A retrial system with two input streams and two orbit queues

Two independent Poisson streams of jobs flow into a single-server service system having a limited common buffer that can hold at most one job. If a type- $i$ job ( $i=1,2$ ) finds the server busy, it is blocked and routed to a separate type- $i$ retrial (orbit) queue that attempts to re-dispatch its jobs at its specific Poisson rate. This creates a system with three dependent queues. Such a queueing system serves as a model for two competing job streams in a carrier sensing multiple access system. We study the queueing system using multi-dimensional probability generating functions, and derive its necessary and sufficient stability conditions while solving a Riemann–Hilbert boundary value problem. Various performance measures are calculated and numerical results are presented. In particular, numerical results demonstrate that the proposed multiple access system with two types of jobs and constant retrial rates provides incentives for the users to respect their contracts.     

Solving convex optimization with side constraints in a multi-class queue by adaptive c\mu rule

We study convex optimization problems with side constraints in a multi-class \(M/G/1\) queue with controllable service rates. In the simplest problem of optimizing linear costs with fixed service rate, the \(c\mu \) rule is known to be optimal. A natural question to ask is whether such simple policies exist for more complex control objectives. In this paper, combining the achievable region approach in queueing systems and the Lyapunov drift theory suitable to optimize renewal systems with time-average constraints, we show that convex optimization problems can be solved by variants of adaptive \(c\mu \) rules. These policies greedily re-prioritize job classes at the end of busy periods in response to past observed delays in each job class. Our method transforms the original problems into a new set of queue stability problems, and the adaptive \(c\mu \) rules are queue stable policies. An attractive feature of the adaptive \(c\mu \) rules is that they use limited statistics of the queue, where no statistics are required for the problem of satisfying average queueing delay in each job class.     

Improved online algorithms for the batch scheduling of equal-length jobs with incompatible families to maximize the weighted number of early jobs

We consider the online scheduling of equal-length jobs with incompatible families on \(m\) identical batch machines. Each job has a release time, a deadline and a weight. Each batch machine can process up to \(b\) jobs (which come from the same family) simultaneously as a batch, where \(b\) is called the capacity of the machines. Our goal is to determine a preemption-restart schedule which maximizes the weighted number of early jobs. For this problem, Li et al. (Inf Process Lett 112:503–508, 2012) provided an online algorithm of competitive ratio \(3+2\sqrt{2}\) for both \(b=\infty \) and \(b<\infty \) . In this paper, we study two special cases of this problem. For the case that \(m=2\) , we first present a lower bound 2, and then provide an online algorithm with a competitive ratio of 3 for both \(b=\infty \) and \(b<\infty \) . For the case in which \(m=3\) , \(b=\infty \) and all jobs come from a common family, we present an online algorithm with a competitive ratio of \((8+2\sqrt{15})/3\approx 5.249\) .     

1/2-Heavy sequences driven by rotation

We investigate the set of \(x \in S^1\) such that for every positive integer \(N\) , the first \(N\) points in the orbit of \(x\) under rotation by irrational \(\theta \) contain at least as many values in the interval \([0,1/2]\) as in the complement. By using a renormalization procedure, we show both that the Hausdorff dimension of this set is the same constant (strictly between zero and one) for almost-every \(\theta \) , and that for every \(d \in [0,1]\) there is a dense set of \(\theta \) for which the Hausdorff dimension of this set is \(d\) .     

On an extension of the Frobenius^{\prime } theorem about p-nilpotency of a finite group

Suppose that \(G\) is a finite group and \(H\) is a subgroup of \(G\) . \(H\) is said to be \(s\) -quasinormally embedded in \(G\) if for each prime \(p\) dividing the order of \(H\) , a Sylow \(p\) -subgroup of \(H\) is also a Sylow \(p\) -subgroup of some \(s\) -quasinormal subgroup of \(G\) . We fix in every non-cyclic Sylow subgroup \(P\) of \(G\) some subgroup \(D\) satisfying \(1<|D|<|P|\) and study the \(p\) -nilpotency of \(G\) under the assumption that every subgroup \(H\) of \(P\) with \(|H|=|D|\) is \(s\) -quasinormally embedded in \(G\) . Some recent results and the Frobenius \(^{\prime }\) theorem are generalized.     

Note on prehomogeneous vector spaces

Let \(V\) be a complex prehomogeneous vector space under the action of a linear algebraic group \(G\) . Assume the poset of orbit closures in the Zariski topology \(\{\overline{Gx}:x\in V\}\) coincides with a (partial) flag \(V_0=0<V_1<\dots <V_k=V\) in \(V\) . Then for any Borel subgroup \(B\) of \(G\) , the poset \(\{\overline{B x}:x\in V\}\) coincides with a full flag in \(V\) .     

On weakly closed subgroups of finite groups

Suppose that \(G\) is a finite group and \(H\) , \(K\) are subgroups of \(G\) . We say that \(H\) is weakly closed in \(K\) with respect to \(G\) if, for any \(g \in G\) such that \(H^{g}\le K\) , we have \(H^{g}=H\) . In particular, when \(H\) is a subgroup of prime-power order and \(K\) is a Sylow subgroup containing it, \(H\) is simply said to be a weakly closed subgroup of \(G\) or weakly closed in \(G\) . In the paper, we investigate the structure of finite groups by means of weakly closed subgroups.     

Endomorphism rings of bimodules

Let \(M\) be an \(R\) - \(R\) -bimodule over a semi-prime right and left Goldie ring \(R\) . We investigate how non-singularity conditions on \(M_R\) are related to such conditions on \(_RM\) . In particular, we say an \(R\) - \(R\) -bimodule \(M\) such that \(_RM\) and \(M_R\) are non-singular has the right essentiality property if \(IM_R\) is essential in \(M_R\) for all essential right ideals \(I\) of \(R\) , and investigate several questions related to this property.     

Some totally geodesic submanifolds of the nonlinear Grassmannian of a compact symmetric space

Let \(M\) and \(N\) be two connected smooth manifolds, where \(M\) is compact and oriented and \(N\) is Riemannian. Let \(\mathcal {E}\) be the Fréchet manifold of all embeddings of \(M\) in \(N\) , endowed with the canonical weak Riemannian metric. Let \(\sim \) be the equivalence relation on \(\mathcal {E}\) defined by \(f\sim g\) if and only if \(f=g\circ \phi \) for some orientation preserving diffeomorphism \(\phi \) of \(M\) . The Fréchet manifold \(\mathcal {S}= \mathcal {E}/_{\sim }\) of equivalence classes, which may be thought of as the set of submanifolds of \(N\) diffeomorphic to \(M\) and is called the nonlinear Grassmannian (or Chow manifold) of \(N\) of type \(M\) , inherits from \( \mathcal {E}\) a weak Riemannian structure. We consider the following particular case: \(N\) is a compact irreducible symmetric space and \(M\) is a reflective submanifold of \(N\) (that is, a connected component of the set of fixed points of an involutive isometry of \( N\) ). Let \(\mathcal {C}\) be the set of submanifolds of \(N\) which are congruent to \(M\) . We prove that the natural inclusion of \(\mathcal {C}\) in \(\mathcal {S}\) is totally geodesic.     

Scattered linear sets generated by collineations between pencils of lines

Let \(A\) and \(B\) be two points of \(\mathrm{{PG}}(2,q^n)\) , and let \(\Phi \) be a collineation between the pencils of lines with vertices \(A\) and \(B\) . In this paper, we prove that the set of points of intersection of corresponding lines under \(\Phi \) is either the union of a scattered \(\mathrm{{GF}}(q)\) -linear set of rank \(n+1\) with the line \(AB\) or the union of \(q-1\) scattered \(\mathrm{{GF}}(q)\) -linear sets of rank \(n\) with \(A\) and \(B\) . We also determine the intersection configurations of two scattered \(\mathrm{{GF}}(q)\) -linear sets of rank \(n+1\) of \(\mathrm{{PG}}(2,q^n)\) both meeting the line \(AB\) in a \(\mathrm{{GF}}(q)\) -linear set of pseudoregulus type with transversal points \(A\) and \(B\) .     

On extensions of completely simple semigroups by groups

An example of an extension of a completely simple semigroup \(U\) by a group \(H\) is given which cannot be embedded into the wreath product of \(U\) by \(H\) . On the other hand, every central extension of \(U\) by \(H\) is shown to be embeddable in the wreath product of \(U\) by \(H\) , and any extension of \(U\) by \(H\) is proved to be embeddable in a semidirect product of a completely simple semigroup \(V\) by \(H\) where the maximal subgroups of \(V\) are direct powers of those of \(U\) .     

Independence Under the G-Expectation Framework

We show that, for two non-trivial random variables \(X\) and \(Y\) under a sublinear expectation space, if \(X\) is independent from \(Y\) and \(Y\) is independent from \(X\) , then \(X\) and \(Y\) must be maximally distributed.     

Characterization of ellipses as uniformly dense sets with respect to a family of convex bodies

Let \(K\subset \mathbb R ^N\) be a convex body containing the origin. A measurable set \(G\subset \mathbb R ^N\) with positive Lebesgue measure is said to be uniformly \(K\) -dense if, for any fixed \(r>0\) , the measure of \(G\cap (x+r K)\) is constant when \(x\) varies on the boundary of \(G\) (here, \(x+r K\) denotes a translation of a dilation of \(K\) ). We first prove that \(G\) must always be strictly convex and at least \(C^{1,1}\) -regular; also, if \(K\) is centrally symmetric, \(K\) must be strictly convex, \(C^{1,1}\) -regular and such that \(K=G-G\) up to homotheties; this implies in turn that \(G\) must be \(C^{2,1}\) -regular. Then for \(N=2\) , we prove that \(G\) is uniformly \(K\) -dense if and only if \(K\) and \(G\) are homothetic to the same ellipse. This result was already proven by Amar et al. in 2008 . However, our proof removes their regularity assumptions on \(K\) and \(G\) , and more importantly, it is susceptible to be generalized to higher dimension since, by the use of Minkowski’s inequality and an affine inequality, avoids the delicate computations of the higher-order terms in the Taylor expansion near \(r=0\) for the measure of \(G\cap (x+r\,K)\) (needed in 2008).     

The k-error linear complexity distribution for 2^n-periodic binary sequences

The linear complexity and the \(k\) -error linear complexity of a sequence have been used as important security measures for key stream sequence strength in linear feedback shift register design. By using the sieve method of combinatorics, we investigate the \(k\) -error linear complexity distribution of \(2^n\) -periodic binary sequences in this paper based on Games–Chan algorithm. First, for \(k=2,3\) , the complete counting functions for the \(k\) -error linear complexity of \(2^n\) -periodic binary sequences (with linear complexity less than \(2^n\) ) are characterized. Second, for \(k=3,4\) , the complete counting functions for the \(k\) -error linear complexity of \(2^n\) -periodic binary sequences with linear complexity \(2^n\) are presented. Third, as a consequence of these results, the counting functions for the number of \(2^n\) -periodic binary sequences with the \(k\) -error linear complexity for \(k = 2\) and \(3\) are obtained.     

Approximation faible et principe de Hasse pour des espaces homogènes à stabilisateur fini résoluble

Let \(K\) be a global field and \(G\) a finite solvable \(K\) -group. Under certain hypotheses concerning the extension splitting \(G\) , we show that the homogeneous space \(V=G'/G\) with \(G'\) a semi-simple simply connected \(K\) -group has the weak approximation property. We use a more precise version of this result to prove the Hasse principle for homogeneous spaces \(X\) under a semi-simple simply connected \(K\) -group \(G'\) with finite solvable geometric stabilizer \({\bar{G}}\) , under certain hypotheses concerning the \(K\) -kernel (or \(K\) -lien) \(({\bar{G}},\kappa )\) defined by \(X\) .     

On the commuting probability and supersolvability of finite groups

For a finite group \(G\) , let \(d(G)\) denote the probability that a randomly chosen pair of elements of \(G\) commute. We prove that if \(d(G)>1/s\) for some integer \(s>1\) and \(G\) splits over an abelian normal nontrivial subgroup \(N\) , then \(G\) has a nontrivial conjugacy class inside \(N\) of size at most \(s-1\) . We also extend two results of Barry, MacHale, and Ní Shé on the commuting probability in connection with supersolvability of finite groups. In particular, we prove that if \(d(G)>5/16\) then either \(G\) is supersolvable, or \(G\) isoclinic to \(A_4\) , or \(G/\mathbf{Z}(G)\) is isoclinic to \(A_4\) .     

Bounding the exponent of a verbal subgroup

We deal with the following conjecture. If \(w\) is a group word and \(G\) is a finite group in which any nilpotent subgroup generated by \(w\) -values has exponent dividing \(e\) , then the exponent of the verbal subgroup \(w(G)\) is bounded in terms of \(e\) and \(w\) only. We show that this is true in the case where \(w\) is either the \(n\text{ th }\) Engel word or the word \([x^n,y_1,y_2,\ldots ,y_k]\) (Theorem A). Further, we show that for any positive integer \(e\) there exists a number \(k=k(e)\) such that if \(w\) is a word and \(G\) is a finite group in which any nilpotent subgroup generated by products of \(k\) values of the word \(w\) has exponent dividing \(e\) , then the exponent of the verbal subgroup \(w(G)\) is bounded in terms of \(e\) and \(w\) only (Theorem B).     

SFT-stability and Krull dimension in power series rings over an almost pseudo-valuation domain

Let \(R\) be an APVD with maximal ideal \(M\) . We show that the power series ring \(R[[x_1,\ldots ,x_n]]\) is an SFT-ring if and only if the integral closure of \(R\) is an SFT-ring if and only if ( \(R\) is an SFT-ring and \(M\) is a Noether strongly primary ideal of \((M:M)\) ). We deduce that if \(R\) is an \(m\) -dimensional APVD that is a residually *-domain, then dim \(R[[x_1,\ldots ,x_n]]\,=\,nm+1\) or \(nm+n\) .     

A sufficient condition for zero entropy

We prove that a diffeomorphism \(f\) defined on a compact manifold has zero topological entropy if there are \(d\in {\mathbb {N}}\) and \(K>0\) such that \(\Vert Dg^{n_x}(x)\Vert \le Kn^d_x\) for every diffeomorphism \(g\) that is \(C^1\) close to \(f\) and every periodic point \(x\) of least period \(n_x\) of \(g\) .