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1.
M. Pellegrini 《Discrete and Computational Geometry》1994,12(1):203-221
We show some combinatorial and algorithmic results concerning finite sets of lines and terrains in 3-space. Our main results include:
- An $O(n^3 2^{c\sqrt {\log n} } )$ upper bound on the worst-case complexity of the set of lines that can be translated to infinity without intersecting a given finite set ofn lines, wherec is a suitable constant. This bound is almost tight.
- AnO(n 1.5+ε) randomized expected time algorithm that tests whether a directionv exists along which a set ofn red lines can be translated away from a set ofn blue lines without collisions. ε>0 is an arbitrary small but fixed constant.
- An $O(n^3 2^{c\sqrt {\log n} } )$ upper bound on the worst-case complexity of theenvelope of lines above a terrain withn edges, wherec is a suitable constant.
- An algorithm for computing the intersection of two polyhedral terrains in 3-space withn total edges in timeO(n 4/3+ε+k 1/3 n 1+ε+klog2 n), wherek is the size of the output, and ε>0 is an arbitrary small but fixed constant. This algorithm improves on the best previous result of Chazelleet al. [5].
2.
LetS be a locally compact (σ-compact) group or semi-group, and letT(t) be a continuous representation ofS by contractions in a Banach spaceX. For a regular probability μ onS, we study the convergence of the powers of the μ-averageUx=∫T(t)xdμ(t). Our main results for random walks on a groupG are:
- if μ is adapted and strictly aperiodic, and generates a recurrent random walk, thenU n (U-I) converges strongly to 0. In particular, the random walk is completely mixing.
- If μ×μ is ergodic onG×G, then for every unitary representationT(.) in a Hilbert space,U n converges strongly to the orthogonal projection on the space of common fixed points. These results are proved for semigroup representations, along with some other results (previously known only for groups) which do not assume ergodicity.
- If μ is spread-out with supportS, then $\left\| {\mu ^{n + K} - \mu ^n } \right\| \to 0$ if and only if e $ \in \overline { \cup _{j = 0}^\infty S^{ - j} S^{j + K} } .$ .
3.
LetX be ann-element set and letA and? be families of subsets ofX. We say thatA and? are crosst-intersecting if |A ∩ B| ≥ t holds for all A ∈A and for allB ∈ ?. Suppose thatA and ? are crosst-intersecting. This paper first proves a crosst-intersecting version of Harper's Theorem:
- There are two crosst-intersecting Hamming spheresA 0,? 0 with centerX such that |A| ≤ |A 0| and|?| ≤ |? 0| hold.
- Suppose thatt ≥ 2 and that the pair of integers (|A) is maximal with respect to direct product ordering among pairs of crosst-intersecting families. Then,A and? are Hamming spheres with centerX.
- Ifn + t = 2k ? 1 then |A| |?| ≤ max \(\left\{ {\left( {K_k^n + \left( {_{k - 1}^{n - 1} } \right)} \right)^2 ,K_k^n K_{k - 1}^n } \right\}\) holds, whereK l n is defined as \(\left( {_n^n } \right)\left( {_{n - 1}^n } \right) + \cdots + \left( {_l^n } \right).\)
- Ifn + t = 2k then |A| |? ≤ (K k n )2 holds.
4.
5.
For a given graph G and integers b,f ≥0, let S be a subset of vertices of G of size b+1 such that the subgraph of G induced by S is connected and S can be separated from other vertices of G by removing f vertices. We prove that every graph on n vertices contains at most $n\left( {_b^{b + f} } \right)$ such vertex subsets. This result from extremal combinatorics appears to be very useful in the design of several enumeration and exact algorithms. In particular, we use it to provide algorithms that for a given n-vertex graph G
- compute the treewidth of G in time O(1.7549 n ) by making use of exponential space and in time O(2.6151 n ) and polynomial space
- decide in time O(n 5· $_{k + 2}^{\left\lceil {(2n + k + 8)/3} \right\rceil } $ ) if the treewidth of G is at most k
- list all minimal separators of G in time O(1.6181 n ) and all potential maximal cliques of G in time O(1.7549 n ).
6.
Svetoslav Ivanov Nenov 《Annali dell'Universita di Ferrara》1996,42(1):121-125
The existence and the uniqueness (with respect to a filtration-equivalence) of a vector flowX on ? n ,n≥3, such that:
- X has not any stationary points on ? n ;
- all orbits ofX are bounded;
- there exists a filtration forX are proved in the present note.
7.
Antonio Granata 《Analysis Mathematica》2010,36(3):173-218
In Part II of our work we approach the problem discussed in Part I from the new viewpoint of canonical factorizations of a certain nth order differential operator L. The main results include:
- characterizations of the set of relations $$ f^{(k)} (x) = P^{(k)} (x) + o^{(k)} (x^{\alpha _n - k} ),x \to + \infty ,0 \leqslant k \leqslant n - 1, $$ where $$ P(x) = a_1 x^{\alpha _1 } + \cdots + a_n x^{\alpha _n } and \alpha _1 > \alpha _2 > \cdots > \alpha _n , $$ by means of suitable integral conditions
- formal differentiation of a real-power asymptotic expansion under a Tauberian condition involving the order of growth of L
- remarkable properties of asymptotic expansions of generalized convex functions.
8.
Suppose K is a skew field. Let K m×n denote the set of all m×n matrices over K. In this paper, we give necessary and sufficient conditions for the existence and explicit representations of the group inverses of the block matrices in the following three cases, respectively:
- $\mathrm{rank}(S)=\mathrm{rank}(B^{\pi}A)$ ;
- $\mathrm{rank}(S)=\mathrm{rank}(AB^{\pi})$ ;
- $\mathrm{rank}(S)=\mathrm{rank}(B^{\pi}A)=\mathrm{rank}(AB^{\pi})$ ,
9.
Rolf Trautner 《Analysis Mathematica》1988,14(2):111-122
Рассматриваются слу чайная величина \(\mathfrak{X} = (X_n (\omega ))\) , удовлетворяющая усл овиюE(X n 4 )≦M, и соответствующ ий случайный степенн ой ряд \(f_x (z;\omega ) = \mathop \sum \limits_{n = 0}^\infty a_n X_n (\omega )z^n\) . Устанавливаются тео ремы непродолжимост и почти наверное:
- дляf x при условиях с лабой мультипликати вности на \(\mathfrak{X}\) ,
- для \(f_{\tilde x}\) , где \(\mathop \mathfrak{X}\limits^ \sim = (\mathop X\limits^ \sim _n )\) есть подп оследовательность в \(\mathfrak{X}\) ,
- для по крайней мере од ного из рядовf x′ илиf x″ , где \(\mathfrak{X}'\) и \(\mathfrak{X}''\) — некоторые п ерестановки \(\mathfrak{X}\) , выбираемые универс ально, т. е. независимо от коэффициентовa n .
10.
K. Tandori 《Analysis Mathematica》1979,5(2):149-166
Пусть {λ n 1 t8 — монотонн ая последовательнос ть натуральных чисел. Дл я каждой функции fεL(0, 2π) с рядом Фурье строятся обобщенные средние Bалле Пуссена $$V_n^{(\lambda )} (f;x) = \frac{{a_0 }}{2} + \mathop \sum \limits_{k = 1}^n (a_k \cos kx + b_k \sin kx) + \mathop \sum \limits_{k = n + 1}^{n + \lambda _n } \left( {1 - \frac{{k - n}}{{\lambda _n + 1}}} \right)\left( {a_k \cos kx + b_k \sin kx} \right).$$ Доказываются следую щие теоремы.
- Если λn=o(n), то существуе т функция fεL(0, 2π), для кот орой последовательность {Vn (λ)(?;x)} расходится почти вс юду.
- Если λn=o(n), то существуе т функция fεL(0, 2π), для кот орой последовательность $$\left\{ {\frac{1}{\pi }\mathop \smallint \limits_{ - \pi /\lambda _n }^{\pi /\lambda _n } f(x + t)\frac{{\sin (n + \tfrac{1}{2})t}}{{2\sin \tfrac{1}{2}t}}dt} \right\}$$ расходится почти всю ду
11.
Г. Г. ГЕВОРКЯН 《Analysis Mathematica》1990,16(2):87-114
In this paper some basis properties are proved for the series with respect to the Franklin system, which are analogous to those of the series with respect to the Haar system. In particular, the following statements hold:
- The Franklin series \(\mathop \Sigma \limits_{n = 0}^\infty a_n f_n (x)\) converges a.e. onE if and only if \(\mathop \Sigma \limits_{n = 0}^\infty a_n^2 f_n^2 (x)< + \infty \) a.e. onE;
- If the series \(\mathop \Sigma \limits_{n = 0}^\infty a_n f_n (x)\) , with coefficients ¦a n ¦↓0, converges on a set of positive measure, then it is the Fourier-Franklin series of some function from \(\bigcap\limits_{p< \infty } {L_p } \) ;
- The absolute convergence at a point for Fourier—Franklin series is a local property;
- If an integrable function (fx) has a discontinuity of the first kind atx=x 0, then its Fourier-Franklin series diverges atx=x 0.
12.
G. G. Gevorkjan 《Analysis Mathematica》1982,8(4):239-256
Пусть (gW, ?,P) - вероятност ное пространство, ?1??2?...?? n ?...,? n ?? -последовательност ь σ-алгебр и ?∞ - порожден ная ими минимальная σ-алгебра. В статье указано необ ходимое и достаточно е условие на последовательность σ-алгебр {? n }, при выполнении кото рого для любой ?∞-измер имой функцииf(x) существует ряд \(\mathop \sum \limits_{n = 1}^\infty \varphi _n (x)\) центрированных отн осительно {? n } функций {? n } n=1 ∞ такой, что
- \(\mathop \sum \limits_{n = 1}^\infty \varphi _n (x)\) абсолютно почти вс юду сходится кf(x) на множестве {x: ¦f(x)¦<+∞};
- \(\mathop \sum \limits_{n = 1}^\infty \varphi _n (x)\) сходится по мере кf(x) на множестве {х: ¦f(х)¦=+∞ }.
13.
Alexander Pott 《Geometriae Dedicata》1994,52(2):181-193
We consider projective planes Π of ordern with abelian collineation group Γ of ordern(n?1) which is generated by (A, m)-elations and (B, l)-homologies wherem =AB andA εl. We prove
- Ifn is even thenn=2e and the Sylow 2-subgroup of Γ is elementary abelian.
- Ifn is odd then the Sylow 2-subgroup of Γ is cyclic.
- Ifn is a prime then Π is Desarguesian.
- Ifn is not a square thenn is a prime power.
14.
Gaston Casanova 《Advances in Applied Clifford Algebras》2001,11(2):287-292
Letx andy be orthogonal coordinates of a pointM(u=ax+iby orax+εby) of a plane whereasx′ andy′ are orthogonal coorditanes of a pointM′(v=ax′+iby′ orax′+εby′) inverse ofM in the elliptic or hyperbolic inversion $u\bar \upsilon = k$ (k positive) $\bar \upsilon $ designating the conjugate ofv whilei andε are Clifford numbers such thati 2=?1 andε 2 = 1 (a andb are real)O is the origin of axises,Ox is the axis of the inversion.
- Two inverse points are aligned withO.
- The inverse of a conic having a centrec is a conic but this inverse is a straight line if the origin is on the conic.
15.
V. A. Baskakov 《Analysis Mathematica》1983,9(1):9-22
Для линейных методов суммирования рядов Ф урье (1) $$L_n (f;x) = \frac{1}{\pi }\mathop \smallint \limits_{ - \pi }^\pi f(x + t)\left( {\frac{1}{2} + \sum\limits_{k = 1}^n {\lambda _{k,n} } \cos kt} \right)dt$$ на классах $$C(\varepsilon ) = \{ f:E_n (f) \leqq \varepsilon _n ;\forall n \geqq 0\} ,\varepsilon = \{ \varepsilon _n \} _{n = 0.}^\infty \varepsilon _n \downarrow 0,$$ доказываются:
- оценки для порядка р оста норм ∥{Ln∥, если из вестен порядок приближения операторами (1) некоторого классаС (?) (при этом, если опера торы (1) приближают класс С(е) с наилучшим порядком, то находится точная а симптотика возрастания норм {∥ Ln∥);
- сравнительные оцен ки порядков приближе ния классовС(?) операторами (1), если известен порядок при ближения ими некотор ого более узкого класса С(?*).
16.
Johan Philip 《Mathematical Programming》1972,2(1):207-229
We consider a convex setB inR n described as the intersection of halfspacesa i T x ≦b i (i ∈ I) and a set of linear objective functionsf j =c j T x (j ∈ J). The index setsI andJ are allowed to be infinite in one of the algorithms. We give the definition of theefficient points ofB (also called functionally efficient or Pareto optimal points) and present the mathematical theory which is needed in the algorithms. In the last section of the paper, we present algorithms that solve the following problems:
- To decide if a given point inB is efficient.
- To find an efficient point inB.
- To decide if a given efficient point is the only one that exists, and if not, find other ones.
- The solutions of the above problems do not depend on the absolute magnitudes of thec j. They only describe the relative importance of the different activitiesx i. Therefore we also consider $$\begin{gathered} \max G^T x \hfill \\ x efficient \hfill \\ \end{gathered} $$ for some vectorG.
17.
The subject of the present research is the M/M/n + G queue. This queue is characterized by Poisson arrivals at rate λ, exponential service times at rate μ, n service agents and generally distributed patience times of customers. The model is applied in the call center environment, as it captures the tradeoff between operational efficiency (staffing cost) and service quality (accessibility of agents). In our research, three asymptotic operational regimes for medium to large call centers are studied. These regimes correspond to the following three staffing rules, as λ and n increase indefinitely and μ held fixed: Efficiency-Driven (ED): $n\ \approx \ (\lambda / \mu)\cdot (1 - \gamma),\gamma > 0,$ Quality-Driven (QD): $n \ \approx \ ( \lambda / \mu)\cdot (1 + \gamma),\gamma > 0$ , and Quality and Efficiency Driven (QED): $ n \ \approx \ \lambda/ \mu+\beta \sqrt{\lambda/\mu},-\infty < \beta < \infty $ . In the ED regime, the probability to abandon and average wait converge to constants. In the QD regime, we observe a very high service level at the cost of possible overstaffing. Finally, the QED regime carefully balances quality and efficiency: agents are highly utilized, but the probability to abandon and the average wait are small (converge to zero at rate 1/ $\sqrt{n}$ ). Numerical experiments demonstrate that, for a wide set of system parameters, the QED formulae provide excellent approximation for exact M/M/n + G performance measures. The much simpler ED approximations are still very useful for overloaded queueing systems. Finally, empirical findings have demonstrated a robust linear relation between the fraction abandoning and average wait. We validate this relation, asymptotically, in the QED and QD regimes. 相似文献
18.
Yuri Bilu 《Israel Journal of Mathematics》1995,90(1-3):235-252
LetK be an algebraic number field,S?S \t8 a finite set of valuations andC a non-singular algebraic curve overK. Letx∈K(C) be non-constant. A pointP∈C(K) isS-integral if it is not a pole ofx and |x(P)| v >1 impliesv∈S. It is proved that allS-integral points can be effectively determined if the pair (C, x) satisfies certain conditions. In particular, this is the case if
- x:C→P 1 is a Galois covering andg(C)≥1;
- the integral closure of $\bar Q$ [x] in $\bar Q$ (C) has at least two units multiplicatively independent mod $\bar Q$ *.
19.
Let $\mathcal{K}$ be the family of graphs on ω1 without cliques or independent subsets of sizew 1. We prove that
- it is consistent with CH that everyGε $\mathcal{K}$ has 2ω many pairwise non-isomorphic subgraphs,
- the following proposition holds in L: (*)there is a Gε $\mathcal{K}$ such that for each partition (A, B) of ω1 either G?G[A] orG?G[B],
- the failure of (*) is consistent with ZFC.
20.
G. D. Allen 《Results in Mathematics》1988,14(3-4):211-222
Let \(\bar x\) , \(\bar y\ \in\ R_n\) be vectors which satisfy x1 ≥ x2 ≥ … ≥ xn and y1 ≥ y2 >- … ≥ yn and Σxi = Σyi. We say that \(\bar x\) is power majorized by \(\bar y\) if Σxi p ≤ Σyi p for all real p ? [0, 1] and Σxi p ≥ Σyi p for p ∈ [0, 1]. In this paper we give a classification of functions ? (which includes all possible positive polynomials) for which \(\bar\phi(\bar x) \leq \bar\phi(\bar y)\) (see definition below) when \(\bar x\) is power majorized \(\bar y\) . We also answer a question posed by Clausing by showing that there are vectors \(\bar x\) , \(\bar y\ \in\ R^n\) of any dimension n ≥ 4 for which there is a convex function ? such that \(\bar x\) is power majorized by \(\bar y\) and \(\bar\phi(\bar x)\ >\ \bar\phi(\bar y)\) . 相似文献