共查询到20条相似文献,搜索用时 31 毫秒
1.
Yu. A. Aminov 《Journal of Mathematical Sciences》1999,94(2):1141-1144
Immersions of domains of the n-dimensional Lobachevski space Ln in the (2n−1)-dimensional Euclidean space E2n−1 are studied. It is shown that the problem of isometric immersion of domains of Ln in E2n−1 is reduced to the study of a certain system of nonlinear partial differential equations, yielding the sine-Gordon equation
as one of the special cases.
Published inZapiski Nauchnykh Seminarov POMI, Vol. 234, 1996, pp. 11–16. 相似文献
2.
Equations for the Missing Boundary Values in the Hamiltonian Formulation of Optimal Control Problems
Vicente Costanza Pablo S. Rivadeneira Ruben D. Spies 《Journal of Optimization Theory and Applications》2011,149(1):26-46
Partial differential equations for the unknown final state and initial costate arising in the Hamiltonian formulation of regular
optimal control problems with a quadratic final penalty are found. It is shown that the missing boundary conditions for Hamilton’s
canonical ordinary differential equations satisfy a system of first-order quasilinear vector partial differential equations
(PDEs), when the functional dependence of the H-optimal control in phase-space variables is explicitly known. Their solutions are computed in the context of nonlinear systems
with ℝ
n
-valued states. No special restrictions are imposed on the form of the Lagrangian cost term. Having calculated the initial
values of the costates, the optimal control can then be constructed from on-line integration of the corresponding 2n-dimensional Hamilton ordinary differential equations (ODEs). The off-line procedure requires finding two auxiliary n×n matrices that generalize those appearing in the solution of the differential Riccati equation (DRE) associated with the linear-quadratic
regulator (LQR) problem. In all equations, the independent variables are the finite time-horizon duration T and the final-penalty matrix coefficient S, so their solutions give information on a whole two-parameter family of control problems, which can be used for design purposes.
The mathematical treatment takes advantage from the symplectic structure of the Hamiltonian formalism, which allows one to
reformulate Bellman’s conjectures concerning the “invariant-embedding” methodology for two-point boundary-value problems.
Results for LQR problems are tested against solutions of the associated differential Riccati equation, and the attributes
of the two approaches are illustrated and discussed. Also, nonlinear problems are numerically solved and compared against
those obtained by using shooting techniques. 相似文献
3.
A. A. Duyunova 《Journal of Mathematical Sciences》2011,177(5):654-667
We consider a three-web W(1, n, 1) formed by two n-parametric family of curves and one-parameter family of hypersurfaces on a smooth (n + 1)-dimensional manifold. For such webs, the family of adapted frames is defined and the structure equations are found,
and geometric objects arising in the third-order differential neighborhood are investigated. It is showed that every system
of ordinary differential equations uniquely defines a three-web W(1, n, 1). Thus, there is a possibility to describe some properties of a system of ordinary differential equations in terms of the
corresponding three-web W(1, n, 1). In particular, autonomous systems of ordinary differential equations are characterized. 相似文献
4.
Christian Bär 《Inventiones Mathematicae》1999,138(1):183-202
Consider a nontrivial smooth solution to a semilinear elliptic system of first order with smooth coefficients defined over
an n-dimensional manifold. Assume the operator has the strong unique continuation property. We show that the zero set of the solution
is contained in a countable union of smooth (n−2)-dimensional submanifolds. Hence it is countably (n−2)-rectifiable and its Hausdorff dimension is at most n−2. Moreover, it has locally finite (n−2)-dimensional Hausdorff measure. We show by example that every real number between 0 and n−2 actually occurs as the Hausdorff dimension (for a suitable choice of operator). We also derive results for scalar elliptic
equations of second order.
Oblatum 22-V-1998 & 26-III-1999 / Published online: 10 June 1999 相似文献
5.
Timothy M. Chan 《Discrete and Computational Geometry》2012,47(4):661-690
We revisit one of the most fundamental classes of data structure problems in computational geometry: range searching. Matoušek
(Discrete Comput. Geom. 10:157–182, 1993) gave a partition tree method for d-dimensional simplex range searching achieving O(n) space and O(n
1−1/d
) query time. Although this method is generally believed to be optimal, it is complicated and requires O(n
1+ε
) preprocessing time for any fixed ε>0. An earlier method by Matoušek (Discrete Comput. Geom. 8:315–334, 1992) requires O(nlogn) preprocessing time but O(n
1−1/d
log
O(1)
n) query time. We give a new method that achieves simultaneously O(nlogn) preprocessing time, O(n) space, and O(n
1−1/d
) query time with high probability. Our method has several advantages:
• |
It is conceptually simpler than Matoušek’s O(n
1−1/d
)-time method. Our partition trees satisfy many ideal properties (e.g., constant degree, optimal crossing number at almost
all layers, and disjointness of the children’s cells at each node). 相似文献
6.
The purpose of the paper is to find explicit formulas describing the joint distributions of the first hitting time and place
for half-spaces of codimension one for a diffusion in ℝ
n + 1, composed of one-dimensional Bessel process and independent n-dimensional Brownian motion. The most important argument is carried out for the two-dimensional situation. We show that this
amounts to computation of distributions of various integral functionals with respect to a two-dimensional process with independent
Bessel components. As a result, we provide a formula for the Poisson kernel of a half-space or of a strip for the operator (I − Δ)
α/2, 0 < α < 2. In the case of a half-space, this result was recently found, by different methods, in Byczkowski et al. (Trans Am Math Soc 361:4871–4900, 2009). As an application of our method we also compute various formulas for first hitting places for the isotropic stable Lévy process. 相似文献
7.
Two natural extensions of Jensen’s functional equation on the real line are the equations f(xy) + f(xy
−1) = 2f(x) and f(xy) + f(y
−1
x) = 2f(x), where f is a map from a multiplicative group G into an abelian additive group H. In a series of papers (see Ng in Aequationes Math 39:85–99, 1990; Ng in Aequationes Math 58:311–320, 1999; Ng in Aequationes Math 62:143–159, 2001), Ng solved these functional equations for the case where G is a free group and the linear group
GLn(R), R=\mathbbZ,\mathbbR{{GL_n(R), R=\mathbb{Z},\mathbb{R}}} , is a quadratically closed field or a finite field. He also mentioned, without a detailed proof, in the above papers and
in (see Ng in Aequationes Math 70:131–153, 2005) that when G is the symmetric group S
n
, the group of all solutions of these functional equations coincides with the group of all homomorphisms from (S
n
, ·) to (H, + ). The aim of this paper is to give an elementary and direct proof of this fact. 相似文献
8.
Symmetries of the first integrals for scalar linear or linearizable secondorder ordinary di?erential equations (ODEs) have already been derived and shown to exhibit interesting properties. One of these is that the symmetry algebra sl(3, IR) is generated by the three triplets of symmetries of the functionally independent first integrals and its quotient. In this paper, we first investigate the Lie-like operators of the basic first integrals for the linearizable maximally symmetric system of two second-order ODEs represented by the free particle system, obtainable from a complex scalar free particle equation, by splitting the corresponding complex basic first integrals and its quotient as well as their associated symmetries. It is proved that the 14 Lie-like operators corresponding to the complex split of the symmetries of the functionally independent first integrals I1, I2 and their quotient I2/I1 are precisely the Lie-like operators corresponding to the complex split of the symmetries of the scalar free particle equation in the complex domain. Then, it is shown that there are distinguished four symmetries of each of the four basic integrals and their quotients of the two-dimensional free particle system which constitute four-dimensional Lie algebras which are isomorphic to each other and generate the full symmetry algebra sl(4, IR) of the free particle system. It is further shown that the (n + 2)-dimensional algebras of the n + 2 first integrals of the system of n free particle equations are isomorphic to each other and generate the full symmetry algebra sl(n + 2, IR) of the free particle system. 相似文献
9.
Gabriel Koch Nikolai Nadirashvili Gregory A. Seregin Vladimir Šverák 《Acta Mathematica》2009,203(1):83-105
We study bounded ancient solutions of the Navier–Stokes equations. These are solutions with bounded velocity defined in R
n
× (−1, 0). In two space dimensions we prove that such solutions are either constant or of the form u(x, t) = b(t), depending on the exact definition of admissible solutions. The general 3-dimensional problem seems to be out of reach of
existing techniques, but partial results can be obtained in the case of axisymmetric solutions. We apply these results to
some scenarios of potential singularity formation for axi-symmetric solutions, and obtain extensions of results in a recent
paper by Chen, Strain, Tsai and Yau [4]. 相似文献
10.
We present some exponential inequalities for positively associated unbounded random variables. By these inequalities, we obtain
the rate of convergence n
−1/2
β
n
log 3/2
n in which β
n
can be particularly taken as (log log n)1/σ
with any σ>2 for the case of geometrically decreasing covariances, which is faster than the corresponding one n
−1/2(log log n)1/2log 2
n obtained by Xing, Yang, and Liu in J. Inequal. Appl., doi: (2008) for the case mentioned above, and derive the convergence rate n
−1/2
β
n
log 1/2
n for the above β
n
under the given covariance function, which improves the relevant one n
−1/2(log log n)1/2log n obtained by Yang and Chen in Sci. China, Ser. A 49(1), 78–85 (2006) for associated uniformly bounded random variables. In addition, some moment inequalities are given to prove the main results,
which extend and improve some known results. 相似文献
11.
Ervin Győri 《Combinatorica》1981,1(4):377-380
The problem is the following: How many questions are necessary in the worst case to determine whether a pointX in then-dimensional Euclidean spaceR
n
belongs to then-dimensional unit cubeQ
n, where we are allowed to ask which halfspaces of (n−1)-dimensional hyperplanes contain the pointX? It is known that ⌌3n/2⌍ questions are sufficient. We prove here thatcn questions are necessary, wherec≈1.2938 is the solution of the equationx log2
x−(x−1) log2 (x−1)=1. 相似文献
12.
Let B = (B
1(t), . . . , B
d
(t)) be a d-dimensional fractional Brownian motion with Hurst index α ≤ 1/4, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals
of B is a difficult task because of the low H?lder regularity index of its paths. Yet rough path theory shows it is the key to
the construction of a stochastic calculus with respect to B, or to solving differential equations driven by B. We intend to show in a forthcoming series of papers how to desingularize iterated integrals by a weak singular non-Gaussian
perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved by using “standard” tools
of constructive field theory, in particular cluster expansions and renormalization. These powerful tools allow optimal estimates
of the moments and call for an extension of the Gaussian tools such as for instance the Malliavin calculus. This first paper
aims to be both a presentation of the basics of rough path theory to physicists, and of perturbative field theory to probabilists;
it is only heuristic, in particular because the desingularization of iterated integrals is really a non-perturbative effect. It is also meant to be a general motivating introduction to the subject, with some insights into quantum field theory
and stochastic calculus. The interested reader should read for a second time the companion article (Magnen and Unterberger
in From constructive theory to fractional stochastic calculus. (II) The rough path for
\frac16 < a < \frac14{\frac{1}{6} < \alpha < \frac{1}{4}}: constructive proof of convergence, 2011, preprint) for the constructive proofs. 相似文献
13.
The main aim of this paper is to prove the existence and uniqueness of the solution for neutral stochastic functional differential
equations with infinite delay, which the initial data belong to the phase space ℬ((−∞,0];ℝ
d
). The vital work of this paper is to extend the initial function space of the paper (Wei and Wang, J. Math. Anal. Appl. 331:516–531,
2007) and give some examples to show that the phase space ℬ((−∞,0];ℝ
d
) exists. In addition, this paper builds a Banach space ℳ2((−∞,T],ℝ
d
) with a new norm in order to discuss the existence and uniqueness of the solution for such equations with infinite delay. 相似文献
14.
In this paper,the dimension of invariant subspaces admitted by nonlinear systems is estimated under certain conditions.It is shown that if the two-component nonlinear vector differential operator F=(F 1,F 2) with orders {k 1,k 2 } (k 1 ≥ k 2) preserves the invariant subspace W 1 n 1 × W 2 n 2 (n 1 ≥ n 2),then n 1 n 2 ≤ k 2,n 1 ≤ 2(k 1 + k 2) + 1,where W q n q is the space generated by solutions of a linear ordinary differential equation of order n q (q=1,2).Several examples including the (1+1)-dimensional diffusion system and Ito 's type,Drinfel'd-Sokolov-Wilson's type and Whitham-Broer-Kaup's type equations are presented to illustrate the result.Furthermore,the estimate of dimension for m-component nonlinear systems is also given. 相似文献
15.
Buchin Su 《Annali di Matematica Pura ed Applicata》1917,26(1):177-197
Sunto. The author establishes the projective differential geometry of a curve inS
n
at a point of inflexion by locating a certain pyramid of geometrical characterization and finding the canonical equations
of the curve in terms or1/2 (n
2
−n−4)−1 projective invariants. 相似文献
16.
Simona Bonvicini Marco Buratti Gloria Rinaldi Tommaso Traetta 《Designs, Codes and Cryptography》2012,62(1):63-78
A Steiner triple system of order v (briefly STS(v)) is 1-rotational under G if it admits G as an automorphism group acting sharply transitively on all but one point. The spectrum of values of v for which there exists a 1-rotational STS(v) under a cyclic, an abelian, or a dicyclic group, has been established in Phelps and Rosa (Discrete Math 33:57–66, 1981), Buratti (J Combin Des 9:215–226, 2001) and Mishima (Discrete Math 308:2617–2619, 2008), respectively. Nevertheless, the spectrum of values of v for which there exists a 1-rotational STS(v) under an arbitrary group has not been completely determined yet. This paper is a considerable step forward to the solution
of this problem. In fact, we leave as uncertain cases only those for which we have v = (p
3−p)n + 1 ≡ 1 (mod 96) with p a prime,
n \not o 0{n \not\equiv 0} (mod 4), and the odd part of (p
3 − p)n that is square-free and without prime factors congruent to 1 (mod 6). 相似文献
17.
We deal with (n−1)-generated modules of smooth (analytic, holomorphic) vector fieldsV=(X
1,..., Xn−1) (codimension 1 differential systems) defined locally on ℝ
n
or ℂ
n
, and extend the standard duality(X
1,..., Xn−1)↦(ω), ω=Ω(X1,...,Xn−1,.,) (Ω−a volume form) betweenV′s and 1-generated modules of differential 1-forms (Pfaffian equations)—when the generatorsX
i are linearly independent—onto substantially wider classes of codimension 1 differential systems. We prove that two codimension
1 differential systemsV and
are equivalent if and only if so are the corresponding Pfaffian equations (ω) and
provided that ω has1-division property: ωΛμ=0, μ—any 1-form ⇒ μ=fω for certain function germf. The 1-division property of ω turns out to be equivalent to the following properties ofV: (a)fX∈V, f—not a 0-divisor function germ ⇒X∈V (thedivision property); (b) (V
⊥)⊥=V; (c)V
⊥=(ω); (d) (ω)⊥=V, where ⊥ denotes the passing from a module (of vector fields or differential 1-forms) to its annihilator.
Supported by Polish KBN grant No 2 1090 91 01.
Partially supported by the fund for the promotion of research at the Technion, 100–942. 相似文献
18.
N. J. A. Sloane Vinay A. Vaishampayan Sueli I. R. Costa 《Discrete and Computational Geometry》2011,46(3):472-478
It is shown that, given any (n−1)-dimensional lattice Λ, there is a vector v∈ℤ
n
such that the orthogonal projection of ℤ
n
onto v
⊥ is, up to a similarity, arbitrarily close to Λ. The problem arises in attempting to find the largest cylinder anchored at
two points of ℤ
n
and containing no other points of ℤ
n
. 相似文献
19.
Swastik Kopparty Vsevolod F. Lev Shubhangi Saraf Madhu Sudan 《Journal of Algebraic Combinatorics》2011,34(3):337-355
For a finite vector space V and a nonnegative integer r≤dim V, we estimate the smallest possible size of a subset of V, containing a translate of every r-dimensional subspace. In particular, we show that if K⊆V is the smallest subset with this property, n denotes the dimension of V, and q is the size of the underlying field, then for r bounded and r<n≤rq
r−1, we have |V∖K|=Θ(nq
n−r+1); this improves the previously known bounds |V∖K|=Ω(q
n−r+1) and |V∖K|=O(n
2
q
n−r+1). 相似文献
20.
Abdul Basit Nabil H. Mustafa Saurabh Ray Sarfraz Raza 《Discrete and Computational Geometry》2010,44(3):637-644
The so-called first selection lemma states the following: given any set P of n points in ℝ
d
, there exists a point in ℝ
d
contained in at least c
d
n
d+1−O(n
d
) simplices spanned by P, where the constant c
d
depends on d. We present improved bounds on the first selection lemma in ℝ3. In particular, we prove that c
3≥0.00227, improving the previous best result of c
3≥0.00162 by Wagner (On k-sets and applications. Ph.D. thesis, ETH Zurich, 2003). This makes progress, for the three-dimensional case, on the open problems of Bukh et al. (Stabbing simplices by points
and flats. Discrete Comput. Geom., 2010) (where it is proven that c
3≤1/44≈0.00390) and Boros and Füredi (The number of triangles covering the center of an n-set. Geom. Dedic. 17(1):69–77, 1984) (where the two-dimensional case was settled). 相似文献
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