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1.
Given a ratio , >>0, and a triangle ABC, on the sides and , using ratios , and , three circles of Apollonius are denned. In this paper, we will show that the three centers are collinear, the circles are coaxal and develop a necessary and sufficient condition that these circles intersect. J. A. Hoskins, W. D. Hoskins and R. G. Stanton obtained these results in a recent paper using algebraic computation. Our aim is to establish all these results using only results from elementary Euclidean geometry and thereby uncovering more geometric insights and avoid lengthy calculations.  相似文献   

2.
In this paper equivalent classes of the classes M' and S' p r, p >1, 0,r {0,1,2,...,[]} defined by Sheng [5] are obtained. Then it is shown that the classes of Fourier coefficients S p, S' p(case r==0) and S p(), p>1, defined by . V. Stanojevi, V. B. Stanojevi Sheng and the author of the present note are identical. As a corollary of this result, the L 1-estimate for cosine series, obtained in [10], is refined.  相似文献   

3.
Consider an exponential family such that the variance function is given by the power of the mean function. This family is denoted by ED if the variance function is given by (2-)/(1-), where is the mean function. When 0<1, it is known that the transformation of ED() to normality is given by the power transformationX (1-2)/(3-3), and conversely, the power transformation characterizes ED(). Our principal concern will be to show that this power transformation has an another merit, i.e., the density of the transformed variate has an absolutely convergent Gram-Charier expansion.  相似文献   

4.
For linear forms of regularized solutions (x, c)=Re c' · Re[I + i)+A'An –1]–1 A'nb of systems of equations Ax=b, where A is an n×m matrix, x, c, b are vectors, and n is a sequence of constants, we propose the estimator , where is any measurable solution of the equation ()Re[1+1a(())]2+ (12)(1+1(gq()))=, a(y)=n–1 Sp[Iy+–1Zs'Zs+ iI]–1, , i=nn 2n –1sn –1, n=mIn 2n –1sn –1, Xi are independent observations on the matrix A. Under certain conditions, it is proved that G8 is a consistent estimator for n and 0.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 66, pp. 111–119, 1988.  相似文献   

5.
We prove that for an arbitrary endomorphism of a ring R the group K1(R[t]) splits into the direct sum of K1(R) and Ñil (r;). Moreover, for any such R and Ñil (R; ) is isomorphic to Ñil (R ; ) for some ring R with : R R – an isomorphism.  相似文献   

6.
We consider the numberN A (r) of subgroups of orderp r ofA, whereA is a finite Abelianp-group of type =1,2,..., l ()), i.e. the direct sum of cyclic groups of order ii. Formulas for computingN A (r) are well known. Here we derive a recurrence relation forN A (r), which enables us to prove a conjecture of P. E. Dyubyuk about congruences betweenN A (r) and the Gaussian binomial coefficient .  相似文献   

7.
Let be a ring of sets, X a normed space, : X ( ) a bounded family of triangular functions. The following generalized Nikodym theorem is established: the family {} is uniformly bounded on if and only if it is bounded on every sequence of pairwise disjoint sets of which the union is a part of some set in . An analogous criterion is established also for semiadditive functions. In addition, it is shown that uniform boundedness of a family of triangular functions is preserved in passing from a ring to the -ring it generates.Translated from Matematicheskie Zametki, Vol. 23, No. 6, pp. 855–861, June, 1978.  相似文献   

8.
Let S be a cancellative semigroup which is a semilattice of left reversible semigroups S, . This article studies the relationship between the group of quotients G of S and the groups of quotients G of S, . It is shown that G is the maximum group homomorphic image of an inverse semigroup which is a semilattice of groups G (up to isomorphism).The technique used here which involves the use of Ore's quotients also applies to the study of the maximum group homomorphic image of a semigroup which is a semilattice of inverse semigroups.  相似文献   

9.
It is known that the value function in an unconstrained Markov decision process with finitely many states and actions is a piecewise rational function in the discount factor a, and that the value function can be expressed as a Laurent series expansion about = 1 for close enough to 1. We show in this paper that this property also holds for the value function of Markov decision processes with additional constraints. More precisely, we show by a constructive proof that there are numbers O = o <1 <... < m–1 < m = 1 such that for everyj = 1, 2, ...,m – 1 either the problem is not feasible for all discount factors in the open interval (j–1, j) or the value function is a rational function in a in the closed interval [j–1, j]. As a consequence, if the constrained problem is feasible in the neighborhood of = 1, then the value function has a Laurent series expansion about = 1. Our proof technique for the constrained case provides also a new proof for the unconstrained case.  相似文献   

10.
Given a context-free grammarG with a cycle of productionsA 1 A 2 1,A 2 A 3 2, ...,A m A 1 m, it is shown that there is a grammarG 1 in which the only left-recursive rules are of the formA 1 A 1 , andL(G 1)=L(G). Using this normalization it is then shown that a Boolean matrix used in the analysis phase of some syntax-directed translation schemes may be partitioned into rectangular and upper-triangular submatrices.  相似文献   

11.
In this note we shall prove that there exist infinitely many cyclic biquadratic fieldsK whose integral bases are neither {1, , 2, } nor {1, , , 3) for any numbers , inK. Next, we shall construct infinitely many cyclic biquadratic fieldsK which have the index 1, but still have not the integral basis {1, , 2, 3) for every inK. Finally we shall give a class of biquadratic fields for a problem of Hasse concerning an integral basis.  相似文献   

12.
It is shown that nonlinear void-fraction waves constitute exact solutions of the two-fluid equations for bubbly flow, if and only if the virtual-mass coefficientm() is a definite function of the void fraction. In the case of spherical bubbles the functional dependence is given bym()=(1/2)(1–a)(1–3). A similar expression was obtained recently by means of an elaborate linear stability analysis of a uniform two-phase flow. It is remarkable that the greater part of the physical quantities of a nonlinear void-fraction wave depends linearly on. The virtual-mass coefficient may be derived directly from a special form of Hamilton's variational principle.
Zusammenfassung Es wird gezeigt daß nichtlineare Konzentrationswellen exakte Lösungen der Zweifluiden Gleichungen einer Blasenströmung darstellen dann und nur dann, wenn der Koeffizientm() der virtuellen Masse eine bestimmte Funktion der Volumenkonzentration der Blasen ist. Die funktionelle Abhängigkeit ist im Falle kugelförmiger Blasen gegeben durchm()= (1/2)(1–) (1–3). Ein gleicher Ausdruck wurde neulich mittels einer umständlichen linearen Stabilitätsanalysis einer uniformen Zweiphasen-Strömung erhalten. Es ist bemerkenswert daß der größere Teil der physikalischen Größen einer nichtlinearen Konzentrationswelle in linearer Weise von abhängt. Der Koeffizient der virtuellen Masse kann in direkter Weise von einer speziellen Form des Hamiltonschen Variationsprinzips hergeleitet werden.
  相似文献   

13.
Consider the axisymmetric, inertialess cone- and -plate flow of a viscoelastic fluid. A perturbation method is used to obtain more tractable equations that describe the flow when the gap angle is small. A linear stability analysis of the base viscometric flow shows that there is a loss of stability when an elasticity parameter =DeWe, increases past a critical value. This purely elastic instability is of the oscillatory type. We obtain expressions for the critical elasticity number, frequency and wave number. The critical Deborah number varies as , the wave length as . and the wave speed as , where . is the gap angle. The most unstable mode exhibits infinitely many logarithmically-spaced roll cells which propagate inward towards the apex of the cone. These results are in agreement with experimental and numerical results.  相似文献   

14.
Let 1 and let (X, d, ) be an -homogeneous metric measure space with conformal Assouad dimension equal to . Then there exists a weak tangent of (X, d, ) with uniformly big 1-modulus.  相似文献   

15.
A general approach is proposed to the interpolation of x -analytical functions of a complex variable with an arbitrary ,+[Basis x -analytical functions whose imaginary pan is a polynomial in x, and y are obtained in explicit form.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 3–9, 1986.  相似文献   

16.
In this paper we give the connection between the zeros of the -function and sequences(g(p)), p prime, mod 1 ifg(x)=x for 0, >0 or ifg(X) is a polynomial in .  相似文献   

17.
Zusammenfassung Die Methode der inneren Parallelen von Makai und Pólya wird hier auf das Randwertproblem (3) angewendet. Die sich ergebenden Schranken (11) und (11) für das FunktionalQ() enthalten als Grenzfälle die von Pólya [9] erhaltenen Schranken (2) und (1) für die Torsionssteifigkeit (Fall=0) bzw. für den Grundton einer Membran (Fall 1.
Summary We apply here the method of interior parallels of Makai and Pólya to the boundary value problem (3). The resulting bounds (11) and (11) for the functionalQ() contain as limit cases the bounds (2) and (1) obtained by Pólya [9] for torsional rigidity (case=0) and for the lowest eigenfrequency of a membrane (case 1).
  相似文献   

18.
Summary We say that the discD()R 2, of radius , located around the origin isp-covered in timeT by a Wiener processW(·) if for anyzD() there exists a 0tT such thatW(t) is a point of the disc of radiusp, located aroundz. The supremum of those 's (0) is studied for which,D() isp-covered inT.  相似文献   

19.
H={h 1,I } — , . : , I ¦(I)¦=¦I¦, ¦I¦ — I. H H ={h (I),I} . , , . L p .

Dedicated to Professor B. Szökefalvi-Nagy on his 75th birthday

This research was supported in part by MTA-NSF Grants INT-8400708 and 8620153.  相似文献   

20.
A Sturmian word is a map W : {0,1} for which the set of {0, 1}-vectors F n(W) {(W(i), W(i + 1),...,W(i + n – 1)) T : i } has cardinality exactly n + 1 for each positive integer n. Our main result is that the volume of the simplex whose n + 1 vertices are the n + 1 points in F n(W) does not depend on W. Our proof of this motivates studying algebraic properties of the permutation ,n (where is any irrational and n is any positive integer) that orders the fractional parts {}, {2},...,{n}, i.e., 0 < {,n (1)} < {,n (2)} < ··· < {,n (n)} < 1. We give a formula for the sign of ,n , and prove that for every irrational there are infinitely many n such that the order of ,n (as an element of the symmetric group S n) is less than n.  相似文献   

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