Cartan Subalgebras in C*-Algebras of Haus dorff étale Groupoids

The reduced C*-algebra of the interior of the isotropy in any Hausdorff étale groupoid G embeds as a C*-subalgebra M of the reduced C*-algebra of G. We prove that the set of pure states of M with unique extension is dense, and deduce that any representation of the reduced C*-algebra of G that is injective on M is faithful. We prove that there is a conditional expectation from the reduced C*-algebra of G onto M if and only if the interior of the isotropy in G is closed. Using this, we prove that when the interior of the isotropy is abelian and closed, M is a Cartan subalgebra. We prove that for a large class of groupoids G with abelian isotropy—including all Deaconu–Renault groupoids associated to discrete abelian groups—M is a maximal abelian subalgebra. In the specific case of k-graph groupoids, we deduce that M is always maximal abelian, but show by example that it is not always Cartan.     

Neumann heat flow and gradient flow for the entropy on non-convex domains

For large classes of non-convex subsets Y in \(\mathbb R^n\) or in Riemannian manifolds (M, g) or in RCD-spaces (X, d, m) we prove that the gradient flow for the Boltzmann entropy on the restricted metric measure space \((Y,d_Y,m_Y)\) exists—despite the fact that the entropy is not semiconvex—and coincides with the heat flow on Y with Neumann boundary conditions.     

An axiomatic look at the Erdős-Trost problem

The Erd?s-Trost problem can be formulated in the following way: “If the triangle XY Z is inscribed in the triangle ABC—with X, Y, and Z on the sides BC, CA, and AB, respectively—then one of the areas of the triangles BXZ, CXY , AY Z is less than or equal to the area of the triangle XY Z.” There are many different solutions for this problem. In this note we take up a very elementary proof (due to Szekeres) and deduce that the class of ordered translation planes is the level in the hierarchy of affine planes where the Erd?s-Trost statement still holds true. We also look at the conditions an absolute plane needs to satisfy for the validity of the Erd?s-Trost statement.     

Stabilization by output of continuous and pulse-modulated uncertain systems

The system ? _i = ? _i (?) + x _i+2, \(i \in \overline {1,n - 2} \), ? _n?1 = ? _n?1(?) + u ₁, ? _n = ? _n (?) + u ₂,where ? _i (·) are nonanticipating functionals of an arbitrary nature with the following properties—\(\left| {{\varphi _i}\left( \cdot \right)} \right| \leqslant c\sum\nolimits_{k = 1}^i {\left| {{x_k}\left( t \right)} \right|} \), \(i \in \overline {1,n} \), c = const—and u ₁ and u ₂ are the controls is considered. It is assumed that only the outputs x ₁ and x ₂ are measurable. The problem of synthesis of both continuous and impulsive controls u1 and u2, which make the system globally asymptotically stable, is solved. The solution of the problem is based on the construction of the observer-based equations, the quadratic Lyapunov function, and the averaging method.     

The characterization problem for designs with the parameters of AG<Subscript>d</Subscript>(<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)

We start a new characterization of the geometric 2-design AG _d (n,q) among all simple 2-designs with the same parameters by handling the cases d ∈ {1,2,3,n — 2}. For d ≠ 1, our characterization is in terms of line sizes, and for d = 1 in terms of the number of affine hyperplanes. We also show that the number of non-isomorphic resolvable designs with the parameters of AG₁(n,q) grows exponentially with linear growth of n.     

Stochastic inventory system with lead time flexibility: offered by a manufacturer/transporter

This paper studies lead time flexibility in a two-stage continuous review supply chain in which the retailer uses the (R, Q) inventory system: when his inventory position reaches R, the retailer places orders with size Q to the manufacturer, who uses a transportation provider to deliver them with different lead time options. According to the contract, the manufacturer is able to expedite or postpone the delivery if the retailer makes such a request. Hence, the retailer has the flexibility to modify the lead time by using the most up-to-date demand information. The optimal lead time policy is found to be a threshold-type policy. The sensitivity analysis also shows that R is much more sensitive to the change of lead time than Q, and thus, the paper is primarily focused on finding optimal R. We also provide a cost approximation which yields unimodal cost in R. Furthermore, we analyze the order crossing problem and derive an upper bound for the probability of order crossing. Finally, we conduct an extensive sensitivity analysis to illustrate the effects of lead time flexibility on supply chain performance and discuss the managerial insights.     

Translation Invariant Asymptotic Homomorphisms and Extensions of <Emphasis Type="Italic">C</Emphasis>*-Algebras

Let A and B be C*-algebras, let A be separable, and let B be σ-unital and stable. We introduce the notion of translation invariance for asymptotic homomorphisms from S A = C₀(?) ? A to B and show that the Connes—Higson construction applied to any extension of A by B is homotopic to a translation invariant asymptotic homomorphism. In the other direction we give a construction which produces extensions of A by B from a translation invariant asymptotic homomorphism. This leads to our main result that the homotopy classes of extensions coincide with the homotopy classes of translation invariant asymptotic homomorphisms.     

An axiomatization of the Choquet integral in the context of multiple criteria decision making without any commensurability assumption

An axiomatization of the Choquet integral is proposed in the context of multiple criteria decision making without any commensurability assumption. The most essential axiom—named Commensurability Through Interaction—states that the importance of an attribute i takes only one or two values when a second attribute k varies. When the importance takes two values, the point of discontinuity is exactly the value on the attribute k that is commensurate to the fixed value on attribute i. If the weight of criterion i does not depend on criterion k, for any value of the other criteria than i and k, then criteria i and k are independent. Applying this construction to any pair i, k of criteria, one obtains a partition of the set of criteria. In each block, the criteria interact one with another, and it is thus possible to construct vectors of values on the attributes that are commensurate. There is complete independence between the criteria of any two blocks in this partition. Hence one cannot ensure commensurability between two blocks in the partition. But this is not a problem since the Choquet integral is additive between subsets of criteria that are independent.     

Improving integrability via absolute summability: a general version of Diestel’s Theorem

A classical result by J. Diestel establishes that the composition of a summing operator with a (strongly measurable) Pettis integrable function gives a Bochner integrable function. In this paper we show that a much more general result is possible regarding the improvement of the integrability of vector valued functions by the summability of the operator. After proving a general result, we center our attention in the particular case given by the \((p,\sigma )\)-absolutely continuous operators, that allows to prove a lot of special results on integration improvement for selected cases of classical Banach spaces—including C(K), \(L^p\) and Hilbert spaces—and operators—p-summing, (q, p)-summing and p-approximable operators.     

Characteristic modules and tensor products over quasi-hereditary algebras

Let A be a monomial quasi-hereditary algebra with a pure strong exact Borel subalgebra B.It is proved that the category of induced good modules over B is contained in the category of good modules over A;that the characteristic module of A is an induced module of that of B via the exact functor-(?)_B A if and only if the induced A-module of an injective B-module remains injective as a B-module.Moreover,it is shown that an exact Borel subalgebra of a basic quasi-hereditary serial algebra is right serial and that the characteristic module of a basic quasi-hereditary serial algebra is exactly the induced module of that of its exact Borel subalgebra.     

Möbius disjointness for models of an ergodic system and beyond

Given a topological dynamical system (X, T) and an arithmetic function u: ? → ?, we study the strong MOMO property (relatively to u) which is a strong version of u-disjointness with all observable sequences in (X, T). It is proved that, given an ergodic measure-preserving system (Z, \(\mathcal{D}\), к, R),the strong MOMO propertly (relately to u) of a uniquely ergodic midel (X, T)of R yields all other uniquely ergodic midel of R to be u-disjiont. It follows that all uniquely ergodic models of: ergodic unipotent diffeomorphisms on nilmanifolds, discrete spectrum automorphisms, systems given by some substitutions of constant length (including the classical Thue—Viorse and Rudin—Shapiro substitutions), systems determined by Kakutani sequences are Möbius (and Liouville) disjoint. The validity of Sarnak5s conjecture implies the strong MOMO property relatively to μ in all zero entropy systems; in particular, it makes μ-disjointness uniform. The absence of the strong MOMO property in positive entropy systems is discussed and it is proved that, under the Chowla conjecture, a topological system has the strong MOMO property relatively to the Liouville function if and only if its topological entropy is zero.     

The strong <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-<Emphasis Type="Italic">greedy</Emphasis> property of the Walsh system

For any 0 < ? < 1 one can find a measurable set E ? [0, 1] with the measure |E| > 1 ? ? such that for each function f(x) ε L ¹ (0, 1) a function g(x) ε L ¹ (0, 1) exists such that it coincides with f (x) on E, its Fourier—Walsh series converges to it in the metric of L ¹ (0, 1), and all nonzero terms of the sequence of Fourier coefficients of the new function obtained by the Walsh system have the modulo decreasing order; consequently, the greedy algorithm for this function converges to it in the L ¹ (0, 1)-norm.     

A note on Linnik’s approach to the Dirichlet <Emphasis Type="Italic">L</Emphasis>-functions

Abstract—Let χ (mod q), q > 1, be a primitive Dirichlet character. We first present a detailed account of Linnik’s deduction of the functional equation of L(s, χ) from the functional equation of ζ(s). Then we show that the opposite deduction can be obtained by a suitable modification of the method, involving finer arithmetic arguments.     

Mortality Risk and Life Expectancy

Changing the mortality risks we face would change human life expectancy. As a special case, one could imagine adding a fixed increment R to all the age-specific mortality rates from age zero upwards. For this case we seek a constant K(A) such that K(A) x R approximates the resulting change in life expectancy remaining at age A, at least for small values of R. The formula for K(A) derived here corrects a heuristic argument that appeared in JORS earlier. An estimate of K(0) suggests that the permanent addition of a one-in-a-million risk at each year of life would reduce life expectancy at birth by about 1 day—a useful fact for risk communication.     

Factorization of the reaction-diffusion equation,the wave equation,and other equations

We investigate equations of the form D _t u = Δu + ξ? _u for an unknown function u(t, x), t ∈ ?, x ∈ X, where D _t u = a ₀(u, t) + Σ _k=1 ^r a _k (t, u)? _t ^k u, Δ is the Laplace-Beltrami operator on a Riemannian manifold X, and ξ is a smooth vector field on X. More exactly, we study morphisms from this equation within the category PDE of partial differential equations, which was introduced by the author earlier. We restrict ourselves to morphisms of a special form—the so-called geometric morphisms, which are given by maps of X to other smooth manifolds (of the same or smaller dimension). It is shown that a map f: X → Y defines a morphism from the equation D _t u = Δu + ξ? _u if and only if, for some vector field Ξ and a metric on Y, the equality (Δ + ξ?)f*v = f*(Δ + Ξ?)v holds for any smooth function v: Y → ?. In this case, the quotient equation is D _t v = Δv + Ξ?v for an unknown function v(t, y), y ∈ Y. It is also shown that, if a map f: X → Y is a locally trivial bundle, then f defines a morphism from the equation D _t u = Δu if and only if fibers of f are parallel and, for any path γ on Y, the expansion factor of a fiber translated along the horizontal lift γ to X depends on γ only.     

Variable interval sampling in economical designs for online process control of attributes with misclassification errors

Online process control consists of inspecting a single item at every mth items produced, where m is an integer greater than two. Based on the results of the inspection, one decides if the process is in-control (the fraction of conforming item is p₁—state I) or out-of-control (the fraction of conforming item is p₂—state II). If the inspected item is non-conforming, the process is designated as out-of-control and production is stopped for possible adjustment; otherwise, production goes on. In this paper, a contribution to online process control is presented, where the inspection system is considered to be subject to classification errors. After every adjustment, the sampling interval is L units (L?m), and in the case of non-adjustment, the sampling interval is m units. The expression for the average cost per produced item is calculated, and optimum parameters (the sampling intervals L and m) are obtained by a direct search. The procedure is illustrated by a numerical example.     

Asymptotic behavior of the spectrum of one-dimensional vibrations in a layered medium consisting of elastic and Kelvin–Voigt viscoelastic materials

For every algebraically closed field k of characteristic different from 2, we prove the following: (1) Finite-dimensional (not necessarily associative) k-algebras of general type of a fixed dimension, considered up to isomorphism, are parametrized by the values of a tuple of algebraically independent (over k) rational functions of the structure constants. (2) There exists an “algebraic normal form” to which the set of structure constants of every such algebra can be uniquely transformed by means of passing to its new basis—namely, there are two finite systems of nonconstant polynomials on the space of structure constants, {f_i}i∈I and {b_j}j∈J, such that the ideal generated by the set {f_i}i∈I is prime and, for every tuple c of structure constants satisfying the property b_j(c) ≠ 0 for all j ∈ J, there exists a unique new basis of this algebra in which the tuple c′ of its structure constants satisfies the property f_i(c′) = 0 for all i ∈ I.     

Multi-peak positive solutions for nonlinear Schrödinger equations with critical frequency

We study the nonlinear Schrödinger equations: \(-\epsilon^{2}\Delta u + V(x)u=u^p,\quad u > 0\quad \mbox{in } {\bf R}^{N},\quad u\in H^{1} ({\bf R}^{N}).\) where p > 1 is a subcritical exponent and V(x) is nonnegative potential function which has “critical frequency” \(\inf_{x\in{\bf R}^{N}} V(x)=0\). We also assume that V(x) satisfies \(0 < \liminf_{|x|\to\infty}V(x)\le \sup_{x\in{\bf R}^{N}}V(x) < \infty\) and V(x) has k local or global minima. In critical frequency cases, Byeon-Wang [5,6] showed the existence of single-peak solutions which concentrating around global minimum of V(x). Their limiting profiles—which depend on the local behavior of the potential V(x)—are quite different features from non-critical frequency case. We show the existence of multi-peak positive solutions joining single-peak solutions which concentrate around prescribed local or global minima of V(x). Moreover, under additional conditions on the behavior of V(x), we state the limiting profiles of peaks of solutions u _ε(x) as follows: rescaled function \(w_\epsilon(y)=\left(\frac{g(\epsilon)}{\epsilon}\right)^{\frac{2}{p-1}} u_\epsilon(g(\epsilon)y+x_\epsilon)\) converges to a least energy solution of ?Δw + V ₀(y) w = w ^p , w > 0 in Ω₀, \(w\in H^{1}_0(\Omega_0)\). Here g(ε), V ₀(x) and Ω₀ depend on the local behaviors of V(x).     

Stability and the immediate acceptance rule when school priorities are weak

In a model of school choice, we allow school priorities to be weak and study the preference revelation game induced by the immediate acceptance (IA) rule (also known as the Boston rule), or the IA game. When school priorities can be weak and matches probabilistic, three stability notions—ex post stability, ex ante stability, and strong ex ante stability—and two ordinal equilibrium notions—sd equilibrium and strong sd equilibrium—become available (“sd” stands for stochastic dominance). We show that for no combination of stability and equilibrium notions does the set of stable matches coincide with the set of equilibrium matches of the IA game. This stands in contrast with the existing result that the two sets are equal when priorities are strict. We also show that in the presence of weak priorities, the transition from the IA rule to the deferred acceptance rule may, in fact, harm some students.