}} \rightarrow {\mathbb{R}}$$ , and they show that, if certain arithmetical conditions and inequalities for certain parameters are fulfilled, f has to be a polynomial provided that f is continuous at some point x0. This result is derived here under the weaker condition that for some x0 the limit $${\rm lim}_{x \rightarrow x_0} f(x)$$ exists. Moreover, another system of inequalities is given leading to the same result on the nature of f. The methods used also give natural explanations for the fact that Bernoulli numbers play an important role in this context.
- Published
- 2009
3. Plenary Papers A Priori Truncation Error Bounds for Continued Fractions
- Author
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Lisa Lorentzen
- Subjects
Sequence ,40A15 ,convergence ,Truncation error (numerical integration) ,General Mathematics ,Mathematical analysis ,complementary error function ,incomplete gamma function ,30B10 ,Upper and lower bounds ,Limit periodic continued fractions ,Combinatorics ,Error function ,truncation error estimates ,Fraction (mathematics) ,Limit (mathematics) ,Incomplete gamma function ,Gamma function ,Mathematics - Abstract
Most of the known continued fraction expansions of special functions are limit periodic. This means that the classical approximants S n (0) are normally not the best ones to use for approximations. In this paper we suggest a number of approximants S n (w n ) which converge faster. The estimation of the improvement and bounds for the error |f - Sn(wn)| (which we still call the truncation error) are mainly obtained by means of Thron's parabola sequence theorem and the oval sequence theorem.
- Published
- 2003
4. Remarks on the paper: 'Basic calculus of variations'
- Author
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John M. Ball
- Subjects
Combinatorics ,Sobolev space ,General Mathematics ,Linear space ,Bounded function ,Mathematical analysis ,Domain (ring theory) ,Boundary (topology) ,Calculus of variations ,Quadratic form (statistics) ,Convex function ,Mathematics - Abstract
iF(y)=( where G c R* is a bounded domain, y: G -» R , y'(x) = (dyydx), and F: M -> R is continuous. Here M denotes the linear space of real N X k matrices. We suppose throughout that K > 2, N > 2. In [7] F is called T-conυex if there exists a convex function /, defined on R, r = (t) ~ 1, such that F(p) = f(τ(p)) for all/? e M, where τ(p) denotes the minors of p of all orders j , 1 y uniformly on G with supx χGG\yj(x) ~ yj(x)\ < C < oo for ally. (Equivalently, if G has sufficiently regular boundary then IF is lsc if and only if IF is sequentially weak* lower semicontinuous on the Sobolev space W(G; R).) A consequence of [7? Theorem 3.6] is that IF lsc implies F polyconvex; that this conclusion is false was pointed out implicitly by Morrey [4, p. 26]. Morrey's remark is based on an example due to Terpstra [8] of a quadratic form
- Published
- 1985
5. A Correction to the Paper 'Semi-Open Sets and Semicontinuity in Topological Spaces' by Norman Levine
- Author
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T. R. Hamlett
- Subjects
Combinatorics ,Topological manifold ,Isolated point ,Connected space ,Topological algebra ,Applied Mathematics ,General Mathematics ,Topological tensor product ,Mathematical analysis ,Topological space ,Homeomorphism ,Mathematics ,Zero-dimensional space - Abstract
A subset A of a topological space is said to be semi-open if there exists an open set U such that U C A C Cl(U) where Cl(U) denotes the closure of U. The class of semi-open sets of a given topological space (X, J) is denoted S.O. (X, J). In the paper Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36-41, Norman Levine states in Theorem 10 that if J and V* are two topologies for a set X such that S.O.(X, 3) C S.O.(X, J*), then 'J C P. In a corollary to this theorem, Levine states that if S.O.(X, if) = S.O.(X,Jf*), then _T= f*. An example is given which shows the above-mentioned theorem and its corollary are false. This paper shows how different topologies on a set which determine the same class of semi-open subse,ts can arise from functions, and points out some of the implications of two topologies being related in this manner. In [6] Norman Levine defines a set A in a topological space X to be semi-open if there exists an open set U such that U C A C Cl (U), where Cl(U) denotes the closure of U. The class of semi-open sets for a given topological space (X, i) is denoted S.O. (X, sT). Levine states in Theorem 10 of [6] that if Jf and * are two topologies for a set X such that S.O. (X, i) C S.O. (X, J9 then iT C *. In a corollary to this theorem, Levine states that if S.O. (X, ) = S.O. (X, iT*), then Jf = -" The following example which is due to S. Gene Crossley and S. K. Hildebrand [1, Example 1.1] shows the above-mentioned theorem and its corollary are false. Example. Let X = la, b, cl, J; = t0, sal, la, bl, la, cl, XI, ;* = 10, sal, la, bl, XI. An exhaustion of all possibilities shows that S.O. (X, ) = S.O. (x, 5j*). Crossley and Hildebrand [3] defined two topologies iT and T* on a set X to be semi-correspondent if S.O. (X, J) = S.O. (X, 5f*). It is shown in [3] that semi-correspondence is an equivalence relation on the collection of Received by the editors March 3, 1974. AMS (MOS) subject classifications (1970). Primary 54B99; Secondary 54C10.
- Published
- 1975
6. Higher order Turán inequalities for the Riemann $\xi$-function
- Author
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Dimitar K. Dimitrov, Fábio Rodrigues Lucas, Universidade Estadual Paulista (Unesp), and Universidade Estadual de Campinas (UNICAMP)
- Subjects
Degree (graph theory) ,Applied Mathematics ,General Mathematics ,Entire function ,Mathematical analysis ,Short paper ,Function (mathematics) ,Maclaurin coefficients ,Riemann ξ function ,Combinatorics ,Riemann hypothesis ,symbols.namesake ,Jensen polynomials ,symbols ,Order (group theory) ,Shape function ,Laguerre-Pólya class ,Turán inequalities ,Mathematics - Abstract
Submitted by Vitor Silverio Rodrigues (vitorsrodrigues@reitoria.unesp.br) on 2014-05-27T11:25:28Z No. of bitstreams: 0Bitstream added on 2014-05-27T14:41:41Z : No. of bitstreams: 1 2-s2.0-79951846250.pdf: 494002 bytes, checksum: 56b6ee8beddda3e7dae971355d44a19f (MD5) Made available in DSpace on 2014-05-27T11:25:28Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-03-01 Item merged in doublecheck by Felipe Arakaki (arakaki@reitoria.unesp.br) on 2015-12-11T17:28:11Z Item was identical to item(s): 71803, 21370 at handle(s): http://hdl.handle.net/11449/72321, http://hdl.handle.net/11449/21804 Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) The simplest necessary conditions for an entire function ψ(x) =∞ ∑ k=0 γk xk/k! to be in the Laguerre-Pólya class are the Turán inequalities γ2 k- γk+1γk-1 ≥ 0. These are in fact necessary and sufficient conditions for the second degree generalized Jensen polynomials associated with ψ to be hyperbolic. The higher order Turán inequalities 4(γ2 n - γn-1γn+1)(γ2n +1 - γnγn+2) - (γnγn+1 - γn-1γn+2) 2 ≥ 0 are also necessary conditions for a function of the above form to belong to the Laguerre-Pólya class. In fact, these two sets of inequalities guarantee that the third degree generalized Jensen polynomials are hyperbolic. Pólya conjectured in 1927 and Csordas, Norfolk and Varga proved in 1986 that the Turán inequalities hold for the coefficients of the Riemann ψ-function. In this short paper, we prove that the higher order Turán inequalities also hold for the ψ-function, establishing the hyperbolicity of the associated generalized Jensen polynomials of degree three. © 2010 American Mathematical Society. Departamento de Ciências de Computação e Estatística IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP Departamento de matemática Aplicada IMECC UNICAMP, 13083-859 Campinas, SP Departamento de Ciências de Computação e Estatística IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP FAPESP: 03/01874-2 FAPESP: 06/60420-0 CNPq: 305622/2009-9 CAPES: DGU-160
- Published
- 2011
7. Morse index of solutions for a critical weighted elliptic problem
- Author
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Liu ZhongYuan
- Subjects
Combinatorics ,law ,General Mathematics ,Mathematical analysis ,Nabla symbol ,Characterization (mathematics) ,Morse code ,Critical exponent ,Prime (order theory) ,law.invention ,Mathematics - Abstract
In this paper, we study the following critical weighted elliptic problem:\begin{equation*}\begin{cases}-\text{div}(|x|^\theta\nabla u) =(N^\prime+\tau)(N^\prime-2)|x|^\ell u^{p_\tau}, \text{in}\ \mathbb{R}^N,\\ u>0, \text{in}\ \mathbb{R}^N,\end{cases}\end{equation*} where $N^\prime=N+\theta>2,$ $\tau=\ell-\theta>-2,$ $ p_\tau=\frac{N^\prime+2+2\tau}{N^\prime-2}.$ Du and Guo (2015) showed the above problem possesses a family of finite Morse index solutions. Our main aim in this paper is to give its exact characterization of Morse index, which depends only on $N,\tau$ and $\theta$.
- Published
- 2017
8. Existence of solutions for a critical fractional Kirchhoff type problem in ℝ N
- Author
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MingQi Xiang, BinLin Zhang, and Hong Qiu
- Subjects
General Mathematics ,Operator (physics) ,010102 general mathematics ,Mathematical analysis ,Degenerate energy levels ,Zero (complex analysis) ,Multiplicity (mathematics) ,Lambda ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Variational principle ,Mountain pass theorem ,0101 mathematics ,Laplace operator ,Mathematics - Abstract
This paper concerns with the existence of solutions for the following fractional Kirchhoff problem with critical nonlinearity: $${\left( {\int {\int {_{{\mathbb{R}^{2N}}}\frac{{{{\left| {u\left( x \right) - u\left( y \right)} \right|}^2}}}{{{{\left| {x - y} \right|}^{N + 2s}}}}dxdy} } } \right)^{\theta - 1}}{\left( { - \Delta } \right)^s}u = \lambda h\left( x \right){u^{p - 1}} + {u^{2_s^* - 1}} in {\mathbb{R}^N},$$ where (−Δ) s is the fractional Laplacian operator with 0 < s < 1, 2 s * = 2N/(N − 2s), N > 2s, p ∈ (1, 2 s *), θ ∈ [1, 2 s */2), h is a nonnegative function and λ a real positive parameter. Using the Ekeland variational principle and the mountain pass theorem, we obtain the existence and multiplicity of solutions for the above problem for suitable parameter λ > 0. Furthermore, under some appropriate assumptions, our result can be extended to the setting of a class of nonlocal integro-differential equations. The remarkable feature of this paper is the fact that the coefficient of fractional Laplace operator could be zero at zero, which implies that the above Kirchhoff problem is degenerate. Hence our results are new even in the Laplacian case.
- Published
- 2017
9. Solving the general split common fixed-point problem of quasi-nonexpansive mappings without prior knowledge of operator norms
- Author
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Jing Zhao and Songnian He
- Subjects
Weak convergence ,General Mathematics ,Operator (physics) ,Mathematical analysis ,Linear operators ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Bounded function ,Common fixed point ,0101 mathematics ,Prior information ,Mathematics - Abstract
Let $H_1$, $H_2$, $H_3$ be real Hilbert spaces, let $A:H_1\rightarrow H_3$, $B:H_2\rightarrow H_3$ be two bounded linear operators. The general split common fixed-point problem under consideration in this paper is to $$\text{find}\ \ x \in \cap_{i=1}^p F(U_i),\ \ y \in \cap_{j=1}^r F(T_j)\ \ \text{such that}\ \ Ax = By,\eqno{(1)}$$ where $p$, $r\geq 1$ are integers, $U_i:H_1\rightarrow H_1$ $(1\leq i\leq p)$ and $T_j:H_2\rightarrow H_2$ $(1\leq j\leq r)$ are quasi-nonexpansive mappings with nonempty common fixed-point sets $\cap_{i=1}^pF(U_i)=\cap_{i=1}^p\{x\in H_1:U_ix=x\}$ and $\cap_{j=1}^rF(T_j)=\cap_{j=1}^r\{x\in H_2:T_jx=x\}$. Note that, the above problem (1) allows asymmetric and partial relations between the variables $x$ and $y$. If $H_2=H_3$ and $B=I$, then the general split common fixed-point problem (1) reduces to the general split common fixed-point problem proposed by Censor and Segal $\cite{C}$. In this paper, we introduce simultaneous parallel and cyclic algorithms for the general split common fixed-point problems (1). We introduce a way of selecting the stepsizes such that the implementation of our algorithms does not need any prior information about the operator norms. We prove the weak convergence of the proposed algorithms and apply the proposed algorithms to the multiple-set split feasibility problems. Our results improve and extend the corresponding results announced by many others.
- Published
- 2017
10. О ГИПЕРБОЛИЧЕСКОЙ ДЗЕТА-ФУНКЦИИ ГУРВИЦА
- Subjects
Mathematics::Number Theory ,General Mathematics ,Analytic continuation ,Mathematical analysis ,Hankel contour ,Riemann zeta function ,Bernoulli polynomials ,Hurwitz zeta function ,Combinatorics ,symbols.namesake ,Improper integral ,symbols ,Dirichlet series ,Mathematics ,Real number - Abstract
The paper deals with a new object of study --- hyperbolic Hurwitz zeta function, which is given in the right \(\alpha\)-semiplane \( \alpha = \sigma + it \), \( \sigma> 1 \) by the equality $$ \zeta_H(\alpha; d, b) = \sum_{m \in \mathbb Z} \left(\, \overline{dm + b} \, \right)^{-\alpha}, $$ where \( d \neq0 \) and \( b \) --- any real number. Hyperbolic Hurwitz zeta function \( \zeta_H (\alpha; d, b) \), when \( \left\| \frac {b} {d} \right\|> 0 \) coincides with the hyperbolic zeta function of shifted one-dimensional lattice \( \zeta_H (\Lambda (d, b) | \alpha) \). The importance of this class of one-dimensional lattices is due to the fact that each Cartesian lattice is represented as a union of a finite number of Cartesian products of one-dimensional shifted lattices of the form \( \Lambda (d, b) = d \mathbb{Z} + b \). Cartesian products of one-dimensional shifted lattices are in substance shifted diagonal lattices, for which in this paper the simplest form of a functional equation for the hyperbolic zeta function of such lattices is given. The connection of the hyperbolic Hurwitz zeta function with the Hurwitz zeta function \( \zeta^* (\alpha; b)\) periodized by parameter \(b\) and with the ordinary Hurwitz zeta function \( \zeta (\alpha; b) \) is studied. New integral representations for these zeta functions and an analytic continuation to the left of the line \( \alpha = 1 + it \) are obtained. All considered hyperbolic zeta functions of lattices form an important class of Dirichlet series directly related to the development of the number-theoretical method in approximate analysis. For the study of such series the use of Abel's theorem is efficient, which gives an integral representation through improper integrals. Integration by parts of these improper integrals leads to improper integrals with Bernoulli polynomials, which are also studied in this paper.
- Published
- 2016
11. L k -biharmonic Hypersurfaces in Space Forms
- Author
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S. M. B. Kashani and M. Aminian
- Subjects
Combinatorics ,Conjecture ,Hypersurface ,Principal curvature ,General Mathematics ,Mathematical analysis ,Simply connected space ,Biharmonic equation ,Regular polygon ,Context (language use) ,Mathematics::Differential Geometry ,Space (mathematics) ,Mathematics - Abstract
In this paper, we introduce L k -biharmonic hypersurfaces M in simply connected space forms R n+1(c) and propose L k -conjecture for them. For c=0,−1, we prove the conjecture when hypersurface M has two principal curvatures with multiplicities 1,n−1, or M is weakly convex, or M is complete with some constraints on it and on L k . We also show that neither there is any L k -biharmonic hypersurface M n in $ \mathbb {H}^{n+1} $ with two principal curvatures of multiplicities greater than one, nor any L k -biharmonic compact hypersurface M n in $ \mathbb {R}^{n+1} $ or in $ \mathbb {H}^{n+1} $ . As a by-product, we get two useful, important variational formulas. The paper is a sequel to our previous paper, (Taiwan. J. Math., 19, 861–874, 5) in this context.
- Published
- 2016
12. On the strong divergence of Hilbert transform approximations and a problem of Ul’yanov
- Author
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Holger Boche and Volker Pohl
- Subjects
Numerical Analysis ,Sequence ,Conjecture ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,020206 networking & telecommunications ,02 engineering and technology ,01 natural sciences ,Combinatorics ,symbols.namesake ,Uniform norm ,Subsequence ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,Hilbert transform ,0101 mathematics ,Divergence (statistics) ,Finite set ,Fourier series ,Analysis ,Mathematics - Abstract
This paper studies the approximation of the Hilbert transform f ? = H f of continuous functions f with continuous conjugate f ? based on a finite number of samples. It is known that every sequence { H N f } N ? N which approximates f ? from samples of f diverges (weakly) with respect to the uniform norm. This paper conjectures that all of these approximation sequences even contain no convergent subsequence. A property which is termed strong divergence.The conjecture is supported by two results. First it is proven that the sequence of the sampled conjugate Fejer means diverges strongly. Second, it is shown that for every sample based approximation method { H N } N ? N there are functions f such that ? H N f ? ∞ exceeds any given bound for any given number of consecutive indices N .As an application, the later result is used to investigate a problem associated with a question of Ul'yanov on Fourier series which is related to the possibility to construct adaptive approximation methods to determine the Hilbert transform from sampled data. This paper shows that no such approximation method with a finite search horizon exists.
- Published
- 2016
13. Properties associated with the epigraph of the $$l_1$$ l 1 norm function of projection onto the nonnegative orthant
- Author
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Li Wang and Yong-Jin Liu
- Subjects
Convex hull ,Epigraph ,021103 operations research ,General Mathematics ,Mathematical analysis ,Tangent cone ,0211 other engineering and technologies ,02 engineering and technology ,Management Science and Operations Research ,Directional derivative ,01 natural sciences ,Orthant ,Combinatorics ,010104 statistics & probability ,Dual cone and polar cone ,Norm (mathematics) ,Convex cone ,0101 mathematics ,Software ,Mathematics - Abstract
This paper studies some properties associated with a closed convex cone $$\mathcal {K}_{1+}$$ , which is defined as the epigraph of the $$l_1$$ norm function of the metric projection onto the nonnegative orthant. Specifically, this paper presents some properties on variational geometry of $$\mathcal {K}_{1+}$$ such as the dual cone, the tangent cone, the normal cone, the critical cone and its convex hull, and others; as well as the differential properties of the metric projection onto $$\mathcal {K}_{1+}$$ including the directional derivative, the B-subdifferential, and the Clarke’s generalized Jacobian. These results presented in this paper lay a foundation for future work on sensitivity and stability analysis of the optimization problems over $$\mathcal {K}_{1+}$$ .
- Published
- 2016
14. Packing constant for Cesàro-Orlicz sequence spaces
- Author
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Zhenhua Ma, Qiaoling Xin, and Lining Jiang
- Subjects
010101 applied mathematics ,Combinatorics ,Mathematics::Functional Analysis ,General Mathematics ,Ordinary differential equation ,Norm (mathematics) ,010102 general mathematics ,Mathematical analysis ,Banach space ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
The packing constant is an important and interesting geometric parameter of Banach spaces. Inspired by the packing constant for Orlicz sequence spaces, the main purpose of this paper is calculating the Kottman constant and the packing constant of the Cesaro-Orlicz sequence spaces \(({\text{ce}}{{\text{s}}_\varphi })\) defined by an Orlicz function φ equipped with the Luxemburg norm. In order to compute the constants, the paper gives two formulas. On the base of these formulas one can easily obtain the packing constant for the Cesaro sequence space cesp and some other sequence spaces. Finally, a new constant \(\widetilde D\)(X), which seems to be relevant to the packing constant, is given.
- Published
- 2016
15. Remarks on smallness of chemotactic effect for asymptotic stability in a two-species chemotaxis system
- Author
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Masaaki Mizukami
- Subjects
lcsh:Mathematics ,General Mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Boundary (topology) ,lcsh:QA1-939 ,Omega ,Delta-v (physics) ,Combinatorics ,Chemotaxis|Sensitivity function|Logistic term|Asymptotic behavior|Stability ,Exponential stability ,Domain (ring theory) ,Nabla symbol ,Mathematics - Abstract
This paper deals with the two-species chemotaxis system where Ω is a bounded domain in RN with smooth boundary ∂Ω, N∈N; h,Xi are functions satisfying some conditions. Global existence and asymptotic stability of solutions to the above system were established under some conditions [11]. The main purpose of the present paper is to improve smallness conditions for chemotactic effect deriving asymptotic stability and to give the convergence rate in stabilization which cannot be attained in the previous work.
- Published
- 2016
16. Sobolev inequalities for the Hardy–Schrödinger operator: extremals and critical dimensions
- Author
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Nassif Ghoussoub, Frédéric Robert, University of British Columbia (UBC), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), and Université de Lorraine, NSERC
- Subjects
Mean curvature ,General Mathematics ,Star (game theory) ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Boundary (topology) ,01 natural sciences ,Omega ,Sobolev inequality ,010101 applied mathematics ,Combinatorics ,Sobolev space ,35J35, 35J60, 58J05, 35B44 ,Mathematics - Analysis of PDEs ,Compact space ,[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,MSC 35J35, 35J60, 58J05, 35B44 ,Nabla symbol ,[MATH]Mathematics [math] ,0101 mathematics ,Mathematics - Abstract
In this expository paper, we consider the Hardy-Schr\"odinger operator $-\Delta -\gamma/|x|^2$ on a smooth domain \Omega of R^n with 0\in\bar{\Omega}, and describe how the location of the singularity 0, be it in the interior of \Omega or on its boundary, affects its analytical properties. We compare the two settings by considering the optimal Hardy, Sobolev, and the Caffarelli-Kohn-Nirenberg inequalities. The latter rewrites: $C(\int_{\Omega}\frac{u^{p}}{|x|^s}dx)^{\frac{2}{p}}\leq \int_{\Omega} |\nabla u|^2dx-\gamma \int_{\Omega}\frac{u^2}{|x|^2}dx$ for all $u\in H^1_0(\Omega)$, where \gamma, Comment: Expository paper. 48 pages
- Published
- 2015
17. Interval Oscillation Criteria for Second-Order Damped Differential Equations with Mixed Nonlinearities
- Author
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Zhenlai Han, Dian-Wu Yang, Meirong Xu, and Yibing Sun
- Subjects
Mathematics::Commutative Algebra ,Oscillation ,Differential equation ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Order (ring theory) ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Interval (graph theory) ,0101 mathematics ,Quotient ,Mathematics - Abstract
In this paper, we consider the interval oscillation criteria for second-order damped differential equations with mixed nonlinearities $$\begin{aligned} \left( r(t)(x'(t))^\gamma \right) '+p(t)(x'(t))^\gamma +\sum ^n_{i=0}q_i(t)\left| x(g_i(t))\right| ^{\alpha _i}\text {sgn}\ x(g_i(t))=e(t), \end{aligned}$$ where $$\gamma $$ is a quotient of odd positive integers, $$\alpha _0=\gamma , \alpha _i>0, i=1,\ 2,\ldots ,n$$ with $$r,\ p,\ e$$ , and $$q_i\in C([t_0,\infty ),\mathbb {R}), r(t)>0, g_i:\ \mathbb {R}\rightarrow \mathbb {R}$$ are nondecreasing continuous functions on $$\mathbb {R}$$ and $$\lim _{t\rightarrow \infty }g_i(t)=\infty , i=0,\ 1,\ 2,\ldots ,n.$$ Our results in this paper extend and improve some known results. Some examples are given here to illustrate our main results.
- Published
- 2015
18. Fractional Laplacian equations with critical Sobolev exponent
- Author
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Raffaella Servadei and Enrico Valdinoci
- Subjects
General Mathematics ,Integrodifferential operators ,Mathematical analysis ,Existence theorem ,Mountain Pass Theorem ,Linking Theorem ,Critical nonlinearities ,Best fractional critical Sobolev constant ,Palais-Smale condition ,Variational techniques ,Fractional Laplacian ,Lambda ,Omega ,Sobolev space ,Combinatorics ,Elliptic curve ,Mountain pass theorem ,Exponent ,Laplace operator ,Mathematics - Abstract
In this paper we complete the study of the following elliptic equation driven by a general non-local integrodifferential operator $$\mathcal {L}_K$$ $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \mathcal {L}_K u+\lambda u+|u|^{2^*-2}u+f(x, u)=0 &{} \hbox {in } \Omega \\ u=0 &{} \hbox {in } {\mathbb {R}}^n{\setminus } \Omega , \end{array}\right. \end{aligned}$$ that was started by Servadei and Valdinoci (Commun Pure Appl Anal 12(6):2445–2464, 2013). Here $$s\in (0,1),\, \Omega $$ is an open bounded set of $${\mathbb {R}}^n,\, n>2s$$ , with continuous boundary, $$\lambda $$ is a positive real parameter, $$2^*=2n/(n-2s)$$ is a fractional critical Sobolev exponent and $$f$$ is a lower order perturbation of the critical power $$|u|^{2^*-2}u$$ , while $$\mathcal {L}_K$$ is the integrodifferential operator defined as $$\begin{aligned} \mathcal {L}_Ku(x)= \int _{{\mathbb {R}}^n}\left( u(x+y)+u(x-y)-2u(x)\right) K(y)\,dy, \quad x\in {\mathbb {R}}^n. \end{aligned}$$ Under suitable growth condition on $$f$$ , we show that this problem admits non-trivial solutions for any positive parameter $$\lambda $$ . This existence theorem extends some results obtained in [15, 19, 20]. In the model case, that is when $$K(x)=|x|^{-(n+2s)}$$ (this gives rise to the fractional Laplace operator $$-(-\Delta )^s$$ ), the existence result proved along the paper may be read as the non-local fractional counterpart of the one obtained in [12] (see also [9]) in the framework of the classical Laplace equation with critical nonlinearities.
- Published
- 2015
19. Two pairs of families of polyhedral norms versus $$\ell _p$$ ℓ p -norms: proximity and applications in optimization
- Author
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Jun-ya Gotoh and Stan Uryasev
- Subjects
021103 operations research ,Linear programming ,General Mathematics ,Mathematical analysis ,0211 other engineering and technologies ,020206 networking & telecommunications ,Single parameter ,02 engineering and technology ,Limiting ,Combinatorics ,0202 electrical engineering, electronic engineering, information engineering ,Convex combination ,Software ,Dual norm ,Mathematics ,Dual pair - Abstract
This paper studies four families of polyhedral norms parametrized by a single parameter. The first two families consist of the CVaR norm (which is equivalent to the D-norm, or the largest-$$k$$k norm) and its dual norm, while the second two families consist of the convex combination of the $$\ell _1$$l1- and $$\ell _\infty $$l?-norms, referred to as the deltoidal norm, and its dual norm. These families contain the $$\ell _1$$l1- and $$\ell _\infty $$l?-norms as special limiting cases. These norms can be represented using linear programming (LP) and the size of LP formulations is independent of the norm parameters. The purpose of this paper is to establish a relation of the considered LP-representable norms to the $$\ell _p$$lp-norm and to demonstrate their potential in optimization. On the basis of the ratio of the tight lower and upper bounds of the ratio of two norms, we show that in each dual pair, the primal and dual norms can equivalently well approximate the $$\ell _p$$lp- and $$\ell _q$$lq-norms, respectively, for $$p,q\in [1,\infty ]$$p,q?[1,?] satisfying $$1/p+1/q=1$$1/p+1/q=1. In addition, the deltoidal norm and its dual norm are shown to have better proximity to the $$\ell _p$$lp-norm than the CVaR norm and its dual. Numerical examples demonstrate that LP solutions with optimized parameters attain better approximation of the $$\ell _{2}$$l2-norm than the $$\ell _1$$l1- and $$\ell _\infty $$l?-norms do.
- Published
- 2015
20. Trigonometric multipliers on real periodic Hardy spaces
- Author
-
S. Fridli
- Subjects
Combinatorics ,Multiplier (Fourier analysis) ,symbols.namesake ,General Mathematics ,Mathematical analysis ,symbols ,Order (ring theory) ,Trigonometry ,Hardy space ,Lp space ,Mathematics - Abstract
In this paper we are concerned with the boundedness of Hormander–Mihlin multipliers of order $$\,r\, (1\le r1.$$ On the other hand the boundedness extends to $$\,\mathcal H_{2\pi }$$ if $$\,r>1,$$ but not if $$\,r=1.$$ Generalizing this result in the present paper we show that the scale of Hardy spaces $$\,\mathcal H^p_{2\pi }\, (01\,$$ we give a sharp bound $$\,p_rp_r\,$$ then the Hormander–Mihlin condition of order $$\,r\,$$ is sufficient on $$\mathcal H^p_{2\pi }$$ .
- Published
- 2015
21. On Poincaré series of singularities of algebraic curves over finite fields
- Author
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Jhon Jader Mira and Karl-Otto Stöhr
- Subjects
Combinatorics ,Polynomial (hyperelastic model) ,Number theory ,Finite field ,General Mathematics ,Poincaré series ,Mathematical analysis ,Local ring ,Context (language use) ,Algebraic geometry ,Connection (algebraic framework) ,Mathematics - Abstract
Let $${\mathcal{O}}$$ be the local ring at a singularity of a geometrically integral algebraic curve defined over a finite field $${\mathbb{F}_q}$$ , and let m be the number of branches centered at the singularity. In a previous paper the second author extended the notion of partial local zeta-functions, by considering for each pair of $${\mathcal O}$$ -ideals $${\mathfrak{a}}$$ and $${\mathfrak{b}}$$ a Poincare series $${P(\mathfrak{a},\mathfrak{b},t_{1},\ldots ,t_{m})}$$ in m variables, which encodes cardinalities of certain finite sets of ideals. To study the behavior of these power series under blow-ups, we generalize the theory by allowing that $${\mathcal{O}}$$ is a semilocal ring of the curve. In this context we establish an Euler product identity, which provides the connection between the local and semilocal theory. We further present a procedure to compute the Poincare series, and illustrate the method by some examples of local rings. Another purpose of this paper is to study the reduction $${\mod q-1}$$ of $${P(\mathfrak{a},\mathfrak{b},t_{1},\ldots,t_{m}),}$$ which becomes a polynomial if m > 1.
- Published
- 2015
22. Global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant
- Author
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Chunlai Mu, Ke Lin, Liangchen Wang, and Jie Zhao
- Subjects
Combinatorics ,Homogeneous ,Applied Mathematics ,General Mathematics ,Domain (ring theory) ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Neumann boundary condition ,General Physics and Astronomy ,Nabla symbol ,Diffusion function ,Omega ,Mathematics - Abstract
This paper deals with an initial-boundary value problem for the chemotaxis system $$\left\{\begin{array}{ll} u_t = \nabla \cdot (D (u) \nabla u)- \nabla \cdot (u \nabla v), \quad & x\in \Omega, \quad t > 0, \\ v_t= \Delta v-uv, \quad & x \in \Omega, \quad t > 0, \end{array}\right.$$ under homogeneous Neumann boundary conditions in a convex smooth bounded domain \({\Omega\subset \mathbb{R}^n}\) with \({n\geq3}\), where the diffusion function D(u) satisfying $$\begin{array}{ll}D(u)\geq c_Du^{m-1}\quad\text{for all}\,\,u > 0 \end{array}$$ with some cD > 0 and m > 1. The main goal of this paper was to extend a previous result on global existence of solutions by Wang et al. (Z Angew Math Phys 65:1137–1152, 2014) under the condition that \({m > 2-\frac{2}{n}}\) can be relaxed to \({m > 2-\frac{6}{n+4}}\).
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- 2015
23. A note on convexity of convolutions of harmonic mappings
- Author
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Antti Rasila, Yue-Ping Jiang, and Yong Sun
- Subjects
ta113 ,ta112 ,General Mathematics ,Operator (physics) ,Mathematical analysis ,ta111 ,Regular polygon ,Harmonic (mathematics) ,Unit disk ,Convexity ,Combinatorics ,Domain (ring theory) ,Simply connected space ,Convex function ,ta512 ,Mathematics - Abstract
In this paper, we study right half-plane harmonic mappingsf 0 and f, where f 0 is xed and f is such that its dilatation of a conformalautomorphism of the unit disk. We obtain a su cient condition for theconvolution of such mappings to be convex in the direction of the realaxis. The result of the paper is a generalization of the result of by Li andPonnusamy [11], which itself originates from a problem posed by Dor etal. in [7]. 1. IntroductionLet D = fz2C : jzj
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- 2015
24. The maximal operator of Marcinkiewicz-Fejér means with respect to Walsh-Kaczmarz-Fourier series
- Author
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Káaroly Nagy
- Subjects
Series (mathematics) ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Hardy space ,Space (mathematics) ,Shift operator ,Combinatorics ,symbols.namesake ,Multiplication operator ,Bounded function ,symbols ,Order (group theory) ,Fourier series ,Mathematics - Abstract
In the paper [4, Theorem 1] Gat, Goginava and the author proved that the maximal operator σκ,∗ of Marcinkiewicz-Fejer means of Walsh-Kaczmarz-Fourier series, is bounded from the dyadic Hardy space Hp into the space Lp for p > 2/3 . Moreover, Goginava and the author showed that σκ,∗ is not bounded from the Hardy space H2/3 to the space L2/3 [6, Theorem 1]. The main aim of this paper is to show that the maximal operator σκ,∗ f := supn∈P |σκ n f | log3/2(n+1) , is bounded from the Hardy space H2/3 into the space L2/3. Moreover, we prove that the order of deviant behavior of the n th Walsh-Kacmarz-Marcinkiewicz-Fejer mean is exactly log3/2(n+1) in the endpoint p = 2/3 . Mathematics subject classification (2010): 42C10.
- Published
- 2015
25. Oscillation criteria for third order nonlinear delay differential equations with damping
- Author
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Said R. Grace
- Subjects
Third order nonlinear ,Differential equation ,Oscillation ,General Mathematics ,lcsh:T57-57.97 ,Second order equation ,Mathematical analysis ,Zero (complex analysis) ,Delay differential equation ,oscillation ,third order ,Prime (order theory) ,Combinatorics ,lcsh:Applied mathematics. Quantitative methods ,delay differential equation ,Mathematics - Abstract
This note is concerned with the oscillation of third order nonlinear delay differential equations of the form \[\label{*} \left( r_{2}(t)\left( r_{1}(t)y^{\prime}(t)\right)^{\prime}\right)^{\prime}+p(t)y^{\prime}(t)+q(t)f(y(g(t)))=0.\tag{\(\ast\)}\] In the papers [A. Tiryaki, M. F. Aktas, Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping, J. Math. Anal. Appl. 325 (2007), 54-68] and [M. F. Aktas, A. Tiryaki, A. Zafer, Oscillation criteria for third order nonlinear functional differential equations, Applied Math. Letters 23 (2010), 756-762], the authors established some sufficient conditions which insure that any solution of equation (\(\ast\)) oscillates or converges to zero, provided that the second order equation \[\left( r_{2}(t)z^{\prime }(t)\right)^{\prime}+\left(p(t)/r_{1}(t)\right) z(t)=0\tag{\(\ast\ast\)}\] is nonoscillatory. Here, we shall improve and unify the results given in the above mentioned papers and present some new sufficient conditions which insure that any solution of equation (\(\ast\)) oscillates if equation (\(\ast\ast\)) is nonoscillatory. We also establish results for the oscillation of equation (\(\ast\)) when equation (\(\ast\ast\)) is oscillatory.
- Published
- 2015
26. Ostrowski type inequalities for s-logarithmically convex functions in the second sense with applications
- Author
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Mevlut Tunc, Ahmet Ocak Akdemir, Belirlenecek, and Akdemir, Ahmet Ocak -- 0000-0003-2466-0508
- Subjects
Convex analysis ,General Mathematics ,Ostrowski type inequalities ,Mathematical analysis ,Function (mathematics) ,Subderivative ,Combinatorics ,Logarithmically convex function ,Convex optimization ,Order (group theory) ,Interval (graph theory) ,s-logarithmically convex functions ,Convex function ,Mathematics - Abstract
In this paper, we establish some new Ostrowski type inequalities for s logarithmically convex functions. Some applications of our results to P.D.F.s and in numerical integration are given. 1. INTRODUCTION Let f : I [0;1] ! R be a di¤erentiable mapping on I , the interior of the interval I, such that f 0 2 L [a; b] where a; b 2 I with a < b. If jf 0 (x)j M , then the following inequality holds (see [2]): (1.1) f(x) 1 b a Z b a f(u)du M b a " (x a) + (b x) 2 # : This inequality is well known in the literature as the Ostrowski inequality. For some results which generalize, improve and extend the inequality (1.1) see ([2]-[6]) and the references therein. Let us recall some known de nitions and results which we will use in this paper. The function f : I R ! [0;1) is said to be log convex or multiplicatively convex if log t is convex, or, equivalenly, if for all x; y 2 I and t 2 [0; 1] ; one has the inequality (See [7], p.7): f (tx+ (1 t) y) [f (x)] + [f (y)] t) : In [1], Akdemir and Tunc were introduced the class of s logarithmically convex functions in the rst sense as the following: De nition 1. A function f : I R0 ! R+ is said to be s logarithmically convex in the rst sense if (1.2) f ( x+ y) [f (x)] s [f (y)] s for some s 2 (0; 1], where x; y 2 I and s + s = 1: In [8], authors introduced the class of s logarithmically convex functions in the second sense as the following: De nition 2. A function f : I R0 ! R+ is said to be s logarithmically convex in the second sense if (1.3) f (tx+ (1 t) y) [f (x)] s [f (y)] (1 t) 2000 Mathematics Subject Classi cation. 26D10, 26A15, 26A16, 26A51. Key words and phrases. Hadamards inequality, s geometrically convex functions . 1 2 MEVLUT TUNC AND AHMET OCAK AKDEMIR for some s 2 (0; 1], where x; y 2 I and t 2 [0; 1]. Clearly, when taking s = 1 in De nition 1.2 or De nition 1.3, then f becomes the standard logarithmically convex function on I. The main purpose of this paper is to establish some new Ostrowskis type inequalities for s logarithmically convex functions. We also give some applications to P.D.F.s and to midpoint formula. 2. THE NEW RESULTS In order to prove our main results, we will use following Lemma which was used by Alomari and Darus (see [3]): Lemma 1. Let f : I R ! R, be a di¤erentiable mapping on I where a; b 2 I, with a < b. Let f 0 2 L[a; b]; then the following equality holds
- Published
- 2014
27. Methods of analysis of the condition for correct solvability in L p (ℝ) of general Sturm-Liouville equations
- Author
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Nina A. Chernyavskaya and Leonid A. Shuster
- Subjects
Combinatorics ,Implicit function ,General Mathematics ,Ordinary differential equation ,Mathematical analysis ,Order (ring theory) ,Sturm–Liouville theory ,Function (mathematics) ,Auxiliary function ,Mathematics - Abstract
We consider the equation $$ - (r(x)y'(x))' + q(x)y(x) = f(x),x \in \mathbb{R}$$ (*) where f ∈ Lp(ℝ), p ∈ (1,∞) and $$r > 0,q \geqslant 0,\frac{1} {r} \in L_1^{loc} (\mathbb{R}),q \in L_1^{loc} (\mathbb{R})$$ , $$\mathop {\lim }\limits_{|d| \to \infty } \int_{x - d}^x {\frac{{dt}} {{r(t)}}} \cdot \int_{x - d}^x {q(t)dt = \infty } $$ . In an earlier paper, we obtained a criterion for correct solvability of (*) in Lp(ℝ), p ∈ (1,∞). In this criterion, we use values of some auxiliary implicit functions in the coefficients r and q of equation (*). Unfortunately, it is usually impossible to compute values of these functions. In the present paper we obtain sharp by order, two-sided estimates (an estimate of a function f(x) for x ∈ (a, b) through a function g(x) is sharp by order if c−1|g(x)| ⩽ |f(x)| ⩽ c|g(x)|, x ∈ (a, b), c = const) of auxiliary functions, which guarantee efficient study of the problem of correct solvability of (*) in Lp(ℝ), p ∈ (1,∞).
- Published
- 2014
28. Toward a classification of killing vector fields of constant length on pseudo-riemannian normal homogeneous spaces
- Author
-
Fabio Podestà, Joseph A. Wolf, and Ming Xu
- Subjects
Mathematics - Differential Geometry ,General Mathematics ,math-ph ,FOS: Physical sciences ,Context (language use) ,01 natural sciences ,Combinatorics ,Killing vector field ,math.MP ,0103 physical sciences ,FOS: Mathematics ,Generalized flag variety ,0101 mathematics ,Moment map ,Mathematical Physics ,Mathematics ,Algebra and Number Theory ,math.SG ,Simple Lie group ,010102 general mathematics ,Mathematical analysis ,Homogeneous spaces, Killing fields, moment map ,Mathematical Physics (math-ph) ,Pure Mathematics ,Compact space ,math.DG ,Differential Geometry (math.DG) ,Mathematics - Symplectic Geometry ,Homogeneous space ,Symplectic Geometry (math.SG) ,010307 mathematical physics ,Geometry and Topology ,Constant (mathematics) ,Analysis - Abstract
In this paper we develop the basic tools for a classification of Killing vector fields of constant length on pseudo-Riemannian homogeneous spaces. This extends a recent paper of M. Xu and J. A. Wolf, which classified the pairs $(M,\xi)$ where $M = G/H$ is a Riemannian normal homogeneous space, G is a compact simple Lie group, and $\xi \in \mathfrak{g}$ defines a nonzero Killing vector field of constant length on $M$. The method there was direct computation. Here we make use of the moment map $M \to \mathfrak{g}^{*}$ and the flag manifold structure of $\mathrm{Ad} (G) \xi$ to give a shorter, more geometric proof which does not require compactness and which is valid in the pseudo-Riemannian setting. In that context we break the classification problem into three parts. The first is easily settled. The second concerns the cases where $\xi$ is elliptic and $G$ is simple (but not necessarily compact); that case is our main result here. The third, which remains open, is a more combinatorial problem involving elements of the first two.
- Published
- 2017
29. A NOTE ON RECURRENCE FORMULA FOR VALUES OF THE EULER ZETA FUNCTIONS ζE(2n) AT POSITIVE INTEGERS
- Author
-
H. Y. Lee and Cheon Seoung Ryoo
- Subjects
General Mathematics ,Recurrence formula ,Open problem ,Mathematical analysis ,Special values ,Riemann zeta function ,Combinatorics ,Arithmetic zeta function ,symbols.namesake ,symbols ,Euler's formula ,Fourier series ,Prime zeta function ,Mathematics - Abstract
The Euler zeta function is defined by ζ E (s)=P ∞n=1(−1) n−1s .The purpose of this paper is to find formulas of the Euler zeta func-tion’s values. In this paper, for s ∈ N we find the recurrence formula ofζ E (2s) using the Fourier series. Also we find the recurrence formula ofP ∞n=1(−1) n−1 (2n−1) 2s−1 , where s ≥ 2(∈ N). 1. IntroductionThe Euler zeta function is defined by ζ E (s) =P ∞n=1(−1) n n s (see [3, 4]). Inthis paper we investigate the recurrence formula of the Euler zeta function fors = 2n with Fourier series. By this result we can find ζ E (2n) for all n ∈ N.For s ∈ C, the Riemann zeta function or the Euler-Riemann zeta function,ζ(s) is defined byζ(s) =X ∞n=1 1n s (s ∈ C), (see [5, 6])which converges when the real part of s is greater than 1. R. Ap´ery provedthat the number ζ(3) is irrational. But it is still an open problem to proveirrationality of ζ(2k +1) for the long time.As well known special values, for any positive even number 2n,ζ(2n) = (−1) n+1 B 2n (2π) 2n 2(2n)!, (see [1])where B
- Published
- 2014
30. The limits on boundary of orbifold Kähler–Einstein metrics and Kähler–Ricci flows over quasi-projective manifolds
- Author
-
Shin Kikuta
- Subjects
Mathematics::Complex Variables ,Divisor ,General Mathematics ,High Energy Physics::Phenomenology ,Mathematical analysis ,Boundary (topology) ,Omega ,Manifold ,Combinatorics ,symbols.namesake ,symbols ,Mathematics::Differential Geometry ,Einstein ,Mathematics::Symplectic Geometry ,Orbifold ,Mathematics - Abstract
In this paper, we consider a sequence of orbifold Kahler–Einstein metrics \((\omega _{X_m})_m\) and normalized Kahler–Ricci flows \((\omega _{X_m}(t))_m\) on a quasi-projective manifold \(X=\overline{X} {\setminus } \overline{D}\) for a projective manifold \(\overline{X}\) and a divisor \(\overline{D}\) of \(\overline{X}\) with simple normal crossings such that \(K_{\overline{X}}+\overline{D}\) is ample. For sufficiently large \(m, \omega _{X_m}\) or \(\omega _{X_m}(t)\) is the orbifold Kahler–Einstein metric or orbifold normalized Kahler–Ricci flow on \(X\) equipped with the orbifold structure with respect to the divisor \(1/m\overline{D}\), respectively. The main theorem of this paper is that the limit on the boundary \(\overline{D}\) or the inside \(X\) of the sequence \((\omega _{X_m}(t))_m\) as \(m \rightarrow \infty \) is exactly the complete normalized Kahler–Ricci flow on \(\overline{D}\) or \(X\), respectively. Moreover our method of the proof also leads to a result that on the boundary \(\overline{D}\) the sequence \((\omega _{X_m})_m\) converges to the complete Kahler–Einstein metric on \(\overline{D}\) as \(m \rightarrow \infty \).
- Published
- 2014
31. Explicit stationary distribution of the (L, 1)-reflecting random walk on the half line
- Author
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Q. Zhao, Yi Qiang, Wenming Hong, and Ke Zhou
- Subjects
Combinatorics ,Heterogeneous random walk in one dimension ,Stationary distribution ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Structure (category theory) ,Half line ,Expression (computer science) ,Stationary sequence ,Random walk ,Positive recurrence ,Mathematics - Abstract
In this paper, we consider the (L, 1) state-dependent reflecting random walk (RW) on the half line, which is an RW allowing jumps to the left at a maximal size L. For this model, we provide an explicit criterion for (positive) recurrence and an explicit expression for the stationary distribution. As an application, we prove the geometric tail asymptotic behavior of the stationary distribution under certain conditions. The main tool employed in the paper is the intrinsic branching structure within the (L, 1)-random walk.
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- 2014
32. On a class of Kirchhoff type problems
- Author
-
Zeng Liu and Yisheng Huang
- Subjects
Combinatorics ,Class (set theory) ,Kirchhoff type ,General Mathematics ,Mathematical analysis ,Nabla symbol ,Lambda ,Energy (signal processing) ,Mathematics - Abstract
In this paper we consider the following Kirchhoff type problem: $$(\mathcal{K}) \quad \left(1 + \lambda \int\limits_{\mathbb{R}^3}\big(|\nabla u|^2 + V(y)u^2dy\big)\right)[-\Delta u + V(x)u] = |u|^{p-2}u, \quad {\rm in} \, \mathbb{R}^3,$$ where $${p\in (2, 6)}$$ , λ > 0 is a parameter, and V(x) is a given potential. Some existence and nonexistence results are obtained by using variational methods. Also, the “energy doubling” property of nodal solutions of $${(\mathcal{K})}$$ is discussed in this paper.
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- 2014
33. THE HYPONORMAL TOEPLITZ OPERATORS ON THE VECTOR VALUED BERGMAN SPACE
- Author
-
Yufeng Lu, Yanyue Shi, and Puyu Cui
- Subjects
Measurable function ,General Mathematics ,Mathematical analysis ,Hilbert space ,Space (mathematics) ,Toeplitz matrix ,Combinatorics ,symbols.namesake ,Tensor product ,Square-integrable function ,Bergman space ,Bounded function ,symbols ,Mathematics - Abstract
In this paper, we give a necessary and sufficient condition for the hyponormality of the block Toeplitz operators T�, where � = F +G∗, F(z), G(z) are some matrix valued polynomials on the vector valued Bergman space L2(D,Cn). We also show some necessary conditions for the hyponormality of TF+Gwith F + G∗ 2 h∞ Mn×n on L2(D,Cn). Let D and T be the open unit disk and unit circle in the complex plane C re- spectively and dA be the normalized Lebesgue area measure on D. h ∞ denotes the space of all bounded harmonic functions on D. L ∞ (D,dA) and L 2 (D,dA) denote the space of essential bounded measurable functions and the space of the square integrable functions on D with respect to dA, respectively. The Bergman space L 2 consists of all analytic functions in L 2 (D,dA). We denote the space of vector valued square integrable functions on D by L 2 (D,C n ) = L 2 (D,dA)⊗C n and the vector valued Bergman space on D by L 2(D,C n ) = L 2 ⊗ C n , respec- tively, where ⊗ denotes the Hilbert space tensor product. In this paper F T denotes the transpose of the matrix F and G ∗ denotes the adjoint of the matrix G.
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- 2014
34. Quasilinear parabolic variational inequalities with multi-valued lower-order terms
- Author
-
Vy Khoi Le and Siegfried Carl
- Subjects
Applied Mathematics ,General Mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Regular polygon ,Solution set ,General Physics and Astronomy ,Function (mathematics) ,Domain (mathematical analysis) ,Combinatorics ,Elliptic operator ,Directed set ,Obstacle problem ,Variational inequality ,Mathematics - Abstract
In this paper, we provide an analytical frame work for the following multi-valued parabolic variational inequality in a cylindrical domain \({Q = \Omega \times (0, \tau)}\) : Find \({{u \in K}}\) and an \({{\eta \in L^{p'}(Q)}}\) such that $$\eta \in f(\cdot,\cdot,u), \quad \langle u_t + Au, v - u\rangle + \int_Q \eta (v - u)\,{\rm d}x{\rm d}t \ge 0, \quad \forall \, v \in K,$$ where \({{K \subset X_0 = L^p(0,\tau;W_0^{1,p}(\Omega))}}\) is some closed and convex subset, A is a time-dependent quasilinear elliptic operator, and the multi-valued function \({{s \mapsto f(\cdot,\cdot,s)}}\) is assumed to be upper semicontinuous only, so that Clarke’s generalized gradient is included as a special case. Thus, parabolic variational–hemivariational inequalities are special cases of the problem considered here. The extension of parabolic variational–hemivariational inequalities to the general class of multi-valued problems considered in this paper is not only of disciplinary interest, but is motivated by the need in applications. The main goals are as follows. First, we provide an existence theory for the above-stated problem under coercivity assumptions. Second, in the noncoercive case, we establish an appropriate sub-supersolution method that allows us to get existence, comparison, and enclosure results. Third, the order structure of the solution set enclosed by sub-supersolutions is revealed. In particular, it is shown that the solution set within the sector of sub-supersolutions is a directed set. As an application, a multi-valued parabolic obstacle problem is treated.
- Published
- 2013
35. On uniqueness in the extended Selberg class of Dirichlet series
- Author
-
Bao Qin Li and Haseo Ki
- Subjects
Applied Mathematics ,General Mathematics ,Mathematical analysis ,Dirichlet eta function ,Class number formula ,Riemann zeta function ,Combinatorics ,Dirichlet kernel ,symbols.namesake ,Riemann hypothesis ,Selberg trace formula ,symbols ,Selberg class ,Dirichlet series ,Mathematics - Abstract
We will show that two functions in the extended Selberg class satisfying the same functional equation must be identically equal if they have sufficiently many common zeros. This paper concerns the question of how L-functions are determined by their zeros. L-functions are Dirichlet series with the Riemann zeta function ζ(s) = ∑∞ n=1 1 ns as the prototype and are important objects in number theory. The Selberg class S of L-functions is the set of all Dirichlet series L(s) = ∑∞ n=1 a(n) ns of a complex variable s = σ + it with a(1) = 1, satisfying the following axioms (see [7]): (i) (Dirichlet series) For σ > 1, L(s) is an absolutely convergent Dirichlet series. (ii) (Analytic continuation) There is a nonnegative integer k such that (s − 1)L(s) is an entire function of finite order. (iii) (Functional equation) L satisfies a functional equation of type ΛL(s) = ωΛL(1− s), where ΛL(s) = L(s)Q s ∏K j=1 Γ(λjs+μj) with positive real numbers Q, λj and with complex numbers μj , ω with Reμj ≥ 0 and |ω| = 1. (iv) (Ramanujan hypothesis) a(n) n for every e > 0; (v) (Euler product) logL(s) = ∑∞ n=1 b(n) ns , where b(n) = 0 unless n is a positive power of a prime and b(n) n for some θ < 12 . The degree dL of an L-function L is defined to be dL = 2 ∑K j=1 λj , where K, λj are the numbers in axiom (iii). The Selberg class includes the Riemann zeta-function ζ and essentially those Dirichlet series where one might expect the analogue of the Riemann hypothesis. At the same time, there are a whole host of interesting Dirichlet series not possessing a Euler product (see e.g. [3], [8]). Throughout the paper, all L-functions are assumed to be functions from the extended Selberg class of those only satisfying the axioms (i)-(iii) (see [3]). Thus, the results obtained in the present paper particularly apply to L-functions in the Selberg class. Received by the editors October 5, 2011 and, in revised form, February 12, 2012. 2010 Mathematics Subject Classification. Primary 11M36, 30D30.
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- 2013
36. On the existence and non-existence of positive solutions for a class of singular infinite semipositone problems
- Author
-
S. H. Rasouli
- Subjects
Combinatorics ,Class (set theory) ,General Mathematics ,Bounded function ,Domain (ring theory) ,Mathematical analysis ,Zero (complex analysis) ,Nabla symbol ,Function (mathematics) ,Lambda ,Omega ,Mathematics - Abstract
In this paper we consider the existence and non-existence of positive solutions of singular nonlinear semipositone problem of the form $$\left\{ \begin{gathered} - div(|x|^{ - ap} |\nabla u|^{p - 2} \nabla u) = \lambda |x|^{ - (a + 1)p + b} (f(u) - \frac{1} {{u^\alpha }}),x \in \Omega , \hfill \\ u = 0,x \in \partial \Omega , \hfill \\ \end{gathered} \right. $$ where Ω is a bounded smooth domain of RN with 0 ∈ Ω, 1 < p < N, 0 ≤ a < \(\tfrac{{N - p}} {p} \), α ∈ (0, 1), and b, λ are positive parameters. Here f : (0, ∞) → (0, ∞) is C2 function. Our aim in this paper is to establish non-existence of positive solution for λ near zero and existence of positive solution for λ large. We use the method of sub-super solutions to establish our existence result.
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- 2013
37. Generalized Hilbert operators on weighted Bergman spaces
- Author
-
Jouni Rättyä and José Ángel Peláez
- Subjects
Weight function ,Mathematics - Complex Variables ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,Combinatorics ,Compact space ,Type condition ,Operator (computer programming) ,Bergman space ,Bounded function ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Connection (algebraic framework) ,Primary 30H20, Secondary 47G10 ,Mathematics - Abstract
The main purpose of this paper is to study the generalized Hilbert operator {equation*} \mathcal{H}_g(f)(z)=\int_0^1f(t)g'(tz)\,dt {equation*} acting on the weighted Bergman space $A^p_\om$, where the weight function $\om$ belongs to the class $\R$ of regular radial weights and satisfies the Muckenhoupt type condition {equation}\label{Mpconditionaabstract} \sup_{0\le r, Comment: This paper has been accepted for publication in Advances in Mathematics
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- 2013
38. A quantitative version of Krein’s theorems for Fréchet spaces
- Author
-
Manuel López-Pellicer, Albert Kubzdela, Carlos Angosto, and J. Ka̧kol
- Subjects
Mathematics::Functional Analysis ,Compactness ,Bounded set ,General Mathematics ,Mathematical analysis ,Banach space ,Space (mathematics) ,Combinatorics ,Compact space ,Krein's theorem ,Relatively compact subspace ,Fréchet space ,Metrization theorem ,Locally convex topological vector space ,Space of continuous functions ,MATEMATICA APLICADA ,Mathematics - Abstract
For a Banach space E and its bidual space E'', the function k(H) defined on bounded subsets H of E measures how far H is from being σ(E,E')-relatively compact in E. This concept, introduced independently by Granero, and Cascales et al., has been used to study a quantitative version of Krein¿s theorem for Banach spaces E and spaces Cp(K) over compact K. In the present paper, a quantitative version of Krein¿s theorem on convex envelopes coH of weakly compact sets H is proved for Fréchet spaces, i.e. metrizable and complete locally convex spaces. For a Fréchet space E, the above function k(H) has been defined in thisi paper by menas of d(h,E) is the natural distance of h to E in the bidual E''. The main result of the paper is the following theorem: For a bounded set H in a Fréchet space E, the following inequality holds k(coH) < (2^(n+1) − 2)k(H) + 1/2^n for all n ∈ N. Consequently, this yields also the following formula k(coH) ≤ (k(H))^(1/2))(3-2(k(H)^(1/2))). Hence coH is weakly relatively compact provided H is weakly relatively compact in E. This extends a quantitative version of Krein¿s theorem for Banach spaces (obtained by Fabian, Hajek, Montesinos, Zizler, Cascales, Marciszewski, and Raja) to the class of Fréchet spaces. We also define and discuss two other measures of weak non-compactness lk(H) and k'(H) for a Fréchet space and provide two quantitative versions of Krein¿s theorem for both functions., The research was supported for C. Angosto by the project MTM2008-05396 of the Spanish Ministry of Science and Innovation, for J. Kakol by National Center of Science, Poland, Grant No. N N201 605340, and for M. Lopez-Pellicer by the project MTM2010-12374-E (complementary action) of the Spanish Ministry of Science and Innovation.
- Published
- 2013
39. The bounds of restricted isometry constants for low rank matrices recovery
- Author
-
Song Li and HuiMin Wang
- Subjects
Combinatorics ,Matrix (mathematics) ,General Mathematics ,Norm (mathematics) ,Mathematical analysis ,Linear system ,Matrix norm ,Schatten norm ,Schatten class operator ,Upper and lower bounds ,Mathematics ,Restricted isometry property - Abstract
This paper discusses conditions under which the solution of linear system with minimal Schatten-p norm, 0 < p ⩽ 1, is also the lowest-rank solution of this linear system. To study this problem, an important tool is the restricted isometry constant (RIC). Some papers provided the upper bounds of RIC to guarantee that the nuclear-norm minimization stably recovers a low-rank matrix. For example, Fazel improved the upper bounds to δ 4 < 0.558 and δ 3 < 0.4721, respectively. Recently, the upper bounds of RIC can be improved to δ 2 < 0.307. In fact, by using some methods, the upper bounds of RIC can be improved to δ 2 < 0.4931 and δ 2 < 0.309. In this paper, we focus on the lower bounds of RIC, we show that there exists linear maps A with δ 2 > 1/√2 or δ > 1/3 for which nuclear norm recovery fail on some matrix with rank at most r. These results indicate that there is only a little limited room for improving the upper bounds for δ 2 and δ . Furthermore, we also discuss the upper bound of restricted isometry constant associated with linear maps A for Schatten p (0 < p < 1) quasi norm minimization problem.
- Published
- 2013
40. A NOTE ON THE q-ANALOGUE OF KIM'S p-ADIC log GAMMA TYPE FUNCTIONS ASSOCIATED WITH q-EXTENSION OF GENOCCHI AND EULER NUMBERS WITH WEIGHT α
- Author
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Kyoung Ho, Mehmet Acikgoz, Park, and Serkan Aracõ
- Subjects
Combinatorics ,Rational number ,Mathematics::Number Theory ,General Mathematics ,Mathematical analysis ,Prime number ,Field (mathematics) ,Absolute value (algebra) ,Type (model theory) ,Gamma function ,Algebraic closure ,Ring of integers ,Mathematics - Abstract
In this paper, we introduce the q-analogue of p-adic log gamma functions with weight alpha. Moreover, we give a relationship be- tween weighted p-adic q-log gamma functions and q-extension of Genocchi and Euler numbers with weight alpha. Assume that p is a fixed odd prime number. Throughout this paper Z, Zp, Qp and Cp will denote the ring of integers, the field of p-adic rational numbers and the completion of the algebraic closure of Qp, respectively. Also we denote N � = N ( {0} and exp(x) = e x . Let vp : Cp ! Q ( {1} (Q is the field of rational numbers) denote the p-adic valuation of Cp normalized so that vp (p) = 1. The absolute value on Cp will be denoted as |·|, and |x| p = p v p(x)
- Published
- 2013
41. On the inverse diamond kernel of Marcel Riesz
- Author
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D. Maneetus and Kamsing Nonlaopon
- Subjects
Kernel (set theory) ,General Mathematics ,Operator (physics) ,Mathematical analysis ,Diamond ,Function (mathematics) ,engineering.material ,Convolution ,Combinatorics ,Mathematics Subject Classification ,Schwartz space ,engineering ,Order (group theory) ,Mathematics - Abstract
In this paper, we define the diamond Marcel Riesz operator of order (α, β) on the function f by M (f) = Kα,β ∗ f, where Kα,β is diamond kernel of Marcel Riesz, α, β ∈ C, the symbol ∗ designates the convolution, and f ∈ S, S is the Schwartz space of functions. Our purpose of this paper is to obtain the operator N (α,β) = [ M (α,β) ]−1 such that if M (α,β)(f) = φ, then N (α,β)φ = f. Our results generalize formulae appearing in A. Kananthai [On the convolutions of the diamond kernel of Marcel Riesz, Applied Mathematics and Computation, 114(2000), 95 − 101]. Mathematics Subject Classification: 46F10, 46F12
- Published
- 2013
42. Existence of critical elliptic systems with boundary singularities
- Author
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Yimin Zhou and Jianfu Yang
- Subjects
Mean curvature ,Elliptic systems ,critical Hardy-Sobolev exponent ,General Mathematics ,lcsh:T57-57.97 ,Mathematical analysis ,existence ,Boundary (topology) ,nonlinear system ,Omega ,Delta-v (physics) ,Combinatorics ,Domain (ring theory) ,lcsh:Applied mathematics. Quantitative methods ,Exponent ,Gravitational singularity ,compactness ,Mathematics - Abstract
In this paper, we are concerned with the existence of positive solutions of the following nonlinear elliptic system involving critical Hardy-Sobolev exponent \begin{equation*}\label{eq:1}(*) \left\{ \begin{array}{lll} -\Delta u= \frac{2\alpha}{\alpha+\beta}\frac{u^{\alpha-1}v^\beta}{|x|^s}-\lambda u^p, & \quad {\rm in}\quad \Omega,\\[2mm] -\Delta v= \frac{2\beta}{\alpha+\beta}\frac{u^\alpha v^{\beta-1}}{|x|^s}-\lambda v^p, & \quad {\rm in}\quad \Omega,\\[2mm] u\gt 0, v\gt 0, &\quad {\rm in}\quad \Omega,\\[2mm] u=v=0, &\quad {\rm on}\quad \partial\Omega, \end{array} \right. \end{equation*} where \(N\geq 4\) and \(\Omega\) is a \(C^1\) bounded domain in \(\mathbb{R}^N\) with \(0\in\partial\Omega\). \(0\lt s \lt 2\), \(\alpha+\beta=2^*(s)=\frac{2(N-s)}{N-2}\), \(\alpha,\beta\gt 1\), \(\lambda\gt 0\) and \(1 \lt p\lt \frac{N+2}{N-2}\). The case when 0 belongs to the boundary of \(\Omega\) is closely related to the mean curvature at the origin on the boundary. We show in this paper that problem \((*)\) possesses at least a positive solution.
- Published
- 2013
43. Global smooth flows for compressible Navier–Stokes–Maxwell equations
- Author
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Jiang Xu and Hongmei Cao
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Physics and Astronomy ,Type (model theory) ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,symbols.namesake ,Maxwell's equations ,Compressibility ,symbols ,Dissipative system ,Initial value problem ,Navier stokes ,0101 mathematics ,Mathematics - Abstract
Umeda et al. (Jpn J Appl Math 1:435–457, 1984) considered a rather general class of symmetric hyperbolic–parabolic systems: $$A^{0}z_{t}+\sum_{j=1}^{n}A^{j}z_{x_{j}}+Lz=\sum_{j,k=1}^{n}B^{jk}z_{x_{j}x_{k}}$$ and showed optimal decay rates with certain dissipative assumptions. In their results, the dissipation matrices $${L}$$ and $${B^{jk}(j,k=1,\ldots,n)}$$ are both assumed to be real symmetric. So far there are no general results in case that $${L}$$ and $${B^{jk}}$$ are not necessarily symmetric, which is left open now. In this paper, we investigate compressible Navier–Stokes–Maxwell (N–S–M) equations arising in plasmas physics, which is a concrete example of hyperbolic–parabolic composite systems with non-symmetric dissipation. It is observed that the Cauchy problem for N–S–M equations admits the dissipative mechanism of regularity-loss type. Consequently, extra higher regularity is usually needed to obtain the optimal decay rate of $${L^{1}({\mathbb{R}}^3)}$$ - $${L^2({\mathbb{R}}^3)}$$ type, in comparison with that for the global-in-time existence of smooth solutions. In this paper, we obtain the minimal decay regularity of global smooth solutions to N–S–M equations, with aid of $${L^p({\mathbb{R}}^n)}$$ - $${L^{q}({\mathbb{R}}^n)}$$ - $${L^{r}({\mathbb{R}}^n)}$$ estimates. It is worth noting that the relation between decay derivative orders and the regularity index of initial data is firstly found in the optimal decay estimates.
- Published
- 2016
44. On the $r$-th Root Partition Function
- Author
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Yong-Gao Chen and Ya-Li Li
- Subjects
Partition function (quantum field theory) ,Mathematics::Combinatorics ,partition function ,010308 nuclear & particles physics ,General Mathematics ,$r$-th root partition ,010102 general mathematics ,Mathematical analysis ,Root (chord) ,Computer Science::Computational Geometry ,01 natural sciences ,Combinatorics ,05A17 ,Integer ,0103 physical sciences ,11P81 ,11P82 ,11P83 ,0101 mathematics ,Mathematics ,Real number - Abstract
The well known partition function $p(n)$ has a long research history, where $p(n)$ denotes the number of solutions of the equation $n = a_1 + \cdots + a_k$ with integers $1 \leq a_1 \leq \cdots \leq a_k$. In this paper, we investigate a new partition function. For any real number $r \gt 1$, let $p_r(n)$ be the number of solutions of the equation $n = \lfloor \sqrt[r]{a_1} \rfloor + \cdots + \lfloor \sqrt[r]{a_k} \rfloor$ with integers $1 \leq a_1 \leq \cdots \leq a_k$, where $\lfloor x \rfloor$ denotes the greatest integer not exceeding $x$. In this paper, it is proved that $\exp(c_1 n^{r/(r+1)}) \leq p_r(n) \leq \exp(c_2n^{r/(r+1)})$ for two positive constants $c_1$ and $c_2$ (depending only $r$).
- Published
- 2016
45. Series Representation of Power Function
- Author
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Kolosov Petro and Odessa State University [Odessa]
- Subjects
Diophantine equations ,Exponentiation ,Math ,Binomial theorem ,arXiv.org ,Difference Equations ,kolosov_petro ,Number Theory ,Physical Sciences and Mathematics ,Newton's Binomial Theorem ,Binomial expansion ,Preprint ,Binomial coefficient ,Mathematics ,Finite differences ,Calculus ,General Medicine ,kolosov.petro ,Power function ,Cube (Algebra) ,arXiv ,Binomial Distribution ,Maths ,Multinomial coefficient ,Differential equations ,Finite difference ,Power series ,[ MATH.MATH-CA ] Mathematics [math]/Classical Analysis and ODEs [math.CA] ,Science ,[ MATH.MATH-GM ] Mathematics [math]/General Mathematics [math.GM] ,[ MATH.MATH-DS ] Mathematics [math]/Dynamical Systems [math.DS] ,Topology ,Numerical Differentiation ,Education ,Fundamental theorem of calculus ,FOS: Mathematics ,Pascal's triangle ,Newton's interpolation formula ,0000-0002-6544-8880 ,Series (mathematics) ,Classical Analysis and ODEs ,Analysis of PDEs ,High order finite difference ,Differential calculus ,Open access ,Partial derivative ,Divided difference ,Algebra ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Forward Finite Difference ,Multinomial theorem ,Binomial transform ,kolosov-petro ,Binomial series ,Ordinary differential equation ,Taylor's theorem ,Monomial ,Computer science ,Series representation ,Binomial Coefficient ,[ MATH.MATH-CO ] Mathematics [math]/Combinatorics [math.CO] ,Polynomial ,Faulhaber's formula ,Science and Mathematics Education ,Perfect cube ,Theory and Algorithms ,Computer Sciences ,Applied Mathematics ,Representation (systemics) ,Partial differential equation ,Numerical Analysis and Computation ,STEM ,High order derivative ,algebra_number_theory ,Exponential function ,Hypercube ,Analytic function ,petrokolosov ,MSC 2010: 30BXX ,Differentiation ,Polynomial expansion ,symbols ,Open science ,Power series (mathematics) ,Derivatives ,Binomial Sum ,Numerical analysis ,Numercal methods ,General Mathematics ,Kolosov Petro ,Mathematical analysis ,[ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT] ,symbols.namesake ,Mathematical Series ,Euler number ,Petro Kolosov ,Binomial Series ,Partial difference ,KolosovP ,Kolosov ,Pascal triangle ,Functional analysis ,Discrete Mathematics ,kolosov_p_1 ,Finite difference coefficient ,Derivative ,petro-kolosov ,Combinatorics ,Backward Finite Difference ,Central Finite difference ,petro.kolosov.9 ,Power (mathematics) ,Series expansion ,Analysis ,Calculus of variations - Abstract
In this paper we discuss a problem of generalization of binomial distributed triangle, that is sequence A287326 in OEIS. The main property of A287326 that it returns a perfect cube n as sum of n-th row terms over k, 0 or 1 , by means of its symmetry. In this paper we have derived a similar triangles in order to receive powers m=5,7 as row items sum and generalized obtained results in order to receive every odd-powered monomial n2m+1, m as sum of row terms of corresponding triangle. This version might be out of date, proceed by the link to see actual version., 16 pages, 8 figures, 2 tables, typos revised, added missing references, results generalized and shortened, derivations detailed.
- Published
- 2016
- Full Text
- View/download PDF
46. On the sum of powers of terms of a linear recurrence sequence
- Author
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Ana Paula Chaves, Alain Togbé, and Diego Marques
- Subjects
Combinatorics ,Sequence ,Fibonacci number ,Sums of powers ,General Mathematics ,Bounded function ,Mathematical analysis ,Recurrence sequence ,Constant (mathematics) ,Mathematics - Abstract
Let (F n ) n≥0 be the Fibonacci sequence given by F n+2 = F n+1 + F n , for n ≥ 0, where F 0 = 0 and F 1 = 1. There are several interesting identities involving this sequence such as F 2 + F +1 2 = F 2n+1, for all n ≥ 0. In a very recent paper, Marques and Togbe proved that if F + F +1 is a Fibonacci number for all sufficiently large n, then s = 1 or 2. In this paper, we will prove, in particular, that if (G m ) m is a linear recurrence sequence (under weak assumptions) and G + ... + G ∈ (G m ) m , for infinitely many positive integers n, then s is bounded by an effectively computable constant depending only on k and the parameters of G m .
- Published
- 2012
47. Integral inequalities for algebraic and trigonometric polynomials
- Author
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P. Yu. Glazyrina and Vitalii V. Arestov
- Subjects
Combinatorics ,Uniform norm ,Unit circle ,Integer ,General Mathematics ,Mathematical analysis ,Pi ,Order (ring theory) ,Algebraic number ,Complex plane ,Fractional calculus ,Mathematics - Abstract
In this paper, we consider sharp estimates of integral functionals $\int_0^{2\pi } {\phi (L|Lf_n (t)|)dt} $ for functions φ defined on the semiaxis (0, ∞) and operators L on the set T n of real trigonometric polynomials f n of order n ≥ 1 by the uniform norm $\left\| {f_n } \right\|_{C_{2\pi } } $ of the polynomials. We also consider similar problems for algebraic polynomials on the unit circle of the complex plane. P. Erdos, A. Calderon, G. Klein, L. V. Taikov, and others investigated such inequalities. In particular, we show that, for 0 ≤ q < ∞, the sharp inequality $\left\| {D^\alpha f_n } \right\|_{L_q } \leqslant n^\alpha \left\| {\cos t} \right\|_{L_q } \left\| {f_n } \right\|_\infty $ holds on the set T n , n ≥ 1, for the Weyl fractional derivatives D α f n of order α ≥ 1. For q = ∞ (α ≥ 1), this fact was proved by P.I. Lizorkin (1965). For 1 ≤ q < ∞ and positive integer α, the inequality was proved by L.V. Taikov (1965); however, in this case, the inequality follows from results of an earlier paper by A. P. Calderon and G. Klein (1951).
- Published
- 2012
48. Best approximation in polyhedral Banach spaces
- Author
-
Libor Veselý, Joram Lindenstrauss, and Vladimir P. Fonf
- Subjects
Mathematics(all) ,Numerical Analysis ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Banach space ,Hausdorff space ,Metric projection ,Codimension ,Geometric property ,Proximinal subspace ,Combinatorics ,Polyhedral Banach space ,Norm (mathematics) ,Analysis ,Subspace topology ,Quotient ,Mathematics - Abstract
In the present paper, we study conditions under which the metric projection of a polyhedral Banach space X onto a closed subspace is Hausdorff lower or upper semicontinuous. For example, we prove that if X satisfies (*) (a geometric property stronger than polyhedrality) and Y@?X is any proximinal subspace, then the metric projection P"Y is Hausdorff continuous and Y is strongly proximinal (i.e., if {y"n}@?Y, x@?X and @?y"n-x@?->dist(x,Y), then dist(y"n,P"Y(x))->0). One of the main results of a different nature is the following: if X satisfies (*) and Y@?X is a closed subspace of finite codimension, then the following conditions are equivalent: (a) Y is strongly proximinal; (b) Y is proximinal; (c) each element of Y^@? attains its norm. Moreover, in this case the quotient X/Y is polyhedral. The final part of the paper contains examples illustrating the importance of some hypotheses in our main results.
- Published
- 2011
49. A note on a third-order multi-point boundary value problem at resonance
- Author
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Xiaojie Lin, Zengji Du, and Fanchao Meng
- Subjects
Linear map ,Combinatorics ,Degree (graph theory) ,Dimension (vector space) ,Continuous function (set theory) ,Differential equation ,General Mathematics ,Linear space ,Mathematical analysis ,Boundary value problem ,Resonance (particle physics) ,Mathematics - Abstract
Based on the coincidence degree theory of Mawhin, we prove some existence results for the following third-order multi-point boundary value problem at resonance where f: [0, 1] × R3 R is a continuous function, 0 < ξ1 < ⋅⋅⋅ < ξm < 1, αi ∈ R, i = 1, …, m, m ≥ 1 and 0 < η1 < η2 < ⋅⋅⋅ < ηn < 1, βj ∈ R, j = 1, 2, …, n, n ≥ 2. In this paper, the dimension of the linear space Ker L (linear operator L is defined by Lx = x′′′) is equal to 2. Since all the existence results for third-order differential equations obtained in previous papers are for the case dim Ker L = 1, our work is new.
- Published
- 2011
50. Theta dichotomy for the genuine unramified principal series of $${\widetilde{Sp}_2(F)}$$
- Author
-
Christian A. Zorn
- Subjects
Combinatorics ,Number theory ,Series (mathematics) ,General Mathematics ,Mathematical analysis ,Pi ,Order (ring theory) ,Field (mathematics) ,Algebraic geometry ,Sign (mathematics) ,Mathematics - Abstract
Let F be a p-adic field with odd residual characteristic. This work is the continuation of a previous paper that contains some detailed computations of the doubling integral for irreducible constituents \({(\pi, \mathcal{V}_{\pi})}\) of the genuine unramified principal series of \({\widetilde{Sp}_2(F)}\) using various “good test data”. This paper aims to interpret those results in terms of the non-vanishing of local theta lifts. Assuming a technical condition on order of a particular pole for the family of doubling integrals for \({(\pi, \mathcal{V}_{\pi})}\) , we aim to determine the so-called “dichotomy sign” of \({(\pi, \mathcal{V}_{\pi})}\) .
- Published
- 2011
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