On Geary’s theorem for the field of <Emphasis Type="Italic">p</Emphasis>-adic numbers

Let ?_p, where p > 2, be a field of p-adic numbers. We consider two independent identically distributed random variables with values in ?_p and distribution μ with a continuous density. We prove that the sum and the squared difference of these random variables are independent if and only if μ is an idempotent distribution, i.e., a shift of the Haar distribution of a compact subgroup of the additive group of the field ?_p.     

Empirical likelihood for right censored data with covariables

For complete observation and p-dimensional parameterθdefined by an estimation equation,empirical likelihood method of construction of confidence region is based on the asymptoticχ2pdistribution of-2 log(EL ratio).For right censored lifetime data with covariables,however,it is shown in literature that-2 log(EL ratio)converges weakly to a scaledχ2pdistribution,where the scale parameter is a function of unknown asymptotic covariance matrix.The construction of confidence region requires estimation of this scale parameter.In this paper,by using influence functions in the estimating equation,we show that-2 log(EL ratio)converges weakly to a standardχ2pdistribution and hence eliminates the procedure of estimating the scale parameter.     

Fan triangulations of a hyperbolic plane of positive curvature

We study the families (?_λ) of normal partitions of a 3-(1)-contour F of a hyperbolic plane \(\hat H\) of positive curvature into simple 4-contours whose hyperbolic diagonal lines are parallel to the base of F. A 3-(1)-contour with a given partition from a family (?_λ) (or some its normal subpartition) is called a fan. We construct fan partitions P _e, P _h, and P _p of \(\hat H\) whose symmetry groups are generated by a shift along an elliptic (respectively, hyperbolic and parabolic) straight line. It is proved that the partitions P _h and P _p are normal. The partitions P _h and P _p whose cells are trihedrals present examples of the first triangulations of \(\hat H\) .     

Identifiability of the multinormal and other distributions under competing risks model

Let X₁, X₂ ,…, X_p be p random variables with joint distribution function F(x₁ ,…, x_p). Let Z = min(X₁, X₂ ,…, X_p) and I = i if Z = X_i. In this paper the problem of identifying the distribution function F(x₁ ,…, x_p), given the distribution Z or that of the identified minimum (Z, I), has been considered when F is a multivariate normal distribution. For the case p = 2, the problem is completely solved. If p = 3 and the distribution of (Z, I) is given, we get a partial solution allowing us to identify the independent case. These results seem to be highly nontrivial and depend upon Liouville's result that the (univariate) normal distribution function is a nonelementary function. Some other examples are given including the bivariate exponential distribution of Marshall and Olkin, Gumbel, and the absolutely continuous bivariate exponential extension of Block and Basu.     

Perturbation theory for commutative m-tuples of self-adjoint operators

Generalizing the Weyl-von Neumann theorem for normal operators, we show that a commutative m-tuple of self-adjoint operators in a separable Hilbert space may be changed into a diagonal one by adding compact perturbations of class c_p, for p>m. On the other hand it is shown that the absolutely continuous part, defined appropriately, of a commutative m-tuple of self-adjoint operators is stable under perturbations of class c_p, if p < m, m ? 3, or if p = 1, m = 2 (the latter case m = 2 corresponding to the case of one normal operator). For the proof of these Kato-Rosenblum-type theorems a wave operator method for m-tuples is introduced.     

Optimal Linear Bernoulli Factories for Small Mean Problems

Suppose a coin with unknown probability p of heads can be flipped as often as desired. A Bernoulli factory for a function f is an algorithm that uses flips of the coin together with auxiliary randomness to flip a single coin with probability f(p) of heads. Applications include perfect sampling from the stationary distribution of certain regenerative processes. When f is analytic, the problem can be reduced to a Bernoulli factory of the form f(p) = C p for constant C. Presented here is a new algorithm that for small values of C p, requires roughly only C coin flips. From information theoretic considerations, this is also conjectured to be (to first order) the minimum number of flips needed by any such algorithm. For large values of C p, the new algorithm can also be used to build a new Bernoulli factory that uses only 80 % of the expected coin flips of the older method. In addition, the new method also applies to the more general problem of a linear multivariate Bernoulli factory, where there are k coins, the kth coin has unknown probability p _k of heads, and the goal is to simulate a coin flip with probability C ₁ p ₁+? + C _k p _k of heads.     

Normal edge-transitive Cayley graphs on non-abelian groups of order 4p, where p is a prime number

We determine all connected normal edge-transitive Cayley graphs on non-abelian groups with order 4p,where p is a prime number.As a consequence we prove if |G|=2δp,δ=0,1,2 and p prime,then Γ=Cay(G,S) is a connected normal 1/2 arc-transitive Cayley graph only if G=F4p,where S is an inverse closed generating subset of G which does not contain the identity element of G and F 4p is a group with presentation F4p = a,b|ap=b4=1,b-1ab=aλ,where λ2≡-1(mod p).     

The normal distribution and repeated games

For a reperated zero-sum two-person game with incomplete information discussed byZamir, it is proved here that \(\mathop {\lim }\limits_{n \to \infty } \sqrt n v_n (p) = \phi (p)\) whereφ (p) is the normal density function evaluated at itsp-quantile (i.e. \(\phi (p) = \frac{1}{{\sqrt {2\pi } }}e^{ - ({1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2})x^2 } p\) where \(\frac{1}{{\sqrt {2\pi } }}\mathop {\smallint ^p }\limits_{ - \infty }^x e^{ - ({1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2})x^2 } dx = p\) . Here for 0?p?1, (p, 1 ?p) is the a priori probability distribution on two states of nature, the actual state of nature is known to the maximizer but not to the minimizer.v _n (p) is the minimax value of the game withn stages.     

Variations of Landau’s theorem for p-regular and p-singular conjugacy classes

The well-known Landau’s theorem states that, for any positive integer k, there are finitely many isomorphism classes of finite groups with exactly k (conjugacy) classes. We study variations of this theorem for p-regular classes as well as p-singular classes. We prove several results showing that the structure of a finite group is strongly restricted by the number of p-regular classes or the number of p-singular classes of the group. In particular, if G is a finite group with O_p(G) = 1 then |G/F(G)|_p' is bounded in terms of the number of p-regular classes of G. However, it is not possible to prove that there are finitely many groups with no nontrivial normal p-subgroup and kp-regular classes without solving some extremely difficult number-theoretic problems (for instance, we would need to show that the number of Fermat primes is finite).     

Supersolubility of a Finite Group with Normally Embedded Maximal Subgroups in Sylow Subgroups

Let P be a subgroup of a Sylow subgroup of a finite group G. If P is a Sylow subgroup of some normal subgroup of G then P is called normally embedded in G. We establish tests for a finite group G to be p-supersoluble provided that every maximal subgroup of a Sylow p-subgroup of X is normally embedded in G. We study the cases when X is a normal subgroup of G, X = O_p',p(H), and X = F*(H) where H is a normal subgroup of G.     

Compound Geometric Distribution of Order <Emphasis Type="Italic">k</Emphasis>

The distribution of the number of trials until the first k consecutive successes in a sequence of Bernoulli trials with success probability p is known as geometric distribution of order k. Let T _k be a random variable that follows a geometric distribution of order k, and Y ₁,Y ₂,… a sequence of independent and identically distributed discrete random variables which are independent of T _k . In the present article we develop some results on the distribution of the compound random variable \(S_{k} =\sum_{t=1}^{T_{k}}Y_{t}\).     

On pro-p analogues of limit groups via extensions of centralizers

We begin a study of a pro-p analogue of limit groups via extensions of centralizers and call ${\mathcal{L}}$ this new class of pro-p groups. We show that the pro-p groups of ${\mathcal{L}}$ have finite cohomological dimension, type FP _?? and non-positive Euler characteristic. Among the group theoretic properties it is proved that they are free-by-(torsion free nilpotent) and if non-abelian do not have a finitely generated non-trivial normal subgroup of infinite index. Furthermore it is shown that every 2 generated pro-p group in the class ${\mathcal{L}}$ is either free pro-p or abelian.     

A Delay Dependent Queue without Pre-emption with General Linearly Increasing Priority Function

This paper considers the delay dependent priority queueing discipline with P non-preemptive priority classes. The priority at time t of a customer from the pth priority class, who arrives at time T_p, is given by q_p(t) = a _p + b _p (t - T _p ). The result has been derived for the expected waiting time of a customer from the pth priority class.     

Convergence of greedy algorithm in Walsh system in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

Convergence of the greedy algorithm in Walsh system in L _p , p > 1 is studied. It is proved that there exists a function in L _p , 1 < p < 2, with greedy algorithm not converging in measure to that function. A continuous function with divergent in L _p , p > 2, greedy algorithm is constructed and sufficient conditions for convergence of the greedy algorithm in L _p , p > 1 are given.     

Process convergence of self-normalized sums of i.i.d. random variables coming from domain of attraction of stable distributions

In this paper we show that the continuous version of the self-normalized process Y _n,p (t)?=?S _n (t)/V _n,p ?+?(nt???[nt])X _[nt]?+?1/V _n,p ,0?<?t?≤?1; p?>?0 where $S_n(t)=\sum_{i=1}^{[nt]} X_i$ and $V_{(n,p)}=(\sum_{i=1}^{n}|X_i|^p)^{1/p}$ and X _i i.i.d. random variables belong to DA(α), has a non-trivial distribution iff p?=?α?=?2. The case for 2?>?p?>?α and p?≤?α?<?2 is systematically eliminated by showing that either of tightness or finite dimensional convergence to a non-degenerate limiting distribution does not hold. This work is an extension of the work by Csörg? et al. who showed Donsker’s theorem for Y _n,2(·), i.e., for p?=?2, holds iff α?=?2 and identified the limiting process as a standard Brownian motion in sup norm.     

On Subspaces of Cesàro Spaces

We obtain a characterization of subspaces of L _p , with 1 < p < ∞, on which the L _p -norm is equivalent to the norm of the Cesàro space Ces _p . Also, we show that Ces _p has a complemented copy of the Cesàro sequence space ces _p .     

Some related paradoxes of queuing theory: new cases and a unifying explanation

In this paper, we study a number of closely related paradoxes of queuing theory, each of which is based on the intuitive notion that the level of congestion in a queuing system should be directly related to the stochastic variability of the arrival process and the service times. In contrast to such an expectation, it has previously been shown that, in all H _k /G/1 queues, PW (the steady-state probability that a customer has to wait for service) decreases when the service-time becomes more variable. An analagous result has also been proved for p_loss (the steady-state probability that a customer is lost) in all H_k/G/1 loss systems. Such theoretical results can be seen, in this paper, to be part of a much broader scheme of paradoxical behaviour which covers a wide range of queuing systems. The main aim of this paper is to provide a unifying explanation for these kinds of behaviour. Using an analysis based on a simple, approximate model, we show that, for an arbitrary set of n GI/G_k/1 loss systems (k=1,..., n), if the interarrival-time distribution is fixed and ‘does not differ too greatly’ from the exponential distribution, and if the n systems are ordered in terms of their p_loss values, then the order that results whenever c_A<1 is the exact reverse of the order that results whenever c_A>1, where c_A is the coefficient of variation of the interarrival time. An important part of the analysis is the insensitivity of the p_loss value in an M/G/1 loss system to the choice of service-time distribution, for a given traffic intensity. The analysis is easily generalised to other queuing systems for which similar insensitivity results hold. Numerical results are presented for paradoxical behaviour of the following quantities in the steady state: p_loss in the GI/G/1 loss system; PW and W _q (the expected queuing time of customers) in the GI/G/1 queue; and p_K (the probability that all K machines are in the failed state) in the GI/G/r machine interference model. Two of these examples of paradoxical behaviour have not previously been reported in the literature. Additional cases are also discussed.     

Solvability of finite groups

H is called an ? _p -embedded subgroup of G, if there exists a p-nilpotent subgroup B of G such that H _p ∈ Syl _p (B) and B is ? _p -supplemented in G. In this paper, by considering prime divisor 3, 5, or 7, we use ? _p -embedded property of primary subgroups to investigate the solvability of finite groups. The main result is follows. Let E be a normal subgroup of G, and let P be a Sylow 5-subgroup of E. Suppose that 1 < d ? |P| and d divides |P|. If every subgroup H of P with |H| = d is ?₅-embedded in G, then every composition factor of E satisfies one of the following conditions: (1) I/C is cyclic of order 5, (2) I/C is 5′-group, (3) I/C ? A₅.     

On local summability of Riesz potentials in the case Reα>0

The rangeI ^α (L _p ) of the Riesz potential operator, defined in the sense of distributions in the casep≥n/α, is shown to consist of regular distributions. Moreover, it is shown thatI ^α (L _p ) ?L _p ^loc (R ⁿ ) for all 1≤p<∞ and 0<α<∞. The distribution space used is that of Lizorkin, which is invariant with respect to the Riesz operator.     

The maximal Fejér operator on the spacesH p (R×···×R)

It is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from thed-dimensional Hardy spaceH _p (R×···×R) toL _p (R ^d ) (1/2<p<∞) and is of weak type (H ₁ ^?i ,L ₁) (i=1,…,d), where the Hardy spaceH ₁ ^?i is defined by a hybrid maximal function. As a consequence, we obtain that the Fejér means of a functionf ∈H ₁ ^?i ?L(logL) ^d?1 converge a.e. to the function in question. Moreover, we prove that the Fejér means are uniformly bounded onH _p (R×···×R) whenever 1/2<p<∞. Thus, in casef ∈H _p (R×···×R) the Fejér means converge tof inH _p (R×···×R) norm. The same results are proved for the conjugate Fejér means, too.