The problem of differentiating an asymptotic expansion in real powers. Part II: Factorizational theory

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:

1. 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
2. formal differentiation of a real-power asymptotic expansion under a Tauberian condition involving the order of growth of L
3. remarkable properties of asymptotic expansions of generalized convex functions.

     

A spectral approach to polyhedral dimension

Thespectrum spec( ) of a convex polytope is defined as the ordered (non-increasing) list of squared singular values of [A|1], where the rows ofA are the extreme points of . The number of non-zeros in spec( ) exceeds the dimension of by one. Hence, the dimension of a polytope can be established by determining its spectrum. Indeed, this provides a new method for establishing the dimension of a polytope, as the spectrum of a polytope can be established without appealing to a direct proof of its dimension. The spectrum is determined for the four families of polytopes defined as the convex hulls of:

1. The edge-incidence vectors of cutsets induced by balanced bipartitions of the vertices in the complete undirected graph on 2_q vertices (see Section 6).
2. The edge-incidence vectors of Hamiltonian tours in the complete undirected graph onn vertices (see Section 6).
3. The arc-incidence vectors of directed Hamiltonian tours in the complete directed graph ofn nodes (see Section 7).
4. The edge-incidence vectors of perfect matchings in the complete 3-uniform hypergraph on 3_q vertices (see Section 8).

In the cases of (ii) and (iii), the associated dimension results are well-known. The dimension results for (i) and (iv) do not seem to be well-known. General principles are discussed for ‘balanced polytopes’ arising from complete structures.     

Monotonicity and error bounds for networks of Erlang loss queues

Networks of Erlang loss queues naturally arise when modelling finite communication systems without delays, among which, most notably are

1. classical circuit switch telephone networks (loss networks) and
2. present-day wireless mobile networks.

Performance measures of interest such as loss probabilities or throughputs can be obtained from the steady state distribution. However, while this steady state distribution has a closed product form expression in the first case (loss networks), it does not have one in the second case due to blocked (and lost) handovers. Product form approximations are therefore suggested. These approximations are obtained by a combined modification of both the state space (by a hypercubic expansion) and the transition rates (by extra redial rates). It will be shown that these product form approximations lead to

• upper bounds for loss probabilities and
• analytic error bounds for the accuracy of the approximation for various performance measures.

The proofs of these results rely upon both monotonicity results and an analytic error bound method as based on Markov reward theory. This combination and its technicalities are of interest by themselves. The technical conditions are worked out and verified for two specific applications:

• pure loss networks as under (i)
• GSM networks with fixed channel allocation as under (ii).

The results are of practical interest for computational simplifications and, particularly, to guarantee that blocking probabilities do not exceed a given threshold such as for network dimensioning.     

Call Centers with Impatient Customers: Many-Server Asymptotics of the M/M/n + G Queue

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.     

Derivationen und Automorphismen von Algebren,in Denen Die Gleichung X2=0 Nur Trivial Lösbar Ist

Let A be a finite-dimensional algebra over a (commutative) field K of characteristic O, assume that x∈A and x²=0 implies x=0. We shall prove among others:

- The derivations and automorphisms of A are semisimple.

- If K is algebraically closed, then Der A=0 and |Aut A|<∞.

- If K=?, then Aut A (and hence Der A) is compact.

     

Convergence of representation averages and of convolution powers

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:

1. 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.
2. 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.
3. 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} } .$ .

     

On theories of random variables

We study theories of spaces of random variables: first, we consider random variables with values in the interval [0, 1], then with values in an arbitrary metric structure, generalising Keisler’s randomisation of classical structures. We prove preservation and non-preservation results for model theoretic properties under this construction:

1. The randomisation of a stable structure is stable.
2. The randomisation of a simple unstable structure is not simple.

We also prove that in the randomised structure, every type is a Lascar type.     

Flow of a weakly ionized gas between parallel cold walls with parabolic velocity profile

Ambipolar diffusion between flat cold insulating walls of a weakly ionized gas which flows in the direction parallel to the walls with parabolic velocity profile is investigated theoretically. It has been found that:

1. the patched velocity of linear and nonlinear regions tends to 1/√2 of the thermal velocity;
2. the thickness of the nonlinear region with parabolic velocity profile is found to be less than that of Shioda who considered uniform streaming (J. Phys. Soc. Japan, 1969,29, 197); and
3. the number density and the electric potential approximations in the sheath edge do not depend uponx, the coordinate in the streaming direction.

     

Generalized conic curves and their applications in curve approximation

The appriximation properties of generalized conic curves are studied in this paper. A generalized conic curve is defined as one of the following curves or their affine and translation equivalent curves:

1. conic curves, including parabolas, hyperbolas and ellipses;
2. generalized monomial curves, including curves of the form x=y^γ, γ∈R, γ≠0,1, in the x?y Cartesian coordinate system;
3. exponential spiral curves of the form ρ(?)=Ae^γ?, A>0, γ≠0, in the ρ-? polar coordinate system.

This type of curves has many important properties such as convexity, approximation property, effective numerical computation property and the subdivision property etc. Applications of these curves in both interpolation and approximations using piecewise generalized conic segment are also developed. It is shown that these generalized conic splines are very similar to the cubic polynomial splines and the best error of approximation isO(h ⁵) or at leastO(h ⁴) in general provided appropriate procedures are used. Finally some numerical examples of interpolation and approximations with generalized conic splines are given.     

Compactness and the axiom of choice

In the absence of the axiom of choice four versions of compactness (A-, B-, C-, and D-compactness) are investigated. Typical results:

1. C-compact spaces form the epireflective hull in Haus of A-compact completely regular spaces.
2. Equivalent are:
3. the axiom of choice,
4. A-compactness = D-compactness,
5. B-compactness = D-compactness,
6. C-compactness = D-compactness and complete regularity,
7. products of spaces with finite topologies are A-compact,
8. products of A-compact spaces are A-compact,
9. products of D-compact spaces are D-compact,
10. powers X ^k of 2-point discrete spaces are D-compact,
11. finite products of D-compact spaces are D-compact,
12. finite coproducts of D-compact spaces are D-compact,
13. D-compact Hausdorff spaces form an epireflective subcategory of Haus,
14. spaces with finite topologies are D-compact.

1. Equivalent are:
2. the Boolean prime ideal theorem,
3. A-compactness = B-compactness,
4. A-compactness and complete regularity = C-compactness,
5. products of spaces with finite underlying sets are A-compact,
6. products of A-compact Hausdorff spaces are A-compact,
7. powers X ^k of 2-point discrete spaces are A-compact,
8. A-compact Hausdorff spaces form an epireflective subcategory of Haus.

1. Equivalent are:
2. either the axiom of choice holds or every ultrafilter is fixed,
3. products of B-compact spaces are B-compact.

1. Equivalent are:
2. Dedekind-finite sets are finite,
3. every set carries some D-compact Hausdorff topology,
4. every T ₁-space has a T ₁-D-compactification,
5. Alexandroff-compactifications of discrete spaces and D-compact.

     

Standard completions for quasiordered sets

A standard completion for a quasiordered set Q is a closure system whose point closures are the principal ideals of Q. We characterize the following types of standard completions by means of their closure operators:

1. V-distributive completions,
2. Completely distributive completions,
3. A-completions (i.e. standard completions which are completely distributive algebraic lattices),
4. Boolean completions.

Moreover, completely distributive completions are described by certain idempotent relations, and the A-completions are shown to be in one-to-one correspondence with the join-dense subsets of Q. If a pseudocomplemented meet-semilattice Q has a Boolean completion ?, then Q must be a Boolean lattice and ? its MacNeille completion.     

A note on weakly compact subgroups of locally quasi-convex groups

In this note two results about compact subgroups of locally quasi-convex groups are shown:

1. The quotient group of a locally quasi convex Hausdorff group modulo a compact subgroup is again locally quasi-convex.
2. If a subgroup of a locally quasi-convex group is compact in the weak topology, it is also compact in the original topology.

     

Order stars and stability theorems

This paper clears up to the following three conjectures:

1. The conjecture of Ehle [1] on theA-acceptability of Padé approximations toe ^z , which is true;
2. The conjecture of Nørsett [5] on the zeros of the “E-polynomial”, which is false;
3. The conjecture of Daniel and Moore [2] on the highest attainable order of certainA-stable multistep methods, which is true, generalizing the well-known Theorem of Dahlquist.

We further give necessary as well as sufficient conditions forA-stable (acceptable) rational approximations, bounds for the highest order of “restricted” Padé approximations and prove the non-existence ofA-acceptable restricted Padé approximations of order greater than 6. The method of proof, just looking at “order stars” and counting their “fingers”, is very natural and geometric and never uses very complicated formulas.     

Topologies on products of partially ordered sets III: Order convergence and order topology

This paper deals with the question under which circumstances filter-theoretical order convergence in a product of posets may be computed componentwise, and the same problem is treated for convergence in the order topology (which may differ from order convergence). The main results are:

1. Order convergence in a product of posets is obtained componentwise if and only if the number of non-bounded posets occurring in this product is finite (1.5).
2. For any product of posets, the projections are open and continuous with respect to the order topologies (2.1).
3. A productL of chainsL _i has topological order convergence iff all but a finite number of the chains are bounded. In this case, the order topology onL agrees with the product topology (2.7).
4. If (L _i :j ∈J) is a countable family of lattices with topological order convergence and first countable order topologies then order topology of the product lattice and product topology coincide (2.8).
5. LetP ₁ be a poset with topological order convergence and locally compact order topology. Then for any posetP ₂, the order topology ofP ₁?P ₂ coincides with the product topology (2.10).
6. A latticeL which is a topological lattice in its order topology is join- and meet-continuous. The converse holds whenever the order topology ofL?L is the product topology (2.15).

Many examples are presented in order to illustrate how far the obtained results are as sharp as possible.     

О рядах по системе Фра нклина

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:

1. 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;
2. 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 \(\bigcap\limits_{p< \infty } {L_p } \) ;
3. The absolute convergence at a point for Fourier—Franklin series is a local property;
4. If an integrable function (fx) has a discontinuity of the first kind atx=x ₀, then its Fourier-Franklin series diverges atx=x ₀.

     

Remarks on star countable discrete closed spaces

In this paper, we prove the following statements:

1. There exists a Tychonoff star countable discrete closed, pseudocompact space having a regular-closed subspace which is not star countable.
2. Every separable space can be embedded into an absolutely star countable discrete closed space as a closed subspace.
3. Assuming $2^{\aleph _0 } = 2^{\aleph _1 } $ , there exists a normal absolutely star countable discrete closed space having a regular-closed subspace which is not star countable.

     

On a certain type of commutators of operators

LetH be a separable infinite-dimensional Hilbert space and letC be a normal operator andG a compact operator onH. It is proved that the following four conditions are equivalent.

1. C +G is a commutatorAB-BA with self-adjointA.
2. There exists an infinite orthonormal sequencee _j inH such that |Σ _j ⁿ =1 (Ce _j, e_j)| is bounded.
3. C is not of the formC ₁ ⊕C ₂ whereC ₁ has finite dimensional domain andC ₂ satisfies inf {|(C ₂ x, x)|: ‖x‖=1}>0.
4. 0 is in the convex hull of the set of limit points of spC.

     

On the homogenization of noncoercive variational problems

The paper deals with variational problems of the form $$\mathop {\inf }\limits_{u \in W^{1,p} (\Omega )} \int\limits_\Omega {a(\varepsilon ^{ - 1} x)(\left| {\nabla u} \right|^p + \left| {u - g} \right|^p )} dx,$$ where Ω is a bounded Lipschitzian domain in ? ^N , g∈L^p(Ω). The function a(x) is assumed to satisfy the following conditions:

1. a(x) is periodic and lower semicontinuous;
2. 0≤a(x)≤1 and the set {∈? ^N , a(x)>0} is connected in ? ^N Under these conditions, basic properties of homogenization (convergence of energies and generalized solutions) and properties of Г-convergence type are proved. Bibliography: 3 titles.

     

A convexity property and a new characterization of Euler’s gamma function

Our main results are:

1. Let α ≠ 0 be a real number. The function (Γ ? exp) ^α is convex on ${\mathbf{R}}$ if and only if $$\alpha \geq \max_{0<{t}<{x_0}}\Big(-\frac{1}{t\psi(t)} - \frac{\psi'(t)}{\psi(t)^2}\Big) = 0.0258... .$$ Here, x ₀ = 1.4616... denotes the only positive zero of ${\psi = \Gamma'/\Gamma}$ .

1. Assume that a function f: (0, ∞) → (0, ∞) is bounded from above on a set of positive Lebesgue measure (or on a set of the second category with the Baire property) and satisfies $$f(x+1) = x f(x) \quad{\rm for}\quad{x > 0}\quad{\rm and}\quad{f(1) = 1}.$$

If there are a number b and a sequence of positive real numbers (a _n ) ${(n \in \mathbf{N})}$ with ${{\rm lim}_{n\to\infty} a_n =0}$ such that for every n the function ${(f \circ {\rm exp})^{a_n}}$ is Jensen convex on (b, ∞), then f is the gamma function.     

Some results on entire functions of finite lower order

Letf(z) be an entire function of order λ and of finite lower order μ. If the zeros off(z) accumulate in the vicinity of a finite number of rays, then

1. λ is finite;
2. for every arbitrary numberk ₁>1, there existsk ₂>1 such thatT(k ₁ r,f)≤k ₂ T(r,f) for allr≥r ₀. Applying the above results, we prove that iff(z) is extremal for Yang's inequalityp=g/2, then
3. every deficient values off(z) is also its asymptotic value;
4. every asymptotic value off(z) is also its deficient value;
5. λ=μ;
6. $\sum\limits_{a \ne \infty } {\delta (a,f) \leqslant 1 - k(\mu ).} $